On Radiant Heating of Bodies at a Variable Temperature of the Heat Source: First Report
Abstract Generated abstract
This paper examines transient heating of solid bodies when the temperature of the heat source varies with time, with emphasis on radiant heating. Starting from the analogous convective heat conduction problem for an infinite plate with a linearly increasing medium temperature, the authors derive a Laplace transform solution and identify unordered, ordered, and synchronous thermal regimes. They then propose approximate relations for radiant heating by combining the constant source temperature solution with correction terms involving relative temperature and the Kirpichev criterion. Comparisons with zonal numerical calculations for plate heating indicate satisfactory agreement within the ordered thermal regime over the tested range of initial to source temperature ratios.
Full Text
On the Radiant Heating of Bodies with a Variable Heat Source Temperature
V. V. Salomatov and G. I. Boikov
(Presented by Professor, Doctor G. I. Fuks)
A first conception of the heating of bodies at a variable temperature of the medium can be obtained on the basis of solutions for convective heat exchange. However, the available solutions [1] are for the most part given for such conditions when the temperature of the medium changes from the initial temperature of the heated body. Of somewhat greater interest are the conditions expressed by the system of equations
\[ \frac{\partial T(x,\tau)}{\partial \tau} = a \frac{\partial^2 T(x,\tau)}{\partial x^2}, \tag{1} \]
\[ T(x,0)=T_0, \tag{2} \]
\[ \frac{\partial T(0,\tau)}{\partial x}=0, \tag{3} \]
for \(x=R\)
\[ \lambda \frac{\partial T(x,\tau)}{\partial x} +\alpha\left[(T_{c.o}+b\tau)-T(x,\tau)\right]=0. \tag{4} \]
Applying the Laplace integral transform to the system (1)–(4), we obtain the solution in transforms
\[ T(x,s)-\frac{T_0}{s} = A\,\operatorname{ch}\sqrt{\frac{s}{a}}\,x. \tag{5} \]
Satisfying the boundary condition (4), we find the constant \(A\). Then, in general form,
\[ T(x,s) = \frac{T_0}{s} + \frac{(T_{c.o}-T_0)\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {s\left(\operatorname{ch}\sqrt{\frac{s}{a}}\,R +\frac{1}{H}\sqrt{\frac{s}{a}}\operatorname{sh}\sqrt{\frac{s}{a}}\,R\right)} + \frac{b\,\operatorname{ch}\sqrt{\frac{s}{a}}\,x} {s^2\left(\operatorname{ch}\sqrt{\frac{s}{a}}\,R +\frac{1}{H}\sqrt{\frac{s}{a}}\operatorname{sh}\sqrt{\frac{s}{a}}\,R\right)}. \tag{6} \]
The inverse transformation leads to the expression
\[ \begin{aligned} T(x,\tau) ={}& T_o + (T_{c0}-T_o) \left[ 1-\sum_{n=1}^{n=\infty} A_n \cos \nu_n \frac{x}{R} \exp\left(-\nu_n^2 \frac{a\tau}{R^2}\right) \right] + \\ &+ b\tau - \frac{bR^2}{2a} \left[ 1-\left(\frac{x}{R}\right)^2 \right] -\frac{bR^2}{a} \left[ \frac{1}{Bi} - \sum_{n=1}^{n=\infty} \frac{A_n}{\nu_n^2} \cos \nu_n \frac{x}{R} \exp\left(-\nu_n^2 \frac{a\tau}{R^2}\right) \right]. \end{aligned} \tag{7} \]
Here \(\nu_n\) are the roots of the characteristic equation
\[ \nu_n \operatorname{tg}\nu_n = Bi. \]
Solution (7) makes it possible to note the following:
- The entire process of heating solid bodies may be conventionally divided into three regimes:
a) an unordered thermal regime, when it is necessary to take into account several terms of the infinite series; b) an ordered thermal regime, when it is sufficient to retain only one term of the series; c) a synchronous thermal regime, when the temperature of the surface of the body lags behind the temperature of the medium by some constant value. This constant, all other conditions being equal, is the smaller the larger the criterion of convective heat exchange. At \(Bi \to \infty\) the surface temperature in this regime coincides with the variable temperature of the medium.
- The solution of the problem for heating with a nonstationary temperature of the medium is composed of the sum of three functions:
a) the first function is the solution of the problem for a constant source temperature;
b) the second function is determined by the law of increase of the temperature of the surrounding medium and by the criterion of the dimensionless coordinate;
c) the third function, beginning with certain values of the criterion \(Fo\), tends to zero as \(Bi \to \infty\).
