Nomenclature
superficial surface of catalyst, surface per unit volume of bed, sq. ft./cu. ft. = heat capacity of gas a t constant pressure, B.t.u./lb. O R . = heat capacity of solid, B.t.u./lb. O R. = rate of mass flow of gas, lb./sq. ft. hr. = heat transfer coefficient between gas and solid, B.t.u./ sq. ft. hr. O R. = equivalent thermal conductivity of gas based on total cross section of convertor (kea = 6 k g ) , B.t.u./ft. hr. a R. = thermal conductivity of gas, B.t.u./ft. hr. O R. = equivalent point-contact thermal conductivity of solid, B.t.u./ft. hr. R . = depth of catalyst bed, ft. = Prandtl number for gas, &,/kg = heat generation due to a chemical reaction, B.t.u./ cu. ft. hr. = Reynolds number, G/Q,$~). = cross-sectional area of convertor, sq. ft. - “time” ( T = ~ j p & J , hr. O F. cu. ft./B.t.u. = average catalyst bed temperature [ j ;T(T,z)dz], a R . = gas temperature, a R . = gas temperature a t z = 0, O R. = solid temperature, a R . = initial solid temperature, O R . = axial distance, ft. = dimensionless axial distance, z = x / L
= c 9
C, G h kea
X
Z
GREEKLETTERS
6
= void fraction of bed
At
= =
A2
?
=
P *a
= = =
PP
=
Peg
size of “time” step, hr. O F. cu. ft./B.t.u. size of axial interval viscosity of nitrogen, lb.mJft. hr. shape factor for catalyst particles equivalent density of gas, (pea = 6p,), lb./cu. ft. density of catalyst bed [ p e s = (1 - 6 ) p , ] , lb./cu. ft. density of gas, lb./cu. ft.
pa
=
7
= =
density of catalyst particles, Ib./cu. ft.
p s r p = density of gas a t 1 atm., 32’ F., lb./cu. ft. u
time, hr. space velocity a t 1 atm., 32’ F., cu. ft./cu. ft./hr.
SUBSCRIPTS = at axial point i inside packed bed
i
SUPERSCRIPTS I = quantity a t time t
+ At
literature Cited
Anzelius, A., 2.Angew. M a t h . Mech. 6 , 291 (1926). Bird, K. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p. 41 1, W’iley, New York, 1960. Furnas, C. C., Trans. A m . Inst. Chem. Engrs. 24, 142 (1930). Hellums, J. D., Churchill, S. W., “International Development in Heat Transfer,” p. 985, Am. SOC.Mech. Engrs., New York, 1961. Hodgman, C. D., et al., “Handbook of Chemistry and Physics,” p. 2267, Chemical Rubber Publishing Co., Cleveland, 1962. Maga, J . A,, Kinosian, J. R., “Motor Vehicle Standards, Present and Future,” SAE Automotive Engineering Congress, Detroit, 1966. Mickley, H. S.,Sherwood, T. K., Reed, C. E., “Applied Mathematics in Chemical Engineering,” p. 376, McGraw-Hill, New York, 1957. Schumann, T . E. W., J . Franklin Inst. 208, 405 (1929). Smith, J. D., “Chemical Engineering Kinetics,” p. 391, McGrawHill, New York, 1956. State of California Motor Vehicle Pollution Control Board, Los Angeles, Calif., “California Procedure for Testing Motor Vehicle Exhaust Emissions,” 1964. Willmott, A. J., Intern. J . Heat M a s s Transfer 7, 1291 (1964). RECEIVED for review July 11, 1966 RESUBMITTED July 12, 1967 ACCEPTED September 21, 1967 Division of Petroleum Chemistry, 152nd Meeting, ACS, New York, N. Y., September 1966.
A MODEL OF CATALYTIC CRACKING
CONVERSION IN FIXED, MOVING, AND FLUID-BED REACTORS VERN W. WEEKMAN, JR.
