Broadening the polyethylene molecular weight distribution by controlling the hydrogen concentration...

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Broadening the Polyethylene MolecularWeight Distribution by Periodic Variation ofthe Hydrogen Feed Rate

Mohammad Al-haj Ali, Emad M. Ali*

The ability to control the molecular weight distribution of the created polyethylene in gas-phase fluidized-bed reactors is discussed. This objective is achieved via nonlinear modelpredictive controller utilizing the hydrogen feed rate as the only manipulated variable. Theidea for limited usage of manipulated variables is to avoideconomically unfavorable operations such as excessive purgeand/or reduced production rate. The simulation results indi-cated successful implementation of the control algorithm toattain the desired molecular weight distribution. The successdepends on the improved hydrogen activities inside thereactor by employing a modified catalyst that is responsiveto hydrogen variation and allowing a wider range for hydro-gen feed rates.

Introduction

The microstructure, which includes the molecular-weight

distribution (MWD), the long-chain branching distribution

(LCBD) etc., of polyolefins controls to a large extent the end

use properties of thesewidely-used polymers. For example,

it is known that when the polyethylene has a high

molecular weight and a wide molecular weight distribu-

tion, the processing of the polyethylene is made easier as a

result of the low-molecular-weight portion; also the high

molecular weight portion contributes to a good impact

resistance for the polyethylene film. Polyethylene of this

type is processed utilizing less energy with higher

processing yields.

Generally, the molecular weight distribution is defined

using number/weight-average molecular weight and the

polydispersity index. These parameters are usually used as

M. Al-haj Ali, E. M. AliChemical Engineering Department, King Saud University, Riyadh,Saudi ArabiaE-mail: [email protected]

Macromol. React. Eng. 2011, 5, 85–95

� 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim wileyonlin

a measure of polymer quality in the polymerization

reaction literature.[1–4] However, molecular weight

averages can be misleading when the molecular weight

distribution shows bimodalities and/or it has high

molecular weight tails. Moreover, the polydispersity index

value itself does not contain enough information about

the complete differential molecular weight distribution.

It is possible that polymers of different chain length

distribution can have the same polydispersity value but

exhibit significantly different end-use properties. Thus,

controlling the entire molecular weight distribution is

crucial.

At the production level, producing polyolefins with pre-

specified molecular weight distribution can be achieved

elibrary.com DOI: 10.1002/mren.201000039 85

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M. Al-haj Ali, E. M. Ali

through changing the operating conditions during the

production process. This can be achieved using either

multistage processes or a single polymerization reactor. In

multi-stage configuration, two reactors are connected in

series. Usually, in the first reactor the polymerization

reaction takes place in the absence of the chain transfer

agent and/or the comonomer. The polymer is then

transferred to the second reactor where polymerization

goes on in the presence of relatively high concentration of

hydrogen and/or comonomer. This configuration has the

advantage that only one optimized catalyst is required for

the production of various grades. However, this method is

subject tohighoperational costs;[5] besides, thepolymerhas

low homogeneity of the two polymer grades and the final

polymer particle has a core/shell-like structure.[6,7]

The production of polyolefins with a desired MWD in a

single reactor has the advantage of requiring a single

reactor that simplifies process design and reduces the

operational costs. Note that the usage of single reactor

requires operating it cyclically to vary the polymerization

conditions inside it. This type of operation can improve the

performanceof the reacting systemandallowbetter design

and control of the molecular weight distribution.[8,9] The

implementation of a single reactor, compared to imple-

menting two reactors, improves polymer homogeneity and

assures that the ratio of both polymer products in each

particle is equal to the overall ratio of these products.

However, the dynamic operation of the polymerization

reactor is difficult and it is subject toappreciableproduction

of off-specification products. Thus, efficient multivariable

control strategies have to be implemented.

Different control strategies have been developed for

polyolefin polymerization reactors, detailed reviews can be

found in refs.[10–13] Ibrahim and coworkers[14] implemen-

ted neural-network based predictive controller to control

only the temperature of polyethylene polymerization

reactor; however, nothing is mentioned about controlling

polymer properties. Zavala and Biegler[15] optimized the

operation of low-density polyethylene (LDPE) reactor

through deriving a general nonlinear model predictive

control (NLMPC) framework. Thedesignedcontroller,which

is based on a first-principles dynamic model, optimizes

process profitability as well as performs regulation tasks.

