Comparison Between Different Mathematical Models for the Simulation of Industrial Fluid Catalytic Cracking

Mathl. Comput. Modelling Vol. 18, NO. 6, pp. 91-110, 1993 0895-7177/93 $6.00 + 0.00

Printed in Great Britain. All rights reserved Copyright@ 1993 Pergamon Press Ltd

Comparison Between Different Mathematical Models for the Simulation of Industrial Fluid Catalytic Cracking (FCC) Units

S. S. E. H. ELNASHAIE’*

Chemical Engineering Department, King Saud University Riyadh, P.O. Box 800, Saudi Arabia

S. S. ELSHISHINI*

Department of Chemistry, College of Science, King Saud University Riyadh, P.O. Box 22452, Saudi Arabia

(Received and accepted September 1992)

Abstract-h&t of the work dealing with the modelling of industrial fluid catalytic cracking units is based upon a highly empirical approach that helps in building units and operating them, but does not help to elucidate the main features and characteristics of these units for better design, control, and optimization. It is shown in this paper that a better insight into the behaviour of these units can be achieved through the use of less empirical mathematical models coupled with industrial verifications. The present paper emphasizes the bifurcation behaviour of these industrial units.

NOMENCLATURE

A

AC

A CT

Ad

Ai

&B> AGB

ARD, AGD

act , aad, ashc

BR, BG

Dimensionless velocity constant for overall cracking reactions (Equations (32), (34))

Frequency factor in Arrhenius equation for coke burning reaction

Overall frequency factor for cracking

Frequency factor for deactivation

Frequency factor in Arrhenius equation for cracking reactions, i = 1 for cracking to gasoline and i = 2,3 for cracking to coke

Area of bed occupied by bubbles in reactor and regenerator, respectively; m2

Area of bed outside bubbles (dense phase) in reactor and regenerator, respectively; m2

Constants for catalytic carbon, additive carbon, and strippable hydrocarbons

Space velocities in reactor and regenerator

CA,,CA,

CA,f, CAzf

Cpa, Cpf, Cplr cps

c CpG PRI

CR, CG

c f co

Ec, Ei

Concentration of gas oil and gasoline; kg/m3

Feed concentration of gas oil and gasoline; kg/m3

Weight % of catalytic carbon, carbon on spent catalyst, and carbon on regenerated catalyst

Weight ratio of coke necessary for complete deactivation of catalyst

Heat capacities of air, liquid feed, vaporized feed, and solids; J/kg K

Heat capacity of gases in reactor and regenerator; J/mol K

Weight ratio of coke to catalyst in reactor and regenerator; kg/kg

Weight fraction of coke in feed

Concentration of oxygen; mol/m3

Activation energies for coke combustion and cracking reaction, i = 1 for cracking to gasoline, i = 2,3 for cracking to coke; J/mol

‘Author to whom all correspondence should be addressed. *On leave from the Chemical Engineering Department, Cairo University, Cairo, Egypt.

Typeset by AA,@--

91

92

Ed

FAG

FC

FD

F,r

GGFIGAF

HR,HG

Kc,

kn,f

NR,NG

Ofi4

91c

&>pG

Q

Qc, Qr

QEFL, QEG

QL

R

Rc

S. S. E. H. ELNASHAIE. S. S. ELSHISHINI

Overall activation energy for cracking; J/mol

Activation energy for deactivation; J/m01

Gas flowrate in the regenerator;

kg/s

Catalyst circulation rate; kg/s

= FGF (In the model of Iscol- Figure la, FD = FGF in lb/hr)

Fresh feed, mixed feed, and air feed flowrates; kg/s

Coke formation factor of total feed; kg Coke/m3 oil

Volumetric feed flowrate for gas oil and air; m3/s

Volumetric regenerator gas flowrate in bubble phase, dense phase, and overall; m3/s

Volumetric reactor gas flowrate in bubble phase, dense phase, and overall; m3/s

Height of fluidized bed in resctor and regenerator; m

Total inventory of catalyst; kg

Reaction velocity constant for coke combustion and cracking reactions, i = 1 for cracking to gasoline, i = 2,3 for cracking to coke

Overall velocity constant for cracking reactions

Experimentally determined constants

Stoichiometric coefficient

Molecular weights of gas oil, gasoline, gases, coke, and air

Catalyst hold up in reactor and regenerator; kg

Gas hold up in reactor and regenerator; mol

Volume percent of oxygen in flue

gases

Partial pressure of hydrocarbons in stripper; Pa

Pressure in reactor and regenerator; Pa

Heat removed from regenerator by cooling coil; J/s

Heat supplied by combustion and heat removed from system; J/s

Mass transfer interphase coefficient for reactor and regenerator;

l/s

Heat loss for unit; J/s

Universal gas constant

Rate of coke combustion; kg/s

Rcf

&*

S"

t

tc

TGFYTAF

TR,TG

T,r

TS

VGF,VR,VG,

VAF

W.

WC0

WH

XCB

xiB,%D,xirxif

XR,XG

Yi,YiF

YOrYOF

YRB,YRD,YR

YGB,YGD,YG

YGF>YAF

K

Greek Symbols

a,0

PC

A

-ri

AH,, AH,

Rate of coke formation; kg/s

Overall rate of cracking; kg/s

Liquid space velocity; vol/vol s

Time; s

Catalyst residence time; s

Feed temperature of gas oil and air; K

Reactor and regenerator temperature; K

Reference temperature; K

Stripper temperature; K

Molar gas flow rates of feed, reactor gases, regenerator gases, and air; mol/s

Weight fraction of coke to (coke +

gases)

Volume % of CO to (CO + COe)

Weight % of H in coke

Fraction of coke burnt

Dimensionless concentrations of component i in the bubble phase, dense phase, output, and feed, i = 1 for gas oil, i = 2 for gasoline

Coke on catalyst in reactor and regenerator; mol/kg cat

Mole fraction of component i in unit and in feed

Mole fraction of oxygen in unit and in feed

Dimensionless temperature in bubble phase, dense phase, and output of reactor = T/T,f

Dimensionless temperature in bubble phase, dense phase, and output of regenerator = T/T,f

