dynamic numerical simulation of gas-liquid two-phase flows euler/euler versus euler/lagrange

Pergamon Chemical En~lineerin 9 Science, Vol. 52, No. 4, pp. 611-626, 1997 Copyright c) 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved P I I : S0009-2509(96)00425-3 0009 2509,/'97 $17.00 + 0.00

Dynamic numerical simulation of gas-liquid two-phase flows

Euler/Euler versus Euler/Lagrange

A. Sokolichin* and G. Eigenberger Institut fiir Chemische Verfahrenstechnik, Universit~it Stuttgart, B6blingerstr. 72,

D-70199 Stuttgart, Germany

and

A. Lapin and A. Liibbert t Institut fiir Technische Chemie, Universit~it Hannover, Callinstr. 3, D-30167 Hannover,

Germany

(Received 18 January 1996; accepted 3 July 1996)

Abstraet--A dynamical, two-phase flow model in two- and three-space coordinates is presented. The gas-liquid flow is modeled by a Navier-Stokes system of equations in an Eulerian representation. The motion of gas is modeled by a separate continuity equation. The Eulerian approach with UPWIND or TVD discretization and the Lagrangian approach for solving the gas-phase equation are compared with each other on two two-dimensional test problems: the dynamical simulation of a locally aerated bubble column and of a uniformly aerated bubble column. The comparison shows that the results obtained with the TVD-version of the Euler/Euler method and the Euler/Lagrange technique agree quantitatively. On the other hand, it has not been possible to obtain similar agreement even qualitatively with the UPWIND technique, due to the influence of the numerical diffusion effects, which are inherent in the case of UPWIND discretization. Copyright 1997 Elsevier Science Ltd

Keywords: Modeling; simulation; fluid-dynamics; gas-liquid-flow; Euler/Euler; Euler/ Lagrange.

1. INTRODUCTION

Numerical simulation is being recognized as a primary tool for improving the performance of process equipment. In particular, for scale-up of chemical reactors a reliable fluid dynamic reactor model is of great benefit. Dynamic numerical simulation is thus on the agenda of most big chemical companies and many scientific research laboratories.

While the computing power of workstations and mainframe computers, necessary to perform adequate numerical simulations, increased considerably over the last years, the appropriate basic simulation soft- ware is currently lagging behind. This is particularly true for numerical codes which can be used to simu- late gas-liquid two-phase flows.

As demonstrated by Lapin and Liibbert (1994), Sokolichin and Eigenberger (1994) and Devanathan

* Corresponding author. * Present address: Institut fiir Bioverfahrenstechnik,

Martin-Luther-Universifftt Halle-Wittenberg, Weinbergweg 23, D-06120 Halle, Germany.

et al. (1995), it is necessary to consider the dynamics of the two-phase flow and the corresponding transient flow behavior in order to account for the reactor properties as mixing and heat transfer, which are of interest to chemical engineers.

In literature, essentially two basic approaches to dynamic flow simulations of two-phase gas-liquid flows have been discussed. The first is an approach where both the liquid motion and the gas-phase motion are considered in a homogeneous way. These two-fluid approximations are presented in Eulerian representation and thus referred to as Euler/Euler simulations (Torvik and Svendsen, 1990; Sokolichin and Eigenberger, 1994). The second approach treats only the liquid-phase motion in an Eulerian representation and computes the motion of the dispersed gas-phase fluid elements in a Lagrangian way by indi- vidually tracking them on their way through the reactor. This approach has been termed Euler/Lagrange representation (Webb et al., 1992; Lapin and Liibbert, 1994). Several numerical solution schemes which are by no means equivalent have been applied to solve the corresponding differential equation systems.

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Before the available codes can be used for reactor development it is necessary to validate them. Prim- arily, validation should be based on experiments where the flow structures are similar to those of industrial reactors which are the final target of process development. There are considerable difficulties in such a direct validation procedure since the measurement techniques necessary to provide comprehensive data from the turbulent flows prevailing in real chemical reactors are not sufficiently developed. Most available measurement devices provide local fluid velocity data only. Usually, only long-time averaged data are published. Even for bubble columns, which can be regarded as the most simple two-phase reactors, gas-liquid flow patterns are available as long- term averages (Torvik and Svendsen, 1990; Grienber- ger and Hofmann, 1992). Consequently, a direct validation of transient flow structures in bubble columns is presently not possible.

The best one can do at the moment is to make sure that the codes predict at least qualitatively all charac- teristic properties of the flow which are known from experience. In this contribution, such a comparison will be based upon measurements in flat bubble columns with a wafer-type geometry where an essentially two-dimensional flow structure prevails (Tzeng et al., 1993; Becker et al., 1994).

It is the aim of the contribution to compare the results of different codes, based upon the same fluid- dynamical model, for two examples of a locally and uniformly aerated flat bubble column. An Euler/Lag- range code is compared with two versions of an Euler/Euler code where for the gas flow either a first- order UPWIND discretization or a second-order discretization is used. The stability of the second-order discretization is based upon the concept of total variation diminution (TVD). Therefore, this code will be referred to as the TVD method.

