Issues in the Thermal Calculation of Betatron Electromagnets
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Issues in the Thermal Calculation of Betatron Electromagnets
I. V. Shipunov
(Presented by the scientific seminar of the Faculty of Physics and Technology)
The betatron, from being an instrument of the physics laboratory—as it was not so long ago—is increasingly becoming an installation of factory type, where it is used for defectoscopy of thick-walled metal products, and an installation of medical clinics, where it is used for the treatment of serious diseases.
Undoubtedly, in the very near future the serial production of induction accelerators of charged particles must be established.
The designers of accelerator installations are faced with the tasks of creating apparatus that are perfect in every respect and that meet the requirements of their operation both in physics laboratories and in industry, medicine, and other branches of the national economy of the country.
One of the most important requirements for an accelerator installation of the betatron type is the possibility of its operating for a long time without a dangerous rise of temperature at any point of the electromagnet.
A number of works have been devoted to the problems of designing electromagnets of induction accelerators. There is not a single special investigation of the thermal regime of accelerator installations either in the domestic or in the foreign literature.
In works describing the designs of betatrons that have been built, some data are given on the thermal loads of the active surfaces of magnetic cores and magnetizing windings, on the cooling systems adopted, and some data on the temperature rises of the steel of the magnetic core and the copper of the winding. The published data are so few in number, unsystematic, and fragmentary that they do not enable the designer of a new installation to choose and justify technically the temperature regime of a betatron electromagnet—to indicate particularly weak points from the standpoint of heating, and to choose a cooling system.
Usually, for thermal calculations of betatron electromagnets, methods and procedures borrowed from transformer building and electrical-machine engineering are used. In these branches of technology a wealth of experience has been accumulated in the design of electromagnets.
There are a number of factors that distinguish the heating and cooling conditions of a betatron electromagnet from the heating of a transformer electromagnet.
The magnetic core of a betatron has a large air gap for placing within it the vacuum accelerating chamber. This changes the course of the heat—
fluxes in the magnetic core. It should be noted that it is precisely this part of the magnetic core, adjacent to the air gap, that releases the greatest amount of heat per unit volume.
The poles of the electromagnet are assembled from plates of electrical steel arranged radially. This provides the precise circular symmetry of the magnetic field required for betatron acceleration. This circumstance also determines the particular distribution of magnetic fluxes in the yoke of the magnetic core, and of losses in its individual parts, as well as of temperatures.
Most betatrons have an Ш-shaped magnetic core. In this article all calculations are carried out with reference to this type of magnetic core. Experiments were also carried out with betatrons having an Ш-shaped type of magnetic core.
After the construction and commissioning by Kerst of the first betatron (1940), induction accelerators were improved in the direction of increasing the energy of the accelerated electrons. Bulky and heavy electromagnets were built, and projects for still larger installations were proposed. The appearance of fundamentally new charged-particle accelerators has led to the fact that it is now disadvantageous to build betatrons for energies above 50 MeV.
At present, most improvements and developments are aimed at increasing the intensity of the hard radiation produced by the betatron and at reducing the dimensions and weight of the betatron.
Below a number of dependences are given—of the weight of the magnetic-core steel, of energy losses, and of specific heat loads on the main parameters of the betatron. These equations may help in choosing certain initial data for the calculation, in choosing the cooling system, and in the thermal calculation of a betatron electromagnet.
As initial formulas, the equations given in [1] are used.
Let us determine how the weight of the individual parts of the Ш-shaped magnetic core of a betatron depends on its main parameters—the energy of the accelerated electrons \(A\) MeV, the radius of the equilibrium orbit \(r_0\) cm, and the induction in the central part of the air gap.
The central inserts in the betatron designs of the Tomsk Polytechnic Institute and in a number of other designs are cylindrical disks. The weight of such an insert is determined by the expression:
\[ G = \pi \cdot r_c^2 h_c \cdot k_c \cdot 7.6 \cdot 10^{-3}, \ \text{kg}, \]
where \(r_c\) is the radius of the central inserts, cm,
\(h_c\) is the height, cm,
\(k_c\) is the coefficient of filling of the inserts with steel.
The values of the radius and height are determined from the equations:
\[ r_c = A B, \ \text{cm}, \]
where \(A\) is the energy of the accelerated electrons, MeV,
\[
B = \frac{6.7 \cdot 10^{-3}}{B_{\mathrm{cm}} \cdot \sin \alpha_p}
\]
is the betatron constant,
\[ h_c = \frac{\delta_v - \delta_c}{2}, \ \text{cm}, \]
where \(\delta_c\) is the magnitude of the air gap at radius \(r_c\), cm,
\(\delta_v\) is the magnitude of the air gap, determined by the expression
\[ \delta_v=\delta_0\frac{r_c^n}{r_0^n}; \]
in the latter expression:
\(\delta_0\) is the magnitude of the air gap at radius \(r_0\), cm,
\(n\) is the exponent of the degree of decrease of the magnetic field.
