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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Application of the Mushy Cell Tracking Method for Gallium Melting
Problem
S. J. Liang1*, Y. J. Jan2, M. S. Chung3
1
2
3
Department of Water Resources Engineering, Feng Chia University, Taichung, 407 Taiwan, China
Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung, 805 Taiwan, China
Civil, Hydraulics, and Informatics Research Center, Sinotech Engineering Consultants, Inc., Taipei, 105 Taiwan,
China
Email: [email protected]
Abstract Gallium melting in a rectangular cavity heated from the side has been studied experimentally [1] and
numerically [2-4]. However, their findings are not consistent, such as, disagreement between numerics and
experiments, the myth of grid-converged solution, is the multicellar solution physically correct?, etc. We re-examine
this problem with the thermally driven mushy cell tracking method [5, 6]. In the method, computational domain is
separated into solid-phase region and liquid-phase region, with the mushy cells in between. The thermo-fluid field is
expressed in terms of primitive variables and discretized on the fixed grid with the finite-volume formulation, and
solved by the SIMPLE algorithm.The interface tracking equation is based on the mass and energy balance to predict
the movement of mushy cells. Prediction of the moving interface and thermo-fluid field of Gallium melting agrees well
with the experiment data [1] and other numerical solutions [2-4].
Key words: phase-change, mushy cells, tracking algorithm, SIMPLE
INTRODUCTION
Study of phase-change problems encountered in natural environments and industry processing has been active in
computational fluid dynamics communities for the last two decades [7, 8]. The problems are characterized by internal
boundaries or interfaces demarcating regions with different physico-chemical properties. The problem encounters
physical complexities, such as the coupled and nonlinear physical mechanisms involving wide range of time and space
scales, and geometrical complexities, such as multibody configurations and bodies with complex shapes. The moving
interface itself is to be determined as part of the solution of the system of equations which governed the behavior of the
thermo-fluid field. The free and moving boundary problems are often difficult to analyze due to geometric and
physical complexities, among other factors. Analytic solution for such problem is available for limited cases with
simple geometry [9]. Numerical solution is another option and seems very promising. Basically numerical study of the
free/moving boundary problems can be classified into three categories: Enthalpy-based methods [10, 11], moving
particles methods [12], and boundary tracking methods [5, 6, 11],. Viswanath and Jaluria [3] and Shyy [7] give detail
review and comparison of these methods.
When a numerical method is applied to solve the problems such as the solidification of liquid and the melting of solid,
the solutions for the equations describing the conservations of mass, momentum, and energy as well as tracking of
moving interface are required. In this study, the cell-by-cell thermally driven mushy cell tracking algorithm [5, 6] is
used to simulate the 1D solidification of water and 2D Gallium melting. The computational domain is separated into
solid-phase region and liquid-phase region, with the mushy cells in between. The transport phenomenon of solid-phase
region is governed by the stable diffusion process, and the transport phenomenon of liquid-phase region is governed by
the conditionally stable convection-diffusion process. The movement of the interface, which is referred to as the
computational mushy cell in the present study, for such a problem is governed by the principle of mass and energy
balance. Detailed description of the method and its applications can be found in [5, 6].
Two model problems are tested to validate the algorithm: One is the 1D solidification of water without consideration of
convection effect [14], the other is the melting of Gallium in a rectangular cavity with consideration of convection
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effect [1-4, 11, 11, 15, 16]. The first problem is the so called Stefan problem which the position of the interface is
known and can be used to access the accuracy of the model prediction. The second problem has been studied
experimentally [1] and numerically [2-4]. Hannoun, et al. [4] gave a good review of the Gallium melting problem and
tried to resolve the controversies of the problem. We re-examine this problem with the mushy cell tracking method and
hope to shed some light on this controversial problem.
