Optimal Design of Bypass Line for an Industrial-Scale 8-Leg Polyolefin

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Kinetics, Catalysis, and Reaction Engineering

Optimal design of bypass line for an industrial scale 8-leg polyolefin loop reactor to manage slurry dispersion using hydraulic and CFD simulations Yuehao Li, Jielin Yu, Rupesh Reddy, Abhijit Rao, Sameer Vijay, Erno Elovainio, Christof Wurnisch, and Krishnaswamy Nandakumar Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b00100 • Publication Date (Web): 11 Apr 2018 Downloaded from http://pubs.acs.org on April 11, 2018

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Industrial & Engineering Chemistry Research

Optimal design of bypass line for an industrial scale 8-leg polyolefin loop reactor to manage slurry dispersion using hydraulic and CFD simulations Yuehao Li1; Jielin Yu1; Rupesh Reddy1; Abhijit Rao1; Sameer Vijay2; Erno Elovainio2; Christof Wurnisch2; Krishnaswamy Nandakumar1* 1

Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana, 70802, USA 2

Borealis Polyolefin GmbH, St.-Peter-Strasse 25, 4021 Linz, Austria

* Corresponding author. Tel.: +01-225-578-2361; fax: +01-225-578-1476; Email Address: [email protected] Abstract

The pump power oscillation caused by slugs in slurry phase polymerization reactors limits production capacity. A bypass line connecting two locations of a loop reactor is suggested in patent literature to dissipate slugs. The optimal design of such bypass lines is studied through a hierarchy of modelling approaches applied to an industrial-scale reactor. In the design process, 1D hydraulic calculations guide the selection of optimal bypass pipe sizes to maximize slurry velocity. Two-dimensional CFD simulations provide data on slug dissipation processes. Threedimensional simulations guide selection of optimal installation angle to maximize solid intake into the bypass line. By combining these approaches, a bypass line connecting the 2nd and 6th legs by 6 inch pipes at 45-degree angle is found to be optimal for reactor considered here. Such designs achieves high slurry velocity in the bypass line while maximizing slug dissipation and the approach is applicable for other loop reactor configurations.

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1. Introduction The slurry-phase polymerization technology has achieved a great commercial success since its emergence in the last century. Billion pounds of olefin polymer products, i.e., polyethylene (PE) and polypropylene (PP), are being synthesized through this technique annually through the world1. The polymerization reactions are primarily carried out inside apparatuses named as “loop reactors” with the aid of designated catalysts. Depending on the production requirements, loop reactors are usually designed with 4, 6 or 8 vertical pipes which are arranged in a closed loop by 180o-bends or 90o-elbows. In polyolefin industry, they are usually termed as “4-leg”, “6-leg” or “8-leg” loop reactors. Loop reactors are generally operated with high pressure, i.e., in the range of 3 to 4 MPa, and moderate temperature to maintain suitable polymerization rates. Under such operating conditions, reactants including monomer and diluent solutions present in the liquid state while the generated polymer particles suspend inside the liquid medium. The resultant particle-fluid mixture is termed as “slurry”. In order to prevent the particles from settling down, the slurry is circulated around the loop reactor by one or multiple axial flow pumps with proper speeds2. The polymer particles expand their sizes slowly during the circulations; they ultimately grow to diameters ranging from 100 µm to 5 mm depending on the residence time of particles and the reaction kinetics3. One challenge confronted by the slurry-phase polymerization technique is the frequent formation of polymer slugs. As shown in our previous study, the 180o-bends or 90o-elbows used by the loop reactors can induce the solid segregation mechanism that stratify polymer particles into particulate ropes4. After the loop reactors are operated for sufficiently long time, these particulate ropes eventually develop into polymer slugs5. Once the polymer slugs are formed, the solid dispersion mechanism inside the loop reactors cannot dissipate them effectively6. When


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these slugs are circulated around the loop reactors by the axial flow pumps, they can cause the so-called “pump power swelling” phenomenon. As revealed in our previous CFD simulations6 as well as in a European Patent7, the pump power swelling phenomenon can lead to violent fluctuations in pump pressure output. For example, Fouarge et al. have reported that the standard deviation of the pump power consumption is in the order of 1 to 10 kW during the normal operation of a 6-leg loop reactor producing PE particles. Once slugs are detected inside the loop reactor, the associated pump power swelling can increase the standard deviations ten-folds7. If it is not controlled properly, the pump power can rapidly reach the safety threshold that the safety interlock system (SIS) has to shut down the entire production process automatically. Nowadays, commercial plants desire to operate loop reactors with long residence time and high solid concentrations due to economic motivations. Such a strategy can not only improve the product quality but also reduce the separation and recycling cost in the downstream process7. However, this strategy intensifies the slug formation and hence causes production loss frequently7, 8. The conflict of interests has raised a challenging problem for both academic and industrial communities: mitigation methods are in urgent need to handle the slug formation and to improve the slurry-phase polymerization process. One effective mitigation method is to equip loop reactors with bypass lines. A bypass line connects two locations of the loop reactor by an alternative route9-11. As claimed by Fouarge et al. in European Patent EP-A-14108439, the bypass line enhances the mixing in the transversal direction thus improves the homogeneity of circulating slurry inside loop reactors. Once a slug forms, the bypass line splits the slug and dilutes it with the slurry of low particle concentration. The efficiency of a bypass line relies on its design. The primary challenges of designing a bypass line are to optimize the connecting locations and the pipe size. These two parameters determine


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the slurry velocity in the bypass line, which is critical not only to the operations inside the bypass line but also to that of the main loop. On one hand, the pressure difference between the connecting locations provides the driving force of the slurry flow inside the bypass line. On the other hand, the pipe size and the fittings of the bypass line influence the frictional forces exerting on the slurry flow. Improper selections of these two parameters may clog the pipeline or interfere the reactor operations. For example, if the pressure difference between the connecting locations is insufficient, the slurry travels inside the bypass line with a slow speed. Particles settle down from the slurry and eventually clog the bypass line. In contrast, if the pressure force is very strong or the pipe size is excessively large, the slurry flow in the main loop tends to travel through the bypass line preferentially. As a result, the velocity in the main loop decreases noticeably, thus the production in the main loop is interfered by the bypass line. In addition, the velocity determines the efficiency of bypass line in dissipating slugs during process upset. Fouarge et al. have suggested that the residence time inside the bypass line shall be different from the time of slurry to travel through the main loop11. Although Fouarge et al. has provided several general guidelines, i.e., minimum velocity in bypass lines, minimum slope, and diameter ratio of bypass lines to main loop tubes11, to help the design of bypass lines, engineers still need more detailed information to gain confidence and to ensure the performance of the designs. Herein, we demonstrate our recent work in designing a bypass line for an 8-leg polyolefin loop reactor of industrial scale. The design procedure combines the hydraulic calculations along with computational fluid dynamics (CFD) simulations. In the 1st step of our design process, onedimensional (1D) hydraulic calculations were carried out to estimate the velocity and pressure profiles inside the main loop and the bypass lines at various design options. This step considered three types of connections and a wide range of pipe sizes of the bypass line. From the 1st step, an


