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 reactor and pellet scale phenomena (Hanika et al., 1975,

1976; Kim and Kim, 1981; Collins et al., 1985; LaVopa
Trickle-bed reactors are packed beds of catalyst with and Satter"eld, 1988; Harold, 1988; Kheshgi et al., 1992;
concurrent down#ow of gas and liquid that are used Harold and Watson, 1993). Hence, a need exists for more
extensively in the petroleum and chemical industries. comprehensive trickle-bed reactor models that properly
Many gas}liquid}solid catalyzed reactions in these com- account for liquid-phase volatilization under conditions
mercial processes are carried out at elevated temper- that correspond to those typically encountered in com-
atures and pressures where one or more reactants or mercial-scale processes. Table 1 summarizes the studies
products can exhibit signi"cant volatility. Even though of trickle-bed reactors with volatile components, which
substantial progress has been made in the analysis of are discussed in detail by Khadilkar (1998).
trickle-bed reactors, the advances have not reached the In this paper, the predictions of a more advanced
point where a priori design and scale-up of trickle-bed trickle-bed reactor model that is based on rigorous multi-
reactors is possible, particularly when highly volatile component transport on the reactor and pellet scale are
components are present. Previous modeling e!orts have presented and compared with a simpli"ed model. To as-
mainly focused on operation under conditions where sess the accuracy of the model predictions, comparisons
the liquid mixture is non-volatile (Satter"eld, 1975; are made with previously published experimental data for
Dudukovic and Mills, 1986). Only a few studies have the hydrogenation of cyclohexene to cyclohexane.
analyzed the in#uence of phase behavior on both the

2. Model development
*Corresponding author. Tel.: 001 314 935 6082; fax: 001 314-935-
4832; e-mail: dudu@wuche3.wustl.edu.
Present Address: GE Plastics, 1 Lexan Lane Mt. Vernon, IN 47620, Based on the literature studies on trickle-bed reactor
USA. 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), Cyclohexene hydrogenation Experimental Dry and wet rates, multiplicity. Measurements
Hanika et al. (1976) 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 Experimental and modeling E!ect of solvent volatility on rate was shown
hydrodesulfurization to be signi"cant
Kocis and Ho (1986) Hydrodesulfurization of Modeling Assumed that liquid #ow not a!ected by evap-
benzothiophene oration
LaVopa and Satter"eld (1988) Dibenzofuran hydrogenation Experimental and modeling E!ect of solvent and gas/liquid feed ratio was
demonstrated.
Harold (1988) and Hydrazine decomposition Modeling Pellet-scale partial "lling, imbibition, conden-
Harold and Watson (1993) sation 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 Multicomponent e!ects, but no volatility
e!ects
Khadilkar et al. (1997) Alpha-methylstyrene Modeling Multicomponent e!ect, volatility transient
hydrogenation conditions

