Proceedings
of the Tomsk Order of the Red Banner of Labor
Polytechnic Institute named after S. M. Kirov

Vol. 190　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1968

OPTIMIZATION OF THE PARAMETERS

OF THE VENTILATION SYSTEM

OF ASYNCHRONOUS MOTORS

V. A. ZHADAN, D. I. SANNIKOV, M. A. SANNIKOVA

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

Intensification of the cooling of electrical machines is one of the most important means of increasing their power without increasing their dimensions. However, this process is usually accompanied by an increase in the cost of the machine due to complication of the design and an increase in the power expenditure for cooling \(P_{\mathrm{v}}\), which makes it necessary to seek an optimal solution.

For enclosed asynchronous motors cooled by external blowing of the casing, such a criterion for assessing the quality of the ventilation system as the specific power expenditure for cooling [1]

\[ P = \frac{P_{\mathrm{v}}}{P_{\Sigma}}, \]

where \(P_{\Sigma}\) is the losses removed by the air flow, cannot serve as a basis for optimization, since its dependence on the parameters of the ventilation system has no extrema.

The efficiency of a ventilation system can be judged by the change in the actual or reduced cost of the motor as a result of changes in the cooling parameters [2, 3]. Since, as a result of improving the ventilation system, the power of the motor may increase, comparison of different variants should be carried out according to the specific actual cost per 1 kW of rated power

\[ C_{y} = \frac{1}{P}\left(\frac{K}{T_{\mathrm{co}}} + C_{a} + C_{p}\right). \tag{1} \]

Here

\[ \begin{aligned} K &\text{ — the total cost of the motor,}\\ T_{\mathrm{co}} &\text{ — payback period,}\\ C_{a}\ \text{and}\ C_{p} &\text{ — the cost of the active energy of losses and of reactive energy over a year of operation.} \end{aligned} \]

Improving the heat transfer of the motor casing makes it possible, at unchanged winding temperature, to increase the heat energy removed and thereby to increase the specific electromagnetic loads and, consequently, the useful power. This leads to a reduction in the specific cost. At the same time, however, the cost of active losses increases

\[ C_{a} = c_{a} T_{\mathrm{r}} P_{\mathrm{H}}, \tag{2} \]

where

\[ \begin{aligned} C_a &\text{— price of one kWh of electric energy,}\\ T_r &\text{— average number of operating hours per year,}\\ P_{\Sigma f} &\text{— motor losses at incomplete average load, characterized by the load factor } f. \end{aligned} \]

The cost \(C_p\) also increases, but since it amounts to no more than 10 percent of \(C_a\), the change in the ratio \(C_p/P\) may be disregarded.

The increase in the cost of the motor frame, fan, and other parts associated with intensified ventilation is taken into account by a corresponding increase in the cost \(K\).

With a simultaneous increase in \(P\), \(K\), and \(C_a\), expression (1) may have a minimum which, evidently, corresponds to the optimal cooling conditions.

At this stage it is expedient to consider the effect of ventilation on cost without taking account of additional factors such as raising the permissible winding temperature, optimizing the geometry of the active part of the motor, improving the utilization of the machine volume, etc. The change in power \(P\) is considered continuous; that is, the discreteness of the power scale is not taken into account in investigating expression (1) for an optimum. The study is carried out on the basis of comparison with the basic version of the motor, to which the index “0” is assigned in the subsequent derivations.

It should be noted that optimization according to expression (1) is not entirely correct. When power is increased by increasing dimensions while the specific electromagnetic loadings and geometric similarity remain constant, the specific actual cost \(C_y\) decreases inversely proportional to \(P^{0.25}\), as is confirmed by an analysis of motors of existing series. When power is increased without increasing dimensions, however, this cost advantage is lost; this can be taken into account by introducing into expression (1) the ratio \((P/P_0)^{0.25}\).

