Analysis of the temperature field of the heat measuring cell

(1)

ANALYSIS OF THE TEMPERATURE FIELD OF THE HEAT

MEASURING CELL

Stanislav Proshkin

Department of General and Technical Physics, Saint-Petersburg Mining University, St.-Petersburg, Russia E-Mail: [email protected]

ABSTRACT

This paper presents the results of mathematical calculations of the temperature field of a thermophysical cylindrical cell. According to the obtained results, the validity of the choice of geometrical parameters of the thermophysical cell and the calorimeter is proved. These results have been used in automated devices to measure the thermophysical characteristics of a wide variety of materials.

Keywords: thermophysical characteristics, heat equation, temperature field, calorimeter.

1. INTRODUCTION

The successful development of many industries, such as energy, metallurgy, electronics and rocket science, requires an increasing amount of information about the thermal properties of substances and materials [7]. These thermal properties include thermal conductivity, thermal diffusivity and heat capacity.

Despite the successful development of computational and physical methods for calculating thermal properties, the main way to obtain information about the thermal properties of substances remains an experiment. This is largely due to the fact that the calculation methods do not always provide the required accuracy, and often the calculations require such a large amount of input data, the production of which is more difficult than direct measurement of the required value.

Because thermophysical properties of many substances greatly depend on their chemical composition, internal structure and environmental conditions, only experiment could be the most powerful tool for making accurate calculations of such properties. This problem was solved by a group of Russian scientists under the hand of Prof. Evgeniy Platunov [1]. As a result the theoretical basement of non-linear monotonic regime method has been developed. This method underlies the operation of many devices that has been developed [2-5] in last 50 years. These devices are intended for measurement thermophysical properties in the temperature range from 4,2 K to 2000 K. Non-linear monotonic regime method also enables simultaneous measurement of enthalpy, effective thermal capacity, differential and integral heat of phase transitions, initial moisture content and some other heat and humidity characteristics of the materials and food products at different temperature levels. All these factors made it possible to develop and create a series of automated devices for measuring the thermal properties of various materials in a wide range of temperatures.

It should also be noted that many materials under thermal influence may experience phase transitions, which greatly changes their thermal properties. In this case 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.

2. METHOD AND APPARATUS

2.1 Mathematical model of the cell

In our previous articles, we have considered some automated devices for measuring the thermal properties of different materials [2, 5]. Since we mainly studied dispersed substances containing a significant amount of moisture, the micro calorimeter method was used in these devices. This method involves free cooling (heating) of the samples of small size in the environment with constant temperature.

It is well known that the basis of any thermal measurements is the calculation of the temperature field of the samples. To calculate such a field, a nonlinear heat equation is valid [6]:

 

   

div



_

T

grad

T

  

_{ }

c T



w T

 

_{ }



T

, (1)

where

T T x, y,z,









– temperature field of the sample;

 

T



- thermal conductivity;

c T

 

– heat capacity of single-phase product components; w T

 

dh_ph

 

T dT-

temperature derivative of specific enthalpy

h

ph

 

T

of phase transformations in the product. This equation serves as a theoretical basis for the proposed method of studying the thermal characteristics of samples in a wide temperature range, including the crystallization of bound moisture.

(2)

Since our devices are mainly used cylindrical shape of samples and thermal cells, the heat equation can be written as follows:

2 2 eff 2

1 d

d

T

c

r

t

r







 









_



_



_

_







 



_

_





, (2)

In general, the thermophysical coefficients of the samples have an arbitrary functional dependence on temperature, so the heat equation has no analytical solution. The latter is possible only in some special cases, when simplifying restrictions are imposed on the nature of the temperature dependence of the thermophysical coefficients and the heat equation can be reduced to a linear one by simplifying restrictions.

In order to substantiate the monotonic regime method for solving the heat equation, we have chosen the method of successive approximations, because it gives a fast convergence of the solution and is simple and versatile.

