SOLVING NONLINEAR HEAT-CONDUCTION PROBLEMS BY THE METHOD OF EXPONENTIAL APPROXIMATION OF LAWS OF CHANGE IN THERMAL PARAMETERS

A. V. Furman, G. F. Shilin

(Presented by Prof. G. I. Fuks)

In [1] an approximate method is set forth for calculating a nonstationary temperature field in solids when the coefficient of thermal conductivity \(\lambda\) and the heat capacity \(c\) are related to temperature by linear dependences, and \(\rho = \mathrm{const}\). In the present article the method is generalized to the case of more general relationships of \(\lambda\) and \(c\) with temperature for \(\rho = \rho(T)\).

The problem of finding the nonstationary temperature field in a body, in the case when the thermophysical parameters depend on temperature, reduces to solving the nonlinear differential equation of heat conduction:

\[ R(T)C(T)\frac{\partial T}{\partial \tau}=a_0 div\,[L(T)\operatorname{grad} T], \tag{1} \]

with the corresponding boundary conditions.

The temperature functions \(R\), \(C\), and \(L\) in (1) are assumed known. Inverting the function \(L(T)\) in order to obtain \(T(L)\), and then taking \(L\) as the independent variable, we represent equation (1) in the following form:

\[ R(T)C(T)\frac{dT}{dL}\cdot\frac{\partial L}{\partial \tau} = a_0 div\left[L(T)\frac{dT}{dL}\operatorname{grad} L\right]. \tag{2} \]

In equation (2) we set

\[ L(T)\frac{dT}{dL}=\frac{1}{M}=\mathrm{const}, \]

\[ R(T)C(T)\frac{dT}{dL}=\frac{1}{N}=\mathrm{const}, \]

which corresponds to replacing the true relationships \(L(T)\) and \(C(T)\) by exponential functions of the form:

\[ L(T)\simeq A\exp[MT], \tag{3} \]

\[ C(T)\simeq B\exp\left[N\int R(T)\frac{dC/dT}{dL/dT}\cdot dT\right]. \tag{4} \]

Thus, approximation of the equations of state \(L=L(T)\) and \(C=C(T)\) in a prescribed interval of temperature variation \(T\), or, if

this interval is sufficiently large, then on its separate subintervals, using the exponential graphs (3) and (4), respectively, makes it possible to approximately linearize equation (1):

\[ \frac{M}{N}\cdot\frac{\partial L}{\partial \tau}=a_0\nabla^2 L . \tag{5} \]

In this case the boundary conditions for equation (1)

\[ T_{\tau=0}=T_{\text{init.}}; \tag{6} \]

or

\[ \left. \begin{aligned} T_{\text{surf.}}&=f(\tau)\\ -\lambda_0\,[L(T)(\operatorname{grad}T)]_{\text{surf.}}&=q(\tau) \end{aligned} \right\} \tag{7} \]

for the new variable \(L\) will be written as follows:

\[ L_{\tau=0}=L_{\text{init.}}; \tag{8} \]

or

\[ \left. \begin{aligned} L_{\text{surf.}}&=\psi(\tau)\\ -(\operatorname{grad}L)_{\text{surf.}}&=\frac{M}{\lambda_0}\,q(\tau). \end{aligned} \right\} \tag{9} \]

Solving the resulting linear system (5), (8), (9) is a less difficult problem than the original nonlinear one (1), (6), (7). For many specific cases these solutions are known [2].

A linear system can also be obtained for a boundary condition of the third kind, if the dimensionless thermal conductivity depends linearly on temperature,

\[ L(T)=1+mT. \]

For any other form of the function \(L(T)\), the boundary condition will remain nonlinear. However, if a linear dependence on temperature is found for the dimensionless density or heat capacity, namely

\[ R(T)=1+eT, \]

\[ C(T)=1+nT, \]

then, using \(R\) or \(C\) as the variable, the system can be reduced to a linear one.

