Heating of Bodies under the Action of Radiant Heat

(First Communication)

G. P. Boikov

The paper considers the heating of bodies of classical shape under a boundary condition of the second kind, when heat exchange occurs according to the Stefan--Boltzmann law and the radiant flux is symmetric.

A zonal (in time) method is presented for calculating the temperature field in the propagation of heat in one dimension, under the assumption that the thermal characteristics of the substance are unchanged.

Calculated dependences and relations are given, as well as a procedure for selecting the computational time interval.

It is known that at high temperatures of the heat source (approximately above \(800^\circ\mathrm{C}\)) the transfer of heat to the body being heated occurs mainly by radiant energy according to the Stefan--Boltzmann law.

In order to determine the temperature field in the body under such heat exchange, it is necessary to solve the system of equations:

\[ \frac{\partial T_{(x,z)}}{\partial z} = a \frac{\partial^{2} T_{(x,z)}}{\partial x^{2}} \tag{1} \]

\[ T_{(x,0)} = T_{0} \tag{2} \]

\[ -\frac{\partial T_{(R,z)}}{\partial x} = \frac{\varepsilon_{n}\cdot C_{0}}{\lambda} \left[ \left(\frac{T_{c}}{100}\right)^{4} - \left(\frac{T_{(R,z)}}{100}\right)^{4} \right] \tag{3} \]

\[ \frac{\partial T_{(0,z)}}{\partial x} = 0. \tag{4} \]

Obtaining a solution directly from the system (1)--(4) is difficult, since no way has yet been found that would make it possible satisfactorily to reconcile the solution of the differential equation (1) with the boundary condition of the form (3). In view of the fact that determining the temperature field described by conditions (1)--(4) is of great practical and fundamental interest, one has to resort to artificial devices which, in a certain approximation, make it possible to calculate the heating of bodies under the action of radiant heat.

One of the most widespread devices for solving the problem under consideration is the method of reducing boundary conditions of the second kind (3) to boundary conditions of the third kind by introducing a conditional coefficient of heat transfer by radiation. The boundary condition expressed by the law

\[ \lambda \frac{\partial T_{(R,z)}}{\partial x} = \alpha_{\mathrm{rad.}}\left(T_{c}-T_{(R,z)}\right), \tag{3'} \]

allows one to obtain the solution of the system (1), (2), (3′), (4) in the form:

\[ \frac{T_{(x,\tau)}-T_o}{T_c-T_o} = 1-\sum_{n=1}^{\infty} \frac{2\sin\mu_n}{\mu_n+\sin\mu_n\cdot\cos\mu_n} \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot e^{-\mu_n^2\frac{a\tau}{R^2}}; \tag{5} \]

\[ \left( \text{here } \operatorname{ctg}\mu=-\frac{\mu}{Bi} \quad \text{and} \quad Bi=\frac{\alpha_{\text{rad.}}\cdot R}{\lambda}, \ \text{see [5], [6]} \right), \]

which, in a certain approximation, makes it possible to carry out practical calculations.

The proposed method of investigation makes it possible to approach the question under consideration in a different way and to bring the calculated data somewhat closer to practical data. According to this method, the temperature field in space is considered in the form of a continuous functional dependence, giving a smooth change of the function depending on the argument. The calculation of the temperature field in time is carried out zonally, using the principle of finite differences (see [9], [10], [12]). The initial data in obtaining the temperature distribution are the solutions of A. V. Lykov for a plate, cylinder, and sphere under a constant radiant flux (see [6]). The solution, for example, for a plate is given by him in the form:

\[ T_{(x,\tau)} = T_o+\frac{g_{c_1}}{\lambda} \left[ \frac{a\tau}{R} - \frac{R^2-3x^2}{6R} + R\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2} \right. \]

\[ \left. \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot e^{-\mu_n^2\frac{a\tau}{R^2}} \right] \tag{6} \]

\[ (\text{here } \sin\mu=0). \]

Taking expression (6) as the initial one, we assume that the constant radiant flux \(g_{c_1}\) is determined by the law:

\[ g_{c_1} = \varepsilon_n\cdot C_o \left[ \left(\frac{T_c}{100}\right)^4 - \left(\frac{T_o}{100}\right)^4 \right] \tag{7} \]

and acts not throughout the entire heating process, but only during a small finite time interval equal to \(\tau_1\). Substituting \(\tau_1\) into expression (6) instead of \(\tau\), we obtain the temperature distribution after the expiration of the first time interval from the beginning of heating. Here, replacing \(x\) by \(R\), we obtain the temperature value on the surface of the plate, \(T_1(R)\).

