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Mixing-cell method for design of trickle-bed reactors
The Chemical Engineering Journal, 17 (1979) 91 - 99 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
91
Mixing-Cell Method for Design of Trickle-Bed Reactors
P. A. RAMACHAMDRAN and J. M. SMITH University of California, Davis, CA 95616
(U.S.A.)
(Received January 18,1978)
Abstract The performance of a trickle bed is analyzed in terms of a mixing-cell concept in which the three-phase reactor is visualized as a number of mixing cells in series. Results are presented for an iso thermal system involving a reaction of general-order kinetics and incorporating all mass transfer resistances. Limiting cases of very slow, and rapid, gas-liquid mass transfer are also considered. The utility of the new model is illustrated with an application to the oxidation of formic acid and by using the method to compare the performance of slurry and trickle-bed reactors.
INTRODUCTION
Three-phase reactors are used for chemical processes that involve a solid catalyst and gas and liquid reactants, or gaseous reactants dissolved in a liquid phase. The most common reactor types are the stirred tank (slurry) or the trickle bed in which the gas and liquid phases flow downward through a stationary bed of catalyst particles. The main difference in modeling these two types is in the flow pattern of the liquid. In a slurry reactor the liquid tends to be completely mixed, while in a trickle bed the behavior is closer to plug flow, or more accurately described by adding an axial dispersion contribution to the plugflow equations. For the gas phase, the flow is close to plug flow in both types of reactors. A design procedure for the trickle-bed reactor, based on the axial dispersion model, has been developed by Goto et al. [ 1, 21 for a firstorder reaction scheme A * R. In the present work, a different design method is proposed -- tanks in series, or mixing-cell, model. The reactor is visualized as consisting of a number
N of stirred tanks in series, in each of which the liquid is completely mixed and the gas is in plug flow. Thus, plug-flow performance for the whole reactor corresponds to N = 00, while a slurry reactor corresponds to N = 1. A system of continuous slurry reactors in series, employed in some practical applications (e.g. gas purification), can be represented equally well by the model where N is equal to the number of reactors. The model is developed for an irreversible reaction of the general form A + VB --f products
(1)
Both reactants may be in the gas phase, or one may be non-volatile and present only in the liquid. The mixing-cell,model has computational advantages compared with the dispersion model, because the equations for each cell are a set of non-linear algebraic equations, rather than a set of second-order non-linear differential equations. Further, more complex flow patterns for the liquid, such as wall flow or bypassing, can be incorporated easily into the model, although this has not been done in the present work. Numerical solutions are given for the more general kinetics of eqn. (1). Analytical solutions are also derived for pseudo first-order kinetics. The model is illustrated with an application to the removal of water pollutants (formic acid) by air oxidation and also is used to compare the performance of slurry and trickle-bed reactors. In all cases it is supposed that the outer surface of the catalyst particles is completely covered with liquid. In principle, the method could be applied for incomplete liquid wetting. However, the model presented would have to be revised to account for the non-symmetrical concentration profiles existing with the particle.
92 FORMULATION
OF THE PROBLEM
The rate of reaction (l), per unit volume of catalyst, is assumed to be given by s1 = p,kAmB”
(2)
where the usual values for m and n are between 0 and 1. The concentration of a stream (gas or liquid) leaving cell j is represented by Aj and Bj, and that entering the jth cell by A,_, and Bj-1, as noted in the schematic diagram of the model (Fig. 1). Mass balances for the jth cell are then as follows.
Species A, gas phase =
@I&A
[F
L]
--Ai,
(3)
where A&) is the local gas-phase concentration at any point z in cell j and z is the axial position in the jth cell.
AN.L
AN.1
BN,L
‘N,g
Fig. 1. Schematic model.
Species A, liquid phase
-
(bsh(4,L
=
UL(Aj,L-Aj-1.L)
’
+ (ks%)A N
t-41,.-Al,11 (4)
where Aj,, is the concentration ticle surface.
(k,%)A (Aj, L --Ai,,) = p,k(l
1
where Ea is the bed porosity catalyst effectiveness factor.
m+=
(KL~L)B [
‘$ -
(5)
JkLadB [‘$
0
B
--B,,,]
=uL(BS,L-BI-~,L)+(~,~,)B
L]
1 (KL~L)B
and 77 the
Bj,
1- EB)A.~~IB~ IV (8)
1
&,%)A
1 HA
(9)
@&)A
and
Species B equations -4
=-+
WLQL)A
-EB)A~IB~IV
Bj, I) = vppk(
of the mixing-cell
In these eqUatiOnS (K,j.& and (KLuL)B are the overall gas-liquid mass transfer coefficients for the transfer of A and B, respectively. Assuming Henry’s law, they are related to the individual film coefficients as follows:
at the par-
Species A, boundary condition at particle surface
representation
(6)
=
l+ VwdB
1
(10)
~dkg%)B
In eqns. (5) and (8), the catalyst effktiveness factor 71cannot be expressed exactly as an explicit function of Aj,, and Bj,I because the reaction is not first order. However, Bischoff [3] has shown that Q is given, to a good approximation, by the expression
d,z
(11)
NL
(Bj,L -BI,I)
(7)
Here 9 is a generalized fined as
Thiele modulus,
de-
93
dC -l;;2)
VDe,AAj,IIDe.BBj,IG 1 A is the limiting reactant, and the integral in eqn. (12) would be with respect to A. Conversely, if
(21)
A0.S
I
where C denotes the limiting component. In these equations the intraparticle concentration profiles are presumed to be symmetrical in the radial direction. If
4
ag =-
z* =-2
(22)
L
Similar definitions apply for b,, bj, L and bj,i. If species B is introduced only in the incoming liquid, or is non-volatile and present only in the liquid phase, bj,L and bj,I are made non-dimensional with respect to B,,. In terms of dimensionless concentrations, eqn. (12) for the Thiele modulus becomes
component B is the limiting reactant. In terms of dimensionless concentrations Uj and bj, the previous equations may be written
(23) if A is the limiting reactant,
dcg(z*)
-
7
= aA
dz
[a,@*)- aj,Ll
(13)
1
J
aAflA[agtz*)
-aj,Ll
dZ* =aj,L-aj-1,L
-& e*B(bj,, - b)Im
+
(14)
-aj,r)
= k*Tar#J
- db&*) * dz = ag [bg(z*)-
(15)
bj,Ll
(24) if B is the limiting reactant. The dimensionless parameters involving the various rate parameters are defined as follows:
(16)
(25) aB=wLaLbL
1 s
db/-1’2
e.A
0
%,A@j,L
and
~BPB
[b&*)
0
(bj,L - bj,I)
+y
(26)
-bj,1,1 h* = bi,L- bj-1.L +
U,$B
PA
=!$
BB
=-
(27)
(17) @B
a*,i3(bj,L-
bj,I) = #*rl@'I&
(Y,A = (kd%)AL . UL
where ~%,LHA =j,L =A0.r
q.1
(23) UL
(18)
&HA = A0.s
(29)
(19)
(30)
(20)
(31)
(32) and
b.
