Chemical Engineering Science 54 (1999) 2421}2431
Trickle-bed reactor models for systems with a volatile liquid phase
M.R. Khadilkar , P.L. Mills, M.P. Dudukovic *
Chemical Reaction Engineering Laboratory (CREL), Department of Chemical Engineering, Washington University, St. Louis, MO 63130-4899, USA
Reaction Engineering Group, DuPont Central Research and Development Experimental Station, E304/A204, Wilmington, DE 19880-0304, USA
Abstract
A signi"cant number of gas}liquid}solid catalyzed reactions in the petroleum processing and chemical industries are carried out in
trickle-bed reactors under conditions where substantial volatilization of the liquid phase can occur. A review of the limited literature
on experiments and models for trickle-bed reactor systems with volatile liquids is presented "rst. A rigorous model for the solution of
the reactor and pellet scale #ow-reaction-transport phenomena based on multicomponent di!usion theory is proposed. To overcome
the assumptions in earlier models, the Stefan}Maxwell formulation is used to model interphase and intra-catalyst transport. The
model predictions are compared with the experimental data of Hanika et al. (1975, Chem. Engng Commun., 2, 19}25; 1976, Chem.
Engng J., 12, 193}197) on cyclohexene hydrogenation and also with the predictions of a simpli"ed model (Kheshgi et al., 1992, Chem.
Engng Sci., 47, 1771}1777). Rigorous reactor and pellet-scale simulations carried out for both the liquid-phase and gas-phase
reaction, as well as for intra-reactor wet}dry transition (hysteresis and rate multiplicity), are presented and discussed. Comparisons
between various models and pitfalls associated with introducing simplifying assumptions to predict complex behavior of highly
non-ideal three phase systems are also presented. 1999 Elsevier Science Ltd. All rights reserved.
Keywords: Trickle-bed reactors; Volatile liquids; Multicomponent transport; Cyclohexene hydrogenation; Multiplicity; Hysteresis;
Modeling; Three-phase
1. Introduction
Trickle-bed reactors are packed beds of catalyst with
concurrent down#ow of gas and liquid that are used
extensively in the petroleum and chemical industries.
Many gas}liquid}solid catalyzed reactions in these commercial processes are carried out at elevated temperatures and pressures where one or more reactants or
products can exhibit signi"cant volatility. Even though
substantial progress has been made in the analysis of
trickle-bed reactors, the advances have not reached the
point where a priori design and scale-up of trickle-bed
reactors is possible, particularly when highly volatile
components are present. Previous modeling e!orts have
mainly focused on operation under conditions where
the liquid mixture is non-volatile (Satter"eld, 1975;
Dudukovic and Mills, 1986). Only a few studies have
analyzed the in#uence of phase behavior on both the
*Corresponding author. Tel.: 001 314 935 6082; fax: 001 314-9354832; e-mail: [email protected].
Present Address: GE Plastics, 1 Lexan Lane Mt. Vernon, IN 47620,
USA.
reactor and pellet scale phenomena (Hanika et al., 1975,
1976; Kim and Kim, 1981; Collins et al., 1985; LaVopa
and Satter"eld, 1988; Harold, 1988; Kheshgi et al., 1992;
Harold and Watson, 1993). Hence, a need exists for more
comprehensive trickle-bed reactor models that properly
account for liquid-phase volatilization under conditions
that correspond to those typically encountered in commercial-scale processes. Table 1 summarizes the studies
of trickle-bed reactors with volatile components, which
are discussed in detail by Khadilkar (1998).
In this paper, the predictions of a more advanced
trickle-bed reactor model that is based on rigorous multicomponent transport on the reactor and pellet scale are
presented and compared with a simpli"ed model. To assess the accuracy of the model predictions, comparisons
are made with previously published experimental data for
the hydrogenation of cyclohexene to cyclohexane.
2. Model development
Based on the literature studies on trickle-bed reactor
systems with volatile liquids (listed in Table 1), the
0009-2509/99/$ } see front matter 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 5 0 3 - X
2422
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
Table 1
Summary of literature on trickle-bed reactor studies with volatile components
Authors
Reaction system
Nature of the study
Key features/observations
Hanika et al. (1975),
Hanika et al. (1976)
Cyclohexene hydrogenation
Experimental
Dry and wet rates, multiplicity. Measurements
on Reactor scale
Mills and Dudukovic (1980)
Alpha-methylstyrene
Model
Pellet e!ectiveness under gas and liquid
limited conditions with volatile liquid
Kim and Kim (1981)
Cyclohexene hydrogenation
Experimental and modeling
Pellet-scale measurements and model. Three
states observed
Collins et al. (1985)
Benzothiophene
hydrodesulfurization
Experimental and modeling
E!ect of solvent volatility on rate was shown
to be signi"cant
Kocis and Ho (1986)
Hydrodesulfurization of
benzothiophene
Dibenzofuran hydrogenation
Modeling
Harold (1988) and
Harold and Watson (1993)
Hydrazine decomposition
Modeling
Assumed that liquid #ow not a!ected by evaporation
E!ect of solvent and gas/liquid feed ratio was
demonstrated.
