This chapter describes how you can model solidification and melting in FLUENT. Information is organized into the following sections:


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1 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 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 liquidsolid front explicitly, FLUENT uses an enthalpyporosity formulation. The liquidsolid 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 liquidsolid solidification/melting in pure metals as well as in binary alloys c Fluent Inc. November 28,
2 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 Limitations As mentioned in Section , 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 ). 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 c Fluent Inc. November 28, 2001
3 21.2 Theory for the Solidification/Melting Model 21.2 Theory for the Solidification/Melting Model An enthalpyporosity 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 enthalpyporosity method Energy Equation The enthalpy of the material is computed as the sum of the sensible enthalpy, h, and the latent heat, H: where H = h + H (21.21) T h = h ref + c p dt T ref (21.22) and h ref = reference enthalpy T ref = reference temperature c p = specific heat at constant pressure The liquid fraction, β, can be defined as β =0 if T<T solidus c Fluent Inc. November 28,
4 Modeling Solidification and Melting β = β =1 if T>T liquidus T T solidus T liquidus T solidus if T solidus <T <T liquidus (21.23) Equation 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 (21.24) 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 vh) = (k T )+S (21.25) t where H = enthalpy (see Equation ) ρ = density v = fluid velocity S = source term The solution for temperature is essentially an iteration between the energy equation (Equation ) and the liquid fraction equation (Equation ). Directly using equation 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 T solidus and T liquidus are equal, a method based on specific heat, given by [249], is used instead Momentum Equations The enthalpyporosity technique treats the mushy region (partially solidified region) as a porous medium. The porosity in each cell is set equal 214 c Fluent Inc. November 28, 2001
5 21.2 Theory for the Solidification/Melting Model 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 (β 3 + ɛ) A mush( v v p ) (21.26) where β is the liquid volume fraction, ɛ is a small number (0.001) to prevent division by zero, A mush is the mushy zone constant, and v p 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 allows newly solidified material to move at the pull velocity. If solidified material is not being pulled from the domain, v p = 0. More details about the pull velocity are provided in Section 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 ): S = (1 β)2 (β 3 + ɛ) A mushφ (21.27) where φ represents the turbulence quantity being solved (k, ɛ, ω, etc.), and the mushy zone constant, A mush, is the same as the one used in Equation c Fluent Inc. November 28,
6 Modeling Solidification and Melting 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 Consequently, the solid material will have a finite velocity that needs to be accounted for in the enthalpyporosity technique. mushy zone solidified shell v p liquid pool wall Figure : Pulling a Solid in Continuous Casting As mentioned in Section , the enthalpyporosity approach treats the solidliquid 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 ) 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: 216 c Fluent Inc. November 28, 2001
7 21.2 Theory for the Solidification/Melting Model 2 v p = 0 (21.28) 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 liquidsolid interface between the liquid and solidified material), a zerogradient 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 for details 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 , is written as q = (T T w ) (l/k + R c (1 β)) (21.29) where T, T w,andl are defined in Figure , k is the thermal conductivity of the fluid, β is the liquid volume fraction, and R c is the contact resistance, which has the same units as the inverse of the heat transfer coefficient. c Fluent Inc. November 28,
8 Modeling Solidification and Melting wall nearwall cell T w T l T w R c l / k Figure : Circuit for Contact Resistance T 218 c Fluent Inc. November 28, 2001
9 21.3 Using the Solidification/Melting Model 21.3 Using the Solidification/Melting Model 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 ). Define Models Solidification & Melting... Figure : 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 (A mush in Equation ). Values between 10 4 and 10 7 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 c Fluent Inc. November 28,
10 Modeling Solidification and Melting 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 and ), 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 ) based on the specified velocity boundary conditions, as described in Section , 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 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 liquidsolid 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 ), Solidus Temperature (T solidus in Equation ), and Liquidus Temperature (T liquidus in Equation ) for the material being used in your model. Define Materials 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 c Fluent Inc. November 28, 2001
11 21.3 Using the Solidification/Melting Model in Section ), specify a nonzero value, a profile, or a userdefined function for Contact Resistance (R c in Equation ) 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 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 ). Section contains additional information about modeling continuous casting. See Sections and for information about solving a solidification/melting model and postprocessing the results Procedures for Modeling Continuous Casting As described in Sections and , 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 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, islessthan1. c Fluent Inc. November 28,
12 Modeling Solidification and Melting Have FLUENT compute the pull velocities (using Equation ) 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 liquidsolid interface. If you have FLUENT compute the pull velocities, then there are no additional inputs or setup procedures beyond those presented in Section 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 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 nonlinear nature of these problems, however, in most cases an unsteady solution approach is preferred. You can specify the underrelaxation 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 Under Relaxation Factors. This sets the value of α β in the following equation for updating the liquid fraction from one iteration (n) to the next (n + 1): β n+1 = β n + α β β (21.31) c Fluent Inc. November 28, 2001
13 21.3 Using the Solidification/Melting Model where β is the predicted change in liquid fraction. In many cases, there is no need to change the default value of α β. If, however, there are convergence difficulties, reducing the value may improve the solution convergence. Convergence difficulties can be expected in steadystate calculations, continuous casting simulations, and simulations where a large value of the mushy zone constant is used 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 dropdown 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 shows filled contours of liquid fraction for a continuous crystal growth simulation. c Fluent Inc. November 28,
14 Modeling Solidification and Melting 1.00e e e e e e e e e e e+00 Contours of Liquid Fraction Figure : Liquid Fraction Contours for Continuous Crystal Growth c Fluent Inc. November 28, 2001
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