Newton Maharjan 02.docx
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Newton Maharjan 02.docx
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NewtonMaharjan02
Finiteelementanalysis(Thepipeflowproblem)
AssignmentII
April1,2018
Newtonmaharjan
StudentID:
-1793364
Contents
∙SpecificProblemProgramWire
1.ProblemStatement
2.GoverningEquations
3.NumericalMethods
4.Verification
5.NumericalExperiment
∙Results&Discussions
∙Conclusion
ThePipeFlowProblem
Thestiffnessmatrixis:
whereυisvelocity,[m/s]
ρisdensityofthefluid,[kg/m3]
kisthermalconductivity,[Nm/sm°C]
Cpisspecificheat,[Nm/kg°C]
Leislengthoftheelement,[m]
k/Le[N/sm°C],ρCpυ[N/sm°C]
PartA:
Verification
1.Problemstatement
Determineadifferentialequationwithnon-zerocoefficientsA,B,C,andDforwhichthesolutionisalinearfunctionofx(i.e.y=C0+C1x.DefinethreeboundaryvalueproblemsforthisODE:
(1)yknownatbothboundaries.
(2)yknownatleftendboundaryanddy/dxknownatrightendboundary.
(3)dy/dxknownatleftendboundaryandyknownatrightendboundary.Useprogramode2.mtosolvethesethreeproblems.
2.OrdinaryDifferentialEquations
(1)
withtheboundaryconditions:
(2)WewanttofindanequationanditssolutionanduseittoverifytheprogramODE.Thus,wechoosey=3x-8asasolution:
Let
.
Substitutionofthesevaluesaboveintotheordinarydifferentialequationleadsto
Thedifferentialequationis:
(3)InordertosolvethesethreeproblemsfirstweshalldefinevaluesinCOEF.masfollows:
yknownatbothboundaries:
%==============================================
%COEF.mincludefile
%
%Coefficientsforode2.m
%==============================================
AX=Xg^2;
BX=4;
CX=10-30*Xg;
DX=90*Xg^2-270*Xg+68;
Inputdata:
%MESHdatafile
%---------------------------------------------
%NUMNP+3dummynumbers
%---------------------------------------------
5000
%---------------------------------------------
%XORDNPBCUQ
%---------------------------------------------
01-80
2000
4000
6000
81160
Results:
Figure1yknownatbothboundaries
yknownattheleftendboundaryand
knownattherightendboundary:
Inputdata:
%MESHdatafile
%---------------------------------------------
%NUMNP+3dummynumbers
%---------------------------------------------
5000
%---------------------------------------------
%XORDNPBCUQ
%---------------------------------------------
01-80
2000
4000
6000
8003
Results:
Figure2)yknownattheleftendboundaryanddy/dxknownattherightendboundary
knownattheleftendboundaryandyknownattherightendboundary:
Inputdata:
%MESHdatafile
%---------------------------------------------
%NUMNP+3dummynumbers
%---------------------------------------------
5000
%---------------------------------------------
%XORDNPBCUQ
%---------------------------------------------
0003
2000
4000
6000
81160
Results:
Figure3dy/dxknownattheleftendboundaryandyknownatrightendboundary
PartB:
NumericalExperiment
1.Problemstatement
Itisnecessarytopreventwaterinapipefromfreezingduringthewintermonths.Thepiperunsfromanartesianwelltoacabin2.17kmdownahillside.Theinsulationforthepipeispartlygroundcoverandpartlysomecommercialwrappingmaterial.Afairapproximationtotheinsulationcanbegivenby
wherehisthecoefficientofconvectiveheattransfer,andListhetotallengthofthepipe.Theequationforthetemperatureinthepipecanbedescribedbytheequation
where
isthetemperature
istheambienttemperature=
isthevelocityinthepipe(m/s)
isthedensity=
istheheatcapacity=
kisthethermalconductivity=
histhecoefficientofconvectiveheattransfer
Inordertokeepthewaterfromfreezingitisnecessarytomaintainaveragevelocitywhichwillkeepthewaterfromfreezingbeforeitreachesthecabin?
Thewaterentersthepipefromthewellatatemperatureof20
.Itemptiesintoastoragetankinthecabinwhichcantainswateratapproximatelythetemperatureoftheenteringwater.Assume,therefore,that
atthepoint.Take
2.GoverningEquations
StrongForm
withtheboundaryconditions
and
Weakform
FiniteElementMatrixForm
[K]{Y}-{F}=0
ElementStiffnessMatric
ElementForceVector
GlobalStiffnessMatrix
3.ResultsandDiscussion
Тheordinarydifferentialequationis:
Thus,wecandefinetheconstants:
Theboundaryconditionsare:
Thevalueofthevelocityisnotdefinedyet.Inordertodeterminev(minimumaveragevelocityoftheflowinthepipenecessarytokeepthewaterfromfreezingbeforeitreachesthecabin),weshalllookforatemperature
isbiggerthanzero.
Inputdata:
%MESHdatafile
%---------------------------------------------
%NUMNP+3dummynumbers
%---------------------------------------------
8000
%---------------------------------------------
%XORDNPBCUQ
%---------------------------------------------
01200
310000
620000
930000
1240000
1550000
1860000
2170000
Letassumethatυ=1.5m/s.
Inputdata:
%==============================================
%COEF.mincludefile
%
%Coefficientsforode2.m
%==============================================
AX=0.574;
BX=-4211*1000*1.5;
CX=-4.5*(1.3-Xg/(2170));
DX=-30*4.5*(1.3-Xg/(2170));
Results:
Figure4Temperaturevariation,watervelocityυ=1.5m/s
Letυ=0.15m/s
Inputdata:
%==============================================
%COEF.mincludefile
%
%Coefficientsforode2.m
%==============================================
AX=0.574;
BX=-4211*1000*0.15;
CX=-4.5*(1.3-Xg/(2170));
DX=-30*4.5*(1.3-Xg/(2170));
Results:
Figure5Temperaturevariation,watervelocityυ=0.15m/s
Letυ=0.0015m/s
Inputdata:
%==============================================
%COEF.mincludefile
%
%Coefficientsforode2.m
%==============================================
AX=0.574;
BX=-4211*1000*0.0015;
CX=-4.5*(1.3-Xg/(2170));
DX=-30*4.5*(1.3-Xg/(2170));
Results:
Figure6Temperaturevariation,watervelocityυ=0.0015m/s
Byimputingdifferentvaluesofυwecanobservethatwhenυ=0.0036m/s,thetemperatureisveryclosetozeroandwhenwedecreaseυ=0.0035m/s,thetemperatureisunderzero.Thus,inconclusion,theminimumvalueofυis0.0036m/s
Velocityυ=0.0035m/s
Inputdata:
%==============================================
%COEF.mincludefile
%
%Coefficientsforode2.m
%==============================================
AX=0.574;
BX=-4211*1000*0.0035;
CX=-4.5*(1.3-Xg/(2170));
DX=-30*4.5*(1.3-Xg/(2170));
Results:
Figure7Temperaturevariation,watervelocityυ=0.0035m/s
Velocityυ=0.0036m/s
Inputdata:
%==============================================
%COEF.mincludefile
%
%Coefficientsforode2.m
%==============================================
AX=0.574;
BX=-4211*1000*0.0036;
CX=-4.5*(1.3-Xg/(2170));
DX=-30*4.5*(1.3-Xg/(2170));
Results:
Figure8Temperaturevariation,watervelocityυ=0.0036m/s
Conclusion:
Theminimumaveragevelocityofthewaterinthepipewhichwillpreventthewateroffreezingbeforeitreachesthecabinis0.0036m/s.
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