INTRODUCTION
Catalytic cracking is the most important and widely used refinery process for
converting heavy oils into more valuable gasoline and lighter products. The
catalytic cracking processes produce carbon (coke) which remains on the catalyst
particle and rapidly lowers its activity. To maintain the catalyst activity
at a useful level, it is necessary to regenerate the catalyst by burning off
this coke with air. The cracking reaction is endothermic and the regeneration
reaction is exothermic. Recent units are designed to use the regeneration heat
to supply that needed for the reaction and to heat the feed up to reaction temperature
(Gary and Handwerk, 1984). This is accomplished by fluidization,
where the fluidized catalyst carries the regeneration heat to the reactor.
Several studies have been conducted for FCC modeling, simulation, optimization
and control, but very few of them consider the RFCC (Xu
et al., 2006; Gauthier et al., 2000;
Fernandes et al., 2007). Many of the models in
these studies describe the riser reactor with one dimensional mass, energy and
chemical species balances. These models are normally simple to formulate and
to solve. More detailed models, are those that consider the riser reactor in
two or three dimensions. These models are based on phenomenological concepts
and use a simultaneous solution of the conservation equations of mass, momentum,
energy and species for both the gas and particulate phases (Gao
et al., 1999; Mathiesen et al., 1999).
Therefore, Souzaa et al. (2003), combined a 2D
fluid flow field with a 6 lumps kinetic model and used two energy equations
(catalyst and gas oil) to simulate the gas oil cracking process inside the riser
reactor. Das et al. (2003), performed the threedimensional
simulation of an industrialscale fluid catalytic cracking riser reactor using
a novel densitybased solution algorithm. The particlelevel fluctuations are
modeled in the framework of the kinetic theory of granular flow. The reactor
model includes separate continuity equations for the components in the bulk
gas and inside the solid phase.
Berry et al. (2004), modified the twodimensional
hydrodynamic model to make it predictive by incorporating the slip factor for
the calculation of the crosssectionally averaged voidage. The model has been
coupled with the fourlump kinetic model of Gianetto et
al. (1994), to predict the effect of operating conditions on profiles
of conversion, yield, temperature and pressure in the riser. With regard to
reaction and kinetics, Xu et al. (2006) proposed
a 7lump kinetic model to describe residual oil catalytic cracking, in which
products especially coke were lumped separately for accurate prediction. Because
in recent studies, kinetics was developed accounting for coke formation leading
to catalyst deactivation. The reactor block is modeled as a combination of an
ideal Plug Flow Reactor (PFR) and a Continuously Stirred Tank Reactor (CSTR).
In the present work, the 7lump kinetic model is utilized to model the cracking
reactions in the riser to determine the products yield with the required conversion
and other process variables. These are the base information required for the
dynamic model. The result is compared with the plant data from Khartoum Refinery
Company (KRC) in Sudan and with Xu et al. (2006)
results. Also the hydrodynamic model of Malcus and Pugsley
(2002), which is modified by Berry et al. (2004),
is solved in this work for the riser operating conditions from KRC. These models
have to be combined with the steady state material and energy balances of RFCC
unit with residual feedstock from Sudanese crude distilling unit.
Process description: To process the heavy, viscous residual feedstock,
that produce relatively high amounts of coke, the design incorporates an advanced
feed injection system, a unique regeneration strategy and a catalyst transfer
system that produces extremely stable catalyst circulation. Recent technology
advances have been made in the areas of riser termination, reactant vapor quench
and Mix Temperature Control (MTC) (Meyers and Hunt, 2003).
The goal of RFCC unit is to maximize gasoline production. Figure
1 shows the schematic diagram of the reaction  regeneration section (area
of our study in this unit), for which the mathematical model is being solved.
The preheated feed is injected into the riser bottom, atomized into small
droplets and contacted with the high temperature regenerated catalyst from the
second regenerator. It vaporizes immediately, being cracked into products. Cycle
oil is injected into four nozzles above the fresh feed nozzles in order to optimize
product distribution by raising the catalyst to oil ratio. In the middle of
the riser, nozzles for slurry recycle and naphtha are provided. At the riser
out let, the product vapor is separated from the catalyst by the Ramshorn disengager
and four single stage cyclones. The stripped catalyst passes through the spent
catalyst slide valve and enters into the catalyst distributor of the first regenerator,
where partial combustion of the coke take place at the dense bed. The semiregenerated
catalyst from the first regenerator passes through the standpipe and slide valve
and then lifted by compressed air to the second regenerator. The regenerated
catalyst from the second regenerator flows into the hopper and then into the
regenerated catalyst standpipe to the bottom of the riser, thus forming a continuous
catalyst circulation loop. Catalyst cooler can be installed by the first regenerator
to keep balance of the reactor regenerator system and to improve the operating
flexibility.
MATHEMATICAL MODELS
Reaction kinetics: For kinetic modeling, the complexity of charge stocks
make it extremely infeasible to characterize and describe at a molecular level,
so attempts have been made to lump large numbers of chemical compounds into
a kinetic species to describe the complex reactions in the process (Xu
et al., 2006; Wei and Kuo, 1969). The simple
models that describe the cracking kinetics with 3, 4 or 5 lumps have the advantage
that just a few kinetic constants must be estimated for each feedstock, but
depending on the simplicity of the model, the key FCC products cannot be predicted
separately. More sophisticated models, normally with more than 10 lumps, have
basically two advantages; a single group of estimated kinetic constants can
be used for various feedstock and all the most important FCC products can be
predicted separately. The disadvantages of these models are that a large number
of kinetic constants must be estimated and as each lump represents a differential
equation in the mathematical model, the complexity of the numerical solution
may increase exponentially (Souzaa et al., 2003).

