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Phase change Convection_Cavity - BGK and MRT_picturedt3u_phasechangeLBM_MRT-0001_phasechange_MRT-LBM_源码.zip （26个子文件）

Phase change Convection_Cavity - BGK and MRT

Kang Luo ATE.pdf 4.11MB

说明文档.docx 16KB

code

bonding.m 4KB

Raseries.txt 12B

Steps.txt 21B

Temp

ux.mat 65KB

Tc.mat 57KB

rho.mat 56KB

uy.mat 62KB

tequ.m 216B

Htequ.m 293B

MLS.m 11KB

Bd.m 6KB

Fget.m 145B

Result

HLBM-BGK_T&Phase&V_Fo3.998_Ra25000_St0.01_Pr0.02.dat 530KB

TLBM-BGK_T&Phase&V_Fo8.0001_Ra25000_St0.01_Pr0.02.dat 531KB

TLBM-BGK_T&Phase&V_Fo2.001_Ra25000_St0.01_Pr0.02.dat 530KB

TLBM-BGK_T&Phase&V_Fo5.9991_Ra25000_St0.01_Pr0.02.dat 531KB

HLBM-BGK_T&Phase&V_Fo8.0001_Ra25000_St0.01_Pr0.02.dat 531KB

HLBM-BGK_T&Phase&V_Fo1.0005_Ra25000_St0.01_Pr0.02.dat 530KB

HLBM-BGK_T&Phase&V_Fo5.9991_Ra25000_St0.01_Pr0.02.dat 531KB

TLBM-BGK_T&Phase&V_Fo3.998_Ra25000_St0.01_Pr0.02.dat 530KB

TLBM-BGK_T&Phase&V_Fo10.0012_Ra25000_St0.01_Pr0.02.dat 531KB

TLBM-BGK_T&Phase&V_Fo1.0005_Ra25000_St0.01_Pr0.02.dat 530KB

HLBM-BGK_T&Phase&V_Fo2.001_Ra25000_St0.01_Pr0.02.dat 530KB

fequ.m 273B

Grid

Research paper

Lattice Boltzmann simulation of convection melting in complex heat

storage systems ﬁlled with phase change materials

Kang Luo, Feng-Ju Yao, Hong-Liang Yi

, He-Ping Tan

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China

highlights

 Convection melting in multitube heat storage systems are numerical studied by LBM.

 Different lattice Boltzmann models for phase change are systematically compared.

 The inﬂuences of various numbers and arrangements of tubes are investigated.

 Effects of Rayleigh and Stefan numbers on volume melt fraction of PCM are examined.

article info

Article history:

Received 6 February 2015

Accepted 20 April 2015

Available online 6 May 2015

Keywords:

Lattice Boltzmann

Melting

Convection

Shell and tube model

abstract

In the present study, a double-population lattice Boltzmann method is applied to the simulation of

convectionediffusion phenomena associated with solideliquid phase transition processes. The research

focus is the advancement of the lattice Boltzmann method to complex multitube heat storage system

with different numbers and arrangements of tubes. Firstly, a systematic comparison of different lattice

Boltzmann models for thermal and ﬂow ﬁeld in the phase change process is numerically conducted in a

square cavity, and the numerical results are validated by the literature data. Then, a comprehensive

analysis has been performed in order to investigate the inﬂuence of various numbers and arrangements

of tubes on the melting dynamics of shell and tube models with different Rayleigh and Stefan numbers.

The computational results show how the transient phase-change process, expressed in terms of the

volume melt fraction of phase change materials (PCM), depends on the thermal and geometrical pa-

rameters of the system.

1. Introduction

The fundamental of heat transfer and ﬂow in phase change

materials (PCM) has received considerable attention during the

past two decades due to its potential for thermal energy storage

systems. There exists a wide range of applications for such systems

[1e3], such as energy storage in buildings, electronics cooling,

material processing and thermal management of spacecraft.

Theoretical, numerical and experimental studies in the ﬁeld have

yielded extensive literature on various aspects of the phase-change

problems, including basic studies of phase-change phenomena [4],

material properties [5], experimental methods and heat transfer

enhancement [6e 8], mathematical modeling and numerical tech-

niques [9e11]. Among them, numerical simulation is a major focus

for its economy and high efﬁciency, which can signiﬁcantly

improve the understanding of convection melting processes in heat

storage systems. It is concluded in the critical review of Agyenim

et al. [7] that the most common numerical approach has been the

use of enthalpy formulation.

Some well-developed methods have been applied to simulate

the convection melting models, such as ﬁnite difference method

(FDM) [12], ﬁnite volume techniques (FVM) [13,14], ﬁnite element

method (FEM) [15] and lattice Boltzmann method (LBM) [16]. These

numerical simulations are mainly based on two types of grid sys-

tems, namely, ﬁxed grid [17] and adaptive grid [18], in which, the

enthalpy-based LBM is a newly introduced ﬁxed grid based

approach for phase change problems.