Having these data, we obtain the temperature distribution after the expiration of the second identical time interval \(\tau_1\) by solving the system:

\[ \frac{\partial T_{(x,\tau)}}{\partial \tau} = a\cdot \frac{\partial^2 T_{(x,\tau)}}{\partial x^2}; \tag{8} \]

\[ T_{(x,0)} = T_o+\frac{g_{c_1}}{\lambda} \left[ \frac{a\tau_1}{R} - \frac{R^2-3x^2}{6R} + R\sum_{n=1}^{\infty}(-1)^{n+1}\cdot \right. \]

\[ \left. \frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot e^{-\mu_n^2\frac{a\tau_1}{R^2}} \right]; \tag{9} \]

\[ \frac{\partial T_{(R,\tau)}}{\partial x} = \frac{\varepsilon_n \cdot C_0}{\lambda} \cdot \left[ \left(\frac{T_c}{100}\right)^4 - \left(\frac{T_1(R)}{100}\right)^4 \right] = \frac{g_{c_2}}{\lambda}; \tag{10} \]

\[ \frac{\partial T_{(0,\tau)}}{\partial x}=0 \tag{11} \]

and substitution into the obtained solution:

\[ \begin{aligned} T_{(x,\tau)}=T_0+\frac{g_{c_1}}{\lambda} \Bigg[ \frac{a\tau_1}{R} -\frac{R^2-3x^2}{6R} +\frac{a\tau}{R} +R\sum_{n=1}^{\infty}(-1)^{n+1} \cdot \frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot e^{-\mu_n^2\frac{a\tau_1}{R^2}} \cdot e^{-\mu_n^2\frac{a\tau}{R^2}} \Bigg] \\ +\frac{(g_{c_2}-g_{c_1})}{\lambda} \cdot \Bigg[ \frac{a\tau}{R} -\frac{R^2-3x^2}{6R} +R\sum_{n=1}^{\infty}(-1)^{n+1} \cdot \frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot e^{-\mu_n^2\frac{a\tau}{R^2}} \Bigg]. \end{aligned} \tag{12} \]

instead of \(\tau\), the values \(\tau_1\) (see [1], [2], [6], [12]).

Continuing analogous reasoning, one can verify that the temperature distribution in the plate after the expiration of the \(m\)-th time interval from the beginning of heating has the form:

\[ \begin{aligned} T_m(x)= &T_0 +\frac{a\tau_1}{\lambda R} \cdot \left(g_{c_1}+g_{c_2}+\ldots+g_{c_m}\right) -\frac{g_{c_m}\cdot(R^2-3x^2)}{6\lambda R} \\ &+R\sum_{n=1}^{\infty}(-1)^{n+1} \frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot \Bigg[ \frac{g_{c_1}}{\lambda} \cdot e^{-m\mu_n^2\frac{a\tau_1}{R^2}} \\ &\qquad -\frac{(g_{c_1}-g_{c_2})}{\lambda} \cdot e^{-(m-1)\mu_n^2\frac{a\tau_1}{R^2}} -\frac{(g_{c_2}-g_{c_3})}{\lambda} \cdot e^{-(m-2)\mu_n^2\frac{a\tau_1}{R^2}} - \\ &\qquad -\ldots -\frac{(g_{c(m-1)}-g_{c_m})}{\lambda} \cdot e^{-\mu_n^2\frac{a\tau_1}{R^2}} \Bigg]; \end{aligned} \tag{13} \]

or, in abbreviated notation:

\[ \begin{aligned} \frac{T_m(x)}{T_c} = &\frac{T_0}{T_c} +\frac{g_c\cdot R}{\lambda\cdot T_c} \Bigg\{ \frac{a\tau_1}{R^2} \cdot \sum_{i=1}^{i=m}\frac{g_{c_i}}{g_c} - \frac{g_{c_m}}{g_c} \cdot \frac{R^2-3x^2}{6R^2} \\ &\qquad + \sum_{n=1}^{\infty}(-1)^{n+1} \frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot \left[ \frac{g_{c_1}}{g_c} \cdot e^{-m\mu_n^2\frac{a\tau_1}{R^2}} \right. \\ &\qquad\qquad \left. - \sum_{i=1}^{i=m-1} \frac{g_{c_i}-g_{c(i+1)}}{g_c} \cdot e^{-(m-i)\mu_n^2\frac{a\tau_1}{R^2}} \right] \Bigg\}. \end{aligned} \tag{14} \]

