aiming beyond infinity: Python Software Foundation - SfePy : GSoC 2013 : Week 10 & 11

The ﬁnal problem accounting convection-diﬀusion along with phase change has been deﬁned below along with the governing equations and validation case studies.

Problem
Heat transfer in the processing of materials involving solid-liquid phase transformations (melting and solidiﬁcation) is commonplace in such ﬁelds as metallurgy, crystal growth from melts and solutions, puriﬁcation of materials, and solidiﬁcation of metals. The associated density gradients in a gravitational ﬁeld can induce natural convection ﬂows. Convection in the liquid phase inﬂuences the process in two diﬀerent ways, one of which is beneﬁcial and the other of which can be detrimental. During melting convection increases the overall transport rate and, hence, the growth rate of the new phase, which is desirable. On the other hand, during solidiﬁcation convection decreases the growth of the new phase and also seems to aﬀect the morphology of the solid-liquid interface adversely. The nature of the solid is largely determined by what occurs in the vicinity of the solid-liquid interface. The heat release (absorption), density change, and other processes that take place in the vicinity of the transformation front result in nonuniformities along the front that cause its shape to change. The resulting density gradients in the liquid generate buoyancy-driven convection that can aﬀect the transport of heat, constituent chemicals, and the growth rate.
The physical domain considered is shown in the ﬁgure. The vertical side walls of the enclosure are maintained at uniform temperatures, while the connecting horizontal walls are adiabatic. The govering equations are written for the entire domain assuming constant thermophysical properties, Boussinesq approximation, laminar, incompressible, and Newtonian two-dimensional ﬂow. The solid-liquid interface motion due to volume change upon melting or solidiﬁcation is neglected through the assumption

ρ_{s} = ρ_{l}

Nomenclature

$ρ$	Density	kg/m $^{3}$
$μ$	Viscosity	Pa.s
$c_{p}$	Speciﬁc heat capacity	J/kg.K
$k$	Thermal conductivity	W/m.K
$β$	Thermal expansion coeﬃcient	1/K
$L$	Latent heat of fusion	J/kg
$H$	Enthalpy	J/kg
$f$	Volume fraction
$T$	Temperature	$^{\circ}$ C
$t$	Time	s
$\vec{u}$	Velocity	m/s
$p$	Pressure	Pa
$S$	Source term	m/s $^{2}$

Subscript

$l$	Liquid
$s$	Solid
$r e f$	Reference
$e f f$	Eﬀective
$h o t$	Hot
$c o l d$	Cold
$i n t$	Initial
$m e l t$	Melting

Governing Equations

The continuum relations:

g_{l} + g_{s} = 1

f_{l} + f_{s} = 1

f_{l} = \frac{g_{l} ρ_{l}}{ρ}

f_{s} = \frac{g_{s} ρ_{s}}{ρ}

ρ = g_{l} ρ_{l} + g_{s} ρ_{s}

\vec{u} = f_{l} {\vec{u}}_{l} + f_{s} {\vec{u}}_{s}

k_{e} f f = g_{l} k_{l} + g_{s} k_{s}

c_{p} = f_{l} c_{p_{l}} + f_{s} c_{p_{s}}

1. Continuity

\nabla \cdot (\vec{u}) = 0

2. Momentum

ρ "["\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u}"]" = - \nabla p + μ \nabla^{2} \vec{u} + ρ \vec{g} β (T - T_{r e f}) + B \vec{u}

Now the key lies in modelling the source term. The coeﬃcient

B

which should tend to 0 as the liquid volume fraction

g_{l}

approaches unity, and should become a large negative number to anhilate the motion in the ﬂuid region at

g_{l} = 0

. Whereas these asymtotic conditions can be satisﬁed by several functions, we adopt the suggestion of Brent et al. :

B = \frac{- C {(1 - g_{l})}^{2}}{(g_{l}^{3} + b)}

where

C = 1.6 \times 1 0^{6}

b = 0.001

2. Energy
Writing a general equation for conservation of thermal energy for all the zones in the domain is facilitated by focusing on an element undergoing phase change. Below are the energy equation of solid and liquid phases under the thermal equilibrium condition

T_{l} = T_{s} = T

:
Solid

\frac{\partial (ρ_{s} g_{s} H_{s})}{\partial t} + \nabla \cdot (ρ_{s} g_{s} \vec{u_{s}} H_{s}) = \nabla \cdot (g_{s} k_{s} \nabla T) + S_{s}

