Determination of thermal diffusivity by the method of spherical thermal waves
Abstract Generated abstract
This paper presents an experimental implementation of the spherical heat wave method for determining the thermal diffusivity of metals, using a steel 40 spherical specimen subjected to periodic heating and cooling. The apparatus employed symmetrically arranged lamp heaters, forced air cooling, and thermocouple measurements at the surface and center of the sphere to obtain temperature oscillation amplitudes under a quasi-stationary regime. Two experimental series gave thermal diffusivity values of 0.0469 and 0.0460 square meters per hour, in agreement with handbook values for steel 40 near 50 degrees Celsius. The paper also analyzes principal error sources, including measurement errors and displacement of the internal temperature sensor, and concludes that moderate central-placement inaccuracies have limited effect under the chosen optimal conditions.
Full Text
DETERMINATION OF THERMAL DIFFUSIVITY BY THE METHOD OF SPHERICAL HEAT WAVES
Communication II
L. G. Fuks
(Presented by the Department of Theoretical and General Heat Engineering)
To determine the coefficient of thermal diffusivity of metals, the method of spherical heat waves may be applied. The theory of the method and the choice of the most favorable experimental conditions are set forth in Communication I. On this basis an experimental apparatus was constructed.
Description of the apparatus
A sphere machined from steel 40 was used as the specimen. The diameter of the sphere was chosen to be 64.5 mm. A hole 2 mm in diameter was drilled in the sphere in the radial direction to a depth equal to the radius of the sphere. A copper-constantan thermocouple, insulated with thin fused glass tubes, was installed in the hole. The surface thermocouple was fastened to the sphere by contact welding; each electrode was welded separately, at a sufficiently small distance from the other.
The device providing periodic heating at sufficiently large amplitudes was designed as a heater consisting of eight centrally symmetrically arranged 40 W incandescent lamps and a small fan. The specimen was placed at the center of the lamp-heater system. The temperature of the specimen could be varied by means of an autotransformer installed before the heater. To obtain periodic oscillations, a Warren motor was used, which actuated a mechanical interrupter through a reducer. A contact was mounted on the interrupter, which was switched on and off during each half-period. The interrupter ensured periodic operation of the lamp heater. At the moments when the heater was switched off, the fan blowing air over the sphere was switched on.
When operating according to the described scheme, after 2.5–3 hours a quasi-stationary heating regime was established. In this regime the temperature of each point of the specimen oscillated about a certain mean value.
Temperature measurements were carried out using a sensitive galvanometer with a short natural period of oscillation. In this case the constant component of the emf, corresponding to the mean temperature of the specimen, was compensated by means of a source of regulated voltages. Each measurement point was investigated separately. From the maximum deflections of the galvanometer, the relative
...of the amplitudes. A separate measurement of the emf of the compensating voltage source gave the mean temperature of the experiment.
The layout of the experimental apparatus and measurements is shown in Fig. 1.
Fig. 1. Layout of the experimental apparatus. \(O\)—specimen; LH—lamp heater; V—fan; M—Warren motor; R—reduction gear; K—interrupter contact; PR—intermediate relay; P—potentiometer; IRV—regulated-voltage source; G—galvanometer; CJ—cold junctions; 1 and 2—external and internal thermocouples. For simplicity, the diagram does not show all the heater lamps, the intermediate relays that ensure reliable long-term operation of the interrupter, and a number of secondary elements.
The frequency of contact breaking by the interrupter was selected from the condition
\[ Z = \sqrt{\frac{\omega}{a}}\,R = 3.34 . \]
From the approximate value of \(a\) and the radius of the spherical specimen, \(\omega\) was chosen equal to 0.1493 and \(0.1263\ \mathrm{s}^{-1}\) (42 and 50 s/rev).
Experimental Data
Two series of experiments were carried out on the apparatus to determine the thermal diffusivity coefficient of the specimen, in order to verify the reproducibility of the data. Both series gave results agreeing with each other to within 2%.
We present the experimental results:
| Quantity | Unit | No. 1 | No. 2 |
|---|---|---|---|
| Measured value of frequency | \(1/\mathrm{s}\) | 0.1493 | 0.1263 |
| Specimen diameter | mm | 64.5 | 64.5 |
| Amplitude values of temperatures | |||
| a) at the surface | deg | 4.23 | 4.69 |
| b) at the center | deg | 2.54 | 3.14 |
| Predvoditelev criterion | 11.9 | 10.25 |
| Mean temperature of the sphere | degrees | 43 | 52.2 |
| Thermal-diffusivity coefficient from the experiment | $\dfrac{m^2}{\text{hour}}$ | 0.0469 | 0.0460 |
| For steel 40 at a temperature of 50°C according to [1] (p. 258) | " | 0.046 | ÷ 0.048 |
Error of the Experiment
In determining the thermal-diffusivity coefficient by the method of spherical thermal waves, errors will occur that limit the accuracy of the experiment. The sources of errors may be divided into the following groups:
a. Inaccurate measurement of amplitudes and phase angles, geometrical dimensions, and the frequency of temperature oscillations.
b. Incorrect installation of the temperature meters, especially at the center of the sphere, where control of the installation is very inaccurate.
c. Inertia of the meters and of the temperature-measurement system.
d. Nonuniformity of the temperature field over the outer surface of the sphere.
e. Fluctuations in the duration of the periods of heating and cooling of the specimen.
