Mass Transport Enhancement in Modified Supercritical Fluid

Ind. Eng. Chem. Res. 1999, 38, 3505-3518

3505

Mass Transport Enhancement in Modified Supercritical Fluid Kamal Abaroudi, Fakher Trabelsi, Barbara Calloud-Gabriel, and Francesc Recasens* Department of Chemical Engineering, ETS d’Enginyers Industrials de Barcelona, Universitat Polite` cnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain

In this paper, the supercritical-fluid extraction (SCFE) of a packed bed of β-naphthol-impregnated porous pellets was studied. An increasing number of industrial SCFE processes involve the extraction of a solute retained within a porous matrix, usually in the form of seeds or irregular grains. The interest in high-pressure extraction is due to certain advantages of dense gases and near-critical solvents over conventional liquid solvents. In this study, modified carbon dioxide was the fluid studied. The effects of temperature, pressure, fluid velocity, particle size, and gravity were experimentally studied using carbon dioxide, pure or mixed with varying amounts of toluene (6%, and 10%). For the solute, β-naphthol, the solubilities in SC carbon dioxide mixtures (from 0 to 10% toluene) were available from separate experiments. The dispersed plug-flow model was used to describe the nonideal flow. Fitting the experimental data with the model solution allowed the measurements of the fluid-to-particle mass transfer coefficient, the intraparticle diffusivity, and the axial dispersion coefficient (the latter in terms of the axial Peclet number). The influence of cosolvent concentration on the three transport parameters, which were not available so far, is presented. Introduction Supercritical-fluid extraction (SCFE), also known as dense gas or near-critical solvent extraction, is increasingly used in the chemical, agricultural, food, pharmaceutical, and fine chemicals industries as well as in some of the environmental and hazardous waste control processes. In all of these applications, the fluid of choice is a key factor for effectiveness and economic profitability. The SC solvent is often a compromise between commercial availability, cost, safety, and manipulation ease. From this point of view, carbon dioxide is usually the fluid employed in most of the SCFE plants, because its critical state for carbon dioxide (31 °C, 73.8 bar) is readily attainable in industrial practice. A general drawback for SC processes is the relatively high pressures required in comparison to standard processes. So when the pressures for extraction are much above the critical for carbon dioxide (>150 bar) or the polarity of the fluid is not suitable for the extracted solute, the use of a modifier or cosolvent is recommended. If a cosolvent is available, the operating pressure can generally be lowered. In the literature,1 oxygenated organics such as acetone, ethanol and other lower alcohols, aromatics, chlorobenzenes, and the like have proved to be efficient cosolvents. Among the criteria for modifier selection, Sunol et al.2 emphasize the interaction between solute and cosolvent. This is very important not only in the extraction stage but also in the regeneration stage. As pointed out by Brunner,3 three fundamental objectives are met by an effective modifier. First, the solubility of the solute in the SCF is increased. Second, the dependence of solubility on the pressure and temperature is enhanced. And finally, the separation factor in the presence of a modifier is also increased. * To whom correspondence should be addressed. Telephone: +3434016793.Fax: +3434017150.E-mail: [email protected].

In the present study, β-naphthol, a solute of very low polarity, was used; therefore, a nonpolar modifier such as toluene was chosen. For β-naphthol, solubility data and other properties are available, both in pure carbon dioxide at low temperature4 and in toluene-modified SC solution.5,6 For the binary system toluene-CO2, vaporliquid equilibrium data and pressure-volume-temperature (PVT) relations for the mixture are available.7 It is remarkable that the sole presence of cosolvent allows an increase in solubility,8 even in the absence of intermolecular forces such as those due to hydrogen bonding or dipole-dipole interactions. While there is a great deal of data on solubility enhancement due to the presence of a cosolvent, very little information is available on the effects of the modifiers on the intrinsic kinetic parameters. It is clear that the driving force term for dissolution is increased by the enhancement of solubility. In this paper, we wanted to know in turn about the change in the transport coefficients with the presence of cosolvent. If solubilization involves solvation of solute molecules by cosolvent, one would expect mass transfer parameters to either remain the same or be somewhat lowered, since solute is bulkier in the solvated form and hence diffuses more slowly. To model extraction kinetics data, a porous solid with well-defined geometry was required. For this purpose, we used cylindrical porous pellets obtained by sintering steel powder to a given density. To interpret rates at the pellet level, the shrinking core model was employed as has been suggested by Jones9 and used by Goto and co-workers,10 and by ourselves.11,12 This type of approach was first used by Knaff and Schlu¨nder13 for mass transfer in SCF in porous media. Under certain conditions, an analytical solution to the shrinking core model has been given by King and Catchpole,14 but in general, a numerical solution is needed. In this work, we performed the extraction of the cylindrical pellets impregnated with solid β-naphthol,

10.1021/ie990105e CCC: $18.00 © 1999 American Chemical Society Published on Web 07/23/1999

3506 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999

Figure 1. Experimental apparatus.

