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THEORY OF CALORIMETRY

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Page 1: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

THEORY OF CALORIMETRY

Page 2: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

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

Hot Topics in Thermal Analysis and Calorimetry

Volume 2

Series Editor:Judit Simon, Budapest University of Technology and Economics, Hungary

Page 3: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

Theory of Calorimetry

by

Wojciech ZielenkiewiczInstitute of Physical Chemistry,

Polish Academy of Sciences, Warsaw, Poland

and

Eugeniusz MargasInstitute of Physical Chemistry,

Polish Academy of Sciences, Warsaw, Poland

KLUWER ACADEMIC PUBLISHERSNEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

Page 4: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

eBook ISBN: 0-306-48418-8Print ISBN: 1-4020-0797-3

©2004 Kluwer Academic PublishersNew York, Boston, Dordrecht, London, Moscow

Print ©2002 Kluwer Academic Publishers

All rights reserved

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

Visit Kluwer Online at: http://kluweronline.comand Kluwer's eBookstore at: http://ebooks.kluweronline.com

Dordrecht

Page 5: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

Contents

Preface ix

Chapter 1The calorimeter as an object with a heat source

1.1.1.2.1.3.1.4.1.5.1.6.

The Fourier law and the Fourier-Kirchhoff equationHeat transfer. Conduction, convection and radiationGeneral integral of the Fourier equation. Cooling and heating processesHeat balance equation of a simple body. The Newton law of coolingThe heat balance equations for a rod and sphereGeneral heat balance equation of a calorimetric system

12

1014202633

Chapter 2Calorimeters as dynamic objects

Types of dynamic objectsLaplace transformationDynamic time-resolved characteristicsPulse responseFrequential characteristicsCalculations of spectrum transmittanceMethods of determination of dynamic parameters

2.1.2.2.2.3.2.4.2.5.2.6.2.7.

2.7.1.2.7.2.2.7.3.2.7.4.2.7.5.

Determination of time constantLeast squares methodModulating functions methodRational function method of transmittance approximationDetermination of parameters of spectrum transmittance

37394147555861666674767981

Page 6: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

vi CONTENTS

Chapter 3Classification of calorimeters. Methods of determination

of heat effectsClassification of calorimetersMethods of determination of heat effects

3.1.3.2.

General description of methods of determination of heat ef-fectsComparative method of measurementsAdiabatic method and its application in adiabatic and scan-ning adiabatic calorimetryMultidomains methodFinite elements methodDynamic methodFlux methodModulating methodSteady-state methodMethod of corrected temperature riseNumerical and analog methods of determination

of thermokineticsHarmonic analysis methodMethod of dynamic optimizationThermal curve interpretation methodMethod of state variablesMethod of transmittance decompositionInverse filter methodEvaluation of methods of determination of total heat effectsand thermokinetics

Linearity and principle of superposition

Chapter 4Dynamic properties of calorimeters

Equations of dynamicsDynamic properties of two and three-domain calorimeters withcascading structure

Equations of dynamics. System of two domains in seriesEquations of dynamics. Three domains in seriesApplications of equations of dynamics of cascading systems

4.2.1.4.2.2.4.2.3.

4.1.4.2.

3.3.

3.2.11.13.2.11.2.3.2.11.3.3.2.11.4.3.2.11.5.3.2.11.6.3.2.11.7.

3.2.10.3.2.11.

3.2.4.3.2.5.3.2.6.3.2.7.3.2.8.3.2.9.

3.2.1.

3.2.2.3.2.3.

858597

97101

103104109111114114116119

123123124125127128129

131136

139139

143143148151

Page 7: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

CONTENTS vii

4.3. Dynamic properties of calorimeters with concentric configurationDependence of dynamic properties of two-domain calorimeterwith concentric configuration on location of heat sources andtemperature sensorsDependence between temperature and heat effect as a functionof location of heat source and temperature sensorApparent heat capacityEnergy equivalent of calorimetric system

Final remarks

References

4.3.1.

4.3.2.

4.3.3.4.3.4.

154

155

165168171

177

179

Page 8: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

Preface

Calorimetry is one of the oldest areas of physical chemistry. The dateon which calorimetry came into being may be taken as 13 June 1783, theday 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 beendeveloped and the measuring techniques have been improved. At pre-sent, numerous laboratories worldwide continue to focus attention on thedevelopment 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 bedetermined by directly measurement of temperature. Accordingly, todetermine 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 themathematical model of the calorimeter. Depending on the type of calo-rimeter applied, the accuracy required, and the conditions of heat andmass transfer that prevail in the device, the relationship between themeasured and generated quantities can assume different mathematicalforms.

Various methods are used to construct the mathematical model of acalorimeter. The theory of calorimetry presented below is based on theassumption of the calorimeter as an object with a heat source, and as adynamic object with well-defined parameters. A consequence of thisassumption 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

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x PREFACE

dynamic properties of the calorimeter. As the basis on which the thermalproperties of calorimeters will be considered, the general heat balanceequations are formulated and the calorimeter is taken as a system oflinear first-order inertial objects.

The dynamic properties of calorimeters are defined as those corre-sponding to proportional, integrating and inertial objects. Attention isconcentrated on calorimeters as inertial objects. In view of the fact thatthe general mathematical equations describing the properties of inertialobjects contain both integrating and proportional terms, a calorimeterwith only proportional or integrating properties is treated as a particularcase of an inertial object.

The thermal and dynamic properties that are distinguished are usedas a basis for the classification of calorimeters. The methods applied todetermine the total heat effects and thermokinetics are presented. Foranalysis of the courses of heat effects, the equation of dynamics is for-mulated. This equation is demonstrated to be of value for an analysis ofvarious thermal transformations occurring in calorimeters.

The considerations presented can prove to be of great use in studiesintended to enhance the accuracy and reproducibility of calorimetricmeasurements, and in connection with methods utilized to observe heateffects in thermal analysis.

Page 10: Zielenkiewicz W., Margas E., Theory of Calorimetry (Kluwer 2004)

A calorimeter can be treated as a physical object with active heatsources inside it. An analysis of the thermal processes occurring insidethe calorimeter, and those between the calorimeter and its environment,requires utilization of the laws and relations defined by heat transfertheory [1–5].

The relations arising from heat transfer theory are applied to designthe mathematical models of calorimeters, which express the dependenceof the change in temperature measured directly as a function of the heateffect produced. There is an understandable tendency to attempt to ex-press these models in the simplest way. In practice, this is achieved byapplying simplifications to the original formulas. To make use of themwisely, one has to understand precisely the assumptions made. Thischapter 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 non-stationary heat transfer state. An understanding of these processes is ofimportance 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 problemsthat form the subject of this book.

Chapter 1

The calorimeter as an objectwith a heat source

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2 CHAPTER 1

1.1 The Fourier law and the Fourier-Kirchhoffequation

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

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

or

Quantity T in Eq. (1.1) is a function of the coordinates x, y, z andtime t. For any time t the value of T determines the scalar temperaturefield. 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 equalto

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

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 3

defines the amount of heat conducted through unit area in unit time if aunit of temperature gradient exists across the plane in which the area ismeasured. The value of for isotropic materials is a scalar, its valuedepending 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 ofthe 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 useof the heat conduction equation of Fourier-Kirchhoff. Let us derive thisequation. 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 envi-ronment 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 ofthe amount of heat accumulated, and the amount of heat ex-changed,

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4 CHAPTER 1

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

where dS = ndS is a normal vector to a surface element in the externaldirection, 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 thatthe surface integral of a vector is equal to the volume integral of thedivergence 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 inunit 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 isequal to

The amount of heat accumulated, according to Eq. (1.6), is equal

to

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

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 5

According to the First Law of Thermodynamics, the amount of heataccumulated, can be described for each volume element of thebody as

where dh is the increase in specific enthalpy (due to unit volume), and pis pressure.

The increase in specific enthalpy is proportional to the increase dT intemperature T:

where is the specific heat capacity at constant pressure, in

and is the density, expressed in The increase in enthalpy in

the body of volume V is equal to

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

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

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6 CHAPTER 1

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

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

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 beexpressed by

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7THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

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 changesin temperature T are not time-dependent:

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

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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.When and are constants independent of both pressure and

temperature, parameter a is often applied:

This is called the thermal diffusivity coefficient, expressed inIntroduction 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 toobtain 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 temperaturethroughout the body is given arbitrarily at the instant taken as the originof the time coordinate t. It is usually assumed that the temperature at thebeginning of the process is constant. For steady-state processes, thecourse of the temperature changes does not depend on the initial condi-tions.

The boundary conditions prescribe:

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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 cylindricalcoordinates r, x in which the Laplacian has the form

where

b) Physical parameters such as specific heat density and thermalconductivity coefficient which characterize the properties of the bodyand the environment. It is determined that they are constant or a functionof temperature. In all the problems discussed in this book, it is consid-ered that and are constants.c) Surface conditions. These conditions are as follows:

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

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

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

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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 ofboundary 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 physicalproblem of heat conduction is to find adequate solutions of the Fourier-Kirchhoff equation or the Fourier equation.

1.2 Heat transfer.Conduction, convection and radiation

“When different parts of a body are at different temperatures, heatflows from the hotter to the colder parts. The transfer of heat can takeplace in three distinct ways: conduction, in which the heat passesthrough the substance of the body itself; convection, in which heat istransferred by relative motion of portions of the heated body; and radia-tion, in which heat is transferred directly between distant portions of thebody 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 collisionsbetween disorderly moving molecules. The process occurs without anymacroscopic motions in the body. The conductivity of diamond withouttraces of isotope is the highest. The conductivity of a metals is alsohigh. 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 itsincrements around the whole surface S of the body. Accordingly, forpractical reasons the Newton equation is usually applied:

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 11

where coefficient is called the surface conductance or the coefficientof surface heat transfer. It defines the amount of heat transferred in unittime through unit surface when the existing temperature gradient is1 deg. For using of the Newton equation, there is no need to introduceany simplifying assumptions. All the complicated character of the heat-transfer phenomenon is enclosed in the value of coefficient whichdepends on many parameters. When the heat transfer conditions aredescribed, appropriate attention must be paid to the proper choice ofand the definition of the influence of different factors on its value.

For a description of the heat flow phenomenon on the border of thebody, 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 consideredsystem is simple, an exact solution of Eq. (1.47) can easily be deduced.

The value of surface heat transfer coefficient is strongly affectedby the presence of heat bridges and thermal resistances. Imperfect con-tact between touching surfaces which are at different temperaturescauses 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 conditionsof heat conduction through the gas present in the space between thecontact points. Temperature variations appear even though the thicknessof the gas layer may be close to the size of the free distance in the gasmolecule. It has been found [6] that even surface irregularities within therange of tens of microns influence the value of the surface heat transfer

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12 CHAPTER 1

coefficient. The variations in thermal resistance between two solid bod-ies are the cause of errors in the heat determination.

The coefficient of surface heat transfer by conduction also de-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 higherthan that of an insulating substance joins those two objects, then throughthis object (called a “heat bridge”) heat will flow, intensively enhancingthe total heat exchange. Unfortunately, it is often very difficult in calo-rimetric practice to eliminate the existence of unwanted heat bridges andthermal resistances. It is crucial to decrease their contribution to thevalue of

Heat transfer by convection occurs in liquids and gases where thereis a velocity field caused by extorted fluid motion or by natural fluidmotion caused by a difference in density. The former case involvesforced convection, and the latter case free convection. Combined con-vection occurs when both forced and free convection are present. Theconvection coefficient of surface heat transfer, defining the heat ex-change in the contact boundary layer between fluid and solid, is deter-mined. Coefficient is often expressed by equations containing criterianumbers, such as those of Nusselt (Nu), Prandtl (Pr), Reynolds (Re) andGrashof(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 distinctivedimension of the body; w is the distinctive velocity; g is the accelerationdue to gravity; is the difference in temperature, and is thethermal expansivity coefficient.

Coefficient usually depends on the difference in temperature be-tween the body and its environment. In the microcalorimeter describedby Czarnota et al. [7], free convection occurs in the spherical spacearound the calorimeter, and as a result the coefficient is defined bythe equation

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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 todecrease or eliminate the heat transfer by convection. A very effectiveway to achieve this is the creation of a vacuum. The contribution of theheat transfer by convection can also be reduced by increasing the shareof 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-sorptivity. The quantity of heat radiated through a transparent gaslayer from surface at temperature to surface at temperaturecompletely enclosing the first surface, is

where is a substitute emission coefficient depending on the emissionfrom a surface, the geometry and the reflectivity, and is a radiationconstant Introduction of the relation express-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 temperaturedifference

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14 CHAPTER 1

is much lower than the temperatures and can be expressed by

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

The heat transfer in calorimeters is described in terms of the effectiveheat 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 initialtemperature 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 referto the heating or cooling processes of a thermally passive body. Anyassumptions 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) isknown, the particular solution of the Fourier equation, can be found, onthe assumption that the initial conditions are zero, T(x, y, z) = 0. Accord-

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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 proceedthrough the surface S according to the boundary condition of third kind,i.e.

