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Three-dimensional CFD model of the temperature eld for a pilot-plant tubular looppolymerization reactor

Xi Gao, De-Pan Shi, Xi-Zhong Chen, Zheng-Hong Luo

Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China

a b s t r a c ta r t i c l e i n f o

Article history:

Received 17 April 2010

Received in revised form 20 June 2010Accepted 27 June 2010

Available online 6 July 2010

Keywords:

Polyolen

Loop reactor

CFD

Temperatureeld

A three-dimensional (3D) computational uid dynamics (CFD) model, using an EulerianEulerian two-uid

model which incorporates the kinetic theory of granular ow, the energy balance and heat transfer

equations, was developed to describe the steady-state liquidsolid two-phase ow in a loop propylene

polymerization reactor composing of loop and axial ow pump. The entire temperature eld in the reactor

was calculated by the model. The predicted pressure gradient data were found to agree well with the

classical calculated data. Furthermore, the model was used to investigate the inuences of the circulation

ow velocity, the slurry concentration, the solid particle size and the cool water temperature on the

temperature eld in the reactor. The simulation results showed that the whole loop can be divided into four

sections. In addition, the simulation results also showed that the continuous stirred-tank reactor (CSTR)

assumption is invalid for the entireeld in the loop reactor.

2010 Elsevier B.V. All rights reserved.

1. Introduction

Polypropylene is one of the most widespread polymers, and canbeproduced by various technologies including Hypol technology, Unipol

technology, Spheripol technology,etc.[1]. The last one is certainly the

most important at present. The key part of this technology consists of

a loop reactor and a uidized-bed reactor (FBR) [2]. In the loop

reactor, polymerization takes place in a liquid phase and the polymer

matrix is produced as a solid suspension in the liquid stream[35].

Accordingly, the polymerization system is considered a mixture of a

liquid phase (monomer and hydrogen) and a solid phase (polymer

and catalyst), namely a liquidsolid two-phase system. In order to use

the loop reactor more effectively, there is a need to obtain a

fundamental understanding of the liquidsolid two-phase ow

behaviors of such a system, including the temperature eld.

Moreover, the polymerization is a highly exothermic reaction [46],

and it may lead to the appearance of hot spots in the two-phase

system if the heat of polymerization can not be efciently removed

from the reactor[5,6]. These hot spots can inuence the reactor safety

and polymer properties [7,8]. Therefore, it is imperative that good

ow and heat transfer qualities are achieved to ensure a good liquid

solid contact and uniformity of temperature in the loop reactor. For

these reasons, computational uid dynamics (CFD) is becoming more

and more an engineering tool to predict ow and temperature elds

in various types of apparatus on an industrial scale [911].

Furthermore, the CFD is an emerging technique and holds great

potential in providing detailed information of the complex uid

dynamics[11

14].In general, two different categories of CFD models are used: the

Lagrangian and the Eulerian models[1117]. The Lagrangian model

solves equations of motion for each particle taking into account

particleparticle collisions and the forces acting on the particle,

whereas in the EulerianEulerian model, the two phases are both

considered continuous and fully interpenetrating. Up to now,

considerable attention was devoted to the application of CFD to

different reactors [1123]. However, most of past studies were

concentrated on the application of CFD to gassolid uidized-bed

reactors (FBRs) based on the Eulerian model[15,16,18,19,22,23]. Less

attention was paid to the CFD modeling of liquidsolid loop reactor.

Recently,we suggested a three-dimensional (3D) CFDmodel based on

the EulerianEulerian approach to describe the steady-state liquid

solid two-phase ow in the tubular loop propylene polymerization

reactor [24]. The entireow eld in the loop reactor was calculated by

the model. In addition, the effects of the circulation ow velocity and

the sold particle size on the solid holdup in the reactor were also

investigated. Although our previous model can predict the ow eld

described via the solid holdup distribution in the loop reactor, there

were still no any temperature distribution/temperature eld data.

Furthermore, according to our knowledge, so far, there was no open

report regarding the application of CFD to modeling the temperature

eld in the loop propylene polymerization reactor. However, as

described above, obtaining the temperature eld is very important to

use the loop reactor more effectively, especially for the safety

operation of the reactor and polymer properties.

Powder Technology 203 (2010) 574590

Corresponding author. Tel.: +86 592 2187190; fax: +86 592 2187231.

E-mail address:[email protected](Z.-H. Luo).

0032-5910/$ see front matter 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.powtec.2010.06.025

Contents lists available at ScienceDirect

Powder Technology

j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
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In this work, we develop a 3D CFD model based on the

EulerianEulerian approach to describe the steady-state liquid

solid two-phase ow in the tubular loop propylene polymerization

reactor. The model incorporates the kinetic theory of granular

ow, the energy balance and heat transfer equations. The

suggested model can predict the entire temperature eld in the

loop reactor. Furthermore, the model is used to investigate the

inuences of the circulation ow velocity, the slurry concentra-

tion, the solid particle size and the cool water temperature on thetemperature eld in the reactor.