* Corresponding authors. Tel.: þ86 451 86412674.

E-mail addresses: yihongliang@hit.edu.cn (H.-L. Yi), tanheping@hit.edu.cn

(H.-P. Tan).

Contents lists available at ScienceDirect

Applied Thermal Engineering

journal homepage: www.elsevier.com/locate/apthermeng

http://dx.doi.org/10.1016/j.applthermaleng.2015.04.059

Applied Thermal Engineering 86 (2015) 238e250

The existing mathematical models for the phase change prob-

lem are mostly based on continuum approaches. The lattice

Boltzmann method (LBM) is a relatively new approach that uses

simple kinetic models to simulate complicated macroscopic

transport phenomena. Owing to its mesoscopic nature, in com-

parison with conventional ﬂuid dynamics solvers, it offers such

advantages as simple calculation procedure, simple and efﬁcient

implementation for parallel computation, and easy and robust

handling of complex geometries. LBM was ﬁrst proposed by

McNamara and Zanetti [19] in 1988. Chen and Doolen [20] con-

ducted an overview of the lattice Boltzmann method for ﬂuid ﬂows.

Then, the LBM was developed to solve a wide range of heat transfer

problems [21]. Recently, phase change problems have also been

investigated by the LBM [22]. Since LBM is inherently transient, it is

an excellent approach for the investigation of transient phase

change process. Speciﬁcally, the LBM solves the problems by evo-

lution, which agrees well with the real physical melting process. In

recent years, the application of LBM to phase change problems has

been extensively investigated by many researchers [23e33].

It seems that Miller et al. [23] ﬁ rst developed a simple reaction

model for the liquidesolid phase transition in the context of the

lattice Boltzmann method with enhanced collisions. In his work, a

two-dimensional test problem of Ga melting and a two-

dimensional anisotropic growth of dendrites were presented.

Jiaung et al. [24] proposed an extended lattice Boltzmann equation

for the simulation of the phase-change problem governed by the

heat conduction equation incorporated with enthalpy formation.

Chakraborty and Chatterjee, in their outstanding works

[16,22,25e28], applied the lattice Boltzmann technique to numer-

ical simulation of conduction-dominated and convection-

dominated phase change process (melting and solidiﬁcation).

Huber et al. [29] investigated the coupled thermal convection and

pure-substance melting using a lattice Boltzmann method. The

transition from conduction-dominated heat transfer to fully-

developed convection was analyzed. Gao et al. [30] performed a

Lattice Boltzmann simulation of natural convection dominated

melting in a rectangular cavity ﬁlled with porous media. Eshraghi

et al. [31] developed an implicit lattice Boltzmann model for heat

conduction with phase change, in which the latent heat source

term was treated implicitly in the energy equation. Huang et al. [32]

introduced an enthalpy based lattice Boltzmann method for phase

change problems. In their works, the phase interface was traced by

updating the total enthalpy, and the moving interface was treated

by the immersed moving boundary scheme. Fuentes et al. [33]

presented a new LBM-MRT hybrid model to simulate melting

with natural convection in a phase change material. In the work,

energy equation was solved by a ﬁnite difference method, whereas

the ﬂuid ﬂow was solved by the multiple relaxation time (MRT)

lattice Boltzmann method.

As mentioned above, phase change problems were numerically

investigated by different models in a generalized lattice Boltzmann

framework, but no detailed comparison between these models was

provided in any reference work. Besides, most of lattice Boltzmann

simulation for the phase change problems has been done in a

simple cavity, while few studies have concerned more complex

geometries. Therefore, the purpose of the present investigation is

threefold: (i) to evaluate the capability of different lattice Boltz-

mann models for phase change process, i.e., the numerical stability

of lattice Bhatnagar-Gross-Krook (LBGK) and multi-relaxation time

(MRT) models for ﬂow ﬁeld, and the enthalpy-based lattice Boltz-

mann model with basic evolution variable of temperature (TLBM)

and enthalpy (HLBM) for thermal ﬁeld; (ii) to test the heat storage

performance for three conﬁgurations consisting of a shell with

different numbers (one, four and nine) of heat transfer tubes and

different tubes arrangements (inline, staggered and a novel

centrosymmetric design); (iii) to analyze the inﬂuence of the Ray-

leigh and Stefan numbers on the melting dynamics of shell and

tube models with various arrangements.

