Page 82 - 48Fundamentals of Compressible Fluid Mechanics

44 CHAPTER 4. ISENTROPIC VARIABLE AREA FLOW

The important sub case in this Chap-
ter is the flow in a converging–diverging
T T+dT
nozzle is considered here. The control r r+dr
P P+dP
volume is shown in Figure (4.4). There U U+dU
are two models that assumed variable
area flow: isentropic and adiabatic and
the second is isentropic and isother- Fig. 4.4: Control volume inside of a
mal model. Clearly, the stagnation tem- converging-diverging nozzle
perature,
, is constant through the
adiabatic flow because there isn’t heat
transfer. Therefore, the stagnation pressure is also constant through the flow be-
cause the flow isentropic. Conversely, in mathematical terms, equation (4.9) and
equation (4.11) are the same. If the right hand side is constant for one variable is
constant for the other. In the same argument, the stagnation density is constant
through the flow. Thus, knowing the Mach number or the temperature provides all
what is needed to find the other properties. The only properties that need to be
connected are the cross section area and the Mach number. Examination of the
relation between properties is carried out.

4.2.1 The Properties in The Adiabatic Nozzle
When no external work and heat transfer, the energy equation, reads

(4.25)

, and dividing by the

continuity equation reads

Differentiation of continuity equation,
(4.26)

The thermodynamic relationship between the properties can be expressed as

(4.27)

and combining equations (4.25) with (4.27) yields
For isentropic process
(4.28)

Differentiation of the equation state (perfect gas),
, and dividing the
results by the equation of state (
) yields
(4.29)