following features should be incorporated into a compre- particle di!usion using power-law kinetics:
hensive model: (i) interphase transport and vapor}liquid
dy ¸ R¹kPL\
equilibrium e!ects; (ii) multicomponent e!ects due to ! ! yL"0 (1)
dx D
large inter-phase #uxes of mass and energy, as well as the C
in#uence of varying concentration on transport of other with standard boundary conditions. The local reaction
components and the total inter-phase #uxes; (iii) the rate was then evaluated using the reactant #ux at the
in#uence of volatilization and reaction on the variation pore mouth
in holdup and velocity; (iv) complete depletion of liquid
reactants in the reactor that are modeled by either cor- D P dy
rate" C (2)
recting or dropping the liquid-phase equations based on ¸ R¹ dx
! V
computed holdup and temperature; (v) external, internal
or combined external}internal partial catalyst wetting; and the heat generated was obtained directly from the
(vi) the combined e!ects of imbibition, capillary conden- rate. Their model considered di!erent e!ective di!usivity
sation, liquid volatility, heats of vaporization, and values that were based upon the state of their catalyst, as
reaction on the particle scale; and (vii) the existence of well as di!erent rate constants for the liquid and vapor
multiple steady states in accordance with experimental phase reaction. This model clearly explained the multipli-
results reported in literature. city e!ects observed in their experiments.
The model presented here attempts to address the Several other features such as capillary e!ects, incom-
above requirements by relaxing many of the assumptions plete catalyst "ling and evaporation examined by Harold
used in previous models. Level I and Level II models (1988), and Harold and Watson (1993) are incorporated
discussed below are catalyst and reactor level models, in the level III model developed in this study.
which are used as the basis to formulate Level III
model as a combination of reactor and pellet-scale 2.2. Level II: reactor-scale model
models.
Kheshgi et al. (1992) developed a model based on
2.1. Level I: pellet-scale model a pseudo-homogeneous approach using the reaction sys-
tem of Hanika et al. (1976) as a basis. It was coupled with
Kim and Kim (1981) assumed that the macropores of an overall energy balance that accounts for the change
the catalyst are "lled with vapor and described intra- 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 simulta-
neously 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 reac-
tor is

d(F a)
¸  "R f #R (1!f )
dz 5 5 " 5
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 * 2.4. Level III reactor-scale yuid domain equations
(3)

where A two-#uid approach is considered for the reactor-

scale model. Equations are written for conservation of
F "F (1#N!a)#F !F (N!a)/(1!K(¹)), the gas and liquid phase mass, energy, and momentum
*  ! 
transport with source terms representing interphase
(4)
#uxes. Since the multicomponent equations involve the
F "F (1#N!a)#F !F . (5) solution of a large number of non-linear simultaneous
4  ! * equations coupled with the di!erential equations, the
The mole fractions in the vapor phase for components second-order derivatives in the di!erential equations that
A (cyclohexene), B (hydrogen), and C (cyclohexane) are represent axial dispersion are dropped. This reduces the
described in terms of vapor and liquid #ows, which are problem to an initial value system, which is more
then used to calculate liquid-phase compositions using straightforward to solve. For the reactor-level equations,
equilibrium relations. The energy balance for the pseudo- the total number of unknowns are 10nc#13 (as listed in
homogeneous mixture is Appendix A) and corresponding equations are listed
below. The numbers in square brackets given along with
d[F ( x H )#F ( y H )] nd d¹
* G G 4 G G ! R j the equations indicate the number of such equations
dz 4 dz available for a system with nc number of components.
#nd ;(¹!¹ )"0 (6) The continuity equations for the liquid and gas phase
R 5 with total interphase #uxes as the source terms are
with boundary conditions
d(o u e )
at z"0, ¹"¹ , a"0 and at z"¸, d¹/dz"0. * '* * "# N%*a M ! N*1a M [1], (8)
 dz G %* G G *1 G
(7)
d(o u e )
No distinction is made between external wetting and % '% % "! N%*a M ! N%1a M [1], (9)
dz G %* G G %1 G
internal wetting of the catalyst pellets.
The unexpanded form of the momentum equations for
2.3. Level III: reactor and pellet-scale multicomponent the liquid and gas phase with source term contributions
model from gravity, pressure drop, drag due to solid, gas}liquid
interaction and added momentum due to interphase
Level III model proposed here is a combination of transport are
a rigorous multi-component model for the reactor scale
and its extension to the pellet scale. Some simplifying d
assumptions are made to keep the numerical solution (o e u u )
dz * * '* '*
tractable while maintaining the necessary rigor. Hence,
only steady-state axial pro"les are modeled. The catalyst dp
pellets are modeled as half-wetted slabs exposed to liquid "e o g!e !F #K (u !u )
* * * dz " * %* '% '*
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.
#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 Table 2
(o e u u )
dz % % '% '%
Gas}liquid transport calculation vector

