Flow Simulation by 1-D Model
A typical set of simulated velocity and holdup profiles for a total cycle time of
80 seconds and LIQUID ON time of 15 seconds are shown in Figure 5.12 and 5.13. The
liquid flow rate used is 42 ml/min (1.4 kg/m2s) during LIQUID ON time and 2 ml/min (0.067
kg/m2s) during the rest of the cycle with gas flow of 400 cc/min (0.0192 kg/m2s). The inlet
pulse is seen to be sharp with significant spreading as the liquid pulse travels down the
reactor. This is observed even with an order of magnitude reduction in spatial and temporal
step size and corroborates experimental observations.
0.1
z=0.0
0.09
z=0.25
z=0.45
Liquid Holdup
0.08
z=0.65
0.07
z=0.85
z=1.0
0.06
0.05
0.04
0.03
0.02
0.01
0
0
10
20
time (s)
30
40
Figure 5. 1 Transient Liquid Holdup Profiles at Different Axial Locations in Periodic
Flow
0.06
z=0.0
z=0.25
0.05
z=0.45
Liquid Velocity,m/s
z=0.65
z=0.85
0.04
z=1.0
0.03
0.02
0.01
0
0
10
20
30
40
time (s)
Figure 5. 2 Transient Liquid Velocity Profiles at Different Axial Locations in Periodic
Flow
Reaction Simulation Results and Discussion
The model equations proposed in the previous sub-section are tested here for the
alpha-methylstyrene hydrogenation reaction system, for which experimental results are
presented in Section 4.3. This reaction system consists of four components, alphamethylstyrene, cumene, hexane and hydrogen. The reactor and catalyst specifications
used in the simulation are as tabulated in Section 4.3. Even for a four component system,
the number of equations to be solved at each time and axial location can be quite large.
For example, for the three pellet apparent rate approach, a set of equations ranging from
92 to 116 needs to be solved, depending upon the wetting conditions (fully wetted pellets:
92 equations, fully dry pellets: 116 equations). These equations originate from the
implicit finite difference approximation of the partial differential equations and the nonlinear algebraic equations resulting from the Stefan-Maxwell equations. These are then
solved by the multivariable globally convergent Newton method discussed in Section
5.3.4.
Typical simulation runs of liquid flow modulation are carried out in two stages.
The first stage involves simulation of the reaction to steady state at the mean flow rate to
obtain the performance under steady state conditions. This is followed by simulation of
one or more cycles with liquid flow ON and OFF (for ON-OFF flow modulation) or with
liquid high and low flow (for BASE-PEAK flow modulation). During the first part of the
simulation, the inlet velocity and holdup are specified along with the feed concentrations
and temperature. The reaction transport equations are solved at each axial location
followed by velocity and holdup computation of the entire spatial domain. Figures ___
and ____ depict the development of alpha-methylstyrene concentration profiles at
different axial locations with time for a gas limited reaction case (C-MS,
feed
= 1484
mol/m3 at an operating pressure of 1 atm). Since the rate under these conditions is zero
order with respect to alpha-methylstyrene, a linear concentration profile is observed as
steady state is reached. Note that the alpha-methylstyrene concentration profiles reach
steady state values relatively early as compared to the cumene concentration profiles
shown in Figure ____. Cumene concentration profiles reach steady state after about 250 s
for this test case as shown in Figure ___. The hydrogen concentration in the liquid phase
shows a very interesting behavior as presented in Figure ____. The hydrogen
concentration builds to saturation at the exit where low alpha-methylstyrene is present at
initial times, whereas at the inlet, it is consumed faster than can be transferred from the
gas phase and results in much lower concentration. As liquid reactant (alphamethylstyrene) reaches each point in the vector, the hydrogen concentration drops to its
steady state values.