After substituting the values of \(r_c\) and \(h_c\), we have:
\[ G_c=\pi A^2 B^2 \frac{\delta_v-\delta_c}{2}\cdot k_c \cdot 7.6\cdot 10^{-3},\ \text{kg} \]
or
\[ G_c=\pi A^2 B^2 \cdot \frac{\delta_0}{2} \left(\frac{r_c^n}{r_0^n}-\frac{B_{om}}{B_{cm}}\right) \cdot k_c\, 7.6\cdot 10^{-3},\ \text{kg}, \]
where \(B_{om}\) is the value of the magnetic-field strength at the radius of the equilibrium orbit, determined from the equation:
\[ B_{om}=\frac{0.5B_{cm}}{\gamma},\ \text{G}, \]
where
\[ \gamma=\frac{r_0}{r_c}. \]
The expression for the weight of the central inserts takes the form:
\[ G_c=\pi A^2 \frac{a^2}{B_{cm}^2}\cdot \frac{\delta_0}{2} \cdot \left[\left(\frac{1}{\gamma}\right)^n-\frac{0.5}{\gamma}\right] \cdot k_c\, 7.6\cdot 10^{-3},\ \text{kg}, \]
where
\[ a=\frac{6.7\cdot 10^3}{\sin \alpha_p}. \]
The magnitude of the air gap at the radius of the equilibrium orbit can be determined approximately by the formula, which takes into account the ratio of the amplitudes of the vertical and radial oscillations of the beam of accelerated electrons in the betatron chamber:
\[ \delta_0=2b_0\sqrt{\frac{1-n}{n}},\ \text{cm}, \]
where \(b_0\) is the half-size of the working zone (cm), which, in turn, is determined by the equality:
\[ b_0=\beta r_c=\beta A\frac{a}{B_{cm}},\text{ cm}, \]
where
\[ \beta=\gamma-1. \]
After substituting the values of \(\delta_0\) and \(b_0\), we have:
\[ G_c=K_{cv}\frac{A^3}{B_{cm}^3},\ \text{kg}, \]
where
\[ K_{cv}=1.2\pi a^3\sqrt{\frac{1-n}{n}}\cdot \frac{(\gamma-1)(1-0.5\gamma^{\,n-1})}{\gamma^n} \cdot k_c\, 7.6\cdot 10^{-3}. \]
The weight of the pole is expressed sufficiently accurately by the equation:
\[ G_n=\pi r_n^2 h_n k_c \cdot 7.6 \cdot 10^{-3},\quad \text{kg}, \]
where
\[ r_n=r_0+b_0=A\frac{a}{B_{cm}}(1+2\beta),\text{ cm}. \]
The height of the pole \(h_n\) cannot be related by a mathematical dependence to such parameters of the installation as the energy \(A\), the induction \(B_{cm}\), etc.
We shall take the height of the pole to be equal to the height of the “window” for the magnetizing winding, as is usually the case.
The area of the “window” for the winding is determined by the known equation
\[ S_{ok}=h_n a_{ok}=\frac{A W_{2\,ef}}{2\Delta_{ok}},\quad \text{cm}^2, \]
where \(A W_{2\,ef}\) is the required ampere-turns of the magnetizing winding;
\[ \Delta_{ok} \]
is the current density in the window, \(\dfrac{a}{\text{cm}^2}\).
After substituting the value of the ampere-turns, we shall have for the area of the “window” the following expression [1]:
\[ S_{ok}=0.31\cdot 1.2 \cdot \frac{\beta}{\gamma}\cdot \frac{aA}{\Delta_{ok}}\cdot \sqrt{\frac{1-n}{n}},\quad \text{cm}^2. \]
Introduce the notation
\[ c=\frac{a_{ok}}{h_n}, \]
after which the pole height is expressed by the equality:
\[ h_n=\sqrt{\frac{S_{ok}}{c}},\quad \text{cm} \]
or
\[ h_n=0.61\frac{\beta^{0.5}a^{0.5}}{c^{0.5}\Delta_{ok}^{0.5}\gamma^{0.5}},\quad \text{cm}. \]
If we take the average values of the quantities \(a\) and \(k_c\), which vary only slightly, and substitute all the quantities found into the formula for determining the weight of the pole, we obtain the following:
\[ G_n=54.7\cdot 10^6\cdot \frac{(1+2\beta)^2\beta^{0.5}}{c^{0.5}\Delta_{ok}^{0.5}\gamma^{0.5}} \sqrt{\frac{1-n}{n}}\cdot \frac{A^{5/2}}{B_{cm}^2},\quad \text{kg}. \]
In a U-shaped construction of the betatron magnetic circuit, there are two yokes (horizontal parts) and two legs. It is more convenient to calculate the weights separately for the yoke and the legs.
