Modeling Coke Distribution above the Freeboard of a FLUID

Article pubs.acs.org/IECR

Modeling Coke Distribution above the Freeboard of a FLUID COKING Reactor Christopher B. Solnordal,*,† Kevin J. Reid,‡ Larry P. Hackman,‡,§ Ray Cocco,⊥ and John Findlay⊥ †

CSIRO Mathematics Informatics and Statistics, Box 312, Clayton South, Victoria, 3169, Australia Syncrude Canada Ltd., 9421 17th Avenue, Edmonton, AB, Canada ⊥ Particulate Solid Research, Inc., 4201 W 36th Street, Suite 200, Chicago, Illinois 60632, United States ‡

ABSTRACT: In a FLUID COKING unit, reactor cyclone fouling by coke deposits can set the run length of the unit. Over time the coke deposits can grow and obstruct the cyclone which will limit throughput and lead to a shutdown. For this reason, producing a more uniform coke distribution pattern within the reactor horn chamber may lead to an increased interval between turnarounds. An existing pilot-scale experimental model of the coker reactor freeboard, horn chamber, and exit cyclones allows determination of coke distribution to the cyclones, but provides limited understanding of the underlying fluid dynamics within the system. In this work a two-phase computational fluid dynamics (CFD) model of this experimental rig was developed. Coke was modeled as an Eulerian stream of solid particles with monodisperse particle diameter. It was found that predicted coke distributions were sensitive to the choice of coke diameter, but a suitable choice gave good agreement with experimental observations. In the current work this value was 167 μm which was substantially higher than the value of the Sauter mean diameter of 139 μm. It was found that the CFD model could quantitatively predict coke distributions in the freeboard region of a FLUID COKING reactor experimental rig, while providing insight into the flow dynamics. When modeling the particle size distribution with a monodisperse particle diameter, comparison with experimental results is necessary to identify the coke particle diameter that leads to optimal model performance.

1. INTRODUCTION The FLUID COKING unit operation run length can be set by the build-up of coke deposits in the reactor cyclone separators.1,2 The coke deposits continue to grow, causing an uneven blockage on each cyclone and can ultimately block off one or more of the cyclones. Over time if the cyclones plug, unit throughput will be limited and the unit will be shut down for maintenance to remove the coke. One mechanism of reactor cyclone fouling in a FLUID COKING unit is hydrocarbon condensation followed by surface coking. The hydrocarbons evolved from the dense bed are in a pseudovapor−liquid equilibrium close to the dew point, and small temperature or compositional changes to the surroundings can force the mixture below its dew point and allow hydrocarbons to condense. The intent of the scouring coke line (SCL) in the reactor is to provide a superheat to the mixture entering the cyclones, thus allowing the gas to remain above the dew point and not condense. It is thought that an uneven distribution of coke to each cyclone may cause preferential build-up of coke in specific cyclones. A greater understanding of the gas and coke flow dynamics in and around the horn chamber could therefore shed light on this problem. Since cyclone fouling can set the run length for the FLUID COKING unit, producing a more uniform coke distribution pattern within the reactor horn chamber may lead to an increased interval between turnarounds. 1.1. FLUID COKING. The FLUID COKING process was developed by ExxonMobil Research and Engineering (EMRE) and is used to convert heavy residuum into lighter hydrocarbons.3 Syncrude Canada Ltd. (Syncrude) uses FLUID COKING for the processing of Athabasca oil sands in Alberta, Canada.4 The process utilizes two vessels: the reactor and burner. The reactor (Figure 1) © 2012 American Chemical Society

Figure 1. Simplified diagram of a fluid coking reactor.

contains a bed of coke particles fluidized by steam and hydrocarbon vapors. Hot coke enters the vessel in the freeboard and passes down the vessel, while bitumen atomized with steam enters lower down where it impacts and coats the hot coke particles. Endothermic cracking reactions take place on the surface of the coke so that Received: Revised: Accepted: Published: 15337

April 19, 2012 October 16, 2012 November 6, 2012 November 6, 2012 dx.doi.org/10.1021/ie3010176 | Ind. Eng. Chem. Res. 2012, 51, 15337−15350

Industrial & Engineering Chemistry Research

Article

Sundaresan14 discusses the limitations of modeling fluidized bed systems using meshes that are coarse relative to the particle size. When using a coarse mesh, structures smaller than the grid spacing will not be resolved. These mesoscale structures can have an effect on the macroscale behavior of the system. For example, it is well-known that it is difficult to predict the correct expansion for a fluidized bed of Geldart A particles. McKeen and Pugsley15 and Zimmermann and Taghipour16 applied a correction to the drag relationship to better predict the bed expansion of Geldart A particles. In both cases, the agreement with experimental data was improved through the use of the drag correction; however, the correction is problem dependent and was not predictive in either case. The use of a drag correction is equivalent to using a different particle diameter within the simulation. Since an ad-hoc modification of the drag law is not predictive and it is impractical to model a commercial scale fluidized bed using a mesh size that is small enough to allow for the prediction of mesoscale structures, Sundaresan and co-workers17−19 have developed a rational approach to include the subgrid effects of mesoscale structures on coarse grid simulations. Their approach involves conducting highly resolved simulations using a conventional drag law and kinetic theory of granular flow formulation on a relatively small control volume. The predicted flow field is then used to construct relationships for drag, solids pressure, and solids viscosity that include the subgrid effects. These relationships are then used on the coarse grid simulation of the commercial scale vessel. Another technique is being developed by the Chinese Academy of Sciences and is known as the energy minimization multi-scale (EMMS) approach. This approach is similar to that developed by Sundaresan’s group in that a modified drag relationship is formulated, but it differs in how the relationship is derived and in the fact that additional relationships for the solids viscosity and solids pressure are not developed. The energy minimization referred to in the name of this approach refers to the assumption that the component of the system energy required to suspend and transport solids within a fluidized bed will be minimized.20 This energy minimization concept is used to close a set of equations that represent mass and momentum balances for the dense and dilute phases of the fluidized bed.21 Models for both the filtered and the EMMS approach were not used in the current work, as the purpose was to better understand flow dynamics using commercial software and a simple approach to the problem. Instead the correct entrainment rate was predicted by modifying the median particle diameter used in the simulations. This was only possible in this study as there was a complete set of experiments with which to compare the model predictions. 1.3. Current Work. As a first step toward understanding the coke flow in the freeboard and horn chamber of a FLUID COKING reactor, Syncrude commissioned Particulate Solid Research, Inc. (PSRI) to develop a room temperature air/coke laboratory model of the process. The experimental model was used to investigate the coke distribution to each of four cyclones above the freeboard of a fluid bed. The effect of changes to air and coke flow rate, as well as scouring coke line exit geometry, was investigated. However, the rig only allowed for limited observation of the air and coke flow dynamics within the vessel, and could not be used to investigate operation at plant scale and operating conditions. To increase understanding of the experimental results, a computational fluid dynamics (CFD) model of the experimental rig was developed. The CFD approach allows detailed prediction

