Attainable Regions for Polymerization Reaction Systems - Industrial

Alberto Nisoli, Michael F. Doherty, and Michael F. Malone. Industrial & Engineering ... Charles David Immanuel and Francis Joseph Doyle, III. Industri...
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Ind. Eng. Chem. Res. 1997, 36, 1076-1084

Attainable Regions for Polymerization Reaction Systems Raymond L. Smith and Michael F. Malone* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003

A method is described for determining the limits of number-average molecular weights, polydispersities, monomer conversions, residual initiator concentrations, and reactor network residence times that are possible in isothermal polymerizations. The general approach provides a visual representation of the “attainable region” and reactor networks which achieve the attainable limits. The approach is based on an extension of the work of Glasser et al. (1987) to polymerization systems and is illustrated with a specific example of the free-radical polymerization of poly(methyl methacrylate). The results show how design conditions affect the range of attainable number-average molecular weights; the results indicate that much higher molecular weights than those typically produced in industry are feasible. The results also demonstrate that while many reactor networks can produce a desired molecular weight, only a few carefully selected networks can yield the narrowest molecular weight distributions. These include a CSTR reactor, a CSTR reactor with a bypass feed stream, and a CSTR reactor followed by a PFR reactor. Introduction In the design of a plant to manufacture a polymer resin, one has a variety of specifications on the desired product(s). These specifications include molecular weights, polydispersities, purities, tacticities, morphologies, etc. These are actually indirect specifications on the end-use and processing properties needed by the consumer or fabricator. A specification on purity often requires particular residual concentrations of monomer and/or initiator, while many end-use properties are correlated with the molecular weight including tensile strength, impact strength, fatigue life, melting point, and glass transition temperature (e.g., Martin et al., 1972; Nunes et al., 1982). Molecular weight also affects processing properties, which are a concern of fabrications. For example, Tadmor and Gogos (1979) describe how molecular weight, through associated rheological properties such as viscosity, affects fiber spinning and extrusion. In particular, the ease with which a polymer can be fabricated depends on the processing temperature; Tadmor and Gogos (1979) quote a desirable value of more than 100 °C above the glass transition temperature. The processing temperature increases with increasing molecular weight, but if the temperature is too high, the polymer may degrade. Therefore, polymers with high molecular weights can have a limited temperature range for processing. As the molecular weight of a polymer increases, the cost of the reactor network for its manufacture generally increases. For example, in free-radical polymerizations, to increase the molecular weight, one can decrease the temperature and/or the amount of initiator (Rodriguez, 1989). Although both of these changes result in higher molecular weights, they also lead to lower rates of polymerization and thus to an increased size and investment cost for the reactors. Along with the reactor costs, the resin producer is also interested in separation and purification costs. Resins are normally devolatized to remove monomer, and residual initiator is also removed if the amount is too * Author to whom correspondence should be addressed. E-mail: [email protected]. Phone: (413)545-0838. Fax: (413)545-1133. S0888-5885(96)00358-2 CCC: $14.00

high. These separations may be minimized or even eliminated if the reactor network can be designed to give high conversions and low residual concentrations of initiator. Thus, the proper design of the reactor network also has the potential to provide significant savings in separation and purification costs. In polymerization plants the preferred design should take into consideration end-use properties, ease of processing, and plant costs. Hence, it is desirable to determine which specifications can be met at various design conditions. In order to see what average molecular weights, polydispersities, conversions, residual concentrations of initiator, and reactor network residence times can be produced, the “attainable region” concept originally introduced by Horn (1965) and expanded recently by Glasser et al. (1987) is developed to analyze the moments of the molecular weight distribution (MWD). As an example, the attainable region is generated for the isothermal free-radical polymerization of poly(methyl methacrylate). Background The attainable region, described in Glasser et al. (1987) for chemical systems, is a geometric technique used to determine the limits on the distribution of products which one can obtain by any combination of reaction and mixing. In constructing this region, models for the processes of reaction and mixing are employed to obtain a convex region which cannot be extended further. Reactor networks which achieve the limiting product distributions are also found in the course of the construction. Previously, Glasser et al. (1987) described how to generate the attainable region for two quantities which follow linear mixing laws (i.e., two additive quantities, such as concentrations). Hildebrandt and Glasser (1990) and Godorr et al. (1994) described how to construct the attainable region in higher dimensions. A short review of the relevant steps in the construction is presented here. The reader is referred to the papers of Glasser, Hildebrandt, and co-workers and the references therein for a more detailed discussion. The only reactors which must be considered to construct the attainable region are isothermal steady-flow homogeneous continuous stirred-tank reactors (CSTR), © 1997 American Chemical Society

