Heating of Bodies of Finite Dimensions under the Action of Radiant Heat

(First Report)

G. P. Boikov

Presented by Professor G. I. Fuks.

The problem of determining the heating of bodies of finite dimensions has long attracted, and continues to attract, the attention of many scholars. The latter is explained by the fact that the solution of the differential equation of heat conduction of the form

\[ \frac{\partial T}{\partial \tau} = a\left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) \tag{1} \]

is of great fundamental interest, while the ability to calculate the heating of bodies of finite dimensions is an immediate practical requirement.

At the present time there are a number of methods which, despite certain shortcomings, indicate ways of solving the spatial problem and make it possible to find the temperature field in one or another approximation. One such method is the grid method or, equivalently, the method of numerical integration of the differential equation of heat conduction [1]. The application of this method, with a sufficiently large volume of numerical operations, makes it possible to obtain sufficiently accurate calculation results [2].

Of theoretical interest is the solution of the differential equation of heat conduction for a three-dimensional field, obtained by N. Yu. Taits [3], who gave it for a parallelepiped in the form:

\[ t(x,y,z,\tau) = \varphi(\tau) + \frac{1}{BSL} \cdot \sum_{l=1}^{\infty} \cdot \sum_{m=1}^{\infty} \cdot \sum_{n=1}^{\infty} \cos\left(\frac{2l-1}{2s}\pi x\right) \times \]

\[ \times \cos\left(\frac{2m-1}{2B}\pi y\right) \cdot \cos\left(\frac{2n-1}{2L}\pi z\right) \cdot \int_{-S}^{+S} \int_{-B}^{+B} \int_{-L}^{+L} \left[F(x,y,z)-\varphi(0)\right] \times \]

\[ \times \cos\left(\frac{2l-1}{2s}\pi x\right) \cdot \cos\left(\frac{2m-1}{2B}\pi y\right) \cdot \cos\left(\frac{2n-1}{2L}\pi z\right) \,dx\cdot dy\cdot dz \times \]

\[ \times e^{ -\left[ \left(\frac{2l-1}{2S}\right)^2 + \left(\frac{2m-1}{2B}\right)^2 + \left(\frac{2n-1}{2L}\right)^2 \right]\pi^2 a\tau } \]

\[ - \frac{64}{\pi^3} \cdot \sum_{l=1}^{\infty} \cdot \sum_{m=1}^{\infty} \cdot \sum_{n=1}^{\infty} \frac{(-1)^{l+m+n+1}} {(2l-1)(2m-1)(2n-1)} \cdot \cos\left(\frac{2l-1}{2s}\pi x\right) \times \]

\[ {}\times \cos\left(\frac{2m-1}{2B}\,\pi y\right)\cdot \cos\left(\frac{2n-1}{2L}\,\pi z\right)\cdot e^{-\left[\left(\frac{2l-1}{2S}\right)^2+\left(\frac{2m-1}{2B}\right)^2+\left(\frac{2n-1}{2L}\right)^2\right]\pi^2 a\tau} {}\times \]

\[ {}\times \int_0^\tau e^{\left[\left(\frac{2l-1}{2S}\right)^2+\left(\frac{2m-1}{2B}\right)^2+\left(\frac{2n-1}{2L}\right)^2\right]\pi^2 a\tau} \cdot \varphi'(\tau)\cdot d\tau . \]

Here \(2s;\ 2B;\ 2L\) are the sides of the parallelepiped; \(\varphi(\tau)\) is the law of variation of the temperature on the surface; \(F(x,y,z)\) is the initial temperature distribution; \(l,m,n\) are characteristic numbers. The solution presented gives, in general form, the temperature field of a parallelepiped at any instant of time for boundary conditions of the first kind. To obtain temperature values using this relation it is necessary to know the law \(t_n=\varphi(\tau)\) and to carry out a series of integrations, which may be more or less difficult depending on the form of the function \(\varphi(\tau)\).

It should be noted that substantiated solutions for two- and three-dimensional unsteady fields were proposed as early as 1919 by Williamson and Adams [4] for the case of an infinitely large heat-transfer coefficient. Later, in 1932, for short cylinders under boundary conditions of the third kind, a similar problem was investigated by Goldstein [5].

