THERMAL CONDUCTIVITY COEFFICIENTS OF WINDINGS MADE OF PEMS WIRE

L. S. Konovalova, A. S. Lyashkov

(Presented by Prof. Dr. G. I. Fuks)

Manganin “stabilized” enameled wire of the PEMS grade [1] is widely used in instrument engineering as winding wire in the manufacture of electrical coils. To calculate the temperature regime of the latter, data are needed on the thermal conductivity coefficients of the windings.

No information on this question could be found in the literature. The aim of the present work was to obtain experimental data on the thermal conductivity of windings made of PEMS-grade wire.

The experiments were carried out in accordance with a method based on the solution of the problem of the temperature field of a short tube with internal heat sources, dissipating heat through the outer and end surfaces (the inner surface does not participate in heat exchange)—Fig. 1-a. Writing the solution of the differential equation describing the temperature field in such a tube,

\[ \frac{\partial^2 t(r,y)}{\partial r^2} +\frac{1}{r}\frac{\partial t(r,y)}{\partial r} +\frac{\partial^2 t(r,y)}{\partial y^2} +\frac{q_v}{\lambda}=0 \tag{1} \]

in the form of the sum of two functions

\[ t(r,y)=F(r)+\Phi(y) \tag{2} \]

and substituting the partial derivatives for (2) into (1), we obtain

\[ F''(r)+\frac{1}{r}F'(r) =-\left[\Phi''(y)+\frac{q_v}{\lambda}\right]. \tag{3} \]

Equation (3) must hold for any values of \(r\) and \(y\), which is possible only if both sides are equal to a constant value, which we denote by \(-\dfrac{q}{\lambda\varepsilon}\),

\[ F''(r)+\frac{1}{r}F'(r)=-\frac{q_v}{\lambda\varepsilon}, \tag{4} \]

\[ \Phi''(y)=\frac{\varepsilon-1}{\varepsilon}\frac{q_v}{\lambda}. \tag{5} \]

Substituting the integrals of these equations into (2), we obtain

\[ t(r,y)=-\frac{q_v r^2}{4\lambda\varepsilon}+C_3\ln r-\frac{\varepsilon-1}{\varepsilon}\frac{q_v}{2\lambda}y^2+C_1y+C_2+C_4, \tag{6} \]

where \(C_1, C_2, C_3, C_4\) are constants of integration.

Fig. 1. Diagram of the experimental setup

The conditions of insulation of the tube along the inner cylindrical surface and of symmetry with respect to the \(y\) axis,

\[ \frac{\partial t(R_1,y)}{\partial r}=0, \tag{7} \]

\[ \frac{\partial t(r,0)}{\partial y}=0, \tag{8} \]

make it possible to determine \(C_3=\dfrac{q_v R_1^2}{2\lambda\varepsilon}\) and \(C_1=0\), and to write (6) in the form

\[ t(r,y)=\frac{q_v}{2\lambda\varepsilon}\left[-\frac{r^2}{2}+R_1^2\ln r-(\varepsilon-1)y^2\right]+D, \tag{9} \]

where \(D=C_2+C_4\).

We shall assume that the temperatures \(t_1, t_2, t_3\) at points \(1(R_1,0)\), \(2(R_1,y)\), and \(3(R_2,0)\) are measured during the experiment and are known. Then, solving jointly the equations written on the basis of (9) for points 1 and 3, we obtain:

\[ \varepsilon= \frac{ q_v\left(R_2^2-R_1^2-2R_1^2\ln\frac{R_2}{R_1}\right) }{ 4\lambda(t_1-t_3) }, \tag{10} \]

\[ D=t_1-(t_1-t_3)\cdot \frac{R_1^2(2\ln R_1-1)} {R_2^2-R_1^2-2R_1^2\ln\frac{R_2}{R_1}}. \tag{11} \]

Solving equations (10), (11), and (9), written for point 2, with allowance for the fact that \(2Y=L\), \(q_v=\dfrac{4P}{\pi(d_2^2-d_1^2)L}\), gives

\[ \lambda = \frac{A L P}{2\pi\left(d_2^2 - d_1^2\right)\left[A(t_1 - t_2) + 2L^2(t_1 - t_3)\right]}, \tag{12} \]

where

\[ A = d_2^2 - d_1^2 - 2d_1^2 \ln \frac{d_2}{d_1}, \qquad d_2 = 2R_2, \qquad d_1 = 2R_1, \]

\(P\) is the power released throughout the entire volume of the tube.

A comparison, for a particular example, of calculations of the temperature field of a finite tube with internal heat sources, carried out on the Minsk-1 electronic computer by the method of elementary balances, with calculations by (9), (10), and (11), in which the temperatures \(t_1\), \(t_2\), \(t_3\) obtained by the indicated computation on the electronic computer were substituted as experimental temperature values, showed that the calculation results are in complete agreement with one another, taking into account the approximate nature of the elementary-balance method.

It proved to be practically straightforward to carry out an experimental determination of the thermal-conductivity coefficients of windings on the basis of the considerations set forth above and the obtained formula (12).

