An Evaluation of Models for Liquid Backmixing in Trickle-Bed Reactors John G. Schwartz and George W. Roberts*’ Chemical Reaction Engineering Laboratory, Washington University, Saint Louis, Missouri 63130
Proposed models of liquid backmixing in trickle-bed reactors are reviewed, and the utility of the various mixing models in reactor design is assessed by comparing the reactor behavior predicted by each model for a first-order heterogeneous reaction. The results show that, over a range of typical trickle-bed conditions, predictions based on the dispersion model differ very little from those based on more complex, two-parameter models. Significant differences occur only at high degrees of backmixing and at high reactant conversions. Analysis of several previously measured residence-time distributions from pilot-scale and commercial trickle-bed reactors indicates that the assumption of plug flow of liquid frequently represents the actual situation quite well. In those cases where it is desirable to account for liquid backmixing, the dispersion model appears to b e an adequate representation.
I n essence, a trickle-bed reactor consists of a fixed bed of catalyst particles with liquid flowing downward through the bed and vapor flowing either countercurrent or cocurrent to the liquid. The rates of the liquid and vapor are such that the vapor phase is continuous. Trickle-bed reactors are used extensively throughout the chemical and petroleum industry, perhaps the largest single application being in the hydrodesulfurization and hydrocracking of heavy petroleum feedstocks. Furthermore, as a result of the need to meet expected future demand for larger quantities of low-sulfur fuel oil, probably containing even lower levels of sulfur than are presently tolerable, the trickle-bed reactor will play an important role in efforts to improve the quality of the air environment. Unfortunately, the trickle-bed reactor is among the most difficult of commonly used industrial reactors to design in an a priori sense. Several published studies (Ross, 1965; Murphree, et al., 1964) also suggest that the techniques of scaleup from smaller units have not been completely mastered. The potential difficulty involved in designing commercialscale trickle-bed reactors may have led, in some cases, to the use of slurry reactors in place of trickle beds, despite the obvious disadvantages of catalyst recovery and liquid backmixing that are inherent in the slurry reactor. One of the factors that can influence the behavior of tricklebed reactors is backmixing in the liquid phase. The “dispersed plug flow” or “dispersion” model of Levenspiel and Smith (1957) has been used to represent liquid flow in trickle-bed reactors, and axial dispersion coefficients for trickle beds have been measured and correlated by several investigators. However, much recent effort has been directed toward deriving more exact mathematical models of the residence-time distribution (RTD) of the liquid phase (Buffham, et al., 1970; Deans, 1963; Hochman and Effron, 1969; Hoogendoorn and Lips, 1965; Van Swaaij, et al., 1969), and these models do seem to provide a closer fit of experimentally measured RTD curves than the dispersion model. Three of the models that have been developed to supersede the dispersion model, specifically the “modified mixing-cell” 1 Present address, Engelhard Industries Division, Englehard Minerals and Chemicals Corporation, Menlo Park, Edison, N. J.
08817. 262 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
model (Deans, 1963), the “crossflow” model (Hoogendoorn and Lips, 1965; Hochman and Effron, 1969), and the “time delay” model (Buffham, et al., 1970) require two adjustable parameters to describe liquid backmixing, in contrast to the one adjustable parameter involved in the dispersion model. A fourth model (Van Swaaij, et al., 1969) requires three adjustable parameters. Considerably more mathematical and experimental effort is required to accurately extract two or more parameters from measured RTD curves than is needed to obtain the single parameter of the dispersion model. Furthermore, although correlations of the axial Peclet number are readily available (Furzer and LIichell, 1970; Hochman and Effron, 1969; Sater and Levenspiel, 1966), only one attempt has been made to develop general parameter correlations for a two-parameter model (Hochman and Effron, 1969). Therefore, a t least a t present, the practicing engineer is handicapped in the effortto apply the two-parameter backmixing models to reactor design and analysis. I n view of the difficulties involved in developing and applying two-parameter models of liquid backmixing in tricklebed reactors, it is important to inquire whether predictions of reactor behavior, based on these models, will differ significantly from predictions based on the simpler dispersion model. This question is addressed in the present study. Procedure
If the liquid flowing through a continuous reactor is completely segregated, Le., there is no mixing on a molecular scale, then, for a homogeneous reaction, the fractional conversion of reactant A in the effluent from the reactor will be given by (Levenspiel, 1962) X
=
1-
Lrn
C*(t)E(t) dt
(1)
where X is the fractional conversion of reactant A, E ( t ) is the residence-time distribution for the reactor, and C*(t) is the concentration of A that would exist in a batch reactor having the same initial conditions and temperature-conversion profile as an individual element of liquid in the continuous re-
