I n d . Eng. Chem. Res. 1993,32, 2069-2076
2069
Desorption from Concentrated Polymer Solutions in Good and Poor Solvents Sripada P. Rao Chemical Engineering Division, Central Leather Research Institute, Adyar, Madras 600 020, India
Kandukuri S. Gandhi* Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India
Davuluri P. Rao and Rakesh P. Singh Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208 016, India
The objective of the study was to investigate the effects of the nature of solvent and polymer concentration on the mass-transfer coefficients in desorption of solvents and to develop a correlation to predict them. Desorption was experimentally studied in a Lewis cell with concentrated binary solutions of polymer in good and poor solvents. The range of parameters covered are polymer weight fraction between 0.25 and 0.6, Reynolds number between 3 and 100; Schmidt number between 1.4 X lo6 and 2.5 X lo8, and Sherwood number between 3.5 X lo2 and 1.2 X lo4. Desorption from moderately concentrated solutions (polymer weight fraction -0.25) is gas-phase controlled. Studies with more concentrated solutions showed that the effects of solvent and concentration were such that corrections due to concentration-dependent diffusivity and viscosity as well as high flux had t o be applied to the mass-transfer coefficients before they could be correlated.
Introduction Mass transfer in polymer solutions has many practical applications in polymer processing and polymer manufacture. In these instances, a polymer could be dissolved in a good or a poor solvent or a mixture of them. Mass transfer from solutions of a polymer in a single solvent (hereafter referred to as a binary solution) is the elementary situation whose understanding has to form the basis for explainingthe behavior in a multicomponent environment. Solvent flux in desorption from binary solutions exhibits acomplex behavior depending on the nature of the solvent. For instance it is known that as polymer concentration is increased, diffusivity in a polymer-poor solvent system exhibits a minimum (Rehage et al., 1970; Roots et al., 1979a)while that in a polymer-good solvent system exhibits a maximum (Rehage et al., 1970). Viscosities of polymer solutions in poor solvents are lower than those in good solvents at lower concentrations but are much greater at higher concentrations (Tager et al., 1968; Gandhi and Williams, 1971). Thus it would be interesting to examine the influence of these effects on convective mass transfer. Only a very few of the existing studies on mass transfer address concentrated solutions, while the bulk of them have been confined to the extremes of the concentration range. Desorption from concentrated solutions is encountered in preparation of polymeric films cast from solutions. Barr-Howell's (1991) studies on drying of films showed that solvent transport can be anomalous or of type super case-I1 depending upon drying temperature, and this non-Fickian behavior is due to slow relaxation of polymer chains in glassy regions of the film. Kunst and Sourirajan (1970),Pilon et al. (1971),and Francesconi and Castellari (1988) studied evaporative flux of the solvent in membrane formation by phase inversion and suggested mathematical analyses for calculating the flux. All these studies generally cover dilute to very high concentration
range, and it is difficult to correlate the data since the diffusion mechanism is known to depend upon the concentration regime which may be different at different locations of the film. Further, the theoretical studies have not considered the contribution of bulk flowwhich becomes increasinglyimportant as polymer concentration increases, and the possibility of liquid-phase mass-transfer resistance being absent in low polymer concentration regime. Studies on dilute solutions have been mainly concerned with determination of diffusion coefficients. Banerjee et al. (19681, Wasan et al. (1972),Mohr and Williams (19731, Mashelkar and Soylu (19741, and Ravetkar and Kale (1981) studied absorption of a gas into flowing films. Astarita (1966) and Smith et al. (1969) studied the dissolution of a solid into flowing films. Astarita and Apuzzo (1965), Hirose and Moo-Young (19691, Calderbank et al. (19701, Leal et al. (1971), and Zana and Leal (1978) studied the absorption of a gas into polymer solutions from rising bubbles. In some of these studies, the special effects of viscoelasticity on shape and motion of the bubbles were also investigated. Astarita and Mashelkar (1977) present a review of the mass-transfer literature. The mass-transfer studies at very high concentrations have been devoted mainly to removal of solvent (or devolatilization) from melts in screw extruders. Denson (1983)presents a review of this work. The high temperature of melts almost obliterates the effect of the nature of the solvent, while in the dilute range the concentration dependence of transport properties is not sufficiently strong to bring out the effects of high concentration and of solvent character. Thus, on the basis of these studies, it is not possible to predict the mass-transfer behavior in the range of concentrations between the extremes. It should be pointed out that diffusional phenomena in such a range of concentrations are encountered in polyesterification and nylon reactors, solution polymerization of acrylic monomers, finishing stages of manufacture of medium, high molecular weight epoxy resins, etc. In all these examples removal or adjustment of concentration of solvent and monomer may be necessary.
