2.2.1 Slurry-phase and solution processes

The two primary reactors for slurry-phase olefin polymerization are the loop reactor and continuous-stirred tanks. In a typical manner, slurry polymerization processes are carried out in a high-pressure continuous reactor. In the process, components such as one or more monomers, a diluent, and a catalyst system and other reactants are introduced to the polymerization reactor to create a reaction mixture, the solid olefin polymer particles and catalyst particles are suspended and mixed well in a liquid diluent. Meanwhile, for solution polymerization processes, a monomer is dissolved in a nonreactive solvent having a catalyst where the solvent temperature is high enough for the dissolution of the polymer material. The heat freed by the reaction is absorbed by the solvent, thereby reducing the reaction rate. In both processes, concentrations of the monomers are high, and the liquid can well remove the polymerization heat from the polymer particles which is what is the most attractive about these processes. However, separation of the polymer from in the solution process is often an expensive operation (Zacca et al. 1996).

2.2.2 Gas-phase processes

Gas-phase olefin polymerization processes prepare an environment that is conducive for olefin polymerization. The remarkable characteristics of gas-phase olefin polymerization lie in the fact that as the system does not involve any liquid phase in the polymerization zone; the gas phase plays a role in the provision of monomers, the integration of polymer particles and removal of the heat of reaction. In this system, polymerization reaction occurs at the interface between the solid catalyst and the polymer matrix, and as can be seen, it is swollen with monomers throughout the polymerization stage. This process includes the VSBR, HSBR and FBR. In the gas-phase polymerization process, reactors must work closely to the dew point of the monomer mixture to achieve high monomer concentrations and high yields and the catalyst morphology must be very tightly controlled so that particle melting and agglomeration due to the heat transfer limitation of the gas can be prevented.

3.2.1 Flow regimes in fluidization

There are different behaviors noted for the fluidized beds in which the solid particles are fluidized because of the variety in the gas, solid and velocity properties. With a growth in the gas velocity in the minimum fluidization regime, the bed voidage will escalate marginally and the drag force conveyed by the rising gas equals the weight of the particles. Next, the bed will go into the gas fluidization state. Higher levels of gas velocity mean that bubbles will set in and a bubbling fluidized bed would be seen. The bubbles in a bubbling fluidized bed will propagate and coalesce when it rises while the velocity intensifies. In the slugging bed, the height-to-diameter ratio (H/D) of the bed is high enough and the bubble size may become the same as the bed’s diameter. The fluidization of particles will occur at a high enough gas flow rate in a turbulent bed, where the velocity will exceed the particles’ terminal velocity. Instead of the bubbles, the upper surface of the bed vanishes and a turbulent motion of solid masses and voids of gas with different dimensions are seen. Pneumatic transport of solids occurs with higher amount of gas velocity, and the fluidized bed becomes an entrained bed in which dispersed, lean or dilute phase fluidized bed would be found.

3.2.2 Bubbling fluidized beds

Gas fluidized beds are differentiated by the bubbles formed at superficial gas velocities larger than minimum fluidization velocity. In this state, it appears that the bed is distributed into the emulsion phase and the bubble phase. The bubbles are very analogous to gas bubbles in liquid form, and they act similarly and merge while rising through the bed. The movement of particles in fluidized beds receives most influence from the rising bubbles passing through the bed. As the result, extreme consideration is given to these bubbles and their properties. To compare the processes occurring inside an FBR, the fluidization bubbles, bubble formation, the way particles are transported and their path through the bed are not to be taken lightly.

3.3.1 Hydrodynamics

FBRs have several issues. They, for starters, are non-ideal and challenging to elaborate due to complicated transport phenomena, flow patterns, and polymerization reactions. A number of studies have focused on several mixing models to model such non-ideality to characterize FBRs’ behavior. Non-ideal FBR modeling would need the integration of transport phenomena, kinetics, and hydrodynamics equations. Choi and Harmon Ray (1985) put forth a two-phase model with constant bubble. The authors placed it on a plug-flow bubble phase and a fully mixed emulsion phase to find traits of the FBR dynamic behavior. McAuley et al. (1990) further guessed that there is an unlimited heat and mass transfer between the emulsion and bubble phases and recommended a much simpler well-mixed model.

