2654
Ind. Eng. Chem. Res. 2003, 42, 2654-2660
Removal of Residual Monomers from Polymers in Fluidized Beds M. Colakyan* and R. S. Eisinger† The Dow Chemical Company, P.O. Box 8361, South Charleston, West Virginia 25303
We present a model for the removal of residual monomers or “purging” from polymers in fluidized beds. The purging is modeled as a diffusion-controlled process. To track the diffusion of the monomer from the polymer, the diffusion equation is solved by the method of lines coupled with the isothermal mass balance equations for either batch or continuously fed fluidized beds. The model accounts for particle size distribution in the fluidized bed. Bypassing of the gas in the form of bubbles is accounted for by using a simple model derived from the “two-phase” theory of fluidization. The diffusion coefficient of ethylidenenorbornene in granular EPDM (a polymer of ethylene copolymerized with propylene and the diene ethylidenenorbornene) is measured by carrying out separate experiments in a small fluidized-bed apparatus. Henry’s law is used to represent the solubility of the monomer in the polymer. The model was validated with data from a 0.6-m-diameter experimental fluidized bed and is used to simulate a large-scale fluidized bed as an illustrative example. Simulations suggest that the bypassing of purge gas in the form of bubbles may reduce the purging efficiency significantly. 1. Introduction Fluidized beds are commonly used in many drying applications. Fluidized-bed drying literature is relevant to the current work dealing with the removal of dissolved monomers from polymers. Polymers made in gas-phase reactors contain residual monomers that need to be removed before downstream processing.1-3 The devolatilization, which is coined as “purging”, is conventionally done in contactors where an inert gas flows through a bin containing a batch of solids or flows countercurrently to a moving bed of solids. For example, such purging operations include the removal of ethylene from polyethylene.2,4 Zahed et al.5 model batch solids and continuous fluidized dryers by considering the diffusional transport of moisture, interstitial gas-to-particle mass transfer, and interphase transfer resistance between the bubble and emulsion. They solve their model with convective boundary conditions as well as by assuming that the surface moisture of the particles is in equilibrium with their surroundings. The model also allows the surface concentration of the moisture to vary in time. This model is quite comprehensive and divides the bed into two phases: a bubble phase and a dense phase. The total gas flow is then divided into a flow of bubbles and gas flow through the dense, or emulsion, phase. Other workers who divide the fluid bed into two phases include Panda and Rao,6 Lai et al.,7 Palancz,8 and Hoebink and Rietema.9 Most early works on fluidized-bed dryers treat the gas as a homogeneous phase.10 In this paper we attempt to model the purging operation in fluidized-bed vessels. For sticky, amorphous polymers, which are hard to handle, fluidized beds offer processing advantages in that high heat- and masstransfer operations can be achieved and consolidation of the sticky polymer is prevented. * To whom correspondence should be addressed. Tel.: (304) 747-4580. Fax: (304) 747-4760. E-mail:
[email protected]. † Tel: (304) 747-5073. Fax: (304) 747-3928. E-mail: eisingrs@ dow.com.
As described below, to model diffusion operations in fluidized beds, the equations at the particle level need to be solved simultaneously with the heat and material balance equations describing the purging vessel. The batch solids fluidized bed as a purging device further complicates the solution. This difficulty is due mainly to the time-varying nature of the monomer gas concentration and its corresponding equilibrium concentration at the surface bathing the particles and the complication introduced by bubbling phenomena in the fluidized bed. Furthermore, fluidized beds can operate with particles having a wide size distribution. In our model, the surface concentration of the monomer is allowed to change and a particle size distribution is used for the feed and bed inventory. To our knowledge, there is no published model of “purging” operations carried out in fluidized beds, combining gas/solid equilibrium, diffusion, and fluidization phenomena. Here, a rigorous approach is offered for solving problems of this nature in which the solution of diffusion equations is coupled with mass balance equations around the fluidized bed. 2. Results and Discussion 2.1. Diffusion Model. The removal of residual monomers or solvents from polymers can be described by a two-step process involving (a) the transport of monomer or solvent to the surface of the polymer by diffusion and (b) the transfer of monomer or solvent to a gas stream by convection. Depending on operating conditions and the transport and equilibrium properties of the monomers or solvents in the polymer, the process can be either diffusion- or mass-transfer-controlled, or in some instances both resistances may play a role. For the present work, a diffusion-controlled mechanism is used for all subsequent modeling. This decision was based on the following analysis comparing the two resistances for the case at hand. Figure 1 depicts the transport of monomer by diffusion to the surface of a plane sheet of polymer having a thickness ∆X, followed by mass transfer to the surrounding gas through a gas film. For this system the following mass balance equations can be written.