actor. =Ilthough eq 1 is not exact for a reactor in which some molecular mixing does take place, the difference between the predictions of this equation and those based on a more fundamental description of mixing is frequently small. Therefore, for homogeneous reactions, eq 1 probably provides a reasonable basis for analyzing the effect of differences in the residence-time distribution, E(t), on reactor performance. Such an analysis, which would be applicable to heterogeneous systems, is the objective of the present study. Attempts have been made to apply eq 1 directly to tricklebed reactors (Murphree, el al., 1964; Cecil, et d . , 1969). However, these applications appear to involve a misinterpretation of the proper residence-time distribution to be employed for a heterogeneous reactor. For this reason, and in order to emphasize the assumptions that are involved in treating a heterogeneous reactor by means of a segregated-flow model, the analog of eq 1 is now developed for a trickle-bed reactor. First, the reactor is visualized as a system of nonidentical parallel tubes. A part of the total liquid stream passes through each tube in plug flow; there is no exchange of liquid between tubes. The gth tube is a portion of the bed which contains a weight, ?I-,, of catalyst and which has liquid flowing through it a t a volumetric rate, Q,. At steady state, this tube contains a volume, V L j ,of liquid. The volume of liquid inside the catalyst particles is designated VI, and the volume external to the particles is VQ,. Because of assumptions to be introduced later, there is no need to define the volumes and flow rate of the gas phase. The total residence time of the liquid phase in the j t h tube is
tLj
=
VLJ/&~
(24
and the residence time of the liquid external to the particles is
V~5/&5 The space time for the j t h tube is defined as =
T~
E
W j / Q j p= ~ Wj/Mj
=
l/(WHSV)j
(-TA‘)
dWj
-
’:sc,”
(5)
where is the average concentration of reactant A in the overall reactor effluent. Although eq 5 is directly analogous to eq 1 for a homogeneous reactor, its application to an actual heterogeneous reactor is not straightforward. For instance, although the residence-time distribution, E(t), may easily be measured for a trickle-bed reactor, experimental techniques for determining E ( s ) ,the distribution of space times, have yet to be developed. Therefore, a model is needed in order to relate E ( T )and E(t). In the following development, it is assumed that (1) liquid holdup is due only to catalyst particles; holdup on the walls of the reactor is assumed negligible; (2) the thickness of the liquid film surrounding the catalyst particles is constant, and does not vary from point to point on the catalyst surface, or from particle to particle throughout the bed; (3) the bed is uniformly packed. With these assumptions, t can be related to r through the external component of the liquid holdup. The total liquid holdup, HT (volume of liquid/volume of bed), in the overall reactor is composed of external and internal components, where “external” and “internal” refer to the porous catalyst particles in the same sense as used previously with liquid volumes. Thus
HT
=
H E iHI
The volume of liquid external to the catalyst pellets in the j t h tube is related to the external holdup, H E , by vE5 =
HEVT~
The total volume of the j t h tube,
vT5,
is
(2c)
(3)
where -rA’ is the rate equation for the disappearance of A. Following normal practice for heterogeneous catalytic reactions, - r A ’ is based on a unit weight of catalyst, rather than on a unit volume of fluid as for homogeneous reactions. The rate equation is taken to include both internal and external transport effects,so that, in essence, eq 3 is a material balance on the liquid external to the catalyst particles. Integrating eq 3 over the complete length of t u b e j 1V3/Qj =
+
(2b)
where p~ is the density of the liquid, M j is the liquid mass flow rate in the j t h tube, and (WHSV), is the weight hourly space velocity for the j t h tube. .A steady-state material balance on reactant A for a differential element of tube j may be written as &,(-dCA) =
space time, such that the fraction of liquid in the reactor effluent with space times between r j and ( r j dTj) is equal to E ( r j )drj, i t follows that
dCA/ (-rA’)
where CAOis the concentration of reactant -4in the feed to the reactor and C A is ~ the concentration of -4in the effluent from the j t h tube. Dividing by p~ gives
PB
where p~ is the density of the catalyst bed. Thus
Wj V E ,= H E PB
Using eq 2b and 2c in conjunction with that above, r can be related to the external residence time. Thus
or, in differential form
Note that this equation will not result from differentiation of eq 6 unless H E and P B are invariant throughout the reactor. The distribution functions E ( ~ Eand ) E ( r ) are normalized, and therefore
Lm
E ( ~ EdtE )
(4)
=
1
=
or Equation 4 defines CAj as a function of r j . Letting the number of tubes in the reactor become very large, and letting E(7J be the distribution function for the Ind. Eng. Chcm. Process Des. Develop., Vol. 12, No. 3, 1973
263
The above equation is generally true only if
E ( r ) = (-)E(tE) PLHE PB
and therefore
E ( r ) d r = E ( ~ E dtE )
(7)
Despite the appealing simplicity of eq 7, i t should be recognized that this relationship has been developed for a highly idealized system. In an actual reactor, there may not be a unique relationship between r and t , due to packing defects, wall effects, variations in film thickness, partial wetting of catalyst particles, etc. Substitution of eq 6 and 7 into eq 5 gives
small variations in k, with CA, due to variations in the transport properties, are neglected, the pseudo-rate constant K‘ will be concentration independent, and the reaction will continue to appear to be first order, even though internal and external concentration gradients may be significant. For a reaction, whose kinetics obey eq 9, taking place in an isothermal reactor, eq 4 may be integrated and rearranged to yield C A j = CA&
--K’r,pL
= cAOe-K7,
(10)
Substitution for ri in terms of t ~ (eq , 6) and introduction of the resultant into eq 8 gives =
(1 -
CAO
x)=
L-
e-(KPB/HEPL)tEE(tE)
dtE (11)