I.1
~~=~[~(~.)~-1(~)n]1’2 .
METHOD
(33)
OF SOLUTION
For any cell, the concentration in the streams entering taj-l,L, aj-l.g, bj-l,L, bj-l,g) we known from calculations for the previous cell. The unknown concentrations are aj,g, u~,~, % , I, bj,g, bj, L, and bj,I . The six equations, (13) - (18), are sufficient for the solution of the problem. To simplify the computations, all the unknowns are expressed in terms of a single unknown concentration aj,i. This is done in the following way. Equations (13) and (16) can be integrated, since ej,L and bj,L are not functions of z* (the liquid is completely backmixed in each cell). The integrated form of (13) is a&*)
-cj,L
= exp (-~.z*)
(34)
%-i.g-%,L with a similar equation for B. By substituting z* = l/N, the dimensionless concentrations of A and of B in the streams leaving the cell j are obtained: t2j.g
Eliminating the reaction terms in eqns. (15) and (18) and rearranging gives
=aj-l,gt?Xp(-'$)+llj,L[l-eXp(--$)I
-%!
= bj,L
bi,g =
4-h exp(A$)+bj,L
(39)
-aj,A
The procedure illustrated by this development of eqns. (37), (38) and (39) shows that all the unknown concentrations can be expressed in terms of aj,i. The value of aj,i itself is obtained by solving algebraic eqn. (15). Since eqn. (15) is non-linear for all cases except m = 1, n = 0 (or m = 0, n = l), a numerical solution is required. As n is a function of the unknown concentration aj,i, it is convenient to start the computations by assuming a value for n. This starting value can be subsequently corrected by a trial and error calculation, once an initial number for aj,r is calculated.
RESULTS
FOR FIRST-ORDER
REACTIONS
In the following section, results are presented for reactions which are first order in A but independent of the concentration of B (m = 1, n = 0). Examples are oxidation reactions with an excess of oxidizable component in the liquid phase and hydrodesulfurization reactions first order in hydrogen. Equation (15) is linear and analytical solutions are possible. The concentration of A in the liquid and in the gas leaving cell j can be obtained by combining eqns. (15) and (37). The results are 13~[1
(35)
and
(aj,L
%,B
-eXP(-Q/N)]%-1.g
a&L= 1 + LY,,A~~,JN
[l-exp(-$)I
+
aj-1.L
+dA[l-eXp(--A/N)]
(40)
and (36) Equation (34) can be used to evaluate integral in eqn. (14). This gives Oj L =
aj-1.L
+PA 11 -eXPf-e~/N)l
1+
PA [I-
eXp(-
the
aj-1.g + (%,A/N)aj,I ~A/N)I
+
(~S,A/N)
B is b.J,L
bj-1.L +PB[~ -eXp(-a~IN)l =
1 + PB[~
--exP(-@B/N)]
bj-1.g + (%,dN)bj,I
+
(as,~/N)
(38)
95
+Uj,L
Uj,g=Uj_l,gC!XJ3(-$)
1.0
[l-exp(-a$)]
-
-
I
- - Plug-Flow Solulion[2]
(41) 0.1
where QL,S
b’ 1
=I-
0.2
(42)
1 + k*d%,A
It is interesting to examine two limiting cases of eqns. (40) and (41), which are approached in many practical situations. 1. (YAapproaching zero This situation occurs when the gas-liquid mass transfer is rate limiting, when the gas rate ug is very large, or when the solubility of the gas A in the liquid is very low. Under these conditions eqn. (40) can be shown to reduce to a simple form; that is, (%P*/Na0,g 'jvL = 1 + %,AVL,S/N
+aj-1.L
(43)
0.1 1
2
3
5
20
10
30
SO
Fig. 2. Concentration of gas A leaving the reactor as a function of the gas-liquid mass transfer coefficient (PA = 10, k* = 50,a,,A = 50, &, = O.l,uo,L = 0, = 0): broken curve, plug-flow solution [ 21. m=l,n 1.0 r
I ‘0.L
=o
I
I
ao,L=o.5 0.95
I3 A
a.)0
+ ~AOA/N ao L’1.0 0.15
aj,g
N aj-1.g N aO,g
(44)
where
0.00 I 1
I
I
4
1
NUMGER OF MIXING CELLS,
(45) Equation (44) implies that concentration changes in the gas phase are very small. 2. (YAapproaching infinity This case corresponds to very rapid mass transfer between the gas and the liquid phase. Equations (40) and (41) now reduce to @A aj
L
+l)%l,L
=
1
+
9,g =aj.L
Q,,A'?L.~/N + PA
(46) (47)
Equation (47) corresponds to a physical situation such that gas and liquid are in equilibrium everywhere in the reactor; gas-liquid mass transfer no longer influences reactor performance. Numerical calculations show that an approximate criterion for neglecting such gas-liquid resistances iS aA > 4.6N. For intermediate VdUeS Of aA, eqns. (40) and (41) should be used. Numerical results are illustrated in Fig. 2 for a particular case represented by the parameter values shown on
I
10 N
Fig. 3. Influence of inlet liquid concentration of A on conversion for various degrees of backmixing for %,A = 50, k* = 1.0, aA = o.l,pA = 10.0, $0 = 0.1, m=l,n=O.