Fig. 1: 
Process diagram of the residue fluid catalytic cracking unit
RFCC, reaction section, at Khartoum Refinery Company (KRC) 

Fig. 2: 
The 7lump reactions scheme for RFCC (2) 
Recently kinetics was developed accounting for coke formation leading to catalyst
deactivation. Therefore, in the present work the 7lump kinetic model, proposed
by Xu et al. (2006), has been chosen. In this
model coke, dry gas and LPG were lumped separately. The probability of the cracking
reaction from gasoline to coke and gases to coke can be neglected since the
kinetic constants for these reactions are many orders of magnitude smaller than
that for the others, which simplifies the kinetic parameters determination.
So, the reaction scheme for RFCC is shown in Fig. 2.
According to the principle of invariant response, lumps are formed by their
boiling ranges having similar behavior. Also these lumps are not separated by
chemical type with the parameters depending on feedstock and catalyst properties,
which is the limitation of this model (Xu et al.,
2006). The j lumps are Lump 1 for residue R, lump 2 for heavy fuel oil HFO,
lump 3 for light fuel oil LFO, lump 4 for Gasoline G, lump 5 for liquefied petroleum
gas LPG, lump 6 for dry gas and lump 7 for coke C.
An irreversible pseudo first order reaction was accepted for all reactions in this model. The reaction rate of a pseudospecies j, is assumed to be a function of the molar concentration of species j(ρa_{j}), the mass density of catalyst relative to the gas volume (ρ_{c}/ε) and reaction constant (k'_{j} ), as in Eq. 1.
The reaction constant k'_{j} decayed with time as:
The heavy, inert aromatics adsorption function f(C_{Arh}) is described by:
Catalyst deactivation was accounted for by a hyperbolic function Ø(t_{C}) as:
Nitrogen poisoning function f(N) is defined as:
With regard to high catalyst to oil ratio, the nitrogen poisoning deactivation can be neglected because of its insignificance. Taking the ideal gas assumption:
The average molecular weight of all lumps,
, change with distance along the riser as
Therefore, the reaction rate in matrix notation can be expressed by:
where, the reaction rate vector is:
The composition vector consisting of j lumped species is a = [a_{1} a_{2} a_{3} a_{4} a_{5} a_{6} a_{7}] and the rate constant matrix defined as in the appendix.
The reactor block is considered as a combination of two different reactors,
according to its hydrodynamic regime, the riser is a near ideal plugflow displacement
of the catalyst and reaction mixture, while the main vessel (separator) is an
ideal mixing CSTR. Therefore, the kinetic model is incorporated into an isothermal
plug flow reactor model by the continuity equation, with interparticle diffusion
and axial dispersion neglected Xu et al., 2006,
as in Eq. 9:
For the steady state one dimensional equation:
Combining this with equation 8 to get the lumps concentration profiles with the axial dimensionless distance, Z = z/L, along the riser:
where, S_{wh} = G_{v}ε/ρ_{c}L is the true weight hourly space velocity.
The concentration profile obtained above assumes the continuity equation of each lump is well represented by the steady state equation only. However, the dynamic continuity equation must be used to get the actual concentration profile for each lump. Thus a dynamic, two dimensional continuity equation is proposed to reoptimize the kinetic parameters:
By this equation the concentration profile of all the lumps can be optimized to get the correct values of concentration at any point in the riser axially and radially.
Riser hydrodynamic model: Several different riser hydrodynamic models
have been presented in the fluidization literature (Malcus
and Pugsley, 2002; Godfroy et al., 1999).
Many of these models are empirical or semiempirical in nature and are based
on experimental observations from coldflow laboratory and pilotscale risers.
One of these is the recently developed model of Malcus and
