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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 pres- ented. 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 pri- mary 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 repres- entation 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 reac- tor. 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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612
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 indus- trial reactors which are the final target of process
development. There are considerable difficulties in such a direct
validation procedure since the measure- ment techniques necessary
to provide comprehensive data from the turbulent flows prevailing
in real chem- ical reactors are not sufficiently developed. Most
available measurement devices provide local fluid ve- locity data
only. Usually, only long-time averaged data are published. Even for
bubble columns, which can be regarded as the most simple two-phase
reac- tors, gas-liquid flow patterns are available as long- term
averages (Torvik and Svendsen, 1990; Grienber- ger and Hofmann,
1992). Consequently, a direct vali- dation 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
col- umns 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 dis- cretization is used.
The stability of the second-order discretization is based upon the
concept of total vari- ation 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 funda- mental 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 den- sity 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 negli- gible. 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 relation- ship
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
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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 de- veloped 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 simu- lations showed (Becker et al., 1994) that gas-liquid
bubble flow can often be described with good quali- tative 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 col- umns 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 re- sult, 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 essen- tially gas
free. A comparison by the same authors with a constant effective
liquid viscosity also led to reason- able 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 effec- tive viscosity/Lef t of the gas
liquid mixture is equal to the viscosity of water.
Another important problem is the bubble path dis- persion. 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 con- sidered 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 disper- sion tensor.
However, presently there is neither enough knowledge available to
model the tensor ele- ments 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 equa- tion 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 vol- umes, and
the velocity components are calculated for the centers of the
surfaces of the control cells.
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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-esta- blished 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 do- main 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 character- istics. 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 de- scribed in
detail.
A. Sokolichin et al.
(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 diffu- sion in the order
of[ UIAx/2. Usually, with the numer- ical grid resolutions which
can be handled in two- or three-dimensional calculations the true
solutions be- come 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
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Dynamic numerical simulation of gas-liquid two-phase flows
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 impli- cit
second-order central-difference flux approxima- tion
'~ii - - ~;i 1 D(
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616
On the other hand, we can determine the numerical solution of
eq. (28) analogous to an ink-drop disper- sion 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 con- trol 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 ap- proach using eq.
(27) is not the optimal way to solve continuous equations like eq.
(14). In particular, prob- lems 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, de- scribes the random component of the movement of
the GPPs. This approach can lead to unphysical re- sults if only a
small number of GPPs per unit cell are present, because it allows
for a transition from a con-
A. Sokolichin et al.
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
contain- ing 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 obser- vation and
photographic documentation of the multiphase flow. For gas
dispersion a single frit, flush- mounted on the bottom of the
apparatus at the dis- tance 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-dimen- sional 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 com- pared
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 ap- proach
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 simu- lation. 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
verti- cal 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 velo- city 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 ob-
tained 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 solu- tions. 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 agree- ment, whereas the UPWIND solution already leads
to different results at 120 s. The comparison of the evolu- tion 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.
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618
Time: 9.4 s
..-,
' ' , ' t
i!!~i}
:::::::::::::::::::::: :.i,?) i:::':.;:;':;.'.
A. Sokolichin et al.
Time: 11.2 s Time: 13.0 s
'r ',~'.'l I ' ' " ' ""~ ~ ~'" ' -~/ ~ , : < :.:,:,,,,
:,:..:t~:,.,:,:,..:,:+ !~" . "71 ' , '< , h;.. ",'.:,:,:,
.~,:;: ...:.:.:,:.:
, , ,h~ ,..:.:.:.:,:, l~ : . : : : : ' , : : : :!::'- i::: :!::
f~~: : ' , : : ,,.,, ,.._.;,,,,>:,:,:, I '~ / , ; ' , ' ; , ' "
, ' : ;
!!?i i iiiiiiiii!iiii '"'" "-"'"'"""
Time: 15.0 s
~1~; i , :. :-:-:,~,l t d' :: :'I I I ~'/illl ~ ','.'~t ,:I
,,t,,:-;i
,'>,]~l{r6-?,h'--'")ilNi ,,',~t~-::,S;J, u ~ ,i tu.
',~llt,{--_,,, i~..,,rll.>_-'..;,',,,,,,,
I/b-:,II.-:--:-.'.',,,,, P,:::.: ::::: :.: :;:,.::
'..>:. >>>>>>: i.
"." , ' t ' , ' . " ".- .
Time: 19.0 s
~'~ S , : : ;:'I ;! ~'tL ::t'
'-~II~iittb "i:',' I, I :-,-.,#1 J ~!!1[1" % ;'.'--'1
~..',, },,.-:---:..,?
) :,:': ~..:.; . . : . . .
f " ; '7 .~>>1 ' ? i
Time: 22.0 s Time: 27.0 s Time: 35.0 s Time: 42.0 s Time: 48.0
s
It,'.i~t ~.~'~ i ~
ff~fif; I1 [!:J:71
i!ti! '>':::
71 " ><
,:,;.,.
t )
t!
lit,':i~
~-'~l I t~)t,.:, ,
l,j~tll',: : : ' 7#/D}~,,'.", ~////~4,' : :' ~/~,~,..-.,
[~f!2"::',
N\\':::-:';
~_ ,,,'-~?! ~- - ' ,
!t,.
iglIlI I 'J ~I,':' fthttltih] i: ;I ,':i i i
tI!::Nt!L::
dlltlfftl,',:::,'i iltfJl/t!(,:.::+
['r//, #- ' , ' 1
7zi,,]ttlt4 :,~tIlIl[?: ::::!Nltl t,\\\"~.-_"!,Jlttt
iI',t;i,,g -:,,',
7,~.",::::7;17tmI
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 compon- ent at point A. After about 4 rain the
agreement be- tween 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
-
Dynamic numerical simulation of gas-liquid two-phase flows
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 dis- perse the
bubble flow sufficiently. Instead, the disper- sion 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 appropri- ate
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 equa-
tion (14) for a constant liquid velocity field will be solved with
TVD and LAGRANGE methods and the results compared. Note that the
Lagrangian tech- nique 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 direc- tion. 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 follow- ing
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 condi- tion 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, be- cause the positions of
the GPPs [Fig. ll(b)] are continuously changing in time. However,
if the gen- eration rate of GPPs is high enough, the time vari-
ation of the calculated gas holdup distribution varies
only slightly in time, so we can speak of a quasi- stationary
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
Dynamic numerical simulation of gas-liquid two-phase flows
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 ve- locity
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 disper- sion 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 re- actors 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 distur- bance 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 col-
umn with Lagrangian approach (dz = 5 cm/sl/2). Instan-
taneous positions of GPPs at two different times.
A. Sokolichin et al.
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 photo-
graphs 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 uni-
formly 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 aver- aged
vertical liquid velocity profiles at three different heights
calculated with different methods.
develops. If the calculated local velocities are aver- aged 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 cor- responding
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-aver- aged 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 ef- fects are of particular importance for the bubble
col- umn 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 artifi- cial diffusion exceeds the
naturally appearing diffu- sion 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 com- puting power. Even
with the finest grids which can be handled with today's computers,
the numerical diffu- sion effects appearing in the UPWIND solutions
are much larger than the real ones.
Consequently, more sophisticated numerical inte- gration 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 nu- merical 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 con- sidered as a kind of validation of both numerical
codes since both solution procedures are much differ- ent. 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 re- quired. 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 be- cause 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 reason- able 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
A. Sokolichin et al.
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
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