Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner
112 V. Kecman
State-Space Models of L...
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Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner
112 V. Kecman
State-Space Models of Lumped and Distributed Systems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Series Editors M. Thoma • A. Wyner Advisory Board L. D. Davisson • A. G. J. MacFarlane • H. Kwakernaak J. L. Massey • Ya Z. Tsypkin • A. J. Viterbi
Author Vojislav Kecman Faculty of Mechanical Engineering and Naval Architecture University of Zagreb Dj. Salaja 5 Zagreb 41 000 Yugoslavia
ISBN 3-540-50082-0 Springer-Vedag Berlin Heidelberg NewYork ISBN 0-387-50082-0 Springer-Vedag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1988 Printed in Germany Offsetprinting: Mercedes-Druck, Berlin Binding: B. Helm, Berlin 2161/3020-543210
PREFACE
This work is d e v o t e d to the s c i e n c e a n d art of the m a t h e m a t i c a l d e s c r i p t i o n of d y n a m i c s f o u n d in various t e c h n i c a l p r o c e s s e s a n d s y s t e m s with s p a t i a l l y l u m p e d a n d d i s t r i b u t e d p a r a m e t e r s . T h e r e a d e r will find b a s i c a n d i n d i s p e n s a b l e k n o w l e d g e here, a n d master the t e c h n i q u e s a n d s e c r e t s for d e v i s i n g a d y n a m i c m a t h e m a t i c a l m o d e l m a k i n g u s e of the f u n d a m e n t a l laws of nature, the laws of c o n s e r v a t i o n of mass. e n e r g y a n d momentum. But this is the b o o k ' s l e s s e r part, it is not m e r e l y d e s i g n e d to p r o v i d e insight into the g e n e r a l a n d s p e c i a l a s p e c t s of the m a t h e m a t i c a l m o d e l i n g of thermal, m e c h a n i c a l , h y d r a u l i c , p n e u m a t i c a n d c h e m i c a l d y n a m i c systems. Other g o a l s a n d p u r p o s e s also m o t i v a t e d the v o l u m e y o u are holding. Firstly. to s h o w the real r e l a t i o n s h i p s b e t w e e n the p r o c e s s v a r i a b l e s , p h y s i c a l a n d g e o m e t r i c a l p r o p e r t i e s a n d the structure, c o e f f i c i e n t s a n d p a r a m e t e r s of m a t h e m a t i c a l models, i.e. to s h o w the c o n n e c t i o n s b e t w e e n the p h y s i c a l world a n d a b s t r a c t c o n c e p t s of e i g e n v a l u e s , p o l e s a n d z e r o s of s t a t e - s p a c e m o d e l s a n d their L a p l a c e r e p r e s e n t a t i o n s in the s-domain. S e c o n d l y , to p r e s e n t how i n d i s p e n s a b l e a s s u m p t i o n s in the m o d e l i n g p r o c e s s a r e m a p p e d onto a m a t h e m a t i c a l structure a n d the p a r a m e t e r s of s u c h d y n a m i c ( a n d static) models. Thirdly, to s h o w c o n n e c t i o n s , similarities a n d d i f f e r e n c e s b e t w e e n d y n a m i c m o d e l s of l u m p e d a n d d i s t r i b u t e d systems, with p a r t i c u l a r attention to the q u e s t i o n of d i s c r e t i z a t i o n (i.e. m o d e l r e d u c t i o n ) a n d its c o n s e q u e n c e s in the world of both m o d e l structure a n d p a r a m e t e r s , a n d in the r e p r e s e n t a t i o n of real d y n a m i c p r o p e r t i e s . The limitations of a n a l y t i c a l tools a r e p o i n t e d out in e v e n the most e l e m e n t a r y a n d simplest d i s t r i b u t e d systems.
IV To realize th e s e tasks it is pos s i bl e ( e v e n n e c e s s a r y a n d useful) to a n a l y z e the d y nam i c s of p h y s i c a l l y c o m p l e t e l y different p r o c e s s e s using the same unified p r o c e d u r e a n d method. From this stem two important characteristics of the book as a whole - a consistent a p p r o a c h to the d y n a m i c s of different systems using the c o n c e p t s of d y n a m i c variables ( a c c u m u l a t e d variable, effort, flow a n d g e n e r a l a c c e l e r a t i o n ) a n d d y n a m i c coefficients (of c a p a c i t y , r es i s t ance a n d inertia), and state-space representation of all the o b t a i n e d lumped models. The latter feature coul d b e of particular use for control system analysts a n d desi gners. The b o o k is d i v i d e d into thzee c h a p t e r s (followed b y an A p p e n d i x with a presentation of the b a s i c mathematical tools u s e d in the book). The introductory c h a p t e r outlines the overall c o n c e p t of the whole book, It defines the bas i c ideas, gives a p r o c e s s classification, defines the focal d y n a m i c p r o b l e m a n d introduces the basi c c o n c e p t s of d y n a m i c variables a n d coefficients in different t e c hni cal systems. The rest of the book is then d i v i d e d a c c o r d i n g to d e p e n d e n c e of system parameters u p o n the spatial co o r di nat e. Chapter 2 c o n s i d e r s p r o c e s s e s with lumped parameters. The p r o c e s s of mass storage, fluid flow, heat transfer a n d m e c h a n i c a l p r o c e s s e s of rigid b o d y motion are then thoroughly e xa m i ned. It is p o i n t e d to their basic a n d common d y n a m i c properties ( n a m e d he re as proportional, integral a n d derivative). The s e c o n d part of this c h a p t e r l eads to distributed systems a n d is d e v o t e d to the modeling of c o m p l e x systems of a higher order. The last e x a m p l e of this c h a p t e r deal s with the problem of the discretization of partial differential equat i ons (PDE) a n d presents an introduction to Chapter 3 which c o n s i d e r s the question of d y n a m i c models of distributed systems. Following the common p r a c t i c e in mathematical physics, this third a n d last c h a p t e r is d i v i d e d into three sections. The first o n e d e a l s with mass a n d e n e r g y transportation systems which are d e s c r i b e d b y h y p e r b o l i c PDE of the 1st order. The s e c o n d section is d e v o t e d to systems with equalization, which are d e s c r i b e d b y p a r a b o l i c PDE, while the last o n e d e a l s with systems with pezi odi c state changes ( d e s c x i b e d b y h y p e z b o l i c PDE of the 2rid order). This c h a p t e r p r o v i d e s a c o m p l e t e l y analytical (and just partly numerical) study of all the important a s p e c t s of the dynam i cs of distributed systems with particular emphasis on c o n n e c t i o n s and comparisons with l um pe d systems. The last two sections h a v e a m o n o g r a p h i c a l char a c t e r , but are written in quite a r e a d a b l e way.
V Concerning
the notation, it must b e
said
that
it w a s
at times
very
difficult ( i m p o s s i b l e , in fact) to a v o i d u s i n g the s a m e notation, s y m b o l s a n d s u b s c r i p t s for p h y s i c a l l y different v a r i a b l e s . This results from the fact that this b o o k c o n t a i n s m a n y v a r i a b l e s a n d c o e f f i c i e n t s from different t e c h n i c a l d i s c i p I i n e s w h i c h d e v e l o p e d independently of e a c h o t h e r a n d h a v e their o w n s e p a r a t e t e r m i n o l o g y a n d s y m b o l s . So it was. for the p r e s e n t , i m p o s s i b l e to b e c o m p l e t e l y c o n s i s t e n t in this r e g a r d . Finally, it must b e s a i d that in S e r b o C r o a t i a n this b o o k is c a l l e d ' P r o c e s s D y n a m i c s " a n d s o m e of the c o m m e n t s in the i n t r o d u c t i o n refer to this title c o n n e c t e d with a c o u r s e of the s a m e n a m e h e l d at the F a c u l t y of M e c h a n i c a l E n g i n e e r i n g a n d N a v a l A r c h i t e c t u r e . University of Z a g r e b , p a r t of the u n d e r g r a d u a t e a n d p o s t g r a d u a t e curriculum. T h e b o o k w a s written for s c i e n t i s t s , p r a c t i s i n g e n g i n e e r s a n d s t u d e n t s i n t e r e s t e d in the a n a l y s i s of s y s t e m d y n a m i c s . E x p e r i e n c e h a s s h o w n that the v o l u m e is of s p e c i a l v a l u e to a n a l y s t s a n d d e s i g n e r s of c o n t r o l s y s t e m s in m a n y d i s c i p l i n e s of e n g i n e e r i n g . T h e first two c h a p t e r s c a n b e of g r e a t u s e a s a t e x t b o o k for s u b j e c t s from the field of d y n a m i c s a n d control s y s t e m s in u n i v e r s i t y u n d e r g r a d u a t e c o u r s e s , while the third c h a p t e r is i n t e n d e d for m o r e d e t a i l e d g r a d u a t e s t u d y . H a v i n g this in mind, e v e r y s e c t i o n of the b o o k e n d s in m a n y s o l v e d n u m e r i c a l e x a m p l e s . This c a n b e of g r e a t u s e in the c o n t i n u i n g e d u c a t i o n a n d h o m e - s t u d y of all t h o s e w h o a r e c o n c e r n e d with this f a s t - d e v e l o p i n g field. Finally, it must b e s a i d that no b o o k g e t s its final form t h r o u g h the efforts of its author a l o n e . T h e r e a r e a l w a y s m a n y o t h e r s d e s e r v i n g c r e d i t for stimuIation, inspiration, s u p p o r t a n d h e I p a n d without w h o s e efforts b o o k s w o u l d not r e a c h their r e a d e r s . In the c a s e of this b o o k I wish to e x t e n d my most heartfelt g r a t i t u d e to Professor T u g o m i r ~urina, University of Z a g r e b . for his u n e n d i n g s u p p o r t d u r i n g r e s e a r c h in this field, a n d to Professor Petar V. Kokotovid. University of Illinois. for s u p p o r t in the p r o m o t i o n of the English e d i t i o n of this b o o k . T h e e x t e n s i v e a n d s u c c e s s f u l work o n the English translation w a s p e r f o r m e d b y Ms. Nikolina J o v a n o v i d . w h o m I a l s o thank. To Ms. Ellen Elias-Bursad. M. A., t h a n k s for her work on the l a n g u a g e e d i t i n g of the English text. T h e merits for w e l l - d r a w n figures a n d p a t i e n t t y p i n g b e l o n g to Z v o n k o G r g e k a n d D r a g i c a ~poljar. r e s p e c t i v e l y , as d o my thanks. Finally. after e v e r y b o d y e l s e h a d f i n i s h e d their work, the e x a c t i n g job of s e t t i n g the w h o l e text o n a p e r s o n a l c o m p u t e r a n d e d i t i n g w a s p e r f o r m e d b y D o r d e T a s e v s k i , M.Sc., for w h i c h the writer of t h e s e lines is s i n c e r e l y thankful.
VI Naturally, the list of institutions, c o l l e a g u e s and friends who c o n t r i b u t e d to the writing of this book is much longer than the always limited s p a c e allotted to s u ch p r e f a c e s allows. I wil! warmly a n d s i n c e r e l y thank all of them in person. Thus this foreword ends. Let the book begin.
Zagreb. June 1988.
The Author
CONTENTS C H A P T E R I INTRODUCTION l.I BASIC CONCEPTS, N O M E N C L A T U R E A N D CLASSIFICATION 1.2 DYNAMIC VARIABLES AND COEFFICIENTS
C H A P T E R 2 LUMPED PROCESSES
I IT 25
A. BASIC P R O C E S S E S 2.1 MASS STORAGE
27
2.1-1 LIQUID STORAGE TANKS
27
~-. E. E. i. E.
27
I 2 3 4 5
Controlled inflow a n d outflow of liquid Free outflow of liquid The tank u n d e r p r e s s u r e Tank with v a r i a b l e cross-sectional a r e a Tank with v a r i a b l e cross-sectional a r e a
30 34 36 38
2.1-2 GAS AND STEAM STORAGE TANKS
44
E. I Controlled c h a r g e a n d free d i s c h a r g e of tanks
45
POSSIBILITIES OF MASS STORAGE. CAPACITANCE C. E. 2 Outflow of gas. Calculation of time constant T E. 3 Outflow of air u n d e r supezcritical conditions E. 4 Linearity of g a s (steam) outflow from tank E. 5 Charge a n d d i s c h a r g e of tank d u e to p r e s s u r e difference
55
2.2 FLUID F L O W
56 58 59 61 64
2.2-I LIQUID FLOW
65
E. I Dynamics of liquid flow through p i p e s E. 2 Calculation of d y n a m i c coefficients. R, I, T E. 3 Calculation of d y n a m i c coefficients. R, I, T
6674
2.2-2 GAS AND STEAM FLOW
75
T2
VIII E . 1 D y n a m i c s of g a s or s t e a m flow. B o u n d a r y c o n d i t i o n s E . 2 C a l c u l a t i o n of d y n a m i c c o e f f i c i e n t s . R, C, T
75
2.2-3 FLUID FLOW
83
E . 1 D y n a m i c s of fluid flow. P e r i o d i c p r o c e s s e s E. 2 C a l c u l a t i o n of d y n a m i c c o e f f i c i e n t s . R, I, C, T, ~,, Z
84 93
2.3 H E A T P R O C E S S E S
82
96
E. 1 H e a t transfer t h r o u g h a walI E. 2 R a d i a t i o n h e a t transfer E . 3 Direct h e a t e x c h a n g e r
97 102 105 I09
2.4 M E C H A N I C A L P R O C E S S E S E. I T r a n s l a t i o n d y n a m i c s E. 2 Duality of d y n a m i c v a r i a b l e s a n d c o e f f i c i e n t s in mechanics E. 3 D y n a m i c m o d e l s of different m e c h a n i c a l s y s t e m s E . 4 D y n a m i c s of a c o m p o s e d m e c h a n i c a l system. Stability of m o v i n g p e n d u l u m e q u i l i b r i u m s t a t e
112 121 125 131
B. COMPLEX SYSTEMS OF HIGHER ORDER
139
2.5 E X A M P L E S O F C O M P L E X
13Q
SYSTEMS
E. I T h r e e liquid t a n k s in s e r i e s E. 2 T w o - p h a s e fluid t a n k E. 3 G e a r train s y s t e m
140 143 I48
X. 4 D i s c r e t i z a t i o n of d i s t r i b u t e d p r o c e s s
152
C H A P T E R 3 D I S T R I B U T E D PROCESSES 3.1 MASS AND ENERGY TRANSPORTATION PROCESSES •BASIC HYPERBOLIC FIRST-ORDER PARTIAL DIFFERENTIAL EQUATION E. 1 C o n v e y e r s y s t e m for b u l k m a t e r i a l transport. Time response
158 161
163
IX E. 2 Fluid flow through uninsulated pipe. T em perat ure time r e s p o n s e W. 3 Dynamics of parallel-flow a n d counter-flow heat exchangers W_ 4 Fluid flow with heat s t o r a g e in the p i p e wall 3.2 PROCESSES WITH EQUALIZATION BASIC PARABOLIC PARTIAL DIFFERENTIAL EQUATION
E.! E. 2 E. 3 E. 4 E. 5
Heat c o n d u c t i o n through homogeneous insulated body Direct heat e x c h a n g e r with a thick wall Heat c o n d u c t i o n through the e x c h a n g e r wall Heat c o n d u c t i o n through h o m o g e n e o u s uninsulated body Process of diffusion with c o n v e c t i o n
169 173 179 181
182 1ST 190 205 206
3.3 PROCESSES WITH PERIODIC STATE CHANGES 208 BASIC HYPERBOLIC SECOND-ORDER PARTIAL DIFFERENTIAL EQUATION
E . I Longitudinal vibration of a b e a m 210 E. 2 Dynamic p r o c e s s e s in fluid pipes. Equations of mass, e n e r g y a n d momentum c o n s e r v a t i o n 212 E. 3 Analysis of dyna m i c properties. S t e a d y state. Linear model 225 E. 4 Calculations of dynam i c characteristics of an evaporator 240 APPENDIX DIFFERENTIAL EQUATIONS AND STATE SPACE M O D E L LINEARIZATION SELECT FUNCTIONS OF C O M P L E X VARIABLE BASIC DYNAMIC PROPERTIES
245 245 258 268 269
L I S T ' O F SYMBOLS
270
BIBLIOGRAPHY
275
CHAPTER
ONE
INTRODUCTION
t.l BASIC CONCEPTS. NOMENCLATURE AND CLASSIFICATION
Until r e c e n t l y , d y n a m i c s (from t h e G r e e k w o r d d y n a m i s - force, p o w e r ) w a s a c l e a i l y d e f i n e d t e c h n i c a l d i s c i p l i n e a n d s c i e n c e , a n d the m e a n i n g of the w o r d d i d not d e m a n d s p e c i a l e x p l a n a t i o n . It w a s t h e n a m e of a s c i e n c e w h i c h f o r m e d a p a r t of the w i d e r c o n c e p t of m e c h a n i c s a n d s t u d i e d the s p a t i a l motion of a p a r t i c l e or a rigid b o d y d u e to force. T h e f o u n d a t i o n s of d y n a m i c s w e r e N e w t o n ' s laws. w h i c h h a v e s i n c e the y e a r s of their e s t a b l i s h m e n t a l s o b e e n a p p l i e d o u t s i d e the field of c l a s s i c a l m e c h a n i c s , a n d in time u s e d more a n d m o r e in the s t u d y of motion of c o n t i n u a ( e l a s t i c a l l y a n d p l a s t i c a l l y d e f o r m a b l e b o d i e s , liquids a n d g a s e s ) , w h i c h c o n t r i b u t e d to the d e v e l o p m e n t of fluid d y n a m i c s , g a s d y n a m i c s or a e r o d y n a m i c s , a n d t h e t h e o r y of elasticity. Differential e q u a t i o n s , o r d i n a r y a n d partial, w e r e a n d still a r e the c h a r a c t e r i s t i c m a t h e m a t i c a l tool u s e d to d e s c r i b e the p r o c e s s of motion. In time, the field of d i s c i p l i n e s , in w h i c h differential e q u a t i o n s a p p e a r in m a t h e m a t i c a l r e p r e s e n t a t i o n s , e x p a n d e d . This mostly o c c u r e d in p r o b l e m s of control in v a r i o u s t e c h n i c a l a n d n o n t e c h n i c a l p r o c e s s e s - m e c h a n i c a l , e l e c t r i c a l , h e a t , flow, c h e m i c a l . p h y s i c a l , b i o l o g i c a l , s o c i o l o g i c a l a n d the like. As the field of d e s c r i b e d a n d a n a l y z e d p r o c e s s e s g r e w . a l r e a d y e x i s t i n g terms w e r e k e p t . b u t their meaning changed and became more generalized. Today coordinates are not o n l y the g e o m e t r i c a l , s p a t i a l c o o r d i n a t e s d e s c r i b i n g t h e p o s i t i o n of a p a r t i c l e , but all t h e i n d i c a t o r s of the p r o c e s s under observation (temperature. pressure, concentration, speed). Moreover, when we say motion, w e no l o n g e r m e a n o n l y d i s p l a c e m e n t or a c h a n g e of the rigid
2
CHAP.I INTRODUCTION
b o d y ' s s p a t i a l c o o r d i n a t e s , but a n y time c h a n g e the s t a t e of the a n a l y z e d p r o c e s s . In short, a m o r e e x t e n d e d
in c o o r d i n a t e s
defining
definition m a y b e g i v e n b y w h i c h d y n a m i c s
is the s c i e n c e w h i c h d e a l s with time c h a n g e s ( v a r i a t i o n s ) of state, or w h i c h s t u d i e s the c h a n g e s of s t a t e with time. Its m a i n f e a t u r e is that time t is o n e of. or the only. i n d e p e n d e n t v a r i a b l e in the m a t h e m a t i c a l m o d e l that d e s c r i b e s s u c h c h a n g e s . If time t is the o n l y i n d e p e n d e n t v a r i a b l e the d y n a m i c s will b e d e s c r i b e d b y o r d i n a r y differential e q u a t i o n s (ODE). If the i n d e p e n d e n t v a r i a b l e s i n c l u d e s p a t i a l or s o m e other c o o r d i n a t e s (for e x a m p l e , a g e in s o c i o l o g i c a l p r o c e s s e s ) b e s i d e s time. the d y n a m i c s will be described
b y p a r t i a l differential e q u a t i o n s (PDE).
T h e s u b j e c t of this b o o k is m a t h e m a t i c a l m o d e l i n g a n d a n a l y s i s of p r o c e s s d y n a m i c s . H e r e w e c o n s i d e r a p r o c e s s (from the Latin p r o c e s s u s p r o g r e s s i n g , g r o w i n g ) to b e a n a r b i t r a r y q u a l i t a t i v e a n d q u a n t i t a t i v e c h a n g e in time. i.e. a time c h a n g e , r e s h a p i n g or t r a n s f o r m a t i o n in q u a l i t y a n d q u a n t i t y . T h e w o r d d y n a m i c s in the term implies that our interest lies in the time profile of t h e s e c h a n g e s , their m a t h e m a t i c a l d e s c r i p t i o n a n d the a n a l y s i s of the o b t a i n e d m o d e l . T h e w o r d c h a n g e must b e c o n s i d e r e d in its most g e n e r a l i z e d m e a n i n g , it c a n m e a n c h a n g e s in the shaft d i a m e t e r d u r i n g l a t h e m a n u f a c t u r i n g . t e m p e r a t u r e a n d p r e s s u r e c h a n g e s at the exit of a s t e a m g e n e r a t o r . d e n s i t y c h a n g e s in a c h e m i c a l r e a c t o r , a n a i r p l a n e ' s c h a n g e s in altitude. or c h a n g e s in the s h a p e of plants, c h a n g e s in h u m a n r e l a t i o n s a n d so on. This short list a l s o s h o w s s o m e t h i n g e l s e . E a c h of t h e s e p r o c e s s e s t a k e s p l a c e in s o m e s p a c e , e l e m e n t , o b j e c t , p r o c e s s unit or plant. Thus it is not u n u s u a l that t h e r e a r e m a n y b o o k s with different n a m e s that treat, for i n s t a n c e , the d y n a m i c s of o b j e c t s , the d y n a m i c s of s y s t e m s , or e v e n m o r e s p e c i f i c a l l y , the d y n a m i c s of c h e m i c a l r e a c t o r or e n e r g y p l a n t s . Their n a m e s i n c l u d e small t e r m i n o l o g i c a l , but a l s o c o n c e p t u a l , d i f f e r e n c e s a n d traps, a n d w e w o u l d not g e t far in a n effort to s o l v e them. H e r e . a n d for now. this is not our p u r p o s e . In c o n n e c t i o n with the n a m e of this b o o k . the following must b e s a i d . It w a s not r e a c h e d b y c h a n c e . M o r e o v e r . it w a s p u r p o s e l y c h o s e n to s t u d y the d y n a m i c s of s p e c i f i c a n d different p r o c e s s e s with a u n i f i e d a p p r o a c h , m e t h o d a n d c o n c e p t u a l tools. In t e c h n i c a l d e v i c e s in w h i c h o n l y o n e p r o c e s s t a k e s p l a c e it is c o r r e c t to e q u a l i z e d the term of o b j e c t d y n a m i c s with the term of p r o c e s s d y n a m i c s . O n e of the s i m p l e s t e x a m p l e s w h e r e this is fulfilled is a liquid s t o r a g e
SEC. 1.!
3
tank. in w h i c h o n l y the process of s t o r i n g ( a c c u m u l a t i n g ) a mass of fluid t a k e s p l a c e . An i n d i c a t o r of t h e s t a t e in that o b j e c t is h e a d H, a n d the u s e of the term d y n a m i c s of the liquid s t o r a g e t a n k or d y n a m i c s of the p r o c e s s of liquid s t o r a g e wilI not c a u s e s e r i o u s m i s u n d e r s t a n d i n g . This is not .the c a s e in m o r e c o m p l e x t e c h n i c a l o b j e c t s in w h i c h s e v e r a l different p r o c e s s e s t a k e p l a c e s i m u l t a n e o u s l y with u s u a l l y unlike d y n a m i c c h a r a c t e r i s t i c s . Thus it would, for e x a m p l e , b e m o r e c o r r e c t to u s e the term d y n a m i c s of ( h e a t a n d / o r of flow) p r o c e s s e s in a s t e a m g e n e r a t o r i n s t e a d of o n l y the d y n a m i c s of a s t e a m g e n e r a t o r , w h i c h is. i n c i d e n t a l l y , m o r e u s u a l . In this latter c a s e the i m p r e s s i o n m a y b e g a i n e d that o n l y o n e p r o c e s s t a k e s p l a c e in the s t e a m g e n e r a t o r w h o s e d y n a m i c s w e wish to e x a m i n e , while t h e r e is in fact a w h o l e s p e c t r u m of d y n a m i c a l l y different p h e n o m e n a , from the s l o w e r p r o c e s s e s of h e a t transfer o n s u p e r h e a t e r s u r f a c e s to the r a p i d , h i g h - f r e q u e n c y p h e n o m e n a of h y d r a u l i c s h o c k s in e v a p o r a t o r p i p e s . (The c o n c e p t s of s l o w n e s s a n d s p e e d a r e of r e l a t i v e c h a r a c t e r , a n d w h a t w e h e r e c a l l fast p r o c e s s e s of h y d r a u l i c s h o c k s w o u l d c e r t a i n l y not b e fast in c o m p a r i s o n with m a n y mechanical oscillatory processes and especially with p r o c e s s e s in e l e c t r i c a l circuits.)
Table
1.1-1
PROCESS CLASSIFICATION according to
into
A
mutual d e p e n d e n c e of variables
linear
nonlinear
B
parameter change with time
time i n v a r i a n t
time
C
dependence on spatial coordinate
lumped
distributed
D
processing of material or energy
flow
E
randomness of variables
deterministic
stochastic
F
state variable c h a n g e with t/me
static
dynam/c
charge
variant
discrete
4
CHAP.I
The a b o v e definition of a p r o c e s s
INTRODUCTION
is o b v i o u s l y s o m e w h a t wider t h a n
this b o o k c o u l d b e e x p e c t e d to c o v e r , a n d it w o u l d b e of u s e to n a r r o w it d o w n to the field that will b e classification characteristics
of can
processes be
a
the s u b j e c t of s t u d y here. Furthermore, a according
good
to
introduction
some
of
.their
into
fields
that
significant are
to
be
i n v e s t i g a t e d in the c o m i n g c h a p t e r s . B e s i d e s b e i n g e l e c t r i c a l , m e c h a n i c a l . heat. h y d r a u l i c , p n e u m a t i c , c h e m i c a l a n d so on, e a c h of t h e s e p r o c e s s e s c a n b e s u b j e c t e d to a c l a s s i f i c a t i o n like the o n e in T a b l e 1.14. What e a c h
of t h e s e
terms i n c l u d e s
will b e
briefly d e s c r i b e d
in the
following lines:
A
Linear p r o c e s s e s are all p r o c e s s e s in w h i c h the r e l a t i o n s h i p s b e t w e e n p r o c e s s v a r i a b l e s of input u, state x a n d o u t p u t y c a n b e d e s c r i b e d b y linear m a t h e m a t i c a l e x p r e s s i o n of the following t y p e y = alu
÷ a 2,
x' = a l x + a 2 u
+ a3.
x' = a t u I + a 2 u 2 ÷ u 3 + u 4
(l/-l)
(On the c o n c e p t s of input, state a n d o u t p u t s e e in the A p p e n d i x . ) Coefficients al a r e c o n s t a n t s , a n d v a r i a b l e s x. x'. y a n d u a r e time v a r i a n t functions a n d c o u l d , c o n s e q u e n t l y , b e written x(t), x'(t), y(t) a n d u(t). If the d e p e n d e n c e y = alux,
b e t w e e n p r o c e s s v a r i a b l e s are as follows
X' ~ -'-d-X '
x
=
~
or c a n b e r e p r e s e n t e d b y s o m e p r o c e s s e s will b e c a l l e d n o n l i n e a r .
.
X' = u
other
sin x
,
nonlinear
(1.1-2)
equations,
such
sec.
1.1
5
Yl
f
i
u2T /"4, UI÷U2] I
I I I
I I i I
Vi\
I I
Fig.
I I
"t
"t
I . | - I The p r i n c i p l e of s u p e r p o s i t i o n
In p r a c t i c e most p r o c e s s e s a r e of a n o n l i n e a r c h a r a c t e r , a n d in o r d e r to facilitate a n a l y s i s their d e s c r i p t i o n is v e r y often t r a n s l a t e d into linear form. This will b e o n e of the n o t i c e a b l e c h a r a c t e r i s t i c s of this b o o k . O n e of the k e y linear p r o p e r t i e s (the o n e w h i c h h a s m a d e linear m o d e l s the most d e s i r a b l e )
is that the p r i n c i p l e of s u p e r p o s i t i o n is a p p l i c a b l e . This
p r i n c i p l e c a n b e e x p r e s s e d in w o r d s as follows: if input uj r e s p o n d s in Yl a n d input u2 in Yz . then input ul + uz will r e s p o n d in yl , Y2 •
This c a n also b e r e p r e s e n t e d graphically~ t h o s e to w h o m it s e e m s a n o b v i o u s a n d e v e r p r e s e n t p r o p e r t y might c h e c k w h e t h e r it h o l d s true for the "simple" n o n l i n e a r function y = u z.
B
An e s s e n t i a l p r o p e r t y of time invariant p r o c e s s e s is that c h a n g e s in o u t p u t v a r i a b l e y d o not d e p e n d o n the moment in w h i c h the input
0
CHAP.I
function u was g i v e n or. in o t h e r words, the r e s p o n s e
INTRODUCTION
of the p r o c e s s is
i n d e p e n d e n t of the moment in w h i c h the d i s t u r b a n c e b e t w e e n time v a r i a n t a n d time invadant processes.
I
I I i
I I
Time invoriont
I
process
i
i" V Time voriont
I I I I I I
t
process
] F i g . 1.1-2 An illustration of time v a r i a n t a n d time invariant p r o c e s s e s It s h o u l d b e b o r n in mind that. v i e w e d t h r o u g h a l o n g e r p e r i o d of time. most p r o c e s s e s s h o w p r o p e r t i e s of time variability i.e. p r o c e s s e s h a v e the t e n d e n c y to c h a n g e their d y n a m i c p r o p e r t i e s . T h e r e a s o n for this are the w e a r i n g , a g e i n g , s e d i m e n t a t i o n , b r e a k d o w n , a n d so on. of d e v i c e s a n d e q u i p m e n t in w h i c h s u c h p r o c e s s e s take p l a c e . (A p o p u l a r e x a m p l e from e v e r y d a y life a r e c h a n g e s in the d y n a m i c p r o p e r t i e s of c a r s . w h i c h a r e as a rule more
and
more
sluggish
from y e a r
to year.)
For
the
needs
of
control, s u c h slow c h a n g e s c a n b e n e g l e c t e d a n d most p r o c e s s e s c a n b e c o n s i d e r e d time invariant. A t y p i c a l e x a m p l e of a time v a r i a n t p r o c e s s is the flight of a rocket, w h i c h b u r n s u p fuel a n d suffers g r e a t w e i g h t loss as it flies. This loss of mass is no l o n g e r n e g l i g i b l e , a n d must b e t a k e n into account
in
a
mathematical
description.
Consequently.
the
model
r o c k e t flying in a straight line is
d(Mw) dt
= F Ct)
,
Cl.l-3)
of
a
s E c . z.z
T
M(t) ~dw + w(t) d M . F(t).
01-4)
w h e r e F is the thrust. M the r o c k e t ' s mass. w the s p e e d of flight. The time variant c o e f f i c i e n t s M(t) a n d w(t) a r e t y p i c a l of time v a r i a n t p r o c e s s e s .
C F i g u r e l.l-3, s h o w s s o m e simple e x a m p l e s of the d i f f e r e n c e s p r o c e s s e s that a r e l u m p e d or d i s t r i b u t e d in s p a c e .
I
between
x=xtt,z)
- t x=x(t) x I ~ x2 ~ x~
x z = x z = x z ~ ftz) ~.
r
_----- ~-
rig.
ideal
o j - -. ~o~.'.',.
"--"
h
o a'i~7°
o
1.1-3 An illustration of l u m p e d a n d d i s t r i b u t e d p r o c e s s e s
We will treat this point in m o r e detail Icier. At p r e s e n t w e will o n l y briefly s a y that all the e x a m p l e s in w h i c h c h a n g e s in the p r o c e s s v a r i a b l e s are i n d e p e n d e n t of s p a t i a l c o o r d i n a t e s a r e c o n s i d e r e d to b e p r o c e s s e s with l u m p e d p a r a m e t e r s or p r o c e s s e s l u m p e d in s p a c e . Where this is not so. i.e. w h e r e s o m e of the v a r i a b l e s a r e net the s a m e at the same
moment
distributed.
in
the
whole
process
volume,
the
process
is
spatially
8
CHAP.I
INTRODUCTION
D In flow p r o c e s s e s m a s s a n d / o r e n e r g y flow t h r o u g h p r o c e s s units in c o n t i n u o u s c u r r e n t s ( h e a t e x c h a n g e r s , flow c h e m i c a l r e a c t o r s , rolling mills. s t e a m g e n e r a t o r s ) . In c h a r g e p r o c e s s e s , on t h e c o n t r a r y , flow is i n t e r r u p t e d a n d s u c h p r o c e s s e s u s u a l l y h a v e t h r e e s t a g e s - c h a r g i n g p r o c e s s units, the p r o c e s s itself, a n d d i s c h a r g i n g p r o c e s s units ( c h a r g e d c h e m i c a l r e a c t o r s , h e a t p r o c e s s i n g of metal, r o t a t i o n a l f u r n a c e s in the c e m e n t industry). In d i s c r e t e p a r t m a n u f a c t u r i n g , the o b j e c t s that a r e to b e p r o c e s s e d a p p e a r in a d i s c o n t i n u o u s p e r i o d of time ( t h e s e a r e the majority of m a n u f a c t u r i n g p r o c e s s e s in the m e t a l - w o r k i n g a n d similar i n d u s t r i e s , e . g . w o o d . p l a s t i c a n d g l a s s working).
E Deterministic p r o c e s s e s a r e p r o c e s s e s in w h i c h for a g i v e n s e t of i n p u t v a r i a b l e s t h e r e is a c o m p l e t e l y d e t e r m i n e d set of o u t p u t v a r i a b l e s . T h e r e l a t i o n s h i p b e t w e e n i n p u t a n d o u t p u t n e e d not b e u n i q u e , but it is e s s e n t i a l for it to b e d e t e r m i n e d . In their static s t a t e s s u c h p r o c e s s e s a r e d e s c r i b e d b y a l g e b r a / c or elliptic PDE. a n d their d y n a m i c s a r e d e s c r i b e d b y ODE a n d / o r PDE or i n t e g r a l e q u a t i o n s . In s t o c h a s t i c p r o c e s s e s , on the c o n t r a r y , the i n f l u e n c e of r a n d o m factors is s o s t r o n g that r e l a t i o n s h i p s b e t w e e n p r o c e s s v a r i a b l e s c a n o n l y b e e x p r e s s e d b y p r o b a b i l i t y laws. This d i v i s i o n refers m o r e to m a t h e m a t i c a l m o d e l s t h a n to the r e a l p r o c e s s e s t h e y d e s c r i b e , b e c a u s e no r e a l p r o c e s s c a n a v o i d r a n d o m i n f l u e n c e s (noises, d i s t u r b a n c e s , b r e a k d o w n ) . T h e d e c i s i o n o n w h e t h e r a p r o c e s s is to b e c o n s i d e r e d d e t e r m i n i s t i c or not d e p e n d s on h o w m u c h i n f l u e n c e s u c h r a n d o m factors h a v e o n the w a y the p r o c e s s d e v e l o p s a n d on the b e h a v i o r of v a r i a b l e s of interest. F i g u r e 1.1-4. s h o w s t y p i c a l p r o c e s s v a r i a b l e c h a n g e s in d e t e r m i n i s t i c a n d s t o c h a s t i c p r o c e s s e s .
sEc. 1.1
9
Y l
unperiodic
Deterministic
Fig.
7
Stochastic process
process
1.1-4 T y p i c a l p r o c e s s v a r i a b l e c h a n g e s
F It h a s a l r e a d y b e e n s a i d that p r o c e s s e s in w h i c h at least o n e p r o c e s s v a r i a b l e c h a n g e s with time a r e c o n s i d e r e d d y n a m i c . Thus all p r o c e s s e s w h e r e t h e r e is no time c h a n g e of state a r e static. In the m a t h e m a t i c a l s e n s e this m e a n s that the time d e r i v a t i v e s of all p r o c e s s v a r i a b l e s e q u a l zero. As a rule, a p r o c e s s that is a l r e a d y in a static r e g i m e will c o n t i n u e to d e v e l o p u n d e r s t a t i o n a r y c o n d i t i o n s if all the input v a r i a b l e s are time invariant, b e c a u s e the p r i n c i p l e of c a u s a l i t y d e m a n d s a time c h a n g e in the input to result in time c h a n g e s in internal a n d o u t p u t p r o c e s s v a r i a b l e s also. T h e r e are, of c o u r s e , e x c e p t i o n s . O n e of them is the c a s e w h e r e p r o c e s s h a s s e v e r a l inputs a n d w h e r e c h a n g e s c a n o c c u r in o p p o s i t e
directions
concerning
in input v a r i a b l e s
their i n f l u e n c e
o n internal
a n d o u t p u t p r o c e s s v a r i a b l e s , as a result of w h i c h t h o s e v a r i a b l e s r e m a i n unchanged. In m o d e r n
technical
practice,
the
static
properties
of d e v i c e s
and
o b j e c t s in w h i c h s p e c i f i c p r o c e s s e s o c c u r a r e u s u a l l y s h o w n in the form of tables, c h a r a c t e r i s t i c s d i a g r a m s , n o m o g r a m s or in s o m e o t h e r g r a p h i c a l m a n n e r ( s e e for e x a m p l e Figs. 2.2-l.2 a n d 2.2-1.3 for the c e n t r i f u g a l p u m p a n d r e g u l a t i o n v a l v e ) . S u c h illustrations a r e o b t a i n e d one
or
of
a
system
of
algebraic
equations
or
as the solution of
elliptical
PDE,
which
d e s c r i b e the relations b e t w e e n p r o c e s s v a r i a b l e s a n d a r e of g r e a t u s e in designing
complex
plants.
However,
such
illustrations
do
not
provide
I0
CHAP.1 I N T R O D U C T I O N
a n s w e r s a b o u t the d y n a m i c p r o p e r t i e s of p r o c e s s e s that o c c u r i n s i d e s u c h d e v i c e s . We will n o w g i v e a v e r y simple, a n d u n d e r
certain conditions
(which is truly of little i m p o r t a n c e ) linear, e x a m p l e to illustrate the d i f f e r e n c e b e t w e e n a s t a t i c a l l y a n d a d y n a m i c a l l y f o r m u l a t e d p r o b l e m or task. similar to the t y p e e n c o u n t e r e d in t e c h n i c a l p r a c t i c e .
5O
¢¢_,o
*/.0 +30 *20
/.0
mc'~'i [
Heat exchanger"
rnc~o
"10
O=O'~"
30 2O
.Ioi-
10
10
]Fig.
20
30
t.O oC
1.1-5 D i a g r a m of c h a r a c t e r i s t i c s of a h e a t e x c h a n g e r
F i g u r e 1.1-5. s h o w s a h e a t e x c h a n g e r w h o s e inner structure, flow v e l o c i t y , c o e f f i c i e n t of h e a t transfer, n u m b e r of p i p e s , d i m e n s i o n s a n d the like
do
not
interest
us
now.
So.
the
equation
relating
t e m p e r a t u r e e0. input fluid t e m p e r a t u r e el a n d the h e a t b r o u g h t to the e x c h a n g e r or t a k e n from it c a n b e written mc
%0 = mc
output
fluid
flow Q that
el~: O
0~-5)
O
(IJ-~)
is
or
eo= el ~
If flow rate m a n d s p e c i f i c h e a t c a r e c o n s t a n t , (1.I-0) c a n b e e a s i l y r e s o l v e d a n d p r e s e n t e d in the form of a d i a g r a m of c h a r a c t e r i s t i c s . For m = 1 k g / s a n d c =I J/kgK. this d i a g r a m is a g r a p h i c a l r e p r e s e n t a t i o n of the f u n c t i o n a l d e p e n d e n c e
~)0 = 8( el , Q).
For a g i v e n Q ~ 10 J/s a n d el = 20 °C. the d i a g r a m of c h a r a c t e r i s t i c s g i v e s o n e a n d o n l y o n e point 1 for w h i c h e0 = 30 °C. w h i c h is a solution to the static p r o b l e m of the h e a t e x c h a n g e r . If input t e m p e r a t u r e 3j remain~ c o n s t a n t a n d the h e a t flow i n c r e a s e s to O = 30 J/s. point 2
s~'c. l.x
11
d e t e r m i n e s the t e m p e r a t u r e ~)0 p r o b l e m . Similarly, a n i n c r e a s e d e t e r m i n e s s0 - 4 0 °C a s point 3.
=
50 °C, w h i c h a g a i n s o l v e s the static of oi to 30 °C, k e e p i n g Q c o n s t a n t ,
F o r m u | a t e d d y n a m i c a l l y , the p r o b l e m is a s follows: h o w l o n g will the p r o c e s s take, will the t e m p e r a t u r e i n c r e a s e b e fast or slow from 30 °C to 50 °C. a n d w h a t will b e the form of this c h a n g e . T h e d i a g r a m of c h a r a c t e r i s t i c s d o e s not a n s w e r this q u e s t i o n , a n d a n y w a y there is a n infinite n u m b e r of s u c h a n s w e r s . This is b e c a u s e the time r e s p o n s e ~)0 d e p e n d s o n the w a y in w h i c h h e a t flow Q c h a n g e s from I0 to 30 J/s a s time p a s s e s , a n d t h e r e is a n infinite n u m b e r of w a y s in w h i c h this c a n t a k e p l a c e . F i g u r e 1.1-6 s h o w s o n l y t h r e e of all the p o s s i b l e w a y s in w h i c h h e a t flow Q c a n c h a n g e with time. It a l s o s h o w s t h r e e p o s s i b l e r e s p o n s e s to s u c h c h a n g e s in h e a t i n g . T h e solution to the d y n a m i c p r o b l e m i n v o l v e s d e t e r m i n i n g ( a n a l y t i c a l l y , g r a p h i c a l l y or b y n u m e r i c a l simulation) a t i m e - d e p e n d e n t function of r e s p o n s e ~)0(t) for a g i v e n Q(t). This, h o w e v e r , c a n n o t b e d o n e with the h e l p of the m a t h e m a t i c a l m o d e l (1.1-6) s i n c e it r e p r o d u c e s o n l y the statics of the e x c h a n g e r . To s o l v e the d y n a m i c p r o b l e m a m a t h e m a t i c a l m o d e l of d y n a m i c s , or a d y n a m i c m a t h e m a t i c a l m o d e l must b e f o r m u l a t e d . T h e p a g e s of this b o o k a r e c o n c e r n e d with p r o b l e m s of o b t a i n i n g d y n a m i c m a t h e m a t i c a l m o d e l s .
Qt 30-~- I 1 /f J / I 2/
50 oc
f ..... /
0 J,-.L--""
-
..-""
"t Fig.
--
t
I.I-6 Output temperature responses
Finally, w e will g i v e a s u r v e y ( T a b l e 1.1-2) of the r e l a t i o n s b e t w e e n the s p a t i a l a n d time p r o p e r t i e s of p r o c e s s e s and their c o r r e s p o n d i n g mathematical notation and representation. This e n d s the short c l a s s i f i c a t i o n of p r o c e s s e s a c c o r d i n g to their n o t i c e a b l e p r o p e r t i e s . Of c o u r s e , a c l a s s i f i c a t i o n c o u l d a l s o b e c a r r i e d out in s o m e o t h e r w a y or a c c o r d i n g to different criteria. Let that r e m a i n for narrower, m o r e s p e c i a l i z e d works. H e r e w e must m e n t i o n that w e will
]2
CHAP.1 I N T R O D U C T I O N
a n a l y z e , in the following c h a p t e r s , e x c l u s i v e l y the d y n a m i c s of b o t h linear a n d n o n l i n e a r , time i n v a r i a n t , l u m p e d and distributed, continuous, d e t e r m i n i s t i c p r o c e s s e s . Static ( s t a t i o n a r y ) s t a t e s will b e of interest o n l y i n a s m u c h a s t h e y a r e n e c e s s a r y for s p e c i f i c p u r p o s e s (for e x a m p l e , in linearization). T h e solution of d y n a m i c p r o b l e m s h a s b e e n g a i n i n g i n c r e a s i n g l y in i m p o r t a n c e , w h i c h h a s m a d e insight into the d y n a m i c p r o p e r t i e s of processes an unavoidable and necessary p a r t of e n g i n e e r i n g and s c i e n t i f i c p r a c t i c e . When w e s a y this. w e d o not m e a n e m p i r i c a l k n o w l e d g e of d y n a m i c s in t h e s e n s e of a c q u a i n t a n c e with the f a c t s a n d phenomena that t a k e p l a c e i n s i d e p l a n t s in u n s t a t i o n a r y c o n d i t i o n s ( a l t h o u g h this is u s e f u l a n d d e s i r a b l e ) . What w e a r e r e f e r r i n g to a r e e x a c t m a t h e m a t i c a l e x p r e s s i o n s , e q u a t i o n s a n d s y s t e m s of e q u a t i o n s , in short. m a t h e m a t i c a l m o d e l s d e s c r i b i n g the d y n a m i c s w e a r e i n t e r e s t e d in. In this last s e n t e n c e , a s i n d e e d o n t h e p r e c e d i n g p a g e s also. w e u s e d the c o n c e p t of a m a t h e m a t i c a l m o d e l s e v e r a l times. This t e r m will a l s o b e m e t o n the following p a g e s . It is, t h e r e f o r e , n e c e s s a r y to s a y s e v e r a l w o r d s o n the p r o b l e m of t h e m o d e l , the w a y in w h i c h it is built, the forms it c o m e s in. its p r o p e r t i e s , a n d so on. Table
1.1-2
PROCESS MODELS WITH LUMPED PARAMETERS
DISTRIBUTED PARAMETERS
~f(x. y. z)
T
algebric equation representations, tables
I
nomograms
M E
characteristics
STATICS
= f(x. y. z) - onedimensionai ODE -
eliptic PDE
- s y t e m of lst-order ODE DYNAMICS - 1 ODE of n-th o r d e r s e v e r a l ODE v a r y i n g order
1
~.
multidimensional
SPACE
hyperbolic PDE parabolic
9EC. I.I
13
In e n g i n e e r i n g a m o d e l is c o n s i d e r e d to b e a material or a n i d e a l i z e d (symbolic)
object
which
substitutes
or
represents
another
existing
or
i m a g i n a r y o b j e c t of interest. The b a s i c c o n d i t i o n that e v e r y m o d e l must fuIfil is that c o n c l u s i o n s a b o u t the p r o p e r t i e s of the original c a n b e d r a w n from the b e h a v i o r of the model. On the p a g e s of this b o o k w e a r e g o i n g to treat o n l y o n e s u b g r o u p of all the
possible
groups
of s y m b o l i c
models
the
mathematical
model,
a b o u t w h i c h the following c a n b e s a i d : If the relations b e t w e e n p r o c e s s v a r i a b l e s a n d the g e o m e t r i c a l a n d p h y s i c a l p r o p e r t i e s of the s p a c e in w h i c h a p r o c e s s d e v e l o p s c a n be mapped onto a mathematical structure, then such a r e p r e s e n t a t i o n is c a l l e d a m a t h e m a t i c a l model. A m o d e l c a n also b e s a i d to r e p r e s e n t a s y m b o l i z e d h y p o t h e s e s a b o u t the w a y in w h i c h the p r o c e s s u n d e r i n v e s t i g a t i o n will d e v e l o p , a n d its a n a l y s i s ( e x a m i n a t i o n a n d solution] g i v e s a n s w e r s a b o u t the b e h a v i o r of the real p r o c e s s . The m o d e l c a n
be
obtained
in two w a y s : t h e o r e t i c a l l y
(also
called
deductively, analytically, based o n the first, natural, p r i n c i p l e ) or e x p e r i m e n t a l l y ( p r a c t i c a l l y , empirically). The first p r o c e d u r e is c a l l e d
modeling
a n d the s e c o n d
identification,
a n d h e r e a r e their shortest
definitions..
Modeling
is
the
process
of
model
building
using
theoretical
m e a n s , i.e. the f u n d a m e n t a l natural laws of the c o n s e r v a t i o n of mass. e n e r g y a n d momentum. I d e n t i f i c a t i o n is a n e x p e r i m e n t a l p r o c e s s of m o d e l building u s i n g the m e a s u r e d v a l u e s of input a n d o u t p u t v a r i a b l e s of the real p r o c e s s a n d its model. T h e error b e t w e e n p r o c e s s a n d m o d e l r e s p o n s e s h o u l d b e minimised b y p a r a m e t e r or structure c h a n g e . The p a g e s of this b o o k will s h o w a n d treat o n l y the t h e o r e t i c a l m e t h o d of formulating a m a t h e m a t i c a l model, b u t we will n e v e r t h e l e s s p r e s e n t some
basic
features,
advantages
and
shortcomings
of
the
process
m o d e l i n g (M) a n d the p r o c e s s of identification (|). In M, m o d e l structure results d i r e c t l y from natural laws a n d p o s s i b l y from s o m e n e c e s s a r y n e g l e c t i o n in the p r o c e d u r e of m o d e l b u i l d i n g In |, structure must b e p r e d i c t e d in a d v a n c e . M r e p r o d u c e s the
of
|4
CHAP.I
INTRODUCTION
relations b e t w e e n input, o u t p u t a n d internal p r o c e s s v a r i a b l e s while I u s u a l l y results in a " b l a c k b o x " t y p e of m o d e l , i.e. a m o d e l that r e p r o d u c e s o n l y the relations b e t w e e n input a n d output. In M, m o d e l p a r a m e t e r s a r e d i r e c t l y l i n k e d with p h y s i c a l v a l u e s a n d p r o p e r t i e s a n d in I, the p a r a m e t e r s a r e p u r e n u m b e r s , u n c o n n e c t e d with p h y s i c a l v a l u e s . A m o d e l built b y M c a n b e u s e d for m a n y o p e r a t i n g r e g i m e s of the s a m e p r o c e s s a n d also for r e l a t e d p r o c e s s e s a b o u t w h o s e p r o c e s s v a r i a b l e s little is often known. | . o n the c o n t r a r y , g i v e s a m o d e l o n l y for the s p e c i f i c o p e r a t i n g r e g i m e for w h i c h m e a s u r e m e n t s
were carried
out a n d
for that
particular
r e g i m e the d e s c r i p t i o n is reliable. In M. a m o d e l c a n b e f o r m u l a t e d for a p r o c e s s that h a s not yet b e e n r e a l i z e d in p r a c t i c e a n d is still in d e s i g n s t a g e . M o r e o v e r , w h i c h is sometimes of p a r t i c u l a r i m p o r t a n c e , it is p o s s i b l e to b u i l d a m o d e l to e x a m i n e the d y n a m i c s o f . b r e a k d o w n p r o c e s s e s in p l a n t s (malfunction of e l e c t r i c a l p u m p s , loss of c o o l i n g m e d i u m in n u c l e a r r e a c t o r s , b u r n s in the wall of a c h e m i c a l r e a c t o r a n d the like). T h e ! p r o c e d u r e c a n o n l y b e a p p l i e d to e x i s t i n g o b j e c t s , a n d it is not c o n v e n i e n t to simulate e m e r g e n c y c o n d i t i o n s o n a n o b j e c t with the p u r p o s e of c a r r y i n g out i d e n t i f i c a t i o n a l m e a s u r e m e n t . To m a k e u s e of M. all t h e b a s i c internal p r o c e s s e s must b e k n o w n a n d m a t h e m a t i c a l l y d e s c r i b a b l e . This is not a d e m a n d for |. Finally, o n c e a p r o g r a m p a c k a g e for I is prepaKed, it c a n q u i c k l y a n d e a s i l y b e a p p l i e d to o b t a i n m o d e l s for v a r i o u s p r o c e s s e s . In the c a s e of M, the p r o c e d u r e must b e s t a r t e d from the v e r y b e g i n n i n g e v e r y time a n d t a k e s u p m u c h more time.
In
practice,
however,
both
these
methods
very
often
c o m p l e m e n t e a c h other. Similarly as in p r o c e s s c l a s s i f i c a t i o n , a c c o r d i n g to the p r o p e r t i e s of the p r o c e s s u n d e r i n v e s t i g a t i o n , m o d e l s c a n b e either linear or n o n l i n e a r , time v a r i a n t or invariant, c o n t i n u o u s or d i s c r e t e , s t a b l e or u n s t a b l e , to d e s c r i b e s y s t e m d y n a m i c s with l u m p e d (then t h e y a r e in the form of ODE) or with d i s t r i b u t e d p a r a m e t e r s (PDE). H o w e v e r it is important to r e a l i z e that a p r o c e s s or a n o b j e c t c a n h a v e
than one dynamic model. A hierarchy of m o d e l s in fact exists, from those that are dynamically the simplest, of the O-th order more
(which describe only stationary states but which can be very complicated), to very complex dynamic models of a high. and if the process is described by PDE, of infinitely high order. M a n y factors influence a model's degree of complexity, the most important being its
SEC. 1.i
15
final o b j e c t i v e a n d
the t e c h n i c a l - e c o n o m i c
and
h u m a n limitations
in the
p r o c e d u r e s of its formulation a n d solution. This b o o k will s h o w h o w to o b t a i n d y n a m i c m o d e l s for " p u r e a n d s i m p l e " p r o c e s s e s that t a k e p l a c e in o b j e c t s with a s i m p l e g e o m e t r y , w h i c h will m a k e the p r o c e d u r e of m o d e l building somewhat shorter and more condensed t h a n the o n e that g e n e r a l l y h o l d s for m o d e l b u i l d i n g . This g e n e r a l p r o c e d u r e c a n . in p r i n c i p l e , b e d i v i d e d into a s e q u e n c e of s t a g e s as d e l i n e a t e d b e l o w : I. P r o b l e m d e f i n i t i o n , in w h i c h the o b j e c t i v e , g o a l s a n d p u r p o s e s of the m o d e l a r e d e f i n e d c o n s i d e r i n g different c o n s t r a i n t s accuracy, simplicity, h u m a n , e c o n o m i c a l and computational constraints. 2. Process or object analysis, with the p u r p o s e of d e t e r m i n i n g b o u n d a r i e s , s e p a r a t i n g the p r o c e s s from its e n v i r o n m e n t , d e f i n i n g input, o u t p u t a n d internal s t a t e v a r i a b l e s , d i v i d i n g the p r o c e s s or o b j e c t into simpler a n d e l e m e n t a r y s u b p r o c e s s a s or p a r t s .
3. A s s e m b l y
of conservation
equations
for m a s s (M) a m d / o r e n e r g y (E) a n d / o r m o m e n t u m (Ira). w h i c h h a v e the following g e n e r a l form: flow r a t e (M, E. Im)l - flow r a t e (M, E, Im)o
i
i p r o d u c e d flow r a t e (M, E. Im)v d(M, E, Im) absorbed = dt v
(1.1-7)
S u b s c r i p t V s h o w s this e q u a t i o n to b e v a l i d for a s p e c i f i c a n d c o n s t a n t v o l u m e within w h i c h the a c c u m u l a t e d M a n d / o r E a n d / o r Im a r e c o n s i d e r e d e q u a l a n d h o m o g e n e o u s l y d i s t r i b u t e d . In a l u m p e d p a r a m e t e r p r o c e s s this is t h e w h o l e p r o c e s s v o l u m e . In s p a t i a l l y d i s t r i b u t e d p r o c e s s e s the a s s u m p t i o n of e q u a l i t y h o l d s for a n infinitesimally small p a r t of the v o l u m e dV. T h e third m e m b e r o n the left s i d e of e q u a t i o n (1.l-T) h a s a v e r y g e n e r a l i z e d m e a n i n g a n d r e p r e s e n t s , in the e q u a t i o n for e n e r g y c o n s e r v a t i o n for e x a m p l e , the work i n c o m i n g or o u t g o i n g from the p r o c e s s t h e n d i v i d e d into the work of the p r e s s u r e m e c h a n i c a l l a b o r . If the c o n t r o l l e d v o l u m e d e s c r i b i n g this c h a n g e must a l s o b e a d d e d to
v o l u m e . This work is force and so-called changes, a member (1.1-7).
l~
Cl--I A P . I
4.
Assembly
of
physical-chemical
equations
INTRODUCTION
of
state
relating process variables (ideal gas, Bernoully equation, Hooke's law). A s l e m b l y of phenomenologi©al equations describing p h e n o m e n a Cheat transfer, F i c k ' s I a w of diffusion, Arrhenius' l a w of c h e m i c a l r e a c t i o n s a n d s o on).
5.
Solution
and
investigation
of
the
mathematical
m o d e l . A n a l y t i c a l solution ( w h i c h is r a r e l y p o s s i b l e ) or s i m u l a t i o n on c o m p u t e r s . N o n l i n e a r m o d e l s a r e either s i m u l a t e d d i r e c t l y , or t h e y a r e first l i n e a r i z e d a n d t h e n l i n e a r a n a l y s i s is p e r f o r m e d ( m o d a l a n a l y s i s , stability, sensitivity, controllability). As a rule d i s c r e t i z a t i o n is u s e d to r e d u c e PDE to a s y s t e m of ODE.
6. T e s t i n g
a n d v e r i f y i n g o b t a i n e d results a n d c o m p a r i n g t h e m with those that were intuitively expected. Discovering inconsistencies and unexpected m o d e l b e h a v i o r (for e x a m p l e , if h e a t i n g w e r e to result in a d e c r e a s i n g fluid t e m p e r a t u r e , or the like). 7. V a l i d a U o n o f t h e m o d e l - e x p e r i m e n t s a r e c o n d u c t e d with b o t h the m o d e l a n d the m o d e l e d p r o c e s s to e s t a b l i s h that the m o d e l p r e d i c t s a c t u a l p r o c e s s r e s p o n s e s to a s a t i s f a c t o r y d e g r e e . If it is i m p o s s i b l e to e x p e r i m e n t on the w h o l e o b j e c t , its i n d i v i d u a l p a r t s must b e t e s t e d . If n e c e s s a r y , after the results h a v e b e e n c o m p a r e d , a s s u m p t i o n s u s e d to b u i l d the m o d e l , its s t r u c t u r e a n d parameters, are changed. 8. Applying built for.
the
model
in a c c o r d a n c e
with t h e p u r p o s e
it w a s
M o d e l s in this b o o k will a s a rule b e r e p r e s e n t e d b y a s y s t e m of differential a n d a l g e b r a i c e q u a t i o n s . T h e a i m of the b o o k is to s h o w the m a n n e r s a n d m e t h o d s of b u i l d i n g a m a t h e m a t i c a l m o d e l of the d y n a m i c s of a p r o c e s s , the a n a l y s i s of s u c h m o d e l s , a n d the r e l a t i o n s b e t w e e n the r e a l p h y s i c a l c h a r a c t e r i s t i c s of a p r o c e s s a n d f o r m a l i z e d m a t h e m a t i c a l parameters. T h e following c h a p t e r s , h o w e v e r , will. in most c a s e s not s h o w the p r o c e d u r e s for s o l v i n g p a r t i c u l a r differential e q u a t i o n s or their a n a l y t i c a l a n d g r a p h i c a l solutions. T h e r e a d e r is r e f e r r e d to the last p a g e s of this
SEC.
1.t
b o o k , to the A p p e n d i x
which
contains
table
with differential e q u a t i o n s ,
transfer f u n c t i o n s (the r e l a t i o n b e t w e e n L a p l a c e t r a n s f o r m a t i o n s of o u t p u t a n d input), t r a n s i e n t f u n c t i o n s ( r e s p o n s e . or c h a n g e of o u t p u t , in t h e c a s e of a unit s t e p c h a n g e of input), a n d t h e z e r o s a n d p o l e s of b a s i c d y n a m i c b e h a v i o r - p r o p o r t i o n a l , i n t e g r a l a n d d e r i v a t i v e . In the following c h a p t e r s we will c l a s s i f y a n d refer to t h e s e t y p i c a l a n d c o m m o n d y n a m i c p r o p e r t i e s of v a r i o u s p r o c e s s e s , w h i c h will s o o n s h o w the n e e d a n d a d v a n t a g e of f r e q u e n t r e f e r e n c e to that t a b l e . T h e p a g e s of the A p p e n d i x a l s o c o n t a i n a b a s i c s u r v e y of r e l a t i o n s b e t w e e n differential e q u a t i o n s a n d n o t a t i o n in the form of matrix e q u a t i o n s of s t a t e s p a c e , a s e c t i o n o n p r o b l e m s of l i n e a r i z a t i o n , a n d p a g e s with the b a s i c m a t h e m a t i c a l t r a n s f o r m a t i o n s for functions that a r e met m o r e often in this b o o k . It is b e l i e v e d that all this m a y b e of g r e a t u s e to a r e a d e r c o m i n g into c o n t a c t with t h e s e p r o b l e m s for the first time. or after a l o n g p e r i o d of time. T h e r e f o r e , b e f o r e turning to the following c h a p t e r , it w o u l d b e of u s e to l o o k t h r o u g h the p a g e s of the A p p e n d i x a n d r e n e w old or g a i n n e w information, a n d g e t a c q u a i n t e d with the b a s i c c o n c e p t s a n d s y m b o l s that will b e u s e d t h r o u g h o u t this b o o k .
1.2 DYNAMIC VARIABLES AND COEFFICIENTS
An a n a l y s i s of the d y n a m i c s of v a r i o u s p r o c e s s e s r e v e a l s in all of t h e m the e x i s t e n c e a n d p r e s e n c e of c o m m o n d y n a m i c v a r i a b l e s a n d c o e f f i c i e n t s ( a d m i t t e d l y u n d e r still different n a m e s a n d c o m p l e t e l y u n l i k e from the a s p e c t of d i m e n s i o n ) . On the p a g e s of this b o o k t h e y will b e c a l l e d a n d n o t e d d o w n a s follows:
-
dynamic variables accumulated (stored) variable flow effort (potential) dynamic coemcients coefficient of c a p a c i t y - c a p a c i t a n c e coefficient of resistance - r e s i s t a n c e coefficient of inertia - i n e r t a n c e . / n d u c t a n c e
AV F E C R I
CHAP.I
INTRODUCTION
In most c a s e s t h e s e terms a n d n o t a t i o n will m e a n the u s u a l v a r i a b l e s a n d c o e f f i c i e n t s , but s o m e situations will d e m a n d that the r e a d e r a c c e p t s p r e v i o u s l y u n u s u a l t e r m i n o l o g y a n d c o n c e p t s . This effort will, h o w e v e r , b e r e w a r d e d b y m a n y benefits a n d a d v a n t a g e s , w h i c h w a s w h y w e insisted on s e a r c h i n g for a n d r e c o g n i z i n g c o m m o n p r o p e r t i e s a n d i n t r o d u c i n g the m e n t i o n e d v a r i a b l e s a n d coefficients. T h e main r e w a r d is the possibility of p e r c e i v i n g d y n a m i c s integrally, as a s c i e n c e treating c h a n g e s of state with time. The
common
properties
of d y n a m i c
processes
will
manifest
t h e m s e l v e s in the e q u i v a l e n t forms of their m a t h e m a t i c a l m o d e l s , a n d in r e s p o n s e s that will s h o w s e l e c t e d o u t p u t v a r i a b l e s for unit step, or s o m e other, c h a n g e s of c e r t a i n input v a r i a b l e s . A c c o r d i n g to s u c h r e s p o n s e s w e will c l a s s i f y them into t y p i c a l p r o p o r t i o n a l , integral a n d p r o p e r t i e s , w h i c h a r e g i v e n in the last t a b l e of the A p p e n d i x .
derivative
Before g i v i n g a t a b u l a r s u r v e y of the m e a n i n g of d y n a m i c v a r i a b l e s a n d coefficients, we must s a y s o m e t h i n g a b o u t their b a s i c c h a r a c t e r i s t i c s . As a rule a n d a c c o r d i n g to the c h a r a c t e r of the p r o c e s s u n d e r o b s e r v a t i o n , a n a c c u m u l a t e d v a r i a b l e will b e the mass a n d / o r the e n e r g y a n d / o r the m o m e n t u m (the m o m e n t u m in a p r o c e s s with inertial p r o p e r t i e s ) that exist within the s p a c e the p r o c e s s is d e v e l o p i n g in. (Although this b o o k will a n a l y s e p r o c e s s e s o c c u r r i n g within a s p a c e of rigid a n d time invariant
g e o m e t r y , all the d i s c u s s i o n s
and
models
that
follow c a n
be
r e p e a t e d a n d p e r f o r m e d u n d e r c o n d i t i o n s of c h a n g e in the c o n t r o l volume. In
such
situations,
when
the
equations
of
conservation
are
being
formulated, it is n e c e s s a r y to i n c l u d e m e m b e r s - w h i c h d o not exist h e r e that also d e s c r i b e t h o s e c h a n g e s in the p r o c e s s s p a c e volume.)
It will b e s h o w n that time changes of accumulated variables are the result of unbalance in mass and/or energy and/or m o m e n t u m flow. and that changes of effort variables result from changes of accumulated variables directly a n d simultaneously. The flows themselves are either externally imposed variables, a n d thus independent of the processes taking place in the process space, or they are variables that d e p e n d directly on effort variables. It will be easy to check in the following chapters the scenario of causal relations a n d links b e t w e e n specific variables described here. particularly in examples where the graphical representations of variable interrelations are given in the form of block diagrams (see for example Figs. 2.1-3.3.2.1-2.7.2.3-2).
SEC.
IQ
1.2
In this b o o k we will a l w a y s try to s h o w m a t h e m a t i c a l m o d e l s
in the
form of matrix e q u a t i o n s of state s p a c e . In the c a s e of noninertial p r o c e s s e s it is. therefore, c o n v e n i e n t to s e l e c t state v a r i a b l e s x so that they
are
accessible,
measurable
process
variables.
As
a
rule,
an
a c c u m u l a t e d v a r i a b l e is not a n e a s i l y a n d d i r e c t l y m e a s u r a b l e v a r i a b l e , but e a c h of its c h a n g e s is s i m u l t a n e o u s l y r e f l e c t e d in the e a s i l y measurable" variables
of effort. In
such
noninertial
processes
c o n s e q u e n t l y s e l e c t effort v a r i a b l e s ( p r e s s u r e P, h e a d
(I=O) w e
H. c o n c e n t r a t i o n c,
tension u, t e m p e r a t u r e e a n d the like} as the v a r i a b l e s of state x. In inertial p r o c e s s e s (flow, m e c h a n i c a l a n d electrical} t h e r e is d u a l i t y in s e l e c t i n g d y n a m i c v a r i a b l e s , as T a b l e 1.2-1 s h o w s , of w h i c h more will b e s a i d in detail in S e c t i o n 2.4. B e s i d e s the t h r e e m e n t i o n e d d y n a m i c v a r i a b l e s w h i c h a r e p r e s e n t a n d n o t i c e a b l e in all the p r o c e s s e s , it is also n e c e s s a r y to i n t r o d u c e into a process with inertial properties the dynam|c war|able o|
acceleration
A. This v a r i a b l e is r e l a t e d to the v a r i a b l e of flow F in the
following m a n n e r : dF , A = --~-or
(12-I).
t
F = J'A dt
(12-2)
0
In
its
classical
meaning
the
dynamic
variable
of
acceleration
A
o r i g i n a t e s from m e c h a n i c s , w h e r e it is written a a n d d e f i n e d b y a = d w / d t . In a n a l o g y
with this, in o t h e r p r o c e s s e s
the
name
of a c c e l e r a t i o n
has
b e e n g i v e n to the time g r a d i e n t of flow F. The c o n c e p t s of c o e f f i c i e n t of c a p a c i t y C, r e s i s t a n c e R a n d i n e r t a n c e I o r i g i n a t e from e l e c t r o t e c h n i c s , but t h e y c a n also b e s h o w n to b e p r e s e n t in other d i s c i p l i n e s . In all the i n v e s t i g a t e d c o e f f i c i e n t s will b e d e f i n e d as follows C =
A_.._VV E
R = E F E
I.
dF dt
.
accumulated effort
E
A
.
effort flow effort
processes
variable
. gradient of flow
these
dynamic
(1.2-3) (1.2-4} effort
acceleration
(l.2-S)
20
CHAP.I
INTRODUCTION
The above equations are true w h e n the r e l a t i o n s between d y n a m i c v a r i a b l e s AV, F a n d E a r e l i n e a r . In p r o c e s s e s in w h i c h this is not so, i.e. w h e n the r e l a t i o n s b e t w e e n AV, F a n d E a r e n o n l i n e a r , C, R a n d I will, in a s p e c i f i c o p e r a t i n g r e g i m e , b e c a l c u l a t e d from the q u o t i e n t s of small c h a n g e s of d y n a m i c v a r i a b l e s a r o u n d that o p e r a t i n g r e g i m e . Expressed malhematlcally, in nonlinear processes the following is true C = ~dAV dE R=
I ~
dE ~
(1.2-6)
(1.2-7)'
,
dE d(-~-)
il -
dE dA
{1.2-0)
-
T h e e q u a t i o n s (1.2-3) - (1.2-5) c a n a l s o b e f o r m u l a t e d a s follows. T h e c o e f f i c i e n t of c a p a c i t y C e q u a l s the c h a n g e in t h e a c c u m u l a t e d v a r i a b l e r e q u i r e d to m a k e a unit c h a n g e in p o t e n t i a l . T h e c o e f f i c i e n t of r e s i s t a n c e R e q u a l s r e q u i r e d to m a k e a unit c h a n g e in flow.
the
T h e c o e f f i c i e n t of inertia I e q u a l s the c h a n g e to m a k e a unit c h a n g e in a c c e l e r a t i o n .
change
potential
in p o t e n t i a l r e q u i r e d
£
AV
P
Fig.
in
1 . 2 - I S q u a r e of s t a t e or d y n a m i c s q u a r e
s E c . 1.2
21
E v e r y t h i n g that h a s b e e n s a i d s o fa, s h o w s that the relations b e t w e e n d y n a m i c v a r i a b l e s a n d c o e f f i c i e n t s are c o m p l e t e l y d e t e r m i n e d .
This c a n
also b e r e p r e s e n t e d g r a p h i c a l l y (to m a k e them e a s i e r to r e m e m b e r , but not only for that r e a s o n ) in w h a t is c a l l e d h e r e the s q u a r e o f s t a t e or
dynamic
square,
a n d is s h o w n in F i g u r e 1.2-1.
D y n a m i c v a r i a b l e s AV. F. E a n d A a r e a s s i g n e d to the c o r n e r s of the d y n a m i c s q u a r e , a n d the d y n a m i c c o e f f i c i e n t s I a n d C to its u p p e r sides. The vertical d i a g o n a l h a s the c o e f f i c i e n t R. a n d s h o w the relations b e t w e e n v a r i a b l e s A. F a n d AV.
the
b o t t o m two s i d e s
The a d j e c t i v e d y n a m i c h a s not b e e n u s e d so m a n y times in this c h a p t e r b y c h a n c e . The r e a s o n is that all d y n a m i c v a r i a b l e s are as a rule time variant functions a n d t h e y s h o u l d a c t u a l l y b e written AV(t), F(t). Eft] a n d Aft). Of c o u r s e , t h e o r e t i c a l l y t h o s e v a r i a b l e s c a n b e time invariant. c o n s t a n t , in s o m e p r o c e s s . But t h e n the p r o b l e m is no l o n g e r d y n a m i c in character
and
falls
outside
this
book's
field
of
interest.
In
everyday
e n g i n e e r i n g p r a c t i c e the d e s i g n , c a l c u l a t i o n a n d c o n s t r u c t i o n of v a r i o u s d e v i c e s , p r o c e s s units a n d p l a n t s is in p r i n c i p l e b a s e d o n the a s s u m p t i o n that t h o s e d y n a m i c v a r i a b l e s a r e truly invariant. But s i n c e it is o b v i o u s that in most c a s e s this is not fulfilled ( a n d especially where q u a l i t y - d e m a n d s for the finished p r o d u c t a r e high), a v a r i e t y of m e a s u r e m e n t - r e g u l a t i o n - a u t o m a t i o n e q u i p m e n t is a d d e d a n d built into s u c h o b j e c t s with the p u r p o s e of p r e s e r v i n g the e s t i m a t e d o p e r a t i n g c o n d i t i o n s u s e d in c a l c u l a t i o n s in c a s e s of a l w a y s p r e s e n t d i s t u r b a n c e s or noises. T h e s e a r e the n a m e s for d e v i a t i o n s from c a l c u l a t e d , nominal v a l u e s . After linearization h a s b e e n c a r r i e d out in n o n l i n e a r p r o c e s s e s . d y n a m i c c o e f f i c i e n t s C. R a n d I a r e c o n s t a n t v a l u e s a n d d e p e n d o n the o p e r a t i n g p o i n t in w h i c h that l i n e a r i z a t i o n was c a r r i e d out. Finally. we must t a k e a look at T a b l e 1.2-1 w h i c h s h o w s d y n a m i c v a r i a b l e s a n d c o e f f i c i e n t s for a g r e a t m a n y different p r o c e s s e s , a n d the units u s e d to e x p r e s s them. The v a l u e s s h o w n h e r e are results, a n d t h e y o r i g i n a t e from all the following p a g e s of this b o o k . This t a b l e h a s m a n y i n t e r e s t i n g points, but o n e of the things analogy
of
different
physical
a c c u m u l a t e d mass of g a s
it is useful for n o t i c i n g is the
variables.
Thus.
for
example,
the
in a tank is a n a l o g o u s to the a m o u n t of h e a t
e n e r g y in the walls of a h e a t e d p i p e , to the d i s p l a c e m e n t of a rigid b o d y a l o n g its p a t h of motion, or to the m o m e n t u m of a p a r t i c l e of liquid a l o n g
99
CHAP.
1
INTRODUCTION
a streamline. We b e l i e v e
that
references
to
Table
1.2-1 will
be
frequent
as
the
c h a p t e r s of this b o o k a r e r e a d . In that c a s e w e must turn the r e a d e r ' s attention to i n s t a n c e s w h e n relations b e t w e e n d y n a m i c v a r i a b l e s a r e nonlinear
and
when
numerical
coefficients
appear
as
the
result
of
linearization ( w h i c h is, for e x a m p l e , often the c a s e with the c o e f f i c i e n t of r e s i s t a n c e ) . Also, the v a r i a b l e s a n d the c o e f f i c i e n t s we s e l e c t e d h e r e a r e not
the
universally
and
only possible
ones.
For
example,
in
the
first
column, AV c o u l d also h a v e b e e n the v o l u m e the fluid takes u p in the tank. a n d the flow c o u l d h a v e b e e n the v o l u m e flow i n s t e a d of the mass flow. T h e n C w o u l d also c h . a n g e a n d w o u l d e q u a l a r e a A. It is similar in the c a s e of R a n d I. If .the p r e c e d i n g a n d all the following p a g e s h e l p a n d e n a b l e the r e a d e r to r e c o g n i z e a n d u n d e r s t a n d the d y n a m i c p r o p e r t i e s of the p r o c e s s a n d the o b j e c t , then this b o o k will h a v e fulfilled its p u r p o s e . If. therefore. the r e a d e r
is
enabled
to~
- r e c o g n i z e the s a m e d y n a m i c p r o p e r t i e s in v a r i o u s p r o c e s s e s , - learn
to
formulate
special
importance,
equations develop
of c o n s e r v a t i o n
his
skill
and,
in s e l e c t i n g
what
is
of
phenomena
to
n e g l e c t ( p h e n o m e n a that a r e of less i m p o r t a n c e for the task b e f o r e him), - f o r e s e e the c o n s e q u e n c e s of a s s u m p t i o n s , a n d r e a l i z e h o w t h e y will b e t r a n s f o r m e d into the structure a n d the p a r a m e t e r v a l u e s of the m a t h e m a t i c a l m o d e l , - r e c o g n i z e the ( u s u a l l y p r e s e n t ) n o n l i n e a r c h a r a c t e r of the p r o c e s s , a n d in c o n n e c t i o n with that a c c e p t but also the linearization. -
shortcomings
and
the a d v a n t a g e s limitations,
of
and
benefits.
procedures
of
c o m p r e h e n d relations b e t w e e n s p a t i a l l y d i s t r i b u t e d p r o c e s s e s a n d
t h o s e that a r e l u m p e d , i.e. if h e learn m e t h o d s for r e d u c i n g m o d e l s or the w a y s a n d p r o c e d u r e s for transforming PDE into s y s t e m s of ODE, to r e t a i n the k n o w l e d g e a n d a c l e a r f e e l i n g for what h e loses a n d w h a t h e g a i n s in d y n a m i c s a n a l y s i s .
sEc. !.2 relate
23 the
meanings
of c o e f f i c i e n t s
and
the
parameters
of the
m a t h e m a t i c a l m o d e l or, to b e more p r e c i s e , the e i g e n v a l u e s , v e c t o r s , poles, zeros geometrical
and and
other m a t h e m a t i c a l c h a r a c t e r i s t i c s with the i'eal physical parameters of the process under
observation. - a c c e p t a n d g e t c l o s e l y a c q u a i n t e d with the c o n c e p t s of d y n a m i c v a r i a b l e s a n d c o e f f i c i e n t s , time c o n s t a n t s , natural f r e q u e n c i e s , p r o p o r t i o n a l , integral a n d d e r i v a t i v e d y n a m i c p r o p e r t i e s , inertial a n d noninertial p r o c e s s e s , a n d p e r i o d i c a n d u n p e r i o d i c p r o c e s s e s , the effort that h a s g o n e into the writing of t h e s e lines will b e justified.
O n the following pages an analysis of spatially l u m p e d processes begins with the simplest example of the liquid tank. Gradually. turning the pages of the book. more complex processes will be analyzed, and that inductive from
the
path
- from
specific
to
the
the
simple
general
to
the
more
complex
- w e will try to p r e s e r v e
e a c h c h a p t e r , a n d in the b o o k as a whole.
and
within
,,,
dE
c = dAV
E
I
dE
NL
I--'a-F'=--~ A I T = d(dF)~
E
L
I coefficient of inertonce i
R = dE dF
R= E F
J
NL
L
coefficient of resistance
c =~
coefficient of capacity L NL
E effort
F flow
AV occumu[oted variable
,,,,
m
2H
kg m
Ap
--
~ kg/s
.