to measure the intraparticle diffusivity, the external mass transfer coefficient, and the axial dispersion coefficient for a packed bed of pellets. The above porous matrix is justified as a model in which transport parameters for pure CO2 and modified CO2 were previously studied, in particular, tortuosity factors.12 Our objective in this paper was the measurement of the external mass transport coefficients such as the particleto-fluid mass transfer and the axial dispersion coefficient, in terms of the total Sherwood number and the axial Peclet number, respectively. If the shrinking core model for internal diffusion is assumed, the intraparticle diffusivity can also be obtained. These three parameters were studied as a function of pressure, temperature, fluid velocity, flow direction, pellet size, and cosolvent concentration. Experimental Methods Extraction Apparatus. The extraction setup (Figure 1) used has been described in detail elsewhere,15 so only the most relevant features will be given. Carbon dioxide (alone or with cosolvent) was pumped with pump P-1 as a subcooled liquid at 275 K to avoid cavitation of the pump. Pressure oscillations were dampened with a pressure-control loop installed before entering the extractor (Dense Gas Management System from Marc Sims, Berkeley, California). Pressure at all points was measured with Wika (Sabadell, Spain) manometers with an error of less than 2 bar. Temperatures were measured with Pt 100-ohm resistance thermometers (Crison, Alella, Spain). The extractor vessel was made of a large cylindrical shell (25 cm length, 3 cm inner diameter), wherein the small extractor cell could be inserted and threaded. The cell was a stainless-steel cylinder whose ends were made of sintered steel plate. The cell allowed us to work with small samples (about 30 cm3). The fluid was distributed before entering the small cell bypass through a packed bed of glass spheres (3 mm diameter) located into the large vessel. The twovessel set is a commercial design of Marc Sims. Discharge of the fluid from the pump in the oven containing the extraction vessel, ensured the desired P and T conditions. A micrometric valve located at the extractor exit in turn ensured a constant gas flowrate. The oven

temperature was controlled within (0.5 K, and the temperature of the micrometric valve was set at 433 K to avoid unwanted condensations during expansion to atmospheric conditions. Condensation of the stream leaving the valve was done into three U-shaped glass tubes in series submerged in dry ice. To avoid condensate losses, an aluminum-cotton wire mesh demister was installed in the exit from the last U-tube. Gas flowrate was measured accurately with a dry-wet meter (equipped with temperature reading) and a rotameter. To check cosolvent fraction during a series of runs done with the same bottle, the condensate of gas from a blank run (without extractor) was analyzed for toluene. This was found to be about constant for a newly opened bottle and for a half-consumed bottle. Other Experimental Methods. Two types of extraction runs were made. The first type of extractions was at very low flowrate of fluid, thus under quasiequilibrium conditions. From these experiments, solubility data could be measured for β-naphthol in SC carbon dioxide (pure and modified with toluene). A detailed account on this method has been given in Abaroudi.16 The second type of runs was at conditions far from the equilibrium solubility: that is, they were done in the mass transfer-controlled regime (fast fluid flowrate), thus not allowing equilibrium solubility to be reached. The sintered metallic pellets used were supplied by AMES (Sant Vicenc¸ dels Horts, Barcelona, Spain). AMES also supplied the method of oil impregnation of porous pellets. Before impregnation, the pieces were cleaned by boiling trichloroethylene and dried perfectly. The impregnation procedure was based first on the airdisplacement from the pores effected by vacuum and then followed by soaking the pellets in molten β-naphthol under nitrogen.17 The pellet dimensions are given in Table 1 together with other experimental details. The degree of impregnation of the total pellet porosity was about 87% of the total pore volume. Table 2 gives the calculated critical data for pure carbon dioxide and its mixtures with toluene. Critical pressures were estimated using the Spencer et al. method18 as recommended by Reid et al.19 The values so obtained agree well with those reported for the values of the convergence pressures determined experimentally

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3507 Table 1. Scope of Experimental Conditions parameter

value

fluids used solute extracted pressure, bar temperature, K Reynolds number Schmidt number Grashof number

CO2, CO2 + 6% toluene, CO2 + 10% toluene β-naphthol 150-200 358-368 9-88 2.1-4.5 3 × 105-7.5 × 107 bed characteristics

height, m cross section, m2 void fraction pellet porosity (vol fraction) shape pellet length, mm pellet diameter, mm

small pellets

large pellets

0.04 3.46 × 10-4 0.4 0.205 cylindrical 8 8.2

0.0804 7.00 × 10-4 0.39 0.25 cylindrical 20 11.6

Table 2. Critical Properties of Pure CO2 and of Mixtures (CO2-toluene) supercritical fluid pure CO2 CO2 + 6% toluene CO2 + 10% toluene

Tc 106 × Vc Fc 105 × µc Pc (bar)a (K)a (m3/mol)a (kg/m3)b (Pa s)c 73.8 105.0 140.0

