Page 143 - Fluid Mechanics and Thermodynamics of Turbomachinery

124 Fluid Mechanics, Thermodynamics of Turbomachinery
between temperature and pressure is therefore,
p .
1//
T/p D constant.
n
Combined with p/ D RT the above expression gives, on eliminating p, /T D
constant, hence
n
T
D , (4.35)
d T d
where n D
/f p .
1/g 1.
For an infinitesimal temperature drop eqn. (4.33) combined with eqns. (4.34) and
(4.35) gives, with little error,

2n
2 2
dT c x T d P m
D D . (4.36)
dT d c xd T P m d
Integrating eqn. (4.36),
2
P m 2nC1
2nC1
T D T C K,
d
P m d
where K is an arbitrary constant.
To establish a value for K it is noted that if the turbine entry temperature is
constant T d D T 1 and T D T 1 also.
2
Thus, K D [1 . Pm/ P d / ]T 2nC1 and
m
I
" #
2nC1 2 2nC1
T P m T d
1 D 1 . (4.37)
T I P m d T I
Equation (4.37) can be rewritten in terms of pressure ratio since T/T I D
.p/p I / p .
1//
.As2n C 1 D 2
/[ p .
1/] 1 then,
2 p .
1//
1/2
P m 1 .p/p I /
D 2 p .
1//
(4.38a)
P m d 1 .p d /p 1 /
With p D 0.9 and
D 1.3 the pressure ratio index is about 1.8; thus the approxi-
mation is often used
2 1/2
P m 1 .p/p I /
D 2 , (4.38b)
P m d 1 .p d /p I /
which is ellipse law of a multistage turbine.

The Wells turbine

Introduction
Numerous methods for extracting energy from the motion of sea-waves have
been proposed and investigated since the late 1970s. The problem is in finding an