MULTIPURPOSE CALORIMETER TO MEASURE
THERMOPHYSICAL PROPERTIES
Stanislav Proshkin
Department of General and Applied Physics, Saint-Petersburg Mining University, St.-Petersburg, Russia E-Mail: stach@mail.ru
ABSTRACT
The article presents the methodology on the basis of which the calorimeter was made for complex automated measurement of thermophysical properties of a various materials including foodstuff. The main attention is paid to the temperature range which includes phase transitions, when thermophysical characteristics undergo significant changes. An automated experimental setup was made allowing measurements of temperature dependences of enthalpy, effective and true heat capacities, and thermal conductivity coefficients of various materials.
Keywords: enthalpy, effective heat capacity, thermal conductivity coefficient, calorimeter, foodstuff.
1. INTRODUCTION
When calculating the economic and operational parameters of systems used in industry, including food and mining [1-3], thermal properties of different materials are taken into account.
The investigation of thermophysical properties of such samples is complicated because they represent a polydisperse mixture containing water in different states. Thermophysical properties of such mixtures, their structure and consistency directly depend on the interaction of various components, primarily water.
For example, the presence of large amount of water in the composition of foods (up to 80 %) leads to the necessity of taking into account kinetics of phase transitions (ice formation - melting-and boiling - condensation), which strongly affect the change in thermal characteristics.
A large amount of moisture in these materials is related in a complex way to their main components at the molecular, cellular and capillary levels. This results in the fact that in the refrigeration processing occurring phase transformation temperature with blurred boundaries, and thermophysical properties may change irreversibly.
All the above mentioned properties of such substances hinder theoretically an accurate description of the process of cold processing and highlight the experimental methods which determine the main thermophysical properties. It is necessary to know not only the integral quantitative values of thermophysical properties, but their functional dependence on temperature of the researched object.
The above features impose stringent requirements on the conditions of measurement. They must be non-destructive so as not to change the structure and composition of the materials, and they should be carried out in an automatic mode, because the processes, especially during phase transformations, can be fleeting. A variety of structural properties of the materials requires a certain degree of universality of the created equipment.
2. METHOD AND APPARATUS
2.1. The physical basis of the method
The processes of ice formation - melting, boiling - condensation, phase transformation, processes of internal dissipation [4] are accompanied by thermal effects, which are mainly functions of temperature and practically independent of speed of its change. For this reason, when choosing a method of measurement we selected monotonic heating or cooling of the test specimen.
In particular, it relates to a method of microcalorimeter that uses free cooling (heating) of the samples of small size in the environment with constant temperature.
From general considerations it is clear that in order to determine the heat capacity it is necessary to measure the heat flux absorbed by the sample, and the volumetric speed of heating (cooling). It is also necessary to reduce the temperature difference in the sample to a minimum, so as to ensure sufficient uniformity of the temperature field throughout the entire experiment.
On the other hand, when we do a comprehensive study of thermophysical properties, measurement of co-factor of thermal conductivity requires knowledge of the heat flux penetrating through the sample, and a temperature gradient in a section. When the temperature field inside the sample is known, the problem can be reduced to checking of the temperature drop in the specimen and penetrating through its border heat fluxes. In this case, the temperature difference should be noticeable and easily measured.
Thus, a comprehensive measurement of thermophysical properties of substances requires implementation of mutually exclusive assumptions, which impose limitations on the design of thermophysical cell. The result was a fundamental re-design of the calorimeter, which allowed solving the above mentioned problem.
2.2. Experimental setup
thermophysical properties in terms of the phase transformations is depicted in Figure-1. It consists of a massive metal block 6 made of duralumin. Inside the block there is a cylindrical cavity in which the calorimeter 5 and the ampoule with the sample 4 are placed.
Tested substance 2 is located in the cylindrical layer formed by the inner wall of the metal ampoule 4 and a metal cylindrical core 1. Inserted into the ampoule insulating layer 3 provide the constancy of the thickness of the cylindrical layer.
Since the heat capacity of the unit is large, its temperature in the course of carrying out one experiment can be neglected and we can consider the temperature field isothermal.
To study the temperature dependency of thermophysical properties, the unit can measure three temperatures using highly sensitive microthermistors with platinum-iridium conclusions. The first microthermistor is placed directly in the central core and measures the temperature of the inner surface of the sample. The second microthermistor measures the temperature of the outer surface of the sample and is installed in the metal wall of the ampoule. The temperature of a solid block measured with the third microthermistor that is located in the thickness of the block.