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optimal pipe size with respect to each connection was obtained, and these sizes were adopted by the CFD simulations in the next step. In the 2nd step, two-dimensional (2D) simulations using the single-phase flow approximation were performed to understand the dissipation processes of slugs qualitatively. The optimal connection was determined by comparing the slug dissipation process of the three connection types. In the 3rd step, a bypass line using the optimal pipe size was installed on the loop reactor. The effect of the installation angle was studied by performing three-dimensional (3D) simulations using the Eulerian–Eulerian two–fluid model. Based on these steps, an optimal bypass line was designed for the 8-leg loop reactor. The systematic design process shown in this work ensures the efficiency of the bypass line; in addition, they can benefit the polyolefin industry to design bypass lines for other types of loop reactors. 2. Reactor geometry and operating parameters The geometry of the 8-leg loop reactor used in this study is referred from a loop reactor presented by US Patent 2004/0116625 1. The reactor consists of eight vertical legs, seven 180o bends and two 90o bends. The detailed geometry of the loop reactor has been presented in our previous work4, 6. The patent indicates that this reactor is capable of producing about 4.0×104 kg PE per hour 1. The inner diameter of all the pipes (D1) is 0.56 m (22” pipe). All the vertical legs are 60.40 m long except the 1st and 8th legs, which are 3.05 m longer than the others. The 180o bends have radius of 1.83 m while the 90o bends have radius of 1.22 meters. The overall length of this loop reactor is 534.6 m. The other pipe fittings in the main loop were not considered in this study. In this study, the bypass lines were implemented to the loop reactor through three types of connections. For the sake of easy maintenance, the bypass line is preferred to be installed close to ground, connecting two descending legs. Three types of connections are shown in Figure 1


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(A) ~ (C). In the 1st type of connection, the 2nd and 4th legs are connected by the bypass line as seen in Figure 1(A). The total length of the bypass line is assumed to be 15 meters. In the 2nd types, the bypass line connects the 2nd and 6th legs as shown in Figure 1(B). Similarly, the bypass line connects the 2nd and 8th legs in the 3rd type as seen in Figure 1(C). The total lengths of the bypass lines in these two connections are assumed as 30 m and 15 m, respectively. All these bypass lines are assumed to be connected to the main loop through two 120o angle valves, which are designed to shut down the pipeline during process upsets or maintenance. In addition, the bypass line is assumed to include one control valve, one gate valve and multiple bends or elbows depending on the connection type. A suggested flow diagram of the bypass line is shown in Figure 1 (D). The physical properties and operating parameters are listed in Table 1. In this study, the loop reactor has a relatively high solid content that the volume–averaged solid volume fraction over the entire reactor (Cv) is 0.23. The solid phase consists of polymer particles with an average diameter of 2.5 mm, which is a typical particle size of PP product. The liquid and solid phases are circulated around the reactor with a superficial velocity of 7.5 m/s. Table 1 Main properties and operation parameters ρl (kg/m3)

ρs (kg/m3)

µl (Pa s)

dp (m)

Cv

V1 (m/s)

417

900

5.54×10-5

2.5×10-3

0.23

7.5


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Figure 1: (A) the 1st connection option that connects the 2nd and 4th legs; (B) the 2nd connection option that connects the 2nd and 6th legs; (C) the 3rd connection option that connects the 2nd and 8th legs. The numbers shown in (A) ~ (C) indicate the indices of the vertical legs, and the arrows illustrate the directions of slurry flow. (D) Schematic diagram of the bypass line: (1), (8): 120oangle valves; (2), (3), (6), (7): elbows/bends; (4): control valve; (5): block valve. 3. Numerical methods 3.1 One-dimensional (1D) hydraulic calculations The objective of 1D hydraulic calculations is to find out the optimal pipe size which can attain maximum slurry velocity inside the bypass line. Achieving the maximum velocity in the bypass line offers at least two benefits. First, the maximum slurry velocity minimizes the settlement of


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particles thus prevents the bypass line from clogging. Second, since the bypass line is not designed for polymerization, maximum slurry velocity minimizes the residence time of the slurry in the bypass line, which helps to minimize the potential non-homogeneity of the polymer particles due to the bypass line. Three assumptions were made prior to the hydraulic calculations: (1) the velocity and pressure distributions are uniform on the cross sections of pipes perpendicular to the flow direction. Such an assumption is generally adopted by the hydraulic calculations using Bernoulli equations. (2) The bypass line has a negligible slope that the flow inside is driven only by the pressure difference between the entrance and exit of the bypass line. As suggested by Fouarge10, the bypass line is preferred to have a slope of about 9 degrees. The small inclination of the bypass is designated to drain the pipe during the process turn around rather than driving the flow. (3) The particle concentration inside the bypass line is the same as that inside the main loop. The validity of this assumption depends on how the bypass line is attached to the main loop, which is addressed in Section 3.3. In this work, the hydraulic calculations solved four Bernoulli equations and one mass balance equation to estimate the average pressure and velocity profiles of the main loop and the bypass line: − = ∙

(1)

(2)

(3)

(4)

∙

− = ∙ ∙

− = ∙ ∙

− = ∙ ∙

∙ = ∙ + ∙

(5)

where P1 is the discharge pressure of the axial flow pump; P2 is the pressure in the entrance of the bypass line; P3 is the pressure in the exit of the bypass line, and P4 is the suction pressure of


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the pump which is assumed to be 0 Pa; g = 9.81 m/s2 is the gravity constant; V1 is the circulating velocity of slurry in the main loop, which is 7.5 m/s; V2 is the velocity magnitude in those legs that are bypassed by the bypass line; Vb is the velocity magnitude in the bypass line; D1 is the diameter of the pipes in the main loop, which is 0.56 m (22” pipe); Db is the diameter of the bypass line pipes; is the average density of the slurry which is calculated by: = ∙ + ∙ 1 − !

