Solved with COMSOL Multiphysics 4.4 1 | TIN MELTING FRONT Tin Melting Front Introduction This example demonstrates how to model phase transition by a moving boundary interface according to the Stefan problem. It is adapted from the benchmark study in Ref. 1. A square cavity containing both solid and liquid tin is submitted to a temperature difference between left and right boundaries. Fluid and solid parts are solved in separate domains sharing a moving melting front (see Figure 1). The position of this boundary through time is calculated according to the Stefan energy balance condition. Liquid Solid Liquid Solid Convection Figure 1: Square cavity with moving phase interface In the melt, motion generated by natural convection is expected due to the temperature gradient. This motion, in turn, influences the front displacement. Model Definition The geometry presented in Figure 2 shows a square of side length 10 cm filled with pure tin. The left and right boundaries are maintained at 508 K and 503 K,
Transcript
Solved with COMSOL Multiphysics 4.4
T i n Me l t i n g F r on t
Introduction
This example demonstrates how to model phase transition by a moving boundary interface according to the Stefan problem. It is adapted from the benchmark study in Ref. 1.
A square cavity containing both solid and liquid tin is submitted to a temperature difference between left and right boundaries. Fluid and solid parts are solved in separate domains sharing a moving melting front (see Figure 1). The position of this boundary through time is calculated according to the Stefan energy balance condition.
Liquid Solid
Liquid Solid
Convection
Figure 1: Square cavity with moving phase interface
In the melt, motion generated by natural convection is expected due to the temperature gradient. This motion, in turn, influences the front displacement.
Model Definition
The geometry presented in Figure 2 shows a square of side length 10 cm filled with pure tin. The left and right boundaries are maintained at 508 K and 503 K,
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respectively. Because the fusion temperature of pure tin is 505 K, both liquid and solid phases co-exist in the square.
Liquid Solid
Hottemperature
(508 K)
Coldtemperature
(503 K)
Fusiontemperature
(505 K)
Thermal insulation
Thermal insulation
Figure 2: Geometry and boundary conditions at starting time.
The initial temperature distribution is assumed to vary linearly in the horizontal direction as shown in Figure 3. The melting front is the vertical line located at x = 6 cm where the temperature is 505 K.
Figure 3: Initial temperature profile.
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The liquid part on the left is governed by the Navier-Stokes equations in the Boussinesq approximation:
In these expressions, ρ0 (kg/m3) is the reference density of the fluid, ρ (kg/m3) is the linearized density formula used for the gravity force, β (1/K) is the tin coefficient of thermal expansion, and Tf (K) denotes the fusion temperature of tin.
As the metal melts, the solid-liquid interface moves toward the solid side. The energy balance at this front is expressed by
(1)
where ΔH is the latent heat of fusion, equal to 60 kJ/kg, v (m/s) is the front velocity vector, n is the normal vector at the front, and Φl and Φs (W/m2) are the heat fluxes coming from the liquid and solid sides, respectively (see Figure 4).
Liquid Solid
Melting front
Φl
Φs
ΔH
Figure 4: Heat fluxes at the melting front
ρ0 t∂∂u ρ0 u ∇⋅( )+ u ∇p ∇+ μ ∇u ∇u( )T
+( )⋅–= ρg+
∇ u⋅ 0=
ρ ρ0β T Tf–( )=
ρ0ΔHv n⋅ Φl Φs–( ) n⋅=
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Table 1 reviews the material properties of tin (Ref. 2) used in this model.
PARAMETER DESCRIPTION VALUE
ρ0 Density 7500 kg/m3
Cp Heat capacity 200 J/(kg·K)
k Thermal conductivity 60 W/(m·K)
β Coefficient of thermal expansion 2.67·10-4 K-1
ν Kinematic viscosity 8.0·10-7 m2/s
Tf Fusion temperature 505 K
ΔH Latent heat of fusion 60 kJ/kg
Results and Discussion
Figure 5 shows the velocity profile in the fluid domain. The convective cell due to buoyancy increases the melting speed at the upper part of the cavity.
Figure 5: Velocity profile in the fluid at the end of the simulation.
TABLE 1: MATERIAL PROPERTIES OF TIN
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At the end of the simulation, the melting front does not move anymore because balance between left and right adjacent fluxes has been reached.
In Figure 6, the temperature profile is represented jointly by a heat flux arrow plot.
Figure 6: Temperature profile at the end of the simulation.
Notes About the COMSOL Implementation
The quantities Φl and Φs, illustrated in Figure 4, can be computed using the up and down operators. The components of Φl − Φs would then be given by up(nitf.tfluxx)-down(nitf.tfluxx) and up(nitf.tfluxy)-down(nitf.tfluxy). However, this method evaluates the temperature gradient which may lead to imprecisions due to the mesh discretization. Instead, the quantity (Φl − Φs) ⋅ n, involved in Equation 1, is more precisely evaluated through the Lagrange multiplier for temperature, T_lm. This variable is available when weak constraints are enabled in the region of interest, as it is the case here with the fixed temperature constraint at the melting front. For more information about weak constraints, refer to the section Weak Constraint in the COMSOL Multiphysics Reference Manual.
