ON THE CALCULATION OF THE UNSTEADY TEMPERATURE FIELD IN BODIES OF FINITE SIZE SUBJECTED TO RADIATIVE HEATING

V. V. Salomatov

(Presented by the scientific seminar of the Faculty of Thermal Power Engineering)

In engineering practice, thermophysical processes occur in finite-dimensional geometric systems. In this connection, the development of effective methods for solving spatial problems of heat-transfer theory is a basic practical requirement.

In the present work, approximate methods are proposed for solving problems of multidimensional heat conduction with boundary conditions of the radiative type. In the first section, unsteady heat transfer is investigated under the condition of a substantial dependence of thermophysical coefficients on temperature, while in the second section the temperature regime of solid bodies is considered in the presence of constant thermal characteristics.

I. Variable thermophysical characteristics

Heating bodies of finite size by radiant heat, when the coefficient of thermal conductivity and the heat capacity depend on temperature, is the most difficult mathematical problem in the theory of heat conduction, since the system of differential equations describing the process of unsteady heat transfer becomes doubly nonlinear. The internal nonlinearity contained in the differential equation of the process is due to the variability of the thermal coefficients with temperature; the external nonlinearity, occurring in the boundary condition, is caused by the presence of a nonlinear relation between the heat flux on the surface of the body and the temperature of this surface. In an analytical investigation the system is linearized. The traditional linearization techniques consist in assuming the constancy of the thermophysical coefficients \((\lambda(T)=\mathrm{const},\ c(T)=\mathrm{const})\) and in replacing the nonlinear boundary condition of radiative type by a linear boundary condition of the third kind with the introduction of a conditional heat-transfer coefficient for radiation. It is quite natural that the linearized solution describes the picture of the temperature field sufficiently well only in the case of small nonlinearities of the system (practically, at separate stages of the heating process). On the whole, however, the problem under consideration is essentially nonlinear; therefore, for a correct description of the temperature distribution, the nonlinearity of the system must be taken into account. At present the problem under investigation in its general form can be solved only approximately: either by the finite-difference method (although the question of convergence ...

of the grid method in each specific case must be investigated separately), or by modeling the heat-transfer phenomenon under study.

In our previous works [1, 2] it was shown that the solution of one-dimensional heat-conduction problems of this kind can be successfully carried out for certain particular laws of variation of the thermal coefficients with temperature, namely, if

\[ \lambda(T)=\varkappa_1 T^3, \tag{1.1} \]

\[ c(T)=\varkappa_2 T^3. \tag{1.2} \]

The proportionality of \(\lambda(T)\), \(c(T)\) to the cube of the absolute temperature is characteristic of many crystals in certain temperature ranges. We note that, taking (1.1), (1.2) into account, the nonlinear differential equation and the nonlinear boundary condition of radiative type automatically become linear with respect to the temperature analog

\[ \eta(T)=\int_0^T \lambda(T)\,dT. \]

Obviously, for other laws \(\lambda(T)\) and \(c(T)\), different from (1.1), (1.2), in order to obtain a sufficiently accurate approximate solution over the computed interval of temperature variation or, if this interval is sufficiently large, over its separate parts, the thermal parameters can be approximated by segments of cubic parabolas. We point out that, simultaneously and independently, an analogous approach to the study of nonlinear heat conduction in bodies of classical shape was applied in [3]. There, a variant of estimating the calculation error under the condition of parabolic approximation of the thermal coefficients is also given.

Let us extend the proposed solution method to multidimensional problems of radiative heating. As an example, let us investigate the process of nonlinear heat conduction in a rectangular parallelepiped, whose mathematical description is given by the system of equations:

\[ c(T)\rho\frac{\partial T(x,y,z,\tau)}{\partial \tau} = \frac{\partial}{\partial x} \left[ \lambda(T)\frac{\partial T(x,y,z,\tau)}{\partial x} \right] + \]

\[ + \frac{\partial}{\partial y} \left[ \lambda(T)\frac{\partial T(x,y,z,\tau)}{\partial y} \right] + \frac{\partial}{\partial z} \left[ \lambda(T)\frac{\partial T(x,y,z,\tau)}{\partial z} \right], \]

