Entropy-Based Design and Analysis of Fluids Engineering Systems Greg F. Naterer José A. Camberos
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Entropy-Based Design and Analysis of Fluids Engineering Systems Greg F. Naterer José A. Camberos
Boca Raton London New York
CRC Press is an imprint of the Taylor & Francis Group, an informa business
© 2008 by Taylor & Francis Group, LLC
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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑0‑8493‑7262‑9 (Hardcover) This book contains information obtained from authentic and highly regarded sources Reason‑ able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The Authors and Publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Naterer, Greg F. Entropy‑based design and analysis of fluids engineering systems / Greg F. Naterer, José A. Camberos. p. cm. ISBN 978‑0‑8493‑7262‑9 (hardback : alk. paper) 1. Entropy. 2. Heat‑‑Transmission. 3. Fluid dynamics. I. Camberos, José A. II. Title. TJ265.N38 2007 620.1’06‑‑dc22
2007028612
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Dedication To my wife Josie, our children Jordan, Julia, and Veronica, and my mother and father.
G.N.
To my parents for bringing me into this world, my wife Tina, and our children Antonio, Isabella, and Esteban.
J.C.
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Contents Foreword...................................................................................................................xi Preface.................................................................................................................... xiii Acknowledgments................................................................................................... xv Authors...................................................................................................................xvii Chapter 1 Introduction...........................................................................................1 1.1 1.2
Background......................................................................................................1 Governing Equations of Fluid Flowand Heat Transfer....................................4 1.2.1 Vector and Tensor Notations.................................................................4 1.2.2 Mass and Momentum Equations..........................................................5 1.2.3 Energy Transport Equations.................................................................7 1.3 Mathematical Properties of Entropy and Exergy............................................8 1.3.1 Concavity Property of Entropy.............................................................9 1.3.2 Distance Functional with Respect to Equilibrium Conditions........... 14 1.4 Governing Equations of Entropyand the Second Law................................... 17 1.4.1 Closed System..................................................................................... 17 1.4.2 Open System.......................................................................................20 1.5 Formulation of Entropy Production and Exergy Destruction........................ 22 1.5.1 Closed System..................................................................................... 22 1.5.2 Linear Advection Equation (without Diffusion)................................. 23 1.5.3 Linear Advection Equation (with Diffusion)......................................24 1.5.4 Navier–Stokes Equations....................................................................25 References................................................................................................................. 29 Chapter 2 Statistical and Numerical Formulations of the Second Law............... 33 2.1 2.2 2.3 2.4
2.5 2.6 2.7
Introduction.................................................................................................... 33 Conservation Laws as Moments of the Boltzmann Equation........................34 Extended Probability Distributions................................................................ 36 Selected Multivariate Probability Distribution Functions............................. 38 2.4.1 Maxwell–Boltzmann Probability Distribution Function.................... 39 2.4.2 Central Distribution Probability Distribution Function..................... 39 2.4.3 Chapman–Enskog Probability Distribution Function........................40 2.4.4 Skew-Normal Probability Distribution Function................................ 41 Concave Entropy Functions........................................................................... 43 Statistical Formulation of the Second Law....................................................46 Numerical Formulation of the Second Law................................................... 48 2.7.1 Discretization of the Problem Domain............................................... 48
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2.7.2 Discretization of the Conservation Equations.................................... 51 2.7.3 Discretization of the Second Law....................................................... 53 References................................................................................................................. 55 Chapter 3 Predicted Irreversibilities of Incompressible Flows............................ 57 3.1 3.2 3.3
Introduction.................................................................................................... 57 Entropy Transport Equation for Incompressible Flows................................. 58 Formulation of Loss Coefficients in Terms of Entropy Production............... 61 3.3.1 Entropy Production in Bernoulli’s Equation....................................... 61 3.3.2 Loss Coefficients in a Plane Diffuser................................................. 63 3.3.3 Case Study of Channel and Diffuser Design......................................64 3.4 Upper Entropy Bounds in Closed Systems.................................................... 70 3.4.1 Upper Bounds of Thermal Irreversibility........................................... 71 3.4.2 Optimal Aspect Ratio of Upper Entropy Bounds............................... 75 3.4.3 Case Study of Mixing Tank Design.................................................... 76 3.5 Case Study of Automotive Fuel Cell Design................................................. 79 3.5.1 Electrochemical Irreversibilities in a Porous Electrode..................... 79 3.5.2 Formulation of Channel Flow Irreversibilities................................... 82 3.5.3 Proton Exchange Membrane Fuel Cell (PEMFC) and Solid Oxide Fuel Cell (SOFC) Design......................................................... 85 3.6 Case Study of Fluid Machinery Design.........................................................90 References.................................................................................................................92 Chapter 4 Measured Irreversibilities of Incompressible Flows........................... 95 4.1 4.2
Introduction.................................................................................................... 95 Experimental Techniques of Irreversibility Measurement............................ 95 4.2.1 Velocity Field Measurement............................................................... 95 4.2.2 Temperature Field Measurement........................................................97 4.2.3 Postprocessing for Entropy Production Measurement.......................99 4.3 Case Study of Magnetic Stirring Tank Design............................................ 100 4.4 Case Study of Natural Convection in Cavities............................................. 103 4.5 Measurement Uncertainties......................................................................... 105 4.5.1 Bias and Precision Errors................................................................. 105 4.5.2 Velocity Field Uncertainties in Channel Flow.................................. 106 4.5.3 Measurement Uncertainties of Entropy Production......................... 108 4.5.4 Entropy Production of Free Convection in Cavities......................... 109 References............................................................................................................... 109 Chapter 5 Entropy Production in Microfluidic Systems.................................... 111
5.1 5.2
Introduction.................................................................................................. 111 Pressure-Driven Flow in Microchannels..................................................... 112 5.2.1 Continuum Equations and Slip Boundary Conditions...................... 112 5.2.2 Case Study of Exergy Losses in Channel Design............................. 113
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5.3
Applied Electric Field in Microchannels..................................................... 117 5.3.1 Irreversibilities with a Constant Magnetic Field............................... 117 5.3.2 Case Study of Channel Design at Varying Hartmann Numbers...... 122 5.4 Micropatterned Surfaces with Open Microchannels................................... 126 5.4.1 Fluid Flow Formulation.................................................................... 126 5.4.2 Heat Transfer Formulation................................................................ 131 5.4.3 Formulation of Entropy Production.................................................. 132 5.4.4 Case Studies of Surface Micropattern Design.................................. 136 References............................................................................................................... 141 Chapter 6 Numerical Error Indicators and the Second Law............................. 143 6.1 6.2
Introduction.................................................................................................. 143 Discretization Errors of Numerical Convection Schemes........................... 145 6.2.1 Finite Volume Formulation............................................................... 145 6.2.2 Central, Upwind, and Exponential Differencing Schemes............... 147 6.2.3 Case Study of Nozzle Flow Analysis and Design............................ 152 6.3 Physical Plausibility of Numerical Results.................................................. 157 6.3.1 Entropy Correction of Numerical Diffusion..................................... 157 6.3.2 Case Study of Shock Capturing in a Shock Tube............................. 161 6.4 Entropy Difference in Residual Error Indicators......................................... 163 6.4.1 Formulation of Average Entropy Difference.................................... 163 6.4.2 Case Study of Error Indicators in Supersonic Flow......................... 165 References............................................................................................................... 173 Chapter 7 Numerical Stability and the Second Law.......................................... 175 7.1 7.2 7.3
Introduction.................................................................................................. 175 Stability Norms............................................................................................ 176 Entropy Stability of Finite Difference Schemes.......................................... 180 7.3.1 Linear Scalar Advection................................................................... 180 7.3.2 Nonlinear Scalar Advection.............................................................. 189 7.3.3 Coupled Nonlinear Equations........................................................... 197 7.4 Stability of Shock Capturing Methods........................................................202 References............................................................................................................... 210 Chapter 8 Entropy Transport with Phase Change Heat Transfer....................... 213 8.1 8.2 8.3 8.4
Introduction.................................................................................................. 213 Entropy Transport Equations for Solidification and Melting...................... 215 Heat and Entropy Analogies in PhaseChange Processes............................. 220 8.3.1 Irreversibility of Interdendritic Permeability................................... 220 8.3.2 Thermal Recalescence and Dimensionless Entropy Ratio............... 222 Numerical Stability of Phase Change Computations................................... 227 8.4.1 Modeling of Two-Phase Entropy Production................................... 227 8.4.2 Iterative Phase Rules and the Second Law....................................... 230
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8.4.3 Entropy Correction of Numerical Conductivity............................... 232 8.4.4 Entropy Condition for Temporal Stability........................................ 234 8.4.5 Case Study of Melting in an Enclosure............................................ 237 8.4.6 Case Study of Free Convection and Solidification...........................240 8.5 Thermal Control of Phase Change with Inverse Methods........................... 242 8.5.1 Formulation of an Inverse Method................................................... 242 8.5.2 Entropy Correction for Numerical Stability.....................................244 8.5.3 Case Study with Solidification of a Pure Material...........................246 8.6 Entropy Production with Film Condensation.............................................. 250 8.6.1 Formulation of Heat Transfer and Irreversibility Distribution......... 250 8.6.2 Case Study of Flat Plate Condensation............................................. 256 References............................................................................................................... 258 Chapter 9 Entropy Production in Turbulent Flows............................................ 261 9.1 9.2 9.3 9.4 9.5
Introduction.................................................................................................. 261 Reynolds Averaged Entropy Transport Equations....................................... 262 Eddy Viscosity Models of Mean Entropy Production................................. 265 Turbulence Modeling with the Second Law................................................266 Measurement of Turbulent Entropy Production........................................... 268 9.5.1 Formulation of Dissipation Rate....................................................... 268 9.5.2 Large Eddy Particle Image Velocimetry.......................................... 271 9.5.3 Case Study of Turbulent Channel Flow............................................ 273 References...............................................................................................................284 Appendix................................................................................................................ 287 Nomenclature........................................................................................................ 299 Index....................................................................................................................... 303
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Foreword Various aspects of the First and Second Laws of Thermodynamics are used in the design of many systems. They are typically applied to components and not in a truly connected fashion. As the title implies, this book has been written and organized to enable the design of energy-efficient systems of fluid systems. It lays out the theoretical methods to support entropy-based design. Chapters 1 through 3 lay out the governing equations for all related aspects of fluid flow with an emphasis on entropy generation and energy flow. The different sections are organized to support computation of the flow characteristics. Chapter 3 ends with case studies to illustrate the use of the preceding methods. A critical factor in all design processes is the question of how the theories compare with experimental measurements. Chapter 4 starts with a presentation of experimental techniques for energy systems. Then there are case studies followed by consideration of the uncertainties in such measurements. Chapter 5 is an example application of the theoretical methods to microchannel flows. Again, it has case studies for better illustration of the methods. Chapters 6 and 7 address the critically important subject of potential errors in the computational application of the theoretical methods. This subject is typically neglected, but even the exact physical equations cannot be numerically computed with complete accuracy. The discussions address numerical convection, numerical diffusion, linear and nonlinear effects, and so forth. Again, various case studies clearly show the effects being presented. Chapters 8 and 9 end the book with theories and case studies of entropy transport. These dynamic considerations are a critical aspect of any practical design problem. It is my belief that this book lays out the theoretical methods for efficient design of energy fluid systems. It presents case studies for easier understanding of all the methods. A very important contribution is the consideration of both experimental and computational errors. Dr. David J. Moorhouse Director, Multidisciplinary. Technologies Center Air Force Research Laboratory
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Preface The word “entropy” is derived from the Greek verb to chase, escape, or rotate. It is generally interpreted in three semirelated ways, namely, a macroscopic viewpoint (classical thermodynamics), microscopic viewpoint (statistical thermodynamics), and an information viewpoint (information theory). At a philosophical level, some believe that thermodynamic entropy can be interpreted as an application of information theory to a particular set of physical phenomena. From another perspective, entropy is often described as a type of clock, as it is the only quantity in the physical sciences that corresponds to a particular direction for time, sometimes called an arrow of time. As systems operate in time, the Second Law of Thermodynamics requires that entropy of an isolated system can only increase or remain constant, but never decrease. For open systems, entropy production must be nonnegative. The history of entropy began with Lazare Carnot, who in 1803 postulated a “loss of moment activity” in any machine with moving parts, due to energy lost by friction. This led to a basic concept of “transformation energy” or entropy, with an inference that perpetual motion was impossible. Lazare’s son, Sadi Carnot, in 1824 published Reflections on the Motive Power of Fire. He visualized an ideal engine where a “caloric” (now known as heat) converted into work could be reinstated by reversing the motion of the cycle, a concept that became known as thermodynamic reversibility. Building on his father’s work, Sadi postulated that “some caloric is always lost,” thus setting a foundation for the concept of available energy loss through entropy production. In the 1850s, German physicist Rudolf Clausius gave this “lost caloric” a mathematical interpretation and set forth the concept of a thermodynamic system, whereby in any irreversible process, a small amount of thermal energy is dissipated across the system boundary. Afterward, scientists such as Ludwig Boltzmann, Willard Gibbs, and James Maxwell gave entropy a statistical basis. Today the applications of entropy are widespread, from engineering fluid mechanics and thermodynamics, to information and coding theory, economics, and biology. Entropy serves as a key parameter in achieving the upper limits of performance and quality in many engineering technologies. As future technologies press toward these theoretical limits, entropy and the Second Law will have an increasingly significant role of importance. They can shed new light on various flow processes, ranging from optimized flow configurations in an aircraft engine to highly ordered crystal structures (low entropy) in a turbine blade, and many other applications. Entropy-based design (EBD) is an emerging design methodology that incorporates the Second Law with computational fluid dynamics (CFD), computational physics in general, and experimental techniques. This book provides an overview of EBD and applications ranging from aerospace to microfluidics, fluid machinery, and others. It builds on past methods like exergy analysis and entropy generation minimization (EGM), by extending those methods to more complex configurations (not
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having analytical solutions), other applications (such as Second Law compliance in CFD), and measurement techniques. Local irreversibilities in fluids engineering systems (such as friction in fluid machinery) lead to reduced system efficiency. The entropy produced in fluid flow leads to pressure losses or other irreversible conversion of kinetic energy into internal energy. Past methods have often studied these losses of useful energy on a global scale, typically through a single loss coefficient (such as pressure measurements at the inlet and outlet of a valve). In contrast, this book outlines new advances, showing how local irreversibilities can be tracked in complex configurations, both numerically and experimentally, so that engineering devices can be redesigned locally to improve overall performance. An example is EBD with CFD applied to fluid motion through a turbine. In this example, flow losses arise from fluid friction along the blades, viscous mixing in the blade wakes and corner or tip vortices, as well as other flow recirculations with the channels between blades. The regions of highest entropy production at the inlet, blade inception, and wake regions identify the regions where the most substantial design improvements can be made. Examples of possible EBD modifications might include changes to the geometric parameters (blade shape and angles, height, curvature around leading edges), cooling holes (design, number, location), or inflow parameters (cooling mass flow rate, temperature). In these examples and others, computational and experimental techniques are needed to accurately predict the entropy transport processes. This book focuses on development of these techniques for EBD, including processes with turbulence, phase change, microfluidic transport, and other complex transport phenomena. Greg F. Naterer José A. Camberos
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Acknowledgments During the past several years, numerous colleagues and students have contributed in significant ways to the development and preparation of materials in this book. The authors would like to express their sincere gratitude for this valuable input, particularly to Kevin Pope (University of Ontario Institute of Technology, Oshawa, Canada), Olusola Adeyinka (Imperial Oil, Calgary, Canada), and Emmanuel Ogedengbe and Xili Duan (University of Manitoba, Winnipeg, Canada).
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Authors Greg Naterer, Ph.D., is a professor of mechanical engineering and a Canada research chair in Advanced Energy Systems at the University of Ontario Institute of Technology (UOIT), Oshawa, Canada. He is the director of Research, Graduate Studies, and Development in the Faculty of Engineering and Applied Science. He received his Ph.D. in mechanical engineering from the University of Waterloo, Canada, in 1995. His research interests involve design of energy systems, hydrogen technologies, and heat transfer, including more than 160 journal and conference publications in these fields. Dr. Naterer is currently leading an international research team, involving Atomic Energy of Canada, Argonne National Laboratory, and different universities across Ontario and abroad to build a copper–chlorine cycle for producing hydrogen from nuclear energy. The cycle aims to combine steam with intermediate copper and chlorine compounds in a sequence of steps to split water into hydrogen and oxygen. The Cu-Cl thermochemical cycle could be eventually linked with nuclear reactors to achieve higher efficiencies, lower environmental impact, and lower costs of hydrogen production than any other conventional technology. Dr. Naterer authored an earlier book entitled Heat Transfer in Single and Multiphase Systems (CRC Press, 2003). He has codeveloped two patents and supervised numerous M.Sc. and Ph.D. students, as well as research assistants and postdoctoral researchers. He has served in various administrative capacities with the Canadian Society for Mechanical Engineering (CSME), American Institute of Aeronautics and Astronautics (AIAA), and American Society of Mechanical Engineers (ASME). He is a fellow of CSME and an associate fellow of AIAA. José Camberos, Ph.D., works as an aerospace engineer for the U.S. Air Force Research Laboratory at Wright-Patterson Air Force Base, Dayton, Ohio. He is also an adjunct professor at the University of Dayton and the Air Force Institute of Technology. He received his Ph.D. in aeronautical and astronautical engineering from Stanford University, Stanford, California. His research interests include high-. performance computing, numerical analysis, and engineering applications of the Second Law of Thermodynamics. Dr. Camberos is currently working in the Multidisciplinary Technologies Center at the Air Force Research Laboratory to develop systems-level analysis, design, and optimization methods based on entropy production as a unifying element to quantify and improve system performance. He is also leading various efforts seeking to accelerate computational analysis and design, ranging from reconfigurable computing, symbolic computation, and advanced methods in the design of experiments. As an adjunct professor, he has also mentored several M.Sc. and Ph.D. students in aeronautics, hypersonics, and computational physics. He is a member of Sigma Xi, American Society for Engineering Education (ASEE), and an associate fellow of AIAA.
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1
Introduction
1.1 Ba c kg r o u n d Effective thermal and fluid system design often requires a creative, iterative, and open-ended process to meet multifaceted objectives of an engineering system. It provides concepts and specifications that will optimize the function, performance, and value of a system, for the mutual benefit of users and manufacturers. Some common tools for such design include computational fluid dynamics (CFD), computer-aided design (CAD), measurement techniques such as particle image velocimetry (PIV), and others. This book focuses on how entropy and the Second Law of Thermodynamics can enhance conventional design methods by providing an iterative methodology to reduce entropy production in a thermal system, thereby improving its energy efficiency. Industrial design methodologies were first adopted widely in the late 1930s and early 1940s, with prominent industrial designers such as Raymond Loewy, Norman Bel Geddes, and Henry Dreyfuss. The importance of their methods has risen steadily since that time for various reasons. Economics has been a key factor because a manufacturer’s profitability depends on the product price in the marketplace and manufacturer’s cost to produce it. As manufactured products become a commodity, cost savings are more difficult, and better industrial designs are needed to allow a product to gain higher profit margins. Also, good engineering designs can allow products to achieve certain attributes that are important for advertising and marketing purposes. With increased worldwide awareness that the world’s fossil fuel resources are limited, major efforts have focused on the design of more efficient and environmentally sustainable energy devices and processes. Energy systems are often thoroughly scrutinized for possible design improvements. Past conventional technology has generally detected energy losses on a system-wide or global scale, such as a single loss coefficient (i.e., valve loss coefficient). With the current state of this technology, the margins for improving the efficiency of existing devices can be relatively small. In this book, entropy-based design with local loss mapping is presented as a robust tool for reaching higher levels of system efficiency, thereby leading to energy savings in various industrial applications. The fundamental principles governing the design of energy systems are Newton’s law of motion and the laws of thermodynamics. Newton’s Second Law of Motion and the First Law of Thermodynamics are the cornerstones on which virtually all energy systems are built today. The other laws have played a secondary support role. A limitation associated with the First Law of Thermodynamics is that it tracks only the quantity of energy. In contrast, the Second Law tracks “quality” of energy, or its work-producing potential. Thus, the Second Law has the unique advantage of offering a systematic tool
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Entropy-Based Design and Analysis of Fluids Engineering Systems
for optimal energy usage and choice of technologies. The unique capabilities of the Second Law can be used to scrutinize flow irreversibilities locally, rather than globally. In this way, the problem regions can be clearly identified by the high entropy production rates, so designers can focus on those regions for improvements. A useful analogy is a sick patient telling a doctor that he or she is sick, without knowing the part of the body that is causing the ailment. Doctors can often use diagnostic tools to pinpoint the source. Similarly for a complex engineering system, large rates of entropy production within a device can identify problematic areas of concern because a commonly desired goal of devices is improving the efficiency through reduced entropy production. This goal is generally desired regardless of application, flow conditions, system parameters, and so on. Local exergy, or the work potential of a device, can be more readily interpreted physically than entropy production because it contains the same dimensional units as energy. It can be related directly to economic indicators. For example, multiplying the local cost of electricity (per kilowatt hour) by exergy destroyed by moving fluid through a valve over a year can indicate a yearly expense of wasted energy therein. This expense can be interpreted directly in terms of lost revenue. Thus, an economic framework can be based on local entropy production rates or exergy losses in a fluids engineering system. Furthermore, there exists a need for a standard metric from which the energy efficiency of all devices can be characterized. For example, fuel efficiency in a car is defined differently from that of a water heater’s efficiency, while still different than how a diffuser’s efficiency is defined, and so on. As a result, it is difficult for regulatory and government agencies to identify a standard method for identifying the energy wasted by a given device. Entropy production gives a single, measurable quantity that is directly related to the efficiency of any device that transforms energy because it characterizes degradation of useful (mechanical) energy to less useful (internal) energy. The utility of entropy and the Second Law have been widely documented in various disciplines, ranging from engineering fluid mechanics, to information and coding theory, economics, and biology. It will be emphasized frequently throughout this book how entropy serves as a key parameter in achieving the upper limits of performance and quality in many technologies. It can shed new light on various flow processes, ranging from optimized flow configurations in an aircraft engine to highly ordered crystal structures (low entropy) in a turbine blade, and other applications (Bejan, 1996). It is likely not possible to find any other law of nature, wherein a proposed violation would bring more skepticism than violation of the Second Law of Thermodynamics. Consider the implications of the Second Law in the thermal design of aircraft subsystems, involving work potential (Camberos, 2000a). Past authors have observed that there is no current systematic method for tracking work potential usage in the design of aircraft subsystems (Roth and Mavris, 2000). Exergy and entropy calculations can identify the loss of work potential within each subsystem and fluid flow process during an aircraft’s operation. These calculations can enable designers to identify key locations that incur the most significant losses. Moorhouse and Suchomel (2001) have discussed how flow exergy provides a unifying framework and set of metrics to more effectively analyze aircraft subsystems.
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Introduction
Various analytical methods have been developed over the past several decades for Second Law analysis. Notable examples include (i) estimation of the theoretical ideal operating conditions of a proposed design (called exergy analysis, or EA) and (ii) minimization of the lost available work or entropy generation by design modifications (called entropy generation minimization, or EGM; Bejan, 1996). Exergy quantifies the capacity of an energy source to perform useful work. It is a measure of the maximum capacity of an energy system to perform useful work as it proceeds to a specified final state in equilibrium with its surroundings. Exergy analysis focuses on closing the gap between maximum exergy and the actual work delivered by a device, through careful examination of the thermodynamic processes involved in a series of energy conversion steps (Dincer and Rosen, 2004). Subsequently, the exergy values at each point are used to evaluate Second Law efficiencies, which quantify the magnitude of irreversibilities (or exergy destruction) associated with the energy conversion process (Bejan, 1997; Rosen and Dincer, 2004). The method of EGM involves fluid mechanics, heat transfer, material constraints, and geometry, in order to obtain relationships between entropy generation and the optimal configuration. Typically, a functional expression for the entropy production in a process is derived (Poulikakos and Bejan, 1982; Zubair et al., 1987). Then the extremum of the derived expression that guarantees a minimum entropy generation is determined by methods of calculus. Because analytical methods are often limited to simplified geometries, this book extends analytical EGM to numerical and experimental methods. Opportunities for design optimization based on the Second Law can be enhanced through CFD as a design tool for complex problems and geometries. Entropy production can be obtained by postprocessing of the predicted flow fields (Sciubba, 1997). Many industrial problems in metallurgy, power generation, energy storage, aerodynamics, and other applications have been successfully solved by CFD. A designer can choose an optimum design from many possible alternatives at a remarkable speed using CFD. Combined EGM with CFD provides an emerging technology with promising potential for design optimization of practical industrial problems. For example, an application involving the design of air-cooled gas turbine blades was presented by Natalini and Sciubba (1999). The full Navier–Stokes equations of motion for turbulent viscous flow and the energy equations were solved with a finite element approach and a two-equation turbulence closure. By identifying the entropy generation rates corresponding to the fluid friction and heat transfer irreversibility, the authors determined which configurations had minimal thermodynamic loss in a turbine cascade. The computed flow field for pitched turbine blades (Kresta and Wood, 1993) can be postprocessed to identify regions of high local losses, thereby guiding engineers in local redesign of the blade profile to reduce such losses. Predictions of entropy production have been used in various other applications such as free convection in inclined enclosures (Baytas, 2000), mixed convection in a vertical channel with transverse fin arrays (Cheng et al., 1994), laminar and turbulent flow through a smooth duct (Demirel, 1999; Sahin, 2000, 2002), flow in concentric cylinder annuli with relative rotation (Mahmud and Fraser, 2002), and diffusers (Adeyinka and Naterer, 2005). These studies are examples of how entropy production computations can successfully complement CFD technology.
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Entropy-Based Design and Analysis of Fluids Engineering Systems
Industrial flow problems usually involve turbulence. Numerical predictions of entropy production in a turbulent flow were given by Moore and Moore (1983). Moore’s work was the first documented effort to develop a numerical model for turbulent entropy production. The Moore model assumes that turbulent fluctuations of the heat flux and viscous dissipation in the positive definite entropy equation can be modeled by the addition of a turbulent conductivity and turbulent viscosity to the molecular conductivity and viscosity, respectively. It has been used to predict the mean local entropy production in a bent elbow (Moore and Moore, 1983), turbulent plane oscillating jet (Cervantes and Soloris, 2002), and a jet impinging on a wall (Drost and White, 1991). A finite volume method for predicting the mean viscous dissipation and entropy production in turbulent flows, based on the time-averaged turbulence equations, was described by Kramer-Bevan (1992). In addition to the previous physical characteristics of entropy production, it can be interpreted alternatively in computational terms. Physical processes of viscous dissipation and heat transfer lead to entropy production. Past Second Law studies have shown how numerical procedures may also produce or destroy entropy, due to discretization errors, artificial dissipation, and nonphysical numerical results (Cox and Argrow, 1992; Naterer, 1999). Solutions of differential equations that do not satisfy an “entropy condition” may be characterized by a lack of uniqueness, oscillations, and other unusual behavior (Adeyinka, 2002; Hughes et al., 1985; and others). Cox and Argrow (1992) computed local entropy production with a finite difference method for compressible flow. Jansen (1993) and Hauke (1995) applied an entropy-based stability analysis to turbulent flows. Jansen (1993) showed that the exact Navier–Stokes equations for compressible flow could lead to an entropy inequality, through a linear combination of equations. The study determined what constraints the Second Law places on modeling of the averaged equations by linking entropy production to the solution variables. A major difficulty with numerical predictions can be the inability to ascertain error bounds. Solutions can be very sensitive to various parameters associated with the numerical algorithm (Naterer, 1999). This can make it difficult to judge the extent to which the computed results agree with reality. In numerical predictions of complex industrial flows, limited or no experimental data may be available for validation purposes. In these cases, checking where predicted entropy production rates are positive (realistic) or negative (unrealistic) is a valuable tool for verification.
1.2 G o v er n in g Eq u a t io n s o f Fl u id Fl o w a n d Hea t Tr a n sf er 1.2.1 Vec t o r
a n d Tenso r No t a t io ns
In this book, conventional notations for vectors and tensors will be used. A vector will be denoted by boldface font or a vector hat. A unit vector is a vector of unit magnitude. For example, i and j refer to the unit vectors in the x and y coordinate directions, i.e., (1, 0) and (0, 1), respectively. The symbol | v | designates the magnitude
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Introduction
of the indicated vector. When performing operations with matrices in this book, matrices are contracted when their individual entries are multiplied by each other and summed. For example, if a11 A= a21
a12 b11 ; B= a22 b21
b12 b22
(1.1)
then A : B = a11b11 + a12 b12 + a21b21 + a22 b22
(1.2)
Tensors are generalized notations for scalars (rank of zero), vectors (rank of 1), matrices (rank of 2), and so on. A tensor is denoted by a variable with subscripts. For example, aij represents the previously described matrix, where the range of subscripts is i = 1, 2 and j = 1, 2. When tensors use indices in this way, the notation is called indicial notation. The summation convention of tensors requires that repetition of an index in a term denotes a summation with respect to that index over its range. For example, in the previously cited case (dot product) involving two vectors, ui vi = u1v1 + u2 v2
(1.3)
The range of the index is a set of specified integer values, such as i = 1, 2 in the previous equation. A dummy index refers to an index that is summed, whereas a free index is not summed. The rank of a tensor is increased for each index that is not repeated. For example, aij contains two nonrepeating indices, thereby indicating a tensor of rank 2 (i.e., matrix).
1.2.2 Ma s s
an d
Mo men t u m Eq u a t io ns
The governing equations of fluid flow and heat transfer can be expressed in either vector or tensor notations. For two-dimensional flows, the mass conservation equation is given by
∂ρ ∂( ρu ) ∂( ρv) + + =0 ∂t ∂x ∂y
(1.4)
For incompressible flows, this equation may be simplified wherein that the divergence of the velocity field (∇ ⋅ v) equals zero. The divergence of velocity may be interpreted as the net outflow from a control volume (fully occupied by fluid), which must equal zero at steady state, because any inflows are balanced by mass outflows. The momentum equations represent a form of Newton’s law. Forces on a fluid element like pressure and shear forces balance the particle’s mass times its acceleration (i.e., total, or substantial derivative of velocity). The x-direction and y-direction
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momentum equations can be expressed as
∂( ρu ) ∂( ρuu ) ∂( ρvu ) ∂σ xx ∂τ yx + + = + + Fbx ∂t ∂x ∂y ∂x ∂y
(1.5)
(1.6)
∂ ( ρv ) ∂ ( ρuv ) ∂ ( ρvv ) ∂τ xy ∂σ yy + + Fby + + = ∂t ∂x ∂y ∂x ∂y
where Fb refers to a body force. These equations cannot be solved in this form because there are more unknowns (i.e., stresses, velocities, and pressure) than available equations. As a result, additional relations called constituitive relations between the stresses and velocities are needed. In Newtonian fluids, the stresses are proportional to the rate of deformation (or strain rate). For incompressible flows of Newtonian fluids, we have the following two-dimensional constitutive relations for stresses in terms of the pressure and velocity fields:
σ xx = - p + 2 µ
σ yy = - p + 2 µ
∂u ∂x
∂v ∂y
∂u ∂v = τ xy τ yx = µ + ∂y ∂x
(1.7)
(1.8)
(1.9)
Substituting these constitutive relations into the previous x-momentum equation and using continuity (mass conservation) to rewrite the left side,
ρ
∂2 u ∂2 u ∂u ∂u ∂u ∂p + ρu + ρv =+ µ 2 + 2 + Fbx ∂t ∂x ∂y ∂x ∂y ∂x
(1.10)
together with a similar y-momentum equation represents the two-dimensional Navier–Stokes equations. Analytical solutions of these equations are usually limited to simplified geometries because of the difficulties inherent in the nonlinear and coupled (with continuity equation) nature of the equations. Fluid flow regions are generally classified as viscous or nearly inviscid regions. In a viscous region, such as a boundary layer, frictional forces are significant. A boundary layer refers to the thin diffusion layer near the surface of a solid body, where the fluid velocity decreases from its freestream value to zero at the wall over a short distance. In contrast to viscous regions, frictional forces are often small in comparison to fluid inertia in regions far from a surface or boundary layer. The Euler equations are a special form of the Navier–Stokes equations for frictionless (or inviscid) flow. An inviscid fluid refers to an idealized fluid with no viscosity. In this situation, the terms involving viscosity are absent from the governing equations.
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Introduction
The fluid motion can be characterized as a potential flow, whereby the reduced governing equations can be written in terms of a scalar potential function.
1.2.3 En er g y Tr a nsp o r t Eq u a t io ns In addition to the fluid flow equations, energy is another transported quantity of key importance in the analysis of thermal and fluid systems. The mechanical energy equations can be obtained by multiplying each ui momentum equation by ui (where i = 1, 2 for two-dimensional flows) and adding them together. Using the substantial derivative notation, we obtain ∂τ xy ∂τ yy ∂τ yx 1 D 2 ∂p ∂p ∂τ (u + v 2 ) = - u + uFx + vFy +v -v + u xx + u +v ∂x ∂y 2 Dt ∂x ∂y ∂x ∂y
(1.11)
Using the product rule and generalizing to a vector notation, the following mechanical energy equation is obtained: 1 D 2 ρ (V ) = -[∇ ⋅ ( pv) - p∇ ⋅ v ] + [∇ ⋅ (τ ⋅ v) - τ : ∇v ] + v ⋅ F 2 Dt
(1.12)
where V = u 2 + v 2 refers to the total resultant magnitude of the velocity. The first term (left side) represents the rate of increase of kinetic energy of a fluid element with respect to time. On the right side, the second term gives the flow work done by pressure on the differential control volume to increase its kinetic energy. The third term represents an energy sink due to fluid compression in the mechanical energy equation, and it becomes zero for incompressible flows. The difference between the fourth and fifth terms on the right side gives the net fluid work done by viscous stresses to increase the kinetic energy of the fluid within the control volume. The latter portion represents work lost through viscous dissipation, which is a degradation of mechanical energy into internal energy through viscous effects. This viscous dissipation is represented by t: ∇ v, which refers to the viscous stress tensor contracted with the velocity gradient. For two-dimensional incompressible flows of a Newtonian fluid, it can be shown that the viscous dissipation term can be written as
2 ∂u 2 ∂v 2 ∂u ∂v τ : ∇v = 2 µ + + µ + ≡ µΦ ∂y ∂y ∂x ∂x
(1.13)
where Φ refers to the positive-definite viscous dissipation function. This function is greater than or equal to zero. As a result, the conversion of mechanical energy into internal energy through viscous dissipation is an energy sink in the mechanical energy equation. Thus, mechanical energy is not conserved, but instead a portion of this energy is degraded and lost to internal energy through viscous dissipation.
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It is degraded in the sense that a certain quality of energy is lost in the irreversible transformation, as internal energy normally has less ability than kinetic energy to perform useful work. The conservation of total energy (internal plus mechanical energy) is called the First Law of Thermodynamics. Performing a total energy balance on a differential control volume within the fluid stream, it can be shown that the total energy equation can be written as
ρ
D 1 eˆ + V 2 = -∇ ⋅ q - ∇ ⋅ ( pv - τ ⋅ v) + F ⋅ v + S Dt 2
(1.14)
where ê refers to internal energy and S is a source term. The rate of increase of total energy within the control volume equals the rate of energy addition by conduction, plus work done by pressure, viscous and external forces, plus internal energy generated per unit volume (S ). The internal energy equation can be derived by subtracting the mechanical energy equation from the First Law (total energy equation). Performing this subtraction and writing the results in a general vector form, we have
ρ
Deˆ = -∇ ⋅ q - p∇ ⋅ v + τ : ∇v + S Dt
(1.15)
where the fourth term (right side) refers to the viscous stress tensor contracted with the velocity gradient. It represents an internal energy source because it arises from the conversion of mechanical energy to internal energy through viscous dissipation. In the thermal energy equation, viscous dissipation represents an energy source, which corresponds to the energy sink previously observed in the mechanical energy equation. In other words, its magnitude is identical, but its sign changes in transposing from the mechanical to internal energy equations.
1.3 Ma t hema t ica l Pr o per t ies o f En t r o py a n d Ex er g y Numerous past studies have examined the significance of exergy as a measure of work potential or maximum useful work (Boehm, 1989b, 1992). A common aspect in all of these analyses is the identification of exergy with useful work potential. For example, Szargut et al. (1988) define exergy as “the amount of work obtainable when some matter is brought to a state of thermodynamic equilibrium.” Similar definitions were documented by Bejan (1996) and Kotas (1985). Although engineers have accepted the capacity to do work as a measure of quality of energy, this does not invalidate another, less anthropomorphic approach. By conceptualizing “exergy” as a distance functional, one eliminates the need to introduce additional terms also found in the literature (e.g., “anergy,” “essergy,” etc.) or to fragment exergy into multiple forms as often done with energy. Concise and critical reviews of the origins and history of exergy have been reported by Bejan (1996), Haywood (1974), Kotas (1985), and Szargut et al. (1988).
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Introduction
The previous section has outlined the governing equations for conserved variables of mass, momentum, and total energy. In this section, entropy and the Second Law will be formulated, particularly fundamental properties associated with the nonconserved variables of entropy and exergy. Thermodynamics began as the science of heat, intended to provide extended mechanics that would account for a common experience, namely, that doing work on a body sometimes makes it hotter, and sometimes heating a body causes it to do work (Truesdell, 1985). Common experience shows that mechanical action does not always result in a mechanical response, so we need to add the concept of heating alongside the concept of working or power. The Second Law is often expressed in terms of “work potential” or exergy. The balance of exergy equation represents a synthesis of the First and Second Laws. Exergy places all thermodynamic processes in a given system on the same basis by providing a common reference and metric. This section examines the essence of the Second Law of Thermodynamics as a statement involving the existence of entropy, with particular mathematical properties, from which a corresponding statement for the existence of exergy follows. It will be shown that exergy represents an abstract, mathematical distance functional. The concept of exergy will be interpreted as a thermodynamic functional representing the distance of a given system from the state of equilibrium at a reference state.
1.3.1 C o n c avit y Pr o per t y
o f En t r o py
The Second Law of Thermodynamics represents a natural foundation for thermophysical processes. The concept of entropy, however, is often viewed as abstract. A fundamental feature of the Second Law reflects a concavity property of entropy (Camberos, 2000a). Given a set of thermodynamic variables, ξ and ζ, there exists a functional, entropy, S = S (ξ, ζ) such that S is a concave function of its arguments. This framework can be useful to unify various formulations of the Second Law, including the principle of nonnegative entropy generation itself (Lavenda, 1991). Consider an example of a rigid material body at some temperature T immersed in a thermal reservoir at temperature T0 (e.g., a hot rock inside a cold room). Suppose T > T0 and we let the cooling process proceed from the initial time, t, to t0 when the body reaches thermodynamic equilibrium with its surroundings. The transfer of energy as the body cools equals
∫
t
t0
Qdt = U 0 - U
(1.16)
where U0 = U (T0) at a final time t0 and U = U (T) at the initial time t. The variables Q and U refer to heat transfer rate and internal energy, respectively. The Second Law of Thermodynamics requires that entropy is produced, but never destroyed, in an isolated system. Thus, Sgen ≥ 0 for an isolated system, where Sgen refers to the entropy generation. In the current example, the entropy flow associated with heat transfer is –Q/T0, so the entropy balance equation is Sgen = S0 - S
1 T0
∫
t
t0
Q dt
(1.17)
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where S = S(T) and S 0 = S(T0). Substituting Equation 1.16 into Equation 1.17, Sgen = S0 - S -
1 (U 0 - U ) T0
(1.18)
To write the change of energy in terms of temperature, we can use the definition of the specific heat (CV = ∂U/∂T). The entropy generated during the cooling process is then Sgen = S0 - S -
CV (T0 - T ) T0
(1.19)
Using standard thermodynamic relations between the specific heat and entropy (CV/T = ∂S/∂T), the expression for entropy generation becomes Sgen = S0 - S -
∂S (T0 - T ) ∂T 0
(1.20)
This expression indicates a concavity property of entropy as a function of T. To clarify the meaning of the concavity property, consider some arbitrary function F = F (X) such that F′′ < 0, where the inequality indicates that F is a concave function of its argument. Integration by parts requires
∫
X2
X1
( X - X1 )F ′′( X )dX = F ( X2 ) - F ( X1 ) - F ′( X2 )( X2 - X1 )
(1.21)
The result on the right-hand side has a geometric interpretation. Figure 1.1 illustrates the right side of the equation with a vertical line. Geometrically, we have
F ( X2 ) - F ( X1 ) - F ′( X2 )( X2 - X1 ) ≥ 0
(1.22)
where the equality holds if and only if X2 = X1. Comparing this result with Equation 1.20, it can be observed that positive entropy generation (the Second Law) is equivalent to asserting the concavity property of entropy when S = S(T). Consider another example of a simple compressible substance, subject to both heat transfer, Q, and work, W, when relaxing to equilibrium with an environment at T0, P0, where P0 = P(T0, V0). Solving for the heat flow from an energy balance and writing the net compression/expansion work of the gas in terms of pressure and a volume difference,
∫ Q dt = (U
0
- U ) + P0 (V0 - V )
(1.23)
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Introduction
S(T )
S(T0)
S(T0) – S(T) – S´0(T0 – T )
T
T0
F ig u r e 1.1 Downward concave function (entropy).
The entropy balance then becomes
Sgen = S0 - S -
P 1 (U 0 - U ) - 0 (V0 - V ). T0 T0
(1.24)
Alternatively, by substituting the appropriate thermodynamic relations and using Sgen ≥ 0, S0 - S
∂S ∂S (T0 - T ) (V0 - V ) ≥ 0. ∂T 0 ∂V 0
(1.25)
The inequality asserts the concavity of entropy as a function of T and V. Equality holds if and only if (T, V) = (T0, V0). Exergy represents the maximum work potential when bringing the system to equilibrium with its surroundings. In this example, it is given by
X = TO ( SO - S ) - CV (TO - T ) - PO (VO - V ).
(1.26)
Standard thermodynamic relations provide
CV ∂S = T ∂T
P ∂S = T ∂V
(1.27)
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Multiplying the entropy inequality in Equation 1.24 leads to
T0 ( S0 - S ) - (U 0 - U ) - P0 (V0 - V ) ≥ 0,
(1.28)
where CV (T0 - T ) can be interchanged with (U 0 - U ) . Identifying the left side as exergy and taking the time rate of change,
X = -T0 S + CV T + P0V .
(1.29)
Also, from the entropy balance equation for this problem,
∂S ∂S T+ V. S = Sgen + ∂T ∂V
(1.30)
Substituting Equation 1.30 into Equation 1.29 and replacing terms defined by and ∂S/∂V = P/T yields
∂S / ∂U = 1 / T
T P X - 1 - 0 CV T - P0 - T0 V = -T0 Sgen . T T
(1.31)
From the definition of exergy, it can be shown that the following thermodynamic relations hold:
P ∂X = P0 - T0 ; T ∂V
∂X T0 = 1 - CV . T ∂T
(1.32)
Because the entropy generation is nonnegative, the previous relations yield
∂X ∂X X TV ≤ 0. ∂T ∂V
(1.33)
This result asserts the mathematical property of convexity for X = X (T, V). Thus, the concavity of entropy is equivalent to the convexity of exergy. Figure 1.2 shows an example of exergy as a convex function of temperature. A geometric complementary relation exists between entropy and exergy, as shown by the concavity inequality for entropy and the line segment that defines exergy (see Figure 1.3). Exergy has an absolute minimum at the point of equilibrium. The tangent slope at this point coincides with the horizontal axis of zero exergy. The straight vertical line from an arbitrary initial state in Figure 1.2 represents the corresponding distance to equilibrium. The distance to equilibrium is represented equally well by the vertical line shown in Figure 1.1 (concavity of entropy) or the horizontal line in Figure 1.3 (geometric representation of exergy).
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X (T )
Introduction
X(T )
X‚ X´(T0) = 0
T
T0
F ig u r e 1.2 Exergy function as a convex function of T.
S´(CVT0)
S (T )
S0 – S
CV (T0 – T )
X
CVT
CVT0
F ig u r e 1.3 Geometric representation of entropy.
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1.3.2 D is t a n c e Fu n c t io n a l
w it h Resp ec t t o Eq u il ibr iu m Co n d it io ns
In addition to its convexity property, exergy may be interpreted as a thermodynamic metric or distance functional. Define a metric, x, based on the Hessian of entropy (second-order tensor of derivatives with respect to temperature and volume) and the following inner product,
( x, x ) ≡ x T ⋅ ( - Sxx )0 ⋅ x
(1.34)
where
∂2 S 2 ∂T ( Sxx )0 = ∂2 S ∂T ∂V
∂2 S ∂T ∂V 2 ∂ S ∂V 2 0
(1.35)
represents the Hessian. The superscript T refers to matrix transpose and x T ≡ (T , V )
(1.36)
represents an algebraic vector of the corresponding thermodynamic variables. The inner product of (x, x) results in nonnegative values, as guaranteed by the concavity property of entropy. A general mathematical metric can be defined by the following distance functional:
|| x - x0 || ≡ ( Dx, Dx )1/2
(1.37)
where Dx = x - x0. To clarify the importance of these equations with respect to exergy, consider the construction of the norm || x ||, which requires evaluation of the second-derivative terms of the Hessian. Thermodynamic relations for a simple compressible substance (Bejan, 1996) give ∂2 S C = - V2 2 ∂T T
(1.38)
∂2 S 1 (κ P - βT ) =κT 2 ∂T ∂V ∂2 S 1 =κT 2 ∂V 2
κ P 2 - 2 βTP CP T + CV CV V
(1.39)
(1.40)
where
β≡
1 ∂V V ∂T
(1.41)
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Introduction
and
κ ≡
1 ∂V V ∂P
(1.42)
represent the volumetric coefficients of thermal expansion and isothermal compressibility of the gas, respectively. Using these definitions to calculate the Hessian yields the following inner product: ( x, x ) =
C V2 1 κ H 2 - 2 β HV + P κ CV T0 T0 V0
(1.43)
where H = CVT + P0V. Consider two cases: an ideal gas and an incompressible substance. For an ideal gas,
κP =1
βT = 1
(1.44)
and the inner product simplifies to 2
T P 2V 2 C P V 2 ( x, x ) = CV - 0 2 + P 0 CV T0V0 T0 CV T0
(1.45)
Simplifying further with the ideal gas equation of state, ˆ PV = RT
(1.46)
together with the following relations: CV =
γ Rˆ Rˆ C CP = γ ≡ P γ - 1 γ - 1 CV
(1.47)
yields
2 2 V Rˆ T ˆ ( x, x ) = + R γ - 1 T0 V0
(1.48)
where Rˆ = mR/M and m, M, and R are the mass, molecular weight, and universal gas constant, respectively. The subscript O denotes reference conditions for pressure and temperature. Replacing (U, V) with the corresponding differences (U - U 0 , V - V0 ) yields the following square of a true mathematical distance functional, || x0 - x ||2 =
2 2 Rˆ T ˆ V + 1 1 R V γ - 1 T0 0
(1.49)
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Following the same procedure for an incompressible substance, where κ , β → 0, and CP , CV → C , yields T || x0 - x || = C 1 - T0
2
2
(1.50)
This result of a mathematical distance functional can be directly related with exergy. Consider a system near equilibrium conditions and expand the entropy function in a Taylor’s series. Using the previous definition of xT = (T, V) and neglecting higher-order terms, S ≈ S0 +
∂S ∂S T (T - T0 ) + (V - V0 ) + 1 2 ( x - x0 ) ⋅ Sxx 0 ⋅ ( x - x0 ) ∂T 0 ∂V 0
(1.51)
Using the definition of exergy from the previous section, entropy derivatives in terms of exergy and the previous result for the distance functional, it can be shown that
2 X≈1 2 T0 || x - x0 ||
(1.52)
This formulation of exergy as a distance functional with respect to equilibrium conditions provides a more systematic and mathematically rigorous interpretation than various definitions of “work potential” in undergraduate textbooks. The previous results show that exergy represents a physical measure of the distance from equilibrium conditions for a system at some arbitrary state. The convexity and distance functional properties of exergy have been presented here to aid understanding of exergy. The concept of exergy has been interpreted through a connection between a system and its environment. Standard textbooks often introduce and discuss “availability” or exergy in the context of “a system’s potential to do work in a reversible manner.” Many modern texts (such as Cengel and Boles, 1989; Müller, 1985) also introduce a number of work terms (reversible work, available work, etc.) in an effort to clarify and expand on the subject. However, this can lead to more confusion and cluttering of terminology. This situation was observed more than 30 years ago (Haywood, 1974a,b). Second Law analyses have found well-deserved attention (Szargut et al., 1988), but the cluttering of terminology and obscurity in the definitions often remain. By providing fundamental mathematical properties of exergy as a state variable, this section has provided a valuable alternative interpretation. The Second Law has deep and significant implications for engineering systems. As future machines become increasingly complex and sophisticated in their ability to transform energy into various forms, exergy and the Second Law will have an increasingly important role in prescribing their upper limits of performance. Since the industrial revolution, the Second Law served only a secondary role by prescribing what the real physical world allowed. Complex machines of the future will require a more interconnected relationship, as they press toward the maximum limits
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Introduction
of engineering design, precisely where the true power of the Second Law becomes evident. Therefore, it is essential that fundamental properties of entropy and exergy as state variables are well understood. Systematically developing the foundations of the Second Law from the essence of entropy as a concave function of state variables, we can advance that concept, together with a simplicity that will make it possible for future engineers and scientists to achieve what we can now only imagine.
1.4 G o v er n in g Eq u a t io n s o f En t r o py a n d t he Sec o n d La w The First and Second Laws are physical principles governing all thermophysical processes, and the addition of constitutive relations describes the response of various classes of materials (Truesdell, 1984, 1985). As discussed in the previous section, a general axiom of thermodynamics postulates the existence of a concave thermodynamic variable called entropy. The Second Law then stipulates that the rate of entropy generation must be nonnegative in all thermophysical processes, that is, S gen ≥ 0. The mathematical property of concavity implies certain restrictions on the constitutive relations for any material body. This section will use this property to develop the governing equations for entropy and the Second Law.
1.4.1 C l o se d Sys t em For a closed system, let xk represent independent variables in the constitutive functional relation, such that U = U (xk), W = W (xk), etc. Then the mathematical expression for the Second Law can be written as S
∂S
∑ ∂ξ ξ - S
gen
=0
k
(1.53)
For S = S (U , V ), the Second Law becomes
∂S ∂S S UV - Sgen = 0 ∂U ∂V
(1.54)
Simplifying this result by using thermodynamic relations for the derivatives (CV/T = ∂S/∂T, P/T = ∂S/∂T, and CV = ∂U/∂T) and the First Law,
Q +W P S - V - Sgen = 0 T T
(1.55)
Relating the work term and third term yields
Q S - - Sgen = 0 T
(1.56)
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which represents the entropy balance for a closed system from classical thermodynamics. This result is well known, although the previous derivation has shown that it follows from the concavity property of entropy. High-quality energy refers to energy from which a great amount of useful work can be extracted, so exergy is used to refer to the work potential of that energy. Lower quality energy like internal energy can produce less work and therefore reflects also lower exergy. Thus, exergy quantifies a qualitative aspect of energy. In standard practice, to derive an equation representing the balance of exergy, one typically considers a closed system at some uniform arbitrary state, (P, T), relative to ambient conditions at (P0, T0). To measure the distance of a system from the reference or so-called “dead state,” imagine a reversible process whereby the system relaxes to thermodynamic equilibrium with the surroundings. The energy balance equation simplifies to U in - U out = DU
(1.57)
where the total change in energy is
∫
DU = U (t2 ) - U (t1 ) =
t0
U dt
t
(1.58)
A closed system relaxes to equilibrium with its surroundings through work and heat transfer. Integrating the balance of energy equation over time,
∫
t
t0
Q dt +
∫
t
t0
W dt = DU
(1.59)
with the integral limits defined at an initial time when the system is at (P, T) and the final time when the system has reached equilibrium with the surroundings at (P0, T0). To replace the heat interaction term with a state variable, one can use the following definition of entropy: 1
∫ S dt = T ∫ Q dt
⇒
T0 ( S0 - S ) =
0
∫ Q dt
(1.60)
For the work term, the energy quality directly relates to the useful work extracted. It is the maximum amount of work done during a thermodynamically reversible process. For a simple compressible substance, - P V dt )Vdt ∫ W dt = -∫ PV dt = -∫(P-P ∫ 0
Wuseful
(1.61)
0
The first term on the right-hand side defines the maximum “useful work” available, and the second term represents the work done by the ambient pressure acting on a
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Introduction
moving boundary. Substituting the previous two equations into the energy balance, T0 ( S0 - S ) - Wuseful - P0 (V0 - V ) = U 0 - U
(1.62)
where the subscripts indicate the value at reference conditions. Solving for the useful work term gives Wuseful = T0 ( S0 - S ) - (U 0 - U ) - P0 (V0 - V )
(1.63)
which is equivalent to the exergy defined earlier as X ≡ T0 ( S0 - S ) - (U 0 - U ) - P0 (V0 - V )
(1.64)
Dividing by the total mass gives the specific exergy (in other words, exergy per unit mass):
φ ≡ T0 ( s0 - s ) - (u0 - u ) - P0 ( v0 - v)
(1.65)
Generalizing to include kinetic and gravitational potential energy requires only that we replace u with e, which is the total specific energy given by e = u + 12 V 2 + gz . Typically, V0 = 0 and z0 = 0 at the reference state. With the exergy defined in this manner, one can study the change in exergy when the state of a system changes. As a system undergoes a process from one thermodynamic state to another, a corresponding change in exergy occurs. Combining the First and Second Laws as expressed by the energy and entropy balance equations for a compressible substance of fixed mass leads to an exergy balance equation. In integral form, the First and Second Laws become
∫
∫
L1 : Q dt + W dt = U 2 - U1
(1.66)
and L2 :
Q
∫ T dt + S
gen
= S2 - S1
(1.67)
Combining by taking L1 - T0 L2 gives Q
∫ Q dt + ∫ W dt - T ∫ T dt - T S 0
0 gen
= U 2 - U1 - T0 ( S2 - S1 )
(1.68)
Collecting terms and replacing the right-hand side with equivalent terms using the definition of exergy gives
∫ 1 - T Q dt + {∫ W dt + P (V - V )} - T S
T0
0
2
1
0 gen
= X2 - X1
(1.69)
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Other than a difference in sign on the work term, Equation 1.69 is a typical result found in textbooks on elementary thermodynamics (Cengel and Boles, 1989). If we identify the first term on the left side as “exergy transfer due to heat interaction” in the same sense that we identify entropy transfer due to heat interactions, and the second term (in brackets) as the “exergy transfer due to work interaction,” then Equation 1.69 reduces to the following result: I X - Xdes = DX
(1.70)
where Ix is the exergy current due to work and heat transfer and Xdes = T0 Sgen is the exergy destruction (called the Gouy–Stodola identity). Alternatively, it is useful to interpret the quantity expressed by Xdes as the distance from which the system approaches thermodynamic equilibrium with its environment. Recognizing the Second Law through the increase of entropy principle, it is required that > 0 T0 Sgen = 0 < 0
Real World Ideal World Impossible
(1.71)
Because we have associated exergy as equivalent to a measure of work potential, this term can be described as “exergy degeneration” or a loss of potential work due to real-world, irreversible effects. Entropy generation has a corresponding destruction of exergy: Xdes = T0 Sgen
(1.72)
A system in the real world undergoes spontaneously a process that brings it closer to thermodynamic equilibrium with its surroundings.
1.4.2 O pen Sys t em During an unsteady process where a substance goes from an initial (inlet; subscript “in”) to a final (exit; subscript “out”) state, the quality of energy changes, and a corresponding change occurs in its thermodynamic distance from equilibrium. Combining the First and Second Laws as expressed in the energy and entropy balance equations for an unsteady process, one may obtain the balance equation for exergy. This derivation is commonly provided in undergraduate thermodynamics textbooks. The First Law for a control volume can be expressed as
∑Q
k
+W +
∑ m h + 2 V 1
2
+ gz in
∑ m h + 2 V 1
2
+ gz = E CV out
(1.73)
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h, and Ecv refer to the mass flow rate, enthalpy and time rate of change of where m, exergy in the control volume, respectively. The corresponding form of the Second Law is
∑ T + ∑ ms - ∑ ms Qk
in
k
out
+ Sgen = SCV
(1.74)
Combining these equations yields
∑ 1 - T Q + W +∑ m (h + 12 V T0
k
∑ m (h + 12 V
2
+ gz
)
out
2
+ gz
)
in
(1.75)
d - T0 S gen = ( E - T0 S )CV dt
If we define a specific “flow exergy,” y, in the same sense that enthalpy represents a “flow energy,” then the exergy balance equation simplifies to
∑ 1 - T Q T0
k
+ (W + P0V ) +
∑ m ψ -∑ m ψ in
out
- T0 S gen = X CV
(1.76)
where
ψ = φ + ( P - P0 )v
(1.77)
and f is the specific exergy, (e - e0) + P0 (v - v0) - T0 (S - S 0). Identifying the first two terms on the left side of Equation 1.76 as the transfer of exergy due to work and heat transfer, respectively, and the third and fourth terms as exergy transfer due to mass flow, the exergy balance equation for a control volume reduces to
X in - X out - T0 S gen = X CV
(1.78)
where “in” and “out” terms represent the flow of exergy into and out of the control volume. In addition to its role in determining the direction of natural processes and a criterion for thermodynamic equilibrium, the Second Law can also characterize the efficiency of engineering devices (Bejan, 1996). Carnot (1960, English translation from French by R.H. Thurston) conceived and developed the Second Law to account for the performance and limits of heat engines. Isentropic efficiency characterizes the performance of various engineering devices, such as turbines and compressors. In the context of exergy, the Second Law defines a more general measure of performance that applies not only to turbines and compressors, but heat exchangers, mixing processes, and other devices. A measure of performance for any engineering device should compare its efficiency, relative to the efficiency of an ideal device (no irreversible losses) operating under the same conditions. This measure of performance is called the “Second Law efficiency” or effectiveness, which can be
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defined as follows, ηII ≡ 1 - X des X supp
(1.79) where the subscripts “des” and “supp” refer to destroyed and supplied, respectively. Past literature has interpreted this measure of performance as a “rational efficiency” of a process or device (Müller and Ruggeri, 1998). Essentially, the effectiveness of any process equals the fractional change in the exergy relative to the exergy supplied. The concept of effectiveness applies to any thermophysical process, including heat engines, refrigerators, heat exchangers, mixing, throttling, and so forth. It is always bounded between zero and one. The Second Law of Thermodynamics will: (i) determine the direction of change for spontaneous, natural processes; (ii) establish criteria for equilibrium in thermodynamic systems; and (iii) provide the theoretical limits for the performance of engineering systems and processes. Items (i) and (ii) identify the role of the Second Law as a limiter in abstracting the differences in response of different materials via the constitutive relations. Item (iii) identifies the role of the Second Law enumerated by the concept of effectiveness, as a limiter to indicate how a system relaxes to equilibrium conditions with its surroundings while producing or consuming work.
1.5 F o r mu l a t io n o f En t r o py Pr o d u c t io n a n d Ex er g y Des t r u c t io n In the previous section, formulations of entropy transport and the Second Law were developed. In those equations, entropy production and exergy destruction were key parameters that characterized the efficiency of the thermal system or device. In this section, detailed expressions for these parameters will be developed, from which design methodologies can be established to reduce and minimize entropy production, thereby optimizing system performance.
1.5.1 C l o se d Sys t em From Section 1.3.1, for a closed system exchanging energy with its surroundings through work and heat transfer, the exergy balance equation can be expressed as
. . . T0 1 - Q + (W + P0 V ) - T0 S gen = X T
(1.80)
Substitution for the heat flow and work term (relating compression/expansion work to pressure and change of volume) leads to
. . . . . T0 . 1 - (U + P V ) - P V + P0 V - T0 S gen = X T
(1.81)
Using thermodynamic relations for the exergy gradients leads to the following similar result as Equation 1.33,
∂X ∂X -T0 S gen = X UV ∂U ∂V
(1.82)
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which is similar to results derived previously for the convexity property of exergy in Section 1.3.1. The convexity property of exergy is intrinsically linked with the Second Law.
1.5.2 L in ea r Ad vec t io n Eq u a t io n (w it h o u t Dif f u s io n ) Entropy is transported throughout a problem domain through advection of scalar quantities like fluid momentum and internal energy. For example, scalar transport of fluid momentum leads to frictional irreversibilities, while transport of internal energy involves convective heat transfer and thermal irreversibilities. In this section, the exergy balance equation is developed with respect to scalar transport of a general scalar quantity, h(x,t). The governing equation for the one-dimensional scalar advection equation without diffusion is given by ∂η ∂f + =0 ∂t ∂x
(1.83)
which represents pure advection and f(h) equals the “flux of h.” According to the entropy concavity principle, the corresponding balance of entropy is given by ∂S ∂η - S′ ≥0 (1.84) ∂t ∂t Substituting for the second term using Equation 1.83 and applying the chain rule, ∂S ∂η + S ′f ′ ≥0 ∂t ∂x
(1.85)
The one-dimensional form of the entropy transport equation can also be expressed as S gen =
∂S ∂F + ∂t ∂x
(1.86)
where F represents the “transfer of entropy with h.” It is a term arising from pure convective transport of h. Subtracting Equation 1.86 from Equation 1.85 and using the chain rule for ∂F/∂x gives
( F '- S ' f ')
∂η ≥0 ∂x
(1.87)
The strict equality must be enforced to avoid violation of the Second Law, so a compatibility condition, F ′ = S ′f ′, is obtained as a constitutive restriction required by the Second Law. This result implies
∂S ∂F + =0 ∂t ∂x
(1.88)
and it follows that Sgen = 0. This result is well known that reversible processes have zero entropy generation, although the previous derivation shows an additional requirement of compatibility between derivatives of entropy and its flux, F.
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The corresponding balance of exergy equation is given by ∂X ∂η - X′ ≤ 0. ∂t ∂t
(1.89)
Substituting for the second term using Equation 1.83 and applying the chain rule, ∂X ∂η + X ′f ′ ≥0 ∂t ∂x
(1.90)
When the balance of exergy is written in an analogous form as entropy transport, X gen =
∂X ∂G + ∂t ∂x
(1.91)
where G represents the “transfer of exergy with h.” It is a term resulting from the purely convective transfer of h across boundaries. Subtracting Equation 1.91 from Equation 1.90 and using the chain rule for ∂G/∂x gives
(G '- X ' f ')
∂η ≤0 ∂x
(1.92)
In this case, the strict equality to satisfy the Second Law leads to an exergy compatibility condition, G ′ = X ′f ′, which is a constitutive restriction required by the Second Law. Also, it leads to
∂X ∂G + =0 ∂t ∂x
(1.93)
and it follows that X des = 0. In the next section, the previous procedure will be extended to scalar advection, including diffusion.
1.5.3 L in ea r Ad vec t io n Eq u a t io n (w it h Dif f u s io n ) In this section, a similar procedure will be used to derive the exergy destruction rate corresponding to scalar advection with diffusion. The governing equation for onedimensional advection with diffusion is
∂η ∂f ∂ 2η + =D 2 ∂t ∂x ∂x
(1.94)
where F = ch and c equals a constant advection velocity. The variable D refers to a diffusion coefficient. The corresponding balance of exergy is given by
∂X ∂η - X′ ≤0 ∂t ∂t
(1.95)
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Substituting for the second term using Equation 1.94 gives
∂X ∂f ∂ 2η + X′ - DX ′ 2 ≤ 0 ∂t ∂x ∂x
(1.96)
Applying the compatibility condition and the chain rule, 2
∂X ∂G ∂ ∂η ∂η + - D X ′ + D X ′′ ≤ 0 ∂t ∂x ∂x ∂x ∂ x X dees
(1.97)
where G = cX. The exergy destruction term is labeled because it is the only term that must be nonnegative. To preclude any violation of the Second Law, the strict equality must be enforced because the magnitude of all terms on the left side is not known beforehand. Because X is convex in h, then X″ < 0. Hence, we arrive at two expressions for the Second Law corresponding to scalar advection with diffusion: 2
∂η X des = D X ′′ ∂x
(1.98)
and X des = D
∂ ∂η ∂X ∂F + X′ - ∂x ∂x ∂t ∂x
(1.99)
When imposing the principle of nonnegative exergy destruction, the first expression represents a constitutive restriction on the diffusion parameter: D ≥ 0 . The second expression represents the true exergy balance equation for this process. The third term contains the effects of the diffusive flux (diffusive transport of h such as fluid friction or heat conduction).
1.5.4 N avier –S t o k es Eq u a t io ns Since the Euler equations represent inviscid fluid motion, they are limiting cases of the Navier–Stokes equations, which describe the dynamic motion of a viscous, heatconducting fluid. The Navier–Stokes equations can be expressed in the following tensor form, ∂ρ ∂ρV j + =0 ∂t ∂x j
(1.100)
∂E ∂ [ EV j + PV j - τ jiVi - q j ] = 0 + ∂t ∂x j
(1.101)
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Entropy-Based Design and Analysis of Fluids Engineering Systems
∂E ∂ [ EV j + PV j - τ jiVi - q j ] = 0 + ∂t ∂x j
(1.102)
where Vj refers to the velocity component in the j-coordinate direction and where E = ρu + 12 ρV 2 represents the total energy (internal plus kinetic energy). These equations are underdetermined, because they contain more unknowns than equations. Consequently, additional information is required. The constitutive relations provide the additional closure information. Typically, these include the ideal gas law, P = ρ RT , the assumption of a thermally perfect gas (cv depends only on temperature, T), and Fourier’s relation for heat conduction, qj = - k
∂T ∂x j
(1.103)
Also, the following constitutive relations will be used for the viscous stress tensor of a Newtonian fluid,
∂V j ∂Vi ∂V j δ ji + +λ ∂x x ∂x j ∂ i j
τ ji = µ
(1.104)
where d ij is the Kronecker delta function. The entropy transport equation associated with processes modeled by the Navier–Stokes equations is S gen =
∂S ∂Fj ∂ qj + + ∂t ∂x j ∂x j T
(1.105)
where S = ρs and Fj = ρsV j. Because one more unknown has been added (specific entropy, s), another constitutive relation is needed, namely, the functional relation between entropy and the other field variables. This relation must satisfy the concavity property of entropy. For an ideal gas, the entropy functional (from thermodynamics; written in nondimensional form) is s ( ρ, T ) =
1 ln T - ln ρ γ -1
(1.106)
where g is the ratio of specific heats. Because the state variables include mass, momentum, and total energy, it is convenient to define an algebraic state vector, qT ≡ {ρ, ρV1 , ρV2 , ρV3 , ρ E}, so that the entire set of conservation equations reduces to
∂q ∂fk + =0 ∂t ∂xk
(1.107)
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where f is an algebraic flux vector. The functional relation for the entropy is S = S(q), and the concavity property is expressed as S0 - S
∂S ⋅ ( q0 - q ) ≥ 0 ∂q 0
(1.108)
where equality holds if and only if q = q0. The transient generalization of Equation 1.108 is the following extension of Equation 1.53 for the state variables: ∂S ∂S ∂q ⋅ ≥0 ∂t ∂q ∂t
(1.109)
Replacing ∂q/∂t and rearranging terms yields the following entropy generation rate: S gen =
τ ji ∂Vi q j ∂T T ∂x j T 2 ∂x j
(1.110)
This result may be obtained by other means (Müller, 1985), but the approach here aims to emphasize the intrinsic connection between entropy concavity and the entropy generation equation in the Second Law of Thermodynamics. Given the Fourier relation and the formula for the viscous stress tensor, the Second Law requires that
µ≥0
k≥0
λ + 23 µ ≥ 0
(1.111)
These results are well known, and they have been documented in past literature dealing with thermodynamics and kinetic theory (Bird, 1976, 1994; Chapman and Cowling, 1990; Müller and Ruggeri, 1998). The origin of the inequalities arrives from the mathematical property of entropy concavity, as a function of the field variables. Two expressions were obtained for the entropy generation: one that places restrictions on the constitutive relations; the other represents the entropy transport equation, Equation 1.105. The corresponding exergy destruction can be obtained by the Gouy–Stodola theorem. It may also be obtained directly by defining exergy from the concavity of entropy and then constructing the appropriate balance equation. Consider the one-dimensional Navier–Stokes equations with the flux vectors separated into convective and dissipative parts as follows:
∂q ∂f ∂f v + + =0 ∂t ∂x ∂x
(1.112)
where 0 f v = -τ -V τ + q
τ=
4 ∂V µ 3 ∂x
∂T q = -k ∂x
(1.113)
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From Equation 1.106 and S = ρs, the row vector of entropy derivatives is γ V2 V 1 S ,q = s + ,- , 2T γ - 1 T T
(1.114)
Then the exergy can be written as X ( q ) ≡ T0 [S0 - S - S,q0 ⋅ ( q0 - q )]
(1.115)
where the subscript comma notation refers to differentiation. The exergy balance equation and exergy destruction rate can be obtained by starting with the convexity relation for exergy in Section 1.3.1, and substituting for the time derivatives to give ∂X ∂q ∂X ∂f ∂f v - X ,q ⋅ = + X ,q ⋅ + ∂x ∂x ∂t ∂t ∂t
(1.116)
where X ,q = T0 ( S,q0 - S,q ). Invoking the chain rule and the corresponding compatibility condition leads to X ,q ⋅
fS ∂q ∂q ∂G ⋅ = G ,q ⋅ = ∂q ∂x ∂x ∂x
(1.117)
Also, note that X ,q ⋅
∂f v ∂f v ∂f v = T0 S,q0 ⋅ - T0 S,q ⋅ ∂x ∂x ∂x
(1.118)
and
0 ∂ Tγ ∂ T0 S,q0 ⋅ f v x = T0 s0 - 0 , 0, 1 ⋅ -τ = (q - V τ ) γ -1 ∂x ∂x -V τ + q
(1.119)
In addition, the following equation can be derived:
S ,q ⋅
∂f v q ∂T ∂ q τ ∂V = + ∂x ∂x T T ∂x T 2 ∂x
(1.120)
Substituting these results into Equation 1.116 leads to
∂X ∂S ∂V τ ∂ T0 T0 ∂V T0 ∂T + + 1 - q + τ - 2q =0 T ∂t ∂U ∂x ∂x T ∂ x T ∂ x exergy destruction
(1.121)
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This identifies the exergy destruction term, and by comparison with Equation 1.110, note that X des = T0 S gen . The formula for the exergy destruction becomes
T0 ∂ ∂X ∂G ∂V τ + X des = - 1 - q + T ∂t ∂x ∂x ∂x
(1.122)
Because the equations of fluid flow presume local thermodynamic equilibrium, there is no inconsistency when applying classical thermodynamic principles as represented by entropy concavity and exergy convexity. These principles are mathematical properties, not limited to thermodynamics. To apply the Second Law of Thermodynamics for availability analyses in practice requires the balance of exergy equation and a functional formula for exergy. The construction of the entropy and exergy balance equations has been derived without specifying an entropy formula, except for the case of an ideal gas. To obtain the proper formula for S = S (ξ ), general optimization principles can be applied. For example, Jaynes’ Maximum Entropy Principle (Jaynes, 1991; Levine and Tribus, 1979) is based on a generalization of the Second Law, when applied to constrained equilibria. Kapur and Kesavan (1992) provide a comprehensive and detailed procedure for generalized entropy optimization principles. If the domain of the dependent variable x is known, then the Maximum Entropy Principle obtains the proper form of entropy for a probability distribution function that quantifies fluctuations in that variable about its mean value. For example, if ξ ∈[0, ∞ ) , then the MaxEnt principle prescribes S = ln ξ . If the dependent variable u ∈ ( - ∞,∞), then MaxEnt prescribes S = -u 2 /2σ 2 , where 2σ 2 = 1 is set without loss of generality. The Second Law, in essence, provides a way to (i) obtain a formula for entropy, and (ii) construct the balance equation for entropy (Liu, 1972; Müller, 1967). The mathematical property of entropy concavity has served multiple purposes, including restricting the types of constitutive relations allowed for modeling of realworld processes. The Second Law is a powerful concept that determines how physical processes can be modeled, so that mathematical models reflect physical reality. This chapter has developed formulae for the balance of entropy and exergy, as required to enforce the restrictions prescribed by the Second Law. The advantage of using exergy balances (instead of entropy) is that they provide a concept that unifies the First and Second Laws into a single principle. This unified approach provides the basis for constructing a single metric across the spectrum of possible thermophysical systems and processes.
