HEATING OF BODIES UNDER THE ACTION OF RADIANT HEAT

(Second communication)

G. P. BOIKOV

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

A method is set forth for zonal (in time) calculation of the temperature field in the propagation of heat in one dimension, with allowance for the variability of the thermal characteristics of the substance.

Computational dependencies and relations are given.

In the first communication on this same question (see the first communication), relations were given that make it possible to calculate the temperature field in the heating of bodies of classical form. In these relations it was assumed that the thermal characteristics of the substance \(\lambda\), \(c\), \(\varepsilon\) do not depend on temperature and remain unchanged throughout the entire heating process. Such an assumption considerably lowers the practical value of the results obtained, despite their mathematical rigor, since in reality the thermal characteristics of the substance \(\lambda\) (thermal conductivity), \(c\) (heat capacity), \(\varepsilon\) (degree of blackness of the surface) do not remain constant in the process of heating a body, but change in some way with the change in the temperature of the latter. Therefore, in the proposed work an attempt is made to give a method for calculating the temperature field that makes it possible, to a certain extent, to take into account the variability of the thermal characteristics of the substance as functions of the temperature of the body.

Let us have an infinite plate of thickness \(2R\) at temperature \(T_0\). Suppose also that the laws are specified:

\[ \lambda = f_{\lambda}(t) \qquad (a) \]

\[ c = f_c(t) \qquad (б) \]

\[ \varepsilon = f_{\varepsilon}(t) \qquad (в) \]

At the initial instant of time the plate is placed between two infinite parallel heat sources, whose temperature changes according to the law

\[ T_c = f_c(\tau) \qquad (г) \]

Let us find the temperature distribution over the thickness of the plate at various instants of time. We first determine the temperature distribution after the lapse of a sufficiently small interval of time \(\tau_1\) from the start of heating; to do this, we preliminarily find:

\[ \lambda_1 = f_{\lambda}(t_0) \]

\[ c_1 = f_c(t_0) \]

\[ \left. \begin{array}{l} \lambda_1 = f_{\lambda}(t_0)\\[4pt] c_1 = f_c(t_0) \end{array} \right\} \qquad a_1 = f_a(\lambda_1, c_1) \]

\[ \begin{aligned} \varepsilon_1 &= f_{\varepsilon}(t_0),\\ \varepsilon_2 &= f_{\varepsilon}(t_{c_1}) \end{aligned} \qquad\} \qquad \varepsilon_{n_1}=f_{\varepsilon n}(\varepsilon_1,\varepsilon_2) \]

\[ T_{c_1}=f_c(0), \]

which we regard as unchanged over this first interval of time, after which we solve the system of equations:

\[ \frac{\partial T(x,\tau)}{\partial \tau} = a_1\frac{\partial^2 T(x,\tau)}{\partial x^2}; \tag{1} \]

\[ T(x,0)=T_0; \tag{2} \]

\[ \frac{\partial T(R,\tau)}{\partial x} = \frac{\varepsilon_{n_1}\cdot C_0}{\lambda_1} \left[ \left(\frac{T_{c_1}}{100}\right)^4 - \left(\frac{T_0}{100}\right)^4 \right] = \frac{g_{c_1}}{\lambda_1}; \tag{3} \]

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

The solution of the system (1)—(4) is available in the literature (see [5], [6]) in the form:

\[ T(x,\tau)=T_0+\frac{g_{c_1}}{\lambda_1} \left[ \frac{a_1\tau}{R} - \frac{R^2-3x^2}{6R} + R\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) e^{-\mu_n^2\frac{a_1\tau}{R^2}} \right]. \]

The temperature distribution after the first time interval from the start of heating will be

\[ T_1(x)=T_0+\frac{g_{c_1}}{\lambda_1} \left[ \frac{a_1\tau_1}{R} - \frac{R^2-3x^2}{6R} + R\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) \cdot e^{-\mu_n^2\frac{a_1\tau_1}{R^2}} \right]. \tag{D} \]

Using expression (D), we find the temperature on the surface of the plate

\[ T_1(R)=T_0+\frac{g_{c_1}}{\lambda_1} \left[ \frac{a_1\tau_1}{R} + \frac{1}{3}R - R\sum_{n=1}^{\infty}\frac{2}{\mu_n^2} e^{-\mu_n^2\frac{a_1\tau_1}{R^2}} \right] \]

and the mean temperature of the entire mass of the plate

\[ t_1^{\mathrm{cp}} = \frac{1}{R}\int_{0}^{R} t_1(x)\,dx = t_0+\frac{a_1\tau_1}{\lambda_1 R}\cdot g_{c_1}, \]

established after the expiration of the first time interval \(\tau_1\) from the beginning of heating.

