On the Problem of Comprehensive Determination of Thermophysical Parameters

L. G. Fuks, V. N. Shmandina

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

At present, a whole series of methods is known that make it possible to carry out the simultaneous determination of thermophysical parameters. The particular value of such experiments lies in the fact that the object of investigation is one and the same specimen, about which sufficiently broad information is obtained in a single experiment.

The determination of volumetric heat capacity \(c\rho\) (\(c\) is heat capacity, \(\rho\) is density), of the coefficients of thermal conductivity \(\lambda\) and thermal diffusivity \(a\), in most methods is carried out comprehensively. Usually, two of these parameters are determined in the experiment, and the third is found from the relation

\[ \lambda = ac\rho . \tag{1} \]

Sometimes the coefficient of thermal activity is included among the parameters being determined:

\[ b = \sqrt{\lambda c\rho}. \tag{2} \]

Then any one of the three parameters can be found by calculation from relations (1) and (2).

When these methods are used, there is no possibility of monitoring and checking the quantities obtained, and errors in the determination of two experimental parameters are transferred to the calculated one.

An experiment appears highly desirable in which three thermophysical parameters are determined simultaneously and independently, with relations (1) and (2) existing between them. In this case, by means of the latter one can check the result of the experiment and also obtain an idea of the possible experimental errors.

For simultaneous determination of the set of parameters, we have developed a method based on the procedure set forth in [1]. According to this procedure, from two temperature determinations over a known time interval on a semi-infinite-body model, the coefficient of thermal diffusivity is found, and then the coefficient of thermal conductivity is found from the known and constant heat flux.

The calculation relation of method (1) has the form:

\[ \vartheta(x,\tau)=\frac{2b\sqrt{a\tau}}{\lambda}\operatorname{ierfc}\frac{x}{2\sqrt{a\tau}} . \tag{3} \]

Here \(\vartheta(x,\tau)\) is the temperature excess, \(q\) is the heat-flux density, \(\tau\) is time, \(x\) is the distance from the heater to the point of measurement, \(a\) is the thermal diffusivity, and \(\lambda\) is the thermal conductivity.

Let us take the derivative of temperature with respect to time:

\[ \vartheta'(x,\tau)=\frac{q}{c\rho\sqrt{\pi a\tau}}\exp\left(-\frac{x^{2}}{4a\tau}\right), \tag{4} \]

where \(c\rho\) is the volumetric heat capacity.

The value of the derivative is determined most accurately in the experiment at the inflection point, where

\[ \vartheta''(x,\tau)=0. \]

This condition gives the Fourier number for the inflection point

\[ Fo=\frac{a\tau}{x^{2}}=\frac{1}{2}, \tag{5} \]

and dependence (4), after transformations with allowance for (5), can be written in the form

\[ c\rho=\frac{2q}{x\sqrt{2\pi e}\,\vartheta'(x,\tau_{\mathrm{inf}})}. \tag{6} \]

Here \(\vartheta'(x,\tau_{\mathrm{inf}})\) is the value of the derivative at the inflection point.

To verify the method, an experiment was carried out for the simultaneous determination of thermophysical parameters.

Well-studied Plexiglas was selected as the material to be tested. The specimen was a cylindrical body 29 mm in diameter and 60 mm long, in which, at a distance \(x=2.75\) mm from the heater, a measuring copper–constantan thermocouple was embedded. The flat heater, wound from constantan wire 0.1 mm in diameter, had a resistance of 28.2 ohm and an area of 1156 mm\(^2\). A constant current from a storage battery was used for heating. The current was measured with an M502 ammeter of class 0.1. The thermocouple emf, according to a galvanometer and an electric stopwatch, was recorded simultaneously by photographic means.

The results of temperature measurements in one of the experiments are given in Table 1.

Table 1

Dependence of temperature on time

The heat-flux density in the experiment was \(q=787\) W/m\(^2\); the measurements were carried out at a temperature of 22°C.

Choosing the 1st and 11th temperature readings (with the ratio \(\tau_2/\tau_1=1.95\simeq2\)), we have

\[ a=1.03\cdot10^{-7}\ \text{m}^{2}/\text{sec}, \]

\[ \lambda=0.196\ \text{W}/(\text{m}\cdot\text{deg}). \]

Hence, by calculation, we obtain

\[ c\rho=1.90\cdot10^{6}\ \text{J}/(\text{m}^{3}\cdot\text{deg}). \]

From condition (5) we find that the time corresponding to the inflection point is

\[ \tau_{\mathrm{inf}}=\frac{x^{2} Fo}{a} =\frac{(2.75\cdot 10^{-3})^{2}\cdot 0.5}{1.03\cdot 10^{-7}} \approx 37\ \text{s}. \]

The value of the derivative, determined at this point from readings 1–4, will be

\[ \vartheta'(x,\tau_{\mathrm{inf}})=0.0725^\circ\mathrm{C}/\text{s}. \]

Substituting the found value of the derivative into (6), we obtain

\[ c\rho=1.91\cdot 10^{6}\ \mathrm{J}/\mathrm{m}^{3}\cdot \mathrm{deg}. \]

The slight discrepancy between the values of the volumetric heat capacity confirms the correctness and reliability of the thermophysical-parameter values obtained in the experiment.

REFERENCES

1. A. B. Verzhinskaya, D. N. Novichenok. IFZh, No. 9, 1960.