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2007, Chemical Engineering Science

International Journal of Chemical Reactor Engineering

Modeling of Fluid Catalytic Cracking Riser Reactor: A Review2010 •

This work aims at compiling the important works on the modeling of a fluid catalytic cracking (FCC) riser reactor. The modeling of a riser reactor is very complex due to complex hydrodynamics and unknown multiple reactions, coupled with mass transfer resistance, heat transfer resistance and deactivation kinetics. A complete model of the riser reactor should include all the important physical phenomena and detailed reaction kinetics. As the computational fluid dynamics (CFD) is emerging as a powerful tool for modeling the FCC riser, various works on riser modeling using CFD are also included in the paper.

Chemical and Process Engineering Research

Riser Reactor Simulation in a Fluid Catalytic Cracking Unit2013 •

2005 •

Chemical Engineering Science

Model for the performance of a fluid catalytic cracking (FCC) riser reactor: effect of feed atomization2001 •

2019 •

This work presents a one-dimensional adiabatic mathematical model for the riser reactor of a commercial fluid catalytic cracking unit, FCCU. The cracking reactions in the riser reactor were based on six-lump kinetics of the catalytic cracking of vacuum gas oil, taking cognizance of diffusion resistance, which is a departure from the general norm in the literature. Moreover, two-phase hydrodynamic model for the riser reactor, coke-on-catalyst deactivation model as well as heat transfer resistance between the fluid and solid phases were considered. Two vaporization approaches (the instantaneous and one-dimensional vaporization) of the feedstock were investigated. The developed model was a set of eleven highly non-linear, coupled and stiff ordinary differential equations, ODEs, which was numerically solved with an implicit MATLAB built-in solver, ode23tb, designed deliberately for handling stiff differential equations to circumvent the problem of instability associated with explicit me...

2019 •

Sādhanā

Seventeen-lump model for the simulation of an industrial fluid catalytic cracking unit (FCCU)Computers & Chemical Engineering

Three-dimensional modeling of fluid catalytic cracking industrial riser flow and reactions2011 •

2019 •

This study developed and simulated a dynamic mathematical model for a Fluid Catalytic Cracking Unit (FCCU) riser reactor with consideration of coke deposition on the catalyst in the overall mass balance of the system. It also proposed a mathematical model that accounts for the dynamic effect of coke deposition on the hydrodynamics of the FCCU riser. The spatial derivatives are discretized using finite difference to obtain the dependency of the state variables on time. The resulting ordinary differential equations were simulated using Java Application Programme Interface, ODEToJava. An approach proposed in this study is in good agreement with experimental and numerical data available in the literature. The results showed an outlet vapor density of 2.76 kg/m3 which gave a +7.39 % deviation when compared with plant data. Also the temperature outlet of the riser predicted by the model deviated by -5.49% when compared with real plant data. The model also confirmed the acceptability of th...

