I
OCTAVE LEVENSPIEL and KENNETH B. BISCHOFF Illinois institute of Technology, Chicago 16, 111.
Backmixing in the Design of Chemical Reactors This article provides a useful perspective on the importance of backmixing in several types of reactors. Design charts are given which set limits for the practicing engineer in the field of reactor kinetics and design
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A
'4' ADEQ~ATE design of chemical flow reactors rests on knowledge of two factors-the over-all rate a t which the reaction proceeds and the extent of backmixing occurring in the reactor. T h e first factor involves the determination of the rate equation, which for homogeneous reactions may be found most conveniently by using small scale laboratory batch equipment. As backmixing does not occur in such setups, the effect of reaction rate may be isolated and examined alone. Backmixing may also be studied independently, as in hydrodynamic studies in which the progress of the fluid is examined as it passes through the reactor vessel. h-umerous techniques have been employed
and the results of the studies reported in many different ways. I n flow reactors backmixing, reaction rate and degree of conversion are tied together in a complex fashion. Because the role of backmixing is difficult to evaiuate in such situations, much present design is based on the assumptions that no backmixing occurs or that the contents of the system are well mixed and uniform in composition. Complete Backmix and Nonbackmix Reactors T h e idealized situation in which there is no backmixing is called plug flow, piston flow, slug flow, tubular flow, or
nonbackmix flow, and is characterized by the fact that flow through the reactor is orderly and the residence times of all fluid elements are alike. For this situation, the volume of reactor required to effect a fractional conversion, x , of reactant A is given by
where r is the reaction rate of A and F is the feed rate of A into the reactor. T h e other extreme in flow conditions is characterized by so great a n extent of backmixing that any fluid in the reactor has an equal chance of being found a t the reactor outlet. This idealized situation is called complete or total backmix VOL. 51, NO. 12
DECEMBER 1959
1431
Backmixing and Local Longitudinal Dispersion Number
Flow patterns in reactors may vary greatly; however, the resulting backmixing may often be characterized by a VrO -
IO2
V.
single dimensionless group--" the locaI longitudinal dispersion number" defined by Dlud, where D is the longitudinal dispersion coefficient. In the local longitudinal dispersion number, u is the fluid velocity and d is
I
STREAMLINE FLOW
IN PIPES D
Udt 0 01 f = I- X
01
IO
= FRACTION OF REACTANT REMAINING
Figure 1. The volume of a backmix reactor i s greater than that of the plug flow reactor required for the same duty. The size ratio rises rapidly with increase in both fractional conversion and reaction order
I
R e . Sc
E ?
I
I
lo3
IOZ
lo4
d t UP .
10'
=- udt
PDV
Dv
IO flow, or stirred tank flow; the volume of reactor required for conversion x of reactant A is given by V , = Fx/r
(2 1
T h e progress of many reactions may be approximated by the simple rate law r = kC"
TURBULENT FLOW IN PIPES
D udt
I
(3)
where n is the order of the reaction. For such cases, when fluid density remains unchanged, the comparison of sizes of reactors for a given feed rate is found from these equations to be
0.I
I
'
lo4
lo3
IO
IO6
I
I
1
I
(4)
This result (Figure 1) shows that except for a zero-order reaction the complete backmix reactor always requires a larger volume than a plug flow reactor for a given feed rate? and that the effect of backmixing becomes increasingly important for higher reaction order and for approach to complete conversion of reactant. Hence approximations to ideality which may be permissible a t low fractional conversions would lead to large errors a t high fractional conversions.
1432
INDUSTRIAL AND ENGINEERING CHEMISTRY
I
GASES
,
011 10-
lo-'
I
I
IO
IO2
lo3
dP G o P
IO,
I
I
I
PACKED BEDS
D UdpE
LIQUIDS
Partial Backmix Reactors Because a real reactor exhibits some degree of backmixing, the requiredvolume for a given duty should lie somewhere between the two extremes given by Equations l and 2. T h e problem then is twofold: to determine the extent of backmixing by a quantitative measure and then to use this measure with rate data to determine the necessary reactor size.