--
Nlm z kg/s
m
25P
kg N/m 2
v
Rp~
N
w
z
M
w _ dFIdt
m N
L A
1__ m
K JIs
kg
N m/s
mls N
D
m N
N/m z kg/s
e
1
~=~-
N
F
S
m
w
m
z
w F
kg
Im=M
s
m
w
N
F
S
kgm
I,~=Mw
1
kgm 2
J
Nms rod
Dt
rod Nm
k0
~=~
Nm
M=
S
rod
m
rod
%0
Ct
rod Nm
C
m N
I
rod Nms
Dt
m/s N
1
1
kgm z
J
rod
w
Nm
M=
s
O
kg
M
m
w
N
F
kgm s
~Im=Mw Lz=Jm
transl, rotation transl. [rotation
Nechonicol proce.sses
26P m
kg N/m z
p
M
N
~
P
S
kg
m
kg
M
FlQid flow
5~
J K
Mc
K
P
H
m=
"~"
S
rn
S
kg
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e
S
m
m
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E
Heat processes
kg
kg
M
kg
M
Storage .... processes fluid gas
~=Li
R
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C
V
u
A
i
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H
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A
i
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Electrical processes
H
L
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PROCESSES AND THEIR DYNAMICVARIABLES AND COEFFICIENTS
-
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I= IT i=n Q
CHAPTER
TWO
LUMPED
PROCESSES
L u m p e d p r o c e s s e s a r e all p r o c e s s e s in w h i c h the spatial d e p e n d e n c e of the v a r i a b l e s
under consideration can
be
neglected,
change in t h o s e v a r i a b l e s is c o n s i d e r e d t h r o u g h o u t the v o l u m e for w h i c h the laws of e s t a b l i s h e d . It is o b v i o u s , h o w e v e r , that clue to distuibances and variable changes propagate,
i.e. in w h i c h
a
equal and simultaneous conseIvation have been the finite s p e e d at w h i c h all p r o c e s s e s are in fact
distributed in s p a c e . In the a n a l y s i s of e v e r y i n d i v i d u a l p r o c e s s ( a n d in e s t a b l i s h i n g differential e q u a t i o n s d e s c r i b i n g its d y n a m i c s ) w e must, thus. q u e s t i o n how c o r r e c t an a n a l y s i s is if it n e g l e c t s this s p a t i a l d e p e n d e n c e . Because
of
the
great
diversity
and
multitude
of
completely
different
d e v i c e s a n d plants, it is i m p o s s i b l e to g i v e a g e n e r a l criterion v a l i d in all c a s e s to tell us w h e n l u m p e d p a r a m e t e r s a r e a c o r r e c t substitution for the p r o c e s s e s o c c u r r i n g in s u c h o b j e c t s . N e v e r t h e l e s s , the following definition is sufficiently w i d e a n d i m p r e c i s e to s e r v e as a g u i d e in most c a s e s , E v e r y p r o c e s s , in w h i c h a c h a n g e in the state v a r i a b l e u n d e r c o n s i d e r a t i o n for an o r d e r - o f - m a g n i t u d e takes l o n g e r than it takes for the d i s t u r b a n c e of that state v a r i a b l e to p r o p a g a t e t h r o u g h o u t the w h o l e p r o c e s s volume, c a n b e c o n s i d e r e d a p r o c e s s with l u m p e d v a r i a b l e s . On the foilowing p a g e s
we
will s h o w
h o w to build
a
mathematical
model for the d y n a m i c s of p h y s i c a l l y different p r o c e s s e s . We will s h o w that the m a t h e m a t i c a l forms are similar a n d that the p r o c e s s e s t h e m s e l v e s . in spite of d i f f e r e n c e s b e t w e e n them, all s h o w the s a m e or similar d y n a m i c
2~
¢~HAP,2 L U I ~ P ~ ' I ~ P R O C ~ " c t c ~ S
p r o p e r t i e s . What will b e fulfilled without e x c e p t i o n , h o w e v e r , is that all the m o d e l s will b e in the form of o r d i n a r y differential e q u a t i o n s (ODE) a n d this form of p r e s e n t i n g d y n a m i c b e h a v i o r is o n e of the k e y c h a r a c t e r i s t i c s of all l u m p e d p r o c e s s e s . At the v e r y b e g i n n i n g w e must stress the following. Most t e c h n i c a l p r o c e s s e s are n o n l i n e a r a n d d e s c r i b e d in the form of n o n l i n e a r ODE. T h e d i r e c t c o n s e q u e n c e of n o n l i n e a r i t y o n the d y n a m i c s of the p r o c e s s is that its d y n a m i c p r o p e r t i e s c h a n g e d e p e n d i n g on the o p e l a t i n g point (on the c o n d i t i o n s u n d e r w h i c h the p r o c e s s is d e v e l o p i n g ) . In this b o o k , h o w e v e r , we will a l w a y s try to l i n e a r i z e the differential e q u a t i o n s a n d then transform s u c h l i n e a r i z e d e q u a t i o n s into matrix n o t a t i o n in the form of s t a t e - s p a c e e q u a t i o n s or transfer functions. This narrows clown the field of v a l u e s c o v e r e d b y the results ( s u c h m o d e l s a I e v a l i d o n l y for s p e c i f i c o p e r a t i n g points), but we g a i n v e r y m u c h from the possibility of a p p l y i n g the powerful m a t h e m a t i c a l tools d e v e l o p e d for linear systems. In the following e x a m p l e s we will point out the c h a n g e s in d y n a m i c p r o p e r t i e s resulting from the n o n l i n e a r i t y of the p r o c e s s , a n d also s h o w in w h i c h v a l u e s the input v a r i a b l e s a n d st~Lte v a r i a b l e s c a n c h a n g e , l i n e a r i z e d m o d e l s to still b e sufficientlLy c l o s e to the real p r o c e s s . From the a s p e c t of p r o c e s s linearity a n d
field of for the
the s y m b o l s we will u s e in
this b o o k . the following must b e said. In all the c a s e s w h e n the differential t h e y c a n also b e c o n s i d e r e d e q u a t i o n s of t h e d e v i a t i o n of s t a t e , i n p u t a n d o u t p u t v a r i a b l e s from s o m e initial s t e a d y state. When we d e r i v e transfer functions, if w e set the initial equations obtained are linear,
c o n d i t i o n s e q u a l to z e r o , this m e a n s that the initial v a I i a b l e d e v i a t i o n s from the s t e a d y state e q u a l zero. The transfe, f u n c t i o n s a n d matrix e q u a t i o n s of s t a t e o b t a i n e d in this w a y a r e v a l i d b o t h for a b s o l u t e v a l u e s of v a r i a b l e c h a n g e from the initial state e q u a l to zero, a n d also for v a r i a b l e d e v i a t i o n c h a n g e s from the initial s t e a d y state. In t h e s e o r i g i n a l l y linear c a s e s we will. therefore, not e s p e c i a l l y e m p h a s i z e this linearity b y i n t r o d u c i n g the d e v i a t i o n s y m b o l b e s i d e the v a r i a b l e s y m b o l (foi e x a m p l e , we will not write Am, AH, AP ..... but just m, H, P ..... ).
s E c . 2.1
27
A. BASIC P R O C E S S E S
2.1 MASS STORAGE
Plants often c o n t a i n liquid, g a s a n d s t e a m tanks in w h i c h the
liquid
l e v e l or h e a d H. or the g a s (steam) p r e s s u r e P, a r e of e v e r y d a y interest. The t e c h n o l o g i c a l r e a s o n s for k e e p i n g t h e s e v a l u e s c o n s t a n t differ. Sometimes a c o n s t a n t H is r e q u i r e d to k e e p the liquid mass fixed for the m a i n t e n a n c e of a r e a c t i o n t a k i n g p l a c e in the tank, a n o t h e r time this insures c o n s t a n t liquid s u p p l y u n d e r fixed p r e s s u r e into other parts plant. Sometimes the tank d e c r e a s e s v i b r a t i o n s of the liquid mass in ( d e c r e a s i n g w a t e r - h a m m e r effects) or, like in s t e a m g e n e r a t o r insures the s u p p l y of the liquid p h a s e into e v a p o r a t i n g a n d the
of the pipes drums, steam
p h a s e into s u p e r h e a t e r s e c t i o n s . It is similar in the c a s e of g a s (steam) tanks in w h i c h a c o n s t a n t p r e s s u r e P insures the a c c u m u l a t i o n of a c e r t a i n mass of g a s (steam) or its u n i n t e r r u p t e d s u p p l y to other parts of the plant. Although
liquids
and
gases
are
media
with
essentially
different
p r o p e r t i e s , it will b e s h o w n that the d y n a m i c s of their s t o r a g e in tanks is similar, a n d that the p h y s i c a l m e a n i n g of the liquid level H is e q u i v a l e n t to the m e a n i n g of p r e s s u r e P. Both v a r i a b l e s s h o w the a m o u n t of mass, liquid or g a s , s t o r e d in the tank. Thus the m a t h e m a t i c a l m o d e l s for b o t h media
will b e
obtained
conservation
by
formulating
only
equation
one
for
the
of mass.
2.1-I LIQUID STORAGE TANKS
Example Consider
1 C o n t r o l l e d inflow a n d outflow of liquid the
tank in F i g u r e
2.1-1.1. into a n d
out
of w h i c h
controlled
a m o u n t s of liquid a r e p u m p e d , ml ,~ ml (H). me ~ me(H). D e r i v e a m o d e l to describe
the
dynamics
of the
change
in
liquid
level
H. The
tank
is
28
CHAP.P L u M P E a PROCESSES
cylindrical. A p = const.
=
const.
The
compressibility
of
liquid
is
neglected.
Fig. 2.1-1.1 Tank with controlled flow mi a n d m, The equation for the conservation of liquid mass in the tank is
m i - m,
=
dM = dt
(2.1-1.1)
This is a n ordinary linear DE of first order with constant coefficients.
(In these, a n d in many following, equations w e will not stress every time that the variables H. mi a n d m, are time functions H(t). mdt) a n d m,(t). Since the whole book treats dynamic processes this is always implied, a n d to make notation simpler this time-variability will not b e specially noted.) In the s t e a d y state. a n d that is the state where there is n o c h a n g e in which from (2.1-1.2) yields h e a d H, it must be valid that dff/dt=~.
Since H d o e s not influence the mass of liquid flowing through the tank (the functions mi(t) a n d m,(t) are arbitrary a n d externally imposed), the tank shows only properties of mass storage, a n d the measure for that accumulation is expressed in capacity constant C
s E c . 2.1
29
A more d e t a i l e d e x p l a n a t i o n of the c o n c e p t of c a p a c i t y c o n s t a n t C is g i v e n in C h a p t e r 2.1-2 in the e q u a t i o n s (2.1-2.39) - (2.1-2.44). The tank thus s h o w s "pure" i n t e g r a l b e h a v i o r . If e q u a t i o n (2.1-1.3) was not satisfied, the liquid l e v e l H w o u l d c o n v e r g e t o w a r d s a t h e o r e t i c a l l y infinite v a l u e (m I > me) or to z e r o (me > m0 • All p r o c e s s e s in w h i c h the s t o r e d v a l u e (mass, e n e r g y , momentum) d o e s not i n f l u e n c e inflow a n d outflow (mass. e n e r g y a n d m o m e n t u m flow) from the a c c u m u l a t i o n - p r o c e s s v o l u m e s h o w t h e s e integral p r o p e r t i e s . It is not difficult to transform E q u a t i o n (2.1-1.2) into a matrix DE of state space =
(2.1-1.5) Ap
X ° =AX4,
A~)
me
BU
S i n c e o n l y o n e m a s s - s t o r a g e tank is p r e s e n t , the s y s t e m matrix A is of first order. As (2.1-1.5) s h o w s , it is also a null matrix b e c a u s e the liquid level H has no f e e d b a c k a c t i o n o n ml a n d me a n d thus also, a c c o r d i n g to (2.1-1.1). o n the rate of its o w n c h a n g e . T h e e i g e n v a l u e of matrix A is z e r o ),
=
0
.
(2.1-1.6)
In p r o p o r t i o n a l
systems
of the
first o r d e r
and
systems
with
integral
p r o p e r t i e s the e i g e n v a l u e s of s y s t e m matrix A h a v e distinct p h y s i c a l m e a n i n g s ( s e e also E q u a t i o n (2.1-1.2S). Their d i m e n s i o n is s -I , a n d t h e y a r e r e l a t e d with the time c o n s t a n t T in the following m a n n e r -1 T = --
(2.1-1.7)
We know that the time c o n s t a n t T is a m e a s u r e for the rate of c h a n g e in the v a r i a b l e s o b s e r v e d after a unit s t e p d i s t u r b a n c e u(t) = 1 for t > O. In p r o p o r t i o n a l s y s t e m s the r e s p o n s e r e a c h e s a b o u t 95Z of its n e w s t e a d y state a l r e a d y after 3 T. For the tank with c o n t r o l l e d ml a n d me. E q u a t i o n s (2,1-1.0) a n d
(2.1-1.7) y i e l d
an
infinitely
large
constant
T.
Therefore.
if
E q u a t i o n (2.1-1.3) is not satisfied liquid l e v e l H will r e a c h its n e w s t e a d y state after a n infinite p e r i o d of time, or, in other words, the liquid l e v e l H will not r e a c h a n e w s t e a d y state in a finite p e r i o d of time. This p r o p e r t y ,
30
CHAP.2
LUMPED
PROCESSES
that a n e i g e n v a l u e lies in the origin of the s - p l a n e , c h a r a c t e r i z e s integral
processes. In the following e x a m p l e w e wilI s h o w that this tank s h o w i n g integral b e h a v i o r is in fact a b o u n d a r y c a s e of a tank of p r o p o r t i o n a l c h a r a c t e r , in w h i c h a finite c h a n g e in liquid level H d o e s not i n f l u e n c e the a m o u n t flowing out of the tank me, It will b e s h o w n that in this b o u n d a r y c a s e the r e s i s t a n c e to outflow R b e c o m e s infinitely great.
Example
2 F r e e outflow of liquid
Formulate a n e q u a t i o n d e s c r i b i n g u n s t e a d y c h a n g e s
in water l e v e l H
for the tank in F i g u r e 2.1-1.2. Inflow is c o n t r o l l e d b y a p u m p a n d outflow is free, t h r o u g h a control v a l v e . All the a s s u m p t i o n s from the p r e c e d i n g e x a m p l e are fulfilled h e r e also. e x c e p t that me is no l o n g e r a n i m p o s e d function, but d e p e n d s on H.
C)
1 mi(t)
L me(t}
/ A
]Fig. 2.1-1.2
Ao(t)
Tank with outflow t h r o u g h a c o n t r o l v a l v e
T h e e q u a t i o n for the c o n s e r v a t i o n of mass is the s a m e as (2.1-1.1)
dH mj- me = A o ~
(2.1-I.9)
In the t e c h n i c a l l y most u s u a l c a s e of turbulent flow t h r o u g h p i p e s , orifices, v a l v e s a n d fittings, we c a n a p p l y the q u a d r a t i c r e s i s t a n c e law. A c c o r d i n g to this law, p r e s s u r e d r o p is p r o p o r t i o n a l to the s q u a r e of flow. For outflow t h r o u g h a r e g u l a t i o n v a l v e , this law is u s u a l l y s h o w n in the following form
me ~ Kv Ao p ~
(2.1-1.10)
sEc. 2.1
31
T h e c o n s t a n t Kv is g i v e n for e v e r y v a l v e b y its m a n u f a c t u r e r , a n d Ao is the v a l v e ' s v a r y i n g c r o s s - s e c t i o n a l a r e a . For a s p e c i f i c liquid (p is known), Equation (2.1-1.10) c a n b e s h o w n in the form of functional d e p e n d e n c e me = mo(Ao,H)
(2.1-1.11)
Substitution of (2.l-l.lO) into (2.14.9) y i e l d s A p--~--dH + KvAo021/~-H
= mi
(2.1-H2)
Equation (2.1-I.12) is a n o n l i n e a r ODE. Its n o n l i n e a r i t y is the result of flow t h r o u g h the v a l v e a n d is not i n c l u d e d only in ~ but also in the p r o d u c t Ao l/-~. The s t e a d y state is d e t e r m i n e d b y ml = me = K v X o O ~ - ] ~
(2.1-1.13)
To l i n e a r i z e (2.1-1.12) we will d i f f e l e n t i a t e it a n d r e p l a c e infinitesimally small d e v i a t i c n s df b y finite d e v i a t i o n s Af. The n o n l i n e a r relations a r e s h o w n in (2.1-1.10), w h o s e differentiation y i e l d s dine = ,~Ao ~,-,o + OH ~'"
(2.1-1.14)
The v a l u e s of the partial d e r i v a t i v e s s h o u l d b e t a k e n in the s t e a d y state in which l i n e a r i z a t i o n is p e r f o r m e d . E q u a t i o n (2.1-i.13) y i e l d s me
(2.1-1.15)
Referring to (2.1-1.15), after differentiation (2.1-1.13) y i e l d s omo
=
me
8Ao a~ o OH
= KA .
(2.1-1.16)
I
(2.1-1.17)
Xo
-
-
=-~--
=R
(2.1-I.18)
32
C~-IAp.2 LUI~P~I:~ P R ~ E ~ K ~
If w e m a k e u s e of t h e s e last e x p r e s s i o n s , e q u a t i o n (2.1-1.12) g e t s l i n e a r i z e d form for v a r i a b l e deviation from the steady state dAH Ap daY-- + - ~ - - A H = C
Am i
~AA
0
(2.1-l.IQ)
l R
T h e r e s i s t a n c e R is defined as the ratio of effort c h a n g e change. In view of (2.1-I.17).w e can write
R
this
-
-
d[~ dmo
2~ mo
--
Considering written
TdAH
the
to flow
(2.1-i.2o)
already
introduced
capacitance
C, (2.1-1.19) c a n
+ AH = R A m j - RKAAAo
be
(2.1-1.21)
T h e c o n s t a n t T h a s the d i m e n s i o n of time a n d is thus c a l l e d the t i m e Its a n a l y s i s l e a d s to i n t e r e s t i n g c o n c l u s i o n s
constant.
T = CR
=
2 A~,p ~ me
=
2'-~--"~= 2To me
[s]
(2.I-1.22)
T h e time c o n s t a n t To is r e a l l y the ratio of the total m a s s s t o r e d in the tank in the s t e a d y s t a t e to the m a s s flow t h r o u g h the tank To ; ~ me
= stored (accumulated) mass m a s s flow r a t e
(2.1-1.23)
In the form of s t a t e s p a c e matrix e q u a t i o n s , (2.1-1.21) b e c o m e s
t Hj. f- -lE H] •
fk ~ A o Jl
(2.1-1.~.4)
X' = A X ' ~ B U T h e e i g e n v a l u e of the s y s t e m matrix A is o b t a i n e d d e t e r m i n a n t IA-xll--O, w h e n c e X =-
1
T
by
solving
(2.1-1.25)
the
sEc.
2.1
33
Equation (2.1-1.25) confirms the a l r e a d y m e n t i o n e d relation b e t w e e n the e i g e n v a l u e a n d the time c o n s t a n t in (2.1-1.7). Here it w o u l d b e useful to r e m e m b e r a n d point out that the p r e c e d i n g e x a m p l e , in w h i c h the p r o c e s s of mass s t o r a g e was integral, is really a b o u n d a r y c a s e of this e x a m p l e . From (2.1-1.20) we c a n therefore, write
(2.1-1.26)
dmo = dR~
The d e m a n d for mo not to b e a function of H really m e a n s that c h a n g e s of liquid level dH will not result in c h a n g e s of flow too. i.e. dmo=O. This is p o s s i b l e o n l y if the r e s i s t a n c e , as d e f i n e d in (2.1-1.20). is infinite. From T = CR it then follows that the time c o n s t a n t T is infinite, a n d the e i g e n v a l u e of the s y s t e m matrix A is zero,
If H is a c o n t r o l l e d e q u a t i o n is
variable,
i.e. a
variable
of interest,
[ Am, ]
the
output
(2.1-1.2"1")
[AH] = [13 [AH] + [O O] LAAoJ
Y=CX+DU The matrix D is a null matrix, b e c a u s e
t h e r e is no d i r e c t i n f l u e n c e of
the input v a r i a b l e s on liquid level H. The c h o i c e of sets of state or o u t p u t v a r i a b l e s order to s h o w that let us n o w c o n s i d e r a n e x a m p l e
is not u n i q u e
and
w h e r e we c a n
in
vary
the s e l e c t i o n of o u t p u t v a r i a b l e s , resulting in c h a n g e s of the o u t p u t e q u a t i o n . Let the o u t p u t flow Amo b e the c o n t r o l l e d (output) v a r i a b l e , The (2.1-1.14). (2.1-I.16) a n d (2.1-1.1"[) y i e l d
[,,,,]. [o
r"",
[AhoJ
(2.1-1.28)
Y--CX+DU If there is no outflow control v a l v e that c a n i n f l u e n c e the liquid level H t h r o u g h c h a n g e s of its outflow c r o s s - s e c t i o n Ao, then we must put aAo=O in all the p r e c e d i n g e q u a t i o n s . C o n s e q u e n t l y , matrices B a n d D a n d v e c t o r U c h a n g e their d i m e n s i o n - the matrices lose their last c o l u m n s a n d the v e c t o r U the last line.
-'44,
CHAP,2
LUk~PEI~
PRO~E~SE~
If the L a p l a c e transformation is a p p l i e d to (2.1-1.21). u s i n g (2.1-1.29) w e c a n show the d y n a m i c relations b e t w e e n s p e c i f i c v a r i a b l e s with the h e l p of b l o c k d i a g r a m s . I AH(s)+ All(s) = - Ts
Fig.
R AMi(s)-~s
KAR Ts
aAo(s)
"
(2.1-1.29)
2.1-1.3 Block d i a g r a m for a liquid s t o r a g e tank
To c o n c l u d e , o n e of the b a s i c p r o p e r t i e s of p r o c e s s e s that s h o w p r o p o r t i o n a l b e h a v i o r (which is c o n t r a r y to the p r o p e r t i e s we met in p r o c e s s e s with integral b e h a v i o r ) is that the v a r i a b l e s s t o r e d in them (mass. e n e r g y , momentum) s h o w feedback action proportional to t h e m s e l v e s on the flows (M. E. I m ) e n t e r i n g or l e a v i n g the v o l u m e observed.
Example
3 T h e tank u n d e r p r e s s u r e
We will a n a l y z e the d y n a m i c b e h a v i o r of liquid l e v e l H for the liquid s t o r a g e tank s h o w n in F i g u r e 2.1-1.4. w h i c h is in c o n t a c t with other parts of the plant a n d w h i c h has c o n s t a n t p r e s s u r e Pin in the air s p a c e . All the other a s s u m p t i o n s from the former e x a m p l e a r e fulfilled h e r e also.
sEc.
35
~'.x
...
!:';L:,,.
:'&';kor~;t':
","
] M:------~-M-~-
Z---Z-------Jt ...... ,', mi( .-" ~ -----_ .-_ -------------------~-. --.- - _ ~ -
~t
mo{t)
- -- =-
Ao(t) Fig.
2.1.1.4 C l o s e d tank u n d e r c o n s t a n t p r e s s u r e Pin
The law for the c o n s e r v a t i o n of mass is the s a m e as in the p r e c e d i n g e x a m p l e s , i.e. like in e q u a t i o n s (2.1-1.1) a n d (2.1-1.9). D i f f e r e n c e s a p p e a r in the e q u a t i o n for outflow
/ mo
= KvAop~/
V
2gH
• v._~__ p
If we i n t r o d u c e p r e s s u r e ~P
the
height
SP
= Pin
of liquid
" Pout
Hp as
~P p = 2grip
(2.1-1.30)
an
equivalent
of
the
(2.1-1.31)
e q u a t i o n (2.1-1.30) b e c o m e s mo = K v A o p l / 2 g ( H
(2.1-1.32)
÷ Hp)
In p r a c t i c e , two b o u n d a r y c a s e s a r e p o s s i b l e : a) Very h i g h p r e s s u r e Pin, i.e. Hp >> H S i n c e the i n f l u e n c e of H o n the a m o u n t mo c a n b e n e g l e c t e d . E q u a t i o n (2.1-1.32) n o w b e c o m e s
(2.1-1.33)
mo = KvAop 2~--Hp
S i n c e m o is n o l o n g e r a function of H, i.e. of the mass s t o r e d in the tank. w e c a n e x p e c t the tank to h a v e integral p r o p e r t i e s in this c a s e also. Truly, (2.1-1.9) a n d (2.1-1.33) y i e l d
dH Ap~T- = m i - KoAo
Ko = K v p ~ p
(2.1-1.34)
3~
CHAP.2
K0 is c o n s t a n t b e c a u s e
LUMp~r~
PROC',::'c~S'a'S
Pin is kept c o n s t a n t in the air s p a c e . E q u a t i o n
(2.1-1.34) is a n a l o g o u s to (2.1-1.2), e x c e p t that the s e c o n d term on the fighthand
side
has
changed
its
appearance
slightly.
The
tank
will
show
i n t e g r a l p r o p e r t i e s a n d will retain them as l o n g as initial a s s u m p t i o n Ho >> H is fulfilled. b) P r e s s u r e SP is of the s a m e o r d e r of m a g n i t u d e as liquid level H, i.e. aP = 2pgH. In this c a s e
all e q u a t i o n s in E x a m p l e 2 hold, e x c e p t that H ° must b e
i n s e r t e d in all the e x p r e s s i o n s u n d e r s q u a r e toots i n s t e a d of H. a n d in the linear m o d e l aH" must b e i n s e r t e d e v e r y w h e r e i n s t e a d of AH , H" = H + H p ,
(2.1-1.35)
AH"
(2.1-1.36)
= AH
Now the tank b e h a v e s
as a p r o p o r t i o n a l s y s t e m of the first o r d e r a n d
its d y n a m i c s is d e s c r i b e d b y E q u a t i o n (2.1-1.21), i.e. b y (2.1-1.24) a n d (2.1-1.27), e x c e p t that in this c a s e r e s i s t a n c e R a n d ' t h e time c o n s t a n t T will b e l a r g e r b e c a u s e H* is larger that H.
In all the
preceding
examples
the tank was
p e r p e n d i c u l a r l y on its b a s e , so that
cylindrical
the c r o s s - s e c t i o n a l
and
area
placed
A did
not
c h a n g e with c h a n g e s in h e a d H. Very often tanks are s p h e r i c a l or c o n i c a l , or c y l i n d r i c a l but p l a c e d h o r i z o n t a l l y (tanker trucks). In s u c h c a s e s the a s s u m p t i o n A = c o n s t , is no l o n g e r fulfilled a n d the tank's c r o s s - s e c t i o n a l a r e a d e p e n d s on the liquid level, A = A(H). The a p p r o a c h to a n a l y s i s a n d the m e t h o d s remain the same, e x c e p t that n o n l i n e a r forms of f u n c t i o n a l dependence
on liquid level n o w a p p e a r in e q u a t i o n s for s t o r e d mass.
Example
4 T a n k with v a r i a b l e c r o s s - s e c t i o n a l a r e a
C o n s i d e r the h o r i z o n t a l l y p l a c e d
cylindrical tank in F i g u r e 2.1-1.5, for
w h i c h a n e q u a t i o n d e s c r i b i n g u n s t e a d y c h a n g e s in liquid l e v e l H must b e formulated.
If the
tank
is filled
with
kerosine
up
to
the
top
and
the
s E c . 2.1
37
c o m p l e t e l y o p e n v a l v e (Kv = O.b) h a s t h e c r o s s - s e c t i o n Ao = O.01 m 2 . f i n d the time n e c e s s a r y
for all t h e k e r o s i n e to flow o u t D = 2R = 2,b m, L = 12
m.
(2.1-1.37)
r = ~
milt)
Fig.
2 . 1 - 1 . 6 Horizontal cylindrical tank. A = A(H)
The equation for mass conservation is in the usual form mi - mo =
dM dt
(2.1-1.38)
If t h e l i q u i d l e v e l c h a n g e s
b y dH, t h e a m o u n t of s t o r e d m a s s d M is
dM = LA(H)p = Lp2rdH ,
(2.1-1.3Q)
d M = 2Lpy/-~2R-H) d H .
(2.1-1.40)
If w e substitute d M (2.1-I.10) for rag. w e g e t 2 L p ~
from
(2.1-l.40)
d H + KvAop 2~--/-H-- = m,
into
(2.1-].38)
and
apply
Equation
(2.1-l.41)
at(H) This is a n o n h o m o g e n e o u s nonlineal ODE. Nonlinearity is contained in the square root and in the product a1(H)dH/dt.
38
CNAP.2
For.the Equation
second
part
(2.1-1.41) a n d
of t h e
problem
we
must
LUMPED
substitute
t i a n s f o r m it i n t o a form s u i t a b l e
PROCESSES
m~ = 0 i n t o
for i n t e g r a t i o n
after
which we get D
4LD~D--
Id,-- [2L t
o
JI%Ao2 ~
3AoI~2~
O
t = 2 5 2 0 s -- 42 min Note that the same water, alcohol not appear
or a n y
a m o u n t of t i m e w o u l d
be needed
other similar liquid, because
for t h e o u t f l o w of
liquid density
~, d o e s
in t h e f i n a l e x p r e s s i o n s .
w-xample Consider
S T a n k with v a r i a b l e the tank shaped
with water, shown
cross-sectional
like a truncated
area
cone
and
in F i g u r e 2.1-1.6. We m u s t c a l c u l a t e
completely
t h e time n e e d e d
filled for
a l l t h e w a t e r to flow o u t of it. D I ; 0.8 m, De = 0.3 m, Hm = 1 m, d ~- 0.03 m. = 0.62.
!
Fig.
2.1-1.§ T a n k
shaped
like a t r u n c a t e d
cone,
A : A(H)
s E c . 2.1
30
The equation dM dt = "mo
for t h e c o n s e r v a t i o n
of m a s s for ml = 0 is n o w (2.1-1.42)
•
H ] ~dH dM = pA(H)dH = {)~-"[D2 + (D, - D2)~--~m
(2.1-].43)
If w e s u b s t i t u t e (2.1-1.43) a n d (2J-l.10) ( t h e v a l v e replaced
by
the
outflow
coefficient
~ ) into
c o e f f i c i e n t Kv h a s b e e n
(2.1-1.42), a n d
arrange
the
variables, we get
Hm
~ ,-
H ~z
. (>~-LD2+(D,-D21--H--~j
dH =
2~,,rn ~ u / O - - " C3Dt 15d2~ 2)/2g
* 4D,Dz + 8Dz) 2
0
(2.1-1.44)
t =194 s = 3 min I4 s e c o n d s Similarly as
in t h e
.
preceding
example,
t does
not depend
on
liquid
truncated
cone
density 0 here either.
The typical
upper
example
nonlinear
understanding difficulties. example
of a
ease
of t h e
Thanks
to
tank
and
basic
shaped also
dynamic
that,
on
the dependence constant
T in
of t h e d y n a m i c proportional
following
characteristics properties
systems
of t h e
a
simple
properties
the
to s h o w o n e of t h e b a s i c
a
like
process
in
which
does
not
present
pages
we
will
of n o n l i n e a r
(which
are
on
a an
great
use
this
processes
expressed
first o r d e r )
is
the
-
in t i m e operating
(steady) state.
DEPENDENCE O F DYNAMIC PROPERTIES ON OPERATING STATE
O n e of t h e v e r y i m p o r t a n t p r o p e r t i e s that t h e i r d y n a m i c of
operation.
independent momentum
This
properties is
devices
are not the same throughout
true
of t h e s y s t e m storage
of t e c h n i c a l
of
completely
different
o r d e r , i.e. of t h e n u m b e r
elements.
The
exception
are
the
and
the whole
processes of e n e r g y , rare
p l a n t s is range and
is
mass and
processes
with
40
CHAP.2
"purely"
linear
behavior.
The
above-mentioned
LUMPED
dependence
PR(DCIm-~SES
does
not
o c c u r o n l y in t e c h n i c a l , b u t a l s o in s o c i o l o g i c a l , b i o l o g i c a l a n d other p r o c e s s e s . (The e x a m p l e of a n a t h l e t e w a r m i n g u p b e f o r e his turn a n d p r e p a r i n g to b r i n g his b o d i l y a n d m e n t a l " s y s t e m " into p e a k o p e r a t i n g c o n d i t i o n in w h i c h h e c a n a c t a n d r e a c t faster, m o r e d e p e n d a b l y , s t r o n g l y or p r e c i s e l y , is o n l y o n e of the m a n y e v e r y d a y n o n t e c h n i c a l e x a m p l e s of h o w c h a n g e s in o p e r a t i n g s t a t e affect the d y n a m i c p r o p e r t i e s of the process.) In p r o p o r t i o n a l s y s t e m s of the first order, like the o n e s that h a v e a p p e a r e d so far, the m e a s u r e a n d the c h a r a c t e r i s t i c of t h o s e dynamic p r o p e r t i e s i s in the first p l a c e time c o n s t a n t T. In this last e x a m p l e w e will, t h e r e f o r e , s h o w this time d e p e n d e n c e o n the o p e r a t i n g ( s t e a d y ) s t a t e in w h i c h l i n e a r i z a t i o n w a s p e r f o r m e d . It will b e useful to b e a r in mind the results s h o w n in the following lines w h e n w e a n a l y z e the d y n a m i c s of c o m p l e t e l y different or m o r e c o m p l e x p r o c e s s e s . For the case of ml = 0 the differential equation describing n o n s t a t i o n a r y c h a n g e s of liquid l e v e l in the e x a m p l e of F i g u r e 2.1.-1.6 c a n a l s o b e written in this w a y dH pA(H) ~
+ ~pAo~ q
(2.1-1.45)
= m] p
f1(H)
f2(H)
Differentiating
Equation
(2.1-1.45),
which
is
the
way
to
linearization,
yields
df,__d_~d~ dr,-
+
~
[, d(dH)dt ÷ dfz -- dml
(2.1-1.46)
dH
(2.1-1.47)
a~'2 dr2 = y H - - d H
dlq
- - =
dt
.
o
Referring to the last t h r e e e q u a t i o n s a n d r e p l a c i n g (2.1-1.40) turns into
(2.1-1.48) (2.1-1.49)
d with the finite A
sEc. 2.1
41
~XCH) d a r t ~pAo 2-/~-AH dt + 2~/~[" = Aml
~r~
(2.1-1.50)
~o
21q Arranging this e x p r e s s i o n g i v e s oH
2.PX(H)lq -
dAH + A H = ~
-
mo
dt
(2.1-1.51)
Aml mo
Time c o n s t a n t T is the term b e s i d e the d e r i v a t i v e of AH', a n d if we insert the v a l u e s for X(H) a n d ~o(~), we g e t 2/~F m "- dZHm 2 . 2~
[I~(D,-D2) • D2Hm] 2
(2.1-1.52)
The last e x p r e s s i o n is clumsy, but it also c l e a r l y confirms that the time c o n s t a n t d o e s truly d e p e n d on the o p e r a t i n g state, t h e functional d e p e n d e n c e of T o n H is
(2.1-l.53)
T(~) = ~::[-(Ko + KI~ * K21q2)
For the d a t a from the p r e c e d i n g e x a m p l e K 0 - T 2 . 8 3 , Ki = 242.T5, K2 = 202.3. a n d the c h a n g e T(F[) o b t a i n e d from (2.1-1.53), is s h o w n in F i g u r e 2.1-1.7. T 500 $
&oo 3oo 200 too .
03
Fig.
,
0.2
|
0.3
0./~
0,5
0.6
0.7
0,8
0,9
t.Om
2 . 1 - I . 7 D e p e n d e n c e of time c o n s t a n t T on o p e r a t i n g s t a t e for the tank s h a p e d like a t r u n c a t e d c o n e
In E x a m p l e 5 ( w h e r e no l i n e a r i z a t i o n was p e r f o r m e d ) we s h o w e d that
42
Ctl--IAP.2 LUR~PI~"I~ p R O C t ~ ' S ~ E ~ :
all the liquid w o u l d run out of the tank in 1(~4 s e c o n d s . H o w e v e r , if we linearize
the e q u a t i o n d e s c r i b i n g
the s a m e outflow we g e t
w h o s e r i g h t - h a n d s i d e w e must put ~mi
(2.1-1.51), into
= O. If T is, thus, d e t e r m i n e d from
the initial o p e r a t i n g state 1~ = lm (z~H(O) = 1), liquid flows out as p r o p o r t i o n a l s y s t e m of the first order, a c c o r d i n g to the e x p o n e n t i a l
t~HCt) = e
l - - t 518
in a
(2.1-1.54)
As H d e c r e a s e s , time c o n s t a n t T also d e c r e a s e s g r e a t l y , a n d the error we w o u l d g e t from w o r k i n g with a c o n s t a n t T is more than o b v i o u s . A c o m p r o m i s e w o u l d b e to take the time c o n s t a n t for lq = 0.Sin, but e v e n then the linear solution w o u l d not b e s a t i s f a c t o r y . F i g u r e 2.1-1.8 s h o w s this well.
It s h o w s
the
response
of the
original
nonlinearized
l i n e a r i z e d m o d e l with five different time c o n s t a n t s 0.75. 0.5. 0.25, 0 m. Figure
2.1-I.8 s h o w s
the
complete
disproportion
calculated
between
and
of the
for lq = I,
the
linear
m o d e l r e s p o n s e a n d the real r e s p o n s e , r e g a r d l e s s of w h i c h time c o n s t a n t is s e l e c t e d . This, h o w e v e r , is not u n e x p e c t e d . T h e linear m o d e l c a n n o t b e expected to c o v e r the w h o l e field in w h i c h the v a r i a b l e u n d e r c o n s i d e r a t i o n c h a n g e s , from O to I00~ of its initial v a l u e , e s p e c i a l l y w h e n the p r o c e s s h a s s u c h a p r o n o u n c e d nonlinearity. What is e s s e n t i a l is that the linear a n d the n o n l i n e a r r e s p o n s e s c o r r e s p o n d q u i t e well for the c a s e in w h i c h l i n e a r i z a t i o n was p e r f o r m e d , at the initial state ~q ; 1 m, i.e. w h e n T = 518 s, a n d for the first 10% c h a n g e from the initial lq. In the m e a s u r e of F i g u r e 2.1-1.8, t h e r e is almost n o d i f f e r e n c e in this first lO~ c h a n g e initial s t e a d y v a l u e .
from the
The p u r p o s e of linear m o d e l s is to g i v e a sufficiently g o o d r e p r e s e n t a t i o n of d y n a m i c p r o p e r t i e s in the n e i g h b o r h o o d of the o p e r a t i n g state u n d e r o b s e r v a t i o n . Most d e v i c e s a n d p l a n t s are m a d e to work u n d e r c e r t a i n c o n d i t i o n s . That is h o w t h e y are d i m e n s i o n e d a n d their e f f i c i e n c y a n d the q u a l i t y of the final p r o d u c t s a r e e n s u r e d in t h o s e nominal o p e r a t i n g regimes, w h i c h we e n d e a v o r to e n s u r e with the h e l p of other d e v i c e s ( a u t o m a t i o n , c o m p u t e r s , r e g u l a t o r s ) or s u p e r v i s i n g p e r s o n n e l . Only in r a r e c a s e s (starting the p l a n t up, b r e a k d o w n , shutting the p l a n t d o w n a n d the like) a r e c h a n g e s
in o p e r a t i n g s t a t e s g r e a t . In s u c h c a s e s , if w e
n e e d m o d e l s that i n c l u d e the d y n a m i c s of t h o s e p r o c e s s e s , w e primarily d e v e l o p n o n l i n e a r m o d e l s or w e u s e linear m o d e l s but c h a n g e a n d
SEC. 2.1
43
recalculate
their
dynamic
substantial c h a n g e
characteristics
(parameters)
after
every
more
of o p e r a t ! n g point, to m a k e them c o r r e s p o n d with the
o p e r a t i n g point that has just b e e n r e a c h e d . H t.0 m
0.~ 0,5 0,4. 0.2 100 200 300 z00 500 500 s Fig.
t
2.1-1.8 D y n a m i c s of c h a n g e in l i q u i d l e v e l H in the c a s e of free
outflow ..... n o n l i n e a r m o d e l r e s p o n s e linear m o d e l r e s p o n s e with 5 different time c o n s t a n t s -
-
0,6 0,4
"
L
0 tl\ L '
Fig.
100 200 300 I.O0 500 6~s
t
2.1-1.9 D y n a l a i c s of c h a n g e in l i q u i d l e v e l H in the c a s e of free
outflow ..... n o n l i n e a r m o d e l r e s p o n s e linear m o d e l r e s p o n s e with a different T for 5 o p e r a t i n g states The e x a m p l e of outflow s h o w n in F i g u r e 2.1-1.8 is a p r o c e s s with v e r y great c h a n g e s in o p e r a t i n g state. F i g u r e 2.1-I.q shows the n o n l i n e a r m o d e l
44
CHAP. 2 LUMP]~D P R O C E S S E S
r e s p o n s e a n d also how c h a n g e s
in l e v e l H w o u l d look if simulation was
p e r f o r m e d on a linear model, but if the time c o n s t a n t T was c h a n g e d
and
r e c a l c u l a t e d for five different o p e r a t i n g states (lq = I, 0.8, 0.6. 0.4, 0.2 m). This
linear-model
response
is
an
obvious
improvement
on
the
r e s p o n s e s s h o w n in F i g u r e 2.!-I.8. It is c l e a r that the "linear" r e s p o n s e w o u l d b e quite c l o s e to the "nonlinear" o n e if time c o n s t a n t T was c h a n g e d after e v e r y 10% c h a n g e in liquid level H. (In c a s e s w h e n the g a i n c o n s t a n t K a p p e a r s in the linear m o d e l too, w h e n e v e r the o p e r a t i n g r e g i m e is c h a n g e d K must also b e c h a n g e d if it d e p e n d s on the o p e r a t i n g state.) The a n a l y s i s w e h a v e just c a r r i e d out on the simple p r o c e s s outflow is important for m a n y c o m p l e t e l y different d e v i c e s in w h i c h different forms of n o n l i n e a r relations b e t w e e n v a r i a b l e s a p p e a r . Of c o u r s e , the results we o b t a i n e d h e r e a r e not c o m p l e t e l y g e n e r a l in the s e n s e that t h e y c a n b e a p p l i e d to all c a s e s . T h e y a r e of more v a l u e as i n d i c a t o r s of s o m e b a s i c relations b e t w e e n
dynamic properties and
the o p e r a t i n g
state,
and
also
point to the different r e s p o n s e s of the original n o n l i n e a r m o d e l s a n d t h o s e o b t a i n e d t h r o u g h linearization.
2.1-2 GAS AND STEAM STORAGE TANKS
Gas a n d s t e a m s t o r a g e tanks a r e s t a n d a r d parts of p r o c e s s i n g a n d e n e r g y plants. In m a n y w a y s t h e y a r e similar to liquid s t o r a g e tanks, a n d g a s p r e s s u r e P c a n b e c o n s i d e r e d a n a l o g o u s to liquid level H. The b a s i c d i f f e r e n c e b e t w e e n g a s a n d liquid s t o r a g e tanks is that g a s e s a n d s t e a m are c o m p r e s s i b l e so that the a s s u m p t i o n p = const, is no l o n g e r valid. When the e q u a t i o n s
are f o r m u l a t e d it is also v e r y important to c o n s i d e r
the c o n d i t i o n s u n d e r w h i c h the tank is c h a r g e d
a n d d i s c h a r g e d , a n d the
t y p e of g a s (steam) flow. In other words, we s h o u l d d i s t i n g u i s h b e t w e e n s u b c r i l i c a l a n d s u p e r c r i t i c a l inflow a n d outflow, isothermal a n d a d i a b a t i c (or p o l y t r o p i c ) c h a n g e s in the s t a t e of g a s (steam). laminar a n d turbulent flow. In technical work conditions w e usually have t u r b u l e n t that assumption holds throughout this whole chapter.
[low
and
SEC. 2.1
45
From the
aspect
of h o w
the
state
of the
gas
(steam)
in
the
tank
c h a n g e s , the following w o r d s would, in p r i n c i p l e , b e sufficient. If the tank walls are metal a n d
uninsulated
and
the c h a n g e
slow. the g a s
(steam)
t e m p e r a t u r e in c h a n g e a b l e o p e r a t i n g r e g i m e s c a n b e c o n s i d e r e d c o n s t a n t . S u c h p r o c e s s e s c a n thus b e c o n s i d e r e d isothermal. In tanks that a r e well insulated and where heat cannot be c o n d u c t e d
from the tank in c a s e s of
e x p a n s i o n , p r o c e s s e s within the tank are a d i a b a t i c is n = 1.41). In p r a c t i c e , m e a s u r e m e n t s h a v e s h o w n and temperature changes in fact result as s o m e w h e r e b e t w e e n isothermal a n d a d i a b a t i c . For
(for aiz the c o e f f i c i e n t that the u s u a l p r e s s u r e polytropic processes, most s t o r a g e tanks the
c o e f f i c i e n t n is b e t w e e n 1 a n d 1.2. In this c h a p t e r , u n l e s s s t a t e d differently, we will p r o c e e d as if g a s e o u s (steam) c h a n g e s o c c u r i s o t h e r m a l l y . Finally, w h e n we formulate the e q u a t i o n s we will take into a c c o u n t whether the tank was c h a r g e d a n d d i s c h a r g e d u n d e r s u p e r c r i t i c a l or
subcritical
flow conditions.
We must a l s o stress that we will work with the e q u a t i o n
ideal
of state
for
g a s . In most c a s e s g a s e s a n d s t e a m satisfy this e q u a t i o n for low
p r e s s u r e a n d for v a p o r s that a r e not too c l o s e
to the line of saturation.
The a p p r o a c h a n d m e t h o d s d o not c h a n g e for real g a s e s a n d steam. In s u c h c a s e s it is o n l y n e c e s s a r y to r e p l a c e the e q u a t i o n of s t a t e for i d e a l g a s with the e q u a t i o n s of state for t h o s e g a s e s a n d steam, a n d verify the results a c c o r d i n g l y . Finally, preceding
in this c h a p t e r a l s o ( i s o t h e r m a l c h a n g e s of state), one. m o d e l s of d y n a m i c p r o c e s s e s will b e o b t a i n e d
establishing o n l y
Example
one equation
for the conservation
as by
of mass.
I C o n t r o l l e d c h a r g e a n d free d i s c h a r g e of tanks
Derive a m a t h e m a t i c a l m o d e l for the d y n a m i c p r o c e s s e s o c c u r r i n g w h e n the state of g a s (steam) in the tank in F i g u r e 2.1-2.1 c h a n g e s . Determine the e q u a t i o n s a n d transfer functions for s u b c r i t i c a l and s u p e r c r i t i c a l d i s c h a r g e . Gas (steam) is fed into the tank from a c o m p r e s s o r a n d the a m o u n t mi(t) d o e s not d e p e n d on the p r e s s u r e P. The c h a n g e s in the tank a r e slow e n o u g h to b e c o n s i d e r e d i s o t h e r m a l .
state
in
4(5
CHAP.2
~ rig.
LUMPED
PROCESSES
mo(t]
g --
2 . 1 - 2 . 1 G a s ( s t e a m ) t a n k with c o n t r o l l e d c h a r g e
T h e e q u a t i o n for t h e c o n s e r v a t i o n
mi(t)
of m a s s in .the c a s e of t h e g a s in the
t a n k is
ml - m o
dM d(Vp) = V d---l- = dt
=
If t h e p u i p o s e (2.1-2.1) w o u l d b e
dp
(2.1-2.1)
dt
h a d b e e n to d e t e r m i n e c h a n g e s in d e n s i t y p. t h e n t h e final form of t h e m o d e l . In p r a c t i c e , h o w e v e r ,
p r e s s u r e P a n d t e m p e r a t u r e ~ a r e u s u a l l y r e g u l a t e d v a r i a b l e s a n d (2.1-2.1) must b e t ~ a n s f o r m e d to s h o w P a n d s. For this w e will u s e t h e e q u a t i o n of state
for g a s
p = p ( P. ~ )
(2.1-2.2)
from w h i c h .
dp dt
.
.
ap . aP
dP dt
ap a3
÷
d3
(2.1-2.3)
dt
With (2.1-2.3). E q u a t i o n (2.1-2.1) g i v e s m I - m o ~" V
ap aP
dP
• V
dt
ap a~
d3
dt
(2.1-2.4)
Equation (2.1-2.4) contains both pressure a n d
temperature c h a n g e s
and
is thus generally not sufficient for a d y n a m i c
analysis. T o complete the for the conservation of energy w o u l d also h a v e to b e formulated. However. as w e h a v e already said, gas
dynamic
description the e q u a t i o n
(steam) sto~age processes slow,
and
usually
in tanks that are not very
occur
in
isothermal
well insulated are 3 = const, a n d
conditions,
d~/dt = 0. This leads to ap V (-~)
dP dt
= ml-
mo.
(2.1-2.5)
SEC. 2.1
47
For i d e a l g a s e s we h a v e (R p
. . .gas c o n s t a n t )
P
=
(2 .i-2.6)
R%
ap
C--~-)
=
l
(2.1-2.7)
R~
The final result is V
dP
RO
dt
= mi-
mo
(2.1-2.8)
Flow rate mi(t) is d e t e r m i n e d
b y the work of the c o m p r e s s o r a n d
its
d e p e n d e n c e on P is n e g l e c t e d . It is thus n e c e s s a r y to d e t e r m i n e too(t). which o b v i o u s l y d e p e n d s o n w h e t h e r the tank is b e i n g d i s c h a r g e d in s u b c r i t i c a l or s u p e r c r i t i c a l c o n d i t i o n s . a) S u b c r i t l c a l
outflow
In p r a c t i c e , for s u b c r i t i c a l flow (which is d e f i n e d s o m e w h a t more "roughly" b y the c o n d i t i o n Po > P/2) t h r o u g h v a l v e s , orifices a n d similar parts of the p i p e l i n e , we u s e the formula m o = KoAo{Po(P-Po)
Po > P-2
(2.1-2.9)
The e x a c t a n a l y t i c a l formula for c a l c u l a t i n g mo is g i v e n
in E q u a t i o n
(2.1-2.4"/), a n d in (2.1-2.9) Ko c o n t a i n s b o t h the v a l v e c h a r a c t e r i s t i c Kv a n d the g a s c h a r a c t e r i s t i c in the s t e a d y state u n d e r c o n s i d e r a t i o n (~. O. x) . Kv is
a
valve
characteristic
known
for
every
valve,
and
Ao is
the
c r o s s - s e c t i o n a l a r e a t h r o u g h w h i c h the g a s (steam) flows. In orifices or o r d i n a r y n o n r e g u l a b l e v a l v e s Ao is c o n s t a n t , but in control v a l v e s Ao = Ao(t). i.e. Ao is a n arbitrary lime function. If g a s (steam) flows out the tank t h r o u g h a n orifice, outflow c h a r a c t e r i s t i c I* is g i v e n i n s t e a d of Kv. This c h a r a c t e r i s t i c i n c l u d e s the s h a p e of the orifice e d g e s , the w a y in w h i c h the fluid is b r o u g h t to the orifice a n d so on. (The s a m e is true of short parts of the pipeline.) Substitution of (2.1-2.9) into (2.1-2.8) y i e l d s a nonlinear, n o n h o m o g e n e o u s ODE with v a r i a b l e c o e f f i c i e n t s V R~
dP dt
* KoAo~/Po(P-Po)
= mi
(2.1-2JO)
48
C}4AP.2
The downstream pressure change
Po(t) is a l s o
variable
LUMPED
and
can
PROCESSES
in p r a c t i c e
arbitrarily with time.
For the n e e d s of further a n a l y s i s , it will b e the most useful to find the linear form of e q u a t i o n (2.1-2.10). Differentiation of (2.1-2.8) g i v e s V d(dP) dt
= dml-
dine
(2.1-2.11)
Re
V a r i a b l e d m o is of k e y i m p o r t a n c e b e c a u s e mo h o l d s all n o n l i n e a r i t i e s . If (2.1-2.9) is written in the form of f u n c t i o n a l d e p e n d e n c e m o = mo (P, Po, Ao)
,
the
(2.1-2.12)
then dmo is B
dmo -- ~ d P I"
0mo ~ ÷'-O~o d r °
0mo ÷ 0-~-o dA°
(2.1-2.13)
In the s t e a d y s t a t e in w h i c h l i n e a r i z a t i o n is p e r f o r m e d we h a v e mo
Ko =
(2.1-2.14)
E q u a t i o n (2.1-2.9), t o g e t h e r with (2.1-2.14), g i v e s the partial d e r i v a t i v e s of (2.1-2.13)
0~o = ~ mo = ~I 0too aPo ,~mo. aA °
=
,
81~ = 15 - 15o
~o(15-215o) I 215o81 ~ = R ~
mo
=
KA
P-2Po ~o
1 = R---~
(2.1-2.15)
'
(2.1-2.1b) (2.1-2.I7)
In the u p p e r e q u a t i o n s w e h a v e r e i n t r o d u c e d s y m b o l s for flow r e s i s t a n c e c o r r e s p o n d i n g with the definition of r e s i s t a n c e a l r e a d y g i v e n in the p r e c e d i n g s e c t i o n - s e e E q u a t i o n (2.1-1.20). A c c o r d i n g to that definition, r e s i s t a n c e R e q u a l s the ratio of c h a n g e in effort to c h a n g e in flow, a n d in this c a s e of g a s ( s t e a m ) flow it is R .
015
. . ~mo
.
2sl5 mo
(2.1-2.18)
F i g u r e 2.1-2.2 s h o w s the d e p e n d e n c e of mo o n p r e s s u r e P in the tank, k e e p i n g I~o = c o n s t . T h e t a n g e n t s l o p e e q u a l s the i n v e r s e v a l u e of
s E c . 2.1
49
r e s i s t a n c e R. It is important to n o t e that the s l o p e of the t a n g e n t is a l w a y s positive, i.e. a n i n c r e a s e in P results in a n i n c r e a s e in too.
2.1-2.2
A brief e x a m i n a t i o n of Fig. a n d a c o m p a r i s o n of E q u a t i o n s (2.1-2.13) a n d (2.1-2.15) s h o w that the first m e m b e r o n the r i g h t - h a n d s i d e of Equation (2.1-2.13) (partial d e r i v a t i v e a~o/aP) r e p r e s e n t s this s I o p e of a plot of mo v e r s u s P with Po a n d Ao at their s t e a d y state v a l u e s Po a n d Xo. The flow rate mo d r o p s to z e r o w h e n the tank p r e s s u r e P is the s a m e as the outlet p r e s s u r e Po. Similarly, the s e c o n d partial d e r i v a t i v e in E q u a t i o n (2.1-2.13) is the s l o p e of a plot of me v e r s u s Po with P a n d Ao at their s t e a d y state v a l u e s l~ a n d •o. This c u r v e is r e p r e s e n t e d in F i g u r e 2.1-2.3.
y
mo
,
[V]
.... ~' ~1'1 ~ supercritical l P=
]Fig.
~
.r----
P
Pw
P[P~]
2 . 1 - 2 , 2 Variation in flow rate mo with tank p r e s s u r e P. Po = const.
A n a l o g o u s l y to the r e s i s t a n c e of flow to variation in the p r e s s u r e P, Equation (2.1-2.16) d e f i n e s r e s i s t a n c e Ro in the c a s e of variation in outlet p r e s s u r e Po Ro = '~I~° = R P o ~mo P-2Po
= ~R. 0~ < O .
(2.1-2.19)
It is useful to r e m e m b e r that in s u b c l i t i c a l flow (P - 2Po) < O. therefore. the c o e f f i c i e n t = < O. a n d c o n s e q u e n t l y r e s i s t a n c e Ro is n e g a t i v e . The physical
meaning
of
this
is
clear
for
constant
P,
an
increase
of
S0
CHAP.
Po (APo
> 0) leads to a d e c r e a s e
This relation c a n
of m o ( A m o < 0) a n d
PROCESSES
vice versa.
b e s e e n in Figure 2.]-2.3.
"°t/o~ |
c2
~. , I~=O,,Ro,==-,,~--~
~0F . . . . . .
P = k on st.
~ - - - - - ,~,~
I~,,erc""='~
', \\
Pk
Fig.
2 LUMPED
P0
P
P0[P~]
2 . | - 2 . 3 Variation in flow rate m o with outlet p r e s s u r e Po • ~ = const.
E q u a t i o n (2.1-2.19) s h o w s a n i n t e r e s t i n g relation b e t w e e n r e s i s t a n c e Ro a n d p r e s s u r e ~o for a g i v e n p r e s s u r e P in the tank. As ~o d e c r e a s e s , Ro c o n v e r g e s to - ,~, a n d this is s h o w n in F i g u r e 2.1-2.4. When Ro b e c o m e s infinite c h a n g e s in ]5o no l o n g e r i n f l u e n c e c h a n g e s in flow. i.e. ,~mo = O. This p h e n o m e n o n is well k n o w n a n d o c c u r s w h e n s u p e r c r i t i c a i flow begins.
.RO.S~
-
_
~o
-2R -3R -&R -5R
-6R -?R
Fig.
2.1-2.4 Dependence
of r e s i s t a n c e Ro on o u t l e t p r e s s u r e Po. = const.
sEc. 21
51
If t h e p r e s s u r e
difference
Sp ~- P - Po is s m a l l ( u p
to 10% P), w e
can
take 2sis
Ro = -R = -
mo
Substituting
(21-216)
- (2.1-2.17) i n t o (2.1-2.13) a n d
replacing
d
with
the
finite A. g i v e s
Amo
1 1 aPo + KAAAo = -..-~.aP - IRol
We i n t r o d u c e d that a n
increase
(amo
0).
<
Rearranging
the notation
IRol a n d
the minus sign
to s t r e s s
in a d e c r e a s e
of flow
in Po (APo > O) r e s u l t s
denotes
capacitance,
and
v
I~,~ = ~
T dAP
mo
v
v~
(2.1-2.21) will
the equality
R
= -]5"- = - - ~
(2.1-2.22)
M u l t i p l y i n g (2.1-2.21) b y R w e g e t t h e w e l l - k n o w n a proportional
rate
t h e j u s t i f i c a t i o n of this s y m b o l
b e s h o w n a little l a t e r . F o r t h e p r e s e n t , w e h a v e C =
fact
(2.1-2.10)
1 1 + ---~--AP = a m i * ][-~ol a P o - KAAAo .
C again
the
(2.1-2.11) a n d r e f e r r i n g to (2.1-2.20), w e g e t t h e l i n e a r form of
the initial equation
cdAP dt
(2.1-2.20)
form of a n e q u a t i o n
of
s y s t e m o f t h e first o r d e r
R + A P = R a m l + [--R-~o[a P o - R K A A A o
(2.1-2.23)
If w e c o m p a r e this e q u a t i o n to E q u a t i o n (2.1-2.21) for a l i q u i d s t o r a g e t a n k . w e c a n s e e h o w s i m i l a r t h e d y n a m i c p r o p e r t i e s of t h o s e t w o d i f f e r e n t tanks are. Time constant preceding
T will a l s o
section.
Here
T
show is
i t s e l f to b e the
substitution of the relevant equalities T = CR =
V R~
2~15 = 2 ~15 ~ mo -~- ~ o
product
s i m i l a r to t h e CR,
which
one
in t h e
through
becomes
= 2
T"
(2.1-2.24)
the
.52
CHAP.2
LLI"I~Pc;'D
PROC:Ec:::c~E c:
T* a g a i n r e p r e s e n t s the ratio of the m a s s of g a s (stream) s t o r e d in the tank to the m a s s flow r a t e t h r o u g h
the
tank in the
observed
operating
state
T" = ~
}~ m o
=
stored mass mass
(2.1-2.25)
flow rate
T h e c o e f f i c i e n t 281s/~ is d i m e n s i o n l e s s . Notation in the form of s t a t e - s p a c e e q u a t i o n s for the c a s e w h e n the v a r i a b l e s of interest a r e p r e s s u r e d e v i a t i o n s in the tank t~P a n d d e v i a t i o n s of flow r a t e t h r o u g h the v a l v e /~mo t a k e s the term
+][
[~P]'" [ X'
1
R
~P]" [c"
KA ] / ,I-~'mi ~Po 1 / c ~*o"
i
tRol T
(2.1-2.26)
- - A X ÷ B U
--
° IF-']
[~p]+
-I
Am°
J-~oJ
(2,1-2.27)
I~P° I KA LAAoJ
Y = C X ÷ D U
T h e p r o c e s s u n d e r o b s e r v a t i o n , with two output (AP, amo) a n d t h r e e input (Ami, APe, AAo) v a r i a b l e s , h a s 6 transfer functions. From their forms we c a n c o n c l u d e a b o u t the d y n a m i c p r o p e r t i e s of the p r o c e s s , i.e. a b o u t the w a y in w h i c h the o u t p u t v a r i a b l e s will v a r y with the input v a r i a b l e s . T h e matrix transfer function @(s) is o b t a i n e d in t h e u s u a l , m a n n e r for matrix s t a t e - s p a c e e q u a t i o n s , w h i c h w e will s h o w h e r e in s h o r t e n e d form
OCs) = CCsI - A)-(B + D
@(s) =
(2.1-2.28)
1
I
s +-- I
T
"
~
R
I
KA
I Rol
T
~
+
-1
~
KA (2.1-2.29)
If w e p e r f o r m the i n d i c a t e d matrix o p e r a t i o n s w e g e t G(s) a n d w e c a n write
sEc. 2.1
53
R
T-~+l
R IRoi Ts+I R -IRol
v (T = CR. C = ~ - .
C2.l-Z.3o)
Cs Ts+I
2~ . R : ~mo
LAAo(s)J
TKAs Ts+I
Ro = ~R)
I n d i v i d u a l transfer functions are e l e m e n t s of the matrix transfer f u n c t i o n G(s). T h e y a r e all, e x c e p t G~2(s) a n d G23(s), transfer functions of p r o p o r t i o n a l s y s t e m s of the first order, w h i c h c o u l d h a v e b e e n e x p e c t e d s i n c e there is o n l y o n e mass s t o r a g e tank. It is e s p e c i a l l y important to note that in the transfer function G~z(s), w h i c h relates AP(s) to APo(s), the gain c o e f f i c i e n t is smaller than unity, s i n c e R < IRol. (For e x a m p l e , if we use F i g u r e 2.1-2.4 we c a n s h o w that if 15 = I0 b a r a n d ~o =" 6 bar. a n d if ~0 i n c r e a s e s
to 7 b a r
(i.e.
AP = 1 bar),
the
pressure
in
the
tank
will
i n c r e a s e o n l y b y R/Ro = R/3 R = 0.333 bar.) Figure 2.1-2.4 also l e a d s to the c o n c l u s i o n that c h a n g e s in p r e s s u r e d e v i a t i o n in the tank a n d in outlet flow rate will b e all the smaller as ~o d e c r e a s e s a n d a p p r o a c h e s critical p r e s s u r e Pk. S i n c e it is o b v i o u s that, when Po c o n v e r g e s to Pk, IRol c o n v e r g e s to infinity, the i n f l u e n c e of variations in ~o will c o m p l e t e l y d i s a p p e a r for ~o • Pk. In the following lines, w h e n we a n a l y z e s u p e r c r i t i c a l outflow, "this will b e c o n f i r m e d . The transfer functions Gzz(s) a n d Gz3(s) s h o w that w h e n APo a n d ~,Ao vary. outflow will r e s p o n d as a d e r i v a t i v e s y s t e m with a time-lag of 1st order. This c h a r a c t e r of transfer b e h a v i o r is t y p i c a l for all
tanks
in the
c a s e of s o - c a l l e d d o w n s t r e a m c h a n g e s . It is not difficult to s e e that in the c a s e of d i s t u r b a n c e in AAo. the flow rate c h a n g e ~mo from the p r e c e d i n g e x a m p l e of a liquid tank will also r e s p o n d like d e r i v a t i v e systems. Similar to the c a s e of p r e s s u r e c h a n g e Ap, the transfer function G~z(s) s h o w s that the i n f l u e n c e of APo o n the flow rate Amo d e c r e a s e s the c l o s e r ~o is to the critical p r e s s u r e Pk. In the c a s e of ~o ~ Pk this i n f l u e n c e d i s a p p e a r s completely. b) S u p e r c r i t i c a l
outflow
Often outlet p r e s s u r e Po is smaller t h a n the critical p r e s s u r e , a n d then
54
CHAP.2
gas
flow
through
characteristic of
orifices
is
supercritical
supersonic
LUMPED
PROCESSES
(supercritical).
The
outflow is that the a m o u n t of g a s
flowing out of the tank d e p e n d s
on pressure
basic (steam)
P in the tank. but not o n
p r e s s u r e Po- It is c a l c u l a t e d from me = KvAo~ R / ~~- P Kv is
the
valve
(2.1-2.31) characteristic,
and
for
orifices
c o e f f i c i e n t ~, ~ is the g a s (steam)
characteristic
0.685
diatomic
and
0.669
for
monoatomic,
r e s p e c ;vely. If c h a n g e s
and
and
of s t a t e in the tank a r e
it
is
the
outflow
its v a l u e
more
is 0.726.
complex
isothermal,
gases,
equation
(2.1-2.31) b e c o m e s me = KoAoP
(2.1-2.32)
T h e n E q u a t i o n (2.1-2.8) b e c o m e s V IR,9 The
dP dt * KoAoP = mi last
equation,
for
(2.1-2.33) Ao = const.,
is
a
linear
DE
with
constant
coefficients, For v a r i a b l e Ao the p r o d u c t Ao P m a k e s (2.1-2.33) a n o n l i n e a r e q u a t i o n w h i c h we c a n t i n e a r i z e to o b t a i n m
V
daP
dt
+
me
T
m
AP
= Am i -
me
&Ao
(2.1-2.34)
The c o e f f i c i e n t s b e s i d e AP a n d AAo a r e a l r e a d y k n o w n a n d r e p r e s e n t the i n v e r s e r e s i s t a n c e R a n d the c o e f f i c i e n t of transfer KA R-
-- - ~ p me KvAod~ '
me
KA = %
•
(2.1-2.35) (2.1-2.3b)
R e a r r a n g i n g (2.1-2,34) for g a s (steam) s t o r a g e p r o c e s s e s in tanks in the c a s e of s u p e r c r l t i c a l outflow we get T dAP
+ AP = R A m i - RKAAAo
(21-2.37)
E q u a t i o n (2.1-2.37) is i d e n t i c a l to E q u a t i o n (2.1-1.21) for v a r i a b l e liquid level in a tank if w e r e p l a c e AH b y AP. It differs from (2.1-2.23) for v a r i a b l e AP in the c a s e of s u b s o n i c g a s (steam) outflow b e c a u s e (2.1-2.3"/) d o e s not
sEc. 2.1
55
c o n t a i n the term s h o w i n g the i n f l u e n c e of outlet p r e s s u r e d e v i a t i o n processes
within
the
tank.
It c o u l d
also
be
said
that
the
L~P o
case
on of
s u p e r s o n i c outflow is a limiting c a s e of the p r e c e d i n g o n e , w h e n Po c o n v e r g e s to Pk. Thus all the e q u a t i o n s w e d e r i v e d for s u b c r i t i c a l flow are c o m p l e t e l y v a l i d here, with two important d i f f e r e n c e s : - resistance R = ~/~o the m i d d l e term on the r i g h t - h a n d s i d e of E q u a t i o n (2.1-2.23) d i s a p p e a r s , resulting in the d i s a p p e a r a n c e of the m i d d l e c o l u m n in matrices B. D a n d G(s) a n d the d i s a p p e a r a n c e of the m i d d l e row of the input v e c t o r s U a n d U(s) in the matrix s t a t e - s p a c e e q u a t i o n s (2.1-2.26) a n d (2.1-2.27), a n d in the matrix transfer functions (2.1-2.30). Further a n a l y s i s of the p r o c e s s is r e d u c e d
to earlier c o m m e n t s w h i c h
we n e e d not r e p e a t . We s h o u l d , h o w e v e r , turn o n c e more to the r e s i s t a n c e Re to g a s (steam) outflow in the c a s e of Po ~ Pk, w h i c h F i g u r e 2.1-2.3 shows to b e infinite. This m e a n s that for a finite c h a n g e in ~Po the i n c r e a s e Amo e q u a l s zero, i.e. that the r e s i s t a n c e Re to c h a n g e in flow is infinite. From the definition of r e s i s t a n c e as the ratio of effort i n c r e a s e to flow i n c r e a s e this is e v e n more o b v i o u s [Rol
=
AP° Amo
=
AP° 0
=
(2.1"2.38)
oo
P O S S I B I L I T I E S OF MASS STORAGE. C A P A C I T A N C E C. Up to now we h a v e p a i d a lot of attention to the r e s i s t a n c e
certain
elements of the s y s t e m offer to c h a n g e s in mass flow in the c a s e c h a n g e s in effort (potential). Less has b e e n s a i d a b o u t possibilities mass s t o r a g e , c h a r a c t e r i z e d b y c a p a c i t a n c e C.
of of
C a p a c i t a n c e C is d e f i n e d as the ratio of i n c r e a s e i n c r e a s e in effoIt
to
C--
dM d~P-
in s t o r e d
mass
(2.1-2.39)
In the liquid s t o r a g e tank effort is the h e i g h t of liquid H, a n d the total mass s t o r e d in the tank is M = AH~,. From (2.1-2.39) we g e t
6(5
ci-iAp.2 I_Uh,lpi~D P R O C E S S E S
C --
dH
A p = -H- ,
(2.1-2,40)
In t h e g a s ( s t e a m ) s t o r a g e t a n k t h e e q u i v a l e n t of h e i g h t H is p r e s s u r e P in t h e tank, b u t in this c a s e also depends aspect
t h e p o s s i b i l i t y of g a s ( s t e a m ) a c c u m u l a t i o n
on t h e c o n d i t i o n s u n d e r w h i c h that fluid is s t o r e d . F r o m that
we
distinguish
capacitance
C. As h a s
between already
isothermal
been
C
and
polytropic
s a i d . in this t e x t w e
Cn
will u s u a l l y
a n a l y z e i s o t h e r m a l p r o c e s s e s a n d u s e the i d e a l g a s e q u a t i o n . T h e r e f o r e dM
C In
d(Vp)
dP
the
-
case
independent
V
dP
=
of
~
M = -P-
polytropic
kg
(2.1-2.41)
'[--~]
temperature
change
% is
no
longer
of p r e s s u r e P s o w e h a v e
Pv n , , c o n s t .
(2.1-2.42)
that is dv d----P =
1 n
v P
(2.1-2.43)
C o n s e q u e n t l y , the p o l y t r o p i c c a p a c i t a n c e
Cn is o b t a i n e d a s follows
C, =
I C
dM
d~
V
d--ft---Vdp = - ~
Thus c a p a c i t a n c e
dv
1 V I
d--~--n v P - - n
(2.1-2.44)
C is t h e g r e a t e s t in the c a s e
of i s o t h e r m a l c h a n g e s
of s t a t e in g a s ( s t e a m ) .
Example
2 . Outflow of g a s . C a l c u l a t i o n of time c o n s t a n t T
C o n s i d e r a t a n k of v o l u m e V = B0 m 3 f i l l e d with g a s at t h e t e m p e r a t u r e of ,~ : 2Q3 °C with a g a s c o n s t a n t IR = 150 J/kgK a n d u n d e r t h e p r e s s u r e P = I0 b a r (x = 1.4. # : 0.685). At t h e t a n k e x i t is a v a l v e of a r e a Ao : 0.002 m 2 with v a l v e
characteristic
Kv = 0.75.
Determine
c h a r a c t e r i z e s t h e d y n a m i c s of p r e s s u r e c h a n g e
time
constant
T that
in t h e t a n k if t h e o u t f l o w
is into, a) t h e a t m o s p h e r e , b) into a tank with c o n s t a n t p r e s s u r e Po = 6 b a r .
24
sea.
57
a) The critical p r e s s u r e for a p r e s s u r e of P ,, lO b a r in the tank is Pk = 5.28 bar. S i n c e Po = 1 bar, outflow is s u p e r c r i t i c a l . From (2.1-2.31) it follows that dmo dP
=
KvAo4J~ ..t
-_ I__
~
Assuming
an
(2.1-2.45)
R
isothermal c h a n g e
in gas
state in the tank. w e
have
(2.1-2.41), w h i c h t o g e t h e r with (2.1-2.45) d e t e r m i n e s T T
-
CR
=
V RB
__~ KvAo~
-- 232.1
s.
(2.1-2.46)
b) In the c a s e of Po = 6 b a r flow is s u b c r i t i c a l so time c o n s t a n t T c a n b e d e t e r m i n e d from (2.1-2.24). T = CR =
V ~a
2~P mo
In the u p p e r e q u a t i o n the o n l y u n k n o w n v a r i a b l e is flow rate mo, w h i c h is d e t e r m i n e d from (2.1-2.0) for s u b c r i t i c a l flow. But in this c a s e we will u s e the more c o m p l e x a n d c o r r e c t m a t h e m a t i c a l e x p r e s s i o n
m o = KvXoP
Equation
x
x-
1
I
(2.1-2.9), w h i c h
practical calculations,
(
Ra
and
we
)-~--. (
used
constant
until
) x
(2.1-2.47)
now, is s a t i s f a c t o r y
Ko c a n , after
the
flow rate
for most mo h a s
b e e n c a l c u l a t e d from (2.1-2.47). b e o b t a i n e d "easily from (2.1-2.14). The u p p e r e q u a t i o n g i v e s mo = 4.84 k g / s . from w h i c h it is not difficult to g e t T -- 188.04 s.
C o m m e n t i n g o n E q u a t i o n (2.1-2.33) w e s a i d that if the v a l v e a r e a Ao is c o n s t a n t , the d i s c h a r g e of the tank u n d e r s u p e r c r i t i c a l c o n d i t i o n s is linear if g a s - s t a t e c h a n g e s in the tank are isothermal. (In that c a s e t e m p e r a t u r e ~) b e s i d e the d P / d t term is c o n s t a n t . ) T h e following e x a m p l e will s h o w that it is then r e l a t i v e l y e a s y to d e s c r i b e the d y n a m i c s of p r e s s u r e c h a n g e in the tank.
,~
CI4AP.2 L U M P E D
Example
PR{3C'2S~ES:
3 . O u t f l o w of air u n d e r s u p e r c r i t i c a l c o n d i t i o n s
Air flows i n t o t h e
atmosphere
with a p z e s s u I e P = 4 0 b a r a n d
o u t of a
t a n k of v o l u m e
temperature
V = 400
d m 3,
,~ = 15 °C, t h r o u g h a n o r i f i c e
Ao = 20 mm 2 a n d ~ = I. Po = 1 b a r . D e t e r m i n e h o w l o n g it will t a k e to r e a c h s u b c r i t i c a l o u t f l o w . We a s s u m e that t h e p r o c e s s to b e c o n s i d e r e d
i n t h e t a n k is s l o w e n o u g h
i s o t h e r m a l . For air. # = 0 . 6 8 5 a n d R = 287 ] / k g K .
For Po = 1 bar. Pk = Po/0.528 = 1.8Q4 bar. E q u a t i o n (2.1-2.33) with mi = 0 gives T dP --~ ,
(2.1-2.48)
0
P =
E q u a t i o n (2.1-2.46) d e t e r m i n e s time c o n s t a n t T. T =
V
I
= 101.54
s
~Ao~ T h e s o l u t i o n of (2.1-2.48) is s i m p l e
I P = 40 - e ' T
t
= 40 e
-o.ooges t
T h e time after w h i c h P will e q u a l 1.8Q4 is o b t a i n e d 1
tkrit = 0.00Q85
from
40 In 1.89----4- = 310 s
(2.1-2.49)
T h e assumption of temperature constancy in the tank c a n b e m a d e "slow"
processes
of
state change.
Many
tanks
are
charged
discharged so quickly that heat cannot b e completely e x c h a n g e d
for and
with the
surroundings in that short time a n d the process in the tank is polytropic. T h e purpose of the following e x a m p l e isothermal model
is to s h o w the error if w e
use an
for a process that is in fact po]ytropic. (0J course, in
every particular case a similar procedure should b e repeated b e c a u s e testing.)
of
S'~C. 2.1
5Q
4. L i n e a r i t y of g a s ( s t e a m ) o u t f l o w from t a n k
Xxample Consider
an
air
tank
of v o l u m e
V = 60
aggregate.
The air pressure
aggregate
is s t a r t e d u p . t h e a i r f l o w s t h r o u g h
into a c y l i n d e r 0.6. a n d
where
the pressure
the air pressure
For air w e
have
for t h e p r e s s u r e change
is 5 0 b a r a n d
4, = 0 . b 8 5 ,
a) polytropic
starting
the temperature
up
Diesel
293 °C. As t h e
a n o r i f i c e 5 m m in d i a m e t e r
is l b a r . T h e o u t f l o w c o e f f i c i e n t
in t h e t a n k d e c r e a s e s
polytropically
R = 287 J / k g K . D e t e r m i n e
in t h e t a n k to d e c r e a s e
graphically
d m 3 for
to 12 b a r
the
and
is t* =
with n = 1.25.
time necessary
show
the pressure
for a:
process.
b) isothermal process. a)
For
Po
supercritical
=
1 bar,
Pk
=
1.894
bar
a l l t h e time. F o r ml = O, u s i n g
and
the
flow
(2.1-2.1) a n d
conditions
are
(2.1-2.44). w e c a n
write
dM
d-t
"
dM ~
dP . c - d P n dt
-~
Since outflow occurs
= " mo
under
(2.1-2.50)
supercritical
conditions,
w e t a k e (2.1-2.31) for
mo a n d g e t
V nR~
dP = vAo~b
P
dt
(2.1-2.51) n-1
Using the known relation we see that dt = -
~,(_P_P_)n
, after arranging
P1 dP 3n-I p --2-fi--
of the upper
(in c o r r e s p o n d e n c e
could write • instead for t i m e t
=
(2.1-2.51)
n-1 "2W
V n~Ao0 ~ ,
Integration pressure
~)
equation with the
(2.1-2.52)
from PI to P, w h e r e notation
of P1) a n d P t h e p r e s s u r e
we
used
PI is t h e up
initial
to n o w ,
we
at t, w e g e t t h e f i n a l r e s u l t
CHAP.2 LUMP~'r~ PRC~CE~SES
~0
v 2[ 2n
]
n-1
t = vAoq~~-~t n-I
(
)
Thus. the time n e e d e d
l
(2.1-2.53)
for the p I e s s u r e in the tank to fail from 60 to 12
bar, if the c h a n g e is p o l y t r o p i c , is from (2.1-2.53)
t =36s.
6O bar
40
\\
30
zo
tO
\ 10
rig.
20
~ 30
' 40
~ 50
60 s
70 t
2 . 1 - 2 . 5 P r e s s u r e d e c r e a s e in the tank ..... p o l y t r o p i c (real) c h a n g e . NL. isothermal c h a n g e . L.
b) As in the p r e c e d i n g e x a m p l e , we h a v e T dP dt
,P=O
where T = CR =
V
~
R01 ~Ao¢
= 25.64 s
For the initial c o n d i t i o n Po = P, = bO bar, the solution of the u p p e r DE is
-it P = 60 e T
= 60 e
-0.039t
The last e q u a t i o n for P ffi 12 b a r yields.
(2.1-2.54)
s E c . 2.1
61
1
t =
60
0.03----'-9
In._-:-- = 41.27 s
T h e e r r o r in t h e t i m e n e c e s s a r y to 12 b a r , if t h e c h a n g e shows
the curves
for p r e s s u r e
is c o n s i d e r e d
in t h e t a n k to fall from 6 0
i s o t h e r m a l , is +14.64Z. F i g u r e
of t h e r e a l p o l y t r o p i c
(nonlinear
t h e i s o t h e r m a l ( l i n e a r - full l i n e ) p r e s s u r e
- broken
fall o b t a i n e d
2.1-2.5
line) and
of
from (2.1-2.53) a n d
(2.1-2.~4). These
curves
show
that the error due
to t h e a s s u m p t i o n
state change
in t h e t a n k is n o t v e r y g r e a t .
this e x a m p l e
are not generally
i l l u s t r a t i o n for t h e p o s s i b l e
To e n d
the preceding
discharged
under
examples,
constant
- s e e E q u a t i o n (2.1-2.46).
Example Consider
S. Charge the
gas
and discharged
pressure
difference.
changes
of g a s
moment,
and discharge tank
through
as an
t h e t a n k is
is l i n e a r o n l y if a n
since
of tank due shown
on
the
the time
temperature
to p r e s s u r e
Figure
~ is
difference
2.1-2.6.
which
an inlet and outlet (control) valve due
We m u s t f o r m u l a t e
(steam) pressure
a n d Po. a n d o n t h e a r e a
that when
the process
w i t h i n t h e t a n k . In t h a t c a s e
in e v e r y
(steam)
charged
we must repeat
takes place
T is u n c h a n g e d
is to s e r v e
in
of s o m e a s s u m p t i o n s .
conditions,
constant
of i s o t h e r m a l
the results given
valid. Their main purpose
consequences
supercritical
isothermal state change
However.
the equations
in t h e
tank
of t h e c o n t r o l v a l v e
describing
depending
on
Ao. T h e c h a n g e s
is to
unsteady
pressures
PI
in t h e t a n k
a r e i s o t h e r m a l . Ai ; c o n s t . Unlike Example independent
I, t h e
of t h e
amount
pressure
l i n e a r form (2.1-2.20). in t h e c a s e mi ~- KiAi~--P~-P) .
P ~
of g a s
at
P. A n a l o g o u s l y
Pi2
of s u b c r i t i c a l
the
tank
inlet
to E q u a t i o n
inflow
is n o
longer
(2.1-2.9), i.e. its we see that (2.1-2.55)
~2
~HAP.2
Aml : - ~ A P I -
8ml R"
=
LUMPED
(21-2.56)
AP
(2,1-2.57)
mi
-~ p~mi
Ri pi-2i -p 5
=
PROCESSEE
=
~iRi
,
ai <
(2.1-2.58)
O.
v
F i g . 2 . 1 - 2 . 6 Gas (steam) tank with u n c o n t r o l l e d inflow mi(t) For small p i e s s u r e d i f f e r e n c e s b e t w e e n P1 a n d P, ~,~ is c l o s e to unity a n d w e h a v e JR'I - IRI.. The s a m e d i a g r a m from F i g u r e 2.1-2.4 c a n b e u s e d for ~i. o n l y n o w R° is o n the o r d i n a t e a n d p r e s s u r e P, e x p r e s s e d in parts of the input p r e s s u r e Pj, on the a b s c i s s a . All the other e q u a t i o n s a n d m e t h o d s u s e d in E x a m p l e 1 a r e v a l i d h e r e also. a n d n e e d not b e r e p e a t e d . For a g i v e n mi(t), E q u a t i o n (2.1-2.21) y i e l d s ~ C dAP ÷ +
Ap - ~
Ap o = R 1~ . a p i -
IRI~IAp
KAAAo
(2.1-2.59)
If an e q u i v a l e n t r e s i s t a n c e Re is i n t r o d u c e d in the following m a n n e i
1 Re
s
1 R
* ~
1
,
(2.1-2.60)
E q u a t i o n (2.1-2.59) is t r a n s l a t e d into a form v e r y similar to (2.1-2.23) TdAP
÷
T = CRe
Re APa ÷ AP ~, --~i
-
1
R
C
I
IR'I
APo _ ReKAAA o
(2.1-2.61)
(2.1-2.62)
ssc. 2.1
63
We will not
rewrite
the
state-space
equations
from E x a m p l e
1. T h e
E q u a t i o n s (2.1-2.26). (2.1-2.27) a n d (2.1-2.30) g i v e n t h e r e a r e c o m p l e t e l y v a l i d h e r e also, with the following c h a n g e s R=Re V R3 T = CRe =
1
1 4..--
R
Aml = API
IR'I
. ~Mi(s) = API(s)
Ri
Ri
We witl not s p e c i a l l y a n a l y z e c a s e s w h e n c h a r g i n g a n d d i s c h a r g i n g o c c u r u n d e r s u p e r c r i t i c a l c o n d i t i o n s either. T h e u p p e r e q u a t i o n s a r e g e n e r a l e n o u g h a n d c a n a l s o b e u s e d in s u p e r c r i t i c a l c a s e s , w h e n it is sufficient to m a k e the r e s i s t a n c e infinite. For e x a m p l e , if the tank is c h a r g e d u n d e r s u p e r c r i t i c a l c o n d i t i o n s , i.e. for P < PI/2, the c o e f f i c i e n t R* =. co. Re = R. a n d Ri a l s o c h a n g e s , Ri = Pl/ml, s o that (23-2.61) b e c o m e s
dAP T-~--
+ aP
R =-~-i aPl ÷
nPo - RKAnAo
(2.1-2.03]
T = CR E q u a t i o n (2.1-2.61) s h o w s
that n o w time c o n s t a n t
T is not d e t e r m i n e d
only b y c o e f f i c i e n t R. a n d a c c o r d i n g to (2.1-2.18) b y the p r e s s u r e s 1~ a n d 15o, but a l s o b y R" a n d c o n s e q u e n t l y , a l s o b y p r e s s u r e ~l. This d e p e n d e n c e on Pj c e a s e s to e x i s t o n l y w h e n the t a n k is c h a r g e d u n d e r s u p e r c r i t i c a l c o n d i t i o n s . Also, if APi or APo c h a n g e . Ap will not c h a n g e for the s a m e v a l u e but its c h a n g e is d e t e r m i n e d b y the g a i n c o e f f i c i e n t s Kpi = Re/Ri a n d Kpo = Re/IRol.