Chemical Engineering Science 62 (2007) 4510 – 4528
www.elsevier.com/locate/ces
A new generic approach for the modeling of fluid catalytic cracking (FCC)
riser reactor
Raj Kumar Gupta a , Vineet Kumar a,∗ , V.K. Srivastava b
a Department of Chemical Engineering, Thapar University, Patiala 147 004, India
b Department of Chemical Engineering, Indian Institute of Technology, Delhi 110 016, India
Received 17 April 2006; received in revised form 4 November 2006; accepted 14 May 2007
Available online 21 May 2007
Abstract
A new kinetic model for the fluid catalytic cracking (FCC) riser is developed. An elementary reaction scheme, for the FCC, based on
cracking of a large number of lumps in the form of narrow boiling pseudocomponents is proposed. The kinetic parameters are estimated using
a semi-empirical approach based on normal probability distribution. The correlation proposed for the kinetic parameters’ estimation contains
four parameters that depend on the feed characteristics, catalyst activity, and coke forming tendency of the feed. This approach eliminates the
need of determining a large number of rate constants required for conventional lumped models. The model seems to be more versatile than
existing models and opens up a new dimension for making generic models suitable for the analysis and control studies of FCC units. The
model also incorporates catalyst deactivation and two-phase flow in the riser reactor. Predictions of the model compare well with the yield
pattern of industrial scale plant data reported in literature.
䉷 2007 Elsevier Ltd. All rights reserved.
Keywords: Fluid catalytic cracking; Mathematical modeling; Simulation; Cracking kinetics; Pseudocomponents; Computational fluid dynamics
1. Introduction
Fluid catalytic cracking (FCC) is a process in which
the heavy hydrocarbon molecules are converted into lighter
molecules. The hydrocarbon feed enters a transport bed tubular reactor (riser) through feed atomizing nozzles and comes
in contact with the hot catalyst coming from the regenerator. The feed gets vaporized and cracks down to the lighter
molecules as it travels upwards along with the catalyst. As a
result of cracking, the velocity of the vapors increases along
the riser height. Coke, the byproduct of cracking reactions,
gets deposited on the catalyst surface thus causing the catalyst
to loose its activity. The cracked hydrocarbon vapors are separated from the deactivated catalyst in a separator; the vapors
adsorbed onto the surface of the catalyst are also stripped off
using steam in the catalyst stripper. The cracked hydrocarbon
vapors are sent to the main distillation column for further
∗ Corresponding author. Tel.: +91 175 2393063.
E-mail address: vikumar@tiet.ac.in (V. Kumar).
0009-2509/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ces.2007.05.009
separation into various fractions, and the deactivated catalyst
flows into the regenerator. In the regenerator, the coke deposited
on the catalyst surface is burned off to regenerate the catalyst.
The catalyst also becomes hot during the regeneration process.
This hot-regenerated catalyst is recycled back to the riser reactor. Thus the catalyst acts as a heat carrier also and provides
the heat required for endothermic cracking reactions in the riser
reactor as well as the heat required for the vaporization of feed.
Detailed modeling of the riser reactor is a challenging task
for theoretical investigators not only due to complex hydrodynamics and the fact that there are thousands of unknown hydrocarbons in the FCC feed but also because of the involvement
of different types of reactions taking place simultaneously. It
is believed that catalytic cracking begins with the formation of
+
carbenium ion (R1–CH+
2 –R2 or R–CH3 ) by the interaction of
olefin molecules with the acidic site on the catalyst followed
by the beta scission of the carbenium ion. In the beta scission reaction, the bond of carbenium ion breaks to form an
olefin and a new carbenium ion. The carbenium ion formed
by beta scission can undergo further cracking reaction. The
olefin can also be cracked further after being converted to a
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
carbenium ion through hydrogen addition. Thus, large hydrocarbon molecules can be cracked repeatedly producing successively smaller hydrocarbons. However, as the hydrocarbon
chain becomes smaller, cracking rates become slower. At the
same time, a short, less reactive ion can transfer its charge to a
larger, more reactive molecule by hydrogen transfer that makes
the cracking reaction of larger molecules still faster (Wilson,
1997). Such detailed chemistry of catalytic cracking coupled
with a large number of unknown compounds present in the
feedstock is very difficult to be used in the mathematical modeling of an industrial scale FCC riser reactor because of the
analytical and computational limitations.
The traditional and global approach of modeling of cracking kinetics is based on lumping of compounds. Mathematical
models dealing with riser kinetics can be categorized into two
main types. In one category, the lumps are made on the basis of boiling range of feed stocks and corresponding products
in the reaction system. This kind of model has an increasing
trend in the number of lumps of the cracked components. The
other approach is that in which the lumps are made on the basis of molecular structure, characteristics of hydrocarbon group
composition in reaction system. This category of models emphasizes on more detailed description of the feedstock (Wang
et al., 2005).
These both categories of models do not include chemical
data such as type of reaction and reaction stoichiometry. The
number of kinetic constants in these models increases very
rapidly with the number of lumps. All these models assume
that FCC feed and products are made of a certain number of
lumps, and kinetic parameters for these lumps are estimated
empirically considering the conversion of one lump to the other.
In both of these categories, however, reaction kinetics being
considered is that of ‘conversion’ of one lump to another and
not the ‘cracking’ of an individual lump.
As the reactivity of the feed depends on the composition of
the feed, and catalyst composition and reactivity is important
in determining conversion and product selectivity, the values
of kinetic constants obtained by the above discussed models
depend on the particular pair of feedstock and catalyst for which
they are obtained. Hence, these kinetic constants cannot be used
for different feed and catalyst pairs.
Other modeling schemes include, models based upon reactions in continuous mixtures (Aris, 1989), structure oriented
lumping (Quam and Jaffe, 1992), and ‘single-events’ cracking
(Feng et al., 1993). Nevertheless, the application of these models to catalytic cracking of industrial feedstocks (vacuum gas
oil), is not realized because of the analytical complexities and
computational limitations. Liguras and Allen (1989a) proposed
a lumped kinetic model so as to utilize the pure components
cracking data for the catalytic cracking of oil mixtures. The
authors in their subsequent work (Liguras and Allen, 1989b)
divided the petroleum feedstock into a number of pseudocomponents. These pseudocomponents were characterized by
grouping the feedstock components into compound classes and
selecting a set of representative compounds in a compound
class and then assigning the concentrations to the representative
compounds. This pseudocomponents characterization method
4511
requires extensive use of analytical procedures and suffers from
the same drawbacks that are cited above. A more detailed compilation of various lumping schemes, and their use in advanced
modeling and control of FCC unit are presented in our recent
work (Gupta et al., 2005).
Even today, in the advanced models of the FCC riser (models
considering various aspects of reactor modeling) three lump or
four lump kinetic schemes are being used by the investigators
to avoid the mathematical complexities and load of computation that will be there if more number of lumps are considered
in the kinetic scheme as the number of cracking constants increases rapidly with the number of lumps. With this in view, in
the present work, a new approach of kinetic scheme, considering a large number of lumps (the lightest being methane and
the heaviest containing all compounds of the feedstock having
boiling point close to its end-point), for the FCC riser is introduced. These lumps are characterized using constant Watson
characterization factor, which is an indicator of the feedstock
composition. Characterization was made in such a way that all
physico-chemical properties of the lumps are known and they
can be treated as hypothetical pure components. The proposed
model considers that each lump on cracking gives two other
lumps in one single reaction step. The proposed model falls
under the category of models in which lumps are formed on
the basis of boiling point, but in this approach, each individual
lump is considered as a pure component with known physicochemical properties. A separate coke lump is also considered
and it is assumed that when one mole of a lump cracks down
it gives one mole each of two other lumps and the balance material gives the coke.
The proposed model also incorporates two-phase flow and
catalyst deactivation. Since a new cracking reaction mechanism
is introduced, a new semi-empirical approach based on normal
probability distribution is also proposed to estimate the cracking reactions’ rate constants. The proposed correlation contains
four parameters that depend on the feed characteristics, catalyst activity, and coke forming tendency of the feed. This eliminates the need of estimating the rate constants by regression
analysis. Stangeland (1974) had proposed a kinetic model, for
the prediction of hydrocracker yields, in which the feedstock
