ARTICLE pubs.acs.org/EF
Kinetic Model Considering Reactant Oriented Selective Deactivation for Secondary Reactions of Fluid Catalytic Cracking Gasoline Xiaowei Zhou, Tao Chen, Bolun Yang,* Xuedong Jiang, Hailiang Zhang, and Longyan Wang Department of Chemical Engineering, State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China ABSTRACT: The lumped scheme with consideration of catalyst deactivation was adopted to simulate catalytic cracking of gasoline for predicting the product distribution of secondary reactions. A reactant oriented selective deactivation model was developed using a new strategy of selective deactivation coupled with nonselective independence deactivation. Catalyst deactivation was correlated with the time on stream rather than coke content. Furthermore, a hybrid self-adaptive genetic algorithm (termed AGA/SA), which incorporated evolution strategies and simulated annealing into a genetic algorithm, was developed and applied in parameter estimation of the proposed model. Results suggest that the lumped kinetic scheme incorporating a deactivation function could enhance its fundamentality and the predication accuracy. AGA/SA exhibits the desired improvements such as rapid convergence, high efficiency, strong in hill-climbing, and effective in escaping the local optimum. Good agreement between the predicted results and experimental data indicates that the proposed kinetic model for secondary reactions of fluid catalytic cracking (FCC) gasoline is well established and AGA/SA is reliable.
1. INTRODUCTION Feeding fluid catalytic cracking (FCC) gasoline into a second reaction zone or riser reactor, where the feedstock undergoes a series of reactions, termed “secondary reaction of FCC gasoline”, such as cracking, hydrogen transfer, isomerization, aromatization, alkylation, condensation, etc., is effective to lower the olefins and sulfur content in gasoline and increase propylene and isobutylenes significantly. Great efforts have been made in this field over the past decade since it is a new way of producing environment-friendly gasoline to meet the increasing stringent specification for clean fuel and the increasing heavy crude oil. Xu and Wang1 investigated the reaction mechanism for catalytic cracking of FCC gasoline in the presence of acid catalysts based on their experimental study. Verstraete et al.2 reported direct and indirect naphtha recycling schemes around an existing resid FCC unit, in which gasoline was cracked over a commercial Y zeolite based equilibrium catalyst. Corma et al.3 reported that recycling light naphtha in the FCC process was an interesting alternative to increase the yield of propylene and to produce clean gasoline. Liu et al.4 investigated secondary cracking of gasoline and diesel from the catalytic pyrolysis of heavy oil in a fluidized bed reactor. Yang et al.5 investigated the effects of reaction conditions on coke formation and olefins conversion in the process of FCC gasoline olefin reformulation in a continuous pilot riser-type FCC unit. Wang et al.6 reprocessed FCC gasoline in a secondary riser of the FCC unit as to increase propylene yield in a continuous pilot plant. Shao et al.7 developed a structural catalyst to increase propylene yield in the naphtha cracking process. Zhao et al.8 reported a nanoscale HZSM-5 zeolite, which was hydrothermally treated with ammoniawater and then loaded with La2O3 and ZnO, for the FCC gasoline upgrading process. Experimental study of FCC gasoline secondary reactions in the riser reactor,9 kinetic modeling,10 molecular simulation r 2011 American Chemical Society
using the method of structure oriented lumping combined with Monte Carlo,11 multiscale modeling of riser reactor based on the multiscale modeling method and multidomain strategy,12 and a prediction model for increasing propylene13 have been reported in our previous work. It can be noted from above review that past efforts devoted to secondary reactions of FCC gasoline have been limited to the development of new technology, experimental study, kinetic modeling, and reactor simulation. However, catalyst deactivation in this process has not been taken into account yet. In fact, catalyst deactivation is an enormously important aspect that affects the secondary reactions of FCC gasoline. In this process, coke deposits on the catalyst surface as catalytic cracking proceeds, and then coke formation causes active site coverage and pore blockage, and consequently, loss of catalyst activity, termed “deactivation”, is observed. Although coke formation deactivates the catalyst and lowers the activity, it affects conversion and selectivity, contributing to the variation of product distribution along the riser reactor, and also provides the whole unit with the necessary heat to keep thermal balance. What’s more, catalyst deactivation in secondary reactions of FCC gasoline is quite different from that in the conventional FCC process since its feedstock is much lighter and contains less heteroatoms such as sulfur, oxygen, nitrogen, and metals. The reacting species, reactions, the characteristics of deactivation, and its effects on the conversion of the reactants and the selectivity of the products would also be quite different. Thus, it is necessary and significant to have a good understanding of catalyst deactivation in secondary reactions of FCC gasoline. Received: February 28, 2011 Revised: May 3, 2011 Published: May 03, 2011 2427
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Energy & Fuels Generally, catalyst deactivation during its operation in the conventional FCC process is caused by coke formation, poison deposition, and solid state transformation,14 while coke formation is the dominant factor affecting catalyst activity for the secondary reaction of FCC gasoline. Great efforts have been addressed to modeling of catalyst deactivation due to coke in the FCC process. Kinetic models considered the effects of catalyst deactivation on the cracking kinetics can be categorized into two types. One is a nonselective deactivation model that assumes the deactivation of catalyst affects each one of the reactions in the reaction network in the same way.1520 Generally, two methods have been used to represent the effects of the coking phenomena on the nonselective dependent catalyst deactivation. One is to relate catalyst deactivation with coke content. Several empirical relations of this type have been reported to be capable of fitting well with the experimental data. However, coke on catalyst cannot be routinely measured or accurately correlated