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Statistical and Numerical Formulations of the Second Law
2.1 In t r o d u c t io n The foundations of the Second Law of Thermodynamics are based on statistical thermodynamics of intermolecular exchange. However, its impact extends from the microscopic scale to the largest scales of engineering systems and, more generally, the environment and the earth. Dincer (2001) presented a detailed study that outlines the key role of exergy in environmental sustainability of energy systems. The environmental impact of waste emissions and power generation systems can be effectively characterized by methods of exergy analysis (Rosen and Dincer, 1997). At the microscopic level, a close relationship exists between the concepts of entropy and probability, the most well known of which is associated with Ludwig Boltzmann. The concavity property of entropy is directly related to a given probability distribution function (PDF) for a fluid. In this chapter, statistical formulations of the Second Law and the Clasius–Duhem (C-D) inequality will be described. The C-D inequality represents the irreversible increase of entropy required by the Second Law of Thermodynamics, and it is a supplementary equation in fluid mechanics. By relating entropy directly to a PDF, one can show that a nonequilibrium version of the entropy function (and also a modified C-D inequality) can be obtained. These probability functions will be outlined in this chapter. Some of the concave entropy functions obtained for the nonequilibrium functions will be shown to be less than or equal to the entropy associated with the equilibrium value, in accordance with the Second Law. Entropy and probability are intrinsically related. However, no general agreement exists among scientists as to what this relation means or even exactly what is the relationship. Jaynes asserts that probability is a “logic of science.” In this way, probability theory (as logic) may be applied to any field of science. Posing questions or problems from one scientific field in terms of concepts and principles from another can prove fruitful, if properly directed. This chapter attempts to frame some fundamental questions regarding the statistical aspects of fluid motion, in terms of the logic of probability. Some of the issues include establishing criteria for numerical techniques based on physical principles, within the logical framework of probability theory, as well as deriving practical mathematical expressions and formulae for implementing the results. Because kinetic theory utilizes many concepts and principles of probability and statistics, there is a wealth of ideas to gain from this
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field, as well as inspiration. This chapter will derive macroscopic rules, principles, and formulae by statistical averaging, rather than constructing constitutive relations based on one particular microscopic model or process. Unlike moment methods that suffer from closure problems, this chapter will assume given constitutive relations (which become constraints on the moments) that fully specify a distribution of probability in the molecular velocity variable. The well-known conservation laws of fluid mechanics can then be obtained by taking subsequent moments of the Boltzmann equation, with a given PDF and set of collisional invariants that include molecular mass, momentum, and energy. This chapter will also present a given probability distribution obtained with Jaynes’s maximum entropy principle. This distribution is then modified to accommodate velocity and temperature gradients in a gas. It will be rewritten to highlight Gauss’s error law (Lavenda, 1991). Some interesting conclusions follow when the macroscopic entropy is obtained by taking the moment of - ln( pdf ) . A function will be obtained that essentially represents entropy associated with the PDF. This function is different for equilibrium (i.e., Euler equations) and nonequilibrium (i.e., Navier–Stokes) conservation laws. This difference will be evident from the entropy associated with each kind of PDF.
2.2 C o n se r va t io n La w s a s Mo men t s o f t h e Bo lt z ma n n Eq u a t io n The dynamics of an ideal, monatomic, dilute gas in the absence of external forces are theoretically governed by the following Boltzmann equation (neglecting body forces):
∂( ng ) ∂( ng ) ∂( ng ) + vk = ∂t ∂xk ∂t coll
(2.1)
where n is the number density and g is the PDF for the molecular velocity vk in an inertial frame. The function g is not a “velocity distribution function” because it is the probability that is distributed, not the velocity. Hence, gdvk equals the probability that the molecular velocity lies between vk and vk + dvk . The repeated index k denotes a sum, and the right-hand side represents the time-rate of change in g, due to molecular collisions. Moment equations can be generated by multiplying the Boltzmann equation by any function of molecular velocity, Q(vi), and integrating over velocity space as follows:
∂ ∂ ( n < Q >) + ( n < vk Q >) = D[Q] ∂t ∂xk
(2.2)
The first operator in Equation 2.2 is the expectation < Q > =
∞
∞
∞
-∞
-∞
-∞
∫ ∫ ∫
Qg dv1dv2 dv3 ,
(2.3)
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35
and the second operator is the collision integral given by
D[Q] =
∞
∞
-∞
-∞
∞
∂( ng ) Q dv1dv2 dv3 . - ∞ ∂t coll
∫ ∫ ∫
(2.4)
Moments of the collision integral are identically zero when the arbitrary function of molecular velocity, Q(vi), is one of the five collisional invariants, ψ ( vi ) ≡ Q INV = m {1, vi , v 2 /2} , where m is the molecular mass and v2 represents the square of the velocity magnitude. This general result holds for any distribution function g and for any molecular interaction law. Taking moments of the Boltzmann equation, given the set of collisional invariants, yields the following conservation laws of gas dynamics:
∂ ∂ ( n < ψ >) + ( n < vkψ >) = 0 ∂t ∂xk
(2.5)
Expanding Equation 2.5 using each of the collisional invariants in turn gives a set of equations for the conservation of mass, momentum, and energy. By introducing a set of relative velocity components, Ci = vi - ui , where ui = is the expected value of the corresponding velocity variable, the following results are obtained for the central moment stress tensor, thermodynamic pressure, viscous stress tensor, internal energy, and heat flux vector, respectively, as follows:
σ ij = ρ < CiC j >
(2.6)
p = 13 σ kk
(2.7)
τ ij = -σ ij + pδ ij ;
(i ≠ j ) (i = j )
eint = < 12 C 2 >
0 δ ij = 1
qi = ρ < 12 CiC 2 >;
C 2 = C12 + C22 + C32
(2.8) (2.9) (2.10)
where C 2 = C12 + C22 + C32 . The conservation laws for gas dynamics can then be written in the following familiar form:
∂Q ∂Fk + =0 ∂t ∂xk
(2.11)
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where the algebraic state and flux vectors are, respectively,
ρ ρu 1 Q = ρu2 ρu3 ρE
ρuk ρuku1 + σ k1 ρuku2 + σ k 2 Fk = ρuku3 + σ k 3 ρuk E + σ kiui + qk
(2.12)
The internal energy, eint, and heat flux vector, qi, are given for a monatomic gas. When the gas has a different internal structure, the procedure may be modified to accommodate the intermolecular degrees of freedom. A common approach will assume that all internal molecular energy modes exist in equilibrium, both internally and with translational degrees of freedom. The total internal energy, eint, can be expressed in terms of the temperature, T, by the following equilibrium relation:
eint =
ξ ξ RT = σ 2 , 2 2
(2.13)
where x equals the number of degrees of freedom, R is the gas constant, and σ 2 = RT is the variance that specifies the second central moment for some PDF, g = g( vk | uk , σ 2 ), to be determined. In this notation, it is understood that the probability function g is conditional on knowing the parameters uk and s 2, so that it is fully specified. The fluid velocity is the first moment, whereas the variance (temperature) is the second central moment. When the first and second moments of a probability distribution are known and the variable exists in the range ( -∞, ∞ ), then the equilibrium or maximum entropy distribution is a central probability distribution (Kapur and Kesavan, 1992). Also, when the off-diagonal correlation coefficients are zero, it is known as a normal or Gaussian PDF. The additional internal energy is introduced through the parameter x. For a monatomic gas, x = 3, whereas for a diatomic gas with two additional degrees of freedom (due to molecular rotation), x = 5. Diatomic gas molecules have a dumbbell structure, so the energy associated with axial rotation is negligible.
2.3 Ex t en d ed Pr o ba bilit y Dis t r ibu t io n s To account for the amount of energy carried by a particle with a certain internal structure, the kinetic energy mv 2 /2 must be replaced by ( 12 mv 2 + ε ), where ε is the additional internal energy per particle. The collisional invariants are then
{
ψ = m, mvi ,
}
1 2 mv + ε 2
(2.14)
To include the effects of internal structure, one can use the mathematical expression for the probability distribution. The additional degrees of freedom can be expressed by defining the variables ω k , with k = 1, 2 for the two rotational degrees of freedom,
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corresponding to a diatomic molecule with a linear structure. The expectation operator is then also modified to include integration over ω k ∈( - ∞, + ∞), with an expectation < ω k > equal to zero. One can therefore set ε = 12 mω 2 with ω 2 = ω 12 + ω 22 in Equation 2.14 and use a multivariate PDF, g = g( vi , ω i | ui , σ 2 ). Assuming Equation 2.1 continues to hold for the extended distribution function, g( vi , ω i | ui , σ 2 ) , and the collision integral on the right side is interpreted properly, then an additional integral over ω k is required to generate Equation 2.2 as moments of Equation 2.1. The quantities in Equation 2.14 must continue to be conserved in a collision. Consequently, Equation 2.4 becomes zero and Equation 2.5 remains unchanged. Evaluating the left side of Equation 2.5 for the five different quantities in Equation 2.14 gives identical results to those obtained for the monatomic gas, for all quantities that contain polynomials in vi alone. This can be shown because integration over the ω k variables can be taken independently from integration over vi. Therefore, the conservation laws for mass and momentum are recovered as written. The same observation also applies to the first term in the quantity 12 m( v 2 + ω 2 ),, and the conservation of energy can be written as ∂ 1 1 ρ < v2 > + ρ < ω 2 ∂t 2 2
∂ > + ∂xk
1 1 ρ < vk v 2 > + ρ < vkω 2 2 2
> = 0
(2.15)
Substituting the central moments, Equation 2.6 to Equation 2.10 into Equation 2.15 and using the previous results for the monatomic gas gives
∂ ∂ ( ρE ) + ( ρuk E + σ ki ui + qk ) = 0 ∂t ∂xk
(2.16)
where E ≡ eint + 12 u 2 ,
and
1 2 (C + ω 2 ) > 2
(2.17)
1 1 Ci C 2 > + ρ < Ci ω 2 > 2 2
(2.18)
eint = <
qi = ρ <
The conservation law Equation 2.11 continues to hold, provided Equation 2.9 is replaced by Equation 2.17, and Equation 2.10 by Equation 2.18. Therefore, the conservation law Equation 2.11 can be used with the state and flux vectors as they occur in Equation 2.12, provided definitions in Equation 2.17 and Equation 2.18 are used when a state of equilibrium exists between the internal modes and the translational degrees of freedom (Vincenti and Kruger, 1965). Since the conservation laws in Equation 2.11 can be developed for a general fluid through phenomenological principles alone, the set is more general than implied by a kinetic theory derivation, because they are also the conservation equations for fluid dynamics. For an ideal gas flow, the kinetic theory approach is necessary in that it shows that Equation 2.11 is valid for any degree of translational nonequilibrium,
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i.e., any PDF for a translational velocity. If equilibrium conditions are assumed, then the Maxwell–Boltzmann PDF (MB-PDF), gMB, can be used, and the conservation laws are the Euler equations because the viscous stresses and heat flux are zero. If instead a Chapman–Enskog (CE-PDF) distribution gCE is chosen, then the set of moment equations can be interpreted as the Navier–Stokes equations because the corresponding expressions describe the dynamics of a viscous, heat-conducting fluid. Note that a PDF can be used in Equation 2.11. As long as g is fully specified, the set becomes closed. If g remains general, then there is a closure problem when using a moment method with kinetic theory because t ij and qi are unknown quantities in the equations. The conservation equations in Equation 2.11 are not the Navier–Stokes equations, until one introduces gCE or some other PDF that incorporates terms to account for the viscous, heat-conducting effects. In the conservation law, Equation 2.11, specifying Fi completely in three dimensions requires the evaluation of 15 quantities. But the task is simplified for a finite volume together with Gauss’s divergence theorem. Equation 2.11 can be written in the following integral form:
∂ ∂t
∫
V
Q dV +
∫
S
Fn dS = 0
(2.19)
where S encloses the volume V, and Fn is the projection of Fi onto the unit outward normal for the surface element dS. If V is a rectangular volume in Cartesian coordinates, then only five quantities need to be calculated for each planar surface, provided Fn can be evaluated directly. Using the notation of Equation 2.5, the state and flux vectors can be generated from moments as follows: Q = n <ψ > Fn = n < vnψ > (2.20) where vn is the molecular velocity component normal to the planar surface. In Equation 2.20, the scalar quantity Q is transported across a fixed surface by vn, thus creating a flux in that quantity. The five fluxes defined in Equation 2.20 are total fluxes. In the general case, they contain both the inviscid (Euler) fluxes, as well as the nonequilibrium effects due to viscous stresses and heat conduction. Expressions for the viscous stresses and heat conduction terms can be obtained readily from classical fluid dynamic theory (Vincenti and Kruger, 1965). Together with the ideal gas equation of state, these nonequilibrium terms complete the constitutive relations for the field equations. They will serve to establish the nonequilibrium PDFs described in the following section.
2.4 S elec t ed Mu lt iva r ia t e Pr o ba bilit y Dis t r ibu t io n Fu n c t io n s This section summarizes four important probability distributions: the MB-PDF, the central distribution PDF, the CE-PDF, and the skew-normal PDF, which have been analyzed by Camberos (1997a,b) in terms of the Second Law. The notation in this section is standard in probability theory.
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2.4.1 Ma x w el l –Bo l t z ma n n Pr o ba bil it y Dis t r ibu t io n Fu n c t io n First consider the MB-PDF. This equilibrium distribution function is a special case of the central probability distribution. It is fully specified by the first moment and the second central moment (a temperature parameter), s 2, known as the variance. Consider a monatomic gas for simplicity. The equilibrium probability distribution for a set of ξ = 3 variables is represented by the MB-PDF, g MB ( z | Co ) =
{
}
1 | Co-1 |1/2 exp - zT ⋅ Co-1 ⋅ z ( 2π )3/2 2
(2.21)
where z = (z1,z2,z3) is the set of standardized velocity variables written relative to expectation values such that zk ≡ ( vk - uk ) / σ . The matrix Co contains the central moments. For an equilibrium distribution, it is given by Co = I3×3
(2.22)
where I3×3 is the identity matrix, sized appropriately for ξ = 3 variables. The matrix in Equation 2.22 has no off-diagonal terms, so the multivariate probability distribution could have been expressed as the product of three univariate central probability distributions in each variable. However, the form given in Equation 2.22 will be retained for consistent notation, when compared with PDFs in following sections.
2.4.2 C en t r a l Dis t r ibu t io n Pr o ba bil it y Dis t r ibu t io n Fu n c t io n The second function selected is the central distribution PDF. The central distribution (CD) for the number of variables sufficient to describe the molecular model of interest is
g(z bar{z}, C)
The second function selected is the central distribution (CD) PDF. The CD for the number of variables sufficient to describe the molecular model of interest is gCD. It has the same form as the right side of Equation 2.21 for the MB-PDF, but now the more general covariant matrix C contains all of the central moments. Components of C are given by x - xi xj - x j cij = rij ≡ i σi σ j
(2.23)
which are known in probability theory as the correlation coefficients. Note that rii = 1, because σ 2jj = < ( x j - x j )2 > . The MB-PDF is a special case of Equation 2.23, whereby all of the mixed central moments are zero. Taking the expectation value of ψ using gMB gives the following algebraic vector of macroscopic state variables:
Q = n < ψ > MB =
∞
∞
-∞
-∞
∫ ∫
nψ g MB ( vi | ui , σ 2 ) dx1dx2 dx3
(2.24)
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which is given in Equation 2.12 with a total specific energy E = 12 (u 2 + 3σ 2 ), where ui are the fluid velocity components in Cartesian coordinates. The algebraic vector for the flux of particles, F, across a volume boundary is given by the expectation value of the state function multiplied by the velocity components normal to the boundaries:
FkEE = n < vkψ > MB
(2.25)
On evaluating the expectation values for each component, with vk = zk σ k + uk , one finds that Equation 2.25 produces the familiar inviscid flux vectors for the Euler equations, as noted by the superscript. If, instead, expectation values are taken by the central PDF with nonzero correlation coefficients, one obtains the same state vector, but
FkCD = n < vkψ > CD
(2.26)
which now contains the viscous terms due to velocity gradients. These fluxes could be used, for example, to represent the Navier–Stokes equations.
2.4.3 C h a pma n –Ensk o g Pr o ba bil it y Dis t r ibu t io n Fu n c t io n Next, consider the CE-PDF. To accommodate heat-conduction effects, the equilibrium MB-PDF or the CD-PDF must be modified. A well-known approach to modify the MB-PDF is to apply the perturbation technique employed by Chapman and Enskog (1939). This technique yields a pseudo-PDF that incorporates the effects of velocity and temperature gradients when deviations from equilibrium are not too severe. One can write the so-called Chapman–Enskog pseudoprobability distribution in different ways. To maintain consistency with the notation used in this chapter, the CE-PDF is written as
g CE ( z,| q, T , σ 2 ) = 1
1 T 1 ( z ⋅ T ⋅ z ) + q ⋅ z - z 3 g MB 2 5
(2.27)
where 1 2 3
z z= z z
z13 z 3 ≡ z23 z33
(2.28)
The effects of heat conduction are included in the following parameter: q = ( qˆ 1, qˆ 2, qˆ 3), qˆ k ≡
qk , ρσ 3
(2.29)
which are called skewness coefficients in probability theory because of their effects on the PDF.
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Statistical and Numerical Formulations of the Second Law
Define the following nondimensional stress matrix
τ 11 τ ≡ τ 21 τ 31
τ 12 τ 22 τ 31
τ 13 τ 23 ; τ 33
τ ij = τ ji
(2.30)
This accounts for the effects of velocity gradients, because we define the dimensionless viscous stresses by tki ≡
τ ki . ρσ 2
(2.31)
Evaluating the expectation value in Equation 2.24 using gCE instead of the equilibrium MB-PDF gives the same state vector, but now the flux vectors in Equation 2.25 will include both the viscous and heat-conducting terms. Hence, the full Navier– Stokes equations are represented by Equation 2.11, when gCE is used together with the constitutive relations for t ki and qk .
2.4.4 S k ew -N o r ma l Pr o ba bil it y Dis t r ibu t io n Fu n c t io n Finally, with a skew-normal PDF, we can choose any probability distribution when generating the moment equations. One may consider alternatives to the CE-PDF, as long as the appropriate field equations are obtained. One approach is to use the PDF directly and a probabilistic approach when constructing it. To incorporate the effects of viscosity, which lead to second-order moments with nonzero correlation coefficients, a multivariate central probability distribution is sufficient. This is the maximum entropy probability distribution, when the first and second moments are specified. However, to include the effects of heat conduction, one needs to specify the third-order central moments. If only these are specified and no others, then a resulting maximum entropy analysis becomes infeasible since the exponential function cannot be normalized when third-order powers are included. The only recourse is the Chapman–Enskog results, with skewness coefficients in the construction of the PDF, which will be a modification of the central distribution. This leads to a skew-normal pseudo-PDF (SN-PDF). In this PDF, a nonzero third-order moment is obtained, although the symmetry of a multivariate central distribution is retained. The skew-normal distribution is defined by variable tii from Equation 2.31.
1 g SN ( z |θ , C ) = 1 - θ ⋅ z - z 3 g CD ( z | C ) 3
(2.32)
where zˆ i = z/ 1 - tii . The skewness coefficients θ = (θ1 , θ 2 , θ 3 ) must be determined from the moment constraints, thereby leading to the full Navier–Stokes fluxes. The PDF on the right side of Equation 2.32 is the multivariate CD-PDF with correlation coefficients given by Equation 2.23.
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The moments n SN produce the Navier–Stokes fluxes, which are obtained by specifying gSN with the following covariant matrix: r11σ 12 C = r21σ1σ 2 r31σ 3σ1
r13σ1σ 3 r23σ 2σ 3 . r33σ 32
r12σ1σ 2 r22σ 22 r32σ 2σ 3
(2.33)
where σ k2 ≡ 1 - τ kk and 1 rji = rij = τ ij (1 - τ ii )(1 - τ jj )
i= j (2.34) i ≠ j.
The structure of the covariant matrix implies that the molecular translational modes are not at equilibrium with the fluid temperature. To obtain the correct energy fluxes, the skewness coefficients q k must be related to the heat-conduction terms. Hence, the third-order moments are equated to the expectation of the flux of thermal energy, relative to fluid velocity. The resulting constraints provide a unique solution, but the solution gives a complicated set of equations for the three unknown parameters, q1, q 2, q 3. A more convenient form of the result is qˆ 1 θ = (θ1 , θ 2 , θ3 ) = K ⋅ qˆ 2 qˆ 3 -1
(2.35)
where qˆ k expressions were previously defined in Equation 2.29. Also, K can be expressed by
σ1 K≡ 0 0
0 σ2 0
0 r11 0 r12 σ 3 r13
r13 a1 r23 0 r33 0
r12 r22 r23
0 a2 0
0 0 a3
(2.36)
where 3
ai =
∑r σ 2 ik
2 k
(2.37)
k =1
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and u,t ij and qi are given by local conditions and constitutive relations. The skewnormal distribution is then fully specified by the parameters q k and covariance matrix C. This then yields the full Navier–Stokes fluxes after evaluating the moments, n < vkψ > , with g = g SN.
2.5 C o n ca v e En t r o py Fu n c t io n s From elementary thermodynamics, the specific entropy for an ideal gas with constant specific heats can be written in the following nondimensional form: s EQ =
1 ln T ′ - ln ρ ′ γ -1
(2.38) where T ′ and ρ ′ are the nondimensionalized temperature and mass density ratios, with respect to some reference state. As a function of the fluid state variables, the specific entropy is concave. The concavity property of entropy can be interpreted through proper probability distributions. The probabilistic approach to statistical physics developed by Lavenda (1991) asserts that “the connection between entropy and probability is through a law of error for extensive thermodynamic variables and Boltzmann’s principle is a consequence of it.” This “law of error” is realized as an inequality expressing the concavity property of entropy. Concavity is directly related to the logarithm of a probability distribution as follows: - ln g( x ) = s( x ) - s( x ) - s ′( x )( x - x ) + constant (2.39) where the prime notation (v′) denotes a derivative. Physical phenomena that are characterized by a probability density, g(x), for some relevant variable, x, can be examined in terms of Gauss’s principle, where one identifies the average value, x , with the most probable value. The function g(x) is a general, unknown probability density, not limited to conditions at equilibrium. The concavity property of entropy requires that s( x ) - s( x ) - s ′( x )( x - x ) > 0 (2.40) The inequality defines a strictly concave function. In thermodynamics, the average value x is uniquely determined at the equilibrium state. Note that Equation 2.40 does not assert that the entropy, s(x), of the nonequilibrium state is always less than the entropy of the equilibrium state, s( x ). A key feature of Equation 2.40 lies in the realization that the entropy function is a constrained maximum, where the derivative s ′( x ) has the role of a Lagrange multiplier for the corresponding constraint, obtained directly as a function of the conserved state variable or variables. Thus, the entropy tends to increase only when the state variable differs from its value at equilibrium. Taking the average of Equation 2.40 gives s( x ) - s( x ) > 0, which implies a principle of nondecreasing entropy. To obtain the entropy associated with a PDF, the expectation value of the negative of the natural logarithm is taken as follows:
s = <- ln g>
(2.41)
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where the functional dependence is given with respect to a set of standardized variables. This allows one to define
1 T -1 z ⋅ C ⋅ z + ln | C | 2
ϕ=
(2.42)
which is “-1n g” with the constant omitted, such that CD
s PDF - s EQ = <- ln g >
(2.43)
where sEQ is the thermodynamic (equilibrium) entropy. The expectation value is taken relative to the standardized variables, so the right side of Equation 2.43 will be either negative for a nonequilibrium PDF or zero for an equilibrium PDF. With the MB-PDF, a constant is obtained, although it does not change the implications. This analysis can be extended for the Maxwell–Boltzmann entropy. For the MB-PDF,
ϕ MB =
1 T -1 z ⋅ Co ⋅ z + ln | Co | 2
(2.44)
where Co is the identity matrix so | Co | = 1. Taking the expectation value yields
s MB - s EQ = < ϕ > MB =
1 T 1 3 < z ⋅ z > = < z12 + z22 + z32 > = 2 2 2
(2.45)
As anticipated, the entropy associated with the MB-PDF is equivalent to the thermodynamic entropy, within a constant. For the Chapman–Enskog entropy, with the CE-PDF,
ϕ CE = ϕ MB + ln{1 + ε}
(2.46) where ε contains the nondimensional parameters associated with velocity and temperature gradients, as expressed in Equation 2.27. The expectation value of the second term in Equation 2.46 cannot be evaluated analytically. One possibility is to expand the term, assuming that ε << 1. But this gives a result indicating that the nonequilibrium entropy is greater than the equilibrium entropy, which is not possible in the context of physical theory (see Figure 2.1). Thus, for the CE-PDF, sCE - s EQ = s MB - s EQ =
3 2
(2.47) The entropy associated with the equilibrium Euler equations is the same as the function associated with the nonequilibrium Navier–Stokes equations. Correspondingly, for the central distribution entropy, with the following central distribution PDF,
ϕ CD =
1 T -1 z ⋅ C ⋅ z + ln | C | 2
(2.48)
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1.0 ˆq
0.5
0.0 –0.5 –1.0
0.4 SCE–S0 0.2
0.0 –1.0 –0.5 t11
0.0 0.5 1.0
Fig u r e 2.1 Chapman–Enskog expectation value for entropy difference relative to equilibrium entropy.
Taking the expectation value yields
<ϕ> CD =
3 + ln | C | 2
(2.49)
where ln | C | is the logarithm of the determinant of the covariance matrix. For simplicity, one can assume only t11 ≠ 0 . For this case, the constraint is t22 = t33 = -t11 /2. Then
sCD - s EQ =
3 1 + ln {(1 - t11 )(1 - t11 /2 )2 } 2 2
(2.50)
Under certain conditions restricting possible values of the nondimensional shear stress t11, the entropy associated with the CD-PDF is always less than the equilibrium value, as shown in Figure 2.2. The range implies that -2 < t11 < 1, which includes values typically found in actual physical processes, such as shock waves. Finally, for the skew-normal entropy, the SN-PDF is
ϕ SN = ϕ CD + ln{1 + ε}
(2.51)
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0.5
0.0 –0.5 –1.0 0.0 –0.5 –1.0 SCD–S0 –1.5 –2.0 –2 –1 t11
0 0.5 1
Fig u r e 2.2 Central distribution function expectation value for entropy difference relative to thermodynamic entropy.
where ε contains the parameters associated with thermal and velocity gradients. This presents the same difficulty as the CE-PDF, because this term cannot be evaluated analytically. Using the same analysis mentioned previously, an expression can be obtained by assuming that ε << 1. But again for some combination of values of t11 and qˆ 1, the result is nonphysical. For the range typically found in practice ( -1 < t11 < 0 ), the entropy associated with the SN-PDF is less than the equilibrium value (see Figure 2.3). Due to this uncertainty, it is assumed that s SN - s EQ = sCD - s EQ (2.52) In the next section, these results will be used to develop a statistical formulation of the Second Law.
2.6 S t a t is t ica l Fo r mu la t io n o f t h e Sec o n d La w Performing an entropy balance over a differential control volume for inviscid flows (Euler equations), it can be shown that S gen =
∂( ρs ) ∂( ρuk s ) + ∂t ∂xk
(2.53)
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47
1 ˆq
0
–1 –2 2
SSN–S0
0
–2 –1.0 –0.5 t11
0.0 0.5 1.0
Fig u r e 2.3 Skew-normal expectation value for entropy difference relative to thermodynamic entropy.
where the first term on the right side is the time-rate-of-change of entropy in a differential control volume. For processes modeled by the Euler equations, which are generated as moments of the Boltzmann equation with the MB-PDF, the entropy generation is zero. For the Navier–Stokes equations, it was shown in the previous chapter that S gen =
κ µ ∇T ⋅∇T + F T2 T
(2.54) where F is the viscous dissipation function (strictly nonnegative). For processes governed by the Navier–Stokes equations, entropy generation is always nonnegative, as the coefficient of thermal conductivity, k, and viscosity, m, is nonnegative. Motion of a viscous, heat-conducting fluid will yield a net production of entropy. A statistical approach for deriving the entropy production equation is to extend the technique for conservation laws, as moments of the Boltzmann equation, to the Second Law. The entropy generation rate can be expressed as S gen =
∂ ∂ ( n < ϕ >) + ( n < vkϕ >) ∂t ∂xk
(2.55)
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which is consistent with kinetic theory, because the collision integral for the quantity represented by ϕ = - ln g is nonzero. It remains to determine under what conditions the entropy generation rate is nonnegative, to satisfy the Second Law of Thermodynamics. For the MB and CE probability distributions, this is already established because both yield the same functional expression for the entropy. For the CD and SN distributions, the expression for entropy in these distributions differs from the equilibrium entropy, so extra terms appear when the entropy generation rate is evaluated using Equation 2.55 with either of these distributions. However, it is evident that a nonequilibrium version of the Clausius-Duhem inequality emerges. The expression in Equation 2.55 may be considered a generalized version of the standard Clasius-Duhem expression for the entropy production rate. The equation will contain additional terms due to the nonequilibrium effects of velocity and temperature gradients. As expected, these effects appear not only in the constitutive relations for the stress tensor, but also in the entropy function itself. This satisfies the requirement of the Second Law of Thermodynamics, as the nonequilibrium entropy is less than its corresponding equilibrium value. But the result is different from the result obtained with the Chapman–Enskog formalism, when constructing the pseudoprobability density (a perturbation) for the Navier–Stokes equations.
2.7 N u mer ica l Fo r mu la t io n o f t h e Sec o n d La w The previous sections have developed statistical formulations of entropy and the Second Law. From this basis and the governing equations developed previously for the Second Law, numerical solutions of the governing equations can be determined. This section develops a numerical formulation of the Second Law. Many types of numerical methods exist for the solution of the Navier–Stokes equations, such as finite differences, elements, volumes, and so forth. This section uses a particular method (called a CVFEM; control volume-based finite element method) to illustrate how discretization of the Second Law can be accomplished. Similar procedures can be readily developed with other methods, by postprocessing of the computed velocity and temperature fields to determine the entropy production rates throughout the flow field.
2.7.1 Disc r et iz a t io n
o f t he
Pr o bl em Do ma in
A typical two-dimensional domain in Figure 2.4 is subdivided into linear, quadrilateral finite elements. The grid is collocated, so that the velocity components, pressure, and temperature are obtained at nodes located at every element corner. The finite element discretization uses a local ( s, τ ) coordinate system that defines the shape functions and other element properties. A local numbering scheme, ranging from 1 to 4, is used within each element, so that the finite element equations can be developed locally and independently of the mesh configuration. Following a conventional assembly procedure (Schneider, 1988) for finite elements, the local nodal equations are assembled into the global system of equations involving global nodes. The conservation principle is applied over an “effective” control volume defined by all subvolumes from elements surrounding a particular node in the mesh. Each element is subdivided into four subcontrol volumes (SCVs), with internal SCV boundaries
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Statistical and Numerical Formulations of the Second Law Local Node Number
Node SCV 2 SCV 3
2 SCV 1
ip 1 ip 2 (SS2)
t
1
ip 4 s ip 3
SCV 4
49
Finite Element
4 Subvolume
Control Volume
Fig u r e 2.4 Schematic of a finite element and control volume discretization.
coincident with the local coordinate surfaces defined by s = 0 and t = 0 (see Figure 2.4; note origin of axes at the center of an element). Transported quantities across the edges (surfaces) of a control volume are approximated from values at the midpoint of a subsurface, called the integration point, where “ip” refers to the integration point in Figure 2.4. Interpolation within each element yields 4
x=
∑Nx
i i
(2.56)
i =1 4
y=
∑Ny
i i
(2.57)
(2.58)
i =1 4
φ=
∑NF i
i
i =1
where xi, yi, and Fi are nodal values of the spatial coordinates and f, respectively. For quadrilateral, isoparametric elements, the linear shape functions, Ni, are given by N1 =
1 (1 + s )(1 + t ) 4
N2 =
1 (1 - s )(1 + t ) 4
(2.60)
N3 =
1 (1 - s )(1 - t ) 4
(2.61)
N4 =
1 (1 + s )(1 - t ) 4
(2.62)
(2.59)
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The spatial derivatives of the scalar function are evaluated according to
∂φ = ∂x ∂φ = ∂y
4
∑ ∂∂Nx F i
i
(2.63)
i
(2.64)
i =1 4
∑ ∂∂Ny F i
i =1
To obtain the x and y derivatives of the shape functions in Equation 2.63 and Equation 2.64, the chain rule for partial derivatives is applied as follows:
∂N i ∂N i ∂x ∂N i ∂y = + ∂s ∂x ∂s ∂y ∂s
(2.65)
∂N i ∂N i ∂x ∂N i ∂y = + ∂t ∂x ∂t ∂y ∂t
(2.66)
Solving for the x and y derivatives,
∂N i ∂y ∂x 1 ∂t = ∂N i J -∂x ∂y ∂t
-∂y ∂N i ∂s ∂s ∂x ∂N i ∂s ∂t
(2.67)
where J is the determinant of the Jacobian of transformation given by J=
∂x ∂y ∂y ∂x ∂s ∂t ∂s ∂t
(2.68)
The derivatives of the global coordinates with respect to local coordinates in Equation 2.67 are obtained from the x and y nodal values as follows: ∂x = ∂s
4
∑ ∂∂Ns 4
∑
∂y = ∂s
4
∂y = ∂t
xi
(2.69)
i =1
∂x = ∂t
i
i =1
∂N i xi ∂t
∑ ∂∂Ns
i
(2.70)
yi
(2.71)
i =1 4
∑ i =1
∂N i yi ∂t
(2.72)
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Statistical and Numerical Formulations of the Second Law
The local derivatives of the shape functions required in Equation 2.72 can be found by differentiating Equation 2.59 through Equation 2.62. In the upcoming approximation of transient and source terms, the area of a two-dimensional control volume bounded by a specific range of s and t is given by dA = Jdsdt
2.7.2 Disc r et iz a t io n
(2.73)
o f t he Co nse r va t io n Eq u a t io ns
The discrete conservation equations are obtained by integrating the differential equations over a discrete control volume (or two-dimensional area, encompassed by a surface, S). Using the Gaussian theorem, the standard form of the integral equation for a conserved quantity, f, can be expressed as
∫
∂( ρφ ) dA + ∂t
A
∫ (ρvφ ) ⋅ dn - ∫ (Γ∇φ ) ⋅ dn = ∫ S
S
PdA
A
(2.74)
The term on the right side refers to a source term, where v and dn refer to the velocity and differential normal vector at the surface, respectively. Equation 2.74 applies to each control volume, as well as the solution domain as a whole. To discretize Equation 2.74 in two dimensions, a particular finite element illustrated in Figure 2.4 is considered. Let the variable Q represent the flow of f across the edge of an element. The flows consist of a diffusive component and convective component. The integral forms of the components are given by the second and third terms on the left side of Equation 2.74, convective and diffusive. The first and second subscripts on S and Q will denote the subsurface and nodal point numbers, respectively. The subscripts e1 and e2 will refer to flows into the control volume through the surfaces, which lie on the exterior edge of the element. Therefore, the equation governing the conservation of f over SCV1 (subquadrant 1 of element in Figure 2.4) can be written as Q1,1 + Q4,1 + Qe1,1 + Qe 2,1 +
∫
scv1
PdV =
∂ ∂t
∫
scv1
ρφ dV
(2.75)
To complete the discretization of the integral conservation equation, the surface and volume integrals need to be approximated. For example, the diffusive component of Q4,1 is approximated by Q4,1 =
∫
S4 ,1
4
( Γ∇φ ) ⋅ dn =
∑ j =1
Γx
∂N i ip ∂N i ip |4 Dx4 F j |4 Dy4 - Γ y ∂y ∂x
(2.76)
where the gradient functions have been evaluated using the shape functions. Note that the surface integral in Equation 2.76 is approximated by the product of the flux evaluated at the surface integration point and the length of the surface (see Figure 2.4).
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Similarly, the convective component is given by Q4,1 =
∫
S4 ,1
( ρvφ ) ⋅ dn = ρv4ip Dx4φ4ip - ρu4ip Dy4φ4ip
(2.77)
where the lower case variables, f ip and uip, denote the integration point values. Also, Dx4 and Dy4 are respective changes in the x and y directions, as subsurface 4 is traversed in a counterclockwise direction. Some common schemes for obtaining the integration point variable, f ip, in terms of the nodal values are the upwinding differencing scheme (UDS), central differencing scheme (CDS), exponential differencing scheme (EDS), and a physical influence scheme (PINS) (Naterer, 1999; Schneider, 1988). UDS uses an upstream value to approximate the scalar at the integration point. CDS uses a linear interpolation between adjacent nodes, and EDS is a “hybrid” scheme, which obtains a smooth transition from the CDS scheme, for low Peclet numbers (Pe = ρui Dxi /Γ), to the UDS scheme for high Peclet numbers (Patankar, 1980). The value taken for the convected variable is determined based on an interpolated value at the nodal points, related by integration point coefficients. PINS predicts the integration point value of the scalar by a local approximation of the governing equation at the integration point. Thus, each integration point equation becomes an approximation to the appropriate partial differential equation, including all physical influences on the upwind value. For the transient term, a lumped mass approximation is adopted. The approach assumes that f is uniformly equal to the nodal value over the whole control volume. The transient term is represented in the following form:
∂ ∂t
∫
scv1
ρφ dV = ρ J
( F1n +1 - F1n ) Dt
(2.78)
where the superscripts n and 0 refer to the new and old time levels, respectively, and J denotes the area of SCV1. Finally, for a given source-type term, such as the pressure gradient, body force, or the contribution from viscous stress terms in the momentum equations, the source term is evaluated as
∫
scv1
P dV = P | 1 , 1 J 2 2
(2.79)
where a midpoint integration has been used in evaluating P at ( 12 , 12 ). The twodimensional domain considered in this discretization is assumed to have a unit depth normal to the plane of interest. Thus, the volume and area integrals reduce to area and line integrals, respectively. Each of the aforementioned components of the discrete equation can be assembled into Equation 2.74 and represented in a final matrix form. If the contributions of the four separate control volume equations are taken together, the algebraic equations can be written as
[ Aφφt + Aφφd ]{φ} + [aφφa ]{φ} = {Bφt } + {Bφ p}
(2.80)
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Statistical and Numerical Formulations of the Second Law
53
where the superscripts f on A denote a f equation and a multiplier of a f variable. The superscripts t, a, d, and p refer to the transient, advection, diffusion, and source, respectively. The square brackets denote a 4 × 4 matrix, and the braces denote a column vector. A row in the matrix represents the SCV contribution to the corresponding control volume equation, and the column indicates the integration point (or nodal point) of the variable multiplied. In the CVFEM, the element stiffness matrix is generated from a control volume formulation.
2.7.3 Disc r et iz a t io n
o f t he Sec o n d La w
The entropy production rate can be calculated numerically from the entropy transport equation (Cheng et al., 1994; Merriam, 1988). Alternatively, after simplifying and using the Gibbs equation, which relates entropy to the temperature, pressure, mass, density, and internal energy, an alternative positive definite form of the entropy production equation can be obtained (Bejan, 1996; Naterer and Camberos, 2001). This positive definite form was presented in previous sections, and it will be further developed in this section with a numerical formulation and CVFEM. For the numerical discretization of entropy production, let Vj denote the volume associated with node j, so the integral form of the Second Law can be written as Vj
∂S j + ∂t
∫
Sj
F ( q ) ⋅ ds ≥ 0.
(2.81)
where q represents the vector of conserved variables. The entropy flux (second term) results from the contribution of four different SCVs within four different elements sharing node j. The resulting equation for the effective control volume surrounding node j becomes ∂S j Vj + ∂t
8
∑ DF
i
(2.82)
≥0
i =1
where the summation (over node i) refers to 8 integration points of the effective control volume. Two alternative temporal discretizations are considered in this formulation of the Second Law. The first way is a semidiscrete approach, whereby the entropy time derivative is transformed by the chain rule, i.e., ∂S j ∂S ( q ) ∂q j = |j (2.83) ∂t ∂q ∂t Now, substitute expressions of the conservation equations in to ∂q j / ∂t and place the resulting form of ∂S j /∂t into Equation 2.82, thereby giving
∂S ( q ) |j ∂q
8
∑ i =1
i Df +
8
∑ DF i =1
i
(2.84)
≥0
In this approach, no temporal differencing is applied.
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The second approach uses a backward-difference in time to evaluate the entropy storage term. For two-dimensional quadrilateral elements, the Second Law becomes Vj
S nj +1 - S nj + Dt
8
∑ DF
i
≥0
(2.85)
i =1
where the superscript n + 1 denotes evaluation at the current time step and the superscript n refers to the previous time step. Equation 2.85 is the fully discrete form of the Second Law. For an implicit time advance, the semidiscrete and fully discrete entropy production rates become ( P s )nj +1 =
∂S n+1 |j ∂q
8
∑
i ,n+1 Df +
i =1
8
∑ DF
i ,n +1
≥0
(2.86)
i =1
and ( P s ) Ij = V j
S nj +1 - S nj + Dt
8
∑ DF
i ,n +1
≥0
(2.87) i =1 respectively. In Equation 2.86 and Equation 2.87, the notation expressing the functional dependence of S on q, S ( q ), has been dropped for brevity. For a variable, h, in the range of n ≤ η ≤ n + 1, the relationship between Sn+1 and Sn can be found from a Taylor’s expansion as follows: ∂ ∂ n +1 S nj = S nj +1 - q1n,+j 1 - q1n, j + ... + q4n,+j1 - q4n, j Sj ∂q1 ∂q4
(
)
(
)
(
)
(
(2.88)
2
1 ∂ ∂ η + q1n,+j 1 - q1n, j + ... + q4n,+j1 - q4n, j Sj ∂q4 2 ∂q1
)
The transport equation for the conserved variables can be written similarly as Equation 2.85 using an implicit time advance as follows: Dt q nj +1 - q nj + Vj
8
∑ Df
i ,n +1
≥0
i =1
(2.89)
The value of ( q nj +1 - q nj ) can be replaced in Equation 2.88 by using Equation 2.89. The resulting form of S nj may then be substituted into Equation 2.87 to obtain an expression for ( P s ) Ij as follows:
( P s ) Ij =
∂S n+1 |j ∂q
8
∑
8
Df i ,n+1 +
i =1
∑
DF i ,n+1
i =1
(2.90) 2
V j n +1 ∂ ∂ η n + ... + q4n,+j1 - q4n, j q1, j - q1, j Sj ≥ 0 2 Dt ∂q1 ∂q4
(
)
(
)
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Statistical and Numerical Formulations of the Second Law
Substituting the expression for the semidiscrete equation into Equation 2.90 using Equation 2.87, it can be shown that ( P s ) Ij = ( P s )nj +1
Vj 2 Dt
4
4
∑∑h α α kl
k =1
k
l
(2.91)
l =1
where hkl denotes the entries of the Hessian matrix, H =∂2 S/∂q 2 , and α k = n +1 n ( q k , j - q k , j ) . Note that hkl depends on the entropy functions, S ( q ) and F ( q ), and the problem parameters under consideration. The Hessian matrix entries (second derivatives of entropy) can be derived analytically for specific cases such as compressible flows, subject to the Navier–Stokes equations of motion. The entries require derivatives of entropy, with respect to the vector of conserved variables. For example, the entropy derivative with respect to q for one-dimensional compressible flows can be written as (Camberos, 1995; Merriam, 1988; Naterer, 1999)
(γ - 1) (γ - 1) ρu 2 (γ - 1) ρu S,q = s + cv -γ + , - cv , ρcv P 2 P P
(2.92)
Physically, the result S,q dq represents the cumulative effect of changes in all conserved quantities on the entropy change. Because H is convex (negative definite), then the quadratic form given by the double sum in the semidiscrete entropy production becomes negative for all (α1 , α 2 , α 3 , α 4 ). As a result,
( P s ) Ij ≥ ( P s )nj +1
(2.93)
Thus, the fully discrete entropy production rate for a numerical scheme is equal to or greater than the semidiscrete entropy production. From Equation 2.91, the effects of entropy production due to temporal and spatial discretization may be separated from each other. In the upcoming chapters, numerical simulations of entropy production that utilize this result and formulation will be presented.
R ef er en c es Bejan, A. 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Time Systems and Finite-Time Processes. CRC Press, Boca Raton, FL. Camberos, J.A. 1994. Probabilistic Approach to the Computational Simulation of Gasdynamic Processes. Doctoral dissertation, Department of Aeronautics and Astronautics (SUDA AR No. 668), Stanford University, Stanford, CA, 102–105. Camberos, J.A. 1997a. Comparison of Split-Fluxes Generated from Selected Probability Distributions (preprint). AIAA Paper 97-2095. Camberos, J.A. 1997b. Comparison of Selected Probability Distributions for Gas Dynamic Simulations Inspired by Kinetic Theory. AIAA Paper 97-0340.
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Camberos, J.A. 1999. Nonlinear Time-Step Constraints Based on the Second Law of Thermodynamics. 37th Aerospace Sciences Meeting and Exhibit. AIAA Paper 99-0558, Reno, NV. Chapman, S. and T.G. Cowling. 1960. Mathematical Theory of Non-Uniform Gases. Cambridge University Press, London. (Reprint of 1939 original.) Chapman, S. and T.G. Cowling. 1939. The Mathematical Theory of Non-Uniform Gases. University Press, Cambridge, U.K. (Reprint Edition 1990.) Cheng, C.H., Ma, W.P., and W.H. Huang. 1994. Numerical predictions of entropy generation for mixed convective flows in a vertical channel with transverse fin arrays. Int. J. Heat Mass Transfer, 21: 519–530. Dincer, I. 2001. Exergy and the environment: A global perspective. Int. J. Global Energy Issue, 15(3/4): 363–374. Kapur, J.N. and H.K. Kesavan. 1992. Entropy Optimization Principles with Applications. Academic Press, New York. Lavenda, B.H. 1991. Statistical Physics. John Wiley & Sons, New York. Merriam, M.L. 1988. An Entropy-Based Approach to Nonlinear Stability. Ph.D. thesis, Stanford University, Stanford, CA. Naterer, G.F. 1999. Constructing an entropy-stable upwind scheme for compressible fluid flow computations. AIAA J., 37(3): 303–312. Naterer, G.F. and J.A. Camberos. 2001. The Role of Entropy and the Second Law in Computational Thermofluids. AIAA 35th Thermophysics Conf. AIAA Paper 2001-2758, June 11–14. Anaheim, CA. Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere, New York, 79–101. Rosen, M.A. and I. Dincer. 1997. On exergy and environmental impact. Int. J. Energy Res., 21(7): 643–654. Schneider, G.E. 1988. Elliptic systems: finite-element method 1, in W.J. Minkowycz, E.M. Sparrow, G.E. Schneider, and R.H. Pletcher, Eds., Handbook of Numerical Heat Transfer. Wiley Interscience, New York, chap. 10. Vincenti, W.G. and C.H. Kruger, Jr. 1965. Introduction to Physical Gas Dynamics. John Wiley & Sons, New York.
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3
Predicted Irreversibilities of Incompressible Flows
3.1 In t r o d u c t io n Entropy production is a key parameter in determining the maximum theoretical limits of energy efficiency of engineering devices. Inefficiencies in a fluids engineering system arise from thermal, friction, and other thermodynamic irreversibilities. The Second Law of Thermodynamics can provide a systematic way of establishing optimal performance in these systems. For example, the Carnot cycle efficiency is based on processes that require the least amount of heat input, to deliver the maximum power output by giving ideal performance without irreversibilities. Actual heat engines are often compared against this Carnot limit. The rate of entropy generation, or any related measure of efficiency based on the Second Law, such as the Second Law efficiency, can be used to quantify the magnitude of irreversibilities in thermofluid applications. Power-generation devices (such as steam turbines) deliver maximum power output, and power-consuming devices (i.e., compressors, pumps) consume the least power when the rate of entropy generation is minimized. In this chapter, dissipative energy losses will be characterized through local rates of entropy production, which could be alternatively expressed in terms of exergy destruction. Past studies have shown the importance of such exergy tracking in various industrial applications, such as fluid machinery, transportation (Dincer et al., 2004), cogeneration district energy systems (Rosen et al., 2004), and others. Many other past efforts have been devoted to the design of highly efficient energy devices and systems. Consequently, these devices have been thoroughly scrutinized for possible design improvements. With the current state of this technology, the margins of increasing such performance further are often relatively small. This chapter discusses how the Second Law can offer new ways of reaching the upper limits of technological performance, based on local predictions of thermofluid irreversibilities. Past studies have described various analytical and empirical techniques for entropybased optimization of engineering systems, most notably the method of entropy generation minimization (Bejan, 1996). Some typical examples include extended fins in forced convection (Poulikakos and Bejan, 1982) and two-phase heat exchangers (Zubair et al., 1987). An analytical approach typically involves the derivation of a functional expression for the entropy generation in a process. The extrema of the functional expression, which characterize the minimum entropy production, are then determined by analytical methods of calculus. The rate of entropy generation is a derived quantity, which is predicted from postprocessing of the temperature and velocity distributions. For complex flow configurations, this typically requires numerical methods like computational fluid
57
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dynamics (CFD). In this chapter, it will be shown that an entropy-based local loss analysis remains consistent with the usual global loss distribution given by traditional loss correlations. A component-level design methodology can be enhanced by the capability of identifying the source and specific location of highest entropy production. This approach can be more valuable than examining global losses like a duct’s end-to-end pressure loss, because the desired overall performance can be improved by redesigning a component locally. Unlike past methods involving global coefficients characterizing the overall performance, this chapter discusses a new entropy-based metric for characterizing local losses of available energy. Also, methods are developed to predict upper bounds on entropy, thereby allowing designers to use the Second law to develop systems with higher performance and efficiency.
3.2 En t r o py Tr a n sp o r t Eq u a t io n f o r In c o mpr es s ibl e Fl o w s In tensor notation, the conservation form of the general scalar transport equation in multidimensions can be written as ∂( ρφ ) ∂ ∂ + ( ρu j φ ) = ∂t ∂x j ∂x j
∂φ Γ ∂x + Sφ j
(3.1)
where j = 1,2,3 and f is a general scalar quantity or dependent variable, such as temperature, velocity, or concentration transported throughout the flow field by diffusion or convection. The terms on the left side of Equation 3.1 represent the transient storage and convective flux. The first term on the right side is the diffusive flux. The last term represents production or sources of f in the volume. In the modeling of Equation 3.1, Γ and Sφ are generalized variables representing the diffusion coefficient and source terms, respectively. The conservation equations involve equalities, whereas the Second Law involves an inequality. As discussed in previous chapters, the entropy transport equation can be written as
∂S ∂Fi + ≡ P s ≥ 0 ∂t ∂xi
(3.2)
where P s is the entropy production rate and S = rs represents the entropy per unit volume. The component of the entropy flux in the xi direction, Fi, may be expressed in terms of the velocity component and heat flux in that direction, vi and qi, as follows: Fi = ρvi s +
qi . T
(3.3)
The specific entropy, s, in the flux term of Equation 3.2 can be obtained from the Gibbs equation as follows: ds =
p 1 de + 2 d ρ T ρT
(3.4)
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where e is the internal energy per unit mass, r represents density, and p is pressure. Integration of the Gibbs equation leads to Ds =
∫
Ts
Tr
cv
dT + T
∫
ρs
ρr
p dρ ρ 2T
(3.5)
where the subscripts r and s denote a specified initial (or reference) state and the current state, respectively. The variable cv represents the specific heat, which will be assumed to be constant (formulation is limited to liquid flows or incompressible gas flows over small to moderate temperature differences). For an incompressible fluid, Equation 3.5 becomes T Ds = s - sr = cv ln s Tr
(3.6)
For an ideal gas, ρ T s = cv ln s - Rln s + sr Tr ρr
(3.7)
Substituting the ideal gas law into Equation 3.7, s = cv ln
p∗ ( ps / pr ) + sr = cv ln ∗g + sr g ( ρs / ρr ) ρ
(3.8)
where g is the ratio of specific heats. When combined with the Gibbs equation, the entropy transport equation provides a way of calculating the local entropy generation for an open system. As discussed in the previous chapter, an alternative way of formulating P s is (Bejan, 1996) 2
P s =
k ∂T τ ij ∂ui + ≥0 T 2 ∂xi T ∂x j
(3.9)
where k is the thermal conductivity and τ ij is the viscous stress arising from velocity gradients in the fluid motion, ∂u j 2 ∂uk ∂u τ ij = m i + δ ij ∂xi 3 ∂xk ∂x j
(3.10)
In Equation 3.10, m and δ ij refer to the dynamic viscosity and Kronecker delta, respectively. The last divergence term in Equation 3.10 will vanish under the assumption of flow incompressibility.
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In Equation 3.9, Fourier’s law has been used to represent heat conduction. Also, a Newtonian fluid was assumed for the viscous stress term. Based on these models, Equation 3.9 becomes a positive definite expression for the entropy generation rate, as it represents a sum of squared terms. Temperature, T, is expressed in absolute (Kelvin) units. The positive definite equation applies to both compressible and incompressible Newtonian fluids. In Equation 3.9, the first term on the right side represents entropy generation due to heat transfer across a finite temperature difference, whereas the second term represents the local entropy generation due to viscous dissipation (i.e., conversion of kinetic energy into internal energy through fluid friction). The vector form of the positive-definite equation for entropy production can be expressed as follows: P s =
k (∇T ⋅ ∇T ) mF + ≥0 T2 T
(3.11)
where F is the viscous dissipation function, which involves velocity gradients in the fluid motion. In Equation 3.11 the first term represents entropy generation due to heat transfer across temperature gradients in the fluid. The second term is the local entropy generation due to viscous dissipation. For turbulent flows, the effective thermal conductivity can be approximated by the sum of the molecular eddy conductivities, whereas effective viscosity is the sum of the molecular viscosity and eddy diffusivity. An upcoming chapter will focus on detailed modeling of entropy transport in turbulent flows. For a nearly isothermal process, the thermal contribution to entropy generation is neglected. The resulting form of the equation, representing the viscous dissipation contribution alone to flow loss, is given by the second term on the right side of the previous equation, that is,
P s =
2 ∂u 2 ∂v 2 m ∂u ∂v + + 2 + ≥ 0 T ∂y ∂x ∂x ∂y
(3.12)
where the expression in square brackets is the viscous dissipation function, F. This result is directly related to the mechanical power needed to transport fluid through a system. Unlike velocity or temperature, the measurement of entropy cannot be performed directly. However, the previous equation can be used as an indirect way of characterizing the flow irreversibility. For example, the entropy produced by friction irreversibility can be estimated by measured gradients of velocity. In upcoming chapters, these velocity gradients will be obtained through postprocessing of experimental velocity data or numerical results obtained from a CFD solution of the Navier–Stokes equations.
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3.3 F o r mu l a t io n o f Lo s s Co ef f ic ien t s IN TER ms o f En t r o py Pr o d u c t io n Conventional loss parameters, such as a global head loss or pressure recovery coefficient, typically cannot identify the specific locations and sources of flow losses within a fluid system. This section presents a formulation that allows local irreversibilities to be scrutinized and converted to local distributions of the loss coefficient. In this way, a designer could use local loss mapping to detect locations of high entropy production (or flow irreversibility), thereby allowing local design changes of geometrical or other parameters to improve system efficiency. It will be shown that local rates of entropy production for an incompressible flow can be converted to local loss parameters, thereby leading to a more generalized approach to loss analysis.
3.3.1 En t r o py Pr o d u c t io n
in
Ber n o u l l i ’s Eq u a t io n
Consider incompressible viscous flow through a streamtube. A streamtube refers to a three-dimensional tube that encompasses a fluid streamline within a channel or other flow configuration. The Bernoulli equation identifies the head loss along this flow path as follows: p1 1 p 1 + gz1 + V12 = 2 + gz2 + V22 + hl ρ1 ρ2 2 2
(3.13)
where hl is the head loss and the subscripts 1 and 2 refer to different points along the streamline. Also, p, g, z, and V refer to pressure, gravitational acceleration, elevation, and total velocity, respectively. It can be shown that Bernoulli’s equation represents an integrated form of the following differential mechanical energy equation (Naterer, 2002):
ρ
D 1 2 V = - v ⋅ ∇p + ∇ ⋅ τ ⋅ v - τ : ∇v + Fb ⋅ v Dt 2
(3.14)
where D/Dt, τ , Fb, and v refer to the total (substantial) derivative, shear stress tensor, body force, and fluid velocity vector, respectively. The colon symbol (:) represents matrix contraction between the shear stress and velocity gradient matrices (yielding the viscous dissipation function). The temporal portion of the substantial derivative on the left side vanishes for steady-state conditions. The previous equation requires that the net convection of kinetic energy (first term) balances the sum of flow work (second term), net work of viscous stresses (third term), plus the net work done by body forces to increase kinetic energy (fifth term), minus the viscous dissipation (fourth term). Rewriting the gravitational body force term, integrating over a streamtube control volume, V, and expressing the vector gradient in the streamwise direction, s, it can be shown that
∫
V
ρV
∂ 1 2 p V + + gz dV = ρ ∂s 2
∫
V
∇ ⋅ τ ⋅ vdV -
∫
V
τ : ∇vdV .
(3.15)
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ds 1
1´
2
2´
Fig u r e 3.1 Flow losses along a streamtube in recirculating flow.
The net viscous work term (first term on right side) is the work done by viscous stresses in the fluid element against the surroundings to change the kinetic energy of the fluid. Consider a control volume, A(ds), of finite width in the cross-stream direction and differential length in the streamwise direction (see Figure 3.1). Integrating over this control volume and assuming a uniform mass flow rate through the streamtube encompassing the control volume, it can be shown that
m m R
∫
2
1
1 1 P d V 2 + + gz = ρ 2 m R
∫
V
∇ ⋅ τ ⋅ vdV -
∫
V
τ : ∇vdV
(3.16)
where m R is a reference global mass flow rate. The last term on the right side refers to viscous dissipation within the control volume. It represents a loss term in Equation 3.16, which can be directly related to the entropy generation, based on Equation 3.18. Performing that substitution and comparing to Bernoulli’s equation, the head loss becomes hl =
1 m R
∫
V
TPs dV
(3.17)
Alternatively, this result can be expressed in terms of the local rate of exergy destruction, X d , due to friction irreversibilities of viscous dissipation at ambient temperature, T0, hl =
1 m R
∫
V
X d
T dV T0
(3.18)
This result represents a valuable local alternative to conventional global loss characterization. It can be observed that the available energy loss within a fluid element is a local volumetric phenomenon involving exergy destruction.
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In contrast to other methods characterizing the flow losses through global empirical coefficients, an entropy-based approach allows local tracking of flow losses, because V can be taken as an arbitrarily located discrete volume. Entropy production encompasses all flow irreversibilities (thermal, friction, chemical, and so forth), unlike other variables such as pressure, which are commonly used in loss analysis. Reduced entropy production is a common objective in fluids engineering systems while changes of individual flow variables are generally problem dependent. For example, higher pressure losses with added baffles may be helpful to increase heat transfer rates in a heat exchanger, but reduced pressure losses are needed in pipe flows, as they entail lower pumping input power. Thus, tracking local pressure changes does not generally identify the problem areas. On the other hand, lower entropy production rates are desired in both cases, and they provide a more robust and common design objective.
3.3.2 L o s s Co ef f ic ien t s
in a Pl a n e Dif f u s er
Consider an example of incompressible viscous flow through a diffuser of unit depth, as shown in Figure 3.2. Assuming a uniform velocity profile between the outlet (subscript 2) and the inlet (subscript 1) of the duct, a balance of total energy, E, over the entire duct gives dE 1 1 = m 1 e1 + pυ1 + gz1 + V12 + Q - m 2 e2 + pυ 2 + gz2 + V22 - W (3.19) dt 2 2
and W refer to the mass flow rate (constant throughout the stream e, v, Q, where m, tube), internal energy (per unit mass), specific volume (per unit mass), heat transfer, and boundary work, respectively. For steady-state conditions without boundary work, Equation 3.19 becomes
p1 1 p 1 m ( e2 - e1 ) - Q . + gz1 + V12 = 2 + gz2 + V22 + ρ1 ρ2 m 2 2
(3.20)
Also, applying an entropy balance to a differential section in Figure 3.2 and using the Gibbs equation, ˆ = Tm de + p dv - T dP ˆ dQ i s T T
Q
(3.21)
Out
dQ In
2 W
z
1
Fig u r e 3.2 Streamtube for diffuser analysis.
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ˆ is used rather than dQ , since Q represents a process, not a where the notation of dQ thermodynamic state variable. Summing over n sections throughout the entire duct from the inlet (i = 1) to the exit (i = n), n
Q = m ( e2 - e1 ) -
∑ T dˆ P i
(3.22)
s ,i
i =1
comparing Equations 3.13, 3.20, and 3.22, By hl =
m ( e2 - e1 ) - Q 1 = m m
n
∑ T dPˆ i
s ,i
i =1
≥0
(3.23)
This result confirms that the head loss in Bernoulli’s equation is a measure of irreversibility, which represents a loss of mechanical energy per unit mass of the flowing fluid. It represents the irreversible dissipation of kinetic energy into internal energy of the fluid. As discussed previously, current design technology usually detects a loss of useful energy on a global scale using a single loss parameter, such as a valve loss coefficient. The previous results suggest that flow losses can be tracked locally based on the entropy production rate. A measure of the overall loss in the bulk fluid entails a summation of local entropy production rates in fluid elements centered on a streamline through the domain. In this way, entropy generation can be used as an alternative metric of flow loss in fluid systems. The information provided to the designer by this entropy-based metric can be more valuable than global data characterizing the end-to-end flow loss. Unlike the conventional loss characterization with a global head loss or pressure recovery coefficient characterizing an entire device, local loss characterization with the entropy-based metric allows the designer to identify the source and specific location of head losses.
3.3.3 C a s e St u d y
o f Ch a n n el a n d Dif f u s er Des ig n
In this section, results will be presented to link entropy generation with conventional loss parameters for channel and diffuser flow problems. The case study considers an incompressible viscous flow between two horizontal plates with a length of L. The plates are spaced 2w apart (y-direction). If the plates are very wide, the fully developed velocity profile does not change in the z-direction, so that y 2 u = uc 1 - w
(3.24)
where the centerline velocity, uc, can be expressed in terms of the channel pressure drop, Dp, and average (mean) velocity, u , as follows: uc =
- w 2 Dp 3 = u 2m L 2
(3.25)
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The total head loss over the entire channel, hl, can be obtained from the integrated frictional irreversibility of entropy production as follows: hl =
1 m
∫
V
1 TPS dV = m
∫
V
2
∂u 4 muc2 m dV = w4 ∂y
∫
w
-w
y 2 DLdy
(3.26)
which gives hl =
2uc L m ρw 2
(3.27)
p1 - p2 ρ
(3.28)
Using the previous expression for uc, hl =
Thus, the entropy-based formulation of head loss for channel flow reduces to the expected result of Dp/ρ , which is the required head loss between two wide horizontal flat plates. Alternatively, the friction factor and the mechanical energy loss can be related to entropy production for the channel flow as follows:
f =
2WT u2 mL
∫
V
P sdV
(3.29)
By using P s as a metric of evaluation, the equivalent friction factor becomes a product of the local entropy generation integrated over the domain and a constant, based on averaged values of the flow variables. Alternatively, exergy is defined as the work potential that can be extracted from an energy source. The exergy destroyed in a process reflects the extent to which the operation of an actual system departs from the theoretical limit of the ideal system. This departure is proportional to the entropy generation. For the purpose of extending this analysis to more complex geometric configurations, a validation study with a numerical simulation was performed. Using a control-volume-based finite element method (CVFEM) (Naterer, 2002), validation of the numerical model was carried out through comparisons with the previous analytical solution of incompressible, viscous flow between two horizontal plates (representing a channel flow). The fully developed velocity profile does not change in the streamwise direction. The analytical solution of the velocity distribution is differentiated to find the spatial variation of entropy production in the channel. The predicted results of velocity and local entropy production, for water flow at 290 K with an average velocity of 0.0504 m/s through a duct with a width of 0.02 m, are illustrated in Figure 3.3a and Figure 3.3b, respectively. The numerical and analytical results show close agreement. The entropy production rate is maximum near the wall
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Entropy-Based Design and Analysis of Fluids Engineering Systems 0.08 0.07
Velocity (m/s)
0.06 Analytic
0.05
Computed
0.04 0.03 0.02 0.01 0 0.000
0.005
0.010
0.015
0.020
0.025
0.015
0.020
0.025
y (m) (a)
Local Entropy Generation (W/m3K)
90
Analytic
80
Computed
70 60 50 40 30 20 10 0 0.000
0.005
0.010 y (m) (b)
Fig u r e 3.3 Channel flow results: (a) velocity and (b) entropy production.
due to viscous effects. It becomes zero at the center of the channel, where the crossstream velocity gradient is zero. To link the entropy generation with traditional loss parameters, the local entropy generation is integrated over the domain and related to the friction factor based on Equation 3.29 as follows:
2 mLu Ps = f 2 wT
(3.30)
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where f , m , L , and T are the friction factor, mass flow rate, plate length, and the average temperature of the fluid, respectively. When Equation 3.30 is integrated over the volume, it gives the total entropy produced in viscous laminar flow between the inlet and exit locations of any two-dimensional expansion section of the same inlet width. For laminar flow between parallel plates, it is known that f = 48 / ReD , so Equation 3.30 becomes 2 12 mLu Ps = (3.31) ReD wT
Equation 3.31 shows how the total entropy generated in a fully developed flow between parallel plates can be evaluated based on geometrical and flow data. An entropy-based formulation used for optimization purposes could be benchmarked against this value for validation. The entropy production results for u = 0.0504 m/s, L = 15 cm, and w = 2 cm with water at 290 K from Equation 3.31 and the numerical formulation are 8.522 × 10 -7 W /K and 8.519 × 10 -7 W /K, respectively. This close agreement provides useful validation of the formulation. Consider another example of gas flow through a subsonic diffuser, which is widely encountered in aerospace and other applications. In this particular configuration, an incoming flow experiences an area expansion. Flow losses arising from an area expansion lead to reduced diffuser effectiveness. The flow characteristics are highly dependent on the area expansion ratio and the Reynolds number. Entropy production can be used as a basis of correlating this optimal flow configuration, with respect to different flow conditions and system parameters. The two-dimensional geometry of the expansion section in the numerical simulation is shown in Figure 3.4. A fully developed velocity profile is prescribed at the inlet, and a Neumann condition is applied at the outlet. For a given outlet-to-inlet area ratio, the velocity fields were computed for different expansion angles. Reynolds numbers of 301 and 602 were investigated with an area ratio kept at 1.5. The flow remained unstalled until θ = 10 o and θ ≅ 7o for Reynolds numbers of 301 and 602, respectively. When the expansion angle increases, the boundary layer separates from the top wall. A recirculation cell is formed in a similar way as flow past a backward facing step. This separation arises from the unfavorable pressure gradient introduced by the expansion. The predicted velocity distribution (Re = 602) and the corresponding entropy generation contours (values multiplied by 105) at an expansion angle of 60o are shown in Figure 3.5a and Figure 3.5b, respectively. For a Reynolds number of 301,
H = 1 cm 2H Vin
=0 =0
Fig u r e 3.4 Schematic of a plane diffuser.
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(a) 78222
(b)
11/31
0.26105
0.13075 0.39134 0/8222
0.13075
0.26105
Fig u r e 3.5 (a) Velocities and (b) entropy production contours (*105 W/m3K) for Re = 602 with a 60o expansion.
the flow reattaches at approximately 8.5 step heights from the expansion. Figure 3.5b illustrates the flow irreversibilities, as characterized by the contours of local entropy production. For comparison purposes, three major regions of entropy generation and loss distribution are evident. The first region is the channel leading into the expansion. The second region is the diverging region, which starts at the inlet and ends where the diverging section joins the wider channel. The third region continues from that point to the outlet. Three subregions of importance are identified in the diverging section: a recirculation and reattachment region close to the top wall, a region of separation close to sharp corners at the beginning of the expansion, and flow along the bottom wall. Entropy production in the recirculation or attachment zone is not predominant, because the flow is relatively slow in that region and the velocity gradients are small. The entropy production is high, close to the separation region. It diminishes when the flow decelerates to fill the larger channel. The flow near the bottom of the expansion also exhibits high entropy generation, due to the wall shear effects. The entropy generation map provides a useful way of detecting the detailed structure of the mechanical energy loss in the expansion section. The variation of total entropy generation in the unstalled flow regime for the diverging section is shown in Figure 3.6. It is interesting to observe that the loss decreases in the region with less expansion, until approximately 3.5o for Re = 301 and 3.0 o at Re = 602. Higher angles cause an increase in the mechanical energy loss, due to greater channel length. The trend also confirms the dependence of flow losses on Reynolds number, when the flow is laminar. In Figure 3.6, an optimal angle is predicted at the point of minimized entropy production. This entropy-based approach provides a useful alternative and systematic way of establishing the optimized geometrical configuration. The same approach of summing local entropy production rates can be applied to any fluids engineering device. Since this methodology entails tracking of local losses throughout an individual device, rather than global Second Law analysis, the approach provides a valuable component-level energy management tool. Figure 3.7 illustrates results for loss characterization over a larger range of expansion angles. The values have been normalized by Equation 3.31 (divided by the
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Normalized Total Entropy Generation
0.58 Re = 301 Re = 602
0.53 0.51 0.49 0.47 0.45
0
2 4 6 Expansion Angle, Degrees
8
Fig u r e 3.6 Total normalized entropy production in the unstalled region.
Normalized Total Entropy Production
0.90
0.60
0.30
0.00
0.0
20.0 40.0 60.0 Expansion Angle, Degrees
80.0
Fig u r e 3.7 Flow loss characterization in the unstalled region.
volume). The resulting parameter after normalization is directly proportional to the loss coefficient. As observed with previous studies, an optimal angle yielding the least flow losses in the expansion section exists. This optimum corresponds to narrow angles (approximately 3.5o for Re = 301 and 3.0o at Re = 602) and unstalled flow conditions. Based on these results, a new entropy-based metric that locally characterizes the pressure recovery factor (or any other global loss parameter) is defined as follows:
h = 1
Ps,θ Ps,ref
(3.32)
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where Ps,θ is the entropy production for a diffuser with an expansion angle of q. Also, Ps,ref represents a loss that would occur in the absence of the expansion section, given by Equation 3.31. An interesting observation from Figure 3.7 is noticed for expansion angles higher than 10 o, whereby small changes in the expansion angle cause a larger loss. In addition to providing a physically based measure of loss, the entropy-based approach also leads to added insight into the specific location of flow losses and the flow structures leading to those losses.
3.4 U pper En t r o py Bo u n d s in Cl o s ed Sys t ems Upper bounds of system performance provide a useful design parameter for ensuring that maximum system capabilities are not exceeded. For example, upper bounds on cooling capabilities of a heat pipe can ensure that maximum operating temperatures are not exceeded during convective cooling of a microelectronic assembly. Various other examples arise in thermal design of aerospace, manufacturing, automotive, and other applications (Naterer, 2002). In this section, a method of establishing upper entropy bounds for convection problems is developed. These bounds involve both friction and thermal irreversibilities arising during convective heat transfer within an enclosure. Various past methods have been developed for establishing upper bounds in thermal systems. Martins and da Gama (2000) developed an upper bound for solutions of coupled heat conduction and radiative heat transfer problems, subject to nonlinear boundary conditions. An auxiliary function was used to establish upper bounds, while confirming that the Laplacian of the temperature field satisfied certain inequality constraints. In heat conduction problems, an upper bound for thermal shape factors was derived for two-dimensional layers by Hassani et al. (1993). Upper bounds for conduction contact resistances were developed by Bobeth and Diener (1982). These bounds are functions of a two-point correlation function of the local contact resistance. Upper bounds for random arrangements of circular contact spots of equal size can be predicted by variational principles with stochastically varying local contact resistances. In convection problems, upper pressure bounds were developed for establishing criteria in thermal destabilization of wall shear flows (Mikic, 1998). These bounds involved admissible system perturbations, which lead to the onset of turbulence and upper bounds for the wall sublayer scales in fully developed turbulent flows. Entropy transport characterizes the dissipation of kinetic energy in these layers (Adeyinka and Naterer, 2005). For laminar separated flows, upper bounds have been derived to predict laminar instabilities of self-sustaining oscillations and vortex shedding (Mikic, 1998). This section focuses on upper entropy bounds for problems involving internal forced convection in enclosures or tanks (Lui and Naterer, 2007). Numerous past studies have been conducted on forced convection with internal confined flows (such as Eames and Norton, 1998; Homan and Soo, 1998; Naterer, 2001; Sinai, 1985). Entropy bounds can provide useful new insight regarding the dynamics of flow mechanisms in these problems. For example, entropy production characterizes the mixing, flow structures, and magnitude of frictional dissipation in a tank. As a result, it can provide useful guidance for effective design and control of internal flows. It can establish optimal conditions to reduce input power during fluid mixing, while transferring fixed rates of heat transfer to a fluid. The objective of this section is to develop analytical expressions
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for the upper entropy bounds and correlate these limits with different geometrical configurations. In this way, design changes could reduce input power required by energy conversion devices. Alternatively, changes of design parameters to alter entropy bounds can enhance mixing in chemical processing applications.