Pellet-scale partial "lling, imbibition, condensation and evaporation in catalyst pores.
Jaguste and Bhatia (1991)
Cyclohexene hydrogenation
Experimental, analysis
E!ect of capillary condensation
Kheshgi et al. (1992)
Cyclohexene hydrogenation
Modeling
Prediction of Hanika et al. Experimental
results
Toppinen et al. (1996)
Toluene hydrogenation
Modeling
Khadilkar et al. (1997)
Alpha-methylstyrene
hydrogenation
Modeling
Multicomponent e!ects, but no volatility
e!ects
Multicomponent e!ect, volatility transient
conditions
LaVopa and Satter"eld (1988)
Experimental and modeling
following features should be incorporated into a comprehensive model: (i) interphase transport and vapor}liquid
equilibrium e!ects; (ii) multicomponent e!ects due to
large inter-phase #uxes of mass and energy, as well as the
in#uence of varying concentration on transport of other
components and the total inter-phase #uxes; (iii) the
in#uence of volatilization and reaction on the variation
in holdup and velocity; (iv) complete depletion of liquid
reactants in the reactor that are modeled by either correcting or dropping the liquid-phase equations based on
computed holdup and temperature; (v) external, internal
or combined external}internal partial catalyst wetting;
(vi) the combined e!ects of imbibition, capillary condensation, liquid volatility, heats of vaporization, and
reaction on the particle scale; and (vii) the existence of
multiple steady states in accordance with experimental
results reported in literature.
The model presented here attempts to address the
above requirements by relaxing many of the assumptions
used in previous models. Level I and Level II models
discussed below are catalyst and reactor level models,
which are used as the basis to formulate Level III
model as a combination of reactor and pellet-scale
models.
2.1. Level I: pellet-scale model
Kim and Kim (1981) assumed that the macropores of
the catalyst are "lled with vapor and described intra-
particle di!usion using power-law kinetics:
dy ¸ R¹kPL\
! !
yL"0
(1)
dx
D
C
with standard boundary conditions. The local reaction
rate was then evaluated using the reactant #ux at the
pore mouth
D P dy
rate" C
¸ R¹ dx
!
V
(2)
and the heat generated was obtained directly from the
rate. Their model considered di!erent e!ective di!usivity
values that were based upon the state of their catalyst, as
well as di!erent rate constants for the liquid and vapor
phase reaction. This model clearly explained the multiplicity e!ects observed in their experiments.
Several other features such as capillary e!ects, incomplete catalyst "ling and evaporation examined by Harold
(1988), and Harold and Watson (1993) are incorporated
in the level III model developed in this study.
2.2. Level II: reactor-scale model
Kheshgi et al. (1992) developed a model based on
a pseudo-homogeneous approach using the reaction system of Hanika et al. (1976) as a basis. It was coupled with
an overall energy balance that accounts for the change
in enthalpy of the liquid and vapor streams with
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
2423
both reaction and phase change. The resulting model,
represented by Eqs. (3) and (6), was solved simultaneously with algebraic equilibrium and #ow relations to
obtain the velocity, conversion, and temperature pro"les
in the reactor. They assumed the reaction order to be
unity with respect to cyclohexene for the dry pellet, and
unity with respect to hydrogen for the wet pellet. The
mass balance for cyclohexene conversion along the reactor is
¸
d(F a)
"R f #R (1!f )
5 5
"
5
dz
Fig. 1. Partially internally wetted model pellet.
f k F (N!a) (1!f )k F K(1!a)
5 " " 5 5 #
,
hF
(KF #F )
4
4
*
(3)
where
F "F (1#N!a)#F !F (N!a)/(1!K(¹)),
*
!
(4)
F "F (1#N!a)#F !F .
(5)
4
!
*
The mole fractions in the vapor phase for components
A (cyclohexene), B (hydrogen), and C (cyclohexane) are
described in terms of vapor and liquid #ows, which are
then used to calculate liquid-phase compositions using
equilibrium relations. The energy balance for the pseudohomogeneous mixture is
d[F ( x H )#F ( y H )] nd d¹
G G
4
G G ! R j
*
4
dz
dz
#nd ;(¹!¹ )"0
R
5
with boundary conditions
(6)
at z"0, ¹"¹ , a"0 and at z"¸, d¹/dz"0.
(7)
No distinction is made between external wetting and
internal wetting of the catalyst pellets.
2.3. Level III: reactor and pellet-scale multicomponent
model
Level III model proposed here is a combination of
a rigorous multi-component model for the reactor scale
and its extension to the pellet scale. Some simplifying
assumptions are made to keep the numerical solution
tractable while maintaining the necessary rigor. Hence,
only steady-state axial pro"les are modeled. The catalyst
pellets are modeled as half-wetted slabs exposed to liquid
on one face and gas on the other with partially internal
pore "ll-up as shown in Fig. 1. Pressure gradients can
exist in the gas-"lled zone, but not in the liquid-"lled
zone of the catalyst pellet.