Further, using relations \((a)\), \((б)\), \((в)\), \((г)\), we find

\[ \lambda_2=f_\lambda(t_1^{\mathrm{cp}}) \qquad \varepsilon_1=f_\varepsilon[t_1(R)]; \]

\[ c_2=f_c(t_1^{\mathrm{cp}}) \qquad \varepsilon_2=f_2(t_{c_2}); \]

\[ a_2=f_a(\lambda_2;c_2) \qquad \varepsilon_{n_2}=f_{\varepsilon n}(\varepsilon_1;\varepsilon_2); \]

\[ T_{c_2}=f_c(\tau_1), \]

which we consider unchanged over the course of a second time interval of the same length \(\tau_1\).

We now determine the temperature field in the plate after the expiration of the second time interval \(\tau_1\), for which purpose we solve the system of equations\(^1\):

\[ \frac{\partial T(x,\tau)}{\partial \tau} = a_2 \cdot \frac{\partial^2 T(x,\tau)}{\partial x^2}; \tag{5} \]

\[ T(x,0)=T_0+\frac{g_{c_1}}{\lambda_1} \left[ \frac{a_1\tau_1}{R} - \frac{R^2-3x^2}{6R} + R\sum_{n=1}^{\infty}(-1)^{n+1} \cdot \frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R}\right) e^{-\mu_n^2\frac{a_1\tau_1}{R^2}} \right]; \tag{6} \]

\[ \frac{\partial T(R,\tau)}{\partial x} = \frac{\varepsilon_{n_2} C_0}{\lambda_2} \left[ \left(\frac{T_{c_2}}{100}\right)^4 - \left(\frac{T_1(R)}{100}\right)^4 \right] = \frac{g_{c_2}}{\lambda_2}; \tag{7} \]

\[ \frac{\partial T(0,\tau)}{\partial x}=0. \tag{8} \]

Applying a Laplace transform of the form

\[ F(s)=\int_{0}^{\infty} T(x,\tau)e^{-s\tau}\,d\tau \]

to the differential equation (5) and taking (6) into account, we obtain:

\[ T''(x,s) - \frac{s}{a_2}\cdot T(x,s) + \frac{A}{a_2} + \frac{Bx^2}{a_2} + \frac{C}{a_2}\cdot \cos\left(\mu_n\frac{x}{R}\right) =0, \tag{9} \]

where

\[ A=T_0+\frac{g_{c_1}}{\lambda_1}\cdot\frac{a_1\tau_1}{R} - \frac{g_{c_1}\cdot R^2}{6\lambda_1\cdot R}; \]

\[ B=\frac{3g_{c_1}}{6\lambda_1\cdot R}; \]

\[ C=\frac{g_{c_1}\cdot R}{\lambda_1}\cdot \sum_{n=1}^{\infty} (-1)^{n+1}\frac{2}{\mu_n^2} \cdot e^{-\mu_n^2\frac{a_1\tau_1}{R^2}}. \]

Applying a Laplace transform of the form

\[ \Phi(S)=\int_{0}^{\infty}T(x,s)\cdot e^{-Sx}\,dx \]

\(^1\) See [1], [5], [6], [12].

