Theory of Calorimetry (Hot Topics in Thermal Analysis and Calorimetry)

THEORY OF CALORIMETRY

Hot Topics in Thermal Analysis and Calorimetry

Volume 2

Series Editor:

Judit Simon, Budapest University of Technology and Economics, Hungary

The titles published in this series are listed at the end of this volume.

Theory of Calorimetry

by

Wojciech Zielenkiewicz Institute of Physical Chemistry,

Polish Academy of Sciences, Warsaw, Poland

and

Eugeniusz Margas Institute of Physical Chemistry,

Polish Academy of Sciences, Warsaw, Poland

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-48418-8 1-4020-0797-3

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Contents

Preface

ix

Chapter 1 The calorimeter as an object with a heat source 1.1. The Fourier law and the Fourier-Kirchhoff equation 1.2. Heat transfer. Conduction, convection and radiation 1.3. General integral of the Fourier equation. Cooling and heating processes 1.4. Heat balance equation of a simple body. The Newton law of cooling

1.5. The heat balance equations for a rod and sphere 1.6. General heat balance equation of a calorimetric system

1

2

10

14

20 26

33

Chapter 2 Calorimeters as dynamic objects 2.1. Types of dynamic objects 2.2. Laplace transformation 2.3. Dynamic time-resolved characteristics 2.4. Pulse response 2.5. Frequential characteristics 2.6. Calculations of spectrum transmittance 2.7. Methods of determination of dynamic parameters 2.7.1. Determination of time constant 2.7.2. Least squares method 2.7.3. Modulating functions method 2.7.4. Rational function method of transmittance approximation Determination of parameters of spectrum transmittance 2.7.5.

37

39

41

47

55

58

61

66

66

74

76

79

81

CONTENTS

vi

Chapter 3 Classification of calorimeters. of heat effects

Methods

of determination

3.1. Classification of calorimeters 3.2. Methods of determination of heat effects General description of methods of determination of heat ef3.2.1. fects 3.2.2.

Comparative method of measurements 3.2.3. Adiabatic method and its application in adiabatic and scan­

ning adiabatic calorimetry 3.2.4. Multidomains method 3.2.5. Finite elements method 3.2.6. Dynamic method 3.2.7. Flux method 3.2.8. Modulating method 3.2.9. Steady-state method 3.2.10. Method of corrected temperature rise 3.2.11. Numerical and analog methods of determination

of thermokinetics 3.2.11.1 Harmonic analysis method 3.2.11.2. Method of dynamic optimization 3.2.11.3. Thermal curve interpretation method 3.2.11.4. Method of state variables 3.2.11.5. Method of transmittance decomposition 3.2.11.6. Inverse filter method 3.2.11.7. Evaluation of methods of determination of total heat effects

and thermokinetics 3.3. Linearity and principle of superposition

85 85 97

97

101

103

104

109

111

114

114

116

119

123

123

124

125

127

128

129

131

136

Chapter 4 Dynamic properties of calorimeters 4.1. Equations of dynamics 4.2. Dynamic properties of two and three-domain calorimeters with

cascading structure 4.2.1. Equations of dynamics. System of two domains in series 4.2.2. Equations of dynamics. Three domains in series 4.2.3. Applications of equations of dynamics of cascading systems

139

139

143

143

148

151

CONTENTS

4.3. Dynamic properties of calorimeters with concentric configuration 4.3.1. Dependence of dynamic properties of two-domain calorimeter with concentric configuration on location of heat sources and temperature sensors 4.3.2. Dependence between temperature and heat effect as a function of location of heat source and temperature sensor Apparent heat capacity 4.3.3. 4.3.4. Energy equivalent of calorimetric system

vii 154 155 165 168 171

Final remarks

177

References

179

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Preface

Calorimetry is one of the oldest areas of physical chemistry. The date on which calorimetry came into being may be taken as 13 June 1783, the day on which Lavoisier and Laplace presented a contribution entitled ,,Memoire de la Chaleur“ at a session of the Academie Française. Throughout the existence of calorimetry, many new methods have been developed and the measuring techniques have been improved. At pre­ sent, numerous laboratories worldwide continue to focus attention on the development and applications of calorimetry, and a number of compa­ nies specialize in the production of calorimeters. The calorimeter is an instrument that allows heat effects in it to be determined by directly measurement of temperature. Accordingly, to determine a heat effect, it is necessary to establish the relationship be­ tween the heat effect generated and the quantity measured in the calo­ rimeter. It is this relationship that unambiguously determines the mathematical model of the calorimeter. Depending on the type of calo­ rimeter applied, the accuracy required, and the conditions of heat and mass transfer that prevail in the device, the relationship between the measured and generated quantities can assume different mathematical forms. Various methods are used to construct the mathematical model of a calorimeter. The theory of calorimetry presented below is based on the assumption of the calorimeter as an object with a heat source, and as a dynamic object with well-defined parameters. A consequence of this assumption is that the calorimeter is described in terms of the relation­ ships and notions applied in heat transfer theory and control theory. With the aim of a description and analysis of the courses of heat effects, the method of analogy is applied, so as to interrelate the thermal and the

x

PREFACE

dynamic properties of the calorimeter. As the basis on which the thermal properties of calorimeters will be considered, the general heat balance equations are formulated and the calorimeter is taken as a system of linear first-order inertial objects. The dynamic properties of calorimeters are defined as those corre­ sponding to proportional, integrating and inertial objects. Attention is concentrated on calorimeters as inertial objects. In view of the fact that the general mathematical equations describing the properties of inertial objects contain both integrating and proportional terms, a calorimeter with only proportional or integrating properties is treated as a particular case of an inertial object. The thermal and dynamic properties that are distinguished are used as a basis for the classification of calorimeters. The methods applied to determine the total heat effects and thermokinetics are presented. For analysis of the courses of heat effects, the equation of dynamics is for­ mulated. This equation is demonstrated to be of value for an analysis of various thermal transformations occurring in calorimeters. The considerations presented can prove to be of great use in studies intended to enhance the accuracy and reproducibility of calorimetric measurements, and in connection with methods utilized to observe heat effects in thermal analysis.

Chapter 1

The calorimeter as an object with a heat source A calorimeter can be treated as a physical object with active heat sources inside it. An analysis of the thermal processes occurring inside the calorimeter, and those between the calorimeter and its environment, requires utilization of the laws and relations defined by heat transfer theory [1–5]. The relations arising from heat transfer theory are applied to design the mathematical models of calorimeters, which express the dependence of the change in temperature measured directly as a function of the heat effect produced. There is an understandable tendency to attempt to ex­ press these models in the simplest way. In practice, this is achieved by applying simplifications to the original formulas. To make use of them wisely, one has to understand precisely the assumptions made. This chapter will present a detailed consideration of this topic. Selected problems from heat transfer theory are also presented. Spe­ cial attention is paid to a discussion of the processes occurring in a nonstationary heat transfer state. An understanding of these processes is of importance for a proper interpretation of calorimetric measurements. The general heat balance equation is introduced into the considerations. Particular forms of this equation will be applied to consider problems that form the subject of this book.

2

CHAPTER 1

1.1 The Fourier law and the Fourier-Kirchhoff

equation

Heat transfer by conduction in a homogenous, isotropic body is mathematically described by the Fourier law [1]:

which assumes proportionality between the heat flux q and the tempera­ is a vector determin­ ture gradient grad T. The heat flux q, in ing the rate of heat Q transferred through unit surface at point P per unit time dt (Fig. 1.1):

or

Quantity T in Eq. (1.1) is a function of the coordinates x, y, z and time t. For any time t the value of T determines the scalar temperature field. At every point upon it, the temperature at this instant is the same. Such a surface is called the isothermal surface for temperature T. The gradient of temperature, grad T, is a vector:

At points of an isothermic surface, the absolute gradient values are equal to

where denotes differentiation along an outward-drawn line normal to the surface. The value of is an experimentally determined coeffi­ cient called the thermal conductivity, expressed in

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

3

defines the amount of heat conducted through unit area in unit time if a unit of temperature gradient exists across the plane in which the area is measured. The value of for isotropic materials is a scalar, its value depending on pressure and temperature changes. If the range of tem­ perature is limited, the variation in may not be large and as a reason­ able approximation it can be assumed to be constant. The reciprocal of the thermal conductivity of a material is called its thermal resistivity. The relation between the heat energy, expressed by the heat flux q, and its intensity, expressed by temperature T, is the essence of the Fou­ rier law, the general character of which is the basis for analysis of vari­ ous phenomena of heat considerations. The analysis is performed by use of the heat conduction equation of Fourier-Kirchhoff. Let us derive this equation. To do this, we will consider the process of heat flow by con­ duction from a solid body of any shape and volume V located in an environment of temperature [5,6]. When heat is generated in the body, two processes can occur: the ac­ cumulation of heat and heat transfer between the body and its environ­ ment. Thus, the amount of heat generated, dQ, corresponds to the sum of the amount of heat accumulated, and the amount of heat ex­ changed,

4

CHAPTER 1

The heat flux through a surface element dS in time dt, in conformity with Eq. (1.5), corresponds to

where dS = ndS is a normal vector to a surface element in the external direction, so that the amount of heat transferred through the whole surface S in time dt is equal to

On application of the Gauss-Ostrogrodsky theorem, which states that the surface integral of a vector is equal to the volume integral of the divergence of the vector, Eq. (1.8) becomes

The amount of heat generated in the body by the inner heat sources of density g (the amount of heat developed by unit volume in unit time) in element dV of volume V in time dt is equal to

Thus, the heat generated in the total volume V of the body in time dt is equal to

The amount of heat accumulated, to

according to Eq. (1.6), is equal

When Eqs (1.9) and (1.11) are taken into account, Eq. (1.12) can be written in the form

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

5

According to the First Law of Thermodynamics, the amount of heat can be described for each volume element of the accumulated, body as

where dh is the increase in specific enthalpy (due to unit volume), and p is pressure. The increase in specific enthalpy is proportional to the increase dT in temperature T:

where

is the specific heat capacity at constant pressure, in

and is the density, expressed in the body of volume V is equal to

The increase in enthalpy in

The second term on the right-hand side of Eq. (1.14) may be written as

With regard to Eqs (1.16) and (1.17), Eq. (1.12) becomes

Comparison of both sides of Eqs (1.13) and (1.18) gives

Equation (1.19) is valid in any element of the body if

Division of both sides of Eq. (1.20) by dt gives

6

CHAPTER 1

On introduction of Eqs (1.15) and (1.1), Eq. (1.21) can be written in the form

For a solid body with a distribution of temperature T at time t given by

we have

Thus, the substantial derivative of temperature takes the form

Introduction of the velocity vector w

into Eq. (1.25) lends to

In a similar way, pressure changes as a function of (x, y, z, t) can be expressed by

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

7

Substitution of Eqs (1.27) and (1.28) into Eq. (1.22) yields

Equation (1.29) is called the Fourier-Kirchhoff equation. When the process takes place under isobaric conditions, i.e.

the Fourier-Kirchhoff equation (Eq. (1.29)) takes the form

When the velocity vector is equal to zero, i.e.

Eq. (1.31) becomes

Equation (1.33) is called the Fourier equation or the equation of con­ duction of heat. Let us use the Fourier equation to consider the following processes: a) Stationary (steady-state) heat transfer, which occurs when the changes in temperature T are not time-dependent:

and the distribution of temperature is a function only of the Cartesian coordinates (x,y, z). Relation (1.34) is fulfilled, when g = const, b) A non-stationary (non-steady-state) heat transfer process, which takes place when the changes in temperature T are time-dependent:

8

CHAPTER 1

When g = 0, the temperature changes depend only on the initial dis­ tribution of temperature. They characterize the heating or cooling proc­ esses occurring in the thermal by passive body. The investigation of heat processes for which

is the subject of great numbers of calorimetric determinations. and are constants independent of both pressure and When temperature, parameter a is often applied:

This is called the thermal diffusivity coefficient, expressed in Introduction of coefficient a into Eq. (1.33) gives

where is a Laplace operator: The Fourier-Kirchhoff differential equation and the equation of con­ duction of heat describe the transfer of heat in general form. In order to obtain the particular solutions of these equations, it is necessary to de­ termine the initial and boundary conditions. The initial conditions are be to understood in that the temperature throughout the body is given arbitrarily at the instant taken as the origin of the time coordinate t. It is usually assumed that the temperature at the beginning of the process is constant. For steady-state processes, the course of the temperature changes does not depend on the initial condi­ tions. The boundary conditions prescribe:

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

9

a) Dimensionless parameters which characterize the shape and dimen­ sions of the body. An example is a homogenous ball with radius R, where

In the spherical system,

is expressed by

In the case of a cylinder, it is convenient to operate with cylindrical coordinates r, x in which the Laplacian has the form

where

b) Physical parameters such as specific heat density and thermal

conductivity coefficient which characterize the properties of the body

and the environment. It is determined that they are constant or a function

of temperature. In all the problems discussed in this book, it is considered that and are constants.

c) Surface conditions. These conditions are as follows:

1. The prescribed temperature distribution on the surface S of the body:

2. The prescribed distribution of the heat flux or temperature gradient on the surface of S of the body:

3. The defined relation between the temperature and the heat flux on the surface S of the body:

10

CHAPTER 1

where is the surface heat transfer coefficient (see § 1.2). Mixed boundary conditions, which are subsequent to the assumption that particular parts of surface S are characterized by various types of boundary conditions, can also be used. d) The form of function g, which describes the inner heat sources. When the initial and boundary conditions are known, the physical problem of heat conduction is to find adequate solutions of the FourierKirchhoff equation or the Fourier equation.

1.2 Heat transfer.

Conduction, convection and radiation

“When different parts of a body are at different temperatures, heat flows from the hotter to the colder parts. The transfer of heat can take place in three distinct ways: conduction, in which the heat passes through the substance of the body itself; convection, in which heat is transferred by relative motion of portions of the heated body; and radia­ tion, in which heat is transferred directly between distant portions of the body by electromagnetic radiation” [1]. Heat conduction is a type of transfer of heat in solids and liquids, in­ terpreted as the imparting of kinetic energy resulting from collisions between disorderly moving molecules. The process occurs without any macroscopic motions in the body. The conductivity of diamond without traces of isotope is the highest. The conductivity of a metals is also high. The lowest conductivity is that of a gas. Heat transfer by conduction is defined by the Fourier equation (1.1). The application of Eq. (1.1) in calculations encounters difficulties be­ cause the temperature gradient of the wall must be defined, as well as its increments around the whole surface S of the body. Accordingly, for practical reasons the Newton equation is usually applied:

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

11

where coefficient is called the surface conductance or the coefficient of surface heat transfer. It defines the amount of heat transferred in unit time through unit surface when the existing temperature gradient is 1 deg. For using of the Newton equation, there is no need to introduce any simplifying assumptions. All the complicated character of the heattransfer phenomenon is enclosed in the value of coefficient which depends on many parameters. When the heat transfer conditions are described, appropriate attention must be paid to the proper choice of and the definition of the influence of different factors on its value. For a description of the heat flow phenomenon on the border of the body, a differential equation is used:

Equation (1.47) results from the Fourier law [Eq. (1.1)] and the New­ ton equation [Eq. (1.46)], expressed by Eqs (1.48) and (1.49), respec­ tively

by equating the right-hand sides. When the geometry of the considered system is simple, an exact solution of Eq. (1.47) can easily be deduced. The value of surface heat transfer coefficient is strongly affected by the presence of heat bridges and thermal resistances. Imperfect con­ tact between touching surfaces which are at different temperatures causes a lack of thermal equilibrium inside the free space between them. The magnitude of the thermal resistance depends on the surface condi­ tions, the number and shape of surface irregularities and the conditions of heat conduction through the gas present in the space between the contact points. Temperature variations appear even though the thickness of the gas layer may be close to the size of the free distance in the gas molecule. It has been found [6] that even surface irregularities within the range of tens of microns influence the value of the surface heat transfer

12

CHAPTER 1

coefficient. The variations in thermal resistance between two solid bod­ ies are the cause of errors in the heat determination. also de­ The coefficient of surface heat transfer by conduction pends on the existence of heat bridges in the system. If a layer of sub­ stance of low heat conductivity isolates two objects, then the heat ex­ change is not intensive. If a third object with heat conductivity higher than that of an insulating substance joins those two objects, then through this object (called a “heat bridge”) heat will flow, intensively enhancing the total heat exchange. Unfortunately, it is often very difficult in calorimetric practice to eliminate the existence of unwanted heat bridges and thermal resistances. It is crucial to decrease their contribution to the value of Heat transfer by convection occurs in liquids and gases where there is a velocity field caused by extorted fluid motion or by natural fluid motion caused by a difference in density. The former case involves forced convection, and the latter case free convection. Combined con­ vection occurs when both forced and free convection are present. The defining the heat ex­ convection coefficient of surface heat transfer, change in the contact boundary layer between fluid and solid, is deter­ mined. Coefficient is often expressed by equations containing criteria numbers, such as those of Nusselt (Nu), Prandtl (Pr), Reynolds (Re) and Grashof(Gr):

The criteria numbers are calculated by use of material constants such as – thermal conductivity coefficient; a – thermal diffusity coefficient; and v – kinematic viscosity. In the expressions in (1.50), l is a distinctive dimension of the body; w is the distinctive velocity; g is the acceleration is the difference in temperature, and is the due to gravity; thermal expansivity coefficient. Coefficient usually depends on the difference in temperature be­ tween the body and its environment. In the microcalorimeter described by Czarnota et al. [7], free convection occurs in the spherical space is defined by around the calorimeter, and as a result the coefficient the equation

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

13

A frequent goal in calorimetry is to obtain a constant value of irre­ spective of existing temperature gradients. A more common trend is to decrease or eliminate the heat transfer by convection. A very effective way to achieve this is the creation of a vacuum. The contribution of the heat transfer by convection can also be reduced by increasing the share of heat conduction in the total heat transfer. Radiation is the transfer of thermal energy in the form of electro­ magnetic waves; it occurs in processes of emission, reflectivity and ab­ radiated through a transparent gas sorptivity. The quantity of heat layer from surface at temperature to surface at temperature completely enclosing the first surface, is

where is a substitute emission coefficient depending on the emission from a surface, the geometry and the reflectivity, and is a radiation Introduction of the relation express­ constant ing the total radiant energy leaving unit surface area:

yields

or, on analogy with Eq. (1.47):

where

is called the radiation heat transfer coefficient. When the temperature difference

14

CHAPTER 1

is much lower than the temperatures

and

can be expressed by

where higher–power exponents of ratio are neglected. If the value of this fraction is very small, then a simplified formula is often valid:

The heat transfer in calorimeters is described in terms of the effective heat transfer coefficient, involving the heat transfer of convection, con­ duction and radiation. In calorimetry, it is more common to use coeffi­ cient G, called the heat loss coefficient, which is equivalent to the effec­ tive heat transfer coefficient calculated for the whole surface S.

1.3. General integral of the Fourier equation.

Cooling and heating processes

Let us consider the temperature changes in a body, due to the initial temperature difference between the body surface S and the environment [1, 6, 8]. The initial condition is defined as

Additionally, it is assumed that

In the examined case, the solution of the Fourier equation will refer to the heating or cooling processes of a thermally passive body. Any assumptions made will not impose any restrictions on the solution. When heat is generated in the body and the expression of g (x, y, z, t) is known, the particular solution of the Fourier equation, can be found, on the assumption that the initial conditions are zero, T(x, y, z) = 0. Accord­

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

15

ing to the superposition rule, the solution will be the sum of the sepa­ rately determined solutions. Let the heat transfer between the body and its environment proceed through the surface S according to the boundary condition of third kind, i.e.

If it is assumed that the physical parameters are constants and the Fourier equation (Eq. (1.38)) becomes

and can be written in the form

To solve the differential Eq. (1.63), the Fourier method of separated variables will be used. According to this function, T(x, y, z, t) can be described as

Then

and

Substituting the above equations into Eq. (1.63) gives

or, after transformation

16

CHAPTER 1

The right–hand side of Eq. (1.69) is a function only of coordinates (x, y, z), whereas the left–hand side is a function only of the independent variable t. To satisfy Eq. (1.69), both sides must be equivalent to the same constant value. This means that this value should be negative. Thus, when the initial temperature of the body is higher than the envi­ ronment temperature, a cooling process occurs and

whereas, when the initial temperature of the body is lower than the envi­ ronment temperature, the heating process is characterized by

Thus:

In order to fulfil this condition, let us denote by the value of the sides of Eq. (1.69). Thus, instead of one differential equation, we obtain the two following differential equations:

Equation (1.73) may be written as

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

17

and its solution is

which can easily be checked.