The traditional well-mixed and constant-bubble-size models presume solid-free bubbles (ε_b=1) and that the fluidization of the emulsion phase is at the minimum (ε_e=ε_mf). This concept does not gauge the impact of the dynamic gas-solid distribution on the heat/mass transfer and reaction rate in the fluidized beds and has its restrictions when it comes to describing the low-velocity bubbling fluidization. Cui et al. (2000), nonetheless, have been able to prove theoretically and experimentally the presence of solid particles in the bubbles. They also proved that the emulsion phase may have more gas at higher gas velocities and it does not stay at minimum conditions of fluidization. Increasing the superficial gas velocity contributes to the better mixing of the two phases, making more amounts of gas to enter the emulsion phase and further leading to higher solid particles entering the bubbles.

Readers can refer to various works that have collected different hydrodynamic formulations from the literature (Shamiri et al. 2015, Abbasi et al. 2016).

3.3.2 The well-mixed model

McAuley et al. (1990, 1994) made a proper introduction of the fluidized bed approximated by a single-phase continuous stirred tank reactor (CSTR) in the well-mixed model. This assumption is believed valid for very high mass and heat transfer rates between phases (uniform monomer concentration and temperature throughout the bed) or small-size bubbles. The bubble phase does not function in the model and the bed voidage (ε_bed) contemplates on the entire gas volume fraction in the bed. The following is a set of assumptions that would work for the well-mixed model:

• The polymerization reactor is a single-phase (emulsion phase) well-mixed reactor because of high heat and the mass transfer rates between phases or possibly because bubbles are sufficiently small.

• Composition and temperature are the same throughout the bed.

• The emulsion phase stays at minimum fluidization state.

Dynamic mass and energy balances are obtainable for the system based on the above assumptions for hydrogen and monomers. The mole balance is given by:

(Vεmf)d[Mi]dt=U0A([Mi]in−[Mi])−Rvεmf[Mi] −(1−εmf)Ri.

The energy balance is expressed by:

[∑i=1m[Mi]CpiVεmf+V(1−εmf)ρpolCp,po1]dTdt =U0A∑i=1m[Mi] Cpi(Tin − Tref)  −Rv[∑i=1m[Mi]Cpiεmf+(1−εmf)ρpo1Cp,po1](T−Tref) + (1−εmf)ΔHRRp − U0A∑i=1m[Mi]Cpi(T−Tref).

In the energy balance equation, the monomer internal energy is considered insignificant. The conditions needed to address the model equation are as follows:

[Mi]t = 0=[Mi]inT(t=0)=Tin.

For kinetics and hydrodynamics equations, one can take guidance from the literature for added information (McAuley et al. 1990, 1994, Hatzantonis et al. 2000).

3.3.3 The constant bubble size model

In this model, we consider the assumption made by Choi and Harmon Ray (1985) that the fluidized bed comes in two phases depending on the phase where polymerization happens (emulsion or bubble phase). In the constant bubble size model, it is safe to assume that the bubbles, at constant velocity, travel through the bed in plug flow where the spherical size is characterized as fixed and consistent. We also need to consider the fact that the emulsion phase is interchanging heat and mass with the bubble phase and the mixture comes in full mixture. To add, the mass and heat transfer coefficients are constant throughout the bed, and we are led to believe that the heat and mass transfer resistances between the solid polymer particles and the monomer gas in the emulsion phase are trivial (Floyd et al. 1986). The constant bubble size model however does not ignore these assumptions:

• The fluidized bed is made of the emulsion and bubble phases. The former is where the reactions would occur.

• The bubbles are in plug flow and the spherical size is the same among each other. Their velocity is also constant.

• It is believed that the emulsion phase is fixed at the minimum stage of fluidization, and that it the mixture is perfect, exchanging mass and heat with the bubble phase at constant rates over the bed height.

• The gas and the solid polymer particles in the emulsion phase have negligible inter-phase heat and mass transfer resistances (Floyd et al. 1986).

It is through these assumptions that the steady-state mass and energy balances are derived. The mole balance for monomer and hydrogen can be expressed as

[M¯i]b=1H∫0H[Mi]bdh= [Mi]e+([Mi]e,(in)−[Mi]e)UbKbeH(1−exp(−KbeHUb)).

The bubble-phase energy balance can be written as

∑i=1m[Mi]bCpidTbdz=HbeUb(Tb−Te).

The dynamic molar balance for the ith monomer in the emulsion phase is expressed by

(Veεmf)d[Mi]edt=UeAeεmf([Mi]e,(in)−[Mi]e)+VeδKbe(1−δ)([M¯i]b − [Mi]e) −Rvεmf[Mi]e−(1−εmf)Ri.

