Article pubs.acs.org/IECR
Experimental and Theoretical Study on the Desorption of Alcohols and Ketones from Polyolefins Vasileios Kanellopoulos,* Mohammad Al-haj Ali, Sabine Sundvall, and Shital Das InnoTech Process Technology, Process Development PO PDO, Borealis Polymers Oy, P.O. Box 330, Finland ABSTRACT: A comprehensive dynamic desorption model has been developed to estimate the desorption rate of alcohols (i.e., tert-butyl alcohol) and ketones (i.e., 2-propanone (acetone)) from semicrystalline polyolefins under operating conditions of industrial interest. The free volume theory is employed to calculate the penetrant diffusion coefficients and their dependence on temperature and penetrant concentration. The diffusivities of acetone and tert-butyl alcohol in semicrystalline polypropylene (PP) pellets were experimentally measured at different temperatures. It was shown that model predictions are in excellent agreement with the corresponding experimental dynamic measurements of the mass desorption of the sorbed species. Moreover, it was shown that the proposed model predicts correctly the diffusion coefficient of alcohols and ketones in semicrystalline polyolefins. This work can be industrially implemented in any process which involves desorption of gaseous penetrants from solid material.
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INTRODUCTION Polyolefins are widely used in our daily life with applications ranging from microcircuits to cars and from packaging materials to medical applications. Contamination with unreacted monomers, with inert material, and in general with volatile organic compounds (VOCs) is a typical problem particularly when polymers are used for specific applications (e.g., food packaging, biomedical, interior paints, etc.). Moreover, polymers may contain small amounts of nonpolymerizable compounds resulting from impurities in the raw materials or byproducts formed by side reactions during the polymerization process. The presence of such nonpolymerizable compounds results in the increase of the residual VOCs amount in the polymer and can adversely affect its end-use properties. In addition, many polymers are produced in the presence of organic solvents, which are considered VOCs and must be eliminated before use.1 The increasingly strict environmental regulations and the higher market sensitivity to environmental and health issues are pushing the polymer manufacturing industry to reduce the amount of monomers and VOCs in polymers. Reduction of VOCs emission can also lead to better workplace conditions, reduce risks of fire, reduce nuisances, and lead to economic savings. More specifically, applications of polyolefins (polyethylene and polypropylene) in food packaging, pharmaceutical industry, and water pipes impose stringent criteria on the amount of VOCs in the final polymer. During extrusion of polyolefin powders, VOCs may be produced and removed through the vent of the extruder. Moreover, different additives are added (e.g., antioxidants, UV-stabilizers, color compounds, acid scavengers, and antistatic and antifogging agents) to polymer powder during the extrusion process to meet the enduse polymer property requirements. Such additives may affect the quality of the polymeric material and cause problems with organoleptic properties (i.e., odor and taste) that prevent the use of such material in specific applications.2−4 The industrial importance of VOCs removal is reflected in different patents that appear in the literature.5,6 On the other © 2013 American Chemical Society
hand, scientific articles on this topic are scarce in the open literature. Araujo et al.7 summarized the principal methods employed for reducing VOCs from viscous polymers. Biesenberger8 and Albalak9 also reviewed the various aspects of polymer devolatilization technology focusing on the desorption of alkanes from polymers. In a more relevant work, Le Blevec et al.10 developed a method to measure volatile diffusivities in polymers (polypropylene, ethylene−vinyl acetate, and ethylene acrylate copolymers) under conditions relevant to solid-state devolatilization, which is usually carried out in silos of fluidized beds where a flowing gas flushes out the volatile molecules. The authors reported that volatiles diffusivities are almost independent of their concentrations in the solid material. Despite the great effort in process modeling11−21 of olefin polymerization reactors, not so much work has been performed on residual polar compound desorption from polyolefins. In fact, there are many patents dealing with