The latter, in similarity criteria¹, is written in the form

\[ \begin{aligned} \Theta_m(X) = \Theta_o + K_i \Bigg\{& Fo_1 \sum_{i=1}^{i=m} Q_{c_i} - Q_{cm}\cdot \frac{1}{6}(1-3X^2) + \\ &+ \sum_{n=1}^{\infty}(-1)^{n+1}\cdot \frac{2}{\mu_n^2}\cdot \cos(\mu_n X)\cdot \left[ Q_{c_i}\cdot e^{-m\mu_n^2 Fo_1} -\right.\\ &\left. -\sum_{i=1}^{i=m-1}\left(Q_{c_i}-Q_{c(i+1)}\right)\cdot e^{-(m-i)\mu_n^2\cdot Fo_1} \right]\Bigg\}. \end{aligned} \tag{15} \]

For the case of “thin”² bodies \((K_i<0.12)\), carrying out the calculation by expressions (14) and (15) will be irrational. In this case it is more convenient to use the concept of the mean temperature of the entire mass of the body, which, for example, for a plate is determined from the relation:

\[ \frac{T_m^{cp}}{T_c} = \frac{1}{R}\int_{0}^{R}\frac{T_m(x)}{T_c}\cdot dx = \frac{T_o}{T_c} + \]

\[ + \frac{g_c\cdot R}{\lambda\cdot T_c}\cdot \frac{a\tau_1}{R^2}\cdot \sum_{i=1}^{i=m}\frac{g_{c_i}}{g_c}; \tag{16} \]

or in criterion form:

\[ \Theta_m^{cp} = \Theta_o + K_i\cdot Fo_1\cdot \sum_{i=1}^{i=m} Q_{c_i}. \tag{17} \]

Expressions [16] and [17] were obtained as the mean integral of the distributions (14) and (15), respectively.

As is evident from dependences (14) and (16), in calculating the temperature distribution it is necessary to specify the computational time interval \(\tau_1\). Undoubtedly, increasing the numerical value of the computational time interval \(\tau_1\) can considerably reduce the amount of computation. On the other hand, increasing the computational time interval entails an increase in the error; exceeding the permissible value of this error reduces the value of the calculation.

The proposed method for choosing the computational time interval includes three steps. The first step consists in a preliminary determination of the computational time interval from the approximate dependence

\[ \tau_1 = \frac{P\cdot \lambda\cdot T_c\cdot R}{g_c\cdot a}, \tag{18} \]

where \(P\) is a constant number, different for a plate, cylinder, and sphere (for example, for a plate and a cylinder equal, respectively, to 0.03 and 0.025).

The second step consists in checking the computational time interval obtained from formula (18) by estimating the approximation using the inequality

\[ \varphi_m \leqslant f(\tau_1), \tag{19} \]

where $\varphi_m$ is the actual error of the calculation, expressed in terms of radiant fluxes

$$ \varphi_m=\frac{g_{c\,\mathrm{ist}}-g_{c_2}}{g_{c_1}}, $$

$f(\tau_1)$ is a certain function possessing the following features: first, this function reflects to some extent the actual error of the calculation $\varphi_m$, approaching it in its numerical value;

second, the actual error of the calculation $\varphi_m$ is always less than or equal to this function.

The function $f(\tau_1)$ is found from an analysis of the change in temperatures and radiant fluxes on the surface of the body. Let us find this function and prove that the actual error $\varphi_m \leq f(\tau_1)$.

Fig. 1

Suppose we have a certain broken curve $y_1=f_1(x)$ (see Fig. 1), which has the following properties:

1) The breaks of the curve occur at equal intervals of the argument, having some value $\Delta x_1''$.

