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ControlVolumeFiniteDifferenceMethods

BasicConcept:

1.  DevelopmatrixequaBonsbasedonVolume/MassFluxesandVolume/MassconservaBonexpressionsratherthansecondorderPDEs.Forexample:

2.Youdon’tuseTaylorseriesperse.UsesimplefinitedifferencerepresentaBonsofthefluxtermsacrossthefacenormaltothecell.

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ddt

ρφdVV∫ = n ⋅ qρdA

A∫

2DExample

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δei, j =Δxi, j + Δxi, j+1

2δwi, j =

Δxi, j + Δxi, j−12

Ke =Ki,kΔxi, j + Ki, j+1Δxi, j+1

Δxi, j + Δxi, j+1Kw =

Ki,kΔxi, j + Ki, j−1Δxi, j−1Δxi, j + Δxi, j−1

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ne = (1,0) nw = (−1,0)€

Δxi, jΔyi, jSs∂h∂t

=k= e,w,n,s∑ AkKk∇hk ⋅ nk

qwAw ⋅ nw = −KwΔyi, jhi, j − hi, j−1

δw

qeAe ⋅ ne = KeΔyi, jhi, j+1 − hi, j

δe

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Vi, j = Δxi, jΔyi, j

Area=Δy

NodeseparaBondistance

nenw

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2DExample,X‐DirecBon

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Ss∂h∂t

=

Ke

hi, j+1 − hi, jΔxi, j + Δxi, j+1

2

−Kw

hi, j − hi, j−1Δxi, j + Δxi, j−1

2Δxi, j

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SsΔt

hi, jk+1 − hi, j

k( ) =2Ki, j+1/ 2

Δxi, j Δxi, j + Δxi, j+1( )

h i , j+1

k+1

+2Ki, j+1/ 2

Δxi, jΔxi, j + Δxi, j+1+

2Ki, j−1/ 2

Δxi Δxi, j + Δxi,i−1( )

h

i , j

k+1 +2Ki, j−1/ 2

Δxi, j Δxi, j + Δxi,i−1( )

h

i , j−1

k+1