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-63C,thepressureisabout700Pa.Problem Description (2)
Thedesignedtemperaturedifferencebetweenapparatussurfaceandtheenvironmentis50C,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
kkAnalysis 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)PPAnalysis 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 C9.81TTSincethedensityofgasdecreases
withtemperatureincreasing,hence
Thesurfacetemperatureofmodel
shouldbe
2 634095 =4032 CTAnalysis of plan 2 (3)
Itisobviousthatnomaterialscan
beusedtomakethemodelwhich
canenduresuchhightemperature.2 634095 =4032 CTConclusion
Plan1isfeasible,
butplan2isnot.