The authors compare the performance of this framework to

the traditional NLMPC and concluded that the developed

controller can find better strategies to distribute polymer

production inside the reactor. However, operational

limitations as fouling still affect reactor performance

which is the case when NLMPC is implemented. In this

work, different polymer grades were defined using

molecular weight averages, polymer density and long-

chain branching.

Recently, Al-haj Ali et al.[16] dealt with theMWD issue by

special design of the operating condition in a single reactor.

Macromol. React. Eng

� 2011 WILEY-VCH Verlag Gmb

They indicated thatabroadMWDcanbeachievedbyproper

control of the hydrogen to monomer ratio (X) inside the

reactor. It is found that manipulating both monomer and

hydrogen feed rate simultaneously can help altering X to

achieve the required MWD, however undesirable reactor

operation is observed. For example at high monomer feed

rate, an excessive bleed flow is resulted to regulate the

reactor pressure. On the other hand, at low monomer feed

rate a reduced production rate is obtained. Ali and Al-haj

Ali[17] tried to overcome the undesirable operation by

altering X via manipulating the hydrogen flow rate solely.

Relying on the hydrogen flow rate was a difficult task

because it has a slow dynamics and a limited effect on the

attainable range for X. The reason for the shortcoming

stems from the fact that hydrogen consumption is null and

that the allowable hydrogen feed rate is limited. For this

reason and to overcome these weaknesses, they suggested

using a hydrogen absorption agent to facilitate rapid

hydrogen consumption. In addition, they increased the

allowable hydrogen feed rate to increase the attainable

range for X. The mechanism of H2 consumption by

absorption remains artificial.

In this paperwe continue to investigate the possibility of

MWD broadening without producing undesirable operat-

ing conditions. This will be achieved via manipulating the

hydrogen feed rate exclusively at afixedmonomer feedand

avoiding the use of hypothetical hydrogen consumption.

Here we adopt the same concept of elevated capacity of

hydrogen intake, which is physically possible. In addition

we used another catalyst that is more sensitive to H2

content. The main idea is to design a proper nonlinear

control system that can utilize the available facilities to

produce polymers with predefined MWD. For this purpose

NLMPCwill be utilized due to its appealing features such as

handling constraints and superiority for nonlinear pro-

cesses with a large number of manipulated and controlled

variables. Moreover, model predictive controller became

the most widely used control system in the chemical

industries.[18,19]

The On-Line NLMPC Algorithm

In this work, the structure of the NLMPC version developed

by Ali and Zafiriou[18] that directly utilizes the nonlinear

model for output prediction is used. A usual NLMPC

formulation solves the following on-line optimization:

. 2011,

H & Co

minDuðtkÞ;::::;DuðtkþM�1Þ

XP

i¼1

G yðtkþiÞ�RðtkþiÞ½ �k k2

þXM

i¼1

LDuðtkþi�1Þk k2 (1)

5, 85–95

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Broadening the Polyethylene Molecular Weight Distribution by Periodic Variation . . .

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subject to

www.M

ATDU ðtkÞ � b (2)

For nonlinear NLMPC, the predicted output, y over

the prediction horizon P is obtained by the numerical

integration of:

dx

dt¼ f ðx;u; tÞ (3)

y ¼ gðxÞ (4)

from tk up to tkþP where x and y represent the states and

the output of the model, respectively. The specific model

used in this work is given in Appendix A. The differential

equations given in the appendix represent the dynamic

behavior of the polyethylene reactor. Numerical integra-

tion of these equations will define the model output to be

incorporated in the NLMPC formulation. The symbols jj. jjin Equation (1) denote the Euclidean norm, k is the

sampling instant, G and L are diagonal weight matrices

and R¼ [r(kþ 1) �� r(kþ P)]T is a vector of the desired output

trajectory. DU(tk)¼ [Du(tk) . . . Du(tkþM-1)]T is a vector of M

future changes of the manipulated variable vector u that

are to be determined by the on-line optimization. The

control horizon (M) and the prediction horizon (P) are used

to adjust the speed of the response and hence to stabilize

the feedback behavior. G is usually used for trade-off

between different controlled outputs. The input move

suppression, L, on the other hand, is used to penalize

different inputs and thus to stabilize the feedback

response. The objective function [Equation (1)) is solved

on-line to determine the optimumvalue ofDU(tk). Only the

current value of Du, which is the first element of DU(tk), is

implemented on the plant. At the next sampling instant,

the whole procedure is repeated.

To compensate for modeling error and eliminate steady

stateoffset, a regular feedback is incorporatedontheoutput

predictions, y(tkþ1) through an additive disturbance term.