2. FLU ID-DYNAMICAL MODEL

As pointed out by Landau and Lifschitz (1971), the Navier-Stokes equation system, which is of fundamental importance to all single-phase flows, can also be applied to two-phase flows if the dispersed phase elements are small and do not significantly change the overall fluid density and if the momentum of the particles or bubbles can be neglected. Then the density p must be chosen as the effective density of the dispersion and, similarly, the usual viscosity/~ must be replaced by an effective viscosity #eff. This leads to the following model equations:

~p a-7 + v. (pu) = 0

A. Sokolichin et al.

stress tensor:

f ~ui ~u; 2. ?~u,\ (3)

The continuity equation (1) is usually combined with the viscous momentum equation (2) to form the Navier-Stokes equation system. Provided a proper separate model is available for the effective viscosity #elf, the system (1), (2) consists of 4 scalar equations and contains 5 unknown variables (p, ul, Uz, u3, p).

The effective density, p, of the gas-liquid mixture can be taken as the corresponding local average

p = ~pg + (1 - ~)p~ (4)

where e is the volume fraction or the local holdup of the gas phase. The system of equations can be closed with an additional continuity equation for the gas phase:

O(ePo) t?----f- + V. (epouo) = D (5)

where D accounts for dispersive effects in the gas phase due to random fluctuations of the bubble motions. The gas velocity u o can be expressed as the sum of liquid velocity u~ and slip velocity Usnp. For the slip velocity, us~ip, various expressions can be found in the literature depending on the pressure gradient, the drag force, the added mass force, the Basset force, the Magnus force and the Saffman lift force (see e.g. Johansen, 1990). For the gas-liquid flow in bubble columns, we assume the last four effects to be negligible. Then we get a simplified expression for the slip velocity:

Vp Uslip - - Cdrag (6)

where Cd,ag is a drag force coefficient for which a large number of correlations can be found in literature, depending upon whether single bubbles or bubble swarms in stagnant or moving liquids are considered. Ca~g depends primarily on the bubble size. This de- pendency is rather weak for air bubbles of 1-10 mm mean diameter in water. According to Schwarz and Turner (1988),

Car~g = 50 g cm 3 s (7)

can be used, leading to a mean bubble slip velocity of about 20 cm/s, which is in complete accordance with experimental velocity data of air bubbles in tap water. The density of the liquid is assumed to be constant within the bubble column while the density of the gas

(1) phase depends on the local pressure p:

Pt = const. (8)

(2) p (9) Pg = RTo"

Together with the equations representing the relationship between the velocities of both phases and the

Opu dt

- - + V.(puu) = -- Vp + V.T + pg

where u is the velocity vector, g is the acceleration due to gravity, p is the pressure and T is the

Dynamic numerical simulation

gas-liquid mixture,

pu = epgug + (1 - - e)plut (10)

we get a closed system of differential and algebraic equations which describes the dynamical behavior of the two-phase flow. It can be solved numerically, if p~ff and D are specified.

3, EFFECTIVE VISCOSITY AND BUBBLE PATH

DISPERSION

In order to determine the effective viscosity #eff of the gas-liquid mixture, the standard k-s model developed for single-phase flows has been used in the majority of publications on numerical simulations of two-phase flows. However, at present it is not clear how far such turbulence models, which have been developed for single-phase flow, can be applied to two-phase flows. The dispersed phase - - here the rising gas bubbles - - obviously influences the effective viscosity of the gas-liquid dispersion. Previous simulations showed (Becker et al., 1994) that gas-liquid bubble flow can often be described with good qualitative and reasonable quantitative accuracy using two-phase flow models without specific assumptions about turbulence. In cases of insufficient quantitative agreements, a moderate increase of the liquid viscosity led to a substantial improvement (Becker et al., 1994).

On the contrary, in the air-in-water bubble columns discussed here, the standard k-e model would predict an effective viscosity four orders of magnitude larger then the liquid viscosity. This would substan- tially change the flow structure since it would completely dampen out the transient motions and in particular it would eliminate all the vortices, which are well known to be present and can easily be seen in such flows. In the often cited paper of Schwarz and Turner (1988) the standard k-s model was used for the case of a locally aerated bubble column. The authors found a good agreement with the measurements. However, it seems dangerous to generalize their result, since in their experiments the gas bubbles were confined to a small portion in the middle of the reactor only, while the rest of the column was essentially gas free. A comparison by the same authors with a constant effective liquid viscosity also led to reasonable agreement with the experiments.

Since simple single-phase flow turbulence models like the k-t- model turned out to be unsuccessful (Becker et al., 1994), we simply assume that the effective viscosity/Lef t of the gas liquid mixture is equal to the viscosity of water.

Another important problem is the bubble path dispersion. When bubbles start from a point source at the bottom of a bubble column with a sufficiently high frequency, they interact with each other and do not rise straight upwards even if the mean liquid velocity is zero. Bubble wake effects (e.g. Fan and Tsuchiya, 1990) are the main reasons. As is well known, small bubbles are accelerated in the wake of larger ones and others are pushed aside. Hence, there is a con-

of gas-liquid two-phase flows 613

siderable path dispersion on a small scale which appears as a random motion on the larger scale considered in our model.

This path dispersion effect is not restricted to bubble plumes but is also present in bubble columns which are aerated across their entire bottom. The most simple way to consider this random spatially dispersive effect is to extend the continuity equation of the gas phase by a diffusion-like term as has already been done in eq. (5). The corresponding diffusion coefficient has been related to the turbulent eddy viscosity of the liquid phase by Grienberger and Hofmann (1992) and Torvik and Svendsen (1990). This approach assumes an isotropic dispersion. How- ever, since bubbles rise relative to the liquid predomi- nantly in a vertical direction, dispersion will not be an isotropic quantity.

