e 1 Á 2pr ¼ 2pu can be easily obtained in tangential direction, that is
e r ¼ u=r
ð9:52Þ
The deformation of impeller is in the elastic deformation range, so the stress and
strain should conform to Hooke’s law. For plane stress, there are
e r ¼
1
E r r À ur 1
ð
Þ
e 1 ¼
1
E r 1 À ur r
ð
Þ
'
ð9:53Þ
where
E Modulus of Elasticity;
l Poisson’s ratio.
Substituting Eqs. (9.51) and (9.52) into Eq. (9.53), it is obtained,
r r ¼
E
1Àu 2
du
dr þ u
u
r
r 1 ¼
E
1Àu 2
u
r þ u du
dr
9
=
;
ð9:54Þ
Substituting Eq. (9.54) into Eq. (9.50), there is
d
2 u
dr 2 þ
1
r
du
dr
À
u
r 2 þ
1 À u
2
E
qx
2 r ¼ 0
Set
1Àu
2
E qx
2
¼ K, there is
d
dr
1
r
d ur
ð Þ
dr
!
þ Kr ¼ 0
After integrating twice, there is
u ¼ ÀK
r
3
8
þ C 1
r
2
þ C 2
1
r
ð9:55Þ
Substituting Eq. (9.55) into Eq. (9.54), it is obtained
r r ¼ À
3 þ u
8 qx
2 r
2
þ
E
2 1Àu
ð
Þ C 1 À
E
1 þ u
1
r 2 C 2
r t ¼ À
1À3u
8 qx
2 r
2
þ
E
2 1Àu
ð
Þ C 1 þ
E
1 þ u
1
r 2 C 2
)
ð9:56Þ
The integral constants C 1 and C 2 can be obtained by known boundary conditions. Usually, the radial stress r ra around the wheel and the radial stress r ri at the
center hole of the wheel are known values, but the following drawing process can
avoid the tedious solution of C 1 and C 2 .
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9 High-Temperature and High-Speed Gas Turbine Pump …