COMPRESSIBLE FLOW IN A CONVERGENT-DIVERGENT NOZZLE

Lab Report

Name: Neville Lawless

Lecturer: M.J. O' Rourke

Demonstrator: Aidan Breen

Student No: 06523587

Date: 2/11/09

Abstract

This aim of this practical was to investigate compressible flow in a convergent-divergent nozzle. Theoretical calculations were done to find the maximum mass flow rate ( = 0.1186 kg/s) and thisṁ was compared against actual recorded values ( = 0.1102 kg/s.) . Some of the factors that causesṁ different flow patterns that influence the results of the investigation are also explored.

The different pressure distributions that occur at varying lengths in the nozzle were also recorded and analysed.

Introduction Converging-Diverging or " De Laval" Nozzles have been widely used over the last few decades in many engineering contexts, from civil and mechanical to aerospace uses. They are designed to accelerate fluids to supersonic speeds at the nozzle exit. Nearly all rockets make use of this fact to create an effective propulsion system to reach high velocities.

Whilst their operation is of a simple appearance, the combination of flows reached, be that subsonic and supersonic and the subsequent change of properties such as pressure, density and temperature make the underlying investigation of their performance more complicated than first expected.

Their operation relies on the ratio between the inlet stagnation pressure P0 and outlet back pressure Pb. As this ratio is brought down from unity, the mass flow rate increases till a

maximum value is achieved where the Mach number in the throat (see Figure 1) becomes sonic. ( Mach no = 1). This is referred to as "Choked flow". As the ratio is further decreased the flow becomes supersonic in the diverging nozzle, till the design pressure ratio is achieved (0.53 for air) with supersonic flow occurring at the nozzle exit. After this the flow becomes more complicated and normal and oblique shock waves begin to occur inside and outside of the nozzle respectively.

The Purpose of this report is to gain an understanding of the nature of this flow by investigating the pressure ratios effects on the mass flow rate of air through the system and the differing pressure distributions that occur at varying lengths into the nozzle.

Illustration 1: Cross section of De Laval Nozzle

Experimental ProcedureApparatus: Figure 2 below shows the main components used during the lab.

• The air input shown is fed with a high capacity positive displacement air compressor. The use of this results in fluctuations due to its cyclic load/unload cycle, to counter these effects and their distorting of readings, it is necessary to include an adjustable pressure reducing pressure valve.

• Plenum: Holds air at stagnation pressure and temp.• Inlet valve: Controls regulation of air flow into Plenum.• Throttling valve: Controls nozzle back pressure P0

• Thermometer: Measures temp within Plenum.• Pressure probe: Measures pressure along varying lengths in the nozzle• Orifice plate and manometer: Used to measure the mass flow rate of air.

Experimental Methodology: Part 1: Determination of mass flow rate as a function of the applied pressure ratio

1. The atmospheric pressure was recorded using a mercury barometer located near the apparatus

2. The throttle valve downstream of the nozzle was closed

3. The inlet valve to the plenum was adjusted so that the stagnation pressure P0 equalled 400 k Pa.

4. With the throttle valve closed the back pressure read 400 k Pa, the pressure drop across the orifice plate was 0 k Pa and the measured mass flow rate was zero.

Illustration 2: Apparatus cross section

5. The throttle valve was opened slowly until the back pressure was 350 k Pa. The plenum pressure was maintained at 400 k Pa,

6. The pressure drop Δp across the orifice plate, the static temperature Tc downstream of the

orifice plate were recorded in table 1. The mass flow rate was then calculated using EQ 6.

7. The back pressure was reduced by 50 k Pa and the measurements made in step 6 above were repeated.

8. The mass flow rate was plotted as a function of the pressure ratio p b / P o in Graph 1.

9. The maximum theoretical mass flow rate was then calculated using isentropic duct theory.

Part 2: Determination of axial pressure distribution within nozzle over a range of applied pressure ratio.

1. The throttle valve downstream of the nozzle was closed.

2. The inlet valve to the plenum was adjusted so that the stagnation pressure P0 equals 400 k Pa.

3. The back pressure read 400 Pa with the throttle valve closed.

4. With plenum pressure maintained at 400 k Pa, the throttle valve was opened until the back

pressure was 350 k Pa.