304 331 353

93.90 90.13 89.06

468 482 481

3.763 3.909

a Reid et al.19 b Peng-Robinson equation of state, Peng et al.47 k12 ) 0.0751. c Reid et al.,19 method of Chung et al.

by Fink and Hershey7 for the system carbon dioxidetoluene. Operating pressures in our experiments were well above those of Fink and Hershey (see Table 2), so well in the SCF region. Packed Bed Preparation for Extraction. The extraction cell was a cylinder holding about five layers of cylindrical pellets. In the bottom (near fluid entrance), 5 g (1 cm height) of glass beads (3 mm diameter) were deposited. Then, five layers of three pellets each (1.45 g β-naphthol) were deposited on top of the glass beads. The extraction was started as follows. With the micrometric expansion valve correctly set at the desired value, the on-off valve was rapidly opened, thus corresponding to zero time. Minor flow adjustments were then made on the micrometric valve based on the rotameter readings. Exact flowrates were later based on the more reliable totalizer readings. During an extraction run, the fraction extracted was measured by determining the naphthol concentration. At each time interval, the three-U tube was analyzed for solute. Generally, more than 95% recovery of the naphthol initially within the pores was possible. The naphthol analysis was made by gas chromatography using the internal standard technique. The U-tube was washed with internal standard solution (2000 ppm dodecanol), and then injected into a Shimadzu GC-8A gas chromatograph apparatus equipped with a FID detector and a SE-30 column. Naphthol and dodecanol peak ratios were resolved very well so they could be quantitatively analyzed on the SE-30 column under temperature-programmed conditions.16 Extraction Modeling The extraction with supercritical carbon dioxide of metallic porous pellets impregnated with β-naphthol and seeds in fixed bed has been modeled with the shrinking-core model.10,12-14,20 The shrinking core concept is acceptable because the solute remains solid at

the prevailing pressure and temperature and the gas is insoluble in the solid core. Then, a sharp limit between the fluid and the solute within the particle at the core radius is expected. In this model, the solute is supposed to be distributed uniformly in the solid matrix, and the latter does not have affinity for the solute. It is assumed further that the solute filling the pores locally saturates the fluid at the solute core before diffusing through the pores, up to the external surface of the particle. In the external surface, a solid-moving fluid boundary layer is formed in which the solute has to diffuse by mass transfer. Consequently, the nonextracted core shrinks regularly during the extraction process as more solute is drawn by the fluid. Catchpole et al.14 and Goto et al.10 have adapted this model for a spherical geometry to describe the leaching of seeds. Stu¨ber et al.12 solved the case of both spherical and cylindrical pellets with either sealed or opened ends. In most of such papers, the influence of axial dispersion of solute in the packed bed on the extraction kinetic in supercritical conditions is disregarded either to simplify equations or because differential contact was actually achieved.12,15 In this study, the shrinking core model is used together with the hypothesis of cylindrical porous pellets with open ends in an integral packed bed. Mass transfer due to axial dispersion will be considered in view of the relatively large particles used. Consider that extraction is performed in a cylindrical bed of length Le. The following hypotheses are followed: (a) The pellets as well as the packed bed are considered isotropic with regards to diffusion. (b) The solvent flows axially with an interstitial velocity u. (c) Pressure, temperature, flowrate, and bed void fraction are constant along the bed. (d) The external effective mass transfer area is that of pore mouth. (e) Within the cylinders, the solute is supposed to be extracted from the core by diffusion in the radial direction through the cylindrical walls and, axially, from the two core planes through the two lateral flat plate ends. (f) Radial mass dispersion gradients within the bed are negligible. With the above assumptions the conservation of solute in the fluid phase and in the particle, respectively, is

{

1 - B ∂2C ∂C ∂C 2 + u ) - kgp -Dax 2 + [C - Ci(R)] + ∂t ∂z R B ∂z R [C - Ci(L)] (1) 2L

{

}

}

∂q 2 R )  k [C - Ci(R)] + [C - Ci(L)] ∂t R p g 2L

(2)

With initial and boundary conditions,

r ) R at t ) 0

(3)

l ) L at t ) 0

(4)

C ) 0 at t ) 0

(5)

Ci ) Csat at r ) rc

(6)

Ci ) Csat at l ) lc

(7)

and further Dankwerts conditions at z ) Le and z ) 0

∂C uC - Dax ) 0 z ) 0 ∂z

(7a)

3508 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999

∂C ) 0 z ) Le ∂z

(7b)

The initial and boundary conditions now are

Conditions for through the external boundary layers are

( ) ( )

∂Ci De ∂r

∂Ci De ∂l

r)R

l)L

) kg[C - Ci(R)]

(8)

) kg[C - Ci(L)]

(8′)

Extraction Rates. At phase boundaries in terms of driving forces and mass transfer, extraction rates are as follows. (a) Mass Transfer in the Pores.