2.4. Level III reactor-scale yuid domain equations
A two-#uid approach is considered for the reactorscale model. Equations are written for conservation of
the gas and liquid phase mass, energy, and momentum
transport with source terms representing interphase
#uxes. Since the multicomponent equations involve the
solution of a large number of non-linear simultaneous
equations coupled with the di!erential equations, the
second-order derivatives in the di!erential equations that
represent axial dispersion are dropped. This reduces the
problem to an initial value system, which is more
straightforward to solve. For the reactor-level equations,
the total number of unknowns are 10nc#13 (as listed in
Appendix A) and corresponding equations are listed
below. The numbers in square brackets given along with
the equations indicate the number of such equations
available for a system with nc number of components.
The continuity equations for the liquid and gas phase
with total interphase #uxes as the source terms are
d(o u e )
* '* * "# N%*a M ! N*1a M [1],
G %* G
G *1 G
dz
(8)
d(o u e )
% '% % "! N%*a M ! N%1a M [1],
G %* G
G %1 G
dz
(9)
The unexpanded form of the momentum equations for
the liquid and gas phase with source term contributions
from gravity, pressure drop, drag due to solid, gas}liquid
interaction and added momentum due to interphase
transport are
d
(o e u u )
dz * * '* '*
dp
"e o g!e
!F
#K (u !u )
* *
* dz
"*
%* '%
'*
*
#u' N%*a M ! N*1a M
'*
G %* G
G *1 G
[1],
(10)
2424
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
d
(o e u u )
dz % % '% '%
Table 2
Gas}liquid transport calculation vector
N%*!c +[b ][k][C], (x'!x*)!x (Dq)/j
H
*
V
GH G
RJ H
* *
:
dp
"e o g!e
!F
#K (u !u )
% %
% dz
"% %* '*
'%
%
#u' N%*a M ! N%1a M
'%
G %* G
G %1 G
[1],
(11)
e #e "e [1].
(12)
*
%
The species conservation equations written with source
terms for absolute interphase #uxes for gas}liquid,
liquid}solid, and gas}solid transport are
d
(u e C )"!N%*a !N%1a
[nc!1],
G %*
G %1
dz '% % G%
(13)
d
(u e C )"N%*a !N*1a
[nc!1].
G %*
G *1
dz '* * G*
(14)
Energy balance can be written for each of the three
phases with source terms for interphase energy #ux and
for heat losses to the ambient as
d(e u o H )
* '* * * "E%*a !E*1a !E*a
[1],
%*
1*
*
dz
N%* !c +[b ][k][C],
(x'!x*)!x
(Dq)/j
LA\
RJ H
* *
LA\H G
H
LA\
V
N%*!c +[b ][k ], (y%!y*)!y (Dq)/j
H
W
GE
% % *H H
H
:
F "
%*
y'!K x'
L
LA LA
x' #x' #x' #2x' !1
LA
y' #y' #y' #2y' !1
LA
LA
q !q # N%* (H%!H*)
G
G
G
%
*
G
LA>
Table 3
Gas}catalyst}liquid transport calculation vector
N*1!c +[b ][k][C], (x*!x*1')!x (Dq)/j
R'
* *
*H H
H
'
V
H
:
(15)
N*1 !c +[b ][k][C],
(x*!x*1')!x
(Dq)/j
LA
R'
* *
LA\H H
H
LA\
V
H
d(e u o H )
% '% % % "!E%*a !E%1a !E%a
[1],
%*
%1
%
dz
N!.!c +[b ][B]\[C], (
x*1')
R!.
!
H
H
H
:
(16)
E*1a #E%1a "0 [1].
(17)
*1
%1
Auxiliary relations required to complete the set of equations, such as equations for local-phase densities [2], and
relations from which the ncth component concentrations
can be calculated for the liquid and gas phase [2], are
listed in Appendix A for reference.
The above analysis shows that 2nc#6#(4 auxiliary
conditions) equations are available for (10nc#13) unknowns. The additional (8nc#3) equations needed are
obtained from the interphase mass and energy transport
relations between the solid, liquid, and gas phases (given
in Tables 2 and 3) for gas}liquid and gas}solid}liquid
transport, respectively (Taylor and Krishna, 1993;
Khadilkar et al., 1997). The boundary conditions at the
reactor inlet are speci"ed for the various di!erential
equations that describe the model. The multicomponent
e!ects are incorporated while calculating the transport
parameters and correcting them using the so-called
&&bootstrap'' condition given by [b] matrices (see Appendix A) using the energy balance equation at the interface
as the bootstrap for all the interphase transport equations (Taylor and Krishna, 1993; Khadilkar et al., 1997).
The transport coe$cients are also corrected for high #ux
as given by Taylor and Krishna (1993). The activity
correction matrix for [C] is obtained from the Wilson
equation for activity coe$cients.
