Investigation of Heat-Transfer Coefficients on Models of Low-Power DC Machines

R. Ya. Klyayn, D. I. Sannikov

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

The heat-transfer coefficient \(\alpha\) is undoubtedly influenced, to a greater or lesser degree, by the following factors: the peripheral velocity of the air flow, the differing cooling conditions of the active and end portions of the armature, the configuration of the interpole gaps, the microgeometry of the armature surface, the diameter and length of the armature, etc. The study and accounting of these factors will make it possible to significantly improve the accuracy of methods for thermal calculation.

Fig. 1. Motor model.

In the present work, results are given from an investigation of the influence of the first three of the indicated factors on the heat-transfer coefficient. Experimental investigation of the factors affecting \(\alpha\) on real machines presents considerable difficulties because of the complexity of changing the geometry of the machine and measuring losses and temperature.

From this point of view, such experiments are more conveniently carried out on models, which make it possible to measure powers and temperatures accurately, and also to make any design changes easily.

Figure 1 shows one of the models of a DC machine, made on the basis of a series P machine in a 4-pole version. The active part

of the armature was made in the form of a hollow steel cylinder 1, inside which a heater of nichrome wire was built in. The heater is pressed tightly against the inner surface of the cylinder. Grooves imitating the slots of a real armature are milled on the outer surface of the cylinder. The end-winding parts are represented by two heaters 3, secured on the winding holders by bandages. The commutator model is a copper cylinder 4 with a built-in heater 5. The commutator, the end-winding parts, and the active part are insulated from one another by heat-insulating gaskets in order to eliminate heat exchange between them. To eliminate heat loss through the shaft, the space between the shaft and the heater of the active part and the winding holders of the end-winding parts is filled with heat-insulating material. The temperature was measured by means of 24 copper-constantan thermocouples embedded on the surface of the active part, the end-winding parts, and the commutator, uniformly around the circumference.

□ — at $n = 0$
○ — at $n = 1500$ rpm
△ — at $n = 3000$ rpm

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

The pole windings were wound bifilarly, which made it possible to avoid losses in the steel of the structural elements. The temperature of the pole coils was measured by resistance. In this connection, the heat-transfer coefficients for the pole coils $\alpha_{\text{п}}$ were determined with allowance for the thermal resistance of the insulation.

The velocity of the air flow was regulated and measured by means of a forcing centrifugal fan and a measuring tube with total-pressure tubes.

The power released in the heaters and pole coils was regulated in such a way that the temperature difference of the investigated parts of the model in the steady state was minimal. In this way, the influence of heat exchange between the individual parts of the model was reduced to a minimum.

The experiment was carried out at two armature rotational speeds, $n = 1500$ rpm and $n = 3000$ rpm, and with a stationary armature. From the experimental results, the dependences shown in Figs. 2, 3, and 4 were constructed:

$$ \alpha_a = f(v_p), \quad \alpha_l = f(v_{pl}) \quad \text{and} \quad \alpha_{\text{п}} = f(v_p), $$

where $\alpha_a$, $\alpha_l$, and $\alpha_{\text{п}}$ are the heat-transfer coefficients, respectively, of the active part, the end-winding part, and the pole coils;

$v_{\mathrm{p}}$ — discharge velocity of the air in the interpole gaps;

$v_{\mathrm{rl}}$ — discharge velocity of the air in the chamber above the end parts.

By approximating the experimental curves from Figs. 2, 3, and 4, formulas of the following form were obtained:

\[ \alpha = C\left(\sqrt{v_{\mathrm{p}}^{2}+\left(\frac{v_{a}}{\kappa}\right)^{2}+B}\right)^{m}, \tag{1} \]

where

$C,\ B,\ m,\ \kappa$ — coefficients selected according to the condition of minimum discrepancy with the experimental data;

$v_{a}$ — peripheral velocity of the armature.

Fig. 3. Dependence of the heat-transfer coefficient of the end part on air velocity.

Fig. 4. Dependence of the heat-transfer coefficient of the pole coils on air velocity.

Thus, for the active part of the armature, formula (1) has the form

\[ \alpha_{a}=0.77\cdot 10^{-3}\, v_{\mathrm{efa}}^{0.9}. \tag{2} \]

$v_{\mathrm{efa}}$ — effective velocity of the air flow

\[ v_{\mathrm{efa}}=\sqrt{v_{\mathrm{p}}^{2}+\left(\frac{v_{a}}{2}\right)^{2}+10} \tag{3} \]

\[ v_{\mathrm{p}}=\frac{V\cdot 10^{4}}{S_{k}} \tag{4} \]

$V$ — air flow rate, m³

$S_{k}$ — cross-sectional area of the interpole ducts, cm²

For the end parts

\[ \alpha_{\mathrm{l}}=3.7\cdot 10^{-3}\, v_{\mathrm{efl}}^{0.85}. \tag{5} \]

$v_{\mathrm{efl}}$ — calculated discharge velocity of the air above the end parts

\[ v_{\mathrm{rl}}=\frac{V\cdot 10^{4}}{S_{\mathrm{ol}}} \tag{6} \]

\[ S_{\mathrm{ol}}=\frac{\pi}{4}\left(D_{\mathrm{st}}^{2}-D^{2}\right) \]

— area between the armature and the inner surface of the frame, cm²;

$D$ — armature diameter, cm;

$D_{\mathrm{st}}$ — inner diameter of the frame, cm;

In an analogous manner, a formula was obtained for the commutator

\[ \alpha_{k}=5\cdot 10^{-3}\, v_{\mathrm{efk}}^{0.7}, \tag{7} \]

where

\[ v_{\mathrm{eff\,c}}=\sqrt{v_{\mathrm{rl}}^{2}+\left(\frac{v_{\mathrm{c}}}{6}\right)^{2}+2.5} \tag{8} \]

is the effective air velocity at the commutator;

\(v_{\mathrm{rl}}\) is the calculated flow velocity over the commutator, taken equal to the velocity over the end-winding parts;

\(v_{\mathrm{c}}\) is the peripheral velocity of the commutator.

The obtained formulas have been verified on a large number of machines of the П series of a similar frame size and showed agreement within 5%.