( )

(9)

rc

( )

lc

(19)

xi ) 1 at ξ ) ξc

(20)

xi ) 1 at η ) ηc

(21)

( )

∂rc ∂N ) 2πrc2lcqp ∂t ∂t

( ) ( ) ∂xi ∂ξ

∂xi ∂η

(11)

rc

Flat plate surface,

(21b) (21c)

) Bi[x - xi(1)]

(22)

L ) Bi [x - xi(1)] R

(22′)

ξ)1

η)1

xi(1) )

xBip ln ξc - 1 Bip ln ξc - 1

(23)

and for a flat plate

(12)

lc

xi(1) )

(c) External Mass Transfer.

xRLBip(1 - ηc) + r2c

(24)

RLBip(1 - ηc) + r2c

With x from eq 23 substituted into eq 16, the radius of the remaining solid core is obtained as

Cylindrical lateral surface, -∂N/∂t ) 2πR2lckgp[Ci(R) - C]

(13)

Flat plate surface, -∂N/∂t ) 2πrc2kgp[Ci(L) - C]

(14)

Using dimensionless variables, the model equations become

{

1 - B a ∂2x ∂x ∂x ) -2Bip +a + [x - xi(ξ)] + 2 Pe ∂Z ∂Z ∂Fo B R [x - xi(η)] (15) 2L

}

with

{

1 ∂x )0 Pe ∂Z

Equating eqs 9 and 13 and 10 and 14 and using eqs 20 and 22 and 21 and 22′, respectively, gives an expression for xi(1) for a cylinder and a plate. For the sealed end cylinder we get

Cylindrical lateral surface,

( )

x ) 0 at Fo ) 0

At the solid-fluid interphase, we have

(10)

(b) Solid Core Dissolution.

∂lc ∂N ) πr2c qp ∂t ∂t

(18)

at Z ) 0 x -

For the flat plate surface,

-

η ) 1 at Fo ) 0

at Z ) 1 ∂x/∂Z ) 0

∂Ci ∂N ) 2πrc2lcDe ∂t ∂r

∂Ci ∂N ) πr2c De ∂t ∂l

(17)

With the two Dankwerts conditions,

For the cylindrical lateral surface,

-

ξ ) 1 at Fo ) 0

}

R ∂y ) 2Bib [x - xi(ξ)] + [x - xi(η)] ∂Fo 2L

(16)

in which the dimensionless variables and parameters are

x ) C/Csat xi ) Ci/Csat ξ ) r/R η ) l/L Z ) z/Le a ) uR2/DeLe y ) q/q0 Fo ) (De/R2)t b ) Csat/q0 Pe ) Leu/Dax Bi ) kgR/De

]( )

[

∂ξc 1 - B x-1 ) 2Bi b ∂Fo 1 - Bip ln ξc B

(25)

Substituting eq 24 into eq 16, the rate of height change of the nonextracted flat-plate core is obtained as

[

]

( )

1 - B ∂ηc x-1 R ) Bi b ∂Fo L L Bip B (1 - ηc) + 1 2 R ξ c

(26)

Finally, the dimensionless mass balance is obtained:

( )

1 - B ∂x ∂x a ∂2x +a + (1 - x) × ) 2Bip 2 Pe ∂Z ∂Z ∂Fo B 1 R 1 + (27) Bi 1 - Bip ln ξc 2L p L 1+ (1 η ) c R ξ2 c

[

]

Equations 25-27, together with eqs 17-22′ constitute the extraction model for cylindrical porous pellets with open ends using the shrinking core model. Sensitivity to Model Parameters. The study of the sensitivity is made by calculating the deviation of the

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3509

Figure 2. Model sensitivity to parameters.

extraction curves obtained experimentally compared with those inferred from the shrinking core model. To check the effects of parameters, the Biot, Peclet, and the a parameters are varied widely, and their influence is measured in terms of AARD. From Figure 2, we can see that the model is strongly sensitive to the Biot number and to the a parameter. The latter is the ratio between the dimensionless diffusion time and the contact time of the supercritical fluid with the impregnated pellets in the packed bed. On the other hand, Figure 2b shows that the model is less sensitive to the Peclet number; therefore, it is anticipated that less accurate Peclet values will be obtained. Limiting Cases. The first case corresponds to the internal diffusion limitation (Biot number going to infinity). The second one refers to the external mass transfer limitation (Biot number going to zero). To calculate the values corresponding to these two extreme cases, we substitute a and Fo by their expressions in the system of equations.25-27 The following equations are then obtained:

[

]

∂ξc 2b(x - 1) 1 - B RkgDe ) 2 ∂t B De - Rkgp ln ξc R

[

(28)

]

RkgDe ∂ηc b(x - 1) 1 - B (29) ) ∂t RL B L p Rk (1 - ηc) + De R gξ2 c

A new expression is therefore obtained for the fluid phase mass transfer balance:

u ∂2x 2 1 - B u ∂x ∂x + ) +  (1 - x) × PeLe ∂Z2 Le ∂Z ∂t R2 p B RkgDe RkgDe R + (30) De - Rkgp ln ξc 2L p L De + Rkg 2(1 - ηc) R ξc

