ANALYSIS OF THE HEATING OF BODIES UNDER THE ACTION OF RADIANT HEAT ON THE BASIS OF SIMILARITY THEORY

G. P. BOIKOV

Presented by Professor G. I. Fuks

In order to determine the criteria characteristic of the process of heating bodies under the action of radiant heat, we shall use the following system of equations:

1. Differential equation of heat conduction

\[ \frac{d\lambda}{dT}\left[\left(\frac{\partial T}{\partial x}\right)^2\right] +\lambda\frac{\partial^2 T}{\partial x^2} = \frac{\partial T}{\partial \tau}\cdot \left(T\frac{dc}{dT}+c\right)^{*} \tag{1} \]

1. Initial condition:

\[ T=T_0 \quad \text{for } \tau=0 \tag{2} \]

1. Condition at the surface:

\[ \sigma\left[\left(\frac{T_c}{100}\right)^4-\left(\frac{T_n}{100}\right)^4\right] = \lambda\frac{\partial T_n}{\partial x} \tag{3} \]

1. Condition characterizing the properties of the substance:

\[ \lambda=f_{\lambda}(T;\,T_0;\,\lambda_0);\quad c=f_c(T;\,T_0;\,c_0) \tag{4} \]

Here the expressions (4) are assumed to be invariant with respect to scale transformations.

Let us now bring systems (1)–(4) to dimensionless form; for this purpose we introduce the scale transformations:

\[ x=l^{*}X;\quad T=T^{*}\Theta;\quad \lambda=\lambda^{*}\Lambda;\quad c=c^{*}\cdot C; \]

\[ T_0=T^{*}\Theta_0;\quad \sigma=\sigma^{*}\cdot\Sigma;\quad T_c=T^{*}\cdot\Theta_c;\quad T_n=T^{*}\cdot\Theta_n; \]

\[ \lambda_0=\lambda^{*}\cdot\Lambda_0;\quad c_0=c^{*}\cdot C_0;\quad \tau=\tau^{*}\mathrm{T} \tag{a} \]

Substitute the written relations into system (1)–(4). Since condition (2) gives no equation of connection, and conditions (4) are invariant under the introduced scale transformations, we shall have:

\[ \frac{\lambda^{*}T^{*}}{l^{*2}}\cdot \frac{d\Lambda}{d\Theta}\cdot \left(\frac{\partial\Theta}{\partial X}\right)^2 + \frac{\lambda^{*}\cdot T^{*}}{l^{*2}}\cdot \Lambda\cdot \frac{\partial^2\Theta}{\partial X^2} = \frac{T^{*}C^{*}}{\tau^{*}}\cdot \frac{\partial\Theta}{\partial \mathrm{T}}\cdot \left(\Theta\frac{dC}{d\Theta}+C\right) \]

\[ \sigma^{*}\cdot T^{*4}\sum\left[\left(\frac{\Theta_c}{100}\right)^4-\left(\frac{\Theta_n}{100}\right)^4\right] = \frac{\lambda^{*}T^{*}}{l^{*}}\cdot \Lambda\cdot \frac{\partial\Theta_n}{\partial X} \tag{n} \]

In order that system \((n)\) correspond to the laws of physics for dimensionless quantities, the following obvious relation between the scales is necessary:

\[ \text{* See [1].} \]

\[ \frac{\lambda^{*}}{l^{*}}=\frac{c^{*}}{\tau^{*}};\qquad \frac{\lambda^{*}}{l^{*}}=\sigma^{*}\cdot T^{*3} \tag{m} \]

The equations \((m)\) include 6 scales. Consequently, one may arbitrarily specify 4 scales, and determine the remaining two from the connection equations \((m)\).

Taking into account that the dimensions of the arbitrarily chosen scales must be independent, and that only parameters entering the system of equations describing the process should be chosen as scales, we set the following four scales:

\[ l^{*}=R;\quad T^{*}=T_c;\quad \lambda^{*}=\lambda_0;\quad c^{*}=c_0 \tag{б} \]

Substituting the chosen scales into the connection equations \((m)\), we find:

\[ \tau^{*}=\frac{R^{2}c_0}{\lambda_0};\qquad \sigma^{*}=\frac{\lambda_0}{R\cdot T_c^{3}} \tag{в} \]

To find the similarity criteria, we use relations \((a)\); \((б)\); \((в)\). We find:

\[ X=\frac{x}{R};\quad \Theta=\frac{T}{T_c};\quad \Lambda=\frac{\lambda}{\lambda_0};\quad C=\frac{c}{c_0}; \]

\[ \Theta_0=\frac{T_0}{T_c};\quad \Theta_n=\frac{T_n}{T_c};\quad \sum=\frac{\sigma\cdot R\cdot T_c^{3}}{\lambda_0}; \]

\[ \mathrm{T}=\frac{\tau\cdot \lambda_0}{c_0R^{2}};\quad \Theta_c=1;\quad \Lambda_0=1;\quad C_0=1 \]