* To whom correspondence should be addressed. o a a a - ~ a a ~ i ~ ~ i ~ ~ ~ ~0 -1993 ~ o American ~ ~ ~ ~ ~Chemical . o o / Society o
2070 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
1. LErnS CELL 2. WATER BATH 5. ANNULAR PLATE 4. IMPELLER 5. HORSE-SHOE MAGNET 6. BAFFLES 7. ALUMINIUM ROO 8. CAPtLLARY TUBE 9. COPPER TUBE 10. CAS DISTRIBUTER 11 STIRRER 12 STIRRER-HOUSING
Figure 1. Sketch of the experimental setup.
The objective of this work is to investigate desorption of solvents from concentrated polymer solutions, and to develop a correlation for the mass transfer coefficients. The 0-temperature for a polymer-solvent pair is defined as the critical miscibilitytemperature in the limit of infinite molecular mass of polymer. The solvent is also referred to as 0-solvent at that temperature. Hence the polymer is just on the verge of precipitation in a 0-solvent, and thus 0-solvents can be regarded as poor solvents. The effect of the nature of any solvent hence has to lie between that of a 0-solvent and a good solvent, and study was accordingly confined to solutions in these two solvents.
Experimental Work Desorption Data. A schematic diagram of the setup used to collect desorption data is shown in Figure 1. It chiefly consisted of a Lewis mass-transfer cell (1)placed in a large thermostated water bath (2). The Lewis cell was a cylindrical vessel divided into two compartments by a brass annular plate (3). The lower compartment held the desired liquid phase and was well mixed by a mechanism consisting of two impellers (4) and a horseshoe magnet (5). This arrangement could ensure good mixing as revealed by tracer experiments with viscous polymer solutions. The impeller speed could be set at a desired constant level. The cell was suitably baffled (6) to prevent vortex formation at higher speeds. Further, the center portion of the interface, where desorption rate would be the maximum, was blocked with an aluminum rod (7) to ensure a more uniform transfer coefficient over the interface. Nitrogen was passed into the upper compartment through a capillary tube (8) and a sufficiently long coiled copper tube (9) submerged in the bath so that the gas would attain the bath temperature before it enters the cell. The gas phase was well mixed by a fan whose speed was fixed at 750 rpm in all experiments. Uniform distribution of nitrogen a t the interface was ensured by introducing it through a gas distributor (10) a t a flow rate such that it would not disturb the interface. The main geometric parameters of the cell are the following: interfacial area = 1.683 X 10" m2; liquid-phase volume = 4.8 X lo4 m3; gas-phase volume = 1.5 X 10-4 m3.
