ME 309 Fall 2008
Section 2
Homework # 36
Due Wednesday 3 December 2008
Problem 361
a)
Convert the energy equation for an adiabatic flow with no work done
into an equation for the stagnationtostatic temperature ratio as an algebraic function
of Mach number.
Take Station 1 as a location where the temperature is
T
1
, and the
velocity is
u
1
; and Station 2 as a location where the flow has been stopped so that the
velocity is
u
2
= 0, and the temperature has been increased to
T
2
.
Show all pertinent
algebra.
b)
Using the Gibbs relation:
dp
dh
Tds
ρ
1

=
along with an isentropic process
between Stations 1 and 2 (defined as in Part a) to provide a definition of the
stagnation pressure.
Relate this stagnation pressure to the static pressure and the
Mach number.
Show all pertinent algebra;p
c)
Convert the continuity equation in terms of
static
variables,
uA
m
ρ
=
&
, into an
algebraic relation dependent upon the Mach number, stagnation pressure and
stagnation temperature.
As a part of this step, define the Mach number, function,
D.
Show all pertinent algebra.
d)
Using the equation from Part c (after comparing with the notes to ensure it is
correct), develop a table giving the mass flow per unit area for a flow that is being
drawn from a quiescent reservoir at Station 1 where the static pressure is 101325
Pa
and the static temperature is 300
K
and the velocity is zero, to Station 2 where the
Mach number ranges from 0 to 2 in increments of 0.2 and from 2 to 20 in increments
of 1.
What do you observe about the maximum flow per unit area at Station 2 (which
is called
choking
)?
How much flow would you expect per unit area if the Mach
number is increased without bound?
Does this expectation make physical sense?
Solution
:
Assumptions
:
perfect gas,
γ
=1.4, constant specific heats
Find:
Relation for stagnation temperature, relation for Mach number, mass flow
per area vs Mach number
Analysis
:
a)
Given:
T
1
, u
1
,
u
2
= 0
Energy Equation;
No work, no heat addition, constant mass flow:
0
1
0
2
h
h
=
Station 2 is the stagnated condition (velocity = 0) for Station 1:
2
0
2
1
1
0
1
2
0
2
u
h
h
h
h
+
=
=
+
=
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 Spring '08
 MERKLE
 Fluid Dynamics, Thermodynamics, Mach number, Station 2

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