Pugsley (2002), which Berry et al. (2004)
have used in their simulations. They have shown that the Boltzmann function
of Eq. 13 leads to good predictions of radial profiles of
the solids volume concentration at operating conditions typical of FCCU risers.
Berry et al. (2004) made the model of Malcus
and Pugsley, predictive by using the slip factor correlation of Patience
et al. (1992), in Eq. 14:
The average particle velocity in the riser, V_{p}, is defined by:
Combining Eq. 14 and 15 gives:
To complete their hydrodynamic model, Malcus and Pugsley, performed a multiple nonlinear regression on a large set of data collected from the open literature to obtain the following expression for predicting the mass flux at the centerline of the riser.
Combining this with the solids concentration at the riser centerline, obtained from Eq. 13, to get:
A second Boltzmann function is then used to describe the radial solids profile of the solids velocity:
X_{o} stands for the center of the Boltzmann function described as:
Where:
Berry et al. (2004) calculated the radial solids
velocity profile using an iterative procedure. They set a value for the wall
solids velocity (at r = R), then calculated the profile V_{p,r} from
the above equations. They integrated this profile to get the average solids
velocity and compared this value with the average solids velocity obtained from
mass balance. If the mass balance was not satisfied, the value of k_{2}
has to be updated and the calculation has to be performed again. They repeated
this iterative procedure until the integrated value of the solids velocity was
in agreement with the average solids velocity according to the mass balance.
However, this procedure did not take into account the axial velocity, V_{p,z} in calculating the average velocity. Whereas, in 2dimensional modeling V_{p,z} must be included in estimating the average velocity. In order to accomplish this, momentum balance equations for both phases are proposed to get the other profiles V_{p,z}, v_{r} ,v_{z}.
Process variables, material and energy balances: In calculating the
flow rate of streams in the reactor and two regenerators, the following definations
and equations (Gary and Handwerk, 1984; Nelson,
1958 and Fei et al., 1997) were used:
• 
The catalytic cracking efficiency (η) depend on how much
gasoline is yielded from the unit as in Eq. 23: 
• 
The catalyst activity = conversion/(100  conversion) 
• 
The severity factor for highly waxstock with characterization factor
equal to 12.4 was correlated from Nelson (1958), with
coefficient of determination 0.99666 as: 
Then some of the operating variables like space velocity and residence time
can be calculated from:
• 
Space velocity = catalyst to oil ratio/severity factor 
• 
Catalyst residence time=1/[space velocity* catalyst/oil ratio] 
• 
Also delta coke is a very popular index and when increased can cause significant
rise in regenerator temperature, ultimately reducing the amount of residual
feed which can be processed. Therefore in order to maximize the amount of
residual content in RFCC feed, a lowdelta coke catalyst must be employed
(must be less than 1.2), which is defined as: 
where, CSC is Carbon on spent catalyst and CRC is Carbon on regenerated catalyst. And as:
All these definitions and equations were solved simultaneously to calculate the flow rate of the streams that was needed in the steady state material and energy balance calculations where:
Sum of flow rates of input streams = Sum of flow rates of output steams
Reactor duty = Enthalpy of the input streamsEnthalpy of the output streams
RESULTS AND DISCUSSION
To solve the above mathematical models, a program in MATLAB was developed. The chosen 7lump kinetic model was solved giving the concentration profiles of the j components (lumps), as shown in Fig. 3.
From this figure, it is observed that most of the oil cracking occurs in the
feed injection zone, this means that oil fraction cracking is an instantaneous
reaction.

