Page 262 - Fluid Mechanics and Thermodynamics of Turbomachinery

Radial Flow Gas Turbines 243
From the design point velocity triangles, Figure 8.3,
2
c 2 D U 2 cosec a 2 ,w 3 D U 3 cosec ˇ 3 ,c 3 D U 3 cot ˇ 3 ,W D U .
2
Thus substituting all these expressions in eqn. (8.8) and noting that U 3 D U 2 r 3 /r 2 ,
then
" ( )# 1
2
1 T 3 2 r 3 2 2
ts D 1 C N cosec ˛ 2 C . R cosec ˇ 3 C cot ˇ 3 / (8.9)
2
T 2 r 2
1
Usually r 3 and ˇ 3 are taken to apply at the arithmetic mean radius, i.e. r 3 D .r 3t C
2
r 3h /. The temperature ratio .T 3 /T 2 / in eqn. (8.9) can be obtained as follows.
At the nominal design condition, referring to the velocity triangles of Figure 8.3,
2
2
w 2 U D c , and so eqn. (8.2) can be rewritten as
3 3 3
2
2
1
h 3 D .U 2 w C c /. (8.2a)
h 2 2 2 3
2
1 2
This particular relationship, in the form I 2 D h 02 rel U D h 03 can be easily iden-
2 2
tified in Figure 8.4.
Again, referring to the velocity triangles, w 2 D U 2 cot a 2 and c 3 D U 3 cot ˇ 3 ,a
useful alternative form to eqn. (8.2a) is obtained,
2
2
2
1
h 2 h 3 D U [.1 cot a 2 / C .r 3 /r 2 / cot ˇ 3 ], (8.2b)
2 2
where U 3 is written as U 2 r 3 /r 2 . For a perfect gas the temperature ratio T 3 /T 2 can
be easily found. Substituting h D C p T D
RT/.
1/ in eqn. (8.2b)
" #
2
.
1/
T 3 1 2 2 r 3 2
1 D U 2 1 cot ˛ 2 C cot ˇ 3
2
T 2
RT 2 r 2
" #
2 2
∴ T 3 D 1 1 .
1/ U 2 1 cot ˛ 2 C r 3 cot ˇ 3 , .8.2c/
2
2
2
T 2 ˛ 2 r 2
where a 2 D .
RT 2 / 1/2 is the sonic velocity at temperature T 2 .
Generally this temperature ratio will only have a very minor effect upon the
numerical value of ts and so it is often ignored in calculations. Thus,
" ( )# 1

2
2
2
2
ts ' 1 C 1 N cosec ˛ 2 C r 3av . R cosec ˇ 3av C cot ˇ 3av / (8.9a)
2
r 2
is the expression normally used to determine the total-to-static efficiency. An alter-
native form for ts can be obtained by rewriting eqn. (8.6) as
h 3ss / h 3 / h 3s / h 3ss /
h 01 h 03 .h 01 .h 03 .h 3 .h 3s
ts D D
h 01 h 3ss .h 01 h 3ss /
2
2
2
D 1 .c C N c C R w //c 2 .8.10/
3 2 3 0
where the spouting velocity c 0 is defined by,
1 2
h 3ss D c D C p T 01 [1 .p 3 /p 01 / .
1//
] (8.11)
h 01
2 0