N%*!c +[b ][k][C], (x'!x*)!x (Dq)/j

dp  RJ H * * GH G H * V
"e o g!e !F #K (u !u ) :
% % % dz "% %* '* '%
%
N%* !c +[b ][k][C], (x'!x*)!x (Dq)/j


LA\ RJ H * * LA\ H G H LA\ V
#u' N%*a M ! N%1a M [1], (11) N%*!c +[b ][k ], (y%!y*)!y (Dq)/j
'% G %* G G %1 G  GE % % *H H H  W
H
:
e #e "e [1]. (12)
* % N%* !c
LA\ GE
+[b ][k ], (y%!y*)!y
% % GH H H LA\
(Dq)/j
W
F " H
The species conservation equations written with source %* y' !K x'
 ' 
terms for absolute interphase #uxes for gas}liquid, :
liquid}solid, and gas}solid transport are y'!K x'
L LA LA
d x' #x' #x' #2x' !1
   LA
(u e C )"!N%*a !N%1a [nc!1], (13)
dz '% % G% G %* G %1 y' #y' #y' #2y' !1
   LA
LA
d q !q # N%* (H%!H*)
(u e C )"N%*a !N*1a [nc!1]. (14) % * G G G
dz '* * G* G %* G *1 G LA>

Energy balance can be written for each of the three

Table 3
phases with source terms for interphase energy #ux and Gas}catalyst}liquid transport calculation vector
for heat losses to the ambient as
N*1!c +[b ][k][C], (x*!x*1')!x (Dq)/j
d(e u o H )  R' * * *H H H ' V
* '* * * "E%*a !E*1a !E*a [1], (15) H
dz %* 1* * :
N*1 !c +[b ][k][C], (x*!x*1')!x (Dq)/j
LA R' * * LA\ H H H LA\ V
d(e u o H ) H
% '% % % "!E%*a !E%1a !E%a [1],
dz %* %1 % N!.!c
 R!.
+[b ][B]\[C], ( x*1')
! H H
H
(16) :
H
E*1a #E%1a "0 [1]. (17) N!. !c +[b ][B]\[C], ( x*1')
LA\ R!. ! H H
*1 %1
N%1!c +[b ][k ], (y%!y%1')!y (Dq)/j
Auxiliary relations required to complete the set of equa-  R% % % H H H ' W
H
tions, such as equations for local-phase densities [2], and :
relations from which the ncth component concentrations F "
%*1 N%1 !c +[b ][k ], (y%!y%1')!y (Dq)/j
LA\ R% % % LA\ H H H LA\ W
can be calculated for the liquid and gas phase [2], are H
N!.!c +[b ][B]\[C], ( x%1')
listed in Appendix A for reference.  R!. ! H H
H
The above analysis shows that 2nc#6#(4 auxiliary N!. !c :
+[b ][B]\[C], ( x%1')
LA\ R!. ! LA\ H H
conditions) equations are available for (10nc#13) un- H
knowns. The additional (8nc#3) equations needed are (y' !K x' )
   %1
obtained from the interphase mass and energy transport :
relations between the solid, liquid, and gas phases (given (y'!K x' )
L LA LA %1
in Tables 2 and 3) for gas}liquid and gas}solid}liquid (x' #x' #x' #2x' ) !1
   LA %1
transport, respectively (Taylor and Krishna, 1993;
(x' #x' #x' #2x' ) !1
Khadilkar et al., 1997). The boundary conditions at the    LA *1
reactor inlet are speci"ed for the various di!erential (y' #y' #y' #2y' ) !1
   LA %1
equations that describe the model. The multicomponent LA
q !q # N%1 (H%!H1)
e!ects are incorporated while calculating the transport % 1 G G G
G
parameters and correcting them using the so-called LA
q !q # N*1 (H*!H1)
&&bootstrap'' condition given by [b] matrices (see Appen- * 1 G G G
G
dix A) using the energy balance equation at the interface
( N M /o ) "( N M /o )
as the bootstrap for all the interphase transport equa- G G *G *1' G G *G %1' LA>
tions (Taylor and Krishna, 1993; Khadilkar et al., 1997).
The transport coe$cients are also corrected for high #ux 2.5. Level III catalyst-scale equations
as given by Taylor and Krishna (1993). The activity
correction matrix for [C] is obtained from the Wilson The present model extends the approach of Harold
equation for activity coe$cients. 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 The required conditions are obtained from mass and
reaction}di!usion equations for both the gas- and liquid- energy #ux boundary conditions for the dry and wetted
"lled part of the pellet (Taylor and Krishna, 1993; interface of the catalyst. Continuity of mass and energy
