Chapter 21. Modeling Solidification and Melting

Transcription

Chapter 21.
Melting
Modeling Solidification and
This chapter describes how you can model solidification and melting in
FLUENT. Information is organized into the following sections:
• Section 21.1: Overview and Limitations of the Solidification/Melting
Model
• Section 21.2: Theory for the Solidification/Melting Model
• Section 21.3: Using the Solidification/Melting Model
21.1
21.1.1
Overview and Limitations of the
Solidification/Melting Model
Overview
FLUENT can be used to solve fluid flow problems involving solidification
and/or melting taking place at one temperature (e.g., in pure metals) or
over a range of temperatures (e.g., in binary alloys). Instead of tracking
the liquid-solid front explicitly, FLUENT uses an enthalpy-porosity formulation. The liquid-solid mushy zone is treated as a porous zone with
porosity equal to the liquid fraction, and appropriate momentum sink
terms are added to the momentum equations to account for the pressure
drop caused by the presence of solid material. Sinks are also added to
the turbulence equations to account for reduced porosity in the solid
regions.
FLUENT provides the following capabilities for modeling solidification
and melting:
• Calculation of liquid-solid solidification/melting in pure metals as
well as in binary alloys
c Fluent Inc. November 28, 2001
21-1
Modeling Solidification and Melting
• Modeling of continuous casting processes (i.e., “pulling” of solid
material out of the domain)
• Modeling of the thermal contact resistance between solidified material and walls (e.g., due to the presence of an air gap)
• Postprocessing of quantities related to solidification/melting (i.e.,
liquid fraction and pull velocities)
These modeling capabilities allow FLUENT to simulate a wide range
of solidification/melting problems, including melting, freezing, crystal
growth, and continuous casting. The physical equations used for these
calculations are described in Section 21.2, and instructions for setting up
and solving a solidification/melting problem are provided in Section 21.3.
21.1.2
Limitations
As mentioned in Section 21.1.1, the formulation in FLUENT can be used
to model the solidification/melting of pure materials, as well as alloys.
The liquid fraction versus temperature relationship used in FLUENT is
the lever rule—i.e., a linear relationship (Equation 21.2-3). Other relationships are possible [232], but are not currently available in FLUENT.
The following limitations apply to the solidification/melting model in
FLUENT:
• The solidification/melting model can be used only with the segregated solver; it is not available with the coupled solvers.
• The solidification/melting model cannot be used for compressible
flows.
• You cannot model species transport or any type of reacting flow in
conjunction with the solidification/melting model.
• Of the general multiphase models (VOF, mixture, and Eulerian),
only the VOF model can be used with the solidification/melting
model.
• You cannot specify material properties separately for the solid and
liquid materials.
21-2
21.2 Theory for the Solidification/Melting Model
21.2
Theory for the Solidification/Melting Model
An enthalpy-porosity technique [247, 248, 249] is used in FLUENT for
modeling the solidification/melting process. In this technique, the melt
interface is not tracked explicitly. Instead, a quantity called the liquid
fraction, which indicates the fraction of the cell volume that is in liquid
form, is associated with each cell in the domain. The liquid fraction is
computed at each iteration, based on an enthalpy balance.
The mushy zone is a region in which the liquid fraction lies between 0
and 1. The mushy zone is modeled as a “pseudo” porous medium in
which the porosity decreases from 1 to 0 as the material solidifies. When
the material has fully solidified in a cell, the porosity becomes zero and
hence the velocities also drop to zero.
In this section, an overview of the solidification/melting theory is given.
Please refer to [249] for details on the enthalpy-porosity method.
21.2.1
Energy Equation
The enthalpy of the material is computed as the sum of the sensible
enthalpy, h, and the latent heat, ∆H:
H = h + ∆H
(21.2-1)
where
Z
h = href +
and
href
Tref
cp
=
=
=
T
Tref
cp dT
(21.2-2)
reference enthalpy
reference temperature
specific heat at constant pressure
The liquid fraction, β, can be defined as
β = 0 if T < Tsolidus
21-3
β = 1 if T > Tliquidus
T − Tsolidus
β=
if Tsolidus < T < Tliquidus
Tliquidus − Tsolidus
(21.2-3)
Equation 21.2-3 is referred to as the lever rule. Other relationships between the liquid fraction and temperature (and species concentrations)
are possible, but are not considered here.