The dynamic energy balance for emulsion phase can be written as

[∑i=1mVeεmf[Mi]eCpi+Ve(1−εmf)ρPo1Cp,po1] dTedt=−∑i=1mVeεmfCpid[Mi]edt(Te−Tref) + UeAeεmf∑i=1m[Mi]e,(in)Cpi(Te,(in)−Tref) − UeAeεmf∑i=1m[Mi]eCpi(Te−Tref)−VeδHbe(1−δ)(Te−T¯b) + Rv((1−εmf)ρpo1Cp,po1+εmf∑i=1m[Mi]eCpi)(Te−Tref) + (1−εmf)ΔHRRp.

The boundary and initial conditions for solving the model equations are as follows:

[Mi]b,z=0=[Mi]inTb(z=0)=Tin[Mi]e,t=0=[Mi]inTe,t=0=Tin.

To estimate the gas velocities for bubble phase and emulsion phases, the bubble phase and emulsion phases voidage, the bed bubble volume fraction, and heat and mass transfer coefficients for the constant bubble size mode, the correlations needed for that purpose can be found from diverse sources in the literature (Shamiri et al. 2012, 2015).

3.3.4 The bubble-growth model

In the bubble-growth model, the constant bubble size model was extended to account for the varying bubble size with respect to the bed’s height (Hatzantonis et al. 2000). Spurred by the developments of Kato and Wen (1969), in this model, an assumption is that the bubble phase is divided into “N” well-mixed sections in series and the emulsion phase is mixed perfectly and at developing fluidization conditions (ε_bed=ε_mf). The bubble-phase sections’ size is fixed to be equal to the bubble diameter at the equivalent bed height. The local mass and heat transfer coefficients between the emulsion and bubble phases, the bubble rise velocity, and the local bubble volume fraction are found by the bubble diameter and thus the equivalent bed height. This is because the bubble size has its minimum value at the gas distributor and it enlarges to its largest constant size while it moves through the bed.

This model does not weigh the heat and mass transfer restrictions between the solid particles and the surrounding gas in the emulsion phase. Nevertheless, these limitations can become something that should not be taken lightly for high rates of polymerization.

If the bubbles are solid-free (no reaction), we can write the molar balance for the i monomer in the n compartment:

[Mi]b,n=[1+(Kbe,ndb,n/ub,n)]−1 ×{δb,n−1ub,n−1δb,nub,n[Mi]b,n−1+Kbe,ndb,nub,n[Mi]e},

where d_b,n is the bubble size corresponding to the n compartment.

In effect, the energy balance for the n compartment can be written as

Tb,n=(∑i=1Nm[Mi]b,nCpMi+Hbe,ndb,nub,n)−1 × {δb,n−1ub,n−1δb,nub,n∑i=1Nm[Mi]b,n−1CpMi(Tb,n−1−Tref) + ∑i=1Nm[Mi]b,nCpMiTref+Hbe,ndb,nub,nTe}.

The dynamic mass and energy equations can also be derived and this had been discussed by Hatzantonis et al. (2000).

3.3.5 Other mathematical approaches

Fernandes and Ferrareso Lona (2001) had contemplated on gas in bubble and emulsion phases plus solid polymer particles, all as plug-flow phases, to suggest on their three-phase heterogeneous model. Jafari et al. (2004) brought to comparison the performance of some available models of the time such as simple two-phase model and generalized bubbling/turbulent model. They summed up that the later model gives the most fitted results to experimental data. Luo et al. (2009) had worked further on a method to model the PP process based on Hypol Technology. The authors adopted the Polymer Plus and Aspen Dynamics to foresee process behavior and physical properties in the steady-state and dynamic modes. In something similar, Zheng et al. (2011) developed a steady-state and dynamic method to model the propylene process with the aid of the Spheripol Technology. Their kinetic model leaned on both single and multi-site catalyst and their molecular weight distribution results were fitted with the help of the actual gel permeation chromatography (GPC) data.

