Page 162 - Advanced thermodynamics for engineers

148 CHAPTER 7 GENERAL THERMODYNAMIC RELATIONSHIPS

and hence
vv R

¼ (7.33)
vT
p a 2b
p 2 1
v v
This can be differentiated again to give
v v v vv
2
vT 2 ¼ vT vT
p p
which results in
2
2
v v a 2ab vv 2a 6ab ¼ 0
vT 2 p v 2 þ v 3 þ vT v 3 v 4
p p
giving
2
vv 2a 6ab
v v p
2 vT v 3 v 4
¼
vT 2
p a 2ab
p 2 þ
v v 3
vv

which, on substituting for becomes
vT
p
4
v
v
v v (7.34)
2 R 2 2a 3 6ab
vT 2 ¼ 3
p a 2b
p 2 1
v v
Hence, for a van der Waals gas, c p ¼ f(p).
This means that allowance would have to be taken of this when evaluating the change of specific
heat capacity at constant pressure for a process in which the pressure is changed. This can be evaluated
from
p
Z 2 2
v v
c p ¼ T 2 dp; (7.35)
vT
p
p 1
2
v v
where is calculated from Eqn (7.34).
vT 2
p
7.3 Tds RELATIONSHIPS
Two approaches have been used previously in performing cycle calculations. When evaluating the
performance of steam plant and refrigeration equipment, much emphasis was placed on the use of
tables and if the work done between two states was required, enthalpy values at these states were
evaluated and suitable subtractions were performed. When doing cycle calculations for gas turbines
and internal combustion engines, the air was assumed to be an ideal gas and the specific heat capacity