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STABILIZED VOLUME AVERAGING FINITE ELEMENT METHOD FOR FLOW IN POROUS MEDIA AND BINARY ALLOY SOLIDIFICATION

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STABILIZED VOLUME AVERAGING FINITE ELEMENT METHOD FOR FLOW IN POROUS MEDIA AND

BINARY ALLOY SOLIDIFICATION

Nicholas Zabaras and Deep Samanta

Materials Process Design and Control Laboratory

Sibley School of Mechanical and Aerospace Engineering188 Frank H. T. Rhodes Hall

Cornell University

Ithaca, NY 14853-3801

Email: [email protected]

URL: http://www.mae.cornell.edu/zabaras/

Materials Process Design and Control Laboratory

OUTLINE OF THE PRESENTATION

- Overview
- Complexities in solidification simulation
- Review of the previous related work
- Salient features of the current model
- Volume averaged governing transport equations
- Stabilized equal-order velocity-pressure formulation for

porous media flows.

- Computational techniques and solution methodology
- Applications to combined thermal, fluid and solutal transport

in porous media and solidification of a binary mixture

- Conclusions

Materials Process Design and Control Laboratory

OVERVIEW

Non-equilibrium

effects

Mass Transfer

Phase Change

Solidification

Simulation

Fluid flow

Shrinkage

Heat Transfer

Deformation

Microstructure

evolution

Materials Process Design and Control Laboratory

COMPLEXITIES IN SOLIDIFICATION SIMULATION

1. Morphological and micro-structural complexities

2. Variable time and length scales involved

3. Different physical phenomenon on different scales

- fluid flow, convective-conductive heat transfer,
- macro-segregation, solid movement and deformation
- on macroscopic scale

- interdendritic flow, latent heat release, nucleation
- and microstructure formation on microscopic scale

4. Formation of two phase mushy zone

5. Diffuse solid-liquid interface difficult to model

6. Non-equilibrium effects on global and local

scales

Materials Process Design and Control Laboratory

liquid

~ 10-2 m

solid

q

(a) Macroscopic scale

~ 10-4 – 10-5m

liquid

(b) Microscopic scale

solid

SCHEMATIC OF THE DIFFERENT LENGTH SCALES

IN A TYPICAL SOLIDIFICATION PROCESS

Materials Process Design and Control Laboratory

PREVIOUS WORK ON SOLIDIFICATION AND POROUS MEDIA TRANSPORT SIMULATIONS

- Continuum mixture approach (Incropera and co-workers) – Finite difference approach using SIMPLER algorithm
- Double diffusive and natural convection in porous media (Nithiarasu et al.) – Volume averaged finite element method (fractional step method for fluid flow problems)
- Solidification phenomenon at high Ra number melt flows (Heinrich, Poirier et al.) – Finite element method using penalty based approach for fluid flow
- Simulation of darcy flows (Hughes and Masud) – Stabilized finite element approach using mixed and equal order velocity-pressure elements.
- Volume-averaged two phase model for solidification and porous media (Beckermann et al.) – Finite difference approach

Materials Process Design and Control Laboratory

SALIENT FEATURES OF THE CURRENT MODEL

1.Single domain solidification model based onvolume averaging

2.Single set of transport equation for mass, momentum, energy and

species transport.

3. No need to track the boundaries between phases

4. Single grid and single set of boundary conditions.

5. Volume averaging tracks solid, mushy and liquid regions

6. Solidification microstructures not modeled.

Microscopic governing transport equations

General form:

Mass :

Momentum :

Energy :

Species :

Materials Process Design and Control Laboratory

Volume-

averaging

process

Macroscopic

governing

equations

Microscopic

transport

equations

Volume averaged transport equation

Microscopic deviation term

=

Interfacial flux term due to transport

=

Interfacial term due to phase change

=

Materials Process Design and Control Laboratory

solute transport

and

and

ASSUMPTIONS AND DIMENSIONLESS VARIABLES

Assumptions in single-phase model

Important dimensionless variables

- Microscopic deviation terms neglected

Prandtl number Pr

- Interfacial terms not modeled in single phase

Lewis number Le

Darcy number Da

- Interfacial drag term in momentum modeled
- using Darcy law with isotropic permeability

Thermal Rayleigh

number RaT

Solutal Rayleigh

number RaC

- Material properties uniform (μ, k etc.)in an
- averaging volume dVk but can globally vary

Conductivity Ratio Rk

- Negligible solutal diffusion in solid phase
- (Ds = 0)

Capacity Ratio Rc

- Solid phase stationary (vs = 0)

Materials Process Design and Control Laboratory

VOLUME AVERAGED DIMENSIONLESS GOVERNING EQUATIONS

Initial conditions :