10.1021/ie020825c CCC: $25.00 © 2003 American Chemical Society Published on Web 05/17/2003
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2655
(
)
∂Qmi ∂2Qmi 2 ∂Qmi + ) Dm(T) ∂t r ∂r ∂r2
Figure 1. Diffusion and mass transfer through a polymer particle.
For mass transfer: Nm,molar )
kg (p - pg) Rg T s
(1)
Nm,molar is the molar flux of the monomer, kg is a masstransfer coefficient, pg is the bulk partial pressure of the monomer, and ps is the partial pressure in equilibrium with the surface concentration.
For diffusion: Nm,molar =
DmFe
(Q h i - Q*) 106∆XMw
(2)
The factor 106 is used to convert to a consistent unit basis, ppmw. The driving force for diffusion inside the particle is Qmi - Q*, the difference between the monomer concentration within the particle and that at the particle surface. The term Q* is replaced with the equilibrium relationship employing Henry’s law constant H, Q* ) ps/KH, where the constant K is used for consistency of units. If Q* is expressed in ppmw and ps and H are in atmospheres, then the value of K is 10-6. Note that for a sphere of particle diameter dp, ∆X is on the order of the particle radius, ∆X ≈ dp/2. After rearrangement, the flux of the residual monomer becomes
Q hi Nm,molar )
6
-
pg H
10 RgT (dp/2)Mw + kg H DmFe
(3)
The values of the two terms of the denominator determine the relative resistance of the film and diffusion in the polymer. With the following typical values of the parameters kg11 ) 0.6 m/s, dp ) 500 × 10-6 m, Dm ) 4 × 10-11 m2/s, Mw ) 120, H ) 0.43 atm, Fe ) 900 kg/m3, T ) 343 K, and Rg ) 0.082 05 m3‚atm/K‚kmol, these two terms become
RgT = 110 kgH
and
(dp/2)Mw = 830 000 m2‚s/kmol DmFe (4)
Because the second term is so much larger than the first, the dominating resistance is the intraparticle diffusion, and the removal of residual monomers can be simply described by a diffusion process. The diffusion of a monomer out of a spherical polymer particle of diameter 2Ri is governed by the following differential equation:
(5)
where Dm is the diffusion coefficient of the monomer diffusing in the polymer matrix. Qmi is the monomer concentration (unit weight of monomer/unit weight of polymer) in the polymer of size cut i at time t and at radial position r, measured from the center of the particle. In this work Qmi is expressed in ppmw for convenience. The diffusion coefficient, Dm, is usually temperature-dependent, and its temperature dependence can be represented by an Arrhenius-type expression:
Dm ) Dm0 exp(-Ed/RgT)
(6)
Rg in the above expression is the gas constant expressed in kJ/kmol‚K. The diffusion coefficient used in the model was derived by fitting experimental data obtained from a shallow fluidized bed to the model described below. Details of this procedure can be found in the Appendix. The boundary conditions representing the diffusion process are
t ) 0 and any r
Qmi ) Q0i
∂Qmi )0 ∂r
r)0 r ) Ri
Qmi ) Q*(t)
where Q0i represents the initial monomer concentration and Q* is the concentration at the surface of the particle in equilibrium with the concentration of the monomer in the gas bathing the particle. For batch solids operations, because the amount of monomer in polymer continuously decreases, Q* becomes a function of time, Q*(t). For continuous operations where a steady state is established, Q* is constant. In other words, for continuous operations, the surface of the particles sees a time-invariant gas concentration, whereas for batch operations, it sees a continually decreasing Q*. As seen below, the constant Q* assumption simplifies the solution of the partial differential equation considerably. Given the monomer partial pressure in the gas phase, one can use Henry’s law to estimate Q*. The average monomer concentration Q h i for a particle of size i is given by integrating Qmi over the particle radius.
