THEORY OF CALORIMETRY

THEORY OF CALORIMETRY

Hot Topics in Thermal Analysis and Calorimetry Volume 2 Series Editor: Judit Simon, Budapest University of Technology and Economics, Hungary The titles published in this series are listed at the end of this volume.

Theory of Calorimetry by Wojciech Zielenkiewicz Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland and Eugeniusz Margas Institute of Physical Chemistry, Polish Academy of Sciences, Warsaw, Poland KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

ebook ISBN: 0-306-48418-8 Print ISBN: 1-4020-0797-3 2004 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print 2002 Kluwer Academic Publishers Dordrecht 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: and Kluwer's ebookstore at: http://kluweronline.com http://ebooks.kluweronline.com

Contents Preface ix Chapter 1 The 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 equation Heat transfer. Conduction, convection and radiation General integral of the Fourier equation. Cooling and heating processes Heat balance equation of a simple body. The Newton law of cooling The heat balance equations for a rod and sphere General heat balance equation of a calorimetric system 1 2 10 14 20 26 33 Chapter 2 Calorimeters as dynamic objects 2.1. Types of dynamic objects 2.2. Laplace transformation 2.3. Dynamic time-resolved characteristics 2.4. Pulse response 2.5. Frequential characteristics 2.6. Calculations of spectrum transmittance 2.7. Methods of determination of dynamic parameters 2.7.1. 2.7.2. 2.7.3. 2.7.4. 2.7.5. Determination of time constant Least squares method Modulating functions method Rational function method of transmittance approximation Determination of parameters of spectrum transmittance 37 39 41 47 55 58 61 66 66 74 76 79 81

vi CONTENTS Chapter 3 Classification of calorimeters. Methods of determination of heat effects 3.1. 3.2. 3.3. Classification of calorimeters Methods of determination of heat effects 3.2.1. 3.2.2. 3.2.3. 3.2.4. 3.2.5. 3.2.6. 3.2.7. 3.2.8. 3.2.9. 3.2.10. 3.2.11. General description of methods of determination of heat effects Comparative method of measurements Adiabatic method and its application in adiabatic and scanning adiabatic calorimetry Multidomains method Finite elements method Dynamic method Flux method Modulating method Steady-state method Method of corrected temperature rise Numerical and analog methods of determination of thermokinetics 3.2.11.1 Harmonic analysis method 3.2.11.2. Method of dynamic optimization 3.2.11.3. Thermal curve interpretation method 3.2.11.4. Method of state variables 3.2.11.5. Method of transmittance decomposition 3.2.11.6. Inverse filter method 3.2.11.7. Evaluation of methods of determination of total heat effects and thermokinetics Linearity and principle of superposition Chapter 4 Dynamic properties of calorimeters 4.1. Equations of dynamics 4.2. Dynamic properties of two and three-domain calorimeters with cascading structure 4.2.1. 4.2.2. 4.2.3. Equations of dynamics. System of two domains in series Equations of dynamics. Three domains in series Applications of equations of dynamics of cascading systems 85 85 97 97 101 103 104 109 111 114 114 116 119 123 123 124 125 127 128 129 131 136 139 139 143 143 148 151

CONTENTS vii 4.3. Dynamic properties of calorimeters with concentric configuration 4.3.1. Dependence of dynamic properties of two-domain calorimeter with concentric configuration on location of heat sources and temperature sensors 4.3.2. Dependence between temperature and heat effect as a function of location of heat source and temperature sensor 4.3.3. Apparent heat capacity 4.3.4. Energy equivalent of calorimetric system Final remarks References 154 155 165 168 171 177 179

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

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