Ind. Eng. Chem. Prod. Res. Dev. 1986, 25, 58-64
58
Solvent Removal from Ethylene-Propylene Elastomers. 1. Determination of Diffusion Mechanism Frank J. Matthews, James R. Fair,' Joel W. Barlow, and Donald R. Paul Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712
Charles Cozewith Exxon Chemical Company, Linden, New Jersey 07036
I n the manufacture of ethylene-propylene elastomers, a critical step involves the removal of residual solvent after polymerization is complete. I n such removal, the solvent must diffuse through the solid matrix, through the pore structure, and across the phase boundary into a stripping medium. I n the pilot-scale study reported here, the transport mechanisms and rates were determined for the typical case of hexane solvent being removed by steam. The overall rate of removal was found to be controlled by particle structure, with surface-connected pores playing a prominent role.
Devolatilization is an important step in the commercial production of many polymeric materials. Devolatilization is usually practiced without a fundamental understanding of the mechanism of mass transport in the process. In this paper the devolatilization step in one commercial process for producing ethylene-propylene elastomers is examined. The objectives are to elucidate the mechanism of mass transport and to point out opportunities for increasing commercial process efficiency. Process Description In a typical commercial process, ethylene-propylene rubber (EPR) is produced by solution polymerization using a Ziegler-Natta catalyst. A t the conclusion of the polymerization, the homogeneous solution contains 8-10% rubber in a mixed hexane solvent. The rubber is recovered from the hexane by steam flocculation (Figure 1). The solution is pumped through orifices into a violently agitated pressure vessel full of boiling water. The hexane vaporizes, and the polymer precipitates in the water as a floc or crumb. This crumb takes the form of porous particles, lf4-1 in. in diameter, containing approximately 50% water w/w. After precipitation, the crumb contains approximately 10 w t 70 hexane dissolved in the rubber itself. It is pumped to a second vessel for further steam-stripping; the crumb is then dried by extrusion, which squeezes out and ultimately vaporizes any remaining water. It is the steam-stripping process that is of interest here. Literature Review There are numerous references relating to diffusion of solvent vapors and gases in elastomers. Extensive listings of these references are provided in the review article by Amerongen (1964) and in the dissertation by Matthews (1983). The most pertinent studies are those that deal with diffusion of hydrocarbon penetrants in essentially nonpolar polymers. In all cases, these studies have been limited to diffusion in dry polymers. Despite extensive searching, we have not been able to locate any studies relating to diffusion of organic penetrants in wet, hydrophobic rubbers such as the ethylene-propylene elastomers considered in our work. This is an important point because it will be shown that the presence of water in rubbers affects diffusion of organic penetrants significantly. Care must be 0196-4321J86J1225-0058$01.50/0
taken in generalizing results obtained with dry rubber in the laboratory to include wet rubbers encountered in steam-stripping applications. A number of reports on investigations of solvent removal from synthetic elastomers have originated in Russia (Bazhenov et al., 1974, 1979; Belousov et al., 1977; Iermakov et al., 1976; Shein et al., 1979). Most of the Russian investigators appear to be associated with a production facility and are concerned with both stages of the solvent removal process. They do not characterize the crumb that they produce, to any significant extent, and their methods are largely empirical. The conclusions of the Russian workers may be summarized as follows: 1. For styrene-butadiene rubber (SBR), the main parameter determining the rate of solvent removal in the flash drum is the pressure of the solution before the initial throttling. The Russian workers appear to be controlling the superheat of the hexane solvent. 2. For the second stage,,the size and porosity of the crumb have a decisive influence on the stripping rate. The Russian workers demonstrate this by comparing stripping rates of crumb formed by different devices (i.e., spinneret, steam injector, steam injector with feed preheat). The crumb produced was in the range of 0.3-1.5-cm particle size, and the finest crumb was stripped the fastest. Steam injection with preheating of the polymer solution produced the finest crumb. Iermakov et al. (1976) imply that the finer crumb also had higher porosity, but the crumb is not well characterized and this may not be the case. Another commercial application that has been the subject of investigation is the steam stripping of vinyl chloride monomer (VCM) from poly(viny1 chloride) (PVC) resins. On the basis of laboratory and plant studies, Mantel1 et al. (1975) conclude that mass-transfer resistances external to the particle are observed and that increases in porosity and temperature increase the stripping rate. The resins used by the Mantel1 group are not well characterized, and a quantitative model is not presented. Berens (1974) presents electrobalance data for diffusion of VCM in PVC powders at concentrations of 2000 ppm or less. At these low concentrations, the diffusivity may be considered constant and simplified equations apply. Berens did characterize his particles using BET surface area measurements and scanning electron microscopy. He dem0 1986 American
Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986 59 S T E A M + HEXANE
STEAM t HEXANE
Table I. Properties of Ethylene-Propylene Rubbers mol wt
M,
MJM,
STEAM
P R E C I PITATOR
STEAM STRIPPER
Figure 1. Process for ethylene-propylene rubber stripping/devolatilization.