In the meantime, some researchers focused on particle size distribution studies in fluidized beds rather than kinetic or property estimation (Khang and Lee 1997, Immanuel et al. 2002, Ashrafi et al. 2008). That said, fluidization regimes have also been studied in vast literature. Different methods that can find the fluidization regimes in gas-solid FBRs have been applied on these reactors to study several hydrodynamic aspects (Makkawi and Wright 2002, Sederman et al. 2007, Tamadondar et al. 2012). Alizadeh et al. (2004) put forth a pseudo-homogeneous tanks-in-series model to guess on the behavior of industrial-scale gas-phase PE production reactor. Kiashemshaki et al. (2006) were inspired by this model and they suggested a two-phase model to describe the fluidized-bed ethylene polymerization reactor. Their model was a dynamic model except in terms of the computation of the temperature and co-monomer concentrations.

Shamiri et al. (2010, 2011, 2013, 2014) analyzed different dynamic and non-dynamic modeling and control approaches for gas phase homopolymerization or copolymerization of olefin in FBRs.

Abbasi et al. (2016) proposed a dynamic two-phase ethylene copolymerization model which considers particle carryover in the model. Their model showed to be a more realistic model where the accuracy was shown by confirming the results with industrial data. The main equations in the developed model are as follows:

Mass balance for emulsion phase:

ddt(Veεe[Mi]e)=[Mi]e,(in)UeAe−[Mi]eUeAe−Rvεe[Mi]e+Kbe([Mi]b−[Mi]e)Ve(δ1−δ)−(1−εe)Rie−KeVeεeAe[Mi]eWe.

Mass balance for bubble phase:

ddt(Vbεb[Mi]b)=[Mi]b,(in)UbAb−[Mi]bUbAb−Rvεb[Mi]b−Kbe([Mi]b−[Mi]e)Vb−(1−εb)AbVPFR∫Ribdz−KeVeεeAe[Mi]eWe.

Energy balance for emulsion phase:

UeAe(Te.(in)−Tref)∑i=1m[Mi]e,(in)Cpi−UeAe(Te−Tref)∑i=1m[Mi]eCpi −Rv(Te−Tref)[∑i=1mεeCpi[Mi]e+(1−εb)ρpolCp.pol] + (1−εe)RpeΔHR−HbeVe(δ1−δ)(Te−Tb) − Veεe(Te−Tref)∑i=1mCpiddt([Mi]e) − KeAeWe(Te−Tref)(∑i=1mεeCpi[Mi]e+(1−εb)ρpolCp.pol) = Ve(εe∑i=1mCpi[Mi]e+(1−εe)ρpolCp.pol)ddt(Te−Tref).

Energy balance for bubble phase:

UbAb(Tb.(in)−Tref)∑i=1m[Mi]b,(in)Cpi−UbAb(Tb−Tref)∑i=1m[Mi]bCpi−Rv(Tb−Tref)(∑i=1mεbCpi[Mi]b+(1−εe)ρpolCp.pol)+(1−εb)AbΔHRVPFR∫Rpbdz+HbeVb(Te−T)b−Vbεb(Tb−Tref)∑i=1mCpiddt([Mi]b)−KbAbWb(Tb−Tref)(∑i=1mεbCpi[Mi]b+(1−εb)ρpolCp.pol)=(Vb(εb∑i=1mCpi[Mi]b+(1−εb)ρpolCp.pol))ddt(Tb−Tref).

Solid elutriation constants gathered from the literature are obtained from Rhodes (2008) and are as follows:

Ke=23.7ρgU0AWeexp(−5.4UtU0)Kb=23.7ρgU0AWbexp(−5.4UtU0)We=AH(1−εe)ρpolWb=AH(1−εb)ρpolUt=Ut∗[μρg−2(ρpol−ρg)g]0.33Ut∗=[18(dp*)−2+(2.335−1.744∅s)(dp∗)−0.5]−1dp∗=dp[μ−2ρg(ρpol−ρg)g]0.33  for 0.5<∅s≤1.

These equations can be solved using the following initial conditions:

[Mi]b,t=0=[Mi]inTb,t=0=Tin[Mi]e,t=0=[Mi]inTe,t=0=Tin.

3.3.6 Condensed mode cooling modeling

Injection of a quench liquid into the reactor is an accepted heat removal method in olefin polymerization FBRs. Although in PE the “quench” liquid is usually an inert alkane (often referred to as an induced condensing agent, ICA), but it is the liquefied monomer in the case of PP. Introduction of propylene copolymers and heavier monomers such as hexane or butene gives the possibility of condensing these monomers and inserting them in liquid form in a gas-phase process.

