2850
Ind. Eng. Chem. Res. 1998, 37, 2850-2863
Purification of Polymers from Solvents by Steam or Gas Stripping G. P. Quadri† Tecnimont, Milan 20124, Italy
A mathematical model is developed of the steam-stripping process in polymer-solvent-water systems under agitation, describing it in terms of the current theories in the field of polymersolvent solutions, interphase equilibrium, mass transfer, and agitation. The validity of the model is verified with the results of previous experimental work on the steam stripping of ethylenepropylene-diene monomer rubber slurries in water and is extended to other polymer-solventstripping vapor or gas systems, in or not in the presence of water, where spheroidal polymer particles are under homogeneous mixing conditions. From a laboratory-scale experimental investigation of the purification from monomer and solvent residues, by batch steam stripping, of ethylene-propylene-diene monomer (EPDM) rubber slurries in water, Galli and co-workers12 found that the relevant stripping kinetics can be expressed in the form C/C0 ) exp(-at). By correlating the experimental data, they also found empirical equations expressing the functional dependence of the kinetic exponent a from the two main operating parameters, the polymer particle size and the specific steam flow rate, and proposed a rule to extrapolate the value of a found for a given monomer or solvent to different monomers or solvents, giving an interpretation of the kinetic mechanisms of the process. Matthews et al.14,15 also investigated the steam stripping of EPDM slurries in water and presented a mathematical model of the process for the case where the diffusion of the solvent within the polymer particles is playing the dominant role and the mass transfer from the polymer particle surface may be neglected. In addition, they investigated the effect of the internal structure of the rubber particles on the diffusion kinetics and did observations on the phenomenon of nucleation, that is, the formation of vapor nuclei at the surface of the particles. The EPDM particles considered by Galli were produced in the reaction stage of a suspension process in liquid momomers, whereas those considered by Matthews were originated by precipitation by steam stripping from a polymer solution in hexane: this difference of origin may have caused a significant diversity of the porous structure of the particles, presumably more compact and regular in the first case. With the support of the works mentioned above and particularly of the experimental data of Galli, the present study has been carried out to derive the kinetics of the process from a theoretical approach, taking into account both the influence on the stripping kinetics of the solvent diffusion inside the polymer particles, the mass transfer at the interfaces, and the nucleation phenomenon. This has made possible an analytical understanding of the kinetic mechanisms involved in the stripping process and the extension of the results to a larger field of conditions and applications than that which the abovementioned investigations were aimed at. With such extensions the results of the study can be applied to the † Address for correspondence: via Maroncelli 10, 20052 Monza, Milan, Italy.
Figure 1. Stripper.
finishing stage of many polymer production processes of industrial interest, in rubbers and plastics field, carried out by steam stripping in an aqueous slurry or by desorption with steam or inert gas in fluid or agitated beds in the absence of a liquid phase, either continuously or in batch. The extensions can be considered particular cases of the general treatment developed in the following pages and are shortly dealt with in a final paragraph. Process Description From the macroscopic point-of-view, the process is represented in Figure 1. A stirred vessel (stripper), containing a suspension in water of solvent-contaminated polymer particles, is fed by steam from the bottom. The steam, uniformly dispersed in bubbles, goes up through the suspension, stripping the solvent from the polymer particles, and exits from the top of the vessel along with the removed solvent. The operation can be carried out either continuously or in batch. From a microscopic point-of-view the stripping is a mass-transfer process, which can be represented as in Figure 2, and be divided into the following elementary steps: Step 1. Solvent diffusion from the inside to the surface of the polymer particle (from 0 to 1). Step 2. Mass transfer of the solvent through the liquid limit layer LL1 surrounding the polymer particle (from 1 to 2). Step 3. Transfer of the solvent through the bulk liquid mass by forced convection induced by agitation (from 2 to 3).
S0888-5885(97)00364-3 CCC: $15.00 © 1998 American Chemical Society Published on Web 06/09/1998
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2851
Figure 2. Main mass-transfer mechanism.
neglected in the following mathematical treatment. The solvent concentration in the nuclei is indeed of some order of magnitude lower than that of the water, whereas the water mass-transfer rate from the stripping vapor to the nuclei, equal to the water flow rate conveyed by the nuclei detaching from the particles, is of the same order of magnitude as the mass-transfer rate of the solvent through LL1, LL2, and LL3, because the overall driving force for the water and the solvent mass transfer is identical and the relevant overall masstransfer coefficient of the same order of magnitude. Batch Stripping In the batch stripping, polymer and water are loaded into the vessel together at the start-up. The zero time for the stripping process corresponds to the beginning of the steam feeding, which goes on continuously at a constant flow rate until the solvent content in the polymer has been reduced to the desired level. If the polymer particles can be assimilated to spherical beads of identical radius r, the diffusion of the solvent from the inside to the surface of the particles can be described by the following equations. Infrapolymer Diffusion Equations.