Fig. 3: 
Axial lumps concentration profiles along riser 

Fig. 4: 
Radial solids velocity profile and mass flux profile in the
riser 
The products yield increase rapidly in the first onethird of the riser height
and then vary gently. The heavy fuel oil yield reaches a maximum at about 20%
of the riser height and then declines due to overcracking reactions. The products
yield at the end of the riser length Z = 1, is consistent with the design value
for this yield. A very good agreement is shown with Xu et
al. (2006), because if the two heavy products are combined together
and treated as the unreacted products lump, the same trend is obtained.
Also in solving the hydrodynamic model, the radial solid velocity profile is
predicted as Berry et al. (2004) have done by
initially setting the wall solid velocity equal to the terminal settling velocity
of a single particle. Then, from the simulator the actual wall solid velocity
is fixed. So, the hydrodynamic model was solved giving the following radial
solids velocity profile and solids mass flux profile, as shown in Fig.
4. It is assumed no axial changes appears in both velocity and flux, with
the average mass flux = 598 kg m^{2} sec^{1} and the superficial
gas velocity = 6.6 m sec^{1}.
There is a good agreement between these results and that of Berry
et al. (2004), who used in their model prediction and validation,
commercial data from three publications in the open literature for two different
commercial risers.
The percent difference between in and out streams balances the reactor, first and second regenerators and gives the optimum values for the process variables. Changing input variables and saving the output gives the relation between process variables and some of the input parameters. It can be noticed from Fig. 5 and 6, if the task is lower residence time or higher space velocity in the reactor then the unit can easily be designed with higher cracking efficiency, but this will be at the cost of severity and % conversion, where both of them decrease with the increasing efficiency. The effect of efficiency on catalyst/oil ratio and delta coke is not big. Therefore, the best cracking efficiency ranges between 7585%.

Fig. 5: 
Changes with cracking efficiency 

Fig. 6: 
Changes with cracking efficiency 

Fig. 7: 
Changes with gasoline yield 
Yield calculations depend on crude and residue specifications so changing gasoline
yield will affect the energy balance slightly. Figures 79
shows how process variables change with the change in gasoline yield. Increasing
gasoline yield will increase % conversion which is obviously true. But an increase
in conversion does not necessarily mean an increase in gasoline yield because
coke and gases yield increase at the expense of gasoline yield. This agrees
with Gary and Handwerk, (1984) and Fei
et al. (1997). Also increasing gasoline yield will increase catalyst/oil
ratio and severity. This increase is sharp and not preferable above 0.6 gasoline
yield. Besides, at this region the energy is not balanced any more. Delta coke
and residence time increase with the increase in gasoline yield till an optimum
point then decrease. The decrease becomes sharp after 0.55 gasoline yield and
lower delta coke gives higher Carbon on Regenerated Catalyst (CRC) which leads
to lower catalyst activity.

