The joint use of a digital and anulog computer gave solutions of process simulations that were 60 times faster than real time. With the faster reaction of the computer, the muhachine interactions were improved in designing control schemes
Hybrid Computer Simulation of a Moving-Bed Cata1yst Regenerator V. W. WEEKMAN, JR. M. D. HARTER G. R. MARR, JR.
s part of a digital computer control project to A analyze and control a commercial thermofor catalyst cracking (TCC) process, it was necessary to study the dynamics of and to synthesize control schemes for the process. The kiln, which continuously burns coke deposits from the catalyst, is an important dynamic element; it transforms coke disturbances to catalyst temperature disturbances, which are recycled in the process. Thus, the goal of the control design was to minimize the disturbance recycle by maintaining the catalyst at a constant temperature as it flows to the reactor. Because it is not practicable to use commercial-scale process units to experiment with control schemes, we decided to study dynamics and control with computer simulation. To do so, we used a steady-state kiln model previously developed by Prater, Moulthrop, and Wei (4, in which the diffusion-limited kinetics of coke burning was accounted for. They had also shown that this model satisfactorily agreed with commercial kiln data. By adding the proper capacitive terms, it was possible to convert this model to dynamic form for use in control simulation. The resulting model is in the form of simultaneous nonlinear partial differential equations. At first, it was necessary to study dynamics for real time periods of 1 to 2 hours. To do this, digital simulation required large amounts of computer time (1 hour of real time required 2 hours of digital time) and allowed only a limited amount of control synthesis. The nature of the digital solution also made it difficult to visualize the dynamics and precluded any direct man-machine interaction in designing the control schemes. Because of these difficulties, simulation with a hybrid computer (integrated analog and digital) was adopted to speed solutions and improve the man-machine interaction. With an EA1 Hydac 2000, these simulations ran 60 times faster than real time. That is, the hybrid gave a 120-fold advantage in machine time, compared to a digital solution employing the method of characteristics. A recent article by Herron and von Rwenberg (3) has demonstrated a numerical method for solving equations similar to those in this wdrk. This method appears to be potentially more efficient than the characteristics method. The work reported here (completed May 1965) represents one of the first uses of hybrid computers to simulate the dynamics of a complex petroleum process.
Description of Process Model-The
TCC P r o c r r
Figure 1 shows a simplified flow diagram of the ,atalytic seetion of the TCC unit. The catalyst flows through the reactor and kiln as a compact moving bed in essentially plug flow. The oil contacts the catalyst in the reactor, and the products of reaction are disengaged from the catalyst at the bottom of the reactor. The coke, which is deposited on the catalyst during the reaction, is removed by burning with air in the kiln. After it passes through the kiln, the catalyst is blown by INDUSTRIAL A N D ENGINEERING CHEMISTRY
-1 = -1 + -1
an air lift up to a surge separator, from which it flows back to the reactor. There are three sections in the kiln: the plume burner, where adhering liquids are burned off; a countercurrent section, where the bulk of the coke is burned off; and a cocurrent section, where the remainder of the coke is removed. Both the flow rate and the temperature of the combustion air may be manipulated, as well as the distribution of air up or down in the kiln. The primary load disturbance on the catalytic section is the variance of the coking properties of the oil feedstock. As feedstock varies, so does the coke-on-catalyst leaving the reactor, causing a change in the heat released in the kiln, which in turn causes the temperature of the catalyst leaving the kiln to vary. The catalyst then passes back to the reactor, where any variation in its temperature causes a fluctuation in the rate and quality of cracked products. The results of a change in the coking properties of the feed are therefore fed back to the reactor as temperature variations. Catalyst temperature variations at the reactor inlet cause a change in coke formation (coking is activated by temperature), and so on. The kiln is the key dynamic (exothermic and amplifying) element in the reactor-kiln loop. It transforms coke disturbances to temperature disturbances, which are passed back to the reactor and can result in additional yield disturbances. For these reasons, the kiln was chosen for detailed process simulation and control.
where y s refers to the normalized slow coke concentration and Qs is the activation energy for the slow coke burning. The time required for oxygen to diffuse to the burning surface is short compared to the dynamics in the slowmoving catalyst flow and the slowly moving coke boundary; it can be assumed that the steady-state effectiveness factor is adequate for normal process disturbances (2). Further simplifying assumptions made in the model are plug flow of gas and solid phase, equal gas and solid temperature, and negligible heat capacity of kiln walls and internals. Comparisons to plant data have shown these to be reasonable assumptions.
