Approximate Methods for Thermal Calculations of Electrical Machines with Natural Cooling

M. N. Ulyanitskii

(Presented by the scientific seminar of the Electromechanical Faculty)

In large-scale electrical-machine construction, the problem of determining winding temperature rises is usually solved by developing methods of thermal calculation applicable to one definite type of machine [1, 2]. This makes it possible to take fuller account of the features and regularities of machines of different series, which helps to obtain the necessary accuracy in thermal calculations.

In individual and small-series micromachine construction, because of the great variety of machine types, the determination of winding temperature rises requires methods of a more general character.

It should be noted that the great complexity of the overall physical picture of the distribution of heat flows in a machine makes it difficult to take into account all the conditions on which the thermal process depends. Therefore, the performance of a completely accurate thermal calculation does not appear possible. However, even an approximate calculation makes it possible to estimate the thermal loading of the insulation and considerably facilitates and shortens the development time for a new machine. In this connection, in engineering practice simplified methods of thermal calculation fully justify themselves, taking into account the most essential features of the heating process, the results of analytical investigations, and experimental-statistical data.

The physical aspect of the heating process of enclosed electrical machines is most accurately described by the theory of two-stage heating [3], according to which the temperature rise of each winding is determined as the sum of the temperature rises of the winding over the frame and of the frame over the surrounding medium.

For low-power machines with natural cooling, the value of the frame temperature rise \((\Theta_k)\) can be obtained by using the characteristics \(\Theta_k = f(q)\), preliminarily constructed for a definite ambient-air temperature \(t_{\mathrm{oc}}\), according to the following formulas:

\[ q = \Theta_k \alpha = \frac{\sum P}{S}; \tag{1} \]

\[ \alpha = \alpha_l + \alpha_k; \tag{2} \]

\[ \alpha_{\ell}=\frac{\varepsilon C_0}{\Theta_{\mathrm{k}}}\left[\left(\frac{T_1}{100}\right)^4-\left(\frac{T_2}{100}\right)^4\right]; \tag{3} \]

\[ \alpha_{\mathrm{k}}=0.94\,\frac{\lambda}{d_{\mathrm{k}}^{0.332}} \left(\frac{\beta g\Theta_{\mathrm{k}}}{\nu^2}Pr\right)^{0.226}. \tag{4} \]

Formulas (1) and (3) are well-known relations. Expression (4) was obtained by the author in studying the heat transfer of free convection for horizontally and vertically oriented smooth housings of electrical machines in the range of values

\[ GrPr=10^5-10^9, \]

\[ l_{\mathrm{k}}/d_{\mathrm{k}}=1.1-2.0. \]

Figure 1 presents the characteristics \(\Theta_{\mathrm{k}}=f(q)\), constructed at \(t_{\mathrm{oc}}=+35^\circ\mathrm{C}\) for a wide range of values of \(\Theta_{\mathrm{k}}\) and \(d_{\mathrm{k}}\) encountered in practice.

Fig. 1. Dependence \(\Theta_{\mathrm{k}}=f(q)\) at \(t_{\mathrm{oc}}=+35^\circ\mathrm{C}\).
\(1\)—\(d_{\mathrm{k}}=(30\text{–}40)\) mm;
\(2\)—\(d_{\mathrm{k}}=(40\text{–}50)\) mm;
\(3\)—\(d_{\mathrm{k}}=(50\text{–}70)\) mm;
\(4\)—\(d_{\mathrm{k}}=(70\text{–}100)\) mm;
\(5\)—\(d_{\mathrm{k}}=(100\text{–}150)\) mm;
\(6\)—\(d_{\mathrm{k}}=(150\text{–}220)\) mm;
\(7\)—\(d_{\mathrm{k}}=(210\text{–}330)\) mm;
\(8\)—\(d_{\mathrm{k}}=(330\text{–}500)\) mm.

In each of the ranges of \(d_{\mathrm{k}}\) indicated in Fig. 1, the characteristic \(\Theta_{\mathrm{k}}=f(q)\) was constructed for the mean value of the diameter. Within the accepted ranges of values of \(d_{\mathrm{k}}\), neglecting the dependence of \(\alpha_{\mathrm{k}}\) on the magnitude of the housing diameter has no significant effect on the accuracy of determining \(\Theta_{\mathrm{k}}\).

