Equivalent Reactor Network Model for the Modeling of Fluid Catalytic

Aug 24, 2015 - First, a CFD model is run that accounts for the reactor hydrodynamics. ... For saving cost, the two risers can also share a common dise...
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Equivalent Reactor Network Model for the Modeling of Fluid Catalytic Cracking Riser Reactor Yupeng Du,† Hui Zhao,† An Ma,‡ and Chaohe Yang*,† †

State Key Laboratory of Heavy Oil Processing, China University of Petroleum (East China), Qingdao 266580, China Petrochina Petrochemical Research Institute, Beijing 100195, China

‡

S Supporting Information *

ABSTRACT: Modeling description of riser reactors is a highly interesting issue in design and development of fluid catalytic cracking (FCC) processes. However, one of the challenging problems in the modeling of FCC riser reactors is that sophisticated flow-reaction models with high accuracy require time-consuming computation, while simple flow-reaction models with fast computation result in low-accuracy predictions. This dilemma requires new types of coupled flow-reaction models, which should own time-efficient computation and acceptable model accuracy. In this investigation, an Equivalent Reactor Network (ERN) model was developed for a pilot FCC riser reactor. The construction procedure of the ERN model contains two main steps: hydrodynamic simulations under reactive condition and determination of the equivalent reactor network structure. Numerical results demonstrate that with the ERN model the predicted averaged error of the product yields at the riser outlet is 4.69% and the computation time is ∼5 s. Contrast to the ERN model, the predicted error with the plug-flow model is almost three times larger (12.79%), and the computational time of the CFD model is 0.1 million times longer (6.7 days). The superiority of the novel ERN model can be ascribed to its reasonably simplifying transport process and avoiding calculation divergences in most CFD models, as well as taking the back-mixing behavior in the riser into consideration where the plug-flow model does not do so. In summary, the findings indicate the capabilities of the ERN model in modeling description of FCC riser reactors and the possibilities of the model being applied to studies on the dynamic simulation, optimization, and control of FCC units in the future.

1. INTRODUCTION Fluid catalytic cracking (FCC) is one of the most important conversion processes in a petroleum refinery. It is widely used to convert heavy oil to valuable vehicle fuel such as gasoline and diesel, as well as improving light olefin production to meet the increasing demand of ethylene and propylene worldwide. In the FCC unit, the riser reactor is always regarded as one of the most complex parts, because of its inherent nonlinear correlations between complicated hydrodynamics, unknown multiple reactions, and deactivation kinetics, as well as heattransfer and mass-transfer resistances.1 So far, detailed experimental investigation of industrial risers is a challenging task. Therefore, modeling description of riser reactors becomes a useful approach to explore the complex process in petrochemical industry.2,3 In the open literature, numerous models for FCC riser reactors are available with various degrees of simplifications and assumptions. Most of the work on modeling FCC riser reactors was focused on either reactor hydrodynamics or catalytic cracking kinetics. As for the hydrodynamic models, Pugsley et al.4 classified most applied models into three types: (i) onedimensional (1-D) plug flow models, normally with simplified formulation and solution; (ii) semiempirical 2-D models, usually described as core-annulus models; and (iii) computational fluid dynamics (CFD) models deduced from phenomenological concepts with a comprehensive and generic character but not easy to formulate and solve. Regarding the reaction kinetic models, three types of models are commonly used:5 (i) lumped kinetic models; (ii) molecular level kinetic © 2015 American Chemical Society