0.005
1.815
5.815
1600
1400
11.853
19.805
Concentration, mol/m3
1200
39.833
1000
74.82
99.93
800
124.817
600
149.811
174.822
400
199.818
200
225.011
250.007
0
0
0.2
0.4
0.6
Axial Location,(z/L)
0.8
1
275.01
299.922
Figure 5.3 Transient alpha-methylstyrene concentration profile development with time
(shown in seconds in the legend table)
Concentration, mol/m3
0.005
900
1.815
800
5.815
11.853
700
19.805
600
39.833
74.82
500
99.93
400
124.817
300
149.811
174.822
200
199.818
100
225.011
0
250.007
0
0.2
0.4
0.6
Axial Location,(z/L)
0.8
1
275.01
299.922
Figure 5.4. Axial Profiles of Cumene Concentration at Different Simulation Times
(shown in the legend table in seconds)
Periodic Flow Modulation: Full Cycle Reaction Transport Simulation
Several test case simulations were conducted to study the effect of flow
modulation to demonstrate the effect of unsteady state operation on species concentration
and hence the performance of the trickle-bed. Results of one such test case are presented
here to show the effects of flow modulation on reactant and product concentrations over
one cycle period. The liquid reactant (alpha-methylstyrene) feed concentration is 1484
mol/m3 and mass velocity is 0.21 kg/m2s (corresponding to an interstitial liquid velocity
of 0.009 m/s and feed holdup of 0.03211). The gas superficial velocity used was 3.8 cm/s
at 1 atm. operating pressure. The total cycle period () chosen for this case is 60 s with a
cycle split () of 0.33.
Figure ___ shows supply of alpha-methylstyrene to a previously dry pellet during
the liquid ON part of the cycle (0 to 20 s). This is followed by consumption of the alphamethylstyrene by enhanced supply of hydrogen during the liquid OFF part of the cycle
(20-60 s). The figure also shows that high alpha-methylstyrene concentrations are also
possible at downstream locations when the reactor is operated under unsteady state
conditions, which would not be possible in steady state operation. Production rates of
cumene are shown in Figure ____ during the same cycle in the same previously dry pellet
at different axial locations. At times from 0 to 20 s, cumene production is small due to
lower gaseous reactant supply to the pellet, which is followed by high production rates
corresponding to enhanced supply of hydrogen to the pellet during the OFF cycle.
Figure 5. 5 Intra-catalyst alpha-methylstyrene Concentration Profiles during Flow
Modulation for a Previously Externally Dry Pellet at Different Axial Locations
Figure 5.6. Intra-catalyst Cumene Concentration Profiles during Periodic Flow
Modulation
The supply of hydrogen to the catalyst pellet is not easy to show due to its
complete consumption as the limiting reactant. Hence, this is shown for a pellet in which
reaction rates are low. The hydrogen concentration profile at the beginning of the cycle
shows a low concentration (~ 3.5 mol/m3) in the pellet. Hydrogen supply is enhanced due
to lower film thickness at high liquid flows. This is followed by significantly higher
supply of hydrogen during the OFF part of the cycle, enhancing the reaction rate further,
which is reflected in the higher cumene concentration (as shown in Figure ___). Bulk
cumene concentration profiles also corroborated the enhanced rate as shown in the Figure
___ for the same set of conditions. This should however not be used in comparison of the
steady state profiles since the holdup and velocity changes need to be considered. The
correct approach to compare these profiles is to evaluate flow averaged conversion as
done in the experimental results. The other rigorous approach is to compare the time
averaged reaction rate at each point in the reactor and then to obtain an overall rate (for
the cycle) in the entire reactor. This approach is used in the evaluation of simulated
reaction rates done in comparison with steady state reaction rates.
