he significance of distributed residence time in T tubular flow reactors o p e k n g in laminar flow was first emphasized by Bosworth ( Z ) , who also developed some analytical approximations for the effect of radial diffusion. The total literature in this area is relatively small compared with that of axial diffusion. Yet there appears to be an appreciable number of design situations where laminar flow reactors are utilized on the pilot or commercial scale and where segregation effects must be considered. The key ingredients appear to be: Existence of appreciable radial variation of axial velocity Large radial Peclet number (NPJ Requirement for a high degree of completion of reaction In our experience these elements are present and important in scale-up design of both truly laminar flow reactors and two-phase flow reactors where the light phase is nominally in turbulent flow but where appreciable radial variations of two-phase viscosity and/or density produce “pseudolaminar” profiles of axial velocity. Thus, flow need not be in a truly laminar regime, as long as the scale of turbulence is small enough to restrict radial transport. In what follows we present if computed results which show the separate effc
reaction order, degree of radial transport, velocity profile, and extent of conversion upon the deficit in conversion attributable to segregation effects. These idealized analyses are then compared with results of partial differential equation solutions of actual complex design problems to illustrate how the simplified results can be used as guidelines in design. EfFects of Reoclion Order
If diffusion is negligible, complete segregation solutions can be obtained for any reaction order or velocity profile by quadrature, as indicated by Aris ( I ) . Analytical integration leads to solutions which are combinations of analytical functions and the exponential integral Ez and hence are readily evaluated for various reaction orders, whenever the residence time distribution weighting function 2 toZdt/P is valid. This distribution function, although given by Bosworth ( 2 ) for Poiseuille flow, holds for any pattern where axial velocity is a parabolic function of radius, as in Equation 2a. We have extended this treatment to zero- and halforder reactions and to sequential first-order reactions A -t B -t C, where it is desired to minimize B in reactor product. Segregation for first- and second-order reactions in laminar flow have been calculated by Cleland and Wilhelm (3) and Denbigh (4, respectively.
W. M. EDWARDS D. I. SALETAN The lack of complete conversion to a desired product is appreciably greater in a laminar-flow reactor than in a plug-flow tubular reactor at the same average residence time. This effect, termed “slip,” is particularly important when high conversions are required. The authors present results from a study of ”slip,” and indicate their applications to problems of scale-up where pseudolaminar flow is present VOL 5 9
NO. 3 M A R C H 1 9 6 7
37
2i
’
6. StOUEIIIIAL
Flm ORDER. ks=2k. :2k;
10
x)
PLUG FLOW UP 6)
Figure 1. Effectof reaction ordm on slip in segregated lmninm Pow
PLUG FLOW SLIP
K)
__
Figurc 3. Effectof radial di@ion on dip fm a half-wdn rcocfion in lminnrj¶om
Whereas previous authors have tended to evaluate effects in terms of conversion of initial reactant, it een our experience that it is more useful to present mecation effects in terms of lack of conversion. or “ slip,” since recycle or recovery costs are proportional to slip in high conversion reactions. A useful method is to plot slip against plug flow slip at the same nominal residence time. Results for the various reaction orders in segregated flow are shown in Figure 1, compared against a 45’ line representing plug flow or complete radial mixing. For a second-order reaction the effect of segregation on slip is small. It may he shown from Denbigh’s analytical result that slip with a second-order reaction will approach of the plug flow slip in the limit as slips become small. For lower-order reactions, on the other hand, the ratio of slip to plug flow slip grows indefinitely as residence time increases. For a first-order reaction the plot assumes a slope of for low slips, whereas for half- and zero-order reactions the segregated slip approaches pseudoasymptotic values of 0.096 and 0.25, respectively. These are not true asymptotes, since, given enough residence time for the fastest moving center-line filament, zero slip will be obtained for even these-low-order reactions in segregated laminar flow. The log-log plot cannot, of course, show the situation for residence times sufficient to give zero plug flow slips. The results for slip of intermediate product B (relative to reactant A fed) in segregated sequential firstorder reactions, shown on Figure 1, merge into the firstorder case at extended residence times. At medium residence times these curves become double valued,
--
-
0.1
’/
PLUG FLOW SLIP @)
lmninmj¶om 38
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
where the concentration-time curve for the intermediate
-
-
component B in a batch system would be concave downward. Since our own work was completed we have noted that Gianetto and_Berbotto (5)have provided analytical solutions for both first- and secondorder sequential reactions in segregated laminar flow. Effects of Radial Transport
