Investigation of Hexane Diffusion in HDPE Particulates for Drying

0.001 μg of hexane/mL of N2 (or 250 ppb mol at 0 °C and 1 atm) and was more ..... (1995) in their investigation of fluidized bed dryers. .... 1977 (...
0 downloads 0 Views 213KB Size
Download PDF
3422

Ind. Eng. Chem. Res. 1996, 35, 3422-3430

Investigation of Hexane Diffusion in HDPE Particulates for Drying Applications Jian Steven Qi,* Chandrasekhar Krishnan, Joseph A. Incavo, Vivek Jain, and William L. Rueter Technology Center, Occidental Chemical Corporation, 2801 Long Road, Grand Island, New York 14072

Diffusion of n-hexane in high-density polyethylene (HDPE) particulates was studied by nitrogen purging of a hexane-permeated HDPE sample in a small glass cell. By monitoring the decrease in hexane concentration in the cell exit gas, the diffusivity of hexane in HDPE was obtained. The method does not require any alteration of the original HDPE specimen and allows both the intrinsic and effective diffusivities to be determined. Intrinsic diffusion data measured for HDPE powder and pellets using this method were compared with those from HDPE sheets to show the unique diffusion characteristics of the powder. Experimental results at various purge flows revealed that the effective hexane diffusivity in the HDPE powder increased with gas purge velocity before reaching a plateau representing the intrinsic diffusivity. The low effective diffusivity values at low purge velocities can be attributed to particle agglomeration in the powder. The effective diffusivity data for unaltered powder samples have great practical significance in characterizing industrial powder dryers. Introduction Diffusivity of a volatile organic solvent in a specific polymer is a fundamental constant that defines the rate at which the solvent is transported within the polymer under the influence of a concentration gradient. The constant has great significance in elucidating the diffusion mechanism involved in absorption or desorption of a solvent in a polymer and is valuable for the design and analysis of diffusion-controlled polymer dryers. Two steps are usually involved in drying a polymer containing a dissolved substance: (1) diffusion of the dissolved molecules from the polymer’s interior to its surface and (2) removal of the molecules from the surface by an inert gas sweep. Diffusion-controlled drying refers to a drying process where intraparticle diffusion is the slowest ratelimiting step, and the drying kinetics are determined entirely by the diffusion rate. Diffusion-controlled drying is frequently encountered in the polymer industry. For instance, the residual toxic vinyl chloride monomer (VCM) embedded in a PVC resin must be removed to render the polymer product safe to use. Removing low levels of VCM is a process controlled by diffusion. Another common example is the removal of hexane solvent from HDPE resin produced by hexane-based polymerization. When the hexane level in HDPE falls below 4%, surface hexane is depleted and further drying is controlled by hexane diffusion in the HDPE (Incavo et al., 1996). Diffusivity data are therefore essential for theoretical analysis and optimal design of these drying processes. Several methods are available for measuring diffusivities in solids (Zogzas et al., 1994). Schlotter and Furlan (1992) have reviewed the literature on small molecule diffusion in polyolefins and presented a summary of the experimental techniques for polymer systems. In principle, most techniques for organic solventpolymer systems are based on (1) the rate to attain a solvent permeation steady state in a polymer membrane (the time-lag method), (2) the steady-state permeability in a membrane if the sorption isotherm is known, or (3) the kinetics of solvent sorption and desorption by a * To whom correspondence should be addressed.

S0888-5885(96)00067-X CCC: $12.00

polymer sheet in a defined environment (Crank and Park, 1968). These methods, however, require the specimen to be made into a standard geometrical form (such as a membrane). For polymers, this transformation often adds a thermal history and changes its original morphological properties such as orientation, crystalline structure, and internal pores and thereby alters diffusivity. As a result, these methods may not produce the diffusivities truly representative of the original material under study. Consequently, direct measurements on unaltered materials are desirable. The current study focuses on the investigation of hexane diffusion in HDPE by measuring diffusivities in unaltered samples under conditions usually encountered in drying. Although Berens (1977, 1982) has developed a method to measure the diffusivities of VCM in unaltered PVC resin powders, which involves monitoring over time the weight change of a PVC sample undergoing a sorption or desorption process in a VCM vapor-filled vacuum chamber, it is not suited for studying particle interactions and their impact on mass transfer under the influence of a flowing carrier gas in a diffusion-controlled powder drying process. The method also is not suitable for characterizing the problem of reduced mass-transfer surface area caused by particle agglomeration. In this work, a purge cell method similar to the techniques used by Beret et al. (1977) and Hufton and Ruthven (1993) was developed to obtain diffusivities for fine HDPE particulates. The work is motivated by a practical need to evaluate the drying kinetics in several commercial dryers for removing hexane from crude HDPE resins. In the industrial process under study, the hexane-laden crude resin made by hexane-based slurry polymerization is dried in two stagessthe first stage removes the free surface hexane, while the second expels the dissolved hexane. The drying in the second stage is effected by purging the polymer powder at 6093 °C with hot, hexane-lean nitrogen carrier gas. The drying kinetics for removing dissolved hexane are controlled by the diffusion of hexane in HDPE and are usually evaluated through process models based on diffusivity data. However, diffusivity data for the © 1996 American Chemical Society

