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Mitigation of backmixing via catalyst dilution
This communization offers a new proof to an old problem, that of mitigating backmixing in fixed-bed reactors through catalyst dilution. This bed dilution strategy has often been practiced in laboratory and pilot-plant studies, Physically, one can easily see why bed dilution would mitigate backmixing. ~athemati~~~~, however, a rigorous proof of the mitigation effect, to our best knowledge, does not appear to be available in the open literature. The only exception is the work of Mears (19X) who used a perturbative nnatysis to estimate the benefit of bed dilution for a plug-flow reactor with superimposed axial dispersion. His analysis is necessariiy restricted to the case of smatl deviations from plug flow. While Me& result is quite useliil, it is worthwhile to lode into the retime of large deviations from plug flow. The reason is that, when conductmg sm&scale experimmts,one surely wants to avoid large deviations from plug flow. In practice it is seldom possible to estimate kinetic paramerers to on accuracygreaterthan, say, 5-10%. Here we revisit the problem by using a cells-in-series model. That is, we approximate the fixed bed by a series of N continuous stirred tank reactors (CSTRs) of equal volume. The deviations from plug Aow ma]*or may not be anrge. A key assumption is that the dilution will not change the hydradynamic characteristics of the reactor. Qf course, this represents an oversimpIif%ation, especially in %hecase of frickiebed reactors for which bed dilution (with an appropriate amount of inert particles) also improves wetting and reduces charmelting_ In this and similar cases, the present result may be used as a lower bound on the potential benefit from bed difution. In what follows we first provide a proof for firstorder reactions. With thii simple system, one can g& same feet For the structure of the problem. We then proceed with the proof for arbitrary mth-order reactions with m z- 0 and m + 1. A few numerical examples are given to illustrate the results. FIltST-ORDER REACTION A first-order reaction A + B is assumed to take place in an isothermal packed-bed reactor with a total catalyst volume of V,. For the undiluted bed, the concentration of A at the reactor exist, C,, or the concentration of A leaving the Nth CSTR, is (Levenspie!, 1972)
wbGre C, is Ihe ~~R~~~~~tiVR of A irt tile i=%Xd, z the space time in each of the CSTRs, and k the volumetric rate constant. Now consider the diluted bed. Suppose that the eaatalyst particles are uniformly mixed with inert hateriais to give a total volume of V,, we can define a catalyst dilution factor R % V,/V,, with R > 1. We assume that fhe effect of dilution is to lower the volumetric efficiency of each CSTR by a factor of R while increasing the total number of reactors to NR, It
folIows that the concentration c X=, becomes
ofA
leaving the NRth CSTR,
We want to show that C,, 4 C,. written 8s
Equation
(21 can be
Comparing eqs (1) and (3f, one obtains c c R<-P[ C* C, or, in terms of the fractional - C,/C,):
(4) conversion
&, I& r 1
5NRz= &J’ (9 That is, the diluted bed converts more A than the undiluted bed, due to a decreased cuneentraiion gradient along the bed. It can be shown that, as R -* m , the performance of the difuted bed approaches that of the plug-flow reactor:
where f is the overall space velocity of the system. nrTH-ORDER REACTIQN With suitable ~urn~tjon~* we have the ~ullow~ng mars balance equation: C,_,=C,CkK$,
mrO
and
m+L
0)
C, for any N is well detined since C + kaC”’ is strictly monotone for m 4 Q and maps (0, m) to (#, co). (However, this is not the Me when M < & C, d-s not exist for h’ &ger than a threshold value.) Moreover, it is easily seen that {C,} is a decreasing sequence since ktC$ s 0 implies that CM-5 z c,. Equation (7) says that C, can be viewed aa a function of/& and C,, i.e. C, = C,(kr, CA We are to show that, for an intege; R r 1:.
To show inequality scaling. Let
(81, we first reduce the problem by
1862
Shorter
x, = (kr)‘““-“C,. Substitution
Communications (9)
x,_,
= x,
+x;.
offwe
have that
(10)
Note that eq. (IO) no longer depends on k7. Thus X, = X,(X,), a one-parameter family of solutions. In terms of X,, inequality (8) becomes RI/W-
X*
1)~
N.8 Rl/o
1
i
We next express function f; where
inequality J(X)
c X,(X0).