To find out the effect of liquefied co-monomer injection on the reaction rate, the influence of these components on solubility, transport, and other “physical” processes on one hand and their impact on the reaction on other hand need to be found. As an example, the instantaneous rate of ethylene polymerization increased in the presence of an ICA in the study which was done by Namkajorn et al. (2014). The authors related this to the enhancement of the local concentration of ethylene due to the heavier hydrocarbon at the catalyst active sites. They also examined different isomers of pentane and hexane. Alizadeh et al. (2015) concluded that the complex effects were the result of the alkane replacement with a similar alkene. Based on thermodynamics, both alkanes and alkenes increase the rate of ethylene polymerization via the cosolubility effect, but considering that the alkenes are also co-monomers, they have a direct influence on the reaction rate. Moreover, they boost the polymerization rate (called co-monomer effect) at low concentrations; nonetheless, they decrease the reaction rate at higher concentrations despite the cosolubility effect. Nevertheless, the ethylene’s concentration is higher in the existence of a heavier compound rather than ethylene alone.

Although condensed mode cooling is common in industries, only a few studies have focused on modeling this approach to analyze its effect on the hydrodynamics, transport phenomena and polymerization reactions, and there is still a need for models that can accurately account for the entire process considering the condensed mode cooling practice (Jiang et al. 1997, Yang et al. 2002, Mirzaei et al. 2007, Zhou et al. 2013, Alizadeh et al. 2017, Pan et al. 2017).

3.3.7 CFD approaches

CFD is a category of fluid mechanics that uses numerical analysis and data structures to solve and analyze problems that involve flow of fluids (Schneiderbauer et al. 2015). Researchers use computers to do the essential calculations to simulate the fluid and surface interactions defined by boundary conditions. With high-performance computations, better solutions can be completed in less time.

CFD model development is a progressive research area for picturing fundamental phenomena without executing real-time experiments. It can be used to solve momentum and conservation equations in multi-phase flows. For polymerization reactors, an added benefit of CFD is that it can offer information on turbulent zones which is very important because the reactants are mostly inserted within these areas where the reaction yield is superior (Dompazis et al. 2008).

Two methods of CFD models, i.e. Eulerian and Lagrangian, are used to define gas-solid fluidized reactors. Both phases (gas and solid) are counted as continuum (fluid) in the Eulerian model, and momentum and continuity equations are dealt with for both phases. The Lagrangian model, meanwhile, solves Newton’s equations of motion for each particle and particle-particle collisions and applied forces on the particle are considered. The Eulerian-Lagrangian method, which is also known as discrete element method or discrete particle model, studies the fluid as a continuum and considers solids to be dispersed phase (Schneiderbauer et al. 2016). The discrete element method models the continuous phase and particle trajectories using the Eulerian and Lagrangian frameworks, respectively. The continuous phase can be modeled by averaging its properties over an extensive variety of these paths or trajectories. However, to obtain an average of all quantities in a moment, it is recommended that a plentiful of particle trajectories be simulated. Gas and emulsion phases are presumed to be continuous in the Eulerian-Eulerian approach while they are considered completely interpenetrating in all control volumes.

Particle size distribution is a key factor in CFD studies of this system. To define the particle size distribution in a multi-phase flow, the population balance equation (PBE), continuity, momentum, and energy equations need to be solved simultaneously. Researchers have used a combination of PBE solving methods with CFD to discuss particle size distributions and flow patterns within polymerization FBRs (Yan et al. 2012, Akbari et al. 2015a,c, Che et al. 2015a, 2016).

Several researchers also performed advanced investigations on the influence of operating conditions and geometry of the reactor, such as distributor type, size of solid particles, gas velocity and operating pressure on the hydrodynamics of the reactor, for accurate scale-up and design of reactors (Akbari et al. 2014, 2015b; Che et al. 2015b) and some used the results of CFD for further analysis using signal or image processing to study fluid structures during the process (Aramesh et al. 2016).

For more details on CFD modeling of olefin polymerization in FBRs, readers may refer to a comprehensive review by Khan et al. (2014).

4.3.1 Model predictive control

Polymerization plants must function under different grade transition scenarios. This is to fully cater for the many types of product qualities needed for various applications. The best answer for this situation depends on a suitable objective function defined for a minimization problem. This optimization problem depends on time needed to change product quality specifications, process safety limitations and the quantity of off-spec polymer. Since choosing the best control scheme has significant impacts on process quality and process operability optimization, the time optimal transition problem needs to be considered together with this control strategy simultaneously.

