On the Problem of Calculating Steel Heating with Account for the Heat of Recrystallization
Abstract Generated abstract
This paper addresses discrepancies between calculated and observed temperature fields during radiant heating of steel in the temperature interval of structural transformation. It proposes a refinement to earlier unsteady heat conduction formulas by representing the heat absorbed during recrystallization as an average temperature drop over the heated body, with separate expressions derived for an infinite plate, cylinder, and sphere. The method is illustrated for a 300 mm diameter cylinder of 9Kh steel, using published transformation heat and thermophysical data, and the resulting heating curves are compared with experimental measurements. The author concludes that including the heat of transformation improves practical calculations of steel heating to a useful extent.
Full Text
ON THE PROBLEM OF CALCULATING STEEL HEATING TAKING INTO ACCOUNT THE HEAT OF RECRYSTALLIZATION
G. P. BOIKOV
Presented by Professor G. I. Fuks
Earlier, computational relations were proposed for determining the temperature field in solid bodies when they are heated by radiant heat [4], [5]. Thus, for example, the formula for an infinite cylinder was given in the form (1):
\[ \begin{aligned} \frac{T_m(r)}{T_c} &= \frac{T_0}{T_c} + \frac{g_c \cdot R}{\lambda_1 \cdot T_c} \Bigg\{ 2 \frac{a_1 \tau_1}{R^2} \cdot \sum_{i=1}^{i=m} \frac{g_{ci}}{g_c} \cdot \frac{c_1}{c_i} - \frac{1}{4} \cdot \frac{g_{cm}}{g_c} \cdot \frac{\lambda_1}{\lambda_m} \cdot \left(1 - 2 \frac{r^2}{R^2}\right) \\ &\quad - \sum_{n=1}^{\infty} \frac{2}{\mu_n^2 \cdot I_0(\mu_n)} \cdot I_0\!\left(\mu_n \frac{r}{R}\right) \cdot \left[ \frac{g_{c1}}{g_c} \cdot e^{-\mu_n^2 \sum_{i=1}^{i=m} \frac{a_i \cdot \tau_1}{R^2}} \right. \\ &\qquad\left. - \sum_{i=1}^{i=m-1} \left( \frac{g_{ci}}{g_c} \cdot \frac{\lambda_1}{\lambda_i} - \frac{g_{c(i+1)}}{g_c} \cdot \frac{\lambda_1}{\lambda_{(i+1)}} \right) \cdot e^{-\mu_n^2 \sum_{k=1}^{k=m-i} \frac{a_{(i+k)} \tau_1}{R^2}} \right] \Bigg\}, \end{aligned} \tag{1} \]
where the calculated time interval and the radiant flux were determined from the relations:
\[ \tau_1 = \frac{0.022 \cdot \lambda_1 \cdot T_c \cdot R}{g_c \cdot a_1}, \]
\[ \varphi < 1 - \frac{a_2}{a_1} \left\{ 1 - \frac{ 2\left(g_{c1} - g_{c2} \cdot \dfrac{\lambda_1}{\lambda_2}\right) }{ g_{c1} \left[ 1 + \sum_{n=1}^{\infty} e^{-\mu_n^2 \dfrac{a_1 \tau_1}{R^2}} \right] } \right\}, \]
\[ g_{ci} = \varepsilon_n C_0 \left[ \left(\frac{T_c}{100}\right)^4 - \left(\frac{T_{i-1}}{100}\right)^4 \right]. \]
Analogous dependences were also proposed for a number of other bodies.
Investigations showed that the proposed computational formulas reflect the actual heating process sufficiently satisfactorily. However, when steel is heated, an appreciable discrepancy between experimental and calculated data is observed in the interval of structural transformation. This is explained by the fact that the system of differential equations describing the process of unsteady heat conduction does not take into account the absorption of part of the heat when the crystal lattice changes. The indicated
transformation is observed during the period of heating when the temperature reaches a value definite for the given steel.
Since, when bodies are heated, the particles of matter at the surface are heated more rapidly, the absorption of the heat of recrystallization also begins in the layers close to the surface, gradually (as the temperature rises) spreading into the depth of the body. This phenomenon undoubtedly affects the character of the temperature field. We have made an attempt, as far as possible, to take into account the influence of the heat absorbed during recrystallization, and it is shown how to introduce a refinement into the calculation method.
The influence of the heat absorbed in the process of transformation on the heating of steel can to a certain extent be taken into account if certain assumptions are adopted and if one proceeds from the following considerations. Suppose we have some temperature distribution, for example in an infinite plate (Fig. 1a), on which we mark the points \(L, K, M\). As a result of the absorption
Fig. 1
of heat in the layer \(x_i - x_{i+1}\) in the zone of volume \(aa\), the temperature drops from the points \(K\) to the points \(K'\) (Fig. 1b). The lowering of the temperature in the indicated region causes an influx of heat from other parts of the body. This, in turn, causes a decrease in the temperature from the points \(M\) and \(L\) to the points \(M''\) and \(L''\), as well as in other regions. At the same time, in the zone \(x_i - x_{i+1}\), the temperatures rise from \(K'\) to the points \(K''\).