(11) in terms of iterates
of the
inequality
(2t), with X replaced
by
we obtain
(12)
_ o f(X) be the composition times. Letf-‘(X) be the inverse of off-’ with itself N times, and Iet is well defined sincefmaps (0, cc ) increasing. Then eq. (10) becomes
L = fW,l.
X,-
From
(11)
= x + X”.
Let f”(X) =/ o S o f o (iteration) offwith itself N J,f-“(X) the composition p(X) = X. Note thatf-’ to (0, cc) and is monotone
Iteration
monotonicity
of eq. (9) into eq. (7) yields
Combining
eq. (23) and inequality
(25) one gets
(13) which showsinequality(l8)for proved.
of eq. (13) yields X0 = f”(X,)
N + 1. Thusinequality(18)
is
(14)
and hence x, Thus inequality
=f_“(X,).
(15)
(11) may be written
Rw-
1y-“R
~
1
X0
cS-“(X.).
[ R’/‘“-” In terms of X =fmN(X,),
f-"R [
as
inequality
f”(X) RI/‘“-
II
(16) becomes
1
X <--. R”+-”
Now we may operate on inequality to get f NR is monotonic, f”(X)
(16)
< R’““-‘yNR
[
(17)
(17) withfNR(
-&,
.), since
(18)
1
Thus it remains to prove inequality (18), which is equivalent to inequality (8). We first show that for X r 0, N z 1 fN(X)
> x + NX”
(19)
where equality holds if X = 0 or N = 1. Proceeding induction, suppose it holds for N. Then f”“(X)
=/C.Y(X)l
=f”(X)
by
+ Cf”W)l”
>X+NX”+(X+NX”)” >X+NX”+X”
NUMERICAL EXAMPLES Let us now define a dilution efficiency aN = CNR/CN. Figure 1 shows c+ as a function of R for different levels of backmixing, with kr = 1.0. As expected, catalyst dilution is more ellizctive for small N values (severe backmixing) than for large N values. Judging from Fig. 1, a value of R equal to 2 can be recommended for practical applications. Figure 2 shows the corresponding plot for second-order reactions with KzC, = 1.0. As expected, aN is more sensitive to R for secondorder reactions than for first-order reactions. Figure 2 suggests that for practical purposes a value of R = 3 seems appropriate.
CONCLUDING REMARKS What we have offered in this note is a rigorous inequality proving a “length” effect in CSTR cascade systems. The results is then used to in@ that, for positive-order reactions, a fixed-bed reactor with backmixing will produce a higher conversion if the catalyst bed is diluted. As noted before, diluting a fixed bed of catalysts does more than just mitigating the backmixing problems. Thus the bed-efficiency improvements seen in Figs I and 2 should be regarded as a lower bound. In any event, the present analysis should be viewed as a first step towards a more rigorous treatment of
(20) for
N z- 0. X z 0. 5
If?-=,
Thus we have proved inequality (19) with N replaced by N + 1. We can now prove inequality (18) for N = I, i.e. we show that
.fW) < R”‘=-“‘f” for R z=-1, X z 0. Using inequality
R”-ff&)
4
1& 1
54
(19) we have that
R~+&)
= x + X” +R[gTJrn}
=f(X)
‘2->
1 1
(22)
so inequality (21) is proved. We now prove inequality (18) for general N. Proceeding by induction, suppose it has been shown for N, then by
Fig. 1. aN as a function of R for different = 1.0, first-order reactions.
values
of N: kr
Shorter
Communications
1863
1.3or
NOTATION krC,
= 1
C0
C, CNR k ; R S v, K
I
/
2
I
/
I
4
6
6
reactor inlet concentration of A concentration of A leaving the Nth reactor concentration of A leaving the NRth reactor volumetric reaction rate constant reaction order number of CSTRs catalyst dilution factor space velocity of the overall system total volume of catalyst combined volume of catalyst and inert particles
Greek letters defined as % conversion conversion ::R Z space time
I 10
R
L-C,., G &JcN with undiluted bed with diluted bed in each of the CSTRs
Fig. 2. c+ as a function of R for different values of N: krC, = 1.0, second-order reactions. REFERENCES
the problem. The latter undoubtedly realistic Row models.