Nomenclature

speciﬁc heat, kJ/(kg K)

c lattice speed, m/s

e propagation velocity in the lattice, m/s

F Buoyancy, (kg m)/s

Fourier number,

t/l

f external body forces per unit mass, particle

distribution function for velocity ﬁeld

volume fraction of liquid

g the acceleration due to gravity, m/s

g particle distribution function for thermal ﬁ eld

H total enthalpy, kJ/kg

enthalpy of the liquid phase at the melting

temperature, kJ/kg

enthalpy of the solid phase at the melting temperature,

kJ/kg

k thermal conductivity, W/(m K)

L latent heat of melt, kJ/kg

l characteristic length, m

M the transformation matrix

m velocity moments of f

Nu Nusselt number, ql/k

ave

average Nusselt number

P pressure, Pa

Pr Prandtl number,

q heat ﬂux, W/m

Ra Rayleigh number, g

S diagonal matrix of non-negative relaxation rates

Stefan number, C

T/L

T temperature, K

dimensionless temperature

u macroscopic velocity vector, m/s

w weight function

Greek symbols

thermal expansion coefﬁcient, 1/K

density, kg/m

relaxation time for velocity and temperature ﬁelds

dynamic viscosity, kg/(m s)

kinematic viscosity, m

thermal diffusivity, m

fraction of an intersected link of curved wall in the

ﬂuid region

T Temperature difference, T

 T

collision operator

Subscripts

ave average

eq equilibrium distribution

h, c hot and cold wall

the direction of streaming step

a the opposite direction of

K. Luo et al. / Applied Thermal Engineering 86 (2015) 238e250 239

2. Mathematic formulations

In this part, we ﬁrst provided the governing equations for con-

vection melting. Then, we presented the equations of different

lattice Boltzmann models, e.g., BhatnagareGrosseKrook model

(BGK), multiple relaxation time model (MRT), and the enthalpy-

based lattice Boltzmann model with basic evolution variable of

temperature (TLBM) and enthalpy (HLBM), respectively.

2.1. Continuum conservation equations

The fundamental equations for thermo-ﬂuidic transport in the

presence of melting/solidiﬁcation (assuming a Newtonian, laminar

and incompressible ﬂow), can be described as follows [16]:

þ V$ðruÞ¼0 (1)

vðruÞ

þ V$

ruu

¼Vp þ V$

mVu

þ rf (2)





þ V$





¼ V$ðkVTÞþq (3)

In the above equations,

, u, T, p,

, C

, and k are the density,

velocity, temperature, pressure, dynamic viscosity, speciﬁc heat

and thermal conductivity, respectively. f is the external body forces

per unit mass, which can be deﬁned as: f ¼ g

(T  T

ref

) with the

assumption of the Boussinesq approximation, where g is the ac-

celeration due to gravity,

is the volumetric thermal expansion

coefﬁcient, and T

ref

is the reference temperature.

The latent heat source term q in the energy equation shown in

Eq. (3) can be expressed as

q ¼



vðrDHÞ

þ V$ðruDHÞ



(4)

in which, H is the latent enthalpy of a computational cell under-

going phase change. Further, the liquid fraction is denoted by f

given as f

H/L, where L is the latent heat of melting. For a pure

substance, the second term in Eq. (4) can be neglected and the

latent heat source term becomes

q ¼

vðrDHÞ

¼

vðrLf

(5)

2.2. Lattice Boltzmann BGK and MRT equations for density and

velocity ﬁelds

In the LBM, separate particle distribution functions are used to

compute the density (and velocity) and thermal ﬁelds. The distri-

bution function is obtained by solving the lattice-Boltzmann equa-

tion which is a special discretization of the kinetic Boltzmann

equation. The macroscopic quantities of the simulated ﬂuid can then

be derived by calculating the hydrodynamic moments of the dis-

tribution function. Unlike the second-order PDEs in the Naviere-

Stokes approach, the LBM uses only ﬁrst order PDEs, and the

generalized lattice Boltzmann equation can be expressed as [34].

f ðx þeDt; t þ DtÞf ðx; tÞ¼U þ Dt  F (6)

where f is the distribution function, e is the microscopic velocity,

is the collision operator, and F is the external force.

In this work, the D2Q9 model is used for the discretization of

velocity space for the two dimensional case. The nine velocities in

the D2Q9 (Fig. 1(a)) lattices are given by

ð0; 0Þ a ¼ 0

cðcos½ða  1Þp=2; sin½ða  1Þp=2Þ a ¼ 1; 2; 3; 4

ﬃﬃﬃ

cðcos½ð2a  1Þp

4; sin½ð2a  1Þp

4Þ a ¼ 5; 6; 7; 8

(7)

in which,

is the streaming direction, and c is the streaming speed,

deﬁned as c ¼

t, where

x and

t are the lattice cell size and

the lattice time step, respectively.