the figure. The concentration of reactant A in the gas leaving the reactor is given for various values of the gas-liquid mass transfer parameter ffA , and for different degrees of backmixing of the liquid, as characterized by the parameter N. The value of aqL corresponds to no dissolved reactant A in the liquid feed. The influence of liquid backmixing is seen to be small when there is little mass transfer between gas and liquid (low values of (YA),but is significant when eA becomes large. The results for large N correspond to plug flow of the liquid. The analytical solution for the plugflow model (obtained by Goto and Smith in eqns. (78) and (79) of ref. 2) is indicated by the broken line in Fig. 2. When N = 10, plug flow is c!osely approached. Figure 3 shows the conversion of gaseous reactant A for conditions such’ that gas-liquid mass transfer and chemical reaction at the catalyst surface
96
control the overall rate. The corresponding parameter values are (Y,,A = 50, k* = 1.0, aA = 0.1, PA = 10.0, and f#~s= 0.1. Results are given for three concentrations of A in the liquid feed. The influence of liquid backmixing (value of N) is small in all cases. However, it is interesting to note that for (I~,~ = 0 (no dissolved A in the liquid feed) the conversion is a maximum at an intermediate level of backmixing corresponding to N = 4. This result, which was also predicted by Goto et al. [ 11, is due to the possibility of the reactant concentration increasing because of backmixing in the liquid near the reactor inlet. This is possible when there are two flowing streams with mass transfer between them. Such a phenomenon cannot occur in fixedbed reactors with a single fluid phase; plug flow always gives the highest conversion, for other than zero-order kinetics.
RESULTS FOR NON-LINEAR
KINETICS
In this section, the effect of reaction order with respect to A is analyzed, first for n = 0 and variable m, and then for the second-order reaction n = m = 1. Figure 4 shows the effluent concentration of reactant A for m = 0, 1,and 2 for the following values of the parameters: LYAPA= 100; CY,,A= 50; k* z 50; Go = 0.1 and uqL = 0. The results
Fig. 4. Effect of reaction order (with respect to A) on concentration of A in effluent from a trickle-bed reactor; with 10 cells (N = 10) for (YAPA = 10, k* = 50, 40 = O.l,%,A = 50, aO,L = 0, n = 0.
s
0.1
m=O 0.05
\
0.02
0.01 1
2
5
10
20
50
100
-A
Fig. 5. Effect of reaction order (with respect to A) on exit concentration of A in a slurry reactor (N = 1); (IIA~~A = 10, (Y,,A = 50, k* = 50, Qo = 0.1, ao,L = 0, n = 0.
are presented as a function of gas-liquid mass transfer coefficient cA for N = 10, which represents the behavior of a trickle-bed reactor with nearly plug flow of liquid. The other extreme (N = 1) corresponding to a slurry reactor is given in Fig. 5. In both cases the conversion of reactant A decreases with increasing reaction order, although the effect is greater for plug flow (N = 10). These results for a reaction of mth order in A would be approached in practice if the parameter g (eqn. (32)) is small compared with unity; that is, when there is a large excess of B in the liquid phase. An example is an oxidation reaction with a large excess of liquid-phase reactant. The above condition may not be achieved in some practical situations. For example, in the oxidation of organic pollutants in water, the concentration of the liquid-phase reactant (pollutant) may be low so that the reaction rate depends on the concentration of both reactants. The influence of the initial concentration of the two reactants (3 and B), as measured by the parameter q, is shown in Fig. 6 for a second-order reaction, m = n = 1. The computations were made for a nonvolatile. reactant B, so that q is equal to uA&HABO,~. Figure 6 demonstrates that the conversion of A decreases as the ratio AO,g/BO,g increases, owing to depletion of reactant B.
97
I
I
___m_ Tiickla-Bed
I
REACTANT RATIO.q
Fig. 6. Effect of reactant ratio on conversion of A ; a/, = lo,fl~ = 10, k* = 50, (Ys,A = a,,$ = 50, $0 = 1.0, a0.L = 0, b0.L = 1.0, m = n = 1.0.
Fig. 7. Effect of catalyst particle size on the performance of a slurry reactor (first-order reaction, m = 1, n = 0): broken line, trickle-bed reactor, N = 10.
SLURRY
R dum)2. Here, spherical particles are assumed in order to relate particle volume and surfaceto-particle radius. Results for three catalyst loading values (y = volume of catalyst/ volume of reactor) are shown in Fig. 7. The catalyst loading of 0.4 corresponds to that used in the trickle-bed reactor. The figure indicates that in some cases the increase in catalyst effectiveness factor for a slurry reactor more than compensates for the adverse effect of liquid backmixing. For example, for the high catalyst loading of 0.2, the conversion in a slurry reactor is more than that in the trickle-bed reactor if the catalyst particle size in the slurry is less than one-third of the particle size in the trickle-bed reactor. For lower, more reasonable catalyst loadings Rduny would have to be much less. However, such results demonstrate that slurry operation may lead to higher conversions in systems for which intraparticle diffusion resistance is very large. Conversely, for systems with negligible intraparticle diffusion, trickle-bed operation may be advantageous. For reactions of finite order with respect to A, but independent of the concentration of B (e.g., some cases of hydrodesulfurization), slurry operation may be preferable if conditions are such that liquid backmixing improves conversion.