Taking the foregoing into account, the specific cost is now determined by the expression

\[ C=\frac{1}{P^{0.75}P_0^{0.25}}\left(\frac{K}{T_{co}}+C_aT_rP_{\Sigma f}\right). \tag{3} \]

The calculation of the quantities entering into (3) is based on the following principles. The increase in linear load and induction is characterized by the coefficient \(D\)

\[ A=A_0\sqrt{1+D}, \tag{4} \]

\[ B=B_0\sqrt{1+iD}. \tag{5} \]

The coefficient \(i\) \((0\le i\le 1)\) takes into account that, owing to saturation, \(B\) must increase in a smaller proportion than \(A\).

We consider the variable losses \(P_k\) proportional to the square of the linear load, and the constant losses \(P_x\) proportional to the square of the induction. Under certain assumptions,

\[ P_k=P_{m1}+P_{m2}+P_d=P_{k0}(1+D) \tag{6} \]

\[ P_x=P_c+P_r=P_{x0}(1+iD). \tag{7} \]

In (6, 7) the losses in the stator and rotor windings, stray losses, losses in the steel, losses due to rotor friction against air, and bearing losses are taken into account.

The increase in fan power is characterized by the coefficient \(y\)

\[ P_b=P_{b0}\cdot y. \tag{8} \]

On the basis of the energy diagram of the motor, taking into account that the electromagnetic power

\[ P_{\mathrm{e}} = AB = \sqrt{(1 + D)(1 + iD)}, \tag{9} \]

and, expanding (9) in a power series, one can obtain an expression for the useful power as a function of \(D\) and \(y\).

\[ P = P_{0} + P_{\mathrm{bo}}(1 - y) + \frac{D}{2}\left[(P_{0} + P_{\mathrm{bo}})(1 + i) +\right. \]

\[ \left.{} + (P_{\mathrm{to}} - P_{\mathrm{m20}})(1 - i) - \frac{D^{2}}{8}(1 - i)^{2}(P_{0} + P_{\mathrm{to}} + P_{\mathrm{m20}} + P_{\mathrm{bo}})\right]. \tag{10} \]

Losses in the motor at partial load

\[ P_{\Sigma f} = f^{2}(1 + D)P_{\mathrm{ko}} + (1 + iD)P_{\mathrm{x0}} + yP_{\mathrm{bo}}. \tag{11} \]

The coefficient \(D\) for a motor with a squirrel-cage rotor is determined from the condition of constancy of the temperature rise of the stator winding

\[ \theta = R_{\mathrm{b}}(P_{\mathrm{k}} + P_{\mathrm{x}}) + R_{\mathrm{i}}[P_{\mathrm{m1}} + k_{\mathrm{p}}(P_{\mathrm{m2}} + P_{\mathrm{x}})]. \tag{12} \]

Here

\(k_{\mathrm{p}}\) — averaged coefficient of the influence of losses released outside the stator winding on its overheating above the frame;

\(R_{\mathrm{i}}\) — constant internal thermal resistance of the motor — from the stator winding to the shell.

The external thermal resistance depends on the intensity of cooling of the shell; its variation as a function of the cooling intensity is determined by the coefficient \(X\)

\[ R_{\mathrm{b}} = R_{\mathrm{bo}}X. \tag{13} \]

The values \(R_{\mathrm{bo}}\), \(R_{\mathrm{i}}\), and \(K_{\mathrm{p}}\) are determined on the basis of experimental investigations of the basic motor.

From expressions (12) and (13) it follows that

\[ D = \frac{1 - X}{a + bX}, \tag{14} \]

where

\[ a = \frac{R_{\mathrm{i}}}{R_{\mathrm{bo}}}\left[1 - \frac{(1 - k_{\mathrm{p}})P_{\mathrm{m20}} + (1 - ik_{\mathrm{p}})P_{\mathrm{x0}}}{P_{\mathrm{ko}} + P_{\mathrm{x0}}}\right] \tag{15} \]

\[ b = 1 - (1 - i)\frac{P_{\mathrm{x0}}}{P_{\mathrm{ko}} + P_{\mathrm{x0}}}. \tag{16} \]