It turns out that short cylinders, prisms of rectangular cross section, or parallelepipeds can be regarded as bodies formed by the mutual intersection of perpendicular bodies of classical form. Thus, a short cylinder may be formed by the mutual perpendicular intersection of an infinite plate and an infinite cylinder. A parallelepiped is formed by the intersection of three infinite plates, and so forth. The temperature criterion at any point of such short bodies can be found by multiplying the temperature criteria for the corresponding points in bodies of infinite extent whose intersection forms the given body. Thus, for example, for a parallelepiped and a short cylinder the temperature criterion may be calculated according to the expressions:

\[ \Theta_{\text{parall.}}=\Theta_{R_1}\cdot \Theta_{R_2}\cdot \Theta_{R_3} \tag{2} \]

\[ \Theta_{\text{cyl.}}=\Theta_r\cdot \Theta_R \tag{3} \]

where \(\Theta_{R_1};\ \Theta_{R_2};\ \Theta_{R_3};\ \Theta_R\) are known solutions in criteria for infinite plates of thickness \(2R_1;\ 2R_2;\ 2R_3;\ 2R\); \(\Theta_r\) is the known solution for an infinite cylinder of diameter \(2r\). The validity of expressions (2) and (3) was confirmed by experiments of D. V. Budrin and B. A. Krasovskii [6].

If in relations (2) and (3) one substitutes their values for \(\Theta\), then the latter are rewritten, for the case of heating, in the form:

\[ \frac{T_c-T(x,y,z,\tau)}{T_c-T_0} = \frac{T_c-T(x,\tau)}{T_c-T_0}\cdot \frac{T_c-T(y,\tau)}{T_c-T_0}\cdot \frac{T_c-T(z,\tau)}{T_c-T_0}, \tag{2′} \]

\[ \frac{T_c-T(r,z,\tau)}{T_c-T_0} = \frac{T_c-T(r,\tau)}{T_c-T_0}\cdot \frac{T_c-T(z,\tau)}{T_c-T_0}, \tag{3′} \]

which corresponds both to boundary conditions of the first kind and to boundary conditions of the third kind [7]. Here \(T(x,\tau)\), \(T(y,\tau)\), \(T(z,\tau)\), \(T(r,\tau)\) are known solutions for bodies of classical form [7]. Using relations (2′) and (3′), one can find the temperature field in a parallelepiped and a short cylinder for the indicated boundary conditions.

One of the interesting methods for determining the temperature field in a number of bodies of finite dimensions is the method of A. I. Veinik [7]. A. I. Veinik proposed calculation relations for such bodies in the form of known solutions for boundary conditions of the third kind, into the structure of which special coefficients are introduced that take into account the complexity of the configuration of the body under investigation.

For boundary conditions of the second kind under a constant radiant flux, calculation relations were given by G. P. Ivantsov [8] and N. Yu. Taits [9]. Methods for calculating the heating of a square ingot were developed by Semikin [10] and Linchevsky [11]. The theory for solving problems of two- and three-dimensional fields under boundary conditions of the first kind is also set forth in the book by H. S. Carslaw [13].

II.

Analyzing the methods for solving the spatial problem, one may arrive at the following conclusion:

1. Obtaining a solution of the differential equation of heat conduction for various boundary conditions presents great difficulties.

2. A rigorous solution of the problem on the basis of the available methods of mathematical physics gives a very complicated final result and requires knowledge of a number of additional factors (for example, the law of variation of temperature on the surface, etc.).

3. Among the various conditions of heating of bodies when heat propagates in more than one dimension, the least studied question is that connected with calculating the temperature field under boundary conditions of the second kind, expressed by the Stefan–Boltzmann law.

All this makes it possible to express the opinion that it is not always expedient to try to solve the spatial problem purely mathematically in the form in which it is described by differential equation (1). In a number of cases it is more correct to solve this problem technically, attempting in some way to take into account the influence of the second and third dimensions. Therefore we decided to abandon the search for a purely mathematical solution of condition (1), and considered it possible to try to find a solution which, without differing in complexity from the one-dimensional problem [12], could give the temperature distribution in bodies of finite dimensions when they are heated under the action of radiant heat. In doing so, we first of all set as our goal the determination of only a part of the temperature field in the body—for the time being, only that portion which coincides with the line of heat flow in the body. By the line of heat flow in a body is meant the direction from the center of the body to that point of the surface whose temperature remains the lowest over the entire course of the heating process.

The essence of the proposed calculation method consists in the fact that the component of the divergence of the temperature gradient along the line of heat flow is determined as the quotient obtained by dividing the divergence of the temperature gradient by a certain constant coefficient, i.e., the existence of a relation of the following form is assumed:

\[ \frac{\partial^2 T}{\partial x^2} = \frac{\operatorname{div}(\operatorname{grad}\cdot T)}{\xi} \tag{4} \]

Then the divergence of the temperature gradient in formula (1) may be replaced in the form

\[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = \xi \cdot \frac{\partial^2 T}{\partial x^2}. \tag{5} \]

and the original differential equation will take the form

\[ \frac{\partial T}{\partial \tau}=\xi a\cdot\frac{\partial^2 T}{\partial x^2}. \tag{6} \]

That expressions (4), (5), (6) exist can be verified by considering the passage of heat through an elementary volume \(dV=dx\,dy\,dz\) during the time differential \(d\tau\).