(6)

The friction factors f1, f2 and fb are estimated through empirical correlations reported in literatures. As the main loop consists of vertical pipes and 180 degree bends primarily, the friction factors in the main loop, f1 and f2, are predicted by the well-known Newitt’s correlation12, which was developed based on hydraulic conveying of solids inside vertical pipes: " = #," + #," ∙

. 0.0037 *+ - ./ , 0 )

(7)

where i = 1 or 2 denotes the sections of the main loop that are not bypassed or bypassed, respectively; fw,i are the friction factors when pure liquid is flowing in the main loop at velocity of vi. For example, when V1 is 7.5 m/s, the corresponding values of fw,1 and f1 are 0.012 and 0.016, respectively. As the bypass line is almost horizontal, the friction factor in the bypass line fb is predicted by the classic Durand’s correlation13: = #, + #, ∙ 84.9 ∙ 4

5 * / 650

∙ 7.8 9

6.8

(8)

where fw,b is the friction factor when pure liquid is flowing in the bypass line at velocity of vb; CD is the drag coefficient that for settling of an individual particle at its terminal velocity in the quiescent, unbounded liquid. In this study, CD was estimated using the ‘CD-vs-Re’ approach proposed by Haider and Levenspiel14; the corresponding terminal settling velocity (vset) and Re


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were estimated by Heywood’s method15. These methods provide the values of CD and vset as 0.42 and 0.26 m/s, respectively. When vb is 6.5 m/s and Db is 0.15 m (6” pipe), the typical values of fw,b and fb are 0.015 and 0.019, respectively. The length parameters in Eq. (1)~(4), L12, L23, L34 and Lb , are the equivalent lengths of each sections in the main loop and the bypass line, including the pressure losses due to pipe fittings, valves, sudden expansions or contractions caused by the pipe entrance or exit. All the friction factors for the pipes, valves and fittings are cited from the standard values shown in the engineering handbook, Crane TP-41016. As the Eqs. (1) ~ (8) are fully coupled, a Matlab script was prepared to solve the equations simultaneously with respect to Db for each connection option. Various Db ranging from 0.0508 m (2”) to 0.305 m (12”) were investigated by the 1D hydraulic calculations. The optimal Db value was adopted by the two-dimension (2D) CFD model. 3.2 Two-dimensional (2D) CFD model using single-phase flow approximation The objective of the 2D CFD model is to understand the dissipating process of slugs qualitatively. A few assumptions were made in order to simplify the model: (1) the slurry flow is homogeneous throughout the entire loop reactor, thus the flow can be approximated as a single phase. (2) The fluctuations of the velocity and pressure profiles caused by the slug circulation, as shown in our previous study6, are neglected in the model. The slug velocity is assumed to be independent from its location and concentration. (3) The effect of solid dispersion mechanism in dissipating the slug is negligible, which has already been demonstrated in our previous study6. The slug dissipation process relies on the bypass line. (4) The slugs can be treated as passive tracers, which have identical properties as the slurry fluid. (5) The slurry is Newtonian and incompressible.


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Figure 2 Left: Geometry of the simplified loop reactor used by the 2D CFD model. The red color indicates the initial slug location in the loop reactor. Right: the detailed view of a typical bypass line which connects the 2nd and 8th legs of the main loop. The simplified geometry used by the 2D CFD model is shown in Figure 2, which shows the 3rd connection. The total lengths of the bypass lines in the 1st and 3rd types of connections are made the same as 15 m, and they are half of that in the 2nd type. As the 2D domain assumes an infinite length in the third direction, the corresponding diameter of the bypass line used in the 2D model (: ) is translated from the diameter calculated in the hydraulic calculations ( ) through : =

⁄ .


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The 2D model solves a set of single–phase Navier–Stokes equations along with a species transport equation. As the velocity field is assumed to be independent from the concentration and location of the slug, the flow can be regarded as in the steady state. Therefore, the continuity and momentum equations are simplified to: ∇ ∙ >=? = 0

>? ∙ ∇= >? = −∇@ + ∇ ∙ AB∇= >?! + ∇= >?!C D =

(9) (10)

>? is the velocity vector; is the average slurry density calculated by Eq. (6); @ is the in which =

pressure scalar; A is the average viscosity of the slurry estimated by the classic correlation proposed by D.G. Thomas17, as shown in Eq. (11): A = A ∙ B1 + 2.5 + 10.05 + 0.00273 exp16.6 !D

(11)

The slug dissipation process was analogized to the mixing process of a tracer with another miscible fluid. A transient species transport equation was adopted to describe the dissipation process: K

KL

+ >=? ∙ ∇ = ∇

(12)

>? is the velocity vector estimated by Eq. (9) and where is the mass concentration of the tracer, =

(10), and D is the diffusion coefficient which was set as 3×10-10 m2/s. As shown in Figure 2, the axial flow pump was not modeled; instead, the pump discharge was modeled as a velocity inlet while its suction was modeled as a pressure outlet. A constant velocity, V1, was specified to the velocity inlet. The outlet pressure was specified as 0 Pa. The turbulent flow was described by the standard k-ε model using the standard wall function. During the 2D simulations, the wall roughness heights of the main loop and the bypass line were adjusted to match the average velocities of the bypass line and the main loop as those estimated by the 1D hydraulic calculations. An inert tracer solution was initially patched inside the loop reactor, mimicking a large slug with an overall length of 132.20 m formed inside the loop


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reactor. The initial mass concentration of the slug was assigned as 0.6, corresponding to the maximum solid concentration that the slug can reach inside the loop reactor. The initial mass concentration of the slurry was assigned as 0. By using a user defined function (UDF), the averaged concentration in the outlet was specified to the inlet condition during the simulations, through which the slug was circulating inside the loop reactor. The mass concentration of the tracer was monitored with respect of time during the dissipation process. The geometry and the computational mesh of the 2D loop reactor with the bypass lines were generated in ANSYS Design Modeler and ANSYS Meshing, respectively. Each mesh contained approximately 100,000 elements. The typical grid sizes of the main loop and of the bypass line are 37 mm and 5.2 mm, respectively. The simulations were performed with 8 processors. A typical simulation usually took about two to three hours of wall time.