2. 3D Model for the loop polymerization reactor

The Spheripol technology is one of the most widespread

commercial methods used to produce polypropylene. Commonly, its

key part is constituted of two liquid-phase loop reactors and a gas-

phase FBR. In this work, a pilot-plant polypropylene loop reactor of

the Spheripol technology in a Chinese chemical plant shown inFig. 1

was selected as our object and the loop reactor selected is the same as

that reported in our previous work[24].

In the present study, to simulate the 3D reactors, a 3D physical

model of the reactor system must be available. Hence, the 3D physical

models and their meshes were both constructed in Gambit 2.3.16(Ansys Inc., US) rstly.

3. CFD Model

Based on the kinetic theories of granular ow and heat transfer, a

3D EulerianEulerian two-uid model was used to describe the

liquidsolid two-phaseow and temperature elds in the above loop

reactor[25,26].

3.1. EulerianEulerian two-uid equations

The continuity equations for phase n (n= l for the liquid phase and

sfor the solid phase) were written as

llvl= 0; 1

ssvs= 0: 2

The momentum balance equations for the liquid and solid phases

were written as

llvl

vl= lp+ l

PP

+Kslvs

vl+ ll g

3

PP

l = llvl

+

vlT; 4

ssvs

vs

= spps + sPP

+Klsvl

vs

+ ss g

5

PP

s = ssvs

+

vsT+ ss

2

3svs

PP

I: 6

The energy balance equations for the liquid and solid phases were

written as

llhlvl= P

P

l : vl

ql

+Qls 7

sshsvs= P

P

s : vs

qs

+Ss + Qsl: 8

3.2. Kinetic theory of granularow (KTGF)

From the above momentum balance equations, one knows that

these equations include certain unknown parameters, i.e.,s,sand

ps. In order to solve them, the kinetic theory of granular ow must

be introduced to describe the effective stresses in the solid phase

resulting from particle streaming (kinetic contribution and direct

collisions) collision contribution [16,18,2729].

In addition, according to Refs. [3033], one can know that in

the numerical simulations of two-phase ow reactors, the kinetic

theory of granular ow, treats the colliding particles similarly as

colliding molecules in an ideal liquid/gas. For denser ows, where

particles are in a sustained contact, the effective stresses between

particles become much larger than those predicted by the kinetic

theory of granular ow. Thus, the frictional stress models must be

used in combination with the kinetic theory of granular ow todescribe the much larger stresses associated with enduring

particleparticle contact. In the present work, the concentration

of particles is low in the loop reactor, and the ow of particles is

dominated by particleparticle contact. Constitutive models for

the stresses of particles in the loop reactor can be deduced from

the kinetic theory of granular ow. Therefore, the next constitu-

tive equations for the stresses of particles obtained directly from

the kinetic theory of granular ow, namely Eqs. (9)(19), are

applied.

For the kinetic theory of granular ow, the concept of granular

temperature was rstly used to describe the vibration of the particle

rate coming from the collision interaction. The granular temperature

was dened as follows[25]:

s= 1

3

0s

0s : 9

Corresponding granular temperature equation in the solid phase

was described via the following equation[26]:

3

2ssvs

s=psPP

I+ PP

s : vs

+ ksss + ls: 10

There were many similar models for the solid pressure and

bulk viscosity. The selected models in this work were as follows

[28,34]:

Ps = sss1 + 2g0s1 + es; 11Fig. 1.Loop reactor and axial ow pump.

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s = 4

3

2s sdsg01 + es

ffiffiffiffiffiffis

r ; 12

g0 = 1

1s=s;max

1=3:13

The collision dissipation of energy, s, was modeled using the

correlation by Lun et al.[28]:

s = 121e2sg0

dsffiffiffi

p s2s1:5s : 14

In addition, a transport equation for the granular temperature was

also needed and shown as follows[35]:

ks = 150sds

ffiffiffiffiffiffiffiffiffis

p3841 + essg0

"1 +

6

5sg01 + es

#2+ 2s

2s ds1 +esg0

ffiffiffiffiffis

r :

15

Furthermore, there were a number of similar models for the solid

phase dynamic viscosity. The model chosen in the present work was

as follows[28,36,37]:

s = s;col + s;kin+ s;fr; 16

where,

s;col= 4

5ssdsgo1 + es

ffiffiffiffiffiffis

r ; 17

s;kin= 10dss

ffiffiffiffiffiffiffiffiffis

p96s1 + esg0

"1 +

4

51 + essg0

#2; 18

s;fr=

pssin

2ffiffiffiffiffiffi

I2Dp

: 19

As described earlier, Eqs. (9)(19) describe the stresses of particles

obtained directly from the kinetic theory of granular ow. However,

from Eqs.(11), (16)(19), one can know that solid frictional viscosity

is considered and solid pressure is depended on the kinetic part. In

practice,Eqs. (9)(19) areapplied andvalidated in many open reports

[3136].