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3423

hexane/HDPE system reported in the open literature (Kreituss and Frisch, 1981; Rogers et al., 1960) are mostly for low temperatures (0-30 °C) and based on the membrane permeation technique. They cannot be safely extended to the high temperatures usually encountered in the drying process. Besides, the fundamental properties of HDPE are dependent on specific polymerization conditions and thermal histories. Therefore, each particular grade and form of HDPE may have its own unique set of physical characteristics (such as crystallinity, orientation, branching, molecular weight and distribution, etc.). Consequently, to accurately evaluate drying kinetics, diffusivity data should be developed for the specific HDPE material to be dried. In this study, we used a purge cell method to obtain diffusivity data that truly reflect the drying process under study. The results revealed the unique diffusion characteristics of HDPE powder under drying conditions and provided the essential data for developing accurate mathematical dryer models. Experimental Theory The basic experimental theory in the present study is similar to the purging method presented by Beret et al. (1977) and the zero length column method by Karger and Ruthven (1992) and Hufton and Ruthven (1993), but with some modifications. The theory is based on unsteady-state diffusion in a single spherical particle that initially contains a uniform concentration of a diffusant (e.g., solvent or other volatile substances). When this particle is exposed to a sudden drop in the diffusant concentration at its surface, the concentration difference drives the diffusant molecules out of the solid. As a result, a concentration gradient begins to develop in the solid. For a constant diffusivity D, this unsteadystate diffusion process can be described by Fick’s second law as follows (Crank, 1975):

[

]

∂w(r,t) D ∂ 2 ∂w(r,t) ) 2 r ∂t ∂r r ∂r

(1)

where w(r,t) is the local diffusant concentration in the solid after time t and at a radial distance r from the center of the sphere. If the diffusant concentration at the surface of the sphere (i.e., at r ) R) is maintained constant at ws, the following initial and boundary conditions apply:

at t ) 0, w(r,0) ) w0 for 0e r e R

(IC 1)

at r ) R, w(R,t) ) ws for t > 0

(BC 1)

at r ) 0, ∂w(r,t)/∂r ) 0 for t g 0

(BC 2)

The mathematical solution for eq 1 with the above set of initial and boundary conditions is (Crank, 1975)

w(r,t) - ws

2R )

w0 - ws

∞

∑ πr n)1

(-1)n+1 sin

n

( ) ( nπr R

exp -

)

n2π2Dt R2

(2)

Multiplying eq 2 by 3r2/R3 dr and integrating the product from r ) 0 to R yields the expression for the average concentration w as a function of t:

w - ws

6 )

w0 - ws

∞

∑

1

π2n)1n2

(

exp -

)

n2π2Dt R2

(3)

Figure 1. Schematic diagram of a purge cell for diffusivity measurements.

It can be shown mathematically that when Dt/R2 > 0.11, the total of the second and subsequent terms in the infinite summation series is less than 1% of the first term, and these higher order terms can be omitted from eq 3 to give

(

)

w - ws 6 π2D ) 2 exp - 2 t w0 - ws π R

for

Dt > 0.11 (4) R2

The rate of concentration change can be obtained from eq 3:

6D(w0 - ws)

dw )dt

R2

(

∞

∑

exp -

n)1

n2π2D t R2

)

(5)

For eq 5, when Dt/R2 > 0.11, the sum of the second and subsequent terms in the summation series is less than 4% of the first term. Thus, this equation can be approximated by

(

)

6D(w0 - ws) π2D dw )exp t dt R2 R2

for

Dt > 0.11 R2 (6)

which can be rewritten as

( )

ln -

[

]

6D(w0 - ws) dw π2D ) - 2 t + ln dt R R2

(7)

Equation 7 shows that, in principle, the diffusivity can be obtained from the straight-line portion of the ln(dw/ dt) ∼ t plot at t > 0.11R2/D. This can also be done by plotting ln(w - ws) vs t using eq 4 provided that ws is known. Experimentally, both methods require measuring the average concentration in the solids as a function of time, which can be cumbersome and may introduce a disturbance to the diffusion process under study, especially for fine particulates at higher temperatures where diffusion is rapid. However, this problem can be overcome by expressing dw/dt in terms of the gas-phase concentration, which is more readily measurable. Consider a cell containing a small amount of particulates that are being purged by a large stream of inert gas (Figure 1) such that the difference between the concentrations of the inlet and outlet gas streams is very small. In such a case, the surface concentration of the

3424 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

solids is essentially determined by the concentration of the inlet gas, i.e., ws ) KCin, and is nearly constant, where K is the partition coefficient of diffusant between the gas and solid phases (Vergnaud, 1992). When Dt/ R2 > 0.11, each particulate should behave according to eq 7, provided that particle agglomeration does not exist and each particulate is completely surrounded by the gas flow (i.e., perfect ventilation in the interstitial space). A mass balance of the diffusant around the cell gives

-

m dw ) Q(Cout - Cin) F dt

(8)

where m represents the total mass of the particulates in the cell, F the density of the solids, and Q the volumetric gas flow rate. Cin and Cout are diffusant concentrations at the cell inlet and outlet, respectively. Through eq 8, the rate of concentration change in the solid phase can be tracked by monitoring the gas concentrations. Combining eqs 7 and 8 gives

ln(Cout - Cin) ) -

[

]

6QFD(w0 - ws) π2D t + ln (9) 2 R mR2

Thus, by plotting ln(Cout - Cin) vs t, one can obtain D from the slope of the linear portion of the plot (slope ) -π2D/R2). Note that because Q, F, m, w0, and ws are contained only in the intercept of the plot, as long as they are kept constant throughout each experiment, their absolute values need not be known. This greatly simplifies the experiment. The above derivation is for spheres. For particulates that are cubic or cylindrical, an equivalent radius should be used. For slabs (thin chips) or long cylinders (needle shapes), mathematical solutions similar to eq 3 exist (Crank, 1975). Therefore, similar derivations can be performed. For these geometries, the group represented by the slope will change only slightly. For example, for slabs, the slope is equal to -π2D/4l2, where l is the slab thickness. A further simplification results if the purge gas contains no diffusant; i.e., Cin ) 0:

ln(Cout) ) -

(

)

