ENGINEERING METHOD FOR THERMAL CALCULATION OF MAGNETIZING WINDINGS OF BETATRONS WITH AIR COOLING

G. F. Shilin, G. I. Fuks

A reliable selection and calculation of the cooling system for the magnetizing windings of betatrons requires knowledge of the temperature distribution over the surface of the vertical rows of winding bars and of the law of heat generation within its volume.

Fig. 1. On the engineering method for thermal calculation of the magnetizing winding of a betatron.

If the heat generation is constant over the volume of a vertical row of bars, then the law of temperature distribution over its height can be found by the method given in [1]. However, our experiments showed that usu-

but the heat release over the height of a vertical row of busbars depends linearly on the vertical coordinate, that is,

\[ q_0=L+Mx,\ \mathrm{W/m^3}, \tag{1} \]

where \(x\) is the vertical coordinate of the winding in m (Fig. 1).

Consequently [1], using this relation, one can determine the law of temperature variation along the wall of the vertical rows of winding busbars in various cases.

a) Air is supplied into the winding slots from below at \(X=0\) (Fig. 1). Then

\[ t_{\mathrm{st}}(X)=t_{\mathrm{in}}+\frac{2U}{\mathrm{Pe}}\mathrm{Nu}\,\frac{a}{s} \sum_{m=1}^{\infty} A_m \left(\sin\mu_m X-\frac{\mathrm{Bi}}{\mu_m}\cos\mu_m X+\frac{\mathrm{Bi}}{\mu_m}\right)+ \tag{2} \]

\[ +\sum_{m=1}^{\infty} A_m\left(\mu_m\cos\mu_m X+\mathrm{Bi}\sin\mu_m X\right). \]

b) Air is supplied into the winding slots from above at \(X=1\). Then

\[ t_{\mathrm{st}}(X)=t_{\mathrm{in}}-\frac{2U}{\mathrm{Pe}}\mathrm{Nu}\cdot\frac{a}{s} \sum_{m=1}^{\infty} A_m\left[(\sin\mu_m X-\sin\mu_m)- \right. \tag{3} \]

\[ \left. -\frac{\mathrm{Bi}}{\mu_m}(\cos\mu_m X-\cos\mu_m)\right] +\sum_{m=1}^{\infty} A_m\left(\mu_m\cos\mu_m X+\mathrm{Bi}\cdot\cos\mu_m X\right). \]

In formulas (2) and (3):

\[ A_m= \frac{ F(\mu_m)\cdot \operatorname{sh}\dfrac{\mu_m}{\eta} }{ \mu_m\cdot D(\mu_m) \left[ \mu_m \operatorname{sh}\dfrac{\mu_m}{\eta} +\mathrm{Bi}\operatorname{ch}\dfrac{\mu_m}{\eta} \right] }; \tag{4} \]

\[ D(\mu_m)=\frac{1}{2\mu_m} \left\{ \mu_m^2(\mu_m+0.5\sin 2\mu_m) +\mathrm{Bi}^2(\mu_m-0.5\sin 2\mu_m) +2\mu_m\cdot\mathrm{Bi}\cdot\sin^3\mu_m \right\}; \]

\[ F(\mu_m)= \frac{\beta}{\mu_m} \left[\mu_m\cdot\sin\mu_m+\mathrm{Bi}(1-\cos\mu_m)\right]+ \]

\[ +\frac{\gamma}{\mu_m^2} \left[\mu_m\cdot\cos\mu_m(1-\mathrm{Bi}) +\sin\mu_m(\mathrm{Bi}+\mu_m^2)-\mu_m\right]; \]

\[ \beta=4\,\frac{L}{\lambda_{\mathrm{eq}}}\cdot a^2; \qquad \gamma=8\,\frac{M}{\lambda_{\mathrm{eq}}}\cdot a^3; \qquad \eta=\frac{a}{b}; \]

\[ X=\frac{x}{2a}; \qquad \mathrm{Bi}=\frac{\alpha\cdot 2a}{\lambda_{\mathrm{eq}}}; \qquad \mathrm{Pe}=\frac{W\cdot 2s}{a^*}; \qquad \mathrm{Nu}=\frac{\alpha\cdot 2s}{\lambda_{\mathrm{air}}}; \]

\(W\) is the air velocity in the cooling slots of the winding, in m/s;

\(\alpha\) is the heat-transfer coefficient in the winding slot to air with temperature \(t_{\mathrm{in}}\), in \(\mathrm{W/m^2\cdot deg}\);

\(s\) is the width of the cooling slot, in m (Fig. 1);

\(a^*\) and \(\lambda_{\mathrm{air}}\) are the thermal diffusivity coefficient and thermal conductivity of air;

\(\lambda_{\mathrm{eq}}\) is the equivalent thermal-conductivity coefficient of the winding, in \(\mathrm{W/(m\cdot deg)}\);

Table 1

Roots of the equation

\[ \operatorname{tg}\mu_m=\frac{2\mathrm{Bi}\cdot\mu_m}{\mu_m^{2}-\mathrm{Bi}^{2}} \]

Continuation of Table 1

$U$ is the perimeter of the flow cross section of the winding cooling slot, in m;

$\mu_m$ are the roots of the equation

\[ \operatorname{tg}\mu_m=\frac{2\mathrm{Bi}\cdot \mu_m}{\mu_m^2-\mathrm{Bi}^2}. \]

The first three roots of this equation, found on the Promin electronic digital computer, are presented in Table 1.

