INVESTIGATION OF THE DISTRIBUTION OF HEAT TRANSFER OVER THE AIR-COOLED SURFACE OF AN INDUCTION MOTOR HOUSING

V. A. Zhadan, Yu. V. Kopylov, D. I. Sannikov

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

In developing the design of ventilation systems for electrical machines, it is very useful to know the distribution of the heat-transfer effect over the cooled surfaces, so as to be able deliberately to modify the machine design in order to achieve maximum efficiency of the ventilation system.

For measuring local heat-transfer coefficients in electrical machines, $\alpha$-sensors of rather complex design are currently used; they have several layers of electrical and thermal insulation, heating elements, and thermocouples. These sensors make it possible to determine the heat-transfer coefficient for a limited surface area with high accuracy, but, owing to their considerable dimensions, they are suitable only for testing machines of medium and large power. The manufacture of sensors that are just as complex in construction but miniature in size, suitable for small electrical machines with large curvature and complexity of the cooled surfaces, encounters serious technical difficulties.

At the same time, high measurement accuracy is necessary only when the experiment is carried out for the purpose of creating a method for thermal calculation of a definite type of machine. If, however, the task is to rationalize the machine design, then with the aid of sensors it is sufficient to obtain a qualitative picture of the distribution of heat transfer over the cooled surface at relatively low measurement accuracy. For these purposes, sensors of significantly simpler construction are suitable, which makes it possible to reduce their dimensions substantially to values acceptable in testing small machines.

The development of ventilation systems is carried out on the basis of investigations both of actual machines and of thermal models.

The use of models makes it possible to reduce the labor intensity of tests and also to increase the volume of information obtained, since the installation of sensors is not limited by the various conditions associated with maintaining normal electromagnetic processes in the actual machine. The literature describes experience in the use of metal models suitable both for ventilation and for thermal investigations. In them, ready-made parts of electrical machines equipped with built-in heaters are usually used. In most cases this principle is used in investigating the cooling of machine housings of closed design.

There are also reports on the application of wooden models reproducing the configuration of the air path of electrical machines.

These models are used only for ventilation studies. However, they can provide certain advantages in some types of thermal studies. In particular, with their aid it is possible to determine not only the distribution of air flows and air velocities, but also the distribution of blowing intensity over different surfaces and different areas of a single surface. For this purpose, α-sensors with heaters may be used.

The proposed principle was implemented in a study of the cooling of the housings of A02-series induction motors, carried out at the Department of Electrical Machines of the Tomsk Polytechnic Institute. A wooden model of the housing of an A02 motor of the 4th frame size, made full-scale, was equipped with a conventional metal casing and a fan driven by an auxiliary motor. Measurement of the air-velocity field at the surface of the model by means of total- and static-pressure tubes confirmed the identity of the model and the actual machine with respect to ventilation.

The main advantage of the wooden model in studying the distribution of heat-transfer intensity over the surface is that, owing to the low thermal conductivity of wood, the leakage of heat from the α-sensor into the mass of the model is small in comparison with the heat removed through the air. Therefore, for measuring the heat-transfer coefficient, a sensor without compensation for heat removal through the mass may be used.

Measurement of the cooling intensity at different areas of the surface was carried out by means of a flat wire thermal resistance on an insulating backing made of lavsan or cable paper.

The technology for manufacturing the resistance deserves special description. A PEL-grade wire of diameter 0.05 mm was wound in one layer on a winding machine onto a polyvinyl chloride tube; the turns were then bonded with nitrocellulose lacquer, and the resulting miniature single-layer coil was carefully removed from the tube. After this, the inner surface of the coil was coated with a layer of adhesive; the coil was flattened in a direction perpendicular to its axis and dried under a press. As a result, a flat, two-layer, rectangular resistance element was obtained. In particular, to obtain a square element, the diameter of the tube used for winding was taken to be π/2 times smaller than the length of the coil. The total thickness of the two-layer element with lacquer films was up to 0.19 mm, and the surface area of one side was 7.5 × 7.5 mm. The resistance element was glued to an insulating backing 0.2 mm thick, to which the lead ends with soldered current and potential wires were also glued.