6QFDw0 π2D t + ln 2 R mR2

(10)

Based on this equation, experiments can be designed to obtain D in which hexane-permeated HDPE particulates are purged in a cell while the hexane concentration in the outlet gas is monitored. Note that when Dt/R2 ) 0.11, w/w0 ) 0.207. Therefore, the diffusivity measured by this method is for the concentration range below 0.207w0. A key assumption in these derivations is that the gas purge flow rate is sufficiently high to assure that the inlet and outlet gas concentrations are essentially the same so that the boundary condition of constant zero surface concentration is satisfied for every particle (thus allowing eq 3 to apply to each particle). Theoretically, the flow rate should approach infinity in order to rigorously satisfy this assumption, although this is not practical. In the Appendix, we show that the surface concentration is negligibly small (less than 1% of the average concentration in the solids) if the dimensionless group

E ) π2K

( )( ) D m R2 QF

(11)

is less than 0.005. Based on this criterion, the practical experimental conditions that allow the diffusivity experiment to be executed without significant errors are identified in the Appendix. In this work, HDPE particulates were purged by a pure dry nitrogen stream (Cin ) 0), while Cout was measured by GC as a function of time using an automated data acquisition system. It should be pointed out that since the GC peak area is proportional to the concentration, obtaining the slope from the semilog plot does not require the actual concentration to be known. The slope can be obtained by plotting the GC area counts directly, rendering calibration of the GC unnecessary. Experimental Details Experimental Setup. The experimental setup for measuring diffusivities in HDPE particulates is shown in Figure 2. The purge cell consisted of a small glass vial (diameter, 2.5 cm; height, 8.2 cm) with a porous glass frit at its bottom as a flow distributor. The use of a glass cell was conducive to preliminary visual evaluation of the behavior of HDPE particles under various gas flow velocities. Glass fibers were placed at the outlet of the vial to prevent fine HDPE particles from being entrained out of the cell. The cell and the gas line (coiled copper tubing) were housed in a convection oven (Blue-M, Model OV-490A-3) for temperature control. The feed gas temperature, the temperature in the cell, and the temperature in the oven were monitored by thermocouples. The flow rate of the purge gas (N2) was measured by a rotameter installed in the gas feed line. Prior to entering the cell, the nitrogen gas was preheated to a desired temperature using a temperature-controlled oil bath. Hexane concentration in the nitrogen stream at the cell outlet was monitored by an HP5890A gas chromatograph (GC) equipped with a DB-5 fused silica capillary column (15-m × 0.32-mm i.d., 0.25-µm film thickness, J&W Scientific, Folsom, CA) and a flame ionization detector (FID). The sampling system consisted of a 6-port Valco valve and a 500-µL sample loop, which was continuously flushed by the process stream using a bellows pump (Gorman-Rupp). The GC system was programmed to automatically inject an on-line sample every 20 s. This ensured the necessary number of data points for each run, particularly at high temperatures (>80 °C), where fast diffusion led to very short run times (5-10 min). Tests showed that to obtain quantitative injections as well as representative samples, the sample valve should be switched back to the process stream 5 s after every injection. Since the process stream contained only hexane, an isothermal GC condition with a continuously running integrator was configured to quantitate the peaks eluting every 20 s. The sensitivity of the GC system was ca. 0.001 µg of hexane/mL of N2 (or 250 ppb mol at 0 °C and 1 atm) and was more than adequate for our purpose (see the Appendix for estimation of the detection sensitivity requirement). Calibration standards were prepared by diluting a 4990 ppm (mol) hexane standard (Scott Gases). Material Characterization and Sample Preparation. The starting material used in this study was a dry powder of film grade HDPE resin with a broad

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3425

Figure 2. Experimental setup of the purge cell for measuring diffusivity in HDPE particulates.

bimodal molecular weight distribution (MW ∼ 200 000, density ) 0.95 g/cm3, melting index MI2 ) 0.055). The powder was obtained directly from the commercial polymerization process by normal recovery and drying procedures and had an average particle size of about 100 µm. Scanning electron micrographs (SEMs) of the HDPE powder at 10 000 and higher magnifications showed the individual particles to be substantially spherical and nonporous. Porosity measurements by the standard mercury intrusion method at 50-5000 psia showed an intraparticle porosity of 4-9%, which was neglected in this study. The crystallinity of the powder was 60% as measured by X-ray diffraction (XRD). For diffusivity measurements, a powder sample with a narrow particle size distribution (PSD) was prepared by double sieving the HDPE resin: the resin was first sieved into various size fractions and then one of the fractions (125-149-µm size fraction) was resieved. The PSD of the double-sieved material was found by Coulter Counter to be within 10% of the mean (137 µm). Therefore, the sample could be treated as spherical particulates with a roughly identical radius of 68.5 µm. Besides the powder, pellets made from the powder also were used for diffusivity measurements. The pellets were made in the plant by melting the HDPE powder and then extruding the molten polymer at about 200 °C. The extruded polymer was immediately quenched to about 50-70 °C in water while being cut into pellets with an average diameter of about 0.34 cm. The crystallinity of the pellets was 50% as measured by XRD, as opposed to the 60% for the HDPE powder. Experimental Procedure. To measure the diffusivity for HDPE particulate, a sample of the dry HDPE particulates (usually 0.25 g) was placed into the glass cell to form a shallow bed (for 0.25 g of sample, the bed depth was about 0.1-0.2 cm), followed by spiking into the cell a known amount of reagent-grade n-hexane (>99% purity). The cell was then closed and held at a desired temperature for 2-5 h to allow the hexane sorption equilibrium to be established between the gas and the solids (minimum equilibration times required at different temperatures were determined by separate experiments). This was designed to ensure a uniform hexane concentration in the solids prior to the onset of diffusion to satisfy the initial condition for eq 2. Note that for measuring the diffusivity, a knowledge of the starting hexane concentration (w0) is not required. Nonetheless, since it is not uncommon that diffusivity