Fig. 1.

According to Fourier’s law, the amount of heat passing during time \(d\tau\) in the direction of the \(x\)-axis through the face \(ABCD\) (Fig. 1) is equal to

\[ Q'_x=-\lambda\frac{\partial T}{\partial x}\cdot dy\cdot dz\cdot d\tau, \tag{7} \]

and through the face \(EFGH\), which has the temperature \(T+\dfrac{\partial T}{\partial x}\cdot dx\), during the same time is equal to

\[ Q''_x=-\lambda\,\frac{\partial}{\partial x}\left(T+\frac{\partial T}{\partial x}\cdot dx\right)\cdot dy\cdot dz\cdot d\tau. \tag{8} \]

Then the change of heat in the volume, in connection with heat transfer along the \(x\)-axis, will be

\[ Q_x=Q'_x-Q''_x \tag{a} \]

similarly

\[ Q_y=Q'_y-Q''_y \tag{б} \]

\[ Q_z=Q'_z-Q''_z \tag{c} \]

The total change in the amount of heat in the volume element \(dx\cdot dy\cdot dz\) over the period \(d\tau\) is equal to the sum of expressions (a), (б), (c), namely:

\[ dQ=dQ_x+dQ_y+dQ_z, \]

or, what is the same,

\[ c\gamma\cdot\frac{\partial T}{\partial \tau}\,dx\cdot dy\cdot dz\cdot d\tau = dQ_x\cdot\left(1+\frac{dQ_y}{dQ_x}+\frac{dQ_z}{dQ_x}\right). \]

Since heat transfer is considered over the time differential \(d\tau\), then

\[ \frac{dQ_y}{dQ_x}=\beta=\mathrm{const};\qquad \frac{dQ_z}{dQ_x}=\gamma=\mathrm{const}. \]

Then

\[ c\gamma\,\frac{\partial T}{\partial \tau}\cdot dx\cdot dy\cdot dz\cdot d\tau = dQ_x(1+\beta+\gamma)=dQ_x\cdot\xi. \]

According to (7) and (8):

\[ dQ_x=Q'_x-Q''_x=\lambda \frac{\partial^2 T}{\partial x^2}\cdot dx\cdot dy\cdot dz\cdot d\tau, \]

whence

\[ C'_1\frac{\partial T}{\partial \tau}=\xi\cdot\lambda\frac{\partial^2 T}{\partial x^2}, \]

or, what is the same,

\[ \frac{\partial T}{\partial \tau}=\xi\cdot a\frac{\partial^2 T}{\partial x^2}, \]

i.e., expression (6) has been obtained.

III

Solving equation (6) for boundary conditions expressed by the Stefan–Boltzmann law, we obtain the temperature distribution along the line of heat flow \(x\), but taking into account the propagation of heat also in the other dimensions (for the method of solution see [12]).

For the case of constant thermophysical characteristics, the latter has the form:

\[ \frac{T_m(x)}{T_c}= \frac{T_0}{T_c} + \frac{g_c\cdot R_1}{\lambda\cdot T_c} \left\{ \xi\cdot \frac{a\tau_1}{R_1^2} \sum_{i=1}^{i=m}\frac{g_{ci}}{g_c} - \frac{g_{cm}}{g_c}\cdot \frac{R_1^2-3x^2}{6R_1^2} + \right. \]

\[ \left. + \sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2}\cdot \cos\left(\mu_n\cdot\frac{x}{R_1}\right) \cdot \left[ \frac{g_{c1}}{g_c}\cdot e^{-m\mu_n^2\cdot \xi\cdot \frac{a\tau_1}{R_1^2}} - \sum_{i=1}^{i=m-1} \frac{g_{ci}-g_{c(i+1)}}{g_c}\cdot e^{-(m-i)\mu_n^2\cdot \xi\cdot \frac{a\tau_1}{R_1^2}} \right] \right\}. \tag{9} \]

Here the \(x\)-axis coincides with the line of heat flow in the body and varies from 0 to \(R_1\) \((2R_1\) is the smallest dimension of the parallelepiped).