Cycling Parameter Effects on Performance
Several cycling parameters and operating conditions were studied in the
experimentally as discussed in Section 4.3. Some of these effects were simulated using
the three pellet apparent rate approach for the gas limited case to examine whether the
observed trends in the experimental data could be simulated. In all the cases simulated
here, the reaction rate constant is set arbitrarily and only the ratio of the unsteady state
rate to the steady state rate is considered to estimate the enhancement. Figure ____ shows
the effect of space time on the performance enhancement under gas limited conditions at
30 psig operating pressure and feed concentration of 1484 mol/m3. This figure shows
higher enhancement at higher space time as observed and discussed in the experimental
results discussed earlier in Section 4.3. The enhancement in partial wetting at higher
space time causes higher supply of gaseous reactant as discussed earlier and results in
reduction of gaseous reactant performance and corresponding higher enhancement over
steady state performance. Exact quantitative comparison between this enhancement and
the experimentally observed enhancement (depicted in Figure 4.__) is not seen due to
several factors such as mass transfer coefficients, thermal effects, assumptions in the
apparent rate model, which prevent exact comparison. Due to the large computational
time required for each set of data, a sensitivity analysis with respect to above parameters
could not be conducted. Hence only quantitative trends are shown in terms of
enhancement in rate.
Figure ____ shows the total cycle period dependence of the simulated rate at the
same operating conditions (P=30 psig, 1484 mol/m3, cycle split = 0.5). The enhancement
is seen to decrease with increase in cycle time, implying the extent of liquid reactant
starvation can be seen as the decrease in enhancement as the total cycle time is increased.
The gaseous reactant starvation and corresponding decrease in performance enhancement
observed experimentally is not seen here for the range of cycle times investigated. As
discussed above, the exact point at which this phenomenon can be observed depends on
the accuracy of the transport parameters. This means that at further lower cycle times
such a decrease in enhancement can be expected due to frequent supply of liquid
reactants and inadequate time for gaseous reactant supply for complete consumption of
the supplied liquid reactant.
The influence of cycle split was simulated by setting the same mean flow and
varying the liquid ON time to obtain cycle splits of 0.5, 0.33, 0.2 as required. The
simulated unsteady state performance shows the expected trend in performance
enhancement, i.e., increase in enhancement with a decrease in cycle split. This was
observed and explained on the basis of enhanced gaseous reactant access to the catalyst
during the liquid OFF time at the same total cycle time of 60 seconds in all the
simulations. The extent of enhancement observed is not quantitatively comparable to the
experimental enhancement due to factors discussed earlier. The simulation can however
be seen to capture the broad trends in the experimentally observed data. Further
refinements in the estimation of transport parameters will be able to match the
experimentally observed enhancement quantitatively at the expense of large
computational effort to obtain exact fits.
Enhancement (rate(us)/rate(ss))
4
3.5
3
P=30 psig, Cfeed=1484 mol/m3,
L=0.22 kg/m2s, Cycle time=60 s, split=0.5
2.5
2
1.5
1
0.5
0
300
500
700
Space time, s
900
Figure 5.7 Effect of Mean Liquid Flow (Space time) on Simulated Unsteady State
Performance
Enhancement (rate(us)/rate(ss))
3.4
P=30 psig, Cfeed=1484 mol/m3,
L=0.22 kg/m2s, split=0.5
3.2
3
2.8
2.6
2.4
2.2
2
0
20
40
60
80
100
120
Cycle Period, s
Figure 5.8 Effect of Total Cycle Time on Simulated Unsteady State Performance
Enhancement (rate(us)/rate(ss))
3
P=30 psig, Cfeed=1484 mol/m3,
L=0.22 kg/m2s, Cycle time=60 s
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
0
0.1
0.2
0.3
0.4
0.5
0.6
Cycle Split ()
Figure 5.9 Effect of Cycle Split on Simulated Unsteady State Performance
Flow Simulation using CFDLIB (2-D Model)
Predicting the complex fluid dynamics in trickle bed reactors is important for their
proper design and scale-up. Previous studies have resorted primarily to prediction of
overall phase holdup and pressure drop based on an empirical or phenomenological
approach (Saez and Carbonnell, 1985; Holub, 1990). Recent advances in understanding
of multiphase flow and development of robust codes that can simulate multi-fluid and
multi-dimensional problems have made simulation of complex flows such as those
observed in trickle beds feasible. Fluid dynamic studies reported in the literature and
conducted at CREL (using CFDLIB of Los Alamos) on trickle bed reactors have
focussed on predicting steady state phase holdups and velocities to compare with
experimental data obtained in steady state operation (Khadilkar, 1997). Recent
experimental investigations with unsteady state flow modulation (induced flow
modulation or periodic operation) have shown marked improvement in liquid reactant
conversion over steady state operation. This is attributed primarily to enhanced supply of
liquid reactant to the catalyst for a preset time interval of increased liquid flow rate
followed by complete absence of liquid whereby gaseous reactant supply is enhanced.
