Computations of Flows with Solidification and Melting

1
Computations of Flows with Solidification and
Melting

• by
• Mahesh Bathina
• M.Sc.-Student of Chemical Engineering
• Institute of Fluid Mechanics
• Friedrich Alexander Universität,
  Erlangen-Nürnberg

Title of Presentation
2
The Presentation Overview

• Physics of Solidification and Melting
• Numerical Computations Using
• Fixed Grid Method
• Moving Grid Method
• Computations of One and Two/Three Dimensional
  Solidification and
• Melting
• Examples of Computations
• Outlook and Conclusion

Contents
3
Melting and Solidification in Nature
Examples
4
Physics of Solidification and Melting
Physics-1
5
Physics of Solidification and Melting

• Temperature at the interface remains
• constant (For Pure Substances)
• Phenomena occurs due to latent heat
• transfer at the interface between liquid
• and solid phases
• Melting occurs when latent heat is added to the
  system

• Solidification occurs when latent heat is
  removed from the system
• The energy in solid is distributed by conduction
  and in liquid primarily
• by conduction, is sometimes superimposed by
  convection

Physics-2
6

• For Pure substances the following relationship
• holds

• By applying the first law of thermodynamics,
  i.e.,
• energy balance at the interface

Tm
u is the velocity at the interface l is the
latent heat
This is known as Classical Stefan Problem

• Stefan number

Physics-3
7
Numerical Computations

• Fixed Grid Method
• Usually referred to Enthalpy Porosity method
  (EPM)
• Thermodynamically inconsistent
• In general, sharp interface is observed, but in
  EPM a mushy zone results

Enthalpy
Enthalpy
Tm
Tm
Temperature
Temperature
Enthalpy Temperature Relationships
Fixed Grids-1
8
The Enthalpy Porosity Formulation

• Enthalpy rather than Temperature is the main
  dependent variable in Energy equation
• The Energy equation is rewritten for both phases
  as
• If enthalpy where h is the
  local enthalpy due to phase change the source
  term is modified as
• For Pure substances the following relationship
  holds

Fixed Grids-2
9

• This discontinuity in Enthalpy is treated as
• The Enthalpy in a control volume where phase
  change takes place is defined as
• Where
• Thus Enthalpy in the phase change control volume
  is updated with the help of liquid fraction
• To account solidification in the momentum
  equation Darcy-type law is used to suppress
  velocity

Fixed Grids-3
10
Idea of Moving Grids

• We need to treat the interface in a manner
  consistent with Thermodynamics
• To cope up with the changes in the solution
  domain (internal / external)
• Grid will move with the applied
• boundary conditions
• When grids are moving, relative velocity in
  convective terms should be considered

Interface
Moving Grids-1
11
Modelling of One Dimensional Melting

• One phase Stefan problem with moving grid
• Semi-infinite Solid slab (TM )
• Thermophysical properties are constant
• Melting initiated by imposing TL at x0
• Solves only liquid domain

Boundary condition
1-D Solidification-1
12

• Lagrangian and Eularian Solution

• Discretisation of the governing equation

1-D Solidification-2
13
Solution Algorithm
Initial melt length at and temperature Ts
(Analytical Solution)
Calculate starting time ts (by analytical
solution)
Advance by time ?t
Create / Recreate grids in the domain Xs (t)
Solve Temperature Equation with B.C
Obtain the Inteface velocity (Stefan condition)
Calculate the new position of the Interface
n times
1-D Solidification-3
14
Moving Grid Formulation for Melting and
Solidification

• The Space Conservation Equation
• The Mass Conservation Equation
• The Momentum Conservation Equation
• The Energy Conservation Equation

General Formulation-1
15

• Additional heat conduction equation is solved for
  the solid
• Boundary Conditions
• Temperatures at Interface are equal
• Stefans Balance at the Interface
• Mass Flux is balanced at the Interface
• No Slip condition on the solid walls

General Formulation-2
16

• General form of Navier-Stokes equation
• Discretisation of the Space conservation equation
  
• Approximation of time integrals

2-D Solidification-1
17

• Discretised form of convective flux
• Discretised form of diffusive flux term
• Approximation of volume integrals
• Boundary conditions

2-D Solidification-2
18
Final form of the Discretised Equation The
algebraic equations relates the value of the
dependent variable at the CV center with the
neighboring values contribtes to
the convective and diffusive flux
transient term knowm terms like
source terms, deferred correction

• Pressure Velocity Coupling
• Obtained on the basis of the iterative SIMPLE
  algorithm developed by
• Patankar and Spalding