Equation (1.74) may be written in the form

Solving this equation gives a set of values

equivalent to the set of solutions of the function

whereas the compatible function

is equal to

The magnitudes are called the eigenvalues of the differential Eq. (1.65). The functions in Eq. (1.80) corresponding to the particular eigenvalues are called eigenfunctions. Since Eq. (1.62) is a linear differ­ ential equation, its solution is also the sum of eigenfunctions:

or, with regard to Eq. (1.80), it can be written in the form

On introduction of the notation

Equation (1.82) can be written as

18

CHAPTER 1

or

We see that the course of the temperature T (x, y, z, t) changes at any point of the body is the sum of infinite number of exponential functions. Whereas the sequence of eigenvalues is an increasing sequence, the monotonously decreases [9]. As a sequence of constants higher accuracy of determination of temperature T (x, y, z, t) is needed, a greater number of exponential terms must be used for a proper descrip­ tion of the changes in temperature of the body within time. In the limiting case, it is possible to neglect all the exponential terms, excluding the one having in the exponent the smallest value of the constant Equa­ tion (1.84) then transforms to an equation equivalent to the mathemati­ cal expression of Newton’s law. This type of description is mostly used in calorimetry and thermal analysis. The expression given by Eq. (1.85) was used for the first time in microcalorimetry to describe short–duration heat processes investigated in a Calvet microcalorimeter [10–13]. The temperature T rise caused by heat effect was expressed by the following equation:

where the adopted relations made possible the determination of T for an exponential temperature course of second or third order. The excellent monograph by Camia [14] is one of the most important works on this field. A number of methods are currently used to determine the heat effects and thermokinetics of short–duration processes, based on the assumption of a multiexponential course of temperature changes. These methods are disscused in Chapter 3. The question of the determination of total heat effects for a multiex­ ponential temperature course in time was investigated by Hattori et al. [15] and Tanaka and Amaya [16].

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

19

Hattori et al. [15] consider the calorimeter in terms of a onedimensional model of distributed parameters. The objects distinguished are the calorimetric vessel A, the heat conductor B and the medium C, at constant temperature, that surrounds the calorimeter (Fig. 1.2). For the solution of the Fourier equation, the following assumptions were made: heat power is generated in the calorimetric vessel at homogenous tem­ perature and with constant heat capacity; the heat conductor along which the heat flows has a well-insulated lateral surface. Its ends are defined as x = 0 and x = L; there exists a heat bridge of the calorimetric vessel with the conductor; the environment is kept at constant temperature. The heat transfer takes place only through the cross-section for x = 0. Tanaka and Amaya [16] consider the calorimeter as a concentric and (Fig. 1.3), which are solids. model of three domains, in which heat q heat is generated or adsorbed, is surrounded Domain by domain Domain is surrounded by domain at constant tem­ perature. These three domains represent the calorimetric vessel, and the heat conductor between the vessel and the shield. In both of the above papers, it was shown that, independently of the

20

CHAPTER 1

number of exponential terms that describe the temperature course in time, it is possible to calculate the total heat effect determining the sur­ face area between the registrated temperature course in time and the time axis below the temperature course in time if the measurement starts from equilibrium conditions and the initial and final temperatures are equal.

1.4. Heat balance equation of a simple body.

The Newton law of cooling

Linear differential equation of first order called the heat balance equation of a simple body, has found wide application in calorimetry and thermal analysis as mathematical models used to elaborate various methods for the determination of heat effects. It is important to define the conditions for correct use of this equation, indicating all simplifica­ tions and limitations. They can easily be recognized from the assump­ tion made to transform the Fourier-Kirchhoff equation into the heat bal­ ance equation of a simple body. Let us consider [1, 6, 8, 17] that the heat transfer process takes place under isobaric conditions, without mass exchange and that the thermal parameters of the body are constant. The Fourier-Kirchhoff equation can then be written as

Integration of both sides of Eq. (1.87) with respect to the volume V of the body of extremal surface S gives

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

21

The left–hand side of Eq. (1.88) can be transformed as follows:

where

is the average temperature of the body of volume V. The first term on the right–hand side of Eq. (1.88) can be trans­ formed by using the Gauss-Ostrogradsky theorem for the vector gradi­ ent:

where dS is an oriented element of the surface S of the body. Applica­ tion of the average value theory gives

where is the average flux of heat across the surface S of the body. Application of the boundary condition of the third kind leads to

where is the average temperature of the external surface S of the body, is the heat transfer coefficient, and is the temperature of the environment. The integral of the second term on the right–hand side of Eq. (1.88):

corresponds to the change in the heat power dQ/dt within time t. Thus, the second term on the left–hand side of Eq. (1.88) can be written in the form

22

CHAPTER 1

From Eqs (1.89) – (1.95) and Eq. (1.88), and putting

where C is the total heat capacity of the body, we have

Equation (1.97) is accurate, but does not give an explicit solution, and the temperature on the because the relation between temperature surface is not defined.

If it is assumed that

Equation (1.97) becomes

or

Equations (1.99) and (1.100) are commonly known as the heat bal­ ance equations of a simple body. From the above considerations, it is clear that these heat balance equations and the Fourier-Kirchhoff equa­ tion [Eq. (1.87)] are equivalent to each other when: 1. the temperature in the total volume of the body is homogenous and only a function of time; 2. the temperature on the whole surface is homogenous and only a function of time; 3. the above temperatures are identically equal to one another at any moment of time; 4. the heat capacity C and the heat loss coefficient G are constant and not functions of time and temperature.

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

23

When

Equation (1.99) becomes

If it is assumed that

Equation (1.102) becomes

Equation (1.104) is equivalent to Newton’s law of cooling. The coef­ ficients and of Eqs (1.97) and (1.104) can be characterized as fol­ lows [9]. Coefficient determines the amount of heat transfer by unit surface in unit time when the temperature difference is equal to 1 deg. Constant is called the cooling constant and characterizes the rate of body cooling. This quantity is the reciprocal of the body time constant It has the same value for any point of the body. Constant does not depend on the initial temperature field, but depends on the shape and dimensions of the body, the thermal parameters of the body (e.g. the thermal diffusivity coefficient) and the conditions of heat transfer. Coefficient is a quantity describing the measure of the ability of the given body to react to cooling or heating of the environment. The influence of this environment on the body is characterized by the heat transfer coef­ ficient and/or the heat loss coefficient G. The dependence between quantities G, C and given by Eq. (1.103) When these temperatures are is appropriate only when different, Eqs (1.97) and (1.105) can be supplemented by a new parame­ ter, the coefficient of heterogenity of the temperature field

24

CHAPTER 1

Thus, Eq.( 1.97) becomes

In order to solve this non–stationary differential equation, it would This condition is difficult to be necessary to know the function fulfil. To solve the theory of the problem of ordered state heat transfer [9], two periods of cooling or heating process of the body are specified.

The first period is characterized by a disordered course of temperature field changes in time. The second, the well-ordered heat transfer period, comes after a certain period of time (Fig. 1.4). It is assumed that, for the ordered state heat transfer, the relation between heat capacity C, heat loss coefficient G, cooling constant and coefficient is expressed as follows:

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

25

Coefficient is a dimensionless quantity; its value can vary between one and zero. The more differs from unity, the greater is the inequality in the field of temperature. This coefficient is a function of Biot’s num­ Coefficient decreases to zero as constant Bi increases ber to infinity. Let us assume that the same body is observed under various conditions of heat transfer from the body to the environment and that as a result there are various values of the surface heat transfer coefficient For the sequence of increasing values of there is a sequence of increasing values of the constants

According to the discussed theory of the ordered state heat transfer, together with the increase in the value the reaction of the body be­ comes weaker and the value of the cooling constant drifts towards the limit value (Fig. 1.5). Only in the case of certain small values of coefficient can it be assumed that a homogenous temperature distribu­ exists in the examined body. It also indicates that the range tion where the heat balance equation can be accurately used for determina­ tion of the heat power changes within time (the P(t) function called thermokinetics) is limited. However, this equation is also used as a mathematical model in determining the P(t) function in instruments with

26

CHAPTER 1

different values of coefficient It is applied as a mathematical model in conduction microcalorimeters, in which heat exchange between the body and environment is extensive and coefficient is significant. For these reasons, it is very important to verify the accepted model experi­ mentally. If a maker of a calorimetric system decides to apply the method of determining the heat effect resulting from Eq. (1.99), it would be most convenient to establish a set of parameters (e.g. such that the is generated fulfils the conditions calorimeter in which the heat needed to apply this equation. A probe to determine such a set of pa­ rameters was undertaken by Utzig and Zielenkiewicz [18] for a simple physical model as an approximation of a real calorimetric system. The the physical pa­ relation between the dimensionless parameter rameters of the system and was elaborated. In the parameter the value corresponds to the time interval after which the body temperature changes can be described by one exponential term, while is a time constant.

1.5. The heat balance equations

for a rod and sphere

A real calorimeter is composed of many parts made from materials with different heat conductivities. Between these parts one can expect the existence of heat resistances and heat bridges. To describe the heat transfer in such a system, a new mathematical model of the calorimeter was elaborated [8, 19] based on the assumption that constant tempera­ tures are ascribed to particular parts of the calorimeter. In the system discussed, temperature gradients can occur a priori. Before defining the general heat balance equation, let us consider the particular solutions of the equation of conduction of heat for a rod and sphere. For a body treated as a rod in which the process takes place under isobaric condi­ tions, without mass exchange, the Fourier-Kirchhoff equation may be written as

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

Consider the above equation for

27

(Fig. 1.6):

If we put

Eq. (1.109) becomes

If we expand the function T(x, t) as a Taylor series in the neighbor­ hood of the point and neglect terms of higher order than the sec­ ond, we have

Consideration of the function T(x, t) at points gives the following set of equations:

and

28

CHAPTER 1

If we put

this set of equations becomes

The solutions of the equation set defined by Eq. (1.115) are

Substitution of Eq. (1.116) in Eq. (1.111) gives

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

29

Let the volume of the n-th element be

where F is the cross–sectional area of the rod. Multiplication of both sides of Eq. (1.118) by gives

where

defined by

is the heat capacity of domain n, whereas

defined by

is the heat power generated in domain n. The quantities

are the heat transfer coefficients between domains n and n+l, and do­ mains n–1 and n, respectively. The quantities

denote the heat loss coefficients. Thus, Eq. (1.120) may finally be writ­ ten as

30

CHAPTER 1

where

denote the amounts of heat exchanged between domains n and n+1 and between domains n–1 and n in time interval dt, respectively. Equa­ tion (1.125) is the desired heat balance equation for a rod considered as a system of domains arranged in a row. The same procedure is applied to deduce the equation of heat conduction for a homogenous sphere of radius r, where an isobaric process without mass exchange takes place. The Fourier-Kirchhoff equation written with spherical coordinates be­ comes

for

Let us consider the above equation with

The expansion of the function T(r, t) into a Taylor series in the neglecting terms of higher order than neighborhood of the point the second, leads to

Substitution of

and

gives the set of equations

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

Let us put

and neglect derivatives of higher order than the second:

The solutions of the above set of equations are

31

CHAPTER 1

32

From Eqs (1.132), (1.134), (1.135) and (1.128) we have

Multiplication of both sides of Eq. (1.13 6) by

where

and putting

gives the following differential equation

The heat balance equations for the rod and sphere described as Eqs (1.125) and (1.138) are identical in form. They are derived on the basis of the same assumptions: in the examined bodies several elements (parts, domains) are distinguished; each is characterized by a constant heat capacity and homogenous temperature in the total volume; the heat exchange between these parts is characterized by heat loss coef­ ficient G. The first term on the left–hand side of these equations deter­ mines the amount of accumulated heat in the domain of the body of indicator n; the second and third terms are the amounts of heat ex­ changed between this part and the neighboring domains of indicators

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

33

n+1 and n–1. The heat exchange between the body and environment is characterized by the boundary condition of the third kind.

1.6. General heat balance equation

of a calorimetric system

Let us assume Eqs (1.126) and (1.139) as the basis on which to de­ rive the generalized heat balance equation of the calorimeter treated as a multidomain (elements, parts) system of various configurations [19-21]. Let us also assume that the heat transfer in the body can take place not only between the neighboring domains, but also between any domains characterized by heat capacities and temperatures Each of the separate domains has a uniform temperature throughout its entire volume; its temperature is a function only of time t, and the heat capacity of domain is constant. The domains are separated by centers characterized by loss coefficient and the heat exchange between the domains and between the domains and the environment of temperature takes place through these centers. Temperature gradients appear only in these centers and between the domains and the environment; their heat capacities are, by assumption, negligibly small. Furthermore, a heat source or temperature sensor may be positioned in any of the domains. The amount of heat exchanged between domains j and i in the time interval dt is proportional to the temperature difference of these domains; the heat loss coefficient is the propor­ tionality factor:

The condition is fulfilled. The amount of heat ex­ changed between domain j and the environment in time interval dt is equal to

Thus, the total amount of heat dQ”(t) exchanged between domain j and the remaining domains and the environment is equal to

34

CHAPTER 1

Taking into account Eqs (1.140) and (1.141), we have

The amount of heat dQ´(t) accumulated in domain j is equal to

The amount of heat generated in domain j in time interval dt is equal to the sum of the heat accumulated in this domain dQ´(t) and the heat exchanged between this domain and the remaining domains and the environment dQ´´(t). Thus:

From Eqs (1.143) and (1.144), we have

Dividing both sides of the above equation by dt gives

or

where

THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

35

v–n,

is the heat power generated in domain j. In Eq. (1.148), the first term on the right–hand side determines the amount of heat accumulated in domain j, the second term the amount of heat exchange between domain j and the environment, and the last term the amount of heat exchange between domain j and remaining N–1 do­ mains. The differential equation Eq. (1.148) is called the general heat bal­ ance equation of domain j, and the set of these equations is the general heat balance equation of a calorimetric system. The heat balance equa­ tion Eq. (1.147) describes in general form the courses of the heat effects in a calorimeter of any configuration of domains and any localization of heat sources. The general heat balance equation corresponds to the formalization of the general calorimetric model by means of the set of equations with lumped parameters. To consider the thermal properties of a calorimeter, the detailed form of this equation has to be derived. It is necessary to define in it the number and configuration of the distinguished domains and the centers which separate these domains and where heat transfer takes place. The representation of the calorimeter by mathematical models de­ scribed by a set of heat balance equations has long traditions. In 1942 King and Grover [22] and then Jessup [23] and Churney et al. [24] used this method to explain the fact that the calculated heat capacity of a calo­ rimetric bomb as the sum of the heat capacities of particular parts of the calorimeter was not equal to the experimentally determined heat capac­ ity of the system. Since that time, many papers have been published on this field. For example, Zielenkiewicz et al. applied systems of heat balance equations for two and three distinguished domains [25-48] to analyze various phenomena occurring in calorimeters with a constant– temperature external shield; Socorro and de Rivera [49] studied microeffects on the continuous-injection TAM microcalorimeter, while Kumpinsky [50] developed a method or evaluating heat-transfer coeffi­ cients in a heat flow reaction calorimeter.

36

CHAPTER 1

The calorimeter as a system of several domains has often been pre­ sented by using a method based on thermal-electric analogy [51–53]. This is based on the similarity of the equations describing heat conduc­ tion and electric conduction. Such a study generally involves both the use of circuit theory and the principle of dimensional similarity. In a corresponding network analog, the thermal resistances and capacities are simulated by electric resistors and capacitors, respectively. In construct­ ing such circuits, it is considered that there is an equivalence between the quantities: electric current and heat flow; electric voltage difference and temperature difference; electric resistance and thermal resistance; and electric capacity and heat capacity. The thermal–electric analogy method has been used to analyze and represent the operation mode of a number of calorimetric and thermal analysis devices. For example, Rouquerol and Boivinet [54] presented analogs of electric circuits for the following heat-flux and power com­ pensation devices: the conventional DTA instrument of Mauras [55], the differential scanning microcalorimeter of Arndt and Fujita [56], and the Tian-Calvet microcalorimeter [57]. Wilburn et al. [58] used both passive and active analogs to investigate the effect of the holder design on the shape of DTA peaks. The thermal-electric analogy was applied by Nico­ laus [59] to analyze the problem of reconstruction of the heat flux curve. Ozawa and Kanari [60] illustrate the discussed heat balance equations for constant heating rate DSC by an equivalent electrical circuit. The method of analogy is also used in the papers of Claudy et al. [61–63]. The thermal-electric analogy method is useful to represent the ac­ cepted structures, which distinguish the domains and the modes of their connection with themselves and the environment. It is also useful to formulate a suitable system of the balance equations for the bodies (do­ mains). The increase in the number of its applications is related to the development of analog computer calculation methods. It will be demonstrated below that the extended information range relating to the investigated calorimeter allows analysis of both its ther­ mal and dynamic properties. The use of the method based on the anal­ ogy of the thermal and dynamic properties of the investigated systems is profitable in this case.

Chapter 2

Calorimeters as dynamic objects

Calorimeters are physical objects that can be described in different ways. The steering theory treats a calorimeter as a dynamic object, in which the generated heat effects (input signals) are transformed to the quantity measured directly in the calorimeter, e.g. temperature (output signal). Let us describe the input signals by the functions and the output signals by the functions In fact, in calorimetry the output function never reproduces the input function directly. The calorimeter is a kind of transducer, which transforms the input functions into the output functions. Thus, to reproduce with the required accuracy on the basis of the relation between these two types of functions should be determined:

and the procedure should be selected so as to allow the inverse opera­ tion, i.e.

which permits determination of the input function on the basis of a given course of the output function of the calorimeter. When the unchanging relation between the input and output functions is determined, it fur­ nishes an explicit description of the dynamic properties of the calo­ rimeter. A calorimeter is usually regarded as an object that can be described by one differential equation or a system of linear differential equations with constant coefficients. These equations are treated as the mathemati­ cal models of calorimeters. If there are many output functions, then the dynamic object (calorimeter) is described by a system of n differential equations. Assuming linearity and applying the superposition rule, one

38

CHAPTER 2

may reduce two or n objects to one. On analysis of an object described by one input and one output function, we do not lose generality. A convenient method for solving linear equations is to apply the Laplace or Fourier transform. In the transfer function (transmittance), the dynamic properties are encoded. The function is defined as the quo­ tient of the Laplace transforms of the output and input functions. The spectrum transmittance is defined as the quotient of the Fourier trans­ forms of the output and input functions under zero initial conditions. The transmittance (transfer function) of the object has the form

where s is a complex variable; x(s) and y(s) are the Laplace transforms of the output signal x(t) and the input signal y(t), respectively. Equation (2.3) written in the form

can be visualized by a block diagram (Fig. 2.1). The objects of the

structure shown in Fig. 2.1 are called open objects in the steering theory.

Let us consider the quantities H(s), x(s) and y(s) of Eq. (2.4). Each of them can be obtained by using the two others: 1. The transmittance H(s) can be determined only on the basis of ex­ perimentally determined y(s) and x(s). The calorimeter is then treated as a “black box” and knowledge of their physical parame­ ters is neglected. It is assumed that the calorimeter can be treated as a linear, stationary and invariant object. 2. The transmittance H(s) of the calorimeter is based on a mathe­ matical model of the calorimeter with distinguished physical pa­ rameters and mutual relation between them. The experimental de­

CALORIMETER AS DYNAMIC OBJECT

39

mination of the functions y(s) and x(s) is applied for evaluation of the values of the dynamic and static parameters of the transmit­ tance. This procedure is equivalent to the calibration of the calo­ rimeter. 3. The transform x(s) is determined in the knowledge of the form of the transmittance H(s) and the values of their parameters. Input function y(s) is determined experimentally. This procedure is convenient to verify the agreement between experimentally de­ termined and calculated functions x(s). 4. The transform y(s) is evaluated on the basis of previously known H(s) and experimentally determined function x(s). It corresponds to the determination in the domain of the complex variable of the course of the investigated heat effect. The form of the transmittance depends on the type of the dynamic object.

2.1. Types of dynamic objects Among the open systems, the following types of dynamic objects can be distinguished [64, 65]: Proportional type, when the input function y(t) is proportional to the output function x(t):

where k is the proportionality coefficient, and the transmittance H(s) has the form

while the spectrum transmittance

is expressed by Eq. (2.7):

Integrating type, when the input function is proportional to the derivative of the output function:

40

CHAPTER 2

while

First-order inertial type, when the input function is a linear combination of the output function and its derivative:

while

Let us compare the equations describing the dynamic properties of the distinguished types of objects with suitable heat transfer equations: 1. We compare Eq. (2.5), which describes the dynamic properies of a proportional object, with the Newton heat transfer equation [Eq. (1.46)]; 2. We compare Eq. (2.8), which describes the dynamic properties of an integrating object, with the first term of the left-hand term of Eq.(1.99); 3. We compare Eq. (2.11), which describes the dynamic properties of an inertial object, with the heat balance equation of a simple body [Eq. (1.99)]. It can readily be imagined that a calorimeter in which the course of heat power within time corresponds (with the accuracy of the factor) to the course of the changes in the output function (e.g temperature) has the properties of the proportional object. The properties of integrating

CALORIMETER AS DYNAMIC OBJECT

41

objects are those of adiabatic calorimeters, inside which the accumula­ tion of heat occurs. Calorimeters that are inertial objects comprise the most numerous group of calorimeters. Let us assume that a calorimeter has only the integrating or proportional properties of the object, as a certain idealization. Of course, this idealization is well-founded in many cases. The relation describing the dynamic properties of the object is equivalent to the mathematical model of the calorimeter. It is expressed as a function of time, frequency or a complex variable domain.

