HEATING OF BODIES OF FINITE DIMENSIONS UNDER THE ACTION OF RADIANT HEAT

(Second communication)

G. P. BOICHKOV

Presented by Professor G. I. FUKS

In communication 1 it was shown that the differential equation of heat conduction, describing the propagation of heat in any bodies,

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

can be reduced to the form:

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

The latter makes it possible to determine the temperature field along the line of heat flow in a body when heat propagates in more than one dimension. Since expression (1′) describes a process occurring in any bodies, the differential equation (2′) also gives the temperature field along the line of heat flow in any body, if the line of flow and the value of the coefficient \(\xi\) are known.

Thus, the problem for boundary conditions corresponding to radiant heat exchange may be formulated as follows:

We have the differential equation of heat conduction:

\[ \frac{\partial T(x,\tau)}{\partial \tau} = \xi \cdot a \cdot \frac{\partial^2 T(x,\tau)}{\partial x^2}. \tag{1} \]

Initial condition:

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

Boundary condition:

\[ \frac{\partial T(R_1,\tau)}{\partial x} = \frac{\varepsilon_n \cdot C_0}{\lambda} \cdot \left[ \left(\frac{T_c}{100}\right)^4 - \left(\frac{T_0}{100}\right)^4 \right] = \frac{g_{c1}}{\lambda} \tag{3} \]

Condition of symmetry:

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

It can be shown that the expression:

\[ \frac{T(x,\tau)}{T_c} = \frac{T_0}{T_c} + \frac{g_{c1}\cdot R_1}{\lambda \cdot T_c} \cdot \left\{ \xi \cdot \frac{a\tau}{R_1^2} - \left[ \frac{\xi}{2(\xi+2)} - \frac{1}{2}\cdot \frac{x^2}{R_1^2} \right] + \right. \]

\[ \left. + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2\xi}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R_1}\right) \cdot e^{-\mu_n^2 \xi \frac{a\tau}{R_1^2}} \right\} \tag{5} \]

satisfies differential equation (1). To do this, we differentiate it once with respect to time \(\tau\) and twice with respect to the coordinate \(x\):

\[ \frac{\partial T}{\partial \tau} = \frac{g_{c1}\cdot R_1}{\lambda} \cdot \left\{ \xi\frac{a}{R_1^2} - \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2\xi}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R_1}\right) \times e^{-\mu_n^2\xi\cdot\frac{a\tau}{R_1^2}} \cdot \mu_n^2\xi\cdot\frac{a}{R_1^2} \right\} \tag{c} \]

\[ \frac{\partial^2 T}{\partial x^2} = \frac{g_{c1}\cdot R_1}{\lambda} \left\{ \frac{1}{R_1^2} - \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2\xi}{\mu_n^2} \cdot \cos\left(\mu_n\frac{x}{R_1}\right) \times e^{-\mu_n^2\xi\cdot\frac{a\tau}{R_1^2}} \cdot \frac{\mu_n^2}{R_1^2} \right\} \tag{d} \]

Substitution of \((c)\) and \((d)\) into equation (1) leads to an identity.

To verify that expression (5) corresponds to the boundary conditions (3) and to the symmetry conditions (4), we find the first derivative with respect to \(x\):

\[ \frac{\partial T}{\partial x} = \frac{g_{c1}\cdot R_1}{\lambda} \cdot \left\{ \frac{x}{R_1^2} - \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2\xi}{\mu_n^2} \cdot \sin\left(\mu_n\frac{x}{R_1}\right) \cdot e^{-\mu_n^2\cdot\xi\frac{a\tau_1}{R_1^2}} \cdot \frac{\mu_n}{R} \right\}. \]

Putting \(x=R_1\), we find:

\[ \frac{\partial T(R_1\tau)}{\partial x} = \frac{g_{c1}}{\lambda}, \quad \text{since } \sin\mu_n=\sin n\pi=0 \]

Putting further \(x=0\), we verify that

\[ \frac{\partial T(0,\tau)}{\partial x}=0; \]

Thus, expression (5) strictly satisfies the differential equation of heat conduction (1), the boundary conditions (3), and the symmetry conditions (4). But since differential equation (1) applies to bodies of any shape and dimensions, the written relation (5) represents the temperature field along the heat-flow line in various bodies, i.e., it claims a certain universality. The universality of formula (5) can also be confirmed by the following considerations.