THE KINETICS MODEL
THE KILN MODEL
The kiln model used was a modified form of the one developed by Prater, Moulthrop, and Wei ( 4 ) . Since a bead catalyst of approximately 1/8-inch diameter is used in the process, the diffusion of oxygen into the bead plays an important role in the kinetics. Two types of coke are recognized: slow (graphitic carbon) coke and fast coke, which is adsorbed material with high hydrogen content that burns much more rapidly than the slow coke. Fast coke burns off mostly in the top part of the kiln where the temperature is low (950' F,); no diffusion model is required for the fast coke. Weisz and Goodwin (5) have shown that the slow coke removal follows a shell burning mechanism, with the rate of burning limited at high temperatures by the rate of oxygen diffusion to the burning zone. At low temperatures, however, the slow coke burns evenly throughout the bead, and the rate is low enough that oxygen diffusion does not limit intrinsic burning. Prater, Moulthrop, and Wei ( 4 ) showed that the transition between intrinsic burning and shell burning may be adequately represented by a series resistance model. Thus the overall effectiveness of the catalyst 17 (ratio of diffusion-limited burning rate to intrinsic burning rate) is dependent on the intrinsic effectiveness factor qI and the shell burning effectiveness ? f & The total resistance to burning is then the sum of the intrinsic resistance and the shell burning diffusion resistance.
Heat and material balances yield the following dynamic kiln model equations.
V
VI
(11
?fsb
The V s b is dependent on the diffusion of oxygen into the bead, which is described by the solution of the oxygen diffusion equation. I t has been experimentally proved (5) that the intrinsic burning is first order in oxygen and coke concentration and follows the Arrhenius relation. Coupling these kinetics to the shell burning model with diffusion gives :
CQ
=
(ys2/* 0 - y J e -Q J R T Thus, since 71 = 1 by definition, the overall effectiveness becomes Vsb
~
1
These equations are badly nonlinear because of the Arrhenius terms and the high heat release of coke burning. They are also coupled by the burning rate terms.
W . Weekman is with the Systems Research Group of the Applied Research and Development Division at Mobil Oil Corp.'s Paulsboro, N . J., laboratory. Michael D. Harter and George R. Marr are both with Electronic Associates, Inc., Research and Computation Division, Princeton, N . J . The authors wish to acknowledge the advice and assistance of M . J . Depasquale of Mobil's Engineering Department, the contribution of Leroy Dahm and Irwin Etter of EA1 to the organization of the computer programs, and the editorial assistance of Judith Gorog of Mobil in the organization of the manuscript. AUTHORS Vern
VOL. 5 9
NO. 1
JANUARY 1967
85
I 0.2
FMCNOM OKIIYCE WWI KlU Figure 2.
Typical steady-state solution of kiln maid
The initial conditions on the fast coke, slow coke, and temperature equations are given; however, the introduction of combustion air one third of the way down the kiln gives a split boundary condition for the oxygen equation in the c o u n t m u m n t section. I t is necegsary to assume a value for the oxygen concentration z at the top of the countercurrent section, and, upon solution of the equations, the calculated oxygen concentration must match the known inlet concentration. This convergence requirement makes the model considerably more difficult to solve. The cOcurrent kiln section represents an initial value problem once the countercurrent section has been calculated. The steady-state version of the kiln model-i.e., capacitive terms zerc-gives typical profiles as shown in Figure 2. The cooling effect of the combustion air is assumed to take place only in the plane of the air inlets, thus the temperature profile exhibits a discontinuity. Since the thermal time constant of the catalyst particles is very small compared with their residence time in the kiln, this assumption is adequate for modeling the cooling efFect of the combustion air. From Figure 3 it can be seen that the model predicts coke profiles which compare favorably with full-scale plant data obtained by vertically probing the kiln. Digital Computer Solutions
Initially it was attempted to solve the dynamic model with digital techniques. To speed solution the equations were solved numerically by the method of characteristics (7). F m Equation 3 we see that the characteristics are OISlANCr D O H KIM
Figwe 3. Cornpm'mn of kiln modal lo plant &a
The characteristics of Equations 3a and 3b represent the ratio of coke-to+atalyst velocity, which is unity. That of Equation 3c is the ratio of catalyst-to-air velocity, approximately zero for most kilns. The characteristic of Equation 3d is essentially the ratio of enthalpy flow of the catalyst to that of the air plus (plus for the cocurrent zone; minus for the countercurrent zone) the catalyst. With more than 90% of the heat contained in the catalyst phase, this characteristic is approximately equal to one. Along each characteristic we solve only an ordinary differential equation, thereby increasing the speed and stability of the solution. For example, along the characteristic of Equation 3d the following equation holds:
E6
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
All characteristics are constant and three are equal, so the digital solution was greatly speeded.
