Computation of gas–solid flows by finite difference Boltzmann equation

Applied Mathematics and Computation 173 (2006) 33–49

www.elsevier.com/locate/amc

Computation of gas–solid flows byfinite difference Boltzmann equation

Sheng Chen *, Zhaohui Liu,Baochang Shi, Chuguang Zheng

State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology,

Wuhan 430074, China

Abstract

In this paper, we will discuss in detail how to use a finite difference lattice Boltzmann

equation model in which an external force term is involved to simulate two-way cou-

pling gas–solid flows. The numerical results are found to be in good agreement with ana-

lytical data and some other numerical results.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Lattice Boltzmann equation; Gas–solid flow; Two-way coupling

1. Introduction

Gas–solid flows are commonly found in energy systems, in material-han-

dling equipment, and in the environment. Frequently, such as in pneumatic

transport or dump combustors, these flows undergo separation and reattach-

ment in which the particle and flow dynamics can become quite complex.

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.02.060

* Corresponding author.

E-mail addresses: [email protected] (S. Chen), [email protected] (Z. Liu).

mailto:[email protected]

mailto:[email protected]

34 S. Chen et al. / Appl. Math. Comput. 173 (2006) 33–49

Two phenomena of interest in these types of flows are the particle dispersion

and gas phase modification due to the presence of particles. And the perfor-

mance of fluid machinery is greatly influenced by the occurrence of the flow

separation and reattachment phenomena. Consequently, enormous efforts,

including those employing lattice Boltzmann method (LBM) which appears

as an alternative efficient tool for simulating fluid flows and modelling complexphysics in fluids, have been done to describe such flows. The pioneering works

using LBM to simulate gas–solid flows belong to Filippova and Hanel [1] and

Masselot and Chopard [2]. In spite of the significant differences in their appear-

ances and technical details, these schemes share one common drawback in their

constructions for gas–solid flows: the two phases are non-coupled, namely the

gas phase modification due to the presence of particles is neglected. However,

as it is well known, the applicable area of this simplification is very limited,

even though in dilute gas–solid two-phase flows [3]. In order to remedy thisimportant fault, the present authors developed a coupled LBM scheme involv-

ing an external force term to implement two-way coupling of gas–solid flow.

Nevertheless, though it can reflect the actual phenomena of gas–solid flows

adequately accurately, this coupled LBM scheme encounters a great obstacle:

its computational efficiency suffers from the limit of the uniform mesh, which is

common in the existing LBM models, especially in such problems where a large

gradient exists only in a small region while the domain of interest is large, for

example flow over a backward-facing step [4]. Recently LBM is also widely ex-tended into another kind of gas–solid flow simulation, named as ‘‘particulate

suspensions’’, Ladd is the pioneer in that field [5]. We shall not further discuss

the second kind of gas–solid flow simulation because it is beyond the scope of

the present work.

The finite difference lattice Boltzmann equation (FDLBE) model is origi-

nally proposed as the remedy for the bad computational efficiency of LBM.

The first FDLBE was perhaps due to Reider and Sterling [6], and was exam-

ined by Cao et al. [7] in more detail. But surprising, though FDLBE has beenfurther extended to simulate thermal flows, compressible flows, porous media

flows, liquid-vapor flows and so on [8–13], the study of using FDLBE on gas–

solid flows simulation is little. Therefore in this paper we will discuss in detail

how to use a two-way coupling FDLBE model to simulate gas–solid flows.

2. Coupled FDLBE model

In conventional numerical methods, to describe the interactions between the

fluid and solid phases is through an additional source term F in the momentum

equation [3,4]:

otqua þ obquaub ¼ �oap þ obqmðobua þ oaubÞ þ F a; ð1Þ

S. Chen et al. / Appl. Math. Comput. 173 (2006) 33–49 35

where p is the pressure, u is the fluid velocity, q is the fluid density and m is thekinematic viscosity. The source term F is derived from the transfer of momen-

tum between phases, in dilute gas–solid flows it is mainly due to the Stokesian

drag force and excluded volume effects of solid particles can be neglected [3].

The principle of using source terms to describe the interaction between phases

was proposed by Migdal et al. and developed into the particle-source-in-cell(PSIC) method [4]. The PSIC method has become a well-established method

for calculating gas–solid flows particularly for engineering problems [3]. Note

that, in the present study, the subscripts a and b represent Cartesian coordi-

nates and the summation convention is applied to these subscripts.