lighter hydrocarbons are created. These pass through the freeboard and into a scrubber, while the coke exits the vessel from below. The cold coke is transferred to the burner where it is partially combusted and heated. The hot coke from the burner then passes back into the reactor. As the lighter hydrocarbons exit through the freeboard they pass through a number of cyclones in parallel, to remove any entrained coke from the gas stream (Figure 1). The diplegs of the cyclones are submersed in the reactor dense bed to ensure the coke stays in the system. The entrained coke may come either from the fluid bed itself, or else from the stream of hot coke entering the reactor freeboard from the burner. Mallory et al. show the typical locations of build-up being at the cyclone entry, the gas outlet tube, and the dipleg outlet.2 It is these deposits that can lead to complete blockage of the cyclones and shut-down of the plant. To try and eliminate the coke deposits, a separate scouring coke transfer line is sometimes positioned within the horn chamber leading to the cyclones. Scouring coke is used in an attempt to prevent deposits from forming by scouring the surfaces of the cyclone with additional coke, and also by raising the temperature of the hydrocarbon stream to reduce the likelihood of physical condensation occurring. Difficulties arise, however, in delivering the scouring coke centrally to the horn and cyclones, due to the need for bends in the transfer line. It is often not known if the scouring coke is delivered evenly to each cyclone and whether this affects the likelihood of coke build-up. 1.2. Literature Review. Published literature investigating the operation of FLUID COKING units has concentrated on dynamics within the reactor and stripperthat is, the reaction regions in the upper and lower part of the fluid bed, respectively. Utilizing a pilot-scale cold model of the Syncrude unit, Knapper et al.5 investigated the overall flow patterns within the fluid bed, and identified limitations in the traditional core-annulus model of flow. Bi et al.6,7 investigated hydrodynamics and flooding behavior within the stripper of the same model and identified conditions under which flooding was likely to occur. Further work on the rig investigated the gas and solid mixing within the stripper region using both gas and solid tracers,8 and mixing in the reactor using a gas tracer.9 The intensive mixing in these regions was observed and studied under a variety of conditions, and the core-annular flow structure was observed to disappear near the bottom of the reactor section. House et al.10 investigated the steam/bitumen injection point in the coker and proposed a new nozzle technology for improved liquid/solid mixing. The role of coke properties in the FLUID COKING unit has been investigated using a number of approaches. Darabi11 investigated the agglomerability of the coke itself, and developed a mathematical model to predict the tendency to agglomerate. The model suggested that agglomeration was exacerbated when temperatures rose above 503 °C. Mallory et al.2 specifically studied the formation of coke accretions within the exit cyclones of the coker by deposition from the vapor phase. The results indicated that coke can be deposited directly from the vapor phase, and such deposition can be minimized by maintaining the vapor temperature at the same temperature as that of the fluid bed surface. CFD modeling techniques have been applied to many fluid bed systems at many scales. Detailed computational analysis of a two-dimensional fluid bed riser was performed by Benyahia,12 while more recently Schwarz and Lee13 presented a model of a fluid bed catalytic cracking unit that included the simulation of coke combustion and heat transfer in the bed. 15338

dx.doi.org/10.1021/ie3010176 | Ind. Eng. Chem. Res. 2012, 51, 15337−15350


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of the flow fields within the experimental rig, thus adding a new level of understanding to the experimental results. Furthermore, the experimental data allow some limited validation of the CFD model, giving confidence in its accuracy before extending it to plant scale and/or plant conditions. This paper presents the development and validation of a computational model of an experimental FLUID COKING reactor. Model performance is optimized against experimental data using base case conditions, and the predicted flow fields are presented. The ability of the optimized model to predict coke distributions in the rig following two geometrical changes is then investigated, and the results are discussed.

scouring coke transfer line (SCL). As the model operated at room temperature, the simulated “hot coke” line is referred to here as the “returned coke” line, or RCL. Slide valves were used to control the rate of coke extraction from the bed for each transfer line, while air conveyed the coke vertically back into the vessel. The solids flow to each cyclone was determined by closing a butterfly valve near the bottom of each cyclone dipleg and measuring the solids collection rate in the line over a short period of time. The butterfly valve was located just above the bed height in the dipleg, and closing the valve did not result in a change in the dipleg pressure as it was already sealed by the dipleg bed. The distribution of coke flowing through the horn chamber was measured by using an extraction probe. Solids mass fluxes were measured at 10 radial positions across the diameter of the horn chamber in two directions, each 90° apart. By integrating the resulting solids flux profiles over the area of the horn chamber, it was possible to calculate the total solids flow rate and compare the entrained flux with the rates collected by the four cyclones. The fluidizing and conveying air were supplied by a 2500 scfm rated blower at a supply pressure of approximately 12 psig. The air exiting the four cyclones was fed through a common line into a bag house rated at 14.15 m3/s (30000 scfm) before it was vented to atmosphere.

2. EXPERIMENTAL MODEL To investigate the coke distributions to the exit cyclones above the freeboard of the coker, a pilot-scale cold laboratory model of the system was built and operated by PSRI (Figure 2).