The weight of one yoke is determined by the formula:
\[ G_y=b_y h_y L_y k_c \cdot 7.6\cdot 10^{-3}\ \text{kg}, \]
where \(b_y\) is the width of the yoke, cm,
\(h_y\) is the height of the yoke, cm,
\(L_y\) is the length of the yoke, cm,
\(k_c\) is a coefficient.
The length of the yoke is determined from the equation
\[ L_y=2(r_n+a_{ok}+h_{cm}),\quad \text{cm}, \]
where \(h_{cm}\) is the horizontal dimension of the leg along the plates.
The height of the yoke is determined as follows:
\[ h_y=\frac{S_y}{b_y}=\frac{\sigma_{ob}\Phi_n}{2B_y b_y} =\frac{\sigma_{ob}B_{n\,av}S_n}{2B_y b_y},\ \text{cm}, \]
where \(\sigma_{ob}\) is the leakage coefficient of the winding magnetic flux,
\(\Phi_n\) is the magnetic flux of the pole, Mx;
\(B_y\) is the average induction in the yoke, G;
\(B_{n\,av}\) is the average induction in the pole, G;
\(S_n\) is the pole cross section, \(\text{cm}^2\).
Introduce the notation \(\dfrac{B_{n\,av}}{B_y}=d\) and substitute the values of the remaining quantities into the expression for the height of the yoke:
\[ h_y=\frac{\pi}{4}\sigma_{ob}da(1+2\beta)\frac{A}{B_{c.m}},\ \text{cm}. \]
Usually \(h_y=h_{st}\). Thus, the expression determining the length of the yoke takes the form:
\[ L_y=\left[2a(1+2\beta)+\frac{\pi}{2}\sigma_{ob}da(1+\beta)\right]\frac{A}{B_{c.m}}+ \]
\[ +1.22\left[\frac{a\beta c}{\Delta_{ok}\gamma}\right]^{0.5} \sqrt[4]{\frac{1-n}{n}}\cdot A^{\frac12},\ \text{cm}. \]
After this, the weight of one yoke is determined from the equation:
\[ G_y=88\cdot 10^8(1+2\beta)^2d(1+0.87d)\frac{A^3}{B_{c.m}^3}+ \]
\[ +6.4\cdot 10^7 d(1+2\beta)^2 \left(\frac{c\beta}{\Delta_{ok}\gamma}\right)^{0.5} \cdot \sqrt[4]{\frac{1-n}{n}}\cdot \frac{A^{2.5}}{B_{c.m}^2}, \]
where averaged values of \(a\) and \(\sigma_{ob}\) have been adopted.
The weight of one upright is determined analogously to the calculation given above for the weight of the yoke:
\[ G_{st}=l_{st}h_{st}b_{st}k_c\cdot 7.6\cdot 10^{-3},\ \text{kg}, \]
where \(b_{st}\) is the width of the upright, cm, with
\[ b_{st}=b_y-2r_n=2a(1+2\beta)\frac{A}{B_{c.m}},\ \text{cm}, \]
\(h_{st}\) is the height of the upright, cm,
\[ h_{st}=h_y=\frac{\pi}{4}\sigma_{ob}ad(1+2\beta)\frac{A}{B_{c.m}},\ \text{cm}, \]
\(l_{st}\) is the length of the upright,
\[ l_{st}=2h_n+\delta_0 =2\cdot 0.61\left(\frac{a\beta}{c\Delta_{ok}\gamma}\right)^{0.5} \sqrt[4]{\frac{1-n}{n}}\cdot A^{\frac12}+ \]
\[ +2a\beta\sqrt{\frac{1-n}{n}}\cdot \frac{A}{B_{c.m}},\ \text{cm}. \]
The weight of the support in kg is obtained after substituting all values \((l_{st}, h_{st}, b_{st})\), with all linear dimensions expressed in cm, energy in MeV, and induction in gauss.
\[ G_{st}=88\cdot 10^{8} d\beta(1+\beta)^2 \sqrt{\frac{1-n}{n}\,\frac{A^3}{B_{cm}^3}} + \]
\[ +63.5\,d(1+2\beta)^2 \left(\frac{\beta}{c\,\Delta_{ok}\,\gamma}\right) \sqrt[4]{\frac{1-n}{n}}\, \frac{A^{2.5}}{B_{cm}^{2}}. \]
By a formula similar to the last one, the weight of the entire magnetic circuit of the betatron may be written:
\[ G_{mp}=40\cdot 10^9 \gamma^5\frac{A^3}{B_{cm}^{3}} + 22\cdot 10^6 \gamma^4\frac{A^{2.5}}{B_{cm}^{2}}, \qquad \text{kg}, \]
where the following quantities have been averaged: \(c=2\); \(\Delta_{ok}=30\,\dfrac{a}{cm^2}\); \(n=0.7\), and \(d=1\).