Boundary conditions :

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NUMERICAL SCHEME FOR FLUID FLOW

- Stabilized equal-order velocity-pressure formulation for fluid flow

- Derived from SUPG/PSPG formulation

- Additional stabilizing term for Darcy drag force incorporated

- Darcy stabilizing term necessary for convergence in even basic problems

Galerkin formulation for the fluid flow problem

Materials Process Design and Control Laboratory

NUMERICAL SCHEME FOR FLUID FLOW

Stabilized formulation for the fluid flow problem

Advection

stabilizing term

Pressure

stabilizing term

Diffusion

stabilizing term

Darcy drag

stabilizing term

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such that for all

are defined as

where

and

EQUAL ORDER FE FORMULATION

Finite element function spaces for velocity and pressure :

the following holds :

Materials Process Design and Control Laboratory

STABILIZING PARAMETERS FOR FLUID FLOW

Stabilizing parameters

Stabilizing terms

advective

viscous

Darcy

pressure

continuity

- Convective and pressure stabilizing terms
- modified form of SUPG/PSPG terms

- Darcy stabilizing term obtained by least
- squares, necessary for convergence

- Viscous term with second derivatives
- neglected

- A fifth continuity stabilizing term added
- to the stabilized formulation

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modified

UNIFIED APPROACH TOWARDS STABILIZING PARAMETERS

- Sub-grid scale or multi-scale approach for determining stabilizing parameters

- Single expression to account for all effects

- Limiting behavior similar to one discussed before

Unified expressions for stabilizing parameter

Darcy dominant regime

Diffusion dominated flows

Convection dominated flows

for 2D problems

including transient effects

Materials Process Design and Control Laboratory

NUMERICAL SCHEME FOR THERMAL AND SOLUTAL TRANSPORT

- Consistent SUPG formulation for thermal and solutal transport

- Non-dimensional liquid enthalpy hl expressed as

- Gradient of temperature expressed as

- Use of consistent instead of lumped matrices

Governing Equations

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NUMERICAL SCHEME FOR THERMAL AND SOLUTAL TRANSPORT

Petrov-Galerkin shape function

Weak formulation for energy equation:

Weak formulation for solute equation:

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and

θ =

and updating liquid concentration as Cl =

Cl = Ce = 0 and ε =

- if hSolidus < h < he (region 3), solidification at eutectic point,

θ = θeutectic,

- Lever rule assumption can be replaced by Scheil rule, ε =

UPDATE FORMULAE FOR THERMODYNAMIC QUANTITIES

- Supplementary relationships between enthalpy, liquid concentration, volume fraction and temperature

- Phase diagram divided into four regions

- if h > hLiquidus (region 1), pure liquid region. θ determined from thermodynamic relation,
- Cl = 1.0 and ε = 1.0

- if he < h < hLiquidus (region 2), mushy zone, θ and ε determined iteratively by solving

- if hSolidus < h (region 4), solid region, θ determined from thermodynamic relations, Cl = Ce and ε = 0.0

Materials Process Design and Control Laboratory

All fields known

at time tn

- Multi-step predictor-corrector
- method for energy and solute
- equations

- Standard Gauss-elimination used
- for both energy and solute transport

n = n +1

Solve for the

enthalpy field

(energy equation)

- Solution typically obtained in
- 2 – 3 steps except at initial times

Solve for velocity

and pressure fields

(momentum equation)

- Newton – Raphson method for
- fluid flow problem

Solve for the

concentration field

(solute equation)

- Preconditioned BICGSTAB
- algorithm employed for fluid flow

Yes

- LU factorization is done for few
- time steps only

Is the error

in temperature, liquid

concentration and liquid

volume fraction less

than tolerance

Solve for temperature,

liquid concentration and

volume fraction

(Thermodynamic relations)

- Line search employed to ensure
- global convergence.

No

Materials Process Design and Control Laboratory

1. Double diffusive convection in a constant porosity medium

Governing equations

Momentum Transport:

Energy Transport:

Solute Transport:

Physical parameters

Thermal Rayleigh number, RaT = 2x108

Solutal Rayleigh number, RaC = -1.8x108

Darcy number, Da = 7.407x10-7

Prandtl number, Pr = 1.0

Lewis number, Le = 2.0

Liquid volume fraction, ε = 0.6

Materials Process Design and Control Laboratory

(a) Temperature

(b) Concentration

(c) Streamfunction

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(b) Streamlines

(a) Finite element mesh - 50x50 Q4 elements

(c) Isotherms

(d) Iso-concentration lines

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determines extent of porous region