devolatizers and similar downstream processing unit designs; however, to our knowledge there is no fundamental modeling approach dedicated to the elucidation of the mass transfer phenomena taking place during polar molecule desorption from polyolefins. The scope of the present study is to evaluate both experimentally and theoretically the desorption dynamics of alcohols and ketones (e.g., tert-butyl alcohol and acetone, respectively) in semicrystalline nonporous PP pellets at different temperatures. A comprehensive unsteady-state diffusion model, accounting for the effect of penetrant concentration, is developed to predict the transport of alcohols and ketones in polymers. A modified free volume model is employed to calculate the penetrant’s diffusion coefficient in terms of temperature, the concentration of sorbed species, and Received: Revised: Accepted: Published: 15855
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polymer crystallinity. Dynamic desorption measurements were carried out for the online measurement of penetrant mass desorption rate from the polymer pellets; these measurements were subsequently employed to calculate the diffusion coefficients of representative alcohols and ketones in semicrystalline, nonporous PP pellets. PP is considered in this work because it has to be further processed after production to control its rheological behavior. This is usually performed by adding peroxides (e.g., 2,5dimethyl-2,5-bis(tert-butylperoxy)hexane, etc.) during the extrusion process. However, thermal decomposition of peroxides to acetone, tert-butyl alcohol, methane, ethane, and tert-amyl alcohol may have a negative impact on final properties such as long-term aging and organoleptic properties (e.g., taste and odor), which is critical in food applications.22 It should be emphasized that acetone and tert-butyl alcohol are the major components produced during the decomposition of peroxides. Finally, it has to be highlighted that the present work provides the framework for assessing the internal mass transfer limitations taking place during desorption of VOCs from polymer pellets. The developed diffusion model can be implemented into an integrated multiscale process unit model to properly design storage and aeration silos or other stripping and devolatization plant-scale units employed in downstream processing of polyolefins.
Boundary conditions: ∂Yi /∂z = 0
at
Yi = 0
z=1
at
z=0
(3) (4)
In the above equations Yi(t, z) (i.e., Yi(t, z) = Ci(t, x)/Ci,0) and z(t) (i.e., z(t) = x/Lx(t)) denote dimensionless space and concentration variables. In addition, Dpi and Ci are the diffusion coefficient (see also Appendix) and the concentration of the “i” penentrant species at time t and Ci,0 is the initial concentration of the “i” penentrant species. One can easily show that the total mass of desorbed species at time t, Mi(t), will be given by the following integral: M i(t ) = M i,0
∫0
1
Yi(z)dz
(5)
where Mi,0 is the total initial mass of sorbed species “i” . It should be mentioned that for the studied systems (i.e., acetone and tert-butyl alcohol desorption from PP pellets) and the selected operating conditions (i.e., temperature and pressure), the effect of polymer pellet deswelling due to the component desorption can be assumed as negligible. Moreover, the density of the polymer−penetrant mixture is approximately constant during the whole desorption process, because only a relatively small amount of gas penetrant is desorbed from the polymer phase (i.e., smaller than 0.5% w/w). Therefore, the variation of the pellet thickness, Lx(t), as well as the convection velocity, u(t) = L̇ x(t) = (dLx(t)/dt)), caused by polymer deswelling, are not considered in eq 1.23 Moreover, the degree of swelling caused by the sorption of alcohols and ketones in the polymer phase was estimated by employing a fundamental thermodynamic model (i.e., Sanchez Lacombe EOS), and it was found to be below 1% in terms of polymer pellet volume change. The above unsteady-state diffusion model consists of a stiff nonlinear partial differential equation (eq 1) and a number of initial and boundary conditions (eqs 2−4). In the present study, the partial differential equation was solved by the global orthogonal collocation method. Specifically, the partial differential equation was first discretized with respect to the spatial coordinate, z, at selected collocation points, while the unknown variable, Yi, was approximated by a Lagrange interpolation polynomial.23 The resulting differential equations were then integrated using the Petzold−Gear method.