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plug flow reactors (PFR), and differential side-stream reactors (DSR). The DSR is a PFR with a second feed stream added continuously along the length of the reactor. The CSTR and PFR reactors represent two extremes of mixing and are the only reactors needed in the construction of the attainable region for two additive quantities (Glasser et al., 1987). For constructing the attainable region in higher dimensions one must also consider the DSR reactor, which may or may not be needed depending on the detailed kinetics (Godorr et al., 1994). Since reaction and mixing are used in generating the attainable region, geometric interpretations are needed for these processes. Reactor-model solutions parametrized by time give curves called profiles. To obtain the profiles for PFR reactors, one integrates the design equation

dci ) ri(c) dt

(1)

where ci is a component of the concentration vector c, ri is the rate of formation of component i, and t is the reactor residence time. The vector c can also include the reactor network residence time (the total residence time for the entire reactor network) and the enthalpy as described by Hildebrandt et al. (1990). In contrast, the compositions in a CSTR change according to the difference in the feed and exit concentrations

ci - ci,0 ) ri(c) t

(2)

where ci,0 is the reactor feed composition. Simple solutions for the CSTR (i.e., presuming no isolas or oscillations) will produce a locus of solutions parametrized by the residence time, which is the CSTR reactor profile. It happens that DSR reactors are not needed for this work, and readers are referred to Godorr et al. (1994) for a description of these reactors. Mixing of two streams results in an average composition given by

F1c1 + F2c2 ) FTcavg

(3)

where ci is the concentration vector of stream i, cavg is the concentration vector of the combined stream, Fi is the flowrate of stream i, and FT is the flowrate of the combined stream. Geometrically, this mixing process is simply represented by a straight line joining the mixed compositions on a graph of concentrations. The attainable region is generated by using reactor profiles and mixing lines to extend the geometric construction until the the largest possible convex region is found. Two necessary conditions to generate the attainable region for systems without isolated solutions are as follows: (1) The region must be convex; otherwise, a mixing line will enlarge the region. (2) The boundaries of the region must have rate vectors (vectors made up of the rates of formation) that point into the region from the boundary, are tangent to the boundary, or are zero. If the boundary of the region had a rate vector pointing out of the region, then a reactor placed at that point would extend the region. Note that a rate vector along a PFR profile is tangent to the profile and that a rate vector along a CSTR profile is collinear with the reactor solution and feed point. An example attainable region for a series reaction is shown as the shaded area in Figure 1. Profiles and rate vectors for CSTR and PFR reactors are also shown. Points inside the region can

Figure 1. Attainable region for an isothermal series reaction of k1

k2

1 98 2 98 3, where k1 ) k2 ) 1, showing CSTR and PFR profiles and rate vectors.

be obtained by mixing one or more pairs of points along the boundaries; many different reactor networks can achieve an internal point. The production of polymers by a free-radical route can include reactions of initiator decomposition, initiation, propagation, chain transfer, and termination. A reaction scheme for free-radical polymerization (Bamford, 1988), for instance, the polymerization of methyl methacrylate (MMA) monomer with azobis(isobutyronitrile) (AIBN) initiator, can be represented by kd

I 98 2R ki

R + M 98 P1 kp

Pn + M 98 Pn+1 ktr

Pn + M 98 P1 + Dn kt

Pm + Pn 98 Dm + Dn

(4) (5) (6) (7) (8)

where I is an initiator, R is a primary radical, M is a monomer, and Pn and Dn are growing or “live” chains and terminated or “dead” chains, respectively, of length n. These symbols are used above to represent the molecules in the reaction scheme and later as concentrations. The quasi-steady-state assumption (QSSA) is applied to the primary radicals, so that the formation of growing chains depends only on the rate of decomposition in eq 4. Also, the conversion of monomer is assumed to occur only by propagation, i.e., the longchain approximation applies (Ray, 1972). As the conversion of monomer increases, the reaction mixture becomes more viscous, and the rate of the termination reaction (eq 8) and eventually the rate of the propagation reaction (eq 6) decrease. These changes in reaction rates are termed the “gel” (or “Trommsdorff”) effect and the “glass effect”, respectively. These effects are often described empirically or semiempirically with correlations (e.g., Ross and Laurence, 1976; Schmidt and Ray, 1981), which are based on a physical picture wherein the rate of diffusion of the reactants decreases when less free volume is available. This decrease in the diffusion of reactants limits the apparent reaction rate for termination (and propagation), as represented by a decrease in the apparent reaction rate coefficient at

