Page 166 - 48Fundamentals of Compressible Fluid Mechanics

128 CHAPTER 8. ISOTHERMAL FLOW
8.5 Supersonic Branch

Apparently, this analysis/model is over simplified for the supersonic branch and
does not produced reasonable results since it neglects to take into account the
4
heat transfer effects. A dimensionless analysis demonstrate that all the common
materials that this author is familiar which create a large error that the fundamental
assumption of the model breaks. Nevertheless, this model can provide a better
understanding so the trends and deviations from Fanno flow model can be under-
stood.
To add figure from the pro-
grams with comparison with
Fanno flow. In the supersonic flow, the hydraulic entry length is very large as shown
below. However, the feeding diverging nozzle somewhat reduces the required entry
length (as opposed to converging feeding). The thermal entry length is in the order
of the hydrodynamic entry length (Look at the Prandtl number, (0.7-1.0), value for
the common gases.). Most of the heat transfer is hampered in the sublayer thus the
core assumption of isothermal flow (not enough heat transfer so the temperature
5
isn’t constant) breaks down .
The flow speed at the entrance is very large, over hundred of meters per
second. For example, a flow gas in a tube with for
the required Mach
number is over 200. Almost all the perfect gas model substances dealt in this

book the speed of sound is a function of temperature. For this illustration, most
. For example, even with low
. So, for even for
gas cases the speed of sound is about

1
relatively small tubes with
the inlet speed is over 56 [km/sec]. This
temperature like

the speed of sound of air is %'
1
requires that the entrance length to be larger than the actual length of the tub for
book under construction per- air. Remember from Fluid Dynamic book

haps to general ref at this
stage

(8.51)

The typical values of the the kinetic viscosity, , are 0.0000185 kg/m-sec at 300K

and 0.0000130034 kg/m-sec at 200K. Combine this information with our case of

'

'3

'
results in

!(3 3% )(

The fact that the actual tube length is only less 1% than the entry length means that
On the other hand with typical value of friction coefficient
to insert the figure from Van the assumption of the isothermal also breaks (as in control large time respond).
Dreist also figure 15-4 in Kays
"
#
and Crawford.
4
This dimensional analysis is a bit tricky, and is based on estimates. Currently and ashamedly this

author is looking for a more simplified explanation. The current explanation is correct but based on
hands waving and definitely dose not satisfied this author.
5 see Kays and Crawford “Convective Heat Transfer” (equation 12-12).