requires
de Bruijn, I. Naka and Sonnemans, J. W. M., 1981, Effect of the noncylindrical shape of extrudates on the hydrodesulfurization of oil fractions. Ind. Engng Chem. Process Des. Dev. 20, 4CL45. Mears, D. E., 1971, The role of axial dispersion in trickleflow laboratory reactors. Chem. Engng Sci. 26,1361-1366. Levenspiel, O., 1972, Chemical Reaction Engineering. John Wiley, New York. van Klinken, J. and van Dongen, R. H., 1980, Catalyst dilution for improved performance of laboratory trickleflow reactors. Chem. Engng Sci. 35, 59.
the use of more
T. C. HO Il. S. WHITE Exxon Research and Engineering Co. Annandale, NJ 08801, U.S.A.
Chemical Engineering Science, Printed in Great Britain.
Vol. 46, No. 7, pp. 1863-1868,
1991.
A generalized spherical multi-particle
axw2_%?9/91 13.00 + aoo 0 1991 Pcrgmm Pressplc
model for particulate systems: fixed and fluid&d b&S
(Received 10 April 1990; accepted@
There exist a number of models describing the dissipation of mechanical energy by a fluid flowing through either a fixed or a Auidized bed. While some of them are widely accepted and have a theoretical background others stand as reliable but empirical ones upon which realistic designs can bemade. But there are not many attempts at generalizing fixed-bed percolation as well as fluidized-bed expansion. Significant among them are the works of Barnea and Mizrani (1973), Barnea and Mednick (1979), Molerus (1980), Foscolo et al. (1983), etc. who all, through adopting different approaches, could achieve partial success in arriving at either a coherent representation or a generalization of these two flow cases. To develop a single expression for the dissipation of mechanical energy and to frame a generalized spherical multi-particle model for both of these cases are the objectives of this communication. A fixed bed, when the upward velocity of the flow through it is increased, becomes fluidized and the porosity or voidage, the characteristic property of the bed, which is approximately 0.4 in the former case, increases in the latter case and reaches unity when the bed is fully expanded, which means
publication IO October 1990)
that the particles suspended in the fluid behave like single isolated particles. Therefore, any relation describing a Auidized bed should be extrapolated, at unit voidage, to one applicable to an isolated particle. On the other hand, at the minimum-fluidization velocity, it is assumed that flow relations for the fixed bed would be applicable to the fluid&d bed. The approach in this work is based upon the flow under the minimum-fluidization condition first and then extrapolation of the results to fixed and fluidized beds. Under minimum-fluidization condition: The well-known pressure drop equation is AP 7 = (0, - P)(l - s)g.
(p. - p) can be replaced number.
by Ga 5,
where Ga is the Galileo
Then eq. (1) becomes AP = Ga Ifl(l L pd3
- e).
(2)
wbGre C, is Ihe ~~R~~~~~tiVR of A irt tile i=%Xd, z the space time in each of the CSTRs, and k the volumetric rate constant. Now consider the diluted bed. Suppose that the eaatalyst particles are uniformly mixed with inert hateriais to give a total volume of V,, we can define a catalyst dilution factor R % V,/V,, with R > 1. We assume that fhe effect of dilution is to lower the volumetric efficiency of each CSTR by a factor of R while increasing the total number of reactors to NR, It
folIows that the concentration c X=, becomes
ofA
leaving the NRth CSTR,
We want to show that C,, 4 C,. written 8s
Equation
(21 can be
Comparing eqs (1) and (3f, one obtains c c R<-P[ C* C, or, in terms of the fractional - C,/C,):
(4) conversion
&, I& r 1
5NRz= &J’ (9 That is, the diluted bed converts more A than the undiluted bed, due to a decreased cuneentraiion gradient along the bed. It can be shown that, as R -* m , the performance of the difuted bed approaches that of the plug-flow reactor:
where f is the overall space velocity of the system. nrTH-ORDER REACTIQN With suitable ~urn~tjon~* we have the ~ullow~ng mars balance equation: C,_,=C,CkK$,
mrO
and
m+L
0)
C, for any N is well detined since C + kaC”’ is strictly monotone for m 4 Q and maps (0, m) to (#, co). (However, this is not the Me when M < & C, d-s not exist for h’ &ger than a threshold value.) Moreover, it is easily seen that {C,} is a decreasing sequence since ktC$ s 0 implies that CM-5 z c,. Equation (7) says that C, can be viewed aa a function of/& and C,, i.e. C, = C,(kr, CA We are to show that, for an intege; R r 1:.