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higher conversions and lower temperatures. The onset of the glass effect is described by the correlations of Schmidt and Ray (1981), at which point the reaction rates will be considered to be zero. To describe the MWD, it is generally unnecessary to solve n equations for the concentration of chains of each length, and trying to graph them in n-space would prove even harder. Therefore, the distribution of chain lengths is modeled in terms of the moments of the MWD, which are written as

λi )

∑n niPn

(9)

µi )

∑n niDn

(10)

where λi is the ith moment of live chains and µi is the ith moment of dead chains. The zeroth moments are the concentrations of chains, and the first moments are the concentrations of monomer units which have been converted to polymer. With these definitions the numberaverage chain length is the ratio of the first moment to the zeroth moment, and the weight-average chain length is the ratio of the second moment to the first moment. Number-average and weight-average molecular weights are obtained by multiplying the average chain lengths by the molecular weight of the repeat unit, and the polydispersity is determined by the ratio of the weight to the number-average molecular weights. The rates at which the moments develop are

dλ0 ) 2fkdI - ktλ02 dt

(11)

dλ1 ) kpMλ0 - ktλ0λ1 - ktrM(λ1 - λ0) dt

(12)

dλ2 ) kpM(2λ1 + λ0) - ktλ0λ2 - ktrM(λ2 - λ0) (13) dt dµ0 ) ktλ02 + ktrMλ0 dt

(14)

dµ1 ) ktλ0λ1 + ktrMλ1 dt

(15)

dµ2 ) ktλ0λ2 + ktrMλ2 dt

(16)

where f is the efficiency with which decomposed initiator forms a growing chain. Changes in density with polymerization have not been considered. Reaction rate coefficients have been taken from Schmidt and Ray (1981), with the transfer constant to monomer (CM ) ktr/kp) taken from Gopolan and Santhappa (1957). Reactor Model Equations Before generating the attainable region for the polymerization of MMA, we need reactor model equations for the additive quantities, the moments of the distribution of dead chains, in autonomous form. To find these equations, two assumptions are needed. One assumption is that the QSSA can be applied to the live moments of the MWD, and the second assumption relates the moments to the concentrations of monomer and initiator. Other assumptions which affect the amount of

detail in the modeling can be added or removed as desired (e.g., chain transfer). The first assumption, that the QSSA applies to the live moments of the MWD, assumes that the moments change slowly with time, so that the time derivatives in eqs 11-13 can be set to zero. While the QSSA approximation is often used for the concentration of live chains, λ0, the assumption is less common for λ1 and λ2. An example where the QSSA is applied to all of the live moments is Ross and Laurence (1976), where they develop a correlation for the gel effect for poly(methyl methacrylate). Bamford (1988) also made this assumption in another form, saying that the concentrations of all the live chains (i.e., the chain-length distribution) can be approximated as constant. To analyze the accuracy of this approximation, equations for I, M, λi, and µi were integrated and compared with the approximate results to yield number- and weight-average molecular weights within 5%. With the QSSA the live moments can be found explicitly in terms of I, M, and the kinetic parameters, which are functions of conversion and temperature. The live moments can then be substituted into eqs 14-16, so that the dead moments are functions only of the concentrations and the kinetic parameters. A second assumption relates the dead moments of the MWD to the concentrations of monomer and initiator, which makes the equations autonomous. Since the concentration of monomer units in dead chains is much larger than the concentration of monomer units in live chains, the first moment of the dead chains, µ1, can be related to the concentration of monomer by

µ1 ) Mf - M

(17)

where Mf is the feed concentration of monomer (8.83 mol of MMA/L for this work). Similarly, the concentration of dead chains is much larger than the concentration of live chains, so the zeroth moment of dead chains can be approximately related to the concentration of initiator. This is done by considering the amount of initiator which decomposes, If - I, to form 2f chains per molecule. By assuming that the chains formed all become dead chains, terminated by disproportionation in eq 8, we find

µ0 ) 2f(If - I)

(18)

and f has been set to 0.5 here. The relations in eqs 17 and 18 permit the moment equations to be written in terms of µ0 and µ1 instead of I and M. This approximation for I is also used when chain-transfer reactions are included. Although chain transfer increases µ0 and another term for chains formed due to transfer could be added to the right-hand side of eq 18, the effect on the approximation for I is small. With these approximations the moment equations can be written as

dµ*0 ) ktλ*02Mf + ktr(1 - µ* 1)λ* 0Mf dt

(19)

dµ* 1 ) ktλ*0λ*1Mf + ktr(1 - µ* 1)λ* 1Mf dt

(20)

dµ* 2 ) ktλ*0λ*2Mf + ktr(1 - µ* 1)λ* 2Mf dt

(21)

where * indicates dimensionless moments normalized by Mf. From eqs 17 and 18, µ*1 is simply the conversion