since some kinds of coke may not contribute to catalyst deactivation. The other method is to describe the catalyst activity as a function of time on the stream. Various models for time dependent catalyst deactivation have been proposed for different lengths of contact time in the catalytic cracking of heavy oil. This approach has been widely accepted and probably used beyond the original purposes, as commented by Forzatti and Lietti.21 Nonselective deactivation model is simple and requires a few parameters. However, it is empirical and does not represent the intrinsic phenomena taking place on the catalyst. Additionally, its prediction accuracy is low and its assumption is unreasonable, especially for a complex reaction system. The other is selective deactivation model, in which catalyst deactivation is assumed to affect each one of the reacting species and reactions in a complex reaction network with different manners and contributes to the different variations with time on the stream to the yield to each product. The first rigorous selective deactivation kinetic model was proposed by Corella and Asua.22 Corella23 reported several concepts on modeling of selective deactivation and developed a selective deactivation kinetic model for the FCC process using the commercial catalysts and feedstocks. Bollas et al.24 proposed a kinetic model which incorporated a five-lump reaction network to describe the kinetics of cracking reactions and adopted a selective deactivation model to study the paths of catalyst deactivation. This kind of model is more realistic and accurate. Also it is able to trace the reaction paths of the key components. However, it also requires more experimental data support, and the overparameterization problem that selective deactivation introduces to parameter estimation is difficult to handle with. What’s more, “These facts, together with the model’s complexity, are the reasons these selective deactivation kinetic models have not been used much until now”, as commented by Corella.23 On the basis of the above considerations, in this work, a new strategy that combines selective deactivation with nonselective independent deactivation is proposed. Also, a reactant oriented selective deactivation model is developed to describe the deactivation behavior of catalyst and its effects on the reactions of the lump scheme. The lumped model considering the kinetics and the catalyst deactivation in secondary reactions of FCC gasoline is developed. To estimate the kinetic parameters of the proposed model, a hybrid technique (AGA/SA), which incorporates evolution strategies (ES) and simulated annealing (SA) into a genetic algorithm (GA), is proposed. This
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optimization technique reserves the heuristically global search of GA,2527 the local search ability of SA,28,29 self-adaptive, and multipattern evolution of ES.30,31 It effectively overcomes the premature convergence and shows rapid convergence and high precision. AGA/SA is evaluated by the comparisons of its computational performance with general GA on several testing problems and parameter estimations. The reliability and flexibility of the proposed kinetic model are verified by sets of experiments.
2. ESTABLISHMENT OF KINETIC MODEL 2.1. Experiments. Experiments were performed in a catalytic cracking unit with a riser reactor. Three feedstocks, three samples of catalytically cracked gasoline, were taken from industrial FCC units of China. Y zeolite catalyst, CC-20D (produced by Sinopec Changling Catalyst Factory, China), was taken from the circulating inventory of a commercial FCC unit. The detailed experimental procedures, the properties of feedstocks and catalysts, reaction conditions, analytical data of feeds and products have been reported in our previous work.10 2.2. Lumped Scheme and Reaction Network. Though secondary reactions of FCC gasoline are not as complex as the primary catalytic cracking of heavy oil, they still contain hundreds of components with a wide distribution of boiling points and thousands of reactions. In this work, the lumped concept is adopted and the reacting species are plotted into five pseudocomponent lumps from the viewpoint of boiling range. These lumps include dry gas lump (DG, H2, and C1C2), liquefied petroleum (LPG, C3C4), gasoline lump (GL, C5477 K), light cycle oil lump (LCO, >477 K), and coke lump (COKE). Since GL represents both feedstock and the products, the PONA compositions of gasoline will be changed as the reactions are occurring. Therefore, GL is further divided into a paraffin lump (GP), olefin lump (GO), naphthene lump (GN), and aromatic lump (GA) based on the hydrocarbon group structure. Secondary reactions between these lumps include cracking, hydrogen transfer, isomerization, aromatization, alkylation, condensation, and coke formation reactions. Assumptions are (1) all the reactions are regarded as first-order reactions, (2) GP, GO, and GN can be converted into each other and these reactions are considered reversible reactions, (3) LPG only converts into dry gas, LCO converts into coke only, and there is no interaction between dry gas and coke, (4) there are few C3þ alkyl aromatics in FCC gasoline, aromatics do not form LPG directly, and lump GP does not produce LCO directly either, (5) lump GA does not yield lump GN directly since the reaction proceeds at nearly atmospheric pressure, (6) the effects of sulfur and nitrogen in feedstock on secondary reactions are neglected. Following these assumptions, the reaction network is schematically represented in Figure 1. 2.3. Mathematic Modeling. Since vapor residence time is very short and the reactions are far away from chemical equilibrium under usual operating conditions, it is thus reasonable to treat all the reactions as first-order reactions. The effects of internal and external diffusion on the reactions have been eliminated. Thus, the continuity equation in riser reactor for the secondary reaction of FCC gasoline can be expressed as
DFaj Daj þ GV ¼ rj Dt x Dx t 2428
ð1Þ
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Table 1. Average Molecular Weights and Stoichiometric Coefficients lumps
GP
GO
GN
GA
LCO
LPG
DG
COKE
Mj, g/mol
102
100
100
90
187
48
21
950
2.13
4.86
0.11
0.53
2.08
4.76
0.11
0.53
2.08
GPf
1.02
GOf GNf
0.98
1.0 1.0
GAf
1.11
0.48
4.76
0.11
4.29
0.09
LCOf
0.20
LPGf
2.29
For the GP lump, dyGP P j tv ½ ðk12 φ12 þ k15 φ15 þ k16 φ16 ¼ RT c=o dX þ k18 φ18 ÞyGP þ 0:98k21 φ21 yGO
ð8Þ
For the GO lump, Figure 1. Reaction network of eight-lump kinetic model.