Thus, through the variable \(R\), equation (1) is rewritten as:

\[ R(T)\,C(T)\frac{dT}{dR}\cdot\frac{\partial R}{\partial \tau} =a_0\,\operatorname{div}\left[L(T)\frac{dT}{dR}\operatorname{grad}R\right]. \]

Putting

\[ L(T)\frac{dT}{dR}=\frac{1}{M'}=\operatorname{const}, \tag{10} \]

\[ R(T)C(T)\frac{dT}{dR}=\frac{1}{E}=\operatorname{const}, \tag{11} \]

we obtain a linear equation with constant coefficients. Conditions (10) and (11) correspond to approximating the actual equations of state \(L(T)\) and \(R(T)\) by exponential graphs of the following form:

\[ L(T)\approx A'\exp\left[M'\int \frac{dL/dT}{dR/dT}\cdot dT\right], \]

\[ R(T)\approx D\exp\left[E\int C(T)\,dT\right]. \]

The nonlinear boundary condition of the third kind for equation (1)

\[ -\lambda_0 [L(T)(\operatorname{grad} T)]_{\text{surf.}} = \alpha (T_{\text{surf.}}-T_c) \]

for the new variable \(R\) is written linearly as

\[ -(\operatorname{grad} R)_{\text{surf.}} = M' \frac{\alpha}{\lambda_0} \left(\frac{R_{\text{surf.}}}{e}-\frac{1}{e}-T_c\right). \]

In an analogous way the system is linearized if \(C\) is used as the variable. For this, approximations of the following form are necessary:

\[ L(T) \approx A'' \exp \left[ M'' \int \frac{dL/dT}{dC/dT}\, dT \right], \]

\[ C(T) \approx B' \exp \left[N' \int R(T)\, dT\right]. \]

As an example, convective heating of a plate was considered
(\(T_{\text{init.}} = 273^\circ\mathrm{K}\), \(T_c = 833^\circ\mathrm{K}\)), made of austenitic steel of type ÉYa1T. The dependences of the dimensionless thermal parameters on temperature were taken according to the data of [3]:

\[ L(T)=1+0.1271\cdot 10^{-2}\,T, \]

\[ C(T)=1+0.6978\cdot 10^{-3}\,T-0.2108\cdot 10^{-6}\,T^2, \]

\[ R(T)=1-0.0056\cdot 10^{-2}\,T. \]

The results of calculations carried out on an electronic computer are presented in Table 1 for two values of the Biot number \(Bi=\dfrac{\alpha l}{\lambda_0}\): 0.5 and 5.0. There are also given

Table 1

Comparison of analytical calculation data for convective heating of a plate made of austenitic steel with numerical data obtained on an electronic computer

results obtained approximately by the method of approximating the true curves of the state \(L(T)\) and \(C(T)\) in the interval of variation of \(T\) from 273 to 833° K by exponential functions of the form:

\[ L(T) \approx A \exp[MT] = 1.11190 \exp[0.74991 \cdot 10^{-3} \cdot T], \]

\[ C(T) \approx B \exp\left[ N \int R(T)\frac{dC/dT}{dL/dT}\cdot dT \right] = 1.00510 \exp\left[0.10882 \times 10^{-2} \left(0.54901T - 0.18123 \cdot 10^{-2}T^{2} + 0.61919 \cdot 10^{-8}T^{3}\right)\right]. \]

Comparison shows that the maximum error of the approximate calculation over the given interval of variation of \(T\) does not exceed 3%. With increasing \(Bi\), the accuracy of the calculation improves.

Notation:

\[ R=\frac{\rho}{\rho_{0}}, \qquad C=\frac{c}{c_{0}}, \qquad L=\frac{\lambda}{\lambda_{0}} \quad \text{and} \quad F_{0}=\frac{a_{0}\tau}{l^{2}} \]

are, respectively, the dimensionless density, heat capacity, thermal conductivity, and time;

\(\rho_{0}, c_{0}, \lambda_{0}, a_{0}\) are characteristic values of the corresponding quantities;

\(2l\) is the thickness of the plate;

\(m, n, e\) are constants depending on the nature of the substance.

REFERENCES

1. V. V. Ivanov, A. V. Furman. IZhF, vol. IX, 1965.
2. A. V. Lykov. Theory of Heat Conduction. GITTL, 1952.
3. Handbook edited by Prof. N. B. Vargaftik. Thermophysical Properties of Substances. GEI, 1956.