Accounting for Nonuniform Heat Transfer Along the Length of a Finned Blown Body

D. I. Sannikov, V. A. Zhadan

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

Studies of the heat transfer of finned blown bodies of electrical machines [2, 3, 4, 5] show that the variation of the heat-transfer coefficients along the length of the frame has a rather complex character. However, for convenience in calculations it can, where possible, be approximated by a simple expression. The most acceptable for the present case is the exponential law of heat-transfer distribution

\[ \alpha=\alpha_m e^{-\frac{x}{L}}, \tag{1} \]

where \(L\) is the exponential constant, which may be called the “heat-transfer attenuation length”;

\(x\) is the distance from the beginning of the path of the cooling flow (from the edge of the frame);

\(\alpha_m\) is the maximum value of the heat-transfer coefficient at \(x=0\).

The heat-transfer attenuation length \(L\) can be obtained experimentally. However, the results of measuring heat-transfer coefficients by means of simplified-type \(\alpha\)-sensors [5] cannot serve as a basis for determining \(L\). The heat-transfer attenuation length can be determined on the basis of the experimental temperature distribution along the length of the frame. For this purpose it is sufficient to characterize the distribution along the length of the frame by two parameters: the mean frame temperature \(\Theta_0\) and the half-difference of the temperatures at the ends of the frame \(\Delta \tau\).

The calculation method is as follows: assigning various values of \(L\), i.e., the character of the heat-transfer distribution along the length of the frame, we calculate the temperature distribution along it by some numerical method. As the actual value of \(L\) we take that for which \(\Delta \tau/\Theta_0\) agrees with experiment.

We consider the frame as a heat-conducting rod to which a heat flux is supplied, having a symmetric parabolic distribution, and from which heat is removed in accordance with the exponential law of distribution of the heat-transfer coefficient. As an example, we give the results of a calculation for A02-42-4 motors. The frame was divided into 12 sections along its length for reasons of sufficient calculation accuracy. For each section, the following were calculated: thermal conductance to the external air, heat inflow, and longitudinal thermal resistance. The calculation results are summarized in Table 1.

Here, too, the average experimental value of \(\Delta \tau/\Theta_0\) is given from the results of thermal tests of 8 AO2-42-4 motors in steady thermal conditions at rated load. Interpolation of the calculated

Table 1

curve \(\Delta \tau = f\!\left(\dfrac{L}{D_{\mathrm{st}}}\right)\) makes it possible to determine that \(\Delta \tau = 25\%\) corresponds to

\[ \frac{L}{D_{\mathrm{st}}}=1.02. \]

Calculations showed that, when the distribution of heat flux along the length is changed, the value found for \(\dfrac{L}{D_{\mathrm{st}}}\) practically does not change. The numerical value of the ratio \(\dfrac{L}{D_{\mathrm{st}}}\), obtained for AO2-42-4 motors, is also confirmed for other motors with similar geometry of the air duct and finning.

Thus, knowing \(L\), one can calculate the dependence of the total thermal conductance “housing–medium” on the length of the frame. It is assumed here that the average heat-transfer coefficient of each end shield is equal to the heat-transfer coefficient of the adjacent section of the frame

\[ G_0=\frac{S_{\mathrm{sh}}}{2}\alpha_m+ \int_0^{L_{\mathrm{st}}}\Pi_\alpha \alpha_m e^{-\frac{x}{L}}\,dx+ \frac{S_{\mathrm{sh}}}{2}\alpha_m e^{-\frac{L_{\mathrm{st}}}{L}} = \]

\[ =\alpha_m\left[ S_{\mathrm{sh}}\frac{1+e^{-\frac{L_{\mathrm{st}}}{L}}}{2} +\Pi_\alpha L_{\mathrm{st}}\frac{L}{L_{\mathrm{st}}} \left(1-e^{-\frac{L_{\mathrm{st}}}{L}}\right) \right]. \tag{2} \]

Here \(S_{\mathrm{sh}}\) is the total cooling surface of the bearing end shields, \(\Pi_\alpha\) is the perimeter of the heat-transfer surface of the frame, and \(L_{\mathrm{st}}\) is the length of the frame.