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DIFFUSION OF ALCOHOLS AND KETONES IN SEMICRYSTALLINE POLYMER PELLETS To calculate the transport of polar compounds (i.e., alcohols and ketones) from semicrystalline polymer pellets, the dynamic sorption model developed by Kanellopoulos et al.22,23 was employed. The full model description along with its equations and details are presented in Appendix. Let us assume that a nonporous polymer pellet of thickness Lx is exposed at time t = 0 to a gaseous inert atmosphere (see Figure 1). Assuming that
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EXPERIMENTAL MEASUREMENTS Desorption Measurements. A new experimental setup has been built to carry out dynamic desorption measurements of tert-butyl alcohol and acetone in polymer pellets; see Figure 2. Desorption measurements were carried out under isothermal (i.e., two temperatures were considered in this work, namely 90 °C and 120 °C) and isobaric conditions (i.e., 1 bar pressure). Nonporous PP pellets (i.e., Tm = 164.3 °C) of an approximate thickness of 0.345 cm were employed to perform the desorption measurements. As shown in Figure 2, the desorption setup consists mainly of (i) a desorption vessel where the polymeric material is added, (ii) an oil bath to control the temperature of the desorption vessel, (iii) flow and temperature measurements, and (iv) a sampling point. As a first step in these measurements, all components were removed from polymer pellets through heating the pellets inside the desorption vessel at a temperature around 120 °C for about 1 h. Then polymer pellets were
Figure 1. Desorption in a nonporous polymer pellet.
the initial penetrant concentration value at the polymer pellet is either the thermodynamic (i.e., Ci(0, x) = Ci,eq) or an initial one (i.e., Ci(0, x) = Ci,0), the following dimensionless dynamic mass balance equation accounting for the diffusion of the penetrant molecules from the polymer pellet to the gas phase can be obtained: p ∂Yi D p ∂ 2Y 1 ∂D ∂Yi = 2 i + i2 2i ∂t Lx ∂z ∂z Lx ∂z
(1)
Initial condition: Yi = 1
at
t=0
(2) 15856
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Figure 3. Measured and predicted desorption acetone curve in PP pellets (T = 90 °C, P = 1.0 bar).
Figure 2. Experimental unit used for conducting the dynamic desorption measurements.
soaked with acetone or tert-butyl alcohol overnight, at temperatures of 25 °C and 40 °C, respectively; so it is expected that equilibrium is achieved. Moreover, the Sanchez− Lacombe equation of state11,23 has been utilized to estimate the initial thermodynamic concentration of adsorbed materials in polymer pellets. After the completion of the soaking step, the process temperature was set to the target temperature. When the polymer temperature reaches the target temperature (the temperature was measured by a thermometer), nitrogen is fed to the desorption vessel at a constant rate of about 10.5 mL/ min. Gas samples were taken through the sampling point; these samples were analyzed using a GC (HP 9850). The mass fraction of the crystalline polymer phase, ωc, was measured using a differential scanning calorimeter (DSC). Specifically, by measuring the heat of fusion of the polymer sample and dividing it by the heat of fusion of 100% crystalline polymer (e.g., 187.0 kJ/kg), the value of ωc could be determined.24 From the measured values of ωc and the densities of crystalline (ρc = 0.946 g/cm3) and amorphous (ρam = 0.855 g/cm3) polymer, the density of the semicrystalline polymer (ρ) was determined:23 ωc =
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ρc ⎛ ρ − ρam ⎞ ⎟⎟ ⎜⎜ ρ ⎝ ρc − ρam ⎠
Figure 4. Measured and predicted desorption acetone curve in PP pellets (T = 120 °C, P = 1.0 bar).
