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Esterification in reactive extrusion.

INTRODUCTION

Reactive extrustion (REX) is the use of extruders for processes such as polymerization, polymer modification or compatibilization of polymer blends (1). The term may be applied when the reactive aspect is not the sole objective, but one of many. The amalgamation of several processes on an extruder is one of the compelling economic reasons for its commercial interest. Additional reasons include the ability to perform reactions with little or no solvent, which entails the pumping and mixing of highly viscous media and at an elevated temperature, which usually favors increasing reaction speeds. For the greater part, twin-screw extruders (TSEXs) have gained the largest industrial interest for REX.

Although one can trace the roots of REX as far back as the last century (2, 3), one may consider the exploitation of TSEX as a relatively young technology, particularly when coupled to the fabrication of polymer blends. The simultaneous physico-chemical phenomena occurring in the TSEX, though currently being studied intensively, remain largely unexplored.

A parallel development in the polymer industry, which has added to the interest of REX, is the recognition of the possibility to attain synergetic effects by mixing two or more types of polymers. Nevertheless, different polymers are rarely miscible, and amphiphilic polymers, or so-called compatibilizers, must be used to obtain reasonable mechanical properties. The compatibilizers may be directly synthesized and added to the polymer blend as an additional ingredient, or they may be created in situ. If the latter is possible, it is preferable, since it avoids an additional processing step.

The research described here aims at the creation of a grafted copolymer using a condensation reaction, and to investigate its relationship with mixing. The focus is on esterification, with the aim of grafting small molecules and oligomers onto a previously maleated polyolefin.

Ratzsch et al. (4-8) have extensively studied esterification in solution. The rates of reaction are slow, on considering the rates required to be commercially viable for REX. Nevertheless, considering the Arrhenius-type of relationship for kinetics, it is not unexpected to find significantly higher rates at elevated temperatures (4, 9).

Above 120 [degrees] C the equilibrium nature of esterification becomes apparent: the reverse reaction becomes significant (4, 10). A kinetic model involving an association of the cyclic anhydride (CAn) group with several alcohol groups had been proposed by Ratzsch and Wohlfahrt (4), although a classical approach was sufficient to explain the experimental results obtained by Hu and Lindt (11).

Lambla et al. (12-15) have studied the esterification in the melt with mono- and di-functional oligomeric alcohols, the latter leading to gels. Elevated molecular weights of the alcohols were shown to be important in causing a decrease in the grafting yield. Kinetic parameters were determined in the context of a model for cross-linking. Frerejean et al. (16) are also studying the melt esterification, but have not presented a kinetic model to explain their data. It was evident that a preliminary study on the kinetics is desirable before moving onto the extruder.

EXPERIMENTAL

Materials

Maleated ethylene-propylene rubber ([EPR.sub.MA]) was supplied by Exxon Chemical International, grafted with an equivalent of 0.5 wt% maleic anhydride to give between 2 and 3 cyclic anhydride (CAn) functionalities on average per chain. The viscosity as a function of temperature and shear rate is shown in Fig. 1.

The monofunctional alcohol used was a nonylphenylethoxylate (NP8), having the molecular formula [C.sub.9][H.sub.19]-Ph-O-[([CH.sub.2][CH.sub.2]O).sub.8]H, where the Ph is an aromatic phenyl ring. On the left side is a polyolefinic chain, and on the right an ethyoxylate chain containing ether linkages. The polyolefinic chain is non-polar and lipophilic, in contrast to the ethoxylate chain, which is relatively polar and hydrophilic: it is commonly used as a surfactant. The NP8 used for the studies has been kindly supplied by ICI, known as Synperonic NP8. The NP8 was used as delivered, with a viscosity of 60 mPa [multiplied by] s at 50 [degrees] C (17), and an equivalent OH content of 0.00163 mol OH/g (as determined by titration (18)).

FTIR Spectrometry

Samples were analyzed using a Nicolet 60SX FTIR spectrometer.

