On the Exact and Approximate Solution of the Temperature Field Equation for a Rectangular Bar

Yu. A. Korolenko, G. I. Boikov

(Presented by Professor, Doctor G. I. Fuks)

Earlier [1] it was suggested that, in a number of cases, in solving the differential equation of heat conduction one may use the representation

\[ \frac{\dfrac{\partial^2 v}{\partial y^2}} {\dfrac{\partial^2 v}{\partial x^2}} = \xi - 1 . \tag{1} \]

For the heat-conduction equation

\[ \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{w}{\lambda} =0 \tag{2} \]

with boundary conditions of the third kind, under the condition of symmetry of the temperature field, on the basis of the assumption made an exact solution can be obtained.

Taking into account (1) and the symmetry condition, the solution of equation (2) takes the form:

\[ v_{xy}=B\cos kx\cdot \cos k\sqrt{\xi-1}\,y \tag{3} \]

or

\[ v_{xy}=B\cos k_n x\cdot \cos k_m y . \tag{3-a} \]

Using the usual procedure, we determine \(k_m\), \(k_n\), and \(B\). After substitution we find (see also [2])

\[ v_{xy} = \frac{w}{\lambda} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{A_n\cdot A_m} {\dfrac{\mu_n^2}{R_1^2}+\dfrac{\mu_m^2}{R_2^2}} \cos \mu_n \frac{x}{R_1} \cos \mu_m \frac{y}{R_2}. \tag{4} \]

Here

\[ \left. \begin{aligned} \operatorname{ctg}\mu_n &= \frac{\mu_n}{\mathrm{Bi}_n} = f\!\left(\mathrm{Bi}_n=\frac{\alpha R_1}{\lambda}\right), \\[4pt] \operatorname{ctg}\mu_m &= \frac{\mu_m}{\mathrm{Bi}_m} = f\!\left(\mathrm{Bi}_m=\frac{\alpha R_2}{\lambda}\right) \end{aligned} \right\}. \tag{5} \]

\[ \left. \begin{aligned} A_n &= \frac{2\sin\psi_n}{\psi_n+\sin\psi_n\cdot\cos\psi_n},\\[4pt] A_m &= \frac{2\sin\psi_m}{\psi_m+\sin\psi_m\cdot\cos\psi_m}. \end{aligned} \right\} \tag{6} \]

According to (1), the differential equation (2) is reduced to a system of equations of the form

\[ \frac{\partial^2 v}{\partial x^2}\,\xi=-\frac{w}{\lambda}, \]

\[ \frac{\partial^2 v}{\partial y^2}\,\frac{\xi}{\xi-1}=-\frac{w}{\lambda}, \tag{7} \]

which make it possible to obtain the solution

\[ v_{xy}=D-\frac{w x^2}{2\xi\lambda}-\frac{\xi-1}{\xi}\frac{w y^2}{2\lambda}. \tag{8} \]

The latter satisfies the symmetry conditions and the differential equation of heat conduction.

The values of \(\xi\) and \(D\) can be obtained only approximately from energy considerations, if the temperatures on the surfaces of the body are replaced by the corresponding mean-integral temperatures. Then

\[ \xi=\frac{\dfrac{R_1+R_2}{\alpha}+\dfrac{R_1^2+R_2^2}{3\lambda}} {\dfrac{R_2}{\alpha}+\dfrac{R_2^2}{3\lambda}}, \tag{9} \]

\[ D=\frac{wR_1}{\alpha\xi}+\frac{wR_1^2}{2\xi\lambda} +\frac{\xi-1}{\xi}\frac{wR_2^2}{6\lambda}. \tag{10} \]

Calculations show that the temperature fields of a rectangular bar, found from the approximate relations (8–10) and the exact relations (4–6), practically coincide for a very wide range of engineering problems.

Thus, for the center of the body cross section \((x=y=0)\), the deviations \(\Delta\), given by formula (8) in comparison with the values according to (4), are in the same direction and have the values shown in Table 1.

Table 1

Thus, for the central points of the body, the approximate solution can give sufficiently reliable results.

For \(Bi<0.5\), the approximate formulas can also be used for points close to the surface of the body.

REFERENCES

1. G. N. Boikov. Heating of bodies of finite dimensions under the action of radiant heat. Izvestiya TPI, vol. 101, Tomsk, 1958.
2. Schneider P. Engineering Problems of Heat Conduction. Foreign Literature Publishing House, Moscow, 1960.