Integrating expression (9) from 0 to \(R_1\), we obtain the mean value of the temperature along the line of heat flow in the body

\[ \frac{T_m^{cp}}{T_c} = \frac{T_0}{T_c} + \frac{g_c\cdot R_1}{\lambda\cdot T_c}\cdot \xi\cdot \frac{a\tau_1}{R_1^2} \sum_{i=1}^{i=m}\frac{g_{ci}}{g_c}. \tag{10} \]

In this case the calculated time interval is determined from the analytic–empirical relations

\[ \tau_1= \frac{0.03\cdot\lambda\cdot T_c\cdot R_1} {g_c\cdot\xi\cdot a} \tag{11} \]

\[ \varphi_m< \frac{(\xi+2)\cdot(g_{c1}-g_{c2})} {g_{c1}\left[ \xi+2\cdot\sum_{n=1}^{\infty} e^{-\mu_n^2\cdot\xi\cdot\frac{a\tau_1}{R_1^2}} \right]}. \tag{12} \]

For \(\xi=1\), (11) and (12) pass into the corresponding analytic expressions for an infinite plate [12].

Expressions analogous to (9)–(12) for the case when the thermophysical characteristics of the substance vary as a function of temperature can be represented in the form (9′)–(12′).

\[ \frac{T_m(x)}{T_c} = \frac{T_0}{T_c} + \frac{g_c R_1}{\lambda_1 T_c} \left\{ \xi \frac{a_1\tau_1}{R_1^2} \cdot \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c}\cdot\frac{c_1}{c_i} - \frac{g_{cm}}{g_c}\cdot\frac{\lambda_1}{\lambda_m}\cdot \frac{R_1^2-3x^2}{6R_1^2} +\right. \]
\[ \left. +\sum_{n=1}^{\infty}(-1)^{n+1}\frac{2}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R_1}\right) \cdot \left[ \frac{g_{c1}}{g_c}\, e^{-\mu_n^2\xi\cdot \sum_{i=1}^{i=m}\frac{a_i\tau_i}{R_1^2}} \right.\right. \]
\[ \left.\left. - \sum_{i=1}^{i=m-1} \left( \frac{g_{ci}}{g_c}\cdot\frac{\lambda_1}{\lambda_i} - \frac{g_{c(i+1)}}{g_c}\cdot\frac{\lambda_1}{\lambda_{(i+1)}} \right) \cdot e^{-\mu_n^2\xi\cdot \sum_{k=1}^{k=m-i}\frac{a_{(i+k)}\tau}{R_1^2}} \right] \right\}. \tag{9′} \]

\[ \frac{T_m^{\mathrm{cp}}}{T_c} = \frac{T_0}{T_c} + \frac{g_c R_1}{\lambda_1 T_c}\cdot \xi\cdot \frac{a_1\tau_1}{R_1^2} \cdot \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c}\cdot\frac{c_1}{c_i} \tag{10′} \]

\[ \tau_1= \frac{0.03\,\lambda_1 T_c R_1}{g_c\,\xi\,a_1} \tag{11′} \]

\[ \varphi_m < 1-\frac{a_2}{a_1} \cdot \left\{ 1- \frac{ (\xi+2)\left(g_{c1}-g_{c2}\frac{\lambda_1}{\lambda_2}\right) }{ g_{c1} \left[ \xi+2\cdot\sum_{n=1}^{\infty} e^{-\mu_n^2\cdot\xi\cdot\frac{a_1\tau_1}{R_1^2}} \right] } \right\}^{*} \tag{12′} \]

Thus, using expressions (9)–(12), with a known coefficient \(\xi\), one can find the temperature distribution along the line of heat flow in a parallelepiped. An approximate value of the quantity \(\xi\) can be found from the following considerations.

At present, the problem of heating a parallelepiped under boundary conditions of the first and third kinds has been solved. It should be noted that the solution of the problem for a parallelepiped, given in the form of the product of the corresponding temperature criteria (formula (2′)), has been mathematically proved [7] and confirmed by experiments [6]. According to formula (2′),

\[ T(x,y,z,\tau) = T_c - \frac{ [T_c-T(x,\tau)]\,[T_c-T(y,\tau)]\,[T_c-T(z,\tau)] }{ (T_c-T_0)^2 }, \tag{15} \]

where \(T(\varphi,\tau)\), the solutions for unbounded plates, are known for boundary conditions of the first and third kinds [7]. For convenience these solutions may be represented in the following abbreviated notation:

\[ T(x,\tau) = T_c - \sum_{n=1}^{\infty} f_n(\tau)\cdot \cos\left(\mu_n\cdot\frac{x}{R_1}\right) \tag{a} \]

\[ T(y,\tau) = T_c - \sum_{m=1}^{\infty} f_m(\tau)\cdot \cos\left(\mu_m\cdot\frac{y}{R_2}\right) \tag{б} \]

\[ T(z,\tau) = T_c - \sum_{k=1}^{\infty} f_k(\tau)\cdot \cos\left(\mu_k\cdot\frac{z}{R_3}\right) \tag{в} \]

\[ {}^{*}\ \text{See [12].} \]

In this case, for boundary conditions of the 1st and 3rd kinds, the quantities \(f_n(\tau)\) and \(\mu_n\); \(f_m(\tau)\) and \(\mu_m\) will be different.