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

and can be written in the form

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

Then

and

Substituting the above equations into Eq. (1.63) gives

or, after transformation

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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 independentvariable t. To satisfy Eq. (1.69), both sides must be equivalent to thesame 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 thesides of Eq. (1.69). Thus, instead of one differential equation, we obtainthe two following differential equations:

Equation (1.73) may be written as

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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 differentialEq. (1.65). The functions in Eq. (1.80) corresponding to the particulareigenvalues 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

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18 CHAPTER 1

or

We see that the course of the temperature T (x, y, z, t) changes at anypoint of the body is the sum of infinite number of exponential functions.Whereas the sequence of eigenvalues is an increasing sequence, thesequence of constants monotonously decreases [9]. As ahigher accuracy of determination of temperature T (x, y, z, t) is needed, agreater number of exponential terms must be used for a proper descrip-tion of the changes in temperature of the body within time. In the limit-ing case, it is possible to neglect all the exponential terms, excluding theone 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 usedin calorimetry and thermal analysis.

The expression given by Eq. (1.85) was used for the first time in mi-crocalorimetry to describe short–duration heat processes investigated ina Calvet microcalorimeter [10–13]. The temperature T rise caused byheat effect was expressed by the following equation:

where the adopted relations made possible the determination of T for anexponential temperature course of second or third order. The excellentmonograph by Camia [14] is one of the most important works on thisfield. A number of methods are currently used to determine the heateffects and thermokinetics of short–duration processes, based on theassumption of a multiexponential course of temperature changes. Thesemethods 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].

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 19

Hattori et al. [15] consider the calorimeter in terms of a one-dimensional model of distributed parameters. The objects distinguishedare the calorimetric vessel A, the heat conductor B and the medium C, atconstant temperature, that surrounds the calorimeter (Fig. 1.2). For thesolution 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 whichthe heat flows has a well-insulated lateral surface. Its ends are defined asx = 0 and x = L; there exists a heat bridge of the calorimetric vessel withthe conductor; the environment is kept at constant temperature. The heattransfer takes place only through the cross-section for x = 0.

Tanaka and Amaya [16] consider the calorimeter as a concentricmodel of three domains, and (Fig. 1.3), which are solids.Domain in which heat q heat is generated or adsorbed, is surroundedby domain Domain is surrounded by domain at constant tem-perature. These three domains represent the calorimetric vessel, and theheat conductor between the vessel and the shield.

In both of the above papers, it was shown that, independently of the

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20 CHAPTER 1

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

1.4. Heat balance equation of a simple body.The Newton law of cooling

Linear differential equation of first order called the heat balanceequation of a simple body, has found wide application in calorimetryand thermal analysis as mathematical models used to elaborate variousmethods for the determination of heat effects. It is important to definethe 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 placeunder isobaric conditions, without mass exchange and that the thermalparameters of the body are constant. The Fourier-Kirchhoff equation canthen be written as

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

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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 thebody, is the heat transfer coefficient, and is the temperature ofthe 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 theform

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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,because the relation between temperature and the temperature on thesurface 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 isclear that these heat balance equations and the Fourier-Kirchhoff equa-tion [Eq. (1.87)] are equivalent to each other when:

the temperature in the total volume of the body is homogenousand only a function of time;the temperature on the whole surface is homogenous and only afunction of time;the above temperatures are identically equal to one another at anymoment of time;the heat capacity C and the heat loss coefficient G are constantand not functions of time and temperature.

1.

2.

3.

4.

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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 unitsurface in unit time when the temperature difference is equal to 1 deg.Constant is called the cooling constant and characterizes the rate ofbody cooling. This quantity is the reciprocal of the body time constantIt has the same value for any point of the body. Constant does notdepend on the initial temperature field, but depends on the shape anddimensions of the body, the thermal parameters of the body (e.g. thethermal diffusivity coefficient) and the conditions of heat transfer. Coef-ficient is a quantity describing the measure of the ability of the givenbody to react to cooling or heating of the environment. The influence ofthis 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)is appropriate only when When these temperatures aredifferent, Eqs (1.97) and (1.105) can be supplemented by a new parame-ter, the coefficient of heterogenity of the temperature field

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24 CHAPTER 1

Thus, Eq.( 1.97) becomes

In order to solve this non–stationary differential equation, it wouldbe necessary to know the function This condition is difficult tofulfil. 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 temperaturefield 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 theordered state heat transfer, the relation between heat capacity C, heatloss coefficient G, cooling constant and coefficient is expressed asfollows:

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 25

Coefficient is a dimensionless quantity; its value can vary betweenone and zero. The more differs from unity, the greater is the inequalityin the field of temperature. This coefficient is a function of Biot’s num-ber Coefficient decreases to zero as constant Bi increasesto infinity. Let us assume that the same body is observed under variousconditions of heat transfer from the body to the environment and that asa 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 thelimit value (Fig. 1.5). Only in the case of certain small values ofcoefficient can it be assumed that a homogenous temperature distribu-tion exists in the examined body. It also indicates that the rangewhere the heat balance equation can be accurately used for determina-tion of the heat power changes within time (the P(t) function calledthermokinetics) is limited. However, this equation is also used as amathematical model in determining the P(t) function in instruments with

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26 CHAPTER 1

different values of coefficient It is applied as a mathematical model inconduction microcalorimeters, in which heat exchange between thebody and environment is extensive and coefficient is significant. Forthese reasons, it is very important to verify the accepted model experi-mentally.

If a maker of a calorimetric system decides to apply the method ofdetermining the heat effect resulting from Eq. (1.99), it would be mostconvenient to establish a set of parameters (e.g. such that thecalorimeter in which the heat is generated fulfils the conditionsneeded to apply this equation. A probe to determine such a set of pa-rameters was undertaken by Utzig and Zielenkiewicz [18] for a simplephysical model as an approximation of a real calorimetric system. Therelation between the dimensionless parameter the physical pa-rameters of the system and was elaborated. In the parameter thevalue corresponds to the time interval after which the body temperaturechanges can be described by one exponential term, while is a timeconstant.

1.5. The heat balance equationsfor a rod and sphere

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

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 27

Consider the above equation for (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 andgives the following set of equations:

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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

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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 bothsides 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

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30 CHAPTER 1

where

denote the amounts of heat exchanged between domains n and n+1 andbetween domains n–1 and n in time interval dt, respectively. Equa-tion (1.125) is the desired heat balance equation for a rod considered asa system of domains arranged in a row. The same procedure is appliedto deduce the equation of heat conduction for a homogenous sphere ofradius 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 theneighborhood of the point neglecting terms of higher order thanthe second, leads to

Substitution of and gives the set of equations

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 31

Let us put

and neglect derivatives of higher order than the second:

The solutions of the above set of equations are

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CHAPTER 132

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 basisof the same assumptions: in the examined bodies several elements(parts, domains) are distinguished; each is characterized by a constantheat 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 ofindicator n; the second and third terms are the amounts of heat ex-changed between this part and the neighboring domains of indicators

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THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE 33

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

1.6. General heat balance equationof 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 amultidomain (elements, parts) system of various configurations [19-21].Let us also assume that the heat transfer in the body can take place notonly between the neighboring domains, but also between any domainscharacterized by heat capacities and temperaturesEach of the separate domains has a uniform temperature throughout itsentire volume; its temperature is a function only of time t, and the heatcapacity of domain is constant. The domains are separated by centerscharacterized by loss coefficient and the heat exchange between thedomains and between the domains and the environment of temperature

takes place through these centers. Temperature gradients appear onlyin these centers and between the domains and the environment; theirheat capacities are, by assumption, negligibly small. Furthermore, a heatsource or temperature sensor may be positioned in any of the domains.

The amount of heat exchanged between domains j and i in thetime interval dt is proportional to the temperature difference of thesedomains; 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 isequal to

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

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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 isequal to the sum of the heat accumulated in this domain dQ´(t) and theheat exchanged between this domain and the remaining domains and theenvironment dQ´´(t). Thus:

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

Dividing both sides of the above equation by dt gives

or

where

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35THE CALORIMETER AS AN OBJECT WITH A HEAT SOURCE

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 ofheat exchange between domain j and the environment, and the last termthe 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 generalheat balance equation of a calorimetric system. The heat balance equa-tion Eq. (1.147) describes in general form the courses of the heat effectsin a calorimeter of any configuration of domains and any localization ofheat sources.

The general heat balance equation corresponds to the formalizationof the general calorimetric model by means of the set of equations withlumped parameters. To consider the thermal properties of a calorimeter,the detailed form of this equation has to be derived. It is necessary todefine in it the number and configuration of the distinguished domainsand the centers which separate these domains and where heat transfertakes place.

The representation of the calorimeter by mathematical models de-scribed by a set of heat balance equations has long traditions. In 1942King and Grover [22] and then Jessup [23] and Churney et al. [24] usedthis 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 thecalorimeter was not equal to the experimentally determined heat capac-ity of the system. Since that time, many papers have been published onthis field. For example, Zielenkiewicz et al. applied systems of heatbalance equations for two and three distinguished domains [25-48] toanalyze various phenomena occurring in calorimeters with a constant–temperature external shield; Socorro and de Rivera [49] studied micro-effects on the continuous-injection TAM microcalorimeter, whileKumpinsky [50] developed a method or evaluating heat-transfer coeffi-cients in a heat flow reaction calorimeter.

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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 theuse of circuit theory and the principle of dimensional similarity. In acorresponding network analog, the thermal resistances and capacities aresimulated by electric resistors and capacitors, respectively. In construct-ing such circuits, it is considered that there is an equivalence betweenthe quantities: electric current and heat flow; electric voltage differenceand temperature difference; electric resistance and thermal resistance;and electric capacity and heat capacity.

The thermal–electric analogy method has been used to analyze andrepresent the operation mode of a number of calorimetric and thermalanalysis devices. For example, Rouquerol and Boivinet [54] presentedanalogs of electric circuits for the following heat-flux and power com-pensation devices: the conventional DTA instrument of Mauras [55], thedifferential scanning microcalorimeter of Arndt and Fujita [56], and theTian-Calvet microcalorimeter [57]. Wilburn et al. [58] used both passiveand active analogs to investigate the effect of the holder design on theshape 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 equationsfor constant heating rate DSC by an equivalent electrical circuit. Themethod 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 theirconnection with themselves and the environment. It is also useful toformulate a suitable system of the balance equations for the bodies (do-mains). The increase in the number of its applications is related to thedevelopment of analog computer calculation methods.

It will be demonstrated below that the extended information rangerelating 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 isprofitable in this case.

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Chapter 2

Calorimeters as dynamic objects

Calorimeters are physical objects that can be described in differentways. The steering theory treats a calorimeter as a dynamic object, inwhich the generated heat effects (input signals) are transformed to thequantity measured directly in the calorimeter, e.g. temperature (outputsignal). 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 functiondirectly. The calorimeter is a kind of transducer, which transforms theinput functions into the output functions. Thus, to reproduce

with the required accuracy on the basis ofthe 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 givencourse of the output function of the calorimeter. When the unchangingrelation 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 describedby one differential equation or a system of linear differential equationswith constant coefficients. These equations are treated as the mathemati-cal models of calorimeters. If there are many output functions, then thedynamic object (calorimeter) is described by a system of n differentialequations. Assuming linearity and applying the superposition rule, one

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38 CHAPTER 2

may reduce two or n objects to one. On analysis of an object describedby one input and one output function, we do not lose generality.

A convenient method for solving linear equations is to apply theLaplace 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. Thespectrum 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 transformsof 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 thestructure 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 ofthem can be obtained by using the two others:

The transmittance H(s) can be determined only on the basis of ex-perimentally determined y(s) and x(s). The calorimeter is thentreated as a “black box” and knowledge of their physical parame-ters is neglected. It is assumed that the calorimeter can be treatedas a linear, stationary and invariant object.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-

1.

2.

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CALORIMETER AS DYNAMIC OBJECT 39

mination of the functions y(s) and x(s) is applied for evaluation ofthe values of the dynamic and static parameters of the transmit-tance. This procedure is equivalent to the calibration of the calo-rimeter.The transform x(s) is determined in the knowledge of the form ofthe transmittance H(s) and the values of their parameters. Inputfunction y(s) is determined experimentally. This procedure isconvenient to verify the agreement between experimentally de-termined and calculated functions x(s).The transform y(s) is evaluated on the basis of previously knownH(s) and experimentally determined function x(s). It correspondsto the determination in the domain of the complex variable of thecourse of the investigated heat effect.