Since CFDLIB is capable of simulating transient multiphase flows (Kashiwa et al., 1994;
Kumar 1995), it is the ideal tool for examining the flow distribution under induced flow
modulation in trickle bed reactors.
The objective of this study is to simulate the effect of induced liquid flow
modulation on the time dependent flow distribution inside the trickle bed reactor and to
demonstrate that better reactor scale liquid distribution is possible in trickle beds operated
under flow modulation. This can validate the hypothesis that maldistribution effects can
be nullified by liquid flow modulation. Some modification of the original CFDLIB code
is required for simulation of the operation under consideration. These include
modification of conventional drag and interfacial exchange terms implemented in
CFDLIB using drag formulations developed at CREL or those available in the trickle bed
literature. The introduction of interfacial tension term for phasic pressure difference
allows computation of the influence of liquid spreading. This test case simulation will
serve as a benchmark for comparison with experimental velocity and phase holdup data
under unsteady state operation, which have not yet been reported in the open literature.
Original Equations: Modeling Interphase Exchange and Interfacial Tension Terms
The underlying equations for the CFDLIB code have been discussed in detail in
earlier reports by Kumar (1995) and can be found in Kashiwa et al. (1994).
CFDLIB developed by Los Alamos National Laboratory, has been used to obtain the
results for comparison with the DCM predictions. It is a collection of hydro-codes that
share a common numerical solution algorithm, and a common data format. The common
solution algorithm is a cell-centered finite-volume method applied to the time-dependent
conservation equations (Kashiwa et al., 1994). The governing equations that serve as the
basis for the CFDLIB codes are:
Equation of continuity:
 k
 .  k uk   k k 
t
(10)
Equation of momentum:
 k u k
 .  k u k u k  .   k  0 u' k u' k   k p   k g 
t
 [( p0  p) I   0 ].  k     0 u0  k  
(11)
.   k  0   k ( p0  p)
The special case of one fixed phase (the catalyst bed) has also been incorporated in the
code for single and two phase flow simulation. The important terms in simulating trickle
bed reactors are the interphase drag term and the influence of phasic pressure difference
due to interfacial tension. Phenomenological models developed at CREL by Holub
(1990) are incorporated in simulating the drag between the stationary solid phase and
each of the flowing phases. The code models the drag force as a product of a user defined
exchange coefficient, phase volume fractions, and relative velocity of the two phases k
and l as
FD( k l )   k  l X kl (uk  ul )
where the Xkl is modeled by the modified Ergun equation (Holub, 1990; Saez and
Carbonell, 1985) with Ergun constants either determined by single phase experiments or
using universal values. The exchange coefficient between liquid and solid phase (L-S)
and gas and solid phase can then be written as
3
X ( L S)
 (1   S )   E 1 Re L E 2 Re 2L 
Lg
 
 


Ga L  | u LS | (1   S )
  L   Ga L
X ( G  S)
 (1   S )   E 1 Re G E 2 Re G2 
G g
 
 


Ga G  | u GS | (1   S )
  G   Ga G
3
For gas-liquid drag, either no interaction is assumed or interaction based on a drag
coefficient is used as
X ( G  L) 
0.75L CD |uGL |
 Ld p
For modeling interfacial tension, the well known Leverett’s J function
(Dankworth et al., 1990) is used to yield the difference between the gas and liquid
pressure calculated in terms of the interfacial tension (), bed permeability (k), and phase
fractions as
 1  S 
p L  pG  

 k 
1/ 2

1  S   L 
 0.48  0.036.ln(
)
L


The bed permeability (k) is related to Erguns constant E1 and equivalent particle diameter
(de) as
((1   S ) / k ) 1/ 2 
( S ) E1
(1   S )d e
The simulations are conducted by incorporating the above equation (5) in the pressure
calculation step in the code.
