Page 42 - Fluid Mechanics and Thermodynamics of Turbomachinery

CHAPTER 2

Basic Thermodynamics,

Fluid Mechanics:

Definitions of Efficiency

Take your choice of those that can best aid your action. (SHAKESPEARE,
Coriolanus.)

Introduction
THIS chapter summarises the basic physical laws of fluid mechanics and ther-
modynamics, developing them into a form suitable for the study of turbomachines.
Following this, some of the more important and commonly used expressions for the
efficiency of compression and expansion flow processes are given.
The laws discussed are:
(1) the continuity of flow equation;
(2) the first law of thermodynamics and the steady flow energy equation;
(3) the momentum equation;
(4) the second law of thermodynamics.
All of these laws are usually covered in first-year university engineering and tech-
nology courses, so only the briefest discussion and analysis is give here. Some
fairly recent textbooks dealing comprehensively with these laws are those written
by Cengel and Boles (1994), Douglas, Gasiorek and Swaffield (1995), Rogers and
Mayhew (1992) and Reynolds and Perkins (1977). It is worth remembering that
these laws are completely general; they are independent of the nature of the fluid
or whether the fluid is compressible or incompressible.

The equation of continuity

Consider the flow of a fluid with density , through the element of area dA,
during the time interval dt. Referring to Figure 2.1, if c is the stream velocity the
elementary mass is dm D cdtdA cos , where is the angle subtended by the normal
of the area element to the stream direction. The velocity component perpendicular
to the area dA is c n D c cos and so dm D c n dAdt. The elementary rate of mass
flow is therefore
dm
d Pm D D c n dA. (2.1)
dt
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