TEMPERATURE REGIME OF PARTS OF ELECTROTECHNICAL DEVICES COOLED BY RADIATION

V. V. Ivanov, A. V. Furman

(Presented by Prof. G. I. Fuks)

In calculating temperature fields in electric heaters, coils, magnetic cores, etc., cases are often encountered in which all the heat released is removed from the surface by radiation. Under such conditions operate certain parts of electrotechnical devices located in vacuum, as well as heaters for which, because of the high surface temperature, heat removal is effected predominantly by radiant emission, etc.

Finding an exact analytical solution of the temperature field in these cases is difficult because of the nonlinearity of the boundary condition, expressed by the Stefan–Boltzmann law

\[ -\lambda(\operatorname{grad} T)_{\text{surf}}=\sigma_{\mathrm{v}}\left(T_{\text{surf}}^{4}-T_{\mathrm{c}}^{4}\right). \]

In the present article, using as an example the cooling by radiant emission of a coil of solid circular cross section, within which Joule heat is generated, an approximate but sufficiently simple and reliable method is proposed for finding the nonstationary temperature field.

The mathematical formulation of the problem has the following form:

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

\[ 0\leq F_0\leq \infty,\qquad -1\leq \rho\leq 1, \]

\[ \frac{\partial \Theta(1,F_0)}{\partial \rho} + Sk_{*}\left[\Theta^4(1,F_0)-1\right]=0, \tag{2} \]

\[ \frac{\partial \Theta(0,F_0)}{\partial \rho}=0, \tag{3} \]

\[ \Theta(\rho,0)=\Theta_{\mathrm{n}}. \tag{4} \]

We represent the desired solution for \(\Theta(\rho,F_0)\) in the form of the sum

\[ \Theta(\rho,F_0)=U(\rho,F_0)+\omega(\rho)-D, \]

in which the quantity \(\omega(\rho)\) is the solution of the system (1)–(4) for \(F_0\to\infty\), that is, the stationary temperature,

\[ \omega(\rho)=P_0/4\,(1-\rho^2)+D,\qquad D=\sqrt[4]{1+P_0/2Sk_{*}}. \]

The function \(U(\rho,F_0)\) is found from the conditions

\[ \frac{\partial U(\rho,F_0)}{\partial F_0} = \frac{\partial^2 U(\rho,F_0)}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial U(\rho,F_0)}{\partial \rho}, \tag{5} \]

\[ \frac{\partial U(1,F_0)}{\partial \rho} + Sk_{*}\left[U^4(1,F_0)-D^4\right]=0 \tag{6} \]

\[ \frac{\partial U(0,F_0)}{\partial \rho}=0, \tag{7} \]

\[ U(\rho,0)=\Theta_{\mathrm{n}}+P_0/4\,(\rho^2-1). \tag{8} \]

If we introduce a new variable \(Z(\rho,F_0)\), defined by the equation

\[ Z(\rho,F_0)= \frac{1}{2D^3} \left[ \operatorname{Arth}\frac{U(\rho,F_0)}{D} + \operatorname{arctg}\frac{U(\rho,F_0)}{D} \right], \tag{9} \]

then the boundary condition (6) is linearized:

\[ \frac{\partial Z(1,F_0)}{\partial \rho}=Sk_{*}. \tag{10} \]

The symmetry condition does not change:

\[ \frac{\partial Z(0,F_0)}{\partial \rho}=0, \tag{11} \]

and the initial condition is written as follows:

\[ Z(\rho,0)= \frac{1}{2D^3} \left[ \operatorname{Arth} \frac{\Theta_{\mathrm{n}}+P_0/4\,(\rho^2-1)}{D} + \operatorname{arctg} \frac{\Theta_{\mathrm{n}}+P_0/4\,(\rho^2-1)}{D} \right] =F(\rho) \tag{12} \]

Let us note that, since \(\operatorname{Arth} a\) exists for \(|a|<1\), substitution (9) may be used in the case where

\[ |\Theta_{\mathrm{n}}-P_0/4|<D. \]

The heat-conduction equation (5) for \(Z(\rho,F_0)\) takes the form

\[ \frac{\partial Z(\rho,F_0)}{\partial F_0} = \frac{\partial^2 Z(\rho,F_0)}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial Z(\rho,F_0)}{\partial \rho} - f, \]

where

\[ f= 4\, \frac{ \left[\dfrac{\partial U(\rho,F_0)}{\partial \rho}\right]^2 \cdot U^3(\rho,F_0) }{ \left[D^4-U^4(\rho,F_0)\right]^2 }. \]

