Heating of Metal Products by Radiative-Convective Heat Transfer Considering Variable Thermal Coefficients and the Thermal Effect of Structural Transformations
Abstract Generated abstract
This paper examines unsteady heating of steel plates and cylinders under combined radiative and convective heat transfer, with temperature dependent thermal conductivity and heat capacity and, for carbon steels, the heat effect of structural transformations. The authors formulate the resulting nonlinear heat conduction problem with symmetry and surface heat exchange boundary conditions, then solve it numerically on a finite difference mesh using empirical temperature dependences for carbon and austenitic steels. Calculations over industrially relevant ranges of heating parameters produce tables and nomograms for surface and center temperatures, and comparisons with experimental heating data show generally satisfactory agreement, supporting the practical use of the model for estimating temperature distributions and transient surface heat fluxes in steel products.
Full Text
HEATING OF METAL PRODUCTS BY RADIANT-CONVECTIVE HEAT WITH ALLOWANCE FOR THE VARIABILITY OF THERMAL COEFFICIENTS AND THE THERMAL EFFECT OF STRUCTURAL TRANSFORMATIONS
V. V. Salomatov, A. A. Torlopov
(Submitted by the Department of Boiler Construction and Boiler Installations)
The study of processes of unsteady heat conduction in solid bodies subjected to sudden heating from sources with a high temperature is a topical problem [1, 2, 3], owing to the important applications of the theory of these processes in the practice of metal heating. A characteristic feature of internal heat transfer in a steel ingot is that the course of this process is accompanied by a change in the physicochemical nature of the material. In addition, the thermal coefficients of steel products vary with the heating temperature so significantly that they exert a substantial influence on the character of the process.
In order to reflect the kinetics of heat transfer in solid bodies most correctly, the mathematical model of the phenomenon under study must take these factors into account. However, in such a rigorous formulation, the problem considered reduces to the solution of doubly nonlinear differential equations, exact solutions of which have not been found at present. The mathematical difficulties encountered in obtaining analytical relations naturally compel researchers to introduce certain conventional linearization procedures, the most widespread of which may be named as the following:
a) replacement of the true laws of variation of thermal conductivity and heat capacity by their mean integral values;
b) neglect of the thermal effect of phase transformations;
c) replacement of heat transfer by radiation with the laws of convective heat exchange.
Such simplifications lead only to approximate solutions covering a narrow range of practical application in metallurgical heat engineering [4, 5, 6].
The present study is carried out with the aim of determining the unsteady temperature distribution in metal products in the form of a plate and a cylinder having a constant initial temperature. In this case, the resulting heat flux passing through the surface of the body being heated is formed by heat exchange with an external medium of constant temperature according to the laws of radiation and convection. The character of the variation of the coefficients of thermal conductivity and heat capacity with temperature and the phenomenon of structural transformations are taken into account.
The equation for the unsteady temperature distribution in this case has the form
\[ c(\Theta)\frac{\partial \Theta}{\partial Fo} = \frac{\partial}{\partial \xi}\left[\lambda(\Theta)\frac{\partial \Theta}{\partial \xi}\right] + \frac{\kappa}{\xi}\cdot \lambda(\Theta)\cdot \frac{\partial \Theta}{\partial \xi}, \tag{1} \]
where the coefficient \(\kappa\) in the plane and cylindrical coordinate systems is equal to 0 and 1, respectively.
It is required to find the solution of this equation, taking into account the initial and boundary conditions
\[ \Theta(\xi,0)=\Theta_0=\mathrm{const}, \tag{2} \]
\[ \frac{\partial \Theta(0,Fo)}{\partial \xi}=0, \tag{3} \]
\[ \lambda(\Theta)\frac{\partial \Theta(1,Fo)}{\partial \xi} = S_k\left\{[1-\Theta^4(1,Fo)]+\frac{Bi}{S_k}[1-\Theta(1,Fo)]\right\}. \tag{4} \]
Owing to the nonlinearity of equations (1) and (4), it has not yet been possible to obtain an analytical solution of the system (1)–(4). Therefore this system was solved by a numerical method with the aid of the “Minsk-1” electronic computer, using the following relations:
1) for the relative temperature of the surface
\[ \Theta_{\mathrm{surf},j+1} = \Theta_{\mathrm{surf},j} + 2\frac{\Delta Fo}{h^2}\cdot \frac{\lambda(\Theta_{\mathrm{surf},j})}{c(\Theta_{\mathrm{surf},j})} \left[ p\cdot \frac{\partial \Theta_{\mathrm{surf},j}}{\partial \xi} + \frac{h^2}{2\lambda(\Theta_{\mathrm{surf},j})} + \right. \]
\[ \left. + \frac{\partial \lambda(\Theta_{\mathrm{surf},j})}{\partial \Theta_{\mathrm{surf},j}} \cdot \left( \frac{\partial \Theta_{\mathrm{surf},j}}{\partial \xi} \right)^2 + (\Theta_{1,j}-\Theta_{\mathrm{surf},j}) \right], \tag{5} \]
where the parameter \(p\) for a plate and a cylinder is respectively equal to
\[ h \quad \text{and} \quad h\left(1+\frac{1}{2}h\right), \]
\[ \frac{\partial \Theta_{\mathrm{surf},j}}{\partial \xi} = \frac{S_k}{\lambda(\Theta_{\mathrm{surf},j})} \cdot \left[ (1-\Theta_{\mathrm{surf},j}^4) + \frac{Bi}{S_k}(1-\Theta_{\mathrm{surf},j}) \right], \]
2) for the relative temperature at the internal nodes of the mesh
\[ \Theta_{i,j+1} = \Theta_{i,j} + \frac{\Delta Fo}{h^2}\cdot \frac{1}{c(\Theta_{i,j})} \left[ \lambda(\Theta_{i,j})(b\Theta_{i+1,j}-2\Theta_{i,j}+l\Theta_{i-1,j}) + \right. \]
\[ \left. + \frac{1}{4}\cdot \frac{\partial \lambda(\Theta_{i,j})}{\partial \Theta_{i,j}} \cdot (\Theta_{i+1,j}^{2}-2\Theta_{i+1,j}\cdot \Theta_{i-1,j}+\Theta_{i-1,j}^{2}) \right], \tag{6} \]
where \(b=l=1\) for a plate, and \(b=1+\dfrac{h}{2\xi_i}\), \(l=1-\dfrac{h}{2\xi_i}\) for a cylinder;
3) for the relative temperature on the axis of symmetry
\[ \Theta_{\mathrm{cen},j+1} = \frac{ \Theta_{\mathrm{cen},j} + m\cdot \dfrac{\Delta Fo}{h^2}\cdot \dfrac{\lambda(\Theta_{\mathrm{cen},j})}{c(\Theta_{\mathrm{cen},j})} \Theta_{N-1,j+1} }{ 1+ m\dfrac{\Delta Fo}{h^2}\cdot \dfrac{\lambda(\Theta_{\mathrm{cen},j})}{c(\Theta_{\mathrm{cen},j})} }, \tag{7} \]
where the coefficient \(m\) for a plate and a cylinder respectively takes the values 2 and 4.
