Page 35 - Fluid Mechanics and Thermodynamics of Turbomachinery

16 Fluid Mechanics, Thermodynamics of Turbomachinery
Stagnation temperature and pressure

If the fluid is a perfect gas, then h D C p T, where C p D
R/.
1/, so that the
stagnation temperature can be defined as

1 2
T 0 D T C c /C p ,
2
c 2
T 0 1 1 2
D 1 C .
1/ D 1 C .
1/M , .1.13a/
T 2
RT 2
p
where the Mach number, M D c/a D c/
RT.
The Gibb’s relation, derived from the second law of thermodynamics (see
Chapter 2), is
1
Tds D dh dp.

If the flow is brought to rest both adiabatically and isentropically (i.e. ds D 0), then,
using the above Gibb’s relation,

dp
dh D C p dT D RT
p
so that
dp C p dT
dT
D D .
p R T
1 T
Integrating, we obtain

ln p D ln constant C ln T,

1

and so,

/.
1/
/
1
p 0 T 0
1 2
D D 1 C M (1.13b)
p T 2
From the gas law density, D p/.RT/, we obtain 0 / D .p 0 /p/.T/T 0 / and hence,
1/.
1/ 1/.
1/
0 T 0
1 2
D D 1 C M . (1.13c)
T 2

Compressible fluid analysis

The application of dimensional analysis to compressible fluids increases, not unex-
pectedly, the complexity of the functional relationships obtained in comparison with
those already found for incompressible fluids. Even if the fluid is regarded as a
perfect gas, in addition to the previously used fluid properties, two further char-
acteristics are required; these are a 01 , the stagnation speed of sound at entry to