5. The nozzle static pressure at the axial locations extending from upstream of the nozzle to

downstream of the nozzle exit were measured and entered into Table 2.

6. The back pressure was reduced by 50 k Pa and the measurements made in step 6 above were

repeated and recorded in Table 2.

7. Each pressure ratio was plotted against the axial distribution in the static pressure in Graph 2.

8. The design pressure ratio was determined.

Results:Recorded experimental results:

Theoretical Maximum Mass flow rate for isentropic flow calculations:

Ratio of local static to stagnation pressure:

Equation 1

where γ is the ratio of specific heats.In this case air is the only fluid being examined so this is set at γ = 1.4.And setting the Mach number to 1 to achieve desired pressure ratio at the nozzle exit yields:

Equation 2

The theoretical mass flow rate through a duct may be determined from the following equation:

Equation 3

where T (K) and T 0 (K) are the static and total temperatures respectively and R is the Gas Constant for air (R = 287.1 J/kg K).

The static to stagnation temperature ratio may be determined using:

Equation 4

Then substitution Eq 2 and 4 into Eq 3 results in:

Equation 5

Substituting previously recorded values above give us our theoretical maximum mass flow rate of:= 0.1186 kg/sṁ

Calculated mass flow rates for differing pressure ratios:

Equation 6

where the discharge coefficient Cd = 0.62, the orifice area Ao = 0.00787 m2 (orifice diameter 50.05 mm).

Graph 1: Mass Flow Rate v Pressure Ratio

Illustration 3: Measurement of mass flow rate within converging-diverging nozzle

P0 plemum T0 Plenum Pb Delta p Tc Po plenum abs pb abs Pb/P0 ṁ400000 290 400000 0 290 502000 502000 1 0400000 290 350000 75 290 502000 452000 0.88 0.08400000 290 300000 130 290 502000 402000 0.7500 0.1062400000 290 250000 140 290 502000 352000 0.6250 0.1102400000 290 200000 130 290 502000 302000 0.5000 0.1062400000 290 150000 130 290 502000 252000 0.3750 0.1062400000 290 100000 130 290 502000 202000 0.2500 0.1062400000 290 50000 140 290 502000 152000 0.1250 0.1102400000 290 0 130 290 502000 102000 0.0000 0.1062

Axial pressure distribution within nozzle over a range of applied pressure ratios:

Table 1: Axial Pressure distibution within converging-diverging nozzle

Applied Pressure Ratio 1 0.875 0.75 0.625 0.5 0.375 0.25 0.125 0Measurement Location 400 350 300 250 200 150 100 50 0

1 400 400 400 400 400 400 400 400 4003 400 400 400 400 400 400 400 400 4005 400 400 400 400 400 400 400 400 4007 400 400 400 400 400 400 400 400 4009 400 390 390 390 390 390 390 390 390

11 400 340 340 290 290 290 290 290 29013 400 325 325 180 180 180 180 180 18015 400 325 325 130 130 130 130 130 13017 400 325 325 110 110 110 110 110 11019 400 325 325 90 90 90 90 90 9021 400 330 330 180 80 80 80 80 8023 400 330 330 200 70 70 70 70 7025 400 330 330 220 60 60 60 60 6027 400 340 340 230 140 50 50 50 5029 400 350 350 240 160 50 40 40 3031 400 350 350 240 180 50 40 40 2033 400 350 350 250 200 120 100 50 035 400 350 350 250 200 140 100 50 037 400 350 350 250 200 150 100 50 039 400 350 350 250 200 150 100 50 0

Graph 2: Axial Pressure Distribution

0 5 10 15 20 25 30 35 40 450

50

100

150

200

250

300

350

400

450

Measurement Location Vs Pressure Ratio

10.8750.750.6250.50.3750.250.1250

Measurement Location

Appl

ied

Pre

ssur

e R

atio

Design condition pressure ratio

As can be seen on graph 2 above the design pressure ratio lies between the ratios 0 and 0.125.By inspection a value of 0.0625 would be the best fit line and so that is the design pressure ratio.