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1079

of monomer, and µ*0 is proportional to the concentration of decomposed initiator. The moment equations (19)-(21) are autonomous and depend only upon the moments, initial concentrations, and temperature. These equations describe the rates of formation of the moments and therefore PFR reactors (as in eq 1). The CSTR model equations for the dead moments can be written as 2 µ*0 - µ* 0,0 ) (ktλ* 0 Mf + ktr(1 - µ* 1)λ* 0Mf)t

(22)

µ*1 - µ* 1,0 ) (ktλ* 0λ* 1Mf + ktr(1 - µ* 1)λ* 1Mf)t

(23)

µ*2 - µ* 2,0 ) (ktλ* 0λ* 2Mf + ktr(1 - µ* 1)λ* 2Mf)t

(24)

where µ*i,0 is the value of the ith moment for the reactor feed. The most significant difference between the CSTR and PFR reactor model equations for free-radical polymerization with termination is the constancy or variability of the reactant composition (Denbigh, 1947). Whereas in a CSTR reactor the concentrations and moments are at a steady state, the PFR reactor and the mathematically similar stirred-tank batch reactor have compositions that vary with time. Denbigh concluded that for live chains which are active for a short time compared to the mean residence time the residence time distribution in a CSTR reactor does not affect the MWD. Tadmor and Biesenberger (1966) affirmed these results and also determined the effects of segregation on CSTR reactor models. More recently, Nauman (1994) pointed out that segregation in stirred-tank polymerization reactors has not been confirmed by experiments and that perfect mixing can be approximated in stirred-tank reactors. We have not considered segregation in our CSTR reactor models. More information can be gained from the reactor model equations in terms of the reactor residence time. Equations for the reactor residence time can be related to the zeroth moment equations for both PFR and CSTR by assuming that the initiator concentration is constant. This is a common approximation, e.g., Rodriguez (1989), which allows the PFR reactor equation for µ* 0 (eq 19) to be integrated to give the same form as the CSTR model equation for the residence time

t)

µ*0 - µ* 0,0 2fkd(I* f - µ* 0/2f)

(25)

where the QSSA of eq 11 has been used in eq 19 without the chain-transfer term. This reactor residence time, t, is also equal to the change in the reactor network residence time, τ - τ0. The reactor network residence time can therefore be written as

τ)

µ* 0 2fkd(I*f - µ* 0/2f)

(26)

where τ depends only on µ*0. This relationship is useful because an additional dimension is not needed to visualize the reactor network residence time since it is simply related to the zeroth moment. Therefore, in the attainable region figures to follow, the reactor network residence time is roughly proportional to the zeroth moment. Reactor model equations for other polymerization reaction systems can be developed in a manner similar

Figure 2. First step of constructing the attainable regionsprofiles of CSTR and PFR reactors. T ) 70 °C, If ) 0.50 wt % AIBN.

Figure 3. Second and third steps in completing the constructing of the attainable regionsaddition of mixing lines from the feed to the CSTR profile and between the CSTR and PFR profiles. T ) 70 °C, If ) 0.50 wt % AIBN. Points inside the region, such as A, can be obtained by mixing.

to the one used here for free-radical polymerization of poly(methyl methacrylate). Attainable Region for Number-Average Molecular Weights Generating the attainable region for the zeroth and first moments of the MWD provides information on the number-average molecular weights, conversions, residual initiator concentrations, and reactor network residence times. The construction can be done in steps. The first step is to graph the reactor profiles for both a PFR and a CSTR beginning from the feed composition of monomer and initiator (at the origin), as shown in Figure 2. As the conversion increases, µ*1 increases with respect to µ* 0 due to the gel effect. When the conversion gets high (approximately 0.80 in the figure), the onset of the glass effect occurs, the reaction rates are considered to be zero, and the reactor profiles stop. A second step in the construction of the attainable region is the addition of mixing lines. Figure 3 shows a mixing line joining the feed composition and the CSTR solution which gives the maximum slope. This makes this part of the construction convex. However, mixing of polymer and monomer seems counterintuitive since normally after polymerization one actually devolatilizes the product to remove monomer from the polymer. The only apparent reasons to add monomer to polymer are to perform further polymerization, perhaps to create a copolymer, or to form a product which is sold in solution. Nevertheless, the mixing line shown in Figure 3 is the limit of what can be attained. For instance, by mixing

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Figure 4. Completed attainable region using the correlation of Friis and Hamielec (1976) with the maximum conversion (onset of the glass effect) taken from Schmidt and Ray (1981). T ) 70 °C, If ) 0.50 wt % AIBN.