onstrated that porous particles of suspension PVC are agglomerates of primary particles or subparticles 1-5 ym in diameter. He then showed that sorption curves of the porous agglomerates were in good agreement with those predicted by a uniform sphere model based on the size of the primary particles. This means that diffusion of VCM through the pores is rapid relative to diffusion in the primary particles. A third commercial application of interest is removal of ethylene from low-density polyethylene (LDPE) pellets. Beret et al. (1977) developed a model for purging LDPE make bins of residual ethylene to avoid a possible explosion hazard. This model allows for spherical pellets with a distribution of residence times in the bin. Again, diffusivities are assumed constant because of the low concentrations of residual ethylene in the pellets. All aspects of ethylene-propylene rubber and its properties are reviewed by Baldwin and Verstrate (1972). This is an excellent reference for more general information concerning ethylene-propylene rubber synthesis, manufacture, and end-use applications as well as transport properties; a recent update is provided by Borg (1979). Frensdorff (1964) reports diffusion coefficients in ethylene-propylene rubbers for predominantly nonpolar penetrants. Finally, Schirber (1983) presents a bench-scale study of stripping fabricated EPR articles in laboratory glassware.
SPARGE
% ethylene
+ HEXANE)
PURGE WATER
STEAM
C. C O N D E N S E R
COOLING WATER
-.0
-
specific gravity
D
Distinguishing Features of the EPR Stripping Process The stripping of ethylene-propylene rubber has several features that distinguish it from some of the stripping processes discussed above: (1)The EPR crumb particles are large. Consequently, concentration gradients within the particles must be taken into account when stripping rates are calculated. (2) The initial hexane content of crumb particles is as high as 10 wt % . Consequently, the concentration dependence of the diffusion coefficient must be taken into consideration when stripping rates are calculated. (3) Hexane is insoluble in water for all practical purposes. Hexane appearing at the rubber-water interface will nucleate to form a bubble. Nucleation at the interface could affect profoundly the liquid-phase mass transfer. (4) Water is a small molecule that can penetrate rubber. It is conceivable that the presence of water in the rubber could affect hexane diffusion. Experimental Work Materials. Two ethylene-propylene rubbers were studied. Their properties are outlined in Table I. Solid cross-linked sheets were stripped to determine diffusion coefficients of n-hexane in the rubber. Cross-linking conditions are outlined by Baldwin (1972). The solvent used in all cases was chromatographic-grade n-hexane. It is important to note that the sheets had no visible porosity, in contrast to the crumb. The compression-
where M,/M, = fractional weight loss, D ( 0 ) = diffusion coefficient at zero penetrant concentration, cm2/s, and L = the sheet half-thickness, cm. (2) A linear dependence of concentration was assumed:
D ( C ) = D(0)(1 + mC)
(2)
The parameter m was obtained by fitting diffusion equa-
60
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986
Table 11. Effect of Water on D ( 0 ) n ( 0 ) x io7, cm'js .__-
temp, 105 I15
O C
stream stripper 1.796 2 181
€35 extrapolation from 25
convection oven Frensdorff (1964)" 23.47 22.41
.52.0 81.0
O('
tion solutions to stripping-rate data taken a t two different initial hexane concentrations. The diffusion equation and boundary conditions that were used were dx
(3)
with boundary conditions as follows: no. 1, x = 0, dCldx = 0 for all times t ; no. 2, t = 0. C = C, for all distances x; no. 3, x = AI,,C = C, for all times t. The initial condition requires that the concentration in the sheet is uniform at the beginning of the strip. The second condition requires that the concentration profile be symmetrical about the sheet center line. The third boundary condition requires that the concentration of hexane just inside the sheet surface is zero a t all times. This implies that there is no resistance to liquid-phase transfer, as discussed in the next section. Liquid-Phase Transport. Hexane is insoluble in water for all practical purposes. The solubility is on the order of mole fraction at 100 "C (Braun et al., 1976). When hexane appears at the surface of the rubber, it cannot be transported away by film diffusion through the liquid phase because only a small concentration gradient can be established before the solubility limit is reached. Instead, hexane bubbles nucleate and grow a