Test Case Simulation Results and Discussion
A test case with a possibility of significant liquid maldistribution was chosen for
investigating the effects of induced liquid flow modulation. A two dimensional
rectangular model bed of dimensions 29.7 cm x 7.2 cm was considered with pre-assigned
porosity values to different cells (33 in the Z direction and 8 in the X direction as shown
in Figure 1). Thus 264 values of porosity were generated (with the mean porosity of
0.406 and a variance of 0.04) to form a pseudo random pattern of porosities in the bed (as
shown in Figure 1). Liquid flow was introduced at the two central cells at the top of the
bed at mean interstitial velocity of 0.1 cm/s, while gas flow was introduced in the rest of
the cells at an interstitial velocity of 10.0 cm/s in simulations of both steady and unsteady
state operation. Steady state simulation shows evidence of significant maldistribution,
particularly at the top and bottom of the reactor (Figure 2a (right) and Figures 3a-3g).
Complete absence of liquid is seen in zones near the bottom of the reactor (Figures 2a,
3a, and 3b). Some spreading effect due to surface tension is seen as reported in earlier
studies (Khadilkar, 1997), but is not enough to overcome inherent maldistribution effects
due to central liquid inlet and the choice of porosity profiles.
The liquid flow distribution observed in the above mentioned steady state case
was compared with transient simulations carried out with a liquid flow ON time of 15
seconds and a total cycle time of 60 seconds (45 seconds liquid OFF). Snapshots of liquid
flow distribution were taken at several time intervals (t= 15, 25, 40, 55 seconds from
beginning of liquid ON time) to compare with the steady state liquid holdup data
obtained in the earlier simulation of the steady state case. Liquid holdup variation over
the reactor cross section is depicted at several axial locations at different times in a
typical flow modulation cycle (Figures 3a-3g). These figures clearly demonstrate that
unsteady state operation ensures better uniformity in liquid distribution at all locations
over that observed in steady state operation. This improved uniformity, although not
perfect, does ensure enhanced liquid supply to all locations not previously possible
during steady state (in particular, the bottom zones shown in Figures 3a and 3b). These
are also plotted as contour plots at t= 15, 25 and 40 seconds shown in Figures 2a, 2b, and
2c, respectively. These clearly show that induced flow modulation results in better liquid
spreading and even distribution of liquid over the entire cross section at each axial
location at some point in time in the cycle. This also indicates that although the average
liquid holdup at each location may not exceed the steady state holdup, the reactor
performance may still be enhanced due to higher than steady state holdup for a subinterval of the entire cycle. This time interval of enhanced liquid supply can allow
exchange of liquid reactants and products with the stagnant liquid and with the catalyst
pellets present in any particular zone. Another observation that can be made from Figures
2a-2c is that for some time interval, all zones in the reactor become almost completely
devoid of liquid, and can allow enhanced access of the gaseous reactant to externally dry
catalyst during this time interval. Temperature rise and internal drying of catalyst and
faster gas phase reaction may also occur in this interval, which can be quenched by the
liquid in the next cycle. This demonstrates the possibility of controlled rate enhancement
due to induced flow modulation.
Title:
Creator:
TECPLOT
Preview :
This EPS picture w as not saved
w ith a preview included in it.
Comment:
This EPS picture w ill print to a
PostScript printer, but not to
other ty pes of printers .