It is not difficult to see that the function \(f>0\) and varies from zero at \(\rho=0\) to its greatest value at \(\rho=1\). For sufficiently thin and heat-conducting bodies, the quantity \(f\) may be neglected, and then

\[ \frac{\partial Z(\rho,F)}{\partial F_0} \simeq \frac{\partial^2 Z(\rho,F_0)}{\partial \rho^2} + \frac{1}{\rho}\frac{\partial Z(\rho,F_0)}{\partial \rho}. \tag{13} \]

The general solution of problem (10)–(13) was obtained in [1]:

\[ \begin{aligned} Z(\rho,F_0)=&\;2\int_0^1 \xi F(\xi)\,d\xi +2\int_0^{F_0} Sk_*\,d\eta \\ &+2\sum_{m=1}^{\infty} \frac{I_0(\mu_m\rho)}{I_0^2(\mu_m)} \exp(-\mu_m^2F_0) \int_0^1 \xi F(\xi) I_0(\mu_m\xi)\,d\xi \\ &+2Sk_*\sum_{m=1}^{\infty} \frac{I_0(\mu_m\rho)}{I_0(\mu_m)} \exp(-\mu_m^2F_0) \int_0^{F_0}\exp(-\mu_m^2\eta)\,d\eta, \end{aligned} \tag{14} \]

where \(\mu_m\) are the roots of the equation \(I_1(\mu_m)=0\), given in [2].

Figure 1 presents the dependence of \(U/D\) on \(Z\cdot D^3\). As is evident from the graph, in the region \(Z\cdot D^3\leqslant 0.6\) the function \(U/D\) has a form close to

Fig. 1. Dependence

\[ Z(\rho,F_0)\cdot D^3 = \frac{1}{2}\left[ \operatorname{Arth}\frac{U(\rho,F_0)}{D} + \operatorname{arctg}\frac{U(\rho,F_0)}{D} \right]. \]

to a straight line directed at an angle \(\varphi=45^\circ\). Therefore, in this region the initial temperature distribution may be represented as

\[ F(\rho)\cdot D^4 \simeq \Theta_{\mathrm{n}} + P_0/4(\rho^2-1). \]

Then solution (14) takes the form

\[ Z(\rho,F_0)\cdot D^4 = \Theta_{\mathrm{n}} -\frac{P_0}{8} +\left(Sk_*+\frac{P_0}{2}\right) \left[ 2F_0-\frac{1}{4}(1-2\rho^2) \right] - \]

\[ -\,Sk_*\sum_{m=1}^{\infty} \frac{2}{\mu_m^2 I_0(\mu_m)} I_0(\mu_m\rho)\times \exp(-\mu_m^2F_0). \]

Numerical calculations carried out by the finite-difference method [3] showed that for \(P_0\leqslant 3.5\) and \(Sk_*\leqslant 0.02\), the error in computing the temperature by the proposed method does not exceed 5%. With a decrease in \(P_0\) or \(Sk_*\), or in both \(P_0\) and \(Sk_*\) simultaneously, the accuracy of the calculation increases.

For the region \(Z \cdot D^3 > 0.6\), the integrals entering into solution (14) can be determined graphically.

Table 1 compares temperature values calculated on the basis of the proposed method and by method [3] for the cases: \(\Theta_{\mathrm{n}}=1\), \(Sk_* = 0.02\), \(P_0=3.2\), and \(P_0=2.1\).

Table 1

Conclusions

The results obtained in this work can be used for calculating temperature fields in parts of electrotechnical devices cooled by radiation, and also for estimating the time required for individual parts to warm up to a specified temperature, which is especially important for the start-up period and in intermittent operating conditions.

Notation

\[ \Theta(\rho,F_0)=T(\rho,F_0)/T_c \]
— relative temperature,

\[ \rho=r/R \]
— relative radius,

\[ T_c \]
— ambient temperature,

\[ Sk_*=\sigma_{\mathrm{B}}T_c^3R/\lambda \]
— Stark radiation criterion,

\[ F_0=a\tau/R^2 \]
— Fourier criterion,

\[ P_0=q_vR^2/\lambda T_c \]
— Pomerantsev criterion,

$\lambda, a$ — coefficients of thermal and temperature conductivity,

$R$ — radius of the coil,

$\tau$ — heating time,

$qv$ — Joule heat.

REFERENCES

1. Yu. A. Samoilovich, E. P. Blokhin. An engineering method for calculating the heating of ingots and billets in flame furnaces. Collection of works of VNIIMT, No. 10, 1960.

2. A. V. Lykov. Theory of Thermal Conductivity. Gostekhizdat, 1952.

3. V. K. Saul’ev. Integration of Equations of Parabolic Type. Fizmatgiz, 1960.