Heat Chap05043.docx

Heat Chap05043.docx

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HeatChap05043

Two-DimensionalSteadyHeatConduction

5-43CForamediuminwhichthefinitedifferenceformulationofageneralinteriornodeisgiveninitssimplestformas

:

（a）Heattransferissteady,（b）heattransferistwo-dimensional,（c）thereisheatgenerationinthemedium,（d）thenodalspacingisconstant,and（e）thethermalconductivityofthemediumisconstant.

5-44CForamediuminwhichthefinitedifferenceformulationofageneralinteriornodeisgiveninitssimplestformas

:

（a）Heattransferissteady,（b）heattransferistwo-dimensional,（c）thereisnoheatgenerationinthemedium,（d）thenodalspacingisconstant,and（e）thethermalconductivityofthemediumisconstant.

5-45CAregionthatcannotbefilledwithsimplevolumeelementssuchasstripsforaplanewall,andrectangularelementsfortwo-dimensionalconductionissaidtohaveirregularboundaries.Apracticalwayofdealingwithsuchgeometriesinthefinitedifferencemethodistoreplacetheelementsborderingtheirregulargeometrybyaseriesofsimplevolumeelements.

5-46Alongsolidbodyissubjectedtosteadytwo-dimensionalheattransfer.Theunknownnodaltemperaturesandtherateofheatlossfromthebottomsurfacethrougha1-mlongsectionaretobedetermined.

Assumptions1Heattransferthroughthebodyisgiventobesteadyandtwo-dimensional.2Heatisgenerateduniformlyinthebody.3Radiationheattransferisnegligible.

PropertiesThethermalconductivityisgiventobek=45W/m°C.

AnalysisThenodalspacingisgiventobex=x=l=0.05m,andthegeneralfinitedifferenceformofaninteriornodeforsteadytwo-dimensionalheatconductionisexpressedas

where

Thefinitedifferenceequationsforboundarynodesareobtainedbyapplyinganenergybalanceonthevolumeelementsandtakingthedirectionofallheattransferstobetowardsthenodeunderconsideration:

where

Substituting,T1=280.9°C,T2=397.1°C,T3=330.8°C,

（b）Therateofheatlossfromthebottomsurfacethrougha1-mlongsectionis

5-47Alongsolidbodyissubjectedtosteadytwo-dimensionalheattransfer.Theunknownnodaltemperaturesaretobedetermined.

Assumptions1Heattransferthroughthebodyisgiventobesteadyandtwo-dimensional.2Thereisnoheatgenerationinthebody.

PropertiesThethermalconductivityisgiventobek=45W/m°C.

AnalysisThenodalspacingisgiventobex=x=l=0.01m,andthegeneralfinitedifferenceformofaninteriornodeforsteadytwo-dimensionalheatconductionforthecaseofnoheatgenerationisexpressedas

Thereissymmetryaboutthehorizontal,vertical,anddiagonallinespassingthroughthemidpoint,andthusweneedtoconsideronly1/8thoftheregion.Then,

Therefore,therearethereareonly3unknownnodaltemperatures,

andthusweneedonly3equationstodeterminethemuniquely.Also,wecanreplacethesymmetrylinesbyinsulationandutilizethemirror-imageconceptwhenwritingthefinitedifferenceequationsfortheinteriornodes.

Solvingtheequationsabovesimultaneouslygives

DiscussionNotethattakingadvantageofsymmetrysimplifiedtheproblemgreatly.

5-48Alongsolidbodyissubjectedtosteadytwo-dimensionalheattransfer.Theunknownnodaltemperaturesaretobedetermined.

Assumptions1Heattransferthroughthebodyisgiventobesteadyandtwo-dimensional.2Thereisnoheatgenerationinthebody.

PropertiesThethermalconductivityisgiventobek=20W/m°C.

AnalysisThenodalspacingisgiventobex=x=l=0.02m,andthegeneralfinitedifferenceformofaninteriornodeforsteadytwo-dimensionalheatconductionforthecaseofnoheatgenerationisexpressedas

（a）Thereissymmetryabouttheinsulatedsurfacesaswellasaboutthediagonalline.Therefore

and

aretheonly3unknownnodaltemperatures.Thusweneedonly3equationstodeterminethemuniquely.Also,wecanreplacethesymmetrylinesbyinsulationandutilizethemirror-imageconceptwhenwritingthefinitedifferenceequationsfortheinteriornodes.

Also,

Solvingtheequationsabovesimultaneouslygives

（b）Thereissymmetryabouttheinsulatedsurfaceaswellasthediagonalline.Replacingthesymmetrylinesbyinsulation,andutilizingthemirror-imageconcept,thefinitedifferenceequationsfortheinteriornodescanbewrittenas

Solvingtheequationsabovesimultaneouslygives

DiscussionNotethattakingadvantageofsymmetrysimplifiedtheproblemgreatly.

5-49Startingwithanenergybalanceonavolumeelement,thesteadytwo-dimensionalfinitedifferenceequationforageneralinteriornodeinrectangularcoordinatesforT（x,y）forthecaseofvariablethermalconductivityanduniformheatgenerationistobeobtained.