Radial diffusion will tend to mitigate segregation effects. For relatively low levels of radial diffusion the “Taylor diffusion” treatment is satisfactory, converting the problem into an equivalent plug flow system with axial diffusion (10). The treatment would be exact for a first-order reaction, but appears to be untested for other orders. The dimensionless group which serves as an index of the degree of radial mixing relative to reaction is the Damkohler number, a,which for a firstorder reaction is: a = D/kP
(1)
This may be rearranged to exhibit better the design variables:
I
where the “number of reaction units” k0 will be in the range 2-10 for our considerations, L / R will be 10 to 1000. Significant effect of radial mixing (0.01 < a < 1) therefore implies a range of radial Peclet number Ru.,/D of roughly 10 to 1000. Such values of Peclet number are consistent with Reynolds numbers in the laminar flow range, whether we are speaking of Schmidt numbers, p/pD, of 1 (characteristic of a gas) or 1 loa (characteristic of liquids). Such radial Peclet numbers are typical of two-phase “pseudolaminar” transport reactions as well (13, 15). Cleland and Wilhelm (3) obtained the effect of a on a first-order reaction in laminar flow by a numerical technique. Their results are plotted in Figure 2. Lauwerier (6) showed that the first-order reaction with radial diffusion could be solved by separation of variables and Wissler and Schechter (74)have furnished adequate eigenvalues for implementation, as well as showing how the result can be extended to sequential first-order reactions. For a second-order reaction the results of Vignes and Trambouze (72) would indicate that a (modified) Damkohler number D / k P C = 0.3 (where C = initial concentration times plug flow slip) will demarcate about half way between zero and perfect mixing (a Damkohler number defined on the basis of inlet concentration does not give as satisfactory an index). However, the effect of even complete segregation in a secondorder reaction is so small that consideration of degree of radial mixing need not be extensive. We have obtained digital computer solutions for both zero- and half-order reactions with radial diffusion, where segregation effects are highly significant to design. These results are plotted in Figures 3 and 4. As in the case of the first-order reaction, it appears that an a
2
4
10
B
PLUG FLOW SLIP (%)
Figure 4. Effect of radial diffusion on slip for lflmiflWpOZ#
n
uro-order reation in
-1
2 4 IO PLU6 FLOW SLIP 6)
20
Figwe 5. Effect of 0sloCity pro& on slip for afirst-order reaction in sepgated lominmJ?aw .VOL 59
NO. 3 M A R C H 1 9 6 7
39
I i
1
2 4 PLUG FLOW SLIP
F i w e 6. Efcct of vclocify pro@ on slip for segrqated laminarflow
10 a
20
half-order reaction in
of 0.1 corresponds to a slip which is roughly half way between complete segregation and plug flow and thus is a useful criterion for design purposes. An a of 10 is sufficiently large to give essentially plug flow conversions for all these reaction orders. The reader should be cautioned that there is, in practice, no such thing as a zero- or half-order reaction maintained all the way to completion of these reactions. Mathematically, zero slip can be reached in finite time, but in reality some other mechanism will intervene before this is achieved. One of our purposes in treating each of the elementary reaction orders is to be able to predict qualitatively how segregation will affect slip in a complex reaction, which changes apparent order over the range of conversion. Effect of Velocity Profile
Departures from cylindrical Poiseuille flow (where U m s x / U ~=v 2) can be as significant as segregation itself in causing increases in slip over what would he expected in plug flow at the nominal residence time. For example, in a half-order reaction where a plug flow slip of 0.01 would he expected, segregated Poiseuille flow gives a slip of 0.131, and a somewhat sharpened profile where center line velocity is 3.0 times average velocity gives a slip of 0.244. An a of 0.1 makes the slip with this sharpened profile 0.128, about the same as completely segregated Poiseuille flow. Results from the IBM 7094 digital computer solutions for sharpened velocity profiles in first-, half-, and zero-order reactions are plotted in Figures 5, 6 , and 7, respectively. Except in the last of these, the curve for X = 3, a = 0.1 would superimpose closely on the X = 2, a = 0 curve. Although the radial profile of axial velocity will, in a partitular case, be derived exactly from a momentumbalance differential equation, we have found it useful in digital computation to fit the departures from Poiseuille flow by the quartic profile equation:
Here X represents the ratio of center line velocity to average velocity and can vary from 1.5 (flattened) to 3.0 (sharper than Poiseuille flow). A different approximating relationship, whereby velocity is parabolic in radius but finite wall velocities are postulated
x
u/u,
+ 2 (1 - X)(r/R)*
('W
is particularly useful for the flatter profiles. The effect of flattened profiles caused by power-law non-Newtonian flow behavior on various reaction orders for complete radial segregation has recently been presented by Novosad and Ulbrecht (7).
