PROCEEDINGS
OF THE TOMSK ORDER OF THE OCTOBER REVOLUTION
AND ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Vol. 284　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

THERMAL CALCULATION OF ELECTROHYDRAULIC THRUSTERS

L. N. VERESHCH, E. M. GUSELNIKOV, V. A. ZHADAN, D. I. SANNIKOV

(Presented by the joint seminar of the Department of Electrical Machines and Apparatus
and General Electrical Engineering)

On the basis of the general principles of the method of equivalent thermal circuits, and proceeding from the features of the temperature field, one may construct an equivalent thermal substitution circuit for an electrohydraulic thruster (Fig. 1). In the equivalent circuit, five bodies are considered: 1 — the stator winding; 2 — the squirrel-cage winding together with the rotor teeth; 3 — the oil filling the internal space of the electrohydraulic thruster; 4 — the motor frame; 5 — the hydraulic-thruster housing. The first three bodies have internal heat generation; moreover, for oil-filled motors, mechanical losses account for a considerable share.

The power of the heat sources is:

\[ \begin{aligned} P_1 &= P_{\mathrm{m}1} + \frac{P_{\mathrm{st}}}{2};\\ P_2 &= P_{\mathrm{m}2} + P_{\mathrm{d}};\\ P_3 &= P_{\mathrm{mech}} + P_{\mathrm{g}};\\ P_4 &= \frac{P_{\mathrm{st}}}{2}, \end{aligned} \tag{1} \]

where

\[ \begin{aligned} P_{\mathrm{m}1} &\text{ — losses in the stator winding;}\\ P_{\mathrm{m}2} &\text{ — losses in the rotor winding;}\\ P_{\mathrm{mech}} &\text{ — mechanical losses in the motor;}\\ P_{\mathrm{d}} &\text{ — additional losses;}\\ P_{\mathrm{g}} &\text{ — losses in the hydraulic thruster;}\\ P_{\mathrm{st}} &\text{ — losses in the stator steel.} \end{aligned} \]

The thermal connections between the bodies are determined by thermal conductances \(\lambda\), the general expression for which is:

1) in the case of heat transfer from a surface —

\[ \Lambda_k = \alpha_k \cdot S_k, \tag{2} \]

2) in the case of transfer of heat flow through a solid body —

\[ \Lambda_i=\frac{\lambda_i \cdot S_i}{l_i}, \tag{3} \]

where

\(\alpha_{\mathrm{k}}\) is the heat-transfer coefficient;
\(S_{\mathrm{k}}\) and \(S_i\) are the heat-flow areas;
\(\lambda_i\) is the thermal conductivity of the material;
\(l_i\) is the length of the flow path.

The process of heat transfer from the housings of the motor and the hydraulic thruster to the surrounding medium takes place by convection and radiation. The coef-

Fig. 1. Equivalent thermal substitution circuit of an electrohydraulic thruster for calculating the average temperatures of individual parts

ficients of convective heat transfer of the motor housing \(\alpha_{\mathrm{kd}}\) and of the hydraulic thruster \(\alpha_{\mathrm{kg}}\) were determined experimentally, since calculation of these coefficients is laborious. As a result, the dependences \(\alpha_{\mathrm{kd}}=f(\theta)\) and \(\alpha_{\mathrm{kg}}=f(\theta)\) were constructed (Fig. 2). The heat-transfer coefficients due to radiation are determined by known methods [2].

Heat transfer from the end surfaces of the rotor to the oil is expressed by the criterion equation

\[ Nu=0.021\,Re^{0.8}\cdot Pr^{0.43}\left(\frac{Pr_{t_1}}{Pr_{t_2}}\right)^{0.25}. \tag{4} \]

For ease of use, formula (4) is presented in the form of the dependence \(\alpha_t=f(v_p,d_a)\) (Fig. 3),

Fig. 2. Dependence of the convective heat-transfer coefficients of the motor housing and the hydraulic pusher on overheating \(\alpha_{kD}=f(\theta_k)\) and \(\alpha_{kT}=f(\theta_k)\)

Fig. 3. Heat transfer from the rotor end surface to oil
\(\alpha_t d_a=f(v d_a)\)

where

\[ v_p \]

is the peripheral speed of the rotor,

\[ d_a \]

is the rotor diameter (the defining dimension).