Dp0,i at two different temperatures (i.e., 90 °C and 120 °C) for both acetone and tert-butyl alcohol are presented in Table 1. Note that the estimated values of Dp0,i are in very good agreement with those reported in the open literature.23,25−29 The numerical values of all the physical and transport Table 1. Numerical Values of the Parameters Used in the Model parameter Dp0,i (cm2/s) (at T = 90 °C) Dp0,i (cm2/s) (at T = 120 °C) Dp0,i (reported in the literature)25−29 ωc f b39−41 Vi* (cm3/g)36,37 ξip αi (1/K)47 αi047,48 αp (1/K)47 αp047,48 γi27 ρam(g/cm3)23 ρc (g/cm3)23 Tg (K)49,50 Lc (cm)39−41
(6)
MODEL PREDICTIONS In Figures 3 and 4, experimental acetone desorption curves in PP pellets (shown by the discrete points) are plotted with respect to time at two different temperatures (i.e., 90 °C and 120 °C, respectively). It should be noted that the initial measured acetone concentration in PP pellets was 4500 and 3500 ppm at 90 °C and 120 °C, respectively. Continuous lines represent the predicted values obtained via the solution of the developed diffusion model (i.e., eqs 1−5 and eq A13). It should be pointed out that the only fitted parameter in the above model calculations was the numerical value of the preexponential constant, Dp0,i, in eq A13. The estimated values of 15857
polypropylene
acetone
tert-butyl alcohol
−
8.0 × 10−5
3.0 × 10−4
−
8.0 × 10−5
3.0 × 10−4
−
3.0 × 10−5 to 3.00 × 10−4 − − 0.83 0.51 1.4 × 10−3 1.0 × 10−2 − − 0.75 − − 100 −
1.0 × 10−4 to 5.0 × 10−4 − − 1.00 0.79 1.0 × 10−3 1.0 × 10−2 − − 0.75 − − 180 −
0.50 0.1 1.04 − − − 12.0 × 10−3 1.0 × 10−2 − 0.855 0.946 253.15 1.0 × 10−6
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worth mentioning that these results agree with Le Blevec et al. findings,10 who studied the desorption of volatile organic compounds (VOCs, i.e., reaction byproducts and degradation compounds) from PP particles at conditions and initial penetrant concentrations similar to the present study. On the basis of their dynamic experimental desorption isotherms, it was found that VOCs diffusion coefficients are almost constant during the desorption process. On the basis of the simulation results carried out in the present study, it was demonstrated that it is possible to fit accurately the experimentally measured isotherms using a single value for Dpi (see Figures 3−6). Therefore, one can conclude that at low pressures (∼1.0 bar which is typical operating pressure in industrial silos), the value of Dpi,eff does not vary significantly with time, which means that a single value for Dpi,eff can be used to describe the transport of acetone and tert-butyl alcohol in semicrystalline, nonporous polymer pellets. On the basis of the findings of Figure 5 and 6, it can be seen that in the case of tert-butyl alcohol the temperature change does not play a significant role in the effective diffusion coefficient value. Thus, by increasing temperature from 90 °C to 120 °C, less than 10% increase in Dpi,eff is observed. On the other hand, during desorption of acetone from PP, the effect of temperature on the effective diffusion coefficient is quite significant. In this case, a temperature increase from 90 °C to 120 °C results in approximately 25% increase in Dpi,eff (see Figures 3 and 4). Finally, in Figures 7 and 8, the values of the effective diffusion coefficients of acetone and tert-butyl alcohol calculated by the
parameters used in the theoretical calculations are also presented in Table 1. As can be seen from the results of Figures 3 and 4, model predictions are in excellent agreement with the experimental desorption curves. Based on the proposed dynamic desorption model, the values of the local diffusion coefficient, Dpi (x), in the polymer pellet can be calculated. Accordingly, the overall (effective) diffusion coefficient, Dpi,eff, can be calculated by integrating the local diffusion coefficient, Dpi (x), with respect to the polymer pellet’s thickness, Lx: p Di,eff =
∫L Dip(x)dx/∫L dx x
x
(7)
In Figures 5 and 6, model predictions are compared with representative experimental desorption measurements on tert-
Figure 5. Measured and predicted desorption tert-butyl alcohol curve in PP pellets (T = 90 °C, P = 1.0 bar).
Figure 7. Comparison of acetone diffusion coefficients at different temperatures, calculated by three different methods (i.e., initial slope, half-time, free volume).
Figure 6. Measured and predicted desorption tert-butyl alcohol curve in PP pellets (T = 120 °C, P = 1.0 bar).