\[ (\tau>0,\quad -R_1<x<+R_1,\quad -R_2<y<+R_2,\quad -R_3<z<+R_3), \tag{1.3} \]

\[ T(x,y,z,0)=T_0=\mathrm{const}, \tag{1.4} \]

\[ \pm \lambda(T)\frac{\partial T(\pm R_1,y,z,\tau)}{\partial x} = \sigma_{\mathrm{B}}\left[T_c^4-T^4(\pm R_1,y,z,\tau)\right], \tag{1.5} \]

\[ \pm \lambda(T)\frac{\partial T(x,\pm R_2,z,\tau)}{\partial y} = \sigma_{\mathrm{B}}\left[T_c^4-T^4(x,\pm R_2,z,\tau)\right], \tag{1.6} \]

\[ \pm \lambda(T)\frac{\partial T(x,y,\pm R_3,\tau)}{\partial z} = \sigma_{\mathrm{B}}\left[T_c^4-T^4(x,y,\pm R_3,\tau)\right]. \tag{1.7} \]

Taking conditions (1.1) and (1.2) into account, we shall show that the solution of problem (1.3)—(1.7) can be represented in the form

\[ \frac{T_c^4-T^4(x,y,z,\tau)}{T_c^4-T_0^4} = \frac{T_c^4-T^4(x,\tau)}{T_c^4-T_0^4} \cdot \frac{T_c^4-T^4(y,\tau)}{T_c^4-T_0^4} \cdot \frac{T_c^4-T^4(z,\tau)}{T_c^4-T_0^4}, \tag{1.8} \]

where \(T(x,\tau)\), \(T(y,\tau)\), \(T(z,\tau)\) are known solutions for three infinite plates, by whose intersection the parallelepiped is formed. Accord-

relation (1.8), in essence, expresses the rule, well known in the theory of heat conduction, for multiplying temperature criteria, where the factor
\(\dfrac{T_c^4 - T^4}{T_c^4 - T_0^4}\) represents the degree of incompleteness of heating under radiative heat exchange. In the proof, naturally, we assume that the initial and boundary conditions for the unbounded plates remain the same as for the parallelepiped, namely:

\[ \mathrm{T}(x,0)=\mathrm{T}(y,0)=\mathrm{T}(z,0)=\mathrm{T}_0=\mathrm{const} \tag{1.9} \]

\[ \mp \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(\pm R_1,\tau)}{\partial x} = \sigma_{\mathrm{в}}\left[\mathrm{T}_c^4-\mathrm{T}^4(\pm R_1,\tau)\right], \tag{1.10} \]

\[ \mp \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(\pm R_2,\tau)}{\partial y} = \sigma_{\mathrm{в}}\left[\mathrm{T}_c^4-\mathrm{T}^4(\pm R_2,\tau)\right], \tag{1.11} \]

\[ \mp \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(\pm R_3,\tau)}{\partial z} = \sigma_{\mathrm{в}}\left[\mathrm{T}_c^4-\mathrm{T}^4(\pm R_3,\tau)\right]. \tag{1.12} \]

Let us satisfy differential equation (1.3). As a result we have:

\[ \begin{aligned} &[\mathrm{T}_c^4-\mathrm{T}^4(y,\tau)] [\mathrm{T}_c^4-\mathrm{T}^4(z,\tau)] \left\{ \rho \varkappa_2 \mathrm{T}^3 \frac{\partial \mathrm{T}(x,\tau)}{\partial \tau} - \frac{\partial}{\partial x} \left[ \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(x,\tau)}{\partial x} \right] \right\} +\\ &\quad+ [\mathrm{T}_c^4-\mathrm{T}^4(x,\tau)] [\mathrm{T}_c^4-\mathrm{T}^4(z,\tau)] \left\{ \rho \varkappa_2 \mathrm{T}^3 \frac{\partial \mathrm{T}(y,\tau)}{\partial y} - \frac{\partial}{\partial y} \left[ \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(y,\tau)}{\partial y} \right] \right\} +\\ &\quad+ [\mathrm{T}_c^4-\mathrm{T}^4(x,\tau)] [\mathrm{T}_c^4-\mathrm{T}^4(y,\tau)] \left\{ \rho \varkappa_2 \mathrm{T}^3 \frac{\partial \mathrm{T}(z,\tau)}{\partial z} - \frac{\partial}{\partial z} \left[ \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(z,\tau)}{\partial z} \right] \right\} =0. \end{aligned} \]

The expressions in braces are equal to zero, since they are the differential equations of the process for infinite plates. Thus, relation (1.8) is transformed into identity (1.3).