models; and (iii) empirical correlation models. Combining various flow models (i.e., hydrodynamics models) with different reaction kinetic models comprises the conventional coupled flow-reaction models.6−18 It is noticed that the empirical correlation models do not need to be combined with hydrodynamic models when they are utilized.2,5 Therefore, as far as the authors’ knowledge is concerned, five categories of coupled flow-reaction models, as illustrated in Figure 1, are commonly used in academia/industry. The main features and applications of each category have been summarized and reported recently by the authors.19 Since the base mathematical equations for these models are varying, each type of coupled model has its own advantages and disadvantages. In particular, while sophisticated flow-reaction models with high accuracy require long computational time, simple coupled flow-reaction models with short computational period obtain low-accuracy results. This point can be best illustrated, on one hand, with the example of CFD-based coupled models1,15 (e.g., the fourthcategory models shown in Figure 1), highly detailed flow field structures can readily be achieved with CFD models. Such detail, however, is time-consuming and requires quite a lot of computational efforts, resulting in low accessibility in real-time dynamic simulation process. On the other hand, the assumption of plug flow in the riser (e.g., the first-category Received: Revised: Accepted: Published: 8732

June 11, 2015 August 22, 2015 August 22, 2015 August 24, 2015 DOI: 10.1021/acs.iecr.5b02109 Ind. Eng. Chem. Res. 2015, 54, 8732−8742

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Industrial & Engineering Chemistry Research

the subsequent section, the construction procedure of the ERN model and determination of its key model parameters are addressed. In section 5, verification and comparisons between the three models are elaborately discussed. Finally, some conclusions on the work are presented.

2. TWO-STAGE-RISER FLUID CATALYTIC CRACKING (TSRFCC) TECHNOLOGY AND PILOT SETUP The concept of two-stage-riser fluid catalytic cracking (TSRFCC) was proposed by State Key Laboratory of Heavy Oil Processing (China), based on the fact that the residence time of oil vapor in a conventional riser reactor for FCC reaction is 2−3 s, and FCC catalysts deactivate rapidly during this time.31 Indeed, the conventional riser serves more as a conveyer than as a reactor in the second half of the riser, because of the fast reactions occurring in the entrance zone of the riser, and most of the catalyst has already deactivated in the first half of the riser.32 In TSRFCC technology, the conventional single riser reactor is divided into two stages, whose diameters and lengths are different from the conventional ones.31 The fresh feedstock is introduced into the first-stage riser and subjected to a certain degree of cracking reactions. The coked catalysts with low activity and selectivity are separated from the oil products. Then, the oil products continue cracking reactions over the regenerated catalysts with good activity and selectivity up to the final conversion in the second-stage riser.33 Each riser is equipped with a stripper and a regenerator. For saving cost, the two risers can also share a common disengager and regenerator. A series of TSRFCC derivative technologies have been developed to achieve various production goals.34 TSRFCC for Maximizing Propylene (TMP),35 one of the latest TSRFCC derivative technologies, was developed to meet the increasing demand of propylene in the worldwide market. Hitherto, the TSRFCC technologies have been applied in 15 commercial FCC units. In our laboratory, most of the TSRFCC experiments are conducted in a pilot-scale setup. Figure 2 shows the schematic diagram of the XTL-5 typed pilot FCC unit, which is similar to the commercial ones and includes a riser, a disengager, and a regenerator. The oil vapor from the top of the disengager goes into the condensing system, where the gas and liquid products can be collected. However, the two-stage riser experiments cannot be conducted simultaneously on the XTL-5 typed pilot FCC unit, because it is almost impossible to realize online separation corresponding to the reaction system for a feeding rate on the scale of 1−2 kg. Consequently, in our laboratory, the two-stage riser results are obtained by the simulated calculation of two-time independent riser experiments.17,35 To test the numerical models developed in this investigation, experimental study on the TMP process carried out by Liang36 with the pilot-scale unit was utilized here for verification and comparisons.