3.3. Drag force model

In the present work, the transfer of forces between the liquid

and solid phases was described by empirical drag laws based on

Gidaspow et al.'s model[36]. Gidaspow et al.'s model incorporates

Wen et al.'s model [38] into the Ergun's equation [39] and was

shown as follows:

at l N0:8; Ksl =3

4CD

Sll

vsvl ds

2:65l ; 20

where

CD = 24

lRes1 +

3

20 lRes 0:687

; 21

Res =lds

vs

vl

l

;

22

at l0:8; Ksl = 150s1ll

ld2s

+ 7

4

sl

vsvl ds

: 23

3.4. Heat exchange coefcient

Theheat transfer rate between differentphases wasthe function of

temperature difference and was described as follows[26]:

Qsl = hslTsTl; 24

hsl = 6lslNus

d 2s; 25

at Res105,

Nus =710f + 52f1 + 0:7Re 0:2s Pr1=3

+1:332:4f + 1:2 2fRe 0:7s Pr1 =3:

26

3.5. Turbulence model

The standard k- model was used in this study and solved thetransport equations forkand[14,29]. Thek-model was written as:

mk vm

=

t;m

k

!+ Gk;mm; 27

m vm

=

t;m

!+

kC1Gk;mC2m; 28

where,

m = N

i = 1ii; 29

vm

=

N

i =1ii

vm

N

i =1ii

; 30

t;m = mCk2

; 31

Gk;m = t;m

vm

+

vm

T!: vm

: 32

3.6. Boundary conditions

As described in our previous work [24], the effects of the inlet andoutlet of theloop reactor on thehydrodynamics were ignored dueto the

high-circulationux in theloop comparingto theinlet/outlet ux. Both

theinletand outlet of thereactor were considered a wall. In addition,the

continuous phase/liquid was assumed to obey the no slip boundary

condition at the wall. For the solid phase, a partial slip model suggested

by Johnson et al.[40,41]was used and shown as follows:

svs;wn

= ssg0

ffiffiffiffiffiffis

p2ffiffiffi

3p s;m

vs;w; 33

where, is the reection coefcient of the wall, which describes the

interaction between the solid phase and the wall. And, its value is the

range of 01. Among them, 0 describes the action of free slip and 1

describes the action of no slip.

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The general boundaryconditions for granulartemperatureat the wall

follow the equation suggested by Johnson et al. [40,41]. In this work,

their equation was also applied as corresponding boundary conditions.

qs =

ffiffiffi3

p sg0s

ffiffiffiffiffiffis

p6s;m

"v

2s;w

3

2

1e

2w

#; 34

where, qs represents the exchange of pseudo-thermal energy between

the particles and the wall.

On theother hand, thepolymerization is a highlyexothermic reaction,

and theheat must be removedfromthe loop reactor as soon aspossible inorder to keep thetemperature in thereactorconstant. In practice, there is

a jacket installed on the straight pipe of the reactor. Most of the

polymerization heat can be removed via the cool medium in the jacket.

Therefore, the boundary condition for the straight pipe of the reactor

follows the convection heat transfer equation shown in Eq.(35)[42]:

q= TfTc

1 = hi + b = k+ 1 = ho: 35

For the curve section and the axial ow pump, the boundary

condition follows the adiabatic heat transfer equation shown as

Eq.(36)[42]:

q= 0: 36

During the polymerization in the loop reactor, the growth rate

of the polymer particles is very slow, their growth in diameter is

mainly determined by the residence time of the polymer particles

in the reactor. Accordingly, under the steady-state conditions, the

assumptions of the constant particles in volume corresponding to

a constant volume fraction of the solid phase and the constant

temperature for any particle were used in this work. Therefore,

the heat of polymerization was described by Eq. (37). Here, we

pointed out that the heat value of polymerization equals that of

the convection heat transfer [43].

Ss= rPH; 37

where, rpis the polymerization rate and is described by a mechanismmodel based on Zacca et al.'s equation[43]. Zacca et al.'s equation is

shown as follows:

rP= kP0exp

E

R273:15 + tb

!MC4gcat: 38

4. Simulations

4.1. CFD modeling strategy

As discussed earlier, the CFD with the EulerianEulerian approach

was used to study the liquidsolid interactions in this work. The

standard k- model was used to take into account the turbulence,

whereas the kinetic theory of granular ow was used to close themomentum balance equation for the solid phase. The above equations

were solved by thecommercial CFD code FLUENT6.3.26(Ansys Inc., US)

in a double precision mode. The phase coupled SIMPLE algorithm was

used to couple pressure and velocity[24,44], the multiple reference

frame model (MRF) was used to simulate theaxial ow pump and they

Table 1

Model parameters[945].