t the rubber surface. Nucleation increases the rate of transfer of hexane away from the surface by a factor of 5 or 6, relative to film diffusion alone. A question arises as to the proper boundary condition to use when nucleation occurs a t the surface. Since it is not possible to write a rigorous boundary condition a t the surface because of the complexity of the nucleation process, an alternative approach has been taken. Under some circumstances the rate of liquid-phase transfer due to nucleation is so high relative to the rate of transfer in the rubber that the liquid-phase resistance may be neglected. In such cases, a constant surface boundary condition has been justified. Solid-Sheet Stripping. Values of D(0) were determined from eq 3 by stripping sheets both in water and in air. Results are shown in Table 11. Solid sheets were stripped in air in a laboratory convection oven. Frensdorffs data (1964) at room temperature have been extrapolated to the temperature of interest by using an Arrhenius relation with an activation energy of 13 kcal/(g mol). In the presence of water, D(0) decreases by a factor of 13. Evidently, water penetrates the sheet and interferes with the diffusion of hexane through the rubber. The mechanism whereby D ( 0 ) is reduced is unknown. The liquid-phase resistance is negligible since these D(0) values were determined a t hexane concentrations between 100 and 1000 ppm in the rubber sheet. Typically, 5 h of steam stripping i s required to reduce the hexane concentration from 1000 to 100 ppm. The following phenomena have been observed with the ethylene-propylene elastomers used in this work. Water is visible in the sheet removed from the steam stripper. The wet sheet is cloudy, and a receding core of wet rubber is visible if a piece of sheet is allowed to dry on a laboratory bench. This cloudiness is due to the formation of water
0
0.2
0.4
[Do
0.6 t/
0. 8
I O
L2]
Figure 3. Desorption data for thin V-2504 sheets
clusters. This observation has been confirmed by Southern and Thomas (19801, and they have demonstrated that hydrophilic impurities in a hydrophobic rubber will encourage clustering. Clusters can form as a result of hydrogen bond formation between molecules, according to Hoeve (1980), and cluster formation has been associated with the decreasing solubility of water in a polymer as temperature is decreased (Johnson et al., 1980). Here, clustering refers to the full range of possible behavior from simple nonrandom mixing of water molecules in the rubber to gross agglomeration of water molecules that interfere with the transmission of light. The rubber used in our work is a standard commercial product, and it could contain between 0.01 and 0.06 wt % calcium stearate, which is added as a processing aid. It is not known if these impurities are responsible for water adsorption. The amount of adsorption is typical of that observed by Southern and Thomas (1980) for solution-polymerized elastomers which they describe as uncontaminated. Again, we are not familiar with any work that reports diffusion coefficients of organic penetrants in wet, hydrophobic polymers with or without clustering of water, so the actual mechanism whereby water reduces D(0) is unknown. Figure 3 shows desorption data for thin (Le., 40 mil) sheets a t 105 "C. When the rubber sheet is thin, hexane can diffuse quickly to the surface of the rubber, and consequently transfer through the liquid phase could limit the rate. The triangles represent data taken a t low starting concentrations. A value of the parameter m is found which fits the solution of eq 3 by using the boundary conditions 1-3 to the triangle locations. The initial concentration in the calculation is then increased, and the upper solid curve is calculated. The sheets are observed to strip more slowly than calculated. This is due to the presence of a liquidphase resistance. When the experiment is performed with thick (i.e., 77 mil) sheets, calculated and experimental values coincide much more closely, as shown in Figure 4. This indicates that most of the liquid-phase resistance has been eliminated. The stripping behavior is controlled predominantly by the solid phase. The resulting diffusivity-concentration relationship is D ( C ) = 1.796(10-7)(1+ 300C) cm2/s
(4)
When thin and thick sheets are compared, stripping rates calculated from thin-sheet data underestimate the strip-
Ind. Eng. Chem. Prod. Res. Dev.. Vol. 25. No. 1. 1986
/
VI VI
80
-
GO
-
o
m
VI
0
0
J
J
ap
61
8
THICK V-2504 SHEETS
40
77 m i l , Mm = 0 , 0 9 4
-
20
A
70 m i l , M, = 0 . 0 9 5
o
77 mil
, M, =
0.029
" 0
0.2
0.4
0.6
1.0
0.8
40
60
80
100
..