3. MATHEMATICAL MODEL The model was formulated using the commercial computational package ANSYS-CFX11, which solves the Reynolds averaged Navier−Stokes equations. The model was set up as a two phase Eulerian-Eulerian simulation, with coke modeled as a single Eulerian phase of particles flowing through a second phase of air. To calculate the flow field, the time-dependent Reynolds averaged Navier−Stokes eqs 1 and 2 are solved for each of the two Eulerian phases.22 ∂ (rαρα ) + ∇·(rαρα uα) = 0 ∂t

(1)

∂ (rαρα uα) + ∇·(rα(ρα uαuα)) = −rα∇pα ∂t + ∇·(rα(μα ∇uα + ρ u′αu′α)) + ρα g + Mα

(2)

In these equations the subscript α refers to each of the two phases (α = a for the continuous phase, air; α = c for the discrete particle phase, coke), while rα is the volume fraction of phase α. Additionally it is required that the volume fractions (r) sum to unity, and that each phase shares the same pressure field, p, so ra + rc = 1;

pa = pc = p

(3)

The presence of Reynolds stress terms on the right-hand side of eq 2 means that the above equations are not closed. To obtain values for the Reynolds stress terms and close the equation set a turbulence model is used. Furthermore, the term Mα on the righthand side of eq 2 represents interphase momentum transfer terms. These equations as written do not allow for the possibility of interphase mass transfer, which was not considered in the current work. All additional source terms of momentum (apart from buoyancy, denoted by the term ραg) are neglected. The most widely used turbulence model is the k-ε model of Launder and Spalding,23 which is based on the Boussinesq approximation. In the current work the k-ε model was implemented

Figure 2. Schematic diagram of pilot scale cold experimental rig.

The rig consisted of a 0.9144 m (3 ft) internal diameter vessel that simulated the reactor bed and freeboard. The bed contained plant coke supplied by Syncrude, and was fluidized with air. The upper region of the vessel narrowed to a 0.279 m (11 in) diameter horn chamber, leading to four cyclones equispaced around the horn and oriented to the north, south, east and west. Coke entered the vessel and horn chamber through two pipes: a 0.152 m (6 in) diameter pipe representing the hot coke transfer line, and a 0.076 m (3 in) diameter pipe that simulated the 15339



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for the air phase so that α = a. Reynolds stress terms are approximated by ρa u′αu′α = μTa ∇ua −

2 ρ kaδ 3 a

It was noted that, in very local regions of the flow, coke volume fraction became large so that the coke flow could no longer be considered disperse. Under these circumstances the Wen and Yu26 drag model is more appropriate. However, the regions of higher coke particle volume fraction were small and localized and comparison of results generated using each of the two models showed little difference, so the Schiller and Naumann model was used for this work. The solids particle collisions were implemented via a solids pressure force using the methodology of Gidaspow.27 This approach does not determine the solids pressure directly, but instead determines the solids pressure gradient. Noting that subscript “c” represents the coke phase, the solids pressure gradient, ∇pc, is given as shown in eq 11.

(4)

where μTa is the turbulent or eddy viscosity for air, obtained from

⎛k 2⎞ μTa = Cμρa ⎜ a ⎟ ⎝ εa ⎠

(5)

An effective viscosity can then be defined (μa,eff = μa + μTa). Equations 6 and 7 are solved to obtain ka and εa, which are the turbulent kinetic energy and turbulence energy dissipation rate, respectively, for the continuous phase. ⎛ ⎛ ⎛ μ ⎞ ⎞⎞ ∂ (raρa ka) + ∇·⎜⎜ra⎜⎜ρa uaka − ⎜μ + T a ⎟∇ka⎟⎟⎟⎟ ∂t σk ⎠ ⎠⎠ ⎝ ⎝ ⎝ = ra(Pa − ρa εa)

∇pc = G0 e K (rc − rc,max)∇rc

In eq 11, G0 is the reference elasticity modulus, K is the compaction modulus, and rc,max is the maximum packing parameter. There are no universal values of G0 and K. However, ANSYS22 recommend K = 1 and G0 in the range 20 to 600 Pa. Smaller values of G0 give greater sensitivity to packing, and since the regions of high coke packing were limited to small volumes of the flow field (basically at bends in the coke transfer lines and at the top of the horn) the smaller value of G0 = 20 Pa was chosen. A value of K = 1 was used, as recommended. The use of eq 11 did not have a significant impact on the flow calculation since the vast majority of the flow had very lean particle volume fraction. However, it was initially included because it was not known what the distribution of coke would be a priori, and then left in as it was expected to have minimal impact on the run time. 3.1. CFD Model Geometry and Mesh. The CFD model considered the flow from the upper mean surface of the fluidized bed through to the entrance to each cyclone, while the SCL and RCL were modeled approximately 10 diameters upstream of their final bends, as shown in Figure 3. The mesh of the model was generated using unstructured meshing software ANSYS Cad2Mesh. Mesh refinement was implemented in the horn chamber and entrances to the cyclones, while inflation layers were applied to all solid boundaries to enable modeling of boundary layers, where they existed. The total number of elements in the grid was approximately 296 000 for the base case geometry. Additional refinement around the SCL exit geometry modifications (runs E2 and E3) necessitated the use of up to 373 000 cells, which allowed detail of the longer SCL exit pipe (run E2) and the impact plate (run E3) to be resolved. 3.2. Boundary Conditions. The model geometry and inlet boundary conditions for all runs are summarized in the upper portion of Table 1. A series of six computational runs were conducted (M1a−d, M2, and M3), representing a total of three experimental conditions (E1, E2, and E3). Each experimental condition used a different exit geometry for the SCL (see “SCL geometry”, Table 1). The model had three inlets: the SCL and RCL inlets, as well as the bed surface (fluid bed upper surface). For the SCL and RCL inlets the normal velocity and volume fraction of each phase was directly specified. For modeling runs M1a−M1d the flow rates were taken from those used under the experimental base case conditions (experiment E1). Details of the mass flow rate of coke (ṁ c), particle volume fraction (rc), and the normal velocity of air and coke (Uin) are given in Table 1. Runs M2 and M3 modeled the experimental runs E2 and E3, respectively. These runs had

(6)

⎛ ⎛ ⎛ μ ⎞ ⎞⎞ ∂ (raρa εa) + ∇·⎜⎜ra⎜⎜ρa uaεa − ⎜μ + T a ⎟∇εa⎟⎟⎟⎟ ∂t σε ⎠ ⎠⎠ ⎝ ⎝ ⎝ ε = ra a (C1Pa − C2ρa εa) ka

(7)

Shear production, Pa, is defined in eq 8. Pa = μa,eff ∇ua ·(∇ua + (∇ua)T ) −

2 ∇· 3

ua(μa,eff ∇·ua + ρa ka)

(8)

Constants for the standard k−ε model are given by Launder and Sharma.24 The k−ε model is suitable for use in the continuous phase of the model. However, it is not suitable for the dispersed particle phase. Instead, an algebraic eddy viscosity model is employed for turbulence of the dispersed particles, as recommended by ANSYS.22 The dispersed phase zero equation model effectively sets the dispersed phase viscosity to be proportional to the continuous phase eddy viscosity, as in eq 9. ρ⎛μ ⎞ μTc = c ⎜ T a ⎟ ρ⎝ σ ⎠ (9) a