The form of the transmittance depends on the type of the dynamicobject.

2.1. Types of dynamic objects

3.

4.

Among the open systems, the following types of dynamic objects canbe distinguished [64, 65]:

Proportional type, when the input function y(t) is proportional to theoutput function x(t):

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

while the spectrum transmittance is expressed by Eq. (2.7):

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

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40 CHAPTER 2

while

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

while

Let us compare the equations describing the dynamic properties ofthe distinguished types of objects with suitable heat transfer equations:

We compare Eq. (2.5), which describes the dynamic properies ofa proportional object, with the Newton heat transfer equation [Eq.(1.46)];We compare Eq. (2.8), which describes the dynamic properties ofan integrating object, with the first term of the left-hand term ofEq.(1.99);We compare Eq. (2.11), which describes the dynamic propertiesof an inertial object, with the heat balance equation of a simplebody [Eq. (1.99)].

1.

2.

3.

It can readily be imagined that a calorimeter in which the course ofheat power within time corresponds (with the accuracy of the factor) tothe course of the changes in the output function (e.g temperature) hasthe properties of the proportional object. The properties of integrating

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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 numerousgroup of calorimeters. Let us assume that a calorimeter has only theintegrating or proportional properties of the object, as a certainidealization. Of course, this idealization is well-founded in many cases.The relation describing the dynamic properties of the object isequivalent to the mathematical model of the calorimeter. It is expressedas a function of time, frequency or a complex variable domain.

2.2. Laplace transformation

The Laplace transformation [66] is the operation of changing oneexpression into another by integration. In this transformation, thefunction f(t) of the real variable is changed into the complex functionF(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) thecondition 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 theintegral formula. This operation is abbreviated as

The Laplace transformation is very convenient to use. Its advantagesinclude: 1) The Laplace transforms of simple functions can be deter-

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42 CHAPTER 2

mined by direct integration or integration by parts. In most cases, thesimple function f(t) and the transform F(s) representing the functiontransform pairs are tabulated; 2) the Laplace transformation is a lineartransformation for which superposition holds; 3) by application of theLaplace transformation, an ordinary differential equation is reduced tothe algebraic equation of the transform, called the subsidiary equation ofthe differential equation.

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 ofthe 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

Let us apply the Laplace transform to the heat balance equation of asimple body (Eq. 1.99):

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43CALORIMETER AS DYNAMIC OBJECT

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. Whenthe heat effects are not generated, and and thus T(t)depends only on the initial temperature difference T(0) between thecalorimeter and its environment. Then:

In a similar way, the Laplace transform can be applied to obtain thesolution 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 objectis 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 changesin the course of the output function. Let us assume that the objects arearranged in series in such a way that the input function of the next iner-

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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 ofobjects ranging from one to six, caused by input function forcingcorresponding to the unit step function, is shown in Fig. 2.4.

The block diagrams presented in Figs 2.1 - 2.3 are characteristic forthe open systems and differ from one another only in the number ofinertial objects. The dynamic objects are not always arranged in series.In many cases, as a result of self-arrangement of the objects and theirconfigurations, we must consider the set of differential equationspresented 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 calorimeterdescribed by the following set of differential equations:

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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, thecalorimeter transmittance will have a more complicated form. For Nobjects, 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 thefollowing set of differential equations:

where

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46 CHAPTER 2

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 zeroinitial conditions can be written in the form Eq. (2.4):

where

is the transmittance of the analyzed system; m < N; and N denotes thesystem 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) thepolynomial has only single zero values, we can write

The roots of Eq. (2.37)

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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), thetransmittance H(s) can be written in the form

The poles and zeros of transmittance express the inertial propertiesof 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 timecaused by the action of the input function is given by the dynamic time-resolved characteristics. In calorimetry, the same input functions areused for their description as in control theory [64]. However, the termi-nology used for this purpose is different. Thus,

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48 CHAPTER 2

The input function described by a short-duration heat pulse ofrelatively high amplitude, called in control theory a unit pulsefunction (impulse function, Dirac function) (Fig. 2.6), isexpressed by

1.

The unit pulse whose surface area is equal to one has a Laplacetransform y(s) equal to one.

The input function described by a constant heat effect in timecorresponds to the unit step function (Fig. 2.7):

2.

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CALORIMETER AS DYNAMIC OBJECT 49

The unit step function corresponds to the integral of the unit pulsefunction with respect to time. The Laplace transform of the unit stepfunction is

3. The input function described by a heat effect that is constant intime over a determined interval of time corresponds to the inputstep function of amplitude b and time interval a, called therectangular 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 isconstant in time for a defined duration. The exceptions to the rule arethose instruments in the calibration of which the frequency characteris-tics are used.

The Laplace transform of the rectangular pulse is expressed by

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50 CHAPTER 2

4. The input function described by a heat effect rising linearly in timeis presented in Fig. 2.9 and expressed by Eq. (2.49):

The generation of such a forcing function is used in steering theoryas well as in adiabatic and scanning calorimetry. The ramp function hasthe following Laplace transform:

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

The Laplace transform of which is

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CALORIMETER AS DYNAMIC OBJECT 51

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

where A is the amplitude of oscillation, is the oscillation frequency,and

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 objectand the Laplace transform of the input function y(t) furnishes the outputfunction x(s) = y(s)· H(s). With the inverse transformation, we obtainy(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 2.1show that the shapes of the output functions depend strongly on the typeof the dynamic object. For proportional objects, the output and inputfunctions have the same shape, while their values are equal to each otherwith the accuracy of the factor.

This means that in a calorimeter with the dynamic properties of aproportional object the output function gives direct information on thecourse of the output function; in other words, the course of the experi-mentally determined function T = T(t) corresponds to the course of thechanges in heat power P in time t.

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52 CHAPTER 2

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For integrating objects, the course of the output function correspondsto the accumulation process and to the operation of integration. Theobject responds to a generated unit pulse with an output signal, which isequivalent to the unit step function; the response of the object to the unitstep function is a linearly rising function; production of the ramp forcingfunction stimulates the response of the object according to the relation

For inertial objects, the course of the output function is induced bythe inertial properties of the object (calorimeter). This results from thetransmittance form H(s), which is expressed by the operator 1/(Cs+k) orthe operator while the relation between the input function y(s),the output function x(s) and the transmittance H(s) is presentedgraphically by a block diagram (Fig. 2.12)

When the trasmittance is represented by the single symbol H(s), it isdepicted by the block diagram presented in Fig. 2.1.

The form of the transmittance H(s) indicates that the time constantis a decisive parameter for characterizing the inertial properties of theobject (calorimeter). This also means that the value of the time constantdetermines the course of the output function, the character of which isapproached more closely for either proportional or integrating objects.Simply, the values of control the inertial, damping properties of theobject. 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 first-order objects are also presented in Table 2. 1. It is seen that, for a propor-tional object, only the value of the sinusoidal (harmonic) oscillationfrequency changes. For an integral object, the sinusoidal input functionis transformed by the object to a cosinusoidal function. The amplitude ofthe output function is then inversely proportional to the frequency of the

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CALORIMETER AS DYNAMIC OBJECT 55

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

where

the factor expresses in terms of frequency the ratio of theoutput and input function amplitudes. The values of the factor and thesize of the phase shift characterize the dynamic properties of the iner-tial object. For a frequency close to zero, when sinusoidal changes havelow frequency, the course of the output function is similar to that of theinput 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 inthe output function course relative to the input function results from therise in frequency. For infinitely high frequency, the shift in the course ofx(t) is expressed as radians or –90°. When the frequency is related tothe conversed time constant then the oscillation frequency is 0.707and the phase shift is radians or 45°. This is the frequency related tothe transfer function of an inertial object, given by the operatoror

2.4. Pulse response

The pulse response function is the output function h(t) caused by theaction of the input impulse function (Dirac function). It is applied fordetermination of the particular forms of the Laplace transmittance. It canbe obtained by applying the Laplace inverse transformation to thetransmittance Eq. (2.41):

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56 CHAPTER 2

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 theoriginal, 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 responseof the calorimetric system:

a) to a rectangular heat pulse of short duration (“experimental” Diracpulse) expressed as a sequence of discrete values

where and is the sampling period; orb) to the unit step function lasting a sufficiently long time to achieve

the stationary state of heat transfer. In this case, the pulse response isexpressed by the derivative of the calorimetric response

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With the measured values of the response at times andrespectively

numerical derivation has to be performed in order to obtain discretevalues of the pulse response h(t).

In the first case, the accuracy of obtaining the values of the unit pulseresponse depends on the degree to which the “experimental” pulse ap-proximates to the “ideal” Dirac function and on the accuracy of themeasurement of the calorimetric signal. In the second case, the proce-dure of numerical derivation influences the accuracy of obtaining thepulse response. Since the experimentally obtained response of the sys-tem does not fulfil the condition [Eq. (2.49)], it is necessary to calculatethe integral from the course obtained for the calorimetric signal anddivide all the values of the signal by the value of this integral. Thus, weobtain a new course that fulfils the condition needed.

From Eqs (2.59) and (2.60), the dependences between the amplitudesand time constants can be obtained:

b) for the second-order inertial system:

c) for the second-order inertial system whose transmittance containsone zero

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

a) for the first-order system:

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In Fig. 2.13, the plots of pulse responses of calorimeters of variousorders are shown. As the order of the system becomes higher, the pulseresponse 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 realvariable can be written as follows:

or in the form

If it is assumed that

where and are the real (even) and imaginary (odd) parts ofthe transmittance respectively, we can write

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Because

from Eq. (2.69) we have

Magnitude is called the amplitude. Division of Eq. (2.69) leadsto

Thus

Magnitude is called the phase and is equal to the argument oftransmittance The phase describes the relative amounts of sineand cosine at a given frequency.

The spectrum transmittance of an N-order inertial system hasthe form

Equation (2.75) is equivalent to Eq. (2.67), assuming thatIn 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-mittance is equal to the product of the amplitudes of particular

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transmittances. Thus, the phase of the transmittance is equal to the sumof the particular transmittances of the phases.

According to Eqs (2.75) and (2.79), the phase is described bythe 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 areused. The amplitude characteristic is a relation expressing the ratio ofthe amplitude of the output function to that of the input function. Thedependence 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 plotsare usually used to draw the amplitudes: one in the coordinates

and the other in the coordinates and two typesof plots for the phase: in the coordinates or The plot inthe coordinates gives the amplitude-phase characteristics.

Let us analyze the amplitude-phase characteristics for a few types ofspectrum transmittance.

According to Eqs (2.72) and (2.75), the amplitude is describedby the function

For sufficiently large values of the function can beapproximated by the equation

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61CALORIMETER AS DYNAMIC OBJECT

Taking logarithms of both sides of the above equation gives

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

2.6. Calculations of spectrum transmittance

The spectrum transmittance can be obtained as the Fouriertransform of the pulse response or as the quotient of the Fouriertransform 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 tothis 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

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and are the real (even) and imaginary (odd) parts of theFourier transform of the response function, respectively.

The result of the calorimetric measurement is obtained as the numbersequence temperature data [Eq. (2.66)]

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

Then, Eqs (2.87) and (2.88) become

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63CALORIMETER AS DYNAMIC OBJECT

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

In this way we obtain the real and imaginary parts ofthe Fourier transform of the response function to a heat pulse. TheFourier transform of the input function should also be determined.

Let us determine the transforms for the following input functions: astep 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 transformof the input pulse.

According to Eqs (2.85), (2.86) and (2.92) – (2.94), we have

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or

where

When the calorimeter is calibrated by a unit pulse of amplitudeand sufficiently long duration, the spectrum transmittance can beexpressed as

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

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

where

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65CALORIMETER AS DYNAMIC OBJECT

On the basis of the above method, the algorithm of the calculation ofthe Fourier transforms expressed by Eqs (2.85) and (2.100) is calculatedby using the Fast Fourier Transform Method [68].

The Fourier transform of the impulse response h(t) can be obtainedin the case of a sinusoidal input function (Fig. 2.11):

becomes

where

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

The Fourier transform is equal to

Hence

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2.7. Methods of determinationof dynamic parameters

2.7.1. Determination of time constant

The dynamic properties of a calorimeter treated as an inertial objectof 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 thedetermination of by applying the input step function [Eq. (2.45)] underconditions where the initial temperatures of the calorimeter and isother-mal shield are the same. Equation (2.19) can then be written in the form

Selected methods will now be discussed that are used to determinethe dynamic properties of calorimeters as inertial objects. Several ofthese methods are similar to those used in steering theory.

Different methods are presented for determining the time constant ofa calorimeter treated as an inertial object of first order. The methodsused to determine the dynamic parameters of calorimeters that areinertial objects of order higher than one are also discussed. All thesenumerical methods, algorithms and listing programs are described indetail in [67].