Applied Research & Development Division, Mobil Research and Development Corp., Paulsboro, N . J .
deal of exploratory work in catalytic cracking takes place in fixed-bed reactors, which are cheap and easy to operate on a laboratory scale. Most potential industrial applications, however, lie in moving or fluid-bed reactors. T h e present work provides a method for relating catalytic cracking conversion from one reactor configuration to another, based on the principles of reaction kinetics. Within the last 20 years various mathematical models have been proposed for catalytic cracking. Voorhies (1945) presented a model of coking and conversion behavior. Blanding (1953) gave a model of cracking behavior based on simple reaction kinetics. Using the Voorhies expressions, Andrews
A
90
GREAT
l&EC PROCESS DESIGN A N D DEVELOPMENT
(1959) compared the various reactor types. Froment and Bischoff (1961) treated catalyst decay in fixed-bed systems which result from series or parallel fouling reactions. Masamune and Smith (1966) described theoretically the interaction between catalyst fouling and diffusion. This paper describes the development of mathematical models that account for the temporary decay of the catalyst during use. The models are presented in terms of a dimensionless decay group and a dimensionless reaction group. Comparisons among isothermal, fixed, fluid, or moving-bed reactors reduce simply to comparisons among the appropriate dimensionless groups. Finally the proposed model is shown
The conversion of gas oils during catalytic cracking may be represented by a pseudo-second-order reaction coupled with a first-order decay of the catalyst activity. For plug flow in the gas phase and for vapor residence times small in relation to the catalyst decay times, it was possible to solve the defining partial differential equation to yield isothermal models for conversion in fixed, fluid, and moving bed reactors. Comparisons of these models and cracking results show that they successfully represent the experimental data over wide ranges of operating conditions. The models are presented in terms of two dimensionless groups: an extent of reaction group and an extent of catalyst decay group. Comparison of various reactor types simply becomes a comparison of the appropriate dimensionless groups.
to represent experimental catalytic cracking data very closely. The conversion of gas oil is defined here as the fraction of the original charge material which has been cracked to the gasoline molecular weight range or lighter. Since most catalytic cracking feed stocks have had the gasoline constituents distilled off, conversion reflects fairly well the actual “work” of the catalyst.
The nonlinear coefficient: B, represents the ratio of oil transit time to the time of catalyst decay, t,. For many gaseous reactions, the time of vapor transit will be negligible compared to the time of the fixed bed experiment-i.e., the time of catalyst decay-and thus B S 0. If the characteristic B is essentially zero, we may integrate the following ordinary differential equation along the characteristic 0 = a constant, provided that the catalyst decay is not extremely rapid.
Discussion
Defining Equation. An isothermal reaction taking place in piston flow through a reactor containing a time-decaying, non-diffusion-limited catalyst may be described in terms of the weight fraction unconverted, y, by the following continuity equation:
T h e reaction rate-decay term R(y,t) will be a function of the amount remaining unconverted and the time during which decay has taken place. If there is molar expansion, as during catalytic cracking, both the vapor density, p o and the velocity, Lrt, will be functions of the fraction of unconverted charge, y. For steady-state moving-bed and fluid-bed reactors, the time partial derivative is, of course, zero. For fixed beds one must, however, treat the full partial differential equation. FIXEDBEDS. Equation 1 may be recast in dimensionless form by defining a normalized time, based on the catalyst decay time, and a normalized distance, based on the over-all reactor length. If the molecular weight of the reactants always bears a fixed relation to that of the products, then (in terms of the ratio of reactant to product molecular weights, a, and the initial vapor density, PO) the actual vapor density is simply
T h e normalized version of Equation 1, including Equation 2 , may be written in terms of liquid hourly space velocity, s = FO/W7Pl)
B
bY be
bY = -- R ( y , 0 ) + bx p,s --
fP0
(3)
where (4)
Inspection of Equation 3 shows that the characteristic direction in the x , 0 plane is
d0 - = B dx
(5)