This method assumes that any thermophysical coefficients can be decomposed into a Taylor series by the temperature drop θ in the vicinity of the base temperature. As a base point of the sample is selected, which is convenient to directly measure the temperature dependence.

This relies on the assumption about monotonicity of the temperature dependence of thermophysical properties. In this regard the analysis has limitation

2 ₂

2

d

1 0 02

d

T

,

T

r









 







_



_



_



_









, (3)

whereby equation (2) takes the form

2

eff 2

1 T

T

c

r























 







. (4)

3. EXPERIMENTALRESULTSANDDISCUSSIONS

The conditions presented in equations (3) and (4) required us to carry out a more serious mathematical analysis of temperature fields in thermophysical measuring cells.

Let’s consider a cylinder with a diameter

2 R

and length 2l, surrounded by a thin thermal insulation shell, the thickness of which

h

_ins. The thermal

conductivity coefficient of the cylinder material is



, and

the thermal insulation shell is



_ins. The outer surface of

the shell is maintained at a constant temperature

T

. The initial temperature of the cylinder is

T

₀. The temperature field

T r, z,







of the cylinder is described by the unsteady heat equation at boundary conditions of the III kind:

2 2

1

1 T

T

,

r

a

r

z









 



(5) 0 0

T

_



T

, (6)

0

r

T

r

_



_



, (7)





0

R _{r R}

r R

T

H

T

r

_ 













, (8)

0

z

T

r

_







, (9)



ср



0

l _z _l

z l

T

H T

T

z

_ 













, (10)

where

H

R and

H

l – the relative heat transfer coefficients of the cylinder with the environment through the thermal insulation shell: ins ins

H

h







. (11)

In our experiments we have taken that

H

R



H

l.

The solution of the equation (5) under the conditions (6) - (10) in the case of heating is given in [6] and has the form:





0



0







T r, z,

 

T



T



T



r, z,



. (12)





 







 

1 2 2 0 2

2 2

2

2 2 2 2

1 1 0

4

1 Bi Bi

Bi

1 cos

exp -

Fo

Bi

k _n

R l l k

k n k

n k k R n l l k n

r

J

z

R

r , z,

J

l

    



_







_

_



 

_

_

_

_







  

_



_

_

_

  

_

_









 



 















, (13)

where

Bi

_R



H R

_R and

Bi

_l



H l

_l - Bio numbers; 2

(3)

 

tg

Bi

l

  

;



n - roots of the equation

 

1

Bi

R 0

J



 



;

J

₀ and

J

₁ – Bessel functions of the first kind of zero and first order.

As a result of mathematical calculations under the following conditions:

Bi

R



Bi

l



0 001

,

, Fo=100,

2

l R, we obtained the results for the temperature distribution



T



T

₀



over the radius and height of the cylindrical sample, which are shown in the Figures 1 and 2. Also in Figure-3 the dependence of the temperature difference in radius at different values of the Fourier number is given.

From the graph in Figure-1 it can be seen that the unevenness of the temperature field in the cylinder does not exceed 0.025°C at



T



T

₀





50

K. This value of the temperature drop does not exceed the instrumental error of temperature measurement in our experiments. Therefore, changes in the temperature field along the radius of the measurements in the sample can be neglected.

Now we analyze the temperature field in the insulating shell around the sample (calorimeter), which we assume to be a hollow infinitely long cylinder. The initial temperature of the outer surface of the calorimeter

T

has a environmental temperature and remains almost unchanged during the experiment. The temperature of the

inner surface abruptly decreases to the temperature of the cell

T

_cell at the initial time. Differential equation of thermal conductivity and boundary conditions of this problem:

2

1

1 T

T

r

a

r



_



_







, (14)

0

T

_



T

, (15)

1 cell

r R

T

_



T

, (16)

2

r R

T

_



T

. (17)

The solution of the equation (14) is as follows:

 

1 cell 2

 

2

1

ln

r

T

R

T r ,

S r ,

R



























 

