If the cross-sectional area of the riser reactor and mass velocity of the material stream are invariant, then GV ¼ FU ¼ constant
For the GN lump, dyGN P j tv ½k23 φ23 yGO ðk32 φ32 þ k34 φ34 ¼ RT c=o dX þ k35 φ35 þ k36 φ36 þ k37 φ37 þ k38 φ38 ÞyGN
dyGA P j tv ½1:11k34 φ34 yGN ðk45 φ45 þ k47 φ47 ¼ RT c=o dX þ k48 φ48 ÞyGA ð11Þ For the DG lump, dyDG P j tv ½4:86k15 φ15 yGP þ 4:76k25 φ25 yGO ¼ RT c=o dX þ 4:76k35 φ35 yGN þ 4:29k45 φ45 yGA þ 2:29k65 φ65 yLPG ð12Þ For the LPG lump,
ð5Þ
dyLPG P j tv ½2:13k16 φ16 yGP þ 2:08k26 φ26 yGO ¼ RT c=o dX þ 2:08k36 φ36 yGN k65 φ65 yLPG ð13Þ
Treating the stream vapor in the riser reactor as an ideal gas, then F¼
P MW RT
ð10Þ
For the GA lump,
ð4Þ
When the reaction process in the riser reactor is kept stable, the partial derivative item on the left side of eq 4 is equal to zero. By the definition of GV, ðF =εÞL ðFc =εÞL GV ¼ c ¼ jc=o tv jc=o ðL=uÞ
ð9Þ
ð2Þ
The disappearance rate of lump j is in direct proportion to its mole concentration Faj(Fc/ε), F rj ¼ kj ðFaj Þ c φ ð3Þ ε where φ is catalyst deactivation function. Therefore, eq 1 can be rewritten DFaj Daj F þ GV ¼ kj Faj c φ Dt x Dx t ε
dyGO P j tv ½1:02k12 φ12 yGP ðk21 φ21 þ k23 φ23 ¼ RT c=o dX þ k25 φ25 þ k26 φ26 þ k27 φ27 þ k28 φ28 ÞyGO þ k32 φ32 yGN
For the LCO lump, ð6Þ
dyLCO P j tv ½0:53k27 φ27 yGO þ 0:53k37 φ37 yGN ¼ RT c=o dX þ 0:48k47 φ47 yGA k78 φ78 yLCO ð14Þ
Replacing the actual distance by dimensionless relative distance X, eq 4 becomes,
For the coke lump, daj P MW jc=o tv kj aj φ ¼ RT dX
ð7Þ
where X = x/L and tv = L/u. Introducing the stoichiometric coefficient from the i lump to the j lump, which are the molecular weight ratios of lump i over lump j to satisfy the global mass balance, as listed in Table 1, the rate laws of eight lumps can be written as follows
dyCOKE P ¼ j tv ½0:11k18 φ18 yGP þ 0:11k28 φ28 yGO dX RT c=o þ 0:11k38 φ38 yGN þ 0:09k48 φ48 yGA þ 0:20k78 φ78 yLCO
ð15Þ where tv is time on stream and kij and φij represent the reaction constants and catalyst decay function for the reaction of lump 2429
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Energy & Fuels i to j, respectively. The temperature dependence of the kinetic parameters in the above equations are described by the Arrhenius expression, (k = k0 exp(E/RT)), and the activation energy of reactions is calculated by the reaction rate constant determined with different temperatures. For the secondary reaction of FCC gasoline, it needs 21 parameters to describe the different effects of catalyst deactivation on each reaction in the lumped scheme for the selective deactivation model and one parameter for the nonselective deactivation model. Obvious, it is an extremely difficult regression problem to simultaneously optimize 42 parameters depending on the eight equations above. In addition, it is unreasonable that 21 reactions are affected by the deactivation catalyst in the same manner in such a complex reaction network. Moreover, its prediction accuracy would be unsatisfactory. On the basis of the above considerations, a new strategy of selective deactivation coupling with nonselective independence deactivation is proposed. With this strategy, a reactant oriented selective deactivation model is developed. The model assumes that catalyst deactivation affects the reactions of each reactant in the same way but affects different reactants (GP, GO, GN, GA) in different manners, and the effects of catalyst deactivation on each reactant can be described by a nonselective independence deactivation, thus leading to four deactivation functions. 8 > > φ1j ¼ φ1 > > > : φ4j ¼ φ4 where φ1, φ2, φ3, and φ4 are the deactivation functions of reactants GP, GO, GN, and GA, respectively. The main reactions that occurred to GP are cracking, isomeration, and dehydrocyclization, GO undergoes cracking, alkylation, dimerization, isomeration, aromatization, dehydrocyclization, and hydrogen transfer, the main reactions of GN are dehydrogenation, substituted group breaking, ring-breaking cracking, and hydrogen transfer, and the reactions of GA are substituted group breaking, ring-breaking cracking, consersation followed by hydrogen transfer and dehydrogenation. Moreover, it is assumed that all active sites of the catalyst have the same strength and uniformly distribute on the surface; pores blocking due to coke formation is the dominant effect of deactivation on the reactivity, and diffusion phenomena are important and the molecular size of each reactant plays a dominant role on the deactivation rate. One additional important issue involved here is the model equation to describe catalyst deactivation. It is reported by Corella23 that the kinetics of cracking of the feedstock (heavy oils) are much more affected by the catalyst deactivation than the cracking of gasoline. For gasoline secondary reactions, coke formation is the most important factor affecting the catalyst activity, and catalyst poisoning caused by impurities (S, N, O, Ni, V, etc.) can be neglected since its gasoline feedstock is much cleaner and lighter than heavy oil, the feedstock for primary catalytic cracking. Therefore, only the effects of coke formation on catalyst deactivation are considered in this work. Moreover, coke cannot be routinely measured or accurately correlated because some kinds of coke may not contribute to the deactivation, while collecting and processing experimental data concerning time on steam is much easier and more accurate.
