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)