, (18)

where

 





 





1

0 0

2 2

cell 2 0 0 0 0

1 ₁ 2 2 2

0 0

2

exp

Fo

n n

n n n n n

n

n n

R

J

R

r

S r ,

T

J

Y

J

R

J

R















_

_

_

_

_

_





 



_

_



_

 

_



_



_



























































,

0 0

J , Y

– Bessel functions of the first and second kind of zero order;



n – roots of the equation:

 

в в

0 0 0 0

н н

0

n n n n

R

J

Y

J

R











 



 

















;

2 2

Fo

a

R





– Fourier number.

Figure-1. Graph of the temperature distribution (

T



T

₀) along the radius of the cylinder when:

Bi

_R



Bi

_l



0 001

,

; Fo=100. 0 0,2 0,4 0,6 0,8 11

10, 12 10,

13 10,

R r K

0,

T

(4)

T



T

0) at the height of the cylinder when:

Bi

R



Bi

l



0 001

,

; Fo=100.

Figure-3.Graph of the temperature difference over the radius of the cylinder at different values of the

Fourier number.

T



T

₀) by the radius of the calorimeterat different values of the

Fourier number: 1 -Fo=1; 2 -Fo=10; 3 -Fo=100.

Figure-5.Graph of the temperature distribution (

T



T

₀) at the height of the calorimeterat different values of the

Fourier number: 1 -Fo=1; 2 -Fo=10; 3 -Fo=100. 0 0,2 0,4 0,6 0,8

 z

11 10,

12 10,

13 10,

К

0,

T T

Fo

10 100

025 0,



02 0,



01 0,



К , T 

015 0,



005 0,



K

0, T

T

40

30

20

10

0

84

0, 0,86 0,88 0,9 0,92 0,94 0,96 0,98 826

0,

н

R z 1

3

2

K

0, T T

45

40

35

30

25

20

0

 z 1

0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

(5)

Figure-6. Graph of the temperature difference



T

over the radius of the calorimeter as a function of the

Fourier number.

The graph in Figure-5 clearly proves that the unevenness of the temperature field in the calorimeter in height for all the time of measurements can be neglected, since it does not go beyond the temperature error of temperature measurement.

The analysis of the temperature field and temperature drops along the radius and axis of the heat measuring cell confirmed the correctness of its size and configuration when creating the device.

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.

4. CONCLUSIONS

Thus, the conducted mathematical analysis confirmed the validity of the application of the method of monotone mode in order to create automated devices for measuring the thermal characteristics of different materials. The results obtained mathematically determine the geometric dimensions of the samples in the thermophysical cell.

REFERENCES

[1] E. Platunov, I. Baranov, S. Burawoy and V. Kurepin. 2010. Thermophysical measurements (St.-Petersburg State University of Low Temperature And Food Technology, St-Petersburg, Russia).

[2] S. Proshkin. 2019. Multifunctional Device for Measuring Thermal Properties under Phase Transitions. Journal of Physics: Conf. Series. 1172: 012061.

[3] I. Baranov and S. Proshkin. 2006. A Multifunctional Device for Thermophysical Measurements. In Sixteenth Symposium on Thermophysical Properties, (Boulder, Colorado, USA).

[4] S. Proshkin. 2016. Calorimeter for Measurement Internal Heat Release in Foodstuff. APRN Journal of Engineering and Applied Sciences. 11(15): 9138-9143.

[5] S. Proshkin. 2018. Multipurpose Calorimeter to Measure Thermophysical Properties. APRN Journal of Engineering and Applied Sciences. 13(5):1827-1832.

[6] A. Luikov. 1978. Heat and Mass Transfer (Energia, Moscow, Russia (in Russian).

[7] V. Bezyazychnyi and M. Szcerek. 2018. Thermal Processes Research Development in Machine-Building Technology. Journal of Mining Institute. 232, 395-400.

K

, T



40

30

20

10

0 1