Liquid

\frac{\partial (ρ_{l} g_{l} H_{l})}{\partial t} + \nabla \cdot (ρ_{l} g_{l} \vec{u_{l}} H_{l}) = \nabla \cdot (g_{l} k_{l} \nabla T) + S_{l}

where

S_{s}

and

S_{l}

are the interphase energy terms, having the same magnitude but being opposite in sign.
A single governing enthalpy equation results:

\frac{\partial (ρ H_{m})}{\partial t} + \nabla \cdot ρ (f_{s} \vec{u_{s}} H_{s} + f_{l} \vec{u_{l}} H_{l}) = \nabla \cdot (k_{e f f} \nabla T)

where

H_{m} = f_{s} H_{s} + f_{l} H_{l} = H_{s} + f_{l} (H_{l} - H_{s})

The latent heat content of the element is due to the fraction of liquid converted to, or from, the corresponding quantity of solid. Hence we write

Δ H = f_{l} (H_{l} - H_{s}) = f_{l} L

H_{m} = H_{s} + f_{l} L

H_{s} = c_{p} T

H_{l} = H_{s} + L

where L is the latent heat of fusion.
The zones where

T > T_{m e l t}

, the entire element is in the liquid state and

f_{l} = 1

.
The zones where

T < T_{m e l t}

, the entire element is in the liquid state and

f_{l} = 0

.
It is the elements undergoing phase change at

T = T_{m e l t}

, where

f_{l}

varies between 0 and 1. Substituting

H_{m}

and

H_{l}

in equation we get

\frac{\partial (ρ H)}{\partial t} + \nabla \cdot (ρ \vec{u} H) = \nabla \cdot (\frac{k_{e f f}}{c_{p}} \nabla H) S_{e}

where

\vec{u} = f_{l} \vec{u_{l}}

S_{e} = - \frac{\partial (ρ Δ H)}{\partial t} = - ρ L \frac{\partial (f_{l})}{\partial t}

Initial Conditions

f_{l} = 0

everywhere

T = T_{i n t}

everywhere

\vec{u} = 0

everywhere.

Boundary Conditions

f_{l} = 1

Left wall

f_{l} = 0

Right wall

f_{l} [i] [j] = f_{l} [i - 1] [j]

Top

f_{l} [0] [j] = f_{l} [1] [j]

Bottom

T = T_{h o t}

Left wall

T = T_{c o l d}

Right wall

\vec{u} = 0

Walls

Validation Cases
Melting Gallium : Gau and Viskanta
Melting Gallium : Brent et al.
Melting Calcium chloride : Zivkovic and Fujii

References

Zivkovic, B., Fujii, I., 2001. “An Analysis of Isothermal Phase Change of Phase Change Material within Rectangular and Cylindrical Containers”. Solar Energy, 70, pp. 51-61.
Brent, A.D., Voller, V.R., Reid, K.J., 1988. “Enthalpy-porosity Technique for Modeling Convection-diﬀusion Phase Change: Application to the Melting of a Pure Metal”. Numerical Heat Transfer, 13(3), pp. 297-318.
Gau, C., Viskanta, R., 1986. “Melting and Solidiﬁcation of a Pure Metal on a Vertical Wall”. Journal of Heat Transfer, 108(1), pp. 174-181.
Rajeev, K., Das, S., 2010. “A Numerical Study for Inward Solidiﬁcation of a Liquid Contained in Cylindrical and Spherical Vessel”. Thermal Science, 14(2), pp. 365-372.
Vreeman, C. J., Krane, M. J. M., Incropera, F. P. , 2000. “The Eﬀect of Free-Floating Dendrites and Convection on Macrosegregation in Direct Chill Cast Aluminum Alloys. Part 1: Model Development”. Int. J. Heat Mass Transfer, 43, pp. 677-686.
Flemings, M. C., 1974. Solidiﬁcation Processing. McGraw-Hill, New York.
Kumar, A., Walker, M. J., Sundarraj, S., Dutta, P., 2011. “Grain Floatation During Equiaxed Solidiﬁcation of an Al-Cu Alloy in a Side-Cooled Cavity: Part II-Numerical Studies”. Metallurgical and Materials Transactions, 42(B), pp. 783-799.
Voller, V.R., 2006. Handbook of Numerical Heat Transfer, 2nd ed.. Wiley, New York, NY, pp. 593-622.

Pages

Tuesday, September 3, 2013

Python Software Foundation - SfePy : GSoC 2013 : Week 10 & 11

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