PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV, ORDER OF THE OCTOBER REVOLUTION AND ORDER OF THE RED BANNER OF LABOR

Vol. 226　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1976

A SIMPLIFIED METHOD FOR CALCULATING THE ERROR IN DETERMINING THERMOPHYSICAL CHARACTERISTICS

Yu. A. ZAGROMOV, B. Z. TOKAR, L. G. FUKS, V. N. SHMANDINA

(Presented by Professor G. I. Fuks)

When determining the coefficients of thermal conductivity and thermal diffusivity by the method of a plane source of constant power, the solution of the classical problem for heating a semi-bounded body is used [1]:

\[ \vartheta^{*}=\frac{2q\sqrt{a\tau}}{\lambda}\operatorname{ierfc}\frac{x}{2\sqrt{a\tau}}, \tag{1} \]

where

\[ \vartheta^{*} \]

is the temperature excess relative to the initial temperature, identical for all points of the body at a distance from the source of constant power; \(q\) is the heat-flux density; \(\lambda, a\) are, respectively, the coefficients of thermal conductivity and thermal diffusivity.

Usually in an experiment a semi-infinite body is simulated by a body of finite dimensions in the form of a rod with a circular or square cross section. When such a rod is heated from the lateral surface, heat transfer to the surrounding medium occurs, distorting the temperature field of the rod. The influence of heat transfer was taken into account in works [2, 3], using the solution of the problem of heating a semi-infinite cylinder by a constant heat flux from the end face in the presence of heat exchange with the lateral surface. However, the solutions obtained by the authors in the form of slowly convergent series are very complicated for practical use. In addition, the presence of a large number of variable quantities in the solution makes its analysis in analytical and graphical form difficult.

As a rule, the magnitude of the error introduced is small in ordinary cases. Therefore it appears possible to restrict oneself to an approximate solution which makes it possible, first, to estimate the error with sufficient accuracy and, second, to choose such experimental conditions under which the error arising from heat exchange with the lateral surface can be neglected.

Formulation of the Problem

A semi-bounded rod is given, located along the \(x\)-axis. Between the lateral surface of the rod and the surrounding medium, heat exchange occurs according to Newton’s law. The initial temperature of the rod and the temperature of the surrounding medium are constant and equal to \(t_0\). At the initial instant \(\tau=0\), at the boundary of the rod \(x=0\), a constant-

constant heat flux \(q=\mathrm{const}\). Find the temperature distribution along the length of the rod at any instant of time.

Since the dimensions of the rod cross section are small in comparison with its length, heat conduction over the cross section may be neglected \(\left(\dfrac{\partial t}{\partial y}=\dfrac{\partial t}{\partial z}=0\right)\), and heat transfer from the lateral surface may be taken into account in the differential equation itself in the form of a negative heat source:

\[ c\rho \frac{\partial t(x,\tau)}{\partial \tau} = \lambda \frac{\partial^2 t(x,\tau)}{\partial x^2} -w; \qquad (\tau>0,\;0\leq x<\infty). \tag{2} \]

Here \(w\) is the amount of heat given off to the surrounding medium by a unit volume of the rod per unit time.

Denoting the cross-sectional area of the rod by \(S\), and the perimeter of the section by \(P\), one may write

\[ w=\alpha [t(x,\tau)-t_0]\cdot \frac{1}{h}, \tag{3} \]

where

\[ \alpha \text{ is the heat-transfer coefficient,} \]

\[ h=\frac{S}{p} \quad \left( \begin{array}{l} \text{for a cylindrical rod of radius } r_0\\[2pt] h=\dfrac{1}{2}r_0;\ \text{for a rod of square cross section with square side } r_0\\[2pt] h=\dfrac{1}{4}r_0 \end{array} \right). \]

After substituting (3), the differential equation is written as

\[ \frac{\partial t(x,\tau)}{\partial \tau} = a\frac{\partial^2 t(x,\tau)}{\partial x^2} - \frac{\alpha}{c\rho h}[t(x,\tau)-t_0] \tag{4} \]

with boundary conditions

\[ t(x,0)=t_0; \tag{5} \]

\[ q=-\lambda \frac{\partial t(0,\tau)}{\partial x}; \tag{6} \]

\[ t(\infty,\tau)=t_0;\qquad \frac{\partial t(\infty,\tau)}{\partial x}=0. \tag{7} \]

We transform the system (4–7) with respect to the variable \(\tau\) by the Laplace transform. The differential equation for the transform \(T(x,s)\) has the form

\[ T''(x,s)-\frac{s}{a}T(x,s)+\frac{t_0}{a} -\frac{\alpha}{\lambda h}T(x,s)+\frac{\alpha t_0}{\lambda h s}=0. \tag{8} \]