100,
80
20
Figure 6. Temperature dependence of stripping rate for 40-mil sheet
Figure 4. Desorption data for thick V-2504 sheets.
I
0
77 m i l V - 2 5 0 4 SHEETS
A
-
. 0
v)
GO
-
40
-
VI
0 J
8
M,
0.085
20
0.094
-
0.029 0.024 0.026
0
20
40
, [ $1 '12
60
80
100
$2
Figure 5. Temperature dependence of stripping rate for 77-mil V-2504 sheets.
ping rate of thick sheets. This indicates that thin sheets are always liquid-phase limited to some extent. Figure 5 illustrates the temperature dependence of the stripping rate for thick sheets. In this case, diffusion in the rubber is controlling the rate and there is no increase in the rate when the temperature is raised 10 "C. Figure 6 illustrates the temperature dependence of the stripping rate for thin sheets. A t 105 "C, the liquid phase is limiting the rate and a 10 O C increase in temperature results in a large increase in rate. Nucleation is the predominant phenomenon affecting liquid-phase transfer. The driving force for nucleation is related to the hexane partial pressure in the vapor in equilibrium with the rubber surface. This increases rapidly with temperature if the system is above the boiling point of hexane. Consequently, the nucleation rate increases quickly with temperature and the liquidphase resistance is reduced. Porous C r u m b Stripping. The properties of porous crumb particles taken from a commercial stripper are shown in Table 111. Figure I shows a cross section of a typical crumb particle viewed with a scanning electron microscope. Much of the porosity in these particles results from encapsulated water. Only a few pores are really
Figure 7. Crcrss section of a crumb particle. maynitimtim :Xlx,har length = 10110 ~ m . Table 111. Porositv and Rubber Content of V6600 Crumb porosity w l 70 riihher 0.384 55.5 0.396 55.2 0.343 59.1
surface connected. This was discovered by drying the particles in a vacuum oven. After several days at full vacuum and 40 O C , the particles were cut open and liquid water was still present in the voids. The stripping behavior of the particles is shown in Figure 8. The solid curve has been calculated by using eq 3 for a 0.63-cm diameter solid sphere. Evidently, porous particles strip much more quickly than their external diameter would indicate. Two mechanisms are available to explain this, as shown by Schirber (1983) in Figure 9. First, crumb particles could have a surface-connected pore structure that divides the particles into small subregions. Hexane molecules would diffuse through the rubber into a surface-connected pore and then migrate through the pore to the surface. Alternatively, hexane could diffuse through rubber- and water-filled voids in a series-parallel fashion. The effective diffusion coefficient for the heterogeneous medium would be greater than the diffusion coefficient of hexane in the solid alone.
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986
62
IO0
Table IV. Effective Radii for V-5600 Crumb Particles external diameter, cm 0.63 1.27 long-time sorption 0.0953 0.138 wet profiles 0.0745 0.091 superposition 0.0750 0.108
00
v)
60
v)
100,
0
1
I
_I
V A 0
0
40
0.
0.63cm POROUS CRUMB
1.27cm POROUS CRUMB 20
Ma
O C
PREDICTED,0.63 cm SOLID SPHERE
o 0
V
0
20
40 f 52
60
[s3
80
100
'12
Figure 8. Crumb stripping a t 105 "C and initial concentration of 0.09 g of hexane per g of rubber.
,\--, \ %
1
0
HEXANE'
, MOLECULE
\
c
0,025 0,027 0,090
0.093 0.086
115
I .27 c m DIAMETER C R U M B PARTICLES
I
0 '
\
\'\'
20
A A
105 115 105 105
60
30 t
.
'4
[s3 4
90
I20
Figure 10. Temperature dependence of stripping rate for 1.27-cm porous crumb particles.
WATER
Figure 9. Potential mechanisms for n-hexane transfer through porous crumb particles (Schirber, 1983).
