(180g) 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.

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)

                                                                                                 (1)

Initial condition

Y_i = 1    at    t = 0                                                                                                                     (2) 

Boundary conditions

 = 0   at     z = 0                                                                                                                (3)                
 

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)

                                                                                 (5)

Initial condition

Y_i = 1    at    t = 0                                                                                                                     (6)                                

Boundary conditions

   at    z = 0                                                                                                                (7)

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:

                                                                                                       (9)

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: 

                                                  (10)

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).

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:

                                                                                          (11)

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 ^oC 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.

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., C₁₂), 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.

Literature Cited

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