Q hi )
∫0R
4πr2Qmi dr
i
(4/3)πRi3
(7)
The average monomer concentration for all sizes can be expressed by Ncuts
Q C )
∫0 ∑ i)1
Ri
4πr2Qmi dr wi (4/3)πRi3
(8)
where wi is the weight fraction for each size cut i. Each size cut corresponds to an average particle radius, usually calculated as the arithmetic or geometric average of two adjacent sieve openings in the particle size measurement analysis. For the diffusion model, we
2656
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003
Table 1. Weight and Size Distribution of Particles Used in the Illustrative Calculations
dt
screen size, U.S. mesh
screen opening, m × 106
average dp, m × 106
Ri , m
wt %
6 10 18 35 60 120 -120
3360 2000 1000 500 250 125 0
3500 2680 1500 750 375 187.5 62.5
0.001 75 0.001 34 0.000 75 0.000 38 0.000 19 0.000 09 0.000 03
1 4 20 50 20 4 1
chose to use seven cuts (Ncuts ) 7) to represent the particle size distribution. The average size for particles of radius i was obtained by the arithmetic average of two adjacent screen openings. This value was used to estimate properties such as the minimum fluidization velocity.12 The size distribution shown in Table 1 was used to carry out the simulations. For constant temperature and surface concentration, the above partial differential equation has a well-known analytical solution.11 For a particle of size i, the solution is given by
Q h i - Q* Q0i - Q*
)
6 π
∞
∑ 2 n)1
1 n2
[
d(WQout)
exp -(nπ)2
]
Dmt Ri2
(9)
2.2. Fluidized Bed Model. In this section, a model for batch solids and continuous fluidized-bed systems is described. The main assumptions and methods used in the model can be summarized as follows: (i) The particles are Geldart Class B,13 and the fluidized bed operates in the bubbling bed regime. (ii) The fluidized bed and the particles are isothermal. Even though a heat balance is not included, the model allows an arbitrarily prescribed temperature profile. This feature is necessary for comparing some of the experimental batch purging data against the model. (iii) The particles are assumed to be spherical, and the model allows for a particle size distribution. (iv) A single average bubble diameter corresponding to the bubble diameter at the top of the bed is used for the entire bed. This assumption provides a conservative approach for design purposes. The initial bubble size for the simulations is assumed to be 15 cm for the large bed of the example and is estimated to be 5 cm for the 0.6-m-diameter experimental unit. (v) We estimate umf according to the correlation of Wen and Yu.12 (vi) The fluidized bed can operate as either a continuous or batch solids fluidized bed. (vii) Henry’s law adequately describes the equilibrium of the monomer in the gas phase and the surface of the particle. (viii) Solids are uniformly mixed. (ix) The gas entering the fluidized bed can contain some monomer. The governing mass balance equations around the fluidized bed can be simply written as
accumulation of mass ) input - output d(WQout) ) FinQin - FoutQout + u0At(Cin - Cout) dt
(10)
)
d dt
(
∑ i)1
W iQ h i) )
∑ i)1
FinQin )
Wi
dQ hi
+
dt
∑ i)1
Q hi
dWi dt
(11)
∑i FiQh i,in
FoutQout )
∑i FiQh i
with i ) 1, 2, ..., number of size cuts Ncuts or i ) 1, 2, ..., 7. The fluidized polymer particles used in this study, under atmospheric conditions, behave like Geldart Class B type particles,13 and with u0 > umf, the fluidized particles do not see all of the available gas for purging because a part of the gas travels through the bed in the form of bubbles. The particles generally contact the emulsion gas, and very little or no particles are contained in the bubble phase.14 During batch or continuous operations, residual monomers are stripped from the polymer particles by undergoing a diffusion process as described above. Because the particles mainly see the emulsion gas, the monomer concentration in the emulsion gas determines the driving force for the transfer of monomer to the gas. Therefore, if the concentration of the monomer in the emulsion gas can be estimated, the governing differential equations can be readily solved and the residual monomer in the polymer can be determined. To this end, in this work, the monomer concentration in the emulsion gas was estimated by using the “twophase” theory of fluidization11,14 by first calculating a bubble size and then the gas interchange between the bubble and the emulsion. This model assumes that the emulsion gas is well mixed and that the excess gas travels in plug flow in the form of bubbles. Once the amount of gas traveling as bubbles is estimated, the emulsion concentration of the monomer can readily be obtained from a mass balance. To find the volume of the gas traveling as bubbles, first the maximum stable diameter of the bubbles is estimated using the correlation of Mori and Wen15 0.4 π dbm ) 0.65 dt2(u0 - umf) 4
[
]
(12)
where dt is the bed diameter. The bubble size is then assumed to vary from the distributor to the top of the bed according to
dbm - db ) e-0.3z/dt dbm - db0
(13)