The cell was filled with the desired solution and allowed to equilibrate with the bath. The bath was maintained at 308 f 0.2 K for all the experiments. Experiment was started by introducing nitrogen and setting the fan and impeller at the desired speed. Nitrogen would pick up solvent vapors, get mixed, and leave the cell through the gap between stirrer (11)and its housing (12). Desorption occurred only at the gas-liquid interface since bubbling does not occur in the solution at the temperature and pressure employed in the study. Makeup solvent was continually added without disturbing the interface to maintain the liquid level and composition constant. After steady state was achieved, the gas was sampled by a warm syringe and analyzed for its composition, y?, by a gas chromatograph (HP-5890) using a thermal conductivity detector with hydrogen as carrier. The flow rate of nitrogen entering the cell was determined from the pressure drop in the capillary tube using a calibrated manometer. The molar flux of solvent could be calculated from the relation
In experiments where the gas flow rate was too high (for a given liquid mixing) or liquid mixing was too low (for a given gas flow rate), solvent flux was observed to be unusually low. In some of these instances, a thin polymer crust was also observed at the interface. Therefore, the very low solvent fluxes could be due to the formation of either a rubbery solid or a glassy polymer layer. For a given bulk concentration of polymer, low mixing rates and high gas flow rates tend to increase the weight fraction of polymer at the interface and lead to the formation of glassy or rubbery layers. Neither of these states are of interest to the present work, and hence suitable ranges of gas flow rates and impeller speeds were selected to avoid this phenomena. Calculations based on the correlation developed later in the work showed that the lowest interface polymer concentration at which unusually low solvent fluxes were observed is about 0.9. Interestingly, an estimate (Foxand Flory, 1954;Ferry, 1970)suggested that glass transition temperature of the solution will be less than 308 K when the polymer weight fraction a t the
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2071 Table I. Range of Conditions Employed in Measurements systems studied
w3
w2/ (wl+wd
impeller speed (m3s-l) (rpm)
QxlOe
Desorption cyclohexane-nitrogen toluene-nitrogen cyclohexane + polystyrenenitrogen toluene + polystyrenenitrogen
0.25-0.6
4.5-10.5 4.5-12.5 1.1-4.0
48-240
0.2E-O.6
1.2-5.3
16-120
Diffusion cyclohexane-polystyrene 0.1-0.4 0.1-0.5 toluenepolystyrene Viscosity 0.4-0.7 0.0-1.0 cyclohexane-toluenepolystyrene
interface is above 0.91. Thus the observed anomaly in solvent fluxes could possibly be due to formation of a glassy layer at the interface, although formation of a rubbery layer cannot be ruled out. Solution Properties. Cyclohexane is known to be a 8-solvent at 307 K. Thus at the operating temperature of 308 K, cyclohexane was selected as the poor solvent for the polymer. Toluene is a good solvent for polystyrene and has a saturated vapor pressure sufficiently different from that of cyclohexane. Thus toluene was selected as the good solvent. Diffusivities of polymer solutions were measured with a diaphragm cell. The construction details of the cell and other procedural details are available elsewhere (Gangadhar, 1987). Solutions of known but slightly different polymer mass concentrations (mass fraction difference about 0.05) were placed in upper and lower compartments, and diffusion was allowed to take place for a specific time (6-15 h). Final compositionsof upper and lower solutions were then determined by measuring the density and polymer mass fraction. The former was measured with a calibrated 15-mL specific gravity bottle using a Mettler balance (Model H-20) while the latter was determined gravimetrically. Diffusivity, taken to be at the average composition, was calculated from the equation (Cussler, 1976)
where 0 is the cell constant and Ap3(t) is the difference in mass concentration of polymer between the top and bottom compartments at any time t. The cell constant was determined by calibrating the diaphragm cell with a cyclohexane-toluene mixture using its diffusivity value reported by Cussler and Lightfoot (1965). The range of composition covered in the experiments is given in Table I. Diffusivity data are discussed in the Appendix. Viscosity of polymer solutions was measured at shear rates in the range 10-1OOO s-l with a Haake viscometer (RV-12). The range of composition covered in the experiments is given in Table I. The viscosity data are discussed in Appendix. Density data were collected over the polymer mass fraction range 0.1-0.5. The correlations obtained for cyclohexane-polystyrene and toluene-polystyrene solutions respectively are = 757.3 2 5 1 . 3 ~ ~ p Z 3 = 847.2 + 2 0 1 . 3 ~ ~
pI3
(3)
(4) The measured and calculated values agree with each other
to within f0.2%. At higher polymer concentrations density was calculated by extrapolation. Materials Used. All the solvents used in the present experiments were of analytical reagent grade (supplied by E. Merck Ltd., Bombay). Polystyrene (M,= 6120) was prepared in the laboratory by solution polymerization of styrene in carbon tetrachloride using benzoyl peroxide as an initiator. A low molecular mass polymer was chosen since its low solution viscosities allow experimentation up to high polymer concentrations, and it ensures that nonNewtonian and viscoelasticeffects of the solutions on mass transfer need not be considered.