Guessed Pressure field
Solve velocity field from momentum Eq.
check velocity field by Continuity Eq. Use
pressure correction
2-D Solidification-3
19

• Natural Convection flow
• Due to density changes with temperature, combined
  with gravity, produce buoyancy-driven flow in the
  liquid.
• With Boussinesq approximation (density constant
  in the unsteady and convective terms) density
  contribution was allowed to vary only in the
  body-force term.
• An additional term appears as a source in the
  momentum equations as

• Grid Generation
• Once the velocities of the interface nodes and
  time step were known, the nodes were then moved
  in the mass flux direction
• The cell vertices of interface control volumes
  were subsequently placed by linear interpolation
• The interface nodes are shared by both the liquid
  and the solid domains

2-D Solidification-4
20
Solution Algorithm
Initial grid for S and L domains, values of the
dependent variables
At dt, solve Solid and Liquid domains
independently.
Solve momentum eq. for velocity components,
Solve for pressure correction
Solve energy eq. for Temperature
Determine the new position of the interface, by
using Stefan condition
Generate grid in Liquid and Solid domains
yes
increment dt
no
Calculate mass fluxes and volume change
nltn?
2-D Solidification-5
21
Important Points to be Considered

• For the wall boundaries, the mass fluxes in the
  pressure correction equation were enforced to
  zero however, at the solid-liquid interface,
  since mass is assumed to pass through the
  interface, the mass fluxes were equated to mass
  fluxes due to grid velocities for physical
  consistency
• Natural convection often leads to numerical
  instabilities and the solutions diverge after a
  few time steps due to grid skewness
• Hence a Sliding Technique was used to find new
  locations of the interface nodes
• The Sliding Technique ensured that the geometry
  of the interface in the physical space remained
  unaltered

2-D Solidification-6
22
Sliding grid points at the interface

• Buoyancy driven convection leads to the presence
  of non-uniform phase change
• To overcome these problem and to avoid grid
  skewness and clustering at the nodes on the
  interface, the nodes were allowed to slide on the
  interface

Grid without Sliding Grid with
Sliding
2-D Solidification-7
23
Algorithm to Slide Grid Points

• Criteria While recreating the grid, the nodes at
  the same proportional distances with respect to
  other as they were placed for the initial
  configuration should be recommended
• For 2D problems, the interface is located at one
  of the ends of the computational domain
• Interface (xj , yj ) j1,2,...n
• New interface (xnj , ynj )
• Distance between two points
• The new parametric coordinate along the curve

2-D Solidification-8
24

• Normalised distance

• Polynomial fitting routine

solved by first derivative equation continous
along the boundary

• With known (xj , yj ), (xnj , ynj ) are solved by
  polynomial fitting and the Variable Snj

2-D Solidification-9
25
Application to Tin Solidification

• Tin solidication in a rectangular cavity
• Thermophysical properties constant
• Solidification temperature T 231.9C
• Grid size 80X60 (80X40 in each domain)

Application-1.1
26
Grid distribution
0.077h 0.165h
0.529h 1.462h
Application-1.2
27
Streamlines and Isotherms
0.077h 0.165h 1.462h
Application-1.3
28
Application to Crystal Growth
Isothermal Wall 1066 K
Crystal
Pull

• Melt Al2O3
• Grashof No 3 x 106
• Prandtl No 0.34
• Stefan No(Liq.) 0.15
• Grid 40X60 in melt
• 20X40 in solid (Crystal)
• Incorporation of Crystal pulling velocity in the
  Stefan condition

Radiation
Radiation
2316 K
Melt
g
(2444.5 K )
(2444.5 K )
Crucible Wall (2444.5 K )
Application-2.1
29
Prediction of Czochralski Crystal Growth
Material Al2O3 Prandtl No. 0.34 Grashoff
No.5.84x106
Case1 With Crystal Rotation (42 RPM )
Case2 With Pure Buoyancy
Figures show Thermal and Velocity fields
Case3 With Crystal Rotation (90 RPM )
Application-2.2
30
Outlook and Conclusion

• Melting and Solidification occurs in many
  engineering processes.
• Moving grid formulation provides an easy way for
  thermodynamically
• consistent implementation of Stefan condition
  at the interface
• Incorrect modelling can lead to incorrect
  prediction of results
• The current modelling can also be extended to
  treat other physical
• phenomena like surface tension forces occurring
  at the interface
• Help us to understand and optimize industrial
  applications like
• Czochralski Process
• Model can be extended for other industrial
  applications like Casting
• Forging

Final Remarks