I n less formal terms we may say that the members of any given group of oil molecules traverse the bed so fast that they see catalyst of essentially the same age. Thus decay of the catalyst is slow relative to the vapor residence time. Equation 6, then, describes the conversion of the reactant in terms of both time and axial distance. For very low vapor velocities and/or very short catalyst contact times, the characteristic direction B may not be a p proximated by zero and Equation 6 can then be integrated along the nonlinear characteristic given by Equation 4. Fortunately, most gas-phase cracking reactions are run under conditions for which the characteristic direction B is well approximated by zero. REACTION RATE EXPRESSION.As the catalyst activity changes with time, the kinetic rate expression R(y, 0) will be a function of both normalized time of exposure to oil, 0, and the fraction of unconverted reactant, y. T h e rate expression is complicated further because the gas oil reactivity is itself a function of conversion. This depth-of-cracking effect is caused by the multiplicity of molecular types present in the original charge. Thus, the first molecules to crack have a much higher reaction velocity than molecules that crack subsequently. The net effect of such a phenomenon on a pseudo-first-order reaction is to increase the apparent order of the reaction, since the average reactivity of the unconverted feed will decrease asy decreases. If the vapor density is treated as a constant, the apparent order also increases, because the actual density decreases as the fraction unconverted gas oil, y,decreases. A pseudo-second-order reaction could satisfactorily account for these nonlinear effects; Blanding (1953) also found that similar kinetics described his data. I t was found that the rate of decay of catalyst activity was proportional to the remaining activity-Le., simple firstorder decay in time. I t was also the simplest decay function that gave a good representation of experimental data. Figure 1 compares the first-order decay function to some hexadecane cracking data obtained by Nace (1965) in a fixed-bed reactor. After approximately 20 seconds the decay is essentially firstorder. Defining a n intrinsic reaction velocity constant as k, and a decay velocity constant as a gives the following form VOL. 7
NO. 1
JANUARY 1968
91
n-HEXADECA
Stock
~
0
20
40
60
100
80
7
-
0
120
Figure 2.
TI ME-ON- STREAM, SECS
Figure 1.
5
IO 15 20 25 30 E X T E N T OF REACTION, A = K,/S
40
35
Time-averaged conversion for fixed beds
Rate of catalyst decay
Data obtained from Nace ( 1 9 6 5 )
of the reaction rate term:
R(y,e) = k,y2e-’B
(7)
x = at, The reaction velocity constant, k,, and the decay velocity constant, a, depend on the charge stock-catalyst combination but are independent of the conversion level. The initial charge concentration has been adsorbed into the reaction velocity constant, ko, since for any given charge material ko must be determined experimentally. The dimensionless decay group, A, is the product of the Thus it decay velocity, a, and the total time of decay, t,. represents the “length” of decay. Equation 6 may now be written as:
no replacement of solid phase but with continuous piston gas flow-can also be characterized by Equation 10. As the instantaneous reaction rate term, Equation 7, is independent of reactor configuration, this term applies equally well to moving or fluid beds. Of course, the time of decay now becomes the catalyst residence time in moving beds and the mean catalyst residence time in fluid beds. MOVING BEDS. For moving beds the time of catalyst decay at any position is the catalyst residence time for flow through the entire reactor multiplied by the fractional distance, x, traversed. Thus, for a steady-state moving bed piston flow reactor Equation 3 becomes:
As before, the decay group, A, is the decay velocity, a, multiplied by the catalyst residence time, t,, for transit through the entire reactor. Under the boundary condition, y(0) = 1, the solution of Equation 1 1 in terms of the instantaneous conversion (e = 1 - y) is
where
E -
A = -fPok0 = K- O PlS
A(I -e-””) X A ( l - e-””)
+
At the bed exit (x = 1) this expression becomes
s
The dimensionless reaction velocity group, A , is the reaction velocity multiplied by the vapor phase residence time and represents the “length” of reaction. Solution of Equation 8 along the characteristic (de/&) = 0, with boundary conditiony (0) = 1 for all e, gives:
A(l
€ =
X
- e-’)
+ A ( l - e-’)
(13)
Figure 3 presents a working plot of Equation 13. FLUIDBEDS. By assuming one has piston gas phase flow and perfectly mixed solids flow, an analytic solution of Equation l 0
9
The time-averaged conversion obtained by collecting and mixing the product flowing from the reactor (x = 1) is:
,-=1-g=1-
L’
Yde
(9.4)
Evaluation of this integral yields the desired solution for time-averaged conversion in fixed beds in terms of the dimensionless decay and reaction groups.
0
d
8
IW
7
a w
E
6
0
4
t, a
3
lA
2
sz
5
I .” 5
Figure 2 presents the time-averaged conversion as a function of the decay and reaction groups. A fixed fluidized bed-i.e., 92
I&EC PROCESS DESIGN A N D DEVELOPMENT
IO 15 20 25 30 E X T E N T OF REACTION, A = K,/S
Figure 3.