Determination of Mass-Transfer Coefficients Definition. Mass-transfer coefficients are generally defined relative to a reference velocity in the framework of the film model (Bird et al., 1960;Cussler, 1976). Taking molar average velocity as the reference and noting that nitrogen is practically insoluble in the solvent, the definitions for gas-phase mass-transfer coefficients, K * , a t interface condition may be given as
4 = Ni(1 - yf)
(5) CK*i4(Yf- $) Value of K*i4 so defined depend upon the magnitude of flux apart from hydrodynamic conditions, system properties, and geometry. The dependency of ~*i4on flux can be eliminated by applying a correction, S; defined as E
= K*/K (6) and can be obtained using the film model (Bird et al., 1960) by
The mass-transfer coefficient for polymer solution, k*di3, may be similarly defined at bulk conditions with reference to mass average velocity: The value of k*di3, apart from the geometry and hydrodynamic conditions, also depends on the magnitude of mass flux, and the sensitivity of diffusivity to changes in concentration. The superscripts * and d indicate the dependence on high flux and variable (i.e., concentration dependent) diffusivity. A correction factor, 8, for accounting these effects on k*$3 can be defined as
lim k*d
lim k*d
%+I
nto
LhLP
k
and evaluated from the Film model in a manner anologous to that indicated by Bird et al. (1960) to give
e=p#=
where R, is
R, = (w? - ~ ; ) / ( l -w?)
(11) Evaluation. When pure solvents are used as the liquid phase, the mass-transfer resistance lies entirely in the gas phase. Hence, the gas-phase coefficient, K * , can be independently determined from eq 5 using desorption data with pure solvents, Ni, as a function of y? (or 8). The low
2072 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
flux coefficient, K , was then determined from eq 6. The saturated vapor pressure required to obtain y: was estimated by Antoine equation (Reid et al., 1977). The interface temperature for the calculation of yt was taken to be same as the bulk (or bath) temperature since interface cooling due t o solvent evaporation was found to be unimportant. As the fan speed was kept constant, the value of K would have the same value in all experiments. For evaluating the liquid-phase coefficient,mass transfer resistance in both the phases is to be considered. In these experiments, Nj from polymer solution was measured as a function of w: and y.: The known K was used with appropriate high flux correction to determine y: from eq 5. w: was then determined by simultaneously solving the phase equilibrium relation at the interface: y i p = a:pi and the thermodynamic model based on the segment interaction equation of Heil and Prausnitz (1966) for relating ai and Wi. These data would allow the calculation of uncorrected coefficient from eq 13 which was obtained by combining eq 5 with eq 1 rewritten in mass units:
Table 11. Desorption from Pure Solvents QX106 K* x 1@ run (m3 8-1) (m 8-11 I Cyclohexane-Nitrogen System 170 171 172 173
4.59 6.05 8.15 10.49
169 166 167 155
4.65 6.44 8.10 12.49
5.0 5.3 5.4 5.4
Kx1@
(m 8-1)
0.96 0.96 0.95 0.95
5.2 5.5 5.7 5.7
Toluene-Nitrogen System 4.4 4.6 4.7 4.5
0.99 0.99 0.98 0.98
4.4 4.6 4.8 4.8
Table 111. Desorption from Binary Polystyrene Solutions impeller speed Q X 106 k * d X 106 kx106 run w? (rpm) (mas-1) (ms-1) e* ed (m 8-1) Cyclohexane-Polystyrene System 241 244 246 247 223 226 222 224 230 218 249 253 254 256 258 260
0.416 0.416 0.416 0.416 0.489 0.489 0.489 0.489 0.489 0.489 0.571 0.571 0.571 0.571 0.571 0.571
80 80 48 48 240 240 216 216 144 144 240 240 144 144 80 80
Results and Discussion
266 268 269 270 271 272 274 275 277 278 279 280 281 282 283 284
0.526 0.526 0.526 0.526 0.526 0.526 0.526 0.526 0.526 0.579 0.579 0.579 0.579 0.579 0.579 0.579
80 48 48 48 26.7 26.7 26.7 16 16 120 120 80 80 80 48 48
Desorption experiments were carried out with polystyrene-cyclohexane and polystyrene-toluene solutions as well as with the pure solvents. The range of Q, and S employed in the study are given in Table I. All experiments were conducted at the ambient pressure and 308 K. At these conditions, as mentioned earlier, toluene is a good solvent for polystyrene while cyclohexane acts as a 8-solvent (Bandrup and Immergut, 1966). Mass-transfer coefficientsin the gas phase were obtained from independent studies involving desorption of pure solvents. The experimental results are given in Table I1 for both the solvents. The high flux corrections were found to be negligible and the low flux coefficients were determined to be 5.4 X m s-l for a cyclohexane-nitrogen mixture and 4.6 X 1 W m s-l for a toluene-nitrogen mixture. The values of the coefficients agree with those reported by Hikita et al. (1975) and Tamir and Merchuk (1978). Preliminary experiments with polymer solutions showed that mass-transfer resistance is absent in liquid phase at moderate polymer concentrations. Typical results for polystyrene solution in toluene (w! = 0.25) showed that