to expression (9), after a number of algebraic transformations, we shall have:

\[ \begin{aligned} \Phi(S,s)={}& -\frac{S\cdot T(0,s)} {\left[S^2-\left(\sqrt{\frac{s}{a_2}}\right)^2\right]} +\frac{T'(0,s)}{\sqrt{\frac{s}{a_2}}}\cdot \frac{\sqrt{\frac{s}{a_2}}} {\left[S^2-\left(\sqrt{\frac{s}{a_2}}\right)^2\right]} \\[4pt] &-\frac{A}{a_2}\cdot \frac{1} {S\left[S^2+\left(\sqrt{\frac{s}{a_2}}\right)^2\right]} -\frac{C}{a_2}\cdot \frac{S} {\left[S^2+\left(\frac{\mu_n}{R}\right)^2\right]\cdot \left[S^2+\left(\sqrt{\frac{s}{a_2}}\right)^2\right]} \\[4pt] &-\frac{B}{a_2}\cdot\frac{\Gamma(3)}{S^3}\cdot \frac{1}{\sqrt{\frac{s}{a_2}}}\cdot \frac{\sqrt{\frac{s}{a_2}}} {\left[S^2-\left(\sqrt{\frac{s}{a_2}}\right)^2\right]} . \end{aligned} \tag{10} \]

Applying to expression (10) the Riemann–Mellin inversion formula of the form:

\[ f(x)=L^{-1}\left[F(S)\right] =\frac{1}{2\pi i}\cdot \int_{\sigma-i\infty}^{\sigma+i\infty} e^{Sx}\cdot F(S)\cdot dS, \]

we find

\[ T(x,s)=A_2\cdot Ch\left(\sqrt{\frac{s}{a_2}}\,x\right) +B_2\cdot Sh\left(\sqrt{\frac{s}{a_2}}\,x\right) +\frac{A}{s}+\frac{Bx^2}{s} + \]

\[ +\frac{2a_2\cdot B}{s^2} +\frac{C}{a_2}\cdot \frac{\cos\left(\mu_n\frac{x}{R}\right)} {\frac{s}{a_2}+\left(\frac{\mu_n}{R}\right)^2}, \]

where \(A_2\) and \(B_2\) are constants determined from conditions (8) and (7). By the symmetry condition (8), \(B_2=0\).

From the boundary condition (7) we find:

\[ A_2= \frac{g_{c_2}-2BR\lambda_2} {\lambda_2\cdot s\sqrt{\frac{s}{a_2}}\cdot sh\left(\sqrt{\frac{s}{a_2}}\cdot R\right)} . \]

The solution for the transform is now represented as

\[ T(x,s)=\frac{A}{s}+\frac{Bx^2}{s} +\frac{2a_2\cdot B}{s^2} +\frac{C}{a_2}\cdot \frac{\cos\left(\mu_n\frac{x}{R}\right)} {\frac{s}{a_2}+\left(\frac{\mu_n}{R}\right)^2} + \]

\[ +\frac{\left(g_{c_2}-2\lambda_2BR\right)\cdot \operatorname{Ch}\left(\sqrt{\frac{s}{a_2}}\cdot x\right)} {\lambda_2\cdot s\cdot \sqrt{\frac{s}{a_2}}\cdot \operatorname{Sh}\left(\sqrt{\frac{s}{a_2}}\cdot R\right)}; \tag{11} \]

Applying the inverse transformation to image (11) and replacing \(A, B, C\) by their values, we find the original in the form:

\[ T(x,\tau)=T_0+\frac{g_{c_1}}{\lambda_1} \left[ \frac{a_1\tau_1}{R}-\frac{R^2-3x^2}{6R}+\frac{a_2\tau}{R} + R\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2}\cdot \cos\left(\mu_n\frac{x}{R}\right)\cdot e^{-\mu_n^2\frac{a_1\tau_1}{R^2}}\cdot e^{-\mu_n^2\frac{a_2\tau}{R^2}} \right]+ \]

\[ +\left(\frac{g_{c_2}}{\lambda_2}-\frac{g_{c_1}}{\lambda_1}\right)\cdot \left[ \frac{a_2\tau}{R}-\frac{R^2-3x^2}{6R} + R\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2}\cdot \cos\left(\mu_n\frac{x}{R}\right)\cdot e^{-\mu_n^2\frac{a_2\tau}{R^2}} \right]. \tag{12} \]