Figures 1 and 2 show the dependence of the weight of the magnetic circuit on the values of \(B_{cm}\) and \(\gamma\) for betatrons with accelerated-electron energies \(A=15\) MeV and \(A=25\) MeV. With the aid of these graphs it is easy to select the value of the induction at the center of the air gap if the weight of the electromagnet is specified in advance.
Fig. 1. Dependence of the weight of the magnetic circuit of a betatron with accelerated-particle energy \(A=15\) MeV on the magnitude of the induction \(B_{cm}\).
For the thermal calculation of a betatron electromagnet, it is essential to know the energy losses in the separate parts of the magnetic circuit.
The losses in one of the two central inserts are readily determined from the equation:
\[ P_c=p_{ud}G_c,\ \text{W}, \]
where \(p_{ud}\) is the specific loss in the steel of the central inserts, \(\dfrac{\text{W}}{\text{kg}}\).
The specific losses are related to the magnitude of the induction in the central part of the interpole space by the following relation:
\[ p_{ud}=p_{10} \left(\frac{f}{50}\right)^{1.3} \left(\frac{B_{cm}}{k_c\,10000}\right)^2, \ \frac{\text{W}}{\text{kg}}, \]
where \(p_{10}\) is the specific loss in steel at an induction of 10000 gauss and frequency \(f=50\) Hz.
After substitution of the corresponding quantities, an expression was obtained that determines the losses in one of the inserts:
\[ P_c = p_{10}\left(\frac{f}{50}\right)^{1.3} \frac{1.2\pi a^3}{k_c\cdot 10^8} \sqrt{ \frac{1-n}{n}\, \frac{(\gamma-1)(1-0.5\gamma^{\,n-1})}{\gamma^n} } \times 7.6\cdot 10^{-3}\frac{A^3}{B_{cm}}, \ \text{W}. \]
Fig. 2. Dependence of the weight of the betatron magnetic circuit with accelerated-particle energy \(A=25\) MeV on the magnitude of the induction \(B_{cm}\).
Energy losses in the steel of the pole are calculated in a similar manner:
\[ P_n = P_{y0}G_n = p_{10}\left(\frac{f}{50}\right)^{1.3} \left[ \frac{\sigma_n}{k_c(2\gamma-1)\cdot 10^4} \right]^2 \times \]
\[ \times 54.7\cdot 10^6 \frac{(1+2\beta)^2\beta^{0.5}} {c^{0.5}\Delta_{ok}^{0.5}\gamma^{0.5}} \sqrt[4]{\frac{1-n}{n}}\, A^{2.5},\ \text{W}. \]
The average value of the induction in the pole is determined not only by the value \(B_{cm}\), but also by the magnitude \(\gamma=\dfrac{r_0}{r_c}\), since
\[ B_{n\,av} = \frac{2\sigma_n}{k_c} \frac{r_0}{r_n} B_{om}, \ \text{Gs}. \]
In turn,
\[ \frac{r_0}{r_n} = \frac{\gamma}{2\gamma-1} \quad\text{and}\quad B_{om}=0.5\frac{B_{cm}}{\gamma}. \]
Thus,
\[ B_{n\,av} = \frac{\sigma_n B_{cm}}{k_c(2\gamma-1)}, \ \text{Gs}. \]
It is characteristic that the magnitude of the losses in the pole steel does not depend on the value of the induction in the central part of the air gap \(B_{cm}\) for \(A=\mathrm{const}\), and changes very sharply with a change in the energy of the accelerated electrons.
We determine the energy losses in the steel of the yoke and of the upright of the magnetic circuit:
\[ P_y=p_{y0}\cdot G_y \quad \text{and} \quad P_{st}=p_{y0}G_{st}. \]
After substituting the values of all the quantities, we have:
\[ P_y=230\,\frac{(1+2\beta)(1+0.87d)}{d}\,\frac{A^3}{B_{cm}}+ \]
\[ +1.67\,\frac{1}{d}\left(\frac{c\beta}{\Delta_{ok}\gamma}\right)^{0.5} \sqrt[4]{\frac{1-n}{n}}\,A^{2.5}, \quad \mathrm{W}, \]
\[ P_{st}=230\,\frac{\beta}{d}\sqrt{\frac{1-n}{n}}\,\frac{A^3}{B_{cm}}+ \]
\[ +1.67\cdot\frac{1}{d}\left(\frac{\beta}{c\cdot\Delta_{ok}\gamma}\right)^{0.5} \sqrt[4]{\frac{1-n}{n}}\,A^{2.5}, \quad \mathrm{W}. \]
And, finally, the total losses in the magnetic circuit of the betatron are determined from the equation:
\[ P_{mn}=\frac{2100\gamma-1200}{\sqrt{d}}\cdot\frac{A^3}{B_{cm}} +\frac{2.53\gamma^{1.5}}{d^{0.8}\Delta_{ok}^{0.5}}\,A^{2.5}, \quad \mathrm{W}. \]
Fig. 3. Dependence of the energy losses in the steel of the magnetic circuit of a betatron \((A=15\ \mathrm{MeV})\) on the induction value \(B_{cm}\).