Analytical and experimental investigations carried out by the author, as well as an analysis of calculated and experimental data [1, 4, 5], make it possible to conclude that, in asynchronous motors, synchronous inductor generators, synchronous reactive and hysteresis motors with a nonsalient-pole stator, and also in other machines of similar design with natural cooling, the values

\(\Theta_k\) amount to approximately 70–90% of the average temperature rises of the stator winding.

The slight difference between the temperature rise of the stator windings and \(\Theta_k\) is explained by the fact that the radial heat flows in the stator do not encounter substantial thermal resistances on their way to the frame. Thus, in impregnated windings, the total temperature drop across the turn and slot insulation is about 5–8%; in the teeth, back, and frame, 5–7%; and between the stator and the frame, 5–7% of the temperature rise of the stator winding.

Thus, for estimating the thermal loading of the stator windings of the indicated types of machines in practical calculations, the following expression may be used:

\[ \Theta_c = 1.25\,\Theta_k . \tag{5} \]

With good impregnation of the windings, relation (5) will also be valid under reduced-pressure conditions. In this case the value of the convective heat-transfer coefficient of the frame is determined by the formula

\[ \alpha'_k = \alpha_k \left(\frac{H'}{H}\right)^{0.452}, \tag{6} \]

which is obtained by elementary transformations from expression (4).

The value of \(\Theta_k\) at reduced pressure can be found either by the method of successive approximations using formulas (1)–(4) and (6), or from the characteristics \(\Theta_k = f(q)\), preliminarily constructed for \(H'\). As can be seen from Fig. 2, the calculated and experimental values of \(\Theta_c\) and \(\Theta_k\) agree well.

Fig. 2. Dependence of the average temperature rises of the frame and stator winding of a synchronous-hysteresis motor on atmospheric pressure at \(\sum P_{\mathrm{const}}\).

\(1\) — frame temperature rise according to (1)–(4), (6); \(2\) — stator-winding temperature rise according to (5); \(0\) — experimental values of temperature rises.

A generalization of the results of experimental investigations of naturally cooled direct-current machines makes it possible to recommend, for determining winding temperature rises, a number of approximate relationships:

for shunt-field windings without insulating spacers between the poles, the frame, and the coils,

\[ \Theta_{\mathrm{v}} = 1{,}4\,\Theta_{\mathrm{k}}; \tag{7} \]

for shunt-field windings with insulating spacers between the poles, the frame, and the coils,

\[ \Theta_{\mathrm{v}} = 1{,}95\,\Theta_{\mathrm{k}}. \tag{8} \]

As experiments have shown, the relationship between the average temperature rises of the frame and of the armature winding depends to a large extent on the distribution of electromagnetic loads. In rationally designed enclosed direct-current machines, the armature winding heats up more than the pole winding. In an experimental investigation of such machines under rated-load conditions, the following approximate expressions were obtained:

for machines with \(d_{\mathrm{k}} : d_{\mathrm{ya}} = 2 - 2{,}2\),

\[ \Theta_{\mathrm{ya}} = 2{,}25\,\Theta_{\mathrm{k}}; \tag{9} \]

for machines with \(d_{\mathrm{k}} : d_{\mathrm{ya}} = 2 - 1{,}8\),

\[ \Theta_{\mathrm{ya}} = 1{,}9\,\Theta_{\mathrm{k}}. \tag{10} \]

For machines with an internal fan, the temperature-rise values of the windings established from (7), (8) should be reduced by 15–20%, and the temperature-rise values established from (9), (10), by 10–15%.

Formulas (1)–(10) make it possible to determine preliminary values of winding temperature rises even at the initial stage of machine design. In this case, the total losses \(q\) needed for the calculation can be found from the rated power and the efficiency, which is preliminarily selected according to State Standard or technical specifications, while the surface area of the frame can be determined on the basis of statistical data on the ratio of the principal and overall dimensions of existing machines.