VERSUS
TRICKLE-BED
REACTORS
Since the same type of equations are used to represent both slurry and trickle-bed reactors, the mixing-cell model provides a simple basis for comparing the performance of the two modes of operation. Such comparisons are given here in terms of expected conversions. However, practical factors are also important. For example, the small-size catalyst particles used in slurries reduce the importance of intraparticle diffusion resistance. Also the fluid-particle mass transfer area is large per unit mass of catalyst. In contrast, the residence time distribution of the liquid in a slurry reactor will normally be close to that for complete mixing, which will reduce the rate of reaction under some circumstances. However, from a practical viewpoint separation of catalyst particles from the effluent liquid can outweigh the net result of factors affecting conversion. As an illustration of expected conversions, we consider a first-order reaction (m = 1, n = 0) for which liquid backmixing adversely affects reactor performance. Chosen parameter values are oA = 10, fl_&, = 10, (Y,,A= 50, k* = 50, $0 = 10, and ao,L = 0.0. The predicted conversion in a trickle-bed reactor for these conditions is 0.58 for N = 10. The performance of a slurry reactor with particle sizes lower than those employed in a trickle bed is shown in Fig. 7. This figure gives the conversion in a slurry reactor as a function of the particle size ratio Rtrickle/Rdurry for the same values of the parameters, except Go = 1O(&~,,,/&i~e) and %,A = 50&ic,,/
FORMIC ACID OXIDATION
Goto and Smith [4] have measured conversions for the air oxidation of dilute aqueous
98
NOMENCLATURE
Feed Gate. IFume
A.1,&f
Acid)
A.I.1 LIQUID FLOW RATE,
cm3/s,c
al 25 “C, 1 rim
Fig. 8. Predicted conversion by mixing-cell model for the air oxidation of aqueous solutions of formic acid at 240 “C and 40 atm.
A&)
A0.g A.1-L
bi solutions of formic acid in a trickle-bed reactor. A commercial CuO-ZnO catalyst was employed at 40 atm and 240 “C. Mass transfer coefficients could be estimated from independent measurements, and reaction kinetics data were available for the same system. Hence, conversions could be predicted by the model presented here and compared with the experimental results. Plug flow was assumed for the gas phase. The mass transfer coefficients KLaL and ksas used were those reported by Goto and Smith (ref. 4, Part I). The effective diffusivity employed to account for intraparticle diffusion and the intrinsic rate of oxidation were available from the experiments of Baldi et al. [ 51. Using these data, conversions of formic acid were predicted by the mixing-cell model for N = 1,4, and 10. The results are presented in Fig. 8 as a function of the liquid flow rate. The experimental conversions are seen to lie somewhere between plug flow (N = 10) and complete mixing (N = 1) of the liquid. However, the operating conditions were such that the gas-liquid mass transfer resistance was relatively large. Hence, predicted results are sensitive to changes in Kr,aL, as noted by Goto and Smith [4] . Further, this parameter is difficult to establish accurately at 240 “C and 40 atm. Hence, the deviation between observed conversions and those predicted for plug flow may not be due entirely to the effects of liquid backmixing.
B.J,B
B.I.1 B.I.L
D, HA, k k*
kg kL ks KL L N 4
R
HB
dimensionless concentration of A in the jth cell, AjHA/AO,g gas-liquid mass transfer area per unit column volume, cm2/cm3 effective liquid-particle mass transfer area per unit column volume, cm2/cm3 concentration of A in the gas phase leaving the jth cell, mol/cm3 concentration of A at the catalyst surface in the jth cell, mol/cm3 local gas concentration at any point in the jth cell, mol/cm3 concentration of A in the feed gas, mol/cm3 concentration of A in the liquid phase leaving the jth cell, mol/cm3 dimensionless concentration of B in the jth cell, BjHB/Bo,g concentration of species B in the gas phase leaving the jth cell, mol/cm3 concentration of B on the catalyst surface in the jth cell, mol/cm3 concentration of B in the liquid phase leaving the jth cell, mol/cm3 effective intraparticle diffusivity of the species indicated by the subscript, cm2/S Henry’s law solubility coefficient of A and B, Ag/AL or Bg/BL reaction-rate constant, [ cm3/g s] [ cm3/mol] m+ n -’ dimensionless rate constant,
gas-film mass transfer coefficient, cm/s liquid-film mass transfer coefficient, cm/s solid-liquid mass transfer coefficient, cm/s overall gas-liquid mass transfer coefficient, cm/s height of the catalyst bed, cm total number of mixing cells into which the column is divided reactant ratio parameter defined by eqn. (32) radius of the catalyst particle, cm
99
external surface area of a catalyst particle, cm* superfic,ial gas velocity, cm/s superficial liquid velocity, cm/s volume of a catalyst particle, cm3 conversion of reactant A in the reactor axial position in the jth cell, cm dimensionless axial position, z/L
b UIA
VP xA z
.z*
7)
v PP
4J 40
Sl
catalyst effectiveness factor stoichiometric coefficient of B in the reaction density of the particle, g/cm3 Thiele modulus, defined by eqn. (12) parameter defined by eqn. (33) reaction rate per unit volume of catalyst particle, mol/cm3 s
Gl'eek aA,aB
%,A
PA,
7
EB
7 %.B
BB
dimensionless gas-liquid mass transfer coefficients defined by eqns. (25) and (26) dimensionless solid-liquid mass transfer coefficients defined by eqns. (29) and (30) parameters defined as UgHA/U= and ugHB/uL catalyst loading, volume of catalyst/volume of reactor void fraction in the bed
REFERENCES
S. Goto, S. Watabe and M. Matsubara, Can. J. Chem. Eng., 54 (1976) 126. S. Goto and J. M. Smith, A.Z.Ch.E. J., 24 (1978) 386. K. B. Bischoff, A.Z.Ch.E. J., 11 (1965) 351. S. Goto and J. M. Smith, A.Z.Ch.E. J., 21 (1975) 706, 714. G. Baldi, S. Goto, C. K. Chow and J. M. Smith, Znd. Eng. Chem., Proc. Des. Deu., 13 (1974) 447.