The latent heat content can now be written in terms of the latent heat
of the material, L:
∆H = βL
(21.2-4)
The latent heat content can vary between zero (for a solid) and L (for a
liquid).
For solidification/melting problems, the energy equation is written as
∂
(ρH) + ∇ · (ρ~v H) = ∇ · (k∇T ) + S
∂t
where
H
ρ
~v
S
=
=
=
=
(21.2-5)
enthalpy (see Equation 21.2-1)
density
fluid velocity
source term
The solution for temperature is essentially an iteration between the energy equation (Equation 21.2-5) and the liquid fraction equation (Equation 21.2-3). Directly using equation 21.2-3 to update the liquid fraction
usually results in poor convergence of the energy equation. In FLUENT,
the method suggested by Voller and Swaminathan [250] is used to update the liquid fraction. For pure metals, where Tsolidus and Tliquidus are
equal, a method based on specific heat, given by [249], is used instead.
21.2.2
Momentum Equations
The enthalpy-porosity technique treats the mushy region (partially solidified region) as a porous medium. The porosity in each cell is set equal
21-4
to the liquid fraction in that cell. In fully solidified regions, the porosity
is equal to zero, which extinguishes the velocities in these regions. The
momentum sink due to the reduced porosity in the mushy zone takes the
following form:
S=
(1 − β)2
Amush (~v − ~vp )
(β 3 + )
(21.2-6)
where β is the liquid volume fraction, is a small number (0.001) to
prevent division by zero, Amush is the mushy zone constant, and ~vp is the
solid velocity due to the pulling of solidified material out of the domain
(also referred to as the pull velocity).
The mushy zone constant measures the amplitude of the damping; the
higher this value, the steeper the transition of the velocity of the material to zero as it solidifies. Very large values may cause the solution to
oscillate.
The pull velocity is included to account for the movement of the solidified
material as it is continuously withdrawn from the domain in continuous
casting processes. The presence of this term in Equation 21.2-6 allows
newly solidified material to move at the pull velocity. If solidified material is not being pulled from the domain, ~vp = 0. More details about the
pull velocity are provided in Section 21.2.4.
21.2.3
Turbulence Equations
Sinks are added to all of the turbulence equations in the mushy and
solidified zones to account for the presence of solid matter. The sink
term is very similar to the momentum sink term (Equation 21.2-6):
S=
(1 − β)2
Amush φ
(β 3 + )
(21.2-7)
where φ represents the turbulence quantity being solved (k, , ω, etc.),
and the mushy zone constant, Amush , is the same as the one used in
Equation 21.2-6.
21-5
21.2.4
Pull Velocity for Continuous Casting
In continuous casting processes, the solidified matter is usually continuously pulled out from the computational domain, as shown in Figure 21.2.1. Consequently, the solid material will have a finite velocity
that needs to be accounted for in the enthalpy-porosity technique.
mushy zone
solidified shell
vp
liquid pool
wall
Figure 21.2.1: “Pulling” a Solid in Continuous Casting
As mentioned in Section 21.2.2, the enthalpy-porosity approach treats
the solid-liquid mushy zone as a porous medium with porosity equal to
the liquid fraction. A suitable sink term is added in the momentum
equation to account for the pressure drop due to the porous structure of
the mushy zone. For continuous casting applications, the relative velocity
between the molten liquid and the solid is used in the momentum sink
term (Equation 21.2-6) rather than the absolute velocity of the liquid.
The exact computation of the pull velocity for the solid material is dependent on the Young’s modulus and Poisson’s ratio of the solid and
the forces acting on it. FLUENT uses a Laplacian equation to approximate the pull velocities in the solid region based on the velocities at the
boundaries of the solidified region:
21-6
∇2~vp = 0
(21.2-8)
FLUENT uses the following boundary conditions when computing the
pull velocities:
• At a velocity inlet, a stationary wall, or a moving wall, the specified
velocity is used.
