Determination of the Thermal Center in Solids

(First communication)

N. N. Vodopyanova, G. P. Boikov

(Presented by Professor, Doctor G. I. Fuks)

It is known that the propagation of heat in solids is described by a system of differential equations of heat conduction. Most of the solutions found in the literature have been obtained for the so-called symmetric conditions of heat exchange, when the maximum (minimum) temperature of the system is located at the geometric center of the body. Cases in which the minimum and maximum temperatures (the thermal center) are located at any point of the body or may move with time have so far been studied insufficiently. The latter is connected with difficulties of investigation of both an analytical and an experimental nature. In our view, these difficulties will be considerably reduced if, in studying a given problem, the thermal center is determined in advance in some way. From the standpoint of mathematics, the thermal center is noteworthy because the first derivative of the temperature field there becomes zero; owing to this, the solution can be represented by an even function. In essence, the problem of asymmetric heating (cooling), when the thermal center is known, can be reduced to a symmetric problem, the solution of which is considerably simpler. The possibilities arising under these circumstances are conveniently illustrated by the example of stationary heat conduction in a plane wall with uniform internal heat generation.

Let us have a plane wall of thickness \(2R\), with coefficient of thermal conductivity \(\lambda\), whose internal heat generation is \(\omega\). The wall is under conditions such that on one of its sides the temperature of the medium is \(T_{\mathrm{cp}1}\), the heat-transfer coefficient is \(\alpha_1\), and on the other, respectively, \(T_{\mathrm{cp}2}\), \(\alpha_2\) (we shall assume that \(T_{\mathrm{cp}2} > T_{\mathrm{cp}1}\); \(\alpha_2 < \alpha_1\)).

The process of heat conduction of this wall is described by the system of the following equations:

\[ \frac{d^2 T}{dx^2} = -\frac{1}{\lambda}\,\omega, \tag{1} \]

\[ -\lambda \frac{dT(+R)}{dx} = \alpha_1 \left[T(+R)-T_{\mathrm{cp}1}\right], \tag{2} \]

\[ \lambda \frac{dT(-R)}{dx} = \alpha_2 \left[T(-R)-T_{\mathrm{cp}2}\right]. \tag{3} \]

The solution of this system of equations has the form:

\[ T=-\frac{w x^{2}}{2\lambda} -\frac{\dfrac{wR}{\alpha_{1}}-\dfrac{wR}{\alpha_{2}}+T_{\mathrm{cp}1}-T_{\mathrm{cp}2}} {\dfrac{\lambda}{\alpha_{1}}+\dfrac{\lambda}{\alpha_{2}}+2R} \cdot\left(x-R-\frac{\lambda}{\alpha_{1}}\right) +\frac{wR^{2}}{2\lambda}+\frac{wR}{\alpha_{1}}+T_{\mathrm{cp}1}. \tag{4} \]

Using this solution as exact, one can find the thermal center in the body from the condition

\[ \frac{dT(-b)}{dx}=0, \tag{5} \]

whence

\[ L=0.5\, \frac{\dfrac{1}{Bi_{1}}-\dfrac{1}{Bi_{2}}+\dfrac{1}{P_{0}}} {\dfrac{1}{Bi_{1}}+\dfrac{1}{Bi_{2}}+2}, \tag{6} \]

where

\[ L=\frac{b}{2R} \]

is the dimensionless coordinate of the thermal center (the origin of coordinates is located at the middle of the plate);

\[ Bi_{1}=\frac{\alpha_{1}R}{\lambda} \]

is the Biot criterion on the side of the colder medium;

\[ Bi_{2}=\frac{\alpha_{2}R}{\lambda} \]

is the Biot criterion on the side of the hotter medium;

\[ P_{0}=\frac{wR^{2}}{\lambda\left(T_{\mathrm{cp}1}-T_{\mathrm{cp}2}\right)} \]

is the Pomerantsev criterion.

When the thermal center is known, the system under consideration can be divided into two separate systems, for which ideal insulation takes place in the plane of the thermal center. The solution for each system is represented, respectively, as

\[ T_{1}(x)=T_{\mathrm{cp}1}+\frac{w l_{1}^{2}}{2\lambda} \left[1+\frac{2\lambda}{\alpha_{1}l_{1}}-\left(\frac{x}{l_{1}}\right)^{2}\right]; \tag{7} \]

\[ T_{2}(x)=T_{\mathrm{cp}2}+\frac{w l_{2}^{2}}{2\lambda} \left[1+\frac{2\lambda}{\alpha_{2}l_{2}}-\left(\frac{x}{l_{2}}\right)^{2}\right]; \tag{8} \]

here \(l_{1}=R-b\); \(l_{2}=R+b\) (the origin of coordinates is in the plane of ideal insulation).