3.4.1 U pper Bo u n d s
o f Th er ma l Ir r ev er s ibil it y
Consider three-dimensional convective heat transfer in a bounded domain, V, governed by the following equations of fluid motion and energy transport: ∂v ∇p (3.33) + v ⋅ ∇v = + ν∇ 2 v ρ ∂t ∂T + v ⋅ ∇T = a∇ 2T ∂t
(3.34)
∂T =0 ∂n
(3.35)
No-slip conditions ( v = 0) and Neumann boundary conditions are applied along the walls of a closed domain, that is,
where n refers to the unit outward normal direction on the surface, Ω, which encompasses the volume of the problem domain, V. . A “temperature excess” is the difference between the actual temperature at some position, T, and the average initial temperature. It is defined as follows:
τ =T
1 |V |
∫
V
T (t = 0 )dV
(3.36)
Both actual and average temperatures satisfy the governing equations, so ∂τ + v ⋅ ∇τ = a∇ 2τ ∂t
(3.37)
subject to Neumann boundary conditions. It can be shown that t has a zero average at all times. Entropy production arising from heat transfer over a finite temperature difference involves a squared temperature and squared temperature gradient (as part of the thermal irreversibility), so the previous equations will be squared. Multiplying Equation 3.37 by t and then integrating over V,
∫
V
τ
∂τ dV + ∂t
∫
V
τ ( v ⋅ ∇τ ) dV = a
∫
V
τ∇ 2τ dV
(3.38)
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which can be rewritten to give 1 ∂ 2 ∂t
∫
V
τ 2 dV = -a
∫
V
| ∇τ |2 dV
(3.39)
From Poincare’s inequality (Zeidler, 1985) for any function, s, with first derivatives that are square integrable, there is some positive constant c < 1 such that
∫
σ 2 dV +
V
∫
V
1 c
| ∇σ |2 dV ≤
∫
V
| ∇σ |2 dV +
∫
V
2 σ dV
(3.40)
Using this inequality, Equation 3.39 becomes
∂ ∂t
∫ τ dV = -2a ∫
2
2
v
v
∇ τ dV ≤ -2a C
∫ τ dV 2
v
(3.41)
because t has a zero average at all times, where C = c/(1 - c). Integrating this result,
where
∫
V
τ 2 dV ≤ C0 e -2aCt
(3.42)
∫ τ (t = 0)dV
(3.43)
C0 =
2
v
This result indicates that the squared temperature excess decreases exponentially over time. This result will be used to show that the temperature excess gradient (needed in the thermal irreversibility of entropy production) also decreases exponentially. Consider a practical example of a closed domain (such as a tank) with fluid containing a fixed initial amount of total energy. For example, consider a magnetic stirrer for thermochemical processing to generate uniform mixtures in a rectangular tank (see Figure 3.8). In this example, uniformly distributed magnitudes of entropy production and high total entropy are desired to maximize mixing. It is worthwhile to consider how different geometrical configurations affect the steady-state entropy field in the tank, particularly the upper bound of total entropy, so maximum mixing could be achieved with minimal power input to the system. After the mixing stops, the velocity field approaches zero in the steady state ( v → 0 as t → ∞). Also, the first derivatives of velocity approach zero at the steady state, when the fluid motion stops. The mixing process within the enclosure decays over time. Numerous past studies have considered time decay of diffusive mixing. For example, for diffusive mixing associated with oscillation of a plane wall, the velocity gradient changes exponentially with time (White, 1974). This oscillation is similar to the problem considered here, whereby a mixer suddenly stops and the fluid velocities decrease over time.
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Volume of Tank (V)
Velocity Decay after the Mixer Stops (Time F)
Surface of Tank (Ω)
Fluid Mixer in Heated Tank
Fig u r e 3.8 Schematic of a tank mixing problem.
Thus, consider that the velocity gradients decrease exponentially over time in this current problem (note: other decay trends, such as decays bounded by a constant multiple of t-2, would yield analogous results). For exponential decay, consider that F ≤ d1e - d2t for positive constants d1, d2, and T (t = 0 ) ≥ Tmin (minimum temperature), where F refers to the viscous dissipation function. The total entropy production includes thermal irreversibilities (due to heat transfer) and friction irreversibilities (due to viscous dissipation). First, consider the thermal irreversibilities, which involve squared temperatures and squared temperature gradients. Applying integration by parts to calculate the time derivative of the squared temperature gradient, d dt
∫
V
| ∇τ |2 dV = 2
∫
Ω
∂τ ∂τ dΩ - 2 ∂n ∂t
∫
∂τ ∇ 2τ dV ∂t
(3.44)
∫
| v |2 | ∇τ |2 dV
(3.45)
V
The right side satisfies the following inequality: RHS ≤ -2a
∫
V
(∇ 2τ )2 dV + a
∫
V
(∇ 2τ )2 dV +
1 a
V
Since the velocity magnitude approaches zero in the steady state and it remains bounded below a throughout the time period t ≥ F, RHS ≤ -a
∫
V
(∇ 2τ )2 dV + a
∫
V
| ∇τ |2 dV
(3.46)
For the first term on the right side, the following inequality can be used:
∫
V
| ∇τ |2 dV = -
∫
V
τ ∇ 2τ dV ≤
1 2
∫
V
τ 2 dV +
1 2
∫
V
(∇ 2τ )2 dV
(3.47)
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Combining this result with the previous right side, d dt
∫
V
| ∇τ |2 dV ≤ C0a e -2aCt - a
∫
| ∇τ |2 dV
V
(3.48)
Integrating this inequality over time from t = F to some other arbitrary time, t, yields
∫
V
| ∇T |2 dV ≤ e -a ( t - F )
C0 -2Ca t | ∇T (t = F ) |2 dV + e - e -a ( t - F + 2 FC ) 1 - 2C
∫
V
(3.49)
This result will establish bounds and an exponential decay of the thermal irreversibility of entropy production. The entropy transport equation can be written as ∂s F ∇ 2T + ρv ⋅ ∇s = k +m ∂t T T
ρ
(3.50)
The total entropy within the domain, V, is given by S (t ) =
∫ ρs( x, t )dV
V
(3.51)
This total (spatially integrated) entropy within the domain becomes a function of time only, and its derivative (with respect to time) is greater than or equal to zero, that is, dS =k dt
∫
V
| ∇T |2 dV + m T2
∫
V
F dV ≥ 0 T
(3.52)
This result represents an integrated form of the Second Law. Using the previous result of the bounded thermal irreversibility, it follows that the total entropy is bounded according to S(t) ≤ M, where M = S( F ) +
k 2 a Tmin
∫
V
| ∇T (t = F ) |2 dV +
C0 e -2Ca F 2C
m d1 | V | e - d2 F + Tmin d2
(3.53)
Also, the total derivative of entropy on the left side of Equation 3.52 approaches zero in the steady state, i.e., lim S ′(t ) = 0. In a case with a constant initial temperature t →∞ throughout the domain, the temperature will remain at that same constant value for all times, and thus ∇T ≡ 0 and C0 = 0 . The upper bound of the total entropy simplifies considerably for this case, that is, M = S( F ) +
m d1 | V | e - d2 F Tmin d2
(3.54)
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In Equation 3.53, the analysis indicates that C should be made as large as possible, in order to minimize the value of M. Fluids having small values of k and m and large values of a lead to maximized upper bounds on the total entropy, defined through M. The geometrical configuration can be modified to maximize the value of the constant C. From Equation 3.41, this constant satisfies C w 2 dV ≤ | ∇w |2 dV , V V where w is any function whose normal derivative vanishes at the boundary and satisfies ∫v wdv = 0. From a variational formulation of the eigenvalue problem, it can be shown that C is given by the smallest eigenvalue of the following problem,
∫
∇ 2 w + λ w = 0,
dw =0 dn
on
∫
Ω
(3.55)
This problem has a solution of l = 0, and w is any nonzero constant function. The solution will be the second eigenvalue of the eigenvalue problem. Of all domains with a fixed volume, it is known that a disk in two dimensions and a sphere in three dimensions yield the largest value of C (Weinberger, 1956). In the next section, these results and analysis will be used to determine the optimal aspect ratio to minimize entropy bounds associated with mixing in the tank.
3.4.2 O pt ima l Asp ec t Ra t io
o f Upper
En t r o py Bo u n d s
Consider the problem of determining the optimal aspect ratio of the rectangular mixing tank (Figure 3.8), which gives the largest possible value of C. Let the crosssectional dimensions of the rectangle be L and L -1. A basis for the space of functions on the rectangle with the left corner at the origin is
{
}
πx πx 2π x 2π x cos π yL, , cos π yL, cos cos π yL, cos , cos 2π yL, cos L L L L (3.56) These functions are also eigenfunctions with the following corresponding eigenvalues: B = cos
E=
{
}
π2 2 2 π2 4π 2 4π 2 , π L , 2 + π 2 L2 2 , 4π 2 L2 , 2 + π 2 L2 , 2 L L L L
(3.57)
Consequently, the minimum eigenvalue is given by min(π 2 L-2 , π 2 L2 ). The goal is to find the value of L that maximizes this eigenvalue. In other words, find
π2 Emax = max min 2 , π 2 L2 L >0 L
(3.58)
It can be shown that the maximum occurs at L = 1, which gives a square. The maximum value of C becomes π 2 ≈ 9.87.
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Consider a disk of unit area (radius of π -1/2 ). The eigenvalues are given by (ζ mn π )2 , where J′n(zmn) = 0. In other words, zmn is the mth zero of the Bessel function derivative J′n. The smallest value of these zeros is approximately 1.8412, which is the first zero of J′1. This leads to C ≈ 10.65. Hence, a disk yields a larger value of C than a square with the same area. In three dimensions, a cube is the domain which yields the eigenvalue π 2, as it gives the largest value of C among all rectangular solids of unit volume. The sphere of unit volume gives a value of C ≈ 11.26. This result suggests that disks and spheres are better domains than rectangles and rectangular solids for minimizing the upper bound of total entropy in the steady state. As an example of a Dirichlet case, consider a one-dimensional problem where the temperatures at the left and right boundaries (x = 0 and x = 1) are fixed at T0 and T1, respectively, with T0 > T1. Suppose that the initial temperature satisfies T (t = 0 ) ≤ min(T0 , T1 ). Heat transfer is initiated and after some time, the temperature approaches a linear profile, with a temperature of T0 on the left side and T1 on the right boundary. This means that |∂T/∂x|2 will approach a positive constant. In this case, the previous analysis yields S ′(t ) ≥ k
∫
1
0
2
1 ∂T dx T 2 ∂x
(3.59)
The entropy derivative is larger than a positive constant for all times. The total entropy grows unboundedly over time, because the heat supply from the left boundary provides a continued entropy flow into the domain over time. This boundary condition is more suitable for mixing problems if a goal is to maximum mixing and entropy.
3.4.3 C a s e St u d y
o f Mixin g Ta n k Des ig n
Consider another example of fluid mixing in a tank, but with nonuniform initial profiles of velocity and temperature. Nondimensional numerical results were obtained by numerical integration with a spectral method and integration over the spatial domain to give the net entropy production (Lui and Naterer, 2007). Spectral methods n seek solutions of the form ∑ j =1 a j φ j for certain basis functions denoted by {f j}. Consider a rectangular domain, V, bounded by [-L -1, L -1] × [-L, L] with the following nonuniform initial temperature and velocity distributions: y2 TL (t = 0 ) = T0 + g (1 - L2 x 2 ) 1 - L
π Lx π y π y π e - nt 1 2 2 vL = - L cos 2 sin L , L sin(π Lx ) cos 2 L 2 2 2 L +L
(3.60)
(3.61)
where T0, g, and h are positive constants. Note that the velocity field satisfies conservation of mass (zero divergence), decays exponentially, and vanishes at the boundary of the domain. After normalizing with x = x/L and y = y/L , the new domain
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becomes a square bounded by [- 1,1] × [- 1,1]. In the numerical simulations, a Chebyshev spectral collocation method was used to solve the temperature equation (Lui and Naterer, 2007). Fluid motion is initially generated by the fluid mixer, but then the mixer within the tank is turned off, and the fluid velocity approaches zero after a period of time. An example of a magnetic stirrer for chemical mixing is illustrated in Figure 3.8. The upper bound of total entropy has practical significance because it characterizes the effectiveness of mixing and the system input power required to achieve certain levels of mixing. This mixing is dependent on the initial temperature and velocity profiles, which characterize the fluid motion and heat transfer leading to entropy production during the mixing process. The upper entropy bounds can be predicted without solving the detailed transient equations in the tank with CFD. Analytical results for the upper entropy bound are developed independently of this transient motion, although they depend on the initial level of fluid mixing in the tank. A simulation was performed with mixing of methane inside the tank. Based on a nonuniform initial temperature with T0 = 873 K, g = 10, h = 0.0155, L = 3, and a time step of 0.00065 s, the predicted results are shown in Figure 3.9 for 17, 19, and 21 Chebyshev modes of a spectral method, respectively. In Figure 3.10, 10 Chebyshev modes were used with a time step of 0.28 s. The initial temperature is fixed at 373 K, and the total entropy at equilibrium is plotted for various aspect ratios of the rectangle. It can be observed that the total entropy increases at larger values of L. In Figure 3.11, the total entropy at equilibrium is minimized for a square (L = 1). The total entropy at equilibrium increases at higher aspect ratios, namely, the higher ratio of surface
0.025 17 21 19
Total Entropy (J/K)
0.02
0.015
0.01
0.005
0
0
100
200
300 400 Time (sec)
500
600
700
Fig u r e 3.9 Total entropy for methane for L = 3.
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9
× 10–3 L=3
8
Total Entropy (J/K)
7 6
L = 2.5
5 4
L=2
3 L = 1.5
2 1 0
L=1 0
2000
4000
6000 8000 Time (sec)
10000
12000
14000
Fig u r e 3.10 Effects of tank width on total entropy change for methane.
9
× 10–3
Total Entropy at Equilibrium (J/K)
8 7 6 5 4 3 2 1
0
0.5
1
1.5 L(m)
2
2.5
3
Fig u r e 3.11 Total entropy at equilibrium (373 K) at varying tank widths.
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area to volume for the domain. Frictional irreversibilities along the wall constitute a major component of the total entropy production, so larger surface areas lead to higher entropy production and total entropy at equilibrium. The results in this case study provide examples of how entropy bounds can give useful insight for better understanding of energy conversion, mixing, and fluid dynamics in tanks and enclosures.
3.5 C a s e St u d y o f Au t o mo t iv e Fu el Cel l Des ig n The next case study in this chapter involves entropy-based design of fuel cells (Naterer and Tokarz, 2006). It will be shown that entropy and the Second Law provide a valuable design tool for achieving higher efficiency of fuel cells. This includes friction and thermochemical irreversibilities of gas flow through the fuel cell channels.
3.5.1 El ec t r o c h emic a l Ir r ev er s ibil it ies
in a Po r o u s
El ec t r o d e
In a fuel cell, entropy production of irreversible chemical reactions, diffusion, and ohmic heating leads to voltage losses (or polarization). During operation of a solid oxide fuel cell (SOFC), fuel and oxidant are continuously supplied to the anode and cathode, respectively (see Figure 3.12). Oxygen molecules combine with free electrons from the external circuit to produce negative oxygen ions (O-), which migrate through the electrolyte and generate ohmic heating with entropy production. Hydrogen molecules diffuse simultaneously through the anode. They combine with the oxygen ions and liberate electrons, while producing H2O and heat. As a result, free electrons flow through the external circuit as an electrical current. They return to the fuel cell at the cathode, where they combine with oxygen molecules to again produce oxygen ions. A proton exchange membrane fuel cell (PEMFC) operates in a similar manner, except that hydrogen ions flow to the cathode where water molecules are produced. The chemical balances for an SOFC and PEMFC are given by SOFC:
H 2 ( g ) + O = ⇒ H 2O( g ) + 2 e -
( anode)
(3.62)
( cathode)
(3.63)
( anode) 1 O ( g ) + 2 H + ( aq ) + 2 e - ⇒ H O (l ) ( cathode) 2 2 2
(3.64)
1
PEMFC:
2
O2 ( g ) + 2 e - ⇒ O =
H 2 ( g ) ⇒ 2 H + ( aq ) + 2 e -
(3.65)
Irreversibilities within a fuel cell lead to voltage losses and lost power to auxiliary devices like blowers to sustain cyclical operation of the fuel cell. The Nernst equation gives the ideal (reversible) performance of a fuel cell, in terms of the ideal standard potential, E 0, ideal equilibrium potential, E, and nonstandard product or reactant temperatures and pressures. For a PEMFC, the maximum theoretical voltage is given by
RT PH2 RT PO2 + E = E0 + ln ln 2 F PH O 4 F P0 2
(3.66)
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Current Flow
End Plate Anode Electrolyte Matrix Cathode
Fuel Flow
Separator Plate
Repeating Unit
Oxidant Flow Anode y=h
Solid Wall (Bipolar Plate) Gas Flow
y
Porous Electrode Interface v(x, 0) = Suction Velocity
x (a)
Bipolar Plate
e– (+)
Electric Current Flow
O2
H+
H2
Gas Channel
Bipolar Plate
(–)
Anode
Gas Channel Cathode 0.5O2 + H+ + 2e– H2O
Electrolyte
Fig u r e 3.12 Schematic of a PEMFC (a) stack, channel flow and (b) operating components.
where P0 refers to the standard pressure (1 atm) and other variables are defined in the “Nomenclature.” Voltage losses are often characterized by the polarization, h. The total polarization is the potential difference, DE, between the reversible voltage and the cell voltage when current flows through the circuit, that is,
h = DE = Erev - Ecurrent
(3.67)
The reversible voltage computed at the wall (subscript w) and bulk value (subscript b) can be expressed in terms of the ideal standard potential for the chemical reaction, E 0, that is,
( )
(3.68)
( )
(3.69)
Ew = E 0 +
RT ln CHI2 2F
Eb = E 0 +
RT ln CH2 2F
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The difference between these voltages is called the concentration loss or concentration polarization. It represents the difference of the reversible potential computed from the gas concentration at the wall and bulk value. The concentration polarization at the anode (subscript conc,a) can be written as (Naterer and Tokarz, 2006, and references therein)
hconc,a = -
RT δ H2 la RT pa pa ln - - pHI2 ⋅ exp ⋅ ⋅ i pHI2 (3.70) 2 F δ H2 δ H2 2 F DH2 ( eff ) pa
The electrochemical polarization is related to entropy production, Ps, according to
h=
T ⋅ Ps 2F
(3.71) Thus, entropy production due to concentration irreversibilities within the anode can be written as p p RT δ H2 la Ps,conc = - R ⋅ ln a - a - pHI2 ⋅ exp ⋅ ⋅ i pHI2 (3.72) δ H2 δ H2 2 F DH2 ( eff ) pa This result was derived for the anode of a PEMFC. An analogous result for the cathode polarization can be derived with the same procedure: Ps,conc =
R In 2
RTlc 1 I 8 FD c ( eff ) p o2
i
RTlc 1 + 4 FD I c ( eff ) p H 2 o
i
(3.73)
In addition to these concentration irreversibilities, the total entropy production within the electrodes includes activation and ohmic irreversibilities. These losses can be written as (Naterer and Tokarz, 2006)
Ps,act =
4R i 4R i i2 i2 ln + 2 + 1 + ln + 2 + 1 ne i0 c i0 c i0 a ne i0 a Ps,ohmic = -
2 F 2CH + Dm i ln 1 - Tlm nd iL
(3.74)
(3.75)
The expression for the activation polarization of an SOFC is identical to this result, except that i0c is replaced by 2i0c (first term on right side) and i0a is replaced by 2i0c (second term on the right side). These changes occur due to the different half-cell reactions. The voltage losses become independent of the limiting current density of the anode, i0a, for a small anode thickness in a PEMFC (or a small cathode thickness in
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an SOFC). The anode concentration polarization diminishes significantly when the anode thickness becomes much smaller than the cathode thickness. For PEMFCs, the following TAP-TCS model (Thin Anode for PEMFC, or Thin Cathode for SOFC approximation) neglects the anode concentration polarization. After combining all irreversibilities, Ps =
4R i 2 F 2CH - Dm i 4R i i2 i2 ln + 2 + 1 + ln + 2 + 1 ln 1 - Tl n i ne i0 a i0 a n i i 0c e 0c m d L -
RTlc R lnn 1 i 8FDc,eff pOI2 2
RTlc 1 + 8FD p I i c ,eff O2
(3.76)
where ne and i0c refer to the number of moles of electrons produced per half-cell reaction and the cathode exchange current density, respectively. On the right side, the five terms represent the activation irreversibility (first and second terms; anode plus cathode), ohmic irreversibility (third term), and concentration irreversibility (fourth term), respectively. For an SOFC, a similar entropy production equation as the PEMFC result can be derived, except the derivation involves different half-cell reactions, so pa, d H2O, pH2, la, Da(eff), Dc(eff), and pO2 in a PEMFC are replaced by pc, d O2, pO2, lc, DO2(eff), Da(eff), and pH2, respectively, for an SOFC.
3.5.2 F o r mu l a t io n
o f Ch a n n el Fl o w Ir r ev er s ibil it ies
To calculate the total entropy production within a fuel cell stack, additional friction and thermal irreversibilities within the gas channels must be formulated. Consider gas flow through a uniform fuel channel involving either hydrogen in the anode-side channel or oxygen in the cathode-side channel (see Figure 3.12). The steady-state gas motion is bounded between a solid wall (bipolar plate) and a porous wall (gasdiffusion layer), where suction flow occurs due to the permeable interface. As a result, the fuel and oxidant concentrations will decrease along the channel, and the bulk gas velocity increases to conserve mass. In the following simplified integral analysis, the gas concentration will be assumed as uniform across the channel, but varying in the streamwise direction. Assuming a parabolic variation of velocity across the channel,
u( x, y) = A( x ) y 2 + B( x ) y + G ( x )
(3.77)
A no-slip condition is applied along the upper wall (y = h), while a slip condition of u( x, 0 ) = xu ( x ) is applied at the bottom wall (see Figure 3.12), where x represents the slip coefficient. This coefficient depends on the permeability of the porous electrode, and it generally varies between 0.1 and 1. Furthermore, the mean velocity is defined as u ( x) =
1 h
∫
h
0
u( x, y)dy
(3.78)
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Applying the boundary conditions and integrating the resulting profile to establish the mean velocity gives the unknown coefficients (A, B, and G), thereby yielding
u y2 y = (3x - 6) 2 + 2(3 - 2x ) + x u h h
(3.79)
For two-dimensional laminar gas flow under steady-state conditions through a fuel channel, the reduced mass and x-momentum equations are given by
∂( ρu ) ∂( ρv) + =0 ∂x ∂y ∂u ∂p ∂u ∂2 u ρu +v =+m 2 ∂x ∂y ∂y ∂x
(3.80)
(3.81)
where the density can be expressed as the product of gas concentration, C(x), and molecular weight, M. Although the gas density varies due to gas concentration changes, it remains an incompressible flow in terms of the Mach number. Integrating the continuity equation across the channel and applying the slip boundary condition,
h
d (Cu ) = CV0 ( x ) dx
(3.82)
where V0(x) is the suction velocity at the base of the fuel channel, defined by 1 i( x ) V0 = nF C ( x )
(3.83)
The current density at a particular x-position is related to the gas concentration by the following Tafel equation: g
i( x ) C ( x ) aF exp h = RT i0 C0
(3.84)
Nondimensionalizing the velocity ( u = u /u0 ) and gas concentration (C = C/C0), the integrated continuity equation becomes
) d (Cu + K1C g = 0; dx
K1 =
i0 aF h exp RT nFhC0 u0
(3.85)
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Substituting the u-velocity profile and solving the continuity equation, subject to appropriate boundary conditions along the top and bottom walls,
v y3 y2 y = ( 2 - x ) 3 - 3(3 - 2x ) 2 - x - 1 V0 h h h
(3.86)
Then, substituting the velocity profiles into the mass and momentum equations, assuming a constant pressure gradient in the streamwise direction and integrating the momentum equation across the channel,
K2
2 ) d (Cu + K 3 + K 4C g u + K 5u = 0 dx
(3.87)
where
2 2 1 6 x - x+ 15 5 5
(3.88)
P MC0 u02
(3.89)
xi0 aF h exp RT nFhC0 u0
(3.90)
1 m K 5 = 12 1 - x 2 Mh C0 u0 2
(3.91)
K2 =
K3 = K4 =
Solving the coupled mass and momentum equations with a series solution, sub 0 ) = 1, yields the following linearized 0 ) = 1 and C( ject to inlet conditions of u( channel flow approximation (LCF model), C ( x ) = C0 (1 - K 6 x );
K6 =
2 K1 K 2 - K 3 - K 4 - K 5 K2
u ( x ) = u0 (1 + K 7 x );
K7 =
K1 K 2 - K 3 - K 4 - K 5 K2
(3.92)
(3.93)
In the LCF model, higher-order terms have been neglected in a series solution for short channels, microfuel cells, or small values of x (near the inlet section of a fuel cell channel). LCF refers to a linearized approximation of channel flow profiles, including the gas concentration and velocity profiles. The purpose of these simplifications is to allow closed-form approximations for entropy production, particularly so net irreversibilities can be analytically
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minimized to illustrate how entropy-based design can be used as a predictive tool to improve the fuel cell performance. The combined friction and thermal irreversibilities within the fuel channel can be expressed as 2
k m ∂u Ps''' = + 2 T ∂y T
2
∂T ∂y
(3.94)
Substituting the differentiated velocity profile, integrating across the channel, and expressing the total entropy production due to friction within a channel of length L, per unit depth, 4 mu 2 L (3 - 3x + 3x 2 - 6hx + 4 h 2 ) Ps = hT
(3.95)
Alternatively, this result can be written in terms of the channel Reynolds number, Reh. Combining all irreversibilities yields the following total entropy production within the fuel cell. It can be shown that (Naterer and Tokarz, 2006) Ps =
i i 2 RT aF a F 2 RT h + h sinh -1 0 (1 - K 6 x )g exp sinh -1 0 (1 - K 6 x )g exp RT RT ne F ne F i0 a i0 c -
p RT δ H2 la RT pa aF h i0 (1 - K 6 x )g exp - I a - 1 exp ln I RT 2 F pH2 δ H2 pH2 δ H2 2 F Da,eff
-
RT RTlc i0 (1 - K 6 x )g aF h ln 1 exp I RT 4 F 8FDc,eff pO2
-
2 F 2 CH + Dm i a F 4 mu 2 L h + (3 - 3x + 3x 2 - 6hx + 4 h 2 ) ln 1 - 0 (1 - K 6 x )g exp RT Tlm nd hT iL
RTlc i0 (1 - K 6 x )g aF exp h 1 + I RT 8FDc,eff pO2
(3.96)
In the TAP-TCS model, the concentration polarization (third term on the right side) becomes negligible. In the following section, numerical predictions will be studied to outline how entropy-based design can provide a valuable tool for improving fuel cell performance.
3.5.3 Pr o t o n Exc h a n g e Membr a n e Fu el Cel l (PEMFC ) a n d So l id Oxid e F u el Cel l (SOFC ) Des ig n In this section, numerical results for PEMFCs and SOFCs will be presented (Naterer and Tokarz, 2006). Problem parameters are presented in Table 3.1 and adopted from Chan and Xia (2002), Chen et al. (2004), Ghadamian and Saboohi (2004), and Kim et al. (1999). Entropy production involves electrochemical reactions at the electrode surfaces, mass transfer ohmic heating, and frictional losses within the fuel channel.
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T a bl e 3.1 O perating Conditions and Problem Parameters Proton Exchange Membrane Fuel Cell Channel temperature, T (K) Inlet gas velocity, u0 (m/s) Inlet gas pressure, P0 (atm) Inlet gas concentration, C0 (mol/m3) Exchange current density, i0 (A/m2) Activation overpotential, h (V) Reaction order, g Electrons transferred in reaction, n Charge-transfer coefficient, a Molar weight, M (kg/mol) Viscosity, m (kg/ms) Channel height, h (m) Standard equilibrium potential, E0 (V) Pressure gradient (Pa/m) Slip coefficient, x
353.15 0.7 2.0 69.0 0.00001 0.3 0.5 4 2.0 0.032 0.00002 0.001 1.167 250.0 0.1
S olid Oxide Fuel Cell Operating temperature, T (oC) Operating pressure, p (atm) Electrolyte resistance, Ri (Wcm2) Concentration resistance, Rconc (Wcm2) RT/4F Exchange current density, i0 (A /cm2) Effective diffusion coefficient, Da,eff (cm2/s) Cathode thickness, lc (m) Average pore radius, (mm) Electrolyte thickness, le (mm) Anode thickness, la (mm) Partial pressure ratio, ph2/ph2o
750 2.0 0.092 0.297 0.02204 0.113 0.166 0.00005 0.5 40.0 750.0 32.352
The overall efficiency of a fuel cell system depends on other losses, such as gas blowers, pumps, electrical losses (DC power conversion to AC), electrolysis, fuel storage, and others. Unlike conventional methods of characterizing fuel cell losses that use overpotential or polarization curves, the current entropy-based method provides a useful alternative by encompassing all losses of available energy. It strives toward the upper limits of performance imposed by the Second Law. Entropy production provides a useful parameter for systematic optimization of design parameters in fuel cells. Irreversibilities and inefficiencies are important factors in evaluating feasibility of energy conversion processes. This is particularly evident when comparing fuel cells against other possible methods of future power generation, such as advanced diesel engines for automobiles. For example, automotive PEMFCs can consume 10% or more power to drive pumps, blowers, heaters, and controllers (Bossel, 2003). DC power is converted to voltage-adjusted DC or frequency-modulated AC, and the electrical efficiency of the electric drive train can be about 90%. The process of generating hydrogen for fuel cells has numerous irreversibilities. For electrolysis, an overall efficiency of 70% for power plant to hydrogen production can be achieved.
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Furthermore, hydrogen gas needs to be compressed (about 90% efficient) or liquefied (65% efficiency) for transportation purposes. Hydrogen gas can be delivered to filling stations by pipelines, which takes about 10% energy (higher heating value, or HHV) for gaseous hydrogen, or about 6% for liquid hydrogen. Also, about 3% energy is needed to transfer gaseous hydrogen from a large storage tank into a car’s tank. When combined with the efficiency for conversion of electricity in fuel cells, Bossel (2003) has reported a “power plant to wheel efficiency” of about 22% for typical operating conditions in a PEMFC, compared with advanced diesel (25%) and hybrid electric with SOFC range extension (33%). Although promising advances like thermochemical hydrogen production will significantly improve the power plant to wheel efficiency, the improvement of fuel cell efficiency will continue to be an important issue for their widespread adoption in the transportation sector. An entropy-based design provides a more powerful design tool for this purpose of improving fuel cell efficiencies. In a PEMFC, hydrogen fuel is consumed at the electrode surface when it reacts, releases electrons, and creates hydrogen ions (or protons). Electrons produced at the anode pass through an electrical circuit to the cathode, while protons diffuse through the electrolyte. The oxygen concentration decreases due to chemical reactions along the electrode surface. In Figure 3.13, the predicted results with the LCF model show close agreement with past data reported by Chen et al. (2004). Due to fuel consumption in the x-direction of the channel, the gas density decreases. From requirements of mass conservation, the gas velocity then increases. A slip-flow boundary condition is applied along the anode or channel interface, which affects the magnitude of gas velocity. Along this interface, entropy production arises from friction and viscous dissipation of kinetic energy to internal energy within the gas stream. As a result, added blower power is consumed from the cell output voltage, to overcome pressure losses created by gas friction. This energy conversion differs from electrochemical irreversibilities characterized by
Concentration (C), Velocity (u)
1.6 1.4
Proton Exchange Membrane Fuel Cell (P = 400 Pa/m, h = 1 mm, = 0.1)
1.2 1.0 0.8
C (Chen et al. 2004)
0.6
C (LCF Model) u (Chen et al. 2004)
0.4 0.2 0.000
u (LCF Model) 0.002
0.004
0.006
0.008
0.010
x(m)
Fig u r e 3.13 Velocity and concentration profiles in the fuel channel.
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PEMFC (T = 353 K, P = 1 atm) 1.0
h = 0.001 m h = 0.002 m
0.8
h = 0.003 m 0.6 0.4 0.2 0.0
0.0
0.1
0.2
0.3
0.4 0.5 0.6 0.7 Slip Coefficient ( )
0.8
0.9
1.0
Fig u r e 3.14 Entropy production at varying slip coefficients in a PEMFC.
conventional polarization methods. An entropy-based design can compare all irreversibilities directly against each other, in terms of their lost work potential. In the LCF model, a slip velocity and slip coefficient (x) were applied in boundary conditions at the porous electrode interface, when predicting the gas velocity profile in the fuel channel. In Figure 3.14, entropy production is shown at varying slip coefficients and channel heights, with a minimum point in each case at a slip coefficient of x = 0.5. For a fixed mass flow rate through a fuel channel, the gas velocity and entropy production increase with smaller channel heights. At low slip coefficients, additional friction at the wall yields higher entropy production. On the other hand, higher slip coefficients affect the suction flow through the porous interface. The momentum balance alters the velocity profile and near-wall velocity gradient along the top wall, as well as skewing of the velocity profile in the lateral (y) direction. The entropy production rises, and the optimal point is reached midway, when net viscous dissipation within the channel is minimized. Figure 3.15 illustrates the combined ohmic, concentration, and activation irreversibilities within the electrode. The total entropy production increases at higher interface surface resistances, R(i) (units of kΩ/cm2), due to higher ohmic heating. Electrode materials with higher electrical conductivity could reduce these ohmic losses. Entropy production rises rapidly at low current densities, due to high activation losses. Activation losses also contribute to higher entropy production at larger current densities. Some possible ways of reducing these irrversibilities include different materials, higher reactant concentrations (possibly using oxygen instead of air), or higher operating temperatures. Furthermore, larger electrode surface roughnesses would increase the effective surface area, while increasing the exchange current density and reducing overall entropy production. Figure 3.16 illustrates close agreement between predicted results with the current TAP-TCS model and past data. Voltage losses increase at higher electrode
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Entropy Production (J/kg K)
250 200 150 R(i) = 0.00003
100
R(i) = 0.00006 R(i) = 0.00009
50
PEMFC (T = 373 K, P = 1 atm) 0
0
100
200
300
400
500
Current Density (mA/cm2)
600
700
800
Fig u r e 3.15 PEMFC entropy production.
surface resistances, due to additional ohmic heating. The TAP-TCS voltage losses were calculated based on entropy production, rather than empirical polarization methods documented by Ghadamian and Saboohi (2004). The results provide useful validation of the current predictive model of entropy production. As mentioned previously, an entropy-based approach provides a useful alternative for characterizing voltage losses, as it encompasses all types of irreversible losses within the fuel cell. For example, power consumed by the fuel and air blowers, due to channel entropy production, comes at the expense of output voltage generated by the fuel cell. Thus, channel 1.2 PEMFC (T = 373 K, P = 1 atm, iL = 800 mA)
Voltage (V)
1.0 0.8 0.6 0.4
TAP/TCS Model
0.2
TAP/TCS Model
0.0
Ghadamian, Saboohi (R = 0.00001) Ghadamian, Saboohi (R = 0.0005) 0
200
400 600 Current Density (mA/cm2)
800
Fig u r e 3.16 PEMFC voltage profile.
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Voltage (V)
1.8 1.6
Experimental (T = 750°C)
1.4
Predicted (Kim, 1999)
1.2
Predicted (TAP/TCS Model)
1.0
i0s = 4.95 A/cm2, i0 = 0.113 A/cm2
0.8 0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0 2.5 3.0 3.5 Current Density (A/cm2)
4.0
4.5
5.0
5.5
Fig u r e 3.17 Voltage profile (SOFC; T = 750°C).
flow, ohmic heating, diffusion, and concentration losses are irreversibile losses that can all be characterized consistently according to their rates of entropy production. Their exergy losses will reduce the overall efficiency of the fuel cell. Unlike past methods characterizing system losses through a “polarization” or “overpotential,” the entropy-based approach can provide a more robust way of calculating all losses of available energy including frictional, thermal, electrochemical, and so forth. Previous figures have investigated PEMFCs, whereas Figure 3.17 illustrates results for SOFCs. Voltage losses were calculated based on the entropy formulation and compared successfully against measured data in Figure 3.17. The results of voltage losses were derived from the predicted entropy production, which could also be expressed in terms of exergy destruction, after multiplying by the operating temperature of the fuel cell. This provides another useful parameter for design purposes, as exergy losses have equivalent units of power. Thus, power lost to irreversibilities could be calculated directly, or converted to economic losses after multiplying by the local cost of electricity per kilowatt hour of operation of the fuel cell.
3.6 C a s e St u d y o f Fl u id Ma c h in er y Des ig n This last case study applies the method of entropy-based design to loss coefficients and analysis of power generation from fluid machinery (particularly turbines in this case study). The mechanical power generated by steam, gas, or wind turbines is highly dependent on the shape of blades, velocity field, and other factors (Leclerc et al., 1999). The turbine power output is related to the change of kinetic energy, internal energy, and heat transfer rate from the system encompassing the turbine. Consider a control volume including a turbine, with the inlet, outlet, and other boundaries sufficiently far from the turbine to permit uniform conditions along those boundaries
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Out
V1
V2
A2
A1 1
2 dQ
Fig u r e 3.18 Schematic of a flow channel for turbine analysis.
(see Figure 3.18). For example, a water turbine submerged below a free surface of water is considered. A standard undergraduate thermodynamic analysis would yield the following energy balance for the control volume: dE 1 1 = m 1 e1 + pυ1 + gz1 + V12 + Q - W - m 2 e2 + pυ2 + gz2 + V22 (3.97) dt 2 2 where m 1, e, υ , Q , and W refer to mass flow rate (constant throughout streamtube), internal energy (per unit mass), specific volume (per unit mass), heat transfer, and boundary work, respectively. The left side of the equation becomes zero under steady-state conditions. Applying an entropy balance to a differential section in Figure 3.18 and using the Gibbs equation,
dυ de dQ = Tm +p - TdSgen T T
(3.98)
In this equation, the heat transfer differential represents a process, not a thermodynamic property or state variable. Summing over n sections throughout the entire channel from the inlet (i = 1) to the exit (i = n), it can be shown that
Q = m ( e2 - e1 ) -
∑
n i =1
Ti dSgen,i
(3.99)
Substituting this result into the energy balance, the following result is obtained for incompressible flows:
1 W = m ( gz1 + V12 - gz1 - V22 ) 2
∑
n i =1
Ti dSgen,i
(3.100)
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This result suggests that the power output is maximized when the net entropy production over the control volume encompassing the turbine is minimized. Thus, a key objective is to minimize the energy availability lost on the right side of the equation from entropy production due to viscous mixing, flow separation, and other flow irreversibilities. This goal can be achieved through design modifications (such as modifications of the blade shape, gap spacing, thickness, and so forth), using CFD or experimental techniques like particle image velocimetry (PIV), which would provide whole-field data for local entropy production rates (Adeyinka and Naterer, 2004). These methods of calculating and measuring whole-field distributions of entropy production will be presented in upcoming chapters.
Ref er en c es Adeyinka, O.B. and G.F. Naterer. 2004. Numerical and Experimental PIV/PLIF Studies of Entropy Production in Natural Convection. AIAA 42nd Aerospace Sciences Meeting and Exhibit. Jan. 5–8, Reno, NV. Adeyinka, O.B. and G.F. Naterer. 2005. Modeling of entropy production in turbulent flows. ASME J. Fluid Eng., 126(6): 893–899. Bejan, A. 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Time Systems and Finite-Time Processes. CRC Press, Boca Raton, FL. Bobeth M. and G. Diener. 1982. Upper bounds for the effective thermal contact resistance between bodies with rough surfaces. Int. J. Heat Mass Transfer, 25(8): 1231–1238. Bossel, U. 2003. Efficiency of Hydrogen Fuel Cell, Diesel-SOFC-Hybrid and Battery Electric Vehicles. European Fuel Cell Forum (October 20). Morgenacherstrasse, Germany. Cengel, Y.A. and M.A. Boles. 2002. Thermodynamics: An Engineering Approach. McGraw-Hill, New York. Chan, S.H. and Z.T. Xia. 2002. Polarization effects in electrolyte/electrode-supported solid oxide fuel cells. J. Appl. Electrochem., 32: 339–347. Chen, F., Wen, Y.Z., Chu, H.S., Yan, W.M., and C.Y. Soong. 2004. Convenient two-dimensional model for design of fuel channels for proton exchange membrane fuel cells. J. Power Sources, 128: 125–134. Dincer, I., Hussain, M.M., and I. Al-Zaharnah. 2004. Energy and exergy utilization in transportation sector of Saudi Arabia. Appl. Thermal Eng., 24(4): 525–538. Eames, P.C. and B. Norton. 1998. The effect of tank geometry on thermally stratified sensible heat storage subject to low Reynolds number flows. Int. J. Heat Mass Transfer, 41(14): 2131–2142. Ghadamian, H. and Y. Saboohi. 2004. Quantitative analysis of irreversibilities causes voltage drop in fuel cell (simulation and modeling). Electrochimica Acta, 50: 699–704. Hassani, A.V., Hollands, K.G.T., and G.D. Raithby. 1993. A close upper bound for the conduction shape factor of a uniform thickness, 2D layer. Int. J. Heat Mass Transfer, 36(12): 3155–3158. Homan, K.O. and S.L. Soo. 1998. Laminar flow efficiency of stratified chilled-water storage tanks. Int. J. Heat Fluid Flow, 19(1): 69–78. Kim, J.W., Virkar, A.V., Fung, K.Z., Mehta, K., and S.C. Singhal. 1999. Polarization effects in intermediate temperature, anode-supported solid oxide fuel cells. J. Electrochem. Soc., 146: 69–78. Leclerc, C., Masson, C., Ammara, I., and I. Paraschivoiu. 1999. Turbulence Modeling of the Flow around Horizontal Axis Wind Turbines, Wind Engineering. Multi-Science Publishing, Essex, U.K., 279–294.
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Lui, S.H. and G.F. Naterer. 2007. Upper entropy bounds for transient forced convection. Heat Mass Transfer, 43: 295–308. Martins, R. and S. da Gama. 2000. An upper bound estimate for a class of conduction heat transfer problems with nonlinear boundary conditions. Int. Commun. Heat Mass Transfer, 27(7): 955–964. Mikic, B.B. 1998. On destabilization of shear flows: concept of admissible system perturbations. Int. Commun. Heat Mass Transfer, 15: 799–811. Naterer, G.F. 2001. Establishing heat-entropy analogies for interface tracking in phase change heat transfer with fluid flow. Int. J. Heat Mass Transfer, 44(15): 2903–2916. Naterer, G.F. 2002. Heat Transfer in Single and Multiphase Systems. CRC Press, Boca Raton, FL. Naterer, G.F. and C.D. Tokarz. 2006. Entropy based design of fuel cells. ASME J. Fuel Cell Sci. Technol., 3(2): 165–174. Poulikakos, D. and Bejan, A. 1987. 1982 (Nov.). Fin geometry for minimum entropy generation in forced convection. ASME J. Heat Transfer, 104: 616–623. Rosen, M.A., Le, M.N., and I. Dincer. 2004. Exergetic analysis of cogeneration-based district energy systems. IMechE-Part A: J. Power Energy, 218(6): 369–376. Sinai, Y.L. 1985. Fundamental sloshing frequencies of stratified two-fluid systems in closed prismatic tanks. Int. J. Heat Fluid Flow, 6: 142–144. Weinberger, H.F. 1956. An isoperimetric inequality for the n-dimensional free membrane problem. J. Rational Mech. Anal., 5: 633–636. White, F.M. 1974. Viscous Fluid Flow. McGraw-Hill, New York. Zeidler, E. 1985. Nonlinear Functional Analysis and Its Applications. 2A. SpringerVerlag, Heidelberg. Zubair, S.M., Kadaba, P.V., and Evans, R.B. 1987. Second Law-based theroeconomic optimization of two-phast heat exchangers. ASME J. Heat Transfer, 109(2): 287–294.
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4.1 In t r o d u c t io n This chapter presents experimental methods to enable measurements leading to the spatial distribution of entropy production within a flow field. Measured entropy production provides a valuable diagnostic tool from which economic impact of exergy losses (losses of work potential) could be determined. Rosen and Dincer (2003) have developed exergoeconomic methods to assess economic impact of exergy losses in various industrial systems, such as power plants operating on various fuels and thermal energy storage systems (Dincer and Rosen, 2000). Linking exergy losses directly with financial losses is a powerful tool for driving changes within energy systems to reduce losses of useful work, which would be otherwise underestimated without understanding their economic impact on system viability. In this chapter, the experimental techniques will focus on the combined use of particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF). PIV measures the spatial distribution of fluid velocity, whereas PLIF is used to acquire temperature data in a flow field. The combined PIV-PLIF method offers certain advantages over standard methods of anemometry for experimental studies of entropy production. Previous methods, limited by single-point measurement techniques, can only measure single-point entropy production or averaged entropy production over a finite volume. On the other hand, PIV-PLIF methods provide whole-field methods, while allowing nonintrusive and time-varying measurements of the instantaneous velocity and temperature distributions within a flow field. This chapter presents a detailed description of methods to collect physical data on the detailed structure of entropy production throughout a flow field. The PIV and PLIF techniques provide multipoint instantaneous data, so they enable measured data for local variations of the entropy production rates. In this chapter, the experimental techniques will give whole-field measurements of entropy production with these nonintrusive, optical methods.
4.2 Ex per imen t a l Tec h n iq u es o f Ir r ev er s ibil it y Mea su r emen t 4.2.1 Vel o c it y Fiel d Mea s u r emen t Whole-field velocity data are needed before the local entropy production rates can be determined. PIV is a widely used experimental method based on light scattering by small particles in a flow fluid, which are illuminated by two laser light pulses at very short intervals. The scattered light has the same frequency as incident laser light at 95
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Optics Test Apparatus
Light Sheet
PIV Seeding Particles
Camera Incoming Flow Base of Plexiglass Flow Chamber
Fig u r e 4.1 Schematic of particle image velocimetry configuration.
low wavelengths. In contrast, laser-induced fluorescence (LIF) does not result from a scattering process, but rather an absorption and wavelength conversion process. The light emitted by molecules and atoms in a de-excitation process, induced by absorption of a photon of higher energy (from a laser source), is red-shifted to longer wavelengths. These combined features of PIV-PLIF allow synchronization of measurement techniques for both thermal and friction irreversibility measurements, without duplication of hardware. The optical configuration for a typical PIV-PLIF setup consists of a light source, light sheet optics, fluorescent dye for PLIF, processor with software, and tracer particles for PIV and CCD or CMOS cameras (see Figure 4.1). In two-dimensional PIV, the pulsed laser illuminates a planar cross section in the center of the flow region of interest, parallel to the flow and perpendicular to the camera. The camera captures the image of the illuminated particles in successive frames at each instant when the light sheet is pulsed. The two successive images are processed, subdivided into small interrogation regions, and matched based on a correlation analysis to determine the displacement of a group of particles, elapsed time, and the local fluid velocity. Denoting M as the magnification of the camera, the velocity is given by a first-order estimate as follows: M Ds U= (4.1) Dt where Ds is a displacement vector in the image plane and Dt is the pulse time interval (Willert and Gharib, 1991). Interrogation analysis is an important element of the PIV technique. The spatial velocity distribution is obtained over a regular grid of small subregions using statistical methods. The recorded image frame is divided into small areas, called interrogation areas. Correlation-based techniques are used within each interrogation region to produce a vector representing the average particle displacement. Autocorrelation and cross-correlation techniques are used for high particle density image analysis, whereas other methods like particle tracking and particle pairing are limited to relatively
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low-density images. A high-density image arises when the number of particles is between 7 and 10 image pairs per interrogation area. In autocorrelation, an interrogation area is correlated with itself. In this section, the analysis will correlate an interrogation area with a second area, which is offset in the mean flow direction. The cross-correlation employed within each interrogation area allows a precise determination of the direction of displacement to give instantaneous values of both components of fluid velocity in two dimensions. Westerweel (1997) and Adrian (1991) provide additional details regarding interrogation analysis in PIV methods. The PIV resolution becomes more important for high Reynolds number experiments that attempt to resolve small-scale variations embedded with in a large-scale motion. Such scenarios exist in turbulence measurements and cases where smallscale flow structures around large objects must be resolved. Two key velocity resolution issues arise in these types of problems (FlowMap, 1998), namely, (1) the dynamic velocity range, which relates to the ability to resolve very small velocity displacements between particle image pairs, and (2) the dynamic spatial range, which relates to the size of the smallest velocity structure that can be resolved in the flow field. The dynamic spatial range is defined as the field of view in the object space, divided by the smallest resolvable spatial variation (Adrian, 1997). This range coincides with the number of independent vectors obtained from the interrogation analysis (without overlapping). The smallest-length scale that can be resolved is given by
λmin =
NIdp M
(4.2)
where Lo is the physical dimension of the field of view in the x direction, L I is the corresponding pixel dimension of the camera, N is the number of interrogation areas, and dp is the pixel pitch of the CCD array. For a 32 × 32 pixel interrogation area, each flow field is resolved to a factor of approximately 32 in the field of view. This dynamic spatial range would be low for turbulence measurements. A decrease in the resolved length scale, λmin, would require the reduction of the view area size to a fixed number of interrogation cells. Higher resolution can be achieved by a higher magnification of the measurement area, such as extension rings between the lens and the camera. However, higher magnification of the image may lead to higher velocity bias errors. Better modifications include a higher resolution CCD (higher number of pixels) or higher format recording media with physical dimensions on the order of 1 cm. The dynamic range is the ratio of the maximum velocity to the minimum velocity resolvable by a particular PIV system. The minimum resolvable velocity occurs in the order of the root mean square (rms) error, when determining the displacement of the particle image.
4.2.2 T emper a t u r e Fiel d Mea s u r emen t Spatial variations of temperature within the flow field are needed to determine the thermal irreversibilities of entropy production, and they can be determined from PLIF. In PLIF, molecules and atoms of a fluorescent dye are excited to a higher electronic energy state, by pulsed laser absorption and fluorescence. The local fluorescence intensity, I, varies with intensity of excitation light, Ie; concentration of the
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fluorescent dye, C; quantum efficiency as a function of temperature, f; and the molar absorptivity, e, as follows: I = fI e εCφ (T )
(4.3)
where f is a factor corresponding to the optical setup. For a known concentration and excitation energy, the quantum energy decreases at higher temperatures. This dependence constitutes the basis for PLIF temperature measurements. The temperature is determined as follows: T − Tref =
DI fI e εC Dφ
(4.4)
Thus, quantitative analysis is based on temperature calibration images that correlate the variation of intensity of the image with the local temperature and laser energy. The first step in the PLIF calibration procedure is to find the optimum concentration resulting in the maximum temperature resolution with low absorption phenomena. The corresponding absorption, A, can be calculated from A = e − lηRhodC
(4.5)
where hRhod is the extinction coefficient of Rhodamine B in water and l is the optical path length. The experimental procedure would involve running a series of trials at a fixed energy level to determine the optimum concentration at which the temperature resolution is maximum, while maintaining linearity between the gray level and temperature. The measurement precision of a particular concentration value is indicated by the slope of the curve obtained in the preliminary experiment. Typically, the temperature resolution approaches an asymptotic minimum at an optimum concentration, and then it increases thereafter. Signal processing consists of a final translation of the recorded images to temperatures via the calibration maps. The final calibration relates the response of every pixel of the CCD camera to varying temperature, laser energy levels, and concentration. The temperature at discrete locations in an actual measurement region is determined from T − Tref =
I − Iref β
(4.6)
where Iref is the intensity of the fluorescent signal at the reference temperature, Tref . The denominator is statistically determined during calibration. The wavelength of the fluorescence emitted from PLIF is longer than the wavelength of the reflected laser light, thereby making simultaneous measurements of both velocity and temperature possible. An optical filter can be attached to the front of the camera for the fluorescent image to cut off reflected light from the PIV particles.
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The camera and image capturing systems detect particle images and fluorescent images successively at two different instants. Postprocessing of the velocity and temperature measurements will allow the estimation of entropy production. In addition, the temperatures can be resampled with spatial resolution of the PIV vectors, ensuring maximum correlation between the thermal and friction irreversibilities. Experimental correlations between velocity and temperature will provide useful data for turbulent entropy transport modeling in upcoming chapters.
4.2.3 Po s t pr o c es s in g
fo r En t r o py Pr o d u c t io n
Mea su r emen t
Unlike velocity or temperature, the measurement of entropy cannot be performed directly. But the entropy production equation can be used in an indirect way to characterize the flow irreversibility. The measured velocities and temperatures are extracted over a discrete grid in the PIV software. The velocity and temperature fluids at grid position (i,j) are denoted by u(i,j), v(i,j), and T(i,j), respectively. From Section 3.2, a positive definite expression for entropy production rate was derived in terms of a sum of squared terms representing the frictional irreversibility (viscous dissipation) and thermal irreversibility (due to heat transfer). Discretizing that result for two-dimensional flows yields the following expression for entropy production in terms of measured velocity and temperature gradients, centered about the point (i,j):
P s =
k T ( i, j ) 2 +
2 k T (i, j + 1) − T (i, j − 1) T (i + 1, j ) − T (i − 1, j ) + Dx Dy T ( i, j ) 2
µ u(i, j + 1) − u(i, j − 1) v(i + 1, j ) − v(i − 1, j ) + T (i, j ) Dy Dx
+2
2
2
2 2 µ u(i + 1, j ) − u(i − 1, j ) v(i, j + 1) − v(i, j − 1) + Dx T (i, j ) Dy
(4.7)
where ∆x and ∆y refer to the grid spacing in the x and y directions. When calculating the previous derivatives of velocity, errors can occur as follows: (1) bias error associated with the displacement measurement, and (2) a propagated uncertainty due to spatial differentiation of the velocity field. For a smaller grid size, the bias error decreases. The bias error associated with the fast Fourier transform-based cross-correlation algorithm in commercial PIV software can been minimized by a subpixel resolution of the PIV images. The entropy production algorithm contains multiple products of velocity derivatives. Hence, it is imperative to minimize the error associated with the determination of spatial derivatives. Two approaches can be taken in this regard. A twice-differentiable empirical function could be fitted to the data. The spatial derivative is then obtained directly through the differential of the empirical function. This approach requires an elaborate, often
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difficult, interpolation routine for multidimensional output of PIV. Another approach is a local piecewise smoothing of the experimental data followed by the application of forward differences, central differences, or a Richardson central difference scheme over an adaptive window to calculate the derivatives. Smoothing or filtering of experimental data reduces the noise in terms of experimental scatter, and it performs a least-squares approximation through a path that minimizes error for all data points in the field. In commercial PIV software, an average filter can usually be implemented in the form of a top-hat Gaussian filter with uniform weighting. The size of vectors in the neighborhood of a position (i,j) is specified by odd numbers, m and n. The filter calculates an average of vectors in a rectangular domain of size m × n surrounding a vector. The average value is substituted for all entries in the initial matrix. The average can then be calculated by the following formula: x + n2−1
y + n2−1
∑ ∑
1 u ( x, y ) = mn i = x − n−1 2
u(i, j)
j = y − n2−1
(4.8)
addition to the average filter, a spline fit based on a second-order polynomial In least-squares algorithm can also be used for data smoothing. Smoothing algorithms mitigate against error in the calculation of derivatives and resulting entropy production. They provide better approximations to an actual flow loss distribution. The interpolation of smooth curves or surfaces should be limited to flow structures present in the raw data from which they were obtained.
4.3 C a s e St u d y o f Ma g n et ic St ir r in g Ta n k Des ig n This section applies the previous techniques to a case study involving measured entropy production of fluid mixing induced by a magnetic stirrer in a cuvette cube. In this case study, a PIV camera views a magnetic stirrer commonly used in chemical processing laboratories (see Figure 4.2). The rotational speed of the stirrer is
Camera
Illuminated Plane of Measurement
Light Sheet
Magnetic Stirrer
Fig u r e 4.2 Schematic of magnetic stirrer.
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Fig u r e 4.3 Measured velocity field.
90 r/min, which allows the camera, running at 30 Hz, to resolve each rotation with 20 image frames. Other problem parameters are summarized as follows: Cube side length: 60 mm Camera: 30 Hz Laser: Double-pulsed Nd:YAG laser at 10 mJ per pulse Light sheet entering the cuvette that is approximately 5 mm below the free surface • Seeding: 50-mm polyamide particles • Background: Ambient light used to capture the magnet stirrer in the images • • • •
Using Dantec Flow Map software, Figure 4.3 illustrates the measured velocity field within the plane of the light sheet used to illuminate the particles. Based on this velocity field, entropy production rates are determined and plotted in Figure 4.4. The regions of high mixing yield the highest rates of entropy production. In this example, the practical application of a magnetic stirrer involves mixing of chemicals to provide uniform mixtures. As a result, uniformly distributed magnitudes of entropy production would be desired to maximize mixing, rather than minimal entropy production in other applications like fluid machinery or power generation. Based on the
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Fig u r e 4.4 Surface profiles of entropy production.
measured entropy production rates, the impeller could be redesigned to extend the diffusive effects induced by mixing. By summing the local entropy production measurements, the results provide a useful basis from which the energy efficiency of fluids engineering devices can be effectively characterized. Using the First Law of Thermodynamics, the thermal efficiency of a heat exchanger is defined differently from a water heater’s efficiency, and still different from a diffuser’s efficiency (in terms of pressure), and so on. Due to such inconsistencies, difficulty arises when trying to establish a standard way of identifying a device’s energy wastefulness. Unlike methods based on the First Law, local or summed entropy production rates can provide a single, measurable quantity that is directly related to the efficiency of any energy-consuming or energyproducing device. The magnetic stirrer example in this section represents a single application where entropy production measurements can provide useful insight for design purposes. The practical utility of the method can be extended to numerous other applications, such as aerospace, automotive, power generation, turbomachinery, sprays, combustion, indoor ventilation, processing industries, and others.
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4.4 C a s e St u d y o f Na t u r a l Co n v ec t io n in Cav it ies In this second case study, the measurement technique will be applied to natural convection in an enclosure. Free convection in enclosures occurs in various practical applications. Some examples include cooling of microelectronic assemblies, heat transfer between panes of glass in double-pane windows, solar collectors, and gasfilled cavities surrounding a nuclear reactor core. Although the physical processes of free convection have been widely documented in the literature, fewer studies have considered the related significance of entropy and the Second Low. For example, convective cooling within a microelectronic assembly entails free convection, whereas pressure losses occur with forced convection of air past internal components. In this instance, each unit of entropy produced leads to a corresponding unit of heat flow which is desired to be removed, but is not removed due to entropy production. This entropy production leads to pressure losses when kinetic energy is dissipated to internal energy, which works against the desired objective of component cooling. Consider two-dimensional free convection within a square enclosure. The experimental setup involves PLIF for measuring temperatures within the test cell, as well as PIV for velocity measurements. An experimental study was conducted by Adeyinka and Naterer (2005) to measure entropy production in a 39 × 29-mm test cell. The cell depth of 59 mm was designed to minimize three-dimensional variations of thermal and flow fields along the plane of symmetry. Values of temperature at discrete locations in the measurement domain were obtained from the method of PLIF. In the commercial PLIF software, statistical averages are available to establish whole-field statistics of the LIF data. Further details regarding the experimental setup are described by Adeyinka and Naterer (2005). The cavity is illuminated from above at the vertical plane of symmetry by an Nd:YAG pulsed laser. A CCD camera captures the sequence of image maps. The temperatures are recorded after their steady-state conditions are reached in both velocity and temperature fields. The Rayleigh number is controlled by adjusting fluid temperatures into the aluminum heat exchanger side walls. The PIV images are postprocessed by a fast Fourier transform based on a cross-correlation scheme in the Dantec Flow Map software. The PLIF images are resampled by a calibration map with a spatial resolution corresponding to the velocity map. As discussed in previous sections, the measured velocity vectors are displayed by the PIV software over a discrete grid. Using the velocity measurements and PLIF temperature measurements, the conversion algorithm for determining entropy production is then applied. The PLIF measurements are used for temperatures in the expression for entropy production in Equation 4.7. For this buoyancy-driven problem, the temperature field varies spatially, thereby affecting the frictional entropy production in Equation 4.7. The nonintrusive method of PIV is used for whole-field measurements of velocity, which are then postprocessed by spatial differencing to yield local rates of entropy generation. In the experimental studies, the working fluid was water (Pr = 8.06). The left hot and right cold walls were maintained at 20 and 10 oC, respectively, thereby yielding a Rayleigh number of 5.35 × 106. The measured velocities indicate that a single clockwise recirculation cell developed with highest velocities near the side walls. The fluid velocities diminish rapidly at locations farther from
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the wall. The velocities become too small for PIV vectors to be displayed in the central region of the cavity. The U-velocity results and V-velocity results along the vertical and horizontal midplanes, respectively, were also obtained. In each case, the velocities were nondimensionalized with respect to the maximum velocity, while the spatial coordinate was nondimensionalized with respect to the cavity width. Close agreement between predicted and measured results was established. The measured velocity field is slightly skewed to the right side of the cavity, so some discrepancy between predicted and measured results was observed near the right wall. The numerical simulation assumes a perfectly insulated boundary on both horizontal walls of the cavity, which leads to complete symmetry without skewing of the velocity field. The experimental apparatus closely approaches this idealization, but any slight heat gains through the horizontal boundaries could potentially lead to asymmetry of the buoyancy-driven flow. Velocity measurements were obtained within 1 mm from the wall. In view of their importance in subsequent spatial differencing for entropy production at the wall, additional measurements were obtained by resolving the velocity field closer to the wall. Surface plots of U-velocity values across the entire cavity were also obtained. The maximum horizontal velocity occurs near the top corner of the cold wall. Unlike fluid flow of air at Pr = 0.71, where the maximum U-velocity is closer to the hot wall in the top corner of the cavity, the predicted and measured results for water (Pr = 8.06) exhibit a maximum magnitude closer to the top corner of the cold wall. Buoyancyinduced acceleration of fluid up the hot wall leads to an adverse pressure gradient and velocity change, when the fluid is redirected horizontally near that corner. This momentum exchange involves a balance between fluid inertia and forces imparted by pressure, friction, and fluid buoyancy. The frictional resistance of the fluid along the wall increases, when the momentum diffusion rate exceeds the rate of heat diffusion (Pr > 1). This affects the overall momentum balance on the fluid, thereby altering pressure gradients near the top corners of the cavity and changing the trends of maximum fluid velocity for air (Pr < 1) and water (Pr > 1). Also, the distance of this maximum velocity point from the wall changes at different Prandtl numbers. Similarity solutions of free convection along a vertical wall confirm that the point of maximum velocity moves closer to the wall at higher Prandtl numbers (Naterer, 2002). Postprocessing of the measured velocity results yields the spatial variation of entropy production throughout the cavity. The peak values occur at the vertical walls, corresponding to the locations of largest spatial gradients of velocity. Away from these points, entropy production decreases sharply to approximately zero close to the wall, which corresponds to the local maximum and zero gradient of V-velocity near the wall. Beyond this local maximum of velocity, entropy production increases to a local maximum and decreases back to nearly zero in the central region of the enclosure. The illustrated results have been normalized, with respect to a reference entropy production, Ps (ref), at the local maximum. The entropy production reaches a minimum value in the center of the cavity, where the stagnation point of the recirculation cell is observed. Near-wall measurements of V-velocity and entropy production in the midregion of the cavity at the cold wall were also obtained. The measured maximum U and V
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components of velocity were 0.611 and 1.69 mm/s, respectively, for this particular study. The predicted maximum U and V components of velocity are 0.632 and 1.89 mm/s, respectively. The close agreement between predicted and measured velocities near the wall are important because near-wall spatial gradients of velocity are needed for the entropy production calculations. Although PIV technology is limited by camera resolution and particle tracing of small-scale structures near the wall, the current experimental study successfully measured velocity and derived entropy production at very close proximity to the wall. A resolution of 0.2 mm was achieved in the wall region, which provided good near-wall accuracy that becomes particularly important for turbulent flows. Measured oscillations of entropy production can be reduced through filtering of the velocity data. In the experimental study, a 3 × 3 average filter was used for smoothing of the raw velocity vectors, before calculating the entropy production. Previous PIV studies (Luff et al., 1999) have shown that filtering does not introduce additional error into the measured velocity, but it serves to mitigate uncertainty by averaging velocities at surrounding grid points. The measured results illustrate the benefit of filtering, particularly for the near-wall raw data points and removing random uncertainty in the measured velocity gradients. This measurement procedure for entropy production provides a useful diagnostic tool for identifying the local flow losses, so that energy conversion devices can be redesigned locally around regions of highest entropy production.
4.5 Mea su r emen t Un c er t a in t ies 4.5.1 Bia s
a n d Pr ec is io n Er r o r s
Uncertainty analysis involves systematic procedures of calculating error estimates for experimental data (Coleman and Steele, 1995). Measurement errors of entropy production arise from various sources. They can be broadly classified as bias errors and precision (or random) errors. Bias errors remain constant during a set of measurements. They are often estimated from calibration procedures or past experience. This section will assess both bias and precision errors in the entropy production measurements. Elemental bias errors arise from calibration procedures or curve-fitting of calibrated data. Also, “fossilized” bias errors arise when measuring and tabulating thermophysical properties. Although such errors are usually less than ±1%, Coleman and Steele (1989) describe cases involving higher levels of fossilized bias errors. Moffat (1988) defines a “conceptual bias,” which includes a residual uncertainty due to variability arising in the true definition of the measured variable. For example, if point measurements are used to approximate bulk temperatures at the inlet and exit of a duct, then the difference between these temperatures and the bulk mean temperature contributes to a conceptual bias error, because point measurements cannot fully capture the spatially averaged bulk value. In contrast to bias errors, precision errors appear through scattering of measured data. Such errors are affected by the measurement system (i.e., repeatability, resolution) or spatial and temporal variations of the measured quantity. Also, the procedure itself may lead to precision errors arising from variations in operating conditions. If an error can be estimated statistically, then it is usually considered to
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be a precision error. Otherwise, it is generally assumed to be a bias error. Anticipated precision errors are often used to guide experimental designs and procedures, in view of collecting data within a desired range of measurement uncertainty. Gui et al. (2001) outline precision errors and other PIV measurement uncertainties in a towing tank experiment. Precision errors are reduced by increasing the number of measurement samples. Alekseeva and Navon (2002) found temperature uncertainties based on first- and second-order adjoint equations. An adjoint formulation of an inverse heat transfer problem leads to uncertainty indicators for the corresponding direct problem. Hessian maximum eigenvalues from the second-order adjoint equations can be used to evaluate the uncertainty indicators (Alekseeva and Navon, 2002). Pelletier et al. (2003) show how sensitivity equations provide key information regarding which parameters most affect the flow response. Measurement uncertainties of flow parameters depending on input data errors (such as initial and boundary conditions) can be effectively calculated with adjoint equations. Alekseeva and Navon (2003) use adjoint temperatures to calculate the transfer of uncertainties from such input data. Propagated uncertainties (Kline and McClintock, 1953) are often classified according to zero-order or higher-order uncertainties. In the former case, all parameters affecting the measurements are assumed to be fixed, except for the procedure of the experiment. Thus, data scattering arises from instrumentation resolution alone. In the latter case (higher-order uncertainty), control of the experimental operating conditions is considered, so factors such as time are included. The degree of variability of operating conditions can be expressed by the standard deviation. Measurement uncertainties of primary variables (such as fluid velocity) with various experimental techniques have been widely reported previously, i.e., Kline (1985), Lassahn (1985), Moffat (1982), and others. Postprocessing of measured data, such as measured vorticity from postprocessed PIV data, entails additional uncertainties in the conversion algorithm. Conventional error indicators (AIAA Standard, 1995) can be extended to the scalar variable of entropy production. In this case, bias errors must be specifically correlated with sensitivity coefficients of the measured entropy production. Equation 4.7 expressed the measured entropy production as a postprocessed variable. Before assessing the experimental uncertainties in this method, the first step is assessing the uncertainties of measured velocities.
4.5.2 Vel o c it y Fiel d Un c er t a in t ies
in Ch a n n el
Fl o w
PIV incurs certain errors from statistical correlations in the interrogation areas when determining the fluid velocities. For example, consider the problem of laminar channel flow where the average fluid velocity, U, for an interrogation area at any instant is measured by the following equation: U=
DsLo DtLI
(4.9)
where ∆t is the time interval between laser pulses, ∆s is the particle displacement from the correlation algorithm, Lo is the width of the camera view in the object plane, and L I is the width of the digital image.
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The total error, ε , in a measured quantity is a sum of the bias component, B, and a precision component, P. The bias error of the measured velocity is related to the elementary bias errors based on the sensitivity coefficients, that is,
2 B2 + η2 B2 Bu2 = ηD2s BD2s + ηD2t BD2 t + ηLo Lo LI LI
(4.10)
where the sensitivity coefficients with respect to an arbitrary variable, c , is given by
ηχ = ∂u / ∂χ
(4.11)
The elementary bias limits (t, ∆s) are usually specified by the manufacturer. The width of the camera view in the object plane, Lo, depends on distances and configurations related to the experimental setup, so the bias limit for Lo is determined from calibration procedures, not manufacturer’s specifications. In this calibration, the physical dimensions and spatial resolution of the camera view in the measurement plane are determined. Then the width of the digital image can be determined by the number of pixels corresponding to these dimensions. The width of the camera view in the object plane and bias limit for Lo are determined. Then, the uncertainty associated with this bias limit can be reduced with a more refined procedure for measuring Lo. The PIV image pairs are cross-correlated with an interrogation window, which yields a value of ∆s in the centerline. Combining the contributions of each bias error and the sensitivity coefficient, the velocity error can then be determined. The precision error (P) of an average value, χ , measured from N samples is given by tσ N
P=
(4.12)
where t is the confidence coefficient, t equals 2 for a 95% confidence level, and s is the standard deviation of the sample of N images. The standard deviation is defined as follows:
σ=
1 N −1
N
∑ (χ
k
− χ )2
(4.13)
k =1
where the average quantity is defined by the following equation:
χ=
1 N
N
∑χ
k
(4.14)
k =1
Typical values of the standard deviation along the centerline and the near-wall region of the channel can be determined from the procedure, thereby yielding the precision limits and resulting total uncertainty of the measured velocity.
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4.5.3 Mea s u r emen t Un c er t a in t ies
o f En t r o py Pr o d u c t io n
Based on the previous velocity results, the errors of measured entropy production can be estimated. A data reduction equation for entropy production of laminar channel flow is approximated by 2
P s =
2
k DT µ Du + T Dy T Dy
(4.15)
The total uncertainties (B + P) for the U (velocity), T (temperature), and y (position) variables are U i = U i ± εU i Ti = T i ± εTi yi = y i ± ε yi
The uncertainty in ∆U is obtained as follows:
where
ε Du = ± (θ ′ u,i +1ε u,i +1 )2 + (θ ′ u,i −1ε u,i −1 )2
θ ′ u,i −1 =
∂( Du ) ∂ui
(4.16)
(4.17)
Note that q¢u,i = -1 = -1 and q¢u,i = -1 = 1 or vice versa. The uncertainty of ∆T is calculated in the same manner as Equation 4.16 and Equation 4.17, except that the velocity component, U, is replaced by temperature, T. Similarly,
ε Dy = ± (θ ′ y,i +1ε y,i +1 )2 + (θ ′ y,i −1ε y,i −1 )2
(4.18)
where
θ ′ y,i −1 =
∂( Dy) ∂yi
(4.19)
Neglecting the error in reported thermophysical properties, the data reduction equation for entropy production leads to
ε P2s = ηT2 ε T2 + ηD2u ε D2u + ηD2y ε D2y + ηD2T ε D2T
(4.20)
Based on this equation and the previous procedure of calculating individual uncertainties, the experimental uncertainty of entropy production can be determined. The measured uncertainties represent a maximum error bound within the 95% confidence interval.
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4.5.4 En t r o py Pr o d u c t io n
of
Fr ee Co n vec t io n
in Cavit ies
Consider another example involving measurement uncertainties of free convection experiments described in Section 4.4. A similar procedure is adopted for the bias and precision errors. However, certain differences exist due to variations of temperature within the enclosure. Unlike the previous channel flow problem, irreversibilities in this problem vary spatially due to both velocity and temperature variations across the flow field. For this problem, the bias error of the measured velocity is related to the elementary bias errors and sensitivity coefficients as follows:
Bu2 = ηD2s BD2s + ηD2t BD2t + ηL2o BL2o + ηL2I BL2I
(4.21)
where the same definition of sensitivity coefficients is used, i.e., ηχ = ∂U / ∂χ . By combining the contributions from each source of bias and the sensitivity coefficient, a fullscale velocity bias error is obtained. Similarly as described previously, the precision error (P) of an average value, χ , is measured from N samples. The data reduction equation for friction irreversibility of entropy production in this problem then becomes 2 2 µ Duy Dvx Dux Dvx + + Ps = + T Dy Dx Dx Dy 2
2
(4.22)
The same definitions are applied from the previous problem, including the total uncertainties for the U, T, y, ∆U, and ∆y variables. The total uncertainty of entropy production becomes
ε P2s = ηT2 ε T2 + ηD2U ε D2U + ηD2V ε D2V + ηD2y ε D2y
(4.23)
Then, the total uncertainty of measured entropy production can be determined. The reader is referred to past studies by Adeyinka and Naterer (1995), which provide detailed examples of measurement uncertainties of entropy production in various applications.