To apply each of the methods presented below, a knowledge of thephysical properties of the investigated object is not necessary. Thesemethods have been qualified as useful in calorimetry to identify thedynamic 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 heattransfer between the domains themselves and between the domains andthe environment. When we follow this procedure, the dynamic proper-ties of a calorimeter can be determined by using the method of N-domains based on the general heat balance equation [Eq. (1.147)]. Thismethod will be presented in Chapter 3.2.4.

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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 writtenin 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 termon the right-hand side of Eq. (2.116) is related to curve I, and the secondterm on the right-hand side of this equation to curve II. Curves I and IIhave identical shapes, but there is a shift in time between them, relatedto the duration of the heat impulse produced. For this period of time, thecourses of the changes in the calorimeter temperature T in time t arerepresented by the interval 0K of curve I. When t > a, the course of thefunction T(t) from to the shield temperature is presented in curve III.This is the cooling curve of the calorimeter. The changes in temperaturethat occur here are only a result of the existing difference in tempera-tures The interval KM of curve III graphically represents the differ-ence between the values T(t) of the first and second terms on the right-hand side of Eq. (2.116).

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When the heat generation period is long enough for a new state ofthermal equilibrium to be reached, characterized by Eq. (2.19) can bewritten 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 thecalorimeter (curve 0I, Fig. 2.15).

Let us use Eq. (2.118) to present several procedures for determiningthe 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 anasymptote of the curve at the point relating to (Fig. 2.15).

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69CALORIMETER AS DYNAMIC OBJECT

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

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

where

Thus, the length of the subtangent is equal to the time constantvalue

Procedure 3. After time T(t) is equal to

After time the temperature becomes and aftertime we have Proceeding in such a way for theknown values of and T(t) of the heating process, we can obtain thevalue of the time constant or its multiplicity.

Procedure 4. The value of the time constant can be determinedanalytically. The values of T(t) related to time moments and cor-respond to

Hence, Eqs (2.124) – (2.126) become

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Thus

and after rearrangement we have

Procedure 5. To obtain the time constant the integration methodcan be used.

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 usedto determine the time constant In this case, it is assumed that y(t) = 0,

and the cooling process is expressed by equation

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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 ofthe cooling curves are not much different from the procedures presentedabove.

Procedure 1. The same as in the case of the heating curve; the lengthof the subtangent at every point of the cooling curve is equal to the timeconstant (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 similarway, the value of T(t) can be determined for the time constant multiplic-ity. Knowing temperature and choosing a matching value of T(t), onecan 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 definedas a function of time are known. When, after time of the cooling proc-

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ess, the value of T(t) becomes and, after time it becomesthen according to Eq. (2.135)

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

Hence

Procedure 4. The time constant for the cooling process can becalculated 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 timeconstant When the experimentally determined T and the t data for thecooling process have been obtained, the plot in the coordinates (t, lnT)

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73CALORIMETER AS DYNAMIC OBJECT

can be drawn, as shown in Fig. 2.17a [8]. When the dynamic propertiesof the calorimeter are characterized by one time constant the relationlnT = f(t) is expressed by a straight line (Fig 2.17a), which forms anangle 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 bymore than one constant of time [Eq. (2.57)], at the begining of thecooling 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 thehigher time constants. The following procedure is used. The straight lineinterval (Fig. 2.17b) is extended to the point of intersection with the lnTaxis and the first time constant is determined. Next, a new function ofthe form

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

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The number of time constants which can be distinguished depends onthe properties of the calorimeter and the accuracy of the determinationof 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 thetime constants and the zeros of the transmittance nominator to beobtained by approximation of the pulse response of the calorimeter bythe least squares nonlinear curve-fitting procedure described byMarquardt [70].

In order to use this method, it is necessary to assume or determine theorder of the model number of poles and the number of transmittancezeros. According to these assumptions, the pulse response is a func-tion of time t, poles and zeros

where m < N. Introducing vector the components of which are polesand zeros:

Equation (2.147) can be written in the form

As is a nonlinear function of the components, expansion ofthis function as a Taylor series in the neighborhood of point isapplied:

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

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Equation (2.150) then becomes

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

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

where and is the sampling period. From thenecessary 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 polesand 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 oneiteration. Thus, it is necessary to repeat the iteration, assuming thecalculated values as initial values and repeat the iterative procedure untilwe obtain a suitable approximation of the pulse response.

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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 constantcoefficients:

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

for i = 1, 2, ... , N; k = 0, 1,..., N – 1 is also assumed. These functionshave 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 totime t in the interval yields

Putting

leads with respect to the coefficients to the following set of algebraicequations:

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77CALORIMETER AS DYNAMIC OBJECT

The above set of equations can be solved with respect toFrom 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 derivativesof the pulse response function h(t). To avoid the calculations of thesederivatives, 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 amplitudesare unknown. These coefficients can be determined by applying themodulating function procedure given above to the pulse response:

On putting

we obtain the following set of equations:

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The coefficients of the pulse response h(t) can be determined bysolving the above set of algebraic equations after previous calculationsof the coefficients

From the determined time constants and coefficients theLaplace 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 theequation

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 thecalorimetric system has the form

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

where

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The obtained set of Eq. (2.175) has a similar form as that of the set ofEq. (2.162). In the particular case when the calorimeter is calibrated by aconstant heat effect of heat power and duration u smaller than

the relation for coefficients given by Eq. (2.177) (taking intoaccount the conditions Eq. (2.159) and integrating by parts), can besimplified to the following form:

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

2.7.4. Rational function methodof transmittance approximation

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

where D(s) and L(s) are polynomials of degree m and N(m<N) given by

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The rational function L(s)/D(s) is the approximation expression offunction H(s) if the expansion in the power series with respect to s isidentical with the expansion in the power series of function H(s) to thedegree 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 thevalues of the pulse response h(t) and assuming a set of values of thecomplex variable s. In order to adequately define the criterionEq. (2.184), the values of variable s must be real. Function (2.185) is afunction of N + m + 1 unknown parameters and(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:

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

2.7.5. Determination of parametersof spectrum transmittance

To determine the dynamic parameters of the calorimetric system, thetransmittance of the compensating system (numeric or analog)[67, 74] should be matched in such a way that the resultanttransmittance corresponds to the non-inertial system and fulfilsthe following condition:

where is the spectrum transmittance of the calorimeter. As thebasis for determination of the transmittance, the experimental pulse re-sponse of the calorimeter is taken. Next, through use of the Fast FourierTransform (FFT) procedure, the values of the amplitude (modulus)and the phase of the Fourier transform of the pulse response arecalculated. These values are plotted in a Bode plot. From the Bode plot,the value of the direction coefficient of the last slope asymptote, whichpermits an evaluation of the difference between the number of zeros andthe poles of the transmittance is determined. If this coefficient is equalto –1, the transmittance can have one pole, or one zero and two poles, or

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two zeros and three poles, etc. If this coefficient is equal to –2, thetransmittance can have two poles, or one zero and three poles, or twozeros and four poles, etc.

To illustrate our considerations, let us assume, for example, that thevalue of the direction coefficient of the last asymptote is equal to –1.

Having determined the difference between the number of zeros andpoles of the transmittance, on the basis of the Bode plot (Fig. 2.18) wecan find the point at which the first slope asymptote interceptsthe axis, and thus estimate the time constant with he largestvalue, according to the relationship

In the knowledge of the time constant we determine a newtransmittance

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

In the other case, from the relations and it can result thatthe 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

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83CALORIMETER AS DYNAMIC OBJECT

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 ofthe second pole. Thus, we can determine from the slope of theasymptote, and we have

From the value of the horizontal asymptote, we can calculate thesecond 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 smallerthan the value of the second pole. In this case, from the slope of theasymptote we can calculate and thus the value of the second timeconstant is

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From the value of the horizontal asymptote, we can calculate

from the formula

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

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Chapter 3

Classification of calorimetersMethods 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 oftypes of calorimeters. These devices have been given various names bythe 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 oftemperature and pressure under which the measurements are performed.In some cases, the type of process investigated is decisive: calorimetersfor heat of mixing, heat of evaporation, specific heat measurements, andothers. The names of calorimeters often have to contain informationabout their construction features, e.g. labyrinth flow calorimeter, calo-rimetric bomb, drop calorimeter, or stopped-flow calorimeter. The nameof the device sometimes stems from the name of its creator. Exampleshere 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 devicesunambiguously.

Let us first define a “calorimeter” as an instrument devised to deter-mine heat. In any calorimeter, we may distinguish: 1) the calorimetricvessel (often called the cell, container, or calorimeter proper) at tem-perature that is usually in good contact with its contents, in whichthe studied transformation occurs. The contents include the reactantsamples and subsidiary accessories necessary to achieve the investigatedtransformation (e.g. to initiate the reaction, or to mix the reagents) or tocalibrate the device; and 2) the surroundings at temperature of-

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ten called the shield, environment or thermostat. The surroundings forma part of the calorimeter that is functionally distinct from the measuringsystem, with a defined temperature time dependence [82]. It may also beconsidered the first thermostat shield of the calorimetric vessel (in somecalorimeters 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 thecalorimeter is a dynamic object in which heat is generated. Calorimetersare graded by applying the criteria of the temperature conditions underwhich the measurement was made. As an initial basis for further consid-erations, the Fourier – Kirchhoff equation (Eq. (1.29)) has been used inthe following form:

A set of simplifying assumptions is next introduced to shorten thisform of equation. We assume that the process takes place under isobaricconditions. 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 shieldtemperature. Taking into account Eq. (3.3), Eq. (3.2) becomes

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

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Equation (3.5) describes the heat transfer in calorimeters with massexchange. These calorimeters are called open calorimeters. In contrast,calorimeters in which mass exchange does not occur are called closedsystems. 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 desiredmanner, e.g. to impose the temperature conditions of the calorimetricvessel or of the calorimetric shieldconst., or the temperature difference between them. Inthis manner, the following cases of temperature conditions can be dis-tinguished:

On the basis of the criteria listed above, calorimeters can be dividedinto two groups:

I.

II.

Adiabatic calorimeters, in which the temperature gradient be-tween the calorimeter proper and the shield is equal to zero0); during the calorimetric measurement, heat transfer does notoccur between the calorimetric vessel and the shield.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:

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Ia.

Ib.

Calorimeters with constant temperature They can be calledadiabatic-isothermal.Calorimeters where the temperature of the shield T0(t) changes intime. They can be called adiabatic-nonisothermal or simply adia-batic.

Nonisothermal calorimeters involve the following subgroups:IIa.

IIb.

Calorimeters with constant temperature T0(t) (isothermal sur-rounding shield), called isoperibol calorimeters.Calorimeters in which the shield temperature changes intime (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 whosetemperature changes in time. The calorimeters in subgroup IIa are non-adiabatic-isothermal, whereas those in subgroup IIb are nonadiabatic-nonisothermal.

The schedule of the presented calorimeters division according to thetemperature 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 generalmathematical models of the classified calorimeter groups. Particularforms of Eq. (3.5) will be given only for the types of open or closedcalorimeters that are known by the authors.

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CLASSIFICATION OF CALORIMETERS

1. When and Eq.(3.6) takes the form

Calorimeters that are described by Eq. (3.7) are characterized onlyby heat accumulation. They are adiabatic-nonisothermal calorimeters,and usually called adiabatic calorimeters. Their functioning rule is basedon the assumption that the temperatures of the calorimetric vessel andthe shield change in the same manner during the measurement. Theirdynamic properties are those of integral objects.

Adiabatic calorimeters were first used by Richards [76, 86] and thenSwietoslawski [76]. Nowadays, adiabatic calorimeters are used for stud-ies of various transformations in a wide temperature range, from veryhigh temperatures to very low, close to the absolute zero (0.001 K).Mention may be made of [83]: 1) low-temperature calorimeters, likethose 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 calorimeterswere 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 dozengrams). A series of new adiabatic calorimeters was recently designedthat allow measurements on very small amounts of substance (< 1 g),among them the calorimeters constructed by Ogata et al. [103], Matsuoand Suga [104], Kaji et al. [105], and Zhicheng Tan et al. [106].

One of the first scanning adiabatic calorimeters designed in the pastwas the DASM 1M microcalorimeter [107], used to determine the ap-parent molar heat capacity and conformational changes of proteins andnucleic acids. Measurements are performed in the temperature intervalfrom 10 to 100°C, the shield heating rate can vary from 0.1 to

and the sensitivity of the instrument is A

89

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new generation of scanning adiabatic calorimeters is represented by thedevices 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 theterms of the left-hand side of Eq. (3.6) are equal to zero. Then:

Since and are identical during the measurement, the proc-esses in such adiabatic calorimeters take place under isothermal condi-tions. These conditions can be fulfilled, when two heat processes ofopposite sign and occur:

where is the heat power of the process studied and is thecompensating heat power.