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Thus, catalyst deactivation is related with time on stream rather than coke content. Furthermore, it is reported by Bollas et al.24 that “for small time-on-stream, which is the case for typical commercial FCC riser and FCC pilot plants with residence time lower than 10 s, the power function better suits the experimental data”. Therefore, the power function (eq 17) is chosen to describe the catalyst deactivation in secondary reactions of FCC gasoline. φi ¼ t ni
i ¼ 1, 2, 3, 4
ð17Þ
where n1, n2, n3, and n4 are the deactivation constants of GP, GO, GN, and GA, respectively.
3. SOLVING THE MODEL 3.1. Methodology. To overcome GA’s drawbacks such as poor local search ability, premature convergence, sensitive to genetic operators and the parameter setting,10,32,33 a hybrid method is developed by incorporating simulated annealing and evolution strategies into general GA to enhance their coupling performance. In this method, self-adaptive strategy is adopted to dynamically control the genetic parameters, floating-point coding is used, and the genetic operators consist of (μ þ λ) selection with elitist strategy, adaptive multiannealing crossover, and mutation. Detailed implementation of the proposed algorithm is described as follows. 3.1.1. Coding and Fitness Function. Floating-point coding is adopted to directly handle the real valued parameters in this work. Fitness function is designed based on the sum of relative errors between calculated results and experimental ones. To enhance the competition among individuals, power function and reciprocal transformation are taken to amplificatory transfer the errors information into the fitness of each individual. The fitness function is expressed as eq 18. 2 0 !2 31 c d a 6 exp F @ycal yexp 5 ð18Þ ψðK Þ ¼ 4 i =yi i j ¼ 1 i ¼ 1 cd
∑∑
_ where K is the vector of model parameters, a is a positive constant to adjust the maximum of the fitness, c is the number of different temperatures, d is the number of reacting components, subscripts j and i are the jth operation temperature set and the ith lump yield, respectively. 3.1.2. Parameter Control Using Self-Adaptive Strategy. Selfadaptive strategy is adopted to dynamically adjust Pc and Pm. Selfadaptive mechanisms used in this work are given as follows, δc Gen Pc ¼ Pc min þ ðPc max Pc min Þ exp ð19Þ Genmax
δm Gen Pm ¼ Pm min þ ðPm max Pm min Þ exp Genmax
ð20Þ
where Gen is the current generation number, Genmax is the current generation number, and both of them are controlled to decrease with the increase of evolutionary generations. δc and δm are coefficients to reflect the decreasing rate of Pc or Pm with generation, respectively. With this strategy, the difficulty of repeated adjusting of the values of Pc and Pm can be avoided. 3.1.3. (μ þ λ) Selection with Elitist Strategy. In this work, (μþλ) evolution strategy termed (μ þ λ) ES,34 is employed. 2430
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Figure 2. Flowchart of adaptive multiannealing crossover.
It is a multipoint searching method using truncation selection in an extended searching space. (μ þ λ) ES copies the parents and the offspring into a selection pool. The superabundant individuals are reproduced, and the individuals with good fitness are selected to form the next generation. (μ þ λ) ES is also an elitist strategy by which a limited number of individuals with the best fitness are chosen to pass to the next generation, preserving the best individuals from random destruction by crossover or mutation operators and thus accelerating convergence. 3.1.4. Adaptive Multiannealing Crossover Operator. Different crossover strategies (part arithmetic crossover, complete arithmetic crossover, and heuristics crossover) are taken to act on each individual circularly, as shown in Figure 2. The process adopts new individuals based on the Boltzmann rule: when an optimized value is adopted, a depraved value is also adopted with a certain probability. This rule is expressed as follows, Δf g randomð0, 1Þ ð21Þ exp T
Figure 3. Flowchart of adaptive multiannealing mutation.