Measurement of the cold resistance of the element was performed at the temperature of melting ice; for various sensors among those manufactured, the resistance value was

\[ r_0 = 13—14\ \Omega \]

depending on the variation in the number of turns laid during winding.

The thermal resistances obtained are of sufficiently high quality, expressed in their reliability, stability of parameters, uniform distribution of conductors over the resistance field, smooth heat-transfer surface, elasticity, small dimensions, resistance to external influences, and low inertia. They can be successfully used for manufacturing α-sensors of complex design and as temperature detectors.

In studying the distribution of heat transfer over a surface, the thermal resistance serves simultaneously as a heater and as a temperature sensor. In order to increase measurement accuracy, the operating temperature is maintained at the maximum permissible value for class A insulation.

level is 100–105°. The element is powered from a 4.5 V battery through a rheostat; resistance is measured by the ammeter–voltmeter method using a four-wire circuit.

The operating temperature of the resistance-field is calculated from the formula

\[ v_{\mathrm{r}}=\frac{U/I-r_0}{r_0}\cdot 235. \tag{1} \]

The heat-transfer coefficient (without correction for leakage) is

\[ \alpha^1=\frac{UI}{S\left(v_{\mathrm{r}}-v_{\mathrm{v}}\right)}, \]

where \(S\) is the area of one side of the resistance-field surface;
\(v_{\mathrm{v}}\) is the ambient-air temperature.

According to experiment, the time constant of the sensor heating is 15–20 sec.

In order to take into account the correction for heat leakage from the sensor through the mass of the model, it is necessary to know the thermal conductance of the leakage along the following path: transfer of heat through the sensor substrate to the wood, spreading of heat through the wood to the surface of the model, and heat transfer from the surface to the air. At the last stage the heat-transfer coefficient of the model surface, which depends on the velocity of the air flowing over it, apparently affects the overall leakage conductance. In reality, however, this effect reduces to zero, as can be shown mathematically.

A heat source in the form of a disk of radius equal to unity is located on the surface of a semi-infinite body made of a material with thermal conductivity \(\lambda\) (Fig. 1). Between the disk and the body there is a gasket of thickness \(\delta\) with thermal conductivity \(\lambda_\delta\). The heat-transfer coefficient at the surface of the body is \(\alpha\).

Fig. 1. For calculating the leakage conductance to the housing.

For a principled solution of the question of the influence of \(\alpha\) on the thermal conductance, it is sufficient to confine ourselves to an approximate solution of the problem, based on the following assumptions:

1. The thermal conductance of the heat source is infinitely large.
2. The heat-flow lines in the gasket have the form of vertical straight lines, while in the body they have the form of circumferences with centers located on the surface of the body. The beginning of each line is characterized by the coordinate \((0<x<1)\), and the end by the coordinate \((0<y<\infty)\), between which there is a certain functional relation \(y=f(x)\), expressed by a continuous, monotonically increasing function.

An elementary tube of heat flow has an initial width \(dx\) and a final width

\[ dy=f'(x)\cdot dx \]

The initial cross section of the tube is

\[ 2\pi(1-x)\,dx. \]

The final cross section is

\[ 2\pi(1+y)\,dy=2\pi[1+f(x)]\cdot f'(x)\cdot dx. \]

The mean geometric cross section is

\[ 2\pi\sqrt{(1-x)[1+f(x)]\cdot f'(x)\cdot dx}. \]

The length of the tube in the solid is

\[ \pi(x+y)=\pi[x+f(x)]. \]

The thermal resistances of the three successive portions of the heat-flow tube are:
of the gasket—

\[ \frac{\delta}{\lambda\delta}\, \frac{1}{2\pi(1-x)\,dx} = \frac{\rho_1}{dx}, \]

of the solid—

\[ \frac{1}{\lambda}\, \frac{\pi[x+f(x)]}{2\pi\sqrt{(1-x)[1+f(x)]f'(x)}\cdot dx} = \frac{\rho_2}{dx}, \]

of the surface—

\[ \frac{1}{\alpha}\, \frac{1}{2\pi[1+f(x)]\cdot f'(x)\cdot dx} = \frac{\rho_3}{dx}. \]