Substituting (1.8) into the boundary conditions (1.10)—(1.12), after uncomplicated transformations,

\[ \begin{aligned} &\left\{ \mp \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(\pm R_1,\tau)}{\partial x} - \sigma_{\mathrm{в}} \left[\mathrm{T}_c^4-\mathrm{T}^4(\pm R_1,\tau)\right] \right\} \times\\ &\qquad\times \frac{ [\mathrm{T}_c^4-\mathrm{T}^4(y,\tau)] [\mathrm{T}_c^4-\mathrm{T}^4(z,\tau)] }{ (\mathrm{T}_c^4-\mathrm{T}_0^4)^2 } =0, \\[1em] &\left\{ \mp \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(\pm R_2,\tau)}{\partial y} - \sigma_{\mathrm{в}} \left[\mathrm{T}_c^4-\mathrm{T}^4(\pm R_2,\tau)\right] \right\} \times\\ &\qquad\times \frac{ [\mathrm{T}_c^4-\mathrm{T}^4(x,\tau)] [\mathrm{T}_c^4-\mathrm{T}^4(z,\tau)] }{ (\mathrm{T}_c^4-\mathrm{T}_0^4)^2 } =0, \\[1em] &\left\{ \mp \varkappa_1 \mathrm{T}^3 \frac{\partial \mathrm{T}(\pm R_3,\tau)}{\partial z} - \sigma_{\mathrm{в}} \left[\mathrm{T}_c^4-\mathrm{T}^4(\pm R_3,\tau)\right] \right\} \times\\ &\qquad\times \frac{ [\mathrm{T}_c^4-\mathrm{T}^4(x,\tau)] [\mathrm{T}_c^4-\mathrm{T}^4(y,\tau)] }{ (\mathrm{T}_c^4-\mathrm{T}_0^4)^2 } =0 \end{aligned} \tag{1.13} \]

we are convinced that expression (1.8) satisfies the prescribed boundary conditions. It is easy to see that relation (1.8) also corresponds to the initial condition.

Thus, expression (1.8), satisfying all the imposed conditions, by the uniqueness theorem is the exact solution of problem (1.3), (1.7).

Similarly, the solution of the problem for a short cylinder has the form

\[ \frac{T_c^4 - T^4(r,z,\tau)}{T_c^4 - T_0^4} = \frac{T_c^4 - T^4(r,\tau)}{T_c^4 - T_0^4} \cdot \frac{T_c^4 - T^4(z,\tau)}{T_c^4 - T_0^4}. \tag{1.14} \]

For practical engineering calculations by formulas (1.8), (1.14), one may use nomograms for determining the temperature on the surface and at the center of an infinite plate and an infinite cylinder under conditions of convective heat exchange, which are given in [4], replacing the corresponding criteria by their analogues

\[ \mathrm{Bi}^*=\frac{4\sigma_B R}{\varkappa_1}, \qquad \mathrm{Fo}^*=\frac{\varkappa_1 \tau}{\rho\varkappa_2 R^2}, \qquad \Theta^*=\frac{T_c^4-T^4}{T_c^4-T_0^4}. \]

The error of calculation by the proposed method will be determined by how accurately the approximation (1.1), (1.2) reproduces the true law of variation of \(\lambda(T)\) and \(c(T)\) with temperature. Obviously, following this method, reliable data can be obtained only for increasing laws of \(\lambda\) and \(c\) with temperature. The method of analytical solution becomes ineffective in the case of decreasing functions \(\lambda(T)\) and \(C(T)\), although it should be noted that such regularities are encountered more rarely. For engineering practice, cases of increasing dependences \(\lambda(T)\) and \(c(T)\) are the most characteristic.