Conversion in moving beds
35
40
3 may also be obtained for fluid beds. If Id0 represents the d0 fraction of catalyst particles with ages between 0 and B (internal age distribution), then a n average reaction velocity constant may be defined as follows:
+
For perfect mixing the internal age distribution, I @ ) , is simple, e-O. Evaluation of the above integral for perfect mixing gives the following expression for the average reaction velocity constant:
For steady-state fluid beds, Equation 3 now becomes
9 -- -- A A
dx
+1
y2
For the boundary condition, y(0) = 1 Equation 16 gives the instantaneous conversion (E = 1 - y) in terms of decay and reaction groups
Ax ‘=l+A+Ax At the bed exit ( x = 1) this expression becomes simply
A ‘ = l + A + A Figure 4 provides a working plot of Equation 18 in terms of A and A. T h e catalyst decay occurring is assumed to proceed uniformly over the length of the various reactors; however, with high metal content feeds or highly asphaltic feeds, the decay may not be uniform, requiring a modification in the model. Comparison of Fixed, Fluid, a n d Moving-Bed Reactors. If there were no decay of catalyst activity, any vapor plug flow reactor would give the same conversion, regardless of catalyst contact time. For the fixed-bed reactor, time-averaged and instantaneous conversion would be identical for a nondecaying catalyst. Application of 1’Hospital’s rule to Equations 10 and 1 3 and inspection of Equation 18 reveal that the fixed, moving, and fluid-bed models all have the following limit as the catalyst decay goes to zero-i.e., A + 0.
This is also, of course, the steady-state solution of Equation 3. Equations 10, 13, and 18 have been used to predict the ratio of conversion for the different reactors as a function of the length of decay and length of reaction groups. Figure 5 compares fixed and moving beds and reveals that for first-order decay a t the same A and A, moving beds will always give more conversion than fixed beds. As the decay group becomes larger, fixed beds become increasingly less attractive. Figure 6 compares fluid and moving beds and shows that, for first-order decay a t the same A and A, moving beds always give more conversion than fluid beds. For high rates of decay they become almost identical. Likewise, as the reaction group A increases, they give almost identical conversion. Although moving beds give more conversion a t the same A and A, the fluid bed generally can be run at much higher catalyst-oil ratios than moving beds-Le., lower A-and they are also less subject to diffusion limitations-i.e., higher effective A. Figure 7 compares fixed and fluid beds and shows that a t low values of A and A fixed beds with first-order decay give slightly more conversion, but at higher values of the decay group, fluid beds give much higher conversions. As reaction group A gets larger, the curves begin to cross over, since as A + m the conversion in all three reactor types approaches 1.0. This effect can also be seen i n Figures 5, 6, and 7. Rapid decay by metals or high molecular weight aromatics found i n heavy gas oils may change the first-order nature of the decay. T h e generalized dimensionless plots given by Figures 2, 3, and 4 are also convenient for comparing processing conditions required to achieve a given constant conversion level.
IO
9
t ’ iI
I
I
’
I
i
I
”
0
2
6 8 IO 12 EXTENT OF REACTION, A ( I ) Time-averaged conversion
A
Lim Eq. 10, 13, 18 = -
I + A
X+O
i
,
I
4
14
16
Ratio of fixed-bed to moving-bed conversion
Figure 5.
I O
_
0
5
I5
IO EXTENT
Figure 4.
20
25
30
35
40
5
I 0
OF REACTION, A = K,/S
Conversion in fluid beds
,
, 5
~
15
IO
EXTENT OF DECAY,
Figure 6.
I
I 20
1
Ratio of fluid-bed to moving-bed conversion VOL. 7
NO, 1
JANUARY 1968
93
Comparisons of reactor variables reduce to comparisons of the appropriate dimensionless decay and reaction groups. T h e following four examples show just how simple these comparisons are. Compare reactor volumes required for the same conversion, catalyst residence time (constant A), and oil throughput. (Vol. reactor)l = A1 (Vol. reactor)z A* Compare catalyst/oil ratios required for the same conversion and space velocity (constant A ) . (Cat./oil)l = -A* (Cat./oil)* XI Compare space velocities required for the same conversion and catalyst residence time (constant A). 5
0
IO
EXTENT OF DECAY, (I)
x
15
20
(Space ve1ocity)l - A_1 (Space ve1ocity)z A!