Figure 5. Maximum and minimum number-average molecular weights which can be produced at various temperatures and initiator concentrations of AIBN.

the effluents of two reactors, in this case along the CSTR profile, one can obtain interior points such as the one marked A in the figure. Therefore, the mixing line forms the boundary of the attainable region. To generate the uppermost boundary, a third step is to add another mixing line at the top of the present construction. This mixing line, shown in Figure 3, connects the highest conversions obtained in the CSTR and PFR reactors. The construction is now convex, and the boundaries need to be checked to see if any rate vectors point outward. Along the PFR profile the rate vectors are tangent. For the mixing line between the feed and the CSTR profile, the rate vectors point inward. The mixing line connecting the CSTR and PFR profiles has rate vectors which are zero because the reaction rates are zero at the onset of the glass effect. Thus, the shaded area of Figure 3 is the attainable region; it is convex, and everywhere along the boundaries the rate vectors point inward, are tangent, or are zero. A comparison of attainable regions based on different kinetic models can be made by using the correlation of Friis and Hamielec (1976) to generate the attainable region for poly(methyl methacrylate). The region is shown in Figure 4 with the maximum conversion (onset of the glass effect) taken from Schmidt and Ray (1981). By comparing Figures 3 and 4, one can see that models based on the correlation of Friis and Hamielec (1976) yield an attainable region smaller than that found using the model of Schmidt and Ray (1981). The greatest difference in the size of the regions appears because the kinetic model of Schmidt and Ray (1981) predicts that a PFR produces more chains (larger values of µ*0) in Figure 3. The mixing lines on the left of each figure appear nearly identical, and the top mixing lines are at the same value of µ* 1 because of the onset of the glass effect. The greatest difference in the reactor profiles in the two figures is the shape of the CSTR profile at low conversions. Fundamentally, this is because the CSTR reactor profile from the correlation of Schmidt and Ray (1981) has a point where the influence of the gel effect suddenly shows itself, whereas the CSTR reactor profile from the correlation of Friis and Hamielec (1976) has a smooth transition to gel effect behavior. From the attainable region, the feasible numberaverage molecular weights can be found. To visualize the attainable number-average molecular weights, one need only know the limiting values of µ* 1/µ* 0. The slope of any line beginning from the origin (for instance in Figure 3) gives µ* 1/µ* 0, which is proportional to the number-average molecular weight. It follows that the

lines of maximum and minimum slope for any region represent the maximum and minimum number-average molecular weights which can be produced. When Figures 3 and 4 are compared, it is found that the maximum molecular weights are 1.97 × 106 and 1.56 × 106, respectively, and the respective minimum molecular weights are 1.97 × 105 and 1.99 × 105. The apparently small difference in the mixing lines on the left of Figures 3 and 4 creates a 21% difference in the maximum attainable molecular weights because µ* 0, the denominator in a number-average molecular weight calculation, is a small number, so that even small changes in µ* 0 can cause a substantial difference in the molecular weight. In addition, the large spatial difference, due to the PFR reactor profiles, has no effect on the minimum attainable molecular weight because the minimum slope occurs at low conversions where the gel effect has little influence. Particular number-average molecular weights can be achieved by mixing two polymers of differing molecular weights. However, two products with the same numberaverage molecular weights can have very different weight-average molecular weights, and to specify a unique molecular weight distribution, three or more moments may be needed (Ray, 1972). Regarding the mixing of feed (the origin on the figures) with polymer, one should note that the average molecular weight does not change because monomer molecules are not counted as chains in determining the average molecular weight for chain-growth systems. By constructing attainable regions at various temperatures and initiator concentractions, one can examine the limit on the number-average molecular weights and the range of molecular weights that can be produced. Figure 5 shows that the minimum numberaverage molecular weights increase (although not dramatically) with a decrease in temperature. However, the maximum number-average molecular weights are strongly influenced by temperature, and the initiator concentration also has a greater effect at lower temperatures. From the figure one can see that the range of molecular weights attainable is also much greater at lower temperatures and initiator concentrations. The range of molecular weights shown in Figure 5 is much greater than the maximum number-average molecular weight of 200 000 given by Harbison (1985) for noncasting polymerization of MMA. This indicates that higher molecular weights than those reported by Harbison (1985) are feasible. It is possible that the higher molecular weight products could have unexpected prop-