It is known that, starting from a certain value of the Fourier criterion (approximately 0.3 for an infinite plate; 0.25 for an infinite cylinder; 0.2 for a sphere), the series from 1 to \(\infty\) in the calculation formulas may be neglected, i.e., set equal to zero. Then the calculation formulas take a simpler form:

a) For an infinite plate:

\[ \Theta = K_i \left\{ F_0 - \left( \frac{1}{6} - \frac{1}{2}\cdot\frac{x^2}{R^2} \right) \right\} \tag{a} \]

b) For an infinite cylinder:

\[ \Theta = K_i \left\{ 2F_0 - \left( \frac{1}{4} - \frac{1}{2}\cdot\frac{r^2}{R^2} \right) \right\} \tag{б} \]

c) For a sphere:

\[ \Theta=K_i\left\{3F_0-\left(\frac{3}{10}-\frac{1}{2}\cdot\frac{r^2}{R^2}\right)\right\} \tag{c} \]

Neglecting in formula (5) the series from 1 to \(\infty\), let us rewrite it in the form:

\[ \Theta=K_i\left\{\xi F_0-\left[\frac{\xi}{2(\xi+2)}-\frac{1}{2}\cdot\frac{x^2}{R^2}\right]\right\} \tag{5′} \]

We note that for \(\xi=1\) the last expression becomes formula (a) for an infinite plate. For \(\xi=2\), (5′) is transformed into the calculation formula for an infinite cylinder (b). For \(\xi=3\), expression (5′) takes the form of the calculation formula for a sphere (c). The latter gives the right to assert that relation (5′) is a calculation formula also for all intermediate values of \(\xi\) from 1 to 2 and from 2 to 3, i.e., (5′) represents a universal calculation formula for determining the temperature field along the line of heat flow under radiant heating for bodies of any shape and dimensions. However, investigations have shown that dependence (5) (with allowance for the series from 1 to \(\infty\)) agrees satisfactorily with the initial condition only for small values of the coefficient \(\xi\).

Fig. 1. Heating of a sphere at \(g=\mathrm{Const}\). Curves 1 and 2 show the change in dimensionless temperature at the surface and at the center of the sphere, constructed according to the exact formula (see [1]). Curves \(1′\) and \(2′\) are constructed according to formula (5).

For large values of the coefficient \(\xi\) (in the limit \(\xi=3\)), the discrepancy with the initial condition is greatest. But these same investigations show (Fig. 1) that even the greatest nonconformity with the initial conditions, for example, at \(\xi=3\), very quickly ceases to affect the course of the calculation. If one proceeds from the case where \(\xi=3\), then it may be considered that the proposed calculation relation (5) can be used with a sufficient degree of accuracy for Fourier criteria greater than 0.08. The recommended formulas for \(\xi\) are given in Table 1.

Table 1

Formula (5) for the $m$-th instant of time in the heating of bodies by radiant heat, when the thermophysical characteristics vary as a function of temperature, has the form (see [2]):

\[ \begin{aligned} \frac{T_m(x)}{T_c} &= \frac{T_0}{T_c} + \frac{g_c \cdot R_1}{\lambda_1 \cdot T_c} \cdot \Bigg\{ \xi \cdot \frac{a_1 \tau_1}{R_1^2} \cdot \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c} \cdot \frac{c_1}{c_i} - \frac{g_{cm}}{g_c} \cdot \frac{\lambda_1}{\lambda_m} \\ &\qquad\cdot \left[ \frac{\xi}{2(\xi+2)} - \frac{1}{2}\cdot \frac{x^2}{R_1^2} \right] + \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2\cdot \xi}{\mu_n^2} \cdot \cos\!\left(\mu_n \frac{x}{R_1}\right) \cdot \Bigg[ \frac{g_{c1}}{g_c} \\ &\qquad - \mu_n^2 \xi \cdot \sum_{i=1}^{i=m} \frac{a_i \tau_1}{R_1^2} \cdot e - \sum_{i=1}^{i=m-1} \left( \frac{g_{ci}}{g_c}\cdot \frac{\lambda_1}{\lambda_i} - \frac{g_c(i+1)\lambda_1}{g_c \cdot \lambda(i+1)} \right) \\ &\qquad -\mu_n^2 \xi \sum_{k=1}^{k=m-i} \frac{a(i+k)\tau_1}{R_1^2} \Bigg] \Bigg\}. \end{aligned} \tag{6} \]