2) When the numerical value of the interval $\Delta x_1''$ is decreased to $\Delta x_1'$, $\Delta x_1$, and further toward zero, and thus when the number of these intervals is increased toward infinity, the curve $y_1=f_1(x)$, through a series of intermediate broken curves $y_2=f_2(x)$; $y_3=f_3(x)$, etc., tends to a certain limit, which is the “true” one and already represents a smooth curve of the dependence $y=f(x)$.

3) Each subsequent broken curve $y_2=f_2(x)$; $y_3=f_3(x)$, etc., is closer to the “true” curve $y=f(x)$ than the preceding one and has a smoother course.

4) The formation of a smooth transition at the place of a break at any arbitrary point indicates that the point with the greatest break is not sufficiently ...

far from the state of smoothness, and consequently the curve under consideration is not far from being superposed on the true curve.

Taking into account everything stated in points 1–4, let us consider separately the segment of Figure 1 denoted by the points:

\[ 0-1 \begin{cases} 1'-2'\\ 2 \end{cases} \qquad \text{(see Fig. 1a).} \]

From Fig. 1a it is seen that the curve \(0-1-2\) is closer to the true curve than the curve \(0-1'-2'\), since \(\alpha_1\) is, in its magnitude, considerably closer to \(\alpha\) than \(\alpha'_1\), as is indicated by the smoother course of the curve \(0-1-2\).

Fig. 1a

Fig. 1b

Obviously, the degree of discrepancy between the curve \(0-1-2\) and the true curve can be estimated from the ratio

\[ \frac{\operatorname{tg}\alpha-\operatorname{tg}\alpha_1}{\operatorname{tg}\alpha}. \tag{V} \]

The smaller the value of this ratio, the closer the curve \(0—1—2\) approaches the true one, and conversely. If the broken curve consists of separate smooth curves (see Fig. 16), whose functional dependence is known in the first two intervals \([y=\varphi(x);\ y=\psi(x)]\), then expression \((V)\) can be represented in terms of angular coefficients:

\[ \frac{ \dfrac{d\varphi(x)_{x=\Delta x_1}}{dx} - \dfrac{d\psi(x)_{x=0}}{dx} }{ \dfrac{d\varphi(x)_{x=\Delta x_1}}{dx} }. \tag{W} \]

Thus, for our case, when calculating the change in temperature, for example, on the surface of a plate, the dependence analogous to \((W)\) takes the form:

\[ \frac{ \dfrac{dT_{1(R,\tau_1)}}{d\tau} - \dfrac{dT_{2(R,0)}}{d\tau} }{ \dfrac{dT_{1(R,\tau_1)}}{d\tau} } = f(\tau_1). \]

Here, according to expressions (6) and (12),

\[ T_{1(R,\tau)} = T_0+ \frac{g_{c_1}\cdot R}{\lambda} \left[ \frac{a\tau}{R^2} + \frac{1}{3} - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2}\cdot e^{-\mu_n^2\frac{a\tau}{R^2}} \right] \]

\[ T_{2(R,\tau)} = T_0+ \frac{g_{c_1}\cdot R}{\lambda} \left[ \frac{a\tau_1}{R^2} + \frac{1}{3} + \frac{a\tau}{R^2} - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2}\cdot e^{-\mu_n^2\frac{a\tau_1}{R^2}} \right] \cdot e^{-\mu_n^2\frac{a\tau}{R^2}} + \frac{(g_{c_2}-g_{c_1})R}{\lambda} \cdot \left[ \frac{a\tau}{R^2} + \frac{1}{3} - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2}e^{-\mu_n^2\frac{a\tau}{R^2}} \right], \]

therefore

\[ f(\tau_1) = \frac{3(g_{c_1}-g_{c_2})}{ g_{c_1} \left[ 1+2\sum_{n=1}^{\infty} e^{-\mu_n^2\frac{a\tau_1}{R^2}} \right] }. \]