Fig. 8: 
Changes with gasoline yield 

Fig. 9: 
Changes with gasoline yield 
Table 1: 
Process variables 

Also, space velocity is too small at higher yield. This means that gasoline
yield above 0.55 is dangerous.
Obtaining fixed and accurate values for the above process variables is very important in the design of the reactor and two stage regenerators, so with an objective of achieving 80% cracking efficiency the values obtained is in Table 1.
CONCLUSION
It has been observed that optimum cracking efficiency is 7585% and gasoline yield up to 0.55. Designing with gasoline yield above this limit is not preferred. It is recommended that, as a future work, these steady state models have to be expanded to the overall dynamic model for the full riser simulation and optimization.
ACKNOWLEDGMENTS
Authors are grateful to Universiti Technologi PETRONAS (UTP) for providing all support to undertake this research work. Also authors would like to acknowledge the staff at Khartoum Refinery Company (KRC) in Sudan, for their resources.
NOMENCLATURE
a 
: 
Composition vector consisting of j lumped species 
a_{j} 
: 
Concentration of lump j (moles j/g gas) 
C_{arh} 
: 
wt.% of aromatics in residual oil 
D 
: 
Riser diameter (m) 
Fr 
: 
Froude number = u_{o}/(gD)^{0.5} 
Fr_{t} 
: 
Froude number based on the terminal settling velocity = U_{t}/(gD)^{0.5} 
G_{v} 
: 
The superficial mass flow rate of gas mixture (g/(cm^{2}.h)) 
g 
: 
Gravitational constant 9.81 (m sec^{2}) 
G_{s} 
: 
Riser solids mass flux (kg m^{2} sec^{1}) 
G_{s,cl} 
: 
Solids mass flux at the centerline of the riser (kg m^{2} sec^{1}) 
L 
: 
Riser length (m) 
k_{1} 
: 
Parameter in the hydrodynamic model defined by Eq. 21 
k_{2} 
: 
Parameter in the hydrodynamic model defined by Eq. 22 
k_{j} 
: 
Invariant reaction rate constant 
k'_{j} 
: 
Reaction rate constant decayed with time 
K 
: 
Rate constant matrix 
K_{h} 
: 
The heavy aromatic ring adsorption coefficient, (wt. % C_{Arh})^{1},
K_{h} = 0.128 (Xu et al., 2006) 
K_{h} 
: 
Basic nitrogen poisoning adsorption coefficient (g basic N/g catalyst)^{1} 
R 
: 
Reaction rate in matrix notation 
R 
: 
Riser radius (m) 
r 
: 
Radial position in the riser (m) 
r_{j} 
: 
Kinetic expression of species j 
t 
: 
Time (sec) 
t_{c} 
: 
Catalyst residencetime 
u_{o} 
: 
Riser superficial gas velocity (m sec^{1}) 
V_{p} 
: 
Average catalyst velocity (m sec^{1}) 
V_{p,cl} 
: 
Solid velocity at the centerline of the riser (m sec^{1}) 
V_{p,r} 
: 
Radial solid velocity profile (m sec^{1}) 
z 
: 
Axial position in the riser (m) 
Z 
: 
Axial dimensionless distance along the riser 
Greek symbols
ε_{avg} 
: 
Average voidage 
ε_{r} 
: 
Voidage at radial position r 
β 
: 
Catalyst deactivation constant = 162.15 (2) 
γ 
: 
Catalyst deactivation constant = 0.76 (2) 
n 
: 
Slip factor 
ρ 
:: 
Density of the catalyst (kg/m^{3}) 
ρ_{s} 
: 
Gas mixture density 
η 
: 
Catalytic cracking efficiency 
APPENDIX: RATE CONSTANT MATRIX