The second hypothesis was rejected when it was found that crumb particles of different diameters strip at approximately the same rate. External crumb diameter would have a much greater effect if series-parallel diffusion were occurring. Further, hexane has a very small solubility in water, and a water-filled void could support very little flux during series-parallel diffusion. It was concluded that crumb particles must consist of small regions which strip into surface-connected pores. This implies that the stripping rate can be increased by introducing more surface-connected pores in the particles. Effective Radii for Crumb Particles. It is desirable to find a means of relating crumb structure to stripping rate. Since pores divide the particles into small subregions that strip independently, it is reasonable to characterize the average size of a subregion. This was done in three ways. First, effective radii were fit to long-time sorption data by using eq 5, which applies to spheres d - [In (1- M t / M , ) ] = -D(O)T~/(R,)* (5) dt where Re = equivalent radius of the sphere. Second, effective radii were fit to stripping-rate data at two different initial concentrations by using Re as the adjustable parameter in the solution of the diffusion
equation in spherical coordinates, given D(C). Finally, a third value of an effective radius was obtained by a particle-characterization technique designated as a wet-profile measurement. This must be explained further. During preparation of crumb sections for microscopic examination, it was noted that crumb sections do not dry uniformly. When a flat 1.6 mm thick section of crumb, cut from the center of a wet crumb particle, is dried at 50 "C, some regions dry quickly. These regions are apparently connected to the surface of the section by pores or voids that were opened when the section was cut. Other regions of the section remain wet for several days. The difference between wet and dry regions is clearly visible; the wet regions are white in appearance. These regions are irregular in shape and are not distributed uniformly across the section. Some sections are composed largely of wet regions, others display almost no wet regions at all. Examinations of V-5600 crumb with the scanning electron microscope does not lead to the conclusion that wet regions are the less porous regions of the particle. Voids in the particle seem to be uniformly distributed. The structural parameter that influences the drying rate is not the distribution of voids, but the extent to which the voids are interconnected and connected to the surface of a section. It is reasonable to assume that the regions which hold water upon drying also hold hexane upon steam stripping. Therefore, if a suitable model were available for characterizing wet regions, it might also be useful for characterizing the hexane stripping rate. Such models are available, and their application to wet regions of crumb particles is outlined in the Appendix. The number-average radius of the wet regions modeled as spheres appears in Table IV with the other measures of effective radius outlined above. The agreement in the table is good, considering the complexity of the particles and the variety of measurements undertaken. Figure 10 shows the data collected for 1.27-cm, wet V-5600 crumb during steam stripping at 105 and 115 "C. There is no marked effect of temperature on the stripping
Ind. Eng. Chem. Prod. Res. Dev., Vol. 25, No. 1, 1986 03
MODEL
A
MODEL B
Figure 11. Two models for two interpenetrating phases. "
rate. This is in agreement with the 77-mil solid-sheet results which showed no severe changes with an increase in temperature. The implication is that diffusion within the rubber is controlling the crumb stripping rate. Conclusions The significant results of this work are as follows: (1) It has been found that the presence of water in V2504 sheet substantially reduces the D(0) value for nhexane diffusion in the rubber. (2) Effective diffusion coefficients for n-hexane diffusion in wet rubber have been measured for both V-2504 and V-5600 by stripping solid-sheet at 105 "C. Both materials have the same effective diffusion coefficients for hexane diffusion a t this temperature. (3) Wet V-5600 crumb particles have been stripped, and the stripping rate has been found to be weakly dependent on crumb diameter in the range of 0.63-1.27 cm and almost independent of temperature in the range of 105-115 "C, within the accuracy of our measurements. (4)Effective radii have been fit to stripping-rate data for crumb particles, and these radii are on the order of 1 mm for 0.63 and 1.27 cm diameter crumb particles. Apparently some pores penetrate the particles and divide them into subregions that strip into the surface-connected pores. (5) Particle structure has been related to the stripping rate by measurements made on regions of wet rubber remaining in thin sections of partially dried crumb. The number-average radii of these wet regions yield the same radii as determined by crumb stripping. The wet regions represent zones in the particle that are not penetrated by surface-connected pores. (6) The crumb stripping rate is not influenced by the liquid phase surrounding the particle. Liquid-phase resistance can be significant when thin solid sheets are stripped. Nucleation of hexane at the particle surface is the feature determining the liquid-phase transfer coefficient. Circumstances that promote nucleation will reduce the liquid-phase resistance. In one test with solid sheet, the liquid-phase resistance was reduced by increasing the stripping temperature, which increases the driving force for nucleation. (7) Predictions based on laboratory data are in reasonable agreement with plant data (see part 2). We conclude that crumb stripping is controlled by particle structure a t both laboratory and plant scales of operation. Surface-connected pores play an important role by dividing particles into small subregions that strip quickly into these pores. It should be possible to increase the stripping rate of crumb particles by introducing more surface-connected pores into the particles. If a liquidphase resistance is encountered as the stripping rate is increased, it may be possible to eliminate it by increasing the stripper temperature.