Descriptions Values

Thermodynamic and physical parameters

Liquid phase

CP, l(KJ/(kmolK)) 61346384.0T+0.628T2

Mm(g/mol) 44

l(kg/m3) 417

lViscosity (Pas) 5.54 105

l(W/(mK)) 0.2960.0006053T+0.1256107

T2

Solid Phase

CP, s(KJ/(kmolK)) 1022 +3.602T

[c*] (mol/g) 0.025

dp(mm) 0.5

gcat(g/m3) 0.1955

[M] (mol/m3) 7.07 105

cat(kg/m3) 2840

s(kg/m3) 900

s(W/(mK)) 0.205

Model parameters linking to boundary conditions

b(m) 0.01

ho(W/(m2K)) 6583.7

K(W/(mK)) 16.27

Tc(C) 65

Kinetic parametersE(J/mol) 50.4 103

kp0(m3/(mols)) 2.69 103

Table 2

Main parameters used in KTGF[24,45].

es ew s

0.9 0.9 0.35 0.0001

Fig. 2. Comparisons of the CFD simulated data with the classical calculated data

according to the empirical equation.

Fig. 3.Temperature distributions in the loop reactor with the axial ow pump rotating

at 400 rad/s.

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are associated by using an interface. In addition, a commercial grid-

generation tool, GAMBIT 2.3.16 (Ansys Inc., US) was used to generatethe 3D geometries and their grids. Simple grid sensitivity was carried

out, the least cells needed to conserve the mass of solid phase in the

modeling were studied, and totally, a total of 610,000 cells are needed

for the loop reactor. Furthermore, the simulations were executed in a

Pentium 4 CPU running on 2.83 GHz with 4 GB of RAM.

4.2. Model parameter investigation

The actual data depend on the range of parameter values

presented in Eqs.(1)(38). Most of the parameters are directly linked

to the properties of the liquid and solid phases. Some parameters are

polymerization kinetic and heat transfer parameters, respectively. All

above parameter values used in this work were listed inTable 1. In

addition, many researchers[9

42]studied the liquid

gas

solid two-

phase ows, a set of reference values of these parameters could be

selected. In this work, three important parameters including therestitution coefcient (es), particlewall restitution coefcient (ew)

and specularity coefcient () were investigated. We obtained the

same result as that in our previous works[24,45]. Therefore, the same

values of the three important parameters were applied and listed in

Table 2. Unless otherwise noted, the parameters used for the

following simulation were those inTables 12.

5. Results and discussion

5.1. Model verication

Compared with our previous models[24,45], this model incorpo-

rates energy balance equations and heat transfer equations. These

equations couple with the other equations in this model. Therefore,

Fig. 4.Temperature distributions in the ascending straight pipe section at different circulation ow velocities (a: circulation ow velocity of 400 rad/s; b: circulation ow velocity of

500 rad/s; and c: circulation ow velocity of 600 rad/s).

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although our previous model was veried, this model suggested inthis work must be re-veried. Similarly, the slurry pressure gradient

in the loop reactor was selected as the testing parameter. Fig. 2gives

the comparisons between the data obtained by the classical Newitt

model [46,47]and the CFD simulated data at different circulation ow

velocities.Fig. 2shows that the simulated pressure gradient data are

in agreement with the classical calculated data.

5.2. Temperature distribution in the entire loop reactor

The temperature distribution reects directly the heat transfer

efciency of the loop reactor. Therefore, it is one of the most

important parameters in the loop reactor and the entire temperature

eld in the reactor was calculated by the above model in this work.

Fig. 3illustrates the temperate eld in the entire loop reactor at acertain circulation ow velocity. FromFig. 3, one can nd that there is

a uniform temperature distribution in the ascending straight pipe

section of the reactor. Namely, the assumption of continuous stirred-

tank reactor (CSTR) in the ascending straight pipe section of the loop

reactor is reasonable from a heat transport viewpoint. However, an

obvious asymmetrical temperature distribution can be found in the

upper curve section with the liquidsolid two-phase system owing

into the upper curve section. In practice, there is a high temperature

area in the outside of the upper curve section at the circulation ow

velocity of 400 rad/s as shown inFig. 3. Due to the united actions of

the difference between the densities of the solid phase and the liquid

phase and centrifugal force, the solid particles in the upper curve

section are forced into the outside of the upper curve section to form

the second ow [48]. It leads to an obvious increase of the solid

Fig. 5.Temperature distributions in the ascending straight pipe section at different slurry concentrations (a: s=0.3, circulation ow velocity of 400 rad/s; b: s=0.35, circulation

ow velocity of 400 rad/s; and c: s=0.4, circulation ow velocity of 400 rad/s).