Toppinen et al., 1996; Khadilkar et al., 1997). For a #uxes is also imposed at the intra-catalyst gas}liquid
half-wetted pellet with internal evaporation, the reac- interface (located at d). Identical phase temperature and
tion}di!usion problem has to be solved for the gas-"lled thermodynamic equilibrium are also enforced at the
and the liquid-"lled part of the pellet (Harold, 1988; gas}liquid interface. These are augmented by the liquid}
Harold and Watson, 1993) using the continuity condi- phase imbibition equation used to obtain the location of
tions at the intracatalyst interface and boundary condi- the intra-catalyst gas}liquid interface. These conditions
tions at the catalyst-#owing phase interface obtained are summarized in Appendix B.
from Table 3. Thus, the pellet-scale model needs to be
solved in conjunction with the reactor model proposed 2.6. Solution strategy
earlier.
The number of unknowns in this set of equations for For solution of Level III model, an IPDAE solver
an nc component system is nc values of gas and liquid (gPROMS, Oh and Pantelides, 1996) was used. The
#uxes each, nc gas and liquid compositions each, gas catalyst level equations were solved using orthogonal
and liquid temperatures (one each), interface location, collocation on "nite elements (OCFEM). The catalyst
and gas-phase total pressure (total"4nc#4). Some of coordinate was normalized using xc"x/d over the
these unknowns are expressed as di!erential equations wetted zone, and xc"(x!d)/(¸c!d) over the dry zone
(nc #ux transport relations for gas and liquid phase, length to retain invariant bounds on the independent
nc!1 liquid #ux-concentration relations, nc gas #ux- variable. Level II model (Kheshgi et al., 1992) was solved
concentration equations, and two thermal energy equa- similarly using a combination of orthogonal collocation
tions for the gas and liquid temperatures), which yields (for the two di!erential equations) and a Newton solver
(4nc!1) "rst order ODEs, two second-order ODEs, and for the algebraic equations. The rate parameters for the
two auxiliary equations (Appendix B). Thus, (4nc#3) dry and wetted pellet reaction rates were obtained from
boundary conditions are needed with one additional Kheshgi et al. (1992) as listed in Table 4. Continuation of
condition to complete the problem de"nition as listed in the dry branch pro"les for the case of multiple steady
Appendix B. The di!erential equations can be written for states was implemented by choosing the thermal con-
the species and energy #uxes in the gas- and liquid-"lled ductivity (j) for Level II model, and the degree of internal
part of the catalyst as given below (remembering here catalyst wetting (d) for Level III model. Catalyst level
that the individual species #uxes are a combination of multiplicity due to intra- and extra-catalyst heat transfer
Fickian and bulk #uxes). For the gas phase, the dusty gas limitations as reported by Harold and Watson (1993) was
model with bulk di!usion control yields independent encountered, but was not investigated in the present
equations for all the nc component #uxes with a pseudo study.
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 3. Results and discussion
analysis are
d Predictions of Level II and Level III models (referred
(N! )"l R [nc], (18) henceforth as LII and LIII, respectively) are presented
dx G% G G
and compared to literature data for the hydrogenation of
d cyclohexene to cyclohexane (Hanika et al., 1975, 1976).
(N! )"l R [nc], (19)
dx G* G G 