PACKED BEDS
I1 0.I
Figure 2.
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I IO
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IO2
I
lo3
Experimental findings on backmixing b y numerous investigators
local dispersion coefficients vary greatly with Reynolds numbers for flaw in pipes ( A and B) but are rather insensitive over wide ranges of Reynolds numbers far flow through packed beds ( C and D)
B A C K M I X I N G I N REACTORS some size characteristic-i.e., tube diameter for flow in pipes or particle size in packed beds. D/ud varies from 0 for plug flow reactors to for complete backmix reactors; a n increase in D/ud corresponds to a rise in the degree of backmixing. Dispersion groups in general are reciprocals of the corresponding mass transfer Peclet groups. For pipes and packed beds, D / u d correlates simply with the dimensionless groups characterizing the flow conditions. Figure 2 , A , is the correlation for streamline flow in pipes; both Reynolds and Schmidt numbers affect the local dispersion number. This relationship was obtained theoretically and verified by three experimental runs (70). Figure 2, B, the recommended correlation for turbulent flow in pipes, is based on numerous laboratory and field studies with both liquids and gases. Preparation and detailed discussion of Figure 2, A and B, may be found elsewhere (7). Figure 2, C, shows packed bed local dispersion numbers found by various investigators (7-4, 6, 8). T h e results suggest a larger local dispersion number for liquid systems as compared to gas systems. Figurc, 2, D. represents a comprehensive study of liquid dispersion in ordered and randomly packed columns and shows two distinct regimes of flow (5). Reactor longitudinal Dispersion Number Though D/udcharacterizes the intensity of backmixing a t any point in a reactor, the “reactor longitudinal dispersion number,” D/uL, must be used in conjunction with the reaction rate equations to determine the reactor size. L is some measure of the length of fluid path through the reactor. I n cylindrical reactors, this is measured most conveniently by the length of packed or
fluidized bed or of vessel in an all-fluid reaction chamber. For vessels in which the intensity of fluid mixing varies from position to position, such as the spherical reaction chamber or the vessel with large end effects, the effective length has yet to be determined. If the local dispersion number may be considered uniform throughout the reactor, D,’uL may easily be found. For a packed bed reactor D l u L = (D/ud,)(d,/L) (5) and for a tubular reactor D/uL (D/udi)(dt/L) (6) Chemical Reaction and Backmixing T h e differential equation governing the fractional conversion of reactant as a function of axial distance 1 in a reactor of length L is given by D -d2f - u df - - kCon-’f” = 0 (7) dlz dl in which f = 1 - x is the fraction of reactant remaining (9). First-Order Reaction. For a firstorder reaction, or rl = 1, Equation 7 has been solved ( 7 7 ) under the appropriate boundary conditions corresponding to any homogeneous reactor vessel or any catalytic reactor bed in which the intensity of backmixing is uniform (constant D,ud throughout). In dimensionless form the solution relates the fractional conversion with both the reactor dispersion number, D / u L , and the rate group for the first-order reaction, kL,’u. Thus
Figure 3, a graphical presentation of the above result in useful form, was prepared by combining Equations 1 and 8 and compares sizes of reactors required for a given feed rate for nonbackinix and partial backmixing conditions. Second-Order Reactions. T h e efFect of backmixing on reactor size requirement when a second-order reaction is taking place can be found by solving Equation 7 for n = 2. An analytical solution for cases of reaction order other than 1 is not kno1z.n because of the nonlinearity of the differential equation. Therefore, the differential equation was solved numerically on an IBM 650 digital computer. A fourth-order Runge-Kutta method was used, and i t was found that 25 increments were sufficient for the desired accuracy. Because the problem was of a boundary value nature rather than initial valuei.e., both the slope, df,‘dl, and the ordinate, f, were not known a t either boundary-the method of solution was necessarily trial and error. The process was started a t the reactor outlet where the slope was known to be zero ( 7 7 ) . A value of the ordinate a t this position was estimated and the Runge-Kutta method was used to calculate back to the reactor inlet boundary, a t which point the boundary condition
where f ( O + ) is the fraction of reactant remaining just within the reactor entrance, had to be satisfied to within a given error of 0.01-Le.. absolute values 1 - f(0-t) D/uL d f ( O + ) ’ d ( l ’L) ab-
+
200 100
60 40
V -
v 20 V.