To show inequality scaling. Let
(81, we first reduce the problem by
1862
Shorter
x, = (kr)‘““-“C,. Substitution
Communications (9)
x,_,
= x,
+x;.
offwe
have that
(10)
Note that eq. (IO) no longer depends on k7. Thus X, = X,(X,), a one-parameter family of solutions. In terms of X,, inequality (8) becomes RI/W-
X*
1)~
N.8 Rl/o
1
i
We next express function f; where
inequality J(X)
c X,(X0).
(11) in terms of iterates
of the
inequality
(2t), with X replaced
by
we obtain
(12)
_ o f(X) be the composition times. Letf-‘(X) be the inverse of off-’ with itself N times, and Iet is well defined sincefmaps (0, cc ) increasing. Then eq. (10) becomes
L = fW,l.
X,-
From
(11)
= x + X”.
Let f”(X) =/ o S o f o (iteration) offwith itself N J,f-“(X) the composition p(X) = X. Note thatf-’ to (0, cc) and is monotone
Iteration
monotonicity
of eq. (9) into eq. (7) yields
Combining
eq. (23) and inequality
(25) one gets
(13) which showsinequality(l8)for proved.
of eq. (13) yields X0 = f”(X,)
N + 1. Thusinequality(18)
is
(14)
and hence x, Thus inequality
=f_“(X,).
(15)
(11) may be written
Rw-
1y-“R
~
1
X0
cS-“(X.).
[ R’/‘“-” In terms of X =fmN(X,),
f-"R [
as
inequality
f”(X) RI/‘“-
II
(16) becomes
1
X <--. R”+-”
Now we may operate on inequality to get f NR is monotonic, f”(X)
(16)
< R’““-‘yNR
[
(17)
(17) withfNR(
-&,
.), since
(18)
1
Thus it remains to prove inequality (18), which is equivalent to inequality (8). We first show that for X r 0, N z 1 fN(X)
> x + NX”
(19)
where equality holds if X = 0 or N = 1. Proceeding induction, suppose it holds for N. Then f”“(X)
=/C.Y(X)l
=f”(X)
by
+ Cf”W)l”
>X+NX”+(X+NX”)” >X+NX”+X”
NUMERICAL EXAMPLES Let us now define a dilution efficiency aN = CNR/CN. Figure 1 shows c+ as a function of R for different levels of backmixing, with kr = 1.0. As expected, catalyst dilution is more ellizctive for small N values (severe backmixing) than for large N values. Judging from Fig. 1, a value of R equal to 2 can be recommended for practical applications. Figure 2 shows the corresponding plot for second-order reactions with KzC, = 1.0. As expected, aN is more sensitive to R for secondorder reactions than for first-order reactions. Figure 2 suggests that for practical purposes a value of R = 3 seems appropriate.
