Approximate Calculation of the Temperature Field During Heating of Bodies by Radiant Heat
Abstract Generated abstract
This paper develops an approximate method for calculating temperature fields in bodies heated by radiant heat when the influence of initial conditions has largely disappeared. It represents the temperature at any point as the center temperature plus coordinate dependent quadratic increments, extending known one dimensional solutions for plates, cylinders, and spheres to infinite bars, finite cylinders, cubes, and parallelepipeds. Empirical correction factors are introduced for multidimensional cases, with central temperatures supplied by existing formulas, grid calculations, temperature criteria methods, or experiment. Comparisons with numerical and experimental data are reported, and the resulting formulas are recommended for practical calculations for Fourier criterion values greater than 0.25.
Full Text
APPROXIMATE CALCULATION OF THE TEMPERATURE FIELD DURING HEATING OF BODIES BY RADIANT HEAT
G. P. BOIKOV
(Presented by Prof. T. I. Fuks, Doctor of Technical Sciences)
It is known that solutions of the differential equation of heat conduction under boundary conditions of the second kind contain an infinite, rapidly convergent series, which in mathematical form reflects the influence of the initial conditions on the process of heat propagation [1], [2].
There are many practical problems in which the process develops in such a way that the initial conditions very quickly cease to have an effect. In these cases, when calculating the temperature field, the infinite series is usually neglected, owing to which the available solutions for bodies of classical form [1], [2] are greatly simplified. The latter makes it possible, in a certain approximation, to find the temperature field in bodies when heat propagates in more than one dimension.
The essence of the proposed method is that the temperature of any point in space is recommended to be found by means of the corresponding increment to the temperature of the center of the body, i.e., the solution of the problem is represented in the form
\[ T(x,y,z,\tau)=T(0,0,0,\tau)+\Phi . \tag{1} \]
The existing solutions in the literature for bodies of classical form in which the temperature field is characterized by one space coordinate are most simply (and mathematically exactly) reduced to form (1). Thus, the expressions for an infinite plate, an infinite cylinder, and a sphere [1] are
\[ T(x,\tau)=T_0+\frac{q_c R}{\lambda} \left[ \frac{a\tau}{R^2}-\frac{R^2-3x^2}{6R^2} \right]; \]
\[ T(x,\tau)=T_0+\frac{q_c R}{\lambda} \left[ 2\frac{a\tau}{R^2}-\frac{R^2-2x^2}{4R^2} \right]; \]
\[ T(x,\tau)=T_0+\frac{q_c R}{\lambda} \left[ 3\frac{a\tau}{R^2}-\frac{3R^2-5x^2}{10R^2} \right], \]
having center temperatures respectively
\[ T(0,\tau)=T_0+\frac{q_c R}{\lambda} \left[\frac{a\tau}{R^2}-\frac{1}{6}\right]; \]
\[ T(0,\tau)=T_0+\frac{q_c R}{\lambda} \left[2\frac{a\tau}{R^2}-\frac{1}{4}\right]; \]
\[ T(0,\tau)=T_0+\frac{q_c R}{\lambda} \left[3\frac{a\tau}{R^2}-\frac{3}{10}\right], \]
are represented by the following generalized expression:
\[ T(x,\tau)=T(0,\tau)+\frac{q_c x^2}{2\lambda R}. \tag{2} \]
As is evident, this expression, identical for an infinite plate, an infinite cylinder, and a sphere, fully corresponds to the form (1).
Earlier [3] it was shown that the temperature field in any bodies (when heat propagates in more than one dimension) along the heat-flow line is calculated by the formula
\[ T(x,0,\tau)=T_0+\frac{q_c R_1}{\lambda} \left\{ \xi\frac{a\tau}{R_1^2} -\left[ \frac{\xi}{2(\xi+2)} -\frac{1}{2}\frac{x^2}{R_1^2} \right] \right\}, \tag{3} \]
whence the center temperature is
\[ T(0,0,\tau)=T_0+\frac{q_c R_1}{\lambda} \left\{ \xi\frac{a\tau}{R_1^2} -\frac{\xi}{2(\xi+2)} \right\}. \tag{3′} \]
Comparing (3) and (3′), we obtain
\[ T(x,0,\tau)=T(0,0,\tau)+\frac{q_c x^2}{2\lambda R_1}, \]
which corresponds to the form (1).