Dimensionless feed temperature of gas oil and air

Dimensionless feed vaporization temperature

Coke oxygen stoichiometric ratio

Exothermicity factor

Endothermicity factor

Dimensionless activation energies

EiIR. Trr Heat of coke combustion and cracking reactions, i = 1 for cracking to gasoline, i = 2,3 for cracking to coke; J/kg

Overall heat of cracking and heat of vaporization; J/kg

Dimensionless heat losses from reactor and regenerator

Dimensionless heat of cracking and vaporization

Porosity of fluidized bed

Simulation of Industrial FCC Units 93

EC Conversion of gas oil cracking; wt. PI I Pf I Pa Density of liquid feed, vaporized

fraction feed, and air; Kg/m3

x Dimensionless decay velocity “bR, $G Catalyst activity in reactor and

constant (Equations (32), (33)) regenerator

INTRODUCTION

Multiplicity phenomenon in chemically reactive systems has been first observed as far back

as 1918 [l], and in the Russian literature as far back as 1940 [2], especially in relation to the

explosion theory. However, it was not until the 1950’s that the great interest in the investigation

of this phenomenon started, inspired by the Minnesota school of Amundson and Aris [3-51 and

their students [6-91. The Prague school also had a notable contribution to the field [lo-121. Since

then, this phenomenon has been high in the agenda of chemical engineering research in general,

and chemical reaction engineering research in particular. The fascination with this phenomenon

has also spread very widely in the biological and biochemical engineering literature where it is

referred to as “short term memory” [13,14]. In the mathematical literature, this phenomenon is

treated in more general and abstract terms under “bifurcation theory” [15].

A major breakthrough with regard to the understanding of this phenomenon in the field of

chemical reaction engineering was achieved by Ray and co-workers [16,17] when in one stroke they

uncovered a large variety of possible bifurcation behaviours in non-adiabatic continuous stirred

tank reactors. In addition to the usual hysteresis type bifurcation, Uppal et al. [17] uncovered

different types of bifurcation diagrams, the most important of which is the “isola” which is a

closed curve on the bifurcation diagram, disconnected from the rest of the continuum of steady

states.

Isolas were also found by Elnashaie et al. [18-201 for enzyme systems where the rate of reaction

depends non-monotonically upon two of the system’s state variables (substrate concentration

and pH), a situation which was shown to be applicable to the acetylcholinesterase enzyme systems.

Later development in the singularity theory, especially the pioneering work of Golubitsky and

Schaeffer [15], provided a very powerful tool for the analysis of bifurcation behaviour of chemically

reactive systems. These techniques have been used extensively, elegantly, and successfully by Luss

and his co-workers [7,8,21-231 to uncover a large number of possible types of bifurcation.

The Fluid Catalytic Cracking unit for the conversion of heavy gas oil is one of the most

important conversion processes in the petroleum refining industry. Despite the great industrial

importance of this unit and the complexity of its rather interesting steady state and dynamic

behaviour, it has been the subject of very few modelling and simulation investigations [24-271.

Most of these investigations are based on highly empirical models and give conflicting conclusions.

In addition to that, most of these investigations concentrate on the thermal characteristics of the

unit and do not involve any reaction network that allows relating the performance of the unit as

a gasoline (and sometimes olefins) producer to its thermal characteristics. A serious limitation

in these models is the fact that they ignore completely the two-phase nature of the fluidized beds

in both reactor and regenerator.

The models of Elnashaie and El-Hennawi [28,29] and Elshishini and Elnashaie [30-331 involve

a reaction network and recognize the two-phase nature of the reactor and regenerator. These

models take into consideration the main processes taking place within the boundaries of the

system in a rational, albeit simplified form.

The present paper is mainly concerned with the mathematical modelling of industrial fluid

catalytic cracking units with special emphasis on their bifurcation behaviour.

BIFURCATION THEORY-A SIMPLIFIED PRESENTATION BASED ON PHYSICO-CHEMICAL ARGUMENTS

Multiplicity of the steady states and the bifurcation behaviour of chemical reactors in general,

and gas-solid reactors in particular, has been a hot research subject in the chemical reaction

94 S. S. E. H. ELNASHAIE, S. S. ELSHISHINI

engineering literature for the last three decades. With the discovery of chaotic behaviour, interest

in this field is expected to grow tremendously in the coming decade.

Physical Explanation of Bifurcation

Multiplicity of the steady states, which is a phenomenon associated with open systems only,

means that such a system will have more than one stationary non-equilibrium state. The open

system may have multiplicity of the steady states for a certain region of its physico-chemical

parameters if and only if there is at least one process that is taking place within the boundaries

of the system which is a non-monotonic function of at least one state variable of the system.

This is a necessary condition for the existence of multiple steady states, but it is not a sufficient

condition. To discuss the sufficient conditions for the existence of multiple steady states, we must

make it clear that there are certain systems that will never show multiplicity of the steady states

even when the non-monotonic dependence conditon is satisfied, e.g., the adiabatic tubular plug

flow reactor, where no mixing between the layers takes place. There are other systems that will

have multiple steady states if the non-monotonic dependence condition is satisfied, but only for

a certain range of parameters (physico-chemical, operating, and design parameters). No system

will exhibit multiplicity of the steady states for all values of the parameters.

The plug flow adiabatic reactor does not show any multiplicity of the steady states because

the plug flow system does not have any diffusion or mixing. In other words, what is happening

upstream is completely independent of what is happening downstream. This gives a second

necessary conditon for multiplicity, which is the existence of some kind of diffusion or mixing

mechanism that creates what might be called a feedback of information, or in simpler terms,

makes what is happening upstream depend on what is happening downstream. Thus, for the

above plug flow reactor with the non-monotonic dependence condtions satisfied, the existence of a

recycle stream from the output of the reactor to the input creates the possibility for multiplicity of

the steady states. In the case of a non-isothermal plug flow reactor, where an exothermic reaction

is taking place, the reactor will never give multiplicity of the steady states if it is provided with

co-current cooling. However, if the cooling is counter-current, multiplicity of the steady states is

possible.

From a mathematical point of view, the simplest bifurcation problem involves treatment of the

local bifurcation problem with one state variable,

where x is a state variable and X the bifurcation parameter. Many chemical engineering problems

can be reduced to the form of equation (1). Evidence for the existence of multiple steady states

for the industrial FCC units discussed in this paper are shown in Figures 1-3.

