Page 180 - 48Fundamentals of Compressible Fluid Mechanics

142 CHAPTER 9. FANNO FLOW

. Taking

and thus when these equations are divided they yields

The stagnation temperature expresses as

derivative of this expression when
is remains constant yields
(9.32)

In the similar fashion the relationship between the stagnation pressure and the

pressure and substitute in the entropy equation results
(9.33)

The first law requires that the stagnation temperature remains constant,

.

Therefore the entropy change
(9.34)

Utilizing the equation for stagnation pressure the entropy equation yields

(9.35)

9.3 The Mechanics and Why The Flow is Chock?
The trends of the properties can examined though looking in equations (9.24)
through (9.34). For example, from equation (9.24) it can be observed that the
the pressure decreases downstream
are positive. For the same
critical point is when
. When
, the pressure increases downstream.
and
as can seen from equation (9.24) because
This pressure increase is what makes compressible flow so different than “conven-

reasons, in the supersonic branch,
tional” flow. Thus the discussion will be divided into two cases; one of flow with
speed above speed of sound, and, two flow with speed below of speed of sound.
9.3.0.1 Why the flow is chock?
There explanation is based on the equations developed earlier and there is no
known explanation that is based on the physics. First it has to recognized that the
critical point is when
at which show change in the trend and singular by
and mathematically it is a singular point
(see equation (9.24)). Observing from equation (9.24) that increase or decrease

itself. For example,

requires a change in a sign pressure direction. However, the pressure has to be a

.
from subsonic just below one
to above just above one
This constrain means that because the flow cannot “cross–over”
the gas
monotonic function which means that flow cannot crosses over the point of
has to reach to this speed,
at the last point. This situation called chocked
flow.