∂c 1 ∂ ∂c ) Dz2 ∂t z2 ∂z ∂z
(
Figure 3. Nuclei mass-transfer mechanism.
Step 4. Mass transfer of the solvent through the liquid limit layer LL2 surrounding the vapor bubble (from 3 to 4). Step 5. Mass transfer of the solvent through the vapor limit layer LL3 forming the surface of the vapor bubble (from 4 to 5). If vapor nuclei form at the surface of the polymer particles, according to the observations of Matthews et al.,14 the previous mass transfer steps are sided by the following step, occurring when the nuclei come into direct contact with the stripping vapor bubbles: Step 6. Mass transfer of the solvent through the vapor limit layer, separating the polymer particles from the core of the bubbles on the nuclei area (from 1 to 5). The nuclei covering a fraction of the polymer particle surface are supposed to contact the stripping vapor bubbles (Figure 3) during a fraction of the stripping time, the “short-circuit” period, in which the masstransfer rate through the nuclei increases by many orders of magnitude: then indeed the mass transfer resistances of steps 1-5, globally of many orders of magnitude higher than that of step 6, are bypassed. During the remaining time, after a transient period, the mass transfer occurs through the resistances in a series of steps 1-5, the driving force having its ultimate origin in the stripping vapor. A solvent concentration gradient in the direction LL1, LL2, and LL3 is so established and consequently a corresponding and complementary water concentration gradient in the opposite direction LL3, LL2, and LL1, causing water migration from the stripping vapor bubbles to the nuclei and therefore the progressive swelling and periodical detachment of the nuclei from the particles surface, conveying the relevant solvent content. This represents a secondary mechanism of removal of the solvent from the polymer particle, during the nonshort-circuit period. It is, however, of minor entity with respect to the principal mechanism of removal by direct mass transfer of the solvent through the layers LL1, LL2, and LL3 and will therefore be
D
∂c 1 dqs ) ∂z sp dt
)
for z ) r
(1)
(2)
It is here to be noted that D corresponds to the diffusion coefficient of the polymeric material only if the polymer particles are compact. If they have a porous structure, it is to be understood as an empirical diffusion coefficient taking into account also the porosity. It must, moreover, be considered that D is depending on c: an expression of the type D(c) ) D0(1 + mc), with D0 and m constant, as suggested by Matthews et al.14 for the system EPDM rubber-hexane, can be used with sufficient approximation. If the polymer particles are supposed to be identical and to simultaneously undergo the same stripping process, at each instant it is
1 dqs 1 dQs ) sp dt Sp dt
(3)
Limit Layer Mass-Transfer Equations. The mass transfer through the limit layers LL1, LL2, and LL3, according to Fick’s law, can be expressed by
-
dQs ) (1 - n)Spkl(cLs,1 - cLs,2) + (Ns)n dt
(4)
Gs ) (Sv - nSp)k′l(cLs,3 - cLs,4) + (Ns)n
(5)
Gs ) (Sv - nSp)kv(ps,4 - ps,5) + (Ns)n
(6)
where n ) ST; S is the fraction of the polymer particle surface covered by nuclei; T is the fraction of time during which the nuclei are in direct contact (shortcircuit) with the stripping vapor bubbles, taking into due account also the transient period; Sp is intended to include the possible extension of the layer LL1 by the nuclei; (Ns)n is the flow rate of solvent removed from the polymer particle through its interface with the nuclei (step 6) given by
2852 Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998
(Ns)n ) nSpkv(ps,1 - ps,5)
Complementary to these equations are the condition of composition homogeneity, which the agitation is supposed to ensure:
cLs,2 ) cLs,3 ) cLs
(
(7)
(8)
Fv ) Gs + Gw ) F0st + 1 -
(9)
Due to the low solubility of the solvent in water, the accumulation term in eq 9 (first term after -dQs/dt) is of some order of magnitude lower than -dQs/dt (Supporting Information), so that we can write
-dQs/dt = Gs
(9.a)
Supposing, as usual in the mass-transfer theory, that equilibrium conditions exist at the interface between the different phases, solid-liquid or solid-vapor (point 1) and liquid-vapor (point 4), and that the solvent has a very low solubility in water, say less than 1% by weight, as it is usually true for nonpolar hydrocarbons, the following equations can also be written. Interphase Equilibrium Equations.