MATHEMATICAL MODELLING OF INDUSTRIAL FLUID CATALYTIC CRACKING UNITS

A Fluid Catalytic Cracking unit is made up of a reactor, in which gas oil is catalytically

cracked to gasoline, coke, and light hydrocarbon gases, and a regenerator, where the catalyst is

regenerated by using air to burn the coke deposited on the catalyst in the reactor.

All the main reactions taking place in the reactor are endothermic and the kinetics are

monotonic. Therefore, the reactor cannot be the source of the bifurcation behaviour of the

system. It is the regenerator with its highly exothermic carbon burning reaction which is the

source of the bifurcation behaviour of these industrial units. The mathematical modelling of

industrial fluid catalytic cracking units is complicated by the fact that each of the two vessels of

the unit is formed of at least two phases. Two-phase models are the simplest physically accepted

formulation, for certainly the flow patterns in these units are more complicated, especially in


2 0.013

-5 xc! 2 z 0.014

22

“,u. 0 0.011 51 (L urn I

_.I 2 _A

560 50 m‘ U

P. d

2 ‘; 950 49 E

P d

c” 340 630 640 650 660 670

FD, LBlHR

0.2 ‘1 gy__;;;:_,:“, =,‘9’ ,y ( j 0.0 ’ 1-f +q ’ ’ c

0 0.6 \.* 1.8 1.4 3.0 3.6 4.3 4.8 5.4 6.0 6.6 7.2

YAF Dimensionless Air Fe& Temp.10 Regenerator

(a) Iscol (241 (bifurcation diagram with gas oil

feed flow rate as the bifurcation parameter).

(b) Elnashaie and El-Hennawi [28] (bifurcation diagrams

at different gas oil flow rates with air feed temperature as

the bifurcation parameter).

200

19c

18C

Q. mega-Jlrec

1JC

163

150 8

-

-

\ la

Partial Comburtion

TOlOl Combv,lion

- - Heat Generation - - - Heit Removal j 500 925 550 975

Regenerator Bed Temperature. “K 1000 1025

(c) Edwards and Kim [27] (Van Heerden diagram-heat generation, heat

removal diagram-for one set of parameters showing multiplicity of the

steady states).

Figure 1. Examples of diagrams showing multiplicity of the steady states for indus-

trial Fluid Catalytic Cracking units (FCC) obtained by different invesigators.

----- 75’/. COMPLETE COMBUSTION

0.00 0.40 0.80 1.20

Dimensionless Catalyst Circulation Rate Fcl Fcr,t

Figure 2. Effect of degree of coke combustion to CO2 on bifurcation diagrams for

reactor temperature vs. catalyst circulation rate [30].


fluidization regimes beyond the bubbling regimes. One of the most complicated problems is that associated with riser-reaction type units because of the complex flow patterns in the riser. Recent research in fast fluidization regime and circulating fluidized beds helps to elucidate the complex nature of the flow in these risers [34-371. Grace [38,39] argues that it is not difficult to revise two-phase fluid bed reactor models, originally intended for bubbling bed regime, to be applied to other regimes of fluidization.

4.0 - INDUSTRIAL UNIT 1

---- INDUSTRIAL UNIT i -z

‘2 2.0 - x

/

vnporizolion temp.

01 gas oil

1. 00 - I:IDUSTRIAL UNIT 1

___- lNDUSTRlAL UNIT 2

_I

0.00 1.00 2.00 3.00 4.00

Dimensionless Reactor Temperature ( YR )

(a) Heat functions for units 1 and 2. (b) Conversion and gasoline yield for units 1 and 2 1291.

Figure 3.

Seven different models of different degrees of sophistication for the FCC unit will be discussed in this paper. The models differ from each other in some basic characteristics as well as in their structure. The degree of empiricism in expressing the rates of the different processes also differ from one model to the other. Table 1 shows a summary of the basic characteristics of the seven models discussed.

Table 1. Summary of some of the basic characteristics of the models used to simulate industrial Fluid Catalytic Cracking (FCC) units.

Characteristics Two- Kinetics Kinetics Kinetics Predicts Verified

Phase of of of Multi- Against

Models Gasoline Carbon Carbon plicity Industrial

Model Formation Formation Burning Data

L & L [40] No Yes Yes Yes No No

K 1251 No No Yes Yes No No

I ]241 No No Yes Yes Yes Yes

L & K [26] No No Yes Yes No Yes

E & E [28] Yes Yes Yes Yes Yes No

E & K [27] No No Yes Yes Yes Yes

E & E modified [30] Yes Yes Yes Yes Yes Yes

LUYBEN AND LAMB MODEL (L & L MODEL)

Luyben and Lamb model [40] was described by Douglas [41] as being a crude model that predicts the direction of change of the most important dependent variables and at least gives an order of magnitude estimate of the response time of these variables.


The authors assume that the following reaction occurs in the reactor:

A - B+O.lC,

where A, B, and C represent gas oil, gasoline, and coke, respectively. The coke combustion taking

place in the regenerator is given by:

C+mO+P,

where 0 and P represent oxygen and combustion products, and m is an empirical stoichiometric

coefficient depending upon the degree of completion of carbon combustion and the percentage of

hydrogen in coke. The model is also based on the following assumptions:

1. The reactor and the regenerator are perfectly mixed.

2. The heat capacities are independent of composition and temperature.

3. The catalyst holdup remains constant in each vessel.

4. The reaction rates are first order and the rate of coke combustion is independent of the

amount of coke.

5. The reaction network in the reactor is formed of one reaction.

The material and energy balance equations for this model can be summarized as follows:

REACTOR.

dNR - = VGF -1/R,

dt dNRys

dt = VGF y2~ - VR y2 + NR (1 - ~2) Al eFEIIRTR,

dXR MR dt

- = Fc XG - Fc XR + 0.1 Nn (1 - yz) Al e-E1’RTR,

C&k% = &RVGFTGF +c,sFcT~ - C~RVRTR - C,,FCTR

- (AHr) Nn (1 - ~2) A1 eeEIIRTR.

c-4

(3)

(4)

(5)

REGENERATOR.

dNG - = VAF -Vi, dt (6)

dNGyo dt

= VAF you - Vi yo - m NG yo A, e- EcIRTG,

dXc MG- =FcXR-FcXG-NGyoA,e- -&/RR

dt ,

C~~MG% = ~~~~~~~~~ i-CpsFcT~ - CpcG~ TG -C,,FCTG

- (AH2) NG yo A, emEciRTG + Q.

