Thermodynamics of Some Concentrated High Polymer Solutions David C. Bonner,* Norman F. Brockmeier,' and Yang-leh Cheng Chemicai Engineering Department Texas Tech Universify, Lubbock. Texas 79409
We have used the corresponding-states theory of polymer solutions to correlate data for the systems n-decane-high-density polyethylene at 185"C, ethylbenzene-atactic polystyrene at 120°C and 185"C, benzene-atactic polystyrene at 12O"C, n-hexane-polypropylene at 80°C, isooctane-high-density polyethylene from 157 to 237°C at 70 atrn, and n-hexane-high-density polyethylene from 137 to 157'C and 4 to 70 atm. The theory correlated data at low pressures moderately well, and the higher pressure data were qualitatively modelled by the theory. The data correlations are discussed in the light of the estimated reliability of the experimental data.
Introduction There are many industrial situations in which accurate knowledge of thermodynamic properties of concentrated polymer solutions i s invaluable to the process designer. For example, the drying of many paints requires that a solvent evaporate from a polymeric film, often a t ambient conditions. Some polymers that are produced in a slurry are devolatilized a t moderately elevated temperatures in a stream of inert gas. The mass thermal polymerization of styrene with rubber and the high-pressure polymerization of ethylene are situations in which phase equilibria are important. Polymers produced via a solution process may be separated from the solvent and any unreacted monomer in a melt devolatilizer, a step which is sometimes preceded by heating under pressure. Additional volatiles or condensation products may be removed in a vented extruder. These processing steps cover a range of temperatures from ambient to about 250°C and pressures from 0.01 to 1000 atm. For devolatilization a t a given set of conditions, i t is extremely useful to be able to calculate the equilibrium vapor pressure above the polymer. This pressure sets a lower limit to the degree of dryness that can be achieved in a recirculating gas dryer. The relative merit of using reduced pressure in a devolatilizer depends on the solvent activity in a polymer. The construction requirements for a melt heater will be affected by the pressure developed by the solvents in the melt. Temperature and/or pressure changes may cause a homogeneous polymer solution to separate into two or more phases. The designer may be able to make use of this separation to isolate a lean or rich stream for further processing. On the other hand, an unanticipated phase separation may foul a process unit. Finally, the ability to calculate solvent activity will aid in the selection of solvents that are best suited to the process requirements. During the past 30 years or more, many workers have measured solvent activities over polymer solutions, usually employing static equilibrium techniques that tend to be slow and to be limited to moderate and dilute polymer concentrations. The urgent need for rapid data acquisition, especially for highly concentrated solutions, has led to the development of a new gas chromatographic (GC) method for obtaining thermodynamic data from polymer solutions over a wide range of temperature and pressure, Some of the early work with the GC method has been reported by Smidsrdd and Guillet (1969), Patterson, et al. (19711, and Newman and Prausnitz (1972). However, all
' Amoco Chemicals Corporation, Naperville, Ill. 60540
of these results were limited to measurements of the activity coefficient a t infinite dilution of the solvent. Brockmeier and coworkers (1972) extended the GC method to permit measurement of solubility isotherms by using finite concentrations of solvent vapor in the carrier gas. The finite concentration GC method was improved to extend its operating range from near ambient conditions to over 70 a t m and over 240°C (Brockmeier, et al., 1973). These latter investigators employed the Flory-Huggins equation and also the Heil-Prausnitz (1966) equation to correlate most of their activity data. In some additional research, Newman and Prausnitz (1973) used the Flory-Huggins equation to extrapolate their infinite dilution results to finite concentrations. The Flory-Huggins equation, however, has serious deficiencies and often gives poor results in correlating solvent activities. This article discusses the merits of using a newer, corresponding-states solution theory based on the concepts of Prigogine (1957) and Flory (1965) to correlate a number of GC measurements on polymer solutions. The solutions studied include high-density polyethylene, atactic polypropylene, 'and polystyrene in which were dissolved selected aromatic and alkane hydrocarbon solvents. All of the data used in the study have been previously reported (Brockmeier, e t al., 1972, 1973). Experimental Section