The best-known [83] such calorimeters are those of Lavoisier andLaplace (determination of the mass of melted ice) [110], Bunsen (pyc-nometric determination of the volume change of the liquid water-icesystem) [111], Dewar (volumetric determination of the air vaporized)[112] and Jessup (room-temperature operation based on the melting ofdiphenylether) [113]. In all of these devices, determinations are made ofthe changes in the measured quantity that result from the phase trans-formation. However, in this case it is very difficult or impossible to ob-tain conditions that correspond to thermodynamic equilibrium [76].

Progress in the development of adiabatic-isothermal calorimeters wasmade by Tian [114–116], who utilized the Peltier and Joule effect tocompensate the generated heat power. At present, the modern electronicand steering devices applied to generate these effects permit measure-ments with high accuracy. A good illustration of this is the achievementsat Brigham Young University [117], where this type of calorimetry hasbeen developed for almost 40 years. The titration calorimeter [118, 119]and the few high-pressure flow calorimeters [120–122] have been con-structed there.

3. When andconst., Eq. (3.6) becomes

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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 etal. [124, 125], Dzhigit et al. [126], Pankratiev [127, 128], Wittig andSchilling [129], Zielenkiewicz and Chajn [130], Hansen et al. [131] andChristiansen 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 theclosed 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 longtime been the most widely used class of calorimeters. The heat effectthat is generated in these calorimeters is in part accumulated the in calo-rimetric vessel and in part exchanged with its surrounding shield. Theseare dynamic properties of inertial objects. The parameter that is decisiveas concerns their properties is the time constant (or time constants).

In this class of calorimeters, there are instruments that have timeconstants of and others with time constants of severalThey have different constructions and find various applications. Amongthem there are:

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92 CHAPTER 3

a)

b)

c)

d)

e)

Dewar vessel calorimeters in more or less sophisticated form, suchas the well-known Nernst low-temperature calorimeter [133] orPierre Curie and Laborde‘s twin calorimeter [134] used in radiol-ogy. The Dewar-type calorimeters have found application in thedetermination of the average specific heats of organic liquids andtheir mixtures [135, 136], the heat of evaporation [137], the heatof 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],and the microcalorimeters of Wedler [148], et al. [149] andRandzio [150] for measurement of the heats of chemisorption ofgases on thin metal films.Calorimeters with the jacket filled with water or air are mainlyused 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 thenapplied. This group includes, for example, oxygen [151–153] andfluorine [154–156] calorimetric bombs and calorimeters used tostudy thermokinetics and the total heat effects of reactions [157–160].

High-temperature calorimeters such as those of Eckman and Ros-sini [161] or Mathieu et al. [162].Heat-flow or conduction calorimeters, where the heat exchangebetween 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 Calvetmicrocalorimeter [10] and the others that are characterized byvery high sensitivity (they can even detect signals of andare applied for various kinds of investigations [163, 164], e.g. themetabolism in living organisms [165–167], sorption and kineticsof 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].Calorimeters where the heat generated is transferred to the liquidthat flows around the outside surface of the calorimetric vessel, as

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CLASSIFICATION OF CALORIMETERS 93

in the Junkers [178] and labirynth flow calorimeters [76, 179,180].

f)

g)

AC calorimetry, which is applied intensively for the measurementof heat capacity in the region of phase transitions [181–185].A number of “quasi-adiabatic” calorimeters are used in such a waythat, in the calculation of the heat involved, the transfer of heatbetween the calorimetric vessel and the surroundings is neglected.Such are the flash calorimeters of Rosencwaig [186, 187], Calliset al. [188] and Braslawsky et al. [189], and the photoacousticcalorimeter of Komorowski et al. [190].

There are open nonisothermal-nonadiabatic calorimeters like those ofPicker, Jolicoeur and Desnoyers [191–193].

When measurements in an open, flow n-n calorimeter are made aftera relatively long time

Equation (3.12) takes the form

Relationship (3.17) describes the processes occurring in the flow andstopped-flow calorimeters of Roughton [194, 195], Kodama andWoledge [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 inwhich changes occur in the shield temperature. The temperature rise isusually linear and described by the ramp function (Eq. 2.49) as in the

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Calvet scanning microcalorimeter. There are several thermal analysisdevices for which Eq. (3.18) in its simplified form can be considered asmathematical model. They operate via observation of the temperature ofthe samples caused by the forcing function generated on the shield. Thisgroup includes: devices for heating and cooling curve determination(temperature measurement); differential thermal analysis, DTA (meas-urement of the temperature difference between the studied sample andthe standard substance) [198, 199]; heat flux differential scanning calo-rimetry, hf-DSC (determination of the difference in heat flux betweenthe studied sample and the standard substance to the shield [200]; andmodulated temperature differential calorimetry, mt-DSC (the tempera-ture change of the sample described by the frequency and amplitude ofvibration) [116].

Regardless of which measurement method is used, in each of themthe heat effects generated in the sample and in the calorimeter shield aresuperimposed. In consequence of the same type of inertial properties (ofinertial 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 consideringonly the changes in temperature that are caused by linear rise of theshield 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

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CLASSIFICATION OF CALORIMETERS 95

When the calorimeter proper and the shield are initially in thermal

equilibrium, i.e. and are equal to each other, and the measure-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) arepresented in Fig. 3.2. This shows that after a certain period of time

following the start of measurement, the course of change inbecomes linear. From this point, the difference between and isequal to and the shift between the temperature curves is equal to thetime constant Such a temperature course is observed when the sampleis a thermally passive object. For an endothermic reaction, the lineappears to be twisted as a result of decrease of the heating rate. For anexothermic reaction, the twist of the line is linked to the increase ofthe heating rate. This is presented in Fig. 3.3(a). Figure 3.3(b) shows thetemperature change reading determined by differential measurement. Inthe case of a Calvet DSC microcalorimeter, thermogram data are usuallyused for the determination of P(t) and then Q(t).

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6. When andconst., 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 forcingfunctions relating to the heat power generated by the studied proc-ess, the compensating heat power and the heat power gener-ated in the shield in order to keep its temperature in compliance with theprogrammed changes.

7. When and Eq.(3.6) becomes

The authors do not know any examples of calorimeters of this type.

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CLASSIFICATION OF CALORIMETERS 97

The classification of calorimeters presented above conforms withthat given in 1930 by Lange and Miszczenko [1], who distinguished fourgroups of calorimeters:

1) isothermal-adiabatic;2) isothermal-nonadiabatic;3) nonisothermal-adiabatic; and4) 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 principlesand design of the calorimeter proper, including the ranges of tempera-ture and pressure in which measurements can be performed; 3) themeasured quantity and measuring device; 4) the static and dynamicproperties of the calorimeter; the calibration mode and the methods ofmeasurement and determination of heat effects; 5) the operational char-acteristics of the calorimetric device, the sensitivity noise level, themethod 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 methodsof 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 otherspermit the determination not only of the total heat effects, but also of thecourse 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 balanceequation (Eq. (1.148)) has been used [21]:

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In an adiabatic method (§ 3.2.3), it is assumed that only heataccumulates in the calorimeter, which is treated as an integral object. Agenerated heat effect P(t) is then described by the first term on the right-hand 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 distinguisheddomains, and their mutual location. A similar procedure is applied in themethod 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) isthen a heat balance equation of a simple body. The dynamic method is aone of the most frequently used in the determination of total heat effectsand thermokinetics.

In the modulating method (§ 3.2.8), Eq. (1.148) is usually reduced tothe form of a heat balance equation of a simple body [Eq. (1.99)], one ofthe 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 heatin the calorimeter proper is negligibly small.

The steady-state method (§ 3.2.9) is based on the assumption that theheat power generated in the calorimetric vessel is compensated by addi-tionally generated heat power of opposite sign. The calorimeter istreated as a proportional object. The inertial properties of the calorimeterare neglected. The left-hand side of Eq. (1.148) is equal to zero.

The heat balance equation (Eq. (1.148)) reduced to the heat balanceequation of a simple body [Eq. (1.99)] is also the basis of many methodsused to determine total heat effects. One of the best-known is themethod of corrected temperature rise (§ 3.2.10), often called the Reg-nault-Phaundler correction.

A group of methods exist which a priori assume that the calorimeteris an inertial object of N order. A model of the calorimeter is expressed

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CLASSIFICATION OF CALORIMETERS 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 byone equation of N order. Let us derive the equations that are the mathe-matical models of these methods. Equation (1.148) in its matrix formgives

where is the diagonal matrix whose elements are the heat capacities

of the particular domains; is the matrix whose elements are the heatloss coefficients; T(t) is a vector whose components are the tempera-tures of the particular domains; P(t) is a vector whose components arethe heat powers of the distinguished domains; and is the derivativeof the T(t) vector. Laplace transformation of Eq. (1.146) gives

where s is the Laplace operator, is the initial state vector (the initialtemperature condition in the domains), and T(s) and P(s) are the Laplacetransforms 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

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100 CHAPTER 3

where is the sampling period and is the Dirac function, Eq. (3.31)can be rewritten as

If it is additionally assumed that the heat source is located in onedomain only, and that the temperature is measured in one domain onlyand under the zero initial temperature, Eqs (3.30), (3.31) and (3.33) de-fine the mathematical models for reconstruction of the thermokineticsP(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 interpretationmethod (§ 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 dynamicproperties (§ 3.2.11.6) of the calorimeter compensate the transmittancezeros 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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CLASSIFICATION OF CALORIMETERS 101

where S is the gain factor.

3.2.2. Comparative method of measurements

Calorimetric determinations are based on the comparative method ofmeasurements. The International Conferences on Chemistry in Lucernein 1936 and in Rome in 1938 accepted the proposal that all physico-chemical measurements should be divided into two groups, absolute andcomparative [202–205]. The absolute measurements include the deter-mination of absolute values characterizing a given physicochemicalproperty of chemical compounds or a mixture. It is necessary to intro-duce all secondary factors which would change the numerical valuedetermined by the absolute method. Absolute measurements should becarried out exclusively by specialists working in bureaus of measuresand standards or in laboratories sufficiently adapted for high-precisionmeasurements. A comparative method of measurements can be per-formed when possible to assume certain substances as standards and theproperties of the investigated systems can be compared with the identityof 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 themethods 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 arethe same.

The method of comparative measurements is often performed withuse of the differential method of measurement, which is defined [206] as“a method of measurements in which the measurand is compared withthe quantity of the same kind, of known value only slighty differentfrom the value of the measurant, and in which the difference betweenthe 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 oftwo identical calorimeters (I and II) located in a common shield. One ofthem contains the sample in which the heat effect is generated; in thesecond, the thermally passive object (or standard substance) is located.The course of the change in temperature of calorimeter I in which theheat effect is generated is based on the measured temperature differencebetween 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 calorimetersare identical; and b) the influence on calorimeters I and II of the inputforcing functions generated in the surroundings are the same. This man-ner of temperature difference measurement allows determination of thetemperature changes caused by the heat power generated in the sample,and elimination of the influence of other forcing functions. This methodis 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 clearthat the disturbances can be eliminated only if the transmittances ofcalorimeters 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 determinean acceptable range of difference of the time constants for which, for agiven 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 calledtwin calorimeters. In such calorimetric systems, it is necessary to obtainequal temperatures of the inner parts of calorimeters I and II in such away that in calorimeter II a the heat power has the same magnitude andcourse as that in calorimeter I. It is obvious that, for twin calorimetricsystems, 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 applicationin adiabatic and scanning adiabatic calorimetry

The adiabatic method [76, 87–89] is based on the assumption that noheat 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 thecalorimeter 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 theshield are heated at constant power input throughout the whole period ofmeasurement. The course of the temperature change with time is meas-ured, as is the electrical power supplied to the calorimeter heater. Weassume that the thermal power generated is used only to heat the studiedsubstance 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 themeasured quantity used to determine the specific heat at temperature T.The generation of electric energy inside the calorimeter proper does notalways lead to a temperature rise. It can happen that an isothermal riseof the enthalpy occurs during the studied process.

In the second method, the temperature of the calorimeter proper ismeasured with no power input. First, the calorimeter proper is heated atconstant power for a known time interval to attain a certain temperature.After the power is turned off, the temperature is measured. Then, thecurrent supply can again be switched in and next, after switching-off,the temperature will be measured. This cycle can be repeated severaltimes. Such a procedure is called the intermittent-heating method. Bothof the measurement procedures described above are used in practice.The first permits control of the temperature of the calorimetric shield inan easier way and it is certainly more convenient to obtain results in afull range of measurement. The second procedure permits study in aprecise way of these temperature ranges where interesting phenomena(e.g. unknown phase transitions) are to be observed.