A more general representation would be a dispersion tensor. However, presently there is neither enough knowledge available to model the tensor elements nor enough experimental data to measure the tensor elements reliably. We thus assume that the term D in eq. (5j can be expressed as

D=:~ L dxi [ (11)

where D~ (i = 1,2, 3) are some constant generalized diffusion coefficients estimated from experimental data.

4. NUMERICAL SOLUTION PROCEDURE

First, we introduce some simplifications into our model. Since the density of the gas phase is much smaller than the density of the liquid phase, we can assume without significant loss of accuracy that

p = (1 - s)p/ (12)

and

u = ut. (13)

Further, for the rest of this paper we assume the gas phase to be incompressible. Under this assumption, the gas continuity equation (5) simplifies to an equation for the local gas holdup s:

where

Vp Ug = u + u~lip = u - Cdra--g" (15)

In our numerical simulations of eqs (1) and (2) the finite-volume method has been used. In the three- dimensional case, the solution domain is discretized into six-sided, rectangular control volumes. We take the staggered grid formulation first used by Harlow and Welch (1965), which means that the scalar quan- titites are attached to the centers of the control volumes, and the velocity components are calculated for the centers of the surfaces of the control cells.

614

Equations (1) and (2) fully correspond with the mass and momentum balances for the single-phase flow. This means that these equations can be solved in the same way as in the single-phase case and well-established iteration procedures can be applied. We use the SIMPLER technique of Patankar (1980). The only modification required is to update the local density values of the gas-liquid mixture, p, in the space domain at the end of each iteration loop. For this pur- pose we solve the gas holdup equation (14) and substitute its result e into the expression (12).

The accuracy of the solving procedure for the gas holdup equation (14) plays a crucial role in the modeling of gas-driven gas-liquid flows, because the resulting flow pattern directly depends upon the gas holdup distribution in the reactor. The two methods most frequently used to solve this equation are the finite-volume method and the method of characteristics. Depending on which of these two methods is used, the fluid dynamical model is referred to as an Euler/Euler model or as an Euler/Lagrange model. In the next sections these two approaches will be described in detail.


(1/At) ~ ~'~ Dex (xi- 1/2, t) dt based on the data d at time h, where I is equal to n or to n + I, depending on what kind of time integration (explicit or implicit) is used. Unless otherwise stated, it is understood that all data are taken at time t, + 1 and the superscript I will be left out.

The usual method to approximate the convective fluxes is the first-order UPWIND method where

(0 Fup(e;i) = I Uei if U < 0. (18)

U replaces the velocity u72al/2 at the left side of the ith cell Ci.

The UPWIND method leads to a numerical diffusion in the order of[ UIAx/2. Usually, with the numerical grid resolutions which can be handled in two- or three-dimensional calculations the true solutions become strongly smoothed. Hence, the accuracy of the solutions is rather low.

The second-order central-difference method

5. SOLVING THE GAS HOLDUP EQUATION

In this section we will concentrate on the numerical algorithms for solving the gas holdup equation (14). We assume the components of u 0 to be known at the faces of the control volumes at a given time from the solution of eqs (1), (2) and (15). Furthermore, the diffusion coefficients D~ are considered to be known and constant.

5.1. Eulerian approach We start with the finite-volume method for the gas

holdup equation in one spatial direction. The ideas presented here can be extended in a rather straight- forward way to two and three dimensions.

In one dimension, eq. (14) simplifies to

~:, = -- (eU)x + Dexx (16)

where the indices t and x denote derivatives along the corresponding variables. In the following, we will omit the subscripts ofug, a, D1 and Xl for the reason of simplicity. We use a finite-volume method in which e7 represents an approximation to the cell average of e at time t, over the ith cell Ci = [X i -1 /2 , X i+a/2] . The finite-volume formulation for the gas holdup equation can be obtained through the integration of eq. (16) over Ci x It., t.+a] and takes the form

E~' + 1 _ e7 1

At - Ax [F(d; i + 1) - F(fl; i)]

(17) 1 l + ~xx [D(e;i + 1) - D(et;i)]

where F(d;i) is some approximation to the average ,*n+l convective flux (1/At)S,. (eu)(xi_ 1/2, t) dt and D(d;i)

is some approximation to the average diffusive flux

ud(e; i) = U ei- 1 + e l _ ___ i (19) 2

works well in cases where only small t-gradients and very low cell Peclet numbers (i.e. the low values of UAx/D) are to be expected but it has difficulties if e has steep gradients since then it is very dispersive and tends to generate artificial oscillations.

Much better results can be obtained using a hybrid method that uses the second-order flux in smooth regions but involves some sort of limiting based on the gradient of the solution so that near discontinuities it reduces to the monotone UPWIND method. The stability theory of such flux-limiter methods is based on the concept of the total variation diminishing of the solution (for details see e.g. LeVeque, 1990), so we will use the abbreviation TVD for this type of the convective flux approximation. Note that the central- difference flux (19) can be decomposed into the UPWIND flux plus a correction term:

FCd(e;i) = FUP(e;i) + 1Ul(el - el- l) . (20)

This suggest the following flux-limiter method:

FTVD(e;i) : FUp(g;i) q- 1Ul(g~ - ~,-~)~ (21)

where qbi is the limiter which depends on the local nature of the solution. Note that if ~i = 0, then we have the UPWIND method while if tb~ = 1 we have central difference. The limiter we will use here has the form

where

C'I - - e l - 1 ~i = ck(Oi), Oi = - (22)