[

]

Case (a) Internal Diffusion Control. In this case, we have Rkg . De. Taking the limit for Bi f ∞ in the eqs 28-30, the following expressions are obtained:

( )

(31)

( )

(32)

∂ξc 2b(x - 1) De 1 - B ) ∂t B R2 ln ξ p

c

∂ηc ξ2c b(x - 1) De 1 - B ) ∂t B L2(1 - η ) c

-

( )

1 - B u ∂2x u ∂x ∂x + )2 + (1 - x) De × PeLe ∂Z2 Le ∂Z ∂t B

[

]

ξ2c 1 + (33) R2 ln ξc 2L2(1 - ηc)

Case (b) External Mass Transfer Control. In this case we have Rkg , De, a situation corresponding to the beginning of the extraction. Taking the limit of Bi f 0 in eqs 28-30, the following expressions are obtained:

( )

∂ξc 2b(x - 1) kg 1 - B ) ∂t R B

(34)

3510 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999

( ) ( )

∂ηc b(x - 1) kg 1 - B ) ∂t L B

(35)

1 - B u ∂x ∂x u ∂2x + (1 - x) × + ) 2 p PeLe ∂Z2 Le ∂Z ∂t B

(

)

1 1 + (36) R 2L

kg

An analytical solution to eq 36 is possible considering the steady-state extraction regime.21,22 If we let the accumulation term be zero in eq 36, one gets

( )( ) ( )(

1 - B dx Pe 1 d2x 1 - Pe kg + x) - 2p 2 dZ u  R 2L dZ B 1 - B 1 Pe 1 - 2p kg + (37) u B R 2L

)

which is a second-order differential equation with constant coefficients, and this allows analytical solution. This is,

xj )

-β s(s - Rs - β)

β β es1Z es2Z s1(s1 - s2) s2(s2 - s1)

(39)

with

s1,2 )

R ( xR2 + 4β 2

2kgγexp{(Pe - xPe2 + 4kgγ/2)}Le Pe2 + 4kgγ - PexPe2 + 4kgγ

-

2kgγexp{[(Pe + xPe2 + 4kgγ/2)Le} Pe2 + 4kgγ + Pe2xPe2 + 4kgγ

where

γ)

( )(

T (K)

F (kg/m3)a

120 150 150 200 200

348 358 368 358 368

310 449 388 616 554

150 150 200 200

358 368 358 368

150 150 200 200

358 368 358 368

105 × µ (Pa s)

108 × D (m2/s)b

Csat (kg/m3)c

4.29 2.69 3.27 1.78 2.12

0.139 1.048 1.187 2.002 2.21

CO2 + 6% Toluene 440 3.62e 387 3.32e 582 4.73e 528 4.30e

2.83 3.36 1.92 2.25

2.162 2.679 4.046 4.883

CO2 + 10% Toluene 483 3.94e 423 3.55e 614 5.13e 560 4.62e

2.68 3.24 1.87 2.18

2.577 2.744 4.736 5.670

Pure CO2 2.78d 3.26d 3.42d 3.80d 3.97d

a From the Peng-Robinson EOS.47 b Calculated with the Catchpole-King correlation,48 for the β-naphthol diffusivity in CO2 + toluene, as D ) (D1D2)1/2. (Wakao and Kaguei49). c Calculated with Reisenberg.50 d From Reid et al.19 e Abaroudi et al.6

for kg once Pe is known. Initial values of kg and Pe were calculated by the formulas (mass transfer correlations) given in Lim et al.23 and Tan et al.,24 respectively. On the other hand, taking the value of the initial slope (dm/ dt)0 of the experimental extraction curve, the flow rate of the gas, and the concentration of β-naphthol in the supercritical fluid, we can infer the x value (mass fraction extracted) as follows:

m(t) )

Methods of Parameter Optimization and Variable Reduction. The external mass transfer coefficient and the Peclet number during short time intervals are related for a given initial extracted rate. By substituting all of the parameters in eq 39 with their definitions, a useful relation between kg and Pe is obtained, provided that an initial value is available for the extraction rate. Thus, during the fitting process, one of the parameters of the model can be eliminated, rendering the problem bivariate instead of trivariate in the optimization procedure. This allows adjusting only two from the initial three parameters (a, Pe, kg). This provides more confidence to the final values supplied. The final expression of x at short times is

x)1-

P (bar)

(38)

2

whose solution is:

x)1-

Table 3. Physical Properties of Carbon Dioxide, Carbon Dioxide-Toluene Mixture, and Saturated Concentration of β-Naphthol in Them

)

1 - B 1 1 Pe 2p + u B R 2L

(40)

The following methods for parameter fitting were implemented. Equation 40 was solved by an iterative method. The regula falsi algorithm was used to solve