where Δf is the difference of fitness value between the child generation and the parent generation, and T is the annealing temperature. Adaptive crossover mechanism is used to dynamically control Pc. Annealing temperature evolution strategy is the fastest annealing strategy: TðtÞ ¼
T0 1 þ Rt
R ∈ ½0, 1
ð22Þ
3.1.5. Adaptive Multiannealing Mutation Operator. As shown in Figure 3, adaptive multiannealing mutation is used, which adopts nonuniform mutation, multistep mutation, and Gaussian mutation, together with adaptive mechanism to dynamic control Pm. These mutation operators act on every individual circularly. 3.2. Solution Scheme of the Model. The proposed AGA/SA is used herein for parameter estimation of the model. As shown in Figure 4, the whole procedure is described as follows. Step 1: Set parameters (population size, number of elite individuals, maximum generation number, parameters for adaptive crossover and mutation, etc.) and start running. Step 2: Generate initial population randomly. Step 3: Evaluation: calculate the
Figure 4. Flowchart of AGA/SA.
objective function value and find the fitness of each individual. Step 4: Stop condition, if a predefined maximum generation number is reached, stop and output the optimal results; otherwise, go to next step. Step 5: Selection operator using (μ þ λ) selection with the elitist strategy. Step 6: Apply adaptive multiannealing crossover operator. Step 7: Apply adaptive multiannealing mutation operator. Step 8: Generate new population and go to step 3. 2431
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Figure 5. Results of FCC gasoline secondary reactions on feed no. 1: (a) lump yield vs temperature at C/O = 13.0, t = 2.0 s; (b) GO yield vs temperature; (c) the selectivity of gasoline to LPG, DG, LCO, and COKE vs GO yield; (d) the yield and selectivity of LPG; (e) the yield and selectivity of DG; and (f) the yield and selectivity of COKE.
4. RESULTS AND DISCUSSION 4.1. FCC Gasoline Secondary Reaction Results. The secondary reaction of FCC gasoline was carried out in a riser reactor under a wide range of operation conditions. Figure 5 shows the typical results of the FCC gasoline secondary reaction. As shown in Figure 5a, with rising temperature, the yields of GP, GO, LCO, and COKE decreased, while the yields of GN, GA, and DG increased. However, the LPG yield first increased then declined slightly since the overcracking of LPG into DG at high temperature. The high catalyst to oil ratio (C/O) and long reaction time not only reduced GO (Figure 5b) but also increased LPG (Figure 5c). Nevertheless, it led to the undesired increase in DG yield (Figure 5d) and COKE yield (Figure 5e). Higher catalyst to oil ratio together with shorter reaction time under a certain temperature means a higher catalyst activity. The highest selectivity of both LPG (Figure 5c) and COKE (Figure 5e) was obtained in the case with 4 s reaction time, whereas the lowest selectivity of LPG and COKE was observed at C/O = 8.0 and t = 2.0s with a moderate catalyst activity. Figure 5d shows that DG selectivity was slightly influenced by the catalyst
Table 2. Computational Parameters of AGA/SA and General GA Used in the Optimization parameters
AGA/SA
general GA
maximum generation number
100
100
population size number of elite individuals
120 2
120 0
crossover probability, Pc
Pc = 0.5 þ 0.4
0.8
exp(3 Gen/100) mutation probability, Pm
Pm = 0.01 þ 0.09
0.02
exp(3 Gen/100)
to oil ratio and reaction time. It was an interesting phenomenon that, as shown in Figure 5f, the selectivity of LCO and COKE declined as the GO yield decreased. This fact indicates that olefins might be the main source of LCO and COKE formation. Furthermore, with the decreasing GO yield, DG selectivity keeps climbing, while LPG selectivity first deceased slowly then dropped fast. 2432
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4.2. Verification of AGA/SA. To verify the effectiveness and reliability of AGA/SA, comparison of AGA/SA with general GA is performed. Both the procedures are implemented in C Programming Language, and the computational parameters used in both algorithms are summarized in Table 2. 4.2.1. Verification of AGA/SA Using Test Functions. Four test functions are adopted to compare the computational performance for each algorithm. The Easom function is formulated as follows, " # n n Y 2 cosðxi Þ exp ðxi πÞ ð23Þ f ðxÞ ¼
∑
i¼1
i¼1
with the constraint, 10 e xi e 10
ð24Þ
The global minimum of the Easom function is 1, corresponding to xi = π, i = 1:n, where n is the number of design variables which defines the dimension of the test function. Rosenbrock function (eq 25) is a classic single peak function and is formulated as finding a vector x that minimizes the objective function, f ðxÞ ¼
n
∑ ð100ðxi1 xi 2Þ2 þ ðxi 1Þ2Þ i¼1
ð25Þ
with the constraints, C1 ðxÞ ¼ 1
n Y
!2 e0
ð26Þ
n C2 ðxÞ ¼ n xi ¼ 0 i ¼ 1
ð27Þ
10 e xi e 10
ð28Þ
xi
i¼1
∑
It is a two-dimensional and single hump function that the global minimum optimum is 0 at the point xi = 1, i = 1:n. The global optimum lies inside a long, narrow, parabolic shaped flat valley. It is trivial to find the valley but is difficult to converge to the global optimum. This function is widely used to evaluate the efficiency of the proposed algorithm in handling constrained optimization problems. Schaffer function is a multimodal test function with two variables and its mathematic expression is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 sin x1 2 þ x2 2 0:5 ð29Þ f ðxÞ ¼ 0:5 1:0 þ 0:001 x1 2 þ x2 2 Þ 2 with the constraint, 100 e x1 , x2 < 100
ð30Þ
The global maximum of the Schaffer function is 1.0 at the point of x1 = x2 = 0. The Griewank function is commonly used to test the ability of different solution procedures to find local optima and the convergence of algorithms because the number of minima grows exponentially as its number of dimensions increases. The function