Therefore, the output prediction is corrected by adding to it

the disturbance estimates. The latter is set equal to the

difference between plant and model outputs at present

time k as follows:

dðkÞ ¼ ypðkÞ�yðkÞ (5)

Thedisturbance estimate,d is assumed constant over the

prediction horizon due to the lack of an explicit means of

predicting the disturbance. However, for severe modeling

errors, oropen-loopunstableprocesses the regular feedback

is not enough to improve theNLMPC response. Hence, state

orparameterestimation isnecessary toenhance theNLMPC

performance in the face of model-plant mismatch. In this

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work, Kalman filtering (KF) will be incorporated to correct

the model state and thus, to address the robustness issue.

Utilization of the NLMPC with KF requires adjusting an

additional parameter, s. More details on the integration of

KF with the NLMPC algorithm are given elsewhere.[18] In

addition to state estimationbyKF, thepredictedoutputwill

be also corrected by the additive disturbance estimates of

Equation (5).

Themainobjectiveof theNLMPC is to control theMWD. It

is also necessary to maintain an acceptable polymer

production rate. Process stability is another important

issue which is handled through regulating the total gas

pressure and the bed temperature. These two controlled

variables are adapted via separate proportional integral (PI)

control loops. The design and tuning parameters of these

loops are given elsewhere.[19] NLMPC is the most suitable

algorithmfor this case because theMWDcontrol problem is

challenging. First, the process is highly nonlinear. More-

over, the control variable does not have a conventional set

point, but ratheranentireprofile. Thepoints comprising the

profile are interrelated and they are the result of a

continuous polymer formation process. It should be noted

that entire simulations including solution of ordinary

differential equations (ODEs) and optimization are carried

out using Matlab software. In the following simulation

bothNLMPC and PI algorithmswill be implemented. NMPC

will control the MWD by manipulating the hydrogen feed

rate and PI will control the reactor pressure and tempera-

ture by manipulating the bleed flow and cooling water

temperature respectively.

Results and Discussion

It is well known that a specific hydrogen to monomer ratio

(X) produces a certain polymer grade in terms of density,

average molecular weight or melt index. In this paper the

polymer grade is defined by the entire polymer molecular

weight distribution. Moving the MWD from a specific

narrow profile to another one with a different number-

average molecular weight (peak value) is somewhat easy

because it is required tomove X from one value to another.

However, moving the MWD from an initial narrow

distribution to a specific broad one is much more complex

because it requires to switch X between two different

values over a specific period of time. This is essential to

capture the formation of two different polymer grades and

blend them together during synthesis.[16] This broadening

process is relatively easy when metallocene catalysts are

used since it produces polyolefins with narrow molecular

weight distribution. However, the broadening is much

more complicated when Ziegler-Natta (Z-N) catalysts are

utilizedbecause thesemult-sites catalystsproducebroadly-

distributed polymers.

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The previous work indicated that MWD can be adjusted

by altering X. The ratio X can be varied by either

manipulating ethylene (C2) and hydrogen (H2) concentra-

tion simultaneously or manipulating hydrogen solely.

Using both C2 and H2 simultaneously helps to obtain the

required MWD as shown in Figure 1. The simulation in

Figure 1 is reproduced from Ali and Al-haj Ali.[17] However

this approach leads to undesirable plant operations. For

example, the obtained MWD requires altering X, and

consequently C2 between high and low values. At low C2

values the reaction rate diminishes leading to minimal

production rate. On the other hand, high C2 concentration

mandates large C2 intake leading to propagation of the

reactor pressure. To regulate the reactor pressure back to

normal, the PI controller increases the bleed flow rate

dispatching valuable gases in the flare.

a)

0 17 34 51 680

100

200

300

F M1, m

ole/

s

0 17 340

2

4

F H2, m

ole/

s

0 17 34 51 680

0.5

1

1.5

X

0 17 340

10

20

30

BT,

mol

e/s

0 17 34 51 68260

280

300

320

T W, K

Time, hr0 17 34

0

2

4

6

OP

, kg/

s

Time

b)

2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

GP

C

log(Mw*j)

Figure 1. High average MW using FM1 and FH2; (a) MWD, dotted: initiatarget, dashed: controlled output. (b)Manipulated variables and produline: reference value.