∫0tQCg dt

It is assumed by hypothesis that the gas flow is constant and saturated with solute at the beginning of the extraction. Iterations are stopped when a good precision is reached. If the values of Pe and kg are known, the system of the partial differential equations (eqs 25-27) are then solved to obtain the concentration of the β-naphthol in the fluid and therefore the mass fraction extracted from the metallic sintered pellets in the packed bed versus time. The numerical solution of the model equations was performed using a MOLCH,25 a computer program based on the method of lines.26 Results and Discussion The rate of SCFE of a packed bed of porous particles depends on six variables: pressure, temperature, gravity, pellet size, fluid velocity, and composition of the fluid phase. In this work, external and internal mass transfer coefficients were studied as a function of the experimental conditions. Extraction rate depends strongly on solubility, thus providing the driving force necessary for dissolution in all cases. Solubility, being a equilibrium property, also depends on some of the above variables. These are gas composition, pressure, and temperature. The other variables affect extraction in a hydrodynamic manner (fluid velocity, gravity, and pellet size) or by intraparticle diffusion (pellet size), Therefore, before proceeding with the mass transfer study, solubilities were determined in separate experiments, using the quasiequilibrium method.16 Table 3 reports the solubilities16 of β-naphthol as a function of temperature,

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3511 Table 4. Percentage Increase of Extraction Parameters at 368 K Relative to the Base Temperature (358 K) at Constant Pressure (200 bar), and Flow Rate (1 L/min)a

supercritical fluid pure CO2 CO2 + 6% toluene CO2 + 10% toluene a

increase in increase in solubility, increase increase fraction % in mass, % in De, % extracted, % 23.0 33.0 22.5

16.6 28.8 34.4

(184.9) 28.49 19.48

21.5 26.9 24.6

STP conditions.

Figure 3. Effect of temperature on extraction (pressure ) 200 bar; flowrate ) 1 L/min; small pellets, upflow). For (a) pure CO2 and (b) CO2 + 10% toluene.

pressure and toluene fraction, together with molecular and transport properties. For pure carbon dioxide, interphase mass transport coefficients were reported by several authors.12,13,15,27-30 In carbon dioxide modified with cosolvents, no data are available for the interphase mass transport. Influence of Temperature on Extraction. Figure 3 shows the effect of temperature on extraction of β-naphthol at constant pressure (200 bar), constant flowrate (1 L/min), and two temperatures (85 and 95 °C) using pure and modified carbon dioxide with 10% toluene. In these experiments the hydrodynamic regimes are approximately constant on the basis of the calculated change in dimensionless numbers. So, the change in the Reynolds, Schmidt, and Peclet numbers is very small (Re ) 14-19, Sc ) 3.4-4.5, and Pe ) 3.13.3). The increase in temperature (85-95 °C) increases the extracted quantity for the three fluids (pure CO2 and 6% and 10% toluene-modified carbon dioxide). See Figure 3. It is seen that in our case the retrograde solubility does appear, because conditions are far removed from the critical state of the fluids (see critical pressure values in Table 2). The observed increase in extraction rate is interpreted in terms of the solubility itself (see Table 3) and, to a lesser extent, to the increase of the effective diffusivity with temperature (see Table 4). In view of the results reported on Figure 3 and Table 4, it is concluded that for other variable remaining constant it is better to operate at 95 °C than at lower temperatures for a fast extraction process. Table 4 summarizes our results. It shows the increase of the fitted external mass transfer coefficient with temperature and with cosolvent fraction. It also shows that the effective diffusivity increases with temperature,

Figure 4. Effect of pressure on extraction (temperature ) 368 K; flowrate ) 1 L/min; small pellets, upflow). For (a) pure CO2 and (b) CO2 + 10% toluene.

but less rapidly in the presence of cosolvent. Given in Table 4 are the solubilities for pure and modified carbon dioxide. The highest temperature effect on solubility is observed for 6% toluene-modified CO2. Influence of Pressure. Figure 4 shows the extraction behavior when temperature and flowrate are held constant and pressure is changed from 150 to 200 bar (for pure carbon dioxide, and 10% toluene-carbon dioxide). In these runs the hydrodynamic regime is changed only slightly (Re ) 14-21, Sc ) 2.6-3.6, and Pe ) 2.83.3). We thus conclude that in these and the above runs the hydrodynamic pattern would not change very much with either pressure or temperature (see above). An increase in pressure allows one first to increase the extraction rate during the first points because of solubility increases11,15,28 and a decrease in the external mass transfer coefficients and the effective diffusivity (see Table 5). These observations were reported previously.12,20,31 See solubilities in Table 3. The effect of pressure is very important in the pure CO2 case, but it levels off for increasing toluene concentrations (see Table 5). We therefore conclude that it is necessary to work at high pressure in the case of a pure fluid, but the pressure can be lowered if the fluid

3512 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 Table 5. Effect of Pressure Increase (from 150 to 200 bar) in the Extraction Behavior Relative to 150 Bar at Constant Temperature (368 K) and flowrate ) (1 L/min)a

supercritical fluid

increase decrease decrease increase in in in in the fraction solubility, % mass, % De, % extracted, %

pure CO2 CO2 + 6% toluene CO2 + 10% toluene a

51.2 33.6 45.4

2.5 7.7 34.1

65.4 81.7

10.4 2.0

STP conditions.