When calculating by the method of parabolic approximation, apparently, the worst case will be that in which the thermophysical constants remain unchanged throughout the entire heating process. Consequently, in order to establish the maximum possible calculation error by the method set forth, it is necessary to compare with the case of heating at constant thermal coefficients, since for any increasing laws \(\lambda(T)\), \(c(T)\) the error will naturally be smaller, because the accuracy of the approximation increases substantially.

For the purpose of comparing the calculated variants, it is necessary to average the variable thermal coefficient over the specified interval of temperature variation. Using the usual mean-value theorem, we have

\[ \bar{\lambda}_{\Delta T} = \frac{1}{\Delta T} \int_{T_i}^{T_i+\Delta T} \lambda(T)\,dT = \mathrm{const}; \qquad \bar{c}_{\Delta T} = \frac{1}{\Delta T} \int_{T_i}^{T_i+\Delta T} c(T)\,dT = \mathrm{const}. \]

Taking into account that \(\lambda(T)\) and \(c(T)\) are determined by (1.1) and (1.2), one can find the values

\[ \varkappa_1 = \frac{4\bar{\lambda}_{\Delta T}} {4T_i^3+6T_i^2\Delta T+4T_i(\Delta T)^2+(\Delta T)^3} \tag{1.15} \]

and

\[ \varkappa_2 = \frac{4\bar{c}_{\Delta T}} {4T_i^3+6T_i^2\Delta T+4T_i(\Delta T)^2+(\Delta T)^3}, \tag{1.16} \]

which will correspond to the necessary approximation condition in the range of temperature increase \(\Delta T\).

In order not to exceed a reasonable error, it is expedient to carry out the entire calculation in stages. At each stage it is necessary to choose an admissible value of \(\Delta T\), which will guarantee the prescribed acc—

ness of the calculation. The choice of an admissible value for bodies of classical shape can be made according to the recommendation given in [3]. Additional control can reasonably be carried out by comparing the data of the nomographic calculation using Fig. [4], which have sufficiently high accuracy. As for bodies of finite dimensions, the magnitude of the admissible temperature increase was tested by us by comparing the data of the analytical calculation with the results of numerical integration on a computer. As a result it was established that the maximum possible error of 3–4% will not be exceeded in all calculated variants if
\(\Delta \Theta_{\mathrm{I}} = 0.60,\ \Delta \Theta_{\mathrm{II}} = 0.25,\ \Delta \Theta_{\mathrm{III}} = 0.15.\)

Numerical example illustrating the developed solution method

Let us study the character of the change in temperature at the center and at the middle of the lateral surface of a short cylinder of dimensions \(2R = 2H = 0.3\ \text{m}\), which is heated symmetrically under conditions of constant radiator power \((T_c = 1273^\circ\mathrm{K})\) from an initial temperature \(T_0 = 293^\circ\mathrm{K}\).

The thermophysical characteristics are numerically equal to: \(\lambda = 34.9\ \dfrac{\text{W}}{\text{m}\cdot\text{deg}}\),

\[ a = 0.0225\ \frac{\text{m}^2}{\text{hour}}, \qquad \rho = 7800\ \frac{\text{kg}}{\text{m}^3}, \qquad \varepsilon_{\mathrm{B}} = 4.24\ \frac{\text{W}}{\text{m}^2\text{deg}^4}. \]

The results of the calculation using the figures in [4] and formulas (1.14)—(1.16) are given in Table 1. Here, too, the data of the calculation by the zonal method [5] are presented and a comparison is made of the results obtained by the two different methods.

Table 1

Temperature field of a short cylinder during heating

II. Constant thermophysical parameters

If it is assumed that during the heating process the thermal coefficients remain constant, then the formal mathematical difficulties associated with solving the nonlinear differential equation of heat conduction are removed. The nonlinearity is preserved only in the boundary condition. But even in this case the classical solution methods prove powerless.

Together with V. V. Ivanov [7], an effective method is proposed for solving the multidimensional problem of radiant heat conduction, which is valid for bodies that are not too “massive” in the thermal sense.