Comparing now

\[ \varphi_m = \frac{g_{c\,\mathrm{inst}}-g_{c_2}}{g_{c_1}} \quad\text{and}\quad \frac{3(g_{c_1}-g_{c_2})}{ g_{c_1} \left[ 1+2\sum_{n=1}^{\infty} e^{-\mu_n^2\frac{a\tau_1}{R^2}} \right] } = f(\tau_1), \]

we conclude that for a plate the inequality always holds

\[ \varphi_m < \frac{3(g_{c_1}-g_{c_2})}{ g_{c_1} \left[ 1+2\sum_{n=1}^{\infty} e^{-\mu_n^2\frac{a\tau_1}{R^2}} \right] }, \tag{20} \]

because

\[ g_{c1}=\varepsilon_n C_0\left[\left(\frac{T_c}{100}\right)^4-\left(\frac{T_o}{100}\right)^4\right]\geqslant g_{c\,\mathrm{ist}}=\varepsilon_n\cdot C_0\left[\left(\frac{T_c}{100}\right)^4-\left(\frac{T_{\mathrm{ist}}}{100}\right)^4\right] \]

(since \(T_o\) is the initial temperature of the body, and \(T_{\mathrm{ist}}\) is the temperature at the surface after a certain interval of time has elapsed from the beginning of heating).

It should be noted that the series from 1 to \(\infty\) in expressions (14) and (15) (the third term in braces) converges rapidly; with increasing time its value becomes very small in comparison with the first two terms in braces. Therefore, beginning from a certain moment of time \((m>1)\), it may be neglected. Then the subsequent calculation, involving only the first two terms in braces, is considerably simplified.

The specific heat consumption for the plate may be determined from the formula:

\[ \Delta g_s^{(m)}=\tau_1\sum_{i=1}^{i=m} g_{ci}. \tag{21} \]

Expressions analogous to (14)—(21) for the cylinder have the form:

\[ \frac{T_m(r)}{T_c} = \frac{T_o}{T_c} + \frac{g_c R}{\lambda T_c} \cdot \left\{ 2\cdot \frac{a\tau_1}{R^2}\cdot \sum_{i=1}^{i=m}\frac{g_{ci}}{g_c} - \frac{1}{4}\cdot\frac{g_{cm}}{g_c}\left(1-2\frac{r^2}{R^2}\right) -\right. \]

\[ \left. - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2\cdot J_0(\mu_n)} \cdot J_0\left(\mu_n\frac{r}{R}\right) \left[ \frac{g_{c1}}{g_c}\cdot e^{-m\mu_n^2\frac{a\tau_1}{R^2}} - \sum_{i=1}^{i=m-1} \frac{g_{ci}-g_{c(i+1)}}{g_c} \cdot e^{-(m-i)\mu_n^2\frac{a\tau_1}{R^2}} \right] \right\}; \tag{14′} \]

\[ \Theta_m(R) = \Theta_o + K_i \left\{ 2F_{o1}\sum_{i=1}^{i=m}Q_{ci} - \frac{1}{4}Q_{cm}(1-2R^2) -\right. \]

\[ \left. - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2\cdot J_0(\mu_n)} \cdot J_0(\mu_n R) \left[ Q_{c1}\cdot e^{-m\mu_n^2 F_{o1}} - \sum_{i=1}^{i=m-1}(Q_{ci}-Q_{c(i+1)}) \cdot e^{-(m-i)\mu_n^2 F_{o1}} \right] \right\}; \tag{15′} \]

\[ \frac{T_m^{cp}}{T_c} = \frac{2}{R^2}\cdot \int_{0}^{R} r\cdot \frac{T_m(r)}{T_c}\cdot dr = \frac{T_o}{T_c} + 2\frac{g_c\cdot R}{\lambda\cdot T_c} \cdot \frac{a\tau_1}{R^2} \cdot \sum_{i=1}^{i=m}\frac{g_{ci}}{g_c}; \tag{16′} \]

\[ \Theta_m^{cp}=\Theta_0+2K_i\cdot F_{0_i}\sum_{i=1}^{i=m} Q_{c_i}; \tag{17′} \]

\[ \varphi_m \le \frac{2\left(g_{c_1}-g_{c_3}\right)} {g_{c_1}\left[1+\sum_{n=1}^{\infty} e^{-\mu_n^2\frac{a\tau_1}{R^2}}\right]}; \tag{20′} \]

\[ \Delta g_v=\frac{2}{R}\cdot \tau_1\cdot \sum_{i=1}^{i=m} g_{c_i}; \tag{21′} \]

Similar relations can also be written for a sphere. In Fig. 2 the curve of the dimensionless temperature of the center of a cylinder is shown

\[ \Theta=\frac{T(\rho,\tau)}{T_c},\qquad K_i=\frac{g_i\cdot R}{\lambda\cdot T_c},\qquad \Theta_0=\frac{T_0}{T_c}. \]

Fig. 2. Dimensionless curves for calculating the heating of a cylinder.
— — — curve of the change in the dimensionless temperature of the center of a cylinder as a function of the Fourier criterion, constructed by formula (15′) for the Kirchhoff criterion equal to 0.206 and the initial-temperature criterion 0.214;

—— calculated data for the same conditions according to Schack’s graphs.