3.3 Three dimensional (3D) CFD model using the Eulerian-Eulerian two fluid method In the hydraulic calculations and the 2D CFD model, we made the assumption that the average solid volume fraction inside the main loop is the same as that inside the bypass line. However, the average solid volume fraction in the bypass line is generally lower than that in the main loop. As seen in Figure 1, a bypass line connects the two descending legs where the slurry flows downward. Due to the strong inertial effect of the solid phase, the polymer particle particles tend to remain its trajectory rather than making a turn and entering the bypass line. In addition, as discussed in our previous work4, the bends in a loop reactor induces the solid segregation mechanism, which results in the spatial distributions of solid phase and impacts the withdrawal rate of the bypass line as well. Therefore, a proper incline angle is important to the bypass to


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withdraw the solid phase. The objective of the 3D model is to find an optimal incline angle to install the bypass line so that it can withdraw as much solid phase as possible. As shown our previous papers4, 6, the full-scale simulations of the 8-leg loop reactor are very expensive and time consuming. Therefore, the 3D CFD model adopted a truncated geometry, which includes a U-bend and a bypass line as shown in Figure 3. The corresponding dimensions are listed in Table 2. The Eulerian-Eulerian two fluid model coupled with the kinetic theory of granular flow was adopted to describe the liquid-solid two-phase flow inside the loop reactor. Due to the dynamic solid segregation mechanism, the slurry flow in the U-bend is intrinsically dynamic. The model solves two sets of Navier-Stokes equations for the liquid and solid phases, respectively, using transient simulations. The governing equations are shown in Eqs. (13) ~ (18). M

MN M

MN M

MN

O ! + ∇ ∙ O P ! = 0

(13)

O ! + ∇ ∙ O P ! = 0

(14)

O P ! + ∇ ∙ O P P ! = −O ∇@ + ∇ ∙ Q + O R + S P − P !

Q = O A B∇P + ∇P !C D M

MN

(15) (16)

O P ! + ∇ ∙ O P P ! = −O ∇@ − ∇@ + ∇ ∙ Q + O R + S P − P !

Q = O A B∇P + ∇P !C D + O T − A ∇ ∙ P !U

(17) (18)

in which l and s denote the liquid and solid phases; α is the volume fraction; ρ is the density; P is the velocity vector; Q and Q are the stress-strain tensors of the liquid and solid phases; S is the

interphase momentum exchange coefficient described by the Gidaspow’s drag law18; g is the gravity vector; @ is the solid pressure; T is the bulk viscosity of the solid phase; A is the shear

viscosity of the solid phase; U is the identity matrix. The parameters related to the solid phase,


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i.e., @ , T and A , are estimated by the kinetic theory of granular flow. A transport equation is adopted to estimate the granular temperature V throughout the domain as well as time:

M

W O V ! + ∇ ∙ O P V !X = −@ U + Q !: ∇P + ∇ ∙ Z[\/ ∇V ] − ^\/ + _

MN

(19)

in which [\/ is the diffusion coefficient for the granular temperature, which is estimated by the

Symlal’s model19; ^\/ is the collisional dissipation of energy derived by Lun et al.20; _ is the energy exchange in particle velocity between the solid and liquid phases, which is estimated by Gidaspow’s model18. The detail expressions of these parameters can be found in our previous

papers4, 6.


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Figure 3 (A) The truncated geometry used by the 3D simulations. The arrows indicate the flow directions in the main loop. (B) The detail view of the bypass line. (C) Three different connecting angles. Table 2 Dimensions of the geometry used in the 3D simulations


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Dimensions of pipes & fittings

Notation

Size (m)

Length of the entrance pipe (belong to the 2nd leg)

L1

30.48

Length of the exit pipe (belong to the 3rd leg)

30.48

Radius of the 3rd bend

R1

1.83

Diameter of the pipes in the main loop

D1

0.56

Length of the entrance portion of the angle valve

L2

0.65

Diameter of the entrance section of the angle valve

D2

0.20

Length of the 1st section of the bypass line

L3

0.91

Radius of the 60o bend

R2

0.46

Length of the 2nd section of the bypass line

L4

1.83

Radius of the 90o bend

R3

0.46

Length of the 3rd section of the bypass line

L5

3.05

Diameter of the pipes of the bypass line

D3

0.15

In the simulations, the slurry was injected with a velocity of 7.5 m/s from the left inlet, which is shown by red color in Figure 3(A). The solid volume fractions were varied as 0.0490, 0.104, 0.166, 0.236, and 0.317, corresponding to the solid mass fraction as 0.10, 0.20, 0.30, 0.40 and 0.50. The exits of the U-bend and the bypass line were specified as pressure outlets. The pressure of the bypass line exit was tuned so as to attain the average velocities in the bypass line as those estimated by the 1D hydraulic calculations. The other boundaries were specified as the wall boundary conditions, of which the solid phase was specified with a specularity coefficient of 0.0001. In this study, the bypass line was connected to the main loop through three types of installation angles: 90o, 60o and 45o, of which the detail views are shown in Figure 3(C). During the simulations, the average solid volume fractions in the cross sections of the main loop and the bypass line were monitored with respect to time.


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The geometries and the computational meshes of the U bend with the bypass line were generated in ANSYS Design Modeler and ANSYS Meshing, respectively. Each mesh contained 899295 elements. A mesh dependence study was performed to compare the effect of grid resolution on the numerical predictions. The details are provided in Appendix A. The results of the mesh dependence study indicate that the numerical results can be regarded as being independent from grid resolution using the selected mesh. The simulations were performed with 48 processors in the HPC Facilities of Louisiana State University. The simulations were run for at least 50 s such that both the average solid volume fractions in the main loop and in the bypass line reached to the quasi-steady-state. A typical simulation took about 24 hours of wall time.

4. Results and Discussion 4.1 Validation of the 1D hydraulic calculation In this study, the accuracy of the 1D hydraulic calculations was validated by comparing the predicted pressure drop per unit length with the existing plant data, which has been reported in US Patent 7,033,54521. The corresponding pressure drop per unit length using Eq. (20) is 424. 4 Pa/m. The patent indicates that a typical pressure drop per unit length in a polyolefin loop reactor ranges between 345 ~ 513 Pa/m 21. `⁄`a = 1⁄2 !b !

(20)

in which `⁄`a is the pressure drop per unit length in the main loop; = 0.016 is the predicted

friction factor of the slurry flow in the main loop, which is estimated by Eq. (7); b = 7.5 m/s is the average velocity in the main loop; is the average slurry density estimated by Eq. (6). Since the predicted pressure drop from the hydraulic calculations is close to the one reported from

plant data, the numerical model is deemed to predict the velocity and pressure values inside the loop reactor and the bypass line with certain accuracy.