the overall mass-transfer coefficient was unchanged with the impeller speed and was equal to the gas-phase coefficient obtained when pure toluene was used. The absence of mass transfer resistance, in spite of relatively high viscosity (about 10 mPa s) and low diffusivity (about m2 of the solution, is due to the fact that 1.5 X solvent’s activity in polymer solutions is very close to that of a pure solvent up to fairly high polymer concentration. For example, at a polymer mass fraction of 0.5, the drop in toluene activity is only about 10% from its pure state value of unity. Calculations using the slope of the equilibrium curve, daildwi, showed that if w: L 0.4,wr will be sufficiently large (in the range of experimental conditions employed) and the liquid phase resistance becomes large enough to be measurable. Experiments were, therefore, carried out in the range of high polymer concentration such that liquid-phase masstransfer resistance varied between 35 and 75 % of the total resistance. The results are presented in Table I11 for both cyclohexane-polystyrene and toluene-polystyrene systems.
The correction due to high flux and variable diffusivity, 8, was estimated from eq 10 using the concentrationdependent diffusivity as described in the Appendix, and the corrected coefficient ki3 obtained from eq 9. The assumption of equilibrium in eq 12 is justified in view of the precautions taken to ensure that interface solution was above the glass transition temperature.
Error Estimation Uncertainties in the measurement Q and y: give rise to errors in the values of K and k and were estimated using the theory of propagation of errors (Pratt, 1965). Typically the uncertainties are CQ = 3 X 1O-e m3 s-I; t q = 0.001; and tq = 0.001-0.004. The resultant error in K is only about i 5 5% while the error ink was estimated for each experiment since it varies from case to case depending upon the relative magnitude of the liquid-phase resistance.
wB,
1.88 1.61 1.52 1.65 4.00 2.37 3.19 3.99 2.71 3.04 3.12 3.45 2.28 1.88 1.46 1.14
0.70 0.72 0.88 0.83 1.66 1.44 1.23 1.28 1.02 0.98 2.07 1.96 1.33 1.48 0.91 0.96
0.70 5.00 0.72 4.24 0.75 3.14 0.73 3.86 0.76 6.41 0.78 5.52 0.75 6.86 0.74 7.53 0.74 7.31 0.73 7.76 0.83 3.84 0.81 3.95 0.81 4.06 0.83 3.74 0.80 4.27 0.83 3.74
0.20 0.24 0.37 0.30 0.34 0.33 0.24 0.23 0.19 0.17 0.66 0.61 0.41 0.48 0.27 0.31
Toluene-Polystyrene System 5.25 3.71 3.08 2.50 2.50 1.88 1.25 1.20 1.46 4.45 3.65 3.00 2.43 1.83 1.31 1.94
0.96 0.68 0.38 0.41 0.22 0.42 0.48 0.35 0.27 1.15 1.06 0.54 0.43 0.40 0.21 0.22
0.86 0.85 0.81 0.83 0.78 0.85 0.89 0.86 0.83 0.89 0.90 0.86 0.85 0.86 0.83 0.82
0.64 0.61 0.54 0.58 0.48 0.63 0.72 0.66 0.58 0.67 0.69 0.59 0.58 0.61 0.54 0.50
1.75 1.32 0.87 0.85 0.59 0.79 0.75 0.65 0.56 1.92 1.70 1.06 0.88 0.78 0.46 0.54
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2073
It can be seen from Table I11 that for mass transfer under similar conditions of wg and S (e.g., runs 260 and 281), cyclohexane-polystyrene system exhibits much higher mass-transfer coefficients, k*d, than those in the other system. This is a surprising result since, in the present study, cyclohexane-polystyrene solutions are more viscous, and diffusivities in this system are smaller than in toluene-polystyrene solutions (see data in the Appendix). This result is even more striking for the runs 260 and 281 since the w! were also nearly the same (0.83 and 0.79, respectively). This observation can be rationalized once it is noted that, for the composition range of interest (w3 0.4-0.9) and the polymer sample used in the present study, diffusivity increases with polymer concentration for the cyclohexane-polymer system while the opposite is true for the toluene-polymer system. Mass flux at 7 = 0 can be expressed as
-