cLs,1 ps,j )
) Λc1
γsfs0xs,j
=
(10)
γsfs0
L cs,j Fw
(11)
At the operating conditions we are considering, the fugacity f 0s can be usually approximated with the pure solvent vapor pressure p0s , as this is relatively low and far from the critical point. The partial pressure of the solvent inside the vapor bubbles, ps,5, due to the composition homogeneity supposed to be ensured by the agitation, is equal to the partial pressure of the solvent in the vapor stream coming out of the top of the stripper:
ps,5 )
Gs Pt Pt ) Gs Gs + Gw Fv
(
F0st ) Fst 1 +
+
(13.a)
)
γs = 1/(xs)sol
(14)
The partition coefficient Λ, if not known experimentally, can be calculated from polymer solutions theories, as exemplified in ref 2, using eq 14 for the activity coefficient of the solvent in the water phase, and a suitable expression for the fugacity of the solvent in the polymer phase.3 For amorphous polymers and nonpolar solvents, when the Flory-Huggins expression for solvent fugacity can reliably be used down to low solvent concentrations, we can write
f Ps,1 ) f 0s Φs,1 exp(Φp,1 + χΦp,12)
(15)
Φs,1 ) c1/Fs
(15.a)
Φp,1 ) 1 - Φs,1
(15.b)
f Ls,1 ) f 0s γsxs,1 cLs,1 xs,1 = Fw
+ ∆Hloss
from which the total flow rate of the mixed vapor leaving the stripper can be derived:
(16) (16.a)
the last approximation being justified by the very low solubility of the solvent in water. The condition of equilibrium between the phases at the interface 1 requires that
f Ps,1 ) f Ls,1
(17)
Combining eqs 14-17 we finally obtain
Λ) Gs∆hvap s
∆Hag - ∆Hloss ∆h* + vap ∆hw ∆hvap w
We also have
+ ∆h*) + ∆Hag ) Gw∆hvap w
(13)
being F0st the equivalent internal steam flow rate of the stripper. As Gs, in the stripping processes under consideration, is of some order of magnitude smaller than F0st and the term between parentheses in eq 13 is usually very small, the stripping vapor flow rate Fv can be approximated with F0st, constant, over all of the stripping process. Polymer-Solvent-Water Interaction Parameters. These are the activity coefficient of the solvent in the aqueous phase, γs, the partition coefficient of the solvent between polymer and water, Λ, and the polymersolvent interaction parameter χ. For solvents with very low solubility in water, (xs)sol, the activity coefficient of the solvent in the aqueous phase, γs, can be approximated by1
(12)
If the agitation is not sufficient to ensure the composition homogeneity of the stripping vapor an average value can be used for ps,5 obtained multiplying the above given value by a correcting factor between 0.5 and 1 depending on the agitation and to be determined experimentally, the lower figure corresponding to no bubbles back-mixing (average between 0, for the steam fed to the stripper, and 1 for the vapor leaving the stripper) and the higher corresponding to full bubbles back-mixing. However the value given in eq 12, being conservative, can be safely used for design purposes. Energy Balance. The energy balance around the stripper is expressed by
Fst(∆hvap w
Gs
∆hvap w
where
and the overall mass-balance equation:
dcLs dQs Gs ) Vl + Gs ) dt dt 1+ξ
)
∆hvap s
cLs,1 Fw ) (xs)sol exp(Φp + χΦp2) c1 Fs
(18)
The value of the Flory interaction parameter, χ, for most amorphous polymer-solvent systems in the range 0.250.55, can be determined experimentally from vapor-
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2853
liquid equilibrium data. For the EPDM rubber-solventwater system it can be obtained on the basis of the work of Di Drusco and Galli:4 the value so determined for the hexane solvent, 0.28, matches with that given by Matthews et al.14 If both the polymer and the solvent are nonpolar, χ can also be estimated by the following equation:
vj s (δ - δp)2 χ ) χS + χH = 0.34 + RT s
-dQs/dt ) RSpc1 ) KVpc1
(20)
with
R ) KVp/Sp ) Kr/3
[
(20.a)
]
R1 + R2 + R3 + R4 ωn
R1 =
-1
(20.b)
1 r 3Λkl(1 - n)
(20.d)
jcsl/Fjp 1 R3 ) Λkvl(Sv - nSp)/Vl 1 - cjsl/Fjp
(20.e)
]
kvl ) kv
ps,j L cs,j
for z ) r
(22)
with R defined by eqs 20.a-20.h and reduced to a constant. In each stripping batch, indeed, all the terms in eqs 20.c-20.h, except r and f 0s , may be considered constant , insofar as they are functions of the physical properties of the system in a narrow temperature range and of the operating conditions, which are fixed. For r the time average value can be taken during the stripping run, given by rav ) r0/(1 - Cav/Fs)1/3, with Cav ) (C0Cf)1/2. An average value may also be taken for f 0s , equal to its value at the water boiling temperature at Pt, this approximation being justified by the fact that the temperature variation during a stripping batch, at the strong interphase off-equilibrium conditions associated with the stripping flow rates of industrial concern, is of only a few degrees centigrade. For the equation system 21 and 22, with Dm and R constant and c ) C0 for all z at t ) 0, as it is our case, Crank5 gives the solution
2Lr z