Ref er en c es Adeyinka, O.B. and G.F. Naterer. 2005. Particle image velocimetry based measurement of entropy production with free convection heat transfer. ASME J. Heat Transfer, 127(6): 615–624. Adrian, R.J. 1991. Particle-imaging techniques for experimental fluid mechanics. Annu. Rev. Fluid Mechanics, 23: 261–304. Adrian, R.J. 1997. Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Measurement Sci. Technol., 8: 1393–1398. AIAA-Standard-S017-1995, Assessment of Experimental Uncertainty with Application to Wind Tunnel Testing. American Institute of Aeronautics and Astronautics. Washington, D.C. Alekseeva, A.K. and M.I. Navon. 2002. On estimation of temperature uncertainty using the second order adjoint algorithm. Int. J. Computational Fluid Dynamics, 16(2): 113–117.
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Alekseeva, A.K. and M.I. Navon. 2003. Calculation of uncertainty propagation using adjoint equations. Int. J. Computational Fluid Dynamics, 17(4): 283–288. Coleman, H.W. and W.G. Steele. 1989. Experimentation and Uncertainty Analysis for Engineers. John Wiley & Sons, New York. Coleman, H.W. and W.G. Steele. 1995. Engineering application of experimental uncertainty analysis. AIAA J., 33: 1888–1896. FlowMap Particle Image Velocimetry Instrumentation: Installation and User Guide. Dantec Dynamics, Demark, 1998. Gui, L., Longo, J., and F. Stern. 2001. Towing tank PIV measurement system, data and uncertainty assessment for DTMB Model 5512. Exp. Fluids, 31: 336–346. Kline, S.J. 1985. The purpose of uncertainty analysis. ASME J. Fluids Eng., 107: 153–160. Kline, S.J. and F.A. McClintock. 1953. Describing uncertainties in single-sample experiments. Mechanical Eng., 75: 3–8. Lassahn, G.D. 1985. Uncertainty definition. ASME J. Fluids Eng., 107: 179. Luff, J.D., Drouillard, T., Rompage, A.M., Linne, M.A., and J.R. Hertzberg. 1999. Experimental uncertainties associated with particle image velocimetry (PIV) based vorticity algorithms. Exp. Fluids, 26: 36–54. Moffat, R.J. 1982. Contributions to the theory of single-sample uncertainty analysis. ASME J. Fluids Eng., 104: 250–260. Moffat, R.J. 1988. Describing the uncertainties in experimental results. Exp. Thermal Fluid Sci., 1: 3–17. Naterer, G.F. 2002. Heat Transfer in Single and Multiphase Systems. CRC Press, Boca Raton, FL. Pelletier, D., Turgeon, E., Lacasse, D., and J. Borggaard. 2003. Adaptivity, sensitivity and uncertainty: Toward standards of good practice in computational fluid dynamics. AIAA J., 41(10): 1925–1933. Rosen, M.A. and I. Dincer. 2003. Exergoeconomic analysis of power plants operating on various fuels. Appl. Thermal Eng., 23(6): 643–658. Westerweel, J. 1997. Fundamentals of digital particle image velocimetry. Measurement Sci. Technol., 8: 1379–1392. Willert, C.E. and M. Gharib. 1991. Digital particle image velocimetry. Exp. Fluids, 10: 181–193.
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5
Entropy Production in Microfluidic Systems
5.1 In t r o d u c t io n Microfluidic irreversibilities of friction, heat transfer, and electrodynamic transport have significance in the achievement of various technological goals involving microand nanoenergy systems, for biodevices, micropower sources, and other applications (Gad-el-Hak, 1999). Microelectromechanical systems (MEMS) have promising applications to aerodynamics, drag reduction, and slow control. For example, embedded surface microchannels can take advantage of local slip-flow behavior to reduce wall friction and entropy production of external flows (Naterer, 2004). In these applications, pressure losses arising from flow irreversibilities affect the power consumption and performance of microsystems. This chapter examines how entropy and the Second Law have importance in the design and optimization of microdevices. Fluid flow through microchannels has been studied extensively by many authors (Cho et al., 2001; Ng and Tan, 2004; Zhao et al., 2001), including experimental, numerical, and theoretical studies. Ng and Tan (2004) developed a three-dimensional finite volume model of fluid motion within rectangular microchannels based on the Navier–Stokes equations, including an electric double layer (EDL) along the walls. Electromagnetic effects of the EDL can be modeled as a type of body force and source term in the momentum equation. Cho et al. (2003) developed a condition for electrowetting on dielectric (EWOD) in microfluidic motion through parallelplate channels. In past studies, some conflicting opinions have arisen with regard to the apparent viscosity of fluids in microchannels. This debate involves whether the apparent (required or measured) viscosity of a microchannel flow equals the bulk viscosity at large distances away from the wall. For thin films, Israelachvili (1986) reported values of apparent viscosity that were much larger than the bulk viscosity. Migun and Prokhorenko (1987) reported that the apparent viscosity increases for capillaries smaller than a micron in diameter. However, other researchers (Anderson and Quinn, 1972) have reported that the apparent and bulk viscosities are nearly equal for flows in capillaries. The bulk viscosity is generally determined from classical thermodynamics, whereby curve fits of measured data to interpolation polynomials are used to estimate the variations of viscosity with temperature and pressure. Using methods of statistical thermodynamics, a velocity or temperature distribution function can be used to include the effects of intermolecular interactions (Ferziger and Kaper, 1972). Avsec (2003) applied statistical methods to include translational, rotational, vibrational, and electron excitation effects on property evaluation
111
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for liquids. Gad-el-Hak (1999) included a Knudsen number (Kn) dependence in the velocity distribution function. Pfahler et al. (1990, 1991) measured the friction factor of liquids (isopropyl alcohol, silicon oil) and gases (nitrogen, helium) flowing through microchannels etched in silicon. The liquid flow rate was measured as a function of channel size, pressure drop, and type of fluid. It was observed that the fluid’s apparent viscosity was smaller than values predicted for macroscale flows. Although some speculation regarding rarefaction and thermal effects were noted, the authors “do not have a satisfactory theoretical explanation for the phenomena observed” (Pfahler et al., 1991). A trend of decreasing friction factor at lower Reynolds numbers was observed, although macroscale theory predicts a constant friction factor for laminar flow. Entropy production includes both frictional and thermal irreversibilities, which lead to pressure losses in microchannel flows. Additional irreversibilities of phase transition, electromagnetic transport, and radiative heat transfer have been reported previously (Bejan, 1996; Naterer, 2001). Camberos (2003) predicted the electromagnetic irreversibilities in compressible flows with computational fluid dynamics (CFD) applications of improved aircraft design. This chapter develops models of entropy production including such irreversibilities, but focuses on applications to microdevices. The main objectives of this chapter involve showing how entropy and the Second Low provide key insight regarding fluid friction, pressure losses, and energy conversion in microdevices.
5.2 Pr es su r e-D r iv en Fl o w in Mic r o c h a n n el s 5.2.1 Co n t in u u m Eq u a t io ns
a n d Sl ip
Bo u n d a r y Co n d it io ns
Microchannel flows can be subdivided into different flow regimes. Depending on the Knudsen number (Kn), different methods of CFD are needed. The ratio of the mean free path of the fluid to a characteristic length scale of the problem is called the Knudsen number. The flow regimes include the continuum flow ( 0 ≤ Kn ≤ 10 -3 ) , slip flow (10 -3 ≤ Kn ≤ 10 -1 ), transition flow (10 -1 ≤ Kn ≤ 10 -1 ) , and free-molecule regimes (101 ≤ Kn ≤ ∞ ). At 10-3 < Kn < 10-1, the flow lies within the slip-flow regime where the continuum-based equations (Navier–Stokes equations) and a slip boundary condition are applied. In this regime of fluid motion, a linear relation between applied stress and strain rate is used for Newtonian fluids. Consider the following Navier–Stokes equations for gas flow in the slip-flow continuum regime,
∂( ρ ) ∂( ρu ) ∂( ρv) + + =0 ∂t ∂x ∂y
(5.1)
∂( ρu ) ∂p + ∇ ⋅ ( ρvu ) = + ∇ ⋅ ( µ∇u ) ∂t ∂x
(5.2)
∂( ρv) ∂p + ∇ ⋅ ( ρvv) = + ∇ ⋅ ( µ∇v) ∂t ∂y
(5.3)
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subject to slip boundary conditions at the walls of the microchannel. The symbols, ρ and µ denote the density and dynamic viscosity of the fluid, respectively. These properties will be assumed uniform throughout the channel. To calculate the temperature field, p = ρ RT
(5.4)
which represents the ideal gas equation of state. Maxwell’s first-order slip velocity (Maxwell, 1879) will be used for boundary conditions at the walls of the microchannel. This boundary model incorporates two coefficients involving velocity and temperature gradients at the wall, i.e., ugas - uwall = ξ1
∂u ∂T + ξ2 ∂y wall ∂x
(5.5) wall
3µ 2 -σ where ξ1 = σ λ and ξ1 = 4 ρT λ . It is unusual to observe streamwise temperature gas gradients affecting the slip velocity at the wall. A higher component of thermally induced wall slip occurs at smaller gas densities, as the internal energy of gas molecules has greater impact on a near-wall region of intermolecular interactions when fewer molecules occupy the region. These mechanisms of near-wall energy exchange can be characterized through the entropy production rate. The frictional dissipation of kinetic energy to internal energy, including near-wall friction associated with velocity slip, occurs at a rate given by the entropy production rate. When this entropy production rate is multiplied by temperature, it becomes the local rate of exergy destruction, X d′′′, per unit volume. Using the Gibbs equation, it can be shown that the rate of exergy destruction for near-isothermal microchannel flows can be written directly in terms of the velocity gradients as follows:
∂u ∂v 2 ∂u 2 ∂v 2 + + 2 + Xd ′′′ = µ ∂y ∂y ∂x ∂x
(5.6)
Thus, wall slip affects the velocity profile and resulting exergy destruction. The frictional dissipation of kinetic energy leads to pressure losses in microchannels, which depend on cross-stream velocity and streamwise temperature gradients, according to Equation 5.5.
5.2.2 Ca s e St u d y
of
Ex er g y Lo s s es
in
Ch a n n el Des ig n
A numerical study of microfluidic entropy production was conducted by Ogedengbe et al. (2006), with a finite volume discretization of the continuum governing equations. In this section, numerical results of nitrogen gas flow through microchannels (from that study) are examined at varying mass flow rates, pressure ratios, Reynolds numbers, and channel aspect ratios. Figure 5.1 shows a schematic of the microchannel flow configuration, and Table 5.1 outlines the problem parameters and thermophysical properties
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Wall Slip (Boundary Condition) Outlet
Rectangular Microchannel
Fully-Developed Velocity Profile (Near Outlet)
Fig u r e 5.1 Schematic of microchannel flow problem.
of the gas. Sample results of slip-flow velocity profiles are illustrated in Figure 5.2. It can be observed that the wall velocity increases at lower momentum accommodation coefficients, due to a higher resulting slip coefficient (note: x2 = 0 in Figure 5.2). In Figure 5.3, the cross-stream profile of the entropy production rate (per unit volume) is illustrated at varying pressure ratios in the fully developed section of the channel. The pressure ratio refers to the inlet pressure divided by the outlet pressure. The area covered under each curve represents the entropy production per unit area at a specific location, x*. It can be observed that the entropy production rate rises when the pressure ratio falls from 2.7 to 1.34. The lowest entropy production occurs at the pressure ratio of 3.0. The diffusion layer grows faster at lower pressure gradients, thereby leading to higher entropy production. The minimum entropy production rate (per unit volume) occurs at the midpoint of the channel, due to the zero transverse velocity gradient at that position. In Figure 5.4, the predicted entropy production rate (per unit area) and varying slip coefficients (ζ2) in the streamwise direction (x* = x/L) are illustrated. At each x* position, the entropy production is calculated based on an integrated profile across the channel. The inlet velocity profile is uniform, so a developing flow region leads to higher exergy destruction (per unit area) in the x-direction, when the diffusion layer propagates inward to the core of the microchannel. It was confirmed that the velocity profile reaches a fully developed condition upstream of the outlet. As a result, the entropy production rate (per unit area) reaches a peak value near the outlet and remains constant thereafter to the outlet. The pressure declines along the microchannel, so temperature decreases according to the ideal gas law. As a result, the slip coefficient rises in the x-direction.
T a bl e 5.1 Problem Parameters and Fluid Properties Length (μm) Height (μm) Dynamic viscosity (Ns/m2) Gas constant (J/kg K) Outlet pressure (kPa) Pressure ratio (Pin / Pout) Reynolds number (Re)
1560, 2560, 3560, 4560, and 5560 1.0 0.000016 296.8 100.8 1.34, 2.70, 3.00, and 3.34 0.001, 0.002, and 0.003
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0.8
Kn = 0.0579 Re = 9.9 × 10–4 Nitrogen (Pout = 100.8 kPa) Pin/Pout = 3.00 1/ε = 1,560
y/H
0.6
0.4
σ = 1.0
σ = 0.8
σ = 0.6
σ = 0.4
σ = 0.2
No-Slip
0.2
0.0
0.0
0.4
0.8 u/Uinlet
1.2
1.6
Fig u r e 5.2 Predicted velocity profiles at varying slip coefficients.
Entropy Production Rate (per Unit Volume), W/m3K
600 P˝s
P(in)/P(out) = 1.34 P(in)/P(out) = 2.701
500
P(in)/P(out) = 3.00 P(in)/P(out) = 3.34
400
Nitrogen (Pout = 100.8 kPa) Kn = 0.0579 ε = 19 × 10–4 Re = 19 × 10–4
300
200
100
0
0.0
0.2
0.4
y*
0.6
0.8
1.0
Fig u r e 5.3 Predicted entropy production rate (per unit volume) at varying pressure ratios.
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6.2E – 05 Entropy Production Ps˝ (W/m2K)
1.21E – 08
Nitrogen (Pout = 100,800 Pa) Kn = 0.0579 ε = 19 × 10–4 Re = 19 × 10–4 Pin/Pout = 19 × 10–4
6.2E – 05 6.1E – 05
1.21E – 08 1.20E – 08
6.1E – 05 6.1E – 05
1.20E – 08
6.1E – 05 1.19E – 08
6.1E – 05 6.1E – 05
Entropy Production
6.1E – 05
1.19E – 08
Slip Coefficient
6.1E – 05 0.0
0.2
0.4
x*
0.8
0.6
Velocity Slip Coefficient ( 2)
6.2E – 05
1.0
1.18E – 08
Fig u r e 5.4 Streamwise exergy destruction rate (per unit area) with a varying slip coefficient.
Entropy Production Rate (per Unit Area), W/m2K
The total entropy production rate within the microchannel is calculated based on the sum of individual rates from all control volumes within the domain. This includes control volumes in both developing and fully developed sections of the microchannel. In Figure 5.5, the total entropy production rate (per unit area) is illustrated at varying Reynolds numbers and channel aspect ratios, ε (note: fixed pressure ratio of 2.7). 16E – 06 P˝s 1.2E – 06
6.0E – 07
Re = 0.00099 Re = 0.0019 Re = 0.00282 Pin/Pout = 3.34
4.0E – 07
0.E + 00 1,000
2,000
3,000
1/ε
4,000
5,000
6,000
Fig u r e 5.5 Changes of entropy production rate (per unit area) at varying Reynolds numbers.
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The entropy production rate (per unit area) increases linearly with 1/ε, because the microchannel volume increases when 1/ε rises. Also, it increases at higher Reynolds numbers, because the mass flow increases and fluid friction rises. Because entropy production characterizes the frictional dissipation of kinetic energy and resulting pressure losses, the current results show that entropy production can serve as a key parameter to improve energy efficiency of microsystems.
5.3 Appl ied El ec t r ic Fiel d in Mic r o c h a n n el s 5.3.1 I r r ev er s ibil it ies
w it h a
Co ns t a n t Ma g n et ic Fiel d
Surface and electromagnetic forces are key differences that distinguish microchannel flows from fluid motion in large-scale channels. A common method of flow control in microchannels involves electromagnetic forces that are exerted on the fluid. Manipulating different charge patterns along the walls of a microchannel will affect the speed and direction of electrokinetic flow. A charged surface of a microchannel can attract ions of the opposite charge in the surrounding fluid. The resulting spatial gradient of ions can lead to an EDL. This EDL contains an immobile inner layer and an outer layer, which can be affected by an external electric field. The EDL reduces the liquid velocity and affects the frictional losses. For example, the friction coefficient increases when the ionic concentration of an aqueous solution decreases. In this section, spatial changes of the electromagnetic forces on the fluid will be considered when predicting entropy production of microchannel flows. Consider a nonpolarized thermomagnetic field that is exerted on a steady-state fully developed flow in a rectangular microchannel (see Figure 5.6). The separation between the walls, 2b, is assumed to be much larger than the distance of 2a. An electromagnetic wave is polarized if the electric field vibrates in only one direction.
Diffusive Layer (Mobile)
Bulk Solution
u(z)
b Wall E
i
B q y
x z
a
Inner Layer (Immobile)
Positive Ion Negative Ion Neutral Molecule
Fig u r e 5.6 Schematic of an applied electric field in a microchannel.
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The electromagnetic waves transmitted through the microchannel are nonpolarized, when the applied electric field is assumed to vibrate in many directions, simultaneously. Polarization is a phenomenon associated with transverse electromagnetic waves. Longitudinal waves, such as sound, are nonpolarized. Ordinary light is another example of nonpolarized electromagnetic waves, because the electric field vibrates in multiple directions at the same time. The nonpolarized waves traveling in the y-direction of the microchannel are a superposition of many waves. For each wave, the electric field is perpendicular to the y-axis, and the angle it makes with the x-axis varies for different waves. For polarized waves, the angle that the electric field makes with the x-axis would be unique. The general form of the momentum equation for electrohydrodynamic flow is
ρ
∂v + ρv ⋅ ∇v = -∇p + ∇ ⋅ ( µ∇v ) + i × B ∂t
(5.7)
where the last term represents the electromagnetic force. The variables i and B refer to the current density and magnetic field strength, respectively. The following reduced form of the momentum equation is approximated for steady-state microchannel flow at small Reynolds numbers (see Figure 5.6):
0 = -∇p + ∇ ⋅ ( µ∇v ) + i × B
(5.8)
Consider fluid velocity, magnetic field, and current density fields that are mutually orthogonal. The net force exerted by the magnetic field on the fluid is perpendicular to the direction of the fluid velocity. The applied electric field is nonpolarized, and the cross-product of the electromagnetic source term in Equation 5.8 is simplified to give the following reduced form of the x-momentum equation: 0=
dp d 2u + µ 2 + iy Bz dx dz
(5.9)
The terms represent pressure, viscous, and electromagnetic forces on the liquid. Using Ohm’s law to express the current density in terms of fluid velocity yields
µ
d 2u dp + σ e Bz2 u = 2 dz dx
(5.10)
where σ e and Bz refer to the electrical conductivity and magnetic field strength (z-direction), respectively. For fully developed flow in the microchannel, the pressure gradient becomes constant and independent of the magnetic field strength. In terms of the Hartmann number, M (where M = aBz σ e / µ ), Equation 5.10 becomes
µ
d 2u M 2 µ dp - u= dz 2 a 2 dx
(5.11)
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Applying no-slip boundary conditions at z = a and z = -a (note: origin of coordinates at the midplane of Figure 5.6), the analytical solution of Equation 5.11 becomes u( z ) =
a 2 ( dp/dx ) cosh( Mz / a ) - 1 µ M 2 cosh M
(5.12)
After nondimensionalizing the z-coordinate (z* = z/a) and velocity (u* = u/u where b ub refers to the mean velocity), it can be shown that u* =
M cosh M - M cosh( Mz* ) M cosh M - sinh M
(5.13)
Using a large Hartmann number approximation (LHA model), the velocity becomes u* = 1 - e M ( z* -1)
(5.14)
Using similar assumptions for the energy equation, the reduced form of the energy equation becomes k
2 iy2 d 2T du + =0 + µ dz σe dz 2
(5.15)
Solving this equation subject to the boundary conditions of a uniform wall heat flux, on θ ( z* ) =
2C 2 ( z*2 - 1) + 8C (cosh Mz* - cosh M )/ M + cosh 2 Mz* - cosh 2 M k (T - Tw ) = qw* 2 4C 2 + 8C sinh M + 2 M sinh 2 M µub
(5.16)
where C = M(K – 1) cosh M – K sinh M and K represent a nondimensional load factor (ratio of the applied electric field strength to the product of the mean velocity and magnetic field strength). It can be readily verified that this result is symmetrical about the midplane of the microchannel. In the case of a constant wall temperature, the thermal boundary conditions are given by
θ ( ±1) = 0
(5.17)
Solving the internal energy equation subject to these boundary conditions, it can be shown that
θ=
M2 ( M cosh M - sinh M )2 1 2C C2 × (1 - z*2 ) + (cosh M - cosh Mz*) + (cosh 2 M - cosh 2 Mz*) 4 2 M
(5.18)
This result shows that both the Hartmann number and load factor affect the temperature profile.
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Under the assumptions outlined previously, the corresponding reduced form of the exergy transport equation can be expressed as (Camberos, 2002): DX ∂ T ∂T ∂(uτ ) + = - Xd k 1 - 0 ∂z Dt ∂z T ∂z
(5.19)
where DX/Dt refers to the total convective derivative of exergy, X. The rate of exergy destruction, X d, can be represented by the reference temperature, T0, multiplied by the local entropy production rate, Ps . Thus, entropy production is needed to calculate the exergy destroyed by friction and thermal and electromagnetic irreversibilities. The entropy production rate can be expressed as a sum of positive-definite terms corresponding to friction, thermal, and electromagnetic irreversibilities, that is,
k∇T ⋅ ∇T µΦ σ e ( E + v × B) ⋅ ( E + v × B) + + Ps = T2 T T
(5.20)
The terms on the right side represent a sum of squared terms, so the entropy production is positive, thereby complying with the Second Law of Thermodynamics. Based on the previous assumptions in the fluid flow and heat transfer formulations, it can be shown that the reduced form of the entropy production equation can be written as
2 2 iy2 µ du k dT + + Ps = 2 σ eT T dz T dz
(5.21)
Define the following nondimensional entropy production, Ps*, and wall temperature: P Ps* = s 2 k/a
θw =
(5.22)
kTw µub2
(5.23) Using these variables, the nondimensional entropy production equation becomes
iy*2 ( dθ /dz* )2 ( du/dz* )2 + + Ps* = (θ + θ w )2 θ + θw θ + θw
(5.24)
In an upcoming case study, a detailed analysis of this entropy production (particularly the electrohydrodynamic term) will be presented.
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Substituting the velocity and temperature profiles, the nondimensional entropy production at the wall becomes 2
q *2 1 M2K 2 M 2 sinh M + Ps*,w = w2 + θ w M cosh M - sinh M θ w θw
(5.25)
This expression for the entropy production rate can be simplified for large Hartmann numbers. For example, the exponential e-M becomes less than about 0.03% of eM for values of the Hartmann number above 3. In this case, the entropy production at the wall becomes 2
q *2 µ u 2 M4 M2K 2 Ps*,w = w2 b + + 2 θ w kTw ( M - 1) θ w θw
(5.26)
This result can be rearranged in terms of the Reynolds number (Re), magnetic Prandtl number (Prm), and thermomagnetic number (N), respectively, as follows, 2
1 q ′ m 2 ρ 2 Ps*,w = w 3 5 Re -8 + N (1 + K 2 ) Prm2 Re 2 4 16β µ θ w
(5.27)
where Re =
Prm =
N=
ρub a µ
(5.28)
µµeσ e ρ H e2 σ e kTw
(5.29) (5.30)
and m e and He refer to the magnetic permeability and electric field strength, respectively. Also, b and q´w are the microchannel aspect ratio (b/a) and wall heat flux per unit length, respectively. Differentiating the previous result with respect to the Reynolds number and setting the result equal to zero yields the following optimal Reynolds number: ReL ,opt =
0.574 B1/ 5 [ N (1 + K 2 )β 6 ]1/10
where the duty parameter, B, is given by q ′m 2 ρ 2 B= Prm µ 5θ w
(5.31)
(5.32)
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The irreversibility distribution ratio (or entropy generation number), Ns, is defined as the ratio of the actual entropy generation to the minimum rate of entropy generation, i.e., P Ns = s Ps,min
(5.33)
After substituting the previous expression for ReL,opt, it can be shown that
Ns =
1 ReL 5 ReL ,opt
-8
+
4 ReL 5 ReL ,opt
2
(5.34)
The first term on the right side outlines the rate of change due to the thermal irreversibility, whereas the second term includes the combined friction and electromagnetic irreversibilities. The result suggests that the entropy generation number changes faster at low Reynolds numbers (below ReL,opt), when the thermal irreversibility is the largest portion of the total irreversibility.
5.3.2 Ca s e St u d y
of
Ch a n n el Des ig n
a t Va r y in g Ha r t ma n n Nu mber s
Using the previous formulation, the section will present a case study with entropy production in a microchannel (Naterer and Adeyinka, 2005). Predicted results will be compared against computational simulations with a finite element volume formulation. Predicted results will be compared against past data reported by Bejan (1996), Salas and coworkers (1999), and Adeyinka and Naterer (2004). The predicted results will be presented in terms of nondimensional variables described in previous sections. These nondimensional variables include the cross-stream coordinate (z* = z/a), velocity (u* = u/ub), Hartmann number (M), temperature (θ), Reynolds number (Re), load factor (K), and duty parameter (B). In Figure 5.7 and Figure 5.8, the predicted nondimensional velocity and temperature fields are shown at varying Hartmann numbers. It can be observed that the temperature and near-wall temperature gradient decrease at lower Hartmann numbers. Also, they decrease at larger values of the load factors. The electromagnetic resistance of fluid motion decreases at larger load factors, when the magnetic field strength decreases. As a result, the fluid velocity increases in the denominator of the nondimensional temperature, thereby reducing the temperature of the fluid. As expected, the temperature decreases when the wall heating rate is reduced. In Figure 5.9, the rate of exergy destruction is illustrated for a microchannel half-width of 43 mm, load factor of 0.5, Hartmann number of 20, and varying magnetic field strengths. It can be observed that exergy destruction decreases at lower magnetic field strengths, due to smaller electromagnetic irreversibilities. The exergy destruction reaches a peak value at the wall. Then it decreases to a local minimum and rises to a uniform nonzero value in the core of the microchannel. The exergy
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u/ub
0.8 0.6 0.4 0.2
Tillack, Morley (M = 40) LHA Model (M = 40) Tillack, Morley (M = 100) LHA Model (M = 100)
0.0 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 z/a
Fig u r e 5.7 Predicted velocity profile at varying Hartmann numbers.
destruction increases for wider microchannels because added surface area increases the friction irreversibilities. Also, the electromagnetic irreversilibity decreases at higher load factors due to a lower magnetic field strength. As a result, the higher load factor of K = 1.0 in Figure 5.10 yields a monotonically decreasing exergy destruction toward zero in the midplane of the microchannel, without a local minimum and rising trend observed in Figure 5.9. The results in Figure 5.9 and Figure 5.10 indicate the friction irreversibility is highest near the walls, whereas the electromagnetic
Nondimensional Temperature
1.2 q* = 2; M = 4 q* = 2; M = 2 q* = 1; M = 4 q* = 1; M = 2
1.0 0.8 0.6 0.4 0.2 0.0
K = 0.5 Constant Wall Heat Flux 0.0
0.2
0.4
z/a
0.6
0.8
1.0
Fig u r e 5.8 Nondimensional temperature at various heating rates.
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M = 20 (Salas, 1999) LHA Model (M = 20; B = 15 kV/m) Predicted (B = 11 kV/m) Predicted (B = 8 kV/m)
Xdest
400 300 K = 0.5 a = 43 +m
200 100 0
0.0
0.1
0.2
1 – z/a
0.3
0.4
1.5
Fig u r e 5.9 Exergy destruction at varying magnetic field strengths.
irreversibility is dominant within the core of the microchannel. If the fluid friction is sufficiently small at low Reynolds numbers and the Hartmann number is sufficiently large, the maximum exergy destruction may occur at the midplane. Unlike classical problems involving convective heat transfer without electromagnetic forces, the point of maximum exergy destruction may not be located at the wall. In this case, local loss coefficients would be better represented in terms of local exergy destruction, rather than friction coefficients at the wall, which may not best reflect the most relevant location of dissipative losses in electrohydrodynamic flows. 1,000 M = 20 (Salas, 1999) LHA Model (a = 43 +m; M = 20) Predicted (a = 35 +m) Predicted (a = 30 +m)
Xdest
800
K = 1.0
600 400 200 0 0.00
0.05
0.10
0.15
0.20 0.25 1 – z/a
0.30
0.35
0.40
Fig u r e 5.10 Exergy destruction at varying channel widths.
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Electrokinetic Flow (Aspect Ratio = 5) Electrokinetic Flow (Aspect Ratio = 10) Electrokinetic Flow (Aspect Ratio = 25) Film Condensation (Adeyinka, 2004) Turbulent Boundary Layer (Bejan, 1996)
1.0E + 07 1.0E + 06 ReL,opt
1.0E + 05 1.0E + 04 1.0E + 03 1.0E + 02 1.0E + 01 1.0E + 00 1.0E – 01
1
10 B
100
Fig u r e 5.11 Optimal Reynolds number at varying magnetic field strengths.
In Figure 5.11, the optimal Reynolds number, ReL,opt, which minimizes the net exergy destruction, is plotted at varying duty parameters, aspect ratios, and flow configurations. The optimum for electrokinetic flow is lower than other configurations, and it increases at higher duty parameters and smaller aspect ratios. This optimum involves a balance, which minimizes the net exergy destruction arising from combined effects of friction and thermomagnetic irreversibilities. The friction irreversibility is reduced with a smaller surface area and net friction, but higher, larger irreversibilities occur due to a higher temperature (between fluid and wall) needed to transfer a specified heat flow (q´) over a smaller area. However, the thermal irreversibility decreases with a larger surface area, because a smaller temperature difference between the fluid and wall is needed to transfer the fixed heat flow. This reduction comes at the expense of higher friction irreversibilities, when a larger surface area (i.e., larger Reynolds number) contributes to added surface friction. Furthermore, the electromagnetic irreversibility increases at higher Reynolds numbers, because Ohm’s law implies that this irreversibility is proportional to the velocity squared. These trends contribute to the physical mechanisms that minimize the net rate of exergy destruction at ReL,opt in Figure 5.11. Operating at other conditions below or above ReL,opt implies that additional electric input power is needed to transfer fixed rates of mass and heat flow through the microchannel. Additional power is needed to offset higher internal irreversibilities, which dissipate kinetic energy into internal energy, rather than transferring power for mass transport. In the case of electro-osmotic flow, additional input power is needed to generate sufficient charge distributions along the wall for the specified mass flow rate. Other methods of microfluidic flow control, such as pressure or thermocapillary driven flow, would also entail wasted power input to overcome system irreversibilities. The previous results of entropy production (or exergy destruction) have practical significance in electrokinetic flow control in microchannels. Exergy losses characterize the friction, pressure losses, and kinetic energy dissipated to internal energy within the microchannel. As a result, they have an important role in the performance
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of microdevices. For example, consider cooling of an electronic assembly with microchannels. Each unit of exergy destroyed corresponds to an amount of internal energy, which could have been removed by convective cooling, but was not removed due to energy dissipated by the thermomagnetic irreversibilities. Additional power input is needed to offset these irreversibilities. As future microdevice technologies become more complex in terms of energy conversion between various subsystems, the spatial tracking of entropy production throughout these networks will become an increasingly valuable tool in reaching the highest levels of performance and device efficiency.
5.4 Mic r o pa t t er n ed Su r fa c es w it h Open Mic r o c h a n n el s 5.4.1 F l u id Fl o w Fo r mu l a t io n Controlled surface roughness has importance in various fluids engineering applications. Surface roughness affects the boundary layer formation in aerodynamics of aircraft, vehicles, and so forth. Extended surfaces (fins), modified surface profiles, and other passive techniques of heat transfer enhancement are commonly used in industrial heat exchangers. Random microscale features of a surface are often modeled as a lumped or overall surface roughness. Recently, advances in micromachining fabrication can allow surface profiles to be carefully designed for various purposes. In this section, the effects of embedded surface microchannels on boundary layer flow and convective heat transfer will be examined. It will be shown that Entropy-Based Surface Microprofiling (EBSM) enables drag reduction and lower entropy production of convective heat transfer, due to slip-flow conditions within the embedded microchannels. Using EBSM, the power consumption to transfer specified rates of mass and heat flow across a surface can be reduced. Consider external flow past a flat surface with embedded open microchannels (see Figure 5.12 and Figure 5.13). This flow configuration closely resembles a flat plate boundary layer flow, with a Blasius similarity solution for streamwise changes of flow variables. But open microchannels are aligned parallel to the incoming flow along the surface, with micron or submicron scale depth. Unlike random surface roughness, the well-controlled profiles of these embedded microchannels
Slip-Flow Region
No-Slip Region
u∞ T∞ L d
Wns
q˝
u∞ T ∞ y x
Tw
W
L Ws
Diverging Microchannels
Fig u r e 5.12 Schematic of embedded microchannels.
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L No-Slip Region No-Slip Region
L No-Slip Region SlipFlow Region
δ = Lcotθ2 δ(x) = xcotθ1
Slip-Flow Region
h(y) = ytanθ1
x θ1 θ2
θ1
x
h(y) = (Ltanθ1cotθ1)ηa y Tangent at Profile Base
θ2
y (a)
f(η, a)
(b)
Fig u r e 5.13 Top view of microchannels with (a) converging and (b) diverging profiles.
can allow geometrical optimization for drag reduction and heat transfer enhancement. A basic geometry of flat plate flow is considered in this section, although, the technique can be readily extended to more complex geometries. In the x-z direction (i.e., side view within a microchannel), a Couette-type flow is encountered. A Couette flow refers to a one-dimensional shear layer enclosed by fixed velocities at both edges of the layer. For example, a linear variation of velocity occurs between a moving plate and a stationary wall below the plate. In the case of the open microchannel, a nonzero Blasius velocity and slip velocity are encountered at the top and base of the microchannel. Diffusion-dominated transport of momentum in the z-direction yields a similar linear profile between both edge velocities. The slope of this profile decreases in the x-direction, as the top velocity changes in the slip-flow pattern of the boundary layer development. Considering a front view of the plate in the z-y direction at a fixed x-location, the cross-stream flow is neglected, and the z- and y-velocities will be assumed to be negligible, relative to the streamwise (x-direction) velocity component. Transition between slip-flow and no-slip regimes occurs at the top corners of the embedded microchannels. Consider a slip-flow embedded microchannel, with a Knudsen number and characteristic length based on both depth and width. At the top corners, the near-wall slip-flow profile approaches no-slip behavior before reaching the top edge of the microchannel. This transition occurs because the local Knudsen number decreases at the top edges of the microchannel. This transition may produce a small submicron semicircular type of zone of influence at the top corners. Little or no experimental data has been reported in the technical literature regarding such slip-flow variations arising from this transition. Past studies have mainly reported slip-flow coefficients based on measured mass-flow rate slopes against various pressure differences in closed microchannels. The current transition regime and mixed Knudsen numbers would not arise in those cases. Also, past measurements yield a single (net) coefficient, without spatial variations across the microchannel. In this method, a single coefficient simulates a spatial variation and transition regime in a similar fashion.
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Analytical solutions of laminar boundary layer flow can be determined from the method of similarity selection. The predicted velocity field, u, from this method for flat plate boundary layer flow can be expressed in terms of a similarity variable, h, freestream velocity, u∞, and a stream function derivative as follows:
u = f ′ (η ) u∞
(5.35)
u∞ νx
(5.36)
where
η=y
Transforming the two-dimensional, steady, laminar boundary layer equations with this similarity variable, the governing equation for mass and momentum transport within the boundary layer becomes the following well-known Blasius equation,
f ′′′(η ) +
1 f (η ) f ′′(η ) = 0 2
(5.37)
This nonlinear ordinary differential equation will be solved by a Runge-Kutta method, subject to boundary conditions of f′ (∞) = 1 and f (0) = 0. The no-slip condition at the wall is f′ (0) = 0. The analytical solution can be obtained from successive integrations as follows: η
exp 0 u u* = = f ′ (η ) = ∞ u∞ exp 0
∫
∫
∫
η
∫
η
0
0
f /2dηˆ dη f /2dηˆ dη
(5.38)
It can be shown that the functional forms of the momentum and thermal energy equations are equivalent, so a similar procedure yields the following result for the nondimensional temperature within the boundary layer: exp - Pr 0 T - Tw = θ (η ) = ∞ T∞ - Tw exp - Pr 0
∫
η
∫
∫
η
∫
η
0
0
f /2dηˆ dη f /2dηˆ dη
(5.39)
When the Prandtl number of the fluid is close to 1 (common fluids such as air), the previous expressions for nondimensional velocity and temperature become identical.
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For the case of a slip-flow condition at the wall (within an embedded microchannel), the boundary condition involves the nonzero wall velocity and spatial gradient of velocity (second derivative of the stream function at the wall), so that
f ′ (0) = K1 f ′′ (0)
(5.40)
where 2 -σ K1 = Kn Re 1/2 σ x x
(5.41)
In Equation 5.41, Knx and Rex are the local Knudsen and Reynolds numbers, respectively. The boundary condition implies that wall slip increases with higher velocity gradients at the wall. For no-slip conditions, f′′ (0) = 0.3321. It can be shown that f′′ (0) varies with K1 for the slip-flow problem, that is, f ′′ (0) =
1.39 4.185 + 0.96K11.11
(5.42)
After the third-order Blasius equation is solved, subject to the slip-flow boundary condition, the resulting stream function, f(h), can be numerically differentiated to yield the velocity field and wall shear stress, τw, distributions, i.e.,
τw =
ρ1/2 µ1/ 2 u∞3/ 2 f ′′(0) x1/ 2
(5.43)
Rearranging this result in terms of the local Reynolds number,
τw = f ′′ (0) ⋅ Rex -1/2 ρu∞2
(5.44)
Unlike macroscale systems with a no-slip condition at the wall, the slip-flow conditions and in a microchannel can lead to lower shear stresses along the walls. These trends have been investigated previously for gases (Martin and Boyd, 2001) and liquids (Choi et al., 2002). Choi et al. (2002) have reported higher water flow rates induced by a different surface coating along a microchannel wall, thereby leading to a variation of slip velocity and shear stress along the wall. Such slipflow effects increase when the channel height decreases and the wall shear stress increases. The percentage reduction of wall shear stress due to the slip-flow condition, as compared to the no-slip solution, can be determined from the previous similarity solutions. The result follows from the difference of 100% (0.3321 - f′′ (0)),
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where f′′ (0) refers to the result from the slip-flow solution. The value of 0.3321 corresponds to the classical Blasius solution with a no-slip velocity at the wall. Martin and Boyd (2001) have examined the validity of key assumptions adopted in this similarity solution. At all locations except the leading edge, the streamwise velocity gradient is assumed to be much smaller than the cross-stream (ydirection) gradient. In the slip-flow condition, the Knudsen number, Kn, characterizes the degree of rarefaction of fluid motion. As discussed previously, the continuum assumption of fluid flow is considered valid when Kn ≤ 10-3, whereas free molecular flow occurs when Kn ≥ 10. Between these two limits, the slip-flow regime exists within the range of 10-3 ≤ Kn ≤ 10, and a transition region occurs for 10-3 ≤ Kn ≤ 1. A similar condition of temperature discontinuity exists for the thermal problem, but “no-jump” (thermal problem) replaces the condition of “no-slip” (flow problem). The boundary between the slip-flow and transition regimes is problem- and geometry-dependent. The principles underlying the no-slip, no-jump conditions for velocity and temperature require that there cannot be any finite discontinuities of velocity and temperature at the wall. Such discontinuities would entail infinite velocity and temperature gradients, thereby leading to infinite viscous stresses and heat fluxes. Based on continuum theory, the no-slip, no-jump conditions require an infinitely high number of collisions between the fluid and solid surface. In practice, such assumptions lead to reasonably accurate predictions, provided that Kn < 0.001 for gases. For flows at higher Knudsen numbers, the mean free path of molecules is no longer sufficiently small relative to a characteristic length of the micron or submicron device (such as the microchannel height). Slip-flow conditions entail direct momentum exchange of intermolecular interactions near the wall. The probability of a fluid molecule striking another fluid molecule within an embedded microchannel, rather than a wall, decreases at higher Knudsen numbers. A molecule may reflect from several walls before colliding with another fluid molecule traveling in the principle flow direction. Some molecules reflect specularly, and others reflect diffusely from the surface of the walls. Thus, a portion of momentum of incident molecules is lost to the wall, while the remaining portion is retained by the reflected molecules. The tangential momentum accommodation coefficient is used to represent the fraction of incident molecules that is reflected diffusely. This coefficient typically varies between 0.2 and 0.8, and it depends on the fluid properties, solid wall, and the surface finish. For an idealized smooth wall, the incident angle exactly matches the reflected angle of impacting molecules. The molecules conserve tangential momentum, thereby not exerting shear on the wall. This process of specular reflection leads to perfect slip at the wall. But for an actual wall with surface roughness, the molecules reflect at some random angle, which is uncorrelated with their incident angle. Perfectly diffuse reflection requires zero tangential momentum for the reflected fluid molecules to be balanced by a finite slip velocity, to account for the shear stress transmitted to the wall. A near-wall force balance requires that the difference between the slip velocity and wall velocity balances the product of mean free path and velocity gradient perpendicular to the wall. This balance will be applied as the slip-flow
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boundary condition in the previous similarity solution of boundary layer flow. Slip occurs only when the mean velocity of molecules changes significantly over a distance of about one mean free path.
5.4.2 H ea t Tr a nsf
er Fo r mu l a t io n
For convective heat transfer analysis in the thermal boundary layer, the reduced form of the governing energy equation is simplified as follows for the current problem:
ρc p u
∂T ∂T ∂2T + ρc p v =k 2 ∂x ∂y ∂y
(5.45)
subject to the following boundary equations at the edge of the boundary layer and wall, respectively, T ( x, y → ∞ ) = T∞
-k
∂T ∂y
(5.46)
= qw′′
(5.47)
0
In this problem, the wall heat flux is specified, and the wall temperature, Tw(x), is unknown. When the Prandtl number of the fluid is close to 1 (common fluids such as air), the functional form of the boundary layer equations for velocity and temperature become analogous. When the velocity solution is obtained from the momentum equations, it can be modeled as a known coefficient when solving the temperature boundary layer equation. Define the following variable as the wall temperature difference:
θ w ( x ) = Tw ( x ) - T∞
(5.48)
After solving the energy equation, subject to the boundary conditions and evaluating the result at y = 0 for the wall temperature difference, it can be shown that (Kays and Crawford, 1980)
θw ( x ) =
0.623 -1/ 3 -1/ 2 Pr Rex qw′′ k
x
∫ [1 - (ξ / x) 0
3/ 4 -2 / 3
]
dξ
(5.49)
A thermal boundary condition with a constant heat flux has been applied at the wall. In the problem configuration, the y-direction is perpendicular to the wall. Thus, both the solid side (y → 0-) and fluid side of the wall (y → 0+) are assumed to have an equivalent heat flux, qw′′,, passing through an infinitesimal control volume along the wall (y = 0), due to conservation of energy. After the boundary layer equation is solved, subject to the constant heat flux condition at the wall, the wall temperature can be obtained as a function of x, qw′′, and T∞. In the no-slip (no-jump) case, the
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spatial temperature profiles on both sides of the wall must meet at a common wall temperature. Each profile exhibits a certain slope at the wall to match the Fourier heat flux corresponding to the specified flux of the boundary condition. But in the slip-flow case, these profiles are assumed to be offset at the wall, due to a temperature jump condition. In addition, the near-wall slopes of temperature are assumed to match the corresponding slopes of the equivalent no-slip case, when the same specified flux condition is applied at the wall. More specifically, qw′′, is specified and T(x, y → 0+) is solved in the slip-flow problem. Then T(x, y → 0-) can be obtained from a thermal jump condition, while simultaneously matching the required Fourier heat flux at the solid side of the wall (y → 0-). The same constant wall flux can be obtained in both no-slip and slip-flow cases, provided that the near-wall temperature slopes are equivalent. Due to the similarity between molecular diffusion of heat and momentum near the wall at fluid Prandtl numbers close to 1, the magnitude of temperature jump at the wall can be approximated with the momentum accommodation coefficient. But the previous convective heat transfer analysis does not need this accommodation coefficient in the solution procedure, when a specified flux boundary condition is used at the wall. It is only needed if spatial temperature variations within the wall are required, or a conjugate (conduction and convection) analysis is required to find the wall heat flux. The local convection coefficient can be evaluated based on the result from the temperature field, thereby leading to the following expression for the local Nusselt number (Kays and Crawford, 1980): Nux = θ 0 Pr 1/ 3 Re1x/ 2
(5.50)
where q 0 = 0.453 for the current case of a specified flux boundary condition. It can be shown that the same Nusselt number is obtained for the case of a constant wall temperature, except that the leading coefficient becomes q 0 = 0.3321. This heat transfer coefficient is about 36% lower than the value at the same point on the plate with a constant wall flux. But the average Nusselt number and average q w are only about 2% lower than the case of a constant wall heat flux at x = L.
5.4.3 F o r mu l a t io n
of
En t r o py Pr o d u ct io n
The total entropy production over a plate of length L and a width of W consists of a thermal irreversibility and a friction irreversibility, which can be expressed in integral form as follows:
q ′′ Sgen = T∞
2
W
∫ ∫ 0
L
0
dxdy u∞ + h T∞
W
∫ ∫ 0
L
0
τ w dxdy
(5.51)
The previous correlations for the convection coefficient (based on the Nusselt number) and wall shear stress will be substituted into this equation. In this section, the entropy production will be analyzed for the following three cases: (i) diverging and converging embedded microchannels; (ii) unspecified (exponential) profile of microchannels; and (iii) unspecified cross-stream variation of the microchannel geometrical profile.
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C ase (i): Diverging and Converging Microchannels, Laminar Flow, U niform Wall Heat Flux For diverging (or converging) microchannels, the predicted entropy production involves a parallel section and varying slip areas. Using the previous expressions for the heat transfer coefficient and wall shear stress, it can be shown that the total entropy production over the plate becomes q ′′ 2ν 2 S gen = 2 2 n kT∞ u∞ u 2µ + ∞ n T∞
ReL
∫ ∫
0
0
ReL
∫ ∫ 0
Ws + 2 d
Ws + 2 d
0
Rex1/ 2 dyd Rex + θ s′ ( 0 )
fs′′( 0 ) dyd Rex + Rex1/ 2
ReL
∫ ∫ 0
ReL
0
∫ ∫ 0
W - nWs
W - nWs
0
Rex1/ 2 dyd Rex θ ns′ ( 0 )
fs′′( 0 ) dyd Rex 1 2 / Rex
+ Sgen, f + Sgen,h
(5.52)
where the subscripts s and ns refer to slip-flow and no-slip regions, respectively. The number of microchannels is denoted by n and other geometrical parameters are illustrated in Figure 5.12. The latter two integrals, Sgen, f and Sgen,h, refer to the wall friction irreversibility difference and wall thermal irreversibility difference due to slip minus no-slip conditions, that is, q ′′ 2ν 2 2n Sgen,h = Re∞2 u∞2
u 2µ Sgen, f = ∞ 2 n T∞
ReL
∫ ∫
0
0
ReL
∫ ∫ 0
δ
δ
0
Rex1/ 2 Rex1/ 2 θ ′ ( 0 ) - θ ′ ( 0 ) dyd Rex ns s
fs′′( 0 ) fns′′( 0 ) Re 1/ 2 - Re 1/ 2 dyd Rex x x
(5.53)
(5.54)
It can be shown that the total entropy production over the plate becomes a sum of entropy production rates for parallel microchannels (subscript p) and irreversibility difference integrals for diverging and converging microchannels ( Sgen, f plus Sgen,h), i.e., (5.55) Sgen = Sgen, p + Sgen,h + Sgen, f The second and third terms in Equation 5.52 represent a parallel microchannel term in Equation 5.55. The result in Equation 5.55 applies to diverging microchannels. An analogous result is obtained for converging microchannels, after subtracting the latter two terms (rather than adding the terms). Without the latter two terms, the result represents the entropy generation over a surface with interspersed parallel microchannels. C ase (ii): Unspecified (Exponential) Profile, Laminar Flow, U niform Wall Heat Flux Consider another case where the best geometrical profile of embedded microchannels is unknown (or unspecified). The profile is characterized by an unknown
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function, h(y), which is determined based on minimization of entropy production along the microprofiled plate. Define a nondimensional profile variable as follows:
η=
y ; L cot θ 2
0 ≤η ≤1
(5.56)
The varying distance between the microchannel base and edge point (see Figure 5.13) becomes h = ( L tan θ1 cot θ 2 )η
(5.57)
An exponentially varying profile shape is defined by h = ( L tan θ1 cot θ 2 )η a
(5.58)
From this definition, it is required that h(0) = 0 and h(1) = Ltan(q1) cot(q 2). In terms of these variables, the thermal irreversibility difference integral becomes q ′′ 2ν 1/ 2 1 1 Sgen,h = 2 1/ 2 2 n kT∞ u∞ θ s′ ( 0 ) θ ns′ ( 0 )
1
∫ ∫ 0
L
h
L cot θ 2 x1/ 2 dxdη
(5.59)
Performing the integrations with the varying geometrical profile, q ′ 2ν 2 Sgen,h = 2 0.921nK11.11 cot θ 2 - cot 5/ 2 θ 2 tan 3/ 2 θ1 ReL1/ 2 (5.60) 3a + 2 kT∞ u∞ Similarly, the friction irreversibility integral becomes u 5/2 ρ1/ 2 µ1/ 2 Sgen, f = ∞ 2 n ( fs′′( 0 ) - fn′′s ( 0 )) T∞
1
∫ ∫ 0
L
h
L cot θ 2 x -1/ 2 dxdη
(5.61)
which yields u µ2 5.56 cot θ - 2 cot 3/ 2 θ 2 tan1/ 2 θ1 Re 3/ 2 (5.62) Sgen, f = ∞ . 328 n × 1. 2 L ρT∞ 4.185 + 0.96 K11.11 a+2 The same result is obtained as the previous case (embedded linearly converging and diverging microchannels), except that the factor 2/5 is replaced by 2/(3a + 2) in the thermal irreversibility integral. Also, 2/3 is replaced by 2/(a + 2) in the friction
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irreversibility integral. When a value of a = 1 (linear profile) is substituted into these expressions, the generalized results match the special geometrical case for linearly converging and diverging microchannels. The previous irreversibility difference integrals are combined into the total entropy production over the plate (per unit width). Then, the optimal shape profile of the embedded microchannels can be obtained by differentiating that expression with respect to the profile parameter, a, and setting the result equal to zero. For a diverging profile, 6A 2B + =0 (3a + 2 )2 ( a + 2 )2
(5.63)
where q ′ 2ν A= 0.921nK11.11 cot 5/ 2 θ 2 tan 3/22 θ1 ReL1/ 2 ρu∞T∞2
(5.64)
u µ2 5.56 B= ∞ - 1.328 n cot 3/ 2 θ 2 tan1/ 2 θ1 ReL 3/ 2 . 1 11 ρT∞ 4.185 + 0.96 K1
(5.65)
For a converging profile, a minus sign is placed before each expression for the coefficients A and B. Solving the previous algebraic equation for the optimal profile coefficient,
(
2 -3 A - 3B + 9( A + B)2 - 3( A + 3B)(3 A + B) a= 3 A + 9 B
)
(5.66)
When substituted into the profile distribution for h(y), the resulting shape of the embedded microchannels minimizes the entropy production over the plate. C ase (iii): Unspecified Cross-S tream Profile Variation, Laminar Flow, U niform Wall Heat Flux Leaving out the integration of total entropy production in the y-direction, the minimization of entropy production yields a detailed variation of microchannel profile in that direction. The thermal and friction irreversibility difference integrals become q′2 Sgen,h = 2 0.461K11.11 (1 - f 3/ 2 ( a, η ) cot 3/ 2 θ 2 tan 3/ 2 θ1 ) ReL -1/ 2 kT∞
(5.67)
u 2µ 2.78 Sgen, f = ∞ - 0.664 (1 - f 1/ 2 ( a, η ) cot1/ 2 θ 2 tan1/ 2 θ1 ) ReL1/ 2 T∞ 4.185 + 0.96 K11.11
(5.68)
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Combining these irreversibility integrals with the portions arising from parallel embedded microchannels, the total entropy production over the plate becomes q′2 Sgen = 2 4.016 + 0.461K11.11 + 0.461K11.11 (1 - f 3/ 2 ( a, η ) cot 3/ 2 θ 2 tan 3/ 2 θ1 ) kT∞ u 2µ 2.78 + 0.664 ReL -1/ 2 + ∞ T∞ 4.185 + 0.96 K11.11 2.78 1/ 2 ( a, η ) cot1/ 2 θ tan1/ 2 θ ) Re 1/ 2 f + . ( 0 664 1 L 2 1 4.185 + 0.96 K11.11
(5.69)
The first, second, fourth, and fifth terms represent the irreversibility contributions from the parallel microchannel profile. The remaining third and sixth terms, involving the trigonometric factors, represent the contributions arising from profile corrections (due to deviations of the profile width in the streamwise direction).
5.4.4 Ca s e St u d ies
o f Su r f a ce Micr o pa t t er n Des ig n
In this section, EBSM results illustrate how the method can provide an effective design tool for reducing drag and entropy production in external flows along a flat plate. Numerical results for air (300 K) are considered. In Figure 5.14, the change of optimized profile parameter, a, at varying Reynolds numbers, wall heat fluxes, and slip coefficients, is presented. Geometrical and surface parameters are shown in the figure. Each profile parameter minimizes the combined entropy production of thermal and friction irreversibilities under each set of flow conditions. This parameter affects the relative proportion of surface area containing slip-flow and no-slip conditions. The friction irreversibility
Optimal Profile Parameter (a)
6.0
q' = 14 W/m (K1 = 0.1)
5.0
q' = 14 W/m (K1 = 0.5) q' = 16 W/m (K1 = 0.1)
4.0
q' = 16 W/m (K1 = 0.5)
3.0
Air (300 K) n = 1,200 θ1 = 0.1 θ2 = 0.8 d/W = 1.0E – 06 (Diverging Microchannels)
2.0 1.0 0.0 800
1,200
1,600
2,000
2,400
2,800
3,200
Reynolds Number (ReL)
Fig u r e 5.14 Change of optimized profile parameter with Reynolds number.
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increases for larger surface areas exposed to no-slip conditions, due to added surface friction. But the thermal irreversibility decreases, because a smaller temperature difference (between the wall and surrounding fluid) is needed to transfer a fixed rate of heat flow, q´. Slip-flow conditions within the embedded microchannels lead to lower friction irreversibility, but only over a certain range of conditions, because they contribute simultaneously to additional surface area with friction. Slip-flow conditions affect the momentum exchange of intermolecular interactions near the wall. The probability of a fluid molecule striking another molecule within an embedded microchannel, rather than a wall, decreases at higher Knudsen numbers. From results obtained in this section, the Knudsen number varies between about 0.02 and 0.07. These values fall within 0.001 < Kn < 0.1, which represents the range governed by the Navier–Stokes equations with slip-flow boundary conditions (Gad-el-Hak, 1999). In Figure 5.14, the optimized surface profile parameter decreases at higher Reynolds numbers. At a fixed freestream velocity, the surface area increases at higher Reynolds numbers. Also, smaller profile parameters lead to a decreasing slip-flow area. At higher Reynolds numbers, the minimal entropy production moves to lower values of the profile parameter. The friction irreversibility rises earlier at those lower values, due to larger surface area. Also, Figure 5.14 shows that the profile parameter increases at higher wall heat fluxes. More slip-flow area is needed to overcome added thermal irreversibility at those higher heat fluxes. Air flow at 300 K past a surface with 2800 parallel microchannels and a surface heat transfer rate of 50 W/m is considered in Figure 5.15. The figure shows predicted trends of entropy production over a range of Reynolds numbers. The benchmark solution refers to the asymptotic no-slip limit, when correlations for the Blasius similarity solution can be integrated directly to yield the net entropy production. This case without microchannels represents the classical boundary layer flow and convective heat transfer from a flat nonprofiled surface. It can be observed that the current numerical 100.00
Numerical (K1 = 0) Benchmark (No-Slip Limit) Experiment (K1 = 0; Czarske et al.) Microchannels (K1 = 1, d/W = 0.00001) Microchannels (K1 = 2, d/W = 0.00001) Microchannels (K1 = 2, d/W = 0.00002) (Air, 300 K, q´ = 50 W/m, n = 2,800)
Sgen (W/m3K)
10.00
1.00
0.10
0.01 1.0E+01
1.0E+02
1.0E+03
1.0E+04
1.0E+05
1.0E+06
ReL
Fig u r e 5.15 Reduced entropy production with embedded surface microchannels.
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slip-flow formulation approaches this benchmark solution properly in the no-slip limit, when the slip coefficient becomes K1 = 0. The close agreement between predicted and benchmark results provides useful validation of the numerical modeling. Experimental data have been reported by Czarske et al. (2002) regarding wall shear stresses in the friction irreversibility portion of total entropy production. This data represent measured changes of skin friction coefficients at varying Reynolds numbers in the no-slip limit case. In Figure 5.15, this measured data (filled circle markers) is used for comparisons against the numerical modeling (solid line) and benchmark data (open circle markers) in the no-slip limit case. Close agreement is reached in these comparisons, thereby providing additional useful evidence regarding the current model’s reliability. The entropy production increases at low Reynolds numbers, when the smaller surface area leads to a high thermal irreversibility. When the surface area decreases, a larger temperature difference (between the wall and surrounding fluid) is needed to transfer a fixed rate of heat transfer from the wall, q´. On the other hand, friction irreversibilities increase at higher Reynolds numbers, due to more friction over a larger surface area. Thus, an optimal Reynolds number occurs at a certain intermediate range, where the entropy production rate is minimized. The predicted results show that the embedded microchannels allow lower values than the minimum entropy production without microchannels, due to slip-flow conditions within the microchannels. As a result, the adaptive microprofiling provides a useful technique of reducing entropy production in external flows. In Figure 5.15, this entropy production decreases at higher slip coefficients and shallower microchannels. Drag reduction occurs at the higher slip coefficients, whereas less microchannel depth reduces the friction irreversbility, due to less overall surface area. In Figure 5.15, it can be observed that the plate without embedded microchannels exhibits the lowest entropy production up to the critical Reynolds number. But this trend changes appreciably at larger Reynolds numbers. When the plate length and surface area become larger, the thermal irreversibility decreases, and added area leads to greater surface friction. The resulting entropy production becomes lower for cases with microchannels, because the added friction irreversibility is more noticeably reduced by slip-flow conditions when the surface area increases. The beneficial impact of drag reduction by slip-flow conditions is not noticeable at lower Reynolds numbers, as thermal irreversibilities constitute a larger portion of the total entropy production. Additional surface area of embedded microchannels appears to raise friction irreversibilities more than frictional reduction by slip-flow conditions. When analyzing the external flow conditions in these problems, the Reynolds number is characterized by the streamwise coordinate, x, and plate length, L. All geometrical and external flow parameters were selected so that the Reynolds number remains below the point of transition to turbulence at ReL = 5 × 105. The formulation could be extended to external turbulent flows, provided that turbulence equations are supplied for the convective heat transfer and wall friction correlations. The open microchannel flow depends on the microchannel depth (or hydraulic diameter), so similarities exist with closed microchannel flows. According to Sharp and Adrian (2004), who performed measurements of pressure drops in microtubes; they confirmed that transition to turbulence occurs at Reynolds numbers of about 1800.
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Microtube transition to turbulence has similarities with rectangular channels of the same hydraulic diameter. The transition to turbulence was estimated by the authors when pressure drops exceeded macroscopic Poiseuille flow results for laminar flow resistance. Also, micro-PIV measurements of mean velocity and root mean square (rms) velocity fluctuations at the centerline were monitored at the transition point. Experimental uncertainties of ±1% systematic and ±2.5% rms random errors were reported by Sharp and Adrian (2004). For microchannel depths and external flow velocities considered in this section, the Reynolds numbers are well below the transition point of 1800. The velocity required in the Reynolds number is best represented by the velocity at the top of the open microchannel (not the freestream velocity). Because this corresponds to the base of the boundary layer in external flow, it is approximately equal to the slip-flow velocity at the wall. At low slip coefficients, this becomes much smaller than the freestream velocity. For example, the similarity solution of f' (0) suggests that the wall velocity is about 0.02% of the freestream velocity at K1 = 0.3. This produces much lower estimates of the microchannel Reynolds number, as compared with the freestream velocity. Thus, the open microchannel flow is assumed to be fully laminar. In Figure 5.16, the ratio of actual entropy production to the minimum entropy production (called the entropy generation number, Ns) is plotted at varying length ratios (L/L opt) and expansion angles of the microchannels. Linearly converging microchannels and airflow at 300 K are considered. Other problem parameters are depicted in Figure 5.16. For small surface areas (low values of L/L opt), the net entropy production occurs mainly from the thermal irreversibility. The varying expansion angles have minor effects on Ns at low values of L/L opt, as those characteristics mainly affect the friction irreversibilities. On the other hand, the slip-flow friction irreversibilities rise faster than the no-slip case for all the expansion angles in Figure 5.16. For a specified surface length, the entropy production increases faster at smaller base and exit expansion angles, relative to the corresponding minimum entropy production, which decreases with added slip-flow area.
NS
100
Plate; without Microchannels Expansion Angles: 0.1, 0.4 (rad) Expansion Angles: 0.1, 0.9 (rad) Expansion Angles: 0.3, 0.4 (rad) Expansion Angles: 0.3, 0.9 (rad) Air (300 K) q´ = 10 W/m d/W = 1.0E – 06 K1 = 0.1 (Converging Microchannels)
10
1 1E – 03
1E – 02
1E – 01
1E + 00
1E + 01
1E + 02
1E + 03
L/Lopt
Fig u r e 5.16 Comparison of predicted entropy generation number with benchmark result.
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Optimized Profile Parameter (a)
–0.6 q´ = 100 W/m (Re = 200,000) q´ = 100 W/m (Re = 400,000)
–0.7
q´ = 200 W/m (Re = 200,000) q´ = 200 W/m (Re = 400,000)
Air (300 K) n = 2,400 U = 40 m/s θ2 = 0.9 d/W = 2.0E – 06 (Converging Microchannels)
–0.8
–0.9
–1.0
0.0
0.2
0.4
0.6 0.8 1.0 Slip Coefficient (K1)
1.2
1.4
1.6
Fig u r e 5.17 Sensitivity to slip coefficient (converging microchannels).
Unlike previous cases with diverging microchannels, the predicted results in Figure 5.17 consider converging microchannels. The microchannel converges into the central parallel section, rather than expanding outward from it. On the vertical axis of Figure 5.17, the slip-flow area decreases at lower values of the profile parameter. For example, because the geometrical configuration represents converging microchannels, the slip-flow area at a = -0.7 exceeds the slip-flow area at a = -0.6. In Figure 5.17, the profile parameter decreases at higher slip coefficients. A higher slip coefficient overcomes the added friction irreversibility of less slip-flow area. Also, the profile parameter decreases at lower Reynolds numbers, which also entails reduced friction irreversibilities with a smaller surface area. The corners of the embedded microchannels represent a transition connecting the no-slip regime (above microchannel; Kn < 0.001) and slip-flow regime (within a microchannel; 0.001 < Kn < 0.1). When calculating the local Knudsen number, the corresponding length scale must accommodate both microchannel depth and width, or a hydraulic diameter-based length. Otherwise, no-slip conditions could be erroneously predicted near the corners. For example, a wide microchannel with a submicron or nanoscale depth could produce an unrealistically small Knudsen number if the width alone were used. It is expected that the local Knudsen number decreases below 0.001 and moves into the no-slip regime at some point near the top corner. This arises with diminished effects of side walls on the intermolecular interactions near the corners. This transition to no-slip conditions is assumed to produce a small submicron semicylindrical type zone of influence at the top corners. This zone penetrates mainly into the open microchannel, as fully no-slip conditions are expected outside of the microchannels. The previous results have demonstrated a new method of surface-embedded microchannels for reducing wall friction, while simultaneously improving heat transfer effectiveness by reducing the overall entropy production. It is shown that local slip-flow conditions within the surface microgrooves can reduce the net
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entropy generation below minimum values achieved without such embedded microchannels. Entropy production is reduced when less kinetic energy is dissipated to internal energy, due to the slip-flow surface behavior. The method of surface microprofiling takes advantage of optimally placed slip-flow regions interspersed in the cross-stream direction across the surface. In contrast to other conventional methods that optimize macroscale parameters of surfaces, such as the width or aspect ratio of plates in external flow, this section has optimized the microscale features of a surface. The optimal spacing between microchannels and microchannel aspect ratios were predicted with a newly developed technique called EBSM. These conditions establish the most effective compromise between friction and heat transfer irreversibilities. It was shown that embedded open microchannels within a surface can sufficiently reduce wall friction through slip-flow conditions, to overcome added friction from the larger surface area of these added microchannels. Similar enhancements of added thermal effectiveness can be achieved with the new technique, thereby offering a useful alternative over conventional methods of heat transfer enhancement, such as baffles, fins, and spiraling.
R efer en c es Adeyinka, O.B. and G.F. Naterer. 2004. Optimization correlation for entropy production and energy availability in film condensation. Int. Commun. Heat Mass Transfer, 31(4): 513–524. Anderson, J.L. and J.A. Quinn. 1972. Ionic mobility in microcapillaries. Faraday Trans. I, 68: 744–748. Avsec, J. 2003. Calculation of equilibrium and nonequilibrium thermophysical properties by means of statistical mechanics. J. Tech. Phys., 44: 1–17. Bejan, A. 1996. Entropy Generation Minimization. CRC Press, Boca Raton, FL. Camberos, J.A. 2002. On the Construction of Exergy Balance Equations for Availability Analyses. AIAA/ASME 8th Joint Thermophysics Conference, AIAA Paper 20022880 (24–27 June). St. Louis, MO. Camberos, J.A. 2003. Quantifying Irreversible Losses for Magnetohydrodynamic (MHD) Flow Simulation. AIAA 36th Thermophysics Conference, AIAA Paper 2003-3647 (23–26 June). Orlando, FL. Cho, S.K., Moon, H., and C.J. Kim. 2003. Creating, transporting, cutting and merging liquid droplets by electrowetting based actuation for digital microfluidic circuits. J. Microelectromech. Syst., 12: 70–80. Choi, C.H., Westin, K.J.A., and K.S. Breuer. 2002. To Slip or Not to Slip — Water Flows in Hydrophilic and Hydrophobic Microchannels. International Mechanical Engineering Conference and Exposition. Proceedings of IMECE 2002-33707 (Nov. 13–16). New Orleans, LA. Czarske, J., Buttner, L., Razik, T., Muller, H., Dopheide, D., Becker, S., and F. Durst. 2002. Spatial Resolved Velocity Measurements of Shear Flows with a Novel Differential Doppler Velocity Profile Sensor. 11th International Symposium on Applications of Laser Techniques to Fluid Mechanics (July 8–11). Lisbon, Portugal. Ferziger, J. and H.G. Kaper. 1972. Mathematical Theory of Transport Processes in Gases. North-Holland, London. Gad-el-Hak, M. 1999. The fluid mechanics of microdevices — the freeman scholar lecture. ASME J. Fluids Eng., 121(March): 5–33.
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Israelachvili, J.N. 1986. Measurement of the viscosity of liquids in very thin films. J. Colloid Interf. Sci., 11(1): 263–271. Kays, W. M. and M.E. Crawford. 1980. Convective Heat and Mass Transfer. McGraw-Hill, New York. Martin, M.J. and I.D. Boyd. Blasius Boundary Layer Solution with Flip Flow Conditions. Rarefied Gas Dynamics 22nd International Symposium. Sydney, Australia, July 9–14, 2000. Maxwell, J.C. 1879. On stresses in rarefied gases arising from inequalities of temperature. Phelps Trans. R. Soc., 170: 231–256. Migun, N.P. and P.P. Prokhorenko. 1987. Measurement of the viscosity of polar liquids in microcapillaries. Colloid J. USSR, 49(5): 894–897. Naterer, G.F. 2001. Establishing heat-entropy analogies for interface tracking in phase change heat transfer with fluid flow. Int. J. Heat Mass Transfer, 44(15): 2903–2916. Naterer, G.F. 2004. Adaptive surface micro-profiling for microfluidic energy conversion. AIAA J. Thermophys. Heat Transfer, 18(4): 494–501. Naterer, G.F. and O.B. Adeyinka. 2005. Microfluidic energy loss in a non-polarized thermomagnetic field. Int. Journal Heat Mass Transfer, 48: 3945–3956. Naterer, G.F. and J.A. Camberos. 2003. Entropy and the second law in fluid flow and heat transfer simulation. AIAA J. Thermophys. Heat Transfer, 17(3): 360–371. Ng, E.Y.K. and S.T. Tan. 2004. Computation of three-dimensional developing pressure-driven liquid flow in a microchannel with EDL effect. Numerical Heat Transfer: Part A: Appl., 45(10): 1013–1027. Ogedengbe, E.O.B., Naterer, G.F., and M.A. Rosen. 2006. Slip flow irreversibility of dissipative kinetic energy and internal energy exchange in microchannels. J. Micromechanics Microengineering, 16: 2167–2176. Pfahler, J., Harley, J., and H. Bau. 1990. Liquid transport in micron and submicron channels. Sensors Actuators, A21–A23: 431–434. Pfahler, J., Harley, J., Bau, H., and J.N. Zemel. 1991. Gas and liquid flow in small channels. Symp. Microelectromech. Sensors Actuators Syst., 32: 49–60. Salas, H., Cuevas, S., and M.L. de Haro. 1999. Entropy generation analysis of magnetohydrodynamic induction devices. J. Phys. D: Appl. Phys., 32: 2605–2608. Sharp, K.V. and R.J. Adrian. 2004. Transition from laminar to turbulent flow in liquid filled microtubes. Exp. Fluids, 36(5): 741–747. Zhao, B., Moore, J.S., and D.J. Beebe. 2001. Surface-directed liquid flow inside microchannels. Science, 291(5506): 1023–1026.
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6.1 In t r o d u c t io n As discussed in previous chapters, entropy provides a valuable design tool for analyzing performance of fluid engineering systems and designing alternatives that improve energy efficiency. This book has been primarily focusing on local entropy tracking within individual components of a system, while other authors such as Rosen and Dincer (1999) have developed comprehensive methods of overall exergy analyses of a system. This chapter focuses on the role of entropy and the Second Law of Thermodynamics in numerical simulations, particularly involving error indicators for computational fluid dynamics (CFD). Entropy indicates the degree of molecular chaos or randomization, and this disorder can be interpreted in a physical sense (a traditional view), as well as a computational sense (a more recent view). The traditional view may be traced back to pioneering developments by the German mathematical physicist, Rudolf Clausius, in 1850, on the importance of entropy in steam engine performance. Computational modeling of entropy has arisen more recently with the advent of digital computers. It relates entropy and the Second Law with discretization errors (Naterer and Schneider, 1987), artificial dissipation (Hughes et al., 1986), and nonphysical numerical results (Majda and Osher, 1979). This chapter will focus on numerical errors, whereas the following chapter will describe the role of entropy and the Second Law in solution uniqueness and numerical stability of CFD simulations. In early pioneering work, Lax (1971) implemented a discrete entropy equation to identify physically relevant and unique solutions in finite difference compressible flow simulations. Harten (1983) then symmetrized the governing equations through a change of variables (entropy gradient variables) to improve the performance of iterative algebraic solvers. Merriam (1987) has shown that satisfaction of the Second Law is sufficient, in some cases, to ensure stability of compressible flow computations. This numerical stability also suggested that entropy could serve as an effective error indicator and criterion for solution convergence. Camberos (1998) showed that entropy provides an effective measure of residual error and convergence because of its physical significance with a full functional dependence on all fluid state variables. Naterer and Schneider (1994) have demonstrated that solution errors and nonphysical phenomena, such as numerical oscillations, coincide with a discrete violation of the Second Law. In this way, solution accuracy and entropy production are closely related in a numerical sense. Conventional error indicators with Taylor 143
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series expansions can only indicate the limiting behavior of numerical errors, when the grid spacing approaches zero. A mathematical Taylor series analysis generally lacks a physical interpretation. It usually cannot identify errors for coarse grids. It also cannot identify nonphysical aspects of the numerical results. Entropy and the Second Law provide a useful alternative because they establish a physical basis that relates physical plausibility of results, numerical accuracy, convergence, reliability, and stability of simulations, all within the scope of the Second Law. MacCormack’s second-order time-split scheme (MacCormack and Baldwin, 1975) and the ARC2D and ARC3D codes of Pulliam and Steger (1980) were key pioneering developments of Navier–Stokes solvers for three-dimensional viscous compressible flows. Subsequent advances were made for boundary layer and shock wave interactions, unstructured grids (finite elements; Lohner et al., 1984), and conservation-based methods (finite volumes; Karki and Patankar, 1989). However, much effort and difficulty arose from general error analysis and robustness of the numerical codes. Solutions were sensitive to time steps, grid spacing, and empirical constants in the schemes. Rigorous order accuracy often could not be established. Complicated problems required specialized “tuning” of coefficients, but the tuning would be altered for each new problem. More grid points and faster computers could achieve more accurate solutions, but they could not necessarily bring a more robust or stable algorithm. As many numerical methods lacked a unified approach to error analysis, subsequent developments implemented entropy and the Second Law for this purpose in CFD codes for viscous compressible flows. Hughes and coworkers (1986) developed finite element schemes that satisfied the Second Law in a global sense. However, numerical oscillations may still occur in individual elements because the Second Law was not enforced at a local level. Merriam (1987) presented a general methodology for satisfying the entropy inequality on a cell-by-cell basis. A general method of analysis, rather than a specific finite element or finite volume scheme, was presented. Entropy-based corrections for error reduction were later implemented for compressible flows (Naterer and Schneider, 1994) and duct flows (Nellis and Smith, 1997). Finite volume methods are widely used for compressible flow simulations because of their capabilities, conservation properties, and physically based discretization (Patankar, 1980). The discrete equations are obtained by integration of the governing equations over discrete control volumes. Approximations of the convection and diffusion terms are usually made at the midpoint of the volume surface (integration point). Unlike the conservation equations, entropy is governed by an inequality (Second Law). This chapter focuses on how entropy and the Second Law can effectively characterize the accuracy and numerical errors inherent in discrete modeling of the conservation equations, such as convective upwind schemes (next section). Also, this chapter will present a novel entropy-based approach for calculating the residual error in steady-state problems. The technique calculates the difference in entropy (averaged over the computational domain) as a metric to analyze solution convergence. Several steady-state calculations of viscous compressible flow fields will be presented, together with an averaged metric based on entropy. This metric is the difference in the averaged entropy from one time step to the next step. Convergence is reached when the entropy-based residual is reduced by several orders of magnitude. The attractive feature of an entropy-based residual is that it provides
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a global measure of changes in the numerical solution, and it implicitly depends on all problem variables simultaneously (mass, momentum, and energy). Typically, conventional error residuals and convergence indicators depend on a single-state variable or certain metrics that are limited to a selected set of problems. In contrast, the universality of entropy does not suffer from these limitations. Thus, it makes an ideal candidate as a robust error indicator and criterion for convergence.
