L = length of pipeline, ft N = volume of gas (at standard conditions) in downstream surge, ft' P = pressure, lb, ft-' sec-' Q = volume flow rate (at standard conditions), ft' sec-' R = gas constant, lb, f t -' sec-' K - ' T = absolute temperature, K liquid phase velocity, ft sec-' volume of surge space, ft3 liquid level, f t Y = 2XnIDo GREEKLETTERS
u= v = x =
y = ratio of specific heats (= 1.4 for air) = density, lb, f t - 3 0 = time, seconds
p
Baird, M. H. I., Brit. Chem. Eng. 11, 20 (1966). Baird, M. H. I., Brit. Chem. Eng. 12, 1877 (1967). Baird, M. H. I.. Gloyne, A. R., Meghani, M. A. N., Can. J . Chem. Eng. 46, 249 (1968). Lemlich, R., Chem. Eng. 68, (lo), 171 (1961). McGurl, G. U., Maddox, R. N., IND.ENG.CHEM.PROCESS DES. DEVELOP.6, 6 (1967). Martinelli, R. C., Boelter, L. M. K., Weinberg, E. B., Yakahi, S., Trans. A.S.M.E. 65, 789 (1943). Massimilla, L., Volpicelli, G., Raso, G., Chem. Eng. Progr. Symp. Ser. 62 (62), 63 (1966). Milburn, C. R., M. Eng. thesis, McMaster University, Hamilton, Canada, 1969. Thornton, J. D., Chem. Eng. Progr. Symp. Ser. 13, 39 (1954).
SUBSCRIPTS
A D L G
literature Cited
Van Dijck, W. J. D., US.Patent 2,011,186 (August 1935). Ziolkowski, Z., Filip, S., Int. Chem. Eng. 3, 433 (1963).
= = = =
atmospheric conditions downstream liquid phase gas phase o = orifice T = test section L' = upstream I = connecting pipes
RECEIVED for review January 12, 1970 ACCEPTED May 15, 1970 62nd Annual Meeting, A.I.Ch.E., Washington, D. C.. November 16-20, 1969. The authors are grateful t o the National Research Council of Canada for financial support of this work.
Carbon Distribution Functions in a Fluid Catalytic Cracker S. M. Jacob Research Department, Mobil Research & Development Corp., Paulsboro, N . J . 08066
An iterative computational method was developed to obtain the carbon distribution functions for a fluid catalytic cracker. The FCC unit was idealized as a three-vessel system consisting of a backmixed regenerator, a plug flow riser reactor, and a backmixed reactor.
I n a TCC unit the catalyst is burned essentially clean of carbon on leaving the regenerator. I n contrast, because of extensive catalyst mixing, particles leaving an FCC regenerator vary in carbon content. Thus, samples of circulating FCC catalyst show wide carbon distributions, while those from a TCC unit have more uniform carbon content. The practical significance of such carbon distributions is that the yield, selectivity, and aging characteristics of a catalyst particle are affected by the amount of carbon present. Furthermore, the heat balance in an FCC unit is more sensitive to coke on catalyst than in a T C C unit which has catalyst cooling coils as a heat sink. An analytical solution was available in the literature for carbon distribution in a system of two perfectly stirred vessels, one acting as a carbon former and the other as a carbon remover. The current work extends this analysis to the case where a riser or plug flow reactor section is also present. The addition of a riser is important,
since carbon distributions are profoundly affected by the presence of this plug flow section. This paper provides a tool for describing the carbon distribution in fluid catalytic cracking units. The regenerator is considered t o be a perfectly mixed vessel in which the carbon on the catalyst is burnt off. Carbon is deposited in a plug flow riser reactor and a backmixed reactor, in that order. Carbon on each catalyst particle grows or is removed according to some prescribed differential law. I n the reactor, the rate of carbon deposited is represented by the coking law for a specific crude. I n the regenerator, carbon removal is measured by the rate at which it is burnt Off. Discussion
Mathematical Development. For a perfectly mixed vessel continuously fed and depleted by a stream of particles, Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 19iU 635
Behnken et al. (1963) have shown that the exit and inlet carbon distribution functions are related by the following differential equation,
DEPOSITION
where f(c,
t)
= distribution of carbon concentrations in ves-
sel and in exit stream, a t time t (concn)
'
g(c, t ) = distribution of carbon concentrations in feed stream at any time t , (concn) t = time, sec
c = concentration of carbon on catalyst R ( c ) = rate a t which carbon grows while in vessel, (concn)(sec) -' N = inventory of particles in vessel, particles Q = vessel feed rate (or take-off), (particles) (sec) fl = N I Q = mean holdup time of particles in vessel, sec Distributions f and g are formally probability densities in c, and 1 f ( c , t ) d c is the fraction of particles in the exit stream having concentrations in the range (X,Y ) a t time t. For a plug flow reactor the unsteady-state equation may be derived from somwhat similar considerations. The distribution functions are now specified as functions of concentration, time. and position in the reactor. f(c, Z ' , t ) = distribution of carbon concentration a t any time t and position 2' within plug flow reactor, (concn) 2'= Z / L = dimensionless length
and the related differential equation is (see Appendix for derivation)
CONCENTRATION
Figure 1. Exit and inlet carbon distribution functions for plug flow reactor
fi(C)
=0
for c > e* for c < c-
where c* = aG. Essentially, Equation 7 represents f 3 ( c ) as a translation of f l (c) in concentration space, a shrinkage of the distribution f l ( e ) in concentration space, and an expansion of the ordinate f l(c). By algebraic means it may be demonstrated that Equation 7 satisfies differential Equation 3. If, in addition, the rate of coke deposition R i ( c ) is represented by the coking law (Voorhies, 1945).