The quantity (KPBIHEPL) in the above expression is a pseudohomogeneous first-order rate constant. Equation 8 shows that a segregated-flow model may be applied to trickle-bed reactors, providing that certain assumptions concerning the liquid holdup are satisfied, and providing that the residence-time distribution used in the calculation is that of the liquid external to the catalyst particles, i.e., the residence-time distribution that would be observed with a nonporous packing. The latter restriction has apparently not been recognized in several previous studies (Murphree, et al., 1964; Cecil, et al., 1969), where total residence-time distributions, which include not only the external residence time, but also the residence time in the pores of the catalyst, were used. The calculations that follow are based on the assumptions that the intrinsic kinetics of the disappearance of reactant A are first order, that reactant A is essentially confined to the liquid phase (negligible vapor pressure), and either that the reaction rate does not depend significantly on the concentration of any species found predominantly in the gas phase, or that the gas composition is essentially constant throughout the reactor. A number of industrially important reactions, for which trickle beds are currently employed, appear to follow first-order kinetics, with respect to a liquid-phase reactant (Murphree, et al., 1964; Ross, 1965; Cecil, et al., 1969; Bondi, (1971)), providing some justification for these assumptions. If the reaction rate does depend on the concentration(s) of components which reside primarily in the gas phase, and if these concentrations are not constant throughout the reactor, proper formulation of reaction kinetics requires that gasliquid interchange be considered in order to account for replenishment of depleted reactant from the gas phase. Under this circumstance, eq 8, which considers an isolated element of fluid, will not be appropriate. For a first-order reaction obeying these assumptions, eq 8 is rigorously true and independent of the actual scale of mixing. If temperature gradients inside the catalyst particles and between the bulk liquid and the catalyst surface are negligible, the overall rate equation is given by
where 7 is the effectiveness factor of the catalyst particle, K, is the intrinsic first-order rate constant, Wp is the weight of a catalyst particle, a is the external surface area of a catalyst particle, k , is the mass transfer coefficient, based on concentration, between the bulk liquid and the catalyst surface, and CAis the concentration of A in the bulk liquid. For a first-order reaction with no intraparticle temperature gradients, 9 is independent of concentration. Therefore, if 264 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
Residence-Time Distribution Models
A. Dispersion Model. The dispersion or dispersed plug flow model (Levenspiel, 1962) is based on the assumptions that (1) liquid flows through the reactor with a flat velocity profile, and (2) the rate of exchange of component A in the axial (x) direction between adjacent fluid elements is given by -D dCA/dx, where D is the axial dispersion coefficient, For this model, the degree of backmixing is characterized by the dimensionless variable D/uLL, commonly called the dispersion number, which is also the inverse of the axial Peclet number. For an ideal plug flow reactor, D/uLL = 0, while for an ideal backmix reactor, D/UL = m . The value of D/uLL for a given reactor may be obtained from the variance, uO2,of the experimentally measured residence-time distribution for the reactor, as shown by ugZ = ( l / t m ) 2 ~ m -(tm)2E(t) t dt = ~ ( D / u L L ) 0
2 ( D / u ~ L ) ~ [-1 exp(-uLL/D)]
(12)
In eq 12, t , is the average residence time of the liquid in the reactor, as given by the normalized first moment of the RTD curve. Correlations of the particle Peclet number, uLd/D, which is also called the Bodenstein number, have been developed for trickle-bed reactors by a number of investigators (Sater and Levenspiel, 1966; Furzer and Michell, 1970; Hochman and Effron, 1969; Van Swaaij, et al., 1969). In general, the observed particle Peclet numbers for trickle-bed reactors are significantly lower than for single-phase liquid flow through packed beds. At low Reynolds numbers (NR, < lo), the particle Peclet number approaches an asymptotic value of about 0.5 with single-phase flow, whereas, with trickle flow a t the same Reynolds numbers, the particle Peclet number is in the region of 0.1-0.2. An analytical expression for the fractional conversion of a first-order reaction, taking place a t steady state in a dispersed plug flow reactor, has been developed by Wehner and Wilhelm (1956), and is given by
l - X = - =C A CAO
4a’ exp(u~L/2D)
+ a ’ ) 2exp(a’u~L/2D)- (1 - a’)2 exp(-a’u~L/2D) where a’ = dl + 4 K r ( D l u ~ L ) . (1
(13)
B. One Model with Three Names. (1) Modified Mixing-Cell Model. Deans (1963) suggested that the
longitudinal mixing which accompanies flow through porous media could be accurately described by modifying the “perfectly mixed cells in series” model to take into account the possibility of mass transfer between stagnant and flowing regions within a given cell. As N , the number of cells in series, is increased a t constant total reactor volume, each cell becomes an element on the order of magnitude of one particle diameter, d . For this case, the modified mixing cell model may be represented by the differential equations (1
+ UL-bCA + ~ ( C A- CA*) = 0 dX
-j)
(14a)
points along their path. The delayed molecule eventually rejoins the flowing stream after a period of time, the time delay, has elapsed. An individual molecule may experience a number of delays in the course of traversing the bed. The developers contend that such a probabilistic model more realistically describes the physical basis for the spread of residence-time distributions in trickle-bed reactors than diff usion-based models. If i t is assumed that the delay times are distributed ex, the response to ponentially, with a mean delay time, t ~then a pulse input of tracer may be shown by a rather detailed stochastic argument to be
E(t) = 0 t I n these equations, CA is the concentration of species A in flowing liquid, CA* is the concentration of A in stagnant liquid, j is the fraction of liquid which is stagnant, and k is the mass-transfer coefficient between flowing and stagnant liquid. For an experiment in which a tracer is injected as a sharp pulse a t the inlet to the reactor (z = 0), eq 14a and 14b are subject to the boundary and initial conditions CA(0, t )
= 6(t)
CA(x, 0 ) = CA*(x, 0 ) = 0 where 6 is the Dirac delta function. The solution 15, evaluated a t x = L , consists of two parts
E(t) = 0 t E(t) = exp(-kt,)d(t
- ti)
< ti
+ X
where Il(z) is the first-order Bessel function of the first kind with imaginary argument and ti is the time a t which tracer first appears in the reactor effluent, Le., the initial breakthrough time for the tracer. This time is related to the stagnant fraction, f, by
