Page 47 - Fluid Mechanics and Thermodynamics of Turbomachinery

28 Fluid Mechanics, Thermodynamics of Turbomachinery

FIG. 2.4. Control volume for a generalised turbomachine.

Euler’s pump and turbine equations

For a pump or compressor rotor running at angular velocity , the rate at which
the rotor does work on the fluid is

A  DPm.U 2 c 2 U 1 c 1 /, (2.12)
where the blade speed U D r.
Thus the work done on the fluid per unit mass or specific work, is

P W c A 
W c D D D U 2 c 2 U 1 c 1 > 0. (2.12a)
P m P m
This equation is referred to as Euler’s pump equation.
For a turbine the fluid does work on the rotor and the sign for work is then
reversed. Thus, the specific work is

P W t
W t D D U 1 c 1 U 2 c 2 > 0. (2.12b)
P m
Equation (2.12b) will be referred to as Euler’s turbine equation.
Defining rothalpy

In a compressor or pump the specific work done on the fluid equals the rise in
stagnation enthalpy. Thus, combining eqns. (2.8) and (2.12a),

W c D P W c / Pm D U 2 C 2 U 1 c 1 D h 02 h 01 . (2.12c)
This relationship is true for steady, adiabatic and irreversible flow in compressor or
1 2
in pump impellers. After some rearranging of eqn. (2.12c) and writing h 0 D h C c ,
2
then
1 2
1 2
h 1 C c U 1 c 1 D h 2 C c U 2 c 2 D I. (2.12d)
2 1 2 2
According to the above reasoning a new function I has been defined having the
same value at exit from the impeller as at entry. The function I has acquired the