/ --__
a-0 - -- - - - - a-0.1; -u,/.u,
-
~~~
PLUG FLOW SLIP (5)
~~~
Physical Systems Where Segregation Effech Are Significant
" =
Figurc 7. E&# of u 8 1 ~ ~pr&i t ~ on slip for a zmo-order in lominorflow 40
= h
iNDUSTRlAL AND ENGINEERING CHEMISTRY
In true laminar flow, segregation will he the usual case, molecular diffusion rarely being great enough to give a values even as large as 0.1. Vignes and Trambouze (72) found some segregation even where natural convection considerably increased radial transport. Although it is frequently possible to arrange for a tur-
bulent flow regime on the commercial scale, interpretation of pilot results may still require dealing with segregation effects. Polymerization flow reactors combine laminar flow with low diffusivities. Moreover, the higher viscosity of the more completely polymerized material near the tube wall serves to sharpen the axial velocity profile considerably beyond Poiseuille flow., In our experience center line velocities as much as eight times average have been observed. Strong coupling of heat effects with reaction in these reactors makes our analysis less applicable to these reactors, however, than to two-phase transport reactors, where the presence of a heavy phase damps out temperature effects. In two-phase vertical transport reactors, despite the superficial appearance of vigorous turbulence, radial transport is really rather slow. For example, van Zoonen (15) found radial Peclet numbers of about 300 in a gas-solid riser about the same as in gas flow alone. Furthermore, preferential accumulation of the heavy phase near the wall gives a light (usually gas) phase velocity profile which is quasiparabolic rather than flat, center line velocities of 1.5 times ?verage and higher not being unusual (17). Our pseudolaminar segregation analysis will apply to reaction taking place in the light phase in such transport reactors. %
*
I
0
Hiach
in Scaleup
In scale-up to larger diameter flow reactors, Damkohler number, a, will normally decrease. At the same time, for a given radial pattern of physical property -i.e., effective viscosity-variation, the axial velocity profile may well grow sharper (unless we are dealing with true laminar flow at the pilot scale’s becoming turbulent in commercial design). The effect of scale on velocity profile can be indicated roughly by the occurrence of the Froude number in normalized formulation of the momentum balance (9) :
~
where y = u = q = r =
“reduced” radius squared = ( r / @ reduced axial velocity reduced viscosity reduced axial pressure gradient
= (I/PO)(P
+ bP/bz)
Sharpening of the velocity profile will depend, therefore, on how L/R is changed in scale-up, assuming that Peclet number changes little with scale. I n terms of reactor design we may be facing a situation where either:
W. M . Edwards and D . I. Sabtan are on the staff of the Engineering Department, Shell Chemical Go. The authors acknowledge the helpful discussions with C. H. Barkelew, C. V . Sterding, and J . R. Street of the Shell Developmmt Go.
AUTHORS
I
I
___
I
PILOT P1ANT;ESIlMAllOM A-1.6, a-0.1
rn COMMERCIAL DESIGN;E5TIYAIION
~
%regdon
I
A-2.5, a-0.001
%OR011 SEGREGATED REACUON, A-2.5 1
2
4
PLUG ROW SLIP (R)
Figure 8. Result5 of analog simulation gcct of scale on slip in a transport mctor m’th kinctics umying from =/t- to ‘/wrdcr (approx.)
( a ) Radial mixing on the pilot scale is adequate to give equivalent plug flow, but segregation is significant on the commercial scale, or (b) The pilot scale is partially segregated, but the commercial scale will be almost completely segregated The situation in ( a ) is more tractable, since fitting of kinetic parameters may be done with ordinary differential equations, while the design procedures, which require partial differential equations, can be done afterward for a limited number of explicit design cases. For the situation in (b), partial differential equations must be handled at the outset, but the final design calculations are relatively simple. It was our misfortune to encounter situation (b) in a design problem involving a two-phase transport reactor. Kinetics were complex, but approximated = / a order initially, approaching ‘/P order later because one of the reactants was present in excess. Figure 8 shows results of an analog computer simulation of two reactor scales. Solution of the partial differential equations involved was done by coarsemesh finite-differencing of the radius, while keeping length as a continuous variable. It will be noted that the commercial scale corresponds fairly closely to complete segregation, if allowance is made for variation of velocity profile from Poiseuille flow and inexactness of half-order kinetics. VOL. 5 9
NO. 3
MARCH 1967
41
tions. While it may be necessary to obtain exact solutions of the appropriate partial differential equations in a new "pseudolaminar" flow reactor design, the guidelines developed here can assist the experienced reactor designer in making initial estimates of the significance of segregation effects in his situation. Reactions which do not follow a simple kinetic order can be qualitatively estimated by sketching in a transition from one kinetic order to another. The curves of Figure 8, for example, could have been estimated by sketching in a curve initially intermediate between first- and second-order, asymptotically approaching the half-order result. There are strong parallels between the work presented here and the extensive literature which has developed over the past few years on dispersed phase reactions (8). There, too, the significance of segregation effects for lower order reactions has been stressed. NOMENCLATURE
D E .. .
