ON THE ERROR IN DETERMINING THERMOPHYSICAL PARAMETERS BY THE CONSTANT-POWER SOURCE METHOD

V. N. Shmandina, L. G. Fuks

To determine thermophysical parameters by the constant-power source method [1], the theoretical solution of the problem of heating a semi-infinite body is used. In the experiment, the semi-infinite body is represented in the form of two sufficiently long cylinders or prisms, between which a flat, inertialess heater is clamped.

From the conditions of minimizing the measuring apparatus and the desire to reduce the dimensions of the specimens, the question arises of the error that occurs with such an imitation.

We shall restrict the range of problems to the study of thermophysical parameters of insulating materials \((\lambda = 0.2—0.25\ \text{W}/(\text{m}\cdot\text{deg}))\). The experiment time is usually 1–3 minutes. Then calculation shows that, for heat fluxes of the order of \(1000\ \text{W}/\text{m}^2\), admissible from the conditions of preservation of the heater and the specimen, the temperature increase at a distance of \(20—25\ \text{mm}\) from the heater will be less than \(0.01^\circ\text{C}\), i.e., it is practically not detected by the meter.

The specimens usually used in these cases, with a length of more than \(30—40\ \text{mm}\), realize the idea of an “infinite” body with sufficient accuracy.

In the present article an estimate is made of the influence of lateral heat exchange of the body that replaces the semi-infinite medium in the investigations. The calculation is simplest for a cylindrical specimen. If the cross section of the latter differs from a circle, then all the reasoning may be applied to the cylinder whose base is a circle of the greatest radius inscribed in the base of the noncircular specimen.

The solution of the problem of heating a semi-infinite body by a constant heat flux at the surface has the form [2]

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

where \(\vartheta\) is the temperature rise at a distance \(x\) from the heater at time \(\tau\),

\(q\) is the heat-flux density at the boundary,

\(\lambda, a\) are the coefficients of thermal conductivity and thermal diffusivity of the material.

If a cylindrical specimen is used, then in the case when the cylinder axis coincides with the direction \(x\), and in the plane \(x=0\)

a constant-power source is located, the problem obtains the following mathematical formulation:

\[ \frac{\partial \vartheta}{\partial \tau} = a\left( \frac{\partial^2 \vartheta}{\partial z^2} + \frac{\partial^2 \vartheta}{\partial r^2} + \frac{1}{r}\frac{\partial \vartheta}{\partial r} \right), \tag{2} \]

\[ \vartheta(0,x,r)=0, \tag{3} \]

\[ -\lambda \frac{\partial \vartheta(\tau,0,r)}{\partial x}=q, \tag{4} \]

\[ \frac{\partial \vartheta(\tau,\infty,r)}{\partial x}=0, \tag{5} \]

\[ -\lambda \frac{\partial \vartheta(\tau,x,r_0)}{\partial r} = \alpha \vartheta(\tau,x,r_0), \tag{6} \]

\[ \frac{\partial \vartheta(\tau,x,0)}{\partial r}=0. \tag{7} \]

Here, in addition to the quantities introduced earlier, the following are denoted:

\(r_0\)—the radius of the cylinder,
\(\alpha\)—the heat-transfer coefficient from the lateral surface of the cylinder,
\(r\)—the radial coordinate.

Equation (2) with the initial condition (3) and mixed boundary conditions (4)—(7) was solved by means of the double Laplace and Hankel transform.

Omitting the intermediate calculations, we give the complete solution of the system, expressed in dimensionless quantities

\[ Bi=\frac{\alpha r_0}{\lambda} \text{—the Biot criterion,} \]

\[ Fo=\frac{a\tau}{r_0^2} \text{—the Fourier criterion,} \]

\[ R=\frac{r}{r_0} \text{—dimensionless radius,} \]

\[ X=\frac{x}{r_0} \text{—dimensionless coordinate:} \]

\[ \vartheta(Fo,X,R) = \frac{q r_0}{\lambda} \sum_{i=1}^{\infty} \frac{I_1(\mu_i) I_0(\mu_i R)} {I_0^2(\mu_i)(Bi^2+\mu_i^2)} \times \]
\[ \times \left[ \exp(-\mu_i X)\operatorname{erfc} \left( \frac{X}{2\sqrt{Fo}}-\mu_i\sqrt{Fo} \right) - \right. \tag{8} \]
\[ \left. -\exp(\mu_i X)\operatorname{erfc} \left( \frac{X}{2\sqrt{Fo}}+\mu_i\sqrt{Fo} \right) \right]. \]

The eigenvalues \(\mu_i\) of the problem are determined from the equation

\[ \frac{I_0(\mu)}{I_1(\mu)}=\frac{\mu}{Bi}, \tag{9} \]

and \(I_0\) and \(I_1\) are Bessel functions of the first kind of order zero and one.

Equation (8) was checked by taking the limiting transition at \(Bi=0\) (\(\alpha=0\)) and, in this case, it identically reduced to (1).

The temperature was calculated at the point \(x=3\) mm and \(r=0\) by formula (8) for the following conditions:

\[ \begin{aligned} Bi &= 0.7\quad (\alpha \simeq 9.5\ \mathrm{W}/\mathrm{m}^{2}\!\cdot\!\mathrm{deg}), \\ X &= 0.2, \\ Fo &= 0.0289,\quad (\tau \simeq 65\ \mathrm{s}) \end{aligned} \qquad \begin{aligned} r_0 &= 15\ \mathrm{mm}, \\ a &= 10^{-7}\ \mathrm{m}^{2}/\mathrm{s}, \\ \lambda &= 0.2\ \mathrm{W}/\mathrm{m}\!\cdot\!\mathrm{deg} \end{aligned} \]

gives the value

\[ \vartheta = 0.0547\,\frac{q r_0}{\lambda}. \tag{10} \]

The temperature rise according to (1) for a semi-infinite body under these conditions is

\[ \vartheta^{*} = 0.0558\,\frac{q r_0}{\lambda}, \tag{11} \]

which differs from (10) by 1.9%.

Using relation (8), one can determine the error arising from neglecting heat exchange from the surface of the specimen, or choose the dimensions of the latter so that the error lies within permissible limits.

REFERENCES

1. A. B. Verzhinskaya, L. N. Novichenok. IFZh, No. 9, 1960.
2. A. V. Lykov. Theory of Heat Conduction. Moscow, “Vysshaya shkola,” 1967.