Page 48 - Fluid Mechanics and Thermodynamics of Turbomachinery

Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 29
widely used name rothalpy, a contraction of rotational stagnation enthalpy, and is
a fluid mechanical property of some importance in the study of relative flows in
Ł
rotating systems. As the value of rothalpy is apparently unchanged between entry
and exit of the impeller it is deduced that it must be constant along the flow lines
between these two stations. Thus, the rothalpy can be written generally as
1 2
I D h C c Uc . (2.12e)
2
The same reasoning can be applied to the thermomechanical flow through a
turbine with the same result.

The second law of thermodynamics entropy

The second law of thermodynamics, developed rigorously in many modern ther-
modynamic textbooks, e.g. ¸Cengel and Boles (1994), Reynolds and Perkins (1977),
Rogers and Mayhew (1992), enables the concept of entropy to be introduced and
ideal thermodynamic processes to be defined.
An important and useful corollary of the second law of thermodynamics, known as
the Inequality of Clausius, states that for a system passing through a cycle involving
heat exchanges,
I
dQ
5 0, (2.13)
T
where dQ is an element of heat transferred to the system at an absolute temperature
T. If all the processes in the cycle are reversible then dQ D dQ R and the equality
in eqn. (2.13) holds true, i.e.
I
dQ R
D 0. (2.13a)
T
The property called entropy, for a finite change of state, is then defined as
2
Z
dQ R
S 1 D . (2.14)
S 2
1 T
For an incremental change of state
dQ R
dS D mds D , (2.14a)
T
where m is the mass of the system.
With steady one-dimensional flow through a control volume in which the fluid
experiences a change of state from condition 1 at entry to 2 at exit,

Z
2 P
dQ
5 Pm.s 2 s 1 /. (2.15)
T
1
Ł A discussion of recent investigations into the conditions required for the conservation of rothalpy
is deferred until Chapter 7.