Page 86 - 48Fundamentals of Compressible Fluid Mechanics

48 CHAPTER 4. ISENTROPIC VARIABLE AREA FLOW

Equation (4.43) is very important be-
cause it relates the geometry (area) M, A
with the relative velocity (Mach num-
ber). In equation (4.43), the factors

and are positive re- A, cross section
or . There-

fore, the only factor that effects relation-
%

gardless to the values of
ship between the cross area and the

M, Much nubmer
the Mach number is varied opposite to
Mach number is
. For
the cross section area. In the case of

x

the Mach number increases with
cross section area and vise versa. The Fig. 4.5: The relationship between the cross
special case is when
, in that section and the Mach number on the

. This con- subsonic branch

2
dition imposes that internal flow has to
case requires that
pass a converting diverging device to
obtain supersonic velocity. This mini-
mum area is referred to as “throat.”
Again, the opposite conclusion that when
implies that
is

not correct because possibility of

. In subsonic flow branch, from the
mathematical point of view, on one hand, decrease of the cross section increase
the velocity and the Mach number. On the other hand, increase of the cross section
decreases the velocity and Mach number (see Figure 4.5).
4.2.2 Examples
Example 4.1:
Air allowed to flow from a reservoir with temperature of 21
C and with pressure
of 5[MPa] through a tube. It was measured that air mass flow rate is 1[kg/sec].
At some point on the tube static pressure pressure was measured to be 3[MPa].
Assume that process is isentropic and neglects the velocity at the reservoir, cal-
culate the Mach number, velocity, and the cross section area at that point where
the static pressure was measured. Assumed that the ratio of specific heats is
.

SOLUTION
The stagnation conditions at the reservoir will be maintained through out tube be-
cause the process is isentropic. Hence the stagnation temperature can be written
and both of them are known (the condition at the
reservoir). For the point where the static pressure is known, the Mach number can

and

2 This condition does not impose any restrictions for external flow. In external flow, an object can be
moved in arbitrary speed.