• At all other boundaries (including the liquid-solid interface between
the liquid and solidified material), a zero-gradient velocity is used.
The pull velocities are computed only in the solid region.
Note that FLUENT can also use a specified constant value or custom field
function for the pull velocity, instead of computing it. See Section 21.3.2
for details.
21.2.5
Contact Resistance at Walls
FLUENT’s solidification/melting model can account for the presence of
an air gap between the walls and the solidified material, using an additional heat transfer resistance between walls and cells with liquid fraction
less than 1. This contact resistance is accounted for by modifying the
conductivity of the fluid near the wall. Thus, the wall heat flux, as shown
in Figure 21.2.2, is written as
q=
(T − Tw )
(l/k + Rc (1 − β))
(21.2-9)
where T , Tw , and l are defined in Figure 21.2.2, k is the thermal conductivity of the fluid, β is the liquid volume fraction, and Rc is the contact
resistance, which has the same units as the inverse of the heat transfer
coefficient.
21-7
wall
near-wall cell
Tw
●
T
l
T
Tw
●
●
Rc
●
l/k
Figure 21.2.2: Circuit for Contact Resistance
21-8
21.3 Using the Solidification/Melting Model
21.3
Using the Solidification/Melting Model
21.3.1
Setup Procedure
The procedure for setting up a solidification/melting problem is described below. (Note that this procedure includes only those steps necessary for the solidification/melting model itself; you will need to set up
other models, boundary conditions, etc. as usual.)
1. To activate the solidification/melting model, turn on the Solidification/Melting option in the Solidification and Melting panel (Figure 21.3.1).
Define −→ Models −→Solidification & Melting...
Figure 21.3.1: The Solidification and Melting Panel
FLUENT will automatically enable the energy equation, so you do
not have to visit the Energy panel before turning on the solidification/melting model.
2. Under Parameters, specify the value of the Mushy Zone Constant
(Amush in Equation 21.2-6).
Values between 104 and 107 are recommended for most computations. The higher the value of the Mushy Zone Constant, the steeper
the damping curve becomes, and the faster the velocity drops to
21-9
zero as the material solidifies. Very large values may cause the solution to oscillate as control volumes alternately solidify and melt
with minor perturbations in liquid volume fraction.
3. If you want to include the pull velocity in your simulation (as
described in Sections 21.2.2 and 21.2.4), turn on the Include Pull
Velocities option under Parameters.
4. If you are including pull velocities and you want FLUENT to compute them (using Equation 21.2-8) based on the specified velocity
boundary conditions, as described in Section 21.2.4, turn on the
Compute Pull Velocities option and specify the number of Flow Iterations Per Pull Velocity Iteration.
!
It is not necessary to have FLUENT compute the pull velocities.
See Section 21.3.2 for information about other approaches.
The default value of 1 for the Flow Iterations Per Pull Velocity Iteration indicates that the pull velocity equations will be solved after
each iteration of the solver. If you increase this value, the pull
velocity equations will be solved less frequently. You may want to
increase the number of Flow Iterations Per Pull Velocity Iteration if
the liquid fraction equation is almost converged (i.e., the position
of the liquid-solid interface is not changing very much). This will
speed up the calculation, although the residuals may jump when
the pull velocities are updated.
5. In the Materials panel, specify the Melting Heat (L in Equation 21.2-3),
Solidus Temperature (Tsolidus in Equation 21.2-3), and Liquidus Temperature (Tliquidus in Equation 21.2-3) for the material being used
in your model.
Define −→Materials...
6. Set the boundary conditions.
Define −→Boundary Conditions...
In addition to the usual boundary conditions, consider the following:
• If you want to account for the presence of an air gap between a wall and an adjacent solidified region (as described
21-10
in Section 21.2.5), specify a non-zero value, a profile, or a userdefined function for Contact Resistance (Rc in Equation 21.2-9)
under Thermal Conditions in the Wall panel.