was divided into a series of 50 ◦ F boiling range cuts without
characterizing these cuts. In the present work we have characterized the pseudocomponents as discussed in Appendix A.
2. Riser model
The riser is modeled as a vertical tube comprising of a
number of equal sized compartments (or volume elements)
of circular cross section. Volume elements are designated by
symbol j (j = 1, 2, . . . , NC ) and numbering of the volume elements is done from bottom (inlet) to top (outlet). Each volume
element is assumed to contain two phases (i) solid phase (catalyst and coke) and (ii) gas phase (vapors of feed and product
hydrocarbon, and steam). In one volume element, each phase is
assumed to be well mixed so that heat and mass transfer resistances can be ignored. Model equations are written for both the
phases in each j th volume element for all pseudocomponents
4512
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
PCi (i = 1, 2, . . . , N ). Cracking reactions’ rate constants in
each volume element are evaluated at the local temperature of
the j th volume element and are subsequently used for calculating the change in molar concentration of each component and
the heats of reactions. This approach of finite volume method
(FVM) is widely used in computational fluid dynamics (CFD)
and has been elaborated elsewhere for membrane separation
process by Kumar and Upadhyay (2000). It is assumed that
known amount of reactants enter the j th volume element, cracking reactions take place in the element for a period equal to the
residence time of the hydrocarbons entering the element. The
concentration of the reacting mixture at the outlet of j th volume
element serve in defining the feed for the (j + 1)th volume element. To start computation from volume element number one,
i.e., the inlet of the riser reactor, the known process parameters
at the inlet of the riser are used. To simplify this FVM based
approach for solving material and energy balance equations
in FCC riser reactor, following commonly used assumptions
were made.
2.1. Assumptions
• At the riser inlet, hydrocarbon feed comes in contact with
the hot catalyst coming from the regenerator and instantly
vaporizes (taking away latent heat and sensible heat from
the hot catalyst). The vapor thus formed moves upwards in
thermal equilibrium with the catalyst (Corella and Frances,
1991; Martin et al., 1992; Fligner et al., 1994; Ali et al.,
1997; Derouin et al., 1997).
• There is no loss of heat from the riser and the temperature of the reaction mixture (hydrocarbon vapors and catalyst) falls only because of the endothermicity of the cracking reactions (Corella and Frances, 1991; Ali et al., 1997;
Theologos et al., 1999; Gupta and Subba Rao, 2001).
• Ideal gas law is assumed to hold while calculating gas
phase velocity variation on account of molar expansion due
to cracking and gas phase temperature (Gupta and Subba
Rao, 2001).
• Catalyst particles are assumed to move as clusters to account for the observed high slip velocities (Gupta and
Subba Rao, 2001).
• Heat and mass transfer resistances are assumed as negligible (Corella and Frances, 1991; Martin et al., 1992; Ali
et al., 1997; Derouin et al., 1997; Theologos et al., 1999).
• Both phases are assumed in plug flow condition hence back
mixing in both phases is neglected.
2.2. Kinetic modeling
Most of the kinetic schemes currently being used for the
modeling of FCC riser reactor are based on the specified number of lumps (four lump and 10 lump schemes being the most
common). In the proposed model, any suitably large number of
lumps can be considered. However, in the present work number
of lumps, N, is taken as 50. Each of these lumps is characterized
on the basis of average boiling point and specific gravity, and
treated as hypothetical pure component with all physical, thermodynamic, and critical properties known. A brief description
for generating these pseudocomponents is given in Appendix
A. For the generation of these lumps, the feedstock is divided
into 12 lumps and average boiling point of each lump is determined by area averaging of the true boiling point (TBP) curve
of the feedstock. Thus one individual lump is a collection of
all compounds having boiling point close to each other. The
temperature range between boiling point of the lightest lump of
the feedstock and the boiling point of propane is divided into
31 equal intervals which were considered to be boiling points
of 31 lumps of liquid products to be formed after cracking.
Molecular weight of each lump is also calculated using empirical correlation proposed by Edmister and Lee (1984) assuming
constant Watson characterization factor for product and feed
lumps. Constituents of dry and wet gas are also considered as
seven lumps consisting of methane, ethane, propane, butene,
butane, pentene, and pentane. Properties of these seven lumps
were taken equal to that of pure components due to the fact that
empirical correlations fail to predict properties of very light hydrocarbons and also due to the fact that gaseous products are
rather known mixtures of these hydrocarbons. Thus we have
lumps of seven pure components and 43 hypothetical components. Henceforth, all these components are collectively called
pseudocomponents and abbreviated as PCs. Apart form these
50 components, a separate coke lump is also considered which
is formed as a byproduct of the cracking reactions.
The pseudocomponent based approach for design and simulation of crude distillation unit is highly successful. Sufficient
empirical correlations are available in literature to predict almost all physico-chemical, thermodynamic, and critical properties of pseudocomponents. Vapor–liquid equilibrium of these
pseudocomponents are predicted fairly accurately without resolving these components in aromatics, olefins, naphthenes, and
paraffins. For applying this approach to FCC modeling, it is assumed that one mole of a pseudocomponent on cracking gives
1 mole each of two other smaller pseudocomponents and some
amount of coke may also form. A schematic diagram of the reaction mechanism is given in Fig. 1. There are total N blocks
in each column of the diagram and each block in a column
represents one pseudocomponent (blocks in a row represents
same pseudocomponent). Pseudocomponents are numbered in
increasing order of normal boiling point, which also ensures increasing order of molecular weight. According to the proposed
scheme, there are several possible ways through which one particular pseudocomponent (say ith pseudocomponent, PCi ) can
crack down to give a pair of pseudocomponents PCm and PCn
along with some amount of coke as cracking byproduct according to the following pseudoreaction mechanism:
ki,m,n
PCi → PCm + PCn + i,m,n ,
(1)
where i, m, and n are pseudocomponents’ numbers, i,m,n is
the amount of coke formed (kg) when one kmol of ith pseudocomponent cracks to produce one kmol each of mth and nth
pseudocomponents.
4513
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
Reactant
Feed
Product 1
+
Product 2
Mol. Wt.
MW1
MW2
MW3
1
2
3
1
2
3
1
2
3
MWm
m
m
m
MWn
n
n
n
MWi
i
{
MWN-11
MWN
PCi
ki , m ,n
PC m + PC n + coke
N-11
N-11
N-11
N
N
N
+
coke
Amount of coke (gram)
=MWi - (MWm+ MWn)
Fig. 1. Schematic diagram of reaction mechanism.
The proposed pseudoreaction mechanism is different from
the conventional reaction mechanism. In Eq. (1) PCi is the ith
pseudocomponent consisting of several hundreds of compounds
having molecular weight close to MWi . After cracking, each
individual molecule of PCi gives at least two product molecules
of molecular weight less than the cracking molecule. There is a
wide range of possibilities in which one particular molecule of
PCi can crack. In one case the products may be of widely different molecular weights (i.e., one molecule is of small molecular
weight and other of high molecular weight close to the original
molecule). In other case both the product molecules can be of
almost equal molecular weight. Also, amongst these reactions,
some may be coke forming and some may not be coke forming
reactions. Eq. (1) represents the collection of only those reactions in which each compound of molecular weight MWi gives
two molecules of molecular weights MWm and MWn along
with the associated coke formed in this process. Here it should
be noted that for incorporating all possible cracking reactions
of ith pseudocomponent, value of m varies from 1 to i as no
product can be heavier than the reactant. Similarly, value of n
ranges from 1 to m only as the products PCm and PCn are interchangeable. Therefore, in the proposed mechanism, there are
(i + 1) × i/2 possible ways in which an ith pseudocomponent
can crack.
For the law of mass conservation, it is required that mass
of reactant should be equal to the mass of products in one
reaction step. Therefore, when MWi kg of PCi cracks to give
MWm kg of PCm and MWn kg of PCn , rest of the material
gives coke (i,m,n ). Thus, the value of i,m,n can be calculated
by taking difference in molar masses of reactant and product
hydrocarbons as follows:
i,m,n = MWi − (MWm + MWn ).
(2)
In Eq. (1) there are (i + 1) × i/2 possible ways in which
cracking reaction for one pseudocomponent PCi can be written. Only two of such possible reactions are shown by solid
line in Fig. 1. The dashed line represents the same reaction, as
the values of m and n in Eq. (1) are interchangeable. Thus the
total number of reactions that could be written for the complete set of reactions becomes N × (N − 1) × (N − 2)/3 (for
N = 50, total number of reactions become 39 200). However,
all of these reactions are not feasible. The feasibility of a reaction is found using the stoichiometry of the cracking reaction
(Eq. (1)). Only those reactions are considered feasible for which
the value of i,m,n calculated by Eq. (2) is either zero or positive. This natural constraint greatly reduces the number of feasible cracking reactions but still the number is too large (more
