Ind. Eng. Chem. Res. 1996, 35, 4457-4463
4457
Measurement of Diffusivity and Solubility of Carbon Dioxide in Gelatinized Starch at Elevated Pressures Bhajmohan Singh,† Syed S. H. Rizvi,*,† and Peter Harriott‡ Institute of Food Science and School of Chemical Engineering, Cornell University, Ithaca, New York 14853
The effect of pressure on diffusivity in binary or multicomponent systems such as gas-liquid or gas-solid systems has rarely been reported. The diffusivity of carbon dioxide in extruded gelatinized starch has been measured in this study at pressures of up to 117 bar (1700 psi). Such data are fundamentally not only useful in the understanding of the supercritical carbon dioxide (SC-CO2)/starch systems but also can be useful for the design and control of processes utilizing carbon dioxide injection or mixing in starch-based matrices. The methodology developed here was an improvement over a previously reported technique, enabling high-pressure data to be obtained. The diffusivity of carbon dioxide in the melt was found to be a strong function of pressure but not of moisture content in the range of 34.5-39% (w/w) studied. This diffusivity value decreased from 7.5 × 10-10 to 0.9 × 10-10 m2/s as pressure was increased from atmospheric to 115 bar. The low-pressure diffusivity value was only an order of magnitude lower than that reported for a carbon dioxide in water system and comparable to reported values of diffusivity of CO2 in softened polymers. These diffusivity values are also the same order of magnitude as the reported values of the diffusivity of water in starch, suggesting similar mechanisms of diffusion for carbon dioxide and water diffusivity in starch. The observed pressure dependency of the diffusivity may be due to the melt’s high compressibility at these pressures. The solubility of carbon dioxide in the starch melt was proportional to the product of the solubility of carbon dioxide in water and the melt’s moisture content. Introduction The effect of pressure on diffusivity in liquids or solids has received little attention in the literature (Reid et al., 1987). In general, even though binary or multicomponent diffusivity data in gas-liquid or gas-solid systems have been reported, the effect of pressure on diffusivity has been ignored. There has also been a general lack of experimental work for measurement of diffusivities in gas/supercritical fluid-solid systems for biopolymers. Experimental protocols for measuring solubility and diffusivity of carbon dioxide and other gases in molten polymers at elevated temperatures were found in the works of Durrill and Griskey (1966) and Lundberg et al. (1963) but only for pressures of up to 20 bar (300 psi). These researchers brought a known quantity of gas in a closed system in contact with a solid constrained in a cavity and allowed diffusion to occur. By monitoring the pressure of the gas phase over time, diffusivity values were estimated from the pressuredecay data, and the equilibrium pressures yielded equilibrium solubility data. The researchers noted that, for pressures above 20 bar, meaningful data could not be obtained because of higher errors. The diffusivity of carbon dioxide in extruded gelatinized starch has been measured in this study at pressures of up to 117 bar (1700 psi). Such data are not only fundamentally useful in the understanding of the supercritical carbon dioxide (SC-CO2)/starch system but also can be useful for the design and control of the carbon dioxide injection or mixing in starch-based processes such as the supercritical extrusion process (SCFX) (Mulvaney and Rizvi, 1993). Modeling of cell growth in extruded starch systems using gas or SC-CO2 or comparing diffusional mass-transfer effects with * Author to whom all correspondence should be addressed. † Institute of Food Science. ‡ School of Chemical Engineering.