(Dq)/j
N%* !c +[b ][k ], (y%!y*)!y
H
LA\
W
LA\
GE H
% % GH H
y' !K x'
' :
N!. !c
LA\
R!.
H
+[b ][B]\[C], (
x*1')
!
H
H
N%1!c +[b ][k ], (y%!y%1')!y (Dq)/j
R%
% % H H
H
'
W
H
:
F "
%*1
N%1 !c +[b ][k ],
(y%!y%1')!y
(Dq)/j
LA\
R%
% % LA\H H
H
LA\
W
H
N!.!c +[b ][B]\[C], (
x%1')
R!.
!
H
H
H
:
N!. !c +[b ][B]\[C],
(
x%1')
LA\
R!.
!
LA\H
H
H
(y' !K x' )
%1
:
(y'!K x' )
L
LA LA %1
(x' #x' #x' #2x' ) !1
LA %1
(x' #x' #x' #2x' ) !1
LA *1
(y' #y' #y' #2y' ) !1
LA %1
LA
q !q # N%1 (H%!H1)
%
1
G
G
G
G
LA
q !q # N*1 (H*!H1)
*
1
G
G
G
G
( N M /o ) "( N M /o )
G G *G *1'
G G *G %1'
LA>
2.5. Level III catalyst-scale equations
The present model extends the approach of Harold
and Watson (1993), and Jaguste and Bhatia (1991) by
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
2425
considering the multicomponent matrix form for the
reaction}di!usion equations for both the gas- and liquid"lled part of the pellet (Taylor and Krishna, 1993;
Toppinen et al., 1996; Khadilkar et al., 1997). For a
half-wetted pellet with internal evaporation, the reaction}di!usion problem has to be solved for the gas-"lled
and the liquid-"lled part of the pellet (Harold, 1988;
Harold and Watson, 1993) using the continuity conditions at the intracatalyst interface and boundary conditions at the catalyst-#owing phase interface obtained
from Table 3. Thus, the pellet-scale model needs to be
solved in conjunction with the reactor model proposed
earlier.
The number of unknowns in this set of equations for
an nc component system is nc values of gas and liquid
#uxes each, nc gas and liquid compositions each, gas
and liquid temperatures (one each), interface location,
and gas-phase total pressure (total"4nc#4). Some of
these unknowns are expressed as di!erential equations
(nc #ux transport relations for gas and liquid phase,
nc!1 liquid #ux-concentration relations, nc gas #uxconcentration equations, and two thermal energy equations for the gas and liquid temperatures), which yields
(4nc!1) "rst order ODEs, two second-order ODEs, and
two auxiliary equations (Appendix B). Thus, (4nc#3)
boundary conditions are needed with one additional
condition to complete the problem de"nition as listed in
Appendix B. The di!erential equations can be written for
the species and energy #uxes in the gas- and liquid-"lled
part of the catalyst as given below (remembering here
that the individual species #uxes are a combination of
Fickian and bulk #uxes). For the gas phase, the dusty gas
model with bulk di!usion control yields independent
equations for all the nc component #uxes with a pseudo
component #ux (for the catalyst pore structure) for which
a zero value is assigned and used as the bootstrap.
The "nal set of model equations that emerge from this
analysis are
The required conditions are obtained from mass and
energy #ux boundary conditions for the dry and wetted
interface of the catalyst. Continuity of mass and energy
#uxes is also imposed at the intra-catalyst gas}liquid
interface (located at d). Identical phase temperature and
thermodynamic equilibrium are also enforced at the
gas}liquid interface. These are augmented by the liquid}
phase imbibition equation used to obtain the location of
the intra-catalyst gas}liquid interface. These conditions
are summarized in Appendix B.
d
(N! )"l R
[nc],
G G dx G%
(18)
d
(N! )"l R
[nc],
G G
dx G*
(19)
Predictions of Level II and Level III models (referred
henceforth as LII and LIII, respectively) are presented
and compared to literature data for the hydrogenation of
cyclohexene to cyclohexane (Hanika et al., 1975, 1976).
C!
LA\
d
N! " G* N! ! [B ]\[C]
(C! )
[nc!1],
G* C! G*
*
dx G*
G*
GH
H
(20)
C!
LA
d
N! " G% N! ! [B ]\
(C! )
G% C! G%
%
dx G%
G%
GH
H
d
d N! H!
G% G%"0 [1],
k
(¹! )!
C% dx %
dx
d
d N! H!
G* G*"0 [1].
k
(¹!)!
