Gopal The specific heat of matter at low temperatures by Taria Analytical calorimetry. Kraftmakher Methods in enzymology. Volume five hundred and sixty seven, Calorimetry by Andrew L. Feig Recent progress in microcalorimetry by E.

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Edouard Calvet Dosimetry of ionizing radiation. Volume 2 by Kenneth R. Zielenkiewicz Calorimetry fundamentals, instrumentation and applications by Stefan M. Sarge Fast scanning calorimetry by Christoph Schick Heat capacity and thermal expansion at low temperatures by T. The calibration consists in generating a rectangular pulse [Eqs 2. If the generation of the rectangular heat pulse is stopped, then the calorimeter becomes a thermally inertial object. Cooling of the calorimeter then occurs up to the moment when the temperatures of the calorimeter and the shield are the same.

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More precise equations expressing the relation between P t and T t should then be used. The calibration procedure presented here is not the only one used in the dynamic method. This assumption is the basis of the flux method.

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The amount of heat transferred between the calorimeter proper and the shield is then directly proportional to the temperature difference. Under isoperibol conditions, this method is called AC calorimetry [—]. Reading et al. In this case, the response of the calorimeter as a linear system would be a superposition of two input functions: 1 the ramp function [Eq.

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When the periodic function is sinusoidal [Eq. Temperature-modulated differential scanning calorimetry is a new analytical technique used to obtain information on the heat capacity in the range close to the phase transformation. It is a method applied in many instruments, e. Let us apply Eq. Integration of both sides of Eq.

The method is based on the heat balance equation of a simple body in the form If we put Eq. The amount of heat generated in the calorimetric vessel corresponding to the temperature increment can then be expressed as The determination of is the aim of the calorimetric measurement. In the calorimetric measurement, three periods can be distinguished Fig.

During the initial period the temperature changes before the examined heat process are measured.

## Theory of Calorimetry

The beginning of the main period is the initial moment of the generation of heat by the process studied. As the end of the main period, the time moment is assumed as the end of the studied process. In the method, it is assumed that, independently of the heat process studied, during the whole period of measurement a constant heat effect can be produced in the calorimeter by secondary processes for example, the process of evaporation of the calorimetric liquid, the friction of the stirrer on the calorimetric liquid, etc.

If we assume that where is the increment of temperature of the calorimetric vessel, which corresponds to the constant heat effect, involve by secondary process Eq. On determining 3. Roth [] used the least squares method for approximation of the initial and final periods by straight lines. Planimetry at first was used to obtain this integral. Numerical and analog methods of determination of thermokinetics Harmonic analysis method In this method, first used by Navarro et al.

Method of dynamic optimization In the dynamic optimization method [—], Eq. This method assumes the existence of one input function T t and one output function P t. The impulse response H t is determined as a derivative with respect to time of the response of the calorimetric system to a unit step.

Such a task can be solved provided that the temperature response T t and the impulse response H t of the calorimeter are known in the analytical form. Because of this, the integrals in function 3. 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.

These values are referred to as state variables. Three types of state variables are distinguished: physical variables, canonical variables and phase variables.

get link The vector the components of which are state variables, is referred to as the state vector. Thus, the calorimeter transformation equation combines the state variables with the parameters of the calorimetric system and the thermal power produced: The state equation Eq. Because of the available calorimetric information, the state variables should be transformed in such a way as to obtain a relationship between one input function P t and one output function. The relationship between the state variables and the output function has the following form: The method of state variables for the determination of thermokinetics was proposed by Brie et al.

Use of Eq. Using Eq. These works extended the applications of the inverse filter method to linear systems with variable coefficients. In many cases [—], as in the multidomains method, as a basis of consideration the mathematical models used were particular forms of the general heat balance equation. Evaluation of methods of determination of total heat effects and thermokinetics The methods discussed above discussion are based on particular forms of the general heat balance equation.

There are also new methods e. Improved conditions in calorimetric experiments lead to the creation of more precise mathematical calorimeter models and methods used for the determination of heat effects. Generally, this is due to the compromise between the available information and the possibility of applying it.

The purpose was to compare different methods independently proposed to reconstruct thermokinetics from the experimental calorimetric data. The dynamic optimization, harmonic analysis, state variable and numerical correction methods were tested. These responses concerned normalized series of heat pulses, generated by the Joule effect in a heater. The sampling period was taken as 0. The values of amplitudes were given in relative units, equal to 10, and Three types of measurements were made: 1, 3 or 7 pulses of various durations were generated one after another.

The optimisation method shows its advantage especially in the cases when the number of experimental data need not be bigger than Let us consider the same problem on using amplitude characteristics [].

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As given in Eq. In the multidomains method, this selecting scheme was included in the procedure of determining the particular form of the calorimeter model. Another manner of selecting is used when harmonic analysis method is applied. In this method, the transmittance is obtained numerically, using the Fast Fourier Transform. The boundary frequency is the second limitation of the frequency spectrum, which can be applied in the reconstruction of the thermokinetics for a given calorimetric system, and The high precision of calorimetric measurements today permits a less strict description of the calorimetric system itself.

When the mathematical model is known, the inverse filter method is used to determine the thermokinetics.

The course of the thermal power change can be determined by using different calorimeters. The choice of the heat effect determination method does not depend on the calorimeter type. In a calorimeter with a vacuum jacket, the thermokinetics can be determined successfully by means of the dynamic method. In calorimeters whose inertia is very small conduction calorimeters , use of the flux method is not always suitable. The forcing function in the calorimeter proper is always the heat effect generated as a result of the studied process.

However, the review of the methods has revealed that there can be forcing functions striving to compensate the generated heat effect of the process; a rectangular pulse or a series of rectangular pulses; periodic heating rate of the sample etc. Additionally, the forcing functions acting in the shield influence the calorimeter. For an isoperibol calorimeter, this is a forcing function, tending to achieve the stability of the temperature of the shield; for scanning calorimeters, it is a ramp function.