91
Mixing-Cell Method for Design of Trickle-Bed Reactors
P. A. RAMACHAMDRAN and J. M. SMITH University of California, Davis, CA 95616
(U.S.A.)
(Received January 18,1978)
Abstract The performance of a trickle bed is analyzed in terms of a mixing-cell concept in which the three-phase reactor is visualized as a number of mixing cells in series. Results are presented for an iso thermal system involving a reaction of general-order kinetics and incorporating all mass transfer resistances. Limiting cases of very slow, and rapid, gas-liquid mass transfer are also considered. The utility of the new model is illustrated with an application to the oxidation of formic acid and by using the method to compare the performance of slurry and trickle-bed reactors.
INTRODUCTION
Three-phase reactors are used for chemical processes that involve a solid catalyst and gas and liquid reactants, or gaseous reactants dissolved in a liquid phase. The most common reactor types are the stirred tank (slurry) or the trickle bed in which the gas and liquid phases flow downward through a stationary bed of catalyst particles. The main difference in modeling these two types is in the flow pattern of the liquid. In a slurry reactor the liquid tends to be completely mixed, while in a trickle bed the behavior is closer to plug flow, or more accurately described by adding an axial dispersion contribution to the plugflow equations. For the gas phase, the flow is close to plug flow in both types of reactors. A design procedure for the trickle-bed reactor, based on the axial dispersion model, has been developed by Goto et al. [ 1, 21 for a firstorder reaction scheme A * R. In the present work, a different design method is proposed -- tanks in series, or mixing-cell, model. The reactor is visualized as consisting of a number
N of stirred tanks in series, in each of which the liquid is completely mixed and the gas is in plug flow. Thus, plug-flow performance for the whole reactor corresponds to N = 00, while a slurry reactor corresponds to N = 1. A system of continuous slurry reactors in series, employed in some practical applications (e.g. gas purification), can be represented equally well by the model where N is equal to the number of reactors. The model is developed for an irreversible reaction of the general form A + VB --f products
(1)
Both reactants may be in the gas phase, or one may be non-volatile and present only in the liquid. The mixing-cell,model has computational advantages compared with the dispersion model, because the equations for each cell are a set of non-linear algebraic equations, rather than a set of second-order non-linear differential equations. Further, more complex flow patterns for the liquid, such as wall flow or bypassing, can be incorporated easily into the model, although this has not been done in the present work. Numerical solutions are given for the more general kinetics of eqn. (1). Analytical solutions are also derived for pseudo first-order kinetics. The model is illustrated with an application to the removal of water pollutants (formic acid) by air oxidation and also is used to compare the performance of slurry and trickle-bed reactors. In all cases it is supposed that the outer surface of the catalyst particles is completely covered with liquid. In principle, the method could be applied for incomplete liquid wetting. However, the model presented would have to be revised to account for the non-symmetrical concentration profiles existing with the particle.
92 FORMULATION
OF THE PROBLEM
The rate of reaction (l), per unit volume of catalyst, is assumed to be given by s1 = p,kAmB”
(2)
where the usual values for m and n are between 0 and 1. The concentration of a stream (gas or liquid) leaving cell j is represented by Aj and Bj, and that entering the jth cell by A,_, and Bj-1, as noted in the schematic diagram of the model (Fig. 1). Mass balances for the jth cell are then as follows.
Species A, gas phase =
@I&A
[F
L]
--Ai,
(3)
where A&) is the local gas-phase concentration at any point z in cell j and z is the axial position in the jth cell.
AN.L
AN.1
BN,L
‘N,g
Fig. 1. Schematic model.
Species A, liquid phase
-
(bsh(4,L
=
UL(Aj,L-Aj-1.L)
’
+ (ks%)A N
t-41,.-Al,11 (4)
where Aj,, is the concentration ticle surface.
(k,%)A (Aj, L --Ai,,) = p,k(l
1
where Ea is the bed porosity catalyst effectiveness factor.
m+=
(KL~L)B [
‘$ -
(5)
JkLadB [‘$
0
B
--B,,,]
=uL(BS,L-BI-~,L)+(~,~,)B
L]
1 (KL~L)B
and 77 the
Bj,
1- EB)A.~~IB~ IV (8)
1
&,%)A
1 HA
(9)
@&)A
and
Species B equations -4
=-+
WLQL)A
-EB)A~IB~IV
Bj, I) = vppk(
of the mixing-cell
In these eqUatiOnS (K,j.& and (KLuL)B are the overall gas-liquid mass transfer coefficients for the transfer of A and B, respectively. Assuming Henry’s law, they are related to the individual film coefficients as follows:
at the par-
Species A, boundary condition at particle surface
representation
(6)
=
l+ VwdB
1
(10)
~dkg%)B
In eqns. (5) and (8), the catalyst effktiveness factor 71cannot be expressed exactly as an explicit function of Aj,, and Bj,I because the reaction is not first order. However, Bischoff [3] has shown that Q is given, to a good approximation, by the expression
d,z
(11)
NL
(Bj,L -BI,I)
(7)
Here 9 is a generalized fined as
Thiele modulus,
de-
93
dC -l;;2)
VDe,AAj,IIDe.BBj,IG 1 A is the limiting reactant, and the integral in eqn. (12) would be with respect to A. Conversely, if
(21)
A0.S
I
where C denotes the limiting component. In these equations the intraparticle concentration profiles are presumed to be symmetrical in the radial direction. If
4
ag =-
z* =-2
(22)
L
Similar definitions apply for b,, bj, L and bj,i. If species B is introduced only in the incoming liquid, or is non-volatile and present only in the liquid phase, bj,L and bj,I are made non-dimensional with respect to B,,. In terms of dimensionless concentrations, eqn. (12) for the Thiele modulus becomes
component B is the limiting reactant. In terms of dimensionless concentrations Uj and bj, the previous equations may be written
(23) if A is the limiting reactant,
dcg(z*)
-
7
= aA
dz
[a,@*)- aj,Ll
(13)
1
J
aAflA[agtz*)
-aj,Ll
dZ* =aj,L-aj-1,L
-& e*B(bj,, - b)Im
+
(14)
-aj,r)
= k*Tar#J
- db&*) * dz = ag [bg(z*)-
(15)
bj,Ll
(24) if B is the limiting reactant. The dimensionless parameters involving the various rate parameters are defined as follows:
(16)
(25) aB=wLaLbL
1 s
db/-1’2
e.A
0
%,A@j,L
and
~BPB
[b&*)
0
(bj,L - bj,I)
+y
(26)
-bj,1,1 h* = bi,L- bj-1.L +
U,$B
PA
=!$
BB
=-
(27)
(17) @B
a*,i3(bj,L-
bj,I) = #*rl@'I&
(Y,A = (kd%)AL . UL
where ~%,LHA =j,L =A0.r
q.1
(23) UL
(18)
&HA = A0.s
(29)
(19)
(30)
(20)
(31)
(32) and
b.