We illustrate the scheme of the method by the example of heating a rectangular bar. The propagation of heat in the specimen under study is described by the system of equations

\[ \frac{1}{a}\frac{\partial \Theta(x,y,\tau)}{\partial \tau} = \frac{\partial^2 \Theta(x,y,\tau)}{\partial x^2} + \frac{\partial^2 \Theta(x,y,\tau)}{\partial y^2}, \]

\[ \Theta(x,y,\tau)=\frac{T(x,y,\tau)}{T_c},\qquad \tau \geq 0,\qquad -R_1<x<+R_1,\qquad -R_2<y<+R_2, \tag{II.1} \]

\[ -\frac{\partial \Theta(R_1,y,\tau)}{\partial x} + \frac{\sigma_B T_c^3}{\lambda} \left[1-\Theta^4(R_1,y,\tau)\right]=0, \tag{II.2} \]

\[ -\frac{\partial \Theta(x,R_2,\tau)}{\partial y} + \frac{\sigma_B T_c^3}{\lambda} \left[1-\Theta^4(x,R_2,\tau)\right]=0, \tag{II.3} \]

\[ \frac{\partial \Theta(0,y,\tau)}{\partial x} = \frac{\partial \Theta(x,0,\tau)}{\partial y} =0, \tag{II.4} \]

\[ \Theta(x,y,0)=\Theta_0=\mathrm{const}. \tag{II.5} \]

We shall show that, with sufficient approximation, the solution of the system (II.1)—(II.5) can be represented in the form

\[ \operatorname{Arth} \frac{\Theta(x,y,\tau)+\Theta_0}{1+\Theta(x,y,\tau)\Theta_0} + \operatorname{arctg} \frac{\Theta(x,y,\tau)+\Theta_0}{1-\Theta(x,y,\tau)\Theta_0} \]

\[ = \operatorname{Arth} \frac{\Theta(x,\tau)+\Theta(y,\tau)}{1+\Theta(x,\tau)\Theta(y,\tau)} + \operatorname{arctg} \frac{\Theta(x,\tau)+\Theta(y,\tau)}{1-\Theta(x,\tau)\Theta(y,\tau)} , \tag{II.6} \]

where \(\Theta(x,\tau)\) and \(\Theta(y,\tau)\) are known solutions for infinite plates whose mutual intersection forms a bar of rectangular cross section. In the proof we take into account the identity of the uniqueness conditions for two infinite plates of dimensions \(2R_1\) and \(2R_2\) with the uniqueness conditions for the rectangular prism (II.2)—(II.5). By direct substitution of expression (II.6) into the boundary and initial conditions for two infinite plates,

\[ - \frac{\dfrac{\partial \Theta(R_1,\tau)}{\partial x}} {1-\Theta^4(R_1,\tau)} + \frac{\sigma_B T_c^3}{\lambda} =0, \]

\[ - \frac{\dfrac{\partial \Theta(R_2,\tau)}{\partial y}} {1-\Theta^4(R_2,\tau)} + \frac{\sigma_B T_c^3}{\lambda} =0, \]

\[ \frac{\partial \Theta(0,\tau)}{\partial x} = \frac{\partial \Theta(0,\tau)}{\partial y} =0, \]

\[ \Theta(x,0)=\Theta(y,0)=\Theta_0 \]

we find that relation (II.6) completely satisfies all the imposed boundary conditions of the one-dimensional problems.

Substitute expression (II.6) into the differential equation of heat conduction (II.1). As a result we obtain:

\[ \left\{ \frac{1}{a}\frac{\partial \Theta(x,\tau)}{\partial \tau} - \frac{\partial^2 \Theta(x,\tau)}{\partial x^2} \right\} + \left\{ \frac{1}{a}\frac{\partial \Theta(y,\tau)}{\partial \tau} - \frac{\partial^2 \Theta(y,\tau)}{\partial y^2} \right\} - \]

\[ - \frac{4\Theta^3(x,\tau)}{1-\Theta^4(x,\tau)} \cdot \left( \frac{\partial \Theta(x,\tau)}{\partial x} \right)^2 - \frac{4\Theta^3(y,\tau)}{1-\Theta^4(y,\tau)} \cdot \left( \frac{\partial \Theta(y,\tau)}{\partial y} \right)^2 \cong 0. \tag{II.7} \]