C* dx *
dx
[nc],
2.6. Solution strategy
For solution of Level III model, an IPDAE solver
(gPROMS, Oh and Pantelides, 1996) was used. The
catalyst level equations were solved using orthogonal
collocation on "nite elements (OCFEM). The catalyst
coordinate was normalized using xc"x/d over the
wetted zone, and xc"(x!d)/(¸c!d) over the dry zone
length to retain invariant bounds on the independent
variable. Level II model (Kheshgi et al., 1992) was solved
similarly using a combination of orthogonal collocation
(for the two di!erential equations) and a Newton solver
for the algebraic equations. The rate parameters for the
dry and wetted pellet reaction rates were obtained from
Kheshgi et al. (1992) as listed in Table 4. Continuation of
the dry branch pro"les for the case of multiple steady
states was implemented by choosing the thermal conductivity (j) for Level II model, and the degree of internal
catalyst wetting (d) for Level III model. Catalyst level
multiplicity due to intra- and extra-catalyst heat transfer
limitations as reported by Harold and Watson (1993) was
encountered, but was not investigated in the present
study.
3. Results and discussion
Table 4
Parameter values used in the LII and LIII models
LII model
LIII model
(22)
k "1.5;10\ mol/s
"
k "0.14 mol/s
5
¸"0.18 m
d "0.03 m
R
;"2.8 J/m s K
(23)
j"0.44 J/m s K
e "0.4
k "0.8 1/s
TQ5
k "30 1/s
TQ"
a "150 m/m
%*
a "a "300 m/m
*1
%1
k "0.15 J/msK,
C*
k "0.1 J/msK
C%
¸ "2;10\
A
R "10;10\ m
N
(21)
2426
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
The simulation results for multiplicity of reaction rates,
the corresponding temperature pro"les, wet (liquidphase) and dry (gas-phase) reaction, and wet}dry
transition are also discussed.
3.1. Multiplicity behavior of reaction rate
The most interesting observation of Hanika et al.
(1976) for this reaction system was reaction rate multiplicity. Referring to Fig. 2, when either the hydrogen to
cyclohexene molar ratio (parameter"N) or feed temperature are increased, the reaction progresses along the
fully wetted catalyst branch and then abruptly shifts to
the high rate or dry catalyst branch. However, if the
reactor is operated at the high-rate state and the hydrogen to cyclohexene molar ratio is reduced, the reaction
continues along the high-rate branch until it reaches the
extinction point, where it abruptly shifts to the low-rate
branch. This is the location where the hydrogen molar
#ow rate cannot support the generation of cyclohexene
and cyclohexane vapor due to equilibrium constraints.
Both branches were simulated successfully using the
present model (LIII) as well as the pseudo-homogeneous
model (LII) of Kheshgi et al. (1992). Fig. 2 also shows that
conversion along both branches is well-predicted by the
present model (LIII) when compared to the experimental
data and the pseudo-homogeneous (LII) model.
3.2. Ewect of hydrogen to cyclohexene molar ratio (N) on
temperature rise in wet and dry operation
At low hydrogen to cyclohexene feed ratios (N(6),
the lower branch in Fig. 3 shows that the catalyst
remains in an internally fully wetted condition throughout the reactor. The reactant conversion corresponds
almost entirely to the contribution of wetted pellets,
Fig. 3. E!ect of hydrogen to cyclohexene ratio (N) on temperature
pro"les (LIII model).
which results in lower rates and hence a lower temperature rise. In contrast to this, the upper branch in Fig. 3
shows that at high hydrogen to cyclohexene ratios
(N'8), the catalyst in the entire reactor is dry. This leads
to much higher rates and a greater temperature increase,
which corresponds to the "ndings reported by Hanika
et al. (1976). The temperature pro"les in wet operation
observed by Hanika et al. (1976), and predicted by the
LIII model as shown in Fig. 3, both decrease with increasing N. This is expected since the higher hydrogen
#ow rate enhances evaporation of some of the liquid and
cools the liquid, even though it is slightly heated by the
heat of reaction. As shown in Fig. 4, both the LII and
LIII models give fairly accurate predictions of the dry
branch temperature pro"le for a particular set of conditions (F "8;10\ mol/s, N"11) for which experi
mental data is published by Hanika et al. (1975). The dry
branch pro"les (in Fig. 3) showed a decrease in the maximum temperature rise with an increase in N, which
implies that some of the heat of reaction is removed by
the excess hydrogen at large N values, thus resulting in
a smaller temperature rise.
3.3. Wet-branch simulation (LIII model)
Fig. 2. Multiplicity behavior: conversion dependence on hydrogen to
cyclohexene ratio.
The reactor-scale equations for the LIII model allow
for the variation of phase holdup and velocity as illustrated in Fig. 5, which become more signi"cant as the
bubble point of the reaction liquid mixture is approached. As shown in Figs. 6 and 7, the predicted
liquid-phase concentration pro"les on both the reactor
and the pellet scale indicate hydrogen limitation is present, which was also observed in the experimental studies
of Hanika et al. (1975). It must be noted here that the
imbibition equation (Appendix B) yielded complete internal wetting of the catalyst pellet (dP¸ ). Fig. 7 also
A
shows that the intra-catalyst temperature rise for the
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
Fig. 4. Comparison of experimental and predicted temperature pro"les
in dry operation (LIII model).
