Dynamic Modeling and Experimental Validation of the MMA Cell-Cast

Ind. Eng. Chem. Res. 2006, 45, 8539-8553

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Dynamic Modeling and Experimental Validation of the MMA Cell-Cast Process for Plastic Sheet Production Martıń Rivera-Toledo,†,‡ Luis E. Garcıá-Crispıń,† Antonio Flores-Tlacuahuac,*,† and Leopoldo Vı´lchis-Ramı´rez§ Departamento de Ingenierıá y Ciencias Quı´micas, UniVersidad Iberoamericana, Prolongacio´ n Paseo de la Reforma 880, Me´ xico D.F. 01210, Me´ xico, Departamento de Ingenierıá Quı´mica, Facultad de Quı´mica, UniVersidad Nacional Auto´ noma de Me´ xico (UNAM) Conjunto E, Cd. UniVersitaria, Me´ xico D. F. 04510, Me´ xico, and Centro de InVestigacio´ n y Desarrollo Tecnolo´ gico, AV. De Los Sauces 87-6, Lerma Edo. de Me´ xico 52000, Me´ xico

The dynamic modeling and experimental validation of the model response of the poly(methyl methacrylate) (PMMA) cell-cast process for plastic sheet production is presented. It is based on the straightforward initiation, propagation, and termination reaction polymerization mechanism. The sheet molding process was modeled by a two-dimensional dynamic mathematical model able to predict conversion, temperature, and molecular weight averages. The mathematical model is cast as a partial differential equations (PDEs) system that is discretized using the numerical method of lines. The resulting set of ordinary differential equations, representing the heat and mass balances for this polymerization system, are then solved by standard ordinary differential equations (ODE) solvers. Comparison of the model prediction capabilities against experimental temperature measurements taken at the extremes of the PMMA sheet being produced are presented. 1. Introduction Global economy and high commercial pressure demand manufacturing better and cheaper products, reducing capital and operating costs. From a process systems engineering (PSE)1,2 point of view, there is a wide scope for process improvement by using advanced mathematical modeling techniques. Once a reliable model is available, it can be used for several purposes such as simulation, control, process synthesis, steady-state and dynamic process optimization, etc. Polymerization reaction engineering constitutes a field where process improvements can be achieved trough appropriate process modeling. A common disadvantage of polymerization products relates to product heterogeneous features due to variations in raw material quality, changing process operating conditions, etc. Product heterogeneous characteristics tend to reduce process profit margins. By developing reliable polymerization models, process operating conditions could be determined, leading to homogeneous product quality. Of course, the details embedded in a model will depend on its economical impact and the available process knowledge. Working along the above ideas, in this work, our aim is to derive a first principles distributed dynamic mathematical model of an industrial polymerization system and to validate it by comparing its dynamic response against experimental pilot-plant data. The reported work represents a long-term research effort aiming to improve polymerization reaction operating conditions, hence leading to better product characteristics and increased profit margins, through advanced modeling, optimization, and control techniques. In particular, in this work, we address the poly(methyl methacrylate) (PMMA) industrial manufacturing process. PMMA is used, for instance, in the rear lights of cars. The spectator protection in ice hockey stadiums is made of PMMA, * To whom correspondence should be addressed. E-mail: [email protected]. Phone/Fax: +52(55)59504074. http:// 200.13.98.241/∼antonio. † Universidad Iberoamericana. ‡ Universidad Nacional Autońoma de Me´xico (UNAM) Conjunto E. § Centro de Investigacioń y Desarrollo Tecnolo´gico.

as are the largest windows and aquariums around the world. The material is also used to produce laser disks, and sometimes for DVDs. PMMA has a good degree of compatibility with human tissue and can be used for replacement of intraocular lenses in the eye when the original lens has been removed in the treatment of cataracts. In orthopedics, PMMA bone cement is used to affix implants and to remodel lost bone. Dentures are often made of this material, too. Commercial production of PMMA sheets can be carried out by processes such as cell casting (either in batch or continuous bulk polymerization), melt calendering, and melt-extrusion. The cell-casting process is becoming more important for PMMA production because of its flexibility in producing PMMA sheets with diverse physical and mechanical properties.3 In Figure 1, one of the flowsheets for the industrial manufacture of MMA is shown. In this, process monomer and small amounts of initiators react in a semibatch reaction system where a prepolymerization step takes place. The aim of the prepolymerization step is to mix the reactants, to remove the inhibitors normally contained in commercial MMA, and to heat the reaction mixture until polymerization reaction conditions are reached. Normally, in the prepolymerization step, only modest monomer conversion values (∼15-20%) are obtained. The prepolymer material is introduced into the casting mold in the form of a viscous liquid where polymerization reactions will proceed until most of the remaining monomer is consumed. The polymerization of the material contained between the glass plates is then carried out by heating the molds while the glass plates are clamped together. Initially, when the reaction heat produced at low conversion is small (because of the very low polymerization rate at those conditions), heat is provided to the molds by inserting them inside hot water baths. There, PMMA plastic sheet polymerization takes place. It has been reported4,5 that this process tends to produce nonuniform (i.e., measured in terms of molecular weight averages) plastic sheets, because polymerization does not take place at the same rate inside the PMMA polymerization mixture. Temperature variation is, perhaps, the main reason to explain polymer nonuniformity. Therefore, tight control of the operating conditions should help to diminish the

10.1021/ie060206u CCC: $33.50 © 2006 American Chemical Society Published on Web 09/28/2006

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Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006

Figure 1. Traditional plastic sheet production flowsheet.