6.2 Disc r et iz a t io n Er r o r s o f Nu mer ica l C o n v ec t io n Sc h emes 6.2.1 Fin it e Vo l u me Fo r mu l a t io n This section presents a procedure that applies entropy principles to a numerical scheme that satisfies the Second Law for a component of the overall formulation, namely, the convection scheme. The governing equations for viscous compressible flow and heat transfer are the Navier–Stokes equations. These equations have been presented in earlier chapters, but they will be rewritten here in a generalized transport form for a subsequent entropy analysis. Define a vector of conserved quantities, q, and a corresponding flux, f, with an advective component, fa, and a pressure (p) and diffusive component, fd.
ρu ρ ρuu ρu q = and f = ρvu ρv ( ρe + p )u ρe
0 ρv - p + τ xx ρuv - τ yx ρvv ( ρe + p )v uτ xx + vτ xy - jx
0 τ xy - p + τ yy uτ xy + vτ yy - jy
(6.1)
The heat flux vector, j, in Equation 6.1, can be related to temperature, T, by Fourier’s law. For each conserved quantity, there exists a corresponding transported scalar, f . For example, x-momentum is conserved (q2 = ru), and the scalar f = u is transported by the flow in the momentum equation. The governing equations can be written in the following conservation form or a nonconservation transport form (excluding continuity): ∂q + ∇ ⋅ f a + ∇ ⋅ f d = 0 ( Conservation Form ) ∂t
(6.2)
∂φ + ρv ⋅∇φ + ∇ ⋅ f d = 0 ( Nonconservation Form ) ∂t
(6.3)
ρ
The components of the stress tensor, t, in Equation 6.1 are
τ xx = 2 µ
∂u 2 ∂u ∂v - µ + ∂x 3 ∂x ∂y
(6.4)
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∂u ∂v τ xy = µ = τ yx + ∂y ∂x
τ yy = 2 µ
(6.5)
∂v 2 ∂u ∂v - µ + ∂y 3 ∂x ∂y
(6.6)
where m is the dynamic viscosity. In addition, the ideal gas law, p = r RT, and the relations g = cp/cv and R = cp - cv, where cp and cv refer to specific heats, allow calculations of pressure as follows:
p = (γ - 1) ρ e
1 2 1 2 u - v 2 2
(6.7)
Consider a numerical discretization with a problem domain subdivided into finite volumes and elements. In one dimension, Figure 6.1a illustrates the grid structure, whereas Figure 6.1b shows the appropriate schematic definitions for two dimensions. In one dimension, a control volume is defined by the two adjacent half-elements surrounding each node, and the integration points (ip) are located at the control volume surfaces. The integration point resides at the element midpoint. Linear interpolation functions are used within each element to represent the variation of dependent scalars, in terms of nodal variables. Integrating Equation 6.2 over a control volume and time step,
∫
V
q(t + ∆t )dV -
∫
V
q(t )dV +
(a)
t +∆t
∫ ∫ f ⋅ ndAdt = 0 t
i–1/2 i
i+1/2
i+1
Element
Integration Points at Control Volume Surfaces Local Node Number
1
2 SCV 2 Flow Direction
(6.8)
Control Volume i–1
(b)
S
SCV 1 SCV 4
iP
4 Subvolume Node
Upstream Point Upwind Difference
Finite Element
Control Volume
Fig u r e 6.1 (a) One-dimensional, and (b) two-dimensional schematic of a finite volume.
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The discrete equations can be obtained by integration over a specific time interval t n ≤ t ≤ t n+1 and one-dimensional volume xi -1/2 ≤ x ≤ xi +1/2. The discretized conservation and nonconservation (transport) forms of the governing equations, Equation 6.2 and Equation 6.3, respectively, then become
φin +1 - φin 1 + ∆xi ∆t
q in +1 - q in fi +1/2 - fi -1/2 + =0 ∆t ∆xi
∑ m φ
ip ip
ip
( Conservation Form )
(6.9)
∂φ - Γ ip = sources ( Nonconservation Form ) (6.10) ∂x ip
where m refers to the mass flow rate, Γ represents a general diffusion coefficient, and “sources” refers to the remaining source-type terms in the governing equation. Conventional methods of interpolation are used to approximate the transient, diffusion, and source terms. To specify a well-posed algebraic system, the advection terms in Equation 6.10 at the integration points need to be related to nodal variables. This requires modeling for the transport of a scalar quantity, f, across a control volume surface, or integration point, such as f = u in the momentum equation.
6.2.2 Cen t r a l , Upw in d , a n d Ex po n en t ia l Differ en cin g Sche mes Various methods can be used in numerical schemes to estimate an integration point value such as φi+1/2. For example, the approximation φi +1/2 = φi represents an upwind differencing scheme (UDS). In two-dimensional problems, an analogous procedure is the skew upwind differencing scheme (SUDS), which uses the local flow direction to determine the appropriate upstream location for the scalar variable approximation. Without the influence of pressure forces on the integration point velocity, there can be a nonphysical decoupling between pressure and velocity. For example, if a large pressure gradient in a flow field has no direct influence on the integration point velocity, it could lead to direct violation of the Second Law. In contrast to UDS, the central differencing scheme (CDS) uses linear interpolation between adjacent nodal values to find the integration point variable, that is, φi +1/2 = (φi + φi +1 )/2 . In CDS, the convective flux dependence on downstream variables may have nonphysical trends when the Peclet number is high and upstream convection influences are dominant. Convection models may use some combination of adjacent nodal values for the integration point approximations. For example, hybrid schemes such as the exponential differencing scheme (EDS) provide the correct balance between UDS and CDS influences, based on the local grid Peclet number ( Pe = ρui ∆xi / Γ) (Minkowycz et al., 1988). EDS obtains a smooth transition from CDS for Pe → 0 to UDS for Pe → ∞. Neglecting transient, pressure, and source terms in Equation 6.3 and solving the resulting equation subject to specified values of f at the nodes yield the EDS solution. Evaluating f at the integration point with this EDS solution,
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1+α 1-α φi +1/2 = φ + φ 2 i 2 i +1
(6.11)
where
α = 1
2( e Pe /2 - 1) Pe 2 ≈ Pe 5 + Pe 2 e -1
(6.12)
The latter approximation is an alternative to reduce the computational expense of frequent exponential calculations (Minkowycz et al., 1988, see Chapter 7 by Raithby, G.D. and Schneider, G.E.). This scheme has first-order accuracy, in terms of the Taylor series truncation error. Higher-order schemes, such as quadratic upstream interpolation for convection kinetics (QUICK) (Leonard, 1979), reduce the discretization errors through quadratic interpolation for integration point values. The finite element differential scheme (FIELDS) solves an approximation to the governing equations at the integration point to incorporate the local fluid physics, such as local pressure and source term effects (Schneider and Raw, 1987). It will be useful to determine whether FIELDS, UDS, and other schemes comply with the requirements of the Second Law at a local (control volume) level. Such schemes will be denoted as “entropy-stable” schemes. Solutions that obey the Second Law will exhibit proper physical characteristics. A flow field governed only by the conservation laws in Equation 6.2, but not necessarily the discrete form of the Second Law, could display unusual physical behavior. For example, it is highly improbable that heated fluid elements could become sufficiently organized to independently produce a cold fast fluid stream that converts internal energy to kinetic energy. Although possible through the First Law, the statistical probability of observing this process is extremely small, according to the Second Law. The Second Law states that entropy, which is a property of matter that measures the degree of disorder at the microscopic level, can be produced, but never destroyed in an isolated system. These observations also apply to numerical computations, wherein numerical approximations should not produce nonphysical results that violate the Second Law. The Second Law of Thermodynamics is written below in a form similar to Equation 6.2:
Ps = S,t + F,x ≥ 0
(6.13)
where the subscript notation with a comma refers to differentiation. For example, the subscript “x” refers to a partial derivative with respect to x in one dimension, or the divergence operator in multidimensions. Also, P s refers to the entropy production rate and S(q) and F(q) represent the thermodynamic entropy and entropy flux, respectively, that is,
S = ρs
(6.14)
F = ρus
(6.15)
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and s represents the specific entropy. For an ideal gas, p/ p0 s = cv log γ ( ρ/ρ0 )
(6.16)
where the subscript 0 refers to values at a specified initial state. In Equation 6.13, the equality refers to reversible processes, and the inequality refers to irreversible processes. The entropy and entropy flux must obey two important mathematical properties:
S,qq < 0 ( convexity )
S,q f,q = F,q ( compatibility)
The convexity condition requires irreversible processes to produce entropy. It ensures that entropy is bounded from above, because S,qq must be a negative definite matrix. The entropy distribution typically reaches a maximum value at thermal and mechanical equilibrium. In the compatibility criterion, F,q represents the entropy flux derivative matrix (a second-order tensor) with a vector component in each of the three coordinate directions. Also, f,q is a third-order tensor that denotes a derivative of four fluxes in three directions with respect to four conservation variables. The compatibility condition guarantees the existence of an entropy flux satisfying the Second Law, whenever an entropy conservation principle holds for reversible processes. For a discrete volume, the Second Law can be expressed as
P s =
Sin+1 - Sin Fi +1/2 - Fi -1/2 + ≥0 ∆t ∆xi
(6.17)
After the solution of the conservation equations is obtained, an additional step is required to find q(x,t) from the nodal and integration point values so that S(q) and F(q) can be properly integrated. An approach that does not violate the Second Law during this reconstruction step is needed. In this way, if a negative entropy production rate arises in the numerical analysis, it can be attributed to the discretized conservation equations, rather than the entropy inequality, Equation 6.17. Thus, assume that q = qi within the control volume, where the subscript i refers to node i. This assumption meets the previous requirement because a piecewise constant distribution maximizes the entropy within each control volume with respect to the choice of qi. A fundamental result of thermodynamics states that for all processes at a constant total volume and energy, the entropy increases or remains constant. Thus, when a system reaches thermal and mechanical equilibrium, then its entropy must be a maximum. Because the state transition from q(x,t) to qi is physically irreversible (in practice), the entropy contained within an isolated control volume must increase and achieve a maximum value at the equilibrium state, q(x,t) = qi. It will be approximated that q is piecewise constant, at its integration point value, along each control surface.
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The entropy is expanded with a truncated Taylor series as follows: Si = Sin + S,t (t - t n ) +
1 S,tt (η )(t - t n )2 2
(6.18)
where t n ≤ η ≤ t . Similarly, defining q i ≤ ζ ≤ q i +1/2 and expanding the entropy flux about an integration point, Fi +1/2 = Fi + F,q (q i +1/2 - q i ) +
1 F,qq (ζ )(q i +1/2 - q i )2 2
(6.19)
If q is scalar, then the squared term in Equation 6.19 represents a scalar multiplication. Otherwise, when q is a vector, then the term is evaluated by the product of the vector and its transpose. In a similar fashion, we can expand the entropy flux about the other integration point. Fi -1/2 = Fi + F,q (q i -1/2 - q i ) +
1 F,qq (ζ )(q i - q i -1/2 )2 2
(6.20)
Substituting these relations into the expression for the entropy production, (q i +1/2 - q i )2 - (q i - q i -1/2 )2 q i +1/2 - q i -1/2 1 + S,tt ∆t + F,qq P s = S,t + F,q ∆x ∆x 2
(6.21)
Using the compatibility condition, and simplifying the first term in Equation 6.21, (q i +1/2 - q i )2 - (q i - q i -1/2 )2 q i +1/2 - q i -1/2 1 + S,tt ∆t + F,qq P s = S,q q,t + f,q ∆x ∆x 2 (6.22) This equation expresses the entropy production rate in terms of several problem variables, so it is difficult to implement or verify the positive definite character of each individual term. Simplifying the expression, in the first term with another Taylor series,
q i +1/2 = q i + q,x ( xi -1/2 - xi ) +
1 q,xx ( xi +1/2 - xi )2 2
(6.23)
Writing another similar expansion about x = xi -1/2 and substituting the results into Equation 6.22,
P s = S,q (q,t + f,x ) +
1 S,tt ∆t ≥ 0 2
(6.24)
The row vector S,q in Equation 6.24 represents a rate of change of entropy with respect to the conserved quantity. If the conservation equations are solved in an exact
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fashion, then the first term in Equation 6.24 vanishes, and the second term remains positive to satisfy the entropy inequality. It should be noted that Equation 6.24 remains valid at both the control volume level (where the overall conservation equations are applied) and the integration point level, where modeling of convective terms, such as φi +1/2 = φi (UDS), are made. In the latter case, violation of the inequality in Equation 6.24 may lead to nonphysical errors such as false diffusion (in the case of UDS) or oscillations (in the case of CDS). The subgrid modeling of convection at the integration points should be governed by the same entropy requirements as the overall conservation equation. The bracketed component of the previous entropy inequality contains the scalar conservation equation. Discretizing that equation by standard differencing techniques, φ - 2φi +1/2 + φi φi +1/2 - φi P - Pi φ n +1 - φ n + a3 i +1/2 + a4 Γ i +1 L (q ) + δ = a1 ρ + a2 ρu ∆xi2 ∆t ∆xi / 2 ∆xi / 2
(6.25) In this form, the exact equation, L(q) = 0, Equation 6.3, is replaced by a discrete approximation, L(q ) + δ = 0, where L() refers to the differential operator (left side) in Equation 6.3 and δ refers to discretization errors at xi+1/2. It is known that δ → 0 as the grid and time step are refined. The conventional models for integration point approximations can be extracted from Equation 6.25 as follows:
1. CDS for a1 = a2 = a3 = δ = 0 and a4 ≠ 0.
2. UDS for a1 = a3 = a4 = δ = 0 and a2 ≠ 0.
3. EDS for a1 = a3 = δ = 0, a2 = α / Pe, and a4 = (1 - α ) / 2.
Figure 6.2 shows these coefficients and their dependence on Pe. As Pe → ∞, it can be shown from Equation 6.25 that | a4 | << | a2 .| Thus, the UDS upstream values dominate the downstream influences, even though a2 becomes small. 0.6
a2 (CDS) a2 (EDS) a2 (UDS)
Upwind Coefficients
0.4
a4 (CDS) a4 (EDS) a4 (UDS)
0.2 0
–0.2 –0.4 –0.6 0.001
0.01
0.1
1 10 Peclet Number
100
1000
Fig u r e 6.2 Conventional upwind coefficients.
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Shock Wave
5.0
Pe
5.0
Fig u r e 6.3 Schematic of converging-diverging nozzle.
The following case study aims to determine whether these coefficients violate the entropy inequality in Equation 6.24. The first term after the equality in Equation 6.24 in a product between each component of the row vector S,q and a conservation (or transport) equation represented by each component of a column vector, i.e., S,q1 (continuity) + S,q2 (momentum transport equation) + S,q3 (energy transport equation). The numerical entropy production of each individual contribution should be high enough to prevent an overall negative sum and a violation of Equation 6.24. The effect of a particular transport equation can be isolated within Equation 6.24 to determine its impact on entropy stability of the numerical method.
6.2.3 Ca se St u d y
o f No z zl e
Fl o w An a l ysis a n d Desig n
This case study examines the entropy stability of convection schemes for convergingdiverging nozzle and channel flows (see Figure 6.3). Numerical simulations were conducted with a control volume-based finite element formulation of the viscous compressible flow equations (Naterer, 1999). Figure 6.4 shows the steady-state density, velocity, pressure, and temperature distributions, respectively, for the three example cases. For the fully subsonic flow, the flow accelerates in the converging section until a point of maximum velocity and minimum pressure is reached at the throat. Then the flow decelerates in the diverging section until it reaches the prescribed outlet condition. However, in the supersonic flow example, the maximum Mach number (Ma = 2.2) occurs at the outlet because the flow continually accelerates through the diverging section of the nozzle to meet the prescribed outlet condition. The example involves compressible air flow through a converging–diverging nozzle with a specified cross-sectional area, A(x): A( x ) = Ath + ( Ae - Ath ) 1
2
x ,x ≤ 5 5
(6.26)
2
x A( x ) = Ath + ( Ae - Ath ) - 1 , x ≥ 5 5
(6.27)
where 0 ≤ x ≤ 10[m] and Ath and Ae represent the throat and exit areas (Ae = 2.035 Ath; see Figure 6.3). Gas properties for air include g = 1.4, cp = 1004[J/kgK], and R = 287 [J/kgK]. In this problem, the stagnation pressure (P0 = 93.75 [kPa]), temperature
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1.50
Subsonic Outlet Supersonic Outlet
1.00 0.75 0.50
0
1
2
600.00
3
4 5 6 Position (m) (a) Density
7
8
9
60.00 40.00
Flow with Shock Supersonic Outlet
400.00
0
1
2
3
4 5 6 Position (m) (c) Pressure
7
8
9
10
450.00
Subsonic Outlet
500.00
300.00 200.00 100.00 0.00
0.00
10
Temperature (kg)
Velocity (m/s)
Supersonic Outlet
80.00
20.00
0.25 0.00
Subsonic Outlet Flow with Shock
100.00
Flow with Shock
Pressure (kpa)
Density (kgm3)
1.25
Subsonic Outlet
400.00
Flow with Shock
350.00
Supersonic Outlet
300.00 250.00 200.00 150.00
0
1
2
3
4
5
6
7
Position (m) (b) Velocity
8
9
10
100.00
0
1
2
3
4 5 6 7 Position (m) (d) Temperture
8
9
10
Fig u r e 6.4 Profiles of (a) density; (b) velocity; (c) pressure; and (d) temperature.
(T0 = 300 [K]), and inlet Mach number (Mai = 0.3) are specified. The inlet conditions are used as initial conditions throughout the problem domain. No-slip conditions are defined along the nozzle walls. Numerical simulations were conducted with a time-marching scheme from the initial conditions to the final steady-state solution. Three different back pressures are specified, and these three values produce the following results in the diverging section of the nozzle: (1) subsonic flow (Pe = 80 [kPa]); (2) mixed flow with a shock wave (Pe = 50 [kPa]); and (3) supersonic flow (Pe = 8.8 [kPa]). If the back pressure is not low enough to induce sonic conditions at the throat, then the flow remains subsonic throughout the nozzle (Case 1). As the back pressure is further reduced, the throat eventually becomes sonic, and the mass flux through the nozzle reaches a maximum value. If the back pressure is reduced below this critical condition, then the throat remains choked at the sonic value, and a normal shock wave occurs in the diverging section to meet the outlet condition (Case 2). The design pressure ratio is achieved when the back pressure is further reduced until the diverging flow is entirely supersonic (Case 3). This design condition is often used for efficient operation of a rocket exhaust. For accelerating flow in a converging nozzle, the row vector S,q in Equation 6.24 refers to a rate of change of entropy with respect to a conserved quantity. For example, if the x-momentum (q2 = ru) increases by an incremental amount, dq2, when the flow accelerates through a converging nozzle, then the term S ,q2dq2 represents the associated entropy increase. The term Sq dq would represent the cumulative effect
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4
Subsonic (Ma = 0.5)
Subsonic (Ma = 1.5)
Sq1/1000[J/kgK] Sq2 [m/sK] Sq3*1000 [1/K]
Sq1/1000[J/kgK] Sq2 [m/sK] Sq3*1000 [1/K]
Entropy Derivative
3 2
Positive–Definite
1 Alternating Signs
0
–1
Negative–Definite
–2 1
2
3
4 Pressure (atm)
5
6
7
Fig u r e 6.5 Rate of entropy change.
of changes in all conserved quantities on the entropy change. From Equations 6.1, 6.7, 6.14, and 6.16, the entropy derivative becomes (γ - 1) ρu 2 (γ - 1) ρu (γ - 1) S,q = s + cv -γ + , - cv , ρcv P P 2 P
(6.28)
The components of S,q are illustrated in Figure 6.5 for the case of air with g = 1.4, T = 300[K], r = 1.16[kg/m3], cv = 717.4[J/kgK], and P0 = 101[kPa]. In Equation 6.26 for one-dimensional flows from left to right, the sign of S,q1 may be positive or negative, but S,q2 ≤ 0 and S,q3 ≥ 0. These properties can be observed in Figure 6.5. In physical terms, this implies that the entropy would decrease with a positive change, dq2. In Figure 6.5, it can be observed that |S,q2 | for supersonic flow is larger than the corresponding subsonic value at a fixed pressure. For subsonic converging-diverging nozzle flow, from Equation 6.28, numerical entropy production associated with discretization of the momentum equation is ( S,q f,x ) 2 =
- cv (γ - 1) ρu P
P - Pi ui +1/2 - ui u - 2ui +1/2 + ui + a3 i +1/2 + a4 Γ i +1 a2 ρ u ∆xi / 2 ∆xi / 2 ∆xi2
(6.29)
This equation indicates a set of necessary constraints on conventional upwind schemes to satisfy the entropy inequality. For example, consider an accelerating flow in a subsonic converging-diverging nozzle (i.e., ∂u / ∂x > 0 ), upstream of the throat. If the flow remains subsonic, then it will decelerate upstream of the throat due to the upcoming area expansion in the duct. As a result, the concavity of the velocity profile changes in the region upstream of the throat, from concave upward to concave downward. Under these conditions with CDS, UDS, SUDS, or EDS ( a3 = 0 ), it can be observed that S,q f,x < 0 in Equation 6.29 and the entropy inequality in Equation 6.24 is violated. In practice, viscous terms are negligible outside the boundary layer in
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this case, but nevertheless, Equation 6.29 still remains negative for accelerating flows. Thus, the upwinding must be modified to prevent negative entropy production. For example, the schemes may apply the approximation to decelerating flows instead, or modify the upwinding with additional terms. Consider a modified upwind scheme that imposes a momentum constraint on the pressure terms in Equation 6.29, to satisfy Equation 6.24. A pressure influence at the integration point is introduced to prevent the problem in the previously mentioned example with an accelerating flow. For steady flow, Bernoulli’s equation can be written in the following form (Fox and McDonald, 1992): Pi + ρ
ui2 u2 = Pi +1/2 + ρ i +1/2 + losses 2 2
(6.30)
where losses ≥ 0 represent frictional losses. The velocity terms in Equation 6.30 can be factored in the following form: Pi - Pi +1/2 + ρu ( ui - ui +1/2 ) = losses
(6.31)
where u = (ui + ui +1/2 )/2. It is evident from Equation 6.30 and Equation 6.31 that setting a2 = a3 in the physical influence scheme (PINS) and a4 ≈ 0 (negligible downstream influences for high Pe number case) allows the losses to be isolated. Then Equation 6.27 may be rewritten as follows: ( S,q f,x ) 2 = 2
a2 cv (γ - 1) ρu (losses ) ≥ 0 ∆xi P
(6.32)
Using this formulation, the Second Law requirement in Equation 6.24 is then satisfied. Consider another example for more general duct flows with friction. A detailed comparison between PINS (subscript pins) and other conventional schemes (subscript o), such as CDS, UDS, SUDS, and EDS, will be examined and interpretated in the context of general duct flows with friction. The wall shear stress is defined as τ w = f ρu 2 /2, for incompressible flows where f refers to the friction factor (Fox and McDonald, 1992). Defining G = (S,q f,x) and Ec = u2/(cp ∆ T) (Eckert number), the difference between upwind schemes can be computed using the ideal gas law with the following result: G pins - Go = 2
a2,o cv (γ - 1) ρu a2, pins 2(γ - 1) D 1 -1 (losses ) (6.33) L fEc ∆xi P γ a2,o
where D and L refer to diameter and length, respectively. The critical points, Ec and D /L, where this difference changes signs, occur when Ec =
D a2, pins a2,o 2(γ - 1) D γ and = fEc - 1 L a2, pins - a2,o γf L a2,o 2(γ - 1)
(6.34)
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G(pins)-G(o) [J/mˆ3Ks]
2
Only PINS Ensures Entropy Stability
0
All Schemes Satisfy Entropy Constraint
–2
D/L=0.01: f=0.010 D/L=0.01: f=0.013 D/L=0.01: f=0.016 D/L=0.02: f=0.010 D/L=0.02: f=0.013 D/L=0.02: f=0.016
–4 –6 –8
0
0.05
0.1 Eckert Number
0.15
0.2
Fig u r e 6.6 Regions that satisfy the entropy constraint (in terms of the Eckert number).
Thus, conventional upwind schemes will exhibit entropy-stable behavior for low Eckert numbers below the critical points. These trends are illustrated in Figure 6.6, where the following sample constants have been selected: g = 1.4 (air), a2,pins = a3,pins = 1, and Pi ≡ cv (γ - 1) ρui (losses )( a2, pins - a2,o ) / ∆xi . Below the critical points, G pins < Go but G pins ≥ 0 for all Eckert numbers, so Go ≥ 0 is guaranteed also in this region. However, above the critical points, PINS satisfies the entropy inequality, Equation 6.24, but the other schemes may violate it. Both Figure 6.6 and Figure 6.7 show that the difference, G pins - Go , increases with the friction factor at a specific Ec or D/L (diameter per length) ratio. This indicates that higher wall friction produces more entropy for a pressure-weighted scheme than a scheme (such as UDS) 2
G(pins)-G(o) [J/mˆ3Ks]
1
G(pins)>0 but-c
0 G(pins)>0 and G(o)>0
–1
Ec=0.2; f=0.010 Ec=0.2; f=0.013 Ec=0.2; f=0.016 Ec=0.4; f=0.010 Ec=0.4; f=0.013 Ec=0.4; f=0.016
–2 –3 –4
0
0.02
0.04
D/L
0.06
0.08
0.1
Fig u r e 6.7 Regions that satisfy the entropy constraint (in terms of D/L).
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without this dependence. Friction, pressure loss, and entropy production are thus closely related to the integration point approximations. Because the current analysis only considers the momentum transport parts of Equation 6.24, the overall accuracy and entropy stability of an entire finite volume method has not been formally confirmed. Each integration point discretization is generally constructed independently of the other transport equations. In closing, this case study has shown that pressureweighted upwinding with PINS leads to entropy stability in the convective formulation of the momentum equation.
6.3 Ph ys ica l Pl a u s ibil it y o f Nu mer ica l Resu l t s 6.3.1 En t r o py Co r r ect io n
o f Nu mer ica l Diffu sio n
In the previous section, compliance with the Second Law was outlined for individual components of the numerical formulation (specifically the convective upwind scheme). In the absence of preventative measures to ensure physically plausible results, an alternative is a corrective measure that uses errors based on computed negative entropy production to recalculate results, therein striving to satisfy the Second Law. This section describes a corrective procedure that first detects anomalous flow patterns in the flow field (like numerical oscillations) using the local entropy production rates, then performs a corrective procedure by applying a required diffusion coefficient to ensure positive entropy production. The analysis will be performed with a control volumebased finite element method (Naterer and Schneider, 1994). Let Vj denote the volume associated with node j, so the integral form of the Second Law can be written as ∂S ( q ) (6.35) Vj + F ( q ) ⋅ ds ≥ 0 ∂t sj
∫
perform the integration, a typical four-noded, quadrilateral finite element was To illustrated in Figure 6.1b. The element comprises four subcontrol volumes, each of which is associated with the node located at its outermost corner. The subcontrolvolume boundaries are defined by the element external surfaces and lines corresponding to local coordinate values of s = 0 and t = 0 (origin at the center of the element). Considering the shaded subcontrol-volume of Figure 6.1b, integration of Equation 6.35 over a subcontrol volume results in
∫
Sj
F ( q ) ⋅ ds =
4
∑ F (q j =1
ipj
) ⋅ ∆s ipj
(6.36)
where in each term, ∆s is an outward facing normal at the midpoint of the appropriate edge. Two alternative temporal discretizations will be considered in the formulation of the Second Law. The first approach is an explicit backward difference for the transient term. This backward difference may potentially lead to a violation of the Second Law inequality if the temporal discretization is inadequate. The second
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approach is a “semidiscrete” method, wherein the entropy time derivative is rewritten in terms of spatial derivatives through the chain rule, and no discrete time step is introduced (Naterer, 1989). Both methods were described previously in Section 2.7.3. Although the semidiscrete formulation is exact in one dimension (Merriam, 1988), it too may violate the Second Law through its midpoint approximations of entropy and conserved variable fluxes in multidimensions. It is desirable to use a discretization that minimizes the numerical entropy production rate such that entropy-violating solutions are not concealed by artificial entropy production through temporal differencing. As shown in Section 2.7.3, the difference between entropy production rates obtained from the semidiscrete and fully discrete formulations is a linear function of ) j < 0 ) can the entries of the Hessian of S ( q ). This remainder term (denoted by ( Ps be written in the following summation form (Naterer, 1989): ( P s ) Rj =
Vj 2 ∆t
4
4
l =1
m =1
∑∑h αα lm
l
(6.37)
m
where hlm denotes the entries of the Hessian matrix, H = ∂2 S/∂q 2 , and α l = ( qln, +j 1 - qln, j ). Because H is convex (negative definite), the quadratic form given by the double sum in Equation 6.37 must be negative for all (α1 , α 2 , α 3 , α 4 ), and it follows that ( P s ) Rj ≤ 0 (6.38) This result shows that the fully discrete entropy production rate is less than or equal to the semidiscrete entropy production rate, independent of the time step or control volume size. At steady state, ( q nj +1 - q nj ) vanishes, and the entropy production rate is entirely determined by the spatial discretization. In this case, the equality in Equation 6.38 holds. In a similar manner, it can be shown that the volumetric entropy production rate formed by implicit time advance is less than or equal to the semidiscrete entropy production rate. These results are equally valid for both Euler and Navier–Stokes equations, except that the entropy function, S ( q ), and entropy flux F ( q ) will be different. The Euler equations do not provide any natural dissipation mechanism (such as viscosity in the Navier–Stokes equations) to diffuse numerical oscillations resulting from inadequate mesh refinement in regions of large gradients like shock waves. Instead, smoothing algorithms such as methods of Lohner et al. (1984) and MacCormack (1975) have been used to add artificial dissipation terms. Early pioneering studies of Von Neumann and Richtmyer (1950) developed various techniques like the user-specified constants to control numerical oscillations and stability. Upwind differencing introduces an implicit artificial viscosity into a scheme. The effect of artificial viscosity reduces the spatial flow gradients. It arises from even derivative terms in the truncation error (called numerical dissipation). When odd derivative terms appear in the truncation error, the properties of various waves are distorted. This quasiphysical effect is called dispersion. A key benefit of an entropy-based error analysis is that the Second Law is sensitive to both of these errors, because the resulting effects are both nonphysical.
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From previous chapters, the following entropy transport and entropy production equations have been derived:
ρ
Ds ∇ ⋅ q ′′ µΦ =+ Dt T T
(6.39)
k∇T ⋅ ∇T µΦ + T2 T
(6.40)
P s =
where Φ refers to the velocity gradient portion of the dissipation function. Because ∇ T ∙ ∇ T, Φ, and the fluid properties in Equation 6.40 are all greater than or equal to zero, then Ps ≥ 0. However, discretization errors and nonphysical solution behavior in the numerical solution may lead to local discrete violations of the Second Law, thus potentially ( Ps ) j < 0 in some control volume j. If the Second Law is violated locally, then a quantitative indication of the artificial viscosity required to correct the solution may be expressed in terms of P s from Equation 6.40. Using the Prandtl number (Pr = v/a), an “artificial viscosity” can be factored out from the previous entropy production equation to give
µ=
P s c p ∇T ⋅ ∇T /( Pr T 2 ) + Φ / T
(6.41)
In Equation 6.40, the local entropy production rate will be greater than or equal to zero, both analytically and numerically, because it is a sum of squared terms. However, temporal and spatial differencing of the entropy transport equation may lead to nonphysical numerical results and negative entropy production rates within a discrete control volume. If P s is computed as a negative value, then the numerical solution behavior is not physically correct because the Second Law is violated, and Equation 6.41 would imply a negative viscosity, which would steepen gradients rather than smooth them. If μ and k are computed in Equation 6.40 using the magnitude of P s from the entropy transport equation, then the “entropy-based” diffusion could prevent potentially nonphysical solution behavior, such as rarefaction shocks and numerical oscillations, because additional diffusion is a “smoothing” process. To implement this approach, the corrective procedure should only be applied in regions containing the nonphysical solution behavior because there is no physical justification for modifying the solution elsewhere. Also, a mechanism is needed to determine how much is a sufficient amount of diffusion. The Second Law can be used as the required corrective mechanism. It is sensitive to the nonphysical numerical results. Also from Equation 6.41, it can be used to provide a quantitative measure of the amount of diffusion required in the numerical procedure to correct any nonphysical results. Following each time step, the entropy production rate is computed based on the entropy transport equation. If a nodal value of P s is negative, then the local solution is not physically correct. Therefore, instead of proceeding to the next time step, a corrective iteration of the Navier–Stokes equations is performed to find the entropy production needed to prevent the computed entropy destruction. The entropy
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production rate for a control volume is given by the left-hand side of Equation 6.35. For a desired production rate, given a prescribed temperature and dissipation distribution, the viscosity needed to supply this production rate can be computed from the discrete representation of Equation 6.41. Given that Φ and (∇ T ∙ ∇ T) can be represented in terms of nodal velocities and temperatures, l
Φj =
∑
l
C su j ,kU k +
k =1
∑C
V
sv j ,k k
(6.42)
k =1
l
(∇T ⋅ ∇T ) j =
∑C
T
TT j ,k k
(6.43)
k =1
where the coefficients are determined through appropriate discretization of velocity and temperature gradients. The required amount of viscosity can then be determined by (µ) j =
( - P s ) j Den j
(6.44)
where Den j =
cp Pr
l
∑ k =1
2 C TT j ,k Tk / T j +
l
l
∑
C ej ,skuU k +
k =1
∑ k =1
C esv j ,k Vk / T j
(6.45)
where l = 4, 2, and 1 for interior, boundary, and corner control volumes, respectively. This procedure attempts to overcome entropy destruction, but does guarantee that the Second Law is satisfied in a single iteration of the corrective procedure. The previous method has not rigorously proven that the entropy-corrected viscosity will ensure compliance with the Second Law. Multiple iterations might be needed to ensure sufficient numerical diffusion. As a result, the numerical viscosity, μe, with a single iteration is not always sufficient to completely remove nonphysical solution behavior, so
µ e ← cm µ e
(6.46)
can be used as an alternative to accelerate iterations, where cm is a correction factor. It typically has the range 1 < cm < 10. Because the numerical solution with cm = 1 does not guarantee local satisfaction of the Second Law, cm > 1 may be required. Past studies have indicated that cm ≈ 1 for subsonic and transonic flows, 1 < cm < 5 for supersonic flows, and 1 < cm < 10 for hypersonic flows (Naterer and Schneider, 1994). Equation 6.44 provides a distribution of viscosity, as well as the conductivity through the Prandtl number, for the correction iteration. However, because these distributions could be digital in nature, due to the elimination of all positive values of P s, it is possible that the distribution could lead to unexpected results. Entropy production alone, caused by local irreversibilities in the flow field or spatial
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differencing or both, may occur in regions of nonphysical solution behavior. But in this case, there is no conclusive evidence that the solution is not possible. Also, in the presence of rapid motion, such as fast shock propagation, it is also possible that the detection-to-application delay may cause the addition of entropy production to “miss its target.” For these reasons, the formulation can further “diffuse” the viscosity distribution calculated from Equation 6.44 through Jacobi iterations using a diffusion operator. The maximum value of the calculated viscosity (after diffusion) is returned to its prediffusion value. In this way, the magnitude of the required viscosity is retained, while the distribution becomes “smoother.” Once the viscosity distribution has been smoothed, the control surface diffusive flows are reevaluated using the shape functions. The discrete Navier–Stokes equations are then resolved with these entropy diffusion terms to correct the nonphysical solution behavior.
6.3.2 Ca se St u d y
of
Sh o ck Ca pt u r in g
in a
Sh o ck Tu be
The method of entropy-based correction with a numerical viscosity (developed in the previous section) will be applied to a shock tube problem in this case study. Consider shock tube flow with initial conditions illustrated in Figure 6.8. The problem involves a 1-m-long duct containing air (assumed perfect gas) that is initially at rest and divided by a diaphragm into a high pressure region (1032 kPa) and a low pressure region (101.3 kPa). The diaphragm is located at x = 0.5 m. A finite element method is used with grid Courant numbers (a ∆ t/∆ x), in the low pressure region varying between 0.24 and 0.07. The boundary conditions are given by zero normal velocity and zero tangential stress at the walls and ends of the shock tube. The conservation of mass and energy equations are completed at the boundaries by using nodal representations of the required boundary surface flows. In Figure 6.9, the predicted results with a control volume-based finite element method (Naterer and Schneider, 1994) indicate that numerical oscillations develop at the initial interface and shock front. The prediction of the rarefaction waves, shock speed, and positioning, as well as the shock resolution, is shown. The dip in the pressure solution, which occurs at the original pressure interface, can be removed through an artificial viscosity stability term of the form q = α | ∇v |
(6.47)
where a = -r(cLL)c and c is the local speed of sound. Also, L is a local characteristic mesh length scale. The above term was added to the momentum equations, and it represents an artificial diffusion term. The results of this addition are shown in 1m High-Pressure Gas 4 1032 kPa
Low-Pressure Gas 101.3 kPa
1 Air
Diaphragm
Fig u r e 6.8 Schematic of shock tube problem.
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Legend c1 – 0.0 c1 – 1.0 c1 – 3.0
10.0
Pressure (Pa)
8.0 6.0 4.0 2.0 0.0
0.0
0.2
0.4 0.6 Distance (m)
0.8
1.0
Fig u r e 6.9 Pressure profiles at varying cl coefficients.
Figure 6.9 for three different values of cL: 0.0, 1.0, and 3.0. This stability damping yields good accuracy at the original high-low pressure interface. However, the numerical oscillations at the shock front are not diminished by this stability damping. To remove the over- and undershoots at the shock wave, the method of entropycorrection of the numerical viscosity from the previous section was used. The viscosity distribution predicted by the Second Law formulation is shown in Figure 6.10. It provides a highly localized viscosity. Six Jacobi iterations were used in the viscosity smoothing operations. The results indicate that the computed entropy production 5.0 Legend cm – 2.0 cm – 5.0
Artificial Viscosity (kg/ms)
4.0 3.0 2.0 1.0 0.0 –1.0
0.0
0.2
0.4 0.6 Distance (m)
0.8
1.0
Fig u r e 6.10 Artificial viscosity distribution.
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500.0
Velocity (m/s)
400.0 300.0 200.0 100.0 0.0 –100.0
0.0
0.2
0.4 0.6 Distance (m)
0.8
1.0
Fig u r e 6.11 Predicted velocity profiles at varying cm coefficients.
provides a useful error indicator, which can be used in a corrective manner to improve shock capturing in compressible flows. The results of the applied viscosity field are shown in Figure 6.11 for values of cm = 0.0, 2.0, and 5.0. For cm = 3.0, the over- and undershoots are significantly diminished. The distribution is determined entirely from Second Law considerations, and there is little or no smearing of the shock front.
6.4 En t r o py Dif f er en c e in Res id u a l Er r o r In d ica t o r s 6.4.1 Fo r mu l a t io n
o f Aver a g e En t r o py Differ en ce
The previous section has shown that negative entropy production can provide a useful error indicator for fluid flow simulations. More generally, fluid entropy differences (averaged over the computational domain) can provide a general measure of residual “error” in fluid flow simulations. In many ways, this measure or metric has key advantages over other conventional methods. When solving the equations of fluid flow with a numerical technique, one would like to know when the solution has reached a steady state, presumably, the correct solution. A numerical solution of a steady-state problem is converged when further calculations will have little or no effect on the flow-field results. Whether the converged solution is correct is another issue. For any given flow variable, like mass density, momentum, or energy, versus time or iteration number, the solution convergence is indicated by a flat line after some iterations. In principle, for any flow variable x, the limit
lim
n→∞
∂ξ =0 ∂n
(6.48)
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indicates convergence where n refers to the iteration number. This limit indicates a change in the flow variable with each iteration or time step. In numerical simulations, the algorithm is limited to a finite number of iterations. This requires a “residual” that effectively represents the finite version of a tangent slope given by Equation 6.48. A residual is any nonnegative indicator of changes in the solution with time (or iteration). A finite difference representation of Equation 6.48 is
ξ n+1 - ξ n = ξ n+1 - ξ n n +1- n
(6.49)
ρ ρ ln ρo ρo
(6.50)
A residual may be defined by Equation 6.49. Any flow or solution variable can be “representative” of the overall solution. Consider the following definitions that are indicative of particular features of the flow field. Mass Density:
ρ* = -
The density r at a reference state is given at some standard conditions (for air, stano dard sea-level conditions can be used). A corresponding reference temperature and pressure To, Po are also used. Kinetic Energy: kε * = -
ρ ε ρo RTo
(6.51)
In this equation, ε = 12 ∑ uk2 , where uk are the velocity components and R is the gas constant. Internal Energy: ie* =
ρ T ln ρo To
(6.52)
S pecific Entropy: P P 0 s - s0 = Cv ln γ ( ρ ρ0 )
(6.53)
This equation holds for an ideal gas (Sonntag and van Wylen, 1982). Using the ideal gas law, P = rRT, together with the definition of the ratio of specific heats, g = Cp /Cv, the following nondimensional formula for the entropy is obtained:
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N ondimensional Entropy: S≡
ρij s - so ρ 1 ρij T ln - ln . = ρo ρo γ - 1 To ρo R
(6.54)
Mathematically, residuals are abstract measures of a “distance” between elements in an abstract space. The abstract space represented in fluid flow simulations involves the flow variables, which are typically mass, momentum, and energy. The residual is a measure of the distance between the flow variables, at some point in a calculation, to their steady-state values. In computational fluid dynamics, it is common to either take the maximum value of this distance function, or take its average over the computational domain. For a two-dimensional domain with grid cells of variable dimensions, the average can be expressed symbolically by the following operation: 〈ξ 〉 =
∑ ∆x ∆y ξ ∑ ∆x ∆y i
j ij
i
j
Average Mass Density Difference:
RESρ ≡ 〈|ρ n +1 - ρ n|〉
(6.55)
(6.56)
The 〈 〉 operator refers to averaging over the computational domain. RMS Mass Density Difference: RESρ 2 ≡
〈( ρ n +1 - ρ n )2 〉
(6.57)
RES p ≡
〈( p n +1 - p n )2 〉
(6.58)
RMS Pressure Difference:
A verage Entropy Difference:
RES∆S ≡ 〈|S n +1 - S n|〉
(6.59)
Other variations are also possible, but Equation 6.56 through Equation 6.58 represent the most common examples (Anderson, 1984). In the next section, sample results of the entropy-based residual error will be investigated for a specific case study.
6.4.2 Ca se St u d y
o f Er r o r In d ica t o r s in
Su per so n ic Fl o w
This section presents a case study that uses an entropy-based residual as an error indicator for compressible flow simulations. The explicit numerical scheme solves the Euler equations with a technique described by Camberos (1995). For a nozzle
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M∞ = 2.0
x 10°
Fig u r e 6.12 Flow-field contours of constant density for two-dimensional supersonic wedge flow.
flow problem, an implicit scheme is used with a Gauss–Seidel line relaxation technique for the thin-layer Navier–Stokes equations (MacCormack, 1985). This section focuses on how flow variables and residuals change with each iteration. The first example represents supersonic flow of an ideal gas over a two-dimensional wedge. The configurations and contours of constant density are shown in Figure 6.12. Because the method is explicit and first-order accurate, the oblique shock wave is quite thick, but oriented at the correct location, as predicted by theoretical gas dynamics. In Figure 6.13, the iteration history for the representative flow Entropy-Based Norm
0.2
Representative Flow Variables
Kinetic Energy Mass Density
0.1
0.0
–0.1
Internal Energy
–0.2 Entropy Difference –0.3
250
500 n
750
1000
Fig u r e 6.13 Iteration history for representative flow variables: two-dimensional supersonic wedge flow.
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(S)n+1_(S)n (∆norm) (|∆p|) (∆p2)
10–3 10–5
(∆p2)
Residual
10–7 10–9
10–11 10–13 10–15 250
500 n
750
1000
Fig u r e 6.14 Iteration history for residual metrics: two-dimensional supersonic wedge flow.
variables is shown. Note that a flat line is evident after about 300 time iterations (or time steps). According to this indicator, the solution is essentially converged. In addition, visible changes in the flow-field solution (not shown) represented in Figure 6.12 are no longer evident. From Figure 6.14, it appears that after 300 iterations, the residual metric based on an average mass density difference has dropped about two orders of magnitude. This is true for other metrics as well, although they are about one order of magnitude less (down to 10-4, compared with 10-3 for density). The sudden drop in the residual that appears at around n = 350 is a spurious but benign result after the oblique shock wave reaches a stable location. The initial conditions were uniform incoming flow at a 10-degree flow angle toward a solid surface, so the oblique shock appears at the leading edge. It gradually propagates through the grid to its final location. Figure 6.14 shows that nearly machine zero is reached after about 750 iterations. A large number of iterations is typical of explicit numerical solutions to steady-state fluid flow problems. Contours of constant Mach number are shown in Figure 6.15 for supersonic flow over a convex corner, which leads to a centered Prandtl-Meyer expansion fan. The numerical method is explicit and first-order accurate, so the expansion fan is quite thick, but oriented at the correct location, as predicted by theoretical gas dynamics. Figure 6.16 shows the iteration history for the representative flow variables. A flat line for all of the variables is evident after about 250 iterations. The solution (from this indicator) is essentially converged. Visible changes in the flow field solution (as represented in Figure 6.15) are no longer evident. From Figure 6.16, not all of the flow variables have reached steady state. In particular, the line representing the kinetic
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M = 3.55
Fig u r e 6.15 Flow-field contours of constant Mach number: two-dimensional Prandtl– Meyer expansion.
energy appears to flatten out at around 100 iterations. This indicates that one should choose a representative flow variable that takes into account all of the state variables to guarantee that convergence has been reached. From Figure 6.17 at 250 iterations, the residual based on the average mass density difference has dropped about two orders of magnitude. This also holds for other residuals as well, although their values are about one order of magnitude less (down to 10-4 compared with 10-3 for the density). Figure 6.17 also shows that machine zero is reached after about 400 iterations, which is again typical of explicit numerical solutions for steady-state fluid flow problems. Supersonic flow over blunt bodies involves the formation of bow shock waves. It provides the challenge of “shock capturing” when the equations of gas dynamics
0.3 Entropy Difference
Representative Flow Variables
0.2
0.1
Internal Energy Entropy-Based Norm
0.0
Mass Density
–0.1
–0.2
–0.3
Kinetic Energy 250
500 n
750
1000
Fig u r e 6.16 Iteration history for representative flow variable for two-dimensional Prandtl– Meyer expansion.
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(S)n+1_(S)n K0||qn_qave||2 |∆|MAX (∆p2)
10–3 10–5
(∆p2)
Residual
10–7 10–9
10–11 10–13 10–15 250
500 n
750
1000
Fig u r e 6.17 Residual error metrics for two-dimensional Prandtl–Meyer expansion.
are solved in conservation law form. Because the method used in these examples is first-order accurate and explicit, the bow shock wave is thick, but located approximately at the correct location as expected from theoretical gas dynamics. The pressure jump obtained by the numerical solution, from the leading edge of the shock wave to the stagnation point at the square block, is within 5% of the predicted value from normal shock wave theory. Contours of constant Mach number are shown in Figure 6.18, where the blunt body is the square block shown in black.
M=3
Fig u r e 6.18 Mach contours for two-dimensional supersonic flow.
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Entropy-Based Design and Analysis of Fluids Engineering Systems 0.10 Kinetic Energy
0.8
Entropy-Based Norm
Representative Flow Variables
0.6 0.4
Mass Density
0.2 0 –0.2 Internal Energy
–0.4 –0.6 –0.8
Entropy Difference 0
500
n
1500
1000
Fig u r e 6.19 Iteration history of flow variables.
Figure 6.19 shows the iteration history for the representative flow variables of mass density, kinetic energy, internal energy, and entropy difference. Note that a flat line is evident after about 600 iterations. The solution (based on this indicator) is essentially converged. In addition, visible changes in the flow-field solution (represented in Figure 6.18) are no longer evident. From Figure 6.20, at 600 iterations, the residual 10–1
(s)n+1–(s)n
–3
Residual
10
K0||qn–qave||2
10–5
|(Δp)|
10–7
(Δp2)
(Δp2)
10–9
10–11 10–13 10–15
1000
2000 n
3000
4000
Fig u r e 6.20 Iteration history of flow variables.
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(S)n+1_(S)n K0||qn_qave||2 |(∆p)| (∆p2) (∆p2)
10–2 10–4
Residual
10–6 10–8
10–10 10–12 10–14 10–16
250
500 n
750
1000
Fig u r e 6.21 Residual metrics for 1/10 size Blunt body.
based on the average mass density difference has dropped about three or four orders of magnitude. Similar results are obtained for other residuals. Figure 6.20 shows that machine zero is essentially reached after about 3400 iterations. For the blunt body problem, a grid size of 100 × 100 cells was used, with the blunt body occupying 10 × 10 grid cells. By reducing the size of the body to a single grid cell, and thereby moving the outflow boundaries farther from the simulated solid surface, the number of iterations required to reach machine zero is reduced to 700 (Figure 6.21). Compared with 3400 shown in Figure 6.20, this is a significant decrease. In Figure 6.22, two-dimensional flow through a converging-diverging nozzle is examined. The thin-layer Navier–Stokes equations are solved with Gauss–Seidel line relaxation and an implicit technique described by MacCormack (1985). In Figure 6.22, the lower portion of the nozzle contour is shown, with the centerline indicated at the upper portion of the figure. The inlet conditions are subsonic. The velocity flow field is shown in Figure 6.22. For this problem, the solution is approximately second-order accurate, and the line relaxation technique is implicit. In Figure 6.23, only the residual metric based on the average entropy difference is shown. For this case, there are two methods for imposing the wall-boundary condition. One method is specifying the normal flux term equal to the pressure at the wall. The other method imposes a flux-splitting technique, with a layer of cells adjacent to the wall to create a layer of ghost cells with the normal velocity component reflected. As shown in Figure 6.23, the two techniques have different convergence histories. Machine zero is reached after about 500 iterations, but the solution appears to converge much earlier, at about 50 to 70 iterations. No major difference between the techniques for imposing the wall-boundary condition was observed.
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0.00
–0.02
–0.04
–0.06
–0.08
–0.10
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Fig u r e 6.22 Converging–diverging nozzle flow implicit of thin-layer, Navier–Stokes equations (TL-NSE) solution.
10–2
Entropy Residual
10–4 10–6
Flux = Pressure at Wall
10–8 10–10 10–12 Flux-Splitting at Wall
10–14 10–16
0
100
200
n
300
400
500
Fig u r e 6.23 Residual error metrics for two-dimensional thin-layer Navier–Stokes implicit solution.
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It is beneficial to reduce the number of iterations without compromising solution quality. A possible method would monitor the residual by calculating the difference in the tangent slope for entropy, as it changes with each iteration. This can be accomplished once the residual has dropped two or three orders of magnitude from its initial value, to ensure that the iteration curve for the representative variable has flattened out sufficiently. An automatic procedure could be developed to reduce user input and monitoring of a fluid flow calculation that requires many iterations. The results indicate that entropy can provide several advantages over other conventional methods for characterizing solution residuals. Fluid entropy, S = S(Q), is functionally dependent on all of the fluid state variables, Q = (q1, q2 , q3 , q4 , q5), namely, mass density, momentum density, and total energy density. Also, entropy has a physical significance embodied by the Second Law of Thermodynamics, that requires nonnegative entropy production.
R ef er en c es Anderson, D.A., Tannehill, J.C., and R.H. Pletcher. 1984. Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, New York. Camberos, J.A. 1995. A Probabilistic Approach to the Computational Simulation of Gas Dynamic Processes. Ph.D. thesis. Stanford University, Stanford, CA. Camberos, J.A. 1998. Calculation of residual error in explicit and implicit fluid flow simulations based on generalized entropy concept. Proceedings of the Joint American Society of Mechanical Engineers/Japan Society of Mechanical Engineers Meeting (PVP377-2). San Diego, CA, 279–286. Fox, R.W. and A.T. McDonald. 1992. Introduction to Fluid Mechanics. 4th ed. John Wiley & Sons, New York, 123–124. Harten, A. 1983. On the symmetric form of systems of conservation laws with entropy. J. Computational Phys., 49: 151–164. Huebner, K. and E. Thornton. 1991. The Finite Element Method for Engineers. 2nd ed. John Wiley & Sons, Toronto, Canada. Hughes, T.J.R., Franca, L.P., and M. Mallet. 1986. A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier– Stokes equations and the Second Law of Thermodynamics. Computer Methods Appl. Mechanics Eng., 54: 223–234. Karki, K.C. and S.V. Patankar. 1989. Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J., 27(9): 1167–1174. Lax, P.D. 1971. Shock Waves and Entropy. Contributions to Non-Linear Functional Analysis. Academic Press, New York, 603–634. Leonard, B.P. 1979. A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Computer Methods Appl. Mechanics Eng., 19: 59–98. Lohner, R., Morgan, K., and O.C. Zienkiewicz. 1984. The solution of the non-linear hyperbolic equation systems by the finite element method. Int. J. Numerical Methods Fluids, 4: 1043–1063. MacCormack, R.W. 1985. Current Status of Numerical Solutions of the Navier-Stokes Equations. AIAA Paper 85-0032. 23rd AIAA Aerospace Sciences Meeting. Reno, NV. MacCormack, R.W. and B.S. Baldwin. 1975. A Numerical Method for Solving the Navier– Stokes Equations with Application to Shock–Boundary-Layer Interactions. AIAA Paper 75-1, 13th AIAA Aerospace Sciences Meeting. Pasadena, CA. Majda, A. and S. Osher. 1979. Numerical viscosity and the entropy condition. Commun. Pure Appl. Math., 32: 797–838.
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Merriam, M.L. 1987. Smoothing and the second law. Computer Methods Appl. Mechanics Eng., 64: 177–193. Merriam, M.L. 1988. An Entropy Based Approach to Nonlinear Stability. Ph.D. thesis. Stanford University, Stanford, CA. Minkowycz, W. J., Sparrow, E. M., Schneider, G. E., and R. H. Pletcher. 1988. Handbook of Numerical Heat Transfer. John Wiley & Sons, New York, 252–253. Naterer, G.F. 1999. Constructing an entropy-stable upwind scheme for compressible fluid flow computations. AIAA J., 37(3): 303–312. Naterer, G.F. and G.E. Schneider. 1994. Use of the second law for artificial dissipation in compressible flow discrete analysis. AIAA J. Thermophysics Heat Transfer, 8(3): 500–506. Nellis, G.F. and J.L. Smith. 1997. Entropy-based correction of finite difference predictions. Numerical Heat Transfer B, 31(2): 177–194. Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere, New York, 30–31. Pulliam, T.H. and J.L. Steger. 1980. Implicit finite difference simulations of three dimensional compressible flow. AIAA J., 18(2): 159–167. Rosen, M.A. and I. Dincer. 1999. Exergy analysis of waste emissions. Int. J. Energy Res., 23(13): 1153–1163. Schneider, G.E. and M.J. Raw. 1987. Control-volume finite-element method for heat transfer and fluid flow using co-located variables. Part 1. Computational procedure. Numerical Heat Transfer, 11(4): 363–390. Sonntag, R.E. and G. van Wylen. 1982. Introduction to Thermodynamics. John Wiley & Sons, New York. Von Neumann, J. and R.D. Richtmyer. 1950. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 21: 232–237.
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Numerical Stability and the Second Law
7.1 In t r o d u c t io n The classic paper by Courant et al. (1928) marked a milestone for numerical analysis. The archived literature often refers to the “CFL” (Courant, Friedrichs, Lewy) condition as a criterion that restricts the time step for linear differential equations, to achieve numerical stability. However, the CFL condition originally was not related to numerical stability, because that term was not phrased until the 1940s by a group associated with John von Neumann. Nevertheless, the terminology remained, and today we understand the CFL condition as a necessary, and in some cases sufficient, condition for both numerical stability and convergence of nonlinear equations. The basic question of numerical stability deals with discretization error and round-off error. Discretization errors are analogous to systematic errors that arise in experimental measurements, whereas round-off errors are analogous to the unpredictable and unavoidable errors that occur in a measurement process itself. Minimizing discretization errors requires very accurate approximations of the differential equations. Round-off errors have a significant impact on the stability of a numerical method. Numerical stability deals with the growth of an overall roundoff error. The growth of a single round-off error is a question most frequently studied because it can be answered more easily and practically than the overall error (Anderson et al., 1984). In the modern use of computational fluid dynamics (CFD) codes, heuristic arguments and rules of thumb are often used to establish a restriction on the time step for explicit methods and time-accurate solutions. Unfortunately, ad hoc trial and error are often needed to determine a method’s stability bounds. This chapter examines how the logic intrinsic to the Second Law of Thermodynamics can be used to establish stability of a numerical algorithm. Pioneering numerical analysis of the Second Law provided mathematical constraints that determine physically relevant solutions to the differential equations. However, these equations may exhibit nonunique and discontinuous solutions (Oleinik, 1957, 1959). Expanding the essence of the Second Law will show that an entropy-based alternative to linear stability analysis exists. Due to the universality of concepts associated with the Second Law, an entropy-based method can be applied to any of the governing differential equations of thermal and fluid dynamics. This chapter will address the question of strong numerical stability, by extending the “modified equation” technique of Warming and Hyett (1974) and others. A modified equation for the balance of entropy will provide a powerful method for gauging a numerical method’s stability
175
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properties, because numerical stability is directly related to the overall generation of entropy (Camberos, 1995). In addition, the Second Law provides a way to gauge the local quality of the solution. For example, spurious oscillations in the numerical solution often arise under conditions of entropy destruction (Merriam, 1989). Transient numerical solutions of the equations of fluid dynamics often require a time-step constraint. The constraint can be reduced to an inequality relating the time step, grid spacing, and some reference wave velocity. Historically, the technical literature in numerical analysis refers to this parametric cluster as the “Courant number” and the condition for the linear case as the “CFL condition.” Classically, numerical analysis relies on linearization and von Neumann’s use of a Fourier series to derive the CFL condition. In contrast, this chapter will use the Second Law to impose a restriction on the time step, for linear and nonlinear equations, as well as systems of equations like the equations of gas dynamics. By transforming the truncation error for the governing equation into an equation representing the balance of entropy, one can obtain an inequality that restricts the time step to satisfy the Second Law in a weak sense. In this chapter, the Second Law will be applied to the linear advection equation, then a nonlinear equation, and finally a system of equations representing the one-dimensional equations of gas dynamics. It will be shown that the results agree with the classical approach for linear equations, but they differ for others, thereby showing that the Second Law has valuable utility in numerical analysis beyond its role in thermodynamics. This chapter will develop entropy criteria for explicit numerical algorithms with truncation errors. Generalized results will be established for implicit and higher-order methods, due to the universality of the Second Law and the concept of entropy.
7.2 St a bil it y No r ms Nonlinear stability can be analyzed in the context of the Second Law, whereby the difference equation must satisfy a global form of the entropy balance equation to guarantee numerical stability. Stability requires that the solution remain bounded in some norm, meaning that this norm either decreases or remains constant for the duration of the calculation. Suppose that the initial value problem for a set of conservation laws has a unique equilibrium solution, q, defined as the average value of the state variables over the computational domain. For a closed domain, this average state remains constant, and it can be computed from the initial conditions. Statistical thermodynamic analysis can be used to establish that the average state can be represented by an invariant probability distribution for the state variables. In equilibrium, this probability density is constant. Define the following averaging operator as the volume integral over the computational domain, Ω. The average state of an arbitrary variable, x, that establishes equilibrium is
ξ= ξ ≡
∫
Ω
ξ dV /
∫
Ω
dV
(7.1)
Concavity of entropy implies a balance of entropy containing a term indicative of nonnegative entropy generation. For an inviscid adiabatic fluid flow, the Second
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Law can be used to prescribe local restrictions on the time step for explicit numerical simulations that give physically relevant solutions. Globally, the Second Law leads to the existence of some probability measure intrinsic to the distribution of state variables in their proper solution space. In this regard, the Second Law can be expressed in terms of the vector of conserved quantities, q, as follows,
- ln g( q ) = S ( q ) - S ( q ) - S,q ⋅( q - q ) ≥ 0
(7.2)
This principle may be interpreted as an expression for the existence of a thermodynamic equilibrium state, where entropy is a maximum. Statistically, this is also a statement about a probability distribution, for which the average state is associated with a maximum probability. Equation 7.2 indicates the existence of some equilibrium distribution of the state variables, which maximizes a probability distribution for those variables. The concavity property of entropy guarantees that the g(q) functional is a probability distribution with a maximum probability that corresponds to the equilibrium state. This is the essence of the Second Law, in a form known as Gauss’s principle or Gauss’s law of error (Lavenda, 1991). The third term in Equation 7.2 vanishes on integration, due to the definition of the average state q and the conservation of state variables. As a result,
S(q ) - S(q) ≥ 0
(7.3)
This expresses the principle of nondecreasing entropy, for an isolated thermodynamic system. Because the state of equilibrium corresponds to the state of maximum entropy,
S(q ) ≥ S(q)
t = t2
≥ S(q)
t = t1
(7.4)
for t2 > t1. This result provides an upper bound to a state metric or norm, which is a direct measure of the mean-square variation in the state variables over the domain. An approach to the construction of a suitable norm is to begin with a series expansion of the local entropy about the equilibrium state, when variations in the state variables are very small, that is, | ξ - ξ |<< 1. A Taylor series expansion, up to the second order, is given by
S ( q ) ≈ S ( q ) + S ,q ⋅ ( q - q ) +
1 ( q - q )T ⋅ S,qq ⋅ ( q - q ) 2
(7.5)
where the row vector of first derivatives S,q and the Hessian S,qq are both evaluated at the average state. The determinant for the Hessian is det [S,qq ] = -
(γ - 1) p3
(7.6) where p is the pressure. The Hessian constitutes a quadratic form that is negative definite and nonsingular when g > 0 and P > 0.
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In the small fluctuation limit, the probability measure for a distribution of state variables in the global domain is approximately normal. It can be shown that the functional form, combining previous equations, becomes g( q ) ≈
- | S,qq | ( 2π )
3
exp
{
}
1 ( q - q )T ⋅ S,q q ⋅ ( q - q ) 2
where | S,qq | indicates the determinant. The negative inverse of the Hessian S,qq equals the covariance matrix for the state variables q. The covariance matrix determines the statistical dependence or independence of two or more variables. In particular, it is a useful measure of mean-square variations in a data set, such as the calculated values of q from a numerical solution. Mathematically, this leads to the following definition of a norm for q in terms of a domain integral as follows: q
2
≡ - q T ⋅ S ,q q ⋅ q
(7.7)
The mean-square variations in the solution will be expressed relative to equilibrium conditions. Given the norm defined by the previous equation, write a distance functional as follows: q-q
2
≡ - ( q - q )T ⋅ S,qq ⋅ ( q - q )
(7.8)
Expanding the right side and simplifying, q-q
2
= - qT ⋅ S,qq ⋅ q + q T ⋅ S,qq ⋅ q = q
2
- q
2
(7.9)
An alternative formulation of the Second Law that explicitly relates the state of a thermodynamic system to its state at equilibrium introduces the concept of exergy or availability (Camberos, 2000a,b,c). This thermodynamic function represents an abstract functional that quantifies the thermodynamic distance from the state of equilibrium. Exergy, for the present analysis, is defined as
X ≡ S ( q ) - S ( q ) - S ,q ⋅ ( q - q )
(7.10)
The concavity of entropy then translates into a convexity condition on exergy, such that
X ( q ) - X ( q ) - X ,q ⋅ ( q - q ) ≤ 0
(7.11)
X (q ) - X (q) ≤ 0
(7.12)
After averaging,
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Subsequently for an isolated system, the principle of monotonic exergy decrease becomes
X (q ) ≤ X (q)
t = t2
≤ X (q)
t = t1
(7.13)
for t2 > t1. This is consistent with the Second Law as a statement that minimum exergy determines the state of equilibrium. From the definition of the distance functional and the near-equilibrium conditions, 2 X (q) ≈ q - q
2
(7.14)
which can be expressed as 2 X (q) + q
2
≈ q
2
(7.15)
For a bounded solution, q
t = t2
≤ q
t = t1
,
(7.16)
which is a statement of stability, given t2 > t1. The inequality may be enforced by postulating the existence of a scaling constant K0, determined from initial conditions, such that 2 X ( q) + 〈 q 〉2 ≥ K 0 〈 q〉2
(7.17)
The scaling constant may be calculated as K0 =
2 〈 X ( q0 )〉 + 〈 q 〉2 〈 q0 〉2
(7.18)
where q0 = q (t = t0). Because exergy is a maximum at the initial state and nonincreasing thereafter, it can be concluded that
2 X0 + q
2
≥ K0 q
2
(7.19)
implying a uniform bound on the mean-square variation in the data for all time, t > t0. In general, mean-square variations will fluctuate, but they cannot fluctuate beyond the upper bound in the previous equation, when a numerical scheme satisfies the Second Law. As a result, satisfaction of the Second Law of thermodynamics provides a sufficient condition for nonlinear numerical stability.
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7.3 En t r o py St a bil it y o f Fin it e Dif f er en c e Sc h emes 7.3.1 Lin ea r Sc a la r Ad v ec t io n The previous section used concavity properties of entropy to establish general stability criteria, without referring to discrete parameters obtained from a numerical solution of the governing equations. In this section, specific stability criteria will be derived in terms of the discretization parameters (time step and grid spacing). Consider the following linear equation for one-dimensional scalar advection: ∂η ∂f + = 0 ∂t ∂x
(7.20)
where f(h) = ch represents the advection of a conserved scalar quantity, h, at a constant wave speed, c. Mathematically, one can readily obtain the solution to this equation, given initial and boundary conditions. For this problem, there exists an “entropy” S(h) and an “entropy flux” F(h) such that S gen =
∂S ∂F + ∂t ∂x
(7.21)
which represents a balance of entropy. The “entropy” and the “entropy flux” are constructed for the dependent variable h by first postulating that there exists a functional S(h) such that S" < 0, which is a sufficient condition for concavity. Integration by parts yields the following necessary condition, Gauss’s principle (Lavenda, 1991):
S (η ) - S (η ) - S ′(η )(η - η ) ≥ 0
(7.22)
The “maximum entropy” formalism of statistical optimization (Jaynes, 1991; Kapur and Kesavan, 1992), combined with the property of concavity given in Equation 7.22, results in an “entropy” that is a logarithmic function of the dependent variable:
S(η ) = η ln(η/η )
(7.23)
Other possibilities exist, like using -η 2 or η . The important functional relation is given by a sufficient condition for concavity. Using the compatibility condition for the “entropy flux” and applying the chain rule gives
F (η ) = cS (η )
(7.24)
Setting η = 1 incurs no loss in generality. With the entropy pair defined, let us proceed to analyze discrete formulae for approximating solutions to the differential equation. A control volume formulation equates the value of the dependent variable fluxes with the change of the integral
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average over the cell the volume. Mathematically, the “weak” or integral solutions may admit discontinuities that can develop for nonlinear equations, even from smooth initial data. Weak solutions may lose their uniqueness; therefore some additional criteria must pick out the solution that best reflects what would be considered physically relevant or observable. It is known that the “entropy condition” (compliance with the Second Law) picks out the correct (physically relevant) weak solution (Lax, 1954, 1973; Oleinik, 1959). Using a standard, one-dimensional explicit discretization of the scalar advection equation,
η nj +1 = C +η nj -1 + C 0η nj + C -η nj +1
(7.25)
This represents an integral approximation of the hyperbolic conservation law in Equation 7.20, based on a three-point numerical stencil. As written in this format, it contains a set of coefficients {C+, C0, C-} that must satisfy certain conditions. First, for the numerical solution to approximate the conservation law, the sum of the transition coefficients must equal unity:
C + + C 0 + C - = 1
(7.26)
This must hold true for a single nonlinear equation and systems of equations, also. Second, the numerical solution must approximate the mathematical behavior of the original differential equation, called consistency (Anderson et al., 1984). This leads to the following second requirement:
∆x + (C - C - ) η = f (η ) ∆t
(7.27)
A third “monotonic” condition (Crandall and Majda, 1980), although not necessary, may suffice for numerical stability:
1 ≥ {C + , C 0 , C - } ≥ 0
(7.28)
The transition coefficients C± may or may not satisfy monotonic criteria. But the coefficient C0 must satisfy it; otherwise the numerical solution will systematically violate the conservation principle. For linear equations, the CFL condition also implies satisfaction of the monotonic requirement in some cases. However, for nonlinear equations, monotonicity can lead to over- or underpermissive time-step limitations. Solutions obtained with a time step close to the limit imposed by the monotonic requirement may be stable, but also erroneous. To expand the error incurred by the discrete numerical formula, assume a uniform, equally spaced mesh with an exact solution available at time t, so that
η nj +1 = C +η(t, x - ∆x ) + C 0η(t, x ) + C -η(t, x + ∆x )
(7.29)
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The required Taylor series expansions are
η(t, x ± ∆x ) = η(t, x ) ± ∆x
∂η ∆x 2 ∂2η ∆x 3 ∂3η + + ± ∂x 2 ∂x 2 6 ∂x 3
(7.30)
∂η ∆t 2 ∂2η ∆t 3 ∂3η + + + ∂t 2 ∂t 2 6 ∂t 3
(7.31)
for the spatial terms and
η(t + ∆t, x ) = η(t, x ) + ∆t
for the temporal terms. Collecting terms gives
η ∂η ∆x + ∂η L∆ = (1 - C + - C 0 - C - ) + (C - C - ) + ∆ t t t ∂ ∆ ∂x (7.32) ∂2η ∆t ∂2η ∆x 3 + ∆x 2 + ∂3η ∆t 2 ∂3η (C + C ) 2 + (C - C ) 3 + + + 2 ∆t 2 ∂t 2 6 ∆t ∂x 6 ∂t 3 ∂x Substituting Equation 7.26 and Equation 7.27, the conservation and consistency conditions, reduce the expression to L∆ =
∂η ∂f ∆x 2 + ∂2η ∆t ∂2η (C + C - ) 2 + + 2 ∂t 2 ∂t ∂x 2 ∆t ∂x ∆x 3 + ∂3η ∆t 2 ∂3η (C - C - ) 3 + + + 6 ∆t 6 ∂t 3 ∂x
(7.33)
Eliminating the higher-order time derivatives using the original conservation equation, Equation 7.20, gives ∂η ∂f ∆x 2 + c 2 ∆t 2 ∂2η ∆x 3 c∆t c3 ∆t 3 ∂3η L∆ = + + (C + C - ) - 2 2 + 6 ∆t ∆x ∂t ∂x 2 ∆t ∆x 3 ∂x 3 ∆x ∂x (7.34) Truncating up to the first-order leading error by setting L∆ ≈ 0 and rearranging,
∂η ∂f ∆x 2 + c 2 ∆t 2 ∂2η + ≈ ( C + C - ) - 2 2 ∂t ∂x 2 ∆t ∆x ∂x
(7.35)
This yields a “modified equation” for the numerical method, which prescribes the transition coefficients in Equation 7.25. This technique originated with Warming and Hyett (1974). The procedure retains the transition coefficient format, so that
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any numerical method is already represented by Equation 7.34, without having to rederive the modified equation each time. Transition coefficients that give a firstorder approximation will have a smoothing effect, due to the second-order derivative on the right side of Equation 7.35. From this formula, we will define a parameter (essentially a numerical viscosity) that has a critical role in formulating the corresponding “modified equation” for the balance of entropy. The following analysis extends Warming and Hyett’s “heuristic stability analysis” by giving it a solid foundation in the Second Law. Multiplying both sides of the “modified equation” by S' (derivative of S with respect to q) and using the chain rule of differentiation, plus the compatibility condition, yields the following balance of entropy equation: ∂S ∂F ∂ 2η + ≈ εS′ 2 ∂t ∂x ∂x
(7.36)
where
ε=
∆x 2 2 ∆t
2 + c∆t ( C + C - ) - ∆x
(7.37)
defines the numerical viscosity parameter. Using the chain rule, the right-hand side of Equation 7.36 expands to 2
S′
∂2η ∂ ∂η ∂η = S ′ - S ′′ ∂x 2 ∂x ∂x ∂x
(7.38)
Integrating both sides of the entropy balance equation, using Equation 7.38 on the right side, over a small interval in space and time yields
∫∫
∫
S gendx dt ≈ ε
x
∂S 2 dt ∂x x1
∫∫
2
∂η ε S ′′ dx dt ∂x
(7.39)
After taking the limit [ x1 , x2 ] × [t1 , t2 ] → 0, the first term on the right side vanishes, but the second term will not vanish if the limits of integration contain a discontinuity. The balance of entropy for the one-dimensional advection equation will satisfy the Second Law only for nonnegative values of the parameter obtained from the leading error terms in the numerical approximation. Examining the transition coefficients for a given numerical method, one can establish the corresponding relation for the time step, grid size, and wave velocity. Table 7.1 contains the transition coefficients for several well-known finite differencing methods for Equation 7.25, which approximates the solution to the advection equation, Equation 7.20. Table 7.2 presents the results of the numerical viscosity parameter, Equation 7.37, given the transition coefficients in Table 7.1. The constraint
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T a bl e 7.1 T ransition Coefficients for Various Numerical Schemes for Linear Scalar A dvection T ransition Coefficients C+
C 0
C-
- (| ν | - ν)
1+ | ν |
- (| ν | + ν)
Method Downwind
1 2
Central Upwind
1 2
Lax–Friedrichs Lax–Wendroff Note: ν
ν
1
(| ν | + ν)
1- | ν |
(1 + ν)
0
(1 + ν) ν
1 - ν2
1 2
1 2
1 2
1 2
- 12 ν 1 2
(| ν | - ν) 1 2
(1 - ν)
- 12 (1 - ν) ν
≡ c∆t/∆x .
matches the classical CFL condition where | c | ∆t / ∆x equals the Courant (or CFL) number. The “downwind” and central methods, from a Second Law perspective, destroy entropy intrinsically for any finite time step. The upwind and Lax–Friedrichs methods both approximate linear advection to first order, but the Lax–Friedrichs formula dissipates the solution more than the upwind method. Comparing the numerical viscosity parameter for these two methods explains why (for a given CFL number, like 0.9) the Lax–Friedrichs method gives 2 ε ∆ t/∆ x = 0.19, whereas the same CFL number gives a value of 0.09 for the upwind method. The Lax–Friedrichs method generates close to twice the amount of entropy as the upwind method (per time step). The last method (Lax–Wendroff) in the table contains transition coefficients that give a second-order approximation in both space and time, so the first-order numerical viscosity parameter equals zero. For these methods, the monotonic requirement on the C0 transition coefficient establishes the time-step constraint shown in the table. Inspecting the next-order leading error terms (third order) justifies the choice.