TIME-AVERAGED CONVERSION
Figure 7.
Compare catalyst residence times required for the same conversion and space velocity (constant A ) .
Ratio of fixed-bed to fluid-bed conversion
(Cat. res. time)l (Cat. res. time)2
"
9
% W a:
8 7
z 6 >
8
5 w 0 W
a 4
W a: > 3 a W
-!H-
EXPERIMENTAL RUN TIME, MIN
2
0
I25
0
0 0
IO
20 40
-3
V
0 25
I
5
0
20 30 40 50 60 S, Liquid H o u r l y Space Velocity, V/(V)(hr)
8
0
70
Figure 8. Comparison of fixed-bed model and fixedbed zeolite catalyst data Charge. Mid-Continent gas oil Catalyst temp. 900' F. Equation 10
-
MODEL CONSTANTS
.9 1 0
2-
.%
g
.7
5
.6
9 W
3
0 Q E
.5 .4
W
p $
+
.3 .2 .I
0 0
10
20
30
40
50
60
70
80
90
S, Liquid Hourly Spoce Velocity, V/(V)(hr)
Figure 9. data
Comparison of fixed-bed model and fixed-bed zeolite catalyst Charge. Mid-Continent gas oil Catalyst temp. 900' F. Equation 10
94
IhEC PROCESS DESIGN A N D DEVELOPMENT
LI XP
As the reaction velocity constant, KO,and the decay velocity constant, a, are intrinsic parts of Figures 2, 3, and 4, they may also be used to compare catalysts and charge stocks. Both constants reflect the combined effect of charge stock and catalyst. For example, the amount of deactivation a catalyst has undergone during operation could be determined by making two identical runs: one on the fresh and one on the aged catalyst. The fractional activity loss would simply be the ratio of the activity groups, A , read from the appropriate reactor chart. Rating of catalysts based only on their activity can be hazardous, particularly when the test is made in a fixed bed unit and the application will be in a moving or fluid bed system. Table I shows how two hypothetical catalysts with realistic values of a and KO rank differently in terms of conversion depending on the type of reactor used. A more complete description of cracking catalyst must include its decay as well as its reaction velocity.
IW
i
-
Table 1.
Hypothetical Comparison of Two Catalysts in Fixed, Moving, and Fluid Bed Reactors Catalyst 1. K O = 30 hr.-l a = 20 hr.-l Catalyst 2. K , = 2 . 5 hr.-l a = 2 hr.-' Catalyst Fraction Conversion .vo. s t,n A X Fixeda Mocing Fluid 30 10 0.34 0.75 1 1 0.5 0.73 0.61 0.55 2 1 0.5 2.5 1 0.59 a Time-aceraged.
The moving-bed form of the model was checked against data from a laboratory moving-bed unit. Table I1 compares the model with the experimental data. Again the two required constants were obtained with a fitting technique. The fixed-bed data were obtained by operating the movingbed reactor as a fixed-bed reactor. Using only the constants obtained from the moving-bed runs, the fixed-bed model predicted the actual time-averaged experimental fixed bed reasonably well ; the maximum deviation was only 47, conversion. Nomenclature
Comparison of Models to Experimental Data. Comparison of the fixed bed model with a large number of fixed-bed catalytic cracking runs revealed that the analytic model was capable of correlating a wide range of conditions. T h e fixedbed reactors employed were the same as those used for routine catalyst evaluations in this laboratory and have been described (Plank et al., 1964). Only two fitted constants were necessary -namely, those for the reaction and the decay velocity. Figure 8 compares the experimental fixed-bed gas oil cracking data (Wojciechowski, 1964) with the fixed-bed model and a zeolite catalyst. The model successfully correlates the data over a 16-fold range in space velocity and a 64-fold range in run time. T h e t\vo required constants were fitted by a nonlinear optimization technique using a least squares criterion (Wheeling and Kelly, 1962). Figure 9 presents fixed-bed gas oil cracking data (Coonradt. 1964) plotted as space velocity us. the catalyst-oil ratio for a slightly different zeolite catalyst. The catalyst-oil ratio is the total catalyst volume divided by the total volume of oil pumped during the run time-i.e., t, = 1/P S. Again, the model successfully correlates the data over a wide range of conditions. I t also predicts a maximum in conversion as space velocity is increased a t constant catalyst-oil ratio. This phenomenon is typical of highly active yet rapidly decaying catalysts. Comparison of the two catalysts shows that the second one (Figure 9) is slightly less active and decays a t a sloner rate.