f
=
1 - (ti/trn)
(17)
Equation 16 presupposes a typographical error in the original paper, and is assumed correct since it is consistent with the calculations presented therein (Deans, 1963). This model requires two arbitrary parameters, f and k , to describe the extent of liquid backmixing. The relationships between f and k and the characteristics of the experimental residence-time distribution curve have been developed by Van Deemter (1961) and interpreted by Hoogendoorn and Lips (1965). Equation 17 shows that f may be calculated from the observed initial breakthrough time and the first moment of the R T D curve. Furthermore, since the variance of the R T D is given by *.z
= 2f2/ktm
(18)
the value of k can also be determined from the experimental R T D curve. (2) Time Delay Model. The time delay model (Buffham, et al., 1970) is based on the abstraction that liquid would flow in plug flow through a trickle-bed reactor, except for the fact that molecules have a chance of being “delayed” a t many
< ti
t
2
ti
(19b)
where n is the total number of delays that a tracer molecule experiences in traversing the bed. Two parameters, t i and t ~ are , required to describe liquid backmixing with this model. The parameter ti may be estimated as the time taken for initial breakthrough of tracer, or may be related to the time taken to reach a normalized concentration of 0.05 (Buffham, et al., 1970). The parameter t~ can be estimated from the peak height (Buffham, et al., 1970) or from the variance of the experimental RTD curve. Although the forms of the residence-time distribution for the time delay and modified mixing cell models appear to be quite different, these solutions may be shown to be identical by suitable redefinition of parameters and by recognizing that I ( z ) = 2,,l”(z/2)2n-1/n! (n - l)!. Thus, the time delay and the modified mixing cell models are mathematically equivalent. (3) Crossflow Model. A model which is based on the assumption that the liquid phase is divided into two regimes, stagnant and plug flow, has been used by Hoogendoorn and Lips (1965) and more recently by Hochman and Effron (1969). Possible areas where stagnant liquid might exist in trickle beds are a t the top of packing particles, in the interstices between tightly packed particles, and between the packing and the wall of the reactor. I t is assumed in this “crossflow” model that mass can be exchanged between the stagnant and flowing portions of liquid. Mass balances on the plug flow and stagnant liquid volumes, over a differential element of reactor, yield
where $ is the fraction of liquid which is in plug flow arid k is the mass transfer coefficient between stagnant and flowing liquid. There is an obvious similarity between the equations of the crossflow model and those of the modified mixing cell model. With suitable redefinition of parameters, it can be easily shown that these differential equations are mathematically identical. Thus, the solution for the modified mixing cell model is identical with the solution of the crossflow model, for the same set of boundary conditions. Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
265
IO
0.a
0 OISPLRS10N YOOLL, A CROS8FLW YOOtL
0 Dl8PCRalou MOOCL. D/U,L. A ClOnSfLOw YODEL
0.8
0.1nzo
n r u n FLOW 11 -01
0 P L U O I L W 11.01
E
E:
0.08
:
-
001 0.0 0 0
Figure 1 . Comparison of dispersion and crossflow models for us2 = 0.05 I O
05
0 Dl~PLR11101MODEL. O/u,L A CRO88rLOW Y W E L P L u e ~ ~ oII.OI w
-
0 00
.: Y
00s
Figure 2. Comparison of dispersion and crossflow models for ug2 = 0.1 0
+
The parameters of the crossflow model, and k , can be obtained from the experimental RTD curve in essentially the same manner as previously described for the two parameters of the modified mixing cell model. It has been shown (Hochman and Effron, 1969; Hoogendoorn and Lips, 1965) that the parameter @, as calculated from the RTD curve, agrees within experimental error with the ratio of the dynamic to the total external liquid holdup. This agreement lends physical support to the crossflow model. I t is evident from the previous discussion that the three models (the modified mixing cell model, the time delay model, and the crossflow model) are mathematically equivalent in that they all lead to the same expression for the residencetime distribution, E ( t ) . For convenience in the subsequent discussion, these three equivalent models will be referred to as the "crossflow" model. C. Conversion Calculations for Crossflow Model. The fractional conversion, X, for a first-order reaction taking place in an isothermal trickle-bed reactor obeying the crossflow model may be calculated by substituting the expression for E ( t ) , eq 16, into eq 11 and performing the indicated integration. As shown in the Appendix, the result is
266
tt.0 o
aa.0 o
40
aa.0 o
n.0 no
7.0 7 0
nn.0o
90
10.0 100
Figure 3. Comparison of dispersion and crossflow models for as2 = 0.30
As f,the fraction of the fluid which is stagnant, approaches zero, (1 - X ) + exp(-Kr), which is the expression for an ideal, plug flow reactor. This result is expected in light of the fact that, in the crossflow model, all of the flowing liquid is assumed to be in plug flow. As f approaches 1.0, an almost completely stagnant liquid volume, (1 - X ) + exp{ -I/ [(l/ktm) (l/Kr)]]. Clearly, this expression represents two resistances in series: a resistance to mass transfer from flowing to stagnant liquid and a reaction resistance. When kt, is very large, implying that mass transfer is very rapid between stagnant and flowing liquid, (1 - X ) -P exp(-KT), which again is the ideal plug flow expression. This is reasonable, since, with very rapid mass transfer, the reactant concentration is the same in the flowing and stagnant liquid. Conversely, when K r is very large, (1 - X)+ exp(-kt,), suggesting that mass transfer from stagnant to flowing liquid controls the reaction rate. It should be noted, however, that the case off = 1.0 (mainly stagnant liquid in the reactor) is never observed under normal trickle-bed operation and could only occur if the liquid mass velocity were extremely small.