AnalysisWeconsideravolumeelementofsize

centeredaboutageneralinteriornode（m,n）inaregioninwhichheatisgeneratedataconstantrateof

andthethermalconductivitykisvariable（seeFig.5-24inthetext）.Assumingthedirectionofheatconductiontobetowardsthenodeunderconsiderationatallsurfaces,theenergybalanceonthevolumeelementcanbeexpressedas

forthesteadycase.Againassumingthetemperaturesbetweentheadjacentnodestovarylinearlyandnotingthattheheattransferareais

inthexdirectionand

intheydirection,theenergybalancerelationabovebecomes

Dividingeachtermby

andsimplifyinggives

Forasquaremeshwithx=y=l,andtherelationabovesimplifiesto

Itcanalsobeexpressedinthefollowingeasy-to-rememberform:

5-50Alongsolidbodyissubjectedtosteadytwo-dimensionalheattransfer.Theunknownnodaltemperaturesandtherateofheatlossfromthetopsurfacearetobedetermined.

Assumptions1Heattransferthroughthebodyisgiventobesteadyandtwo-dimensional.2Heatisgenerateduniformlyinthebody.

PropertiesThethermalconductivityisgiventobek=180W/m°C.

Analysis（a）Thenodalspacingisgiventobex=x=l=0.1m,andthegeneralfinitedifferenceformofaninteriornodeequationforsteadytwo-dimensionalheatconductionforthecaseofconstantheatgenerationisexpressedas

Thereissymmetryaboutaverticallinepassingthroughthemiddleoftheregion,andthusweneedtoconsideronlyhalfoftheregion.Then,

Therefore,therearethereareonly2unknownnodaltemperatures,T1andT3,andthusweneedonly2equationstodeterminethemuniquely.Also,wecanreplacethesymmetrylinesbyinsulationandutilizethemirror-imageconceptwhenwritingthefinitedifferenceequationsfortheinteriornodes.

Notingthat

andsubstituting,

Thesolutionoftheabovesystemis

（b）Thetotalrateofheattransferfromthetopsurface

canbedeterminedfromanenergybalanceonavolumeelementatthetopsurfacewhoseheightisl/2,length0.3m,anddepth1m:

5-51"!

PROBLEM5-51"

"GIVEN"

k=180"[W/m-C],parametertobevaried"

g_dot=1E7"[W/m^3],parametertobevaried"

DELTAx=0.10"[m]"

DELTAy=0.10"[m]"

d=1"[m],depth"

"Temperaturesattheselectednodesarealsogiveninthefigure"

"ANALYSIS"

"（a）"

l=DELTAx

T_1=T_2"duetosymmetry"

T_3=T_4"duetosymmetry"

"Usingthefinitedifferencemethod,thetwoequationsforthetwounknowntemperaturesaredeterminedtobe"

120+120+T_2+T_3-4*T_1+（g_dot*l^2）/k=0

150+200+T_1+T_4-4*T_3+（g_dot*l^2）/k=0

"（b）"

"Therateofheatlossfromthetopsurfacecanbedeterminedfromanenergybalanceonavolumeelementwhoseheightisl/2,length3*l,anddepthd=1m"

-Q_dot_top+g_dot*（3*l*d*l/2）+2*（k*（l*d）/2*（120-100）/l+k*l*d*（T_1-100）/l）=0

k[W/m.C]

T1[C]

T3[C]

Qtop[W]

10

5134

5161

250875

30.53

1772

1799

252671

51.05

1113

1141

254467

71.58

832.3

859.8

256263

92.11

676.6

704.1

258059

112.6

577.7

605.2

259855

133.2

509.2

536.7

261651

153.7

459.1

486.6

263447

174.2

420.8

448.3

265243

194.7

390.5

418

267039

215.3

366

393.5

268836

235.8

345.8

373.3

270632

256.3

328.8

356.3

272428

276.8

314.4

341.9

274224

297.4

301.9

329.4

276020

317.9

291

318.5

277816

338.4

281.5

309

279612

358.9

273

300.5

281408

379.5

265.5

293

283204

400

258.8

286.3

285000

g[W/m3]

T1[C]

T3[C]

Qtop[W]

100000

136.5

164

18250

5.358E+06

282.6

310.1

149697

1.061E+07

428.6

456.1

281145

1.587E+07

574.7

602.2

412592

2.113E+07

720.7

748.2

544039

2.639E+07

866.8

894.3

675487

3.165E+07

1013

1040

806934

3.691E+07

1159

1186

938382

4.216E+07

1305

1332

1.070E+06

4.742E+07

1451

1479

1.201E+06

5.268E+07

1597

1625

1.333E+06

5.794E+07

1743

1771

1.464E+06

6.319E+07

1889

1917

1.596E+06

6.845E+07

2035

2063

1.727E+06

7.371E+07

2181

2209

1.859E+06

7.897E+07

2327

2355

1.990E+06

8.423E+07

2473

2501

2.121E+06

8.948E+07

2619

2647

2.253E+06

9.474E+07

2765

2793

2.384E+06

1.000E+08

2912

2939

2.516E+06

5-52Alongsolidbodyissubjectedtosteadytwo-dimensionalheattransfer.Theunknownnodaltemperaturesaretobedetermined.

Assumptions1Heattransferthroughthebodyisgiventobesteadyandtwo-dimensional.2Thereisnoheatgenerationinthebody.

PropertiesThethermalconductivityisgiventobek=20W/m°C.

AnalysisThenodalspacingisgiventobex=x=l=0.01m,andthegeneralfinitedifferenceformofaninteriornodeforsteadytwo-dimensionalheatconductionforthecaseofnoheatgenerationisexpressedas

（a）Thereissymmetryaboutaverticallinepassingthroughthenodes1and3.Therefore,

and

aretheonly4unknownnodaltemperatures,andthusweneedonly4equationstodeterminethemuniquely.Also,wecanreplacethesymmetrylinesbyinsulationandutilizethemirror-imageconceptwhenwriting