MATHEMATICAL MODEL
The governing equation for solid phase is
ρscp
∂Ts
= k s ∇2Ts
∂t
(1)
and the governing equations for the liquid phase are
r r
∇ ⋅U = 0
(2)
r
r
r
⎡ ∂U r r r ⎤
r
+ ∇ ⋅ UU ⎥ = −∇P + μ∇ 2U − ρ l g β T (Tl − Tref )
ρl ⎢
⎣ ∂t
⎦
( )
(3)
⎡ ∂Tl r r ⎤
+ ∇ ⋅ UTl ⎥ = kl ∇ 2Tl
∂
t
⎣
⎦
( )
ρl cp ⎢
(4)
where subscript s and l denote the solid phase and liquid phase, respectively; ρ density, c p specific heat, k thermal
r
r
conductivity, T temperature; μ dynamic viscosity, βT coefficient of thermal expansion, g gravity acceleration, U
velocity vector, P pressure, and Tref is the reference temperature, taken to be the melting temperature Tm . The
boundary condition of the solid and liquid interface is expressed as
r
r r
r r
r
ρ s ( ΔH )Vm ⋅ n = ks ∇Ts ⋅ n − kl ∇Tl ⋅ n
(5)
r
r
r
where ΔH denotes the latent heat of solidification; n is the outward and normal vector to the moving front; ∇Ts ⋅ n
r
r
r
and ∇Tl ⋅ n denote the normal derivative of T in the solid region and liquid region, respectively; Vm is the velocity of
the moving front.
NUMERICAL METHOD
Mushy cell tracking algorithm is based on the mass and energy balance at the mush cells which separate the
solid-phase region and liquid-phase region. Comparison of the mush cell tracking approach with other numerical
free/moving boundary approaches, such as the Lagrangian moving particles method and the enthalpy method, and
applications of the method to phase-change problems can be found in Jan [5, 6].
1. Mushy Cell Tracking Equation The equations of mass and energy conservation of mushy cells can be written in
the integral form as follows
G
G G
∂
ρ d Λ + v∫ ρ V − W ⋅ dA = 0
∫
Ω
∂Ω
m
∂t m
(
)
(6)
G
G G
G G
∂
ρ H d Λ + v∫ ρ H V − W ⋅ dA = v∫ q ⋅dA
(7)
∫
Ω
∂Ω
∂Ω
m
m
∂t m
r
r
where W is the control surface velocity. Ω m is the fixed control volume. q is the heat flux due to conduction. Two
(
)
assumptions are made: (1) There is no flow between two mushy cells, such as, the mushy cells are treated as a porous
medium with very high flow resistance in contrast with the fluid flow in the liquid region; (2) There is no temperature
gradient across two mushy cells, because the temperature of the mushy cell is specified as Tm . The control volume
formulation of Eqs. (6) and (7) for the mushy cells becomes
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∂ ρ
Ωm
r r
r
+ ∑ ⎡ ρ l Vml − 0 ⋅ Aml ⎤ = 0
⎣
⎦
(
Ωm
∂t
∂ ρH
Ωm
Ωm
∂t
)
(8)
r r
r
r
r
r
r
+ ∑ ⎡ ρ l H ml Vml − 0 ⋅ Aml ⎤ = ∑ k s ∇Ts ⋅ Ams + ∑ kl ∇Tl ⋅ Aml
⎣
⎦
(
)
(
)
(
)
(9)
r
where Aml is the face normal vector from the mushy cell Ω m pointing outwards and towards the neighboring cell of
r
liquid phase. The associated enthalpy H ml is evaluated at the center of this face. Vml is the convection velocity at the
face between mushy and liquid cells. The volume-averaged solid fraction F m is defined as the volume average of the
local solid fraction F of the mushy cell
F
m
1
Ωm
≡
∫
Ωm
Noted that ρ
r
∑ρ V
l ml
F dΛ
Ωm
(10)
is a time-dependent property due to F
r
∂
ρ
⋅ Aml = − Ωm
∂t
m
is a function of time. Hence, Eq. (8) can be rewritten as
(11)
Ωm
To complete the derivation of the mushy cell tracking equation, the additional condition for the enthalpy of solid or
liquid commonly used in thermodynamics is required:
H + Cl (Tl − Tm ), Ω l + ∂Ω ml
⎧⎪ H l ≡ CsTm + Δ
{
Latent Heat
⎨
⎩⎪ H s ≡ CsTs , Ω s + ∂Ω ms
(12)
With Eqs. (11) and (12), Eq. (9) becomes
⎛ ∂ ρH
Ωm ⎜
⎜
∂t
⎝
Ωm
− H ml
∂ ρ
∂t
Ωm
⎞
r
r
r
r
⎟ = ∑ k s ∇Ts ⋅ Ams + ∑ kl ∇Tl ⋅ Aml
