ASYMMETRY OF HEATING OF THE FRAME OF ELECTRIC MACHINES WITH EXTERNAL VENTILATION

V. A. Zhadan, D. I. Sannikov

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

The widespread use of electric machines of the closed, ventilated design, especially asynchronous motors, and the need to reduce their weight per unit of power require an in-depth study of a number of special problems of their cooling.

One of the features of cooling the frame of a machine with external one-sided ventilation consists in the considerable nonuniformity of the distribution of heat transfer and of the shell temperature along its length. At the entrance to the inter-rib channels the air flow has a high velocity and a high degree of turbulence. Then, as it advances along the stator ribs, the flow velocity decreases owing to dispersion of the flow and decay of vortices. Measurements of air velocities carried out on the A02-42-2 motor using total- and static-pressure tubes showed that the air velocity decreases from the beginning to the end of the channel by a factor of 3–4. In other types of motors, predominantly of higher power, with more successfully chosen parameters of the ventilation system, this decrease is, according to published data, 1.5–2, which is also a significant value.

As a result of the decrease in the heat-transfer coefficient of the stator frame from the fan toward the drive side, a considerable skewing of the temperature-distribution curve arises (Fig. 1). A skewing of the temperature curve is also observed along the stator winding, but it is much less pronounced because of the high axial thermal conductivity of the conductors. Thus, the frontal parts of the winding on the drive side in machines of the A02 series of frame sizes 3 and 4 are heated by 5–9 percent more than on the fan side.

The question of what determines the nonuniformity of heating of a machine, and to what degree, and how it affects the cooling conditions of the machine as a whole, deserves special consideration.

For analysis of this phenomenon, in addition to experimental data, an analytical expression for the temperature curve is necessary. Considering the shell of the machine as a heat-conducting rod, to which from one side (from inside the machine) a heat flux distributed according to a definite law is supplied, while from the other side heat is removed in accordance with the distribution of the heat-transfer coefficient, one can write the following differential equation of heat conduction:

\[ \frac{d^{2}\tau}{di^{2}} = R [G(i)\cdot \tau(i) - Q(i)]. \tag{1} \]

Fig. 1. Distribution of overheating along the length of the frame.
1 — thermal model,
2 — electric motor AO2-42-4,
○ — experiment,
● — calculation by the finite-segment method,
— — according to formulas (8), (12)—(16).

Here $\tau$ is the overheating of the frame above the ambient air;

$i$ is the coordinate, measured from the fan side and expressed in relative units (the length of the rod is taken as unity);

$R$ is the total axial thermal resistance of the rod;

$G(i)$ is the law of distribution of the thermal conductance from the rod to the cooling air;

$Q(i)$ is the law of distribution of the heat input.

The heating of the air in small machines may be neglected; in medium-power machines it can be taken into account by introducing an additional skewing of the temperature curve obtained from the solution of equation (1).

According to the proposed method, the influence of the bearing shields is taken into account by adding, to the frame length $l_c$, segments from each side

\[ l_{\mathrm{sh}}=\frac{S_{\mathrm{sh}}}{\Pi_c}, \tag{2} \]

where \(S_{\text{sh}}\) is the cooling surface of the end shield,
\(\Pi_c\) is the cooled perimeter of the frame.

In this case, the additional axial thermal resistance of the end shields assumes a somewhat understated value; however, because of the smaller nonuniformity of the temperature curve in the end sections of the bar, the resulting error is insignificant.

Thus, the length of the bar is

\[ l = l_c + 2l_{\text{sh}} . \tag{3} \]

Owing to the adopted assumption, there is no need to consider the heating of the shell by sections, and the boundary conditions take the most convenient form:

\[ \frac{d\tau}{di}_{(i=0)} = 0;\qquad \frac{d\tau}{di}_{(i=1)} = 0 . \tag{4} \]

For consideration of the problem in a first approximation, it is expedient to adopt a linear law of distribution of heat transfer along the length:

\[ G(i)=\alpha l\Pi_c =G_{\text{av}}\left[\left(1+\frac{\kappa}{2}\right)-\kappa i\right] \tag{5} \]

on the basis of an approximation of the actual distribution law.

