METHODS FOR ESTIMATING HEAT GENERATION IN THE VOLUME OF MAGNETIZING WINDINGS OF BETATRONS

G. F. Shilin

(Presented by the scientific seminar of the Department of TOT)

One of the basic questions in solving the problem of temperature distribution in betatron windings is knowledge of the law of distribution of heat generation \(q_v\) throughout the entire volume of this winding, which makes it possible to identify the section most heavily loaded thermally. The problem is complicated by the fact that heat release in the winding is caused not only by the current flowing in it (active losses in the copper), but also by eddy currents arising because of the very large stray magnetic fields in the betatron window [1].

It is difficult to solve the stated problem theoretically, and it is possible only with special experimental studies [2]. Below, engineering methods are proposed for estimating heat generation in the windings of both operating and newly designed betatrons.

1. Heat generation throughout the entire winding volume is a constant quantity

Such a case occurs in those sections of the magnetizing windings of betatrons that are removed from its poles [1] by fairly considerable distances (closer to the posts). In this case, the absolute value of the heat generation \(q_v\) from the magnetizing current considerably exceeds the heat generation from eddy currents.

If, in addition, the section of winding under consideration is reliably thermally insulated, then the temperature distribution in it during heating will be described by the differential equation

\[ \frac{1}{a}\frac{\partial t(\tau)}{\partial \tau}=\frac{q_v}{\lambda} \]

or

\[ q_v=c\cdot \rho \frac{\partial t(\tau)}{\partial \tau}, \tag{1} \]

where

\[ a=\frac{\lambda}{c\rho} \]

is the thermal diffusivity coefficient of the betatron winding;

\(\lambda\) is the equivalent coefficient of thermal conductivity of the winding;

\(c\) and \(\rho\) are, respectively, the heat capacity and density of the winding material.

It follows from this equation that if, experimentally, one measures the rate of change of temperature with time, \(\dfrac{\partial t(\tau)}{\partial \tau}\), on this section of the winding, then the heat generation \(q_v\) in it can be determined, at least with sufficient approximation.

2. Heat generation throughout the entire volume of the winding is a linear function of temperature

If the electrical resistance of the winding material depends on temperature, that is,

\[ R_t = R_0(1+\gamma \cdot t), \ \text{ohm}, \]

where \(R_0\) is the resistance of the winding material at \(0^\circ\text{C}\);
\(\gamma\) is the proportionality coefficient,

then, in the case of thermal insulation of this winding, the nonstationary temperature field in it is described by the system of equations:

\[ \frac{1}{a}\frac{\partial t(\tau)}{\partial \tau} = \frac{q_{v0}[1+\gamma \cdot t(\tau)]}{\lambda}; \tag{2} \]

\[ t(0)=0. \tag{3} \]

The solution of (2)—(3) is given by the dependence [3]:

\[ t(\tau)=\frac{1}{\gamma} \left[ \exp\left(\frac{q_{v0}\cdot a}{\lambda}\gamma \cdot \tau\right)-1 \right]. \tag{4} \]

From the experiment one can obtain the curve of temperature variation with time for the investigated (thermally insulated) section of the winding. For some fixed time interval \(\tau^*\), from this graph one can find the corresponding temperature value \(t^*(\tau^*)\). Substituting these quantities into (4), we obtain

\[ q_{v0}=c\cdot \rho \cdot \frac{\ln\left[t^*(\tau^*)\cdot \gamma + 1\right]} {\gamma \cdot \tau^*}. \tag{5} \]

3. Heat generation in the volume of the winding is a linear function of one coordinate

Consider the nonstationary temperature field in an infinite rectangular bar \(2R_1 \times 2R_2\), when its faces are thermally insulated. Inside the bar there acts a heat source that depends linearly on the coordinate \(X\), that is,

\[ q_v = q_{v0}+\beta \cdot X. \]

The law of variation and distribution of temperature is described by the system of equations:

\[ \frac{\partial t(X,\mathrm{Fo})}{\partial \mathrm{Fo}} = \frac{\partial^2 t(X,\mathrm{Fo})}{\partial X^2} + B\cdot X + A; \tag{6} \]

\[ \frac{\partial t(1,\mathrm{Fo})}{\partial X}=0; \qquad \frac{\partial t(0,\mathrm{Fo})}{\partial X}=0; \tag{7} \]

\[ t(X,0)=0. \tag{8} \]

The solution of this system is given in [3]:

\[ t(X,\mathrm{Fo})=\mathrm{Fo}(A+0.5B)+\sum_{m=1}^{\infty}B\left[\frac{2\cos \mu_m X}{\mu_m^2}\cdot\frac{\cos \mu_m-1}{\mu_m^2}\right]\times \]

\[ \times \left[1-\exp(-\mu_m^2\mathrm{Fo})\right]. \tag{9} \]

Here:

\[ t(X,\mathrm{Fo})=\frac{t(X,\mathrm{Fo})-t_0}{t_0} \]

is the dimensionless temperature;

\[ t_0 \]

is a fixed value of the temperature;

\[ \mathrm{Fo} \]

is the Fourier criterion;

\[ A=4R_1^2 q_{v0}/\lambda t_0;\quad B=4R_1^2\beta/\lambda t_0;\quad X=\frac{x}{2R_1};\quad \mu_m=m\cdot\pi,\quad m=1,2,3\ldots \]

For some fixed point of the bar with coordinate \(X^*\), taking into account one term of the series \((m=1)\), we obtain:

\[ t(X^*,\mathrm{Fo})=\mathrm{Fo}(A+0.5B)+B\varkappa\left[1-\exp(-\pi^2\mathrm{Fo})\right], \tag{10} \]

where

\[ \varkappa=-\frac{4\cos \pi X^*}{\pi^4}. \]

The dependence (10) is presented graphically in Fig. 1 (curve I). At sufficiently large Fourier criteria \((\mathrm{Fo}\geq 0.3)\), curve I passes into a straight line. If from the origin of coordinates one draws a straight line II parallel to the latter, then its equation will be \(\Theta^*=\mathrm{Fo}(A+0.5B)\). For some fixed \(\mathrm{Fo}^*>0.3\) (Fig. 1) one may write that

\[ \Theta_1=B\cdot\varkappa \quad\text{and}\quad \Theta_2=\mathrm{Fo}^*(A+0.5B). \]

Fig. 1. On the problem of estimating heat generation in the volume of a betatron winding, when it depends linearly on one coordinate.