Figure 1. Conventional flow-reaction models for FCC riser reactors.

models), which was commonly accepted in dynamic modeling and optimization studies of FCC units,2,6 could not fully describe the real hydrodynamic behavior inside the riser as the back-mixing was always neglected in the assumption of the model. Therefore, developing new flow-reaction models with relatively high accuracy and short computation time is necessary to the modeling description of FCC riser reactors, especially in the case of dynamic modeling, optimization, and control of FCC units. To make a compromise between the complicated CFD model and the simple plug-flow model, hybrid approaches have emerged recently.20 In these approaches, CFD is employed only for hydrodynamic simulation, while the chemical phenomena are resolved in a custom-built compartmental model, such as the Equivalent Reactor Network (ERN) model. The ERN concept was introduced in the 1950s, but it started to flourish in the late 1990s.21,22 From then on, the approach has been applied to bubble columns,20,23 pack-bed reactors,24 industrial boilers,25 and biological reactors.26 Most recently, its use has been extended to gas turbine combustors21,27 and gasifiers.22,28 Generally, a CFD-based ERN model is developed through a procedure with three steps. First, a CFD model is run that accounts for the reactor hydrodynamics. Sequentially, an image analysis or algorithm is applied to the CFD-generated flow field, to create an ensemble of connected zones or compartments. Finally, each zone or compartment is considered as an ideal chemical reactor, and calculations are run with detailed kinetics. The superiority owned by the ERN model lies in its reasonably simplifying transport process and avoiding calculation divergences of CFD models, especially for situations where many types of reactants and complicated and nonlinear reaction paths are involved in the process.22 Using this methodology, many successful investigations were carried out for several types of reactors where model predictions were in very good agreement with the experimental setup results.20,22,25,27,29,30 Based on the above, the main objective of this investigation is to develop a CFD-based ERN model for a pilot FCC riser reactor and to make comparisons with the conventional complicated CFD model and the simple plug-flow model in terms of the model accuracy and computation time. The present paper is organized as follows: In section 2, an FCC process named TSRFCC is briefly described and the pilot-scale setup to carry out the process is presented. The threedimensional (3D) CFD simulations of the pilot-scale riser reactor for the FCC process then are detailed in section 3. In

3. COMPUTATIONAL FLUID DYNAMICS (CFD) MODEL 3.1. Conservation Equations of CFD Model. As is wellknown, two CFD approaches are widely utilized to simulate gas−solid flows: the Euler−Euler model and the Euler− Lagrange model. The base assumption of the former is that the gas and solid phases are interpenetrating continua. In the Euler−Euler model, the solid phase, similar to the gaseous phase, is also treated as a continuum media. As for the Euler− Lagrange model, trajectories of individual particles are tracked 8733

DOI: 10.1021/acs.iecr.5b02109 Ind. Eng. Chem. Res. 2015, 54, 8732−8742

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Industrial & Engineering Chemistry Research a=

1+

3.68N 100CTO

⎛ 1 ⎜ ⎜ 1 + 2.10Ah ⎝ 100CTO

⎞ ⎟(1 + 14.36C )−0.20 c ⎟ ⎠

(2)

