TEMPERATURE REGIME OF A RADIATION-TRANSMITTING PLATE WITH AN INTERNAL HEAT SOURCE

V. V. IVANOV, A. V. FURMAN

Presented by Professor G. I. Fuks

The mathematical formulation of the problem under consideration has the following form:

\[ \frac{\partial \Theta(X,F_0)}{\partial F_0} = \frac{\partial^2 \Theta(X,F_0)}{\partial X^2} + P_0, \tag{1} \]

\[ \frac{\partial \Theta(1,F_0)}{\partial X} = S_{k_*}\,[1-\Theta^4(1,F_0)], \tag{2} \]

\[ \frac{\partial \Theta(0,F)}{\partial X}=0, \tag{3} \]

\[ \Theta(X,0)=\Theta_{\mathrm{H}}, \tag{4} \]

where

\[ -1 \le X=\frac{x}{R} \le 1,\qquad 0 \le F_0=\frac{a\tau}{R^2}<\infty, \]

\[ \Theta(X,F_0)=\frac{T(X,F_0)}{T_c},\qquad S_{k_*}=\frac{\sigma_{\mathrm{B}}T_c^3}{\lambda}\cdot R, \]

\[ P_0=\frac{q_v\cdot R^2}{\lambda\cdot T_c}. \]

We represent the sought temperature distribution \(\Theta(X,F_0)\) in the form of the sum

\[ \Theta(X,F_0)=\vartheta(X,F_0)+t(X)-D, \tag{5} \]

in which the function \(t(x)\) is the solution of the system (1)–(4) as \(F_0\to\infty\), i.e., the steady-state temperature,

\[ t(X)=\frac{P_0}{2}(1-X^2)+D, \]

\[ D=\sqrt[4]{1+P_0/S_{k_*}}. \]

To determine the nonstationary temperature \(\vartheta(X,F_0)\), it is necessary to integrate the heat-conduction equation

\[ \frac{\partial \vartheta(X,F_0)}{\partial F_0} = \frac{\partial^2 \vartheta(X,F_0)}{\partial X^2}, \tag{6} \]

with boundary conditions

\[ \frac{\partial \vartheta(1,F_0)}{\partial X} = S_{K_*}\,[D^4-\vartheta^4(1,F_0)], \tag{7} \]

\[ \frac{\partial \vartheta(0,F_0)}{\partial X}=0, \tag{8} \]

\[ \vartheta(X,0)=\Theta_{\mathrm{H}}+P_0/2\,(X^2-1). \tag{9} \]

Finding an exact analytical solution of the system (6)–(9) is difficult because of the nonlinearity of the boundary condition (7). Linearization of condition (7) can be carried out with the aid of a new variable \(W(X,F_0)\), defined by the relation

\[ W(X,F_0)= \frac{1}{2D^3} \left[ \operatorname{Arth}\frac{\vartheta(X,F_0)}{D} + \operatorname{arctg}\frac{\vartheta(X,F_0)}{D} \right]. \tag{10} \]

For the new variable \(W(X,F_0)\), the system (6)–(9) is written in the form:

\[ \frac{\partial W(X,F_0)}{\partial F_0} = \frac{\partial^2 W(X,F_0)}{\partial X^2} - \varphi(X,F_0); \tag{11} \]

\[ \varphi(X,F_0) = 4\, \frac{ \vartheta^3(X,F_0) \left[ \dfrac{\partial \vartheta(X,F_0)}{\partial X} \right]^2 }{ \left[D^4-\vartheta^4(X,F_0)\right]^2 }, \]

\[ \frac{\partial W(1,F_0)}{\partial X}=S_{K_*}, \tag{12} \]

\[ \frac{\partial W(0,F_0)}{\partial X}=0, \tag{13} \]

\[ W(X,0)= \frac{1}{2D^3} \left[ \operatorname{Arth} \frac{\Theta_{\mathrm{H}}+P_0/2\,(X^2-1)}{D} + \operatorname{arctg} \frac{\Theta_{\mathrm{H}}+P_0/2\,(X^2-1)}{D} \right] = f(X)^{1)}. \tag{14} \]

The quantity \(\varphi(X,F_0)\) entering equation (11) is always positive; moreover, \(\varphi(0,F_0)=0\) (condition 7), \(\varphi(1,F_0)=4\vartheta^3(1,F_0)S_{K_*}^2\) (condition 8). We estimate \(\varphi(X,F_0)\) by the expression

\[ \varphi(X,F_0)\simeq \frac{1}{2}\,[\varphi(0,F_0)+\varphi(0,0)] = 2\Theta_{\mathrm{H}}^{3}S_{K_*}^{2}. \]

\(^{1}\) We note that, since \(\operatorname{Arth}\,D^{-1}\left[\Theta_{\mathrm{H}}+P_0/2\,(X^2-1)\right]\) exists when

\[ \left| \frac{\Theta_{\mathrm{H}}+P_0/2\,(X^2-1)}{D} \right|<1, \]

substitution (10) may be used in the case when

\[ |\Theta_{\mathrm{H}}-P_0/2|<D. \]

\(^{2}\) The error caused by this assumption will be indicated below.