The adopted laws of variation \(\lambda(T)\) and \(c(T)\) were based on experimental data [7, 8, 9, 10], processed by A. V. Ka-
Vaderov and Yu. A. Samoilovich [11]. However, the temperature interval of structural transformations in carbon steels adopted by the authors of [11] proved to be somewhat overestimated, which did not give satisfactory agreement between the calculated data and the experimental results. Analyzing the possible temperature intervals of phase transformations according to experimental studies of the heating of products made of various grades of steel [6, 12, 13], it is necessary to take it within 80°K.
Taking into account the analysis carried out, the temperature dependence of \(\lambda\) and \(c\) is written in the form:
a) for carbon steels in the temperature interval from 273°K to 998°K
\[ \frac{\lambda(T)}{\lambda_0}=1-0.6\cdot 10^{-3}(T-273), \]
\[ \frac{c(T)}{c_0}=1+0.5\cdot 10^{-3}(T-273); \]
in the interval of structural transformations \((998 \div 1078^\circ K)\)
\[ \frac{\lambda(T)}{\lambda_0}=0.58,\qquad \frac{c(T)}{c_{998}}=1+0.048(T-998); \]
outside the range of structural transformations
\[ \frac{\lambda(T)}{\lambda_0}=0.58,\qquad \frac{c(T)}{c_0}=1.4; \]
b) for steels of the austenitic type in the temperature interval from 273°K to 973°K
\[ \frac{\lambda(T)}{\lambda_0}=1+0.9\cdot 10^{-3}(T-273), \]
\[ \frac{c(T)}{c_0}=1+0.5\cdot 10^{-3}(T-273) \]
and for \(T>973^\circ K\)
\[ \frac{\lambda(T)}{\lambda_0}=1.63,\qquad \frac{c(T)}{c_0}=1.35. \]
From the stability condition of the numerical scheme, the quantity \(\dfrac{\Delta F_o}{h^2}\) was taken in all cases to be equal to \(\dfrac{1}{4}\), with the exception of the calculation of the heating of a plate made of austenitic steel, for which the ratio \(\dfrac{\Delta F_o}{h^2}\) was taken equal to \(\dfrac{1}{5}\). The error of the adopted numerical scheme was checked by the Runge method [14]. The error of the calculations does not exceed 1% in the initial stage of heating and amounts to tenths and hundredths of a percent beyond it.
To obtain numerical results, the variation of the heating regime parameters was taken in the range of values encountered in production practice,
\[ Sk=0.25\div 4,\qquad Bi/Sk=0\div 2,\qquad T_0=293\div 873^\circ K,\qquad T_c=1273\div 1773^\circ K. \]
Part of the numerical material obtained is given in Appendices 1 and 2, and also in the form of nomograms in Figs. 4, 5, 6, 7.