The purpose of the results shown in Figure 1 is to set the

motivation for the rest of the work. It is obvious that

allowing monomer feed to change disturbs the process

operation. Thus it is thoughtful to control X viamanipulat-

ing FH2 exclusively at fixed FC2. However, earlier investiga-

tion[17] indicated that, using FH2 alone, the desired MWD

can be either unachievable or hard to be achieved for two

reasons. The possible attainable range for X is restricted

because the permissible range for H2 feed flow rate is

physically limited. For this reason, some specific X and

consequently some specific MWDs cannot be achieved.

Even within the attainable range for X, moving from one

value of X to another takes long time due to the H2 slow

dynamic inside the reactor. The H2 slow dynamics is

referred to its large residence time caused by the H2

minimal feed flow rate and trivial consumption. For this

51 68

51 68

51 68, hr

l condition, solid:ction rate; dotted

. 2011, 5, 85–95

H & Co. KGaA, Weinhe

reason specific attainable MWD may be

hard to achieve.

In the following we investigate the

ability to control MWDby adjusting only

FH2 using the proposed modification to

overcome the aforementioned limita-

tions. A wider range for the hydrogen

feed rate is considered. In fact, FH2will be

constrained between 0 and 11.6mol � s�1,

which is almost 7 times larger than the

nominal value. In all the following

simulations, Fc, FN2, FM1, and FM2 will be

kept constant. Furthermore, the target

MWD is designed by separate open-loop

analysis. Details of the open-loop analy-

sis can be found elsewhere.[16] In reality,

such desired MWD can be tailored by

process experience, process historical

measurement and design data.

Figure 2 shows the results for tracking

a target MWD with medium average

molecular weight represented by dotted

line. The targeted MWD consists of 104

points where only four representative

points are chosen as set points for the

control system. These points are denoted

by black dots in the figure. The number of

set points is intentionally minimized to

reduce the computational effort required

by NLMPC without sacrificing the latter

performance. Determination of these

points is somewhat iterative in nature;

however few points in the center of the

MWD profile is found to be sufficient. In

this simulation, NLMPC is tuned toM¼ 1,

P¼ 2, L¼ 0, G¼ [10, 10, 10, 10] and a

sampling time of 1 h is employed.

Obviously, NLMPC managed to steer the

im www.MaterialsViews.com

2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

GP

C

log(Mw *j)

a)

0 15 30 430

5

10

15

F H2, m

ole/

s

0 15 30 430

1

2

X

0 15 30 430

5

10

15

BT,

mol

e/s

Time, hr0 15 30 43

3

4

5

OP

, kg/

s

Time, hr

b)

Figure 2. Medium average MW using only FH2; (a) MWD, dotted:initial condition, solid: target, dashed: controlled output. (b)Manipulated variables and production rate; dotted: referencevalue.

2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

GP

C

log(Mw*j)

a)

b)

0 50 100 105-5

0

5

10

15

F H2, m

ole/

s

0 50 100 1050

0.2

0.4

0.6

0.8

X0 50 100 105

0

5

10

BT,

mol

e/s

Time, hr0 50 100 105

3

4

5

OP,

kg/

sTime, hr

Figure 3. High average MW using only FH2; (a) MWD, dotted:initial condition, solid: target, dashed: controlled output. (b)Manipulated variables and production rate; dotted line: referencevalue.


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polymer grade from the initial condition shown by the

dotted line in Figure 2a to the target. The corresponding

manipulated variable response is demonstrated in

Figure 2b, where FH2 is switched between its extreme

values. The latter produced a cyclic reaction in X enough to

bring the formed MWD within the target shape. Figure 2b

shows the transient behavior for the bleed flow induced by

the PI controller to regulate the reactor pressure due to the

variation in the hydrogen partial pressure. Fortunately, the

adaptation in thebleed streamlieswithinacceptable range.

Moreover, Figure 2b illustrates how the production rate

remained very close to the reference value of 3.7 kg � s�1

because the monomer and catalyst feed rates were kept

constant.

Figure 3 depicts another set point tracking case where a

MWD with higher number-average molecular weight is

sought. The NLMPC tuning parameter values are M¼ 1,

P¼ 1, L¼ 0, G¼ [10, 10, 10, 10]. The target MWD and the

significant set points are determined as discussed earlier. It

www.MaterialsViews.com



is clear that NLMPC was successful in moving the polymer

molecular weight distribution form the initial curve to the

desired one. The associated adaptation of the process input

is shown in Figure 3b manifested by swinging action

betweenminimumandmaximumallowablevalues forFH2.