for PMMA plastic sheet manufacturing purposes. Similarly to our work, Zhou et al.6 have reported a sheet-molding process for the production of PMMA that involves a prepolymerization batch reactor followed by polymerization in a sheet reactor. However, they did not compare their results against experimental data, and their formulation for heat energy balance considers only unidirectional heat transfer. Moreover, the simulation results presented by the authors do not reflect true operating times, since their reported dynamic responses tend to be quite faster when compared to characteristic industrial response times. The authors showed temperature changes of ∼70 °C achieved in only 30 s, while, according to our industrial experience, such a response is achieved in ∼25 min. Wanwanichai et al.7 report one- and two-step isothermal sheet-casting processes for production of PMMA sheets with a thickness of 3 mm, based primarily on water and water-air systems. In the present work, we propose a dynamic twodimensional mathematical model able to reproduce the industrial dynamic response of the cell-cast PMMA process. The dynamic model is cast in terms of a two-dimensional partial differential equation (PDE) system whose numerical solution was addressed using the method of lines, giving rise to a system of ordinary differential equations. The model considers free radical MMA bulk polymerization kinetics and heat conduction transport within the polymerization mixture. The model response is validated against experimental pilot-plant data. Good agreement between model response and experimental plant data was observed. We also present some simulation results for certain sheet plastic widths for which experimental results were not available. 2. Modeling

Figure 2. Cell-cast process for PMMA plastic sheet manufacture.

polymer nonhomogeneity problem. However, the abovedescribed traditional PMMA polymerization process using warm water baths features some shortcomings that could not be lowered by using even advanced control systems. Therefore, recently there has been interest in devising new and efficient plastic sheet polymerization processes featuring homogeneous polymer properties. An interesting variation of the PMMA polymerization process described above consists of using a furnace like that shown in Figure 2, where heat is provided by circulating warm air. There, the air temperature is modified by the cooling and heating exchangers. During start-up, there is a radiator through which cooling water is circulating. There are many thin fins on the tubular surface, steam enters into the tubes, and the heat is exchanged to the air. Using the convective oven process for plastic sheet production, PMMA polymer homogeneity features could be more easily enforced. A warm air temperature profile along the plastic sheet could be imposed and tracked by a control system. In pilot-plant experiments, it has been found that convective ovens, when properly operated, increase the PMMA polymer homogeneity characteristics and reduce the operating time allowed for complete polymerization. Moreover, temperature profile variations are faster to achieve when using air rather than water as the heating medium. This operating aspect becomes important if an on-line control system is used for quality control. It could also be important for temperature profile variations required to manufacture plastic sheets of widely different widths. In the research literature, there have been relatively few research papers dealing with the modeling of convective ovens

2.1. Description of the System. Casting is the process whereby a liquid is poured into a mold and allowed to react, cure, or harden to form a rigid object that takes the shape of the mold cavity. Many resin systems may be utilized in the process, e.g., acrylic, diethylene glycol bis(allyl carbonate), epoxy, phenol-formaldehyde, polystyrene, polysulfide, and silicone. Methyl methacrylate is the principal monomer used in the casting, or bulk polymerization, process of acrylic plastics. Acrylic sheets featuring high molecular weights are the primary casting product; however, rods, tubes, spheres, lenses, and other intricate shapes may also be produced in batch casting. Plastic sheet is conventionally cast between two parallel heat resistance glass plates, separated by a compressible gasket to allow for shrinkage. The prepolymer material which may be mixed with a mold release (stearic acid) is introduced into the casting mold in the form of a viscous casting liquid. The polymerization between the glass plates is then carried out by subjecting the mold to heat while the glass plates are clamped together. The glass plates are stacked inside a so-called sheet reactor, as shown in Figure 2. Depending upon the intended end use, plastic sheets of different widths can be manufactured. As the width of the plastic sheet rises, so does the polymerization time because of heat transfer resistance and viscous effects. It is well-known that careful control of the polymerization operating conditions is necessary in order to obtain a bubble-free product of good optical clarity. When the polymerization is about to begin, the charge molds are inserted inside a furnace which is heated by circulating warm air, as displayed in Figure 2. Industrial experience has demonstrated that the best results are obtained if the plates are extended in a horizontal plane during polymerization so as to negate as much as possible the hydrostatic pressure of liquid monomer or prepolymer, which has a tendency to cause outward building of the glass plates. During the

Ind. Eng. Chem. Res., Vol. 45, No. 25, 2006 8541

polymerization step, the temperature is maintained at such a value that rapid polymerization takes place without, however, permitting the temperature to run away so as to lose effective control over the thermal conditions. If the temperature rises too quickly, the polymerization may proceed like a chain reaction resulting in undesirable characteristics of the final product. Therefore, heating profiles may become an important manipulated variable to enforce tight control on the polymer properties. This issue will be deeply explored in future work and is one of the reasons why we undertake the derivation, and experimental validation, of a dynamic model able to mimic industrial cellcast PMMA polymerization operating conditions. 2.2. Process Mathematical Model. The cell-cast PMMA mathematical model was derived assuming that the heating source resulting from polymer reactions is a function of the local temperature. It was also assumed that polymer properties changed along the length (x-axis) and width (y-axis) of the plastic sheet as depicted in Figure 3. For simplicity, the glass plates are considered homogeneous and isotropic, i.e., the properties are the same in all directions (thermal conductivity, heat capacity, and density), and the average values were taken for MMA and PMMA. To derive the two-dimensional dynamic energy balance, the total heat entering and leaving at the x and y coordinates was modeled by the Fourier heat conduction law, and the rate of change of energy in the control volume was obtained by applying the shell energy balance method.8 Because of symmetry considerations, only half of the sheet width (from the center to the external surface) was taken into account. From a reaction engineering point of view, the PMMA plastic sheet process can be considered as taking place in a constant-volume batch reactor. Dynamic mass and energy balances coupled through polymerization kinetics9 describe polymer conversion and molecular weights dynamic time evolution. Air is circulated through the oven to provide the required energy to rise up the plastic sheet temperature until a point where significant polymerization rates take place. We assume that heat is transported by the forced convection mechanism. Inside the monomer, the dominant heat transfer mechanism is conduction. Therefore, heat is transferred along the x and y axes, giving rise to the following two-dimensional dynamic heat transfer equation.

(

)

∂T ∂2T ∂2T Q )R 2+ 2 + ∂t F ∂x ∂y MCp,M

(1)

which is subject to the following initial and boundary conditions

t)0 x)0

y)H

-kM

T ) T0 ∂T ) ha(T - Ta) ∂x

(2)

Figure 3. Control volume for energy balance.