I
2
3
4
5
6
CLASS
7
8
9
1
0
1
1
1
2
INTERVAL
Figure 12. Profile histograms and histogram completion.
Acknowledgment We thank Exxon Chemical Co. for its financial support of this work. Appendix Particle Characterization by Stereology. Stereology is a body of mathematical methods relating the three-dimensional parameters defining the structure of an object to two-dimensional measurements obtainable on sections of the structure. Weibel (1979) presents two models for representing a material composed of two interpenetrating phases shown in Figure 11. In model A, the elements of the object phase may have any shape; they can be connected and show as many nonconvexities as is conceivable. In model B, the elements of the object phase are phase are discrete particles, and one in general has to assume convex shape. If all the particles in model B are assumed to be spheres, then diffusion calculations can be performed readily. Hence model B was adopted in the present work. Profile Histograms. The profile size distribution corresponding to the wet regions of a particle must be determined quantitatively and reproducibly. A procedure that can be applied directly to the particle sections without microscopy is outlined below. A practical and very useful measure of profile size is the diameter of a circle having an equal area, called the "area equivalent diameter", de,, which is de, = 2(a/7r)'/' where a = profile area. The practical procedure for estimating d, is very simple: one requires a set of test circles of increasing diameter. A convenient tool is a transparent plastic stencil as used for graphic work. The diameter diof the set of test circles increases from circle to circle by a constant amount. The smallest size class extends from 0 to h/2. The remaining classes extend from di- h/2 to d, + h/2 as i takes on the values 1to n. The class sizes used in this study were based on h equal to 0.0397 cm. The profiles are counted and assigned to the appropriate size class. The resulting frequency distribution is plotted as a histogram in Figure 12. All histograms are deficient in the lower part of the plot because of the loss of small profiles. Small profiles are difficult to count accurately. Completion of the histogram is always necessary since an unknown number of small profiles is always missing. It is not necessary to measure the lower quarter of the histogram, since it is certainly in error and the correction procedure is independent of measurements made in this region. The histograms are corrected as follows: (1)Smooth the histograms by hand, eliminating excessive peaks and class-to-classfluctuations. (2) Determine the mode of the distribution (peak of the
04
Ind. Eng.
Chem. Prod. Res. Dev., Vol. 25, No.
1, 1986
smoothed curve), and mark a point a t half the height of this peak. (3) Extrapolate linearly from this point toward the origin. (4) Draw histogram bars a t the intersections of the extrapolated line with class midpoints for all classes in which the actual profile count is less than the extrapolated profile count. ( 5 ) Smooth the lower part of the histogram by hand using the extrapolated histogram bars. The resulting smoothed distribution is the observed profile distribution corrected for the loss of small profiles. It must be transformed mathematically into a sphere size distribution. Calculation of True Particle Size Distributions from Profile Distributions Observed in Thick Sections. One of the problems encountered in measuring the size of wet regions is the thickness of the section required. The section must be thick enough to prevent rapid drying of the particles and thin enough to prevent overlap of regions which makes identification of individual regions impossible. Obviously, a suitable section thickness will be on the order of the diameter of the wet regions. The effect of using a section this thick relative to the wet regions themselves is to increase the number of profiles in the larger profile size classes derived from a particular region. A method outlined by Goldsmith (1967) is available for transforming profile distributions to particle size distributions when section thickness is of the same order as the particle diameter. This involves solution of a system of linear algebraic equations by successive substitution, which is easily done on a computer. The true frequencies f,, f 2 , ..., f, in each histogram interval are calculated from the observed frequencies g,, g,, ..., g, by means of the equations