~i -- E i - 1

i -1 if U~>0 I = (23) +1 if U


We see that 0~ is the ratio of the slope at the neighbor- ing interface in the upwind direction to the slope of the current interface. One standard limiter we use in our calculations is the superbee limiter

0(0) = max(0,min(1,20),min(2,0)). (24)

The implicit TVD method described above is uncon- ditionally stable, while the explicit one is stable only if the condition

UAt 1 Ax

is satisfied for each i. If the implicit TVD method is used, eq. (17) leads to a system of non-linear algebraic equations, which has to be linearized and to be solved iteratively. Violation of condition (25) may lead to negative central coefficients in the resulting system of linear equations which may cause severe convergence problems. That is why we prefer to use the explicit approximation to the convective fluxes if the TVD method is applied. At high space resolution of the solution domain, condition (25) requires very small time steps, which leads to a considerable increase of computation time. The computation time can be dras- tically reduced if one uses a finer time mesh only for solving the gas holdup equation and keeps larger time increments for solving the other model equations.

For the diffusion term we exclusively use the implicit second-order central-difference flux approximation

'~ii - - ~;i 1 D(

616

On the other hand, we can determine the numerical solution of eq. (28) analogous to an ink-drop dispersion experiment by releasing M particles with asso- ciated volume Ax~ Ax2Ax3/M at some initial time t = 0 from the point x = 0 (the discrete analogon of the 6-function) and let them disperse, i.e. changing their coordinates in space and time according to eq. (27). The numerical solution e(x, t) calculated from the particle distribution at time t will then converge to the probability distribution function of single gas-phase particle position x(r) in the limit for Ax--, 0 and M ~ ~. The position of a single particle x(t) at time t = n At is defined by

x, (nAt) = (~ + ~2 + ... + / , ? ) ,~t td , . (33)

The distribution of x(t) converges for fixed t and n ~ ~, At = t in --, 0 to the three-dimensional Gauss distribution function by the central limit theorem of the probability theory, with

= (r2(x,t)) = ~2(d 2 + d2 2 + d~). (34) t~2(t)

From eqs (32) and (34) we are now able to express the relationship between d~ and D~:

di = 2~t ' . (35)

From the numerical point of view it is not necessary or even not correct to associate a dispersed gas-phase particle with a single gas bubble. The number of GPPs and the number of gas bubbles in the reactor may be different. If the volume of a single gas bubble is much smaller than the volume of a control cell and many of them are within this single cell, then the number of the GPPs can be taken smaller than the number of the gas bubbles. In this case, one GPP represents a bubble cluster.

On the other hand, if the gas holdup is low and the volume of a single gas bubble is larger than the control volume element, then it might be of advantage to represent such big bubbles by a number of GPPs in order to obtain a more continuous distribution of the gas across the numerical grid.

The total number of the GPPs in the solution domain is controlled by the particle generation rate, which depends on the grid resolution and the bubble size distribution, but in every case it must match the predefined superficial gas velocity.

It should be mentioned that the Lagrangian approach using eq. (27) is not the optimal way to solve continuous equations like eq. (14). In particular, problems with the number of GPPs arise through the task of representing the gas diffusion terms in eq. (14) by means of the Lagrangian approach. Physically, the diffusion approach describes a gas transport from the regions with high gas concentration to regions with lower gas concentration. Equation (27), however, describes the random component of the movement of the GPPs. This approach can lead to unphysical results if only a small number of GPPs per unit cell are present, because it allows for a transition from a con-


trol cell with lower gas holdup into the control cell with the higher one. As an example we can imagine a random jump of one GPP from a control cell containing a single GPP to the adjacent cell containing two GPPs. In the following, examples of two- phase flow with many small bubbles are considered. Then it is no problem to represent the diffusional component in eq. (14) adequately with a Lagrangian approach.

6. TEST CASE

In order to compare the simulation results pro- duced with both methods we use the example of a partially aerated flat bubble column. This test case is described in detail in Becker et al. (1994), hence only a brief description will be given here. The apparatus has a rectangular cross-section with the following dimensions: width 50cm, depth 8 cm and height 150 cm.

Glass walls on the front and the back allow observation and photographic documentation of the multiphase flow. For gas dispersion a single frit, flush- mounted on the bottom of the apparatus at the distance of 14.5 cm from the left side of the column has been used. In this flat column, an essentially two- dimensional flow structure develops, depending on the gas flow rate used. At superficial gas velocities below 1.5 mm/s, the flow depicts a transient character. This is shown by results obtained with a gas through- put of 0.66 mm/s. Several liquid circulation cells can be observed in the column. They continuously change

y~.. '~, , .

. ~Y,,'.::-

~. !~.

oq.:. ,e4-.~.l,, ..t '~..':.. L' " " -~:.'r ' " " '

: 4,,'." ~" . . ".., _ . ,~ ~.~: .:. ;' : "~"~"2 "

/g-i::

~ l , I

Fig. l. Locally aerated bubble column: binary and inverted photographs of the oscillating bubble swarm at two different

times (Becker et al., 1994).

Dynamic numerical simulation

their location and their size. The bubble swarm motion is influenced by these vortices and therefore rises in a meander-like way (Fig. 1).