exp(-βnDmt/r2) sin(βnz/r)
∞
∑
n)1
[βn2 + L(L - 1)]
sin βn
(23)
with
L ) Rr/Dm ) Kr2/3Dm βn cot βn + L - 1 ) 0
(20.f)
n kvl R2 + R3 1+ = 1 - n kl R1 n kvl kl Sp - nSp 1+ 1+ (20.g) 1 - n kl k′l Sv - nSp
[
∂c 1 dQs ) = -Rc ∂z Sp dt
c ) C0
cjsl/Fjp 1 R2 ) 1 cjsl/Fjp Λk′l(Sv - nSp)/Vl
[
Dm
(21)
(20.c)
cjsl/Fjp kvPt R4 ) ΛkvlFv/Vl 1 - cjsl/Fjp ωn ) 1 +
∂c ∂2c 2 ∂c ) Dm 2 + Dm ∂t z ∂z ∂z
(19)
The values of vjs, δs, and δp can be found in the literature or be estimated by the group contribution method. This method, however, could give scarce approximation at low solvent concentrations.2 Simplified Solution of the Equation System. Making a system of eqs 4-9.a and 10-12 and solving for dQs/dt, we get
K)
value Dm in the field of C where the stripping is performed. Therefore, the equation system simplifies to
]
kvf 0s exp(Φp + χΦp2) ) ) ΛFs Fw(xs)sol (20.h) kvf 0s
Regarding the coefficient of nucleation ωn, it is to be remarked that as the ratio R3/R2 ) k′l/kvl, from an order of magnitude estimate, results to be lower than 10 -2, this has been neglected in the last part, between brackets, of eq 20.g. It is also clear that ωn ) 1 when there is no nucleation, being in such the case n ) S ) 0. Moreover, it seems reasonable to suppose that T, then n, is increasing with Sv (i.e., with Fv/Vl). If this increase is quasi-proportional, eq 20.g can justify a slight increase of ωn with Fv/Vl. The real form of dependence, however, can only be determined experimentally. When C is low enough (say (K)1. We can then conclude that the scale-up problem can safely be solved adopting the Deckwer criterion of constant Pow/Vl and Fv/Vl, which involves the constancy of the stripping kinetic exponent K. (3) Change of Slurry Concentration. A last interesting case of extrapolation we are taking into consideration is occurring when the slurry concentration is varied, keeping fixed the polymer-solvent system, the stripping vessel, the agitation, and all the other operating conditions. The presence of the term cjsl in eqs 20.d-20.f gives an indication of how to perform this extrapolation. It can indeed be noted that if it is made keeping Fv/Vl proportional to (cjsl/Fjp)/(1 - cjsl/Fjp) or, what is the same, Fv proportional to Qp, R4 remains unchanged. On the other hand, nk′lSv/Vl and nkvlSv/Vl increase with Sv/Vl and perhaps with cjsl (possible bubble splitting by impact with polymer particles causing an increase of Sv). The actual variation of (R2 + R3)/ωn depends on how nk′lSv/Vl and nkvlSv/Vl increase with Fv/Vl and cjsl, when they are simultaneously varied according to the criterion indicated above. If they vary proportionally to Fv, with Fv ∝ Qp, (R2 + R3)/ωn also remains unchanged. Regarding R1/ωn, at least for low values of nkvl/kl (solvents or monomers with relatively high solubility and low volatility), eqs 20.c and 20.g make it reasonable to assume that it is little or nonaffected by a change of Fv/Vl. The conclusion is that if the extrapolation is made by keeping Fv ∝ Qp, the stripping kinetics exponent K remains unchanged. This hypothesis is confirmed by the experimental data of Galli, but as these were
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2859
obtained, as far as the effect of the slurry concentration is concerned, only for the system EPDM-ENB (ethyliden norbornene) at fixed r and Fv/Qp, varying cjsl from 0.01 to 0.14 g/cm3, further experimental confirmation is recommendable. For the time being it will be assumed as a working hypothesis. Discussion. Starting from the stripping kinetics equations, we have developed correlation methods and extrapolation procedures of the experimental data for three cases: (A) change of polymer-solvent system; (B) change of stripper size (scale-up); (C) change of slurry concentration; to which we have to add a fourth case already treated previously, that is, (D) extrapolation from batch to continuous operation. Whereas the extrapolation rule developed for case (A) does not require any additional assumption, over those upon which we have based our model, the extrapolation rules for cases (B) and (C) do require additional assumptions, that for the time being must be considered as working hypotheses waiting for experimental confirmation. The additional hypothesis for case (B), that is, the validity of Deckwer’s scale-up criterion, is since now wellsupported experimentally and only needs definitive validation; besides this, it can be safely adopted for design purposes, because if not valid, it would be conservative. Although justified by some experimental data, the additional hypothesis for case (C), the independence of the stripping kinetics from slurry concentration at fixed Fv/Qp, should be supported by further experimentation to delimit its field of validity . For the time being, therefore, it is recommended to perform the extrapolations starting from experimental data obtained in proximity of the target