2.3. Kinetic Model. The two major problem sources encountered in industrial mass polymerization processes are the heat released by the highly exothermic reactions and the large increase in viscosity of the reacting mixture over the course of the polymerization. For a typical addition polymerization, the heat of polymerization ranges from 10 to 20 kcal/mol, which can result in an adiabatic temperature rise of ∼200-400 °C. This large release of heat, coupled with the low thermal diffusivity of the reacting mixture, often leads to thermal runaway conditions. Therefore, proper control of the process is difficult. A temperature rise generally lowers the degree of polymerization. Hence, large temperature variations in the course of the reaction broaden the polymer molecular weight average, with accompanying deterioration of mechanical properties of the polymer. A quantitative cell-cast PMMA model able to describe free-radical polymerization over the entire course of the reaction under both isothermal and nonisothermal conditions has not been fully established. It is especially desirable to develop such a mathematical model to predict the performance and outcome of the cell-cast PMMA polymerization process and to correlate the product specifications with various operating parameters such as reactor temperature trajectory, initiator loading, etc. The reaction mechanism adopted here consists of the wellknown free-radical polymerization kinetics featuring straightforward initiation, propagation, and termination reactions. Chaintransfer reactions are neglected for convenience. The PMMA free-radical polymerization model taking place in a batch reactor is available elsewhere.9 However, we should mention that there are more complete kinetics polymerization models proposed in the literature.10 For instance, good reviews on this topic have been published by O’Neil et al.11 and Dube´ et al.12 Although Vivaldo-Lima et al.13 have compared the effectiveness of the Chiu et al.9 and Marten and Hamielec14 models, we found that the model proposed in ref 9 is simple enough and able to reproduce PMMA polymerization characteristics. Therefore, in this work, we use the polymerization kinetics and gel effect models as proposed by Chiu et al.9 The model is given as follows

(3)

dI I ) -kdI λ k (1 - X) dt 1 + X 0 p

(7)

x)L

∂T )0 ∂x

(4)

dX ) λ0kp(1 - X) dt

(8)

y)0

∂T )0 ∂y

(5)

dλ0 λ02 )(1 - X)kp + 2fkdI - ktλ02 dt 1 + X

(9)

-kM

∂T ) ha(T - Ta) ∂y

(6)

where T is the polymer temperature, T0 is the initial monomer temperature, Ta is the surrounding temperature, L is the sheet length, H is the sheet thickness, k is the average thermal conductivity, F is the density, Cp is the heat capacity, R is the thermal diffusivity, ha is the heat transfer coefficient, and Q stands for the heat released by the polymerization reactions.

dλ1 λ1λ0 )(1 - X)kp + 2fkdI - ktλ0λ1 dt 1 + X

1-X (10) + kpλ0M0 1 + X dλ2 λ2λ0 )k (1 - X) + 2fkdI - ktλ0λ2 + dt 1 + X p 1-X (2λ + λ0) (11) kpM0 1 + X 1

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dµ0 µ0λ0 1 )k (1 - X) + ktdλ02 + ktcλ02 dt 1 + X p 2

(12)

dµ1 µ1λ0 )k (1 - X) + ktdλ0λ1 + ktcλ0λ1 dt 1 + X p

(13)

dµ2 µ2λ0 )k (1 - X) + ktdλ0λ2 + ktc(λ2λ0 + λ12) (14) dt 1 + X p where represents the volume expansion factor determined by

)

(Fm - Fp) Fp

Figure 4. Notation used to identify grid points.

(15)

Molecular weight averages were calculated from the moments method15 as follows,

Mn )

(

)

(

)

µ1 + λ 1 m µ2 + λ 2 m Mw , Mw ) M µ0 + λ 0 µ1 + λ1 w

(16)

I ) I0, X ) X0, λ0 ) λ00 ,λ1 ) λ10, λ2 ) λ20, µ0 ) µ00, µ1 ) µ10, µ2 ) µ20 (17) 2.4. Model Solution. The application of the method of lines16 is an alternate approach for solving the set of partial differential equations describing the PMMA cell-cast process. We use this method for addressing the spatial discretization of both the PDE, which models temperature variations along the plastic sheet, as given by eq 1, and the boundary conditions (eq 2). The discretized equations thus obtained form a set of differentialalgebraic equations which can be numerically integrated by standard integration routines17 as an initial value problem. When the left-hand side of eq 1 is approximated by the method of lines, one obtains

)

Ti-1j - 2Tij + Ti+1j Tij-1 - 2Tij + Tij+1 dTij + + )R dt ∆x2 ∆y2 Qij , i ) 2, ..., Nx - 1, j ) 2, ..., Ny - 1 (18) FMCp,M similarly, the boundary and initial conditions are transformed as follows

kM

3T1j - 4T2j + T3j kM ) ha(T1j - Ta) 2∆x

(19)

3TNxj - 4TNx-1j + TNx-2j )0 2∆x

(20)

-3Ti1 + 4Ti2 - Ti3 )0 2∆y

(21)

3TiNy - 4TiNy-1 + TiNy-2 ) ha(TiNy - Ta) 2∆y

t ) 0 Tij ) T0

initial monomer & initiator conc. initial monomer conversion initial living & dead moments polymer density initial temperature initiator efficiency

9.98 & 0.0258 X)0 λ 0 ) 0 & µ0 ) 0 1200 343.15 0.58

mol/dm3 mol/dm3 kg/m3 K

Table 2. Simulation Data for Sheet Reactora

where λ0, λ1, and λ2 are the zeroth and first moments of the growing radicals; µ0, µ1, and µ2 are the zeroth, first, and second moments for the dead polymer, respectively; and Mm w is the monomer molecular weight. The above set of equations are subject to the following initial conditions:

(

Table 1. Simulation Data for Batch Reactor

(22)

i ) 1, ..., Nx, j ) 1, ..., Ny (23)

monomer & initiator initial conc. monomer conversion living & dead moments heat of reaction polymer thermal conductivity polymer heat capacity air temperature initial temperature sheet length sheet thickness