than 10 000 in the present case with N = 50).
To handle such a large number of reactions in riser reactor,
special considerations for the estimation of reaction rate constants and technique for solving material and energy balance
equations are required. In the present work, a new semiempirical scheme for the estimation of rate constants is developed. This scheme makes the kinetic model more versatile. In
the beginning, six parameters were introduced to adjust more
than 10 000 reaction rate constants (ki,m,n ) needed to explain
complete reaction mechanism in a typical FCC riser reactor.
Later, it was observed that only four of these parameters are
significant. However, for the pedagogical point of view, this
correlation is discussed in detail in Appendix B in the same
sequence as it was developed.
The final form of this correlation is given by the following
equation:
2
ki,m,n = [k0 MWi ]e−1 (MWm −MWn )
−i,m,n /2
− e−MWi
−E0 MWi /RT e
×e
.
1 − e−MWi
(3)
In the above equation, parameters to be estimated by using
experimental data are k0 , , E0 , , 1 , and 2 . Later it was
observed that two parameters (correlating frequency factor
in terms of molecular weight of pseudocomponent) and 1
(an indicator of cracking tendency of pseudocomponents from
4514
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
middle or from sides) are insignificant, thus the correlation
reduces to the following four parameter equation:
−i,m,n /2
− e−MWi
e
ki,m,n = k0 e−E0 MWi /RT
.
(4)
1 − e−MWi
Therefore, the parameters required to be estimated are k0 ,
E0 , , and 2 . All these parameters are constant for a particular pair of feed and catalyst. Parameter 2 correlates the coke
forming tendency of the feed and parameter correlates activation energy of individual pseudocomponent in terms of its
molecular weight.
The theoretical kinetic scheme developed in the present work
assumes that all the cracking reactions are of first order. This is
in agreement with the reported cracking rate of gas lump and
gasoline lump. Many researchers have reported the cracking of
heavier fractions of feed such as gas oil lump to be of second
order due to deeper cracking action. In the present case feed
is broken into large number of narrow boiling lumps (pseudocomponent), therefore, first order cracking can be assumed
safely even for heavier lumps.
Thus for first order reactions, rate of disappearance of ith
pseudocomponent due to cracking in j th volume element
through one reaction as indicated in Eq. (1) is given by
ri,m,n = j · ki,m,n · Ci,j · (Mcat · t Cat j ),
(5)
equations. Using higher order numerical techniques, such as
Runge–Kutta method, is also not feasible for solving these
ODEs due to excessive computational load. Therefore, an FVM
based approach was used for computing material balances. As
discussed earlier, in this FVM approach it is assumed that the
riser reactor is made up of a large number of volume elements
of circular cross-section and of very small height, placed one
over the other. Volume elements are numbered from bottom to
top, i.e., from inlet to outlet (Fig. 2).
Both gas phase and solid phase are moving through each volume element in upward direction, with different velocity. Since
the height of a typical j th volume element zj is very small,
therefore the residence-time for the gas phase, tj , and the
residence-time of the solid phase, t Catj , is also very small.
Applying concepts of finite volume approach, it is assumed that
all reactions take place for tj time at a constant rate determined by the prevailing temperature, pressure, and concentration at the inlet of j th volume element. After elapse of time
tj the temperature and concentration of the stream (which is
now outgoing stream from j th element) are determined by the
following material balance equations.
Material balance over j th volume element for the gas
phase:
Rate of mass in from the (j − 1)th element
− rate of mass out from j th element
where the concentration of ith pseudocomponent at the inlet of
j th volume element is given by
Ci,j =
Pi,j
.
(ug,j · g,j )Ar
(6)
B +1
,
B + exp(A · Ccj )
or
N
i=1
Pi,j in Eq. (6) is kmol of ith component entering per second
into the j th volume element. Ar , ug,j , and g,j are area of
cross-section of the riser, local gas velocity, and gas phase volume fraction in the j th volume element. t Cat j is the residence time of catalyst in the j th volume element, hence Mcat ·
t Cat j in Eq. (5) is the mass of catalyst present in j th volume
element.
Non-selective deactivation of catalyst, because of coke deposition, is assumed. The activity coefficient (j ) depends on
the coke concentration on the catalyst. Pitault et al. (1995) proposed the following correlation for the estimation of activity
coefficient:
j =
= rate of mass converted to coke in j th element
(7)
The values for deactivation constants A and B reported
by the authors are 4.29 and 10.4, respectively; the same are
used in this work. Ccj is the concentration of coke on catalyst
surface (wt%).
2.3. Material balance
Calculation of material balances with several thousands of
parallel reactions is practically impossible by analytical solution of ordinary differential equations (ODE) representing rate
=
Pi,j −1 MWi −
N
m
i
i=1 m=1 n=1
N
(8)
Pi,j MWi
i=1
ri,m,n i,m,n
.
(9)
for all feasible reactions
Similarly, material balance over j th volume element for solid
phase:
Rate of mass out from j th element
− rate of mass in from the (j − 1)th element
= rate of mass accumulated in j th element
= rate of mass of coke formed
or (Mcat + Mcokej ) − (Mcat + Mcokej −1 )
N i m
=
ri,m,n i,m,n
i=1 m=1 n=1
(10)
,
(11)
for all feasible reactions
where Mcokej is the cumulative mass of coke formed upto
j th volume element. For the first volume element (j = 1),
Mcokej −1 is the mass of coke on regenerated catalyst.
In Eqs. (9) and (11) summation for i is done from 1 to N.
However, summation for m is done from 1 to i only because
no product species can have molecular weight greater than the
reactant species, whereas summation over n is done from 1 to
m only as the products PCm and PCn are interchangeable.
4515
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
Fig. 2. A typical volume element in the riser reactor.
2.4. Heat balance
It is assumed that the hydrocarbon feed, after coming into
contact with hot catalyst from the regenerator, vaporizes completely in the first volume element of the riser. Afterwards the
gas phase is in thermal equilibrium with the catalyst all along
the reactor. At the entrance (the first volume element of the
riser reactor) the equilibrium temperature, Tin , is calculated by
taking into account the specific heat of catalyst and the latent
heat of vaporization of the feed.
The pseudocomponents are chemically stable hypothetical
components, and according to the proposed reaction mechanism one pseudocomponent produces two other chemically
stable pseudocomponents. This indicates that the proposed reaction mechanism includes all possible reactions taking place
during cracking (such as hydrogen-transfer, condensation,
isomerization, etc.) that leads to the formation of chemically
stable products in the form of pseudocomponents. Therefore, the overall heat of reaction (Hr ), of Eq. (1) can be
estimated by finding the difference between the heat of combustion of all products and the heat of combustion of the
reactant.
Heat of combustion of pseudocomponents, H comb, is estimated by the following equations in terms of the API gravity
of the hydrocarbon:
H combi = 133.976 APIi + 41170
for APIi 25,
(12)
H combi = − 0.4017 API2i + 57.859 APIi + 1953.14 ln(APIi )
+ 37037.09 for 25APIi 50,
(13)
H combi = 55.824 APIi + 43775.32
for APIi 50.
(14)
Eqs. (12)–(14) were obtained by curve fitting of graphical data
proposed by Maxwell (1968). Curve fitting was done in such
a way that there is no discontinuity in the heats of combustion
values predicted by these three equations.
Thus for the cracking of ith pseudocomponent, giving mth
and nth pseudocomponents, heat of reaction becomes
H r i,m,n = i,m,n · H coke + (MWm H combm
+ MWn H combn − MWi H combi ).
(15)
Thus the energy balance equation for the j th volume element
can be written as
Mcat · Cpcat + Mcokej −1 · Cp coke + Mst · Cp st
+
N
i=1
=
Pi,j −1 MWi · Cp i (Tj −1 − Tj )
i
m
n
ri,m,n · H r i,m,n
(16)
.
for all feasible reactions
Rearranging the above energy balance equation, we get
Tj = Tj −1 −
( i m n ri,m,n · H r i,m,n )for
(Mcat · Cp cat + Mcokej −1 · Cp coke + Mst
all feasible reactions
· Cp st + N
i=1 Pi,j −1 MWi
· Cp i )
.
(17)
4516
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
The temperature (Tj ) thus calculated was used for the estimation of kinetic parameters in the next volume element and other
temperature dependent properties of the pseudocomponents.
Re =
2.5. Hydrodynamics
Eq. (19) can be solved as a difference equation and the value
of solid phase velocity in the next volume element can be calculated with the initial condition of solid phase velocity at the
riser entrance calculated by the following equation:
In the proposed riser model two phases (cluster phase and
gas phase) are considered. Cluster phase includes the loosely
held particles of catalyst and the coke. The cluster phase and
gas phase hold up vary along the riser height. Solid particles
spend more time in the riser than hydrocarbon vapor due to
slip between the two phases. The slip velocities observed in the
riser are higher than the terminal settling velocity of a single
particle. The reason for the higher slip velocities is attributed
to particles moving in clusters (Subba Rao, 1986; Fligner et al.,
1994; Horio and Kuroki, 1994). The clusters are agglomerates
of loosely held particles (Fligner et al., 1994). Cluster voidage
( c ) is assumed to be 0.5, in line with the two-phase theory of
