Indirect method for determining the equivalent heat transfer coefficient of roller table motors
Abstract Generated abstract
This paper addresses discrepancies between calculated and experimentally determined dynamic constants of roller table induction motors operating under frequent starting and braking. It derives an indirect method for estimating an equivalent heat transfer coefficient from type test data, treating the coefficient as a combined measure of frame cooling, internal thermal resistance, and related factors affecting stator winding temperature rise. Statistical processing of AR series motor tests shows that heat transfer coefficients vary by frame size and stator stack length, making use of a single series wide value unjustified. The paper proposes averaged coefficients for each frame length and a regression formula based on cooling surface and the ratio of stack length to stator bore diameter, with calculated values differing from experimental means by no more than 3 percent.
Full Text
INDIRECT METHOD FOR DETERMINING THE EQUIVALENT HEAT TRANSFER COEFFICIENT OF ROLLER-TABLE MOTORS
V. V. DNEPROVSKII
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
One of the main indicators characterizing the operation of roller-table motors under frequent starting and braking conditions is the dynamic constant.
The dynamic constant measures the acceleration work performed over the course of an hour and is a measure of the motor’s loading capacity in dynamic modes. Experimentally, the dynamic constant is determined from the expression [1]
\[ D = k \sum GD^2 \cdot z, \quad [\mathrm{kg}\cdot \mathrm{m}^2/\mathrm{h}.], \tag{1} \]
where
\(\sum GD^2\) — the total flywheel moment, equal to the sum of the flywheel moment of the rotor and the additional load flywheel moment reduced to the motor shaft, \([\mathrm{kg}\cdot \mathrm{m}^2]\);
\(z\) — permissible number of cycles per hour;
\(k\) — coefficient characterizing the type of braking;
\(k = 4\) — for plugging braking;
\(k = 2\) — for dynamic braking;
\(k = 1\) — for self-braking.
The permissible number of reversals per hour at a specified duty factor (DF%) is established so that, during acceleration of the required flywheel mass to the synchronous speed of the motor, the temperature rise of the stator winding for the corresponding insulation class reaches the maximum permissible value.
By calculation, the dynamic constant is determined from the formula
\[ D = \frac{ 3600\,\Theta_{\mathrm{cu}} \cdot S_{\mathrm{cool}}\,\alpha - 3600\,P_0\,\dfrac{\mathrm{DF}\%}{100} }{ \dfrac{n_c}{730} \left[ 1+\dfrac{r_1}{r_2} \left( 1+\dfrac{2I_0}{D} \right) \right] }, \tag{2} \]
where
\(\Theta_{\mathrm{cu}}\) — permissible temperature rise of the stator winding for insulation class “H”; \(\Theta_{\mathrm{cu}} = 120^\circ \mathrm{C}\);
\(S_{\mathrm{cool}}\) — cooling surface of the stator core, determined from the expression
\[ S_{\mathrm{cool}} = \pi D_a \left( \frac{D_a}{2} + l_1 \right), \tag{3} \]
where
\(D_a\) and \(l_1\) are the outside diameter and length of the stator stack;
\(P_0\) — no-load losses;
\(n_c\) — synchronous rotational speed of the motor;
\(r_1\) and \(r_2'\) — respectively, the resistance of the stator winding and the reduced resistance of the rotor squirrel cage for the hot state of the motor;
\(I_0\) — no-load current;
\[ D=\frac{u_1}{x_k} \]
is the diameter of the circle of the circle diagram,
where
\(u_1\) — phase voltage supplied to the motor;
\(x_k\) — short-circuit reactance;
\(\alpha\) — heat-transfer coefficient, \([\text{W}/\text{m}^2\,^\circ\text{C}]\).
As the results of statistical processing have shown, the actual values of the dynamic constant, determined from data of type tests for motors of the same type, have a certain scatter. The mean value of half the scatter field for the AR series is 21.2% [2]. The scatter of the dynamic constant is explained by the instability of the properties of the materials used, technological deviations in the manufacture of the motors, and a number of other factors. These deviations are the cause of the scatter of the heat-transfer coefficient.