At the e n d of this e x a m p l e it will b e of u s e to s h o w the i n t e r d e p e n d e n c e of v a r i a b l e s from E q u a t i o n (2.1-2.61) in a b l o c k d i a g r a m . We d o this not o n l y b e c a u s e a g r a p h i c a l r e p r e s e n t a t i o n is c o n v i n c i n g , but also b e c a u s e the d i a g r a m i m p o s e s c o m p a r i s o n b e t w e e n the d y n a m i c s of h e a t transfer t h r o u g h the wall of the h e a t e x c h a n g e r a n d t h e s e p r o c e s s e s of m a s s s t o r a g e . T h e s t r u c t u r e of the b l o c k s in F i g u r e s 2.I-2.T a n d 2.3-2 is c o m p l e t e l y i d e n t i c a l . T h e o n l y small d i f f e r e n c e is that in F i g u r e 2.1-2.7 the r e s i s t a n c e s in the f e e d b a c k link differ from the r e s i s t a n c e b e s i d e the i n p u t
~Z~
~HAP.2
LUMPE
r~ P R O C E S S E S
v a r i a b l e s . (If the d i f f e r e n c e s b e t w e e n p r e s s u r e s PI. P a n d Po a r e not g r e a t , t h o s e r e s i s t a n c e s b e c o m e the s a m e . i.e. w e will h a v e Re = R a n d R" ; RI, so the d i a g r a m s b e c o m e i d e n t i c a l . )
~P
Fig.
2 . 1 - 2 . T Block d i a g r a m of p r e s s u r e d y n a m i c s in a g a s ( s t e a m ) t a n k
In the p r e v i o u s d i s c u s s i o n of t a n k s that c o u l d o n l y s t o r e m a s s . the m a t h e m a t i c a l m o d e l of d y n a m i c s w a s o b t a i n e d flora the e q u a t i o n for the c o n s e r v a t i o n of m a s s . In the following s e c t i o n we will s h o w h o w the s a m e or similar d y n a m i c p h e n o m e n a a n d their n o t a t i o n a p p e a r in the p h y s i c a l l y c o m p l e t e l y different p r o c e s s e s of fluid flow t h r o u g h p i p e s . Their m o d e l s will either b e o b t a i n e d o n l y from the e q u a t i o n f o r t h e c o n s e r v a t i o n of momentum or from the s i m u l t a n e o u s formulation of the e q u a t i o n for the conservation o f m o m e n t u m a n d the e q u a t i o n for the
conservation
of mass.
2 . 2 FLUID FLOW
Fluid s u p p l y a n d t r a n s p o r t a n d the c o n n e c t i o n of i n d i v i d u a l p r o c e s s units runs t h r o u g h p i p e s ( s t e a m p i p e s , g a s p i p e s , w a t e r p i p e s , p i p e l i n e s ) , w h i c h t o g e t h e r with p u m p s , c o m p r e s s o r s , v a l v e s a n d o t h e r fittings a r e the u n a v o i d a b l e c o m p o s i t e p a r t s of p l a n t s . T h e p r o c e s s of fluid flow r a t e s t h r o u g h p i p e s is u s u a l l y a c o n t r o l l e d p r o c e s s in the s e n s e that p r e s s u r e s
sEc. 2.2
05
a n d flows a r e k e p t within p r e d e t e r m i n e d b o u n d a r i e s t h r o u g h the a c t i o n of control
devices
on
various
executive
control
elements
or
actuators
(usually r e g u l a t i o n v a l v e s a n d p u m p s or c o m p x e s s o r s ) . It is thus a l s o important to k n o w the d y n a m i c p r o p e r t i e s of t h o s e flow p r o c e s s e s . The b a s i c e q u a t i o n s d e s c r i b i n g flow p r o c e s s e s are i d e n t i c a l for liquids, g a s e s a n d steam. N e v e r t h e l e s s . for p r a c t i c a l n e e d s , to simplify the often c o m p l i c a t e d c a l c u l a t i o n s that must b e u n d e r t a k e n if we u s e the most g e n e r a l laws a n d e q u a t i o n s of fluid m e c h a n i c s , it is useful to a n a l y z e liquid flow s e p a r a t e l y from g a s a n d s t e a m flow. In the first c a s e we c a n usually n e g l e c t the c o m p r e s s i b i l i t y of liquids . (p ; const.) but momentum, i.e. the inertia of liquid in the p i p e . In the c a s e of g a s e s
not and
steam the p r o c e d u r e will b e the o p p o s i t e . It will u s u a l l y b e p o s s i b l e to n e g l e c t their momentum, but not also the effects of c o m p r e s s i b i l i t y , i.e. we will h a v e to c o u n t o n c h a n g e s in d e n s i t y p in u n s t e a d y c o n d i t i o n s . On the following p a g e s , w h e n we d e v i s e m o d e l s for the d y n a m i c s of flow p r o c e s s e s , we will start from simpler a n d more s p e c i f i c c a s e s of liquid a n d g a s (steam) flow m a k i n g u s e of the u p p e r h y p o t h e s e s . G r a d u a l l y w e will r e a c h the most g e n e r a l d y n a m i c m o d e l for the flow of an arbitrary fluid that has b o t h the p r o p e r t y of c a p a c i t a n c e (p ,~ const., C ,, O) a n d the p r o p e r t y of inertia (l
,, 0).
2.2-I LIQUID FLOW
The b a s i c a s s u m p t i o n in this part of t h e b o o k is that liquid is i n c o m p r e s s i b l e (~ = const.), w h i c h in fact m e a n s n e g l e c t i n g liquid s t o r a g e in p i p e s , i.e. we c o n s i d e r that c a p a c i t a n c e
C = O. In this way, at the v e r y
b e g i n n i n g , we dismiss the possibility of o b s e r v i n g the w e l l - k n o w n water hammer effects in p i p e s . In S e c t i o n 2.2-3 we will l e a v e out this a s s u m p t i o n a n d g e t a m o d e l for the flow of a n arbitrary fluid, for w h i c h the m o d e l s in this c h a p t e r (C ; O) a r e o n l y a s p e c i a l c a s e . Unlike in the p r e c e d i n g section, h e r e we n e g l e c t e d c o m p r e s s i b i l i t y so w e will n o w o b t a i n a m o d e l b y formulating a n e q u a t i o n
for the conservation
of momentum.
66
Cl.-.IAP.2
• ..xample
LUMPED
PROCI~C~c~lZg
| D y n a m i c s of liquid flow t h r o u g h p i p e s
F i g u r e 2.2-I.I s h o w s a h o r i z o n t a l p i p e of l e n g t h L. with solid walls a n d of c o n s t a n t c r o s s - s e c t i o n a l a r e a A, w h i c h c o n n e c t s two tanks. A c e n t r i f u g a l p u m p p u m p s liquid t h r o u g h the p i p e a n d a control v a l v e in the p i p e r e g u l a t e s flow. D e r i v e a m o d e l for u n s t e a d y flow c h a n g e s t h r o u g h p i p e . T h e liquid is c o n s i d e r e d i n c o m p r e s s i b l e (p = const., m # m(z)).
the
II _ Iz ~ A 13 [_-7.-_-_--'_ --------------l~...... ~)~')----~-----------.--------_,~---'----"------I r
l
t
'~J
i,
6g
5~ P=
,
Z
F i g . 2.2-1.1 L i q u i d t r a n s p o r t b e t w e e n two t a n k s T h e law for the c o n s e r v a t i o n of m o m e n t u m (Ira = M w) for liquid in a p i p e h a s the w e l l - k n o w n form n
dlm
Z
= d--V-
(2.2-H)
i=l Fi denotes all the forces that act on the total mass ( M = ALp ) of liquid in the pipe. In the general unsteady case pressures PI a n d P2. pressure changes
in the valve (drop ~Pv) in the pipe (drop, decrease
~Pc) a n d
in
the p u m p (increase SPp) are time functions. If w e specify the forces that act on the mass of liquid M in Equation (2.2-I.I),w e get (PI * ~Pp - ~Pv - ~Pc)A - P2A =
d(Mw) dt
=
ALp d w d-[-" "
(2.2-1.2)
T h e liquid flow r a t e ( m a s s flow m) is m = Awp
.
(2.2-1.3)
s~c. 2.2
67
a n d if w e insert (2.2-1.2), we g e t
velocity
w from the
upper
equation
L dm PI + aPp - aPv - SPc - P2 = A dt This e q u a t i o n
describes
the c h a n g e
into
Equation
(2.2-1.4) of mass
flow t h r o u g h
d e p e n d i n g on p r e s s u r e s P, a n d P2 a n d p r e s s u r e c h a n g e s
the
pipe
~Pp. ~Pv a n d ~Pc.
The g r e a t e s t
m a t h e m a t i c a l difficulties in o b t a i n i n g m o d e l s h i d e in the d e t e r m i n a t i o n of t h o s e p r e s s u r e d e c r e a s e s a n d i n c r e a s e s . The s a m e result c a n b e f o u n d in S e c t i o n 3.3, w h e n the fluid flow is o b s e r v e d as a p r o c e s s with d i s t r i b u t e d p a r a m e t e r s ( s e e E q u a t i o n (3.3-29)). It is a k n o w n fact. a n d in the c a s e of v a l v e s it was s h o w n in the p r e c e d i n g s e c t i o n , that there is a d i r e c t r e l a t i o n s h i p b e t w e e n flow rate m a n d p r e s s u r e c h a n g e ~P in i n d i v i d u a l parts of the fittings. For p u m p s a n d v a l v e s t h e s e c h a n g e s d e p e n d on m a n y other factors as well. In the c a s e of p u m p s the s p e e d of rotation n is of b a s i c i m p o r t a n c e , a n d for v a l v e s their c r o s s - s e c t i o n a l a r e a Av. In short, if we want to g e t a final e x p r e s s i o n for d y n a m i c flow c h a n g e s , we must know the following a n a l y t i c a l relations b e t w e e n flow a n d p r e s s u r e c h a n g e
~Pp = ~Pp(m. n) ~Pv = ~Pv(m. Av)
.
(2.2-I.5) .
(2.2-1.6)
~Pc-- ~Pc(m)
(2.2-1.7)
As a rule the u p p e r a n a l y t i c a l e x p r e s s i o n s a r e not a v a i l a b l e , but t h e r e is ( a l w a y s s u p p l i e d with the e q u i p m e n t ) a g r a p h i c a l p r e s e n t a t i o n of p u m p and valve characteristics. Figures 2.2-1.2 a n d 2.2-1.3 s h o w those c h a r a c t e r i s t i c s , a n d t h e y also s h o w h o w to d e t e r m i n e g r a p h i c a l l y the v a l u e s of the partial d e r i v a t i v e s of p r e s s u r e ~Pp a n d ~Pv b y the
independent variables. For turbulent flow t h r o u g h p i p e s we h a v e the c h a r a c t e r i s t i c resistance law, a c c o r d i n g to w h i c h L
w2
L
~iPc- x-~---~--t~ = Xd
m 2 . , Kcm 2 2AZp
(2.2-1.8)
~8
(2HAP.2
LUI'~PED
PROCESS~'S
6%,
......
I÷)
I
'q, !\ o=
iN
i
~,
k
N
Fig.
2 . 2 - 1 . 2 Static c h a r a c t e r i s t i c s for a c e n t r i f u g a l p u m p
increoseAv/
&Av
Fig. 2.2-1.3Static
c h a r a c t e r i s t i c s for a , e g u l a t i o n v a l v e
The curves s h o w n and Equation (2.2-1.8) m a k e it obvious that Equation (2.2-1.4) is nonlinear, a n d if w e want to analyze unsteady flow it should b e ]inearized. This narrows d o w n the field of process variable changes for which
the model
gives
a
faithful dynamics
description,
but
linearized
models m a k e it simpler to understand the character of transient processes, to carry out analyses and to c o m p a r e the dynamics of flow processes with other, complelely different, processes.
Sg'C.
69
2.2
Differentiating E q u a t i o n s (2.2-1.4) - (2.2-1.7) we g e t
L d(dm) dP1 + d&Pp - d~Pv - dSPc - dP2 = A dt
(2.2-1.9)
dBPp = aSPp dm + a&Pp d n
(2.2-1.10)
am
~n
Rp
bp
dSPv = ,--.a,sPv --,a-----m-d m + ~aaPv dAy Rv
(2.2-1.11)
bv
dSPc = ~l~c dm
(2.2-132)
om
Rc In the last three e q u a t i o n s , in c o r r e s p o n d e n c e definition
for
resistance
to
flow
R, w e
with the a l r e a d y g i v e n
introduced
symbols
for
pump
r e s i s t a n c e Rp, v a l v e r e s i s t a n c e Rv a n d p i p e r e s i s t a n c e Rc. If a n a l y t i c a l e x p r e s s i o n s in the form of E q u a t i o n s (2.2-1.5} - (2.2-1.7) exist, it is in most c a s e s not difficult to c a l c u l a t e c o e f f i c i e n t s R a n d b. If t h e r e are no s u c h e x p r e s s i o n s , h o w e v e r , c o e f f i c i e n t s R a n d b in the s t e a d y state a r e d e t e r m i n e d g r a p h i c a l l y , as the p r e c e d i n g figures show. On the b a s i s of p r e v i o u s a n a l y s i s , we c a n write without r e p e a t i n g the derivation 2~v
Rv -- ~ by =
-
(2.2-1.13)
m 2~5Y
X~
(2.2-1.14)
'
Rc = ~2~I5c =
2i~cr~
(2.2-1.15)
m
It is important to point out that as flow t h r o u g h the p u m p i n c r e a s e s , the pump h e a d d e c r e a s e s . Thus r e s i s t a n c e Rp is n e g a t i v e . Finally, we c a n write
a~Pp : -IRplam ", bpan ~&Pv = RvAm - Ibv[~Av
.
(2.2-1.16) (2.2-1.17)
70
el-lAP.2 L U M P ~ ' D
PROCESS~'S
(2.2-1.18)
a~Pc = R c a m
If we substitute the last t h r e e e q u a t i o n s into (2.2-1.9), w e g e t U dam
A
dt
dt
÷ Rc ÷ [Rpl)Am -
÷
÷ Am = ---ff-,sP, -
bp n • [b IAA
APz ÷ - - ~ z x n ÷
-
z~Av
(2.2-i.19)
(2.2-1.20)
The u p p e r e q u a t i o n s h o w s that u n s t e a d y c h a n g e s of mass flow of a n i n c o m p r e s s i b l e liquid t h r o u g h a p i p e h a v e the p r o p e r t i e s of the first-order ( p r o p o r t i o n a l ) system. This is a g e n e r a l e q u a t i o n with four input v a r i a b l e s on the f i g h t - h a n d side. In p r a c t i c e , if the v a l v e c a n n o t b e r e g u l a t e d the term with AAv o n the r i g h t - h a n d s i d e d i s a p p e a r s .
Similarly, if the p u m p
rotates with a c o n s t a n t n u m b e r of r e v o l u t i o n s the term with An is lost, a n d SO o n .
It is e s p e c i a l l y important to a n a l y z e the time c o n s t a n t T. We must know how it c a m e into b e i n g a n d what its v a l u e is. Before that, h o w e v e r , we must g e t a c q u a i n t e d with the c o n c e p t of i n e r t a n c e I of the mass of liquid. Let liquid flow ideally, without friction, t h r o u g h a s m o o t h p i p e of l e n g t h L a n d c r o s s - s e c t i o n a l a r e a A, T h e n M = ALp d e n o t e s the total mass of liquid in the p i p e . a n d m = Awp the mass flow rate of that liquid t h r o u g h the p i p e in k g / s . The law for the c o n s e r v a t i o n of momentum is ~PA = d(Mw)~ ¢. M--dw = M dm = L dm. dt dt A¢~ dt dt
(2.2-1.21)
L dm ~P = - - - A dt
(2.2-1.22)
The u p p e r e q u a t i o n follows d i r e c t l y from (2.2-1.4) if w e n e g l e c t p r e s s u r e d r o p s a n d i n t r o d u c e the s y m b o l (sP ; PI Pz) for the p r e s s u r e d i f f e r e n c e that c a u s e s flow c h a n g e with g r a d i e n t dm/clt. The c o e f f i c i e n t b e s i d e d m / d t is the m e a s u r e o f that c h a n g e , a n d it is c a l l e d i n e r t a n c e I. The l a r g e r I is, the smaller the i n c r e a s e in mass flow for a g i v e n p r e s s u r e d i f f e r e n c e ~P. ]~ertance I is thus d e f i n e d as the ratio of effort (potential) ~P to the g r a d i e n t of c h a n g e in fluid flow d m / d t . 1 =
~P
~
dt
L
-- - A
(2.2-1.23)
sec.
71
2.2
It is not difficult to s e e that the ratio I/R h a s the d i m e n s i o n of time, a n d thus w e i n t r o d u c e d in to E q u a t i o n (2.2-1.20) the s y m b o l for c o n s t a n t T T =- I__ = __L = ~LAm = LAoAw= R AR ARZ~m AASP If we e x p a n d
MAW Az~SP
(2.2-1.24)
the numerator a n d the d e n o m i n a t o r in the last e q u a t i o n
by Am, we g e t the m e a n i n g of the time c o n s t a n t T -- MA___ww: AASP
c h a n g e in m o m e n t u m c h a n g e in f o r c e
If d e p e n d e n c e not q u a d r a t i c
(2.2-1.25)
b e t w e e n p r e s s u r e d r o p a n d mass flow was linear, a n d
as in the c a s e of v a l v e s a n d
pipes,
we w o u l d
have
the
following e x p r e s s i o n for r e s i s t a n c e
8P
R = ~ -
(2.2-1.20)
m
T h e n we c o u l d s h o w that
[ T
=
I
--.~.
L =
m
- -
=
ALp w
A(R,÷R~,IRpl) E~ M~ = ~
momentum
=
(2.2-1.27)
force
This e n d s the a n a l y s i s of liquid flow d y n a m i c s a n d we will not attempt to g i v e the s t a t e - s p a c e e q u a t i o n s . That will b e d o n e at the e n d of S e c t i o n 2.2 for the most g e n e r a l c a s e of an arbitrary fluid flow, for w h i c h this flow in E x a m p l e 1 is a s p e c i a l c a s e with the a s s u m p t i o n C = O. It is. h o w e v e r , useful to g i v e a b l o c k d i a g r a m of v a r i a b l e interrelations o b t a i n e d after the L a p l a c e transformation of E q u a t i o n (2.2-1.19). This h a s b e e n d o n e o n F i g u r e 2.2-1.,4.
~m' ,,P,
Fig.
2 . 2 - | . 4 Block d i a g r a m of the d y n a m i c s of liquid flow
~(Mw)
"/2
CI.-IAP.2 L U I ~ P E ~
The
following two n u m e r i c a l
examples
will s h o w
PRC)c'EESEE
the v a l u e s
of time
c o n s t a n t s c h a r a c t e r i z i n g liquid flow p r o c e s s e s . We will s h o w that t h e s e are fast p r o c e s s e s with a small T, w h o s e d y n a m i c s c a n u s u a l l y a n d most f r e q u e n t l y b e n e g l e c t e d in c o m p a r i s o n with the d y n a m i c s of the m u c h slower final c o n t r o l e l e m e n t s (actuators).
Before a n a l y z i n g n u m e r i c a l e x a m p l e s we must r e p e a t that in the c a s e of d y n a m i c p r o c e s s e s the most s e n s i t i v e part is, as a rule, to d e t e r m i n e the c o e f f i c i e n t s ( p a r a m e t e r s ) of the d y n a m i c m o d e l u s i n g statical ( d e s i g n ) c a l c u l a t i o n s . Here this c o n c e r n s d e t e r m i n i n g the c o e f f i c i e n t s Rp, Rv, Rc, bp a n d by. To c a l c u l a t e them c o r r e c t l y it is n e c e s s a r y to know v e r y well the m e t h o d s of statical c a l c u l a t i o n , as the following lines will show.
Example
2 C a l c u l a t i o n of d y n a m i c c o e f f i c i e n t s . R, I, T
Cold water (0 = IOO0 k g / m 3, n = O.001 k g / m s ) runs at a rate of O.417 k g / s (1.5 m3/h) t h r o u g h a p i p e of inner d i a m e t e r 25 mm a n d l e n g t h a) 100 m b) 1000 m. The p i p e is smooth, with two e l b o w s a n d control v a l v e NO 25 in w h i c h the p r e s s u r e d r o p for the g i v e n flow is 0.5 bar. The flow is not f o r c e d b y p u m p but is m a i n t a i n e d b y Determine time c o n s t a n t s T.
pressure
difference
between
the
tanks.
a) L = I00 m, A = d 2 ~ / 4 - 0 . 0 0 0 5 m 2 T
L =
(2.2-1.28)
ACR~+R~+IRpJ)
The a b o v e e x p r e s s i o n s h o w s that it is still n e c e s s a r y to d e t e r m i n e the r e s i s t a n c e s . As there is n o p u m p , Rp --- O. From (2.2-1.13) Rv = 2 ~ r s v m
2.3981
bar kg/s
For the p i p e a n d the built-in e l b o w s formula for p r e s s u r e d r o p c a l c u l a t i o n
there
is
the
already
known
ssc. 2.2
73
pw 2
~Pc = (x~Ld ~ ~ ) ~
X~÷~ 2A2p
=
Kc is the coefficient u s e d expression
for ~Pc is the
Reynolds n u m b e r
(2.2-1.29)
m2
in Equation (2.2-1.8). The
coefficient
X. which
for a smooth pipe a n d
unknowlL
depends
only
in the on
the
is calculated from the Blasius
formula Re = __wd~ = 0.85.0.025.100
= 21250
> 2320
O.001 x = The
0.3164
= 0.O261
individual
resistances
coefficient
for t h e
elbow
is 0.2. a n d
can
be
written as follows
0.0261 ~ Kc =
]00
= 217466
2. 1 0 0 0 . 0 . 0 0 0 5 2
~Pc = 2 1 7 4 6 6 Referring
~z
,
N
= 37815 m---7 -
to (2.2-1.15) w e h a v e
Rc = 2~Pc m
_ 2. 37815 O.417
Now we can T=
+ 2.0.2
= 1.814 ba----Zr kg/s
g e t f r o m (2.2-1.28)
L ACRv÷Rc)
I 0 0 . IO -5 0.0005(2.398÷1.814)
= 0.474 s
b) L - I000 m. A = 0.0005 m 2 The
values
for Rp a n d
Rv d o
not change,
as follows
0.0201 Kc =
I000 + 2" 0.2 0.025
= 20.9,105
,
2" 0 . 0 0 0 5 2 . 1 0 0 0 ~Pc = 363219 N m2
. Rc
2~c m
17.42 bar kg/s
and
Rc a n d
T are
calculated
74
CHAP.
T
lO00
=
=
0.0005(2.3981÷17.42)I0
1.009
2 LUMP'~D
PRCbCI:;'c'-~EcI
s
s
Therefore, a l t h o u g h the l e n g t h
of
the p i p e i n c r e a s e d I0 times, the time
c o n s t a n t T is o n l y t w i c e b i g g e r . T h e r e a s o n for this is that as l e n g t h L i n c r e a s e s ( i n c r e a s e d inertia} so d o e s also the r e s i s t a n c e of the p i p e
Rc,
w h i c h d e c r e a s e s the g r o w t h of the time c o n s t a n t T.
]Example
3 C a l c u l a t i o n of d y n a m i c c o e f f i c i e n t s . R. I. T
Water flows at 3 m/s t h r o u g h a p i p e 300 mm in d i a m e t e r a n d 300 m l o n g . The relative r o u g h n e s s of the p i p e is d d = 0.002, a n d the kinematic v i s c o s i t y of water v = 9.10 -7 m2/s, p = 1000 k g / m 3. C a l c u l a t e the time c o n s t a n t T. As there is n o p u m p a n d v a l v e in the p i p e T
z
d
I Rc
~
L ARc r e s i s t a n c e Rc w e must first c a l c u l a t e the pipe, A = 0.0707 m 2. m = Awp = 212 k g / s , c o e f f i c i e n t x for r o u g h p i p e s c a n b e d e t e r m i n e d
To d e t e r m i n e the p i p e pressure drop a l o n g the Re = w d / v = 106. the
either from the Altshul or the M o o d y formula 1
Moody:
o.oo55[i (2oooo@.
)3 ] =o.o245
68,0.25 ), = 0.11 -a{-':'-+"~~ = 0.02346
Altshuh
For this field of Re a n d relative r o u g h n e s s the d i f f e r e n c e is only 4Z, a n d in the further c a l c u l a t i o n s we will u s e the v a l u e X = 0.02346. L Kc
= x d
1 2Ato
= 2.346
S~c = Kc~2 = 2.346" 2122 = 105438 Rc = 25Pc m
T
2" 105438 212
300 =
0.0707. 994.7
=
994.7
4.265 s
Pa
Pa kg/s
75
SEC. 2.2
2 . 2 - 2 GAS A N D STEAM FLOW
As in the c a s e with liquids, h e r e w e will a l s o b e g i n b y m a k i n g as m a n y a s s u m p t i o n s a s p o s s i b l e to o b t a i n a s i m p l e d y n a m i c m o d e l for the flow r a t e a n d p r e s s u r e in g a s p i p e s . If we p r u n e the p h y s i c a l p r o c e s s , in this w a y . of p h e n o m e n a that a r e at p r e s e n t s e c o n d a r y , insight into its b a s i c d y n a m i c p r o p e r t i e s will b e e a s i e r . T h e a n a l y s i s that follows is true of r e l a t i v e I y short p i p e s t h r o u g h w h i c h g a s ( s t e a m ) flows with the u s u a l t e c h n i c a l v e l o c i t i e s of flow ( m u c h smaller than critical), a n d a l o n g w h i c h p r e s s u r e d r o p r e s u l t i n g from v a r i o u s r e s i s t a n c e s n e v e r e x c e e d s 10% of the p r e s s u r e . In this c a s e p r e s s u r e d r o p c a n a l w a y s b e c a l c u l a t e d as in the c a s e of liquid. T h e s e a r e the introductory
assumptions
that
will b e
valid
for
the
complete
following
analysis. Unlike i n the c a s e of liquids, c h a n g e s in g a s d e n s i t y 0 © a n n o w n o t be neg|e©ted, but b e c a u s e of low g a s ( s t e a m ) d e n s i t y t h e i n e r t i a o f the gas (steam) mass in the pipe can be neglected. Therefore. we l e a v e inertia e f f e c t s out of our a n a l y s i s (I = 0), a n d t a k e into c o n s i d e r a t i o n p o s s i b i l i t i e s of m a s s s t o r a g e in the p i p e b e c a u s e of the c o m p r e s s i b i l i t y of the m e d i a (p ~, const., C ,, const.). As in the c a s e of liquids, g a s flow a l s o h a s r e s i s t a n c e s a l o n g the p i p e : v a l v e s , f l a n g e s a n d the like. From the a s p e c t of g a s s t a t e c h a n g e s in the p i p e , w e will a s s u m e h e r e the c a s e of i s o t h e r m a l c h a n g e . T h e r e is n o s p e c i a l difficulty if w e want to c o n s i d e r p o l y t r o p i c c h a n g e . In that c a s e w e must s i m p l y insert in all the e q u a t i o n s , i n s t e a d of the i s o t h e r m a l c a p a c i t a n c e C, the p o l y t r o p i c c a p a c i t a n c e Cn. As in the c a s e of the tank. h e r e a l s o we will u s e the s t a t e e q u a t i o n s for i d e a l g a s .
Example
I D y n a m i c s of g a s or s t e a m flow. B o u n d a r y c o n d i t i o n s
A c o m p r e s s o r (air p u m p ) d r i v e s m k g / s of g a s t h r o u g h a c o n t r o l v a l v e b e t w e e n two tanks. D e r i v e the m a t h e m a t i c a l m o d e l d e s c r i b i n g the d y n a m i c c h a n g e s in m a s s flow a n d p r e s s u r e in the p i p e .
7(5
CHAP.2
LUIMPED
PRd~CES~ES
...~..~.~.'..~.....~.~...~....~.~...~:.~j~;..~..~.~...~.~.~.;..:..~+.~:.:.~
~,~."---".-'.:::'.2~ p
I
"~'I
'
~---::.~I
]~p,
'
q Fig.
T
Z
2 . 2 - 2 . 1 G a s (steam) t r a n s p o r t b e t w e e n two tanks
T h e p i p e h a s two p r o p e r t i e s . T o g e t h e r with all the fittings a n d the p u m p ; it resists g a s flow. This r e s i s t a n c e is c h a r a c t e r i z e d b y total r e s i s t a n c e R. It a l s o e n a b l e s s t o r a g e of a m a s s of g a s , i.e. it h a s t h e p r o p e r t y of c a p a c i t a n c e C. Thus this p i p e c a n b e r e p r e s e n t e d a s a c o n n e c t i o n in s e r i e s of r e s i s t a n c e a n d c a p a c i t y , w h i c h h a s b e e n d o n e on the following two figures.
Fig.
2 . 2 - 2 . 2 P i p e a p p r o x i m a t i o n in
m,I
-
Im, l
-
R-C
series
Ira,
]FIG. 2.2-2.3 P i p e a p p r o x i m a t i o n in C-R s e r i e s
s ~ c . 2.2
77
In the u p p e r figures, R r e p r e s e n t s all the p o s s i b l e r e s i s t a n c e s that i n f l u e n c e p r e s s u r e c h a n g e o n the o b s e r v e d l e n g t h of p i p e L, a n d C r e p r e s e n t s the total c a p a c i t a n c e of the w h o l e s y s t e m . Although this d o e s not s e e m i m p o r t a n t at first g l a n c e , we must n e v e r t h e l e s s e m p h a s i z e the following. T h e o r d e r in the s e r i e s i s n o t a w b i t , , a r y but d e p e n d s on the g i v e n b o u n d a r y c o n d i t i o n s at the e n d s of the p i p e . T h a t this is s o will b e s h o w n in the following lines w h e n we formulate e q u a t i o n s for b o t h the a r r a n g e m e n t s . As the p o s s i b i l i t i e s of m a s s s t o r a g e a r e not n e g l e c t e d , w e will g e t the d e s i r e d d y n a m i c m o d e l s b y formulating equations for the conservation of mass and
momentum,
and
in the latter w e
will neglect the m o m e n t u m
of the m a s s
of g a s in the pipe.
For the R-C o r d e r o n F i g u r e 2.2-2.2 w e h a v e m!
PI
-
+
m 2
=
- -
dM dt
=
~Pp
-
~Pv
-
V
dP2
IR8
dt
~Pc
-
P2
=
(2.2-2.1)
d(Mw) =0 dt
(2.2-2.2) (2.2-2.3)
PI " P2 = SPy + ~Pc " SPp
After w e n e g l e c t the i n f l u e n c e of c h a n g e s in the n u m b e r of r e v o l u t i o n s of the c o m p r e s s o r a n a n d c h a n g e s in the c r o s s - s e c t i o n a l a r e a of the v a l v e AAv ( w h i c h d o e s not m a k e us l o s e a n y t h i n g e s s e n t i a l but o n l y m a k e s it e a s i e r to d e r i v e the e q u a t i o n s ) , the l i n e a r i z a t i o n of the u p p e r equations gives am,
- Am 2 = C
dAP2
dt
(2.2-2.4)
AP, - A P 2 = (Rv ÷ Rc ÷ [Rp[)Am, - R A m ,
T h e last e q u a t i o n , e x c e p t for the a s s u m p t i o n s m a d e AAv = O, I = L/A = O). is i d e n t i c a l to E q u a t i o n (2.24.19).
(2.2-2.5)
here
(An : O,
For the C-R o r d e r we a n a l o g o u s l y g e t from F i g u r e 2.2-2.3 Aml - Am2
= cdAPI dt
(2.2-2.~)
78
CHAP.2 LLIIvfP'~'D PROCESSES
APt - ~P~ = Rt~m2
(2.2-2,7)
The s y m b o l s C a n d R h a v e
the s a m e
meaning
as
in the p r e c e d i n g
chapters C =
V
~
~,~
-~-
R --
Rv
÷ Rc + IRp] =
25~v
m
2~i~c
m
E q u a t i o n s (2.2-2.4) - (2.2-2.7) are pairs of i n t e r c o n n e c t e d e q u a t i o n s with four v a r i a b l e s - aml. am2. APt a n d aP2. To s o l v e t h o s e pairs of e q u a t i o n s it is n e c e s s a r y to k n o w two v a r i a b l e s , a n d then it is no p r o b l e m to d e t e r m i n e the other two. For fluid flow p r o c e s s e s we c a n m a k e a n arbitrary s e l e c t i o n of i n d e p e n d e n t b o u n d a r y c o n d i t i o n s (inputs). e x c e p t that m a s s not
be
flow given
rate at
the
and
(both b o u n d a r y c o n d i t i o n s ) m u s t
pressure
same
end,
at
the
lame
( W e will s a y
time.
more a b o u t b o u n d a r y c o n d i t i o n s in C h a p t e r 3, w h e n we talk a b o u t partial differential e q u a t i o n s . For the p r e s e n t it is e n o u g h to s a y that if both the b o u n d a r y c o n d i t i o n s a r e g i v e n at the s a m e e n d , the other e n d
remains
c o m p l e t e l y u n d e t e r m i n e d b e c a u s e there is no information a b o u t the w a y in w h i c h that e n d of the p i p e " c o n t a c t s " the s u r r o u n d i n g s ) . F i g u r e 2.2-2.4 s h o w s the four p o s s i b l e c a s e s of b o u n d a r y c o n d i t i o n s that c a n a p p e a r in p r a c t i c e . The most h e q u e n t a n d most i n t e r e s t i n g are e x a m p l e s a) a n d b). a n d h e r e we will a n a l y z e t h e s e c a s e s in more detail.
z~rn~ =
. I
I _ z~rnz I-
z~,~ ]
Q)
~-n!
,,,m z -
b)
,~,ml
,*,m z
c)
d)
] F i g . 2 . 2 - 2 . 4 P o s s i b l e b o u n d a r y c o n d i t i o n s in fluid flow The following d e r i v a t i o n s will s h o w the truth of the s t a t e m e n t that in a d y n a m i c r e p r e s e n t a t i o n the o r d e r of r e s i s t a n c e d e t e I m i n e d b y the g i v e n b o u n d a r y c o n d i t i o n s .
and
capacity elements
is
SEC.
79
2.2
a ) LXP1 a n d
-
&m z a r e g i v e n . D e t e r m i n e
R-C o r d e r . F i g u r e
2.2-2.2
AP a a n d
&ml.
.
E q u a t i o n s (2.2-2.4) a n d (2.2-2.5) g i v e T
d a m t dt
+ Amt
=
_
Am2 * C
_ dd& tPt
'
(2.2-2.8)
T d&P2 dt + AP2 = " RAmz * AP t
(2.2-2.Q)
- C-R o r d e r . F i g u r e 2.2-2.3 . Equations
(2.2-2.b) a n d (2.2-2.7) g i v e
&mr = &m2 + C
d/xPt dt
'
(2.2-2.10)
AP2 = -RAm2 ÷ &P, .
(2.2-2.11)
T --CR T h e l a s t four e q u a t i o n s 2.2-2.4a, d e p e n d i n g completely
show
different models
is o b v i o u s l y
that the
boundary
o n t h e o r d e r of r e s i s t a n c e for p r e s s u r e
not satisfactory
lost.
Figure
a n d flow d y n a m i c s .
T h e C-R o r d e r
in it t h e
and
conditions
we can draw similar conclusions.
-
are
on
elements, give
Td&mt/dt
b ) Am, a n d
Td~Pz/dt
because
conditions
and capacity
For
LXP2 a r e g i v e n . D e t e r m i n e
the
components
second
Am2 a n d
pair
of t i m e of
boundary
aP,.
R-C o r d e r , F i g u r e 2.2-2.2 .
Am~ = .~mt - C d A P 2 dt
(2.2-2.12)
APt = R a m l + AP~ .
(2.2-213)
-
lag,
C-R o r d e r . F i g u r e 2.2-2.3 .
d&m2 = T --~ * Am2 &mr - C T daP, dt
+ apt = Ramt+
N o w t h e R-C o r d e r d o e s
ddt &Pz
(2.2-2.]4)
z~P2 not reproduce
(2.2-2.15) the dynamics
faithfully enough.
BO
CHAP.2
b e c a u s e time l a g e l e m e n t s h a v e d i s a p p e a r e d Therefore, and
when
we
approximate
c a p a c i t y , w e must take
pressure change
care
a
LUMPED
PROCK~S~S
from its d y n a m i c m o d e l .
pipe
with e l e m e n t s
of the b o u n d a r y
of r e s i s t a n c e
c o n d i t i o n s . Where a
is g i v e n we must put a n e l e m e n t of r e s i s t a n c e , a n d on
the e n d w h e r e flow rate is g i v e n we put c a p a c i t y . In p r a c t i c e we often u s e the solution that the total p i p e r e s i s t a n c e is d i v i d e d into input a n d o u t p u t p i p e r e s i s t a n c e with a c a p a c i t y e l e m e n t in b e t w e e n . This a r r a n g e m e n t is s h o w n o n F i g u r e 2.2-2.5.
I:{:!:.:.{:!:|
m,
m2
Fig.
"~--
2 . 2 - 2 . 5 Pipe a p p r o x i m a t e d with input a n d o u t p u t r e s i s t a n c e a n d capacity element
The e q u a t i o n for the c o n s e r v a t i o n of mass a n d two e q u a t i o n s
for the
c o n s e I v a t i o n of m o m e n t u m for the o r d e r s h o w n o n F i g u [ e 2.2-2.5, in linear form, a r e as follows Aml - Am~
= C dAPs dt
APt - ~ P s = R ~ A m ,
APs
'
,
aP2 : R~am2
c} A ml a n d AP2 a r e g i v e n . Determine ~m2 a n d AP I. - RI - C - R2 o r d e r , F i g u r e 2.2-2.5 . R e a r r a n g i n g the last t h r e e e q u a t i o n s we g e t
(2.2-2.16) (2.2o2.17)
(2.2-2.18)
2.2
SEC.
81
dams
T~
~
daPt
T2
=
.
Aml
C dAP~dt
dAml
~-~
T2
=
+ Am2
+ a P , = R,T2 ~
(2.2-2.19)
+ (R, + R2)am, + ~P2
(2.2-2.20)
R2C
If w e
compare
the
last
two
equations
with
Equations
(2.2-2.15), w e s e e t h e f o l l o w i n g d i f f e r e n c e s . T i m e c o n s t a n t T, a n d t h e r e is a l s o a d e r i v a t i v e rate
ml. W e
thus
Resistance
have
division
something
a
is
dependence
property
already
that a
t h a t is in f a c t d i s t r i b u t e d
book lumped
in o n e
place.
Thus
lost
along
in
the
towards
and pressure
P1 o n m a s s f l o w R-C
combination.
distribution,
towards
t h e p i p e , b u t in this p a r t of t h e
we must believe
that the approximation
in F i g u r e 2.2-2.5 is b e t t e r t h a n t h a t in F i g u r e 2.2-2.2, a n d form of t h e l a s t t w o e q u a t i o n s
and
T2 is s m a l l e r t h a n
of p r e s s u r e
is
step
(2.2-2.14)
the model
c l o s e r to t h e r e a l b e h a v i o r
in t h e
of flow r a t e ",m2
APt.
d ) AP, a n d
A m z a r e g i v e n . D e t e r m i n e AP 2 a n d
Am1.
- o r d e r Rt - C - Rz. F i g u r e 2.2-2.5 . After r e a r r a n g e m e n t
E q u a t i o n s (2.2-2.16) - (2.2-2.18) g e t t h e d e s i r e d
Ti d A ~ m I ÷ Aml = Am2 + C d dAt P t P2 Tt d A ~-~
'
form
(2.2-2.21)
+ AP2 = APt _ (R~Tt d a m s
÷ (R, + R2)Am2)
(2.2-2.22)
Tl = RIC N o w T, is s m a l l e r t h a n T in E q u a t i o n s c) t h e r e change
is a
derivative
(2.2-2.8) a n d
of p r e s s u r e
(2.2-2.Q), a n d
P2 o n
mass
l i k e in
flow
rate
feeling
that
mr.
Here
we
must
approximation resistance elements there
dependence
are
elements
will
and
completely
be
better
capacity
would no
be
increase
exact
specific
we
elements, the order
analytical
pipes
if
clear
should
divide but
the
express the
about
divided.
our
pipe
into
addition
of t h e s y s t e m .
criteria be
and
when
of
We must and
In t h e s e
even
new also
into
more
capacity say
how
contemplations
that many we
~'~
CHAP.2
LUMPED
PROCESSES
c a n u s e the fact that time c o n s t a n t s l i n k e d with g a s a n d s t e a m flow are small a n d in p r a c t i c e r o u g h e r a p p r o x i m a t i o n s (R-C or C-R), or e v e n statical relations, will satisfy. What h a s just b e e n s a i d is true u n l e s s we h a v e p e r i o d i c (oscillatory) p r o c e s s e s , w h i c h a r e not i n c l u d e d in this m o d e l b e c a u s e w e n e g l e c t e d inertia (I = 0). Finally, it is also useful to r e p r e s e n t the d y n a m i c relations b e t w e e n input a n d o u t p u t v a r i a b l e s in the m o d e l g i v e n b y E q u a t i o n s (2.2-2.21) a n d (2.2-2.22) b y matrix transfer functions.
Ts]i
1
AP2(s)
- R Tls*I - -
Tls*l =
Cs T,s+I
AM1Cs)
Y(s)
=
API(S)
._/___1 Tis*l
(2.2-2.23)
AM2(s)
O(s) U(s)
R = RI + R2
. T -- R2 R
T~
The u p p e r e x p r e s s i o n , if we put R1 = R a n d R2 ; O, from w h i c h follows T = O. is a matrix p r e s e n t a t i o n of c a s e at in this e x a m p l e .
Before w e d e t e r m i n e the n u m e r i c a l v a l u e of the time c o n s t a n t for a n air p i p e in the following e x a m p l e , we must s a y that t h e r e is a c o m p l e t e a n a l o g y b e t w e e n the d y n a m i c s of a g a s tank a n d a p i p e t h r o u g h w h i c h g a s flows if t h e r e are v e r y small p r e s s u r e d i f f e r e n c e s in the p i p e a n d if the flow is far from critical. With r e g a r d to the v a l u e of the time c o n s t a n t , it will b e smaller than the c o n s t a n t that c h a r a c t e r i z e d
the g a s
or s t e a m
tank d y n a m i c s , not o n l y b e c a u s e of the u s u a l l y smaller c a p a c i t y but also b e c a u s e of the m u c h smaller r e s i s t a n c e the p i p e offers to flow.
E x a m p l e 2 C a l c u l a t i o n of d y n a m i c coefficients. R, C, T Air of d e n s i t y p = 1.16 k g / m 3 a n d kinematic v i s c o s i t y v = 15.7'I0 -6 m2/s flows at a v e l o c i t y of 20 m/s t h r o u g h a p i p e length
100 m. The
pressure
in the
pipe
of d i a m e t e r
equals
external
200 rum a n d pressure
and
sEc. 2.2
83
temperature constant
e = 0 *C, t h e
pipe
T that characterizes
roughness
the process
is ~ = 0.I mm. D e t e r m i n e of flow a n d
pressure
T will b e
determined
time
change
in
only
by
the pipe. As t h e
pipe
resistance For
has
no
to flow t h r o u g h
flow
calculation
through
x is c a l c u l a t e d
"
X
the
Rc
=
pipe
and
drop
for
due
small
pressure
to f r i c t i o n in t h e
of liquid. The calculation
differences pipe
runs
the
in t h e
i t s e l f f o l l o w s , in w h i c h
from t h e A l t s h u l f o r m u l a .
L
m~
2A2~
d
or v a l v e ,
t h e p i p e . T = RcC.
of t h e p r e s s u r e
s a m e w a y a s in t h e c a s e
~Pc
pump
=
Kc ~
2 ~iPc m
Re
= wd
=
20.0.2
= 0,255
lO s
15.7" 10 -5 A = d2~ - 0,0314 m 2 4 m
=
Awp
=
0.7285 kg
.
S
68 )0.25 X = 0.11 ( - ~ • Re = 0.0174
0.2
.
N m z
N/m 2 kg/-----~
Rc = 5533.8 C . __V t~)
0.72852 - 2015.68 2 ' 0 . 0 0 3 1 4 ~'1.16
I00
SPc = 0.0174
i
3.14 287.2(~2
= 3.747.10-s kg m 2 N
T = Rc C = 0.2073 s
2 . 2 - 3 FLUID FLOW
In t h e p r e v i o u s were
obtained
application
and
analysis
after
of l i q u i d , g a s
assumptions
values.
However.
that those
and
steam dynamics
narrowed models
clown made
the models
their sense
field and
of can
~4
CHAP.
answer
questions
concerning
basic
dynamic
2 LUMPED
processes
PROCESSES
for
very
many
devices a n d plants. Liquids, h o w e v e r , d o p o s s e s s the p r o p e r t y of c o m p r e s s i b i l i t y , w h i c h w a s n e g l e c t e d in 2.2-1 b y m a k i n g C = O. T h e consequences of that p r o p e r t y a r e the k n o w n p h e n o m e n a of p e r i o d i c o s c i l l a t i o n s of the liquid c o l u m n , i.e. w a t e r h a m m e r effects. G a s e s . in s p i t e of their small d e n s i t y p, n e v e r t h e l e s s p o s s e s s kinetic e n e r g y , a n d m o m e n t u m (Mw) d u r i n g their flow, w h i c h w a s m a d e e q u a l to z e r o in 2.2-2, c a n n o t a l w a y s b e n e g l e c t e d . In short, it is d e s i r a b l e to d e r i v e a m o d e l for d y n a m i c c h a n g e s of m a s s flow r a t e a n d p r e s s u r e in a p i p e for a n arbi{rar F fluid, not n e g l e c t i n g either its c o m p r e s s i b i l i t y or its m o m e n t u m , i.e. the kinetic e n e r g y a flowing m a s s of fluid p o s s e s s e s . This will b e d o n e in the following lines, a n d it will b e s h o w n that the p r e c e d i n g m o d e l s a r e o n l y s p e c i a l c a s e s of the o n e o b t a i n e d for a n a r b i t r a r y fluid with p r o p e r t i e s of c a p a c i t a n c e (C), r e s i s t a n c e (R) a n d i n e r f a n c e (I). Also. w h e n the p r o c e s s e s of fluid flow a r e later r e g a r d e d a s p r o c e s s e s with d i s t r i b u t e d p a r a m e t e r s in S e c t i o n 3.3, w e will g e t m o d e l s w h o s e first a n d r o u g h e s t a p p r o x i m a t i o n will b e e q u a l (or, at least, similar) to the e q u a t i o n s d e r i v e d h e r e , for w h i c h all the p r o p e r t i e s w e r e c o n s i d e r e d l u m p e d in o n e p o i n t in s p a c e .
Example
I Dynamics of fluid flow. Periodic processes
T h r o u g h a p i p e with solid walls, of l e n g t h L a n d c r o s s - s e c t i o n a l a r e a A. flows m k g / s fluid at a v e l o c i t y w, d e n s i t y p a n d t e m p e r a t u r e a. l) D e r i v e the linear m o d e l for m a s s flow r a t e a n d p r e s s u r e c h a n g e s the inlet a n d outlet c r o s s - s e c t i o n for the following b o u n d a r y c o n d i t i o n s :
at
a) AP, a n d Am 2 a r e g i v e n , b) AP2 a n d Am1 a r e g i v e n . 2) For c a s e a) d e r i v e a m o d e l in the form of matrix e q u a t i o n s , a n d a matrix transfer function r e p r e s e n t a t i o n . la) For the given input variables the pipe can be arrangement of elements as presented on Figure 2.2-3.1.
state-space
shown
by
an
St=C. 2.2
85
u,=A,. ml
Fig.
R
i
-i
i
I
P=
-
Imll-
r:
.
I
P, -_ uz=mz
2 . 2 - 3 . 1 Fluid flow with the p r o p e r t i e s of r e s i s t a n c e , inertance and capacitance
Here also, a n a l o g o u s l y to the earlier p r o c e d u r e s , e q u a t i o n s for the c o n s e r v a t i o n of mass a n d m o m e n t u m m, - m2 =
dM d(Vp2) d----~ = dt -
V R.3.
(P, * ~Pp" ~Pv - ~Pc " P2) A =
dP2 dt
we
formulate
(2.2-3.11
d(Mw,) dt
dwl dmt = ALp-..--:--- = L dt dt
(2.2-3.21
T h e u p p e r e q u a t i o n s a r e n o n l i n e a r a n d if t h e y a r e l i n e a r i z e d taking ,~n - O, dAy = 01, w e g e t the following two e q u a t i o n s Amt - Am2
= C dAP2 dt
(2.2-3.31
AP, - ~Pz = RAm, ÷ L A
I
(again
d dt ~ml
(2.2-3.41
L A
=--
In the g e n e r a l c a s e R = Rc + Rv ÷ JRph c o m p a r e the d e r i v a t i o n s for liquid flow from S e c t i o n 2.2-1. T h e u p p e r two e q u a t i o n s g i v e the d e s i r e d model in the following form IC d 2 A P 2
dt 2
* RC
IC d2~ml
+ RC
dt 2
For
the
equations
first
dAP2
dt
* ~P2
=
API
-
(I d•m2 dt
+ R~m2)
d~ml d~P1 ~ * am, - C ~ + z~m2
time
in
this
of the second
book order.
we
have
(2.2-3.5) (2.2-3.6)
obtained
differential
T h e r e a r e thus t w o p o s s i b l e m a s s
or energy storage elements. Truly, fluid is c o m p r e s s i b l e a n d fluid mass c a n b e s t o r e d in the g i v e n v o l u m e V. This p r o p e r t y is r e p r e s e n t e d
a~
by
CHAP.
element
C on Figure
2.2-3.1. But
fluid
mass
2 LUMP~'D
storage
also
PROCESSES
represents
p o t e n t i a l e n e r g y s t o r a g e , as will b e s e e n from further d i s c u s s i o n . E q u a t i o n (2.2-3.1) c a n a l s o b e written m,
-
m~
-
d(VPz) dt
1 R~
(2.2-3.7)
T h e term VP~ h a s the d i m e n s i o n of e n e r g y Nm a n d is a m e a s u r e for p o t e n t i a l e n e r g y c h a n g e in the p i p e v o l u m e V r e s u l t i n g from p r e s s u r e c h a n g e P2Similarly, a n a n a l y s i s of the e q u a t i o n for the c o n s e r v a t i o n of m o m e n t u m i n d i c a t e s c h a n g e s in kinetic e n e r g y . If w e i n t r o d u c e n e w s y m b o l s . (2.2-3.2) c a n a l s o b e written ~PA = F = Ivl dw-----L dt
(2.2-3.8)
~P d e n o t e s the total p r e s s u r e s d i f f e r e n c e a c t i n g o n the fluid m a s s in the p i p e . a n d F the f o r c e that c a u s e s the d i s p l a c e m e n t of that fluid. If the fluid flows at the v e l o c i t y wl. t h e n it p a s s e s the p a t h d z = w~ dt in a time p e r i o d dt. If the last e q u a t i o n is m u l t i p l i e d b y d z , w e g e t dw, I Fdz = M ~ w~dt = M(dwl) wl = -~- Md(wl 2)
(2.2-3.9)
Let the fluid m a s s , w h i c h is in p o s i t i o n Zo at the m o m e n t to. m o v e into p o s i t i o n z u n d e r the i n f l u e n c e of f o r c e F, a n d let its v e l o c i t y c h a n g e s from w,0 to wl. I n t e g r a t i o n of the u p p e r e q u a t i o n g i v e s Z
j'Fdz Zo
= ~U --
w I
j'd(w$)
(2.2-3.10)
Wl0
W = Mw~2 Mwl°2= 2 " 2 AEk
(2.2-3 .ll)
T h e i n t e g r a l o n the l e f t - h a n d s i d e is work W that the p r e s s u r e f o r c e s w o u l d r e a l i z e if m a s s M m o v e d from Zo to z, a n d that w o r k w o u l d b e t r a n s f o r m e d into kinetic e n e r g y . In the c a s e of a p i p e of l e n g t h L, c h a n g e s in a n y of the p r e s s u r e s ( w h i c h r e p r e s e n t a c h a n g e in f o r c e F that a c t s on the fluid) d o not c a u s e the m a s s of fluid M to m o v e . b u t l e a d to the c o m p r e s s i o n or e x p a n s i o n of the fluid in p i p e v o l u m e V. This, in turn. l e a d s to a n e n e r g y - f o r m c h a n g e - k i n e t i c into p o t e n t i a l for c o m p r e s s i o n a n d p o t e n t i a l into kinetic for e x p a n s i o n .
sEc. 2.2
87
It is not difficult to s e e that this is a n o s c i l l a t o r y ( p e r i o d i c ) p r o c e s s . Let m2 i n c r e a s e
by
~m~ ,, 1 at
t ,, O. From (2.2-3.7) p r e s s u r e
gradient
P~
b e c o m e s n e g a t i v e a n d P2 (the p o t e n t i a l e n e r g y of the fluid in the p i p e ) d e c r e a s e s . This m a k e s the total p r e s s u r e d i f f e r e n c e SP i n c r e a s e , a n d from (2.2-3.8)
the v e l o c i t y w! of the
fluid
at
the
entrance
increases
(i.e. its
kinetic e n e r g y i n c r e a s e s ) . B e c a u s e of an i n c r e a s e in m~ g r a d i e n t d ~ P 2 / d t remains n e g a t i v e , but its a b s o l u t e v a l u e d e c r e a s e s a n d e q u a l s z e r o at the moment w h e n m~ = m + Am2. i.e. for Am~ = Amz. The p r o c e s s d o e s not e n d here b e c a u s e p r e s s u r e P2 has d e c r e a s e d b y AP2 a n d ~P is still positive, c o n s e q u e n t l y the v e l o c i t y wl (the kinetic e n e r g y of the fluid) c o n t i n u e s to grow. Kinetic e n e r g y
will i n c r e a s e
until
to its initial
P2 returns
steady
value. At that moment ~P ; ~ , or ,~sP -- 0 a n d d(Aw~)/dt = O. H o w e v e r , the p r o c e s s d o e s not e n d h e r e either, b e c a u s e now ml k g / s of fluid flows in from the o u t s i d e with maximum v e l o c i t y w~ ; ~ ÷ Aw~. S i n c e m~ > m~, a c c o r d i n g to (2.2-3.7) we h a v e an i n c r e a s e in Pz. This p r o c e s s c o n t i n u e s to r e p e a t itself until friction r e d u c e s e n e r g y c h a n g e s to z e r o a n d Am~ b e c o m e s e q u a l to ~mz. Figure variables
2.2-3.2 g i v e s a q u a l i t a t i v e p r e s e n t a t i o n of c h a n g e s in Am1 a n d AP2 in zeal c o n d i t i o n s (friction p r e s e n t ) for
the th'e
d e s c r i b e d c a s e of i n c r e a s e in fluid c o n s u m p t i o n in the outlet s e c t i o n for Am2 ; 1 k g / s . F i g u r e 2.2-3.3 g i v e s the c o u r s e of the s a m e v a r i a b l e s in the c a s e of p r e s s u r e i n c r e a s e riP1 = 1 bar at the p i p e inlet.
•
_ o
.
,
~TI!
.
.
.
.
.
.
.
.
.
.
.
.
.
~u2=1
,,~ =0
F i g . 2.2-3.2 P e r i o d i c c h a n g e s in mas flow rate at p i p e inlet nml a n d p r e s s u r e at p i p e outlet APz in the c a s e of i n c r e a s e d fluid c o n s u m p t i o n Amz = 1 k g / s
~8
t~HAP. 2 L U m P e D
PROCESSES
z~
am2=0
rig'.
-t
2 . 2 - 3 . 3 P e r i o d i c c h a n g e s in m a s s flow r a t e at p i p e inlet Am1 a n d p r e s s u r e at p i p e outlet Z~P2 in the c a s e of i n c r e a s e d inlet p r e s s u r e API
T h e following shorter d e r i v a t i o n s h o w s that if t h e r e is n o |wi©|ion, i.e. no e n e x g y loss. s t a t e c h a n g e s of fluid in a p i p e r e a l l y c o n s e r v e the e n e r g y (it o n l y c h a n g e s in form) the fluid c o n t a i n s within it. With R ; O, E q u a t i o n s (2.2-3.3) a n d (2.2-3.4) b e c o m e Am,-
A m 2 = C d dt AP2
'
aP, - AP~ = I d A m l dt
(2.2-3.12) (2.2-3.13)
If w e d i v i d e the u p p e r e q u a t i o n s o n e b y the o t h e r w e g e t Am,Ap,-
Am2 Ap 2
=
C l
dAP2 dam,
(2.2-3.14)
If o u t s i d e d i s t u r b a n c e s d o not i n f l u e n c e fluid in the p i p e . i.e. with Am2 = 0 a n d AP, = O. (2.2-3.14) c a n a l s o b e written !
C
d(~m$) + ~
d ( a P 2 2) ., 0
(2.2-3.15)
In the last e q u a t i o n w e d i v i d e b o t h s i d e s b Y fluid d e n s i t y ~ in the s t e a d y state. This, of c o u r s e , d o e s not c h a n g e the result, but l e a d s to the d e s i r e d form for the final e x p r e s s i o n . From that e q u a t i o n it follows d i r e c t l y that
89
sEc. 2.2 Ipam~ 2
*
CpAP2 ~ 2 ,, const '-----'
KE
(2.2-3.16)
,, Eo I
PE
C
(I ~ -- 7- ' c~ - T )
T h e u p p e r e q u a t i o n is the law for the c o n s e r v a t i o n of e n e r g y . A d i m e n s i o n a l a n a l y s i s s h o w s that b o t h the terms o n the l e f t - h a n d s i d e h a v e the d i m e n s i o n of e n e r g y (Nm), the first term r e p r e s e n t i n g kinetic e n e r g y c h a n g e , a n d the s e c o n d p o t e n t i a l e n e r g y c h a n g e in the fluid. In this frictionless c a s e the p e r i o d i c r e s p o n s e s s h o w n in F i g u r e s 2.2-3.2 a n d 2.2-3.3 would not b e d a m p e d a n d o s c i l l a t i o n s w o u l d c o n t i n u e , s i n c e the initial e n e r g y Eo w o u l d a l w a y s b e p r e s e r v e d in the fluid, with u n d a m p e d a m p l i t u d e s until infinity. E q u a t i o n s (2.2-3.5) a n d (2.2-3.6) a r e the most g e n e r a l form of the m o d e l for fluid flow d y n a m i c s from w h i c h m o d e l s for g a s or s t e a m flow (I ~ O) or for fluid flow (C = O) a r e o b t a i n e d d i r e c t l y . If w e insert I = O - i n t o t h o s e two e q u a t i o n s we g e t E q u a t i o n s (2.2-2.8) a n d (2.2-2.9) from the s e c t i o n on g a s flow. If w e w a n t to o b s e r v e c h a n g e s in liquid flow d u e to inlet a n d outlet p i p e p r e s s u r e c h a n g e s , i n s e r t i n g C = O into (2.2-3.5) g i v e s us a m o d e l like in E q u a t i o n (2.2-1.19). lb) If Am, a n d AP2 a r e g i v e n , the following s e q u e n c e of e l e m e n t s m a k e s it e a s i e r to g e t a d y n a m i c m o d e l for fluid s t a t e c h a n g e s .
P,
_
.,=m,~
rig.
,..
P,
"- I m . - ]
,
'
o
I
-1
~'l
u,=~
m,
2 . 2 - 3 . 4 Fluid flow with the p r o p e r t i e s of c a p a c i t a n c e , inertance and resistance
Using the s a m e a s s u m p t i o n s as in c a s e E q u a t i o n s (2.2-3.3) a n d (2.2-3.4), w e c a n write
a),
and
analogously
with
(~E)
CHAP.
2 LUMPED
C d~_P, A m ~ - ~m2 = - dt
"
API
AP2 = RAm2
+
PRE)CEEEEE
(2,2-3,17) L
A
dam2
dt
(2.2-3.18)
R e a r r a n g e m e n t of t h o s e e q u a t i o n s g i v e s the d e s i r e d e x p r e s s i o n s I,.. d 2 a P , daPi ,,.~ + RC dt
dam1 ÷ aPI = AP2 + I T ÷ Ram1
icdSam2 + RC d adim 2 + Am 2 - Am~ - cdAP2dt
(2.2-319) (2.2-320)
2) To r e p r e s e n t the m o d e l from la) in the form of s t a t e - s p a c e e q u a t i o n s w e must first d e t e r m i n e w h i c h v a r i a b l e s a r e c o n s i d e r e d inputs a n d w h i c h o u t p u t s , a n d w h i c h r e p r e s e n t s t a t e v a r i a b l e s . Now this s e l e c t i o n is not difficult as it is c o m p l e t e l y n a t u r a l for the input v e c t o r to b e c o m p o s e d of aPi a n d Am2, a n d the s t a t e a n d o u t p u t v e c t o r of AP 2 a n d Am I. We will thus h a v e
X=Y=
U=
LAm~
L~m,J
In formulating m a t r i c e s A, B a n d C (D = 0 b e c a u s e
t h e r e is no d i r e c t
a c t i o n of U on Y), it i s b e s t to start from E q u a t i o n s (2.2-3.3) a n d (2.2-3.4) from w h i c h w e i m m e d i a t e l y g e t e x p r e s s i o n s for AP~' a n d Am,'. Written differently, t h o s e e q u a t i o n s a r e
o
1
[o
I I
R I
I
!
LAm~J X'
Lam, J
T
1 (2.2-3.21) Lamz_l
= A X ÷ B U
AP21
[
1 (2.2-3.22)
Lt, m,J Y=CX
0
1 Lamd
ql
S E C . 2.2
T h e e i g e n v a l u e s of matIix A point to transient p h e n o m e n a a n d it w o u l d b e useful to d e t e r m i n e e x p r e s s i o n s from w h i c h the}, c a n b e c a l c u l a t e d . From E q u a t i o n I),| - A I = 0 w e g e t I
)' 1
"-C
= X2 ÷
X + TC
+ RC~ + 1 = 0
= IC
(2.2-3.23)
R
T
~+T
It s h o u l d b e o b s e r v e d that the u p p e r e q u a t i o n is the c h a r a c t e r i s t i c e q u a t i o n of the ODE (2.2-3.5) a n d (2.2-3.b).
same
as
the
T h e transient functions from F i g u i e s 2.2-3.2 a n d 2.2-3.3 a n d the differential e q u a t i o n s (2.2-3.5) a n d (2.2-3.b) s h o w that in this c a s e w e h a v e the s t a n d a r d b e h a v i o r of a s e c o n d - o r d e r ( p r o p o r t i o n a l ) s y s t e m that shows p e r i o d i c c h a r a c t e r i s t i c s . We will, t h e r e f o r e , i n t r o d u c e s y m b o l s for natural oscillation f r e q u e n c y (~n a n d the d a m p i n g c o e f f i c i e n t ~. ~n = ~
1
RC
,
(2.2-3.24)
RC = - ~n
(2.2-3.25)
With t h o s e v a r i a b l e s the c h a r a c t e r i s t i c e q u a t i o n (2.2-3.23) c a n b e written in the u s u a l form for 2 n d - o r d e r p r o c e s s e s (2.2-3.2~)
),2 + 2{¢OnX + Un2 = 0
T h e e i g e n v a l u e s a r e d e t e r m i n e d b y the e x p r e s s i o n
R
XI'2 = " 2"7- ~
~i~ ( -R2C ~" ~-
l) " "F,(an * (an 1 / ~ 2 - ~ - -l = 0 t
(OJ (2.2-3.27)
Here
will
not
enter
features
because
the
key
solution
of
negative
we
the
tea]
upper part.
f r i c t i o n is n e g l e c t e d
into
a
propexties
equation.
For
o < O, b e c a u s e
more are real E~n
detailed
analysis
completely cases > O. In
X will
of
obvious always
theoretic
dynamic from have
cases,
the a
when
(R = 0), ~ e q u a l s z e r o a n d so d o e s , t h e r e f o r e , t h e z e a l
part o = O, i.e. )~l,z = ~nJ = ~:(IC)-°'S J. T h e eigenvalues are c o n j u g a t e - c o m p l e x pairs o n the i m a g i n a r y axis a n d the s y s t e m o s c i l l a t e s without d a m p i n g with f r e q u e n c y ~n. In the p r a c t i c a l l y most f r e q u e n t c a s e
Q~
~HAP.
'~ L U I ~ P X I D
PR(:DCI=~I=~
w e h a v e 0 < ~ ~ 1, a n d t h e s e are the a l r e a d y s h o w n d a m p e d Increasing
the
damping
coefficient
leads
to
a
decrease
oscillations.
in
oscillation
f r e q u e n c y , a n d in the c a s e of ~ , l b o t h mass flow a n d p r e s s u r e c h a n g e s are
no longer periodical.
The e x p r e s s i o n
under
the
square
root is no
l o n g e r i m a g i n a r y a n d the solution a r e two n e g a t i v e a n d real e i g e n v a l u e s X. This
boundary
case
when
the
periodic
character
of
the
transient
p r o c e s s e n d s . i.e. w h e n ~ ~ 1, is d e t e r m i n e d b y R =2I¢'~-C . Finally. as p r e s e n t a t i o n in the form of transfer functions has b o t h clarity and
conviction,
the
model
given
by
Equations
(2.2-3.213 a n d
(2.2-3.22)
s h o u l d b e t r a n s l a t e d into the c o m p l e x r e g i o n u s i n g the a l r e a d y m e n t i o n e d transformation G(s] = C ( s i - A ] -1 B , D . Matrices C, B a n d D h a v e a l r e a d y b e e n i n v e r s e matrix of matrix j(sl - A)J is u n k n o w n .
(sl-
A)-'
detCsI-A]=
,, a d j C s I - A ) detCsI-A)
,,
R s 2 • --~-s *
1 det(sl-A)
determined
R
1
s"i-
"E
"T
1
and
o n l y the
(2.2-3.28) s
I IC
(2.2-3.29)
The final e x p r e s s i o n , after the n e c e s s a r y multiplications d e m a n d e d
by
the u p p e r transformation for o b t a i n i n g G(s). is as follows
1 ICs2~RCs÷I
nM,(s)J
Cs "ICs2+RCs÷I
ICs2,RCs÷I 1
(2.2-3.303 L z~M2(sll
ICs2+RCs,I
YCs) - GCs) UCs) This is the most g e n e r a l form of matrix transfer functions G[s) for fluid flow d y n a m i c s . From it c a n e a s i l y b e o b t a i n e d the s p e c i a l c a s e s from the p r e c e d i n g two s e c t i o n s . If w e n e g l e c t fluid mass inertia (I = O), the u p p e r e q u a t i o n g i v e s matrix ~ s ) from E q u a t i o n (2.2-3.23]. into w h i c h w e must insert R1 -- R, Rz ; O, T ; O to m a k e it v a l i d for c a s e a) from E x a m p l e I. S e c t i o n 2.2-2. o n g a s a n d s t e a m flow. If, h o w e v e r , we n e g l e c t the possibility of fluid mass s t o r a g e (C = 03 in the last e q u a t i o n , we g e t AP2 = AP1 - (Is ÷ R]Alvfz
,
(2.2-3.313
sEc. 2.2
q3
~M, ~ AM~ ~ AM , 1 R ~M(s) ; Ts+---'~ (AP,(s) - AP2(s)) T
-.
(2.2-3.32) (2.2-3.33)
I R
T h e last e q u a t i o n is c o m p l e t e l y t h e s a m e a s t h e L a p l a c e t r a n s f o r m a t i o n of E q u a t i o n (2.2-3.20) from t h e s e c t i o n o n l i q u i d flow. w h e r e t h e p o s s i b i l i t y of m a s s s t o r a g e
was also neglected
because
we worked
with 0 = c o n s t . ,
i.e. C = O. In t h e u p p e r e x p r e s s i o n , u n l i k e in (2.2-3.20), c h a n g e s in t h e n u m b e r of r e v o l u t i o n s of p u m p An a n d in t h e c r o s s - s e c t i o n a l a r e a of t h e c o n t r o l v a l v e AAv a r e m i s s i n g . T h e s e a r e v a r i a b l e s that w e r e n e g l e c t e d at the b e g i n n i n g
of t h e p r e s e n t s e c t i o n .
T h e d e n o m i n a t o r of all t h e t r a n s f e r f u n c t i o n s Gi.j (i ,, 1,2, j = 1,2) in E q u a t i o n (2.2-3.30) is t h e s a m e a n d e q u a l to t h e c h a r a c t e r i s t i c e q u a t i o n of differential E q u a t i o n s (2.2-3.5) a n d (2.2-3,6), a n d a l s o to E q u a t i o n (2.2-3.23) for d e t e r m i n i n g t h e e i g e n v a l u e s of t h e s y s t e m matrix A. T h e w e l l - k n o w n fact t h a t t h e p o l e s of t r a n s f e r f u n c t i o n s r e p r e s e n t t h e e i g e n v a l u e s of t h e s y s t e m matrix A is c o n f i r m e d .
Examplo
2 C a l c u l a t i o n of d y n a m i c
c o e f f i c i e n t s . R, I. C, T. ~. Z
In t h e c a s e of airflow t h r o u g h a p i p e , u s i n g t h e d a t a in E x a m p l e 2, S e c t i o n 2.2-2. d e t e r m i n e the dynamic coefficients characterizing that process. In E x a m p l e 2 from t h e p r e c e d i n g
s e c t i o n , g a s or s t e a m flows t h r o u g h a
p i p e of l e n g t h L - IOO m a n d c r o s s - s e c t i o n a l already determined
capacitance
a r e a A - 0.0314 m 2 . We h a v e
and resistance
C - 3.7474.10 -5 kgm2/N, N - 5533.78 ( N / m 2 ) / ( k g / s ) I n e r t a n c e I is n o t difficult to o b t a i n from I - L/A - 3184.7 m -~ .
.
Q4
CHAP.
2
LUMP~
PROC~S
Natural f r e q u e n c y is ~n
"
( I C ) -°'s
=
2 . 9 s -I .
The d a m p i n g c o e f f i c i e n t is =
RC
~n12
0.301
=
The e i g e n v a l u e s of s y s t e m matrix A a n d also the p o l e s of the transfer functions are XI,~ = -0.8719 i 2.425 j . It is not difficult to s e e to w h i c h p h y s i c a l p r o c e s s the natural f r e q u e n c y ~n is related, a n d what its real m e a n i n g is. We must first r e m e m b e r that the w h o l e a n a l y s i s was c a r r i e d out for the c a s e of isothezmal s t a t e c h a n g e s
in w h i c h the v e l o c i t y of s o u n d ( a n d that is the
s p e e d with w h i c h p r e s s u r e a n d mass flow d i s t u r b a n c e s p r o p a g a t e t h r o u g h the p i p e ) is d e t e r m i n e d from the ( g e n e r a l l y known) e x p r e s s i o n
cs
= ~
=
In p r a c t i c e conditions.
2)r2-87.292 = 2 8 9 . 5
fast
This,
disturbances
however,
does
m/s
in not
fluid
state
change
propagate anything
in
in
the
adiabatic present
a n a l y s i s . As we s a i d earlier, w e then work with p o l y t r o p i c c a p a c i t a n c e
Cn
= C / n i n s t e a d of with isothermal c a p a c i t a n c e C. In the a d i a b a t i c c a s e for n = x a n d for the v e l o c i t y of s o u n d we h a v e cs = ~ • The time n e c e s s a r y for the d i s t u r b a n c e , p r o p a g a t i n g with the v e l o c i t y of s o u n d c s to p a s s t h r o u g h the w h o l e l e n g t h of the p i p e L, is TL
=
L cs
=
0,345 s
[2.2-3.34)
The natural f r e q u e n c y oscillation is e q u a l to the i n v e r s e v a l u e of time TL ~n = ~
l
= 2.9
S-1
sEc. 2.2
05
T h a t the u p p e r e x p r e s s i o n satisfies c a n a l s o b e s e e n from the following derivation 1
l
. yr~ v
l/X
I
= c~.
(2.2-3.35)
,.
R.O.
In e n g i n e e r i n g literature w e v e r y often e n c o u n t e r , besides the c o e f f i c i e n t s w e h a v e a l r e a d y i n t r o d u c e d a n d u s e d a n d the j u s t - m e n t i o n e d time for d i s t u r b a n c e transfer a l o n g the p i p e TL, s o - c a l l e d impedance d e f i n e d as follows for fluid t r a n s p o r t
cs./
Z = A
(2.2-3.36)
It is not difficult to s h o w that I - Z TL C = TL Z IC
(2.2-3.37) (2.2-3.38)
'
(2.2-3.39)
= TL 2
Finally, to e n a b l e
comparison
p r o c e s s e s for the c o n d i t i o n s matrix transfer f u n c t i o n s
with
belonging
the
case
of s p a t i a l l y
to F i g u r e
distributed
2.2-3.4, w e
aP,(s)] _-
1 It s 2+RCs+I
Is+R ICs z÷RCs+l
aP2(s) ]
aM2(slJ
-Cs [Csz+RCs+I
I ICs~+RCs*I
aM,Cs)J
give
(2.2-3.40)
the
Q6
CHAP.
2 LUMPED
PROCESSES
2 . 3 HIEAT PROCESSES
The field c o v e r e d b y this term is v e r y w i d e a n d the v a r i e t y of d e v i c e s . o b j e c t s a n d s y s t e m s in w h i c h p r o c e s s e s of h e a t p r o d u c t i o n , transfer a n d a c c u m u l a t i o n o c c u r is so g r e a t that we must limit o u r s e l v e s to the b a s i c p r o c e s s e s we m e e t in t e c h n i c a l p r a c t i c e . D e r i v i n g m a t h e m a t i c a ! m o d e l s a n d a n a l y z i n g the d y n a m i c s of s u c h b a s i c p r o c e s s e s is a s t e p in the d i r e c t i o n of c o m p r e h e n s i v e insight into m u c h more c o m p l e x p h e n o m e n a . Like
on
the
preceding
pages,
here
too
only
processes
with
lumped
p a r a m e t e r s will b e s t u d i e d , but w e will a l s o try to s h o w |hat the m o d e l s o b t a i n e d a r e in fact o n l y r o u g h e r ( w h i c h d o e s not in a n y w a y m e a n b a d , but v e r y often of g r e a t u s e ) a p p r o x i m a t i o n s p r o p e r t i e s c h a n g e t h r o u g h o u t a s p a c e volume. Heat e x c h a n g e r s
of c a s e s
in
which
heat
a r e o n e of the most f r e q u e n t l y e n c o u n t e r e d parts of
plants, a n d m u c h of this c h a p t e r will treat the d y n a m i c s of h e a t e x c h a n g e . Heat e x c h a n g e r s
are u s u a l l y d e v i c e s with metal walls w h o s e t h i c k n e s s is
kept as small as p o s s i b l e , only as m u c h as is n e c e s s a r y to g i v e strength. S i n c e metals a r e v e r y c o n d u c t i v e , in our d e r i v a t i o n s w e will often a s s u m e that the thermal c o n d u c t i v i t y c o e f f i c i e n t ), is infinite. T h e result of this a s s u m p t i o n is that the t e m p e r a t u r e of the wall is the s a m e a c r o s s its w h o l e t h i c k n e s s (~)w(Z) =' const.). The d e r i v a t i o n s for h e a t transfer t h r o u g h a wall will in most c a s e s b e v a l i d b o t h for p l a n e a n d for c u r v e d walls of p i p e e x c h a n g e r s . In them the wall t h i c k n e s s is so small c o m p a r e d with the p i p e r a d i u s thai n e g l e c t i n g c u r v a t u r e effects will not i n t r o d u c e e s s e n t i a l errors into the m o d e l s o b t a i n e d . An e x c e p t i o n a r e h i g h - p r e s s u r e , t h i c k - w a l l e d p i p e s for w h i c h the a b o v e a s s u m p t i o n d o e s not hold. As a n e x a m p l e of a m a r k e d l y n o n l i n e a r p r o c e s s w e will s t u d y r a d i a t i o n h e a t transfer, a n d after l i n e a f i z a t i o n of the initial m o d e l the linear a n d the n o n l i n e a r time r e s p o n s e s will b e c o m p a r e d . Finally, at
the
end
of this
section
direct
heat
exchangers
will
be
m o d e l e d . T h e s e a r e u s u a l l y tanks (well or not-so-well i n s u l a t e d , a n d with thicker or thinner walls, d e p e n d i n g on the p r e s s u r e u n d e r w h i c h the p r o c e s s d e v e l o p s ) in w h i c h s e v e r a l mass flows a r e m i x e d a n d the mixture h e a t e d b y h e a t i n g e l e m e n t s (electric, s t e a m a n d so on). In the p r o c e s s of
modeling
we
will b e g i n
with the
most e l e m e n t a r y
example
with
many
a p p r o x i m a t i o n s , a n d b y g r a d u a l l y d i s c a r d i n g t h o s e initial a s s u m p t i o n s will b e s h o w n how the m o d e l of d y n a m i c t e m p e r a t u r e c h a n g e
it
g r o w s in
complexity.
I H e a t transfer t h r o u g h a wall
Example
F i g u r e 2.3-1 s h o w s a p r o c e s s of h e a t transfer t h r o u g h a wall b e t w e e n fluids 1 a n d 2.
Fig.
a) Derive a
model
2 . 3 - I Heat transfer t h r o u g h a wall describing
depending on changes a s s u m p t i o n s are..
in
the d y n a m i c s temperatures
of the wall t e m p e r a t u r e 3,
and
32.
The
basic
the thermal c o n d u c t i v i t y c o e f f i c i e n t t h r o u g h the wall ), ,: co. w h i c h m e a n s that 3w is c o n s t a n t a c r o s s the w h o l e width of the wall, - h e a t transfer o n b o t h s i d e s is c o n v e c t i v e . - the c o e f f i c i e n t s of c o n v e c t i v e h e a t transfer ~l a n d
functions of t e m p e r a t u r e ,
"2 are not
Q8
CHAP.
2 LUk~PEE)
PROCI:'S~EE
- mass fluid flow rates ml a n d mz a r e c o n s t a n t (therefore, ~xl a n d cx~ a r e a l s o c o n s t a n t ) . b) Determine the transfer functions r e l a t i n g h e a t flow t r a n s f e r r e d o n t o fluid 2 with t e m p e r a t u r e . a)
The
equation
for
the
conservation
of
energy
for
a
wall
of
s u r f a c e - a r e a A is el
dE dt
- e~ =
(2.3-I)
e~ a n d e~ a r e h e a t ( e n e r g y ) flow rates b r o u g h t to the wall a n d l e d from it, a n d E h e a t ( e n e r g y ) s t o r e d in the wall. el
-- A q l
= A(x,(,% - ,~w)
e2 = Aq~ = A~2(ew - e2)
.