This made the process to operate in a cyclic fashion for X

between 0 and 0.6 leading to the desired MWD. The cyclic

operation produced periodic spikes in the bleed flow

necessary to remove thehydrogenexcess inside the reactor.

Nevertheless, the bleed flow remains within acceptable

range. More important, the process regulation produced

steady polymer production. It should be noted though that

an exact match of the target function is not necessary

especially when we know that the relative error in gel

permeation chromatography (GPC) measurements is

around 10%. An average absolute error of less than 5% in

the MWD profile is considered acceptable in this study. A

long simulation time is used in this plot to demonstrate

howNLMPC recognized thenecessity to operate the process

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in a periodic fashion to keep producing the same desired

MWD over extended period. The first batch of the required

MWD can be collected as early as 40h which is twice the

time required when FM1 is used as shown in Figure 1. This

delay is attributed to the slow dynamic of hydrogen as

mentioned earlier especially when low X is needed as it is

the case in this specific simulation where high average

molecular weight is sought.

To further study the feasibility of achieving the objective

of widening the MWD while maintaining favorable

operation, additional servo problem is investigated. Speci-

fically, a broad MWD with smaller average molecular

weight is examined.Theoutcomeof this test is illustrated in

Figure 4. The value of the NLMPC tuning parameters is

M¼ 1, P¼ 10, L¼ 0, G¼ [10, 10, 10, 10]. Once again, the

hydrogen feed rate was successfully altered to excessively

regulate themolar ratioX in amanner sufficient to produce

the necessary output. Similarly, the purge stream was

2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

GP

C

log(Mw*j)

a)

b)

0 25 500

5

10

15

F H2, m

ole/

s

0 25 500

0.5

1

1.5

2

X

0 25 500

5

10

15

BT,

mol

e/s

Time, hr0 25 50

3

4

5

OP,

kg/

s

Time, hr

Figure 4. Low averageMWusing only FH2; (a) MWD, dotted: initialcondition, solid: target, dashed: controlled output. (b) Manipu-lated variables and production rate; dotted line: reference value.



manipulated inresponse to thehydrogencontentvariation.

As usual no excessive drainage of the gases is observed and

steady polymer production is guaranteed. The simulations

show that this distribution is obtainable 23h after the

starting of the change in feed streams flow rate. This is

faster thanwhat it is required for the case of higher average

molecular weight. In fact, lower average molecular weight

demands high X and thus higher H2 partial pressure. It is

common that elevating the hydrogen concentration takes

less time than diminishing because the rate of hydrogen

consumption is marginal.

NLMPC is a model-based controller, thus the accuracy of

the model used influence the feedback performance. To

assess the controller effectiveness in the presence of

modeling error the above cases are repeated. In this case,

�20% error in the catalyst activation (ac),�20% error in the

reaction rate constant, kp1 andþ15% error in the bed mass

(Bw) are injected in the reactormodel. It is expected that this

amount of model uncertainty deteriorates the model

predictions and hence the controller performance. The

simulation results for the same three previous cases under

the impact of the proposed modeling error are shown in

Figure 5. Obviously, themodeling error had its influence on

the resulted MWD in the sense of slight distortion in the

obtained MWD function. Despite the minor distortion in

the resulted distribution, the average error is within

acceptable margin (i.e. 5%). The corresponding variation

in the hydrogen feed rate and production rate for those

three cases is depicted in Figure 6. Thefigure illustrateshow

reasonable variation in the bleed flow and a satisfactory

production rateareobtained.Theconsequencesprovedthat

a Kalman filter was useful to help NLMPC to overpower the

model uncertainty and to drive the process towards the

required target MWD.

All previous tests revealed that the desired MWD can be

attained with consistent polymer production rate and

modest use of the purge stream solely by regulating the

hydrogen feed rate. This successful outcome is obtained

provided that the modified responsive catalyst is utilized

and constant hydrogen consumption by polymerization is

considered.

Despite the satisfactory conclusion, the transient beha-

vior is slower than that when FM1 and FH2 are used

collectively. The speed of response largely depends on the

dynamic activities of the hydrogen and the polymer

formation process. The extent of broadness of the MW

depends primarily onmoving between two extreme values

for X within a specific critical time during polymer

formation. As the extreme values go apart, the MWD

becomes wider and vice versa. The switching time plays a

different rule. Within the critical switching time, a broad

MWD can be obtained. When the switching time departs

away from the critical time either increasing or decreasing,

the formedMWDmoves towardswhatwould be formed at

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2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8G

PC

log(Mw*j)

a)

b)

c)

2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

GP

C

log(Mw*j)

2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

GP

C

log(Mw*j)

Figure 5.MWD using only FH2 in the presence of modeling errors;(a) medium average molecular weight, (b) high average molecu-lar weight, (c) low average molecular weight. Solid line: targetMWD.