⎟
⎠
(
The volume-averaged properties ρ H
)
Ωm
and ρ
(
Ωm
)
(13)
are given as follows
⎧ ρH
= F m ( ρ s csTm ) + (1 − F m ) ρ l ( csTm + ΔH )
Ωm
144244
3 14444244443
⎪
Enthalpy of solid part
Enthalpy of liquid part
⎨
⎪ ρ
= F m ρ s + (1 − F m ) ρ l
⎩ Ωm
(14)
Under these settings, Eq. (13) can be simplified as
Ω m ( ρ s ΔH )
∂ F
m
∂t
= Σ ( K s ∇Ts ⋅ Ams ) + Σ ( K l ∇Tl ⋅ Aml )
(15)
Further rearranging of Eq. (15) gives
Ωm
∂ F
∂t
m
=
⎡ − Ki
∑ ⎢ ρ ΔH
f
i =l ,s
⎣
s
r
r ⎤
∇Ti ⋅ Af ⎥
⎦
(16)
r
where Af is the face normal vector of face from the mushy cell Ω m pointing outwards, towards the neighboring
f
cell with the state of “i” (solid or liquid) phase. Only the thermal diffusion fluxes from liquid and solid phases are
included in the light of the above two assumptions.
The mushy cell tracking equation is used to determine the minimum time-stepping size such that the advancing length
of the moving front is limited by the length of a cell (thus, in a cell-by-cell manner), and then to update F m for every
mushy cell. The candidate of mushy cells can be easily identified by examining the neighbors (in liquid state) of the
newly solidified mushy cells. Thus, the mushy cells advance in a cell-by-cell manner.
2. Numerical Algorithm The computational domain is separated into solid-phase region and liquid-phase region, with
mushy zone in the present mushy cell tracking approach. All the formulation is based on the non-staggered
cell-centered, collocated control-volume method. The resulting nonlinear and coupled system of equations is handled
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by the SIMPLE algorithm [17, 18]. The pre-conditioned conjugate gradient (P-CG) method is used for the pressure
correction equation due to its symmetric and positive definite coefficient matrix. The P-BiCGSTAB method is selected
for the momentum equations and the thermal equation with convection because their coefficient matrices are
non-positive and non-symmetric [19].
RESULTS AND DISCUSSIONS
Two model problems are tested to validate the present method: 1D solidification of water without consideration of the
convection effect, and the Gallium melting in a rectangular cavity heated from the side with consideration of the
convection effect. The first problem is the so called Stefan problem [14] which the exact solution exists and can be used
to access the accuracy of the present method. The second problem has been studied experimentally [1] and numerically
[2-4, 11, 11, 15, 16]. However, the comparison of numerical prediction and experiment data is not consistent, such as,
disagreement between numerics and experiments, the myth of grid-converged solution - Prediction of coarse-grid and lower-order
scheme agrees better with the experimental results than prediction of fine-grid and high-order discretization, is the multicellar
solution physically correct, etc. We re-study the problem systematically with the mushy cell tracking method, and try to
shed some light on this controversial problem.
1. Example 1: 1D Solidification of Water The domain and boundary conditions of the 1D Solidification of water is
depicted in Fig. 1. The problem is referred to as the one-region problem which the exact solution of the moving front
and the associated temperature derivatives are known [14]. The position of the moving interface can be expressed
analytically as S ( t ) = η thr , where thr has the unit of “hour”, and constant η is function of the surface temperature.