7. S IMULATION RESULTS WITHOUT BUBBLE PATH

DIFFUSION

All numerical simulations assume a two-dimensional rectangular geometry with height 150 cm and width 50 cm. A regular numerical grid with 150 x 50 grid points was used. The simulation results obtained with three different numerical algorithms were compared with each other: the Eulerian approach with UPWIND discretization of the gas holdup equation (short: UPWIND), the Eulerian approach with TVD discretization (short: TVD) and the Lagrangian approach for the gas equation (short: LAGRANGE).

Let us first neglect the path diffusion effects in the gas phase. This means that the coefficients Dj in eq.(14) and di in eq. (27) are assumed to be zero. Figure 2 depicts the gas holdup pattern in the bubble column, 5 s after the onset of the aeration. A tremen- dous influence of the numerical diffusion in the Euler- ian solution obtained with the UPWIND discretization technique can be recognized. This is not due to the Eulerian approach as the results obtained with the TVD method demonstrate. The results obtained with the TVD method look much more similar to the gas distribution which results from the Lagrangian simulation. We thus can conclude that the UPWIND technique leads to strong numerical diffusion effects. The amount of numerical diffusion in vertical and horizontal directions is proportional to the local vertical and horizontal gas velocity components. In the

of gas liquid two-phase flows 617

first 10 s after the onset of the aeration the vertical velocity component prevails over the horizontal velocity component in the region where the gas phase is present, leading to a much higher numerical diffusion in the vertical direction than in the horizontal one (see Fig. 2, left). The evolution of the velocity field during the first 48 s after the beginning of the aeration obtained with the Lagrangian approach (Fig. 3) shows, however, a continuously changing velocity pattern, leading to different local numerical diffusion effects at each time step. This means that the effect of numerical diffusion of the UPWIND method is completely un- controllable and its influence on the distribution of the gas phase has an unpredictable character.

The comparison of the liquid velocity patterns 60 s after the onset of the aeration (Fig. 4) shows a very good agreement between the TVD and the LAG- RANGE results. Also the UPWIND solution shows a good qualitative agreement with the other two solutions. For the better quantitative comparison between the simulation results obtained with all three methods, the vertical liquid velocity profiles at height 100 cm are presented in Fig. 5. We see that the TVD and the LAGRANGE solutions are close to each other, whereas the velocity variation in the UPWIND solution is about a factor 2 smaller.

Figures 6 and 7 show the comparison of the liquid velocity patterns at t = 120 and t = 180 s. Even 180 s after the onset of the aeration, the TVD and the LAGRANGE solutions are in good qualitative agreement, whereas the UPWIND solution already leads to different results at 120 s. The comparison of the evolution of the vertical liquid velocity component in time,

UPWIND TVD LAGRANGE

[]

0.28% [] 0.42% I 0.56%

0.69%

0.97% []

1.25% N 1.39% []

Fig. 2. Locally aerated bubble column. Distribution of the gas holdup 5 s after the beginning of the aeration calculated with three models. Diffusion term is assumed to be zero.

618

Time: 9.4 s

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!!?i i iiiiiiiii!iiii '"'" "-"'"'"""

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iglIlI I 'J ~I,':' fthttltih] i: ;I ,':i i i

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7zi,,]ttlt4 :,~tIlIl[?: ::::!Nltl t,\\\"~.-_"!,Jlttt

iI',t;i,,g -:,,',

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Fig. 3. Locally aerated bubble column. Evolution of the liquid velocity field during the first 48 s after the beginning of the aeration. Lagrangian approach without diffusion. (Here and subsequently velocity vectors are shown only at each 8th grid point; the vertical vector in the left bottom corner of each plot corresponds

to the velocity of 10 cm/s.)

plotted for some fixed position A in the reactor (Fig. 8), also shows quantitative agreement between the TVD and the LAGRANGE solutions. This is a rather striking result since different numerical solutions of an intrinsically unstable dynamical system tend to devi- ate more and more from each other as time proceeds. As a matter of fact, such a deviation can also be observed for longer simulation times. Figure 9 shows the long-time behavior of the liquid velocity component at point A. After about 4 rain the agreement between the TVD and the LAGRANGE solution vanishes. Later on, however, a quasiperiodic solution is established which is again in close accordance for

the two methods whereas the UPWIND solution shows a completely different single-periodic behavior.

Let us now look at the void fraction distribution calculated with the three methods for t = 60s (Fig. 10). If we compare the TVD and the LAGRANGE solution with the photographs in Fig. 1, we must state that the radial dispersion in the gas phase cannot be reproduced by both methods, whereas the UPWIND solution seems to perform much better. Even under strongly fluctuating flow conditions (see Fig. 3) the spread of the bubble plume calculated with TVD or LAGRANGE method is much smaller than observed experimentally. This shows that the different medium


UPWIND TVD LAGRANGE

619

Fig, 4. Instantaneous liquid velocity field at 60 s after beginning of the aeration calculated with different methods (no diffusion is considered).

20

10

i -10

-20

-30 0

" "N

\ ~'/ ...." ........................ ~,,"*~ ..... / ,.-' \ .....

/ .-" \ .... , .... / , . "

-'I I .... x \ ....

............... i ! ] LAGRANGE UPWIND

i i i b

10 20 30 40 DISTANCE FROM THE LEFT WALL [cm]

50

Fig. 5. Vertical liquid velocity profiles at height 100 cm. Time = 60 s after beginning of the aeration calculated with

different methods (no diffusion is considered).

size vortices of the computed flow field do not disperse the bubble flow sufficiently. Instead, the dispersion is caused by numerous small vortices and flow variations caused by the liquid flow around individual bubbles or bubble clusters. Since our model does not resolve these small-scale phenomena, some appropriate corrections become necessary. In the following, diffusion term D in the gas holdup equation (5) will be used as a first approximation.