A brief description of the experimental method follows, because the details have been published elsewhere (Brockmeier, et al., 1972, 1973). The GC columns were prepared using acid-washed Chromosorb W, 80/100 mesh. Selected polymers were coated on the support by slowly cooling and then evaporating a dilute solution. The finished columns were made of 1/4-in. stainless steel tubing 10 ft long and contained about 1 to 5 wt '?LO polymer on about 16 g of support. The GC retention times were measured with a dual katharometer detector, using nitrogen carrier gas containing a n accurately preset concentration plateau of the chosen solvent. After the system reached steady state, an additional small amount of a gas mixture of solvent and helium was injected. The primary GC data consisted of the retention time difference for the two small peaks (helium and solvent) eluted on top of the relatively large plateau. D a t a Reduction The treatment of the raw GC data is essentially the method of Conder and Purnell (1969). The retention volume 1 V of the solvent a t the conditions in the GC column is calculated from the recorded peak times Ind. Eng. Chem.. Process Des. Develop..Vol. 1 3 , No. 4, 1974
437
where t~ - t M is the difference in the retention times of the solvent and helium respectively, T is the column temperature, and T F is the flowmeter temperature. The mix, PF and po are the presture compressibility factor is ;Z sures a t the flowmeter outlet (atmospheric) and column outlet, respectively. F M is the correction to the flowmeter reading required due to water saturation vapor pressure and VF is total flow rate measured a t the outlet of the column. The AV results must be corrected for the sorption effect of the solvent in the substrate using the parameter $. The value of $ was usually approximately equal to the mole fraction of solvent in the carrier and was calculated by the method of Conder and Purnell (1969). Therefore, A V corrected for sorption is related to the plateau concentration c AV, =
AV 1-4
One of the fundamental properties calculated is the solubility q ( p ) , defined as the gram-moles of solvent sorbed per gram of dry polymer at a mean column pressure p
where Js2 is a correction for pressure drop across the column, m2 is the mass of polymer in the stationary phase, and c is solvent concentration in the carrier gas. The method requires that AVc be measured a t several chosen plateau concentrations and q ( p ) is calculated graphically from eq 3 a t constant temperature T and pressure p . The weight fraction of solvent w1 a t equilibrium in the polymer solution follows from eq 3
(4) where .I41 is the solvent molecular weight. The results of eq 1-4 are used to calculate the weight-fraction solvent activity coefficient in the polymer R 1 from the definition of Patterson, et al. (1971)
(5) The solvent fugacity a t mean column conditions is f1, is the fugacity of pure, saturated solvent a t column temperature (the standard state), and a1 is solvent activity a t mean column conditions. The weight-fraction activity coefficient, Q1, is convenient for polymer solution calculations because it requires no knowledge of the polymer molecular weight or molecular-weight distribution, so R 1 is easily applied to polydisperse systems. The above method for obtaining the solvent activity does not require the use of a solution theory. Theoretical Correlation of Phase Equilibria Results in Polymer Solutions. The first solution theory for polymer solutions which qualitatively correlated many of the phenomena associated with polymer phase equilibria was proposed independently by Flory (1941) and Huggins (1941). The Flory-Huggins theory proposed that there are two major, independent contributions t o the thermodynamic properties of binary, polymer-solvent solutions: an entropy of athermal mixing and an enthalpy of mixing which is due to differences in intermolecular forces. The contributions of intermolecular forces are approximated in the Flory-Huggins theory by an empirical van Laar term.