where db0 is the bubble size at the distributor plate and z is the height in the bed. This equation suggests that, for very tall beds, the bubble size reaches the limiting value of db ) dbm. Other correlations predict a comparable bubble diameter.11 Figure 2 displays how the bubble diameter varies in a 3-m-diameter fluidized bed as a function of the height. For illustrative purposes, the initial bubble diameter is chosen to be 10 cm. The initial bubble diameter depends on the orifice size of the distributor plate. In this work, for all calculations, the value of the bubble diameter corresponding to the top of the bed is chosen; in other words, z is set to the height of the fluidized bed, Lf. For design purposes, this value gives
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2657
Cbout ) Ce + (Cin - Ce)e-Γ
(19)
The emulsion concentration of the monomer can be readily obtained by combining eqs 17 and 18 and by solving for Ce:
Ce )
Figure 2. Bubble diameter as a function of the height for a 10cm initial bubble diameter.
a conservative estimate of the residual monomer concentration because it represents the maximum bubble size attained in the bed as a function of the bed height. A more rigorous approach would have been to vary the bubble diameter with the height. The amount of monomer exchanged between the emulsion and the bubble phases was estimated using interchange coefficients between the bubble and the cloud and between the cloud and the emulsion. The cloud is the denser phase immediately surrounding the bubble. The reader is referred to refs 11 and 14 for further details. From the physical standpoint, the interchange coefficient Kbe can be looked upon as a flow of gas from the bubble to the emulsion, with an equal flow in the opposite direction. The two other interchange coefficients, Kbc and Kce, have similar meanings. These coefficients are also called “cross-flow rates”. The interchange coefficients Kbe, Kce, and Kbc were calculated using the following:11
Interchange from bubble to cloud:
() (
)
DM-N20.5g0.25 umf Kbc ) 4.5 + 5.85 db d 5/4 b
(14)
Interchange from cloud to emulsion: Kce = 6.77
(
)
DM-N2mfubr db3
0.5
(15)
The overall interchange coefficient is then defined as
1 1 1 ) + Kbe Kbc Kce
(16)
The gas interchange between the bubble and the rest of the bed may also be expressed as a dimensionless cross-flow ratio, Γ, defined as
γ)
Kbeh ub
(17)
where Γ is the number of times the bubble gas is replaced as the bubble passes through the bed. Assuming that the bubble gas superficial velocity is u0 - umf, the overall mass balance becomes15
u0Cout ) umfCe + (u0 - umf)Cbout
(18)
The concentration of the monomer in the bubbles exiting the fluidized bed can be written as
-(Cinu0 - CouteΓu0 - Cinumf) -u0 + eΓu0 + umf
(20)
Once Ce is calculated, C* can readily be calculated using Henry’s law. To calculate the residual monomer concentration in the polymer, the following iterative procedure was used: (1) Guess the exit gas monomer concentration, Cout. (2) Using Cout, Cin, and eq 20, calculate the concentration of the monomer Ce bathing the surface of the particles. Calculate Q* by using Henry’s law. (3) Take a small time step to solve the partial differential equation system described above by using the method of lines.17 Obtain Qmi(r,t,Q*,T) for each cut at time t and calculate Qout by numerically integrating 7
Qout )
∫0 ∑ i)1
Ri
4πr2Qmi dr wi (4/3)πRi3
(21)
(4) Calculate d(WQout)/dt numerically. (5) Check to see if the mass balance, equation 10, is satisfied. (6) Recalculate Cout. Use a Newton-Raphson procedure to accelerate convergence. Go to step 1. Repeat the procedure until the mass balance in step 5 is satisfied. (7) When it converges, save monomer concentration profiles for each cut for the next time step. (8) Increment time step and restart. The same solution can apply for semibatch, batch, or continuous fluidized beds. For example: (a) If Fin ) finite and Fout ) 0 or Fin ) 0, Fout ) finite, and W ) variable, it is a semibatch solution. (b) If Fin ) Fout ) 0 and W ) constant, it is a straight batch solution. (c) If Fin ) Fout and no accumulation of monomer, this is the case for a continuous fluidized bed. For this instance, to obtain the monomer concentration leaving the bed, one can use the analytical solution11 shown in eq 9 with
t ) mean residence time ) weight of the polymer in the fluidized bed (22) Fin and account for the residence time distribution (RTD) as discussed below. The above equation must be summed over all particle sizes to obtain the average monomer concentration in the polymer. Figure 3 displays the solution of a case study using the above outlined model and numerical procedure. This figure shows the residual monomer profile and the effect of gas “bypassing” on the purging efficiency in a 3-mdiameter batch fluidized bed. For illustration purposes, two different bubble sizes were used: (a) an average bubble size of 40 cm and (b) an average bubble size of 80 cm. As stated above, bubbles were assumed to have the same size throughout the bed. The same figure also displays the case of very
2658
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 Table 2. Size Distribution of the 0.6-m-Diameter Fluidized-Bed Material screen size, U.S. mesh
mean radius, m
composition of the bed material, wt %
6 10 18 35 60 120 -120
0.001 75 0.001 34 0.000 75 0.000 38 0.000 19 0.000 09 0.000 03
1.50 0.20 4.40 87.61 4.50 1.40 0.39
Figure 3. Effect of gas bypassing in the form of bubbles on the purging efficiency.