Labels in the plot: \(P_{mn}\), W; \(B_{cm}\), kGs; \(A=15\ \mathrm{MeV}\); \(\gamma=1.45\); \(c=2\); \(\Delta=33\ \mathrm{A/cm^2}\); \(d=1\).
It follows from the last formula that increasing the induction in the air gap between the inserts \((B_{cm})\) leads to a reduction of losses for constant \(\gamma\), \(d\), \(\Delta_{ok}\), \(A\) (the quantity \(c\) has a very insignificant effect on the losses). To illustrate this dependence, Fig. 3 shows the graph
\[ P_{mn}=f(B_{cm}). \]
The energy losses in the magnetic core determined above still do not characterize the thermal regime of the electromagnet of an induction accelerator. To judge the heating of one or another part of the electromagnet, it is necessary to know the cooling conditions: which surfaces remove the heat being liberated, and how actively. For the thermal calculation it is necessary to determine the specific heat load, i.e., the ratio of the magnitude of the losses to the dimensions of the cooling surface.
In the general case the specific heat load, as is known, is determined by the expression:
\[ q=\frac{P}{S_{\text{cool}}},\ \frac{\text{W}}{\text{cm}^2}, \]
where \(P\)—energy losses, W,
\[ S_{\text{cool}} \] — active cooling surface, \(\text{cm}^2\).
For the central inserts, the cooling surfaces should be taken as the vertical ones, since disks of nonmagnetic material (cardboard, textolite) are placed between the inserts, as well as between the poles and the inserts. The thickness of the layer of insulating material depends on the size of the air gap, and this thickness is always sufficient to thermally insulate the inserts from the poles.
For the specific heat load of one insert we have:
\[ q_c=\frac{p_{\text{ud}}\cdot G_c}{2\pi r_c h_c} =3.8\,p_{\text{ud}}\,k_c r_c,\ \frac{\text{W}}{\text{cm}^2} \]
or
\[ q_c=3.8\cdot 10^{-11}p_{10}\left(\frac{f}{50}\right)^{1.3} \frac{a}{k_c}\cdot A\cdot B_{cm},\ \frac{\text{W}}{\text{cm}^2}. \]
Thus, for inserts that are in the poorest cooling conditions, the relation between the product of the energy of the accelerated electrons and the value of the induction in the central part of the air gap, and the specific heat load, is entirely unambiguous. Knowing the heat-transfer coefficient for the selected cooling system and the magnitude of the maximum steady-state heating temperature \((\tau_m)\), it is easy to calculate the permissible heat load from the relation:
\[ q=\tau_m \alpha,\ \frac{\text{W}}{\text{cm}^2}, \]
where \(\alpha\)—heat-transfer coefficient,
\[ \frac{\text{W}}{\text{cm}^2\,^\circ\text{C}}. \]
After this, the choice of the magnitude of the induction in the central part of the air gap \((B_{cm})\) presents no difficulty, since the energy of the accelerated electrons is specified when the accelerator is being designed.
For example, for a betatron with accelerated-electron energy \(A=15\) MeV, the heat-transfer coefficient from the surfaces of the inserts has been determined experimentally and is
\[ \alpha=0.8\cdot 10^{-3}\ \frac{\text{W}}{\text{cm}^2\,^\circ\text{C}}. \]
We take the overheating of the insert steel to be \(\tau_m=60^\circ\text{C}\). We determine the value of the permissible specific heat load:
\[ q_c=0.8\cdot 10^{-3}\cdot 60=0.048\ \frac{\text{W}}{\text{cm}^2}. \]
The magnitude of the permissible induction \(B_{c.m}\) under natural cooling is then determined by the formula:
\[ B_{c.m}=\frac{q_c}{45\cdot 10^{-8}\cdot A} =\frac{0.048}{15\cdot 45\cdot 10^{-8}}=7100\ \text{gauss}. \]
The specific heat load for the cooling surface of the pole is determined by the equality:
\[ q_n=\frac{P_n}{\pi\left(r_n^2-r_c^2\right)},\quad \frac{\text{W}}{\text{cm}^2}. \]
For a Ш-shaped magnetic core, the profiled part of the pole should be taken as the cooling surface. In such a design the pole is a separate part and is fastened to the yoke by means of bolts and studs. Several layers of cardboard are necessarily placed between the pole and the yoke. The pole is tightened to the yoke with considerable force. Most of the radially arranged pole plates intersect the yoke plates. The cardboard between the pole and the yoke prevents the yoke plates from short-circuiting, which is inadmissible because of the increase in eddy currents in the iron of the yoke.