+
01
8
e
IO
Ind. Eng. Chem. Process Der. Develop., Vol. 12, No. 3, 1973
Results
A. Parametric Studies. The equations for predicting the fractional conversion via the dispersion and crossflow models, eq 13 and 21, respectively, were programmed for evaluation on an IBM 360/50 computer. For the dispersion model, the fraction of unconverted reactant, (1 - X ) , was calculated for a wide range of values of K r and D / u d . For the crossflow model, (1 - X) was calculated as a function of the three parameters, K r , f , and kt,. Figures 1-5 show the results of these calculations. Each of these figures is a plot of In (1 - X ) us. K T a t constant variance ue'. For the dispersion model, reactor behavior is completely defined by K T and uo2.However, for the crossflow model, f must be specified in addition to K r and (rez. Therefore, Figures 1-5 show families of curves for different values off. Figures 1-5 cover a range of uo2 from 0.05, a situation very close to plug flow, to 0.97, almost complete backmixing. This range is considered to be more than sufficient to encompass all possible values of variance from actual trickle-bed reactors. The experimental reactors examined in this study were found to have variances ranging from 0.015 to 0.132. Unfortunately, all of the RTD data from commercial reactors that has been published to date includes intraparticle diffusion, and therefore exhibits higher variances than would be expected for the
Table 1. Values of f for which Dispersion a n d Crossflow Models are Nearly Superimposible f
u:
0.1-1.0
0.01 0.05 0.10
f
0:
0.6 0.7 0.9
0.30 0.50 0.97
0.4-0.6 0.5
0 OISPIISION Y O D I L . O/vLL. 0.014 A CnoasfLw YOOLL. 1.0.44
0 PLUO FLOW 11.01
KT
Figure 4. Comparison of dispersion and crossflow models forue2 = 0.50
oa
0 DISPCMION YODEL. 0 b L L . 10.18 CRWSFLOW YOOEL
A
0 PLU@FLOW lf.0)
K I
e
s E
0.1
Figure 6. Comparison of dispersion and crossflow models for experimental residence-time distribution data of Buffham, et a/. (1 970),run no. 1
005
L
Figure 5. Comparison of dispersion and crossflow models foruo2 = 0.97
external liquid distribution alone. However, the total R T D variances for commercial reactors are of the order of 0.60, which is within the range of this parametric study. A significant result of these graphs is that, over the entire range of 6 g 2 , the conversion predicted from the dispersion model always lies within the range of that predicted from the crossflow model. A more detailed examination suggests that, for any value of ug2, there is a value of f for which the dispersion model and the crossflow model are nearly superimposible. Table I shows the approximate value off which gives a reasonably close fit to the dispersion model as a function of ug2. These values were determined by visual curve matching, and not by any statistical procedure. Since the ratio of the static holdup to the total holdup, which in theory is equal to f, has been found to lie in the region of 0.1-0.5 in a number of experimental studies, the results shown in Table I suggest that the predictions of the two models will agree quite closely when the adjustable constants in each model are determined by fitting experimental RTD data, a t least a t low values of u02. B. Case Studies. (1) Experimental Reactors. Experimental R T D data from laboratory or pilot-scale trickle-flow reactors were obtained from two previous publications (Buffham, et al., 1970; Hoogendoorn and Lips, 1965).