To show how an external force term F can be introduced into FDLBE, we

begin with a concise review of the similar previous works because there have

been plentiful publications discussing how to handle external force in LBM.

Despite the significant differences in their appearances and technical details,the existing LBM schemes involving an external force term may be classified

into three categories: The first and the simplest one is introducing the force

term in a manner similar to that adopted for lattice-gas model, that is, by add-

ing a term to the collision function that modifies the distribution function fi[14,15]. The second appears as a heuristic scheme, the derivation in which is

mainly accomplished by constructing a phenomenological equilibrium distri-

bution function �f i, which can reflect the effect produced by a force [16]. The

last one that represents the force term starts from kinetic equation with rigor-ous proof [17,18]. Martys et al. [17] proved the latter two ways are equivalent.

And the accuracy and performance of above schemes were compared in Ref.

[19].

It seems that the third way also is a straightforward method to construct a

coupled FDLBE model because FDLBE only is a particular discretization

form of Boltzmann equation [7]. However, we construct our coupled FDLBE

model along the line of the second category since this heuristic scheme can be

extended more easily to non-standard distribution functions, for example, thedistribution function defined in Ref. [4]. For simplicity, we take the 2D 9-veloc-

ity model for example, in which the particle velocities are defined by e0 = 0,

ei = c(cos[(i � 1)p/2], sin[(i � 1)p/2]) for i = 1–4 and ei ¼ffiffiffi2

pcðcos½ði� 5Þp=2þ

p=4�; sin½ði� 5Þp=2þ p=4�Þ for i = 5–8.

The equilibrium distribution function of the present coupled FDLBE model

otfi þ eiaoiafi ¼ �s�1ðfi � �f iÞ ð2Þ

is defined as follows:

�f i ¼ xiq 1þ eiau�ac2s

þu�au

�bðeiaeib � c2sdabÞ

2c4sþMab

" #; ð3Þ


where the weight coefficients are x0 = 4/9, xk = 1/9 for k = 1–4 and xk = 1/36

for k = 5–8. cs is the sound speed and u�a is the ‘‘equilibrium velocity’’:

u�a ¼ ua þ nF a=q ð4Þand the form of Mab is:

Mab ¼Cabðeiaeib � c2sdabÞ

2c4s. ð5Þ

in above equations n and Cab are coefficients that determined by hydrodynamic

equations. Through the multiscaling analysis the exact Navier–Stokes equa-

tions involving the force F can be recovered provided n = s, Cab = �2sFaub/qor �s(Faub + Fbua)/q, s is the relaxation time in Eq. (2). Its derivation is given

in Appendix A.

It must be indicated that the physical interpretations of u�a and Cab are also

clear. One can see the main difference between our equilibrium distribution

function and that in Ref. [16] is the additional term Mab. This additional term

used here is motivated by Ref. [19]. In that paper, the authors pointed out: in

LBM, to recover the exact Navier–Stokes equations involving a force term, in

the evolving equation

giðxþ ei; t þ 1Þ � giðx; tÞ ¼ �s�1½giðx; tÞ � �giðx; tÞ� þ F i; ð6Þthe force term must be written as follows:

F i ¼ xi Aþ Kaeiac2s

þ H abðeiaeib � c2sdabÞ2c4s

� �; ð7Þ

where A, Ka and Hab are coefficients that determined by hydrodynamic equa-

tions. For clarity in Eq. (6) the particle distribution and local equilibrium dis-

tribution are represented by gi and �gi respectively. The original form of �gi is [19]

�gi ¼ xiq 1þ eiauac2s

þ uaubðeiaeib � c2sdabÞ2c4s

� �. ð8Þ

If we make some transformation, Eq. (6) can be rewritten as follows

giðxþ ei; t þ 1Þ ¼ ð1� s�1Þgiðx; tÞ þ s�1�g�i ðx; tÞ; ð9Þ

where �g�i ðx; tÞ ¼ ð�giðx; tÞ þ sF iÞ can be thought as a ‘‘equivalent equilibrium dis-

tribution’’ in which the influence of F is absorbed into the equilibrium distribu-

tion function.

Rewriting �g�i ðx; tÞ as power series form

�g�i ðx; tÞ ¼ xi ðqþ sAÞ þ eiaðqua þ sKaÞc2s

þ ðquaub þ sH abÞðeiaeib � c2sdabÞ2c4s

� �.

