Experimental Verification of the Law of the Relationship Between Excess Temperatures in the Winding of a PMB-6 Betatron

V. S. Loginov, A. A. Geizer

(Presented by the Research Institute of Nuclear Physics, Electronics, and Automation)

In the thermal calculation of a betatron, it is necessary to know the distribution of temperatures over the individual elements (rows of busbars, the magnetic core, etc.) of the electromagnet. Within the framework of operational monitoring it is not always possible to observe the course of temperature changes at points that are difficult to reach with temperature sensors, while carrying out a calculation using analogous expressions for bodies of complex shape presents certain difficulties.

From experience [1] it is known that the winding of a betatron is the most heavily loaded thermally. It may be represented as a hollow cylinder of finite dimensions. We take the thermophysical characteristics of the body under consideration to be known and constant, and the heat-transfer coefficients to be unequal on the inner and outer surfaces but constant in time. The internal heat sources are assumed to be prescribed and to be a function depending on the coordinates and time.

Fig. 1. For verification of the law of the relationship between excess temperatures in the cylindrical coil of a betatron

Under the adopted assumptions, this problem can be solved by the method of A. G. Grinberg. This solution is given in the appendix. We note that in the subsequent calculations the analytical expression obtained is not used by us, but with its aid it is possible to obtain an approximate relationship between excess temperatures and to check it against experimental values. As applied to the betatrons known to us, in solution (II) one may restrict oneself to one term of the series (this assertion does not apply to small time intervals of the thermal process under consideration and to discontinuous functions \(\operatorname{Po}(X_2Fo)\). In this case an approximate relationship between temperatures is observed

\[ \theta(R,Z,Fo)\simeq \frac{\theta(R^{*},Z,Fo)\theta(R,Z^{*},Fo)} {\theta(R^{*},Z^{*},Fo)}, \tag{1} \]

Comparison of experimental and calculated temperature values
Table 1

where \(R^*, Z^*\) are fixed coordinates.

In its structure, relation (1) is analogous to the dependence given in [2]. We shall write expression (1) in a form convenient for practical calculations:

\[ \nu(r,z,\tau) \simeq \frac{\nu(r,z^*,\tau)\nu(r^*,z,\tau)} {\nu(r^*,z^*,\tau)}, \tag{2} \]

where

\[ \nu(r,z,\tau)=t(r,z,\tau)-t_f \]

is the excess temperature, °C.

Table 1 gives the temperature values obtained experimentally, and the calculations carried out for point 4 (Fig. 1), by the formulas

\[ \nu_4 \simeq \frac{\nu_1\cdot \nu_3}{\nu_2}, \tag{3} \]

and

\[ \nu_4 \simeq \frac{\nu_6\cdot \nu_3}{\nu_5}. \tag{4} \]

Comparison of the temperatures found experimentally and calculated from relations (3), (4) shows that there is a relationship between the temperatures in the betatron winding (a multicore cable [1]) over a sufficiently long time interval (28 min). For small time values, relations (3), (4) give a deviation of up to 50% compared with what is actually observed.

Appendix

The original system of differential equations describing the process of heat conduction in a hollow cylinder of finite dimensions, under the accepted assumptions, has the form:

\[ \frac{\partial \theta}{\partial Fo} = \frac{\partial^2\theta}{\partial R^2} + \frac{1}{R}\frac{\partial\theta}{\partial R} + \frac{\partial^2\theta}{\partial Z^2} + Po(R,Z,Fo); \tag{5} \]

\[ R_1<R<R_2,\qquad 0<Z<1,\qquad Fo>0; \]

\[ \frac{\partial\theta(R_1,Z,Fo)}{\partial R} - Bi_1\,\theta(R_1,Z,Fo)=0; \tag{6} \]

\[ \frac{\partial\theta(R_2,Z,Fo)}{\partial R} + Bi_2\,\theta(R_2,Z,Fo)=0; \tag{7} \]

\[ \frac{\partial\theta(R,0,Fo)}{\partial Z} - Bi_3\,\theta(R,0,Fo)=0; \tag{8} \]

\[ \frac{\partial\theta(R,1,Fo)}{\partial Z} + Bi_4\,\theta(R,1,Fo)=0; \tag{9} \]

\[ \theta(R,Z,0)=\varphi(R,Z). \tag{10} \]