Solution of the system:

\[ \frac{d\Lambda}{d\Theta}\left(\frac{\partial \Theta}{\partial X}\right)^2 +\Lambda\cdot \frac{\partial^2\Theta}{\partial X^2} = \frac{\partial\Theta}{\partial \mathrm{T}}\cdot \left(\Theta\frac{dC}{d\Theta}+C\right) \tag{1a} \]

\[ \Theta=\Theta_0\ \text{at}\ \mathrm{T}=0 \tag{2a} \]

\[ \sum\left[\left(\frac{\Theta_c}{100}\right)^4- \left(\frac{\Theta_n}{100}\right)^4\right] = \Lambda\frac{\partial\Theta_n}{\partial X} \tag{3a} \]

\[ \Lambda=f_{\lambda}(\Theta;\Theta_0);\quad C=f_c(\Theta;\Theta_0) \tag{4a} \]

will be represented in the form:

\[ \frac{T}{T_c} = f\left( \frac{x}{R}; \frac{\tau\lambda_0}{c_0R^2}; \frac{\lambda}{\lambda_0}; \frac{c}{c_0}; \frac{\sigma\cdot RT_c^3}{\lambda_0}; \frac{T_0}{T_c}; \frac{T_n}{T_c} \right). \tag{5} \]

A similarity criterion can be transformed. For example, the criterion

\[ \sum=\frac{\sigma\cdot T_c^3\cdot R}{\lambda_0} \]

can be represented in the form of the Kirchhoff criterion:

\[ \frac{g_c\cdot R}{\lambda_0\cdot T_c}=K_i, \]

the criterion \(\Theta_n=\frac{T_n}{T_c}\) in the form:

\[ \frac{\sigma}{\sigma}\cdot\left(\frac{T_n}{T_c}\right)^4 = \frac{g_{cn}}{g_c} \quad \text{or} \quad Q_{ci}=1-\frac{g_{cn}}{g_c}=\frac{g_{ci}}{g_c} \quad \text{etc.} \]

Then (5) is rewritten as follows:

\[ \Theta=f(X;\,F_0;\,\Lambda;\,C;\,K_i;\,\Theta_0;\,Q_{ci}) \tag{5′} \]

Fig. 1. Heating of a cylinder of 9Kh steel with a diameter of 300 mm:
1. Change in surface temperature according to experimental data [2].
2. Change in surface temperature under the same conditions, constructed by formula (8).

Fig. 2. Heating of a cylinder made of ball-bearing steel, 125 mm in diameter:
1. Change in temperature at the surface according to experimental data [3].
2. Change in temperature at the surface under the same conditions, constructed from formula (8).

In view of the fact that

\[ Q_{ci}=\varphi(K_i;\ F_0;\ \Theta_0;\ \Lambda;\ C) \]

relation (5′) can be simplified:

\[ \Theta=f(X;\ F_1;\ \Lambda;\ C;\ K_i;\ \Theta_0) \tag{6} \]

The latter is in agreement with the \(\Pi\)-theorem, according to which the number of complex criteria is

\[ n-k=6-4=2, \]

and the number of simplex criteria is:

\[ N-n=11-6=5. \]

Here \(N\) is the total number of quantities entering into system (1)–(4):

\[ (\lambda;\ T;\ x;\ \tau;\ c;\ T_0;\ \sigma;\ T_c;\ T_n;\ \lambda_0;\ c_0), \]

\(n\) is the total number of scales with unlike dimensions:

\[ (T;\ \tau;\ x;\ \lambda;\ c;\ \sigma) \]

\(k\) is the number of scales with independent dimensions:

\[ (R;\ T_c;\ \lambda;\ c_0) \]

For identical materials and corresponding points of space we obtain:

\[ \Theta=f(\Theta_0;\ K_i;\ F_0). \tag{7} \]

Analysis of dependence (7) confirms the correctness of its representation. Thus, for example, for the surface of a steel infinite cylinder, within the range of variation of the complex \(\Theta_0^2+2K_iF_0\) from 0 to 0.5, dependence (7) is confirmed quite satisfactorily by the expression:

\[ \Theta_n=\left[\Theta_0^2+2K_i\cdot F_0\right]^{0.5} \tag{8} \]

Figures 1 and 2 show curves of the change in the surface temperature of a steel cylinder. Curves (1) are plotted from experimental data [2]–[3]. Curves (2) are according to formula (8).

CONCLUSION

1. The heating of bodies under the action of radiant heat, with variable thermophysical characteristics of the substance, is described by two complex criteria and five simplex criteria [expression (6)].

2. For the same materials, the qualitative dependence (7) remains the same both for processes with variable and for processes with constant [4] thermophysical characteristics. Quantitatively, the influence of changes in the thermophysical characteristics can be accounted for by empirical coefficients.

LITERATURE

1. Eigenson L. S.—Modeling, Moscow, 1952, State Publishing House “Soviet Science.”

2. Degtyarev V. M.—High-speed heating in the heat treatment of products of large cross sections. Kiev—Moscow, 1953, Mashgiz.

3. Nemchinsky A. L.—Thermal calculations of heat treatment, Sudpromgiz, 1953.

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