The temperature difference in the pole and the yoke is usually small; the cardboard layer represents a considerable thermal resistance. Calculations show that practically all the heat is removed only by the surfaces facing the air gap.
The vertical surfaces of the pole remove such an insignificant part of the heat released in the pole that they cannot be included in the calculation as cooling surfaces.
After substituting the values of the quantities entering into the expression for the specific heat load of the pole surface, we obtain:
\[ q_n= \frac{ 4.65\cdot 10^{-11}\cdot p_{10}\cdot \left(\frac{f}{50}\right)^{1.3} \sigma_n^2\,\beta^{0.5}\,a^{0.5}\, \sqrt[4]{\frac{1-n}{n}} }{ \left[(1+2\beta)^2-1\right]\,k_c^{0.5}\, \Delta_{ok}^{0.5}\,\gamma^{0.5} } A^{0.5}B_{c.m}^2,\quad \frac{\text{W}}{\text{cm}^2}. \]
The permissible induction in the central part of the interpole space, as determined by heating of the pole under natural cooling, is found from the equality:
\[ B_{c.m}=2.6\cdot 10^3 \frac{ \left[(1+2\beta)^2-1\right]^{0.5} \left(c\Delta_{ok}\gamma\right)^{0.25} }{ \beta^{0.25} } \cdot \frac{1}{A^{0.25}}, \quad \text{gauss}, \]
where \(f=50\ \text{Hz}\) and the steel is grade E42, \(0.35\ \text{mm}\) thick.
In calculating the specific heat load of the yoke we adopt the following assumptions, which substantially simplify the thermal calculation:
1) the yoke is heated independently of the pole and the magnetizing winding;
2) only the horizontal surface of the yoke is considered to be cooling; the effect of the end surfaces can be taken into account by increasing the heat-transfer coefficient;
3) it is assumed that the induction and the energy losses are uniformly distributed over the entire yoke.
The value of the heat load after these simplifications can be expressed as follows:
\[ q_y=\frac{P_y}{S_{oxl}} =\frac{p_{yd}G_y}{S_{oxl}} =p_{yd}k_c\cdot 7.6\cdot 10^{-3}h_y, \quad \frac{\text{W}}{\text{cm}^2}. \]
or
\[ q_y = 7.6 \cdot 10^{-11} \cdot \frac{\pi}{4} \cdot p_{10} \cdot \left(\frac{f}{50}\right)^{1.3} \cdot \frac{\sigma_n \cdot \sigma_{ob} a(1 + 2\beta)}{d k_c (2\gamma - 1)^2} \cdot A B_{cm}, \qquad \frac{\mathrm{W}}{\mathrm{cm}^2}. \]
For E42 steel of thickness \(0.35\ \mathrm{mm}\), at a frequency of the current feeding the electromagnet \(f = 50\ \mathrm{Hz}\), and for average values of the quantities \(k_c = 0.95\), \(\sigma_n = 1.15\), \(\sigma_{ob} = 1.05\), \(a = 7.2 \cdot 10^3\), and \(d = 1\), we have
\[ q_y = 81.5 \cdot 10^{-8} \frac{1}{2\gamma - 1} \cdot A \cdot B_{cm}, \qquad \frac{\mathrm{W}}{\mathrm{cm}^2}. \]
The heat-transfer coefficient in still air from the horizontal surface of the yoke is approximately
\[ \alpha = 1 \cdot 10^{-3}, \frac{\mathrm{W}}{\mathrm{cm}^2 \cdot {}^\circ\mathrm{C}}, \]
but, if the heat transfer from the end surfaces is taken into account, one should adopt:
\[ \alpha = 1.2 \cdot 10^{-3}, \frac{\mathrm{W}}{\mathrm{cm}^2 \cdot {}^\circ\mathrm{C}}. \]
After this, the value of the induction at the center of the air gap \(B_{cm}\) is determined from the equation:
\[ B_{cm} = 0.9 \cdot 10^5 \cdot \frac{2\gamma - 1}{A}, \qquad \mathrm{G}, \]
where \(\tau_m = 60^\circ\mathrm{C}\).