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3.2.4. Multidomains method

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

where N is the number of distinguished domains; is the heat losscoefficient between domains i and the j; is the heat loss coefficientbetween domain i and the shield; and is the heat power.

Equation (3.41) describes the “changeable” part of the calorimeter; itis assumed that is the heat capacity of the calorimetric vessel withthe 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 heatcapacities there is no need to change the mathematical model of thecalorimeter; a change in calls only for the introduction of new data inthe 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 theheat source and temperature sensor. This means that the mathematicalform of the equations is related to the thermal and geometrical parame-ters of the calorimeter and to the location of the sensors and heat

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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. Theamount of heat exchanged between the domains through these media isproportional to the difference between their temperatures. The propor-tionality factors are the corresponding heat loss coefficients, and a heatsource and temperature sensor can be located between them. The systemof domains is located in a shield of temperature, which is treated as thereference temperature.

To elaborate the dynamic model, it is necessary to calculate the heatcapacities of defined domains and the heat loss coefficients, and to de-termine the structure of the model. For example, qualitative analysis ofthe heat exchange in the BMR calorimeter [157] (Fig. 3.4) on the basisof the calorimeter construction allowed the distinction of 19 domains.

The distinguished domains are individual parts of the calorimeter. Ablock diagram of a such system, elaborated by Cesari et al. [211], ispresented in Fig. 3.5, where are the interaction coefficients betweenthe domains. The calorimeter was divided into upper U and lower Bparts, 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 ofthe calorimetric vessel, the following domains were chosen for the partsof the calorimeter: 1, 2 – the calorimetric heater R; 3, 4– the shield A ofthe heaters; 5–8–the block vessel, 9, 10–the cylindrical parts of thevessel shield; 19 – the bottom part of the shield; 11,2,15,16–thermocouples; 13, 14–thermocouple supports; 17–inner thermostatblock; 18– vessel shield support. It was assumed that the calorimeter isideally 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 geometryand its parameters: the specific heat and density of the domain material;coefficients were calculated from the equation where isthe thermal conductivity; F is the surface area of heat exchange; andis the distance between the centers of the neighboring domains. Thisrelation was applied when the surface dividing the domains was a plane.For a cylindrical surface of heat transfer, the equation

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CLASSIFICATION OF CALORIMETERS 107

was used, where l is the height of the cylinder; and and are the radiiof the cylindrical surfaces that pass through the centers of the neighbor-ing domains.

The elaboration of the model allows determination of the set of heatbalance equations. The procedure of calculating the calorimetric re-sponse starts from the heat balance equation of the domain in which theheat source is located, with the assumption of zero initial conditions forall the temperatures, for i = 1,2, ... , N. On substituting thederivatives 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 themodel responses to a given heat effect. There are as many equations asdomains distinguished and all calculations are carried out in one calcula-tion loop.

The algorithm of the thermokinetics calculation is very similar tothat for thermogram calculations. The beginning of the procedure is thetemperature of the sensor. Next, the temperature of the neighboringdomain is calculated, assuming zero initial temperature for the remain-ing domains. For a system of domains, Eq. (3.42) for can be writ-ten in the form

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

where z is the index of the heat source domain, and z – 1 is the index ofthe domain next to z.

For a model of more complicated configurations, when one domaincan have more than two neighbors, one calculation loop has as manyequations as neighbors. For example, domain j has r neighbors. The heatexchange between these domains is characterized by heat loss coeffi-cients Coefficients and are the heat loss coefficients betweendomain j and its neighbors and between domain j and the shield, respec-tively; The calculation loop will then be a set of r equations ofthe form

where r is the index of the neighboring domain to the j domain. Thefollowing loops are related to the number of domains and the configura-tion between them, up to the last loop for calculating the course of heatpower 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 numeratorof the transmittance) and is the pole of the transmittance (theroot of the dominator of the transmittance); S is the static amplificationfactor. The finding of the real form of the transmittance is the same asdetermination of the dynamic properties of the examined calorimetricsystem. 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 stabilityconditions of the numerical solution are fulfilled. These conditions de-mand the admission of the value of a sampling period obtained from theamplitude characteristics of the calorimeter, taking into account thenoise-signal ratio [see § 3.2.11.7]. The optimal sampling period limitsthe smallest value of the time constants. If any number of time constantsof the calorimeter does not satisfy the stability condition, the a newmodel of the system must be worked out, decreasing the number of do-mains and calculating new parameters. The maximum order of the newmodel 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 meansthat a model useful for the determination of thermokinetics has beenelaborated.

3.2.5. Finite elements method

The finite elements method was proposed and described by Davisand Berger [212]. This method is based on the following assumptions. Acertain number of elements are distinguished, between which the heatexchange is characterized by the heat loss coefficients For each ele-ment, the heat capacity is calculated. It is assumed that temperaturefor each of the distinguished element is known at moment t and nextcalculated for For each i-th element, the amount of heatexchanged between the element i and element i+1 is determined by theequation

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 of temperature the heat powerP 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 representsthe course of the changes in time of the temperature of the element inwhich the sensor was placed.

Thermokinetics is determined on the basis of deconvolution of thepulse response of the examined system and the measured thermogram.For calculations, the first nonzero value of the pulse responseis taken. The increase in temperature of the element (in which theheat effect is generated) is determined on the basis of the equation

where is the increase in temperature of the element in which thesensor is placed. The magnitude R plays the role of the multiplier, whichallows transformation of the sensor signal to the signal in the reactionvessel. In order to determine the thermokinetics at moment it isnecessary to know the temperature distribution of all elements and theheat power P at moment t. In each sampling period the increase intemperature is caused by two effects: 1) the heat generated by thereaction (or pulse), which causes the temperature increase; and 2) theheat transferred, which causes the temperature increase Thus, theincrease 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 determinedon the basis of the thermogram obtained, and the increase in temperature

is calculated as the difference

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The value of the increase in temperature is calculated on the ba-sis of Eq. (3.53). Thus, the temperature of the reaction vessel foris

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. Thismethod is very sensitive to the errors in the determination of the value ofthe pulse response and coefficient R.

3.2.6. Dynamic method

The dynamic method [10] of measurement is used for the determina-tion of: a) the course of the thermal power change in time, b) thecourse of the temperature change time, which characterizes thecourse 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 assumethat the calorimeter proper at temperature is surrounded by a shieldat constant temperature while the results of the temperature differ-ence are expressed by function T(t). This assumption enablesus to discuss this method by using the procedure proposed by Calvet[10].

To describe the relation between the generated heat effect Q(t) andthe difference T(t) in the calorimeter proper and the shield temperatures,Eq. (3.13) is used in the form

or in the form

vs.

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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) isequivalent 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 properand the shield.

Measurements of the temperature are usually made by using sensorssuch as a resistance thermometer and thermistor, where the resistance isa function of temperature, or a thermocouple and thermopile, where theelectromotive force is a function of temperature. Accordingly to expressT it is convenient to use the quotient with where is themeasured 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 intime and the time axis t. The values of and F are the results of calo-rimetric measurement. The values of C/g, G/g, and are determined viathe 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 timelong enough for a new state of stationary heat exchange to be achieved.In practice, it is characterized by a constant temperature difference, de-

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scribed as and simultaneously by constant value Allowing a smallerror, we can assume that it occurs in the time period equal to Thesolution 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 todetermine 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 isconstant in the interesting range of changes in the temperature of thecalorimeter. 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 thecalorimeter becomes a thermally inertial object. Cooling of thecalorimeter then occurs up to the moment when the temperatures of thecalorimeter and the shield are the same. This process has been describedby Newton‘s cooling law [Eq. (1.104)] and is used to determine thevalue 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 importantinformation is provided by observation of the course of the changesfor the calorimeter as a thermally inertial object. When the changes= f(t) are nonlinear during the initial period of the cooling or heatingprocess, it is necessary to analyze whether application of the simplebody heat balance equation to the calculations is correct. A nonlinearcourse of means that the function is multiexponen-tial. More precise equations expressing the relation between P(t) andT(t) should then be used. The calibration procedure presented here is notthe 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 assumptionthat 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), itcan be assumed that the quantity of heat accumulated is extremely small.This assumption is the basis of the flux method. The amount of heattransferred between the calorimeter proper and the shield is then directlyproportional to the temperature difference. Thus, the course of ob-tained from the measurement resembles that of P(t), and its value isdetermined 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. Underisoperibol conditions, this method is called AC calorimetry [213–215].The first AC calorimetry experiments were performed in 1962 by

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CLASSIFICATION OF CALORIMETERS 115

Kraftmacher [213], who measured the heat capacities of metals in thehigh-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 bea 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 andv is the angular frequency. The solution of Eq. (3.68) in the case ofsteady-state temperature modulation of the sample, according to Eqs.(2.55) and (3.26), is

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

The resultant signal is then analyzed by using the combination of aFourier transform to deconvolute the response of the sample to the un-derlying ramp from its response to the modulation that can give rise toan 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 theheat 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 newanalytical technique used to obtain information on the heat capacity inthe range close to the phase transformation. It is a method applied inmany instruments, e.g. as the Modulated DSC of TA Instru-

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

3.2.9. Steady-state method

In order to obtain a constant gradient of temperature between thecalorimeter proper and its environment, the steady-state method is used.The method is based on the assumption that the following condition isfulfilled:

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; compensatingDSC).

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-

terval gives

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 theaverage temperature. Thus, the temperature increment in the consid-ered time period is

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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, wecan put

and Eq. (3.80) simplifies to the form

For adiabatic-isothermal calorimeters, it is assumed, that0 and the first term on the left-hand side of Eq. (3.82) is neglected. Thethermal power is used to compensate

For adiabatic-nonisothermal calorimeters when to de-termine the amount of heat it is necessary to know the time intervalof the process, the average temperature the heat loss coefficient Gand the amount of heat delivered to the calorimeter proper in orderto hold the desired value with minimum oscillations.

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The thermal power is produced by several pulses of Peltier orJoule effects by:

a) Application of pulse compensation consisting in the generation ofquantized thermal power pulses as shown in Fig. 3.6, where A is apulse 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 thermalpower

or

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

where is the integration constant.

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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 methodis 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 heatgenerated in the calorimetric vessel corresponding to the temperatureincrement can then be expressed as

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 (segmentBC); and the final period (segment CD).

During the initial period the temperature changes before theexamined heat process are measured. The beginning of the main period

is the initial moment of the generation of heat by the processstudied. As the end of the main period, the time moment is assumed asthe 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 processstudied, during the whole period of measurement a constant heat effectcan be produced in the calorimeter by secondary processes (for example,the process of evaporation of the calorimetric liquid, the friction of thestirrer on the calorimetric liquid, etc.). The course of the temperaturechanges is then described by

expressing Newton’s law of cooling by magnitudes u corresponding tothe 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 secondaryprocess 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

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and in the final period, we have

Because of the small changes in temperature in these periods ofmeasurement, we can approximate them as linear changes in time. Thus,

and can be treated as constant, and can be ascribed to the average

temperatures and of the initial and final periods, respec-tively:

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

Let us assume that the constant heat effect caused by the secondaryprocess exists in the main measurement period too. In Eq. (3.87), thetemperature must be substituted by the term thus, wehave

The value of the integral

can be approximated by the expression

T(t)

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where are the values determined during the subsequent read-ings of temperature T(t) in the main period; and Thismethod involves the assumption that, in small time intervals of the mainperiod, temperature T(t) changes linearly, and the average temperatureof the main period is the arithmetic average of the average temperaturesobtained in these small time intervals. On determining from Eq.(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

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Equations (3.105) and (3.107) for the temperature correction arecalled the Regnault-Phaundler correction. They assumed the samplingperiod to be equal to unity, and thus This correction ismostly used in isoperibol calorimetry to calculate the corrected tempera-ture 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 initialand final periods by straight lines. Planimetry at first was used to obtainthis integral.