2427
Fig. 7. Intracatalyst hydrogen concentration and temperature pro"les
at di!erent axial locations in wet-branch operation.
Fig. 5. Axial variation of phase holdup and velocity in wet-branch
operation.
Fig. 8. Intra-catalyst #uxes at di!erent axial reactor locations in wetbranch operation (cyene: cyclohexene, cyane: cyclohexane).
Fig. 6. Axial variation of liquid-phase concentration of components.
liquid-full zone, which corresponds to the low rate
branch operation, did not exceed 5 to 63C at all reactor
locations.
Fig. 8 shows the multicomponent intra-catalyst #uxes
for the various species at several axial locations. The
hydrogen #uxes indicate that the supply occurs from
both the externally wetted side and the internally dry side
with zero #ux in the central core as a result of complete
hydrogen consumption (see also Fig. 7). The cyclohexene
#ux pro"les in the pellet are similar to those of hydrogen
in shape, but exhibit negative values at the reactor entrance due to condensation on the internally dry side and
transport to the liquid}solid interface. Only positive cyclohexene #ux values are seen downstream in the reactor
where the pellet contains a high concentration of the
product. Higher temperatures at this location enhance
2428
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
internal evaporation within the catalyst as observed
in Fig. 8 for the reactant (cyclohexene) and product
(cyclohexane). This represents the initiation point of
intracatalyst drying, since the temperature exceeds the
boiling point of the liquid mixture.
3.4. Dry-branch simulation
At high hydrogen to cyclohexene molar ratios
(N'6.3), the reaction occurs completely in the vapor"lled pores of the catalysts with very high reaction rates
(Table 4). This was simulated in the present model (LIII)
by setting the intracatalyst gas}liquid interface location
to d"0, eliminating the liquid-phase conservation equations and corresponding exchange terms, and setting the
gas holdup equal to bed porosity. To simplify the numerical solution, the pellet-scale equations were solved by
imposing symmetry conditions. Figs. 9 and 10 show that
the gas velocity, pressure, and concentration pro"les
undergo signi"cant variation along the reactor. Among
these, the gas velocity undergoes the greatest variation
due to mass transfer e!ects and heat generation due to
reaction.
The concentration pro"les on the reactor scale and the
pellet scale are shown in Figs. 10 and 11, respectively.
These suggest that cyclohexene is limiting when a high
hydrogen to cyclohexene molar ratio is used (N"8) and
only the gas-phase reaction is occuring. Fig. 12 shows
that a moderate pressure buildup occurs inside the pellet
during dry-branch operation so that a non-isobaric
condition exists. For reactions having a net reduction in
the total number of moles, a decrease in the centerline
pressure over the bulk pressure is expected (Taylor and
Fig. 11. Intracatalyst cyclohexene concentration and temperature
pro"les at di!erent axial locations in dry-branch operation.
Fig. 9. Reactor-scale pro"les of gas velocity and pressure in dry-branch
operation.
Fig. 10. Axial concentration pro"les for dry branch simulation
(LIII model).
Fig. 12. Intra-catalyst pressure pro"les at di!erent axial locations
(LIII model).
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
2429
Krishna, 1993). However, in the present case, a moderate
increase in pressure near the reactor inlet (Fig. 12) is
caused by the high reaction rates and corresponding
temperature rise. This pressure buildup is seen to diminish at downstream locations where the reaction rate
and corresponding intracatalyst temperature rise is
negligible.
3.5. Intra-reactor wet}dry transition
This intra-reactor transition from wet to dry branch
operation (at N'6.3) is not straightforward to predict with a heterogeneous (LIII) model, since the liquidphase conservation and exchange equations collapse
at the transition point. Numerical problems were encountered during Level III model simulation of the
abrupt transition from d"¸ to d"0 as reported by
A
Hanika et al. (1976) and Harold and Watson (1993). Since
very little experimental data is available for comparison
with the transition pro"les at the pellet scale, this aspect
was not pursued in further detail here using Level III
model.
The phenomena of interest associated with intra-reactor phase transition were simulated using Level II model
by introducing the feed in the reactor under transition
conditions (N"7) and examining the change from the
liquid to the vapor-phase reaction. Fig. 13 shows that the
liquid #ow rate approaches zero near the reactor inlet
because the heat of reaction and the high hydrogen #ow
rate, resulting in complete vaporization of the liquid
reactants and products. The temperature rise until this
point was also negligible (corresponding to a near
isothermal phase change), after which the gas-phase
reaction proceeded downstream at a much higher rate.
Fig. 14. Simulated vapor-phase compositions and catalyst wetting for
intra-reactor wet-to-dry transition (LII model) (cyane: cyclohexane,
cyene: cyclohexene).
The maximum temperature rise in this case was between
that observed for the wet and dry branch due to usage of
some of the heat of reaction for evaporation of the liquid
and the transition to dry operation.