Samples from the extruder were simply pressed into a film of about 150 [[micro]meter] thickness. Pressing the film to a precise thickness was not important, as a reference peak at 4325 [cm.sup.-1] was used. Although peaks at 712 [cm.sup.-1] (19, 20) and between 2210 and 2250 [cm.sup.-1] (21) have been used in other studies, the peak at 4325 [cm.sup.-1] was commensurate with peaks of CAn. Films could be pressed much thicker causing a good part of the spectrum to be saturated, but enlarging the reference and CAn peaks. This was important due to the relatively low concentration of CAn group. These reveal peaks at 1865 and 1790 [cm.sup.-1], of which the first was used, as the latter could interfere with the ester peak at 1738 [cm.sup.-1].

Thermally Controlled Cell

For the kinetic study, a sample was placed in a thermally controlled cell as shown in Fig. 2. The critical piece is the 266 [[micro]meter] spacer at the center, machined to be quite fiat ([+ or -] 10 [[micro]meter]), which maintained the sample at the same thickness independent of temperature changes in the cell.

Twin-Screw Extruder

The extruder was a co-rotating self-wiping twin-screw extruder (Werner & Pfleiderer ZSK 30). Its length-to-diameter ratio was 42. The temperature profile, screw geometry, and other processing factors are given in Fig. 3. The [EPR.sub.MA] was weight-loss fed, while the NP8 was injected using a GPC pump. The tube between the pump and the extruder was flexible, so that injections could be performed at any location along the barrel where barrel plugs were located with standard threaded holes for sensors.

PROCEDURE

Since peak absorbances are a function of temperature, this dependence was characterized to determine the CAn conversion in a reactive system.

The [EPR.sub.MA] was mixed with the NP8 in a batch mixer, such that the initial stoichiometric ratio OH/CAn was 1.5. A sample was placed in the thermally-controlled cell and left to equilibrate. Step changes in temperature were performed, each time gathering the spectra until equilibrium had again been reached. The CAn conversion was determined in dynamic and equilibrated conditions. A model was then used to correlate the data.

The NP8 was injected into the various ports shown in Fig. 3. Samples were obtained at each plug downstream, at the die exit, and at the tip of the screws with the die head removed. Before sampling, the process was allowed to attain steady state. Samples were then taken directly from the screws of the running extruder, and immediately submersed in liquid nitrogen.

Residence time distributions were determined by using a tracer which was [EPR.sub.MA] grafted with a reactive dye called Fast Blue BB (22, 23). It was thus possible to make a comparison with the results obtained from the FTIR cell.

RESULTS AND DISCUSSION

Calibration

The absorption of the peaks in the IR spectrum followed the Beer-Lambert law. The molar absorption coefficients were estimated to be 1710 [+ or -] 80, (9.4 [+ or -] 0.4) x [10.sup.4], and (1.3 [+ or -] 0.1) x [10.sup.4] [cm.sup.2]/g at 1865, 1790, and 1712 [cm.sup.-1], respectively (22). They appeared in reasonable agreement with prior determined values in the literature (24-26). The phenylether linkage in the NP8 gave a peak at 1512 [cm.sup.-1], whose molar absorption coefficient was estimated to be 35 [m.sup.2]/mol.

The weight fraction of CAn groups in the rubber could be expressed as

[W.sub.CAn] = [[Xi].sub.CAn][DOP[prime].sub.1865] (1)

where [DOP[prime].sub.1865] is the optical density (the peak height given n terms of the baseline between 1882 and 1848 [cm.sup.-1]) divided by that of the reference (baseline 4380 to 4280 [cm.sup.-1]), and [[Xi].sub.CAn] a derivative of the molar absorption coefficient, which has a value of 1.79 [+ or -] 0.06 equivalent MA wt%. Similarly [[Xi].sub.NP8] = 0.054 [+ or -] 0.003.