I.1
~~=~[~(~.)~-1(~)n]1’2 .
METHOD
(33)
OF SOLUTION
For any cell, the concentration in the streams entering taj-l,L, aj-l.g, bj-l,L, bj-l,g) we known from calculations for the previous cell. The unknown concentrations are aj,g, u~,~, % , I, bj,g, bj, L, and bj,I . The six equations, (13) - (18), are sufficient for the solution of the problem. To simplify the computations, all the unknowns are expressed in terms of a single unknown concentration aj,i. This is done in the following way. Equations (13) and (16) can be integrated, since ej,L and bj,L are not functions of z* (the liquid is completely backmixed in each cell). The integrated form of (13) is a&*)
-cj,L
= exp (-~.z*)
(34)
%-i.g-%,L with a similar equation for B. By substituting z* = l/N, the dimensionless concentrations of A and of B in the streams leaving the cell j are obtained: t2j.g
Eliminating the reaction terms in eqns. (15) and (18) and rearranging gives
=aj-l,gt?Xp(-'$)+llj,L[l-eXp(--$)I
-%!
= bj,L
bi,g =
4-h exp(A$)+bj,L
(39)
-aj,A
The procedure illustrated by this development of eqns. (37), (38) and (39) shows that all the unknown concentrations can be expressed in terms of aj,i. The value of aj,i itself is obtained by solving algebraic eqn. (15). Since eqn. (15) is non-linear for all cases except m = 1, n = 0 (or m = 0, n = l), a numerical solution is required. As n is a function of the unknown concentration aj,i, it is convenient to start the computations by assuming a value for n. This starting value can be subsequently corrected by a trial and error calculation, once an initial number for aj,r is calculated.
RESULTS
FOR FIRST-ORDER
REACTIONS
In the following section, results are presented for reactions which are first order in A but independent of the concentration of B (m = 1, n = 0). Examples are oxidation reactions with an excess of oxidizable component in the liquid phase and hydrodesulfurization reactions first order in hydrogen. Equation (15) is linear and analytical solutions are possible. The concentration of A in the liquid and in the gas leaving cell j can be obtained by combining eqns. (15) and (37). The results are 13~[1
(35)
and
(aj,L
%,B
-eXP(-Q/N)]%-1.g
a&L= 1 + LY,,A~~,JN
[l-exp(-$)I
+
aj-1.L
+dA[l-eXp(--A/N)]
(40)
and (36) Equation (34) can be used to evaluate integral in eqn. (14). This gives Oj L =
aj-1.L
+PA 11 -eXPf-e~/N)l
1+
PA [I-
eXp(-
the
aj-1.g + (%,A/N)aj,I ~A/N)I
+
(~S,A/N)
B is b.J,L
bj-1.L +PB[~ -eXp(-a~IN)l =
1 + PB[~
--exP(-@B/N)]
bj-1.g + (%,dN)bj,I
+
(as,~/N)
(38)
95
+Uj,L
Uj,g=Uj_l,gC!XJ3(-$)
1.0
[l-exp(-a$)]
-
-
I
- - Plug-Flow Solulion[2]
(41) 0.1
where QL,S
b’ 1
=I-
0.2
(42)
1 + k*d%,A
It is interesting to examine two limiting cases of eqns. (40) and (41), which are approached in many practical situations. 1. (YAapproaching zero This situation occurs when the gas-liquid mass transfer is rate limiting, when the gas rate ug is very large, or when the solubility of the gas A in the liquid is very low. Under these conditions eqn. (40) can be shown to reduce to a simple form; that is, (%P*/Na0,g 'jvL = 1 + %,AVL,S/N
+aj-1.L
(43)
0.1 1
2
3
5
20
10
30
SO
Fig. 2. Concentration of gas A leaving the reactor as a function of the gas-liquid mass transfer coefficient (PA = 10, k* = 50,a,,A = 50, &, = O.l,uo,L = 0, = 0): broken curve, plug-flow solution [ 21. m=l,n 1.0 r
I ‘0.L
=o
I
I
ao,L=o.5 0.95
I3 A
a.)0
+ ~AOA/N ao L’1.0 0.15
aj,g
N aj-1.g N aO,g
(44)
where
0.00 I 1
I
I
4
1
NUMGER OF MIXING CELLS,
(45) Equation (44) implies that concentration changes in the gas phase are very small. 2. (YAapproaching infinity This case corresponds to very rapid mass transfer between the gas and the liquid phase. Equations (40) and (41) now reduce to @A aj
L
+l)%l,L
=
1
+
9,g =aj.L
Q,,A'?L.~/N + PA
(46) (47)
Equation (47) corresponds to a physical situation such that gas and liquid are in equilibrium everywhere in the reactor; gas-liquid mass transfer no longer influences reactor performance. Numerical calculations show that an approximate criterion for neglecting such gas-liquid resistances iS aA > 4.6N. For intermediate VdUeS Of aA, eqns. (40) and (41) should be used. Numerical results are illustrated in Fig. 2 for a particular case represented by the parameter values shown on
I
10 N
Fig. 3. Influence of inlet liquid concentration of A on conversion for various degrees of backmixing for %,A = 50, k* = 1.0, aA = o.l,pA = 10.0, $0 = 0.1, m=l,n=O.