Here \(t(r,z,\tau)\) is the current temperature, \(\tau\) is time; \(t_f\) is the temperature of the surrounding medium, which remains unchanged over time, °C; \(a_i\) \((i=1,2,3,4)\) are heat-transfer coefficients, W/m\(^2\)·deg; \(\lambda\) are thermal-conductivity coefficients, W/m·deg; \(r_1, r_2, h\) are the geometric dimensions of the hollow cylinder; \(\theta(R,Z,Fo)=\dfrac{t(r,z,\tau)-t_f}{t_m}\) is the dimensionless temperatu-

ra; \(t_{\mathrm{m}}\) is the temperature scale; \(Bi_i=\dfrac{\alpha_i h}{\lambda}\) is the Biot criterion; \(Fo=\dfrac{a\tau}{h^2}\) is the Fourier number; \(R=\dfrac{r}{h}\), \(Z=\dfrac{z}{h}\) are dimensionless coordinates.

The solution of (5)—(10) by the method of finite integral transforms [3] has the form

\[ \theta(R,Z,Fo)= \sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{T(\mu_n,\gamma_m,Fo)K(\mu_nR)W(\gamma_m,Z)} {\displaystyle \int_{R_1}^{R_2}\int_{0}^{1} RK^2(\mu_nR)W^2(\gamma_m,Z)\,dR\,dZ}, \tag{11} \]

where

\[ \begin{aligned} T(\mu_n,\gamma_m,Fo) &=\exp[-(\mu_n^2+\gamma_m^2)Fo] \biggl[ \int_{R_1}^{R_2}\int_{0}^{1} \varphi(R,Z)K(\mu_nR)W(\gamma_m,Z)\,dR\,dZ + {}\\ &\quad +\int_{0}^{Fo}\int_{R_1}^{R_2}\int_{0}^{1} \exp[(\mu_n^2+\gamma_m^2)Fo'] P_0(R,Z,Fo')K(\mu_n,R)W(\gamma_m,Z)\,dR\,dZ\,dFo' \biggr], \end{aligned} \]

\[ \begin{aligned} K(\mu_n,R) &=[\mu_nY_1(\mu_nR_1)-Bi_1Y_0(\mu_nR_1)]I_0(\mu_nR)+{}\\ &\quad +[Bi_1I_0(\mu_nR_1)-\mu_nI_1(\mu_nR_1)]Y_0(\mu_nR)- \end{aligned} \]

is the kernel of the finite integral transform with respect to the variable coordinate \(R\),

\(I_1(\mu_nR), I_0(\mu_nR), Y_1(\mu_nR), Y_0(\mu_nR)\) are Bessel functions,

\(\mu_n\) are eigenvalues, which are determined from the expression

\[ [\mu Y_1(\mu R_2)+Bi_2Y_0(\mu R_2)] [Bi_1I_0(\mu R_1)-\mu I_1(\mu R_1)] = [\mu I_1(\mu R_2)+Bi_2I_0(\mu R_2)] [Bi_1Y_0(\mu R_1)-\mu Y_1(\mu R_1)], \tag{12} \]

\[ W(\gamma_m Z)=\gamma_m\cos\gamma_mZ+Bi_3\sin\gamma_mZ \]

is the kernel of the finite integral transform with respect to the variable \(Z\), and \(\gamma_m\) are the roots of the equation

\[ \operatorname{ctg}\gamma=\frac{\gamma(Bi_3+Bi_4)}{\gamma^2-Bi_3Bi_4}. \tag{13} \]

LITERATURE

1. V. S. Loginov, A. A. Geizer, V. L. Chakhlov. Evaluation of electrical losses in the electromagnet of a PMB-6 betatron with pulsed power supply by increased-frequency current. Present collection.

2. G. P. Boikov. The law of the relationship between excess temperatures of finite-size bodies. IFZh, vol. V, 3, 1962.

3. A. T. Grinberg. Selected Problems in the Mathematical Theory of Electrical and Magnetic Phenomena. Publishing House of the USSR Academy of Sciences, 1948.