The apparently good performance of the UPWIND solution (Fig. 10, left) is of course a consequence of the numerical diffusion which in our case happens to have about the right order of magnitude. If the space grid is further refined however, the numerical diffusion of the UPWIND solution decreases and the spread of the bubble plume would also decrease.

S. S IMULAT ION RESULTS WITH D IFFUS ION

The preceding example showed that a gas-phase dispersion model is necessary to obtain a physically reasonable distribution of the gas phase. We will therefore use the procedure described in Section 5.2 to obtain compatible dispersion parameter values for the TVD and LAGRANGE solutions. Before doing so, a validation of the algorithm, eq. (27), as well as of the equivalence relation, eq. (35), is necessary since they have been obtained on more or less intuitive argu- ments. For the validation only the gas holdup equation (14) for a constant liquid velocity field will be solved with TVD and LAGRANGE methods and the results compared. Note that the Lagrangian technique is used here in order to solve the continuous gas dispersion term in eq. (14). Computationally this is not the most efficient way of using this technique, however, this allows for a direct comparison of the results obtained with different techniques.

620 A. Sokolichin et al.

UPWIND TVD LAGRANGE

Fig. 6. Instantaneous liquid velocity field at 120 s after beginning of the aeration calculated with different methods (no diffusion is considered).

UPWIND TVD LAGRANGE

t t I

Fig. 7. Instantaneous liquid velocity field at 180 s after beginning of the aeration calculated with different methods (no diffusion is considered).

Dynamic numerical simulation of gas-liquid two-phase flows 621

20

10

-10

-20

-30 0

1

3O

~ . ' ] 'wr jD ,

.. / 1500 t

t ......... UPWIND i

i" i~ : /01 ..,. [tl A 90o . , ..... , , 35]1_

I i I I

60 90 120 150 180 G TIME [s]

Fig. 8. Vertical liquid velocity at position A calculated with different methods (no diffusion is considered).

>- [..,

O

>

15

0

-15

-30

15

0

-15

-30

15

0

-15

-30

0 500 1000 1500 2000 2500 3000 TIME [s]

Fig. 9. Long-time vertical liquid velocity fluctuation at position A (see Fig. 8) calculated with different methods (no diffusion is considered).

8.1. Test p rob lem Let us consider the gas holdup equation (14) in two

dimensions. We assume the components of ug and the diffusion coefficients Di to be known and constant, and consider uniform convection only in the vertical direction and diffusion only in the horizontal direction. Under these assumptions eq. (14) simplifies to

e t = - - (~U)x + Deyy (36)

where x and y denote the vertical and horizontal coordinate directions, u the vertical velocity and D the

horizontal diffusion coefficient. We assume the following values for u and D: u = 20 cm/s, D = 4.1(6) cm2/s; the latter corresponds to the disturbance coefficient d = 10 cm/s t/2. This means that two GPPs with the same coordinates at time t = to can be a maximum distance of 10 cm apart at time t = to + 1, if time step At = 1 s is used.

We solve eq. (36) in the same calculation domain as described in Section 7 and with the initial condition e(t = 0, x, y )= 0, and the boundary condition e(t, x, y = 0) = 0.05 for x~[13 em, 16 cm]. After 7.5 s

622 A. Sokolichin et al.

UPWIND TVD LAGRANGE

0.13%

0.25% n 0.38%

0.63%

0.76%

1.01% I 1.14% [] 1.27% []

Fig. 10. Locally aerated bubble column. Distribution of the gas hold-up 60 s after the beginning of the aeration calculated with three models. Diffusion term is assumed to be zero.

a b c d

~.44

0.89 m 1.33

1.77

2.66

3.10

4.43 []

Fig. 11. Simulation results for convection-diffnsion test problem with constant vertical velocity and constant horizontal diffusion: (a) stationary TVD solution; (b) instantaneous positions of 45,630 GPPs in Lagrangian method; (c)Lagrangian solution with 45,630GPPs; (d) Lagrangian solution with

180,566 GPPs.

of real time, the gas front reaches the top of the calculation domain and the solution of eq. (36) be- comes stationary. The corresponding numerical stationary solution calculated with TVD method is shown in Fig. 11 (a). In the frame of the LAGRANGE method no stationary solution can be reached, because the positions of the GPPs [Fig. ll(b)] are continuously changing in time. However, if the generation rate of GPPs is high enough, the time variation of the calculated gas holdup distribution varies

only slightly in time, so we can speak of a quasistationary solution of the Lagrangian approach. One such quasistationary distribution of the gas holdup is shown in Fig. 1 l(c). At that time the total number of GPPs in the calculation domain equals 45,630. Al- though a good qualitative agreement can be found between Figs 11 (a) and (c), the Lagrangian solution is not very smooth. Only after increase of the generation rate of GPPs by a factor 4 can we get the smooth solution presented in the Fig. 11 (d) (corresponding to

t.0

0.8

E

~ 0.6 0.4

0.2

0,0 0


20 A, ~TVD [-~.-~, LAGRANGE ( 180566 GPP~ )~

. 630 GPPs )l

10 20 30 40 DISTANCE FROM THE ~ WALL [crn]

t0

-10

> -20

-313 50

623

t t

.:.,, I m ~ . ,rv D :.