In eq 9 σ is a turbulent Prandtl number relating the dispersed phase eddy viscosity (μTc) to the continuous phase eddy viscosity (μTa). The interphase momentum transfer term, Mα, in eq 2, may consist of many components. ANSYS22 provides standard models for transfer due to drag, lift, virtual mass, wall lubrication, turbulent dispersion, and solid particle collisions. Most of these terms were not applicable to the current work and so were set to zero. However, the interphase drag, as well as solids pressure due to solids particle collisions were both specified. The commonly used drag coefficient for interphase drag between dispersed particles in a continuous fluid is that specified by Schiller and Naumann,25 which was implemented as shown in eq 10. The “maximum” function is employed to ensure values of CD remain sensible in the limits of high particle Reynolds number, Re. ⎛ 24 ⎞ C D = max⎜ (1 + 0.15Re 0.687), 0.44⎟ ⎝ Re ⎠

(11)

(10) 15340



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Figure 3. (a) Elevation view of CFD model, looking north. (b) Plan view of CFD model, showing the approach directions of the scouring coke and returned coke lines (SCL, RCL) and the entry ducts to the four cyclones (N, S, E, and W). The vertical RCL and SCL planes are also defined. (c) Isometric view of CFD model, looking south. The surface grid is shown.

slightly different air and coke flow rates (Table 1), most notably, the RCL coke rate for these runs was increased by 44% over that used in experiment E1. For the bed surface boundary, it was known that the air flow through this boundary came into the flow domain, while the net coke flow through the boundary lef t the flow domain. The coke flow in the lower freeboard and upper bed surface is shown in Figure 4. Thick arrows represent the flow of RCL coke with approximately 12% of this coke being entrained into the freeboard and out to the cyclones, and the remaining coke falling back to the bed. At the bed surface itself there are smaller rates of

coke entrained from the bed and re-entering the bed. However, the net flow of coke at the bed surface is back into the bed and, hence, out of the flow domain. To accommodate this behavior the flow rate of air through the boundary was directly specified at the bed surface (Uin,a,B, Table 1). The mass flow rate of coke leaving through this boundary was estimated based on the results of experiment E1. Assuming all the scouring coke was entrained into the horn chamber, the observed entrainment rate of 12% in combination with the RCL coke rate of ṁ c, R = 6.2 kg/s leads to a rate of coke entering the bed of 5.5 kg/s (hence ṁ c, B = −5.5 kg/s). 15341



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Table 1. Boundary Conditions, Entrainment, and Coke Distribution to Cyclones for Both Experiment and CFD Simulationsa run:

M1a

expt

SCL geometry RCL

dc (μm)

standard 6.2 0.016 12.7 1.6 0.024 9.0 −5.5 (approx 0.005) 1.1 −1.3 44−825 150

coke entrainment, E (%), from eq 12 coke distribution to % north each cyclone (from % west eq 14) % east % south

12 11 18 28 43

SCL

bed surface

a

E1

experiment or CFD:

ṁ c,R (kg/s) rc,R (−) Uin,R (m/s) ṁ c,S (kg/s) rc,S (−) Uin,S (m/s) ṁ c,B (kg/s) rc,B (−) Uin,a,B (m/s) Uin,c,B (m/s)

M1b

M1c

M1d

CFD CFD CFD CFD Geometry and Boundary Conditions

160

170

180

E2 expt

M2 CFD

deflection 8.9 0.022 13.6 1.7 0.026 9.1 −8.1 (approx 0.005) 1.1 −1.3 44−825 167

Observations (from Experiment) and Predictions (from CFD) 18 14 11 9 9 13 12 14 9 52 26 23 17 15 23 24 26 28 30 20 37 40 41 40 5

15 37 34 22 8

E3 expt

M3 CFD

impact plate 8.9 0.023 13.4 1.9 0.029 9.1 −8.2 (approx 0.005) 1.1 −1.3 44−825 167

8 19 11 44 27

15 41 19 34 6

Note: All runs use coke fluidized by air at temperature = 20 °C, pressure = 1 atm. Symbols are defined in the nomenclature.

there was no solids flow out, and no effect on the flow field. Preliminary modeling runs using a value of Uin,c,B that was smaller in magnitude than that specified here led to prediction of a buildup of coke in the bottom of the flow domain. The value reported in Table 1 however provided adequate drainage of coke through the bed surface. The model had a total of four outlets corresponding to the inlets to the four cyclones. Owing to the transient nature of the simulation it was necessary to model these outlets as constant pressure openings, where flow could momentarily be drawn into the flow domain despite the overall flow outward. Each outlet had a constant pressure of 0 Pag specified. All walls were smooth and were given a no slip boundary condition (velocities equal to zero at the wall). The logarithmic law of the wall was implemented in the near-wall region. Given the prediction of particle flow down the wall, a free slip condition would have been more appropriate. However, other than reducing the rate of drainage at the wall itself, this condition is thought to have little effect on the otherwise dilute flow that dominated within the vessel freeboard. The coke particle size used in the experiments, as well as that assumed for each modeling run, is shown in Table 1 as dc. 3.3. Gas and Coke Properties. All CFD runs used air as the fluidizing gas. The air was assumed to behave as an ideal gas with standard air density and viscosity at room temperature. The coke phase was plant coke supplied by Syncrude. The coke had a size distribution varying from 44 to 825 μm, with a Sauter mean diameter of 139 μm, a median diameter of 156 μm, and a volume/weight mean diameter of 168 μm. Particle size distributions were measured before and after the experiments, and no significant difference was found, indicating that the effects of attrition in the system were negligible for the time period of the experiments. The particle size distribution is shown in Figure 5. A single particle size was used for the coke phase of the simulation, the size being determined using a sensitivity analysis. The density of the coke was 1600 kg/m3 (100 lb/ft3).

Figure 4. Schematic diagram of coke flow to and from the bed surface.

