Heat Conduction through Composite Walls.pdf

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1. Heat Conduction through
Composite Walls
1

2. Significance
•Inindustrialheattransfer
problemsoneisoften
concernedwithconduction
throughwallsmadeupof
layersofvariousmaterials,
eachwithitsown
characteristic thermal
conductivity
2

3. Objective
•Toshowhowthevariousresistancestoheattransferare
combinedintoatotalresistance
3

4. Diagram
Fluid Fluid
Distance, x
Temperature, T
0 x
o
x
1x
2 x
3
T
o
T
1
T
2
T
3
T
a
k
o1 k
12 k
23
Δx
H
Substance 01
Substance 12
Substance 23
T
b
4

5. Nomenclature
•Threematerialsofdifferentthicknesses,x
1–x
o,x
2–x
1,andx
3–x
2
•Thermalconductivitiesk
01,k
12,andk
23
•T
a=AmbientTemperature
•T
b=FluidTemperature
•Theheattransferattheboundariesx=x
oandx=x
3,isgivenbyNewton's
"lawofcooling"withheattransfercoefficientsh
0andh
3
5

6. Energy Balance
•ForaslabofvolumeWHΔx:
Heat entering at x = q
x|
xWH
Heat leaving at x + Δx = q
x|
x+Δx WH
6

7. Energy Balance
•For Region 01:
q
x|
xWH-q
x|
x+Δx WH = 0
•Dividing by WHΔx:
•Taking limit Δx→0,
7

8. Integration
•Integratingthepreviousequation:
•q
o=Heatfluxattheplanex=x
o
•Similarly,forregions12and23:
•Withcontinuityconditionsonq
xatinterfaces,sothattheheatflux
isconstantandthesameforallthreeslabs
8

9. Applying Fourier’s Law
•Region 01:
•Region 12:
•Region 23:
9

10. Integration over the entire thickness
•Wenowassumethatk
o1,k
12,andk
23,areconstants.Thenwe
integrateeachequationovertheentirethicknessoftherelevantslab
ofmaterialtoget
•Region01:
•Region12:
(Eq-1)
(Eq-2)
10

11. Integration over the entire thickness
•Region 23:
•In addition we have the two statements regarding the heat transfer
at the surfaces according to Newton's law of cooling
•At Surface 0:
•At Surface 3:
(Eq-3)
(Eq-4)
(Eq-5)
11

12. Simplification
•Adding Eq-1 through Eq-5
12

13. Final Expression
13

14. General Expression
•SometimesthisresultisrewritteninaformreminiscentofNewton'slawof
cooling,eitherintermsoftheheatfluxq
o(J/m
2
-s)ortheheatflowQ
o
(J/s):
•ThequantityU,calledthe"overallheattransfercoefficient,"isgiven
thenbythefollowingfamousformulaforthe"additivityofresistances“
•Herewehavegeneralizedtheformulatoasystemwithnslabsofmaterial
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