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The solvent activity a1 given by the Flory-Huggins theory for a binary, polymer--solvent solution (1 = solvent; 2 = polymer) is
where rL = number of segments per molecule of i; Q, = segment fraction of i = x L r L / Z L x L xr L ;= mole fraction of i in solution; and x = empirical parameter characteristic of binary intermolecular force interactions between molecules of types 1 and 2. The empirical parameter x is assumed to be independent of composition. However, when that parameter is calculated from experimental activity data it is found to vary considerably with concentration for most types of polymer solutions. The variation of x with concentration is a result, in part, of the crude nature of the intermolecular force (van Laar) term in the theory. The Flory-Huggins theory, assuming constant x, gives approximate representation of the activities of nonpolar polymer solutions using only one adjustable parameter and allows approximate calculation of liquid-liquid phase equilibria in binary, nonpolar polymer solutions. Since the Flory-Huggins theory is based on a rigid lattice model, it gives no equation of state. It assumes that isobaric, isothermal mixing of solvent and polymer occurs with no change of volume. Corresponding-States Theory of Polymer Solutions. The thermodynamic properties of pure fluids and mixtures obtained from statistical mechanics can be divided, to good approximation, into two categories: combinatorial and noncombinatorial (Prigogine, 1957). The entropy of athermal mixing is a combinatorial property, while pressure-volume-temperature (PVT) properties due to intermolecular forces are noncombinatorial. The combinatorial contribution in the Flory-Huggins theory (the entropy of athermal mixing) is the simplest formulation available for practical use. However, the Flory-Huggins estimate of noncombinatorial properties is crude. Prigogine (1957) developed a corresponding-states theory for polymer solutions which attempts to account for noncombinatorial contributions in a more rigorous and realistic way than does the Flory-Huggins theory. The major feature of the Prigogine theory is that the energy modes that give rise to internal and external degrees of freedom are explicitly treated. Internal degrees of freedom are those which are (essentially) unaffected by the presence of neighbors, while external degrees of freedom are those which are significantly affected by the presence of neighbors. For example, translational motions of the molecules are external degrees of freedom a t all densities, and internal rotations tend to be external degrees of freedom a t high densities. High-frequency vibrations, however, are unaffected by the presence of neighbors, and therefore they are considered to be internal degrees of freedom. Prigogine (1957) postulates that an assembly of N chain molecules, of r segments each, behaves as if it were an assembly of rcN spherical molecules containing one segment each; 3c is the number of external degrees of freedom per segment. Several workers have used the Prigogine concept of a corresponding-states theory for polymer solutions (Simha and Hadden, 1956; Hijmans and Holleman, 1969, and earlier papers cited therein; Patterson, 1969, and earlier papers cited therein; Flory, 1970, and earlier papers cited therein; Bonner and Prausnitz, 1973; and Bonner, et al., 1974). Using the Flory (1965) version of the correspondingstates theory, one may calculate the activity ai of the solute in the polymer (Bonner and Prausnitz, 1973)
Table I. Characteristic Parameters for Solvents and Polymers Substance
['SD
*
3
cm3/g
T*. ~
Benzenea -Decane Et hylbenzene' iz -Hexane Isooctane a Pol yet hylene, high -density" Polypropylenea Polystyrene ' 12
We have chosen the standard state for al to be that of pure, saturated solvent a t solution temperature T. The parameters U I * ~ ~ ,p1*, , and TI* are characteristic reducing parameters of the solvent. Similar parameters for the polymer are denoted by a subscript 2. The segment fraction of the solvent in the mixture, *1(=l - *2), is given by
=
*l
w l v l * s p / ( w l ~ l * s p + w2v2*sp)
(8
where w1 (= 1 - w2) is the weight fraction of solvent in the mixture. R is the gas constant and T is the absolute temperature. The characteristic pressure p l z * is given by
where S is a measure of the deviation of the binary interaction parameter plz* from the geometric mean of p1* and p z * . The segment ratio r 1 / r g is given by
as suggested by Flory (1965). The reduced volume of pure solvent 51 is equal to ulsp/ul*sp, where u l s r , is the solvent specific volume. The quantity 5 is obtained by solving the equation of state for pure solvent a t zero pressure (formally the same as eq 11). The mixture reduced volume 5 is obtained from the equation of state of the mixture a t zero pressure
