3. Berger 3
MATHEMATICAL SYMBOLS
R = universal gasconstant
M = molecularmass
T = temperature
P = pressure
Cp = constant pressure specificheat
Cv = constantvolume specificheat
K = ratioof specificheats(Cp/Cv)
FT = thrust
𝑚̇ = massflowrate
v = velocity
NM = Mach number
ρ = density
A = area
a = speedof sound
AR = arearatio
d = diameter
%d = percentdifference
4. Berger 4
ABSTRACT
Nozzle design is of great importance in for many fluid flow applications. Missiles, rockets, jet engines,
hoses,Jacuzzis,spraybottles,andahostof otherdevicesusenozzles.The de Laval nozzle,or“converging-
diverging” (CD) nozzle, is of special importance in high-speed flows – most notably in jet/rocket
propulsion. Analyzing CD nozzles and developing useful relationships has been an important field of
researchinaerospace engineeringformanyyears.
This experimentwill firstanalyze the fluidflow of one smooth-bore CDnozzle andthenanalyze the flow
of several differentnozzlesinsubsonicflow todetermine relationshipsbetweenflow characteristicsand
geometrythatcan be useful inthe manufacture of CD nozzlesforspecificperformancecharacteristics.
5. Berger 5
INTRODUCTION
This experiment will consist of two parts. For one part, a single
nozzle will be given with known geometry and inlet conditions.
Outletconditionswill be calculated,andexperimental datawill be
collectedtocompare to the theoretical values. Figure 2showsthe
general behaviorof flowthrougha CD nozzle. The goal of Part 1 is
to developsome familiaritywithanalyzingsubsonicflowsthrough
a CD nozzle. The second part of the experiment will involve the
comparison of several different nozzles in order to find
relationships between nozzle geometry and flow characteristics.
The nozzles to be tested will have different area ratios with the
lengths of the convergent and divergent sections being held
constant(meaningthatconvergenceanddivergenceangleswillbe
analyzed). Figure 1 shows one application of CD nozzle design –
rocketry. The goal of Part 2 is to define important parameters for
the designandmanufacture of CD nozzles.
Figure 2: Diagram of a CD nozzle showing
approximate flow velocity (v),
temperature (T), and pressure (P), with
Mach number (M) [2].
Figure 1: F-1 rocket engine from 1960 [1]
6. Berger 6
THEORY
Thisexperimentwillbe conductedunderthe followingassumptions:
1. Steady-state,steadyflow conditions.
2. Ideal gas.
a. R = 8.314462 J/mol-K[3].
b. Mair = 28.97 g/mol-K[3].
3. Standardtemperature andpressure.
a. T = 20o
C = 293.15 K [4].
b. Patm = 101.325 kPa [4].
4. Negligiblechangesinpotential energy.
5. Constantspecificheats.
a. For air: Cp = 1.01 kJ/kg-K,Cv = 0.718 kJ/kg-K k = 1.40 [5]
One of the mostimportantperformancecharacteristicsforanozzle isthe outletflow speed.Forexample,
thrustprovidedbya jetengine canbe describedbythe velocitiescomingintoandoutof the nozzle,
𝐹𝑇 = 𝑚̇ ( 𝑣𝑒 − 𝑣𝑖).
By combiningthe Lawof Conservationof Massand the Law of Conservationof momentumforisentropic
flow,the followingequationshowsthe behaviorof CDnozzleswithrespecttoflow velocity[6]:
(1 − 𝑁 𝑀
2 )
𝑑𝑣
𝑣
= −
𝑑𝐴
𝐴
The Mach number, Nm iscalculatedby usingthe speedof sound a:
𝑁 𝑀 =
𝑣
𝑎
These equationsshowthat,for subsonicflows(Nm < 1), an increase inarea (dA > 0) causesa decrease in
flowvelocity,while forsupersonicflows(Nm > 1), an increase inarea causesan increase inflow velocity.
This result is used to great effect in ramjets,scramjets,and rockets to achieve veryhigh speeds [6]. For
thisparticular experiment,onlysubsonicflowsare analyzed.Therefore,the flow speedcanbe expected
to decrease comingoutof the nozzle.
In most situations, the inlet velocity is either known or easily calculated. The exit velocity ve for a fluid
flowing through a nozzle at inlet temperature T, molecular weight M, ratio of specific heats k, inlet
pressure pi,andexitpressure pe can be calculatedusingthe followingequation:
𝑣𝑒 = √
𝑇𝑅
𝑀
∗
2𝑘
𝑘 − 1
∗ [1 − (
𝑝𝑒
𝑝𝑖
)
𝑘−1
𝑘
]
8. Berger 8
FACILITY ANDAPPARATUS
The apparatusneededforthisexperimentare the several CDnozzles,the JetStream500windtunnel,and
the accompanyingdata acquisitionsoftware. The appropriate sensorsmustbe addedtothe windtunnel
to acquire temperature,pressure,andspeedandthe nozzle inletandexit.