 
C! LA\ d
N! " G* N! ! [B ]\[C] (C! ) [nc!1], Table 4
G* C! G* * dx G* Parameter values used in the LII and LIII models
G* H GH
(20) LII model LIII model

 
C! LA d
N! " G% N! ! [B ]\ (C! ) [nc], (21) k "1.5;10\ mol/s
"
k "0.8 1/s
TQ5
G% C! G% % dx G% k "0.14 mol/s k "30 1/s
G% H GH 5
¸"0.18 m
TQ"
a "150 m/m
%*
d d N! H! d "0.03 m a "a "300 m/m
k (¹! )! G% G%"0 [1], (22) R *1 %1
C% dx % dx ;"2.8 J/m s K k "0.15 J/msK,
C*
k "0.1 J/msK
C%
d d N! H! j"0.44 J/m s K ¸ "2;10\
A
k (¹!)! G* G*"0 [1]. (23) e "0.4 R "10;10\ m
C* dx * dx N
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 (liquid-
phase) 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 multipli-
city. Referring to Fig. 2, when either the hydrogen to
cyclohexene molar ratio (parameter"N) or feed temper-
ature 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 hydro-
Fig. 3. E!ect of hydrogen to cyclohexene ratio (N) on temperature
gen to cyclohexene molar ratio is reduced, the reaction pro"les (LIII model).
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
which results in lower rates and hence a lower temper-
#ow rate cannot support the generation of cyclohexene
ature rise. In contrast to this, the upper branch in Fig. 3
and cyclohexane vapor due to equilibrium constraints.
shows that at high hydrogen to cyclohexene ratios
Both branches were simulated successfully using the
(N'8), the catalyst in the entire reactor is dry. This leads
present model (LIII) as well as the pseudo-homogeneous
to much higher rates and a greater temperature increase,
model (LII) of Kheshgi et al. (1992). Fig. 2 also shows that
which corresponds to the "ndings reported by Hanika
conversion along both branches is well-predicted by the
et al. (1976). The temperature pro"les in wet operation
present model (LIII) when compared to the experimental
observed by Hanika et al. (1976), and predicted by the
data and the pseudo-homogeneous (LII) model.
LIII model as shown in Fig. 3, both decrease with in-
creasing N. This is expected since the higher hydrogen
3.2. Ewect of hydrogen to cyclohexene molar ratio (N) on #ow rate enhances evaporation of some of the liquid and
temperature rise in wet and dry operation cools the liquid, even though it is slightly heated by the
heat of reaction. As shown in Fig. 4, both the LII and
At low hydrogen to cyclohexene feed ratios (N(6), LIII models give fairly accurate predictions of the dry
the lower branch in Fig. 3 shows that the catalyst branch temperature pro"le for a particular set of condi-
remains in an internally fully wetted condition through- tions (F "8;10\ mol/s, N"11) for which experi-
out the reactor. The reactant conversion corresponds 
mental data is published by Hanika et al. (1975). The dry
almost entirely to the contribution of wetted pellets, branch pro"les (in Fig. 3) showed a decrease in the max-
imum 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)

The reactor-scale equations for the LIII model allow

for the variation of phase holdup and velocity as illus-
trated in Fig. 5, which become more signi"cant as the
bubble point of the reaction liquid mixture is ap-
proached. 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 pres-
ent, 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 inter-
nal wetting of the catalyst pellet (dP¸ ). Fig. 7 also
Fig. 2. Multiplicity behavior: conversion dependence on hydrogen to A
cyclohexene ratio. shows that the intra-catalyst temperature rise for the
 M.R. Khadilkar et al. /Chemical Engineering Science 54 (1999) 2421}2431 2427

Fig. 4. Comparison of experimental and predicted temperature pro"les

in dry operation (LIII model). 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 wet-

branch operation (cyene: cyclohexene, cyane: cyclohexane).