0 20 40 630

5

10

15

F H2, m

ole/

s

0 20 40 630

1

2

X

0 20 40 630

5

10

15

BT,

mol

e/s

Time, hr0 20 40 63

3

4

5

OP

, kg/

s

Time, hr

a)

b)

c)

0 35 700

10

20

F H2, m

ole/

s

0 35 700

0.5

1

X

0 35 700

5

10

15B

T, m

ole/

s

Time, hr0 35 70

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4

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, kg/

s

Time, hr

0 20 40 600

5

10

15

F H2, m

ole/

s

0 20 40 600

1

2

X

0 20 40 600

5

10

15

BT,

mol

e/s

Time, hr0 20 40 60

3

4

5

OP

, kg/

s

Time, hr

Figure 6.Manipulated variables and production rate for the threecases in Figure 5; dotted line: reference value.


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the extreme value, i.e. narrower. The switching time is very

crucial and essentially difficult to determine. At nominal

production rate, it takes 5 h to produce 7 t of polymer.

However, the rate of polymerization varies during the

operation, especially when periodic forcing is involved.

Furthermore, the amount produced over a certain time

cannot be isolated frompolymers thathas beenwithdrawn

from the reactor as product and what has being created

during that long period of time. Therefore, the perfect time

for switching between two corresponding end values for X




to produce specific broad MWD is difficult to determine.

This makes the control objective challenging. The situation

gets more complex when model-plant mismatch exists.

Conclusion

Nonlinear model predictive control is used to regulate the

entire MWD of polyethylene formed in a single fluidized

bed reactor. A broadMWD is obtained even in the presence

. 2011, 5, 85–95


Figure A1. Scheme of the polyethylene reactor.

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of model-plant mismatch by manipulating the hydrogen

feed rate exclusively. The control objective is obtained

while maintaining the bleed flow and production rate

within an acceptable range. This is achieved through

improved hydrogen dynamic pace and wider range for

hydrogencontent inside the reactorbyutilizinganelevated

capacity of the hydrogen feed rate.Moreover, the hydrogen

impact is further improvedbyemployingaZ-Ncatalyst that

is sensitive to small changes in hydrogen concentration

inside thepolymerization reactor. The simulations revealed

the ability of NLMPC to capture the high nonlinearity of the

process and consequently the need to generate cyclic

reactor operation. The periodic operation was found

necessary in order to produce the desired MWD. The cyclic

operation is physically acceptable because it does not

involvevery frequentalterationof thehydrogen feedrate.A

somewhat long time to reach steady state is observed

because the hydrogen content still has slow dynamics

compared to the other process variables.

The main scientific contribution of this paper is the

success in handling this control objective, which requires

forcing the input into cyclic operation, through feedback

control configuration. The prediction feature ofNLMPCwas

themain factor in identifying theprocessbehaviorandthus

anticipates theneed toproduceperiodic feedflowrates. The

second contribution comes from the success in altering X

simply and solelyby regulating thehydrogen feed. Thiswas

achieved bymodifying feed conditions and using a catalyst

with suitable characteristics.

Appendix A: Model Equations

Polyethylene is produced in a fluidized-bed reactor,

Figure A1. Model equations for this reactor were developed

earlier byMcAuley et al.[20] Thismodel is chosen because its

kinetic parameters were validated against plant data.[21]

The definition of the various states and parameters of the

model is given in the section ‘Nomenclature’.

VgdCM1

dt¼ FM1�xM1Bt�RM1 (A.1)

VgdCM2

dt¼ FM2�xM2Bt�RM2 (A.2)

VgdCH2

dt¼ FH2�xH2Bt�FsxH2�RH2 (A.3)


. 2011,
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VgdCN

dt¼ FN�xNBt (A.4)

dYc

dt¼ Fciac�kdYci�OpYci=Bw (A.5)

ðMrCp;rþ BwCp;pÞdT

dt

¼ HFþHG�HR�HT�HP (A.6)

MwCpwdTgdt

¼ FgCpg ðTgi�TgÞ þ FwCpwðTwi�TwoÞ (A.7)