2.2. Laplace transformation The Laplace transformation [66] is the operation of changing one expression into another by integration. In this transformation, the function f(t) of the real variable is changed into the complex function F(s) of the complex variable s. The Laplace transform is defined by Eq. (2.14):

and abbreviated as

where while for the existence of the transform F(s) the condition must be fulfilled. The inverse Laplace transformation is defined by Eq. (2.16):

where C is a contour that outlines all extremes of the function in the integral formula. This operation is abbreviated as The Laplace transformation is very convenient to use. Its advantages include: 1) The Laplace transforms of simple functions can be deter­

CHAPTER 2

42

mined by direct integration or integration by parts. In most cases, the simple function f(t) and the transform F(s) representing the function transform pairs are tabulated; 2) the Laplace transformation is a linear transformation for which superposition holds; 3) by application of the Laplace transformation, an ordinary differential equation is reduced to the algebraic equation of the transform, called the subsidiary equation of the differential equation. Let us apply the Laplace transform to the heat balance equation of a simple body (Eq. 1.99):

On dividing by G and putting

we obtain

Use of the Laplace transformation for Eq. (2.19):

gives the solution for Eq. (2.19) in the complex domain

After simple rearrangement, Eq. (2.21) becomes

Equation (2.22) is called the subsidiary equation of differential Eq. (1.99); T(s) is the response transform of the output function; F(s) and are driving forces; and is the characteristic function of the object. The first and third terms on the right-hand side of Eq. (2.22), the function of the initial conditions, are the transforms of the transient

CALORIMETER AS DYNAMIC OBJECT

43

solution. The second term, which is independent of the initial condi­ tions, represent the transform of the steady-state solution. The inverse Laplace transformation defines the function T(t) charac­ terizing the course of the temperature changes of the calorimeter. When the heat effects are not generated, and and thus T(t) depends only on the initial temperature difference T(0) between the calorimeter and its environment. Then:

In a similar way, the Laplace transform can be applied to obtain the solution of Eq. (2.20) for the other input functions. If it is assumed that two first-order inertial objects in series are dis­ tinguished in the calorimeter, while the output function of the first object is at the same time the input function of the second object (Fig. 2.2), then the calorimeter transmittance H(s) has the form

In the time domain, the system is described by the following differ­ ential equations:

Increase of the number of inertial objects causes significant changes in the course of the output function. Let us assume that the objects are arranged in series in such a way that the input function of the next iner­

44

CHAPTER 2

tial first-order object is the output function of the previous object (Fig. 2.3).

A graphical presentation of the output functions for numbers of objects ranging from one to six, caused by input function forcing corresponding to the unit step function, is shown in Fig. 2.4.

The block diagrams presented in Figs 2.1 - 2.3 are characteristic for the open systems and differ from one another only in the number of inertial objects. The dynamic objects are not always arranged in series. In many cases, as a result of self-arrangement of the objects and their configurations, we must consider the set of differential equations presented by the block diagrams of closed-loop systems. For example, for the calorimeter described by the differential equation

the resulting block diagram is a in Fig. 2.5, while for the calorimeter described by the following set of differential equations:

CALORIMETER AS DYNAMIC OBJECT

45

where

the resulting block diagram is b in Fig. 2.5.

When a larger number of inertial objects are distinguished, the calorimeter transmittance will have a more complicated form. For N objects, the transmittance has the form

and we obtain the form of the calorimeter transmittance H(s):

In this case, in the time domain the calorimeter is described by the following set of differential equations:

where

CHAPTER 2

46

When the physical parameters of the system are neglected, Eq. (2.30) takes the form

On application of the Laplace transformation, Eq. (2.33) under zero initial conditions can be written in the form Eq. (2.4):

where

is the transmittance of the analyzed system; m < N; and N denotes the system rank. Determination of the transmittance expressed by Eq. (2.34) is eqivalent to calculation of the polynomials

and

The equation

is called the characteristic equation and its roots are the “eigenvalues” or “poles” of transmittance. If it is assumed that in Eq. (2.35) the polynomial has only single zero values, we can write

The roots

of Eq. (2.37)

CALORIMETER AS DYNAMIC OBJECT

47

are named the “zeros” of transmittance, and the polynomial in this equa­ tion is expressed by

On substitution of Eqs (2.36) and (2.38) into Eq. (2.34), the transmittance H(s) can be written in the form

The poles and zeros of transmittance express the inertial properties of the calorimetric system as a dynamic object. With the introduction of

transmittance H(s) becomes

where

is a constant called the static factor.

2.3. Dynamic time-resolved characteristics The relation that describes the output function changes in time caused by the action of the input function is given by the dynamic timeresolved characteristics. In calorimetry, the same input functions are used for their description as in control theory [64]. However, the termi­ nology used for this purpose is different. Thus,

48

CHAPTER 2 1. The input function described by a short-duration heat pulse of relatively high amplitude, called in control theory a unit pulse function (impulse function, Dirac function) (Fig. 2.6), is expressed by

The unit pulse whose surface area is equal to one has a Laplace transform y(s) equal to one. 2. The input function described by a constant heat effect in time corresponds to the unit step function (Fig. 2.7):

CALORIMETER AS DYNAMIC OBJECT

49

The unit step function corresponds to the integral of the unit pulse function with respect to time. The Laplace transform of the unit step function is 3. The input function described by a heat effect that is constant in time over a determined interval of time corresponds to the input step function of amplitude b and time interval a, called the rectangular pulse (Fig. 2.8): where u(t) is determined by

while u(t–a) is expressed by

This is the shape of the input function that is applied when the cali­ bration of the calorimeter consists in generation of a Joule effect that is constant in time for a defined duration. The exceptions to the rule are those instruments in the calibration of which the frequency characteris­ tics are used. The Laplace transform of the rectangular pulse is expressed by

CHAPTER 2

50

4. The input function described by a heat effect rising linearly in time is presented in Fig. 2.9 and expressed by Eq. (2.49): The generation of such a forcing function is used in steering theory as well as in adiabatic and scanning calorimetry. The ramp function has the following Laplace transform:

5. Generation of the heat effect of the first-order kinetic reaction is expressed by the exponential function (Fig. 2.10)

The Laplace transform of which is

CALORIMETER AS DYNAMIC OBJECT

51

6) To evaluate the dynamic properties of calorimeters and calibrate the instruments, period input functions have also been used (Fig. 2.11).

where A is the amplitude of oscillation, and

is the oscillation frequency,

The Laplace transform of the sinusoidal input function is

These periodic heat forcing functions are the basis for some calo­ rimetric methods, e.g. those used in modulated scanning calorimetry. Determination of both the transmittance of the investigated object and the Laplace transform of the input function y(t) furnishes the output function x(s) = y(s)· H(s). With the inverse transformation, we obtain y(t). Output functions x(t) of proportional, integrating and inertial ob­ jects for various input functions are collected in Table 2.1. The time-resolved dynamic characteristics presented in Table 2.1 show that the shapes of the output functions depend strongly on the type of the dynamic object. For proportional objects, the output and input functions have the same shape, while their values are equal to each other with the accuracy of the factor. This means that in a calorimeter with the dynamic properties of a proportional object the output function gives direct information on the course of the output function; in other words, the course of the experi­ mentally determined function T = T(t) corresponds to the course of the changes in heat power P in time t.

52

CHAPTER 2

CALORIMETER AS DYNAMIC OBJECT

53

54

CHAPTER 2

For integrating objects, the course of the output function corresponds to the accumulation process and to the operation of integration. The object responds to a generated unit pulse with an output signal, which is equivalent to the unit step function; the response of the object to the unit step function is a linearly rising function; production of the ramp forcing function stimulates the response of the object according to the relation For inertial objects, the course of the output function is induced by the inertial properties of the object (calorimeter). This results from the transmittance form H(s), which is expressed by the operator 1/(Cs+k) or the operator while the relation between the input function y(s), the output function x(s) and the transmittance H(s) is presented graphically by a block diagram (Fig. 2.12)

When the trasmittance is represented by the single symbol H(s), it is depicted by the block diagram presented in Fig. 2.1. The form of the transmittance H(s) indicates that the time constant is a decisive parameter for characterizing the inertial properties of the object (calorimeter). This also means that the value of the time constant determines the course of the output function, the character of which is approached more closely for either proportional or integrating objects. Simply, the values of control the inertial, damping properties of the object. Different values of the function x(t), depending on the values of are responses to the same heat forcing (Fig. 2.13). The courses of the output functions caused by the generation of the sinusoidal input function for proportional, integrating and inertial firstorder objects are also presented in Table 2. 1. It is seen that, for a propor­ tional object, only the value of the sinusoidal (harmonic) oscillation frequency changes. For an integral object, the sinusoidal input function is transformed by the object to a cosinusoidal function. The amplitude of the output function is then inversely proportional to the frequency of the

CALORIMETER AS DYNAMIC OBJECT

55

sinusoidal input function. The frequency phase lag is 90° relative to the input function. For inertial objects, the sinusoidal input function is trans­ formed by the object to an other sinusoidal function that has different phase and amplitude. In the following relation, expressing the output function x(t) for the steady state

where

the factor expresses in terms of f requency the ratio of the output and input function amplitudes. The values of the factor and the size of the phase shift characterize the dynamic properties of the iner­ tial object. For a frequency close to zero, when sinusoidal changes have low frequency, the course of the output function is similar to that of the input function y(t). The phase shift is then close to zero and the proper­ ties of inertial and proportional objects become very similar. The shift in the output function course relative to the input function results from the rise in frequency. For infinitely high frequency, the shift in the course of x(t) is expressed as radians or –90°. When the frequency is related to then the oscillation frequency is 0.707 the conversed time constant radians or 45°. This is the frequency related to and the phase shift is the transfer function of an inertial object, given by the operator or

2.4. Pulse response The pulse response function is the output function h(t) caused by the action of the input impulse function (Dirac function). It is applied for determination of the particular forms of the Laplace transmittance. It can be obtained by applying the Laplace inverse transformation to the transmittance Eq. (2.41):

CHAPTER 2

56

where coefficients

have the form

from the Cauchy theorem of residues. The pulse response function is a positive function of real argument t. It fulfils the condition given by Eq. (2.43):

Taking into account Eq. (2.57) and integrating, we have

If we take advantage of the theorem of the initial value of the original, i.e. the function h(t), the initial value of the pulse response h(t) on the basis of Eq. (2.34) can be given by

The pulse response can be obtained experimentally as the response of the calorimetric system: a) to a rectangular heat pulse of short duration (“experimental” Dirac pulse) expressed as a sequence of discrete values

where and is the sampling period; or b) to the unit step function lasting a sufficiently long time to achieve the stationary state of heat transfer. In this case, the pulse response is expressed by the derivative of the calorimetric response

CALORIMETER AS DYNAMIC OBJECT

With the measured values of the response respectively

57

at times

and

numerical derivation has to be performed in order to obtain discrete values of the pulse response h(t). In the first case, the accuracy of obtaining the values of the unit pulse response depends on the degree to which the “experimental” pulse ap­ proximates to the “ideal” Dirac function and on the accuracy of the measurement of the calorimetric signal. In the second case, the proce­ dure of numerical derivation influences the accuracy of obtaining the pulse response. Since the experimentally obtained response of the sys­ tem does not fulfil the condition [Eq. (2.49)], it is necessary to calculate the integral from the course obtained for the calorimetric signal and divide all the values of the signal by the value of this integral. Thus, we obtain a new course that fulfils the condition needed. From Eqs (2.59) and (2.60), the dependences between the amplitudes and time constants can be obtained:

a) for the first-order system:

b) for the second-order inertial system:

c) for the second-order inertial system whose transmittance contains

one zero

It results from Eqs (2.65) and (2.66) that the values of coefficient and depend not only on the time constants, but also on the zeros of transmittance.

CHAPTER 2

58

In Fig. 2.13, the plots of pulse responses of calorimeters of various orders are shown. As the order of the system becomes higher, the pulse response is “flattened”, and its maximum value drifts more in time.

2.5. Frequential characteristics To analyze the dynamic properties of calorimeters, frequential char­ acteristics, similarly as time-resolved characteristics, are determined [8, 67]. To obtain the frequency characteristic, Fourier transforms are used. The Fourier transform which is a complex function of the real variable can be written as follows: or in the form If it is assumed that

where and the transmittance

are the real (even) and imaginary (odd) parts of respectively, we can write

CALORIMETER AS DYNAMIC OBJECT

59

Because

from Eq. (2.69) we have

Magnitude

is called the amplitude. Division of Eq. (2.69) leads

to

Thus Magnitude is called the phase and is equal to the argument of transmittance The phase describes the relative amounts of sine and cosine at a given frequency. The spectrum transmittance of an N-order inertial system has the form

Equation (2.75) is equivalent to Eq. (2.67), assuming that In the spectrum transmittance described by Eq. (2.75), let us distinguish the component transmittances and

If this becomes

and for

It results from Eqs (2.78) and (2.79) that the amplitude of the trans­ is equal to the product of the amplitudes of particular mittance

CHAPTER 2

60

transmittances. Thus, the phase of the transmittance is equal to the sum of the particular transmittances of the phases. is described by According to Eqs (2.75) and (2.79), the phase the equation

If the magnitude approaches infinity, then, according to Eq. (2.80), the value of the limit phase approaches

To analyze the dynamic properties of the object, the amplitude char­ acteristics, phase characteristics and amplitude-phase characteristics are used. The amplitude characteristic is a relation expressing the ratio of the amplitude of the output function to that of the input function. The dependence of a phase shift in frequency is called the phase characteris­ tic of the object. The amplitude-phase characteristic presents the ampli­ tude changes and the phases of the output function. Two types of plots are usually used to draw the amplitudes: one in the coordinates and the other in the coordinates and two types of plots for the phase: in the coordinates or The plot in gives the amplitude-phase characteristics. the coordinates Let us analyze the amplitude-phase characteristics for a few types of spectrum transmittance. is described According to Eqs (2.72) and (2.75), the amplitude by the function

For sufficiently large values of approximated by the equation

the function

can be

CALORIMETER AS DYNAMIC OBJECT

61

Taking logarithms of both sides of the above equation gives

It results from the above equation that the plot of the amplitude in the asymptotically approaches a straight line coordinate system with direction coefficient equal to –(N – m). In this way, the asymptotic plot permits an estimation of the difference between the number of poles and the number of zeros of the transmittance. The plot of helix shape which passess through the –(N – m) guater of the system in the coordi– nates will be obtained. The number of –(N – m) is related with the phase shift [Eq. (2.81)].

2.6. Calculations of spectrum transmittance The spectrum transmittance can be obtained as the Fourier transform of the pulse response or as the quotient of the Fourier transform of the system response to a known heat effect.

where is a heat pulse of constant heat power and time interval u (input step function)(Fig. 2.14a); is the temperature response to this heat effect (Fig. 2.14b). In order to determine the spectrum transmittance it is neces­ sary to calculate the integrals on the right-hand side of Eq. (2.85). Ac­ cording to the Euler formula:

where

CHAPTER 2

62

and are the real (even) and imaginary (odd) parts of the Fourier transform of the response function, respectively. The result of the calorimetric measurement is obtained as the number sequence temperature data [Eq. (2.66)]

at times respectively. Thus, the integrals of the calorimetric [Eqs (2.87) and (2.88)] are to be calculated numerically by signal applying a convenient approximation, e.g.

Then, Eqs (2.87) and (2.88) become

CALORIMETER AS DYNAMIC OBJECT

63

Integration and rearrangement of Eqs (2.90) and (2.91) gives

In this way we obtain the real and imaginary parts of the Fourier transform of the response function to a heat pulse. The Fourier transform of the input function should also be determined. Let us determine the transforms for the following input functions: a step input function, a pulse function, and a periodic (sinusoidal) func­ tion). These functions are often used for spectrum transmittance deter­ mination. The Fourier transform of the heat pulse can be written in the form

where

are the real and imaginary parts, respectively, of the Fourier transform of the input pulse. According to Eqs (2.85), (2.86) and (2.92) – (2.94), we have

CHAPTER 2

64

or

where

When the calorimeter is calibrated by a unit pulse of amplitude and sufficiently long duration, the spectrum transmittance can be expressed as

where is the calorimetric response to a unit pulse heat effect. Ap­ plication of the approximation of the calorimetric response given by Eq. (2.89) yields

and, after integration, the spectrum transmittance obtained can be writ­ ten in the form

where

CALORIMETER AS DYNAMIC OBJECT

65

On the basis of the above method, the algorithm of the calculation of the Fourier transforms expressed by Eqs (2.85) and (2.100) is calculated by using the Fast Fourier Transform Method [68]. The Fourier transform of the impulse response h(t) can be obtained in the case of a sinusoidal input function (Fig. 2.11):

The Fourier transform is equal to

Hence

becomes

where

It results from Eqs (2.96)–(2.99) that the accuracy of the determina­ depends on the accuracy of the tion of the spectrum transmittance determination of and Errors connected with their determi­ nation are due to the approximation of the thermograms or and also connected with the accuracy and precision of the measurements made.

CHAPTER 2

66

2.7. Methods of determination

of dynamic parameters

Selected methods will now be discussed that are used to determine the dynamic properties of calorimeters as inertial objects. Several of these methods are similar to those used in steering theory. Different methods are presented for determining the time constant of a calorimeter treated as an inertial object of first order. The methods used to determine the dynamic parameters of calorimeters that are inertial objects of order higher than one are also discussed. All these numerical methods, algorithms and listing programs are described in detail in [67]. To apply each of the methods presented below, a knowledge of the physical properties of the investigated object is not necessary. These methods have been qualified as useful in calorimetry to identify the dynamic parameters and to study thermokinetics. To obtain the most information about the properties of the calorime­ ter, it is recommended to determine the physical properties of the par­ ticular domains of the calorimeter and quantities characterizing the heat transfer between the domains themselves and between the domains and the environment. When we follow this procedure, the dynamic proper­ ties of a calorimeter can be determined by using the method of Ndomains based on the general heat balance equation [Eq. (1.147)]. This method will be presented in Chapter 3.2.4.

2.7.1. Determination of time constant The dynamic properties of a calorimeter treated as an inertial object of first order are characterized unambiguously by the time constant To evaluate the time constant on the basis of the heat balance equa­ tion of a simple body, different input functions are used. Consider the determination of by applying the input step function [Eq. (2.45)] under conditions where the initial temperatures of the calorimeter and isother­ mal shield are the same. Equation (2.19) can then be written in the form

CALORIMETER AS DYNAMIC OBJECT

67

where

is determined by Eqs (2.45) and (2.47).

The Laplace transform of the function y(t) corresponds to

while after the Laplace transformation Eq. (2.111) becomes

After simple rearrangement, Eqs (2.113) and (2.114) can be written in the form

Inverse Laplace transformation of Eq. (2.115) gives

The function T(t) expressed by Eq. (2.116) can be presented graphi­ cally by using the curves I, II and III shown in Fig. 2.15. The first term on the right-hand side of Eq. (2.116) is related to curve I, and the second term on the right-hand side of this equation to curve II. Curves I and II have identical shapes, but there is a shift in time between them, related to the duration of the heat impulse produced. For this period of time, the courses of the changes in the calorimeter temperature T in time t are represented by the interval 0K of curve I. When t > a, the course of the to the shield temperature is presented in curve III. function T(t) from This is the cooling curve of the calorimeter. The changes in temperature that occur here are only a result of the existing difference in tempera­ The interval KM of curve III graphically represents the differ­ tures ence between the values T(t) of the first and second terms on the righthand side of Eq. (2.116).