varies with concentration, the amount of hexane spiked in the HDPE sample should be defined. The starting hexane concentration in this study was such that the diffusivity value in the range of 3500-5000 ppm was measured. This range was representative of the typical concentration range in the drying process under study. The nitrogen gas was initially set to bypass the cell (see Figure 2) to purge all the lines leading to the GC. During this bypass period, the flow rate was adjusted to and set at a desired rate using the rotameter. The temperature of the feed gas was monitored to ensure that it matched the cell temperature. After a blank GC run using the bypass nitrogen to show a clean and stable baseline on the chromatogram, the flow was switched to the cell to initiate purging. The cell outlet gas was sampled by the automatic valve every 20 s to generate a chromatograph output showing a series of successively declining peaks. The run was terminated when the peak sizes on the integrator had diminished below the detection limit. Both the temperature and the nitrogen flow rate were held constant throughout the run. Membrane Permeation Method. Diffusivity measurements using a conventional permeation technique (with a setup similar to the standard ASTM D1434-82 method) were conducted to study the fundamental diffusion characteristics of the HDPE material. The data also established a baseline for comparing results from different methods. The HDPE sheets used for the experiments were made by a compression-molding process based on the standard method given in ASTM D1928-90 Procedure A. In the molding process, an HDPE powder was first melted and then pressed into a thin sheet while being slowly cooled over more than 30 min. The ASTM D1928-90 methods contained several different cooling procedures. The slow cooling and low shear method used here afforded a high crystallinity of about 60%, close to the crystallinity of the powder. In the permeation experiment, an HDPE sheet was sandwiched between two circular PTFE-coated aluminum cells. Hexane was first permeated into the HDPE sheet (area 126 cm2) to equilibrium by exposing both sides of the sheet to a nitrogen atmosphere containing 0.5 or 2.0 mol % hexane. After permeation equilibrium was reached, a pure helium flow was introduced into the cells to sweep both sides of the polymer sheet. The hexane concentration in the helium as a result of its desorption from the polymer sheet was tracked by a GC. By plotting the hexane concentrations in the helium

3426 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

over time, both permeability and diffusivity were obtained (time-lag method). The theory for the method can be found in Crank and Park (1968). Results and Discussion In this section, experimental data for HDPE sheets using the conventional permeation method are first presented to elucidate the general diffusion characteristics of n-hexane in the HDPE material. The results are then compared with those for HDPE pellets and powders obtained using the purge cell method. Diffusivity Data for HDPE Sheets. Using the membrane permeation technique, diffusivity data for HDPE sheets were obtained. Data collected at five different sheet thicknesses (0.54, 0.77, 0.89, 1.02, and 1.32 mm) indicated that sheet thickness had no effect on diffusivity measurements. Although diffusivities for polymer-organic solvent systems often exhibit concentration dependence, data obtained at two different hexane concentrations (obtained by permeation using 0.5 and 2 mol % hexane in nitrogen) did not show any significant difference. Diffusivity data using the HDPE sheets were obtained at five different temperatures (40, 60, 70, 80, and 93 °C). These temperatures were representative of the typical range for commercial HDPE drying processes. The data, which had a relative standard deviation of 4.5%, are presented graphically in Figure 3 as an Arrhenius plot along with two literature data points at lower temperatures. The linearity of the plot indicates that the diffusion mechanism does not change with temperature. The apparent activation energy for the diffusion obtained from the Arrhenius plot is 16 kcal/ mol. The dependence of diffusivity (in cm2/s) on temperature in the range of 40-93 °C can be expressed as follows:

ln D ) -6.8008 -

8040.7 T

(12)

Note that the two literature data points in Figure 3 are close to the straight line represented by eq 12, even though they were for different grades of HDPE. Diffusivity in HDPE Pellets. For comparison, the diffusivity in the HDPE pellets was measured at 93 °C using the purge cell method shown in Figure 2. The study on pellets is important because the molten HDPE product from solution polymerization is usually pelletized before being dried. Even for processes based on slurry polymerization, where the crude polymer is dried as a powder, the final resin must be pelletized for storage and shipping. The pellets often are further purged in the storage silos by air circulation to expel the residual hexane and prevent a buildup of explosive mixtures in closed containers. About 25 pellets (0.5 g in total weight) with an average diameter of about 0.34 cm were used. Figure 4 is a typical result obtained at 93 °C under 300 cm3/ min nitrogen purge rate (sufficient based on the analysis in the Appendix) and shows the linear portion at long times. The D obtained from the slope of the linear portion is 4.1 × 10-7 cm2/s, significantly higher than the 2.5 × 10-7 cm2/s obtained for the HDPE sheets (see Figure 3). This is probably due to the difference in their thermal histories as a result of different sample preparation processes. The crystallinity for the HDPE sheets was 60%, compared to 50% for the pellets, and therefore should result in a lower diffusivity since the crystalline

Figure 3. Arrhenius plot of n-hexane diffusivity vs 1/T measured by the membrane permeation method. The apparent activation energy is obtained from the slope of the straight line following the Arrhenius equation. The two literature points are from Rogers et al. (1960) and Kreituss and Frisch (1981) for HDPE with the following properties: HDPE density ) 0.954 g/cm3, crystallinity ) 75% (Rogers et al., 1960). HDPE density ) 0.952 g/cm3, crystallinity ) 65% (Kreituss and Frish, 1981).