small bubbles, or a well-mixed gas. The superficial velocity, u0, was kept the same for all three cases. As seen from this figure, when a part of the purging gas bypasses the bed as bubbles, the purging efficiency is lowered. For the case studied in this example, the exit monomer concentrations in the polymer are approximately 218 and 126 ppmw for 80 and 40 cm bubbles, respectively, whereas the exit monomer content of the “mixed” case (no gas bypasses in the form of bubbles) is 43 ppmw. 2.3. Accounting for the RTD in a Fluidized Bed. In a batch solids fluidized bed, the time to reach a given monomer concentration in the polymer is the same for all particles having a given particle size. In continuous operation, the purging residence time of individual particles may vary. A particle may travel quickly from the entrance to the discharge and leave the fluidized bed incompletely purged. Particles may also spend a period of time much longer than necessary for purging. Thus, for estimating the average monomer concentration of the polymer properly, the exit age distribution or the RTD of the particles needs to be considered. Treating the fluidized-bed contents as a macrofluid, we obtain the average monomer concentration in the polymer in the following manner:
(
Q h i - Q* 1 ) Q0i - Q* τ
Q - Q*
)
∫0∞ Q0ii - Q* e-t/τ dt
(23)
where t is the average residence time defined as the weight of the bed/flow rate in ) weight of the bed/flow rate out. Integrating and rearranging, we get
Q h i - Q* Q0i - Q*
)
2
πτ
1
∑ n)1
2
π Dmn
(24)
4
+
n2 τ
or ∞
6 2
(Q0i - Q*)
πτ
that all particles reside in the fluidized bed and are purged at the mean residence time (eq 22), whereas eq 25 accounts for an age distribution. 2.4. Comparison of the Model with a 0.6-mDiameter Experimental Unit. Batch solids purging experiments were carried out using EPDM containing ethylidenenorbornene and a 0.6-m-diameter fluidized bed and by using nitrogen as the purging gas. The fluidized bed contained about 110 kg of polymer, and the average starting monomer concentration was about 800 ppmw. Table 2 summarizes the particle size distribution for the bed material. Figure 4 shows the simulated monomer concentration profile and the measured values of the monomer concentration in the polymer. For this test, the bed was heated to 70 °C. The solids temperature is also displayed in the figure. The monomer concentration in the polymer levels out at 30 ppmw. This asymptote is due to the trace amount of ethylidenenorbornene contained in the incoming gas. The initial bubble diameter was assumed to be 5 cm. As seen from this figure, the model does an adequate job of representing the physical situation. 3. Conclusion
∞
6
Ri2
Q hi )
Figure 4. Model comparison with the data of a 0.6-m-diameter fluidized bed.
∑ n)1
1
π Dmn4
+ Q*
2
Ri2
+
(25)
n2 τ
Thus, for continuous fluidized beds, eq 25 must be used instead of eq 9. The use of eq 9 instead of eq 25 means
A model for the removal of residual monomers or “purging” from polymers in fluidized beds was developed. The purging was modeled as a diffusion-controlled process. To track the diffusion of the monomer from the polymer, the diffusion equation was solved by the method of lines coupled with the isothermal mass balance equations for either batch or continuous fluidized beds. Bypassing of the gas in the form of bubbles was accounted for by using a simple model derived from the “two-phase” theory of fluidization. This model assumes that the emulsion gas is well mixed and that the excess gas travels in plug flow in the form of bubbles. The diffusion of ethylidenenorbornene from EPDM was used as a vehicle for simulations. The diffusion coefficient was measured by carrying out separate