j ; = -pki3(dwi/d7),,=o= pki38(wg- w:) (14) This is equivalent to the definition of uncorrected masstransfer coefficient in eq 8 with the correction factors given in eqs 9-11. It is only the corrected coefficient ki3 that should decrease as diffusivity decreases and viscosity increases. However, coefficient k*$3 includes the contributions from both 8* and ed. In the present study the values of 8* were nearly the same for both the systems. The value of ed depends upon the nature of concentration dependence of Di3. It may be seen from eq 10 that ed > 1 for cyclohexane-polystyrene system for which Di3 increases with w3, while ed < 1 for toluene-polystyrene system for which Di3 decreases with w3. The same observation may be inferred from the calculations of steady-state concentration profiles in the film when Di3 is composition dependent (Crank, 1975). The observed discrepancy in k*di3 can be therefore attributed to 8d. This observation reemphasizes the fact that corrected coefficients, ki3, should be used for developing any common correlation for the desorption data in both the systems. Lewis cell data are typically correlated by an equation of the type (Hikita et al., 1975)
Sh = alRea2Sca3 (15) where Sh,Re, and Sc are Sherwood,Reynolds, and Schimdt numbers respectively, and are defined according to eq 16: Sh
(k,d)/Dz
(16a)
The power of Re in eq 15was found to be 0.5. This square root dependence of Re is consistent with observations in several studies on low Reynolds number studies. The correlation in eq 15 is satisfactory when viscosity is insensitive to composition. In polymer solutions viscosity varies strongly with composition, and its effect on the diffusion film thickness should be accounted for. In fact it was found that eq 15 could not correlate the present data. In heat-transfer correlations, it is customary to apply a correction factor, (pI/pB)co"t, to account for viscosity variation in the film due to temperature changes (Kern, 1976). A similar approach was tried in the present work also. The following correlation was obtained: Sh = 1-27~ R ~ ' . ' S C(pl/pB) '.~ (17) We found that a value of 0.4 for the exponent of Sc gave a better correlation than a value of 0.33. The latter value
2800I
Sh = 1.272 Reo" St!'4(p'/9B)
-0'15
2400'
2000.
=%
16001
l200f
800
'
400
'
CYCLOHEXANE
1/
0 400
TOLUENE 800
-
1200
- POLYSTYRENE
SYSTEM
I
POLYSTYRENE SYSTEM 1600
ZOO0
2400
2800
Shexp
Figure 2. Comparisonof Sh obtained from desorption experiments with those calculated from the correlation (eq 17).
is typical of boundary layer type of models. In the Lewis cell, although the gas-liquid interface is shear stress-free, it is formed (in the present work) from averyviscous liquid. This perhaps is the reason as to why the observed exponent is close to l / 3 but slightly different from it. The calculated and experimental values of Sh are compared in Figure 2. The correlation represents the data with an average error of -*lo% and a maximum error of -*20%, except for a few points for which the deviation is larger. The uncertainty in the correlation arises due to that in determining wi which in turn affects viscosity correction and 8. The uncertainty in the determination wi is typically about 3%. For example, in run 254, w! = 0.57, wi = 0.86 and (p1/pB)-0.15 = 0.32. The uncertainty in wi would make it either 0.83 or 0.89, and the corresponding viscosity correction factors would be either 0.38 or 0.29. This accounts for about 1&15% uncertainty in the correlation. Typical calculations showed that B would entail about 5-10% error depending upon the composition range involved, and this accounts for the rest of the uncertainty in the correlation. In those runs (266,282,283,246, and 247) which showed larger deviations, the liquid-phase and mass-transfer resistance was smaller as also indicated by the fact that the values of wi were closer to w!. Division of fluxes by such small driving forces to obtain ki3 will enhance the uncertainty in the correlation. Further, in these Runs the uncertainty in wi was higher due to relatively less sensitivity of ai to composition at lower polymer concentration. In view of these factors, the binary correlation may be considered satisfactory. Irreversible thermodynamics suggests the use of molar fluxes and the gradient of chemical potential (or logarithm of activity) as driving force to describe diffusion rates. The use of V ln(ai) separates the thermodynamic contribution and hence is expected to simplify the concentration dependence of diffusivity (Cussler, 1976). It is therefore useful to test if a better correlation for the desorption data can be obtained in this framework. For this purpose mass-transfer coefficients and correction factors for the polymer phase were defined in an appropriate manner analogous to the definition in eqs 8-11 but based on molar units and A(ln ai). The results showed that the correlation for Sh was inferior to and subject to more scatter than that based on weight fraction driving force.
