250 Ë 7 Numerical simulations of HTS Maglev
Eq. (7.75) can be rewritten in the following form when Eq. (7.77) is taken into account:
∇×휌s(∇×T)=휌ab ̈−휕(^2) Tx
휕x^2
−훼휕
(^2) Tx
휕y^2
−휕
(^2) Tx
휕z^2
+(훼− 1 )
휕^2 Ty
휕x휕y©x̂+휌ab ̈−훼휕^2 Ty
휕x^2−
휕^2 Ty
휕y^2−
휕^2 Ty
휕z^2+(훼− 1 )
휕^2 Tx
휕x휕y©
ŷ+휌ab¤−휕^2 Tz
휕x^2−
휕^2 Tz
휕y^2−
휕^2 Tz
휕z^2¥ẑ. (7.78)Besides,∇耠=
1
R(P,P耠)=
휕
휕x耠^1
R(P,P耠)
x̂+ 휕
휕y耠^1
R(P,P耠)
ŷ+ 휕
휕z耠^1
R(P,P耠)
ẑ, (7.79)and
Be=Bexx̂+Beyŷ+Becẑ. (7.80)Finally, the 3D PDEs for governing the electromagnetic behavior of HTSC were derived
after Eqs. (7.78) to (7.80) are substituted into Eq. (7.73) and휎abis replaced by its
reciprocal휎ab,
1
휎ab¬−
휕^2 Ty
휕x^2−훼
휕^2 Ty
휕y^2−
휕^2 Ty
휕z^2+(훼− 1 )
휕^2 Ty
휕x휕y+휇 0 C(P)휕Tx
휕t+휇 0
4 휋
X
S휕(n耠⋅T耠)
휕t휕
휕x耠^1
R(P,P耠)
dS耠+휕Bex
휕t= 0 , (7.81)
1
휎ab¬−훼−
휕^2 Ty
휕x^2−
휕^2 Ty
휕y^2−
휕^2 Ty
휕z^2+(훼− 1 )
휕^2 Tx
휕x휕y+휇 0 C(P)
휕Ty
휕t+휇^0
4 휋
X
S휕(n耠⋅T耠)
휕t휕
휕y耠^1
R(P,P耠)
dS耠+휕Bey
휕t= 0 , (7.82)
1
휎ab¤−휕
(^2) T
z
휕x^2
−휕
(^2) T
z
휕y^2
−휕
(^2) T
z
휕z^2
¥+휇 0 C(P)휕Tz
휕t
+휇^0
4 휋
X
S휕(n耠⋅T耠)
휕t휕
휕z耠^1
R(P,P耠)
dS耠+휕Bez
휕t= 0 , (7.83)
where휎abis the conductivity in thea-bplane. It is worth noting that compared with
the traditionalT-Ωmethod, the complexity of the governing equations is reduced due
to the omission of the variableΩ.