Under practical conditions, the cases of radiative heating with a variable temperature of the heat source are of greatest importance. The solution of such a problem is associated with mathematical difficulties. In the literature [2, 3] one may find a number of devices which, to some approximation, give a solution of the posed problem.
We believe that a technical solution of the indicated problem can be obtained on the basis of idealized conceptions of the process of convective heat exchange. Thus, for example, for an infinite plate the solution, analogous to (7), may be represented in the form
\[ T(x,\tau)=T^*(x,\tau)+b\tau -\frac{bR^2}{2a} \left[ 1-\left(\frac{x}{R}\right)^2 \right] -\frac{bR^2}{a}F(\Theta^*,Ki). \tag{8} \]
Here \(T^*(x,\tau)\) is the solution of the problem of radiant heating at a constant temperature of the heat source;
\[
F(\Theta^*,Ki)
\]
is a certain function of the relative temperature and of the criterion of radiant heat exchange \(Ki\);
An appropriate choice of the form of the function \(F(\Theta^*, Ki)\) makes it possible to recommend formula (8) as a first approximation. Investigations have shown that the results of calculation according to (8) agree satisfactorily with calculations performed by a numerical method, if one takes
\[ F(\Theta^*, Ki)=\frac{\Theta^*}{Ki\left(e^{\sqrt[4]{Ki}}+1\right)}, \]
where
\[ \Theta^*=\frac{T^*-T_0}{T_{co}-T_0} \]
is the known criterion in heating for a constant temperature of the heat source.
It should be noted that there can be no complete analogy between (7) and (8), since the criterion of radiant heat exchange (the Kirpichev criterion) is, in the present case, a variable quantity.
It is known [4] that the temperature field inside a body can be determined on the basis of the expression
\[ \Theta(X,Fo)=\int_0^{Fo} Q(Fo)\,dFo -2\sum_{n=1}^{n=\infty}(-1)^n \cos n\pi X\, e^{-n^2\pi^2 Fo} \int_0^{Fo} e^{n^2\pi^2 Fo} Q(Fo)\,dFo . \tag{9} \]
Here
\[ X=\frac{x}{R}, \]
\[ \Theta=\frac{T(x,\tau)-T_0}{T_{co}-T_0}, \]
\[ Q=\frac{q(\tau)R}{\lambda\left(T_{co}-T_0\right)} . \]
However, for this purpose, from external sources it is necessary to know the law of variation of the heat flux \(q(\tau)\) at the surface of the body.
Generalizations of graphical constructions carried out on the basis of zonal calculation lead to the criterion expressions
\[ Q_n=\frac{q_n(\tau)}{q_c(\tau)}=e^{-d^2}, \]
\[ Q_{\lambda}=\frac{q_{\lambda}(\tau)}{q_c(\tau)}=1-e^{-d^2}. \]
According to the latter, the surface temperature is
\[ \Theta_n=\frac{T_n}{T_c(\tau)}=\sqrt[4]{1-e^{-d^2}} . \tag{10} \]
Here \(d\) is the criterion complex
\[ f(\Theta_0)+2Ki^m Fo . \tag{11} \]
Fig. 1 — calculated data according to formula (8)
— calculated data by the zonal method
— calculated data according to formula (10) (coefficient \(m = 1.2\))
For an infinite plate, an infinite cylinder, and a sphere the coefficient \(\xi\) is equal, respectively, to 1, 2, and 3.
Figures 1 and 2 show the change in surface temperature of an infinite plate of thickness \(2R = 200\) mm, for \(\lambda = 39\ \mathrm{kcal}/\mathrm{m}\cdot\mathrm{h}\cdot\mathrm{deg}\), \(a = 0.045\ \mathrm{m}^2/\mathrm{h}\), \(\varepsilon_n c_0 = 4.0\), constructed by various methods.
Fig. 2 — calculated data by formula (8); calculated data by the zonal method; calculated data by formula (10) (coefficient \(m = 1, 2\)).
The proposed relations were checked under conditions of an ordered thermal regime within the limits of the ratio
\[ \Theta = \frac{T_0}{T_{c0}} = 0.6 \div 0.1 . \]
Literature
- Lykov A. V. Theory of Heat Conduction. GITL, Moscow, 1952.
- Boikov G. P. Proceedings of the Tomsk Polytechnic Institute, vol. 89, 1957.
- Shumakov N. V. ZhTF, No. 4, 1957.
- Samoilovich Yu. A. IFZh, No. 11, 1961.