In formula (5)

\[ \alpha=f(i) \]

is the heat-transfer coefficient,

\[ G_{\text{av}}=l\Pi_c\int_0^1 \alpha\,di \tag{6} \]

is the total conductance,
\(\kappa\) is the coefficient of heat-transfer skewness.

Since the heat input to the shell is greater at the point where it is in contact with the stator core, for \(Q(i)\) it is expedient to adopt a symmetrical parabolic distribution

\[ Q(i)=Q\left[(1-2m)+12mi-12mi^2\right], \tag{7} \]

where \(Q\) is the total power of the heating losses,
\(m\) is the coefficient of nonuniformity of heat release.

Substitution of (5) and (7) into (1) gives an inhomogeneous differential equation with a variable coefficient, the solution of which is not expressible in tabulated functions. In theoretical works on the study of heating of electric machines, the solution of an equation of this kind, as far as is known, has not been considered.

For practical purposes it is quite sufficient to obtain an approximate solution of the equation, having an admissible error within the actual limits of variation of the parameters.

Experimental data, the results of a numerical solution of equation (1), and also general considerations lead to the conclusion that the curve of the distribution of overheating \(\tau(i)\) can have no more than three extrema: at the end points, according to the boundary conditions, and in the middle part. On this basis, the principal features of this distribution can be represented in simplified form by a fourth-degree polynomial satisfying the boundary conditions (4):

\[ \tau(i)=\tau_0+(30\tau_{\text{av}}-18\tau_0-12\tau_1)i^2 \]
\[ {}-(60\tau_{\text{av}}-32\tau_0-28\tau_1)i^3 +(30\tau_{\text{av}}-15\tau_0-15\tau_1)i^4, \tag{8} \]

where \(\tau_0\) and \(\tau_1\) are the values of the overheating at the ends of the bar,

\[ \tau_{\text{av}}=\int_0^1 \tau(i)\,di \]

is the average overheating.

After differentiation and substitution of (8) into equation (1), in the left-hand side a polynomial of the second degree is obtained, of the form

\[ M_2(i)=a_0+a_1i+a_2i^2, \]

and in the right-hand side—a polynomial of the fifth degree,

\[ M_5(i)=b_0+b_1i+b_2i^2+b_3i^3+b_4i^4+b_5i^5. \]

Let us separate from \(M_5(i)\) the terms with degree higher than the second:

\[ M_\partial(i)=b_3i^3+b_4i^4+b_5i^5, \]

and select for \(M_\partial(i)\) an approximating trinomial of the second degree in the interval \(0<i<1\),

\[ M_a=b'_0+b'_1i+b'_2i^2, \]

using the method of least squares.

The square of the total root-mean-square discrepancy

\[ D=\int_0^1 [M_a(i)-M_\partial(i)]^2\,di \]

has a minimum for values of the coefficients of the approximating trinomial satisfying the equations

\[ \frac{\partial D}{\partial b'_0}=0;\quad \frac{\partial D}{\partial b'_1}=0;\quad \frac{\partial D}{\partial b'_2}=0. \tag{9} \]

The solution of system (9) gives the following values of the coefficients:

\[ b'_0=\frac{1}{20}b_3+\frac{3}{35}b_4+\frac{3}{28}b_5, \]

\[ b'_1=-\frac{3}{5}b_3-\frac{32}{35}b_4-\frac{15}{14}b_5, \tag{10} \]

\[ b'_2=\frac{3}{2}b_3+\frac{12}{7}b_4+\frac{25}{14}b_5. \]

Thus, equation (1) is transformed into the following identity:

\[ a_0+a_1i+a_2i^2=(b_0+b'_0)+(b_1+b'_1)i+(b_2+b'_2)i^2. \]