3.2.11. Numerical and analog methodsof determination of thermokinetics

3.2.11.1. Harmonic analysis method

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

where is the Fourier transform of the response As a criterionof 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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function calculated. The practical frequency is determined on thebasis of the dependence

where is the value of the spectrum transmittance amplitude for v

= 0, 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 asa mathematical model of the calorimeter, and thus appropriate zero ini-tial conditions are assumed. This method assumes the existence of oneinput function T(t) and one output function P(t). The impulse responseH(t) is determined as a derivative with respect to time of the response ofthe calorimetric system to a unit step. As a criterion of accordance be-tween the measured temperature change T(t) and the estimated course oftemperature x(t), the integral of the square of the difference betweenthese 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 atask can be solved provided that the temperature response T(t) and theimpulse response H(t) of the calorimeter are known in the analyticalform. However, in calorimetric measurements, numerical values

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are usually available on the course of the temperature changesT(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 coefficientsThe 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, becausewe 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), weobtain the values of coefficients as

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

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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 toas 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 statevector. Thus, the calorimeter transformation equation combines the statevariables with the parameters of the calorimetric system and the thermalpower produced:

The state equation (Eq. (3.124)) written in this way is a system offirst-order equations. Because of the available calorimetric information,the state variables should be transformed in such a way as to obtain arelationship between one input function P(t) and one output function.The relationship between the state variables and the output function hasthe following form:

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

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3.2.11.5. 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, isdescribed 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 timedomain corresponds to the differential equation

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

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

basis of we calculate the coefficients and differential equa-

tion (3.131), and on the basis of This enables us todecrease the number of derivations of m and is very useful when thedifference between the degree of the denominator and the degree of thenominator 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 accordingto Eq. (2.13) in the form of the following differential equation:

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

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

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 thedesired heat power:

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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 oranalog method) of terms providing an inverse function to theterms For an individual corrector, we have

For two correctors, we have

For an ideal correction, this would lead to the case when the inputfunction corresponds to the output function, as represented by the rela-tionship

The practical performance of the inverse filter method is different indifferent systems [246–257], but advantage is always taken of electronicelements active under the form of an operational amplifier and imped-ance divider of the feedback, by which the required function isachieved.

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 Toeach of these terms there corresponds a linear differential equation offirst order of the form

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

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CLASSIFICATION OF CALORIMETERS 131

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

3.2.11.7. Evaluation of methods of determinationof total heat effects and thermokinetics

The methods discussed above discussion are based on particularforms of the general heat balance equation. Methods based on the simplebody balance equation or its simplified form have been used in calo-rimetry for a very long time: the method of corrected temperature risefor more than 100 years, and the adiabatic method for about 100 years.There are also new methods (e.g. the modulating method) or thosewhich have become very important (e.g. the conduction method) in thepast 20–30 years.

Improved conditions in calorimetric experiments lead to the creationof more precise mathematical calorimeter models and methods used forthe determination of heat effects.

From the methods in which the set of heat balance equations wasused, the multidomains method (§ 3.2.3) and the finite elements method(§ 3.2.4) were elaborated. Numerical methods for the determination ofthermokinetics in n-n calorimeters (§ 3.2.11.1 - § 3.2.11.6) were alsodeveloped.

In the multidomains method and in the numerical methods, in accor-dance with the detailed solution of Fourier’s heat conduction equation, itwas assumed that the impulse response of the calorimeter is describedby an infinite sum of exponential functions [Eq. (2.57)]:

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132 CHAPTER 3

Thus, in these methods an ideal model for reconstruction of thethermokinetics should be a model of infinite order. In fact, it is neces-sary to limit the number of exponential terms (domains). Generally, thisis due to the compromise between the available information and thepossibility 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 differentmethods independently proposed to reconstruct thermokinetics from theexperimental calorimetric data. The dynamic optimization, harmonicanalysis, state variable and numerical correction methods were tested.To eliminate the uncertainties caused by the individual features of thevarious calorimeters in different laboratories, it was generally approvedthat the responses of all calorimeters should be distributed to all partici-pants of the program. These responses concerned normalized series ofheat 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 as0.002–0.003 of the largest constant of the calorimeter, which corre-sponds to 1/300 of the half-period of cooling. The values of amplitudeswere given in relative units, equal to 10, 100 and 1000. The amplitudeequal to 10 was defined as corresponding to the heat power of the pulseof duration equal to 8 sampling periods, for which the maximal tempera-ture increase was 10 times larger than the amplitude of the measurementnoise (the change in baseline when the thermally inert object is placed inthe calorimeter). Three types of measurements were made: 1, 3 or 7pulses 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 ofthe data reconstruction by dynamic optimization and harmonic analysisin Fig. 3.8.

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134 CHAPTER 3

The results of the investigations indicated that the examined methodsreveal considerable progress in the reconstruction of thermokinetics. Itwas stated [265] that “it is clear that the objective is not complete recon-struction of thermokinetics (thermogenesis), but rather decrease of theinfluence of thermal lags in calorimetric data; thus, after reconstruction,the residual time constant which appears in the data is ~ 200 timessmaller 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 caseswhen 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 dataand that the amplitude of the heat effect under study, compared to thatof 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 thenumber of certain digits in the measurement of the calorimetric signal.In order to estimate the values of the sampling period h instead ofthe approximate values

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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 samplingperiod 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 thenumber of zeros of the transmitance (q = N – m), the formula for theoptimal sampling period takes the form

A scheme for choosing the optimal signal-to-noise ratio has been thesubject of many papers, e.g. [267–270]. In the multidomains method,this selecting scheme was included in the procedure of determining theparticular form of the calorimeter model. Another manner of selecting isused when harmonic analysis method is applied. In this method, thetransmittance 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 mustcover the whole interval of time from “initial zero” to “final zero” of thetemperature calorimetric response. The discrete measurement of tem-perature limits the upper bounds of frequency which can be applied forreconstruction of the thermokinetics. The value of this frequency,resulting from Shannon’s theorem is a function of the sampling periodand can be expressed by the relation It is thereforeimpossible to use the complete spectrum of the frequency. There alsoexists a boundary frequency

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136 CHAPTER 3

which is related to the measurement noise. The boundary frequency isthe second limitation of the frequency spectrum, which can be applied inthe reconstruction of the thermokinetics for a given calorimetric system,and

The high precision of calorimetric measurements today permits a lessstrict description of the calorimetric system itself. The comparison ofreconstructions of the thermokinetics obtained with the multibodymethod and with other methods demonstrates the convergence of theresults [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 balanceequation of second or higher order are frequently used in isoperibol andDSC calorimetry. When the mathematical model is known, the inversefilter method is used to determine the thermokinetics.

The course of the thermal power change can be determined by usingdifferent calorimeters. The choice of the heat effect determinationmethod does not depend on the calorimeter type. In a calorimeter with avacuum jacket, the thermokinetics can be determined successfully bymeans of the dynamic method. In calorimeters whose inertia is verysmall (conduction calorimeters), use of the flux method is not alwayssuitable. Better results in the reconstruction of thermokinetics are ob-tained from the use of methods where it is assumed that the calorimeteris 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

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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 quantitativemeasurement.

Regardless of what method of heat effect determination is applied, ithas 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 thecorrect 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 touse 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 forcingfunction in the calorimeter proper is always the heat effect generated asa result of the studied process. However, the review of the methods hasrevealed that there can be forcing functions striving to compensate thegenerated heat effect of the process; a rectangular pulse or a series ofrectangular pulses; periodic heating rate of the sample etc. Additionally,the forcing functions acting in the shield influence the calorimeter. Foran isoperibol calorimeter, this is a forcing function, tending to achievethe stability of the temperature of the shield; for scanning calorimeters,it is a ramp function. Analysis of the output functions courses generatedin the shield is quite simple to deal with and usually consists inconsidering the course of the system response, if there is a thermallyinertial 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 thegeneral 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 equationof 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 overallcoefficient of heat loss, the time constant of the domain, the interactioncoefficient and the forcing function.

The overall coefficient of heat loss for each of the domains is de-fined as

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140 CHAPTER 4

This coefficient characterizes the heat exchange between domain jand 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 of the domain:

The time constant 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

coefficient to the overall coefficient of heat loss

This is a measure of the heat interaction of domain i with the domainj relative to the interactions of the remaining domains and surroundingswith domain j. The interaction coefficients essentially affect the thermalinertia of the calorimeter and allow us to establish the structure of thedynamic model of a given calorimeter. If the value of the interactioncoefficient is negligibly small, it may be assumed that there is nothermal interaction between domains i and j or, more exactly, that thethermal interaction between domains i and j is small enough to be ig-nored in comparison with the interactions between domains i and j andother domains and the surroundings.

Furthermore, the notion of the forcing function (taken from con-trol theory) was introduced into these considerations. This function isdefined 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

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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 aredimensionless, chosen in such a way that an increment of the tempera-ture in a stationary state is equal to an increment of the forcing

function Defined in this manner, the coefficients are especiallyuseful in applying the principle of superposition to the system of do-mains.

The set of differential equations Eq. (4.1) can be written in matrixform 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 statevector, 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) isthe 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 transmittanceswhich have the form

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142 CHAPTER 4

where is the determinant of the matrix is the

corresponding minor of the matrix and is the determinant of

the matrix These determinants, after development into a power serieswith 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 heatsources in relation to the temperature sensors. When the N-domainmethod is used to describe the heat effects, determination of the trans-

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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 effectgenerated in the calorimeter.

In many cases, the general equation of dynamics can be simplified toa few-domain system, as will be pointed out below, in order to analyzethe courses of different heat effects in calorimeters.

4.2. Dynamic properties of two and three-domaincalorimeters 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, characterizedby heat capacities and respectively. The heat transfer betweenthese domains is described by the coefficient while that between thedomains and the environment (surroundings) is described by the coeffi-cient The set of heat balance equations of the system is as follows:

If we put

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144 CHAPTER 4

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,and the temperature change of domain 2 is measured. The

domain 1 output function here is at the same time the input func-tion for domain 2. Equations (4.22) and (4.23) then become

If we additionally assume that Eqs (4.24) and (4.25) can bewritten in the form

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:

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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, amongothers, 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) onThis 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 forvarious 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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146 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, bothfunctions P(s) and G(s) must be determined if the function is to bedefined.

Let us now define the courses of function in the cases when 1) aunit pulse function; 2) an input step function of amplitude b and timeinterval 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 thetime period a, the time interval between two successive pulses being Z.

Let us suppose that a heat effect that is constant in time (an inputpulse 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 sys-tem at the moment t = 0, if the initial conditions are zero.

If we assume and with Eq. (4.40), after inverse Laplacetransformation Eq. (4.34) becomes

If the constant heat effect occurs in domain 1 for a period of time aand amplitude (unit step function of amplitude the function f(t)assumes the form

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DYNAMIC PROPERTIES OF CALORIMETERS 147

The changes in temperature in time, are then expressed by

When n rectangular pulses occur in domain 1, each pulse lasting fora 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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148 CHAPTER 4

4.2.2. Equations of dynamics.Three domains in series

In numerous cases, the determination of the output function with agiven accuracy requires that the two-domain cascading system should beenlarged 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 differentialequations of dynamics describing the courses of the temperaturechanges in time in the domains considered. In Eq. (4.49), f(t) is the inputfunction. It may be assigned an arbitrary course. Function is thedomain 1 output function, which at the same time constitutes the inputfunction for domain 2. The output function of domain 2 constitutes theinput function of domain 3, whose output function is

For the solution of the above system of differential equations, theLaplace transform method was used, yielding

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DYNAMIC PROPERTIES OF CALORIMETERS 149

where

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

When f(t) = 0, and

a solution of Eqs (4.49)–(4.51) in relation to is given by the equa-tion

where

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150 CHAPTER 4

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 conditionsdescribed by Eq. (4.55), the temperature changes are expressed bythe equation

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Forms of the particular equation of dynamics developed for the sys-tems of two domains and three domains in series are presented only withselected input functions. This selection was made by taking as criteriontheir application in the analysis of the courses of the heat effects. Theparticular 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 dynamicsof cascading system

The equations of dynamics of cascading systems have been utilizedin calorimetry for many years [25–35]. For example, these equations canbe applied as follows:

1)

a)

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. Theoutput function of domain 2 is measured (Fig. 4.1).

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 andgeneration time of the rectangular pulse and the time constantsof domains 1 and 2 are known. For the calculations, one canapply Eq. (4.43). These calculations allow determination of theoutput function of the system, which is of special importancein measurements of heat effects of short duration.The input function is a ramp function. Domain 1 is a thermo-stat, while domain 2 consists of the calorimetric vessel withsample. Let us assume that the calorimetric vessel containssamples with the same volume, but with differing heat capaci-

ties The time constants of domain 2 there-

fore change. This causes differences in the values of the outputfunction. They can be calculated after determination of theparticular form of Eq. (4.43), values characterizing the ramp

function and the values of

b)

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152 CHAPTER 4

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 ofdomain 2, which in turn is the input function of domain 3. Theoutput function of domain 3 is measured. The input function f(t) isthe input step function. Domain 1 consists of the calorimetric ves-sel filled with the substance studied, domain 2 is the inner shieldof the calorimeter, and domain 3 is in isolation the temperaturesensor located on the outer surface of the inner calorimetricshield. The amplitude and generation time of the rectangular pulseand the time constants of domains 1–3 are known. To calculatethe output function, we can apply Eq. (4.61). The calculations al-low us to determine the influence of the inertia of the inner shieldand the temperature sensor on the recording of the heat effectgenerated in the calorimetric vessel.

In the discussed examples, the responses of the system to only oneforcing 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 operationof several forcing functions. This can be demonstrated via the followingexamples.