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holdup, which was validated by Huang's experimental work [49].

Huang applied a sound wave measurement to record the oweld ina similar loop reactor and found that the slurry concentration in the

outside of the curve section is higher than that in the inside of the

curve section[49]. Furthermore, fromFig. 3, one nds that with the

ow system owing into the descending straight pipe section and

the lower curve section of the reactor continuously, the temperature

eld changes to uniform and itschange extent decreases. Based on the

above discussion, as a whole, for the entire ow eld in the loop

reactor, the CSTR assumption is invalid from a heat transport

viewpoint.

According to theabove discussion andthe physical model of theloop

reactor shown inFig. 1, one knows that the reactor can be divided into

the ascendingstraight pipe section, thedescendent straight pipe section

and the curve section. The curve section can be also divided into the

upper curve section and the lower curve section. Obviously, there are

different temperature elds in these sections. In order to obtain more

detailed information, the temperatureelds in these sections labeled inFig. 3 and the inuences of the circulation ow velocity, the slurry

concentration, the solid particle size and the cool water temperature on

them are also simulated.

5.3. Temperature distribution in the ascending straight pipe section

In the ascending straight pipe section, three planes (P-2, P-3

and P-4) were selected and their temperature distributions under

different conditions were simulated in this work.

5.3.1. The effects of the circulation ow velocity on the temperature

distribution in the ascending straight pipe section

In order to simulate the effect of the circulation ow velocity

on the temperature distribution, three different cases were

Fig. 6. Temperature distributions in the ascending straightpipe section at different temperatures ofthe cool water in the jacket (a: Tc=336.15 K, circulation ow velocity of 400 rad/

s; b:Tc=338.15 K, circulation ow velocity of 400 rad/s; and c: Tc=340.15 K, circulation ow velocity of 400 rad/s).


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selected in this work. Namely, the axial ow pump rotating speeds

selected are 400 rad/s, 500 rad/s, and 600 rad/s, respectively.Fig. 4 illustrates the temperature distributions in the ascending

straight pipe section at three different cases. FromFig. 4, one knows

that an obvious temperature gradient can be found in all planes at a

different circulation ow velocity. For all the cases in the ascending

straight pipe section as described inFig. 4, the heat of polymerization

is removed via the pipe wall with the owing of the cool water in the

jacket installed on the straight pipe of the reactor, accordingly, the

temperature near thepipe wall is lower than that in thepipe center. In

addition, the temperature in the center decreases due to the

continuous removal of heat with the ow system owing along the

straight pipe at a certain constant velocity. However, with theincrease

of the heat of polymerization due to the increase of the slurry

concentration/solid holdup near the wall driven by the centrifugal

force, the temperature near the wall increases with the ow system

owing along the straight pipe at a certain constant velocity.

Furthermore, fromFig. 4, one can also nd that the temperature inthe center increases with the decrease of the circulationow velocity

in the same plane. In practice, the decrease of the circulation ow

velocity leads to the increase of the polymerization residence time,

accordingly, the heat of polymerization increases and the

corresponding temperature increases.

5.3.2. The effects of the slurry concentration on the temperature

distribution in the ascending straight pipe section

Fig. 5illustrates the temperature distributions in P-2, P-3 and P-4

at different slurry concentrations. According to Fig. 5, one can nd that

an obvious temperature gradient can be still found in all planes. In

addition, Fig. 5 also shows that at a certain constant slurry

concentration, the temperatures in the center and near the wall

both increase due to the continuous polymerization with the ow

Fig. 7. Temperature distributions in the ascending straight pipe section at different solid particlesizes (a: d =0.1103 m, circulation ow velocity of 400 rad/s;b: d =0.5103 m,

circulationow velocity of 400 rad/s; and c: d =0.9103 m, circulation ow velocity of 400 rad/s).


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systemowing along the straight pipe. Furthermore, fromFig. 5, one

nds that the temperature in the center and near the wall bothincrease with the increase of the slurry concentration in the same

plane. However, it also must be pointed out that the slurry

concentration can not increase abandonedly. At the slurry concen-

tration of 0.4 corresponding to the slurry concentration of beyond 0.5

in the outside of the curve section, the polymer density of the solid

phase in theloop reactor is saturated, andit canlead to wall stickiness.

