A Numerical Investigation of Directional Binary Alloy Solidification

1
A Numerical Investigation of Directional Binary
Alloy Solidification Processes Using A
Volume-Averaging Technique
Lei Wan Sibley School of Mechanical and
Aerospace Engineering Cornell University
2
Outline of the presentation

• Introduction
• Review of related previous works
• Volume averaging technique
• Computational methods and solution schemes
• Numerical examples
• Suggestions for future studies

3
Background of the study

• Why study solidification of metal alloys?
• --- wide engineering application
• --- economic significance
• --- physical richness
• Our objective
• --- study the effect of fluid flow on the
• distributions of chemical species and
• grain structures in binary alloy
• solidification processes

4
Basic aspects of alloy solidification
a) Pure substance
b) Binary alloy
Liquidus interface
Solid/liquid interface
S O L I D
M U S H
Solidus interface
LIQUID
SOLID
Liquid
T
T
Tliq
Tm
Tsol
x
x
5
Important physical mechanisms in solidification
Capillarity
Dendrites movement
Melt
Double diffusive convection
Solute Segregation
Mushy
Morphological instability
Interfacial thermodynamics
Solid
Shrinkage
Diffusion
6
Macrosegregation pattern in cast ingots
V-segregation
A-segregation

• Macrosegregation refers
• to mal-distribution of
• alloy constituents
• Length-scales ranging
• from millimeters to
• centimeters or even
• meters
• A consequence of
• species advection in the
• liquid

Mold
Casting
Bottom negative cone segregation
7
Causes of melt flow in alloy solidification

• Thermal and concentration gradients
• Surface tension gradients at a free surface
• Density changes due to solidification
• Drag force from solid motion
• Residual force due to filling of the mold
• Electromagnetic field
• External forces such as rotation of the mold

8
Thermosolutal convection in the melt
N RaC / RaT
RaT gßT?TL3/?a RaC gßC?CL3/?a
9
Relate previous works two-domain model

• Pioneering work by Flemings and coworkers in
  the
• mid-1960s linking macrosegregation to
• interdendritic fluid flow.
• Mehrabian et al. extended Flemings work by
• incorporating an equation for
  buoyancy-driven flow
• in mushy region.
• Fuji et al. refined the model by coupling the
• momentum and energy equations.
• The first model to couple the bulk melt and
  mushy
• regions and to include macrosegregation was
  
• reported by Ridder et al.

10
Limitation of the two-domain model

• Explicit tracking of the phase
• interfaces involved matching
• variables at the boundaries.
• Requires tracking of a
• distinct solid/liquid interface,
• which is very complex for
• most alloy solidification
• systems
• Cannot predict phenomena
• occurring in the mushy zone
• such as channel formation,
• remelting and double
• diffusive interfaces.

q

• Macroscopic
• scale (10-2100m)

L
S
(b) Microscopic scale (10-510-4m)
11
Relate previous works single-domain model

• Recent numerical investigation of metal alloy
  
• solidification using single-domain model has
  been
• done by Incropera et al. and Beckermann et
  al.
• Advantages of single domain model include
• --- only a single set of equations to be
  solved in a single, fixed
• numerical grid
• --- phase interfaces are implicitly
  determined by the solution
• of temperature and concentration
  fields
• --- able to model freckle formation and
  remelting of solid
• Two approaches to derive the single domain
  model
• --- volume averaging technique
• --- classical continuum theory

12
Volume averaging technique
Volume averaged quantity Important
theories Mixture quantity ? lt?lgtel
lt?sgtes
Solid
Liquid
Solid
Liquid
Definition of averaging volume ? (l) ltlt ?(d) ltlt
?(L)
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Assumptions for the volume averaged model

• Only solid and liquid phases may be present
• Variation in material properties are neglected
  in
• averaging volume, although globally they may
  vary.
• The flow is laminar and we assume Newtonian
  fluid.
• The Boussinesq approximation is made
• The solid phase is stationary
• All the phases in the averaging volume are in
• thermodynamical equilibrium
• No back diffusion in the solid phase

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Macroscopic governing equations
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Supplementary relationships

• Mixture enthalpy- liquid enthalpy relation
• h hl el hs es
• Mixture concentrationliquid concentration
• relation (Lever rule)
• C Cl el Cs es
• Thermodynamic relations (phase diagram)
• C (T - Tm)/(Te - Tm) Ce

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Finite element analysis solution procedure
start
Go to next time step tn1
Solve energy equation
Decoupled solution methodology for
various sub-problems at each time step
Solve solute equation
Solve momentum equations
Update temperature, liquid concentration and
volume fraction
errors lt tolerance?
No
yes
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Finite element analysis momentum equation

• Stabilized finite element method (Tezduyar et
  al., 1990)
• Darcy stabilizing term is added to the
  standard
• SUPG/PSPG stabilizing method, velocity and
  pressure
• are solved simultaneously, computationally
  intense
• Penalty method (Brooks and Hughes, 1982)
• eliminate pressure with the help of penalty
  function
• and incorporate mass and momentum equations
• into one equation
• Fractional step method (Partankar, 1980)
• solve velocity and pressure iteratively by
  projecting
• velocity into divergence free function space

18
Stabilized finite element method
The trial and test function spaces are defined as
The stabilized Galerkin formulation of the mass
and momentum equations can be stated as follows
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Stabilized finite element method
Where tSUPG and tPSPG are the standard SUPG and
PSPG stabilizing parameters
20
Stabilized finite element method

• The spatial discretization of the weak form
  leads to
• a set of nonlinear ordinary differential
  equation of
• velocity and pressure.
• The backward Euler method is utilized for time
• stepping algorithm.
• The resulting nonlinear algebraic equations are
  
• solved by Newtons method.
• Due to the sparse nature of the resulting
  linear
• system, an iterative solver
  (PrecondBiCGStab) is
• used to reduce the computation cost.