Whence

\[ A=-\frac{\Theta_1-2\Theta_2\varkappa\cdot\frac{1}{\mathrm{Fo}^*}}{2\varkappa}, \qquad B=\frac{\Theta_1}{\varkappa}. \tag{11} \]

Therefore, having determined \(\Theta_1\) and \(\Theta_2\) for a fixed value of \(Fo^*\) (Fig. 1), one can determine \(A\) and \(B\) from (11), and hence the heat generation as a function of the coordinate \(X\).

4. Approximate method for estimating the law of heat formation

As is known [4], the magnetizing windings of betatrons are made in the form of a spiral of rectangular insulated busbars. The vertical rows of busbars are separated by cooling channels.

Experiments have shown that the heat generation in the vertical rows of busbars may be regarded as a function only of one vertical coordinate \(x\) along the height of the row. In this case the method for finding \(q_v(x)\), described in Sec. 1, may be used. For this it is necessary to measure the wall temperatures at several points along the height of the vertical rows of busbars in the absence of heat exchange with the surrounding medium, i.e., with the cooling channels closed (Fig. 2). The resulting heating curves

Fig. 2. Curves of temperature variation along the height of a vertical row of busbars of a betatron winding at points from 1 to 5.

(Fig. 2) make it possible to determine, for each of them, the tangent of the angle of inclination to the time axis, i.e.,

\[ \left[\frac{\partial t(\tau)}{\partial \tau}\right]_{x_i} \]

for fixed coordinates \(x_i\) of points on the surface of the row of busbars, and to calculate the heat generation from (1) at each point studied. The obtained values of heat generation at each point are plotted on a graph (Fig. 3), from which the equation of the curve is found

\[ q_v = f(x). \]

Figure 3 shows the results of processing the data from a thermal test of the magnetizing winding of the B-25-2 betatron. It follows from it that

heat generation \(q_v\) in its different parts varies with height practically according to a linear law.

Fig. 3. Heat generation over the height of the vertical rows of busbars of the winding of the B-25-2 betatron: ● — thermal-test data for row No. 3 counting from the pole; + — the same for row No. 5; / — the same for row No. 7; ○ — data from modeling on conducting paper.

5. Determination of additional eddy-current losses by the method of mathematical modeling

As already noted, considerable stray magnetic fields of the poles [1] cause very strong local heating of the winding in regions with a higher density of magnetic lines of force.

A simple experimental way of finding heat losses from eddy currents is to model the stray field of a betatron on conducting paper, as described in [5].

From a sheet of conducting paper with known electrical resistivity, a figure is cut out that is geometrically similar to the cross section of the betatron window (Fig. 4). The resulting figure is placed on a thin sheet of dielectric. At the place of the model corresponding to the air gap in the po-

across the height of the betatron laminations, a copper busbar 4 is placed, to which current is supplied. On the other side of the dielectric (glass), in positions corresponding to the vertical rows of busbars, electrodes made of aluminum foil are glued on with silicate adhesive. The electrodes are made to the scale of the model. The model is supplied from an audio-frequency voltage source.

Fig. 4. Setup for modeling the eddy stray field of betatrons:
1—glass sheet; 2—electrically conductive paper; 3—electrodes made of aluminum foil, corresponding to the magnetizing winding of the betatron; 4—copper electrode corresponding to the nonmagnetic gap; 5—current lines; 6—equipotential lines.

The relationship between the electrical quantities of the model and the electromagnetic quantities of the betatron, as well as the modeling coefficients, are given in [5].

As a result of measuring the voltage field on the model, one obtains patterns of the distribution of the components of the induction vector \(B_x\) and \(B_y\) along the height of the vertical rows of winding busbars, which make it possible, according to [6], to estimate the heat generation in the winding due to eddy currents.

By the indicated method, heat releases were determined in the vertical rows of busbars Nos. 3, 5, 7, counted from the pole of the B-25-2 betatron. These data are plotted in Fig. 3 with circles. The same figure also shows the data obtained according to item 4.

Conclusions

1. For the thermal calculation of betatron windings it is necessary to know the law of heat generation throughout the entire volume of the winding. The methods proposed in items 1–4 make it possible to obtain these laws experimentally only on already manufactured betatrons.

1. The method for modeling the stray eddy-current field of a betatron (Sec. 5) can also be used to estimate heat generation in the design of betatrons.

REFERENCES

1. I. V. Shipunov. Investigation of the heating and cooling of electromagnets of induction accelerators. Dissertation. Tomsk, 1959.

2. G. I. Babat. Induction heating of metal and its industrial application. Energiya Publishing House, 1965.

3. A. V. Lykov, Yu. A. Mikhailov. Theory of heat and mass transfer. Gosenergoizdat, 1963.

4. V. V. Ivanov, G. F. Shilkin. Thermal calculation of an air-cooled magnetizing winding of a betatron. Izvestiya VUZov—Elektromekhanika, No. 8, 1964.

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

6. E. A. Man’kin, D. N. Morozov, A. V. Alferova. Additional eddy-current losses in transformer windings. Elektrotekhnika, No. 10, 1965.