fluidization.
As proposed by Tsuo and Gidaspow (1990), solid phase
momentum balance along the riser height may be written as
d( c c uc uc )
= Cf (ug − uc ) + 2fs c
dz
2
c uc
−
c c g.
(18)
Assuming the change in the mass of solids along the riser height
as negligible (this assumption is valid as the change in the mass
of solid phase from riser entrance to riser outlet due to coke
deposition is typically less than 1%, hence in a volume element
of the riser the change in the mass of solids will be negligibly
small) solid phase continuity equation can be written as
2fs c c u2c
duc
= Cf (ug − uc ) +
−
c c uc
dz
D
c c g.
(19)
The frictional force per unit volume at the gas particle interphase due to differing phase velocities can be calculated by the
following expression (Markatos and Shinghal, 1982):
F = 0.5CD AP
g |ug
− uc |(ug − uc ) = Cf (ug − uc ),
(20)
where AP is total projected area of particles per unit volume,
CD is interphase friction coefficient between the two phases.
Projected area per unit volume can be calculated based on
equivalent spherical diameter as
AP = 1.5 c /dc .
(21)
In the above equations dc is cluster diameter, 6.0 × 10−3 m
(Fligner et al., 1994), and c is cluster density. The cluster
density can be approximated by the following expression:
c=
p (1 −
c) +
g c
p (1 −
c ).
(22)
The empirical correlations for CD used by Arastoopour and
Gidaspow (1979) are
uc0 =
g g |ug
− uc |dc
g
.
Mcat
,
c Ar c0
(25)
(26)
where uc0 and c0 are the values of solid phase velocity and
solid phase volume fraction at the entrance of first volume
element.
The value of cluster volume fraction for the next volume
element is calculated by the equation
cj =
Mcat
.
c Ar ucj
(27)
Gas phase volume fraction is obtained using the relation
c + g = 1.
(28)
Having obtained the values of the solid phase velocity and
cluster volume fraction for the second volume element, the total
pressure drop for the first volume element can be calculated.
The pressure drop is assumed to be composed of four main
components (Pugsley and Berruti, 1996):
dP
dP
dP
dP
dP
=
+
+
+
,
dz total
dz s
dz acc
dz f s
dz fg
(29)
where (dP /dz)s is the pressure drop due to the hydrostatic
head of the solids, (dP /dz)acc is the pressure drop due to solids
acceleration, and (dP /dz)f s and (dP /dz)fg are the pressure
drops due to solids friction (defined as the frictional force per
unit volume between solids and wall) and gas friction (defined
as frictional force per unit volume between the gas and the
solids), respectively. These components can be calculated using
the following relations:
dP
= c gc ,
(30)
dz s
dP
c u2c
,
(31)
= c
dz acc
2 z
2fs c c u2c
dP
,
(32)
=
dz f s
D
dP
dz
fg
=
2
g ug
fg
D
.
(33)
Blasius friction factor given by the following empirical equation
is used as the gas friction factor
fg = 0.316Re−1/4 .
(34)
24
CD =
(1 + 0.15Re0.687 ) for Re < 1000,
Re
(23)
Konno’s correlation is used to calculate the solids friction factor
CD = 0.44
(24)
fs = 0.0025u−1
c .
for Re1000,
(35)
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
The pressure in the next volume element is obtained by the
following relation:
dP
Pj = Pj −1 −
z.
(36)
dz total
The gas phase density is calculated from the ideal gas law as
Pj N
i=1 yi,j MWi
(37)
gj =
RTj
and the gas phase velocity is calculated by
Mst + N
i=1 Pi,j MWi
.
ugj =
Ar g,j gj
(38)
Residence time of catalyst in volume element j is
t Catj =
(z)j
.
uc,j
(39)
2.6. Model solution
Model simulation is done on a P-IV computer, which took
less than 5 min for giving the simulation results. Although we
can take variable height of volume elements, however, in the
present work the height of each volume element of the riser
was kept 10 mm. Further decrease in the height of the volume
element had no appreciable effect on the results.
3. Results and discussions
The material balance equations were combined with reaction
kinetics and the hydrodynamic model equations to obtain the
moles of each pseudocomponent coming out of any volume
element j (=1 to Nc) of the riser reactor. Thus, the model
could predict the yield pattern along the riser height. Model
validation is done as three case studies by using industrial data
reported in the literature. Results of the proposed model can
be obtained by adjusting six parameters as discussed earlier.
Numerical values of these parameters were obtained separately
for each case study. Results of the simulator are being discussed
in the following case studies.
Due to the unavailability of the feed TBP data for each case,
boiling point characteristic of feed (simulated distillation, SD)
reported by Pekediz et al. (1997) is used (Table A1) for all the
case studies. Various other parameters common to all cases are
given in Table 1 and plant data used in these cases are presented
in Table 2.
3.1. Case study 1
Industrial FCC plant data reported by Ali et al. (1997),
presented in Table 2, was used in this case study. In this
case five industrial data at the riser outlet—gasoline yield,
gas yield, unconverted hydrocarbon, coke yield, and riser
outlet temperature—are available to obtain the rate constant
parameters.
4517
Model results were obtained by adjusting four rate constant
parameters and two tuning parameters of Eq. (3) in such a
way that the deviation in predicted and actual data, at the riser
outlet, is minimum. To achieve this, these parameters (k0 , ,
E0 , , 1 , and 2 ) were determined by line-search technique
followed by golden-section method. To estimate the value of
these parameters, an iterative method was adopted in which one
parameter was varied at a time (keeping other five constant).
Starting with an initial guess value of all six parameters, a linesearch was made (by changing one parameter with a constant
increment/decrement) to find a condition where the sum of
absolute value of deviation in the predicted and experimental
values of product yield and temperature at riser outlet are minimum. Then the value of this parameter was fine tuned between
two consecutive values considered during line-search by the
method of golden-section. Since the system of equations under
consideration is highly nonlinear, on-line graphical observation of reported and predicted data was made almost after each
iteration.
The simulator results, yield pattern and temperature profile,
are compared with the industrial data at the riser outlet in
Figs. 3 and 4, respectively. The values of six parameters are
k0 = 0.01, = 0.01, E0 = 1540, = 0.43, 1 = 0.0, 2 = 17.0.
Fig. 4 indicates that as the hydrocarbons and catalyst mixture
travel upwards, the temperature inside the FCC riser reactor
decreases because of the endothermic cracking reactions. The
catalyst temperature at the inlet of the riser (960 K) falls sharply
to 880 K because sensible heat of catalyst coming from the regenerator is utilized in providing heat for raising the sensible
heat of the feed, for vaporizing the feed, and for further heating
of the vaporized feed. Afterwards, within first 10 m height inside the riser reactor the temperature drops from 880 to 790 K,
as most of the cracking takes place within first 10 m of the
riser height. The temperature at the outlet of the riser is 774 K
(Fig. 4). The decrease in reaction mixture’s temperature and
catalyst activity along the riser height cause a decline in the
reaction rate, hence the temperature gradient falls appreciably
with the increasing riser height.
Sensitivity analysis: Sensitivity analysis of these parameters
is presented in Figs. 5–7. Sensitivity analysis is the process of
varying the parameters over a wide range about the mean value
and recording the relative change in predicted gas, gasoline,
and coke yields (Figs. 5–7). The sensitivity of one parameter
relative to other is also demonstrated in these figures. Such
analysis is useful in cases when the experimental data are few
in number and statistical analysis cannot be applied to predict
confidence interval. A detailed discussion about the sensitivity
analysis is given by Saltelli (2000). Figs. 5–7 show that in the
proposed model, gas yield and coke yield, are strong functions
of E0 , , and 2 , and gasoline yield is very sensitive to E0 , and
and moderately sensitive to k0 , and 2 , whereas, is almost
insensitive parameter. It is evident from these figures that
has virtually no effect on any product yield. This indicates
that the frequency factor (k0,i ) is independent of molecular
weight. Also, the value of parameter 1 is zero. Therefore only
four parameters may be used to match the model results with
the industrial data, with 1 = 0, and = 0. Hence for all the
4518
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
Table 1
Parameters used for the simulation of riser reactor
Parameter
Value
Source
Heat of combustion of coke
Molecular weight of coke
Volume fraction of clusters at inlet
Specific heat of catalyst
Specific heat of steam
Mass flow rate of steam
Feed temperature at the riser inlet
Latent heat of feed vaporization
Catalyst particle density
Catalyst particle diameter
Specific gravity of feed
Cluster diameter
−32950 kJ/kg
12 kg/kmol
0.5
1.15 kJ/kg K
2.15 kJ/kg K
1.33 kg/s
494 K
96 kJ/kg
1200 kg/m3
75 m
0.9292 g/cm3
6 mm
Austin (1984)
Arbel et al. (1995)
Gupta and Subba Rao (2001)
Ali et al. (1997)
Blasetti and de Lasa (1997)
Blasetti and de Lasa (1997)
Ali et al. (1997)
Gupta and Subba Rao (2001)
Gupta and Subba Rao (2001)
Gupta and Subba Rao (2001)
Pekediz et al. (1997)
Fligner et al. (1994)
Table 2
Plant data used for simulation of riser reactor
Riser height
Riser diameter
Riser pressure
Catalyst temperature
Feed rate
Feed temperature
C/O ratio
Ali et al. (1997) (Case study—1)
Derouin et al. (1997) (Case study—2)
Theologos and Markatosa (1993) (Case study—3)
33 m
0.8 m
2.9 atm
960 K
20 kg/s
496 K
7.2
(32 m)
1.0 m
3.15 atm
(960 K)
85 kg/s
650 K
5.53
50 m
1.24 m
(2.9 atm)
1025 K
17.5 kg/s
568 K
8.0