Figure 4. A typical hexane concentration decay curve in cell outlet nitrogen gas for HDPE pellets at 93 °C [HDPE sample weight ) 0.5 g; average radius ) 0.17 cm; nitrogen flow ) 300 cm3/min; GC response factor ≈ 6 × 10-7 (µg/mL)/(GC area count)]. D obtained from the slope of the line ) 4.1 × 10-7 cm2/s. Linear portion begins at 100 min (Dt/R2) 4.1 × 10-7 × 6000/0.172 ) 0.085).

structure behaves like impermeable microcrystalline islands within a continuous amorphous matrix (Crank and Park, 1968). It is well-known that quenching HDPE usually produces a large amorphous content and thereby a large diffusivity (Schlotter and Furlan, 1992). Detailed discussions on the effect of crystallinity can be found in Kreituss and Frisch (1981), Schlotter and Furlan (1992), and Michaels and Bixler (1961). The difference in diffusivities between the pellets and the polymer sheets exemplifies the significance of directly measuring the diffusivity on unaltered HDPE samples. Since it is the original material, not the film or sheets made from it, that undergoes drying, the diffusivity value for the original unaltered material has a much greater value. For example, in the hexanebased solution HDPE technology, the crude polymer product is a mixture of a molten HDPE and hexane. After the majority of hexane is flashed off through a series of evaporators, the molten HDPE is pelletized and dried. In such a process, diffusivity in pellets dictates the drying kinetics. Diffusivity in HDPE Powder. According to the analysis in the Appendix, a minimum purge rate of 500

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3427 Table 3. Biot Numbers Calculated for 80 °C and 1 cm/s Flow Velocity Ka

D, cm2/s

R, cm

k,b cm/s

Bi

7 7

1.1 × 10-7 1.1 × 10-7

0.006 0.06

0.9 0.3

6 400 21 400

a The K value was measured directly by first allowing an HDPE sample with a known amount of hexane to equilibrate in a closed container and then assaying the hexane concentration in the headspace above the sample using a GC (Incavo et al., 1996). b The mass-transfer coefficient k was obtained using the Wilke and Hougen correlation for a packed bed (Perry and Green, 1984) in conjunction with the following properties: hexane diffusivity in nitrogen ) 0.07 cm2/s; kinematic viscosity of nitrogen ) 0.2 cm2/s (Vargaftik, 1975); the void fraction for a loosely packed HDPE bed was about 0.6.

Figure 5. Typical hexane concentration decay curve in cell outlet nitrogen gas for HDPE powder at 60 °C [HDPE sample weight ) 0.25 g; mean particle size ) 137 µm; nitrogen flow ) 1000 cm3/ min; GC response factor ≈ 6 × 10-7 (µg/mL)/(GC area count)]. D obtained from the slope of the line )1.9 × 10-8 cm2/s. Linear portion begins at about 3 min (Dt/R2 ) 1.9 × 10-8 × 180/0.006 852 ) 0.073). Table 1. Effective Diffusivity Values for Powders at 60 and 70 °Ca 108D (cm2/s) at 60 °C 108D (cm2/s) at 70 °C N2 flow, 3 cm /min run 1 run 2 run 3 mean run 1 run 2 run 3 mean 500 1000 1500 2000 2500 3000 2000b

0.65 1.9 2.8 3.1 3.2 3.2 3.1

0.64 1.9 3.1 3.0 3.9 3.0

0.65 1.8 2.8 3.3 3.1 3.2

0.65 1.9 2.9 3.1 3.1 3.1 3.1

0.74 2.2 3.7 6.0 5.9 5.8 5.5

0.73

5.5 5.9

5.9 5.7

0.74 2.2 3.7 6.0 5.8 5.8 5.5

a All runs used 0.25 g of powder with a mean particle size of 137µm unless otherwise indicated. b This run used a powder with a mean particle size of 74 µm. Starting concentration: 35005000 ppm hexane in the powder. Relative standard deviation < 5%.

Table 2. Intrinsic Diffusivity Values of HDPE in Different Formsa pellets D at 40 °C D at 60 °C D at 70 °C D at 80 °C D at 93 °C crystallinity by X-ray

4.1 × 10-7 50%

powder 0.31 × 10-7 0.57 × 10-7 1.1 × 10-7 60%

sheets 0.061 × 10-7 0.28 × 10-7 0.55 × 10-7 1.24 × 10-7 2.50 × 10-7 60%

a

Diffusivity values for pellets and powders were obtained by the purge cell method. Diffusivity values for HDPE sheets were obtained by the membrane permeation method. All D values in units of cm2/s.

cm3/min should be used for 0.25 g of the powder sample. Figure 5 is a typical concentration decay curve for an HDPE powder, where the linear portion is clearly evident. In Table 1, experimental data under various purge velocities and at two temperatures (60 and 70 °C) are summarized to show the good reproducibility at each flow velocity. The data indicate that the measured diffusivity values increased with the purge flow rate before reaching a plateau where further increase in flow no longer had any effect. The plateau region signifies the attainment of the intrinsic diffusivity value that is not affected by gas flow velocity. Note that the intrinsic diffusivities measured at two different particle sizes (74 and 137 µm) are almost identical. The diffusivity values in the plateau region are compared in Table 2 with those from the pellets and sheets. The comparison shows that

