Page 78 - 48Fundamentals of Compressible Fluid Mechanics

40 CHAPTER 4. ISENTROPIC VARIABLE AREA FLOW

noted by the subscript “
.”
First, the stagnation tempera-
ture is calculated. The energy conser-
vation can be written as

(4.1)

Perfect gas is an ideal gas with a constant heat capacity, . For perfect gas

equation (4.1) is simplified into
(4.2)

Again it common to denote
as the stagnation temperature. Recalling from

thermodynamic the relationship for perfect gas
(4.3)

than the thermodynamics relationship obtains the form

and denoting
(4.4)

yield

is a specific constant. Dividing equation (4.2) by
and where
(4.5)

equation (4.5) changes into

Now, substituting
or
(4.6)

Utilizing the definition of by equation (4.4) and inserting it into equation (4.6)

yields
(4.7)

It very useful to convert equation (4.6) into a dimensionless form and de-

note Mach number as the ratio of velocity to speed of sound as
(4.8)

Inserting the definition of Mach number, (4.8) into equation (4.7) reads

(4.9)