HEAT TRANSFER COEFFICIENTS OF THE ARMATURE OF LOW-POWER DC MACHINES

R. Ya. Klyain, D. I. Sannikov

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

The experimental determination of the heat-transfer coefficients of the armature was carried out on an experimental setup [1], in which a mock-up of a motor served as the test object.

The mock-up was built on the basis of a series P direct-current machine. A distinctive feature of its design is that, instead of the armature winding as the heat source, heaters in the form of spirals made of nichrome wire are built into the active and end-winding parts. The heater of the active part is built into a steel cylinder simulating the armature core. The heaters of the end-winding parts are located on winding supports. The commutator is represented by a solid copper cylinder with a heater on its inner surface.

The indicated armature elements are insulated from one another by thermal-insulation spacers in order to eliminate heat exchange between them.

The bifilar winding of the pole coils made it possible to avoid additional heat sources in the massive structural parts.

The heaters were supplied with direct current through slip rings.

Such a design of the mock-up armature makes it possible to regulate and measure the power of the heat sources in each element independently of one another. In this case, the power on the heaters and pole coils was set so that the temperature rise of the armature elements was identical, or at least differed only slightly [2]. This experimental condition made it possible to reduce to a minimum the influence of heat exchange between the poles and the armature and between the armature elements in the steady state.

The flow rate of the cooling air was varied with the aid of an auxiliary fan; the temperature rise was determined from thermocouples built into the surfaces of the armature elements.

It is known that the heat-transfer process in an electrical machine is influenced by many factors that are not equivalent in their role. Our task included studying the influence of the most important of them, such as the peripheral speed of the armature, the axial velocity of the air flow, the difference in heat transfer in machines differing in frame sizes under otherwise equal conditions, and also the influence of the cross-sectional area of the interpole channels.

For this purpose, three mock-ups of machines of frame sizes 1, 3, and 5 were subjected to tests, with axis-of-rotation heights of 112, 150, and

180 mm. In the last two, the area of the interpole channel varied due to the different filling of the pole coils.

The study of three standard sizes of machines makes it possible, with sufficient justification, to extend the experimental results to the entire segment of the P 1–6 size series.

The heat-transfer coefficients were determined according to [1]. The nature of the influence of the peripheral speed \(v_a\) of the armature on the heat-transfer coefficient \(L\) is shown in Fig. 1, using the model of the 5th size as an example, in the form of the dependences

Fig. 1. Dependence of the heat-transfer coefficient of the active and end-winding parts on the peripheral speed of the armature

\[ \alpha_a = f(v_a) \]
—the heat-transfer coefficient of the active part of the armature \([ \mathrm{W}/(\mathrm{m}^2\,{}^\circ\mathrm{C})]\),

\[ \alpha_l = f(v_a) \]
—the heat-transfer coefficient of the armature end-winding part at different values of the cooling-air flow rate.

The results of studying the influence of the axial air velocity \(v_p\) on the heat-transfer coefficient of the armature elements are presented in Figs. 2–4 for a model of one standard size, \(H = 180\). For the other models, the character of the dependences \(\alpha = f(v_p)\) is analogous. Selected data for the remaining models are given in Table No. 1. Here \(v_p\) is the air velocity in the interpole channel, determined from the relation

\[ v_p = \frac{V}{S_k}, \tag{1} \]

where \(S_k\) is the cross-sectional area of the interpole channel, \(\mathrm{m}^2\); \(v_{p.l}\) is the air velocity in the chamber above the end-winding parts, \(\mathrm{m/sec}\).

\[ v_{p.l} = \sqrt{\frac{V^2}{S_k \cdot S_{k.l}}}, \tag{2} \]

\(S_{kl}\) — cross-sectional area of the chamber above the end parts;
\(v_{pk}\) — air velocity in the chamber above the commutator

\[ v_{pk}=\frac{V}{S_{kl}}. \tag{3} \]

Fig. 2. Dependence of the heat-transfer coefficient of the active part of the armature on the air velocity in the interpole channels

Fig. 3. Dependence of the heat-transfer coefficient of the end part on the air velocity in the chamber above the end parts

The size of the interpole-channel window is taken into account by the coefficient \(k_0\)

\[ k_0=\frac{4\cdot S_k}{\pi\left(D_{\mathrm{st.in}}^2-D_a^2\right)}, \tag{4} \]

where \(D_{\mathrm{st.in}}\) is the inner diameter of the frame,
\(D_a\) is the armature diameter.

