METHOD FOR CALCULATING THE TEMPERATURE OF FINNED AIR-COOLED HOUSINGS OF ELECTRICAL MACHINES

V. A. ZHADAN, D. I. SANNIKOV

(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering).

The operating temperature of the windings of enclosed, air-cooled electric motors depends to the greatest extent on the temperature rise of their housing above the ambient medium. However, the accuracy of calculating this temperature rise for low-power machines by the methods available in the literature is extremely low. At present, the accumulated experimental data on the heating of asynchronous motors with power up to 10 kW make it possible to develop a method for calculating the temperature rise of the housing that has the necessary accuracy.

Heat transfer from the surface of the housing, consisting of a finned frame and bearing shields, is affected by a large number of factors, the main ones being the air velocity, the rotational speed of the motor, the drop in air velocity along the length of the frame, air heating, the temperature difference over the height of the fin, the difference between the heat-transfer coefficients of the frame and the bearing shields, etc. A certain portion of heat is removed from the housing by radiation and, although this portion is insignificant under forced convection, it should be taken into account when calculating the heat-transfer coefficient.

The equation of convective heat transfer is represented by a power-law relation of the form

\[ \mathrm{Nu} = c \mathrm{Re}^{m}, \tag{1} \]

where

\[ \mathrm{Nu} = \frac{a \cdot d_{\mathrm{g}}}{\lambda} \quad \text{— Nusselt criterion,} \]

\[ \mathrm{Re} = \frac{v d_{\mathrm{g}}}{\nu} \quad \text{— Reynolds criterion.} \]

For the characteristic dimension in calculating these criteria, the hydraulic diameter of the channel between two adjacent fins was adopted,

\[ d_{\mathrm{g}} = \frac{4S_{\mathrm{k}}}{P_{\mathrm{k}}}, \tag{2} \]

where \(S_{\mathrm{k}}\) is the cross section of the channel,
\(P_{\mathrm{k}}\) is the “wetted” perimeter of the channel,
\(\lambda\) and \(\nu\) are the thermal conductivity and kinematic viscosity of air at ambient temperature.

The investigations were carried out on a mock-up specimen of a D-132 electric motor, which represents the housing of this motor with

built-in sectionalized heaters. By regulating the power of individual sections of the heater, it is possible to achieve complete identity of the temperature field of the motor housing and the model. The total power of the heater is equal to the heating losses of the motor under study. An increase in the air flow rate was achieved by various improvements to the ventilation, and also by blowing air from an external fan. A reduction in the fan flow rate was accomplished by increasing the aerodynamic resistance of the inlet to the enclosure. The average temperature of the housing was determined from 40 thermocouples arranged uniformly on the cylindrical surface of the housing.

To determine the resulting heat-transfer coefficient $\alpha_{\Sigma}$, which takes into account heat transfer by convection and radiation, we first find the conditional heat-transfer coefficient without taking into account the temperature drop over the height of the rib

\[ \alpha_y=\frac{P_{\Gamma}}{\theta \cdot S_o}, \tag{3} \]

where $P_{\Gamma}$ is the heating losses of the motor, W;
$\theta$ is the average overheating of the housing according to the thermocouples, deg.;
$S_o=S_{\text{gl}}+S_p$ is the total cooling surface of the housing, cm$^2$;
$S_{\text{gl}}$ is the surface of the smooth part of the frame and bearing shields, cm$^2$;
$S_p$ is the surface of the ribs, cm$^2$.

In the Reynolds criterion we first substitute the air-flow velocity at the outlet from under the enclosure,

\[ v_p=\frac{V_p}{S_{\text{out}}}, \tag{4} \]

where $V_p$ is the air flow rate, m$^3$/s;
$S_{\text{out}}$ is the outlet cross-sectional area, m$^2$.

The actual heat-transfer coefficient is

\[ \alpha_{\Sigma}=\frac{P_{\Gamma}}{\theta\left(S_{\text{gl}}+k_p \cdot S_p\right)}, \tag{5} \]

where

\[ k_p=\frac{\beta}{\operatorname{tg}\beta} \tag{6} \]

is the rib efficiency coefficient,

\[ \beta=h\sqrt{\frac{2\alpha_{\Sigma}}{\lambda_i b_{\text{avg}}}} \tag{7} \]

is the coefficient of propagation of the temperature field over the height of the rib,

$h$ is the rib height,
$\lambda_i$ is the thermal conductivity of the rib material,
$b_{\text{avg}}$ is the average rib thickness.