Heat transfer in the gap between the rotor and the stator is determined to a significant extent by the flow regime of the liquid. It is calculated from the criterion dependence for the turbulent flow regime of a liquid during rotation of the inner cylinder [4]

\[ Nu_{\delta}=Re_{\delta}\frac{v_{2}}{v_{1}}, \tag{5} \]

where

\(v_{1}\) and \(v_{2}\) are the peripheral velocities of the liquid at the surfaces of the cylinders.

By transformation, equation (5) is brought to the form

\[ \alpha_{\delta v}=\alpha_{\delta}\frac{v_{p}}{5.5}. \tag{6} \]

The dependence \(\alpha_{\delta}=f(\vartheta_{\mathrm{cp}})\) is presented in Fig. 4, where \(\vartheta_{\mathrm{cp}}\) is the oil temperature in the gap.

The value of the heat-transfer coefficient of the end portion of the stator winding \(\alpha_{\ell 1}\) is determined as a function of the velocity of liquid flow and of temperature. In general form this dependence is expressed by the criterion equation

\[ Nu=CRe^{n}Pr^{m}. \tag{7} \]

The dependence \(\alpha_{\ell 1}=f(v_{p})\) at various oil temperatures is given in Fig. 5.

In an analogous way, the dependence of the coefficient of heat transfer from the oil to the housing in the region of the end portion, \(\alpha_{ak}=f(v_{p})\), shown in Fig. 6, was obtained.

Fig. 4. Heat transfer in the gap \(\alpha_{\delta}=f(\theta_{\mathrm{cp}})\)

It is convenient to determine the mean temperatures of the individual parts of the hydraulic pusher by the method of transformations of diagram \(a\) (Fig. 1), by successively eliminating nodes 2, 1, 3. From the final diagram \(d\), the temperature rises of the housings of the motor and of the hydraulic pusher are calculated

Fig. 5. Heat transfer from the end portion of the winding to the oil
\(\alpha_{\ell 1}=f(v)\)

\[ \theta_{kd}= \frac{P_{4}''(\Lambda_a+\Lambda_{50})+P_{5}'\Lambda_a} {\Lambda_a(\Lambda_{40}+\Lambda_{50})+\Lambda_{40}\Lambda_{50}}, \tag{8} \]

\[ \theta_{kg}= \frac{P_{4}''\Lambda_a+P_{5}'(\Lambda_a+\Lambda_{40})} {\Lambda_a(\Lambda_{40}+\Lambda_{50})+\Lambda_{40}\Lambda_{50}}. \tag{9} \]

Then, from diagrams d and c, the temperature rises of the oil and of the stator winding are determined:

\[ \theta_m= \frac{P_{3}''+\theta_{kg}\Lambda_{35}+\theta_{kd}\Lambda_{44}'} {\Lambda_{35}+\Lambda_{34}'}, \tag{10} \]

\[ \theta_o= \frac{P_{1}'+\theta_{kd}\Lambda_{14}+\theta_m\Lambda_{13}'} {\Lambda_{14}+\Lambda_{13}'}. \tag{11} \]

Fig. 6. Heat transfer from oil to the housing in the region of the end part
\(\alpha_{ak}=f(v)\)

Example of calculation of a TGM-50 hydraulic thruster.

Initial data: \(P_n=0.2\ \mathrm{kW}\); \(\eta=0.67\); \(n=3000\ \mathrm{rpm}\); \(d_a=0.0552\ \mathrm{m}\); \(P_{m1}=16.3\ \mathrm{W}\); \(P_{m2}=10.8\); \(P_{st}=24\ \mathrm{W}\); \(P_{mekh}=18\ \mathrm{W}\); \(P_d=2.9\ \mathrm{W}\); \(P_g=66\ \mathrm{W}\).

Cooling surfaces of the ribbed housing of the motor and hydraulic thruster for calculating convection and radiation: \(S_{kd}=0.107\ \mathrm{m^2}\); \(S_{ld}=0.057\ \mathrm{m^2}\); \(S_{kg}=0.198\ \mathrm{m^2}\); \(S_{lg}=0.137\ \mathrm{m^2}\).

Areas of the end surfaces of the rotor, the gap, the end portions of the stator, and the internal surfaces of the housing of the motor and hydraulic thruster: \(S_t=0.0088\ \mathrm{m^2}\); \(S_{\delta}=0.0087\ \mathrm{m^2}\); \(S_l=0.0239\ \mathrm{m^2}\); \(S_{ak}=0.0332\ \mathrm{m^2}\); \(S_{mg}=0.0852\ \mathrm{m^2}\).