Even so, the characteristic method was time-consuming; on an IBM 7040 computer, 2 houn of machine time were required to simulate 1 hour of real process time. These characteristic solutions exhibited the same dynamic features as were later observed with the hybrid simulation. Still by use of digital computers, an entirely different method was attempted. The kiln was divided into 30 stirred tanks to simulate the dynamics, which were described by a total of120simultaneous ordinary differential equations. Solution times were considerably longer than with the characteristic method, and even 30 stirred tanks were not sufficient to give convergence to a known steady state. Because of the long solution time required for the digital computer using the method of characteristics and the additional time needed by the control algorithm equations, hybrid computer simulation was adopted. Hybrid Compukr Solution.
Two basic methods are available for the solution of simultaneous partial differential equations ofthe form: bu - v du - = f(u,c)
de
+
bx
where u(x,8) is a state vector, v and c a& vectors, and f(u,c) is an arbitrary, autonomous vector function. The first is the so-called parallel method, whese the distance variable is discretized and the partial differential equation is approximated by the set of differenccdifferential equations: du
.
dea=
This scheme may be implemented using analog computer components alone by interconnecting n similar cells-each integrating in parallel a member of the family of differential Equation 8. When the rate terms of Equation 7 are complex and need many analog components for their generation, the total equipment becomes prohibitive (roughly n times that for each cell). The number of cells required depends on the accuracy needed to reproduce the axial profiles, and in this problem was estimated as at least 30. Boundary conditions (initial value with respect to time, mixed value with respect to distance) are readily imposed without need for iteration. This advantage has to be balanced against the large component count per cell. In the TCC kiln simulation, the rate terms alone required 50 ampliien and many nonliiear function generators and, consequently, would have used more than 1500 ampliers or about 10 consoles of analog equipment had the purely analog, parallel, method been adopted. Hybrid techniques allow time-sharing of complex nonlinear circuits, thus significantly reducing component requirements. The second, the serial method, discretizes the time variable, and the new differencdierential equation set is
- u(ui - u t i ) + f ( U d Ax
where ui = u(iaX,O); i = 1, 2, x 5 L.
. . .n;
(8)
naX = L; 0 5
.
where u’ = u(x,jA8); j = 1,2, . .N; NAO = time span of investigation. These equations may be solved by allowing the continuous analog integration to taLe place with respect to distance and by using the currently evolving distance p d e u’(x) and an “historic” profile uf-*(x) to obtain the required time derivative. Thus a single cell, used repeatally, suffices; however, the method requirea function storage and playback to produce the time derivative, which requirea a hybrid facility. V O L 5 9 NO. 1
JANUARY 1967
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Profiles are sampled at 100 equally spaced distance points, which are converted to digital form, retained in a bulk memory device until required, and later reconverted to analog form by a suitable interpolation scheme. Figure 4 illustrates a typical sequence of events. During the jth pass down the kiln, ~ ' ( 0 )to u'(L), the variable u'"(x) is played back, and u'(x) is simultaneously sampled for use on the (j 1)st pass. On any given time pass, the solution of the equations was straightfomard in the first and third sections of the kiln model because it was initial-value specified in all variables. However, the oxygen concentration is unknown at the entrance to the countercurrent section, and an iterative search is required to match the computed value at the combustion air inlet to the true oxygen condition at that point. During transient operation, this value changes relatively slowly, and a standard proportional-plus-integral error correction technique assured convergence. However, since not all transients require the same number of iterations, the solution time for a single time step may be expected to vary during a disturbance. The implication here is that there is a changing time scale relation between elapsed computation time and real plant time. Such a variation presents no