Data given in the parentheses are those used in the present work in place of data either not reported or reported in ranges.
a Literature data.
1000
120
Plant data
Gas yield
Gasoline yield
Coke yield
Unconverted
100
Model prediction
Gas yield
Gasoline yield
Coke yield
Unconverted
Plant data (Source: Ali et al., 1997)
950
Riser temperature (K)
80
Yields (wt%)
Model predictions
60
40
900
850
20
800
0
750
0
5
10
15
20
Riser height (m)
25
30
35
0
5
10
15
20
Riser height (m)
25
30
35
Fig. 3. Case study 1, comparison with the data reported by Ali et al. (1997).
Fig. 4. Axial temperature profile along the riser height.
subsequent simulations only four parameters (k0 , E0 , , and 2 )
are used.
Case study 1 with four parameters: The results for case study
1 using the four parameters are presented in Figs. 8–11. The
values of the four parameters are k0 =0.01, E0 =1540, =0.43,
and 2 = 17.0. The product yield profiles given in Figs. 3 and 8
match very closely. Also, the temperature profiles obtained in
Figs. 4 and 9 are similar.
4519
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
Fig. 7. Sensitivity analysis for coke yields.
Fig. 5. Sensitivity analysis for gas yields.
120
Plant data
100
Yields (wt%)
80
Model prediction
Gas yield
Gasoline yield
Coke yield
Gas yield
Gasoline yield
Coke yield
Unconverted
Unconverted
60
40
20
0
0
5
10
15
20
25
30
35
Riser height (m)
Fig. 6. Sensitivity analysis for gasoline yields.
The activity of the catalyst also decreases rapidly as byproduct (coke) of the cracking reactions gets deposited on the catalyst surface (Fig. 10). Fig. 11 shows an initial decline in the gas
velocity because of the sharp increase in the gas void fraction
due to increase in the moles of the gas as a result of cracking.
After this initial decline, the gas velocity starts increasing as
the cracking reactions along the riser height continues to increase the moles of the gas causing a continuous decline in the
gas phase density. The initial sharp increase in the catalyst velocity is due to the sharp fall in the solid volume fraction and
drag exerted by the gas. After this initial sharp increase the
Fig. 8. Case study 1, comparison with the data reported by Ali et al. (1997)
with four parameters.
catalyst velocity keeps on increasing gradually all along the
riser height.
Also, high values of slip factor are predicted in riser entry zone which gradually decreases along the riser height
and finally reaches at 1.8 (Fig. 11). Catalyst volume fraction
falls from about 0.5 to 0.1 in first few meters of riser height
(Fig. 12). This sharp decline can be attributed to the fact that
most of the cracking takes place within first few meters of the
riser height. Furthermore, the catalyst volume fraction along
the riser height is plotted for two initial values (0.5 and 0.3).
4520
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
0.55
14
Plant data
12
0.50
Model prediction
2
950
10
0.45
Velocity (m/s)
Riser temperature (K)
Gas phase molar flux (kmol/m s)
1000
900
8
0.40
6
0.35
850
4
Gas phase velocity
Catalyst velocity
2
800
0.30
Gas phase molar flux
0
0.25
0
5
750
10
15
20
25
30
35
Riser height (m)
0
5
10
15
20
25
30
35
Fig. 11. Predicted catalyst and gas velocity profiles and gas phase molar flux
along the riser height.
Riser height (m)
Fig. 9. Axial temperature profile along the riser height with four parameters.
0.50
1.2
Initial cluster volume fraction = 0.5
0.45
Initial cluster volume fraction = 0.3
1.0
Catalyst volume fraction
0.40
Catalyst activity
0.8
0.6
0.4
0.35
0.30
0.25
0.20
0.15
0.10
0.2
0.05
0.00
0.0
0
0
5
10
15
20
25
30
35
5
10 15 20 25 30 35
5
Riser height (m)
Riser height (m)
Fig. 12. Predicted catalyst volume fraction along the riser height.
Fig. 10. Predicted catalyst activity along the riser height.
The plots for both the values are almost similar because this
value gets adjusted very quickly at the riser entrance itself
(in the first 2 m of the riser height itself the catalyst volume
fraction value reaches 0.12 for both the cases) and hence the
yield profiles remain unaffected.
Since plant data for the product yields were available at the
riser outlet only, few more comparisons were made in subsequent case studies. Even without changing the values of the
parameters the results of the simulation were encouraging.
However, for better comparison, the parameters were adjusted
further in both the following case studies. It is observed that
the parameters k0 , and 2 are the two parameters those were
needed to be adjusted for different cases.
3.2. Case study 2
In this case FCC plant data (Table 2) reported by Derouin
et al. (1997) was used to compare the simulator predictions.
Authors have reported the product data for gasoline yield and
conversion at different positions along the riser height (Fig. 13).
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
ous transport equations. Riser operating conditions given in
Table 2 (Theologos and Markatos, 1993) were used for the
simulation. A comparison of the gasoline yield from the model
presented in this work, with the gasoline yield from 3-D, twophase flow, heat transfer and reaction model of Theologos and
Markatos (1993) is made. Only two parameters k0 , and 2 were
changed appreciably. The final values of the parameters were
k0 = 0.0012, E0 = 1540, = 0.4, and 2 = 13.0. Comparison
of results predicted by present approach and those reported by
Theologos and Markatos (1993) is given in Fig. 14. Although
the model of Theologos and Markatos (1993) is mathematically complex, its yield prediction profiles are similar to the
models that use constant values for the gas and solid velocities
and four to six lumps.
The present model overpredicts the product yields in the first
few meters of riser height. This may be attributed to the fact
that in this region the heat and mass transfer resistances are not
negligible.
80
Conversion or yield (wt%)
70
60
50
40
Model gasoline
30
Plant gasoline
Model conversion
20
Plant conversion
10
0
0
5
10
15
20
25
30
4521
35
4. Conclusion
Riser height (m)
Fig. 13. Case study 2, comparison with the plant data reported by Derouin
et al. (1997).
100
Theologos and Markatos model prediction
Present model prediction
Conversion or yields (wt%)
80
Conversion
60
Gasoline
40
Gas+coke
20
0
0
10
20
30
40
50
60
Riser height (m)
Fig. 14. Case study 3, comparison with the simulator data reported by
Theologos and Markatos (1993).
The results of simulation with k0 = 0.045, E0 = 1540, = 0.43,
and 2 = 17.0, are presented in Fig. 13. Model predictions for
the gasoline yield along the riser height matches satisfactorily
with the plant data.
3.3. Case study 3
The objective of this case study is to compare the results
from the present work with other models based on rigor-
A new technique for modeling the FCC riser has been developed. The model incorporated a more realistic kinetic scheme
for the cracking reactions, and a new correlation to evaluate
Arrhenius type reaction rate constants. The rate constant parameters can easily be obtained for each combination of feed and
catalyst. Although there is significant variation in the yield pattern of different case studies, activation energy parameter (E0 )
remained same for all cases. However, to account for different characteristics of the feedstock and catalyst, only frequency
factor parameter (k0 ) and feed coking tendency parameter 2
were required to be adjusted to compare the yield patterns of
different case studies with the model results.
The proposed model is capable of predicting overall conversion, products yields, temperature, and catalyst activity along
the riser height. The model results are in close agreement with
the industrial data reported in the literature and the data predicted by other simulators.
The predictions of the FCC riser reactor model are dependent
on the values of cracking reactions’ rate constants, which can
easily be obtained with the help of proposed kinetic model for
different characteristics of the feedstock, type of catalyst, activity of catalyst, and operating parameters. Therefore, it seems
to be more appropriate to use these rate constant parameters
obtained for a pair of feedstock and catalyst in place of using
the kinetic constants from the literature which are obtained for
a different combination of feedstock and catalyst by regression
analysis. Further, this detailed kinetic model can be easily used
for the other advanced studies (such as control and optimization) of FCC modeling.
Notation
Ar
Cc
Ci,j
cross-sectional area of riser, m2
coke concentration on catalyst surface, wt%
concentration of ith pseudocomponent in j th
volume element, kmol/m3
4522
Cpcat
Cpi
Cpmix,j
Cpst
Ei
H coke
H comb
Hr
k0,i
ki,m,n
Mcat
Mcokej
Mst
MWi
N
NC
p
Pi,j
ri,m,n
R
tj
t Cat j
Tin
Tj
u
uc
ug
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
specific heat of catalyst, kJ/kg
specific heat of ith pseudocomponent, kJ/kg
specific heat of the gas and solid mixture in j th
volume element, kJ/kg
specific heat of steam, kJ/kg
activation energy for cracking of ith pseudocomponent giving kmax,i , kJ/kmol
heat of combustion of coke, kJ/kg
heat of combustion of pseudocomponents, kJ/kg
heat of reaction, kJ/kmol
frequency factor for cracking of ith pseudocomponent giving kmax,i m3 /(kg Cat s)
rate constant for the cracking of ith pseudocomponent to produce mth and nth pseudocomponents, m3 /(kg Cat s)
mass flow rate of catalyst, kg/s
mass flow rate of coke at the outlet of j th volume
element, kg/s
mass flow rate of steam, kg/s
molecular weight of ith component, kg/kmol
total number of components, including pure
components and pseudocomponents
total number of hypothetical volume elements
in the riser
pressure, atm
molar flow rate of ith component,PCi , through
j th volume element, kmol/s
rate of disappearance of ith pseudocomponent
giving mth, and nth pseudocomponents, kmol/s
gas constant, atm m3 /(kmol K)
residence time of gas phase in j th volume
element, s