5.3.3. The effects of the temperature of the cool water in the jacket on the

temperature distribution in the ascending straight pipe section

Fig. 6illustrates the temperature distributions in P-2, P-3 and P-4

at different cool water temperatures in the jacket installed on the

ascending straight pipe. From Fig. 6, one knows that an obvious

temperature gradient can be still found in all planes. In addition, Fig. 6

shows that the temperatures in the center and near the wall both

increase with the increase of the cool water temperature due to the

different cool capabilities at different cool water temperatures andexcellent cool capability at a low cool temperature.

5.3.4. The effects of the solid particle size on the temperature distribution

in the ascending straight pipe section

Fig. 7shows the temperature distributions in P-2, P-3 and P-4 at

different solid particle diameters. From Fig. 7, one can obtain that

there is an obvious temperature gradient in all sub-gures, which is

similar to those inFigs. 46. In addition, one can also nd that the

temperature in the center increases with the ow system owing

along the ascending straight pipe as shown in these sub-gures of

Fig. 7at a certain constant solid particle size. In the same plane, Fig. 7

shows that with the increase of the solid particle size, the above

temperature gradient decreases. In practice, the small particles in the

reactor lead to a more uniform mix and excellent heat transfer

Fig. 8.Temperature distributions in the curve section at different circulation ow velocities (a: circulation ow velocity of 400 rad/s; b: circulation ow velocity of 500 rad/s; and c:

circulationow velocity of 600 rad/s).


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efciency. Therefore, the temperature gradient increases with the

decrease of the solid particle size.

5.4. Temperature distribution in the curve section

As described in Section 5.2, the curve section can be divided

into the upper curve and lower curve sections. Meanwhile, from

Section 5.2, one knows that the temperature eld in the lower

curve section is more uniform than that in the upper curve

section. In addition, the temperature eld in the lower curve

section is similar to that in the descending straight pipe section.

The latter would be discussed in Section 5.5. Here, only the

temperature eld in the upper curve section is obtained due to

limited space. In the upper curve section, three planes (P-5, P-6

and P-7) were selected and their temperature distributions under

different conditions were simulated.


distribution in the curve sectionFig. 8illustrates the temperature distributions in the upper curve

section at different circulation ow velocities. Compared with the

temperature shown in Fig. 4, the temperature in the curve section

shown inFig. 8is higher as a whole due to the absence of the jacket

installed on the curve section. Furthermore, the temperature

distribution in the curve section is more uniform and not centrosym-

metric yet.

On the other hand,Fig. 8shows that with the ow systemowing

along the curve section, the temperature gradient decreases and the

high temperature region extends from the c entre of the

corresponding plane at a certain constant velocity, which is due to

thedischarge of theheat of polymerization andthe absence of thecool

jacket. In addition,Fig. 8also shows that the temperature increases

with the decrease of the circulationow velocity in the same plane. In

Fig. 9.Temperature distributions in the curve section at different slurry concentrations (a: s=0.3, circulation ow velocity of 400 rad/s; b:s=0.35, circulation ow velocity of

400 rad/s; and c: s=0.4, circulation ow velocity of 400 rad/s).


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practice, as to the point, it is the same as that obtained fromFig. 4.

Namely, the decrease of the circulation ow velocity leads to the

increase of the polymerization residence time, accordingly, the heat ofpolymerization increases and the corresponding temperature

increases.


distribution in the curve section

Fig. 9illustrates the temperature distributions in the upper curve

section (P-5, P-6 and P-7) at different slurry concentrations. As

described inSection 5.4.1, as a whole, the temperature in the curve

section shown inFig. 9is higher than that in the ascending straight

pipe section shown inFig. 5. And its distribution is more uniform and

not centrosymmetric yet. On the other hand, from Fig. 9, one knows

that the temperature increases with the increase of the slurry

concentration in the same plane. The above change is due to the

resulting higher heat of polymerization at a higher slurry concentra-

tion. Certainly, as described inSection 5.4.1, the slurry concentration

can not increase abandonedly in order to avoid the accident of wall

stickiness.


in the curve section


different solid particle diameters. As a whole, one can obtain that the

similar result as obtained in Sections 5.4.1 and 5.4.2comparingFig. 10

with Fig. 7. Namely, the temperature in the curve section is higher and

its distribution is more uniform and not centrosymmetric even if at

different solid particle diameters.

On the other hand, Fig. 10 shows that the temperature gradient

decreases and the high temperature region in the same plane extends

from the centre with the ow system owing along the upper curve

section at a certain constant solid particle size. In practice, the above

results areresulted from the dischargeof theheat of polymerization and

Fig. 10. Temperature distributions in the curve section at different solid particle sizes (a: d =0.1103 m, circulation ow velocity of 400 rad/s; b: d =0.5103 m, circulation ow

velocity of 400 rad/s; and c: d =0.9103 m, circulation ow velocity of 400 rad/s).

http://localhost/var/www/apps/conversion/tmp/scratch_10/image%20of%20Fig.%E0%B1%B0

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the absence of the jacket of the removing heat on the curve section.