Placing a sample in the thermally controlled cell [ILLUSTRATION FOR FIGURE 2 OMITTED] resulted in the changes of absorbances as shown in Fig. 4.

A correction factor [e.sub.T] was defined as

[e.sub.T](T) = [DOP[prime].sub.1865](20 [degrees] C)/[DOP[prime].sub.1865](T) (2)

whose variation with temperature is shown in Fig. 5. The solid line was the result of an empirical correlation, given by

[e.sub.T] = 1.12104 - 7.3480 x [10.sup.-3]T + 4.2073 x [10.sup.-5][T.sup.2] - 8.9978 x [10.sup.-8][T.sup.3] (3)

The conversion was based on the CAn groups. Given the equilibrium nature of the reactants, the CAn conversion was based on the system with no ester content, rather than at the initial conditions of each experiment. This allowed the direct comparison between all data obtained. The CAn conversion from the IR spectra was thus given by

X = [DOP[prime].sub.1865, o] (20 [degrees] C) - [e.sub.T] [DOP[prime].sub.1865](T)/[DOP[prime].sub.1865, o] (20 [degrees] C) (4)

Kinetics

The mathematics for the kinetics of a particular mechanism are given in a number of standard textbooks, e.g. Smith (27) and Perry (28). The kinetic details for an equilibrium like esterification is rare (one approach is given in (11)), and is hence given here.

Assuming a homogeneous media, that the reaction occurs according to the functional groups present, and that no volatilization occurs, the equilibrium reaction can be expressed as

[Mathematical Expression Omitted]

where the A stands for the alcohol, C for the cyclic anhydride, and E for the ester. The net rate of consumption of the CAn groups is given by the combination with the alcohol plus the break-up of the ester, i.e.

- d[C.sub.c]/dt = [k.sub.RE][C.sub.A][C.sub.c] - [k[prime].sub.RE][C.sub.E] (6)

The concentration of the various functional groups can be put in terms of only one of them, so

[C.sub.A] = [C.sub.Ao] - ([C.sub.Co] - [C.sub.C]) (7)

[C.sub.E] = [C.sub.Eo] + [C.sub.Co] - [C.sub.C] (8)

Substituting Eqs 7 and 8 in Eq 6, dividing by [k.sub.RE] and rearranging yields

[Mathematical Expression Omitted]

where [K.sub.E] is the equilibrium constant [k.sub.RE]/[k[prime].sub.RE]. For a constant temperature, the only variable is [C.sub.C] on the right-hand side of this equation. To simplify the equation then, it can be expressed as

[Mathematical Expression Omitted]

where the parameters [c.sub.i] are given by the corresponding terms in the prior equation. Equation 10 can be analytically integrated, using [C.sub.C](t = 0) = [C.sub.Co],

1/[c.sub.k3] ln([c.sub.k4] 2[C.sub.C] + [c.sub.k2] - [c.sub.k3]/2[C.sub.C] + [c.sub.k2] + [c.sub.k3]) = -[k.sub.RE]t (11)

where

[Mathematical Expression Omitted]

and

[c.sub.k4] = 2[C.sub.Co] + [c.sub.k2] + [c.sub.k3]/2[C.sub.Co] + [c.sub.k2] - [c.sub.k3] (13)

Solving for [C.sub.C] from Eq 11 gives

[C.sub.C] = 1/2 [([c.sub.k2] + [c.sub.k3])[e.sup.-[c.sub.k3][k.sub.RE]t] + ([c.sub.k3] - [c.sub.k2])[c.sub.k4]/[c.sub.k4] - [e.sup.-[c.sub.k3][k.sub.RE]t]] (14)

where the equation has been rearranged to reflect the diminishing terms with increasing time.

