Page 45 - Fluid Mechanics and Thermodynamics of Turbomachinery

26 Fluid Mechanics, Thermodynamics of Turbomachinery
a blade in a compressor or turbine cascade caused by the deflection or acceleration
of fluid passing the blades.
Considering a system of mass m, the sum of all the body and surface forces acting
on m along some arbitrary direction x is equal to the time rate of change of the total
x-momentum of the system, i.e.
d
F x D .mc x /. (2.9)
dt
For a control volume where fluid enters steadily at a uniform velocity c x1 and leaves
steadily with a uniform velocity c x2 , then
c x1 / (2.9a)
F x DPm.c x2
Equation (2.9a) is the one-dimensional form of the steady flow momentum equation.

Euler’s equation of motion

It can be shown for the steady flow of fluid through an elementary control volume
that, in the absence of all shear forces, the relation
1
dp C cdc C gdz D 0 (2.10)

is obtained. This is Euler’s equation of motion for one-dimensional flow and is
derived from Newton’s second law. By shear forces being absent we mean there
is neither friction nor shaft work. However, it is not necessary that heat transfer
should also be absent.
Bernoulli’s equation

The one-dimensional form of Euler’s equation applies to a control volume whose
thickness is infinitesimal in the stream direction (Figure 2.3). Integrating this equa-
tion in the stream direction we obtain
Z
2
1 1 2 2
dp C .c 2 c / C g.z 2 z 1 / D 0 (2.10a)
1
1 2

FIG. 2.3. Control volume in a streaming fluid.