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2659
The diffusion coefficient was assumed to have an Arrhenius form, and the constants thus obtained were represented by
Dm ) 3 × 10-11e-3258(1/T-1/333) m2/s
Figure 5. Normalized ethylidenenorbornene concentration for the diffusion experiments.
experiments in a small fluidized-bed apparatus. Henry’s law was used to represent the solubility of the monomer in the polymer. The model was compared to the data obtained from a 0.6-m-diameter fluidized bed. Simulations done with the model indicate that the bypassing of purging gas as bubbles may reduce the purging efficiency considerably. For example, the simulation of a 3-m-diameter fluidized bed suggests that the exit monomer concentrations are approximately 218 and 126 ppmw for 80 and 40 cm bubbles, respectively. The exit monomer content of the “mixed” case (no gas bypasses as bubbles) is 43 ppmw, however. Acknowledgment The authors acknowledge DuPont-Dow Elastomers R&D for their support and valuable suggestions. Appendix Measurement of Diffusion Coefficients Used in the Diffusion Model. The intraparticle diffusion coefficient was measured in a small fluidized bed where resistances to devolatilization other than through intraparticle diffusion were negligible. The purge cell consisted of an 11-cm-diameter shallow fluidized bed with a porous, high-pressure-drop distribution plate at its bottom. The bed height was approximately set at 7 cm. The particles were vigorously fluidized well above umf by using nitrogen as the purge gas. The temperatures both of the fluidizing gas and of the bed walls were controlled. Prior to each experiment, polymer was spiked with monomer, heated to the test temperature, and then charged to the heated bed. Samples of the solids were taken from the fluidized bed and analyzed for monomer content by using headspace gas-chromatographic analysis. Each experiment lasted about 1 h. Because for these measurements the concentration of the monomer in the gas was very low, eq 9 could be used to fit the data with small error. Granular EPDM with a narrow particle size distribution having a mean particle diameter of 0.5 mm was used. Experiments were carried out at three temperatures to determine the temperature dependence of the diffusion coefficient. The data are displayed in Figure 5. The data fitting was done by minimizing no. of datapoints
∑1
(Q h obsd - Q h calcd)2
(26)
(27)
Measurement of Henry’s Law Coefficients Used in the Diffusion Model. Henry’s law constants, which are the partitioning values at equilibrium of the absorbate between the polymer and vapor phases, were calculated from measurements using a static headspace gas-chromatographic technique. The headspace technique is similar to that described by Robbins et al.18 and by Ramachandran et al.19 It consists of adding the same weight of absorbate to vials containing differing amounts of the polymer. Calculation of H from the measured peak area differs from that of the previous references because of differences in the operation of the gas chromatographs. Henry’s law constants were measured at two different temperatures, namely, at 60 and 70 °C, and were found to be 0.25 and 0.43 atm, respectively. To find values at intermediate temperatures, values