Page 46 - Fluid Mechanics and Thermodynamics of Turbomachinery

Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 27
which is Bernoulli’s equation. For an incompressible fluid, is constant and
eqn. (2.10a) becomes
1
.p 02 p 01 / C g.z 2 z 1 / D 0, (2.10b)

2
1
where stagnation pressure is p 0 D p C c .
2
When dealing with hydraulic turbomachines, the term head H occurs frequently
and describes the quantity z C p 0 /. g/. Thus eqn. (2.10b) becomes
H 2 H 1 D 0. (2.10c)
If the fluid is a gas or vapour, the change in gravitational potential is generally
negligible and eqn. (2.10a) is then
2
Z
1 1 2 2
dp C .c 2 c / D 0. (2.10d)
1
1 2
Now, if the gas or vapour is subject to only a small pressure change the fluid density
is sensibly constant and
p 02 D p 01 D p 0 , (2.10e)

i.e. the stagnation pressure is constant (this is also true for a compressible isentropic
process).

Moment of momentum

In dynamics much useful information is obtained by employing Newton’s second
law in the form where it applies to the moments of forces. This form is of central
importance in the analysis of the energy transfer process in turbomachines.
For a system of mass m, the vector sum of the moments of all external forces
acting on the system about some arbitrary axis A A fixed in space is equal to the
time rate of change of angular momentum of the system about that axis, i.e.
d
A D m .rc /, (2.11)
dt
where r is distance of the mass centre from the axis of rotation measured along the
normal to the axis and c the velocity component mutually perpendicular to both
the axis and radius vector r.
For a control volume the law of moment of momentum can be obtained. Figure 2.4
shows the control volume enclosing the rotor of a generalised turbomachine.
Swirling fluid enters the control volume at radius r 1 with tangential velocity c 1
and leaves at radius r 2 with tangential velocity c 2 . For one-dimensional steady
flow
A DPm.r 2 c 2 r 1 c 1 / (2.11a)
which states that, the sum of the moments of the external forces acting on fluid
temporarily occupying the control volume is equal to the net time rate of efflux of
angular momentum from the control volume.