Illustration of application of similarity theory

  • 格式:pps
  • 大小:558.50 KB
  • 文档页数:17

An Illustration

Application of Similarity Theory

What is the weather on Mars ?Problem Description (1)

ScientistsdesignedanapparatusofmomentumandheattransportforaMarsdetectortoinvestigatetheatmosphereofMars.Amodelexperimentisrequiredtocheckthedesign.Itisestimatedthat,intheworkingcircumstanceonMars,thetemperatureisabout-63C,thepressureisabout700Pa.Problem Description (2)

Thedesignedtemperaturedifferencebetweenapparatussurfaceandtheenvironmentis50C,sobothofforcedandfreeconvectionarenotnegligible.Basedontheconditionsoftheirlaboratory,thecharacteristiclengthofmodelshouldbe1/6ofthatofrealapparatusandagasmixturesimilartotheMarsatmosphereisusedastheexperimentalfluid.Tokeepthedynamicalandthermalsimilaritybetweenmodelexperimentandestimatedworkingsituation,twoplansareproposed:Problem Description (3)

Plan 1

Withthetemperaturedifference

betweenapparatussurfaceand

environmentinexperimentkeptthe

sameasthatinestimatedworking

situation,thepressureofgasand

thevelocityofflowareadjustedto

makethesimilaritynumbersalike.Problem Description (4)

Plan 2

Withthepressureofgasin

experimentkeptthesameasthatin

estimatedworkingsituation,the

temperaturedifferencebetween

apparatussurfaceandenvironment

andthevelocityofflowareadjusted

tomakethesimilaritynumbersalike.The Task for Us

Analyze and verify

the feasibility of the two plans.

Provide a theoretical basis

for making decision.General Considerations

Becauseforcedandfreeconvection

havethesameorderofmagnitudein

thisproblem,itisrationaltoomitthe

viscousdissipationthatBrcanbe

neglected.Andbecausethereisno

curvedinterfacebetweenimmiscible

fluids,itisnotnecessarytoconsiderFr.Thentherequiredsimilarity

numbersareRe,Pr,Gr.Analysis of plan 1 (1)

1)Sincetheexperimentalgasis

similartotherealgasonMarsandµ,k,Cp,βcanbetakenasconstants

underlowpressureandsmall

temperaturechange,wehave

11221212PrPrppCC

kkAnalysis of plan 1 (2)

2)Bysimilaritycondition,

232311111222221212GrGr gTLgTL



Accordingtotheassumptionin1),

thisidentityreducesto

31123122gL

gL

Analysis of plan 1 (3)

Let“1”beontheearthand“2”theMars,

1

29.811.623.72g

g

1.5322113112211.620.11026PgL

PgL

Assumingthestateequationofidealgasisapplicableunderlowpressure,wehaveAnalysis of plan 1 (4)

Therefore,

Gr1=Gr2canbesatisfiedby

adjustingP1andP1isstillinthe

rangeoflowpressure.Thatis,the

aboveassumptionsarevalid.12/0.1102700/0.20256352 (Pa)PPAnalysis of plan 1 (5)

3)Bysimilaritycondition,

Accordingtotheassumptionin1)

and2),thisidentityreducesto1112221212ReRe vLvL



122

2110.11026=0.6612vL

vL

Analysis of plan 1 (6)

ThenRe1=Re2canbesatisfiedby

adjustingv1andv1hasavalueof

thesameorderofmagnitudeas

thatofv2.

Itisfeasibletosatisfyallthree

identitiesofsimilaritynumbersby

adjustingsomeparametersin

experimentwithinrationalranges.Analysis of plan 2 (1)

Assumingconstantpropertiesvalid

232311111222221212GrGr gTLgTL



2321222232211113.72 2169.81TgL

TgL

Analysis of plan 2 (2)

213.72 > 2160.3792216504095 C9.81TTSincethedensityofgasdecreases

withtemperatureincreasing,hence

Thesurfacetemperatureofmodel

shouldbe

2 634095 =4032 CTAnalysis of plan 2 (3)

Itisobviousthatnomaterialscan

beusedtomakethemodelwhich

canenduresuchhightemperature.2 634095 =4032 CTConclusion

Plan1isfeasible,

butplan2isnot.