The total cooling surface of the housing is

\[ S_\Sigma=S_{\mathrm{sh}}+\Pi_\alpha L_{\mathrm{st}}. \tag{3} \]

Introduce the following notation for the relative dimensions of the housing:

\[ l_{\mathrm{sh}}=\frac{S_{\mathrm{sh}}}{D_{\mathrm{st}}\Pi_\alpha}, \tag{4} \]

\[ \frac{L_{\mathrm{st}}}{D_{\mathrm{st}}} =1+\frac{L_{\mathrm{st}}-D_{\mathrm{st}}}{D_{\mathrm{st}}} =1+y, \tag{5} \]

\[ \gamma=\frac{D_{\mathrm{st}}}{L}. \tag{6} \]

Expressing (1) and (2) through the relative quantities (4)—(6), we obtain

\[ D_0=\alpha_m\Pi_\alpha D_{\mathrm{st}} \left[ l_{\mathrm{sh}}\frac{1+e^{-\gamma(1+y)}}{2} +\frac{1-e^{-\gamma(1+y)}}{\gamma} \right], \tag{7} \]

\[ S_{\Sigma}=\Pi_a D_{cm}(1+l_{\text{щ}}+y). \tag{8} \]

The mean value of the heat-transfer coefficient is

\[ \alpha_{cp}=\frac{G_0}{S_y} = \frac{\alpha_m}{1+l_{\text{щ}}+y} \left[ l_{\text{щ}}\frac{1+e^{-\gamma(1+y)}}{2} + \frac{1-e^{-\gamma(1+y)}}{\gamma} \right]. \tag{9} \]

The change in the mean heat-transfer coefficient as a function of the length of the machine in the thermal calculation [1, 6] can be taken into account by means of a coefficient, which we shall call the “length coefficient” \(\kappa_l\). The length coefficient \(\kappa_l\) is the ratio of the mean heat-transfer coefficient for a “unit machine” \(\alpha_{cp\,1}\), for which the length of the frame is equal to the diameter \(L_{cm}=D_{cm}\) (that is, \(y=0\)), to the mean heat-transfer coefficient for a machine with an arbitrary frame length:

\[ \kappa_l= \frac{ \dfrac{1}{1+l_{\text{щ}}} \left( l_{\text{щ}}\frac{1+e^{-\gamma}}{2} + \frac{1-e^{-\gamma}}{\gamma} \right) }{ \dfrac{1}{1+l_{\text{щ}}+y} \left[ l_{\text{щ}}\frac{1+e^{-\gamma(1+y)}}{2} + \frac{1-e^{-\gamma(1+y)}}{\gamma} \right] }. \tag{10} \]

Expanding \(\kappa_l=f(y)\) in a Maclaurin series, we obtain

\[ \kappa_l=\kappa_l(0)+\frac{d\kappa_l(0)}{dy}\cdot \frac{y}{1!}+\cdots . \tag{11} \]

For \(y=0\), \(\kappa_l(0)=1\),

\[ \frac{d\kappa_l(0)}{dy} = \frac{1}{1+l_{\text{щ}}} - \frac{ e^{-\gamma}\left(1-\dfrac{\gamma l_{\text{щ}}}{2}\right) }{ \dfrac{1-e^{-\gamma}}{\gamma} + l_{\text{щ}}\dfrac{1+e^{-\gamma}}{2} }. \tag{12} \]

Since it is known that

\[ \frac{L}{D_{cm}}=\frac{1}{\gamma}=1.02, \]

then

\[ \frac{d\kappa_l(0)}{dy}=0.375 \]

and

\[ \kappa_l=1+0.375y=1+0.375\left(\frac{L_{cm}}{D_{cm}}-1\right). \tag{13} \]

The error of formula (13) in comparison with the exact formula (10) does not exceed 1.5% when \(L_{cm}/D_{sm}\) varies within the limits \(0.7\text{—}1.5\), which cover practically all possible designs of asynchronous motors of a single series.