(7)

(8)

(9)

Using the ideal gas law and simplifying the equations, the authors reduced the model to a set

of six coupled, non-linear, ordinary differential equations which can be solved numerically.

Evaluation of the Model

The model was not tested experimentally, and using such a simplified model for estimating the

dynamic characteristics of the system may give misleading results. However, by introducing some

modifications on the kinetic scheme used for the cracking reactions and on the rate of carbon

burning, it can provide a starting point for developing an approximate model.


KURIHARA MODEL (K MODEL)

Denn, in his recent book [42], considers Kurihara model [25,43] to be the best model available

in the open literature.

The model considers two different carbon species. The carbon that is deposited on the catalyst

during the course of the cracking reaction is denoted catalytic carbon (C,,,). The remaining

(additive) carbon is generally present in gas oil and is deposited without reaction.

Carbon that has passed into or through the regenerator on the spent or regenerated catalyst

has a different chemical activity from the catalytic carbon. This species is identified as C,, or C,,,

depending on whether the catalyst has been regenerated or spent.

The msss balance on total carbon on the spent catalyst in the reactor is given in this model

by: dcs,

MR- dt

= Fc (Cm - C’s,) + R,f. (10)

The term Fc C,, is simply the rate at which catalytic carbon leaves the reactor, while FC C,, is

the rate at which carbon remaining on the regenerated catalyst is supplied to the reactor through

the catalyst circulation stream. The specific rate of carbon formation is described by:

R,r = PRMRK~ +ptrG~F, (11)

Kz = c “~,,,s exp[-&/RTn]. (12) cat rc

The first term in (11) represents the rate of production of catalytic carbon from the cracking

reaction, and the second term represents the rate of formation of additive carbon, which is

proportional to the feed rate GGF.

The remaining reactor mass balance is written for C&t, which is the catalytic carbon (in

weight %) obtained from the cracking reaction,

dC,,t MR-=

dt -Fc &at + MRPRK~. (13)

The energy equation for the reactor is written as:

CPS MR~ = C,,FC(TG-TR)+C~~P~GGF(T GF-TR)+(-AHv)P~GGF+(-AH~~)R~~. (14)

The term containing the enthalpy of vaporization accounts for the fact that the gas oil enters as

a liquid and is vaporized inside the system. The overall specific rate of cracking reaction R,, is

given by:

Rcr = MRPRP~Kc,

1 +MR&&@GF' and

K,, = c A~,,,, exp[-&/RT’]. cat CT

(15)

(16)

The regenerator mass and energy balance equations are given by:

Gc MG-

dt = Fc (Cs, - Cr,) - R,,

C,,, MG? = C,, Fc (TR - TG) + Cpa FAF (TAF - TG) + (-AHc) Rc.

(17)

(18)

The carbon burning rate R, is given by:

where

R c = FAF@~ -of,)

1oocr '

Of' = 21 exp

MGPG/FAF

(104/3.15 Fx,) + (100/C,, A, exp(-&/RTG)) 1 ’ (19)

(20)

Cr is a stoichiometric coefficient that reflects the selectivity of the combustion reaction to CO2

and CO products, and (21 - Or,)/100 is the oxygen conversion, since the volume fraction in the

air feed is 0.21.


Evaluation of Kurihara Model

The model is distinguished from other models by the fact that it differentiates between catalytic

carbon which deposits on the catalyst due to catalytic cracking, and non-catalytic carbon which

is generally present in gas oil and is deposited without reaction. In the solution of the model,

Kurihara assumes additive carbon to be zero (Ftf = 0).

Other than that, the model is, in principle, very similar to the model of Iscol [24] and that

of Lee and Kugelman [26] discussed later, especially with regard to the model structure and the

degree of empiricism involved in the different generation, and consumption functions inserted in

the model equations.

Kurihara model [25,43] does not recognize the two-phase nature (bubble and dense phases)

of the two fluidized beds in the reactor and the regenerator. Although the solid and gas in the

dense phase can be considered as having the same conditions (pseudo-homogeneous model for the

dense- or emulsion-phase), the differentiation between dense phase as a whole and bubble phase

as a whole, in such bubbling or turbulent fluidized beds, can be quite important for accurate

mass and heat balance formulation of the system.

Kurihara [25,43] uses a very simplified kinetic scheme given by the following equation:

Cracking Gas oil-Product gas+Coke (additive + catalytic).

This network is similar to that used by Luyben and Lamb [40] and has the disadvantage of

lumping gasoline, coke, and light hydrocarbons. Consequently, there is no way to obtain the

gasoline yield which is the most important design variable.

The most serious limitation in this model is that the constants AZ (for the rate of catalytic

carbon formation) and A1 (for the rate of catalytic cracking) have been fitted for a certain assumed

steady state. Kurihara [25,43] assumes that the reactor is operated at a certain probable steady

state condition (e.g., Mn = 54432 kg, PR = 275 kPa, tn = 499”C, GGF = 0.184 m3/s, conversion

on fresh feed = 0.6), together with a certain probable steady state condition of the regenerator,

and he evaluates AZ and Al by setting the time derivatives to zero and solving the steady

state mass balance equations. This was justified for optimal control, since the investigation was

concentrating on the problem of small disturbances around the steady state condition, as Kurihara

puts it in his thesis: %‘ince it is not only dificult but also uesless for our purpose of simulation to

estimate accurately these constants by purely chemical analysis of feed composition and catalyst

condition, this kind of closed-loop method was used” [25]. However, these rate equations are

not valid at any other conditions away from the assumed steady state (simple calculations show

that for a reactor temperature of 593”C, the percent carbon formed is much higher than 100%).

Consequently, they cannot be used to investigate multiplicity of the steady states, since the model

will always give one steady state (the assumed steady state).

ISCOL MODEL (I MODEL)

Iscol [24] carried out his modelling exercise to investigate the steady state and dynamic char-

acteristics of Model IV Fluid Catalytic Crackers with special emphasis on trying to explain the

continuous drift of a commercial unit away from its operating point at high gasoline yield.