(2.3-2)
,
(2.3-3)
(2.3-4)
E = Mu = ASpcw~w
The last three e q u a t i o n s , t o g e t h e r with the first, g i v e pSCw d 3 w
cxl
tx2
(2.3-s)
We h a v e o b t a i n e d a linear ODE so there is no n e e d
for l i n e a r i z a t i o n
a n d i n t r o d u c i n g the s y m b o l s a. E q u a t i o n (2.3-5). as l o n g as the initial a s s u m p t i o n s a r e true. is v a l i d b o t h for real t e m p e r a t u r e c h a n g e s a n d also for c h a n g e s in the d e v i a t i o n of t h o s e t e m p e r a t u r e s from the initial s t e a d y state. As in the preceding c a s e s , h e r e w e c a n also s h o w that the c o n s t a n t b e s i d e ew has the d i m e n s i o n of time a n d , therefore, r e p r e s e n t s the time c o n s t a n t T of h e a t transfer. Its p h y s i c a l m e a n i n g b e c o m e s c l e a r from the following formulas, in w h i c h the s a m e definitions for c a p a c i t a n c e r e s i s t a n c e R a r e u s e d as in earlier d i s c u s s i o n . C a p a c i t a n c e C e q u a l s the ratio of c h a n g e
C and
in s t o r e d h e a t ( e n e r g y ) E to
c h a n g e in h e a t effort - t e m p e r a t u r e ,~w, so (2.3-4) g i v e s dE C = --= dOw
Asoc = Mc
(2.S-6)
sEc.
2.3
99
R e s i s t a n c e R e q u a l s the ratio of c h a n g e
in h e a t effort ( t e m p e r a t u r e ) to
c h a n g e in h e a t flows el a n d e2. As r e s i s t a n c e to h e a t transfer exists b o t h on the s i d e of fluid I a n d o n t h e s i d e of fluid 2, t h e total r e s i s t a n c e will d e p e n d on b o t h the i n d i v i d u a l r e s i s t a n c e s R~ a n d R2. E q u a t i o n s (2.3-2) a n d (2.3-3) g i v e d~w
RI
=
de---:"
I
=
"
dOw R2 = de-'--2- = The
fact that
(2.3-7a)
A~,-----~ l A=---~
(2.3-7b)
Ri is n e g a t i v e
represents
the p h y s i c a l
fact that
when
t e m p e r a t u r e ew i n c r e a s e s , t h e h e a t flow r a t e et o n t h e wall d e c r e a s e s . Now we c a n s h o w the m e a n i n g of c o n s t a n t T. T =
s ~ c = ABpc ~i+~z A~I÷A~
=
C ~ + l___ IRiI Rz
C = RC = __T R
(2.3-8)
E q u a t i o n (2.3-5) c a n c o n v e n i e n t l y b e s h o w n in b l o c k d i a g r a m form a n d this s t r u c t u r e of b l o c k s is p r e s e n t in all first-order ( p r o p o r t i o n a l ) s y s t e m s . It is useful to c o m p a r e this p r e s e n t a t i o n with F i g u r e 2.1-2.7.
B
~...~.~ .
~l
l
i.
-i
D
Fig.
2 . 3 - 2 Block d i a g r a m of t e m p e r a t u r e c h a n g e in p i p e wail ew
dynamics
b) T h e transfer f u n c t i o n s r e l a t i n g h e a t flow r a t e ez to fluid 2 d e p e n d i n g on t e m p e r a t u r e c h a n g e a r e o b t a i n e d after c a r r y i n g out a L a p l a c e t r a n s f o r m a t i o n of E q u a t i o n s (2.3-3) a n d (2.3-5). If the e x p r e s s i o n for e(s) from the t r a n s f o r m e d E q u a t i o n (2.3-5) is i n s e r t e d into the t r a n s f o r m e d E q u a t i o n (2.3-3), after r e a r r a n g e m e n t w e g e t
|00
(::HAP.
K Q2(s) = Tns÷]
t r a n s f o r m a t i o n of h e a t flow r a t e e2(t).
with d i s t r i b u t e d p a r a m e t e r s , for t h e c a s e Here
we
that
e2(t) will h a v e
~)2 of t h e h e a t e d
those transfer processes
is o b s e r v e d
as a p r o c e s s
of a finite X. E q u a t i o n (3.2-b2) will
transfer function identical
must indicate
temperature change
c~
(2.3-9)
In S e c t i o n 3.2. w h e n t h e h e a t t r a n s f e r p r o c e s s an approximated
PRQCESSE
K(Tbs+l) o2(s) Tns+I
O1(s}-
Q2(s) is a s y m b o I for t h e L a p l a c e
be
2 LUI~PEI3
to u p p e r
Equation
a derivative
(2.3-Q).
dependence
fluid. A m o r e d e t a i l e d
on
p r e s e n t a t i o n of
is g i v e n i n S e c t i o n 3.2.
C o n s t a n t s K. Tn a n d Tb a r e g i v e n b y t h e e x p r e s s i o n s K =
c(,c(s A
(2.3-10)
0el÷C(2
Tn =
8pc
,
(2.3-II)
0Ct+(X2
Tb =
8pc
(2.3-]2)
(x!
From E q u a t i o n depending
(2.3-8)
for T
we
can
see
on other v a r i a b l e s , also d e p e n d s
that
this
constant,
besides
on heat transfer c o e f f i c i e n t s
=~ a n d =2. The d e t e r m i n a t i o n of these c o e f f i c i e n t s is the most c r i t i c a l part of the c a l c u l a t i o n s i n c e the v a l u e s of the other constants in (2.3-8) are not difficult
to
calculating
obtain.
Here.
coefficients
of
however,
we
convective
wii] heat
b e e n written on this subject, t h e r m o d y n a m i c
not
enter
transfer.
the
Whole
problem books
atlases, t a b l e s a n d
of
have
the like,
so for these n e e d s w e refer the r e a d e r to s p e c i a l i z e d literature. But it is, nevertheless, necessary
tc s t u d y s o m e t y p i c a l t e c h n i c a l c a s e s
r a n g e w i t h i n w h i c h time c o n s t a n t T c h a n g e s Let
80
"C t e m p e r a t u r e
water
flows
and see the
for s u c h p r o c e s s e s .
through
an
uninsulated
hot-water
p i p e w h o s e w a l l s a r e 8 = 5 mm thick. If t h e h e a t t r a n s f e r c o e f f i c i e n t to t h e s u r r o u n d i n g air is 0q = IO W / m 2 ,. a n d to t h e i n t e r n a l s u r f a c e of t h e p i p e
~z
,. 230 W/m 2 . for p = 7 8 5 0 k g / m ~ . c = 0 . 5 0 kJ/kg,, t h e time c o n s t a n t T is T =
8~c 7850. O.005.500 = ,.,)+(x2 230+10
The superheater
= 81.77 s
p i p e of a n e l e c t r i c p l a n t s t e a m g e n e r a t o r
of t h i c k n e s s
8 = 0 . 0 0 5 . c = 5 0 0 . a n d with c o e f f i c i e n t s ~'t = bO. ~'2 = 2 4 0 0 . h a s t h e time
sEc.
101
2.3
constant
T
7 8 5 0 . 0 . 0 0 5 . 500 =
=
7.98
60+2400
T h e s e two simple n u m e r i c a l
s
examples c l e a r l y s h o w that T will d e c r e a s e
with a n i n c r e a s e in the h e a t transfer coefficients, i.e. with a d e c r e a s e of the r e s i s t a n c e s to h e a t transfer. It must also b e o b s e r v e d that in the c a s e of great, d i f f e r e n c e s in c o e f f i c i e n t s e,. the time c o n s t a n t is c h i e f l y d e t e r m i n e d b y the o n e that is b i g g e r . If we n e g l e c t "l in the s e c o n d c a s e , T b e c o m e s T ; 8.177, w h i c h is o n l y 2.5% more that the original time constant. Up to now. o n the b a s i s
of the first a s s u m p t i o n that ), is infinite, w e
c o m p l e t e l y n e g l e c t e d the r e s i s t a n c e to h e a t c o n d u c t i o n t h r o u g h the wall. a n d T was d e t e r m i n e d o n l y b y r e s i s t a n c e to h e a t c o n v e c t i o n . If b o u n d a r y condition-, in the form of f o r c e d h e a t llow a r e g i v e n on b o t h b o u n d a r i e s of the wall, a n d if thelmal c o n d u c t i v i t y c o e f f i c i e n l ), is finite. in p r a c t i c e we u s e a transfer f u n c t i o n s h o w i n g the d e p e n d e n c e of h e a t flow rate ez(t) o n h e a t flow rate e~(t) o n the "warmer" s i d e of the wall O2(s)
1 Tws+l
- -
=
Tw "
(~c~2 2),
Ol(s)
-
-
(2.3-13)
'
(2.3-14)
This transfer f u n c t i o n a n d the time c o n s t a n t of the p i p e Tw will b e d e r i v e d in S e c t i o n 3.2. In the d y n a m i c s e n s e , the wall b e h a v e s like a first-order ( p r o p o r t i o n a l ) system, a d d p i p e with ), = 46.5 W/InK Tw
=
Pc~2 2), ,, 1.055
Finally.
in the
common phenomena
for a
superheater
steam
generator
s
wealth
of p o s s i b l e
different
in p i p e h e a t e x c h a n g e r s
cases
one
of the
more
is that the fluid mass flow
rates m~ a n d mz a r e not c o n s t a n t , but c a n , in u n s t e a d y o p e r a t i o n , s u b s t a n t i a l l y c h a n g e in m a g n i t u d e . In s u c h c a s e s the fourth a s s u m p t i o n is no l o n g e r fulfilled a n d w e c a n n o t work with c o n s t a n t v a l u e s for c o n v e c t i v e h e a t transfer c o e f f i c i e n t s (,~ a n d ~z. s i n c e t h e y d e p e n d to a g r e a t e x t e n t o n the rate of fluid flow. In p r a c t i c e we will, thus, v e r y often u s e the following e x p r e s s i o n to d e t e r m i n e (, for s i n g l e - p h a s e fluids flowing through pipes
102
CHAP.
X_-Km
2 LUh4PED
PROCESSES
n
(2.3-15)
n - 0.8 .....liquids n
=
.....g a s e s
0.(5
If the fourth a s s u m p t i o n is not fulfilled. E q u a t i o n
(2.3-5) is no l o n g e r
linear, a n d if we insert into it the e x p r e s s i o n s (2.3-15) i n s t e a d of (~, after linearization we g e t the following m a t h e m a t i c a l formulation for the d y n a m i c s of wall t e m p e r a t u r e c h a n g e for c h a n g e s in the four p o s s i b l e disturbance variables ~pC
dd._~'~w+ A~)w--
0~1+0{2
~----.LA~), + 0{1÷~2
_%,'~_wC , A m , + _~)2-%w _ C2Am 2
~____~2A~)2 + O{I+(X2
O~÷I{X2
(XI+(X2
(2.3-10) Ci "= K i n i m i n i ' l
If we
cancel
. i = l, 2
the
(2.3-17)
fourth a s s u m p t i o n ,
the
expression
for T d o e s
not
c h a n g e , but b e c o m e s d e p e n d e n t o n the initial s t e a d y state a n d two n e w d i s t u r b a n c e v a r i a b l e s a p p e a r - mass flow rates mt a n d m2.
Many more different c a s e s c o u l d b e a n a l y z e d , w h i c h will not b e d o n e here because
we c o n s i d e r that this w o u l d not b r i n g a n y e s s e n t i a l l y n e w
d i s c o v e r i e s . If the r e a d e r
w a n t s to g a i n d e e p e r
insight into the m o d e l s
s h o w n here. he s h o u l d turn his attention to an a n a l y s i s of the results i n S e c t i o n 3.2. T h e r e the different p h e n o m e n a
are examined
in more detail
b e c a u s e of different b o u n d a r y c o n d i t i o n s o n the s u r f a c e s of the walls. In this s e c t i o n , to c o n t i n u e our a n a l y s i s of p r o c e s s e s of h e a t transfer, w e wiIl s h o w the t y p i c a l n o n l i n e a r c a s e of h e a t r a d i a t i o n a n d c o m p a r e the r e s p o n s e s of the original n o n l i n e a r m o d e l with the r e s p o n s e s of the linear model.
E x a m p l e 2 R a d i a t i o n h e a t transfer The
two
nearby
parallel
walls
in
Figure
2.3-3
exchange
heat
by
r a d i a t i o n after, at t ,, O, the t e m p e r a t u r e of wall I u n d e r g o e s step i n c r e a s e from O °C to 85 °C. At t ,, 0 the t e m p e r a t u r e of wall 2 is 0 °C. T h e mass of
s E c . 2.3
103
the wall 2 is 0.1 k g a n d its s p e c i f i c h e a t c = 1000 ] / k g ,. T h e c o e f f i c i e n t of e m i s s i o n a r e e q u a l a n d a r e ~1 = ~2 = 0.065. Both the walls a r e i n s u l a t e d t o w a r d s their s u r r o u n d i n g s a n d o n l y e x c h a n g e h e a t with e a c h other. A, = A2 = I m 2. a) D e t e r m i n e the time c o n s t a n t s h o w i n g the s p e e d increase.
of wall t e m p e r a t u r e
b) C o m p a r e the r e a l n o n l i n e a r r e s p o n s e with the r e s p o n s e of the linear m o d e l for c a s e s of s t e p i n c r e a s e in wall I t e m p e r a t u r e by: 8.5 °C a n d 8 5 °c.
xX\'%
O=
Fig.
,-~,,
2.3-3 Radiation heat exchange
b e t w e e n p a r a l l e l walls
a) T h e c o n s e r v a t i o n of e n e r g y e q u a t i o n for wall 2 is Oz = dE2 dt
(2.3-18)
'
Oz = C12 (~I 4 - ~24)
(2.349)
I
1004
(2.3-20)
E2 = M2c2~2 T h e t h r e e u p p e r e q u a t i o n s g i v e the m o d e l d e m a n d e d M
da'2
2c2d--U-
Ct2 , i-5-6z
4 2
CI2 =
4
(2.3-21)
The result is an O D E of pronounced n o n l i n e a r i t y d e s c r i b i n g the dynamics of wall 2 temperature c h a n g e ~)2. Linearization of the upper equation gives
104
CHAP.
M2c2 ~23 4C 12'i-0--~4
dAB2 dt
2
LUMPSO PRC~CEaa~S
(2.3-22)
8~
T Time c o n s t a n t T, if b o t h the n u m e r a t o r a n d e x p a n d e d b y 82. h a s a c l e a r p h y s i c a l m e a n i n g I T = ~
M2c2~2 To
. To =
CI~
accumulated
~24
=
heat
the
denominator
heat
flow rate
are
(2.3-23)
1004
If wall l is a b l a c k b o d y , i.e. C ~ = Cz, the e x p r e s s i o n in the d e n o m i n a t o r r e a l l y d o e s r e p r e s e n t the h e a t flow r a t e b e t w e e n s u r f a c e s 1 a n d 2~ For the g i v e n v a r i a b l e s it is not difficult to d e t e r m i n e that C12 will b e CI~ = 0.1904, a n d T = 0.I. I 0 0 0 . 2 7 3 = b46 s 4.0.1904.2.734 b) T h e a n a l y t i c n o n l i n e a r solution of E q u a t i o n (2.3-21) h a s the following e x p l i c i t form for time t
t
M2c210s ( a r c t g 82 Clz" 2813 ~
32 - arctg ~
~ *2In
St - 82 81 + 82-- } .
(2.3-24)
T h e l i n e a r s o l u t i o n is in the a l r e a d y k n o w n form for a first-order s y s t e m 1 a,% = A,% (I - e b 4 5 t ) (2.3-25)
If w e u s e E q u a t i o n s (2.3-24) a n d (2.3-25) to c a l c u l a t e the i e s p o n s e s for i n c r e a s e s ~8 = 8.5 °C a n d 85 °C , a n d s h o w them g r a p h i c a l l y , we g e t the d i a g r a m o n F i g u r e 2.3-4. From the following r e p r e s e n t a t i o n , w h i c h is a g r a p h i c a l solution of E q u a t i o n s (2.3-24) a n d (2.3-25) for t h e g i v e n d i s t u r b a n c e s , w e c a n s e e that the d i f f e r e n c e b e t w e e n the n o n l i n e a r a n d the linear r e s p o n s e s i n c r e a s e s with a n i n c r e a s e of the d i s t u r b a n c e v a l u e .
I05
S E C . 2.3
~J 10 +•
oc
100
oc
?.5,
,:..~.[.......
o'"'"""""'"
2.5
rig.
i
2 . 3 - 4 R e s p o n s e of the n o n l i n e a r a n d the linear m o d e l in r a d i a t i o n h e a t e x c h a n g e NL z~,% = 85 ° C . . . . . NL z~,S+ .......
L
........
=
8.5
°C
L
The o r d i n a t e h a s v a l u e s for b o t h the d i s t u r b a n c e s a n d as the s c a l e is the s a m e ( o n l y I0 times larger) o n e c u r v e is sufficient for b o t h c a s e s of linear m o d e l r e s p o n s e . For the n o n l i n e a r model, h o w e v e r , w e g e t two different c u r v e s a n d . as we e x p e c t e d , the d i f f e r e n c e from the linear r e s p o n s e is g r e a t e r for the g r e a t e r d e v i a t i o n Ae, = 85 °C from the s t e a d y state.
Example
3 Direct h e a t e x c h a n g e r
In d i r e c t e x c h a n g e r s , h e a t is t r a n s f e r r e d b y the d i r e c t mixing of s e v e r a l mass flows of different t e m p e r a t u r e . In the tank itself a d d i t i o n a l h e a t i n g also takes p l a c e b y electric, s t e a m or other h e a t e r s . F i g u r e 2.3-5 s h o w s o n e s u c h e x c h a n g e r , a n d the s i g n for a mixer s y m b o l i z e d the c o m p l e t e stirring of the liquid in the tank a n d a h o m o g e n e o u s h e a t field, fie. it is c o n s i d e r e d that the t e m p e r a t u r e is the s a m e in t h e w h o l e v o l u m e of the e x c h a n g e r a n d that liquid of that t e m p e r a t u r e flows out of the tank. Derive a m o d e l of d y n a m i c t e m p e r a t u r e c h a n g e at the tank outlet with the following a s s u m p t i o n s : - inlet a n d outlet mass flow rates a r e c o n s t a n t , i.e. m i ,, mo = m,
lOb
CHAP.
-
2
LIJK~Pn'D
PROCESSES
t h e t a n k is i n s u l a t e d ,
- | h e r e is i d e a l m i x i n g e ,, eo, the liquid has a constant -
specific
t h e t e r m P v in t h e e x p r e s s i o n
heat cp = const.,
for i n t e r n a l s p e c i f i c
h e a t is n e g l e c t e d
for t h e l i q u i d , u = i - Pv. i.e. u = i = c p s .
°Qe I o_-o
mo~0
2.3-5
]Fig.
Insulated
The equation
for t h e c o n s e i v a t i o n
el
=
direct heat exchanger
of h e a t h a s t h e u s u a l form
dE + Oel
" eo
dt
(2.3-26)
'
el -- m c p ei
(2.3-27)
eo = mcp eo
(2.3-28)
E = MCpeo
(2.3-29)
When we arrange T
*
So
T = M/re. a n d
(2.3-30) if b o t h t h e d e n o m i n a t o r
upper
(proportional) longer
assumptions processes.
linear because
expressions.
and
the numerator
we get
T = MCpeo = s t o r e d ( a c c u m u l a t e d ) mcp,~o h e a t flow r a t e The
we get
I -- ~i ÷ - Oel mCp
Time constant are expanded,
the last expression,
give
a
heat
standard
(2.3-31)
linear
ODE
for a
first-order
If m ; c o n s t , is n o t s a t i s f i e d , t h e e q u a t i o n
the products
m~oCp a n d
m~lCp a r e n o l o n g e r
is n o linear
107
~EC. 2.3
To e n a b l e c o m p a r i s o n with the a p p r o x i m a t e d transfer S e c t i o n 3.2, E q u a t i o n (2.3-30) will b e t r a n s f o r m e d into
functions
in
1
oo(s) =
Ml m
el(s) + s+l
MmCp s+! m
Qel(S)
(2.3-32)
T h e following lines will s h o w how "insignificant" c h a n g e s in a s s u m p t i o n s c h a n g e e s s e n t i a l l y the m a t h e m a t i c a l model, thus s h o w i n g all the i m p o r t a n c e of a c a r e f u l c h o i c e of simplifications w h e n the m o d e l is b e i n g formulated. D e r i v e a d y n a m i c m o d e l u s i n g all the initial a s s u m p t i o n s e x c e p t that the s e c o n d o n e is c h a n g e d a n d two n e w o n e s a r e a d d e d the tank is not i n s u l a t e d , the a c c u m u l a t i o n of (he tank wall is n e g l e c t e d (a thin-walled tank), - the c o e f f i c i e n t of thermal c o n d u c t i v i t y t h r o u g h the wall is infinite.
Oo
o
Fig.
2.3-6 Uninsulated direct heat exchanger
T h e e q u a t i o n for the c o n s e r v a t i o n of h e a t has the following form mCp~'i
+ Qel
clso
" ~vAv(,~o - ~v) " mCpSo = ]ViCp dt
After a r r a n g e m e n t
~t o T T =
we get
mc p + 8o =
MCp mcp+~vAv
(2.3-33]
mCp+=vAv ,~1 +
l mcp+=vAvQel +
~xvA__ ._~v mcp+=vAv~V
(2.3-34) (2.3-35)
108
CHAP.
2 LIJMP'¢'I~
PROCESSES
A g a i n this is the d y n a m i c s of the u s u a l first-order ( p r o p o r t i o n a l ) system. e x c e p t that the c o n s t a n t T h a s c h a n g e d . If w e l e a v e out the a s s u m p t i o n a b o u t the n e g l i g i b t e a c c u m u l a t i o n of the
exchanger
wail
(and
keep
all
the
other
assumptions),
we
must
formulate t w o b a l a n c e equations since w e n o w h a v e two p o s s i b l e h e a t tanks - the liquid in the tank a n d the wall of the h e a t e x c h a n g e r
itself, - liquid
mCpSi + Oel " "iAl(,9'o - 8w) - mCpSo = Mcp dSo
(2.3-36)
dt
- wall d8 w
(2.3-37)
=iAi(,.q'o" aw)- cxvAv(,~w- ,~v) = Mwcw dt
As l o n g as the c o n v e c t i v e h e a t transfer c o e f f i c i e n t s ~v a n d oq d o not d e p e n d on t e m p e r a t u r e , the u p p e r s y s t e m of e q u a t i o n s is linear a n d c a n b e written d i r e c t l y in the form of matrix state s p a c e e q u a t i o n s
mco+oqAi
,
(xiAI Mwcw
aw
m
cxiAi
oqAi+cxvAv .... Mwcw J L a w J
1
0
~y~oy w
Lay J (2.3-38)
we
To c o n c l u d e our a n a l y s i s of the d y n a m i c s of d i r e c t h e a t e x c h a n g e r s . wil! n o w s h o w a m o d e l w h i c h a s s u m e s that liquid stirring is v e r y
intense. This insures
a
large
heal
transfer
coefficient
,,j on
the
interior s u r f a c e of the w a i l ew = 80, a n d with ), = ,= w e a s s u m e that t h e r e is o n l y o n e e q u a t i o n for the c o n s e r v a t i o n of h e a t mcp'9'i
+ Oel
-
d,9,o mCp~o = ( M c p + Mwcw) dt
[2.3-39)
After the L a p l a c e transformation we g e t the following form
eoCs) =
1
1
Mcp+Mwcws + 1
mcp i
p
Gis)
(e~Cs) * --~-~-2elCS))
(2.3-40)
s E c . 2.3
109
It is useful to o b s e r v e that the transfer f u n c t i o n o b t a i n e d G(s) is the s a m e a s the a p p r o x i m a t e d
transfer f u n c t i o n g i v e n in E q u a t i o n (3.2-27) in
the s e c t i o n o n t r e a t i n g the d i r e c t h e a t e x c h a n g e r
with thick walls as a
s y s t e m with d i s t r i b u t e d p a r a m e t e r s .
At the e n d of our a n a l y s i s of the d y n a m i c s of b a s i c h e a t p r o c e s s e s we must stress that t e m p e r a t u r e c h a n g e s a r e d e s c r i b e d as a rule b y the a l r e a d y k n o w n e q u a t i o n s for proportional s y s t e m s . Time c o n s t a n t T is h e r e a l s o e q u a l to the p r o d u c t of thermal c a p a c i t a n c e
C a n d r e s i s t a n c e R, T
CR. From this a s p e c t h e a t p r o c e s s e s , w h i c h a r e in the p h y s i c a l s e n s e r e d u c e d to p r o c e s s e s of h e a t s t o r a g e , a r e d y n a m i c a l l y similar to p r o c e s s e s of mass s t o r a g e in tanks, a n d the main d i f f e r e n c e b e t w e e n t h e s e a n d flow p r o c e s s e s is that h e a t p r o c e s s e s h a v e no inertia. Thus t h e i e will b e p e r i o d i c , o s c i l l a t o r y state c h a n g e s in their c a s e .
no
2.4 MECHANICAL PROCESSES
The motion d y n a m i c s of the rigid b o d y , mass point ( p a r t i c l e ) or more c o m p l e x m e c h a n i s m s is a c l a s s i c a l t e c h n i c a l s c i e n c e a n d a d i s c i p l i n e that is s t u d i e d in m a n y institutions. T o d a y t h e r e is a v e r y v a r i e d s e l e c t i o n of t e x t b o o k s , h a n d b o o k s , c o l l e c t i o n s of e x a m p l e s a n d b o o k s from that field in the world, w h i c h s h o w both b a s i c a n d also v e r y e l a b o r a t e m e t h o d s a n d m o d e l i n g p r o c e d u r e s . As a rule, all t h e s e b o o k s c o n t a i n n u m e r o u s e x a m p l e s (from the simplest to v e r y c o m p l e x m e c h a n i s m s ) a n d a r e h i g h l y s p e c i a l i z e d , so it must i m m e d i a t e l y b e e m p h a s i z e d that this s e c t i o n of our b o o k is not in a n y w a y i n t e n d e d as a s u p p l e m e n t , substitute or p e r h a p s c o m p e t i t i o n to s u c h b o o k s a n d c o l l e c t i o n s . Its b a s i c p u r p o s e , like that of the p r e c e d i n g s e c t i o n s , is to d e s c r i b e the d y n a m i c s of m e c h a n i c a l p r o c e s s e s of the rigid b o d y motion in m a t h e m a t i c a l forms u s u a l in this b o o k , a n d to thus s h o w unity in the d y n a m i c s of t h e s e a n d other processes.
With
capacitance
that
in
mind,
the
dynamic
coefficients
of
resistance,
a n d i n e r t a n c e will b e r e d e f i n e d a n d their p h y s i c a l m e a n i n g
given. In this w a y we will d e v i a t e to a c e r t a i n e x t e n t from the u s u a l form of mathematical
models
in c l a s s i c a l
dynamics
to w h i c h
some
readers
are
U0
CHAP. 2 LUMPHD
PROCESSHS
p r o b a b l y m o r e t h a n a c c u s t o m e d , but all the e x a m p l e s will b e p r e s e n t e d in the ( h e r e w i d e l y u s e d ) form of matrix s t a t e - s p a c e e q u a t i o n s or, simply, b y s y s t e m s of first-order DE. This n o t a t i o n h a s s h o w n itself to b e of g r e a t u s e in d e s i g n i n g c o n t r o l p r o c e d u r e s for m o d e r n r o b o t m e c h a n i s m s a n d m a n i p u l a t o r s . For that p u r p o s e , n a m e l y , to s y n t h e s i z e the w h o l e c o n t r o l l i n g assembly from s e n s o r s and logical devices that a r e built of m i c r o p r o c e s s o r s or small c o m p u t e r units, to a c t u a t o r s ( e l e c t r i c a l . h y d r a u l i c or p n e u m a t i c motors) - it is e s s e n t i a l to h a v e a m o d e l in the v e r y form that is u s e d in m o d e r n c o n t r o l theory, a n d that is n o t a t i o n in the form of state-space equations. T h r e e b a s i c p r o c e d u r e s h a v e b e c o m e s t a n d a r d in o b t a i n i n g m o d e l s for the d y n a m i c s of the rigid b o d y motion in m e c h a n i c s : - the u s e of N e w t o n ' s law for t h e c o n s e r v a t i o n of m o m e n t u m - the u s e of d ' A l e m b e r t ' s p r i n c i p l e , - deriving
Lagrange's e q u a t i o n s of motion.
H e r e w e must s a y that d ' A l e m b e r t ' s p r i n c i p l e is in fact o n l y a skilful t r a n s f o r m a t i o n of N e w t o n ' s law, while L a g r a n g e ' s e q u a t i o n s of motion a r e b a s e d on c o n s i d e r a t i o n s c o n c e r n i n g the c o n s e r v a t i o n of e n e r g y . It c a n b e s h o w n that L a g r a n g e ' s e q u a t i o n s c a n b e d e r i v e d from N e w t o n ' s laws, a n d without c a r r y i n g out a critical c o m p a r i s o n of the m e t h o d s , w e will o n l y a d d that in d e r i v i n g a m o d e l for the d y n a m i c s of motion of m o r e c o m p l i c a t e d s y s t e m s it is a d v i s a b l e to d o s o b y the s i m u l t a n e o u s u s e of L a g r a n g e ' s e q u a t i o n s a n d of N e w t o n ' s laws. Both m e t h o d s must g i v e the s a m e result. B e f o r e b e g i n n i n g the a n a l y s i s of s p e c i f i c e x a m p l e s , w e must p o i n t to the a n a l o g y b e t w e e n the two b a s i c forms of motion in m e c h a n i c s - the similarities b e t w e e n t r a n s l a t i o n a n d rotation. Without a n y m o r e e x t e n s i v e e x p l a n a t i o n , t h e s e a n a l o g i e s a r e g i v e n in the following t a b l e .
SEC.
111
2.4
T a b l e 2.4-!
Rotation
Translation Z w-a
z'
= w'
= z"
M F Im=Mw
displacement velocity acceleration mass force momentum
angular angular angular moment torque
displacement velocity acceleration of inertia
angular momentum
~, ~ - 9' ~ = w'= ~," J Mz I.~ = J - - ~ t~
|
!
T -- ~ - MW ~ W =
= Mz
.[Fdz
P -- F w
kinetic e n e r g y
T = -~
work
W = .[Mzd~
power
P -- IVfz~
In m e c h a n i c a l s y s t e m s t h e r e is d u a l i t y in the s e l e c t i o n of the v a r i a b l e s of effort (potential), flow a n d s t o r e d v a r i a b l e s , a n d thus a l s o d i f f e r e n c e s in the p h y s i c a l m e a n i n g of d y n a m i c c o e f f i c i e n t s . This d u a l i t y results from w h e t h e r w e s e l e c t f o r c e or v e l o c i t y in translation ( t o r q u e or a n g u l a r v e l o c i t y in rotation) for the v a r i a b l e of effort, a n d it will b e s h o w n o n a n e x a m p l e of a s i m p l e m e c h a n i c a l system., m a s s - s p r i n g - d a m p e r . We will b e g i n this c h a p t e r b y d e s c r i b i n g the motion d y n a m i c s of a b o d y u s i n g the s i m p l e s t e x a m p l e of translation a n d the u s u a l m a t h e m a t i c a l tools. E v e r y t h i n g that is d o n e in further e x a m p l e s for c a s e s of t r a n s l a t i o n c a n , without a n y r e s e r v a t i o n s , b e above analogy table.
applied
to rotation a s well. u s i n g
the
B e f o r e s e t t i n g u p the e q u a t i o n s it must b e p o i n t e d out that this s e c t i o n will in most c a s e s a n a l y z e linear p r o c e s s e s . This m e a n s that the v a r i a b l e c h a n g e s will b e c o n s i d e r e d small e n o u g h to m a k e n o n l i n e a r i n t e r r e l a t i o n s u n n e c e s s a r y . It follows that t h e r e will b e no l i n e a r i z a t i o n , w h i c h a l s o m e a n s that t h e r e will b e no d e v i a t i o n n o t a t i o n ,~ b e s i d e l i n e a r i z e d v a r i a b l e s . In c a s e s w h e r e w e d o c a r r y out l i n e a r i z a t i o n , h o w e v e r ,
112
CHAP. 2 LUmPeD
previously u s e d symbols that the original m o d e l
IExamplo
PROCESSES
will b e retained but then it must not b e forgotten w a s in fact nonlinear.
I Translation dynamics
C o n s i d e r a b o d y of m a s s M, u n d e r the i n f l u e n c e of f o r c e F(t), in t r a n s l a t i o n a l motion on a flat s u r f a c e . [n the g e n e r a l c a s e the following f o r c e s resist motion: the f o r c e of d r y (sliding) friction F~. a f o r c e p r o p o r t i o n a l to the v e l o c i t y of the b o d y (the f o r c e of v i s c o u s friction) F3 a n d a f o r c e p r o p o r t i o n a l to t h e b o d y ' s d i s p l a c e m e n t F4. D e r i v e a m o d e l that d e s c r i b e s the d y n a m i c s of d i s p l a c e m e n t z a n d v e l o c i t y w if at t ; 0 there was:
1st IC z = 0 dz 2 n d IC ~
-~ w = w o (IC ... initial condition)
a n d for the following cases:
a) i d e a l frictionless motion, b) motion with the p r e s e n c e c) motion with the p r e s e n c e v e l o c i t y w.
of s l i d i n g friction. of a r e s i s t a n c e
f o r c e p r o p o r t i o n a l to the
d) r e s i s t a n c e f o r c e to motion p r o p o r t i o n a l to b o d y d i s p l a c e m e n t z.
z
I-~-.I I
Fig.
2.4-1
w- ~i-
A b o d y in motion with d r i v i n g a n d r e s i s t a n c e f o r c e s
For e a c h c a s e the m o d e l will b e o b t a i n e d from the following b a s i c e x p r e s s i o n d e s c r i b i n g N e w t o n ' s law for the c o n s e r v a t i o n of m o m e n t u m
sEc.
113
2.4
4 ~=.z~F I = d ( M w ) dt
= M d._~._w dt
a) in the c a s e (2.4-1) b e c o m e s . dw G!
M--w-,
of i d e a l motion without a n y
(2.4-])
f o r c e s , i.e. FI = 0 (i=l.4),
(2.4-2)
= 0
This is a n e q u a t i o n with a "pure" i n t e g r a l term fol v e l o c i t y w c h a n g e , w h i c h with the g i v e n initial c o n d i t i o n s b e c o m e s w
=
Wo
=
const.
Since the generally-known relation between velocity d i s p l a c e m e n t z is ( a l s o a n i n t e g r a l m e m b e r without time l a g ) = w
dz
dt
w
and
(2.4-3)
it follows that z will c o n t i n u o u s l y i n c r e a s e to infinity, i.e. Z = Wot
b) T h e r e is a r e s i s t a n c e f o r c e of d r y (sliding, C o u l o m b ) friction, s o that (2.4-1) b e c o m e s dw
M ~ - - = - ~Mg
(2.4-4)
A s y s t e m d y n a m i c a l l y e q u i v a l e n t to this e x a m p l e of t r a n s l a t i o n is g i v e n b y E q u a t i o n (2.2-1.2) for the c a s e w h e n the input p u m p c e a s e s o p e r a t i o n , i.e. for m = O. Like liquid l e v e l H in that e x a m p l e , h e r e v e l o c i t y l i n e a r l y d e c r e a s e s to zero. It c a n e a s i l y b e s h o w n that the following e x p r e s s i o n is the solution of (2.4-4) for the g i v e n IC w
dz = d---[- = Wo - # g t
(2.4-5)
E q u a t i o n (2.4-5) g i v e s the a n a l y t i c a l e x p r e s s i o n for d i s p l a c e m e n t z z = wot
- 2-~t 2
(2.4-b)
114
CHAP. 2 L U M P ~ This p r o c e s s of free translation with the p r e s e n c e
PROCesses
of d r y friction c a n
also b e s h o w n in b l o c k d i a g r a m form.
z--y
w
i,i
-1MI Fig. If we
L
2 . 4 - 2 Block d i a g r a m of translation with d r y friction
select
displacement
z
,,
x~ a n d
velocity
w
= x~
for
state
v a r i a b l e s , the s y s t e m of E q u a t i o n s (2.4-3) a n d (2.4-4) c a n e a s i l y b e g i v e n in the form of matrix s t a t e - s p a c e e q u a t i o n s '
0
(2.4-'i')
X'
= A X ÷ B U
The e i g e n v a l u e s of matrix A a r e e q u a l to zero, x l = ;~2 = 0, i n d i c a t i n g the p r e s e n c e of the two i n t e g r a l terms without a time l a g s h o w n on F i g u r e 2.4-2. c) Driving f o r c e F a n d a c t on the b o d y
a f o r c e of r e s i s t a n c e
p r o p o r t i o n a l to v e l o c i t y
Fa - - Dw
(2.4-8)
D is the c o e f f i c i e n t of that linear d e p e n d e n c e
a n d is also c a l l e d
the
c o e f f i c i e n t of v i s c o u s friction. Now (2.4-1) b e c o m e s dw M - ~ . • Ow = F This
equation
(2.4-9) is
the
classical
and
already
several
times
obtained
e x a m p l e of a first-order p r o p o r t i o n a l s y s t e m for the d y n a m i c s of a b o d y ' s v e l o c i t y w. in w h i c h time c o n s t a n t T e q u a l s M T = y
(2.4-10)
SEC. 2.4
115
At initial moment t = 0 the v e l o c i t y is We a n d
if the n u m e r a t o r
and
d e n o m i n a t o r a r e e x p a n d e d b y that v e l o c i t y , the p h y s i c a l m e a n i n g of the constant T b e c o m e s clearer. T = MwQ = I m = Dwo F
momentum force
(2.4-11)
T is thus the ratio of the total m o m e n t u m s t o r e d to the f o r c e that a c t s o n the b o d y in the s t e a d y s t a t e o b s e r v e d . If d i s p l a c e m e n t z is important, it c a n . as in the p r e c e d i n g c a s e , b e o b t a i n e d b y the simple i n t e g r a t i o n of v e l o c i t y w. If
(2.4-9)
is
compared
with
the
other
equations
of
proportional
first-order s y s t e m s in p r e c e d i n g s e c t i o n s , it c a n b e s e e n that v e l o c i t y w is the v a r i a b l e of effort or potential, a n d f o r c e F the v a r i a b l e of flow. T h e s t o r e d v a r i a b l e is m o m e n t u m Im = Mw. In this c a s e the c a p a c i t a n c e a n d r e s i s t a n c e are C = A__~V =
M__._~w= M w
E R = E__ = : F F
= I___ D
(2.4-12)
(2.4-13) '
a n d the time c o n s t a n t , as until now, is T=RC=
M D
If c o e f f i c i e n t s C a n d R that h a v e just b e e n i n t r o d u c e d are u s e d . the following b l o c k d i a g r a m s h o w s translational motion in the c a s e of a r e s i s t a n c e force p r o p o r t i o n a l to v e l o c i t y .
FCtl •
Z I_
_
Fig.
-6"
I
--
_
2 . 4 - 3 Block d i a g r a m of translational motion if the r e s i s t a n c e f o r c e is linearly d e p e n d e n t on v e l o c i t y
c ~ A p . 2 I-UMP~D P R O C E S S E S T h e motion d e s c r i b e d
b y two OLDE of the first o r d e r (2.4-3) a n d (2.4-9)
c a n b e g i v e n as follows b y s t a t e e q u a t i o n s !
(2.4-14)
X' = A X + B U It is not difficult to s o l v e the c h a r a c t e r i s t i c e q u a t i o n to o b t a i n e i g e n v a l u e s of matrix A. T h e y differ from e a c h other a n d a r e ~1 :
0
)~2:
" - - : " M
D
the
1 T
T h e first e i g e n v a l u e s h o w s the i n t e g r a l c h a r a c t e r of the p r o c e s s of p o s i t i o n c h a n g e in time, i.e. d i s p l a c e m e n t z. T h e s e c o n d s h o w s a s t a b l e transient p r o c e s s of a first-order ( p r o p o r t i o n a l ) s y s t e m d e s c r i b i n g v e l o c i t y w c h a n g e d u e to c h a n g e s in f o r c e F a c t i n g on the b o d y . If we s u b s t i t u t e (2.4-3) into (2.4-9), the m o d e l is t r a n s f o r m e d into the following OLDE of the s e c o n d order, w h i c h is the u s u a l m a t h e m a t i c a l presentation of displacement z d2z M-~-
the
dz + D-~- = F
dynamics
of
translational
motion
and
shows
(2.4-15)
This w a y of f o r m u l a t i n g the m o d e l with s e c o n d - o r d e r d e r i v a t i v e s of the d i s p l a c e m e n t v a r i a b l e z is u s u a l a n d w i d e s p r e a d in c l a s s i c a l m e c h a n i c s . T h e r e is, n e v e r t h e I e s s , s o m e s i m i l a r i t y in the different w a y s in w h i c h m o d e l s a r e f o r m u l a t e d b y the s t a t e - s p a c e m e t h o d a n d in c l a s s i c a l m e c h a n i c s . It lies in the fact that b o t h a p p r o a c h e s try to o b t a i n a s y s t e m c o m p r i s i n g DE, o n l y in m e c h a n i c s it is u s u a l for t h e m to b e DE of the s e c o n d , a n d in the s t a t e - s p a c e m e t h o d DE of the first o r d e r . T h e m e t h o d w e u s e d u p to n o w to p r e s e n t the d y n a m i c s of a b o d y in t r a n s l a t i o n a l motion is in a w a y u n u s u a l in c l a s s i c a l m e c h a n i c s . T h e r e f o r e , this last c a s e will b e p r e s e n t e d in the m a n n e r u s u a l in that d i s c i p l i n e .
sEc. 2.4
I17
As a rule, t h r e e s t a n d a r d
elements
are used
to r e p r e s e n t
mechanical
structures. With their h e l p it is p o s s i b l e to c o m b i n e a n d r e p r e s e n t d y n a m i c a l l y e v e n the most c o m p l e x m e c h a n i s m s , a s h a s b e e n s h o w n , to a s a t i s f a c t o r y d e g r e e . T h e y are: a n inertia e l e m e n t of m a s s M, a s p r i n g of c o n s t a n t c a n d a d a m p e r of c o e f f i c i e n t of v i s c o u s friction D. ( S o m e t i m e s a n d p a r t i c u l a r l y in a n g l o s a x o n literature the s p r i n g c o n s t a n t is d e n o t e d with S, b u t in t h e s e lines w e will r e t a i n the s t a n d a r d s y m b o l u s u a l in m e c h a n i c s - c.) T h e b a s i c p r o p e r t i e s of e a c h of t h e s e e l e m e n t s a r e r e p r e s e n t e d o n t h e following figures a n d g i v e n with the e x p r e s s i o n s b e s i d e them. • Z~ _
Zz
-
Spring
Damper
~
.
.
F
I--'---
I__
.
5
.
_
~
l
|
T
II ,
l.._t_ ~
I
F
~..__
F :
oz
=
cCz,- z2) (2.4-I(5)
F = D(w~-
D(z',
w2)
dZz
_
. d2z F - Ma = M--T-~- (2.4-18) at ~
Mass
All the e l e m e n t s s h o w n h e r e a r e a l s o p r e s e n t in the c a s e a c c o r d i n g to the a n a l o g i e s g i v e n in T a b l e 2.4-1.
Torsional spring
~°t
ct
~z
Mz
Mz
Torsional damper
GNz B o d , w, h m o m . n of inertia J
(2.4-17)
- z'2)
of rotation,
= CI.({PI "ll)2)" (2.4-19)
" Dt(ol " (~2) (2.4-20)
Mz
z, ,d2.d (2.4-21)
118
CMAP.
2 LUMPED
PROCHSSHS
(ct .... torsional s p r i n g c o n s t a n t , D t .... torsional c o e f f i c i e n t of v i s c o u s friction). The
above
elements
are
linear
approximations
of
real
mechanical
o b j e c t s a n d a r e a s a rule v a l i d for small d i s p l a c e m e n t s . In the c a s e of the s p r i n g the a p p r o x i m a t i o n is c o r r e c t if w e d o not c r o s s limit. The resistance resistance
the p r o p o r t i o n a l
v i s c o u s d a m p e r is a g o o d substitute for e n e r g y loss a n d f o r c e s o n l y in the r e g i o n of real linear d e p e n d e n c e of f o r c e to motion o n v e l o c i t y . S i n c e this is not a b o o k w h o s e
p u r p o s e is to e x a m i n e the validity of i n t r o d u c i n g t h e s e linear e l e m e n t s into an a n a l y s i s of m e c h a n i c a l systems, nor to s t u d y the r a n g e within w h i c h s u c h linear a p p r o x i m a t i o n s are s a t i s f a c t o r y , t h e s e e l e m e n t s will in the
further
text
be
used
in
the
way
usual
in
mechanics,
taking
into
a c c o u n t the fact that their a p p l i c a t i o n in fact a l r e a d y r e p r e s e n t s l i n e a r i z a t i o n of the task. T h e last c a s e c a n thus b e s h o w n b y the following c o m b i n a t i o n of e l e m e n t s of inertia a n d d a m p i n g - Fig. 2.4-4.
I'-------
Z -
F(t]--I
-I °
I
M
dz
~.~
0
~--
ol "'
"////7////////////////////,
Fig.
2 . 4 - 4 T r a n s l a t i o n s h o w n as a c o m b i n a t i o n of e l e m e n t s of inertia a n d d a m p i n g
d) Driving f o r c e F a n d a r e s i s t a n c e f o r c e p r o p o r t i o n a l to d i s p l a c e m e n t z a c t on the b o d y . ~4
= CZ
This c a s e , w h e n r e s i s t a n c e is p r o p o r t i o n a l to d i s p l a c e m e n t z, is s h o w n b y the simplest element, a spring, a n d the following figure r e p r e s e n t s this t y p e of translation.
SEC.
2.4
119
,._~ I I I
, F(t)
--Z
dz
r/
/11/i11/1II/I/111/11111111111/I/'/~,
Fig.
2 . 4 - 5 T r a n s l a t i o n s h o w n as a c o m b i n a t i o n of e l e m e n t s of inertia a n d s p r i n g
For small d i s p l a c e m e n t s E q u a t i o n (2.4-1) b e c o m e s
where
the
principle
of
linearity
is
valid.
dw Mdt
(2.4-22)
dw M-~-- ÷ c z : F
(2.4-23)
FI
-
F4
:
i.e.
T h e m o d e l of motion n o w c o m p r i s e s E q u a t i o n s (2.4-3) a n d (2.4-23). with w h o s e h e l p it is e a s y to s h o w that M d2z tit 2
+ cz = F
(2.4-24)
It is not difficult to c o m b i n e matrix s t a t e - s p a c e form
[0
these
two OLDE of the first o r d e r
into
[0]
A T h e e i g e n v a l u e s of matrix A a r e n o w a c o m p l e x c o n j u g a t e p a i r x 1.2 = ~/--~- J
(2.4-26)
[20
CHAP.
2 LUMPED
PROCESSES
w h i c h s h o w s that the t r a n s i e n t p r o c e s s will b e p e r i o d i c . T h e r e a l p a r t e q u a l s z e r o ( w h i c h is a d i r e c t c o n s e q u e n c e of the a s s u m p t i o n that t h e r e a r e n o r e s i s t a n c e friction forces), s o this is the a l r e a d y k n o w n c a s e of u n d a m p e d o s c i l l a t o r y motion. E q u a t i o n (2.4-24) is a DE of a s e c o n d - o r d e r ( p r o p o r t i o n a l ) s y s t e m ( p e r i o d i c s y s t e m ) in w h i c h the d a m p i n g p a r t e q u a l s z e r o . T h e fact that the p r o c e s s is o s c i l l a t o r y i n d i c a t e s that t h e r e must b e a p o s s i b i l i t y for c h a n g e in e n e r g y form in this c o m b i n a t i o n . Intuitively, this is c l e a r . T h e kinetic e n e r g y c o n t a i n e d in the motion of m a s s M b y v e l o c i t y w c h a n g e s into the p o t e n t i a l e n e r g y s t o r e d in the s p r i n g b y its c o m p r e s s i o n a n d s t r e t c h i n g , i.e. b y c h a n g e s in d i s p l a c e m e n t v a r i a b l e z. H o w e v e r , the d y n a m i c c o e f f i c i e n t , i n e r t a n c e I, must a l s o b e found. This i n e r t a n c e is p a r t of the o s c i l l a t o r y p r o c e s s b e c a u s e , as w e a l r e a d y s h o w e d in S e c t i o n 2.2. s u c h p r o c e s s e s a r e p o s s i b l e o n l y if t h e r e is a link b e t w e e n e l e m e n t s of capacitance a n d i n e r t a n c e . T h e p a t h l e a d i n g to that i n e r t a n c e I will, n e v e r t h e l e s s , not b e intuitive s i n c e it is m o r e s u i t a b l e to u s e m a t h e m a t i c a l e x p r e s s i o n s a n d d e f i n i t i o n s from w h i c h it follows d i r e c t l y I
=
~
=
dF dt
Ed__.~t = wd___t = d__~z = dF
dF
Fit)--l" ' ]Fig.
dF
I__.
(2.4-27)
c
I , 1:.=+1, 1 - I l l :1 :1 -I : l : l
Z
2 . 4 - 6 Block d i a g r a m of t r a n s l a t i o n a l motion of a b o d y if r e s i s t a n c e d e p e n d s l i n e a r l y on d i s p l a c e m e n t
We h a v e , thus, u s e d the d e f i n i t i o n b y w h i c h I e q u a l s the ratio of effort to g r a d i e n t of flow, a n d a little c a l c u l a t i o n g a v e us that c o e f f i c i e n t . T h e d y n a m i c p r o c e s s of c h a n g e in p o s i t i o n z d u e to f o r c e F is g i v e n in the b l o c k d i a g r a m as s h o w n in F i g u r e 2.4-6. C o e f f i c i e n t s C a n d I a r e g i v e n b y (2.4-12) a n d (2.4-27) .
$EC.
2.4
T h e p r e c e d i n g e x a m p l e h a s b r o u g h t us c l o s e r to c o n s i d e r i n g a s o n e w h o l e the d y n a m i c s of m e c h a n i c a l a n d o t h e r p r o c e s s e s . E x a m p l e s h a v e b e e n p r e s e n t e d in the form of s t a t e - s p a c e e q u a t i o n s a n d b l o c k d i a g r a m s , c o e f f i c i e n t s of c a p a c i t y , r e s i s t a n c e a n d inertia h a v e b e e n g i v e n , a n d in this w a y w e d e v i a t e d from the u s u a l a n a l y t i c a l tools of c l a s s i c a l mechanics. The following e x a m p l e will s h o w the earlier m e n t i o n e d d u a l i t y w h i c h a p p e a r s in m e c h a n i c s as a result of the s e l e c t i o n of d y n a m i c v a r i a b l e s a n d c o e f f i c i e n t s . A similar d u a l i t y a l s o e x i s t s in e l e c t r i c a l circuits.
• -xample
2
Duality of mechanics
dynamic
variables
and
coefficients
in
F i g u r e 2.4-7 s h o w s two m a s s - s p r i n g - d a m p e r s y s t e m s . D e r i v e DE d e s c r i b i n g their d y n a m i c s , d e t e r m i n e their d y n a m i c v a r i a b l e s a n d d y n a m i c c o e f f i c i e n t s , a n d d r a w their b l o c k d i a g r a m s . T h e d r y friction r e s i s t a n c e is n e g l e c t e d , i.e. F2 ; 0. The systems shown in F i g u r e 2.4-7 are similar in dynamic characteristics. The essential d i f f e r e n c e is that in F i g u r e a) the d i s p l a c e m e n t s z, i.e. v e l o c i t i e s w = z' a r e e q u a l , w h i c h d o e s not h o l d for F i g u r e b). On that F i g u r e the v e l o c i t i e s a r e different but the f o r c e s a r e equal.
I
I
"~I-ill
Zo
I !
~
c
!
0
Zd
Y/i/i/i/i/////////////////////////////.,,
Zm = Z0 = Zd
z,,. F" z o ~ z d
z ~ : z~ : z~
a]
Fig.
bl
2 . 4 - 7 M e c h a n i c a l s y s t e m s with~ a) e q u a l v e l o c i t i e s b) e q u a l f o r c e s
~
Zm
192
CHAP. 2 LUk,fPED P R O C E S S E S App]ying Newton's law for the conservation of m o m e n t u m
to the system
on Figure 2.4-7a (where Z m = zo = Zd = z) yields F
d2z
- cz - ~dD-~--~ = M L-T£,2
(2.4-28)
U.[
F,
F+
F3
*
D dz dt * cz
i.e. M d2z dtz
= F
(2.4-29)
In the s y s t e m on F i g u r e 2.4-Yb the v e l o c i t i e s , i.e. the d i s p l a c e m e n t s , a r e no l o n g e r e q u a l . H e r e the f o r c e s a r e e q u a l . If w e d e r i v e d ' A l e m b e r t ' s e q u a t i o n s for the equJ]ibrium of f o r c e s for p o i n t A a n d for m a s s M. w e g e t the following two e q u a t i o n s C ( Z o " Zd) = D ( Z ' d - Z'm)
,
D ( z ' d - Z'm)= M Z r ~
(2.4-30)
(2.4-31)
Rearrangement yields M |1 M i " ~ - W m + "-~'W m + W m = W 0
(2.4"32)
T h e c a s e o n f i g u r e 2.4-7a. d e s c r i b e d b y (2.4-29). h a s a l r e a d y b e e n a n a l y z e d in the p r e c e d i n g e x a m p l e a n d m a t h e m a t i c a l l y d e s c r i b e d b y (2.4-24), o n l y h e r e the r e s i s t a n c e f o r c e Dw, r e p r e s e n t e d b y a d a m p e r , is a l s o i n c l u d e d . T h e d y n a m i c v a r i a b l e s a n d c o e f f i c i e n t s d e f i n e d earlier a l s o satisfy in this c a s e . Now t h e y must b e d e t e r m i n e d for the s y s t e m o n F i g u r e 2.4-Tb. It h a s a l r e a d y b e e n m e n t i o n e d that f o r c e F is a c o m m o n a n d e q u a l v a r i a b l e for all the t h r e e e l e m e n t s , a n d h e r e it is the v a r i a b l e of effort. T h e v a r i a b l e of flow, a l s o a n input v a r i a b l e , is n o w v e l o c i t y w. This is the c h a n g e from the p r e c e d i n g e x a m p l e , w h e r e the d y n a m i c v a r i a b l e s w e r e the o p p o s i t e - the v a r i a b l e of effort w a s v e l o c i t y w a n d of flow f o r c e F. T h e s t o r e d v a r i a b l e is the v a r i a b l e of s p r i n g s t r e t c h i n g a n d c o m p r e s s i o n ~Z
=
Z o
The
-
Zd-
dynamic
coefficients (capacitance. resistance a n d
obtained directly from their definitions
inertance) are
s E c . 2.4
123
z F
AV C
= - -
E
E
I c
F
R . . . . . F w
E . dF dt
l .
.
.
.
[2.4-33)
D
(2.4-34)
F . . d._ww dt
F z"
M
(2.4-35)
The T a b l e 2.4-2 s h o w s the d y n a m i c v a r i a b l e s a n d coefficients, i.e. their p o s s i b l e d u a l i t y in m e c h a n i c s (the c a s e of rotational motion is also presented). Table
2.4-2
Translation
E
Im= MW F w
C
M
AV
F
R
l D I
I
Rotation 2 w F
L z
I
j
c D
=
Jw
%o
M z
Mz
1 Ct
l Dt
Dt
! M
C
Ct
I
F i g u r e 2.4-8 r e p r e s e n t s a b l o c k d i a g r a m for the d e s c r i p t i o n s g i v e n b y E q u a t i o n s (2.4-2Q) a n d (2.4-32). The e x a m p l e in w h i c h f o r c e F is the flow v a r i a b l e is more natural a n d e a s i e r to u n d e r s t a n d , a n d this is the c a s e for w h i c h w e will d e r i v e s t a t e - s p a c e e q u a t i o n s a n d a n a l y z e e i g e n v a l u e s . If state v a r i a b l e s a n d inputs are s e l e c t e d as follows X(=Z X 2 =
u
W
= F
= Z'
.
two first-order O L D E
XI ~ z
Z~ =
W
=
X2
ate obtained
124
CHAP.
= Z" = -
which
c ~-Z
D
,
1
"
4-
it is easy
to
set
--
c
down
D
+
.in the
2 LUMP':D
PROCESSES
of
state-space
I
form
matlix
[:] [0 ][:] [el
equations
F=F~
- 'I' --%
GF
!
Z..Mw
1_.
w
o)
Wm
Wo
-IL
6w
~
!
6z
!
~
F
Im=MW
m
b)
Fig.
2 . 4 - $ B l o c k d i a g r a m s of t h e m e c h a n i c a l mass-spring-damper
system
It is not difficult to e s t a b l i s h that (2.4-T), (2.4-14) a n d (2.4-25) a r e o n l y s p e c i a l c a s e s of t h e u p p e r e q u a t i o n s . T h e e i g e n v a l u e s a r e o b t a i n e d b y s o l v i n g E q u a t i o n (X I - A) = 0 s o that Xl.2 =
-D , ~/D ~ - 4 c M 2M
An a n a l y s i s of t h e last e x p r e s s i o n v a l u e s for t h e e i g e n v a l u e s ),:
(2.4-37)
points
to
three
possible
sets
of
s E c . 2.4
125
a) D 2 < 4
cM,
x l,~ is a
pair
of c o m p l e x
n e g a t i v e r e a l part, the r e s p o n s e is of d a m p e d b) D 2 = 4 cM, ),~ = ),2 is a p a i r r e s p o n s e is u n p e r i o d i c .
conjugate
The natural bequency 60 n
i
coefficients
of u n d a m p e d
with
a
periodic character.
of e q u a l
negative
c) D ~ > 4 cM~ ),1 " ),2, the roots a r e real. n e g a t i v e r e s p o n s e is u n p e r i o d i c a n d s t l o n g l y d a m p e d . T h e following s t a n d a r d often u s e d in p r a c t i c e .
values
for p e r i o d i c
real
and
roots,
the
different, the
processes
aIe
very
(D = O) o s c i l l a t i o n
(2.4-38)
s
T h e d a m p i n g ratio actual damping coefficient = critical d a m p i n g c o e f f i c i e n t
D 2-~c--M
(2.4-39)
S u b s t i t u t i n g t h e s e v a r i a b l e s into (2.4-2"/) for e i g e n v a l u e s
(2.4-40) T h e a n a l y s i s that w a s c a r r i e d out for E q u a t i o n (2.2-3.27) is v a l i d h e r e also. e x c e p t that in this c a s e undamped o s c i l l a t i o n s (D = 0). a n d (2.4-13) m a k e s r e s i s t a n c e R infinite. In the m e n t i o n e d a n a l y s i s w e must. t h e r e f o r e . t a k e the c o e f f i c i e n t of v i s c o u s friction D i n s t e a d of R in s u c h c a s e s .
Example
3 D y n a m i c m o d e l s of different m e c h a n i c a l s y s t e m s
F i g u r e 2.4-9 s h o w s t h r e e s i m p l e m e c h a n i c a l s y s t e m s with g i v e n i n p u t a n d o u t p u t v a r i a b l e s . D e r i v e a d y n a m i c m o d e l for t h o s e s y s t e m s a n d d e t e r m i n e the e i g e n v a l u e s of s y s t e m matrix /~. T h e b a s i c a s s u m p t i o n s for the i n d i v i d u a l s y s t e m s a r e . a) the c y l i n d e r rolls without sliding.
12b
CFIAP. 2 LUMP~'I~ P R ~ C E S S K S
b) the d r i v i n g torque is transferred, without sliding, from drum 1 to drum 2, which lifts a l oad by winding a stiff, u n s t r e t c h a b l e rope, c) the mass a nd moment of inertia of the lever are n e g l e c t e d . ~
i
'- Z I : U
ol T.//////,~/2//////////////.
b)
~z = tJI
M=U 2 Z ~.tj [ . . . . . .
-4
¢3
"////////it,
F i g . 2 . 4 - 9 Three simple m e c h a n i c a l systems a) The e q u a t i o n d e s c r i b i n g relations b e t w e e n d i s p l a c e m e n t z of the c y l i n d er an d d i s p l a c e m e n t z~ of the e n d of the spring are o b t a i n e d by deriving the equations of equilibrium for horizontal forces a n d torques
sEc. 2.4
127
a r o u n d the c e n t r e of the c y l i n d e r (d'Alembert's p r i n c i p l e ) with the h e l p of Figure 2.4-10.
~ F,=clzl-z} }/////////,~W.X.Z~ X.~:.Z/////////W//F
N
Fig. 2 . 4 - I 0 The f o r c e s a c t i n g o n the c y l i n d e r The equilibrium of f o r c e s y i e l d s Mz" ; F o - F = c(z~ -z) - F
,
(2.4-41)
a n d the equilibrium of t o r q u e s a r o u n d the c y l i n d e r c e n t r e y i e l d s J,p" = Fr
(2.4-42)
S i n c e t h e r e is no s l i d i n g z = r,~, so (2.4-42) b e c o m e s
J z " = Fr 2 For the c y l i n d e r
(2.4-43) J = Mr2/2. so if F is e x p r e s s e d from the last equation.
(2.4-41} b e c o m e s
2c 2c z" , -~-~-z = ~ - - - ~ z l With the state v a r i a b l e s
(2.4-44) xt = z a n d
x2 = w , it follows that
|
[z,]
(2.4-45)'
~2~
CHAP.
2 LUMPED
PROCESSES
T h e e i g e n v a l u e s of matrix i~ a r e a c o m p l e x c o n j u g a t e p a i r
X1,2
= • 2~32~ ~ .j
(2.4-4~)
w h i c h s h o w s that this is a t y p i c a l second-order (proportional) system.
periodic
undamped
process
of
b) F i g u r e 2.4-11 s h o w s all the p o s s i b l e f o r c e s a n d t o r q u e s on e a c h the p a r t s of the load-lifting s y s t e m .
of
C
z,z~zN F
N]E
%0'1~'11~ )
Mg
Mz"
l~l,=U ] F i g . 2 . 4 - U F o r c e s a n d t o r q u e s on a l o a d - l i f t i n g s y s t e m T h e e q u a t i o n s for t o r q u e e q u i l i b r i u m a r o u n d the c e n t r e s of rotation A a n d B a n d for the e q u a l i t i e s of the v e r t i c a l f o r c e s a c t i n g on m a s s M a r e Jl ~01"
=
Mz
-
Frl
.
(2.4-4T)
J2~2" = Fz2 - Sr~
(2.4-48)
Mz" = S - Mg
(2.4-49)
Point C of d r u m 2, a s s u m i n g t h e r o p e is stiff a n d u n s t r e t c h a b l e , m o v e s at the s a m e v e l o c i t y as the r a i s i n g l o a d w = z °. a n d s i n c e t h e r e is n o sliding, the v e l o c i t i e s of p o i n t s E a n d D will b e e q u a l . T h e following r e l a t i o n s a r e thus fulfilled
sEc. 2.4
129 :
r2~ ~
Z'
]'2~a~
: Z"
r|o!
z2G)~ = ziG,}I'
I'2%o2" :
Z"
(2.4-50) r2G) 2 :
r2~o2" :
]l~1 'i
Using the a b o v e e q u a l i t i e s a n d e l i m i n a t i n g the e x p r e s s i o n s for F a n d S from the p r e c e d i n g t h r e e e q u a t i o n s , g i v e s the final m o d e l fox the l o a d ' s h e i g h t c h a n g e in the form of the following s e c o n d - o r d e r OLDE
J_.~
(M ,
r~
J2
(2.4-51)
+ '~--o) r t z " = Mz - M g r l T
al
or the s t a t e - s p a c e e q u a t i o n i
0Jill [0 al
A
bz~
(2.4-52}
a!
bzz
T h e e i g e n v a l u e s of matrix A a r e e q u a l Xl
:
X2
:
0
(2.4-53)
Thus w e h a v e a s e r i e s of two " p u r e " i n t e g r a l s y s t e m s with no time l a g . T h e s e i n t e r r e l a t i o n s a r e c l a r i f i e d in the b l o c k d i a g r a m p r e s e n t a t i o n in F i g u r e 2.4-12.
G Z"
Z
M
Fig.
2.4-12 B l o c k d i a g r a m of load-lifting s y s t e m
F i g u r e 2.4-12 s h o w s q u i t e c l e a r l y that a l o a d of m a s s M will rest o n l y if Mzb2t i.e. for
+
Mb22 = 0
130
CHAP.
2
LUMPED
Mz = M g r,
PROCEEEEE
(2.4-54)
In all o t h e r c a s e s the l o a d r a i s e s or d e s c e n d s c o n t i n u o u s l y to ( t h e o r e t i c ) infinity, w h i c h is a t y p i c a l a n d g e n e r a l l y k n o w n p r o p e r t y of integral systems. c) T h e f o r c e s c a u s e d b y motion of this m e c h a n i c a l s y s t e m a r i s e in the d a m p e r a n d in the s p r i n g . A c c o r d i n g to (2.4-17), the f o r c e in the d a m p e r is p r o p o r t i o n a l to the v e l o c i t y d i f f e r e n c e b e t w e e n the p i s t o n a n d the cylinder
(2.4-55)
Fp = D ( z ' - L~') T h e f o r c e in the s p r i n g is p r o p o r t i o n a l to its c o m p r e s s i o n Fo = cL~,
(2.4-56)
T h e e q u a l i t y of t h o s e two f o r c e s g i v e s D ( z ' - L~') = cL~
(2.4-57)
i.e. -D -%0
c
,
+ @
=
._D Z' c]/
If w e use the .first column of Table 2.4-2 a n d coefficients of inertia and resistance (2.4-~8), b e c o m e s I
I I
--~-,p' ÷ ,~ = --~--~z'
(2 4-58) introduce dynamic
(2.4-59)
i.e. d~o 1 , T--~-- + ~ = ---~-Tz
(2.4-60)
The time constant T = I/R has already b e e n obtained (see (2.2-1.24)), and the right-hand side of Equation (2.4-60) shows that the relationship between the displacement of cylinder piston z and angle of displacement %0 is given by a derivative system with first-order lag. In the dynamic sense this relationship is completely analogous to that existing between outflow mo and pressure Po or the cross-sectional area of valve A o, given by Equation (2.2-2.30) in the case of the gas storage tank. There is also a dynamic analogy with the process of heat conduction, with the
s E c . 24
131
r e l a t i o n s h i p b e t w e e n h e a t flow rate q2 a n d t e m p e r a t u r e of h e a t e d fluid 0f, w h i c h is s h o w n with an a p p r o x i m a t e d transfer function (3.2-51), i.e. (3.2-61). Eigenvalue
),1 =
I/T.
but
the
dynamics
of
this
process
is
also
c h a r a c t e r i z e d b y the z e r o v a l u e of the transfer function nominator, w h i c h is now in the origin, n, = 0. We must also s a y that d y n a m i c p r o c e s s e s a l w a y s b e of a d e r i v a t i v e c h a r a c t e r d i s p l a c e m e n t of the d a m p e r piston.
when
in m e c h a n i c a l s y s t e m s will the
input
variable
The next a n d last e x a m p l e in this s e c t i o n s h o w s a c a s e
is
the
that in fact
b e l o n g s to the following part of this c h a p t e r a b o u t p r o c e s s e s with l u m p e d p a r a m e t e r s , the part in w h i c h c o m p o s e d p r o c e s s e s of h i g h e r o r d e r will b e shown. This e x a m p l e c a n s e r v e as a kind of transition to the d e s c r i p t i o n of s u c h p r o c e s s e s , w h i c h c a n
be
b r o k e n d o w n into s e v e r a l
basic
ones
a n d w h o s e m o d e l is (at least) of s e c o n d or h i g h e r order. B e s i d e s , w e also want to s h o w a n e x a m p l e h e r e of a p r o c e s s that is o r i g i n a l l y n o n l i n e a r and
has
two states
of equilibrium.
One
of t h o s e
equilibrium
points
of
o p e r a t i o n is u n s t a b l e , a n d the other m a r g i n a l l y s t a b l e in this i d e a l i z e d frictionless c a s e , but s t a b l e in p r a c t i c e . We a l s o wish to s h o w that the same
models
obtained
for the
if we
use
motion of c o m p o s e d Lagrange's
equations
mechanical of motion
systems or
can
Newton's
be
laws.
Although the e x a m p l e will b e rather i d e a l i z e d a n d s e e m i n g l y o n l y of theoretic i m p o r t a n c e , w e must s a y that t h e s e are c a s e s that a p p e a r as p r o b l e m s of c o n t r o l in m o d e r n t e c h n i c a l p r a c t i c e .
Example
4 D y n a m i c s of a c o m p o s e d m e c h a n i c a l system. Stability of m o v i n g p e n d u l u m e q u i l i b r i u m state
F i g u r e 2.4-13 s h o w s an i d e a l i z e d p e n d u l u m w h o s e mass is c o n c e n t r a t e d at its e n d , a t t a c h e d to a cart that c a n m o v e in the v e r t i c a l z - y p l a n e u n d e r the i n f l u e n c e of d r i v i n g f o r c e F in the d i r e c t i o n of the h o r i z o n t a l z axis. Two
basic
positions
maintained by changes the simplest
are
shown
in
which
the
pendulum
must
be
in f o r c e F. (The first, u p r i g h t position r e p r e s e n t s
one-dimensional
model
that is u s e d
for c o n t r o l l i n g
rockets
132
CHAP.
2 UUMP~D
PROCESSES
d u r i n g takeoff, w h e n t h e y must b e m a i n t a i n e d in a n u p r i g h t position. T h e s e c o n d is the c a s e of a s u s p e n d e d p e n d u l u m , t o d a y often u s e d as a m o d e l for p l a n n i n g c o n t r o l p r o c e d u r e s w h e n l o a d s a r e t r a n s p o r t e d b y c r a n e a n d the task of their o p t i m u m c o n t r o l c a n b e r e g a r d e d either as a q u e s t i o n of the s m a l l e s t p o s s i b l e s w i n g i n g of the l o a d , or a s c o n l r o l l i n g l o a d t r a n s p o r t in minimum time.) L o s s e s d u e to friction in the joints a n d the b o d y ' s r e s i s t a n c e to motion a r e n e g l e c t e d . D e r i v e the n o n l i n e a r m o d e l of motion d y n a m i c s for the m o v i n g pendulum, linearize it, formulate state-space equations, determine e i g e n v a l u e s of the s y s t e m matrix A a n d on their b a s i s c o n c l u d e a b o u t the c h a r a c t e r i s t i c s of the t r a n s i e n t p r o c e s s a n d the stability of the e q u i l i b r i u m point. D e r i v e the m o d e l s : a - using Newton's laws (upright pendulum) b - formulating L a g r a n g e ' s e q u a t i o n s of motion ( s u s p e n d e d
M 2
o)
:
=
pendulum)
Z
b)
Fig.
2.4-13 Moving pendulum
a) F i g u r e 2.4-14 s h o w s the g e o m e t r i c a l r e l a t i o n s w h i c h will b e u s e d to d e r i v e the laws of motion. We must i m m e d i a t e l y o b s e r v e that if the c a r t m o v e s a l o n g the p a t h z, the d i s p l a c e m e n t of the p e n d u l u m in the d i r e c t i o n of the z a x i s e q u a l s z + L sin ~ .
133
SEC. 2.4
z
Lsin~O
LCOS~p
mn~
F L
A!
M,
[
z
I
"///..H/~/////H,/~//,/H,,, Fig. 2.4-14 Geometrical relations in The application of Newton's direction of the z axis yields d2z h/i,~-~
d2 + M 2 ~
(z + Lsin~,) = F
For rotation of the p e n d u l u m of m o m e n t u m
laws
an upright moving p e n d u l u m
of motion for translation in the
(2.4-61)
about point A the law for the conservation
is
d2
d2
[Mzd---~-(z ÷ L s i n e ) ] L c o s e - [Mz dt--d~--(2Lcose)]Lsine = MzgLsine (2.4-62) Before differentiating the a b o v e expressions we must mention that the pendulum's angular displacement is a time function (a d y n a m i c variable), which means d
dt
sine = cose'e'
d 2
d t 2 s i n e = -sin,p.q, '~ + c o s e e "
d d!
cose = -sinq,.e'
d2 dt2cose
= -cos~o.e .2 - sin~o.~p"
(2.4-63)
|~4
CHAP.
Referring to the a b o v e yields
expressions
and
2 LUh~fP~'D
differentiating the
PROC~'SS~'S
equations
(M~ + M 2 ] z " - M 2 L ( s i n ~ ) ~ '2 + M2L(cos~)~0" = F
(2.4-64)
Mzz"cose
(2.4-65)
÷ MzLe" = M2gsine
T h e s e two e q u a t i o n s are the n e n | | n o a l " m o d e l for the motion d y n a m i c s of an u p r i g h t m o v i n g p e n d u l u m u n d e r the i n f l u e n c e of f o r c e F. The simplest w a y to l i n e a r i z e
this m o d e l
is to u s e
the fact that for
small d e v i a t i o n s from the upright, equilibrium position sin,p = ~, cos~0 ; I. a n d b e c a u s e of its small v a l u e , the p r o d u c t ~0 • ,p,2 c a n b e m a d e e q u a l to zero. This g i v e s us the following linear m o d e l for the motion of this system, w h i c h is s a t i s f a c t o r y foi small a n g u l a r d i s p l a c e m e n t s . (MI ÷ M z ) z " ÷ M 2 L e " = F
.
(2.4-66)
M~z" * M2L~" = M2g~
(2.4-b7)
To o b t a i n m o d e l s in the form of matrix s t a t e - s p a c e e q u a t i o n s the state v a r i a b l e s must b e s e l e c t e d . As until now, in m e c h a n i c a l p r o c e s s e s of motion the most natural s e l e c t i o n will also b e m a d e . Here state v a r i a b l e s are the d i s p l a c e m e n t s a n d v e l o c i t i e s of the c a r t a n d the p e n d u l u m . Thus for XI = Z
x2 = z' = w X3
=
(2.4-68]
~
the linear m o d e l o b t a i n e d c a n b e t r a n s f o r m e d into this final form Z
r!w
0
1
0
0
0
0
0
0
0
--~g
0-
"z-
0
I
W
0 MI+M2 M1L g
1
0
,
0
-~
l
]
!,
[F]
(2.4-6Q)
SEC.
135
2.4
X'
=AX4"BU
S o l v i n g e q u a t i o n (XI e q u a t i o n for that m o d e l MI+M2
)2()2
E
A)
;
0
gives
the
following
, gJ
(2.4-'ro)
0
"
characteristic
from w h i c h the following e i g e n v a l u e s follow d i r e c t l y k1=0
~2 = 0
/ MI÷M2 (2.4-71)
/ MI÷M2 -
As until n o w in the c a s e of t r a n s l a t i o n , the e i g e n v a l u e s XI = k 2 - 0 b e l o n g to c h a n g e s in the p o s i t i o n z a n d v e l o c i t y w of the cart. T h e e i g e n v a l u e X3 i s p o s i ( i v e , a n d s h o w s that the e q u i l i b r i u m position of the upright pendulum is unstable in the s e n s e that if o n e v a r i a b l e (in this c a s e the a n g u l a r d i s p l a c e m e n t of p e n d u l u m ,p) is d i s t u r b e d for e v e n the s m a l l e s t a m o u n t it will i r r e v e r s i b l y d e p a r t further a n d further a w a y f r o m the e q u i l i b r i u m p o s i t i o n ,p = O. k4 is a r e a l a n d n e g a t i v e v a r i a b l e a n d r e l a t e d to c h a n g e s in a n g u l a r v e l o c i t y . b) T h e L a g r a n g e e q u a t i o n s of motion a r e g i v e n as follows d
3L
- ~,.~, ' ~o~. ) .1 L
...
T
...
V D O q k
.
aL
aD
" ~q--/- ÷ ~
'~ O I ,
(i = 1. 2 . . . . .
k)
(2.4-72)
Lagrangian. L - T - V kinetic e n e r g y ... p o t e n t i a l e n e r g y ... R a y l e i g h ' s d i s s i p a t i o n function ... e x t e r n a l ( g e n e r a l i z e d ) f o r c e in the d i r e c t i o n of the i-th c o o r d i n a t e ... g e n e r a l i z e d coordinate _. d e g r e e of f r e e d o m of the m e c h a n i c a l s y s t e m
E q u a t i o n (2.4-72) s h o w s a s y s t e m of k (in the g e n e r a l c a s e n o n l i n e a r ) differential e q u a t i o n s of the s e c o n d o r d e r .
13~
CHAP.
The
2 LUMPED
PROCESSES
moving
suspended p e n d u l u m is a n e x a m p l e of a m e c h a n i c a l of f r e e d o m . T h e g e n e r a l i z e d c o o r d i n a t e s of that motion a r e v a r i a b l e z for t h e t r a n s l a t i o n a n d the a n g u l a r d i s p l a c e m e n t ,p for the rotation of the p e n d u l u m .
s y s t e m with two d e g r e e s
T h e kinetic e n e r g y of the s y s t e m is T=
l
~-M,w,
2
1
2
+ ~-M2w.
(2.4-73)
T h e v e l o c i t y of the c a r t wt = z, a n d w2 is the a b s o l u t e v e l o c i t y of m a s s M2 w h i c h is o b v i o u s l y c o m p o s e d of v e l o c i t y c o m p o n e n t s in the d i r e c t i o n of the z a x i s a n d the y axis. T h e r e f o r e . F i g u r e 2.4-14 a n d E q u a t i o n (2.4-63) yield w22 - (z + L s i n ~ ) '2 * ( L c o s ~ ) '2 = z '2 ÷ LZ~ 'z ÷ 2 z ' L c o s ~ . , p '
(2.4-74)
S u b s t i t u t i n g this e x p r e s s i o n for v e l o c i t y into (2.4-73) y i e l d s 1
T = ~-Mtwl
2
l.
.
+ -~mzlZ
,2
+ L2~o'2 + 2z'Lcos~o'~')
(2.4-75)
T h e p o t e n t i a l e n e r g y of the s y s t e m is V -- MzgL(1 - c o s e )
(2.4-75)
It is c l e a r that in the e q u i l i b r i u m s t a t e V = O. w h i c h results from the u p p e r e q u a t i o n for ,p = O. T h e L a g r a n g i a n is n o w not difficult to c a l c u l a t e 1 , / L = T - V ---~-M,wl +
M 2 ( z ' 2 + L2,p '2 ÷ 2 z ' L c o s ~ 0 . , p ' ) - M 2 g L ( I - cosec) (2.4-77)
T h e e q u a t i o n s of m o t i o n d i r e c t l y from (2.4-72)
dt
d dt
az'
az
aL
(__3
the
generalized
coordinates
follow
'
aL -
along
=
0
(2.4-78)
SEC. 2.4
137
S u b s t i t u t i n g E q u a t i o n (2.4-77) for L into this s y s t e m o f e q u a t i o n s a n d c a r r y i n g out the n e c e s s a w o p e r a t i o n s g i v e s the d e m a n d e d d e s c r i p t i o n of the motion d y n a m i c s of a m o v i n g s u s p e n d e d p e n d u l u m (M, * M2)z" - M2L(sin~o)~o'2 * M2L(cos~o)~o" = F
(2.4-79)
Mzz"cos~o + M2L~"
(2.4-80)
:
- Mzgsin,p
A g a i n this is a m o d e l c o m p o s e d of two n o n l i n e a r s e c o n d - o r d e r ODE, w h i c h differs from the m o d e l of the u p r i g h t p e n d u l u m in the s i g n of the f i g h t - h a n d s i d e of the s e c o n d e q u a t i o n . H o w e v e r . this "small'* d i f f e r e n c e l e a d s to e s s e n t i a l l y different results in the w a y in w h i c h this s y s t e m moves. L i n e a f i z a t i o n a n d the s e l e c t i o n of s t a t e v a r i a b l e s is c a r r i e d out in the s a m e w a y a s in the p r e c e d i n g c a s e . w h i c h g i v e s the following linear m o d e l for a m o v i n g s u s p e n d e d p e n d u l u m in the form of matrix s t a t e - s p a c e equations ["Z
[ i W I i : '#
=
(.,O
0
l
0
O-
0
0
-~-g
0
0
0
0
1
0
0
MI*M2 M~L g
o 1
-z w
0
M~
*
[F]
(2.4-81)
f~
T h e c h a r a c t e r i s t i c e q u a t i o n of matrix ~ is n o w X2(X 2 - a,3) = 0
(2.4-82)
from w h i c h follow the e i g e n v a l u e s
for a m o v i n g s u s p e n d e d
pendulum
},l=O x2=O
/ M,*M2 (2.4-83)
/ Ml*Mz = _ l / : g
J
138
CHAP. 2 I..UI~P'~'I~ PROC~'SSES
T h e e i g e n v a l u e s )`~ a n d )`2 b e l o n g to translational motion of the cart a n d h a v e not c h a n g e d , which was to b e e x p e c t e d s i n c e t h e r e was no c h a n g e in that motion. H o w e v e r . the c h a n g e a n d ),4 a r e a c o m p l e x part e q u a l s zero), which transient p r o c e s s similar
in the s e c o n d pair of e i g e n v a l u e s is essential. )`3 c o n j u g a t e pair ~on the i m a g i n a r y axis (their zeal means they characterize an undamped periodical to p r o c e s s e s that h a v e a l r e a d y b e e n e n c o u n t e r e d .