Fig. 14 shows how the fraction of catalyst particles
that are wetted changes from being almost completely
wetted to a vapor-"lled state. The mole fraction of both
cyclohexene and cyclohexane in the vapor phase increased slightly at the reactor inlet as a result of liquid
evaporation and reaction followed by transition to the
gas phase, with a decrease in cyclohexene and hydrogen
mole fraction. This is the expected behavior in the dry
rate branch of the reaction.
4. Summary and conclusions
Fig. 13. Simulated #ow and temperature pro"les for intra-reactor wetto-dry transition (LII model).
A rigorous trickle-bed model was proposed on the
basis of reactor and pellet-scale phenomena reported in
literature for systems with volatiles. This model incorporated rigorous multicomponent mass and energy transport on the reactor and catalyst level, and was shown to
give accurate predictions of the conversion and temperature pro"les on the reactor scale. The ability to model
both reactor-scale variation of the phase velocities and
#uid holdups, as well as the rigorous interphase #ux and
vapor}liquid equilibrium e!ects, was also demonstrated.
Pellet-scale multicomponent reaction-transport equations that accounted for evaporation}condensation
e!ects were also rigorously computed. The rigorous
approach was thus shown to be comprehensive enough
in modeling both reactor- and pellet-scale phenomena
and is recommended for future models for complex reaction systems with volatiles.
2430
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
Acknowledgements
Subscripts and superscripts
The authors acknowledge the support of the industrial
sponsors of the Chemical Reaction Engineering Laboratory (CREL).
A
ambient
A, B, C components
B
bed voidage
C
catalyst
D, G, < dry, gas, or vapor phase
i
species number
I
interface
¸, = wet, liquid phase
S
solid phase
t
total
Notation
a
[B]
C
D
C
d
R
E
f
U
F
B
F
g
h
h
H
[k]
k
C
k
K
K
%*
¸
M
G
N
N
P
q
R
R
G
S
V
¹
u
'
;
x
y
z
interfacial area per unit volume of reactor
mass transfer coe$cient matrix
concentration
e!ective di!usivity
reactor diameter
energy transfer #ux
fractional wetting
#uid}solid drag
molar #ow
gravitational acceleration
heat transfer coe$cient
Henry's constant
enthalpy
mass transfer coe$cient
e!ective thermal conductivity of catalyst
reaction rate constant
equilibrium constant
interphase momentum transfer coe$cient
length
molecular weight of species i
hydrogen to cyclohexene molar feed ratio
#ux at the interface
pressure
heat #ux
apparent reaction rate
intrinsic reaction rate in catalyst pellet
catalyst external surface area
temperature
interstitial (actual) velocity
reactor to wall heat transfer coe$cient
mole fraction, intra-pellet spatial coordinate
mole fraction in the gas phase
axial coordinate
Greek symbols
a
[b]
c, [C]
d
e
l
G
o
j
j
V
conversion (based on cyclohexene)
bootstrap matrix in Maxwell}Stefan formulation
activity coe$cient, matrix form
intracatalyst gas}liquid interface location
phase holdup
stoichiometric coe$cient of component i
density
reactor e!ective thermal conductivity
latent heat
References
Collins, G.M., Hess, R.K., & Ackgerman, A. (1985). E!ect of volatile
liquid phase on trickle-bed reactor performance. Chem. Engng
Commun., 35, 281}291.
Dudukovic, M.P., & Mills, P.L. (1986). Contacting and hydrodynamics
of trickle-bed reactors. Encyclopedia of -uid mechanics, (pp.
969}1017). Huston: Gulf Publishing Company.
Hanika, J., Sporka, K., Ruzika, V., & Krausova, J. (1975). Qualitative
observations of heat and mass e!ects on the behavior of a tricklebed reactor. Chem. Engng Commun., 2, 19}25.
Hanika, J., Sporka, K., Ruzika, V., & Hrstka, J. (1976). Measurement of
axial temperature pro"les in an adiabatic trickle-bed reactor. Chem.
Engng J., 12, 193}197.
Harold, M.P., & Watson, P.C. (1993). Bimolecular exothermic reaction
with vaporization in the half-wetted slab catalyst. Chem. Engng Sci.,
48, 981}1004.
Harold, M.P. (1988). Steady-state behavior of the non-isothermal partially wetted and "lled catalyst. Chem. Engng Sci., 43, 3197}3216.
Jaguste, D.N., & Bhatia, S.K. (1991). Paritial internal wetting of catalyst
particles: Hysterisis e!ects. A.I.Ch.E. J., 37, 661}670.
Khadilkar, M.R. (1998). Performance studies of trickle-bed reactors.
D.Sc. Dissertation, Washington University, St. Louis, Missouri, USA.
Khadilkar, M.R., Al-Dahhan, M.H., & Dudukovic, M.P. (1997). Simulation of unsteady state operation in trickle-bed reactors. A.I.Ch.E.
Annual Meeting, Los Angeles, CA.
Kheshgi, H.S., Reyes, S.C., Hu, R., & Ho, T.C. (1992). Phase transition
and steady-state multiplicity in a trickle-bed reactor. Chem. Engng
Sci., 47, 1771}1777.