ð10Þ


Now we can find the contribution of the force to local equilibrium distribu-

tions can be split into three parts: variance of fluid at density q + sA, momen-

tum qua + sKa and momentum flux quaub + sHab due to the presence of F. If

some of above contributions are neglected, correspondingly unphysical terms

will inevitably appear in the recovered macrodynamic equations [19]. For

incompressible flow, fluid density is independent of F (i.e. A = 0), so we onlyconsider the latter two effects. The same as LBM, the phenomenological equi-

librium distribution function of FDLBE should also represent above effects to

consist with the hydrodynamic equations, therefore u�a and Cab are added. The

shortcoming of the phenomenological equilibrium distribution function in Ref.

[16] is obvious since in which only the modification on momentum is

considered.

In the present model, the macroscopical quantities, density and momentum,

are obtained by:

q ¼Xi

fi; qua ¼Xi

eiafi. ð11Þ

3. Results and discussion

In this part, the discrete scheme recently proposed by Chen et al. [4] and

Guo and Zhao [11] is used for solving Eq. (2) together with the non-equilib-

rium extrapolation scheme for boundary condition.

3.1. Verification of present model

As a first step toward validation of the performance of the present FDLBEdeveloped to simulate gas–solid flows, we conduct here a test problem which is

amenable to the analytic solution. The test case is that of the Poiseuille flow,

which is a channel flow driven by a force F between two parallel plates [19].

It has the following steady analytical solution:

ua ¼ u0 1� y2

L2

� �; v ¼ 0; ð12Þ

where u0 = F L2/(2m) is the peak velocity and L is the half-length of the distance

between walls. The mesh distribution and numerical results are shown in Fig. 1.

And the grid-dependence study of the present model is given in Fig. 2. It can be

seen from these results that the present FDLBE performs very well. The error

is defined by

E ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

ðu� uaÞ2q ffiffiffiffiffiffiffiffiffiffiffiffiX

u2a

q�. ð13Þ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Y

u/uo

Fig. 1. Mesh distribution and velocity profiles of the Poiseuille flow with F = 0.001. Solid line,

analytical solution; dot, numerical result.

0.011

0.010

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

Ave

rage

d er

ror

0.001

0.00020 40 60

Grid Number, Ny

80 100 120 140

Fig. 2. Numerical error of the Poiseuille flow at different F as a function of the number of grid in y

direction.


3.2. Results of gas–solid flows

The problem domain which simulated in this paper, as Fig. 3 illustrated, is a

backward-facing step proposed by Barton since it is a benchmark of flows

undergoing separation and reattachment, where h is the step height. The figure

X

Y

X1

X2

X 3

h

h

u=0v=0

u parabolicv=0

u'x = 0v'x = 0

u=0 v=0

u=0 v=0

Primary votex

Uppersecondary votex

30h

g

inlet outlet

Fig. 3. Sketch of backward-facing step, boundary conditions, and standing vortices.


also illustrates the reattachment and separation lengths x1, x2 and x3 [3]. The

non-uniform mesh generating functions for this case can be found in our pre-

vious works [20,21]. In this study the grid density is 100 · 80. The force F which

is derived from the transfer of momentum between phases mainly due to the

Stokesian drag force is calculated by the method used in Ref. [4], which is

widely used in gas–solid flows simulations:

F ¼ �XMp¼1

fp; ð14Þ

where M is the number of particles within a control volume. fp is the modified

drag force acting on a particle, which can be calculated by the following equa-

tion if the particle is heavier than the surrounding fluid [3]

fp

mp

¼ dup

dt¼ f

Stkðu� upÞ þ

1

Fr1� 1

q

� �; ð15Þ

where q is the density ratio of a particle to the fluid, in this paper q = 10. up isthe velocity of particle and the coefficient f that describes the influence of the

ultra-Stokesian drag is f ¼ 1þ 0.15Re0.687p . The term Rep is the particle Rey-

nolds number. When Rep < 1000, it can be calculated by Rep = Dpju � upjRe,where Dp = dp/(2h), dp is the diameter of particle.

Gas phase is simulated by the present model and solid phase is calculated

through the equation of motion for a single particle because Eulerian–

Lagrangian scheme is more favor with the engineering community [3]. In order

to solve the particle�s motion equation Eq. (15) which is a stiff equation whenStokes number Stk � 1, the recently developed exponential Lagrangian track-

ing scheme is adopted [3,4]. Reynolds number Re ¼ uavð2hÞm , Stokes number

Stk ¼ qd2puav18mð2hÞ, Froude number Fr ¼ u2av

gð2hÞ and inlet void fractions ainlet = Vp/V,


are the same as those defined in Refs. [3,4]. Where uav is the average inlet veloc-

ity, V is a unit volume in the fluid field, Vp is the volume occupied by the par-

ticles displacement in a unit volume, and g is the gravitational force which acts

downwards as shown in Fig. 3.