T h e f r e q u e n c y ~, of the oscillation ( s w i n g i n g ) is / M,÷M2
: 1/~g
(2.4-84)
T h e s e e i g e n v a l u e s , n e i t h e r of w h i c h is in the r i g h t - h a n d s e m i p l a n e of the c o m p l e x p l a n e of c h a r a c t e r i s t i c E q u a t i o n (2.4-82) solutions, s h o w that the position of equilibrium for the s u s p e n d e d p e n d u l u m (~0 = O) is no l o n g e r u n s t a b l e . All the same, s i n c e all p o s s i b l e r e s i s t a n c e s h a v e b e e n n e g l e c t e d , w e must s a y that this position is not a s y m p t o t i c a l l y s t a b l e in the s e n s e that after a d i s t u r b a n c e the a n g u l a r d i s p l a c e m e n t of the p e n d u l u m will g r a d u a l l y return to its initial v a l u e ~ = O. With this c h o i c e of a s s u m p t i o n s , after d i s t u r b a n c e the p e n d u l u m will swing with f r e q u e n c y around the initial position of equilibrium without damping.
SEc. 2.s
13Q
B. COMPLEX SYqTEMS OF H I G H E R ORDER
2 . 5 E X A M P L E -q OF COMPLEX SYSTEMS In all the p r e c e d i n g s e c t i o n s of this c h a p t e r o n p r o c e s s e s with t u m p e d p a r a m e t e r s (with the e x c e p t i o n of the last e x a m p l e in S e c t i o n 2.4, w h i c h r e p r e s e n t s a transition to d e s c r i p t i o n s of h i g h e r - o r d e r p r o c e s s e s ) w e met with p r o c e s s e s w h o s e d y n a m i c s w a s d e s c r i b e d b y DE of no m o r e t h a n s e c o n d o r d e r . S u c h p r o c e s s e s h a v e b e e n n a m e d b a s i c , w h i c h a l l o w s us to c o n c l u d e that t h e y c o u l d b e p a r t s of e v e n the most c o m p l e x s y s t e m s , i.e. that c o m p l e x s y s t e m s c a n b e r e d u c e d to v a r i o u s c o m b i n a t i o n s of t h o s e b a s i c p r o c e s s e s . This is v e r y f r e q u e n t l y d o n e in p r a c t i c e w h e r e m o d e l s of e x t r e m e l y c o m p l i c a t e d t e c h n o l o g i c a l s y s t e m s a r e e n c o u n t e r e d . For c e r t a i n p u r p o s e s of a n a l y s i s or s y n t h e s i s of c o n t r o l a l g o r i t h m s s u c h s y s t e m s a r e d e s c r i b e d b y m o d e l s of fourth, third, a s well as s e c o n d or first order. D e p e n d i n g on the d e m a n d s a n d g o a l s that must b e satisfied, o p p o s i t e situations a l s o e x i s t w h e n a s e e m i n g l y s i m p l e o b j e c t is d e s c r i b e d b y a d y n a m i c m o d e l of h i g h order. T h e c o n c e p t s of s i m p l e a n d c o m p l e x a r e o b v i o u s l y r e l a t i v e a n d w e must d i s t i n g u i s h , e s p e c i a l l y from the a s p e c t of simplicity, the p r o c e s s from its m o d e l . All c o m b i n a t i o n s a r e p o s s i b l e , c o m p l e x p r o c e s s - s i m p l e m o d e l , s i m p l e p r o c e s s - c o m p l e x m o d e l , a n d so on. This s e c t i o n will. t h e r e f o r e , a n a l y z e c a s e s w h e n the m o d e l is of s e c o n d or of m o r e t h a n s e c o n d order, a n d the p r o c e d u r e for o b t a i n i n g m o d e l s a n d the u s u a l a n a l y s i s will b e s h o w n o n s e v e r a l s i m p l e c a s e s . C o m p l e x p r o c e s s e s d e s c r i b e d b y h i g h e r - o r d e r m o d e l s a r e met in v a r i o u s situations. In m o d e l i n g the d y n a m i c s of a l a r g e p l a n t it is u s u a l to b r e a k its s t r u c t u r e d o w n into e l e m e n t a r y d e v i c e s : p u m p s , r e a c t o r s , p i p e s , v a l v e s a n d the like, a n d t h e n m o d e l e a c h of t h o s e p a r t s s e p a r a t e l y . I n t e r c o n n e c t i n g all the b a s i c m o d e l s g i v e s a h i g h - o r d e r m o d e l for the w h o l e plant. Another c a s e w h e n a h i g h - o r d e r m o d e l is o b t a i n e d is for p r o c e s s e s that o c c u r in o b j e c t s of s i m p l e g e o m e t r i c a l s t r u c t u r e within which simultaneous mass, e n e r g y a n d momentum c h a n g e s take place. D e r i v i n g all the d e m a n d e d e q u a t i o n s l e a d s to a h i g h e r - o r d e r m o d e l . Finally. in t e c h n i c a l p r a c t i c e the d y n a m i c s of p r o c e s s e s with d i s t r i b u t e d
140
CHAP.
2 LUMPED
PROCESSES
p a r a m e t e r s is u s u a l l y i n v e s t i g a t e d u s i n g s p a t i a l d i s c r e t i z a t i o n . This m e a n s that a c o n t i n u o u s p r o c e s s is d i v i d e d into e l e m e n t s of finite d i m e n s i o n a n d then
balance
(conservation)
equations
are
derived
for
each
of
those
e l e m e n t s . T h e r e is a l s o a n o t h e r w a y : first PDE a r e f o r m u l a t e d d e s c r i b i n g the dynamics of the distributed process and then mathematical d i s c r e t i z a t i o n is p e r f o r m e d with r e s p e c t to the s p a t i a l v a r i a b l e s . Both p r o c e d u r e s l e a d to the s a m e result in the s e n s e that i n s t e a d of o n e or s e v e r a l PDE a s y s t e m of ODE is o b t a i n e d , w h i c h c a n then b e e l a b o r a t e d further a n a l y t i c a l l y or b y n u m e r i c a l p r o c e s s i n g on a c o m p u t e r . Thanks
to c a l c u l a t i n g
order are derived
aids
computers,
and processed.
today
models
of v e r y
H o w e v e r . for u n d e r s t a n d i n g
high
the b a s i c
p r o p e r t i e s of a m o d e l it is irrelevant w h e t h e r it is of 5th. 55th, 105th or lO05th order. With v e r y h i g h o r d e r s n e w p r o b l e m s c o n n e c t e d with c o m p u t e r p r o c e s s i n g a r e e n c o u n t e r e d , but t h e s e are not the c o n c e r n of this b o o k . T h e r e f o r e . the following e x a m p l e s will remain simple so as to b e c l e a r a n d e a s y to u n d e r s t a n d . T h e y c e r t a i n l y i n c l u d e the e x a m p l e of the m o v i n g p e n d u l u m from the p r e c e d i n g s e c t i o n .
Example Derive a
I T h r e e liquid tanks in series mathematical
model
of d y n a m i c s
for three
liquid
tanks
in
series, F i g u r e 2.5-1, a n d after linearization d e t e r m i n e s y s t e m matrix A input matrix ]8 a n d find the e i g e n v a l u e s . N e g l e c t liquid inertia in tanks a n d p i p e s . The following d a t a are g i v e n : Ai
= I m2
dl
= 0.045
m
LI
= 40
m
)`t = 0 . 0 4
mh
= 3
kg S
A2 = 0.5 m 2
d2 = 0.O5 m
L~ = 1 m
X~ = 0.04
mi2 = 2 k g s
A3 = 1 m 2
d3 = 0.06 m
L3 = 20 m
),3 = 0.04
mi3 = 2 kg S
The p r e s s u r e d r o p o n the v a l v e is ~Pv = 1 bar c r o s s - s e c t i o n a l a r e a of the v a l v e is av " O.OOl m 2. The
dynamic
model
is
obtained
in
the
usual
and
way
by
the
mean
deriving
e q u a t i o n s for the c o n s e r v a t i o n of mass for e a c h of the tanks. B e s i d e s t h o s e e q u a t i o n s , h o w e v e r , it is also n e c e s s a r y to d e r i v e a l g e b r a i c
sEc. 2.5
141
e x p r e s s i o n s for liquid flow t h r o u g h t h e s y s t e m u s i n g Bernoulli's e q u a t i o n s .
_
rail
rot2
1
l
---
I,.II
.
:_ __..
mi3
.--_~
h.o, H_. L! It= -~1 =0
AH!
Iz=O
~Hz
~2
I3= L-'I O) =0
~'t3
,e}"J3
. o ... . . _
,
/
Fig.
L..IK.H /
2 . 5 - 1 T h r e e liquid t a n k s in s e r i e s a n d a b l o c k d i a g r a m of the i n t e r r e l a t i o n s
mh - mot = A ,p. dH: ~-
(2.5-I)
dH2 mi2 + too, - mo~ = A 2 p ~ mi3 ÷ too2
- mo3
=
w~ pgH3 = - ~ p
(2.5-2)
dH3 A3p dt
L, p g ( H i " H2) = ~Pc: = ),ldl pg(Hz " H3) -- ~Pv
.
(2.5-3)
1 2 2a~pmol
(2.5-4)
.
m2o3 L~ + ~Pc3 = ~ + X3d3
(2.~-6) I pm2o3 2-
(2.5-b}
The s t e a d y s t a t e for the g i v e n m a s s flow r a t e v a l u e s a n d the tank s y s t e m must first b e d e t e r m i n e d , a n d t h e n l i n e a r i z a t i o n a b o u t that o p e r a t i n g p o i n t p e r f o r m e d . E q u a l i z i n g the r i g h t - h a n d s i d e s of the first three e q u a t i o n s with z e r o y i e l d s
142
CHAP.
'2 L U M P E D
PROCESSES
mi, = mo~ = 3 -kg S
mo2 =
mil ÷ mi~ = 5 kg S
m.o3 = ml, ÷ mI2 ÷ mi3 -- T k g S
T h e f o l l o w i n g l i q u i d l e v e l s will b e e s t a b l i s h e d I:I3 = (),,
:}
~
Pg
+ ~
~2 = ~Pv + ~3 = 14.77 m pg I
mZ°'
= 4.57 m
,
+ R2 = 21.21
LineaHzing the model following linear medel
in t h e t a n k s
as shown
m
in S e c t i o n s
2.1-I a n d
2.2-] g i v e s
A m i l - Amol = C~ d A H i dt
(2.5-7)
Ami2
= C 2 d ~d't H ~
(2.5-8)
= C3 d A dtH 3
(2.5-9)
+ Z~mo, - A m o 2
Ami3 + A m 0 2
- Am03
aH~ - AH2 = RCXmol AH2
- AH3
= Rzamo2
(2.5-10) -
R2KAAav
(2.5-li)
~H3 = R3~mo3
The
(2.5-12)
resistance
and
capacitance
are
obtained
expressions RI =
2(lqr[q2}
= 4.3
CI -- A10 ;
R~ =
2(~'~3)
= 4.08
C z = 500
Tz = 2040
C3 = I 0 0 0
T3 = 1300
mot
too2
R3 = ~21213
= 1.3
mo3
KA -
mo2 av
the
= 5000
I000
Ti = 4300
from
the
known
sEc. a.s
143
If the internal v a r i a b l e s tool, too2 a n d mos a r e e x c l u d e d from the last y i e l d s this s t a t e - s p a c e s y s t e m of e q u a t i o n s , a little r e a r r a n g e m e n t description I
: ~H,I
AH21 "
i
I
°
l
l
TI
T~
R~ R)
.(_=~=i +i)
I
T2
T2
T2
R3
-(,~--÷I)
R2 T3
T3
0
i
AH~
AH2
AH3
A
I
0
1
0
-C~
O
0
0 0
O
Ami!
KA
Aml2
(2.s-13)
C2
]
KA
C3
C~
Amia Aav
After s u b s t i t u t i n g into matrix A the v a l u e s of the c o e f f i c i e n t s , a c o m p u t e r c a n b e u s e d to o b t a i n the n u m e r i c a l v a l u e s of the e i g e n v a l u e s for this c a s e ),; = - 0 . 0 0 0 0 8 6 .
),2 = - 0 . 0 0 0 7 3 6 ,
X3 = - 0 . 0 0 1 3 7 8
T h r e e n e g a t i v e , r e a l e i g e n v a l u e s s h o w that this is a s t a b I e e q u i l i b r i u m state of t h r e e first-order p r o p o r t i o n a l s y s t e m s in series, w h e r e t h e r e is f e e d b a c k a c t i o n of e a c h s u c c e s s i v e m e m b e r on the p r e c e d i n g o n e . This c a n b e s e e n o n the b l o c k d i a g r a m o n F i g u r e 2.5-1.
Example
2 T w o - p h a s e fluid t a n k
F i g u r e 2.5-2 s h o w s a t w o - p h a s e fluid t a n k with c o m p l e t e l y s e p a r a t e d p h a s e s . It is f e d b y mj~ k g / s s a t u r a t e d w a t e r a n d mi2 w a t e r a n d s t e a m
14,4.
CHAP. 2 L U M P E D
m i x t u r e with s t e a m
quality
x
and
enthalpy
h2. mm
kg/s
PROCESSES
dry
saturated
s t e a m a n d me2 k g / s s a t u r a t e d w a t e r a r e t a k e n from t h e tank. D e v i c e s that a r e not s h o w n on F i g u r e 2.5-2 c o m p l e t e l y s e p a r a t e t h e m i x t u r e miz into the liquid and
the v a p o r
phase.
D e r i v e t h e m o d e l for u n s t e a d y
changes
of
p r e s s u r e P a n d the liquid p h a s e
l e v e l H, i.e. t h e w a t e r v o l u m e Vw. All the
thermodynamic
for
state
equations
saturation
variables
are
known,
i,e.
t h e r e a r e a n a l y t i c e x p r e s s i o n s of t h e t y p e h' = h'(P), e' = 0' (P), v" = v"(P) a n d so on. T h e b a s i c a s s u m p t i o n , b e s i d e s t h e m e n t i o n e d p h a s e s e p a r a t i o n , is that t h e r e is n o m a s s t r a n s f e r o n t h e c o n t a c t change,
and
that b o t h p h a s e s
surface due
are homogeneous,
to p r e s s u r e
i.e. t h e r e a r e n o w a t e r
d r o p s in t h e s t e a m , nor c a n s t e a m b u b b l e s form in t h e w a t e r . The
model
will b e
derived
by
the
same
method
that
was
E x a m p l e I. S e c t i o n 2.1-2. w h e n a d y n a m i c m o d e l w a s o b t a i n e d s t o r a g e tank. In that c a s e , h o w e v e r , b e c a u s e
used
in
for a g a s
isothermal processes
in the
t a n k w e r e a s s u m e d , no e n e r g y e q u a t i o n w a s f o r m u l a t e d , S i n c e t e m p e r a t u r e dynamics
was
neglected
in
dynamics
was
of first o r d e r
Equation and
(2.1-2.4)
obtained
the
only
model
from
the
for
pressure
law
for
the
c o n s e r v a t i o n of m a s s a n d t h e g a s e q u a t i o n . H e r e the s i t u a t i o n is s o m e w h a t more c o m p l e x . T h e total m a s s M of the fluid a n d
its i n t e r n a l e n e r g y
U, i,e. t h e h e a t
within the t a n k of c o n s t a n t v o l u m e V, a r e c o m p l e t e l y d e t e r m i n e d
by
two
v a r i a b l e s : p r e s s u r e P a n d t h e v o l u m e of l i q u i d p h a s e Vw, M = g l ( P , V w ) . U = g2(P,Vw). In c o n d i t i o n s of m a s s a n d
energy
constant
unbalanced
values.
If the
flows
describe
them
occur
and
to
mass
and
energy
become
equations
for
flow e q u i l i b r i u m t h e s e a r e
the
unsteady
phenomena
conservation
of
must b e f o r m u l a t e d . For t h e t a n k u n d e r c o n s i d e r a t i o n
here they are dM mi,
+ mi2 -
mol
- mo2 =
mi2h2 - ( m o z - m i l ) h '
(2.5-14)
dt
- mmh"
= ddU t
(2.5-I5)
T h e total mass a n d energy in the tank comprises the mass a n d energy of the liquid a n d the vapor p h a s e ,
(2.5-1~)
U = Vwp'U' + (V - Vw)p"u"
(2.5-17)
M = Vwp' * (V - Vw)p"
sEc.
2.s
145
I m°lh,
..:....-:::,:i.!;i.i-:z.
~~...'.'~:.'.'.;:,':...:".':,".,:p.'.."
- - - - - - - V,; = '
Fig.
l H
---------
2.5-2 Two-phase
fluid tank
Substituting the known equality eu = 0h - P
.
(2.5-18)
into t h e p r e c e d i n g
expression,
we get
U ; Vwp'h' * (V - Vw)p"h" - VP Here
we
must
point
substituted by enthalpy, lead
to t h e
out
that
(2.5-19) in
litezature,
internal
w h i c h w o u l d ( h a d it b e e n
disappearance
of
the
last
term
done
on
the
energy
is
often
in t h i s d e f i ; ~ a t i o n ) right-hand
side
of
E q u a t i o n (2.5-19). T h i s s u b s t i t u t i o n l e a d s
to t h e a p p e a r a n c e
e r r o r in t h e " d y n a m i c "
t e r m of E q u a t i o n
(2.5-15) ( t h e t e r m o n its r i g h t - h a n d
side)
the
thus
influencing
transient
process
but not
the
of a " d y n a m i c " steady
state
as
well. For the variables
needs
of f u r t h e r
on the boundary
derivation curves
we
must
are functions
remember of p r e s s u r e
that
the
state
only, and
all of t h e m w e c a n w r i t e d d---t =
0 aP
dP dt
(2.5-20)
for
146
CHAP.
2 LUMPED
PROCESSES
S u b s t i t u t i n g (2.5-161 a n d (2.5-191 into the first two e q u a t i o n s of this e x a m p l e , i e f e n i n g to (2.5-201. after a little r e a r r a n g e m e n t w e g e t t h e m o d e l for the d y n a m i c s of p r e s s u r e a n d v o l u m e of the liquid p h a s e in the form of the folIowing two n o n l i n e a r DE
d___P=(mi,+mi2-mo,-mos)({)'h'-~"h") (mi2h 2-(mos-mh)h'-mo,h")(9'-p") dt
N
N (2.5-21) I
dVw (mi'+m~'m°"m°2] dt
II
" ( - ~00 -Vw
p'-
÷ - ~ ( p V-Vw)} dP dt
p'°
N = N~ - N2
(2.5-22)
(2.5-23)
N, = (0'h'- p"h")(--~--Vw a~' ÷
~'__~ _ (V-Vw)1
,
N"2=~(~-~ , , ,,~r ap',, ,_,+--~FVw() ah', , . ~ " (V-Vw)h"+-~-~!~ Oh" V-Vw)g"-V ] &--~-Fvwn
(2.5-24) (2.5-25)
E q u a t i o n s (2.5-211 a n d (2.5-22) a r e the d e m a n d e d n o n l i n e a r m a t h e m a t i c a l m o d e l of d y n a m i c p r o c e s s e s o c c u r r i n g in the t w o - p h a s e fluid tank. As the g i v e n e x p r e s s i o n s show, to s o l v e t h e m it is n e c e s s a r y to h a v e all the f u n c t i o n a l d e p e n d e n c i e s of t h e r m o d y n a m i c s a t u r a t i o n v a r i a b l e s a n d their d e r i v a t i v e s with r e f e r e n c e to p r e s s u r e P. By their n a t u r e t h e s e a r e c l u m s y e x p r e s s i o n s , a n d e v e n in the c a s e of their s i m p l e s t linear a p p r o x i m a t i o n t h e a b o v e s y s t e m of DE is p r a c t i c a l l y i n s o l v a b l e without a digital computer. But in this c a s e the s t e a d y s t a t e c a n a l s o b e c a l c u l a t e d a n a l y t i c a l l y . After the r i g h t - h a n d s i d e s of (2.5-141 a n d (2.5-15) a r e m a d e e q u a l to z e r o . t h e s e o b v i o u s e q u a l i t i e s result mi,
*
ml2
=
tool
÷
moz
mi2h'2 = (mo2 - ~..)E' + ~'otE"
(2.5-20) (2.5-27)
The last e x p r e s s i o n s are completely clear and represent the e q u i l i b r i u m of m a s s a n d e n e r g y flows in the g i v e n s t e a d y state. H e r e it is i m p o r t a n t to p o i n t out that a n e w s t e a d y s t a t e will e s t a b l i s h itself o n l y in the c a s e of limited c h a n g e in h 2. In the c a s e of a finite c h a n g e in a n y other i n p u t ( d i s t u r b a n c e ) v a r i a b l e (mil, mi2. mol. mo2) no s t e a d y s t a t e c a n b e e s t a b l i s h e d a n d the t a n k s h o w s p r o p e r t i e s of a n i n t e g r a l s y s t e m . In such cases mass and energy equilibrium are permanently disturbed and
s E c . 2.s
147
the v a r i a b l e Vw c o n v e r g e s to t h e o r e t i c a l l y infinite c h a n g e , w h i c h w o u l d in p r a c t i c e l e a d to the c o m p l e t e d i s a p p e a r a n c e of the liquid or the v a p o r p h a s e in the tank. The s a m e c o n c l u s i o n is r e a c h e d after the m o d e l I s l l n e r l z e d . The functional d e p e n d e n c e of v a r i a b l e s Vw a n d P g i v e n in the m o d e l c a n a l s o b e e x p r e s s e d a s follows dVw dt : f~ (Vw. P. rail. mlz. mot. moz. h2}
(2.5-28)
dP dt = f2 (Vw. P. mi,, ml~. tool, mo2, h2}
(2.5-29)
q
w
statevariable sinput L i n e a r i z a t i o n is c a r r i e d out in the u s u a l m a n n e r b y d i f f e r e n t i a t i n g the a b o v e e q u a t i o n s ( s e e A p p e n d i x } . but h e r e w e will not write out the c o m p l e t e a n a l y t i c a l e x p r e s s i o n s for e a c h s p e c i f i c c o e f f i c i e n t b e c a u s e of their a w k w a r d n e s s . It is, n e v e r t h e l e s s , useful to s h o w w h a t m a t r i c e s A a n d B a r e built of in this c a s e of a t w o - p h a s e fluid tank. AVw
[Ap ]
:
/
aVw
~P
AV
[~-V-ja~ ~r~p o Ap, j A a{',
a~',
air~
air,
amis
aml2
amo,
amo2
amit
ami2
amot
amoz a/2 ahz O
~h2
Amol l
amo21
(2.5-3o]
/,h2 J
B T h e partial d e r i v a t i v e s s h o w n in E q u a t i o n (2.5-30) a r e the d i f f e r e n t i a t i o n of the r i g h t - h a n d s i d e of E q u a t i o n (2.5-22} for the first row of m a t r i c e s A a n d B, a n d of the r i g h t - h a n d s i d e of E q u a t i o n (2.5-21) for the s e c o n d row of t h e s e m a t r i c e s , with r e s p e c t to the v a r i a b l e s of s t a t e a n d input. T h e line a b o v e the functions a n d the s u b s c r i p t O s h o w that the partial d e r i v a t i v e s must b e c a l c u l a t e d in a s p e c i f i c o p e r a t i o n a l ( s t e a d y ) state. It must a l s o b e s a i d that the first c o l u m n of matrix A is c o m p o s e d of z e r o s . i.e. the c o e f f i c i e n t s a,, a n d a2, a r e e q u a l to z e r o . w h i c h s h o w s that o n e of
14~
CHAP.
the matrix e i g e n v a l u e s
2 LUMPED
will b e z e r o a n d to it will c o r e s p o n d
PROCESSFS
a n infinitely
g r e a t time c o n s t a n t . In other w o r d s , the v a r i a b l e to w h i c h that e i g e n v a l u e is r e l a t e d n e e d s a n infinitely l o n g time to r e a c h a n e w s t e a d y state. Or to put it e v e n m o r e c l e a r l y - that v a r i a b l e c a n n o t r e a c h a n e w s t e a d y v a l u e in a finite p e r i o d of time. That v a r i a b l e , as h a s a l r e a d y b e e n s a i d . is the v o l u m e of the liquid p h a s e Vw. T h e s e c o n d e i g e n v a l u e e q u a l s the c o e f f i c i e n t a~2 a n d in p r a c t i c e this is a n e g a t i v e real v a l u e c h a r a c t e r i s t i c of a p e r i o d i c p i o p o r t i o n a l first- o r d e r s y s t e m s . T h e p r o c e s s of p r e s s u r e P c h a n g e s in the t a n k will s h o w t h e s e d y n a m i c c h a r a c t e r i s t i c s . Finally, it must b e r e p e a t e d that s i n c e b o t h m a s s a n d e n e r g y a r e s t o r e d in the tank. this is a s e c o n d - o r d e r m o d e l for c h a n g e s in the v o l u m e of liquid p h a s e Vw (or in the liquid l e v e l H. b e c a u s e t h o s e two v a r i a b l e s a r e u n i q u e l y l i n k e d b y a l g e b r a i c e x p r e s s i o n s ) a n d p r e s s u r e P in the tank. In this c a s e , Vw s h o w s p r o p e l t i e s of a n i n t e g r a l a n d P of a p r o p o r t i o n a l system.
w - - x a m p l e 3 G e a r train s y s t e m
J,i
°'I N Fig.
'21
U @
U
2 . 5 - 3 G e a r train s y s t e m
D e r i v e a m o d e l for d y n a m i c c h a n g e s of a n g u l a r d i s p l a c e m e n t s a n d a n g u l a x v e l o c i t i e s for the g e a r train s y s t e m s h o w n o n F i g u r e 2.5-3. if the interrelations a r e linear. All the g e o m e t r i c v a r i a b l e s a n d the c o e f f i c i e n t s ct a n d Dt (torsional s p r i n g c o n s t a n t a n d torsional c o e f f i c i e n t of v i s c o u s friction) a r e g i v e n . Loss d u e to the friction of shaft 1 a n d the stiffness of
s E c . 2.5
149
shaft 2 a r e n e g l e c t e d . T h e i n p u t d i s t u r b a n c e v a r i a b l e s a r e d r i v i n g t o r q u e Mzp a n d l o a d t o r q u e Mzt. T h e m o d e l is to b e s h o w n in the form of state-space equations. To
derive
the
dynamic
model
of this
mechanical
system
we
must
formulate e q u a t i o n s for m o m e n t u m c o n s e r v a t i o n for the rotation of shafts 1 a n d 2. For shaft l ]1 'v~' =
" ct~ol " M z l + Mzp
.
(2.5-31)
w h e r e Mzl is the t o r q u e t r a n s m i t t e d to shaft 2. for w h i c h J2~o~ = - D t ~
- M z t ÷ Mz2
(2.5-32)
Mz2 is t r a n s m i t t e d to shaft 2 t h r o u g h g e a r s .
T h e r e is a l s o he,
= r2 ,(,2
(2.5-33)
M z l e~ = Mz~ e~,
(2.5-34)
T h e last e q u a t i o n e x p r e s s e s
the e q u a l i t y of the work d o n e
by gear
I
a n d g e a r 2. T h e a b o v e e q u a t i o n s y i e l d
Mzl
~o2
Mz~
~ot
(2.5-35)
rl r2
Referring to (2.5-35). w e s e e that (2.5-32) b e c o m e s (2.5-30)
]2qo~ + Dt~o~ + Mzt = Mz2 = M z 1 ~
If Mz, is e x p r e s s e d from (2.5-3I) a n d s u b s t i t u t e d into the a b o v e expression, after s u b s t i t u t i n g ~P2 = (rl/r2) ~i. t h e DE d e s c r i b i n g the d y n a m i c s of the g e a r train s y s t e m is o b t a i n e d
(It * ( ]
) 12)q~'* Dr(
) ~'+
ct~
= Mzp-
Mzt •
(2.5-37)
l~O
CHAP.
2
LU}~P~'D
PRC)CEe:SES
T h e last e q u a t i o n , with the e q u i v a l e n t substitution of mt b y ~'z g i v e s a m o d e l for the d y n a m i c s of the g e a r train s y s t e m e x p r e s s e d d i s p l a c e m e n t of the s h a f t mz
(], + (
) t~)'p~+ Dr(
) ~+
ct'P~:bLMT~,
-(
r2
) Mzt
b y the a n g u l a r
(2.5-38)
It s h o u l d b e o b s e r v e d that the d y n a m i c m o d e l of this g e a r train s y s t e m is of the s e c o n d o r d e r , b u t is e x p r e s s e d in two w a y s - t h r o u g h the a n g u l a r d i s p l a c e m e n t of shaft I a n d t h r o u g h the a n g u l a l d i s p l a c e m e n t of shaft 2. T h e m o d e l is thus e i t h e r E q u a t i o n (2.6-37) o r E q u a t i o n (2.5-38). a n d it w o u l d b e w r o n g to c o n s i d e r that b o t h e q u a t i o n s at o n c e r e p r e s e n t the m o d e l d e m a n d e d . In that c a s e this w o u l d b e a f o u r t h - o r d e r system, w h i c h it is not. We c a n c h o s e either t h e pair ~, a n d ,p', or the p a i r ,#2 a n d 'P'2 for the s t a t e v a r i a b l e . If w e c o n s i d e r that the d i s p l a c e m e n t a n g l e s ,Pi a n d ~2 a n d the a n g u l a r v e l o c i t i e s ~'1 a n d ~'2 a r e o u t p u t v a r i a b l e s , the following two n o t a t i o n s in the form of s t a t e - s p a c e e q u a t i o n s a r e p o s s i b l e .
XI
=
@!
• )[2 =
G)|
'0~ =
State equation:
I:l [;;] I0
1
0
0
ct "Dt(~)~
l
r__L r2
J
J
!
J
(2.5-39) Mzt ]
Output equation:
b31
"PZ
"
I 0
0 l
rl
0
r2 ~2 XI
=
rl I2
0 %o2. X 2
=
]E::I
~2'
=
~2
(2.5-40)
sEc.
15]
2.s
State e q u a t i o m |
0
ct
J
.Dt(-~2) 2 j
+
~2
r_L jr~
(2.5-4l)
Output e q u a t i o n :
[
L r~
~ol
[i
0
=
0 r2
(2.5-42)
r!
~2
1
0
(.')2
0
1
~2
Both t h e s e s e l e c t i o n s of s t a t e v a r i a b l e s a r e a c c u r a t e a n d this e x a m p l e is useful to s h o w h o w the s e l e c t i o n s c a n often b e m a d e in s e v e r a l w a y s . There is m o r e on this s u b j e c t in the A p p e n d i x , a n d h e r e we must e m p h a s i z e that in this c a s e matrix A h a s r e m a i n e d u n c h a n g e d , w h i c h is not a l w a y s so. H o w e v e r , the b a s i c p r o p e r t i e s of t h e p r o c e s s s h o w n b y the e i g e n v a l u e s a r e a l w a y s p r e s e r v e d . Here, in matrix A of E q u a t i o n s (2.5-39) a n d (2.6-41), t h e y h a v e o b v i o u s l y r e m a i n e d the s a m e .
The following e x a m p l e will s h o w h o w a h i g h - o r d e r m o d e l c a n b e o b t a i n e d b y the d i s c r e t i z a t i o n of a s p a t i a l l y d i s t r i b u t e d p r o c e s s , w h o s e m o d e l is of infinite order. This infinite o r d e r results from the fact that e v e r y ( e v e n the simplest) d i s t r i b u t e d p r o c e s s is d e s c r i b e d b y a partial DE s h o w i n g the c h a n g e of a c e r t a i n v a r i a b l e in e v e r y p o i n t in the s p a c e in which that p r o c e s s o c c u r s , a n d t h e r e is a n unlimited n u m b e r of s u c h points. T h e o r d e r of the p r o c e s s e s t h e m s e l v e s , for the s i m p l e s t PDE, is no p r o b l e m w h e n a n a n a l y t i c solution c a n b e o b t a i n e d . An e x a m p l e of h o w s u c h a solution is o b t a i n e d in a c l o s e d form is g i v e n in the following c h a p t e r , but it must b e s a i d that s u c h c a s e s axe rare. Usually this is not the w a y to o b t a i n a solution a n d the u s e of c o m p u t e r s is u n a v o i d a b l e . The first s t e p is in most c a s e s to r e d u c e the m o d e l , i.e. to d e c r e a s e its order. D i s c r e t i z a t i o n with r e s p e c t to the s p a t i a l v a r i a b l e is t h e u s u a l w a y to transform o n e or s e v e r a l PDE into a finite s y s t e m of ODE.
I~2
(~[-IAP. 2 LUMPWD
PROCESSES
In the following e x a m p l e a h y p e r b o l i c first-order PDE (3.1-28) from the first s e c t i o n of the f o l l o w i n g c h a p t e r h a s b e e n s e l e c t e d for d i s c r e t i z a t i o n . It d e s c r i b e s the d y n a m i c s of c o n v e c t i v e h e a t transfer to a fluid flowing t h r o u g h a n u n i n s u l a t e d p i p e . This is a linear PDE, w h i c h is s u i t a b l e b e c a u s e p r o b l e m s of d i s c r e t i z a t i o n will not u n n e c e s s a r i l y b e c o m p l i c a t e d b y p r o b l e m s of linea~ization. T h e s e two p r o c e d u r e s are completely i n d e p e n d e n t from the a s p e c t of the o r d e r in w h i c h t h e y a r e p e r f o r m e d , a n d w h e n w e n e e d to f o r m u l a t e a n o n l i n e a r s p a t i a l l y d i s t r i b u t e d p r o c e s s in the fo~m of s t a t e s p a c e , this c a n b e clone e i t h e r b y first p e r f o r m i n g d i s c r e l i z a t i o n a n d then the l i n e a r i z a t i o n of the s y s t e m of ODE o b t a i n e d , or b y p r o c e e d i n g in the o p p o s i t e o r d e r . After the PDE h a s b e e n l i n e a r i z e d , a s y s t e m " of ODE is o b t a i n e d t h r o u g h d i s c r e t i z a t i o n , t r a n s f o r m e d into matrix s t a t e - s p a c e notation. We must
say
is t h e n
easily
discretization will b e p e r f o r m e d b y the method of backward difference w h i c h is o n l y o n e of the m a n y w a y s in w h i c h it c a n b e d o n e .
Example
also
which
that
4 D i s c r e t i z a t i o n of d i s t r i b u t e d process
For p u r p o s e s of n u m e r i c a l simulation a h y p e r b o l i c first-order PDE (3.1-28) must b e m a t h e m a t i c a l l y d i s c r e t i z e d . T h e e q u a t i o n d e s c r i b e s the d y n a m i c s of the c o n d e n s e r heat exchanger (ew # f(z). sw = ~w(t)) developed a n d s h o w n o n F i g u r e 3.1-5 in the form of a p i p e . For d i s c r e t i z a t i o n into four e l e m e n t s , s h o w the r e d u c e d m o d e l in the form of state-space equations. Also s h o w h o w the s a m e m o d e l c a n b e o b t a i n e d if the p r o c e s s is p h y s i c a l l y d i v i d e d first a n d the c o n s e r v a t i o n e q u a t i o n s f o r m u l a t e d for e a c h of the thus o b t a i n e d finite e l e m e n t s . T h e d i s c r e t i z a t i o n of the following h y p e r b o l i c PDE 88
--
~t
+ w
88 ~z
+ k~ = k~w
'
(k =
=U ) Apc
(2.5-43)
of backward difference by w h i c h the g r a d i e n t of a s p e c i f i c f u n c t i o n with r e s p e c t to the s p a c e v a r i a b l e z , i n t h e s e c t i o n n , is s u b s t i t u t e d in the following m a n n e r will b e c a r r i e d
out b y the m e t h o d
'
sEc. 2.s
153
(~f)
fn fn-I function v a l u e in the s e c t i o n n
fn fn o
(2.5-4,43
"
function v a l u e in the s e c t i o n n-I
l
Sz
d i s t a n c e b e t w e e n s e c t i o n s n-1 a n d n (or, l e n g t h of element obtained through discretization)
B e s i d e s this m e t h o d there a r e also m a n y other m e t h o d s (for e x a m p l e the m e t h o d s of forward a n d c e n t r a l d i f f e r e n c e ) a n d m a n n e r s w h i c h essentially no different. We will not enter into a n a n a l y s i s of advantages and disadvantages methods b e s h o w n here.
of s u c h
procedures
nor
will a n y
are the other
A p p l y i n g (2.5-44) to the g i v e n PDE. the following s y s t e m of n first-order ODE is o b t a i n e d d0ndt + w- OT~ ~z-en-~
+ ken = kew , n = 2, NE + 1
(2.5-45)
NE is the n u m b e r of e l e m e n t s o b t a i n e d b y d i s c r e t i z a t i o n ( s e e F i g u r e 2.5-4). T h e s y s t e m of four ODE in the c a s e of the d i v i s i o n of the h e a t e x c h a n g e r into four e l e m e n t s follows de~ dt
~"
ki82
=
k23t * kew
de3 d ~ + k,~)3 = k2e2 * kew (2.5-46)
dt
* k,84 = k2~)3 + ka-w
des dt
+ k,es
(k, = - -w+ k . ~z
=
k284 + kew
k2=
..~_z)
If v a r i a b l e s w. k a n d ew d e p e n d e d o n spatial c o o r d i n a t e z, t h e y w o u l d also h a v e s u b s c r i p t n in (2.5-45), or a c o r r e s p o n d i n g s u b s c r i p t in the system of e q u a t i o n s
(2.5-46). This s y s t e m is not difficult to s h o w
form of matrix s t a t e - s p a c e
equations
if the o n l y t e m p e r a t u r e
t e m p e r a t u r e es, is s e l e c t e d as the o u t p u t v a r i a b l e
in the
of interest.
154
CHAP. 2 L U M P E D
° o ol ]
!
~3
k2
=
PROCESSES
O, OS
-k,
0
0
o3
k2 0
-k,
0
04
-kl
S
o
l]
k2
(2.5-47)
X' - - A X + B U
[o~]--[
o
o
I
~'
~4
•
[o
o
5
l[:,]
(2.~-48~
Y=CX+DU This s h o w s h o w d i s c r e t i z a t i o n transforms a n infinite-dimensional p r o c e s s w h o s e d y n a m i c s is d e s c r i b e d b y PDE into a m o d e l of finite d i m e n s i o n s . Intuitively it is c l e a r
that a s y s t e m of ODE will d e s c r i b e
a p r o c e s s , the
more faithfully the h i g h e r the d e g r e e of d i s c r e t i z a t i o n , i.e. the "thicker" the division. But in that c a s e the m o d e l o b t a i n e d is of h i g h e r o r d e r a n d more difficult to p r o c e s s , so it is u p to the m o d e l b u i l d e r to d e c i d e on its final size.
1 I I
LI
°'°."'~,~ -I
1
i o,
?,
?,
2
3
/.
1~0.
1~
1~
u,:~, [ - 7 - ' ] ~, F - T - I ~, F T - l NE=&
Fig.
xl
x:
z
I o,~.,,°o,
! 5
10. ~, F T - I x3
,~,--y x4=y
2 . 5 - 4 D i s c r e t i z a t i o n of a s p a t i a l l y d i s t r i b u t e d p r o c e s s a n d i e p r e s e n t a t i o n of state, input a n d o u t p u t v a r i a b l e s
s ~ c . 2.s
155
T h e p a t h l e a d i n g - t o the m o d e l (2,5-45) c o u l d h a v e b e e n different. It is p o s s i b l e to d i v i d e the p i p e into e l e m e n t s of finite v o l u m e first, a n d t h e n formulate e n e r g y c o n s e r v a t i o n e q u a t i o n s for e a c h of t h o s e e l e m e n t s s e p a r a t e l y , or for o n e t y p i c a l e l e m e n t if all the s u b s y s t e m s a r e alike. This will b e d o n e in the following lines for a n e l e m e n t b e t w e e n c r o s s - s e c t i o n s n-I a n d n a n d that d e r i v a t i o n will b e d i s c r e t i z a t i o n b y the m e t h o d of b a c k w a r d
a s u i t a b l e illustration of w h a t d i f f e r e n c e r e a l l y r e p r e s e n t s , i.e.
w h e r e a n error is c o n s c i o u s l y m a d e in the a p p l i c a t i o n of that m e t h o d .
1
L V///~ F v~-,, '
B~
I
%
_j,\ } -i_~,/
'
] F i g . 2 . 5 - 5 Finite p i p e e l e m e n t of the c o n d e n s e r
heat exchanger
F i g u r e 2.5-5 s h o w s the (n-l)st e l e m e n t , for w h i c h a n e q u a t i o n for the c o n s e r v a t i o n of e n e r g y must b e f o r m u l a t e d . A b o v e it is s h o w n a n e x p o n e n t i a l fluid t e m p e r a t u r e c h a n g e a l o n g that e l e m e n t , g i v e n b y E q u a t i o n (3,1-29) in the s t e a d y state. T h e e n e r g y c o n s e r v a t i o n e q u a t i o n for the e l e m e n t is mcan., + ~U~z(aw- afsr) - mCan = IV[cdOfsr dt
(2.5-4Q)
8fsr d e n o t e s the m e a n fluid t e m p e r a t u r e b e t w e e n c r o s s - s e c t i o n s n-I a n d n. a n d the error r e s u l t i n g from d i s c r e t i z a t i o n o c c u r s in the s e l e c t i o n of this t e m p e r a t u r e . It is p o s s i b l e to m a k e m a n y different s e l e c t i o n s a n d write Ofsr
= ~'n-I
Ofsr
=
,
Or
~)n-! + ~n
2
, or
15~)
CHAP.
afsr
= ~n
afs r
= asrlog
= ~n-!
LUMPED
PRCDCESC:E
c:
, Or
....
The thicker the the c r o s s - s e c t i o n s d i s c r e t i z a t i o n . At elements has b e e n afsr
2
division, the smaller the t e m p e r a t u r e d i f f e r e n c e b e t w e e n n-I a n d n. w h i c h r e d u c e s the error i n t r o d u c e d into the limit w h e n division into a n infinite n u m b e r of m a d e . we h a v e
= 8n
= vasrlog
,
a n d the error b e c o m e s e q u a l to zero. A s y s t e m of a n infinite n u m b e r of first- o r d e r ODE (2.5-49) has b e e n (2.S.-43).
o b t a i n e d , w h i c h is e q u i v a l e n t
to PDE
If the n u m b e r of e l e m e n t s is finite (I. 3. 9, IT) a n d the t e m p e r a t u r e at the output of the n-th c r o s s - s e c t i o n c h o s e n for the m e a n fluid t e m p e r a t u r e , i.e. a#sr
=
(2.5-50)
an
the s a m e s y s t e m of ODE is o b t a i n e d as the o n e d e r i v e d b y the m e t h o d of b a c k w a r d
difference
and
s h o w n in E q u a t i o n
(2.5-45). Substitution
of
(2.5-50) into (2.5-49) for m = Awp a n d M = Ap~z a n d shorter r e a r r a n g e m e n t y i e l d s the e q u a t i o n d e s c r i b i n g the d y n a m i c s of t e m p e r a t u i e of the (n-1)st element d~n
d~
+ w
8n
-
~z
8n-
+ kan = kaw
(2.5-51)
Therefore, the m e t h o d of b a c k w a r d d i f f e r e n c e r e p r e s e n t s a substitution of m e a n fluid t e m p e r a t u r e in a finite e l e m e n t obtained through d i s c r e t i z a t i o n , with the o u t p u t fluid t e m p e r a t u r e of that element. T h e mathematical
discretization s h o w n
here can, as a rule. b e applied
to all the P D E that appear in this book. If it is, for example, applied to the system of three P D E (3.3-53) - (3.3-55), a system of 3. N E first-order O D E obtained which c a n then b e processed (simulated) on a digital computer.
is
sEc. 2.s
157
Here w e h a v e c o m p l e t e l y left out an a n a l y s i s of all the m a n y p r o b l e m s linked with m e t h o d s of d i s c r e t i z a t i o n t h e m s e l v e s , a b o u t w h i c h m a n y ~ p e c i a l i z e d works, s t u d i e s a n d b o o k s h a v e b e e n written. We must at least mention that this primarily c o n c e r n s the a c c u r a c y of the m o d e l (in the s e n s e of it a p p r o a c h i n g
as c l o s e l y as p o s s i b l e the real d y n a m i c s of the
u n d i s c r e t i z e d s y s t e m of PDE) its stability a n d size, w h i c h is c o n n e c t e d with the c o s t s of simulation a n d the difficulties of n u m e r i c a l p r o c e s s i n g .
This e n d s the d e s c r i p t i o n of d y n a m i c m o d e l s of p r o c e s s e s with l u m p e d parameters, a n d the last e x a m p l e is a transition to the m o d e l i n g of p r o c e s s e s in w h i c h s o m e v a r i a b l e s s h o w s p a t i a l d e p e n d e n c e as well. Most of them c a n n o t b e c o m p l e t e l y d e s c r i b e d b y o n e or more ODE, so from now o n partial DE will a p p e a r . This will b e e x p l a i n e d in more d e t a i l o n the following p a g e s .
CHAPTER
THREE
DISTRIBUTED
PROCESSES
In p r e v i o u s d i s c u s s i o n we a n a l y z e d a n d d e s c r i b e d the d y n a m i c p r o p e r t i e s of p r o c e s s e s in w h i c h t h e d e p e n d e n c e of s t a t e v a r i a b l e s on the s p a t i a l c o o r d i n a t e w a s n e g l e c t e d , i.e. in w h i c h w e c o n s i d e r e d that a n y c h a n g e in s t a t e v a r i a b l e o c c u r s s i m u l t a n e o u s l y a n d e q u a l l y in the w h o l e p a r t of the p l a n t u n d e r o b s e r v a t i o n . Strictly s p e a k i n g , in b y far the g r e a t e s t n u m b e r of c a s e s that a p p e a r in e n g i n e e r i n g p r a c t i c e the v e l o c i t i e s , t e m p e r a t u r e , flows, c o n c e n t r a t i o n s , forces, p r e s s u r e s a n d so on c h a n g e differently b o t h in time a n d in the s e p a r a t e p a r t s of the plants. T h e d y n a m i c s of c e r t a i n v a r i a b l e s thus d e p e n d s o n the s p a t i a l c o o r d i n a t e a n d in t h e s e c a s e s , w h e n the time n e e d e d for d i s t u r b a n c e p r o p a g a t i o n C c h a n g e s in s t a t e v a r i a b l e s ) t h r o u g h the p l a n t c o r r e s p o n d s to the time c o n s t a n t s c h a r a c t e r i z i n g o t h e r t r a n s i e n t p r o c e s s e s , that s p a t i a l distribution must not b e n e g l e c t e d . T h e r e a r e m a n y e x a m p l e s of s u c h s y s t e m s in w h i c h t h e r e is o b v i o u s s p a t i a l distribution a n d d e p e n d e n c e of v a r i a b l e s : pipe heat exchangers, gas pipes and pipelines, transport conveyers and so on, a r e e x a m p l e s in w h i c h s t a t e v a r i a b l e s d e p e n d o n o n e s p a t i a l c o o r d i n a t e (the s p a t i a l c o o r d i n a t e z), a n d the o s c i l l a t i o n of m e m b r a n e s a n d shells, h e a t i n g s l a b s , s p r e a d i n g of Oil p a t c h e s o n the s e a or forest fires a r e t y p i c a l c a s e s w h e r e t h e r e is p r o c e s s v a r i a b l e d e p e n d e n c e on two s p a t i a l c o o r d i n a t e s . (The last two c a s e s h a v e r e c e n t l y b e c o m e e s p e c i a l l y i n t e r e s t i n g a n d theii d y n a m i c s is the s u b j e c t of i n t e n s e s t u d y in m a n y c o u n t r i e s . ) Finally. d y n a m i c t e m p e r a t u r e c h a n g e s in h e a t i n g l a r g e c a s t s (ingots) or the d y n a m i c s in s t e a m g e n e r a t o r e v a p o r a t o r p i p e s a r e
s x c . 3.1
15q
t o d a y often a n a l y z e d b y o b s e r v i n g v a r i a b l e c h a n g e of the c a s t coordinates.
or
pipe.
which
means
dependence
in the w h o l e v o l u m e on
three
spatial
In the following text we h a v e limited o u r s e l v e s to p r o c e s s e s that d e p e n d on o n l y o n e s p a t i a l c o o r d i n a t e (z) a n d o n time (t), For most t e c h n i c a l d e v i c e s a n d p l a n t s this a p p r o a c h c o m p l e t e l y satisfies the d e m a n d s p l a c e d b e f o r e the a n a l y s i s of d y n a m i c p r o c e s s e s . It will b e s h o w n that e v e n for t h o s e " s i m p I e s t " c a s e s of s p a t i a l d e p e n d e n c e the m a t h e m a t i c a l a p p a r a t u s a n d m e t h o d o l o g y of solution a r e of a h i g h e r d e g r e e of c o m p l e x i t y a n d with less p o s s i b i l i t i e s for g e n e r a l solution t h a n was the c a s e u p to now. H e r e too the d e s c r i p t i o n of d y n a m i c p h e n o m e n a a n d the solution of the partial differential e q u a t i o n s (PDE) o b t a i n e d from that d e s c r i p t i o n will b e b e g u n from the b a s i c a n d s i m p l e s t p r o c e s s e s a n d m a t h e m a t i c a l structures, after w h i c h w e will p r o c e e d to m o r e c o m p l e x phenomena and m a t h e m a t i c a l notation. In c a s e s w h e r e c o m p l e x a n d v e r y e x t e n s i v e e x p r e s s i o n s a n d s t r u c t u r e s a r e not o b t a i n e d , w e will a l s o s h o w the a n a l y t i c a l p r o c e d u r e s for o b t a i n i n g solutions of PDE. At t h e v e r y b e g i n n i n g of this third c h a p t e r it will b e of a d v a n t a g e to r e m e m b e r s o m e b a s i c a n d g e n e r a l c h a r a c t e r i s t i c s of PDE. T h e y d e s c r i b e time c h a n g e s "of s t a t e v a r i a b l e s ( c o n c e n t r a t i o n . t e m p e r a t u r e s , flows, p r e s s u r e s , d e n s i t i e s a n d so on) t h r o u g h o u t the d e v i c e or p l a n t o b s e r v e d . i.e. in e v e r y a r b i t r a r y p o i n t in s p a c e , of w h i c h t h e r e a r e of c o u r s e infinitely m a n y . PDE t h e r e f o r e d e s c r i b e p r o c e s s e s of infinite order. T h e f a c t that in p r a c t i c e a s t a t e v a r i a b l e c h a n g e is i m p o r t a n t on the o u t p u t (input) or in o n l y s e v e r a l s e c t i o n s , a n d not in a n infinite n u m b e r of them. is not of g r e a t h e l p in this c a s e b e c a u s e s t a t e v a r i a b l e c h a n g e s in the s e c t i o n u n d e r o b s e r v a t i o n d e p e n d on the v a r i a b l e g r a d i e n t s with r e s p e c t to the s p a t i a l c o o r d i n a t e , w h i c h m a k e s it n e c e s s a r y to k n o w the v a r i a b l e s that are infinitesimally c l o s e to that s e c t i o n also. A n a l o g o u s l y , for that infinitesimally c l o s e s e c t i o n the p r o b l e m b e c o m e s (or r e m a i n s ) of infinitely high o r d e r . For e v e r y d a y t e c h n i c a l p r a c t i c e , t h e r e f o r e , the task of d e c r e a s i n g the m o d e l ' s o r d e r will b e o n e of p r i m a r y i m p o r t a n c e , a n d will as a rule b e r e d u c e d to the p o s s i b i l i t y of a c c u r a t e l y c a l c u l a t i n g the g r a d i e n t of s t a t e v a r i a b l e s with r e s p e c t to the s p a t i a l c o o r d i n a t e with the h e l p of a finite n u m b e r of points.
T h e following must a l s o b e s a i d in c o n n e c t i o n with PDE that will b e the b a s i c tools u s e d to d e s c r i b e a n d a n a l y z e d y n a m i c p r o c e s s e s o c c u r r i n g in d e v i c e s a n d p l a n t s a l o n g w h i c h p a r a m e t e r s c h a n g e . This p a r t of the b o o k is not i n t e n d e d to b e , nor s h o u l d it b e c o n s i d e r e d , a s h o r t e n e d v e r s i o n or the r e p e t i t i o n of b o o k s from m a t h e m a t i c a l p h y s i c s or s p e c i a l c h a p t e r s from t h e r m o d y n a m i c s , fluid m e c h a n i c s a n d m e c h a n i c s , in the s e n s e that it g i v e s a n s w e r s a n d s h o w s m e t h o d s for c l a s s i f y i n g PDE. a n a l y z e s p r o b l e m s of e x i s t e n c e a n d u n i q u e n e s s of solutions, s h o w s different m e t h o d s for s o l v i n g c e r t a i n t y p e s of PDE a n d the like. H e r e . o n m a n y e x a m p l e s that a r e e n c o u n t e r e d in e v e r y d a y t e c h n i c a l p r a c t i c e , w e will primarily try to i n d i c a t e s o m e c h a r a c t e r i s t i c s of p r o c e s s e s d i s t r i b u t e d
basic dynamic properties and in s p a c e a n d s h o w m e t h o d s of
d e r i v i n g m o d e l s for u n s t e a d y s t a t e c h a n g e s . To d o this, in a n a l o g y with l u m p e d p a r a m e t e r p r o c e s s e s , w e will u s e L a p l a c e t r a n s f o r m a t i o n a n d transfer functions. Wishing e v e r y m o d e l o b t a i n e d to b e s o l v a b l e a n d usal~le in p r a c t i c e , in m a n y e x a m p l e s w e will a l s o s h o w h o w a m o d e ! of finite d i m e n s i o n is o b t a i n e d , w h a t is lost w h e n the o r d e r of the m o d e l is r e d u c e d a n d h o w t r a n s c e n d e n t a l transfer functions a r e a p p r o x i m a t e d b y transfer f u n c t i o n s in the form of p r o p e r rational function. From the a s p e c t of p r o c e s s linearity, the u s e of L a p l a c e t r a n s f o r m a t i o n a n d the f o r m a l i z a t i o n of s y m b o l s , w e must r e p e a t a n d b e a r in m i n d the r e m a r k that w a s a l r e a d y m a d e at the b e g i n n i n g of C h a p t e r 2. In all the following c a s e s w h e n the P D E o b t a i n e d a r e l i n e a r , for the s t a t e v a r i a b l e s o b s e r v e d t h e y c a n a l s o b e c o n s i d e r e d as the e q u a t i o n s of the deviation of t h o s e v a r i a b l e s from s o m e s t e a d y state. In the p r o c e s s of o b t a i n i n g transfer functions, w h e n the initial c o n d i t i o n s a r e t a k e n to e q u a l z e r o this a l s o m e a n s that t h e initial d e v i a t i o n s e q u a l zero. i.e. that the s y s t e m is in s t e a d y state. T r a n s f e r f u n c t i o n s o b t a i n e d in this w a y a r e v a l i d b o t h for a b s o l u t e v a l u e s of v a r i a b l e c h a n g e from the initial z e r o s t a t e (from initial c o n d i t i o n s e q u a l to zero), a n d a l s o for c h a n g e s of v a r i a b l e d e v i a t i o n from the initial s t e a d y state. In s u c h linear c a s e s this linearity will. t h e r e f o r e , not b e s p e c i a l l y s t r e s s e d b y i n t r o d u c i n g d e v i a t i o n s y m b o l A b e s i d e the v a r i a b l e s y m b o l s (for e x a m p l e , we will not write a m . z~. Ae. a n d s o on, b u t s i m p l y m. p. e). Finally. w e must a d d will also b e
mass.
derived
that the m a t h e m a t i c a l m o d e l for t h e s e p r o c e s s e s
from the e q u a t i o n s
for the
conservation of a s s u m p t i o n of s t a t e
energy and momentum, but s i n c e the v a r i a b l e e q u a l i t y t h r o u g h o u t the w h o l e finite v o l u m e o b s e r v e d
no l o n g e r
sEc. ~.1
101
holds, t h e s e e q u a t i o n s small
part
will b e
formulated
for
the
infinitesimally
of the v o l u m e dV for w h i c h the u p p e r a s s u m p t i o n is fulfilled•
3.1 M A S S A N D E N E R G Y BASIC HYPERBOLIC PARTIAL
TRANSPORTATION FIRST-ORDER
DIFFERENTIAL
PROCESSES
EQUATION
T h e t r a n s p o r t a t i o n of b u l k material on c o n v e y e r b e l t s a n d the p r o c e s s of fluid flow t h r o u g h p i p e s , w h e r e h e a t e n e r g y is c a r r i e d c o n v e c t i v e l y from the p i p e inlet to its outlet, a r e f r e q u e n t , a n d the s i m p l e s t e x a m p l e s of p r o c e s s e s with s p a t i a l l y d i s t r i b u t e d p a r a m e t e r s . F i g u r e s 3.1-1 a n d 3.1-2 s h o w s u c h p r o c e s s e s of m a s s a n d e n e r g y t r a n s p o r t a t i o n , a n d in the following lines w e will s h o w that t h e d y n a m i c p r o p e r t i e s of t h o s e p r o c e s s e s a r e d e s c r i b e d b y the s a m e PDE.
A(z) rn
Z=O
z*dz
1 I I I
3.1-1 Bulk m a t e r i a l t r a n s p o r t a t i o n
•.
Fig.
.
Z=~ Fig.
m÷ -~-dz
•°, ...•.
•°,°.,
-.-j,,-
.•
.•. • -°-•
3.1-2 C o n v e c t i v e h e a t transfer b y liquid flow
In d e r i v i n g the m o d e l the a s s u m p t i o n s for F i g u r e 3.1-1 a r e : - t h e r e is no s i d e d i s s i p a t i o n .
1~2
CHAP.
3
DISTRI~LITED
P~(:DC~"SS;';S
t h e t r a n s p o r t v e l o c i t y w is c o n s t a n t . The
equation
between
for
the
conservation
of
mass
in
the
volume
segment
z a n d dz is dz]
~m(z. 0 ~z
re(z, t) - [m(z, t) +
= dM dt
(3.1-I)
M a s s flow rate m a n d m a s s b e t w e e n s, a n d dz are m
= A(z) ow
(3.1-2)
d M : A(z) o d z
(3.1-3)
The last equations ~m(z.t) at
* w
Before
deriving
give a hyperbolic
am(z.t) ,~z
:
the
f i r s t - o r d e r PDE
0
(3.1-4)
model
that describes
the
propagation
of
heat
disturbances along the pipe. the following assumptions are m a d e : -
the pipe
is i n s u l a t e d ,
the heat
capacity
of t h e p i p e
the liquid flowing through
w a l l s is n e g l e c t e d .
the pipe
is i n c o m p r e s s i b l e ,
i.e.
u = i - Pv ; i ; ce, - t h e flow v e l o c i t y w is c o n s t a n t . The equation a n d dz is
u(z.
t) - [ u ( z ,
for t h e c o n s e r v a t i o n
~u(z. t) dz] az
t) +
of e n e r g y
for t h e
dU = dt
H e a t flow r a t e u a n d i n t e r n a l e n e r g y
(3.l-S) U ate given
by
u = mc,~ U = Mc~
(3.1-6)
(3.1-7)
= Adz0c~)
Finally. the same hyperbolic
,~(z,t) -
-
at
+
fluid between
w-
~o(z.t) ~z
=
0
f i r s t - o r d e r PDE is o b t a i n e d (3.1-8)
z
sEc.
3.,
In =
163
both
cases
const, a n d ,~
Equations
the =
steady
(3.1-4) a n d
(3.1-8) s h o w
fact d i f f e r e n t , d y n a m i c a l l y some variables spatial
they
(mass. heat
coordinate,
first-order
to b e r e g u l a t e d
is
determined
by
that
although
obviously are
processes
the
,)/,~t = O,
the
same.
transferred are
processes
i.e.
In b o t h
described
by
(for t h e c a s e
w = c o n s t . ) it is d o n e
advantageous
systems along
a
m a t e r i a l , or t h e variables
by acting
to f i n d t h e
in
hyperbolic
a
i n s e c t i o n z = L is i m p o r t a n t , a n d if t h e s e
thus be
are
convectively
As a r u l e . o n l y t h e a m o u n t of u n l o a d e d
liquid temperature, ,~(O,t). It w o u l d
are
energy)
so both
PDE.
state
const.
are
o n m(O,t) or
relation
between
o u t for t h e
case
those
two s e c t i o n s . The discussion material
in E x a m p l e
transport,
but
F i g u r e 3.1-2, it is e n o u g h Before obtaining how
whether
much
want
to
to r e p l a c e
material
carried get
expxessions
for mCL,t), w e
from z = L d o e s is p l a c e d
on
the
must
initial c o n d i t i o n s conveyer
Mathematically
expressed,
(IC) m(z,t0), i.e. t h e
belt at the moment
u s u a l l y s t a n d in t h e p l a c e
of b u l k
the
not depend
case
on
out that
the
o n l y o n m(O,t),
belt, but
of t h e
we usually
also
o n it w h e n
it is n e c e s s a r y
distribution
to. ( S i n c e
of to a n d
point
conveyer
t h e r e w a s a lot, a little or n o m a t e r i a l a l r e a d y
put into operation.
for
m b y ~) in t h e m o d e l s o b t a i n e d .
an expression
amount of material unloaded i.e. o n
1 will b e
if w e
to k n o w
the
on
the
t a k e to = O, z e r o
will
this will d e s c r i b e
material
on
it w a s
t h e IC, for e x a m p l e
m(z,O).)
Example
I Conveyer
The process
s y s t e m for b u l k m a t e r i a l t r a n s p o r t . T i m e r e s p o n s e
of b u l k m a t e r i a l t r a n s p o r t
t h e f o l l o w i n g PDE a n d t h e g i v e n b o u n d a r y
~m(z,13 ~t
* w
~m(z,t) ¢~z
= 0
,
BC= z = O. m(O. t)= zCt). t > to IC, t = to, m C z . t o )
= pCz).
z ~ Co.
on Figure
3.1-I is d e s c r i b e d
and initial conditions (3.1-9)
by
l~,4
CI--IAP.
Velocity
w
= const.,
the
general
r(t)
and
p(z)
are
"~ I ~ I ~ T R I B I . I T I ; I ~
given
functions
P R C ) O I:'c~c~':'c~
of time
and
space. a) F i n d
analytical
form) a n d p r e s e n t it g r a p h i c a l l y al) r(t) = h = I
solution
(or t h e
solution
in a
closed
for t h e f o l l o w i n g c o n d i t i o n s
for t * O. p ( z ) = O.
a2) r(t) = 0, p(z) = I. 0 '~ z "~ L
b) F i n d the t r a n s f e r f u n c t i o n G(s) = M(L.s)/M(O,s) = M(L,s)/R(s). T h e s i m p l e s t w a y to o b t a i n s o l u t i o n s for the h y p e r b o l i c first-order PDE is to u s e L a p l a c e t r a n s f o r m a t i o n that is c a r r i e d o u t with r e s p e c t to time t, s o {hat the initial PDE is t r a n s f o r m e d into ODE with r e s p e c t to z in the s - d o m a i n . T h e n the ODE o b t a i n e d is s o l v e d in the s - d o m a i n b y the u s u a l p r o c e d u r e s a n d its s o l u t i o n r e t u r n e d b y i n v e r s e L a p l a c e t r a n s f o r m into the t - d o m a i n . T h e r e is ~.{am(z, t) j- _ a M ( z . s) = M ' ( z . s) c~z
dz
A p p l i c a t i o n of the a b o v e
t r a n s f o r m to (3.l-Q) y i e l d s
(3.1-1o)
sM(z. s) - p(z) + w M ' ( z , s) = 0
M' (z. s) = - S---M(z. s) + p ( z ) W
(3.1-I1)
W
T h e last e q u a t i o n
is a n o n h o m o g e n e o u s
first-order ODE with r e s p e c t
to
z in the s - d o m a i n , a n d it c a n b e s o l v e d b y the u s u a l m e t h o d of c o n s t a n t s v a r i a t i o n . T h e g e n e r a l s o l u t i o n will thus b e the s u m of t h e h o m o g e n e o u s and the particular solution
M(z. s) = Mh(z, s) + Mp(z. s)
(3.1-12)
Mh(Z. s) = C e -~~z
(3.1-12a) Z
Mp(Z. s) = e w
-~- p ( { ) e
dI~
(3.1-12b)
165
sEc. 31
Substituting the e x p r e s s i o n s for Mh a n d Mp into (3.1-12) a n d e s t a b l i s h i n g with the h e l p of the BC M(O.s) = R(s) that the c o n s t a n t C = R(s), w e g e t the final e x p r e s s i o n which must b e r e t u r n e d into the time d o m a i n b y i n v e r s e Laplace transform Z
M(z.
s) = e-"--~Z JR(s)
J ~--1 p(~)e-~ d~]
(3.l-131
D e n o t i n g the e x p r e s s i o n in the b r a c k e t s F(s) a n d its original fCt) y i e l d s
M(z, s) = e --~-~z V(s)
(3.I-14)
On the b a s i s of the time tianslation t h e o r e m b y w h i c h translation in the d o m a i n of the original function f(t) c o r r e s p o n d s to d a m p i n g of function F(s) in the s - d o m a i n (which is the c a s e a b o v e ) , the i n v e r s e transform yields Z m(z, t) = f(t - -~-)
C3.l-15)
H o w e v e r . ~(t) is still u n k n o w n a n d is o b t a i n e d as follows f(t) = ~.'l{R(s) ÷ F,(s)] = r(t) + ~-l{F~(s)} = z
(3.1-10)
z
= r(t)+ ~'liJwl---p(~)e~'~d~]--r(,)
+J-~lw p(~)..~'l/e%S/, , dl~ . ~(t
,/w-w)
The solution of the integral on the right-hand side follows z
i
- ~ p(~) a c t . ~ )
d~
(3.1-m
Substitution of ~ = w~ a n d dl~ : wd~ into (3.1-17) y i e l d s z
I p(w~)~(~ • t)d~
= p(-wt)
(3.1-18)
Inserting the last e x p r e s s i o n into (3,1-16) i n s t e a d of the i n t e g r a l y i e l d s
~(t) fCt) = r(t) + pC-wt) .
(3.1-IQ)
A c c o r d i n g to Equation (3.1-15) the solution is translated, so the final e x p r e s s i o n for m(z,t) is
I(~')
CHAP.
3 DISTIqlBUTED
Z
m(z, t) = r(t - -~--) ÷ p ( z - wt)
PROCESSES
(3.1-20)
F i g u r e s (3.1-3) a n d (3.1-4) a r e a g r a p h i c a l r e p r e s e n t a t i o n of the solution in t h r e e - d i m e n s i o n a l s p a c e (m,z,t) for c o n d i t i o n s al) a n d a2) C o n s e q u e n t l y , for t > Tt c o n v e y e r , i.e. m(z.t > Tt) = O.
there
will
be
no
more
material
on
the
b) By definition, transfer function G(s) r e p r e s e n t s the ratio of output s i g n a l transform to input s i g n a l transform, with initial c o n d i t i o n s e q u a l zero. T h u s G(s) is not difficult to o b t a i n . It is e n o u g h to s u b s t i t u t e p(~) = 0 into E q u a t i o n (3.1-13), after w h i c h the w h o l e e x p r e s s i o n u n d e r the i n t e g r a l b e c o m e s e q u a l to z e r o a n d r e m a i n s z --'~s
M(z, s) = e
z
R(s) - lvl(O, s ) e ~ ' s
(3.1-21)
If w e insert z = L. w e g e t G(s) :
M(L. s) = e - ~ S
(31-22)
M(O, s)
Ratio L / w h a s the d i m e n s i o n of time a n d r e p r e s e n t s the time n e e d e d for a p a r t i c l e t r a v e l l i n g at v e l o c i t y w to m o v e from z ,, 0 to z = L. It is u s u a l l y g i v e n the s u b s c r i p t t ( t r a n s p o r t e d ) L Tt = W-
(31-23)
(This time is a l s o often c a l l e d " d e a d " time b e c a u s e it is a p e r i o d of time d u r i n g w h i c h n o t h i n g will a p p e a r in the s e c t i o n z = L if the IC e q u a l zero, m(L. t ( T t ) = O. i.e. it is a p e r i o d of time d u r i n g w h i c h e v e r y t h i n g will b e " d e a d " in the o u t p u t s e c t i o n . S e e F i g u r e 3.1-3.) T h e n e x t e q u a t i o n is the w e l l - k n o w n t r a n s c e n d e n t a l function ( a n d no l o n g e r in the form of a p r o p e r rational function) for t r a n s p o r t p r o c e s s e s G(z. s) = M(z, s) . ~ T t s M(0, s)
z (Tt = ~ - )
(3.1-24)
For t h e c a s e of liquid flow w h e r e w e s e e k c o n n e c t i o n s b e t w e e n input a n d o u t p u t t e m p e r a t u r e s , r e p l a c i n g M b y e in the last e q u a t i o n d i r e c t l y y i e l d s E q u a t i o n (3.1-25).
sEc.
3.z
167
a l ) BC rCt) = h = I for t • 0 IC pC:') = o. 0
h ... H e a v i s l d e
unit-step
function
~ z ~ L
in(z, t) .. h(t - --~)
miz,t)
'-•'• I tV_
L=.wTt_
,~ z = w t
-Z
~[h./h(z,.:0
/
/#i '~
T,--~,W
~,"
,/
]Fig. 3,|-3
Changes
in quantity
a 2 ) BC r(t) = 0 IC p ( z ) = 1. re(z, t) = h ( z
of m a t e r i a l m(z,t) for r(t) = h a n d
p(z) = 0
for t 2 0 0 ~ z • L - wt)
m(z,t} p(z}'=1
o, ~ r ' , .
;
d
!.l~
,/°(z.,,--°-.J: I l" l L. // "--L IL," T,o÷ . / .......
"-,I,"
,Y Fig.
3.1-4 Changes
in quantity
of m a t e r i a l In(z.t) for r(t) = 0 a n d
p(z) = 1
I/",~
CHAP.
e(z. s) GCz. s) = e(0. s)
3 E:)IC~TRII~JTC'E~
=. ~ T t s
If e -Tts is e x p a n d e d
PR~C:Ec~a:E c:
(3.1-25)
into a n
infinite s e r i e s , t h e transfer f u n c t i o n
for
t r a n s p o r t p r o c e s s e s o b t a i n s the following form G(~', s)
= eT T t s
=
I
I + Tt's" +(Tts)2+ ( T t s ) 3 , ...
11
21
(3.I-26)
31
The d e n o m i n a t o r of E q u a t i o n (3.1-26) c o n t a i n s a p o l y n o m i a l in s of infinite o r d e r t h a t h a s a n infinite n u m b e r of roots a n d thus confirms the s t a t e m e n t in the i n t r o d u c t i o n to this third c h a p t e r a b o u t PDE d e s c r i b i n g p r o c e s s e s of infinitely h i g h o r d e r . S i n c e this d e n o m i n a t o r ( w h i c h is a l s o the c h a r a c t e r i s t i c e q u a t i o n of the initial PDE (3,1-9)) c a n a l s o b e written eTt s = eTt( ° + ~j)
= e T t O eTt~aj
it c a n b e s e e n that all t h o s e roots lie o n the "straight line" o = .oo in the c o m p l e x s - p l a n e a n d that t h e y all h a v e a n a r b i t r a r y m e a n i n g for the i m a g i n a r y part. On the b a s i s of e v e r y t h i n g that h a s b e e n
spatially presented
distributed in the form
s a i d , it follows c l e a r l y that
processes can not possibly of state-space equations, which
be
was a l w a y s p o s s i b l e u p to n o w for p r o c e s s e s with l u m p e d p a r a m e t e r s . In this c a s e the d i m e n s i o n o f t h e s t a t e v e c t o r x w o u l d b e i n f i n i t e . It must a l s o b e s a i d that in t h e s e p r o c e s s e s the h i g h e s t o r d e r of d e r i v a t i v e s with r e s p e c t to time d o e s not e q u a l the o r d e r of the s y s t e m , w h i c h w a s a l w a y s the c a s e in l u m p e d p r o c e s s e s .
In E x a m p l e I w e a n a l y z e d the r e a l l y s i m p l e s t p r o c e s s e s with s p a t i a l p a r a m e t e r d e p e n d e n c e a n d with a s e l e c t i o n of a s s u m p t i o n s that " i n s u r e d " a m o d e l in the form of a linear h o m o g e n e o u s h y p e r b o l i c first-order PDE. In the following e x a m p l e w e will s h o w how, if t h e a s s u m p t i o n s a r e " m a r r e d " just a little (which is often c l o s e r to the r e a l c a s e ) , m o r e c o m p l e x m a t h e m a t i c a l forms a r e o b t a i n e d to s o l v e a n d a n a l y z e . (This c h a n g e of a s s u m p t i o n s c o u l d m e a n that m a t e r i a l a l s o s l i d e s d u r i n g transport, or that it fails off the s i d e of the c o n v e y e r belt, or that the p i p e is not i n s u l a t e d , a n d so on.)
sEc.
IOQ
3.1
Example
2
Fluid
flow t h r o u g h
uninsulated
pipe.
Temperature
time
response Derive a m o d e l for h e a t transfer b y liquid flow t h r o u g h a n u n i n s u l a t e d pipe, w h e r e the wall t e m p e r a t u r e is g i v e n b y .~w(Z, t). Heat s t o r a g e in the p i p e wall is n e g l e c t e d a n d the mass flow is c o n s i d e r e d c o n s t a n t ( a n d thus also flow v e l o c i t y w). T h e c o n s e q u e n c e
of the last a s s u m p t i o n is that the
coefficient of c o n v e c t i v e h e a t transfer ~ is c o n s t a n t as well.
"0"w(z,t} i
I
v,,
I
I
i
Io
,
'
J
I
I "7q"-~ "v
Z=0
Fig.
z z
= w = ko,st.---~.
I
_
I
~
~
l
z=L
3.1-5 Liquid flow t h r o u g h p i p e a n d h e a t e x c h a n g e
with wall of
t e m p e r a t u r e ew(z, t) This c a s e
is p r e s e n t in c o n d e n s e r h e a t e x c h a n g e r s
the a s s u m p t i o n fulfilled.
of
constant
wall
temperature.
a n d in their c a s e
Sw = const.,
is
usually
Formulation of the e q u a t i o n for the c o n s e r v a t i o n of e n e r g y for the fluid b e t w e e n s e c t i o n s z a n d z+dz y i e l d s d(Mc,~) mc,~ - (mce + mc-~-z d r ) • 0cAu(,~w - ,~) = - dt
(3.1-27)
The a r e a t h r o u g h w h i c h h e a t is e x c h a n g e d is Au = 2r~dz = Udz, w h e r e U r e p r e s e n t s the c i r c u m f e r e n c e of the p i p e . M = Apdz is the mass of fluid in the o b s e r v e d dV. A r r a n g e m e n t of the last e q u a t i o n y i e l d s
~Cz. ~t
t)
+ w
,~Cz. t) ,~z
+ ka(z,
t) = k ~ w ( Z ,
t)
(3.l-28) k-
=U Apc
170
CI4AP.
T h e m o d e l is a n o n h o m o g e n e o u s
"~ D I S T R I B U T E D
PROC:'~..qSKS
linear h y p e r b o l i c first-order PDE a n d
linear r e l a t i o n s will b e r e t a i n e d as l o n g as the a s s u m p t i o n a b o u t the c o n s t a n t flow v e l o c i t y w is fulfilled. To o b t a i n a n a n a l y t i c solution of E q u a t i o n (3.1-28) it is n e c e s s a r y to k n o w the b o u n d a r y c o n d i t i o n s function e(O,t), the initial c o n d i t i o n s function e(z,O) a n d the d i s t u r b a n c e function
ow(z. t). Coefficient k is a n i m p o r t a n t d y n a m i c c o e f f i c i e n t a n d will b e met in all p i p e h e a t e x c h a n g e r s . T h e d i m e n s i o n of k is s -1 a n d its i n v e r s e v a l u e is c a l l e d the h e a t ( t h e r m a l ) lime constant o f fluid T = Ilk. For the c a s e of t h e c o n d e n s e r h e a t e x c h a n g e r w h e r e ew(Z, t) = const., the s t e a d y s t a t e is c h a r a c t e r i z e d b y the k n o w n e x p o n e n t i a l i n c r e a s e of fluid t e m p e r a t u r e e(z) a l o n g the p i p e . This e x p r e s s i o n is o b t a i n e d b y s o l v i n g a n ODE with r e s p e c t to z, w h i c h is o b t a i n e d from (3.1-26) b y i n s e r t i n g a,~/at = O. T h e b o u n d a r y c o n d i t i o n is g i v e n b y ~(z=O) = ,~(0). This g i v e s the fluid t e m p e r a t u r e c h a n g e in the s t e a d y s t a t e k ~(z) = ,~w" a~oe " ' ~ z
. ~,~o = ,~w - ,~(0)
(3.1-29)
If ~(o) = o k
~ ( z ) = ~ , ( I - e "-~z )
(3.1-30)
a n d if b o t h ~(O) = 0 a n d ~w(O) = O, the e q u a l to z e r o is o b t a i n e d , i.e. ,~(z) = O.
often u s e d
initial c o n d i t i o n
If w e n e e d to k n o w the r e l a t i o n b e t w e e n fluid t e m p e r a t u r e c h a n g e in a n a r b i t r a r y s e c t i o n a l o n g the p i p e a n d the input t e m p e r a t u r e ( b o u n d a r y c o n d i t i o n ) %(O,t), or the t e m p e r a t u r e of the p i p e wall ew(z. t), it is a d v a n t a g e o u s to d e t e r m i n e the transfer f u n c t i o n s that d e f i n e that relation. A p p l y i n g the L a p l a c e transform to (3.1-28), for IC e q u a l to z e r o , w e g e t the following ODE with r e s p e c t to z in the s - d o m a i n w
dO[z. s) dz
*
(s
÷
k)e(z,
s)
= kew(z, s)
S o l v i n g that DE g i v e s the d e m a n d e d
(3.1-31)
relation between temperatures
. ~--~kz
s+k I- e e(z. s) = e --w-z 8(0. s) + s+k i G,Cs) G2(~)
kew(z, s)
(3.1-32)
s~c. 3.1
171
It is important, b e c a u s e of c o m p a r i s o n with the c a s e w h e n the p i p e is insulated as s h o w n o n F i g u r e s 3.1-3 a n d 3.1-4, to r e p r e s e n t the r e s p o n s e of the s y s t e m (fluid t e m p e r a t u r e c h a n g e at the p i p e outlet) in the c a s e w h e n the inlet d i s t u r b a n c e is an impulse fluid t e m p e r a t u r e c h a n g e , a(O, t) = a(t). (The wall t e m p e r a t u r e remains u n c h a n g e d a n d e q u a l to zero.) With e(O, s) = 1 the a b o v e e q u a t i o n y i e l d s s+k ~
z
Z
,c;~Z. ,~. cL-l{e'--w'-z / = e-wk ~.i{ew-s} An impulse r e s p o n s e b y fluid flow) in w h i c h impulse that p r o p a g a t e s of z / w s e c o n d s after amplitude d e c r e a s e s b y
Z
Z . = e" ~ ~(t _ ~-)
(3.1-33)
for the c o n v e c t i v e h e a t transfer p r o c e s s (transfer t h e r e is h e a t e x c h a n g e with the wail is thus a n b y v e l o c i t y w a l o n g the p i p e (with a time d e l a y the o c c u r r e n c e of the d i s t u r b a n c e ) a n d w h o s e the d a m p i n g factor e -zk/w, ~(z.t)
:\ ~\ {" /Z ~ ~,/,,,,," /1" ,. /
]Fig.
7
..........
L
/,i-"
t
<',,,,
3 . | - 6 P r o p a g a t i o n of impulse t e m p e r a t u r e d i s t u r b a n c e a l o n g the u n i n s u l a t e d p i p e
S p e c i a l a t t e n t i o n must b e p a i d to this d e c r e a s e in a m p l i t u d e s h o w n on Figure 3.1-6 b e c a u s e it d i d not exist in the c a s e of the i n s u I a t e d p i p e . T h e r e the d i s t u r b a n c e p r o p a g a t e d a l o n g the p i p e (tzansportez) u n c h a n g e d . For b e t t e r c o m p a r i s o n with F i g u r e 31-3 (which is also v a l i d for the i n s u l a t e d p i p e , as was m e n t i o n e d in the first e x a m p l e ) we s h o u l d o b t a i n an a n a l y t i c a l e x p r e s s i o n for t e m p e r a t u r e c h a n g e a l o n g the p i p e if e(O. t) =
I'T2
CHAP.
3 DISTRIBUTED
rCt) = h(t) ,, 1 for t > O. Substituting O(O. s) = l/s a n d (3.1-32) y i e l d s
eCz. s)
.
e" ~ k
whence
PROCESSES
Ow(Z, s)
= 0 into
z g~s ! 5
[
oCz, t)- -~kse. I ew
/"
I__ :;~k hCt-~-) S
(3.1-34)
F i g u r e 3.1-7 shows fluid t e m p e r a t u r e c h a n g e . O{z,t)
I
/
L /f
\ ./
/
/
6h ~ TI= L
,/ Fig.