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
Fig. 6. Axial variation of liquid-phase concentration of components. #ux pro"les in the pellet are similar to those of hydrogen
in shape, but exhibit negative values at the reactor en-
liquid-full zone, which corresponds to the low rate trance due to condensation on the internally dry side and
branch operation, did not exceed 5 to 63C at all reactor transport to the liquid}solid interface. Only positive cyclo-
locations. hexene #ux values are seen downstream in the reactor
Fig. 8 shows the multicomponent intra-catalyst #uxes where the pellet contains a high concentration of the
for the various species at several axial locations. 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 the gas velocity, pressure, and concentration pro"les
in Fig. 8 for the reactant (cyclohexene) and product undergo signi"cant variation along the reactor. Among
(cyclohexane). This represents the initiation point of these, the gas velocity undergoes the greatest variation
intracatalyst drying, since the temperature exceeds the due to mass transfer e!ects and heat generation due to
boiling point of the liquid mixture. reaction.
The concentration pro"les on the reactor scale and the
3.4. Dry-branch simulation pellet scale are shown in Figs. 10 and 11, respectively.
These suggest that cyclohexene is limiting when a high
At high hydrogen to cyclohexene molar ratios hydrogen to cyclohexene molar ratio is used (N"8) and
(N'6.3), the reaction occurs completely in the vapor- only the gas-phase reaction is occuring. Fig. 12 shows
"lled pores of the catalysts with very high reaction rates that a moderate pressure buildup occurs inside the pellet
(Table 4). This was simulated in the present model (LIII) during dry-branch operation so that a non-isobaric
by setting the intracatalyst gas}liquid interface location condition exists. For reactions having a net reduction in
to d"0, eliminating the liquid-phase conservation equa- the total number of moles, a decrease in the centerline
tions and corresponding exchange terms, and setting the pressure over the bulk pressure is expected (Taylor and
gas holdup equal to bed porosity. To simplify the numer-
ical solution, the pellet-scale equations were solved by
imposing symmetry conditions. Figs. 9 and 10 show that

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 Fig. 12. Intra-catalyst pressure pro"les at di!erent axial locations
(LIII model). (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 dimin-
ish 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 pre-
dict with a heterogeneous (LIII) model, since the liquid-
phase conservation and exchange equations collapse
at the transition point. Numerical problems were en-
countered during Level III model simulation of the
Fig. 14. Simulated vapor-phase compositions and catalyst wetting for
abrupt transition from d"¸ to d"0 as reported by
A intra-reactor wet-to-dry transition (LII model) (cyane: cyclohexane,
Hanika et al. (1976) and Harold and Watson (1993). Since cyene: cyclohexene).
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 maximum temperature rise in this case was between
The phenomena of interest associated with intra-reac-
that observed for the wet and dry branch due to usage of
tor phase transition were simulated using Level II model
some of the heat of reaction for evaporation of the liquid
by introducing the feed in the reactor under transition
and the transition to dry operation.
conditions (N"7) and examining the change from the
Fig. 14 shows how the fraction of catalyst particles
liquid to the vapor-phase reaction. Fig. 13 shows that the
that are wetted changes from being almost completely
liquid #ow rate approaches zero near the reactor inlet
wetted to a vapor-"lled state. The mole fraction of both
because the heat of reaction and the high hydrogen #ow
cyclohexene and cyclohexane in the vapor phase in-
rate, resulting in complete vaporization of the liquid
creased slightly at the reactor inlet as a result of liquid
reactants and products. The temperature rise until this
evaporation and reaction followed by transition to the
point was also negligible (corresponding to a near
gas phase, with a decrease in cyclohexene and hydrogen
isothermal phase change), after which the gas-phase
mole fraction. This is the expected behavior in the dry
reaction proceeded downstream at a much higher rate.
rate branch of the reaction.

