PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE, HOLDER OF THE ORDER OF THE OCTOBER REVOLUTION AND THE ORDER OF THE RED BANNER OF LABOR, named after S. M. KIROV

Vol. 205 1972

DETERMINATION OF THERMAL DIFFUSIVITY BY THE METHOD OF SPHERICAL THERMAL WAVES

Communication I

E. I. GONCHAROV, L. G. FUKS

(Submitted by the Department of Theoretical and General Heat Engineering)

At the present time, methods for determining thermophysical coefficients based on analyses of the temperature field of a specimen placed in a medium with periodically varying temperature are of great importance. Such a problem of the propagation of thermal waves makes it possible to determine the thermal diffusivity of a specimen comparatively simply and accurately, without measuring the heat flux.

The first to use the principle of periodic heating and cooling of a rod to determine its thermophysical properties was Ångström [3]. Subsequently his method underwent considerable improvements. Weber [4] developed Ångström’s ideas by using thermal waves in short specimens with simultaneous periodic heating of both ends of the rod. King [5] used an identical technique, working with specimens made in the form of rods and determining the thermal diffusivity coefficient from the phase shift between the temperatures at two points along the rod. Later, many investigators [13, 14, 18, 17, 19] applied Ångström’s method, using it for a wide variety of substances, up to and including liquid metals.

Experimental determination of thermophysical coefficients by means of plane temperature waves was carried out by I. Tadokoro [20], E. Shvidkovskii [6], E. Kravchuk [9], and, most recently, by O. Kraev and A. Stelmakh [21], as well as by Z. Geller and E. Koval’skii [11]). In doing so, the most varied methods of heating the specimens were used, including electronic heating. A noteworthy circumstance is that the temperature-wave method makes it possible to work up to very high temperatures.

Experimental installations based on theoretical premises found for radial thermal waves have become widespread in modern experimentation. V. Vodichka [10] gave a theoretical analysis used later in the works of Van Zee and Babcock [15], A. Gordon [8], and in a series of investigations carried out by Yu. Kirichenko [7]. On the basis of this method, specimen apparatus is being created for determining thermal diffusivity up to 700°C, with the immediate prospect of increasing the measurement limits to 1500°C [12]. For a complete characterization of the latter method it is necessary to point to the detailed investigation

errors of the experiment carried out by F. Horn and P. Wilski [16], and of the accuracy of the measurements made by Yu. Kirichenko [7].

We have not been able to find in the literature any indications of the use of the thermal-wave method for determining the thermal diffusivity of spherical specimens, which in a number of cases may prove successful. In spherical specimens made of solid materials, the necessary boundary conditions are easily provided. A liquid, filling a spherical volume, can be investigated by means of thermal waves, since the influence of convection, with a proper choice of experimental parameters, will be minimal. The thermal diffusivity of spherical specimens can be determined over a wide temperature range and with materials having very different physical properties. Certain difficulties arise in designing heating devices that ensure a periodic change of temperature, and also in installing internal temperature sensors. These difficulties can be overcome by a rational design of the elements of the experimental setup.

Theory of the method

If a sphere having, at the initial instant, a constant temperature is placed in a medium whose temperature varies according to a simple harmonic law, then after some time [1] the temperature of each point of the sphere will oscillate with the same frequency. By heating the sphere for a sufficiently long time, one may consider a simpler problem in which the law of variation of the temperature of the surface of the sphere is given by a harmonic function:

\[ \vartheta = A_R \sin \omega\tau . \tag{1} \]

The initial temperature of the sphere is equal to zero.

In this case, as shown in [2], the temperature distribution in the sphere after prolonged heating is represented by the function:

\[ \vartheta = A_R \frac{RA}{r}\sin(\omega\tau+\varphi). \tag{2} \]

Here the following notation is used:

\[ A = \left| \frac{ \operatorname{Sh}\left(\sqrt{\frac{\omega i}{a}}\,r\right) }{ \operatorname{Sh}\left(\sqrt{\frac{\omega i}{a}}\,R\right) } \right|, \tag{3} \]

\[ \varphi = \arg \left\{ \frac{ \operatorname{Sh}\left(\sqrt{\frac{\omega i}{a}}\,r\right) }{ \operatorname{Sh}\left(\sqrt{\frac{\omega i}{a}}\,R\right) } \right\}, \tag{4} \]

\[ \omega \]

— angular frequency of oscillations,

\[ a \]

— coefficient of thermal diffusivity,

\[ r \]

— current radius of the sphere,

\[ A_R \]

— temperature amplitude on the surface of the sphere,

\[ R \]

— external radius.

Having obtained experimentally the dependence of the temperature of two points of the sphere on time, one can determine the thermal diffusivity using expressions (3) or (4).

In an experiment it is usually convenient to place one measurement point on the surface of the sphere and the other at the center. For determining the calculated

of the dependences, we shall carry out the following transformations. The real and imaginary parts of the hyperbolic functions entering into expressions (3) and (4) will be expressed in terms of hyperbolic-trigonometric functions:

\[ A_s(Z)=\operatorname{Sh}\frac{Z}{\sqrt{2}}\cdot \operatorname{Cos}\frac{Z}{\sqrt{2}}, \]

\[ B_s(Z)=\operatorname{Ch}\frac{Z}{\sqrt{2}}\cdot \operatorname{Sin}\frac{Z}{\sqrt{2}}. \]

Here and below the following notation is used:

\[ Z=\sqrt{\frac{\omega}{a}}\,R,\qquad Z_r=\sqrt{\frac{\omega}{a}}\,r. \]