difficulty if the partial differential equations alone are being solved. One of the main purposes of the simulation, however, was to study various control systems for the regenerator. These controllers are modeled as ordinary differential equations coupled to the partial differential equations of the kiln, so it was necessary to select a constant time scale. This was done by fixing the number of iterations performed in each real-time step, regardless of whether closure was obtained on the first iteration or not. Seven iterations were experimentally determined to be sufficient to ensure closure under the most severe disturbances. As a result of adopting the serial method substantially less analog equipment was required, although more
elaborate digital control programs were necessary; a function storage and playback facility was required; and iteration was necessary to satisfy the split boundary value conditions. To satisfy these requirements, a HYDAC 2000 was chosen as a hybrid computing configuration. I t provided a parallel-logic and serial memory capability in the digital section, and a general purpose analog computer, including individual integrator mode controls. The programs for both machines existed as removable patched panels. The organization of the various functions of the simulation is shown in Figure 5. The operator's interface consists of push buttons, function switches, dials, and preset counteradjustmenethus permitting flexible use of the simulation. Mode controls include a static parameter setting, initialization (including the preliminary writing into memory of the first "historic" profile prior to a dynamic run), run, and region selection (solution for just one regenerator region). The speed controls were used during volume production runs (Fast), transient profile examination (Freeze), and in the checkout stages (Slow). The rest of the programming was designed for automatic operation through a hierarchy of subroutines, both logical and analog. Apart from status monitoring through indicator lights, the outputs included high-speed, eight-channel analog recorders, and an oscilloscope display. The simulation was made quite flexible for two
+
reasons: -To permit an efficient system checkout, a fairly difficult task with any fint-generation hybrid configuration. This problem is much less significant with the present generation of hybrid computen -To allow for a broad variety of experimentation or of interactive designs, many of which could not be anticipated before the sensitivity experiments had been made
Figwe 5. W o n a l organization (Hydoc 2wo TCC simulation)
-
OPIUTORS INTERFACE MODI CONTROL PAKAMEIER SEI SPIED CONIRDL
...._... .7..I"..
DIGITAL COMPUTIR
INTERFACE
A I D CDNVERERS
EXBUTIVE PROGRAM
MULTIPLEXER
D/A CONVERlIRi
REGION COUNTER
CONIRDL LINES ITIRATION COUNTER
DATA LINES FUNCTION STDRAGI AND QLAYBACK
FASl
I
SERIALMEMORY M I A TIMING
INTIGRATOR MODI CONTROL
HIGH SPIED LOW W I D IHTIRPOLATORS TRACK /STOR~lPOINTI , . PARAMEIERS TIME-SCALE MAGNIIUDI SCALIS ADJUSIABLI "CONSTIUP' SWITCHING PARAMEIIRS FORCING FUNCTION LINEAR, NONLI-'- -"NCTION GINIRATORS 8-CHANNIL RICORDIRS OUTPUT OS~1110SCOPi DISPLAY 'ER
88
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
R.g.n.rotor Paformonce: Open Loop
In simulation studies of process dynamics and c o n t d , one generally uses the first period of simulation operation to obtain a general understanding of the dynamics of the plant. Much of this activity can, and should, be planned. It is especially important when studying open-loop performance under typical load disturbances. However, because the first period of simulation operation is also the first time that the mathematical model of the physical system is made visual and manipulatable, some unplanned experiments are bound to be made. The more direct the display of system performance, the easier it is for the process engineer to devise additional experiments. With the overall object of achieving control system design, the obvious first step was to test the simulated
NM
-
SINCE
nARI OF UM(MIK)
1.6 1.4
E 1.2
==1300 1.0
Y 0
81250
2
0
-15
15
Figurs 6. RU@K
".,
75
60
105
90
nm OF uw (urvraj to coke purSa
".7
".L
45
30
nyI NCSIE
V."
".V
(A1 uolt = A8.4 OF.)
I."
FRE9puWO [OaS/HL)
Figure 7. Response fa pulse in combustion air frmparatura (A1 wlf A8.4 OF., 73 V = 1300 OF.)
F i p 8 8. Eo& plot af coke s&8 facing ( l O . 2 urt.
1,EYTEUlUolE ABOVE C O O L M C O U
1250
c
-I5
__
,_._
om '.GL UOStD .,
Figure 70. il300 Comparison of conirol s c h s 3 (A1 mlf A8.4 OF.)