Differentiating expression (15) twice with respect to \(x\), we obtain:

\[ \frac{\partial^2 T(x,y,z,\tau)}{\partial x^2} = \frac{ \dfrac{\partial^2 T(x,\tau)}{\partial x^2} \,[T_c - T(y,\tau)]\,[T_c - T(z,\tau)] }{ (T_c - T_0)^3 }. \]

Taking further the second derivative of expression (15) with respect to \(y\), we shall have:

\[ \frac{\partial^2 T(x,y,z,\tau)}{\partial y^2} = \frac{ [T_c - T(x,\tau)]\, \dfrac{\partial^2 T(y,\tau)}{\partial y^2}\, [T_c - T(z,\tau)] }{ (T_c - T_0)^2 }. \]

Then:

\[ \frac{ \dfrac{\partial^2 T}{\partial y^2} }{ \dfrac{\partial^2 T}{\partial x^2} } = \frac{ [T_c - T(x,\tau)]\cdot \dfrac{\partial^2 T(y,\tau)}{\partial y^2} }{ \dfrac{\partial^2 T(x,\tau)}{\partial x^2}\cdot [T_c - T(y,\tau)] }. \]

Taking into account expressions (a), (b), (c), we represent the last relation as:

\[ \frac{ \dfrac{\partial^2 T}{\partial y^2} }{ \dfrac{\partial^2 T}{\partial x^2} } = \frac{ \displaystyle \sum_{n=1}^{\infty} f_n(\tau)\cdot \cos\!\left(\mu_n \frac{x}{R_1}\right) \cdot \sum_{m=1}^{\infty} f_m(\tau)\cdot \cos\!\left(\mu_m \frac{y}{R_2}\right) \cdot \frac{\mu_m^2}{R_2^2} }{ \displaystyle \sum_{n=1}^{\infty} f_n(\tau)\cdot \cos\!\left(\mu_n \frac{x}{R_1}\right) \cdot \frac{\mu_n^2}{R_1^2} \cdot \sum_{m=1}^{\infty} f_m(\tau)\cdot \cos\!\left(\mu_m \frac{y}{R_2}\right) }. \]

For convenience of notation, we rewrite this relation in the following form:

\[ \frac{ \dfrac{\partial^2 T}{\partial y^2} }{ \dfrac{\partial^2 T}{\partial x^2} } = \frac{ \displaystyle \sum_{n=1}^{\infty} A_n \cdot \sum_{m=1}^{\infty} B_m \frac{\mu_m^2}{R_2^2} }{ \displaystyle \sum_{n=1}^{\infty} A_n \frac{\mu_n^2}{R_1^2}\cdot \sum_{m=1}^{\infty} B_m } = \beta . \]

Expanding the series, we obtain:

\[ \beta = \frac{ (A_{1n}+A_{2n}+\ldots)(B_{1m}\mu_{1m}^{2}+B_{2m}\mu_{2m}^{2}+\ldots)R_1^2 }{ (A_{1n}\mu_{1n}^{2}+A_{2n}\mu_{2n}^{2}+\ldots)(B_{1m}+B_{2m}+\ldots)R_2^2 }. \]

Carrying out the division according to the rule for dividing a polynomial by a polynomial, we shall have:

\[ \frac{ B_{1m}\cdot \mu_{1m}^{2}+B_{2m}\mu_{2m}^{2}+\ldots }{ B_{1m}+B_{2m}+\ldots } \simeq \mu_{1m}^{2}. \]

Similarly:

\[ \frac{ A_{1n}+A_{2n}+\ldots }{ A_{1n}\mu_{1n}^{2}+A_{2n}\mu_{2n}^{2}+\ldots } \simeq \frac{1}{\mu_{1n}^{2}}; \]

Then

\[ \beta = \frac{ \dfrac{\partial^2 T}{\partial y^2} }{ \dfrac{\partial^2 T}{\partial x^2} } \simeq \frac{ \mu_{1m}^{2}\cdot R_1^2 }{ \mu_{1n}^{2}\cdot R_2^2 }. \tag{16} \]

In exactly the same way we find:

\[ \gamma = \frac{\dfrac{\partial^2 T}{\partial z^2}} {\dfrac{\partial^2 T}{\partial x^2}} \simeq \frac{\mu_{1k}^{2}\cdot R_1^{2}} {\mu_{1n}^{2}\cdot R_3^{2}} . \tag{17} \]

