Modeling, Numerical Methods and Simulation for Particle-Fluid Two

Phase Flow Problems

Qiang Zhang1, Mengping Zhang2, Guodong Jin3, Dayou Liu4 and Chi-Wang Shu5

Abstract

In this paper we study the issues of modeling, numerical methods and simulationwith comparison to experimental data for the particle uid two phase ow probleminvolving a solid-liquid mixed medium. The physical situation being considered is apulsed liquid uidized bed. The mathematical model is based on the assumption ofone dimensional ows, incompressible in both particle and uid phases, equal particlediameters, and the wall friction force on both phases being ignored. The modelconsists of a set of coupled di�erential equations describing the conservation of massand momentum in both phases with coupling and interaction between the two phases.We demonstrate conditions under which the system is either mathematically well-posed or ill-posed. We consider the general model with additional physical viscositiesand / or additional virtual mass forces, both of which stabilizes the system. Twonumerical methods, one of them is �rst order accurate and the other �fth orderaccurate, are used to solve the models. A change of variable technique e�ectivelyhandles the changing domain and boundary conditions. The numerical methods aredemonstrated to be stable and convergent through careful numerical experiments.Simulation results for realistic pulsed liquid uidized bed are provided and comparedwith experimental data.

Key Words: Particle- uid two phase ow, uidized bed, two phase models,experiments, WENO scheme

1Department of Mathematics, University of Science and Technology of China, Hefei, Anhui

230026, China. E-mail: zhqiang@ustc.edu.cn. The research of this author is supported by NSFC

grant 10028103.2Department of Mathematics, University of Science and Technology of China, Hefei, Anhui

230026, China. E-mail: mpzhang@ustc.edu.cn. The research of this author is supported by NSFC

grant 10028103.3Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China. E-mail:

gd jin@263.net. The research of this author is supported by the State Key Laboratory of Non-

linear Mechanics, China.4Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China. E-mail:

dyliu@mail.imech.ac.cn. The research of this author is supported by the State Key Laboratory

of Nonlinear Mechanics, China.5Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:

shu@cfm.brown.edu. The research of this author is supported by NSFC grant 10028103 while he

is in residence at the Department of Mathematics, University of Science and Technology of China,

Hefei, Anhui 230026, China. Additional support is provided by ARO grant DAAD19-00-1-0405 and

NSF grant DMS-9804985.

1

1 Introduction

In this paper we study the issues of modeling, numerical methods and simulationwith comparison to experimental data for the particle uid two phase ow probleminvolving a solid-liquid mixed medium. The physical situation being considered is apulsed liquid uidized bed. The mathematical model is based on the assumption ofone dimensional ows, incompressible in both particle and uid phases, equal particlediameters, and the wall friction force on both phases being ignored. The modelconsists of a set of coupled di�erential equations describing the conservation of massand momentum in both phases with coupling and interaction between the two phases.We demonstrate conditions under which the system is either mathematically well-posed or ill-posed. We consider the general model with additional physical viscositiesand / or additional virtual mass forces, both of which stabilizes the system. Twonumerical methods, one of them is �rst order accurate and the other �fth orderaccurate, are used to solve the models. A change of variable technique e�ectivelyhandles the changing domain and boundary conditions. The numerical methods aredemonstrated to be stable and convergent through careful numerical experiments.Simulation results for realistic pulsed liquid uidized bed are provided and comparedwith experimental data.

The equations we are solving in this paper are in the form of conservation laws

@u

@t+@f(u)

@x= G (1.1)

or in general a �rst order system

@u

@t+ A(u)

@u

@x= G (1.2)

where G contains forcing and possibly viscosity terms. Notice that (1.1) is a special

case of (1.2) with A(u) = @f(u)@u

. The system (1.2) is called hyperbolic (when G doesnot contain viscosity terms) if the eigenvalues of the matrix A(u) are all real and thereis a complete set of independent eigenvectors. Hyperbolic systems are mathematicallywell-posed, meaning that their solutions depend continuously on the initial conditions.This can be proven for the linear case and also for some nonlinear cases. On the otherhand, if the eigenvalues of A(u) have non-zero imaginary parts, the system becomeselliptic. Typically, the system becomes elliptic only in certain regions of u and remainshyperbolic in other regions. Such systems are called mixed-type. Elliptic and mixed-type systems are not mathematically well-posed. Perturbations in high modes growin an unbounded way. Such systems are very diÆcult to simulate numerically.

In recent years many high order, high resolution numerical methods have beendeveloped in the literature to solve hyperbolic systems. The main applications arein computational uid dynamics, but there are also applications in other physicaland engineering areas. In this paper we apply both low order and high order �nite

2

di�erence WENO (weighted essentially non-oscillatory) schemes [8], [3] to solve boththe well-posed and ill-posed models. In particular, �fth order WENO schemes in [8]and splitting techniques in [13] to treat mixed type systems are used. The numericalprocedures are summarized in section 3. These numerical methods are found to bevery useful in computational uid dynamics and in other applications, because oftheir simultaneous high order accuracy and non-oscillatory property in the presenceof shocks and other discontinuities or sharp gradient regions in the solution, or ingeneral for convection dominated problems. For a review of such methods, see [14].