hn = gn/%n h, = (B, - ai,i+lhL+l- a,,,+zh,+z -aL,nhn)/%,, **a
where i = n - 1, ..., 1,f, = (C,"gi/CIRhJhi, i = 1, 2, ..., n, and the coefficients are given by =u
+
+ (i - 3/4)'/2
where j = i + 1, i 2, ...,n,u = d / h , d = section thickness, and h = particle size class width. The true particle size distribution is given by fl, ..., f,,. Nomenclature ai,, = coefficients defined in Appendix C = solvent concentration at any point in the polymer, g/cm3 C, = arbitrary constant concentration in the polymer, g/cm3 d = section thickness, cm de, = area equivalent diameter, cm D ( C ) = diffusion coefficient of solvent in the polymer system or polymer/water system, cm2/s
D(0) = diffusion coefficient at zero penetrant concentration, cm'/s f l , ..., f,= true sphere size distribution expressed as frequencies. f i is determined as the frequency of occurrence of a sphere of the ith class size g,, ..., g, = observed profile size distribution expressed as frequencies. gi is defined as the frequency of occurrence of a profile of the ith class size h = class interval width h,, ..., h, = variables defined in Appendix L = one-half of the thickness of a solid sheet, cm. The sheet is defined to lie in the region -L < x < +L M , = solvent content of polymer at the beginning of a steam strip, g of solvent/g of polymer M , = number-average molecular weight Mt = solvent content of polymer at any time, g of solvent/g of polymer M t / M , = fraction of solvent remaining at any time M , = weight-average molecular weight n = number of class intervals r = radial coordinate in spherical coordinates, cm Re = effective radius of a porous particle, cm t = time, s u = number of class intervals spanned by the section thickness = d/h, cm x = x coordinate in rectangular coordinates, cm Registry No. Hexane, 110-54-3.
Literature Cited Amerongen. G. J. Rubber Chem. Technoi. 1964, 3 7 , 1065. Baldwin, F. P.; Verstrate, G. Rubber Chem. Technoi. 1972, 4 5 , 709. Bazhenov, V. D.; Shein, V. S.;Velousov, N. S.;Minaev, V. G. Zh. Prikl. Khim. (Leningrad) 1974, 4 7 , 407. Bazhenov, V. D.; Shein, V. S.; Konoalenko. N. A. Int. Polym. Sci. Technol. 1979, 6(8),12. Belousov, N. S . ; Sherban, G. T. Zh . Prikl. Khim. (Leningrad) 1977, 5 0 , 355. Berens, A. R. Polym. Prepr. ( A m . Chem. SOC.,Div. Polym. Chem.) 1974, 15, 203. Beret, S. E.; Mulhke, M. E.; Viliamii, I.A. Chem. Eng. Progr. 1977, 73(12), 44. Borg, E. P. "Kirk-Othmer Encyclopedia of Chemical Technology"; Grayson, M., Ed; Wiley-Interscience: New York, 1979; Vol. 8, p 492. Braun, W. G.; Danner, R. P.; Daubert, T. E., Eds. "API Technical Data Book-Petroleum Refining"; American Petroleum Institute: New York. 1976. Crank, J. "The Mathematics of Diffusion"; Oxford University Press: Cambridge, 1975. Frensdorff, H. K. J . Polym. S d . , PartA 1964, A 2 , 341. Goldsmith, P. L. B r . J , Appi. Phys. 1967, 18, 813. Hoeve, C. A. J. ACS Symp. Ser. 1980, No. 127, 135. Iermakov, V. I.; Mamiedov, U. A.; Dubuzhskii, B. E. Teor. Osn. Khim. Tekhno/. 1976, 10, 137. Johnson, G. E.; Bair, H. E.; Matsuoka, S.;Anderson, E. W.; Scott, J. E. ACS Symp. Ser. 1980, N o . 127, 451. Manteli, G. J.; Barr, J. T.; Chan, R. K. S. Chem. Eng. Progr. 1975, 71(9),54. Matthews, F. J. Ph.D. Dissertation, The University of Texas at Austin, 1983. Schirber, P. C. M.S. Thesis, The University of Texas at Austin, 1983. Shein, V. S.; Grib, A. P.; Kozlov, A. U. Int. Polym. Sci. Technol. 1979, 6 , 83. Southern, E.; Thomas, A. G. ACS Symp. Ser. 1980, No. 127, 375. Weibel, E. R. "Stereological Methods"; Academic Press: New York, 1979.
Received f o r review August 16, 1984 Accepted August 29, 1985