T a bl e 7.2 T ime-Step Constraints Imposed by the Second Law for One–D imensional Scalar A dvection Method
2 ε ∆ t/ ∆ x 2
C onstraint
Downwind
- |c∆|∆xt - (
)
N/A
Central
- ( c ∆t / ∆x )
N/A
Upwind Lax–Friedrichs Lax–Wendroff
|c|∆t ∆x
c∆t 2 ∆x
- ( c∆∆xt )
1- (
)
c∆t 2 ∆x
0
2
2
|c|∆t ∆x
≤1
|c|∆t ∆x
≤1
|c|∆t ∆x
≤1
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By extending the Taylor series expansion to include second- and third-order terms, it can be shown that ∂η ∂f ∂ 2η ∂3η ∂ 4η + ≈ ε 2 + ε2 3 + ε3 4 + ∂t ∂x ∂x ∂x ∂x
(7.40)
where ν = c∆t / ∆x,
ε2 = and
∆x 3 (ν - ν 3 ) 6 ∆t
(7.41)
∆x 4 [(C + + C - ) - ν 4 ] (7.42) 24 ∆t For the nth-order derivative, multiplying by S' and using the chain rule gives
ε3 =
S′
∂nη ∂ ∂n-1η ∂η ∂n-1η . = S ′ n-1 - S ′′ n ∂x ∂x ∂x ∂x ∂x n-1
(7.43)
The general expression for the balance of entropy becomes S gen =
∂ ∂η ∂2η ∂3η + ε 2 2 + ε 3 3 + S ′ ε ∂x ∂x ∂x ∂x
(7.44) ∂η ∂η ∂2η ∂3η - S ′′ + ε 2 2 + ε 3 3 + . ε ∂x ∂x ∂x ∂x The first nonzero e k indicates the order of accuracy for a given method. A higherorder numerical method may still satisfy the entropy-generation inequality in the limit, but not necessarily for finite space and time increments. Entropy generation will depend on the sign of the wave speed and the local distribution of data. According to the Warming–Hyett technique for obtaining the modified equation from the numerical formula, one should not use the original differential equation when replacing temporal derivatives with spatial ones, because a solution to the partial differential equation does not necessarily satisfy the difference equation. Using the Warming–Hyett (1974) approach to obtain the modified equation leaves the firstorder parameter e unchanged, but gives
ε2 = -
ε3 =
∆x 3 [1 - 3(C + + C - ) + 2ν 2 ] 6 ∆t
(7.45)
∆x 4 {(C + + C - - 4ν 2 ) [1 - 3 (C + + C - )] - 6ν 4 } 24 ∆t
These are second- and third-order parameters that are substituted into the leadingerror terms in Equation 7.40. These parameters differ from the results obtained by
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replacing the higher-order temporal derivatives using the original conservation law. In any case, formulae for e k do not change the conclusions reached by a Second Law analysis. The second- and third-order terms associated with the third- and fourthorder derivatives may yet satisfy the inequality in Equation 7.39. Standard Fourier stability analysis and Warming–Hyett’s technique relate these terms with dispersion and dissipation of the solution, respectively. Contrary to this approach, the analogy to the Second Law indicates that errors associated with these two terms may both destroy or generate entropy. These effects are usually attributed only to numerical dispersion or dissipation. By examining the sign of local derivatives in the second term of the right side of Equation 7.44, one may pinpoint regions where a numerical method might fail to satisfy the Second Law, thus providing a means for predicting entropy destruction and the subsequent adverse effects on solution error and numerical stability. Figure 7.1 shows sample results of the numerical approximation of linear advection with the upwind method. This solution is obtained with a CFL number of 0.5 using the updated expression in Equation 7.25 with the transition coefficients presented in Table 7.1. The figure shows the exact (initial) data advected to the right (solid line) and the numerical approximation (circles connected by lines) after 10 time steps. Numerical dissipation reduces the magnitude of the peaks and spreads the data in the spatial domain. Compare this solution with results presented for the
2.2 2 1.8 1.6 η
1.4 1.2 1 0.8
0
0.25
0.5 x
0.75
1
Fig u r e 7.1 Exact (–) and numerical (– –) solution of linear advection equation with upwind method at n = 10.
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1.4 1.2 1 0.8
0
0.25
0.5 x
0.75
1
Fig u r e 7.2 Exact (–) and numerical (– –) solution of linear advection equation with Lax–Wendroff method at n = 10.
Lax–Wendroff method in Figure 7.2. This method is accurate to second order in both space and time, but the solution quality suffers from spurious oscillations that appear near sharp discontinuities, which is often observed with higher-order techniques. Both numerical solutions satisfy the CFL condition, and they are numerically stable. An explanation for the spurious oscillations in the Lax–Wendroff method is shown in Figure 7.3, illustrates entropy destruction at the first time step, as calculated from the discrete approximation to Equation 7.21. For the linear advection equation, any method that destroys entropy locally suffers from the same defect. In Figure 7.3 and Figure 7.4, peaks in entropy generation and destruction occur at the first time step. A net destruction of entropy in the total domain leads to in numerical instabilities that cause the solution to diverge. The downwind and central differencing methods, as represented by the transition coefficients in Table 7.1, exhibit this behavior. The Lax–Wendroff method does not suffer this defect because sufficient entropy generation occurs to offset the effects of entropy destruction. In conclusion for linear advection: (i) a net destruction of entropy results in numerical instability; (ii) local violations of the Second Law, depending on severity, introduce spurious oscillations in the solution; and (iii) the Second Law approach reproduces the CFL condition established from conventional linear analysis.
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12
8
(Sgen)nj
4 0
–4
–8
–12
0
0.25
0.5 x
0.75
1
Figure 7.3 Entropy generation for linear advection equation with Lax–Wendroff method at the time step n = 1.
12 8
(Sgen)nj
4 0
–4 –8 –12
0
0.25
0.5 x
0.75
1
Figure 7.4 Entropy generation for linear advection equation with upwind method at the time step n = 1.
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The leading-error numerical viscosity parameter exhibits a critical role that determines whether a numerical method satisfies the Second Law. Satisfying the Second Law by prescribing nonnegative entropy generation is a robust way to establish timestep restrictions for the numerical scheme.
7.3.2 N o n lin ea r Sc a la r Ad v ec t io n This section extends the previous analysis from linear to nonlinear problems involving the scalar advection equation. Consider the variable u(t, x) (note: general scalar variable, not velocity field) with the following governing equation: ∂u ∂f + = 0 ∂t ∂x
(7.46)
where u(t, x ) ∈( -∞, +∞ ), and the flux function is defined as 1 2 u 2
(7.47) Using the same construction strategy as the linear case in the previous section, a pair of functions will represent the “entropy” and “entropy flux” for this case. The entropy functional will be defined as follows: f (u ) =
S (u ) = - u 2
(7.48)
Applying the compatibility condition S ′(u ) f ′(u ) = F ′(u ) and integrating give the following entropy flux function: F (u ) =
2 uS (u ) 3
(7.49)
The balance of entropy equation for this case is S gen =
∂S ∂F + ∂t ∂x
(7.50)
Using an explicit time advance, the discretized transport equation can be expressed as
u nj +1 = (C + u ) nj -1 + (C 0 u ) nj + (C - u ) nj +1
(7.51)
where the solution-dependent transition coefficients must satisfy consistency and conservation as follows:
∆x + (C - C - ) u = f (u ) ∆t
(7.52)
C+ + C0 + C- =1
(7.53)
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T a bl e 7.3 T ransition Coefficients in Various Numerical Methods for a Nonlinear A dvection Equation
T ransition Coefficients C+
Method Lax–Friedrichs Baganoff Upwind flux Lax–Wendroff
1 2
C 0
(1 + 2∆∆tx u )
0
r + εu ∆t 4 ∆x ∆t 4 ∆x
∆t ∆x
1 2
(1 - 2∆∆tx u ) r - εu
1 - 2r
(| u | +u )
(1 +
C-
)
uˆ j - 12 u
11-
1 4
( ∆∆xt )
2
∆t 2 ∆x
(uˆ
|u| j + 12
)
+ uˆ j - 12 u
∆t 4 ∆x
(
(| u | -u )
- 4∆∆tx 1 -
∆t ∆x
)
uˆ j + 12 u
For this nonlinear case, the numerical approximation expands in a Taylor series with the transition coefficients included in the expansion. Some methods prescribe complex formulae for the transition coefficients. For first-order methods that utilize a three-point stencil whereby the transition coefficients depend only on upstream values of u, the technique for obtaining the modified equation is the same as the linear case, except that the transition coefficients must be included in the Taylor series expansion. The first three methods in Table 7.3 are shown of this kind. The second method listed in the table as “Baganoff” refers to an artificial dissipation technique derived by a statistical approach (Baganoff, 1983). The amount of dissipation is controlled by the parameters (r, ε ). For consistency with the hyperbolic equation, Equation 7.46, the time step is calculated by the formula ∆t = 4ε∆x for Baganoff’s method. In addition, other criteria apply to satisfy the monotonic condition and stability. The condition obtained by the present technique agrees with Baganoff’s statistical analysis (Baganoff, 1983). The modified equation for Lax and Wendroff’s technique requires a separate derivation (presented below). For the first-order methods, spatial derivatives for the terms on the right side of the discretized transport equation lead to C u(t, x ± ∆x ) = C u(t, x ) ± ∆x
∂C u ∆x 2 ∂2C u ∆x 3 ∂3C u + ± + (7.54) ∂x 2 ∂x 2 6 ∂x 3
Using these expressions and the series expansion for u(t + ∆t, x ), collecting terms and simplifying the result by using the conservation and consistency conditions give the following discretized advection equation (up to second order):
L∆ ≈
∂u ∂ f ∆ x 2 ∂ d ∂u ∆t ∂2 u ∆ x 2 ∂2 d f ∂u ∆t 2 ∂3u + + [(C + + C - )u] + + ∂t ∂ x 2 ∆ t ∂ x du ∂ x 2 ∂t 2 6 ∂ x 2 du ∂ x 6 ∂t 3
(7.55)
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The chain rule was used to modify the second- and third-order derivative terms. Similar to the linear case (previous section), with the Warming–Hyett technique, direct substitution of the original governing equation to eliminate second-order time derivatives lead to the same result for the leading first-order parameter. Using the chain rule and Equation 7.46 twice, ∂2 u ∂ df df ∂u = ∂t 2 ∂x du du ∂x
(7.56) Truncating Equation 7.55 to the leading order and moving terms to the right side lead to the following modified equation: ∂ ∂u ∂u ∂f + ≈ ε ∂t ∂x ∂x ∂x
where (by definition)
ε (u ) ≡
∆x 2 2 ∆t
(7.57)
2 d df ∆t + + C - )u] - [( C du ∆x du
(7.58)
This result reflects the nonlinearity of the governing equation. The corresponding modified equation for the balance of entropy is obtained by multiplying both sides of Equation 7.57 by S´(u) and using the chain rule on the left side: ∂ ∂u ∂S ∂F + ≈ S′ ε ∂t ∂x ∂x ∂x
(7.59)
Applying the chain rule to the right side of Equation 7.59 gives 2
S′
∂ ∂u ∂ ∂u ∂u ε S ′ - ε S ′′ ε = ∂x ∂x ∂x ∂x ∂x
(7.60)
The numerical entropy generation becomes
2
S gen ≈
∂ ∂u ∂u ε S ′ - S ′′ε ∂x ∂x ∂x
(7.61)
Integrating this expression over a discrete space and time interval, the first term on the right side vanishes in the limit. The second term will not vanish if the interval contains a discontinuity. Therefore, entropy generation remains nonnegative, if and only if the leading-error parameter remains nonnegative. Similar to the linear case (previous section), selected methods can be used to see what restrictions are imposed on the time step or other parameters. Before deriving those results, the modified equation for the Lax–Wendroff method must be obtained.
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The second-order modified equation for the Lax–Wendroff method, is ∂2 ∂u ∂f + ≈ 2 ∂t ∂x ∂x
∂u ε 2 ∂x
(7.62)
where
ε 2 (u ) ≡ -
∆x 3 6 ∆t
u∆t u∆t 3 - ∆x ∆x
(7.63)
This result can be derived by directly replacing the third-order time derivative with the original governing equation and using the chain rule. The corresponding balance of entropy equation reduces to ∂ ∂ ∂u ∂2 u ∂u ∂u S ′ ε 2 - S ′′ ε 2 + ε 2 2 ∂x ∂x ∂x ∂x ∂x ∂x 2
S gen ≈ with the derivative
2 dε 2 ∆x 2 u∆t =1 - 3 6 du ∆x
(7.64)
The sign for S gen in the second-order Lax–Wendroff method therefore depends on the sign of the local wave speed and the sign of the first and second derivatives. The method does not guarantee satisfaction of the Second Law as stipulated by S gen ≥ 0. It appears evident from Equation 7.63 to set | u | ∆t / ∆x ≤ 1, which is the same CFL condition given by linear analysis. Fortunately for this method, local violations of the Second Law are offset by sufficient entropy generation in adjacent regions, resulting in a net production of entropy so the method remains numerically stable. However, the upcoming results will show that solution quality is compromised. Besides providing a means for nonlinear numerical analysis, an entropy-based approach also provides a way of developing numerical methods consistent with the Second Law. The dependence of the local first-order numerical viscosity on the solution in Equation 7.58 can be used to construct a second-order method comparable to the Lax–Wendroff technique. Reordering the terms in Equation 7.58 to derive the following differential equation for the transition coefficients:
2 ∆t u 2 ∆t 2 d ε+ [(C + + C - )u] = 2 ∆x du ∆x 2
(7.65)
Integrating and solving for C + + C - gives C+ + C- =
u 2 ∆t 2 2 ∆t ε + ∆x 2 3∆x 2
(7.66)
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Combining the result with the consistency condition in Equation 7.52 gives two equations with two unknowns. The solution is obtained as
2u∆t u∆t ε∆t C+ = + 1 + 3∆x 4 ∆x ∆x 2
(7.67)
2u∆t u∆t ε∆t C- = - 1 + 3∆x 4 ∆x ∆x 2
(7.68)
Substituting the resulting transition coefficients into the discretized transport equation yields a second-order accurate method, if e = 0. By controlling the parameter e based on local gradients, the numerical dissipation can be minimized in a way that satisfies the Second Law. The numerical methods selected for comparison represent a sampling of a variety of techniques available. The few methods chosen clearly indicate the advantages of the Second Law analogy for establishing a nonlinear restriction on the time step. Numerical analysis based on local linearization yields the standard CFL condition. The results in Table 7.4 confirm this condition as consistent with the Second Law analogy. It guarantees nonnegative entropy generation, due to the concavity property of entropy over the solution domain. In the artificial dissipation technique, the constant coefficient r lies between zero and two. It yields the same condition as other methods. The Lax–Wendroff technique gives e = 0, so the time step is set by inspection of e 2 and the monotonic restriction on C0. An interesting result for the nonlinear case shows that the monotonic requirement may yield an overpermissive time step condition. Examining the transition coefficients and the monotonic requirement leads to a CFL-like condition with ν ≤ 2 for the Lax–Friedrichs, Baganoff, and upwind methods. For the Lax–Wendroff methods, the monotonic requirement gives ν ≤ 2 . The results in Table 7.4 indicate that a CFL condition based on monotonicity would destroy entropy for these methods. Numerical simulations confirm that the quality of the solution degrades with entropy destruction, even if the method remains numerically stable.
T a bl e 7.4 T ime-Step Constraints Imposed by the Second Law for One-D imensional Nonlinear Scalar A dvection Method
2 ε ∆ t/ ∆ x 2
Lax–Friedrichs
1 - ( u∆∆xt )
Baganoff
2r - (
Upwind Lax–Wendroff
|u|∆t ∆x
)
u∆t 2 ∆x
-( 0
2
)
u∆t 2 ∆x
C onstraint |u|max ∆t ∆x |u|max ∆t ∆x
≤1
≤ 2r
|u|max ∆t ∆x
≤1
|u|max ∆t ∆x
≤1
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Sample numerical results for the nonlinear scalar equation were obtained for a domain x ∈[0, 1] with the following initial conditions:
1 u( 0, x ) = (10 x + 2 )/3 0
0.0 ≤ x ≤ 0.1 0.1 < x < 0.4 0.4 ≤ x ≤ 1.0
For the finite volume formulation, 100 cells were used with uniform grid spacing of ∆x = 0.01 (Camberos, 1995). The time increment was evaluated at each calculation by searching data for the largest local wave speed, then obtaining a numerical value for the maximum time step allowed, according to the condition in Table 7.4. The theoretical solution for this case predicts that the characteristic waves will coalesce into a shock wave at tn = 0.3 (nondimensional time), which subsequently moves at a constant speed of cs. The scalar form of the Rankine–Hugoniot jump condition is f(uL) - f(uR) = cs(uL - uR), where the subscripts L and R represent the downstream and upstream values, respectively. Solving for the given initial conditions yields a shock propagation speed of cs = 1/2. The weak form of the governing equation admits nonunique solutions. As mentioned previously, a separate “entropy condition” is required to mathematically enforce the desired solution. The desired solution corresponds to a compression shock wave. For convex flux-functions, f(u), one version of the entropy condition in the literature is f ′(uL ) > cs > f ′(uR ) (LeVeque, 1992). The theoretical shock solution satisfying this condition requires that the shock propagates in the direction indicated from umax to umin (left to right in Figure 7.5). Another form of the entropy condition for scalar equations corresponds to the weak form of Equation 7.49, and the inequality S gen ≥ 0 is enforced. Numerical methods satisfying this inequality will admit only discontinuities consistent with the desired solution. Numerical results for two methods are presented in Figure 7.5 through Figure 7.8. Various numerical simulations revealed that the upwind method consistently gave the best results, unmatched even by second-order methods like the Lax–Wendroff method. Figure 7.5 shows the exact and numerical solution (with the upwind method) at the point of shock formation. Characteristic lines are shown in Figure 7.6, clearly indicating coalescence at the predicted time. The slope also gives dt/dx = 2, consistent with the theoretical value for shock propagation. The numerical solution for the upwind method is again shown in Figure 7.7, superimposed with the numerically predicted entropy generation. The peak in the entropy generation corresponds to the cell where the shock wave is “captured.” No regions of negative entropy generation are evident, and the solution quality is reasonably good. Figure 7.8 shows the numerical solution for the Lax–Wendroff method, superimposed with the numerical entropy generation. The peak in the entropy generation is higher than results from the first-order upwind method, but it also results in a region of entropy destruction (lagging the discontinuity). The cause of the spike in the solution is the entropy destruction occurring in the corresponding cell. Both methods converge to the physically relevant solution in the limit (by virtue of satisfying the weak form of the Second Law). However, only the upwind method matched the theoretical solution
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Numerical Exact, n = 29
1.2 1.0
unj
0.8 0.6 0.4
Initial Distribution
0.2 0.0
– 0.2
0
0.25
0.5 x
0.75
1
Fig u r e 7.5 Numerical solution of the nonlinear scalar equation with the upwind method at tn = 0.3.
1 0.9 0.8 0.7 Shock Motion at Steady Speed
0.6 tn 0.5 0.4
Shock Formation
0.3 0.2 0.1 0
0
0.2
0.4
x
0.6
0.8
1
Fig u r e 7.6 Space-time contours of the numerical solution of the nonlinear scalar equation with upwind method.
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22 (Sgen)nj
1.6
18
1.4
16
1.2 unj
20
14
1
12
0.8
10
0.6
Shock Motion
0.4
8 6 4
0.2
2
0
0
–0.2 0.0
0.2
0.4
x
0.6
0.8
–2 1.8
Fig u r e 7.7 Numerical solution and entropy generation for nonlinear scalar equation with upwind method.
for finite time and space increments. Thus, for the nonlinear case, the nonnegative “entropy generation” criterion establishes a restriction on the time step for numerical calculations, which differs from monotonic criteria, but matches quasilinear analysis corresponding to the CFL condition. 1.6
35 (Sgen)nj
1.4
30
1.2
25
1
20
0.8
15
unj 0.6
Shock Motion
0.4
10
0.2
5
0
0
–0.2
–5
–0.4 0.0
0.2
0.4
x
0.6
0.8
–1.0 1.0
Fig u r e 7.8 Numerical solution and entropy generation for nonlinear scalar equation with Lax–Wendroff method.
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7.3.3 C o u pled No n lin ea r Eq u a t io n s This third case will focus on a system of coupled equations that represent the conservation of mass, momentum, and energy for an ideal gas. Consider the following form of the Euler equations of gas dynamics in conservation law form, ∂q ∂f + =0 ∂t ∂x
(7.69)
where ρ q = ρu , ρe
ρu f = ρu 2 + P ρu( e + P/ρ )
represent the algebraic state and flux vectors, respectively. Define the flux Jacobian function as follows: A( q ) =
∂f ∂q
(7.70)
It has certain properties that arise from the first-order homogeneous nature of the flux vector as a function of the state variables. Note that A ⋅ q = f and A can be decomposed as A = Y ⋅ Λ ⋅ Y -1 where Y contains the eigenvectors and L has the eigenvalues of A. For an ideal gas, the thermodynamic entropy functional is known. Assuming an ideal gas with constant specific heats, the nondimensional formula for the specific entropy is s=
1 ln T - ln ρ γ -1
(7.71)
where T and ρ are the normalized temperature and mass density, respectively. The balance of entropy equation is S gen =
∂S ∂F + ∂t ∂x
(7.72)
where the functional relation for entropy is S ( q ) = ρs, and F ( q ) = ρus for the entropy flux. The flux function F satisfies the following compatibility condition required by the Second Law:
F ,q - S ,q ⋅ f ,q = 0
(7.73)
The subscripts represent derivatives with respect to the algebraic vector of state variables. The ideal gas formula for the entropy leads to a concave function of the state
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variables. Writing the formula in terms of q and taking the Hessian (a matrix of second derivatives) demonstrate its negative definite property. Using a standard one-dimensional explicit discretization of the system of equations, q nj +1 = (C + ⋅ q ) j -1 + (C 0⋅ q )nj + (C - ⋅ q )nj +1 n
(7.74)
This equation reflects a three-point numerical stencil in a transition coefficient form. In this case, matrix transition coefficients arise instead of scalars. The transition coefficients must still satisfy the conservation and consistency requirements. From conservation requirements, the transition coefficient matrices must add up to the identity matrix of the same size, C+ + C0 + C- = I
(7.75)
For consistency purposes, we require that ∆x + (C - C - ) ⋅ q = f ( q ) ∆t
(7.76)
As described previously in the scalar case, the transition coefficient C ± may or may not satisfy monotonicity, but for conservation it is necessary that I ≥ C 0 ≥ 0. Expanding the numerical discretization in Equation 7.74 to the known solution at time level, t, in a Taylor series leads to C q(t, x ± ∆x ) = C q ± ∆x
∂C ⋅ q ∆x 2 ∂2C ⋅ q ∆x 3 ∂3C ⋅ q + + ± ∂x 3 ∂x ∂x 2 2 6
(7.77)
and similarly for q(t + ∆t, x ). Substituting this result into the previous equation, collecting terms, and simplifying by using the conservation and consistency conditions gives (up to second order)
L∆ ≈
∂q ∂ f ∆ x 2 ∂ ∂ ∆t ∂2 q ∆ x 2 ∂2 ∂ q ∆ t 2 ∂3q + + [(C + + C - ) ⋅ q] + A⋅ + ∂t ∂ x 2 ∆ t ∂ x ∂x 6 ∂ x 2 ∂ x 6 ∂t 3 2 ∂t 2
(7.78)
In contrast to the nonlinear scalar equation, the first-order error term with spatial derivatives can be evaluated as follows: ∂ ∂ ∂q (C + + C - ) ⋅ q = (C + + C - ) ⋅ q + (C + + C - ) ⋅ ∂x ∂x ∂x
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The transition coefficients are constructed by the flux vector, f, which is homogeneous of degree one. Thus, ∂A ⋅q = 0 ∂x
(7.79)
The sum of the transition coefficients, C + + C -, then obeys the following equation:
∂ (C + + C - ) ⋅ q = 0 ∂x
(7.80)
Using this result and replacing the second-order time derivatives with the original differential equation, ∂2 q ∂ 2 ∂q = A ∂t 2 ∂x ∂x
(7.81)
∂q ∂f ∂ ∂q + ≈ [ ε ] ⋅ ∂t ∂x ∂x ∂x
(7.82)
gives
where the numerical viscosity parameter is [ε ]=
{
}
∆x 2 ∆t 2 (C + + C - ) - A2 2 2 ∆t ∆x
(7.83)
The modified entropy balance equation is obtained after multiplying Equation 7.82 by S,q and using the chain rule. The result becomes ∂ ∂S ∂F ∂q + ≈ S,q ⋅ [ε ] ⋅ ∂t ∂x ∂x ∂x
(7.84)
which represents the modified equation for the balance of entropy. Expressing the result in terms of the entropy generation rate gives S gen ≈
∂ ∂q ∂q T ∂q ⋅ ( - S,qq ⋅ [ ε ]) ⋅ S,q ⋅ [ ε ] ⋅ + ∂x ∂x ∂x ∂x
(7.85)
For the entropy variable, S,q equals an algebraic row vector and S,qq represents a second-order tensor, which is the Hessian of S with respect to q. The Hessian of the entropy for an ideal gas is symmetric and negative definite. The following result was
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derived by Merriam (1989):
S,qq = - (Y -1 )T ⋅ Y -1
(7.86)
where the superscript T represents the matrix transpose operation. The condition stipulated by the Second Law from Equation 7.85 requires that
x T ⋅ ( - S,qq ⋅ [ε ]) ⋅ x ≥ 0
(7.87)
If - S,qq⋅ [ ε ] is positive definite, then equality holds if and only if x = 0. This effectively imposes a limit on the time step, given a particular grid size and initial distribution. After substituting Equation 7.86 for the Hessian, the matrix core of Equation 7.87 becomes - S,qq ⋅ [ ε ] = (Y -1 )T ⋅ Y -1⋅ [ ε ] ⋅ Y ⋅ Y -1 Defining a new vector z = Y ⋅ x reduces Equation 7.86 to
zT ⋅ (Y -1 ⋅ [ ε ] ⋅ Y ) ⋅ z ≥ 0
(7.88)
(7.89)
The matrix Y -1 ⋅ [e] = Y is mathematically similar to [ε ], and hence they have the same eigenvalues. Furthermore, the construction of transition coefficients satisfying the consistency requirement guarantees that Y -1 ⋅ [ε ] ⋅ Y is diagonal and therefore also symmetric. This implies that
δ max zT ⋅ z ≥ zT ⋅ (Y -1 ⋅ [ε ] ⋅ Y ) ⋅ z ≥ δ min zT ⋅ z
(7.90)
where δ max ≥ δ min are the largest and smallest eigenvalues of the symmetric matrix. From Equation 7.83, for a given set of transition coefficients, the minimum eigenvalue δ min coincides with the maximum eigenvalue of the flux Jacobian A. Thus, δ min ≥ 0 will guarantee the condition given by Equation 7.89, and hence also satisfy the Second Law as required by S gen ≥ 0 in Equation 7.85. The transition coefficients for various methods are presented in Table 7.5. The upwind method includes the classical Steger–Warming flux vector splitting method (FVS) (Steger and Warming, 1981), Roe’s “flux-difference splitting” (FDS) (Roe, 1981), and other related methods. To obtain the correct formula for the error parameter, an equivalent differentiable term replaces the dissipation nondifferentiable error term for Steger–Warming FVS. The time step limit observed in numerical simulations to keep the Steger–Warming method stable places an upper value of 0.7 for the Courant number. This differs from results of either linear analysis or the monotonic condition, both of which limit the Courant number to an upper value of one. Steger–Warming’s method exhibits numerical instability when the Courant number lies too close to its maximum (linearly obtained) value. The method labeled “arithmetic averaging” is a variant of FVS and flux-difference splitting. The transition coefficients contain flux Jacobians evaluated at intermediate
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T a bl e 7.5 T ransition Coefficients for the One-D imensional Euler Equations with Various Numerical Schemes T ransition Coefficients C
Method
C 0
+
Lax–Friedrichs
1 2
Steger–Warming Arithmetic averaging Roe FDS
∆t ∆x
0
A)
(| A | + A)
I-
∆t 2 ∆x
(| A | + A ) j - 1
∆t 2 ∆x
(| Aˆ |
Kinetic split flux Lax–Wendroff
(I +
∆t 2 ∆x
∆t 2 ∆x
∆t 2 ∆x
(
∆t ∆x
+A
)
I-
∆t 2 ∆x
| A|
∆t ∆x
2
I-
)
I-
A j - 1 + I ⋅ A
∆t 2 2 ∆x 2
∆t ∆x
2
2
⋅A
A)
(| A | - A)
(| A | - A )
∆t 2 ∆x
(| Aˆ |
∆t 2 ∆x
j + 12
∆t ∆x
∆t 2 ∆x
Aσ
A j - 1 + A
(I -
∆t 2 ∆x
| Aˆ | j - 1 + | Aˆ | j + 1
( Aσ + A) 2
1 2
I - (C + + C - )
2
j - 12
C-
∆t 2 ∆x
(
∆t ∆x
j - 12
j + 12
+A
)
( Aσ - A)
)
A j + 1 - I ⋅ A 2
values of the state variables: q j +1/2 = ( q j + q j +1 )/2. The fifth method listed in Table 7.5 represents the “kinetic split flux” technique. Construction of the split fluxes (or, equivalently, the transition coefficients) relies on generating one-sided moments of the Boltzmann equation from kinetic theory (Camberos, 1997a,b). The technique utilizes a given probability distribution function that reflects local values of the density, velocity, and temperature. For the Euler equations, the probability distribution reflects the local equilibrium assumption and hence equals a Maxwell–Boltzmann probability distribution in the molecular velocity variable, with the mass density, macroscopic velocity, and temperature as constraints. The following abbreviations are used in Table 7.6: 2σ 2 π
Iσ ≡ I
(7.91)
T a bl e 7.6 T ime-Step Constraints Imposed by the Second Law for the One-D imensional Euler Equations Method
Y -1 ⋅ [ε ] ⋅ Y
Lax–Friedrichs Steger–Warming Roe FDS & AA K–Split flux Lax–Wendroff
C onstraint
2∆ t ∆ t2
I - Λ 2 ( ∆∆xt )
|λ|
2
Iσ
∆t ∆x
- Λ 2 ( ∆∆xt )
2
Λ
∆t ∆x
- Λ 2 ( ∆∆xt )
2
Λσ
∆t ∆x
- Λ 2 ( ∆∆xt ) 0
2
λ2
∆t ∆x
|λ| λ2
∆t ∆x
|λ|
∆t ∆x
≤1
≤σ ∆t ∆x
2 π
≤1
≤ λ αβ ∆t ∆x
≤1
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where I equals the identity matrix. Also
2 Λσ = αΛ + βσ 0 0
γ -2 γ -1 0
with
0 0 2
0
{ }
(7.92)
2 u2 u α ≡ erf , β≡ exp - 2 π 2σ σ 2
(7.93)
λαβ = α | λ |max + 2 βσ
(7.94)
and
The parameter σ defines a characteristic thermal velocity as follows:
γ -1 P 2 q1q3 - q22 ) = . ( 2 2 q1 ρ
σ 2 (q) =
(7.95)
Table 7.6 presents results of the Second Law analysis for the equations of gas dynamics. From the table, the Lax–Friedrichs method requires a time-step constraint, essentially equivalent to a CFL condition based on linear theory. For the Steger–Warming method, the constraint gives a CFL number of 0.674 for uniform initial conditions of zero velocity. This number is surprisingly close to the limit observed in numerical simulations, thereby suggesting a possible theoretical explanation. For uniform initial conditions of zero velocity, a = 0 and β = 2 / π , the Second Law constraint for the kinetic split flux method reduces to 2 λmax
∆t 2 ≤ 2σ π ∆x
(7.96)
which equals twice the value of the CFL number imposed on the Steger–Warming method by ad-hoc arguments. This result suggests that a connection between the two methods may exist.
7.4 St a bil it y o f Sh o c k Ca pt u r in g Met h o d s Differences in the numerical solution of the Euler equations become evident by testing methods against various cases. Consider a shock-structure problem (Liepmann et al., 1962) governed by the viscous Navier–Stokes equations. Although the Euler equations of gas dynamics contain no mathematical terms representing viscous dissipation or heat conduction, their numerical solution does contain diffusive effects. These effects of numerical dissipation and heat conduction will be made evident by solving the shock-structure problem.
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Gas dynamics theory provides a solution to the “jump equations” across a planar shock wave (Liepmann and Roshko, 1957). They are given by the following Rankine– Hugoniot relations for an ideal gas:
ρ2 u1 (γ + 1) M12 = = ρ1 u2 (γ - 1) M12 + 2
(7.97)
P2 2γ = 1+ ( M12 - 1) P1 γ +1
(7.98)
T2 P2 ρ1 = T1 P1 ρ2
(7.99)
s2 - s1 =
P 1 ln 2 γ - 1 P1
ρ2 ρ 1
γ
(7.100)
where M1 and M2 refer to the upstream and downstream Mach number, respectively. The initial and boundary conditions are set according to these relations with the jump centered in the computational domain, assuming a linear connection across 10 grid cells. Camberos (1995) solved these equations numerically with a finite-volume algorithm and piecewise constant data using the expression of Equation 7.74 and the various transition coefficients in Table 7.5. An idealized solution is given by the exact jump relations with a shock width of zero thickness. Theoretically, independent of the dissipative mechanisms, the entropy change across the shock wave has a maximum value (Pike, 1985). The maximum value of the specific entropy for the shock wave depends only on the upstream Mach number and the ratio of specific heats, just like other variables in the Rankine–Hugoniot relations. This maximum can be found from the conservation equations and setting the velocity equal to u1u2 , where u1 and u2 represent the upstream and downstream velocities, respectively. The formula for the entropy peak is given by
Smax where
γ +1 2 2 ρmx 2 γ - 1 = ln M1 + γ - 1 γ + 1 (γ + 1) M12
ρmx = ρ1
(γ + 1) M12 (γ - 1) M12 + 2
(7.101)
(7.102)
equals the mass density at the same conditions which yield the entropy maximum. Note that Equation 7.101 is a maximum for all Mach numbers, only for the specific entropy. Consider a stationary shock wave, as described previously for supersonic flow at a Mach number of 1.5. Gas dynamics theory predicts a positive jump in the pressure, temperature, and mass density, whereas a negative jump is predicted for the velocity
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(the flow slows down to subsonic conditions). The entropy also undergoes a positive jump because a shock wave represents an intrinsically irreversible (entropy-generating) process. Thus, a finite amount of entropy generation occurs, and the entropy balance equation can be used to predict its magnitude. The numerical value of the entropy maximum for M1 = 1.5 equals 0.2214, which is 38% greater than the entropy jump of 0.1356 given by Equation 7.100. In the following examples, Equation 7.69 is solved numerically with the approximation of Equation 7.74 together with the transition coefficients for various methods listed in Table 7.5. The spatial domain was ξ ∈[0, 1], with initial and boundary conditions given by fixed upstream and downstream values of the mass density, pressure, and velocity according to Equation 7.97 to Equation 7.99, along with a linear variation spanning 10 cells for ξ ∈[0.4, 0.6]. The grid consists of 100 cells that give ∆ξ = 0.01 uniformly. Time increments are set according to the constraints prescribed in Table 7.6. Numerical solutions converge in about 1000 time increments, giving a steady-state solution that satisfies the Rankine–Hugoniot jump conditions for a stationary, planar shock wave. As shown in Figure 7.9 and Figure 7.10, the Lax–Friedrichs method solves the shock structure problem with a smooth transition from upstream (ahead of the shock) to downstream values. The result is close to the solution of the Navier–Stokes equations for a viscous, heat-conducting gas. For the gas dynamic equations, the shock structure is entirely due to numerical effects, indicating the numerical dissipation present in this method. In Figure 7.9, no entropy peak is present, and Equation 7.98 is not satisfied. Entropy generation has a very low peak, spread over 24 to 26 cells. For this example, the number of cells with nonzero entropy generation can be used as a measure of the shock width. To represent an entropy peak, a minimum of 3 points 2.6 0.20
P/P 1
2.2 2 1.8
0.14
S – S1
u/u1
Entropy
Pressure, Temperature, Velocity
2.4
1.6
0.08
1.4 T/T1
1.2
0.02
1 0
0.25
0.5
0.75
1
Fig u r e 7.9 Numerical “shock-structure” solution of Euler equations with Lax–Friedrichs method.
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Numerical Stability and the Second Law 14 0.25
12 10
0.2
(Sgen)nj
8
0.15
S – S1
6 0.1
4
0.05
2 0
0 0
0.25
0.5
0.75
1
Fig u r e 7.10 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with Lax–Friedrichs method.
is required, so an ideal method would have a shock width of 2 cells. The Lax–Friedrichs method (with this metric) gives a very wide shock. A solution to the stationary shock problem given by the Steger–Warming method is shown in Figure 7.11 and Figure 7.12. The transition from upstream to downstream 2.6 0.20
P/P 1
2.2 2 u /u1
1.8
0.14
S – S1
Entropy
Pressure, Temperature, Velocity
2.4
1.6
0.08
1.4 T/T1
1.2
0.02
1 1
0.25
0.5
0.75
1
Fig u r e 7.11 Numerical shock structure solution of Euler equations with Steger–Warming flux-vector splitting.
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12
(Sgen)nj
10
0.2
8
S – S1
0.15
6 0.01
4
0.05
2 0
0 0
0.25
0.5
0.75
1
Fig u r e 7.12 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with Steger–Warming flux-vector splitting.
values appears to have a sharp discontinuity (Figure 7.11) followed by a smooth transition to downstream conditions. Nonzero entropy generation occurs in the shock interior, as displayed in Figure 7.12, indicating a shock thickness of about 8 or 9 cells. An entropy peak is evident in the figure, but it is much smaller than the value predicted by Equation 7.101. Also, the peak in entropy generation appears to have a value of 4 (nondimensional units), whereas theory predicts a value of 6.5 for a minimumwidth shock wave of 2 cells required to capture the entropy peak. Figure 7.13 and Figure 7.14 present the results of Roe’s FDS method. Although not shown, the arithmetic averaging and Lax–Wendroff methods give converged results identical to those shown in Figure 7.13 and Figure 7.14 (Lax–Wendroff entropy generation peaked at about 12). The discontinuity is well resolved by these methods, changing from upstream to downstream conditions within one cell spacing. Unfortunately, none of these methods captures the peak in the entropy predicted by Equation 7.101. Judging from the nonzero entropy generation shown in Figure 7.14, the shock wave is actually 2 cells wide. The peak entropy, as it changes from upstream to downstream conditions across the shock wave, is predicted by ideal gas dynamics. This is consistent with past studies by Pike (1985), where “any scheme for obtaining steady solutions of the Euler equations, which conserves mass and energy and obeys the equation of state, will be correct in exhibiting an entropy maximum at the shock wave.” But it differs from a standard view that successful shock capturing should produce monotonic results by means of a minimum of intermediate values. Changes in the entropy across a shock wave are not monotonic, but these are not made evident in the test cases typically presented. Only the kinetic split flux technique appears to approximately satisfy Equation 7.101. Results with this method are presented in Figure 7.15 and Figure 7.16.
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P/P 1
2.2 2 u/u1
1.8
0.14
S – S1
Entropy
Pressure, Temperature, Velocity
2.4
1.6
0.08
1.4 T/T1
1.2
0.02
1 0
0.25
0.5
0.75
1
Fig u r e 7.13 Numerical shock-structure solution of Euler equations with Roe’s fluxdifference splitting.
14 0.25
12
(Sgen)nj
10
0.2
8
S – S1
0.15
6 0.1
4
0.05
2
0
0 0
0.25
0.5
0.75
1
Fig u r e 7.14 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with Roe’s flux-difference splitting.
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0.26
2.2 2
0.18 u/u1
1.8
Entropy
Pressure, Temperature, Velocity
2.4
S – S1
1.6 0.10
1.4 T/T1
1.2
0.02
1 1
0.25
0.5
0.75
1
Fig u r e 7.15 Numerical shock-structure solution of Euler equations with kinetic split-flux method.
14 0.25
12
(Sgen)nj
10
0.2
8
S – S1
0.15
6 0.1
4
0.05
2
0
0 0
0.25
0.5
0.75
1
Fig u r e 7.16 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with kinetic split-flux method.
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12
(Sgen)nj
10
0.2
8
0.15
S – S1
6 0.1
4
0.05
2
0
0 0
0.25
0.5
0.75
1
Fig u r e 7.17 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution with Steger–Warming FVS at 100 time-step iterations.
The transition from upstream to downstream conditions is smooth, but not as smeared as the Lax–Friedrichs method. In addition, the entropy clearly peaks inside the shock wave, although the value obtained is about 28% higher than results predicted by Equation 7.101. The peak entropy generation appears to be centered in the shock interior, and it has a value of about 4.5, compared with 6.5, predicted by theory. The shock wave is about 12 cells wide, as noted by nonzero entropy generation in Figure 7.16. The positive results obtained with this method suggest that combining the kinetic split flux technique with some type of averaging of the flux Jacobians in the transition coefficients (like Roe’s FDS) may produce an optimal method that captures both the entropy peak and the shock wave with minimal spreading. Transient results are displayed in Figure 7.17 and Figure 7.18. Figure 7.17 shows the transient entropy and entropy production after 100 time increments for the Steger–Warming FVS method. The initial and boundary conditions led to a spike in the state variables. Figure 7.18 shows the same result for the Lax–Wendroff method, which clearly exhibits the oscillations of this technique. As reported previously for the scalar cases, regions of spurious oscillations in the solution coincide with entropy destruction. This highlights the difference between entropy generation and entropy change, which are often confused in the literature. Spurious, nonphysical oscillations in the solution of the gas dynamics equations occur from local violations of the Second Law, which stipulates that entropy generation must be nonnegative. Peaks in the entropy change across a control volume are not violations of the Second Law. The example in this section has shown that numerical methods exhibiting a maximum in the entropy at the shock wave are not entirely spurious.
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0.2
12 0.15
10 8
0.1
(Sgen)nj
6
S – S1
4
0.05
2 0
0
–2 –0.05
–4 0
0.25
0.5
0.75
1
Fig u r e 7.18 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution with Lax and Wendroff’s method at 100 time-step iterations.
In closing, this chapter has demonstrated the utility of the Second Law as a reliable method for establishing time-step constraints and numerical stability. It also serves as a time-step indicator for implicit calculations in transient simulations. One could well interpret the results of a Second Law approach as a logical component of numerical or mathematical analysis that establishes numerical stability, convergence, existence, and uniqueness of weak solutions to a given problem. By the same argument, one could extend the conjecture to apply the same conclusions to systems of equations, regardless of equation type (parabolic, elliptic, or hyperbolic). This would follow early pioneering efforts to fully understand the implications of entropy and the Second Law in numerical analysis. It would also firmly establish the utility of the Second Law beyond its role in thermodynamics, or an analogy to that role. Instead, it would become an essential element in numerical modeling itself, not limited to its physical and historical origins.
Ref er en c es Anderson, D.A., Tannehill, J.C., and R.H. Pletcher. 1984. Computational Fluid Mechanics and Heat Transfer. Hemisphere, New York, 70–71. Baganoff, D. 1983. Stochastic Processes in Aeronautics. Department of Aeronautics and Astronautics. ©1983 by D. Baganoff, Stanford University, Stanford, CA (unpublished), 37–39. Camberos, J.A. 1995. Probabilistic Approach to the Computational Simulation of Gasdynamic Processes. Doctoral dissertation, Department of Aeronautics and Astronautics (SUDAAR No. 668), Stanford University, Stanford, CA, 102–105.
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Camberos, J.A. 1997a. Comparison of Split Fluxes Generated with Selected Probability Distributions Functions. Presented at the 13th Computational Fluid Dynamics Conference, AIAA Paper No. 97–2095. Snowmass Village, CO (unpublished). Camberos, J.A. 1997b. Comparison of Selected Probability Distribution Functions for Gasdynamic Simulations Inspired by Kinetic Theory. Presented at the 35th Aerospace Sciences Meeting AIAA Paper No. 97-0340. Reno, NV (unpublished). Camberos, J.A. 2000a. On the Construction of Entropy Balance Equations for Arbitrary Thermophysical Processes. Paper draft in preparation for submission to the 39th AIAA Aerospace Sciences Meeting in January 2001 (unpublished). Camberos, J.A. 2000b. Nonlinear time-step constraints based on the Second Law of Thermodynamics. AIAA J. Thermophysics Heat Transfer, 14(3): 231–244. Camberos, J.A. 2000c. An alternative interpretation of work potential in thermophysical processes. AIAA J. Thermophysics Heat Transfer, 14(2): 177–185. Courant, R., Friedrichs, K.O., and H. Lewy. 1928. On the partial difference equations of mathematical physics. IBM J. Res. Dev. (1967), 11: 215-234; originally published in Mathematische Annalen, 100: 32–74. Crandall, M.G. and A. Majda. 1980. Monotone difference approximation for scalar conservation laws. Math. Computation, 34(149): 1–21. Jaynes, E.T. 1991. Probability theory as logic. In Maximum-Entropy and Bayesian Methods, P.F. Fougére, Ed., Kluwer, Dordrecht, 1–16. Kapur, J.N. and H.K. Kesavan. 1992. Entropy Optimization Principles with Applications. Academic Press–Harcourt Brace Jovanovich, San Diego, CA, 66–67. Lavenda, B.H. 1991. Statistical Physics: A Probabilistic Approach. John Wiley & Sons, New York. Lax, P.D. 1954. Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math, 7: 159–193. Lax, P.D. 1973. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conference Series in Applied Mathematics. LeVeque, R.J. 1992. Numerical Methods for Conservation Laws. Birkhäuser-Verlag, Berlin, 36–40. Liepmann, H.W. and A. Roshko. 1957. Elements of Gasdynamics. John Wiley & Sons, New York, 59–60. Liepmann, H.W., Narasimha, R., and M.T. Chahine. 1962. Structure of a plane shock layer. Phys. Fluids, 5(11): 1313–1324. Merriam, M.L. 1989. An Entropy-Based Approach to Nonlinear Stability. Ph.D. thesis. Stanford University, Stanford, CA. Oleinik, O.A. 1957. Discontinuous solutions of non-linear differential equations. AMS Translation Series 2, 26: 95–172 (1963). Russian original: Uspehi Mat. Nauk (N.S.), 12(2): 3–73. Oleinik, O.A. 1959. Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. AMS Translation Series 2, 33: 285–290 (1963). Russian original: Uspehi Mat. Nauk (N.S.), 14(2): 165–170. Pike, J. 1985. Notes on the structure of viscous and numerically-captured shocks. Aeronaut. J. R. Aeronaut. Soc., November: 335–338. Steger, J.L. and R.F. Warming. 1981. Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. J. Computational Phys., 40: 263–293. Warming, R.F. and B.J. Hyett. 1974. The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Computational Phys., 14: 159–179.
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Entropy Transport with Phase Change Heat Transfer
8.1 In t r o d u c t io n This chapter examines the role of entropy and the Second Law of Thermodynamics in selected problems involving phase change heat transfer, particularly with solidification, melting, and film condensation. It is not intended to cover all aspects of multiphase problems, but rather a sample of selected topics in specific areas where entropy and the Second Law have particular importance. Typical applications arise in heat exchangers, multiphase processing in chemical equipment, and so forth. Rosen and coworkers (1999, 2004) have developed methods of exergy analysis for applications to systems like industrial steam process heaters and thermal energy storage systems. This chapter examines the role of entropy and exergy as design tools for developing improvements to such engineering systems, particularly involving phase change and multiphase flows. Solidification and melting arise in many engineering applications, including materials processing, ice accretion on structures, and thermal energy storage in electronic assemblies. The design and prediction of these phase change processes typically involve solutions of the conservation equations (mass, momentum, energy, and species equations). A variety of numerical procedures, such as finite differences (Salcudean and Guthrie, 1979), finite elements (Pardo and Weckman, 1990), finite volumes (Bennon, Incropera, 1988), and combined finite volume-element methods (Naterer and Schneider, 1996), have been developed for these problems. Numerical models provide effective tools for better understanding of transport processes during solidification and melting. This includes solute segregation, thermosolutal convection, and interdendritic and shrinkage flows. Flood and Davidson (1994) observed the formation of centerline macrosegregation in aluminum cast ingots including the sensitivity to ingot thickness and casting speed. Rady and coworkers (1997) used a finite volume method to predict thermal and solutal buoyancy during solidification of hypereutectic and hypoeutectic binary alloys. Additional phenomena involving interdendritic flows, including solute redistribution in the mushy zone, were examined by Maples and Poirier (1984). Modeling developments in these phase change problems were summarized and discussed in a comprehensive review by Salcudean and Abdullah (1988). In solid–liquid phase change problems, entropy can serve as an effective parameter for understanding and describing various physical processes. For example, interface properties like interface “roughness” are determined from the entropy change during phase transition. At microscopic scales, a rough or “nonfaceted” surface exhibits a low 213
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entropy of fusion. During solidification, dendritic arms grow in a direction corresponding to the maximum thermal irreversibility, because it is aligned with the heat flow direction (i.e., direction of the local temperature gradient). As a result, entropy is an important characteristic of the two-phase permeability, when describing how pockets or channels of liquid are formed within the solid matrix. Another complex process is thermal recalescence, which involves a transient temperature rise during cooling of a freezing crystal it occurs from a latent heat release that exceeds the other modes of cooling. In this case, a positive rate of entropy change can indicate the duration and magnitude of the local reheating. These processes have been observed by many researchers, but less attention has been given to the role of entropy during the processes. Bejan (1996) applied minimization of entropy generation to various multiphase systems, including refrigeration, energy storage systems, and power generation. Charach and Rubinstein (1989) investigated entropy generation during phase change heat conduction. Much additional opportunity exists for incorporating entropy and the Second Law into phase change analysis. As discussed in previous chapters, past computational fluid dynamics (CFD) studies have shown that the Second Law can improve solution accuracy (Lax, 1971) and upwinding accuracy (Naterer, 1999) in fluid flow simulations. In the context of phase change heat transfer, these results can be extended to establish the uniqueness of interface resolution, subject to different convergence tolerances imposed on a numerical model. Entropy production can also establish an optimal and convergent phase distribution during numerical iterations, without randomly cycling through phases, because only entropy-producing solutions are physically possible. Arbitrary convergence tolerances can be reduced or eliminated with the Second Law. Numerous techniques have been developed for convergence acceleration in nonlinear problems, including relaxation factors and multigrid methods (Minkowycz et al., 1988). Interface tracking by sequential steps was proposed by Schneider and Raw (1984), whereby two phase rules were used to coordinate the orderly progression of phase transition between adjacent control volumes. In this chapter, it will be shown that these iterative procedures are closely linked to the Second Law. This chapter will derive an alternative entropy-based framework (or heat-entropy analogies) for various transport processes during phase change. This includes interphase momentum exchange and recalescence phenomena. The intrinsic generality of entropy as an abstract concept provides opportunities for deeper insight into complicated phenomena. Previous analogies have established connections between heat transfer and friction coefficients (i.e., Reynolds analogy between heat and momentum), and similar opportunities can be realized with entropy. It will be shown that transport phenomena involving one variable (temperature) may be inferred through consideration of the other analogy variable (entropy). This type of similarity can be particularly useful if prediction of a certain variable is difficult or time-consuming, whereas analysis involving the other analogy variable may be more readily implemented. Additional benefits arising from the Second Law in computational models, such as numerical stability, may be realized. In addition to past numerical studies, experimental data provides vital insight for detailed understanding of phase change processes. Previous experimental studies
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of solidification and melting have often used aqueous mixtures, such as sodium chloride and water (Hayashi and Komori, 1979) or ammonium chloride and water (Szekely and Jassal, 1978), due to their translucency, low melting temperatures, and similar behavior to dendritic solidification in metals. Important transport processes have been observed in these studies (Burton et al., 1995; Clyne and Kurz, 1981; Voller and Brent, 1989), including dendritic formations (Yoo and Viskanta, 1992), planar interface movement, microgravity formations, and solid matrix permeability (Naterer, 2000). This chapter will investigate the importance of entropy and the Second Law in these processes. Applications ranging from materials processing to energy storage will be considered. An example is phase change materials (PCMs) for thermal management of electronic assemblies (Vesligaj and Amon, 1999). The temperature difference between an electronic component and the PCM, ∆T, at a fixed heat transfer rate, Q, is reduced when the entropy production rate, Q∆T 2 /T 2, is minimized. In this example and others, the unique insight provided by the Second Law will be examined.
8.2 En t r o py Tr a n sp o r t Eq u a t io n s f o r So l id if ica t io n a n d Mel t in g The governing equations for solid–liquid phase transition are the conservation equations (mass, momentum, and energy), in conjunction with an appropriate phase diagram, equations of state, and supplementary equations relating microscopic and macroscopic quantities. Continuum equations can be written for the conserved quantities, x k , where x k refers to a vector of conserved quantities, including mass and energy, and the subscript k refers to phase k, that is, k = 1 (solid) and k = 2 (liquid). The mixture equations are obtained by summing the individual continuum equations over both phases within a control volume, including solid and liquid phases, and rewriting the variables in terms of mixture variables. A mixture quantity is defined as the mass fraction-weighted sum of individual phase components. For example,
v = fl v l + fs v s
(8.1)
k = fl kl + fs ks
(8.2)
refer to the mixture velocity and thermal conductivity, respectively. If a conserved quantity is written without a subscript involving a phase, then it refers to a mixture quantity. After performing the summation of conservation equations over both phases, the mixture equations for mass and momentum, respectively, can be expressed in the following manner:
∂ρ + ∇ ⋅ ( ρv ) = 0 ∂t
(8.3)
(8.4)
ρ ∂( ρv ) + ∇ ⋅ ( ρvv ) = -∇p + ∇ ⋅ l µl ∇v + Fb + Fp ∂t ρ
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where Fb and Fp refer to body forces and phase interaction forces, respectively. In the present analysis, the body forces, Fb, are given by Fb = ρgβT (T - T0 ) + ρgβC (C - C0 )
(8.5)
where g, b T, and b C represent the gravity vector and thermal and solutal expansion coefficients, respectively. The phase interaction forces, Fp, will be determined from appropriate supplementary relations. The conservation equations for species and energy, respectively, are given by
∂( ρC ) + ∇ ⋅ ( ρvC ) = ∇ ⋅ ( ρl fl Dl Cl + ρs fs DsCs ) + ∇ ⋅ ρ(C - Cl )v ∂t ( ∂ρh ) + ∇ ⋅ ( ρvh ) = ∇ ⋅ ( k∇T ) + ∇ ⋅ ρ( h - hl )v ∂t
(8.6) (8.7)
where h refers to enthalpy. In phase k, this enthalpy is written as hk (C , T ) =
∫
T
T0
cr ,k (ζ )dζ + hr ,k (C , T )
(8.8)
In Equation 8.8, cr,k (T) refers to the reference specific heat of phase k. The final terms in Equation 8.6 and Equation 8.7 are written in a way that simplifies their evaluation as source terms, Sc and Se, respectively, in a conventional numerical model. The previous governing equations must be solved in conjunction with a phase equilibrium diagram (see Figure 8.1). The following assumptions will be used in the
Temperature
Tm
Liquid
Mushy Liquidus
Te
Solid
(Eutectic)
Solidus CS Composition
CL
Fig u r e 8.1 Binary alloy phase diagram.
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Solid
Entropy Production Due to Shear Action Heat Flow
Vs,i
dfsdV Liquid
dn dA n
Fig u r e 8.2 Schematic of phase interface.
upcoming analysis: continuous liquid–solid mixture without internal gas voids; twodimensional, incompressible, laminar, Newtonian flow; and a stationary solid phase during phase transition. In addition to the conservation equations, interfacial constraints will be required for balances of conserved quantities and entropy across the moving phase interface. In numerical models, interface tracking typically requires iterative solutions, because the interface position is generally unknown and its motion has a nonlinear behavior. The interfacial constraints will be utilized with entropy as a basis for effective interface tracking. The binary phase diagram will be used to determine the equilibrium temperature and concentration at the solid–liquid interface. In Figure 8.2, a typical schematic of the solid–liquid interface is illustrated (note: n, Vi, dA, and dfsdV refer to the normal direction, interface velocity, area, and solid fraction increment multiplied by a change in volume, respectively). The heat transfer from the liquid phase into the phase interface, HTl, consists of conduction (Fourier’s law) and advection components, HTl = - kl dA
dT dt + ρlVl el dAdt dn l
(8.9)
A similar heat transfer expression can be written in the solid phase. Consider a control volume at the phase interface with a thickness dn. Then the change of energy that accompanies the advance of the interface arises from the energy difference of an entirely liquid volume, dAdn, and a final solid volume, so
dE ≡ HTl - HTs = ρl el dAdn - ρs es dAdn
(8.10)
Based on the results in Equation 8.9 and Equation 8.10, it can be shown that the following interfacial energy constraint is obtained - kl
dT dT + ks = -Vs ρs De f + ρs De f Vi dn l dn s
(8.11)
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where Def = el - es is the latent heat of fusion and Vi refers to the interface velocity. In the present analysis, the solid phase will be assumed stationary, so the first term on the right side of Equation 8.11 is neglected. Similarly, the interfacial constraints for the other conserved variables, such as mass, solute concentration, and momentum, can be obtained from appropriate balances across the phase interface. For example, the interfacial relation for the concentration of component c in a multicomponent mixture, Cc, is obtained as follows: - ρl Dl
dCc dC + ρs Ds c = -Vs ρsCc,ls + ρsCc,lsVi dn l dn s
(8.12)
where the difference between phase concentrations, Cc,ls = Cc,l - Cc,s, is obtained from the binary phase equilibrium diagram (see Figure 8.1). In the case of entropy transport, the following transport equation in phase k is obtained: D ( fk ρk s k ) f k ∇T + ∇⋅ = ∇ ⋅ k k T Dt
N
∑ c =1
fkζ c,k jc,k + P s,k T
(8.13)
where P s,k refers to the entropy production rate and ζ c,k = ∂ek /∂Cc,k is the chemical potential of constituent c in phase k. It can be interpreted as an increase of work potential in the fluid, if dCc,k of constituent c is added to the mixture. On the right side of Equation 8.13, the terms represent the entropy flow associated with the heat flux and species (mass) flux, jc,k, and the entropy production rate, respectively. Entropy is not measured directly, so an additional relationship involving entropy and the conserved quantities, such as energy (or temperature), solute concentration, etc., is needed. The following Gibbs equation for a multicomponent mixture in terms of phase k will be used: dsk =
dek pdvk + T T
N
∑ c =1
ζ c,k dCc,k T
(8.14)
where vk represents specific volume of phase k. A latent heat term is not included in Equation 8.14 because the equation is written within a single phase k. Assuming an incompressible substance in each phase, rewriting Equation 8.14 in terms of a substantial derivative and rearranging terms, T
D( fk ρk sk ) D( fk ρk ek ) = Dt Dt
N
∑g
c ,k
c =1
D( fk ρk Cc,k ) Dt
(8.15)
Substituting terms from Equation 8.6 and Equation 8.7 into this equation, D ( fk ρ k s k ) 1 1 = kk ∇ 2T + τ : ∇v k + P e,k Dt T T
{
}
N
∑ζ
c ,k
{-∇ ⋅ jc,k + P c,k} (8.16)
c =1
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Expanding the divergence terms in this equation, using the product rule and comparing the result to Equation 8.13, we obtain the following result for the entropy production in phase k:
P s,k =
fk kk (∇T )2 µΦ P e,k 1 + + T2 T T T
N
∑
jc,k ⋅ ∇ζ c,k +
c =1
1 T2
N
∑
N
ζ c,k jc,k ⋅ ∇T -
c =1
∑ c =1
ζ c,k P c,k T (8.17)
The production terms on the right side of Equation 8.17 vanish after summation over the phases because production or destruction of energy or species in one phase is accompanied by destruction or production in the other phase. However, this does not apply to entropy because processes such as heat transfer, viscous dissipation, and fluid mixing are irreversible and thus produce entropy within an individual phase. The entropy interfacial constraint can also be written in terms of the local entropy production rate at the phase interface. The entropy transfer from the liquid phase into the interface, ETl, may be written as ETl = - kl
dA dT dt + ρlVl sl dAdt T dn
(8.18)
which consists of entropy flow arising from heat conduction, as well as advection, because liquid motion carries entropy into the interface. A similar expression, ETs, can be obtained for the solid phase. Similarly, as the energy analysis, the entropy difference between a liquid volume, dAdn, at the interface and a subsequent solidified volume can be written as
dS = ρl sl dAdn - ρs ss dAdn = ETl - ETs + ρl Ps,i dAdn
(8.19)
where the interfacial entropy production, Ps,i, accounts for the entropy produced per unit mass due to heat transfer and shear action along the dendrite arms, when the dendrite moves a distance dn during the time interval dt. Combining the previous equations and rearranging terms, ( ρl sl - ρs ss )
dn k dT k dT dn =- s + s + ρlVl sl - ρsVs ss + ρl Ps,i dt Tl dn l Ts dn s dt
(8.20)
Using the continuity equation, the following result for the entropy production at the phase interface is obtained: Ps,i =
ρs ρl
De f kl dT 1 1 Ds f - T + ρ V dn T - T l l l i s
(8.21)
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where Dsf = sl - ss refers to the entropy of fusion. The entropy of fusion is approximately constant for most metals and alloys, Dsf ≈ 8.4 [J/molK]. The lowercase symbol for entropy, s, refers to the intensive (specific) variable, whereas the upper-case notation refers to the extensive variable. Richard’s rule states that the entropy of fusion is approximately equal to the heat of fusion divided by the phase change temperature.
8.3 Hea t a n d En t r o py An a l o g ies in Pha s e Cha n g e Pr o c es s es The Reynolds analogy between momentum and heat flow relates the Nusselt number to the friction coefficient. This section develops similar analogies between entropy and heat flow. Transport phenomena involving one variable (temperature) will be inferred through behavior of the other analogy variable (entropy). If predicting a certain variable is difficult or time-consuming, but analysis with the other analogy variable is more readily implemented, then the analogy can be particularly beneficial. Two specific examples of heat and entropy analogies will be given, involving processes of interdendritic permeability and thermal recalescence.
8.3.1 Ir r ev er s ibil it y
of
In t er d en d r it ic Per mea bil it y
Modeling of the two-phase momentum equations requires a supplementary relation for the momentum phase interactions, Fp, for closure of the solid–liquid phase change equations. For fluid flow through a porous medium, Darcy’s law (Bird et al., 1960), FpK = vf l (vl - vs), may be used for the phase interaction force dependence on the porous medium permeability, K, and liquid fraction, f l. In Darcy’s law, a fixed dendritic section (porous medium) is required. The assumption of a stationary solid material (i.e., vs = 0) in the governing equations is needed for consistency with Darcy’s law. Previous models of solid–liquid phase change have often used the following Blake-Kozeny equation (Bird et al., 1960) for the solid permeability: fl3 K = K0 2 (1 - fl )
(8.22)
where K0 is an empirical coefficient. This model was developed from a physical analogy between interdendritic flow and Hagen-Poiseuille viscous flow (Bird et al., 1960), through a noncircular tube with an equivalent hydraulic radius based on the local liquid fraction (see Figure 8.3). At high values of f l (i.e., f l > 0.5), a crossflow perpendicular to the dendrite arms may produce a higher pressure difference than the results of the Blake-Kozeny prediction, because of wake interactions and interdendritic viscous effects. The following alternative model can be used to account for the crossflow effects (Naterer, 2000): fl K = K0 1 - fl
(8.23)
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Solid fL
Liquid
Fig u r e 8.3 Permeability analogy for interdendritic axial conditions.
It can be shown that both Equation 8.22 and Equation 8.23 approach the proper limits as f l Æ 0 (solid) and f l Æ 1 (liquid). Consider a flow alignment factor, c, to represent a weighting factor between the axial relation, Equation 8.22, and the crossflow relation, Equation 8.23, fl fl3 K = K0 χ + K 0 (1 - χ ) 2 1 - fl (1 - fl )
(8.24)
where the first and second terms represent axial and crossflow permeabilities, respectively. For example, c = 0 corresponds to a crossflow, and c = 1 represents an axial flow, with the appropriate permeability factors used in each limiting case. Entropy and the Second Law may provide key insight about how c can be best calculated to ensure physically plausible trends of interdendritic flow. Because the direction of dendritic growth is related to the local thermal irreversibility, entropy can be used to interpret axial and crossflow contributions to the interdendritic permeability. Consider a weighting between v and the dendritic growth direction, based on thermal irreversibility and entropy. In dendritic solidification, the primary dendrite arms grow in the direction of the local temperature gradient (Flemings, 1974). This gradient gives the direction of steepest ascent (or descent for a negative gradient vector). Because the primary dendrite arm growth is aligned with the local temperature gradient, this growth occurs toward the steepest temperature slope. This direction corresponds to the maximum thermal irreversibility. This thermal irreversibility, P s,t , can be subdivided into x- and y-direction components, P s,tx and P s,ty , respectively, in the following manner: 2
P s,t =
k ∂T k + 2 T 2 ∂x T
2
∂T ∂y = P s,tx + P s,ty
(8.25)
Although entropy production is a scalar (not a vector), we may define an equivalent entropy vector, Pˆ s,t, consisting of the above components of entropy production in the x- and y-directions, given by Ps,tx ˆi and Ps,tx ˆj (note: unit vectors ˆi, ˆj), respectively.
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For example, a large component P s,tx (in comparison to P s,ty) suggests that entropy production arises mainly from heat flow in the x-direction. Based on this concept, a flow alignment weighting factor is defined as follows:
χ=
| v ⋅ Pˆ s,t | ; | v || Pˆ s,t |
Pˆ s,t ≡
ˆ ˆ P s,tx i + P s,ty j
(8.26)
where c can be interpreted in terms of thermal irreversibility (or temperature gradient) relative to the direction of local interdendritic flow. The square root in Equation 8.26 is used for a more direct analogy between entropy production in Equation 8.25 and temperature gradient (or heat flow). The equivalent vector Pˆ s,t is similar to the heat flux vector, with two exceptions: (i) minus sign for Fourier heat flux, and (ii) magnitude of k | ∇T 2 | /T 2, rather than k | ∇T |. A higher crossflow weighting is given when the interdendritic flow is normal to the direction of maximum thermal irreversibility (direction of dendritic growth). Conversely, the axial permeability is adopted when the velocity is aligned with the direction of the maximum thermal irreversibility. Using a physical interpretation based on entropy and the Second Law, the alignment weighting factor will be shown (in an upcoming case study) to provide a robust method for accurate interface tracking of phase change processes.
8.3.2 T he r ma l Rec a l es c en c e a n d Dimens io n l es s En t r o py Ra t io This second example demonstrates that another multiphase process (thermal recalescence) can be interpreted or modeled through analogies and insight provided by the Second Law, which would not otherwise be gained through the conservation equations alone. During dendritic solidification, latent heat is released to the surrounding liquid–solid mixture. If it exceeds the rate of external cooling, then a local temperature rise, or recalescence, may be observed. Together, with solute diffusion, this reheating may initiate melting of smaller dendrite arms at the expense of growing primary arms (i.e., dendritic coarsening). Coarsening during crystal formation and sedimentation in solidification processes may contribute to recalescence and latent heat release. In both cases, thermodynamic irreversibilities with entropy production arise during the heat transfer processes. Because reheating and coarsening may affect the final material properties, such as material strength, these processes have significance during solidification processing of materials. In the following analysis, it will be shown that entropy serves as an important variable in these processes. Consider a simplified heat balance for a crystal (or dendrite) during solidification. The transient temperature change arises from the net heat exchange with the surrounding solid–liquid mixture (described by an overall heat transfer coefficient, h) and release of latent heat from the freezing crystal, that is,
ρVc p
∂T ∂f = hA(T - T f ) - ρV De f l ∂t ∂t
(8.27)
where V, A, and Tf refer to a characteristic crystal (or dendrite) volume, corresponding surface area, and surrounding mixture temperature, respectively. Expanding the
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last derivative in Equation 8.27 through the chain rule, we obtain
∂T hA = ∂t ρVc p
T - Tf 1 + ( ∂fl /∂T ) De f /c p
(8.28)
A characteristic length scale, Lc, may be represented by the ratio V/A (reciprocal appears in the above equation). Equation 8.28 can be written in a dimensionless form by selecting appropriate reference scales for temperature and time (i.e., tref = L2c /α ). The dimensionless entropy, temperature, time, and Stefan number (c p DT /De f ) will be denoted by s*, θ, t*, and Ste, respectively. Writing Equation 8.28 in dimensionless form, it can be shown that
θ h ′Lc = kl θ
kl kl k ≡ Nu k s s
(8.29)
where 1 h ′ = h 1 1 + Ste/( ∂fl /∂θ )
(8.30)
refers to a modified heat transfer coefficient that accounts for simultaneous external cooling and release of latent heat from the solidified crystal to the interdendritic fluid. Also, the overdot in Equation 8.29 represents differentiation with respect to time. Comparing Equation 8.29 with the rate of entropy change arising from the sensible heat portion of the Gibbs equation (i.e., Maxwell-type relation where cdT = Tds*), k ∂s∗ θ = = Nu l ∗ ∂t θ ks
(8.31)
The change of liquid fraction with temperature is calculated based on a supplementary relation, such as an approximated linear variation of f l with q through the two-phase region. In a similar way as analogies between heat and momentum transport (i.e., Reynolds analogy), this result suggests a type of analogy between heat and entropy. The process of dendritic coarsening is closely related to recalescence during solid–liquid phase change. During coarsening, small or secondary dendrite arms (or crystals) shrink or melt at the expense of heat transferred from the large and growing main dendrites. The previous results suggest that entropy may serve as an effective variable in characterizing the coarsening and recalescence. Because entropy cannot be measured directly, it requires measurements indirectly through other means. The magnitude of recalescence is observed by a measured temperature rise. The corresponding dimensionless temperature gradient at the dendrite arm (characterized by the Nusselt number) can be interpreted in terms of the local rate of entropy change. This entropy change (measured indirectly) incorporates heat flow from the interdendritic fluid, as well as latent heat released from the crystal or dendrite.