The simulation and experiments reported in this paper are for the uidized beds.Fluidized beds are common and important reactors in process engineering such ascatalytic reactions and bio-reactions because of the high contact eÆciency between the uid and solid. It is well known that there are concentration non-uniform structuressuch as bubbles and slugs in uidization. Bubbling and slugging are undesirable foreÆcient operations, because they can reduce the contact eÆciency. Lots of studiesshow that uidization quality can be signi�cantly improved by externally appliedvibrations or pulses, e.g. Baird [2], Kobayashi et al. [10] and Pence and Beasley [12].However, the theoretical understanding of pulsed uidized beds is far from complete.

The organization of the paper is as follows. In section 2 we present the mathemat-ical models of the two phase ow under consideration. We emphasize two featuresin the general model to stabilize the system. One is the additional physical viscosi-ties, and the other is the additional virtual mass force parameterized by C and �,the details of which will be provided in section 2. The additional physical viscositiesrender the system to be incomplete parabolic; and the additional virtual mass forcerenders the system hyperbolic under certain combinations of C and �. In section 3we present two numerical methods to be used for the simulations in this paper. The�rst method is a �rst order Lax-Friedrichs type scheme, which is adapted as in [13]to e�ectively simulate both hyperbolic and mixed type systems. The second one is a�fth order WENO type method [8], suitable for simulating hyperbolic or incompleteparabolic systems. In section 4 we describe our experimental apparatus and method.Section 5 contains our simulation results and comparisons with experimental data.Concluding remarks are provided in section 6.

3

2 Models of the two phase ow

The continuity and momentum equations for the solid phase and the uid phase are,respectively (e.g. [11]),8>>>>>>>>>>>><

>>>>>>>>>>>>:

@@t(�p�p) +

@@x(�p�pup) = 0

@@t(�f�f ) +

@@x(�f�fuf) = 0

@@t(�p�pup) +

@@x(�p�pu

2p) + �p

@P@x

= ��p�pg + Fp + "1@2

@x2(�p�pup)

@@t(�f�fuf) +

@@x(�f�fu

2f) + �f

@P@x

= ��f�fg � Fp + "2@2

@x2(�f�fuf)

�p + �f = 1

�pup + �fuf = U(t)

(2.1)

Here �p and �f are the volume fractions of particles and uids, respectively, in themixture. up and uf are the velocities, and �p and �f are the (constant) densities ofparticles and uids, respectively. P is the pressure, g is the gravity acceleration, U(t)is a given function of t, measuring the instantaneous uid inlet rate (or uidizationvelocity), controlled by a time-lag relay and electromagnetic valve in our experiment.Fp is the interaction force per unit volume between the solid and uid phases, givenby Fp = Fo+Fc, where Fo represents a drag force being a function of the local relativevelocity (uf � up) and is often expressed as (Tsinontides and Jackson [16], Glasser etal. [7]):

Fo =�p � �fuT

g�p�1�nf (uf � up); (2.2)

where, for uniform sized particles, the index n depends on the particle Reynoldsnumber, Rep =

uT dp�f�f

with dp denoting the diameter of the particles and �f denoting

the viscosity coeÆcient of the uid, n decreases with an increase of Rep, reachingn = 1:39 when Rep > 500, see Wallis [17]. uT is the terminal velocity of a singleparticle in a static uid (when the drag force of the uid acting on the particlebalances with the weight of the particle in the uid) and can be calculated as

uT =d2p(�p � �f )

k�fg:

For non-uniform sized particles, the expression above is just for a reference. Fc

represents the virtual (or added) mass force related to the local relative acceleration.A good expression for this force has not been well researched yet. The followingexpression was given by Drew et al. [5]

Fc = �C�p�f [@up@t

+ uf@up@x

�@uf@t

� up@uf@x

� (1� �)(uf � up)@(uf � up)

@x]: (2.3)

Drew et al. [5] did not explain the clear physical meaning of C and �, and treatedthem as model parameters being functions of �p. According to their opinion, the range

4

of these parameters should be 0 < C < 0:5 and 0 < � < 2, and when �p ! 0 (as thelimiting case of a single particle moving in an in�nite ow �eld), then C = 0:5 (thiscan be proved by the theory of uid mechanics) and � = 2 (this is more heuristic).

The physical motivation for the additional physical viscosities in the momen-tum equations in (2.1) parameterized by "1 and "2 is the normal stresses �pxx and�fxx, whose x-derivatives are added to the two momentum equations. In a non-

homogeneous bed, the normal stress can be expressed as �pxx = �p@up@x, where �p is

the particle viscosity. No theory or experimental data about �p is available. Batchelor[4], Anderson et al. [1] and Glasser et al. [6, 7] gave two di�erent conjectures for �p.Clearly, "1 and "2 have similar meanings as those for �p and �f . We have taken themas constants in this paper for simplicity.

We use the dimensional form of the equation (2.1) with meter-kg-second measuringsystem. As we mentioned before in the introduction, this model is based on theassumption of one dimensional ows, incompressible in both particle and uid phases,equal particle diameters, and the wall friction force on both phases being ignored.