Thermal conductances, calculated by the method of [1, 3]: in the gap between the stator and the housing \(\Lambda_{\lambda}=52\ \mathrm{W/deg}\), between the housings of the motor and the hydraulic thruster \(\Lambda_{\Delta1}=0.79\ \mathrm{W/deg}\), of the yoke and stator teeth \(\Lambda_j=42\ \mathrm{W/deg}\), \(\Lambda_z=8.8\ \mathrm{W/deg}\), of the insulation of the stator winding in the slot and end portion \(\Lambda_i=9.5\ \mathrm{W/deg}\), \(\Lambda_{il}=16.3\ \mathrm{W/deg}\).

Leakage conductance to the mounting plate from the motor housing: \(\Lambda_y=0.1\ \mathrm{W/deg}\).

Taking preliminarily the temperature rises of the motor and hydraulic-thruster housings as \(\Theta_{kd}=\Theta_{kg}=50^\circ\mathrm{C}\), we determine from Fig. 2 the heat-transfer coefficients \(\alpha_k=6.05\ \mathrm{W/(m^2\,deg)}\) and \(\alpha_{kg}=5.15\ \mathrm{W/(m^2\,deg)}\).

Radiative heat-transfer coefficient of the motor and the hydraulic pusher:

\(T_1 = 273 + 27 = 300^\circ K;\)

\(T_2 = 300 + 50 = 350^\circ K;\)

\[ \alpha_{ld} = \alpha_{lg} = \frac{5.77 \cdot 0.72}{50}\left(3.5^4 - 3.0^4\right) = 5.68\ \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{deg}. \]

Total heat-transfer coefficients

\[ \alpha_{\Sigma d} = 6.05 + 5.68\frac{0.0508}{0.107} = 9.06\ \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{deg}; \]

\[ \alpha_{\Sigma g} = 5.15 + 5.68\frac{0.137}{0.198} = 9.05\ \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{deg}. \]

Total conductances to the surrounding air of the motor—

\[ \Lambda_{kd} = 9.06 \cdot 0.107 = 0.97\ \mathrm{W}/\mathrm{deg}, \]

of the hydraulic pusher—

\[ \Lambda_{kg} = 9.05 \cdot 0.198 = 1.79\ \mathrm{W}/\mathrm{deg}. \]

We preliminarily set the oil temperature at \(\vartheta_{\mathrm{avg}} = 90^\circ\) and, from Figs. 3–6, determine the heat-transfer coefficients
\(\alpha_t = 1810\ \mathrm{W}/\mathrm{m}^2 \mathrm{deg};\quad \alpha_b = 605\ \mathrm{W}/\mathrm{m}^2 \mathrm{deg};\quad \alpha_{l1} = 184\ \mathrm{W}/\mathrm{m}^2 \mathrm{deg};\quad \alpha_{ak} = \alpha_{mg} = 151\ \mathrm{W}/\mathrm{m}^2 \mathrm{deg}.\)

The corresponding conductances are—

\[ \Lambda_{l2} = 1810 \cdot 0.0088 = 16\ \mathrm{W}/\mathrm{deg}; \]

\[ \Lambda_b = 605 \cdot 0.0087 = 5.25\ \mathrm{W}/\mathrm{deg}; \]

\[ \Lambda_{lm} = 184 \cdot 0.0239 = 4.4\ \mathrm{W}/\mathrm{deg}; \]

\[ \Lambda_{ak} = 151 \cdot 0.0332 = 5.05\ \mathrm{W}/\mathrm{deg}, \]

\[ \Lambda_{mg} = 151 \cdot 0.0852 = 12.8\ \mathrm{W}/\mathrm{deg}. \]

Power of the heat sources—

\[ P_1 = 16.3 + 12 = 28.3\ \mathrm{W}; \]

\[ P_2 = 10.8 + 2.9 = 13.7\ \mathrm{W}; \]

\[ P_3 = 18 + 66 = 84\ \mathrm{W}; \]

\[ P_4 = 12\ \mathrm{W}. \]