It is not difficult to see that the expressions in braces are equal to zero. Then it follows from (II.7) that relation (II.6) does not satisfy equation (II.1). However, analyzing the value of the residual

\[ \Psi(x,y,\tau)= -\frac{4\Theta^{3}(x,\tau)}{1-\Theta^{4}(x,\tau)} \left(\frac{\partial\Theta(x,\tau)}{\partial x}\right)^{2} -\frac{4\Theta^{3}(y,\tau)}{1-\Theta^{4}(y,\tau)} \left(\frac{\partial\Theta(y,\tau)}{\partial y}\right)^{2}, \]

one may note that \(\Psi(x,y,\tau)=0\) for \(x=y=0\) (condition (II.4)); hence for the coordinate of the center the solution (II.6) is exact, and \(\Psi(x,y,\tau)\to 0\) when \(\mathrm{Sk}\to 0\). Consequently, up to certain values of the radiation heat-exchange criterion, the value of the residual may be neglected; then the solution (II.6) satisfies equation (II.1) with an admissible approximation. Similarly, for a short cylinder and a rectangular parallelepiped the solutions are written in the form:

\[ \operatorname{Arth}\frac{\Theta(r,z,\tau)+\Theta_{0}}{1+\Theta(r,z,\tau)\Theta_{0}} +\operatorname{arctg}\frac{\Theta(r,z,\tau)+\Theta_{0}}{1-\Theta(r,z,\tau)\Theta_{0}} = \]

\[ = \operatorname{Arth}\frac{\Theta(r,\tau)+\Theta(z,\tau)}{1+\Theta(r,\tau)\Theta(z,\tau)} +\operatorname{arctg}\frac{\Theta(r,\tau)+\Theta(z,\tau)}{1-\Theta(r,\tau)\Theta(z,\tau)}; \tag{II.8} \]

\[ \operatorname{Arth}\Theta(x,y,z,\tau) +\operatorname{arctg}\Theta(x,y,z,\tau) +2\left(\operatorname{Arth}\Theta_{0}+\operatorname{arctg}\Theta_{0}\right) = \]

\[ = \operatorname{Arth} \frac{\Theta(x,\tau)+\Theta(y,\tau)+\Theta(z,\tau)+\Theta(x,\tau)\Theta(y,\tau)\Theta(z,\tau)} {1+\Theta(x,\tau)\Theta(y,\tau)+\Theta(x,\tau)\Theta(z,\tau)+\Theta(y,\tau)\Theta(z,\tau)} + \]

\[ +\operatorname{arctg} \frac{\Theta(x,\tau)+\Theta(y,\tau)+\Theta(z,\tau)+\Theta(x,\tau)\Theta(y,\tau)\Theta(z,\tau)} {1+\Theta(x,\tau)\Theta(y,\tau)-\Theta(x,\tau)\Theta(z,\tau)-\Theta(y,\tau)\Theta(z,\tau)}, \tag{II.9'} \]

if

\[ \frac{\Theta(x,\tau)\Theta(z,\tau)+\Theta(y,\tau)\Theta(z,\tau)} {1-\Theta(x,\tau)\Theta(y,\tau)} <1; \]

\[ \operatorname{Arth}\Theta(x,y,z,\tau) +\operatorname{arctg}\Theta(x,y,z,\tau) +2\left(\operatorname{Arth}\Theta_{0}+\operatorname{arctg}\Theta_{0}\right) = \]

\[ = \operatorname{Arth} \frac{\Theta(x,\tau)+\Theta(y,\tau)+\Theta(z,\tau)+\Theta(x,\tau)\Theta(y,\tau)\Theta(z,\tau)} {1+\Theta(x,\tau)\Theta(y,\tau)-\Theta(x,\tau)\Theta(z,\tau)-\Theta(y,\tau)\Theta(z,\tau)} + \]

\[ +\pi+\operatorname{arctg} \frac{\Theta(x,\tau)+\Theta(y,\tau)+\Theta(z,\tau)+\Theta(x,\tau)\Theta(y,\tau)\Theta(z,\tau)} {1+\Theta(x,\tau)\Theta(y,\tau)-\Theta(x,\tau)\Theta(z,\tau)-\Theta(y,\tau)\Theta(z,\tau)}, \tag{II.9''} \]

if

\[ \frac{\Theta(x,\tau)\Theta(z,\tau)+\Theta(y,\tau)\Theta(z,\tau)} {1-\Theta(x,\tau)\Theta(y,\tau)} >1. \]