2074 Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993
Prediction of fluxes in multicomponent systems from correlations of mass-transfer coefficients in constituent binary systems has been a longstanding objective of mass transfer theories. Work is currently under progress to extend the existing methodologies to predict fluxes in ternary polymer solutions on the basis of correlations developed here for the binary systems.
Conclusions Desorption studies from concentrated solutions of polymer in a good and a poor solvent were carried out under nonbubbling conditions. Mass-transfer rates were found to depend upon the nature of the solvent. Masstransfer Coefficients for both the systems could be correlated by a single equation only after corrections due to variable diffusivity and viscosity and high flux were applied. Nomenclature a: thermodynamic activity in liquid phase A: interfacial area, m2 c: molar concentration gas mixture, kmol ma d: diameter of the impeller, m D: binary diffusivity in liquid mixture, m2 s-l j : mass flux with respect to mass average velocity, kg m-2 s-1 J: molar flux with respect to molar average velocity, kmol m-2
g-l
k: mass-transfer coefficient in liquid phase, m s-1 L: film thickness, m
M molecular mass n: mass flux with respect to fixed coordinates, kg m-2 s-1
I t molar flux with respect to fixed coordinates,kmol m-2s-1 pa: saturated vapor pressure, kPa
P: pressure, kPa
8: nitrogen flow rate, m3 s-1 Re: Reynolds number S speed of the impeller, s-1 Sc: Schmidt number Sh: Sherwood number w: mass fraction in liquid phase y: mole fraction in gas phase Greek Letters
error or uncertainty fi correction to gas-phase mass-transfer coefficient, eq 6 7: dimensionless distance coordinate 8: correction to liquid phase mass-transfer coefficient, eq 9 K : mass-transfer coefficient in gas phase, m s-1 I.L: viscosity of polymer solution, m Pa s-1 p: density of polymer solution, kg m-3 t:
Subscripts
1: cyclohexane 2: toluene 3: polystyrene 4: nitrogen i3: polystyrene solution in solvent i i4: gas mixture comprising nitrogen and solvent i Superscripts
B: bulk cal: calculated value d: diffusivity correction exp: experimental value I interface *: flux correction
Appendix Diffusivity Data. Binary diffusivities for polymersolvent systems, 0 2 3 and 0 1 3 , were determined at 308 K
-0
1x10
I
I
I
,
-@
1x10
tt \1 \
- I
\
.1
I
0
0.2
I
I
0.4
0.6
'\
0.8
1.0
w3
Figure 3. Diffusivities in polystyrenetoluene system.
as function of polymer concentration in the composition range given in Table I. Diffusivity data at higher concentrations than indicated in the table were unreliable because of problems connected with transferring viscous polymer solutions in to and out of the diffusion cell. Hence, the data had to be extrapolated to higher concentrations, and this is discussed below. Vrentas and Duda (1979) suggest the following equation to correlate or predict the binary diffusivity in polymer solutions:
Di3 = Dp'x3ri3 where r is thermodynamic factor, x 3 is polymer mole is the self-diffusivity of the solvent in fraction, and the solution. Free volume parameters reported by Vrentas and Chu (1989) were used to estimate the self-diffusivity. The segment interaction equation of Heil and Prausnitz (1966) was employed for the estimation of Pi3 (=xi d(ln ai)ldxi). The present data in toluene-polystyrene system along with the data of other investigators are shown in Figure 3. It may be seen that present data are in agreement with the literature data. The diffusivity was also estimated from the free-volume theory as described above. The calculated values are consistently lower than the observed values as shown in Figure 3, and this discrepancy may be because of the invalidity of eq A1 or due to the low molecular weight of the sample used in the present study. Hence an empirical approach was used to correlate the data. Diffusivities were corrected for thermodynamic nonideality to obtain the generalized Maxwell-Stefan diffusivity, a23, from eq A2: (A21 This generalized Maxwell-Stefan diffusivity displayed a gradual decrease with increasing concentration as shown in Figure 3. This observation is similar to the trends found by Rehage et al. (1970)for polystyrene-ethylbenzene (also a good solvent) system. The diffusivity at higher conD23
=
r23a)23
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2075 Table IV. Polynomial Coefficients in Viscosity Correlation
Cyclohexan- polystyrene
I
Reference 0
A
Rehage e t aL(1970) Munch e t aL(1983)
.