Iscol model consists of the following mass and energy balance equations together with the

necessary generation and consumption functions.

REACTOR ENTHALPY BALANCE.

MRC’~~~ = Fc C,, (TG - TR) - FGF (335oT~ - 814100) - 8141000 R,f - 2.93 x 106. (21)

REGENERATOR ENTHALPY BALANCE.

dTG MG C,,,- = dt

3.02 x lo7 R, - Fc C,, (TG - TR) - 75172 (TG - 288.6) - 1.2 x 107. (22)

KM 16:6-H


REACTOR COKE BALANCE.

dCR MR-

dt = Fc(CG - cR)+%f*

REGENERATOR COKE BALANCE.

dk MG-

dt =Fc(&-CG)-%.

COKE MAKE RATE(REACTOR).

Z&f = FGF [0.0754 + 2.297 x 10e6 x (7'~ - 772)3].

(23)

(24)

(25)

COKE BURN RATE CORRELATION (REGENERATOR).

& = 6.45 [l - eXp[-CG (747 + 14.94 (??G - 894) + 0.107 (TG - 894)2)]]. (26)

The coke make and burn rates are given by empirical correlations, which as Iscol states in

conjunction to the coke burning rate: ‘it should be noted that equation (26) will vary among

different FCC units and depends upon catalyst type, chemical composition of the coke and actual

catalyst flow pattern, and thus vessel geometry. ”

Evaluation of Iscol Model

The empirical nature of the coke forming, coke burning, and heat losses from the vessels is

apparent. Also, the coke make rate is expressed in terms of the input variable FGF (gas oil feed

flow rate) which is not very rigorous, since all processes in a physical model should be expressed

in terms of state variables in the model formulation. As for the rate of coke burning, although it

is expressed in a semi-empirical form, it is actually expressed as a function of the regenerator’s

state variables CG and TG, with no input variables appearing in the equations.

The Iscol model does not involve any reaction network entailing all the consequences discussed

earlier, and it also does not recognize the two-phase nature of the fluidized beds in the reactor

and the regenerator.

Iscol [24] investigated the steady state characteristics of his model and showed that multiplicity

of the steady states (bifurcation behaviour) does exist over a relatively wide range of parameters

(Figure la).

He also concluded that the operating steady state is the middle unstable steady state, and that

the instability of the system manifests itself as a slow drift from this unstable steady state due

to the very high heat capacity of the system.

LEE AND KUGELMAN MODEL (L & K MODEL)

The model of Lee and Kugelman [26] was published three years after Iscol model to reinvestigate

the problem of multiplicity of the steady states in fluid catalytic cracking units. The model

consists of the following equations:

REACTOR CARBON BALANCE.

d(MRCR)

dt = Fc(CG - cR)+%f. (27)

REACTORENERGY BALANCE,

~(MRC,,TR)

dt = FC CPS (TG - TR) + FGM C,l (TGF - TR) - FGM (AH” + tc AH,,). (28)


REGENERATOR CARBON BALANCE.

d(MG CG)

dt = Fc (CR - c,) - R,.

REGENERATOR ENERGY BALANCE.

d(MG c,s TG)

dt = Fc Cps (TR - TG) + FAF Cp, 0” AF - TG) + R, AH,. (30)

The carbon formation in the reactor is given by Lee and Kugelman [26] as:

&r = act emE21RTR t: Fc + a,d FGM + ashc F&. (31)

Lee and Kugelman [26] state that the first term on the right hand side of (31) corresponds to the cracking reaction and is based on the assumption that coke formation follows the so-called

“Voorhies relationship” [44], in which the weight fraction of coke on catalyst is proportional to

the nth power of the catalyst residence time in the reactor. The coke formation rate constant ucc

is taken by Lee and Kugelman [26] to be a function of feedstock and catalyst. Lee and Kugel-

man [26] determine ucc, Ez, and n experimentally to fit laboratory and plant data, and therefore

their values strongly depend upon catalyst, feedstock, and the characteristics of the particular

FCC unit. In addition to catalytic coke, Lee and Kugelman consider additive coke (e.g., Conradson-type

coke) which is included in the second term of (31). They also take into consideration the efficiency

of the stripper by adding a term which depends upon the fact that the hydrocarbons interspersed

among the catalyst particles are to be completely stripped by countercurrently flowing steam. However, due to the poor efficiency of the stripper, some fraction of the strippable hydrocarbons remains and is carried over to the regenerator. This effect is included in the last term of (31), the

amount of strippable hydrocarbons to be burned being proportional to an lth power of catalyst

circulation rate. In fact, this term should not be included in the carbon formation equation, for it does not

affect the catalyst activity the same way carbon does. It should be accounted for in the heat

balance of the regenerator, for, in actual fact, it gets burned in the regenerator and contributes to the amount of heat released in the regenerator.

The gas oil overall cracking conversion cc is given by the model as:

A (1 - emX)

EC=X+A(l-e-X)’

where

X = t, Ad edEdIRTR, (33)

A _ f pf ACr ,-&IRTR PI S”

(34)

and Ad, A,,, E,,, Ed are experimentally determined parameters characterizing the combination of the particular catalyst and feedstocks. The coke burning rate is given by

R = FAG Mc (21 -Of,), c 100 (Y,O MA

(35)

where Of, is the oxygen mole fraction at the exit of the regenerator which is computed from the relation:

Of, = 21 exp -A, e -Eel RT~ CG I6f~ MG CY,O

0.21 FAG MC I* Lee and Kugelman [26] used this model to investigate the steady state characteristics of the system, and they reached an opposite conclusion to that of Iscol [24] with regard to multiplicity. Lee and Kugelman [26] attributed the multiplicity results obtained by Iscol [24] to the cubic temperature dependence in the expression for R,f used. Denn [42] comments on this contradiction by emphasizing the need to study the sensitivity of the model predictions to structural as well as parametric changes.


Evaluation of Lee and Kugelman Model

The model does not consider the two-phase nature of the two fluidized beds which is similar to previous models. However, it distinguishes between catalytic carbon and additive carbon, for which is in principle comparable to Kurihara model [25,43].