Thus, for boundary conditions of the first and third kinds there exist dependences (16) and (17), completely identical in form. This makes it possible to suppose that, for boundary conditions expressed by the Stefan–Boltzmann law, these relations should also hold. On the other hand, for boundary conditions of the first and second kinds the characteristic numbers do not depend on the parameters characterizing the process and are respectively equal to:

\[ \mu_n = (2n-1)\frac{\pi}{2} \qquad (n=1,2,3\ldots) \]

\[ \mu_n = n\pi \qquad (n=1,2,3\ldots) \]

Therefore, for boundary conditions of the first and second kinds, relations (16) and (17) will take the form:

\[ \frac{\dfrac{\partial^2 T}{\partial y^2}} {\dfrac{\partial^2 T}{\partial x^2}} \simeq \frac{R_1^2}{R_2^2}; \qquad \frac{\dfrac{\partial^2 T}{\partial z^2}} {\dfrac{\partial^2 T}{\partial x^2}} \simeq \frac{R_1^2}{R_3^2} \tag{18} \]

Using relation (18), the divergence of the temperature gradient may be replaced according to the scheme:

\[ \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = \left( 1+ \frac{\dfrac{\partial^2 T}{\partial y^2}} {\dfrac{\partial^2 T}{\partial x^2}} + \frac{\dfrac{\partial^2 T}{\partial z^2}} {\dfrac{\partial^2 T}{\partial x^2}} \right) \frac{\partial^2 T}{\partial x^2} = \]

\[ = (1+\beta+\gamma)\frac{\partial^2 T}{\partial x^2} \simeq \left( 1+\frac{R_1^2}{R_2^2} +\frac{R_1^2}{R_3^2} \right) \cdot \frac{\partial^2 T}{\partial x^2}, \]

therefore,

\[ \xi \simeq 1+\frac{R_1^2}{R_2^2}+\frac{R_1^2}{R_3^2} \tag{19} \]

The last expression, being an approximate quantitative relation for the coefficient \(\xi\), well reveals its qualitative aspect. It shows that the required coefficient is determined by the ratio of the squares of the body dimensions. Practical calculations carried out for boundary conditions of the second kind confirmed the correctness of the structural form of expression (19), but under the condition that \(R_2^2\) and \(R_3^2\) are understood as the mean values of the body dimensions, calculated from the relations:

\[ R_{2\,\mathrm{av}}^2 = \frac{R_2^2+R_{2\,\mathrm{ident.}}^2}{2} = \frac{R_2^2+R_2^2+R_1^2}{2} = R_2^2+0.5R_1^2 \]

\[ R_{3\,\mathrm{av}}^2 = \frac{R_3^2+R_{3\,\mathrm{ident.}}^2}{2} = \frac{R_3^2+R_3^2+R_1^2}{2} = R_3^2+0.5R_1^2 \]

Then, for a parallelepiped, the formula for the coefficient \(\xi\) takes the form:

\[ \xi = 1+\frac{R_1^2}{R_2^2+0.5R_1^2}+\frac{R_1^2}{R_3^2+0.5R_1^2} \tag{20} \]

From formula (20) the value of this coefficient can be obtained for a long bar of rectangular or square cross section, a cube, and an infinite plate. The corresponding formulas are given in Table 1.

Table 1

Verification by substitution of expressions (9) and (9′) into the equations describing the process showed that they satisfy the differential equation of heat conduction (6) and the corresponding uniqueness conditions.

Figure 2 shows the heating of a steel parallelepiped with sides \(0.2\times0.4\times0.6\ \mathrm{m}\). Curve 1 (change in the surface temperature) and curve 2 (center) are constructed from formula (9). Curves \(1'\) and \(2'\) are constructed for the same conditions by the method of multiplying temperature criteria at \(\alpha_{\mathrm{rad}}=225\ \mathrm{kcal}/\mathrm{m}^2\cdot\mathrm{h}\cdot\mathrm{deg}\).

IV

The differential equation (6) for a short cylinder can be written in the form

\[ \frac{\partial T}{\partial \tau} = \xi\cdot a \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\cdot\frac{\partial T}{\partial r} \right)\ldots \tag{21} \]

\[ 0\le r\le R_1<R_2;\qquad -R_2\le z\le R_2, \]

where \(2R_2\) is the length of the cylinder.

The recommended formula for determining the coefficient \(\xi\) is given in Table 2.