residence time of catalyst in j th volume
element, s
temperature of reaction mixture in the riser
inlet, K
temperature of reaction mixture leaving j th
volume element, K
superficial gas velocity, m/s
cluster velocity, m/s
actual gas velocity, m/s
Subscripts
i, m, n
j
ith, mth, and nth component
j th volume element in the riser starting from the
bottom
Greek letters
i,m,n
g,j
mass of coke formed when 1 kmol of pseudocomponent PCi cracks to give 1 kmol each of
PCm and PCn , kg coke/kmol PCi
volume fraction of gas in j th volume element
exponent of molecular weight for frequency
factor
exponent of molecular weight for activation
energy
cat
coke
g,j
1 , 2
j
density of catalyst, kg/m3
density of coke, kg/m3
density of gas phase in j thvolume element
tunable parameters
catalyst activity coefficient
Appendix A.
Petroleum fractions are mixtures of innumerable components which are difficult to be identified individually. However,
Watson characterization factor can be treated as an indicator of
the composition of various groups of compounds (such as paraffin, olefin, naphthene, aromatic, etc.) present in the petroleum
fraction. Watson and Nelson (1933) made a remarkable ob1/3
servation that the factor KW (=Tb /sg), known as Watson
characterization factor, is closer to 12 for paraffins and olefins,
approximately 10 for aromatics, and between 11 and 12 for
naphthenes when the normal boiling point of the component,
Tb , is in Rankin and sg is the specific gravity at 60◦ /60 ◦ F.
The characterization factor of the mixture of hydrocarbons is
given by KW = MeABP1/3 /sg, where MeABP is the mean average boiling point of the mixture (API Data Book, Chapter 2,
Characterization of Hydrocarbons, 1976). Using this, Miquel
and Castells (1993) proposed a method along with a computer
program (Miquel and Castells, 1994) that can represent an oil
fraction by an equivalent mixture of small number of hypothetical components or pseudocomponents. To use this approach,
atmospheric TBP distillation curve and the entire fraction density is required. This method assumes that if the difference in
final boiling point (FBP) and initial boiling point (IBP) of a
petroleum oil is not too high (i.e., < 300 K) then the Watson
characterization factor of any narrow-boiling fraction (boiling
range between 15 and 25 K) of this oil remains equal to that of
original petroleum oil.
In the present case, due to unavailability of TBP curve for
the FCC feed, SD curve reported by Pekediz et al. (1997) was
used. The SD curve was first converted to ASTM-D86 curve
and then to TBP curve by the correlation proposed by Daubert
(1994). The two-step conversion of SD data to TBP data is given
in Table A1. To generate pseudocomponents, the TBP curve of
the feed was divided into 12 parts, out of which four were of
5 vol% each and eight of 10 vol% each (shown as vertical bars
in Fig. A1 ). These vertical bars represent 12 pseudocomponents of the feed. The boiling point of each individual pseudocomponent was determined by area-averaging of the TBP curve
(clearly visible in Fig. A1). Considering constant Watson characterization factor, specific gravity of each pseudocomponent
was determined by the equation
1/3
T
sg = 1.21644 b
KW
(where Tb is in K).
(A.1)
Molecular weights of these pseudocomponents were then
calculated by the following equation proposed by Edmister and
Lee (1984), which requires knowledge of boiling point and the
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
4523
Table A1
Distillation data of hydrocarbon feed (Pekediz et al., 1997)
Vol. % distilled (wt%)
SD (K)
ASTM-D86a (K)
TBPa (K)
IBP
10
30
50
70
90
FBP
532
587
621
650
683
730
800
585.5
615.5
632.8
652.3
680.5
721.4
756.6
558.9
604.3
636.4
665.3
700.6
743.9
808.1
Density of feed (at 15 ◦ C) = 929.20 kg/m3 .
a Estimated using correlation proposed by Daubert (1994).
The lightest seven components were taken as pure components
which are the major constituent of gases. Initially, volume fraction of all these seven pure components and 31 pseudocomponents, which are not present in feed, were taken zero. Thus
total 50 components (seven pure components and 43 pseudocomponents) were considered in the present approach for the
simulation of FCC riser reactor.
After determining normal boiling point, specific gravity,
and molecular weight of all pseudocomponents, heat capacities were determined using the correlations of Kesler and
Lee (1976). Pseudocomponents thus generated are listed in
Table A2.
Predicted concentrations of pseudocomponents in the product stream are given in Fig. A2. Also, various product streams,
viz., gas, gasoline, LCO, and residue are marked on the basis
of boiling points.
Appendix B.
Fig. A1. Pseudocomponents generated from feed TBP.
specific gravity of a hydrocarbon fraction:
MW = 204.38 · e(0.00218·Tb ) · e(−3.07·sg) · Tb0.118 · sg1.88 . (A.2)
Having known values of the volume fraction, boiling point,
specific gravity, and molecular weight, each individual bars
of Fig. A1 can be treated as a pure component (of course,
hypothetical pure component or pseudocomponent). To make
use of the Eqs. (A.1) and (A.2), an iterative method has to
be adopted as the value of Watson characterization factor is
not known beforehand. Miquel and Castells (1993, 1994) have
explained this iterative approach in detail.
After breaking the FCCU feed into 12 pseudocomponents,
and determining the exact value of Watson characterization factor, properties of other 31 pseudocomponents were also determined by using Eqs. (A.1) and (A.2). Boiling points of these
31 pseudocomponents were taken at equal intervals between
the boiling point of n–pentane and the boiling point of first
pseudocomponent of FCCU feed (570 K in the present case).
The reaction rate constant, k, is normally determined by
Arrhenius equation in terms of frequency factor, k0 , and energy
of activation E. In most of the cases of FCC kinetic modeling, these parameters are determined empirically, using experimental data. In the present case, however, all data available in
literature relevant to calculate the rate constants are for lumped
reaction mechanism which, in fact are reaction rate data for
conversion of one lump to other and not for cracking of one
lump giving two other lumps. Due to lack of experimental data,
a purely hypothetical correlation for predicting Arrhenius type
rate constant is being used. The proposed scheme can be perfected in future after performing more and more experimental
work (may be in different laboratories).
Hypothetical scheme: It is well observed fact that almost any
physical, thermodynamic, or transport properties of hydrocarbons of a particular group have similar behavior, and properties of these hydrocarbons can be well correlated empirically
(Daubert, 1998). In the present case, we are dealing with pseudocomponents, which do not fall under a particular group of
hydrocarbon, but are mixtures of large number of hydrocarbons of almost equal boiling point but widely different properties. In fact, pseudocomponents are neither paraffin, nor olefin,
and not aromatic either. However, the average characteristic
4524
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
Table A2
Properties of pseudocomponents
Component ID
Component name
Boiling point (K)
Molecular weight
H comb (kJ/kmol)a
PC1
PC2
PC3
PC4
PC5
PC6
PC7
PC8
PC9
PC10
PC11
PC12
PC13
PC14
PC15
PC16
PC17
PC18
PC19
PC20
PC21
PC22
PC23
PC24
PC25
PC26
PC27
PC28
PC29
PC30
PC31
PC32
PC33
PC34
PC35
PC36
PC37
PC38
PC39
PC40
PC41
PC42
PC43
PC44
PC45
PC46
PC47
PC48
PC49
PC50
Methane
Ethane
Propane
Butene
Butane
Pentene
Pentane
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
Pseudocomponent
111.65
184.50
231.09
266.90
272.65
303.11
309.21
317.37
325.54
333.70
341.86
350.03
358.19
366.35
374.52
382.68
390.84
399.01
407.17
415.33
423.50
431.66
439.83
447.99
456.15
464.32
472.48
480.64
488.81
496.97
505.13
513.30
521.46
529.62
537.79
545.95
554.11
562.28
570.44
593.12
612.48
628.50
643.76
658.25
674.30
691.91
711.55
733.25
760.13
792.20
16.043
30.070
44.097
56.108
58.124
70.135
72.150
88.563
91.443
94.402
97.443
100.569
103.781
107.083
110.478
113.969
117.558
121.249
125.045
128.948
132.964
137.094
141.343
145.713
150.210
154.835
159.595
164.491
169.530
174.714
180.049
185.538
191.187
197.000
202.982
209.138
215.474
221.995
228.706
248.399
266.496
282.446
298.497
314.565
333.352
355.231
381.318
412.311
454.185
509.673
62764.79
58622.94
51983.86
50464.83
49960.31
49073.02
48952.84
47318.37
47226.62
47137.89
47052.01
46968.81
46888.17
46809.94
46734.00
46660.24
46588.54
46517.73
46445.89
46373.59
46300.89
46227.80
46154.34
46080.57
46006.47
45932.07
45857.39
45782.42
45707.18
45631.67
45555.89
45479.85
45403.53
45326.93
45250.05
45172.88
45095.39
45017.59
44939.46
44720.49
44531.20
44353.16
44187.51
44034.96
43871.15
43697.31
43510.14
43311.36
43075.64
42808.56
PC1 to PC7 are pure components constituting gas; PC8 to PC29 constitute gasoline fraction; PC30 to PC41 constitute light cycle oil fraction; PC42 to PC50
constitute residual fraction whereas feed contains PC39 to PC50 .
a Heat of combustion values are calculated by using Eqs. (12), (13), and (14).
of these mixtures of hydrocarbons with almost equal boiling
point is characterized by Watson characterization factor KW (as
discussed in Appendix A). Therefore it can safely be assumed
that the over all cracking behavior of all pseudocomponents
should follow similar trend, as all the pseudocomponents are
generated with exactly the same value of KW . The Watson char-
acterization factor is an indicator of an average characteristic in
terms of paraffinicity, as well as aromaticity of the hydrocarbon
mixtures (pseudocomponents). Therefore, these pseudocomponents are treated just as pure components with specific characteristics, represented by KW , which helps in determining all
their physico–chemical properties.
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
