258876 Development of a Comprehensive Dynamic Diffusion Model for the Desorption of Low and High Molecular Weight Components From Polyolefins
Abstract
Degassing-desorption of hydrocarbons (i.e., monomers, comonomers and oligomers) from polyolefins (POs) is one of the most challenging issues in polymer downstream processing. There is a number of studies dealing with degassing of volatile organic components (VOCs) from POs. However, there is no detailed modelling framework in parallel with experimental studies on that subject for thoroughly understanding of the effect of mass transfer phenomena in the polymer matrix.
To predict the transport of low and high molecular weight a-olefins in semi-crystalline polymer films and powders an unsteady-state diffusion model was employed^{1-3}. Let us assume that a non-porous polymer film of thickness L_{x} (see Figure 1) is exposed at time t= 0 to a gaseous a-olefin atmosphere.
Figure 1. Sorption and desorption process in a polymer film. |
Assuming that at the solid-gas interface, the penetrant concentration in the film is either at equilibrium (i.e.,C_{i}(t,L_{x})=C_{i,eq}) or at an initial value (i.e., C_{i}(t,L_{x})=C_{i,0}), one can derive the following unsteady-state dimensionless mass balance equation accounting for the diffusion of the penetrant molecules in the polymer film (see Figure 1):
Continuity equation (planar coordinates)
Initial condition
Y_{i} = 1 at t = 0 (2)
Boundary conditions
Y_{i} = 0 at z = 0 (4)
Where D_{i}^{p}, z(t) and Y(t,z) are the diffusion coefficient, the dimensionless space variable and a dimensionless concentration, respectively. Similarly, the unsteady-state diffusion of penetrant molecules in a spherical polymer particle can be described by the following continuity equation:
Continuity equation (spherical coordinates)
Initial condition
Y_{i} = 1 at t = 0 (6)
Boundary conditions
Y_{i} = 1 at z = 1 (8)
Both diffusion models consist of a stiff non-linear partial differential equation (eq 1 or eq 5) and a number of initial and boundary conditions (eqs 2-4 or 6-8). In both cases, the partial differential equations were solved by the global collocation method.^{4} It can easily be shown that the mass of the sorbed species at time t, M_{i}(t), will be given by the following integral:
Where M_{i,eq} is the total mass of the sorbed species “i” at equilibrium.
To calculate the diffusion coefficient of the penetrant molecules, D_{i}^{p}, in a semi-crystalline, non-porous polymer matrix, the free volume theory was employed.^{ }Thus, following the original developments of Vrentas and Duda the diffusion coefficient of a-olefins in a semi-crystalline, non-porous polyolefin (e.g., film, powder) was expressed as follows^{5-6}:
^{ }
where the subscripts i and p refer to the penetrant molecules and the polymer, respectively. Eq 10 refers to the local diffusion coefficient, D_{i}^{p}(x). Accordingly, the overall (effective) diffusion coefficient, D_{i,eff}^{p}, in the polymer film can be calculated by integrating the local diffusion coefficient, D_{i}^{p}(x), with respect to the polymer film's thickness, L_{x}.^{3}
The diffusion of the penetrant species from the bulk gas-phase to the amorphous polymer phase in a spherical, porous, semi-crystalline polymer particle is assumed to occur via a dual-mechanism that comprises molecular diffusion of the penetrant species through the particle's pores and the amorphous polymer phase. In case of monomer diffusion through the amorphous polymer phase, the diffusion coefficient of the penetrant species will depend on the temperature, the concentration of sorbed species as well as the degree of polymer crystallinity (eq 10).