LAGRANGE : i UPWIND

30 60 90 120 150 180 TIME lsl

Fig. 12. Calculated gas holdup profiles at height 100 cm for convection~liffusion test problem with constant vertical velocity and constant horizontal diffusion: stationary TVD solution, Lagrangian solutions with different numbers of

GPPs.

Fig. 13. Vertical liquid velocity at position A (see Fig, 8) calculated with different methods (d2 = 5 cm/sa'Z).

15

0

~ -15 TVD

15

0

-15

LAGRANGE ] 15

0

-15

0 500 1000 1500 2000 TIME [s]

Fig. 14. Long-time vertical liquid velocity fluctuation at position A (see Fig. 8) calculated with different methods (d2 = 5 cm/st'2).

the total number of GPPs = 180,566). For better quantitative comparison between the solutions, the gas holdup profiles at the height 100 cm calculated with different models are presented in Fig. 12.

As a result it can be stated that gas-phase dispersion can be modeled with equal accuracy by both the Eulerian-TVD as well as the Lagrangian approach provided that a rather high number of GPPs is used in the latter case. The required big number of GPPs may present computational problems if industrial-scale reactors have to be simulated, since the computational time is roughly proportional to the GPP number,

8.2. Two-phase .flow with diffusion Let us now compare the simulation results for the

whole two-phase system in the presence of diffusion in the gas phase. Since a comparison of Fig. 1 l(b) with the photographs in Fig. 1 shows a somewhat larger spread in the calculated flow, only half of the disturbance coefficient of Fig. 11 (b) is used in the horizontal direction: d2 = 5 cm/s 1/2 (resp. D2 = 1.041(6)cruZ/s). The diffusion in the vertical direction is assumed to be negligible (dl = 0, D1 = 0). Figures 13 and 14 now correspond to Figs 8 and 9 (without diffusion). We can see that the TVD and the LAGRANGE solutions

624

Fig. 15. Simulation results for a locally aerated bubble column with Lagrangian approach (dz = 5 cm/sl/2). Instan-

taneous positions of GPPs at two different times.


again show very similar long-time behavior and the UPWIND method gives a totally different solution. Figure 15 shows the distribution of the GPPs in LAGRANGE solution at two different times, and we can now observe a great similarity with the photographs in Fig. 1.

9. UNIFORM AERATION

The next test example is the dynamical simulation of a bubble column which is aerated uniformly over its entire bottom. Visual observation shows that at low superficial gas velocity a so-called homogeneous flow structure prevails, where the bubbles rise uniformly through an essentially stagnant liquid. As the superficial gas velocity is increased, an instationary flow structure develops, where vortices are created close to the gas distributor and move upwards and sideways in a rather irregular way. Long-term measurements of the gas holdup distribution and of the liquid velocities show the well-known picture of an increased gas holdup in the middle of the column, leading to an overall liquid circulation with upflow in the center and downflow near the walls (Grienberger and Hofmann, 1992).

The simulation results for the same flat column as specified in section 6 with uniform aeration over the entire bottom obtained with the Lagrangian approach (d2 = 5 cm/s 1/2) are given in Fig. 16. The simulations show that above a minimum value of the superficial gas velocity of about 2 cm/s an unsteady flow structure

Fig. 16. Uniformly aerated bubble column: instantaneous (left, middle) and long-time-averaged (right) simulation results of liquid velocity field. Lagrangian approach with diffusion (d2 = 5 cm/sl/2). Superficial

gas velocity equals 2 cm/s.

Dynamic numerical simulation of gas-liquid two-phase flows 625

20

0

-20

-40

20

0

-20

-40

2 1 0 . . . . . . . -20 --40 0 10 20 30 40 50

DISTANCE FROM THE LEFT WALL [cm]

Fig, 17. Uniformly aerated bubble column: long time averaged vertical liquid velocity profiles at three different heights

calculated with different methods.

develops. If the calculated local velocities are averaged over a longer time period, as is done in the usual bubble column measurements, a regular flow structure with one overall circulation cell results (Fig. 16, right).

It is not possible to make a direct quantitative comparison between instantaneous flow pattern re- suits obtained with all three methods, because of the chaotic character of the solution. However, we can make an indirect comparison through the calculation of the long-time-averaged velocity patterns. The corresponding vertical liquid velocity profiles at three different heights calculated with LAGRANGE, TVD and UPWIND methods are shown in Fig. 17. As in the case of a locally aerated bubble column we can state a very good quantitative agreement between the TVD and the LAGRANGE solutions. The UPWIND method leads again to quantitatively different results. The influence of the numerical diffusion in the case of a uniformly aerated bubble column is, however, not so high as in the case of a locally aerated bubble column, due to the smoother distribution of the gas phase.

For the transport phenomena in the two-phase flow, the instantaneous and not the long-time-averaged velocities are of decisive importance. If we now take a look at the representative instantaneous liquid velocity patterns calculated with the LAGRANGE (Fig. 16, middle), the TVD (Fig. 18, left) and the UPWIND (Fig. 18, right) methods, we can see that the UPWIND solution depicts a qualitatively different behavior with a much lower number of vortices as in the other two solutions. So we can state that also in the case of a uniformly aerated bubble column, the UPWIND method leads to a qualitatively different solution compared to the LAGRANGE and the TVD approaches.

Fig. 18. Uniformly aerated bubble column: instantaneous simulation results of liquid velocity field calculated with

TVD (left) and UPWIND (right) methods.