As has already been noted [4], the ratio of the components of the divergence of the temperature gradient
\[ \frac{\partial^2 T}{\partial y^2}:\frac{\partial^2 T}{\partial x^2}=\beta \]
may be regarded as a quantity independent of \(y\). Therefore, using (3) and carrying out double integration with respect to the variable \(y\), we obtain
\[ T(x,y,\tau)= \beta\frac{q_c y^2}{2\lambda R_1} +\left[F_1(x,\tau)+C_1\right]y +\left[F_2(x,\tau)+C_2\right]. \]
By the symmetry condition,
\[ \left[F_1(x,\tau)+C_1\right]=0. \]
According to the central conditions (for \(y=0\)),
\[ [F_2(x,\tau)+C_2]=T(x,0,\tau). \]
Then
\[ T(x,y,\tau)=T(x,0,\tau)+\beta \frac{q_c y^2}{2\lambda R_1} = T(0,0,\tau)+\frac{q_c x^2}{2\lambda R_1} +\beta \frac{q_c y^2}{2\lambda R_1}, \]
or, equivalently,
\[ T(x,y,\tau)=T_0+\frac{q_c R_1}{\lambda} \left\{ \xi \frac{a\tau}{R_1^2} - \left[ \frac{\xi}{2(\xi+2)} - \frac{1}{2}\frac{x^2}{R_1^2} \right] \right\} +\beta \frac{q_c y^2}{2\lambda R_1}. \tag{4} \]
Fig. 1. Graph of heating and cooling of a ceramic cylinder \(d=40\) mm.
— — — change in the rib temperature according to formula (6)
(central conditions taken from experimental data);
—— the same according to experimental data [6].
If \(\beta=\xi-1\) is assumed, then (4) satisfies the differential equation of heat conduction for a two-dimensional field and may serve as the basis for calculating the temperature in an infinitely long bar of square and rectangular cross sections, for which the values of the coefficient \(\xi\) are determined by the formula
\[ \xi=1+\frac{R_1^2}{R_2^2+0.5R_1^2}\,[3]. \]
An expression analogous to (4) for a variable heat flux (ibid.) has the form
\[ T_m(x,y)=T_0+\frac{R_1}{\lambda} \left\{ \xi\frac{a\tau_1}{R_1^2} \sum_{i=1}^{i=m} q_{ci}\frac{c_1}{c_i} \right. \]
\[ \left. -\, q_{cm}\frac{\lambda_1}{\lambda_m} \left[ \frac{\xi}{2(\xi+2)} - \frac{1}{2}\frac{x^2}{R_1^2} \right] \right\} +\beta \frac{q_{cm}y^2}{2\lambda_m R_1}. \tag{4′} \]
Referring to a certain degree of approximation, (4′) in this form could already be recommended for calculations of an infinite bar.
However, with such a choice of \(\beta\) and \(y \ne 0\), satisfactory results are obtained for cases in which \(R_1\) and \(R_2\) are close to each other and \(x = 0.5 R_1\). In this connection it is further proposed to depart from the analytical method of investigation and to carry out the calculation by the formula
Fig. 2. Graph of heating and cooling of a ceramic parallelepiped \(40 \times 80 \times 160\) mm.
— — — — change in the temperature of the edge and corner according to formula (7)
(central conditions taken from experimental data);
——— the same according to experimental data [6].
\[ T(x,y,\tau)=T(x,0,\tau)+\beta^*\,\frac{q_{cm}y^2}{2\lambda_m R_1}, \tag{5} \]
and to regard the factor \(\beta^*\) as empirical. For an infinitely long bar of square cross section, the value of this factor is recommended to be chosen on the basis of the available relation
\[ \beta=\xi-1=\frac{R_1^2}{R_2^2+0.5R_1^2}=\frac{1}{1+0.5}, \]
where, instead of the coefficient \(0.5\) (in the denominator), the ratio \(\dfrac{x}{R_1}\) is substituted,
\[ \beta^*_{kv}=\frac{1}{1+\left(\dfrac{x}{R_1}\right)}. \tag{5′} \]
In the case of an infinitely long bar of rectangular cross section (with a constant temperature of the heat source), satisfactory...
results are obtained when the coefficient \(\beta^*\) is chosen in the form of the function
\[ \beta^*_{pr} = \left[ \left(0.5-\frac{R_1^2}{R_2^2+R_1^2}\right) \frac{a\tau}{R_1^2} + \frac{1}{2-\dfrac{R_1^2}{R_2^2}} + 0.666 \left(1-\frac{x^2}{R_1^2}\right) \left(1-\frac{R_1^2}{R_2^2}\right) \right] \cdot \frac{R_1^2}{R_2^2+\left(\dfrac{x}{R_1}\right)R_1^2}, \tag{5″} \]
which, for \(R_1=R_2\), passes into expression (5′).