From the solution of the system of equations

\[ b_0+b'_0-a_0=0, \]

\[ b_1+b'_1-a_1=0, \]

\[ b_2+b'_2-a_2=0. \]

With respect to \(\tau_{\mathrm{cp}}, \tau_0\), and \(\tau_1\), after a number of transformations, one obtains

\[ \Delta\tau=\frac{\tau_1-\tau_0}{2} =\tau_y \frac{ k\left(\dfrac{5}{12}-\dfrac{1}{3}\dfrac{m}{1+42n}\right) }{ 1+10n-k^2\left(\dfrac{1}{12}+\dfrac{5}{84}\cdot\dfrac{1}{1+42n}\right) }, \tag{12} \]

\[ \tau_{\mathrm{cp}}=\tau_y+\Delta\tau\frac{k}{5}, \tag{13} \]

\[ \tau_{01}=\frac{\tau_0+\tau_1}{2} =\tau_{\mathrm{cp}}-\frac{1}{1+42n} \left(\frac{7}{5}\tau_y m-\Delta\tau\frac{k}{4}\right), \tag{14} \]

where

\[ \tau_y=\frac{Q}{G_{\mathrm{cp}}}. \tag{15} \]

the conventional average overheating of the shell without taking nonuniformity into account,

\[ \mathrm{n}=\frac{1}{R G_{\mathrm{cp}}} \tag{16} \]

is a dimensionless coefficient, representing, in meaning, a value inverse to the Biot criterion.

As checking calculations show, approximation by the least-squares method gives the minimum error in comparison with other methods of approximation: by points, by mean values over intervals, etc.

Figure 1 presents the results of calculation by the proposed method (curves), in comparison with the results of the numerical solution of the equation (·) and with experimental data (o) for the A02-42-4 motor (curve 1) and for a thermal model (a shell with a built-in heater) of an A02 machine of the 4th frame size (curve 2).

The agreement with experiment is quite satisfactory. The error of the approximate analytical method of solving equation (1), as compared with the exact one—as was established by comparison with the results of numerical integration in the real range of variation of the parameters—is proportional to the coefficient \(K\) and increases as the parameter \(n\) decreases, that is, with increasing machine size and cooling-air velocity. However, for machines of limiting frame sizes, with the maximum possible nonuniformity of blowing, the greatest discrepancy between the results of the solution at individual points does not exceed \(3\%\,\tau_{\mathrm{cp}}\), and for \(\tau_{\mathrm{cp}}\) it is no more than \(0.5\%\). Thus, the proposed method of solving the heat-conduction equation is practically exact.

Analysis of the formulas obtained shows that nonuniform cooling of the frame leads to an increase in its average overheating, in comparison with the conventional value, by several percent; nonuniform heat generation reduces the effect of nonuniform blowing and does not cause an increase in the average overheating.

The relationships obtained can be used for solving the inverse problem: from an experimental plot of the temperature distribution along the frame, knowing the losses \(Q\) and the thermal resistance \(R\), to determine the coefficients of heat-transfer skewness, the nonuniformity of heat generation, and to refine the value of the total thermal conductance \(G_{\mathrm{cp}}\). For this purpose, on the basis of formulas (9)—(13), the following was obtained:

\[ \mathrm{k}= \frac{\tau_{\mathrm{cp}}+\dfrac{1}{10p}} {\dfrac{\Delta \tau}{5}+\dfrac{\tau_{01}+0.75\,\tau_{\mathrm{cp}}}{42p\Delta\tau}}, \tag{17} \]

\[ \tau_y=\tau_{\mathrm{cp}}-\Delta\tau\,\frac{\mathrm{k}}{5}, \tag{18} \]

\[ \mathrm{m}=\frac{5}{7}\cdot\frac{1}{\tau_y} \left[ (\tau_{\mathrm{cp}}-\tau_{01})(1+42\mathrm{n})+\Delta\tau\,\frac{\mathrm{k}}{4} \right], \tag{19} \]

\[ G_{\mathrm{cp}}=\frac{Q}{\tau_y}, \tag{20} \]

where

\[ \mathrm{p}=\frac{1}{RQ}=\frac{\mathrm{n}}{\tau_y}. \tag{21} \]