Pt ¼ ðCM1 þ CM2 þ CH þ CNÞRT (A.8)

Tgi ¼ ð PtPt þ DP

ÞT (A.9)

FwCpwðTwi�TwoÞ

¼ 0:5UA½ðTwo þ TwiÞ�ðTgi þ TgÞ� (A.10)

5, 85–95

. KGaA, Weinheim www.MaterialsViews.com

Table A.2. Process parameters.


www.mre-journal.de

where

Parameter Value

Bw 70� 107 g

Cpp 0.85 cal � g�1 �K�1

E 9000 cal �mol�1

Vg 500m3

Tref 360K

kh 0.005m3 � s�1

DHr �894 cal � g�1

Tab

Par

CM

CM

CH

CN

FM1

FM2

FH

FN

Yc

T

Bt

Fw

Tg

Two

Twi

Fc

www.M

HF ¼ ðFM1CpM1 þ FM2CpM2 þ FHCpH

þ FNCpNÞðTf�TrefÞ (A.11)

HG ¼ FgCpgðTg�TrefÞ (A.12)

HT ¼ ðFg þ BtÞCpgðT�TrefÞ (A.13)

HP ¼ OpCppðT�TrefÞ (A.14)

kd 0 s�1

MrCpr 1 400 kcal �K�1

Fg 8 500mol � s�1

UA 1.263� 105 cal � s�1 �K�1

kp1 85 L �mol�1 � s�1

kp2 3 L �mol�1 � s�1

Tf 293K

DP 3 atm

ac 0.548mol � kg�1

CpH 7.7 cal �mol�1 �K�1

CpM1 11 cal �mol�1 �K�1

CpN 6.9 cal �mol�1 �K�1

CpM2 24 cal �mol�1 �K�1

Cp 18 cal �mol�1 �K�1

HR ¼ Mw1ðRM11 þ RM12ÞDHr (A.15)

Op ¼ Mw1ðRM1Þ þMw2ðRM2Þ (A.16)

RM1 ¼ CM1Yckp1e�E

Rð1=T�1=TrefÞ (A.17)

RM2 ¼ CM2Yckp2e�E

Rð1=T�1=TrefÞ (A.18)

RH2 ¼ CH2kh (A.19)

Cpg ¼X4

j¼1

xjCpj (A.20)

w

Themodel equations listed above are slightly modified

from those given by McAuley et al.[20] For simplicity,

le A.1. Steady-state operating conditions.

ameter Value

1 297.06mol �m�3

2 116.17mol �m�3

105.78mol �m�3

166.23mol �m�3

131.13mol � s�1

3.5100mol � s�1

1.6000mol � s�1

2.5200mol � s�1

5.849mol

82.7 8C

10.39mol � s�1

3.11� 104mol � s�1

324.7K

308K

293K

2 kg �h�1

aterialsViews.com



the energy balance around the cooler considers the

dynamic of the recycle temperature explicitly instead of

the heat removal as used by McAuley et al.[20] In due

course, the cooling process is modeled as a well mixed

system. The thermal effect of the recycle compressor

is also included in this model. Note that the partial

pressure of the reactants can be calculated directly

from the reactant concentrations using ideal gas law.

The steady state operating condition for the plant is

given in Table A.1 and Table A.2. This operating condition

is found by optimization and is discussed elsewhere.[19]

The given steady state corresponds to the initial operat-

ing points.

The Molecular-Weight Distribution Model

The instantaneous molecular weight distribution for each

type of active sites is given by the following equation:

. 2011,

H & Co

ydj ¼ j � q2 � expð�j � qÞ (A.21)

5, 85–95

. KGaA, Weinheim93

94

www.mre-journal.de


While the cumulative distribution is given by:

aThe Z

dyjdt

¼Op � ðydj �yjÞ

Bw(A.22)

Finally the gel permeation chromatography (GPC) read-

ing of the MWD is calculated by the following:

GPC ¼ j � ydj � lnð10Þ (A.23)

In the above equations, j is the number of repeating units

and q is the chain termination probability, it is defined as

q¼ [sum of chain-termination rates]/[chain propagation

rate] and can be computed from:

q ¼ ktm þ kthX2

ktm þ kthX2 þ kp(A.24)