Kim, D.Y., & Kim, Y.G. (1981). Experimental study of multiple steady
states in a porous catalyst particles due to phase transition. J. Chem.
Engng Japan, 14, 311}317.
Kocis, G.R., & Ho, T.C. (1986). E!ects of liquid evaporation on the
performance of trickle-bed reactors. Chem. Engng Res. Dev., 64,
288}291.
LaVopa, V., & Satter"eld, C.N. (1988). Some e!ects of vapor} liquid
equilibria on performance of a trickle-bed reactor. Chem. Engng Sci.,
43, 2175}2180.
Mills, P.L., & Dudukovic, M.P. (1980). Analysis of catalyst e!ectiveness
in trickle-bed reactors processing volatile or non-volatile reactants.
Chem. Engng Sci., 35, 2267.
Oh, M., & Pantelides, C.C. (1995). Modeling and simulation language
for combined lumped and distributed parameter systems. Comput.
Chem. Engng, 20, 611}633.
Satter"eld, C.N. (1975). Trickle bed-reactors A.I.Ch.E. J., 21, 209}228.
Taylor R., & Krishna, R. (1993). Multicomponent mass transfer.
New York, Wiley.
Toppinen, S., Aitamma, J., & Salmi, T. (1996). Interfacial mass transfer
in trickle-bed reactor modeling. Chem. Engng Sci., 51, 4335}4345.
M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431
2431
Appendix A. reactor-scale model equations
Appendix B. Catalyst level equations
The number of unknowns are: gas and liquid velocities
(2), holdups (2), pressure (1), nc liquid and gas phase
concentrations, 3 temperatures (gas, liquid, and solid),
densities of gas and liquid (2). Total unknowns"
2nc#10.
Interphase transport (3 interfaces, G}¸, G}S, and
¸}S): For each interface, the unknowns are nc #uxes, nc
liquid, and nc vapor interface compositions and the interface temperature. Total"3nc#1 (G}L)#3nc#1
(G}S)#2nc#1 (L}S)"8nc#3.
Number of equations at the catalyst level"4nc#4.
Boundary conditions (at the catalyst-bulk -uid boundary): Liquid}solid boundary
Auxiliary equations:
C R¹ "P,
(A.1)
G%
E
C M /o "1,
(A.2)
G* G *G
C M "o ,
(A.3)
G% G
%
C M "o .
(A.4)
G* G
*
Bootstrap matrix [b] ( for liquid phase based on energy
flux):
[b ] "d !x K ,
GI *
GI
G I
K "(j !j )/j ,
GI
I
LA V
j " x j , j "x (H4 (@¹ )!H* (@¹ )).
V
G G G
G G
%
G
*
G
Mass transfer coe.cient matrix:
y
y
[B ]" G # I (for i"j),
GH
k
k
GLA
I HI
1
1
!
(for iOj).
[B ]"!y
GH
G k
k
GH
GLA
Enthalpy of gas and liquid phase:
L
H " C (DH*#C (¹ !¹ ))/o ,
*
G*
G
.G *
0D *
G
L
H " C (DH%#C (¹ !¹ ))/o .
%
G%
G
.G %
0D %
G
Activity correction matrix:
* ln c
G.
C "d #x
GH
GH
G *x
H
Interface energy transport equation:
L
h (¹ !¹ )# N%* H* (¹ )
* '
*
G
G *
G
L
"h (¹ !¹ )# N%* H% (¹ ).
% %
'
G
G %
G
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
N*
"N*1 [nc!1],
!V
Similarly the energy #ux boundary condition
(B.14)
d¹!
L
*# N!* H* (¹!)
!k
C* dx
G
G *
G
L
"h (¹ !¹!)# N*1 H* (¹ ) [1].
(B.15)
*1 *
*
G G *
G
Gas}solid boundary (nc conditions can be used due to
dusty gas model)
N%
"N%1 [nc].
!V*A
Energy #ux boundary condition
(B.16)
d¹!
L
%# N!* H* (¹! )
!k
C% dx
G
G %
G
L
"![h (¹ !¹! )# N%1 H% (¹ )] [1]. (B.17)
%1 %
%
G
G %
G
Relationships between variables at the gas}liquid
intracatalyst interface:
"N%
[nc],
N*
!VB
!VB
C!
C!
2p< cos h
G% "K exp !
G*
[nc],
G
C!
R R¹! C!VB
G%
G*
N *
¹!
"¹!
[1],
* VB
% VB
L
d¹!
%# N!* H* (¹! )
!k
G
G %
C% dx
G
d¹!
L
*# N!* H* (¹!) [1].
"!k
C* dx
G
G *
G
¸iquid imbibition velocity:
(B.18)
(B.19)
(B.20)
(B.21)
v"N! /C!
R* R*
"(R/8dk )(P
!P
#2p cos h/R ) [1].
N
* V*A
VB
.
(B.22)