3.2.1. Single-phase flow

We firstly simulated the single-phase flow where there is no particle. The

behavior of the flow is briefly summarized as following: The flow separates

at the step and reattaches downstream at position x1. The position x1 increases

almost linearly. For higher Reynolds numbers the adverse pressure gradient is

strong enough to create an upper recirculation region which is illustrated in

Fig. 3. The upper recirculation region increases in size with increasing Rey-

nolds number and its core moves downstream. The reattachment and separa-

tion positions are summarized in Fig. 4 together with the experimental dataof Armaly et al., the results of Guj and Stella [3]. For low Reynolds numbers

the present predictions are in good agreement with the experimental data, but

for higher Reynolds numbers the numerical predictions start to deviate from

the experimental data. The differences are probably caused by three-dimen-

sional effects present in the experiment [3]. In this case we also make a compar-

ison between the results obtained by the present FDLBE model and those

obtained by our previous coupled LBM scheme, the differences are very small,

less than 1%. For clarity, the results got by the coupled LBM scheme are notplotted in the figure.

Fig. 4. Variation in reattachment and separation lengths with inlet Reynolds number for single-

phase flow.


3.2.2. Particle-laden flow

Particles are introduced into the flow at the inlet. Stokes number is the most

important parameter on which gas–solid flows depend though the behaviors of

the gas–solid flows also depend on Re and ainlet. The behaviors of the various

results are considered by assuming one of these parameters to be a constant

value and varying the others. Because the emphasis of this study is to validatethe feasibility of this model rather than deepen our understanding of gas–

particle flow, we do not discuss the relationship between phenomena of the

flow and above parameters particularly, for further details can see Ref. [3].

Variation in Stokes number. As Barton pointed out: low Stokes number par-

ticles, which tend to follow the flow, has the effect of reinforcing the freestream

of the flow that increases the lower recirculation region. High Stokes number

particles are dominated by the effect of gravity that increases downward move-

ment, so they will compress the lower recirculation region and increase the sizeof the upper recirculation region [3]. The behavior of particles with different

Stokes numbers is examined for Re = 450,700 and inlet void fraction ainlet =3 · 10�3. The reattachment and separation lengths for the single-phase flow

are also shown in Fig. 5 (dash lines) for a comparison with the particle-laden

flow. The particle-laden flow has either larger or smaller lengths x1 and x2depending on the Stokes number, and the particle-laden flow length x3 is al-

ways bigger than the single-phase flow length. The critical Stokes number,

where the values tend to intersect, is between 1 · 10�3 and 1 · 10�2, agree wellwith the results in Refs. [3,4]. Quantitative comparison is listed in Table 1.

Variation in inlet void fraction. As Barton pointed out: The differences

caused by varying the inlet void fraction are, however, dependent on the Stokes

number. Varying the inlet void fraction simply has the effect of increasing the

differences which are dependent on the Stokes number [3]. As the critical

Stokes number is between Stk = 1 · 10�3 and 1 · 10�2, the variation in the

reattachment and separation lengths is examined for Stk = 1 · 10�3 and

Fig. 5. Variation in reattachment and separation lengths with Stokes number for Re = 450,700;

ainlet = 3 · 10�3.

Table 1

Comparison of the critical Stokes number between the coupled LBM and the present model,

ainlet = 3 · 10�3

Critical Stokes number

(coupled LBM[4]) · 10�3

Critical Stokes number

(present model) · 10�3

% Error

x1(Re = 450) 4.384 4.206 4.06

x2(Re = 450) 2.987 3.114 4.25

x1(Re = 700) 5.015 5.187 3.43

x2(Re = 700) 4.396 4.468 1.64


1 · 10�2 in Fig. 6 for Re = 450. Obviously, low-void-fraction results tend to

single-phase results and inlet void fractions of 1 · 10�6 were found to give

the same solution as the single-phase results. The figure shows how the differ-

ences increases with inlet void fraction, where the Stk = 1 · 10�3 particles

drive the lower recirculation region and the Stk = 1 · 10�2 particles shorten

the lower recirculation region and increase the size of the upper recirculation

region, which also agree well with the results in Refs. [3,4].

