Solution manual for Heat Convection 2nd edition by Latif Jiji

All Chapters Covered
m m




SOLUTION MANUAL
m

, PROBLEMm1.1


Heatmismremovedmfrommamrectangularmsurfacembym conve L
ctionmtomanmambientmfluidmatmTm.m Themheatmtransfermco
efficientmismh.mSurfacemtemperaturemismgivenmby
A 0 x W
Tsm = 1/m2
x

wherem Am ism constant.
Determinem them steadym statem he
atmtransfermratemfrommthemplate.
L
(1) Observations. (i)mHeatm ismremovedmfrommthemsurface
bym convection.m m Therefore,m m Newton'sm m lawm ofm coolingm is dqs
applicable. (ii)m Ambientm temperaturem andm heatm transfer 0 x W
coefficientm arem uniform.m m (iii)m Surfacem temperaturem varies
alongmthemrectangle. dx
(2) ProblemmDefinition.m Findm themtotalmheatm transfermratembymconvectionmfrommthemsurfacem
ofmamplatemwithmam variablemsurfacemareamandmheatmtransfermcoefficient.

(3) SolutionmPlan.m Newton'smlawmofmcoolingmgivesmthemratemofmheatmtransfermbymconvection.m
However,minmthismproblemm surfacemtemperaturemismnotmuniform.m Thismmeansmthatmthemratemof
mheatmtransfermvariesmalongmthemsurface.m Thus,m Newton’msmlawmshouldmbemapplied mto manminfin
itesimalmareamdAsmandmintegratedmovermthementiremsurfacemtomobtainmthemtotalmheatmtransfer.

(4) PlanmExecution.

(i) Assumptions.
(1)m Steadym state,m (2)m negligiblem radiation,m (3)m uniformm heatm transferm coefficientma
ndm(4)muniformmambientmfluidmtemperature.

(ii) Analysis.m Newton'smlawmofmcoolingmstatesmthat
qsm=mhmAsm(Tsm -mT) (a)
where
Asm=msurfacemarea,mm2
hm=mheatmtransfermcoefficient,mW/m2-oC
qsm=mratemofmsurfacemheatmtransfermbymconvection,mW
Tsm=msurfacemtemperature,moC
Tm=mambientmtemperature,moCmAppl
yingm(a)mtomanminfinitesimalm aream dAs
dq s m m =mhm(Tsm -mT)mdAs (b)
Themnextmstepmismtomexpressm Tsm(x)minmtermsmofmdistancemxmalongmthemtriangle.m Tsm(x)m ismspecifiedmas
A
Tsm = 1/m2 (c)
x

, PROBLEMm1.1m(continued
)
TheminfinitesimalmareamdAsmismgiven
mby
dAsm =mWmdx (d)

where
xm=maxialmdistance,mm
Wm=mwidth,mmmSubstitut
ingm(c)mandmintom(b)
A
dq s =mh 1/m2 -m T)m Wdx (e)
( x
Integrationmofm(f)mgivesm
qs

 dq
L
qm =m )dx (f)

=mhWm ( mAxm1/m2
m

T
m m
s s 
0
Evaluatingmthemintegralminm(f)


qsmmhWm 2mAL1/m2mmL


Rewritemthemabov 
e 
Tm mqsm mhWL 2 AL
m (g)

1/m2mmT 
m

Notemthatmatmxm=mLmsurfacemtemperaturem Tsm(L)mismgivenmby
m(c)mas
(h)
Tsm(L)mm AL1/m2
(h)mintom(g 
) qsmmhWLm2Tsm(L)mm (i)
Tm

(iii) Checking.mDimensionalmcheck:mAccordingmtom(c)munitsmofmCmaremomC/m1/m2m.mThereforemunits
qsminm(g)maremW.
Limitingmchecks:m If
hm=m0mthenm qsm=m0.m Similarly,m ifm Wm=m0mormLm=m0mthenm qsm=m0.m Eq
uationm(i)msatisfiesmthesemlimitingmcases.

(5) Comments.m Integrationm ism necessarym becausem surfacem temperaturem ism variable..m Them s
amemproceduremcanmbemfollowedmifmthemambientmtemperaturemormheatmtransfermcoefficientmismn
on-uniform.

, PROBLEMm1.2m(continued
Amrightmanglemtrianglemismatmamuniform
) msurface mtemperaturemTs.m Heat mismremovedmbymconvect

ionmtomanm ambientm fluidm m atm Tm.m Them heatm transferm coefficientm hm variesm alongm them surfac
em accordingm to

hm= Cm
x2m1/m

wherem Cm ism constantm andm xm ism distancem alongm them basem measuredm fromm them apex.m Determ
inem themtotalmheatmtransfermratemfrommthemtriangle.

(1) Observations.
(i)mHeatm ismremovedmfrommthemsurfacembymconvection.m Therefore,m Newton'smlawm ofm cool
ingm maym bem helpful.m (ii)m Ambientm temperaturem andm surfacem temperaturem arem uniform.
(iii)mSurfacemareamandmheatmtransfermcoefficientmvarymalongmthemtriangle.

(2) ProblemmDefinition.
Findm themtotalmheatm transfermratembymconvectionmfrommthemsurfacemofmamplatemwithmam var
iablemsurfacemareamandmheatmtransfermcoefficient.

(3) SolutionmPlan.m Newton'smlawmofmcoolingmgivesmthemrate
mofmheatmtransfermbymconvection.mHowever, minmthismproblem dqs W
m surfacemareamandmheatmtransfermcoefficientmaremnotmuniform
.m Thismmeansmthatmthemratemofmheatmtransfermvariesmalongmth
emsurface.m Thus,mNewton’msmlawmshouldmbemappliedmtomanmi x
nfinitesimalmareamdAsmandmintegratedmovermthementiremsurface dx
mto mobtainmthemtotal mheatmtransfer.

L
(4) PlanmExecution.

(i) Assumptions.m (1)m Steadym state,m (2)m negligiblem radiationm andm (3)m uniformm ambient
m fluidmtemperature.

(ii) Analysis.m Newton'smlawmofmcoolingmstatesmthat
qsm=mhmAsm(Tsm -mT) (a)
where
Asm=msurfacemarea,mm2
hm=mheatmtransfermcoefficient,mW/m2-oC
qsm=mratemofmsurfacemheatmtransfermbymconvection,mW
Tsm=msurfacemtemperature,moC
Tm=mambientmtemperature,moCmAppl
yingm(a)mtomanminfinitesimalm aream dAs
dq s m m =mhm(Tsm -mT)mdAs (b)
ThemnextmstepmismtomexpressmhmandmdAsminmtermsmofmdistancemxmalongmthemtriangle.mThemheatmt
ransfermcoefficientmhmismgivenmby
mby Cm
hm= x1/m2m
TheminfinitesimalmareamdAsmismgiven