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

AE 401 – Spring 2005 

Compressible Flows and Nozzle Performance 

J. A. Taylor ∗ 

Mech. & Aero. Eng. Dept. 

Clarkson University Box 5725 

Potsdam, NY 13699-5725 

January 5, 2005 

Purpose: 

This experiment is intended to support the Propulsion 

Systems and Intermediate Fluid Dynamics 

courses. The student will investigate the behavior 

of converging and converging – diverging nozzles 

with sufficient pressure ratios to produce transonic 

exit flow velocities. The student will perform measurements 

of the mass flow rate and the thrust and 

compare his or her measurements to the trends predicted 

using their knowledge of isentropic flows. 

Background: 

This experiment deals with the flow of a compressible 

fluid through nozzles. A nozzle is a suitably 

shaped passage in which a fluid is accelerated to 

a high velocity while its static pressure decreases. 

They are frequently used as thrust-producers for 

jet and rocket engines. The discussion below will 

provide a brief overview of compressible flows. For 

detailed derivations of the equations shown below, 

the reader is referred to: White, Frank M., Fluid 

Mechanics, 2nd edition, 1986, McGraw-Hill, Chap. 

9. 

Compressible fluids can not be analyzed in the 

same manner as incompressible flows. As a compressible 

flow passes through devices such as nozzles 

its temperature, T , pressure, P , and density, ρ, 

are all free to vary. Variations in these fields provides 

additional unknowns that must be accounted 

for. An additional two equations must be used to 

describe the flow. To simplify the analysis of the 

flows in this experiment, the nozzles will be modeled 

according to isentropic theory. 

Isentropic theory assumes that the entropy of 

the fluid remains constant throughout the nozzle. 

Hence, the temperature of the fluid should not 

change appreciably from one side of the nozzle to 

the other. It also predicts that a nozzle can be used 

both to increase the velocity of a compressible flow 

∗ taylorja@clarkson.edu, (315) 268-6683, and CAMP 266 

from sub to supersonic or to slow a flow from super 

to subsonic. This contradicts common intuition. 

This is a property of compressible flow, and can be 

infered from the equation: 

dV 

V 

= dA A 

1 

Ma 2 − 1 = − dP 

ρV 2 (1) 

Equation 1 also dictates that the flow can only reach 

Mach 1 when the change in the area, dA, equals 

zero. This can only occur at a throat, a section 

where the walls of the nozzle are parallel. If there 

is a slope to the walls dA will not equal zero. Another 

item to note about this equation is that, assuming 

the pressure ratio is great enough to produce 

sonic conditions, the velocity of the flow will 

increase as the flow approaches the throat of the 

nozzle, because dA is less than zero. As the flow 

passes through the throat it will reach Mach 1. For 

the flow to continue accelerating there must be a 

section of the nozzle where dA is greater than zero. 

In a converging nozzle there is no section where dA 

is greater than zero. The nozzle terminates at the 

throat and immediately thereafter dA goes to infinity. 

The flow can not react to such an abrupt event 

and simply remains at a value of Mach 1. 

The Mach number of a flow, is the ratio of the velocity 

of the flow, V , divided by the speed of sound, 

a. Written algebraically: 

Ma = V a 

(2) 

There are five commonly used classes of flows, each 

corresponding to a range of Mach numbers: 

Mach # 

Classification 

0.0 < Ma < 0.3 Incompressible Flow 

0.3 < Ma < 0.8 Subsonic Flow 

0.8 < Ma < 1.2 Transonic Flow 

1.2 < Ma < 3.0 Supersonic Flow 

3.0 < Ma < ∞ Hypersonic Flow 

This experiment will not study any flows in the hypersonic 

range. We will briefly investigate a flows 

1


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. 

2

Experiment Apparatus: 

These tests will be conducted using the Hilton F790 

Nozzle Performance Test Unit, which is located in 

the undergraduate lab. This apparatus, see Figure 

2, is essentially a fluid flow loop with suitable instrumentation 

for measurement of the mass flow rate, 

ṁ, static thrust, F , and the exit velocity of the air 

being discharged from a nozzle. The rig includes a 

regulator and an inlet control valve to control the 

total pressure, P 1 , upstream of the nozzle, and a 

chamber pressure valve to control the back pressure 

in the manner to which the nozzle discharges. 