10. CONCLUSIONS

From the chemical reaction engineering point of view, fluid dynamical models are required for a proper description of fluid mixing and contacting patterns, i.e. they model the way by which materials flow through the reactor and contact each other in order to react chemically (e.g. Levenspiel, 1989). Hence, the local transport properties are of primary importance. In this light it is essential that numerical diffusion effects which corrupt the numerical simulation results are kept under control. Such numerical diffusion effects are of particular importance for the bubble column reactors considered in this paper, since flow in bubble columns is essentially buoyancy driven. Strong diffusional transports, however, may degradate the density gradients. Numerical diffusion will, thus, lead to incorrect driving forces in the simulations.

Simple numerical solution techniques such as the commonly applied UPWIND technique may lead to unacceptable numerical diffusion effects. This artificial diffusion exceeds the naturally appearing diffusion considerably, often by orders of magnitude. In principle it would be possible to compensate for this deficiency by using finer numerical grids. Practically, this counter measure is limited by the available computing power. Even with the finest grids which can be handled with today's computers, the numerical diffusion effects appearing in the UPWIND solutions are much larger than the real ones.

Consequently, more sophisticated numerical integration schemes must be applied which are much more immune to numerical diffusion. Here we discussed the TVD as a reasonable alternative. It is

626

shown that it provides results which are in the same order of accuracy as the solutions obtained with the Euler/Lagrange method which is not affected by numerical diffusion.

A comparison of the numerical solutions of the model equations obtained with the TVD-technique which can be used in Euler/Euler representations and the LAGRANGE technique showed that the resulting flow patterns agree quantitatively over a surprisingly long period of simulation time. This is particularly interesting since the model equations are capable of instable chaotic solutions where two solutions with slightly different initial conditions will not lead to the same long-time results.

Consequently, the results obtained with the TVD and the LAGRANGE technique can be regarded to be equivalent. The results presented can also be considered as a kind of validation of both numerical codes since both solution procedures are much different. On the other hand, it has not been possible to obtain similar agreement even qualitatively with the UPWIND technique, which is the common approach to handle the gas-phase motion in the Euler/Euler approach.

The implementations of the TVD and the LAGRANGE techniques differ considerably from each other with respect to the computing times required. In systems which are not sensitive to diffusion, the LAGRANGE technique is considerably faster than the TVD technique, since for reasons of stability, the TVD method requires much smaller time steps for the integration of the gas holdup equation than the LAGRANGE method. In cases where diffusion effects cannot be neglected, the LAGRANGE method be- comes less effective if the dispersion effects are to be modeled by a diffusion-type continuous equation.

When large bubble numbers are to be considered, the LAGRANGE method might become slower because of the big number of GPPs to be handled. However, it proved to be possible to follow the trajec- tories of individual bubble clusters instead of single bubbles.

In cases where the gas holdup is too large (> 10%) none of the presented techniques can provide reliable results since the bubble-bubble interactions must then be taken into account. Presently, there is no reasonable physical model available for such situations.

Acknowledgement Support of this work through Deutsche Forschungs-

gemeinschaft is gratefully acknowledged.

Cdrag d D g P t T U

NOTATION drag force coefficient, g/(cm 3 s) disturbance coefficient, cm/s 1/2 diffusion coefficient, cm2/s acceleration due to gravity, 981 cm/s 2 pressure, dyn/cm 2 time, s stress tensor, dyn/cm 2 velocity vector, cm/s


Greek letters 6(x) three-dimensional Dirac's delta function e gas holdup, dimensionless /~ viscosity, g/(cm s) p density, g/cm 3 V gradient operator, cm-1

Subscripts eft effective g gas phase l liquid phase

REFERENCES

Becker, S., Sokolichin, A. and Eigenberger, G. (1994) Gas-liquid flow in bubble columns and loop reactors. Part II. Comparison of detailed experiments and flow simulations. Chem. Engng Sci. 49, 5747-5762.

Devanathan, N., Dudukovic, M. P., Lapin, A. and Ltibbert, A. (1995) Chaotic flow in bubble column reactors. Chem. Engng Sci. 50, 2661-2667.

Fan, L S., Tsuchiya, K. (1990) Bubble Wake Dynamics in Liquids and Liquid-Solid Suspensions. Butter- worth-Heinemann, Boston, U.S.A.

Grienberger, J. and Hofmann, H. (1992) Investiga- tions and modelling of bubble columns. Chem. Engng Sci. 47, 2215-2220.

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Patankar, Suhas V. (1980) Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York, U.S.A.

Schwarz, M. P. and Turner, W. J. (1988) Applicability of the standard k-e turbulence model to gas-stirred baths. Appl. Math. Modelling 12, 273-279.

Sokolichin, A. and Eigenberger, G. (1994) Gas liquid flow in bubble columns and loop reactors. Part I. Detailed modelling and numerical simulation. Chem. Engng Sci. 49, 5735-5746.

Torvik, R. and Svendsen, H. F. (1990) Modelling of slurry reactors: a fundamental approach. Chem. Engng Sci. 45, 2325-2332.

Tzeng, J.-W., Chen, R. C. and Fan, L.-S. (1993) Visual- ization of flow characteristics in a 2-D bubble column and three-phased fluidized bed. A.I.Ch.E.J. 39, 733 744.

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dynamic numerical simulation of gas-liquid two-phase flows euler/euler versus euler/lagrange

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