Three different values of η (0.0225, 0.0315, and 0.0385, respectively) are chosen for simulation. The corresponding
surface temperatures are Tsurf = −10o C , −20o C and −30o C , respectively.
Figure 1: Computational domain and the associated boundary conditions of the 1D solidification of water
Table 1 shows the physical properties of water and ice [21]. It is assumed that the heat transfer coefficient between the
atmosphere and the surface of lake is very large such that the atmosphere temperature ( Tatm ) is close to the surface
temperature Tsurf of the lake. The temperature of water is 0o C initially.
Table 1 Physical properties of water.
symbol
ρ
cp
k
ΔH
(latent heat)
Tm
units
kg m3
kcal kg -K
kcal kg -hr -K
kcal kg
K
solid
920
0.487
1.9
liquid
1000
1.01
0.489
80
273
Three different meshes, with 5748, 9270, and 20594 meshes, respectively, are used for computations. The computed
S ( t ) at various time instances are shown in Fig. 2. The discrepancy between the numerical solution and analytic
solution, defined as ε ( t ) ≡ S ( t )num − S ( t )exact , is used to access the accuracy of the method, i.e., ε ≈ C ( t ) h n , where h is
the characteristic length of the cells. Comparison of S ( t ) , ε ( t ) , and CPU with the three meshes is shown in Fig. 3. The
numerical prerdiction of S ( t )num agree excellently with the analytical solutions S ( t )exact , both lines are
undistinguishable, as demonstrated in Fig. 3. Assessment of the order of accuracy is illustrated in Fig. 4. It is found that
the value of n is in the range of 1.5 ~ 3.0. This finding is consistent with the result of Udaykumar, et al. [12], which
claims the second-order accuracy of the prediction of moving interface can be achieved.
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(a) Tsurf = −10D C
(b) Tsurf = −20D C
(c) Tsurf = −30D C
Figure 2: Moving front of the 1D solidification of water at various time instances
2. Example 2: Melting of Gallium Gallium melting in a rectangular cavity heated from the side has been
experimentally [1] and numerically [2-4, 11, 11, 15, 16] studied. The experimental result of Gau and Viskanta [10] is
selected as the reference because it has been widely cited for the verification of the numerical models. The geometrical
configuration and boundary conditions is sketched in Fig. 5. Initially, a solid Gallium block is kept at T = 28.3o C . The
temperature at the left wall is increased instantly to T = 38.0o C , while the right wall is maintained at the initial
temperature T = 28.3o C . The physical properties used in the calculations are listed in Table 2 [4, 20].
Table 2 Physical properties of Gallium.
symbol
ρ
k
Cp
μ
units
kg
m3
w
m−K
J
kg -K
Ns
m
solid
6095
33.5
395.15
0
liquid
6095
33.5
395.15 1.8×10
ρ gβ
β
ΔH
C
kg m 1
m3 s 2 K
1
K
J
kg
29.78
7.0
1.173×10−4
8×104
Tm
o
−3
Hannoun, et al. [4] employed the enthalpy method with various discretization schemes and grid resolution to study this
problem. He gave excellent viewpoints of the controversy of the problem – The numerical prediction of the dynamics
of interface and thermo-fluid of the coarse meshes matches better with the experiment data than the fine meshes does.
In order to shed some light on the mesh dependency on the prediction, several grid resolution of unstructured and
structured meshes are employed, examples are depicted in Fig. 5. The particular arrangement of the coarse meshes
around the region of interface is very important to the numerical results, as noted in Hannoun, et al. [4]
⎯ 690 ⎯
(a) Tsurf = −10D C
(b) Tsurf = −20D C
(c) Tsurf = −30D C
Figure 3: Comparison of CPU, error ε ( t ) , and moving front S ( t ) of the 1D solidification of water
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Figure 4: Assessment of the order of accuracy of the 1D solidification of water
(a) Unstructured/coarse meshes
(b) Structured/fine meshes
Figure 5: Geometry, boundary conditions and computational meshes of Gallium melting
Figure 6: Comparison of the moving interface of Gallium melting at various time
the special arrangement of coarse meshes around the region of interface can produce better prediction with the
experiment data [1] and other numerical results [2, 3, 11], as shown in Fig. 6. However, obvious discrepancy is
observed between the predicted interface and the experiment data at t = 1140 sec, especially near the top wall region.