c = at:
+b
where b is a constant. Then,
And
I n the steady state, this may be written as
For the plug flowriser study the exit stream distribution, f$(c).must be obtained in terms of the inlet distribution. f1 ( c ) . From Figure 1,f l ( c ) d e represents the fraction of particles having carbon concentration in the range c and c + dc. With the reactor being plug flow then in time Oi (the residence time in the plug flow reactor). these particles are shifted to new concentrations c' and e' + de, which are prescribed by the rate law. Since the number of particles remains the same, f i
CARBON
(c)dc = f (c')dc'
where e* = [a' & + b' '1". Three-Vessel Steady-State Problem. The development is now advanced sufficiently to represent a combination reactor system in the steady state. This consists of two perfectly mixed vessels and a plug flow reactor. If we use the nomenclature indicated in Figure 2 , the following equations may be written using Equations 1 and 8.
(4)
Using a coking law given by Equation 5
c = at"
(5)
where a is the coking constant and t, is the catalyst residence time (Voorhies, 1945). Then, c' = [e' + a' IO,]^ or c = [(e')' '1 -a' " H (6) '1
1)'
and Solution of the problem requires obtaining functions f l , f?, and f l that will satisfy these three equations. A 636 Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4,1970
-
A
P E R F ECTLY MIXED REACTOR
f,
P L U G FLOW REACTOR R,'
83
so that
(C)
f , (C)
Figure 2. Idealized representation of FCC unit specific case is treated for which
f ? ( X )= 0 Substituting Equation 19 into Equation 15 and remembering that f 2 ( X )= Ofor X < c*,
Ri(c) = K ~ C dc c n - l R2(c) = - = a n dt a
ft(c")
and
fof c
These represent first-order burn-of€ of coke in the regenerator (Weisz, 1966) and coking characteristics in the reactor as indicated by Equation 5. The coking rate in the mixed reactor differs from that in the plug flow riser, in that the additional term b is absent. Equation l a may now be written as
< c* and
.[XL
'-b
1--(2'
~ f l ( Z ) e [ 1 / 0 2 ] (b () iZa ]+l 'dZdX
(20)
for c 2 c*. Computation Method. T o perform the desired computations, the cycle loop is broken and a function fll(c) is introduced. The problem is now reduced to solving the single integral Equation 20 with f l i ( c ) in place of f l ( c ) on the left-hand side of the equation. This entails iterating about a distribution f l ( c ) until
and
fil
Similarly, for Equation 9
1 O2an(c a )
f 2
=
( e ) = f l 'cc)
The problem is nontrivial because the computational scheme involves matching an entire function. However, in the absence of the plug flow reactor, and with b = 0, Equation 2 may be written as
and
fl(y)e(1/S2)b'/a)' " dvdX for X for X
f ? ( X )= 0
> e" < cx
(21)
For such a two-vessel problem, an analytic solution of Equation 2 1 is given by
Substitute for f 3 ( y ) from Equation 11 into Equation 17
x (1 - n ) / n e-(l/02)(xm). " f ? ( X )=
ne2(a)' ''
X
and is termed a modified gamma function. The modified gamma function is used as an initial guess for the numerical iterative scheme. The iterations are performed using
f i - ' (c)
=
fii(C)
for i > 1, where i represents the iteration number. Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970 637
400
Convergence is established when fA+'(C)
f:-'(C)
The system converges rapidly and solutions have been obtained for a number of different cases. Once f l ( c ) is calculated, f 2 ( c ) and fi(c) may be easily computed. Special Cases. CASEA. K181 > 1 as c 0 For Kl8l > 1 fi(c) m However, function f i ( c ) is still integrable-that is, J f i ( c ) d c is finite. I n fact, all the distributions are normalized so that f ' f l ( c ) d c= 1. CASEB. Klel = 1
-
n a
= 0.015 LB CARBON/LB CAT-(SE(
b
= 0.0015 LB
= 0.2
K,
300
-
-
CARBON / LB CAT
e , = 2.50
S2= 0.30 MINUTE
e3=
0.00 M I N U T E
E,=
0.01 127
c,= 0.00337 200
t
IO0
fl(C)
for c 2 c" for c < c"
= fl(C*)
0 O.(
30 LB CARBON / LB CATALYST
CASEC. KlOl < 1, f l ( c )+ 0 as c -+ 0 CASED. O2 = 0 (no stirred tank reactor)
Figure 3. Carbon distribution functions in reactor a n d regenerator Two-vessel problem
Substituting for f l from Equation 11 and letting 2 = [XI -a' &y - b so that C ( l ~ K ~-H1 ~ )
flk) =
KiOi IC'
X
fl(Z) [(Z + b ) ' "
+ a' f103]-(n'K101)dZ
(25)
-a' ' A , I - b
for c
2
c" and
for c