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4.2 Estimations of pressure and velocity profiles from hydraulic calculations

Figure 4 The relationship between the bypass line diameter (Db) with the pressure values of the pump discharge (P1), the bypass line entrance (P2), and the bypass line exit (P3) for (A) Connection 1: the leg 2 and 4 are connected by the bypass line; (B) Connection 2: the leg 2 and 6 are connected by the bypass line; (C) Connection 3: the leg 2 and 8 are connected by the bypass line.

Figure 5 The relationship between the bypass line diameter (Db) with the average velocity in the main loop that is bypassed by the bypass line (V2) and the average velocity in the bypass line (Vb) for (A) Connection 1: the leg 2 and 4 are connected by the bypass line; (B) Connection 2: the leg 2 and 6 are connected by the bypass line; (C) Connection 3: the leg 2 and 8 are connected by the bypass line.


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As mentioned in the Introduction, the connection locations of the bypass line determine the driving force of the slurry flow inside the bypass line while the bypass line diameter (Db) influences the frictional force. These two parameters simultaneously determine the velocities inside the main loop (V2) and the bypass line (Vb), which are critical for the operation of loop reactors. The hydraulic calculations predicted the profiles of pressure and velocity as a function of Db for these three types of connections, of which the results are shown in Figure 4 and 5, respectively. Comparing the pressure profiles shown in Figure 4, one may notice that Db has a marginal effect on the pressure distributions in Connection 1 but noticeable impacts on those in Connection 2 and 3. As shown in Figure 4(A), the pressure values of the pump discharge (P1), bypass line entrance (P2) and bypass line exit (P3) remain almost constant with respect to the investigated Db values in Connection 1. Since only two legs (which are the 3rd and 4th legs) of the main loop are bypassed in this connection, the pressure difference between the entrance (P2) and the exit (P3) of the bypass line is not significant to result in high velocity in the bypass line (Vb). For example, the pressure difference is around 6×104 Pa in Connection 1. The insufficient pressure difference cannot drive the slurry flow with high speed inside the bypass line. As seen in Figure 5(A), the maximum Vb is 3.80 m/s appearing in the Db size of 0.102 m (4” pipe). Consequently, the velocity in the main loop (V2) remains almost constant for the investigated range of Db. In addition, bypass line designs of different pipe sizes do not show significant difference in Vb. By increasing Db size from 0.102 m (4” pipe) to 0.152 m (6” pipe), Vb shows a marginal change to 3.59 m/s for Connection 1. When the bypass line connects the 2nd and 6th legs of the main loop through Connection 2, the bypass line can attain moderate pressure drop. As shown in Figure 4(B), P1, P2, and P3 decrease


20

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noticeably when Db increases; the pressure difference between P2 and P3 is around 1.1×105 Pa in this type of connection. Such pressure difference can withdraw a considerable amount of slurry from the main loop; as a result, V2 decreases noticeably, which is clearly shown in Figure 5(B). In this type of connection, the slurry can travel through the bypass line with a moderate speed. The maximum Vb that can be achieved by this connection option is approximately 6.5 m/s. When Connection 3 is adopted that the bypass line connects the 2nd and 8th legs, P1, P2 and P3 decrease drastically with the increase of Db as seen in Figure 4(C). The pressure drop between P2 and P3 can attain 1.5×105 Pa. Such significant pressure drop can drive the slurry in the bypass line with a high speed, which can even exceed that in the main loop as seen in Figure 5(C). On the other hand, the high speed in the bypass line can cause a remarkable reduction of flow rate in the main loop. The profile of V2 versus Db shown in Figure 5(C) indicates that the velocity in the main loop decreases quickly with Db. For example, if the bypass line with Db = 0.203 m (8” pipe) is installed, the velocity of the bypassed main loop decreases from 7.50 m/s to 6.42 m/s. As the slurry flow preferentially through the bypass line, the production process inside the main loop is interfered. 4.3 Selection of pipe size for the bypass line In order to attain optimal performance of the bypass line, the pipe size of the bypass line shall be chosen in such a way that the velocity in the bypass line can attain its maximum value. From the velocity profiles of the three types of connections shown in Figure 5, one may notice that the profiles of Vb versus Db exhibit parabolic shapes clearly for Connection 2 and 3. Interestingly, the maximum velocity of these three connections all appear in the Db range between 0.13 m (5” pipe) and 0.18 m (7” pipe). Since 6” pipe is commonly used in industry, such pipe size is adopted for the following analyses. For Connection 1, although 4” pipe provides maximum


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slurry velocity of 3.80 m/s, the resultant volumetric flow rate in the bypass line is small, which is 3.079*10-2 m3/s. Using the 6” pipe, the slurry velocity drops slightly to 3.59 m/s, but the volumetric flow doubles, which becomes 6.545*10-2 m3/s. Since increasing the flow rate benefits the splitting and diluting mechanisms, which is discussed later on, 6” pipe is also selected for Connection 1. When the 6” pipe with Db = 0.15 m is used by the bypass line, the maximum velocities can be attained by the Connection 1, 2 and 3 are 3.59 m/s, 6.50 m/s and 8.31 m/s, respectively. As explained in the Introduction, such velocities must satisfy several criterions in order to ensure proper operations inside both the main loop and the bypass line. The first criterion is that such velocities must be above the critical value bc so as to maintain the particles in the suspension state. If the velocity in the bypass line is less than bc , the slurry flow is governed by the so-called

“saltation flow” regime. In this regime, particles settle down from the slurry and may eventually clog the pipe line. There are several correlations in literature that can be used to estimate bc . Based on the correlation proposed by Durand13, bc can be estimated by:

bc = d e2 ⁄ − 1!

(21)

in which d is given graphically as a function of particle diameter, which ranges between 0.4 and

1.5; Db is the pipe diameter. For a bypass line with Db = 0.15 m (6” pipe), the bc estimated from

Durand’s expression is close to 2.80 m/s. According to the correlation proposed by Turian and Yuan22, bc can also be estimated by the expression: 67. 67.h bc = f2.411 7.g #, ⁄ − 1!

