Page 43 - Fluid Mechanics and Thermodynamics of Turbomachinery

24 Fluid Mechanics, Thermodynamics of Turbomachinery

FIG. 2.1. Flow across an element of area.

Most analyses in this book are limited to one-dimensional steady flows where
the velocity and density are regarded as constant across each section of a duct
or passage. If A 1 and A 2 are the flow areas at stations 1 and 2 along a passage
respectively, then
P m D 1 c n1 A 1 D 2 c n2 A 2 D c n A, (2.2)

since there is no accumulation of fluid within the control volume.

The first law of thermodynamics internal energy

The first law of thermodynamics states that if a system is taken through a complete
cycle during which heat is supplied and work is done, then
I
.dQ dW/ D 0, (2.3)

H H
where dQ represents the heat supplied to the system during the cycle and dW the
work done by the system during the cycle. The units of heat and work in eqn. (2.3)
are taken to be the same.
During a change of state from 1 to 2, there is a change in the property internal
energy,
Z
2
E 1 D .dQ dW/. (2.4)
E 2
1
For an infinitesimal change of state
dE D dQ dW. (2.4a)

The steady flow energy equation

Many textbooks, e.g. ¸Cengel and Boles (1994), demonstrate how the first law of
thermodynamics is applied to the steady flow of fluid through a control volume so
that the steady flow energy equation is obtained. It is unprofitable to reproduce this
proof here and only the final result is quoted. Figure 2.2 shows a control volume
representing a turbomachine, through which fluid passes at a steady rate of mass