For approximating the experimental dependences \(\alpha=f(v_a)\) and \(\alpha=f(v_p)\), a formula of the following form was chosen:

\[ \alpha=\left(\alpha_0^{1/f}+A\cdot v_a^m+B\cdot v_p^k\right)^f, \tag{5} \]

where \(\alpha_0\) is the heat-transfer coefficient at \(v_a=0\) and \(v_p=0\).

Fig. 4. Dependence of the heat-transfer coefficient of the commutator on the air velocity in the chamber above the commutator

The second term of the sum takes into account the influence of the circumferential velocity of the armature at \(v_p=0\). The coefficient \(A\) depends on the circumferential velocity and on the armature diameter.

The third term of the sum takes into account the influence of the axial air velocity. The coefficient \(B\) is a function of the area of the interpole channel \(S_k\), the diameter, and the circumferential velocity.

The coefficients in formula (5) were determined from the condition of the minimum root-mean-square deviation from the experimental data.

Table 1

| Type | \(v_{\mathrm{p}(a,k)}\), m/s | \multicolumn{3}{c}{\(n=0\)} | \multicolumn{3}{c}{\(n=1500\) rpm} | \multicolumn{3}{c}{\(n=3000\) rpm} |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | | \(\alpha_{\mathrm{a}}\) | \(\alpha_{\mathrm{l}}\) | \(\alpha_{\mathrm{k}}\) | \(\alpha_{\mathrm{a}}\) | \(\alpha_{\mathrm{l}}\) | \(\alpha_{\mathrm{k}}\) | \(\alpha_{\mathrm{a}}\) | \(\alpha_{\mathrm{l}}\) | \(\alpha_{\mathrm{k}}\) |
| P12
\(\kappa_0=0.274\)

\(D_a=\)
\(=0.083\) m | 0 | 2.3 | 4.7 | 4.8 | 3.2 | 7.6 | 5.4 | 4.6 | 11.3 | 6.4 |
| P12
\(\kappa_0=0.274\)

\(D_a=\)
\(=0.083\) m | 2 | | 6.4 | 7.2 | | 9.0 | 7.6 | | 12.9 | 7.9 |
| P12
\(\kappa_0=0.274\)

\(D_a=\)
\(=0.083\) m | 4 | 3.45 | 9 | 10.5 | 3.8 | 11.7 | 10.7 | 5.0 | 15.4 | 11.0 |
| P12
\(\kappa_0=0.274\)

\(D_a=\)
\(=0.083\) m | 6 | | 12.1 | | | 15.0 | | | 18.0 | |
| P12
\(\kappa_0=0.274\)

\(D_a=\)
\(=0.083\) m | 8 | 5.4 | | | 5.7 | | | 6.25 | | |
| P12
\(\kappa_0=0.274\)

\(D_a=\)
\(=0.083\) m | 12 | 7.7 | | | 7.95 | | | 8.2 | | |
| P32
\(\kappa_0=0.18\)

\(D_a=\)
\(=0.12\) m | 0 | 2.3 | 4.7 | 4.8 | 4.3 | 7.8 | 6.0 | 6.7 | 10.4 | 7.2 |
| P32
\(\kappa_0=0.18\)

\(D_a=\)
\(=0.12\) m | 2 | | 5.7 | 7.7 | | 8.1 | 8.1 | | 10.8 | 8.5 |
| P32
\(\kappa_0=0.18\)

\(D_a=\)
\(=0.12\) m | 4 | 3.2 | 7.7 | 12.7 | 4.9 | 9.5 | 13.0 | 7.4 | 11.6 | 13.3 |
| P32
\(\kappa_0=0.18\)

\(D_a=\)
\(=0.12\) m | 6 | | 10.1 | | | 12.0 | | | 13.1 | |
| P32
\(\kappa_0=0.18\)

\(D_a=\)
\(=0.12\) m | 10 | 5.9 | | | 7.6 | | | 9.3 | | |
| P32
\(\kappa_0=0.18\)