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Penalty method

• Standard Garlerkin formulation is applied to
  the
• modified momentum equation with the penalty
  term.
• A 109 penalty number is chosen for all the
  examples.
• Backward Euler time-marching scheme and
• Newtons method are used for solving the set
  of
• nonlinear ODEs.
• As the pressure is eliminated from the momentum
  
• equation, the linear system to be solved is
  much
• smaller than that of the stabilized method,
  so
• Gaussian elimination is used for all examples.

22
Fractional step method
Part one obtain approximate pressure field Step
1 calculate fictitious velocity by dropping
the pressure term in the momentum
equation Step 2 extract pressure field from
the fictitious velocity using
Poisson equation
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Fractional step method
Part two correct velocity and pressure
iteratively Step 3 solve momentum equation for
velocity using the newly
obtained pressure field Step 4 solve Poisson
equation for pressure correction and
correct pressure as Step 5
velocity is correctly using the pressure
correction
24
Finite element analysis energy and solute
equations

• Energy equation
• Solute equation
• Standard form of scalar advection-diffusion
  equation

25
Example1 Natural convection in uniform porosity
media
36 x 36 grid
u v 0
y
T 1 u v 0
T 0 u v 0
g
e 0.8
x
u v 0
Pr 1 RaT 108 Da 7.8x10-8
a) streamline contour
b) temperature contour
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Example 2 double diffusive convection in
uniform porosity media
50 x 50 grid
u v 0
y
T 1 C 1 u v 0
Pr 1 RaT 2x108 RaC -1.8x108 Da 7.4x10-7
T 0 C 0 u v 0
g
e 0.6
u v 0
x
c) isoconcentration
a) streamline
b) isotherm
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Example 3 natural convection in variable
porosity media
u v 0
y
36 x 36 grid
ew0.4
Free Fluid e 1
T 1 u v 0
T 0 u v 0
x
u v 0
Pr 1 RaT 106 Da 6.6x10-7
a) streamline contour
b) temperature contour
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Example 4 directional binary alloy solidification
y
Material system Aqueous solution of ammonium
chloride (NH4Cl-H2O) Finite element mesh 41 x
41 bilinear elements 1849 nodes Governing
physical parameters Prandtl number 9.025 Lewis
number 27.84 Darcy number 8.9x10-8 Thermal
Rayleigh number 1.9x106 Solutal Rayleigh number
-2.1x106
u v 0 ?T/?n 0 ?C /?n 0
u v 0 T Tinital ?C /?n 0
u v 0 T Tcold ?C /?n 0
H
x
L
u v 0 ?T/?n 0 ?C /?n 0
Geometry, FE mesh and boundary conditions
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Alloy solidification without convection
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Alloy solidification with convection
t 0.036
a) streamline
b) isotherm
c) isoconcentration
a) streamline
b) isotherm
c) isoconcentration
t 0.018
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Alloy solidification with convection
a) streamline
b) isotherm
c) isoconcentration
Macrosegreation pattern at time t
0.14 Cinital 0.7, Ceutectic 0.803
t 0.07
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Alloy solidification with convection
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Effect of anisotropic permeability
R Kx / Ky 4 Kx permeability in the x
direction Ky permeability in the y direction
t 0.036
a) streamline
b) isotherm
c) isoconcentration
a) streamline
b) isotherm
c) isoconcentration
t 0.018
34
Example 5 pure aluminum solidification
q 0
Bi Ta
q 0
g
Melt
Solid
t 0.05
q 0
1.0
Dimensionless physical parameters
t 0.35
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Conclusions for the numerical analysis of binary
alloy solidification processes

• Thermal-solutal convection is responsible
• for macrosegregation
• Permeability of the mushy zone will affect the
• strength of flows in both mushy and bulk
  liquid
• region as well as the shape of liquidus
• Tested for a limiting case of pure substance
• solidification, the single domain model
  demon-
• strated great versatility

36
Suggestions for future studies

• Additional physical phenomena
• Thermo/diffusocapillary effect
• Shrinkage
• Electromagnetic field
• Rotation and vibration
• Multi-constituent alloys
• Computational issues
• Simulation of solidification system with
  higher
• Rayleigh and Lewis number
• Extend the model to 3D simulation

37
Acknowledgements
Thesis advisor Professor Nicolas J.
Zabaras Committee member Professor Stephen A.
Vavasis Discussions and motivations Ganapathysub
ramanian Shankar Xiaoyi Li Research funded by a
grant from NASA Additional support from Sibley
School of Mechanical and Aerospace Engineering,
Cornell University