Appendix 1
Values of the dimensionless temperature \(\Theta \cdot 10^4\) of the surface and center of an infinite cylinder under simultaneous heating by radiation and convection, taking into account the temperature dependence of \(\lambda\) and \(C\) (calculation data from a computer)
a) carbon steels
\(T_c = 1273^\circ\mathrm{K}\), \(S_k = 0{,}5\)
| \(F_0\) | \multicolumn{4}{c}{\(T_0=293^\circ\mathrm{K}\)} | \multicolumn{4}{c}{\(T_0=873^\circ\mathrm{K}\)} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | \(Bi/S_k=0\), center | \(Bi/S_k=0\), surface | \(Bi/S_k=0{,}5\), center | \(Bi/S_k=0{,}5\), surface | \(Bi/S_k=0\), center | \(Bi/S_k=0\), surface | \(Bi/S_k=0{,}5\), center | \(Bi/S_k=0{,}5\), surface |
| 0,05 | 2302 | 3547 | 2302 | 3948 | 6858 | 7774 | 6858 | 7904 |
| 0,25 | 2714 | 5085 | 2817 | 5755 | 6926 | 8381 | 6933 | 8531 |
| 0,4 | 3250 | 5802 | 3449 | 6499 | 7080 | 8613 | 7098 | 8735 |
| 0,6 | 3969 | 6543 | 4256 | 7201 | 7316 | 8821 | 7346 | 8930 |
| 1,0 | 5216 | 7646 | 5571 | 8121 | 7734 | 9014 | 7772 | 9142 |
| 1,5 | 6404 | 8344 | 6720 | 8751 | 8010 | 9252 | 8037 | 9336 |
| 2,0 | 7231 | 8840 | 7530 | 9029 | 8172 | 9385 | 8206 | 9464 |
| 2,5 | 7788 | 9060 | 7936 | 9277 | 8325 | 9476 | 8371 | 9549 |
| 3,0 | 8036 | 9271 | 8126 | 9410 | 8643 | 9560 | 8799 | 9630 |
| 3,5 | 8196 | 9402 | 8288 | 9505 | 9022 | 9663 | 9136 | 9728 |
| 4,0 | 8350 | 9491 | 8545 | 9588 | 9279 | 9753 | 9373 | 9804 |
| 4,5 | 8720 | 9574 | 8990 | 9683 | 9468 | 9819 | 9544 | 9853 |
| 5,0 | 9068 | 9679 | 9268 | 9771 | 9608 | 9867 | 9669 | 9898 |
| 5,5 | 9313 | 9765 | 9469 | 9834 | 9710 | 9902 | 9759 | 9926 |
| 6,0 | 9494 | 9828 | 9614 | 9880 | 9786 | 9929 | 9824 | 9944 |
\(T_c = 1273^\circ\mathrm{K}\), \(S_k = 2\)
| \(F_0\) | \multicolumn{4}{c}{\(T_0=293^\circ\mathrm{K}\)} | \multicolumn{4}{c}{\(T_0=873^\circ\mathrm{K}\)} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | \(Bi/S_k=0\), center | \(Bi/S_k=0\), surface | \(Bi/S_k=0{,}5\), center | \(Bi/S_k=0{,}5\), surface | \(Bi/S_k=0\), center | \(Bi/S_k=0\), surface | \(Bi/S_k=0{,}5\), center | \(Bi/S_k=0{,}5\), surface |
| 0,05 | 2304 | 6662 | 2305 | 7251 | 6858 | 9017 | 6858 | 9146 |
| 0,1 | 2399 | 7797 | 2414 | 8242 | 6860 | 9223 | 6860 | 9306 |
| 0,25 | 3301 | 8800 | 3369 | 8978 | 6988 | 9508 | 6995 | 9566 |
| 0,4 | 4263 | 9108 | 4344 | 9230 | 7229 | 9610 | 7242 | 9660 |
| 0,6 | 5293 | 9342 | 5372 | 9430 | 7558 | 9695 | 7574 | 9731 |
| 0,8 | 6082 | 9477 | 6155 | 9542 | 7831 | 9735 | 7845 | 9765 |
| 1,0 | 6692 | 9547 | 6755 | 9602 | 7979 | 9764 | 7988 | 9796 |
| 1,5 | 7709 | 9725 | 7757 | 9759 | 8222 | 9825 | 8233 | 9846 |
| 2,0 | 8094 | 9803 | 8115 | 9828 | 8502 | 9864 | 8573 | 9881 |
| 2,5 | 8323 | 9846 | 8348 | 9866 | 9098 | 9905 | 9129 | 9918 |
| 3,0 | 8836 | 9881 | 8904 | 9899 | 9420 | 9939 | 9443 | 9948 |
| 3,5 | 9258 | 9922 | 9303 | 9934 | 9626 | 9961 | 9643 | 9966 |
| 4,0 | 9522 | 9950 | 9554 | 9958 | 9758 | 9975 | 9770 | 9978 |
Continuation of Appendix 1
| $F_0$ | \multicolumn{4}{c}{$T^\circ=293^\circ K$} | \multicolumn{4}{c}{$T^\circ=873^\circ K$} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | $\mathrm{Bi}/\mathrm{Sk}=0$, center | $\mathrm{Bi}/\mathrm{Sk}=0$, surface | $\mathrm{Bi}/\mathrm{Sk}=0.5$, center | $\mathrm{Bi}/\mathrm{Sk}=0.5$, surface | $\mathrm{Bi}/\mathrm{Sk}=0$, center | $\mathrm{Bi}/\mathrm{Sk}=0$, surface | $\mathrm{Bi}/\mathrm{Sk}=0.5$, center | $\mathrm{Bi}/\mathrm{Sk}=0.5$, surface |
| | \multicolumn{4}{c}{$T_c=1773^\circ K$} | \multicolumn{4}{c}{$\mathrm{Sk}=0.5$} |
| 0.05 | 1653 | 2907 | 1653 | 3349 | 4924 | 5944 | 4924 | 6064 |
| 0.25 | 2048 | 4529 | 2148 | 5312 | 4998 | 7067 | 5018 | 7373 |
| 0.4 | 2555 | 5320 | 2744 | 6022 | 5163 | 7480 | 5192 | 7796 |
| 0.6 | 3230 | 5999 | 3487 | 6862 | 5432 | 7867 | 5486 | 8153 |