A system comprising the calorimetric vessel with the sample(domain 1), the internal shield (domain 2) and the temperaturesensor (domain 3). In the calorimetric vessel, heat effect isgenerated, while the compensating heat effect expressed byis produced in the internal shield of the calorimeter.

3 b)

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

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DYNAMIC PROPERTIES OF CALORIMETERS 153

This leads to the conclusion that, for the two-domain system wheninput 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) supplementedby the term.

For the three-domain system, when forcing functions andare produced at the same time, the output function is expressed byEq. (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 ruleto linear systems.

Equations of dynamics are also useful in analysis of the courses ofthe 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 firstorder, placed in a common shield. The sample is situated in one of them,and the reference substance in the other. The forcing input function isthe ramp function generated in the thermostat. Let us assume that thisfunction is at the same time the input function of both calorimeters. Theinfluence of the forcing function on a differential calorimeter is shown

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154 CHAPTER 4

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

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

4.3. Dynamic properties of calorimeterswith concentric configuration

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

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DYNAMIC PROPERTIES OF CALORIMETERS 155

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

4.3.1. Dependence of dynamic properties oftwo-domain calorimeter with concentricconfiguration on location of heat sourcesand temperature sensors

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

This is characterized by Eq. (4.18):

and the equation

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156 CHAPTER 4

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

The time constants and of the distinguished domains 1 and 2

are

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

On putting we can write

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

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 inthe form

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

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DYNAMIC PROPERTIES OF CALORIMETERS 157

concentric configuration [44, 45]. It may be noted that the left-handsides 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 ofEqs (4.74) and (4.75) referring to the heat balance equation of a simplebody, 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 temperaturesof domains 1 and 2, respectively, and that of the temperature of theshield at time = 0; and are the Laplace transforms of tempera-tures and respectively; and are the Laplace trans-forms of the forcing functions and

When

Eqs (4.76) and (4.77) become

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158 CHAPTER 4

For the transform from Eq. (4.80) and the transform fromEq. (4.79), we have

where

are the transmittances of domains 1 and 2, respectively. These functionscharacterize 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. Inthis 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:

1.

2.

3.

4.

A heat power characterized by forcing function is generated,and the changes in temperature in time are measured.A heat power is measured and the changes is temperature

in time are measured.A heat power characterized by function is generated, and thechanges in temperature in time are measured.A heat power characterized by function is generated, and thechanges 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 andinteraction 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 Table4.2. 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 presentedin Table 4.3.

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DYNAMIC PROPERTIES OF CALORIMETERS 159

If we introduce the time constants and of the calorimeter, de-fined as

which satisfy the equation

the transmittances given in Table 4.3 can be written in the form given inTable 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).

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160 CHAPTER 4

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

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

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DYNAMIC PROPERTIES OF CALORIMETERS 161

We can demonstrate this on examples of 1) the pulse response of thecalorimeter; and 2) the temperature changes of the calorimeter as athermally inert object.

When the inverse Laplace transformation is applied to the transmit-tances given in Table 4.4, the pulse responses of the calorimetric systemgiven 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 temperaturesensor are situated in the same domain, the shape of the pulse responseof the calorimetric system is reminiscent of the response of the calo-rimetric system of first order. From the shapes of the curves given inFigs 4.8d and 4.8c, it is clear that, when the heat source and temperaturesensor are situated in different domains, the pulse response of the calo-rimetric system at the initial time moment is equal to zero, next in-creases to a certain maximum value and respectively), andthen decreases to zero.

Thus, the pulse responses depend on the mutual locations of the heatsource and temperature sensor, and on the parameters of the calorimetricsystem, 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 arethermally inert objects, which is equivalent to the conditions

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162 CHAPTER 4

The temperature changes of the calorimeter are caused only by theinitial 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-peratures and when at least one of the temperatures of thedistinguished 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

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DYNAMIC PROPERTIES OF CALORIMETERS 163

where the time constants and of the calorimeter are given byEq. (4.84). When

the plots of the functions (4.92) and (4.93) are shown in Figs. 4.9a and4.9b.

From the plot in Fig. 4.9a, it is seen that domain 1, at a temperaturehigher by than the temperature of the environment, undergoes cool-ing, the course of the temperature changes in time being reminiscent inshape of a one-exponential course. Domain 2, whose temperature at theinitial moment is equal to the environment temperature, first undergoesheating to temperature by a certain time and then cools to theenvironment temperature. At time moment temperatures andare equal. When the initial temperature of domain 1 is smaller bythan the environment temperature (Fig. 4.9b), the domain is heated tothe environment temperature; the temperature of domain 2 cools to

by time next returning to the environment temperature. Theextremum of temperature of domain 2 is accepted as the time moment atwhich 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 environmenttemperature, and the course of the temperature changes in time arereminiscent in shape of a one-exponential curve.

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164 CHAPTER 4

Domain 1 with an initial temperature equal to the environment tem-perature heats up to temperature not reaching the temperature ofdomain 2, and then cools to the environment temperature. The tempera-ture of domain 1 is always lower than the temperature of domain 2. Ifthe initial temperature of domain 2 is lower than the environmenttemperature (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-ronment temperature, decreases to a certain temperature and next

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DYNAMIC PROPERTIES OF CALORIMETERS 165

increases to the environment temperature. The extremal value of thetemperature of domain 1 depends on the value of interaction coefficientk. As the value of interaction coefficient comes nearer to unity, theneight 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 anddomain 2 decrease in time, but the temperature of domain 1 decreasesfaster than that of domain 2. When the initial temperatures are smallerby than the environment temperature (Fig. 4.9f), both domains heatup to the environment temperature, but domain 1 does so faster thandomain 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 isrelated to the mutual locations of the heat sources and temperature sen-sor.

The presented equations were defined on the assumption that theform 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 effectas a function of location of heat sourceand temperature sensor

Let us define particular forms of the equation of dynamics [Eqs(4.74) and (4.75)] expressing the dependence between temperature andheat 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 theequation 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

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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 followingcases:

1. The temperature is dependent on the function In thiscase, the elimination of temperature from Eqs (4.97) and (4.98)furnishes

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

With the assumption that

Eq. (4.102) can be written in the form

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DYNAMIC PROPERTIES OF CALORIMETERS 167

Equation (4.105) connects the changes in temperature of domain 1with the heat effect generated in the same domain. It results from thisequation that the changes in temperature of this domain are determinedby the relation of the heat power and its derivative. This equation cannotbe directly applied to determine an unknown heat power, because it isnecessary to know the derivative of the value we are looking for.

2. The temperature depends on the function In this case, byeliminating the temperature from Eqs. (4.97) and (4.98) and fol-lowing the above routine, we can establish the relationship between thetemperature changes of domain 2 and the heat effect generated indomain 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 In this case,the equation of dynamics has the form

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168 CHAPTER 4

Equation (4.107) has a form similar to that of Eq. (4.104). Particularforms of the equation of dynamics show that, for the two-domain calo-rimeter, the temperature sensor should be situated in another domainthan 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 eachdomain. 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 andtemperature sensors can form the basis of the explanation of severalobserved 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 theKRM calorimeter [282], it was found that the calculated value treated asheat capacity changes in time.

Changes in the heat capacity of the calorimeter were noted, whenwas calculated by using the heat balance equation of a simple body, inwhich it is assumed that the heat capacity C depends neither on time tnor on the geometrical distribution of the heat power source and thelocation of the temperature sensor; it is equal to the sum of the heat ca-pacities of all the parts i of the calorimeter:

On the assumption that the heat capacity C of the system correspondsto 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

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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 intervalleads 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 intervalyields

or

Comparison of Eqs (4.109) and (4.114) gives

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

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170 CHAPTER 4

and

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

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

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

It follows from the above equations that the effective heat capacitydepends on various parameters: the heat capacities of the distin-

guished domains, the heat transfer coefficients between these domainsand the environment, the character of the changes in the heat effects intime and their derivatives with respect to time, the changes in particulartemperatures 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 = 1,2for a system of two interacting domains.

or

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DYNAMIC PROPERTIES OF CALORIMETERS 171

4.3.4. Energy equivalent of calorimetric system

When the corrected temperature rise method was applied for thecalibration of the calorimetric bomb, it was established [22–24, 284] thatthe calculated heat capacity of the device was not equal to the sum of theheat capacities of the calorimeter parts.

King and Grover [22] described the calorimetric bomb in terms of atwo-domain model with a concentric configuration. As a result, in themethod of corrected rise the heat capacity C was replaced by a correctedterm, called the energy equivalent of the calorimeter:

where and are the heat capacities of domains 1 and 2, is theheat loss coefficient between domains 1 and 2, and isequal to the smaller of the cooling constants of the calorimeter.

Let us analyze [283, 285] the influence of the mutual locations of theheat 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-tinguished during the experiment: and

In the initial and final periods, the calorimeter is a ther-mally inert object; in the main period, the heat effect is generated in thecalorimeter.

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 ofthese equations in the time intervalsimilarly as for Eqs (4.110), (4.113), (4.116) and (4.119), leads to

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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 whenheat 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

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DYNAMIC PROPERTIES OF CALORIMETERS 173

If the temperature changes in the initial and final periods result fromthe 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 secondterm 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

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174 CHAPTER 4

Putting

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

Let us define the energy equivalent R of the calorimetric system asthe ratio of the generated heat amount Q to the corrected temperaturerise as a function of the mutual location of the heat source

and the temperature sensor. Then, taking into account Eq. (4.146), wefind that

when the temperature of domain 1 is measured and the heat effectis generated in domain 1, the energy equivalent is equal to

when the temperature of domain 1 is measured and the heat effectis generated in domain 2, the energy equivalent is equal to

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DYNAMIC PROPERTIES OF CALORIMETERS 175

when the temperature of domain 2 is measured and the heat effectis generated in domain 1, the energy equivalent is equal to

when the temperature of domain 2 is measured and the heat effectis generated in domain 2, the energy equivalent is equal to

It can be seen from Eqs (4.151)–(4.153) that the energy equivalent isthe same, equal to the product of the largest time constant of the calo-rimetric system and the heat loss coefficient In Eq. (4.154), theenergy equivalent is smaller than in the first three cases.

In calorimetric measurements, the location of the temperature sensoris generally fixed, while those of the calibrating heat source and the heatsource of the examined process can differ. If the temperature of theother domains is measured, it is without difference in that one in whichonly the calibrating effect and the heat effect process are situated be-cause there is equivalence of the heat sources and On the otherhand, 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 constantof 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 energyequivalent has the dimension of heat capacity, but is not the sum of theheat capacities and of the distinguished domains. When the calo-

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176 CHAPTER 4

rimeter is thermally insulated which corresponds to adiabaticconditions, the value of the equivalent is equal to the sum of the heatcapacities of the distinguished domains. When ideal heat exchange oc-curs with the distinguished domain both domains can betreated as simple domains and the energy equivalent is equal to the sumof the heat capacities.

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Final Remarks

The present theory of calorimetry is a result of the authors’ ownwork. Its essential feature is the simultaneous application of the relation-ship and notions specific to heat transfer theory and control theory. Thepresent 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 ofvarious types. Calorimeters have been assumed to constitute linear sys-tems. This assumption allowed the principle of superposition to be usedto 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 temperatureof the shield. It includes calorimeters in which the flow of heat betweenthe calorimeter proper and its surroundings is quite intense, and alsothose 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 whoseprinciple of operation is based on the assumption that there is no heattransfer (adiabatic calorimeters) or that, by definition, the heat transferprocess is stationary (the generated heat effect is compensated).

The considerations presented are based on the general heat balanceequation of N-domains. A majority of the methods used to determineheat effects have been shown to rely on the simplest, particular form ofthis equation. This makes the calculations very convenient, but implies anumber of simplifying assumptions. These give rise to several limita-tions, which have been duly pointed out. Of the methods described and

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178 FINAL REMARKS

used to determine heat effects and thermokinetics, the method of N-domains 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 volumesand heat capacities of the samples examined. This method is advanta-geous in that it allows the identification of calorimeter parameters byspecifying the “exchangeable” part of the calorimeter as contrasted withthe “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 valuewhen the construction of the calorimeter permits the instrument to beused for various applications. This is true of the numerous calorimetersproduced by the major manufacturers.

The references reviewed illustrate that most investigators have beenlooking for ways to enhance the accuracy of measurements by makinguse of the conclusions deduced from the analysis of the course of heateffects proceeding in the calorimeter. This is of particular concern asregards calorimetric determinations carried out with the recent DSCtechniques. In such investigations, the application of the proceduresdescribed in the book and based on the general equation of dynamicsand on the method of analogy of the thermal and dynamic properties ofthe 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 calorimetersshould encourage investigators to undertake work to improve the theoryof calorimetry still further and to develop more practical applications.

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