\(D_a=\)
\(=0.12\) m | 14 | 8.6 | | | 10.2 | | | 11.6 | | |
| P32
\(\kappa_0=\)
\(=0.278\) | 0 | 2.3 | 4.7 | 4.8 | 4.3 | 7.8 | 6.0 | 6.7 | 10.4 | 7.2 |
| P32
\(\kappa_0=\)
\(=0.278\) | 2 | | 6.3 | 7.7 | | 8.6 | 8.1 | | 11.5 | 8.5 |
| P32
\(\kappa_0=\)
\(=0.278\) | 4 | 3.5 | 9.1 | 12.7 | 5.3 | 10.6 | 13.0 | 8.0 | 13.5 | 13.3 |
| P32
\(\kappa_0=\)
\(=0.278\) | 6 | | 12.0 | | | 13.1 | | | 15.6 | |
| P32
\(\kappa_0=\)
\(=0.278\) | 10 | 6.9 | | | 9.0 | | | 10.8 | | |
| P32
\(\kappa_0=\)
\(=0.278\) | 14 | 9.8 | | | 11.8 | | | 13.2 | | |

For the active part

\[ \alpha_{\mathrm{a}}= \left( 5.29+ \frac{0.206v_{\mathrm{a}}^2}{1+0.00314v_{\mathrm{a}}^2} + 6.28\kappa_0^{0.98}v_{\mathrm{p}}^{2.02}D_{\mathrm{a}}^{-1.37} \frac{v_{\mathrm{a}}}{1+0.0273v_{\mathrm{a}}} \right)^{0.5}\cdot 10^{-3} \tag{6} \]

for the end-winding part

\[ \alpha_{\mathrm{l}}= \left( 13.24+ \frac{3.3v_{\mathrm{a}}^2D_{\mathrm{a}}^{-1.08}}{D^{-0.2}+0.00353v_{\mathrm{a}}^2} + 1.13\kappa_0^{0.75}v_{\mathrm{p.l}}^{1.65} \frac{v_{\mathrm{a}}}{1+0.0318v_{\mathrm{a}}} \right)^{0.6}\cdot 10^{-3} \tag{7} \]

for the commutator

\[ \alpha_{\mathrm{k}}= \left( 11.2+ \frac{0.169v_{\mathrm{k}}^2}{1+0.009v_{\mathrm{k}}^2} + 1.53v_{\mathrm{pk}}^{1.55}D_{\mathrm{k}}^{0.64} \frac{0.35v_{\mathrm{k}}+3}{v_{\mathrm{k}}} \right)^{0.65}\cdot 10^{-3}. \tag{8} \]

The approximation error is characterized by a root-mean-square scatter \(\sigma=3.6\%\).

The formulas obtained were checked by the method of thermal calculation [3] on motors of the P2 series with shaft-center height \(H=112\) mm and \(H=160\) mm.

The rated data of the motors being calculated are:

The calculation results are summarized in Table 2.

Adopted notation:

\(M_{\mathrm{a}}\); \(M_{\mathrm{l}}\); \(M_{\mathrm{k}}\)—specific internal thermal resistances of the insulation, referred to the cooling surface of the active part, end-winding part, and commutator, respectively,

\(\zeta_a, \zeta_l, \zeta_k\) are the fractional values of the lengths of the armature sections,

\(\alpha_{ae}, \alpha_{le}, \alpha_{ke}\) are the equivalent heat-transfer coefficients of the sections,

\(k_p\) is the loss reduction coefficient,

\(\alpha_{eq}\) is the equivalent heat-transfer coefficient of the armature,

Table 2

\(S_a\) is the cooling surface of the armature,

\(P_a\) is the total heat-generating loss in the armature,

\(\theta_f\) is the average heating of the air in the machine,

\(\theta_a\) is the temperature rise of the armature above the ambient air.

Conclusions

1. Experimental dependences have been obtained that take into account the influence on the heat-transfer coefficients of the following factors: the axial velocity of the air, the circumferential velocity of the armature, the area of the interpole channels, and the diameters of the armature and commutator.

2. Approximation of the experimental dependences

\[ \alpha = f(v_a, v_p) \]

by formulas of the form (5) is acceptable.

1. The formulas obtained may be used for the thermal calculation of DC machines of series II and the new P2 series with an armature diameter up to 200 mm.

References

1. D. I. Sannikov, R. Ya. Klyain, M. A. Sannikova. Installations for studying heat-transfer parameters in low-power electrical machines. — Proceedings of TPI, vol. 212, Tomsk, 1971.

2. R. Ya. Klyain, D. I. Sannikov. Investigation of heat-transfer coefficients on models of low-power DC machines. — Proceedings of TPI, vol. 211, TSU Publishing House, Tomsk, 1970.

3. D. I. Sannikov, R. Ya. Klyain. Thermal calculation of protected-design DC machines of the P2 series. — Proceedings of TPI, vol. 242, Tomsk, 1972.