Heating of Carbon Steel Products under Conditions of Combined Radiative and Convective Heat Exchange

A. A. Torlopov

(Presented by the Department of Boiler Construction and Boiler Installations)

A characteristic feature of ingots and blanks made of carbon steel during their heating to high temperatures is the presence of structural transformations in the range of critical temperatures \((A_{c1} \div A_{c3})\). During the pearlite transformation in steel, the rise of temperature in it ceases, and with further heating the temperature rise will be slowed owing to the expenditure of heat on the dissolution of ferrite.

The influence of the thermal effect of structural transformations on the heating dynamics of steel products may be taken into account approximately by varying the specific heat capacity in the range of critical temperatures.

In [1], taking into account experimental data for carbon steel, generalized dependences of the thermophysical characteristics \(\lambda\) and \(C\) on temperature are given. We adopted these dependences as the basis for solving the problem of heating ingots and blanks in flame furnaces with a medium temperature constant in time.

The problem of heating steel products in the form of an infinite plate and a cylinder under conditions of radiative-convective heat exchange and with thermal coefficients depending on temperature is formulated mathematically as follows:

the differential equation of heat conduction

\[ C(\Theta)\frac{\partial \Theta}{\partial \mathrm{Fo}} = \frac{\partial}{\partial X} \left[ \Lambda(\Theta)\frac{\partial \Theta}{\partial X} \right] + \frac{k}{X}\Lambda(\Theta)\frac{\partial \Theta}{\partial X}, \tag{1} \]

where \(k\) is equal to 0 and 1, respectively, for a plate and a cylinder; the boundary conditions are

at \(X = 1\)

\[ \Lambda(\Theta)\frac{\partial \Theta}{\partial X} = \mathrm{Sk} \left[ 1-\Theta^{4}+\frac{\mathrm{Bi}}{\mathrm{Sk}}(1-\Theta) \right], \tag{2} \]

at \(X = 0\)

\[ \frac{\partial \Theta}{\partial X}=0, \tag{3} \]

at \(\mathrm{Fo}=0\)

\[ \Theta=\Theta_{0}. \tag{4} \]

Under the stated conditions, the solution was obtained by the grid method [2], using the relations given in [3], where the stability condition and the method for checking the error of the adopted numerical calculation scheme are shown.

Fig. 1. Dependence of the relative temperature of the mid-plane of the plate on the value of the Fourier criterion \(Fo\) and the regime parameters of heating.

Fig. 2. Dependence of the relative temperature of the surface of the plate on the value of the Fourier criterion \(Fo\) and the regime parameters of heating.
\(a — Sk = 2,\ b — 1,\ v — 0.5,\ g — 0.25\).

Fig. 3. Dependence of the relative temperature of the center of a cylinder on the value of the criterion \(Fo\) and the heating regime parameters

Fig. 4. Dependence of the relative temperature of the surface of a cylinder on the value of the criterion \(Fo\) and the heating regime parameters. \(a\)—\(Sk = 2\), \(b\)—1, \(v\)—0.5, \(g\)—0.25.

Numerical results for the solution of the system (1)–(4) were obtained over a wide range of variation of the heating-regime parameters using the “Minsk-1” electronic computer.

Part of the numerical material obtained has been reduced to the graphs shown in Figs. 1, 2, 3, and 4.

In Figs. 1 and 3 the dependences are given of the relative temperature of the center of a plate and a cylinder on the value of the criterion \(\mathrm{Fo}\), for the criterion value \(\mathrm{Sk}=0.25\), the ratio \(\mathrm{Bi}/\mathrm{Sk}=0\), and different specifications of \(\Theta_0\). For cases in which the values of the criterion \(\mathrm{Sk}\) and the ratio \(\mathrm{Bi}/\mathrm{Sk}\) differ from the above values, it is necessary to use the curves located in the right-hand part of the graphs. These curves determine the time scale as a function of the value of the criterion \(\mathrm{Sk}\) and the ratio \(\mathrm{Bi}/\mathrm{Sk}\). For the plate they are constructed at the criterion value \(\mathrm{Fo}=4\), and for the cylinder at \(\mathrm{Fo}=2\), and have the same form as the curves of the dependence of the relative temperature of the center of the product on the value of the criterion \(\mathrm{Fo}\) to which they refer.

The key to the graphs is shown for the case when the criterion value is \(\mathrm{Sk}=1.2\), the ratio \(\mathrm{Bi}/\mathrm{Sk}=0.5\), \(\Theta_0=0.5\), and the criterion value \(\mathrm{Fo}=1.5\) for the plate and \(\mathrm{Fo}=0.5\) for the cylinder. Along the ordinate axis we plot the value of the criterion \(\mathrm{Sk}\) (point \(m\)); then, by interpolation with respect to the values of \(\mathrm{Bi}/\mathrm{Sk}\) and \(\Theta_0\), we find point \(n\), project the latter onto the abscissa axis (point \(l\)), and, knowing that the scale curves have been constructed only for the criterion value \(\mathrm{Fo}=4\) and 2 for the plate and cylinder, respectively, we graduate the time axis in the new scale. Next, from the values of the criteria \(\mathrm{Fo}\) and \(\Theta_0\), we find, as shown by the arrows on the graphs, the value of the relative temperature at the center of the product.

In Figs. 2 and 4 the dependences are given of the relative temperature of the surface of a plate and a cylinder on the value of the criterion \(\mathrm{Fo}\) for various values of the criterion \(\mathrm{Sk}\), the ratio \(\mathrm{Bi}/\mathrm{Sk}\), and \(\Theta_0\). Use of these graphs presents no special difficulties if one takes into account that, for ease of interpolation, three different graphs (with respect to \(\Theta_0\)) have been combined into one.

As a comparison, let us compare the calculated data with experimental data on heating a forging billet of \(>100\) mm made of carbon steel (0.4% C) in a chamber furnace at an ambient temperature of \(1473^\circ\mathrm{K}\) [4].

After 6 and 26 min, according to experimental data, the temperature of the billet center reaches, respectively, 788 and \(1413^\circ\mathrm{K}\), and according to calculation, 795 and \(1422^\circ\mathrm{K}\).

Notation:

\(T(x,\tau)\) — temperature, \(^\circ\mathrm{K}\);

\(T_c\) — temperature of the heating medium, assumed constant during the heating process, \(^\circ\mathrm{K}\);

\[ \Theta=\frac{T}{T_c} \]
— relative temperature;

\[ \mathrm{Sk}=\frac{\sigma_b T_c^{3} R}{\lambda_0} \]
— Stark radiation criterion;

\[ \mathrm{Bi}=\frac{\alpha R}{\lambda_0} \]
— Biot criterion.

References

1. A. V. Kavaderov, Yu. A. Samoilovich, IFZh, No. 1, 1960.
2. I. S. Berezin, N. P. Zhidkov. Computational Methods, Fizmatgiz, 1959.
3. A. A. Torlopov, V. V. Salomatov. Izvestiya TPI, vol. 150, 1967.
4. V. N. Sokolov. Heating of Steel and Furnaces, collection of articles, No. 19, 1949.