S0888-5885(96)00295-3 CCC: $12.00
convective mass-transfer effects would also require such data as input. The method developed herein improves upon that of Durrill and Griskey (1966). This allowed better accuracy of the data and higher pressures to be used. However, pressures above 117 bar could still not be used because of increasing magnitude of errors at the higher pressures. A constant temperature of 343 K was chosen for these studies because it represents the typical temperature encountered in the mixing zone of a twinscrew extruder in a supercritical fluid extrusion process. The matrix used (extruded gelatinized starch) was chosen since it is typical of most melts used in a cooking extruder. The melt is a homogeneous mass of amylose and amylopectin molecules bound with water and is known to display a viscoelastic behavior and whose glass transition temperature is affected by its moisture content. The diffusivity behavior of carbon dioxide in this melt was found to be a strong function of the pressure in the range studied but not of the moisture content. This was attributed to either the melt’s high compressibility at these pressures or other physicochemical changes of the melt. Experimental Design and Theory Apparatus. The apparatus used for the study is shown in Figure 1. Carbon dioxide from a commercially available 22.7 kg siphon tank (99.98% purity, Empire Airgas Inc., NY) was chilled and pumped by a liquid pump to the desired pressure. Carbon dioxide at that pressure was then brought into a temperature-controlled oven set at 343 K, fitted with a circulating fan for uniformity of temperatures within the oven. Carbon dioxide was then isolated between needle valves 1 and 2, which could be manipulated through handles from outside the oven. Between needle valves 1 and 2, a high-accuracy absolute pressure transducer (Omega © 1996 American Chemical Society
4458 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
radial and only a small amount of axial diffusion from the top and bottom surfaces occurred which was ignored. Experimental Procedure
Figure 1. Apparatus used for the diffusivity and solubility measurement experiments.
Engineering, PX-620, rated accuracy of 0.1%) was placed to monitor the pressure of the system which comprised of the needle valves, the 0.0032 m (1/8 in.) coiled highpressure piping, and the diffusion cell. The output of the pressure transducer was directly input to a computer programmed to measure the transducer output at defined intervals. A temperature probe measured temperature fluctuations in the oven which were on the order of 0.5 K over a 24 h period. The needle valve 2 was connected through piping to a high-pressure leakproof diffusion cell, which was a two-ended tubing reactor (9.85 mL capacity, Autoclave Engineers, CCSS20). It was desirable to minimize the amount of tubing required in the system to obtain the maximum change in the pressure drop of the system. This experimental setup had a lower total system volume, a single temperature environment, better instrumentation accuracy, and a better control of temperature fluctuations than that of Durrill (1965). A Wenger twin-screw TX-52 extruder, as previously described (Mulvaney and Rizvi, 1993) was used for extruding cornmeal. A high-pressure-generating screw profile was used, which gave specific mechanical energy inputs of between 90 and 110 W h/kg to the different samples. The extruder operating conditions are shown in Table 1. Materials. Degermed yellow cornmeal (Lauhoff Grain Co., Dainville, IL) was used. Water and steam rates were adjusted to give three levels of final moisture contents of the melt at 34.5, 36.0, and 39.2% wet basis. Final moisture contents of samples were measured by a loss in weight during the drying (AACC, 1991) method and taking the average of three sample measurements. The extruded starch samples were initially refrigerated (1 week) and then cut into cylindrical shapes (0.0254 m length, 0.0032 m diameter) with a size-1 cork borer. Eight of these cylindrical noodle-shaped samples were placed in the diffusion cell which was then sealed to the rest of the tubing. The previous investigators (Durrill and Griskey, 1966) had constrained one cylindrical-shaped sample of L/D of 1 (25.0 mm each) in a cylindrical cavity of similar dimensions and assumed that the diffusion occurred only axially when the gas was exposed to the sample’s top surface. However, it would have been difficult to prevent gas from penetrating the small clearances between the sample sides and the cavity wall especially since the sample surfaces used in this study were flexible and higher pressures were used. Thus, a higher number of samples, but each with a smaller diameter and thus larger L/D, were used in this study, and it was assumed that the diffusion was