For a cylinder, the calculation dependences are established in exactly the same way as for an infinite bar. Expressions analogous to (5) and (5″) are written here as
\[ T(r,z,\tau)=T(r,0,\tau)+\beta^*\cdot\frac{q_{cm}z^2}{2\lambda_m R_1} \tag{6} \]
and
\[ \beta^* = \left[ \left(0.5-\frac{R_1^2}{R_2^2+R_1^2}\right) \frac{a\tau}{R_1^2} + \frac{1}{2-\dfrac{R_1^2}{R_2^2}} + 0.666 \left(1-\frac{r^2}{R_1^2}\right) \left(1-\frac{R_1^2}{R_2^2}\right) \right] \cdot \frac{R_1^2}{R_2^2+\left(\dfrac{r}{R_1}\right)R_1^2}. \tag{6″} \]
For \(R_1=R_2\)
\[ \beta^*=\frac{1}{1+\dfrac{r}{R_1}}. \tag{6′} \]
To obtain relations for calculating the temperature field in a cube and in a parallelepiped, one may proceed from the following solution
\[ T(x,y,z,\tau)=T(x,y,0,\tau)+\gamma^*\frac{q_{cm}z^2}{2\lambda_m R_1}, \]
where \(T(x,y,0,\tau)\) are the central conditions (at \(z=0\)), and \(\gamma^*\) is the analog of \(\beta^*\). Expanding \(T(x,y,0,\tau)\), we obtain
\[ T(x,y,z,\tau) = T(x,0,\tau) + \beta^*\cdot\frac{q_{cm}y^2}{2\lambda_m R_1} + \gamma^*\cdot\frac{q_{cm}z^2}{2\lambda_m R_1}. \tag{7} \]
For a cube \((R_3=R_2=R_1)\):
\[ \gamma^*=\beta^*=\beta^*_{kv}=\frac{1}{1+\dfrac{x}{R_1}}. \tag{7′} \]
For a parallelepiped \(\beta^*=\beta^*_{pr}\) (see formula 5″)
\[ t^*=\left[\left(0.5-\frac{R_1^2}{R_3^2+R_1^2}\right)\frac{a\tau}{k_1^2} +\frac{1}{2-\dfrac{R_1^2}{R_3^2}} +0.666\left(1-\frac{x^2}{R_1^2}\right)\left(1-\frac{R_1^2}{R_3^2}\right)\right] \frac{R_1}{R_3^2+\left(\dfrac{x}{R_1}\right)R_1^2}. \tag{7''} \]
Here, as in the preceding formulas,
\[ T(x,0,\tau) \]
is the temperature field along the line of heat flow in the body [4] (from the center of the body along the smallest dimension);
\[ \lambda_m \]
is the average coefficient of thermal conductivity of the body at the given instant of time;
\[ q_{\mathrm{st}} \]
is the specific heat flux at the surface at the given instant of time, directed along the \(x\)-axis;
\[ 2R_1 < 2R_2 < 2R_3 \]
are the dimensions of the body.
Figure labels: \(t^\circ\mathrm{C}\); surface point, \(d''\); diagonal point, \(a''\); \(2R_1\); \(2R_2\); \(2R_3\); \(\tau\), min.; \(\mathrm{Fo}_1=0.287\).
Fig. 3. Heating curve of a steel cube with side \(457\ \mathrm{mm}\).
— — — temperature variation of the points according to formula (7)
(the central conditions were found by the network method);
—— the same by the network method [5].
Expressions (5), (6), (7) can be used as calculation formulas in a whole series of problems when the central conditions are known. The latter may be found from formula \((4')\), by the network method, by the method of multiplying temperature criteria, or on the basis of experiment.
The specific heat flux at each instant of time must be determined by the law of convection or by the law of radiation.
To test the calculation procedure set forth above, the grid method (data of V. N. Sokolov [5]), the method of multiplying temperature criteria, and some experimental data of Budrin and Krasovskii [6] were used. In all, about fifty comparisons were made. Some of them are presented in Figs. 1–4.
Fig. 4. Heating curve of a steel parallelepiped \(200 \times 400 \times 400\) mm.
— — — change in the temperature of the corner and of the middle of the small surface according to formula (7) (the central conditions were found by the method of multiplying temperature criteria);
—— the same by the method of multiplying temperature criteria.
Formulas (5), (6), and (7) are recommended for practical calculations as approximate ones for values of the Fourier criterion
\[ \frac{a \tau}{R_1^2} > 0.25. \]
The heat consumption in this case is calculated according to the principles of the theory of heat conduction [1]
\[ Q = c \gamma \left[\frac{1}{v}\int_{v} T\,dv - T_0\right],\ \frac{\text{kcal}}{\text{m}^3}. \]
Conclusion
The temperature distribution in an infinite bar of square and rectangular cross-section, in a cylinder of finite dimensions, in a cube, and in a parallelepiped under conditions of process stabilization can be obtained approximately by making the appropriate addition to the temperature at the center of the body.
References
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A. V. Lykov. Theory of Heat Conduction, GITTL, Moscow, 1952.
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G. P. Boikov. Heating of bodies under the action of radiant heat, Izvestiya TPI, vol. 89, Tomsk, 1957.
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G. P. Boikov. Heating of bodies of finite dimensions under the action of radiant heat (second communication), Izvestiya TPI, vol. 101, Tomsk, 1958.
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G. P. Boikov. Heating of bodies of finite dimensions under the action of radiant heat (first communication), Izvestiya TPI, vol. 100, Tomsk, 1958.
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V. N. Sokolov. Calculation of the heating of bodies by the mesh method, TsNIIMASH. (Heating of large ingots), GITIML, Moscow, 1954.
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D. V. Budrin, B. L. Krasovskii. Heating and cooling of bodies of various shapes, Transactions of the Ural Industrial Institute named after S. M. Kirov, issue XVII, GNTIL, Sverdlovsk–Moscow, 1941.