4525
800
Residue
Boiling Temperature (K)
700
Feed
600
LC O
500
Fig. B1. Schematic representation of cracking of a pseudocomponent: (a)
cracking from middle of the molecule (without coke formation), (b) cracking
from side of the molecule (without coke formation), (c) cracking from middle
of the molecule (along with coke formation), (d) cracking from side of the
molecule (along with coke formation).
Gasoline
400
300
Gas
200
100
0
20
40
60
80
100
Liquid Volume % (in Product)
Fig. A2. Mass fraction of pseudocomponents in the product.
In the absence of experimental data, the rate constant of a
particular reaction can be considered as the probability of reaction to take place. A higher probability of the reaction will
correspond to higher cracking rate and hence a higher rate constant. According to the proposed reaction scheme (Eq. (1)) a
pseudocomponent PCi , cracks to give two other pseudocomponents PCm , and PCn and some amount of coke () is also
formed. For the cracking of one specific PCi (i.e., for one fixed
value of i between 1 to N) there are a large number of parallel ‘feasible’ reactions taking place through which two components PCm and PCn are formed such that m and n can lie
between 1 to i only (Fig. 1). However, the rate constants of all
these feasible reactions (with different values of i, m, and n)
must be different from one another. There are only three possible ways through which variation in the magnitude of cracking
rate constant of ith pseudocomponent with changing molecular weight of PCm and PCn can occur, (i) the rate constant of a
cracking reaction is maximum when molecular weights of PCm
and PCn are almost equal (Fig. B1(a)), (ii) the rate constant is
highest when molecular weights of PCm and PCn are widely
apart (Fig. B1(b)), and (iii) the rate constant is almost constant
for all values of m and n. These three possibilities can be expressed in terms of probability distribution as, (i) probability
of cracking of a pseudocomponent from its middle is highest,
(ii) cracking from its sides is more probable than from the middle, and (iii) cracking from anywhere is equally probable. It
was later observed that cracking reactions follow the last case
in which rate constant is almost equal for all possible combinations of cracking of a particular pseudocomponent. However, for the pedagogical point of view, following discussion
explains how the present model was developed. First of all we
consider the first case, i.e., when cracking from middle of the
molecule has highest probability. Pitault et al. (1994) has also
supported this assumption. Here it should be noted that ‘middle
of the molecule’ means the molecular weight of PCm and PCn
are equal, since there is no differentiation between ring-chain
and straight chain molecules of pseudocomponents.
Thus in this first case, cracking of pseudocomponents from
the middle has highest probability, therefore, numerical value
of the rate constant should also be highest when molecular
weights of the two product pseudocomponents are equal. It
implies that for the cracking of ith pseudocomponent (PCi )
giving two other pseudocomponents PCm and PCn, the rate
constant (ki,m,n ) is maximum (kmax,i ) when molecular weight
of mth and nth components are equal. In the present scheme,
however, this is possible only when m=n since there are no two
pseudocomponents having equal molecular weight. For cases
when m = n (Fig. B1(b)), a function f (x) is defined to predict
ki,m,n in such a way that the function value approaches kmax,i
when x (=MWm − MWn ) tends to zero, and f (x) is less than
kmax,i for all |x| > 0. This function could be any even function
with a maxima at x = 0. However, in the present case, it is
assumed that the probability of cracking of a pseudocomponent
may follow a normal distribution. Therefore the function f (x),
the normal distribution function in standard form, becomes
1
2
2
f (x) = √ e−x /2 .
2
(B.1)
At x = 0, this function has maximum value which corresponds
to the rate constant of the most probable cracking reaction,
kmax,i . Therefore
1
f (0) = √ = kmax,i .
2
Hence 2
2
=
1
(kmax,i )2
(B.2)
.
(B.3)
Substituting Eqs. (B.2) and (B.3) in (B.1) we get the normal
distribution function
f (x) = kmax,i e−x
2
(kmax,i )2
.
(B.4)
Here it should be noted that x is the difference in the molecular
weights of the two product components and it is possible that
the value of x is same for different combination of PCm and
PCn .
Another parameter in Eq. (1) is the amount of coke formed
during cracking reaction. The point to ponder is, for a fixed
value of x whether the cracking reaction rate shall increase,
decrease, or remain unchanged when coke formation increases?
4526
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
Table B1
Kinetic data for the cracking reactions reported by Arbel et al. (1995)
Cracking reaction
HFO to LFO
HFO to gasoline
HFO to coke
LFO to gasoline
LFO to coke
Gasoline to coke
Activation energy (kJ/mol)
Frequency factor (h−1 )
Rate at 538 ◦ C (h−1 )
Molecular weight of cracking lump
60.7086
23.0274
73.269
23.0274
73.269
41.868
1.422 × 107
1760.4
3380.4
712.8
2707.2
356.4
172.8
380
380
380
255
255
120
1.026 × 105
3.704 × 107
8.215 × 104
1.852 × 107
8.555 × 104
HFO: heavy fuel oil; LFO: Light fuel oil.
Table B2
Kinetic data for the cracking reactions reported by Lee et al. (1989)
Cracking reaction
Activation energy (kJ/mol)
Frequency factor (h−1 )
Rate at 538 ◦ C (h−1 )
Molecular weight of cracking lump
Gas oil to gasoline
Gas oil to gas
Gas oil to coke
Gasoline to gas
Gasoline to coke
68.2495
89.2164
64.5750
52.7184
115.4580
7.978 × 105
4.549 × 106
3.765 × 104
3.255 × 103
7.957 × 101
39.364
9.749
6.012
2.470
1.364
380
380
380
120
120
The formation of coke can be depicted as in Fig. B1(c) and
(d), and the mass of coke formed (in kg), when one kmol of
reactant (PCi ) cracks, can be given by
i,m,n = MWi − (MWm + MWn ).
(B.5)
It appears from Table B.1 and Table B.2 that the rate constant for the conversion of a lump to coke is significantly lower
than the rate constants for the conversion of same lump to other
hydrocarbon lumps. It indicates that the formation of coke is
not favored during the cracking reaction. From this fact an inference can be drawn that the probability (or the rate constant)
of a cracking reaction decreases when coke formation (i,m,n )
is increased. Since we are considering cracking of pseudocomponents, which itself are mixtures of large number of actual
compounds, it seems that the fractional decrease in rate constant with increasing coke formation should be a continuous
function say g(), where is the amount of coke formation
(i,m,n ). Thus the general correlation for the rate constant can
be expressed as the product of these two functions as
ki,m,n = f (x)g().
(B.6)
e−i,m,n − e−MWi
,
1 − e−MWi
such that,
g(i,m,n ) = 1 at i,m,n = 0 (i.e., no coke formation)
and
g(i,m,n ) = 0 at i,m,n = MWi
(i.e., all hydrocarbon mass is converted to coke).
2
2
ki,m,n = kmax,i e− (MWm −MWn ) (kmax,i )
e−(MWi −(MWm +MWn )) − e−MWi
×
.
1 − e−MWi
(B.8)
A schematic three dimensional surface of ki,m,n as functions
of coke formation (i,m,n ) and x (the difference between MWm
and MWn ) for the cracking of a typical pseudocomponent is
shown in Fig. B2. It can be observed that at different values of
coke formation, the grid-line parallel to x-axis follows the normal distribution function of different curvature, and the function has a maxima at x = 0.
Up to this point we have considered only hypothetical correlations, without any experimental verification. Eq. (B.8) is
quite rigid to accommodate experimental data as the surface
shown in Fig. B2 can change only by changing kmax,i . Therefore, at this stage, we introduced two tunable parameters 1 and
2 which can be used to adjust the span (variance) of the normal
distribution function and the curvature of the decay function,
respectively, such that
2
In the present work, we hypothesize that the function g() is
an exponential decay function of the form
g(i,m,n ) =
Combining Eqs. (B.4), (B.6) and (B.7), we get
(B.7)
ki,m,n = [kmax,i · e−1 [(MWm −MWn )·kmax,i ] ]
×
e−(MWi −(MWm +MWn ))/2 − e−MWi
1.0 − e−MWi
.
(B.9)
The parameter 2 is an indicator of coking tendency and its
value will depend on the nature of the feed. Parameter 1 is
an indicator of cracking tendency of pseudocomponents from
middle or from sides.
By increasing 2 decay of exponential function g() is reduced. The thick solid line on the grid surface (Fig. B2) is
elevated vertically upward to thick dashed line when 2 is increased. Hence by increasing 2 reaction rate constant can be
increased even with higher coke formation. On the other hand,
R.K. Gupta et al. / Chemical Engineering Science 62 (2007) 4510 – 4528
4527
the following relationship with molecular weight of cracking
lump:
k0,i = k0 MWi ,
(B.11)
Ei = E0 MWi .
(B.12)
Therefore, the parameter kmax,i used in Eq. (B.9) can be
expressed in terms of the frequency factor (k0,i ) and the energy
of activation (Ei ) with four rate constant parameters k0 , ,
E0 , and . After the sensitivity analysis, as discussed in the
main body of the text, it was observed that the parameter is
insignificant; therefore it can be eliminated from the list of six
parameters.
References
Fig. B2. Schematic representation of the probability distribution function f (x)
(on the vertical plane) and the reaction rate constant ki,m,n as a function of
coke formation and the difference in molecular weights of the products.
by decreasing value of 1 the function f (x) (hence the entire
surface of Fig. B2) can be made more flat. If the value of 1
is reduced to zero, the intersection of the surface with ki,m,n –x
plane becomes a straight line. By further decreasing the value
of 1 the surface becomes concave upward. In Fig. B2, the intersection of such surface with ki,m,n –x plane is shown by a
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