Finally, the points \(L^*, K^*, K^*, M^*\) (Fig. 1c) lie on the line \(L^* K^* K^* M^*\), which is described by an equation analogous to that of the original temperature distribution \(LKKM\). Further heating proceeds in the usual way and is calculated by the same relations, until a structural transformation again occurs in some part of the body and, in connection with this, a new portion of heat is lost. Under symmetric heating the phenomenon described occurs simultaneously in regions to the right and to the left of the axis of symmetry of the body.
In order to be able to use the calculation formulas with such a representation of the process, it is necessary to know the value of the average temperature drop \(\Delta T_{\mathrm{pr}}\). This value will be different for bodies of different configuration. If, for example, we are dealing with the heating of an infinite plate, then the amount of heat that was
absorbed in the volume of the plate bounded by the cross sections \(x_i\) and \(x_{i+1}\) can be determined as
\[ Q_{pr}=g_{pr}(x_i-x_{i+1})\cdot 1\cdot 1\cdot \gamma\ \text{kcal}, \]
where \(g_{pr}\ \dfrac{\text{kcal}}{\text{kg}}\) is the heat absorbed by a unit weight of the substance, determined on the basis of special experiments [1].
The amount of heat \(Q_{pr}\) determined in this way goes toward lowering the temperature of the entire body under consideration. Therefore
\[ Q_{pr}=c_{cp}\cdot \gamma \cdot \Delta T_{pr}\cdot R\cdot 1\cdot 1\ \text{kcal}. \]
Comparing these two equations, we find:
\[ \Delta T_{pr}^{pl}=\frac{g_{pr}}{c_{cp}}\cdot \frac{(x_i-x_{i+1})}{R}. \]
The same reasoning makes it possible to find the average temperature drop for any other bodies as well. Thus, for an infinite cylinder and a sphere we obtain:
\[ \Delta T_{pr}^{cyl}=\frac{g_{pr}}{c_{cp}}\cdot \frac{(r_i^2-r_{i+1}^2)}{R^2}, \]
\[ \Delta T_{pr}^{sphere}=\frac{g_{pr}}{c_{cp}}\cdot \frac{(r_i^3-r_{i+1}^3)}{R^3}. \]
In the formulas presented for determining the average temperature drop, the quantities \((x_i-x_{i+1})\) and \((r_i-r_{i+1})\) are the portions of the body in which heat absorption has occurred as a result of the transformation;
\[ c_{cp}\ \frac{\text{kcal}}{\text{kg deg}} \]
is the average heat capacity of the substance at the given moment;
\(2R\,[m]\) is the thickness or diameter of the body.
Figure 2 shows the heating of a cylinder \(d=300\ \text{mm}\) made of 9Kh steel. Curves 1, showing the change in the temperature of the surface and center of the cylinder, were constructed by formula (1) without taking account of the absorption of the heat of recrystallization. Curves 2, showing the change in the temperature of the surface and center of the cylinder, were constructed by formula (1) with allowance for the absorption of the heat of recrystallization. In the calculations, the numerical value of the heat of transformation (14.53 kcal/kg) was taken according to the data of A. I. Lazarev [1].
Allowance for the variation of the thermophysical characteristics as a function of temperature was made using the graphs given in the book by A. L. Nemchinsky [2]. The curves of the temperatures at the surface and center of the cylinder constructed from the experimental data of V. M. Degtyarev [3] are denoted by number 3.
Fig. 2. Heating of a cylinder with \(d = 300\) mm made of steel 9Kh.
1. Change in temperature at the surface and at the center of the cylinder, plotted according to formula (1) without taking into account absorption of the heat of recrystallization.
2. The same, taking into account absorption of the heat of recrystallization.
3. Change in temperature at the surface and at the center of the cylinder, plotted from the experimental data of V. M. Degtyarev.
CONCLUSION
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The method presented makes it possible, to a certain extent, to take into account the influence of the heat of transformation on the temperature field in steel bodies when they are heated under the action of radiant heat.
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The proposed method can be recommended for practical calculations.
LITERATURE
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A. I. Lazarev — New methods and instruments for studying the heat of transformation and the true heat capacity of metals at high temperatures, (Dissertation), Leningrad, Institute of Precision Mechanics and Optics, 1955.
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A. L. Nemchinskii — Thermal calculations of heat treatment, GIZL, 1953.
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V. M. Degtyarev — Rapid heating in the heat treatment of articles of large cross sections, Mashgiz, 1953.
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G. P. Boikov — Heating of bodies under the action of radiant heat (first communication), Izv. TPI, vol. 89, 1957, Tomsk.
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G. P. Boikov — Heating of bodies under the action of radiant heat (second communication), Izv. TPI, 89, 1957, Tomsk.