At this stage, three principal factors influencing the intensity of shell cooling are considered: the cooling surface \(S\), the fan head \(H\), and the hydraulic resistance of the air duct \(z\). As parameters of the ventilation system subject to optimization, the relative values of the indicated quantities are taken:

\[ \sigma = \frac{S}{S_{0}}, \tag{17} \]

\[ h = \frac{H}{H_{0}}, \tag{18} \]

\[ \zeta = \frac{z}{z_{0}}. \tag{19} \]

Flow rate of the cooling air

\[ V = \sqrt{\frac{H}{z}} = V_0(h\zeta)^{0.5}. \tag{20} \]

Experiments show that the heat-transfer coefficient of the casing \(\alpha\) is, on average, proportional to the square root of the air velocity. Taking the outlet cross section to be constant, one may assume the same dependence of \(\alpha\) on the air flow rate. Then

\[ R_b=\frac{1}{S\alpha}=\frac{R_{b0}}{\sigma h^{0.25}\zeta^{0.25}}. \tag{21} \]

Thus, the coefficient of change of the external thermal resistance is

\[ x=\frac{1}{\sigma h^{0.25}\zeta^{0.25}}. \tag{22} \]

The change in ventilation losses at constant fan efficiency is

\[ y=\frac{P_b}{P_{b0}}=\frac{VH}{V_0H_0}=h^{1.5}\zeta^{0.5}. \tag{23} \]

The cost of the motor when the ventilation system is changed can be expressed as follows:

\[ K=K_0(1+d)+K_\sigma+K_h+K_\zeta. \tag{24} \]

Here

\(K_0\) is the cost of the basic motor;

\(d\) is a coefficient allowing for the additional savings from a reduction in the costs of transporting and installing the motor, a reduction in the volume and weight of the production equipment at the motor installation site, etc., as a consequence of reducing the weight and volume of the motor per unit power.

\[ K_\sigma=f(\sigma),\quad K_h=f(h),\quad K_\zeta=f(\zeta)— \tag{25} \]

the increase in the cost of the motor parts pertaining to the ventilation system, in this case respectively the casing, fan, and guide shroud.

We obtain analytical expressions for (25) on the basis of the following assumption: with an unlimited increase in the cost of some part \(m\), for example the fan, the corresponding parameter of the ventilation system \(h\) increases, tending to some maximum possible value \(h_M\), the growth taking place according to an exponential law with constant \(M\),

\[ h=h_M\left(1-e^{-\frac{m}{M}}\right). \tag{26} \]

With an initial cost equal to \(m_0\), \(h=1\); hence

\[ K_h=m-m_0=m_0\frac{\ln(h_M-1)-\ln(h_M-h)}{\ln h_M-\ln(h_M-1)}. \tag{27} \]

For \(K_\zeta\) and \(K_\sigma\) the expressions are analogous. Of course, expression (27) is of a particular character, and the question of the actual dependence of costs on the ventilation parameters requires further study.

As an example, let us consider the results of calculating the specific cost of motors A02-42-4 and A02-42-2 under variation of the ventilation ...

parameters. The initial data for the calculation were taken on the basis of tests of the basic version and the factory cost calculation.

The calculation was performed at the limiting values of the coefficient \(i = 0\) and \(1\), that is, at constant induction and with its variation in the same proportion as the linear load. In addition, the cost was calculated without taking into account additional savings in operation (\(d = 0\)) and with them taken into account (\(d = 0.5\)).

Each of the parameters \(\sigma\), \(h\), and \(\zeta\) was varied within the range from 1 to 2, with the other two equal to unity.

The results of the calculation are shown in Fig. 1; the specific cost \(C\) is given as a percentage of its value for the basic version \((\sigma = h = \zeta = 1)\), which is:

for motor A02-42-4—2.476 rubles/kW at \(d = 0\) and 3.191 rubles/kW at \(d = 0.5\);

for motor A02-42-2—2.178 rubles/kW at \(d = 0\) and 2.766 rubles/kW at \(d = 0.5\).