is defined as follows:
n 1 n 2 Y xi xi cos pffi þ 1 f ðxÞ ¼ 4000 i ¼ 1 i i¼1
∑
ð31Þ
with the constraint, 600 e xi e 600
ð32Þ
Its global minimum is 0, which corresponds to xi = 0, i = 1:n. Comparison of AGA/SA and general GA on solving the above four test problems are shown in Figure 6. As shown in Figure 6, the generation number of AGA/SA to obtain the best solutions is smaller and the optimized objective function value is much better than those of general GA. Figure 6a presents a faster convergence rate, and Figure 6b illustrates a better optimization result. Figure 6c shows that AGA/SA converges to a minimum objective function value of 1.0 after the 65th generation for the Schaffer function. However, general GA oscillates at a lower value of 0.55 and falls into local extremes. When it comes to the Griewank function (Figure 6d), the proposed method is able to find a much lower value of 0.022 (near the global optimum 0) after 17th generation, while general GA traps into premature convergence to 0.43 after the 80th generation. These results suggest that AGA/SA exhibits a better computational performance, and it is robust to find the global optimum solutions with a fast convergence rate. 4.2.2. Verification via Parameter Estimation. For AGA/SA, R, c, and d are set as 60, 7, and 8, respectively. Figure 7 shows a typical evolution process of averaged fitness with generation for AGA/SA and general GA. The fitness value is calculated by eq 18 in each generation. As shown in Figure 7, general GA obtains its maximum average fitness value of 325 after 400 generations, while AGA/SA improved the average fitness value to 1050 after 500 generations, which means the average relative error between calculated results and experimental data for each lump is less than 3%. Therefore, the maximum iteration number is set as 500 to terminate the procedure. Above comparisons indicate that AGA/SA shows significant improvements in terms of solution quality, convergence rate, and hill-climbing ability. It is reliable to adopt AGA/SA method for parameter estimation in this work. 4.3. Parameter Estimation Results. Kinetic parameters of the proposed model are estimated by AGA/SA based on over 30 sets of experimental data from feed no. 1 and feed no. 3. Table 3 shows the estimated expressions of kinetic constants and deactivation functions for the reactant oriented selective deactivation model. As shown in Table 3, the 95% probability intervals indicate that the average activation energies and the parameters of deactivation functions are precisely estimated. The sum of rate constants related to olefins (k2j) is the highest since olefins are the most active content in the secondary reaction system. k25 > k15 and k26 > k16 mean the cracking rate of olefins is faster than that of paraffins. Small k65 indicates that light olefins in LPG are stable and difficult to crack into lower-molecular hydrocarbons, which follows the carbonium ion mechanism of catalytic cracking. k78 > k27 means LCO condensation into coke is faster than olefins condensation into LCO. The reason is that diesel is an unstable intermediate and easily undergoes secondary reactions to form its final product coke. Coke is mainly formed from the condensation of olefins and aromatics, but with low contribution 2433
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Figure 6. Comparison of AGA/SA with general GA on four test functions: (a) Easom function, (b) Rosenbrock function, (c) Schaffer function, and (d) Griewank function.
of paraffins. Moreover, activation energies of the cracking reaction such as EA15, EA25, EA35, EA45, and EA55 are high, which means cracking reactions need to overcome large energy barriers and can be significantly affected by reaction temperature, while relatively low EA12, EA23, and EA34 correspond to hydrogen transfer. Low temperature would be favorable for these endothermic reactions. Kinetic parameters obtained in this work are reasonable and show consistency with the reaction laws of catalytic cracking of hydrocarbons since it takes into account the reactant oriented selective deactivation. 4.4. Verification of the Model. Verification of the proposed model is performed by comparing its predicted results with the experimental data obtained under a wide range of reaction conditions. Feed no. 2 is used to check model reliability and to evaluate the prediction accuracy and extrapolation capability of the proposed model. The operation conditions for the five verification runs on feed no. 2 are summarized in Table 4. The time on stream in Table 4 is equal to the residence time. Table 5 gives the results of run 1. The gasoline yield calculated from the total yields of GP, GO, GN, and GA is 68.92%. A large catalyst to oil ratio and short reaction time resulted in a good catalyst activity in this case. As the key lumps, olefins were greatly reduced by 28.35%, and LPG was produced with yield of 20.46%. The low yield of DG is attributed to the low temperature. Table 6 shows that in run 2 the main conversions were the cracking of olefins and the production of LPG, while aromatics and naphthenes were much less converted. Catalyst activity decreased rapidly due to the long residence time, which led to a low gasoline yield and a high coke yield of 5.7%. As shown in Table 7, the gasoline yield in run 3 is 75.11%. The conversions were mainly related with the cracking of paraffins and severe cracking of olefins, together with the production of aromatics. Catalyst activity might be in a good condition due to the short reaction time and middle temperature. Olefins were
Figure 7. Comparison of AGA/SA with general GA on parameter estimation.