2. Natural convection in a fluid saturated variable porosity medium

Governing equations

Momentum Transport:

Energy Transport:

- Central liquid core surrounded by porous medium

- Porosity varying uniformly from wall to core

Physical parameters

Thermal Rayleigh number, RaT = 1x106

Solutal Rayleigh number, RaC = 0

Darcy number, Da = 6.665x10-7

Prandtl number, Pr = 1.0

Wall porosity, εw = 0.4

= 0.3

Width ratio,

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(c) Streamlines

(b) Isotherms

(a) 50x50 finite element mesh

(c) Streamlines

(b) Isotherms

(a) 100x100 finite element mesh

Materials Process Design and Control Laboratory

= 0.2)

STEADY-STATE FE SOLUTION OF THE NATURAL CONVECTION PROBLEM

Physical parameters

Thermal Rayleigh number, RaT = 1x106

Darcy number, Da = 6.665x10-7

Prandtl number, Pr = 1.0

Wall porosity, εw = 0.4

= 0.2

Width ratio,

(c) Streamlines

(b) Isotherms

(a) 80x80 finite element mesh

Materials Process Design and Control Laboratory

3. Solidification of a binary aqueous solution

Physical parameters

Thermal Rayleigh number, RaT = 1.938x107

Solutal Rayleigh number, RaC = -2.514x107

Prandtl number, Pr = 9.025

Heat conductivity ratio, Rk = 0.84

Heat capacity ratio, Rc = 0.576

Dimension slope of liquidus, m = 0.905

Darcy number, Da = 8.896x10-8

Lewis number, Le = 27.84

Initial and boundary conditions

Temperature of hot wall, Thot = 311 K

Temperature of cold wall, Tcold = 223K

Initial temperature, T0 = 311K

Initial concentration, C0 = 0.7

- Presence of a dendritic mushy zone during solidification

Eutectic concentration, Ce = 0.803

- Thermal and solutal buoyancy forces opposing each other

Eutectic temperature, Te = 257.75K

- Highly irregular liquid interface

Solutal flux on all boundaries = 0 (adiabatic flux condition)

- Temperature and concentration fields distorted by
- advection effects.

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FE SOLUTION AT DIFFERENT TIME STEPS (t* = 0.009)

(c) Isotherms

(d) Liquid

concentration

(b) Streamlines

(a) Velocity vectors

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FE SOLUTION AT DIFFERENT TIME STEPS (t* = 0.018)

(c) Isotherms

(d) Liquid

concentration

(b) Streamlines

(a) Velocity vectors

Materials Process Design and Control Laboratory

FE SOLUTION AT DIFFERENT TIME STEPS (t* = 0.036)

(c) Isotherms

(d) Liquid

concentration

(b) Streamlines

(a) Velocity vectors

Materials Process Design and Control Laboratory

FE SOLUTION AT DIFFERENT TIME STEPS (t* = 0.071)

(c) Isotherms

(d) Liquid

concentration

(b) Streamlines

(a) Velocity vectors

Materials Process Design and Control Laboratory

FE SOLUTION AT DIFFERENT TIME STEPS (t* = 0.142)

(c) Isotherms

(d) Liquid

concentration

(b) Streamlines

(a) Velocity vectors

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NUMERICAL COMPARISON AND MACROSEGREGATION

Macrosegregation patterns at steady state

(t = 0.142)

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TRANSIENT FE SOLUTION OF THE SOLIDIFICATION PROBLEM

(a) Temperature

(b) Liquid concentration

(c) Liquid volume-fraction

(d) Streamfunction

Materials Process Design and Control Laboratory

CONVERGENCE STATISTICS FOR FLUID FLOW SOLVER

Residual norm

Iteration number

Example 3

Example 2

Example 1

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CONCLUSIONS AND FUTURE RESEARCH PLANS

CURRENT ACHIEVEMENTS

- Stabilized formulation for high Rayleigh number porous media and solidification problems
- Single domain, enthalpy based models incorporating mushy zone
- Porous media simulations with convergence studies
- Solidification of a binary aqueous solution at high thermal and solutal Rayleigh numbers

FUTURE RESEARCH INTERESTS

- Incorporation of a unified stabilizing parameter for all regimes
- 3D solidification simulation and development of a parallel simulator
- Solidification of metal alloys with phase constituents of different densities
- Sensitivity analysis and parametric studies
- Magneto-convection and solidification in the presence of magnetic fields.
- Marangoni convection and effects of thermo-capillary convection on solidification
- Combined analysis of solidification and deformation, and modeling of residual stresses
- Inverse design and control of solidification processes with mushy zones

Materials Process Design and Control Laboratory

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