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Figure 6. Schematic diagram depicting areas of low and high conversion (X), initiator concentration (I), and reactor network residence time (τ) for T ) 70 °C and If ) 0.50 wt % AIBN.

erties. An example where new high molecular weight properties were found is given by Pennings et al. (1973) for polyethylene. Bhateja and Andrews (1983) discuss the properties of ultrahigh-molecular-weight linear polyethylene and report that the ultimate tensile strength for ultrahigh-molecular-weight polyethylene is nearly twice that of a normal-molecular-weight polymer. Knowing that the high molecular weights in Figure 5 are feasible, it may be interesting to solve the problems of mixing, heat transfer, etc., which limit the production of high-molecular-weight poly(methyl methacrylate). This generation of the attainable region and comparison of the results to molecular weights typically produced could prove useful for other polymerization systems as well. Once the attainable region is constructed, one can also determine feasible reactor network residence times and the concentrations of residual initiator and monomer from the moments. As the zeroth moment increases, the reactor network residence time increases, and as the zeroth and first moments increase, the residual concentrations of initiator and monomer decrease. Thus, in the lower left corner of the region in Figure 6, the reactor network residence time is low, as is the conversion of monomer, and a large amount of initiator remains in the product. In the upper left corner of the region the reactor network residence time is low, the conversion is high, and a large amount of initiator still remains. In the upper right corner of the region both the reactor network residence time and conversion are high and little initiator remains. The boundaries limiting the attainable region and the reactor networks which produce these boundaries are shown in Figure 7. A CSTR reactor can produce the highest number-average molecular weight and conversion. Adding a bypass stream to the CSTR reactor (which represents the limiting case of a small parallel reactor being replaced by a bypass stream) produces the same high molecular weight as a CSTR but lowers the conversion. A PFR reactor produces the lowest numberaverage molecular weights and so is on the lower boundary of the region. Reactor networks have not been specified for the top mixing line in Figure 7 because a number of networks can attain the limiting conversions located along this mixing line. For example, the CSTR and PFR reactors shown in Figure 7 could produce the limiting conversion and their products could be mixed; CSTR reactors in series, a CSTR followed by a PFR, or combinations of networks could also be used to attain

Figure 7. Outline of the attainable region with reactor networks which form the boundary. T ) 70 °C, If ) 0.50 wt % AIBN.

Figure 8. Attainable region with PFR profiles (blue), CSTR profile (red), mixing surfaces (green), and mixing lines on the back surface of the volume (light green). T ) 70 °C, If ) 0.50 wt % AIBN.

the limiting conversion. While two or more of these networks could produce the same number-average molecular weight product, the MWD could vary to a large degree. This points toward examining higher moments of the MWD. Attainable Region for Polydispersity Constructing the attainable region for the first three moments of the MWD will allow us to examine the breadth of the distribution. While the number-average molecular weight is proportional to µ*1/µ*0, the weightaverage molecular weight is proportional to µ* 2/µ* 1. The construction is more difficult than that for the twodimensional region because in two dimensions reactor profiles and mixing lines separate areas as being either inside or outside of the region, while in three dimensions, volumes must be determined. The construction of the attainable region in three dimensions can be accomplished in steps similar to those for the two-dimensional region. First, a PFR reactor profile starting from the origin is generated. This profile is shown on the lower right of Figure 8 and the upper right of Figure 9 as a blue line on the edge of the volume. In Figure 8, which is a view similar to the twodimensional Figure 3, the PFR reactor profile from the origin initially coincides with the red (CSTR) line. Following the blue PFR reactor profile, one can see that the conversion (first moment) initially increases slowly but eventually increases rapidly as the zeroth moment

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Figure 10. Reactor networks found on the boundaries of the attainable region.