The test specimen consisted of a textolite frame with wound wire and four thermocouples 1, 2, 2′, 3. Thermal insulation at \(R_1\) was provided by guard heaters made in the form of copper coils of diameter \(2R_1\), with a heating winding and thermocouple 2′. With the aid of the heaters, during the experiment equality of temperatures at points 2 and 2′ was maintained. Identical heat-exchange conditions at both ends were automatically fulfilled when the coil axis was horizontal during experiments under conditions of natural convection.

The diagram of the experimental setup is given in Fig. 1.

Adjustment of the readings of thermocouples 2′ until they coincided with the readings of the corresponding thermocouples 2 was carried out by changing the current in the guard heaters with rheostats and proved easy to implement within \(\pm 0.02\) mV, which corresponds to 0.5°C.

Measurement of the thermocouple emf and the current in the specimen winding was performed under steady-state thermal conditions. The duration of an experiment was 2–3 hours.

It should be noted that during the experiment thermocouple 3 was located not on the upper generatrix of the outer cylindrical surface of the winding, but at a point deviating from the upper generatrix by \(\varphi = 70 \div 75\) degrees. With such a position, an average integral value of the thermal-conductivity coefficient is obtained. This result was obtained on the basis of processing specially conducted experiments to determine \(\lambda\) at different positions of thermocouple 3 in the range \(\varphi = 0 \div \pi\).

The experimental determination of the thermal-conductivity coefficients was carried out for unimpregnated and impregnated windings made of PEMS wire of all diameters from 0.05 mm to 0.5 mm included in the specification TU 017, 109—65. The impregnation was vacuum impregnation with epoxy compound 1N, TU OAT 504 005 of the Tomsk Branch of VNIIEM, with drying in accordance with the plant’s technology.

For each specimen, 6–7 experiments were conducted under different conditions in the temperature range \(50 \div 100^\circ\)C.

After the experiments and their processing, dependences \(\lambda = f(t)\) were constructed for each specimen; these made it possible to draw the following conclusions. For unimpregnated windings made of thin wire (up to Ø 0.1 mm), the winding \(\lambda\) does not depend on temperature, whereas for windings made of wire Ø \(0.1 \div 0.5\) mm a linearly increasing dependence of \(\lambda\) on \(t\) is observed, weakly expressed for windings made of Ø 0.1 mm wire (the change in \(\lambda\) is 3–5%), and more noticeable for windings made of Ø 0.5 wire (8–10%);

For impregnated windings the dependence of \(\lambda\) on \(t\) appeared for windings made of wire \(\varnothing 0.15\) and larger, but in all cases it does not exceed 3–5%. Such a weak influence of temperature on \(\lambda\) made it possible to average the values of \(\lambda\) for each specimen with sufficient reliability, using as the averaged quantities the value of \(\lambda\) at \(t = 75^\circ\mathrm{C}\), and also to find the arithmetic mean \(\lambda\) for several specimens made of wire of the same diameter (the scatter of the averaged data from their arithmetic means did not exceed 3–4%). The averaged \(\lambda\) values thus obtained for unimpregnated and impregnated windings made of wire of different diameters are presented in Fig. 2.

Fig. 2. Averaged values of the thermal-conductivity coefficients of windings made of PEMS wire. 1—unimpregnated windings, 2—impregnated windings.

It follows from Fig. 2 that the thermal-conductivity coefficient of unimpregnated windings made of wire with a diameter greater than 0.1 mm, and for impregnated windings made of wire with a diameter greater than 0.2 mm, is practically independent

Table 1

Thermal-conductivity coefficients \(\lambda\) of windings made of PEMS wire, W/(m·deg)

of the wire diameter. The slight scatter of the points from the averaging curves 1 and 2 in Fig. 2 is most likely associated with possible non-

by the identical nature of the manual winding of the specimens, their impregnation, and other random factors. Therefore, as the thermal-conductivity coefficients of windings made of PEMS wire of various diameters, it is advisable to use the most probable values of $\lambda$, obtained by dissecting curves 1 and 2, Fig. 2, which are given in Table 1.

As a result of the evaluation carried out of systematic and random errors due to possible displacement of the thermocouples from the specified coordinates, and also due to heat leakage through the flanges of the frame because of the possible discrepancy between the temperatures at points 2 and $2'$ by 0.5°C ($\pm 0.02$ mV), the limiting error of the method was $\pm 11\%$.

It should be noted that the satisfactory agreement of the results for non-impregnated windings (Table 1) with the results of [3], obtained for windings made of the same wire by means of the “tube” method, confirms the reliability of the data presented on the thermal-conductivity coefficients of windings made of PEMS wire.

LITERATURE

1. Cables, wires, and electrical cords. T.U., part III. T.U. 017.109—65. Department of Scientific and Technical Information, Standardization, and Normalization in Electrical Engineering, Moscow, 1966.

2. A. N. Zaidel. Elementary Estimates of Measurement Errors. Izd. “Nauka,” Leningrad, 1967.

3. A. S. Lyalikov. Experimental data on the thermal-conductivity coefficient of windings made of PEMS wire. In: collection of articles Research on Electrical Measuring Instruments. Proceedings of VNIIEP, Ts.B.T.N., Moscow, 1961, issue 4.