the figure. The concentration of reactant A in the gas leaving the reactor is given for various values of the gas-liquid mass transfer parameter ffA , and for different degrees of backmixing of the liquid, as characterized by the parameter N. The value of aqL corresponds to no dissolved reactant A in the liquid feed. The influence of liquid backmixing is seen to be small when there is little mass transfer between gas and liquid (low values of (YA),but is significant when eA becomes large. The results for large N correspond to plug flow of the liquid. The analytical solution for the plugflow model (obtained by Goto and Smith in eqns. (78) and (79) of ref. 2) is indicated by the broken line in Fig. 2. When N = 10, plug flow is c!osely approached. Figure 3 shows the conversion of gaseous reactant A for conditions such’ that gas-liquid mass transfer and chemical reaction at the catalyst surface
96
control the overall rate. The corresponding parameter values are (Y,,A = 50, k* = 1.0, aA = 0.1, PA = 10.0, and f#~s= 0.1. Results are given for three concentrations of A in the liquid feed. The influence of liquid backmixing (value of N) is small in all cases. However, it is interesting to note that for (I~,~ = 0 (no dissolved A in the liquid feed) the conversion is a maximum at an intermediate level of backmixing corresponding to N = 4. This result, which was also predicted by Goto et al. [ 11, is due to the possibility of the reactant concentration increasing because of backmixing in the liquid near the reactor inlet. This is possible when there are two flowing streams with mass transfer between them. Such a phenomenon cannot occur in fixedbed reactors with a single fluid phase; plug flow always gives the highest conversion, for other than zero-order kinetics.
RESULTS FOR NON-LINEAR
KINETICS
In this section, the effect of reaction order with respect to A is analyzed, first for n = 0 and variable m, and then for the second-order reaction n = m = 1. Figure 4 shows the effluent concentration of reactant A for m = 0, 1,and 2 for the following values of the parameters: LYAPA= 100; CY,,A= 50; k* z 50; Go = 0.1 and uqL = 0. The results
Fig. 4. Effect of reaction order (with respect to A) on concentration of A in effluent from a trickle-bed reactor; with 10 cells (N = 10) for (YAPA = 10, k* = 50, 40 = O.l,%,A = 50, aO,L = 0, n = 0.
s
0.1
m=O 0.05
\
0.02
0.01 1
2
5
10
20
50
100
-A
Fig. 5. Effect of reaction order (with respect to A) on exit concentration of A in a slurry reactor (N = 1); (IIA~~A = 10, (Y,,A = 50, k* = 50, Qo = 0.1, ao,L = 0, n = 0.
are presented as a function of gas-liquid mass transfer coefficient cA for N = 10, which represents the behavior of a trickle-bed reactor with nearly plug flow of liquid. The other extreme (N = 1) corresponding to a slurry reactor is given in Fig. 5. In both cases the conversion of reactant A decreases with increasing reaction order, although the effect is greater for plug flow (N = 10). These results for a reaction of mth order in A would be approached in practice if the parameter g (eqn. (32)) is small compared with unity; that is, when there is a large excess of B in the liquid phase. An example is an oxidation reaction with a large excess of liquid-phase reactant. The above condition may not be achieved in some practical situations. For example, in the oxidation of organic pollutants in water, the concentration of the liquid-phase reactant (pollutant) may be low so that the reaction rate depends on the concentration of both reactants. The influence of the initial concentration of the two reactants (3 and B), as measured by the parameter q, is shown in Fig. 6 for a second-order reaction, m = n = 1. The computations were made for a nonvolatile. reactant B, so that q is equal to uA&HABO,~. Figure 6 demonstrates that the conversion of A decreases as the ratio AO,g/BO,g increases, owing to depletion of reactant B.
97
I
I
___m_ Tiickla-Bed
I
REACTANT RATIO.q
Fig. 6. Effect of reactant ratio on conversion of A ; a/, = lo,fl~ = 10, k* = 50, (Ys,A = a,,$ = 50, $0 = 1.0, a0.L = 0, b0.L = 1.0, m = n = 1.0.
Fig. 7. Effect of catalyst particle size on the performance of a slurry reactor (first-order reaction, m = 1, n = 0): broken line, trickle-bed reactor, N = 10.
SLURRY
R dum)2. Here, spherical particles are assumed in order to relate particle volume and surfaceto-particle radius. Results for three catalyst loading values (y = volume of catalyst/ volume of reactor) are shown in Fig. 7. The catalyst loading of 0.4 corresponds to that used in the trickle-bed reactor. The figure indicates that in some cases the increase in catalyst effectiveness factor for a slurry reactor more than compensates for the adverse effect of liquid backmixing. For example, for the high catalyst loading of 0.2, the conversion in a slurry reactor is more than that in the trickle-bed reactor if the catalyst particle size in the slurry is less than one-third of the particle size in the trickle-bed reactor. For lower, more reasonable catalyst loadings Rduny would have to be much less. However, such results demonstrate that slurry operation may lead to higher conversions in systems for which intraparticle diffusion resistance is very large. Conversely, for systems with negligible intraparticle diffusion, trickle-bed operation may be advantageous. For reactions of finite order with respect to A, but independent of the concentration of B (e.g., some cases of hydrodesulfurization), slurry operation may be preferable if conditions are such that liquid backmixing improves conversion.