are interpolated using an exponential function. Nomenclature At ) cross-sectional area of the fluidized bed, m2 Cout ) exit monomer concentration in the gas, kg/m3 Cbout ) exit monomer concentration in the bubble, kg/m3 Cin ) inlet monomer concentration, kg/m3 Ce ) emulsion monomer concentration, kg/m3 Dm ) diffusion coefficient of the monomer in the polymer particle, m2/s DM-N2 ) monomer-N2 diffusion pair, m2/s db ) effective bubble diameter, m dbm ) maximum stable bubble diameter, m dp ) particle diameter, m dt ) diameter of the fluidized bed, m Ed ) activation energy of diffusion, kJ Fout ) outlet flow rate of the polymer from the fluidized bed, kg/s Fin ) Inlet flow rate of the polymer to the fluidized bed, kg/s g ) gravitational constant, m/s2 H ) Henry’s law constant, atm K ) constant used for consistency of units, ppmw-1 Kce ) coefficient of interchange between the cloud-wake region and the emulsion phase, m/s Kbc ) coefficient of interchange between the bubble and the cloud-wake region, m/s Kbe ) overall coefficient of gas interchange between the bubble and emulsion phases, m/s kg ) mass-transfer coefficient, m/s Lf ) height of the fluidized bed, m Mw ) molecular weight, kg/kmol Nm,molar ) molar flux of the monomer, kmol/m2‚s Ncuts ) number of cuts (sieves) used in the particle size measurement pg ) partial pressure of the monomer in the gas, atm ps ) partial pressure of the monomer in equilibrium with the surface concentration, atm Qmi ) concentration of monomer in the polymer particle of size i at a radial point, ppmw Q0i ) initial average concentration of the monomer in the polymer of size i, ppmw Q*, Q*(t) ) surface concentration of the monomer in equilibrium with the gas, ppmw Q h i ) average concentration of the monomer in polymer particle of size i, ppmw
2660
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003
Q h i,in ) average concentration of the monomer in the polymer of size i at the inlet of the fluidized bed, ppmw Q C ) concentration of the monomer in the polymer averaged over the radial distance and all weight fractions, particle sizes, ppmw Qin ) average monomer content (all sizes) of the polymer at the inlet of the fluidized bed, ppmw Qout ) average outlet monomer content (all sizes) of the polymer from the fluidized bed, ppmw Rg) gas constant as specified in the text, either in m3‚atm/ kmol‚K or kJ/kmol‚K Ri ) particle radius, m r ) radial distance, m T ) temperature, K ub ) velocity of a bubble rising through a bed, m/s ubr ) rise velocity of a bubble with respect to the emulsion phase, m/s umf ) minimum fluidization velocity, m/s u0 ) superficial gas velocity, m/s W ) total weight of fluidized particles, kg Wi ) weight of polymer for each size fraction i in the fluidized bed, kg wi ) weight fraction of particles of size i z ) distance away from the bottom of the fluidized bed of solids, m Greek Letters Γ ) cross-flow ratio ∆X ) thickness of a flat polymer particle, m mf ) bed voidage at minimum fluidization conditions Fe ) effective density of a polymer particle, kg/m3
Literature Cited (1) Burdett, I. D.; Eisinger, R. S.; Cai, P.; Lee, K. H. Gas-Phase Fluidization Technology for Production of Polyolefins. In Fluidization X; Kwauk, M., Li, J., Yang, W. C., Eds.; United Engineering Foundation: New York, 2001. (2) Beret, S. E.; Muhle, M. E.; Villamil, I. A. Purging Criteria for LDPE Make Bins. Chem. Eng. Prog. 1977, 23, Dec, 44. (3) Burdett, I. D. A Continuing Success: The UNIPOL Process. CHEMTECH 1992, 22, 616.