We remark that the last equation in (2.1) can be obtained from the �rst twocontinuity equations and the second last equation, and hence is not an independentequation. Notice that there are only two independent evolution equations in (2.1),due to the last two constraints. We choose to write out the equations for �p and upand eliminate all the other variables, resulting in the following non-conservative formof systems: 8<

:@�p@t

+ A1@�p@x

+B1@up@x

= 0;

@up@t

+ A2@�p@x

+B2@up@x

= D + E;(2.4)

where

A1 = up;

B1 = �p;

A2 = ��r(uf � up)

2[�f + C(1� �)]

�f (�f + �p�r) + C�r; (2.5)

B2 =�p�f�r(2uf � up) + �2

fup + C�r[2uf � up � �(uf � up)]

�f(�f + �p�r) + C�r;

D = �g +�r(g + U 0(t))(�f + C)

�f(�f + �p�r) + C�r+�2�nf (1� �r)g(uf � up)

�f (�f + �p�r) + C�r�1

uT

If we using the model with C > 0, then we set "1 = "2 = 0; otherwise, we need theviscosity terms to make the system well-posed, thus

E ="1�f + "2�p�r�p(�f + �p�r)

@2

@x2(�pup):

Here �r =�f�p. Notice that it is impossible to write the momentum equation in

conservation form if we eliminate the pressure P . We thus have adopted this non-

5

conservative form for the simulation. This is allowed as long as no shocks or otherdiscontinuities appear.

To check if the system (2.4) is mathematically well-posed (when the viscosity termE = 0), we look at the eigenvalues of the Jacobian0

@ A1 B1

A2 B2

1A ;

which are given by

�1;2 =Q

2R�4u

R

��r

�f�p

�1=2

H1=2;

where 4u = uf � up, and

R =�r�f

+1

�p+

�r�2f�p

C;

Q =2�ruf�f

+2up�p

+C�r�2f�p

[uf + up + (1� �)4u];

H = �1 +C

�f

(1� �)(�r � 1) +1

4

�r�f�p

�C

�f

�2

f1 + 2(�f � �p)(1� �) + (1� �)2g:

Notice that these eigenvalues are real only if H � 0. This for example is never truewhen C = 0. Thus the system without physical viscosity is always ill-posed if noadditional virtual mass force characterized by C and � is added.

We remark that preliminary simulation results using a Lax-Wendro� type schemewere provided in Jin et al. [9], for the model with C = 0, with also comparison withexperimental results. The study in this paper is however more exhaustive both in thegenerality of the models (including those with C > 0 and the physical viscosities) andin the accuracy and robustness of the numerical methods. Moreover, the experimentalresults for both the bed height h(t) and the particle concentration �p are obtained indata form and compared with the simulation results.

3 Numerical methods

In the simulation presented in section 5, the computational domain is given by 0 �x � h(t), with an initial value given as8>>>>>><

>>>>>>:

�p(x; 0) = 1� (U(0)=uT )1=(n+1);

up(x; 0) = 0;

uf(x; 0) = U(0)=(1� �p(x; 0));

h(0) = h0:

(3.1)

6

The boundary condition is given by8<:

x = 0 : up(0; t) = 0; �p(0; t) = 1� (U(t)=uT )1=(n+1);

x = h(t) : dh(t)dt

= up(h(t); t):(3.2)

In the simulation, the length of the computational domain [0; h(t)] is changing withthe time t according to the boundary condition (3.2) which describes the evolutionof the bed height h(t) when the velocity up at the right (top) boundary is given. AneÆcient method for the computation is through a transformation

x =h(t)

h0�; and t = �;

where (x; t) is the original physical coordinate and (�; �) is the computational coor-dinate in which the spatial mesh size is kept uniform. If we de�ne

C0 =h0h(t)

; C1 =h0(t)

h(t)=up(h(t); t)

h(t);

then we have@

@t=

@

@�� C1�

@

@�;

@

@x= C0

@

@�;

and the eigenvalues of the Jacobian in the new coordinate system (�; �) are given by

�1;2 = C0�1;2 � C1�: (3.3)

The �nal system to be solved numerically is thus

@�p

@�+ ~A1

@�p

@�+ ~B1

@up@�

= 0; (3.4)

@up@�

+ ~A2@�p

@�+ ~B2

@up@�

= D

where

~A1 = C0up � C1�;~B1 = C0�p;

~A2 = �C0�r(uf � up)

2[�f + C(1� �)]

�f(�f + �p�r) + C�r;

~B2 =C0f�p�f�r(2uf � up) + �2

fup + C�r[2uf � up � �(uf � up)]g

�f(�f + �p�r) + C�r� C1�;

for the case C > 0, where D is still given by (2.5), and

@�p

@�+@(C0�pup � C1��p)

@�= �C1�p; (3.5)

@up@�

�C0�r

�f + �r�p

@(�pu2p + �fu

2f)

@�+@(C0

2u2p � C1�up)

@�= �C1up + ~E + F;

7

where

F = (�f + �r�p)�1f�rU

0(t) + (�r � 1)�fg[1� ��nf (uf � up)=uT ]g;

and

~E =C20 ("1�f + "2�p�r)

�p(�f + �p�r)

@2

@�2(�pup)

for the case C = 0. Notice that, for the simpler case (3.5), we have written the systemgrouping as many terms in a conservation form as possible.