3.1-7 P r o p a g a t i o n of s t e p t e m p e r a t u r e d i s t u r b a n c e a l o n g the u n i n s u l a t e d p i p e
T h e t e m p e r a t u r e profile a l o n g the p i p e is s h o w n at the moment Tt. Outlet t e m p e r a t u r e is e q u a l to e "Lk/w - e "Ttk. w h i c h m e a n s that it is c o n s i d e r a b l y smaller than inlet t e m p e r a t u r e .
P i p e h e a t e x c h a n g e r s (parallel-flow. c o u n t e r - f l o w a n d cross-flow) are frequent parts of plants and plant units and typical examples in which the dynamics of temperature c h a n g e between the warmer and the colder current is described b y a sysltelm of hyperbolic first-order PDE. In continuation, after accepting m a n y assumptions, w e will s h o w h o w to
ssc.
a.1
173
d e r i v e a m o d e l for the p a r a l l e l - f l o w a n d c o u n t e r - f l o w h e a t e x c h a n g e r .
For
the former w e will a l s o s h o w h o w to o b t a i n the matrix transfer f u n c t i o n relating input a n d o u t p u t t e m p e r a t u r e s .
Examplo
3
Dynamics of exchangers
parallel-flow
and
counter-flow
heat
F i g u r e 3.1-8 s h o w s a s k e t c h of c o u n t e r - f l o w a n d p a r a l l e l - f l o w (the d i r e c t i o n of flow is g i v e n b y a b r o k e n line) h e a t e x c h a n g e r s for w h i c h m a t h e m a t i c a l m o d e l s for u n s t e a d y fluid t e m p e r a t u r e c h a n g e s must b e d e r i v e d . T h e a s s u m p t i o n s are: b o t h fluids a r e i n c o m p r e s s i b l e liquids. p r e s s u r e s , m a s s flow rates, flow v e l o c i t i e s , c r o s s - s e c t i o n a l a r e a s a n d h e a t transfer c o e f f i c i e n t s a r e c o n s t a n t , t h e h e a t c a p a c i t a n c e of the p i p e walls, h e a t c o n d u c t i o n in the d i r e c t i o n of the z a x i s a n d l o s s e s d u e to friction a r e neglected. - the c o e f f i c i e n t of h e a t c o n d u c t i o n infinite.
through
the
pipe
2iI I0,o
I'
m) -
-
-
-
I
t
W///A
Fig. 3.1-8 S c h e m a t i c p r e s e n t a t i o n of c o u n t e r - f l o w a n d parallel-flow heat exchanger
wall is
174
CHAP.
paratteL-ftow
/
pzAzw2 mzu r
mzU2-
Bz
'..~'~";:'~..~?..;'.'.-~'.':.~.2~"..:.~.:'~'/..:'.'.:.~':+:':(::." ~.....-.-...-.... • -.-...-.- • ,.-.- - ..-.-+...,.,....-.._.+.+.~ . . . . . .
---"1
Pl Aiwl mlu I iJtPzAzwzl dz FJAzwz- az
~z
l
dz
3 IDISTRIBUT~'D
'
xk~,
.
'a" (
/
,~\~2~
r--'-
+~,+'~.'~
L__
81P. A w PzArwz+ ~ z z 2
I
dz
al,,z,2~
PzAzwz counter-ftow
=1::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: .............. '
PROCESSES
°'°'
. . . . . . . . . .
z'-dz
Fig. 3.1-9 Sketch of oontzol volume dV with heat inflow and heat outflow E q u a t i o n s for t h e c o n s e r v a t i o n of e n e r g y (power equations) are f o r m u l a t e d for infinitesimal v o l u m e s of b o t h t h e fluids b e t w e e n s e c t i o n s z a n d z+dz. ECE for fluid I.. mlul + PIAIws + U d z q - (mlu, + a ( m ' u 0 d z ) az
(PIAlwl + a(PIAIWl) d z ) = az
(3.1-3,~]
= a-~-(A1dzplul) yields ~-~(mlu+ + P1Aiwl) +
(3.1-36)
(Alplul) = Uq
C o n s i d e r i n g that mlul + P1AIwl ,= mlul ÷ Ptv~m~ = m~(ul * Pivl) " mlil plu, = p t i l - P. A~ ~p' = ~ml ~t
~z
'
,
P = const.
d i f f e r e n t i a t i o n of E q u a t i o n (3.1-3b) y i e l d s
-all- + at
~il wl~-~
=
U q= Alp,
T"'--U=(ez ~1Pl
For l i q u i d s t h e r e is i - c~. s o t h e e x p r e s s i o n for fluid 1 t e m p e r a t u r e c h a n g e
(3.1-37) last
equation
yields
the
final
,==c.
175
s.l
~,~lCz, t) , aelCz, t) U~ (~2 -el) at + --1 ~ = A~plcl
(3.1-38)
T h e ECE for the warmex fluid 2 (the u p p e r parallel-flow h e a t e x c h a n g e r ) is
signs
are
valid
for the
m2u2 + P2Azwz - Udzq - (mzu ~ * ~(m2u2)dz) - (P2A2w2 i a(PzAzW2)dz) = ~z az = - ~ ' (A 2dzp2u2)
(3.1-3q)
In this e q u a t i o n attention must b e p a i d to the n e g a t i v e s i g n in front of the h e a t t r a n s f e r r e d to fluid 1, a n d in front of the partial d e r i v a t i v e s with r e s p e c t to z in the c o u n t e r - f l o w e x c h a n g e r . T h e s e signs are what distinguishe E q u a t i o n (3.1-3q) (for the h e a t i n g fluid) from E q u a t i o n (3.1-351 (for the h e a t e d fluid) a n d t h e y a l w a y s reflect the fact that fluid 2 g i v e s heat to fluid 1, a n d that the e n e r g y l e v e l of fluid 2 in the c o u n t e r - f l o w e x c h a n g e r d e c r e a s e s in the n e g a t i v e d i r e c t i o n of the z axis. If the s a m e p r o c e d u r e as the o n e c a r r i e d out with the e q u a t i o n for fluid I is p e r f o r m e d , w e g e t
~t
-z
~
=
(~,-
(3.l-40)
~2)
Equations (3.1-38) a n d (3.1-40) a r e a g y = t o m of h y p e r b o l i c first-order PDE, a n d to s o l v e them it is n e c e s s a r y to d e f i n e ( w o b o u n d a r y and two lnit|al cond|t|onm. In the c o u n t e r - f l o w e x c h a n g e r t h e s e c o n d i t i o n s are of the form BC: ,%(0, t) = rift). IC, ~,(z. to) = p~(z).
,8'2(L. t) = r2(t) ,%(z. to) " P2(z)
The u p p e r a s s u m p t i o n s m a k e the s y s t e m p r o c e s s o c c u r s without d i s t u r b a n c e a n d conditions c h a n g e . If the matrix differential coefficients kl = U ~ / A I p l C I , i = 1. 2 are e q u a t i o n s c a n b e written as follows
, et(z.
[-wl~"z - kl t)
= L
kz
k~ *w2~'-ff- kz
.
of e q u a t i o n s h o m o g e n e o u s , the o n l y b e c a u s e the b o u n d a r y o p e r a t o r Az is d e f i n e d a n d the reintroduced, the s y s t e m of
] aCz, t) - Az~Cz, t)
(3.1-41)
176
CHAP.
3
DISTRIBUTED
PROCESSES
O b t a i n i n g c l o s e d form solutions for this s y s t e m of e q u a t i o n s
is n o w a
r a t h e r m o r e c o n s i d e r a b l e task t h a n w h e n t h e r e w a s o n l y o n e e q u a t i o n . T h e p r o c e s s h a s i n c r e a s e d in c o m p l e x i t y . H e r e w e will, n e v e r t h e l e s s , s h o w h o w to o b t a i n the matrix transfer function w h i c h r e l a t e s t e m p e r a t u r e s a l o n g the p a r a l l e l - l l o w h e a t e x c h a n g e r as f u n c t i o n s of b o u n d a r y c o n d i t i o n s ; t e m p e r a t u r e s ~ ( 0 . t) a n d ~z(0. t}. With IC e q u a l to z e r o . after L a p l a c e t r a n s f o r m a t i o n with r e s p e c t to time t the s y s t e m of e q u a t i o n s (3.1-41) b e c o m e s the following s y s t e m of ODE with r e s p e c t to z in the s - d o m a i n dem(z.dz s) = de2(z.
s}
,
dz To
make
s Wl + k~ el(z. s) + --~le2(z. s)
.
(3.1-42)
k2e1(z, s) -s + k2 ez(z. s) w z w 2
it easier to perform
that the f l u i d s
are
(he
(3.1-43)
the following
lamo
derivation
(p, =~z. ci - cz) a n d
w I = w 2 ~-w. This results in kt = k2 =k. After introducing -(s+k)/w a n d
B = k/w. the a b o v e
system
of equations
we
will a s s u m e
that A, = Az the s y m b o l s
shown
and A
=
in matrix form
becomes I
(3.1-44)
e2(z. The
s).Jz
e2(z,
solution of system (3.1-44) is
~(z. s) = e A Z e ( o ,
s) - ,~(z)e(o,
T h e t r a n s i e n t or f u n d a m e n t a l matrix ~ ( z ) formula ,~(z) = e xlz A
- ),2I + eXzZ A
)~t- Xz ),1 a n d ~2 a r e t h e e i g e n v a l u e s (A-)~I) = 0. Xl " A + B - .s__
- X,I
~,2" ),i
(3.1-4s)
s) is c a l c u l a t e d
according
to the
(3.1-4{~)
of matrix A o b t a i n e d from the e q u a t i o ns
,
(3.1-4Y)
s+2k ),z = A - B = - - -
(3.1-48)
W
w
see.a.z
177
Equation (3.1-46), after determining matrices IF1 and Fz, yields
[chBz ~,(z) - eAz|
LshBz
= C z [ eBZ+e'Bz
shB
z]
eBZ.e-Bz ] (3.1-40)
chBzJ
eBZ.e -Bz
eBZ,e -Bz
Temperature interrelations can be shown clearly in a block diagram. and this has been done in Figure 3.1-10 aocozding to Equations (3.1-45) and (3.1-4g).
%(0,s)
I
-[chaz
~
= e,(z,s)
s
shBz [ ~ _ _1
Oz{O,s)
÷wk
hBz
_ chBz
B=
I
-k_v
:1e ~
"- ez(Z,S)
Fig. 3.1-10 Block diagram of interrelations between input and output temperatures in the parallel-flow heat exchanger (kt = k2 = k. w, -w 2 =w) Equations (3.1-45) and (3.1-49) make it temperatures along the whole heat exchanger changes e,(O, t) and e2(O, t). If ~1 undergoes at(O, t)= h(t). the temperature change of both solution of the following equation k k
Le2(z. s)
2
e"~'z- e-~--z
possible to determine for given temperature unit step increase, i.e. fluids is given by the
k__z k e w , e ~ 'z
][o]
(3.l-5O)
Rearrangement and inverse Laplace transformation yields
~,(z. t) = 1 * e
2
h(t - z ) w
(3.1-51)
178
CHAP. 3 DISTRIBUTED
PROCESSES
z -2~.k 8{z. t) - l - e
h(t - z__) w '
2
For the case
o f k = 1. p i p e
length
(3.1-52)
L = l m and
flow
velocity
w ,, 1 m / s .
F i g u r e 3.1-II s h o w s t e m p e r a t u r e c h a n g e s of b o t h the fluids a l o n g the w h o l e e x c h a n g e r after T t ffi L / w ,, I s e c o n d . It must b e o b s e r v e d that w h e n z - . oo. ~h a n d 82 c o n v e r g e to the v a l u e 0.5. a n d this differs from the results in F i g u r e 3.1-7 (~)w ffi const.) w h e r e the fluid t e m p e r a t u r e 3 c o n v e r g e s to zero. Olz.I) O,[z.O)=l
._1
0.5
1 !
0.2
Fig.
0.~,
0.6
0.8
L=t
Z
3.1-11 T e m p e r a t u r e profile a l o n g h e a t e x c h a n g e r p i p e after Tt s e c o n d s in the c a s e 31(0. t) = h(t) = I
This last example shows h o w technical devices with relatively simple geometry a n d for which a whole series of assumptions has b e e n made. have complicated mathematical models which m a k e it difficult to obtain analytically the d e m a n d e d responses in the case of disturbed boundary conditions. Therefore it. a n d also the m a n y examples that follow, shows w h y everyday technical practice aspires to methods of direct numerical simulation of P D E in the examination of dynamic properties of specific p,ocesses, or to approximation of transcendental transfer functions which always appear in processes that are distributed in space.
==c.
a.l
ITQ
In the p r e v i o u s d i s c u s s i o n of h e a t e x c h a n g e r s
the p o s s i b i l i t y of h e a t
a c c u m u l a t i o n in the walls of the p i p e w a s n e g l e c t e d . In a g r e a t n u m b e r of c a s e s for t h i n - w a l l e d p i p e s this is c o m p l e t e l y justified. H o w e v e r . w h e n t h e wall of the p i p e is thicker, or w h e n the h e a t c a p a c i t a n c e of the wall is of the s a m e o r d e r of m a g n i t u d e a s the h e a t c a p a c i t a n c e of the fluid ( w h i c h the m o d e l b u i l d e r must e v a l u a t e at the b e g i n n i n g ) , the p r o c e s s of h e a t s t o r a g e in the p i p e wall must b e t a k e n into a c c o u n t a s well. T h e following e x a m p l e s h o w s c h a n g e s in the m o d e l of u n s t a t i o n a r y t e m p e r a t u r e c h a n g e for that c a s e .
]Example
4 Fluid flow with h e a t s t o r a g e in the p i p e wall
Fluid flows through a thicker pipe, externally insulated. Derive the transfer function relating output fluid temperature c h a n g e with input fluid temperature change, taking into account possibilities of heat storage in the wall. Assume that the coefficient of heat conduction in the radial direction is infinite, which m e a n s that the temperature of the wall is the s a m e in its w h o l e t h i c k n e s s . N e g l e c t h e a t c o n d u c t i o n in the d i r e c t i o n of the z axis. T h e o t h e r a s s u m p t i o n s a r e the s a m e as in t h e p r e c e d i n g example.
A O(O.t} - ~ = -
~
a
....
,.~','.
m
. j'2~2".1,...,..
'.
Aw
~
r
I
;
::
...,...
.....
./....
I
-~--~
-~ ( L . t )
.....
.1
z
z*dz
]Fig. 3.1-12 Convective heat transfer through an insulated pipe with heat storage in the pipe wall Because of the appearance of a n e w possible heat storage tank equations for the conservation of energy both for the fluid and for the pipe wall must n o w be derived. ECE for the fluid:
180
C H A P . 3 DISTRIBUTED P R O C E S S E S
inca - ( i n c a
a(mca) d z ) - cxUdz(a- - 3 w) = a ( A d z p c a )
+
at
8z
(3.1-63)
The above equation yields ~8 --
at
~8
+ w-az
ECE
aU =
(aw-
Apc
for the wall:
A w p w c-w - a z -0~w -~-=
a~)w at
(3.1-54)
,~)
aU
aUdz(a
-
(3.[-55)
aw)
(e - ,~w)
(3.1-56)
Aw~wCw
=
T h e s y s t e m c o m p r i s i n g E q u a t i o n s (3.1-54) a n d (3.1-56) is a m o d e l for u n s t e a d y fluid a n d wall t e m p e r a t u r e c h a n g e s a n d to s o l v e it w e m u s t d e f i n e the following b o u n d a r y a n d initial c o n d i t i o n s ,
a(0. t) - r(0 a(z. 0) - p,(z). ~,,,,,(z. 0 ) = pz(z)
BC..
IC:
T h e transfer function r e l a t i n g fluid t e m p e r a t u r e a l o n g the p i p e with the given boundary t e m p e r a t u r e at z = 0 is o b t a i n e d after L a p l a c e t r a n s f o r m a t i o n of E q u a t i o n s (3.1-54) a n d (3.1-56) with IC e q u a l to z e r o , a n d after s o l v i n g the ODE o b t a i n e d with r e s p e c t to z in the s - d o m a i n for fluid t e m p e r a t u r e a. s2,(k,÷k~)s
G(z. s)
-
e(z, s)
.
e(o. s)
e
"
w(s+k~)
z
(3.l-s~')
T h e c o e f f i c i e n t s k h a v e the s a m e m e a n i n g as b e f o r e ,,U kl = Apc
(3.1-58)
=U kz = AwpwCw
(3.1-59)
Their i n v e r s e v a l u e s a r e fluid t h e r m a l time c o n s t a n t TF a n d wall thermal time c o n s t a n t Tw. TF-
1 Apc k-7 = =U
'
(3.1-60)
lal
=ec. a.1
1 Tw = kz
AwPwCw =U
(3.l-hi)
T h e transfer function G(z.s) g i v e n in E q u a t i o n (3.1-57) is a m o r e e n c o m p a s s i n g a n d g e n e r a l form of transfer function for the p i p e h e a t e x c h a n g e r . From it w e c a n o b t a i n a s b o u n d a r y c a s e s the transfer functions d e r i v e d earlier if w e m a k e a s s u m p t i o n s that w e r e left out in this c a s e . Thus, for e x a m p l e , if w e allow the c o e f f i c i e n t = to c o n v e r g e to z e r o , i.e. if t h e r e is n o h e a t transfer o n t o the wall, the transfer function (3.1-57) b e c o m e s the a l r e a d y k n o w n transfer f u n c t i o n that w a s d e r i v e d for the "pure" t r a n s p o r t p r o c e s s in the c a s e of the i n s u l a t e d p i p e in E x a m p l e 1, g i v e n b y E q u a t i o n C3.1-25).
This e n d s the p r e s e n t a t i o n of the d y n a m i c s of c o n v e c t i v e m a s s or e n e r g y transfer, b e t t e r k n o w n as t r a n s p o r t p r o c e s s e s , w h o s e u n s t e a d y s t a t e c h a n g e s w e d e s c r i b e d in the form of a h y p e r b o l i c firstorder PDE or b y a s y s t e m of h y p e r b o l i c first-order PDE. T h e s e a r e a l s o the s i m p l e s t p r o c e s s e s d i s t r i b u t e d in s p a c e a n d their a n a l y s i s will b e of u s e for u n d e r s t a n d i n g more c o m p l e x p h e n o m e n a .
3.2 P R O C E ~ S E | WITH EQUALIZATION BABIC PARABOLIC PARTIAL DIIFYERENTIAL EQUATION
This name d e s c r i b e s processes in w h i c h d i f f e r e n c e s i n p o t e n t i a l result
in a n e q u a l i z a t i o n of c o n d i t i o n s t h r o u g h o u t the s y s t e m u n d e r o b s e r v a t i o n . The most t y p i c a l e x a m p l e s met in e v e r y d a y p r a c t i c e a r e h e a t c o n d u c t i o n , diffusion a n d the motion of v i s c o u s (sticky) liquids. C o m m o n to all t h e s e p h y s i c a l l y different p r o c e s s e s is a g a i n their m a t h e m a t i c a l notation., their d y n a m i c s is d e s c r i b e d b y p a r a b o l i c P D E . In this s e c t i o n w e will c o m p l e t e l y a n a l y z e the p r o c e s s of h e a t c o n d u c t i o n , b u t t h e d e r i v a t i o n of e q u a t i o n s , m e t h o d s for s o l v i n g t h e m a n d the s o l u t i o n s t h e m s e l v e s a r e i d e n t i c a l for this w h o l e c l a s s of p r o b l e m s , a n d to o b t a i n a s a t i s f a c t o r y p h y s i c a l i n t e r p r e t a t i o n it is e n o u g h to g i v e c o e f f i c i e n t s their r e a l m e a n i n g .
182
CHAP.
If c e r t a i n a s s u m p t i o n s a r e m a d e , i! c a n b e
PROCESSES
S DISTRIBUTED
s h o w n that all the p r o c e s s e s
m e n t i o n e d a r e d e s c r i b e d b y the following PDE, 8u ~u --~-- a ~z--~z-
(3.2-I)
D e p e n d i n g o n the nature of the process, u a n d physical meaning~ Heat conduction.- u - temperature, a = ),/co -
a have
the following
coefficient of thermal
conductivity
diffusion~ u - c o n c e n t r a t i o n , a = D - c o e f f i c i e n t of diffusion motion of v i s c o u s liquid: u -
p a r t i c l e v e l o c i t y , a - ~, -
c o e f f i c i e n t of
kinematic v i s c o s i t y Of c o u r s e , as a s s u m p t i o n s c h a n g e so d o e s E q u a t i o n (3.2-1). It g e t s new terms a n d e x p a n d s . N e v e r t h e l e s s , the form it is s h o w n in is b a s i c for all p r o c e s s e s with e q u a l i z a t i o n .
Example
1 Heat c o n d u c t i o n t h r o u g h h o m o g e n e o u s i n s u l a t e d b o d y
F i g u r e 3.2-1 s h o w s a prismatic b o d y i n s u l a t e d Derive the e q u a t i o n d e s c r i b i n g c o o r d i n a t e a n d in time, a n d boundary conditions:
o n all its lateral sides.
t e m p e r a t u r e c h a n g e a l o n g the spatial express m a t h e m a t i c a l l y the following
a) g i v e n t e m p e r a t u r e s a r e m a i n t a i n e d at the e n d s of the b o d y (frontal cross-sections), b) a g i v e n h e a t flow rate is e x t e r n a l l y s u p p l i e d body.
to the
ends
of the
c] t h e r e is c o n v e c t i v e h e a t transfer from the e n d s of the b o d y to the environment. Also d e r i v e transfer functions c h a r a c t e r i z i n g h e a t c o n d u c t i o r ~ for the b o u n d a r y c o n d i t i o n s for c a s e a), a n d for a c o m b i n a t i o n of b o u n d a r y c o n d i t i o n s for c a s e s a) a n d b).
s~c. a.2
1l~3
Assumptions,
-
temperature is uniform o n a cross-section of the body.
- the b o d y is h o m o g e n e o u s . -
heat conduction is observed along the z axis only. the coefficient of thermal conductivity )~ is not a function of temperature.
.dM = Adzp
zt I I [ I I l X I ~ J
/
//" I
gdz
/ / Y / / / '/./'q~q(
8
I I I l l l l l l l l l l . ^ ~ t t l *
u;
I I I I I I!11M-.I--L-J.-J-~I I l d I I ~ -i-+ H - t - I - -MI--I-t ~ A H - - / r - - / ~ / IIIII.
/
~
I t
I
-t. H- . ~. . . . . - - - - ~4
,~/YZdlII
I I II/1
1114111 I1'11111
z m
I
i
_Mj z+dz
I I ,
,
I
--i --i
z=0
Fig.
3.24
/
I I
z=L
Heat c o n d u c t i o n t h r o u g h h o m o g e n e o u s i n s u l a t e d b o d y
The b a l a n c e of h e a t e n e r g y is f o r m u l a t e d for a infinitesimal v o l u m e dV of the b o d y qCz. t)A - [ q ( z , t) * bq(z. l)dz]A az
,, dU dt
,,
,
as.(z, t)
^pc ~ - - ~ - - - d z
(3.2-2)
R e a r r a n g e m e n t of (3.2-2) y i e l d s aq(z. az
t) =
pc a~(z. at
t) (3.2-3)
The d e n s i t y of h e a t flow is k n o w n to b e p r o p o r t i o n a l to the spatial g r a d i e n t of t e m p e r a t u r e a n d is e x p r e s s e d a s follows (Fourier's law) qCz,
t)
-- - x ~(z,
az
t)
(3.2-4)
Substitution of (3.2-4) into (3.2-3) a n d differentiation y i e l d
a~(z. t) at
a2~(z, t) = a ~
(3.2-5)
~4
CHAP.
3 " D I S T R I B U T n ' D P R O C E S S E ¢~
C o n s e q u e n t l y . the d y n a m i c s of t e m p e r a t u r e c h a n g e
a l o n g the b o d y is
d e s c r i b e d b y the linear p a r a b o l i c PDE (3.2-5). If we want to k n o w the t e m p e r a t u r e
in a s p e c i f i c
c r o s s - s e c t i o n at a
g i v e n m o m e n t w e must k n o w the t e m p e r a t u r e distribution at to = O, w h i c h is c o n s i d e r e d the b e g i n n i n g of c a l c u l a t i o n (i.e. w e must know the initial c o n d i t i o n s ) , a n d also the c o n d i t i o n s at the e d g e s ( b o u n d a r i e s ) of the b o d y ( b o u n d a r y c o n d i t i o n s ) , i.e. h o w the b o d y " c o n t a c t s " its e n v i r o n m e n t . The
initial
condition
is
identical
to
the
one
for
the
hyperbolic
first-order P O E IC: ~(z. to) " p(z) . 0 ¢ z < L
a n d it d e s c r i b e s the t e m p e r a t u r e profile a l o n g the b o d y at to. The b o u n d a r y conditions, according to b e g i n n i n g of the task, c a n b e of t h r e e kinds,
a) b)-xA
~(0. t) = f,(t). ,~(0, t)
at
the
(3.2-e,) .
(3.2-7)
bz
c) ~Co, t) = ~ [~(o. O- f,Ct)].~(u. t) 8Z
made
~(L. t) - f2(t) .
= q~(t). ),A,'~(L' t) = q2(t)
~z
demands
A
-~[~(t,.O- f2(t)]
~Z
(3.2-8) The f u n c t i o n s p(z), fl(t), f2(t), ql(t) a n d q2(t) a r e g i v e n , f~(t) a n d fz(t) are t e m p e r a t u r e s of the e n v i r o n m e n t at the b o u n d a r i e s z = O a n d z = L. a n d q,(t) a n d q2(t) are h e a t flow r a t e s (i.e. the a m o u n t of h e a t in a unit of time) f o r c e d on the e n d s of the b o d y . It is o b v i o u s , but s h o u l d n e v e r t h e l e s s b e m e n t i o n e d , that b e s i d e s the t h r e e a b o v e p a i r s of b o u n d a r y c o n d i t i o n s . their c o m b i n a t i o n s a r e also p o s s i b l e , for e x a m p l e , t e m p e r a t u r e f,(t) c a n b e g i v e n in z = O. a n d c o n v e c t i v e h e a t e x c h a n g e in z ,, L, i.e. b o u n d a r y c o n d i t i o n c). a), b) a n d c) d o not s h o w all the w a y s in w h i c h h e a t c a n b e t r a n s f e r r e d to the e n d c r o s s - s e c t i o n s . For e x a m p l e , t h e r e c a n also b e r a d i a t i o n or, if there is firm c o n t a c t with s o m e o t h e r s u r f a c e at z = O a n d z = L combined conduction.
with low thermal
resistance,
heat
will b e
transferred
by
s=c. ~.2
The
!/}5 boundary
conditions
shown
by
c)
are
the
most
general
and
c o n d i t i o n s a) a n d b) c a n a r i s e a s b o u n d a r y c a s e s of E q u a t i o n (3.2-8). If the c o n v e c t i v e h e a t transfer c o e f f i c i e n t - is v e r y l a r g e , t h e o r e t i c a l l y e v e n infinite, in other w o r d s if it is m u c h g r e a t e r t h a n ),, c o n d i t i o n c) b e c o m e s b o u n d a r y c o n d i t i o n a) b e c a u s e 0~ e ~ l e a d s to e(O, t) ,, fl(t) a n d %(h. t) ~- f2(t). But if ,, is insignificant, in o t h e r w o r d s if h e a t transfer on the e d g e s is c o m p l e t e l y i m p o s s i b l e , c o n d i t i o n c) b e c o m e s b o u n d a r y c o n d i t i o n b) for q,(t) = qz(t) ; O. w h i c h r e a l l y satisfies the c a s e of i n s u l a t e d frontal cross-sections. The transfer function for the c o n d i t i o n p(z) ,, O will a g a i n b e o b t a i n e d after L a p l a c e transformation of E q u a t i o n (3.2-5) with r e s p e c t to time t a n d s o l v i n g the ODE o b t a i n e d with r e s p e c t to z d2e(z, s)
dz z
s a
- --e(z,
s) = 0
(3.2-Q)
S o l v i n g E q u a t i o n (3.2-9) y i e l d s e(z. s) - c , :
•
=
(3.2-1o)
The law a c c o r d i n g to w h i c h t e m p e r a t u r e c h a n g e s a l o n g the z axis. i.e. the transfer function that s h o w s that law, o b v i o u s l y d e p e n d s o n l y o n c o n d i t i o n s at the b o u n d a r y of the b o d y . Thus C~ a n d C2 d e p e n d o n l y o n the m a n n e r in w h i c h the p r o c e s s o c c u r s at the b o u n d a r i e s , s o t h e y must b e s p e c i a l l y d e t e r m i n e d for the c o n d i t i o n s g i v e n at the b e g i n n i n g example. TEMPERATURE
IS GIVEN AT B O T H
of this
ENDS
B o u n d a r y c o n d i t i o n s , e(O, t) - f,(t), z ,. 0 (left e n d ) ,~(L. t) - fz(t), z ,, L (right e n d ) Substituting these boundary expressions for C, and Cz
e~-~ L C, ,,
2 sh( ]/s~ L)
conditions
into
(3.2-10)
yields
] Fl(s) -
2 sh()~
L)
F2(s) ,
(3.2-11)
the
180
CHAP.
C2 = - e" "/E~L
2 s h ( ~ L)
Fl(s) +
I
2 s h ( ~ L)
3 DISTRIBUTED
PROCESSES
43,2-12)
Fz(s) .
Inserting C, a n d Cz into (3.2-10) a n d substituting b o u n d a r y temperatures eGO, s) a n d e(L. s) for F,(s) a n d Fz(s) yields
e(z.
s)÷
sh(/:-::L)
sh(~O
e(L, s)
(3.2-13)
or
e(z. s) = G~(z, s)e(O, s) + G2{z. s)e(L, s)
TEMPERATURE
G I V E N A T L E F T END. R I G H T E N D
INSULATED
Boundary conditions: o(o,
t) = f,(t),
a,~(L, t) ~z
=0.
z
-
o
z=L
The coefficients C1 a n d Cz can b e o b t a i n e d from Equation 43.2-10) and its differentiation with r e s p e c t to z (to satisfy the s e c o n d b o u n d a r y condition). It is not difficult to p r o v e that
C~ =
C2 =
e•L
o(0, s)
(3.2-14)
e O(O, s) 2ch(~-a L)
(3.2-15)
2 ch( ~ a L)
Equation (3.2-10). referring to (3.2-14) and (3.2-15). yields
G,(~. s) - o(z, s) = ch[)~(L-~)] e(O. s)
ch(~a L)
(3.2-I~)
~c.
3.2
187
It must b e o b s e r v e d
that the e q u a t i o n s fez transfer f u n c t i o n (3.2-13) a n d
(3.2-1(5) a r e f u n c t i o n s of b o t h v a r i a b l e z a n d v a r i a b l e s. T h e r e is a n unlimited n u m b e r of s u c h transfer functions, a n d it is p o s s i b l e , u s i n g the c o r r e s p o n d i n g z, to d e t e r m i n e for e v e r y c r o s s - s e c t i o n t h e r e l a t i o n b e t w e e n its t e m p e r a t u r e a n d the g i v e n b o u n d a r y c o n d i t i o n s . The transfer f u n c t i o n (3.2-16) will b e of u s e to a n a l y z e t h e following e x a m p l e , w h i c h is t y p i c a l for a w h o l e c l a s s of p r o c e s s d e v i c e s - d i r e c ! heat e x c h a n g e r s with a thick wall.
Example
2 Direct h e a t e x c h a n g e r
with a thick wall
F i g u r e 3.2-2 s h o w s a n e x t e r n a l l y i n s u l a t e d t a n k with a thick wall in w h i c h a liquid is h e a t e d a n d i n t e n s e l y m i x e d , i n s u r i n g its uniform t e m p e r a t u r e t h r o u g h o u t the w h o l e v o l u m e . T h e p x o c e s s d e m a n d s that the output t e m p e r a t u r e must b e c o n t r o l l e d . For c o n s t a n t i n p u t a n d o u t p u t m, the o n l y d i s t u r b a n c e v a r i a b l e is the input fluid l e m p e r a l u r e e~j. T h e h e a t s u p p l i e d b y the e l e c t r i c a l h e a t e r Qel is in this c a s e a m a n i p u l a t i n g (controling) v a r i a b l e . D e t e r m i n e the transfer f u n c t i o n s r e l a t i n g liquid t e m p e r a t u r e ~) in the tank with input t e m p e r a t u r e ell a n d h e a t QelAssumptions: p e r f e c t a n d v i g o r o u s m i x i n g s o that the c o e f f i c i e n t of h e a t transfer o n t o the wall c a n b e c o n s i d e r e d v e r y l a r g e ( = ~ ~) or, w h i c h is the s a m e , the wall t e m p e r a t u r e o n the inner s u r f a c e c a n b e c o n s i d e r e d to e q u a l t h e fluid t e m p e r a t u r e in the t a n k ew(O, t) m e(fl. m.cf Ow
_-_-__--~ -_ -__-____-
. _
-
o
Fig. 3.2-2
H e a t e r with a thick wall ( d i r e c t h e a t e x c h a n g e r )
188
CHAP.
3 DISTRIBUTED
PROCESSES
T h e e q u a t i o n for the c o n s e r v a t i o n of (heat) e n e r g y y i e l d s
do mcf~)fl ,~ Oel " mcf~ - O = Mcfd--~
(3.2-17)
O is the h e a t flow rate o n the tank wall a n d it c a n
be
e x p r e s s e d as
the time c h a n g e in h e a t e n e r g y s t o r e d in the wall dE O = d-T
(3.2-18)
Knowing the law g o v e r n i n g the t e m p e r a t u r e c h a n g e of the wall ew(Z, t) a l o n g z, E c a n b e e x p r e s s e d as
L (3.2-19)
E = IApwcwew(z, t) dz 0
Substituting E q u a t i o n (3.2-19) into (3.2-18) y i e l d s
L
d I Apwcw3w(Z. t)
Q(t) =~-~-
(3.2-20)
dz
0
If w e c o n s i d e r that E(O) = O at t = O, L a p l a c e transformation of (3.2-20) yields L
OCs) = s IApwcwew(z. s) d z
(3.2-21)
o
Inserting expression
ew(Z. s) from E q u a t i o n
O(s) = Apwcws
th( ~
(3.2-1b) into
(3.2-21) g i v e s
L) e(s)
Transformation of Equation (3.2-]7) into the s-domain, (3.2-22) for Q(s) a n d shorter rearrangement yield e,s) = G(s)er,Cs) •
1 --
roof
Gfs)OenCs)
the
final
(3.2-22) substitution
(3.2-23)
of
s=c. a.',
lfiq
1
S(s) = ---Ms÷ A p ~ , , s th(~L) m ~ mcf
(3.2-24)
+ ]
The transfer function G(s) c o m p l e t e l y d e t e r m i n e s c h a n g e s in the fluid t e m p e r a t u r e ~ in the tank for a r b i t r a r y c h a n g e s of 3fl a n d Oel. This e x a m p l e has not b e e n c h o s e n o n l y to s h o w that t r a n s c e n d e n t a l forms a p p e a r in G(s). (That this is so in s p a t i a l l y d i s t r i b u t e d s y s t e m s h a s p r o b a b l y a l r e a d y b e e n a c c e p t e d a n d is clear.) H e r e w e want to s h o w that the transfer function (3.2-24) is a g e n e r a l form that c o m p r i s e s (as b o u n d a r y c a s e s ) transfer functions that w e r e o b t a i n e d w h e n this h e a t e r was o b s e r v e d as a s y s t e m with l u m p e d p a r a m e t e r s ( E q u a t i o n s (2.3-32) a n d (2.3-40).
a) T h e tank walls a r e m a d e of an insulating material for w h i c h ), = O. In that c a s e
G(s)
1
1 m
=
Ms+ m
~
ApwcwS Ws
th(
s L)+ l
--s
m
mcf
+ I
(3.2-25)
0 E q u a t i o n (3.2-23) h a s thus b e c o m e (2.3-32).
b) T h e tank walls a r e m a d e of a material with v e r y g r e a t c o n d u c t i v i t y ( x -. co a n d 4)w(Z. t) = ~)(t). 0 < z < L). In this case. w h e n
~ -',.co. it c a n b e s h o w n
thermal
that the middle term in the
denominator of the transfer function (3.2-24) is
lira ),... ~o
ApwCwS ~/F'~--w S mcf
th( ~C---w- s
L)
A1
.
.,-.
--pwv____w_
mcf
s
(3.2-26)
Referring to (3.2-26). (3.2-24) b e c o m e s
G(s)
=
1 --s +M m
MwCws mcf
+ I
-
1 Mcf * Mwcw mc[
s * I
(3.2-27)
IQO
CHAP.
3
DISTRIBUTED
PROCESSES
Thus G(s) in E q u a t i o n (2.3-40) is the limiting c a s e of E q u a t i o n (3.2-24) for )~ --* ®. a n d e q u a l s the a b o v e E q u a t i o n (3.2-27).
T h e n e x t s t e p in the a n a l y s i s of h e a t c o n d u c t i o n d y n a m i c s w o u l d b e to s o l v e PDE (3.2-5) for some g i v e n g e o m e t r y of the b o d y (slab, c y l i n d e r , semi-infinite solid b o d i e s , s p h e r e a n d so on) a n d g i v e n initial a n d b o u n d a r y c o n d i t i o n s . S u c h a n a n a l y s i s w o u l d s u r p a s s the p l a n n e d framework a n d p u r p o s e of this b o o k . It is the s u b j e c t of w h o l e c h a p t e r s a n d b o o k s o n t h e r m o d y n a m i c s a n d the e q u a t i o n s of m a t h e m a t i c a l p h y s i c s that treat the more limited p r o b l e m s of u n s t e a d y h e a t c o n d u c t i o n , diffusion p r o c e s s e s a n d the like. a n d g i v e v e r y d e t a i l e d solutions a n d m e t h o d s for s o l v i n g p a r a b o l i c PDE. N e v e r t h e l e s s , this b o o k d o e s c o v e r a n a n a l y s i s of the d y n a m i c s of h e a t c o n d u c t i o n t h r o u g h p i p e walls of the t y p e f o u n d in c l a s s i c a l h e a t e x c h a n g e r s , in e v a p o r a t o r s u r f a c e s of s t e a m g e n e r a t o r a n d the like. A c o m m o n f e a t u r e of most m u c h smaller t h a n the p i p e s u r f a c e c u r v a t u r e ) s o that o b s e r v e d as h e a t c o n d u c t i o n
p i p e e x c h a n g e r s is that the wall t h i c k n e s s is r a d i u s (which m a k e s it p o s s i b l e to n e g l e c t h e a t c o n d u c t i o n t h r o u g h the wall c a n b e t h r o u g h a flat s u r f a c e .
T h e following e x a m p l e will s h o w how to o b t a i n a transfer function, how to a n a l y z e the d y n a m i c p r o p e r t i e s a n d r e l a t e d c h a r a c t e r i s t i c s ( p o l e s a n d z e r o s of transfer functions) of h e a t c o n d u c t i o n p r o c e s s e s t h r o u g h the p i p e of the h e a t e x c h a n g e r , a n d e s p e c i a l l y s h o w how to r e d u c e the o r d e r of the s y s t e m a n d o b t a i n transfer functions in the form of p r o p e r rational function.
Example
3 H e a t c o n d u c t i o n t h r o u g h the e x c h a n g e r wall
For the p i p e wall of the h e a t e x c h a n g e r s h o w n o n F i g u r e d e t e r m i n e transfer functions d e s c r i b i n g c h a n g e s in the h e a t flow rate t r a n s f e r r e d from the wall o n t o the h e a t e d fluid d e p e n d i n g o n h e a t rate q~(t) b r o u g h t to the p i p e b y h e a t i n g g a s e s a n d the t e m p e r a t u r e of the h e a t e d fluid. T h e d y n a m i c s of t h r e e c a s e s will b e analyzed~
3.2-3 q~(t) flow el(t)
~t:~.
3.:2
1QI
a) -
forced radiation heat flow rate ql(t) on the side of the heating gases
very high coefficient of c o n v e c t i v e heat transfer ~2 onto the h e a t e d fluid b) - f o r c e d r a d i a t i o n h e a t flow i a t e q,(t) on the s i d e of the h e a t i n g
gases finite c o e f f i c i e n t h e a t e d fluid
of
convective
heat
transfer
~2 o n t o
the
c) convective heat
transfer o n b o t h s u r f a c e s
of the
wall
with
are
often
finite h e a t transfer c o e f f i c i e n t s '*t a n d ~z-
(%1
G2
ql
~,(tl
~ f ( t ) : Oz(t)
~-A
z=O
Fig.
z=l
3 . 2 - 3 Heat transfer t h r o u g h e x c h a n g e r wall
The c o n d i t i o n s a) a n d b) c o r r e s p o n d to r e a l i z e d in s t e a m g e n e r a t o r e v a p o r a t o r p i p e s .
conditions
that
a) F o r c e d h e a t flow rate q,(t) a n d a n infinite h e a t transfer c o e f f i c i e n t ~z g e t the following m a t h e m a t i c a l form for b o u n d a r y c o n d i t i o n s in z ,~ 0 and
z =
Ist B C
.~,A ae(O, t) az = q1(t)
, z ,, 0
IQ2
CHAP. 3 DISTRIBUTEID
2ndBC
o(~. t) = of(t)
PROCESSES
. z ,, s
S u b s t i t u t i n g the B C in c r o s s - s e c t i o n z = 0 into E q u a t i o n (3.2-10). w h i c h h a s first b e e n d i f f e r e n t i a t e d with r e s p e c t to z. a n d the BC for z - ~ into E q u a t i o n (3.2-10). y i e l d s c o n s t a n t s C! a n d C2
el(s) *
e ~--~ ~
Of(s) ),A
C,.,
(3.2-28)
2ch(1/gX ~ s 8) C2 = C, -
(3.2-29)
O,(s)
S T h e final e x p r e s s i o n for t e m p e r a t u r e distribution a l o n g the z axis in the wall is o b t a i n e d after the a b o v e e x p r e s s i o n s a r e i n s e r t e d into (3.240). T h e following e q u a t i o n d e t e r m i n e s h e a t flow r a t e from the wall o n t o the h e a t e d fluid in the s - d o m a i n O2(s) = - ),Ade,s.r s) dz
After z = ~ is
(3.2-30]
inserted
into
d0(z, s)/dz,
multiplication
by
-),A a n d
rearrangement yields c h C ¥ ~l" s ~ )
O2(s) " i
O1(s)" P
s,Cs)
s
thC(e- s 4) e,Cs)
~
(3.2-31)
P
2(s)
T h e transfer f u n c t i o n G,(s) d e s c r i b e s the d y n a m i c s of h e a t c o n d u c t i o n t h r o u g h the p i p e wall. a n d G2(s) t h e d y n a m i c s of h e a t flow r a t e c h a n g e r e s u l t i n g from t e m p e r a t u r e c h a n g e s of the h e a t e d fluid. To i n v e s t i g a t e t h e d y n a m i c p r o p e r t i e s of h e a t c o n d u c t i o n w e must a n a l y z e t h e d e n o m i n a t o r of the transfer f u n c t i o n Gl(s). T h e term cp82/), h a s the d i m e n s i o n of time. s o w e will r e p r e s e n t it a s the time c o n s t a n t Tw (it is useful to c o m p a r e Tw with the time c o n s t a n t from E q u a t i o n (2.3-14)).
~=c.
~.~
103
= cpS---~ X
Tw
(3.2-32)
The d e n o m i n a t o r of G~(s) n o w b e c o m e s
c h T~wS. For a finite
s
this
function h a s no singularities, so the roots of c h Tyr~-ws = 0
(3.2-33)
are the p o l e s of the transfer function Gt(s). I n t r o d u c i n g the substitution T~-~ws = x + jy
.
(3.2-34)
E q u a t i o n (3.2-33) b e c o m e s c h x c o s y , jshx siny - 0 .
(3.2-35)
This is satisfied o n l y w h e n b o t h the real a n d
i m a g i n a r y parts
equal
zero. w h i c h is fulfilled for x = 0 a n d for all y thai satisfy cosy
= 0
(3.2-36)
i.e° Yk = (2k - l)-;-
, k = O, ±1. *2 . . . .
(3.2-37)
Inserting t h e s e v a l u e s into (3.2-34) y i e l d s the e x p r e s s i o n for the p o l e s of the transfer f u n c t i o n Gl(s) ( 2 k - 1)2, 2 Sk = -
l
4
k = 1. 2. 3 ....
Tw
(3.2-38)
( S i n c e for k < O t h e r e is So = sl a n d Sk • Sk + I. it is e n o u g h to take only p o s i t i v e v a l u e s for k. w h i c h is sufficient to d e t e r m i n e all the poles.) The
expression
number
(3.2-38)
o| negative,
real
indicates
and
that
Gl(s)
different
has
polos.
an u n l i m i t e d The fact that the
p o l e s h a v e n o i m a g i n a r y p a r t s s h o w s that the p r o c e s s of h e a t c o n d u c t i o n (and
this
described
will
also
by
a
be
true
parabolic
(periodic) character.
of PDE
all
other
(3.2-1)
)
processes cannot
with
have
an
equalization oscillatory
IQ4
CHAP. 3 DISTRIBUTED PROCESSES
When the p o l e s h a v e
been
determined,
the transfer
function c a n
be
written in the form of a n infinite p r o d u c t l Gt(s) = c h Ti/~w s
~ Sk sl =..=I (s - Sk) " = (s-sl)
s2
s3
[s-s2) (s-s3)
"'"
(3.2-30)
In other words. G,(s) c a n b e r e p l a c e d b y a n unlimited n u m b e r of p I o p o r t i o n a l first-order systems in selies with d i f f e r e n t time constants, w h i c h c a n b e c a l c u l a t e d from
1
Tk = - - Sk
. k = I. 2. 3 . . . .
(3.2-40)
T a b l e 3.2-1 for the c a s e of a s t e e l e v a p o r a t o r p i p e of wall t h i c k n e s s 5 mm (X = 4b.5 W/mK. c = 500 J/kgK, p = 7850 k g / m 3) g i v e s the v a l u e s for the first five p o l e s a n d their c o r r e s p o n d i n g time c o n s t a n t s .
Table k
3.2.1 Sk
Tk [sl
1 -1,169 2 -10,523 3 -29.232 L,-ST.zgs 5 -9~.714
0.855 0.095 O.03L 0,0~8
O,Oll
F i g u r e 3.2-1 s h o w s the distribution of the p o l e s in the s - p l a n e .
s¢
ss ',
.=. :
q
;
:V.
-I00
Fig.
s3 . Sz~ ~l joJ l
-50
:
=
~-
~
J
I
o
3 . 2 - 4 Position of transfer f u n c t i o n p o l e s for p r o c e s s e s with e q u a l i z a t i o n
Here it is v e r y important to o b s e r v e the g r a d i e n t o! d e c r e a s e constants
(and
this
is
not
a
function
of
the
thickness
or
of time physical
==c. "~.2
195
p r o p e r t i e s of the wall) w h e r e the 2ncl, 3rd, 4th ... time c o n s t a n t s a r e a b o u t 10, 30, bO a n d 100 times smaller t h a n the first. T h e y d e c r e a s e ~2(2k-1)2/4 times. This distribution of time c o n s t a n t s s h o w s that it is p o s s i b l e to r e p l a c e the t r a n s c e n d e n t a l transfer function G~(s) with a rational transfer function of a p r o p o r t i o n a l first-order s y s t e m w h o s e time c o n s t a n t is similar to the first, d o m i n a n t time c o n s t a n t in E q u a t i o n (3.2-39). I n this w a y w e would, h o w e v e r , n e g l e c t all the v e r y r a p i d transient p r o c e s s e s w h i c h a r e c h a r a c t e r i z e d b y the 2nd a n d h i g h e r time c o n s t a n t s in the p r o d u c t (3.2-39). This was a l r e a d y clone in the s e c t i o n o n l u m p e d p a r a m e t e r s (Equation (2.3-14)), a n d h e r e it remains for us to s h o w the kind of thinking that m a d e it p o s s i b l e . Besides being expanded into the p r o d u c t m a t h e m a t i c a l l y - a l s o b e e x p a n d e d as follows 1
shown.
G=(s)
can
I
~,(s) = ch T~/~;~s =
(T-,/f-~-~s)~ ( Ty"f~s)" 1
÷
~
21
÷
~
41
(3.2-41) ÷
....
If all the terms of 4th and higher order are neglected, rearrangement of the upper equation yields the approximated transfer function G~a(S) which has already a p p e a r e d in (2.3-14) G,a(s) =
1 cp~ 2
~s*l
(3.2-42)
T h e time c o n s t a n t c h a r a c t e r i z i n g E q u a t i o n (3.2-42) for the a b o v e d e f i n e d p i p e wall is 1.055 s, which is o n l y slightly more than the v a l u e of TI = 0.855 s. Finally. w e will g i v e a g r a p h i c a l r e p r e s e n t a t i o n of c h a n g e s in h e a t flow r a t e qz(t) if the flow r a t e q~(t) of the h e a t i n g g a s e s s u d d e n l y i n c r e a s e s b y a unit v a l u e . F i g u r e 3.2-5 s h o w s two c u r v e s : c u r v e a shows the zeal c h a n g e s of q2(t) g i v e n b y the transfer function (3.2-39), a n d c u r v e b s h o w s the r e s p o n s e q~(t) g i v e n b y the a p p r o x i m a t e m o d e l , E q u a t i o n (3.2-42).
1(~6
CHAP, 3 IDISTR1BUTED PROCESSES
q~t)~
0~
T=1.055 . . . . . .
Fig.
3
4s
t
3 . 2 - S C h a n g e in h e a t flow r a t e qz(t) in the c a s e of a s t e p i n c r e a s e in q,(t) on the h e a t e r s i d e
T h e c u r v e s s h o w that t h e r e . a r e d i f f e r e n c e s at the b e g i n n i n g of the transient p r o c e s s (for small t) that result from n e g l e c t i n g p o l e s that a r e m o r e d i s t a n t in the n e g a t i v e d i r e c t i o n of the real axis. i.e. b e c a u s e fast. h i g h - f r e q u e n c y r e s p o n s e c o m p o n e n t s h a v e b e e n n e g l e c t e d . Of c o u r s e , the n e w l y a c h i e v e d s t e a d y s t a t e will b e the s a m e in b o t h c a s e s , a n d for t ..,,o qzCt)'-'qi(t). T h e transfer f u n c t i o n Gz(s) s h o w s h o w the h e a t flow r a t e that is t r a n s f e r r e d from the p i p e wall o n t o the e v a p o r a t i n g fluid is a f f e c t e d b y t e m p e r a t u r e c h a n g e ~f(t) of the fluid (in the e v a p o r a t o r this is c a u s e d b y changes in the p r e s s u r e of e v a p o r a t i o n ) . To a n a l y z e the d y n a m i c p r o p e r t i e s of that p r o c e s s further. Gz(s) c a n b e s h o w n a s follows
O2(s) G2(s) -
sh(x~-~s) -
ch(#
(3.2-43)
T h e d e n o m i n a t o r of this transfer f u n c t i o n e q u a l s that of Gl(s). s o p o l e s will b e the s a m e a s the o n e s that h a v e just b e e n d e r i v e d e x p r e s s e d b y E q u a t i o n (3.2-38). G2(s). h o w e v e r , a l s o h a s z e r o s a n d i n f l u e n c e the t r a n s i e n t p r o c e s s a s well. T h e y c a n b e o b t a i n e d equation
the and they from
sEc.
107
a.a
If Kp
=
),A/8
W/K is i n t r o d u c e d a n d
the a b o v e
equation
divided
by
this Kp, r e f e r r i n g to (3.2-32), E q u a t i o n (3.2-44) y i e l d s
Tv/T'~-wssh T~'~wS , 0
(3.2-45)
T h e s u b s t i t u t i o n g i v e n b y E q u a t i o n (3.2-34) is a l s o u s e d last e q u a t i o n b e c o m e s
h e r e . s o the
(3.2-46)
(x + jy)(shx c o s y + j c h x siny) = 0
This p r o d u c t e q u a l s z e r o if a n y of the f a c t o r s e q u a l z e r o , w h e n c e the d e m a n d x = 0 a n d siny = O. T h e final e x p r e s s i o n for the z e r o s of the transfer f u n c t i o n G2(s) is k2~ 2
nk - -
Tw
,
k = O, I, 2 . . . .
(3.2-47)
F i g u r e 3.2-6 s h o w s the first 5 z e r o s a n d p o l e s of the transfer function Gz(s) in the s - p l a n e . For the g i v e n s t e e l p i p e from (3.2-47) w e g e t , n~ - O, n2 = -4.677. ns = -18.71, n4 = - 4 2 I , ns = -T4.83. ja~ SIS
S&
-lOO
Fig.
3.2-6
S3
$2
-50
Z e r o s a n d p o l e s of transfer function G~(s)
-
Oz(s)/Of(s)
On the b a s i s of the a n a l y s i s a n d p r o c e d u r e a n a l o g o u s to the o n e w e u s e d to o b t a i n E q u a t i o n (3.2-39). the transfer f u n c t i o n Gz(s) c a n b e written in the form of the following infinite p r o d u c t
Gis)
=
-
sh T~-~wS = - Kp c h Tl/~-sws = . K p ,l,k l ~
(S-Sk)
(3.2-48)
Kp sCs-n,)Cs-nz)...Cs-nn)...
Cs-s,)Cs-s2)...Cs-s.)... Two
essential
facts
derivative dependence
are
o b v i o u s , t h e h e a t flow r a t e qz(t) s h o w s o n fluid t e m p e r a t u r e c h a n g e ~fCt} (the a p p e a r a n c e
CHAP. 3 DISTRIBUTED P R O C E S S E S
IQB
of an independent s in the numerator of G2(s)). a n d since all the poles are negative a n d real. q2(t) has no oscillatory properties in this case also. If w e w a n t to s h o w h o w q2(t) c h a n g e s with time if s t e p unit d e c r o a l o , t h e following e q u a t i o n g i v e s a n solution
ch T ~ w S
Or(t)
undergoes a exact analytical
(3.2-49)
After inverse Laplace transformation w e
get q2(t) in the form of an
infinite s u m of exponential functions .k2Tw
q2(O
-
K /-VW (l ÷ 2 -'7.(-1) k e ~V
lit
k=l
-V-
)
(3.2-so)
I n v e s t i g a t i o n into q2(t) h a s s h o w n that for t = O. qz(t) is infinite a n d as t a p p r o a c h e s infinity the d e n s i t y of h e a t flow r a t e qz(t) c o n v e r g e s to zero. On F i g u r e 3.2-T c u r v e a s h o w s c h a n g e s of the transfer function q2(t). %(0
2K
o
.....
il T=1,055
~,
t
•
Flg. 3.2-7 C h a n g e in heat flow rate qz(t) in the case of a unit step temperature ~)f(t)decrease Just after (t÷ = o) the step declease of 8f heat energy stored in the surface layer of the wall is immediately released. This amount, however, is infinitesimally small (the heat flow rate q2Ct) must not be confused with the heat released 02(0) a n d as all the other layers of the wall in the z direction offer thermal resistance, the heat flow rate qz(t) decreases very quickly. For small t the gradients of c h a n g e q2 are very great, a n d only after the d e e p layers of the wall have b e e n included in the plocess of
~C,
I¢~Q
a.2
h e a t c o n d u c t i o n a n d transfer t o w a r d s the fluid of lower t e m p e r a t u r e ef (which t a k e s a c e r t a i n finite p e r i o d of time) d o e s the c h a n g e q2(t) b e c o m e less s t e e p . H e r e a g a i n the c o m p l e x i t y of GzCs) invites u s to try to a p p r o x i m a t e this t r a n s c e n d e n t a l transfer function b y a s i m p l e r f u n c t i o n in the form of rational p o l y n o m i a l s . T h e s i m p l e s t a p p r o x i m a t i o n o b t a i n e d from E q u a t i o n (3.2-48) is to r e t a i n in the e x p a n s i o n of the function G2(s) o n l y p r o c e s s e s with the g r e a t e s t time c o n s t a n t , i.e. to n e g l e c t all terms of a third a n d higher o r d e r in the e x p a n s i o n of t r a n s c e n d e n t a l h y p e r b o l i c functions. T h e a p p r o x i m a t e function G2a(S) is o b t a i n e d from the following e x p a n s i o n
~.(s)
-
-Kp T./f~s ( T/T-~5 l + ( T~wS)2 21
,
+
...)
_
.
s KpTw cp~2 2), s
"'"
(3.2-513 +
I
This h a s g i v e n us t h e w e l l - k n o w n transfer function of the d e r i v a t i v e s y s t e m with first-order l a g . w h o s e r e s p o n s e to unit s t e p d e c r e a s e in the t e m p e r a t u r e of the h e a t e d fluid is s h o w n b y c u r v e b on F i g u r e 3.2-7. As in Figure 3.2-5. h e r e a l s o the s a m e r e m a r k h o l d s that n e g l e c t i n g fast p r o c e s s e s ( h i g h e r - o r d e r terms of the d e v e l o p m e n t ) l e a d s to different r e s p o n s e s at the b e g i n n i n g of the t r a n s i e n t p r o c e s s , b u t that this d i f f e r e n c e b e t w e e n real a n d a p p r o x i m a t i o n - g e n e r a t e d response curves d e c r e a s e s with the i n c r e a s e of time t. This e n d s the a n a l y s i s of the c a s e w h e n the c o e f f i c i e n t of c o n v e c t i v e h e a t transfer o n the s i d e of the h e a t e d fluid c a n b e c o n s i d e r e d infinite. The c l o s e s t to this a r e e v a p o r a t i o n p r o c e s s e s in s t e a m g e n e r a t o r p i p e s . T h e r e the c o e f f i c i e n t s ~,~ a r e r e a l l y high. b u t not infinite, s o it is of interest to e x a m i n e the c a s e ( g i v e n by b) in this e x a m p l e ) w h e n the c o e f f i c i e n t of c o n v e c t i v e h e a t transfer ~2 is finite. b) F o r c e d h e a t flow r a t e q~(t) a n d a finite c o e f f i c i e n t of h e a t transfer ~z l e a d to the following m a t h e m a t i c a l formulation of b o u n d a r y c o n d i t i o n s
1st BC 2nd BC
- x A .~'s'O' - (~z
~,9(~, t) ~z
t) =
-
-
q1(t)
'~ (~(~. t) A
z
-
=
O
~rCt)).
200
CHAP. 3 DISTRIBUTED P R O C E S S E S
Like in the p r e c e d i n g the b o u n d a r y c o n d i t i o n s
c a s e , Ct a n d C2 a r e d e t e r m i n e d
depending
on
T=2ef(s) + D(s) O,(s)
E(s) ~,A
C,-
B(s} + D(s)
-~
el,s)
-
B(s)
'
(3.2-52}
O,(s)
E(s) XA
C2 =
B(s) * D(s)
s(s) - (-~-;,
(3.2-53)
'
-f) ~;
(3.2-54)
(3.2-55)
E(s) - X ~
(3.2-56)
T h e t e m p e r a t u r e of the p i p e wall a l o n g the z a x i s e(z, s) is d e t e r m i n e d b y E q u a t i o n (3.2-10). a n d z = s g i v e s the t e m p e r a t u r e o n the s u r f a c e f a c i n g the h e a t e d fluid. If w e k n o w e(~, s), the h e a t flow t o w a r d s the fluid is determined by
O2(s)
O(~. s3
-
-
,,2A[eCs. s) - efCs)]
S u b s t i t u t i n g e(~, s) expression demanded
into
O2Cs) =
this
(3.2-57)
equation
1
and
arranging
it
yields
the
OiCs) -
sh
ch T~/~-wS+ X i
p
G;Cs)
(3.2-58)
sh T~ws
Kp ch T][~wS + )'
el(s)
shT ws
I
G,Cs3 In the further t e x t w e will not r e p e a t the m e t h o d s a n d p r o c e d u r e s u s e d in the p r e c e d i n g c a s e , e s p e c i a l l y s i n c e t h e s e c a l c u l a t i o n s a r e m u c h m o r e
=Re.
a.2
201
c o m p l e x a n d the results o b t a i n e d d o not s h o w a n y e s s e n t i a l l y n e w quality. T h e b a s i c c h a r a c t e r i s t i c s of the p r o c e s s for the a b o v e b o u n d a r y conditions a r e the following. The p o l e s of the transfer f u n c t i o n s G3(s) a n d G4(s) a r e still n e g a t i v e a n d real a n d t h e r e is a n infinite n u m b e r of them, b u t n o w t h e y c a n n o t b e g i v e n in e x p l i c i t m a t h e m a t i c a l form. T h e y are, t h e r e f o r e , o b t a i n e d g r a p h i c a l l y - a n a l y t i c a l l y a n d g i v e n in t a b u l a r form. T h e first p o l e , w h i c h also c h a r a c t e r i z e s the p r o c e s s with t h e g r e a t e s t time c o n s t a n t , is d e t e r m i n e d b y the following a p p r o x i m a t e formula for the c a s e of (,',2~/X) > O.l.
~2 1 1 sl - - ~ I + 2.24(w = ~ s ) _~1 . 0 2 - 1 ~
(3.2-59)
T h e r e s p o n s e s q2Ct) t o s t e p changes qs(t) a n d ~f(t) a r e still of u n p e r i o d i c c h a r a c t e r a n d a r e c o m p l e t e l y similar in form to the p r e c e d i n g c a s e , a n d t h e transfer f u n c t i o n q2(t) a g a i n s h o w s a d e r i v a t i v e r e s p o n s e in the c a s e of d i s t u r b a n c e in 8f(t). Unlike the p r e c e d i n g c a s e , h o w e v e r , q2(t) is no l o n g e r infinite for t = to. it is i n t e r e s t i n g that v e r y g o o d a p p r o x i m a t i o n s of transfer functions G3(s) a n d G4(s) c a n a l s o b e r e a l i z e d n o w if all the terms of third a n d higher o r d e r in the e x p a n s i o n of the t r a n s c e n d e n t a l f u n c t i o n s c h T./~wS a n d shT~ws a r e n e g l e c t e d . T h e r e s p o n s e s o b t a i n e d from t h e s e t r a n s f o r m e d functions a r e c l o s e to the real r e s p o n s e s , but b e c a u s e h i g h e r - o r d e r terms h a v e b e e n n e g l e c t e d , w h i c h m e a n s all the fast. h i g h - f r e q u e n c y p a r t s of the r e s p o n s e s that p a r t i c i p a t e in the d y n a m i c s of the p r o c e s s , t h e r e a r e insignificant d i f f e r e n c e s at the b e g i n n i n g of the t r a n s i e n t p r o c e s s . T h e a p p r o x i m a t e d transfer function Gsa(S) is o b t a i n e d a s follows G3aCs) =
l
C] +
( T~-~)~ 2
) *
,'
~ ~
(x2~
C3.2-00)
TCT~s
I
c,~(!=2
÷
s
--~-)s ÷ I
This is o b v i o u s l y a transfer f u n c t i o n of the p r o p o r t i o n a l s y s t e m of first
202
CHAP. 3 DISTRIBUTED PROC~'SSES
o r d e r a n d it is i m p o r t a n t to n o t e that the time c o n s t a n t o b t a i n e d is in fact equal to the e x p a n d e d (with the term I/,,z) time c o n s t a n t that c h a r a c t e r i z e d h e a t c o n d u c t i o n w h e n it w a s o b s e r v e d as a p r o c e s s with l u m p e d p a r a m e t e r s : E q u a t i o n s (2.3-14) a n d (3.2-42). If the h y p e r b o l i c f u n c t i o n s c o m p r i s i n g the e x p a n d e d into a s e r i e s a n d the h i g h e r - o r d e r obtained G4aCs) = - KpTw
transfer function G4(s) a r e terms n e g l e c t e d . G4a(s) is
s
cr)~(jc£-2
(3.2-61)
8 + --~--)s + I
T h e transfer f u n c t i o n s G3a(S) a n d G4a(S) a r e c o m p l e t e l y i d e n t i c a l with the a p p r o x i m a t e d f u n c t i o n s Gla(S) a n d G2a(S) s h o w n b y E q u a t i o n s (3.2-42) a n d (3.2-51). d i f f e I i n g o n l y in the time c o n s t a n t . It is i m p o r t a n t to n o t e that G~a(S) a n d G2a(S) a r e s p e c i a l b o u n d a r y c a s e s of E q u a t i o n s (3.2-60) a n d (3.2-b1) with t h e a s s u m p t i o n of a n infinite ~2. K e e p i n g to o u r u s u a l ( i n d u c t i v e ) m a n n e r of p r e s e n t i n g a n d a n a l y z i n g d y n a m i c p r o p e r t i e s a n d p r o c e e d i n g from the s p e c i a l a n d s i m p l e to the g e n e r a l a n d m o r e c o m p l e x , it still r e m a i n s to a n a l y z e the c a s e w h e n
there in convective heat transfer with a finite value for the coefficient at i n z = 0 a l s o . H e r e w e will l e a v e out c o m p l e t e l y the d e t a i l e d p r o c e d u r e s of o b t a i n i n g t r a n s c e n d e n t a l transfer f u n c t i o n s a n d o n l y s h o w their a p p r o x i m a t i o n s . c) Finite c o e f f i c i e n t s of h e a t transfer ~1 a n d (x~ In this c a s e of c o n v e c t i v e h e a t transfer with finite transfer c o e f f i c i e n t s in z = 0 ({x,) a n d z ffi S (c(~). the b o u n d a r y c o n d i t i o n s b e c o m e e q u a l to t h o s e g i v e n in c) of E x a m p l e 1. Ist BC
~ C 0 . t)
2nd BC
~eCs. t}
az
~z
= ~, (4(0. t) - e,(t)) x
. .
.
z = 0
(xz (e(~. t) - e2(t3) ),
z = '
el(t) a n d ez(t) a r e n o w t h e t e m p e r a t u r e s of t h e m e d i a f l o w i n g o n the o u t s i d e a n d the i n s i d e of the wall w h e r e e, > ~2.
~=zc. :~.:=
203
T h e a p p r o x i m a t e d transfer f u n c t i o n s in E q u a t i o n (3.2-b2) s h o w h o w h e a t flow r a t e q2(t), t r a n s f e r r e d o n t o the h e a t e d fluid, d e p e n d s o n t e m p e r a t u r e c h a n g e s of b o t h the h e a t i n g a n d the h e a t e d fluid. K Or(s) = Tns • l
O,(s) - K(Tbs + l)o2(s) ~ 7 T q
G~.(s) K =
=1~ tx I ÷
¢x? ,=-
T b = cp$(
P
G5~(s)
S e¢le¢Z-~-
A
(3,2-b3)
÷ 2--~-)
(3.2-b4)
8
I + (=, + =2)-'~
$2
* =1=2 6X~
T n = cpS
(3.2-05)
S ~l
In the
(3.2-62)
above
÷ (%2 ÷ (~1C(2--~"
equations
we
have
assumed
that
the
surface
areas
thzough which heat is transferred are the s a m e on the outez a n d on the inner side of the wall. i.e. w e w o r k e d with a m e a n surface area A = A, = A2.
"t w d I U U U
K I= 58.2
|
~
T~z=Z~G
t
|
I
I 10
I
!
|
20 s
I
tr
t
Tm=8.98
Fig.
3 . 2 - 8 H e a t flow r a t e c h a n g e qz(t) in t h e c a s e of. - s t e p t e m p e r a t u r e i n c r e a s e eiCt] - a.c - s t e p t e m p e r a t u r e d e c r e a s e e2Ct) - b , d
T h e transfer f u n c t i o n s Gsa(S) a n d G~a(S) s h o w that q2(t) will r e s p o n d a s a p r o p o r t i o n a l first-order s y s t e m to a d i s t u r b a n c e on the h e a t i n g s i d e ( a
204
C H A P . 3 DISTRIBUTI=D PROC]~SS;:S
disturbance
that
disturbances
of t h e
passes
through
heated
the
medium
pipe
wall).
it will c o n t i n u e
To
temperature
to s h o w
derivative
p r o p e r t i e s , o n l y n o w qz(t) will n o l o n g e r c o n v e r g e to z e r o after a d i s t u r b a n c e of ez(t). This p r o p e r t y f u n d a m e n t a l l y d i s t i n g u i s h e s this transient p r o c e s s from t h e r e s p o n s e in t h e p r e c e d i n g two c a s e s w h e n t h e r e w a s a f o r c e d h e a t flow r a t e q~(t) in z = O. w h e n ql(t) w a s not a f u n c t i o n of the wall t e m p e r a t u r e e(O. t). i.e. w h e n t h e r e w a s n o f e e d b a c k a c t i o n of the temperature
of t h e h e a t e d
m e d i u m e2(t) o n t h e a m o u n t of h e a t b r o u g h t to
t h e p i p e wall ( a n d t h r o u g h it to t h e h e a t e d
fluid).
T h e t r a n s f e r f u n c t i o n G~a(S) h a s q u a l i t a t i v e l y d i f f e r e n t t r a n s f e r p r o p e r t i e s s o w e will s h o w its r e s p o n s e s in this c a s e w h e n t h e c o e f f i c i e n t s of h e a t t r a n s f e r o n b o t h s i d e s of t h e p i p e a r e finite. For s t e a m g e n e r a t o r s u p e r h e a t e r p i p e s w h o s e d a t a h a v e a l r e a d y b e e n g i v e n in t h e s e c t i o n on l u m p e d p a r a m e t e r s . F i g u r e 3.2-8 s h o w s t h e r e s p o n s e s o b t a i n e d o n the b a s i s of E q u a t i o n (3.2-62) in t h e c a s e of unit s t e p t e m p e r a t u r e i n c r e a s e el(t) ( c u r v e a) a n d d e c r e a s e
ez(t) ( c u r v e b).
T h e f o l l o w i n g d a t a a r e v a l i d for t h e m e t a l wall, c - 5 0 0 J/kgK. p - 7 8 5 0 k g / m 3. ~ - 0 . 0 0 5 m. ), - 4 6 . 5 W/mK. =, ,, 60 W/m~K. ~xz - 2 4 0 0 W/m2K. A - 1 m ~ . The
coefficients
characterizing
the
transfer
process
are
K z 58.1704
W/K. Tb " 328.138 s, Tn ~ 8.98 s.
It is o b v i o u s decrease
that
heat
in oz. w h i c h
stored (accumulated)
flow
is t h e
rate
q2(t)
consequence
in t h e p i p e
increases of u s i n g
much up
wall. W h e n el i n c r e a s e s
more
heat
for
that
a
was
n o t o n l y is no
s t o r e d h e a t r e l e a s e d from t h e wall. b u t t h e g r e a t e r h e a t flow is u s e d to i n c r e a s e the e n e r g y l e v e l of t h e wall itself, w h i c h r e s u l t s in a m u c h s m a l l e r g r o w t h of qz(t). N e v e r t h e l e s s , this d i f f e r e n c e in t h e v a l u e of q2(t) r e s u l t s from t h e m u c h g r e a t e r h e a t r e s i s t a n c e in t h e c r o s s - s e c t i o n z = O (~, is a b o u t 4 0 times s m a l l e r t h a n =z). Thus. F i g u r e 3.2-8 a l s o s h o w s the t r a n s f e r p r o c e s s e s if t h e c o e f f i c i e n t ¢1 is m u c h g r e a t e r t h a n a b o v e a n d e q u a l to =z, ¢~ " ~x~ = 2 4 0 0
W/m2K. T h e n K - 1062.8 W/K. Tb - q.232 s. a n d
Tn - 4.6 s. C u r v e c c o r r e s p o n d s temperature decrease To e n d
this p a r t
to t e m p e r a t u r e
incIease
el(t), c u r v e
d to
ez(t). w e must still s h o w
that
the
analysis
in S e c t i o n
2.3
s=c. ~.2
205
g a v e time c o n s t a n t s that a r e b o u n d a r y c a s e s of E q u a t i o n s (3.2-63), (3.2-64) a n d (3.2-65) for a n infinite ~. If t h e s e time c o n s t a n t s a r e c o m p a r e d it c a n b e s e e n that the d i f f e r e n c e s a r e not v e r y g r e a t . H o w e v e r , it must b e r e p e a t e d that if the c o e f f i c i e n t s of h e a t transfer ,xl a n d ~2 a r e l a r g e , w h i c h m e a n s that the c o n v e c t i v e p a r t of the h e a t r e s i s t a n c e is small, t h e n the part of the r e s i s t a n c e r e s u l t i n g from h e a t c o n d u c t i o n t h r o u g h the wall m a k e s u p a l a r g e p a r t of the total time c o n s t a n t T a n d thus E q u a t i o n s (2.3-10). (2.3-11) a n d (2.3-12) a r e not s a t i s f a c t o r y .
In all the p r e v i o u s e x a m p l e s w e u s e d the b a s i c form of the p a r a b o l i c PDE s h o w n b y E q u a t i o n (3.2-1) to a n a l y z e the d y n a m i c p r o p e r t i e s of p r o c e s s e s with e q u a l i z a t i o n , a n d a n y d i f f e r e n c e s a n d difficulties w e e n c o u n t e r e d in s o l v i n g it r e s u l t e d from the n e c e s s i t y of s a t i s f y i n g different c o n d i t i o n s of h e a t transfer on the b o u n d a r i e s of the s y s t e m u n d e r observation. The preceding pages show the complexity of the m a t h e m a t i c a l tools a n d the n e c e s s i t y of turning to simpler forms of transfer functions. All the solutions that we o b t a i n e d h e r e w e r e solutions of the simplest forms of p a r a b o l i c PDE, but it is s e v e r a l times m o r e difficult to g e t solutions a n a l y t i c a l l y if the p r o c e s s o c c u r s in a b o d y with a m o r e c o m p l i c a t e d g e o m e t r y , or if a m o r e " e x t e n d e d " form of E q u a t i o n (3.2-1) d e s c r i b e s the p r o c e s s . In s u c h c a s e s a digital c o m p u t e r a n d n u m e r i c a l m e t h o d s of s o l u t i o n must b e u s e d . T h e following two e x a m p l e s will s h o w how a v e r y small c h a n g e in a s s u m p t i o n s ( w h o s e e x i s t e n c e a n d h e l p a r e c r u c i a l w h e n the e q u a t i o n s a r e f o r m u l a t e d ) results in a different ( m o r e c o m p l e x ) m a t h e m a t i c a l form of the b a s i c p a r a b o l i c PDE (3.2-I}.
Example
4 Heat conduction through homogeneous
uninsulated body
D e r i v e a n e q u a t i o n d e s c r i b i n g u n s t e a d y t e m p e r a t u r e c h a n g e s for the b o d y o n F i g u r e 3.2-1 if all its s i d e s u r f a c e s e x c h a n g e h e a t with a n e n v i r o n m e n t of t e m p e r a t u r e eo(t). R e s i s t a n c e to h e a t c o n d u c t i o n in a c r o s s - s e c t i o n is n e g l e c t e d a n d the m o d e l is f o r m u l a t e d for the c a s e w h e n the s u r r o u n d i n g t e m p e r a t u r e is g r e a t e r t h a n the t e m p e r a t u r e of the b o d y . if the c i r c u m f e r e n c e of the b o d y is U a n d if a is the c o e f f i c i e n t of h e a t transfer o n the s i d e - s u r f a c e b o u n d a r y of the b o d y . the I e f t - h a n d s i d e of
206
CHAP. 3 DISTRIBUTED
PROCESSES
the e q u a t i o n for the c o n s e r v a t i o n of e n e r g y (3.2-2) is e x p a n d e d b y terms c o n t a i n i n g the a m o u n t of h e a t b r o u g h t to v o l u m e dV t h r o u g h the side surfaces
q(z. ,)A =Apc
-
[q(z. t) , ~qC:,. t)az]A ~z
- ,,u[sc,,. t)- ~0(t)]d,, =
ae(z. ,3 dz at
(3.2-66)
A r r a n g i n g this u p p e r e q u a t i o n g i v e s the final PDE
a~(z, t)
at
x a2e(z, t) cp
=U • ~Cz,
~xu t) - - ~ o C O
(3.2-67)
If the c o e f f i c i e n t = c a n b e c o n s i d e r e d i n d e p e n d e n t of t e m p e r a t u r e , a n o n h o m o g e n e o u s l i n e a r p a r a b o l i c PDE is o b t a i n e d . T h e b o u n d a r y a n d initial c o n d i t i o n s d e f i n e d for E x a m p l e 1 a r e c o m p l e t e l y v a l i d for E q u a t i o n (3.2-67) also, but to g e t a u n i q u e solution of the e q u a t i o n it is n e c e s s a r y to k n o w the law of t e m p e r a t u r e c h a n g e So(t) in the e n v i r o n m e n t as well.
r.xample S P r o c e s s of diffusion with c o n v e c t i o n C o n s i d e r a p i p e of c r o s s - s e c t i o n a l a r e a A t h r o u g h w h i c h a solution flows with v e l o c i t y w. T h e solution c o n t a i n s a C c o m p o n e n t with v a r y i n g c o n c e n t r a t i o n c a l o n g the p i p e . D e r i v e a n e q u a t i o n d e s c r i b i n g d y n a m i c changes along the p i p e in the concentration of c o m p o n e n t C. C o n c e n t r a t i o n c is c o n s i d e r e d c o n s t a n t o n the c r o s s - s e c t i o n a l a r e a a n d Fick's law is v a l i d for diffuse flow.
dV
°'°'H
"z z.dz
Fig.
.A
3 . 2 - 9 Diffusion with c o n v e c t i o n
F i g u r e 3.2-9 s h o w s a n e l e m e n t a r y ' p i p e v o l u m e dV for w h i c h the e q u a t i o n for m a s s c o n s e r v a t i o n of the C c o m p o n e n t in the solution is to b e
s=c. z.a
207
d e r i v e d . T h e following n o t a t i o n s h a v e b e e n u s e d : n ... C c o m p o n e n t molar flow t r a n s p o r t e d b y the flow of the solution ( c o n v e c t i v e molar flow), n = c a w tool C / s . c ... c o n c e n t r a t i o n of C c o m p o n e n t tool C / m 3, N ... a m o u n t of C c o m p o n e n t . N = cV tool C. nd ... diffuse molar flow (results from c o n c e n t r a t i o n d i f f e r e n c e a l o n g z). n d = .... c o e f f i c i e n t of diffusion mZ/s.
-DA a c / a z tool C / s (Fick's law). D
[n(z. t) • nd(z. t)] - [n(z. t) • nd(z. t) +
a[n(z, t) + nd(z, t)] dz]= aN az at (3.2-b8)
a [ n ( z , t) ,, nd(z. t ) ] d z = aCcCz, t)dV) = ac(z. t)Ad z az
at
(3.2-b9)
at
.r ~ , a C ( Z . 01
ac(z. t)Aw _ aL'UA a--~?--J = ac(z. t____)k az
az
(3.2-70)
at
R e a r r a n g e m e n t of the last e q u a t i o n for a c o n s t a n t diffusion D. w h i c h m e a n s it is not a function of c. y i e l d s ac(z. t) ac(z. t) _ Da~C(Z._ t) + w = 0 at az az ~
coefficient
of
(3.2-71)
This is a linear h o m o g e n e o u s parabolic PDE. C o m p a r e d to the basic equation for processes with equalization (3.2-I). it is e x p a n d e d by the second term on the left-hand side which contains the so-called convective part of the transport of component C along the z axis, the part that is carried with the solution flow by velocity w.
T h e s e last two e x a m p l e s c o m p l e t e the p r e s e n t a t i o n of s o m e b a s i c c h a r a c t e r i s t i c s of p r o c e s s e s with e q u a l i z a t i o n . T h e following s e c t i o n treats p e r i o d i c p r o c e s s e s w h o s e b a s i c m a t h e m a t i c a l n o t a t i o n (for p r o c e s s e s with d i s t r i b u t e d p a r a m e t e r s ) is a s e c o n d - o r d e r h y p e r b o l i c PDE. T h e major d i f f e r e n c e c h a r a c t e r i z i n g the d y n a m i c s of e a r l i e r p r o c e s s e s in c o m p a r i s o n with t h e s e p r o c e s s e s is that the p o l e s of the transfer f u n c t i o n s will n o w b e c o m p l e x c o n j u g a t e p a i r s . This distribution of p o l e s is the result of p r o p e r t i e s of inertia p o s s e s s e d by continua, which appear in the m o m e n t u m e q u a t i o n w h e n the c o n s e r v a t i o n e q u a t i o n s a r e f o r m u l a t e d . S u c h p r o p e r t i e s a r e not s h o w n in p r o c e s s e s with e q u a l i z a t i o n so t h e r e w a s n o n e e d to formulate the e q u a t i o n for the c o n s e r v a t i o n of m o m e n t u m for them.
208
CHAP.