Figure-1. The scheme of a thermal cell for measurement of thermophysical properties.
1- cylindrical core; 2 - studied substance; 3 - insulating layer; 4 - metal ampoule;
5 - insulating layer of calorimeter; 6 - metal block.
Measurement of all temperatures was produced automatically with a given time-step. Job time step and total time of experiment were carried out by a specialized controller. Commonly used time intervals of 10 and 20 sec, with a total duration of each experiment 30 to 40 min is equal to 180...240 steps. The controller “collects”
information about the three temperatures as functions of time and transmits the data into the computer memory.
Thus, the original experimental data is a time dependence of the three temperatures recorded in the computer memory, which are further processed using the programs "EXCEL" and "ORIGIN".
As follows from the Figure-1, the method of cylindrical layer is implemented the unit. In his works Prof. Eugeny Platunov found the necessary requirements to the cells for measurement of thermal conductivity in a cylindrical layer [5]:
a) The heat capacity of the core should be known.
b) The total heat capacity of the sample should be small as compared with the heat capacity of the core.
c) The measurement of thermophysical properties of liquid sample is required to account for convective heat transfer.
d) The block and core must have a coefficient of thermal conductivity, hundreds of times greater than the thermal conductivity of the sample.
e) Block, ampoule and core should ensure the uniformity of the temperature field on the outer and inner surfaces of the sample.
2.3. Mathematical model of the cell
When considering mathematical model of the thermal cell in the case of a cylindrical layer temperature field in the sample was taken to be radial, i.e. tt
r, , it obeys the heat conduction equation
c t
r t t r t r r
t
eff 2
2 2
d d 1
, (1)
where:
t – coefficient of thermal conductivity depending on temperature; hh
t – specific enthalpy;
tceff – effective heat capacity.
The thermal coefficients of the given samples generally have arbitrary functional dependence on temperature, thus equation (1) has no rigorous analytical solution.
When method of successive approximations is most often used solving such problems concerning application of the methods of monotone mode. This relies on the assumption about monotonicity of the temperature dependence of thermophysical properties. In this regard the analysis has limitation
r t r r
t r
t t
1 02
, 0 d
d
2 2 2
whereby equation (1) takes the form
c t
r t r r t eff 2 2 1
. (3)
In the stationary regime the solution of this equation for the coefficient of thermal conductivity is well known [6]
in out
in out 2 ln t t h R R Q
, (4)
where: Q –heat flux passing through the sample; Rin - the inner radius of the cylindrical layer; Rout - the outer radius of the cylindrical layer; h - the height of the sample; tin - the temperature on the inner surface of the sample; tout - the temperature on the outer surface of the sample.
Since of great interest is the case when phase transition goes inside the sample, accompanied by release (absorption) of heat, it was necessary to define more exactly the concept of heat flux Q, which is included in the formula (4).
The cell could directly measure heat flux passing through the inner element and the outer surface of the sample, Qin and Qout. Therefore, the heat flux penetrating through the middle section of the specimen, will be considered as follows
2
out in Q
Q
Q
The result is a solution which can be considered as a first approximation
in out
in out out in 4 ln t t h R R Q Q
. (5)
In the second approximation, equation (3) became rb c t r c r t r
r
eff eff , (6)
where:
t
b – is a rate of change of temperature in
time.
After integration of this equality (assuming that the coefficient of thermal conductivity does not depend on the coordinates), we have the formula that represents the equation of heat balance, if each of the components multiplied by 2h
r R R r br hc r t hrR r t hr in in d 2 22 in eff (7)
As can be seen from the equation of Fourier, and the term, standing to the left, represents the heat flux passing through the cross section of the sample with the coordinate r. The first term on the right side is the heat flux entering through the inner surface of the sample. In expanded terms, ceffb remains virtually constant (according to the terms of experiment), so
2
in 2 eff in 2 in 2 eff in,
, QR c b hr R Q c b hr R
r
Q (8)
The last term is the heat radiated (absorbed) by the sample per unit time.
On the other hand, equation (7) can be converted otherwise r R r b c r t r R r t R 2 in eff in 2 in
, (9)
after the integration we shall get
in 2 in 2 in 2 eff in in in ln 2 2 ln 2 , , R r R R r b c R r h Q R t rt (10)
The last formula describes the temperature field in the sample in the presence of phase transformations.