By suitable adjustments of the latter, it is possible 

to set the back pressure, P 2 , to any value in the 

range P a ≤ P 2 ≤ P 1 , where P a is the atmospheric 

pressure. There are 5 interchangeable nozzles, see 

Figure 1, supplied with the F790 unit, all of which 

are axisymmetric with a minimum (or “throat”) diameter 

of d = 2.02mm. You will be testing nozzle 

no. 1 and nozzle no. 3. The first is a converging 

nozzle and the second is a converging-diverging nozzle 

with an exit area/throat area = 1.4. Each one 

is identified with a number stamped on its outside 

cylindrical surface. 

Stated specifically, the objectives of this experiment 

will be to determine the effect of backpressure 

P 2 upon the mass flow rate and the static thrust. 

This will be done for both nozzles at constant P 1 . 

To set up the unit for the measurements of nozzle 

fluid and thrust. Please proceed as follows: 

1. Close the air inlet valve and make sure the rig 

is not pressurized. 

2. Fully lower the contacts by rotating the micrometer 

screw. 

3. Unscrew the nuts securing the flange at the lefthand 

end of the chamber and withdraw the cantilever. 

4. Unscrew the impact head from the cantilever 

if it is attached, and fit the knurled nozzle 

adapter in its place. Make sure that the 

o-ring at the base of the nozzle threads 

is in place. 

5. Screw nozzle no. 1 into the adapter. 

6. Reassemble cantilever into chamber. Make 

sure that the o-ring on the flange is in 

place. 

7. Zero the micrometer dial so that for zero thrust 

(i.e. no flow through the nozzle) you obtain a 

zero dial reading. To do this, switch on the contact 

circuit, then rotate the micrometer screw 

until contact is just made. This is indicated by 


the voltmeter and lamp. The greatest sensitivity 

is achieved when the micrometer is adjusted 

until the voltmeter indicates approximately 0.5 

V 

8. The indicating dial may now be set to zero by 

loosening the clamping screw and rotating the 

dial on its spindle. 

9. Recheck the zero setting it should be repeatable 

within 0.5 of a division on the dial. If not, 

clean the contacts. 

10. Use the deflector to plug the hole in the right 

hand end of the chamber. Secure it with the 

knurled nut. 

11. Turn the diverter valve to the left 

The unit is now ready for the actual measurements 

of fluid and thrust. The following procedures 

should be followed: 

1. Adjust the air inlet valve and regulator to give 

a constant P 1 of between 500 and 900 kN/m2 

gage, with the chamber pressure control valve 

fully opened. 

2. Record the pressures P 1 and P 2 , the inlet temperature, 

T 1 , the chamber temperature, and 

the mass flow rate indicated by the rotameter. 

3. Rotate the micrometer adjustment until contact 

is made and the voltmeter reads 0.5V. Note 

the corresponding micrometer dial reading. 

4. Increase P 2 by about 80 kN/m 2 and repeat 

steps 2 and 3. Make sure you keep P 1 constant 

throughout. 

5. Take readings for each such P 2 until you finally 

reach P 2 = P 1 . 

When you reach step 5 you have finished taking 

data for nozzle no.1. Now close the inlet valve and 

open the chamber valve. When the chamber has 

been fully is discharged, replace nozzle no. 1 by 

nozzle no. 3. Take the same set of measurements 

(as per steps 1-5 above) for nozzle no. 3, making 

sure you use the same P 1 . 

# Type A exit /A throat P exit /P inlet 

1 Convergent 1.0 1.0 – 0.528 

2 Conv. – Div. 1.2 0.260 

3 Conv. – Div. 1.4 0.185 

4 Conv. – Div. 1.6 0.140 

5 Conv. – Div. 2.0 0.095 

Table 1: Nozzle geometries 

3


1 2 3 4 5 

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10 O 

Thrust Force [ N ] 

4.5 

4 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

Calibration Data 

y = 0.044185x - 0.033807 

-0.5 

0 10 20 30 40 50 60 70 80 90 100 

Micrometer Reading 

Figure 1: Nozzle Geometries 

Figure 3: Micrometer calibration curve. 

A rotameter is used to measure the mass flow 

rate, ṁ, through the nozzle. The scale on the rotameter 

is incremented in millimeters. The calibration 

curve is used to convert from millimeters to the 

mass flow rate. The calibration curve is shown in 

Figure 4. Make sure you apply the density correction 

factor k that can be obtained from Figure 5 or 

Equation 17. 

