AE 401-- Procedure -- Lab: Nozzle Performance - Clarkson University

AE 401 – Spring 2005
in the hypersonic range in the shock structures experiment
later in the semester. The speed of sound,
a, can be calculated using the formula:
a = (γRT ) 1 2
(3)
In Equation 3, γ is equal to the specific heat of the
fluid at constant pressure divided by the specific
heat at a constant volume:
γ = C P
C V
. (4)
For the air flows near STP, we will assume a value
of 1.4 for γ and 287 m 2 /(s 2 k) for R. Using these
assumptions, the speed of sound becomes a function
of only temperature.
The perfect gas law can be used to obtain the
air density, rho, in the calculations for these nozzles.
The perfect gas law can be written as:
ρ =
P RT
(5)
It will become important in calculations to determine
the density of the air entering the nozzle,
which is commonly referred to as the stagnation
density, ρ o . By simply substituting the stagnation
values, P o (the inlet pressure) and T o , into Equation
5, the stagnation density can be determined.
The following equations can be used for calculations
involving the flow of air through the nozzles
in this experiment:
T o
T = 1 + 0.2Ma2 (6)
ρ o
ρ = (1 + 0.2Ma2 ) 2.5 (7)
P o
P = (1 + 0.2Ma2 ) 3.5 (8)
Rearranging these equations, we can obtain the a
series of equations which can be used to obtain the
Mach number:
Ma 2 = 5( T o
T − 1) = 5[( ρ 2
o
rho ) 5
− 1] = 5[( P 2
o
P ) 7
− 1]
(9)
From these equations the mass flow rate, ṁ, from
a nozzle can be calculated. The equation for mass
flow rate is:
ṁ = ρA t Ma · a (10)
The area, A t , and all the values of Equation 10 are
evaluated at the throat of the nozzle.
The mass flow rate, ṁ, will continue to increase
through a nozzle until a sonic velocity is reached
at the throat. When the flow through the nozzle
reaches a velocity of Mach 1, the mass flow rate
becomes choked. This can be explained by equation
14. When the nozzle becomes choked, there is no
way to increase the mass flow without increasing the
throat diameter.
The maximum mass flow rate, ṁ max , at γ = 1.4
can be computed from the following relation:
ṁ max = 0.6847P oA ∗
(RT o ) 1/2 (11)
The design point for a nozzle is the point at
which the back pressure, P 2 , is equal to the exit
pressure, P e . When a nozzle is operated that this
point, it will be most efficient. If the back pressure
is less than the exit pressure, the mass flow could
be increased by increasing the back pressure. If the
back pressure exceeds the exit pressure, the flow will
be choked. The design point can be calculated for a
particular nozzle with a known stagnation pressure,
P o , stagnation temperature, T o , and the ratio of the
throat area to the exit area, A e /A t , of the nozzle.
To compute the design point of the nozzle, the
exit plane Mach number, Ma e , will be needed. For
dry air with γ = 1.4, an iterative approach can be
used to determine the exit plane Mach number from
Equation 12:
A
A ∗ = 1 (1 + 0.2Ma 2 ) 3
Ma 1.728
(12)
Rather than performing the iteration over and over
again, scientists have performed a series of curve fits
to that directly relate the area ratio to the Mach
number. For nozzles with an area ratio where 1.0 <
A
A ∗
< 2.9 the appropriate curve fit is:
Ma ≈ 1 + 1.2( A
A ∗ − 1) 1/2
. (13)
For nozzles where the area ratio is 2.9 < A A
< ∞
∗
the curve fit is described by:
[
Ma ≈ 216 A ( A
) 2/3 ] 1/5.
A ∗ − 254 (14)
A ∗
Once the exit plane Mach number is computed, the
pressure ratio and the mass flow rate corresponding
to the design point can be determined from equations
8 and 11.
A primary objective of this experiment is to
show how static thrust varies with back pressure.
To make any comparisons to the data obtained in
the lab, we must first develop an equation to model
the thrust. This equation is related to the momentum
equation, and can be written as:
F = ṁMa e · a + (P 1 − P 2 )A e (15)
From this equation we see that the thrust, F , produced
by a nozzle is a function of the mass flow
rate, ṁ, the flow velocity, Ma · a, and the pressure
difference, P 1 − P 2 , multiplied by the exit area, A e ,
of the nozzle.
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