Mout-batch & Iout-batch Xout-batch λ0out-batch & µ0out-batch -58.19 0.09 1674 318 298.15 1.8 0.003, 0.006, 0.012, & 0.018

mol/dm3 mol/dm3 kJ/mol W/(m-K) J/(kg-K) K K m m

a The subscript out-batch means that the numerical value of the property is taken as the one at the outlet of the batch reactor.

where ∆x ) L/(Nx - 1) and ∆y ) H/(Ny - 1). Here, we have used the well-known second-order finite difference approximation for the first and second derivatives along the x- and y-axes. The backward difference formula was applied to the boundary conditions at x ) 0 and y ) 0, and the forward difference formula was used at x ) L and y ) H. In the definition of the finite differences, the point of reference is taken to be the point Tij in Figure 4. After some trials, we found that 20 and 8 points suffice to represent the system dynamic response along the longitude and thickness of the plastic sheet, respectively. 3. Results The aim of this section is to validate the response of the underlying cell-cast PMMA mathematical model by comparing it against industrial pilot-plant experiments. To this end, several experiments were run at a local pilot-plant facility using plastic sheet thicknesses of 3 and 6 mm. Because of technical difficulties for carrying experiments using a larger sheet thickness, only open-loop simulation results are presented for 12 and 18 mm plastic sheet thicknesses. Even when a direct experimental comparison was not carried out, good qualitative agreement was also observed. Moreover, it is hoped that the predictive capabilities of the model would allow us to extrapolate its response for larger thicknesses. In Table 1, design data for the prepolymerization step carried out in a batch reactor are shown, while Table 2 contains the corresponding design data for the PMMA plastic sheet reactor. Information regarding the PMMA kinetic rate constants is displayed in Table 3. Pilot-plant experiments were conducted under isothermal conditions; during all the experiments, air was forced to be circulated and kept at 50 °C. To record sheet temperatures, two


Figure 5. Comparison of experimental (o) versus simulation results. The continuous line stands for constant physical properties, while the short dashed line (- - -) represents results obtained allowing temperature-dependent physical properties and the large dashed line (- -) represents dependent conversion besides temperature. (a) and (b) are temperature results for 3 mm sheet thickness at x ) 0 and x ) L, while similarly, (c) and (d) represent temperature results for a 6 mm sheet thickness at x ) 0 and x ) L. Table 3. PMMA Kinetic Rate Constants

(

)

2.3φm 1 1 + θp(T)λ0 exp ) kp kp0(T) A + B(T)φm

apparent propagation coefficient9

(

2.3φm

apparent termination coefficient9

1 1 + θt(T, I0)λ0 exp ) kt kt0(T) A + B(T)φm

monomer diffusion time for propagation stage9

θp ) exp -35.167 +

monomer diffusion time for termination stage9 parameter A in the Fujita-Doolittle parameter B9

equation21

true propagation coefficient9 true termination

coefficient9

initiator coefficient22 volume fraction of the monomer9 kinetic coefficient for termination kinetic coefficient for termination by addition9 glass transition temperature

termocouples were located at the x extremes (x ) 0 and x ) L) of the sheet. A data acquisition system was used for on-line temperature recording using sampling intervals of 7 min. Only polymerization temperatures recorded at the furnace are re-

)

14 000 ( T ) 17 500 θ ) exp(-46.983 + T ) t

A ) -8 × 10-6(T - Tg)2 + 0.1678 B ) 0.03 4353 kp0 ) 2.95 × 107 exp 1.987T 701 kt0 ) 5.88 × 109 exp 1.987T 14 855 kd ) 4.25 × 1016 exp T 1-X φm ) 1 + X kt ) ktc + ktd ktc ) 0 Tg ) 387.15

( ( (

) ) )

ported; no measurements related to the prepolymerization stage were recorded. As depicted in Figure 5a, because of the warm air flow direction, at x ) 0 the temperature rises quicker when compared to the same temperature increase at x ) L, as

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Figure 6. Temperature dynamic profiles for (i) 3, (ii) 6, (iii) 12, and (iv) 18 mm at (a) y ) 0 (interior middle point) and (b) y ) H (external plate surface). Solutions are labeled for the edge (x ) 0) and right extreme (x ) L) along the longitudinal x-coordinate. The rest of the nonlabeled temperature profiles correspond to discretized points along the x-coordinate.

displayed in Figure 5b. After 4 h, the sheet temperatures acquired the steady-state value. Here, steady state means that, after the reaction has been essentially completed, polymer

properties (i.e., conversion, molecular weight averages, etc.) remain constant. However, at x ) L, the temperature dynamic response is slower, but finally it also attains the steady-state


Figure 7. Monomer conversion profile along the longitude of the plastic sheet: s, x ) 0; - -, x ) L/4; - ‚ -, x ) L/2; and ‚ ‚ ‚, x ) L for (a) 3 mm, (b) 6 mm, (c) 12 mm, and (d) 18 mm plastic sheet thickness. Table 4. Temperature- and Conversion-Dependent Properties Fm km Cpm Fp kp Cpp FM kM Cp,M ka νa Pra Rea ha

value when the polymerization reaction has been completed. In both figures, a temperature peak can be clearly seen. The peak is related to the onset of the gel effect, and it is larger at x ) L because, at this point, thermal effects are stronger because of the heat produced by the reaction contribution. At x ) 0, the simulation and experimental results are shifted by 15 min. Moreover, at x ) L, the agreement between simulation and experimental results is good.