the intrinsic values for powder are very close to those obtained from the HDPE sheets by the permeation method, consistent with the fact that they both have an almost identical crystallinity. Note that the slow cooling during the preparation of the HDPE sheets allowed the sheets to possess a crystallinity comparable to that of the original powder. On the other hand, the pellets had been subjected to quenching during pelletization and, therefore, exhibited a lower crystallinity. Effective Diffusivity. As will be seen later, the velocity dependence of measured diffusivities for powder samples is a manifestation of the impact of flow velocity on the dispersion of the particles. Visual studies on the fluidization behavior of a sample of the HDPE powder (mean particle size: 137 µm) in the glass cell indicated that the powder remained packed (stationary) at flow rates up to 300 cm3/min (superficial velocity ) 1 cm/s). As the flow was further increased, a gradual change to an agitated bed occurred, with the top portion of the bed being fluidized. At about 2000 cm3/min (velocity ) 7 cm/s), the entire bed was sufficiently agitated and behaved like a boiling bed. At even higher flow rates (>2000 cm3/min), a significant amount of the powder was entrained into the upper portion of the cell. For diffusivity measurements, the nitrogen flow should be set to impose a zero surface concentration that satisfies the boundary condition for eq 10 (see the Appendix for detailed analysis). A tacit premise here is that the gas phase has a mass-transfer rate so fast that the intraparticle diffusion is the only rate-limiting step. An established way to evaluate this premise is by estimating the Biot number, defined as Bi ) kR/DK, where k is the mass-transfer coefficient. The Biot number is a measure of the external mass-transfer rate relative to internal diffusion. The larger the Biot number, the smaller the external mass-transfer effect. Generally, when the Biot number is greater than 103, the mass-transfer effect can be neglected (Srinivasa Kannan et al., 1994). Table 3 lists the Biot numbers calculated for the n-hexane/HDPE system at two particle sizes. Clearly, the gas-phase mass-transfer effect is more pronounced for smaller particles. However, even for the powder (R ) 0.006 cm) used in this study, the Biot number is still much greater than 103. Therefore, the premise of no gas-phase mass-transfer effect should be valid. The Biot number analysis, however, does not take particle agglomeration into account, since the masstransfer coefficient used for the analysis applies to packed beds of well-ventilated particulates. For the HDPE powder in this study, we have visually observed a tendency to agglomerate. One effect of agglomeration is the reduction of surface area available for mass

3428 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

transfer. Also, since agglomeration creates local hexane entrapment pockets with poor interparticle ventilation, mass transfer in the interstitial gas within each agglomerated cluster is impeded. Consequently, the Biot number within each cluster is reduced. Since estimating the Biot number and diffusion rate within each cluster is difficult, an alternative and practical approach is used in this study. Instead of examining the complicated mass transfer within clusters which involves diffusion in gas and polymer as well as various interparticle geometries, we treat each cluster as if it were a single porous particulate, as illustrated in Figure 6. The mass-transfer resistance in the gas flow around each cluster should be negligible compared to the resistance within the cluster. Since the cluster is heterogeneous containing both polymer particles and gas voids, it as a whole should have an apparent diffusivity Dcluster that is greater than the intraparticle (intrinsic) diffusivity (i.e., Dcluster > D) but smaller than the interparticle diffusivity in the interstitial gas. By treating the cluster as a single particulate, eq 10 will still be valid if D and R in the equation are replaced by Dcluster and Rcluster, respectively. Consequently, the slope obtained from the straight line using eq 10 will be -π2Dcluster/Rcluster2 instead of -π2D/ R2. This approach, however, involves two variables (Dcluster and Rcluster) which are difficult to ascertain accurately. A more convenient and practical approach is to use an effective diffusivity, Deff, that is based on the actual radius of a single particle (R) and is defined as

Deff/R2 ) Dcluster/Rcluster2

(13)

Since this treatment lumps the effects of particle agglomeration into Deff, the difference between Deff and D reflects the degree of agglomeration. Obviously, since R < Rcluster, Deff < Dcluster. Another practical alternative is through an effective radius. In such a treatment, an agglomerated cluster is represented by a fictitious HDPE solid sphere with a radius that would produce an overall diffusion rate equivalent to that from the cluster. This radius can be regarded as the effective radius, Reff, and is related to other treatments as follows:

D/Reff2 ) Deff/R2 ) Dcluster/Rcluster2

(14)

Since Reff > R, D > Deff. Overall, we have

R < Reff < Rcluster

Figure 6. Schematic illustration of an agglomerated cluster. The radius of the cluster and the effective radius are indicated in the drawing.

Deff < D < Dcluster

Thus, the degree of agglomeration can be measured by the ratio of D to Deff, which equals (Reff/R)2. The larger the D/Deff or Reff/R, the more agglomerated the powder. Note that the three different treatments are merely three different ways to look at an agglomerated cluster. Consequently, the slope obtained from eq 10 will produce Deff/R2, D/Reff2, or Dcluster/Rcluster2. Since the actual particle radius is a measured and known quantity, the use of effective diffusivity is most convenient. The phenomenon of increased effective diffusivity with increase in gas flow velocity has been observed before by Srinivasa Kannan et al. (1995) in their investigation of fluidized bed dryers. The diffusivity data shown in Table 1 are effective diffusivities based on the radius of a single particle. Clearly, the low values of effective diffusivity at low flow velocities can be attributed to agglomeration. As the

Figure 7. Ratio of surface concentration to average concentration at short Dt/R2.

gas flow increases, there are more vigorous particle collisions and improved gas/solid contact that tend to break apart the agglomeration, thereby providing better ventilation. When the solids are completely fluidized, each particle has nearly complete access to the gas stream and entrapment pockets with large gas-phase mass-transfer resistance no longer exist, and the intrinsic diffusivity values are reached. Thus, Reff and Deff approach R and D, respectively. The effective diffusivities measured in this study have corroborated data obtained from several commercialscale HDPE dryers in two HDPE plants. In one plant, which uses solution polymerization technology, the residual hexane is removed by drying the pellets. The drying kinetics is correctly predicted based on the intrinsic diffusivity obtained here. In the second plant, in which the slurry technology is used, the residual hexane is removed by drying HDPE powder in a rotary dryer followed by a pneumatic conveying fluidized dryer. For the rotary dryer, which is operated at a low purge gas velocity of about 1-2 cm/s, the dryer efficiency is very low owing to the low effective diffusivity observed in this study (Qi and Krishnan, 1996). On the other hand, the pneumatic conveying dryer operates at a gas velocity greater than 10 m/s and exhibits a high efficiency consistent with the intrinsic diffusivity (Qi, 1996). The current study therefore has provided an excellent foundation for elucidating these dryer characteristics and pointed to particle agglomeration as a major factor undermining the efficiency of the rotary dryer.

Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3429 Table 4. Values of m/QG for Different Diffusivity Values and Particulate Radiia D, cm2/s

K

R, cm

E

m/QF, s

10-7 10-8 10-7

7 7 7

0.006 0.006 0.17

0.005 0.005 0.005

0.026 0.26 21

a

w0 ) starting concentration in solids, mg/cm3 ws ) concentration in a solid particle at its surface, mg/ cm3 w(r,t) ) local concentration in a solid particle at t and radial distance r from the center, mg/cm3 Greek Symbols

K and D are the approximate values for hexane-HDPE system at the temperatures (60-80 °C) under study.

F ) density of solids, g/cm3

Table 5. Cout Estimated at Various Times for w0 ) 1000 mg/cm3 (1050 ppm wt)

Appendix

Dt/R2

K

E

Cout, mg/cm3

Cout,a ppm mol

0.11 0.5 1.0

7 7 7

0.005 0.005 0.005

0.15 0.003 0.000023

37000 781 6

a

Estimation of Surface Concentration. Combining eqs 5 and 8 with Cin ) 0 yields

Cout )

QFR

Conclusion The diffusivity of hexane in HDPE particulates was studied using a method that simulates a drying process and requires no sample alteration. Compared with other methods, this method produced diffusivity data more useful for polymer drying applications. Because the thermal history and morphology of the original sample remained intact during measurements, the measured diffusivity was the actual value for the material in question. The significance of this feature was demonstrated by the differences in intrinsic diffusivities between the HDPE pellets and sheets. Furthermore, this study shows that due to agglomeration, the effective diffusivity displayed by HDPE powder may be lower than the intrinsic value. The problem can be assessed by comparing the effective diffusivities obtained under low gas flows with the intrinsic values at high velocities. Consequently, HDPE drying kinetics at various purge flows can be more precisely predicted. Acknowledgment We thank Dr. M. Markelov of ACS Labs for performing the diffusivity measurements on the polymer sheets.

)

Dt

∑ exp -n2π2 n)1

2

At 0 °C and 1 atm.

(

∞

6mD(w0 - ws)

R2

(A1)

The gas-phase concentration C can be related to the surface concentration of the solids by the partition coefficient K:

ws ) KC

(A2)

Since the hexane concentration in the carrier gas varies from 0 at the cell inlet to Cout at its outlet, the surface concentration of the solids varies as well. The surface concentration at the cell outlet can be estimated from eqs A1 and A2:

ws ) K

6mD(w0 - ws)

(

∞

2

QFR

)

Dt

exp -n2π2 ∑ n)1

R2

(A3)

From eq A3 and eq 3, the following is obtained:

(

∞

ws w - ws

)E

∞

∑ n)1

1 2

n

)

Dt

exp -n2π2 ∑ n)1

(

R2

exp -n2π

)

(A4)

Dt 2 R2

E denotes the following dimensionless coefficient: Nomenclature Bi ) kR/DK, the Biot number C ) concentration of hexane in gas phase, mg/cm3 Cin ) concentration of hexane in carrier gas at cell inlet, mg/cm3 Cout ) concentration of hexane in carrier gas at cell outlet, mg/cm3 D ) intrinsic or intraparticle diffusivity in HDPE, cm2/s Dcluster ) apparent or superficial diffusivity for an agglomerated particle cluster, cm2/s Deff ) effective diffusivity, cm2/s E ) π2KDm/R2QF, dimensionless number K ) partition coefficient, defined as the ratio of concentration in solids to that in gas at equilibrium, (g/cm3)S/(g/ cm3)g k ) mass transfer coefficient, cm/s m ) sample mass loaded in diffusion cell, g Q ) volume flow rate of carrier gas, cm3/s R ) particle radius, cm Rcluster ) radius of an agglomerated particle cluster, cm Reff ) effective radius, cm r ) radial distance from the center of a spherical particle, cm T ) temperature, K t ) diffusion time, s w ) average concentration in solids, mg/cm3

E ) π2K

( )( ) D m R2 QF

(A5)

Equation A4 shows that when t ) 0, ws ) w ) w0. However, when t > 0, the surface concentration ws declines sharply with time. In Figure 7, the ratio of the surface concentration to the average concentration in the solids (ws/w) is plotted against the dimensionless group Dt/R2 using eq A4 for four different values of E. The figure shows that after a certain value of Dt/R2 (depending on the value of E), the ratio decreases to a constant. For E ) 0.005, ws/w is less than 0.02 after Dt/R2 > 0.003 and eventually reaches 0.005 when Dt/ R2 f ∞. This decline is even sharper for lower E values. Here, ws represents the surface concentration of the solids in contact with Cout and therefore is the highest in the cell (note that the solids close to the cell inlet gas have a zero surface concentration). This analysis indicates that for the conditions studied in Figure 7, the surface concentration in the solids becomes less than 1% of w after a very short purging time. Therefore, assuming a zero surface concentration should not introduce significant errors under these conditions. It should be emphasized that for diffusivity measurements, we are merely interested in the diffusion