| 1.0 | 4365 | 7146 | 4668 | 7693 | 5804 | 8352 | 5862 | 8602 |
| 1.5 | 5394 | 7952 | 5663 | 8394 | 6515 | 8724 | 6784 | 8931 |
| 2.0 | 5883 | 8463 | 6083 | 8783 | 7422 | 9047 | 7656 | 9224 |
| 2.5 | 6774 | 8803 | 7261 | 9035 | 8077 | 9307 | 8287 | 9444 |
| 3.0 | 7599 | 9118 | 7996 | 9344 | 8572 | 9496 | 8751 | 9632 |
| 3.5 | 8210 | 9359 | 8538 | 9530 | 8943 | 9632 | 9092 | 9714 |
| 4.0 | 8672 | 9533 | 8936 | 9663 | 9219 | 9732 | 9341 | 9794 |
| 4.5 | 9018 | 9660 | 9227 | 9757 | 9424 | 9804 | 9521 | 9851 |
| 5.0 | 9275 | 9751 | 9439 | 9825 | 9575 | 9856 | 9652 | 9892 |
| 5.5 | 9465 | 9818 | 9592 | 9873 | 9686 | 9894 | 9747 | 9922 |
| 6.0 | 9605 | 9866 | 9704 | 9908 | 9768 | 9922 | 9818 | 9943 |
| | \multicolumn{4}{c}{$T_c=1773^\circ K$} | \multicolumn{4}{c}{$\mathrm{Sk}=2$} |
| 0.05 | 1655 | 5950 | 1655 | 7008 | 4924 | 8073 | 4924 | 8367 |
| 0.1 | 1740 | 7546 | 1752 | 7969 | 4927 | 8699 | 4927 | 8874 |
| 0.25 | 2534 | 8618 | 2595 | 8845 | 5082 | 9203 | 5092 | 9312 |
| 0.4 | 3401 | 8981 | 3478 | 9142 | 5387 | 9382 | 5406 | 9462 |
| 0.6 | 4361 | 9241 | 4442 | 9351 | 5739 | 9510 | 5755 | 9573 |
| 0.8 | 5089 | 9377 | 5170 | 9470 | 5954 | 9590 | 5981 | 9642 |
| 1.0 | 5644 | 9498 | 5687 | 9565 | 6584 | 9645 | 6660 | 9690 |
| 1.5 | 6692 | 9654 | 6825 | 9702 | 7835 | 9767 | 7893 | 9798 |
| 2.0 | 7897 | 9774 | 7989 | 9808 | 8606 | 9852 | 8654 | 9872 |
| 2.5 | 8646 | 9856 | 8715 | 9878 | 9101 | 9905 | 9139 | 9919 |
| 3.0 | 9128 | 9908 | 9178 | 9922 | 9422 | 9939 | 9449 | 9948 |
| 3.5 | 9438 | 9941 | 9473 | 9952 | 9628 | 9961 | 9646 | 9967 |
| 4.0 | 9638 | 9962 | 9662 | 9968 | 9759 | 9975 | 9773 | 9979 |
b) steels of the austenitic type
| $F_0$ | \multicolumn{4}{c}{$T^\circ=293^\circ K$} | \multicolumn{4}{c}{$T^\circ=873^\circ K$} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | $\mathrm{Bi}/\mathrm{Sk}=0$, center | $\mathrm{Bi}/\mathrm{Sk}=0$, surface | $\mathrm{Bi}/\mathrm{Sk}=0.5$, center | $\mathrm{Bi}/\mathrm{Sk}=0.5$, surface | $\mathrm{Bi}/\mathrm{Sk}=0$, center | $\mathrm{Bi}/\mathrm{Sk}=0$, surface | $\mathrm{Bi}/\mathrm{Sk}=0.5$, center | $\mathrm{Bi}/\mathrm{Sk}=0.5$, surface |
| | \multicolumn{4}{c}{$T_c=1273^\circ K$} | \multicolumn{4}{c}{$\mathrm{Sk}=0.5$} |
| 0.1 | 2338 | 3909 | 2352 | 4355 | 6889 | 7687 | 6894 | 7809 |
| 0.2 | 2604 | 4494 | 2702 | 5023 | 7053 | 7957 | 7084 | 8098 |
| 0.3 | 3000 | 4926 | 3207 | 5495 | 7266 | 8151 | 7324 | 8301 |
Continuation of Appendix 1
| \(F_0\) | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface |
|---|---|---|---|---|---|---|---|---|
| 0,4 | 3431 | 5296 | 3735 | 5889 | 7480 | 8317 | 7561 | 8471 |
| 0,8 | 5053 | 6549 | 5588 | 7158 | 8220 | 8850 | 8352 | 9001 |
| 1,2 | 6371 | 7522 | 6955 | 8070 | 8767 | 9222 | 8907 | 9351 |
| 1,6 | 7392 | 8260 | 7927 | 8723 | 9158 | 9477 | 9284 | 9580 |
| 2,0 | 8157 | 8806 | 8613 | 9167 | 9430 | 9650 | 9534 | 9729 |
| 2,4 | 8722 | 9191 | 9085 | 9460 | 9616 | 9766 | 9697 | 9825 |
| 2,8 | 9126 | 9456 | 9402 | 9652 | 9742 | 9843 | 9802 | 9886 |
| 3,2 | 9408 | 9636 | 9611 | 9775 | 9824 | 9894 | — | — |
\(T_c=1273^\circ\mathrm{K}\) \(Sk=2\)
| \(F_0\) | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface |
|---|---|---|---|---|---|---|---|---|
| 0,02 | 2302 | 5022 | 2302 | 5614 | 6858 | 8177 | 6858 | 8332 |
| 0,04 | 2302 | 5860 | 2302 | 6450 | 6859 | 8513 | 6859 | 8663 |
| 0,08 | 2366 | 6787 | 2384 | 7301 | 6906 | 8850 | 6912 | 8980 |
| 0,12 | 2598 | 7321 | 2665 | 7736 | 7035 | 9032 | 7053 | 9146 |
| 0,2 | 3414 | 7921 | 3587 | 8289 | 7398 | 9240 | 7438 | 9333 |
| 0,3 | 4559 | 8387 | 4803 | 8661 | 7848 | 9399 | 7904 | 9476 |
| 0,4 | 5563 | 8713 | 5823 | 8932 | 8234 | 9516 | 8296 | 9580 |
| 0,56 | 6800 | 9096 | 7044 | 9258 | 8720 | 9659 | 8782 | 9707 |
| 0,8 | 8027 | 9461 | 8218 | 9565 | 9215 | 9794 | 9268 | 9827 |