Figure 9. Effectof wlon'ty p&le sharpening in scale-up on predicted ./in
g k L n
N
P
= diffusivity, molecular or eddy = kinemtic viscosity = gravitational constant
= rate constant = reactor length = reaction order = dimensionless group =mlressure
--t
R = radius u, U = axial vdoeity
Wansbrougb (73) found that reactor efficiency decreased as the '/% power of the group k&R/U, in the firstorder decomposition of ozone, catalyzed by hematite in a vertical 2-inch i.d. by %foot long gas-solids transport reactor. His estimated Peclet number of 700 indicates that he was operating at Damkohler numbers of the order of 0.3. However, the fact that the solids which catalyze the ozone decomposition were radially segregated, in a more or less inverse relation to gas flow, greatly compounds the effective degree of segregation. I n Wansbrough's system, at constant Peclet number and constant L / R , variation of the group kpsR/Dg is equivalent to varying the group k0 in our Equation la. Since, for a first-order reaction, k0 is linear in log slip for plug flow, it may readily be shown that Wanshrough's result is exactly what would be expected from the limiting '/% slope noted for curve 3 (segregated first-order reaction) of our Figure 1. I n flow reactors limitations on total length will frequently make it difficult to maintain Froude number in scale-up. If Froude number declines as radius is increased, the resulting sharper pseudolaminar flow profile may be the dominant factor in scale-up (9). Figure 9 compares slips calculated on the analog computer for a (nearly) first-order reaction with the theoretical completely segregated results at the velocity profiles expected for each scale (A = 1.6 on pilot scale and 2.5 on commercial scale). Both the commercial scale and pilot scale are essentially completely segregated for this fast reaction.
0
Discussion of Results
The largely theoretical results presented here have been applied by us in several significant design applica42
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
y
= normalized axial velacity, u/U., = normalized radius, squared (rlR)*
Greek Symbols = radial Damk6hler number, DILRaCa"" a q
0
x
p r
p
= = = = = =
normalized viscosity, p / k u nominal residence time
U,../U, viscosity normalized pressure gradient density
Subscripts a" = average Fr = Fraude ( N F ~ = d U . S / 2 gR) g = gas max = maximum-i.e., center line = inlet or center line 0 Pe = Peclet (Np. = 2 RU,,/E) s = solids REFERENCES (1) Ark> R., "Introduction to the Analysis of Chcmical Rcaefors," pp. 2916,
Prmtxce-Hall, 1965.
( 2 ) Bosworrh, R. C. L., Phil. Mag. J9, 8 4 7 6 1 (1948). 131 Cleland. ,Wilhclm. R. H..A.1.CD.E. J . 2. 489-97 (19561. . . I ~ ,
F.A.. ~. C., J. Ap I C h m 1,227-36 (1951); "Chemical RcnctorThcory,.' pp. 56-62, Cambridge &va& Prcsr, Cambridge, 1965. (5) Glaneffo,A.,Baboffo,G., Ing. Chim. (Ilor.) 1,79-88,111-12 (1966). (4) Denbigh, K.
(6) Lauwaia, H.A., AppLSti. fit. AS, 366-76 (1959). 0 ) Nouorad,Z., Ulbrechf, J . , C k ~ . E ~ g , S " . 2 1 , 4 0 5 - 1 1 (1966). (8) Rictcma, K., "Scgr,qation in Li uid li uid Diapcmionr," Admnco in Chm'col Enginrcing 5, Acadcrmc Press, New 1164. (9) Sfanling, C. V., "Two-Phase Flow-Thcory and E~gin-ng Decision," Seunfrmth Annull I n l i ( u l Larim, A.1.Ch.E. Mcctmg, Philadelphia, Deecmba 6,
?'mi
1965.
110) Tavlor. G.. Pror. ROY.Sm. A219. 186203 119531
(12) Vignea, 1. P., Trambouze, P. J., C h m . E.s. Sti. 1 7 , 7 3 8 6 (1962). (13) Wanrbrough, R. W., "Radial and Axial Reactant Gradimtr in a Vafical Transport Reactor,.' Se.D. The%$ MIT, S q t e m b a 1964. (14) W i s l a , E.H., Schechta, R. S., A p p l . S i . Res. A10,198-204 (1961). (15) Zoonen, D. van, Proc. 1962 Symposium on Infaaction Between Fluid8 ?nd Particier Third Congrcss of Eumpean Federation of Chemical Engincenng, Iartituti& of Chemical Engineerr, p. 64, London, Enghnd, lune 20-22, 1962.