To further address the dependence of the mesh resolution on the prediction of thermo-fluid field and interface
dynamics, 80*60, 90*70 and 100*71 orthogonal, quadrilateral meshes are used. Comparison of the prediction of the
interface of the present method with other numerical result is shown in Fig. 7(a). The interface of present method
matches well with the numerical results of Hannoun, et al. [4] for the early period of simulation (t < 280 sec), and agree
well other numerical results for t > 360 sec. Comparison of the numerical results with experiment data [1] is depicits in
Fig. 7(b). The discrepancy between the numerical result with fine meshes and the experiment data is significant and has
been discussed in Hannoun, et al. [4] – It might attribute to the differences between the experiment setup and
assumptions made in the numerical method.
⎯ 692 ⎯
(a) Comparison with other numerical results
(b) Comparison with experimental data [1]
Figure 7: Comparison of the present result with other numerical results and experimental data
The corresponding streamlines pattern at various time instants are shown in Fig. 8 and Fig. 9. The results of the present
method agree well with the results of Hannoun, et al. [4] for the early period of simulation ( t < 280 sec), and close to
the results of [13] for t > 400 sec.
CONCLUSIONS
A thermally driven mushy cell tracking algorithm has been developed. The method is based on the finite-volume
formulation on the fixed meshes where the domain is separated into solid-phase region and liquid-phase region, with
the mushy cells in between. Mushy cell tracking equation is derived from the principal of mass and energy
conservation and used to predict the movement of mushy cells. We use 1D solidification of water without convection
and 2D Gallium melting with convection to veridate the method. Based on the computed results, the following
conclusions are made:
(1) Simulation of 1D water solidification shows that prediction of the moving interface S ( t ) agrees well with the
analytic solution. Order of accuracy of prediction of the moving interface exhibits a super-linear accuracy, i.e.,
ε ( t ) ≈ C ( t ) h1.5~3 . This finding is close to the order of accuracy found using the moving particles method [12].
(2) Hannoun, et al. [4] demonstrated that prediction of the Gallium melting interface with coarse meshes around the
interface region matches better with experiment data [1] and other numerical results [2, 3, 11]. We believe these
predictions are mesh-dependent – The mesh resolution is not fine enough to resolve the small and complex
thermo-fluid field structure near the top wall close to the interface area. Simulation of Gallium melting with fine
⎯ 693 ⎯
(a) data from [4]
(b) data of the present method
Figure 8: Comparison of the streamlines and interface of the present result with data from [4]
(a) data from [11]
(b) data from the present method
Figure 9: Comparison of the streamlines and interface of the present result with data from [11]
meshes shows that the model prediction of the thermo-fluid field and dynamics of muchy cells agree well with the
experiment data [1] and other numerical results [4, 11, 15, 16] in the early period of simulation (t = 0 ~ 280 sec), and
differs from the experiment data, but are consistent with other numerical results, for the later period of simulation (t >
360 sec). The sidcrepancy between the experiment data and numerical results might attribute to the experiment setup is
not consistent with the assumpitions made in the numercal methods, as elaborated in [4].
The cell-by-cell thermally driven mushy cell tracking algorithm has been successfully applied to some 2D
phase-change problems [5, 6]. It exhibits a great potential for 3D complex phase-change problems. Application of the
method to dendritic processing will be presented in another paper.
Acknowledgements
The second author would like to thank the National Science Council of Taiwan for funding this Research (project no.:
NSC 94-2611-E-022-003).
⎯ 694 ⎯
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