%)
)I
. V
100) LOOP II
regenerator sensitivity. Briefly, sensitivity studies include a quantitative look at how much a dependent variable, such as the output catalyst temperature, varies in response to change of some independent variable, or forcing function, such as combustion air temperature. With a graphic display of system performance, the sensitivity study is an unusual opportunity to observe the dynamics of some key variables. Critical internal ones, such as maximum catalyst temperature, diffusion effectiveness, or coke concentration profile, are observed on a routine basis to assure that nothing important is being overlooked, or to inquire into the physical basis of the overall response being observed. What one seeks is an understanding of the dynamics in a way that will suggest appropriate kinds of control system measurements and manipulations. The present study considered regenerator sensitivity to four forms of disturbances: change in the level of coke on catalyst; change in the combustion air temperature; change in combustion air rate; and change in the fast coke on catalyst (and the corresponding catalyst feed temperature). Step changes were applied to simulate gross changes in process conditions. Single pulses were applied to simulate common upsets of short duration. Sinusoidal variations were applied to simulate periodic disturbances recycling through the reactor-regenerator system. Only a sampling of the typical results is shown here. Figure 6 shows the temperature response of the catalyst at the air inlets and kiln exit to a “pulse” of higher coked catalyst. The first-order increase and decrease in coke simulates the result of a slug of higher coking tendency oil passing through the reactor. The response to this forcing function is a damped oscillation above the combustion air inlet because the countercurrent flow in this region allows the disturbance to be fed back by the counterflowing air. This “echoing” phenomenon was also observed in the digital solutions. The bottom or cocurrent section tends to filter out these secondary disturbances, and the main response above the cooling coils is a pulse of high temperature catalyst that has been widened by the air-flow effect. Figure 7 presents a step decrease, followed by an increase, in the combustion air temperature on the catalyst temperature above the air inlets and above the coils. Above the inlets the temperature rises because of the increased oxygen concentration in the combustion air (the air is heated by direct fuel gas firing), which increases the amount of coke burned in the top or countercurrent zone. However, above the coils the catalyst temperature falls because the total coke burned remains the same, and the total enthalpy input to the kiln has been lowered. Plant tests on a commercial unit have confirmed the dynamics of the combustion air change as simulated by this study. In Figure 8 a summary of the regenerator frequency response at one level of operation is given in the form of a Bode plot. Catalyst temperature sensitivity above the cooling coils, expressed as a magnitude ratio, decreases a t the higher frequencies, since the air has a 90
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
much higher velocity than the catalyst and can transport heat between hot and cold catalyst zones. Results were displayed in a form (17-inch oscilloscope) that allowed interpretation on physical grounds ; several engineers completely unfamiliar with the program details gained useful insights and prescribed important experiments within 1 hour of seeing the simulation for the first time. An artist’s conception of a typical temperature transient is shown in the isometric drawing, Figure 9. The maximum temperature at the inlet to the cocurrent section gets amplified and travels through the zone. As the temperature peak passes out of the zone, temperature drops sharply. Regenerator Performance: Closed Loop
A second phase of operating the regenerator simulation was concerned with the design of closed-loop systems. Various ideas as to how to achieve effective control existed before the computer study began and others were conceived during the study period; all of these were evaluated using simulation techniques. We proposed to apply modern control and computing techniques to a basic processing operation. We chose computer simulation because it was high-speed, low-cost experimentation. Also, it is always desirable to try out any control concept in a low-risk environment, to prove and improve it, and to guarantee that it will work when put to test in the plant. Simulation allowed a choice to be made; the relative effectiveness of control systems was clear. One can always question the quantitative accuracy of a mathematical model, but once the model operates in a qualitatively correct fashion, and one control system is clearly much more effective than another, then the choice of which system to implement is simple enough. I t was significant in this study, for example, that the most sophisticated control system proposed was proved to be inferior to most other systems. Without a simulation evaluation, the sophistication itself could have engendered support and this system might have been tested in the plant at considerable cost. Figure 10 illustrates one system choice. Catalyst temperature responses using two different control systems are shown; one is a little better than no control at all, whereas the other keeps the system in line and effectively prevents temperature disturbances from being passed back through the reactor. Loop I on Figure 10 consisted of predicting (in a feed forward fashion) the exit kiln temperature from a simplified kiln model and manipulating the combustion air enthalpy to hold the kiln exit temperature constant. In addition, Loop I1 manipulated the combustion air rate to hold the oxygen rate to the kiln constant no matter what the combustion air temperature. Steadystate information indicated this to be a desirable mode of operation ; however, the dynamic simulation showed evidence of an interaction between the combustion air rate and the fuel gas rate (to the combustion air line burner). Loop I is clearly to be preferred over Loop 11.
NOMENCLATURE
CATALYST FLOW
r-+--- I
SIGNAL TO CHANGE CONSTANT IN MODEL
1-
-4-
-, I
I
b
= frequency factor of burning, cu. ft. of reactor/(lb. mole
c
= parameter vector
ct
= ___
Oz)(hr.)