where the values of deactivation constants N and Ah, which are related to the feedstock, are 0.10 and 22.64, respectively (data from Daqing AR), and Cc is the coke concentration on the catalysts (Cc = ycoke/CTO, where ycoke represents the mass fraction of coke in the 10 lumps). 3.3. Numerical Considerations. As shown in Figure 2b, the length and diameter of the pilot XTL-5 riser is 3.0 m and 0.016 m, respectively. Since CFD solve partial differential equations that model the fluid dynamics using appropriate numerical methods on a discrete grid in a computation domain, a 3-D mesh with only regularly structured hexahedral elements was constructed. This type of grid division algorithm should provide quick convergence and minimal numerical errors.40 The mass flow rates of the catalysts and the fresh feedstock were 17.6 kg/h and 1.6 kg/h, respectively, with a catalyst-to-oil ratio of 11.0.36 The inlet velocities were set for both the gas phase and the particle phase. The outlet of the riser reactor was set to atmospheric pressure. No-slip boundary conditions were set for the gaseous phase at the wall, while the partial-slip boundary conditions reported by Johnson and Jackson41 were employed for the particulate phase. The set of governing equations given in Table 1 were solved using the finite control volume technique in Ansys Fluent v14.42 The pressure−velocity coupling was obtained using the SIMPLEC algorithm. Simulations were run on a highperformance computing machine in State Key Laboratory of Heavy Oil Processing. The computation was carried out for a total of 30 s in real time to ensure that the simulation duration was long enough to establish the desired operating conditions. Time-averaged distributions of variables were computed covering a period of the last 15 s of the simulation time. 3.4. Analysis on CFD Flow Field. Figure 4 shows the gas velocity profiles in different cross-sectional planes, corresponding to different heights in the riser reactor. It can be seen from Figure 4 that the gas velocity is uniformly distributed at the inlet. Over the increase in riser height, a more and more obvious core-annulus distribution of gas velocity is recorded. This behavior was also observed in the experimental study of Patience and Chaouki.43 Figure 5 shows the axial section profiles of gas velocity at axial sections of x = 0 and y = 0. As illustrated in Figure 5, at the bottom of the riser reactor, the gas velocity is almost equally distributed on radial direction. Closer to the outlet, an obvious distribution of higher gas velocity at the riser center and lower gas velocity near the riser wall is presented. Such a velocity distribution of gaseous phase in the risers was also reported by Vandewalle et al.,44 Li et al.,45 and Lopes et al.46 From a 3D view, the gas velocity distribution in the riser reactor is just like the schematic illustrated in Figure 6a.

Figure 2. Schematic diagram of the XTL-5 typed pilot unit.

by solving their equations of motion, leading to huge computational requirements. In this investigation, the twofluid model (TFM) approach (i.e., the Euler−Euler model) derived from the kinetic theory of granular flows (KTGF) was utilized to simulate the complicated gas−solid flow in the riser reactor. The CFD model equations given in Table 1 represent the conservations of mass, momentum, and energy for the gas and solid phases. 3.2. Reaction Kinetics. A ten-lump kinetic model37 has been developed by our research group for the feedstock of Daqing atmospheric residue (AR) and the catalyst of LTB-2. The same feedstock and catalyst were used in Liang’s experiments.36 The properties of the feed oil and solid catalysts are listed in Table 2. Daqing AR, which is a representative paraffinic base feedstock in China, is a good FCC feedstock for producing propylene, because it has a large amount of longchain hydrocarbons.35,38 According to the kinetic model, the cracking gases were divided into 10 lumps: heavy oil (HO), diesel oil (DO), gasoline olefins (GO), gasoline aromatics (GA), gasoline saturates (GS), [butane+propane] (C3,4), butylene (C4=), propylene (C3=), dry gas (DG), and coke (CK). The reaction schemes are shown in Figure 3. The corresponding kinetic parameters are listed in Table S1 in the Supporting Information. Therein, reaction heat is calculated based on the molecular expansion method. Incidentally, in Section 5, the ten-lump kinetic model was also integrated into the ERN model and the plug-flow model. The reaction rate of lump i (source term ri in Table 1) is expressed as ri = a × CTO × K i × Yi

1

4. ERN MODEL The ERN model describes the reactor as a network of functional compartments or zones spatially localized. Essentially, six parameters for an equivalent multizone network to describe a multiphase reactor are required: (i) the number of compartments, n; (ii) the shape of each compartment, si; (iii) the volume of each compartment, Vi;

(1)

where CTO is the catalyst-to-oil ratio and Ki represents the reaction rate constants. The deactivation function a of the catalysts is expressed as39 8734

DOI: 10.1021/acs.iecr.5b02109 Ind. Eng. Chem. Res. 2015, 54, 8732−8742

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Industrial & Engineering Chemistry Research Table 1. CFD Model Equations name/description

equation

continuity equation (k = g, s)

∂ (αkρk ) + ∇·(αkρk vk) = 0 ∂t

momentum equation (k = g, s; l = s, g)