The modulus of function (3) in this case is written in the form

\[ A=\left\{\frac{A_s^2(Z_r)+B_s^2(Z_r)}{A_s^2(Z)+B_s^2(Z)}\right\}^{1/2}. \tag{5} \]

Repeating the same procedure, we determine the argument of function (4)

\[ \varphi=\operatorname{arctg}\frac{A_s(Z)\cdot B_s(Z_r)-A_s(Z_r)\cdot B_s(Z)} {A_s(Z)\cdot A_s(Z_r)+B_s(Z)\cdot B_s(Z_r)}. \tag{6} \]

Let us find an expression for the amplitude and the phase-shift angle at the center of the sphere. Taking into account that

\[ A_s'(0)=B_s'(0)=\frac{\sqrt{2}}{2}, \]

we have for the amplitude at the center

\[ \frac{A_0}{A_R}\, \frac{Z}{\sqrt{A_s^2(Z)+B_s^2(Z)}}= \frac{Z}{\sqrt{\operatorname{Sh}^2\frac{Z}{\sqrt{2}}+\operatorname{Sin}^2\frac{Z}{\sqrt{2}}}}. \tag{7} \]

In this case the phase-shift angle is equal to

\[ \varphi_0=\operatorname{arctg}\frac{A_s(Z)-B_s(Z)}{A_s(Z)+B_s(Z)}. \tag{8} \]

From the dependences obtained, (7) or (8), the thermal-diffusivity coefficients are calculated from the ratio of the amplitudes and the phase-shift angle of the temperatures at the center and on the surface of the sphere. These expressions can be used for determination and mutual checking. The most convenient way of finding the value \(Z\) from the transcendental equations (7) and (8) is graphical, although in some cases it is possible to simplify somewhat the form of these equations. Graphs of functions (7) and (8) are given in Fig. 1.

Determination of the Optimal Experimental Conditions

The method of spherical thermal waves makes it possible to determine the thermal-diffusivity coefficient of various specimens over a wide temperature range. However, the accuracy, or more precisely the sensitivity, of the method depends strongly on the experimental conditions. As is seen from Fig. 1, the initial portions of curves 1 and 2 up to values \(Z=1.0 \div 1.5\) are hardly advisable to use because of the weak dependence of the phase-shift angle and especially of the amplitude ratio on the parameter \(Z\).

It is also inconvenient to pass to large values of the parameter, since the amplitude in the central part of the specimen rapidly attenuates.

Let us determine the optimal value of the parameter \(Z\), ensuring the greatest sensitivity of the method. In the experiment one finds the phase-shift angle

Fig. 1. The figure shows the dependences:
\(1 — (A_0/A_R)=f_1(z);\quad 2 — \varphi_0=f_2(z);\)
\(3 — (A_0/A_R)'=f_3(z);\quad 4 — (\varphi_0)'=f_4(z)\)
\(*\) — ordinates for curve 2

\(\varphi_0\) or the ratio \(A_0/A_R\), and then \(Z\) is calculated from formulas (8) and (7). From the latter the thermal diffusivity of the specimen is determined.

The derivative \(\varphi_0'\) with respect to \(Z\) from (8) is equal to

\[ \varphi_0'=\frac{1}{\sqrt{2}}\, \frac{ \operatorname{Sin}\frac{Z}{\sqrt{2}}\cdot\operatorname{Cos}\frac{Z}{\sqrt{2}} -\operatorname{Sh}\frac{Z}{\sqrt{2}}\cdot\operatorname{Ch}\frac{Z}{\sqrt{2}} }{ \operatorname{Sh}^2\frac{Z}{\sqrt{2}}+\operatorname{Sin}^2\frac{Z}{\sqrt{2}} }. \tag{9} \]

The error in determining the parameter \(Z\) will be the smaller, the larger the absolute value of derivative (9).

Differentiating (9) and equating the derivative to zero, we find the equation for determining the extremal values

\[ \operatorname{th}^2\frac{Z}{\sqrt{2}}=\operatorname{tg}^2\frac{Z}{\sqrt{2}}. \tag{10} \]

This equation has, as is known, an infinite number of transcendental roots, the first of which, to three significant figures, are:

\[ Z_1=3.34;\quad Z_2=5.56;\quad Z_3=7.78;\quad Z_4=10.0,\ \text{etc.} \]

Let us return to equation (7). Taking the derivative with respect to \(Z\), we obtain

\[ \frac{d}{dZ}\frac{A_0}{A_R} = \frac{ 1-Z\frac{A_s(Z)\cdot A_s'(Z)+B_s(Z)\cdot B_s'(Z)} {A_s^2(Z)+B_s^2(Z)} } {\left[A_s^2(Z)+B_s^2(Z)\right]^{1/2}} . \tag{11} \]

Carrying out analogous reasoning, we determine that the absolute value of expression (11) has a maximum at \(Z=3.34\). The values of the derivatives are given in Fig. 1, curves 3 and 4.

Thus, the value of the parameter \(Z=3.34\), or, what is the same, of Predvoditelev’s criterion,

\[ Pd=11.15 \]

will be optimal in the experiment from the standpoint of the greatest sensitivity of the method. It is seen from Fig. 1 that other values of the roots of equation (10) are unsuitable for the experiment, since with increasing \(Z\) the amplitudes decay sharply and the accuracy of the readings is significantly reduced. At \(Z=5.56\), for example, the ratio is about 0.15.

Conclusions

The method of spherical thermal waves can be used to determine the thermal diffusivity coefficient of samples. In order to achieve the greatest sensitivity of the experiment, the experiment should be carried out under optimal conditions determined by the value of Predvoditelev’s criterion, equal to 11.15.

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