CHAPTER 2

68

When the heat generation period is long enough for a new state of thermal equilibrium to be reached, characterized by Eq. (2.19) can be written in the form

The solution of Eq. (2.117) when T(0) = 0 is

which corresponds to the first term on the right-hand side of Eq. (2.116). Equation (2.118) describes the heating process occurring in the calorimeter (curve 0I, Fig. 2.15). Let us use Eq. (2.118) to present several procedures for determining the time constant Procedure 1. We find the time derivative of the heating curve

From Eq. (2.119), we have that a tangent to the curve 0K crosses an (Fig. 2.15). asymptote of the curve at the point relating to

CALORIMETER AS DYNAMIC OBJECT

69

Procedure 2. At every point of the heating curve characterized by the interval 0K of the curve I (Fig. 2.15), the length of the subtangent determined by its point of intersection with the straight line is related to the time constant From Eqs. (2.117) and (2.118), we have

If we take into account the graphical presentation of the heating curve in (Fig. 2.15), we have

where

Thus, the length of the subtangent is equal to the time constant value Procedure 3. After time

T(t) is equal to

After time the temperature becomes and after we have Proceeding in such a way for the time known values of and T(t) of the heating process, we can obtain the value of the time constant or its multiplicity. Procedure 4. The value of the time constant can be determined analytically. The values of T(t) related to time moments and cor­ respond to

Hence, Eqs (2.124) – (2.126) become

CHAPTER 2

70

Thus

and after rearrangement we have

Procedure 5. To obtain the time constant can be used.

the integration method

Hence, the time constant is equal to

Integral

represents the surface area F1 between the

straight line and the heating curve (Fig. 2.16a). The changes in temperature T(t) in time, caused only by the initial temperature difference between the calorimeter and shield, are also used to determine the time constant In this case, it is assumed that y(t) = 0, and the cooling process is expressed by equation

CALORIMETER AS DYNAMIC OBJECT

71

The solution of Eq. (2.134) is

The course of T(t) is graphically presented by the cooling curve (curve IV) in Fig. 2.15. The procedures used to determine the time constant on the basis of the cooling curves are not much different from the procedures presented above. Procedure 1. The same as in the case of the heating curve; the length of the subtangent at every point of the cooling curve is equal to the time constant (see heating curve, Fig. 2.16, Procedure 2). Procedure 2. After time the decrease in T(t) is 36.8% com­ pared to the initial value This results from Eq. (2.135). In a similar way, the value of T(t) can be determined for the time constant multiplic­ ity. Knowing temperature and choosing a matching value of T(t), one can determine the value of the time constant. Procedure 3. The value of the time constant can be determined ana­ lytically if at least two temperature values of the cooling process defined as a function of time are known. When, after time of the cooling proc­

CHAPTER 2

72

ess, the value of T(t) becomes then according to Eq. (2.135)

and, after time

it becomes

Division of both sides of Eq. (2.136) by Eq. (2.137) gives

Hence

Procedure 4. The time constant for the cooling process can be calculated by an integrating procedure as for the heating process (Fig. 2.16b). On calculation of the integral:

Hence

Procedure 5. Taking logarithms of both sides of Eq. (2.135) gives

Hence

Equation (2.143) is used for the graphical determination of the time constant When the experimentally determined T and the t data for the cooling process have been obtained, the plot in the coordinates (t, lnT)

CALORIMETER AS DYNAMIC OBJECT

73

can be drawn, as shown in Fig. 2.17a [8]. When the dynamic properties of the calorimeter are characterized by one time constant the relation lnT = f(t) is expressed by a straight line (Fig 2.17a), which forms an angle with the t axis, for which

This line cuts the axis at the point of the ordinate. Thus

When the dynamic properties of the calorimeter are characterized by more than one constant of time [Eq. (2.57)], at the begining of the cooling process the relation lnT = f(t) is not linear, as shown in Fig. 2.17b. In this case, the discussed method can be applied to evaluate the higher time constants. The following procedure is used. The straight line interval (Fig. 2.17b) is extended to the point of intersection with the lnT axis and the first time constant is determined. Next, a new function of the form

is created and a new plot in the coordinate system can be drawn. is determined. The iterative procedure is The second time constant repeated until a plot similar to the plot given in Fig. 2.17a is obtained.

CHAPTER 2

74

The number of time constants which can be distinguished depends on the properties of the calorimeter and the accuracy of the determination of the experimental data.

2.7.2. Least squares method The least squares method for determination of the transmittance pa­ rameters was proposed by Rodriguez et al. [69]. This method allows the and the zeros of the transmittance nominator to be time constants obtained by approximation of the pulse response of the calorimeter by the least squares nonlinear curve-fitting procedure described by Marquardt [70]. In order to use this method, it is necessary to assume or determine the order of the model number of poles and the number of transmittance zeros. According to these assumptions, the pulse response is a func­ tion of time t, poles and zeros

where m < N. Introducing vector and zeros:

the components of which are poles

Equation (2.147) can be written in the form As is a nonlinear function of the components, expansion of this function as a Taylor series in the neighborhood of point is applied:

neglecting the derivatives of higher order. As a criterion of fitting the and approximated pulse experimentally determined pulse response it is assumed that response

CALORIMETER AS DYNAMIC OBJECT

75

Equation (2.150) then becomes

where < 0, T > is the time interval in which the changes in pulse response are measured, and

In possession of the discrete values of the pulse response, instead of Eq. (2.152) the following fitting criterion is accepted:

where and is the sampling period. From the necessary condition of the minimum of the function expressed by Eq. (2.154):

we have the following set of equations:

Solving the set of Eqs. (2.156) with respect to leads to the poles and zeros of transmittance according to the equation

Since the linear approximation is applied to the nonlinear function, we can obtain the values of the parameters only with large error in one iteration. Thus, it is necessary to repeat the iteration, assuming the calculated values as initial values and repeat the iterative procedure until we obtain a suitable approximation of the pulse response.

CHAPTER 2

76

2.7.3.

Modulating functions method

The modulating functions method was proposed by Ortin et al. [71]. This method permits determination of the poles and zeros of transmit­ tance. The poles of transmittance are determined on the basis of an ac­ cepted model in the form of a differential equation. As the pulse re­ sponse h(t) is the finite sum of exponential functions [Eq. (2.157)]:

it can be assumed that it satisfies the differential equation with constant coefficients:

where is k times the derivative with respect to time t of the pulse Next, the time interval in response of the system, and which the changes in the pulse response are measured (observed) is assumed as the basis of the modulating functions which fulfils the conditions

for i = 1, 2, ... , N; k = 0, 1,..., N – 1 is also assumed. These functions have continuous derivatives of desired order. The order and the smooth­ ness are connected with the order of differential Eq. (2.159). Multiply­ ing both sides of Eq. (2.159) by and intergrating with respect to time t in the interval yields

Putting

leads with respect to the coefficients equations:

to the following set of algebraic

CALORIMETER AS DYNAMIC OBJECT

77

The above set of equations can be solved with respect to From the values of the coefficients from the characteristic equation

it is possible to determine the eigenvalues (poles) of the transmittance, taking into account that the time constants The expressions in Eq. (2.162) contain the derivatives of the pulse response function h(t). To avoid the calculations of these derivatives, which can result in the introduction of large errors, Eq. (2.162) (after integrating by parts) can be presented in the form

taking into account the conditions given by Eq. (2.160). When we have determined the poles of transmittance and know the value of the pulse response h(t) in Eq. (2.158), only the amplitudes are unknown. These coefficients can be determined by applying the modulating function procedure given above to the pulse response:

On putting

we obtain the following set of equations:

78

CHAPTER 2

The coefficients of the pulse response h(t) can be determined by solving the above set of algebraic equations after previous calculations of the coefficients From the determined time constants and coefficients the Laplace transform to the pulse response gives

where

is the denominator of the transmittance, and

is the m–degree polynomial m < N with respect to s. By solving the equation the zeros of the transmittance H(s) are obtained. The general form of the equation expressing the relation between the generated heat power P(t) and the temperature changes T(t) of the calorimetric system has the form

Multiplying both sides of the above differential equation by the modulating function and integrating with respect to time t in the interval leads to the following set of algebraic equations with respect to coefficients and

where

CALORIMETER AS DYNAMIC OBJECT

79

The obtained set of Eq. (2.175) has a similar form as that of the set of Eq. (2.162). In the particular case when the calorimeter is calibrated by a constant heat effect of heat power and duration u smaller than the relation for coefficients given by Eq. (2.177) (taking into account the conditions Eq. (2.159) and integrating by parts), can be simplified to the following form:

The flow diagram of the modulating functions method, the program algorithm and the listing program are given in [67].

2.7.4. Rational function method

of transmittance approximation

In this method, the transmittance H(s) is approximated by a rational function [67, 72, 73]:

where D(s) and L(s) are polynomials of degree m and N(m
80

CHAPTER 2

The rational function L(s)/D(s) is the approximation expression of function H(s) if the expansion in the power series with respect to s is identical with the expansion in the power series of function H(s) to the degree m +N. As the criterion of fitting, we can assume

or, using Eq. (2.183), we can write

where is the number of points taken into account in the approximation. The values are calculated by numerical integration, having the of the values of the pulse response h(t) and assuming a set of values complex variable s. In order to adequately define the criterion Eq. (2.184), the values of variable s must be real. Function (2.185) is a and function of N + m + 1 unknown parameters (n = 0, 1, ... , N). The necessary condition of the minimum of function (2.185) with respect to parameters and has the form

and gives the following set of algebraic equations:

or, after simplification:

CALORIMETER AS DYNAMIC OBJECT

81

On solving the above set of equations with respect to parameters and we can calculate their values as well as the polynomials D(s) and L(s), the quotient of which gives the expression approximating to the transmittance H(s). The roots of the polynomial L(s) yield the zeros of transmittance, and the roots of the polynomial D(s) yield the poles of the transmittance.

2.7.5. Determination of parameters

of spectrum transmittance

To determine the dynamic parameters of the calorimetric system, the transmittance of the compensating system (numeric or analog) [67, 74] should be matched in such a way that the resultant transmittance corresponds to the non-inertial system and fulfils the following condition: where is the spectrum transmittance of the calorimeter. As the basis for determination of the transmittance, the experimental pulse re­ sponse of the calorimeter is taken. Next, through use of the Fast Fourier Transform (FFT) procedure, the values of the amplitude (modulus) and the phase of the Fourier transform of the pulse response are calculated. These values are plotted in a Bode plot. From the Bode plot, the value of the direction coefficient of the last slope asymptote, which permits an evaluation of the difference between the number of zeros and the poles of the transmittance is determined. If this coefficient is equal to –1, the transmittance can have one pole, or one zero and two poles, or

82

CHAPTER 2

two zeros and three poles, etc. If this coefficient is equal to –2, the transmittance can have two poles, or one zero and three poles, or two zeros and four poles, etc. To illustrate our considerations, let us assume, for example, that the value of the direction coefficient of the last asymptote is equal to –1.

Having determined the difference between the number of zeros and poles of the transmittance, on the basis of the Bode plot (Fig. 2.18) we can find the point at which the first slope asymptote intercepts the axis, and thus estimate the time constant with he largest value, according to the relationship In the knowledge of the time constant transmittance

we determine a new

and its amplitude and phase and draw their plots. If the plot of the amplitude and the plot of the phase are similar to the plots of the relations and for a first-order inertial object in the accepted frequency range, we stop our calculation and assume that our system is the first-order inertial system of transmittance:

In the other case, from the relations and it can result that the number of zeros is equal to the number of poles. We assumed ear­ lier, that the value of the direction coefficient is equal to –1, and we

CALORIMETER AS DYNAMIC OBJECT

83

have to deal with one pole and one zero, or two poles and two zeros, etc. Let us assume additionally that the experimental data enable us to de­ termine only one zero and one pole more. In the first case (Fig. 2.19), as emerges from the previous considera­ tions, the value of the zero of transmittance is larger than the value of from the slope of the the second pole. Thus, we can determine asymptote, and we have

From the value of the horizontal asymptote, we can calculate the second time constant

After calculation of the parameters of the transmittance, we can write

In the second case, the value of the zero of transmittance is smaller than the value of the second pole. In this case, from the slope of the asymptote we can calculate and thus the value of the second time constant is

84

From the value from the formula

CHAPTER 2

of the horizontal asymptote, we can calculate

and write the form of the transmittance as above. If the amplitude plot and the phase plot of the obtained transmittance are similar to the in an appropriate relation that results from the dependence frequency range, we can assume that the parameters of the transmittance have been determined correctly.

Chapter 3

Classification of calorimeters Methods of determination of heat effects 3.1.

Classification of calorimeters

The papers that consider determination of the heat effects that ac­ company physical and chemical processes present a wide spectrum of types of calorimeters. These devices have been given various names by the authors, who made their choices on the basis of different criteria. Names such as low-temperature calorimeters, high-temperature calo­ rimeters and high-pressure calorimeters come from the conditions of temperature and pressure under which the measurements are performed. In some cases, the type of process investigated is decisive: calorimeters for heat of mixing, heat of evaporation, specific heat measurements, and others. The names of calorimeters often have to contain information about their construction features, e.g. labyrinth flow calorimeter, calo­ rimetric bomb, drop calorimeter, or stopped-flow calorimeter. The name of the device sometimes stems from the name of its creator. Examples here include the calorimeters of Lavoisier, Laplace, Bunsen, Calvet, Swietoslawski, Junkers, and others. This diversity of the names of calo­ rimeters justifies an attempt to find features that classify the devices unambiguously. Let us first define a “calorimeter” as an instrument devised to deter­ mine heat. In any calorimeter, we may distinguish: 1) the calorimetric vessel (often called the cell, container, or calorimeter proper) at temperature that is usually in good contact with its contents, in which the studied transformation occurs. The contents include the reactant samples and subsidiary accessories necessary to achieve the investigated transformation (e.g. to initiate the reaction, or to mix the reagents) or to of­ calibrate the device; and 2) the surroundings at temperature

86

CHAPTER 3

ten called the shield, environment or thermostat. The surroundings form a part of the calorimeter that is functionally distinct from the measuring system, with a defined temperature time dependence [82]. It may also be considered the first thermostat shield of the calorimetric vessel (in some calorimeters there are several shields). There are single, differential and twin calorimeters (see 3.2.2.). Various classifications of calorimeters have been presented [75–83]. The classification given here [84] is based on the assumption that the calorimeter is a dynamic object in which heat is generated. Calorimeters are graded by applying the criteria of the temperature conditions under which the measurement was made. As an initial basis for further consid­ erations, the Fourier – Kirchhoff equation (Eq. (1.29)) has been used in the following form:

A set of simplifying assumptions is next introduced to shorten this form of equation. We assume that the process takes place under isobaric conditions. On multiplying both sides of Eq. (3.1) by volume V, we have

where is equal to heat capacity C. Furthermore, assuming bound­ ary conditions of the third kind (Newton’s cooling law), we can write where is the temperature of the calorimetric vessel and is the shield temperature. Taking into account Eq. (3.3), Eq. (3.2) becomes

With the additional assumptions that mass transport takes place only in the x direction, and that temperature T and heat power P are functions of the one coordinate x and time t, e.g. and Eq. (3.4) becomes

CLASSIFICATION OF CALORIMETERS

87

Equation (3.5) describes the heat transfer in calorimeters with mass exchange. These calorimeters are called open calorimeters. In contrast, calorimeters in which mass exchange does not occur are called closed systems. Then, w = 0 and Eq. (3.5) takes the form:

In the construction of a calorimeter, it is possible to provide condi­ tions which make it possible to carry out an experiment in a desired manner, e.g. to impose the temperature conditions of the calorimetric vessel or of the calorimetric shield or the temperature difference between them. In const., this manner, the following cases of temperature conditions can be dis­ tinguished:

On the basis of the criteria listed above, calorimeters can be divided into two groups: I. Adiabatic calorimeters, in which the temperature gradient be­ tween the calorimeter proper and the shield is equal to zero 0); during the calorimetric measurement, heat transfer does not occur between the calorimetric vessel and the shield. II. Nonadiabatic calorimeters, in which the temperature gradient be­ tween the calorimeter proper and the shield is different from zero during the calorimetric measurement, heat transfer oc­ curs between the calorimetric vessel and the shield. Two subgroups of adiabatic calorimeters one can be distinguished:

88

CHAPTER 3

They can be called Ia. Calorimeters with constant temperature adiabatic-isothermal. Ib. Calorimeters where the temperature of the shield T0(t) changes in time. They can be called adiabatic-nonisothermal or simply adia­ batic. Nonisothermal calorimeters involve the following subgroups: IIa. Calorimeters with constant temperature T0(t) (isothermal sur­ rounding shield), called isoperibol calorimeters. changes in IIb. Calorimeters in which the shield temperature time (e.g. scanning calorimeters). In both subgroups IIa and IIb, there are calorimeters with a tempera­ ture gradient that is stable in time, or with a calorimetric vessel whose temperature changes in time. The calorimeters in subgroup IIa are nonadiabatic-isothermal, whereas those in subgroup IIb are nonadiabatic­ nonisothermal. The schedule of the presented calorimeters division according to the temperature conditions by Czarnota and Utzig [85] is shown in Fig. 3.1.

Let us determine particular forms of Eq. (3.6) for distinguished tem­ perature conditions. These equations will then be treated as general mathematical models of the classified calorimeter groups. Particular forms of Eq. (3.5) will be given only for the types of open or closed calorimeters that are known by the authors.

CLASSIFICATION OF CALORIMETERS

1. When (3.6) takes the form

89

and

Eq.

Calorimeters that are described by Eq. (3.7) are characterized only by heat accumulation. They are adiabatic-nonisothermal calorimeters, and usually called adiabatic calorimeters. Their functioning rule is based on the assumption that the temperatures of the calorimetric vessel and the shield change in the same manner during the measurement. Their dynamic properties are those of integral objects. Adiabatic calorimeters were first used by Richards [76, 86] and then Swietoslawski [76]. Nowadays, adiabatic calorimeters are used for stud­ ies of various transformations in a wide temperature range, from very high temperatures to very low, close to the absolute zero (0.001 K). Mention may be made of [83]: 1) low-temperature calorimeters, like those of Westrum Jr et al. [87–89], Furukawa et al. [90], Suga and Seki [91], and Gmelin and Rodhammer [92]; 2) room-temperature calorime­ ters, such as those of Swietoslawski and Dorabialska [93], Zlotowski [94], and Prosen and Kilday [95]; and 3) high-temperature calorimeters, e.g. those of Kubaschewski and Walter [96], West and Ginnings [97], Cash et al. [98], and Sale [99]. High-pressure adiabatic calorimeters were designed by Goodwin [100], Rastogiev et al. [101], Takahara et al. [102] and others. In measurements performed with the use of adiabatic calorimeters, quite large amounts of substances have been applied (even several dozen grams). A series of new adiabatic calorimeters was recently designed that allow measurements on very small amounts of substance (< 1 g), among them the calorimeters constructed by Ogata et al. [103], Matsuo and Suga [104], Kaji et al. [105], and Zhicheng Tan et al. [106]. One of the first scanning adiabatic calorimeters designed in the past was the DASM 1M microcalorimeter [107], used to determine the ap­ parent molar heat capacity and conformational changes of proteins and nucleic acids. Measurements are performed in the temperature interval from 10 to 100°C, the shield heating rate can vary from 0.1 to and the sensitivity of the instrument is A

90

CHAPTER 3

new generation of scanning adiabatic calorimeters is represented by the devices designed by S.V. Privalov et al. [108] and Plotnikov et al. [109]. Each of these instruments is constructed as the differential system. 2. When and the terms of the left-hand side of Eq. (3.6) are equal to zero. Then: Since and are identical during the measurement, the processes in such adiabatic calorimeters take place under isothermal conditions. These conditions can be fulfilled, when two heat processes of opposite sign and occur: where is the heat power of the process studied and is the compensating heat power. The best-known [83] such calorimeters are those of Lavoisier and Laplace (determination of the mass of melted ice) [110], Bunsen (pycnometric determination of the volume change of the liquid water-ice system) [111], Dewar (volumetric determination of the air vaporized) [112] and Jessup (room-temperature operation based on the melting of diphenylether) [113]. In all of these devices, determinations are made of the changes in the measured quantity that result from the phase transformation. However, in this case it is very difficult or impossible to obtain conditions that correspond to thermodynamic equilibrium [76]. Progress in the development of adiabatic-isothermal calorimeters was made by Tian [114–116], who utilized the Peltier and Joule effect to compensate the generated heat power. At present, the modern electronic and steering devices applied to generate these effects permit measurements with high accuracy. A good illustration of this is the achievements at Brigham Young University [117], where this type of calorimetry has been developed for almost 40 years. The titration calorimeter [118, 119] and the few high-pressure flow calorimeters [120–122] have been constructed there. 3. When and const., Eq. (3.6) becomes

CLASSIFICATION OF CALORIMETERS

91

Methods for the determination of heat effects in these types of calo­ rimeters are based on the assumption that where is the heat power generated during the process examined; is the compensating heat power generated additionally to carry out the calorimetric measurements under isothermal but nonadiabatic condi­ tions. This class includes the calorimeters of Olhmeyer [123], Kisielev et al. [124, 125], Dzhigit et al. [126], Pankratiev [127, 128], Wittig and Schilling [129], Zielenkiewicz and Chajn [130], Hansen et al. [131] and Christiansen and Izaat [132]. 4. When and Eq. (3.6) takes the form of Eq. (3.12) for the open system and Eq. (3.13) for the closed system:

It results from Eqs (3.12) and (3.13) that the calorimeter systems de­ scribed by these equations are open and closed nonadiabatic­ nonisothermal (n-n) systems. For the open calorimetric systems de­ scribed by Eq. (3.12), heat and mass exchange occur simultaneously. Closed, nonisothermal-nonadiabatic calorimeters have for a long time been the most widely used class of calorimeters. The heat effect that is generated in these calorimeters is in part accumulated the in calo­ rimetric vessel and in part exchanged with its surrounding shield. These are dynamic properties of inertial objects. The parameter that is decisive as concerns their properties is the time constant (or time constants). In this class of calorimeters, there are instruments that have time and others with time constants of several constants of They have different constructions and find various applications. Among them there are:

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a) Dewar vessel calorimeters in more or less sophisticated form, such as the well-known Nernst low-temperature calorimeter [133] or Pierre Curie and Laborde‘s twin calorimeter [134] used in radiol­ ogy. The Dewar-type calorimeters have found application in the determination of the average specific heats of organic liquids and their mixtures [135, 136], the heat of evaporation [137], the heat of polymerization [138–142], the heats of hydration of cements [143–146], etc. Vacuum jackets of special design are used, for in­ stance, in the accurate instrument of Sunner and Wadsö [147], et al. [149] and and the microcalorimeters of Wedler [148], Randzio [150] for measurement of the heats of chemisorption of gases on thin metal films. b) Calorimeters with the jacket filled with water or air are mainly used to measure heat effects with a duration of a few minutes, calculated by use of the method of corrected temperature rise (§ 3.2.9). The well-known Regnault-Phaundler equation is then applied. This group includes, for example, oxygen [151–153] and fluorine [154–156] calorimetric bombs and calorimeters used to study thermokinetics and the total heat effects of reactions [157– 160]. c) High-temperature calorimeters such as those of Eckman and Ros­ sini [161] or Mathieu et al. [162]. d) Heat-flow or conduction calorimeters, where the heat exchange between the calorimetric vessel and the isothermal shield is excel­ lent. Most of them make use of the thermometric “heat flow­ meters” in a differential assembly, like the well-known Calvet microcalorimeter [10] and the others that are characterized by very high sensitivity (they can even detect signals of and are applied for various kinds of investigations [163, 164], e.g. the metabolism in living organisms [165–167], sorption and kinetics of adsorption [168–171], photochemical reactions [93], calorimet­ ric titration [173–175], investigation of the properties of medica­ ments [176], and the precipitation or crystallization of lysozyme [177]. e) Calorimeters where the heat generated is transferred to the liquid that flows around the outside surface of the calorimetric vessel, as

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in the Junkers [178] and labirynth flow calorimeters [76, 179, 180]. f) AC calorimetry, which is applied intensively for the measurement of heat capacity in the region of phase transitions [181–185]. g) A number of “quasi-adiabatic” calorimeters are used in such a way that, in the calculation of the heat involved, the transfer of heat between the calorimetric vessel and the surroundings is neglected. Such are the flash calorimeters of Rosencwaig [186, 187], Callis et al. [188] and Braslawsky et al. [189], and the photoacoustic calorimeter of Komorowski et al. [190]. There are open nonisothermal-nonadiabatic calorimeters like those of Picker, Jolicoeur and Desnoyers [191–193]. When measurements in an open, flow n-n calorimeter are made after a relatively long time

Equation (3.12) takes the form

Relationship (3.17) describes the processes occurring in the flow and stopped-flow calorimeters of Roughton [194, 195], Kodama and Woledge [196] and Berger [197]. 5. When and Eq. (3.6) takes the form for the closed system:

Equation (3.18) describes the nonisothermal-nonadiabatic system in which changes occur in the shield temperature. The temperature rise is usually linear and described by the ramp function (Eq. 2.49) as in the

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94

Calvet scanning microcalorimeter. There are several thermal analysis devices for which Eq. (3.18) in its simplified form can be considered as mathematical model. They operate via observation of the temperature of the samples caused by the forcing function generated on the shield. This group includes: devices for heating and cooling curve determination (temperature measurement); differential thermal analysis, DTA (meas­ urement of the temperature difference between the studied sample and the standard substance) [198, 199]; heat flux differential scanning calo­ rimetry, hf-DSC (determination of the difference in heat flux between the studied sample and the standard substance to the shield [200]; and modulated temperature differential calorimetry, mt-DSC (the tempera­ ture change of the sample described by the frequency and amplitude of vibration) [116]. Regardless of which measurement method is used, in each of them the heat effects generated in the sample and in the calorimeter shield are superimposed. In consequence of the same type of inertial properties (of inertial objects) of the devices mentioned, the course of the output func­ tion caused by the programmed rise of temperature of the shield is al­ ways the same (see § 3.2.5). Let us confine ourselves to considering only the changes in temperature that are caused by linear rise of the shield temperature When the initial temperature of the shield the temperature of the vessel and the ramp func­ tion is Equation (3.18) then takes the form

Division of both sides of Eq. (3.20) by G yields

The solution of Eq. (3.21) is

CLASSIFICATION OF CALORIMETERS

95

When the calorimeter proper and the shield are initially in thermal are equal to each other, and the measure­ equilibrium, i.e. and ment is performed until conditions of stationary heat exchange between the calorimeter proper and shield Eq. (3.22) takes the form

The courses of the changes in and according to Eq. (3.22) are presented in Fig. 3.2. This shows that after a certain period of time following the start of measurement, the course of change in becomes linear. From this point, the difference between and is equal to and the shift between the temperature curves is equal to the time constant Such a temperature course is observed when the sample is a thermally passive object. For an endothermic reaction, the line appears to be twisted as a result of decrease of the heating rate. For an exothermic reaction, the twist of the line is linked to the increase of the heating rate. This is presented in Fig. 3.3(a). Figure 3.3(b) shows the temperature change reading determined by differential measurement. In the case of a Calvet DSC microcalorimeter, thermogram data are usually used for the determination of P(t) and then Q(t).

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6. When and const., Eq. (3.6) (similarly as previously) takes the form of Eq. (3.10), i.e. The function P(t) is a superposition of at least three heat forcing functions relating to the heat power generated by the studied proc­ and the heat power gener­ ess, the compensating heat power ated in the shield in order to keep its temperature in compliance with the programmed changes. 7. When and Eq. (3.6) becomes The authors do not know any examples of calorimeters of this type.

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The classification of calorimeters presented above conforms with that given in 1930 by Lange and Miszczenko [1], who distinguished four groups of calorimeters: 1) isothermal-adiabatic; 2) isothermal-nonadiabatic; 3) nonisothermal-adiabatic; and 4) nonisothermal-nonadiabatic. Obviously, the experimental description of the device should also contain information such as: 1) the purpose of the instrument (combus­ tion, heat of mixing, heat capacity, sublimation, etc.); 2) the principles and design of the calorimeter proper, including the ranges of tempera­ ture and pressure in which measurements can be performed; 3) the measured quantity and measuring device; 4) the static and dynamic properties of the calorimeter; the calibration mode and the methods of measurement and determination of heat effects; 5) the operational char­ acteristics of the calorimetric device, the sensitivity noise level, the method of calibration, the accuracy, etc.; 6) a description of the experi­ mental procedure used in the calibration and the actual measurements. The mathematical models of calorimeters presented above were ap­ plied as the basis of methods used to determine heat effects.

3.2. Methods of determination of heat effects 3.2.1.

General description of methods

of determination of heat effects

Many methods are used to determine heat effects. Some of them al­ low the determination of the total heat effect Q studied, while others permit the determination not only of the total heat effects, but also of the course of thermal power P in time t (function P(t), called the thermoki­ netics or thermogenesis). For the characteristics of these methods, the general heat balance equation (Eq. (1.148)) has been used [21]:

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In an adiabatic method (§ 3.2.3), it is assumed that only heat accumulates in the calorimeter, which is treated as an integral object. A generated heat effect P(t) is then described by the first term on the righthand side of Eq. (1.148). Several methods are used for nonadiabatic calorimeters. In the multidomains (N-domains) method (§ 3.2.4), particular forms of Eq. (1.148) are applied, depending on the number of distinguished domains, and their mutual location. A similar procedure is applied in the method of finite elements (§ 3.2.5). In the dynamic method (§ 3.2.6), the calorimeter is treated as a sim­ ple body of uniform temperature. The detailed form of Eq. (1.148) is then a heat balance equation of a simple body. The dynamic method is a one of the most frequently used in the determination of total heat effects and thermokinetics. In the modulating method (§ 3.2.8), Eq. (1.148) is usually reduced to the form of a heat balance equation of a simple body [Eq. (1.99)], one of the input forcing functions being a periodic function. In the flux method (§ 3.2.7), the calorimeter is treated as a propor­ tional object. In this method, it is assumed that the accumulation of heat in the calorimeter proper is negligibly small. The steady-state method (§ 3.2.9) is based on the assumption that the heat power generated in the calorimetric vessel is compensated by addi­ tionally generated heat power of opposite sign. The calorimeter is treated as a proportional object. The inertial properties of the calorimeter are neglected. The left-hand side of Eq. (1.148) is equal to zero. The heat balance equation (Eq. (1.148)) reduced to the heat balance equation of a simple body [Eq. (1.99)] is also the basis of many methods used to determine total heat effects. One of the best-known is the method of corrected temperature rise (§ 3.2.10), often called the Regnault-Phaundler correction. A group of methods exist which a priori assume that the calorimeter is an inertial object of N order. A model of the calorimeter is expressed

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99

in the form of a convolution function [42, 43]. The set of linear differen­ tial heat balance equations resulting from Eq. (1.142) is then replaced by one equation of N order. Let us derive the equations that are the mathe­ matical models of these methods. Equation (1.148) in its matrix form gives

where

is the diagonal matrix whose elements are the heat capacities

of the particular domains; is the matrix whose elements are the heat loss coefficients; T(t) is a vector whose components are the temperatures of the particular domains; P(t) is a vector whose components are the heat powers of the distinguished domains; and is the derivative of the T(t) vector. Laplace transformation of Eq. (1.146) gives where s is the Laplace operator, is the initial state vector (the initial temperature condition in the domains), and T(s) and P(s) are the Laplace transforms of state vector T(t) and P(t), respectively. The solution of Eq. (3.27) in the complex domain s is where is the transfer matrix; in the frequency domain

the solution is

and in the time domain t, the solution is

where H(t) is the matrix of fundamental solutions. If it is assumed that

100

where is the sampling period and can be rewritten as

CHAPTER 3

is the Dirac function, Eq. (3.31)

If it is additionally assumed that the heat source is located in one domain only, and that the temperature is measured in one domain only and under the zero initial temperature, Eqs (3.30), (3.31) and (3.33) de­ fine the mathematical models for reconstruction of the thermokinetics P(t). Equation (3.32) written in the form corresponds to the mathematical model of the harmonic analysis method (§ 3.2.11.1). Eq. (3.31) in the form

represents the mathematical model of the dynamic optimization method (§ 3.2.11.2). Equation (3.26) is the basis of the state variable method (§ 3.2.11.4). Equation (3.33) in the form

represents the mathematical model of the thermal curve interpretation

method (§ 3.2.11.3). Under zero initial conditions, Eq. (3.28) in the form

where

represents the mathematical model of the method of transmittance de­ composition (§ 3.2.11.5). Techniques that use analog and numerical correction of the dynamic properties (§ 3.2.11.6) of the calorimeter compensate the transmittance zeros and poles and can be also applied to reconstruct the thermokinet­ ics. In agreement with Eq. (3.41), the transmittance has the form

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101

where S is the gain factor.

3.2.2. Comparative method of measurements Calorimetric determinations are based on the comparative method of measurements. The International Conferences on Chemistry in Lucerne in 1936 and in Rome in 1938 accepted the proposal that all physico­ chemical measurements should be divided into two groups, absolute and comparative [202–205]. The absolute measurements include the deter­ mination of absolute values characterizing a given physicochemical property of chemical compounds or a mixture. It is necessary to intro­ duce all secondary factors which would change the numerical value determined by the absolute method. Absolute measurements should be carried out exclusively by specialists working in bureaus of measures and standards or in laboratories sufficiently adapted for high-precision measurements. A comparative method of measurements can be per­ formed when possible to assume certain substances as standards and the properties of the investigated systems can be compared with the identity of substances. The main principle of this method is the identity of condi­ tions during the measurements and the system calibration. The achieve­ ment of this condition is difficult on determining the heat power by the methods when the consideration of the physical parameters of the sys­ tem is abandoned. In these methods it is assumed a priori that the trans­ mittance of the calorimeter during the calibration and measurement are the same. The method of comparative measurements is often performed with use of the differential method of measurement, which is defined [206] as “a method of measurements in which the measurand is compared with the quantity of the same kind, of known value only slighty different from the value of the measurant, and in which the difference between the two values is measured”.

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The differential method of measurement is the basis of the use of dif­ ferential calorimeters. It is assumed that such devices are constructed of two identical calorimeters (I and II) located in a common shield. One of them contains the sample in which the heat effect is generated; in the second, the thermally passive object (or standard substance) is located. The course of the change in temperature of calorimeter I in which the heat effect is generated is based on the measured temperature difference between calorimeters I and II. The conditions of the differential method of measurement are ful­ filled when: a) the static and dynamic properties of the two calorimeters are identical; and b) the influence on calorimeters I and II of the input forcing functions generated in the surroundings are the same. This man­ ner of temperature difference measurement allows determination of the temperature changes caused by the heat power generated in the sample, and elimination of the influence of other forcing functions. This method is very often used in isoperibol and scanning calorimetry, in DTA de– vices and in some cases in adiabatic calorimetry. From the dynamic properties of differential calorimeters, it is clear that the disturbances can be eliminated only if the transmittances of calorimeters I and II are the same. It is very difficult to fulfill this condi­ tion. Hence, for a given differential calorimeter it is useful to determine an acceptable range of difference of the time constants for which, for a given disturbance and required accuracy of T(t) measurements, the as­ sumption of a differential character of the calorimetric system is satis­ fied [207]. The application of the dynamic equations discussed in Chap­ ter 4 can be very helpful in this type of investigations. An other type of calorimeters used is the group of devices called twin calorimeters. In such calorimetric systems, it is necessary to obtain equal temperatures of the inner parts of calorimeters I and II in such a way that in calorimeter II a the heat power has the same magnitude and course as that in calorimeter I. It is obvious that, for twin calorimetric systems, similarly as for differential calorimeters, the dynamic proper­ ties of the two calorimeters should be the same.

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3.2.3. Adiabatic method and its application in adiabatic and scanning adiabatic calorimetry The adiabatic method [76, 87–89] is based on the assumption that no heat exchange occurs between the calorimetric vessel and the shield. The calorimeter has the dynamic properties of an integral object. “Ide­ ally, the adiabatic shield is always maintained at the temperature of the calorimeter so that there is no temperature differential [208].” To perform the measurement, two methods are used. In the first, called the continuous heating method, the calorimeter proper and the shield are heated at constant power input throughout the whole period of measurement. The course of the temperature change with time is meas­ ured, as is the electrical power supplied to the calorimeter heater. We assume that the thermal power generated is used only to heat the studied substance and the other parts of the calorimeter proper. If this assump­ tion is true, then the three quantities temperature, energy rise and tem­ perature rise (caused by the energy rise) are sufficient to define the measured quantity used to determine the specific heat at temperature T. The generation of electric energy inside the calorimeter proper does not always lead to a temperature rise. It can happen that an isothermal rise of the enthalpy occurs during the studied process. In the second method, the temperature of the calorimeter proper is measured with no power input. First, the calorimeter proper is heated at constant power for a known time interval to attain a certain temperature. After the power is turned off, the temperature is measured. Then, the current supply can again be switched in and next, after switching-off, the temperature will be measured. This cycle can be repeated several times. Such a procedure is called the intermittent-heating method. Both of the measurement procedures described above are used in practice. The first permits control of the temperature of the calorimetric shield in an easier way and it is certainly more convenient to obtain results in a full range of measurement. The second procedure permits study in a precise way of these temperature ranges where interesting phenomena (e.g. unknown phase transitions) are to be observed.

104

3.2.4.

CHAPTER 3

Multidomains method

The multibody (domain) method [67, 209, 210] allows the determi­ nation of thermokinetics as well as the total heat effects of the process examined. The method is based on the general heat balance equation [Eq. (1.148)]. The mathematical model of the calorimeter is described by the following set of equations:

where N is the number of distinguished domains; is the heat loss coefficient between domains i and the j; is the heat loss coefficient between domain i and the shield; and is the heat power. Equation (3.41) describes the “changeable” part of the calorimeter; it is assumed that is the heat capacity of the calorimetric vessel with the contents. The remaining part of the calorimeter described by Eq. (3.40) corresponds to an “empty” calorimeter. This part is called the “nonchangeable” part of the system. The above form of the set of equations indicates that, for various heat capacities there is no need to change the mathematical model of the calls only for the introduction of new data in calorimeter; a change in the deconvolution program. This can also be applied when the heat ca­ pacities of the calorimetric vessel content changes during the experiment (e.g. in the titration process). The form of the heat balance equations for a given calorimeter de­ pends on a number of factors, such as the number of defined domains, the interactions between the domains, and the mutual location of the heat source and temperature sensor. This means that the mathematical form of the equations is related to the thermal and geometrical parame­ ters of the calorimeter and to the location of the sensors and heat

CLASSIFICATION OF CALORIMETERS

105

sources. It is accepted that each of the domains has a uniform tempera­ ture throughout its volume and that its heat capacity is constant. Tem­ perature gradients occur only in the media that separate these domains, and the heat capacities of the media are taken to be negligibly small. The amount of heat exchanged between the domains through these media is proportional to the difference between their temperatures. The propor­ tionality factors are the corresponding heat loss coefficients, and a heat source and temperature sensor can be located between them. The system of domains is located in a shield of temperature, which is treated as the reference temperature. To elaborate the dynamic model, it is necessary to calculate the heat capacities of defined domains and the heat loss coefficients, and to de­ termine the structure of the model. For example, qualitative analysis of the heat exchange in the BMR calorimeter [157] (Fig. 3.4) on the basis of the calorimeter construction allowed the distinction of 19 domains.

The distinguished domains are individual parts of the calorimeter. A block diagram of a such system, elaborated by Cesari et al. [211], is presented in Fig. 3.5, where are the interaction coefficients between the domains. The calorimeter was divided into upper U and lower B parts, because there is no symmetry of the device in the horizontal plane.

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Starting from the vertical symmetrical axis going through the center of the calorimetric vessel, the following domains were chosen for the parts of the calorimeter: 1, 2 – the calorimetric heater R; 3, 4– the shield A of the heaters; 5–8–the block vessel, 9, 1 0–t h e cylindrical parts of the vessel shield; 19 – the bottom part of the shield; 11,2,15,16– thermocouples; 13, 14–thermocouple supports; 17–inner thermostat block; 18– vessel shield support. It was assumed that the calorimeter is ideally differential, and that the other twin battery has been eliminated, because it does not contribute to the transfer function. The heat capaci­ ties of each of the domains were calculated on the basis of its geometry and its parameters: the specific heat and density of the domain material; where is coefficients were calculated from the equation the thermal conductivity; F is the surface area of heat exchange; and is the distance between the centers of the neighboring domains. This relation was applied when the surface dividing the domains was a plane. For a cylindrical surface of heat transfer, the equation

CLASSIFICATION OF CALORIMETERS

107

was used, where l is the height of the cylinder; and and are the radii of the cylindrical surfaces that pass through the centers of the neighbor­ ing domains. The elaboration of the model allows determination of the set of heat balance equations. The procedure of calculating the calorimetric re­ sponse starts from the heat balance equation of the domain in which the heat source is located, with the assumption of zero initial conditions for all the temperatures, for i = 1,2, ... , N. On substituting the derivatives in the set of Eqs (3.40) and (3.41) by differences, we obtain

The temperature changes vs. time are calculated from

When the values of are known, the temperature is cal­ culated, and next the temperatures of the neighboring domains are calcu­ lated from Eq. (3.43). The last equation in the loop calculates the tem­ perature of the sensor. The temperature changes in these domains are the model responses to a given heat effect. There are as many equations as domains distinguished and all calculations are carried out in one calcula­ tion loop. The algorithm of the thermokinetics calculation is very similar to that for thermogram calculations. The beginning of the procedure is the temperature of the sensor. Next, the temperature of the neighboring domain is calculated, assuming zero initial temperature for the remain­ can be writ­ ing domains. For a system of domains, Eq. (3.42) for ten in the form

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Equation (3.44), which is the calculation loop, is used N–1 times to calculate the temperatures of neighboring domains. The final calculation of the heat power gives

where z is the index of the heat source domain, and z – 1 is the index of the domain next to z. For a model of more complicated configurations, when one domain can have more than two neighbors, one calculation loop has as many equations as neighbors. For example, domain j has r neighbors. The heat exchange between these domains is characterized by heat loss coeffi­ Coefficients and are the heat loss coefficients between cients domain j and its neighbors and between domain j and the shield, respec­ The calculation loop will then be a set of r equations of tively; the form

where r is the index of the neighboring domain to the j domain. The following loops are related to the number of domains and the configura­ tion between them, up to the last loop for calculating the course of heat power in time. The transmittance (Eq. 2.41) of the calorimetric system has the form

where is the root of the transmittance (the root of the numerator of the transmittance) and is the pole of the transmittance (the root of the dominator of the transmittance); S is the static amplification factor. The finding of the real form of the transmittance is the same as determination of the dynamic properties of the examined calorimetric system. In this method, it is assumed that synchronous knowledge of the

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real transmittance form and S and will be suitable for determina­ tion of the thermokinetics, but only when the optimization and stability conditions of the numerical solution are fulfilled. These conditions de­ mand the admission of the value of a sampling period obtained from the amplitude characteristics of the calorimeter, taking into account the noise-signal ratio [see § 3.2.11.7]. The optimal sampling period limits the smallest value of the time constants. If any number of time constants of the calorimeter does not satisfy the stability condition, the a new model of the system must be worked out, decreasing the number of do­ mains and calculating new parameters. The maximum order of the new model is limited by the number of time constants which satisfy the sta­ bility condition. Next, the model response to a given heat effect is com­ pared with the experimental response. If the result is positive, this means that a model useful for the determination of thermokinetics has been elaborated.