For single relaxation time (BGK), the collision term in Eq. (6) will

be replaced by the following expression

U ¼½f ðx; tÞf

ðx; tÞ

(8)

where f

is the equilibrium distribution function, and

is the

relaxation time, computed from

(9)

where v is the kinematic viscosity.

The equilibrium distribution function of Eq. (6) is expressed as

Fig. 1. Spatial discretization (a) D2Q9 lattice used in a 2-D geometry and (b) curved boundary treatment.

K. Luo et al. / Applied Thermal Engineering 86 (2015) 238e250240

¼ rw

1 þ

9ðe

$uÞ



(10)

where w

is the equilibrium distribution weight for direction

given as

9 a ¼ 0

1=9 a ¼ 1; 2; 3; 4

36 a ¼ 5; 6; 7; 8

(11)

For the multiple relaxation time (MRT), the collision term in Eq.

(6) can be expressed as

U ¼M

1

Sðm  m

Þ (12)

where M is the transformation matrix, S is the diagonal matrix of

non-negative relaxation rates, and m and m

represent the velocity

moments of the distribution functions f and their equilibria,

respectively.

The basic quantities such as m, m

, M and S are completely

deﬁned in Ref. [35]. Here, we just provide brieﬂy the expressions for

them. The physical signiﬁcance of velocity moments in Eq. (12) is

given in Ref. [36] as

m ¼



r; e; ε; j

; q

; j

; q

; p



(13)

and M is a q  q matrix which linearly transforms the distribution

functions f to the velocity moments m

m ¼ M$f ; f ¼ M

1

$m (14)

S is a non-negative q  q diagonal relaxation matrix

S ¼ diag



0; s

; s

; 0; s

; s



(15)

In order to incorporate buoyancy force in the model, the Bous-

sinesq approximation is applied. By assuming the reference tem-

perature to be zero, the force term in Eq. (6) can be calculated as

below in the vertical direction (y):

¼ 3w

bTðx; tÞrðx; tÞe

(16)

Then, the ﬂuid density

can be evaluated from Eq. (17a),

whereas the velocity u is contained in the momentum ﬂuxes of Eq.

(17b)

r ¼

(17a)

ru ¼

(17b)

2.3. Lattice Boltzmann equation for thermal ﬁeld

Two enthalpy-based lattice Boltzmann methods are used for

solideliquid phase transitions. (1) Use temperature T as basic

evolution variable in the LB equation [29], and the melting term is

introduced as a sink (melting) term in the collision step. In this

method, the temperature can locally decrease below the initial

solid temperature when the sink term dominates at low Stefan

number (St), which is in contradiction with the phase change

associated with pure substance melting. (2) An improved LB

method for solideliquid phase change is introduced by Huang [32],

by directly using the enthalpy H as basic evolution variable in the LB

equation, and avoiding the sink (melting) term in energy equation.

To distinguish these two methods, the lattice Boltzmann methods

with basic evolution variable of temperature T and enthalpy H are

named as ‘TLBM’ and ‘HLBM’, respectively.

For the TLBM, the evolution equation for the temperature dis-

tribution function is expressed as

ðx þ e

Dt; t þDtÞg

ðx; tÞ¼



ðx; tÞg

ðx; tÞ



Dtw

ðf

ðx; tÞf

ðx; t DtÞÞ

(18)

in which, g

is the temperature distribution function in the

di-

rection, g

is the equilibrium distribution function, and

is the

relaxation time, represented as

(19)

where

is the thermal diffusion coefﬁcient. Equilibrium distribu-

tion function for temperature ﬁeld can be expressed as [29].

¼ w

1 þ

$u þ

9ðe

$uÞ



(20)

and the ﬂuid temperature T can be evaluated from

T ¼

(21)

The local enthalpy is obtained by the liquid fraction f

of the

previous iteration

H ¼ C

T þ f

t  Dt

L (22)

The melt fraction is obtained by the linearly interpolating of

enthalpy

ðx; tÞ¼

0 H < H

¼ C

H  H

 H

 H  H

¼ H

þ L

1 H > H

(23)

For the HLBM, enthalpy is used as the evolution variable and the

evolution equation is given as [32].

ðx þ e

Dt; t þ DtÞg

ðx; tÞ¼



ðx; tÞg

ðx; tÞ



(24)

and the modiﬁed equilibrium distribution function for the enthalpy

can be chosen as

H  C

T þ u



1 



a ¼ 0

1 þ

9ðe

$uÞ



as0

(25)

Then, the total enthalpy is calculated as

H ¼

(26)

Finally, the temperature is obtained by the linearly interpolating

of enthalpy

K. Luo et al. / Applied Thermal Engineering 86 (2015) 238e250 241

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