Relations (2) and (3) can be simplified if it is taken into account that the series in them converge rapidly. In engineering calculations, these dependences may be used with one term of the series, $m=1$. Then:

a) air is supplied into the winding slots from below at $X=0$ (Fig. 1),

\[ t_{\mathrm{st}}(X)=t_{\mathrm{in}}+\frac{2U}{\mathrm{Pe}}\cdot \mathrm{Nu}\,\frac{a}{s}\,A_1 \left(\sin\mu_1 X-\frac{\mathrm{Bi}}{\mu_1}\cos\mu_1 X+\frac{\mathrm{Bi}}{\mu_1}\right)+ \tag{5} \]

\[ +\,A_1(\mu_1\cos\mu_1 X+\mathrm{Bi}\sin\mu_1 X), \]

where $A_1$ is calculated from (4), with $m=1$.

The coordinate of the maximum temperature is found from the expression:

\[ X_{\max}=\frac{1}{\mu_1}\operatorname{arctg} \left[ \frac{-\mu_1\left(\mathrm{Bi}-\dfrac{2U}{\mathrm{Pe}}\mathrm{Nu}\,\dfrac{a}{s}\right)} {\mathrm{Bi}\left(\dfrac{\mu_1^2}{\mathrm{Bi}}+\dfrac{2U}{\mathrm{Pe}}\mathrm{Nu}\,\dfrac{a}{s}\right)} \right]; \tag{6} \]

b) air is supplied into the winding slots from above at $X=1$ (Fig. 1),

\[ t_{\mathrm{st}}(X)=t_{\mathrm{in}}-\frac{2U}{\mathrm{Pe}}\,\mathrm{Nu}\cdot \frac{a}{s}\,A_1\left[(\sin \mu_1X-\sin \mu_1)-\frac{\mathrm{Bi}}{\mu_1}(\cos \mu_1X-\cos \mu_1)\right]+ \]
\[ +\,A_1(\mu_1\cos \mu_1X+\mathrm{Bi}\cdot \sin \mu_1X). \tag{7} \]

The coordinate of the maximum temperature is determined from the relation:

\[ X_{\max}=\frac{1}{\mu_1}\operatorname{arctg} \left[ \frac{\mu_1\left(\mathrm{Bi}+\dfrac{2U}{\mathrm{Pe}}\,\mathrm{Nu}\cdot \dfrac{a}{s}\right)} {\mathrm{Bi}\left(\dfrac{\mu_1^2}{\mathrm{Bi}}-\dfrac{2U}{\mathrm{Pe}}\cdot \mathrm{Nu}\cdot \dfrac{a}{s}\right)} \right]. \tag{8} \]

For calculating \(A_1\) from (4), a nomogram has been constructed, which significantly reduces the labor of computational operations (Fig. 2).

Fig. 2. Nomogram for calculating the quantity \(A_1\) by formula (4).

The thermal calculation of the magnetizing winding of a betatron, based on the use of relations (5)—(8), must proceed according to the electro-

Table 2

Engineering procedure for the thermal calculation of the magnetizing winding of a betatron

Continuation of Table 2

Continuation of Table 2

Continuation of Table 2

technical calculation. The latter gives the geometric dimensions and a number of electrical parameters that are necessary for a quantitative assessment of the heat distribution in the vertical rows of the winding busbars by the method of eddy-field modeling [2].

It is convenient to carry out the thermal calculation in the form of Table 2, entering in the first part all the data necessary for the calculation. Ultimately, the air velocity in the cooling ducts of the winding is calculated, at which the maximum temperature on the winding surface does not exceed the permissible value. From the air velocity and the total cross-sectional area of all cooling ducts of the upper and lower windings of the betatron, the total quantity of air required for cooling the winding is calculated.

REFERENCES

1. V. V. Ivanov, G. F. Shilin. Thermal calculation of the magnetizing winding of a betatron with air cooling. Izv. vuzov—Elektromekhanika, No. 8, 1964.

2. S. I. Lur’e. Mathematical modeling of stray magnetic fields of transformers and reactors on electrically conducting paper. Elektrichestvo, No. 10, 1965.

3. F. M. Tarasov. Thin-Layer Heat-Exchange Apparatus. Mashgiz, 1964.

4. S. S. Kutateladze, V. M. Borishanskii. Handbook of Heat Transfer. Gosenergoizdat, 1959.

5. Electrical Engineering Handbook, Vol. 1. Gosenergoizdat, 1962.