An experimental verification of formula (13) was carried out on a thermal model of an AO2-4 frame-size motor. In this experiment the length of the frame was varied within the limits

\[ 0.8 \leq \frac{L_{cm}}{D_{cm}} \leq 1.19; \]

the power of the heaters was selected so that the heat-flux density along the length remained constant in all experiments. The results of the calculations and experiments are given in Table 2.

Thus, the discrepancy between the values calculated by (13) and the experimental values does not exceed 0.5%.

Formula (13) takes into account the influence of the frame length on the mean heat-transfer coefficient \(\alpha_{cp}\) by means of the length coefficient \(\kappa_l\). However, the distribution of heat transfer along the length is also influenced by such factors as the length of the closed part of the inter-rib channel.

\(l_k/D_{\mathrm{st}}\) and the cross section of the annular channel between the stator and the casing \(S_l/D_{\mathrm{st}}^2\). The results of tests of the thermal model of the AO2-42-4 motor with different lengths of the closed part of the channel \(l_k/D_{\mathrm{st}}\) are shown in Fig. 1. Increasing the casing length by 10% leads to a reduction in the overheating

Table 2

of the frame by \(1.6^\circ\mathrm{C}\); however, a further increase of \(l_k\) to half the stator length leads to an additional reduction in overheating of only \(1^\circ\mathrm{C}\); reducing the length \(l_k\) by only 4% of the stator length already leads to an increase in the average overheating of the frame by \(1^\circ\mathrm{C}\) owing to the increase in air scattering. This experiment makes it possible to conclude that increasing \(l_k/D_{\mathrm{st}}\) has little effect on improving the heat transfer of the frame. Therefore one may recommend

\[ l_k/D_{\mathrm{st}} = 0.15\text{--}0.20. \]

Fig. 1.

Another quantity that has a substantial influence on the distribution of air velocity and of the local heat-transfer coefficients along the length of the inter-rib channels is the cross section of the annular channel between the stator and the casing, \(S_l/D_{\mathrm{st}}^2\), or the relative rib height \(h_p/D_{\mathrm{st}}\). For this purpose, thermal and ventilation tests were carried out on a thermal model of the AO4-132M-4 motor with wide variation of \(h_p/D_{\mathrm{st}}\). The test results are shown in Fig. 2. It was established that there exists an optimum value of \(h_p/D_{\mathrm{st}}\), equal to 0.1, at which the average heat-transfer coefficient of the frame is at a maximum. Here

Here the graph is also given of the component of the length coefficient \(K_l''\), depending on \(h_p/D_{cm}\), which is well approximated by the expression

\[ K_l'' = 0.82 + \frac{0.2}{10 \dfrac{h_p}{D_{cm}} + 0.1}. \tag{14} \]

Fig. 2.

Thus, the expression for the length coefficient, taking (14) into account, takes the form

\[ k_l = \left[ 1 + 0.375\left(\frac{L_{cm}}{D_{cm}} - 1\right) \right] \left/ \left( 0.82 + \frac{0.2}{10 \dfrac{h_p}{D_{cm}} + 0.1} \right) \right. . \tag{15} \]

REFERENCES

1. I. M. Postnikov. Design of Electrical Machines. Gostekhizdat of the Ukrainian SSR, 1960.

2. G. G. Schastlivyi, Ya. B. Tubis. Heat transfer of ribbed stators of electric motors of the A010 frame-size series. Elektrotekhnika, 1964, No. 8.

3. E. B. Kovalev, A. T. Tokarenko, A. N. Burkovskii. Investigation of the ribbing of the housings of electric motors of the VAO series. Elektrotekhnika, 1964, No. 12.

4. E. B. Kovalev, A. N. Burkovskii, A. T. Tokarenko. Heat transfer in the inter-rib channels of the housings of enclosed electrical machines. Elektrotekhnika, 1965, No. 11.

5. V. A. Zhadan, Yu. V. Kopylov, D. I. Sannikov. Investigation of the distribution of heat transfer over the blown surface of the housing of an asynchronous motor. Izvestiya TPI, vol. 172, 1967.

6. V. A. Zhadan, Yu. V. Kopylov, D. I. Sannikov. Method for thermal calculation of enclosed low-power asynchronous motors. Izvestiya TPI, vol. 212 (in press).