proposed unsteady-state diffusion model (see eqs 1−5 and eq A13)) and the two classical methods of the initial slope (eq A3) and half-time (eq A4) are plotted with respect to temperature. As can be seen, the results obtained by the present model (triangle points) are in relatively good agreement with the results obtained by the half-time method (solid cycles). On the other hand, the initial slope method does not provide p satisfactory estimates of Di,eff for the studied penetrants, especially at low temperatures (see also Table 2). Thus, the last method should be avoided for calculating the value of Dpi,eff using dynamic desorption measurements. The fact that the estimates of Dpi,eff, obtained by the half-time method, are in a good agreement with the estimates of the present model can be explained from the results of Figures 3−6 on the desorption rate in short times (Mi/Mi,0 vs t). It is apparent that the value of
butyl alcohol at 90 °C and 120 °C, respectively. In these figures, the discrete points represent the experimentally measured mass fraction of the desorbed species. The continuous lines denote the predicted desorption curves using a locally varied diffusion coefficient, Dpi (x). It should be emphasized that Dpi (x) does not change so much with respect to penetrant concentration because of the relatively small initial concentration of sorbed species (i.e., 3500 ppm and 3000 ppm at T = 90 °C and 120 °C, respectively) as well as because of the very slow desorption process. More specifically, it was found that the maximum variations in tert-butyl alcohol and acetone diffusion coefficients with respect to their concentration changes during the desorption processes were 5% and 10%, respectively. It is 15858
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when the slope of the initial part of the desorption curve cannot be well-defined. The present study provides fundamental understanding of polar compound diffusion phenomena during desorption from polymer pellets as well as the adsorption of these compounds into polymer pellets. The proposed model can be used as a tool for determining the maximum amount of polar compounds removed from polymer pellets stored in silos. Moreover, this model can be employed for optimizing the operation of existing plant silos or/and for designing new ones to ensure safe operation. Finally, it should emphasized that the proposed model can be implemented in other industries where sorption/ desorption of gases in solid materials take place, such as in the food and pharmaceutical industries.
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Figure 8. Comparison of tert-butyl alcohol diffusion coefficients at different temperatures, calculated by three different methods (i.e., initial slope, half-time, free volume).
Calculation of the Diffusion Coefficient
In the open literature, a number of mathematical models have been proposed for the calculation of the diffusion coefficient of gaseous penetrant species in semicrystalline polymer pellets. Most models are straightforward applications of Fick’s second law governing the diffusion of low molecular weight gaseous species in a planar pellet of infinite length.23 Assuming that the diffusion coefficient, Dpi , is independent of the concentration of sorbed species, one can show that the total mass of the sorbed species at time t for a planar geometry will be given by the following analytical solution:30
Table 2. Deviations of the Classical Shortcut Diffusion Methods from Model Predictions tert-butyl alcohol
acetone diffusion model
90 °C
120 °C
90 °C
120 °C
half-time (% deviation) initial slope (% deviation)
20 60
20 25
20 65
8 20
APPENDIX
the diffusion coefficient, Dpi,eff, depends on the slope and, because the desorption of the studied compounds from the PP pellets is a very long process (more than 1 h), the slope of the initial part of the desorption curve cannot be defined explicitly. Therefore, a lot of attention should be paid when polar compound diffusion coefficients are estimated by shortcut approaches based on dynamic sorption or desorption processes. From the findings of Figures 7 and 8, it can also be concluded that both shortcut methods underestimate the diffusion coefficients at low temperatures while they give better predictions at higher temperatures.
M i(t )/M i,0 = 2(Dipt /Lx2)1/2 (1/ π ∞
+ 2 ∑ ( −1)n erfc(nLx /2 Dipt )) n=1
(A1)
For small diffusion times, the contribution of the summation term in eq A1 will be negligible, and thus the above solution can be further simplified to:
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M i(t )/M i,0