Figure 9. Another view of the attainable region. T ) 70 °C, If ) 0.50 wt % AIBN.

increases. As discussed earlier, the shape of this profile is due to the changes in reaction rates brought about by the gel effect. This behavior is also shown in Figure 9, which shows the relationship of the second moment to the zeroth and first moments. The PFR reactor profile which starts at the origin forms the boundary on the right of the pictured volume. As µ*0 and µ*1 increase from the origin, µ* 2 increases rather slowly until the gel effect causes a more dramatic increase at higher conversions. The increase in µ*2 with respect to µ*1 indicates an increase in the weight-average molecular weight (proportional to µ* 2/µ* 1). At the same time the numberaverage molecular weight (proportional to µ*1/µ*0) increases more slowly. The result is an increase in the breadth of the MWD due to the formation of long chains. Next, the CSTR reactor profile from the feed is considered. The profile is shown in Figures 8 and 9 as the red line. One can visualize how the moments change with respect to each other and compare the plot of the two-dimensional CSTR profile shown in Figure 3. The moments are profoundly affected by the gel effect, which is visible in the large increase in the second moment. The reason that the increase in the second moment is so pronounced is that in the CSTR reactor all reaction occurs at the same state, which differs from the PFR reactor where the instantaneous reaction rates are integrated over time. Completing the three-dimensional construction is a process similar to that performed for two dimensions. The original PFR and CSTR reactor profiles are connected with mixing lines to form a convex volume. Points along the surface of the volume are checked to determine the direction in which the rate vectors point. When a rate vector that points out is found, a reactor is started from that point and mixing lines from the new reactor profile are added. The reactor profile and mixing lines are then used in constructing a new convex volume, which again must be checked for rate vectors pointing outward. The end result is a completed plot of the attainable region in three dimensions as shown in Figures 8 and 9. Describing the color scheme will help to elucidate Figures 8 and 9: the blue and red lines represent PFR and CSTR reactor profiles, respectively, while green lines denote mixing. The thin, light green lines are mixing lines on the back surface of the volume. They extend from the upper left CSTR solution point to the PFR reactor profile which starts from the origin. The green surfaces in the figures are constructed of mixing

lines. The green surface which touches the origin is made up of mixing lines from the origin to the CSTR reactor profile. Therefore, this surface follows the shape of the CSTR reactor profile, which can be seen most easily in Figure 8. The green surface on top of the volume is generated from mixing lines joining the endpoints of reactor profiles. Since these profiles all stop at the conversion where the glass effect occurs, the resulting surface is a flat plane. This plateau is directly analogous to the top mixing line of Figure 7, where various reactor networks or mixing processes could be used to obtain a desired point. The blue surface shows the shape of the PFR reactor profiles which begin from the CSTR reactor profile (with one beginning from the origin). The networks that form the boundaries of the region in Figures 8 and 9 are shown in Figure 10 with their corresponding letters. These networks include a CSTR (red, a), a CSTR with mixing of feed using a bypass stream (green, b), a PFR (blue line from origin, c), a CSTR followed in series by a PFR (blue surface starting from red, d), a CSTR at the onset of the glass effect in parallel with a PFR which starts at the origin (light green lines, e), and the multiple types which can be used to attain points along the green surface on top of the region (e.g., parallel configurations of high-conversion networks). It is interesting to note that the plateau surface which forms the top of the region has high weight-average molecular weights but low numberaverage molecular weights. This indicates that there are a number of ways to generate a broad MWD but that, to attain points along the surface of the region where the MWD is narrower, one needs the reactor networks a, b, or d. To see that the reactor networks a, b, and d produce the narrowest MWD products, one can study Figures 8 and 9. By selecting a conversion (µ*1) or a reactor network residence time (through µ* 0) and a numberaverage molecular weight (µ* 1/µ* 0), one has defined desired values for µ* 0 and µ* 1. A selection of desired conversion and weight-average molecular weight similarly define µ*1 and µ*2 values. For the selected values of µ*0 and µ*1, one can consider the attainable values of µ* 2 which would correspond to a horizontal line in Figures 8 and 9. The locations where this line would emerge from the attainable region define points where the breadth of the MWD is the largest and smallest, at large and small µ* 2, respectively. By noting the surface color of the attainable region at the point of emergence, one can read the reactor networks which produce the limiting MWD from the figure. For example, consider Figure 8 with µ*1 ) 0.5 and different µ*0 values. If µ*0 ≈ 1.0 × 10-4, then the smallest µ* 2 (and therefore the

Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997 1083

Conclusion

Figure 11. Maximum and minimum polydispersities which can be produced for number-average molecular weights with design conditions of T ) 70 °C, If ) 0.50 wt % AIBN.