Furthermore, from Fig. 10, one can obtain that the temperature inthe centre decreases with the increase of the solid particle size in

the same plane. It is well known that the centrifugal force acting

on the particle in the curve section increases with the increase of

the particle diameter. Accordingly, the slurry concentration in the

centre decreases and the heat of polymerization decreases with the

increase of the particle size. Therefore, the temperature in the centre

decreases.

5.5. Temperature distribution in the descending straight pipe section

In the descending straight pipe section, three planes (P-7, P-8 and

P-9) were selected and their temperature distributions under

different conditions were simulated in this work. In addition, here,

we pointed out that P-7 was also selected due to its location at the

joint of the upper curve section and the descending straight pipe

section.


distribution in the descending straight pipe section

Fig. 11 shows the temperature distributions in the descending

straight pipe section at different circulation ow velocities. From all

sub-gures inFig. 11, one knows that the temperature distribution is

not centrosymmetric and the temperature in the outside is higher

than that in the inside. As shown in Fig. 11, at a certain constant

circulationow velocity, thetemperaturein the centre decreaseswith

the ow system owing along the pipe. In addition,Fig. 11also shows

that the temperature in the whole pipe decreases with the increase of

the circulation ow velocity. In our viewpoint, the increase of the

circulationow velocity leads to the decrease of the polymerization

residence time, accordingly, the heat of polymerization decreases and

Fig. 11.Temperature distributions in the descending straight pipe section at different circulation ow velocities (a: circulation ow velocity of 400 rad/s; b: circulation ow velocity

of 500 rad/s; and c: circulation ow velocity of 600 rad/s).


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the corresponding temperature decreases for the same plane in the

pipe.


distribution in the descending straight pipe section

Fig. 12illustrates the temperature distributions in the descend-

ing straight pipe section at different slurry concentrations. Atrst, according toFig. 12, one can obtain the same result obtained

from Fig. 11. Namely, the temperature distribution is not centro-

symmetric and the temperature in the outside is higher than that

in the inside. In addition, Fig. 12 shows that at a certain constant

slurry concentration, the temperature in the outside increases

with the ow system owing along the straight pipe. Furthermore,

from Fig. 12, one can obtain that the temperature in the outside

increases with the increase of the slurry concentration due to the

high heat of polymerization corresponding to the high slurry

concentration.

5.5.3. The effects of the temperature of the cool water in the jacket on

the temperature distribution in the descending straight pipe section

Fig. 13illustrates the temperature distributions in P-7, P-8 and P-9at different cool water temperatures in the jacket installed on the

ascending straight pipe. From Fig. 12, one knows that an obvious

temperature gradient can be found in all planes. In addition, Fig. 13

shows that the temperature in the pipe increases with the increase of

the cool water temperature as described inFig. 6.


in the descending straight pipe section


different solid particle diameters. From Fig. 14, one can obtainthat the

temperature in the centre decreases with the increase of the solid

particle diameter in the same plane of the descending straight pipe

section. In addition, the above result and its cause is the same as those

described inSections 5.3.4 and 5.4.3.

Fig.12. Temperature distributions in the descending straight pipe section at different slurry concentrations (a:s=0.3, circulationow velocity of 400 rad/s;b: s=0.35, circulation

ow velocity of 400 rad/s; and c: s=0.4, circulation ow velocity of 400 rad/s).


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5.6. The effects of the circulation ow velocity on the temperature

distribution in the lower curve section

In addition, in this work, the effects of the circulation ow

velocity on the temperature distribution in the down curve

section were also simulated and the results were shown in

Fig. 15. P-10 is a plane before the axial ow pump, polypropylene

particles gathered at the side of the loop, the emergence of a high

temperature area makes a uniform temperature distribution. P-11

is a plane at the hub of the ow pump, the area of the high

temperature region becomes signicantly smaller with a more

uniform temperature distribution. As can be seen from Fig. 15,

the temperature distribution of the slurry becomes uniform, axial

ow not only provides the driving force, but also has played a very

good role of mixing, as the axial speed increases, the better the

mixing.

6. Conclusions

In this study, a 3D CFD model wasdeveloped to describe the steady-

state liquidsolid two-phase ow in a tubular loop propylene

polymerization reactor. The model incorporated the kinetic theory of

granular ow, the energy balance and heat transfer equations into the

EulerianEulerian approach. The slurry pressure gradient data calculat-

ed according to the classical Newitt model were supplied to verify the

model. In addition, the entire temperature eld in the loop reactor was

obtained via the model. The model was also used to investigate the

inuences of the circulation ow velocity, the slurry concentration, the

solid particle size and the temperature of thecool water in the jacket on

the temperatureeld in the reactor. The detailed conclusions include:

(1) The predicted pressure gradient data via the model were found

to agree well with the classical calculated data.