(0.973 - 1.164 × 10-3(T - 273)) × 1000 0.13 1.674 (1.9872 - 3.971 03 × 10-3T + 4.4626 × 10-6T2) × 1000 -0.06852 + 8 × 10-4T 1.45 FpX + Fm(1 - X) kpX + km(1 - X) Cp,pX + Cp,m(1 - X) 2 × 10-8T2 + 7 × 10-5T + 0.0243 13.638 e(0.0052T)x10-6 -7 × 10-7T2 - 5 × 10-5T + 0.7145 LV/νa 0.0288Rea4/5Pra1/3ka/L

In all the simulation responses discussed so far, constant physical properties were assumed. In a real industrial environment, it is unlikely that such an assumption can be justified. Commonly, constant physical properties are assumed because this assumption can drastically lower the computational load needed for model solution. However, with the advent of power computer processors and/or parallel computing environments, assumptions such as the one discussed above can be relaxed,

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Figure 8. Initiator concentration profile along the longitude of the plastic sheet: s, x ) 0; - -, x ) L/4; - ‚ -, x ) L/2; and ‚ ‚ ‚, x ) L for (a) 3 mm, (b) 6 mm, (c) 12 mm, and (d) 18 mm plastic sheet thickness.

leading to models with improved prediction capabilities. To get a clear idea about modeling improvements, physical properties were made a function of temperature, using the physical properties correlations as displayed in Table 4. Moreover, some properties were also assumed to be a function of monomer conversion (X). In Figure 5, the temperature dynamic response using temperature-dependent physical properties is displayed by dashed lines. For the 3 mm plastic sheet thickness, the predicted results using either constant or temperature-dependent physical properties look similar. The CPU time on a Pentium 4 PC computer running at 2.6 GHz for the physical properties as a function of temperature was ∼320 min, while for the constanttemperature case, it required only 11 min. Intuitively, because of heat transfer limitations leading to large local temperature spots, we expect that good model prediction capabilities for a larger sheet thickness will require temperature-dependent physical properties. As seen from Figure 5, if the physical properties are both temperature- and conversion-dependent, the predicted results are better than the cases of only temperature-dependent physical properties. However, because physical properties as a function of conversion and temperature require quite large CPU computer times (∼48 h) to get a typical model response, we will keep using constant physical properties that demand

relatively modest computational loads for getting the same type of model response. In parts c and d of Figure 5, temperature dynamic responses for a 6 mm plastic sheet thickness are shown. The experimental temperature response becomes slightly sluggish when compared to the similar 3 mm plastic sheet temperature response. As the sheet thickness increases, so do the heat transfer limitations; this explains the rise in the process time constant. It is also interesting to realize the growth of the temperature peak: as the sheet thickness increases, so does the temperature peak. This behavior can be partially attributed to the availability of larger amounts of monomer that contribute to extend the gel effect outcome. Moreover, this behavior is related to the lower mass/ heat transfer area ratio. Hence, heat accumulation occurs as a consequence of greater amounts of released heat. Another point to highlight is that, regardless of the sheet thickness, the temperature peak occurs at practically the same operating time. Overall, the predicted temperature response is not as good as it was for the 3 mm sheet thickness; even using temperaturedependent physical properties, the model response could not be further enhanced. We decided to use the same numerical values of the physical and kinetic constants in both the 3 and 6 mm sheet thicknesses. This partially explains the worsening in


Figure 9. Number-average molecular weight profile along the longitude of the plastic sheet: s, x ) 0; - -, x ) L/4; - ‚ -, x ) L/2; and ‚ ‚ ‚, x ) L for (a) 3 mm, (b) 6 mm, (c) 12 mm, and (d) 18 mm plastic sheet thickness.

temperature response. Although some model improvement could be achieved by proper parameter fitting, we decided not to do so to test the validity of our model using standard physical and kinetic parameter values. Of course, real time use of the model would require additional off-line and perhaps on-line parameter estimation. Figure 6 displays plastic sheet temperature profiles at y ) 0 (interior middle point of the plate) and y ) H (surface of the plate) along the longitude of the sheet corresponding to the x-coordinate as shown in Figure 3. The results depicted in Figure 6 were obtained by solving the cell-cast PMMA dynamic twodimensional model given by eqs 1-17, and corresponding to 3, 6, 12, and 18 mm plastic sheet thicknesses. Temperature profiles are labeled for the edge (x ) 0) and end (x ) L) parts of the sheet; the rest of the nonlabeled temperature profiles correspond to intermediate longitudinal points. As can be seen, the shape of the temperature profile is very similar for the 3 mm sheet thickness. Although the temperature profile shape looks similar, the difference of the maximum peak temperature tends to rise as sheet thickness is increased. Therefore, the internal temperature profiles (y ) 0) tend to be higher than the external ones (y ) H). Indeed, for the 18 mm sheet thickness, the temperature difference is ∼6 °C. Internal temperature

profiles are larger than the external ones because of the cooling effect of air flowing over the plastic sheet. Moreover, as the polymerization reaction goes on, polymer viscosity increases, leading to heat transfer limitations. Hence, heat dissipation becomes harder to achieve. In Figure 6, we also note that the gel effect suddenly appears at different times for a long plate; this autoacceleration effect causes a conversion rate increment, driving the system (see Figure 7c) from 0.85 to 0.94 conversion degree. This effect suffices to obtain a nonuniform plate thickness (see Figure 12c). In this case, there is a thickness difference of ∼0.2 mm (1.6%) between the plastic sheet edges. The initial edge (x ) 0) of the sheet reactor is closest to the heat source (the air flow rate). In consequence, the temperature rise in this zone is higher than in the rest of the sheet. This is the reason the onset of gel effect appears first there rather than in the rest of the system. Because the initial edge of the sheet is the one with the quickest temperature response, it is also the one with the highest amount of instantaneous heat release because of the exothermic nature of the MMA polymerization reaction; the latter phenomenon clearly contributes to an even higher temperature rise. Even though the presence of this temperature sudden rise, due to the onset of the autoacceleration effect, can act like the driving force for heat transfer to the rest

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Figure 10. Weight-average molecular weight profile along the longitude of the plastic sheet: s, x ) 0; - -, x ) L/4; - ‚ -, x ) L/2; and ‚ ‚ ‚, x ) L for (a) 3 mm, (b) 6 mm, (c) 12 mm, and (d) 18 mm plastic sheet thickness.