CONCLUDING REMARKS What we have offered in this note is a rigorous inequality proving a “length” effect in CSTR cascade systems. The results is then used to in@ that, for positive-order reactions, a fixed-bed reactor with backmixing will produce a higher conversion if the catalyst bed is diluted. As noted before, diluting a fixed bed of catalysts does more than just mitigating the backmixing problems. Thus the bed-efficiency improvements seen in Figs I and 2 should be regarded as a lower bound. In any event, the present analysis should be viewed as a first step towards a more rigorous treatment of
(20) for
N z- 0. X z 0. 5
If?-=,
Thus we have proved inequality (19) with N replaced by N + 1. We can now prove inequality (18) for N = I, i.e. we show that
.fW) < R”‘=-“‘f” for R z=-1, X z 0. Using inequality
R”-ff&)
4
1& 1
54
(19) we have that
R~+&)
= x + X” +R[gTJrn}
=f(X)
‘2->
1 1
(22)
so inequality (21) is proved. We now prove inequality (18) for general N. Proceeding by induction, suppose it has been shown for N, then by
Fig. 1. aN as a function of R for different = 1.0, first-order reactions.
values
of N: kr
Shorter
Communications
1863
1.3or
NOTATION krC,
= 1
C0
C, CNR k ; R S v, K
I
/
2
I
/
I
4
6
6
reactor inlet concentration of A concentration of A leaving the Nth reactor concentration of A leaving the NRth reactor volumetric reaction rate constant reaction order number of CSTRs catalyst dilution factor space velocity of the overall system total volume of catalyst combined volume of catalyst and inert particles
Greek letters defined as % conversion conversion ::R Z space time
I 10
R
L-C,., G &JcN with undiluted bed with diluted bed in each of the CSTRs
Fig. 2. c+ as a function of R for different values of N: krC, = 1.0, second-order reactions. REFERENCES
the problem. The latter undoubtedly realistic Row models.
requires
de Bruijn, I. Naka and Sonnemans, J. W. M., 1981, Effect of the noncylindrical shape of extrudates on the hydrodesulfurization of oil fractions. Ind. Engng Chem. Process Des. Dev. 20, 4CL45. Mears, D. E., 1971, The role of axial dispersion in trickleflow laboratory reactors. Chem. Engng Sci. 26,1361-1366. Levenspiel, O., 1972, Chemical Reaction Engineering. John Wiley, New York. van Klinken, J. and van Dongen, R. H., 1980, Catalyst dilution for improved performance of laboratory trickleflow reactors. Chem. Engng Sci. 35, 59.
the use of more
T. C. HO Il. S. WHITE Exxon Research and Engineering Co. Annandale, NJ 08801, U.S.A.
Chemical Engineering Science, Printed in Great Britain.
Vol. 46, No. 7, pp. 1863-1868,
1991.
A generalized spherical multi-particle
axw2_%?9/91 13.00 + aoo 0 1991 Pcrgmm Pressplc
model for particulate systems: fixed and fluid&d b&S
(Received 10 April 1990; accepted@
There exist a number of models describing the dissipation of mechanical energy by a fluid flowing through either a fixed or a Auidized bed. While some of them are widely accepted and have a theoretical background others stand as reliable but empirical ones upon which realistic designs can bemade. But there are not many attempts at generalizing fixed-bed percolation as well as fluidized-bed expansion. Significant among them are the works of Barnea and Mizrani (1973), Barnea and Mednick (1979), Molerus (1980), Foscolo et al. (1983), etc. who all, through adopting different approaches, could achieve partial success in arriving at either a coherent representation or a generalization of these two flow cases. To develop a single expression for the dissipation of mechanical energy and to frame a generalized spherical multi-particle model for both of these cases are the objectives of this communication. A fixed bed, when the upward velocity of the flow through it is increased, becomes fluidized and the porosity or voidage, the characteristic property of the bed, which is approximately 0.4 in the former case, increases in the latter case and reaches unity when the bed is fully expanded, which means
publication IO October 1990)
that the particles suspended in the fluid behave like single isolated particles. Therefore, any relation describing a Auidized bed should be extrapolated, at unit voidage, to one applicable to an isolated particle. On the other hand, at the minimum-fluidization velocity, it is assumed that flow relations for the fixed bed would be applicable to the fluid&d bed. The approach in this work is based upon the flow under the minimum-fluidization condition first and then extrapolation of the results to fixed and fluidized beds. Under minimum-fluidization condition: The well-known pressure drop equation is AP 7 = (0, - P)(l - s)g.
(p. - p) can be replaced number.
by Ga 5,
where Ga is the Galileo
Then eq. (1) becomes AP = Ga Ifl(l L pd3
- e).
(2)