narrowest MWD) is produced in a CSTR (a). For a lower -4 value, µ* 0 ≈ 0.8 × 10 , the narrowest MWD is produced in a CSTR with bypass of feed (b). For a larger -4 value, µ* 0 ≈ 2.0 × 10 , the narrowest MWD is produced in a series configuration of a CSTR followed by a PFR (d). In all three cases the broadest MWD (largest µ*2) is produced by a parallel configuration of a CSTR operated at the highest conversion with a PFR which starts from the origin (e). Thus, the resulting network for producing the narrowest MWD is either a CSTR, a CSTR with bypass, or a CSTR followed by a PFR. Examples of such reactor networks are provided by Simon and Chappelear (1979) including a CSTR followed by a PFR and a series of CSTR reactors for the polymerization of the analogous resin polystyrene. For poly(methyl methacrylate), example networks are given by Schildknecht (1977), where polymerization is started in a CSTR and completed by cell casting or in continuous sheets (which correspond to PFR reactors when heat-transfer effects are small). These reactors networks can be generalized as a CSTR followed in series by a PFR, and the attainable region figures show that this network produces the narrowest MWD for certain conversions and number-average molecular weights. However, the attainable region does not show whether the best reactor network has been found, for which an optimization should be done, and operating considerations should be taken into account. Once the attainable region for the first three moments of the MWD has been constructed, the breadth of the distribution can be examined. Figure 11 shows the ranges of polydispersities and number-average molecular weights which can be produced. The range of number-average molecular weights along the abscissa in Figure 11 corresponds to a vertical range in Figure 5 at T ) 70 °C and If ) 0.50 wt % AIBN. Two effects which could lead to higher polydispersities are nonisothermal reactors (Nauman, 1994) and segregation (Tadmor and Biesenberger, 1966). Note that Figure 11 is not an attainable region plot and that the numberaverage molecular weight and polydispersity do not both follow linear mixing laws (i.e., when mixing molar concentrations, the number-average molecular weight depends on the chain length linearly while the weightaverage molecular weight depends on the chain length squared). Thus, the figure is not necessarily convex. The minimum polydispersities are generated by a CSTR reactor or a PFR reactor at low conversions. The maximum polydispersities are generated by a parallel network of a CSTR reactor designed at high conversion and a CSTR or PFR reactor designed at low conversion.

This paper has developed the attainable region for polymerization systems. From the attainable region for the example of the free-radical production of poly(methyl methacrylate) the limiting average molecular weights, polydispersities, reactor network residence times, reactant concentrations, and reactor networks which make up the boundaries of the region have been found. The effects of design conditions on the range of achievable molecular weights indicate that temperature has the greatest effect and that initiator concentration has more influence at lower temperatures. The range of attainable number-average molecular weights shows that much higher molecular weights than those typically produced in industry are feasible. These higher molecular weight products could have unexpected properties. By using the moments of the distribution in constructing the attainable region, one can visualize the average molecular weights. One can also see the reactor network residence time, the conversion of monomer, and the residual concentration of initiator through relationships between these quantities and the moments. The ability to easily visualize the molecular weight distribution, residence time, conversion, and residual initiator concentration will allow a plant designer to focus on design conditions which meet desired objectives. Reactor networks which produce narrow molecular weight distributions at desired conversions or reactor network residence times can be read from a figure of the attainable region. These reactor networks include a CSTR reactor, a CSTR reactor with a bypass stream, and a CSTR reactor followed by a PFR reactor. Acknowledgment The authors acknowledge funding provided by sponsors of the Process Design & Control Center, Department of Chemical Engineering at the University of Massachusetts, Amherst, MA, and the work of Pam Stephan on a number of the figures. Nomenclature ci ) concentration of i [mol/L] ci,0 ) reactor feed concentration of i [mol/L] Dn ) dead chains of length n [mol/L] f ) efficiency of initiator decomposition Fi ) volumetric flow rate of i [L/s] I ) initiator [mol/L] If ) initiator process feed [mol/L] kd ) reaction rate coefficient for dissociation of initiator [1/s] kp ) reaction rate coefficient for propagation [L/mol‚s] kt ) reaction rate coefficient for termination [L/mol‚s] ktr ) reaction rate coefficient for chain transfer to monomer [L/mol‚s] M ) monomer [mol/L] Mf ) monomer process feed [mol/L] n ) chain length Pn ) live chains of length n [mol/L] R ) primary radicals [mol/L] ri ) rate of formation of i [mol/L‚s] t ) residence time [s] Greek Letters λi ) ith moment of live chains [mol/L] µi ) ith moment of dead chains [mol/L] τ ) reactor network residence time [s]

1084 Ind. Eng. Chem. Res., Vol. 36, No. 4, 1997

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Received for review June 21, 1996 Revised manuscript received August 21, 1996 Accepted August 22, 1996X IE960358X X Abstract published in Advance ACS Abstracts, February 15, 1997.