Then the solution of the system (11)–(14) takes the form [1]:

\[ \begin{aligned} W(X,F_0) = S_{K_*}\Bigg\{& F_0-\frac{1-3X^2}{6} +\sum_{m=1}^{\infty}\frac{2(-1)^{m+1}}{\mu_m^2}\cdot \cos \mu_m X \cdot \exp[-\mu_m^2 F_0]\Bigg\} \\ &-2\Theta_{\mathrm{H}}^3 S_{K_*}^2 \cdot F_0+\int_0^1 f(\xi)\,d\xi \\ &+\sum_{m=1}^{\infty}2\cos\mu_m X\cdot \exp[-\mu_m^2 F_0]\int_0^1 f(\xi)\cdot \cos\mu_m \xi\cdot d\xi . \end{aligned} \tag{15} \]

The integrals entering equation (15) can be evaluated in the following way.

In the region \(W(X,F_0)\cdot D^3 \leqslant 0.6\), dependence (10) has a linear character (Table 1), i.e.

\[ f(X)\simeq \frac{1}{D^4}\left[\Theta_{\mathrm{H}}\,\frac{P_0}{\varepsilon}(X^2-1)\right]. \]

Dependence

\[ W(X,F_0)D^3=\frac{1}{2}\left[ \operatorname{Arth}\frac{\vartheta(X,F_0)}{D} +\operatorname{arctg}\frac{\vartheta(X,F_0)}{D} \right], \]

Table 1

Then

\[ \begin{aligned} W(X,F_0)\cdot D^4={}&\Theta_{\mathrm{H}}-P_0/2-2\Theta_{\mathrm{H}}^3\left(S_{\kappa_*}^2+P_0S_{\kappa_*}\right)F_0+{}\\ &+\left(S_{\kappa_*}+P_0\right)\left[F_0-\frac{1-3X^2}{6}\right]+{}\\ &+S_{\kappa_*}\sum_{m=1}^{\infty}\frac{2(-1)^{m+1}}{\mu_m^2}\cos \mu_m X\cdot \exp[-\mu_m^2F_0]. \end{aligned} \tag{16} \]

In the region \(W(X,F_0)D^3>0.6\), dependence (14) may be approximated by the function

\[ f(X)\simeq f(0)+[f(1)-f(0)]X^2 . \]

In this case

\[ \begin{aligned} W(X,F_0)\cdot D^3={}&\frac{1}{6}\left(\operatorname{Arth}\frac{\Theta_{\mathrm{H}}}{D}+\operatorname{arctq}\frac{\Theta_{\mathrm{H}}}{D}\right)+{}\\ &+\frac{1}{3}\left(\operatorname{Arth}\frac{\Theta_{\mathrm{H}}-P_0/2}{D} +\operatorname{arctq}\frac{\Theta_{\mathrm{H}}-P_0/2}{D}\right)-{}\\ &-2\left(D\Theta_{\mathrm{H}}^3S_{\kappa_*}^{\,2}F_0 +D^3S_{\kappa_*}\left[F_0-\frac{1-3X^2}{6}\right]\right)+{}\\ &+\left[S_{\kappa_*}D^3-\left(\operatorname{Arth}\frac{\Theta_{\mathrm{H}}}{D} +\operatorname{arctq}\frac{\Theta_{\mathrm{H}}}{D}\right) +\left(\operatorname{Arth}\frac{\Theta_{\mathrm{H}}-P_0/2}{D}\right)+{}\right.\\ &\left.+\operatorname{arctq}\frac{\Theta_{\mathrm{H}}-P_0/2}{D}\right] \sum_{m=1}^{\infty}\frac{2(-1)^{m+1}}{\mu_m^2}\cos\mu_m X\cdot \exp[-\mu_m^2F_0]. \end{aligned} \tag{17} \]

Having determined \(W(X,F_0)\) from (16) or (17), from Table 1 we find \(\vartheta(X,F_0)\).

An investigation of the error of the proposed method was carried out by numerical integration of system (1)–(4) [2] for \(P_0<2.1\), \(S_{\kappa_*}<0.15\). As the numerical calculations showed, in the indicated range of variation of the criteria \(P_0\) and \(S_{\kappa_*}\), the error for all points of the plate does not exceed \(7\text{–}8\%\). With decreasing \(P_0\) and \(S_{\kappa_*}\), the accuracy of the calculation increases.

As an example, let us consider the heating of a ray-absorbing plate \((2R=0.5\ \text{m},\ \lambda=17.45\ \text{W}/\text{m}\cdot\text{deg},\ \sigma_{\mathrm{B}}=4.54\ \text{W}/\text{m}^2\,^\circ\text{K}^4,\ q_v=183460\ \text{W}/\text{m}^3)\) in a medium with \(T_c=313^\circ\text{K}\).

Table 2 compares the values of \(\Theta(X,F_0)\) obtained on the basis of the proposed calculation method and by numerical integration.

Comparison of the results of determining the temperature by different methods.

Table 2

The results obtained in this work may be used to calculate temperature fields in heat-generating elements cooled by radiation, and also to estimate the time required for individual parts to warm up to a specified temperature, which is especially important for the start-up period and for intermittent operation.

REFERENCES

1. M. S. Smirnov. On the study of unsteady heat exchange in baking ovens. Proceedings of MTIIP, issue 6. 78, 1956.

2. L. Kollatz. Numerical Methods for Solving Differential Equations. IL. Moscow, 1953.