Substituting instead of \(\tau\) the value \(\tau_1\) after the transformations, we obtain the temperature distribution at the expiration of the second interval of time from the start of heating, likewise equal to \(\tau_1\):

\[ T_2(x)=T_0+\frac{a_1\tau_1}{R}\cdot \left(\frac{g_{c_1}}{\lambda_1}+\frac{g_{c_2}}{\lambda_1}\cdot\frac{c_1}{c_2}\right) -\frac{g_{c_2}}{\lambda_2}\cdot\frac{R^2-3x^2}{6R}+ \]

\[ +R\sum_{n=1}^{\infty}(-1)^{n+1}\cdot\frac{2}{\mu_n^2}\cdot \operatorname{Cos}\left(\mu_n\frac{x}{R}\right) \left[ \frac{g_{c_1}}{\lambda_1}\cdot e^{-\mu_n^2\left(\frac{a_1\tau_1}{R^2}+\frac{a_2\tau_1}{R^2}\right)} - \left(\frac{g_{c_1}}{\lambda_1}-\frac{g_{c_2}}{\lambda_2}\right)\cdot e^{-\mu_n^2\frac{a_2\tau_1}{R^2}} \right]. \tag{13} \]

Using expression (13), we find the temperature on the surface of the plate

\[ T_2(R)=T_0+\frac{a_1\tau_1}{R} \left(\frac{g_{c_2}}{\lambda_1}+\frac{g_{c_2}}{\lambda_1}\cdot\frac{c_1}{c_2}\right) +\frac{1}{3}R\frac{g_{c_2}}{\lambda_2} -R\sum_{n=1}^{\infty}\frac{2}{\mu_n^2}\cdot \]

\[ \cdot \left[ \frac{g_{c_1}}{\lambda_1}\cdot e^{-\mu_n^2\left(\frac{a_1\tau_1}{R^2}+\frac{a_2\tau_1}{R^2}\right)} - \left(\frac{g_{c_1}}{\lambda_1}-\frac{g_{c_2}}{\lambda_2}\right)\cdot e^{-\mu_n^2\frac{a_2\tau_1}{R^2}} \right] \]

and the average temperature of the entire mass of the plate

\[ t_2^{\mathrm{av}}=\frac{1}{R}\int_{0}^{R}t_2(x)\,dx = t_0+\frac{a_1\tau_1}{\lambda_1\cdot R} \left(g_{c_1}+g_{c_2}\cdot\frac{c_1}{c_2}\right), \]

which will be established (from the start of heating) at the expiration of the second time interval, equal to \(\tau_1\).

Next, using relations \((a)\), \((b)\), \((c)\), \((d)\), we find \(\lambda_3;\ c_3;\ a_3;\ T_{c_3};\ \varepsilon_{n_3}\), which we regard as constant over the course of a third identical

time interval \(\tau_1\). Then, solving again the corresponding system of differential equations, we determine \(T_3(x)^*\), etc.

Carrying out analogous reasoning for \(T_4(x)\), \(T_5(x)\), we find that the temperature distribution after the expiration of any \((m\)-th) instant of time is represented by the relation:

\[ \begin{aligned} \frac{T_m(x)}{T_c} &= \frac{T_0}{T_c} + \frac{g_c \cdot R}{\lambda_1 \cdot T_c} \cdot \Biggl\{ \frac{a_1\tau_1}{R^2} \sum_{i=1}^{i=m} \frac{g_{c_i}}{g_c} \cdot \frac{c_1}{c_i} - \frac{g_{c_m}}{g_c} \cdot \frac{\lambda_1}{\lambda_m} \\ &\qquad\cdot \frac{R^2-3x^2}{6R^2} + \sum_{n=1}^{\infty} (-1)^{n+1} \cdot \frac{2}{\mu_n^2} \cdot \cos\!\left(\mu_n\frac{x}{R}\right) \cdot \Biggl[ \frac{g_{c_1}}{g_c} \cdot e^{-\mu_n^2\sum_{i=1}^{i=m}\frac{a_i\tau_i}{R^2}} \\ &\qquad\qquad - \left( \frac{g_{c_1}}{g_c} - \frac{g_{c_2}}{g_c} \cdot \frac{\lambda_1}{\lambda_2} \right) \cdot e^{-\mu_n^2\sum_{i=2}^{i=m}\frac{a_i\tau_i}{R^2}} - \left( \frac{g_{c_2}}{g_c} \cdot \frac{\lambda_1}{\lambda_2} - \right. \\ &\qquad\qquad\left. - \frac{g_{c_3}}{g_c} \cdot \frac{\lambda_1}{\lambda_3} \right) \cdot e^{-\mu_n^2\sum_{i=3}^{i=m}\frac{a_i\tau_i}{R^2}} -\ldots \\ &\qquad\qquad\left. -\left( \frac{g_{c(m-1)}}{g_c} \cdot \frac{\lambda_1}{\lambda_{(m-1)}} - \frac{g_{c_m}\cdot\lambda_1}{g_c\cdot\lambda_m} \right) \cdot e^{-\mu_n^2\frac{a_m\tau_1}{R^2}} \right] \Biggr\}. \end{aligned} \tag{14} \]