• If you want to specify the gradient of the surface tension with
respect to the temperature at a wall boundary, you can use
the Marangoni Stress option for the wall Shear Condition. See
Section 6.13.1 for details.
• If you want FLUENT to compute the pull velocities during
the calculation, note how your specified velocity conditions
are used in this calculation (see Section 21.2.4).
Section 21.3.2 contains additional information about modeling continuous casting. See Sections 21.3.3 and 21.3.4 for information about solving
a solidification/melting model and postprocessing the results.
21.3.2
Procedures for Modeling Continuous Casting
As described in Sections 21.2.2 and 21.2.4, you can include the pull
velocities in your solidification/melting calculation to model continuous
casting. There are three approaches to modeling continuous casting in
FLUENT:
• Specify constant or variable pull velocities.
To use this approach (the default), do not turn on the Compute
Pull Velocities option.
If you use this approach, you will need to patch constant values
or custom field functions for the pull velocities, after you initialize
the solution.
Solve −→ Initialize −→Patch...
See Section 22.13.2 for details about patching values. Note that
it is acceptable to patch values for the pull velocities in the entire
domain, because the patched values will be used only if the liquid
fraction, β, is less than 1.
21-11
• Have FLUENT compute the pull velocities (using Equation 21.2-8)
during the calculation, based on the specified velocity boundary
conditions.
To use this approach, turn on the Compute Pull Velocities option.
This method is computationally expensive, and is recommended
only if the pull velocities are strongly dependent on the location of
the liquid-solid interface.
If you have FLUENT compute the pull velocities, then there are no
additional inputs or setup procedures beyond those presented in
Section 21.3.1.
• Have FLUENT compute the pull velocities just once, and then use
those values for the remainder of the calculation.
To use this approach, perform one iteration with FLUENT computing the pull velocities, and then turn off the Compute Pull Velocities option and continue the calculation. For the remainder of
the calculation, FLUENT will use the values computed for the pull
velocities at the first iteration.
21.3.3
Solution Procedure
Before solving the coupled fluid flow and heat transfer problem, you
may want to patch an initial temperature or solve the steady conduction problem as an initial condition. The coupled problem can then be
solved as either steady or unsteady. Because of the non-linear nature of
these problems, however, in most cases an unsteady solution approach
is preferred.
You can specify the under-relaxation factor applied to the liquid fraction
equation in the Solution Controls panel.
Solve −→ Controls −→Solution...
Specify the desired value in the Liquid Fraction Update field under UnderRelaxation Factors. This sets the value of αβ in the following equation for
updating the liquid fraction from one iteration (n) to the next (n + 1):
βn+1 = βn + αβ ∆β
21-12
(21.3-1)
where ∆β is the predicted change in liquid fraction.
In many cases, there is no need to change the default value of αβ . If,
however, there are convergence difficulties, reducing the value may improve the solution convergence. Convergence difficulties can be expected
in steady-state calculations, continuous casting simulations, and simulations where a large value of the mushy zone constant is used.
21.3.4
Postprocessing
For solidification/melting calculations, you can generate graphical plots
or alphanumeric reports of the following items, which are all available in
the Solidification/Melting... category of the variable selection drop-down
list that appears in postprocessing panels:
• Liquid Fraction
• Contact Resistivity
• X, Y, Z or Axial, Radial, Swirl Pull Velocity
The first two items are available for all solidification/melting simulations,
and the others will appear only if you are including pull velocities (either
computed or specified) in the simulation. See Chapter 27 for a complete
list of field functions and their definitions. Chapters 25 and 26 explain
how to generate graphics displays and reports of data.
Figure 21.3.2 shows filled contours of liquid fraction for a continuous
crystal growth simulation.
21-13
1.00e+00
9.00e-01
8.00e-01
7.00e-01
6.00e-01
5.00e-01
4.00e-01
3.00e-01
2.00e-01
1.00e-01
0.00e+00
Contours of Liquid Fraction
Figure 21.3.2: Liquid Fraction Contours for Continuous Crystal Growth
21-14

Chapter 21. Modeling Solidification and Melting

Transcription

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