On the basis of the precise calculations carried out, it has been established that the scatter of the dynamic constant for motors of the AR series depends to a large extent on the scatter of the heat-transfer coefficient.
At present, when determining the dynamic constant by calculation from (2), the averaged constant value \(\alpha=45\ \text{W}/\text{m}^2\,^\circ\text{C}\) is taken as the heat-transfer coefficient, which explains the considerable discrepancy between the calculated and experimental values of the dynamic constant. Therefore, determining the actual heat-transfer coefficients from the results of type tests as the average value for each frame size will make it possible to calculate the dynamic constants of motors more accurately.
The value of the heat-transfer coefficient for roller-table motors of the basic design (phase voltage 220 V, at \(PV=25\%\)) can be determined from expression (2)
\[ \alpha= \frac{ D\cdot \frac{n_c}{730} \left[ 1+\frac{r_1}{r_2'} \left( 1+\frac{I_0x_k}{110} \right) \right] +900P_0 }{ 432000\cdot S_{\text{cool.}} }. \tag{4} \]
The short-circuit reactance is determined from the relation
\[ x_k=\sqrt{z_k^2-r_k^2}, \tag{5} \]
where
\[ z_k=\frac{u_1}{I_k} \]
is the total impedance of the motor;
\(I_k\) — short-circuit current for the motor in the hot state;
\[ r_k=\frac{P_k}{3I_k^2} \]
is the active short-circuit resistance in the hot state;
\(P_k\) — short-circuit losses for the motor in the hot state.
All data for calculating the coefficient \(\alpha\) were taken from the type-test reports.
The quantity \(\alpha\), calculated by formula (4), is understood to mean the equivalent heat-transfer coefficient, which determines the cooling conditions of the stator winding, taking into account the heat-transfer coefficient of the frame surface, the internal thermal resistance from the winding to the frame, and other factors.
Table 1 gives the mean values of the heat-transfer coefficient \(\overline{\alpha}\) and quantities characterizing its scatter: \(\sigma\)—the root-mean-square deviation; \(\delta_{\alpha}\)—one-half of the scatter range. The quantities characterizing the scatter of the heat-transfer coefficient, given in Table 1, can be used in calculating tolerances for the dynamic constants for various motor sizes.
Table 1
| Motor type AR | \(\overline{\alpha}\), W/m\(^2\)°C | \(\sigma\), W/m\(^2\)°C | \(\delta_{\alpha}\), % |
|---|---|---|---|
| 43—4 | 47,83 | 2,69 | 16,8 |
| 42—6 | 50,83 | 2,07 | 12,2 |
| 43—6 | 48,16 | 1,73 | 10,7 |
| 42—8 | 56,63 | 1,52 | 8,06 |
| 43—8 | 47,57 | 2,47 | 15,5 |
| 42—10 | 55,10 | 2,92 | 15,9 |
| 43—10 | 50,66 | 5,18 | 30,3 |
| 43—12 | 46,40 | 4,24 | 27,4 |
| 52—8 | 50,29 | 1,49 | 8,9 |
| 53—8 | 51,58 | 2,20 | 15,8 |
| 52—10 | 50,39 | 0,89 | 5,32 |
| 53—10 | 45,53 | 0,98 | 6,41 |
| 52—12 | 52,10 | 1,57 | 9,03 |
| 53—12 | 47,89 | 2,56 | 16,0 |
| 64—10 | 44,50 | 4,59 | 31,0 |
| 63—12 | 47,81 | 3,15 | 19,7 |
| 64—12 | 45,35 | 1,07 | 7,06 |
| 64—16 | 46,57 | 2,22 | 14,3 |
| 73—10 | 44,65 | 3,70 | 25,0 |
| 74—10 | 40,78 | 1,84 | 13,5 |
| 73—12 | 45,84 | 2,74 | 17,9 |
| 73—16 | 46,42 | 2,41 | 15,6 |
| 74—16 | 41,69 | 2,05 | 14,7 |
As can be seen from Table 1, the mean values of the heat-transfer coefficients differ for different motor sizes. Therefore, using the averaged heat-transfer coefficient \(\alpha = 45\) W/m\(^2\) °C, adopted as constant for the entire series when calculating the dynamic constants by expression (2), is unjustified. This is explained by the fact that, for the purpose of standardization, one frame has been adopted for each size for two stator-core lengths. In addition, in motors of the same size but for different rotational speeds, the same stator lamination is also used for standardization purposes.