We now describe the numerical methods used in solving the systems (3.4) and(3.5). We �rst describe spatial discretizations. Two di�erent numerical methods areused.

The �rst method is a simple �rst order accurate Lax-Friedrichs type spatial dis-cretization, adjusted as in [13] to handle mixed type systems. Thus, all the �rstderivative terms in (3.4) and (3.5) are approximated by a second order central di�er-ence formula. For example, @�p

@�in the second equation of (3.4) is discretized by the

formula@�p

@�j�=�j �

(�p)j+1 � (�p)j�12��

:

To correct the instability caused by central approximations, we add numerical viscos-ity terms to the equations in (3.4) and (3.5). To the �rst equation in (3.4) and (3.5),we add a term

C0�1�

��((�p)j+1 � 2(�p)j + (�p)j�1); (3.6)

and to the second equation in (3.4) and (3.5), we add a term

C0�2�

��((up)j+1 � 2(up)j + (up)j�1); (3.7)

where � is the maximum in absolute value of the eigenvalues de�ned in (3.3) takenover the whole � line. �1 and �2 are two constants, which can actually change withthe mesh size ��: as long as they do not grow as fast as 1

��, the terms de�ned above

will approach zero when �� ! 0 and hence can be justi�ed as numerical (ratherthan physical) viscosities. We take �2 = 0 for the second equation in (3.5), since thephysical viscosities already take e�ect in this equation.

The second method is the high order weighted essentially non-oscillatory (WENO)method. In particular, we use the �fth order WENO �nite di�erence method in [8].The advantage of the WENO scheme is that it maintains uniform high order accuracywhile automatically adapting its reliance on local stencils to obtain information froma locally smoothest region, hence avoiding numerical oscillations and instabilitieswhen the solution contains either shocks or sharp gradient regions. The higher orderaccuracy of the method allows us to use fewer grid points to obtain good resolution

8

of the solution. Thus, for any of the �rst derivative terms, such as @g@�

with g = �p,we discretize it by a �fth order conservative WENO di�erence formula

@g

@�j�=�j �

1

��

�gj+1=2 � gj�1=2

�

where the numerical ux gj+1=2 is obtained with the following procedure. We de-scribe only the construction of this numerical ux with a left biased stencil, suitablefor computing the derivative with a positive ow direction. The procedure for the nu-merical ux with a right biased stencil would just be a mirror symmetry with respectto j + 1=2 when computing gj+1=2. We obtain the numerical ux by

gj+1=2 = !1g(1)j+1=2 + !2g

(2)j+1=2 + !3g

(3)j+1=2

where g(m)j+1=2 are three third order uxes on three di�erent stencils given by

g(1)j+1=2 =

1

3gj�2 �

7

6gj�1 +

11

6gj;

g(2)j+1=2 = �

1

6gj�1 +

5

6gj +

1

3gj+1;

g(3)j+1=2 =

1

3gj +

5

6gj+1 �

1

6gj+2;

and the nonlinear weights !m are given by

!m =~!mP3l=1 ~!l

; ~!l = l

("+ sl)2;

with the linear weights l given by

1 =1

10; 2 =

3

5; 3 =

3

10;

and the smoothness indicators sl given by

s1 =13

12(gj�2 � 2gj�1 + gj)

2 +1

4(gj�2 � gj�1 + 3gj)

2

s2 =13

12(gj�1 � 2gj + gj+1)

2 +1

4(gj�1 � gj+1)

2

s3 =13

12(gj � 2gj+1 + gj+2)

2 +1

4(3gj � 4gj+1 + gj+2)

2 :

Finally, " is a parameter to avoid the denominator to become 0 and is taken as" = 10�6 in the computation of this paper.

Thus, for the conservative terms, i.e. the terms @(C0�pup�C1��p)@�

and@(

C02u2p�C1�up)

@�

in (3.5), we perform a Lax-Friedrichs ux splitting

f+ =1

2(f + ��p); f� = f � f+; (3.8)

9

for f = (C0�pup � C1��p), and

f+ =1

2(f + �up); f� = f � f+; (3.9)

for f = (C0

2u2p � C1�up), where � is again the maximum in absolute value of the

eigenvalues de�ned in (3.3) taken over the whole � line, and is a constant controllingthe numerical viscosities in this ux splitting. can change with the mesh size ��.As long as it does not grow as fast as 1

��, the numerical viscosity will vanish together

with �� and the scheme is still at least formally fourth order accurate, justifying thescheme as a high order method. The WENO ux di�erence approximation is thenused, with a left biased ux to approximate the derivative of f+, and a right biased ux to approximate the derivative of f�. The sum of them is then an approximationto the derivative of f .

We now describe the WENO recipe for the non-conservative �rst derivative terms,

such as all the �rst derivative terms in (3.4) and the term � C0�r�f+�r�p

@(�pu2p+�fu2

f )

@�in

(3.5). The approximation for the \diagonal terms", namely ~A1@�p@�

in the �rst equation

of (3.4) and ~B2@up@�

in the second equation of (3.4), is taken as an upwind biasedWENO

approximation. Namely, if the coeÆcient, for example cj = ( ~A1)j, is positive, then

the derivative @�p@�

is approximated by a left biased stencil based WENO; otherwise, itis approximated by a right biased stencil based WENO. For all the other terms, theWENO approximation is central, namely taken as the average of the the left biasedstencil based approximation and the right biased stencil based approximation. Finally,the numerical viscosity terms in (3.6) and (3.7) are added. This seems necessary, notto control any instability, but to maintain the bed height h(t) to reach a periodic (intime) pattern. See section 5 for details.