CONCLUSIONS A comprehensive dynamic desorption model is developed to (i) describe the desorption of polar compounds from polyolefin pellets and (ii) predict the diffusion coefficient of polar penetrants in semicrystalline polymer pellets. The free volume theory is employed to calculate the dependence of the diffusion coefficient on the penetrant concentration and temperature. The diffusivities of acetone and tert-butyl alcohol in semicrystalline polypropylene pellets at atmospheric pressure and high temperatures were measured by carrying out desorption experiments. It was shown that in the case of tert-butyl alcohol, the effect of temperature on the calculation of the effective diffusion coefficient is limited (i.e., less than 10% increase in Dpi,eff for temperature change from 90 °C to 120 °C). On the other hand, when acetone is considered, the effect of temperature on the effective diffusion coefficient is significant (i.e., ∼25% increase in Dpi,eff for temperature change from 90 °C to 120 °C), and it should be accounted for in the accurate calculation of the value of the effective diffusion coefficient. Finally, it was demonstrated that the calculated values of Dpi,eff, obtained by the proposed diffusion model, are in relatively good agreement with the results obtained by the half-time method. Moreover, it was shown that the initial slope method should be avoided for calculating the value of polar compound diffusion coefficients from dynamic desorption measurements,
2 = π
⎛ D pt ⎞1/2 ⎜ i2 ⎟ ⎝ Lx ⎠
(A2)
In general, the diffusion coefficient of the penetrant species can be determined by measuring the change in the mass fraction (Mi(t)/Mi,o) of the sorbed species as a function of time. Thus, from eq A2 and experimental measurements on the mass uptake of the sorbed or desorped species, one can easily calculate the value of Dpi in terms of the initial slope of (Mi(t)/ Mi,0) versus t1/2 curve, S, and the thickness of the polymer pellet, Lx.30
Dip = πS2Lx2/4
(A3)
Dpi
The above estimation method of is known as the initial slope method. Alternatively, the diffusion coefficient can be estimated from the reduced sorption/desorption curves, (i.e., Mi(t)/Mi,o versus t) using the half-time method:30 Dip =
⎛ π ⎞⎛ Lx2 ⎞ ⎜ ⎟⎜ ⎟ ⎝ 16 ⎠⎝ t1/2 ⎠
(A4)
where t1/2 represents the time at which the total mass of the sorbed species becomes equal to the 50% of Mi,0. In the present study, the diffusion coefficient, Dpi , was calculated using the free-volume theory of Vrentas and Duda31−33 and Cussler.34 Accordingly, the diffusion coefficient 15859
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impedance factor (τ) defined by the ratio of the diffusivity of the penetrant molecules in a completely amorphous polymer phase to that corresponding to a semicrystalline one. Thus, by multiplying the right hand side of eq A12 by the term (1/τ), one can account for the presence of crystalline polymer domains and the fact that the penetrant molecules must follow a longer diffusion pathway than the actual thickness of the pellet.38
of low molecular weight penetrants in a semicrystalline, nonporous polyolefin pellet was expressed as follows: p Dip = D0,i exp( −γi(ωiV i* + ωpV p*ξip)/VFH)
(A5)
where the subscripts i and p refer to the penetrant molecules and the polymer, respectively. Dp0,i is the nominal (i.e., at the zero concentration limit) temperature-dependent diffusion coefficient of species “i”. γi is an overlap factor (taking values in the range of 0.5−1), accounting for the amount of free volume shared between different penetrant molecules. ωi and ωp are the weight fractions of the penetrant molecules and polymer, respectively. Vi* denotes the specific critical hole free volume of the “i” sorbed species required for its displacement. Note that the value of V*i can be estimated using the group contribution method of Haward and the molecular composition of the penetrant species.35−37 The parameter ξip can be expressed in terms of Vi* and the critical molar volume of the polymer jumping unit, Vpk. That is, ξip = (V i*MW)/ i Vpk
p Dip = D0,i exp( −γi(ωiV i* + ωpV p*ξip)/VFH)τ −1
There are relatively few studies regarding the calculation of the geometrical impedance factor, τ. Newey et al.39−42 using experimental measurements on the volume fraction of crystalline polymer domains, ϕc, and crystal aspect ratio, proposed the following correlation for the calculation of τ: τ = 1 + (ϕc(0.384 + x 2))/(1.848 − x 2); x = 0785 − Lc/ωc
(A6)
Tgp ≥ 295 K
(A7)
Vpk = (0.0925Tgp + 69.47),
Tgp ≤ 295 K
(A8)
̃ = γ (VFHp − f ϕ ) VFHp i b c
VFHp = (a p0 + a p(T − Tgp)V p*)
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(A9)
AUTHOR INFORMATION
Corresponding Author
(A10)
*Tel.: + 35 358 503794332. E-mail: vasileios.kanellopoulos@ borealisgroup.com.
where αi and αip represent the corresponding differences in the values (above and below the glass transition temperature) of the thermal expansion coefficients of the penetrant molecules and polymer. αi0 and αp0 are the fractional free volumes of the penetrant molecules and polymer at the glass transition temperature, respectively. Thus, the overall free volume of the system, VFH, will be given by the weighted sum of the corresponding free volumes of the penetrant molecules and the amorphous polymer phase.35,36 VFH = VFHiωi + VFHpωp
(A15)
where f b is a parameter accounting for the effect of the degree of crystallinity on the free volume of the polymer phase.38,43−46 Thus, the free volume of the semicrystalline polymer decreases as the polymer crystallinity increases.