| 0,96 | 8571 | 9617 | 8728 | 9694 | 9434 | 9853 | 9480 | 9877 |
| 1,2 | 9123 | 9769 | 9236 | 9819 | 9654 | 9910 | 9688 | 9927 |
| 1,36 | 9368 | 9835 | 9456 | 9872 | 9750 | 9936 | 9778 | 9948 |
\(T_c=1773^\circ\mathrm{K}\) \(Sk=0{,}5\)
| \(F_0\) | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=293^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface |
|---|---|---|---|---|---|---|---|---|
| 0,1 | 1690 | 3237 | 1706 | 3710 | 4963 | 6000 | 4974 | 6224 |
| 0,2 | 1960 | 3807 | 2072 | 4374 | 5179 | 6396 | 5236 | 6670 |
| 0,3 | 2362 | 4231 | 2596 | 4847 | 5465 | 6697 | 5573 | 6997 |
| 0,4 | 2796 | 4597 | 3140 | 5247 | 5759 | 6960 | 5914 | 7277 |
| 0,8 | 4421 | 5864 | 5027 | 6608 | 6854 | 7849 | 7129 | 8176 |
| 1,2 | 5753 | 6963 | 6465 | 7684 | 7740 | 8510 | 8042 | 8799 |
| 1,6 | 6852 | 7848 | 7537 | 8460 | 8416 | 8985 | 8693 | 9218 |
| 2,0 | 7739 | 8510 | 8337 | 8991 | 8909 | 9315 | 9140 | 9494 |
| 2,4 | 8415 | 8984 | 8897 | 9345 | 9257 | 9540 | 9439 | 9673 |
| 2,8 | 8908 | 9314 | 9277 | 9576 | 9498 | 9692 | 9635 | 9789 |
| 3,2 | 9257 | 9510 | 9529 | 9728 | 9662 | 9794 | 9763 | 9864 |
| 3,6 | 9498 | 9692 | 9694 | 9824 | 9773 | 9862 | — | — |
Continuation of Appendix 1
| \(Fo\) | \(T_0=293^\circ K\), \(T_c=1773^\circ K\), \(Bi/Sk=0\), center | \(T_0=293^\circ K\), \(T_c=1773^\circ K\), \(Bi/Sk=0\), surface | \(T_0=293^\circ K\), \(T_c=1773^\circ K\), \(Bi/Sk=0{,}5\), center | \(T_0=293^\circ K\), \(T_c=1773^\circ K\), \(Bi/Sk=0{,}5\), surface | \(T_0=873^\circ K\), \(Sk=2\), \(Bi/Sk=0\), center | \(T_0=873^\circ K\), \(Sk=2\), \(Bi/Sk=0\), surface | \(T_0=873^\circ K\), \(Sk=2\), \(Bi/Sk=0{,}5\), center | \(T_0=873^\circ K\), \(Sk=2\), \(Bi/Sk=0{,}5\), surface |
|---|---|---|---|---|---|---|---|---|
| 0,02 | 1652 | 4334 | 1652 | 4976 | 4924 | 6728 | 4924 | 7038 |
| 0,04 | 1653 | 5166 | 1654 | 5770 | 4925 | 7295 | 4926 | 7613 |
| 0,08 | 1720 | 6114 | 1743 | 6780 | 4991 | 7906 | 5004 | 8188 |
| 0,12 | 1966 | 6750 | 2050 | 7345 | 5178 | 8250 | 5216 | 8495 |
| 0,2 | 2844 | 7520 | 3064 | 7982 | 5721 | 8646 | 5805 | 8840 |
| 0,3 | 4067 | 8090 | 4364 | 8442 | 6430 | 8943 | 6546 | 9097 |
| 0,4 | 5114 | 8485 | 5421 | 8767 | 7053 | 9155 | 7182 | 9282 |
| 0,56 | 6401 | 8950 | 6693 | 9153 | 7851 | 9408 | 7979 | 9503 |
| 0,8 | 7755 | 9380 | 7992 | 9506 | 8675 | 9646 | 8781 | 9707 |
| 0,96 | 8372 | 9560 | 8566 | 9653 | 9043 | 9718 | 9132 | 9793 |
| 1,2 | 9000 | 9736 | 9138 | 9794 | 9414 | 9847 | 9479 | 9877 |
| 1,36 | 9278 | 9811 | 9386 | 9855 | 9578 | 9890 | 9629 | 9913 |
| 1,6 | 9558 | 9885 | 9632 | 9914 | 9742 | 9933 | 9777 | 9948 |
Appendix 2
Values of the dimensionless temperature \(\Theta \cdot 10^4\) of the surface and center of an infinite cylinder under simultaneous heating by radiation and convection, taking into account the temperature dependence of \(\lambda\) and \(c\) (computer-calculated data)
a) Carbon steels
| \(F_0\) | \(T_0=2933^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=2933^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=2933^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=2933^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0\), surface | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), center | \(T_0=873^\circ\mathrm{K}\), \(Bi/Sk=0{,}5\), surface |
|---|---|---|---|---|---|---|---|---|
| \(T_c=1273^\circ\mathrm{K}\) | \(Sk=0{,}5\) | |||||||
| 0,05 | 2305 | 3687 | 2306 | 4126 | 6858 | 7842 | 6858 | 7961 |
| 0,25 | 3309 | 5802 | 3528 | 6522 | 7065 | 8532 | 7084 | 8684 |
| 0,4 | 4316 | 6871 | 4656 | 7542 | 7355 | 8772 | 7430 | 8912 |
| 0,6 | 5488 | 7877 | 5880 | 8242 | 7768 | 9019 | 7809 | 9135 |
| 0,8 | 6425 | 8327 | 6774 | 8772 | 7964 | 9228 | 7995 | 9355 |
| 1,0 | 7118 | 8793 | 7417 | 9015 | 8086 | 9359 | 8122 | 9461 |