slurry concentration. Moreover, for a precise evaluation of the stripping kinetics, R1 and ( R2 + R3 ) should be known separately. The theoretical prediction of R1 and ( R2 + R3 ), on the other hand, would require the calculation of kl and k′lSv/Vl from reliable correlations Sh-Re-Sc, specific for bubbled solid particles suspensions in water under mechanical agitation, which until now are not available or not sufficiently validated. The problem of how to determine R1 and (R2 + R3) from experimental data is beyond the scope of this study and would require, to be solved, a specifically addressed experimentation. Some insight can however be obtained from the stripping experiments carried out by Galli et al.12 with the systems of EPDM rubber with ENB (ethylidene norbornene), MTHI (methyltetrahydroindene), benzene, and heptane in the following field of operating conditions:
C h < 0.12 g/cm3 cjsl ) 0.02-0.15 g/cm3 r ) 0.03-0.6 cm F h 0st/V1 ) (6 × 10-6)-(150 × 10-6) g/cm3 s The experiments were carried out in a 50-L glass stripper operated at 60% filling degree, provided with a baffled two-stage tilted-blade turbine agitator, with dag/d = 0.6 and rotating speed of 450 rpm. The correlation of the experimental data obtained by Galli confirms the exponential form given by eq 26 (Figure 5) and the method from there developed to determine the stripping kinetics exponent K.
Figure 5. Stripping kinetics, C vs t. EPDM-ENB; r ) 0.08 cm; T ) 100 °C; F h 0st/Vl ) 30 × 10-6 g/cm3 s; cjsl ) 0.1 g/cm3.
Figure 6. 1/K vs r at various stripping rates Y. Y ) F h 0st/Vl, 10-6 g/cm3 s.
If the values of K derived from such experiments are plotted on a diagram with r as the abscissa, 1/K as the h v/Vl), a diagram like ordinate, and parameter F h 0st/Vl (=F that represented in Figure 6 is obtained. Two things are shown by this diagram: (1) At fixed F h 0st/Vl (that is of Sv and ωn), the approximate linearity of 1/K with respect to r, allowing one to plot 1/K versus r with straight lines. (2) The decrease of slope of such straight lines at increasing F h 0st/Vl. This behavior is in accordance with the form of eqs 20.b-20.h and the indication from previous research work that, at increasing r, kl should either decrease or remain constant, and k′lSv/Vl should either increase or remain constant, and is illustrated in Figure 7, which represents the functional dependence from r of the mass-transfer resistances Rj and of 1/K ) [(R1 + R2 + R3)/ωn + R4)]. According to this picture the approximate linearity of 1/K with r, which is experimentally proved, can be explained by the exact compensation of the convexity of the curve R1/ωn with the concavity of the curve (R2 + R3)/ωn or vice versa, or most probably with the quasilinearity of both curves R′1 ) R1/ωn and (R′2 + R′3) ) (R2 + R3)/ωn. For R1/ωn this last conclusion is supported by the form of eq 20.c. The decrease of slope of the straight line 1/K at increasing F h 0st/Vl could be explained by an increase of negative steepness of the (R2 + R3)/ωn line or by a decrease of positive steepness of the R1/ωn line at increasing F h 0st/Vl. As to (R2 + R3)/ωn, taking into
2860 Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998
account that according to eqs 20.d and 20.e, (R2 + R3) ∝ 1/(k′lSv/Vl), this explanation is in agreement with some previous research findings evidencing an increase of k′lSv/Vl with r, which is enhanced by increasing F h 0st/Vl. However, as it can be inferred from Figure 6, the only variation of the (R2 + R3)/ωn line steepness with F h 0st/Vl would not be sufficient to justify, for instance, the halfing of the t slope of the straight line 1/K when F h 0st/Vl is increased from 6 × 10-6 to 60 × 10-6 g/cm3 s. A decrease of steepness of the R1/ωn curve, that is (eq 24.c) h 0st/Vl, must an increase of kl or of ωn, at increasing F therefore in any case be called for to explain the experimental data. This is in agreement with eq 20.g, which justifies as probable an increase of ωn with Fv/Vl and can therefore be considered as a clue in favor of the mechanism proposed for the nucleation. The actual variation of k′lSv/Vl, Sv/Vl, and ωn with r and Fv/Vl can only be determined experimentally by means of techniques like those reviewed by Van’t Riet10 and semibatch tests (Supporting Information). Remarkable consequences would arise from the case, not missing some support from previous experimentation, that the influence of r on k′lSv/Vl is quantitatively negligible, at least for the lower values of Fv/Vl and r. Under these conditions, combined with a relatively low density of the polymer particles, the impact between the bubbles and the polymer particles could indeed be insufficient to cause the bubble splitting and the consequent increase of Sv/Vl, with which the increase of k′lSv/Vl with r can be explained. In such a case, at fixed Fv/Vl , [(R2 + R3)/ωn + R4)] would remain constant by varying r and R1/ωn