We now describe the time discretization. For the �rst order Lax-Friedrichs typeapproximation, a simple Euler forward is used in time. For the �fth order WENOscheme, we use the following third order TVD Runge-Kutta time discretization [15]:

�(1) = �n +�tL(�n; tn)

�(2) =3

4�n +

1

4�(1) +

1

4�tL(�(1); tn +�t)

�n+1 =1

3�n +

2

3�(2) +

2

3�tL(�(2); tn +

1

2�t);

where � represents the quantities to be evolved in time, namely �p, up and h(t), andL is the approximation to the spatial derivatives and the forcing terms (the righthand side of the ODE for the time variable t when all spatial derivatives have beendiscretized). Notice that this TVD Runge-Kutta method is just a convex combinationof three simple Euler forward time discretizations.

We �nally summarize the computational procedure as follows. We only describethe procedure for Euler forward in time, since the third order TVD Runge-Kuttamethod is just a convex combination of three such Euler forward steps.

10

1. Given �p, up and h(t) at t = tn, compute �f and uf at t = tn by the last twoequations of (2.1);

2. Compute all the spatial derivatives and forcing terms in (3.4) or (3.5) using oneof the numerical methods described above;

3. Update the bed height h(t) by

h(tn+1) = h(tn) + �tup(h(tn); tn);

4. Update �p, up using Euler forward, and go back to step 1.

4 Experimental apparatus and method

Figure 4.1 is the schematic diagram of the experimental apparatus. The experimentshave been carried out under ambient conditions in a 29 ID and 1150 high perspexcolumn �tted with a porous plate distributor. The variations of bed height andparticle concentration have been recorded using a Sony R digital video camera recorder(DCR-TRV6E).

Figure 4.1: Schematic diagram of the experimental apparatus. 1 Fluidized bed;2 Pressure transducer; 3 Plate distributor; 4 Spheric valve; 5 Time-lag relay andelectromagnetic valve; 6 Lower water tank; 7 Centrifugal pump; 8 Upper water tank;9 Ampli�er; 10 A/D converter; 11 Computer; 12 Water gathering vessel; 13 Ruler.

Particles are uidized by pulsed liquid. The liquid ows through two routes: oneis a steady ow and the other is a pulsed ow. The two routes intersect before liquid ows through a section with packed particles, then into the bed to make the ow

11

uniform. The ux is measured using a stopwatch and a platform balance. The sketchmap of ux vs. time is illustrated in Figure 4.2. The pulse frequency, and the timeof on-period and o�-period are controlled by the graded time-lag relay.

Figure 4.2: Illustration diagram of ow rate varying with time.

We now describe the experimental method. We put a ruler beside the bed inthe experiment and record the variation process of the bed height by the camera.The camera can record 25 frames per second. The variation of bed height vs. time ismeasured from a frame by frame playing of the tape. In order to get the concentrationdistribution, we must calibrate the relation between the particle concentration andthe scale of the photos. We carry out the experiment at night, and use six 500Wiodine-tungsten lamps in vertical direction to get uniform background light. Wemake space corrections considering the non-uniform exposure of the �lm even to auniform light. For example, Figure 4.3 (left) shows that, even for a uniform state,throughout the bed, the scale of the photo is still non-uniform along the bed. Afterwe make the corrections, we can get a much more uniform distribution of the particleconcentration, and the standard deviation is about 0.7% and the system error is about1.0%, Figure 4.3 (right). This is allowable for dense two-phase ow measurement inthe overall ow �eld.

12

Figure 4.3: Left: Non-uniformity of the scale of the photo along the bed for uniformconcentration. Right: Particle concentration after correction.

Table 5.1: Summary of the three test cases.

U2(m/s) U1(m/s) CT;1(s) CT;2(s) n UT (m/s) h0(m)

Case 1: 0.1020 0.0400 4 8 1.44 0.194 0.909

Case 2: 0.1541 0.0376 2 6 1.44 0.194 0.476

Case 3: 0.0550 0.1062 6 9 1.44 0.191 1.365

5 Simulation results and comparison with experi-

ments

In the numerical tests, we consider the initial and boundary conditions (3.1)-(3.2)with

U(t) =

8<:

U1; CT;1 < t � CT;2;

U2; 0 < t � CT;1;

and periodically extended with period CT;2. Note that U(0) = U1 as initial condition.Three cases are studied in the numerical simulation: the corresponding parametersfor these cases are given in Table 5.1.