(22)

For a bypass line with Db = 0.15 m (6” pipe), the bc predicted by Turian and Yu’s is close to 3.24 m/s. Based on their experiences from industry operations, Fouarge et al. have suggested that bc

must be maintained above 3 m/s to avoid sedimentation and the possibly clogging 11. The values


22

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of bc from all these three sources are all less than the predicted velocities in the bypass line when the 6” pipes are used. Therefore, one may conclude that sedimentation can be avoided in all these three connections by using 6” pipes for the bypass line. Table 3 Operating parameters of the loop reactor by adopting 6” pipes for the bypass line

Connection V2 type (m/s)

Vb

1st

7.23

3.59

3.60%

132.28

15

2nd

7.01

6.50

6.50%

264.57

30

3rd

6.88

8.31

8.20%

399.90

15

(m/s)

Qbypass Qtotal

/ Bypassed Bypass line section of main length loop (m) (m)

Besides the first criterion, the second criterion is also important that the velocity in the bypass line cannot be excessively large that interferes with the production inside the main loop. From Figure 5(C), one may have already noticed that the high velocity in the bypass line can cause a noticeable drop of the velocity in the main loop. In US Patent 2007/0255019, Fouarge et al. suggest that the bypass line is preferred to carry a fraction of 1 to 15% of the total flow rate11. The estimated velocity in the bypassed section of the main loop V2 and the carried fraction of slurry in the bypass line are shown in Table 3. Such results indicate that the flow rates carried by the bypass line, which is denoted by Qbypass, are within 10% of the total flow rate Qtotal for all these three connections, which conforms to the guideline proposed by Fouarge. By using 6” pipes, the operations in the main loop are not interfered noticeably in none of these three connections. 4.4 The splitting and diluting mechanisms of the bypass line


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Once the optimal pipe size is selected, the next step is to find out the optimal connection type of the bypass line. Herein we want to first explain the splitting and diluting mechanisms of the bypass line with 2D simulation results. As seen in Figure 6(A), the bypass line acts the splitting mechanism when the slug is passing by the entrance of the bypass line. A portion of the slug is withdrawn from the main slug body and sent to the 6th leg. In the 6th leg, the withdrawn slug part mixes with the slurry having low solid volume fraction, which is named as the “clear slurry”. The splitting mechanism endures until the entire slug body leaves the entrance of the bypass line. Similarly, the diluting mechanism is performed when the slug body travels to the exit of the bypass line. As seen in Figure 6(B), the clear slurry is withdrawn from the 2nd leg by the bypass line and sent to the 6th leg, where the main slug body was diluted continuously by the clear slurry.

Figure 6 (A) The splitting mechanism that the bypass line withdraws a portion slug from the 2nd leg and sends to the 6th leg; (B)The diluting mechanism that the bypass line withdraws a clean slurry from the 2nd leg and mixes it with the slug in the 6th leg. The black arrows indicate the flow direction in the bypass line. The efficiency of the splitting and diluting mechanisms on dissipating the slug relies on two factors. On one hand, the bypass line shall attain high velocities thus large flow rates, thus it can split and dilute as much amount of slug as possible in each circulation cycle of the slug. On the


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other hand, there must be significant residence time difference between the slug parts in the main loop and that in the bypass line. The aim of different residence time is to avoid re-merge of the slug parts that have been split by the bypass line. Using the estimated velocity from the 1D hydraulic calculations shown in Table 3, we performed a time span analysis to estimate if the slug parts will reunite or not. Table 4 lists the spans of time that the slug parts stay in the main loop and bypass line. Starting at the time when the slug reaches the bypass line entrance (t = 0), tm1 standards for the time when the slug part in the main loop leaves the bypass line entrance; tm2 standards for the time when this slug part reaches the bypass line exit; tm3 standards for the time when this slug part leaves the bypass line exit. Similarly, tb1 standards for the time when the slug part in the bypass line reaches the bypass line exit; tb2 standards for the time when this slug part leaves the bypass line exit; tb3 standards for the time when this slug part in the bypass line travels back to the bypass line entrance. A sample calculation for these variables are shown in Appendix B. Table 4 Time span of the slug parts in the main loop and bypass line Connection Type

tm1 (s)

tm2 (s)

tm3 (s)

tb1 (s)

tb2 (s)

tb3 (s)

1st

18.3

18.3

36.6

4.2

22.5

57.8

2nd

18.3

37.7

56.0

4.6

22.9

40.6

3rd

18.3

58.1

76.4

1.8

20.1

19.8

Table 4 suggests that the 1st connection may result in reunion of slug parts. When the slug part in the main loop reaches the bypass line exit, the other slug part has not left the bypass line completely yet, as indicated by tm2 < tb2 < tm3. In the other words, the slug part split by the bypass line reunite with the head of the slug part in the main loop.


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Table 4 also suggests that the 2nd connection does not result in reunion of slug parts. As indicated by tm2 < tb2 < tm3, when the slug part in the main loop reaches the bypass line exit, the other slug part has not left the bypass line completely. Therefore, these two slug parts do not reunite in the bypass line exit. On the other hand, tb3 is much larger than tm1, indicating that when the slug part split by the bypass line circulates back to the bypass line entrance, the slug part in the main loop has already left this location. Therefore, the two slug parts do not reunite in the bypass line entrance either. In addition, Table 4 suggests that the 3rd connection type is vulnerable to reunion of slug parts. As tb3 is very close to tm1, it suggests that the slug part split by the bypass line is very likely to catch up with the tale of the slug part in the main loop. These two slug parts have high possibility to reunion in the bypass line entrance. 4.5 Comparison of slug dissipation process in the three types of connections In order to confirm whether these connection options can cause reunion of slug parts, we performed 2D CFD simulations. During the simulations, one monitor was put in the main loop to record the tracer concentration with respect to time. The resultant tracer concentration profiles mimics the slug dissipation process. Figure 7 shows the tracer concentration profiles of the three types of connections. As a slug circulate around the loop reactor, the monitor records the consecutive spikes. Each time the slug passes the monitor, the concentration profile shows the peak value. The interval between the peaks is about 75 s, which is the mean slug circulation time in the loop reactor. Since the slug is split and diluted by the bypass line, the peak value of each spike decreases gradually. On the other hand, the minimum values of the spike increases as the slug mixes with the clear slurry. When the peak value is close to the minimum value of the spike, the slug is regarded as been dissipated completely.


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By comparing the tracer concentration profiles shown in Figure 7, one may conclude that the 2nd type of connection is more efficient than the other two types. The slug is dissipated completely within 2000 s in the 2nd connection. On the other hand, the 1st connection has the lowest efficiency: the slug still contains high mass fraction after 2500 s.