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To handle the melting front movement, a mesh deformation is necessary. During such a transformation, matter from solid tin is removed while the same amount of liquid tin is added to the fluid. The appropriate tool for deforming the mesh without reflecting any expansion or contraction effects to the material properties is the Deformed Geometry interface.
References
1. F. Wolff and R. Viskanta, “Solidification of a Pure Metal at a Vertical Wall in the Presence of Liquid Superheat,” Int.J. Heat and Mass Transfer, vol. 31, no. 8, pp. 1735–1744, 1988.
2. V. Alexiades, N. Hannoun, and T.Z. Mai, “Tin Melting: Effect of Grid Size and Scheme on the Numerical Solution,” Proc. 5th Mississippi State Conf. Differential Equations and Computational Simulations, pp. 55–69, 2003.
Model Library path: Heat_Transfer_Module/Phase_Change/tin_melting_front
Modeling Instructions
From the File menu, choose New.
N E W
1 In the New window, click the Model Wizard button.
M O D E L W I Z A R D
1 In the Model Wizard window, click the 2D button.
2 In the Select physics tree, select Heat Transfer>Conjugate Heat Transfer>Laminar Flow
(nitf).
3 Click the Add button.
4 In the Select physics tree, select Mathematics>Deformed Mesh>Deformed Geometry
(dg).
5 Click the Add button.
6 Click the Study button.
7 In the tree, select Preset Studies for Selected Physics>Time Dependent.
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8 Click the Done button.
G L O B A L D E F I N I T I O N S
Parameters1 On the Home toolbar, click Parameters.
2 In the Parameters settings window, locate the Parameters section.
3 In the table, enter the following settings:
G E O M E T R Y 1
Square 11 In the Model Builder window, under Component 1 right-click Geometry 1 and choose
Square.
2 In the Square settings window, locate the Size section.
3 In the Side length edit field, type 0.1.
4 Click to expand the Layers section. In the table, enter the following settings:
Material 11 In the Model Builder window, under Component 1 right-click Materials and choose
New Material.
2 Select Domain 2 only.
3 In the Material settings window, locate the Material Contents section.
4 In the table, enter the following settings:
5 Right-click Component 1>Materials>Material 1 and choose Rename.
6 Go to the Rename Material dialog box and type Tin (solid) in the New name edit field.
7 Click OK.
Material 21 Right-click Materials and choose New Material.
2 Select Domain 1 only.
3 Right-click Component 1>Materials>Material 2 and choose Rename.
4 Go to the Rename Material dialog box and type Tin (liquid) in the New name edit field.
5 Click OK.
Before defining the material properties of liquid tin, indicate which is the fluid domain to let COMSOL Multiphysics flag what properties you need to specify.
C O N J U G A T E H E A T TR A N S F E R
Fluid 1Select Domain 1 only.
M A T E R I A L S
Tin (liquid)1 In the Model Builder window, under Component 1>Materials click Tin (liquid).
Property Name Value Unit Property group
Thermal conductivity k k_Sn W/(m·K) Basic
Density rho rho_Sn kg/m³ Basic
Heat capacity at constant pressure Cp Cp_Sn J/(kg·K) Basic
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2 In the Material settings window, locate the Material Contents section.
3 In the table, enter the following settings:
C O N J U G A T E H E A T TR A N S F E R
Initial Values 11 In the Model Builder window, under Component 1>Conjugate Heat Transfer click Initial
Values 1.
2 In the Initial Values settings window, locate the Initial Values section.
3 In the p edit field, type p_ref.
Define the initial temperature as a function of Xg, the first coordinate on the undeformed geometry.
4 In the T edit field, type Th-Xg/0.1[m]*(Th-Tc).
Volume Force 11 On the Physics toolbar, click Domains and choose Volume Force.
2 Select Domain 1 only.
3 In the Volume Force settings window, locate the Volume Force section.
4 Specify the F vector as
Temperature 11 On the Physics toolbar, click Boundaries and choose Temperature.
2 Select Boundary 1 only.
3 In the Temperature settings window, locate the Temperature section.
4 In the T0 edit field, type Th.
Property Name Value Unit Property group
Density rho rho_Sn kg/m³ Basic
Dynamic viscosity mu rho_Sn*nu_Sn Pa·s Basic
Thermal conductivity k k_Sn W/(m·K) Basic
Heat capacity at constant pressure
Cp Cp_Sn J/(kg·K) Basic
Ratio of specific heats gamma 1.4 1 Basic
0 x
rho_Sn*g_const*beta_Sn*(T-Tf) y
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Temperature 21 On the Physics toolbar, click Boundaries and choose Temperature.
2 Select Boundary 4 only.
3 In the Temperature settings window, locate the Temperature section.
4 In the T0 edit field, type Tf.
Show advanced physics options as follows to enable weak constraints on the melting front. This creates the Lagrange multiplier for temperature, which evaluates the heat flux jump between the adjacent liquid and solid domains more accurately.
5 In the Model Builder window’s toolbar, click the Show button and select Advanced
Physics Options in the menu.