4. Summary and conclusions

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 incorpor-
ated rigorous multicomponent mass and energy trans-
port on the reactor and catalyst level, and was shown to
give accurate predictions of the conversion and temper-
ature 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 equa-
tions 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
Fig. 13. Simulated #ow and temperature pro"les for intra-reactor wet- and is recommended for future models for complex reac-
to-dry transition (LII model). tion 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 A ambient

sponsors of the Chemical Reaction Engineering Laborat- A, B, C components
ory (CREL). B bed voidage
C catalyst
D, G, < dry, gas, or vapor phase
i species number
Notation I interface
¸, = wet, liquid phase
a interfacial area per unit volume of reactor S solid phase
[B] mass transfer coe$cient matrix t total
C concentration
D e!ective di!usivity
C
d reactor diameter
R References
E energy transfer #ux
f fractional wetting
U Collins, G.M., Hess, R.K., & Ackgerman, A. (1985). E!ect of volatile
F #uid}solid drag
B liquid phase on trickle-bed reactor performance. Chem. Engng
F molar #ow Commun., 35, 281}291.
g gravitational acceleration Dudukovic, M.P., & Mills, P.L. (1986). Contacting and hydrodynamics
h heat transfer coe$cient of trickle-bed reactors. Encyclopedia of -uid mechanics, (pp.
h Henry's constant 969}1017). Huston: Gulf Publishing Company.
H enthalpy Hanika, J., Sporka, K., Ruzika, V., & Krausova, J. (1975). Qualitative
observations of heat and mass e!ects on the behavior of a trickle-
[k] mass transfer coe$cient bed reactor. Chem. Engng Commun., 2, 19}25.
k e!ective thermal conductivity of catalyst
C Hanika, J., Sporka, K., Ruzika, V., & Hrstka, J. (1976). Measurement of
k reaction rate constant axial temperature pro"les in an adiabatic trickle-bed reactor. Chem.
K equilibrium constant Engng J., 12, 193}197.
K interphase momentum transfer coe$cient Harold, M.P., & Watson, P.C. (1993). Bimolecular exothermic reaction
%* with vaporization in the half-wetted slab catalyst. Chem. Engng Sci.,
¸ length 48, 981}1004.
M molecular weight of species i
G Harold, M.P. (1988). Steady-state behavior of the non-isothermal par-
N hydrogen to cyclohexene molar feed ratio tially wetted and "lled catalyst. Chem. Engng Sci., 43, 3197}3216.
N #ux at the interface Jaguste, D.N., & Bhatia, S.K. (1991). Paritial internal wetting of catalyst
P pressure particles: Hysterisis e!ects. A.I.Ch.E. J., 37, 661}670.
Khadilkar, M.R. (1998). Performance studies of trickle-bed reactors.
q heat #ux D.Sc. Dissertation, Washington University, St. Louis, Missouri, USA.
R apparent reaction rate Khadilkar, M.R., Al-Dahhan, M.H., & Dudukovic, M.P. (1997). Simu-
R intrinsic reaction rate in catalyst pellet lation of unsteady state operation in trickle-bed reactors. A.I.Ch.E.
G Annual Meeting, Los Angeles, CA.
S catalyst external surface area
V Kheshgi, H.S., Reyes, S.C., Hu, R., & Ho, T.C. (1992). Phase transition
¹ temperature
and steady-state multiplicity in a trickle-bed reactor. Chem. Engng
u interstitial (actual) velocity
' Sci., 47, 1771}1777.
; reactor to wall heat transfer coe$cient Kim, D.Y., & Kim, Y.G. (1981). Experimental study of multiple steady