∂ (αkρk vk) + ∇·(αkρk vkvk) = − αk∇pg + ∇·τk + αkρk g + Klk(vl − vk) ∂t

energy equations

∂ (αg ρg CpgTg) + ∇·(αg ρg vg CpgTg) = ∇·[λ ·grad(Tg)] − ∂t

n

∑ rQ i ri + Q sg i=1

∂ (αsρs CpsTs) + ∇·(αsρs vsCpsTs) = − Q sg ∂t component continuity equation

∂ (αg ρg Yi ) + ∇·(αg ρg ug Yi ) = ∇·[εg ρg ·grad(Yi )] + ri ∂t

stress equations

τg = 2μg Sg + αg λg ∇·vg τs = [− ps + αsλs∇·vs] + 2μs Ss

deformation rate

1 1 αk[∇vk + (∇vk)T ] − αk∇·vkI 2 3

Sk =

solid phase pressure

ps = αsρs Θs[1 + 2αsg0(1 + e)]

radial distribution function

−1 ⎡ ⎛ α ⎞1/3⎤ g0 = ⎢1 − ⎜⎜ s ⎟⎟ ⎥ ⎢ ⎝ αs ,max ⎠ ⎥⎦ ⎣

solid phase viscosity

μs = μs ,kin + μs ,col

granular temperature equation

μs ,kin =

αsdsρs Θsπ ⎡ ⎤ 2 ⎢1 + (1 + e)(3e − 1)αsg0⎦⎥ 6(3 − e) ⎣ 5

μs ,col =

Θ 4 αsρ dsg π(1 + e) s 5 s 0 π

⎤ 3⎡ ∂ v Θ ) = τs ⎢ (ραs Θs) + ∇·(ρα s s s s ⎥ ⎦ 2 ⎣ ∂t s where vs − ∇·(κs∇Θs) − γ − 3β Θs

collisional energy dissipation

γ = αs 2ρs g0(1 − e 2)

interphase drag coefficient

12 Θs 3/2 ds π

⎛ α 2μ ⎞ ⎛ α ρ |u − us| ⎞ s g ⎟ + 1.75⎜ s g g ⎟⎟ for αg < 0.74 Ksg = 150⎜⎜ ⎜ 2⎟ ds ⎠ ⎝ ⎝ αg ds ⎠ Ksg =

3 ⎛ αsαg ρg |ug − us| ⎞ ⎟⎟wEMMS CD⎜⎜ ds 4 ⎝ ⎠

for αg > 0.74

where

CD =

24 [1 + 0.15(Res)0.687 ] for Res ≤ 1000 Res

C D = 0.44

(iv) the volume fraction of each phase in a compartment, αi; (v) the mass/volumetric flux rate between adjacent compartments, Fij; and (vi) the type of an elementary reactor (plug flow, perfect mixing reactor, dead volume, etc.) for representing a compartment, ti. Therefore, the Structure of an Equivalent Reactor Network (SERN) can be expressed, in the form of a mathematical function, as SERN = f (n , s , V , α , F , t )

for Res > 1000

Generally, two approaches are commonly utilized to determine these parameters. The two methods differentiate themselves from the other at the beginning step of model construction: one starts from the image analysis on flow field distribution in the reactor.20,22,28,47 Based on the analysis, the shape of each compartment and a limited number of zones the reactor required to be divided into can be preliminarily assumed. The two parameters (i.e., n and si) can also be adjusted in subsequent steps if the model designer seeks higher model accuracy. In most cases, the goal can be achieved by adjusting other yet undetermined parameters, such as the type of an elementary reactor for representing each compartment in a reactor network (ti). The other approach is based on certain

(3)

The six parameters are intercorrelated and indispensable to identify the structure of the equivalent reactor network. 8735

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Industrial & Engineering Chemistry Research Table 2. Properties of Daqing AR and Solid Catalysts parameter Daqing atmospheric residue (AR) density @ 20 °C viscosity @ 80 °C carbon residue SARA saturates aromatics resins asphaltenes solid catalysts particle density diameter

value 905.2 kg/m3 78.84 mm2/s 5.31 wt % 47.54 wt % 29.82 wt % 22.61 wt % 0.03 wt % 1700 kg/m3 76 μm

Figure 5. Axial section profiles of the gas velocity at x = 0 and y = 0.