The behavior of particles is reflected in the concentration plots of void frac-tions in Figs. 7 and 8. Similar to the results in Refs. [3,4]: The Stk = 1 · 10�2

particles penetrate the lower recirculation region. The particles that penetrated

are then pulled upstream by the recirculation region. The penetration of the

lower recirculation region is caused by the gravitational force and by the upper

recirculation region forcing particles downwards. Elsewhere in the channel the

gravitational force has only a slight effect. The Stk = 1 · 10�3 particles tend to

stay in the main body of the flow. The particles are forced slightly downwards

by the upper recirculation region and they fail to recover completely. A smallnumber of particles entered the lower recirculation region near the lower reat-

tachment; they were transported upstream in the recirculation region almost all

the way up to the backward-facing step wall.

Fig. 6. Variation in reattachment and separation lengths with void fraction for

Stk = 1 · 10�3,1 · 10�2;Re = 450.

Fig. 7. Concentration field of void fraction for Re = 450, Stk = 1 · 10�3 and ainlet = 3 · 10�3.

Fig. 8. Concentration field of void fraction for Re = 450, Stk = 1 · 10�2 and ainlet = 3 · 10�3.


Reynolds number. Through this model, we also find the interesting transi-

tional phenomena reported by Chen et al. [4]: in the flows where the Reynolds

numbers are close to 400 while with the high-Stokes-number particles, there

will appear a small secondary upper vortex in the flows and the lower recircu-

lation length will be shortened. However, the secondary upper vortex will not

appear in the single-phase flows that have the same Reynolds numbers. Fig. 9

illustrates the process that the secondary upper vortex appears and then disap-

pears, at Re = 389, ainlet = 3 · 10�3 and Stk = 3 · 10�3. From this figure we cansee that the secondary upper vortex appears about at t = 119.7 and disappears

about at t = 1266.7. Such transitional phenomena are caused by the gravity

force. The downward movement of the particles caused by the gravity force

encourages flow downwards along the channel. The downward movement

Fig. 9. Variation in x1, x2 and x3 with Re = 389, ainlet = 3 · 10�3 and Stk = 3 · 10�3.


produces the upper vortex and reduces the lower recirculation region size. This

effect lasts up to particles having been distributed almost evenly along thechannel. Figs. 10 and 11 show the concentration plots of particles at

t = 119.7 and 1266.7. If the Stokes number or the Reynolds number is too

big or too small, such transitional phenomena will not appear.

3.2.3. Computational efficiency contrast

At last, we make a comparison between this coupled FDLBE model with

grid density 100 · 80 and the coupled LBM model with grid density

Fig. 10. Concentration plot of particles for Re = 389, ainlet = 3 · 10�3 and Stk = 3 · 10�3 at

t = 119.7.

Fig. 11. Concentration plot of particles for Re = 389, ainlet = 3 · 10�3 and Stk = 3 · 10�3 at

t = 1266.7.


1200 · 80 for this kind of gas–solid flows. The purpose of setting the grid num-ber of the coupled LBM scheme along y direction to be 80 is to ensure the accu-

racy of the results obtained by the coupled LBM to be free of the effects of

inadequate grid density [4]. No surprise, the computational efficiency of the

present model is always significant higher than the coupled LBM model since

the great differences on grid density. Fig. 12 only illustrates time differences

between the coupled LBM scheme and the present model. The termination

Fig. 12. Time difference between the coupled LBM and the present FDLBE with various Re;

Stk = 1 · 10�3, ainlet = 3 · 10�3.

Fig. 13. The relative error differences between the coupled LBM and the present FDLBE with

various Re; Stk = 1 · 10�3, ainlet = 3 · 10�3.


condition for computing this kind of two-phase flow is that the residuals of x1,

x2 and x3 are reduced by five orders of magnitude. The simulations were per-

formed on a Pentium 4 (2.4G CPU). In Fig. 12 the unit of time is hour. The

reattachment and separation lengths between the two methods were in agree-

ment with the order of 3% (the maximal error appeared at x1), as shown inFig. 13. Where the superscripts L and P denote the coupled LBM scheme

and the present model respectively.

4. Conclusion

To fill the gap of FDLBE on simulating gas–solid flows, in this paper we dis-

cuss in detail how to developed a coupled FDLBE model for such flows.Numerical results shows the good performance of this model. It can be mod-

ified easily to three dimensional gas–solid flows. Such extension and applica-

tions will be considered in future studies.