0.01 0.0
os
1.0
15
20
2.5
3.0
3.5
4.0
45
50
Kf
Figure 7.Comparison of dispersion and crossflow models for experimental residence-time distribution data of Hoogendoorn and Lips ( 1 965). Data used is from experiment with USL = 0.007
The information provided in these references was sufficient to allow ug2 and f to be accurately determined. Experimentally determined values of the dispersion number, D/u&, and the dimensionless breakthrough time, tilt,, are given in Table I of the paper by Buffham, et al. (1970), for a number of different sets of experimental conditions. Using this data, a value of u g 2 was calculated for each run from eq 12. The value of f for each run was calculated using eq 17, and kt, was then calculated from eq 18. Table I1 of the paper by Hoogendoorn and Lips (1965) contains experimentally determined values of parameters 4 and 12k (= kt,) for several experimental conditions. The parameter f was calculated Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
267
I O
1.0
0 DISPLRSION YODEL, D / r L L * D . l D 1 l
01
A cm)awLow YODCL. f.0.w 0 *tu0 FLOW lf.01
0.1
f
I
01
g
0.01
8
1 0 . 1
I
E
0.01 00
1.0
1.0
3.0
1.0
4.0
6.0
7.0
1.0
9.0
10.0
0'01
0.01 00
IO
2.0
3.0
4.0
an
6n
7.0
0.0
SO
10.0
Kf
Figure 8. Comparison of dispersion and crossflow models for experimental residence-time distribution data of Ross (1 965). Data used is from Figure 13 of the original reference and applies to a commercial hydrotreater. Intraparticle diffusion effects are included in the residence-time distribution
\ 0.01 0.0
IO
20
3.0
40
h 0.0 t
I 00
TO
I 0.0
9.0
10.0
KT
Figure 9. Comparison of dispersion and crossflow models for experimental residence-time distribution data of Murphree, et a / . (1 964).Data used is from Figure 5 of the original reference and applies to a commercial desulfurization reactor. lntraparticle diffusion effects are included in the residence-time distribution
from eq 18. The dispersion number, D/uLL, for each run was calculated from the values of go2 v i a eq 12. Plots of In (1 - X) us. KT]as predicted from both the dispersion and the crossflow models using the above data, are shown in Figures 6 and 7. For each of the above references, only that run with the largest deviation from plug flow is shown. For reference purposes, Figures 6 and 7 also include curves for plug flow. I n both figures, data from the two models superimpose t o within the accuracy of the plots. Furthermore, the deviation between the conversion predicted from both models and that for a plug flow reactor is quite small. For the data of Figure 6, even a t 99% conversion, only about 6% more catalyst is required for the actual reactor than for a plug flow reactor. For the data in Figure 7, however, about 30% more catalyst is required a t 99% conversion. Examination of the other reported runs confirms the observation that the predictions of the two models are essentially 268
Ind. Eng. Chem. Process Der. Develop., Vol. 12, No. 3, 1973
Figure 10. Comparison of dispersion and crossflow models for experimental residence-time distribution data of Murphree, et al. ( 1 964).Data used is from Figure 7 of the original reference and applies to a commercial desulfurization reactor. lntraparticle diffusion effects are included in the residence-time distribution.
superimposed. For the other runs within a given reference] deviations from plug flow are less pronounced than those shown in Figure 6 or 7, whichever is appropriate. (2) Commercial Reactors. Literature data from commercial trickle-bed reactors is not extensive] especially data of sufficient detail and quality to permit accurate estimation of the parameters of the crossflow model. Furthermore, as pointed out in the Procedure section, the R T D data that is needed to analyze reactor behavior must be for the liquid external to the catalyst particles. The commercial reactors for which the RTD has been measured involve porous catalyst particles, and consequently the measured RTD values reflect a large internal liquid holdup. For valid comparison of various liquid backmixing models, the portion of the RTD due to external holdup would have to be separated from the total RTD. The presence of internal holdup undoubtedly causes a long tail on the residence-time distribution curve and thus an increase in both the variance of the curve (Lapidus, 1957) and the skewness. Thus, commercial reactors with large amounts of internal holdup obviously offer an extreme case of large variance, uO2,and large stagnant (internal) fluid fraction,j . Figure 13 of the article by Ross (1965) and Figures 5-7 of the article by Murphree and coworkers (1964) were used to obtain rough estimates of the necessary parameters in the crossflow and dispersion models. Values of ug2were calculated from data points taken directly from these curves. The dispersion number, D / u d , was calculated from ue2 using eq 12. The parameter! was calculated using eq 17 with estimates for the initial breakthrough time, ti, and the mean residence time, t, that were calculated from the same data points. These parameter values were then used to predict reactor behavior. The results are shown in Figures 8-10. Once again, for reference purposes, these figures also show curves for plug flow. A graph for Figure 6 of the Murphree, el al. (1964), article is not presented because the results are very similar t o those shown in Figure 9. There is good correspondence between the predicted conversion of the two models for the data of Ross (1965) in Figure 8. However, calculations for the data of Murphree, et al. (1964), in Figures 9 and 10 do not correspond as well,
Table II. Comparison of Dispersion and Crossflow Models. Catalyst Ratio (Dispersion Model/Crossflow Model)
Conversion level, ReL = 8.3 ReL = 65.8
00
I O
20
10
40
SO
60
10
I O
¶O
100
IC
Figure 1 1. Comparison of dispersion and crossflow models for parameter correlations of Hochman and Effron (1 969), using minimum (f = 0.39) and maximum (f = 0.22) particle Reynolds numbers
although the two model curves lie in the same general location on the graph. The value of the parameterf used in these calculations could be in error, since the initial breakthrough time was difficult to estimate accurately. If the estimate of ti were in error by lo%, the resultant f would also be in error by 10% in the opposite direction. A 10% lower value o f f would bring the two model curves into almost perfect agreement. The difficulties encountered in this calculation exemplify the problem of parameter estimation for the two-dimensional models and underscore the need for very accurate R T D data when dealing with such models. It is also apparent that the model curves from the commercial reactors data are further removed from plug flow than when experimental reactor data are used. The values of uo2 are quite high, since it was not possible to extract that portion of the variance due only to external liquid holdup. As can be seen from the parametric study (Figures 1-5), a lower value of variance reduces the deviation of both models from plug flow. I t is of interest to note that the model curves from the data of Murphree, et al., in Figure 10, which was labeled “good operation” in the original article, lie closer to plug flow than those curves from data labeled ‘‘poor operation’’ in Figure 9. (3) A Priori Reactor Design. Correlations for the parameters of both the crossflow model and the dispersion model as functions of the liquid and gas Reynolds numbers have been provided by Hochman and Effron (1969). The minimum and maximum values of ReL, as shown in Table IV of the Hochman and Effron (1969) article, were used to generate two sets of parameter estimates. Values of 4 were obtained directly from the reference, and the crossflow parameter f was calculated as 1 - 4. The ReL values, corrected for the particular void fraction of the experiments, were then used to determine the particle Peclet number, PeL, from Figure 6 of the reference. Multiplying PeL by an assumed L/d ratio gives the inverse of the dispersion number, D/uLL. In the present study, an L/d ratio of 42 was chosen for calculations based on Re = 8.3, and L/d = 13.5 was chosen for calculations based on Re = 65.8. These choices lead, in both cases, to a value of D/uLL = 0.18, which should correspond to significant backmixing. Mears (1971) has recently shown that an L/d ratio exceeding about 300 is needed to essentially eliminate backmixing effects in trickle-bed reactors. The value of uo2 for both cases was 0.3.