Furthermore, we have �f = 1000 kg=m3 and �p = 2500 kg=m3.In all the simulations we set the CFL number to be 0.1, where the CFL number

is de�ned by

��t

��+C20("1�f + "2�p�r)

�p(�f + �p�r)

�t

��2

13

and � is again the largest eigenvalue (in absolute value) of the Jacobian in (3.3) overthe whole � line and "1 and "2 are the physical viscosity coeÆcients. For the modelwith C = 0, we set the coeÆcients of the physical viscosity as "1 = "2 = 10�5. Forthe model with C > 0, we do not include any physical viscosity ("1 = "2 = 0) andset C = 0:5 and � = 3 for all the test cases. Ideally we would like to keep � between0 and 2, however it seems that the system is still ill-posed when � is in this range.The choices for C and � are taken after extensive numerical experiments to guaranteehyperbolicity and good numerical results.

Numerical simulations are continued for a long time until a steady periodic patternis observed.

Figures 5.1 to 5.12 contain comparisons between the experimental data and thesimulation results for the bed height h(t), for the three cases and two numericalmethods. The square symbols represent the experimental data, the circles representthe numerical simulation results with a coarser mesh, and the lines represent thenumerical simulation results with a re�ned mesh. For the �rst order method, we use300 points for the coarser mesh and 600 points for the re�ned mesh; for the �fth orderWENO method, we use 150 points for the coarser mesh and 300 points for the re�nedmesh.

We summarize the numerical viscosity coeÆcients �1 and �2 in (3.6) and (3.7), andthe coeÆcient in front of the largest eigenvalue in (3.8) and (3.9), in Table 5.2. Wehave made sure to choose these values so that the numerical viscosity vanishes withgrid re�nement, justifying the term \numerical viscosity". Also notice that there isno numerical viscosity added for the second equation when C = 0 because the secondequation comes from the momentum equations and actually contains physical viscos-ity. The results remain close when these parameters take values in a neighborhoodof the chosen values. However, the bed height h(t) does not stay in a time periodicfashion (rather it either increases or decreases without bound for large time) whenthese parameters are in the wrong range. It is not clear a priori mathematically thatthe bed height h(t) to these models should maintain a periodic pattern in time withthe given initial and boundary conditions.

Figures 5.1 to 5.4 are for the �rst case. Among them, Figure 5.1 represents theresult for the model with C = 0 using the �rst order method, and Figure 5.2 representsthe result for the same model with C = 0 using the �fth order WENOmethod. Figures5.3 and 5.4 contain the �rst order and �fth order results, respectively, for the modelwith C > 0. We notice that convergence is observed during mesh re�nement for bothnumerical methods and both models. The numerical results are reasonably close tothe experimental results for both models, however the match seems to be better forthe model with C = 0, especially for the �rst order simulation result.

Figures 5.5 to 5.8 are for the second case. Among them, Figure 5.5 and Figure5.6 represent the results for the model with C = 0 using the �rst order method andthe �fth order method, respectively. Figures 5.7 and 5.8 contain the �rst order and�fth order results, respectively, for the model with C > 0. Again, we notice that

14

Table 5.2: Summary of the numerical viscosity coeÆcients.

C = 0 C = 0:5; � = 3

Test mesh �rst order 5th order �rst order 5th order

Cases �1 �2 �1 �2 �1 �2 �1 �2

Case 1 coarse 0.5 0 0.5 0 2.5 1.75 1.75 0.85 0.85

re�ned 0.5 0 0.5 0 4.0 3.0 3.0 1.2 1.2

Case 2 coarse 0.75 0 0.5 0 0.15 2.0 2.0 0.448 0.448

re�ned 1.25 0 0.5 0 0.4 3.25 3.25 0.59 0.59

Case 3 coarse 0.25 0 0.5 0 4.0 1.25 1.25 0.5 0.5

re�ned 0.375 0 0.5 0 5.0 1.5 1.5 0.9 0.9

convergence is observed during mesh re�nement for both numerical methods andboth models. The numerical results are close to experimental data for both models.It seems that the model with C > 0 �ts the experimental result better for this case.It also seems that the �fth order results are closer to the experimental results thanthe �rst order results for this case, especially for the model with C > 0.

Figures 5.9 to 5.14 are for the third case. Among them, Figures 5.9 and 5.10represent the results for the model with C = 0 using the �rst order method and the�fth order method, respectively. Figures 5.11 and 5.12 contain the �rst order and �fthorder results, respectively, for the model with C > 0. Again, the numerical results arewell converged for both methods and both models, and the agreement between thesimulation results and experimental data is very good. Finally, Figures 5.13 and 5.14contain the comparison between the simulation results and the experimental data forthe volume fraction of particles �p, for �ve equally spaced time snaps during a timeperiod. Figure 5.13 is for the model with C = 0, and Figure 5.14 is for the model withC > 0. The �rst order results are shown on the left, and the �fth order results onthe right. The solid lines represent the simulation results with the re�ned mesh, andthe symbols are the experimental data. We can observe a reasonably good agreementbetween the simulation results and the experimental data for �p. The results obtainedby the �fth order WENO method seem to be smoother (less numerical overshoot) nearthe junction than those obtained by the �rst order method.

15

time(s)

height(m)

3035

4045

501.25

1.3

1.35

1.4E

xp30

060

0

Figu

re5.1:

Case

1with

C=

0.First

order

numerical

meth

od.Squares

arethe

experim

ental

data,

circlesare

thesim

ulation

result

with

300grid

poin

ts,andthe

solidlin

eisthesim

ulation

resultwith

600grid

poin

ts.