would be represented by a straight line parallel to 1/K, whose angular coefficient would be 1/(3Λklωn). So kl, k′lSv/Vl, R1, and (R2 + R3) could be determined plotting 1/K versus r, with parameter Fv/ Vl, after estimating ωn with the method indicated. An example of an approximate determination of ωn and of R1, R2, R3, and R4, under the above-discussed hypothesis, along with a discussion of the conditions for which one mass-transfer resistance is rate-limiting, is provided in the Supporting Information. In particular, for the systems investigated by Galli, at medium or high stripping rates, the diffusion is rate-limiting for particle diameters of the order of magnitude of 1 cm. Extensions The mathematical model developed in this work can be extended both to take into account the deviations of the real conditions from the ideal case on which the model is based and to cover a larger field of processes of industrial interest. For brevity and because the derivations are simple, we will give here only the results of such extensions, with some hint of the way followed to derive them. Nonuniform Granulometry. If the polymer particles are not uniform in shape and size, but do not deviate too much from the ideal case (identical spheres), an equivalent radius req can be used in the equations, given by req ) 3Vtot/Stot, where Vtot and Stot are respectively the total volume and the total surface of the particle population. Composition Dishomogeneity. Perfect agitation conditions as assumed in our ideal model will never be completely attained. We have already discussed how to take into account the partial back mixing of the vapor phase by means of a correcting factor applied to the full
Figure 7. Stripping kinetics scheme.
back-mixing composition and how the omission of such a factor is a conservative assumption for design purposes. It remains to consider how the dishomogeneities of polymer particles slurry concentration can affect the continuous stripping design method developed under the hypothesis of perfect agitation, which involves identical composition of the slurry discharged from the stripper with that within the stripper. For an overview on the matter, reference is made to the investigations of Kuzmanic and Kessler18 and to the previous contributions they are mentioning. The main effect of these dishomogeneities is that the residence time ht in the continuous stripper will be different from that calculated by assumig equal concentration of the slurry discharged from the stripper and of that within the stripper (i.e., higher if the slurry concentration in the stripper is higher than that in the discharged stream, lower if it is lower). The effect will depend on many factors, including the type and intensity of agitation, the geometry of the vessel, the location and the flow rate of the discharge, and the size and the density of the polymer particles. From the investigations of Kazmanic and Kessler on floating polymers, it appears that the concentration is usually higher in the vessel than in the discharged stream: under these conditions the use for design purposes of the value of ht calculated with the assumption of equal concentration is conservative. Multiple Solvents. In case two or more solvents are dissolved in the polymer particle, at low total solvent concentrations, the equations developed in this work can still be used for each component solvent, provided the Flory interaction parameter χ is calculated for the solvent mixture, as for instance indicated by Blanks and Prausnitz.13 In case both the solvent and the polymer are nonpolar, this calculation procedure gives
χ ) 0.34 +
vj s,m (δ - δp)2 RT s,m
vj s,m ) ΣiΦivj i δs,m ) ΣiΦiδs,i where the sums are extended to all the component solvents i making up the solvent mixture and Φi are
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2861
the volumetric fractions, in the absence of a polymer, of said components. Steam Stripping in a Fluid Bed. If the process is carried out in a fluid bed with saturated steam in wet conditions (polymer particles surrounded by a thin layer of water), in batch or continuously, the equations are modified putting R2 ) R3 ) 0 and introducing in eq 20.f the relation Qp/Vl ) cjsl/(1 - cjsl/Fjp), obtaining
K ) (R1/ωn + R4)-1 R1 )
1 r 3Λkl(1 - n)
R4 )
kvPtVp ΛkvlFv
kl,av ) (1 - n)kl + nkvl ωn ) 1 +
n kvl 1 - n kl
If the batch steam stripping is performed in dry conditions (polymer particles in direct contact with steam, without interposition of the water layer), put n ) 1 in the above equations. The stripping kinetics in wet and dry conditions is still given by eqs 23.2, 24, or 26 for the batch case and by eqs 30.a, 33.1, 37, and 37.2 for the continuous case. The expressions for K and L (eq 30.a) in the dry case are useful here to remark
K)
kv exp(Φp + χΦp2)(fs0)2/Fs r/3 + kvPtVp/Fv