In the last equation, X denotes the molar ratio of

hydrogen to monomer inside the reactor. This ratio is a

crucial parameter to vary the value of q and consequently

the molecular weight distribution. Careful adjustment of q

is necessary to achieve desired polymer properties as itwill

be discussed in the results section. The parameters in

Equation (A.24) are determined by confidential data taken

from local industry. Note that the expression of q in

Equation (A.24) and the value of its parameters have been

changed from those used in earlier works[16,17] since a

different Z-N catalyst is used.a

Nomenclature

A C

-N catalyst was ad

onstant matrix for linear con-

straints

ac A
ctive site concentration, mol � kg�1
b V
ector of upper and lower bounds for
the linear constraints

Bw M
ass of the polymer in the bed, g
Bt B
leed flow rate, mol � s�1
CM1, CM2, CN, CH C
oncentration monomer, co-
monomer, nitrogen, and hydro-

gen,mol �m�3

CpM1,CpM2,CpH,CpN H
eat capacity of monomer, co-
monomer, hydrogen and nitrogen,

cal �mol�1 �K�1

Cpg, Cpw H
eat capacity of recycle gas and
water cal �mol�1 �K�1

Cpp H
eat capacity of polymer,
cal � g�1 �K�1

opted from local industry.



E A

. 2011, 5, 85–95

H & Co. KGaA, Weinheim

ctivation energy for propagation,

cal �mol�1

Fc C
atalyst flow rate, kg � s�1
Fw, Fg C
ooling water and recycle flow rate,
mol � s�1

FM1, FM2, FN, FH M
onomer, co-monomer, hydrogen
and nitrogen flow rate,mol � s�1

Fs F
eed flow rate of the hydrogen
absorbent, mol � s�1

GPC G
el Permeation Chromatography
HF, HG, HP S
ensible heat of fresh feed, recycle
gas and product, cal � s�1

HR E
nthalpy generated from ethylene
polymerization, cal � s�1

kd D
eactivation rate constant, 1 � s�1
kp1, kp2 P
ropagation rate constant for
monomer and co-monomer,

L �mol�1 � s�1

kth R
eaction rate constant for chain
transfer to hydrogen, m3 �mol�1 � s�1

Kp P
ropagation reaction rate constant,
m3 �mol�1 � s�1

ktm R
eaction rate constant for chain
transfer tomonomer,m3 �mol�1 � s�1

KF K
alman Filter
LCBD L
ong Chain Branching Distribution
LDPE L
ow Density Polyethylene
M, Mp C
ontrol horizon, constant matrix
Mw W
ater holdup in the heat exchanger,
mol

MrCpr T
hermal capacitance of the reaction
vessel, kcal �K�1

MWD M
olecular Weight Distribution
MPC M
odel Predictive Controller
NLMPC N
onlinear Model Predictive Con-
troller

Op P
olymer outlet rate, kg � s�1
ODE O
rdinary differential Equation
P P
rediction horizon
Pt T
otal pressure, atm
PM1, PM2, PN, PH P
artial pressure of monomer, co-
monomer, nitrogen and hydrogen,

atm

PI P
roportional Integral
q c
hain termination probability
R Id
eal Gas constant, atm
m3 �K�1 �mol�1, also vector of set

points

RM1, RM2, RH C
onsumption rate of monomer, co-
monomer, and hydrogen, mol � s�1

T, Tf, Tref B
ed, feed and reference temperature,
8C
Tgi, Tg T emperature of recycle stream
before and after cooling, 8C



www.mre-journal.de

Twi, Two C


ooling water temperature before

and after cooling, 8C
t T ime, s
UA O
verall heat transfer coefficientmul-
tiplied by the heat transfer area,

cal � s�1 �K�1

Vg G
as holdup in the reactor, m3
x V
ector of states
xM1, xM2, xN, xH M
ole fraction ofmonomer, co-mono-
mer, nitrogen and hydrogen.

X H
ydrogen to monomer ratio
Y, Yp V
ector of future outputs overn and P,
respectively

Yc N
umber ofmoles of catalyst site,mol
y, yp V
ector ofmodel outputs, andof plant
outputs

yj,ydj C
umulative and instantaneous
molecular weight distribution

Greek letters

Du V
ector of manipulated variables
DU V
ector of M-future manipulated variables
DHr H
eat of reaction, cal � g�1
L I
nput weight
G O
utput weight
s T
uning parameter for Kalman filtering
Acknowledgements: The financial support from the ResearchCenter of College of Engineering in King Saud University is greatlyappreciated.

Received: August 17, 2010; Published online: October 21, 2010;DOI: 10.1002/mren.201000039



Keywords: modeling; molecular weight distribution/molar massdistribution; polyethylene (PE)

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