6Concludingremarks

Wehave

studied

agen

eralparticle-

uid

twophase

model

involv

ingasolid

-liquid

mixturemedium.Twophysical

mech

anism

sto

stabilize

thesystem

,onebyaddition

alphysical

viscosity

andtheoth

erbyaddition

alvirtu

almass

forces,are

consid

ered.

Twodi�eren

tnumerical

meth

ods,one�rst

order

Lax-Fried

richstypeandtheoth

er�fth

order

WENO

schem

e,are

develop

edto

solvethisgen

eralsystem

.A

realisticpulsed

liquid

uidized

bed

issim

ulated

andthesim

ulation

results

arecom

pared

with

experim

ental

data.

Itisfou

ndthat

both

numerical

meth

odscon

vergewell

forall

testcases

durin

ggrid

re�nem

ent,

indicatin

gtheir

suitab

ilityfor

thesim

ulation

ofsuch

agen

eralmodel.

The�fth

order

meth

odprov

ides

better

resolution

forthesam

enumber

ofmesh

poin

ts,or

comparab

leresolu

tionusin

gfew

ermesh

poin

ts,when

compared

with

the�rst

order

meth

od.Simulation

results

agreein

general

quite

well

with

experim

ental

data.

References

[1]K.G.Anderson

,S.Sundaresan

andR.Jack

son,Insta

bilitiesandtheform

atio

n

ofbubbles

in uidized

beds,Jou

rnalofFluidMech

anics,

v303

(1995),pp.327-366.

[2]M.H.I.

Baird

,Vibra

tionsandpulsa

tions,

British

Chem

icalEngin

eering,

v11

(1966),pp.20-25.

16

time(s)

height(m)

3035

4045

501.25

1.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

1.4E

xp15

030

0

Figu

re5.2:

Case

1with

C=0.

Fifth

order

WENO

numerical

meth

od.Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

resultwith

150grid

poin

ts,andthe

solidlin

eisthesim

ulation

resultwith

300grid

poin

ts.

[3]D.Balsara

andC.-W

.Shu,Monotonicity

preservin

gweigh

tedessen

tially

non-

oscilla

tory

schem

eswith

increa

singly

high

order

ofaccu

racy,

Journalof

Com

pu-

tationalPhysics,

v160

(2000),pp.405-452.

[4]G.K.Batch

elor,Anew

theory

oftheinsta

bilityofaunifo

rm uidized

beds,Jou

r-nalof

Fluid

Mech

anics,

v193

(1988),pp.75-110.

[5]D.Drew

,L.ChengandR.T.Lahey

Jr.,Theanalysis

ofvirtu

almass

e�ects

in

two-phase

ow,Intern

ationalJou

rnalofMultip

hase

Flow

,v5(1979),

pp.233-242.

[6]B.J.

Glasser,

I.G.Kevrek

idisandS.Sundaresan

,One-a

ndtwo-dim

ensio

naltra

v-

ellingwave

solutio

nsin

gas-

uidized

beds,

Journal

ofFluid

Mech

anics,

v306

(1996),pp.183-221.

[7]B.J.

Glasser,

I.G.Kevrek

idisandS.Sundaresan

,Fully

develo

pedtra

vellingwave

solutio

nsandbubble

form

atio

nin

uidized

beds,Jou

rnalof

FluidMech

anics,

v334

(1997),pp.157-188.

[8]G.-S

.Jian

gandC.-W

.Shu,EÆcien

tim

plem

entatio

nofweigh

tedENO

schem

es,Jou

rnalof

Com

putation

alPhysics,

v126

(1996),pp.202-228.

[9]G.Jin

,Y.Nie

andD.Liu,Numerica

lsim

ulatio

nofpulsed

liquid

uidized

bed

andits

experimentalvalidatio

n,Pow

der

Tech

nology,

v119

(2001),pp.153-163.

[10]M.Kobayash

i,D.Ram

aswam

iandW.T.Brazelton

,Pulsed

-bedapproa

chto

u-

idiza

tion,Chem

icalEngin

eeringProgress

Symposiu

mSeries

{Fluidization

Fun-

dam

entals

andApplication

,v66

(1970),pp.47-57.

17

time(s)

height(m)

3035

4045

501.25

1.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

1.4E

xp30

060

0

Figu

re5.3:

Case

1with

C=0:5

and�=3.

First

order

numerical

meth

od.Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

resultwith

300grid

poin

ts,and

thesolid

lineisthesim

ulation

resultwith

600grid

poin

ts.

[11]D.Liu,Fluid

Dynamics

ofTwo-Phase

System

s,High

erEducation

Press,

Beijin

g,1993

(inChinese).

[12]D.V.Pence

and

D.E.Beasley,

Loca

l,insta

ntaneousheattra

nsfer

inpulse-

stabilized

uidiza

tion,Proceed

ings

oftheASMEHeat

Tran

sferDivision

,v334

(1996),pp.65-75.

[13]C.-W

.Shu,A

numerica

lmeth

odforsystem

sofconserva

tion

lawsofmixed

type

admittin

ghyperbo

lic uxsplittin

g,Jou

rnalof

Com

putation

alPhysics,

v100

(1992),pp.424-429.