The rest of the experimental procedure for gas/fluid introduction in the system was adapted from Durrill’s work (1965). First the samples were saturated with carbon dioxide at a pressure of 2 bar or less. Then carbon dioxide was isolated at a preset pressure between valves 1 and 2 and allowed to equilibrate for 2 h. Thereafter, valve 2 was opened and the gas allowed to come in contact with the samples. The system pressure instantaneously attained a new initial value, and at the same time, pressure and temperature readings were taken at increasing intervals until the pressure change became insignificant and it was evident that only negligible diffusion was taking place. This completed phase one of the test. The sample was allowed to be saturated with the gas for another 4 h before beginning the second phase. This involved shutting valve 2, bringing in carbon dioxide at the next higher pressure, and isolating it between valves 1 and 2 followed by the equilibration stage. The subsequent procedure in this phase was similar to the first phase. A total of four phases at about 20 (300), 55 (800), 90 (1300), and 115 bar (1700 psi) were performed on each set of samples. After the final phase, the diffusion cell was depressurized and the assembly cleaned for the next set of experiments. Theory For diffusion in a long cylinder, Fick’s second law:
(
)
∂C ∂2C 1 ∂C )D 2 + ∂t r ∂r ∂r
(1)
was applied, assuming that diffusion occurred primarily in the radial direction (large L/D), that there were no angular concentration gradients, and that the diffusion coefficient was isotropic and independent of concentration at the low gas concentrations used. The amorphous melt was assumed to have no preferred molecular orientation. Crank (1975) has solved the above diffusivity equations for the case where diffusion occurs from a solution of limited volume, i.e., when the concentration of the gas phase, or the total amount of solvent, is changing at the surface of the samples. Durrill and Griskey (1966) had assumed a constant surface concentration. Crank’s (1975) solution is
Mt M∞
∞
)1-
∑
n)14
4R(1 + R) + 4R + R2qn2
[ ]
exp
-Dqn2t a2
(2)
where Mt is the total amount of solute in the cylinderical sample and M∞ is the amount of solute at infinite time and where a is the radius of each sample and qn are the nonzero roots of the equation:
RqnJ0(qn) + 2J1(qn) ) 0
(3)
and where R is related to the final fractional uptake of the solvent by the solute by the following equation:
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4459 Table 1. Extruder Operating Conditions for the Production of Gelatinized Starch sample
moisture content (%)
feed rate (kg/min)
motor load (%)a
screw speed (rpm)b
steam rate (kg/min)
water rate (kg/min)
SMEc (W h/kg)
high moisture content intermediate moisture content low moisture content
39.2 36.0 34.5
0.99 0.99 0.99
55 57 63
200 200 200
0.21 0.21 0.18
0.30 0.23 0.22
91.8 95.4 106.2
a
For a 30 hp motor and minimum motor load of 4%. b Maximum rpm on the machine: 420 rpm. c Specific mechanical energy input.
Pi/zi - Pf/zf 1 ) 1+R Pi/zi
(4)
and six roots of qn for different values of R are given in Crank’s book (1975). Considering the nonideality of the gas, the solution in terms of pressures in the system was assumed to be:
Pi/zi - P/z Pi/zi - Pf/zf
∞
)1-
∑
n)14
4R(1 + R) + 4R + R2qn2
exp
[ ] -Dqn2t a2
(5)
Diffusivity values were calculated from the experimental pressure-time data as follows. First, the raw pressure decay data were corrected to a fixed temperature basis, knowing the temperature and the pressure values at different times in the temperature-controlled environment. Then, a model pressure-decay curve was generated using six terms of the summation series in eq 3, from a knowledge of the initial and final pressure values, and an estimated value of diffusivity. The model equations were generated on a spreadsheet (Excel 5.2 for Microsoft Windows). The dependency of the compressibility, z, used in the model equation on the predicted pressure made the model equations implicit, but they were solved by Excel’s automatic convergence software. Finally, a value of diffusivity was chosen which gave the best fit of the model pressure-decay curve with the experimental pressure-decay curve, as was done by Durrill and Griskey (1966). From the initial and final pressure values, the temperature of the oven, the total system volume, and sample weight, the solubility of carbon dioxide in the melt was determined. The nonideality of the gas was represented by its compressibility factor taken from IUPAC tables (Angus et al., 1976). Thus the number of grams of carbon dioxide at a given temperature and pressure in the gas phase was expressed as:
PVT m)k zT
(6)
where k is a constant depending upon the units used. Carbon dioxide lowered the total system pressure by ∆P, which was on the order of 0.3 bar (4 psi) or less. The value of z changed by less than 0.1% during this small pressure drop, and therefore an average compressibility, zavg, corresponding to the average pressure during the pressure drop was used for computing the amount of total carbon dioxide solubilized in the system, s, per gram of sample by the following equation:
s) ∆s )
∑∆s
( )
VT k (Pi - Pf) M zavgTo
(7)
where M is the grams of sample loaded, VT is the total gas volume of the system, and To is the temperature of
the oven. The total solubility at any phase was calculated by summing the ∆s for that phase and those of any previous phases. The total gas phase volume, needed in eq 7, comprised of two parts: one was the gauge volume occupied by tubing and valves between valve 1 and valve 2 and the second was the diffusion cell free volume. The total system volume was measured by the following method. A stainless steel coiled tubing was soldered closed at one end and fitted with a ferrule and nut type tube fitting at the other end. The volume of water required to fill this tube was measured, and therefore a good estimate of the volume of the coil was obtained. This coil was attached to valve 2 at the point where the diffusion cell coil is attached. With valve 2 closed, the gauge system was filled with carbon dioxide to a preset pressure of, say, 7 bar (100 psi). The coil initially contained air at atmospheric pressure. Valve 2 was then opened, and the final pressure of the combined systems was noted. Assuming ideal gas at these pressures and no change in temperature of the total system, the volume of the gauge system was calculated by the following equation:
P2 - Pa V1 ) V0 P1 - P2
(8)
where V1 ) volume of the gauge system, P2 ) final pressure of the system, Pa ) initial pressure of the coil, and V0 ) volume of the coil. Having determined the volumes of the gauge system, the volume of the free diffusion cell space, V2, was determined similarly for each series of runs. This was done after loading the samples in the diffusion cell and from the initial pressure of the gauge system and the pressure drop in the system when valve 2 was opened to begin a run. Results and Discussion Pressure-decay curves for 24 data sets (three different moisture contents and four pressure levels in duplicates, set one and two) were obtained. The diffusivity and solubility values calculated for the complete data set are shown in Table 2. Model pressure-decay curves and observed pressure-decay curves for a typical series of pressure levels (39.2% moisture content sample, set one) are shown in Figures 2-5. Plots for other sets were similar in nature to those shown here. Figure 2 shows the pressure-decay curve at the lowest pressure level for the 39% moisture content, set one, sample. The pressure decays by about 0.25 bar (4 psi) in about 3000 s. The model curve, with a diffusivity value of 7.5 × 10-10 m2/s, matches well with the experimental curve. However, the initial pressure value can only be estimated to within 0.007 bar (0.1 psi), and considering the errors introduced by the estimation procedure, the diffusivity value was estimated to be accurate to within 4%. Figure 3 shows the next pressure level (58 bar) of the measurement, which was similar to the first level except
4460 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Table 2. Diffusivity and Solubility Values of All Data Sets sample 39.2% m.c. repeat 1
39.2% m.c. repeat 2
36.0% m.c. repeat 1
36.0% m.c. repeat 2
34.5% m.c. repeat 1
34.5% m.c. repeat 2
stage
initial pressure (×10-5 Pa)
final pressure (×10-5 Pa)
diffusivity (×1010 m2/s)
cumulative solubility (g/100 g of melt)
1 2 3 4
26.66 58.41 92.33 118.51
26.39 58.14 92.11 118.31
7.5 5.7 1.9 0.9
0.37 0.78 1.22 1.65
1 2 3 4
25.74 59.10 93.13 117.80
25.41 58.76 92.82 117.58
7.7 5.7 1.5 0.9
0.37 0.81 1.31 1.72
1 2 3 4
23.21 56.05 90.80 114.41
22.92 55.78 90.56 114.24
7.0 5.7 1.9 0.9
0.35 0.75 1.18 1.54
1 2 3 4
22.32 56.94 92.95 118.21
22.00 56.62 92.67 117.99
7.5 5.7 1.7 0.9
0.37 0.81 1.29 1.74
1 2 3 4
24.37 57.89 91.39 117.04
24.09 57.61 91.14 116.85
7.3 5.7 1.5 0.9
0.35 0.76 1.19 1.59
1 2 3 4
23.32 58.52 94.10 117.02
23.07 58.27 93.88 116.82
7.5 5.7 1.7 0.9
0.35 0.77 1.22 1.70
Figure 2. Pressure-decay curve at the 26 bar pressure level for set one of the sample with 39.2% m.c. at 343 K.