3.2.5.

Finite elements method

The finite elements method was proposed and described by Davis and Berger [212]. This method is based on the following assumptions. A certain number of elements are distinguished, between which the heat For each ele­ exchange is characterized by the heat loss coefficients ment, the heat capacity is calculated. It is assumed that temperature for each of the distinguished element is known at moment t and next calculated for For each i-th element, the amount of heat exchanged between the element i and element i+1 is determined by the equation For each element, we can write

Thus:

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Next, the temperature of each element is calculated after time

For example, if for the first element oftemperature the heat power P is generated during time then after time the change in tempera­ ture of this element is The result of these calculations is the thermogram, which represents the course of the changes in time of the temperature of the element in which the sensor was placed. Thermokinetics is determined on the basis of deconvolution of the pulse response of the examined system and the measured thermogram. For calculations, the first nonzero value of the pulse response is taken. The increase in temperature of the element (in which the heat effect is generated) is determined on the basis of the equation where is the increase in temperature of the element in which the sensor is placed. The magnitude R plays the role of the multiplier, which allows transformation of the sensor signal to the signal in the reaction it is vessel. In order to determine the thermokinetics at moment necessary to know the temperature distribution of all elements and the heat power P at moment t. In each sampling period the increase in is caused by two effects: 1) the heat generated by the temperature reaction (or pulse), which causes the temperature increase; and 2) the heat transferred, which causes the temperature increase Thus, the increase in temperature of the sensor, according to the superposi­ tion principle, is given by the formula The increase in temperature is calculated on the basis of Eqs (3.48)–(3.51). The value of the temperature increase is determined on the basis of the thermogram obtained, and the increase in temperature is calculated as the difference

CLASSIFICATION OF CALORIMETERS

111

The value of the increase in temperature is calculated on the ba­ sis of Eq. (3.53). Thus, the temperature of the reaction vessel for is The heat power P for

is given as

The presented method permits the use of an iterative procedure to de­ termine the values of heat power in sequenced time moments. This method is very sensitive to the errors in the determination of the value of the pulse response and coefficient R.

3.2.6.

Dynamic method

The dynamic method [10] of measurement is used for the determination of: a) the course of the thermal power change in time, b) the course of the temperature change vs. time, which characterizes the course of the thermal power change in time; and c) the total heat effect. As the mathematical model of the calorimeter, we use here the heat bal­ ance equation of a simple body [Eq. (1.99)]. As concerns the explanation of the method principles, let us assume that the calorimeter proper at temperature is surrounded by a shield at constant temperature while the results of the temperature differ­ ence are expressed by function T(t). This assumption enables us to discuss this method by using the procedure proposed by Calvet [10]. To describe the relation between the generated heat effect Q(t) and the difference T(t) in the calorimeter proper and the shield temperatures, Eq. (3.13) is used in the form

or in the form

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while to describe the course of the thermal power change in time ex­ pressed as a function of temperature, the dynamic equation is used:

The first term on the left-hand side of Eqs (3.58) and (3.59) is equivalent to the amount of heat accumulated in the calorimeter proper, while the second term on the left-hand side of these equations is equiva­ lent to the amount of heat exchanged between the calorimeter proper and the shield. Measurements of the temperature are usually made by using sensors such as a resistance thermometer and thermistor, where the resistance is a function of temperature, or a thermocouple and thermopile, where the electromotive force is a function of temperature. Accordingly to express T it is convenient to use the quotient with where is the measured magnitude, and g is the factor of proportionality. Equation (3.58) then becomes

Integration of Eq. (3.73) leads to

where

notes the values of

and

at times

and

while

is equal to the surface area F located between the course of changes in and F are the results of calo­ time and the time axis t. The values of rimetric measurement. The values of C/g, G/g, and are determined via the calibration procedure. Equation (3.61) is often called the Tian– Calvet equation. The calibration consists in generating a rectangular pulse [Eqs (2.45) and (2.46)] of known heat power inside the calorimetric vessel in a time long enough for a new state of stationary heat exchange to be achieved. In practice, it is characterized by a constant temperature difference, de­

CLASSIFICATION OF CALORIMETERS

113

scribed as and simultaneously by constant value Allowing a small The error, we can assume that it occurs in the time period equal to solution of Eq. (3.60) is then

and for Eq. (3.61)

In this way, the value of G/g is defined. Quite often, it is useful to determine a set of values G/g by generating rectangular pulses with dif­ ferent amplitudes. It can then be verified if a value of coefficient G/g is constant in the interesting range of changes in the temperature of the calorimeter. This constant value means that the calorimeter has the lin­ ear properties of a dynamic object. If the generation of the rectangular heat pulse is stopped, then the calorimeter becomes a thermally inertial object. Cooling of the calorimeter then occurs up to the moment when the temperatures of the calorimeter and the shield are the same. This process has been described by Newton‘s cooling law [Eq. (1.104)] and is used to determine the value of the time constant (see § 2.7.1). In the knowledge of and G/g, by means of Eq. (1.103) we can de­ termine the value of C/g:

Besides the determination of the time constant value, very important information is provided by observation of the course of the changes for the calorimeter as a thermally inertial object. When the changes = f(t) are nonlinear during the initial period of the cooling or heating process, it is necessary to analyze whether application of the simple body heat balance equation to the calculations is correct. A nonlinear course of means that the function is multiexponen­ tial. More precise equations expressing the relation between P(t) and T(t) should then be used. The calibration procedure presented here is not the only one used in the dynamic method. The literature on this subject

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contains several modifications of such way of calibration, e.g. [7, 159, 160].

3.2.7.

Flux method

The dynamic method presented above is based on the assumption that the heat effect generated in the calorimeter proper in part accumu­ lates in the calorimetric vessel, and in part is transferred to the calo­ rimetric shield. When excellent heat transfer occurs between the calo­ rimeter proper and the shield (as in conduction microcalorimeters), it can be assumed that the quantity of heat accumulated is extremely small. This assumption is the basis of the flux method. The amount of heat transferred between the calorimeter proper and the shield is then directly proportional to the temperature difference. Thus, the course of ob­ tained from the measurement resembles that of P(t), and its value is determined on the basis of the second term on the left-hand side of Eq. (3.61):

while the total heat effect is

which results from Eq. (3.62).

3.2.8. Modulating method The modulating method is based on the measurement of the tempera­ ture oscillations of a sample heated by oscillating heat power. Under isoperibol conditions, this method is called AC calorimetry [213–215]. The first AC calorimetry experiments were performed in 1962 by

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115

Kraftmacher [213], who measured the heat capacities of metals in the high-temperature region. Reading et al. [216, 217] proposed a method in which a DSC is used. In this case, the response of the calorimeter as a linear system would be a superposition of two input functions: 1) the ramp function [Eq. (2.49)] generated in the calorimetric shield and 2) the periodic function gener­ ated in the sample. When the periodic function is sinusoidal [Eq. (2.53],

where is the initial temperature, is the temperature amplitude and v is the angular frequency. The solution of Eq. (3.68) in the case of steady-state temperature modulation of the sample, according to Eqs. (2.55) and (3.26), is

where is the time constant, which corresponds to phase angle given by

and

is the

The resultant signal is then analyzed by using the combination of a Fourier transform to deconvolute the response of the sample to the un­ derlying ramp from its response to the modulation that can give rise to an underlying and a cyclic heat capacity. For data evaluation, two pro­ cedures are used, for “reversing” and kinetic “nonreversing” compo­ nents [217], the determination of “storage” and “loss” heat capacities [218, 219]. The “reversing” parameter is most readily identified with the heat capacity of the sample, whereas the nonreversing component in­ cludes contributions from irreversible processes such as crystallization, glass transition, and the melting of polymers. Temperature-modulated differential scanning calorimetry is a new analytical technique used to obtain information on the heat capacity in the range close to the phase transformation. It is a method applied in many instruments, e.g. as the Modulated DSC of TA Instru­

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ments [220, 221], the Alternating DSC [222] or the ODSC (Seiko Instruments) [223].

3.2.9.

of Mettler-Toledo

Steady-state method

In order to obtain a constant gradient of temperature between the calorimeter proper and its environment, the steady-state method is used. The method is based on the assumption that the following condition is fulfilled: where is the heat power generated during the process examined; is the compensating heat power; P(t) = 0 (adiabatic-isothermal calorimeters) or P(t)= const. (adiabatic-nonisothermal; compensating DSC). Let us apply Eq. (3.6) to formulate the basis of the method. Integration of both sides of Eq. (3.6) with respect to time in the in­ gives terval

or

Let us assume that the temperature

can be expressed by two terms:

where is the average temperature of the calorimeter proper (the calo­ rimetric vessel), and are the temperature oscillations around the average temperature. Thus, the temperature increment in the consid­ ered time period is

CLASSIFICATION OF CALORIMETERS

117

The integral of Eq. (3.73) can then be written in the form

or

When the temperature of the shield can be assumed to be constant: the third term on the left-hand side of Eq. (3.77) is

and Eq. (3.77) becomes

If it is additionally assumed that the amplitude of temperature oscil­ lations around the average temperature is negligibly small, we can put

and Eq. (3.80) simplifies to the form

For adiabatic-isothermal calorimeters, it is assumed, that 0 and the first term on the left-hand side of Eq. (3.82) is neglected. The thermal power is used to compensate

For adiabatic-nonisothermal calorimeters when

to de­ termine the amount of heat it is necessary to know the time interval of the process, the average temperature the heat loss coefficient G and the amount of heat delivered to the calorimeter proper in order to hold the desired value with minimum oscillations.

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The thermal power is produced by several pulses of Peltier or Joule effects by: a) Application of pulse compensation consisting in the generation of as shown in Fig. 3.6, where A is a quantized thermal power pulses pulse amplitude, is the pulse duration, and F is frequency.

b) Use of proportional compensation; it is assumed that the tempera­ ture change is proportional to the generated compensation thermal power

or

c) Use of proportional-integral compensation; the proportionality of to and to the integral value of is assumed:

where

is the integration constant.

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119

3.2.10. Method of corrected temperature rise The method of corrected temperature rise [224, 225, 8] is used to de­ termine the total heat effects in n-n isoperibol calorimeters. The method is based on the heat balance equation of a simple body in the form

If we put

Eq. (3.87) can be written in the form

where

is called the corrected temperature rise. The amount of heat generated in the calorimetric vessel corresponding to the temperature can then be expressed as increment

The determination of is the aim of the calorimetric measurement. The heat capacity C of the calorimeter is determined during the calibra­ tion in which a heat effect of known value is generated. In the calorimetric measurement, three periods can be distinguished (Fig. 3.7): the initial period (segment AB); the main period (segment BC); and the final period (segment CD). During the initial period the temperature changes before the examined heat process are measured. The beginning of the main period is the initial moment of the generation of heat by the process studied. As the end of the main period, the time moment is assumed as the end of the studied process. During the final period, the temperature

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changes of the calorimeter after the examined heat process are meas­ ured. In the method, it is assumed that, independently of the heat process studied, during the whole period of measurement a constant heat effect can be produced in the calorimeter by secondary processes (for example, the process of evaporation of the calorimetric liquid, the friction of the stirrer on the calorimetric liquid, etc.). The course of the temperature changes is then described by

expressing Newton’s law of cooling by magnitudes u corresponding to the constant heat effect. If we assume that where is the increment of temperature of the calorimetric vessel, which corresponds to the constant heat effect, involve by secondary process Eq. (3.91) can be written in the form

It is assumed that this equation is valid in the initial and the final pe­ riods of measurement. In the initial period, we have

CLASSIFICATION OF CALORIMETERS

121

and in the final period, we have

Because of the small changes in temperature in these periods of measurement, we can approximate them as linear changes in time. Thus, and can be treated as constant, and can be ascribed to the average temperatures tively:

and

of the initial and final periods, respec­

By subtraction from Eqs (3.96) and (3.97), we can calculate the value of the cooling constant as

Let us assume that the constant heat effect caused by the secondary process exists in the main measurement period too. In Eq. (3.87), the temperature T(t) must be substituted by the term thus, we have

The value of the integral

can be approximated by the expression

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where are the values determined during the subsequent read­ This ings of temperature T(t) in the main period; and method involves the assumption that, in small time intervals of the main period, temperature T(t) changes linearly, and the average temperature of the main period is the arithmetic average of the average temperatures from Eq. obtained in these small time intervals. On determining (3.94), we have Similarly, from Eq. (3.95), we obtain Use of Eqs (3.99), (3.101) and (3.102) leads to

Substitution of according to Eq. (3.98) gives

On substituting Eqs (3.101) and (3.103) into Eq. (3.99), we can write

or on substituting

by expression (3.98), we obtain

CLASSIFICATION OF CALORIMETERS

123

Equations (3.105) and (3.107) for the temperature correction are called the Regnault-Phaundler correction. They assumed the sampling period to be equal to unity, and thus This correction is mostly used in isoperibol calorimetry to calculate the corrected temperature rise Depending on the need for precision of the temperature measure­ ment, various approximations of the course of temperature T(t) are used. The graphicel-numerical methods of calculating the corrected tempera­ ture rise include the Dickinson method [226]. White [227, 228] pro­ posed the Simpson rule for calculation of the integral of Eq. (3.100). Roth [229] used the least squares method for approximation of the initial and final periods by straight lines. Planimetry at first was used to obtain this integral.

3.2.11.

3.2.11.1.

Numerical and analog methods

of determination of thermokinetics

Harmonic analysis method

In this method, first used by Navarro et al. [230, 231] and by van Bokhoven [232, 233], the convolution recorded in the frequency domain is accepted as the transformation equation [Eq. (2.8)] of the calorimeter. For the determination of an unknown heat effect, it is assumed that the (§ 2.6) has been determined previously. spectrum transmittance Next, an unknown heat effect is generated and the calorimetric signal is measured. After determination of the spectrum transmit­ tance and the calorimeter response the thermokinetics is obtained as the inverse Fourier transform

where is the Fourier transform of the response As a criterion

of the possibility of reproduction, Shannon’s theorem is applied. In prac­

tice, a frequency is used which is obtained on the basis of the transfer

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CHAPTER 3

function calculated. The practical frequency basis of the dependence

is the value of the spectrum transmittance amplitude for v

where = 0,

is determined on the

is the value of the amplitude for the practical frequency,

and noise is the noise amplitude.

3.2.11.2.

Method of dynamic optimization

In the dynamic optimization method [234–236], Eq. (2.9) is taken as a mathematical model of the calorimeter, and thus appropriate zero ini­ tial conditions are assumed. This method assumes the existence of one input function T(t) and one output function P(t). The impulse response H(t) is determined as a derivative with respect to time of the response of the calorimetric system to a unit step. As a criterion of accordance be­ tween the measured temperature change T(t) and the estimated course of temperature x(t), the integral of the square of the difference between these two courses is taken:

or, on substituting the convolution of functions, we obtain

The task of dynamic optimization consists in selection of the un­ known thermal power P(t) so that Eq. (3.111) attains a minimum. Such a task can be solved provided that the temperature response T(t) and the impulse response H(t) of the calorimeter are known in the analytical form. However, in calorimetric measurements, numerical values

CLASSIFICATION OF CALORIMETERS

125

are usually available on the course of the temperature changes T(t) of the calorimeter. Because of this, the integrals in function (3.111) are approximated by sums, and the function therefore becomes a func­ tion of multivariables: In the search for the minimum of function (3.112), a conjugate gradi­ ent method is used.

3.2.11.3. Thermal curve interpretation method It is known that the response to a Dirac heat pulse of unit ampli­ tude is the pulse response For stationary linear systems, the re­ sponse to a heat pulse generated at time moment is In the method [237, 248], it is assumed that the heat effect P(t) gen­ erated can be approximated by the function

The thermogram T(t) can then be approximated by the expression

or

where is the sampling period, are the values of the heat pulse gen­ erated at moment j, and is the i-th value of the pulse response, which corresponds to the unit pulse generated in the calorimeter at mo­ ment j.

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The aim of the spectrum method is to find the values of coefficients The scheme of searching for the values of these coefficients can be shown as follows. Let us transform the set of functions into the set of orthonormal functions –

The condition of orthonormal functions has the form

where

Orthonormalization of the function [Eq. (3.117)] is difficult, because we orthonormalize very similar functions. Using the basis of orthonormal functions and curve T, the co­ efficients are calculated as

Applying the inverse transformation to transformation (3.117), we

obtain the values of coefficients as

The computer realization of this scheme is very simple under the condition that the orthonormalization does not give problems. The ther­ mal curve interpretation method was proposed by Adamowicz [237– 239].

CLASSIFICATION OF CALORIMETERS

127

3.2.11.4. Method of state variables In the method of state variables [240–242], a certain set of parame­ ters is distinguished: which characterize the calorimetric system. These values are referred to as state variables. Three types of state variables are distinguished: physical variables, canonical variables and phase variables. The vector

the components of which are state variables, is referred to as the state vector. Thus, the calorimeter transformation equation combines the state variables with the parameters of the calorimetric system and the thermal power produced: The state equation (Eq. (3.124)) written in this way is a system of first-order equations. Because of the available calorimetric information, the state variables should be transformed in such a way as to obtain a relationship between one input function P(t) and one output function. The relationship between the state variables and the output function has the following form:

The method of state variables for the determination of thermokinetics was proposed by Brie et al. [239]. The method was applied in the fol­ lowing works [239, 243, 244].

128

3.2.11.5.

CHAPTER 3

Method of transmittance decomposition

In this method [245], the dependence between the measured calo­ rimetric signal and the generated heat effect, in the complex plane, is described by Eq. (2.11). Use of Eq. (2.41) leads to

or, after dividing (m < N), we obtain

where

R(s) is a polynomial of degree smaller than m (remainder of division). Using Eq. (3.127), we have

The expression after the use of Eq. (3.128), in the time domain corresponds to the differential equation

The expression after the use of (3.129), in the time domain corresponds to the convolution function

CLASSIFICATION OF CALORIMETERS

129

where is the inverse Laplace transform of Eq. (3.129). After us­ ing Eqs (3.130–3.132), we obtain the final form of the equation for the desired heat power:

In order to use the above equation to determine the unknown heat ef­ fect, it is necessary to determine first the nominator and denominator of On the the transmittance and next the expressions and basis of

we calculate the coefficients

and differential equa­

tion (3.131), and on the basis of This enables us to decrease the number of derivations of m and is very useful when the difference between the degree of the denominator and the degree of the nominator of the transmittance is small.

3.2.11.6. Inverse filter method This method [244, 246] is based on the assumption that the time con­ stants of the calorimeter as an inertial object have been determined pre­ viously. For example, when the calorimeter is treated as a object of sec­ ond order, the relation between P(t) and T(t) can be expressed according to Eq. (2.13) in the form of the following differential equation:

Using Laplace transformation and replacing the coefficients by the time constants and of the calorimeter: Eq. (3.134) becomes

and

The purpose of the inverse filter method is the application of a pro­ cedure which allows elimination of the influence of inertia units in the

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determination of the relation between T(s) and P(s), and thus that be­ tween T(t) and P(t). It consists in the introduction (by the numerical or analog method) of terms providing an inverse function to the terms For an individual corrector, we have

For two correctors, we have

For an ideal correction, this would lead to the case when the input function corresponds to the output function, as represented by the rela­ tionship The practical performance of the inverse filter method is different in different systems [246–257], but advantage is always taken of electronic elements active under the form of an operational amplifier and imped­ is ance divider of the feedback, by which the required function achieved. Another way to achieve the differential correction is numeric correc­ tion [255–257]. Taking into account Eq. (3.136), it is intended to calcu­ late numerically the functions corresponding to the terms To each of these terms there corresponds a linear differential equation of first order of the form

Thus, for an inertial object of second order, the following equations can be noted:

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131

When the time constants and and T(t) are known, it is possible values consecutively, and thus P(t). to determine the and The numerical differential correction method has also been applied to reproduce the thermokinetics in these calorimetric systems, in which time constant vary in time [258–264], such as the TAM 2977 titration microcalorimeter produced by Thermometric. These works extended the applications of the inverse filter method to linear systems with variable coefficients. In many cases [258–262], as in the multidomains method, as a basis of consideration the mathematical models used were particular forms of the general heat balance equation.