T(K)
x 10' x 10' 1.3 X LO* 2.0 x 10'
313 307.6
2.0
308 308
net work 1.5 X 10' 8.1 x 105
A
+ 0
Cangadhar (1987) 0 Present Work
~~
1.8
Cyclopentane-polystyrene
1 .a
0.6
On\
I 0.4
0.2
ba
0.71 0.59 0.49 0.41
9.256 6.171 4.636 3.340
-2.393 -1.916 -1.963 -1.181
0.810 0.943 1.144 0.843
w 9 w,/(wl + w,) (A41 The coefficients, bi, were obtained by regressing the data, and are given in Table IV for all the four polymer concentrations. The correlation in eq A3 represent the data to an accuracy of f0.276. These data were interpolated or extrapolated to obtain the viscosity a t any other composition, say W* and w*3, using the following procedure: (i) At W*,viscosity was determined for w3 = 0.41, 0.49,0.59, and0.71; (ii) logarithm of viscosities so calculated were correlated with w3 by a polynomial of the form
0.8
,
b2
In p = b, + b,W, + b3W; (A31 where Wz is weight fraction of toluene on polymer-free basis defined as
M3
1.3 io5 5.7 x 106 1.0 x 108
e
bi
measurements at different shear rates indicated Newtonian behavior. The data at each w3 was correlated according to the equation
Munch et a1 (1983),T=293.8K
e
~
wa
0 0
2
4 P
3
4
6
8
5
4 g cni3
10
12
14
Figure 4. Diffusivities in polystyrene-poor solvent systems.
centrations for toluene-polystyrene system were therefore obtained by extrapolating the curve for the generalized Maxwell-Stefan diffusivity coefficient and applying the suitable thermodynamic correction according to eq A2. Diffusivity data for cyclohexane-polystyrene are shown in Figure 4 along with the data of other investigators. As can be seen, the behavior is much more complex in this system with a minimum in diffusivity. Equation A I was once again used to predict diffusivities for this system. However, even the observed qualitative trends could not be predicted. The dependence on concentration could not be made simpler by obtaining the generalizedMaxwellStefan diffusivity, as was possible in the case of toluenepolystyrene system. As was suggested by Munch et al. (1983) for polymer solutions in poor solvents, the diffusivities were normalized by the value at infinite dilution, 0'13, and were plotted against a parameter a = p 3 ( & . l 104)0.5,where 104 is the molecular mass of styrene. The parameter a may be interpreted as the degree of overlap of polymer domains. Figure 4 also includes the data of other investigators on polymer solutions in poor solvents. The data suggest that the normalized diffusivity first decreases and, after reaching a minimum value, increases nearly linearly with a slope of 0.16. Though the data of Rehage et al. (1970) do show a slight curvature, the data of other workers (Adam and Delsanti, 1977; Roots and co-workers, 1979a,1979b,1980)indicate linear dependence with a slope of 0.16. Thus the data obtained in the present work were extrapolated to higher concentrations by using the linear dependence of the normalized diffusivity, 0 1 3 1 D'13, on the parameter a. Viscosity Data. Viscosity of binary and ternary polymer solutions were measured at 308K for four different polymer weight fractions: 0.41,0.49,0.59, and 0.71. The
In p = c1 + ~ 2 + ~c3w32 3 (A5) and (iii) the viscosity at the desired w*3 was calculated from eq A5.
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7
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