sharply reduced by 25.37%, and the undesired products of DG and COKE were less produced; however, a low LPG yield of 16.33% is obtained due to the mild reaction conditions which are unfavorable to cracking reactions. Table 8 shows the results of run 4. High temperature and long reaction time significantly promoted the deep cracking of olefins and paraffins into low-molecular products, resulting in the lowest total yield of GP and GO together with the highest total yield of LPG and DG. Moreover, naphthenes were greatly converted and aromatics were produced with a high yield of 24.36% because the high temperature was favorable to aromatization and condensation reactions. Good agreement between predicted results and experimental data in above comparisons indicates that the proposed reactant oriented selective deactivation model is reliable and can be 2434
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Table 3. Results of Parameters Estimation for the Reactant Oriented Selective Deactivation Model no.
kij = k0 exp(EAij/RT) (m3 3 gcat/s)
Table 5. Comparison of Predicted Results with Experimental Data on Run 1
EAij (J/mol)
lump yield
Exp
Pred
RE (%)
1
k12 = exp(1.4124 1848.7/T)
15 370 ( 46
GO (wt %)
13.45
13.12
2.45
2 3
k15 = exp(0.1566 4205/T) k16 = exp(0.2061 1209/T)
34 960 ( 57 10 052 ( 49
GP (wt %) GN (wt %)
24.78 7.72
24.77 7.80
0.04 1.04 0.52
4
k18 = exp(1.088 3143/T)
26 131 ( 63
GA (wt %)
22.97
22.85
5
k21 = exp(1.3162040/T)
16 961 ( 41
LPG (wt %)
20.46
21.33
4.25
6
k23 = exp(2.269 2052/T)
17 060 ( 58
DG (wt %)
0.96
0.90
6.25
7
k25 = exp(5.375 7260/T)
60 359 ( 134
LCO (wt %)
5.46
5.37
1.65
8
k26 = exp(0.7265 1465/T)
12 180 ( 79
COKE (wt %)
4.11
3.79
7.79
9
k27 = exp(0.4274 913.2/T)
7 592 ( 44
10 11
k28 = exp(2.159 1519/T) k32 = exp(0.3414 1942.5/T)
12 629 ( 65 16 150 ( 61
12
k34 = exp(1.981 1585/T)
13 178 ( 53
13
k35 = exp(5.475 9765/T)
81 186 ( 129
14
k36 = exp(0.3288 895.1/T)
15
k37 = exp(1.639 2115/T)
17 584 ( 60
16
k38 = exp(0.347 1299/T)
17 800 ( 72
17
k45 = exp(0.2956 5604/T)
46 592 ( 101
18 19
k47 = exp(0.5146 1180/T) k48 = exp(2.031 3512/T)
9 811 ( 34 29 199 ( 83
20
k65 = exp(1.405 6739/T)
56 028 ( 112
21
k78 = exp(0.9812 975/T)
8 106 ( 37
no.
deactivation function
confidence interval
22
j1 = t0.4202
0.4202 ( 0.0381
23
j2 = t0.4681
0.4681 ( 0.0476
24
j3 = t0.4884
0.4884 ( 0.0186
25
j4 = t0.7957
0.7957 ( 0.0432
Table 6. Comparison of Predicted Results with Experimental Data on Run 2 lump yield
7 442 ( 26
reaction pressure (MPa)
run 2
run 3
run 4
8.87
8.81
0.68
GP (wt %)
22.83
22.47
1.58
GN (wt %)
6.40
6.31
1.41
GA (wt %) LPG (wt %)
23.62 25.32
23.86 26.20
1.02 3.48
DG (wt %)
1.62
1.61
0.62
LCO (wt %)
5.25
5.24
0.19
COKE (wt %)
5.70
5.49
3.68
lump yield
run 5
1.13
1.13
1.13
1.13
1.13
reaction temperature (K)
793
823
773
853
873
time on steam (s)
4.19
2.06
2.0
3.71
1.82
catalyst to oil ratio
8.20
8.20
13.0
9.10
12.50
extrapolated to predict the production yields under a wide range of operating conditions with a high accuracy. 4.5. Comparison of Models. Comparison of the proposed model with the kinetic model without consideration of deactivation reported by Wang et al.10 on run 5 is carried out and the results are summarized in Table 9. As shown, reactant oriented selective deactivation model shows a better overall performance. The averaged relative error of reactant oriented selective deactivation model is 2.22%, while that of kinetic model by Wang et al. is 6.09%. Moreover, the yields of DG and COKE predicted by the proposed model also show better agreement with the experimental data. Thus, it is convincing that the lumping scheme considered catalyst deactivation could enhance its fundamentality and predication accuracy. 4.6. Prediction of Product Distribution. Figure 8 shows a case prediction for the variation of lump yields with time on stream.