3.2.11.7. Evaluation of methods of determination

of total heat effects and thermokinetics

The methods discussed above discussion are based on particular forms of the general heat balance equation. Methods based on the simple body balance equation or its simplified form have been used in calo­ rimetry for a very long time: the method of corrected temperature rise for more than 100 years, and the adiabatic method for about 100 years. There are also new methods (e.g. the modulating method) or those which have become very important (e.g. the conduction method) in the past 20–30 years. Improved conditions in calorimetric experiments lead to the creation of more precise mathematical calorimeter models and methods used for the determination of heat effects. From the methods in which the set of heat balance equations was used, the multidomains method (§ 3.2.3) and the finite elements method (§ 3.2.4) were elaborated. Numerical methods for the determination of thermokinetics in n-n calorimeters (§ 3.2.11.1 - § 3.2.11.6) were also developed. In the multidomains method and in the numerical methods, in accor­ dance with the detailed solution of Fourier’s heat conduction equation, it was assumed that the impulse response of the calorimeter is described by an infinite sum of exponential functions [Eq. (2.57)]:

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Thus, in these methods an ideal model for reconstruction of the thermokinetics should be a model of infinite order. In fact, it is neces­ sary to limit the number of exponential terms (domains). Generally, this is due to the compromise between the available information and the possibility of applying it. It has been proved by the results of a multina­ tional program [265] on the evaluation of the new methods of determi­ nation of thermokinetics, in 1978. The purpose was to compare different methods independently proposed to reconstruct thermokinetics from the experimental calorimetric data. The dynamic optimization, harmonic analysis, state variable and numerical correction methods were tested. To eliminate the uncertainties caused by the individual features of the various calorimeters in different laboratories, it was generally approved that the responses of all calorimeters should be distributed to all partici­ pants of the program. These responses concerned normalized series of heat pulses, generated by the Joule effect in a heater. The values of am­ plitudes, the time durations of the generated heat effects and the sam­ pling period were fixed as follows. The sampling period was taken as 0.002–0.003 of the largest constant of the calorimeter, which corre­ sponds to 1/300 of the half-period of cooling. The values of amplitudes were given in relative units, equal to 10, 100 and 1000. The amplitude equal to 10 was defined as corresponding to the heat power of the pulse of duration equal to 8 sampling periods, for which the maximal tempera­ ture increase was 10 times larger than the amplitude of the measurement noise (the change in baseline when the thermally inert object is placed in the calorimeter). Three types of measurements were made: 1, 3 or 7 pulses of various durations were generated one after another. For exam­ ple, plots of the series of 7 pulses are given in Fig. 3.7, and the results of the data reconstruction by dynamic optimization and harmonic analysis in Fig. 3.8.

CLASSIFICATION OF CALORIMETERS

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The results of the investigations indicated that the examined methods reveal considerable progress in the reconstruction of thermokinetics. It was stated [265] that “it is clear that the objective is not complete recon­ struction of thermokinetics (thermogenesis), but rather decrease of the influence of thermal lags in calorimetric data; thus, after reconstruction, the residual time constant which appears in the data is ~ 200 times smaller than the actual order time constant of the instrument used” and that “the range of application of the presented methods is varied. The optimisation method shows its advantage especially in the cases when the number of experimental data need not be bigger than 100-150. The harmonic analysis, state variable and inverse filtering are not influ­ enced by such a limitation and may be freely applied for the reconstruc­ tion of effects requiring a great number of data points”. It appeared that all methods are equally sensitive to noise in the data and that the amplitude of the heat effect under study, compared to that of noise in the data, is of particular importance for the quality of recon­ struction. Let us consider the same problem on using amplitude characteristics [266]. As given in Eq. (3.143), amplitude is described by the func­ tion

and for large values of

can be approximated by Eq. (3.144):

As a criterion for choosing the optimal sampling period, it is as­ sumed that

where denotes the noise-to-signal ratio, and q corresponds to the number of certain digits in the measurement of the calorimetric signal. In order to estimate the values of the sampling period h instead of the approximate values

CLASSIFICATION OF CALORIMETERS

135

are assumed in Eq. (3.144):

Raising both sides of Eq. (3.147) to the power 1/m results in

Thus, the values of the constants must be larger than the sampling period with respect to the stability of the numerical solution. In the par­ ticular case when the number of certain digits in the calorimetric me­ surements is equal to the difference between the number of poles and the number of zeros of the transmitance (q = N – m), the formula for the optimal sampling period takes the form

A scheme for choosing the optimal signal-to-noise ratio has been the subject of many papers, e.g. [267–270]. In the multidomains method, this selecting scheme was included in the procedure of determining the particular form of the calorimeter model. Another manner of selecting is used when harmonic analysis method is applied. In this method, the transmittance is obtained numerically, using the Fast Fourier Transform. This procedure uses points, where n = 10, 11, 12 and N = 1024, 2048, 4096, respectively. The number of data used for calculations must cover the whole interval of time from “initial zero” to “final zero” of the temperature calorimetric response. The discrete measurement of tem­ perature limits the upper bounds of frequency which can be applied for reconstruction of the thermokinetics. The value of this frequency, resulting from Shannon’s theorem is a function of the sampling period It is therefore and can be expressed by the relation impossible to use the complete spectrum of the frequency. There also exists a boundary frequency

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which is related to the measurement noise. The boundary frequency is the second limitation of the frequency spectrum, which can be applied in the reconstruction of the thermokinetics for a given calorimetric system, and The high precision of calorimetric measurements today permits a less strict description of the calorimetric system itself. The comparison of reconstructions of the thermokinetics obtained with the multibody method and with other methods demonstrates the convergence of the results [67] when the calorimetric system is described by a linear differ­ ential equation with constant coefficients of the third to sixth order. A review of the published papers leads to the conclusion that mathe­ matical models which are particular forms of the general heat balance equation of second or higher order are frequently used in isoperibol and DSC calorimetry. When the mathematical model is known, the inverse filter method is used to determine the thermokinetics. The course of the thermal power change can be determined by using different calorimeters. The choice of the heat effect determination method does not depend on the calorimeter type. In a calorimeter with a vacuum jacket, the thermokinetics can be determined successfully by means of the dynamic method. In calorimeters whose inertia is very small (conduction calorimeters), use of the flux method is not always suitable. Better results in the reconstruction of thermokinetics are ob­ tained from the use of methods where it is assumed that the calorimeter is an inertial object of first or higher order.

3.3.

Linearity and principle of superposition

In the review of methods of heat effect determination and thermoki­ netics, the linearity of the system has implicitly been assumed. Linearity [8, 271] can be expressed by the equations

CLASSIFICATION OF CALORIMETERS

137

where x and y express quantities to be measured and the function ex­ presses the output signal. Equation (3.141) is the basis of the superposi­ tion principle, and Eq. (3.142) indicates proportionality for quantitative measurement. Regardless of what method of heat effect determination is applied, it has to be known whether measurements were performed in a linear sys­ tem. It is essential to draw a distinction between linear and nonlinear re­ sponses of the system. To correct the response of the system the linear­ ity has to be confirmed by the experimental facts. When we look for the correct answer, the best method is to analyze the response of each forc­ ing input function in the system. To achieve this, it would be helpful to use the dynamic time-resolved characteristics for the typical input func­ tion, given earlier (§ 2.3). The response of the calorimetric system is always a result of super­ position of some forcing functions acting at the same time. The forcing function in the calorimeter proper is always the heat effect generated as a result of the studied process. However, the review of the methods has revealed that there can be forcing functions striving to compensate the generated heat effect of the process; a rectangular pulse or a series of rectangular pulses; periodic heating rate of the sample etc. Additionally, the forcing functions acting in the shield influence the calorimeter. For an isoperibol calorimeter, this is a forcing function, tending to achieve the stability of the temperature of the shield; for scanning calorimeters, it is a ramp function. Analysis of the output functions courses generated in the shield is quite simple to deal with and usually consists in considering the course of the system response, if there is a thermally inertial object inside the calorimeter proper.

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Chapter 4

Dynamic properties of calorimeters

4.1. Equations of dynamics In our considerations relating to the analysis of the course of heat ef­ fects that occur in calorimeters, we have used particular solutions of the general heat balance equation (1.148):

It is frequently more convenient to apply this equation in the tem­ perature dimension [8, 20]:

The set of differential equations (4.1) is called the general equation of dynamics, the assumptions for which are sufficient to allow the calo­ rimeter to have different configurations. The following notions have been introduced in Eq. (4.1): the overall coefficient of heat loss, the time constant of the domain, the interaction coefficient and the forcing function. The overall coefficient of heat loss for each of the domains is de­ fined as

CHAPTER 4

140

This coefficient characterizes the heat exchange between domain j and the surroundings, but also that between domain j and other domains. The time constant of domain j is defined as the ratio of heat ca­ pacity

and the overall coefficient of heat loss

The time constant

of the domain:

of domain j is a measure of the thermal inertia

of this domain in the system of domains. The interaction coefficient is defined as the ratio of the heat loss to the overall coefficient of heat loss coefficient This is a measure of the heat interaction of domain i with the domain j relative to the interactions of the remaining domains and surroundings with domain j. The interaction coefficients essentially affect the thermal inertia of the calorimeter and allow us to establish the structure of the dynamic model of a given calorimeter. If the value of the interaction coefficient is negligibly small, it may be assumed that there is no thermal interaction between domains i and j or, more exactly, that the thermal interaction between domains i and j is small enough to be ig­ nored in comparison with the interactions between domains i and j and other domains and the surroundings. Furthermore, the notion of the forcing function (taken from con­ trol theory) was introduced into these considerations. This function is defined as

or, making use of Eq. (1.149), we have

The function has the dimension of temperature and is propor­ tional to the heat power evolved. The coefficients are defined as

DYNAMIC PROPERTIES OF CALORIMETERS

141

where

is a determinant whose value depends on the interaction coef­

ficient

is the determinant of the matrix obtained by crossing out

the j-th row and the j-th column of matrix D. The coefficients are dimensionless, chosen in such a way that an increment of the tempera­ in a stationary state is equal to an increment of the forcing ture function Defined in this manner, the coefficients are especially useful in applying the principle of superposition to the system of do­ mains. The set of differential equations Eq. (4.1) can be written in matrix form as follows: where

is the diagonal matrix whose elements are the time constants is the diagonal matrix whose elements are the coefficients A is the matrix whose elements are and for is the state vector, and f is the forcing vector Application of the Laplace transformation to Eq. (4.8) under zero ini­ tial conditions gives where T(s) is the Laplace transform of the state vector T(t) and f(s) is the Laplace transform of the forcing vector f(t). The solution of Eq. (4.9) is

or where is the transfer matrix and its elements are the transmittances which have the form

CHAPTER 4

142

where

is the determinant of the matrix

corresponding minor of the matrix

and

is the is the determinant of

the matrix These determinants, after development into a power series with respect to s, give polynomials of N-th and M-th degree (M < N), respectively:

Thus, the transmittance [Eq. (4.13)] can be written in the form

or, on developing the nominator and the denominator of the transmit­ tance (4.16) into first-degree factors:

where

is the static gain,

is the root of the nomina­

tor of the transmittance (the zero of the transmittance) and is the root of the denominator of the transmittance (the pole of the transmit­ tance). The advantage of the use of the presented transmittance is the ex­ pression of both the input and output functions in temperature dimen­ sions. It can be applied to analyze the courses of the heat effects in calo­ rimeters of different constructions, with different locations of the heat sources in relation to the temperature sensors. When the N-domain method is used to describe the heat effects, determination of the trans­

DYNAMIC PROPERTIES OF CALORIMETERS

143

mittance form makes it possible to verify the correctness of the mathe­ matical model of the calorimeter used in the calculations. In such a case, it is sufficient to compare the values of the calculated and experimen­ tally measured function T (t) as the response to the known heat effect generated in the calorimeter. In many cases, the general equation of dynamics can be simplified to a few-domain system, as will be pointed out below, in order to analyze the courses of different heat effects in calorimeters.

4.2. Dynamic properties of two and three-domain calorimeters with cascading structure

4.2.1.

Equations of dynamics.

System of two domains in series

Let us distinguish a system of two domains in series, characterized by heat capacities and respectively. The heat transfer between these domains is described by the coefficient while that between the domains and the environment (surroundings) is described by the coeffi­ The set of heat balance equations of the system is as follows: cient

If we put

CHAPTER 4

144

Eqs (4.18) and (4.19) can be written in the form

Let us consider that the heat power is generated only in domain 1, of domain 2 is measured. The and the temperature change here is at the same time the input func­ domain 1 output function tion for domain 2. Equations (4.22) and (4.23) then become

If we additionally assume that written in the form

Eqs (4.24) and (4.25) can be

As results from Eqs. (4.26) and (4.27), the domain 1 output function, here is at the same time the input function for domain 2. Function is the output function of the system. In the general case, the initial conditions for Eqs (4.26) and (4.27) are

Application of the Laplace transformation to Eqs (4.26) and (4.27) gives the solution in the complex domain:

DYNAMIC PROPERTIES OF CALORIMETERS

145

Equations (4.29) and (4.30) can be rewritten as

Simple rearrangement of Eqs (4.31) and (4.32) with the aim, among

others, of eliminating results in

If we put

Eq. (4.33) reduces to The function P(s) expresses the effect of the input function F(s) on This may be represented by a block diagram (Fig. 4.1) in which the first and second rectangles represent the dynamic properties of do­ mains 1 and 2, respectively.

The function P(s) provides a basis for defining the function for various functions f(t). The functions G(s) [(Eq. 4.35)] will define the course of T(t) at f(t) = 0, the initial conditions being as follows: After simple rearrangement, Eq. (4.35) becomes:

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CHAPTER 4

and after an inverse Laplace transformation this gives

If, at the initial moment of heat generation, there is a temperature dif­ ference between the domains (or domain) and the surroundings, both is to be functions P(s) and G(s) must be determined if the function defined. Let us now define the courses of function in the cases when 1) a unit pulse function; 2) an input step function of amplitude b and time interval a; and 3) a periodic input function is generated in domain 1. This function f(t) is defined by rectangular pulses, each of them over the time period a, the time interval between two successive pulses being Z. Let us suppose that a heat effect that is constant in time (an input pulse function) is producted in domain 1. In this case: This forcing function results in a rise in temperature until a new sta­ tionary heat transfer state is established in the system. This state is char­ acterized by a temperature higher by than the temperature in the system at the moment t = 0, if the initial conditions are zero. If we assume and with Eq. (4.40), after inverse Laplace transformation Eq. (4.34) becomes

If the constant heat effect occurs in domain 1 for a period of time a

and amplitude (unit step function of amplitude the function f(t) assumes the form

DYNAMIC PROPERTIES OF CALORIMETERS

The changes in temperature in time,

147

are then expressed by

When n rectangular pulses occur in domain 1, each pulse lasting for a period of time a, the interval between two successive pulses being Z, the input function f(t) may be written in the form

where

For f(t) expressed by Eq. (4.44), the function F(s) is

and Eq. (4.34) takes the form

The function corresponding to P(s) in Eq. (4.47) was deter­ mined by an inverse Laplace transformation. This function is

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C HAPTER 4

4.2.2. Equations of dynamics.

Three domains in series

In numerous cases, the determination of the output function with a given accuracy requires that the two-domain cascading system should be enlarged with an “additional” first-order inertial object (domain). A cascading system composed of three domains in series can be de­ scribed by the following set of equations of dynamics:

As before, Eqs (4.49)-(4.51) are assumed to be general differential equations of dynamics describing the courses of the temperature changes in time in the domains considered. In Eq. (4.49),f(t) is the input is the function. It may be assigned an arbitrary course. Function domain 1 output function, which at the same time constitutes the input function for domain 2. The output function of domain 2 constitutes the input function of domain 3, whose output function is For the solution of the above system of differential equations, the Laplace transform method was used, yielding

DYNAMIC PROPERTIES OF CALORIMETERS

149

where

The relation between the functions F(s) and P(s) may be represented as in the diagram in Fig. 4.2.

When f(t) = 0, and

a solution of Eqs (4.49)–(4.51) in relation to tion where

is given by the equa­

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For a forcing function f(t) described by Eq. (4.40) and initial condi­ tions described by Eq. (4.55), we obtain the changes in temperature

For a forcing function f(t) described by Eq. (4.43) and initial condi­ tions described by Eq. (4.55), we obtain

For a forcing function described by Eq. (4.45) and initial conditions described by Eq. (4.55), the temperature changes are expressed by the equation

DYNAMIC PROPERTIES OF CALORIMETERS

151

Forms of the particular equation of dynamics developed for the sys­ tems of two domains and three domains in series are presented only with selected input functions. This selection was made by taking as criterion their application in the analysis of the courses of the heat effects. The particular forms of the equations of dynamics for the other input func­ tions can be obtained in a similar way.

4.2.3. Applications of equations of dynamics of cascading system

The equations of dynamics of cascading systems have been utilized in calorimetry for many years [25–35]. For example, these equations can be applied as follows: 1) A system of two domains in series. The input function f(t) of do­ main 1 is at the same time the output function of domain 2. The output function of domain 2 is measured (Fig. 4.1). a) The forcing input function f(t) is the input step function gener­ ated in the calorimetric vessel (in domain 1). Domain 2 con­ sists of the temperature sensor in isolation. The amplitude and generation time of the rectangular pulse and the time constants of domains 1 and 2 are known. For the calculations, one can apply Eq. (4.43). These calculations allow determination of the output function of the system, which is of special importance in measurements of heat effects of short duration. b) The input function is a ramp function. Domain 1 is a thermo­ stat, while domain 2 consists of the calorimetric vessel with sample. Let us assume that the calorimetric vessel contains samples with the same volume, but with differing heat capacities The time constants of domain 2 there­ fore change. This causes differences in the values of the output function. They can be calculated after determination of the particular form of Eq. (4.43), values characterizing the ramp function and the values of

152

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2) The system consists of three domains in series (Fig. 4.2). The in­ put function f(t) of domain 1 is at the same time output function of domain 2, which in turn is the input function of domain 3. The output function of domain 3 is measured. The input function f(t) is the input step function. Domain 1 consists of the calorimetric ves­ sel filled with the substance studied, domain 2 is the inner shield of the calorimeter, and domain 3 is in isolation the temperature sensor located on the outer surface of the inner calorimetric shield. The amplitude and generation time of the rectangular pulse and the time constants of domains 1–3 are known. To calculate the output function, we can apply Eq. (4.61). The calculations al­ low us to determine the influence of the inertia of the inner shield and the temperature sensor on the recording of the heat effect generated in the calorimetric vessel. In the discussed examples, the responses of the system to only one forcing function were considered. In many cases, the course of the out­ put function of the system has to be defined as the result of the operation of several forcing functions. This can be demonstrated via the following examples. 3 a) A system composed of a thermostat (domain 1) and a calorime­ ter (domain 2). In the thermostat, the ramp function f(t) is pro­ duced, whereas the calorimeter gives the periodic input function

3 b) A system comprising the calorimetric vessel with the sample (domain 1), the internal shield (domain 2) and the temperature is sensor (domain 3). In the calorimetric vessel, heat effect generated, while the compensating heat effect expressed by is produced in the internal shield of the calorimeter.

DYNAMIC PROPERTIES OF CALORIMETERS

153

This leads to the conclusion that, for the two-domain system when input functions f(t) and (Fig. 4.3) are produced at the same time, the output function P(s) will be expressed by Eq. (4.36) supplemented by the term.

For the three-domain system, when forcing functions and are produced at the same time, the output function is expressed by Eq. (4.54) supplemented by the term

This means that in the two cases considered the solution of the equa­ tions of dynamics will be the sum

This procedure results from the application of the superposition rule to linear systems. Equations of dynamics are also useful in analysis of the courses of the heat effects in differential and twin calorimeters. Let us consider a system composed of two calorimeters (I and II), characterized by the dynamic properties of inertial objects of the first order, placed in a common shield. The sample is situated in one of them, and the reference substance in the other. The forcing input function is the ramp function generated in the thermostat. Let us assume that this function is at the same time the input function of both calorimeters. The influence of the forcing function on a differential calorimeter is shown

154

CHAPTER 4

schematically in Fig. 4.5, where and are the output functions of calorimeters I and II, respectively and T(t) is the output function of the differential calorimeter.