Similarly, the hole-free volume of the penetrant molecules and polymer can be calculated by the following equations:35,36 VFHi = (a i0 + a i(T − Tgi)V i*)
(A14)
where Lc (in μm) is the average crystal thickness in the semicrystalline polymer. The presence of crystalline domains can also affect the value of the free volume of the polymer phase, VFHp. In the present study, the free volume of the amorphous polymer phase, (Ṽ FHp), was assumed to be a linear function of the volume fraction of the crystallites, ϕc:38,43,44
where MWi denotes the molecular weight of the penetrant molecules. Because of the inherent difficulties in determining the critical molar volume of the polymer jumping unit, an empirical equation was employed for its calculation in terms of the glass transition temperature of the polymer:36 Vpk = (0.6224Tgp − 86.95),
(A13)
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors gratefully acknowledge Borealis Polymer Oy for supporting this work, and they also thank Dr. Sameer Vijay (Group Expert at InnoTech Process Technology Department) for his help and support during the execution of experiments.
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(A11)
By substituting eqs A9 and A10 into eq A11, we obtain the following expression for VFH
NOTATION
General
penetrant concentration (kg·m−3) equilibrium penetrant concentration (kg·m−3) Initial enetrant concentration (kg·m−3) local penetrant-polymer diffusion coefficient (m2·s−1) effective penetrant-polymer diffusion coefficient (m2· s−1) Dp0,i pre-exponential factor (m2·s−1) Lk pellet thickness at each space direction, with k = x, y, z (m) Mi mass of the sorbed species (kg) Mi,0 initial mass of the sorbed species (kg) MWi molecular weight of penetrant (kg·kmol−1) MWp molecular weight of polymer (kg·kmol−1)
Ci Ci.eq Ci,0 Dpi Dpi,eff
VFH = (a i0 + a i(T − Tgi)V i*)ωi + (a p0 + a p(T − Tgp)V p*)ωp (A12)
Effect of Polymer Crystallinity
It is well established that the transport of penetrant molecules in semicrystalline polymers either during sorption or desorption is greatly affected by the degree of crystallinity because crystalline polymers are in general impermeable to most gaseous molecules. Thus, eq A12 needs to be properly corrected so that it can account for the presence of semicrystalline polymer domains in the pellet. This is conveniently done by introducing the so-called geometrical 15860
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Industrial & Engineering Chemistry Research S T Tgi Tgp t u VFH VFHi VFHp V*i V*p Vpk x Yp z
Article
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initial slope of the reduced desorption curve (dimensionless) temperature (K) penetrant glass temperature (K) polymer glass temperature (K) time (s) convection velocity (m·s−1) free volume of the system (m3/g) free volume of the penetrant (m3/g) free volume of the polymer (m3/g) specific critical hole free volume of penetrant (m3/g) specific critical hole free volume of polymer jumping unit (m3/g) specific critical hole free volume of polymer jumping unit (m3/mol) position inside the pellet (m) penetrant concentration (dimensionless) position inside the pellet (dimensionless)
Greek Letters
ai ai0 ap ap0 γi ξi,p ρ ρam ρc τ ϕc ωc ωi
difference of penetrant thermal expansion coefficient above and below the glass transition temperature (K−1) fractional free volume of the penetrant at the glass transition temperature difference of polymer thermal expansion coefficient above and below the glass transition temperature (K−1) fractional free volume of the polymer at the glass transition temperature overlap factor (dimensionless) ratio of the volumes of the diffusing jumping units of the penetrant and the polymer (dimensionless) density of the polymer pellet (kg·m−3) density of the amorphous phase (kg·m−3) density of the crystalline phase (kg·m−3) tortuosity factor (dimensionless) volume fraction of crystalline phase (dimensionless) weight fraction of crystalline phase (dimensionless) penetrant weight fraction (dimensionless)
Subscripts and Superscripts
c i am p
■
crystalline property penetrant property amorphous polymer property polymer property
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dx.doi.org/10.1021/ie402479e | Ind. Eng. Chem. Res. 2013, 52, 15855−15862
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dx.doi.org/10.1021/ie402479e | Ind. Eng. Chem. Res. 2013, 52, 15855−15862