Figure 1. Solid Holdup (THE1 = 1.0 - Bed Porsity) Distribution in the Model Trickle Bed
(note: lighter areas indicate higher porosity)
Title:
Creator:
TECPLOT
Prev iew :
This EPS picture w as not s av ed
w ith a preview inc luded in it.
Comment:
This EPS picture w ill print to a
Pos tSc ript printer, but not to
other ty pes of printers.
Figure 2a. Snapshot of Liquid Holdup (THE2) Contours at t =15 s from Start of the Liquid ON Cycle
(left) in Comparison with Steady State Holdup Contours (right). (Note: lighter areas indicate higher
liquid holdup)
Title:
Creator:
TECPLOT
Prev iew :
This EPS picture w as not s av ed
w ith a preview inc luded in it.
Comment:
This EPS picture w ill print to a
Pos tSc ript printer, but not to
other ty pes of printers.
Figure 2b. Snapshot of Liquid Holdup (THE2) Contours at t = 25 s from the Start of the Liquid ON
Cycle (left) in Comparison with Steady State Holdup Contours (right). (Note: lighter areas indicate
higher liquid holdup)
Title:
Creator:
TECPLOT
Prev iew :
This EPS picture w as not s av ed
w ith a preview inc luded in it.
Comment:
This EPS picture w ill print to a
Pos tSc ript printer, but not to
other ty pes of printers.
Figure 2c. Snapshot of Liquid Holdup (THE2) Contours at t = 40 s from the Start of the Liquid ON
Cycle (left) in Comparison with Steady State Holdup Contours (right). (Note: lighter areas indicate
higher liquid holdup)
0.120
0.080
0.100
0.060
0.040
0.020
0.080
0.060
0.040
0.020
0.000 Z=1.8 cm from bottom
0.0
1.8
0.000
3.6
5.4
7.2
0.0
1.8
X Location,m
5.4
7.2
(b)
0.120
0.120
Steady State
Periodic-t1
Periodic-t2
Periodic-t3
Periodic-t4
Z=9.9 cm from bottom
0.080
Steady State
Periodic-t1
Periodic-t2
Periodic-t3
Periodic-t4
Z=18.9 cm from bottom
0.100
Liquid Holdup
Liquid Holdup
3.6
X Location,m
(a)
0.100
Steady State
Periodic-t1
Periodic-t2
Periodic-t3
Periodic-t4
Z=3.6 cm from bottom
Liquid Holdup
0.100
Liquid Holdup
0.120
Steady State
Periodic-t1
Periodic-t2
Periodic-t3
Periodic-t4
0.060
0.040
0.020
0.080
0.060
0.040
0.020
0.000
0.000
0.0
1.8
3.6
X Location,m
(c)
5.4
7.2
0.0
1.8
3.6
X Location,m
(d)
5.4
7.2
0.120
0.120
Steady State
Periodic-t1
Periodic-t2
Periodic-t3
Periodic-t4
0.080
Steady State
Periodic-t1
Periodic-t2
Periodic-t3
Periodic-t4
0.100
Liquid Holdup
Liquid Holdup
0.100
0.060
0.040
0.020
0.080
0.060
0.040
0.020
Z=21.6 cm from bottom
Z=26.1 cm from bottom
0.000
0.000
0.0
1.8
3.6
5.4
7.2
X Location,m
0.0
1.8
3.6
5.4
7.2
X Location,m
(e)
(f)
0.120
Steady State
Periodic-t1
Periodic-t2
Periodic-t3
Periodic-t4
Liquid Holdup
0.100
0.080
0.060
0.040
0.020
Z=28.8 cm from bottom
0.000
0.0
1.8
3.6
5.4
7.2
X Location,m
(g)
Figure 3. Comparison of Cross Sectional Liquid Holdup Profiles at Different Axial Locations under
Steady and Unsteady State Operation ((a), Z= 0.9 cm; (b), Z= 1.8 cm; (c), Z= 3.6 cm; (d), Z= 18.9 cm;
(e), Z= 21.6 cm; (f), Z= 26.1 cm; (g), Z= 28.8 cm)