Carrying out reasoning analogous to that for the parallelepiped, we obtain the temperature distribution along the line of heat flow \(r\),

Table 2

Fig. 2. Heating of a steel parallelepiped of dimensions \(0.2 \times 0.4 \times 0.6\ \text{m}^3\) at
\(\lambda = 30\ \text{kcal}/(\text{m}\cdot\text{h}\cdot\text{deg})\), \(a = 0.03\ \text{m}^2/\text{h}\); \(C_{\text{rad}} = 4\ \text{kcal}/(\text{m}^2\cdot\text{h}\cdot{}^\circ\text{K}^4)\).

— Calculated data for changes in the temperature of the surface and center of the parallelepiped, obtained by formula (9)

– – – The same, by the method of multiplying temperature criteria at \(\alpha_{\text{rad}} = 200\ \text{kcal}/(\text{m}^2\cdot\text{h}\cdot\text{deg})\).

taking into account the propagation of heat also along the \(z\)-axis. The latter, for the case of constant thermophysical characteristics, has the form:

\[ \frac{T_m(r)}{T_c} = \frac{T_0}{T_c} + \frac{g_c R}{\lambda T_c} \left\{ 2\xi \frac{a\tau_1}{R_1^2} \sum_{i=1}^{i=m}\frac{g_{ci}}{g_c} -\frac{1}{4}\frac{g_{cm}}{g_c} \left(1-2\frac{r^2}{R_1^2}\right) - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2 J_0'(\mu_n)} J_0\!\left(\mu_n\frac{r}{R_1}\right) \left[ \frac{g_{c1}}{g_c} e^{-\mu_n^2\xi\, m\,\frac{a\tau_1}{R_1^2}} - \sum_{i=1}^{i=m-1} \left( \frac{g_{ci}}{g_c} - \frac{g_{c(i+1)}}{g_c} \right) e^{-\mu_n^2 (m-i)\xi\,\frac{a\tau_1}{R_1^2}} \right] \right\}. \tag{9''} \]

Here \(r\) coincides with the heat-flow line in the body and varies from \(0\) to \(R_1\) (\(2R_1\) is the diameter of the cylinder).

Integrating expression \((9'')\) from \(0\) to \(R_1\), we obtain the mean value of the temperature along the heat-flow line:

\[ \frac{T_m^{\mathrm{av}}}{T_c} = \frac{T_0}{T_c} + 2\frac{g_c R_1}{\lambda T_c}\, \xi\,\frac{a\tau_1}{R_1^2} \sum_{i=1}^{i=m}\frac{g_{ci}}{g_c}. \tag{10''} \]

In this case the calculated time interval is determined from the analytic-empirical relations:

\[ \tau_1 = \frac{0.0225\,\lambda\,T_c\,R_1}{g_c\,\xi\,a} \tag{11''} \]

\[ \tau_m \le \frac{(1+\xi+2)(g_{c1}-g_{c2})} { g_{c1} \left[ \xi+1+2\sum_{n=1}^{\infty} e^{-\mu_n^2\xi\,\frac{a\tau_1}{R_1^2}} \right] }. \tag{12''} \]

For \(\xi=1\), \((11'')\) and \((12'')\) pass into the analogous expression for an infinite cylinder [12].

Expressions analogous to \((9'')\)—\((12'')\) for the case when the thermophysical characteristics of the substance vary as a function of temperature may be represented in the form

\[ \frac{T_m(r)}{T_c} = \frac{T_0}{T_c} + \frac{g_c R_1}{\lambda_1 T_c} \left\{ 2\xi\,\frac{a_1\tau_1}{R_1^2} \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c}\cdot\frac{c_1}{c_i} - \frac{1}{4}\cdot \frac{g_{cm}}{g_c}\cdot \frac{\lambda_1}{\lambda_m} \right. \]

\[ \left. \cdot \left(1-2\frac{r^2}{R_1^2}\right) - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2 J_0'(\mu_n)} J_0\!\left(\mu_n\frac{r}{R_1}\right) \left[ \frac{g_{c1}}{g_c}\, e^{-\mu_n^2\xi\sum_{i=1}^{i=m}\frac{a_i\tau_i}{R_1^2}} - \sum_{i=1}^{i=m-1} \left( \frac{g_{ci}}{g_c}\cdot\frac{\lambda_1}{\lambda_i} - \frac{g_{c(i+1)}}{g_c}\cdot\frac{\lambda_1}{\lambda_{(i+1)}} \right) e^{-\mu_n^2\xi\sum_{k=1}^{k=m-i}\frac{a_{(i-k)}\tau}{R_1^2}} \right] \right\}. \tag{9'''} \]

\[ \frac{T_m^{\mathrm{av}}}{T_c} = \frac{T_0}{T_c} + 2\frac{g_c R_1}{\lambda_1 T_c}\cdot \xi\cdot \frac{a_1\tau_1}{R_1^2} \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c}\cdot\frac{c_1}{c_i}. \tag{10'''} \]