For alternating-current machines, approximate values of the overall dimensions can be obtained from the expressions

\[ \left. \begin{aligned} d_{\mathrm{k}} &= (1{,}1)\,d_{\mathrm{st}},\\ l_{\mathrm{k}} &= (2{,}2)\,l_{\mathrm{st}} \end{aligned} \right\}. \tag{11} \]

When considering the ratios between the principal and overall dimensions of a large number of direct-current machines, the following relationships were established:

for machines without insulating spacers between the pole coils, the poles, and the frame,

\[ d_{\mathrm{k}} = (1{,}7 - 1{,}9)\,d_{\mathrm{ya}}; \tag{12} \]

for machines with insulating spacers between the pole coils, the poles, and the frame,

\[ d_{\mathrm{k}} = (2{,}0 - 2{,}2)\,d_{\mathrm{ya}}. \tag{13} \]

The frame length may be determined as

\[ l_{\mathrm{k}} = (1{,}35 - 1{,}75)\,d_{\mathrm{k}}. \tag{14} \]

The lower limit of the constant coefficients in (14) corresponds to values of \(l_{\mathrm{ya}}/d_{\mathrm{ya}}\) close to 0.7, and the upper limit to values of \(l_{\mathrm{ya}}/d_{\mathrm{ya}}\) close to 1.1–1.2.

The preliminary values of the temperature rises of the windings are refined after carrying out the electrical calculation and the final determination of the machine’s overall dimensions.

Conclusions

The proposed relationships make it possible, as a first approximation, to estimate the temperature rise of the frame even at the initial stage of the electrical calculation.

To improve the accuracy in determining the temperature rises of the armature winding of DC machines, it is desirable to obtain the relationship between \(\Theta_k\) and \(\Theta_a\), taking into account the magnitude of the specific heat flux on the armature surface.

Symbols Used

\(q\) — specific heat flux on the frame surface;

\(\sum P\) — sum of the losses in the machine, W;

\(S = S_1 + S_2\) — total cooling surface of the machine, m\(^2\);

\(S_1\) — surface of the feet, m\(^2\);

\[ S_2 = \frac{\pi d_k^2}{2} + \pi d_k l_k \]

— frame surface, m\(^2\);

\(d_k\) — frame diameter, m;

\(l_k\) — total frame length, including the bearing end shields, m;

\(\alpha\) — total heat-transfer coefficient of the frame surface, W/°C·m\(^2\);

\(\alpha_l, \alpha_k\) — heat-transfer coefficients by radiation and free convection, W/°C·m\(^2\);

\(\varepsilon = 0.85\) — degree of blackness of the frame surface;

\(C_0 = 5.77\) — radiation coefficient of an absolutely black body, W/°C·m\(^2\);

\(T_1, T_2\) — values of the absolute temperature of the frame surface and of the surrounding air, °K;

\(\lambda\) — coefficient of thermal conductivity of air, W/°C·m;

\(g\) — acceleration of gravity, m/s\(^2\);

\(\nu\) — coefficient of kinematic viscosity of air, m\(^2\)/s;

\[ \beta = \frac{1}{T_2} \]

— coefficient of volumetric expansion of air, 1/°K;

\[ Pr = \frac{a}{\nu} \]

— Prandtl criterion;

\(a\) — coefficient of thermal diffusivity of air, m\(^2\)/s;

\[ Gr = \frac{\beta g d_k^3 \Theta_k}{\nu^2} \]

— Grashof criterion;

\(H = 760\) mm Hg — normal atmospheric pressure;

\(H'\) — reduced atmospheric pressure, mm Hg;

\(d_{\mathrm{st}}, l_{\mathrm{st}}\) — outside diameter and length of the stator stack.

References

1. A. I. Borisenko, E. I. Yantovskii. Thermal calculation of enclosed asynchronous motors of types MA36 and PÉD, Vestnik elektropromyshlennosti, No. 5, 1958.

2. I. M. Postnikov, G. G. Schastlivyi. Thermal calculation of enclosed asynchronous motors of type AO, Vestnik elektropromyshlennosti, No. 3, 1961.

3. A. B. Ioffe. Traction Electrical Machines. Gosenergoizdat, 1957.

4. E. M. Lopukhina, E. I. Ivonin, N. P. Mushketov. Comparison of experimental heating data for enclosed asynchronous micromotors with thermal calculations, Izv. vuzov “Elektromekhanika”, No. 12, 1959.

5. A. P. Borisov. On the thermal calculation of enclosed micromotors, LPI, Scientific and Technical Information Bulletin, No. 8, 1960.