Table II. Comparison of Models to Experimental Data (Mid-continent gas oil, commercial TCC catalyst) a = 2.96 hr.-I K O = 1069 e-Q'RT = 2 . 9 at 900" F. Q = 9000 cal./g.-mole LHSV, V . / ( V . ) Cat./Oil. Cat. Res. Temp., 6, e, (Hr.) V./V. T i m e , M i n . a F. Model Exptl. MOVING BED 2 4 7.5 900 0.53 0.55 4 7.5 950 0.58 0.58 2 15 950 0.53 0.50 1 30 950 0.47 0.50 1 30 905 0.42 0.41 2 15 903 0.48 0.45 4 7.5 850 0.47 0.49 1 30 950 0.47 0.47 FIXEDBED (Time-Averaged ) 4 0.75 20 900 0.32 0.33 0.375 40 900 0.25 0.27 1.5 10 900 0.36 0.39 2 1.5 20 900 0.47 0.43
ratio of reactant to product molecular wt., M , / M , dimensionless reaction group, ratio of reaction to space velocity, pofk0/plS p0fPalpiLv a(1 - j ) I 2 fraction voids in catalyst bed lb. reactantjhr. coking constants intrinsic reaction velocity constant a t 0 = 0, hr.-I (includes initial charge concentration because of second-order reaction)
+
POfkO/P,
instantaneous kinetic rate expression liquid hourly space velocity, vol./(vol.) (hr.) clock time, hr. catalyst residence time, hr. temperature, R. velocity of oil vapor, ft./hr. volume of reactor, cu. ft. normalized axial distance, z / z , instantaneous weight fraction charge unconverted time-averaged weight fraction charge unconverted axial distance in reactor, ft. total reactor length, ft. Greek letters
decay velocity constant, hr.-l catalyst to oil ratio, vol. of cat./vol. of total oil for fixed beds, (vol. of cat./hr.)/(vol. of oil/hr.) for moving and fluid beds instantaneous weight fraction converted to gasoline and lighter time-averaged weight fraction converted to gasoline and lighter normalized time-on-stream, t / t m dimensionless decay group, C Y / P S or at, initial charge density a t reactor conditions, lb/cu. ft. density of reactor vapor phase, 1b.icu. ft. density of liquid charge at room temperature, lb./cu. ft. Cited
.4ndrews, J. M., Znd. Eng. Chem. 51, 507 (1959). Blanding, F. H., Ind. Eng. Chem. 45, 1186 (1953). Coonradt, H. L., Research Department, Mobil Oil Corp., Paulsboro, N. J., private Communication, 1964. Froment, G. F., Bischoff, K. B., Chem. Eng. Scz. 16, 189 (1961). Masamune, S., Smith, J. M., A.I.Ch.E.J. 12, No. 2, 384 (1966). Nace, D. M., Research Department, Mobil Oil Corp., Paulsboro, N. J., private communication, 1965. Plank, C. J., Rosinski, E. .J., Hawthorne, W. P., IND.END.CHEM. 3, 165 (1964). PROD.RES.DEVELOP. Voorhies, A., Jr., Znd. Eng. Chem. 37, 318 (1945). Wheeling, R. F., Kelly, R. J., Research Department, Mobil Oil Corp., Princeton, N. J., private communication, 1962. LYojciechowski, B. \V., Research Department, Mobil Oil Corp., Paulsboro, N. J., private communication, 1964. RECEIVED for review May 1, 1967 A C C E P ~ ESeptember D 29, 1967 Division of Petroleum Chemistry, 154th Meeting, ACS, Chicago, Ill., September 1967.
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