V O
10
6 4
2 I 0 001
001
01
10
f = I - X = FRACTION OF REACTANT REMAINING
0.0I
1.0
0.I
Figure 3. The volume o f an actual reactor with a given D/uL compared t o that of the ideal reactor required for the same duty
f = I-X=FRACTION OF REACTANT REMAINING Figure 4. The volume of an actual reactor with a given D/uL compared to that of the ideal reactor required for the same duty
For a first-order reaction, analytical solution of the differential equation is possible (J J )
For a second-order reaction, a numerical solution of differential equation was obtained on an IBM 650 computer
VOL. 51, NO. 12
DECEMBER 1959
1433
solute values L 0.01, If it was satisfied, the answers were punched out and a new set of values were calculated for a different value of D/uL. If not satisfied, the initial assumed ordinate was corrected by the computer, and a new curve was calculated. This process was repeated until the boundary condition at the reactor inlet was satisfied. The last correctedvalue of the reactant concentration a t the outlet was used as the total conversion for plotting. Figure 4 compares ideal and actual reactor sizes for a given feed rate. I t was prepared by combining Equation 1 with the result of the trial and error numerical analysis described above. Fractional-Order Reaction. Interpolation between Figures 3 and 4 allows estimation of reactor size to effect a given conversion for fractional order reactions involving a single reactant. Backmixing does not affect a zero-order reaction. Example A gaseous material undergoes a complex series of changes when in contact with a solid catalyst. To investigate the kinetics of this reaction a small laboratory reactor is constructed consisting of a 1-inch schedule 40 pipe packed to a depth of 1.5 inches with solid ‘/l-inch diameter catalyst pellets. For a feed rate of 2.7 cubic feet per hour or GO = 49 Ib./hr.-sq. ft. the reactant is 99% decomposed. T h e reaction is of the first order with respect to the reactant. What depth of V,-inch diameter catalyst is required to yield a 9970 decomposition in a larger reactor to be constructed of 12-inch schedule 30 pipe if the feed rate is 4000 cubic feet per hour or Go = 547 Ib.,’hr.-sq. ft.? Additional information and assumptions: p = 0.01 cp. ; isothermal conditions throughout and no net change in the number of moles of material passing through the reactor; neglect any nonuniform flow patterns due to channeling a t the pipe wall. Solution. FOR I-INCHREACTOR.The particle Reynolds number
From Figure 2, C, the local dispersion number D/udp = 0.5
Thus the reactor dispersion number D / u L = ( D / u d p ) ( d p / L )= (0.5)(0.25/1.5) = 0.0833
iYow from Figure 3
-SVo/SV
E
VIVO
1.35
E
1434
=
CO
= initial concentration of reactant
d,
= particle diameter in a packed
FOR12-INCHREACTOR. T h e particle Reynolds number
d,
= diameter of pipe or tubular
D
= longitudinal dispersion
and from Figure 2, C, the local dispersion number D/ud, = 0.55. At this point the unknown length, L, enters in both variables S V and DjuL, so the solution involves successive trials. Assume to start that no backmixing occurs in this reactor. For this case using the value of S V o found above vel. feed/hr. SVO = 4860 h r . 7 = vol. reactor 4000 (0.797)L
Therefore the reactor depth L = 1.032 feet = 12.4 inches Check the assumption of plug flow made above. D/uL = (Djudp)(d,/L) = (C.55)(0.25/12.4) = 0.0111 From Figure 3