The mean temperature of the body and the calculated time interval should be determined using the relations:

\[ \frac{T_m^{\mathrm{cp}}}{T_c} = \frac{T_o}{T_c} + \frac{g_c \cdot R_1}{\lambda_1 \cdot T_c} \cdot \xi \cdot \frac{a_1 \tau_1}{R_1^2} \cdot \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c} \cdot \frac{c_1}{c_i}; \tag{7} \]

\[ \tau_1 = \frac{0.022 \cdot \lambda_1 \cdot T_c \cdot R_1} {g_c \cdot \xi \cdot a_1}; \tag{8} \]

\[ \varphi_m < 1 - \frac{a_2}{a_1} \cdot \left\{ 1 - \frac{ (\xi+2)\left(g_{c1}-g_{c2}\cdot\frac{\lambda_1}{\lambda_2}\right) }{ g_{c1}\cdot \left[ \xi + 2\cdot \sum_{n=1}^{\infty} e^{-\mu_n^2 \xi \cdot \frac{a_1\tau_1}{R_1^2}} \right] } \right\}. \tag{9} \]

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

— Calculated data for the change in temperature of the surface and center of the cylinder without taking into account changes in the thermophysical characteristics of the substance, obtained from formula (6).

— — — the same, by the method of multiplying temperature criteria, at \(\alpha_{\mathrm{rad}}=200\) kcal/m²·h·deg.

For \(\xi=1, 2, 3\), expressions (7), (8), (9) pass into the corresponding formulas for an infinite plate, an infinite cylinder, and a sphere [2].

If the thermophysical characteristics of the substance are constant, then for

\[ c_1=c_2=\ldots c_i;\quad \lambda_1=\lambda_2=\ldots \lambda_i;\quad a_1=a_2=\ldots a_i \]

expressions (6)–(9) pass into the calculation formulas for the case when the thermophysical characteristics do not depend on temperature [2].

Fig. 3. Heating of a cube of steel (grade 0.34) with side \(0.457\ \text{m}\), for

\[ \lambda=f(t)\ \frac{\text{kcal}}{\text{m hour degree}};\quad C=f(t)\ \frac{\text{kcal}}{\text{kg degree}};\quad a=f(t)\ \frac{\text{m}^2}{\text{hour}} \]

— calculated data for the change in temperature of the surface and the center of the cube, taking into account the change in the thermophysical characteristics of the material, obtained from formula (6) for \(C_{\text{rad}}=3.6\ \text{kcal}/\text{m}^2\text{hour }{}^\circ K^4\);

— — — the same, according to experimental data [3].

In most practical calculations, the value of the series from 1 to \(\infty\) is very small in comparison with the first two terms in braces in formula (6). Therefore it may be neglected. Then the calculation relation is greatly simplified and takes the form:

\[ \frac{T_m(x)}{T_c} = \frac{T_o}{T_c} + \frac{g_c R_1}{\lambda_1 T_c} \cdot \left\{ \xi \cdot \frac{a_1 \tau_1}{R_1^2} \cdot \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c}\cdot\frac{c_1}{c_i} - \frac{g_{cm}}{g_c}\cdot\frac{\lambda_1}{\lambda_m} \cdot \left[ \frac{\xi}{2(\xi+2)} - \frac{1}{2}\cdot\frac{x^2}{R_1^2} \right] \right\}. \tag{6′} \]

Under the indicated condition, formula (6′) is a mathematically exact calculation relation for bodies of any shape and size. In particular, for \(\xi=1,2,3\) it passes into the formulas for an infinite plate, an infinite cylinder, and a sphere (see [2]).

Figure 2 shows the heating of a steel cylinder of diameter \(d=0.3\ \text{m}\) and length \(l=0.3\ \text{m}\). Curve 1 (change in surface temperature) and curve 2 (center) were constructed from formula (6) with unchanged thermophy-

physical characteristics of the substance. Curves \(1'\) and \(2'\) were constructed for the same conditions by the method of multiplying the temperature criteria at

\[ a_{\text{rad}} = 200\ \frac{\text{kcal}}{\text{m}^2\ \text{h}\ \text{deg}} . \]

Figure 3 shows the heating of a steel cube with side \(0.457\ \text{m}\). Curve 1 (change in the surface temperature) and curve 2 (of the center) were constructed from formula (6), taking into account the variation of the thermophysical characteristics of the substance as a function of temperature. Curves \(1'\) and \(2'\) were constructed from the experimental data [3].

CONCLUSION

1. The distribution of the temperature field along the line of heat flow in bodies of various finite shapes can be represented by a single formula, not differing in complexity from the solution for a one-dimensional problem.

2. The calculation relations (6)–(9) may be recommended as approximate formulas for determining the temperature field along the line of heat flow in bodies of various shapes when they are heated under the action of radiant heat. The proposed expressions are applicable for \(F_0 > 0.08\).

REFERENCES

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

2. G. P. Boikov—Heating of bodies under the action of radiant heat, (Izv. TPI, vol. 89, 1957), Tomsk.

3. V. N. Sokolov—Calculation of metal heating by the grid method, TsNIIMASH (Heating of Large Ingots), GNTIML, Moscow, 1954.