of the sheet reactor, the heat transfer resistance in the internal part of the sheet is higher than the one presented by the glass at the interface. Hence, most of the heat flow is exchanged at the interface by convection with the flowing air. Figure 6 shows the final temperature profile once monomer has been almost totally consumed, as can be seen at Figure 7, too. At this time, the polymerization reaction in the sheet reactor is over and the amount of heat generated by the reaction has been totally exchanged with the air, so the system attains thermal equilibrium conditions. Because of the initial and operating conditions of the system, final values of ∼0.9 conversion were obtained. A widely used technique in the polymerization industry is the postcuring stage of the sheet (see Figure 1), well above the glass transition temperature of the polymer, to consume the remaining monomer. The basic chemistry involved in free radical polymerization is by now well-established.18 Highly reactive free radical species trigger fast chain reactions, adding monomers into polymer chains as follows. When a free radical attacks a monomer, it transfers its active center to the monomer itself, initiating a live growing chain, or macroradical, as successive monomers are added in similar fashion. These living chains propagate into the surrounding monomer medium, with their growing ultimately

terminated by interpolymeric reactions in which pairs of living chains annihilate each other. Each termination reaction results in a dead chain; this is the final polymer product. In Figure 7, we see that monomer conversion gradually increases with the polymerization temperature. Note that monomer conversion is lowest at 50 °C; this is consistent with the glass effect, whereby the reaction mixture essentially freezes at a concentration whose glass transition temperature (Tg) corresponds to the reaction temperature. Another important feature characterizing both the experimental temperature data and the predicted curves is the curvature of the conversion profile before the onset of the socalled gel region. The monomer conversion profile is homogeneous (meaning that the conversion profile is almost the same at any point on the plastic sheet) along the 3 mm thickness flat plate, as shown in Figure 7a, while the same conversion profile is slightly heterogeneous for the 6 mm thickness plastic sheet, as seen in Figure 7b. These curves are convex form preceding the sharp rise. When the gel effect appears suddenly, the movements of polymer chains are restricted as a result of the high viscosity of the reaction mixture; this leads to a dramatic decrease in the termination rate constant and significant autoacceleration of the polymerization because of the increased freeradical concentration. Thus, at extremely high conversions, even


Figure 11. Polydispersity profile along sheet longitude: s, x ) 0; - -, x ) L/4; - ‚ -, x ) L/2; and ‚ ‚ ‚, x ) L for (a) 3 mm, (b) 6 mm, (c) 12 mm, and (d) 18 mm plastic sheet thickness.

the movements of small molecules are restricted, lowering the propagation rate constant and possibly leading to a complete termination of polymerization. Therefore, the portion of the conversion curve under examination would be concave. Hence, a major difference exists between the present model and other contemporary ones in that our model incorporates diffusion in the basic sequence of events constituting the termination of growing radicals from the very outset of reaction. Onset of gel effect is only a convenient phrase denoting the region of rapid increase in monomer conversion. There exists, in fact, no sharp demarcation in terms of molecular processes before and after the occurrence of this autoacceleration region. The rapid rise in conversion is merely a natural consequence of the increasing importance of mass transfer limitations.9 Figure 7 clearly displays a critical problem arising in the isothermal cellcast process: as the plastic sheet thickness rises, so do the heterogeneous characteristics of the plastic sheet, leading to sheet zones of different polymer properties (i.e., conversion, molecular weight average, etc.). Heterogeneity could be lowered, to a certain extent, by manipulating the air flow rate temperature. In Figure 8, the dynamic evolution of the initiator concentration is shown. It can be seen that the initiator concentration decreases linearly at the initial edge of the sheet, and it is not

affected by the thickness of the sheet. For sheet adjacent points, the spectacular decrease in initiator concentration is accompanied by rapid temperature rises; this is a consequence of the autoacceleration effect. Also, increasing temperature at a constant initiator loading significantly lowers the molecular weight. At high conversions, when termination rate is greatly reduced, the average molecular weight rises appreciably. The influence of the process variables on the characteristics of the product can be understood from the molecular weight average (Mw) profiles calculated by the method of moments. It is well-known that the material properties (stress-strain, toughness, and fracture and fatigue behavior) depend on the molecular weight average of a given polymer.19 It has been reported that mechanical properties improve with molecular weight up to a number average ∼105. The molecular weight averages for a long flat plate are plotted in Figure 10 for 3 and 6 mm plastic sheet thicknesses. The effect of a runaway scenario on the molecular weight is associated with the gel effect. It can be seen that Mw decreases along the plastic sheet from 3% to 6% as seen in Figure 10a, in agreement with basic rules-of-thumb,20 meaning that the net effect of temperature rise is a decrease in the molecular weight profile. On Figure 10b, the Mw profile decreases compared to the 3 mm plate; Mw falls from 4% and 11% for the middle and final points, respectively. Similar

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Figure 12. Variation of plastic sheet thickness: s, x ) 0; - -, x ) L/4; - ‚ -, x ) L/2; and ‚ ‚ ‚, x ) L for (a) 3 mm, (b) 6 mm, (c) 12 mm, and (d) 18 mm plastic sheet thickness.

behavior is observed for large plastic sheet thicknesses, as depicted in Figure 10 parts c and d. On Figure 11, the polydispersity index is displayed at four points along the longitude of the plastic sheet. Since the polydispersity index measures the breadth of molecular weight average, it is expected to increase when the instantaneous molecular weight undergoes large variations during the course of the reaction. In parts c and d of Figure 11, we can see that the dotted line (x ) L) corresponds to the one showing the highest temperature peak, as depicted in Figure 6. Therefore, gel effect onset and dramatically runaway conditions seem to be accompanied by substantial increases in the polydispersity index. When the peak of maximum temperature is reached, the polydispersity index abruptly levels off, since the reaction is prematurely stopped by the depletion of initiator, as displayed by dotted lines in Figure 8 parts c and d. The plastic sheet zones featuring temperature peaks can have a disastrous consequence on the polydispersity index. As shown by the conversion profiles corresponding to these cases, a significant amount of polymer is still formed beyond the hot spot. The chains formed after a runaway situation, however, have a different molecular weight, since many process variables and characteristics of the polymerizing medium (initiator concentration, temperature, and, consequently, termination rate kinetics) have changed. The growing