| 1,3 | 7803 | 9052 | 7938 | 9303 | 8262 | 9525 | 8324 | 9607 |
| 1,6 | 8050 | 9333 | 8139 | 9480 | 8778 | 9649 | 8978 | 9721 |
| 1,9 | 8226 | 9499 | 8349 | 9619 | 9237 | 9760 | 9358 | 9817 |
| 2,1 | 8369 | 9534 | 8840 | 9695 | 9424 | 9819 | 9522 | 9864 |
| 2,4 | 9036 | 9704 | 9237 | 9797 | 9620 | 9831 | 9692 | 9912 |
| 2,7 | 9374 | 9803 | 9542 | 9870 | 9750 | 9922 | 9812 | 9950 |
| \(T_c=1273^\circ\mathrm{K}\) | \(Sk=2\) | |||||||
| 0,025 | 2302 | 5751 | 2302 | 6408 | 6858 | 8[[unclear: digit]]62 | 6858 | 8829 |
| 0,05 | 2314 | 7032 | 2318 | 7605 | 6858 | 9093 | 6858 | 9208 |
| 0,125 | 2894 | 8420 | 2960 | 8822 | 6898 | 9345 | 6902 | 94[[unclear: digit]]6 |
| 0,25 | 4442 | 9170 | 4549 | 9283 | 7218 | 9[[unclear: digit]]93 | 7233 | 9647 |
| 0,4 | 5858 | 9464 | 5954 | 9549 | 7664 | 9717 | 7684 | 9753 |
| 0,6 | 7072 | 9619 | 7151 | 9722 | 7993 | 9804 | 8004 | 9832 |
| 0,8 | 7841 | 9770 | 7855 | 9802 | 8183 | 9860 | 8197 | 9878 |
| 1,0 | 8062 | 9828 | 8038 | 9855 | 8403 | 9894 | 8443 | 9909 |
| 1,2 | 8247 | 9871 | 8280 | 9890 | 9080 | 9923 | 9127 | 9935 |
| 1,4 | 8683 | 9905 | 8815 | 9920 | 9403 | 9948 | 9435 | 9956 |
| 1,6 | 9212 | 9932 | 9270 | 9944 | 9605 | 9965 | 9628 | 9971 |
| 1,8 | 9483 | 9955 | 9522 | 9963 | 9738 | 9977 | 9755 | 9981 |
| 2,0 | 9657 | 9970 | 9708 | 9979 | 9822 | 9984 | — | — |
| \(T_c=1773^\circ\mathrm{K}\) | \(Sk=0{,}5\) | |||||||
| 0,05 | 1656 | 3049 | 1657 | 3[[unclear: digit]]32 | 4924 | 5986 | 4924 | 6179 |
| 0,25 | 2601 | 5268 | 2805 | 5969 | 5140 | 7313 | 5166 | 7677 |
| 0,4 | 3325 | 6240 | 3818 | 6878 | 5479 | 7886 | 5533 | 8212 |
Continuation of Appendix 2
| $F_0$ | $T_0=293^\circ K$, Bi/Sk = 0, center | $T_0=293^\circ K$, Bi/Sk = 0, surface | $T_0=293^\circ K$, Bi/Sk = 0,5, center | $T_0=293^\circ K$, Bi/Sk = 0,5, surface | $T_0=873^\circ K$, Bi/Sk = 0, center | $T_0=873^\circ K$, Bi/Sk = 0, surface | $T_0=873^\circ K$, Bi/Sk = 0,5, center | $T_0=873^\circ K$, Bi/Sk = 0,5, surface |
|---|---|---|---|---|---|---|---|---|
| 0,6 | 4545 | 7207 | 4853 | 7798 | 5778 | 8414 | 5826 | 8684 |
| 0,8 | 5303 | 7882 | 5593 | 8386 | 6098 | 8775 | 6618 | 9004 |
| 1,0 | 5745 | 8368 | 5900 | 8789 | 7234 | 9058 | 7525 | 9251 |
| 1,3 | 6699 | 8893 | 7351 | 9200 | 8152 | 9388 | 8391 | 9522 |
| 1,6 | 7829 | 9266 | 8282 | 9488 | 8765 | 9598 | 8955 | 9696 |
| 1,9 | 8546 | 9522 | 8884 | 9674 | 9180 | 9738 | 9324 | 9803 |
| 2,1 | 8891 | 9641 | 9165 | 9759 | 9377 | 9803 | 9495 | 9856 |
| 2,4 | 9264 | 9766 | 9461 | 9846 | 9589 | 9871 | 9674 | 9908 |
| 2,7 | 9514 | 9847 | 9652 | 9901 | 9728 | 9915 | 9790 | 9940 |
$T_c=1773^\circ K$
Sk = 2
| $F_0$ | $T_0=293^\circ K$, Bi/Sk = 0, center | $T_0=293^\circ K$, Bi/Sk = 0, surface | $T_0=293^\circ K$, Bi/Sk = 0,5, center | $T_0=293^\circ K$, Bi/Sk = 0,5, surface | $T_0=873^\circ K$, Bi/Sk = 0, center | $T_0=873^\circ K$, Bi/Sk = 0, surface | $T_0=873^\circ K$, Bi/Sk = 0,5, center | $T_0=873^\circ K$, Bi/Sk = 0,5, surface |
|---|---|---|---|---|---|---|---|---|
| 0,025 | 1652 | 4998 | 1652 | 5559 | 4924 | 7688 | 4924 | 8044 |
| 0,05 | 1664 | 6375 | 1667 | 7390 | 4924 | 8246 | 4924 | 8542 |
| 0,125 | 2171 | 8133 | 2225 | 8560 | 4970 | 9020 | 4976 | 9171 |
| 0,25 | 3512 | 9010 | 3605 | 9187 | 5346 | 9395 | 5370 | 9483 |
| 0,4 | 4784 | 9374 | 4876 | 9470 | 5756 | 9589 | 5773 | 9647 |
| 0,6 | 5754 | 9606 | 5795 | 9665 | 6873 | 9731 | 6963 | 9769 |
| 0,8 | 6839 | 9732 | 7028 | 9774 | 7980 | 9820 | 8048 | 9846 |
| 1,0 | 7967 | 9819 | 8085 | 9849 | 8664 | 9881 | 8717 | 9899 |
| 1,2 | 8656 | 9880 | 8742 | 9901 | 9113 | 9922 | 9155 | 9934 |
| 1,4 | 9109 | 9921 | 9171 | 9935 | 9412 | 9948 | 9443 | 9956 |
| 1,6 | 9408 | 9948 | 9453 | 9957 | 9609 | 9966 | 9632 | 9971 |
| 1,8 | 9607 | 9965 | 9639 | 9972 | 9740 | 9977 | 9757 | 9981 |