Expanding $\operatorname{tg}\beta$ in a series in equation (6), we obtain

\[ k_p=\frac{1}{1+\dfrac{\beta^2}{3}-\dfrac{\beta^4}{45}} = \frac{1}{1+\dfrac{\beta^2}{3}\left(1-\dfrac{\beta^2}{15}\right)}. \]

For the average value of $\beta$, obtained for a large number of frames,

\[ 1-\frac{\beta^2}{15}=0.94, \]

and, taking (7) into account, we obtain

\[ k_p=\frac{1}{1+0.625\,\dfrac{h^2}{\lambda_i\cdot b_{\mathrm{av}}}\alpha_\Sigma}. \tag{8} \]

Solving equations (3) and (5) simultaneously, we obtain

\[ \alpha_\Sigma=\alpha_y \frac{S_{\mathrm{gl}}+S_p}{S_{\mathrm{gl}}+\dfrac{S_p}{1+c\cdot\alpha_\Sigma}}, \tag{9} \]

where

\[ c=0.625\,\frac{h^2}{\lambda_i b_{\mathrm{av}}}. \tag{10} \]

Solving the resulting equation (9) with respect to \(\alpha_\Sigma\) and applying the formula for simplified division and extraction of the root, we obtain the solution

\[ \alpha_\Sigma=\alpha_y(1+\chi\alpha_y), \tag{11} \]

where

\[ \chi=0.156\,\frac{h^2}{\lambda_i\cdot b_{\mathrm{av}}} \left(1+\frac{S_p}{S_{\mathrm{gl}}}\right). \tag{12} \]

The dependences \(Nu_\Sigma=\dfrac{\alpha_\Sigma d_\Gamma}{\lambda}=f(Re)\) for various rotational speeds of the D-132 motor are shown in Fig. 1 in a logarithmic anamorphosis. Curves of the same form were obtained by G. G. Schastlivyi for AO motors of the 10th frame size [2]. In both cases it is evident that, at one and the same air flow rate, the heat-transfer coefficient of the housing is greater for higher motor rotational speeds.

Fig. 1. Dependence of heat transfer from the surface of finned housings of D-132 electric motors (without accounting for the effective air velocity)

This phenomenon is evidently explained by the higher level of turbulence of the air stream emerging from the fan at a higher rotational speed, as well as by the cooling conditions of the bearing shield on the fan side. The identical slope of the curves in Fig. 1 makes it possible to assume that a single criterion dependence can be obtained for

of all rotational speeds, if instead of the flow velocity one takes the effective air velocity, which accounts for both the air flow rate and the peripheral speed of the fan \(v_0\).

For calculating the effective air velocity, the following empirical formula is recommended:

\[ v_{\mathrm{eff}} = v_p \left( 1 + \frac{v_0}{70} \sqrt[4]{\frac{v_0}{v_p}} \right). \tag{13} \]

The dependence of \(\mathrm{Nu}_\xi\) as a function of

\[ \mathrm{Re}_{\mathrm{eff}} = \frac{v_{\mathrm{eff}} d_{\Gamma}}{\nu} \]

is shown in Fig. 2 (line 2). The experimental points for different rotational speeds lie satisfactorily on a straight line, which is described by the equation

\[ \lg \mathrm{Nu}_{\xi} = -0.454 + 0.576 \lg \mathrm{Re}_{\mathrm{eff}} \tag{14} \]

or

\[ \mathrm{Nu}_{\xi} = 0.737 \mathrm{Re}_{\mathrm{eff}}^{0.576} \tag{15} \]

in the range

\[ 0.25 \cdot 10^4 \leq \mathrm{Re}_{\mathrm{eff}} \leq 4.5 \cdot 10^4 . \]

The root-mean-square scatter of the experimental points is 2.35 percent, which is quite acceptable for thermal calculations. This method was verified on a model of an AO2 motor of the 3rd frame size. The results of the experiment on this model, together with the experimental data for the D-132 model, are presented in Fig. 2 (line 2). The small scatter of the experimental points, obtained—

Fig. 2. Dependence of heat transfer from the surface of finned housings
(with allowance for the effective air velocity).

● — model of motor D-132, \(n = 3000\) rpm,
○ — model of motor D-132, \(n = 1500\) rpm,
△ — model of motor D-132, \(n = 750\) rpm,
T — model of motor AO2-32, \(n = 3000\) rpm,
× — model of motor AO2-32, \(n = 1500\) rpm,
+ — model of motor AO2-32, \(n = 1000\) rpm,
▲ — model of motor AO2-32, \(n = 750\) rpm.

obtained in tests of model specimens with different geometric dimensions, makes it possible to conclude that the method for taking the rotational speed into account has been chosen correctly, and that further investigations should be carried out taking the effective rotational speed into account.

In the proposed procedure, unlike existing ones, the influence of secondary factors is not calculated separately, such as heat removal by radiation, the difference between the cooling conditions of the end shields and the frame, heating of the air flow by the losses removed, and the drop in air velocity along the length of the frame. All the factors listed are taken into account by the heat-transfer coefficient \(\alpha_{\Sigma}\). This yields a substantial simplification of the calculation procedure; the error allowed as a result of all the indicated simplifications can be estimated from the scatter of the experimental points on the graph (Fig. 2). This scatter lies within permissible limits, although the secondary factors listed varied in the experiments over a fairly wide range. Thus, the ratio of the areas of the cooled surface of the end shields and the frame was \(0.13\text{--}0.17\), and the ratio of the air velocity at the outlet from the frame ducts to the initial velocity was \(0.35\text{--}0.55\).