At equilibrium, the net rate is equal to zero, and the equilibrium constant can be expressed as

[K.sub.E] = [C.sub.E, eqm]/[C.sub.A, eqm][C.sub.C, eqm] (15)

Substituting Eqs 7 and 8 and rearranging to isolate the equilibrium concentration gives

[Mathematical Expression Omitted]

One can also start from Eq 14 and take the limit as time approaches infinity. The terms being multiplied by exp(-[c.sub.k3] [k.sub.RE]t) go to zero, and the equation becomes

[C.sub.C, eqm] = [c.sub.k3] - [c.sub.k2]/2 (17)

The conversion is defined as

X = [C.sub.Co] - [C.sub.C]/[C.sub.Co] (18)

where [C.sub.Co] must be defined as the level of conversion when no ester has been formed, to be able to compare the dynamics between different equilibria using the same reactant system.

A detailed analysis, including error estimations, can be found in (22).

Kinetic Parameters

Using the [EPR.sub.MA]/NP8 mixture in the thermally controlled cell, a large number of step changes in temperature were conducted. Although performed in random order, they may be summarized as those initially equilibrated at 60 [degrees] C and mounting to 80, 100, 120, 140, 160, and 180 [degrees] C [ILLUSTRATION FOR FIGURE 6 OMITTED], starting from 180 [degrees] C and descending to 160, 140, 120, 100, 80, and 60 [degrees] C [ILLUSTRATION FOR FIGURE 7 OMITTED], starting from the various temperature levels listed and descending to 60 [degrees] C, and similarly starting at the various temperatures and raising them to 180 [degrees] C.

The first step was to analyze the data at equilibrium to determine the equilibrium constant [ILLUSTRATION FOR FIGURE 8 OMITTED]. The solid line is given by

[K.sub.E] = 0.025[e.sup.29700/[R.sub.G][T.sub.K]] (19)

which shows the good relationship between the model and equilibrium data. Figure 8 also shows the data of Hu and Lindt (11), where the same activation energy was obtained.

The analytical solution, as given by Eq 11, was initially used to determine the kinetic parameters by plotting the left-hand vs. the right-hand side, but some large deviations from the experimental data were observed. Upon closer examination, the CAn conversion was rapidly advancing before the new temperature set-point was being reached [ILLUSTRATION FOR FIGURE 9 OMITTED]. At temperatures above 160 [degrees] C the conversion appeared to equilibrate as fast as the temperature could be changed (maximum rate 10 [degrees] C/min). It was thus necessary to numerically integrate Eq 6 using the temperature recorded during the experiment at each time step. Optimizing the pre-exponential factor and activation energy of [k.sub.RE] gave a model which well represented the experimental data obtained (e.g. dashed line in [ILLUSTRATION FOR FIGURES 6 AND 7 OMITTED]), i.e. [k.sub.RE, o] = (4.6 [+ or -] 0.1) x [10.sup.4] l/mol[multiplied by]min and [E.sub.A, E] = 34.5 [+ or -] 0.1 kJ/mol. This is a relatively low activation energy and indicates that the system may be catalyzed by the inherently pendant acid group once the CAn opens.

Results From Static Melt

Apart from the kinetic parameters, some further important observations may be made from the data obtained in the IR cell.

Figures 6 and 7 show that the higher the temperature, the lower the CAn conversion, i.e. formation of ester groups. From this view, the esterification should be performed at a temperature as low as possible. Yet there are limitations as given by the ability to pass the polymer through the extruder.

The ester continued to form even at ambient temperatures going to practical completion over extended amounts of time. The reason for this was the highly amorphous nature of the rubber, allowing limited molecular movement although not reaching significant rates of diffusion. Due to the low concentration of the functional groups, mixing must have been very efficient, allowing the alcohol groups to be well distributed among the CAn ones. Considering the EPR to be a non-polar medium, it is not unexpected to find the functional groups having an affinity towards another due to their polar nature. The mixture was viewed as being a non-polar medium dispersed with polar sites on a molecular level.

An important experimental consequence is that reactor samples needed to be immediately quenched in liquid nitrogen to stop the reaction, and to make the IR spectrum without delay. It was thus possible to obtain representative results (22).