Figure 1 shows the temperature distribution in a flat wall, calculated from expression (4). Exactly the same result is obtained if (7) and (8) are used. In the calculation it was assumed that:

\[ w=1000\ \frac{\mathrm{kcal}}{\mathrm{m}^{3}\,\mathrm{h}}; \qquad \lambda=0.1\ \frac{\mathrm{kcal}}{\mathrm{m}\,\mathrm{h}\,\mathrm{deg}}; \qquad T_{\mathrm{cp}1}=273\ \mathrm{K}; \]

\[ T_{\mathrm{cp}2}=373\ \mathrm{K}; \qquad \alpha_2=10\,\frac{\mathrm{kcal}}{\mathrm{m}^{2}\,\mathrm{h}\,\deg}; \]

\[ \alpha_1=1000\,\frac{\mathrm{kcal}}{\mathrm{m}^{2}\,\mathrm{h}\,\deg}; \qquad 2R=0.2\ \mathrm{m}. \]

Investigations based on the theory of similarity show that the dimensionless thermal center, for a large number of heat-conduction phenomena—

Fig. 1. Steady-state heat conduction in a plate with internal heat generation.

—is a function of the heat-transfer criteria \((Bi\) or \(Ki)\) and the criterion \(Po\), which in different cases is expressed through the corresponding quantities. In quantitative meaning, in the present case it represents the ratio of the heat given off from the surface, under identical averaged boundary conditions, to the quantity of heat conducted by steady-state heat conduction through half the thickness of the plate under a temperature difference corresponding to the difference between the temperatures of the first and second media.

Indeed,

\[ \rho_0 = \frac{w R^2}{\lambda (T_{\mathrm{sr}1}-T_{\mathrm{sr}2})} = \frac{wR}{\dfrac{\lambda}{R}(T_{\mathrm{sr}1}-T_{\mathrm{sr}2})} \]
\[ = \frac{a^*(T_n^*-T_{\mathrm{sredn}}^*)}{\dfrac{\lambda}{R}(T_{\mathrm{sr}1}-T_{\mathrm{sr}2})} = Bi^*\,\frac{T_n^*-T_{\mathrm{sredn}}^*}{T_{\mathrm{sr}1}-T_{\mathrm{sr}2}} . \tag{9} \]

Here

\[ a^*=\frac{a_1+a_2}{2}, \]

\(T_n^*\) is the temperature of the body surface in the case of averaged boundary conditions;

\[ T_{\mathrm{sredn}}^*=\frac{T_{\mathrm{sredn}1}+T_{\mathrm{sredn}2}}{2}; \]

\[ Bi^*=\frac{a^*R}{\lambda} = \frac{Bi_1+Bi_2}{2}. \]

Fig. 2. Values of the thermal center under unsteady cooling of an infinite plate.

1 — calculated data according to formula (6)

for \(Bi_1=5;\ Bi_2=1.25;\ \lambda=0.8\,\dfrac{\mathrm{kcal}}{\mathrm{m}\,\mathrm{h}\,\mathrm{deg}};\)

\[ a=0.02\,\frac{\mathrm{m}^2}{\mathrm{h}};\qquad T_{\mathrm{sr}1}=273^\circ\mathrm{K};\qquad T_{\mathrm{sr}2}=173^\circ\mathrm{K}; \]

\[ T_0=873^\circ\mathrm{K}. \]

2 — the same for \(Bi_1=Bi_2=1.25;\)

\[ \lambda=0.8\,\frac{\mathrm{kcal}}{\mathrm{m}\,\mathrm{h}\,\mathrm{deg}};\qquad a=0.02\,\frac{\mathrm{m}^2}{\mathrm{h}};\qquad T_{\mathrm{sr}1}=273^\circ\mathrm{K}; \]

\[ T_{\mathrm{sr}2}=423^\circ\mathrm{K};\qquad T_0=773^\circ\mathrm{K}. \]

1. The same by Schmidt’s method.

For boundary conditions of the third kind (unsteady thermal regime), the heat-transfer criterion remains the criterion \(Bi\), while the criterion \(Po\), according to (9), takes the form:

\[ Po = Bi^{*}\frac{T_{n}^{*}-T_{\text{medium}}^{*}}{T_{\text{avg}1}-T_{\text{avg}2}} \tag{10} \]

\[ = Bi^{*}\frac{ \left(T_{i}-T_{\text{medium}}^{*}\right) \displaystyle\sum_{n=1}^{\infty} A_{n}\cdot \cos(\mu_{n})\cdot e^{-\mu_{n}^{2}Fo} }{ T_{\text{avg}1}-T_{\text{avg}2} }, \]

where the characteristic numbers \(\mu_n\) and the coefficients \(A_n\) may be taken from [1].

Figure 2 shows the values of the coordinates of the thermal center “\(h\),” calculated on the basis of (6) with allowance for (10). The same figure also gives, in dotted lines, the results of a calculation obtained by Schmidt’s finite-difference method. A number of other calculations performed for unsteady heat transfer under boundary conditions of the third kind give analogous convergence.

Thus, the supposition arises that, in a number of heat-conduction phenomena, relation (6) may be used with the corresponding calculation of the heat-transfer criteria (\(Bi\) or \(Ki\)) and with the expression of the criterion \(Po\) according to the indicated interpretation.

In the case of unsteady heat propagation, when the thermal center is known, the system may likewise be divided into two separate systems in which ideal insulation occurs in the plane of the thermal center. In this case, however, the position of the plane of the thermal center in the body depends on time, and therefore it is necessary to solve the problem with a moving coordinate system.

LITERATURE

1. A. V. Lykov, Theory of Heat Conduction, GITTL, Moscow—Leningrad, 1952.