The model takes into consideration the stripping efficiency, which accounts for what is known as “cat-to-oil” coke. This is a measure of unstripped, but potentially strippable, hydrocarbons

that are carried out to the regenerator [45] because of the non-ideal stripping in the commercial process, and they are also burned in the regenerator. Obviously, this is not really coke. The inclusion of this stripping efficiency is an advantage to the model building; however, the authors include this term with the carbon formation rate in the reactor (equation (31))! This causes some

errors in the carbon balance in the reactor which affect the computation of the catalyst activity

and the heat produced in the regenerator.

The first term in (31), which is supposed to be the term for the formation of catalytic carbon,

has a dependence over t,“, which is the catalyst residence time in the reactor. This dependence can

be justified by Voorhies’ work [44] and also by the derivations of Kurihara [25] and Denn [42], who have shown that under certain assumptions and when vapors are flowing isothermally in

plug flow, such a dependence can be obtained. However, as shown in (31), the term is made also dependent on the catalyst circulation rate (F,), which is not a state variable and cannot be

directly related to the rate of catalytic carbon formation. The inclusion of such an input variable in such a manner affects negatively the logical structure of the model and will, therefore, affect

the consistency and the conclusions obtained using this model. The model also represents E, (the conversion) in a manner which is, to a great extent, inde-

pendent of the rate of formation of catalytic carbon. In fact, the rate of formation of catalytic

carbon and the conversion are strongly interlinked.

The model is not expected to show any multiplicity of the steady states, since the authors use

Kurihara rate equations for which the constants are fitted to a specific steady state. As explained

earlier, these rates are not valid at any other steady state, and the solution of the steady state model will always give the assumed steady state.

ELNASHAIE AND EL-HENNAWI MODEL (E & E MODEL)

Elnashaie and El-Hennawi [28] developed a model that overcomes some of the limitations of the previous models in order to check the contradictory results reported in the literature, specifically the opposite conclusions of Iscol [24] on one hand, and Lee and Kugelman [26] on the other hand.

The model of Elnashaie and El-Hennawi takes into consideration the two-phase nature of the

fluidized beds in the reactor and the regenerator, albeit in a simple form. Elnashaie and El-

Hennawi used the three pseudo-components kinetic model of Weekman [46,47]:

Gas Oil (Al> 3 Gasoline (AZ) 3 Coke + Gases (As) i<

K3

The rates of reactions for this consecutive-parallel network are given by Weekman and Nate [48] a%

RATE OF DISAPPEARANCE OF GAS OIL.

(36)

(37)

RATE OF APPEARANCE OF GASOLINE.

Simulation of Industrial FCC Units

RATE OF APPEARANCE OF COKE.

R3 = (1 - E) ps ‘R’ (K&i, +K3C&).

103

(38)

For the coke burning in the regenerator, Elnashaie and El-Hennawi used the continuous reaction

model for solids of unchanging size presented by Kunii and Levenspiel [49] because of its simplicity.

In this simple coke burning model, the gaseous reactant (oxygen) is assumed to be present

throughout the particle at constant concentration (no diffusional resistance) and reacts with

solid reactant (coke deposit) according to the rate equation:

R, = A, exp(-EJRTG) (1 - ZCB) Co. (39)

The model equations, based on the above relations, are given in dimensionless form (details of

the derivations are given in [28]) as follows:

REACTOR DENSE PHASE EQUATIONS.

Gas oil balance:

BR (Zlf - $lD) - ?,!& & 01 e-Y1’YRD + (Y3 eY31YRD >

= 0.

Gasoline balance:

BR (Xzf - X~D)- ?,!JG ( CY~X~D~T-~~'~~~ -(Y~X~D~"'~~~

> = 0.

Coke balance:

Heat balance:

al (YGD - YRD) + BR (Yv - YRD) + ~2 (YGF - K) - (AR” + ARLR) + ABcr = 0.

REACTOR BUBBLE PHASE EQUATIONS.

Gas oil balance:

Gasoline balance:

Heat balance:

(X1B - X1D) = (xlf - 51~) emaR HR.

(ZZB - 22~) = (x2f - 521)) eeaR HR.

(YRB - YRD) = (K - YRD) e- a~ HR .

REGENERATOR DENSE PHASE EQUATIONS.

Coke balance:

Heat balance:

BG (YAF -YGD)+~~(~RD -YGD) --ARLG+A R, =O.

REGENERATOR BUBBLE PHASE EQUATIONS.

Heat balance:

WGB -YGD) = (YAF -YGD)~-~~~~,

(40)

(41)

(42)

(43)

(44

(45)

(46)

(47)

(48)

(4%

104

where

S. S. E. H. ELNASHAIE, S. S. ELSHISHINI

C al = Fc ?H~Cpfpf,

ARB

ARI aR = QER-

GRD'

CPl a2 = FGM - H~C~fpf,

AGB

ARI aG =QEG-

GGD'

as=FcsH~Cn~P~, AGI

and

QI = Al c CAlf, (YZ=A~E, (Y3 = A3ECAIf,

The space velocities BR and BG in the reactor and the regenerator are given by:

BR= GRD +GRB (1 -emaRHR)

ARIHR 7

BG = GGD +GGB(~ -cacHG)

AGIHG

The rate of coke combustion in dimensionless variables is given as:

R, = a, erciYG (1 - $Q),

where

(Y, =A,Co(l -E), &

^(’ = RTrf’

and the exothermicity factor for the coke burning reaction is given by:

p = 6 (-AH4 c

The rate of coke formation is given by:

Rcf= ( ~3e-Y3'YRDZ~D+ff2e-YP'YRD22D .

>

The endothermicity factors for the three cracking reactions are given by:

k=cA,f~m~~f, i=l,2,3, rf pf

(50)

(51)

(52)

(53)

(54)

(55)

and the overall endothermic heat of cracking is given by:

This model has been used by Elnsshaie and El-Hennawi to investigate the multiplicity phe-

nomenon in fluid catalytic cracking units. Since the model involves a reaction kinetic network,

they were able to investigate the gasoline yield of the system. The investigation of Elnashaie and

El-Hennawi [28], using this relatively more rigorous model, confirmed the conclusions of Iscol [24]

and contradicted the conclusions of Lee and Kugelman [26]. Elnashaie and El-Hennawi found

that the multiplicity region covers a very wide range of operating conditions (Figure lb). They

also found that the maximum gasoline yield corresponds to the middle unstable branch of the

bifurcation digaram, and that operation outside this unstable branch, at any of the other stable

steady states, gives very low gasoline yield.