Preliminary modeling runs suggested the average volume fraction of coke across the surface of the bed was approximately 0.005. These values were combined to give an estimated velocity for coke at the bed surface boundary of Uin,c,B = −1.3 m/s. This value ensured that any coke arriving at the boundary would leave the flow domain. If no coke was present at the boundary then 15342



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ṁ c,cyclones is equal to the sum of the coke flow to each of the individual cyclones: ṁ c,north, ṁ c,east, ṁ c,south, and ṁ c,west, so that ṁc,cyclones = ṁc,north + ṁc,east + ṁc,south + ṁc,west

(13)

The coke distribution to each cyclone (% north, % east, % south and % west) is then expressed as a percentage of the coke exiting through all cyclones as shown: %north = %east = Figure 5. Particle size distribution of coke as supplied by Syncrude.

%west =

ṁc, cyclones ṁc,,east

ṁc,cyclones

%south =

3.4. Solution Strategy. Solution of the preceding equations by analytical techniques is not possible. To solve the equations and obtain the flow field the commercial CFD code, ANSYSCFX 11.0 was used.22 ANSYS-CFX solves eqs 1−11 using the finite volume method on a colocated body-fitted grid. To avoid chequer-board oscillations in the pressure field, the Rhie and Chow28 interpolation procedure was used, as modified by Majumdar.29 A second order total variation diminishing discretization scheme30 was employed, as outlined by Barth and Jespersen,31 while a second order backward Euler discretization approach was adopted for transient terms. Further details of the solution procedure are given by ANSYS.22 A transient approach was taken to the solution, as the flow of coke through the system was anticipated to be unsteady. A time step of 0.001 s was specified and led to an RMS Courant number of approximately 1. The model was run on 3.2 GHz Intel Pentium 4 instruments using the Linux operating system. A simulation of 15 s real time, using a time step of 0.001 s and 12 CPUs in parallel, took 40 h. This time frame allowed the bed surface air to pass through to the cyclone entry ducts several times over.

ṁc,north

100

100

ṁc,south ṁc,cyclones ṁc,west ṁc,cyclones

100

100 (14)

4.1. Selection of Optimal Coke Diameter, dc. In the experimental rig, most if not all of the scouring coke was expected to travel up the horn and into the cyclone entry ducts (Figure 3). Coke from the RCL could either fall to the bed surface or else be entrained upward and into the horn. The amount of entrainment was expected to be affected by the single coke particle diameter chosen for the model. Figure 5 shows the coke particle size distribution, with a median diameter of dp50 = 156 μm, a Sauter mean diameter d3,2 = 139 μm, and a volume/weight mean diameter of d4,3 = 168 μm. A coke particle diameter of dc = 150 μm was initially chosen for simulation of experimental run E1, this value being in the vicinity of all three reported diameters. Work by Song32 suggested that using fluid coke with either a mean particle diameter of 133 or 145 μm, or FCC catalyst with mean particle diameter of 99 μm, made no discernible difference to bed dynamics in their cold laboratory coke model, and so the choice was thought to have little effect on the predicted bed dynamics. Comparison of the predicted (M1a) and experimental (E1) coke distributions to each cyclone is made in Table 1, and while the trend is captured (most coke to the south cyclone, 43% measured, cf. 37% predicted; least to the north, cyclone, 11% measured, cf. 13% predicted), the overall quantitative agreement is poor, with the predicted coke distribution being more uniform than measured. Table 1 also shows the amount of returned coke entrained into the horn chamber from the vessel. For CFD run M1a the amount is 18%, while for experiment E1 it is only 12%. It was thought that this additional returned coke entering the horn might have the effect of evening out the coke distribution to each cyclone. Therefore the assumed value of dc was increased to reduce the predicted degree of entrainment until it matched the experimentally observed value of 12%. Thus runs M1b, M1c, and M1d were performed (using dc = 160, 170, and 180 μm, respectively), and their predicted degree of entrainment is shown graphically in Figure 6a and compared to the experimental value from experiment E1. Predicted entrainment of returned coke into the horn chamber decreases as the assumed value of dc increases. The graph implies that using dc = 167 μm would produce a prediction of 12% returned coke entrainment, corresponding to the value observed in experimental run E1. Figure 6b shows the predicted distribution of coke to each cyclone and how these values vary with the assumed value of dc. The experimentally observed coke distribution is shown on the

4. RESULTS AND DISCUSSION Table 1 (lower section) lists results for both the experimental and computational runs performed. The base case conditions are represented by experimental run E1, and CFD runs M1a−M1d were performed to simulate this experiment. Each of these CFD runs used a different assumed value for the coke particle diameter, dc, from 150 to 180 μm. Experimental runs E2 and E3 represent similar flow conditions to E1, but with changes to the scouring coke transfer line exit geometry. These experiments were simulated by CFD runs M2 and M3, using the optimized coke particle diameter of dc = 167 μm. The experimental coke entrainment rate reported in Table 1 was calculated assuming all coke from the SCL was entrained. Thus the remaining coke reporting to the cyclones must come from the RCL or the bed itself since, as was explained in section 3.2, the net rate of coke flow through the bed surface was out of the flow domain. Hence, the coke entrainment normalized using the mass flow of coke entering through the RCL was equal to ṁc,cyclones − ṁc,S 100 E% = ṁc,R (12) In eq 12, E is the coke entrainment rate, ṁ c,cyclones is the combined mass flow rate of coke to all four cyclones, ṁ c,S is the mass flow rate of coke entering through the SCL, and ṁ c,R is the mass flow rate of coke entering through the RCL. 15343



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Figure 7 shows color contours representing the distribution of coke particle volume fraction, rc, in a vertical plane passing

Figure 6. (a) Predicted returned coke entrainment rate (from eq 12) as a function of assumed coke particle size. Note that the experimentally determined entrainment rate of 12% corresponds to a model particle size of 167 μm. (b) Predicted mass flow distribution to each cyclone (from eq 14), as a function of assumed coke particle size. The experimentally determined distribution is also shown for a particle size of 167 μm.

graph at dc = 167 μm, and agrees very well with the curves of predicted coke distribution at this location. Figure 6 also shows that as dc increases and the entrainment of coke from the vessel decreases, more coke is predicted to report to the south and east cyclones, and less to the north and west cyclones. Thus the increase in entrained coke creates a more uniform distribution of coke to the four cyclones. 4.2. Predicted Base-Case Flow Dynamics. Runs M1a− M1d were transient simulations; however, all flow patterns were fairly stable with only occasional slugging predicted in the SCL causing a temporary disruption to the coke flow in the horn. Specifically, in the 10 s of simulation there were three slugs predicted in the SCL, varying between one and four pipe diameters in length. However, these slugs passed rapidly through the horn chamber, after which the coke flow pattern returned to its quasi-steady state distribution. Therefore single images are used to represent the generally stable (but not strictly steady state) flow patterns in the vessel and horn.