b) steels of the austenitic type
$T_c=1273^\circ K$
Sk = 0,5
| $F_0$ | $T_0=293^\circ K$, Bi/Sk = 0, center | $T_0=293^\circ K$, Bi/Sk = 0, surface | $T_0=293^\circ K$, Bi/Sk = 0,5, center | $T_0=293^\circ K$, Bi/Sk = 0,5, surface | $T_0=873^\circ K$, Bi/Sk = 0, center | $T_0=873^\circ K$, Bi/Sk = 0, surface | $T_0=873^\circ K$, Bi/Sk = 0,5, center | $T_0=873^\circ K$, Bi/Sk = 0,5, surface |
|---|---|---|---|---|---|---|---|---|
| 0,1 | 2434 | 4172 | 2482 | 4670 | 6961 | 7829 | 6979 | 7965 |
| 0,2 | 3122 | 5034 | 3386 | 5637 | 7362 | 8234 | 7436 | 8394 |
| 0,3 | 3986 | 5747 | 4424 | 6394 | 7781 | 8552 | 7896 | 8717 |
| 0,4 | 4804 | 6376 | 5360 | 7029 | 8152 | 8818 | 8292 | 8977 |
| 0,5 | 5547 | 6933 | 6166 | 7566 | 8472 | 9034 | 8623 | 9186 |
| 0,6 | 6212 | 7427 | 6849 | 8025 | 8744 | 9219 | 8897 | 9358 |
| 0,8 | 7315 | 8223 | 7906 | 8730 | 9164 | 9491 | 9302 | 9600 |
| 1,0 | 8137 | 8808 | 8638 | 9199 | 9453 | 9670 | 9563 | 9752 |
| 1,2 | 8738 | 9215 | 9132 | 9500 | 9644 | 9787 | 9728 | 9847 |
| 1,4 | 9162 | 9488 | 9454 | 9690 | 9769 | 9863 | 9832 | 9910 |
| 1,8 | 9642 | 9786 | 9788 | 9881 | — | — | — | — |
Continuation of Appendix 2
| $Bi/Sk$
$F_0$ | \multicolumn{4}{c}{ $T_0=293^\circ K$ } | \multicolumn{4}{c}{ $T_0=874^\circ K$ } |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | 0 center | 0 surface | 0,5 center | 0,5 surface | 0 center | 0 surface | 0,5 center | 0,5 surface |
$T_c=1273^\circ K$ $Sk=2$
| $F_0$ | 0 center | 0 surface | 0,5 center | 0,5 surface | 0 center | 0 surface | 0,5 center | 0,5 surface |
|---|---|---|---|---|---|---|---|---|
| 0,05 | 2322 | 6495 | 2329 | 7068 | 6879 | 8773 | 6882 | 8915 |
| 0,1 | 2842 | 7598 | 2969 | 8027 | 7185 | 9162 | 7219 | 9272 |
| 0,15 | 3921 | 8229 | 4186 | 8568 | 7658 | 9375 | 7718 | 9463 |
| 0,2 | 5069 | 8650 | 5397 | 8919 | 8107 | 9519 | 8179 | 9590 |
| 0,25 | 6068 | 8957 | 6399 | 9170 | 8487 | 9628 | 8553 | 9683 |
| 0,3 | 6879 | 9189 | 7184 | 9359 | 8796 | 9707 | 8870 | 9754 |
| 0,35 | 7520 | 9367 | 7788 | 9504 | 9045 | 9770 | 9113 | 9809 |
| 0,4 | 8024 | 9305 | 8260 | 9615 | 9243 | 9819 | 9305 | 9851 |
| 0,5 | 8752 | 9693 | 8927 | 9767 | 9525 | 9887 | 9573 | 9909 |
| 0,6 | 9218 | 9815 | 9343 | 9861 | 9704 | 9931 | 9739 | 9945 |
| 0,8 | 9697 | 9930 | 9756 | 9949 | — | — | — | — |
$T_c=1773^\circ K$ $Sk=0,5$
| $F_0$ | 0 center | 0 surface | 0,5 center | 0,5 surface | 0 center | 0 surface | 0,5 center | 0,5 surface |
|---|---|---|---|---|---|---|---|---|
| 0,1 | 1787 | 3494 | 1841 | 4025 | 5056 | 6197 | 5088 | 6454 |
| 0,2 | 2497 | 4338 | 2782 | 4992 | 5584 | 6806 | 5720 | 7128 |
| 0,3 | 3355 | 5046 | 3846 | 5764 | 6169 | 7316 | 6395 | 7663 |
| 0,4 | 4170 | 5681 | 4796 | 6458 | 6721 | 7759 | 7007 | 8110 |
| 0,5 | 4914 | 6280 | 5610 | 7075 | 7222 | 8143 | 7539 | 8480 |
| 0,6 | 5582 | 6838 | 6320 | 7617 | 7668 | 8476 | 7996 | 8789 |
| 0,8 | 6746 | 7783 | 7494 | 8454 | 8396 | 8986 | 8700 | 9237 |
| 1,0 | 7693 | 8494 | 8351 | 9018 | 8923 | 9335 | 9173 | 9524 |
| 1,2 | 8415 | 9000 | 8941 | 9385 | 9238 | 9568 | 9480 | 9709 |
| 1,4 | 8936 | 9344 | 9331 | 9618 | 9535 | 9721 | 9680 | 9818 |
| 1,6 | 9297 | 9574 | 9581 | 9763 | 9698 | 9820 | 9799 | 9887 |
| 2,0 | 9702 | 9822 | 9824 | 9911 | — | — | — | — |
$T_c=1773^\circ K$ $Sk=2$
| $F_0$ | 0 center | 0 surface | 0,5 center | 0,5 surface | 0 center | 0 surface | 0,5 center | 0,5 surface |
|---|---|---|---|---|---|---|---|---|
| 0,05 | 1674 | 5781 | 1683 | 6503 | 4952 | 7743 | 4959 | 8054 |
| 0,1 | 2228 | 7031 | 2367 | 7677 | 5390 | 8468 | 5459 | 8708 |
| 0,15 | 3383 | 7866 | 3716 | 8322 | 6100 | 8871 | 6228 | 9058 |
| 0,2 | 4594 | 8389 | 4991 | 8742 | 6810 | 9140 | 6967 | 9287 |
| 0,25 | 5619 | 8765 | 6010 | 9041 | 7430 | 9334 | 7594 | 9452 |
| 0,3 | 6459 | 9045 | 6832 | 9263 | 7944 | 9482 | 8101 | 9577 |