For this system, we can determine the changes in the course of the function T(t) due to the changes in the time constants of both calorime­ ters and therefore the changes caused by the different heat capacities and heat loss coefficients. For the observation of such changes, in many cases it is necessary to consider systems having more domains, both in series and with a concentric configuration. As often as not, with a high number of domains present in the system, even the identification of these domains may pose considerable difficulties and the direct determi­ nation of the time constants may be impracticable. The transmittance of the system can then be determined or the system can be approached in terms of a single system with vicarious characteristics. The application of these characteristics does not mean, however, that the accuracy of establishing the output function is somewhat sacrificed.

4.3. Dynamic properties of calorimeters

with concentric configuration

Let us discuss the two-domain model with a concentric configura­ tion, indicating the dependences of the dynamic properties on the mutual locations of the heat sources and temperature sensors (§ 4.3.1 and § 4.3.2); explaining why the calibration of the calorimeter by means of

DYNAMIC PROPERTIES OF CALORIMETERS

155

the dynamic method in some cases is accompanied by apparent changes in heat capacity (§ 4.3.3); and proving the need for use of the equivalent coefficient (§ 4.3.4) instead of the heat capacity value when the method of corrected temperature rise is applied in calculations of heat effects.

4.3.1. Dependence of dynamic properties of

two-domain calorimeter with concentric

configuration on location of heat sources

and temperature sensors

Let us normalize in the dimension of temperature the isoperibol n-n calorimeter characterized by a system of two concentric domains shown in Fig. 4.5 [21, 45].

This is characterized by Eq. (4.18):

and the equation

CHAPTER 4

156

In this case, the overall heat loss coefficients mains 1 and 2 are, respectively

The time constants

and

and

for the do­

of the distinguished domains 1 and 2

are

It may be noted that the thermal inertia of domain 1 is influenced by the heat transfer between domains 1 and 2 and with the shield, whereas the thermal inertia of body 2 is influenced by the heat transfer between domains 1 and 2. The interaction coefficients are, respectively On putting

we can write

In order to determine coefficients and are calculated, which are, respectively

the

determinants

From Eqs (4.7) and (4.72), we have

With regard to Eqs (4.68)–(4.73), Eqs (4.18) and (4.66) can be written in the form

The differential equations (4.74) and (4.75) are called the equations of dynamics of a calorimeter treated as a system of two domains with a

DYNAMIC PROPERTIES OF CALORIMETERS

157

concentric configuration [44, 45]. It may be noted that the left-hand sides of these equations are similar to the equation of dynamics for a one-domain model; the time constants and of the particular do­ mains thus characterize the first-order inertial properties of each, con­ sidered independently of one another.

However, because of the different forms of the right-hand sides of Eqs (4.74) and (4.75) referring to the heat balance equation of a simple body, a new quality is obtained: the mutual interaction expressed by and This expression follows from the block diagram (Fig. 4.7), which may be assigned to the equations of dynamics. To obtain the block diagram shown in Fig. 4.6, the Laplace trans­ formation should be applied. Equations (4.74) and (4.75) then become

where and are the differences between the initial temperatures of domains 1 and 2, respectively, and that of the temperature of the shield at time = 0; and are the Laplace transforms of tempera­ and respectively; and are the Laplace trans­ tures forms of the forcing functions and

When

Eqs (4.76) and (4.77) become

158

For the transform Eq. (4.79), we have

CHAPTER 4

from Eq. (4.80) and the transform

from

where

are the transmittances of domains 1 and 2, respectively. These functions characterize the dynamic properties of the domains that are distin­ guished in the examined calorimeter. It is seen from the block diagram (Fig. 4.7) that the dynamic properties of the system depend on the dy­ namic properties of the domains and on the interaction between them. In this connection, let us consider the dependence of the dynamic proper­ ties of the calorimeter described by Eqs (4.74) and (4.75) on the loca­ tions of the heat sources and temperature sensors in domains 1 and 2. The following cases can be distinguished: is generated, 1. A heat power characterized by forcing function and the changes in temperature in time are measured. 2. A heat power is measured and the changes is temperature in time are measured. 3. A heat power characterized by function is generated, and the in time are measured. changes in temperature 4. A heat power characterized by function is generated, and the changes in temperature in time are measured. For these cases, the calorimeter transmittances as functions of trans­ mittances and [Eq. (4.83)] of the distinguished domains and interaction coefficient k are given in Table 4.1; the relationships be­ tween the input function F(s) and output function T(s) are given in Table 4.2. Taking into account the form of transmittances given by Eq. (4.83) for the distinguished domains 1 and 2, the transmittances of the calo­ rimetric system given in Table 4.1 can be written in the form presented in Table 4.3.

DYNAMIC PROPERTIES OF CALORIMETERS

If we introduce the time constants fined as

159

and

of the calorimeter, de­

which satisfy the equation

the transmittances given in Table 4.3 can be written in the form given in Table 4.4. The time constants and determine the inertia of the calo­ rimeter and depend on the time constants of the distinguished domains and the values of coefficients k. These are the roots of Eq. (4.85).

160

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It can be shown from the transmittances collected in Table 4.4 that depends only on the time constants and of the transmittance calorimeter; transmittance depends on the constants and but also on the interaction coefficient k, which takes into account the tem­ perature differences between domain 2 and domain 1 in steady-state heat and depend on the time constants transfer; transmittances of the distinguished domains as well as on the time constants of the calorimeter.

As a consequence of the different forms of transmittance, there are different courses of the output function for the same forcing function.

DYNAMIC PROPERTIES OF CALORIMETERS

161

We can demonstrate this on examples of 1) the pulse response of the calorimeter; and 2) the temperature changes of the calorimeter as a thermally inert object. When the inverse Laplace transformation is applied to the transmit­ tances given in Table 4.4, the pulse responses of the calorimetric system given in Table 4.5 are obtained; their plots are shown in Fig. 4.8. Fig­ ures 4.8a and 4.8d reveal that, when the heat source and the temperature sensor are situated in the same domain, the shape of the pulse response of the calorimetric system is reminiscent of the response of the calo­ rimetric system of first order. From the shapes of the curves given in Figs 4.8d and 4.8c, it is clear that, when the heat source and temperature sensor are situated in different domains, the pulse response of the calo­ rimetric system at the initial time moment is equal to zero, next in­ and respectively), and creases to a certain maximum value then decreases to zero. Thus, the pulse responses depend on the mutual locations of the heat source and temperature sensor, and on the parameters of the calorimetric system, and are non-negative functions.

Let us also consider the changes in temperature in time of the calo­ rimeter when there are no heat sources; the distinguished domains are thermally inert objects, which is equivalent to the conditions

CHAPTER 4

162

The temperature changes of the calorimeter are caused only by the initial differences in temperature of the bodies with respect to the envi­ ronment temperature, which can be written as When the conditions given by Eq. (4.74) are taken into account, Eqs (4.79) and (4.80) can be written in the form

The determination of

and

from these equations leads to

Equations (4.90) and (4.91) represent the Laplace transforms of tem­ and when at least one of the temperatures of the peratures distinguished domains at the initial moment t = 0 is not equal to the en­ vironment temperature. Application of the inverse Laplace transforma­ tion to Eqs. (4.90) and (4.91) gives

DYNAMIC PROPERTIES OF CALORIMETERS

where the time constants Eq. (4.84). When

and

163

of the calorimeter are given by

the plots of the functions (4.92) and (4.93) are shown in Figs. 4.9a and 4.9b. From the plot in Fig. 4.9a, it is seen that domain 1, at a temperature higher by than the temperature of the environment, undergoes cool­ ing, the course of the temperature changes in time being reminiscent in shape of a one-exponential course. Domain 2, whose temperature at the initial moment is equal to the environment temperature, first undergoes heating to temperature by a certain time and then cools to the and environment temperature. At time moment temperatures are equal. When the initial temperature of domain 1 is smaller by than the environment temperature (Fig. 4.9b), the domain is heated to the environment temperature; the temperature of domain 2 cools to by time next returning to the environment temperature. The extremum of temperature of domain 2 is accepted as the time moment at which equalization of the temperatures of the two domains takes place. When the plots of functions (4.92) and (4.93) are shown in Figs 4.9c and 4.9d. When the initial temperature of domain 2 is higher by than the envi­ ronment temperature (Fig. 4.9c), domain 2 cools to the environment temperature, and the course of the temperature changes in time are reminiscent in shape of a one-exponential curve.

164

CHAPTER 4

Domain 1 with an initial temperature equal to the environment tem­ perature heats up to temperature not reaching the temperature of domain 2, and then cools to the environment temperature. The tempera­ ture of domain 1 is always lower than the temperature of domain 2. If lower than the environment the initial temperature of domain 2 is temperature (Fig. 4.9d), domain 2 heats up to the environment tempera­ ture. The temperature of domain 1, which at first is equal to the envi­ and next ronment temperature, decreases to a certain temperature

DYNAMIC PROPERTIES OF CALORIMETERS

165

increases to the environment temperature. The extremal value of the temperature of domain 1 depends on the value of interaction coefficient k. As the value of interaction coefficient comes nearer to unity, the neight of the peak is smaller. When the initial temperatures of the two domains are equal and dif­ ferent from the environment temperature: plots of the functions (4.92) and (4.93) are shown in Figs 4.9e and 4.9f. From Fig. 4.9e, it can be seen that the temperatures of domain 1 and domain 2 decrease in time, but the temperature of domain 1 decreases faster than that of domain 2. When the initial temperatures are smaller than the environment temperature (Fig. 4.9f), both domains heat by up to the environment temperature, but domain 1 does so faster than domain 2. The course of the temperature changes in domain 1 is remi­ niscent in shape of a one-exponential course. The examples given above clearly show that the output function is related to the mutual locations of the heat sources and temperature sen­ sor. The presented equations were defined on the assumption that the form of the input function and the initial conditions are known. To re­ construct the input function on the basis of the form of the output func­ tion, it is necessary to know the particular forms of the dynamics equa­ tion of the two-domain calorimeter with a concentric configuration.

4.3.2. Dependence between temperature and heat effect as a function of location of heat source and temperature sensor Let us define particular forms of the equation of dynamics [Eqs (4.74) and (4.75)] expressing the dependence between temperature and heat effect as a function of the locations of the heat source and the tem­ perature sensor [8, 30, 40], As a basis of consideration, we will take the equation of dynamics [Eqs (4.74) and (4.75)]. When heat power is gen­ erated only in domain 1, Eqs (4.74) and (4.76) take the form

166

CHAPTER 4

When heat power is generated only in domain 2, Eqs (4.74) and (4.75) take the form

Taking into account Eqs (4.97)–(4.100), let us consider the following cases: 1. The temperature is dependent on the function In this case, the elimination of temperature from Eqs (4.97) and (4.98) furnishes

Division of both sides of this equation by k and taking into account Eq. (4.84) gives

With the assumption that

Eq. (4.102) can be written in the form

DYNAMIC PROPERTIES OF CALORIMETERS

167

Equation (4.105) connects the changes in temperature of domain 1 with the heat effect generated in the same domain. It results from this equation that the changes in temperature of this domain are determined by the relation of the heat power and its derivative. This equation cannot be directly applied to determine an unknown heat power, because it is necessary to know the derivative of the value we are looking for. 2. The temperature depends on the function In this case, by from Eqs. (4.97) and (4.98) and fol­ eliminating the temperature lowing the above routine, we can establish the relationship between the of domain 2 and the heat effect generated in temperature changes domain 1, which has the form

Analogously, by taking into account Eqs (4.99) and (4.100), the de­ pendences between the heat effect generated in domain 2 and the tem­ perature changes can be established for the following cases. 3. The temperature depends on the function In this case, the equation of dynamics has the form

4. The temperature depends on the function the equation of dynamics has the form

In this case,

CHAPTER 4

168

Equation (4.107) has a form similar to that of Eq. (4.104). Particular forms of the equation of dynamics show that, for the two-domain calo­ rimeter, the temperature sensor should be situated in another domain than the heat source. In another case, it is possible to carry out a meas­ urement in such a way that the temperature sensor is situated in each domain. Zielenkiewicz and Tabaka [278–281] showed that correct re­ sults can be obtained by applying multipoint temperature measurement, corresponding to the number of distinguished domains. Considerations as to the mutual locations of the heat sources and temperature sensors can form the basis of the explanation of several observed facts, such as the apparent change in heat capacity of the calo­ rimeter with time.

4.3.3.

Apparent heat capacity

During the calibration of the Calvet microcalorimeter [10] and the KRM calorimeter [282], it was found that the calculated value treated as heat capacity changes in time. Changes in the heat capacity of the calorimeter were noted, when was calculated by using the heat balance equation of a simple body, in which it is assumed that the heat capacity C depends neither on time t nor on the geometrical distribution of the heat power source and the location of the temperature sensor; it is equal to the sum of the heat ca­ pacities of all the parts i of the calorimeter:

to

On the assumption that the heat capacity C of the system corresponds Eq. (3.59) after integration can be written in the form

If we assume that the two-domain model with a concentric configu­ ration is the proper mathematical model for the examined calorimeters

DYNAMIC PROPERTIES OF CALORIMETERS

169

[293], then, in a comparison of Eq. (4.109) with integrating Eqs (4.104)– (4.107), the following cases can be considered: 1. Integration of both sides of Eq. (4.104) in the time interval leads to

Simple rearrangement of Eq. (4.110) gives

A comparison of Eqs (4.109) and (4.111) results in

2. Integration of both sides of Eq. (4.106) in the time interval yields

or

Comparison of Eqs (4.109) and (4.114) gives

3. On integration of both sides of Eq. (4.105) in the time interval we have

CHAPTER 4

170

and

By comparison of Eqs (4.109) and (4.117), we find

4. Integration of both sides of Eq. (4.108) in the time interval leads to

or

On comparing Eqs (4.109) and (4.120), we have

It follows from the above equations that the effective heat capacity depends on various parameters: the heat capacities of the distin­ guished domains, the heat transfer coefficients between these domains and the environment, the character of the changes in the heat effects in time and their derivatives with respect to time, the changes in particular temperatures in time and their time derivatives, and also the time inter­ val in which the heat effects are evaluated. The effective heat ca­ pacity is time-invariant in only a few cases, e.g. when and i = 1,2 for a system of two interacting domains.

DYNAMIC PROPERTIES OF CALORIMETERS

4.3.4.

171

Energy equivalent of calorimetric system

When the corrected temperature rise method was applied for the calibration of the calorimetric bomb, it was established [22–24, 284] that the calculated heat capacity of the device was not equal to the sum of the heat capacities of the calorimeter parts. King and Grover [22] described the calorimetric bomb in terms of a two-domain model with a concentric configuration. As a result, in the method of corrected rise the heat capacity C was replaced by a corrected term, called the energy equivalent of the calorimeter:

where and are the heat capacities of domains 1 and 2, is the and is heat loss coefficient between domains 1 and 2, equal to the smaller of the cooling constants of the calorimeter. Let us analyze [283, 285] the influence of the mutual locations of the heat sources and temperature sensors on the value of the energy equiva­ lent for a calorimeter treated as a system of two domains. As stated pre­ viously, in the corrected temperature rise method three periods are dis­ and tinguished during the experiment: In the initial and final periods, the calorimeter is a ther­ mally inert object; in the main period, the heat effect is generated in the calorimeter. As shown in §4.3.2, in the main period, the calorimeter can be de­ scribed by one of Eqs (4.104)–(4.107), depending on the mutual loca­ tions of the heat sources and temperature sensors, while integration of these equations in the time interval similarly as for Eqs (4.110), (4.113), (4.116) and (4.119), leads to

172

CHAPTER 4

where

and and are the values of the derivatives of the temperatures and with respect to time t at moments and respectively. Since the moments and belong in the periods when heat power is not generated:

From Eqs (4.4) and (4.5), we have

On taking into account Eqs (4.130)–(4.132), after simple transforma­ tion and integration, we find

DYNAMIC PROPERTIES OF CALORIMETERS

173

If the temperature changes in the initial and final periods result from the cooling or heating of the calorimeter as a thermally passive object, the right-hand sides of Eqs (4.104)–(4.107) are equal to zero. The solu­ tions of these equations are

where and are constants. When the second term of Eqs (4.136) and (4.137) decreases quickly and we can write

Equations (4.138) yield the following equations:

For

and

we have

When Eqs (4.123)–(4.134) and (4.140) are taken into account, Eqs (4.123)–(4.126) can be written in the form

CHAPTER 4

174

Putting

Eqs (4.141)–(4.144) can finally be written in the form

Let us define the energy equivalent R of the calorimetric system as the ratio of the generated heat amount Q to the corrected temperature rise as a function of the mutual location of the heat source and the temperature sensor. Then, taking into account Eq. (4.146), we find that when the temperature of domain 1 is measured and the heat effect is generated in domain 1, the energy equivalent is equal to when the temperature of domain 1 is measured and the heat effect is generated in domain 2, the energy equivalent is equal to

DYNAMIC PROPERTIES OF CALORIMETERS

175

when the temperature of domain 2 is measured and the heat effect is generated in domain 1, the energy equivalent is equal to when the temperature of domain 2 is measured and the heat effect is generated in domain 2, the energy equivalent is equal to It can be seen from Eqs (4.151)–(4.153) that the energy equivalent is the same, equal to the product of the largest time constant of the calo­ rimetric system and the heat loss coefficient In Eq. (4.154), the energy equivalent is smaller than in the first three cases. In calorimetric measurements, the location of the temperature sensor is generally fixed, while those of the calibrating heat source and the heat source of the examined process can differ. If the temperature of the other domains is measured, it is without difference in that one in which only the calibrating effect and the heat effect process are situated be­ cause there is equivalence of the heat sources and On the other hand, when the temperature is measured in the inner domain, the cali­ brations and examined heat effects must occur in the same domain be­ cause The larger root of Eq. (4.84), which corresponds to the time constant of the calorimetric system, is equal to

Taking into account the relationships (4.68)–(4.71), we obtain

From Eq. (3.96), we have

It results from Eq. (4.157) and Eqs (4.151)–(4.154) that the energy equivalent has the dimension of heat capacity, but is not the sum of the heat capacities and of the distinguished domains. When the calo­

176

C HAPTER 4

rimeter is thermally insulated which corresponds to adiabatic conditions, the value of the equivalent is equal to the sum of the heat capacities of the distinguished domains. When ideal heat exchange occurs with the distinguished domain both domains can be treated as simple domains and the energy equivalent is equal to the sum of the heat capacities.

Final Remarks

The present theory of calorimetry is a result of the authors’ own work. Its essential feature is the simultaneous application of the relation­ ship and notions specific to heat transfer theory and control theory. The present theory has been used to develop a classification of calorimeters, to discuss selected methods of determining thermal effects and ther­ mokinetics, and to describe the processes proceeding in calorimeters of various types. Calorimeters have been assumed to constitute linear sys­ tems. This assumption allowed the principle of superposition to be used to analyze several constraints acting simultaneously in and on the calo­ rimeter. The present theory of calorimetry is concerned mostly with the in­ struments whose principle of operation is assumed to involve the trans­ fer of heat in the system. This is true for most of the existing calorime­ ters, whether those with a constant or those with a variable temperature of the shield. It includes calorimeters in which the flow of heat between the calorimeter proper and its surroundings is quite intense, and also those in which this flow of heat is very low. On the other hand, the pre­ sent theory is concerned to only a minor degree with calorimeters whose principle of operation is based on the assumption that there is no heat transfer (adiabatic calorimeters) or that, by definition, the heat transfer process is stationary (the generated heat effect is compensated). The considerations presented are based on the general heat balance equation of N-domains. A majority of the methods used to determine heat effects have been shown to rely on the simplest, particular form of this equation. This makes the calculations very convenient, but implies a number of simplifying assumptions. These give rise to several limita­ tions, which have been duly pointed out. Of the methods described and

178

FINAL REMARKS

used to determine heat effects and thermokinetics, the method of Ndomains has been given special attention. This method may be particu­ larly useful when the calorimeter constructed is used to study transfor­ mations of short or long duration, and with constant or varying volumes and heat capacities of the samples examined. This method is advanta­ geous in that it allows the identification of calorimeter parameters by specifying the “exchangeable” part of the calorimeter as contrasted with the “nonexchangeable” part; this makes the changes that have to be in­ troduced into the algorithm of calculations when the measuring condi­ tions are modified rather minor ones. The method is of particular value when the construction of the calorimeter permits the instrument to be used for various applications. This is true of the numerous calorimeters produced by the major manufacturers. The references reviewed illustrate that most investigators have been looking for ways to enhance the accuracy of measurements by making use of the conclusions deduced from the analysis of the course of heat effects proceeding in the calorimeter. This is of particular concern as regards calorimetric determinations carried out with the recent DSC techniques. In such investigations, the application of the procedures described in the book and based on the general equation of dynamics and on the method of analogy of the thermal and dynamic properties of the objects, can be of essential assistance. The numerous illustrative applications of the theory to various prob­ lems in the analysis of heat transfer processes occurring in calorimeters should encourage investigators to undertake work to improve the theory of calorimetry still further and to develop more practical applications.

References

[1] [2] [3] [4]

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