:3
DISTRIBUTED
PROCHSSES
3.3 PROCESSES WITH PERIODIC STATE CHANGES BASIC SECONI~ORDER HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION
O s c i l l a t i o n s ( v i b r a t i o n s ) in p a r t s of d e v i c e s a n d p l a n t s a r e a n e v e r y d a y occurrence in technical practice and many different processes d e m o n s t r a t e t h e s e p e r i o d i c d y n a m i c p r o p e r t i e s . Vibrations of r o d s . b e a m s . shafts, shells a n d w a t e r m a s s e s in h y d r o - e l e c t r i c p o w e r p l a n t f e e d i n g p i p e s , w a t e r h a m m e r e f f e c t s in h y d r a u l i c a n d p n e u m a t i c p i p e s , oscillations of w a t e r l e v e l in c o n n e c t e d tanks, o s c i l l a t i o n s in e l e c t r i c a l circuits. t o r s i o n a l o s c i l l a t i o n s a n d so o n a r e all e x a m p l e s of d y n a m i c a l l y the s a m e or similar p h e n o m e n a w h o s e s t a t e v a r i a b l e s a r e r e l a t e d b y the s a m e t y p e ot differential e q u a t i o n . If p r o c e s s e s w h o s e p a r a m e t e r s c h a n g e a l o n g the s p a t i a l a x i s a r e a n a l y z e d , t h e n the b a s i c a n d s i m p l e s t form of e q u a t i o n d e s c r i b i n g p e r i o d i c s t a t e c h a n g e s will b e the following s e c o n d - o r d e r
hyperbolic ~u
~t~
PDE. 2 ~2u
=
a ~---,r,r,r,r,r,r,r,r~-
(3.3-I)
E q u a t i o n (3.3-1) w a s d e r i v e d after m a k i n g m a n y a s s u m p t i o n s that e x c l u d e d p h e n o m e n a u n i m p o r t a n t for the p u r p o s e s of this a n a l y s i s , a n d it is v a l i d for different p r o c e s s e s in w h i c h the v a r i a b l e s u a n d a h a v e the foilowing p h y s i c a l m e a n i n g . l o n g i t u d i n a l o s c i l l a t i o n of a b e a m u .,. d i s p l a c e m e n t
... v e l o c i t y of d i s t u r b a n c e
E
... m o d u l u s
p
... density
beam.
hydraulic
shocks
of c r o s s - s e c t i o n
a
propagation
along
the
a = of elasticity (Young's
modulus)
in p i p e u
... pressure,
a
... s p e e d
flow. of
propagation)
velocity,
sound
density
(speed
of
disturbance
~=c.
:~.~
200
oscillation in e l e c t r i c a l circuits u
... v o l t a g e , c u r r e n t flow
a = ~ L ... circuit i n d u c t a n c e C ... circuit c a p a c i t a n c e To o b t a i n a u n i q u e solution of E q u a t i o n (3.3-1) it is n e c e s s a r y to k n o w the initial c o n d i t i o n s (IC) a n d the b o u n d a r y c o n d i t i o n s (BC), w h o s e m a t h e m a t i c a l formulation c o n t a i n s d a t a a b o u t whether, a n d h o w much. e n e r g y , mass or m o m e n t u m w a s s t o r e d in the p r o c e s s u n d e r o b s e r v a t i o n at the initial moment (IC), a n d h o w t h o s e v a r i a b l e s a r e e x c h a n g e d with the e n v i r o n m e n t t h r o u g h the b o u n d a r i e s (BC). In m a t h e m a t i c a l form the BC of the s e c o n d - o r d e r h y p e r b o l i c PDE a r e e q u a l or similar to the BC from the p r e c e d i n g s e c t i o n for the p a r a b o l i c PDE. In the c a s e of the IC. s i n c e Equation (3.3-1) now c o n t a i n s the s e c o n d d e r i v a t i v e of v a r i a b l e u with r e s p e c t to time t. b e s i d e s k n o w i n g f u n c t i o n uo ; u(z. O) it is a l s o n e c e s s a r y to know the v a l u e of function u'o ,, du(z, O)/dt. To s o l v e the a b o v e e q u a t i o n it is. therefore, n e c e s s a r y to formulate t w o
initial a n d t w o
b o u n d a r y c o n d i t i o n s in a c c o r d a n c e with the real c o n d i t i o n s u n d e r w h i c h the p r o c e s s o c c u r s . S p e c i a l attention must b e p a i d to the m a t h e m a t i c a l formulation of c o n d i t i o n s u n d e r w h i c h the s y s t e m c o m m u n i c a t e s with the e n v i r o n m e n t t h r o u g h its b o u n d a r i e s . In the c a s e of o n e - d i m e n s i o n a l s p a t i a l distribution ( a n d h e r e w e will treat s u c h c a s e s ) it is a l w a y s n e c e s s a r y to define c o n d i t i o n s o n b o t h the b o u n d a r i e s of the p r o c e s s , z ,, O a n d z ,, L. The p r o b l e m c a n n o t b e s o l v e d if b o t h b o u n d a r y c o n d i t i o n s are g i v e n o n one b o u n d a r y b e c a u s e then the w a y in w h i c h the p r o c e s s o c c u r s at the other e n d of the s y s t e m is not known. This is the simplest e x p l a n a t i o n of the s t a t e m e n t m a d e in the s e c t i o n o n fluid flow r e g a r d e d as a p r o c e s s with l u m p e d p a r a m e t e r s , w h e r e it was s a i d that b o t h flow c h a n g e a n d pressure c h a n g e cannot b e simultaneously given as disturbance variables at the s a m e e n d of the p i p e . In this part of the b o o k . w h e r e we treat p e r i o d i c c h a n g e s of state, most of our attention will g i v e n to the a n a l y s i s of d y n a m i c p r o c e s s e s that o c c u r inside p i p e s for t r a n s p o r t i n g fluid. Oscillations in e l e c t r o m a g n e t i c a n d m e c h a n i c a l s y s t e m s are t r e a t e d in g r e a t d e t a i l in s p e c i a l i z e d b o o k s from t h e s e fields. N e v e r t h e l e s s . a l t h o u g h the a c c e n t in the further lines will b e on h y d r a u l i c a n d
p n e u m a t i c p r o c e s s e s , we will also try to i n d i c a t e
the
d y n a m i c c h a r a c t e r i s t i c s c o m m o n to all p e r i o d i c p r o c e s s e s i n d e p e n d e n t of the s y s t e m in w h i c h t h e y o c c u r . Therefore. we will b e g i n with a c l a s s i c a l e x a m p l e of p e r i o d i c s t a t e c h a n g e s ( l o n g i t u d i n a l v i b r a t i o n Of a b e a m ) a n d
2]0
CHAP.
3 DISTRIBUTED PROCESSES
use it to show the variety of possible b o u n d a r y conditions. Later. when we a n a l y z e h y d r a u l i c oscillations, the similarity of the d y n a m i c p r o p e r t i e s with this c a s e from m e c h a n i c s will b e s e e n .
Zxamplo
| L o n g i t u d i n a l v i b r a t i o n of a b e a m
F i g u r e 3.3-1 s h o w s a prismatic b o d y ( b e a m , bar) w h i c h performs l o n g i t u d i n a l v i b r a t i o n s d u e to the a c t i o n of a force. Assuming that the b e a m is of c o n s t a n t c r o s s - s e c t i o n a l a r e a , h o m o g e n e o u s , that its b e n d i n g is n e g l e c t e d a n d that l o n g i t u d i n a l d i s p l a c e m e n t s o c c u r within the field of elastic d e f o r m a t i o n s , d e r i v e the m o d e l d e s c r i b i n g a n d d e f i n e b o u n d a r y c o n d i t i o n s if:
that p e r i o d i c
process
a - the left e n d is fixed, the right e n d free b - f o r c e F a c t s o n the left e n d . the right e n d is f i x e d c - the left e n d is fixed, the right e n d c a r r i e s mass M.
I i T.
_I
z~dz
Fig. The
equation
I i
"
I
m
z=L
I
3.3-1 S k e t c h of l o n g i t u d i n a l b e a m v i b l a t i o n for the
conservation
of momentum,
in this c a s e
more
u s u a l as the e q u a t i o n of f o r c e equilibrium, is f o r m u l a t e d for a n infinitesimal mass of the b e a m dM w h i c h is at the d i s t a n c e u in the s t r e t c h e d state a n d h a s the l e n g t h (1 + 8 u / a z ] d z . (In the s t e a d y s t a t e that s a m e mass is at the d i s t a n c e z a n d is of l e n g t h dz.) The e q u a t i o n for f o r c e equilibrium is
s=c.
a.3
211 ~~u
- oA + (o +
d z ) A = ~Adz ~-~,2-
(3.3-2)
o d e n o t e s t e n s i o n in the c r o s s - s e c t i o n , of d i m e n s i o n e q u a t i o n is r e d u c e d ~o ~--
N / m z. T h e a b o v e
to
~2U p ~-~-,~-
(3.3-3)
In the r e g i o n of e l a s t i c d e f o r m a t i o n s there is
o
=
Ee
=
E ~--u
~z
w h e r e e d e n o t e s the linear d e f o r m a t i o n , a n d E the m o d u l u s of e l a s t i c i t y
N/m z. T h e last two e q u a t i o n s y i e l d cz~2u
- ~~Zu
.
c -
/_~E
(3.3-4)
T h e a b o v e e q u a t i o n h a s the a l , e a d y - k n o w n form of a s e c o n d - o r d e r h y p e r b o l i c PDE that d e s c r i b e s p e r i o d i c p r o c e s s e s . To s o l v e , it, it is n e c e s s a r y to k n o w t w o b o u n d a r y a n d t w o initial c o n d i t i o n s . H e r e w e will not e n t e r into h o w to o b t a i n solutions b u t will o n l y g i v e the g e n e r a l f o r m for the solution of t h e a b o v e e q u a t i o n u(z. t) - (C, sin~t * C z c o s ~ t ) ( C 3 s i n ~ - z
+ C4cos-~-Wz).
(3.3-5)
T h e c o n s t a n t s C,. Cz. Cs a n d C 4 a r e d e t e r m i n e d from the b o u n d a r y a n d the initial c o n d i t i o n s . To c o r r e s p o n d with the c o n d i t i o n s d e m a n d e d at the b e g i n n i n g of this e x a m p l e , the b o u n d a r y c o n d i t i o n s c a n b e v a r i e d : a) T h e left e n d is fixed, the right e n d free. In this c a s e t h e r e c a n b e no d i s p l a c e m e n t in z = O. a n d in z = L. s i n c e that e n d is free. t h e r e c a n b e no t e n s i o n , s o t h e r e is 1st BC 2nd BC
u(O. t) = 0 ~u(L. "~t_.., = 0 ~z
b) F o r c e F a c t s on the left e n d . the right e n d is fixed.
CHAP. 3 D I S T R I B U T E D
212
In z - 0 a force of tension o p p o s e s
PROCESSES
force F. a n d the conditions
of force equilibrium yields Ist
2nd
BC
F = Aa = A E ~u(0" ,~z
t)
u(L. t) I. 0
BC
c) T h e left e n d is fixed, the right carries m a s s M. In this case causes
the inertia of m a s s
tension along is
M
the whole
obtained
from
in the cross-section z ffi L
cross-sectional area A so the
second
BC
between
the inertia of mass }4 a n d
the
condition
of
equilibrium
the force of tension in the
cross-section
1st
BC
u(O. 0 ffi 0
2nd
BC
M~2(L. t)
This
ends
our
. " " A°L "
presentation
.AE~U(L. t) az
of
the
simplest
example
of
periodic
p r o c e s s e s in s y s t e m s with d i s t r i b u t e d p a r a m e t e r s . Of c o u r s e , it is p o s s i b l e to g i v e m a n y other different b o u n d a r y c o n d i t i o n s , to s e e k for the natural f r e q u e n c i e s of v i b r a t i o n s or for the solution of E q u a t i o n (3.3-4) for g i v e n c o n d i t i o n s a n d the like. We will l e a v e this to more s p e c i a l i z e d textbooks. Our next e x a m p l e will b e o n e - d i m e n s i o n a l fluid flow, w h e r e the s a m e or similar (more c o m p l i c a t e d , in fact) e q u a t i o n s to t h e s e in the a b o v e lines will b e o b t a i n e d for the d e s c r i p t i o n of p e r i o d i c p r o c e s s e s . It is our d e s i r e to s h o w that different p r o c e s s e s h a v e the s a m e d y n a m i c p r o p e r t i e s , a n d also
to d e s c r i b e
in more
detail
processes
of d i s t u r b a n c e
propagation
a l o n g the fluid streamlines.
Example
2
Dynamic
processes
in fluid
pipes.
Equations
of mass,
e n e r g y a n d momentum conservation
by
C o n s i d e r a p i p e of c o n s t a n t c r o s s - s e c t i o n a l a r e a A e x t e r n a l l y h e a t e d h e a t flow of d e n s i t y q J/ms p e r unit length, t h r o u g h w h i c h flows
=Re. ~.:3
213
m k g / s mass of fluid. T h e flow is h o m o g e n e o u s with the s a m e v e l o c i t i e s and t h e r m o d y n a m i c p r o p e r t i e s p e r c r o s s - s e c t i o n . The thermal c a p a c i t y of the p i p e
wall
is n e g l e c t e d .
Derive
the
mathematical
model
describing
u n s t e a d y state v a r i a b l e c h a n g e s of that fluid. Here also. a n a l o g o u s l y to the
previous
process,
we
will o b t a i n
the
model b y d e r i v i n g c o n s e r v a t i o n e q u a t i o n s for the fluid in a c o n t r o l v o l u m e dV. H o w e v e r . unlike the p r e v i o u s p r o c e s s , h e r e there will b e mass. momentum a n d e n e r g y s t o r a g e in the c o n t r o l v o l u m e dV at the s a m e time. so it will b e n e c e s s a r y to formulate all t h r e e c o n s e r v a t i o n e q u a t i o n s . After they h a v e been arranged, reduced to a s u i t a b l e form a n d the t h e r m o d y n a m i c e q u a t i o n s of fluid s t a t e r e f e r r e d to. a m o d e l in the d e s i r e d form will b e o b t a i n e d . As until now. this will b e a differential e q u a t i o n . We must, n e v e r t h e l e s s , m e n t i o n that w h e n p r o b l e m s
from fluid d y n a m i c s
are
a n a l y z e d the c o n s e r v a t i o n e q u a t i o n s a r e often g i v e n in the integral form. which is not the most s u i t a b l e form for the p u r p o s e s of this a n a l y s i s . Therefore. s u c h m o d e l s a r e not d e r i v e d in this b o o k . This remark c a n b e of u s e in a situation w h e n the r e a d e r
e n c o u n t e r s literature in w h i c h the
same laws are d e s c r i b e d u s i n g different m a t h e m a t i c a l tools. EQUATION FOR THE CONSERVATION OF MASS
.!
Fig.
3 . 3 - 2 S k e t c h for d e r i v i n g c o n t i n u i t y e q u a t i o n
If t h e r e a r e n o s o u r c e s or sinks of mass in dV. the d i f f e r e n c e b e t w e e n the a m o u n t of mass that e n t e r s dV a n d the a m o u n t that e m e r g e s from it in the time dt e q u a l s the c h a n g e of the mass dM in dV. If m a n d m÷dm s h o w mass flow rates t h r o u g h c r o s s - s e c t i o n s z a n d z , d z . then the a m o u n t of mass that e n t e r s dV d u r i n g the time dt e q u a l s mdt. a n d the a m o u n t that e m e r g e s is (m÷dm)dt. Thus w e c a n write
2/4
CHAP.
3
DISTRIBUTED
PROCESSES
mdt - (m , d m ) d t = d M
(3.3-6)
In the g e n e r a l l y u n s t e a d y
state
mass
flow rate is v a r i a b l e b o t h a l o n g
the p i p e a n d in time, i.e. m = re(z, t), so that flow rate c h a n g e a l o n g the z c o o r d i n a t e , at time t, c a n b e written am
dm=
~
dz
(3.3-7)
T h e last two e q u a t i o n s y i e l d = dM
. a__m_md z
az
_ d(Apdz)
dt
ap
am
(3.3-6)
= Adz dp
dt
dt
= 0
(3.3-~)
E q u a t i o n (3.3-9) is the law for the c o n s e r v a t i o n of mass. H o w e v e r , as the
same
law
advantageous
is
encountered
to p r e s e n t
those
in
various
various
forms
in
literature,
it
is
forms of the c o n t i n u i t y e q u a t i o n
h e r e also. It must b e r e m e m b e r e d that mass flow rate m is m = Awp
(3.3-10)
a n d for p ,, 1/v from (3.3-0), equation can be obtained
the
following
forms
of the
~a~ + w S ~ ,,._p a w at az az av
at
- -
av
w~-~
4-
~m
v az
em 4-
at
(3.3-11)
8w ~
w-~-
:=
continuity
(3.3-12)
.
A aw
~
m sl
-
aw -
(3.3-13)
-
w at
The last three e x p r e s s i o n s c o n t a i n a d e r i v a t i v e that is c h a r a c t e r i s t i c of fluid flow p r o c e s s e s . It is c a l l e d the material or s u b s t a n t i a l d e r i v a t i v e a n d d e n o t e d D/dr Df af af d---t = a + w ? ~ The material or s u b s t a n t i a l derivative D/dt c o n s e r v a t i o n e q u a t i o n s that d e s c r i b e s t a t e c h a n g e s n a m e itself c o m e s from the fact that e v e r y c h a n g e
(3.3-14) will a p p e a r in all of flowing fluid. The depends
b o t h o n the
a.a
~=c.
local
215 gradient
of
$/~t,
change
g r a d i e n t of c h a n g e , which ( s u b s t a n c e ) " flow wa/az.
and
on
depends
on
the
so-called
the
convective
velocity
of
"material
Using (3.3-14). w e c a n write Do at
ow
Dv
= - p'~z
~w
d--t = V S - z
Dm '
d~- =
m 8w w
~z
In s t e a d y state, in w h i c h t h e r e must b e (3.3-13) y i e l d the following e q u a l i t i e s mz
-
-
= mz*dz
,~/Ot = O, E q u a t i o n s
= m
(3.3-9)
(3.3-15)
m
pW = cons[.
(3.3-16]
For incompressible fluids (sometimes liquids incompressible) p = const., a n d (3.3-10) yields
can
be
considered
(3.3-17]
W z = W z , dz = W = o o n s t .
EQUATION FOR THE CONSERVATION OF MOMENTUM
F i g . 3.3-3 S k e t c h for d e r i v i n g m o m e n t u m e q u a t i o n T h e law for the c o n s e r v a t i o n
of m o m e n t u m s a y s
that the s u m of all
forces a c t i n g on the m a s s of fluid dM in the d i r e c t i o n of flow (the d i r e c t i o n of the z axis) e q u a l the c h a n g e of m o m e n t u m (dMw) of the particle dM in time dt.
CHAP. 3 DISTRIBUTED PROCESS~'S
2/6 Mathematically 4 i-l
7: F, -
d(dMw) dw dt = dM
It must b e o b s e r v e d
C3.3-18)
that, unlike p r e c e d i n g
c a s e w h e n the b a l a n c e of
mass was f o r m u l a t e d for a n e l e m e n t a r y a n d u n c h a n g e a b l e c o n t r o l v o l u m e dV. h e r e the b a l a n c e of m o m e n t u m is f o r m u l a t e d for a n e l e m e n t a r y a n d u n c h a n g e a b l e c o n t r o l mass dM ( w h i c h is in dV at the moment t). T h e fact that the e q u a t i o n of e q u i l i b r i u m is f o r m e d for an u n c h a n g e a b l e (i.e. c o n s t a n t a n d i n d e p e n d e n t of b o t h z a n d t) p a r t i c l e of mass dM m a k e s it possible
to
place
the
symbol
dM
in
front
of
the
operator
of
total
differentiation d / d t in E q u a t i o n (3.3-10). The following f o r c e s a c t in the d i r e c t i o n of the z axis: p r e s s u r e f o r c e in the c r o s s - s e c t i o n z:
pressure force in the cross-section z+dz: component
F l = PA ,
F2 = P A
+
aCPA) dz ,)z
g r a v i t a t i o n a l f o r c e in the z d i r e c t i o n :
of
ah F3 = Gcos,, = d M g ~ - =
= Adzog friction f o r c e of fluid a g a i n s t the p i p e wall:
F4
=
F
ah ~z =
.
fAdz
.
In the a b o v e e q u a t i o n s g is the g r a v i t a t i o n a l a c c e l e r a t i o n a n d f the c o e f f i c i e n t of friction, i.e. it r e p r e s e n t s the p r e s s u r e d r o p d u e to friction aPF per unit l e n g t h of p i p e , of the d i m e n s i o n P a / m . S i n c e force a n d m o m e n t u m a r e v e c t o r s , their d i r e c t i o n s a r e important a n d if the i n c r e a s i n g z d i r e c t i o n is g i v e n with the * sign. (3.3-18) b e c o m e s F,
- F2 - F3 - F4 = A o d z d t
o! aP az
~h
~ga-~-
f = "
~w
~C%V•
The total c h a n g e d w is
w
~w) az
(3.3-19)
~c.
a.a
dw
217
aw aw bw a w dz = ~-~----dt+ ~--~----dz= (--~- + az ~-)dt
(3.3-20)
If fluid flows at velocity w. in period of time dt its particle d M will m o v e along path dz = wdt. so quotient dzldt is flow velocity w. This means that the expression in brackets in the a b o v e equation c a n b e written in the form of material derivative D w / d t a n d Equation (3.3-I@). referring to (3.3-20). after rearrangement b e c o m e s Dw aP ah P"~'- + ~Z + gp~-~
+
APG
f
,, O
(3.3-21)
APF
AP G ... g r a v i t a t i o n a l p r e s s u r e c h a n g e AP F ... frictional p r e s s u r e c h a n g e
per unit length
(drop) per
unit
length
Pa/m Pa/m
The equation for the conservation of m o m e n t u m can also b e s h o w n in different forms, which will be used in further derivations. Multiplying (3.3-21) b y A w yields c~w
c~w
m(---~-[- + w-~-~z) + A w A P G
,~ - A w aP az
- AwAP F
(3.3-22)
S u b s t i t u t i n g w - m v / A for w in E q u a t i o n (3.3-21) a n d r e f e r r i n g to the continuity e q u a t i o n y i e l d s the form of m o m e n t u m e q u a t i o n that i n c l u d e s the g r a d i e n t of m a s s flow rate m with r e s p e c t to time I am A at
I a(m2v) *A~ a~ ""
aP a-~-" A P G " APF
(3.3-23)
Finally, to a n a l y z e p r e s s u r e c h a n g e a l o n g the s t r e a m l i n e of fluid the following form of the e q u a t i o n for m o m e n t u m c o n s e r v a t i o n is useful I a(mv 2) m 2 av + . . . . X~ at E az
aP az
APG-
APF
(3.3-24)
Equations (3.3-21) - (3,3-24) are only different forms of the well-known Navier-Stokes equation for the one-dimensional flow of real fluid, in which the d y n a m i c coefficient of viscosity ~ differs from zero. i.e. in which friction is not neglected. T h e most similar in form to the Navier-Stokes equation is Equation (3.3-21). which b e c o m e s by density p,
identical to it if it is divided
218
CHAP. 3 DISTRIBUTED PROCESSES
1 p
~P = F z + - ~ Bz f
FZ
Z
(3.3-25)
... f r i c t i o n f o r c e p e r u n i t m a s s in t h e d i r e c t i o n o f t h e z a x i s
~ P
Dw dt
complete
,,.
(material) gradient
of v e l o c i t y w c h a n g e
along
the z axis ~h Z = - g ~ ,.. g r a v i t a t i o n
f o r c e p e r u n i t m a s s in t h e d i r e c t i o n
of t h e
z axis F o r i d e a l f l u i d s (to w h i c h r e a l g a s e s of v i s c o s i t y
~ and
(3.3-25) b e c o m e s 1
aP
p
~z
----
The
high
÷ Z =
liquid
respect
Bernoulli's
whole
P2
ha ,
~ = 0 and
equation
equation
w2
g
,
is
for
also
for i d e a l f l u i d
the by
flow
derive
of
from
viscous Equation
dz. after integration
with
2. w e g e t
z2
,
÷
0 W I
to
1 to c r o s s - s e c t i o n z2
,~(~}
unsteady
easy
is m u l t i p l i e d
p i~ t
Equation
(3.3-26)
to z from c r o s s - s e c t i o n
l_
with small coefficients
close)
Dw "" dt
(~ ~, O. 0 = c o n s t . )
(3.3-25). If t h e
vapors
come
the well-known Euler equation
extended
(real)
and
flow velocities
Z I
=0
"~'2
w~ z
l (pP 2
- Pt) + g ( h ~ - hi) + ( 2
w~ z
"
a[,
2 ) + J~t
APF
dz +
p
(zz - zl) = 0
z1
(3.3-27) In t h e f l o w of a n i n c o m p r e s s l b i o function
of z (A = c o n s t . )
equation
is
dW(z ~-~
and
equals
liquid the gradient dw/dt,
so the
a w / a t is n o t a
integral
in t h e
last
equation
for
dw 2 -
z i)
-
Rearrangement
-
dt
L
yields
r e a l l i q u i d in u n s t e a d y
the well-known
conditions
form
of B e r n o u l l r s
P~ -- + ghl
p
+
w~ 2
2
P2 = -- ÷ gh2
p
+
w2 2
-~-
*
dw
-~-L
APF L
,
.
P
(3.3-28)
Multiplying the a b o v e e q u a t i o n b y p. a n d for w = m/Ap. we g e t ( P i - P2) - a P F L 8Pc
* pg(hl
- hi)
+
w~ 2 - w 2 2 2 P=
L dm A dt
(3.3-2q)
8PG
In the h o r i z o n t a l p i p e of c o n s t a n t c r o s s - s e c t i o n the last two terms o n the l e f t - h a n d s i d e of the last e q u a t i o n a r e lost a n d it b e c o m e s e q u a l to Equation (2.2-1.4) from the c h a p t e r on l u m p e d p a r a m e t e r s . EQUATION FOR THE CONSERVATION OF ENERGY
-
\
Fig.
3 . 3 - 4 S k e t c h for d e r i v i n g e n e r g y e q u a t i o n
T h e law for the c o n s e r v a t i o n of e n e r g y (thermal a n d m e c h a n i c a l ) e x p r e s s e s the fact that the d i f f e r e n c e b e t w e e n the a m o u n t s of e n e r g y b r o u g h t to a n d l e d from the fluid p a r t i c l e dM. in the e l e m e n t a r y v o l u m e dV d u r i n g time dt, e q u a l s the c h a n g e of the total a m o u n t of e n e r g y dE c o n t a i n e d in dV eidt
- eodt
= dE
(3.3-30)
A k i l o g r a m of flowing fluid m a s s p o s s e s s e s t h r e e forms of e n e r g y : internal t h e r m a l e n e r g y u, kinetic e n e r g y w2/2 a n d m e c h a n i c a l p o t e n t i a l e n e r g y gh. T h e r e f o r e . the e n e r g y flow r a t e b r o u g h t t h r o u g h c r o s s - s e c t i o n z into dV is ez, = m(u+w2/2+gh) J/s. T h e p o w e r n e c e s s a r y to inject t h e m a s s of fluid m, u n d e r p r e s s u r e P, into s e c t i o n z is ez2 = PAw J/s. As the fluid flows out of c r o s s - s e c t i o n z , d z it t a k e s with it a total e n e r g y flow r a t e of
220
CHAP.
ez+dez
l/s. and
to e j e c t
P A w * d ( P A w ] is n e e d e d . from t h e o u t s i d e
fluid out the pipe
given the following mathematical
the said
powel into dV can
be
folm
w2 ~ , g h ) * PAw + q d z
(3.3-31)
ez2
eo = ez * dez
¢1 = ez * ~-;-(m(u •
dE = dM(u
W2 ~ , gh)=
Substituting
cross-section
w a l l . W h a t h a s just b e e n
ez!
,
same
PROCESSES
F i n a l l y . a h e a t flow r a t e q d z J/s is b r o u g h t
through
ei = e z * Q = m(u ,
of t h e
3 DISTRIBUTED
w2 ~ • gh),
A0(u * ~
the last three
W2
PAw)dz
,
÷ gh)dz
equations
(3.3-32)
(3.3-33)
into (3.3-30)
and
dividing
by dz
yields W2 q " ~z-z( m ( u * T * gh)+
~ PAw) =
W2 (Ap(u + T * gh))
(3.3-34)
R e f e r r i n g to [3.3-I0) t h e r e is Aw = my. s o mu
÷ PAw
= m(u
, Pv)
=
mi
T h e t e r m pu o n t h e r i g h t - h a n d
(3.3-35) side can
be written (3.3-36)
pu = oi - P
S u b s t i t u t i n g (3.3-35) a n d (3.3-36) i n t o (3.3-34) a n d d i f f e r e n t i a t i n g y i e l d s w2 q - az :~--(i
=
A
_•_(i
Referring cancel
q
, - Tw-~ ÷
=
A ai
The last
gh)
,
to c o n t i n u i t y
out,
rearranged
ai ~w ~h * g h ) " m ( ~ z * w~--~- * g i f t - )
* T
so
Equation
ai ,~w Ap(--~-, w-~-)equation (3.3-37),
(3.3-9). with
=
A aP ~ the
[3.3-37)
terms
reference
-i~ml,~z a n d iA,~pl,~t to (3.3-10). c a n be
i n t o this f i n a l form -
AeP
term
6i + m-- * of
this
~w
equation
,~w * w--~-) + AwaP G . was
E q u a t i o n (3.3-21]. in t h e f o l l o w i n g m a n n e r
obtained,
using
[3.3-38]
notation
from
• E,".
221
a.'~
~h ah mg~-~ = A w p g ~ - =
AwAPG
(3.3-3Q)
E q u a t i o n (3.3-38) r e p r e s e n t s o n e of the p o s s i b l e forms of the law for e n e r g y c o n s e r v a t i o n . As in the p r e c e d i n g e x a m p l e s , it is a d v a n t a g e o u s to show this law in different, more suitable, forms. O n e of them is o b t a i n e d using the b a l a n c e of momentum. It c a n b e s h o w n that the last two terms on the r i g h t - h a n d s i d e of E q u a t i o n (3.3-38) e q u a l the l e f t - h a n d s i d e of Equation (3.3-22), so substituting the r i g h t - h a n d s i d e of E q u a t i o n (3.3-22) into (3.3-38) y i e l d s ai + w ~ i ) aP , w a p ) • I (~-taz - v(--~az "~"p(q ÷ AWaPF) Di DP d--'-[- " v - - ~ - -
qu
.
(3.3-40]
] kgs
(3.3-41)
The total h e a t flow rate b r o u g h t to o n e kilogram of fluid mass is d e n o t e d qu. a n d it e q u a l s the sum of h e a t flow rate b r o u g h t from the outside t h r o u g h the p i p e wall q/Ap a n d the h e a t flow rate r e l e a s e d b y friction wAPF/p. E q u a t i o n (3.3-36) y i e l d s i = u + vP. so the e n e r g y c o n s e r v a t i o n E q u a t i o n (3.3-41) b e c o m e s Du pDV "dt + dt = qu
(3.3-42)
Multiplying (3.3-42) b y dt y i e l d s confirmation of the first main law of t h e r m o d y n a m i c s , a c c o r d i n g to which the total h e a t i n t r o d u c e d , d Q = q , d t , is s p e n t on a n i n c r e a s e of the fluid's thermal e n e r g y a n d o n m e c h a n i c a l work dO = Du + PDv
J kg
(3.3-43)
A c c o r d i n g to E q u a t i o n (3.3-14). the total c h a n g e Df(z. t) e q u a l s
Df(z, t) = ( ~af + w.~f)dt ,~z
(3.3-44)
It is c l e a r that the law for e n e r g y c o n s e r v a t i o n s h o u l d satisfy the s e c o n d main law of t h e r m o d y n a m i c s as well, w h o s e g e n e r a l l y - k n o w n form for the e l e m e n t a r y fluid p a r t i c l e is
222
CHAP.
ds = l ( d i
PROCESSES
(3.3-45)
- vdP)
In this c a s e Equation
3 DISTRIBUTED
(3.3-45)
of s p a t i a l
distribution
remains
the
same.
for fluid flow the the
only
difference
law
given
being
that
by a
material, s u b s t a n t i a l i n c r e a s e D is u s e d i n s t e a d of c h a n g e d. T h e r e f o r e Ds I cDi dt" = "~ dt Now the
DP - v--~-)
expression
(3.3-4b) on
the
left-hand
side
(3.3-41), is r e p l a c e d b y Ds/dt. so the law expressed through entropy change, becomes as at
of E q u a t i o n for
energy
(3.3-40).
or
conservation.
as l l W , + w - - = -=-( .----q ÷ - ~ P F ) dz TAp
(3.3-47)
The e q u a t i o n s for mass. e n e r g y a n d m o m e n t u m c o n s e r v a t i o n that we d e r i v e d a r e not sufficient to o b t a i n a solution in the c a s e of u n s t e a d y fluid flow. T h e y c o n t a i n four u n k n o w n v a r i a b l e s that c h a n g e in time a n d in s p a c e (mass flow rate m. s p e c i f i c v o l u m e v. p r e s s u r e P a n d e n t h a l p y a n d / o r e n t r o p y s) so the t h r e e e q u a t i o n s o b t a i n e d a r e not e n o u g h for a solution. That system of e q u a t i o n s must be completed by the t h e r m o d y n a m i c e q u a t i o n of s t a t e that relates a n y state e q u a t i o n with two others. Which form that e q u a t i o n will take ( w h e t h e r v - v(P. s). t~ = p(P. i). i = i(P.s), i = i(P. v) or s - s(P, i). s - s(P. v) a n d so on) d e p e n d s
o n which
v a r i a b l e s from the c o n s e r v a t i o n e q u a t i o n s are s e l e c t e d for d i r e c t numerical calculation
(integration).
Thus,
for
example,
numerical
integration
or
s o l v i n g the s y s t e m of e q u a t i o n s is p o s s i b l e , after e x p r e s s i n g a0/at, am/~t a n d . s/at from E q u a t i o n s (3.3-0). (3.3-23) a n d (3.3-47). After d i s c r e t i z a t i o n with r e s p e c t to the s p a t i a l v a r i a b l e , a n d k n o w i n g the initial state, the first s t e p of i n t e g r a t i o n y i e l d s n e w v a l u e s for 0. m a n d s. H o w e v e r , b e f o r e the s e c o n d step of i n t e g r a t i o n c a n b e u n d e r t a k e n , n e w v a l u e s for v. T a n d P must also b e known, b e c a u s e t h o s e v a r i a b l e s a r e in t h e s y s t e m of e q u a t i o n s b e i n g i n t e g r a t e d . While v is g i v e n d i r e c t l y b y v = l/p. P a n d t must b e d e t e r m i n e d from s t a t e e q u a t i o n s P = P(p. s) a n d T = T(p, s). Only after t h e s e v a r i a b l e s h a v e b e e n f o u n d is it p o s s i b l e to b e g i n the s e c o n d s t e p of integration. In p r a c t i c e , s i n c e m. e a n d P a r e s t a n d a r d c o n t r o l v a r i a b l e s , we u s u a l l y try to g e t a s y s t e m from the e q u a t i o n s for c o n s e r v a t i o n that c o n t a i n s differentiation with r e s p e c t to time of mass flow rate, t e m p e r a t u r e
==~. ~.~
223
(i.e. e n t h a l p y or e n t r o p y ) a n d p r e s s u r e . For the n e e d s
of further d y n a m i c
analysis, the following m o d e l c o n t a i n s d e r i v a t i v e s with r e s p e c t to time of flow rate, e n t r o p y a n d p r e s s u r e . Thus it is n e c e s s a r y to o b t a i n , in a n a n a l y l i c a n d d i f f e r e n t i a b l e foTm the s t a t e e q u a t i o n v = v(P, s)
(3.3-48)
T h e u p p e r e q u a t i o n e a s i l y y i e l d s aP/at. From av a-V =
av aP
aP at
av as
+
as at
(3.3-49)
7B
TA
whence aP
1
as
av
(3.3-so)
"rA a n d "rB a r e d e r i v e d from E q u a t i o n (3.3-48) a n d in the g e n e r a l c a s e they a l s o a r e a n a l y t i c f u n c t i o n s of p r e s s u r e a n d e n t r o p y . c~v
TA = ('~')S = TA(P, S)
(3.3-51)
(-~'-) a v , p = TB(P. s)
(3.3-52)
YB =
(The v a r i a b l e x A is r e l a t e d to the v e l o c i t y of s o u n d in fluid "fA=-V2/CZ.) Finally the m o d e l s o u g h t for is o b t a i n e d from E q u a t i o n s (3.3-23), (3.3-47) a n d (3.3-50) with the h e l p of the c o n t i n u i t y e q u a t i o n , a n d it c o m p r i s e s a s y s t e m of t h r e e first-order h y p e r b o l i c PDE. am
aP
2my am
a-i- " " A az
as at
aP
.
v(
.
1
~%~"
1
. . ~ - t" - q
v 2 am
at " 'TA [A
aZ at
m as A
m 2 av "
~z
+
A
mv -X-f
a'--z-" A ( A P G APF')
÷ nPr)
'
~'8 v ( A ~ q - m as + mv aPF)J .
A az
(3.3-53)
(3.3-54) (3.3-55)
at
T h e s e t h r e e e q u a t i o n s must n e c e s s a r i l y b e a c c o m p a n i e d b y the t h e r m o d y n a m i c s t a t e e q u a t i o n s (3.3-48), (3.3-51) a n d (3.3-52). Unlike in the p r e c e d i n g s e c t i o n w h e r e the d e r i v e d m o d e l s w e r e in most c a s e s linear
224
CHAP.
3 DISTRIBUTED
PROCESSES
after c e r t a i n a s s u m p t i o n s h a d b e e n m a d e , the m o d e l s h o w n b y the a b o v e s y s t e m is n o n l i n e a r . This n o n l i n e a r i t y is not r e f l e c t e d o n l y b y the fact that it c o n t a i n s the s q u a r e s m ~ a n d v ~, but also b y the p r o d u c t s of variables, for e x a m p l e mv,~m/dz a n d so on. T h e s e l e c t i o n of v a r i a b l e s m, s a n d P is o n l y o n e p o s s i b l e selection. H e r e it was m a d e b e c a u s e we w a n t e d to a n a l y z e the e i g e n v a l u e s of the p r o c e s s of d i s t u r b a n c e p r o p a g a t i o n t h r o u g h the fluid. T h a n k s to the fact that the e n e r g y e q u a t i o n is e x p r e s s e d b y e n t r o p y c h a n g e , it is p o s s i b l e to separate the h y d r o d y n a m i c (fast, a n d thus considered adiabatic) p r o p a g a t i o n of d i s t u r b a n c e s in p r e s s u r e a n d flow rate from the slower transport p r o p a g a t i o n of d i s t u r b a n c e s in the thermal state of the fluid. T h e following must also b e s a i d c o n c e r n i n g the s y s t e m of e q u a t i o n s (3.3-53) - (3.3-55), w h i c h w e h a v e c a l l e d a m o d e l . It d e s c r i b e s momentum, e n e r g y a n d mass transfer (i.e. c h a n g e s of mass flow rate, e n t r o p y and p r e s s u r e ) in a n a r b i t r a r y fluid a n d d o e s not i n c l u d e a n y s p e c i a l limitations ( e x c e p t the a s s u m p t i o n s m a d e at the b e g i n n i n g ) c o n c e r n i n g the specific p r o p e r t i e s of the p r o c e s s e s t h e m s e l v e s , the c h a r a c t e r i s t i c s of the o b j e c t in w h i c h t h e y o c c u r or the c o n d i t i o n s u n d e r w h i c h t h e y o c c u r . T h e r e f o r e , this s y s t e m of e q u a t i o n s c a n b e u s e d for a n y arbitrary, o n e - d i m e n s i o n a l flow a n d h e a t p r o c e s s (transport of water a n d o t h e r liquids, g a s p i p e s , e v a p o r a t o r s a n d s t e a m g e n e r a t o r s u p e r h e a t e r s , p n e u m a t i c p i p e s a n d so
on). All t e c h n i c a l p ~ o c e s s e s o c c u r in s y s t e m s with s p e c i f i c g e o m e t r i c a l c h a r a c t e r i s t i c s , in v a r i o u s d e v i c e s , a p p a r a t u s a n d plants w h o s e size limits the d o m a i n of c h a n g e of the spatial c o o r d i n a t e in differential transfer e q u a t i o n s . The p h y s i c a l c h a r a c t e r i s t i c s of the o p e r a t i n g m e d i u m are also factors of limitation in the g i v e n s y s t e m of e q u a t i o n s . B e s i d e s these limitations, to s o l v e the p r o b l e m c o r r e c t l y it is also n e c e s s a r y to k n o w how the s y s t e m i n t e r a c t s with the e n v i r o n m e n t at its b o u n d a r y ( b o u n d a r y c o n d i t i o n s ) a n d the states of the p r o c e s s v a r i a b l e s at the moment we c o n s i d e r the b e g i n n i n g of c a l c u l a t i o n (initial c o n d i t i o n s ) . T h e g e o m e t r i c
and physical characteristics of the system, together with the boundary and the initial conditions, are a set that separetes the observed specific process from the whole class of processes to which we can apply the system of equations (3.3-53) - ( 3 . 3 - 5 5 ) .
= ~ c , ~.'~
Therefore,
225
the
mathematical
model
of a
process
under
observation
c o m p r i s e s a s y s t e m of differential e q u a t i o n s t o g e t h e r with the g e o m e t r i c a n d p h y s i c a l c h a r a c t e r i s t i c s . This is i m p o r t a n t to stress b e c a u s e in this b o o k a c e r t a i n m a t h e m a t i c a l r e c o r d is often c a l l e d the m o d e l of a p r o c e s s , a n d the m e n t i o n e d a d d i t i o n a l c o n d i t i o n s a r e not s p e c i a l l y i n d i c a t e d . In such c a s e s it is t a k e n for g r a n t e d a n d known.
that all t h o s e c o n d i t i o n s
are defined
T h e m o d e l p r e s e n t e d in the p r e c e d i n g e x a m p l e d o e s not b e a r a n y r e s e m b l a n c e to the h y p e r b o l i c PDE (3.3-1), nor d o e s it c o n t a i n s e c o n d d e r i v a t i v e s with r e s p e c t to time a n d s p a c e . It is, h o w e v e r , a c o n n e c t e d system of first-order h y p e r b o l i c PDE a n d with the h e l p of c e r t a i n m a t h e m a t i c a l t r a n s f o r m a t i o n s a s e c o n d - o r d e r PDE in m or P c a n b e o b t a i n e d , w h i c h will b e a m o r e e x p a n d e d form of the a l r e a d y s h o w n simplest t y p e in E q u a t i o n (3.3-I). S i n c e the m a t h e m a t i c a l a p p a r a t u s that h a s b e e n d e v e l o p e d m a k e s it m u c h simpler to a n a l y z e a linear m o d e l , in the following e x a m p l e w e will l i n e a r i z e the m o d e l (3.3-53) - (3.3-55) to o b t a i n a linear s e c o n d - o r d e r h y p e r b o l i c PDE for m a s s flow r a t e m. This linear model will b e u s e d to a n a l y z e the d y n a m i c p r o p e r t i e s of d i s t u r b a n c e p r o p a g a t i o n a l o n g p i p e s for fluid t r a n s p o r t . A n a l o g o u s l y to the p r e c e d i n g s e c t i o n w h e r e this w a s d o n e for p r o c e s s e s with e q u a l i z a t i o n , it will b e s h o w n that the m o d e l s from the s e c t i o n o n l u m p e d p a r a m e t e r s a r e just r e d u c e d forms of the m o d e l s s h o w n h e r e . We m a y a l s o r e p e a t that the linear m o d e l o b t a i n e d r e p r o d u c e s faithfully all the e s s e n t i a l d y n a m i c p r o p e r t i e s of the real n o n l i n e a r p r o c e s s o n c o n d i t i o n that d e v i a t i o n s from the s t e a d y s t a t e o b s e r v e d a r e small.
Examplo
3 Analysis of d y n a m i c p r o p e r t i e s . S t e a d y state. L i n e a r m o d e l
For the p r o c e s s of o n e - d i m e n s i o n a l fluid flow t h r o u g h a p i p e of l e n g t h L w h o s e u n s t e a d y s t a t e c h a n g e s a r e s h o w n b y the n o n l i n e a r m o d e l (3.3-53) - (3.3-55). d e t e r m i n e the s t e a d y s t a t e a n d then: a) d e r i v e a linear m o d e l for d e v i a t i o n fiom the s t e a d y s t a t e o b s e r v e d , b) u s i n g the l i n e a r m o d e l o b t a i n e d , a n a l y z e d y n a m i c p r o p e r t i e s for:
22~
CHAP.
3 DISTRIBUTED
I - in the initial steady state the fluid is at rest a n d
o,
PROCESSES
is not heated (m -
o),
2 - the fluid flows t h r o u g h the p i p e a d i a b a t i c a l l y (q= O}. 3 - the fluid flows t h r o u g h the p i p e a n d is h e a t e d b y a h e a t flow rate of c o n s t a n t d e n s i t y q a l o n g the p i p e . The following b o u n d a r y c o n d i t i o n s a r e g i v e n : 1st BC ... mass flow rate is g i v e n at the input
c r o s s - s e c t i o n m(z-O.t)=
m0Ct) 2nd BG ... pressure is g i v e n at the output cross-section PCz=I...t)= PLCt) 3zd BC ,.. fluid with e n t r o p y So is s u p p l i e d at the input c~oss-section, s(z=O,t) = so(t) The
cross-section
A of the
pipe
is
constant.
All the
variables
are
uniform in a c r o s s - s e c t i o n . P r e s s u r e d r o p s d u e to friction a n d g r a v i t a t i o n a r e n e g l e c t e d , AP F ,, AP G = O. STEADY
STATE
CALCULATION
T h e equation of continuity yields
B2
= 0
(3.3-S6}
After setting (3.3-53} y i e l d s ----= A ,)z
A
the terms
Bml~t.
APG and
AP F e q u a l
to zero.
Equation
(3.3-57)
Oz
Equation (3.3-54) yields
a'-z- = ~ The
last
7q, . three
(3.3-58) equations,
with
a
given
heat
rate
q and
c o n d i t i o n s that a r e c o n s t a n t in the s t e a d y s t a t e too. PL, Eo, y i e l d
boundary
~a.
~.~
re(z) " me = c o n s t .
(3.3-59)
]~(Z) " l~L + - - ~
(3.3-00)
(9 L - 9(Z))
T h e further d e r i v a t i o n of a n a l y t i c a l e x p r e s s i o n s is c o u p l e d with difficulties b e c a u s e t h e r m o d y n a m i c s t a t e v a r i a b l e s a r e i n t e r r e l a t e d a n d without a k n o w l e d g e of their r e l a t i o n s in a n a l y t i c a l form solutions c a n n o t be r e a c h e d e x c e p t in s p e c i a l c a s e s . Thus. for e x a m p l e , to d e t e r m i n e the distribution of s a l o n g z. i.e. to find {(z), d e m a n d s the i n t e g r a t i o n of (3.3-58) with r e s p e c t to z. H o w e v e r , (3.3-581 h a s t e m p e r a t u r e T in the d e n o m i n a t o r , w h i c h is a l s o a f u n c t i o n of z. T(z). O n e of the s p e c i a l c a s e s w h e r e a n a l y t i c a l e x p r e s s i o n s for all the s t a t e v a r i a b l e s in the s t e a d y s t a t e c a n b e o b t a i n e d is the c a s e when fluid,flows through the pipe and evaporates, w h i c h m a k e s it p o s s i b l e to a s s u m e that t e m p e r a t u r e is a function of the p r e s s u r e of e v a p o r a t i o n only, T = T(!~). We will n o w s h o w h o w to d e t e r m i n e the s t e a d y state for t h e c a s e o f t h e e v a p o r a t o r . B e f o r e d e r i v i n g the o t h e r a n a l y t i c a l e x p r e s s i o n s w e must p o i n t out that the s e c o n d term o n the r i g h t - h a n d s i d e of E q u a t i o n (3.3-60) is as a r u l e , very, v e r y small (less t h a n 0.01% ~L) a n d c a n b e n e g l e c t e d . If this is so. then P(z) = ~L. or ~P -
-
~z
= o.
(3.3-61)
B e s i d e c o n s t a n t p r e s s u r e , t e m p e r a t u r e is a l s o c o n s t a n t a l o n g the e v a p o r a t o r . T(z) = c o n s t . Now E q u a t i o n (3.3-58) c a n b e i n t e g r a t e d , w h i c h yields
~Cz)
=
1 ~o • m--~o q z
To d e t e r m i n e ~(z) a n d w(z), it is useful (3.3-61), E q u a t i o n (3.3-43) y i e l d s
|(z) = [o + q-*-z. Io • iCso) m In the e v a p o r a t i o n p r o c e s s w e h a v e
(3.3-623 to
have
[(z). R e f e r r i n g
(3.3-63)
to
Z~.8
CHAP.
aT
ai ~"'
~"'= : " -
3 DISTRIBUTED
~',, v'"(P)
PROCESSES
(3.3-~)4)
The given equation yields
Till ;(z)
=
;o + ~
z
r
(3.3-65)
mo
Finally, r e f e r r i n g to E q u a t i o n (3.3-10), it follows that
+
r
(3.3-o6)
A
T h e r e f o r e , in the e v a p o r a t o r p i p e it is p o s s i b l e (with the a s s u m p t i o n ;5(z) = const.) to o b t a i n a n a l y t i c a l e x p r e s s i o n s for all p r o c e s s v a r i a b l e s in the s t e a d y state. A c h a r a c t e r i s t i c s of the e v a p o r a t o r is that t h o s e v a r i a b l e s ( e x c e p t for flow rate, p r e s s u r e a n d t e m p e r a t u r e w h i c h a r e c o n s t a n t ) c h a n g e l i n e a r l y a l o n g z. In all other c a s e s , w h e n it is not p o s s i b l e s i m p l y to d e t e r m i n e the a n a l y t i c a l e x p r e s s i o n s for v a r i a b l e s in the s t e a d y state, this must b e d o n e with the h e l p of a c o m p u t e r a n d u s i n g iteration. Only then, after o b t a i n i n g a s p e c i f i c s t e a d y o p e r a t i n g c o e f f i c i e n t s in the l i n e a r i z e d e q u a t i o n s b e c a l c u l a t e d . a) M O D E L
state,
can
LINEARIZATION
T h e p r o c e s s of o b t a i n i n g a linear m o d e l from the initial s y s t e m of e q u a t i o n s (3.3-53) - (3.3-55) is b a s e d o n the a s s u m p t i o n that v a r i a b l e s P, s, v, m, T a n d q r e m a i n in the n e i g h b o r h o o d of s t e a d y s t a t e ;L s. v, m. T, a n d q a b o u t w h i c h t h e y c a n b e e x p a n d e d into a T a y l o r s e r i e s in w h i c h h i g h e r - o r d e r terms a r e n e g l e c t e d . (The p r o c e d u r e of l i n e a r i z a t i o n is s h o w n in m o r e d e t a i l in t h e A p p e n d i x . ) H e r e w e will s h o w the l i n e a r i z a t i o n of E q u a t i o n (3.3-53), a n d for the other e q u a t i o n s o n l y the final linear e x p r e s s i o n s will b e g i v e n . With APG -- APF = O, E q u a t i o n (3.3-53) b e c o m e s am at
-A
aP az'
m 2 av A az
2my am A az
T h e g r a d i e n t am/at d e p e n d s am caP av am] at = g a--z-" m, ~--~. v, ~
(3,3-67)
on the o t h e r v a r i a b l e s a s follows (3.3-68)
s=c. :~.3
229
Differentiating
(3.3-O7)
with
respect
d e p e n d s (that f u n c t i o n a l d e p e n d e n c e Teplacing d with A y i e l d s aam a~
"
- A aAP a~
~2 A
2m" am A
to
the
functions
on
which
m
is g i v e n b y the last e q u a t i o n ) a n d
aav (2m av 2v a m . az " A az ÷ A - - ~ - ) A m
-
(3.3-60)
2m~ a A m A
az
az
To o b t a i n the l i n e a r i z e d l e f t - h a n d s i d e w e m a d e u s e o! the following
property, for small deviations, of the operator of differentiation d" a m .
a(dm)
L-~--) =
(3.3-T0)
at
Referring to (3.3-~6) y i e l d s the final, linear e q u a t i o n for the g r a d i e n t of small d e v i a t i o n of flow rate Am with r e s p e c t to time t
aAm - -
at
,,
-
A
aAP ~
,~z
~2
aAv
A
az
L i n e a r i z a t i o n of the o t h e r e q u a t i o n of c o n t i n u i t y y i e l d s a,',s
at
aAP a~v
at
~
~M
aAs
A
az
= A-T-~ q "
rl
( __a~v
2~ aT - - - -
A
two
v
az
2~
aam
A
az
-
as
-A a-z- ~ m
- a~s.
~
"
(3.3-71)
of the
equations
~2 d a m
A
A m
--
A--~q'pAP
model
and
of
Ihe
V--
"
A--~q'sAs (3.3-72) (3.3-73) (3.3-74)
az
The a b o v e e q u a t i o n s r e p r e s e n t the linear m o d e l d e m a n d e d . To o b t a i n the last two terms in E q u a t i o n (3.3-72) it is n e c e s s a r y to h a v e the s t a t e e q u a t i o n T = T(P.s). from w h i c h follows dT =
,)T aP d P , ~p
aT as- ds
(3.3-75)
~S
We must. n e v e r t h e l e s s , s a y that the i n f l u e n c e of t h o s e last two lerms in (3.3-72) is small b e c a u s e their d e n o m i n a t o r s h a v e the v e r y l a r g e v a l u e T 2. The m o d e l o b t a i n e d is a linear s y s t e m of three first-order, h y p e r b o l i c .
2.30
CHAP.
3 DISTRIBUTED
c o n s t a n t c o e f f i c i e n t PDE with r e s p e c t to time. For further following n o t a t i o n must b e i n , o d u c e d for t h o s e c o e f f i c i e n t s K, = A.
K2 =
~z Ks = A - ,
~2 2m A ' K3 = A
av az'
2m~ K4 = T
~ m--v Ks = A--T-' Kz = A = w.
PROCESSES
needs
the
= 2w.
v as K8 = A-a--z-
(3.3-7b) K9 = A--~zqxp, -
Klo = ~ .
Ku = "~--A' K~z = Ks KIo
-
K,3 = A--T,zq , s
In the g e n e r a l c a s e of flow a n d h e a t i n g of a n a r b i t r a r y fluid, all the c o e f f i c i e n t s e x c e p t KI a n d K2 a r e functions of the s p a t i a l v a r i a b l e K] = fi(z),
(3.3-77)
i = 3, 13
T h e a b o v e e x p r e s s i o n s for the c o e f f i c i e n t s Ki s h o w that t h e y are functions of the o p e r a t i n g s t a t e in w h i c h l i n e a r i z a t i o n w a s p e r f o r m e d , that t h e y c h a n g e d e p e n d i n g on it. a n d s i n c e the c o e f f i c i e n t s of a n y p a r t i c u l a r partial d e r i v a t i v e d e t e r m i n e the d y n a m i c p i o p e r t i e s of the p r o c e s s itself, t h o s e d y n a m i c p r o p e r t i e s will a l s o d e p e n d o n the o p e r a t i n g state, i.e. on the c o n d i t i o n s u n d e r w h i c h the p r o c e s s o c c u r s . b) A N A L Y S I S
OF DYNAMIC
PROPERTIES
As p a r t of the a n a l y s i s of d y n a m i c s w e will s h o w h e r e the transfer f u n c t i o n s for m a s s flow r a t e (the e x p r e s s i o n s AM(z. s)/AMo(s) a n d z~M(z,s)/aPL(s)) a n d e x a m i n e their p o l e s a n d z e r o s , w h i c h t o g e t h e r with the g a i n c o e f f i c i e n t s d e t e r m i n e the t r a n s i e n t p r o c e s s . That e n d c a n b e r e a c h e d in v a r i o u s w a y s , but h e r e w e will transform the initial m o d e l into a s e c o n d - o r d e r h y p e r b o l i c PDE for m a s s flow r a t e (or p r e s s u r e ) a n d retain the first-order h y p e r b o l i c PDE for the transfer of e n t r o p y d i s t u r b a n c e s . Before we b e g i n work o n o b t a i n i n g transfer functions w e must s a y that m o d e l (3.3-71) - (3.3-73) r e p r e s e n t s the p r o p a g a t i o n of - , m a l l d i s t u r b a n c e s am. As a n d Ap a l o n g the s t r e a m l i n e s of flow. Earlier. in s y s t e m s with l u m p e d p a r a m e t e r s , it w a s s h o w n that small d i s t u r b a n c e s of flow a n d pressure waves
propagate
through
ol complesslon
and
the
fluid
expansion
in thermal state (propagation
with have
the
velocity
of s o u n d
(the
that velocity), while cha,~ges
of the disturbance
As) occur
convectively
s~c. ~.~
231
(carried b y the v e l o c i t y of fluid flow). T h e s e a r e , t h e r e f o r e , d y n a m i c a l l y two different p r o c e s s e s . , a fast p r o c e s s (for Am a n d ~P) a n d the p r o p a g a t i o n of e n t r o p y d i s t u r b a n c e s . It is k n o w n that p r e s s u r e a n d flow d i s t u r b a n c e s p r o p a g a t e in a d i a b a t i c c o n d i t i o n s (As = 0). T h e v e I o c i t y of p r o p a g a t i o n is s u c h that t h e r e is no time for h e a t to b e e x c h a n g e d with the e n v i r o n m e n t in a n y p a r t of the fluid d u r i n g the time i t t a k e s for the d i s t u r b a n c e to p a s s . T h a n k s to this, the a n a l y s i s c a n b e d i v i d e d into i n v e s t i g a t i o n s into the d y n a m i c s of fast a n d of slow p r o c e s s e s , a n d Equations (3.3-71) a n d (3.3-73), w h i c h c o n t a i n As = O, c a n b e a n a l y z e d s e p a r a t e l y from E q u a t i o n (3.3-72). This is w h a t w e h a v e d o n e . but s i n c e this s e c t i o n is c o n c e r n e d with p e r i o d i c p r o c e s s e s , the a n a l y s i s of the first-order h y p e r b o l i c PDE for c h a n g e s of ,',s will in most c a s e s b e left out. If w e d i f f e r e n t i a t e E q u a t i o n (3.3-71) with r e s p e c t to t a n d r e p l a c e the e x p r e s s i o n a2Av/,~taz in it with the e q u i v a l e n t v a l u e o b t a i n e d from (3.3-74) after d i f f e r e n t i a t i o n with r e s p e c t to z. a s e c o n d - o r d e r PDE is o b t a i n e d for Am(z. t) w h i c h c o n t a i n s the partial d e r i v a t i v e ~2AP/at~z. S u b s t i t u t i n g (3.3-7,4) into (3.3-73) a n d d i f f e r e n t i a t i n g (3.3-73) with r e s p e c t to z y i e l d s that d e r i v a t i v e , w h o s e s u b s t i t u t i o n y i e l d s the final e x p r e s s i o n d e s c r i b i n g c h a n g e s in d e v i a t i o n s ~m(z. t) a2Am ÷ K 8Am K K K "a~Am ~t 2 3 ~ + (K2K5 ÷ 1 ~ lo) ~z-~-~z- +
rw ~Ks
+ ~"2"~- + K,
~
~Am ÷ K ~2Am
)'~'z--
4'~-['~z
(3.3-78) = 0
It will b e e a s i e r to u n d e r s t a n d this PDE if the p h y s i c a l m e a n i n g of the c o e f f i c i e n t s is g i v e n b e s i d e the partial d e r i v a t i v e s . In the g e n e r a l c a s e . all t h o s e c o e f f i c i e n t s a r e functions of the s p a t i a l c o o r d i n a t e a n d the e q u a l i t i e s (3.3-76) y i e l d K3 - K~(z), K1KsKlo -
K2K5 - w2(z), A YA
~
K4 - 2w(z)
(3.3-79)
s
is the fluid flow v e l o c i t y a n d ~ the v e l o c i t y of s o u n d in s t e a d y state. If we i n t r o d u c e the n o t a t i o n m 2
fl(z) = KeK5 + K1KsKto = "~2(z) - ~2(z) v ,~Ks
~, ~KI2
m
[7] (3.3-80)
232
CHAP.
K,.
3 DISTRIBUTED
PROCESSES
[m]
E q u a t i o n (3.3-78) b e c o m e s ,~2Am at z
. , ,6Am + f, ,a2Am ,, , ~Am a2Am 0 + r,.3~,zJ-~-~--I~'zJ az'-~-~- + 12~'z) ~z + f3(z) ata--'--z- =
(3.3-81)
We h a v e o b t a i n e d a s e c o n d - o r d e r h y p e r b o l i c PDE that r e s e m b l e s the b a s i c a n d simplest E q u a t i o n (3.3-1) from the b e g i n n i n g of this s e c t i o n and thus points to p e r i o d i c c h a n g e s of mass fIow rate. Here w e h a v e left out the d e r i v a t i o n of the g e n e r a l PDE for p r e s s u r e c h a n g e . It is not difficult to prove
that
the
dynamic
coefficients, equations,
p o l e s a n d z e r o s a n d so on for p r e s s u r e c h a n g e s
transfer
functions,
are equal
their
to t h o s e foI
flow c h a n g e s , w h i c h will b e s h o w n for the simplest c a s e s . bl) Fluid at zest without h e a t i n g This is a s p e c i a l c a s e a n d the simplest, w h e n the m e d i u m is at rest in a p i p e of l e n g t h L (m = O, ~ --- O) a n d there is no h e a t i n g a l o n g the pipe (q = O), so that (3.3-76) y i e l d s K2 = K3 = K4 = Kz = Ke = Kt3 = 0
(3.3-82)
aK5
-
-
m
,~K~2
,)z
a
O.
0z
Thus E q u a t i o n (3.3-81) o b t a i n s the form of the c l a s s i c a l w a v e e q u a t i o n ~2Am . 52 a~am ~
= 0
(3.3-83)
Further m a t h e m a t i c a l p r o c e d u r e is i d e n t i c a l to m e t h o d s that w e r e u s e d in the p r e c e d i n g
s e c t i o n for p r o c e s s e s with e q u a l i z a t i o n , a n d is r e d u c e d
to the fact that after L a p l a c e transformation of the initial PDE w e g e t an ODE for Alvf(z. s) with r e s p e c t to z in the s - d o m a i n , w h o s e solution is a sum of e x p o n e n t i a l f u n c t i o n s b e s i d e w h i c h a r e c o e f f i c i e n t s that are functions of the b o u n d a r y c o n d i t i o n s . On the b o u n d a r i e s z = O a n d z = L these conditions are ]st BC. 2nd
BC,
z = 0. A m ( 0 . t) = Amo(t) z = L. AP[L.
t] = APL(t)
(3.3-84) (3.3-85]
~=c. ~.a
233
From (3.3-73) a n d (3.3-74). in the c a s e
the second b o u n d a r y ~Am(L. t ) =
of a d i a b a t i c
processes
conditions becomes
1 aAP(L, t) KsK~o at
~z
AS = O,
(3.3-8b)
T h e solution of E q u a t i o n (3.3-83) in the s - d o m a i n is S --=-z
s -~z
AM(z. s) = C~e c
+ C2e c
(3.3-87)
For c o n d i t i o n s o n the b o u n d a r i e s z = 0 a n d z = L, C1 a n d C2 a r e d e t e r m i n e d from E q u a t i o n (3.3-87) a n d its d e r i v a t i v e with r e s p e c t to z (to satisfy the s e c o n d BC s h o w n in (3.3-8b)) L
" ~ S
m
AMo(s)e c
÷ K5~1o APL(S)
L
C, = e
L
c
'
(3.3-88)
c
+ e
L ~s
AMo(S) e c C2=
- -
L e
APL(s)
(3.3-89)
KsKIo L
~S C
+ e
-~5 C
T h e final e x p r e s s i o n for the c h a n g e of m a s s flow r a t e c r o s s - s e c t i o n is o b t a i n e d from (3.3-87) - (3.3-89) 1 ~,M(L, s) =
A
chL_~s
~Mo(s) _ ~
c
c
at the o u t p u t
sh L---s c
chLs
APL(S)
(3.3-90)
c
G,(s)
G2(s)
T h e p o l e s of the transfer functions ( w h o s e p h y s i c a l m e a n i n g is that they r e p r e s e n t n a t u r a l f r e q u e n c i e s of o s c i l l a t i o n of the c o l u m n of fluid) c h a r a c t e r i z e the d y n a m i c p r o p e r t i e s a n d it is not difficult to d e t e r m i n e them b y s e t t i n g the d e n o m i n a t o r s of Gl(s) a n d G2(s) e q u a l to z e r o
ch(Ao •
L
C
C
j) =
chLo c o s ~L C
C
÷ j shLosinLca C
C
- O.
234
CHAP. 3 DISTRIBUTED PROCESSES
Both the real a n d the i m a g i n a r y parts will e q u a l z e r o at the s a m e time o n l y if cos~)
= 0
c
and
sh
c
o -- 0
w h e n c e the p o l e s of GI(s) a n d G2(s) a r e o = 0
,
(3.3-91)
~k = (2k - i ) ~
L' (3.3-92)
= z(2k
+ I) ~~- ~ .
k = O. I. 2, 3 ....
Setting the n u m e r a t o r of G2(s) e q u a l transfer f u n c t i o n ao = O ~ok
to z e r o y i e l d s the z e r o s of that
(3.3-Q3)
= k~--L--,
k = O. II. 12 ....
= +k~-~-,
k = O, I. 2. 3 . . . .
(3.3-~4)
The fact that the p o l e s of the transfer f u n c t i o n s are c o m p l e x c o n j u g a t e pairs w h o s e real parts e q u a l z e r o allows us to c o n c l u d e that the flow rate AML(s), in the c a s e of a n y d i s t u r b a n c e
from the b o u n d a r y , will oscillate
with f r e q u e n c i e s ~k p e r m a n e n t l y a n d without d a m p i n g (o = 0). This is the i d e a l i z e d c a s e of a fluid at rest without e n e r g y e x c h a n g e a n d n e g l e c t i n g all p o s s i b l e losses. The form of PDE that d e s c r i b e s it is e q u a l to the PDE for l o n g i t u d i n a l v i b r a t i o n s of a r o d or b e a m (3.3-4). This similarity is not u n e x p e c t e d b e c a u s e in the o b s e r v e d c a s e . w h e n there is no flow. the p i p e in fact c o n t a i n s a " rod" of fluid with its elasticity m o d u l u s a n d v e l o c i t y of small p r e s s u r e d i s t u r b a n c e s p r o p a g a t i o n a l o n g the p i p e . B e s i d e s , it c a n b e s e e n that in the c a s e of a finite-dimensional s p a c e (limited s p a c e ) the e q u a t i o n s of c o n s e r v a t i o n d o not h a v e a solution that w o u l d satisfy the g i v e n b o u n d a r y c o n d i t i o n s for a n y arbitrary f r e q u e n c y . S u c h solutions exist o n l y for a c o m p l e t e l y d e t e r m i n e d s p e c t r u m of v a l u e s g i v e n b y E q u a t i o n s (3.3-QI) a n d (3.3-92). In other words, in a s p a c e with a finite d i m e n s i o n s , o s c i l l a t i o n s are p o s s i b l e o n l y for a s p e c i f i c set of f r e q u e n c i e s (of w h i c h t h e r e are. true, infinitely m a n y ) which, s i n c e
they
~ = ~ . :i.~
235
uniquely d e s c r i b e the properties of that s p a c e , frequencies.
are c a l l e d the natural
T h e s e results show the main d i f f e r e n c e b e t w e e n the propert i es of t h e s e processes a n d properties of p r o c e s s e s with equalization, w here the poles can b e real a n d n e g a t i v e , while he r e they are c o m p l e x c o n j u g a t e pairs. It was mentioned earlier that the d y n a m i c s of p r e s s u r e c h a n g e al ong a pipe of fluid is d e s c r i b e d by the same PDE that d e s c r i b e s the d y n a m i c s of flow rate. Now this will b e shown. We will start from Equations (3.3-71) and (3.3-73). which, referring to (3.3-74) a n d (3.3-82). in the a d i a b a t i c c a s e (as -- O) obtain the following simpler forms 6Am ~t
SAP
= - A--
~2 l ~ a m A YA ~z
aAP at
(3.3-95)
0z
-
(3.3-96)
Differentiating (3.3-95) with r e s p e c t to z a n d (3.3-96) with r e s p e c t to t yields 82Ap
~2 1 aZam "
2AP
=
A
"~A o t o z
~z I (12~P •, - A -~- ~ A 8z~ (3.3-97)
~2 ~2~P
This e q u a t i o n is c o m p l e t e l y the same as (3.3-83), which shows that the dynamic properties of p r o c e s s e s of flow rate a n d pressure d i s t u r b a n c e s p r o p a g a t i o n are identical. For the g i v e n b o u n d a r y before, we obtain
APe(s} --
l
ch~s
conditions, using
APt(s) + C
-K chLs riMe(S)
c '
S h cL~'-s
the
G3(s ) . . . . . .
same
procedure
as
(3.3-98)
c G4(s ) '
Comparing G3 a n d G( with G1 a n d G2 from (3.3-90) it is o b v i o u s that the poles a n d zeros d e r i v e d there will satisfy h e r e also for d y n a m i c p r e s s u r e c h a n g e s at the input cr os s - s e c t i on /,Pc(s).
CHAP, 3 DISTRIBUTED PROCEE-~E~
236
E q u a t i o n s (3.3-90) a n d (3.3-98) c a n a l s o b e s h o w n in matrix form which will later b e u s e d for a c o m p a r i s o n with the m o d e l s from S e c t i o n (2.2-33 on p r o c e s s e s with l u m p e d p a r a m e t e r s .
Po(]s)
1
~
sh L-'-s c "~" c h L _ s
,,
chL__s c
(3.3-99)
sh L_-s c
A - -~
AML(S)
1 chL~.--s C
chLs c
b2) Fluid flow without h e a t i n g The c a s e of a d i a b a t i c fluid flow t h r o u g h p i p e s is v e r y f r e q u e n t in { e c h n i c a l p r a c t i c e a n d is met in the t r a n s p o r t of liquids, g a s e s and v a p o r s . Unlike in t h e p r e c e d i n g c a s e . n o w the v e l o c i t y a n d m a s s flow rate a r e not e q u a l to z e r o in the s t e a d y s t a t e o b s e r v e d (m ,,, O, w ~, O). but = 0 still holds. A c c o r d i n g to (3.3-76), h e r e we will h a v e __~c~_._~_~ = ~ K l z
K~ = Ke = K~ -- K~3 = 0 .
~z
~z
= 0
.
(3.3-i00)
a n d the initial a n d g e n e r a l PDE (3.3-81) will o b t a i n this s p e c i a l form ~2&m
. (~2
. ~z)a2~m ~
a~Am = 0 + 2 ~ ,~t~,---~-
(3.3-I01)
For the s a m e b o u n d a r y c o n d i t i o n s a n d with the s a m e p r o c e d u r e as in the p r e c e d i n g c a s e c h a n g e s of m a s s flow r a t e at the o u t p u t c r o s s - s e c t i o n are
2L~ AML(S) =
L - ~ s (w+c)e w-c
L - .-~-----s (~-~) e w+c
AMo(S) (3.3-102)
i
G,(s) ~2 ~ 2
L
~(eW'C ~ A
- ~ s
L
- ~ s
- e w+c
L
L
- ~ S
(W+c)eW-C
- ~ S
- (W-c)e
G2Cs)
W+C
APL(s)
~.
a.a
237
Setting
the
denominator
(which
functions) e q u a l to z e r o in the c a s e the p o l e s of Gl(s) a n d Gz(s) s.
52. ~2 2oL
In
~- 5
is
common
of s u b c r i t i c a l
52- ~
1),j.
to
both
the
transfer
flow ( ~ < ~) g i v e s
k - O, 1. 2. 3
(3.3-103) T h e z e r o s of the transfer f u n c t i o n s setting their n u m e r a t o r s e q u a l to zero:
Gl(s) a n d
G2(s) a r e
obtained
by
z e r o s of Gl(s) ... in the s - p l a n e t h e y lie o n t h e "straight line" 001 = -'~ (3.3-104} z e r o s of G2(s) ... 002 = O. ~02k = *k~
~L
(3.3-105)
Unlike in the p r e c e d i n g c a s e bl), w h e r e the real p a r t of the set of e i g e n v a l u e s (the p o l e s of the transfer f u n c t i o n s ) e q u a l l e d z e r o , i.e. w h e n the t r a n s i e n t p r o c e s s o c c u r r e d without d a m p i n g , in this c a s e o is different from z e r o a n d after a n y d i s t u r b a n c e at the p i p e e n d s , the p e r i o d i c changes AmL(t) a r e d a m p e d . Their o s c i l l a t i o n a m p l i t u d e s ( a l t h o u g h i d e a l i z a t i o n w a s p e r f o r m e d b y t a k i n g APF = ,~PG= 0) c o n v e r g e to z e r o with time. Also c h a r a c t e i i s t i c of the n a t u r a l f r e q u e n c i e s is that b o t h the lowest a n d the h i g h e s t a r e e q u a l l y d a m p e d (the s a m e o). T h e a p p e a r a n c e of the r e a l p a r t that is not e q u a l to z e r o is a v e r y important c h a r a c t e r i s t i c of a d i a b a t i c flow. T h e transfer f u n c t i o n s o b t a i n e d differ e s s e n t i a l l y from t h o s e in the c a s e w h e n the flow p r o c e s s w a s o b s e r v e d a s a s y s t e m with l u m p e d p a r a m e t e r s . If R = 0 is s u b s t i t u t e d into the t r a n s f e r f u n c t i o n s s h o w n b y E q u a t i o n (2.2-3.40) ( w h i c h c o r r e s p o n d s to the loss. n e g l e c t e d h e r e also. r e s u l t i n g from friction, AP F = 0), the c l a s s i c a l c a s e of u n d a m p e d o s c i l l a t i o n is o b t a i n e d , with c h a r a c t e r i s t i c c o m p l e x c o n j u g a t e p a i r s of p o l e s o n t h e i m a g i n a r y axis. O b v i o u s l y , t h e d i f f e r e n c e s a r e of a n e s s e n t i a l n a t u r e . For p r a c t i c a l p u r p o s e s , h o w e v e r , it c a n b e s h o w n that if c e r t a i n a s s u m p t i o n s a r e m a d e the m o d e l s h o w n b y Equation (3.3-102) c a n b e t r a n s l a t e d into forms that h a v e their e q u i v a l e n t in p r o c e s s e s with l u m p e d p a r a m e t e r s . In the u s u a l p r o c e s s e s of fluid flow (which is e s p e c i a l l y true of liquids) t h e d i f f e r e n c e b e t w e e n the v e l o c i t y of flow w a n d the s p e e d of s o u n d p r o p a g a t i o n c a r e v e r y g r e a t , s o that w in E q u a t i o n (3.3-102] a n d (3.3-103) c a n b e n e g l e c t e d . With this a s s u m p t i o n just d e r i v e d PDE. transfer functions, p o l e s a n d z e r o s a r e r e d u c e d to the results
238
CHAP. 3 D I S T R I B U T E D
in c a s e
PROCESSES
bl). s o that for w = 0 the formulas (3.3-91) - (3.3-94) will satisfy
h e r e also. a n d s o will the matrix transfer function (3.3-90). e x p r e s s i o n s for h y p e r b o l i c f u n c t i o n s from (3.3-99) a r e e x p a n d e d s e r i e s a n d all the terms of third a n d h i g h e r o r d e r n e g l e c t e d , the c o e f f i c i e n t s L = TL ' -~-
~~
= Z.
IC = TL z,
If the into a
introducing
TL = C. ~
y i e l d s the following first, r o u g h e s t a p p r o x i m a t i o n of the matrix transfer functions
I Po(s)1= AML(S)
2
(3.3-106)
-Cs
1
12C-~-s2+l
2IC s2,I
"
[ MoCs)J
A form h a s b e e n o b t a i n e d that c o r r e s p o n d s c o m p l e t e l y to the form of matrix transfer f u n c t i o n s s h o w n b y E q u a t i o n (2.2-3.40). T h e d i f f e r e n c e is that w h e n flow is o b s e r v e d as a p r o c e s s with l u m p e d p a r a m e t e r s (Section 2.2-3). the c o n s t a n t b e s i d e s 2 is t w i c e as l a r g e a s the c o n s t a n t o b t a i n e d h e r e with the first a p p r o x i m a t i o n of the h y p e r b o l i c functions. H o w e v e r , it must b e b o r n in m i n d that the similarity b e t w e e n Equations (2.2-3.40) a n d (3.3-1Ob) h a s b e e n o b t a i n e d after n e g l e c t i n g flow v e l o c i t y w. From this a s p e c t , the m o d e l s h o w n b y (2.2-3.40) is limited to p r o c e s s e s with small flow v e l o c i t i e s . If the flow v e l o c i t i e s w a r e l a r g e a n d of a similar o r d e r of m a g n i t u d e as the s p e e d of s o u n d c, the initial e q u a t i o n (in the c a s e of l u m p e d p a r a m e t e r s a l s o ) must b e E q u a t i o n (3.3-102) from w h i c h s i m p l e r forms, or f i n i t e - d i m e n s i o n a l m o d e l s , a r e o b t a i n e d by expanding the e x p o n e n t i a l terms into a s e r i e s a n d n e g l e c t i n g the h i g h e r - o r d e r terms in that s e r i e s . We c a n u s e a p r o c e d u r e a n a l o g o u s to the o n e u s e d to o b t a i n Equation (3.3-102) to g e t a r e l a t i o n b e t w e e n p r e s s u r e c h a n g e s in the c r o s s - s e c t i o n z ; 0 a n d c h a n g e s o n the b o u n d a r i e s
s~c.
"~.'~
230
L
APoCs) =
2~
(~÷~) e w-c
APL(S) +
L
( ~ - ~ ) e w÷c
r
p
G~s)
(3.3-1o7)
L
L
- '=~----S
7.2 4-
- -
A
e
-
e
~
S
W+C
L ( W ÷ c ) e W-C
i
-
W-C
I. - (w-c)e
AMo(S)
w÷c
GiCs)
T h e p o l e s a n d z e r o s of the transfer f u n c t i o n Gs(s) a n d G4(s) a r e e q u a l to the p o l e s a n d z e r o s d e t e r m i n e d with the h e l p of (3.3-103) - (3.3-105). Like b e f o r e , h e r e a l s o E q u a t i o n (3.3-10T) will b e t r a n s f o r m e d in the b o u n d a r y c a s e . w h e n w c o n v e r g e s to zero. into E q u a t i o n (3.3-98) o b t a i n e d in the c a s e w h e n the fluid w a s at rest. bl). b3) Fluid flow a n d h e a t i n g S t u d y i n g the d y n a m i c p r o p e r t i e s in this most c o m p l i c a t e d c a s e , w h e n the fluid flows a n d is h e a t e d at the s a m e time b y a h e a t flow of d e n s i t y a l o n g the p i p e (i.e. w h e n t h e r e is m ,, O, ~,, O). is m a d e m u c h m o r e difficult b y the fact that the initial PDE (3.3-81) is n o w a n e q u a t i o n with s p a c e - v a r i a b l e c o e f f i c i e n t s : K3(z). fl(z), f2(z) a n d fs(z). In p r a c t i c e it is no longer p o s s i b l e to o b t a i n a solution in a c l o s e d form, i.e. in the form of a n a l y t i c a l e x p r e s s i o n s , w h i c h w a s e a s y in the p r e c e d i n g c a s e s bl) a n d b2). T h e c o e f f i c i e n t s a r e s o c o m p l i c a t e d that it is n e c e s s a r y to u s e a digital c o m p u t e r a n d try to r e a c h a s a t i s f a c t o r y solution u s i n g the r e l e v a n t a p p r o x i m a t i o n s . T h e r e f o r e . we will not e n t e r further into the a n a l y s i s of |his g e n e r a l c a s e h e r e . T h e following n u m e r i c a l e x a m p l e , h o w e v e r , will s h o w the results o b t a i n e d for a s p e c i a l c a s e . the a n a l y s i s of the d y n a m i c p r o p e r t i e s Of a s t e a m g e n e r a t o r e v a p o r a t o r .