T
= T / T * = ( ~ ' 1 1 3- 1 ) p 3
(11)
where P = (*I2p1* + Q 2 2 p 2 * + 2*1*2p12*)/(91pl*/T1* + * ~ p z * / T 2 * ) .Note that Tz* is the characteristic temperature of the polymer. It can be seen from the equations used to obtain solvent activity that there is only one binary parameter: p l z * . The pure-component parameters are obtained from pressure-volume-temperature data for each of the components of the mixture (Bonner and Prausnitz, 1973); p l z * can be obtained by fitting binary solution data. The reader is referred to Flory (1965) and Bonner and Prausnitz (1973) for further details of the theory. Application of the Corresponding-States Theory. We have used the corresponding states theory described above to correlate results for the systems n-hexane in atactic polypropylene at 80°C and atmospheric pressure, benzene in atactic polystyrene at 120°C and atmospheric pressure, ethylbenzene in atactic polystyrene a t 120 and 185°C a t atmospheric pressure, n-decane in high-density polyethylene a t 185°C and atmospheric pressure, isooctane in highdensity polyethylene a t 157, 200, and 237°C a t 70 atm, and n-hexane in high-density polyethylene a t 137, 143, and 157°C from 4 to 70 atm. The values of the characteristic parameters used in the calculation are listed in Table I. The data and optimal theoretical correlations are shown in Figures 1-6. The solid lines in Figures 1-6 are the optimal correlations of the data using eq 8. The optimal correlations were obtained by a nonlinear regression analysis to determine values of S by minimizing the sum of the squares of per cent errors of solvent activity for each data set.
"K
P*, a t m
~~
0.890 1.136 0.926 1.198 1.13 1.04
4780 5715 5210 4768 4840 7370
5845 3045 5440 4836 4619 4457
1.00
6940 7970
5662 5409
0.817
a Obtained from Bonner and Prausnitz (1973). Calculated for this work in the temperature range 0-200°C based on the correlations of Bondi (1968).
Let us consider the data correlations in two categories: those data obtained a t pressures below 5 atm and those data obtained a t pressures above 5 atm. The low-pressure data are shown in Figures 1-4 and 6, and the high-pressure data are shown in Figures 5 and 6. The low pressure data are all correlated well by the theory except those for ethylbenzene in atactic polystyrene a t 120°C (see Figure 2 ) . The ethylbenzene data a t 185°C are well correlated. We believe that the ethylbenzene data a t 120°C may not be reliable, since it is possible that the carrier gas in the chromatograph may not have been completely saturated with ethylbenzene, causing the data at 120°C to be somewhat in error. The high-pressure data shown in Figures 5 and 6 are not as well correlated by the theory as the low-pressure data. The isooctane data in Figure 5 are moderately well correlated a t 200 and 237°C but not a t 157°C. This may be due to incomplete carrier gas saturation a t 157°C. The highpressure hexane data shown in Figure 6 are rather poorly correlated by the theory. There are a t least two possible explanations for this. First, the data obtained by Brockmeier, et al. (1973), may not be extremely accurate due to the novelty and experimental difficulty of the technique used to obtain the data. Second, the corresponding states theory discussed here is formulated for low pressures, and using the theory at 70 atm may be trying to utilize it a t conditions which are too extreme. The characteristic parameters for solvent and polymer are somewhat pressuredependent a t high pressures, and so is 5 . Thus it may be that the theory should be used in a form which takes pressure into account (see Bonner, et al., 1974). The characteristic paameters used for the calculations presented here are given in Table I. We assume that the characteristic parameters are independent of temperature and pressure. The binary interaction parameter p l z * (and 1)obtained from regression correlation of data are shown in Table 11. Unfortunately, there are insufficient data available to draw conclusions concerning the temperature dependence of p12* (or 1). For nonpolar systems, however, the temperature variation of p12* should be monotonic, but p12* for isooctane-HDPE is not monotonic. This may lend credence to the contention that the high-pressure data may not be extremely accurate. Discussion of Results The ability of the corresponding-states theory to model results of the type presented here is put in better perspective by analysis of the experimental results using the simpler Flory-Huggins theory exemplified in eq 6. This has Ind. Eng. Chem., Process Des. Develop., Vol. 13,No. 4 , 1974
439
07+
I
P
P 0.61
5- 0.51
6a 0.4W
0.3.