kv exp(Φp + χΦp2)(fs0)2r L) FsDm An important consideration to take into account when treating the wet steam stripping in a fluid bed is that values of kl higher than those found for the steam stripping in an aqueous slurry can occur, because the limit layer LL1, depending on the thermal condition of the polymer stream being fed, can be sensibly thinner than that of the slurry case, or be partially dry, due to the hydrophobic nature of the polymer or to an uneven distribution of the water layer, combined with strong turbulent conditions caused by polymer particles collision. The solvent removal from the polymer particles can also be performed with inert gas. This can be considered as a subcase of the steam stripping in a fluid bed under dry conditions treated above, when steam is substituted by inert gas: it includes the polymer drying or purification with inert gas in fluid beds and covers many important applications in industrial polymer production processes. The equations developed for the dry steam stripping in a fluid bed are still valid, provided the subscript w in the equations is referring to the inert gas, Fv is indicating the feed flow rate of the inert gas, and the energy balance (eq 13) is rewritten adequately. When (ceq)z , Cz, it is also possible to treat the case of fluid beds in series or of partially back-mixed fluid
beds, simulating these as a series of perfectly mixed fluid beds, by means of eq 37.2. Conclusion The present article is proposing a mathematical model and a theoretical framework for a class of processing operations largely used in the chemical industry, consisting of the purification of polymers from solvents or monomers by steam or gas stripping. The reached results allow one to solve the problems connected with the strippers design and operation and the extrapolation from one stripping system to another differing for the type of polymer or solvent, or the scale, or the operating conditions. The model covers the stripping in agitated vessels containing polymer particles suspended in water or in another liquid medium and the stripping, purging, or drying in fluid beds. At the base of the mathematical model are the following main assumptions: (1) Spheroidal particles of the same size. (2) Perfect agitation, with complete back mixing. (3) Low solubility of the solvent or monomer in the liquid medium (order of 1% or less). Such limitation is of course not applying to the fluid beds operated in dry conditions. (4) Validity of Fick’s law. (5) Equilibrium conditions at the interface between different phases. (6) Diffusion of the solvent or monomer in the polymer particle representable by means of eqs 1 and 2, where D may be approximated with a constant Dm taking into account both the solvent or monomer concentration and the porous structure of the particle. The work, at the stage it has been developed, is since now usable, for practical purposes, for the range of applications mentioned above, but under the theoretical point-of-view, is demanding some additional confirmation and investigation. In particular, further work should be done to determine and correlate the single boundary layer mass-transfer resistances R1, R2, R3, and R4 and the coefficients n and ωn characterizing the nucleation phenomenon. The latter should be confirmed and better clarified in its origin, conditions, and mechanism: one hypothesis that could be investigated is that the nuclei form where porous channels, through which the solvent diffuses in the vapor phase, reach the particle surface. Another is that the nucleation is taking place only with relatively high volatility and very low solubility solvents, in particular when the system temperature is overcoming the boiling point of the solvent. Research should also be done on the relation between the porous structure of the polymer particles in general and the effective diffusion coefficient, on the example of that made by Matthews et al. for the EPDM crumbs obtained by steam stripping from the rubber solution.14 Acknowledgment The financial support provided by Tecnimont and the permission of Prof. P. Galli to publish experimental results of his research are gratefully acknowledged. Nomenclature Al ) transversal area of the stripper occupied by liquid, cm2