It follows from Table 1 that the averaged values of the heat-transfer coefficients for motors of the first lengths of a given size are higher than for motors of the second lengths. Moreover, the scatter of the heat-transfer coefficients of motors for different rotational speeds of one and the same length of each size is considerably smaller than for motors of both lengths within one size. In this connection, in order to simplify the calculation of dynamic constants, it will be justified to adopt averaged values of the heat-transfer coefficients for each length of one and the same size of all pole numbers. These averaged values of the heat-transfer coefficients are given in Table 2.
Table 2
| Type of AP motors | $\alpha$, W/m$^2$°C | $S_{\mathrm{cool}}$, m$^2$ | $\lambda$ |
|---|---|---|---|
| 42 | 54.2 | 0.1177 | 1.026 |
| 43 | 48.1 | 0.1407 | 1.383 |
| 52 | 50.9 | 0.2020 | 0.920 |
| 53 | 45.0 | 0.2483 | 1.315 |
| 63 | 47.8 | 0.3065 | 0.551 |
| 64 | 45.5 | 0.3528 | 0.734 |
| 73 | 45.6 | 0.4270 | 0.661 |
| 74 | 41.2 | 0.5020 | 0.892 |
To determine the heat-transfer coefficients analytically, we find a correlation dependence of the form
\[ \alpha = b_0 + b_1 S_{\mathrm{cool}} + b_2 \lambda + b_3 \cdot S_{\mathrm{cool}}^2 + b_4 \lambda^2, \tag{6} \]
where
\[ \lambda = \frac{l_1}{D_i} \]
(the dimensions $l_1$ and $D_i$ are taken to be the same);
$D_i$ is the inside diameter of the stator bore;
$b_0, b_1, b_2, b_3, b_4$ are regression coefficients.
Using the calculated data $S_{\mathrm{cool}}$, $\lambda$, and the averaged values of the heat-transfer coefficients $\alpha$, whose values are given in Table 2, after carrying out regression analysis the following regression equation was obtained:
\[ \alpha = 60.03 - 60.19 S_{\mathrm{cool}} + 10.08 \lambda + 40.93 S_{\mathrm{cool}}^2 - 9.39 \lambda^2. \tag{7} \]
The heat-transfer coefficients calculated by (7) have a maximum discrepancy from the experimental mean values of 3%.
For calculating heat-transfer coefficients, the values $S_{\mathrm{cool}}$ and $\lambda$ are taken from Table 2. When developing intermediate modifications, which is typical for roller-table motors, these values are determined from the adopted package lengths to their diameters, respectively.
Conclusions
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The single averaged heat-transfer coefficient for the entire AP series, currently adopted for calculating dynamic constants, is unjustified. A single averaged heat-transfer coefficient may be adopted only for the same stator package lengths in a given frame size at all rotational speeds (Table 2).
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For the analytical calculation of averaged heat-transfer coefficients as functions of frame size and stator package length, a correlation dependence (6, 7) is proposed as a function of the principal dimensions of the magnetic core.
References
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GOST 10283-69. Three-phase squirrel-cage asynchronous roller-table motors of the AP series.
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V. V. Dneprovskii, O. P. Muravlev, S. A. Shelekhov. Selection of nominal parameters of roller-table motors. Electrical Engineering Industry, 1971, No. 6.