%
80 1.03 1.09
90 1.11 1.22
99 1.20 1.43
The a priori parameter estimates were then used to predict reactor behavior. The results are plotted in Figure 11. The chosen L/d ratios did give significant deviations from plug flow, as expected. Moreover, as shown in Figure 11, the deviation between the two models can be significant a t high coqversions. The ratio of the required catalyst weight, predicted from the dispersion model, to that predicted from the crossflow model, a t various conversion levels, is shown in Table 11. The deviation between the two models becomes significant above about 80% conversion. However, the data in Table I1 suggest that the dispersion model is conservative in that i t predicts a greater required catalyst weight for a given conversion. Along the same lines, in analyzing experimental data, use of the dispersion model, as opposed to the crossflow model, will lead to a lower apparent rate constant, and therefore to a conservative commercial reactor design, even if backmixing effects are negligible in the commercial reactor. Thus, assuming that the crossflow model is indeed the correct representation of liquid backmixing, use of the dispersion model as an approximation should lead to somewhat inoptimum, but nevertheless workable, reactor designs. Conclusions
The above calculations suggest that liquid backmixing is frequently not of major importance in trickle-bed reactors in agreement with previous conclusions (Lapidus, 1957; Schiesser and Lapidus, 1961). Deviations from plug flow become significant only for short reactors and high fractional conversion of reactant. In those cases where it is desirable to account for liquid backmixing, the dispersion model appears to be an adequate representation. This conclusion has been experimentally verified by Mears (1971). However, the present study is exclusively based on the assumption that the reaction occurring is first order in a reactant which is essentially confined to the liquid phase. The above conclusions may not be valid for situations where this assumption is not applicable. Appendix
Prediction of Fractional Conversion Using the CrossThe fractional conversion for a first-order reaction in an isothermal trickle-bed reactor which obeys the crossflow model may be calculated by substituting the expression for E(t), eq 16a and 16b, into eq 11, resulting in eq A1 and A2
flow Model.
Ind. Eng. Chem. Process Der. Develop., Vol. 12, No. 3, 1973
269
where
The bracketed term can be expressed in terms of exponentials (Abramowitz and Segun, 1964) giving 1-
x = e-C1 + 2c8e-(C1-C1)
Let
[(~CI
- e-Ca)/2ca1 = exp{ -(G - ~ c I ) }
Substituting the definitions for C1 and Ca
so that
From eq 6 Then eq A3 follows.
= PBtm/HEPL
so that
This result could have been obtained more simply by solving the steady-state form of the equations for this model. However, this approach would not have given an insight into the assumptions involved in applying the segregated-flow model to trickle-bed reactors. Nomenclature
external surface area of catalyst particle, cm2 parameter of the dispersion model, dl ~ K T ( D / u L Ldimensionless ), CA = concentration of reactant A, mol/cm3 of liquid CA = mean concentration of reactant A leaving reactor, mol/cm3 of liquid CA* = concentration of reactant A in stagnant liquid, mol/cm3 of liquid D = dispersion coefficient, cm2/sec d = diameter of catalyst particle, cm E(t) = residence-time distribution of reactor, sec-' f = fraction of liquid which is sta nant, dimensionless The integral portion of the first term on the right is the H = liquid holdup, om3 of liquidkm3 of bed Z1(!) = firsborder Bessel function of the first kind with Laplace Transform of the Dirac delta function and has a value imaginary argument of 1. K, = intrinsic first-order rate constant, cm3/g of catalyst I n order to simplify this expression further, several temsec porary variables are defined K = modified heterogeneous firsborder rate constant, K'PL,g of liquid/g of catalyst sec K' = heterogeneous firsborder rate constant, cm3/g of catalyst sec KpBtm(l HEPL k = mass-transfer coefficient between flowing and stagnant liquid, sec-l CZE [ ( ~ H E P L K P B ~ ) / ~ H E P L ~ ~ ~ ~ ~ k, = mass-transfer coefficient, cm/sec L = total length of catalyst bed, cm and the expression for fractional conversion simplifies to M = mass liquid flow rate, g of liquid/sec N = number of mixing cells, L/d, dimensionless 1 - X = e-C1 e-C1 e-CsY'Il(y) dy n = integral number of delaying stops, dimensionless PeL = liquid particle Peclet number, uLd/D, dimensionless The integral of the Bessel function may be obtained from Q = volumetric liquid flow rate, cm3 of liquid/sec the "Handbook of Mathematical Functions" (Abramowite ReL = liquid Reynolds number, duSLpL/p, dimensionless -TA' = rate of disappearance of reactant A by heteroand Segun, 1964). geneous reaction, mol/g of catalyst sec t = time, sec ti = initial breakthrough time of tracer, sec UL = mean liquid interstitial velocity U ~ L = mean superficial liquid velocity, cm/sec Let V = volume, cm3 W = weight of catalyst, g of catalyst WHSV = weight hourly space velocity, M / W , sec-' X = fractional conversion of reactant, dimensionless z = axial distance, cm so that the above expression becomes GREEKLETTERS 6 = Dirac delta function, dimensionless 17 = effectiveness factor of the catalyst, dimensionless
a
a'
