Basics of Fluid Mechanics, 2014a

422 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
The momentum equation is written as the following
−AdP − τ w dA wetted area = ṁdU (11.115)
where A is the cross section area (it doesn’t have to be a perfect circle; a close enough
shape is sufficient.). The shear stress is the force per area that acts on the fluid by the
tube wall. The A wetted area is the area that shear stress acts on. The second law of
thermodynamics reads
s 2 − s 1
=ln T 2
− k − 1 ln P 2
(11.116)
C p T 1 k P 1
The mass conservation is reduced to
ṁ = constant = ρUA (11.117)
Again it is assumed that the gas is a perfect gas and therefore, equation of state
is expressed as the following:
P = ρRT (11.118)
11.6.2 Dimensionless Representation
In this section the equations are transformed into the dimensionless form and presented
as such. First it must be recalled that the temperature is constant and therefore,
equation of state reads
dP
P
= dρ
ρ
It is convenient to define a hydraulic diameter
(11.119)
4 × Cross Section Area
D H = (11.120)
wetted perimeter
The Fanning friction factor 12 is introduced, this factor is a dimensionless friction factor
sometimes referred to as the friction coefficient as
f =
τ w
1
2 ρU (11.121)
2
Substituting equation (11.121) into momentum equation (11.115) yields
ṁ
−dP − 4 dx ( ) A
1 {}}{
f
D H 2 ρU2 = ρU dU (11.122)
12 It should be noted that Fanning factor based on hydraulic radius, instead of diameter friction
equation, thus “Fanning f” values are only 1/4th of “Darcy f” values.