(dashed line), constructed by formula (15′) for the criteria \(K_i'=0.206;\ \Theta_0=0.214\).
For comparison, the solid line shows the dimensionless calculated curve constructed from Schack’s graphs for the same conditions.

REFERENCES

1. Dikin V. A. and Kuznetsov N. I.—Handbook of Operational Calculus, M.—L., 1951.

2. Carslaw H. S. and Jaeger J. T.—Operational Methods in Applied Mathematics, GITTL, Moscow, 1948.

3. Ivantsov G. P.—Analysis of similarity in the heating of metal in a muffle furnace, collection Industrial Furnaces, Metallurgizdat, 1953.

4. Karpichev M. V., Mikheev M. A., Eigenson L. S.—Heat Transfer, GEI, Moscow, 1940.

5. Lykov A. V.—Thermal Conductivity of Nonstationary Processes, GITTL, M.—L., 1948.

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

7. Nemchinskii A. L.—Thermal Calculations of Heat Treatment, Sudpromgiz, 1953.

8. Ivantsov G. P.—Heating of Ingots, Metallurgizdat, Moscow, 1950.

9. Taits N. Yu.—Technology of Steel Heating, Metallurgizdat, Moscow, 1950.

10. Shvab V. A. Nonstationary temperature fields in solid bodies under changing boundary conditions, Bulletin of Engineers and Technicians, 3, 1935.

11. Eigenson L. S.—Modeling, GI “Soviet Science,” Moscow, 1952.

12. Boikov G. P.—Heating of Bodies under the Action of Radiant Heat (dissertation), Tomsk, 1955.

Principal Notation

$T$ — variable temperature of the body, $^\circ K$.

$T_c$ — temperature of the heat source, $^\circ K$.

$T_0$ — initial temperature of the body, $^\circ K$.

$g_c$ — radiant flux from the heat source into a vacuum, for the degree of blackness of the system $\varepsilon_n$,

\[ \frac{\mathrm{kcal}}{m^2\,\mathrm{h}} . \]

$g_{c_i}$ — radiant flux on the surface of the body at the $i$-th instant of time, directed into the body,

\[ \frac{\mathrm{kcal}}{m^2\,\mathrm{h}} . \]

$\tau$ — time, h.

$x, r$ — current coordinates, m.

$R$ — half-thickness (radius) of the body, m.

$\lambda$ — coefficient of thermal conductivity,

\[ \frac{\mathrm{kcal}}{m\,\mathrm{h}\,^\circ} . \]

$\varepsilon_n$ — reduced degree of blackness of the system.

$C_0$ — black-body radiation coefficient,

\[ \frac{\mathrm{kcal}}{m^2\,\mathrm{h}\,^\circ K^4} . \]

$\alpha_{\mathrm{rad}}$ — coefficient of heat transfer by radiation,

\[ \frac{\mathrm{kcal}}{m^2\,\mathrm{h}\,^\circ} . \]

$\mu_n$ — roots of the characteristic equations: $\sin \mu = 0$; $J_1(\mu)=0$.

$a$ — coefficient of thermal diffusivity, $m^2/\mathrm{h}$.

\[ K_i=\frac{g_c R}{\lambda T_c} \]

— Kirpichev criterion.

\[ F_{o_i}=\frac{a\tau_1}{R^2} \]

— Fourier criterion.

\[ \Theta=\frac{T}{T_c} \]

— dimensionless temperature criterion.

\[ Q_{c_i}=\frac{g_{c_i}}{g_c} \]

— dimensionless radiant-flux criterion.

\[ X=\frac{x}{R} \]

— dimensionless-coordinate criterion.

1. See [3], [4], [11], [12]. ↩

2. On “thin” bodies see [7], [8]. ↩