is modified with cosolvent. The lower pressure limit would be the critical pressure for the prevailing composition and temperature. In our case, it is clearly better to work at lower pressure (150 bar) than at a higher pressure (200 bar). The strong pressure effect in the extraction yield for neat CO2 is explained in terms of the solubility dependence on pressure (Table 3), so that the higher solubility driving force clearly offsets a slight drop in the diffusional properties far from the critical point (see Table 5). For the case of modified carbon dioxide (6 and 10% toluene), the increase in solubility is less apparent than for pure CO2 (Table 3). As shown in Table 5, the external mass transfer coefficient and the intraparticle diffusivity decrease more than they do in pure carbon dioxide. This explains the slight influence of pressure on the extraction rate for carbon dioxide modified with toluene. Influence of Cosolvent. Although experimental studies are available on the effect of cosolvent on solubility and on extraction curves in general,32-34 an interpretation of the extraction rate enhancement in terms of the diffusional parameters has not been made so far. In this work, we have attempted to improve the naphthol extraction rate from porous metal pellets using toluene as a modifier, and we tried to see how toluene concentration affects the values of the transport coefficients. In Figure 5, the effect of cosolvent on extraction is presented as a function of fluid velocity (at 150 bar and 95 °C). Results are also available at 200 bar, for two temperatures (85 and 95 °C), at a constant flowrate (1 L/min). In all cases, the extracted fraction increases with the fraction of cosolvent. According to the solvation theory, the external mass transfer coefficient and intraparticle diffusivity would decrease with the amount of cosolvent, but the increase in solubility is dominant. After the parameter values are fitted, it is seen that both parameters increase as well with modifier concentration but less than solubility does (see Table 6). Table 6 indicates that the mass transfer coefficient increases with an increase in cosolvent concentration. In addition, the increase in solubility and effective diffusivity is less important with more cosolvent. This explains why the extracted fraction is higher at the beginning (in going from 0 to 6% toluene) than afterward when toluene concentration is already fairly large (in going from 6% to 10% toluene). The influence of cosolvent has also been studied at 150 bar and 85 °C and at 200 bar, 95 °C, and 0.5 L/min for upflow conditions. For downflow conditions, carbon dioxide neat and with 10% toluene have been used. From these runs, it is concluded that solubility increase with the fraction of cosolvent. The change in solubility and effective diffusivity is not affected by gravity action (upflow or downflow operation).

Figure 5. Effect of cosolvent for different fluid flowrate (pressure ) 150 bar; temperature ) 368 K; small pellets, upflow): (a) 1 L/min and (b) 4 L/min (gas volumes at STP conditions).

Influence of Fluid Flowrate. Figure 6 illustrates the influence of flowrate for constant pressure and temperature (150 bar, 95 °C) for three flowrates using pure CO2 and modified CO2 with 6 and 10% toluene. In these experiments, the values of the Schmidt and the Peclet numbers are about constant (Sc ) 2.6 and Pe ) 3), whereas Reynolds and Grashof numbers changed widely (Re ) 20-88 and Gr ) 2.2 × 106-5.1 × 106). In general, the result of increasing flowrate is to increase the amount extracted for all fluids. This suggests that the external mass transfer coefficient is responsible for the increase in extraction rate, since the intraparticle diffusivity and solubility remain constant with fluid velocity. An increase in flowrate may also give a better contact pattern, but since the Pe number is about constant, this latter effect is relatively unimportant. The increase in flowrate strongly affects the first experimental data points where the external mass transfer effects on the boundary layer are dominant over intraparticle diffusion. This is more apparent in changing the flow from 1 to 2 L/min. For this reason, the higher flowrate was preferred. Effect of Pellet Size. The influence of the size has been studied using the larger pellets using both pure carbon dioxide and CO2 modified with 6% toluene (Figure 7). In these runs, the Reynolds and Grashof numbers changed because the pellet sizes changed (Re ) 18-25 and Gr ) 4 × 106-73 × 106). On the other hand, the Schmidt and Peclet numbers practically did not change (Sc ) 2.6-3.4 and Pe ) 2.9-3.4). For constant pressure, temperature, and flowrate (150 bar, 95 °C, and 1 L/min), the effect of increasing the limiting particle dimension is to decrease extraction yield (for 6% toluene). Since for larger pellets, the

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3513 Table 6. Relative Increases in Solubility, Mass Transfer Coefficient, and Effective Diffusivity with Cosolvent P (bar)

T (K)

Q (L/min)a

pure CO2

CO2 + 6% toluene ×104

150 150 150 200 200

368 368 368 368 368

1 2 4 1 1

8.1 8.1 8.1 9.9 12.2

CO2 + 10% toluene

solubility

21.1 21.1 21.1 21.2 28.2

19.8 19.8 19.8 23.5 28.8

×105 Mass Transfer Coefficient (kg)c 150 150 150 200 200

368 368 368 358 368

1 2 4 1 1

1.81 3.25 4.79 1.51 1.77

2.03 4.40 6.02 1.46 1.88

3.30 7.77 9.30 1.62 2.17

×108 Effective Diffusivity (De)c 150 150 150 200 200 a

368 368 368 358 368

1 2 4 1 1

0.10 0.10 0.10 (0.14) (0.40)

1.38 1.38 1.44 0.37 0.48

2.69 2.88 2.53 0.41 0.49

At STP. b In mass fractions. c From model fitting.