RE (%)
Table 7. Comparison of Predicted Results with Experimental Data on Run 3
Table 4. Operation Conditions of Five Verification Runs on Feed No. 2 run 1
Pred
GO (wt %)
GO (wt %) GP (wt %)
operation parameters
Exp
Exp
Pred
RE (%)
16.43 25.58
16.20 25.83
1.40 0.98
GN (wt %)
10.03
9.88
1.50
GA (wt %)
23.07
23.11
0.17
LPG (wt %)
16.33
16.44
0.67
DG (wt %)
1.61
1.64
1.86
LCO (wt %)
4.66
4.60
1.29
COKE (wt %)
2.20
2.31
5.00
Table 8. Comparison of Predicted Results with Experimental Data on Run 4 lump yield
Exp
Pred
RE (%)
GO (wt %)
6.25
6.08
2.72
GP (wt %) GN (wt %)
21.09 6.31
21.21 6.23
0.57 1.27
GA (wt %)
24.36
25.01
2.67
LPG (wt %)
27.05
27.00
0.19 2.68
DG (wt %)
4.11
4.00
LCO (wt %)
5.19
5.43
4.62
COKE (wt %)
5.24
5.03
4.01
As shown, GO reduces rapidly and GP keeps a steady decline, while GN first increases slightly and then decreases slowly with time on stream. LPG increases fast and finally reaches a high yield, while GA, LCO, DG, and COKE keep increasing at a slow 2435
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Table 9. Comparison of the Proposed Model with Kinetic Model by Wang et al. on Run 5 lump model of Wang et al. lump yield
Exp
Pred
RE (%)
reactant oriented selective deactivation model Pred
RE (%) 0.12
GO (wt %)
8.27
8.03
2.90
8.26
GP (wt %)
22.16
21.50
2.98
22.64
2.16
GN (wt %)
8.14
8.56
5.16
8.11
0.37
GA (wt %)
24.69
24.98
1.17
23.82
3.52
LPG (wt %)
23.19
23.80
2.63
23.53
1.47
DG (wt %)
4.58
3.76
17.90
4.84
5.68
LCO (wt %)
5.23
5.39
3.06
5.10
2.49
COKE (wt %)
3.63
4.10
12.95
3.70
1.93
and experimental data suggests that the proposed kinetic model is capable of accurately predicting product distribution for secondary reactions of FCC gasoline over a wide range of operating conditions. Results also show that reactant oriented selective deactivation model is exact and it allows for explaining the variation of the distribution of products with time on stream. The proposed AGA/SA exhibits better performance than GA on several testing problems and also shows desired improvements such as rapid convergence, high efficiency, strong ability of hillclimbing, and excellent to escape local optimum. It is concluded that the kinetic model is well established and the hybrid algorithm is reliable.
’ AUTHOR INFORMATION Corresponding Author Figure 8. Production distribution variation with time on stream (feed no. 2; T, 853 K; C/O, 9.10).
rate. The reason might be that the reactions easily take place on GO with high chemical activity and convert it into GA, LPG, and other lumps, leading to a high selectivity of olefin conversions and thus a fast decrease in GO and an increase in product lumps, while a slight change in GA yield is attributed to its chemical stability. Moreover, changes in the yields of lumps are proportional with their corresponding reaction rates. When it comes to LPG, the cracking of gasoline into LPG is a major conversion, and a part of this product can be ascribed to the competitive cracking reactions occurring in the void volume of the riser reactor. As reaction proceeding, catalyst losses its activity due to coke formation, but the diffusion phenomenon of caging product molecules in the catalyst pores might happen due to pore blocking, which could enhance the selectivity of LPG and produce a significant amount of LPG. This is another explanation that can be given to the increase in the yields of LPG, DG, and COKE with time on stream.
5. CONCLUSIONS Catalyst deactivation has significant effects on secondary reactions of FCC gasoline. The effects of coke formation on catalyst activity are described by the power deactivation function based on time on stream. Kinetic model considering reactant oriented selective deactivation for secondary reactions of FCC gasoline has been developed. The proposed model has been validated by sets of experiments and compared with the reported kinetic model. Good agreement between the predicted results
*Telephone: þ86-29-82663189. Fax: þ86-29-82668789. E-mail:
[email protected].
’ ACKNOWLEDGMENT Financial support of this work by the National Basic Research Program of China (973 Program, Grant 2009CB219906), the National Natural Science Foundation of China (Grant 200976144), and the Scholarship Award for Excellent Doctoral Student granted by Ministry of Education of China are gratefully acknowledged. ’ NOMENCLATURE Rj = concentration of jth lump in vapor, mol/g Cal = calculated results EA = activation energy, J/mol Exp = experimental data Gv = vapor mass flow rate cross the riser, g/m2 s kij = kinetic constant for the reaction between lumps i and j, m3/ _ gcat s K = reaction rate vector L = riser reactor length, m MW = average molecular weight of vapor mixture, g/mol n = reaction order P = reaction pressure, Pa Pc = crossover probability, dimensionless Pm = mutation probability, dimensionless Pred = predicted results rij = rate of reaction between lumps i and j, mol/m3 s R = gas constant (8.3143 J/mol K) RE = relative error 2436
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Energy & Fuels t = time on stream, s T = reaction temperature, K u = vapor flow velocity in bed, m/s x = distance from reactor entrance, m yj = yield of lump j, wt % Greek Symbols
ε = void volume fraction of fluidized bed, dimensionless F = gasoline vapor density, g/m3 Fc = catalyst density, g/m3 jc/o = catalyst to oil ratio, dimensionless φ = catalyst deactivation function
ARTICLE
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