The proposed method can be successfully applied in engineering practice only if sufficiently effective methods are available for solving one-dimensional heat-conduction problems with boundary conditions of the radiation type. We have constructed engineering nomograms for calculating the temperature field in bodies of classical shape when they are heated by radiant heat. Such nomograms, respectively for an infinite plate and an infinite cylinder, are given in [6]. The accuracy of the nomographic method of solution does not exceed 2% in the direction of underestimating the calculation results, including the initial stage of the process, which is usually determined most roughly.

Examples of practical use of the proposed method

Example 1. Determine the temperatures at the center and in the middle of the large face of a steel ingot of dimensions \(2R_x\times 2R_y\times 2R_z=0.2\times 0.4\times 0.6\ \text{m}\), which is uniformly heated on all sides at a furnace temperature

1373°K for 0.5 and 1 hour. The initial temperature of the ingot is 273°K, the mean coefficient of thermal conductivity

\[ \lambda = 34.9 \frac{\text{W}}{\text{m}\cdot\text{deg}}, \]

the thermal diffusivity \(a = 0.03 \, \text{m}^2/\text{h}\), the reduced radiation coefficient

\[ \sigma_{\mathrm{r}} = 4.65 \frac{\text{W}}{\text{m}^2\text{deg}^4}. \]

First we calculate the dimensionless complexes

\[ Sk_x = 0.345, \qquad Sk_y = 1.035, \qquad Sk_z = 0.690. \]

\[ Fo_x = 1.5, \qquad Fo_y = 0.167, \qquad Fo_z = 0.375 \; (\tau = 0.5 \text{ h}). \]

\[ Fo_x = 3.0, \qquad Fo_y = 0.335, \qquad Fo_z = 0.75 \; (\tau = 1 \text{ h}). \]

From the nomograms (Figs. 1, 2, [6]) we find:

a) for the first case (\(\tau = 0.5\) h)

\[ \Theta_{x=0} = 0.605, \qquad \Theta_{y=0} = 0.256, \qquad \Theta_{z=0} = 0.348, \qquad \Theta_{x=R_x} = 0.75. \]

b) for the second case (\(\tau = 1\) h)

\[ \Theta_{x=0} = 0.895, \qquad \Theta_{y=0} = 0.373, \qquad \Theta_{z=0} = 0.562, \qquad \Theta_{x=R_x} = 0.931. \]

Calculation by formula (II.9′) gives

a)

\[ \Theta(0,0,0) = 0.752 \; (=0.752 \; \delta = 0\%) \]

\[ \Theta(R_x,0,0) = 0.841 \; (=0.854 \; \delta = 1.5\%) \]

b)

\[ \Theta(0,0,0) = 0.904 \; (=0.961 \; \delta = 5.9\%) \]

\[ \Theta(R_x,0,0) = 0.942 \; (=0.985 \; \delta = 4.3\%) \]

In parentheses are given the calculated data of work [5] for this case, and the magnitude of the error of the calculations by the two different methods is shown.

Example 2. A bar of square cross section is subjected to symmetric radiative heating. The dimensionless regime parameters are specified by the quantities: \(Sk_x = Sk_y = 1.05\); \(\Theta_0 = 0.175\).

What is the value of the relative temperatures at the center, at the middle of the side face, and at the edge of the bar after the lapse of time a) \(Fo_1 = 0.245\); b) \(Fo_2 = 0.815\)?

