Method of three-dimensional optimization design for asymmetric cusp magnetic field in MCZ single crystal furnace
A method of three-dimensional optimization design for an asymmetric cusp magnetic field in an MCZ single crystal furnace is disclosed. An optimization design for structural parameters of the magnetic field includes first establishing a three-dimensional magnetic model by using ANSYS software, varying parameters of the model, determining a span between upper and lower parts of coils, determining a number of transverse turns of the coils and the thickness of a shield. An optimization design for specification parameters of the coils of the magnetic field includes determining a relationship between heat and specification parameters of the coils and between heat transfer of the coils at their copper pipe walls and the specification parameters of the coils, analyzing heat absorbed by cooling water, establishing an optimization model of a system, and optimizing the specification parameters of the coils of the magnetic field using the ANSYS software.
1. A method of constructing electrical coils for generating an asymmetric cusp magnetic field within a single crystal furnace, the method comprising:
establishing a first mathematical expression of electrical coil heat as:
w
1
=
I
2
l
s
ρ
0
(
1
+
at
)
,
where I is a current of the coils, l is the total length of the coils, s is the cross-sectional area of the coils, p 0 is the resistivity of the coils, a is a temperature coefficient of resistivity, and t is the temperature of copper pipe walls of the coils;
establishing a second mathematical expression of heat transfer at the copper pipe walls as:
w
2
=
0.023
λ
A
de
(
Vde
v
)
0.8
Pr
0.4
Δ
t
,
where de is a characteristic size, V is an average flow velocity of cooling water, Δt is the temperature difference between the copper pipe walls and the cooling water, λis the heat conductivity of the cooling water, A is the total heat transferring area of the copper pipe walls, ν is the kinematic viscosity of the cooling water, and Pr is a Prandtl number;
establishing a third mathematical expression of heat absorbed by the cooling water as follows:
w 3 =q m c ( t 2 −t 1)· n ,
where q m is the amount of cooling water flow, c is the specific heat capacity of the cooling water, t2 is the water temperature at a cooling water outlet, t1 is the water temperature at a cooling water inlet, and n is the number of cooling water paths for the copper pipes;
determining, with a processor, values for parameters s and l of the coils for which a minimum value of w1 satisfies the relation w1=w2=w 3 , Δt+t2<40 and hf<20, where Δt+t2 is the temperature of the copper pipe walls and hf is the linear loss along the coils; and
constructing coils in accordance with the values determined for the parameters s and l.