of a second population of chains of different molecular weights is responsible for the drastic increase in the polydispersity index after the highest plastic sheet temperature is reached. In Figure 12, thickness variation behavior for the four cases explained above is displayed. The first case, Figure 12a, has a uniform shrinkage and features an estimated 18% thickness variation between the initial and end time for the polymer reaction. In Figure 12b, the plastic sheet has an 18% thickness variation and 0.7% variation (not shown) along the sheet longitude; this last change is not really important. Thickness variation problems do not decrease as polymer sheet width rises. Indeed, as displayed in parts c and d of Figure 12, plastic sheets of 12 and 18 mm also feature ∼18% thickness variation. Wide thickness variation is a severe problem that could be lowered by proper process operation. We could achieve homogeneous thickness by eliminating the high-temperature behavior at the end edge for both cases. If the heating profile is near to isothermal conditions, uniform shrinkage during the cell-cast process will be promoted. Figure 13 depicts the variation of temperature and monomer conversion profiles at quarter (x ) L/4) and half (x ) L/2) the plastic sheet longitude for three points along the y-coordinate: the center (y ) 0), the surface (y ) H), and half of the thickness (y ) H/2). In Figure 13a, the temperature rises slowly and the


Figure 13. Predicted temperatures and monomer conversion: s, y ) 0; ‚ ‚ ‚, y ) H/2; and - ‚ -, y ) H for (a) and (b) x ) L/4, and (c) and (d) x ) L/2 for an 18 mm thickness plastic sheet.

temperature of the surface is greater than the one at the center, because the air heats the surface of the sheet, and heat is transported into the bulk polymerization through the heat conduction mechanism. At ∼300 min, the gel effect appears and the difference of temperature change becomes maximal, 4 °C at 300 min. After that, the temperature approach, T(y ) 0) - T(y ) H), is zero at 500 min and the temperature is 71.5 °C. From this time on, the inner temperature will be greater than the surface temperature; thus, the heat produced by the reaction is not removed efficiently by the air. The plastic sheet thermal behavior affects the monomer conversion. Indeed, we can see in Figure 13b that, for the first 300 min, the conversion is almost 35%, but suddenly, it becomes too high, 82% and 85% at the inner and surface plate positions, respectively. When the temperature approach is zero, the monomer conversion is 91%; this is near to maximum conversion for final polymer processing. It should be noticed that the thermal response at half and quarter plate is very similar, but now in Figure 13c, the temperature approach becomes too large, 24 °C at 400 min, and so forth; the cross-temperature appears at 435 min and its temperature value is 90 °C, greater than the case of quarter plate. In Figure 13d, the monomer conversion increases slowly from 0 to 350 min, but when the gel effect appears, it grows suddenly and

gets 88% and 90% values at the central and surface points, respectively. Even when, in this work, no parameter estimation was undertaken, a parameter sensitivity analysis was carried out. Model response was sought for small changes on the free volume parameters (A, B, θp, and θt). Chiu et al.9 report that A increases with the temperature, and its range of change is small (0.13-0.16) on the temperature operation range. However, there was not an effect on changing this parameter for the model response. It has been found empirically that B is not sensitive to temperature for many systems and can be treated as a constant. θp and θt reflect primarily the behavior of the diffusion coefficient at the limit of vanishing the volume fraction of the monomer for termination and propagation, respectively, and θt is a function of initial initiator concentration, too. Small changes on θp did not not significantly affect model response. We found that θt mainly affects sheets featuring a larger thickness, because this parameter reflects the mass transport processes for growing radicals and monomer. Figure 14 displays the temperature behavior when θt is (5% changed. In parts c and d of Figure 14, the sensitivity results are shown for 6 mm sheet thickness at x ) 0 and x ) L, respectively. If we change θt by -5%, it was found that the simulation results display a better matching,

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Figure 14. Comparison of experimental (o) versus sensitivity simulation results: (s) physical constant properties, (‚‚‚) physical dependent properties (for the same θt values), (- ‚ -) when θt is affected on changing by -5%, and (- -) when θt is affected on changing by +5%. (a) and (b) are temperature results for 3 mm sheet thickness at x ) 0 and x ) L, respectively, and (c) and (d) represent temperature results for 6 mm sheet thickness at x ) 0 and x ) L, respectively.

because the diffusional effect increases and then the temperature increases faster, showing a larger peak. 4. Conclusions Good agreement between pilot-plant experimental results and model predictions for the cell-cast PMMA plastic sheet process was confirmed. A dynamic two-dimensional model able to predict conversion rate, molecular weight, and temperature distributions in a PMMA cell-cast sheet reactor was developed. From the results discussed in this work, we can conclude that the proposed model is a useful tool to test the advantages of using heating ovens for the sheet polymerization process. In contrast, the traditional PMMA polymerization process using warm water baths tends to produce both nonuniform conversion rates and molecular weight averages. Therefore, by proper operation, heating ovens should lead to obtaining polymerization products with uniform polymerization properties. Comparison against experimental data shows that the model prediction capabilities seem to be satisfactory for industrial application. Hence, we expect to use this model to compute optimal

temperature trajectories able to maintain uniform molecular weight averages to achieve the specified product quality. Once the optimal temperatures profiles have been computed, the problem of how to closed-loop track such optimal transition trajectories, in the presence of modeling errors, remains. Nomenclature Cp ) heat capacity (J/kg‚K) f ) initiator efficiency h ) heat transfer coefficient [W/(m2 K)] H ) sheet thickness (m) I ) molar concentration of initiator (mol/dm3) k ) thermal conductivity [W/(m‚K)] kd ) kinetic coefficient for initiator (1/min) ki ) kinetic coefficient for initiation reaction [dm3/(min‚mol)] kp ) kinetic coefficient for propagation [dm3/(min‚mol)] ktc ) kinetic coefficient for termination by addition [dm3/(min‚ mol)] ktd ) kinetic coefficient for termination disproportionation [dm3/ (min‚mol)]