Fig. 13. Temperature distributions in the descending straight pipe section at different temperatures of the cool water in the jacket (a: Tc=336.15 K, circulation ow velocity of

400 rad/s; b: Tc=338.15 K, circulation ow velocity of 400 rad/s; and c: Tc=340.15 K, circulation ow velocity of 400 rad/s).


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(2) According to the temperature distributions in the studied

reactor, the whole loop can be divided into four sections;namely, the ascending straight pipe section, the descending

straight pipe section, the upper curve section and the lower

curve section.

(3) Although a uniform temperature distribution can be found in

the ascending straight pipe section of the reactor, an obvious

asymmetrical temperature distribution can be found in the

upper curve section. Therefore, as a whole, for the entire ow

eld in the loop reactor, the CSTR assumption is invalid from

heat transport.

(4) The temperature in the whole reactor decreases with the

increase of the circulation ow velocity and it increases with

the increase of the slurry concentration. In addition, the

temperature in the centre of the reactor rises with the increase

of the solid particle diameter. Simultaneously, the temperature

in the straight pipe sections increases with the increase of the

cool water temperature.

Further studies on the 3D CFD model for the liquidsolid two-

phaseow in the loop reactor are in progress in our group.

Notation

b wall thickness, m

Cd drag coefcient, dimensionless

[C*] active catalyst site concentration, kmolkgcat 1

CP, l heat capacity of liquid phase, kjkmol1K1

CP, s heat capacity of solid phase, kjkmol1K1

dp particle diameter, m

D pipe diameters, m

es particleparticle restitution coefcient, dimensionless

ew particle

wall restitution coefcient, dimensionless

Fig. 14.Temperature distributions in the descending straight pipe section at different solid particle sizes (a: d =0.1103 m, circulationow velocity of 400 rad/s; b: d =0.5103 m,

circulation ow velocity of 400 rad/s; and c:d =0.9103 m, the circulationow velocity of 400 rad/s).


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E active energy, kjkmol1

g gravitational acceleration, ms2

gcat catalyst concentration, kgm3

kP0 propagation rate, m3kmol1s1

hi uid side heat transfer at walls, Wm2K1

ho cool water side heat transfer at walls, Wm2K1

h specic enthalpy, kjkg1

hsl heat transfer coefcient between liquid phase and solid

phase, Wm3K1

H heat of polymerization, kjkmol1

IPP

identity matrix, dimensionlessI2D second invariant of the deviatoric stress tensor, dimensionless

Kls interphase exchange coefcient, kgm2s1

[Mb] monomer concentration, molm3

Nus Nusselt number of solid phase, dimensionless

p pressure, Pa

ps particulate phase pressure, Pa

Pr Prandtl number of liquid phase, dimensionlessq heat ux, Wm2

qsl rate of heat transfer between phases, Wm3

rp rate of polymerization, kmolm3

R Prandtl constant, dimensionless

Res particles Reynolds number, dimensionless

Ss heat ux of polymerization, Wm3

Tc cool water side temperature, K

Tf uid side temperature, K

Tl liquid temperature, KTs solid temperature, K

vl liquid velocity, m s1

vs solid velocity, ms1

vs,w solid velocity at wall, ms1

l volume fraction of liquid phase, dimensionless

Fig. 15. Temperature distributions in the lower curve section at different circulationow velocities (a: circulation ow velocity of 400 rad/s; b: circulation ow velocity of 500 rad/s;

and c: circulation ow velocity of 600 rad/s).


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s volume fraction of solid phase, dimensionlesss,m maximum volume fraction of solid phase specularity factor, dimensionlessl thermal conductivity of liquid phase, Wm

1K1

s thermal conductivity of solid phase,W m1K1

k thermal conductivity of steel,W m1K1

l viscosity of liquid phase, Pas

s solid shear viscosity, Pas

s, col solid collisional viscosity, Pass, kin solid kinetic viscosity, Pass,fr solid frictional viscosity, Pas

angle of internal friction, degs granular temperature, m

2s2

s collisional dissipation of energy, m2s2

lPP

shear stress of liquid phase, Nm2

sPP

shear stress of solid phase, Nm2

s solid bulk viscosity, Pascat catalyst density, kgm

3

l liquid density, kgm3

s solid density, kgm3

Acknowledgment

The authors thank the National Natural Science Foundation of

China (No. 20406016) and the China National Petroleum Corporation

for supporting this work. The authors also thank the anonymous

referees for comments on this manuscript.

The simulation work are implemented by advanced software tools

(FLUENT 6.3.26 and GAMBIT 2.3.16) provided by the China National

Petroleum Corporation and its subsidiary company.

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