It also shows the importance of having been able to take the samples directly off the extruder screws. A sampling technique relying on quenching the extruder (e.g. 29 and 30) would not have been sufficient.

Application to Reactive Extrusion

Penetration of NP8

The mixing of the injected oligomer can be divided into two steps: mixing the oligomer into the melt, and mixing within the melt. Since the content of the NP8 in the rubber could be measured, it was possible to determine the rate of penetration per unit screw length.

A single experiment showed that the oligomer penetrated the [EPR.sub.MA] over a very short screw distance. The same screw geometry as in Fig. 3 was used, except that the die head was unmounted and the injection performed at port 1. The distance to the open end of the extruder measured 1.3 L/D, with an average residence time of 11 s. Running the extruder at 50 rpm, the IR spectra showed consistently that all the NP8 had entered the [EPR.sub.MA].

Comparison of Conversions

The residence time distribution was determined at each plug (as indicated in [ILLUSTRATION FOR FIGURE 3 OMITTED]), at the end of the screws and at the die exit by having placed impulses of the tracer at the feed [ILLUSTRATION FOR FIGURE 10 OMITTED]. The axial dispersion model, here given due to a pulse input

E([Theta]) = 1/2 [square root of [Pi][Theta][Pe.sub.L]] [e.sup.-[[(1 - [Theta]).sup.2]/4[Theta][Pe.sub.L]]] (20)

was used to approximate the obtained data, see e.g. (27). For a better fit, not only the value of the Peclet number ([Pe.sub.L]) was optimized, but also that of the mean residence time (Table 1). The parameters thus evaluated are shown as solid curves in Fig. 10.

The data from port 5, just after the first set of kneading blocks, was actually quite noisy, indicating that clumps of rubber were emerging and homogenization incomplete. Moving further downstream there was a good fit with the data, except that the model produced an increasingly symmetrical distribution with downstream distance whereas the data from the extruder tended to remain skewed. Nevertheless, the fit was judged sufficient for subsequent calculations.

As a first approach, it was supposed that all the NP8 instantaneously penetrated the [EPR.sub.MA] and mixed perfectly. Using the estimated kinetic parameters, it was possible to estimate the CAn conversion as a function of time [ILLUSTRATION FOR FIGURE 11 OMITTED]. The experimental data from the extruder was compared by plotting the CAn conversion as a function of the mean residence time. An initial deviation thus became apparent.

Account for the RTD

To find the source of deviation, the complete RTD was to be taken into account. The average conversion is given by

[Mathematical Expression Omitted]

The RTDs given in Fig. 10 are those determined for the polymer flow between the feed hopper and the various sampling locations along the screw, while the RTDs between these sampling points are necessary for the calculations. To determine these, the well-known convolution needs to be considered.

[E.sub.i](t) = [integral of] [E.sub.5](t[prime])[E.sub.j](t - t[prime]) dt[prime] between limits t and 0 (22)

where [E.sub.t] is the RTD at the given sampling location (known), [E.sub.5] that at port 5, and [E.sub.j] that between ports 5 and i (to be estimated). The possibility of using this principle for twin-screw extruders has been mentioned in Ref. 31, although its application has not been realized due to numerical stability problems in rigorously calculating the deconvolution.

The approach taken was to continue using the axial dispersion model, and to find the optimal values of the Peclet number and mean residence times in [E.sub.j], i.e.

[Mathematical Expression Omitted]

The resultant values (Table 2) showed that the axial dispersion model gave very close fits between ports 5 and 3 and further downstream; the predicted and experimentally determined curves fall perfectly on one another. However, between ports 5 and 4 the fit was not so good (as shown by the dashed line in [ILLUSTRATION FOR FIGURE 10 OMITTED]). The axial dispersion model can be used when the difference in residence times between the sampling points is above about 1.5 min, under the conditions studied.
Table 1. Parameters for the Axial Dispersion Model When Using the
Screw Geometry of Fig. 3.