Fig. 1. Change in specific cost with separate variation of ventilation parameters:

\(1\) — \(i = 0,\quad d = 0,\)

\(2\) — \(i = 1,\quad d = 0,\)

\(3\) — \(i = 1,\quad d = 0.5;\)

solid line — motor A02-42-4.

dash-dot line — motor A02-42-2.

From the data presented it is evident that increasing the cooling surface is most effective; in second place is the reduction of hydraulic resistance, and in third is increasing the fan head. The minimum of the specific cost in most cases lies within the considered range of parameter variation, and for the A02-42-4 motor at 1500 rpm, which is the most common, at \(d = 0\) it lies within the limits

\[ \sigma = 1.5—1.75;\quad h = 1.1—1.25;\quad \zeta = 1.5—1.6. \]

Thus, intensification of cooling in this case is economically expedient even without taking additional savings into account.

The A02-42-2 motor at 3000 rpm has the same optimal values of \(\sigma\); however, the reduction in cost in this case is significantly smaller, while increasing \(h\) and \(\zeta\) within the range considered leads to an increase in cost, mainly due to the growth of ventilation losses.

In general, the reduction in specific cost is insignificant and, at \(d = 0\), amounts to a maximum of 3.2 percent.

Taking the additional saving into account \((d \ne 0)\) increases both the optimal values of the ventilation parameters and the resulting effect. For \(i=1\), improvement of cooling is more effective than for \(i<1\). It should be noted that in this case the calculation expressions are substantially simplified:

useful power

\[ P = P_0 \frac{B + C(1 - X)}{1 - A(1 - X)}, \tag{28} \]

where

\[ A = \frac{1}{1 + a}, \tag{29} \]

\[ B = 1 + \frac{P_{bo}}{P_0}(1 - y), \tag{30} \]

\[ C = A \frac{P_{bo}}{P_0} y, \tag{31} \]

losses during operation

\[ P_{\Sigma j} = (iPKO + PXO)\frac{1}{1 - A(1 - X)} + P_{bo}y. \tag{32} \]

The use of approximating formulas and the neglect of factors that have no substantial influence make it possible to reduce expression (3) to a simple form suitable for the analytical determination of the optimal ventilation parameters and for a qualitative analysis of the change in cost; however, this lies beyond the scope of the present article, since in the present case it is sufficient to carry out the analysis on numerical examples.

The determination of the global optimum was carried out, as an example, for the AO2-42-4 motor at \(i=1\) and \(d=0\), by varying the parameters. Some calculation results are shown in Fig. 2. The optimal values of the ventilation parameters in this case are:

\[ \sigma_0 = 1.76; \quad h_0 = 0.88; \quad \zeta = 1.45. \]

Fig. 2. Change in the specific cost of the AO2-42-4 motor with simultaneous variation of the ventilation parameters

The specific cost in the optimal variant is \(C_{\min}=96.6\%\, C_0\), and the power is \(P=131\%\) of the initial value. Thus, intensification of cooling is advisable. The value of \(C\) changes little over a fairly

with a significant deviation of the parameters, especially $\xi$ and $h$, from the optimal values.

It is characteristic that the partial optimum of the relative cooling surface $\sigma$ is practically independent of the parameters of the fan and the casing, $\xi$ and $h$, which substantially facilitates the search for the global optimum.

The partial optima of $\xi$ and $h$ decrease as the cooling surface is increased; likewise, $h_{\mathrm{opt}}$ decreases as $\xi$ increases.

In conclusion, it should be noted that when designing a new machine it is not necessarily advisable to strive precisely for the global optimum of specific cost, the calculation of which, owing to a number of assumptions, is not sufficiently accurate. It is more expedient to increase the parameters $\sigma$, $h$, and $\xi$ to a greater extent, seeking the maximum increase in the machine’s specific power up to such limits at which a significant cost advantage will still be obtained.

REFERENCES

1. I. F. Filippov. Problems of Cooling Electrical Machines, Energiya, 1964.

2. I. M. Postnikov. Design of Electrical Machines, Kiev, 1960.

3. A. S. Konson. Economic Justification of Designs of Electrical Machines, Apparatus, and Instruments, GEI, 1963.