VkPbblCoaO F,
I I
I TI
----I--.
AIR INLElS
I FUEL GAS I
I
DESIRED TEMPERATURE SET POINT
L- +-
I
I ---@--I t-+-i +
Figure ?I.
I
1
P / I CONTROLLER
Adaptive control scheme
An adaptive loop was added to correct for any inadequacy in the simple feed forward predictive model. Loop I plus the adaptive controller is described in Figure 11. The general conclusion drawn from these simulation experiments was that effective control can be achieved by manipulating combustion air temperature from an adaptive feed forward control scheme based on measurements of catalyst temperatures in the countercurrent zone and at the bottom end of the regenerator. Discussion
This simulation problem using hybrid computing techniques was necessarily complex. Before undertaking such computer simulation, one should note that programming and checkout costs are high, probably equal to those of writing and debugging the equivalent digital computer program. However, the total cost of getting specific results is low, because the simulation operates at such high speeds that large exploratory programs can be completed in a very short time. In addition, although the time required for debugging and testing represented over 90% of the total computation time, the rapid results during the simulation phase more than compensate for the initial time expenditure. The current generation of hybrid computers are supported by software systems which make setup and checkout less time consuming, with an attendant increase in productivity. In this study, several possible control systems were conceived and evaluated, and the most promising one is being applied in an on-line computer system. An additional significant benefit of the work was the organized insight of the process gained from operating it and observing responses that are 60 times faster than real time. Watching temperature pulses travel through the system, seeing catalyst (diffusional) effectiveness change in both distance and time was of great importance to the successful and timely conclusion of the work.
= 3 D,a,Mw, G o = bsro2piW2 C O= ~ original oxygen concentration, lb. mole OZ/CU. ft. of reactor Cp, = air specific heat, B.t.u./(lb.)( F.) Cpc = catalyst specific heat, B.t.u./(lb.)( F.) D e = effective catalyst diffusivity, s q . ft./hr.
Cg
O
F, Fa Mw Q
catalyst feed rate, lb./hr. air rate, lb./hr. molecular weight of coke, lb./(lb. mole) activation energy, B.t.u./(lb. mole) R gas constant, B.t.u./(lb. mole)(’ F.) ro radius of catalyst sphere, ft. t = clock time, hr. T = temperature, O F. or R. = state vector, function of x, 0 u = volume of kiln section, cu. ft. vk W o = weight fraction original coke-on-catalyst, lb. coke/lb. catalyst = normalized axial distance in kiln, dimensionless, x = fraction of original coke remaining, dimensionless conceny tration = fraction of original 0 2 remaining, dimensionless concentraz tion * Sign in denominator is in cocurrent section. = = = = = =
r/ro
+
Greek a
= oxygen utilization, lb. mole coke burned/lb.
E
=
AH
=
17
= =
v8b
8
= =
pb
=
vI pa pt
Q
b
bo
= = = = =
mole
0
2
consumed void fraction heat of reaction, B.t.u./lb. coke burned catalyst utilization factor (Eq. 2) shell burning effectiveness factor intrinsic effectiveness factor vI = 1 by definition normalized time, t(Fc/Qropb) bulk catalyst density, lb./cu. ft. of kiln air density, lb./cu. ft. of reactor true catalyst density, lb./cu. ft. of catalyst cross-sectional kiln area, sq. ft. axial distance in kiln, ft. kiln length, ft.
Subscripts f = fast coke s
= slow coke
B I BL IOGRAPHY (1) Acrivos A., “Method of Characteristics Technique,” IND. END. CHEM. 48 (4), 703 (1956). (2) Bischoff K. B., “Accuracy of the Pseudo Steady-State Approximation for Moving Bbundary Diffusion Problems,” Chem. Eng. Sci. 18, 711 (1963). (3) Herron, E. H., von Rosenberg, D. U.,“An Efficient Numerical Method for the Solution of Pure Convective Trans ort Problems with Split Boundary Conditions,” Chem. Eng. Sci. 21, 337-42 (1966). (4) Prater, C. D., Moulthrop, B. L., Wei, J., “Simulation of T C C Kilns in Terms of Fundamental Parameters,” presented a t National A.1.Ch.E. Meeting, Buffalo, New York (May 1963). (5) Weisz P. B. Goodwin R. D., “Combustion of Carbonaceous Deposits within Poroun )Catal;st Particles) I.-Diffusion-Controlled Kinetics,” J. Catalysis 2, 397 (1963).
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