Here \(T_c\) and \(g_c\) are taken at \(\tau=0\).

In criterion form** expression (14) is written as

\[ \begin{aligned} \Theta_m &= \Theta_0 + K_i \Biggl\{ F_{0i}\cdot \sum_{i=1}^{i=m} Q_{c_i}\cdot C_i - Q_{c_m}\cdot\Lambda_m \cdot \frac{1}{6}(1-3X^2) \\ &\qquad + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2}{\mu_n^2} \cdot \cos(\mu_n X) \cdot \Biggl[ Q_{c_1}\cdot e^{-\mu_n^2\sum_{i=1}^{i=m}Fo_i} - (Q_{c_1} \\ &\qquad\qquad - Q_{c_2}\cdot\Lambda_2) \cdot e^{-\mu_n^2\sum_{i=2}^{i=m}Fo_i} - (Q_{c_2}\cdot\Lambda_2 - Q_{c_3}\cdot\Lambda_3) \\ &\qquad\qquad\cdot e^{-\mu_n^2\sum_{i=3}^{i=m}Fo_i} -\ldots - (Q_{c(m-1)}\cdot\Lambda_{(m-1)} - Q_{c_m}\cdot\Lambda_m) e^{-\mu_n^2 Fo_m} \Biggr] \Biggr\}. \end{aligned} \tag{15} \]

It should be noted that the series \(\sum_{n=1}^{\infty}\) in expressions (14) and (15) (the third term in the braces) converges rapidly. Therefore, starting from some instant of time \((m>1)\), it may be neglected, and the subsequent calculation carried out taking into account only the first two terms in the braces, which considerably simplifies it.

Remark. If in expressions (14) and (15) one sets \(\lambda_1=\lambda_2=\ldots=\lambda_m\) and \(c_1=c_2=\ldots=c_m\), then the latter assume the form of the computational formulas for the case of constant thermal characteristics (see the first communication).

* See also [9], [10].
** See [4], [11], [12].

If the plate being heated belongs to the class of “thin” bodies* \((K_i < 0.12)\), having an insignificant temperature difference between the surface and the middle of the body, then it is more expedient to carry out the calculations in terms of the mean temperature of the body, determined from the relation

\[ \frac{T_m^{\mathrm{cp}}}{T_c} = \frac{1}{R}\int_{0}^{R}\frac{T_m(x)}{T_c}\,dx = \frac{T_0}{T_c} + \frac{g_c R}{\lambda_1 T_c}\cdot \frac{a_1\tau_1}{R^2} \sum_{i=1}^{i=m}\frac{g_{c_i}}{g_c}\cdot \frac{c_1}{c_i}; \tag{16} \]

or, in criterion form,

\[ \Theta_m^{\mathrm{cp}} = \Theta_0 + K_i F_{o_1}\sum_{i=1}^{i=m} Q_{c_i} C_i; \tag{17} \]

The specific heat consumption for heating the plate may be determined by the formula

\[ \Delta g_s^m = \tau_1 \sum_{i=1}^{i=m} g_{c_i}. \tag{18} \]

For the correct choice of the computational time interval \(\tau_1\) (see the first communication), for example for a plate, the following sequence must be carried out:

1. Preliminary determination of the computational time interval from the approximate dependence

\[ \tau_1 = \frac{0.03\,\lambda_1 T_c R}{g_c a_1}. \tag{19} \]