x mole fraction, intra-pellet spatial coordinate states in a porous catalyst particles due to phase transition. J. Chem.
y mole fraction in the gas phase Engng Japan, 14, 311}317.
z axial coordinate Kocis, G.R., & Ho, T.C. (1986). E!ects of liquid evaporation on the
performance of trickle-bed reactors. Chem. Engng Res. Dev., 64,
288}291.
LaVopa, V., & Satter"eld, C.N. (1988). Some e!ects of vapor} liquid
Greek symbols equilibria on performance of a trickle-bed reactor. Chem. Engng Sci.,
43, 2175}2180.
a conversion (based on cyclohexene) Mills, P.L., & Dudukovic, M.P. (1980). Analysis of catalyst e!ectiveness
in trickle-bed reactors processing volatile or non-volatile reactants.
[b] bootstrap matrix in Maxwell}Stefan formulation Chem. Engng Sci., 35, 2267.
c, [C] activity coe$cient, matrix form Oh, M., & Pantelides, C.C. (1995). Modeling and simulation language
d intracatalyst gas}liquid interface location for combined lumped and distributed parameter systems. Comput.
e phase holdup Chem. Engng, 20, 611}633.
l stoichiometric coe$cient of component i Satter"eld, C.N. (1975). Trickle bed-reactors A.I.Ch.E. J., 21, 209}228.
G Taylor R., & Krishna, R. (1993). Multicomponent mass transfer.
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V
 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 Number of equations at the catalyst level"4nc#4.
(2), holdups (2), pressure (1), nc liquid and gas phase Boundary conditions (at the catalyst-bulk -uid bound-
concentrations, 3 temperatures (gas, liquid, and solid), ary): Liquid}solid boundary
densities of gas and liquid (2). Total unknowns"
N* "N*1 [nc!1], (B.14)
2nc#10. !V
Interphase transport (3 interfaces, G}¸, G}S, and Similarly the energy #ux boundary condition
¸}S): For each interface, the unknowns are nc #uxes, nc
d¹! L
liquid, and nc vapor interface compositions and the inter- !k *# N!* H* (¹!)
C* dx G G *
face temperature. Total"3nc#1 (G}L)#3nc#1 G
(G}S)#2nc#1 (L}S)"8nc#3. L
"h (¹ !¹!)# N*1 H* (¹ ) [1]. (B.15)
Auxiliary equations: *1 * * G G *
G
C R¹ "P, (A.1) Gas}solid boundary (nc conditions can be used due to
G% E dusty gas model)
C M /o "1, (A.2)
G* G *G N% "N%1 [nc]. (B.16)
C M "o , (A.3) !V*A
G% G % Energy #ux boundary condition
C M "o . (A.4)
G* G * d¹! L
Bootstrap matrix [b] ( for liquid phase based on energy !k %# N!* H* (¹! )
C% dx G G %
flux): G
L
[b ] "d !x K , (A.5) "![h (¹ !¹! )# N%1 H% (¹ )] [1]. (B.17)
GI * GI G I %1 % % G G %
K "(j !j )/j , (A.6) G
GI I LA V Relationships between variables at the gas}liquid
j " x j , j "x (H4 (@¹ )!H* (@¹ )). (A.7) intracatalyst interface:
V G G G G G % G *
G
N* "N% [nc], (B.18)
Mass transfer coe.cient matrix: !VB !VB


C! 2p< cos h C!
y y G% "K exp ! G* [nc], (B.19)
[B ]" G # I (for i"j), (A.8) C! G R R¹! C!VB
GH k k G% N * G*
G LA I HI
[1], (B.20)


¹! "¹!
1 1 * VB % VB
[B ]"!y ! (for iOj). (A.9)
GH G k k d¹! L
GH G LA !k %# N!* H* (¹! )
C% dx G G %
Enthalpy of gas and liquid phase: G
L d¹! L
"!k *# N!* H* (¹!) [1]. (B.21)
H " C (DH*#C (¹ !¹ ))/o , (A.10) C* dx G G *
* G* G .G * 0D * G
G
L ¸iquid imbibition velocity:
H " C (DH%#C (¹ !¹ ))/o . (A.11)
% G% G .G % 0D % v"N! /C!
G R* R*
Activity correction matrix: "(R/8dk )(P !P #2p cos h/R ) [1].
N * V*A VB .
* ln c (B.22)
C "d #x G. (A.12)
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 ). (A.13)
% % ' G G %
G