Acknowledgement

This work was supported by the State Key Development Programme forBasic Research of China (Grant no. G1999022207), and the National Natural

Science Foundation of China (Grant no. 60073044).


Appendix A

The macrodynamic behavior arising form Eq. (2) can be found from a mul-

tiscaling analysis using a small parameter e which is proportion to the Knudsen

number. To do this, the following expansions are introduced [19,22]

fi ¼ f ð0Þi þ ef ð1Þ

i þ e2f ð2Þi ; ð16Þ

�f i ¼ �fð0Þi þ e�f

ð1Þi þ e2�f

ð2Þi ; ð17Þ

ot ¼ eo1t þ e2o2t; oa ¼ eo1a; ð18Þ

F a ¼ eF 1a;Cab ¼ eC1ab. ð19ÞIntroducing Eqs. (16) and (17) into Eq. (2) and sorting in orders of e, then

resulting in the following expressions:

f ð0Þi ¼ �f

ð0Þi þOðe0Þ; ð20Þ

D1ifð0Þi ¼ �s�1ðf ð1Þ

i � �fð1Þi Þ þOðe1Þ; ð21Þ

o2tfð0Þi þ D1if

ð1Þi ¼ �s�1ðf ð2Þ

i � �fð2Þi Þ þOðe2Þ; ð22Þ

where D1i = o1t + eiao1a. Introducing Eq. (19) into Eq. (3), then picking up

terms on the same order and formalizing them in following equations.

�fð0Þi ¼xiq

eiauac2s

þ eiaeibuaub2c4s

� uaua2c2s

� �; ð23Þ

�fð1Þi ¼xi n

eiaF 1a � F 1auac2s

þ eiaeibðF 1aub þ F 1buaÞ2c4s

� �þqC1abðeiaeib � c2sdabÞ

2c4s

� �;

ð24Þ

f ð2Þi ¼ n2xiq

�1 eiaeibF 1aF 1b

2c4s� F 1aF 1a

2c2s

� �. ð25Þ

In above equations n and Cab are the parameters to be determined. From

Eqs. (23)–(25) we can find

Xi

�fð0Þi ¼ q;

Xi

eia�fð0Þi ¼ qua;

Xi

eiaeib�fð0Þi ¼ quaub þ c2sqdab; ð26ÞX

i

�fð1Þi ¼

Xi

�fð2Þi ¼ 0; ð27ÞX

i

eia�fð1Þi ¼ nF 1a;

Xi

eia�fð2Þi ¼ 0; ð28ÞX

i

eiaeib�fð1Þi ¼ nðF 1aub þ F 1buaÞ þ qðC1ab þ C1baÞ=2 ð29Þ


andP

ifð1Þi ¼

Pieiaf

ð1Þi ¼

Pif

ð2Þi ¼

Pieiaf

ð2Þi ¼ 0. Summation over all discrete

velocities in Eq. (21) gives:

o1tqþ o1aqua ¼ 0. ð30ÞWhile multiplying by eib before summing gives:

o1tqua þ o1bPð0Þab ¼ nF 1a=s; ð31Þ

where Pð0Þab ¼ pdab þ quaub, p ¼ c2sq. The same process for Eq. (22) gives:

o2tq ¼ 0; ð32Þo2tqua þ o1bP

ð1Þab ¼ 0; ð33Þ

where Pð1Þab ¼

Pieiaeibf

ð1Þi . With the aid of Eq. (21), we can get

Pð1Þab ¼ ½nðF 1aub þ F 1buaÞ þ qðC1ab þ C1baÞ=2� � sc2sqðo1aub þ o1buaÞ.

ð34ÞCombining equations from Eqs. (30)–(33) gives

otqþ oaqua ¼ 0þOðe2Þ ð35Þ

otqua þ ob½nðF aub þ F buaÞ þ qðCab þ CbaÞ=2� þ obðquaubÞ¼ �oap þ ob½mqðoaub þ obuaÞ� þ nF a=sþOðe2Þ; ð36Þ

where m ¼ c2ss. The velocity and pressure of fluid are obtained by Eq. (11).

Comparing to the momentum equation Eq. (1), we now can get n = s,Cab = �2sFaub/q or Cab = �s(Faub + Fbua)/q.

References

[1] O. Filippova, D. Hanel, Comput. Fluids 26 (1997) 697.

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Computation of gas–solid flows by finite difference Boltzmann equation