Figure 7. Color contours showing predicted coke particle volume fraction for run M1c, in a vertical plane passing through the returned coke line. Red regions represent rc > 0.01.

through the RCL. Red regions represent rc > 0.01, while shades of blue represent the leaner regions of coke flow. The overall flow pattern shows coke exiting the RCL, deflecting off the impact plate and passing to the wall of the vessel. High near-wall values show that coke is predicted to run down the vessel walls. However, the presence of coke in the upper vessel indicates that some coke is entrained into the horn chamber. The entrained coke enters the horn chamber from one side, and so contributes to asymmetry in the coke distribution to the cyclones. The inset in Figure 7 indicates the coke particle volume fraction distribution around the RCL exit and impact plate. The model predicts that coke in the transfer line is centrifuged to the 15344



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Figure 8. Color contour plots in a series of horizontal slices through the model for run M1c, looking into the RCL plane as defined in Figure 3b. (a) Coke particle volume fraction, rc, and (b) vertical component of air velocity.

ascending flow surrounded by descending flow at the vessel walls, where the coke falls back to the bed surface. The upper vessel (above the impact plate) becomes a region of entirely ascending flow. The peak velocity (red/orange region) is shown to spiral around the vessel as the horn chamber is approached. Also of note is the segregation of ascending air and descending coke. The coke falls preferentially down the southwest wall, and consequently the air ascends more to the northeast. Thus the ascending core is not centered in the vessel. The coke distribution in the horn chamber and cyclone entry ducts is shown in greater detail in Figure 9, looking into the SCL plane. As was the case for coke entering from the RCL (Figure 7), the SCL exit has coke concentrated on one side. For this reason the distribution of coke in the horn chamber is skewed toward

side as the coke passes through the line bend, thus causing significant asymmetry of coke flow in the vessel. Similar behavior is predicted at the exit of the SCL (see Figure 9). Figure 8a shows the coke particle volume fraction in a series of horizontal slices through the vessel, horn, and cyclone entry ducts, using the same color scale as Figure 7. This view shows coke traveling preferentially down the southwest side of the vessel wall with some apparent rotational movement of coke in the upper vessel. Figure 8b shows the vertical component of air velocity in a similar fashion to the coke particle volume fraction in Figure 8a. The blue regions represent downward flow of gas, with other colors showing upward flow. The lower vessel (below the impact plate) indicates a core/annulus flow structure, with a central region of 15345



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were selected to test the CFD model’s ability to predict coke distributions to the cyclones using the optimized coke diameter, dc = 167 μm. The geometries investigated are shown in Figure 10. The deflection geometry (experiment E2, Table 1) attempted to redistribute the coke by deflecting the SCL exit toward the north−northwest. The impact-plate geometry (experiment E3, Table 1) positioned a small impact plate over the SCL exit. Modeling runs M2 and M3 simulated these experiments, leading to the predicted coke distributions shown in Figure 11 (see also Table 1). Figure 11 shows a scale drawing of the relative coke flows to each cyclone for the three geometries. The measured coke flow is shown as a white rectangle, while the CFD prediction is shown in black. For runs E1 and M1c (Figure 11a) the agreement between experiment and prediction is excellent, as E1 was the experiment used to optimize the model. Figure 11b shows the deflection geometry. The experimentally determined coke distribution indicates that the coke has been forced toward the north and west cyclones (in accordance with a geometric deflection toward the north−northwest), and 52% of the coke now reports to the north cyclone. The CFD results of run M2 also reflect this change in distribution. The results agree qualitatively with experiment, and good quantitative agreement is shown in the south and east cyclones. The model, however, predicts a relatively even split of coke between the north and west cyclones, rather than a larger amount of coke reporting to the north cyclone. For the impact-plate geometry (experiment E3, model run M3, Figure 11c) agreement in coke rates to the east and west cyclones appears reasonable, while that to the north and south cyclones is poor. However, there is a rotational agreement between the experiment and prediction. If the predicted coke distribution (run M3) is rotated 90° clockwise, then agreement becomes very good. On the experimental rig the impact plate is supported from the SCL by three vertical support plates that are radially aligned with the geometry (Figure 12b). These plates were ignored in the CFD model (Figure 12a) as it was assumed the flow would be entirely axial and radial in that region of the horn chamber (i.e., in the plane of the support plates). However, the flow entering the horn chamber from the vessel has a rotational component which

the southwest side, and an even distribution to the cyclones is not expected. Figure 9 also shows the entrained coke entering the horn chamber from the southwest side of the vessel. As shown in

Figure 9. Color contours showing predicted coke particle volume fraction, rc, for run M1c, looking into SCL plane as defined in Figure 3b. Red regions represent rc > 0.01.

Figure 8, this coke is rotating counter-clockwise toward the south and east. The majority of coke reports to the south (41%) and east (28%) cyclones, suggesting that it is the combination of the direction of the SCL, together with the rotational flow of entrained coke, which affects the coke distribution to the cyclones. 4.3. Change of Geometry. During the experimental campaign, several runs were performed that investigated the effect of changing the SCL exit geometry with the aim of redistributing the coke to the four cyclones. Two of these runs

Figure 10. SCL exit geometries for experiments E1, E2, and E3 (model runs M1, M2, and M3). Plan and elevation views in the SCL plane. 15346



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Figure 11. Graphical representation of the time-averaged coke distributions for the three SCL exit geometries. Locations of SCL and RCL also shown.