that it took longer for equilibration to be achieved, which implies a lower diffusivity value. Diffusivity was estimated to be 5.7 × 10-10 m2/s, and the error in this value was again estimated to be 4%. Also the raw pressure data obtained were corrected to a constant-temperature basis by accounting for the fluctuations in the oven which were on the order of 0.5 K over the duration of the experiment. Note that, as the total pressure is increased in a fixed volume system, small fluctuations in temperature cause increasing magnitude of pressure fluctuations which could then overwhelm the observed pressure decay in the system due to diffusion. Figure 4 shows the third pressure level (92 bar) which was similar to the second pressure level except that it took even longer for equilibrium to be achieved, and
Figure 3. Pressure-decay curve at the 58 bar pressure level for set one of the sample with 39.2% m.c. at 343 K.
there were increasing orders of fluctuations in the observed pressure-decay curve. The estimated diffusivity was 1.9 × 10-10 m2/s, and the error was estimated higher at 10%. Figure 5 shows the final pressure level (118 bar) where diffusivity was estimated at 0.9 × 10-10 m2/s, and again the error was estimated at about 10%. At higher pressures than these, the fluctuations are expected to further increase. To reduce the effect of these fluctuations would require better temperature sensors with an accuracy greater than 0.1 K, which was not possible with the present setup. As mentioned before, other sets of experimental data were similar to those shown in Figures 2-5. Figure 6 shows the diffusivity values for all data sets in an algebraic plot, while Figure 7 shows the same data on a logarithmic plot along with the error bars. The effect
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4461
Figure 7. Measured diffusivity of carbon dioxide in melt for all samples as a function of pressure at 343 K (s) and estimated diffusivity of carbon dioxide in water (-‚-), logarithmic scale.
Figure 4. Pressure-decay curve at the 92 bar pressure level for set one of the sample with 39.2% m.c. at 343 K.
diffusivity value at low pressure was similar to that obtained at the first stage of experiments. This is also consistent with the diffusivity behavior of most binary systems at low pressures. However, the diffusivity value decreased between the first stage and subsequent stages to nearly one-eighth of its low-pressure value at 115 bar (1700 psi). Durrill and Griskey (1969) had found no appreciable effect of pressure on the diffusivity of carbon dioxide in thermally-softened polyethylene, polypropylene, polyisobutylene, polystyrene, and poly(methyl methacrylate) up to 30 bar. Their low-pressure diffusivity values for CO2 in polyethylene, polypropylene, and polyisobutylene at 461 K were between 3.4 × 10-9 and 5.7 × 10-9 m2/s. Explanation of this decrease in observed diffusivity at higher pressures is not conclusive. If one assumes the samples behaved as melts, then simple empirical equations that have been proposed to determine the pressure dependency of diffusivity in viscous liquids as a function of solvent viscosity (Reid et al., 1987) could be applied:
DAB ∝ ηB-0.5
Figure 5. Pressure-decay curve at the 118 bar pressure level for set one of the sample with 39.2% m.c. at 343 K.