1. Verification of the computational time interval found from relation (19) by estimating the approximation expressed by the inequality

\[ \frac{g_{c\,\mathrm{ist}}-g_{c_2}}{g_{c_1}} = \varphi_m < 1 - \frac{a_2}{a_1} \left\{ 1 - \frac{ 3\left(g_{c_1}-g_{c_2}\frac{\lambda_1}{\lambda_2}\right) }{ g_{c_1}\left[1+2\sum_{n=1}^{\infty} e^{-\mu_n^{2}\frac{a_1\tau_1}{R^2}}\right] } \right\}. \tag{20} \]

Here \(\varphi_m\) is the actual error of the calculation.

It is possible that the numerical result of the right-hand side of inequality (20) may turn out to be a negative number (a particular case depending on the laws of the substance \(\lambda=f(t)\), \(c=\psi(t)\), and also on the criterion

\[ F_{o_1}=\frac{a_1\tau_1}{R^2} \]).
]

In this case the minus sign may be disregarded, i.e., the value obtained may be considered positive, and by the quantity \(\varphi_m\) on the left-hand side of the inequality one should understand the ratio

\[ \varphi_m=\frac{g_{c_2}-g_{c\,\mathrm{ist}}}{g_{c_1}}. \]

Expressions analogous to relations (14)—(20) for a cylinder have the form:

\[ \frac{T_m(r)}{T_c} = \frac{T_0}{T_c} + \frac{g_c R}{\lambda_1 T_c} \left\{ 2\frac{a_1\tau_1}{R^2} \sum_{i=1}^{i=m}\frac{g_{c_i}}{g_c}\cdot \frac{c_1}{c_i} - \frac{1}{4}\cdot \frac{g_{cm}}{g_c}\cdot \frac{\lambda_1}{\lambda_m} \cdot \right. \]

* On “thin” bodies see [3], [7].

\[ \cdot \left(1-2\frac{r^{2}}{R^{2}}\right) -\sum_{n=1}^{\infty} \frac{2}{\mu_n^{2}J_{0}(\mu_n)} \cdot J_{0}\left(\mu_n\frac{r}{R}\right) \cdot \left\{ \frac{g_{c_1}}{g_c}\, e^{-\mu_n^{2}\sum_{i=1}^{i=m}\frac{a_i\tau_i}{R^{2}}} -\right. \]

\[ \left. -\sum_{i=1}^{i=m-1} \left( \frac{g_{c_i}\lambda_1}{g_c\lambda_i} - \frac{g_{c(i+1)}\lambda_1}{g_c\lambda_{(i+1)}} \right) \cdot e^{-\mu_n^{2}\sum_{k=1}^{k=m-i}\frac{a_{(i+k)}\tau_1}{R^{2}}} \right\} \right]; \tag{14'} \]

\[ \Theta_m=\Theta_0+K_i \left\{ 2F_{o_1}\sum_{i=1}^{i=m}Q_{c_i}C_i -\frac{1}{4}Q_{c_m}\Lambda_m\cdot(1-2r^2) -\right. \]

\[ \left. -\sum_{n=1}^{\infty} \frac{2}{\mu_n^{2}J_{0}(\mu_n)} \cdot J_0(\mu_n\cdot r)\cdot \left[ Q_{c_1}e^{-\mu_n^{2}\sum_{i=1}^{i=m}F_{o_i}} -\right. \]

\[ \left. -\sum_{i=1}^{i=m-1} \left( Q_{c_i}\Lambda_i-Q_{c(i+1)}\cdot\Lambda_{(i+1)} \right) \cdot e^{-\mu_n^{2}\sum_{k=1}^{k=m-i}F_{o(i+k)}} \right] \right\}; \tag{15'} \]

\[ \frac{T_m^{cp}}{T_c} = \frac{2}{R^2} \int_{0}^{R} r\cdot\frac{T_m(r)}{T_c}\cdot dr = \]

\[ = \frac{T_0}{T_c} + 2\,\frac{g_cR}{\lambda_1T_c} \cdot \frac{a_1\tau_1}{R^2} \cdot \sum_{i=1}^{i=m} \frac{g_{c_i}}{g_c}\cdot\frac{c_1}{c_i}; \tag{16} \]