Evaluation of Elnashaie and El-Hennawi Model

The model of Elnashaie and El-Hennawi recognizes the two-phase nature of the fluidized beds in both the reactor and the regenerator, albeit in a simple manner. The model uses a reaction network that relates the different steps of the cracking reactions to each other and to the reactor model.

From a hydrodynamics point of view, a simplified two-phase model is used and many assumptions were introduced to simplify the overall model of the system.

With regard to the kinetics of the cracking reaction, although the approach adopted, based on lumping hundreds of components into three pseudocomponents, is reasonably sound for the purpose of investigating bifurcation behaviour and gasoline yield, the network used has a strong deficiency in the lumping of coke with light hydrocarbons in one component while they have completely different roles. A four-component network [50] may improve the reliability of the model. However, a much more detailed network is needed for computing product distribution and gasoline octane number [51].

It is also important to notice that the activation energies of the kinetic reaction network are obtained from the work of Weekman [46,47] and Weekman and Nate [48], which gives the rate constants at only two temperatures. Therefore, the reliability of these activation energies is not high, and more experimental work is required to accurately determine the activation energies and rate constants for different gas oil feedstocks.

Another weakness with regard to this model is the fact that it has not been systematically checked against the performance of commercial, or even pilot plant, FCC units.

EDWARDS AND KIM MODEL (E & K MODEL)

The model used by Edwards and Kim [27] is proprietary; however, the authors give a brief summary of the concepts and relations employed by them in the use of the model to investigate the multiplicity phenomenon. The model consists of an overall carbon balance for the whole unit (reactor + regenerator) in the following form:

(57)

In this model, the capacity of the reactor is considered negligible in comparison to the capacity of the regenerator. This is justified by Edwards and Kim [27] by the fact that modern-day FCC units reactor normally amounts to less than 1% of the total catalyst in the system.

The energy balance of the system is given as:

where the heat capacity of capacity of the regenerator

d(lc 2 TG) = Qc _ Q,, (58)

the reactor has been assumed negligible as compared to the heat

Qr = FAF&,(TG -TAF) +Fccp,(TG -TR) +QL. (59)

The rate of carbon formation, which appears in the right hand side of equation (57) is given by:

Rcf = &at(tc, Ti, cc> Fc + &CT,, Phc) Fc + Rad(%, Tci, cf) Fw. (60)

The first term on the right hand side of (60) represents catalytic carbon formed by cracking, the second term represents the rate of coke entrainment in the stripper, and the third term is the rate of laydown of “additive coke.”

A formula for coke burning similar to that used by Iscol [24] and by Lee and Kugelman [26] has been used in this model.


Edwards and Kim [27] use the following pseudo-steady state relation for the computation of the catalyst circulation rate when the gas oil conversion is known,

FCc~s(TG -TR) +FGF%~(TGF -TR.)-FGF (AH, +ecAHcr)= 0. (61)

The gas oil conversion is given by:

EC = %(TR,TG,CG,FC,FGF). (62)

The authors used this model to investigate the behaviour of a commercial FCC unit and concluded that the phenomenon of the multiplicity of the steady state does exist, and they explained some of the pathological behaviour of the unit due to the existence of multiple steady states.

Evaluation of the Model

It is extremely useful to have a view from industry. The model and the paper give lots of extra insight into the behaviour of the complex fluid catalytic cracking units.

The neglect of the carbon and the heat capacity of the reactor makes the model valid only for systems where the inventory of catalyst in the reactor is much smaller than in the regenerator. However, in such systems, the reaction is taking place in the riser where the catalyst and gases are moving in plug flow, and hence, the lumped parameter formulation in terms of CSTR may not be valid. The model does not recognize the two-phase nature of the fluidized beds in both the reactor and the regenerator, and is similar in this respect to the models of Kurihara [25], Iscol [24], and Lee and Kugelman [26].

In the expression for the rate of coke formation R,f, (60), the important first term expressing the rate of catalytic coke formation is made dependent upon Fc, the rate of catalyst circulation. In fact, Fc is neither a state variable nor a capacitance term. It is an input-output parameter, and therefore the rate of a process taking place inside the system should not be expressed as a direct function of it.

The relation (62) for cc seems to be highly empirical, where E, is a function of (TR,TG, CG,

Fc, FGF). In fact, any consistent mass and heat balance formulation will show that cc is not a direct function of FC or TG or FGF. Actually, these parameters, through the overall mass and heat balance, will affect TR in different ways depending upon the configuration, the size, and other conditions of the system. TR will then affect ec through the Arrhenius dependence of the rate constants upon TR.

Edwards and Kim 1271 reach opposite conclusions to those of Lee and Kugelman [26]. They con- clude the existence of multiple steady states; however, their conclusions regarding the bifurcation behaviour of the system need to be clarified in two important points.

1. The relation between multiplicity of the steady states and catalyst circulation rate Fc. Heat is communicated from the regenerator to the reactor by means of the catalyst circulation rate, which is also responsible for transporting the carbon formed in the reactor to the regenerator. Therefore, too low a catalyst circulation rate will deprive the endothermic reactions in the reactor from the necessary heat. The cracking reactions will stop, and consequently, carbon formation in the reactor will stop, which results in a sharp drop of the carbon supply to the regenerator, and the whole system will “quench.”

On the other hand, too high a catalyst circulation rate may supply too much heat to the reactor, raising its temperature and causing overcracking with large carbon formation. This, in turn, will raise the amount of carbon recirculated to the regenerator, and “ignition” occurs, causing complete loss in gasoline yield.

For intermediate values of the catalyst circulation rate, which is the range in which most industrial units are operating, multiple steady states may exist. Actually, there are three steady states: a quenched low temperature steady state with almost no reactions taking place, an


ignited high temperature steady state with almost zero gasoline yield and very high rate of carbon formation, and an intermediate unstable steady state with a moderate temperature and high gasoline yield. This last steady state is the one at which most industrial units are operating. For any fixed value of catalyst circulation rate, in this region, multiple steady states do exist.