supported that finding. The inherent asymmetries in the plant geometry lead to asymmetrical distribution of coke entering the horn chamber. Work by Yang and Kuan33 indicates that pneumatically conveyed solid material negotiating a bend may remain unevenly distributed for up to 9 diameters downstream of the bend. Therefore the SCL and RCL should ideally travel vertically up the centerline of the vessel, but such an arrangement is impractical on the plant. Alternatively, a method of distributing coke evenly within each transfer line could be sought in order to achieve uniform distribution of coke to the cyclones. The current model has given insight into the flow dynamics within the freeboard and horn chamber of the reactor. However, even with a uniform distribution of coke, fouling will initially occur at a single site in a specific cyclone. Once this happens, the pressure distributionand hence the coke distributionwill become unbalanced. It would therefore be beneficial to extend the model for investigation of nonisothermal effects. Such a model could predict temperature distributions between the cyclones and identify cold spots where coke deposition is more likely. 4.5. Relevance to Other Fluid Bed Systems with Cyclones in Parallel. The system studied here is an example of an industrial fluidized bed with multiple parallel cyclones. Many other such systems occur in industry, and work by Grace34,35 has investigated and discussed these systems. Grace35 has shown that for all systems of parallel cyclones there are degrees of freedom allowing for multiple steady state solutions to exist. Furthermore, Grace35 argues that the solution of equal flow/equal pressure drop to/through each cyclone is likely to be an inherently unstable steady state solution. By this it is meant that slight perturbations in the flow conditions, which will inevitably occur in an industrial plant, will disrupt the symmetrical steady state solution and move it to a different, asymmetrical but more stable solution. If such an argument holds true then it is not possible, no matter how carefully a system is designed, to produce a uniform distribution of gas and particles to each cyclone. The results presented in the current work have demonstrated the difficulty in

Figure 12. (a) SCL impact-plate geometry as modeled in M3. (b) Actual geometry.

may be damped by the support plates. Thus the exclusion of the support plates could account for this rotational disagreement between experiment E3 and run M3. The predicted coke particle volume fraction for runs M2 and M3 is shown in Figure 13. As the geometry changes were all made at the exit to the SCL, these changes made little effect on the vessel flow dynamics. However, the increased rate of returned coke in experiments E2 and E3 meant that entrained coke was predicted to enter the horn chamber from the northwest. Figure 13a shows that the model predicts a strong deflection of scouring coke toward the north−northwest. In Figure 13b the impact plate has generated an annular distribution of coke in the horn chamber. However, downstream of the impact plate there is still asymmetry in the coke distribution, leading ultimately to a coke distribution favoring the north and east cyclones. 4.4. Relevance to a Full-Scale Plant. The experimental results had previously suggested that control of the coke distribution to each cyclone is difficult, and the CFD results have 15347



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Figure 13. Color contours showing predicted coke particle volume fraction, rc, for run M2 and M3, in a series of horizontal planes passing through the horn chamber and cyclone entry ducts. Red regions represent rc > 0.01.

back into the bed itself, and under these circumstances the optimized particle diameter may again change. However, such a model would be far more complex to implement, and not necessarily add to the qualitative observations of coke distribution shown in the current work.

producing a uniform solids distribution to four parallel cyclones and provide further evidence to support Grace’s theory. 4.6. Limitations of the Model. The main limitation of the CFD model in this work is its dependence on an Eulerian− Eulerian formulation. Using this method limits simulation of the coke phase to a at most a few discrete particle diameters, since the inclusion of each new particle size represents a new set of differential equations to be solved simultaneously. An alternative approach of using Lagrangian particle tracking techniques for the solid coke phase has its advantages, specifically, a detailed distribution of particle diameters can be incorporated into the simulation. However, the simulation of such tracks is best accomplished for steady state simulations with lean mixtures of solid particles. Another alternative is to employ the multiphase particle-in-cell technique which allows Lagrangian tracking of particle size distributions under nondilute conditions.36 While the experimental rig geometry was kept as simple as possible, it was necessary to have various bracing structures to hold the transfer lines, as well as the impact plates, in place. All these structures were omitted from the model geometry for simplicity. However, it has already been noted that the omission of the support struts for the impact plate in run M3 may have led to inaccuracies in the model prediction. Therefore a more accurate depiction of the geometry could improve the model performance. While the overall flow dynamics predictions did not vary greatly with the assumed single particle size of coke, the predicted entrainment rate and consequently the predicted distribution of coke to each cyclone did. Changing the single particle size has the effect of changing both the weight force on the particle as well as its drag. The specified density of the coke was measured and found to be within 2% of the supplier value of 1600 kg/m3. However, changes in the coke density would lead to changes in this force balance and hence would also modify the optimized single coke particle diameter used in the model. The model assumes equal pressure at the inlet to each cyclone, whereas in practice this in unlikely to be the case. There is a coupling between the gas and solids flow to each cyclone and the respective pressure loss through the cyclone. Thus ideally it would be preferable to model all four cyclones including diplegs

5. CONCLUSIONS Coke fouling in the exit cyclones of FLUID COKING reactors is believed to be affected by uneven distribution of coke to each cyclone. In this work a computational fluid dynamics model of the FLUID COKING reactor freeboard, horn chamber, returned coke, and scouring coke transfer lines, and four cyclone entries was developed and validated against an experimental air/coke model of the system. The model predicted that coke exiting both transfer lines was distributed unevenly due to upstream bends. The coke exiting the returned coke line was deflected to the side of the vessel, with 12−15% entrained into the horn chamber and the remainder running down the side of the vessel into the main coke bed. The scouring coke did not enter the horn symmetrically, and this contributed to the uneven distribution of coke to the cyclones. The computational model employed an Eulerian−Eulerian approach to solving the two-phase air/coke flow using a single particle diameter for the solid coke phase. It was found that varying coke diameter from 150 to 180 μm did not greatly affect the overall flow dynamics in the vessel freeboard. However the entrainment of coke from the freeboard, and consequent distribution of coke to each of the four cyclones, was sensitive to this parameter. A value of 167 μm led to optimal prediction of the coke distribution,. Using the optimized coke diameter, two additional experimental conditions were modeled, each one representing a change to the scouring coke transfer line exit geometry. In the case of the “deflection” geometry, the coke distribution was well predicted, while prediction using the “impact-plate” geometry was less satisfactory. The results of this work have shown that computational fluid dynamics models can be used to qualitatively predict coke distributions in complex air/coke systems. To maximize quantitative accuracy of predictions the assumed value of coke particle diameter should be calibrated against experimental data. 15348