\[ \Theta_m^{cp} = \Theta_0 + 2K_iF_{o_1} \sum_{i=1}^{i=m} Q_{c_i}C_i; \tag{17'} \]

\[ \Delta g_v^m = \frac{2}{R}\cdot\tau_1\cdot \sum_{i=1}^{i=m}g_{c_i}; \tag{18'} \]

\[ \tau_1 = \frac{0{,}025\lambda_1T_cR}{g_ca_1}; \tag{19'} \]

\[ \varphi_m \leq 1-\frac{a_2}{a_1}\left\{1- \frac{2\left(g_{c_1}-g_{c_2}\frac{\lambda_1}{\lambda_2}\right)} {g_{c_1}\cdot\left[1+\sum_{n=1}^{\infty} e^{-\mu_n^2\frac{a_1\tau_1}{R^2}}\right]} \right\}. \tag{20'} \]

Analogous expressions can also be written for a sphere.

Figures 1, 2, 3, and 4 show the heating of a cylinder under various conditions. The dependences \(\lambda=f_\lambda(t)\) and \(c=f_c(t)\) are taken from the work of A. L. Nemchinskii [7]. The experimental data are taken according to [2], [7], [8].

Fig. 1. Heating of a cylinder made of ball-bearing steel
\(d = 125\) mm

— · — · — calculated surface-temperature data by formula No. \(14'\), taking account of the variability of the thermal constants of the material, for \(\varepsilon_n \cdot C_0 = 4\).

— — — calculated surface-temperature data by formula No. \(14'\), without taking account of the variability of the thermal constants of the material, for \(\varepsilon_n \cdot C_0 = 4\).

——— calculated surface-temperature data according to Russell’s tables for \(a_{\mathrm{izl}} = 230\).

· · · · · experimental surface-temperature curve according to the data of N. N. Dobrokhotov.

Fig. 2. Heating of a cylinder made of carbon steel
\((0.4\%\,C)\)
\(d = 25\) mm

— · — · — calculated data for the mean temperature of the whole mass of the body by formula No. \(16'\), taking account of the variability of the thermal constants of the material, for \(\varepsilon_n \cdot C_0 = 3.8\).

— — — calculated data for the mean temperature of the whole mass of the body by formula No. \(16'\), without taking account of the variability of the thermal constants of the material, for \(\varepsilon_n \cdot C_0 = 3.8\).

——— calculated data for the mean temperature of the whole mass of the body by A. V. Lykov’s convective formula, for \(a_{\mathrm{izl}} = 210\).

· · · · · experimental curve of the mean temperature of the whole mass of the body according to the data of V. N. Sokolov.

Fig. 3. Heating of a cylinder of 9Kh steel, \(d = 300\) mm

— · — · — calculated data for the temperature at the surface by formula No. \(14'\), taking into account the variability of the thermal constants of the substance, for \(\varepsilon_n \cdot C_0 = 4\).

— · — — calculated data for the temperature at the surface by formula No. \(14'\), without taking into account the variability of the thermal constants of the substance, for \(\varepsilon_n \cdot C_0 = 4\).

———— calculated data for the temperature at the surface according to the charts of D. V. Budrin for \(\alpha_{\mathrm{rad}} = 220\).

· · · · · experimental curve of the temperature at the surface of the cylinder, according to the data of V. M. Degtyarev.

Fig. 4. Heating of a cylinder of 9Kh steel, \(d = 300\) mm

— · — · — calculated data for the temperature at the center by formula No. \(14'\), taking into account the variability of the thermal constants of the substance, for \(\varepsilon_n \cdot C_0 = 4\).

— · — — calculated data for the temperature at the center by formula No. \(14'\), without taking into account the variability of the thermal constants of the substance, for \(\varepsilon_n \cdot C_0 = 4\).

———— calculated data for the temperature of the center according to the formulas and tables of Heiligenstaedt for \(\alpha_{\mathrm{rad}} = 225\).

· · · · · experimental curve of the temperature of the center, according to the data of V. M. Degtyarev.

References

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3. Ivantsov G. P. — Heating of Metals, Metallurgizdat, Moscow, 1948.

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