Figure 3. Reaction network of the ten-lump kinetic model.

Figure 6. Division scheme of the pilot riser reactor.

values,30 giving rise to numerical difficulties and long computational time in the solution of ERN models. Although integrating those generated small volumes into adjacent large volumes can somehow decrease numerical difficulties,21 the problem remains still unresolved. Consequently, in this investigation, the six parameters were determined with a procedure as follows. First, the number of compartments and the shape of each compartment were assumed based on CFD flow field analysis presented in Section 3.4. Then, the volume and voidage of each compartment and the mass flux rate between adjacent compartments were calculated based on CFD simulations. Finally, the types of elementary reactors to represent all compartments were determined using the RTD approach. 4.1. Determination of n and si. In an ERN model, the number of zones (n) that a reactor has been divided into is closely related to the complexity of the multizone network structure and the computational time consumed to solve the model. Meanwhile, the shape of each zone, which can be either the same as or different from others, determines the interface of

Figure 4. Cross-sectional profiles of the gas velocity at different riser height.

criteria (e.g., velocity, voidage, concentration, fluid age, etc.).21,27,30 The criteria can be used to control the division of the reactor into separated homogeneous zones in which the gradients of the selected criteria are in a limited range. However, this approach may produce a large amount of compartments with rather small volumes on the condition that the tolerances of the selected criteria are set to be quite small 8736

DOI: 10.1021/acs.iecr.5b02109 Ind. Eng. Chem. Res. 2015, 54, 8732−8742

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Industrial & Engineering Chemistry Research adjacent zones, which have a great influence on calculating the mass flux rates between connected zones. Therefore, determination of n and si becomes essential to establishing an equivalent multizone network. Many researchers22,28 considered this to be the first step in the ERN model construction. SImilarly, in the present study, they were first determined based on the analysis on flow fields generated by CFD simulations in Section 3. It is well-known that the riser can always be divided into the acceleration section and fully developed section according to the multiphase hydrodynamic behaviors in the riser.4,48 Herein, the division method of the riser remained based on the above analysis on the CFD flow field of gaseous phase in Section 3.4. Preliminarily, the riser reactor was axially divided into four stages. The first stage represents the acceleration section and the rest represent the fully developed section. For the purpose of reproducing the nonuniform core−annulus distribution of gas velocity at the upper section of the riser reactor (as shown in Figures 3 and 4 and discussed in Section 3.4), each fully developed section was further divided into three sections radially. Consequently, the pilot riser reactor was divided into a total of 10 zones, which were labeled respectively in Figure 6b. The shape of each divided zone was also presented in the figure. 4.2. Calculation of Vi, αi, and Fij. As shown in Figure 6b, once the shape of each compartment is assumed, the interface of adjacent zones was determined. Thereby, the volume (Vi) and voidage (αi) of each zone and the mass flux rates (Fij) between connected zones can be calculated. As is known, while the volume and voidage are scalar parameters, the mass flux rate is a vector. Thus, the approaches for solving these parameters are different. However, both of them can be calculated by making use of the flow field information (e.g., volume fraction, velocity) provided by the CFD simulation.23,24 Figures 7a and

αi =

Fij =

c∈i

(5)