2862 Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 c ) variable solvent concentration along the radius of the polymer particle, gmol/cm3 ceq ) equilibrium concentration, gmol/cm3 c1 ) value of c at the polymer particle surface c1,m ) average value of c1 at the polymer particle surface cνi,j ) concentration of component i, at position j, in phase ν, gmol/cm3; by default i ) solvent, j ) within the polymer particle, and ν ) polymer phase cLs ) solvent concentration in the core of the liquid (water) phase, gmol/cm3 cjsl ) slurry concentration, g/cm3 C ) average solvent concentration in the polymer particle, gmol/cm3 C0 ) value of C at the beginning of the batch stripping Cf ) value of C at the end of the batch stripping Cm ) volumetric average of C in the feed or discharge stream of a continuous stripper d ) stripper diameter, cm dag ) agitator’s impeller diameter, cm D ) variable solvent diffusivity in the polymer particle, cm2/s Dm ) average value of D Ds,v ) diffusivity of solvent through stripping vapor, cm2/s Ds,w ) diffusivity of solvent through water, cm2/s EC(Cz-1) ) polymer particle solvent concentration distribution function at the zth stripper inlet EN(t) ) numeric residence time distribution function EV(t) ) volumetric residence time distribution function f ) fugacity, bar f0 ) fugacity in the state of pure liquid, bar Fst(F h st) ) flow rate of the steam fed to the stripper, gmol/s (g/s) F0st(F h 0st) ) stripper’s internal equivalent steam flow rate, gmol/s (g/s) Fv(F h v) ) stripping vapor flow rate, gmol/s (g/s) G ) vapor flow rate discharged from the stripper top, gmol/s kl ) mass-transfer coefficient of the solvent through the liquid boundary layer LL1 surrounding the polymer particle (Figure 2), cm/s kl,av ) average mass-transfer coefficient of the solvent through the liquid boundary layer LL1 including vapor nuclei, cm/s k′l ) mass-transfer coefficient of the solvent through the liquid boundary layer LL2 surrounding the vapor bubble (Figure 2), cm/s kv ) mass-transfer coefficient of the solvent through the vapor boundary layer LL3 enveloping the vapor bubble (Figure 2), g/s cm2 bar kvl ) value of kv referred to equilibrium concentration in liquid phase, cm/s (eq 20.h) K ) stripping kinetics exponent, s-1 l,l′ ) characteristic lengths in Sh, cm L ) Rr/Dm, stripping kinetics diffusion parameter M ) molecular weight n ) number of strippers in series (Ns)n ) solvent flow rate removed through the vapor nuclei-polymer interface, gmol/s p ) partial pressure, bar p0 ) vapor pressure, bar Pt ) total pressure inside the stripper, bar Pow ) agitation power released inside the stripper, W qi ) weight content of component i in the polymer particle, gmol Qi(Q h i) ) weight content of component i in the stripper, gmol (g) r ) radius of the polymer particle, cm r0 ) radius of the polymer particle without solvent, cm Re ) Reynolds number Rj ) mass-transfer resistance, s
sp ) single-polymer particle surface area, cm2 Sp ) total surface area of the polymer particles in the stripper, cm2 Sv ) total vapor-liquid interfacial area inside the stripper, cm2 Sc ) νw/Ds,w, νv/Ds,v, Schmidt number Sh ) kll/Ds,w, k′ll′/Ds,w, Sherwood number t ) time, s ht ) Q h p/W h p, average residence time, s T ) operating temperature, K vj ) liquid molar volume at boiling point, cm3/gmol Vl ) liquid volume in the stripper, cm3 Vp ) polymer particles volume in the stripper, cm3 Vsl ) slurry volume in the stripper, cm3 W(W h ) ) flow rate discharged from the stripper bottom, gmol/s (g/s) x ) molar fraction in the liquid phase (xs)sol ) solubility of the solvent in water at T, molar fraction z ) radial distance from the center of the polymer particle, cm Greek Letters R ) Kr/3, cm/s βn ) indexed constant in eq 23 γ ) activity coefficient Γ, Γ′ ) parameter defined in eqs 33 and 33.1 δ ) solubility parameter, (cal/cm3)1/2 ∆h* ) overheating heat content of the steam fed to the stripper, cal/gmol ∆hvap ) latent heat of vaporization, cal/gmol ∆Hag ) agitator’s heat input to the stripper, cal/s ∆Hloss ) heat losses from the stripper, cal/s n ) ST S ) fraction of polymer particle surface covered by nuclei T ) fraction of time during which the nuclei are in contact with the stripping vapor η ) viscosity, P Φ ) volumetric fraction in the polymer particle λ ) fraction of the total water stream discharged from a stripper feeding the next stripper, Figure 4 Λ ) partition coefficient of the solvent between polymer and liquid (water) phase ν ) kinematic viscosity, St ν′ ) atomic diffusion volume ξ ) accumulation term, eqs 9 and 35 F(Fj) ) density, gmol/cm3 (g/cm3) χ ) polymer-solvent interaction parameter χS, χH ) entropy and enthalpy components of χ ψz,n(L) ) functions of L defined by eqs 24.a and 24.b (z ) 1, 2) ωn ) nucleation coefficient, eq 20.g Subscripts i ) index of component j ) index of position n ) series current index z ) generic index Values of i l ) liquid p ) polymer s ) solvent sl ) slurry v ) vapor w ) water Values of j j ) 1, 2, 3, 4, 5 (positions indicated in Figure 2) i,j ) when two indexes are used together, the first one refers to the component, the second to the position; by
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2863 default the component is the solvent and the position is a point inside the polymer particle. Superscripts ν ) index of phase assuming the following values L ) liquid phase P ) polymer phase (by default ν ) P) V ) vapor phase
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Received for review May 20, 1997 Revised manuscript received April 13, 1998 Accepted April 14, 1998 IE9703649