It should be said that many conventional control algorithms are not enough in dealing with the strict limitations enforced in a few industrial processes, specifically once a first-rate commodity is needed. This is particularly valid for polymerization processes wherein definite properties like MFI or average molecular weight with the effect on plastic quality should be fulfilled. At this point, a proper way is to adopt a model predictive control (MPC), that makes use of a process tailored dynamic model as an essential part for the process control structure.

The MPC is an optimization-based control strategy which is very suitable for constrained, multi-variable processes. The MPC predicts the actual system’s future behavior over a time interval defined by the prediction horizon. The implication made by the simpler diagram of the MPC algorithm shown in Figure 3 is that a parallel process model to the controller is used in MPC to predict the controlled variable.

Figure 3:

Model predictive control system (Shamiri et al. 2013).

MPC works by minimizing a cost function given by Seborg et al. (2004):

V(k)=∑i=1P∑j=1ny(wjy[yi(k+1)−rj(k+1)])2 +∑i=1M∑j=1nmv{wjΔu[Δuj(k+i−1)]}2,

where P is the prediction horizon, n_y is the number of plant outputs, wjy is the weight for output j, k is the current sampling interval, k+i is a future sampling interval (within the prediction horizon), [y_i(k+i)–r_j(k+i)] is the predicted deviation at future instant k+i, n_mv is the number of manipulate variables (inputs), M is the control horizon, wjΔu is the weight for input j and Δu_j(k+i–1) is the predicted adjustment (i.e. move) in manipulated variable j at future (or current) sampling interval k+i.

The unequal constraints of the manipulated and control variables also restrict the cost function as follows:

Manipulated variable constraint:

umax(k)≥u(k+i−1)≥umin(k)

Manipulated variable rate constraint:

Δumax(k)≥u(k+i−1)≥Δumin(k)

Output variable constraint:

ymax(k)≥u(k+i−1)≥ymin(k).

The problem formulation would decide if several parameters such as the control horizon, prediction horizon and weighting matrices in the optimization formulation should be completed so that the predicted output can perform better.

As mentioned by Campello et al. (2003), MPC algorithms have been used for chemical process control for their usage simplicity and capability to deal with limitations that involve the input and output variables in procedures. Schnelle and Rollins (1997) adopted a continuous polymerization (CP) process design and applied an MPC. They illustrated that MPC technology proved to be a promising substitute for this type of process. Santos et al. (2001) made the effort to control the temperature and liquid level in a pilot plant CSTR reactor and they implemented an on-line nonlinear MPC algorithm. The authors also established several sources of unmeasured disturbances and model mismatch. They leave an impact on the model quality in representing the reactor dynamics. While there are mismatches and disturbances, the closed loop system performed credibly in terms of disturbance rejection and set-point tracking. The application of a nonlinear MPC based on extended Kalman filter to control polymer properties had been analyzed by Park and Rhee (2003), for a semi-batch methyl methacrylate/methacrylate copolymerization reactor. The authors drew a comparison between the experimental results and other techniques to show the superior performance of this control strategy.

Ramaswamy et al. (2005) had sought to control an unsteady-state CSTR bioreactor with the help of an MPC. A vital MPC parameter for its tuning is prediction horizon which was analyzed in this work by concentrating on the variation effects.

The DMC (dynamic matrix control) strategy has the utmost applications in industry between MPC control techniques. It is because of its design and application straightforwardness together with its ability to soundly handle cases where manipulated variable is restricted within a range.

Within chemical industries, MPC has stood out to become the principal method of advanced multi-variable. In another approach, Dougherty and Cooper (2003) had given a multiple model adaptive control approach for multi-variable DMC. This technique blends the output of multiple linear DMCs without adding extra computational intricacy contrasting to the non-adaptive DMC. Adaptive linear DMC (ALDMC) algorithm was developed and employed by Guiamba and Mulholland (2004) in a MIMO pump-tank setup. Where the plant/model mismatch is concerned, ALDMC presented better performance in comparison to the non-adaptive linear DMC (LDMC). Haeri and Beik (2005) considered the handling of MIMO systems within specified circumstances and recommended an approach which extended the nonlinear DMC procedure. The authors have illustrated that the method usefulness is clear by presenting the simulated control results of a MIMO stirred reactor nonlinear model and another MIMO power unit nonlinear model.