Figure 6. Measured diffusivity of carbon dioxide in melt for all samples as a function of pressure at 343 K, algebraic scale.
of pressure on the estimated diffusivity values was interesting. Previous low-pressure experiments performed in our laboratory had indicated that diffusivity values were independent of pressure up to at least 20 bar (300 psi). One run was performed at 8 bar (120 psi) under similar conditions which also confirmed that the
(9)
To apply this equation to the present case, the melt viscosity or melt density for starch needs to be known as a function of pressure at various temperatures, but this was not available in the literature. Vergnes and Villemaire (1987) reported low-shear melt viscosities on the order of 104 Pa‚s for gelatinized corn at 383 K. The pressure effect on melt viscosity for simple polymeric liquids has been investigated and is known to depend on the molecular weight, molecular weight distribution, branching, composition, etc. (Utracki, 1985). The viscosity increase, for example, in polybutylene oil was only 10% between 1 and 100 bar at 372 K. However, since starch at 343 K may be more compressible than polymeric liquids or water, the melt viscosity or melt density may increase more significantly in this pressure range than it does for such liquids. In preliminary investigations using an Instron Universal Testing Machine, the gelatinized starch at 343 K was found to be 12-15% compressible at 100 bar. Such high compressibilities might decrease the hole-free volume (Zielinski and Duda, 1992) available for diffusion of carbon dioxide through the starch matrix and thereby decrease the observed diffusivity. The hole-free volume for an epoxy resin, for example, at a low temperature of 298 K was found to decrease by about 1% in 100 bar (Deng et al., 1992). The diffusivity of carbon dioxide in water at 298 K and atmospheric pressure is 1.96 × 10-9 m2/s (Perry and
4462 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
dioxide in water as a function of pressure. The solubility of carbon dioxide in water follows the solid curve as shown in Figure 8 (Kohlensaure, 1986). The solubility of CO2 in starch on a water basis is correlated with the solubility of CO2 in water. It is about 20% less than the latter value, except at higher pressures. The lower value may be because all the moisture content in the melt is not available for solubilizing carbon dioxide, but only the free water. Conclusions
Figure 8. Measured solubility of carbon dioxide in melt on a water basis (9) and solubility of carbon dioxide in pure water as a function of pressure ([) at 343 K.
Green, 1988). Using the following equation which can be used to model the effect of temperature,
DABηB/T ) constant
(10)
the diffusivity of carbon dioxide in water at 343 K and atmospheric pressure was estimated to be 4.8 × 10-9 m2/s. The viscosity of water changes by less than 0.1% up to 115 bar, and, therefore, the diffusivity of carbon dioxide in water is not expected to change in this pressure range. Interestingly, the diffusivity of carbon dioxide in gelatinized starch was only an order of magnitude lower than that for carbon dioxide in water, even though the melt viscosity of starch is several orders of magnitude higher than that of water. If it is assumed that the carbon dioxide diffusivity in starch is primarily through the absorption of carbon dioxide in water, then it could be related to the diffusivity of water in starch. Karathanos et al. (1991) had reported a decrease in their observed diffusivity of water in starch (3.0 × 10-10-1.0 × 10-10 m2/s at 333 K with pressure increasing from 1 to 40 bar, and this diffusivity is of the same order of magnitude as the values seen here for the diffusivity of carbon dioxide in starch (7.5 × 10-10-0.9 × 10-10 m2/s). This may suggest similar mechanism of diffusion for water and carbon dioxide molecules in starch, and as shown later, the solubility of carbon dioxide in starch is related to the amount of water present in starch. Interestingly, Karathanos et al. (1991) had also postulated that the decrease in bulk porosity (or increase in bulk density) of starch probably caused the observed decrease in their diffusivity values. Another explanation for the observed dependency of the diffusion on pressure may be that the gelatinized starch may have undergone physicochemical changes induced at the higher pressures in the time period that it was tested for causing the diffusivity values to decrease. This would