5 0.2X
y 0.11 /-
:
I
_ i d
005 010 015 WEIGHT FRACTION DECANE
002 0.04 006 WEIGHT FRACTION HEXANE
Figure 4. Sorption of n-hexane by polypropylene a t 80°C.
Figure 1 . Sorption of n-decane by high-density polyethylene a t
I
I
185°C.
097
W 120'
c
A 185'
C
0
I
0
157%. 7 0 0 l m
0
200.C. 237%
A
002 004 006 W WEIGHT FRACTION ETHYLBENZENE Figure 2. Sorption of ethylbenzene by atactic polystyrene. I-
-
7001m
70olm
Theory
I
0.20 0.40 WEIGHT FRACTION ISOOCTANE
Figure 5. Sorption of isooctane by high-density polyethylene.
I
9 0.031 -
'
Theory
002 0.04 006 WEIGHT FRACTION BENZENE
Figure 3. Sorption of benzene by atactic polystyrene a t 120" C.
Table 11. Binary Interaction Parameters for Systems Studied System
Temp, "C
A
pi2*,
o.l
atm
f
0.1 0.2 0.3 0.4
WEIGHT FRACTION HEXANE
Decane-high density polyethylene Ethylbenzenepolystyrene Benzenepolystyrene Hexanepolypropylene Isooctane- high dens it y polyethylene Hexane-high density polyethylene
185
0.01034
3646
120 185 120
-0.003623 0.008546 -0.1143
5444 5378 6266
80
0.02104
5007
157 2 00 237 137 143 157
- 0.006270 - 0.00284
1
-0.05149 0.01834 0.01201 0.007024
4565 4550 4770 4557 4587 4610
been done by Brockmeier, et al. (1972, 1973), for the polymer-solvent systems discussed here. Of the systems presented here, only the results for ndecane-polyethylene are fit by an approximately constant x parameter. For all other systems, the x parameters are 440
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4, 1974
Figure 6. Sorption of n-hexane by high-density polyethylene.
highly concentration dependent a t constant temperature varying from 30% for benzene-polystyrene to 400% for isooctane-high-density polyethylene over the respective concentration ranges. The same data are adequately modelled (with the exceptions noted) by the corresponding-states theory using no concentration-dependent parameters. Both theories require one temperature-dependent parameter each. Conclusion We have used the corresponding-states theory of polymer solutions to correlate data for some concentrated polymer solutions a t a variety of temperatures and pressures. The theory correlates data a t low pressures moderately well, although a t higher pressures (of the order of 70 atm) the data are only qualitatively modelled by the theory. The reasons for the relatively poor performance of the theory may be due to failure of the theory itself, or, more plausibly, to inaccuracies in the data.
Since the parameters used in the corresponding-states theory are concentration independent, the correspondingstates theory represents a significant improvement over the Flory-Huggins theory in data modelling for many binary, polymer-solvent systems. Acknowledgment
The authors thank Amoco Chemicals Corporation for permission to publish these results, and R. W. McCoy for his assistance in designing the GC apparatus. The authors also thank the Texas Tech University Computer Center for the use of their facilities. Literature Cited Bondi. A , , "Physical Properties of Molecular Crystals, Liquids, and Glasses," pp 214-260, Wiley, New York, N. y . , 1968. Bonner, D. C.. Maloney, D. P., Prausnitz, J. M . , lnd. Eng. Chem.. Process Des. Develop.. 13, 91 (1974).
