n01 enthalpy method
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Enthalpy MethodIntroductionM.S Darwish
MECH 636: Solidification Modelling
solid liquidmush
The Mushy Zone
solid liquidmush
�
+ (Interface term)s
�
+ (Interface term)l
�
hl = c ldTTref
T
∫ + L�
∂ ρ shs( )∂t
= ∇ ⋅ ks∇T( )
�
hs = csdTTref
T
∫
�
∂ ρ lCl( )∂t
+ ∇ ⋅ ρ lvlCl( ) = ∇ ⋅ Dl∇Cl( )
�
∂ ρ sCs( )∂t
= ∇ ⋅ Ds∇Cs( )
�
∂ ρ lh l( )∂t
+ ∇ ⋅ ρ lvlh l( ) = ∇ ⋅ k l∇T( )
kCo
Co
Co/k
T
Cmax
Ceut
T1
T2
T3
CS
CL
k=CS/CL
C
A B
Conservation of Energy
Conservation of Specie
solid liquidmush
Averaging
�
rs + rl =1
�
∂ rsρ shs + rlρ lh l( )∂t
+ ∇ ⋅ rs˜ v shs + rlvlh l( ) = ∇ ⋅ rsk s∇T + rlk l∇T( )
�
rs = Vs
V
�
rl = Vl
V
�
∂ rsρ sCs + rlρ lCl( )∂t
+ ∇ ⋅ rs˜ v sCs + rlvlCl( ) = ∇ ⋅ rsk s∇Cs + rlk l∇Cl( )
�
rs = 0
�
rl =1
�
rs =1
�
rl = 0
�
χ s = ms
m
�
χ l = ml
m
�
χ s + χ l =1
Mass and Volume Fractions
mass fraction volume fraction
�
χ s = Cl −Co
Cl −Cs
= ms
m
�
χ s
rs= ms
m× VVs
= ρs
ρ
kCo
Co
Co/k
T
Cmax
Ceut
T1
T2
T3
CS
CL
k=CS/CL
C
A B
�
rs + rl =1
�
rs = Vs
V
�
rl = Vl
V
�
χ s = ms
m
�
χ l = ml
m
�
χ s + χ l =1
�
rs = Vs
V
�
rl = χ lρρl
= Co −Cs
Cl −Cs
ρρl
�
rs = χ sρρs
= Cl −Co
Cl −Cs
ρρs
Equilibrium Relations
�
Cl
Co
= T1 −T2T0 −T2
�
Co −Cs
Co − kCo
= T2 −T3T1 −T3
�
Cs = Co 1− 1− k( ) T2 −T3T1 −T3
⎛
⎝ ⎜
⎞
⎠ ⎟
�
Cl = CoT0 −T2T0 −T1
�
χ s = T1 −T21− k( ) T0 −T2( )
�
rs = T1 −T21− k( ) T0 −T2( )
ρρs
⎛
⎝ ⎜
⎞
⎠ ⎟
kCo
Co
Co/k
T
Cmax
Ceut
T1
T2
T3
CS
CL
k=CS/CL
C
A B
To
�
χ s = Cl −Co
Cl −Cs
= ms
m
Average Energy Equations
mixture
3
1
2
�
rsks∇T + rlkl∇T = km∇T
mixture �
ρm = rsρs + rlρl
�
km = rsks + rlkl
�
∂ rsρshs + rlρlhl( )∂t
+ ∇ ⋅ rsρs˜ v shs + rlρlvlhl( ) = ∇ ⋅ rsks∇T + rlkl∇T( )
�
rsρs˜ v shs + rlρlvlhl
�
= rsρs˜ v s CsdTTref
T
∫ + rlρlvl CsdTTref
T
∫ + δH⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
= rmρmvm CsdTTref
T
∫ + rlρlvlδH
�
vm = rsρs˜ v s + rlρlvl
rmρm
mixture
�
Tref = 0
�
rsρshs + rlρlhl = rsρs c p,sdTTref
T
∫ + rlρl c p,ldTTref
T
∫ + δH⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
�
= ρm cp,sdTTref
T
∫ + rlρlδH�
hl = cp,ldTTref
T
∫ + L
�
= cp,sdTTref
T
∫ + δH
�
δH = cp,l − cp,s( )dTTref
T
∫ + L
�
CdTTref
T
∫ ≈ C dTTref
T
∫
�
Tref = 0
�
∂ ρmcp,sT + rlρlδH( )∂t
+ ∇ ⋅ ρmvmT + rlρlvlL( ) = ∇ ⋅ km∇T( )
�
∂ ρmcp,sT( )∂t
= ∇ ⋅ km∇T( ) − ∂ rlρlδH( )∂t
�
ρmcp,sTPVP − ρm•
cp,s•
TP•
Δt= km∇T ⋅ dS( ) f
f = nb(P )∑ − rlρlδHP − rl
•ρl•δHP
•
Δt
old old
Assumptions
Neglecting Convection
�
aPTP = aNBTNBNB(P )∑ + bP + aP
t TP• − VρlδH
Δtrl,P − rl ,P
•( )
Discretizing
Algebraic Form
Average Specie Equation
�
∂ rsρsCs + rlρlCl( )∂t
+ ∇ ⋅ rsρs˜ v sCs + rlρlvlCl( ) = ∇ ⋅ rsDs∇Cs + rlDl∇Cl( )
1
�
rsDs∇Cs + rlDl∇Cl3
�
= rsDs∇ kCl( ) + rlDl∇Cl
�
= Dm∇Cl
2mixture
�
rsρs˜ v sCs + rlρlvlCl
�
Cm = rsρsCs + rlρlCl
ρm
�
= rsρs˜ v sk + rlρlvl( )Cl
�
= ρmvmCl
�
vm = rsρs˜ v sk + rlρlvl
ρm
�
rsρsCs + rlρlCl = ρmCm
mixture
mixture
�
Dm = rskDs + rlDl
�
∂ ρmCm( )∂t
+ ∇ ⋅ ρmvmCl( ) = ∇ ⋅ Dm∇Cl( )
�
Cm = χ sCs + χ lCl
�
= ms
mCs + ml
mCl
�
⇔mCm = msCs + mlCl
�
mVCm = ms
VCs + ml
VCl
�
ρmCm = ms
VVs
Vs
Cs + ml
VVl
Vl
Cl
�
ρmCm = rsρsCs + rlρlCl
�
= ρmCl − rsρs 1− k( )Cl
�
∂ ρmCl( )∂t
+ ∇ ⋅ ρmvmCl( ) = ∇ ⋅ Dm∇Cl( ) +∂ rsρs 1− k( )Cl( )
∂t
�
= rsρskCl + rlρlCl
�
∂ ρmCl( )∂t
= ∇ ⋅ Dm∇Cl( ) +∂ rsρs 1− k( )Cl( )
∂t
Case 1: no Specie diffusion
�
rs + rl =1
�
χ s + χ l =1
�
∂ ρmcp,sT( )∂t
= ∇ ⋅ km∇T( ) − ∂ rlρlδH( )∂t�
∂ ρmCm( )∂t
= 0
�
rl = f T( ) =
1 T > TLiquidus
T −TLiquidusTLiquidus −Tsolidus
Tsolidus < T < TLiquidus
0 T < Tsolidus
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
�
aPTP = aNBTNBNB(P )∑ + bP + aP
t TP• − VρlδH
Δtrl,P − rl ,P
•( )
Liquid Fraction Update
�
aPTP = aNBTNBNB(P )∑ + bP + aP
t TP• + VρlδH
Δtrl,P − rl ,P
•( )
�
aP f−1 rl .P( ) = aNBTNB
NB(P )∑ + bP + aP
t TP• − VρlδH
Δtrl,P + δrl,P − rl ,P
•( )
�
δrl,P = aPΔtTP − f −1 rl .P( )VρlδH
�
rl,Pn+1 = rl,P
n + δrl ,P
�
0 ≤ rl,Pn+1 ≤1
�
TP = f −1 rl .P( )
Algorithm
Compute T
Compute r
Update Source of T
Converged
Compute Average Properties
case 2: Specie Diffusion
�
∂ ρ C ( )∂t
+ ∇ ⋅ rs˜ v sCs + rlvlCl( ) = ∇ ⋅ rsDs∇Cs + rlDl∇Cl( )
Under equilibrium conditions a discontinuity exist at the solid/liquid interface given by
�
Cl = Cs /k
�
C = χ lCl + χ sCs
�
χ l = ml
m
�
χ l = ml
m= ρlVl
ρV= rl
ρl
ρ
�
χ l = Co −Cs
Cl −Cs
�
ρ = ρl rl + ρsrs
�
ρ = ρlVl
V+ ρs
Vs
V
�
ρV = ρlVl + ρsVs
�
mC = mlCl + msCs
�
ρVC = ρlVlCl + ρsVsCs
�
ρC = ρlVl
VCl + ρs
Vs
VCs
�
= ρl rlCl + ρsrskCl
�
= ρ − ρsrs( )Cl + ρsrskCl
�
= ρCl − (1− k)ρsrsCl
�
ρC = ρCl − (1− k)ρsrsCl
�
∂ ρ C ( )∂t
+ ∇ ⋅ rs˜ v sCs + rlvlCl( ) = ∇ ⋅ rsDs∇Cs + rlDl∇Cl( )
�
rsDs∇Cs + rlDl∇Cl = rsDs + rlDlk( )∇Cl
�
= D∗∇Cl
�
∂ ρ Cl( )∂t
+ ∇ ⋅ ρ v∗Cl( ) = ∇ ⋅ D ∗∇Cl( )
�
∂ ρ C ( )∂t
+ ∇ ⋅ rs˜ v sCs + rlvlCl( ) = ∇ ⋅ rsDs∇Cs + rlDl∇Cl( )
Algorithm
Unknowns
�
rl ,rs
�
χ l ,χ s�
T
Equations�
C,Cl ,Cs
Relations
Conservation of Energy (T)
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