9 

8 

Calibration Data 

y = 0.8895 + 0.0292x + 2.34e-5x 2 

7 

Figure 2: Apparatus for Nozzle Tests 

Instrumentation 

The thrust force is measured using what is effectively 

a beam type load cell. The nozzle is threaded 

into the end of a long section of tubing. The thrust 

force produced as a result of mass being thrown 

from the nozzle, causes the beam to deflect downward. 

From our strengths of materials studies, we 

know that a cantilever beam with concentrated load 

can be described as: 

y = F l3 

3EI 

(16) 

The modulus of elasticity, E, the length of the 

beam, l, and the moment of inertia, I, are all 

constant which indicates that the deflection of the 

beam, y should vary linearly with the thrust force, 

F . Figure 3 demonstrates this and provides an 

equation relating the displacement to thrust force. 

Air Flow Rate / 10 - 3 kg ⋅ s - 1 

6 

5 

4 

3 

2 

1 

0 

0 20 40 60 80 100 120 140 160 180 200 220 240 

Scale [ mm ] 

Figure 4: Calibration curve for 35E rotameter with 

duralumin float. Note: Curve is correct for ρ = 

1.2kg · m −2 . Multiply mass flow rate by correction 

factor, k, from Figure 5. 

The rotameter calibration correction factor, k, 

can be extracted from Figure 5. To obtain a reasonable 

estimate of the correction factor from the plot, 

the atmospheric pressure, P a and ambient temperature, 

T , must be measured. The atmospheric pressure 

can be measured using the mercury barometer 

located on the wall near the safety goggle cabinet. 

4


With these readings, you can locate the appropriate 

pressure on the horizontal axis, and then you 

should be able to linearly interpolate between the 

temperature curves. Equation 17 has been provided 

to reduce the effort associated with determining the 

correction factor. 

Rotameter Correction Factor, k 

1.03 

1.02 

1.01 

1 

0.99 

0.98 

0.97 

10 o C - Air Temp. 




0.96 

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 

Atmospheric Pressure / 100 kN ⋅ m - 2 

Figure 5: Correction factor, k, for 35E rotameter. 

( 2∑ 

k ≈ m i T i) ( 2∑ 

· P a + b i T i) (17) 

i=0 

i=0 

i m i b i 

0 0.52475 0.50301 

1 -0.00665 0.00499 

2 0.00015 -0.00015 

Tasks: 

Now the remainder of your job is to suitably reduce, 

display, and interpret the experimental data. 

1. Plot the measured flow rate, ṁ (gm/s) vs. the 

pressure ratio P 2 / P 1 , making sure that P 2 and 

P 1 are both expressed as absolute pressures. 

or your model be modified to improve the 

agreement? 

(d) Indicate on your graph the ranges of P 2 / 

P 1 over which the two nozzles are choked. 

2. Plot the measured static thrust, F , vs. the 

pressure ratio P 2 / P 1 . 

(a) Plot data from both nozzles on the same 

graph using the same symbols as you did 

on the graph of mass flow rate. 

(b) Plot (as a solid line) the F that is predicted 

by one-dimensional isentropic flow 

theory as a function P 2 / P 1 for nozzle no. 

1. 

(c) Does the theoretical curve agree well with 

your test data for nozzle no. 1? If not, 

why not? 

3. The flow in a converging-diverging nozzle is 

fairly complex. It may, depending upon the 

value of P 2 / P 1 , include normal shock waves 

in the diverging section or complex supersonic 

phenomena which occurs beyond its exit plane. 

In his text, White indicates that it is desirable 

to operate near the so-called design point of the 

nozzle. Calculate the exit plane Mach number, 

Ma, and the static thrust, F , at the design 

point and compare it the value of F that was 

measured. 

References: 

White, F.M., (1979), Fluid Mechanics, 2 n d Edition, 

McGraw-Hill, Inc., New York, New York, USA, 

Chapter 9 

(a) Plot the data you measured for both nozzles 

on the same graph, identifying the 

data points for nozzle no. 1 with one symbol 

and those for nozzle no. 3 with another 

symbol. Use errorbars to indicate 

the uncertainty in your measurements. 

(b) Plot (as a solid line) the theoretical curve 

of ṁ vs. P 2 / P 1 for a converging nozzle 

(like nozzle no.1) which may be predicted 

assuming one-dimensional isentropic flow. 

(c) Does the theoretical curve agree well with 

your test data for nozzle no. 1? If not, 

why not and how could the experiment 

5

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

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