L ) sheet length (m) M ) molar concentration of monomer (mol/dm3) Mn ) number-average molecular weight (kg/kmol) Mw ) weight-average molecular weight (kg/kmol) Mm w ) monomer molecular weight (kg/kmol) Nx ) number of points for discretization on x-direction (dimensionless) Ny ) number of points for discretization on y-direction (dimensionless) Pr ) Prandtl number (dimensionless) Q ) heat released by the polymerization reactions (W/m3) R ) universal gas constant [kJ/(mol‚K)] Re ) Reynolds number (dimensionless) T ) polymer temperature (K) Ta ) air temperature (K) Tg ) glass transition temperature of the pure polymer (K) T0 ) initial monomer temperature (K) t ) reaction time (min) X ) monomer conversion x ) x-axis for the sheet length (m) y ) y-axis for the sheet thickness (m) Greek Letters R ) thermal diffusivity (m2/min) ∆Hr ) heat of polymer reaction (kJ/mol) ) volume expansion factor (dimensionless) λ0 ) zeroth moment of the growing radicals (mol/dm3) λ1 ) first moment of the growing radicals (mol/dm3) λ2 ) second moment of the growing radicals (mol/dm3) µ0 ) zeroth moment for the dead polymer (mol/dm3) µ1 ) first moments for the dead polymer (mol/dm3) µ2 ) second moments for the dead polymer (mol/dm3) ν ) kinematic viscosity (m2/min) F ) density (kg/m3) Subscripts a ) air i ) position in x-direction j ) position in y-direction m ) monomer properties M ) polymer mixture properties p ) methyl methacrylate properties Literature Cited (1) Kiparissides, C. Polymerization Reaction Modeling: A Review of Recent Developments and Future Directions. Chem. Eng. Sci. 1996, 51 (10), 1637-1659. (2) Penlidis, A. Polymer Reaction Engineering: From Reaction Kinetics to Polymer Reactor Control. Can. J. Chem. Eng. 1994, 72, 385-391. (3) McKetta, J. Encyclopedia of Chemical Processing Design; Marcel Dekker Inc.: New York, 1992.

(4) Rosetti, C. Apparatus for Casting Plastic Sheet. U.S. Patent 3,689,022, 1972. (5) Daddona, P. Apparatus for the Production of Case Polymer Sheets. U.S. Patent 3,694,129, 1972. (6) Zhou, F.; Guptam, S.; Ray, A. K. Modeling of the Sheet-Molding Process for Poly(methyl methacrylate). J. Polym. Sci., Part A: Polym. Chem. 2001, 81, 1951-1971. (7) Wanwanichai, P.; Junkasem, J.; Saimaneewong, B.; Mahasan, R.; Tantivess, P.; Magaraphan, R.; Vanichvarakij, Y.; Petiraksakul, P.; Supaphol, P. Sheet-cast Poly(methyl methacrylate): One-step (Water) versus Twostep (Water-Air) Isothermal Processes. Iran. Polym. J. 2005, 14 (1), 6169. (8) Bird, R. B.; Stewart, W. E.; Lighfoot, E. N. Transport Phenomena; John Wiley: New York, 1960. (9) Chiu, W. Y.; Carrat, G. M.; Soong, S. A Computer Method for the Gel Effect in Free Radical Polymerization. Macromolecules 1983, 16, 348357. (10) Achilias, D. S.; Kiparissides, C. Development of a General Mathematical Framework for Modeling Diffusion-Controlled Free Radical Polymerization Reactions. Macromolecules 1992, 25, 3739-3750. (11) O’Neil, G. A.; Torkelson, J. M. Recent Advances in the Understanding of the Gel Erect in free Radical Polmeryzation. Trends Polym. Sci. 1997, 5, 349-355. (12) Dube´, M. A.; Soares, J. B. P.; Penlidis, A.; Hamielec, A. E. Mathematical Modeling of Multicomponent Chain-Growth olymerizations in Batch, Semibatch, and Continuous Reactors: A Review. Ind. Eng. Chem. Res. 1997, 36, 966-1015. (13) Vivaldo-Lima, A.; Hamielec, A. E.; Wood, P. E. Auto-Acceleration Effect in Free Radical Polymerization. A Comparison of the CCS and MH Models. Polym. React. Eng. 1994, 2, 17. (14) Marten, F. L.; Hamielec, A. E. High-conversion diusion-controlled polymerization. In ACS Symposium Series; Henderson, H. N., Bouton, T. C., Eds.; American Chemical Society: Washington, DC, 1979; Vol. 104. (15) Ray, W. H. On the Mathematical Modeling of Polymerization Reactors. J. Macromol. Sci., ReV. Macromol. Chem. 1972, c8 (1), 1-56. (16) Schiesser, W. E. The Numerical Method of Lines. Integration of Partial Differential Equations. Academic Press: New York, 1991. (17) Ascher, L.R.; abd Petzold, U. M. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM 1998. (18) Odian, G. Principles of Polymerization; Wiley-Interscience: New York, 1991. (19) Nunes, R. W.; Martin, J. R.; Johnson, J. F. Influence of Molecular Weight and Molecular Weight Distribution on Mechanical Properties of Polymers. J. Polym. Sci., Part A: Polym. Chem. 1982, 22, 205. (20) Baillagou, P. E.; Soong, D. S. MolecularWeight Distribution of Products of Free Radical Nonisothermal Polymerization with Gel Effect Simulation of Poly(Methyl methacrylate). Chem. Eng. Sci. 1985, 40 (1), 87-104. (21) Stewart, M. D. Catalyst Diffusion in Positive-Tone Chemically Amplified Photoresists. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, 2003. (22) Akzo Nobel Chemicals, Inc. Initiators for Polymer Production Product Catalog; Publication PC-2002-05-b; Akzo Nobel Chemicals Inc.: Chicago, IL, 2002; http://www.akzonobel-polymerchemicals.com/ ProductGroups/.

ReceiVed for reView February 19, 2006 ReVised manuscript receiVed August 21, 2006 Accepted August 21, 2006 IE060206U

Dynamic Modeling and Experimental Validation of the MMA Cell-Cast

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