Figure 7 Tracer concentration profiles of the three types of connections. (A) The 1st connection option that connects the 2nd and 4th legs; (B) The 2nd connection option that connects the 2nd and 6th legs; (C) The 3rd connection option that connects the 2nd and 8th legs. The small figures in the upper right corners show the shapes of the first few spikes in detail. The efficiencies of the bypass lines relies on the connection type. The hydraulic calculation has indicated that the 1st type connection has very low velocity in the bypass line. In other words, the corresponding bypass line can only split a small amount of slug or withdraw a small amount of clear slurry in each circulation cycle. In addition, the reunion of the slug parts were observed in the 2D CFD simulations: the simulations indicate that the 1st type connection withdraws slurry from the slug tail and reunites it in the slug head, which agrees with the time span analysis shown in Table 4. Figure 8(A) captures an instance when these slug parts reunites with each other. At this moment, the slug part in the main loop has left the bypass line entrance, and the slug head is passing through the bypass line exit. Meanwhile, the slurry that is withdrawn from the slug tail reaches the bypass line exit as well. These two parts merge at the bypass line exit,


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leading to the overlapped peaks shown in Figure 7(A). As the reunion occurs in the front tip of the slug, the overlapping area is shown in the front end of the large peak. The 2nd type of connection avoids the reunion of the slug parts. As the velocity in the bypass line is higher than that in the main loop, the slug part travelling through the bypass line reaches the bypass line exit in prior, resulting in a small peak of low concentrations shown in Figure 7(B). The snapshot shown in Figure 8(B) indicate that the slug part in the main loop has not reached the bypass line exit yet. As a result, the corresponding tracer concentration profile shown in Figure 7(B) demonstrates two separate spikes. The moderate velocity in the bypass line of Connection 2 avoids the reunion of the slug parts thus ensures a good performance of the bypass line. Although the hydraulic calculation suggests that the 3rd connection type attains a high velocity in the bypass line, the 2D CFD simulations indicate that this connection type also causes reunion of the slug parts. The slurry is withdrawn from the slug head and reunites with the slug tail, which confirms the time span analysis. As revealed in Figure 8(C), the head of the slug is travelling to the bypass line entrance at this time instance. The bypass line splits a slug part and sends to the bypass line exit. Meanwhile, the slug tail is still travelling through the bypass line exit. Therefore, these two parts reunite, leading to the overlapped peaks as seen in Figure 7(C). As the reunion occurs in the tail of the main slug, the overlapping area in Figure 7(C) appears in the rear end of the spike. The reunion of the slug parts undermines the efficiency of the 3rd type of connection. From this section, one may confirm that the 2nd type of connection that connects the 2nd and 6th legs of the loop reactor can avoid the reunion of the slug parts, thus it provides the highest efficiency among the three types of connections.


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Figure 8 Contour plots of the tracer mass concentration of the three connection options: (A) Connection 1 that connects the 2nd and 4th legs; (B) Connection 2 that connects the 2nd and 6th legs; (C) Connection 3 that connects the 2nd and 8th legs. 4.6 The effect of installation angle on the solid phase withdrawn rate Due to the complexity of slurry flow, one assumption was made in the hydraulic calculation and the 2D simulations to simplify the design process: the solid volume fraction in the bypass line is always the same as that in the main loop. However, such an assumption is not accurate as the solid phase has a higher inertia than the liquid phase. When the slurry travels to the bypass line entrance, it was hypothesized that solid particles tend to retain their trajectories rather than taking a sharp turn and entering the bypass line. As a result, the bypass line could not withdraw as much solid phase as those in the main loop. The withdrawn rate was hypothesized to be influenced critically by the installation angle of the bypass line. In order to verify the hypothesis, a series of 3D simulations were conducted with the U-bend system. The bypass line was installed to the U-bend through a 90-degree angle. The inlet of the U-bend was specified with a variety of solid phase volume fraction, mimicking slurry flow of varying solid concentration. The results shown in Figure 9 indicate that the average solid volume fraction in the bypass line is always lower than that in the main loop. The difference becomes


29


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more significant as the solid concentration in the slurry increases. These results verify the first hypothesis that the strong inertia of the solid phase leads to a lower solid content in the bypass line than that in the main loop.

Figure 9 Comparison of average solid volume fraction in the main loop (α90, main) and in the bypass line (α90, bypass) at different solid inlet concentrations. In the next step, the effect of installation angle on the solid phase withdrawn rate was studied by 3D simulations. The U-bend inlet was specified with a solid volume fraction of 0.317, and the bypass line was installed to the main loop with 90-degree, 60-degree and 45-degree inclining angles, respectively. Figure 10 shows the average solid volume fraction and solid phase velocity in the bypass line with respect to the installation angle. The figure indicates that both the solid volume fraction and solid phase velocity in the bypass line decrease almost linearly as the installation angle increase, which confirms the importance of the installation angle. Converting these two parameters into the mass flow rate of the solid phase, Table 5 suggests that using 45 degree installation angle offers 34% improvement of the solid withdrawal rate than using the 90 degree one. A smaller angle than 45 degree is more favorable in terms of further increasing the


30

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solid phase withdrawn rate; on the other hand, it will add the complexity in installation and maintenance. As revealed in Figure 10, the average solid volume fraction in the bypass line is still lower than that in the main loop even though a 45 degree installation angle is used. This trend is in accordance with intuitive expectation as heavier particles do not change their momentum direction readily. As a result, the bypass line would be less efficient than what the 2D CFD simulations predicted. Such evidence does not refute the validity of the 1D hydraulic calculations and 2D CFD simulations, but it suggests that the bypass line requires longer time to dissipate a slug as it only withdraws less amount of solid phase in each cycle. On the other hand, it emphasizes the importance of designing an optimal bypass line using the methods presented in our work. Table 5 Summary of the predicted bypass line performance with respect to the installation angles Installation angle (degree)

Average solid Average slurry Average solid Improvement pf solid withdrawal volume fraction in velocity in the withdrawal rate compare to the bypass line bypass line rate 90 degree (%) (m/s) (m3/s)