3430 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996

characteristics after Dt/R2 > 0.11. When Dt/R2 > 0.11, only the first terms in the two summation series in eq A4 are significant. This results in ws/w ) E/(E + 1). Thus, to minimize the error caused by a nonzero surface concentration, E should be reduced to a minimum. Clearly, a large purge gas rate would satisfy the zero surface concentration boundary condition more closely but would result in a small Cout that is more difficult to measure. Clearly, the condition that satisfies ws ) 0 rigorously throughout the diffusion process is when E ) 0 or Q/m f ∞. Since this is not viable, finite Q/m values must be selected to achieve a small yet practical E. In Table 4, the values of m/QF needed to maintain E < 0.005 are calculated. The calculations indicate that for R ) 0.006 cm, m/QF should be less than 0.026. Therefore, for 0.25 g of HDPE powder sample and F ) 0.95 g/cm3, a gas flow rate of at least 10 cm3/s (600 cm3/ min) should be used. But for HDPE pellets with R ) 0.17 cm, 1 cm3/s (60 cm3/min) would suffice. Note that a method to minimize the ws/w even for the initial short time is to use a very large gas flow at the beginning period when Dt/R2 < 0.11. Although this may result in an undetectable Cout during this initial period, it is not a concern since determination of the diffusivity does not rely on the data in this period. Upon Dt/R2 reaching 0.11, the gas flow rate can be reduced and maintained at a lower level to allow for a measurable Cout. Although this method should produce the most accurate result, it is deemed not necessary for this study. Gas Concentration Detection Limit Requirement. Since for E < 0.005, ws , w0, when Dt/R2 > 0.11, eq A1 is simplified to

Cout )

6mDw0 QFR2

(

exp -π2

)

(A6)

)

(A7)

Dt R2

or

Cout )

(

6 Dt Ew0 exp -π2 2 Kπ2 R

Using this equation, Cout for w0 ) 1000 mg/cm3 (1050 ppm wt) was estimated and is shown in Table 5. The calculations indicate that for the GC to still be able to capture data at Dt/R2 ) 1 in this case, its lower detection limit should be below 0.000 023 mg of hexane/cm3 nitrogen (or 0.023 µg/mL). For a w0 that is lower or higher than 1000 mg/cm3, the detection limit required will change proportionally.

Literature Cited Berens, A. R. Diffusion and Relaxation in Glassy Polymer Powders: 1. Fickian Diffusion of Vinyl Chloride in Poly(vinyl chloride). Polymer 1977, 18, 697-704. Berens, A. R. Diffusion of Organic Vapors at Low Concentrations in Glassy PVC, Polystyrene, and PMMA. J. Membrane Sci. 1982, 10, 283-303. Beret, S. E.; Muhle, M. E.; Villamil, I. A. Purging Criteria for LDPE Make Bins. Chem. Eng. Prog. 1977 (Dec), 44-49. Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon: Oxford, 1975. Crank, J.; Park, G. S. Diffusion in Polymers; Academic: London, 1968; pp 4-10. Hufton, J. R.; Ruthven, D. M. Diffusion of Light Alkanes in Silicalite Studied by the Zero Length Column Method. Ind. Eng. Chem. Res. 1993, 32, 2379-2386. Incavo, J. A.; Jain, V.; Qi, J. S.; Krishnan, C. Determination of Partition Coefficient and Residual Solvent for Polymer Drying Applications. Chromatographia 1996, in press. Karger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; John Wiley & Sons: New York, 1992. Kreituss, A.; Frisch, H. L. Free-Volume Estimates in Heterogeneous Polymer Systems. I. Diffusion in Crystalline EthylenePropylene Copolymers. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 889-905. Michaels, A. S.; Bixler, H. J. Flow of Gases Through Polyethylene. J. Polym. Sci. 1961, L, 413-439. Perry, R. H.; Green, D. Perry’s Chemical Engineers’ Handbook, 6th ed.; McGraw-Hill: New York, 1984; pp 16-22. Qi, J. S. Mathematical Modeling of Diffusion-Controlled PneumaticConveying Fluidized Dryers. Chem. Eng. Commun. 1996, in press. Qi, J. S.; Krishnan, C. Mathematical Modeling of Continuous Cross-flow Diffusion-Controlled Dryers. Chem. Eng. Sci. 1996, in press. Rogers, C. E.; Stannett, V.; Szwarc, M. The Sorption, Diffusion, and Permeation of Organic Vapors in Polyethylene. J. Polym. Sci. 1960, XLV, 61-82. Schlotter, N. E.; Furlan, P. Y. A Review of Small Molecule Diffusion in Polyolefins. Polymer 1992, 33, 3323-3342. Srinivasa Kannan, C.; Rao, S. S.; Varma, Y. B. G. A Kinetic Model for Drying of Solids in Batch Fluidized Beds. Ind. Eng. Chem. Res. 1994, 33, 363-370. Srinivasa Kannan, C.; Thomas, P. P.; Varma, Y. B. G. Drying of Solids in Fluidized Beds. Ind. Eng. Chem. Res. 1995, 34, 30683077. Vargaftik, N. B. Handbook of Physical Properties of Liquids and Gases: Pure Substances and Mixtures, 2nd ed.; Hemisphere: Washington, DC, 1975; pp 458, 635. Vergnaud, J. M. Drying of Polymeric and Solid MaterialssModeling and Industrial Applications; Springer-Verlag: London, 1992; p 62. Zogzas, N. P.; Maroulis, Z. B.; Marinos-Kouris, D. Moisture DiffusivitysMethods of Experimental Determination. A Review. Drying Technol. 1994, 12, 483-515.

Received for review February 6, 1996 Accepted May 31, 1996X IE960067O X Abstract published in Advance ACS Abstracts, August 15, 1996.