Formula (10) can be considered as the source when calculating the average in the sample conductivity coefficient
in out
in out 2 in 2 out eff in out eff 2 in in out out in 2 ln 1 2 1 ln ln 2 1 t t h R R R R b c h R R b c hR R R Q Q
. (11)
The difficulty of calculation with the use of this formula is that it includes the average effective heat capacity of the sample, which we must know beforehand. For solution of this problem, we had two approaches. Temperature dependence on heat capacity can be obtained
cancel out each other by selecting the appropriate sizes
in
R and Rout.
Therefore, when calculating the average coefficient of thermal conductivity in our case it was possible to use the formula of the first approximation in the form
in out
in out out in
4
ln
t t h
R R Q Q
(12)
At the same time, the mathematical analysis proves the possibility of obtaining the same experimental data information about the average effective heat capacity of a substance as a function of volume temperature. Indeed, formula (8) is converted to the following form
v2 in 2 out
in out eff
b R R h
Q Q c
(13)
where:
v
v
t
b – the rate of change of volume
temperature of the sample.
In the final calculation of the formulae consist of heat fluxes Qin and Qout can be estimated by the ratios
3 2
am 2 c1т
out am ins in
out Q Q Q Q K t t C b Cb
Q (14)
where:
1 in
1
t t
b ;
2 out
2
t t
b .
Thus, during one experiment of measured
2 31 ,t ,t
t we can obtain information about the temperature dependence of the whole complex of thermophysical properties of the test specimen.
In the formulae (11) – (14) to calculate thermophysical properties are included "constants" of thermal cell, Kт, Cam, Cc.
The value of the heat conductivity of the calorimeter Kт can be calculated from the experiment with a sample of known heat capacity. For this purpose we used copper cylinders placed instead of the ampoule with the sample.
A technique similar to that described can be applied to estimate the full heat capacity of the empty ampoule Cam. For this purpose it is necessary to set up the experiment with an empty ampoule and, considering the known value Kт, to find the heat capacity.
Value of the total heat capacity of the core Cc was determined analytically from of known specific heat capacity of brass, of which the core was made.
3. EXPERIMENTAL RESULTS AND DISCUSSIONS Using the described methods and the calorimeter we measured thermophysical properties of several dozens of materials of different origin and structure. Special attention was paid to the field of phase transitions.
As an example let us consider the results obtained for potatoes, this will demonstrate the method of calculation of the main thermal characteristics.
A graph of the dependence of enthalpy h
t on temperature for potatoes is shown in Figure-2. Numbering of the graphs reflects the sequence of operations carried out with the sample: at first sample was frozen (curve 1), then melted (curve 2) and again frozen (curve 3). Because the moisture content in potatoes can reach 80%, the chart is well visible the phase transition water – ice. Comparison of 1 and 3 shows that the re-freezing is accompanied by irreversible changes in the microstructure of the sample. It is seen that in region a sharp jump in the enthalpy curves 1 and 3 are not the same. This fact can be qualitatively explained by the possible mechanical damage of the cell structure of the potatoes during the growth of ice crystals.Another distinctive feature of curves 1 and 3 is the presence of area of where the temperature is lowered to below the crystallization temperature without the occurrence of phase transition. During the initial freezing, this temperature was minus 5, 50°C, during the second freezing the temperature was minus 5, 92 °C. From the graphs for enthalpy it is possible to obtain the temperature dependence on specific heat and initial moisture content of the sample.
Figure-2. The dependence of enthalpy h
t on temperature for potatoes.1-curve of first freezing; 2-curve of melting; 3-curve of second freezing.
Now we may consider the results of a study dynamic thermal characteristics.
Experiments for determination of the coefficient of thermal conductivity was carried out in the same order as for enthalpy: in the beginning the freeze was, then the same sample is melted. Figure-3 shows temperature dependence on the conductivity for the two cases defrost (curves 1 and 2). As noted above, our method allows
obtaining the whole complex of thermophysical properties of stuff. Using the formula (14) we obtained the temperature dependence of the effective specific heat of potatoes (see Figure-4). We want to emphasize that all graphs in Figures 3 and 4 are constructed as dependent on volume temperature.
Figure-3. The dependence of the conductivity on the temperature for the two cases of defrost (curves 1 and 2).
Figure-4. The dependence of the effective specific heat on the temperature for potatoes.
4. CONCLUSIONS
Thus, we proposed method of investigation of various materials under the conditions of freezing and melting when the material is undergoing the intensive phase and structural transitions. This method allows studying the kinetics of the changes of the thermophysical characteristics of materials, as in the mode of monotonic heating and freezing of the samples. Using this method we
created a calorimeter which can measure the equilibrium and dynamic thermal characteristics as a function of temperature in the temperature range from -30 to 20 °C.
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