If analogous reasoning is carried out, we obtain an expression for determining the specific heat load of the surfaces of the legs of the magnetic core:
\[ q_{cm} = 40 \cdot 10^{-8} \cdot \frac{1}{2\gamma - 1} \cdot A \cdot B_{cm}, \qquad \frac{\mathrm{W}}{\mathrm{cm}^2}. \]
One of the cooling surfaces of the leg faces the winding of the electromagnet; therefore its cooling capacity is considerably lower than the cooling activity of the yoke surface. The mean heat-transfer coefficient, determined experimentally, is for the surfaces of the leg
\[ \alpha = 0.7 \cdot 10^{-3}, \frac{\mathrm{W}}{\mathrm{cm}^2 \cdot {}^\circ\mathrm{C}}. \]
If the value of the induction \(B_{cm}\) is determined from the heating of the leg, we obtain the expression:
\[ B_{cm} = 1.05 \cdot 10^5 \frac{2\gamma - 1}{A}, \qquad \mathrm{G}. \]
It follows from the last equality that the cross section of the leg may be taken 10–15% smaller than the cross section of the yoke; the heating of the steel of the leg will then be equal to the heating of the steel of the yoke.
The values of the induction in the central part of the interpolar space may turn out somewhat different if they are determined from the heating conditions of the pole, the central inserts, the yoke, and the legs. In design it is possible to reduce to a minimum the difference in the calculated temperatures of the individual parts of the magnetic core by changing, for example, the height of the pole, the yoke, the size of the “window” for the winding, etc.
Using the formulas given above, one can establish a limit for the energy of electrons accelerated in a betatron up to which construction of betatrons with natural cooling of the electromagnet is possible. Such a limit can be established approximately, since preliminarily
it is necessary to establish a minimum for the induction in the central part of the air gap.
Above 10–15 MeV it is not advisable to build betatrons with natural cooling of the electromagnet, since the utilization of the electromagnet material is very low and the weight of the electromagnet increases excessively.
It should be noted that the sharpest possible dependence, with natural cooling, of the energy of the accelerated electrons on the change in
\[ B_{c.m} \]
exists for poles
\[ \left(A \sim \frac{1}{B_{c.m}^{4}}\right). \]
It follows from this that, by applying artificial cooling to the poles alone, it is possible to take the induction \(B_{c.m}\) to be of the order of 12–14 kGs for an energy of 10–12 MeV.
Using the formula
\[ \alpha = 0.6 \cdot 10^{-3} v^{0.8}, \quad \frac{\mathrm{W}}{\mathrm{cm}^{2}\,{}^\circ\mathrm{C}} \]
to determine the heat-transfer coefficient from the blown surface (\(v\) in m/sec), one can calculate the possible energy of the accelerated electrons at various inductions in the central part of the air gap without impermissible overheating of the pole steel.
The magnetizing winding of the betatron electromagnet is divided into two coils, for which the heating and cooling conditions are approximately the same.
To simplify the calculations, we assume that each of the coils occupies the entire “window.” In reality there are gaps between the winding and the yoke, and between the winding and the pole. This assumption introduces an insignificant error (less than 5%).
As cooling surfaces we consider only the horizontal surfaces of the coils.
The dimensions of the “window” are expressed through the quantities previously adopted in the calculation:
\[ a_{ok}=0.61\, \frac{c^{0.5}(\gamma-1)^{0.5}a^{0.5}}{\Delta_{ok}^{0.5}\gamma^{0.5}}\, \sqrt[4]{\frac{1-n}{n}}\,A^{0.5}, \quad \mathrm{cm}, \]
\[ b_{ok}=0.61\, \frac{a^{0.5}(\gamma-1)^{0.5}}{c^{0.5}\Delta_{ok}^{0.5}\gamma^{0.5}}\, \sqrt[4]{\frac{1-n}{n}}\,A^{0.5}, \quad \mathrm{cm}. \]
The energy losses in the copper of one coil may be expressed by the equation usually used in transformer calculations:
\[ P_{m}=2.4 \cdot 10^{-4}\Delta_{m}^{2}G_{m}, \quad \mathrm{W}, \]
where \(\Delta_{m}\) is the effective current density in the copper, \(\dfrac{\mathrm{A}}{\mathrm{cm}^{2}}\);
\(G_{m}\) is the weight of copper, kg.
The specific heat load is expressed by the equality:
\[ q=\frac{2.4 \cdot 10^{-7}\cdot V\Delta_{m}^{2}\cdot 8.5\cdot k_{z}}{S'\cdot k_{zakr}}, \quad \frac{\mathrm{W}}{\mathrm{cm}^{2}}, \]
where
\(V\) is the volume occupied by the coil, \(\mathrm{cm}^{3}\);
\(k_{z}\) is the coefficient of filling of the coil with copper;
\(k_{zakr}\) is the coefficient of covering of the cooling surface;
\(S'\) is the total cooling surface.