As an example, let us consider how much the calculation procedure is improved if one of the factors is taken into account separately, for example, heat transfer by radiation.

The coefficient of heat transfer by radiation is determined from the known formula

\[ \alpha_{\ell}=\frac{\varepsilon C}{\theta}\left[\left(\frac{T_{2}}{100}\right)^{4}-\left(\frac{T_{1}}{100}\right)^{4}\right], \tag{16} \]

where \(\varepsilon C=C_{1}\) is the radiation coefficient;
\(T_{2}\) and \(T_{1}\) are the absolute temperatures of the casing and of the surrounding medium.

In electrical-machine engineering, according to [1], for most heat-radiating surfaces the radiation coefficient \(C_{1}\) may be taken equal to \(5\ \mathrm{W}/\mathrm{m}^{2}(\mathrm{deg})^{4}\).

The reduced coefficient of heat transfer by radiation is

\[ \alpha'_{\ell}=\alpha_{\ell}\frac{S_{\mathrm{enc}}}{S_{0}}, \tag{17} \]

where \(S_{\mathrm{enc}}\) is the surface enclosing the frame along the tips of the ribs and the bearing end shield not covered by the casing.

The coefficient of heat transfer by convection is

\[ \alpha_{k}=\alpha_{\Sigma}-\alpha'_{\ell}. \tag{18} \]

The dependence

\[ \mathrm{Nu}_{k}=\frac{\alpha_{k}d_{r}}{\lambda}=f(\mathrm{Re}_{\mathrm{eff}}), \]

described by the equation

\[ \mathrm{Nu}_{k}=0.599\mathrm{Re}_{\mathrm{eff}}^{0.606}, \tag{19} \]

is shown in Fig. 2 (line 3). The root-mean-square scatter of the experimental points in this case is \(\delta_{k}=2.8\%\). It is therefore evident that separate allowance for heat transfer by radiation does not reduce the scatter of the experimental points and, consequently, does not increase the accuracy of the calculation.

In this connection the question arises whether such a factor as rib efficiency should be taken into account separately. If the dependences (6--12) are neglected, then the heat transfer of the casing will be calculated through the conditional heat-transfer coefficient \(\alpha_{y}\) (3).

Dependence

\[ \mathrm{Nu}_{y}=\frac{\alpha_{y} d_{\mathrm{g}}}{\lambda}=f(\mathrm{Re}_{\mathrm{eff}}), \]

which is described by the equation

\[ \mathrm{Nu}_{y}=0.826\mathrm{Re}_{\mathrm{eff}}^{0.547}, \tag{20} \]

is shown in the graph in Fig. 2 (line 1). The root-mean-square scatter in this case increased from 2.35 to 2.6 percent. Although the increase in error in this case is insignificant, in view of the fact that the error increases substantially at large values of \(\mathrm{Re}\), and the present-day tendency is to increase the blowing velocity, the fin efficiency should be taken into account in the calculation.

Thus, the overheating of the housing is calculated from the following formula:

\[ \theta=\frac{P_{\mathrm{g}}}{\alpha_{\Sigma}\left(S_{\mathrm{g.l}}+k_{\mathrm{r}}\cdot S_{\mathrm{r}}\right)}, \tag{21} \]

where the total heat-transfer coefficient is

\[ \alpha_{\Sigma}=\frac{\mathrm{Nu}_{\Sigma}\cdot\lambda}{d_{\mathrm{g}}}, \tag{22} \]

\(d_{\mathrm{g}}\) is the hydraulic diameter of the channel, determined by formula (2);

\(\mathrm{Nu}_{\Sigma}\) is determined by formula (15);

\(\mathrm{Re}_{\mathrm{eff}}\) is determined through the effective air velocity (13);

\(k_{\mathrm{r}}\) is according to formula (8).

Conclusions

1. The proposed method for calculating the temperature of the housing is sufficiently simple for engineering calculations and surpasses existing methods in accuracy when they are used for the thermal calculation of motors with a power of up to 10 kW.

2. The method makes it possible, for the first time, to take into account unequal heat-transfer conditions at the same air flow rate but at different motor rotational speeds.

3. Separate calculation of air heating, heat transfer by radiation, heat transfer of the end shields, etc., does not contribute to increasing the accuracy of the thermal calculation as a whole. The indicated factors are sufficiently accounted for in the overall heat-transfer coefficient \(\alpha_{\Sigma}\) (5).

References

1. G. Gotter. Heating and Cooling of Electrical Machines, GEI, 1961.

2. G. G. Schastlivyi. Heating of Enclosed Asynchronous Electric Motors, Naukova Dumka, 1966.