Figure 3: JetStream 500 wind tunnel (equipped with the force balance used in Experiment 11).
PROCEDURE
Part1: CD NozzleOutletConditions
A single nozzleisselectedandplacedinthe windtunnel,whichisthenturnedon. A range of inletspeeds
shouldbe tested,anddata acquiredfor temperature,pressure,andflow speedatthe nozzle outlet.This
data is then compared against the calculated theoretical outlet conditions to determine the success of
Part 1.
Part2: Analysis of CD NozzleDesignParameters
The procedure for Part 2 is similar to Part 1. For each nozzle tested, the following procedure should be
followed.The nozzle isplacedintothe windtunnel,whichisthenturnedon. A range of inletspeedsare
usedto generate datafor the outletconditions(asin Part 1). Then,nozzle efficienciescan be calculated
using the efficiency equation in the theory section. Plots should be made of area ratio vs. change in
velocity,divergence angle vs.nozzle efficiency,andoutletspeedvs.inletareaand outletarea (a surface
plot).The R2
valuesof the fittedcurvesinthe firsttwo plotsare usedto determine the successof Part2,
while the surface plot is used to observe the effects of various inlet and outlet areas on nozzle outlet
speed.
9. Berger 9
STATEMENTOF UNCERTAINTY
For both parts of this experiment, the sensors can be assumed to be exact. The sensors produce a
negligibleamountof randomintrinsicerrorinmeasurement.Itistherefore expectedthatPart1will result
in highlyaccurate data. Successfor Part 1 can be declaredif the percentdifference betweentheoretical
and experimental outletconditionsare all lessthan 5%. For Part 2, the anglesand area ratios are varied
across several nozzles.Anerrorof approximately±0.5o
in divergenceangle willpropagate intothe nozzle
outletarea,andagainintothe equationsusedtocalculate outletspeedandnozzle efficiency,ifdivergence
angle isusedtodescribe the nozzle.If arearatioisused,thenanerrorof ±0.5 mm will propagate.Success
inPart 2 can be declaredif the correlationscoefficients(R2
) forall fittedcurvesare atleast0.95. If Parts 1
and 2 are successful,the experimentcanbe declaredasuccess.
CONCLUSION
The goal of this experiment is to first test the quantitative accuracy of the theoretical equations for
calculatingoutletconditionsforflowthroughaCDnozzle.Fromthere,abasisisestablishedforgenerating
accurate experimental nozzleexit conditions.ForPart2, thisbasisisusedtoanalyze several CDnozzles of
equal length and develop relationships between the input variables of area ratio and divergence angle
andthe outputvariablesof outletspeedandnozzle efficiency.A surface plotof outletspeedvs.inletarea
and outlet area can be plotted to see the effects of changes in area on outlet speed for a given nozzle
lengthandinletspeed.While there are few sourcesof errorinthis experimentdue tothe sensors,there
are some quantitative measuredtodetermine success.The experimentcanbe declaredsuccessful if the
percent difference between all theoretical and experimental valuesin Part 1 are less than 5% and if the
R2
valuesforthe fittedcurvesinPart2 are at least0.95.
10. Berger 10
REFERENCES
[1] Hutchinson,Lee.“NewF-1B rocketengine upgradesApollo-eradesignwith1.8Mpoundsof
thrust,”14 April 2009, Conde Nast,ArsTechnica.15 Nov.2015. Available:
<http://arstechnica.com/science/2013/04/new-f-1b-rocket-engine-upgrades-apollo-era-deisgn-
with-1-8m-lbs-of-thrust/>
[2] N.A.“de Laval Nzzle,”21 August2015, Wikipedia.15Nov.2015. Available:<
https://en.wikipedia.org/wiki/De_Laval_nozzle>
[3] N.A.“The Individual andUniversal GasConstants,”The EngineeringToolbox.16Nov.2015.
Available:<http://www.engineeringtoolbox.com/individual-universal-gas-constant-d_588.html>
[4] N.A.“STP – Standard Temperature andPressure &NTP – Normal Temperature andPressure,”
The EngineeringToolbox.16Nov 2015. Available:< http://www.engineeringtoolbox.com/stp-
standard-ntp-normal-air-d_772.html>
[5] N.A.“Gases – SpecificHeatsandIndividualGasConstants,”The EngineeringToolbox.16Nov
2015. Available:<http://www.engineeringtoolbox.com/specific-heat-capacity-gases-
d_159.html>
[6] Hall,Nancy,ed.“Nozzle Design –Converging/Diverging(CD) Nozzle,”5May 2015, National
AeronauticsandSpace Administration.16Nov2015. Available:
<https://www.grc.nasa.gov/www/K-12/airplane/nozzled.html>
[7] Micklow,Gerald.MAE 3161: FluidMechanics – Lecture Notes.June 2014, FloridaInstitute of
Technology.16 Nov2015.