The total thermal conductance of the leakage is

\[ G_y=\int_0^1 \frac{dx}{\rho_1+\rho_2+\rho_3}. \]

The function \(f(x)\) must be chosen in such a way that the conductance \(G_y\) has the greatest possible value. The exact determination of \(f(x)\) is a problem of the calculus of variations. However, in the present case there is no need for this solution. It is sufficient to investigate the expression obtained for \(G_y\) under the simplest suitable approximation of the function \(f(x)\), namely the fractional-linear function

\[ y=\varphi\,\frac{x}{1-x}, \]

which maps the circle of unit radius onto the entire free surface of the solid. In this case the problem reduces to an ordinary extremal problem; however, the analytical determination of the coefficient \(\varphi\) from the conditions for the maximum of \(G_y\)—

\[ \frac{dG_y}{d\varphi}=0 \]

—is extremely difficult. Therefore, the expression for \(G_y\) was analyzed by a numerical method for various values of \(\varphi\). As a result it was established that \(G_y\) tends to a certain maximum limit as \(\varphi\) increases without bound. In this case the coefficients of thermal resistance of the tube portions \(\rho_1\) and \(\rho_2\) tend to finite values, while \(\rho_3\) tends to zero.

Thus, from the expression for \(G_y\) the term containing the coefficient of heat transfer from the surface of the solid disappears. From this one can de-

draw the conclusion that, in the idealized case considered, the leakage conductivity does not depend on the heat-transfer coefficient.

The conclusion obtained can be extended to the actual operating conditions of the sensor, since the square shape of the heater and the anisotropic structure of the wood are of no fundamental significance. The finite dimensions of the model likewise do not introduce substantial changes into the conclusion obtained, since the thermal resistance of the solid is concentrated mainly near the contact of the heater with the solid. The principal practical conclusion from the foregoing is that the correction to the readings of the $\alpha$-sensor, accounting for heat leakage into the mass, is a constant quantity and can be determined experimentally under one cooling regime, best of all for heat transfer to still air.

First, the heat-transfer coefficient under free convection is determined with the aid of a resistance element placed freely (without a backing) in still air in a vertical plane.

\[ \alpha_0=\frac{U_1 I_1}{2S\left(v_{\mathrm{r}1}-v_{\mathrm{B}1}\right)} . \tag{3} \]

Then the sensor with its backing is glued to the vertical surface of the model. In this case, from one surface of the element heat is removed to still air at the same value of the heat-transfer coefficient $\alpha_0$, while from the other surface it is removed into the mass of the model owing to leakage. The heat-balance equation is then

\[ U_2 I_2=\left(\alpha_0 S+G_y\right)\left(v_{\mathrm{r}2}-v_{\mathrm{B}2}\right) =\alpha'_0 S\left(v_{\mathrm{r}2}-v_{\mathrm{B}2}\right), \]

whence the correction to the experimental value of $\alpha$

\[ \Delta\alpha=\frac{G_y}{S}=\alpha'_0-\alpha_0 =\frac{U_2 I_2}{S\left(v_{\mathrm{r}2}-v_{\mathrm{B}2}\right)}-\alpha_0 . \tag{4} \]

The values of the heat-transfer coefficient obtained in measurements in an air stream under forced convection by formula (2) are refined as

\[ \alpha=\alpha'-\Delta\alpha . \tag{5} \]

Fig. 2. Distribution of the heat-transfer coefficient along the length of the channels.

Figure 2 presents data from measurements of the heat-transfer coefficient in several inter-rib channels of a wooden model of an AO2 motor of the 4th frame size. The measurements were carried out at 4 points around the perimeter of the channel. The values of $\alpha$ differed by no more than 10%. The variation of the coefficient along the length of the channels is more significant. The difference in $\alpha$ is especially large in different channels that differ in the conditions of air flow.

The data obtained indicate the primary need to develop a design that ensures a uniform supply of air flow to all inter-rib channels of the frame, and, secondarily, to reduce the drop in air velocity along the length of the channel.