∑c ∈ i Fcell ·Acell ∑c ∈ i Acell

(6)

where Fij is the area-averaged gas mass flux rate from the ith zone to the jth zone; Acell is the area of the interface of the two adjacent zones, and Fcell is the mass flux rate flowing cross the interface. The calculated results of the volume and voidage of each compartment and mass flux rates between connected zones for the ERN model are listed in Tables S2 and S3, respectively, in the Supporting Information. The values of these parameters would be used in the subsequent section to help determine the last parameter, that is, the type of ideal reactor representing each zone. 4.3. Determination of ti. The choice of the type of an elementary reactor to represent each zone (ti) is essential to an ideal-reactor network during the development of the ERN model. The configuration of a reactor network can be determined in such a way that the ERN gas residence time distribution (RTD) matches the CFD one,25,28 because the RTD value reflects the degree of mixing and the deviation from ideal flow patterns. The RTD curve, in the form of F (t) or E (t), can be determined by a widely used method of stimulusresponse experiments. Most recently, many researchers49,50 have demonstrated the capability of CFD simulations to determine gas or solid RTD in circulated fluidized bed (CFB) risers through virtual tracer experiments. In this study, the CFD approach was employed because of the lack of experimental data for RTD curves. Indeed, it is a challenging task to carry out stimulus-response experiments in a FCC riser reactor under reactive conditions, even if it is a pilot-scale one. The RTD curve of the ERN model can be obtained with the Markov chain stochastic method.28,51 Elementary reactors in the network are treated as a series of states in the stochastic model. The probabilities of appearances of the tracked gas fluid elements in each state at time t + Δt are only related to their last states at time t. Transitions of their states are calculated with a discrete time step Δt. A transition matrix, pij, which can be derived from CFD flow-field analysis (as listed in Tables S2 and S3 calculated in Section 4.2), is used to give the transition probability from state i to state j for a tracked fluid element. As presented in Section 4.1, the XTL-5 pilot riser reactor was assumed to be divided into 10 compartments. If each compartment is represented by either a CSTR or a PFR, then, from a mathematical viewpoint, there exist 1024 (= 210) types of configurations of the ideal-reactor network for the pilot riser reactor. Among them, two extreme cases are the configuration of all CSTRs and that of all PFRs, as illustrated by Case A and Case B, respectively, in Figure 8. Obviously, it is still impractical to assess the other 1022 cases, which consist of both CSTRs and PFRs. However, we do not need to analyze every configuration since the hydrodynamics in each compartment can be estimated or known according to process

7b present the calculation schematic for the scalar variables and for the vector variable, respectively. As a brief description of the calculation method, the volume and the volume-averaged voidage of a specific zone can be calculated by

∑ Vcell

Vi

where αi is the volume-averaged gas volume fraction in each zone which is enclosed and labeled in Figure 6b; αcell and Vcell are the gas volume fraction and the volume of a CFD cell that belongs to the zone, respectively. When it comes to the gas mass flux rate between connected zones, the area-averaged value can be expressed as

Figure 7. Calculation schematic for (a) scalar variables and (b) the vector variables.

Vi =

∑c ∈ i αcellVcell

(4) 8737

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reproducing well the main fluid dynamic behaviors in the riser reactor. Therefore, the ERN Case C model for the pilot riser was adopted in the subsequent section for comparison studies of ERN model against conventional flow-reaction models, including the CFD model developed in Section 3 and the widely used plug-flow model.

5. RESULTS AND DISCUSSION Model equations listed in Table S4 in the Supporting Information for each CSTR and PFR in the ERN Case C model, coupled with the ten-lump kinetic model,37 were solved by using a sequential method with FORTRAN codes. Model equations for the plug-flow model are the same as that for PFRs in the ERN model. 5.1. Verification and Comparison. For verification and comparisons of the three models, Table 3 compares the predicted product yield distributions at the riser outlet against experimental data36 and the computational time consumed to implement each model. As can be seen from columns 2−4 in Table 3, the good agreement of predicted and measured values suggests that the CFD model simulates accurately the performance of the pilot riser reactor. Regarding to the ERN model, the predicted yields of all products approximate to the experimental data closely with relative errors of