"I
+
+
270 Ind. Eng. Chem.
J,-
Process Des. Develop., Vol. 12, NO. 3, 1973
=
=
+
Bondi, A., Chem. Tech., 186 (1971). Buffham, B. A., Gibilaro, L. G., Rathor, M. N., AIChE J., 16, 218 (1970). Cecil, R. R.,’Mayer, F. X., Cart, E. N., Jr., AIChE, 10th Annual Symposium, Chicago, Ill., April 1969. Deans, H. A., SOC.Petrol. Engr. J . , 3, 49 (1963). Furzer, I. A., Michell, R. W., AZChE J., 16, 380 (1970). Hochman. J. M.. Effron., E.., Znd. Ena. Chem.. Fundam.., 8.. 63 (1969).’ Hoogendoorn, C. J., Lips, J., Can. J. Chem. Eng., 43, 125 (1965). Lapidus L., Ind. Eng. Chem., 49, 1000 (1957). Levensp)iel, O., “Chemical Reaction Engineering,” Wiley, New York, N. Y., 1962, pp 257-259. Levenspiel, O., Smith, W. K., Chem. Eng. Sci., 6, 227 (1957). Mears, D. E., Chem. Eng. Sci., 26, 1316 (1971). Murphree, E. V., Voorhies, A., Jr., Mayer, F. X., Ind. Eng. Chem., Process Des. Develop., 3, 383 (1964). Ross, L. D., Chem. Eng. Progr., 61, 78 (1965). Sater, V. E., Levenspiel, O., Znd. Eng. Chem., Fundam., 5, 86
dimensionless time, t/tm, dimensionless reactor bed density, g of catalyst/cm3 of reactor density of the liquid phase, g of liquid/cm3 of liquid variance of the dimensionless RTD, E(O), dimensionless = space time for a heterogeneous system, W/&PL or l/WHSV, sec
0
pB p~
= = = =
SUBSCRIPTS A refers to reactant A D refers to the mean delay time of an exponential distribution of delay times
E refers to the exterior of the catalyst I refers to the interior of the catalyst j
refers to the j t h tubular element of the reactor
m p T 0
refers to mean or average values refers to a single catalyst particle refers to the total quantity refers to initial conditions
L refers to the liquid phase
I\--””,. lcififi)
Schiesser, W. E., Lapidus, L., AZChE J.,7, 163 (1961). Van Deemter. Chem. Ena. Sci.. 13. 190 (1961). Van Swaaij, W. P. M., (?harpenti&, J. C., Villermaux, J., Chem. Eng. Sci., 24, 1083 (1969). Wehner, J. F., Wilhelm, R.H., Chem. Eng. Sci., 6, 89 (1956).
Literature Cited
Abramowitz, M., Segun, I. A., ‘‘Handbook of Mathematical Functions,” Dover Publications, New York, N. Y., 1965, pp 443 and 487.
RECEIVED for review May 24, 1972 ACCEPTED March 16, 1973
Prediction of Ternary Liquid-Liquid Equilibrium from Binary Data J. M. Marina*l and Dimitrios P. Tasrios Newark College of Engineering, Newark, N . J . 07102
Prediction of ternary liquid-liquid equilibrium from binary data is discussed. The NRTL equation is used with the value of a! set according to the rules of Renon and Prausnitz ( 1 968) and according to Marina and Tassios ( 1 973),who recommend the substitution a! = 1. The latter value of a seems to yield better results.
-
v e r y often when working with liquid-liquid equilibrium problems, one finds situations in which data for the binary constituents of the ternary system are available, but data on the ternary itself are unavailable. Hence the importance of predictive techniques which with the aid of a computer may predict ternary data from binary data alone. The problem under consideration requires determination of the binodal curve and tielines of a three-component partially miscible system. Solution of the problem requires the specification of eight variables, six liquid composition variables, the temperature, and the pressure of the system. Now applying the phase rule
G=n+2-@
(1)
where G = degrees of freedom, n = number of components, and @ = number of phases in the system. We find that we have two degrees of freedom, that is, two independent variables. Therefore, we can specify the temperPresent address, 3215 33165. 1
s. W.
127th Avenue, Miami, Fla.
ature and one liquid composition, say (XI’). The remaining variables are then fixed through the following equations 3
CX,!1 =
1
CX,’l = 1
(3)
1
Xt’YE’ = XI””
i
=
1, 2 , 3
(4)
Equations 2 and 3 are just the conditions that in each phase the sum of the mole fractions must equal 1. Equation 4 represents the condition for equilibrium between phases. The nonrandom two liquid equation, S R T L (Renon and Prausnitz, 1968), and the local effective mole fraction equation, LEMP (Marina and Tassios, 19?3), will be employed in this study for the prediction of ternary activity coefficients from binary data. NRTL and LEMF Equations
Renon and Prausnitz (1968) developed the following exInd. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
271
