On the Question of Stationary Temperature Fields in Bodies of Finite Size with Internal Heat Generation

G. P. BOIKOV, V. A. BLYUKHIN, B. S. SKREBUSHEVSKII

(Presented by Prof. G. I. Fuks, Doctor of Technical Sciences)

In work [1] a method is proposed for calculating the temperature distribution along the lines of a heat “sink” when bodies are heated by radiant energy.

In our opinion, the proposed approach can analogously be used also for cases of heat conduction in the presence of internal heat sources. The relationships obtained in this case, within a certain approximation, will give the temperature distribution along the line of the heat “source” in the body.

Let us have an infinite bar of rectangular cross section, the propagation of heat in which is described by the equation

\[ \frac{\partial^2 t}{\partial x^2}+\frac{\partial^2 t}{\partial y^2}+\frac{w}{\lambda}=0. \]

According to [1], for the line of the heat source the last equation is rewritten in the form

\[ \xi \frac{d^2 t}{d x^2}+\frac{w}{\lambda}=0, \tag{1} \]

where the constant coefficient \(\xi\) for an infinite bar of rectangular cross section is determined by the relation

\[ \xi = 1+\frac{R_1^2}{R_2^2+0.5 R_1^2}; \tag{2} \]

here \(0 \le x \le R_1\),
\(2R_2\) is the greatest dimension of the body.

Using (1), as well as the conditions

\[ -\lambda \left(\frac{d t}{d x}\right)_{x=R_1}=\alpha (t_n-t_c), \]

\[ \left(\frac{d t}{d x}\right)_{x=0}=0, \]

we find

\[ t_x=t_c+\frac{wR_1^2}{2\xi\lambda} \left[1+\frac{2\lambda}{\alpha R_1}-\left(\frac{x}{R_1}\right)^2\right]. \tag{3} \]

Formula (3) gives the temperature distribution from the center of the body to the surface along the smallest dimension.

Let us now solve this same problem by the method of elementary balances for the case when \(R_2=R_1\), for which we divide the area of the body’s cross section into 25 elementary sections. (Fig. 1)

Owing to symmetry it is sufficient to set up 6 equations of elementary balances:

\[ \begin{aligned} W+2Q_{2;1}&=2Q_{1;\alpha},\\ W+Q_{3;2}+Q_{7;2}&=Q_{2;\alpha}+Q_{2;1},\\ W+Q_{8;3}&=2Q_{3;2}+Q_{3;\alpha},\\ W+2Q_{8;7}&=2Q_{7;2},\\ W+Q_{13;8}&=2Q_{8;7}+Q_{8;3},\\ W&=4Q_{13;8}. \end{aligned} \tag{4} \]

Fig. 1

Here

\[ W=w\cdot f=w\cdot S\cdot S=w\left(\frac{R}{2.5}\right)^2\ \frac{\text{kcal}}{\text{hour}}, \]

\(Q_{i;k}\) is the amount of heat transferred from element \(i\) to element \(k\), \(\dfrac{\text{kcal}}{\text{hour}}\);

\(Q_{i;\alpha}\) is the amount of heat transferred from element \(i\) to the surrounding medium, \(\dfrac{\text{kcal}}{\text{hour}}\).

Taking the thermal-conductivity coefficient as

\[ \lambda=1\ \frac{\text{kcal}}{\text{m hour deg}} \]

and the heat-transfer coefficient as

\[ \alpha=\infty\ \frac{\text{kcal}}{\text{m}^2\text{ hour deg}}, \]

we find the coefficient of heat transfer from the element to the surrounding medium

\[ K=2\ \frac{\text{kcal}}{\text{m}^2\text{ hour deg}}. \]

If we assume that the temperature of the surrounding medium is \(t_c=0^\circ\text{C}\), then the system of balance equations (4) can be written in terms of the unknown temperatures:

\[ w\left(\frac{R}{2.5}\right)^2+2(t_2-t_1)=4t_1, \]

\[ w\left(\frac{R}{2.5}\right)^2+(t_3-t_2)+(t_1-t_2)=2t_2+(t_2-t_1), \]

\[ w\left(\frac{R}{2.5}\right)^2+(t_8-t_3)=2(t_3-t_2)+2t_3, \tag{5} \]

Table 1

Table 2

\[ w\left(\frac{R}{2.5}\right)^2 + 2(t_8 - t_7) = 2(t_7 + t_2), \]

\[ w\left(\frac{R}{2.5}\right)^2 + (t_{13} - t_8) = 2(t_8 - t_7) + (t_8 - t_3), \]

\[ w\left(\frac{R}{2.5}\right)^2 = 4(t_{13} - t_8). \]

The solution of system (5) is given in Table 1. The values of the temperatures \(t_{13}\), \(t_8\), \(t_3\), obtained for the same conditions from formula (3), are also given there.

Taking the coefficient of thermal conductivity to be

\[ \lambda = 1 \ \frac{\mathrm{kcal}}{\mathrm{m}\,\mathrm{hr}\,\mathrm{deg}}, \]

and the heat-transfer coefficient

\[ \alpha = \frac{2}{S}\ \frac{\mathrm{kcal}}{\mathrm{m}^2\,\mathrm{hr}\,\mathrm{deg}}, \]

we find the coefficient of heat transfer from the element to the surrounding medium \(K = 1\).

If we assume that the temperature of the surrounding medium is \(t_c = 0^\circ\mathrm{C}\), then the system of balance equations (4) can be written in terms of the unknown temperatures as:

\[ w\left(\frac{R}{2.5}\right)^2 + 2(t_2 - t_1) = 2t_1, \]

\[ w\left(\frac{R}{2.5}\right)^2 + (t_3 - t_2) + (t_7 - t_2) = 2t_2 - t_1, \]

\[ w\left(\frac{R}{2.5}\right)^2 + (t_8 - t_3) = 2(t_3 - t_2) + t_3, \tag{6} \]

\[ w\left(\frac{R}{2.5}\right)^2 + 2(t_8 - t_7) = 2(t_7 - t_2), \]

\[ w\left(\frac{R}{2.5}\right)^2 + (t_{13} - t_8) = 2(t_8 - t_7) + (t_8 - t_3), \]

\[ w\left(\frac{R}{2.5}\right)^2 = 4(t_{13} - t_8). \]

The solution of system (6) is given in Table 2. The values of the temperatures \(t_{13}\), \(t_8\), \(t_3\), obtained for the same conditions from formula (3), are also given there.

As can be seen from Tables 1 and 2, the data calculated by formula (3) agree quite satisfactorily with the calculation data obtained on the basis of the method of elementary balances.

REFERENCES

1. Boikov G. P. Heating of bodies of finite dimensions under the action of radiant heat (first and second communications), Izv. TPI, vol. 101, 1958.