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# Fluids Lab

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### Fluids Lab

1. 1. Berger 1 Term Project: Analysis of Converging/Diverging Nozzles MAE 3064: Fluid Mechanics Lab By Thaddeus Berger Submitted: 11/19/2015 Instructor: Muzammil Arshad
2. 2. Berger 2 CONTENTS Figures...............................................................................................................................................2 Mathematical Symbols........................................................................................................................3 Abstract.............................................................................................................................................4 Introduction.......................................................................................................................................5 Theory...............................................................................................................................................6 Facility and Apparatus.........................................................................................................................8 Procedure..........................................................................................................................................8 Part 1: CD Nozzle Outlet Conditions.................................................................................................8 Part 2: Analysis of CD Nozzle DesignParameters...............................................................................8 Statement of Uncertainty....................................................................................................................9 Conclusion .........................................................................................................................................9 References.......................................................................................................................................10 FIGURES Figure 1:.............................................................................................................................................5 Figure 2:.............................................................................................................................................5 Figure 3:.............................................................................................................................................8
3. 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 ๐ฬ = massflowrate v = velocity NM = Mach number ฯ = density A = area a = speedof sound AR = arearatio d = diameter %d = percentdifference
4. 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,orโconverging- divergingโ (CD) nozzle, is of special importance in high-speed flows โ 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. 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 โ 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. 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, ๐น๐ = ๐ฬ ( ๐ฃ๐ โ ๐ฃ๐). By combiningthe Lawof Conservationof Massand the Law of Conservationof momentumforisentropic flow,the followingequationshowsthe behaviorof CDnozzleswithrespecttoflow velocity[6]: (1 โ ๐ ๐ 2 ) ๐๐ฃ ๐ฃ = โ ๐๐ด ๐ด The Mach number, Nm iscalculatedby usingthe speedof sound a: ๐ ๐ = ๐ฃ ๐ 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: ๐ฃ๐ = โ ๐๐ ๐ โ 2๐ ๐ โ 1 โ [1 โ ( ๐๐ ๐๐ ) ๐โ1 ๐ ]
7. 7. Berger 7 In general,the propulsiveefficiencyof anozzle canbe expressedby[7]: ๐ ๐ = ๐๐๐ค๐๐ ๐๐๐๐๐ฃ๐๐๐๐ ๐๐๐ก๐ ๐๐ ๐๐๐๐๐ก๐๐ ๐๐๐๐๐๐ฆ ๐๐๐๐๐ข๐๐ก๐๐๐ Mathematically,thisratiocanbe formulatedas: ๐ ๐ = ๐ฬ ( ๐ฃ๐ โ ๐ฃ๐) ๐ฃ๐ ๐ฬ ( ๐ฃ๐ 2 2 โ ๐ฃ๐ 2 2 ) Thissimplifiesto: ๐ ๐ = 2๐ฃ๐ ๐ฃ๐ + ๐ฃ๐ The area ratio of a CD nozzle iscalculatedby: ๐ด ๐ = ๐๐๐๐ ๐๐ก ๐กโ๐ ๐๐๐ง๐ง๐๐ ๐๐ฅ๐๐ก ๐๐๐๐ ๐ก๐ ๐คโ๐๐โ ๐กโ๐ ๐๐๐๐ค ๐๐๐๐ฃ๐๐๐๐๐  Mathematically, ๐ด ๐ = ๐๐ ๐ 2 4 ๐๐ ๐๐๐๐ฃ 2 4 = ๐ ๐ 2 ๐ ๐๐๐๐ฃ 2 Percentdifference isusedinforthisexperimenttodeterminethesuccessofPart1.The percentdifference betweenthe experimental andtheoretical valuescanbe calculatedby: %๐ = |๐ฃ ๐๐ฅ๐๐๐๐๐๐๐๐ก๐๐ โ ๐ฃ ๐กโ๐๐๐๐๐ก๐ ๐ ๐๐| ๐ฃ ๐กโ๐๐๐๐๐ก๐๐๐๐ โ 100%
8. 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. 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 approximatelyยฑ0.5o in divergenceangle willpropagate intothe nozzle outletarea,andagainintothe equationsusedtocalculate outletspeedandnozzle efficiency,ifdivergence angle isusedtodescribe the nozzle.If arearatioisused,thenanerrorof ยฑ0.5 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. 10. Berger 10 REFERENCES [1] Hutchinson,Lee.โNewF-1B rocketengine upgradesApollo-eradesignwith1.8Mpoundsof thrust,โ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.โde Laval Nzzle,โ21 August2015, Wikipedia.15Nov.2015. Available:< https://en.wikipedia.org/wiki/De_Laval_nozzle> [3] N.A.โThe Individual andUniversal GasConstants,โThe EngineeringToolbox.16Nov.2015. Available:<http://www.engineeringtoolbox.com/individual-universal-gas-constant-d_588.html> [4] N.A.โSTP โ Standard Temperature andPressure &NTP โ Normal Temperature andPressure,โ The EngineeringToolbox.16Nov 2015. Available:< http://www.engineeringtoolbox.com/stp- standard-ntp-normal-air-d_772.html> [5] N.A.โGases โ SpecificHeatsandIndividualGasConstants,โThe EngineeringToolbox.16Nov 2015. Available:<http://www.engineeringtoolbox.com/specific-heat-capacity-gases- d_159.html> [6] Hall,Nancy,ed.โNozzle Design โConverging/Diverging(CD) Nozzle,โ5May 2015, National AeronauticsandSpace Administration.16Nov2015. Available: <https://www.grc.nasa.gov/www/K-12/airplane/nozzled.html> [7] Micklow,Gerald.MAE 3161: FluidMechanics โ Lecture Notes.June 2014, FloridaInstitute of Technology.16 Nov2015.

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