Basics of Fluid Mechanics, 2014a

2.1. BASIC DEFINITIONS 51
Utilizing equation (2.28) and subsisting into equation (2.30) and dividing by dT yields
C p − C v = R (2.31)
This relationship is valid only for ideal/perfect gases.
The ratio of the specific heats can be expressed in several forms as
C v to Spesific Heats Ratio
C v =
R
k − 1
(2.32)
C p to Spesific Heats Ratio
C p =
kR
(2.33)
k − 1
The specific heat ratio, k value ranges from unity to about 1.667. These values depend
on the molecular degrees of freedom (more explanation can be obtained in Van Wylen
“F. of Classical thermodynamics.” The values of several gases can be approximated as
ideal gas and are provided in Table (2.1).
The entropy for ideal gas can be simplified as the following
∫ 2
( dh
s 2 − s 1 =
T − dP )
(2.34)
ρT
Using the identities developed so far one can find that
s 2 − s 1 =
∫ 2
1
1
C p
dT
T − ∫ 2
Or using specific heat ratio equation (2.35) transformed into
1
RdP
P = C p ln T 2
T 1
− R ln P 2
P 1
(2.35)
s 2 − s 1
R
= k
k − 1 ln T 2
T 1
− ln P 2
P 1
(2.36)
For isentropic process, Δs =0, the following is obtained
ln T ( ) k−1
2 P2 k
=ln
T 1 P 1
(2.37)
There are several famous identities that results from equation (2.37) as
Ideal Gas Isontropic Relationships
( ) k−1 ( )
T 2 P2
k
k−1 V1
= =
T 1 P 1 V 2
(2.38)