 Model Actual
 Mean Mean
 Res. Res.
Sampling [Pe.sub.L] Time Time
Location (-) (min) (min)


port 5 0.0277 1.0 1.2
port 4 0.0140 2.0 2.1
port 3 0.0096 3.2 3.3
port 2 0.0079 4.2 4.0
screw 0.0063 5.3 5.0
tips
Table 2. Parameters for the Axial Dispersion Model to Give the RTD
Between Sampling Points.


 Model
 Mean
 Res. CA
Sampling Peclet Time Conversion
Location Number (min) (%)


port 4 0.0140 1.00 29
port 3 0.0154 2.11 40
port 2 0.0114 3.11 44
screw 0.0180 4.00 45
tips


Placing Eqs 14 and 20 into Eq 21 with the processing parameters as given in Fig. 2 and the parameters for the axial dispersion model as given in Table 2 yields estimates of the CAn concentration and its conversion (listed in last column of Table 2). These data were placed on the plot given in Fig. 11.

In comparison with the calculations using only the mean residence time, the CAn conversion was higher when applying the RTD. This first datum lies lower than the experimentally obtained data at the same sampling location, but nevertheless quite close.

Esterification and Mixing

As given in the above text, the NP8 rapidly penetrated the rubber in the screw channels. The question at this point is whether the CAn conversion data as given in Fig. 11 is supportive of the notion whether the NP8 has also been well mixed within the melt pool.

A model was created supposing that all the NP8 injected mixed instantaneously with a partial volume of the melt pool [ILLUSTRATION FOR FIGURE 12 OMITTED]. A molar balance of the alcohol groups in the partial volume which the alcohol occupies in the mixture gives

d[N.sub.A]/dt = [r.sub.A]V (24)

where [N.sub.A] is the number of moles of alcohol, and [r.sub.A] the rate of generation of the alcohol groups.

Differentiating the alcohol concentration

[C.sub.A] = [N.sub.A]/V (25)

with respect to time gives

d[C.sub.A]/dt = 1/V d[N.sub.A]/dt - [N.sub.A]/[V.sup.2] dV/dt (26)

Substituting Eq 24 and rearranging the last term gives

[Mathematical Expression Omitted]

This can be shown to be volume independent by substituting the volumetric fractions and its rate of change with time

[[Phi].sub.A] = V[V.sub.T] (28)

[Mathematical Expression Omitted]

giving

[Mathematical Expression Omitted]

Developing the molar balance for the CAn groups is similar except that they are also introduced into the partial volume; this is reflected in the last term in the equation

[Mathematical Expression Omitted]

where the concentration of CAn groups outside the partial volume was introduced.

The reactive term [r.sub.Ri] was not replaced in the last two equations to show their applicability for any reactive condensation system. For the esterification studied, Eq 6 was substituted for [r.sub.RC] and [r.sub.RA]. The ester concentration is given by

[Mathematical Expression Omitted]

The average conversion of the moisture is given by

[X.sub.app] = [[Phi].sub.A][X.sub.p] (33)

Equations 30 and 31 were numerically solved using the fourth order Runge-Kutta technique (32).

An example was taken where the initial partial volume fraction was 0.05, the temperature 220 [degrees] C, and the global stoichiometric ratio [r.sub.OH/CAn] 1. The volume fraction then increased at a linear rate until it occupied the entire volume at 5 min. This was the approximate residence time in the extruder when the screws turned at 50 rpm. The corresponding apparent CAn conversion profiles show that complete mixing times up to 2 min cannot experimentally be distinguished [ILLUSTRATION FOR FIGURE 13 OMITTED]. Lowering the stoichiometric ratio pulls the curves together, making it experimentally more difficult to distinguish between the curves. The curves also show that with instant and perfect mixing the fastest approach to equilibrium is obtained.

