Basics of Fluid Mechanics, 2014a

11.4. ISENTROPIC FLOW 385
Now, substituting c 2 = kRT or T = c 2 /k R equation (11.21) changes into
1+ kRU2
2 C p c 2 = T 0
T
(11.22)
By utilizing the definition of k by equation (2.24) and inserting it into equation (11.22)
yields
1+ k − 1
2
U 2
c 2 = T 0
T
(11.23)
It very useful to convert equation (11.22) into a dimensionless form and denote
Mach number as the ratio of velocity to speed of sound as
Mach Number Definition
M ≡ U c
(11.24)
Inserting the definition of Mach number (11.24) into equation (11.23) reads
Isentropic Temperature relationship
T 0
T =1+k − 1
2
M 2
(11.25)
The usefulness of Mach number and
equation (11.25) can be demonstrated by the
following simple example. In this example a gas
flows through a tube (see Figure 11.5) of any
shape can be expressed as a function of only
the stagnation temperature as opposed to the
function of the temperatures and velocities.
The definition of the stagnation state
A
T 0
P 0
ρ 0
Q
velocity
Fig. -11.5. Perfect gas flows through a
tube
provides the advantage of compact writing. For example, writing the energy equation
for the tube shown in Figure (11.5) can be reduced to
B
T 0
P 0
ρ 0
˙Q = C p (T 0B − T 0A ) ṁ (11.26)
The ratio of stagnation pressure to the static pressure can be expressed as the
function of the temperature ratio because of the isentropic relationship as
Isentropic Pressure Definition
( ) k
P 0
P = T0
k−1
=
(1+ k − 1 ) k
M 2 k−1
T
2
(11.27)