Processing the test results for a considerable number of motors and thermal models of the A02 series at various rotational speeds

of the 3rd and 4th frame sizes shows that, for these machines, the nonuniformity of the air blast is characterized by the value

\[ k = 1.05—1.15, \]

and the nonuniformity of heat generation is, on average,

\[ m = 0.3. \]

Table 1

Results of processing the test data for the thermal model of the AO2-42 motor

Figure 2 shows the variation of overheating at individual points along the length of the frame for different fan rotational speeds. The data

Fig. 2. Temperature distribution along the length of the model frame at different rotational speeds.

obtained in testing the thermal model of the AO2-42 motor with a 1000 W heater.

Processing the experimental data by formulas (14)--(17), as is seen from Table 1, shows that the nonuniformity coefficient of heat transfer \(k\) does not depend on the rotational speed, while the decrease in the effective conductance

\[ G_{\mathrm{e}}=\frac{Q}{\tau_{\mathrm{av}}} \tag{22} \]

due to asymmetry of cooling, in comparison with the average conductance \(G_{\mathrm{av}}\), increases with increasing speed. Thus, the proposed method gives a definite effect in the analysis of experimental data.

One application of the proposed method may be a preliminary estimate of the change in blowing efficiency when the frame length is increased in order to increase the motor power without changing the height of the shaft centers. Taking into account that the distribution of the heat-transfer coefficient along the length is not actually linear and that the rate of decrease of \(\alpha\) slows as the distance from the fan increases, it may be assumed that, for a small increase in the frame length, the heat-transfer coefficient on the added length segment remains the same as at the end of the main segment. Thus, on the main segment \((0<i<1)\)

\[ \alpha'=\alpha_{\mathrm{av}}\left[\left(1+\frac{k}{2}\right)-ki\right], \tag{23} \]

on the added segment \((1<i<1+\Delta)\)

\[ \alpha''=\alpha_{\mathrm{av}}\left(1-\frac{k}{2}\right), \tag{24} \]

where \(\alpha_{\mathrm{av}}\) is the average heat-transfer coefficient for the main segment.

The average heat-transfer coefficient for the lengthened frame is

\[ \alpha'_{\mathrm{av}}=\frac{1}{1+\Delta} \left(\int_{0}^{1}\alpha'\,di+\int_{1}^{1+\Delta}\alpha''\,di\right) =\alpha_{\mathrm{av}}\left(1-\frac{k}{2}\frac{\Delta}{1+\Delta}\right). \tag{25} \]

The relative increase in the total thermal conductance due to the added segment is

\[ \Delta_G=\frac{\alpha'_{\mathrm{av}}\Pi_c l(1+\Delta)-G_{\mathrm{av}}}{G_{\mathrm{av}}} =\Delta\left(1-\frac{k}{2}\right). \tag{26} \]

The formula obtained makes it possible to calculate approximately that, for example, for motors of the AO2-42 type, for each percent increase in the frame length the increase in heat transfer will amount to only

\[ 1-\frac{1.09}{2}=0.455\%. \]

In this connection, for the given change in the design it is necessary simultaneously to take measures aimed at maintaining a high blowing velocity over the entire length of the frame.

Conclusions

1. The proposed method for the approximate calculation of the temperature distribution along the length of the frame of enclosed machines, taking into account the nonuniformity

uniformity of cooling and heat generation along the length of the machine, ensures good agreement between the calculated results and the experimental data.

1. The use of the least-squares method for an approximate analytical solution of the differential equations of heat conduction makes it possible to successfully solve practical problems in calculating the temperature field.

2. The developed method for calculating the frame temperature can be successfully used for the analysis of experimental data.