It can be shown that \(-\dfrac{V}{S'} = \dfrac{b_{ok}}{2}\), i.e.
\[ q = \frac{6.2 \cdot 10^{-7}\Delta_m^2 k_3(\gamma - 1)^{0.5} a^{0.5}} {\kappa_{\mathrm{zakr}} c^{0.5}\Delta_{ok}^{0.5}\gamma^{0.5}} \sqrt[4]{\frac{1-n}{n}}\, A^{0.5}, \quad \frac{\mathrm{W}}{\mathrm{cm}^2}. \]
The latter equation relates the specific heat load to the energy of the accelerated electrons \(A\), the characteristic of the interpole space \(\gamma = \dfrac{r_0}{r_c}\), and the winding parameters: \(\Delta_m\), \(k_3\), \(c\), \(\kappa_{\mathrm{zakr}}\), \(\Delta_{ok}\). The quantities \(a^{0.5}\) and
\[ \sqrt[4]{\frac{1-n}{n}} \]
may be included in the numerical coefficient, since they differ only very slightly for betatrons in the energy range \(A = 10 \div 40\) MeV.
After this the basic equation is somewhat simplified:
\[ q = 0.425 \cdot 10^{-4} \frac{\Delta_m^2 k_3(\gamma - 1)^{0.5}} {\kappa_{\mathrm{zakr}} c^{0.5}\Delta_{ok}^{0.5}\gamma^{0.5}} A^{0.5}, \quad \frac{\mathrm{W}}{\mathrm{cm}^2}, \]
\[ (\sin \alpha_p = 0.95;\quad n = 0.7). \]
It is easy to show that \(\Delta_{ok} = \Delta_m k_3\). Consequently, for the specific heat load one may write yet another expression:
\[ q = 0.132 \cdot 10^{-4} \frac{\Delta_m \Delta_{ok}^{0.5}\gamma^{1.5}} {\kappa_{\mathrm{zakr}} c^{0.5}} A^{0.5}, \quad \frac{\mathrm{W}}{\mathrm{cm}^2}, \]
where \(\gamma = 1.2 \div 2\).
Thus, other conditions being equal, the specific heat load on the surface of the magnetizing coil of a betatron increases with increasing \(\gamma = \dfrac{r_0}{r_c}\).
From the table below it is seen that the product \(\gamma^{1.5} A^{0.5}\) for betatrons with energies from 5 to 25 MeV changes only slightly [2].
| \(A\), MeV | \(\gamma\) | \(\gamma^{1.5}\) | \(A^{0.5}\) | \(\gamma^{1.5} A^{0.5}\) | Place where the betatron was built |
|---|---|---|---|---|---|
| 5 | 2.09 | 3.02 | 2.24 | 6.77 | USSR Tomsk |
| 15 | 1.45 | 1.75 | 3.86 | 6.77 | USSR Tomsk |
| 20 | 1.33 | 1.54 | 4.46 | 6.90 | USA |
| 25 | 1.31 | 1.50 | 5.00 | 7.50 | USSR Tomsk |
If one takes \(\gamma^{1.5} A^{0.5} \simeq 7\), the error in determining the specific heat load due to this averaging will not exceed a few percent.
The equation for the specific heat load in this case may be written as:
\[ q = 0.925 \cdot 10^{-4} \frac{\Delta_m \Delta_{ok}^{0.5}} {\kappa_{\mathrm{zakr}} c^{0.5}} = 0.925 \cdot 10^{-4} \frac{\Delta_m^{1.5} k_3^{0.5}} {\kappa_{\mathrm{zakr}} c^{0.5}}, \quad \frac{\mathrm{W}}{\mathrm{cm}^2}. \]
The value of the copper fill factor of the “window” has an optimum value and depends on the height of the coil.
The ratio \(c=\dfrac{a_{\text{w}}}{b_{\text{w}}}\) varies within narrow limits \((c=2 \div 3)\).
Fig. 4. Dependence of the magnitude of the specific heat load on the surface of the magnetizing winding on the magnitude of the effective current density in the copper
In Fig. 4 graphs of the function \(q=f(\Delta_m)\) are plotted, from which the current density in the copper of the magnetizing winding of a betatron electromagnet can be selected. The graphs correspond to the values \(k_z=0.3\) and \(c=\dfrac{a_{\text{w}}}{b_{\text{w}}}=2\). In the event that the winding design differs from that for which the graph was plotted, it is necessary to calculate the value of the specific heat load from the formula. Experience has shown that, for \(q<2000\ \dfrac{\text{W}}{\text{m}^2}\), natural air cooling is sufficient.
A calculation with artificial cooling will not differ in any way from that given above; it is only necessary, for each operating condition, to know the heat-transfer coefficient of the winding surface.
LITERATURE
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Filippov M. F. On the choice of the radius of the equilibrium orbit of a betatron. Izvestiya TPI, vol. 87, 1957.
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The Proceedings of the Institution of electrical Engineers. Part A. Vol. 102, 1955.