The same model was used to illustrate the apparent conversion profile if the partial volume increased at a linear rate, but once a critical volume fraction was reached, it remained at this value. This simulated a situation where the alcohol is injected into the melt and is mixed up to a given volume fraction, being in effect not perfectly mixed when it leaves the extruder. The results show that the conversion profiles are quite similar and that for final volume fractions greater than about 0.5, the CAn conversion becomes experimentally difficult to distinguish [ILLUSTRATION FOR FIGURE 14 OMITTED].

CONCLUSIONS

To better understand the potential of condensation reactions for the creation of amphiphilic copolymers, complementary work on grafting reactions between maleated rubber ([EPR.sub.MA]) and monofunctional alcohols and amines was performed. This offered the cyclic anhydride (CAn) functionality as a reactive site.

Esterification was studied using a soluble monofunctional alcohol, namely a nonylphenylethoxylate with an ethoxylate chain length of 8 (NP8). It was mixed with the [EPR.sub.MA] and studied in a thermally controlled cell in which the changes in the IR spectrum could be closely followed. These were interpreted to show that the equilibrium could be successfully modeled according to the scheme given by its chemical equation.

The mixing action in the extruder was viewed in relation to the melt pool in the screws' channels; it was divided between the mixing into the pool and that within it. For the prior the NPs were particularly amenable as a research tool, since their phenyl content could be measured in the IR spectra and thus gave the relative concentration of the NP within the pool. It was shown that the miscible NP8 rapidly penetrated the melt pool ([less than] 1.3 L/D), even using conveying elements which are known for their minimal mixing action.

Injections of the soluble NP8 showed that the increase in the rate of conversion along the screws' length occurred slightly faster than one would suppose making the assumption that the reactants are perfectly mixed at the point of injection.

On the other hand, it is difficult to make deductions about the state of the mixture given the amount of CAn conversion obtained from such a reversible reaction. A corollary of this is that although a product may have the amount of conversion expected from thermodynamic considerations, it does not imply that the reactants in the product have been well mixed. Nevertheless, the model provides a means to comprehend the results obtained.

ACKNOWLEDGMENTS

The authors thank Exxon Chemical International for the financial support upon which this work (and Ref. 22) was based. Texaco and ICI donated their samples in a most expeditous manner. The support of Patrice Petitjean, Caroline Kadri, and Dominique Suhr during the experiments is also greatly appreciated.

NOMENCLATURE

[C.sub.i] = Concentration of species i.

[Mathematical Expression Omitted] = Average concentration species i.

[c.sub.ki] = Constants as defined in text.

D = Diameter of extruder screw.

[DOP[prime].sub.1865] = Base-line compensated peak height at 1865 [cm.sup.-1].

E = Residence time distribution.

[e.sub.T] = Factor to compensate [DOP[prime].sub.1865] for temperature.

[K.sub.E] = Equilibrium constant.

[k.sub.RE] = Forward reaction constant.

[k[prime].sub.RE] = Reverse reaction constant.

[N.sub.i] = Number moles of species i.

[Pe.sub.L] = Peclet number.

[R.sub.G] = Ideal gas constant.

[r.sub.OH/CAn] = Initial stoichiometric ratio of alcohol over cyclic anhydride.

T = Temperature in [degrees] C.

t = Time.

[Mathematical Expression Omitted] = Mean residence time.

[T.sub.K] = Temperature in K.

V = Volume.

[V.sub.T] = Total volume.

[Mathematical Expression Omitted] = Rate of volume change.

[w.sub.CAn] = Equivalent weight fraction maleic anhydride.

X = CAn conversion.

[Theta] = Dimensionless time.

[[XI].sub.CAn] = Factor to determine [w.sub.CAn] from IR spectrum.

[[Phi].sub.A] = Volume fraction of alcohol.

[Mathematical Expression Omitted] = Rate of change in volume fraction of alcohol.

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Author:Maier, C.; Lambla, M.
Publication:Polymer Engineering and Science
Date:Aug 15, 1995
Words:5254
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