Carlos et al. (1989) highlighted critical issues whereby any control system should discuss and review MPC techniques in the light of those issues to highlight their advantages in terms of the design and implementation. Several design techniques such as internal model control and inferential control emanate from MPC; they were put in perspective with respect to one another and the similarity with more traditional methods like linear quadratic control were investigated. The malleable constraint management capabilities of MPC were a prominent asset in the framework of the global operating goals of the process industries and the 1-, 2-, and ∞-norm formulations of the performance goal. The application of MPC to nonlinear systems was examined and its attractions were also studied. The authors also suggested that, although the MPC is not any stronger than classical feedback, its robustness can be adjusted more easily.

The nonlinear model based control was applied by Özkan et al. (2001) to the styrene solution polymerization in a jacketed batch reactor and checked its effectiveness to reach the required molecular weight and monomer conversion. The authors used Hamiltonian optimization to assess the optimal temperature profiles for the properties of polymer quality. Analytical and experimental nonlinear model based control were analyzed to keep tab of the temperature at the trajectory which was believed to be optimal. Two types of parametric and nonparametric models were assessed to control temperature optimally. Nonlinear auto regressive moving average exogenous (NARMAX) gives an association among reactor temperature and heat input to depict the system dynamics. It should also be added that this model served to define the control system as a parametric model. Simulation results were brought for comparison with experimental control data. Authors summed up that the control simulation program is the representative of the behavior of the controlled reactor temperature with some good prediction capabilities. Moreover, nonlinear model based control keeps the reactor temperature stable within the optimal trajectory.

Qin and Badgwell (2003) did a survey on commercially available MPC technology, both linear and nonlinear, based primarily on data provided by MPC vendors and described approaches taken by each vendor for the various aspects of the calculation. A review of the identification technology was needed to enable a list of similarities and differences to be found between the approaches. MPC applications performed by each vendor were summarized following the application area.

Ibrehem et al. (2008) implemented adaptive predictive model-based control to control the system and compared with the conventional PID controller, giving acceptable results.

Mjalli et al. (2009) considered a multi-model adaptive control strategy to handle the variation of operational process parameters and the effect of process disturbances in a biodiesel transesterification reactor.

Ho et al. (2012) had concentrated on the control of a gas phase propylene polymerization model in an FBR, where a two-phase dynamic model was used, and since the process is nonlinear, an advanced control scheme was adopted to ensure an efficient regulation of the process variables. Adaptive predictive model-based control (APMBC) strategy (an integration of the recursive least squares algorithm, RLS and the generalized predictive control algorithm, GPC) have the function to control the emulsion phase temperature and PP production rate by manipulating the reactor cooling water flow rate and catalyst feed rate again. The APMBC in set point tracking was reported to be superior, as compared to the conventional PI controller and the ability of APMBC to capture the effects of monomer concentration, hydrogen concentration and superficial gas velocity on the process variables as efficiently as possible. It is common for the polymerization processes to have a highly nonlinear dynamic behavior leading to an abysmal performance of controllers based on conventional internal models to be poor or for it to need a considerable amount of effort in controller tuning.

Shamiri et al. (2013) took advantage of a two-phase model to delve into the dynamic behavior and process control of a fluidized-bed PP production reactor production rate and temperature. To control the reactor temperature and the PP production rate, a centralized MPC technique was used by making use of the catalyst feed rate and cooling water flow rate, respectively. They reached a conclusion that the MPC could yield some controller moves which not only were subjected to the specified input constraints for both control variables, but also that they are characteristically non-aggressive and sufficiently smooth for practical use.

Broadly speaking, the challenge is to organize the sophisticated chemical processes due to their volatile nature of parameters and structural mismatch. Adaptive linear or nonlinear algorithm is a very important tool to control these kinds of processes. In normal circumstances, the stability of Lyapunov functions serves to design nonlinear adaptive controllers. This algorithm can also deal with other parametric uncertainties throughout the input saturation. Biswas and Samanta (2013) considered the controlling polymerization process with input saturation and parametric uncertainty and worked on an adaptive back stepping methodology. The controller is showed to be robust in modeling uncertainties in a polymerization process and it also showed powerful disturbance rejection ability. The adaptive controller could take both matched and unmatched uncertainties. In addition, to get better results, the authors made use of several parameters in which the controller gave a sturdy performance even around high parametric uncertainty.