| 0,35 | 7155 | 9258 | 7496 | 9432 | 8362 | 9594 | 8506 | 9672 |
| 0,4 | 7725 | 9422 | 8027 | 9560 | 8698 | 9682 | 8827 | 9745 |
| 0,5 | 8558 | 9646 | 8782 | 9734 | 9182 | 9804 | 9279 | 9845 |
| 0,6 | 9095 | 9785 | 9253 | 9842 | 9489 | 9880 | 9559 | 9907 |
| 0,7 | 9436 | 9868 | 9544 | 9904 | 9682 | 9926 | 9731 | 9944 |
| 0,9 | 9782 | 9950 | 9819 | 9962 | — | — | — | — |
In order to verify the correctness of the initial assumptions, the results of numerical integration of system (1)–(4) were compared with data from experimental investigations of the heating of articles [6, 12, 13]. The comparison results are presented in Figs. 1 [6], 2 [12], 3 [13].
Fig. 1. Heating of a cylinder of bearing steel, \(\varnothing = 125\) mm:
1—relative temperature at the surface; 2—relative temperature at the center.
(In the plot: solid line—experiment; dashed line—calculation.)
Fig. 2. Heating of a cylinder of carbon steel \((0.4\%\,\mathrm{C})\), \(\varnothing 100\) mm:
1—relative temperature at the surface of the cylinder; 2—relative temperature at the center.
(In the plot: solid line—experiment; dashed line—calculation.)
Some discrepancy between the experimental and calculated curves may be explained by the fact that, although structural transformations in carbon steels occur within the adopted temperature interval, the moment of the beginning or end of phase transformations for a particular grade of steel may differ somewhat from the calculated one. Nevertheless, despite some error introduced by possible discrepancies, analysis of the comparison shows sufficiently reliable agreement of the calculated data with the experimental data. The latter indicates that the adopted mathematical model corresponds sufficiently well to the real physical process of metal heating.
Fig. 3. Heating of a cylinder of steel 45Х, \(\varnothing 650\):
1—relative temperature at the surface of the cylinder; 2—relative temperature at a depth of 85 mm; 3—relative temperature at the center.
(In the plot: solid line—experiment; dashed line—calculation.)
Using the data presented in the work, it is easy to determine the temperature distribution in steel ingots. In addition, knowing the surface temperature and the temperature of the surrounding medium, one can find the unsteady heat flux at the surface of the article being heated.
Principal notation
\[ \Theta = \frac{T}{T_c} \]
— relative temperature;
\(2S\)—thickness of the plate under symmetric heating, m;
Fig. 4. Nomogram for determining the relative temperature at the center of a plate made of austenitic steel: a — \(S_k = 0.5\), b — \(1.0\), c — \(2.0\), d — \(4.0\).
Fig. 5. Nomogram for determining the relative surface temperature of a plate made of austenitic steel: a — \(S_k = 0.5\); b — \(1.0\); c — \(2.0\); d — \(4.0\).
Fig. 6. Nomogram for determining the relative temperature at the center of a cylinder made of austenitic steel:
a — \(S_k = 0.5\), b — \(1.0\), c — \(2.0\), d — \(4.0\).
Fig. 7. Nomogram for determining the relative surface temperature of a cylinder made of austenitic steel:
a — \(Sk = 0.5\), b — \(1.0\), c — \(2.0\), d — \(4.0\).
\(\lambda_0\) — coefficient of thermal conductivity at 273° K, W/m·deg;
\(c_0\) — heat capacity at 273° K, J/kg·deg;
\[ \xi=\frac{x}{S}\left(\text{or } \frac{r}{R}\right), \qquad h=\frac{\Delta x}{S}=\frac{1}{N}, \]
\(N\) — the number of layers into which the thickness of the plate \((S)\) or the radius of the cylinder \((R)\) is divided;
\[
S_k=\frac{\sigma_{\mathrm{B}}T_c^3S}{\lambda_0}
\]
— the Stark criterion;
\[
F_o=\frac{\lambda_0}{c_0\rho}\cdot\frac{\tau}{S^2}
\]
— the Fourier criterion;
\[
\mathrm{Bi}=\frac{aS}{\lambda_0}
\]
— the Biot criterion.
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