Bonner, D. C . , Prausnitz. J. M., Amer. lnst. Chem. Eng. J., 19, 943 (1973). Brockrneier, N. F . . Carlson. R. E., McCoy, R . W., AlChE J.. 19, 1133 (1973). Brockrneier. N. F., McCoy, R. W.. Meyer. J. A . . Macromolecules. 5, 464 (1972). Conder. J. R., Purnell. J. H., Trans. Faraday Soc.. 65,824 (1969). Flory. P. J . , J. Chem. Phys., 9, 660 (1941). Flory, P. J . . J. Amer. Chem. SOC.,87, 1833 (1965). Flory, P. J.. Discuss. faraday Soc., 49, 7 (1970) Heil, J. F.. Prausnitz, J. M., Amer. lnst. Chem. Eng. J.. 678 (1966) Hijrnans, J., Hollernan, T., Advan. Chem. Phys.. 16, 223 (1969) Huggins, M. L.. J. Chem. Phys., 9, 440 (1941). Newman, R. D., Prausnitz, J. M., J. Phys. Chem., 76, 1492 (1972). Newman, R. D., Prausnitz, J. M., AlChEJ., 19, 704 (1973). Patterson, D.. Macromolecules, 2,672 (1969). Patterson, D., Tewari, Y . B . , Schreiber, H. P., Guillet, J. E.. Macromolecules, 4, 356 (1971). Prigogine, I . , "The Molecular Theory of Solutions " North-Holland, Amsterdam, 1957. Simha. R., Hadden, S. T.. J. Chem. Phys.. 25, 702 (1956). Smidsrfid. O., Guillet, J. E.. Macromolecules. 2,272 (1969)
Received for reoiew M a r c h 13, 1974 A c c e p t e d June 5, 1974
Heat Transfer in Fixed Beds Arcot R . Balakrishnan and David C. T. Pei* Department of Chemical Engineering. University of Waterloo. Waterloo. Ontario. Canada
The use of microwave power in the heating of solids in a packed bed results in a uniform temperature throughout the bed, almost instantaneously. Thus, heat transfer through the particle-particle mode between the pellets was eliminated and fluid-particle heat transfer coefficients alone were determined experimentally. Eight commercial catalysts with different physical, thermal, and transport properties were used as bed materials. Ar/Re,,', defined as the ratio of gravity force to the inertia force and the shape factor were found to be the important parameters for heat transfer in fixed bed systems. A model for the total heat transfer in a fixed bed is presented and comparison of results with the data reported in the literature showed good agreement.
Introduction Considering the many applications of fixed bed gassolid systems, it is not surprising to find a considerable amount of work reported in the literature and summarized in a number of textbooks (Davidson and Harrison, 1971; Zabrodsky, 1966). Therefore, only pertinent references and reviews are given here. Barker (1965) presented a n extensive review of such literature and a more recent review has been presented by Bhattacharyya (1973). In the reviews, it was pointed out that the application of many of these empirical and semiempirical correlations are severely limited as they cannot be safely extrapolated beyond the range of existing data and, secondly, discrepancies exist between the results of many investigations. The cause of the discrepancies among the empirical relations was generally attributed to the definitions of parameters and the experimental techniques used by different researchers. Since heat is transferred by several distinct mechanisms, many models have been presented in the literature which consider each mode or mechanism separately. The assumption here is that the contribution of each mode is independent of each other and they are additive. Yagi and Kunii (1957) proposed a model which predicts the effective thermal conductivity of the bed when there
is no flow of fluid through it. Wakao, et al. (1969), studied radiation heat transfer in packed beds and Wakao and Kat0 (1969) studied the effective conductivity of packed beds. They found that the area of contact has no effect on the effective thermal conductivity of the bed. Yovanovich (1973) studied conductivity of glass microspheres from atmospheric pressure to vacuum conditions. He used the Hertz theory for elastic contact to obtain the actual contact area in the model. Kunii and Smith (1960) used a model in which they assumed that only point contact exists between the spheres. Chan and Tien (1973) also used elastic contacts to obtain the area of thermal transfer. It is now generally accepted that a t very low pressures, the use of a model that takes into account the contact area is mandatory. On the other hand, Galloway and Sage (1970) used existing data and their own data to propose a generalized model for the mechanism of transport in packed, distended and fluidized beds. Bhattacharyya and Pei (1974), working with 0.126-in. diameter Fez03 spheres and 0.2-in. diameter Fen03 cylinders with L = D, i e . , a shape factor of 1, showed experimentally that the total heat transfer coefficient in a fixed bed consists of three parts, namely the effective conductivity of the bed, the effect of the presence of fluid on the conductivity, and finally the conInd. Eng. Chem., Process Des. Develop., Vol. 13, No. 4 , 1974
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