4.3.2 Fuzzy-based MPC

Nascimento Lima et al. (2009) built a predictive control system based on type Takagi-Sugeno fuzzy models for polymerization, where they had a close look into the copolymerization of methyl methacrylate with vinyl acetate to test the ability of the recommended control system. They adopted a nonlinear mathematical model to justify the reaction plant that can generate data and probe deeper into the controller performance. The fuzzy approach adopted in their work suggested that it can predict outputs as a function of dynamic data input. The authors drew a comparison with this fuzzy approach with conventional predictive control in regulatory and servo issues, and as the result, the fuzzy controller was easy to implement, and its response is much more reliable.

Bringing together the capability of fuzzy logic predictive approaches and system characterization is appealing for designing controllers. Roubos et al. (1999) integrated the MPC algorithms with Takagi-Sugeno fuzzy models. In this approach, what happens first is that the model identification using fuzzy logic is given for MIMO architectures. With the establishment of the fuzzy model, it was brought together with MPC. The authors ran a test on this method for a two input-four output MIMO liquid level control case.

With the help of fuzzy-Hammerstein (FH) models, the nonlinear system was identified and controlled and this was described by Abonyi et al. (2000). This model carries a static fuzzy model linked with a series of a linear dynamic model. This model was integrated in a MPC control structure and a new method was introduced to handle constraints. To elaborate further, a simulated water-heater process was adopted. Simulation results have shown not only good dynamic modeling performance but also a well-captured steady-state behavior of the system. The application of fuzzy decision making (FDM) in MPC was studied by da Costa Sousa and Kaymak (2001), and there is a conformity of the results with those obtained from the usual MPC. Experiments with nonlinear dynamics were run on three states, namely an air conditioning system, unstable linear system, and a non-minimum phase. Results have suggested that the MPC performance can be made better using fuzzy criteria in the framework of fuzzy decision making.

To address the many optimization problems that are non-convex which come from the application of MPCs to nonlinear processes, Mendonça et al. (2004) gave the simplified version of fuzzy predictive filters to multi-variable processes. The introduced structure was implemented on a portal crane control. The benefits of the method were presented following the simulation results. The TS fuzzy design for a hybrid fuzzy modeling methodology was an idea proposed by Karer et al. (2007). The authors analyzed an MPC algorithm which was fit for systems with discrete inputs and the results have given a clue on the advantages of the MPC algorithm using the proposed fusion fuzzy model on a batch-reactor simulation example. The hybrid fuzzy predictive control design that is leaning on a genetic algorithm was suggested by Causa et al. (2008). Using two on/off input valves and a discrete-position mixing valve, the batch reactor temperature was controlled. The strategy revealed to be a proper technique to control hybrid systems, delivering the same performance in comparison with conventional hybrid predictive controllers plus large reductions in computation time. To add, the authors also formulated a problem with hybrid predictive adaptive control structure, and the results were believed highly potential.

4.3.3 Generic model controller

Generic model control (GMC) is a type of control scheme which directly uses the nonlinear process model. The dynamic mass, momentum, and energy balances are used to get first-principle models. If the process was still ambiguous, black-box models can be used to represent the unknown parts. Being a process model based control algorithm, the GMC incorporates a process model within the control structure directly. It has been illustrated to show excellent control, despite reasonable modeling errors. Abonyi et al. (2002) looked into these hybrid GMC models. They concluded that the first principles part of these models focuses on the main structure of the controller, while the black-box section plays its role as state/disturbance estimators.

Signal and Lee (1992) had built an algorithm within a GMC framework which seeks to mitigate larger modeling errors by updating the model parameters occasionally. Authors made a strong claim that this adaptive algorithm had the capability of adapting model parameters in a nonlinear model, where the parameters manifest themselves nonlinearly. Several examples were presented to highlight the technique principles.

Ali et al. (2007) had worked on a modified GMC and simulated it to analyze the results. It entails a model-based control of a propylene polymerization reactor in which the melt index and monomer conversion of the polymer are controlled by manipulating the inlet hydrogen concentration and the reactor cooling water flow. Nonlinear control is made, using a simplified nonlinear model, to show the strength of the control strategy. Two model variables are updated on-line to make sure that the outputs of the controlled process and their estimated values are followed carefully. The controller is the static inverse of the process model with set points of the measured process outputs changed to set points for several state variables. The simulation study illustrated that the suggested controller has good set point tracking and disturbance rejection properties and is the best, compared to the conventional GMC and Smith predictor control approaches.