The boundary conditions for the transform are

\[ T'(0,s)=-\frac{q}{\lambda s}, \tag{9} \]

\[ T'(\infty,s)=0. \tag{10} \]

The solution of equation (8) under condition (10) will be

\[ T(x,s)-\frac{t_0}{s} = B\exp\left(-\sqrt{\frac{s}{a}+\frac{\alpha}{\lambda h}}\,x\right). \tag{11} \]

The constant \(B\) is found from boundary condition (9):

\[ B=\frac{q}{\lambda s\sqrt{\dfrac{s}{a}+\dfrac{\alpha}{\lambda h}}}, \]

which, upon substitution into (11), gives

\[ T(x,s)-\frac{t_0}{s} = \frac{q}{\lambda}\, \frac{1}{s\sqrt{\dfrac{s}{a}+\dfrac{\alpha}{\lambda h}}} \exp\left(-\sqrt{\frac{s}{a}+\frac{\alpha}{\lambda h}}\,x\right). \tag{12} \]

Passing from the image to the original, we obtain

\[ t(x,\tau)-t_0 = \frac{1}{2}\, \frac{q}{\lambda\sqrt{\dfrac{\alpha}{\lambda h}}} \left[ \exp\left(-x\sqrt{\frac{\alpha}{\lambda h}}\right) \operatorname{erfc}\left( \frac{x}{2\sqrt{a\tau}}-\sqrt{\frac{\alpha a}{\lambda h}\tau} \right) - \exp\left(x\sqrt{\frac{\alpha}{\lambda h}}\right) \operatorname{erfc}\left( \frac{x}{2\sqrt{a\tau}}+\sqrt{\frac{\alpha a}{\lambda h}\tau} \right) \right]. \tag{13} \]

If it is assumed that heat transfer from the lateral surface of the rod is absent, i.e., \(\alpha=0\), then expression (13) reduces to (1).

Solution (13) can be represented in criterion form for a cylindrical rod as

\[ \vartheta = \frac{q r_0}{\lambda}\, \frac{1}{2\sqrt{2\mathrm{Bi}}} \left[ \exp(-X\sqrt{2\mathrm{Bi}}) \operatorname{erfc}\left( \frac{X}{2\sqrt{\mathrm{Fo}}}-\sqrt{2\mathrm{BiFo}} \right) - \exp(X\sqrt{2\mathrm{Bi}}) \operatorname{erfc}\left( \frac{X}{2\sqrt{\mathrm{Fo}}}+\sqrt{2\mathrm{BiFo}} \right) \right]. \tag{14} \]

where

\[ \mathrm{Bi}=\frac{\alpha r_0}{\lambda} \]

is the Biot criterion,

\[ \mathrm{Fo}=\frac{a\tau}{r_0^2} \]

is the Fourier criterion,

\[ X=\frac{x}{r_0} \]

is the dimensionless coordinate.

Fig. 1. Dependence of the relative error on \(\mathrm{Fo}\), \(\mathrm{Bi}\), and \(X\)

For fixed values of \(\mathrm{Bi}\), \(\mathrm{Fo}\), and \(X\), one can determine the relative error of the experiment,

\[ \frac{\vartheta^*-\vartheta}{\vartheta}, \]

caused by the presence of heat transfer from the lateral surface.

Using the expression obtained, calculations were carried out and a graph was constructed for the dependence of

\[ \frac{\vartheta^*-\vartheta}{\vartheta} \]

on \(\mathrm{Fo}\), \(\mathrm{Bi}\), and \(X\), which were specified from the conditions of a particular experiment on heat-insulating materials (Fig. 1).

It can be shown that solution (14), for small values of the criterion \(\mathrm{Bi}\), is a special case of the exact solutions given in [2, 3].

The error determined using (14), at small values of \(\mathrm{Fo}\), differs insignificantly from the value of the error calcu-

given by the exact dependence. Using Fig. 1, one can find the limits of the admissible values of the criteria $\mathrm{Fo}$ and $\mathrm{Bi}$ for which the error will not exceed the admissible values. In addition, Fig. 1 makes it possible to take into account and introduce into the calculation the distortion of the temperature field arising because of heat exchange with the lateral surface of the rod. The curves $\mathrm{Bi} = \mathrm{const}$ in the region of large $\mathrm{Fo}$ are sufficiently close to straight lines, which greatly facilitates linear interpolation with respect to the limiting $\mathrm{Bi}$ and the dimensionless parameter $X$.

REFERENCES

1. A. V. Lykov. Theory of Heat Conduction. Vysshaya Shkola, Moscow, 1967.
2. A. B. Verzhinskaya. IFZh, No. 7, 1963.
3. V. N. Shmandina, L. G. Fuks. Izv. TPI, vol. 206, 1970.