[14]C.-W

.Shu,Essen

tially

non-oscilla

tory

andweigh

tedessen

tially

non-oscilla

tory

schem

esforhyperbo

licconserva

tionlaws,in

Advanced

Numerica

lApproxim

atio

n

ofNonlin

earHyperbo

licEquatio

ns,B.Cockburn,C.Joh

nson

,C.-W

.ShuandE.

Tadmor

(Editor:

A.Quarteron

i),Lectu

reNotes

inMath

ematics,

volume1697,

Sprin

ger,1998,

pp.325-432.

[15]C.-W

.ShuandS.Osher,

EÆcien

tim

plem

entatio

nofessen

tially

non-oscilla

tory

shock

capturin

gsch

emes,

JournalofCom

putation

alPhysics,

v77

(1988),pp.439-

471.

[16]S.C.Tsin

ontid

esandR.Jack

son,Themech

anics

ofgas uidized

bedswith

an

interva

lofsta

ble uidiza

tion,Jou

rnal

ofFluid

Mech

anics,

v255

(1993),pp.237-

274.

[17]G.B.Wallis,

One-d

imensio

naltwo-phase

ow,McG

raw-Hill,

New

York

,1969.

18

time(s)

height(m)

3035

4045

501.25

1.26

1.27

1.28

1.29

1.3

1.31

1.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

1.4E

xp15

030

0

Figu

re5.4:

Case

1with

C=0:5

and�=3.

Fifth

order

WENO

numerical

meth

od.

Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

result

with

150grid

poin

ts,andthesolid

lineisthesim

ulation

resultwith

300grid

poin

ts.

time(s)

height(m)

2025

3035

400.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92E

xp30

060

0

Figu

re5.5:

Case

2with

C=

0.First

order

numerical

meth

od.Squares

arethe

experim

ental

data,

circlesare

thesim

ulation

result

with

300grid

poin

ts,andthe

solidlin

eisthesim

ulation

resultwith

600grid

poin

ts.

19

time(s)

height(m)

2022

2426

2830

3234

3638

400.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92E

xp15

030

0

Figu

re5.6:

Case

2with

C=0.

Fifth

order

WENO

numerical

meth

od.Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

resultwith

150grid

poin

ts,andthe

solidlin

eisthesim

ulation

resultwith

300grid

poin

ts.

time(s)

height(m)

2025

3035

400.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92E

xp30

060

0

Figu

re5.7:

Case

2with

C=0:5

and�=3.

First

order

numerical

meth

od.Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

resultwith

300grid

poin

ts,and

thesolid

lineisthesim

ulation

resultwith

600grid

poin

ts.

20

time(s)

height(m)

2025

3035

400.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

Exp

150

300

Figu

re5.8:

Case

2with

C=0:5

and�=3.

Fifth

order

WENO

numerical

meth

od.

Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

result

with

150grid

poin

ts,andthesolid

lineisthesim

ulation

resultwith

300grid

poin

ts.

time(s)

height(m)

8085

9095

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98E

xp30

060

0

Figu

re5.9:

Case

3with

C=

0.First

order

numerical

meth

od.Squares

arethe

experim

ental

data,

circlesare

thesim

ulation

result

with

300grid

poin

ts,andthe

solidlin

eisthesim

ulation

resultwith

600grid

poin

ts.

21

time(s)

height(m)

8085

9095

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98E

xp15

030

0

Figu

re5.10:

Case

3with

C=0.

Fifth

order

WENO

numerical

meth

od.Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

resultwith

150grid

poin

ts,andthe

solidlin

eisthesim

ulation

resultwith

300grid

poin

ts.

time(s)

height(m)

8085

9095

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98E

xp30

060

0

Figu

re5.11:

Case

3with

C=0:5

and�=3.

First

order

numerical

meth

od.Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

resultwith

300grid

poin

ts,and

thesolid

lineisthesim

ulation

resultwith

600grid

poin

ts.

22

time(s)

height(m)

8085

9095

0.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98E

xp15

030

0

Figu

re5.12:

Case

3with

C=0:5

and�=3.

Fifth

order

WENOnumerical

meth

od.

Squares

aretheexperim

ental

data,

circlesare

thesim

ulation

result

with

150grid

poin

ts,andthesolid

lineisthesim

ulation

resultwith

300grid

poin

ts.

23

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 5.13: The �p for Case 3 for the model with C = 0. Left: simulation resultsof the �rst order method with 600 points; right: simulation results of the �fth orderWENO method with 300 points. From top to bottom: results at t = 0:0, t = 1:8,t = 3:6, t = 5:4 and t = 7:2 in a period. Solid lines: simulation results; symbols:experimental data.

24

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.2

0.25

0.3

0.35

0.4

0.45

0.5

V10.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

height(m)0.2 0.4 0.6 0.8

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 5.14: The �p for Case 3 for the model with C = 0:5 and � = 3. Left:simulation results of the �rst order method with 600 points; right: simulation resultsof the �fth order WENO method with 300 points. From top to bottom: results att = 0:0, t = 1:8, t = 3:6, t = 5:4 and t = 7:2 in a period. Solid lines: simulationresults; symbols: experimental data.

25