Figure 2. Molecular diffusion of the penetrant species through the particle's pores and the amorphous polymer phase. |
The random pore model of Wacao and Smith was employed to take into account the dual mode of penetrant transport from the bulk gas-phase to the polymer phase (Figure 2)^{2,3,7}. It is apparent that the selected arrangement of the pores in the random pore model of Wacao and Smith (see Figure 2) does not represent the real pore geometry in a polymer particle. Furthermore, the pores are not parallel to the direction of diffusion. Thus, a tortuosity factor, τ_{f}, is often introduced to take into account the tortuous nature of the pores and the presence of random constrictions in the pore geometry. Based on the above random pore configuration and model assumptions, it can be shown that the overall diffusion coefficient, D_{i,eff} , of the penetrant molecules in a semi-crystalline, porous polymer particle can be expressed as follows:
Where D^{p}_{i,eff} is the diffusion coefficient of the penetrant species “i” in the amorphous polymer phase. It should be pointed out that the first term on the right hand side of eq 11 accounts for the penetrant mass transfer via the particle's pores while the second term accounts for the penetrant transport through the amorphous polymer phase.
In Figure 3, the ethylene desorption rate from HDPE films are plotted with respect to time for two different particle sizes (i.e., 400 μm and 800 μm). It is apparent that as the particle size increases the mass transfer limitations are more manifested (i.e., much time for penetrant to be desorbed from the polymer matrix). The numerical values of all the physical and transport parameters used in the theoretical calculations are presented elsewhere^{1-3,7}. In Figure 4, the spatial variation of the penetrant (i.e., ethylene) for different particle diameters (the largest diameter is marked by the broken lines) is depicted at different times. As can be seen, the value of the local penetrant concentration changes with respect to the dimensionless spatial distance and the time. It is evident that at steady-state the concentration of the desorbed species will be equal to zero. However, the small polymer particles attain their steady state sooner due to the less mass transfer limitations in comparison to the largest size particles.
In Figures 5 and 6, the calculated values of sorption curves and diffusion coefficients for low porosity and high porosity HDPE particles of d_{p} = 400 μm, at T = 80 ^{o}C are plotted as a function of time, respectively. It is apparent that in highly porous polymer particles, ethylene desorption process takes less time than in the case of low porosity particles due to less mass transfer resistances.
Figure 3. Predicted desorption ethylene curves in HDPE powders (T = 80 ^{o}C, P = 7.5 bar). | Figure 4. Predicted penetrant concentration profiles in HDPE powders (T = 80 ^{o}C, P = 7.5 bar). | |
Figure 5. Predicted desorption ethylene curves in HDPE powders at different porosity values (d_{p} = 400 μm, P = 6 bar). | Figure 6. Predicted ethylene diffusion coefficients in HDPE powders at different porosity values (d_{p} = 400 μm, P = 6 bar). | |
In Figure 7, the dependence of the penetrant size on the dynamic evolution of the desorption curve is depicted. It is apparent that the low-molecular weight penetrants desorption curve attains its final - steady state - value sooner in comparison to that of a high molecular weight penetrant (e.g., C_{12}), at the same pressure. Figure 8 depicts the variation of penetrants diffusion coefficients in HDPE particles. As can be seen from Figure 8 the diffusion coefficient value of a high molecular weight penetrant may have 2 order of magnitude lower values than that of a low molecular weight penetrant.
Figure 7. Estimated desorption curves for various low and high molecular weight penetrants in HDPE powders. | Figure 8. Estimated diffusion coefficient values for various low and high molecular weight penetrants in HDPE powders. |
1. Kanellopoulos, V. ; Mouratides, D.; Pladis, P.; Kiparissides, C., Ind. & Eng. Chem. Res., 2006, 45, 5870.
2. Kanellopoulos, V.; Mouratides, D.; Tsiliopoulou, E.; Kiparissides, C., Macromolecular Reaction Engineering, 2007, 1, 106.
3. Kanellopoulos, V.; Tsiliopoulou, E.; Dompazis, G.; Touloupides, V.; Kiparissides, C., Ind. Eng. & Chem. Res., 2007, 46, 1928.
4. Villadsen, J.; Michelsen, M.L., Prentice-Hall, 1978.
5. Vrentas, J.S.; Duda, J.L., Macromolecules, 1976, 9, 785.
6. Cussler, E. L. Diffusion Mass Transfer in Fluid Systems; University Press: Cambridge, 1997.
7. Neogi, P., Diffusion in Polymers; Marcel Dekker: New York, 1996.
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