N e v e r t h e l e s s . b e f o r e w e b e g i n to p r e s e n t the following e x a m p l e , this remark must b e m a d e to m a k e the a n a l y s i s c o m p l e t e . In the last e x a m p l e we e x a m i n e d o n l y the d y n a m i c s of fast p r o c e s s e s of p r e s s u r e a n d flow rate d i s t u r b a n c e p r o p a g a t i o n a n d c o m p l e t e l y left out a n a n a l y s i s of the d y n a m i c s of slow p r o c e s s e s of e n t r o p y d i s t u r b a n c e p r o p a g a t i o n , w h i c h was g i v e n b y the first-order h y p e r b o l i c PDE (3.3-72). This t y p e of e q u a t i o n
240
CHAP. 3 DISTRIBUTED
PROCESSES
was a n a l y z e d in S e c t i o n 3.1, a n d s i n c e our p u r p o s e h e r e was to e x a m i n e p e r i o d i c p r o c e s s e s , we d i d not p a y a n y attention to slow p r o c e s s e s . It is necessary h o w e v e r , to p o i n t out that e v e r y d i s t u r b a n c e includes s i m u l t a n e o u s c h a n g e s of mass flow rate a n d e n t r o p y a n d p~assure, so this s e p a r a t i o n of fast p e r i o d i c p r o c e s s e s from slower c o n v e c t i v e p r o c e s s e s of e n t r o p y d i s t u r b a n c e p r o p a g a t i o n m a k e s s e n s e only within the fiamework of the t h e o r e t i c a l a n a l y s i s c a r r i e d out h e r e .
w - - x a m p ] e 4 C a l c u l a t i o n s of d y n a m i c c h a r a c t e r i s t i c s of a n e v a p o r a t o r The e v a p o r a t o r
pipe
of a s t e a m
generator
of l e n g t h
L = 28 m and
d i a m e t e r d = 0.03 m is h e a t e d b y a h e a t flow rate of q = 4688 J/ms d e n s i t y p e r unit length. T h r o u g h the p i p e flows m = 0.4783 k g / s of e v a p o r a t i n g fluid (water-steam) that is s a t u r a t e d a n d at o p e r a t i n g p r e s s u r e 135 b a r at the input c r o s s - s e c t i o n . At the o u t p u t c r o s s - s e c t i o n , for a g i v e n q. it is a mixture with x = 0.25 s t e a m quality. Determine the e i g e n v a l u e s (the z e r o s of the transfer f u n c t i o n d e n o m i n a t o r ) that c h a r a c t e r i z e the d y n a m i c s of c h a n g e in m a s s flow rate. After i n t r o d u c i n g s o m e a p p r o x i m a t i o n s , the solution d e m a n d e d o b t a i n e d n u m e r i c a l l y . This is b e c a u s e
will be
E q u a t i o n (3.3-81) n o w has v a r i a b l e
c o e f f i c i e n t s a n d c a n n o t b e s o l v e d a n a l y t i c a l l y . In the s p e c i f i c c a s e of the e v a p o r a t o r , K3 d e p e n d s o n l y on the s t e a m q u a l i t y x, a n d for a g i v e n x it is c o n s t a n t a l o n g the w h o l e e v a p o r a t o r . T h e e q u a l i t i e s (3.3-79) a n d (3.3-80) s h o w that c o e f f i c i e n t s ft, f~ a n d f3 a r e f u n c t i o n s of c o o r d i n a t e z, but also of s t e a m q u a l i t y x. F i g u r e 3.3-5 s h o w s h o w t h o s e c o e f f i c i e n t s c h a n g e a l o n g the e v a p o r a t o r p i p e d e p e n d i n g on the d e g r e e of e v a p o r a t i o n . (Here we must m e n t i o n that the n u m e r i c a l p r o g r a m u s e d to c a l c u l a t e these c o e f f i c i e n t s c o n t a i n e d all the d e m a n d e d e q u a t i o n s for c a l c u l a t i n g the s t e a d y state (3.3-5Q) - (3.3-66), a n d a l s o the t h e r m o d y n a m i c state functions, for e x a m p l e (3.3-48). (3.3-51), (3.3-52).) It is o b v i o u s that the d e p e n d e n c e with
the
increase
of
the
degree
of t h e s e c o e f f i c i e n t s o n z i n c r e a s e s of
evaporation,
and
that
when
x
c o n v e r g e s to z e r o t h e y o b t a i n the s a m e v a l u e s as the c o e f f i c i e n t s of the PDE (3.3-101) from the preceding example, i.e. Ks = O. fl " ~2 _ ~ , const., f~ - 0 a n d f3 " 2~ = c o n s t . F i g u r e 3.3-5 s h o w s that c h a n g e s in the c o e f f i c i e n t s f~ a l o n g z, d e p e n d i n g on x, a r e not simple a n d
~=c.
~.~
241
for
a suitable description, a second-order function s h o u l d b e u s e d . But with that approximation an analytical solution in a c l o s e d form would not b e p o s s i b l e . From this follows the n a t u r a l c o n c l u s i o n that the solution, after making c e r t a i n a p p r o x i m a t i o n s , s h o u l d b e r e a c h e d n u m e r i c a l l y . T h e r e f o r e , to o b t a i n transfer functions r e l a t i n g o u t p u t flow r a t e with o u t p u t p r e s s u r e a n d input flow r a t e for the g i v e n x = 0.25, i n s t e a d of v a r i a b l e c o e f f i c i e n t s a l o n g z. w e will s e l e c t for c o e f f i c i e n t s fl. f2 a n d f3. the c o n s t a n t s A, C, B a n d u s e t h e m to s o l v e the task d e m a n d e d , i.e. to d e t e r m i n e the e i g e n v a l u e s . T h e s e c o n s t a n t s a r e a l s o the m e a n v a l u e s s h o w n in F i g u r e 3.3-5 b y b r o k e n lines.
fliz).lO ] [m21s 2 ] 0,6
0
,
0,8 ,
k
J
-10
--
g I
L z:Z8k x : O.OZ5 f31zl (mls3
-30 -40 -SO -60
x 05 i
-?0 -80
-90 fz|z).loZ [ m l s 2 ]
0 -5
0(2
K3
Or4
016
I
018
k,
~x:O L z : Z S k x=O.OZS C
x:O,~
-15 -20
x:O,5
0,6. 0.5 0.4. 0.3"
-25
0.2.
-30
o.i
-35 0 -/,-0
Fig.
x=l
S
.i o~z o',4 0',6 o:a ;
3 . 3 - S C h a n g e s of the c o e f f i c i e n t s of PDE (3.3-81) a l o n g the e v a p o r a t o r d e p e n d i n g on the o u t p u t s t e a m q u a l i t y x
T h e i n t r o d u c t i o n of t h o s e c o n s t a n t s turns E q u a t i o n (3.3-81) into a PDE which, after L a p l a c e t r a n s f o r m a t i o n with r e s p e c t to time, b e c o m e s the
242
CHAP.
3 [~IETRIBUTED' P R O C E S S E S
following ODE with r e s p e c t to z in the s - d o m a i n
Ad2AM(z, s) dz z
* (C +
Bs)dAM(z. s) dz + (s~ ÷ K3s)AM(z' s) = 0
(3.3-108)
This e q u a t i o n is s o l v e d in the u s u a l manner, b y substituting /,M(z. s) = ~ e xs. This l e a d s to a c h a r a c t e r i s t i c e q u a t i o n w h o s e roots are no l o n g e r s i m p l e e x p r e s s i o n s , but XI,2 =
-(Bs*C)* v/(BZ-4A)s 2 + (2BC-4AK3)s + C 2" 2A "
(3.3-I09)
With t h e s e ),,,2 t h e r e is
aM(z. s) = C , e X l Z , C2e x2z
(3.3-II0)
As b e f o r e , the g i v e n b o u n d a r y c o n d i t i o n s m a k e it p o s s i b l e to d e t e r m i n e the c o e f f i c i e n t s ~1 a n d C2. w h i c h y i e l d s the following, final e x p r e s s i o n for flow r a t e c h a n g e in the o u t p u t c r o s s - s e c t i o n (z = L)
x,e(Xl*x2)L. ),2e(),l*x 2)L xleXlL - x2e>,2 L
AML(S) =
~Mo(s)
*
P
G,{s)
(3.3-111)
s (eXlL. e),~L) KI2 ~PL(S) X~e),=L . X~e>'2 L
+ q
p
T h e p o l e s of the transfer f u n c t i o n s a r e the e i g e n v a l u e s d e m a n d e d a n d t h e y wiIl b e r e a c h e d n u m e r i c a l I y , as the z e r o s of the transfer function denominator N(s) = ),le x~L- X2eX~L= 0
(3.3-I12)
A s p e c i a l p r o g r a m is n e e d e d for that p u r p o s e that will iteratively. a p p l y i n g N e w t o n ' s m e t h o d for the function of a c o m p l e x v a r i a b l e N(s). d e t e r m i n e a n a r b i t r a r y n u m b e r of e i g e n v a l u e s . F i g u r e 3.3-6 s h o w s those v a l u e s b o t h for the c a s e x = 0.25 a n d for situations w h e n u n h e a t e d s a t u r a t e d w a t e r a n d u n h e a t e d d r y s a t u r a t e d s t e a m flow t h r o u g h the s a m e p i p e (i.e. fluids with x = O a n d x = 1, a n d t h o s e p a i r s of e i g e n v a l u e s a r e r e a l l y the g r a p h i c a l r e p r e s e n t a t i o n of E q u a t i o n (3.3-103)). Also. a s in the
==c.
"~.~
243
qiO
X:O.5+
x=I
m
q=O
%% % % \
I= I
l±~J
I
x=O \\
x= O.ZS
\
%.%
I00
~+\ o
\
\ \
gO
% o
\
\
gO
%
•,
\
o 7O
°[,0o
% I-
\\
~
%
\
\
N
÷\ \
5O \
\
\
o
I.O
o ti
ON theoretical values ) Qdiabatic case(q=O) l J according t0{3,3-103)
",.,~ ",..~. -..
Xout evaporat ion
A 4.n
o.zs-~ O.S !
~z~ o
~r,,..~,
eigenvalues obtained by approximation
30
"~,,
o ZO
m o
I0 o I
-0.¢
i
J
-0o3
!
i
-0,2
o
,
i
-0,1
F i g . 3.3-6 P r e s e n t a t i o n of e i g e n v a l u e s for c a s e s of e v a p o r a t i o n (Xout = 0.25, 0.5, 1) a n d a d i a b a t i c flow with x = 0 a n d x = 1 c a s e w h e n t h e r e was x = 0.25 in the o u t p u t c r o s s - s e c t i o n , pairs of e i g e n v a l u e s o b t a i n e d b y a p p r o x i m a t i o n for the e v a p o r a t o r p i p e with x = 0.5 a n d x = 1 at the o u t p u t c r o s s - s e c t i o n a r e s h o w n . S e v e r a l c o n c l u s i o n s c a n b e d r a w n from F i g u r e 3.3-6. First. the e i g e n v a l u e s a r e c o m p l e x c o n j u g a t e pairs, w h i c h a g a i n s h o w s that c h a n g e s of mass flow rate a r e p e r i o d i c . (It must b e r e m a r k e d that the o r d i n a t e s h o w s b o t h the p o s i t i v e a n d the n e g a t i v e i m a g i n a r y axis. w h i c h results in the c o m p l e x - c o n j u g a t e pair b e i n g
g i v e n b y o n e p o i n t in that s-plane.) S e c o n d , the h i g h e r the
244
CHAP. 3 DISTRIBUTED PROCEfiEttS
d e g r e e of e v a p o r a t i o n the m o r e l e f t w a r d s the w h o l e s p e c t r u m m o v e s , i.e. the g r e a t e r the d a m p i n g or real p a r t o of the e i g e n v a l u e s . Third, b o t h the l o w e s t a n d the h i g h e s t f r e q u e n c i e s a r e e q u a l l y d a m p e d b e c a u s e their o a r e the s a m e a n d all the e i g e n v a l u e s a r e o n the s a m e straight line. (All e x c e p t the first e i g e n v a l u e in u n a d i a b a t i c c a s e s of e v a p o r a t i o n , w h i c h is d i s p l a c e d further left. That d i s p l a c e m e n t is c a u s e d b y the a p p r o x i m a t i o n that w a s just c a r r i e d out.)
This e n d s the p r e s e n t a t i o n of d y n a m i c p r o c e s s e s with d i s t r i b u t e d p a r a m e t e r s , a n d o n c e a g a i n w e must e m p h a s i z e the b a s i c c h a r a c t e r i s t i c s that h a v e a l w a y s b e e n p r e s e n t in this a n a l y s i s . T h e d y n a m i c s is d e s c r i b e d b y PDE a n d thus the m o d e l is of infinite o r d e r . T h o s e e q u a t i o n s c a n b e s o l v e d a n a l y t i c a l l y in t h e c l o s e d form o n l y for c a s e s of v e r y s i m p l e g e o m e t r y a n d if m a n y a s s u m p t i o n s a r e m a d e , w h i c h m e a n a n i d e a l i z a t i o n of the p r o b l e m . In most c a s e s , to s o l v e the d y n a m i c p r o b l e m n u m e r i c a l methods and computer processing are used, which usually gives s a t i s f a c t o r y solutions. T h e b a s i s of t h o s e m e t h o d s a r e v a r i o u s p r o c e d u r e s of d i s c r e t i z a t i o n with r e s p e c t to the time a n d the s p a t i a l v a r i a b l e . T h e r e is g r e a t similarity in the d y n a m i c p i o p e r t i e s of p r o c e s s e s with lumped and distributed parameters. When the s p a t i a l variable is d i s c r e t i z e d the p r o p e r t y of d i s t r i b u t i o n is lost a n d s u c h p r o c e s s e s b e c o m e p r o c e s s e s with l u m p e d c h a r a c t e r i s t i c s . In t h e m a t h e m a t i c a l s e n s e this m e a n s the o r d e r of the m o d e l h a s b e e n r e d u c e d , m a k i n g it finite, a n d n o w the m o d e l d e s c r i b e s the b a s i c ( a n d in the d y n a m i c s e n s e lhe s l o w e s t ) d y n a m i c p r o p e r t i e s . T h e l o w e r the d e g r e e of r e d u c t i o n , i.e. the h i g h e r the o r d e r of the m o d e l thus o b t a i n e d , the b e t t e r i n c l u d e d the h i g h e r e i g e n v a l u e s a r e (smaller time c o n s t a n t s or g r e a t e r n a t u r a l f r e q u e n c i e s ) . T h e r e s p o n s e s s h o w n b y a m o d e l o b t a i n e d b y r e d u c t i o n d e v i a t e from the r e s p o n s e s of the real d i s t r i b u t e d p r o c e s s in the first p a r t s of the t r a n s i e n t p r o c e s s (i.e. for smaller t), w h i c h is the result of the fact that the faster m o d e s , in the d y n a m i c s e n s e , of the total r e s p o n s e w e r e not i n c l u d e d in the m o d e l . T h e lower the d e g r e e of r e d u c t i o n , the smaller that " d y n a m i c m i s t a k e " is.
APPENDIX
A
DIFFERENTIAL AND
EQUATIONS STATE SPACE
O n e of the f u n d a m e n t a l a n d most n o t i c e a b l e c h a r a c t e r i s t i c s of all the processes
that w e r e a n a l y z e d
o n the p a g e s
of this b o o k
was
that
the
m a t h e m a t i c a l tools u s e d to d e s c r i b e their u n s t e a d y states w e r e differential equations. These equations were described b y different a d j e c t i v e s d e p e n d i n g o n their s p e c i f i c p r o p e r t i e s o r d i n a r y , partial, n o n l i n e a r or linear. Their c o e f f i c i e n t s w e r e c o n s t a n t or v a r i a b l e . We must r e m e m b e r that at the e n d we tried ( a n d in most c a s e s
succeeded)
to r e d u c e
all t h e s e
different notations to o n e or s e v e r a l o r d i n a r y linear differential e q u a t i o n s (OLDE). This e q u a t i o n or s y s t e m of OLDE we tried to translate into the form of matrix s t a t e - s p a c e differential e q u a t i o n s . We will these
not
discuss
mathematical
the
tools.
accuracy, Many
advantages
pages
could
or d i s a d v a n t a g e s
be
written
about
of
their
s u c c e s s f u l u s e a n d the w i d e field in w h i c h t h e s e linear m o d e l s c a n b e a p p l i e d . Here it is. p e r h a p s , e n o u g h to s a y o n l y that linear m o d e l s a r e u s e d in v e r y m a n y t e c h n i c a l d i s c i p l i n e s a n d that t h e y h a v e c o n t r i b u t e d to the p r a c t i c a l a n d t h e o r e t i c a l solution of m a n y p r o b l e m s in t h e s e fields. M o d e r n c o n t r o l t h e o r y is c e r t a i n l y o n e of the l e a d i n g d i s c i p l i n e s in the e x t e n s i v e u s e of the m e n t i o n e d m o d e l s , but useful a n d important results have
also
been
obtained
from their
use
in the
theories
of vibrations.
a c o u s t i c s , thermal a n d flow s y s t e m s a n d in d e s i g n i n g e l e c t r i c a l circuits. In the following lines, without the u s u a l m a t h e m a t i c a l strictness, we will s h o w s o m e b a s i c p r o p e r t i e s of differential e q u a t i o n s , c o n n e c t i o n s b e t w e e n o n e n - t h - o r d e r DE a n d a s y s t e m of first-order ODE. a n d h o w to p r e s e n t s u c h a s y s t e m in the form of ( m a n y times a l r e a d y m e n t i o n e d ) matrix s t a t e - s p a c e e q u a t i o n s . E v e r y d a y t e c h n i c a l p r a c t i c e has s h o w n it to b e useful
to translate
every
existing
model
of d y n a m i c
processes
into
a
24b
APPENDIX
s y s t e m of f i r s t - o r d e r DE. ( O n e of t h e m o r e i m p o r t a n t r e a s o n s is
that
mathematical
programs
specially
L e t it b e
libraries
developed
possible
of
modern
computers
w h y t h i s is s o contain
system
for s o l v i n g s y s t e m s of f i r s t - o r d e r DE.)
to w r i t e t h e n o n l i n e a r
(or l i n e a r ) n - t h - o r d e r
DE a s
follows
(A.l)
X (n) = f(t, X. X'. X '°, .... X (n'l))
(In
the
above
function x and
equation
we
its d e r i v a t i v e s
did
not
specially
emphasize
a r e f u n c t i o n s of t h e i n d e p e n d e n t
that
the
variable
t.
i,e. w e d i d n o t w r i t e x(t). x'(t) ... a n d s o o n . ) T h e f o l l o w i n g s u b s t i t u t i o n , i.e. t h e i n t r o d u c t i o n of n e w f u n c t i o n s X
-- X I,
X'
=
X"
X I' -- X 2.
-- X 2' =
X 3.
x (n-l) - Xn_ I - x n x (n)
'
=
= f(t. x,. x2
X 1,.i.
Xn)
. . . . .
y i e l d s this s y s t e m o f n f i r s t - e i d e r DE XI' =
X 2.
X2'
X3,
=
(A.2)
|
x n = f(t. x,. x2 . . . . . Xn) The above
system
is o n l y
a special
case
or a form of t h e
following
system x,' = f, (t. x,, xz ..... x.).
(A.3)
x2' = f~ (t. x,. x2 ..... Xn). •
.
.
,
.
.
,
.
J
Xn = fn (t. X,. xz . . . . . Xn). w h i c h is u s u a l l y that the number
called
a normal
of u n k n o w n s
e n d of this a p p e n d i x ,
equals
s y s t e m of DE. w h e r e the number of equations.
it will b e s h o w n
it is s u p p o s e d Later. at the
t h a t n o t a t i o n in t h e form of d y n a m i c
r'~[I:'I=C'R~'NT|AL
state-space
':'QT.JATIQN~ A N ~
equations
is in
C:TAT~ ~ P A ~
fact
a
matrix
?47
~'
:ep:esentation
of
the
normal
s y s t e m of DE.
The solution
of t h e a b o v e
s y s t e m is a s e t of n f u n c t i o n s x~Ct), x2(t).
x3(t) ..... Xn(t) that s a t i s f y all t h e a b o v e
equations.
A special (or particular) s o l u t i o n is t h e s e t of s u c h f u n c t i o n s that a l s o s a t i s f y all t h e initial c o n d i t i o n s Xj(t=to)
= xl(to),
xz(t=to)
= x2(to) .....
(A.4)
xn(t=t o) = Xn(to),
w h e r e x,(t0) ..... xn(t0) a r e g i v e n n u m b e r s . Here
this will n o t b e
carried
out. b u t
the existence
and
uniqueness
t h e o r e m for t h e s o l u t i o n of a n o r m a l s y s t e m of DE w o u l d not b e difficult to p r o v e . We c o n s i d e r it m o r e u s e f u l to s h o w t h e o p p o s i t e of t h e a b o v e
derivation, i.e. that the n o r m a l s y s t e m of n first-order DE is e q u i v a l e n t o n e n - t h - o z d e r DE. With this in mind, w e will d i f f e r e n t i a t e with r e s p e c t the first e q u a t i o n from t h e s y s t e m (A.3) d2xt
all = at
Replacing
to to t
all dr1 all dx2 af 1 d x n * ~xl dt * -ax~ - - - dt ÷ "'" + -~Xn --- dt the
expressions
dxi/dt
= x'i b y
their
equivalent
functions
fi (t. xl. x2 ..... Xn) y i e l d s d2xl
=
c~fl all ~fl f ~fl fn at + ~xltl ÷-'ax'22 + --. * ~xn
,
dZxt dt ~ - F'2(t. xt. x2 ..... Xn)
(A.5)
D i f f e r e n t i a t i n g t h e last e q u a t i o n a n d r e f e r r i n g to (A.3) y i e l d s d3xl
aF2 + a F 2 . "
d3xl
dt 3
,~t
~F2fn + -.- + <)Xn
= F3(t. xv, x2 ..... Xn)
If this p r o c e d u r e d4xt
~F2-
~xl ~1 + ~tr2
= F4(t,
...............
(A.b)
is c o n t i n u e d , w e o b t a i n
x l . x2 ..... Xn)
(A.7)
248
APPENDIX
dIlx1
= Fn(t.
Xlo X 2 . . . . .
X n)
•
Now it is p o s s i b l e to write the following s y s t e m of e q u a t i o n s
dx, = f, (t. x,. x~. dt
xn) ....
d~x,
d t z. = F~(t, x , . x z . . . . . Xn)
CA.8) dnx~ dt n
= F n ( t , Xl. x2 . . . . . x n )
From the a b o v e s y s t e m (A.8) we c a n
express
(n-l) u n k n o w n functions
xz, x 3 ..... Xn with the h e l p of the function x~ a n d its d e r i v a t i v e s of u p to, a n d i n c l u d i n g , the (n-l)st order. If t h o s e functions a r e s u b s t i t u t e d into the last e q u a t i o n of the a b o v e system, w e finally o b t a i n dnxt = S(t. dt-'-ff-
x , . . . .dx~ . ~ .
dn"x1~ dt n
(A.9)
-
To p r o v e the validity of what h a s just b e e n d e r i v e d w e will u s e the g e n e r a l solution of the last e q u a t i o n , w h i c h i n c l u d e s n u n k n o w n c o n s t a n t s x l = g l (t. C1. C2 . . . . .
Cn)
(A.IO)
Substituting xl a n d all its d e r i v a t i v e s into the e x p r e s s i o n s for x2. x3 ..... Xn. o b t a i n e d from the following solution x l = g l ( t . CI. C2 . . . . .
first n-] e q u a t i o n s
of the s y s t e m (A.8), y i e l d s
the
Cn).
X2 = g2(t. C1. C~ . . . . . Cn).
(A.ll)
Xn = gn(t, Ct. C2 ..... Cn). This e n d s our proof, s i n c e it is not difficult to s h o w that the s y s t e m (All) in fact r e p r e s e n t s the solution of the initial normal s y s t e m of first-order DE (A.3). The a b o v e lines h a v e s h o w n the r e l a t i o n s h i p b e t w e e n o n e n-th-order DE a n d a s y s t e m of first-order DE. w h i c h is often u s e d in p r a c t i c e to s o l v e a d y n a m i c problem. In that t e c h n i c a l p r a c t i c e , h o w e v e r , it has b e e n s h o w n
~|FFE~NTIAL
~QUATION~
A N r'~ S T A T E
SPA~E
2~(~
useful to d i s t i n g u i s h b e t w e e n the v a r i a b l e s , or functions, a n d
to d e n o t e
them b y different letters a p p r o p r i a t e to their p h y s i c a l m e a n i n g . In all the e x a m p l e s w e s h o w e d , w h o s e d y n a m i c s was d e s c r i b e d b y o n e or s e v e r a l first-order ODE, it was a l w a y s p o s s i b l e to d i s t i n g u i s h b e t w e e n t h r e e b a s i c v a r i a b l e s (functions), - input v a r i a b l e s w e r e d e n o t e d u(t) - state v a r i a b l e s w e r e d e n o t e d x(t) - o u t p u t v a r i a b l e s w e r e d e n o t e d y(t) F i g u r e A.1 s h o w s g r a p h i c a l l y the relations a m o n g t h e s e v a r i a b l e s . In m a n y of the e x a m p l e s we p o i n t e d to the relativity of the s e l e c t i o n of state a n d o u t p u t v a r i a b l e s , while the input functions w e r e u s u a l l y k n o w n and completely determined. N e v e r t h e l e s s . state v a r i a b l e s a r e not arbitrary v a r i a b l e s in the p r o c e s s a n d t h e y must b e s e l e c t e d to c o r r e s p o n d with their following definition: State v a r i a b l e s a r e p r o c e s s v a r i a b l e s w h o s e k n o w l e d g e at the moment to. k n o w i n g the laws g o v e r n i n g e v e r y t > to, e n a b l e s for e v e r y t > to.
the b e h a v i o r
of the
input
variables
for
the c o m p l e t e d e t e r m i n a t i o n of the o u t p u t v a r i a b l e s
State v a r i a b l e s a r e time variant (they a r e functions) a n d the set of their m o m e n t a r y v a l u e s s h o w s the s t a t e of the p r o c e s s (system. object). If it is n e c e s s a r y to know n state v a r i a b l e s to d e s c r i b e a p r o c e s s , t h e n the p r o c e s s is s a i d to b e of n-th o r d e r a n d g i v e n b y a s y s t e m of n first-order DE.
its m a t h e m a t i c a l
description
is
Where. then. d o e s the f r e e d o m of c h o i c e for state v a r i a b l e s lie? This is the e a s i e s t to s h o w o n a n e x a m p l e , a n d the simpler the e x a m p l e , the c l e a r e r the proof. Let us e x a m i n e the c a s e of the liquid s t o r a g e tank ( S e c t i o n 2.1, E x a m p l e 2). for w h i c h E q u a t i o n s (2.1-1.27] a n d (2.1-1.28) a l r e a d y s h o w e d that it is p o s s i b l e to s e l e c t o u t p u t v a r i a b l e s freely d e p e n d i n g needs.
In that e x a m p l e
we s e l e c t e d c h a n g e s
in liquid l e v e l
aH
as
on the
state variable, s i n c e c h a n g e s in AH really d i d u n i q u e l y d e t e r m i n e ( t h r o u g h the e q u a t i o n s w e m e n t i o n e d ) c h a n g e s in the o u t p u t v a r i a b l e s for
250
APPENDIX
a r b i t r a r y input v a r i a b l e s Aml[t) a n d AAo(t). H o w e v e r . the s t a t e of that p r o c e s s , i.e. the o p e r a t i n g r e g i m e , is. in fact, d e t e r m i n e d b y the a m o u n t of liquid that is s t o r e d in the t a n k at the m o m e n t to. a n d the liquid l e v e l is o n l y the m e a s u r e of that a m o u n t . Thus the i d e a that c h a n g e s in s t o r e d mass aM could also be selected as the s t a t e v a r i a b l e is not i n c o m p r e h e n s i b l e , e s p e c i a l l y a s for this t a n k of c o n s t a n t s u r f a c e a r e a t h e r e is input v a r i a b l e s [inputs. e x c i t a t i o n s . disturbances, noises)
ul(t } _ uz{t } _
_ r
ZI
um{t} "
-I
u
Fig.
aM
state variables
output variables [outputs, r e s p o n s e s . solutions)
xllt | xz(t )
.
xnlt}
--
:
X
yl(t) YzIt) Yr(t)
y
A . I Illustration of t h e definition of input, s t a t e a n d o u t p u t v a r i a b l e s
= ApAH
(A.12)
If a H is e x p r e s s e d and becomes
[ MJo
from(A.12), the s t a t e e q u a t i o n
taM]. El -KA] Atoll AAoj.
changes
appearance
[A.la)
And. of c o u r s e , if liquid l e v e l is a c o n t r o l l e d v a r i a b l e , the o u t p u t e q u a t i o n n o w a l s o c h a n g e s a p p e a r a n c e a n d i n s t e a d of (2.1-1.27] b e c o m e s
E H]o
e0 0] [ ~m'l AoJ
"
(A.i4)
It is v e r y i m p o r t a n t to n o t e that the o r d e r of the s y s t e m is u n i q u e l y d e t e r m i n e d , a n d w h i c h e v e r v a r i a b l e is s e l e c t e d , the o r d e r of the s y s t e m c a n n o t a n d must not c h a n g e .
DIFF~'R='HT/AL.
I~OUATI(D2q~
It must a l s o b e
AND
mentioned
~TAT
=" ~ P A C
='
that the s t a t e v a r i a b l e s
2~I
need
not h a v e
a
c l e a r l o g i c a l or p h y s i c a l m e a n i n g , a l t h o u g h it is in p r a c t i c e useful to c h o s e t h e m So that t h e y d o h a v e a c l e a r m e a n i n g or. e v e n , so that t h e y a r e a l w a y s m e a s u r a b l e a n d a c c e s s i b l e v a r i a b l e s in the p r o c e s s . ( T h e d e m a n d for t h e m to b e m e a s u r a b l e a n d a c c e s s i b l e c a n often not b e fulfilled.) Thus, for e x a m p l e , m u l t i p l y i n g the w h o l e s t a t e e q u a t i o n (2.1-1.24) b y n a n d i n t r o d u c i n g a n e w s t a t e v a r i a b l e AV ; n a i l ( w h i c h e q u a l s a n n-told c h a n g e in liquid l e v e l ) y i e l d s ['~V]'=
[-+] [ A V ] . [Cn
Aml] K~n] EAAoJ.
(A.15)
H e r e w e c a n c o n j e c t u r e a b o u t the p h y s i c a l m e a n i n g of the s t a t e v a r i a b l e AV, but it is c l e a r that the s e l e c t i o n is n e i t h e r l o g i c a l nor natural. N e v e r t h e l e s s . s o l v i n g (A.15) w o u l d g i v e c o m p l e t e l y v a l i d s o l u t i o n s for aV, with w h o s e h e l p it w o u l d b e p o s s i b l e to d e t e r m i n e the other v a r i a b l e s of interest a s well. It is a l s o a d v a n t a g e o u s to t r e a t the c o n c e p t of s t a t e s p a c e . This term h a s c o m e into p r o c e s s d y n a m i c s a n a l y s i s from m a t h e m a t i c s a n d it h a s a c l e a r p h y s i c a l m e a n i n g . T u r n i n g a g a i n to the same" e x a m p l e of the s t o r a g e tank w h o s e m o d e l is g i v e n b y the s t a t e E q u a t i o n (2.1-1.24) a n d the o u t p u t E q u a t i o n (2.1-1.27), w e c a n r e p e a t that it is a first-order s y s t e m s i n c e it c o n s i s t s of o n l y o n e m a s s s t o r a g e tank. T h e r e f o r e , a s time p a s s e s o n l y o n e v a r i a b l e c h a n g e s d u e to i n p u t v a r i a b l e c h a n g e , a n d that is the liquid l e v e l ~H. ( O b v i o u s l y , o u t p u t flow ~mo a l s o c h a n g e s , but that c h a n g e d e p e n d s b o t h o n the i n p u t v a r i a b l e s u(t) a n d AH. w h i c h m e a n s on that s t a t e v a r i a b I e , w h i l e c h a n g e in AH d e p e n d s o n l y on itself a n d on input v a r i a b l e s . ) T h e p r o c e s s is o n e - d i m e n s i o n a l in the s e n s e of t h e r e b e i n g only o n e d e p e n d e n t variable. C h a n g e s in that s t a t e v a r i a b l e c a n b e s h o w n in two w a y s : in a t - AH d i a g r a m a n d in the o n e - d i m e n s i o n a l t,H s p a c e in w h i c h time t is g i v e n p a r a m e t r i c a l l y . T h a t s p a c e is c a l l e d t h e
space
of state changes
or s t a t e s p a c e a n d is s h o w n in F i g u r e A.2b for the c a s e w h e n at to = 0 w a t e r flow into the t a n k is d i s c o n t i n u e d a n d w a t e r l e v e l in the t a n k e q u a l s I m, w h i c h results in free outflow. ( S i n c e this is a o n e - d i m e n s i o n a l s y s t e m , w e c o u l d a l s o h a v e c a l l e d it a line of state.)
252
APPENDIX
m
X(T) XlO)/t=O
0.5
T
t
a)
Fig
z~H=x I D
b)
A . 2 G r a p h i c a l p r e s e n t a t i o n of state c h a n g e s in a o n e - d i m e n s i o n a l p r o b l e m - line of state
If the v a r i a b l e is s t o r e d in two tanks the p r o c e s s is of s e c o n d
order
a n d there a r e two state v a r i a b l e s . In S e c t i o n 2.2-3 ( E x a m p l e l. F i g u r e 2.2-3.2) the t-x d i a g r a m s h o w s how t h o s e two state v a r i a b l e s c h a n g e (xl = APz a n d xz = am1) a n d F i g u r e A.3 s h o w s the s a m e in the s t a t e s p a c e (plane), i.e. in a z~P2 - Z~ml d i a g r a m w h e r e time t is the p a r a m e t e r . The trajectory of state C h a n g e is t y p i c a l for p e r i o d i c d a m p e d p r o c e s s e s in w h i c h the state v a r i a b l e s , for a g i v e n input Am2 = 1, c o n v e r g e to a n e w s t e a d y state (0,1).
~ Am| = 3 = X2
.&
t=1"7t
Fig.
A . 3 Graphical presentation of state c h a n g e s in a two-dimensional problem - plane of state, p h a s e plane
~II~I~I~REMTIAL
~QUATI(::~Ig~
AIg~
~TJkTE
EPJk~E
25:]
In F i g u r e A.3. as i n d e e d in all the other p r e s e n t a t i o n s of state s p a c e , it must b e o b s e r v e d that o n l y o n e trajectory p a s s e s t h r o u g h o n e point of the state s p a c e . This is a d i r e c t c o n s e q u e n c e of the definition of the s t a t e variables
themselves,
according
uj(t) (j = 1. 2..... m), t h e y d e t e r m i n e
to
which,
for
completely and
given
input
uniquely
the
values future
b e h a v i o r of the s y s t e m for e v e r y t > to. ( E x c e p t i o n s a r e s o - c a l l e d s i n g u l a r points in state s p a c e . ) Finally. in the c a s e of three s t o r a g e
tanks in series
the state
space
w o u l d b e t h r e e - d i m e n s i o n a l a n d the trajectory c o u l d b e s h o w n as a c u r v e in three-dimensional space with coordinates x, - x2 - x3 (AH, - AH2 - AH3, in the e x a m p l e with t h r e e s t o r a g e tanks - S e c t i o n 2.5). Therefore, w e c a n s a y that the d i m e n s i o n of the state s p a c e e q u a l s the n u m b e r of state v a r i a b l e s n e c e s s a r y to d e s c r i b e the d y n a m i c s of the p r o c e s s u n d e r o b s e r v a t i o n . If that d i m e n s i o n is n < 3 the s t a t e s p a c e c a n also b e r e p r e s e n t e d ( g r a p h i c a l l y ) , but for m a t h e m a t i c a l p r o c e s s i n g the impossibility of g r a p h i c a l r e p r e s e n t a t i o n for c a s e s of n > 3 is of no e s s e n t i a l m e a n i n g . (In m e c h a n i c s the s t a t e trajectory in s e c o n d - o r d e r s y s t e m s is v e r y f r e q u e n t l y r e p r e s e n t e d , a n d then the s t a t e s p a c e ( p l a n e ) is called a phase plane.) It must also b e e m p h a s i z e d that in a l a r g e n u m b e r of c a s e s or d i m e n s i o n of the p r o c e s s m o d e l , i.e. the n u m b e r
the o r d e r
of first-order DE that
d e s c r i b e its d y n a m i c s , is not firmly d e t e r m i n e d but g r e a t l y d e p e n d s o n the final p u r p o s e of the m o d e l i n g . Or. in o t h e r words, almost e v e r y p r o c e s s can
be described
b y m o d e l s of different o r d e r - f r o m t h o s e that d e s c r i b e
o n l y s t e a d y states (O-th o r d e r
models)
to m o d e l s in the form of partial
differential e q u a t i o n s (infinite-order models). Therefore, it must a l w a y s b e r e m e m b e r e d that in most c a s e s a d y n a m i c s p r o c e s s m o d e l is in fact a h i e r a r c h y of m o d e l s d e s c r i b i n g that p r o c e s s - from v e r y simple o n e s to t h o s e of g r e a t c o m p l e x i t y . The f i n a l g o a l of m o d e l i n g , the e q u i p m e n t that is to b e u s e d , the m a t h e m a t i c a l m e t h o d s , p r o c e d u r e s a n d p r o g r a m s that h a v e b e e n d e v e l o p e d , e c o n o m i c c o n s i d e r a t i o n s , time a n d p e r s o n n e l - all t h e s e are factors that d e t e r m i n e the c o m p l e x i t y , a c c u r a c y a n d p r i c e of the model derived. Finally. b e f o r e p r e s e n t i n g the most g e n e r a l s y s t e m of first-order DE in the form of matrix DE of state s p a c e , it will b e useful to g i v e a c l e a r a n d simple e x a m p l e of a m o d e l u s i n g the n o t a t i o n u, x a n d y for the v a r i a b l e s
254
APPENDIX
of input, s t a t e a n d output. T h a t transition to the g e n e r a l a n d a b s t r a c t form of notation will b e s h o w n o n E x a m p l e 1 from S e c t i o n 2.1-2 o n g a s s t o r a g e tanks, w h o s e m o d e l is g i v e n b y E q u a t i o n s (2.1-2.10) a n d (2.1-29). We i n t r o d u c e the notation: xl(t)
= P(t)
... the s t a t e v a r i a b l e is the p r e s s u r e in t h e tank
u,Ct) = miCt) u2Ct) = PoCt)
... input v a r i a b l e s
u3(t) = Ao(t)
y,(t) = P(t)
... o u t p u t v a r i a b l e s
Ys(t) = mo(t)
In further e q u a t i o n s w e will no l o n g e r e m p h a s i z e that all t h e s e are t i m e - d e p e n d e n t v a r i a b l e s , so the d y n a m i c m o d e l of t h e p r o c e s s of filling a stQrage tank becomes x,' - - k , u 3 ~ u 2 ( x ,
- u2) ÷ u,
Yl = xl ~" 0 • ul + 0 • uz ÷ 0 • u s Y2 = kju3~'u2(x,
- u2) ÷ 0
(A.I6)
• u,
In this s y s t e m the d e p e n d e n c i e s that d o not e x i s t a r e e s p e c i a l l y e m p h a s i z e d ( b y m u l t i p l i c a t i o n with zero). In g e n e r a l , t h e r e f o r e , the a b o v e s y s t e m h a s the following f u n c t i o n a l form x', - f, (x~, ul. u2. u~).
NL
y, = g, (x. u,. u z. u3).
L
Y2 = g2 (x,, u,. u2. u~).
NL
(A.IZ)
T h e o r i g i n a l m o d e l g i v e n b y (A.17) is n o n l i n e a r a n d in p r i n c i p l e further a n a l y s i s of the p r o c e s s c a n t a k e two different d i r e c t i o n s . T h e first is d i r e c t n u m e r i c a l i n t e g r a t i o n o n a c o m p u t e r , w h i c h p r e s e n t s no s p e c i a l difficulties. T h e s e c o n d h a s b e e n s h o w n in S e c t i o n 2.1-2, it i n c l u d e s l i n e a r i z i n g the o r i g i n a l m o d e l , t r a n s l a t i n g the l i n e a r m o d e l into matrix s t a t e ( s p a c e ) e q u a t i o n n o t a t i o n a n d o b t a i n i n g a matrix transfer function w h i c h c l e a r l y s h o w s the c h a r a c t e r of the o u t p u t v a r i a b l e s ' d y n a m i c d e p e n d e n c e on the input v a r i a b l e s . Without r e p e a t i n g the p r o c e d u r e of l i n e a r i z a t i o n , it is useful h e r e to g i v e a m o r e a b s t r a c t form of t h e l i n e a r m o d e l s h o w n b y the s t a t e - s p a c e e q u a t i o n s (2.1-2.26) a n d (2.1-2.27)
]~II:'~'cRCNTIAL
X'I
=OLIATIQN~
allXI
"
"
bllUl
Yl = l • x~ + 0 Y2
= C2,X~
AND
+
b12u2
+
• ul + 0
+ 0
~TAT
~ P A ~
2~
b13u3,
• u2
" U, + b 2 2
c
+ 0
" Us
(A.18)
• u3,
+ b23
' u3.
It must b e n o t i c e d that t h e z e r o s in t h e l i n e a r i z e d m o d e l a p p e a r same places
where there was no dependence
in t h e
in t h e n o n l i n e a r m o d e l .
N o w it is p o s s i b l e to g i v e t h e m o s t g e n e r a l p r e s e n t a t i o n of the d y n a m i c m o d e l of a p r o c e s s w h i c h h a s m i n p u t s , n s t a t e s ( t h e s y s t e m is of n-th order) and r outputs x', -- f, (x, ..... Xn, u, ..... urn). x'2 = f2 (x~ ..... xn, ul ..... Urn). .......
'. . . .
, .....
(A.19)
.
x'n = fn (xl ..... Xn, ul ..... urn).
In t h e g e n e r a l c a s e • t h e f u n c t i o n s fi (i = 1. 2 . . . . . n] a r e n o n l i n e a r . T h e o u t p u t v a r i a b l e s yj (j = 1. 2 . . . . . r) a r e r e l a t e d to t h e s t a t e v a r i a b l e s a n d t h e i n p u t v a r i a b l e s b y static, a l g e b r a i c Yl
-- g l
(Xl.
X 2 .....
X n, Ul.
Ys = g s ( x , . x s . . . . . .
.
.
.
.
.
.
.
.
.
.
.
U 2 .....
.
.
,
.
°
•
form
Urn).
xn, u,. us ..... .
e q u a t i o n s of this g e n e r a l
urn).
.
,
(A.20)
,
Yr = g r (x,. xs ..... x , , u~. us ..... urn). F u n c t i o n s gj (j = 1. 2 . . . . . r) a r e a l s o in g e n e r a l n o n l i n e a r . E q u a t i o n s (A.16) a r e a n e x a m p l e of o n e s u c h n o n l i n e a r s y s t e m in w h i c h t h e n o n l i n e a r i t i e s a r e of t h e s q u a r e root a n d p r o d u c t of f u n c t i o n s x(t) a n d u(t) t y p e . In most of this b o o k w e m a d e efforts to t r a n s f o r m o r i g i n a l l y n o n l i n e a r m o d e l s g i v e n b y [A.19) a n d (A.20] into a l i n e a r form of t h e f o l l o w i n g k i n d X' I
=
allx
x'2
=
a21x
I
4. a l 2 x 1 +
2
a22x2
. . . . . . . . . . . X'n
Y l
=
=
Y 2
=
.
•
•
Y r
=
aniX~
CIIXl C21X! . . . . CrlX!
+
÷
+
... +
...
+
C12X2
+
C22X2 .
+
alnXn
+
4. a 2 n x
bllUl
n
+
.
.
Cr2X2
...
... +
.
. +
.
+
÷
.
annxn
+
CInXn
+
...
+
C2nXn
.
.
.
...
+
CrnXn
.
.
.
I
.t- b 2 2 u .
+
÷
•
•
drlut
d12u2
. +
+
+ ...
blrnUm, +
b2rnUm,
÷
....i.
bnmum,
..•
+
bn2Ll2
+ .
2
...
(A.21)
. . . . . . . . . +
d21u~ .
÷
.
dllUl
•
b12u2
bnlU!
+ .
+
b21u
o . . . . . . . . . . anzx2
+ .
+
+
d22u2 .
.
•
dr2u2
+ .
•••
dlmum, +
. . . . +
...
dzmum, o
+
drmum.
(A.22)
250
APPENDIX
In v e c t o r n o t a t i o n the a b o v e matrix s t a t e - s p a c e e q u a t i o n s X'
=A
X ÷ B U
equations
get
the
well-known
.
form of
(A.23)
Y'CX~'DU Now it is e a s y to s e e that what w a s s h o w n earlier a b o u t the s e l e c t i o n of state v a r i a b l e s not b e i n g u n i q u e is also v a l i d for the most g e n e r a l c a s e g i v e n b y m o d e l (A.23). It is k n o w n that the c o o r d i n a t e s of a point in n - d i m e n s i o n a l s p a c e d e p e n d on the s e l e c t i o n of the b a s e . i.e. on the s y s t e m of mutually p e r p e n d i c u l a r unit v e c t o r s a l o n g w h i c h the c o o r d i n a t e a x e s a r e d i r e c t e d . By c h a n g i n g
the
base
we c a n
m o v e from the
state
v e c t o r X to t h e state v e c t o r :~ u s i n g the simple transformation •. MX. M ,, O .
(A.24)
M is a n n o n s i n g u l a r matrix of o r d e r (nxn). For e v e r y s u c h transformation there is a l s o a n i n v e r s e transformation X = M "~ ~ .
(A.26)
If. thus. the s y s t e m of state e q u a t i o n s (A.23) is v a l i d for the v e c t o r of state v a r i a b l e s X. then for the n e w v e c t o r o f s t a l e v a r i a b | e a r ~ t h e r e will b e
(A.2~) The old a n d the n e w m a t r i c e s are r e l a t e d as follows
A , . M A M-I mM B
~=C
M -I
It
obvious
is
(A.27)
that
there
is
an
unlimited
number
of
different
transformation matrices M a n d it w o u l d b e difficult to e x p e c t the n e w state v a r i a b l e s to retain their o r i g i n a l l y c l e a r p h y s i c a l m e a n i n g after a n
r'~IF~"c'RCNTIAL ~ ' Q U A T I O N ~
ANI~) ~ T A T ~* S P A ~
arbitrary transformation. T h e r e f o r e . if s u c h they
usually
have
a
deliberately
2~T
transformations a r e
selected
matrix
M
which
performed, serves
to
transform matrix A into a form that is s u i t a b l e for the d y n a m i c a n a l y s i s of the p r o c e s s d e s c r i b e d , a n d w h i c h m a k e s it v e r y m u c h e a s i e r . Finally,
we
must
repeat
that
spatially
distributed
m o d e l is a s y s t e m of partial differential e q u a t i o n s
processes,
whose
(PDE), c a n n o t (without
a d d i t i o n a l p r o c e d u r e s of d i s c r e t i z a t i o n ) d i r e c t l y b e d e s c r i b e d b y a m o d e l in the form of matrix s t a t e - s p a c e e q u a t i o n s . This results from the fact that s u c h a s y s t e m of PDE (or just o n e ) d e s c r i b e s c h a n g e s of p r o c e s s v a r i a b l e s in e v e r y p o i n t in s p a c e i n s i d e w h i c h the p r o c e s s o c c u r s , a n d there are, of c o u r s e , infinitely m a n y s u c h points. Therefore. s y s t e m matrix A w o u l d b e of infinite order. Only after spatial d i s c r e t i z a t i o n h a s b e e n p e r f o r m e d c a n a ( r e d u c e d ) m o d e l of the s p a t i a l l y d i s t r i b u t e d p r o c e s s b e o b t a i n e d in the form of a s y s t e m of ODE like the o n e g i v e n b y (A.1Q) a n d (A.20), from w h i c h it is not far to s t a t e - s p a c e form (A.23). It must also b e r e m a r k e d that, a l t h o u g h s u c h e x a m p l e s were not t r e a t e d in this b o o k , the c o e f f i c i e n t s of matrices A, B. C, a n d D as a rule n e e d not b e c o n s t a n t s , but c a n b e functions of time t. S u c h p r o c e s s e s will b e called time variant processes, a n d to d i s t i n g u i s h them from t h e n o t a t i o n u s e d u p to now, time t will a p p e a r e x p l i c i t l y in the functions f~ (i = I. 2 ..... n) a n d gj (j = I, 2 . . . . . r). as was the c a s e in (A.I). A time-variant s y s t e m would, thus. b e c h a r a c t e r i z e d b y the following n o t a t i o n x'i = fl(t, x,, xz ..... Xn, u,, uz ..... urn) (i = 1, n)
(A:8)
To g e t a solution for s u c h s y s t e m s it is n e c e s s a r y , b e s i d e s k n o w i n g the initial conditions xj (i --- I, 2 ..... n) and the input functions uj(t > to)(j = 1, 2 ..... m), to k n o w also the time t w h e n the d i s t u r b a n c e o c c u r s a n d c a l c u l a t i o n b e g i n s , b e c a u s e n o w the law g o v e r n i n g state v a r i a b l e c h a n g e s will n o l o n g e r b e the s a m e at two different moments for the s a m e initial c o n d i t i o n s a n d inputs.
258
APPENDIX
MODEL LINIEARIZATION
L i n e a r i z a t i o n is the r e p l a c e m e n t of n o n l i n e a r m a t h e m a t i c a l e x p r e s s i o n s r e l a t i n g p r o c e s s v a r i a b l e s with linear o n e s . L i n e a r i z a t i o n h a s b e e n u s e d in this b o o k in a l m o s t e v e r y e x a m p l e , w h i c h s h o w s that i n e v e r y d a y technical practice originally linear models are in most cases the exception a n d that in b y far the g r e a t e s t n u m b e r of p r o c e s s e s the r e l a t i o n s b e t w e e n the input, s t a t e a n d o u t p u t v a r i a b l e s a r e n o n l i n e a r . T h e p e r s i s t e n c e , h o w e v e r , that is s h o w n in d e r i v i n g m o d e l s a n d t r a n s f o r m i n g t h e m into l i n e a r forms p o i n t s to the g r e a t t h e o r e t i c a l a n d p r a c t i c a l v a l u e of s u c h linear m a t h e m a t i c a l forms. T h e m a i n r e a s o n s that justify this p r o c e d u r e
a r e the following:
- for small d e v i a t i o n s from the u s u a l o p e r a t i n g s t a t e s the linear m o d e l b e h a v e s like the o r i g i n a l n o n l i n e a r p r o c e s s , - linear m o d e l s a r e s o l v e d u s i n g the h i g h l y - d e v e l o p e d a n d p o w e r f u l m a t h e m a t i c a l a p p a r a t u s of linear a l g e b r a , w h i c h is t o d a y p a r t of s t a n d a r d m a t h e m a t i c a l c o m p u t e r libraries, - the p r i n c i p l e of s u p e r p o s i t i o n is u s e d , s o it is sufficient to e x a m i n e s y s t e m s with u n i q u e a n d s i m p l e i n p u t s i g n a l s on the b a s i s of w h o s e r e s p o n s e s d y n a m i c p r o p e r t i e s c a n b e t y p i f i e d (into p r o p o r t i o n a l , i n t e g r a l , d e r i v a t i v e ) , w h i c h l e a d s to a g e n e r a l i z e d a n d u n i f i e d a p p r o a c h to the a n a l y s i s of d y n a m i c s of o t h e r w i s e different p r o c e s s e s . - as stability is a p r o p e r t y of the p r o c e s s itself a n d d o e s not depend on the initial conditions or disturbances, the e x a m i n a t i o n of the s t a b i l i t y of a c e r t a i n s t e a d y s t a t e or o p e r a t i n g r e g i m e is r e d u c e d of s y s t e m matrix A.
to a n a n a l y s i s of the e i g e n v a l u e s
N o n e of t h e s e p r o p e r t i e s ( e x c e p t , of c o u r s e , t h e first) c h a r a c t e r i z e n o n l i n e a r m o d e l s , w h o s e b a s i c f e a t u r e is that g e n e r a l solutions, m e t h o d s a n d a p p r o a c h e s c a n n o t b e u s e d . E v e r y n o n l i n e a r p r o c e s s c o n t a i n s within it a different form of n o n l i n e a r i t y that d e m a n d s a s p e c i a l m e t h o d of solution. This i m p o s s i b i l i t y of a u n i f i e d a p p r o a c h (the i m p o s s i b i l i t y of
I~Q I~L,
L1NI~ARIZATIQN
25Q
g e n e r a l i z a t i o n } is o n e of the r e a s o n s w h y n o n l i n e a r t h e o r y is still in the p h a s e of d e v e l o p m e n t a n d i n t e n s e s t u d y . N e v e r t h e l e s s . o n e u n i f i e d a p p r o a c h to s u c h s y s t e m s d o e s exist, a n d that is the p r o c e d u r e of n u m e r i c a l simulation of t h e n o n l i n e a r m o d e l . T o d a y this is the most f r e q u e n t l y u s e d m e t h o d for a n a l y z i n g s u c h p r o c e s s e s a n d it c a n b e a p p l i e d without difficulty to e v e r y e x a m p l e in this b o o k . In p r a c t i c e the s y s t e m v a r i a b l e s in d e v i c e s , p l a n t s a n d p r o c e s s e s a r e connected b y v a r i o u s forms of n o n l i n e a r relations~ limiters, s w i t c h e s . b r e a k e r s , flip-flops, s a t u r a t i o n s , d e a d z o n e s , friction, c l e a r a n c e s and h y s t e r e s e s a r e o n l y a small n u m b e r of t y p i c a l n o n l i n e a r relations. In the text of this b o o k we e n c o u n t e r e d n o n l i n e a r i t i e s that w e r e a n a l y t i c a l l y expressed in the form of p r o d u c t s b e t w e e n v a r i a b l e s , s q u a r e roots, s q u a r e s a n d the like, or n o n l i n e a r r e l a t i o n s w e r e s h o w n g r a p h i c a l l y in d i a g r a m s of c h a r a c t e r i s t i c s . For s u c h v a r y i n g c a s e s of n o n l i n e a r i t y , a n d d e p e n d i n g p u r p o s e of the m o d e l , t o d a y the following d e v e l o p e d linearization are usually used:
o n the final m e t h o d s of
the m e t h o d of t a n g e n t i a l a p p r o x i m a t i o n or l i n e a r i z a t i o n a b o u t the o p e r a t i n g (usually steady) state, in w h i c h nonlinear c h a r a c t e r i s t i c s a r e r e p l a c e d b y linear o n e s (in t w o - d i m e n s i o n a l s p a c e this m e a n s r e p l a c i n g the c u r v e b y the t a n g e n t in the operating point). In t h e m a t h e m a t i c a l sense this means d e v e l o p i n g the f u n c t i o n s into a T a y l o r s e r i e s in w h i c h all the h i g h e r ( n o n l i n e a r ) terms a r e d i s c a r d e d . T h e p r o c e d u r e is the m o r e s u c c e s s f u l , the s m a l l e r the d e v i a t i o n s from the o p e r a t i n g state. the method of the descriptive function or harmonic linearization, which linearizes processes in the f r e q u e n c y d o m a i n a n d not in the time d o m a i n , as in the a b o v e c a s e . In c o n t r o l t h e o r y this m e t h o d is o f t e n a n d s u c c e s s f u l l y u s e d a n d its m a i n f e a t u r e is that o n l y the s i g n a l of the b a s i c f r e q u e n c y is r e c o r d e d in a n a l y s i s while the h i g h e r h a r m o n i c s a r e c o n s i d e r e d d a m p e d . T h e s u c c e s s of the m e t h o d d e p e n d s on h o w similar the p r o c e s s e s a r e to l o w - f r e q u e n c y filters. the m e t h o d of statistical l i n e a r i z a t i o n is a l s o p e r f o r m e d in the
260
APPENDIX
time d o m a i n . H e r e the n o n l i n e a r p r o c e s s is r e p l a c e d b y a n e q u i v a l e n t linear m o d e l a s s u m i n g that the d i s t u r b a n c e h a s a normal ( G a u s s e a n ) distribution, - the c o m b i n e d m e t h o d of h a r m o n i c a n d statistical l i n e a r i z a t i o n . In this b o o k d e t e r m i n i s t i c p r o c e s s e s w e r e a n a l y z e d in the time d o m a i n (smaller e x c u r s i o n s into the d o m a i n of the c o m p l e x v a r i a b l e w e r e p e r f o r m e d to o b t a i n a n d d e m o n s t r a t e transfer functions), so the s u b s e q u e n t lines will s h o w o n l y the m e t h o d of t a n g e n t i a I a p p r o x i m a t i o n b e c a u s e it w a s u s e d in all the e x a m p l e s in the p r e c e d i n g s e c t i o n s a n d is t o d a y still the most f r e q u e n t l y u s e d . (In A n g l o - A m e r i c a n literature it is often c a l l e d p e r t u r b a n c e a n a l y s i s . This n a m e is u s e d to s h o w that the m e t h o d y i e l d s linear m o d e l s that d e s c r i b e small d e v i a t i o n s , p e r t u r b a n c e s , from the o p e r a t i n g state.) T h e b a s i c a s s u m p t i o n that must b e fulfilled if w e w a n t to a p p l y this m e t h o d is that the r e l a t i o n s b e t w e e n v a r i a b l e s must b e g i v e n in a f u n c t i o n a l form that h a s a finite first d e r i v a t i v e in the o p e r a t i n g p o i n t a b o u t w h i c h l i n e a r i z a t i o n is p e r f o r m e d (i.e. that t h e y a r e d i f f e r e n t i a b l e at l e a s t o n c e ) , a n d if the r e l a t i o n s a r e s h o w n b y a c h a r a c t e r i s t i c c u r v e , t h e n it must h a v e a c l e a r l y d e t e r m i n e d t a n g e n t in that o p e r a t i n g point. This m e t h o d , is, a s a rule, not s u i t a b l e for s t e p a n d o t h e r b r o k e n or d i s c o n t i n u o u s n o n l i n e a r relations. T a n g e n t i a l a p p r o x i m a t i o n of a n o n l i n e a r function is p e r f o r m e d b y e x p a n s i o n into a T a y l o r s e r i e s (if the f u n c t i o n is g i v e n in a n a l y t i c a l form) or g r a p h i c a l l y if the r e l a t i o n s a r e s h o w n in d i a g r a m . This s e c o n d m e t h o d w a s s h o w n in d e t a i l in S e c t i o n 2.2 w h e r e the s t a t i c a l c h a r a c t e r i s t i c s of the c e n t r i f u g a l p u m p a n d of the r e g u l a t i o n v a l v e w e r e l i n e a r i z e d . Using F i g u I e s 2.2-1.2 a n d 2.2-1.3, w e o b t a i n e d linear e x p r e s s i o n s for the p u m p , (2.2-1.10) a n d (2.2-1.16), a n d for the v a l v e . (2.2-I.I1) a n d (2.2-1.17). T h e s a m e p r o c e d u r e c a n b e u s e d in all the other c a s e s a n d h e r e it will not b e r e p e a t e d . B e f o r e the m e t h o d of t a n g e n t i a l a p p r o x i m a t i o n is d e m o n s t r a t e d , t h e s t a t e a n d o u t p u t e q u a t i o n s g i v e n b y (A.19) a n d (A.20) s h o u l d a l s o b e g i v e n in v e c t o r n o t a t i o n
X' = f ( X , U)
(A.29)
Y = g ( X , U)
(A.30)
~r~l~ KL
LINKARIZATIC~N
2~l
y=g(u)
Ay = y . y
y
.
.
.
.
.
.
.
t,y g-
/
~
=K~u
R
,,u = u - ~
~U
~
Fig. The
meaning
u
A . 4 T h e m e a n i n g of t a n g e n t i a l a p p r o x i m a t i o n of
linearization
is
the
easiest
to
comprehend
in
two-dimensional space where relations among variables can be shown by a c u r v e a n d a t a n g e n t in the o p e r a t i n g point. In the c a s e of t h r e e - d i m e n s i o n a l r e l a t i o n s the g r a p h i c a l p r e s e n t a t i o n is a s u r f a c e in this s p a c e a n d the p l a n e t a n g e n t i a l to it, a n d in the most g e n e r a l c a s e it is a h y p e r s u r f a c e a n d its t a n g e n t i a l h y p e r p l a n e , w h i c h c a n n o t b e s h o w n g r a p h i c a l l y . Let the o u t p u t v a r i a b l e , t h e r e f o r e , b e r e l a t e d o n l y with the input v a r i a b l e b y the following r e l a t i o n y = g(u)
(A.31)
For a g i v e n u. y is d e t e r m i n e d s h o w n o n F i g u r e A.4
and
the o p e r a t i n g
point
R(u, y) is
E x p a n s i o n into a T a y l o r s e r i e s a b o u t the point R(u, y) y i e l d s y - ~7 + -B--h-(u - ~ )
+ ~h--~u
- ~)2
+ ...
(A.32)
T h e d e v i a t i o n s s h o w n h e r e a r e c a l c u l a t e d in the o p e r a t i n g p o i n t R. If a n a l y s i s is limited o n l y to small d e v i a t i o n s , t h e n the s q u a r e term a n d all the following terms of h i g h e r o r d e r c a n b e n e g l e c t e d b e c a u s e t h e y a r e m u c h smaller t h a n the linear term in the e x p a n s i o n , so w e g e t y - y
: -~-g(u
- ~.)
.
[A.33)
262
APPE~NDIX
a y = KAu.
K = d~ du
(A.34)
Now we h a v e got l i n e a r dependance, no longer of the absolute values of variables u a n d y, b u t o f t h e v a l u e s of their deviation from the operating s t a t e . It is i m p o s s i b l e to form a g e n e r a l criterion a b o u t the validity of the linear model, i.e. to a n s w e I the q u e s t i o n - what is the r a n g e of v a r i a b l e s for w h i c h the linear a n d the n o n l i n e a r m o d e l s g i v e c l o s e results. The a d j e c t i v e " c l o s e " is rather a n u n p r e c i s e formulation for the m o d e l ' s a c c u r a c y . What has to b e d o n e is to c h e c k , for e v e r y p a r t i c u l a r c a s e . what the a l l o w e d d e v i a t i o n of the linear m o d e l from the n o n l i n e a r o n e is. H o w e v e r , the v a l u e of the error c a n a l w a y s b e o b t a i n e d b e ' c a u s e it e q u a l s the m a g n i t u d e of the d i s c a r d e d h i g h e r - o r d e r terms. The following simple e x a m p l e s h o w s the d i f f e r e n c e in results y i e l d e d b y the n o n l i n e a r a n d the linear m o d e l s . y = u 2.
R(2. 4)
u = 2, y = 4
E x p a n s i o n into a s e r i e s g i v e s I
y - ~+ 4(u- h) . ( u - ~ ) ~
nonlinear model
If the last, s q u a r e
is n e g l e c t e d ,
term in the e x p a n s i o n
the following
linear m o d e l is o b t a i n e d y - ~ = 4 ( u -u)
~ y ; 4AU
linear m o d e l
O b v i o u s l y . the d i f f e r e n c e b e t w e e n t h o s e two m o d e l s , or the error m a d e b y n e g l e c t i n g h i g h e r - o r d e r terms, e q u a l s e - YNL " YL " (u - ~)2 . Au 2 T a b l e s h o w i n g the results a n d their g r a p h i c a l p r e s e n t a t i o n
li~l:~ IZI IiL,
I,.,Il k l l A itl";r A T I t ' l hi
9A~
u
YNLIAu
hy
au
Ay
u 0
0
0
-2
-4
-2
-8
I 2
I 4
-1 0
-3 0
-I
-4
3 4
9
1 2
5
I
16
Eiior
Linear m o d e l
Nonlinear m o d e l
0 2
12
6
e - YNL " YL = Auz
YL 1
2
4
-4
1
0
0
4
1
8 8
4
e
4
12
Y, e : ,~,U2
YNL= 9
h.:8
-_:,__I,,,.< Io ,,,f.,.,,,, o:,
U
~--2
Fig.
V
A . 5 Error d u e to the l i n e a r i z a t i o n of the c u r v e u ~
The figure s h o w s the v a l u e of the error. For e x a m p l e , if the input v a r i a b l e u d e v i a t e s b y 50%. i.e. for Au ; 1. the linear m o d e l m a k e s a n error of I1.1%, or the v a l u e g i v e n b y the linear m o d e l is 11.1% smaller than the real v a l u e g i v e n b y the n o n l i n e a r model. Here w e must i m m e d i a t e l y c o n s i d e r w h e t h e r there a r e s u c h p r o c e s s e s in w h i c h the input v a r i a b l e c h a n g e s its v a l u e b y 5OZ, a n d w h e t h e r it matters if the error in c a l c u l a t i o n is of the a b o v e o r d e r of m a g n i t u d e . For smaller d e v i a t i o n s
the errors a r e smaller,
almost n e g l i g i b l e . The
same
procedure
problem can be used
that
has
been
shown
for
the
w h e n y is a function of s e v e r a l
two-dimensional input a n d
state
24)4
APPENDIX
v a r i a b l e s , i.e. w h e n it is g i v e n in the form (A.20) or (A.30). In s u c h h i g h e r - o r d e r p r o c e s s e s , in w h i c h w e d i s t i n g u i s h b e t w e e n s t a t e a n d input v a r i a b l e s , e x p a n s i o n into a T a y l o r s e r i e s will b e s h o w n on the i-th differential s t a t e e q u a t i o n from the s y s t e m (A.19) (A.35]
xl' = fl(xl..... xn, ul ..... urn)
. % :
dr, 'u
cx
.....
xo,
(A.363
.....
The above equation does not c o n t a i n the s q u a r e and other h i g h e r - o r d e r term of the e x p a n s i o n . ~'l is the d e r i v a t i v e of the state v a r i a b l e xi in the o p e r a t i n g point. (It is u s u a l to p e r f o r m l i n e a r i z a t i o n of DE a b o u t the s t e a d y state, a n d t h e n t h e r e is x°l = 0.) E q u a t i o n (A.36) y i e l d s
xi'-
~i ' = A x i ' =
~-~,~x,
+ ... + .drlaXn + 6u~--z~u'~[l. , ... + B[i Au m (A.3T) dXn
~U m
T h e s y s t e m of e q u a t i o n s (A.21) (in w h i c h the s y m b o l L~ h a s b e e n left out) w a s o b t a i n e d from the a b o v e e x p a n s i o n of (A.19), a n d the c o e f f i c i e n t s a and b are aij =
~T~ ~Xj
~Ti bik ..... ~uk
'
i. j = l..... n
T h e v a l u e of the p a r t i a l state observed.
k = I..... m
derivatives
is d e t e r m i n e d
in the
operating
T h e c o e f f i c i e n t s all a n d bik a r e m e m b e r s of the m a t r i c e s A a n d L we c a n write
~f K= (~3 R
,
~f B= (~3
The subscript R denotes the o p e r a t i n g s t a t e R. Exactly the s a m e
R
(A.39)
that the partial d e r i v a t i v e s a r e c a l c u l a t e d
procedure
so
is used
in
to linearize the output equations
CA.20) a n d (A.30), only instead of xl a n d output variables Yi a n d the function gl-
the function fl w e
now
have
l~r~ r~I~I.. I - I N I A R I Z A T I ~ N
26B
T a n g e n t i a l a p p r o x i m a t i o n is a m e t h o d that c a n b e w i d e l y a p p l i e d , but it must a l w a y s b e b o r n in mind that the d o m a i n in w h i c h the v a r i a b l e s c a n c h a n g e after linearization has b e e n p e r f o r m e d is limited to 5. I0 or 20~ of the s t e a d y graphical
values
presentation
of t h o s e v a r i a b l e s .
of h o w
two
The
completely
following figure
different
is a
processes
can
h a v e the s a m e linear m o d e l for a g i v e n ft.
//~liul
f1(u) f2(u)
fz(U)
Fig.
A . 6 Two different p r o c e s s e s with the s a m e linear m o d e l
It must also b e s a i d that in the c a s e of a n o n l i n e a r m o d e l it is v e r y important to d e t e r m i n e the o p e r a t i n g state in w h i c h l i n e a r i z a t i o n was p e r f o r m e d . The n o n l i n e a r m o d e l c a n h a v e one. two or more s t e a d y states. or it c a n h a v e n o n e . so w e must c o n s i d e r p r o b l e m s of d e t e r m i n i n g the s t e a d y state as well. Here we will not e n t e r this a n a l y s i s a n d the r e a d e r is referred, as a small reminder, to the last e x a m p l e of S e c t i o n 2,4, w h e r e it was s h o w n that the m o v i n g p e n d u l u m has two s t e a d y states in w h i c h l i n e a r i z a t i o n g i v e s different matrices l ,
w h i c h a l s o m e a n s different linear
m o d e l s with e s s e n t i a l l y different d y n a m i c p r o p e r t i e s . In p r a c t i c e , o t h e r w a y s for o b t a i n i n g also
often
used.
The
most
widespread
a
tangential
procedure
is
appIoximation are to
substitute
n o n l i n e a r e x p x e s s i o n s the v a l u e s for v a r i a b l e s w h i c h are r e p r e s e n t e d
into as
the sum of the v a r i a b l e s h o w i n g the s t e a d y state a n d that s h o w i n g d e v i a t i o n from it. On the n e x t s i m p l e e x a m p l e we wilt s h o w this m e t h o d , w h i c h a v o i d s partial differentiation.
APPENDIX
266
L i n e a r i z e the following n o n l i n e a r o u t p u t e q u a t i o n y--
XU
CA.40)
,
a b o u t its s t e a d y s t a t e (x. 5. y). Introducing
ya~+Ay. x--x+
(A.41)
AX,
U~'U+AU, (A.40) b e c o m e s
(A.42)
As t h e r e is y ,. xu, the first terms on the l e f t - h a n d s i d e a n d the r i g h t - h a n d s i d e c a n c e l out. For small d e v i a t i o n s the p r o d u c t AxAu is v e r y small, a n d c a n b e n e g l e c t e d . Thus (A.42) y i e l d s the following linear expression Ay = u Ax + x A u Of c o u r s e , the s a m e result w o u l d h a v e b e e n into the T a y l o r s e r i e s Y = ~ + ( a'~--Y-)R(x~x " ~) + ( ~.Y )oCu - u) dU r~
(-~-)R= ~,. ay
(A.43) obtained by expansion
(A.44)
(_~_uy)R=
Ay -- u Ax + X A U
(A.45)
T h e r e is o n e more m e t h o d w h i c h i n d i r e c t l y r e p r e s e n t s e x p a n s i o n into a T a y l o r s e r i e s in which h i g h e r - o r d e r terms h a v e b e e n n e g l e c t e d . It is the , m e t h o d oJ d i f f e r e n t i a t i n g the w h o l e n o n l i n e a r e x p r e s s i o n , after w h i c h infinitesimal c h a n g e s a r e r e p l a c e d b y finite d e v i a t i o n s . On the s a m e e x a m p l e this m e t h o d g i v e s
dy
--
(
.Y]odx dX
÷ ( . Y)Ddu
i~
dy ~ udx + xdu
dU
,
L~
(A.46)
Now the s y m b o l s for infinitesimal c h a n g e d a r e r e p l a c e d b y the s y m b o l s of finite d e v i a t i o n A a n d the s a m e l i n e a r m o d e l as b e f o r e is obtained Ay = ~ A x + ~ A u This m e t h o d of d i f f e r e n t i a t i n g is p r o b a b l y the most s u i t a b l e for use. It h a s b e e n u s e d in a l m o s t all the e x a m p l e s in this b o o k a n d w a s a l s o u s e d to l i n e a r i z e the PDE in S e c t i o n 3.3 - s e e the d e r i v a t i o n g i v e n b y E q u a t i o n s (3.3-67)
- (3.3-71).
T h e r e a r e o t h e r m e t h o d s for o b t a i n i n g linear m o d e l s also. O n e of t h e m w a s s h o w n in the last e x a m p l e of S e c t i o n 2.4, w h e r e the linear m o d e l of the m o v i n g p e n d u l u m w a s r e a c h e d after s o - c a l l e d a l g e b r a i c l i n e a r i z a t i o n the s u b s t i t u t i o n of sine a n d c o s i n e f u n c t i o n s b y functions similar to t h e m for small a n g u l a r d i s p l a c e m e n t s . B e s i d e s this m e t h o d , that of l e a s t s q u a r e s is u s e d in p r a c t i c e to d e l e r m i n e c o e f f i c i e n t s of l i n e a r d e v e l o p m e n t . T o d a y , t h a n k s primarily to p o s s i b i l i t i e s o p e n e d u p b y c o m p u t e r s a n d in c a s e s w h e n it is v e r y difficult ( b e c a u s e of the s i z e a n d c o m p l e x i t y of the m o d e l ) to p e r f o r m differentiation a n a l y t i c a l l y , t h e linear m o d e l is a r r i v e d at numerically.
2t~
APPENDIX
SELECT FUNCTIONS OF COMPLEX VARIABLE
s
=
o
, j~
=
x
+ jy
s
s2
@ =i+~
+-~T- +"'+
e s = eX(cosy
S
= cosx
•
~
sins
= sinx
chs
= l + T
= chx
S 3
-- s + ~ -- s h x
ths
= - -
521"1+1
jcosx
+ "'" ÷
+
jshx
-
s2n (2n)l
• cosy
+ ... ÷ +
jchx
-
S2n+I -
(2n+l)J
.,.
+
e -s
=
e s . e-S +
,.. m
• siny
shs chs
e s +
• siny
S5
÷ ~
(2n+l)l
• shy
S4
' cosy
0o.
, shy
"'" + ('l)n
+
+ -~
e-Js
2j ÷
S5 61
• chy
jsins
(2n)l jsinx
+
=
... • (-l)n ~
chy
S 2
shs
coss+
S 2n
41
S3 sin s = s -
shs
e is=
eJs_
s4
cos
s
kJ
sin S
= 1 - 2--[- +
sk
"'" = ~' k=O
+ e-J s
cos s
ch
+
2
--
S2
s
nl
+ jsiny)
e js COS
n
sn
chs cths
= - -
shs
2~q
BASIC Differentia[
DYNAMIC
Transfer function
equotion
Transient
function
Poles Zeros
°1_ _ , ]
"response to u=h=l It"Oil - -
x o
s...plane
K
Kp
Kp
x = Kpu
PROPERTIES
!
Kp . TS*I
T dx ~- ,, x =Kpu periodic case
T~dZx z ~-~r + T ~dx- ÷x = Kpu
!
Kp
K. ( 1 - • T)
Kp TZ2S . l"s * 1
• sin (~pt * ~0)]
Ce.= 1/T2, ~ = T / 2 T z -
C~n~
!
il---,f
1 ~t
1 Ti S
x= -,;oOl 0 t
ox = +,;ud, T~T+x
. . . . .
TisITs*I)
.±
1 ~[t-T(1-e')]
0
x = ~ -du
T~s
Td6(t)
dx du T~-*x =Ta~ i-
TdS Ts+I
T~-T
x=u+Td~du
I *Tds
t + "Ida|t)
TdS ÷ 1
I * ( TT0 - 1)e'~
!
T ~dx' * x
= u . T a ~du -
T ~
Ts+~
I
-T,s
x=u(t-Tt)
t t
I ( t - T,)
e
I
T~*.x=
x = Ku
+
K,J'udt
T!x'* x = Ku * K,j~udt
1_ 2e'er
K * K"~L or S 1
K,(1 • .-~-_). K~= Kr T~" Kr(l"
T-~]
1.Ti s
Kr(S.
x = KU * K j ' u d t - Kdu'
t
1 -Ts I *Ts
u-T~-~
Kr 'Ki=~ i
Kr{I*~-Ti
K~[~o 1 - ~ 1
I
1-#tn)] /ITs
T, > T--'--'~
t
1 .Tds |
Kr K = Kr, Ki = "~" Kd.K,Td
K r [1 * "~i + Td~(t)]
K-~.~ ---1""
1"; t
I
d
IZ=id
LIST
OF
SYMBOLS
(Numbers in p a r e n t h e s e s indicate sections) A
m2
A ACt3 AV(t} a
m ~
a
m/s 2
a
m by C
C c
J/kgK
as, c
m/s
c
mol/m ~
c
N/m
ct
Nm/rad
D
m
D
Ns/m
m Dt
Nms/rad
d
m
E
J
E
N/m 2
E(t) e
J/s
F
N
F(t} f
e(s) OCs) g 14
hCt)
m
area
system's matzix in state equation acceleration a c c u m u l a t e d (stored} variable a r e a (2.5} acceleration arbitrary coefficient input's matrix in state equation valve coefficient (2.2-1} coefficient of c a p a c i t y output's matrix in output equation specific heat velocity of s o u n d (2.2-3). (3.3} concentration spring constant torsional spring constant diameter viscous friction coefficient input's matrix in output equation torsional viscous friction coefficient diameter energy modulus of elasticity, Young's modulus (3.3} effort (potential} e n e r g y (heat} flow rate force (2.4) flow various function transfer function matrix transfer function various function height of liquid level, h e a d Heaviside function of unit step
271 I
I m = Mw
kgm/s
i
J/kg kgm 2
I K Kv k = I/T L Lz
l/s
I
m
M
kg Nm kg/s too!
Mz m
N n n
m
kgm2/s
S-1
n
nd
moI/s mol/s
P
Nlm 2
n
pCz) Q.q q
]/s J/m2s
R R
m
P.
J/kgK
r
m
,(t) r(P) S
=
J/kg 0 "I" jW
s
J/kgK
T
s
t
s
U
J
U
m
U U
J/s
U
J/kg
Ul
V
m 3
coefficient of inertia momentum enthalpy moment of inertia gain coefficient v a l v e characteristic h e a t (thermal) coefficient (3.1) length a n g u l a r momentum length mass
torque mass flow rate quantity of mater polyt, o p i c e x p o n e n t number of revolution zeros of transfer function molal flow rate diffusiv molal flow rate pressure initial conditions heat flow rate (3.2) heat flux per unit a r e a (2.3) coefficient of r e s i s t a n c e radius g a s constant radius b o u n d a r y conditions s p e c i f i c heat of vaporization c o m p l e x v a r i a b l e of L a p l a c e transformation entropy time c o n s t a n t time c o o r d i n a t e internal e n e r g y (3.1) circumference input v e c t o r h e a t flow rate (3.1) internal e n e r g y input variables, elements of U volume
272 v
m31kg
specific volume
w
m/s
velocity state vector s t a t e v a r i a b l e s , e l e m e n t s of X output vector o u t p u t v a r i a b l e s , e l e m e n t s of Y impedance spatial coordinate
X xi Y Yi Z z
m
SUBSCRIPTS a
approximative pipe d a m p i n g (2.4) electrical external fluid input interna! (2]-1) critical
c d el ex f i in k m n o o
mass
(2.4)
natural (2.4)
output spring pump s o u n d (2.2-3), (3.3) torsional (2.4) valve wall water (2.5)
P s
t v w w
A G F
area
g r a v i t a t i o n (3.3) friction
GREEK
¢,
W/m2K
c o n v e c t i v e h e a t transfer c o e f f i c i e n t
273 (X
c o e f f i c i e n t b e l o n g i n g to r e s i s t a n c e ~v
YA : ( ~ ) . "rB =
~v)
C-FF P m
~(t) A £
rad/s ~ K
E:
3 e(s) x
x x
W/mK
wall t h i c k n e s s s y m b o l of d i f f e r e n c e (e.g. SP = PI - P2) Dirac (impulse) function small d i f f e r e n c e partial d e r i v a t i v e emissivity (2.3) angular acceleration temperature Laplace transform of temperature adiabatic exponent coefficient of flow resistance coefficient of thermal conductivity
eigenvalue g g m2/s
9
P 13
N/m z
0
leakage (outflow) coefficient hiction coefficient (2.4) kinematic viscosity damping coefficient density real part of complex variable s stress (3.3)
~T ~p = (-gF)s =
'ts
(aT) ~-
p
,p q,
tad
to
rad/s S-!
angle gas characteristic i m a g i n a r y part of c o m p l e x v a r i a b l e s angular velocity natural f r e q u e n c y L a p a l a c e transformation a n d its i n v e r s e . respectively
274 SPECIAL
NOTATIONS
DE ODE
PDE L NL h', h". r p, p"
differential e q u a t i o n o r d i n a r y differential e q u a t i o n partial differential e q u a t i o n linear nonlinear s a t u r a t i o n v a r i a b l e s in S e c . 2.5 a n d 3.3
v'. v". v'" T h r o u g h o u t this w h o l e b o o k w e after L a p l a c e transformation, e v e r y funqtion into a n u p p e r - c a s e letter, c o m p l e x s - d o m a i n (for e x a m p l e , after b e c a m e e(s) a n d s o on).
h a v e b e e n c o n s i s t e n t in c h a n g i n g , l o w e r - c a s e letter d e n o t i n g a time to r e p r e s e n t t r a n s f o r m a t i o n into the t r a n s f o r m a t i o n re(t) b e c a m e M(s). e(t)
A line o v e r a s y m b o l r e p r e s e n t s the v a l u e of the p h y s i c a l v a r i a b l e in the s t e a d y s t a t e (for e x a m p l e ~. ~ a n d m a r e p r e s s u r e , s p e c i f i c v o l u m e a n d m a s s flow r a t e in the o p e r a t i n g r e g i m e in w h i c h t h e r e is no time c h a n g e - in the s t e a d y o p e r a t i n g state). B o l d f a c e s y m b o l s d e n o t e m a t r i c e s and vectors.
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•
•
•
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Pzozessmodell-Katalog.
upravliaemost,
nabliudaemost.
s (G),
(R).
8T. W e b e r . T.W.. An Introduction to Process Dynamics and Control, John Wiley&Sons. New York, 1973. T h e n a m e s of the a u t h o r s w h o s e b o o k s a r e the c l o s e s t in c o n t e n t or in a p p r o a c h to s p e c i f i c p a r t s of this b o o k h a v e b e e n p r i n t e d in b o l d f a c e . (CS ... in C z e c h , G ... in G e r m a n , R ... in Russian, SC ... in S e r b o C r o a t i a n )