Therefore, the conclusion of Edwards and Kim [27] that, for an open loop system, there exists one unique steady state solution when the catalyst circulation rate is fixed, is not correct. A unique steady state can occur if the chosen fixed value for the catalyst circulation rate lies outside the multiplicity region, since no system will exhibit multiple steady states over the entire range of parameters. A unique steady state can be obtained in a closed loop system, say, with a feedback proportional controller. The occurence of multiplicity will depend upon the controller gain [48]. For certain values of the controller gain, the middle steady state may become unique and stable, and the multiplicity may disappear altogether.

2. Multiplicity and the degree of combustion completion in the regenerator. It is well-known that there are two sources of multiplicity in chemically reacting systems:

(a) isothermal (or concentration) multiplicity, resulting from the non-monotonic dependence of the rate of reaction upon the concentration of one or more of the reactants.

(b) thermal multiplicity, resulting from the exothermicity of the reaction (the reaction should be highly exothermic and the activation energy of the reaction should be high).

In FCC units, obviously the cracking reactions cannot be the source of multiplicity because the kinetics of the reactions are monotonic and the main reactions are endothermic. Therefore, neither thermal nor isothermal multiplicity is possible in the reactor. The only possible source of multiplicity is the regenerator, where a highly exothermic reaction is taking place, that is carbon burning. Therefore, partial combustion is not the source of multiplicity as stated by Edwards and Kim [27]. Actually, complete combustion of the carbon in the regenerator gives larger regions of multiplicity. This has been proved to be true by Elshishini and Elnashaie [31] (Figure 2).

ELSHISHINI AND ELNASHAIE MODEL (E & E MODIFIED MODEL)

The model developed earlier by Elnashaie and El-Hennawi [28] was further modified [30] to account, for:

1. The change in volumetric gas flow rates between inlet and outlet of reactor and regenerator.

The volumetric gas flow rate at outlet of reactor is:

(1 -xl -x2)(1 -WC) FGM RTR

MG 1 PR .

The volumetric gas flowrate at outlet of the regenerator is:

& RTG GG = (0.5 WH + 0.5 Wco) 100 +

FAFRTG

G 29PG

2. The partial cracking of gasoline and gas oil to lighter hydrocarbons. The heats of cracking reported earlier [29] were based on complete cracking of hydrocarbons to carbon and hydrogen, and thus, were overestimated. In this model, the heats of cracking were modified. The overall heat of cracking obtained after introducing these modifications was found to be very close to the industrial value obtained empirically [51,52].

3. The lumping of light hydrocarbon gases with coke. Weekman [46,47] kinetic scheme lumps light gases with coke. This is not suitable for realistic modelling of commercial FCC units; therefore, the amount of light gases was obtained from the weight ratio of coke to (coke + gases) given in plant data.

4. The recycle stream. Weekman [46,47] kinetic scheme does not account for the formation of HCO (heavy cycle oil), LCO (light cycle oil), and CSO (clarified slurry oil), which are


the fractions produced in addition to gasoline. In the refineries, the only fraction recycled is HCO, and to calculate it, the ratio of HCO/(HCO + LCO + CSO) was obtained from plant data.

Evaluation of Elshishini and Elnashaie Model

The model was successfully compared with two industrial FCC units, and it was shown that

both the units are operated at the middle steady state (Figure 3) and that the multiplicity region

covers a very wide range of parameters. The model was also used to investigate the effect of feed

composition on the performance of FCC units [31,32]. Figure 3 shows the Van Heerden diagrams (Figure 3a) and the unrescted gas oil and gasoline yield profiles (Figure 3b) for both industrial units simulated. It is clear that the maximum gasoline yield for both industrial units occurs at the multiplicity region and, specifically, at the unstable saddle type state. It is also clear from the figure that both units are not operating at their optimum conditions, with the operating point slightly shifted from the maximum gasoline yield. Simple manipulation of the operating

variables can shift the units to their maximum gasoline yield with a considerable improvement

in the productivity of the unit [30,31].

The model needs to be further improved to avoid the empirical evaluation of the ratio of carbon

to light hydrocarbons from industrial measurements, by using a I-lump kinetic network [50]. Also,

the effect of recycle on the kinetics needs to be incorporated into the model in a more rational way than its present empirical form. A more rigorous kinetic model for coke and hydrogen burning in

the regenerator needs to be incorporated, instead of the simple kinetic model used. The model also needs further development to include modern riser type FCC units.

Since the units are usually operated at the middle unstable steady state, extensive effort

needs to be carried out for the analysis of the dynamic behaviour of open loop and closed loop

systems [33].

CONCLUSION

Empirical models could be useful to simulate a unit operating at the conditions for which the empirical model was derived, but these models cannot simulate the performance of the reactor at any other conditions. Such models cannot be used reliably for design purposes or for the investigation of bifurcation behaviour, as was demonstrated in the evaluation of these models.

Furthermore, the introduction of the two-phase model of fluidization is not an unnecessary sophistication, since the role of bubbles is not only to achieve better mixing, but also to supply

reactants to the active dense phase and to remove intermediate products from the active dense phase to the inactive bubble phase. Thus, bubbles have an important and complex effect on conversion, yield, and selectivity of the system. They also affect the heat balance of the system, especially in the regenerator, which is the source of heat to the system.

Much experimental and theoretical work is needed in cooperation between industry and aca-

demia, to develop rigorous models of high-fidelity for this industrially important system. Both the steady state and the dynamic behaviour of the unit need to be investigated.

From a fundamental point of view, such a research will have a strong impact on the understanding of bifurcation behaviour in reactor-regenerator systems. A richness of steady state and dynamic phenomena, exceeding those found for the classical CSTR problem, is certainly lying there waiting to be discovered. Our preliminary unsteady state investigation (331 has shown very interesting bifurcation behaviour including complex hysteresis and pitchfork type multiplicities. The dynamic behaviour also reveals very high sensitivity to initial conditions in the neighbour- hood of the desirable middle steady state as well as high sensitivity to the dimensionality of the dynamic model and the different assumptions involved in it.


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Comparison Between Different Mathematical Models for the Simulation of Industrial Fluid Catalytic Cracking