\[ \text{* See also [12].} \]

\[ \tau_1=\frac{0.0225\cdot \lambda_1\cdot T_c\cdot R_1}{g_c\cdot \xi\cdot a_1} \tag{11''} \]

\[ \varphi_m<1-\frac{a_2}{a_1} \left\{ 1- \frac{(\xi+1+2)\left(g_{c1}-g_{c2}\frac{\lambda_1}{\lambda_2}\right)} {g_{c1}\left[\xi+1+2\sum_{n=1}^{\infty} e^{-\mu_n^2\frac{a_1\tau_1}{R_1^2}\cdot \xi}\right]} \right\} \tag{12''} \]

Figure 3 shows the heating of a steel cylinder of diameter \(d=0.3\ \text{m}\) and length \(l=0.6\). Curve 1 (change in surface temperature) and curve 2 (center) are plotted according to formula \((9'')\). Curves \(1'\) and \(2'\) are plotted for the same conditions by the method of multiplying temperature criteria at \(\alpha_{\text{izl}}=200\ \text{kcal}/\text{m}^2\text{ h deg}\).

Fig. 3. Heating of a steel cylinder \(d=0.3\ \text{m}\), \(l=0.6\ \text{m}\), at \(\lambda=30\ \text{kcal}/\text{m h deg}\),
\(a=0.0225\ \text{m}^2/\text{h}\); \(C_{\text{izl}}=3.65\ \text{kcal}/\text{m}^2\text{ h }{}^\circ\text{K}^4\).

— Calculated data for the change in temperature of the surface and center of the cylinder, obtained from formula \((9'')\)

— — — the same, by the method of multiplying temperature criteria at \(\alpha_{\text{izl}}=200\ \text{kcal}/\text{m}^2\text{ h deg}\).

Conclusion

1. The distribution of the temperature field along the line of heat flow in a body, when the latter propagates in more than one dimension, can be represented by an expression that does not differ in complexity from that for the one-dimensional problem (see [12]).

2. The calculation relations (9)–(12) may be recommended as approximate formulas for determining the temperature field along the line of heat flow in a parallelepiped and a short cylinder when they are heated according to the law of radiant emission.

Principal notation

\(T_c\) — temperature of the heat source, °K,
\(T_0\) — initial temperature of the body, °K,

\(g_c\) — radiant heat flux from the source into the void, \(\dfrac{\text{kcal}}{\text{m}^2\ \text{h}}\),

\(g_{ci}\) — radiant heat flux onto the surface of the body at the \(i\)-th moment of time, determined from the relation,

\[ g_{ci}=\varepsilon_n \cdot C_0\left[\left(\frac{T_c}{100}\right)^4-\left(\frac{T_{i-1}}{100}\right)^4\right] \]

\(T_m(x)\) — distribution of temperature along the line of heat flow in the body at the \(m\)-th moment of time,
\(R_1\) — half of the smallest dimension of the body, m,
\(\tau_1\) — calculation time interval, h,
\(\varphi_m\) — calculation error,
\(a,\ \lambda,\ c\) — thermophysical characteristics of the substance of the body.

Literature

1. D. Yu. Panov, Handbook on the Numerical Solution of Differential Equations, GITTL, Moscow–Leningrad, 1951.

2. V. N. Sokolov, Calculation of the Heating of Bodies by the Grid Method, TsNIIMASH (Heating of Large Ingots), GITTL, Moscow, 1954.

3. N. Yu. Tail, Theoretical Foundations of Metal Heating (dissertation), 1940, Dnepropetrovsk.

4. Williamson and Adams, “Physic Rev.,” (2), 14, 99, 1919.

5. Goldschtein, “Zeitschrift f. Angew. Math. u. Mech,” 1932, No. 4.

6. D. V. Budrin, B. L. Krasovskii, Heating and Cooling of Bodies of Various Shapes, Proceedings of the Ural Industrial Institute named after S. M. Kirov, issue XVII, GNTIL, Sverdlovsk–Moscow, 1941.

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

8. G. P. Ivantsov, Heating of Metal, Moscow, 1948, Metallurgizdat.

9. N. Yu. Tail, Technology of Metal Heating, Metallurgizdat, Moscow, 1950.

10. I. Semikin, “Steel,” 1937, No. 12; 1939, Nos. 4–5.

11. Linchevskii, “Metallurg,” 1938, No. 9.

12. G. P. Boikov, Heating of Bodies under the Action of Radiant Heat [Izv. TPI, vol. 89, 1957], Tomsk, 1955.

13. I. Sneddon, Fourier Transforms, IIL, Moscow, 1955.