Discussion:

• Comparison of expected results with those obtained.

◦ The first part of this lab was to investigate the mass flow rates that were obtained from different pressure ratios by using the Converging-Diverging nozzle.From our initial calculations using equations 1-5 we resulted with a theoretical value of

= 0.1186 kg/s.ṁ

On continuation of the experiment, and completion of table 1 using Eq 6, to calculate our actual mass flow rates it can be seen in table 1, the Max mass flow rate achieved is 0.1102 kg/s.This is a very desirable result as it it only differs by 7% of the Max theoretical value.

However, on plotting out mass flow rate versus the equivalent pressure ratio and using the value of 0.528 obtained in Eq 2 it is found that the corresponding mass flow rate is0.1062 kg/s which is slightly different to our actual maximum.It should be found that the maximum mass flow rate should occur at the design pressure ratio of 0.528.

The discrepancies that bring about these differing results will be further discussed.

◦ The second objective of this lab was to measure the Axial pressure distribution within the nozzle over a range of applied pressure ratios.

In the De Laval nozzle, the isentropic expansion of the fluid to supersonic flow is dependent on the pressure ratios applied to the system.

By measuring the linear variation of pressure at different lengths through the nozzle it can be determined from existing literature what type of flow is occurring. These flow patterns can be seen over in figure 3.During the laboratory, these values were recorded in table 2 and plotted in graph 2.To achieve a flow pattern which can be attributed to the design condition, ( supersonic flow at the nozzle exit which contains no shock waves and is not chocked at the throat) the plotted line must decrease smoothly to a value above the unexpanded flow condition.On examining the resulting graph 2 it is seen that we achieve no value consistent with the design condition. This can be attributed to the pressure drop, step size of 50 k Pa that was used in the lab. If this was reduced to a value of perhaps 20-25kPa the design condition would have resulted.Using a best fit line, ( thick red line on graph 2 ) it is seen that the pressure ratio lies between 0 and 0.125. A value of 0.0625 was felt by the group would suffice.

• Analysis of experimental error.◦ It was felt after the conclusion of the lab, that a number of factors could have caused

discrepancies between the analytical and the theoretical results.◦ The biggest contributor to these, it was felt, was the positive displacement compressor.

The reason for this is the fluctuations that occur because of its method of operating in load and unload cycles. Even with the pressure regulator and plenum being incorporated to the system to smooth out the fluctuations there is still level of inherit error present. A very high level of maintenance and calibration would be necessary to reduce these to a certain extent.

◦ Next the lab demonstrator noted us to the fact that there was possible leaks in the back of the rig which can cause deviations in the actual results obtained for which we could not correct

◦ There possibly may have been errors that went unnoticed in the lab with pressure gauges which could account for a small level of error.

◦ It has to be mentioned that there could have been possible meniscus errors made by students when reading the inverted manometer (although these are unnecessary mistakes they still need to be mentioned)

Illustration 4: Different flow patters for different pressure ratios

Conclusion:

Following this laboratory it is evident that the analysis of a simple device like the converging diverging De Laval nozzle is more involved than was originally anticipated.Using the methods set out above, the challenges involved with designing such nozzles, for the hugely stressful environments that occur in such applications varying from aerospace to civil engineering uses is made very apparent.

For the results obtained for the maximum mass flow rates,it is felt that the theoretical values hold very closely to the analytical ones and so could be used safely for non critical approximations if desired.

The second part of the lab, measuring the pressure distribution along the axis of the nozzle as a function of the pressure ratio gave a good indication of the flow patterns that occur in the nozzle and how they can be used to achieve desired flows if necessary.

References: Engineering applets William J. Devenport [Accessed 8th Nov 2009]. Available from World Wide Web: < http://www.engapplets.vt.edu/fluids/CDnozzle/cdinfo.html >

Dr. Malachy J O' Rourke Lecture notes MEEN40020: Mechanics of Fluids II 2009

Bibliography:

http://en.wikipedia.org/wiki/Choked_flow

Journal of Fluid Mechanics (1969), 35:3:599-608 Cambridge University PressCopyright © 1969 Cambridge University Press