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Both mechanisms are important aspects for understanding recalescence and coarsening, during remelting of small, secondary dendrite arms. Further information regarding the effective heat transfer coefficient, h ′ (and Nu), could be gained by performing many phase change experiments with different initial temperatures (leading to different Grashof numbers) and then seeking a suitable dimensionless correlation for the Nusselt number, Nu, in terms of the Grashof number, Gr, and other parameters. This is similar to the approach when single phase convection correlations are constructed for external or internal flows. For example, the initial and wall temperatures can be measured (with resulting Grashof number), then the entropy changes and other parameters can be found by a specific phase change experiment. The effective heat transfer coefficient can be estimated from Equation 8.31. These experiments can then be repeated for other fluids and presented in a final correlated form. In the previous analysis, a uniform heating or cooling rate was assumed. Entropy can serve as an effective variable for recalescence analysis, because the local heating or cooling rate can be related to the correlation involving heat transfer coefficient and entropy in Equation 8.31. The heat-entropy analogy can provide an improved estimate of the relevant heating or cooling rate, based on the result of how the Nusselt number is correlated with the local rate of entropy change. In particular, the average heat transfer coefficient during this period can be estimated by integration over the range where a positive entropy derivative is observed in the data (i.e., conditions corresponding to recalescence or coarsening). In this way, the time period of recalescence can be expressed in terms of a heat-entropy analogy. Consider the freezing of an ammonium chloride and water mixture in a rectangular enclosure with an initial temperature and solute (ammonium chloride) concentration of 318 K and 0.72, respectively (Naterer, 2001). Thermal and solutal buoyancy in the liquid region generates two counterrotating convection cells on the left and right sides of the cavity, respectively (see Figure 8.4). Upward transport of crystals by these convection cells and sedimentation of crystals along the vertical midplane create a growing mushy layer (NH4Cl dendrites and liquid) along the lower boundary of the domain. The crystals along the vertical midplane descend to create an inverted v-shaped sedimentation layer. Due to different sizes and structures of crystals, the descent of various crystals is initiated at different positions. Recalescence is a thermal phenomenon that occurs when the rate of latent heat release from a freezing crystal (or dendrite) exceeds the rate of heat transfer from the surrounding solid–liquid mixture. This creates a local heating effect that has important impact on the properties of solidified materials, such as mechanical strength. For example, it can lead to material defects such as dendrite tearing through repetitive freezing-melting cycles. A freezing crystal (or detached dendrite arm) may lead to reheating and melting of the surrounding solid back to liquid. Combined with solute transfer, the local melting of smaller dendrite arms may occur at the expense of growing primary arms (called dendritic coarsening). From temperature fluctuations alone, the occurrence of these processes may not be evident. The heat-entropy analogy aims to provide new insight in these regards. Thermal irreversibilities occur by heat transfer and phase change in these processes. The previous analysis gave a relationship between the rate of entropy change and the ratio between the
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* *
* *
*
Convection Cell
*
*
*
*
*
*
*
(a) Thermocouple Wire
Cold Boundaries (Connected to Heat Exchanger)
Test Midplane
Solid Liquid
Thermocouple Grid 1
5
12
16
2
6
11
15
3
7
10
14
4
8
9
13
(b)
Fig u r e 8.4 Schematic of (a) convection cell and (b) thermocouple positions.
transient temperature derivative and temperature itself. Comparing Figure 8.5a and Figure 8.5b at location 3, it can be observed that the temperature rise after about 0.26 h (i.e., period of recalescence; Figure 8.5a) is closely coincident with the crossover to a positive value of entropy change in Figure 8.5b. This analogy between temperature and entropy can shed new light on the associated thermodynamic processes. The magnitude of the highest entropy change indicates the maximum rate of reheating and coarsening, whereas the duration of the positive entropy change indicates the length of time over which the recalescence occurs. The magnitude of the entropy change includes the following thermal irreversibilities: (i) release of latent heat by the freezing material, and (ii) coarsening by heat absorption or melting of the smaller arms, combined with heat release from the
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(1)
(2)
(3)
(4)
60.0
Legend Liquidus
Temperature (°C)
45.0
30.0
15.0
0.0
–15.0
(17) 0
(20) 2300.0
6900.0
4600.0 t(s)
9200.0
(a) 0.40 T0 = 343 K, C0 = 0.68
ds/dt (W/kgK)
0.20
0.00
–0.20 Location (3) Location (4)
–0.40
Location (7) Location (8)
–0.60
0.27
0.46
0.66 Time (hr) (b)
0.86
1.05
Fig u r e 8.5 (a) Measured temperatures, and (b) rate of entropy change for solidification with free convection in a cavity.
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growing larger arms. The magnitude of these combined irreversibilities decreases after the point of maximum ds/dt in Figure 8.5b. At higher initial concentrations of water, the level of initial undercooling of the liquid is lower (i.e., liquidus temperature minus the wall temperature is lower). As a result, slower solid growth occurs with less pronounced temperature fluctuations from thermal recalescence. A period of recalescence (Dt > 0) occurs when the rate of latent heat release within a freezing crystal (or dendrite) exceeds the rate of heat transfer from the solid–liquid mixture around the crystal, thereby creating a local heating effect. Thermal irreversibilities arise during recalescence when heat is released by freezing material and transferred to the surrounding liquid or solid. Similarly, irreversibilities occur during coarsening by heat absorption or melting of smaller arms, as well as heat release by growing larger arms. These effects can be related to the magnitude and slope of the entropy change. The magnitude of the maximum entropy change indicates the highest rate of reheating and coarsening, whereas the duration of positive entropy change indicates the approximate time period of recalescence. As the coarsening time increases, an increasing number of smaller arms disappear while the main dendrite arms grow larger and their spacing increases.
8.4 N u mer ica l St a bil it y o f Pha s e Cha n g e Co mpu t a t io n s 8.4.1 M o d el in g
o f Tw o
-Ph a se En t r o py Pr o d u c t io n
In addition to heat-entropy analogies based on entropy in the previous section, the Second Law can provide unique insight for improving the performance of a numerical formulation. In this section, a Second Law formulation is presented with predictive and corrective capabilities for the improvement of phase change predictions. The formulation can serve as an effective complement to the discretized conservation equations in the overall numerical procedure. This section will focus on numerical modeling and implications of entropy and the Second Law in the numerical formulation. The Second Law for a multiphase mixture can be written in the following form:
P s,k ≡
∂Sk + ∇ ⋅ Fk ≥ 0 ∂t
(8.32)
In Equation 8.32, P s,k , S(f k), and F(f k) refer to the entropy production rate in phase k, entropy (as a function of the vector of conserved quantities, f k), and the entropy flux in phase k, respectively. Expanding the entropy flux in terms of advective and diffusive components, Equation 8.32 can be written as
D ( λ k ρk sk ) λ k ∇T + ∇⋅ = ∇ ⋅ k k T Dt
N
∑ c =1
λk gc,k jc,k + P s,k T
(8.33)
where l k , sk , and gc,k = ∂ek ∂Cc,k refer to the mass fraction, specific entropy and chemical potential of constituent c, in phase k, respectively. The summation includes each chemical potential from c = 1 to c = N constituents, i.e., N = 2 for a binary alloy. The species flux, jc,k, for constituent c in phase k is determined by Fick’s law.
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On the left side of Equation 8.33, the substantial derivative includes both transient and advection terms. The entropy flux (Fk) in Equation 8.32 includes an advective component from the substantial derivative in the left side of Equation 8.33, as well as diffusive (energy and species) components in the first two terms on the right side of Equation 8.33. A positive–definite expression is needed for the evaluation of the entropy production rate. From Section 8.2, this expression can be determined from the Gibbs equation and the entropy transport equation. The following mixture expression can be derived after summing over both solid and liquid phases: 2
P s =
∑ k =1
λk kk (∇T ⋅ ∇T ) µΦ 1 + T2 T T
N
∑
jc,k ⋅ ∇gc,k +
c =1
1 T2
N
∑g
j
c ,k c ,k
c =1
⋅ ∇T
(8.34)
where Φ refers to the viscous dissipation function. From Section 8.2, the interfacial entropy production is, Ps,i =
DE f k dT 1 1 ρs + l ≥0 DS f T ρl ρlVi dn l Tl Ts
(8.35)
where DS f = sl - ss refers to the entropy of fusion (approximately equal to the heat of fusion divided by the phase change temperature). The entropy production in Equation 8.35 includes effects of viscous dissipation due to shear action along a dendrite arm, as it moves a particular distance over a time interval. The discretized form of the Second Law can be written as
Sin +1 - Sin P s ≡ DV + Dt
∑ F (DS ) ≥ 0 ip
ip
(8.36)
ip
where the summation over “ip” integration points refers to surface flux calculations in a finite volume method. A reconstruction step is required in Equation 8.36 because the distribution of conservation variables φ ( x,t ) must be approximated from integration point and nodal values. It will be assumed that f is piecewise constant within a control volume. This quasiequilibrium assumption can complete the reconstruction step, without violating the Second Law. The Gibbs equation will be used to write the transient entropy derivative in Equation 8.36, in terms of variables obtained from solutions of the conservation equations, such as temperature and liquid fraction, as follows:
Sin +1 - Sin ρc p = n Dt Ti
λln,i+1 - λln,i Ti n +1 - Ti n + ρDS f Dt Dt
(8.37)
In this approach, entropy computations can be distinguished between sensible and latent heat contributions.
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An entropy equation of state is required for implementation of the Second Law. For solid–liquid phase transition, the entropy varies with temperature and concentration. Based on the form of the Gibbs equation for an incompressible substance, consider a piecewise logarithmic equation of state in the following form: T s(T , C ) = sr ,k + cr ,k log Tr ,k
(8.38)
where the subscripts r and k denote reference and phase, respectively. In the solid phase (k = 1), the following reference values will be used: sr,1 = 0
cr ,1 = cs Tr ,1 = Te (eutectic)
In the mushy (two-phase) region (k = 2), the reference specific heat must include both sensible and latent entropy contributions. Also, the reference entropy is determined at the solidus line. The following reference values are obtained (Naterer, 2000): sr ,2 = cs log(Tsol /Te )
cr ,2 =
(
csTliq - clTsol Tliq -Tsol
)+
cl - cs +DS f log (T Tliq /Tsol )
Tr ,2 = Tsol
Finally, the Gibbs equation and the binary phase diagram (Figure 8.6) can be utilized to derive the following reference values in the liquid phase (k = 3):
sr ,3 =
(
csTliq - clTsol Tliq -Tsol
) log (T
liq
/Tsol ) + cs log (Tsol /Te ) + cl - cs + DS f cr ,3 = cl
Tr ,3 = Tliq
Examples of typical entropy equations of state are illustrated in Figure 8.6a and Figure 8.6b (note logarithmic scale). Thermophysical properties of the material in Figure 8.6a and Figure 8.6b include cs = 167[J/kg K] and cl = 167[J/kg K]. The melt temperature, eutectic temperature, and liquidus-eutectic intersection (see Figure 8.1) are T = 343[K], T = 341[K], and CL = 0.4, respectively. In Figure 8.6a, L = 32.6[KJ/kg] and CS = 0.2, whereas in Figure 8.6b, L = 3.26[KJ/kg] and CS = 0.38 (see Figure 8.1).
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100
100
10
10
1
1
Entropy
0.1
0.1 0.01 0.35
75 0.3
0.25
0.2 Concentration 0.15
0.1
0.05
0 (a)
68
68.5
69
69.5
0.01
70
Temperature
100
100
10 Entropy 1
10 1
0.1
0.1 0.01 0.35
75 0.3
0.25
0.2 Concentration 0.15
0.1
0.05
0 68 (b)
68.5
69
69.5
0.01
70
Temperature
Fig u r e 8.6 Entropy equation of state with (a) L = 32.6 kJ/kg, CS = 0.2; (b) L = 3.26 kJ/kg, CS = 0.38.
8.4.2 It er a t iv e Ph a se Ru l es
a n d t he Sec o n d La w
Because the phase change process is nonlinear, numerical iterations are needed within a time step to achieve solution convergence. Determining the phases in each control volume requires a solution of the energy equation, but the phase distribution is needed before a solution can be obtained. Thus, a tentative phase distribution is required prior to the solution of the energy equation. If the computed solution leads to a different phase distribution, then further iterations are required until convergence between tentative and computed phases is achieved. The phase iterations will
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231
be established based on discrete analogies of the Second Law (the following three rules): 1. A control volume must pass through a melt region, during phase transition between solid and li quid phases. 2. Phase transition in a control volume cannot occur without a phase transition to the same phase in an adjacent volume first. 3. The tentative phase within a control volume must give a positive entropy production rate for the discrete time step. If the Second Law is violated in a control volume, then an entropy-based correction is required in the numerical formulation. It has been shown that the first two rules can be interpreted as discrete analogies of the Second Law (Naterer, 2001). For example, rule (1) remains consistent with nonnegative entropy production in the interfacial entropy constraint, Equation 8.35. It is not surprising that the Second Law can be interpreted in various different ways. For example, Carnot’s statement of the Second Law requires that heat cannot be converted completely and continuously into work. On the other hand, Kelvin stated that it is impossible by means of an inanimate material to derive a mechanical effect from any portion of the material, by cooling it below the temperature of the coldest of the surrounding objects. In a similar way, the previous phase rules express alternative consequences of the Second Law in the case of a solid–liquid phase change. The previous rules (1) and (2) represent effective procedural guidelines for phasetemperature iterations, as well as discrete analogies of the Second Law. Under certain circumstances, a volume may ultimately exist in a phase different from its neighboring volumes (i.e., supercooled dendrite pocket ahead of a liquidus interface) without violating the second rule during its progression between phases. To clarify this situation, consider a sequence of solution iterations illustrated by steps (0) to (5) for a binary alloy in Figure 8.7 with solution convergence at step (5). In this example, different initial mean solute compositions, C1, C2, and C3 in volumes 1, 2, and 3, respectively, create the liquidus temperature, TL , and solidus temperature, Ts, step functions (see Figure 8.7). After solution (1), rule 2 enforces volumes 3 to 7 to return to a liquid (L) phase because none of their neighbors existed in a melt (M) phase at step (0). A similar result for volumes 4 to 7 occurs after solution (2). After solution (3), rule 1 permits a solid (S) volume 3 due to the nature of the liquidus and solidus temperatures between two melt (M) volumes. Following a subsequent rule 2 correction after solution 4, the final state illustrates a converged solution with a solid volume 3 between two volumes, 2 and 4, with melt (M) phases. The solution converges because the tentative and modified phases agree with one another after the application of the phase rules. This example demonstrates how physical reasoning embodied by phase rules and the Second Law provides a sound basis for numerical iterations of a phase change problem. Although rules 1 and 2 do not ensure positive computational entropy production in rule 3, they permit a closer compliance with the Second Law, in comparison to other iterative techniques based on ad-hoc convergence tolerances or purely numerical manipulations such as underrelaxation factors. Local satisfaction of the Second
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CS = 0.23 T CL = 0.54 M C1 = 0.15 C2 = 0.30 C3 = 0.10
5 4 3 2 1 Solid x = 1 x = 0 Liquid T=1 T=0 Interface
Phase Diagram (Ag-Sn)
C3
C1
CS C2
CL
M M L
M M M M L L
S M M M L M
S S S
M M L
M M M M L M
S M M M S M
S S S
S M M M S M
S S S
TL 0
1 Phase
L
L
L
L
L
L
S
x
Solution Rule 1 Rule 2
TS
Solution Rule 1 Rule 2
3
L L L
M M L
M M M M L L
S M M M M M
S S S
x
L L L
x
TL 5
TS
Solution Rule 1 Rule 2
x
TS x
Solution Rule 1 Rule 2
TL 4
L L L
TL
TL 2
TS x
L L L
M M L
M M M M M M
S M M M S M
S S S
x
Solution TS Convergence
Solution Rule 1 Rule 2
L L L
M M M
M M M M M M
x
Fig u r e 8.7 Binary alloy solidification from side boundary.
Law provides several potential benefits, such as convergence enhancement, and a physically based understanding of discrete error analysis. The third rule aims to ensure compliance with the Second Law. In the next section, a predictive technique, as well as additional entropy-based diffusivity corrections, will be applied in the computations.
8.4.3 En t r o py Co r r ec t io n
o f Nu mer ic a l Co n d u c t iv it y
In this section, an entropy-based approach will apply both corrective steps for accuracy improvements (entropy-based diffusivity), as well as predictive steps (nonlinear time constraint) for stable computations. The magnitude of the computed negative entropy production rate can provide a quantitative measure of the degree of discretization error in the control volume. The third phase rule in the previous section represents a quantitative rule, whereas the first two rules provide qualitative or procedural guidelines. If the Second Law is violated within a discrete control volume, then a quantitative indication of the diffusivity required to correct the solution may be
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233
expressed in terms of the negative entropy production rate (absolute value) from Equation 8.34 and Equation 8.36. This effective diffusivity will be substituted into the discrete equations, and a corrected solution will be obtained. It is anticipated that this corrected solution can achieve closer compliance with the Second Law. Summing over the solid and liquid phases, and rewriting Equation 8.34 in terms of an entropy-corrected conductivity, ks,
T2 T cp ks ≈ P s + ∇T ⋅ ∇T Φ Pr
(8.39)
where cross-diffusion (Soret) effects have been neglected. In the numerical procedure, the solution is obtained by first solving the conservation two-phase equations independently of the Second Law. Following each time step, the entropy production rate is then computed based on the Second Law in Equation 8.36. If a nodal value of P s is negative, then discretization errors occur or the local solution exhibits nonphysical behavior. A computed negative entropy production at the phase interface may indicate a discretization error arising from temporal or spatial differencing or both. Thus, instead of proceeding to the next time step, a corrective diffusion step is taken in the next solution of the discrete equations, to prevent the computed entropy destruction. The required quantitative amount of corrective diffusion is inferred by the magnitude of the discretization error outlined by Equation 8.39. In the implementation of Equation 8.39, temperature gradients and the viscous dissipation function are evaluated from the numerical solution and the interpolation functions. Then, an entropy-based conductivity, ks, is combined with the molecular conductivity in the following manner:
k → k + Max | ks , ε |
(8.40)
where e represents an upper limit on the corrective step. Because this entropy-based approach is decoupled from the implicit solution of the conservation equations, the term e may be interpreted as a relaxation factor in the corrective procedure. Numerical dispersion is a quasiphysical effect that may distort phase relations between various thermal or fluid waves, through odd derivative terms appearing in the truncation error. A comprehensive error analysis, based on a Taylor series truncation for multidimensional problems, is unavailable for nonlinear phase change problems with fluid flow. The current entropy-based approach provides an alternative basis for a unified approach to error analysis. Differences between computed and physical entropy, particularly those differences leading to computational entropy destruction, may allow us to quantify the amplitude and frequency of the discretization error. Entropy-based corrections of the thermal diffusivity were introduced when negative entropy production rates were computed at the current time level. If the entropybased diffusivity coefficients alternated between zero and large nonzero values, then a “smooth” diffusivity distribution could be obtained in an appropriate manner. A concern would be poor results arising from the application of a highly irregular (digital) diffusivity field. A smoothing algorithm could be applied to overcome this potential problem. Positive diffusivity coefficients are computed and located at
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global nodes whenever the Second Law is violated in the corresponding finite control volume. If the resulting diffusivity variations resemble a “digital” distribution (i.e., diffusivity alternates between zero and nonzero values), then diffusion can be applied over the entire mesh to smooth the distribution. For example, a Jacobi iterative procedure can be used to solve the diffusion (of conductivity) equation.
8.4.4 En t r o py Co n d it io n
f o r Tempo r a l St a bil it y
In addition to corrective steps described in the previous section to improve accuracy and convergence of numerical computations, entropy and the Second law can provide key insight for predictive measures to ensure stability of phase change computations. This section presents a stability analysis based on the Second Law, which can be applied to time-step selection in phase change problems. Consider the following one-dimensional transport equation for a general scalar, f: La (φ ) = ρφ,t + ρuφ,x - Gφ,xx - c(φ - φr ) = 0
(8.41)
where the subscript comma notation refers to differentiation and G refers to a reference value. In Equation 8.41, G and c refer to diffusion and source term coefficients, respectively. We will use ξ to refer to the entire source term, i.e., last term on the right side of Equation 8.41. Integrating Equation 8.41 over a discrete volume (one dimensional) and time step, φ n - φ n-1 i Ld (φ ) = ρ i Dx + ( M φ - Gφ ,x ) i +1/2 - ( M φ - Gφ ,x ) i -1/2 - ξDx Dt
(8.42)
where M = ρu refers to the mass flow rate per unit width and depth. The subscripts i - 1/2 and i + 1/2 refer to west and east integration points, respectively. Also, L and the superscripts a and d refer to operator, analytic, and discrete, respectively. In the discrete formulation, Ld (φ ) gives a nonzero residual, because the approximate solution, φ, in general, will be different from the exact solution. In the following analysis, the overbar tilde notation will be subsequently dropped, and it will be understood that the discrete operator acts on the approximate solution, φ . Using standard finite difference approximations for the convection and diffusion terms, φ n - φin-1 φi - φi -1 φi +1 - φi Ld (φ ) = ρ i + M (1 - α ) + Mα Dt Dx Dx φ - 2φi + φi +1 - G i +1 - ξ Dx Dx 2
(8.43)
Here α = 1 represents upwinding, and α = 1 / 2 represents central differencing. This factor may include a weighting bias on the local Peclet number to include the
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proper balance between upstream and downstream influences. Using Taylor series expansions, Dt 2 φin = φin-1 + Dtφ,t + φ + 2 ,tt
(8.44)
Dx 2 φi +1 = φi + Dxφ,x + φ + 2 ,xx
(8.45)
A similar expansion may be written for the scalar value at the upstream location about point i. Substituting these expansions into the discrete operator, G 1-α Ld (φ ) = La (φ ) + ρur Dxφ,xx φ,x Dx i +1 2 1 G α φ,x + ρDtφ,tt - ρur Dxφ,xx 2 Dx i -1 2 i
(8.46)
where ur represents a reference, or characteristic (i.e., lagged), linearization velocity. Also, higher-order terms have been neglected. Also, taking derivatives in Equation 8.41,
ρφ,tt = - ρur φ,xt + Gφ,xxt - ρcφ,x
(8.47)
ρφ,xt = - ρur φ,xx + Gφ,xxx - ρcφ,t
(8.48)
In the present analysis, only transient, convection, and source-type terms will be subsequently analyzed because these terms will predominantly affect the numerical stability. Then, substituting Equation 8.47 and Equation 8.48 into Equation 8.46, Ld (φ ) = La (φ ) +
1 1 ρur (ur Dt - Dx )φ,xx + ρDtc( 2ur φ,x + cφr - cφ ) 2 2
(8.49)
Thus the discrete operator depends on the second-order spatial derivative of the scalar, such as temperature, and higher-order terms (neglected). In addition, the Second Law of Thermodynamics may be written as (8.50) P s = S,t + F,x where S(f) and F(f) represent the entropy and entropy flux, respectively. As discussed in previous chapters, these entropy functions must satisfy two important properties: (i) downward concavity, and (ii) compatibility. Let us now premultiply the discretization error, L d(f), by S,f in Equation 8.49 to give the following expression for the
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entropy-weighted discretization error: 1 1 S,φ Ld (φ ) = S,φ La (φ ) + S,φ ρur (ur Dt - Dx )φ,xx + ρDtc( 2ur φ,x + cφr - cφ ) (8.51) 2 2 Also, from the chain rule of multivariable calculus, S,φ φ,xx = ( S,φ φ,x ),x - ( S,φφ )(φ,x )2
(8.52)
where the last term is positive definite as a result of downward concavity of the entropy function. Expressing f in dimensionless terms with a Taylor series expansion about the phase interface, then seeking an analogous result as Equation 8.52 from the chain rule for the first-order derivative of f, it can be shown that S,φ φ,x = ( S,φ φ ),x - Dx( S,φφ )(φ,x )2
(8.53)
If we integrate Equation 8.52 over a discrete volume, then the first term on the right side becomes the difference between the entropy gradient at the two edges of the control volume, whereas the remaining terms represent average values of entropy derivatives and entropy production. The difference between entropy gradient terms diminishes, in comparison to the remaining terms, when the grid spacing is refined. Approximating the right side (entropy-weighted discretization error) in Equation 8.51 as zero, and using the chain rule to write the first term as the entropy production rate, 1 1 P s = S,φ ρur ( Dx - ur Dt )φ,xx + HOT + S,φ ρDtc( -2ur - cDx )φ,x + HOT (8.54) 2 2 where HOT refers to higher-order terms. Thus, substituting Equation 8.52 and Equation 8.53 into Equation 8.54, it can be shown that P s = -
1 1 ρur ( Dx - ur Dt )S,φφ (φ,x )2 - ρDtc( -2ur - cDx ) DxS,φφ (φ,x )2 ≥ 0 (8.55) 2 2
As a result of the downward concavity property of entropy, and the requirement of positive entropy production in the Second Law, the following result is obtained: Dt ≤
ur Dx (ur + cDx )2
(8.56)
Thus, the Second Law identifies an appropriate time step for the numerical computations. Results from the previous stability constraints are illustrated in Figure 8.8a
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Entropy Transport with Phase Change Heat Transfer 0.4
0.3
Unstable
∆t 0.2
0.1
0
Stable 0
0.01
0.02 ∆x (a)
0.03
0.04
6 5 Unstable
4 ∆t 3 2 1 0
Stable 0
0.1
0.2
c (b)
0.3
0.4
0.5
Fig u r e 8.8 Stability curves for (a) a fixed interface velocity and (b) fixed grid spacing.
and Figure 8.8b. It can be observed that the time step must be reduced when the grid spacing increases, or the phase change coefficient, c, increases (i.e., interface velocity and source terms increase), to maintain stable computations. In a similar manner as the CFL Courant condition for single phase problems, the time step is reduced when ∆x approaches zero because a disturbance would not propagate beyond the extent of the control volume boundaries for stable time advance.
8.4.5 C a se St u d y
o f Mel t in g in a n En c l o su r e
This section presents a case study with several example problems that apply the methods developed in the previous section to demonstrate the valuable utility of the Second Law in phase change analysis. The application problems in this section will illustrate both physical and computational aspects of entropy production, as well as their roles in predictive capabilities of the overall numerical formulation.
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Liquid
Y
Solid Tc X
Fig u r e 8.9 Schematic of melting problem.
A control volume-based finite element method (Naterer and Schneider, 1996; Naterer, 2000) will be used in the numerical analysis. Although the present results will consider heat transfer with melting and solidification, the entropy principles can be extended to other heat transfer problems, such as phase change with boiling or condensation. The following thermophysical properties for materials in the application problems will be given in the liquid (subscript l) and solid (subscript s) phases. For the Ag (silver)–Sn (lead) alloy, cp,s = 250[J/kgK], cp,l = 285[J/kgK], m l = 7.1 × 10-8[kg/ms], ks = kl = 315[W/mK], r = 105000[kg/m3], L = 120[kJ/kg], Te = 494[K], Tm = 1234[K], ce = 0.54, and kp = 0.43. For the Lipowitz (Cerrobend) material, cp,s = cp,l = 167[J/kgK], vl = 3.31 × 10-7[m2/s], ks = 19[W/mK], Kl = 5.5[w/mK], L = 32.6[kJ/kg], and Tm = 374[K]. Consider an application problem with melting from an upper boundary (see Figure 8.9) (Gau and Viskanta, 1984). In this example, an initially solid material at Tc = 68[o C] is suddenly exposed to a hot upper boundary ( Th = 92[o C]). The melting temperature is Tm = 70[o C], and both vertical boundaries are well insulated. Melting of the solid begins near the upper boundary and proceeds downward as time advances. Interface movement results are illustrated in Figure 8.10a. Sharp changes in curvature of the temperature profiles characterize the interface movement over time. Each sharp change of curvature occurs at the melting front. As time advances, the temperature gradient and heat flux decrease in the bulk liquid region, because the interface moves farther inward. In Figure 8.10a, close agreement between the computed results and experimental data (Gau and Viskanta, 1984) is achieved when the grid spacing is reduced.
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0.8
Corrected Computations
0.6 y/Y
With EntropyCorrected Model
0.4
Without EntropyCorrected Model
0.2
0
0
0.2
0.4
0.6
0.8
Fo
1
1.2
1.4
1.6
1.8
(a) 1000
Direct Computations (10 × 07) Corrected Computations
800
Entropy-Stable Result (Nonnegative Entropy Production)
Ps (W/m 3K)
600 400 200 0 –200
0
0.2
0.4
y/Y
0.6
0.8
1
(b)
Fig u r e 8.10 Computed (a) interface position and (b) entropy production.
Entropy-corrected computations of interface movement and entropy production results are illustrated in Figure 8.10a and Figure 8.10b, respectively. Corrective steps in the computations are based on the entropy formulation. Without the entropycorrected model, the direct computations underpredict the interface position at
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large Fourier (Fo) (nondimensional time) values. The entropy-correction technique improves the predictions (see Figure 8.10a). At Fo = 0.32, the entropy production is illustrated in Figure 8.10b. The large positive entropy production occurs near y/Y = 0.3 because the phase interface closely coincides with this position at Fo = 0.32 (see Figure 8.10a). At the interface, an entropy of phase change is absorbed by the melting solid and then transferred outward by heat conduction. In the direct computations, a positive entropy production is observed at the interface, but behind the advancing interface, some computational entropy destruction occurs. This result coincides with discretization errors, due to coarse spatial differencing that leads to overpredicted interface locations in Figure 8.9 with 10 × 7 elements. Improved predictions can be achieved with the entropy-corrected computations (see Figure 8.10b). In the corrected computations, the diffusivity required to correct the direct calculations is based on the magnitude of negative computed entropy production (absolute value) whenever the Second Law is violated in a discrete control volume. When both physical and computational parts of entropy production are combined, then interpretation of the entropy prediction becomes more difficult because nonphysical data associated a nonobservable event may have been combined with positive physical entropy production. In other words, a physically plausible result may occur, even though the computations destroyed entropy, thereby creating a nonphysical situation. Unfortunately, this situation cannot be readily identified when the numerical and physical parts of the computations are not separated. Conventional analysis of errors, such as “diffusive” or “dissipative” errors, would not apply to nonobservable processes. For example, a nonobservable process would be negative kinetic energy. The abstract meaning of entropy computations requires a special interpretation in such cases. The Second Law may reveal nonphysical interactions between thermal or fluid waves, which might otherwise be too complex to assess in terms of physical plausibility. For example, consider a soliton, which represents a wave that behaves like a particle. It can be shown that when high, fast waves are sent behind low, slow waves, then each series of waves may preserve its identity through the interactions, even though these interactions (soliton waves) are nonlinear. The outcome of such complex wave interactions may not be easily understood. The Second Law is a physical principle available to determine the correct behavior. In the current example, the propagation of thermal waves is disturbed through its interaction with the phase interface and its subsequent release or absorption of latent heat. The Second Law can be used to identify nonphysical processes through the manifestation of negative entropy production.
8.4.6 C a se St u d y
o f Fr ee Co n v ec t io n a n d So l id if ic a t io n
In this final example, solid–liquid phase transition with natural convection is predicted in a two-dimensional domain (see Figure 8.11a) (Naterer, 2000). This problem considers solidification of an initially liquid metal at T = 70[o C], subject to Dirichlet boundary conditions along the vertical boundaries and adiabatic boundary conditions along the upper and lower boundaries. Zero velocity conditions are specified along all boundaries. In Figure 8.11a, Y = 8.89[cm], X = 8.89[cm] , Tc = 68[o C], and Th = 92[o C].
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Th
Free Convection Recirculation cell
Y
Tc
Liquid Solid
98.0 95.0 92.0 89.0 86.0 83.0 80.0 77.0
.8 .6
74.0
.4
71.0 68.0
1
.2 0
0.012
0.025 0.038 Width (m) (b)
0.05
0 = Y/Yo
Fig u r e 8.11 (a) Schematic of solidification problem and (b) onset on numerical oscillation.
Solidification begins at the cold right boundary, and the phase interface propagates inward as time advances. Thermal buoyancy creates an upward flow near the left boundary and downward flow near the phase interface and mushy regions in the right section of the domain. This resulting clockwise recirculation cell has a key role in the phase interface advancement and heat transfer characteristics. Recall that the following entropy-based time-step constraint, which was derived for numerical stability, Dt ≤
ur Dx (ur + cDx )2
(8.57)
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where ur and cDx in Equation 8.57 refer to a reference velocity (or characteristic, lagged, linearization velocity) and a characteristic velocity associated with phase change, such as the interface velocity, respectively. In this two-dimensional phase change example, the time-step constraint becomes Dt ≤ 5[s]. This stability condition provides a general order of magnitude for recommended time steps. Temperature results at t = 40[s] from two different time advances (Dt = 20[s] and Dt = 5[s], respectively) were investigated. The results represent two different cases: (i) time step exceeds nonlinear stability constraint, and (ii) time step complies with the stability constraint. Convergence criteria (interequation residual tolerance and maximum iteration cycles) are identical in both of the above cases; only the time step is modified between different simulations. Instead of an ad-hoc time step selection, the Second Law provided guidance in the time-step selection for subsequent numerical stability in the computations. The time step adopted in the latter result (Dt = 5[s]) comes within a close proximity of the above entropy-based time-step guideline, and it yielded stable computations. However, the former case (Dt = 20[s]), which exceeds the time-step constraint, leads to an onset of instability, or oscillation, arising near the phase interface (see Figure 8.11b). The entropy-based time-step constraint provides an approximation of the general order of magnitude required for stable computations with phase change. In these computations, interequation iterations are performed between the momentum and energy equation (i.e., phase-temperature iterations based on discrete analogies of the Second Law). Additional predictive and corrective steps based on the Second Law are performed to ensure numerical stability and convergence.
8.5 Ther ma l Co n t r o l o f Pha s e Cha n g e w it h In v er s e Met ho d s 8.5.1 F o r mu l a t io n
o f an
In v er se Met h o d
The previous sections have examined the importance of entropy and the Second Law in direct problems, where boundary conditions are known and internal temperatures, velocities, and other dependent variables are sought. Inverse problems represent another class of important problems that specify desired behavior of the dependent variables within the domain, then require boundary conditions to be determined. Inverse problems often suffer from greater instability problems than direct solutions because any perturbations in boundary conditions are magnified into the domain to generate numerical oscillations. This section will consider heat and entropy transport in inverse modeling of phase change processes with solidification. Inverse methods provide an effective way to control phase change processes by changing boundary temperatures to achieve a desired progression of isotherms within the domain. A fixed domain method will be used, and the phase interface moves uniformly at a desired progression rate. An entropy-based correction is applied to improve the numerical stability of the inverse computations. The formulation considers local violations of the Second Law (arising from spatial or temporal discretization errors) as a criterion for a corrective strategy in the computations. The magnitude of negative entropy production is used for a quantitative correction of the apparent thermal
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Liquid
Solid (a)
T(t)
Interface Solid
V Tm
Liquid (b)
Fig u r e 8.12 Schematic of (a) multiphase control volume and (b) inverse Stefan problem.
conductivity. It will be shown that this approach provides an effective alternative to previous inverse stabilizing techniques, such as future time stepping. As an example of a typical inverse problem, consider solidification in a closed region (see Figure 8.12) that is initially occupied by a pure liquid of temperature Tin(x), where Tm denotes the melting temperature. The governing equation consists of onedimensional heat conduction with solid–liquid change, as developed previously in this chapter. Because the top, bottom, and right boundaries of the cavity are insulated, no temperature gradients (and no thermal buoyancy) arise in the liquid region. The position of the interface will be controlled by the temperature of the left boundary. The conductivity, k, density, r, and specific heat, c, are independent of temperature. Also, the melting temperature, Tm, is given. In this example, it is assumed that the interface moves in the x-direction, and the shape of the interface remains a vertical straight line (moving into the domain). Also, the velocity of the phase interface can be given as a desired quantity. The temperature at the left boundary is uniform, and it only varies with time, i.e., T0 = T0 (t). Although the domain is illustrated in a two-dimensional geometry (Figure 8.12), this example can be treated as a one-dimensional heat transfer problem. In an inverse problem, with the exception of the boundary having the unknown controlling temperature, all boundary conditions can be treated as a direct problem,
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with Dirichlet, Neumann, or Robin boundary conditions. At the controlling boundary, the temperature distribution remains unknown, and it represents the final solution of the inverse problem. This temperature is obtained through iterations, following an initial estimate. At the beginning of each time step, a temperature estimate is initially taken. For example, it may be set as the temperature value at the previous time step (lagged estimate). Then, during each subsequent step in the simulation, the controlling boundary temperature (at x = 0 in our example) is updated at each iteration until the predicted interface movement agrees, within a given tolerance, with the specified (or desired) phase interface movement. In the current formulation, the following iterative update of the boundary temperature T0, is used: T0( m +1) = T0m +
Tpm++11 - Tpm+1 Rp +1
(8.58)
where m, p + 1 (subscript), and R refer to the iterative counter, nodal point p + 1, and the sensitivity coefficient, respectively. The sensitivity coefficient is defined as follows: Rp +1 =
∂Tp +1 ∂T0
(8.59)
This coefficient essentially measures the influence of changes in the boundary temperature, T0, on the temperature at nodal point p + 1. The range of the sensitivity coefficient is 0 ≤ R ≤ 1. The value of R may be interpreted as a temperature connection between nodal point p + 1 and the boundary value. For example, when the sensitivity coefficient, R, becomes larger, then the influence of the changes at the boundary temperature on the nodal value at point p + 1 becomes stronger. Nodal values closer to the boundary yield larger sensitivity coefficients. This coefficient will be used to update the boundary temperature in the current inverse model. The finite volume equations for the sensitivity coefficient, R, can be derived after taking derivatives with respect to T0 on both sides of the discretized energy equation (Xu and Naterer, 2001). The inverse heat transfer problem is ill-posed, since arbitrarily small errors in the temperature measurements or interface position can be projected back to the boundary as magnified large errors. For inverse phase change problems, when the interface moves farther from the boundary, it becomes more difficult to control the interface by adjusting the boundary temperature. As a result, numerical oscillations may arise in the computations. Voller (1992) developed a future time stepping method to address this problem. In contrast, the following section examines how the Second Law can be used to stabilize computations in inverse problems.
8.5.2 En t r o py Co r r ec t io n
f o r Nu mer ic a l St a bil it y
For an inverse problem with solidification from the left boundary, the interface moves farther away from the left boundary over time, so the effect of the boundary temperature on the interface movement becomes weaker. The thermal influences
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245
may not be effectively carried from the boundary to the interface due to the large distance between them. As a result, it becomes more difficult to control the interface movement solely by the temperature at the left boundary. Numerical oscillations in the inverse solution may start to occur. When the interface moves farther away from the left boundary, the sensitivity coefficients become smaller. As the values of the sensitivity coefficient become small, the resulting roundoff error may lead to numerical instability. Meanwhile, from the iterative update equation for the boundary temperature, it can be shown that when the sensitivity coefficient becomes very small, the updated boundary temperature will exhibit large changes between one iteration and a subsequent iteration. The solution may become unstable if the iterative values change drastically. Consequently, the entropy production rate may become negative whenever numerical instabilities arise. In this situation, the Second Law would be violated. Thus, the algorithm should be modified to stabilize the calculations. The entropy production can be used as a predictive tool in this regard. If the local value of entropy production becomes negative, then a numerical instability may arise, and an entropy-based correction of the computations should be made. The corrective procedure would only be applied when nonphysical solution behavior occurs. In this way, we can reduce overall computational time and avoid corrections of the solution at every time step. The entropy production is computed after the solution of the energy equation is obtained. If a nodal value of P s is negative, then the local solution is not physically plausible. Instead of proceeding to the next time step, a correction is performed based on the magnitude of entropy production within the discrete volume. From the positive-definite expression for entropy production, the conductivity can be expressed in terms of the entropy production, P s , as follows: k=
T 2 | P s | ∇T ⋅ ∇T
(8.60)
In this way, the effective conductivity, k, is related to the local entropy production rate and temperature gradient. If the Second Law is violated locally, then the conductivity can be corrected based on the computed entropy production rate. Then this entropy-based conductivity would be used to calculate the sensitivity coefficients again, from which modified sensitivity coefficients are obtained. These new sensitivity coefficients can be used when updating the boundary temperature during each iteration. This method can prevent potentially nonphysical oscillations during solidification computations in inverse problems. In the inverse problem, the governing equations are identical to the equations of the corresponding direct problem. The difference arises because the interface position is given, instead of unknown, and the temperatures at the controlling boundary are unknown, rather than known boundary conditions. The other boundary conditions are the same conditions arising from the direct problem. Initially, during each time step, a tentative (guessed) temperature at the controlling boundary is used, and the equations are solved as a direct problem. The temperatures at the controlling boundary are then updated continuously in an iterative manner, until the predicted
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movement of the phase interface agrees within a given tolerance with the desired path. Then, the solution at the given time step is completed. The numerical procedure for the inverse solution can be summarized as follows: 1. Specify the movement of the phase interface (interface velocity), and choose a constant time step based on this interface velocity. 2. Based on the velocity of the interface and the time step, a fixed numerical grid is specified, so at any time step the interface moves from one grid point to the next grid point. 3. Within each time step, an estimate of the unknown controlling boundary temperature is given, and the energy conservation equation is solved in a direct manner for the temperature, T. 4. The sensitivity coefficients are obtained, and the unknown (controlling) boundary temperature is updated. Then, the energy equation is solved again. An entropy-based correction of conductivity is made for the sensitivity coefficients whenever the Second Law is locally violated. 5. Repeat steps 3 and 4 for each iteration when solving the energy conservation equation. The solution is terminated when the predicted movement of the interface agrees, within a given tolerance, with the specified (desired) interface movement. Although the Second Law is decoupled from the implicit solution of the energy equation, the current formulation provides an entropy-based corrective step, which strives to achieve Second Law compliance and stable computations in the inverse problem.
8.5.3 C a se St u d y
w it h So l id if ic a t io n o f a
Pu r e Ma t er ia l
Consider a specific case study depicted in Figure 8.12, where solidification of a pure material (initially at the phase change temperature, T0 = 0) begins at the left boundary and proceeds rightward into the domain (Xu and Naterer, 2001). Cooling from the boundary at x = 0 leads to solidification of the liquid metal. The inverse problem requires the estimation of the boundary temperature, T (0,t), at x = 0, which generates a constant interface velocity, V, during advancement of the solid–liquid interface. The liquid portion within the mold remains at (or slightly above) the melting temperature. A slight undercooling is required for the onset of solidification. The numerical results in this section will be compared with an analytic solution (Charach and Rubenstein, 1989) to assess the accuracy and performance of the formulation. The following problem parameters (dimensionless) are adopted in this example: a = 1, c = 1, L = 0.5, V = 2.0, where L, a, and c refer to latent heat, thermal diffusivity, and specific heat, respectively. An analytical solution of this problem, involving solidification with a constant interface velocity, was reported by Carslaw and Jaeger (1967) as follows: T ( x, t ) = T0 =
L c
V V2 1 - exp α t - α
x
(8.61)
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and T ( x, t ) = T0
(8.62)
where x ≥ Vt. At x = 0, the surface temperature required to produce the specified interface motion is T ( 0, t ) = T0 +
L c
V2 1 - exp α t
(8.63)
for t > 0. The analytical solution for this particular case study can be obtained from substitution of the previous parameters into the previous general solution to give T ( 0, t ) =
1 [1 - exp( 4t )] 2
(8.64)
This result becomes unrealistic after long time periods due to the exponential behavior of the time-dependent boundary condition. It would be difficult to achieve in a large casting after long periods of time, because the boundary temperature is eventually required to reach an extremely low temperature at x = 0, to maintain a constant velocity of the solid–liquid interface. However, it represents practical conditions during early stages of solidification in problems of materials processing. In Figure 8.13, the boundary temperature, T(0,t), for three specified interface velocities for pure gallium is illustrated. The phase interface position can be determined
50
Boundary Temperature (°C)
0 –50 –100 –150 V = 0.04 mm/s
–200
V = 0.06 mm/s V = 0.08 mm/s
–250 –300
0
500
1000 Time (s)
1500
2000
Fig u r e 8.13 Boundary temperature at varying interface velocities.
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Boundary Temperature
0 –20 –40 –60 –80 –100
Predicted Results Exact Solution
–120 –140
0
0.5
Time (a)
1
1.5
0 Boundary Temperature
–5 –10 –15 –20 –25 –30
Predicted Results Exact Solution
–35 –40 –45
0
0.2
0.4
Time (b)
0.6
0.8
1
1.2
0
Boundary Temperature
–0.1 –0.2 –0.3 –0.4 –0.5
Predicted Results Exact Solution
–0.6 –0.7
0
0.05
0.1 Time (c)
0.15
0.2
Fig u r e 8.14 Boundary temperature at dimensionless time steps of (a) 0.1, (b) 0.05, and (c) 0.005 s.
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based on the given interface velocity and the specified time level. As time advances, the phase interface moves farther away from the left boundary, and the solution becomes unstable. Oscillations appear and they lead to solution divergence (note: only stable regions are illustrated in Figure 8.14a through Figure 8.14c). It appears that oscillations occur earlier in the case of smaller time steps. The boundary temperature computations become unstable earlier with smaller time steps. In the numerical simulations, whenever the distance between the phase interface and left boundary becomes too large, it becomes difficult to control this interface movement solely through the left boundary temperature. Numerical oscillations appear in Figure 8.15a with Dt = 0.05. Before t = 1.1, the computations proceed well. But after t = 1.1, oscillations arise, and their magnitude 40
Boundary Temperature
20 0 –20 –40 –60 Predicted Results Exact Solution
–80
–100 –120
0
0.2
0.4
0.6
Time (a)
0.8
1
1.2
1.4
Boundary Temperature
0 –50 –100 –150
Predicted Results Exact Solution
–200 –250
0
0.5
Time (b)
1
1.5
Fig u r e 8.15 Boundary temperature (∆t = 0.05) (a) without and (b) with entropy correction.
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increases with further time advancement. The computations from the Second Law give indications of these trends. An entropy-based approach is used to provide a criterion for corrective steps in the formulation. During each time step, the entropy production rate is computed, then the Second Law requires that it must remain positive within a discrete volume. If oscillations appear in violation of the Second Law, the predictive entropy-based correction is applied in the computations. The effective thermal conductivity is corrected based on the local entropy production rate. This value is then adopted, and the sensitivity coefficient in the inverse model is modified accordingly. The results in Figure 8.15b show that the entropy-based correction performs well and improves the numerical stability of results. Oscillations are reduced for subsequent time steps, in comparison to Figure 8.15a. The modifications permit stable computations of boundary temperature for longer time periods. It is worthwhile to compare this approach with another conventional technique for stabilizing inverse computations, namely, future time stepping. In future time stepping, the boundary temperature is assumed fixed for r future time steps, and the system of discrete equations is solved over this time range. Then, the boundary temperature is updated, and iterations continue until a sum of squares difference (involving the interface temperature at a future time level and the phase change temperature) is minimized. This section has indicated that the Second Law provides a physically based alternative to future time stepping. It underlies a physical mechanism that can stabilize results that exhibit nonphysical behavior, such as numerical oscillations, for problems involving solidification and melting.
8.6 En t r o py Pr o d u c t io n w it h Fil m Co n d en sa t io n 8.6.1 F o r mu l a t io n
o f Hea t Tr a nsf er a n d
Ir r ev er s ibil it y Dis t r ibu t io n
The previous sections have analyzed entropy and the Second Law in solidification and melting problems. Before closing this chapter, another multiphase system will be examined, namely, heat transfer with condensation. Previous sections have focused on numerical analysis with the Second Law, whereas this section will use analytical methods and entropy generation minimization. Pioneering work on laminar film condensation along an isothermal surface was conducted by Nusselt (1916). This analysis neglected the advection terms to obtain an approximate solution, in terms of forces and heat balances within the condensate film. Sparrow and Gregg (1959) showed that inertial effects only have a significant influence on the heat transfer rate when the Prandtl number (Pr) is less than 10. Koh and coworkers (1961) obtained an exact boundary layer solution, when the shear forces at the liquid/vapor interface were taken into account. The effects of gravitational forces and interfacial stress were found to be negligible for high Prandtl numbers (Pr > 10). An improved correlation of film thickness was developed by Rohsenow (1956), when thermal advection effects become significant. A modified latent heat of vaporization was introduced, which depends on the Prandtl number (Sadasivan and Lienhard, 1987). Dier and Lienhard (1974) showed that results for a vertical plate
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could be extended to inclined plates, provided that g is replaced by g · cos(q), where q is the angle between the vertical direction and the wall. Although analytical solutions can be derived for laminar film condensation, correlations involving turbulence typically require approximate or empirical methods. Film condensation is accompanied by thermodynamic irreversibilities of fluid friction and heat transfer over finite temperature ranges. There exists a direct relationship between entropy production of a process and the amount of power consumed or lost by the process (Bejan, 1996). A traditional approach to the study of laminar film condensation involves a solution of the governing conservation equations. The Second Law may serve as an important additional tool for optimization in the design of thermal systems involving laminar film condensation, such as finned surfaces in a heat exchanger. The method of entropy generation minimization (EGM) has been applied previously to single-phase convective heat transfer. Bejan (1979) analyzed the entropy production of various configurations and flow regimes. It was shown how certain flow and geometric parameters can be selected to minimize the irreversibilities of the thermofluid processes. In this section, an optimization correlation is presented for film condensation on a flat plate, including results of entropy and the Second Law. Consider laminar film condensation of a pure saturated vapor at Tsat on a vertical isothermal plate held at a temperature of Tw. Uniform thermophysical properties will be assumed. Also, it is assumed that the condensate film flows in the x-direction along the plate (see Figure 8.16) due to either gravitational or shear-driven flow effects. The product of the friction coefficient (based on the interfacial shear stress at the liquid/vapor interface), cf, and the Froude number, Fr, identifies the relative magnitudes of these effects. For cf · Fr << 1, gravitational effects are dominant, whereas shear-driven effects are dominant when cf · Fr >> 1.
y τ
ρv
x
Vapor Region Liquid Condensate Film
g
Case 1 Velocity Profiles Case 2 ρ
q”x
δ Wall
Г
CV
m”v h fg
q”x
dx
Г + (dГ/dx)dx
Tw Tsat
Temperature Profile
Fig u r e 8.16 Schematic of film condensation.
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The first case will examine conditions of cf · Fr << 1 (without effects of interfacial shear stress). The reduced equations for mass, x-momentum, and energy are given by
∂u ∂v + =0 ∂x ∂y
µ
∂2 u = - gx ( ρ - ρv ); ∂y 2
k
∂2T =0 ∂y 2
(8.65)
The boundary conditions are y = 0,
v = 0,
u = 0,
y = δ,
T = Tsat ,
k
T = Tw , (8.66)
∂T • = m v h fg ∂y
where u, v, m, gx, r, k, Tsat, Tw, m v, hfg, and d refer to x-velocity, y-velocity, dynamic viscosity, gravity (x-component; see Figure 8.16), density, thermal conductivity, saturation temperature, wall temperature, vapor mass flow rate, enthalpy of vaporization, and film thickness, respectively. The rate of entropy production, Ps, within the liquid (excluding the interface) is 2
µ ∂u k ∂T Ps = 2 + T ∂y T ∂y
2
(8.67)
with streamwise velocity and temperature gradients assumed to be much smaller than y-direction gradients. Assuming a constant wall heat flux, q´´, and a zero interfacial velocity gradient, T - Tsat = DT =
u=
q ′′ (y - δ ) k
2 gx ( ρ - ρv )δ 2 y 1 y δ 2 δ µ
(8.68)
(8.69)
Combining Equation 8.67 through Equation 8.69 and assuming that T - Tsat is much smaller than Tw or Tsat, 2
( g δ ( ρ - ρv )( y - δ ))2 1 q ′′ Ps = + x k Tsat µTsat
(8.70)
On the right side of Equation 8.70, the first term represents the entropy generated by heat transfer, and the second term represents the fluid friction irreversibility. The entropy production is maximum at the wall, as viscous effects are highest there. Substituting k · Pr/cp for m in Equation 8.70, it can be shown that the second term
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in Equation 8.67, which represents the friction irreversibility due to viscous drag, becomes less significant as the Prandtl number increases. In that case, entropy production is mainly due to heat transfer. This result agrees with earlier findings of Sparrow and Gregg (1959), whereby the inertial effects become negligible for Prandtl numbers of 10 and higher. The velocity boundary layer is comparatively larger than the thermal boundary layer for Pr >> 1, so that inertial effects on the Nusselt number in the condensation analysis become relatively small. Define an irreversibility distribution ratio, f, which represents the ratio of friction irreversibility, Ps,V , to thermal irreversibility, Ps,T , in Equation 8.70. At the wall, this ratio is
P 1 φ = s,V = Ps,T Pr
gx c p ( ρ - ρv )δ 3Tsat ⋅ q ′′ 2
(8.71)
As mentioned previously for Pr >> 1, inertial effects become negligible as the hydrodynamic boundary layer thickness becomes much larger than the thermal boundary layer thickness. In that case, it is expected that the friction irreversibilities, Ps,V, are much lower than the thermal irreversibilities, Ps,T . The inertial term becomes negligible, as the velocity gradients at the wall are reduced when the boundary layer thickness increases. Thus, f << 1 when Pr >> 1, which is confirmed by the functional form of Equation 8.71. Previous researchers (Sparrow and Gregg, 1959) have shown the inertial terms to be significant when Pr << 1. This trend is confirmed by Equation 8.71. If friction irreversibilities are comparatively larger when inertial terms are important, then f >> 1, which requires that Pr << 1 for given problem parameters. The second case will examine conditions with cf · Fr >> 1 (predominant effects of interfacial shear stress). In Figure 8.16, the shear stress, t, acts at the phase interface. The same governing, boundary, and entropy equations, Equation 8.65 through Equation 8.67, are solved except that a condition of ∂2 u/∂y 2 = 0 is applied at the edge of the condensate film, thereby yielding u=
gx ( ρ - ρv )δ 2 µ
y 1 y2 τ δ - 2 δ + µ y
(8.72)
2
( g ( ρ - ρv )( y - δ ) + τ )2 1 q ′′ Ps = + x k Tsat µTsat
(8.73)
It should be noted that Equation 8.73 does not include the entropy change of phase transformation at the interface, because it only applies within the liquid film (not at the phase interface). The last portion of Equation 8.73 arises from irreversibilities due to the interfacial shear stress. Applying mass and energy balances for the control volume, CV, in the liquid (see Figure 8.16),
dG T -T = h fg k sat δ dx
(8.74)
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For this case of cf · Fr >> 1, the interfacial stress term is dominant, so Equation 8.72 becomes u=
τ y µ
(8.75)
Then, the condensate mass flow rate per unit width, G, may be determined as follows: G=
∫
δ
0
ρudy =
ρτ 2 δ 2µ
(8.76)
Using Equation 8.76 in Equation 8.74 and solving for the film thickness, 1/ 3
3k (Tsat - T ) µ x δ= ρτ h fg
(8.77)
For a given condensate thickness, d, the average liquid velocity will be lower when the interfacial stress is applied. This leads to a smaller mass flow rate of condensate. Because the heat transfer rate, q´´, is proportional to the mass flow rate multiplied by the latent heat (mv´´ ⋅ hfg ), the rate of heat transfer will decrease. This heat transfer rate becomes 3k (Tsat - T ) µ q ′′ = k (Tsat - T ) ρτ h fg
-1/ 3
x -1/ 3
the Nusselt number is so
(8.78)
1/ 3
Nux =
( q ′′ / DT ) x Re x ⋅ Pr = k 3Ja
(8.79)
where the Reynolds number is based on the shear velocity, τ /ρ . Consider entropy generation minimization of the geometrical and flow parameters when shear effects are dominant (cf · Fr >> 1; case 2). Based on the previous assumptions and Equation 8.73, the entropy production per unit width becomes
Ps' =
∫
δ
0
2
1 q ′′ τ2 Ps dy = δ+ δ k Tsat µTsat
(8.80)
Substituting d and combining Equation 8.79 and Equation 8.80, it can be shown that
-2 / 3 1/ 3 1 q ′′ 2 τ 2 3Ja ρτ ' Ps = x1/ 3 + µT Pr µ k Tsat sat
(8.81)
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Integrating over a plate length of L, expressing the heat transfer per unit length as q´ = q´´ L, and simplifying in terms of the Reynolds number, ReL , it follows that
3 1 q′ Ps = 4 k Tsat
2
1/ 3 3Ja 1/33 3 µτ 3Ja -2 / 3 + Re ReL4 / 3 L 4 ρTsat Pr Pr
(8.82)
From this result, the surface length is an example of a parameter that can be changed to minimize the entropy production. Setting ∂ Ps /∂ ReL = 0 , we find the following optimum that minimizes Ps : ReL ,opt = 0.707 B
(8.83)
where the duty parameter, B, for a plate width of W is defined by B=
q/W
(kντ Tsat )1/ 2
(8.84)
Equation 8.83 is a criterion to deliver more effective heat exchange in problems involving film condensation. It can be rewritten in terms of the optimal plate length, based on the definition of the Reynolds number. As an example, for a fixed rate of heat transfer within a finned heat exchanger, it gives the optimal length to minimize the thermal irreversibility and required temperature difference to achieve that amount of heat transfer. Alternatively, these results can be expressed in terms of minimized destruction of energy availability (or exergy). The rate of exergy destruction is Tsat Ps , so an equivalent result of Equation 8.83 is obtained for the minimized exergy destruction. If Equation 8.83 is substituted into Equation 8.81, an expression is obtained for the optimal (minimized) entropy production. The ratio of the actual entropy production to the minimized entropy production represents the entropy generation number, Ns, which is determined to be
ReL P N s = s = 0.666 Ps,opt ReL ,opt
-2 / 3
ReL + 0.333 ReL ,opt
4/3
(8.85)
For a fixed rate of heat transfer, q´, within a finned heat exchanger, a large surface length would reduce the temperature difference (between steam and wall) required to achieve q´, thereby reducing the refrigeration power needed to maintain Tw. But the interfacial shear and vapor friction increase for a larger surface length, thereby increasing the input power required to maintain a certain vapor flow rate at steady state. On the other hand, a smaller surface length reduces the total vapor friction and input power, but at the expense of a higher thermal irreversibility and refrigeration power, due to the higher temperature difference needed to maintain q´. The best compromise is reached when the entropy generation number is minimized, thereby minimizing the net power input from both vapor and refrigeration sides.
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Plate, Condensation Plate, Single Phase
100 NS
Internal Flow Tube
10
1 1E–03
1E–02
1E–01
1E+00
1E+01
1E+02
1E+03
L/Lopt, D/Dopt
Fig u r e 8.17 Entropy generation number.
8.6.2 C a se St u d y
o f Fl a t
Pl a t e Co n d ens a t io n
Results will be presented for an example involving condensation of steam at 1 atm (Adeyinka and Naterer, 2004). Comparisons will be made with previous studies involving other optimized flow configurations, as described by Bejan (1996) and Fowler and Bejan (1994). In Figure 8.17, the variation of entropy generation with length (or diameter, D, for the case of a tube) shows that a minimum entropy generation number, Ns, occurs when ReL reaches the optimized value. For the case of film condensation and others shown in Figure 8.18 at low values of L (or D), the friction irreversibility is relatively small, due to the small surface area of fluid friction. But the thermal irreversibility is high because a high temperature difference occurs over a thin region to transfer a given heat flow, q´. On the other hand, the friction
10,000
Plate, Turbulent, Single Phase Air Water
Re L,opt
1,000
Plate, Laminar, Single Phase Air Water
100 10 1
Plate, Laminar, Condensation 1
10
100
1,000
B
Fig u r e 8.18 Optimized Reynolds number.
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257
irreversibility is higher at large values of L (due to the large surface area involving fluid friction), whereas the thermal irreversibility is lower because a smaller temperature difference between the wall and surrounding fluid is needed to transfer the given q´ for a larger surface area. As a result, a minimized Ns exists for each case shown in Figure 8.17. The entropy generation number for flat plate condensation within the film is lower than the case involving tube flow. The surface curvature implies that the area available to heat and momentum transfer decreases in the radial direction when the fluid is heated by the wall. As a result, the near-wall temperature and velocity gradients become higher for the duct flow at a specified Reynolds number, so the entropy production increases. The entropy production for the single phase boundary layer appears smaller than the condensation result. Linear velocity and temperature profiles, with constant shear stress and heat flux values within the condensate film, were derived for case 2, whereas nonlinear profiles with decreasing stresses and heat fluxes (perpendicular to the wall) arise in single phase boundary layers (Fowler and Bejan, 1994). The entropy generation number in Figure 8.17 for single phase laminar flow is symmetric about the minimum, as both friction and thermal irreversibilities appear to have equal contributions to the net entropy production (note: on a relative basis after nondimensionalization). However, a higher contribution of friction irreversibility at high values of ReL , as compared to the thermal irreversibility, leads to asymmetry about the minimum for film condensation. When a given heat flow is transferred over a surface area, this effect is distributed spatially to reduce the required temperature difference and thermal irreversibility. However, the fluid friction is a cumulative effect when the surface area increases, thereby showing different characteristics with variations of the surface length. Similar interpretations can be made for the asymmetry observed in the duct flow with convective heat transfer in Figure 8.17. In Figure 8.18, the minimized ReL (or plate length) for film condensation increases at higher values of the duty parameter, B. The duty parameter for film condensation involves the total heat transfer, q, and interfacial shear stress, t, in Equation 8.84. Its definition for other flow configurations is documented by Bejan (1996). As B increases, then q also increases, so a larger plate is needed to minimize the entropy production (i.e., positive slope in Figure 8.18) by reducing the thermal irreversibility associated with a larger heat flow. For the single phase flow cases in Figure 8.18, the turbulent friction irreversibility rises faster, (steeper velocity gradient), and the wall or fluid temperature difference, ∆T, falls faster than the laminar case. This occurs when turbulent mixing entails a lower ∆T needed to transfer a given q. Thus, ReL for the turbulent case is lower than the laminar case at low values of q and B. But at high values of q and B, the thermal irreversibility component becomes more significant. Then the laminar friction irreversibility rises faster, whereas the thermal irreversibility falls faster. As a result, ReL is lower for the laminar case at low values of B, but crosses and exceeds the turbulent profile at sufficiently high values of B. In Figure 8.18, the slopes of the curves for film condensation and turbulent single phase flow are nearly equal. The relative rates at which the friction irreversibility rises and the thermal irreversibility falls with increasing B (or q) appear closely coincident. However, the condensation slope is lower than the laminar single phase case,
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suggesting that the friction irreversibility increases at a faster rate or the thermal irreversibility falls faster with B (or q) in the former case. Similar observations are made when the slopes of laminar and turbulent flows are compared. Although the film condensation results for ReL were derived independently of Pr, earlier results for single phase flow included a Prandtl number dependence (Fowler and Bejan, 1994). As shown in Figure 8.18, water has a higher viscosity and thermal conductivity than air at 300 K, so the friction irreversibility rises faster and the thermal irreversibility falls faster (i.e., lower DT to meet a given q) for water. Thus, the minimum Ps occurs at a lower ReL for water at a fixed value of B, as shown in Figure 8.18. These results have practical implications for the design of two-phase heat exchangers. An example of a case with cf · Fr >> 1 (case 2) is spacecraft thermal systems in microgravity, that is, heat pipes and capillary pumped loops. In variableconductance capillary pumped loops, the surface length in contact with the condensing vapor is lowered to suppress the temperature rise during high thermal loads, or operation in a hot environment. Similarly, this length increases to impede the temperature drop during low thermal loads or operation in a cold environment (Furukawa, 1999). An entropy-based analysis provides useful insight to improve performance of systems involving condensation heat transfer.
Ref er en c es Adeyinka, O.B. and G.F. Naterer. 2004. Optimization correlation for entropy production and energy availability in film condensation. Int. Commun. Heat Mass Transfer, 31(4): 513–524. Bejan, A. 1979. Study of entropy generation in fundamental convective heat transfer. ASME J. Heat Transfer, 101: 718. Bejan, A. 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Time Systems and Finite-Time Processes. CRC Press, Boca Raton, FL, Chap. 8. Bennon, W.D. and F.P. Incropera. 1988. Numerical analysis of binary solid–liquid phase change using a continuum model. Numerical Heat Transfer, 13: 277–296. Bird, R., Stewart, W., and E. Lightfoot. 1960. Transport Phenomena. John Wiley & Sons, New York. Burton, R., Yang, G., Dong, Z.F., and M.A. Ebadian. 1995. An experimental investigation of the solidification process in a V-shaped sump. J. Heat Mass Transfer, 38(13): 2383–2393. Carslaw, H.S. and Jaeger, J.C. 1967. Conduction of Heat in Solids, Oxford University Press, New York. Charach, C. and I.L. Rubinstein. 1989. On entropy generation in phase-change heat conduction. J. Appl. Phys., 66(9): 4053–4061. Clyne, T.W. and W. Kurz. 1981. Solute redistribution during solidification with rapid solid stated. Metallurg. Trans., 12A: 965. Dhir, V.K. and J.H. Lienhard. 1974. ASME Journal of Heat Transfer, 1971: 93, 97. Flemings, M.C. 1974. Solidification Processing. McGraw-Hill, New York, Chap. 5. Flood, S.C. and P.A. Davidson. 1994. Natural convection in aluminum direct chill cast ingot. Mater. Sci. Technol., 10: 741–751. Fowler, A. and A. Bejan. 1994. Correlation of optimal sizes of bodies with external forced convection heat transfer. Int. Commun. Heat Mass Transfer, 21: 17. Furukawa, M. 1999. AIAA 33rd Thermophysics Conference. Paper 3445. Norfolk, VA.
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259
Gau, C. and R. Viskanta. 1984. Melting and solidification of a metal system in a rectangular cavity. Int. J. Heat Mass Transfer, 27(1):113–123. Hayashi, Y. and T. Komori. 1979. Investigation of freezing of salt solutions in cells. J. Heat Transfer, 101: 459–464. Koh, J.C., Sparrow, E.M., and J.P. Hartnett. 1961. Int. J. Heat Mass Transfer, 2: 69. Lax, P.D. 1971. Shock waves and entropy, in Contributions to Non-Linear Functional Analysis. Academic Press, New York, 603–634. Maples, A.L. and D.R. Poirier. 1984. Convection in the two-phase zone of solidifying alloys. Metallurg. Trans., 15B: 163–172. Minkowycz, W.J., Sparrow, E.M., Schneider, G.E., and R.H. Pletcher. 1988. Handbook of Numerical Heat Transfer. John Wiley & Sons, New York, Chap. 8. Naterer, G.F. 1999. Constructing an entropy-stable upwind scheme for compressible fluid flow computations. AIAA J., 37(3): 303–312. Naterer, G.F. 2000. Predictive entropy based correction of phase change computations with fluid flow. Part 1. Second Law formulation. Numer. Heat Transfer B, 37(4): 393–414. Naterer, G.F. 2001. Applying heat-entropy analogies with experimental study of interface tracking in phase change heat transfer. Int. J. Heat Mass Transfer, 44(15): 2917–2932. Naterer, G.F. and G.E. Schneider. 1996. PHASES model of binary constituent solid– liquid phase transition. Part 1. Numerical method. Numerical Heat Transfer B, 28(2): 111–126. Nusselt, W. 1916. Die Oberflachen Kondensatin des Wasser dampfes. Zeitschrift des Vereines Deutscher Ingenieure, 60: 541. Pardo, E. and D.C. Weckman. 1990. A fixed grid finite element technique for modelling phase change in steady state conduction — Advection problems. Int. Journal Numerical Methods Eng., 29: 969–984. Rady, M.A., Satyamurty, V.V., and A.K. Mohanty. 1997. Thermosolutal convection and macrosegregation during solidification of hypereutectic and hypoeutectic binary alloys in statically cast trapezoidal ingots. Metallurg. Mater. Trans. B, 28: 943–952. Rohsenow, W.M. 1956. Heat transfer and temperature distribution in laminar film condensation. Trans. ASME, 78: 1645. Rosen, M.A. and I. Dincer. 2004. A study of industrial steam process heating through exergy analysis. Int. J. Energy Res., 28(10): 917–930. Rosen, M.A., Pedinelli, N., and I. Dincer. 1999. Energy and exergy analyses of cold thermal storage systems. Int. J. Energy Res., 23(12): 1029–1038. Sadasivan, P. and J.H. Lienhard. 1987. Sensible heat correction in laminar film boiling and condensation. ASME J. Heat Transfer, 109: 545. Salcudean, M. and Z. Abdullah. 1988. On the numerical modelling of heat transfer during solidification processes. Int. J. Numerical Methods Eng., 25: 445–473. Salcudean, M. and R.I.L. Guthrie. 1979. A three dimensional representation of fluid flow induced in ladles or holding vessels by the action of liquid metal jets. Metallurg. Trans. B, 10: 423–428. Schneider, G.E. and M.J. Raw. 1984. An implicit solution procedure for finite difference modelling of the Stefan problem. AIAA J., 22: 1685–1690. Sparrow, E.M. and J.L. Gregg. 1959. A boundary layer treatment of laminar film condensation. ASME J. Heat Transfer, 81: 13. Szekely, J. and A. Jassal. 1978. An experimental and analytical study of the solidification of a binary dendritic solution, Metallurg. Trans., 9b: 389–398. Vesligaj, M.J. and C.H. Amon. 1999. Transient thermal management of temperature fluctuations during time varying workloads on portable electronics. IEEE Trans. Components Packaging Technol., 22(4): 541–550.
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Voller, V.R. 1992. Enthalpy method for inverse Stefan problem. Numerical Heat Transfer, B, 21: 41–55. Voller, V.R. and D. Brent. 1989. The modelling of heat, mass and solute transport in solidification systems. Int. J. Heat Mass Transfer, 32(9): 1719–1731. Xu, R. and G.F. Naterer. 2001. Inverse method with heat and entropy transport in solidification processing of materials. J. Mater. Processing Technol., 112(1): 98–108. Yoo, H. and R. Viskanta. 1992. Effect of anisotropic permeability on the transport process during solidification of a binary mixture. Int. J. Heat Mass Transfer, 35(10): 2335–2346.
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9
Entropy Production in Turbulent Flows
9.1 In t r o d u c t io n This chapter concludes the book by examining entropy and the Second Law for turbulent flows. It presents an overview of modeling and experimental methods for determining entropy production in turbulent flows. The turbulent entropy equation will be derived from the Reynolds averaged Clausius–Duhem equality (Hauke, 1995), which expresses entropy in terms of mean and fluctuating components in the Reynods averaging. A small thermal turbulence assumption (STTAss) will be used in the turbulence analysis (Kramer-Bevan, 1992). Under the STTAss, the fluctuating component of temperature is assumed small compared with the mean temperature, which allows the mean turbulent entropy production to be expressed in terms of viscous mean and turbulent fluctuating parts. Experimental measurement of the turbulent dissipation rate can be obtained with different methods, such as a turbulent kinetic energy balance (Hussein and Martinuzzi, 1995), direct measurement of strain rate tensors (Andreopoulos and Honkan, 1996), turbulent energy spectrum, Taylor’s frozen turbulence hypothesis, dimensional analysis (Kresta and Wood, 1993), or a more recent large eddy particle image velocimetry (PIV) method (Adeyinka and Naterer, 2007). A detailed review of past advances regarding the measurement of turbulence dissipation has been presented by Sheng et al. (2000). A major limitation of pointwise methods is the laborious measurement needed to acquire whole-field data. The whole-field method of PIV offers certain advantages over standard methods of anemometry for entropy-related experimental analysis. For pointwise methods, a direct evaluation of the dissipation rate from its definition would require resolution of the fluctuating strain rate tensor, which is possible only with multiple hot-wire probes in the flow field. In contrast, the PIV method provides a whole-field measurement technique, while allowing nonintrusive and time-varying measurements of instantaneous velocity and temperature fields. Because the PIV technique provides wholefield data for velocity and temperature fields, it can lead to spatial measurements of turbulent entropy production throughout a flow field. Measured velocities by PIV are estimated over finite grids, so the turbulence statistics are influenced by the type of low-pass filter (FlowMap, 1998). For this reason, conventional dissipation rate approximations are limited when analyzing PIV data. The large eddy PIV method does not preclude the possibility of obtaining high resolution velocity measurements, where the detailed turbulent structures are captured (Liu et al., 1991). It can obtain the turbulent dissipation rate in whole-field regions, where the dynamic range of velocity measurements captured by PIV is limited by spatial resolution. This chapter will investigate both modeling and experimental methods for measuring turbulent 261
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entropy production rates throughout an incompressible flow field. Results will be particularly examined for channel flow problems.