45

0.231

7.809

0.0319

34.0

60

0.223

7.297

0.0288

20.9

90

0.213

6.335

0.0238

-


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Figure 10 The relationship between installation angle and average solid volume fraction in the bypass line (αs, bypass) as well as average solid velocity in the bypass line (Vs, bypass). 5. Conclusions The pump swelling phenomenon resulted from the slug circulation has become a bottleneck for the current polyolefin industry. One mitigation used by industry is to install a bypass line to connect two locations of the loop reactor by an alternative route so as to split and dilute the formed slugs. However, the design process encounters challenges due to the complicity of the slurry flow; there are limited discussions in literatures about how the design and optimization of a bypass line are achieved. In this work, we present our systematic approaches to design an optimal bypass line for a polyolefin 8-leg loop reactor. The design procedures combined the 1D hydraulic calculations, the 2D CFD simulations using the single-phase approximation and the 3D CFD simulations using the Eulerian-Eulerian two fluid method. The 1D hydraulic calculations solved a set of Bernoulli equations with a mass balance equation to estimate the pressure and velocity profiles of the main loop and the bypass line with respect to a wide range of pipe sizes used by three types of bypass line connections. The results indicated that all three types of connections could attain maximum velocity in the


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bypass line by using a 6” pipe. This optimal pipe size was then used by the 2D CFD simulations to compare the slug dissipation processes of the three types of bypass line connections qualitatively. The 2D model adopted the single-phase approximation and mimicked slugs by inert tracers. The resultant slug dissipation profiles indicated that the 2nd type of connection that connects the 2nd and 6th legs of the loop reactor offered the highest efficiency in splitting and diluting the slugs. The reunion of slug parts were observed in the 1st and 3rd types of connection, which undermines the efficiencies of the bypass lines. In the next step, a series of 3D simulations using the Eulerian-Eulerian two fluid model was conducted to compare three typical installation angles. The results indicate that installing the bypass line with a 45 degree inclining angle provides the highest withdrawn rate of solid phase into the bypass line. Overall, an optimal bypass line design that connects the 2nd and 6th legs of the loop reactor using 6 inch pipe was recommended to the 8-leg polyolefin loop reactor used in this study. In addition, the loop reactor is suggested to be installed using a 45-degree installation angle so as to ensure effective withdrawn rate of the solid phase. The presented design processes can also be applied to design bypass lines for other types of loop reactors. Acknowledgements The authors are grateful for the financial support from Borealis Group. The supercomputing facility at Louisiana State University and Louisiana Optical Network Initiative (LONI) are acknowledged for their technical support. Supporting Information Mesh dependence study in Appendix A; Sample Calculation of Time Span Analysis in Appendix B.


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References (1) Hottovy, J.; Zellers, D.; Verser, D.; Burns, D. Loop reactor apparatus and polymerization processes with multiple feed points for olefins and catalysts. US 2004/0116625 A1, 2004. (2) Hottovy, J. D.; Zellers, D. A.; Franklin, R. K. Pumping apparatus for slurry polymerization in loop reactors. US 7,014,821, 2006. (3) Hutchinson, R. A. Modelling of particle growth in heterogeneous catalyzed olefin polymerization. Ph.D. Thesis, University of Wisconsin-Madison, 1990. (4) Li, Y.; Ma, Y.; Reddy, R. K.; Vijay, S.; Elovainio, E.; Wurnitsch, C.; Nandakumar, K., CFD investigations of particle segregation and dispersion mechanisms inside a polyolefin 8-leg loop reactor of industrial scale. Powder Technol. 2015, 284, 95-111. (5) Marissal, D. Slurry phase polymerisation process. US 2012/0208960 A1, 2012. (6) Li, Y.; Yu, J.; Reddy, R. K.; Vijay, S.; Elovainio, E.; Wurnitsch, C.; Nandakumar, K., Computational study on the effect of slug dynamics on the operation of a polyolefin 8-leg loop reactor of industrial scale. Powder Technol. 2017, 319, 452-462. (7) Fouarge, L.; Lewalle, A.; Van, D. B. F.; Van, D. A. M. Swell control in slurry loop reactor. EP 1 660 230 B1, 2005. (8) Marissal, D.; Walworth, B., Process for manufacturing an olefin polymer composition. In U.S. Pat. No. 6586537: 2003. (9) Fouarge, L.; Lewalle, A. Slurry loop polyolefin reactor. EP 1410843 A1, 2004. (10) Fouarge, L.; Lewalle, A. Slurry loop polyolefin reactor. US 0094835 A1, 2006. (11) Fouarge, L.; Davidts, S. Polymerization reactors with a by-pass line. US 0255019 A1, 2007. (12) Newitt, D. M.; Richardson, J. F., Hydraulic conveying of solids in vertical pipes. Trans. Inst. Chem. Eng. 1961, (39), 93. (13) Durand, R. In The hydraulic transport of coal and solid materials in pipes, Proc. Colloquium on the Hydraulic Transport of Coal, 1953; National Coal Board, London: 1953; pp 39-52. (14) Haider, A.; Levenspiel, O., Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol. 1989, 58, (1), 63-70. (15) Heywood, H., Calculation of particle terminal velocities. J. Imp. Coll. Chem. Eng. Soc. 1948, 140-257. (16) Flow of Fluids Through Valves, Fittings, and Pipe. Long Beach, CA: Crane Co.: 2009. (17) Thomas, D. G., Transport characteristics of suspension: VIII. A note on the viscosity of Newtonian suspensions of uniform spherical particles. Journal of Colloid Science 1965, 20, (3), 267-277. (18) Gidaspow, D.; Bezburuah, R.; Ding, J., Hydrodynamics of Circulating Fluidized Beds, Kinetic Theory Approach. In In Fluidization VII, Proceedings of the 7th Engineering Foundation Conference on Fluidization, 1992; pp 75-82. (19) Syamlal, M.; Rogers, W.; O'Brien, T. J., MFIX Documentation: Volume 1. Theory Guide. National Technical Information Services, Springfield, VA: 1993. (20) Lun, C. K. K.; Savage, S. B.; Jeffrey, D. J.; Chepurniy, N., Kinetic Theories for Granular Flow - Inelastic Particles in Couette-Flow and Slightly Inelastic Particles in a General Flowfield. J. Fluid Mech. 1984, 140, (Mar), 223-256. (21) Kufeld, S. E.; Reid, T. A.; Tait, J. H.; Burns, D. H.; Verser, D. W.; Hensley, H. D.; Przelomski, D. J.; Cymbaluk, T. H.; Franklin, R. K.; Perez, E. P. Slurry polymerization reactor having large length/diameter ratio. US 7033545 B2, 2006.


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(22) Turian, R. M.; Yuan, T. F., Flow of Slurries in Pipelines. Aiche J. 1977, 23, (3), 232-243.


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For Table of Contents Only.


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Optimal Design of Bypass Line for an Industrial-Scale 8-Leg Polyolefin

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