require further investigation on the nature of the gelatinized starch before and after diffusivity experiments which was beyond the scope of the present study. Since all the moisture content samples and repeats are clustered together within their error ranges at each pressure stage, there was no significant effect of the moisture content in the range studied on the diffusivity of CO2 in gelatinized starch. The solubility of carbon dioxide in gelatinized starch on a moisture basis (grams of CO2/100 grams of water in sample) for all different moisture content samples and repeats is shown in Figure 8. This was done to compare these solubilities with those of the solubility of carbon
Carbon dioxide solubility and diffusivity in gelatinized starch, obtained via cornmeal extrusion, have been measured at 343 K and pressures up to 115 bar as a function of moisture content. The methodology employed has been an improvement over previously reported techniques, enabling high-pressure data to be obtained. The diffusivity of carbon dioxide in starch decreased with pressure but was not affected by moisture content in the 34.5-39% moisture content range studied. This diffusivity value varied from 7.5 × 10-100.9 × 10-10 m2/s as pressure was increased from atmospheric pressure to 115 bar. Higher pressure data were prone to an increasing magnitude in the error terms. The low-pressure diffusivity value was only an order of magnitude lower than that reported for carbon dioxide diffusivity in water, which is independent of pressure in the range studied but the same order of magnitude as the reported diffusivity of water in starch. The observed dependency of diffusivity on pressure may be due to the starch’s compressibility and also suggests a similar mechanism of diffusion of carbon dioxide and water in starch. The solubility of carbon dioxide in the starch melt also was proportional to the product of the solubility of carbon dioxide in pure water and the starch’s moisture content. Literature Cited AACC Approved Methods. American Association of Cereal Chemists, MA, 1991; 45-15A. Angus, S.; Armstrong, B.; deReuck, K. M. IUPAC, Carbon Dioxide, International Thermodynamic Tables of the Fluid State-3; Pergamon Press: Oxford, U.K., 1976. Crank, J. The mathematics of diffusion; Clarendon Press: Oxford, England, 1975. Deng, Q.; Sundar, C. S.; Jean, Y. C. (1992) Pressure Dependence of Free-Volume Hole Properties in an Epoxy Polymer. J. Phys. Chem. 1992, 96 (1), 492-495. Durrill, P. L. The determination of solubilities and diffusion coefficients of gases in polymers at temperatures above the softening point. Ph.D. Thesis, Virginia Polytechnic Institute, Blacksburg, VI, 1965. Durrill, P. L.; Griskey, R. G. Diffusion and solution of gases in thermally softened or molten polymers; Part I; Development of technique and determination of data. AIChE J. 1966, Nov, 1147-1151. Durrill, P. L.; Griskey, R. G. Diffusion and solution of gases in thermally softened or molten polymers; Part II; Relation of diffusivities and solubilities with temperature, pressure and structural characteristics. AIChE J. 1969, 15 (1), Jan, 11471151. Karanthanos, V. T.; Vagenas, G. K.; Saravacos, G. D. Water diffusivity in starches at high temperatures and pressures. Biotechnol. Prog. 1991, 7 (2), 178-184. Kohlensaure-Industrie, e.v. Eigenschaften der Kohlensaure. CO2; GmbH and Co.: KG, Honningen, Germany, 1986. Lundberg, J. L.; Wilk, M. B.; Huyett, M. J. Sorption studies using automation and computation. Ind. Eng. Chem. Fundam. 1963, 2 (1), 37-43.
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4463 Mulvaney, S. J.; Rizvi, S. S. H. Extrusion processing with supercritical fluids. Food Technol. 1993, 47 (12), 74-82. Perry, R. H.; Green, D. Perry's chemical engineers handbook, 6th ed.; McGraw-Hill: New York, 1988. Reid, R. C.; Prausnitz, J. M.; Poling, B. C. The properties of gases and liquids, 4th ed.; McGraw Hill Book Co.: New York, 1987. Utracki, L. A. A method of computation of pressure effect on melt viscosity. Polym. Eng. Sci. 1985, 25 (11), 655-668. Vergnes, B.; Villemaire, J. P. Rheological behaviour of low moisture molten maize starch. Rheol. Acta 1987, 26, 570576.
Zielinski, J. M.; Duda, J. L. Predicting Polymer/Solvent Diffusion Coefficients Using Free-Volume Theory. AIChE J. 1992, 38 (3), 405-415.
Received for review May 28, 1996 Revised manuscript received September 5, 1996 Accepted October 5, 1996X IE960295I X Abstract published in Advance ACS Abstracts, November 15, 1996.