(4) Bobst, R. W.; Garner, B. J.; Frederick, F. W. Degassing For Removing Unpolymerized Monomers From Olefin Polymers. U.S. Patent 4,372,758, 1983. (5) Zahed, A. H.; Zhu, J. X.; Grace, J. R. Simulation of Batch and Continuous Fluidized Bed Dryers. Drying Technol. 1995, 13, 1. (6) Panda, R. C.; Rao, S. R. Dynamic Model of a Fluidization Bed Dryer. Drying Technol. 1993, 11, 589. (7) Lai, F. S.; Chen, Y.; Fan, L. T. Modeling and Simulation of a Continuous Fluidized-Bed Dryer. Chem. Eng. Sci. 1996, 41, 2419. (8) Palancz, B. A Mathematical Model for Continuous Fluidized Bed Drying. Chem. Eng. Sci. 1983, 38, 1045. (9) Hoebink, J. H. B. J.; Rietma, K. Drying Granular Solids in Fluidized BedsDescription on the Basis of Mass and Heat Transfer Coefficients. Chem. Eng. Sci. 1980, 35, 2135. (10) Kannan, C. S.; Thomas, P. P.; Varma, Y. B. G. Drying of Solids in Fluidized Beds. Ind. Eng. Chem. Res. 1995, 34, 3068. (11) Kunii, D.; Levenspiel, O. Fluidization Engineering; Butterworth-Heinemann: Boston, 1991. (12) Wen, C. Y.; Yu, Y. H. A Generalized Method of Predicting the Minimum Fluidization Velocity. AIChE J. 1966, 12, 610. (13) Geldart, D. Types of Fluidization. Powder Technol. 1973, 7, 285. (14) Clift, R.; Grace, J. R. Continuous Bubbling and Slugging. In Fluidization; Davidson, J. F., Clift, R., Harrison, D., Eds.; Academic Press: London, 1985. (15) Mori, S.; Wen, C. Y. Estimation of Bubble Diameter in Gaseous Fluidized Beds. AIChE J. 1975, 21, 109. (16) Lapidus, L.; Amundson, N. R. Chemical Reactor Theory: A Review; Prentice Hall: Englewood Cliffs, NJ, 1977. (17) International Mathematical and Statistical Libraries; IMSL: Houston, 1989. (18) Robbins, G. A.; Wang, S.; Stuart, J. D. Using the Static Headspace Method to Determine Henry’s Law Constants. Anal. Chem. 1993, 65, 3113. (19) Ramachandran, B. R.; Allen, J. M.; Halpern, A. M. The Importance of Weighted Regression Analysis in the Determination of Henry’s Law Constants by Static Headspace Gas Chromatography. Anal. Chem. 1996, 68, 281.
Received for review October 23, 2002 Revised manuscript received February 13, 2003 Accepted February 17, 2003 IE020825C