Therefore the assumption is justified, and L = 12.4 inches. From this example, we see that backmixing usually plays only a minor role in large packed bed reactors, although it may be significant in small laboratory reactors. Precautions A number of precautions must be observed in applying this procedure in the scale-up of process equipment. The first involves the isothermal requirement. As heat effects depend on the surface-volume ratio of the reactor, scale-up will result in larger heat effects and probably nonisothermal temperature distributions which must be accounted for by a complex analysis or a mean reaction rate constant. Another precaution involves the unjudicious extrapolation of backmixing data from small to large equipment. Gross flow patterns could vary considerably; this probably is the case between laboratory and industrial-sized fluidized units with their different degrees of bypassing of the fluid in the form of bubbles. T h e degree of backmixing may not be uniform throughout a reactor because of entrance effects, nonuniform cross section, etc. This may be dealt with by using a n average D/uL or going directly to the distribution functions from which the D values are obtained. Nomenclature = defined by Equation 9, dimensionless A = reactant C = concentration of A , moles A / cu. ft. a
but for the experimental run - = vol. feed/hr. sv vol. reactor [2.7/(7.5 X
Therefore, for the ideal case of plug flow 57’0 = 1.35 (3600) = 4860 hr.-1
3600 hr.-l
INDUSTRIAL A N D ENGINEERING CHEMISTRY
A , moles A/cu. ft. bed, ft. reactor, ft. coefficient or the effective axial diffusion coefficient, sq. ft./ hr. D, = molecular diffusion coefficient, sq. ft./hr. Dlud = local longitudinal dispersion number, a reciprocal Peclet group for mass transfer, dimensionless D/uL = reactor longitudinal dispersion number, a reciwocal Peclet group for mass transfer, dimensionless f = 1 - X, fraction of reactant A remaining, dimensionless F = feed rate of A into reactor, moles A/hr. = superficial mass velocity, 1b.l Go (hr.) (sq. ft.) k = reaction rate constant, moles A l--rr hr,-l = axial distance from entrance of 1 reactor, ft. L = length of pipe or reactor, ft. = order of reaction as defined in n Equation 3 1’ = reaction rate, rate of disappearance of reactant A , moles A / (hr.) (cu. ft.) = F/VCo space velocity, hr.-I U = average velocity of flow, ft./hr. V = volume of reactor, cu. ft. X = fraction of reactant A converted into product, dimensionless e = porosity, dimensionless = viscosity, lb./(ft.) (hr.) J ! = density, lb./cu. ft. P
(2T)
m
SUBSCRIPTS = nonbackmix, slug or plug flow 0 situation = complete backmix or stirred tank situation literature Cited (1) Carberry, J. J., Bretton, R. H., A.I.CI1.E. Journal 4, 367 (1958). (2) Danckwerts, P. V., Chem. Eng. Sci. 2, 1
(1953). (3) Deisler, P. F., Jr., Wilhelm, R. H., IND.END.CHEM.45, 1219 (1958). (4) Ebach, E. A,, Ph.D. dissertation, University of Michigan, Ann Arbor, 1957. (5) Jacques, G. L., Vermeulen, T., U. of Calif., Berkeley, Rept. UCRL-8029 (November 1957). (6) Kramers, H., Alberda, G., Chem. Eng. Sci. 2,173 (1953). (7) Levenspiel, O., IND.ENG.CHEM.50, 343 (1958). (8) McHenry, K. W., Jr., Wilhelm, R . H., A.1.Ch.E. Journal 3, e3 (1957). (9) Smith, J . M., “Chemical Engineering Kinetics.” Chap. 11, McGraw-Hill, New York, 1956. (10) Taylor, G. I., Proc. Roy. Soc. 219A, 186 (1953). (11) Wehner, J. F., Wilhelm, R. H., Chem. Eng. Sci. 6,89 (1956). RECEIVED for review May 1, 1959 ACCEPTED August 28, 1959