9.2 R eyn o l d s Av er ag ed En t r o py Tr an sp o r t Eq u at io n s Using the entropy transport equation from Section 3.2 and subdividing entropy into mean and fluctuating components, the following result can be obtained for the Reynolds averaged Clausius–Duhem equality (Jansen, 1993): 2
t ij ∂ui ∂ k ∂T k ∂T ∂ + (ρs ) + ρui s + ρui′ s ′ − = 2 ∂t T ∂x j T ∂xi T ∂xi ∂xi
(9.1)
where the overbar (i.e., s ) and prime (i.e., s´) notations refer to mean and fluctuating components associated with the Reynolds averaging, respectively. Because T and ui (and consequently the viscous dissipation term) have mean and fluctuating components in the denominator and numerator, it becomes difficult to explicitly express the mean entropy production in terms of other mean flow variables alone. Two main methods for evaluating the mean entropy production will be briefly addressed below. In the first approach, the two sides of Equation 9.1 can be rearranged as follows: P s =
T′ ∂ ∂ ∂ ln T 1 + ≥ 0 (ρs ) + ( ρu is + ρui′ s ′ ) + k T ∂t ∂xi ∂xi
(9.2)
The first term can be simplified by substituting ∂(lnT ) / ∂xi for ( ∂T / ∂xi ) / T before the time averaging. The time-averaged positive definite entropy equation becomes
P s = k
1 ∂u ∂u j ∂u i ∂ ∂ ∂ ∂ (lnT ) (lnT ) + k (lnT )′ (lnT )′ + µ i + ∂xi ∂xi ∂xi ∂xi T ∂x j ∂xi ∂x j
′ ′ ′ ∂u 1 ∂u ′j ∂u 1 ∂ui′ 1 ∂u ′ ∂u j ∂ui′ +µ i + µ i + + 2µ i T ∂x j ∂xi ∂x j ∂x j T ∂xi ∂x j T ∂x j
(9.3)
′ ′ ∂u j 1 ′ ∂ui′ 1 ∂u ′ ∂u j ∂ui′ + µ i + ≥0 T ∂x j ∂xi ∂x j ∂xi T ∂x j A close examination of Equation 9.3 reveals the physical processes leading to turbulent entropy production. The first two terms on the right side are entropy production terms that arise from thermal fluctuations and transport. The terms in the +µ
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first squared brackets represent the entropy production due to mean viscous effects. The terms in the second squared brackets represent entropy produced due to the dissipation of turbulent kinetic energy. The terms in the last squared brackets represent the mechanism of entropy produced by the interaction of fluctuating viscous effects and temperature fluctuations. The remaining terms represent the conversion of entropy production, due to mean viscous effects, to irreversibilities of fluctuating viscous-temperature effects and back. By defining the mean viscous stress and the fluctuating viscous stress, respectively, as
∂u i ∂u j + t ij = µ ∂x j ∂xi
(9.4)
∂u ′ ∂u ′j t ij′ = µ i + ∂x j ∂xi
(9.5)
then Equation 9.3 becomes
P s = k
∂ ∂ ∂ ∂ 1 ∂u 1 ∂u ′ (lnT ) (lnT ) + k (lnT )′ (lnT )′ + t ij i + t ij′ i T ∂x j T ∂x j ∂xi ∂xi ∂xi ∂xi ′
′
(9.6)
′
1 ∂ui′ ∂u i 1 ′ 1 ′ ∂ui′ + t ij + ≥0 t + t T ∂x j ∂x j T ij T ij ∂x j
No models exist at the present time for complete correlations involving the (1/T)´ terms. Any such correlations would be difficult to validate or measure with some degree of accuracy. Modeling of the mean entropy generation can be simplified by the following approach, whereby the Clausius–Duhem equality is averaged. The left side is multiplied by temperature to give 2
T P s =
k ∂T ∂u + t ij i T ∂xi ∂x j
(9.7)
Kramer-Bevan (1992) presented a derivation of the time-averaged form of Equation 9.7, with the following result:
T P s + T ′ P ′s = k
∂u ∂ ∂T ∂u ′ ∂ ∂T ′ (lnT ) (lnT )′ + t ij i + t ij′ i +k ∂x j ∂xi ∂xi ∂x j ∂xi ∂xi
(9.8)
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In Equation 9.8, the physical processes of conversion of entropy production, arising from mean viscous effects, to irreversibilities of fluctuating viscous and temperature effects, have been captured in the T ′Ps′ correlation. Other terms remain as previously described for Equation 9.3. This equation is more straightforward than Equation 9.2 and Equation 9.3, provided that suitable empirical models can be developed for T ′Ps′ and thermal gradient correlations. Kramer-Bevan (1992) proposed a closure approximation for a subset of possible flow fields by using an STTAss. The following section will develop modeling for the T ′Ps′ correlation. To derive a general, combined equation for the mean entropy generation, time averaging is performed to yield ∂ T ′ ∂ ∂ T P s + T ′ P s′ = T ( ρ s ) + ( ρu js + ρu ′ s ′ ) + k ln T 1 + T ∂xi ∂xi ∂t +T ′
∂ T′ ∂ ∂ ∂ ( ρui′ s ) + T ′ ( ρu is ′ ) + T ′ ( ρui′ s ′ ) + kT ′ ln T 1 + ∂xi T ∂xi ∂xi ∂xi
(9.9)
By comparing Equation 9.2 with Equation 9.9, it can be shown that ∂ ∂ T′ ∂ ∂ T ′ P ′s = T ′ ln T 1 + ( ρui′ s ) + T ′ ( ρu is ′ ) + T ′ ( ρui′ s ′ ) + kT ′ ∂xi ∂xi T ∂xi ∂xi (9.10) Using the chain rule of calculus, T ′ P ′s = ρT ′ ui′ + kT ′
∂s ′ ∂s ∂ ∂u ′ ∂u + T′ + ρ s T ′ i + ρT ′ s ′ i + ρ u i T ′ ( ρui′ s ′ ) ∂xi ∂xi ∂xi ∂xi ∂xi T ∂ ln T 1 + T ∂xi
′
(9.11)
By assuming incompressibility, the mean and instantaneous velocity fields are solenoidal, and Equation 9.11 reduces to
T ′ P ′s = ρT ′ ui′
∂s ∂ ∂s ′ T′ ∂ ( ρui′ s ′ ) + kT ′ ln T 1 + + ρ u iT ′ + T′ ∂xi ∂xi ∂xi T ∂xi
(9.12)
This equation provides the full expression for the T ′Ps′ correlation. The following section will consider modeling of individual terms in Equation 9.12.
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9.3 Ed d y Visc o s it y Mo d el s o f Mean En t r o py Pr o d u c t io n A few simplified models, based on the solution of the Reynolds-averaged Navier– Stokes (RANS) equations and an eddy viscosity for mean entropy generation, have been documented in the literature (Adeyinka and Naterer, 2004; Drost and White, 1991; Moore and Moore, 1983). The linear eddy viscosity model assumes a Boussinesq relationship between the turbulent stresses (or second moments) and the mean strain rate tensor, through an isotropic eddy viscosity. These models attempt to reduce complexity, but it is difficult to ascertain whether the essence of relevant irreversibilities has been captured with sufficient accuracy, due to the lack of experimental data. Moore and Moore (1983) suggested that following correlations for mean entropy production, thermal diffusion, and viscous dissipation, respectively: k T P s = T
2 2 ′ ′ ∂T + ∂T + t ij ∂u i + t ij′ ∂ui ∂xi ∂xi ∂x j ∂x j 2
∂T ′ ∂T = kt k ∂xi ∂xi
t ij′
∂ui′ µt ∂u = t ij i µ ∂x j ∂x j
(9.13)
2
(9.14) (9.15)
In Equation 9.14 and Equation 9.15, kt and m t denote the turbulent molecular conductivity and the turbulent molecular viscosity, respectively. This model misses most of the correlation in Equation 9.8, due to the assumption that the temperature fluctuations are small compared with the mean temperature. Inconsistencies with this formulation occur close to the wall, so Kramer-Bevan (1992) proposed a different model for the viscous dissipation correlation,
t ij′
∂ui′ = ε ∂x j
(9.16) ε is the “true” dissipation of turbulent kinetic energy. The definition of ε difwhere fers from the dissipation of turbulent kinetic energy in the standard k - e model. The resulting model of entropy production becomes
k + kt T P s = T
2
∂T ∂u i ∂x + t ij ∂x + ε i
j
(9.17)
In contrast to Moore’s model, which uses the positive definite entropy equation, the STTAss is based on time averaging of the entropy transport equation. It assumes that the fluctuating component of temperature is small compared with the mean temperature. When formulating this model, the fluctuating temperature is replaced by a Taylor series expansion of those functions. The expansions are
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truncated after the linear terms, thereby yielding the following equations for mean entropy production and mean specific entropy: P s =
∂T 1 c µ ∂ ∂ ρu is − v t + k (ρs ) + ≥0 T Prt ∂t ∂xi ∂xi
ρ s = sr + cv ln T s − Rln s ρr Tr
(9.18) (9.19)
The turbulent Prandtl number, Prt, arises in Equation 9.18 because the entropyvelocity correlation has been modeled with a Reynolds analogy. Under the STTAss, extra terms arise in the entropy transport equation, with an increase of the diffusion term. This is equivalent to adding an effective diffusivity, cv m t / Prt, to the thermal diffusivity in the laminar model.
9.4 T u r bu l en c e Mo d el in g w it h t h e Sec o n d Law The exact equation for the dissipation of turbulent kinetic energy (TKE) is useful to understand the meaning and importance of various terms, but usually it cannot be rigorously modeled in its full detailed form (Hanjalic and Jakirilic, 2002). Modeling of the exact equation is traditionally carried out by drastic simplification, and it usually involves a laborious empirical approximation of five or more closure coefficients. This section attempts to obtain the dissipation of TKE using the Second Law under the STTAss. In this approach, the local entropy production in convection-dominated flow can be found based on mean quantities (velocity and temperature) obtained from the solution of the RANS equations, using both the transport and positive definite forms of the entropy equation. Because the dissipation of TKE appears in the positive definite mean entropy production equation, its local value can be computed throughout the flow domain by the Clausius–Duhem equation. A formulation for the proposed model will be presented for the eddy viscosity and second moment turbulent closure. Combining Equation 9.2, Equation 9.8, and Equation 9.12, we obtain the following combined entropy equation for turbulent flow: ∂ T ′ ∂ ∂ ln T 1 + T P s = T ( ρ s ) + ( ρu is + ρu ′ s ′ ) + k T ∂xi ∂xi ∂t =k
∂T ′ ∂ ∂T ∂u ∂ ∂u ′ (lnT )′ + t ij i + t ij′ i +k (lnT ) ∂xi ∂xi ∂xi ∂x j ∂xi ∂x j
− ρT ′ ui′
∂s T′ ∂s ′ ∂ ∂ ( ρui′ s ′ ) − kT ′ − ρ u iT ′ ln T 1 + − T′ ∂xi T ∂xi ∂xi ∂xi
(9.20)
The fourth term on the right side of Equation 9.20 represents the dissipation of TKE. This term, called e, can be interpreted as a physical mechanism by which exergy ( TPs)
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is destroyed in turbulent flow. This interpretation agrees with the traditional interpretation that associates e with the rate at which TKE is converted to internal energy in the flow. The terms after the second equality in Equation 9.20 reveal the physical processes leading to exergy destruction in turbulent flow. The total exergy destroyed in turbulent flow is the sum of exergy destroyed due to irreversible heat transfer (terms 1, 2, and 8), viscous dissipation (terms 3 and 4), turbulent enthalpy transfer (term 5), and the work done by fluctuating temperatures against turbulent entropy transfer by mass exchange (terms 6 and 7). All of these irreversible processes dissipate mechanical energy to internal energy. Equation 9.20 indicates the importance of maintaining positivity of e in the numerical simulations. The time-averaged entropy equation does not shed much light with regard to modeling of e, except when simplified by the STTAss. Complete modeling of the Clausius–Duhem equation can only be achieved through experiments to calibrate closure coefficients, when approximating the nonlinear fluctuating terms. Two following approaches (linear eddy viscosity and differential second moment closures [DSM]) will be described for modeling and simplification of Equation 9.20. The terms in the time-averaged entropy equation, Equation 9.20, can be determined from a linear eddy viscosity model as follows: 2 ∂T t ij ∂u i ∂T 1 ∂ 1 cv µt ∂ + ( k + γ kt ) (ρs ) + + k = ρu is − ∂t T Prt ∂xi ∂xi T ∂x j ∂xi T 2
+
2
k ∂ 1 ∂ ε k ∂T ′ ρc (T ′2 ) − v + + T T 2 ∂xi 2 ∂xi T ∂xi T
∂ ui′T ′2 ∂ T ′2 + ui ∂xi T ∂xi T
ρcv ∂ ′ ′2 ∂ ′2 u T + u i T 2 i 2T ∂xi ∂xi +
(9.21)
On the left side of Equation 9.21, the terms represent the transient change of mean entropy (first term) and the transport of entropy by mass and heat flow (second term in square brackets). On the right side of Equation 9.21, the terms refer to entropy production associated with thermal molecular and turbulent diffusion of the mean temperature field (first term in square brackets), viscous dissipation of the mean velocity field (second term), and irreversibilities through dissipation of TKE (third term). Within the braces, the terms represent entropy production corresponding to irreversible temperature fluctuations (first and second terms) and irreversible interactions between fluctuating velocity and temperature fields (remaining terms). The individual terms in braces can be obtained through the following correlation governing the dynamics of T′ 2 (Tennekes and Lumley, 1972), ui
∂ ∂xi
2 T ′2 ∂T ′ ∂T ∂ 1 ′ ′2 ∂ T ′2 ′T ′ α u α − = u T − − i 2 ∂x 2 i ∂x ∂xi ∂xi 2 i i
(9.22)
where a is the thermal diffusivity.
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The differential second moment closure (DSMC) directly solves the transport equations for the Reynolds stresses in the momentum equation. This approach is used to obtain the scalar fluxes in turbulent flow, involving the transport of passive scalars. The computed turbulent heat flux can then be used directly in Equation 9.20 to give 2
∂s t ij ∂u i ∂ ρcv ′ ′ k ∂T k ∂T ∂ + ρT ′ ui′ + (ρs ) + ui T − ρu is + = ∂xi T ∂x j T ∂t T ∂xi T 2 ∂xi ∂xi
2
+
k ∂ 1 ∂ ρc ε k ∂T ′ + (T ′2 ) − v + 2 2 ∂xi T ∂xi T T T ∂xi
+
ρcv ∂ ′ ′2 ∂ ′2 (ui T ) + u i T 2 2T ∂xi ∂xi
∂ ui′T ′2 ∂ T ′2 ui ∂xi T ∂xi T
(9.23)
This approach dispenses with the eddy viscosity to express the turbulent shear stress in terms of mean flow quantities. Similarities in turbulent irreversibilities can be observed in Equation 9.21 and Equation 9.23. From left to right in Equation 9.23, the terms represent the transient change of mean entropy (first term) and the transport of entropy by mass and heat flow (second term in square brackets). Unlike Equation 9.21, the heat flow is not modeled with a turbulent conductivity in this case. On the right side of Equation 9.23, the terms refer to entropy production corresponding to thermal molecular diffusion of the mean temperature field (first term), diffusive entropy transport in the mean flow due to velocity fluctuations (second term), viscous dissipation of the mean velocity field (third term), and dissipation of TKE (fourth term). In a similar way as previously described, the terms within braces represent entropy production corresponding to irreversible temperature fluctuations (first and second terms) and irreversible interactions between fluctuating velocity and temperature fields (remaining terms).
9.5 Measu r emen t o f Tu r bu l en t En t r o py Pr o d u c t io n 9.5.1 Fo r mu l a t io n
of
Dis s ipa t io n Ra t e
Unlike near-isothermal laminar flows (such as unheated pipe flows) where the only physical process that produces entropy is the mean viscous dissipation, the rate of dissipation of TKE is needed to compute entropy production in turbulent flows. A segment for extracting mean and turbulent quantities from velocity data is needed to measure the turbulent entropy production rates throughout a flow field. This section investigates modeling of the turbulent dissipation rate, for purposes of finding the turbulent entropy production rates. The effect of mean and fluctuating quantities on the total mechanical energy of a turbulent flow can be separated by the Reynolds averaging procedure. By subtracting the balance equation for the kinetic energy of
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the mean motion, the following expression for the balance of kinetic energy of turbulence is obtained, ∂u ′j ∂u j D k2 ∂ ′ p k2 ∂ ′ ∂ui′ ∂u ′j −ν =− uj + ui + − ui′ u ′j +ν ∂xi ∂x j ∂xi ∂xi Dt 2 ∂xi ρ 2 ∂xi
∂ui′ ∂u ′j ∂x + ∂x j i (9.24)
Equation 9.24 requires that the net convection of TKE (term 1) balances the flow work or work done by the total dynamic pressure (term 2), net work of turbulent stresses (term 3 minus term 4), minus the dissipation of TKE (last term). In the absence of periodic oscillation in the flow, the total dissipation in turbulent flows is a sum of mean (viscous shear stress) and random (dissipation of TKE) parts. The viscous shear stress performs deformation work, which increases the internal energy of the fluid at the expense of TKE. Because turbulence consists of a continuous spectrum of scales ranging from more energetic large scales to dissipative small scales, a continuous supply of energy from the large scales or “eddies” is required to maintain turbulence. Otherwise, turbulence decays rapidly, and loss analysis of the fluid system reduces to an analysis involving only the mean viscous dissipation, as in laminar flows. By expansion, the 12-term dissipation of TKE tensor, e, in Equation 9.24 can be expressed as
ε =ν
∂ui′ ∂u ′j ∂u ′j ∂u ′j + ∂x j ∂xi ∂xi ∂xi
(9.25)
Measurement of all terms in Equation 9.25 is difficult. A simplified form will be used based on the theory of homogenous turbulence and isotropy (Hinze, 1975). In homogeneous turbulence, the first term in Equation 9.25 vanishes due to incompressibility, i.e., u ′j ∂2 ui′ / ∂xi ∂x j = 0, resulting in the following 9-term tensor for e,
ε =ν
∂u ′j ∂u ′j ∂xi ∂xi
(9.26)
The essence of homogeneous turbulence is idealized, whereby that the mean properties of turbulence (including the mean velocity) are independent of translations of the coordinate axis. However, it provides a reasonable basis for estimating experimental turbulence quantities (Batchelor, 1982). The assumption of homogeneous turbulence also implies a relationship between the viscosity and mean square vorticity through Equation 9.26, so,
ε = νω kω k
(9.27)
where w k is the vorticity. Equation 9.27 is the entropy-based dissipation described by Tennekes and Lumley (1972).
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Isotropic turbulence assumes that small turbulent scales are statistically independent of rotation and reflection of the coordinate axes at sufficiently high Reynolds numbers. A further simplification with the isotropic turbulence assumption can be obtained from Equation 9.26 in the following two-dimensional form: 2 ′ 2 ′ 2 ∂u ′ ∂u ∂u ∂u ′ ∂u ′ ε = 6ν 1 + 1 + 1 2 = 15ν 1 ∂x1 ∂x2 ∂x2 ∂x2 ∂x1
(9.28)
The Kolmogorov length scale represents the smallest length scale of turbulence, h = (v3/ e)1/4. Another length scale associated with the energy dissipated by turbulent eddies is the Taylor microscale, l, where,
λ2 =
u1′2 ( ∂u1′ / ∂x1 )2
(9.29)
Rearranging Equation 9.28 in terms of the Taylor microscale leads to
ε = 15ν
u1′2 λ2
(9.30)
A similar dimensional analysis based on the integral length scale, l, and an assumption of mechanical equilibrium gives
ε=A
u1′3 l
(9.31)
where A is a proportionality constant of the order of unity. Equation 9.31 can be used to predict the dissipation rate, when only one integral length scale characterizes the flow region. It does not require the dissipation of TKE to be equal to the production of TKE, as its derivation is independent of the presence of turbulence production. Another class of dissipation estimation methods (commonly used in laser doppler anemometry [LDA]) involves uses a time-series analysis and the turbulence energy spectrum. The following homogeneous turbulence relation applies:
ε = 2ν
∫
4
0
k 2 E ( k, t )dk
(9.32)
with a corresponding isotropic version given by
ε = 15ν
∫
∞
0
k12 E1 ( k1 )dk1
(9.33)
where E refers to the power spectrum, k is the wavenumber, and the subscripts “1” denote the one-dimensional values.
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9.5.2 L a r g e Eddy Pa r t ic l e Ima g e Vel o c imet r y There are similarities between the correlation analysis of PIV and large eddy simulation (LES). LES computes the dynamics of the large energy-containing scales of motion, up to a certain cutoff wavelength, while modeling only the effect of the small, unresolved flow structures on the larger resolved scales. The underlying principle is that large-scale motions are affected by the geometry and not universal. The small-scale motions have a weaker influence on the Reynolds stress, and they have a universal character, which can be represented by simple subgrid scale (SGS) models. The approach in LES requires the solution of the Navier–Stokes equations for the filtered velocity field on a computational grid, with the objective of resolving the actual flow field with fewer discrete volumes. In the same way, the correlation techniques in PIV give velocities that are results of a spatial average over a discrete volume or interrogation area. In LES, the filter size is proportional to a cutoff wavelength in the inertial subrange of the turbulence energy spectrum. The size of the interrogation area determines the filter width, which averages the smaller scales of motion in PIV. Because the spatial filtering properties of PIV are similar to LES, the benefits of SGS modeling in LES will be helpful to determine the small-scale turbulence characteristics from PIV data. With the filter in the inertial subrange, the turbulence dissipation rate in LES can be approximated by the following SGS dissipation rate:
ε ≈ ε SGS = −2t ij S ij, S ij =
1 ∂u i ∂u j + 2 ∂x j ∂xi
(9.34)
where S ij is the filtered rate of strain tensor and t ij is the SGS stress. Several SGS stress models have been used in previous LES studies at high Reynolds numbers. The first subgrid model to be widely used was reported by Smagorinsky (1963). Other models that were developed to improve the Smagorinsky model include the dynamic model of Germano et al. (1991; Lilly, 1992; Meneveau et al., 1996), Bardina scale similarity model (Bardina et al., 1980), Clark gradient model (Clark et al., 1979), structure function model of Métais et al. (1992), and the transport equation model (Mason, 1989; Sullivan et al., 1994). The next section will focus on the Smagorinsky and Gradient models, as well as compare the accuracy of different models. The PIV technique permits the measurement of instantaneous velocity data in a whole-field manner, which allows direct calculation of the turbulence dissipation rate from spatial derivatives of velocity. However, the spatial range of PIV cannot usually be extended down to the required near-wall resolution for exact measurements, due to limitations imposed by the hardware, such as the size of recording media and the maximum allowable sampling speed (Adrian, 1997; Saarenrinne et al., 2001). Saarenrinne and Piirto (2000) proposed a restrictive requirement in PIV (depending on the flow), where the size of the PIV interrogation window and the laser light thickness do not exceed 30% of the lateral Taylor’s microscale and five times the Kolmogorov length scale, respectively.
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Dimensional analysis based on equilibrium turbulence assumes local isotropy. It can be used to estimate the turbulence energy dissipation in stirred vessels (Kresta, 1998). The method uses Equation 9.31, and it yields accurate qualitative results despite implementation difficulties due to the variation of length scales in certain flows. In flows where the Taylor microscale can be estimated, dimensional analysis based on Equation 9.30 has been used successfully (Saareninne and Piirto, 2000). The dissipation rate in the turbulent kinetic equation can be determined from terms represented by the mean flow convection, diffusion and production of turbulent energy, and neglected terms of viscous diffusion. The applicability of this method is limited by an appropriate model for the pressure diffusion term, which is difficult to measure experimentally (Turan and Azad, 1989). Although other terms in the TKE equation involve large-scale quantities, the limitation imposed by spatial resolution has restricted the application of the method to simple geometries. Another method for measuring turbulence dissipation uses Taylor’s frozen turbulence hypothesis, which allows Equation 9.28 to be rewritten in terms of a time series differential of the velocity fluctuation, that is,
ε = 15ν ( ∂u1′ / ∂x1 )2 = 15ν ( ∂u1′ / ∂t )2 / u 2
(9.35)
To obtain a reliable value of e, a calibration of the time derivative is necessary. It can be determined based on the energy spectrum function in Equation 9.32. Turan and Azad (1989) developed a “zero-wire-length dissipation method,” which defined the one-dimensional spectrum of the longitudinal velocity fluctuation by the following integral,
∫
∞
0
2 E1 ( k1 )dk1 = u1′
(9.36)
But the sampling rate of a PIV system is often not high enough to allow this spectral analysis. The large eddy PIV method is a promising method for the previous measurements. Sheng et al. (2000) established an appropriate resolution of time and length scales with this method. The authors developed a method to use full-field velocity data to estimate the dissipation rates. The large eddy PIV method is based on a dynamic equilibrium assumption, between the spatial scale that can be resolved by PIV and the subgrid length scales. When the interrogation or filter size is much smaller than the integral length scale of the flow, the turbulence dissipation rate can be approximated by Equation 9.34. In the following case study, the Clark Gradient model and the Smagorinsky model will be used for the SGS stress. For the Gradient model,
t ij =
1 2 ∂u i ∂u j D 12 ∂uk ∂uk
(9.37)
where D is the width of the interrogation area. The Smagorinsky model is given by
t ij = −(Cs D )2 | S | S ij
(9.38)
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where | S | is the characteristic filtered rate of strain, 2 S ij S ij , and Cs is the Smagorinsky coefficient (proportional to D), taken to be 0.07 (Adeyinka and Naterer, 2004). The large eddy PIV method and other simplified models based on the isotropic assumption in this section do not preclude the possibility of obtaining high resolution measurements, where detailed turbulent structures are captured (1991). The method provides a useful estimate of turbulence dissipation in regions where the dynamic range of velocity measurements captured by PIV is limited by the spatial resolution.
9.5.3 C a se St u dy
o f Tu r bu l en t Ch a n n el
Fl o w
In this section, measured results in the previous formulations of mean turbulent entropy production will be compared against past DNS (direct numerical simulation) data. Large eddy PIV is used to determine the velocity, dissipation rate, and entropy production data (Adeyinka and Naterer, 2004). The DNS solution assumes negligible viscous dissipation in the energy equation. Therefore, attention is focused on the positive definite model involving the dissipation of TKE (right side of Equation 9.23), because the entropy transport equation requires inclusion of the viscous dissipation in the energy equation for accurate modeling. Turbulent flow between two parallel plates at four different Reynolds numbers, based on the friction velocity, will be examined. Computations of the friction factor, f, at Ret = 180, 395, and 590, will be compared with DNS data of Moser et al. (1999). The data of Kuroda et al. (1989) were used to compute f at Ret = 100. The computed friction factors are compared in Table 9.1. The present results show excellent agreement with Darcy’s friction factor, computed from the Colebrook equation. The Colebrook equation is documented by White (1991). The results are illustrated at various Reynolds numbers, based on the bulk velocity in Figure 9.1. They confirm that the present turbulence modeling of entropy production (particularly in terms of e) has been accurately formulated. A comparison with Moore’s model is presented (Figure 9.2), with regard to the spatial distribution of entropy production in the channel. The integral value of entropy production computed from Moore’s model in Equation 9.13 and Equation 9.15, based on the production of TKE, is within 1% of the currently formulated model. Figure 9.2 shows that Moore’s model underpredicts the entropy production close to the wall and overpredicts entropy production away from the wall, before it decreases to zero in the middle of the channel. The additional curve in Figure 9.2 shows that the viscous mean dissipation is the main component of entropy production
Tabl e 9.1 F riction Factors at Different Ret Ret f (based on tw) f (based on present modeling)
100
180
395
590
0.0383 0.0388
0.0325 0.0324
0.0260 0.0252
0.0232 0.0225
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1.00
Present Model Colebrook: Laminar Flow
Friction Factor (Turbulent)
0.20 0.15
0.10
0.10
Friction Factor (Laminar)
Entropy-Based: Laminar
0.25
0.05 0.00
100
1000
10000
100000
0.01 1000000
ReD
Fig u r e 9.1 Friction factor based on entropy production correlation.
1.4 Moore Model Nondimensional Entropy Production
1.2
Present Viscous Mean
1 0.8 0.6 0.4 0.2 0 0.00
0.10
0.20
0.30
0.40
y/h
Fig u r e 9.2 Local distribution of integrated entropy production in the channel.
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x/h = 160
1.80
x/h = 160.3
U/Uc
0.8
x/h = 160.8
U/Uc
0.40
0.00
0.4
0
0.4
y/h
0.8
1.2
Reτ = 295 Reτ = 395 Reτ = 187
0.0
0
0.4
0.8 y/h
1.2
Fig u r e 9.3 Mean velocities.
near the wall, but other components become more significant at further distances away from the wall. In particular, the mean viscous dissipation accounts for more than 80% of the total entropy production at approximately y+ < 9, where y+ = yut /v. This percentage decreases to zero in the center of the channel. In Figure 9.3, the mean velocity profile in the fully developed region is presented at Ret = 187, 295, and 395. The velocity profiles (shown in the inset) at different transverse locations collapse onto each another, due to fully developed conditions. The mean velocities are normalized by the centerline velocity in Figure 9.3, and the y-coordinate is normalized by the half-channel height. Figure 9.4 shows the distribution of mean velocity profiles in terms of wall variables. The wall shear stress is determined by the Clauser plot technique, which assumes a universal logarithmic profile in the overlap region. The experimental data confirm the extent of the logarithmic layer, as the Reynolds number increases. The mean profiles for Ret = 295 and 399 agree out to y+ ≈ 250. At Ret = 187, the standard constants (k = 0.4 and B = 5.0) give a logarithmic slope with a slight offset from a best fit (k = 0.4 and B = 5.5), in agreement with DNS data. These results are consistent with previous experimental measurements, which associate such flow behavior with low Reynolds number effects. The spatial resolution of PIV is limited by the size of the interrogation area, so measurements by Anteyinka and Naterer (2007) could not be made any closer to the wall than y+ = 8.18. The data compare well with DNS results, thereby providing useful validation of the formulation.
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Entropy-Based Design and Analysis of Fluids Engineering Systems 30 Reτ = 187 (PIV) Reτ = 295 (PIV)
25
Reτ = 395 (PIV) Reτ = 395 (DNS)
u+
20
Log Law u+ = y+
15
10
5
0
1
10
y+
100
1000
Fig u r e 9.4 Velocities normalized by inner variables.
The turbulent fluctuating velocities are normalized by the friction velocity and plotted in Figure 9.5 at three different cross sections. Figure 9.6 compares the distributions of u+ and v+ obtained at Ret = 187 with the PIV results of Liu et al. (1991) and the DNS results of Kim et al. (1987). Good qualitative agreement is observed among the results. Compared with the DNS results, the peak value for the fluctuating streamwise velocity is underpredicted for the present results at y+ = 13. The peak value shows close agreement with previous PIV results of Liu et al. (1991). The measured data also shows higher values that the DNS results in the channel core. The fluctuating velocities are plotted against y/h in Figure 9.7 at all Reynolds numbers. The u+ profiles collapse onto the Ret = 399 curve away from the wall, at approximately y/h > 0.36 for Ret = 187 and y/h > 0.2 for Ret = 295. All profiles vary nearly linearly in Figure 9.7 for u+ between 0.4 < y/h < 0.9 at the three Reynolds numbers tested and 0.2 < y/h < 0.9 for v+. The linear range for v+ at Ret = 187 is not immediately apparent. This observation is consistent with past studies of Moser et al. (1999) that suggested a collapse of the u+ profiles to a high Reynolds number outerlayer limit at y+ > 80. No such collapse is observed when the inner variables are used as the normalizing quantities. The qualitative trends of fluctuating velocities also compare well with the DNS data shown in the inset. The Kolmogorov length scale, h, estimated from its definitions and DNS data, is between 6 and 18 mm at the highest Reynolds number and between 14 and 38 mm at Ret = 187. With a 32 × 32 PIV interrogation region, the spatial resolution of the PIV measurements is approximately 280 mm. Thus, the spatial resolution is about 16 times the Kolmogorov length scale at the channel core and 48 times close to the wall. The resolution of the velocity field is too small to accurately determine spatial
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Entropy Production in Turbulent Flows 2.5 x/h = 160 x/h = 160.3 x/h = 160.8
u+
2
1.5
1
0.5
0.0
0.40
0.80
1.20
y/h
Fig u r e 9.5 Turbulent velocities at Ret = 395. 4 Reτ = 187 (PIV) Reτ = 180 (DNS)
3.2
Reτ = 183 (Lui et al.)
u+, v+
2.4
1.6
0.8
0
0
0.4
0.8
1.2
y/h
Fig u r e 9.6 Turbulent velocities at Ret = 187.
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3.2 Reτ = 180 Reτ = 395
U+, V+
2.4
2.4
1.6 0.8
U+, V+
0
0
0.4
y/h
0.8
1.2
Reτ = 187 Reτ = 295
1.6
Reτ = 399
0.8
0
0
0.4
0.8
1.2
1.6
y/h
Fig u r e 9.7 Turbulent velocities plotted in outer variables.
derivatives of the fluctuating velocity field and dissipation rate with the total dissipation method. Nevertheless, simplified expressions for e and isotropic conditions can be used to estimate e. The dissipation rate is estimated by the dimensional analysis relation, Equation 9.31, and the large eddy PIV approach. The SGS stress is obtained from the Smagorinsky and Gradient models. The accuracy of the estimation methods can be verified by comparisons with the DNS data of Moser et al. (1999). The measured dissipation rates are compared with the corresponding DNS solution at Ret = 187 in Figure 9.8. The dissipation rate in Figure 9.8 and all subsequent figures is normalized by ut4 /ν . The different methods show close agreement with DNS data, and they give correct distributions of the TKE in the channel. A high dissipation region is concentrated near the wall. The DNS data show an inflection point, not captured by PIV, closer to the wall at y+ = 12. The dissipation rate reaches a minimum in the center of the channel, and it becomes the only mechanism for energy loss in the channel centerline for turbulent flows. Greater deviations from the DNS data are noticed for all estimation methods closer to the wall, due to the anisotropic nature of the flow and smaller dissipation length scales in this region. Dissipation rates computed from the DNS results of Kuroda et al. (1989) at Ret = 100 and Moser et al. (1999) at Ret = 180, 395, and 590 are plotted in Figure 9.9. In Figure 9.10, the dissipation rate has been estimated from the dimensional analysis relation at all Reynolds numbers investigated. The integral length scale, l, is defined as the distance from the wall to a point where the streamwise velocity is 99% of
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Entropy Production in Turbulent Flows 0.15 Dimensional Analysis Smagorinsky Model Gradient Model Reτ = 180 (DNS)
ε+
0.1
0.05
0
0.0
0.4
0.8
1.2
y/h
Fig u r e 9.8 Dissipation of turbulent kinetic energy at Ret = 187.
0.25
Reτ = 100 Reτ = 180 Reτ = 395 Reτ = 590
0.2
ε+
0.15
0.1
0.05
0
0.0
0.4
y/h
0.8
1.2
Fig u r e 9.9 Direct numerical simulation results.
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0.2 Reτ = 187 Reτ = 295 0.15 ε+
Reτ = 399
0.1
0.05
0
0
0.5
1
1.5
y/h
Fig u r e 9.10 Dimensional analysis-based e estimation.
the centerline velocity. The DNS results suggest lower values of e in the middle region, with higher values at the wall and steeper gradients as the Reynolds number increases. The dissipation rate shows similar trends in the wall layer in Figure 9.10. The large eddy PIV dissipation estimates are shown in Figure 9.11 (Smagorinsky model) and Figure 9.12 (Gradient model). The filter size for the correlation analysis is ∆ = 280 mm, whereas the integral length scale is l ≈ 8 mm at Ret = 180. The Kolmogorov length scale is h = 18 mm at the channel centerline. Thus, ∆ << l, and the filter size is sufficiently larger than the Kolmogorov length scale to warrant the use of large eddy PIV. The dissipation rate closely agrees with the DNS result at low Reynolds numbers, but it is underpredicted at higher Reynolds numbers. One would expect better performance with increasing Reynolds numbers because the flow shows higher tendencies toward local isotropy as the Reynolds number increases. This discrepancy is partly due to the overall accuracy of PIV measurements at high Reynolds numbers. At high Reynolds numbers, the PIV dynamic range required to accurately resolve the smaller Kolmogorov length scales becomes very high, and the spatial resolution of the PIV fails to capture certain aspects of the flow structure. Better performance of all estimation methods at Ret = 187 can be attributed to larger Kolmogorov length scales and a consequent higher flow resolution. Measurements too close to the wall suffer from reflections and poor accuracy of the velocity at high Reynolds numbers. The two SGS models show similar predictions, suggesting a weak dependence of the large eddy PIV method on the SGS stress model.
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0.2 Reτ = 187 Reτ = 295
0.15 ε+
Reτ = 399 0.1
0.05
0
0
0.5
1
1.5
y/h
Fig u r e 9.11 Large eddy PIV-based e estimation with Smagorinsky SGS model.
0.25
0.2 Reτ = 187 Reτ = 295 0.15 ε+
Reτ = 399
0.1
0.05
0
0
0.5
1
1.5
y/h
Fig u r e 9.12 Large eddy PIV-based e estimation with Gradient SGS model.
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Reτ = 187 (PIV) Φ+
Reτ = 180 (DNS) 0.4
0
0
0.5
1
1.5
y/h
Fig u r e 9.13 Viscous dissipation at Ret = 187.
Measured oscillations are effectively reduced by filtering of the velocity data. In Figure 9.13, a 3 × 3 average filter is used for smoothing of the velocity vectors, before calculating the viscous dissipation. A comparison of the experimental data with DNS results is presented in Figure 9.14, with regard to the spatial distribution of the mean viscous dissipation and the total entropy production at Ret = 187. All three estimation methods agree closely. The entropy production is scaled by ρut4T /ν . A region of high entropy production is evident close to the wall. The distribution of TKE dissipation is plotted as a percentage of the total mechanical energy loss in Figure 9.15, to provide a Second Law insight into the energy conversion in turbulent flows. The measured results show close agreement with DNS results. The percentage of TKE decreases from a maximum at the channel centerline to approximately 14% of the total mechanical energy loss, just outside the logarithmic region, toward the wall. The percentage of e + is fairly constant in the outer region and logarithmic layers. This distribution implies that the viscous stress due to molecular viscosity dominates at the wall. Unlike laminar flow, where the viscous shear stress increases linearly across the fluid layer from the channel center to the wall and entropy production is distributed evenly over the entire channel, the viscous stress is concentrated in a region between the buffer layer and the wall in turbulent flows, thereby leading to a much higher mean shear stress and entropy production at the wall. This explains the need for higher pumping power in turbulent flows to move fluid through a duct at a given mass flow rate.
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Reτ = 178.12 (DNS) Dimensional Analysis
1.00
Ps+
Gradient Model Smagorinsky Model Viscous Mean (DNS) 0.50
0.00
0
0.5 y/h
1
Fig u r e 9.14 Turbulent entropy production at Ret = 187. 120
Percentage ε+/Ps+
100
80 Reτ = 180 (DNS)
60
PIV Reτ = 100 (DNS)
40
20
0
0
0.5
y/h
1
1.5
Fig u r e 9.15 Percentage of e + in total entropy production across the channel at Ret ≈ 187.
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R ef er en c es Adeyinka, O.B. and Naterer, G.F. 2007. Measured turbulent entropy production with large eddy particle image velocimetry. Exper. in Fluids, 42(6): 881–891. Adeyinka, O.B. and Naterer, G.F. 2004. Modeling of entropy production in turbulent flows. ASME J. Fluids Eng., 126(6): 893–899. Adrian, R.J. 1997. Dynamic ranges of velocity and spatial resolution of particle image velocimetry. Measurement Sci. Technol., 8: 1393–1398. Andreopoulos, Y. and A. Honkan. 1996. Experimental techniques for highly resolved measurements of rotation, strain, and dissipation rate tensors in turbulent flows. Measurement Sci. Technol., 7: 1462–1476. Bardina, J., Ferziger, J.H., and W.C. Reynolds. 1980. Improved Subgrid-Scale Models for Large-Eddy Simulation, AIAA Paper 80–1357, AIAA 13th Fluid and Plasma Dynamics Conference, Snowmass, CO. Batchelor, G.K. 1982. The Theory of Homogeneous Turbulence. Cambridge University Press, London. Clark, R.A., Ferziger, J.H., and W.C. Reynolds. 1979. Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mechanics, 91: 1–16. Drost, M.K. and M.D. White. 1991. Numerical predictions of local entropy generation in an impinging jet. ASME J. Heat Transfer, 113: 823–829. FlowMap Particle Image Velocimetry Instrumentation: Installation and User Guide. 1998. Dantec Dynamics, Demark. Germano, M., Piomelli, U., Moin, P., and W.H. Cabot. 1991. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3: 1760–1765. Hanjalic, K. and S. Jakirilic. 2002. Second-moment turbulence closure modelling. In: Closure Strategies for Turbulent and Transitional Flows Part A. B. Launder and N. Sandham, Eds., Cambridge University Press, London. Hauke, G.A. 1995. A Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation for the K-Epsilon Model. Ph.D. thesis. Stanford University, Stanford, CA. Hinze, J.O. 1975. Turbulence. 2nd ed. McGraw-Hill, New York. Hussein, H.J. and R.J. Martinuzzi. 1995. Energy balance for turbulence flow around a surface mounted cube place in a channel. Phys. Fluids, 8: 764–780. Jansen, K.E. 1993. The Role of Entropy in Turbulence and Stabilized Finite Element Methods. Ph.D. thesis. Stanford University, Stanford, CA. Kim, J., Moin, P., and R. Moser. 1987. Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mechanics, 177: 133–166. Kramer-Bevan, J.S. 1992. A Tool for Analysing Fluid Flow Losses. M.Sc. thesis. University of Waterloo, Waterloo, Ontario, Canada. Kresta, S. 1998. Turbulence in stirred tanks: anisotropic, approximate, and applied. Can. J. Chem. Eng., 76(3): 563–575. Kresta, S.M. and P.E. Wood. 1993. The flow field produced by a pitched blade turbine: characterization of the turbulence and estimation of the dissipation rate. Chem. Eng. Sci., 48: 1761–1774. Kuroda, A., Kasagi, N., and M. Hirata. 1989 (August 28–29). A Direct Numerical Simulation of the Fully Developed Turbulent Channel Flow. International Symposium on Computational Fluid Dynamics, Nagoya, Japan, 1174–1179. Lilly, D.K. 1992. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A, 4: 633–635. Liu, Z.C., Landreth, C.C., Adrain, R.J., and T.J. Hanratty. 1991. High resolution measurement of turbulent structure in a channel with particle image velocimetry. Exp. Fluids, 10: 301–312.
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Mason, P. 1989. Large-eddy simulation of the convective atmospheric boundary layer. J. Atmospheric Sci., 46: 1492–1516. Meneveau, C., Lund, T., and W.H. Cabot. 1996. A Lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech., 319: 353–385. Métais, O. and M. Lesieur. 1992. Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech., 239: 157–194. Moore, J. and J.G. Moore. 1983 (March 27–31). Entropy Production Rates from Viscous Flow Calculations Part I. A Turbulent Boundary Layer Flow. ASME Gas Turbine Conference. ASME Paper 83-GT-70. Phoenix, AZ. Moser, R.D., Kim, J., and N.N. Mansour. 1999. Direct numerical simulation of turbulent flow up to Ret = 590. Phys. Fluids, 11(4): 943–945. Saarenrinne, P. and M. Piirto. 2000. Turbulent kinetic energy dissipation rate estimation from PIV velocity vector fields. Exp. Fluids, 29(7): S300–S307. Saarenrinne, P., Piirto, M., and H. Eloranta. 2001. Experiences of turbulence measurement with PIV. Measurement Sci. Technol., 12: 1904–1910. Sheng, J., Meng, H., and Fox, R.O. 2000. A large eddy PIV method for turbulence dissipation rate estimation. Chem. Eng. Sci., 55: 4423–4434. Smagorinsky, J. 1963. General circulation experiments with the primitive equations. I. The basic equations. Monthly Weather Rev., 91: 99–164. Sullivan, P.P., McWilliams, J., and C.-H. Moeng. 1994. A subgrid-scale model for largeeddy simulation of planetary boundary-layer flows. Boundary Layer Meteorol., 71: 247–276. Tennekes, H. and J.L. Lumley. 1972. A First Course in Turbulence. MIT Press, Cambridge, MA. Turan, O. and R. Azad. 1989. Effect of hot-wire probe defects on a new method of evaluating turbulence dissipation. J. Phys. E: Scientific Instruments, 22(4): 254–261. White, F.M. 1991. Fluid Mechanics, 5th ed. McGraw-Hill, New York.
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Appendix Ta bl e A .1 Conversion of Units and Constants Conversion Factors Acceleration Area Density Dynamic viscosity Energy Force Heat flux Heat transfer coefficient Heat transfer rate Kinematic viscosity Latent heat Length Mass Mass diffusivity Mass flow rate Mass transfer coefficient Pressure, stress Specific heat Temperature Temperature difference Thermal conductivity Thermal diffusivity Thermal resistance Velocity Volume Volume flow rate
1 m/s2 1 m2 1 kg/m3 1 kg/m ⋅ s 1 kJ 1N 1 W/m2 1 W/m2 K 1W 1 m2/s 1 kJ/kg 1m 1 kg 1 m 2/ s 1 kg/s 1 m/s 1 Pa 1 kJ/kgK K R 1K 1 W/mK 1 m2/s 1 K/W 1 m/s 1 m3 1 m3/s
= 4.252 × 107 ft/hr2 = 1550.0 in2 = 0.06243 lbm/ft3 = 1N ⋅ s/m2 = 0.9478 Btu = 1 kg ⋅ m/s2 = 1 kg/s3 = 0.1761 Btu/ft2 hr °F = 3.4123 Btu/hr = 10.7636 ft2/s = 0.4299 Btu/lbm = 39.37 in = 2.2046 lbm = 10.7636 ft2/s = 7936.6 lbm/hr = 1.181 × 104 ft/hr = 1 N/m2 = 0.2388 Btu/lbm ⋅ °F = °C + 273.15 = °F + 459.67 = 1°C = 0.57782 Btu/hr ⋅ ft ⋅ °F = 10.7636 ft2/s = 0.5275° F/hr ⋅ Btu = 3.2808 ft/s = 264.17 gal (U.S.) = 1.585 × 104 gal/min
= 3.2808 ft/s2 = 10.764 ft2 = 2419.1 lbm/ft ⋅ hr =737.56 ft ⋅ lbf = 0.22481 lbf = 0.3171 Btu/hr ⋅ ft2 = 1.341 × 10–3 hp = 104 stokes = 3.2808 ft = 1.1023 × 10–3 US tons = 3.875 × 104 ft2/hr
= 1.4504 × 10–4 lbf/in2 = 0.2389 cal/g ⋅ °C = (5/9)(°F + 459.67) = (9/5) (°K) = (9/5) °F = 3.875 × 104 ft2/hr = 3.6 km/hr = 1000 L = 2118.9 ft3/min
SI Unit Conversions Prefix (Symbol) Tera (T) Giga (G) Mega (M) kilo (k) milli (m) micro (m) nano (n) pico (p) femto (f)
Multiplier 1012 109 106 103 10–3 10–6 10–9 10–12 10–15
287
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Constants Atmospheric pressure (Patm) e Gravitational acceleration (g) 1 mole p Universal gas constant (R)
= 101,325 N/m2 = 2.7182818 = 9.807 m/s2 = 6.022 × 1023 molecules = 3.1415927 = 8.315 kJ/kmol ⋅ K
= 14.69 lbf /in2
= 10–3 kmol = 1.9872 Btu/lbmol ⋅ R
TABLE A .2 Properties of Metals at STPa
Metal
Melting Point (°C)
Boiling Point (°C)
Aluminum Antimony Beryllium Bismuth Cadmium Chromium Cobalt Copper Gold Iridium Iron Lead Magnesium Manganese Mercury Molybdenum Nickel Niobium Osmium Platinum Plutonium Potassium Rhodium Selenium Silicon Silver Sodium Tantalum Thorium Tin Titanium Tungsten Uranium Vanadium Zinc
660 630 1285 271.4 321 1860 1495 1084 1063 2450 1536 327.5 650 1244 -38.86 2620 1453 2470 3025 1770 640 63.3 1965 217 1411 961 97.83 2980 1750 232 1670 3400 1132 1900 419.5
2441 1440 2475 1660 767 2670 2925 2575 2800 4390 2870 1750 1090 2060 356.55 4651 2800 4740 4225 3825 3230 760 3700 700 3280 2212 884 5365 4800 2600 3290 5550 4140 3400 910
a
Thermal Conductivity (W/mK)
Specific Heat, cp (kJ/kgK)
Coefficient of Expansion (×106/K)
Density, r (kg/m3)
237.0 18.5 218 8.4 93 91 69 398 315 147 80.3 34.6 159 7.8 8.39 140 89.9 52 61 73 8 99 150 0.5 83.5 427 134 54 41 64 20 178 25 60 115
0.900 0.209 1.825 0.126 0.230 0.460 0.419 0.385 0.130 0.130 0.452 0.130 1.017 0.477 0.138 0.251 0.444 0.268 0.130 0.134 0.134 0.753 0.243 0.322 0.712 0.239 1.226 0.142 0.126 0.226 0.523 0.134 0.117 0.486 0.389
25 9 12 13 30 6 12 16.6 14.2 6 12 29 25 22
2700
397.8
7150 8860 8960 19300
330.8 276.4 205.2 62.8
7870 11300
272.2 23.0
10200 8900
288.9 297.3
10500
111.0
7280 4500
59.0 418.8
5 13 7 5 9 54 83 8 37 3 19 70 6.5 12 20 8.5 4.5 13.4 8 35
Heat of Fusion (kJ/kg)
Data reprinted with permission from Hewitt et al. (1997) and Weast (1970).
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Appendix
TABLE A .3 Properties of Nonmetalsa
Material Asbestos millboard Asphalt Brick, common Brick, hard Chalk Charcoal, wood Coal, anthracite Concrete, light Concrete, stone Corkboard Earth, dry Fiber hardboard Fiberboard, light Firebrick Glass, window Gypsum board Ice (0°C) Leather, dry Limestone Marble Mica Mineral wool blanket Paper Paraffin wax Plaster, light Plaster, sand Plastics, foamed Plastics, solid Porcelain Sandstone Sawdust Silica aerogel Vermiculite Wood, balsa Wood, oak Wood, white pine a
Density, r (kg/m3) 1400 1100 1750 2000 2000 400 1500 1400 2200 200 1400 1100 240 2100 2500 800 900 900 2500 2600 2700 100 900 900 700 1800 200 1200 2500 2300 150 110 130 160 700 500
Thermal Conductivity, k (W/mK) 0.14 0.71 1.3 0.84 0.088 0.26 0.42 1.7 0.04 1.5 0.2 0.058 1.4 0.96 0.17 2.2 0.2 1.9 2.6 0.71 0.04 0.1 0.2 0.2 0.71 0.03 0.19 1.5 1.7 0.08 0.02 0.058 0.050 0.17 0.12
Specific Heat, cp (kJ/kgK) 0.837 1.67 0.920 1.00 0.900 1.00 1.26 0.962 0.753 1.88 1.26 2.09 2.51 1.05 0.837 1.09 2.09 1.502 0.908 0.879 0.502 0.837 1.38 2.89 1.00 0.920 1.26 1.67 0.920 0.920 0.879 0.837 0.837 2.93 2.09 2.51
Data reprinted with permission from Weast (1970).
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TABLE A .4 Properties of A ir at A tmospheric Pressurea Temperature, T (K) 150 200 250 260 270 275 280 290 300 310 320 330 340 350 400 450 500 550 600 700 800 900 1000 a
Density, r (kg/m3)
Specific Heat, cp (kJ/kgK)
Viscosity, r (kg/ms)
Thermal Conductivity, k (W/mK)
Pr
2.367 1.769 1.413 1.359 1.308 1.285 1.261 1.218 1.177 1.139 1.103 1.070 1.038 1.008 0.882 0.784 0.706 0.642 0.588 0.504 0.441 0.392 0.353
1.010 1.006 1.005 1.005 1.006 1.006 1.006 1.006 1.006 1.007 1.007 1.008 1.008 1.009 1.014 1.021 1.030 1.040 1.051 1.075 1.099 1.121 1.142
10.28 × 10–6 13.28 × 10–6 15.99 × 10–6 16.50 × 10–6 17.00 × 10–6 17.26 × 10–6 17.50 × 10–6 17.98 × 10–6 18.46 × 10–6 18.93 × 10–6 19.39 × 10–6 19.85 × 10–6 20.30 × 10–6 20.75 × 10–6 22.86 × 10–6 24.85 × 10–6 26.70 × 10–6 28.48 × 10–6 30.17 × 10–6 33.32 × 10–6 36.24 × 10–6 38.97 × 10–6 41.53 × 10–6
0.014 0.018 0.022 0.023 0.024 0.024 0.025 0.025 0.026 0.027 0.028 0.029 0.029 0.030 0.034 0.037 0.040 0.044 0.047 0.052 0.058 0.063 0.068
0.758 0.739 0.722 0.719 0.716 0.715 0.713 0.710 0.708 0.705 0.703 0.701 0.699 0.697 0.689 0.684 0.680 0.680 0.680 0.684 0.689 0.696 0.702
Data reprinted with permission from Hewitt et al. (1997) and Weast (1970).
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Appendix
TABLE A .5 Properties of Other Gases (1 atm, 298 K)a
Gas
Density, r (kg/m3)
Specific Heat, cp (kJ/kgK)
Gas Constant (J/kgoC)
Thermal Conductivity, k (W/mK)
Dynamic Viscosity, m (kg/ms)
Acetylene, C2H2 Ammonia, NH3 Argon, Ar n-Butane, C4H10 Carbon Dioxide, CO2 Carbon Monoxide, CO Chlorine, Cl2 Ethane, C2H6 Ethylene, C2H4 Fluorine, F2 Helium, He Hydrogen, H2 Hydrogen Sulfide, H2S Methane, CH4 Methyl Chloride, CH3Cl Nitric Oxide, NO Nitrogen, N2 Nitrous Oxide, N2O Oxygen, O2 Ozone, O3 Propane, C3H8 Propylene, C3H6 Sulfur Dioxide, SO2 Xenon, Xe
1.075 0.699 1.608 2.469 1.818 1.144 2.907 1.227 0.072 0.097 0.164 0.083 10.753 0.662 2.165 1.229 1.147 1.802 1.309 1.965 1.812 1.724 2.622 5.375
1.674 2.175 0.523 1.675 0.876 1.046 0.477 1.715 1.548 0.828 5.188 14.310 0.962 2.260 0.837 0.983 1.040 0.879 0.920 0.820 1.630 1.506 0.460 0.481
319 488 208 143 189 297 117 276 296 219 2077 4126 244 518 165 277 297 189 260 173 188 197 130 63.5
0.024 0.026 0.0172 0.017 0.017 0.024 0.0087 0.017 0.017 0.028 0.149 0.0182 0.014 0.035 0.010 0.026 0.026 0.017 0.026 0.033 0.017 0.017 0.010 0.0052
1.0 × 10–5 1.0 × 10–5 2.0 × 10–5 0.7 × 10–5 1.4 × 10–5 1.8 × 10–5 1.4 × 10–5 9.5 × 10–5 1.0 × 10–5 2.4 × 10–5 2.0 × 10–5 0.9 × 10–5 1.3 × 10–5 1.1 × 10–5 1.1 × 10–5 1.9 × 10–5 1.8 × 10–5 1.5 × 10–5 2.0 × 10–5 1.3 × 10–5 8.0 × 10–5 8.5 × 10–5 1.3 × 10–5 2.3 × 10–5 (Continued)
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TABLE A .5 (Continued) Properties of Other Gases (1 atm, 298 K)a Gas Acetylene, C2 H2 Ammonia, NH3 Argon, Ar n-Butane, C4H10 Carbon Dioxide, CO2 Carbon Monoxide, CO Chlorine, Cl2 Ethane, C2H6 Ethylene, C2H4 Fluorine, F2 Helium, He Hydrogen, H2 Hydrogen Sulfide, H2S Methane, CH4 Methyl Chloride, CH3Cl Nitric Oxide, NO Nitrogen, N2 Nitrous Oxide, N2O Oxygen, O2 Ozone, O3 Propane, C3H8 Propylene, C3H6 Sulfur Dioxide, SO2 Xenon, Xe a
Boiling Point (°C) -75 -33.3 -186 -0.4 -78.5 -191.5 -34.0 -88.3 -103.8 -188.0 4.22 K 20.4 K -60 -23.7 -151.5 -195.8 -88.5 -182.97 -112.0 -42.2 -48.3 -10.0 -108.0
L atent Heat of Evaporation (kJ/kg) 614.0 1373.0 163.0 386.0 572.0 216.0 288.0 488.0 484.0 172.0 23.3 447.0 544.0 510.0 428.0 199.0 376.0 213.0 428.0 438.0 362.0 96.0
Melting Point (°C) -82.2 -77.7 -138 -205 -101 -172.2 -169 -220 -259.1 -84 -182.6 -97.8 -161 -210 -90.8 -218.4 -193 -189.9 -185 -75.5 -140
L atent Heat of Fusion (kJ/kg) 53.5 332.3 44.7
95.4 95.3 120.0 25.6 58.0 70.2 32.6 130.0 76.5 25.8 149.0 13.7 226.0 44.4 135.0 23.3
Heat of Combustion (kJ/kg) 50,200 — — 49,700 — 10,100 — 51,800 47,800 — — 144,000 18,600 5327 — — — — — 50,340 50,000 — —
Data reprinted with permission from Weast (1970).
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TABLE A .6 Properties of Other Gases (Effects of Temperature)a Temperature T (°C)
Density, r (kg/m3)
Specific Heat, cp (kJ/kgK)
Thermal Conductivity, k (W/mK)
0 20 50
0.956 0.894 0.811
2.176 2.176 2.176
0.022 0.024 0.027
9.18 × 10–6 9.82 × 10–6 1.09 × 10–5
-13 -3 7 27 77 227 727 1227
1.87 1.81 1.74 1.62 1.39 0.974 0.487 0.325
0.523 0.523 0.523 0.523 0.519 0.519 0.519 0.519
0.016 0.016 0.017 0.018 0.020 0.026 0.043 0.055
2.04 × 10–5 2.11 × 10–5 2.17 × 10–5 2.30 × 10–5 2.59 × 10–5 3.37 × 10–5 5.42 × 10–5 7.08 × 10–5
Butane, C4 H10
0 100 200 300 400 500 600
2.59 1.90 1.50 1.24 1.05 0.916 0.812
1.591 2.026 2.454 2.812 3.127 3.402 3.642
0.013 0.023 0.036 0.052 0.069 0.090 0.113
6.84 × 10–6 9.26 × 10–6 1.17 × 10–5 1.40 × 10–5 1.64 × 10–5 1.87 × 10–5 2.11 × 10–5
Carbon Dioxide, CO2
-13 -3 7 17 27 77 227
2.08 2.00 1.93 1.86 1.80 1.54 1.07
0.813 0.823 0.832 0.842 0.851 0.898 1.014
0.014 0.014 0.015 0.016 0.017 0.020 0.034
1.31 × 10–5 1.36 × 10–5 1.40 × 10–5 1.45 × 10–5 1.49 × 10–5 1.72 × 10–5 2.32 × 10–5
Carbon Monoxide, CO
-13 -3 7 17 27 77 227
1.31 1.27 1.22 1.18 1.14 0.975 0.682
1.041 1.041 1.041 1.041 1.041 1.043 1.064
0.022 0.023 0.024 0.025 0.025 0.029 0.039
1.59 × 10–5 1.64 × 10–5 1.69 × 10–5 1.74 × 10–5 1.79 × 10–5 2.01 × 10–5 2.61 × 10–5
Ethane, C2 H6
0 100 200 300 400 500 600
1.342 0.983 0.776 0.640 0.545 0.474 0.420
1.646 2.066 2.488 2.868 3.212 3.517 3.784
0.019 0.032 0.047 0.065 0.085 0.108 0.132
8.60 × 10–6 1.14 × 10–5 1.41 × 10–5 1.68 × 10–5 1.93 × 10–5 2.20 × 10–5 2.45 × 10–5
Ethanol, C2H5OH
100 200 300
1.49 1.18 0.974
1.686 2.008 2.318
0.023 0.035 0.050
1.08 × 10–5 1.37 × 10–5 1.67 × 10–5
Gas Ammonia, NH3
Argon, Ar
Dynamic Viscosity, m (kg/ms)
(Continued)
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TABLE A .6 (Continued) Properties of Other Gases (Effects of Temperature)a Gas
Helium, He
Temperature T (°C)
Density, r (kg/m3)
Specific Heat, cp (kJ/kgK)
Thermal Conductivity, k (W/mK)
Dynamic Viscosity, m (kg/ms)
400 500
0.828 0.720
2.611 2.891
0.067 0.086
1.97 × 10–5 2.26 × 10 –5
0 20 40
0.368 0.167 0.156
5.146 5.188 5.188
0.142 0.149 0.155
1.86 × 10–5 1.94 × 10–5 2.03 × 10–5
14.133 14.175 14.226 14.267 14.301 14.506
0.162 0.167 0.172 0.177 0.182 0.272
8.14 × 10–6 8.35 × 10–6 8.55 × 10–6 8.76 × 10–6 8.96 × 10–6 1.26 × 10–5
Hydrogen, H2
-13 -3 7 27 77 727
0.0944 0.0910 0.0877 0.0847 0.0819 0.04912
Methane, CH4
0 100 200 300 400 500 600
0.716 0.525 0.414 0.342 0.291 0.253 0.224
2.164 2.447 2.805 3.173 3.527 3.853 4.150
0.031 0.046 0.064 0.082 0.102 0.122 0.144
1.04 × 10–5 1.32 × 10–5 1.59 × 10–5 1.83 × 10–5 2.07 × 10–5 2.29 × 10–5 2.52 × 10–5
Nitrogen, N2,
77 227 727
1.14 9.75 6.82
1.041 1.042 1.056
0.026 0.030 0.040
1.79 × 10–5 2.00 × 10–5 2.57 × 10–5
Oxygen, O2
-13 -3 7 27 77 227 727
1.50 1.45 1.39 1.35 1.30 1.11 7.80
0.915 0.916 0.918 0.918 0.920 0.929 0.972
0.023 0.024 0.025 0.026 0.027 0.031 0.042
1.85 × 10–5 1.90 × 10–5 1.96 × 10–5 2.01 × 10–5 2.06 × 10–5 2.32 × 10–5 2.99 × 10–5
Propane, C3 H8
0 100 200 300 400 500 600
1.97 1.44 1.14 0.939 0.799 0.694 0.616
1.548 2.015 2.456 2.833 3.159 3.446 3.695
0.015 0.026 0.040 0.056 0.074 0.095 0.118
7.50 × 10–6 1.00 × 10–5 1.25 × 10–5 1.40 × 10–5 1.72 × 10–5 1.94 × 10–5 2.18 × 10–5
a
Data reprinted with permission from Weast (1970).
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TABLE A .7 Properties of L iquids (300 K, 1 atm)a
L iquid Acetic acid Acetone Alcohol, ethyl Alcohol, methyl Alcohol, propyl Benzene Bromine Carbon disulfide Carbon tetrachloride Chloroform Decane Dodecane Ether Ethylene glycol Fluorine, R-11 Fluorine, R-12 Fluorine, R-22 Glycerine Heptane Hexane Iodine Kerosene Linseed oil Mercury Octane Phenol Propane Propylene Propylene glycol Toluene Turpentine Water a
L atent Heat of Fusion (kJ/kgK) 181 98.3 108 98.8 86.5 126 66.7 57.6 174 77.0 201 216 96.2 181 34.4 183 200 140 152 62.2
11.6 181 121 79.9 71.4 71.8 333
Boiling Point (K)
L atent Heat of Evaporation (kJ/kgK)
Coefficient of Expansion (1/K)
391 329 351.46 337.8 371 353.3 331.6 319.40 349.6 334.4 447.2 489.4 307.7 470 297.0 243.4 232.4 563.4 371.5 341.86 457.5
402 518 846 1100 779 390 193 351 194 247 263 256 372 800 180.0 165 232 974 318 365 164 251
0.0011 0.0015 0.0011 0.0014
560 630 398 455 231.08 225.45 460 383.6 433 373
295 298 428 342 914 363 293 2260
0.0013 0.0012 0.0013 0.0013 0.0013
0.0016
0.00054
0.00018 0.00072 0.00090
0.00099 0.00020
Data reprinted with permission from Weast (1970).
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TABLE A .8 Properties of Saturated Watera T (°C) 0.01 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 220 240 260 280 300 320 340 360 373 a
P (kPa)
rf (kg/m3)
rv (kg/m3)
hfg (kJ/kg)
0.612 1.228 2.339 4.246 7.381 12.34 19.93 31.18 47.37 70.12 101.3 143.2 198.5 270.0 361.2 475.7 617.7 791.5 1001.9 1254.2 1553.7 2317.8 3344.7 4689.5 6413.2 8583.8 11279 14594 18655 21799
999.8 999.7 998.2 995.6 992.2 988.0 983.2 977.8 971.8 965.3 958.4 951.0 943.2 934.9 926.2 917.1 907.5 897.5 887.1 876.2 864.7 840.3 813.5 783.8 750.5 712.4 667.4 610.8 528.1 402.4
0.005 0.009 0.017 0.030 0.051 0.083 0.130 0.198 0.293 0.423 0.597 0.826 1.121 1.495 1.965 2.545 3.256 4.118 5.154 6.390 7.854 11.61 16.74 23.70 33.15 46.15 64.6 92.7 143.7 242.7
2501 2477 2453 2430 2406 2382 2358 2333 2308 2283 2257 2230 2202 2174 2145 2114 2082 2049 2015 1978 1940 1858 1766 1662 1543 1405 1239 1028 721 276
cp,f (kJ/kgK)
mf ⋅ 106 (kg/ms)
4229 4188 4182 4182 4183 4181 4183 4187 4196 4205 4217 4233 4249 4267 4288 4314 4338 4368 4404 4444 4489 4602 4759 4971 5279 5751 6536 8241 14686 21828
1791 1308 1003 798 653 547.1 466.8 404.5 355.0 315.1 282.3 255.1 232.2 212.8 196.3 182.0 169.6 158.9 149.4 141.0 133.6 121.0 110.5 101.5 93.4 85.8 78.4 70.3 60.2 46.7
kf (W/mK)
Prf
sf (N/m)
0.561 0.580 0.598 0.615 0.631 0.644 0.654 0.663 0.670 0.675 0.679 0.682 0.683 0.684 0.683 0.682 0.680 0.677 0.673 0.669 0.663 0.650 0.632 0.609 0.581 0.548 0.509 0.469 0.428 0.545
13.50 9.444 7.010 5.423 4.332 3.555 2.984 2.554 2.223 1.962 1.753 1.584 1.444 1.328 1.232 1.151 1.082 1.025 0.977 0.937 0.904 0.857 0.832 0.828 0.848 0.901 1.006 1.236 2.068 18.69
0.0757 0.742 0.0727 0.0712 0.0696 0.0680 0.0662 0.0645 0.0627 0.0608 0.0589 0.0570 0.0550 0.0529 0.0509 0.0488 0.0466 0.0444 0.0422 0.0400 0.0377 0.0331 0.0284 0.0237 0.0190 0.0144 0.0099 0.0056 0.0019 0.0001
Data reprinted with permission from Hewitt et al. (1997).
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TABLE A .9 A tomic Weights of Elementsa Element Hydrogen, H Helium, He Lithium, Li Beryllium, Be Boron, B Carbon, C Nitrogen, N Oxygen, O Fluorine, F Neon, Ne Sodium, Na Magnesium, Mg Aluminum, Al Silicon, Si Phosphorus, P Sulfur, S Chlorine, Cl Argon, Ar Potassium, K Calcium, Ca Scandium, Sc Titanium, Ti Vanadium, V Chromium, Cr Manganese, Mn Iron, Fe Cobalt, Co Nickel, Ni Copper, Cu Zinc, Zn Gallium, Ga Germanium, Ge Arsenic, As Selenium, Se a
kg/ kmol 1.01 4.00 6.94 9.01 10.81 12.01 14.01 16.00 19.00 20.18 22.99 24.31 26.98 28.09 30.97 32.07 35.45 39.95 39.10 40.08 44.96 47.87 50.94 52.00 54.94 55.85 58.93 58.69 63.55 65.39 69.72 72.61 74.92 78.96
kg/ kmol
Element Bromine, Br Krypton, Kr Rubidium, Rb Strontium, Sr Yttrium, Y Zirconium, Zr Niobium, Nb Molybdenum, Mo Technetium, Tc Ruthenium, Ru Rhodium, Rh Palladium, Pd Silver, Ag Cadmium, Cd Indium, In Tin, Sn Antimony, Sb Tellerium, Te Iodine, I Xenon, Xe Caesium, Cs Barium, Ba Lanthanum, La Cerium, Ce Praseodymium, Pr Neodymium, Nd Promethium, Pm Samarium, Sm Europium, Eu Gadolinium, Gd Terbium, Tb Dysprosium, Dy Holmium, Ho Erbium, Er
79.90 83.80 85.47 87.62 88.91 91.22 92.91 95.94 97.91 101.07 102.91 106.42 107.87 112.41 114.82 118.71 121.76 127.60 126.90 131.29 132.91 137.33 138.91 140.12 140.91 144.24 144.91 150.36 151.97 157.25 158.93 162.50 164.93 167.26
Element Thulium, Tm Ytterbium, Yb Lutetium, Lu Hafnium, Hf Tantalum, Ta Tungsten, W Rhenium, Re Osmium, Os Iridium, Ir Platinum, Pt Gold, Au Mercury, Hg Thallium, Tl Lead, Pb Bismuth, Bi Polonium, Po Astatine, At Radon, Rn Francium, Fr Radium, Ra Actinium, Ac Thorium, Th Protactinium, Pa Uranium, U Neptunium, Np Plutonium, Pu Americium, Am Curium, Cm Berkelium, Bk Californium, Cf Einsteinium, Es Fermium, Fm Mendelevium, Md Nobelium, No
kg/ kmol 168.93 173.04 174.97 178.49 180.95 183.84 186.21 190.23 192.22 195.08 196.97 200.59 204.38 207.2 208.98 208.98 209.99 222.02 223.02 226.03 227.03 232.04 231.04 238.03 237.05 239.05 243.06 247.07 249.07 251.08 252.08 257.10 258.10 259.10
Data reprinted with permission from Hewitt et al. (1997).
References Hewitt, G.F., Shires, G.L., and Y.V. Polezhaev, Eds. 1997. International Encyclopedia of Heat and Mass Transfer. CRC Press, Boca Raton, FL. Weast, R.C., Ed. 1970. CRC Handbook of Tables for Applied Engineering Science. CRC Press, Boca Raton, FL.
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Nomenclature B cv, cp c, C d, D e, E Ec f F g G H He I j J Ja k Kn l, ℓ, L M n N Ns p, P Pe Pr q Q R Re s, S Ste u, v, w U vk
Bias and precision components of error Specific heats Wave speed; advection speed coefficient Diffusion coefficient Specific total energy (per unit mass); total energy Eckert number Flux function; mass fraction Body force or source term, flux of entropy, or Faraday’s constant Probability distribution function, gravitational vector Flux of exergy Enthalpy Electric field strength Exergy flow Heat flux compressible flow Jacobian of transformation Jacob number Thermal conductivity; turbulent kinetic energy Local Knudsen number Thickness; reference length; Pixel dimension Molecular weight Number density Thermomagnetic number, finite element shape function Entropy generation parameter Pressure Peclet number Prandtl number Heating, heat flow rate Generic function of molecular velocity Gas constant Reynolds number; local Reynolds number Specific entropy; Entropy Stefan number Cartesian velocity components Total internal energy Molecular velocity 299
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Work Phase fraction Exergy rate
Su bsc r ipt s a n d S u per sc r ipt s 0 a act B c CD CE coll conc CV d des eff f F gen i, j, k J′ m MB min n ns p PDF ref rev supp t T w x, y, z
Initial; exchange Advection quantity; anode; analytic Activation Body interactions Cathode Central distribution Chapman–Enskog Molecular collision Concentration Control Volume Diffusion quantity; discrete quantity Sestruction Effective Friction Faraday constant Generation Coordinate direction indices Bessel function Membrane Maxwell–Boltzmann Minimum value Time step index No slip condition Phase interactions; parallel Probability density function Reference Reversible Supplied Transient; time-dependent; turbulent Matrix transpose Wall value Cartesian coordinate directions
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Nomenclature
Gr eek Symbo l s a b T, b C g G d D e h q k l m, m e x r s t f F c Y wk
Charge transfer coefficient Thermal and solution expansion coefficients Ratio of specific heats; reaction order General diffusion coefficient Kronecker delta function; diffusion ratio Difference operator Total digital image error; turbulent dissipation rate Second-Law effectiveness; polarization Skewness coefficient Isothermal compressibility Interrogation length scale; Eigenvalue; Taylor microscale Dynamic viscosity coefficient; magnetic permeability Constitutive variable; slip coefficient Mass density Mechanical stress Shear stress Specific exergy Viscous dissipation function Flow alignment factor Flow exergy Vorticity
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