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(2) Mallory, D. G.; Mehta, S. A.; Moore, R. G.; Richardson, S. The role of the vapour phase in fluid coker cyclone fouling: Part 1. Coke yields. Can. J. Chem. Eng. 2000, 78, 330−336. (3) Kunii, D.; Levenspiel, O. Fluidization Engineering, 2nd ed.; Butterworth-Heinmann: Oxford, 1991. (4) De Malherbe, R. Synthetic Crude from Oil Sands; VDI-Verlag: Dusseldorf, 1983; Vol. Reihe 3, Verfahrenstechnik, nr. 80., p iii, 103 p. (5) Knapper, B.; Berruti, F.; Grace, J. R.; Bi, H. T.; Lim, C. J. Hydrodynamic characterization of fluid bed cokers. In Circulating Fluidized Bed Technology VII; Grace, J. R., Zhu, J., Lasa, H. d., Eds.; Canadian Society of Chemical Engineers: Ottawa, Canada, 2002; pp 263−270. (6) Bi, H.; Cui, H.; Grace, J.; Kern, A.; Lim, C. J.; Rusnell, D.; Song, X.; McKnight, C. Flooding of gas−solids countercurrent flow in fluidized beds. Ind. Eng. Chem. Res. 2004, 43, 5611−5619. (7) Bi, H. T.; Grace, J. R.; Lim, C. J.; Rusnell, D.; Bulbuc, D.; McKnight, C. A. Hydrodynamics of the stripper section of fluid cokers. Can. J. Chem. Eng. 2005, 83, 161−168. (8) Cui, H. P.; Strabel, M.; Rusnell, D.; Bi, H. T.; Mansaray, K.; Grace, J. R.; Lim, C. J.; McKnight, C. A.; Bulbuc, D. Gas and solids mixing in a dynamically scaled fluid coker stripper. Chem. Eng. Sci. 2006, 61, 388− 396. (9) Song, X. Q.; Grace, J. R.; Bi, H.; Lim, C. J.; Chan, E.; Knapper, B.; McKnight, C. Gas mixing in the reactor section of fluid cokers. Ind. Eng. Chem. Res. 2005, 44, 6067−6074. (10) House, P. K.; Saberian, M.; Briens, C. L.; Berruti, F.; Chan, E. Injection of a liquid spray into a fluidized bed: Particle-liquid mixing and impact on fluid coker yields. Ind. Eng. Chem. Res. 2004, 43, 5663−5669. (11) Darabi, P. Agglomeration of Bitumen-Coated Coke Particles in Fluid Cokers. Int. J. Chem. React. Eng. 2010. (12) Benyahia, S.; Arastoopour, H.; Knowlton, T. M.; Massah, H. Simulation of particles and gas flow behavior in the riser section of a circulating fluidized bed using the kinetic theory approach for the particulate phase. Powder Technol. 2000, 112 (1−2), 24−33. (13) Schwarz, M. P.; Lee, J. Reactive CFD simulation of an FCC regenerator. Asia-Pac. J. Chem. Eng. 2007, 2, 347−354. (14) Sundaresan, S. Modeling the hydrodynamics of multiphase flow reactors: Current status and challenges. AIChE J. 2000, 46 (6), 1102− 1102. (15) McKeen, T.; Pugsley, T. Simulation and experimental validation of a freely bubbling bed of FCC catalyst. Powder Technol. 2003, 129 (1− 3), 139−152. (16) Zimmermann, S.; Taghipour, F. CFD modeling of hydrodynamics and reaction kinetics of FCC fluidized bed reactors. Ind. Eng. Chem. Res. 2005, 44, 9818−9827. (17) Agrawal, K.; Loezos, P. N.; Syamlal, M.; Sundaresan, S. The role of meso-scale structures in rapid gas-solid flows. J. Fluid Mech. 2001, 445, 151−185. (18) Andrews; Loezos, P. N.; Sundaresan, S. Coarse-grid simulation of gas-particle flows in vertical risers. Ind. Eng. Chem. Res. 2005, 44 (16), 6022−6037. (19) Igci, Y.; Andrews, A. T.; Sundaresan, S.; Pannala, S.; O’Brien, T. Filtered two-fluid models for fluidized gas-particle suspensions. AIChE J. 2008, 54 (6), 1431−1448. (20) Yang, N.; Wang, W.; Ge, W.; Wang, L.; Li, J. Simulation of heterogeneous structure in a circulating fluidized-bed riser by combining the two-fluid model with the EMMS approach. Ind. Eng. Chem. Res. 2004, 43 (18), 5548−5561. (21) Yang, N.; Wang, W.; Ge, W.; Li, J. CFD simulation of concurrentup gas−solid flow in circulating fluidized beds with structure-dependent drag coefficient. Chem. Eng. J. 2003, 96 (1−3), 71−80. (22) ANSYS ANSYS-CFX11.0 User Manual; ANSYS Inc.: Pittsburgh, USA, 2007. (23) Launder, B. E.; Spalding, D. B. The numerical computation of turbulent flows. Comput. Method. Appl. M. 1974, 3, 269−289. (24) Launder, D. E.; Sharma, B. T. Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1977, 1 (2), 131−138.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. § Formerly with Syncrude Canada Ltd., now retired.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the cooperation of CSIRO Australia, ExxonMobil Research and Engineering, Syncrude Canada Ltd, and Particulate Solid Research, Inc, for granting permission to publish this work.

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NOMENCLATURE c = compaction modulus CD = drag coefficient C1, C2, Cμ = coefficients in k-ε turbulence model d3,2 = Sauter (area/weight) mean diameter d4,3 = volume/weight mean diameter dc = coke particle diameter dp50 = screen size at which 50% of material passes E = entrainment rate defined in eq 12 % east = mass flow distribution to east cyclone, defined in eq 14 g = acceleration due to gravity G0 = reference elasticity modulus k = turbulent kinetic energy ṁ c = mass flow rate of coke ṁ c,X = mass flow rate of coke to/from location X, where X is one of the following: (cyclones) B (for bed surface); S (for SCL); R (for RCL); north, east, south, or west M = interphase momentum transfer terms % north = mass flow distribution to north cyclone, defined in eq 14 p = pressure P = shear production ∇p = solids pressure gradient rα = mass fraction of species α Re = Reynolds number % south = mass flow distribution to south cyclone, defined in eq 14 t = time u = velocity vector u′u′ = Reynolds stress tensor Uin = inlet velocity % west = mass flow distribution to west cyclone, defined in eq 14 δ = Kronecker delta function ε = turbulence energy dissipation rate μ = viscosity μT = turbulent viscosity μeff = effective viscosity ρ = density σk = turbulent Prandtl number for k σε = turbulent Prandtl number for ε σ = turbulent Prandtl number for dispersed phase eddy viscosity α = subscript indicating phase: α = a for air; α = c for coke

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REFERENCES

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Modeling Coke Distribution above the Freeboard of a FLUID

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