Transformation of the equivalent circuit:

scheme \(b\)

\[ \Lambda_{12} = \frac{2\Lambda_z \cdot \Lambda_b}{2\Lambda_z + \Lambda_b} = \frac{2 \cdot 8.8 + 5.25}{2 \cdot 8.8 + 5.25} = 4.4; \]

\[ \Lambda_{13} = \frac{\Lambda_{il} \cdot \Lambda_{lm}}{\Lambda_{il} + \Lambda_{lm}} = 3.47; \]

\[ \Lambda_{40} = \Lambda_{kd} + \Lambda_y = 1.07; \]

scheme \(c\)

\[ P'_1 = P_1 + P_2\frac{\Lambda_{12}}{\Lambda_{12} + \Lambda_{23}} = 28.3 + 13.7\frac{4.04}{4.04 + 15.9} = 31.47\ \mathrm{W}; \]

\[ P'_3=P_3+P_2\frac{\Lambda_{23}}{\Lambda_{12}+\Lambda_{23}} =84+13.7\frac{15.9}{4.04+15.9}=94.9\ \text{W}; \]

\[ \Lambda'_{13}=\Lambda_{13}+\frac{\Lambda_{12}\cdot\Lambda_{23}}{\Lambda_{12}+\Lambda_{23}} =3.47+\frac{4.04\cdot15.9}{4.04+15.9}=6.69; \]

scheme 2

\[ P''_3=P'_3+P'_1\frac{\Lambda'_{13}}{\Lambda'_{13}+\Lambda_{14}} =94.9+31.47\frac{6.69}{6.69+4.9}=113.11\ \text{W}; \]

\[ P'_4=P_4+P'_1\frac{\Lambda_{14}}{\Lambda'_{13}+\Lambda_{14}} =12+31.47\frac{4.9}{6.69+4.9}=25.5\ \text{W}; \]

\[ \Lambda'_{34}=\Lambda_{34}+\frac{\Lambda'_{13}\cdot\Lambda_{14}}{\Lambda'_{13}+\Lambda_{14}} =5.05+\frac{6.69\cdot4.9}{6.69+4.9}=7.88; \]

scheme d

\[ P''_4=P'_4+P''_3\frac{\Lambda'_{34}}{\Lambda'_{34}+\Lambda_{35}} =25.5+113.1\frac{7.88}{7.88+12.8}=68.5\ \text{W}; \]

\[ P'_5=P''_3\frac{\Lambda_{35}}{\Lambda'_{34}+\Lambda_{35}} =113.1\frac{12.8}{7.88+12.8}=70\ \text{W}; \]

\[ \Lambda_{\text{a}}=\Lambda_{45}+\frac{\Lambda'_{34}\cdot\Lambda_{35}}{\Lambda'_{34}+\Lambda_{35}} =0.786+\frac{7.88\cdot12.8}{7.88+12.8}=5.66. \]

Calculation of temperature rises:

\[ \theta_{\text{kd}}=\frac{68.5(5.66+1.79)+70\cdot5.66}{5.66(1.79+1.07)+1.79\cdot1.07}=50.2^\circ\text{C}; \]

\[ \theta_{\text{kg}}=\frac{70(5.66+1.07)+68.5\cdot5.66}{5.66(1.79+1.07)+1.79\cdot1.07}=47.4^\circ\text{C}; \]

\[ \theta_{\text{m}}=\frac{113.1+50.2\cdot7.88+47.4\cdot12.8}{7.88+12.9}=53.7^\circ\text{C}; \]

\[ \theta_0=\frac{31.47+53.7\cdot6.69+50.2\cdot4.9}{6.69+4.9}=55.2^\circ\text{C}. \]

The temperature rises of the motor housing and the hydraulic pusher, the oil, and the winding, obtained experimentally—\(\Theta_{\text{kd}}=49.4^\circ\text{C}\); \(\Theta_{\text{kg}}=47.6^\circ\text{C}\); \(\Theta_{\text{m}}=54.8^\circ\text{C}\); \(\Theta_0=55.5^\circ\text{C}\)—differ little from the calculated values.

Conclusions

1. The proposed graphical method for determining heat-transfer coefficients for oil-filled electrohydraulic pushers is sufficiently accurate and convenient for practical calculations.

2. The heat-transfer coefficients of the motor housing and the hydraulic pusher have the greatest influence on the accuracy of the thermal calculation.

References

1. G. G. Schastlivyi. Heating of enclosed asynchronous electric motors. Kiev, “Naukova Dumka,” 1966.

2. G. Gotter. Heating and cooling of electrical machines. Moscow, Gosenergoizdat, 1961.

3. V. A. Zhadan, Yu. V. Kopylov, D. I. Sannikov. A method for the thermal calculation of enclosed low-power asynchronous motors. Izvestiya TPI, vol. 212, 1971.

4. L. A. Dorfman. Hydrodynamic resistance and heat transfer of rotating bodies. GIFML, 1960.