We shall dwell on the first two groups of errors. Errors due to nonuniformity of the temperature field and fluctuations in the duration of the heating and cooling periods are described in sufficient detail in [2]. The other errors are investigated below.
a. Inaccurate measurement of experimental quantities
We shall assume that the measurements of the body dimensions, frequency, phase shift, and amplitude values of the temperature are mutually independent, and that the errors in their measurement are random in character. Then, for the relative error of the thermal-diffusivity coefficient, we shall have
\[ \frac{\Delta a}{a} = \sqrt{ \left(\frac{\Delta \omega}{\omega}\right)^2 + 4\left(\frac{\Delta R}{R}\right)^2 + 4\left(\frac{\Delta Z}{Z}\right)^2 }. \tag{12} \]
The relative errors in measuring the frequency and radius of the specimen are determined by the measurement conditions and, in any case, may be reduced to 0.5–1.0%.
As for the error in determining the parameter $Z$, it is associated either with the determination of the phase shift or with the measurement of the amplitude ratio.
Let $Z$ be found from measurement of the phase shift. Then, considering derivative (9) under experimental conditions close to optimal,
\[ Z = 3.34,\qquad B'_s(Z)=0, \]
\[ A_s(Z)\simeq -B_s(Z)\simeq \frac{A'_s(Z)}{\sqrt{2}}, \tag{13} \]
we find that the absolute errors will be related to one another by a relation of the form
\[ \Delta Z = 1.4\,\Delta \varphi. \]
Substituting the latter into (12), we find
\[ \frac{\Delta a}{a} = \sqrt{ \left(\frac{\Delta \omega}{\omega}\right)^2 + 4\left(\frac{\Delta R}{R}\right)^2 + 0.7(\Delta\varphi)^2 }. \tag{14} \]
Relative errors in measuring the temperature amplitude at the center of the sphere and on the surface may be regarded as equal. In this case, carrying out analogous transformations, under optimal experimental conditions,
\[ \frac{\Delta a}{a}= \sqrt{\left(\frac{\Delta \omega}{\omega}\right)^2 +4\left(\frac{\Delta R}{R}\right)^2 +2\left(\frac{\Delta A}{A}\right)^2}, \tag{15} \]
where \(\Delta A/A\) is the relative error in measuring the amplitudes.
6. Estimation of the error associated with the placement of the internal temperature meter
It was established earlier, formulas (2) and (5), that in the quasi-stationary heating regime the ratio of the temperature amplitude at an arbitrary heating point to the amplitude on the surface of the sphere will be equal to:
\[ \frac{A_R}{A_r} = \frac{r}{R} \sqrt{ \frac{A_s^2(Z)+B_s^2(Z)} {A_s^2(Z_r)+B_s^2(Z_r)} }. \tag{16} \]
Choosing the operating region from the optimal ratio and taking the derivative of (16) with respect to \(r\), we obtain
\[ \frac{d}{dr}\frac{A_R}{A_r} = \frac{\sqrt{2}A_s(Z)}{R} \cdot \frac{ 1-Z_r \frac{A_s(Z_r)\cdot A'_s(Z_r)+B_s(Z_r)\cdot B'_s(Z_r)} {A_s^2(Z_r)+B_s^2(Z_r)} } {\left[A_s^2(Z_r)+B_s^2(Z_r)\right]^{1/2}}. \tag{17} \]
If we restrict ourselves to an accuracy of the order of 2%, then it can be shown that, under the condition \(0<Z_r<0.15\), the derivative
\[ \frac{d}{dr}\left(A_R/A_r\right)=0. \]
The latter means that, when the position of the central meter deviates by up to 15% of the radius, the amplitude ratio retains its value with an accuracy of at least up to 2%. This circumstance for a cylindrical specimen was noted earlier in [3].
Similar studies were carried out of the error in the phase shift due to inaccuracy in the placement of the central meter. In this case as well, small inaccuracies in the placement of the central meter may be neglected.
A more detailed analysis shows that, for deviations from the geometric center, the shift angle decreases, while the amplitude ratio increases. This refines the mutual control of experimental data obtained in a single experiment.
Conclusions
Thermal waves in a spherical specimen can be used to determine thermal diffusivity coefficients on an equal footing with other known methods. A certain difficulty in manufacturing the specimens is offset by a number of advantages inherent in this method. Data on the thermal diffusivity coefficient for steel have been obtained.
REFERENCES
- Edited by N. B. Vargaftik. Thermophysical Properties of Matter. Handbook. GEI, 1965.
- F. Hort, M. Wilski. Chem.-Ing. Technik, 35, 1963, 19.
- Yu. M. Kurichenko. Izmeritel’naya tekhnika, No. 5, 29 (1960).