The problem is solved analogously with the aid of the nomograms in Figs. 1, 2, [6]. As a result we obtain:

\[ \begin{aligned} \text{a)}\quad & \Theta(0,0,Fo_1) = 0.3796 \; (\delta = 0.8\%) \qquad && \text{b)}\quad \Theta(0,0,Fo_2) = 0.931 \; (\delta = 2\%) \\ & \Theta(1,0,Fo_1) = 0.758 \; (\delta = 6.7\%) \qquad && \Theta(1,0,Fo_2) = 0.978 \; (\delta = 0.8\%) \\ & \Theta(1,1,Fo_1) = 0.937 \; (\delta = 7.1\%) \qquad && \Theta(1,1,Fo_2) = 0.989 \; (\delta = 0.2\%) \end{aligned} \]

Let us compare with the computer-calculation data for this case:

\[ \begin{aligned} \text{a)}\quad & \Theta(0,0,Fo_1) = 0.3767 \qquad && \text{b)}\quad \Theta(0,0,Fo_2) = 0.9133 \\ & \Theta(1,0,Fo_1) = 0.7096 \qquad && \Theta(1,0,Fo_2) = 0.9701 \\ & \Theta(1,1,Fo_1) = 0.8748 \qquad && \Theta(1,1,Fo_2) = 0.9887. \end{aligned} \]

In parentheses is given the calculation error in comparison with the computer data.

Example 3. What is the temperature at the center and at the middle of the side surface of a short steel cylinder of diameter \(2R = 0.3 \, \text{m}\) and height \(2H = 0.3 \, \text{m}\), symmetrically heated in a radiating medium with temperature 1300°K for 1 hour. The initial temperature is 293°K; the thermophysical coefficients are numerically equal to

\[ \lambda = 34.9 \frac{\text{W}}{\text{m}\cdot\text{deg}}, \qquad a = 0.0225 \, \text{m}^2/\text{h}, \]

\[ \sigma_{\mathrm{r}} = 4.24 \frac{\text{W}}{\text{m}^2\text{deg}^4}. \]

We calculate: \(\mathrm{Sk}_R=\mathrm{Sk}_H=0.401,\ \mathrm{Fo}_R=\mathrm{Fo}_H=1\). Using the nomograms in Figs. 1, 2, 3, 4 [6] and formula (II.8), we obtain

\[ \Theta(0,0,\mathrm{Fo})=0.913,\qquad \Theta(R,0,\mathrm{Fo})=0.959. \]

Let us compare this with the data of the zonal calculation method [5], according to which

\[ \Theta(0,0,\mathrm{Fo})=0.892,\qquad \Theta(R,0,\mathrm{Fo})=0.945. \]

The calculation error in comparison with the data of the zonal method is, respectively,

\[ \delta=1.2\%,\qquad \delta=1.5\%. \]

By comparing the numerical data obtained by the proposed method with the results of computer calculations and with the zonal method, it has been established that the maximum error in calculating the temperature at any point of a body of finite size will not exceed 5% in those cases when

\[ \mathrm{Sk}_{\min}\leq 0.8, \]

and 7% when

\[ \mathrm{Sk}_{\min}\leq 1.0. \]

Conclusions

1. Very effective methods are proposed for calculating the multidimensional problem of nonlinear heat conduction.

2. The established relations (1.8), (1.14), (II.6), (II.8), (II.9) are valid both in the case of symmetric and in the case of asymmetric heating. In this connection, it is possible to take into account asymmetry caused by different temperature levels of the radiant heat source on opposite faces of a body of finite size and by the inequality of the reduced blackness coefficients of the surfaces. The solutions presented are also valid in the case when the thermal coefficients are different in all coordinate directions, i.e., when the body of finite size is anisotropic.

References

1. V. V. Salomatov. Temperature field in a crystal under radiation heating. Izv. vuzov—Fizika, No. 5, 1965.

2. V. V. Salomatov. Heating of bodies by radiation at a variable temperature of the heat source (Candidate dissertation), Tomsk, 1964.

3. L. A. Brovkin. Izv. vuzov—Energetika, No. 3, No. 4, 1965.

4. A. V. Lykov, Yu. A. Mikhailov. Theory of Heat and Mass Transfer. GEI, 1963.

5. G. P. Boikov. Izv. TPI, vol. 101, 1958.

6. V. V. Salomatov, A. A. Torgopov. Izv. vuzov, “Ferrous Metallurgy”, No. 10, 1968.

7. V. V. Ivanov, V. V. Salomatov. Izv. vuzov, “Fizika”, No. 6, 1966.