VERSUS
TRICKLE-BED
REACTORS
Since the same type of equations are used to represent both slurry and trickle-bed reactors, the mixing-cell model provides a simple basis for comparing the performance of the two modes of operation. Such comparisons are given here in terms of expected conversions. However, practical factors are also important. For example, the small-size catalyst particles used in slurries reduce the importance of intraparticle diffusion resistance. Also the fluid-particle mass transfer area is large per unit mass of catalyst. In contrast, the residence time distribution of the liquid in a slurry reactor will normally be close to that for complete mixing, which will reduce the rate of reaction under some circumstances. However, from a practical viewpoint separation of catalyst particles from the effluent liquid can outweigh the net result of factors affecting conversion. As an illustration of expected conversions, we consider a first-order reaction (m = 1, n = 0) for which liquid backmixing adversely affects reactor performance. Chosen parameter values are oA = 10, fl_&, = 10, (Y,,A= 50, k* = 50, $0 = 10, and ao,L = 0.0. The predicted conversion in a trickle-bed reactor for these conditions is 0.58 for N = 10. The performance of a slurry reactor with particle sizes lower than those employed in a trickle bed is shown in Fig. 7. This figure gives the conversion in a slurry reactor as a function of the particle size ratio Rtrickle/Rdurry for the same values of the parameters, except Go = 1O(&~,,,/&i~e) and %,A = 50&ic,,/
FORMIC ACID OXIDATION
Goto and Smith [4] have measured conversions for the air oxidation of dilute aqueous
98
NOMENCLATURE
Feed Gate. IFume
A.1,&f
Acid)
A.I.1 LIQUID FLOW RATE,
cm3/s,c
al 25 “C, 1 rim
Fig. 8. Predicted conversion by mixing-cell model for the air oxidation of aqueous solutions of formic acid at 240 “C and 40 atm.
A&)
A0.g A.1-L
bi solutions of formic acid in a trickle-bed reactor. A commercial CuO-ZnO catalyst was employed at 40 atm and 240 “C. Mass transfer coefficients could be estimated from independent measurements, and reaction kinetics data were available for the same system. Hence, conversions could be predicted by the model presented here and compared with the experimental results. Plug flow was assumed for the gas phase. The mass transfer coefficients KLaL and ksas used were those reported by Goto and Smith (ref. 4, Part I). The effective diffusivity employed to account for intraparticle diffusion and the intrinsic rate of oxidation were available from the experiments of Baldi et al. [ 51. Using these data, conversions of formic acid were predicted by the mixing-cell model for N = 1,4, and 10. The results are presented in Fig. 8 as a function of the liquid flow rate. The experimental conversions are seen to lie somewhere between plug flow (N = 10) and complete mixing (N = 1) of the liquid. However, the operating conditions were such that the gas-liquid mass transfer resistance was relatively large. Hence, predicted results are sensitive to changes in Kr,aL, as noted by Goto and Smith [4] . Further, this parameter is difficult to establish accurately at 240 “C and 40 atm. Hence, the deviation between observed conversions and those predicted for plug flow may not be due entirely to the effects of liquid backmixing.
B.J,B
B.I.1 B.I.L
D, HA, k k*
kg kL ks KL L N 4
R
HB
dimensionless concentration of A in the jth cell, AjHA/AO,g gas-liquid mass transfer area per unit column volume, cm2/cm3 effective liquid-particle mass transfer area per unit column volume, cm2/cm3 concentration of A in the gas phase leaving the jth cell, mol/cm3 concentration of A at the catalyst surface in the jth cell, mol/cm3 local gas concentration at any point in the jth cell, mol/cm3 concentration of A in the feed gas, mol/cm3 concentration of A in the liquid phase leaving the jth cell, mol/cm3 dimensionless concentration of B in the jth cell, BjHB/Bo,g concentration of species B in the gas phase leaving the jth cell, mol/cm3 concentration of B on the catalyst surface in the jth cell, mol/cm3 concentration of B in the liquid phase leaving the jth cell, mol/cm3 effective intraparticle diffusivity of the species indicated by the subscript, cm2/S Henry’s law solubility coefficient of A and B, Ag/AL or Bg/BL reaction-rate constant, [ cm3/g s] [ cm3/mol] m+ n -’ dimensionless rate constant,
gas-film mass transfer coefficient, cm/s liquid-film mass transfer coefficient, cm/s solid-liquid mass transfer coefficient, cm/s overall gas-liquid mass transfer coefficient, cm/s height of the catalyst bed, cm total number of mixing cells into which the column is divided reactant ratio parameter defined by eqn. (32) radius of the catalyst particle, cm
99
external surface area of a catalyst particle, cm* superfic,ial gas velocity, cm/s superficial liquid velocity, cm/s volume of a catalyst particle, cm3 conversion of reactant A in the reactor axial position in the jth cell, cm dimensionless axial position, z/L
b UIA
VP xA z
.z*
7)
v PP
4J 40
Sl
catalyst effectiveness factor stoichiometric coefficient of B in the reaction density of the particle, g/cm3 Thiele modulus, defined by eqn. (12) parameter defined by eqn. (33) reaction rate per unit volume of catalyst particle, mol/cm3 s
Gl'eek aA,aB
%,A
PA,
7
EB
7 %.B
BB
dimensionless gas-liquid mass transfer coefficients defined by eqns. (25) and (26) dimensionless solid-liquid mass transfer coefficients defined by eqns. (29) and (30) parameters defined as UgHA/U= and ugHB/uL catalyst loading, volume of catalyst/volume of reactor void fraction in the bed
REFERENCES
S. Goto, S. Watabe and M. Matsubara, Can. J. Chem. Eng., 54 (1976) 126. S. Goto and J. M. Smith, A.Z.Ch.E. J., 24 (1978) 386. K. B. Bischoff, A.Z.Ch.E. J., 11 (1965) 351. S. Goto and J. M. Smith, A.Z.Ch.E. J., 21 (1975) 706, 714. G. Baldi, S. Goto, C. K. Chow and J. M. Smith, Znd. Eng. Chem., Proc. Des. Deu., 13 (1974) 447.