6 Click to expand the Constraint settings section. Locate the Constraint Settings section. Select the Use weak constraints check box.
Temperature 31 On the Physics toolbar, click Boundaries and choose Temperature.
2 Select Boundary 7 only.
3 In the Temperature settings window, locate the Temperature section.
4 In the T0 edit field, type Tc.
Pressure Point Constraint 11 On the Physics toolbar, click Points and choose Pressure Point Constraint.
2 Select Point 1 only.
3 In the Pressure Point Constraint settings window, locate the Pressure Constraint section.
4 In the p0 edit field, type p_ref.
The model only contains information about the pressure gradient and estimates the pressure field up to a constant. To define this constant, you arbitrarily fix the pressure at a point.
D E F O R M E D G E O M E T R Y
Free Deformation 11 On the Physics toolbar, click Domains and choose Free Deformation.
2 In the Free Deformation settings window, locate the Domain Selection section.
3 From the Selection list, choose All domains.
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Prescribed Mesh Displacement 21 On the Physics toolbar, click Boundaries and choose Prescribed Mesh Displacement.
2 Select Boundaries 1 and 7 only.
3 In the Prescribed Mesh Displacement settings window, locate the Prescribed Mesh
Displacement section.
4 Clear the Prescribed y displacement check box.
Prescribed Mesh Displacement 11 In the Model Builder window, under Component 1>Deformed Geometry click
Prescribed Mesh Displacement 1.
2 In the Prescribed Mesh Displacement settings window, locate the Prescribed Mesh
Displacement section.
3 Clear the Prescribed x displacement check box.
Prescribed Normal Mesh Velocity 11 On the Physics toolbar, click Boundaries and choose Prescribed Normal Mesh Velocity.
2 Select Boundary 4 only.
3 In the Prescribed Normal Mesh Velocity settings window, locate the Normal Mesh
Velocity section.
4 In the vn edit field, type T_lm[W/m^2]/(rho_Sn*DelH).
The variable T_lm is the Lagrange multiplier for temperature.
M E S H 1
1 In the Model Builder window, under Component 1 click Mesh 1.
2 In the Mesh settings window, locate the Mesh Settings section.
3 From the Element size list, choose Fine.
4 Click the Build All button.
S T U D Y 1
Step 1: Time Dependent1 In the Model Builder window, under Study 1 click Step 1: Time Dependent.
2 In the Time Dependent settings window, locate the Study Settings section.
3 Click the Range button.
4 Go to the Range dialog box.
5 In the Step edit field, type 100.
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6 In the Stop edit field, type 10000.
7 Click the Replace button.
8 On the Home toolbar, click Compute.
R E S U L T S
Velocity (nitf)The first default plot shows the velocity profile in the fluid region. To reproduce Figure 5, add arrows of the velocity field to visualize the convective flow direction.
1 Right-click Results>Velocity (nitf) and choose Arrow Surface.
2 In the Arrow Surface settings window, locate the Coloring and Style section.
3 From the Arrow type list, choose Cone.
4 From the Color list, choose Black.
5 On the 2D plot group toolbar, click Plot.
Temperature (nitf)The second default plot shows the temperature profile. To reproduce Figure 6, add arrows of the heat flux field to see the relation between temperature and velocity.
1 In the Model Builder window, under Results click Temperature (nitf).
2 Right-click Results>Temperature (nitf) and choose Arrow Surface.
3 In the Arrow Surface settings window, click Total heat flux (Material)
(nitf.tfluxx,...,nitf.tfluxy) in the upper-right corner of the Expression section. Locate the Coloring and Style section. From the Arrow type list, choose Cone.
4 From the Color list, choose Black.
5 On the 2D plot group toolbar, click Plot.
Finally, plot the mesh deformation as follows.
2D Plot Group 31 On the Home toolbar, click Add Plot Group and choose 2D Plot Group.
2 In the Model Builder window, under Results right-click 2D Plot Group 3 and choose Mesh.
3 In the Mesh settings window, locate the Color section.
4 From the Element color list, choose None.
5 From the Wireframe color list, choose Blue.
6 Right-click Results>2D Plot Group 3>Mesh 1 and choose Filter.
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7 In the Filter settings window, locate the Element Selection section.
8 In the Logical expression for inclusion edit field, type dom==1.
This logical expression restricts the plot to the fluid domain.
9 On the 2D plot group toolbar, click Plot.
10 Right-click Mesh 1 and choose Duplicate.
11 In the Mesh settings window, locate the Color section.
12 From the Wireframe color list, choose Red.
13 In the Model Builder window, expand the Results>2D Plot Group 3>Mesh 2 node, then click Filter 1.
14 In the Filter settings window, locate the Element Selection section.
15 In the Logical expression for inclusion edit field, type dom==2.
This logical expression filters the solid domain.
16 In the Model Builder window, click 2D Plot Group 3.
17 On the 2D plot group toolbar, click Plot.
18 Right-click 2D Plot Group 3 and choose Rename.
19 Go to the Rename 2D Plot Group dialog box and type Mesh deformation in the New
name edit field.
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20 Click OK.
The plot should look like that in the figure below.