Figure 7. Influence of pellet size on extraction rate (small cylinders ) 8 × 8 mm and large cylinders ) 12 × 20 mm).

Figure 6. Effect of fluid velocity for different cosolvent contents (pressure ) 150 bar; temperature ) 368 K; small pellets, upflow): (a) 0%, (b) 6%, and (c) 10%.

internal diffusion path is larger, the observed extraction time is correspondingly longer. In this way, the net effect of increasing particle size is to shift the controlling resistance to intraparticle diffusion. The same effect is observed using pure carbon dioxide (at 200 bar, 95 °C, and 1 L/min). Flow Direction and Free-Convection Effects. We next look at the effects of gravity on the extraction using pure and modified carbon dioxide (10% toluene). Some of the results are given in Figure 8. In these runs, the Reynolds, and Schmidt number were Re ) 8-11 and

Sc ) 3-4, respectively, and the Grashof number varied as Gr ) 2 × 106-8 × 106. A small flowrate was employed in order to be in the zone where the effects of gravity flow should appear. At 200 bar and 95 °C, the effect of inverting the flow direction from upward to downward was to slightly increase the amount extracted, for both pure CO2 and for modified carbon dioxide. At 150 bar and 85 °C, the effect of changing the flow direction from upflow to downflow was to increase the extraction yield for pure CO2 but to decrease it for 10% toluene. From these results, we can say that we are in the range of variable values where gravity and flow instability might both affect our results, and they can do so in opposite manners. There would be competing effects between gravity-driven mass transfer, on one hand, and favorable mixing effects due to flow instability, on the other hand. Since these effects are both very small, they can reversed easily, as shown in Figure 8. Recently, Benneker et al.35,36 have shown that this type of flow instability may change the pattern brought about by free convection, due to a change in the mixing effects. Figure 9a summarizes the values of the Grashof number for our experiments compared with other authors’ values and our previous experimental values. In Figure 9b, we see the fraction of the contribution of

3514 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999

Figure 8. Influence of flow direction on extraction rate for similar values of pressure and temperature (constant fluid flowrate). UF: upflow. DF: down flow. Continuous lines: pure CO2. Interrupted lines: CO2 + 10% toluene.

natural convection to total convective mass transfer (in terms of Sherwood numbers) versus Reynolds number. The Sherwood number is based on a previous correlation developed under strong free convective effects combined with forced convection.11 It is seen that for our data there is a very small contribution of free convection (maybe 10-20% of the total) since our data for the Grashof number are 100-1000 times lower than those of other authors who noted their effects. This may be due to the relatively low solubility in both CO2 and modified CO2; in this manner, the flow direction or gravity would be expected to have little influence on the overall forced flow. A similar observation was made by Knaff and Schlu¨nder13 in comparing the extraction of naphthalene with that of caffeine. The latter compound behaved without apparent free convective effects as compared to the former, due to its much smaller solubility. The effect of flow direction on extraction curves has been addressed by many authors.11,28,36-39 It seems fairly well established that there is an important effect at high Grashof numbers, due to a large gravity force difference between the saturated solution and the fluid (see Figure 9a). An upflow extraction curve above the downflow curve is interpreted by the fact that flow instability is dominant over gravity-driven mass transfer. For downflow conditions, flow instability would tend to induce mixing between elements of fluid already carrying extract, hence a back-mixed flow behavior or (CSTR) would result. Therefore, it would produce a larger dispersion effect. For the upflow case, however, circumstances would be such that no flow instability would be induced by the displacing fluid. The solute thus would tend to

Figure 9. Metais-Eckert regime map for convective mass transfer (adapted from Debenedetti and Reid, 1986). (a) Plot of Re vs Gr × Sc. (b) Contribution of free convection to the overall Sherwood number.

flow in ideal plug flow, hence with zero axial dispersion effects; consequently, the extraction would be more efficient. One of the problem open to further research is to identify the controlling regime as both phenomena exhibited at large Grashof numbers. The phenomenon of flow instability has been made clear by Benneker and co-workers36 in connection with packed-bed reactor behavior but at moderate pressures (20-30 bar). Their particular fluid properties allowed them to measure and correlate the Bodenstein number as a function of the Reynolds and Grashof numbers, under well-defined tracer conditions. In our runs, we could not isolate an experimental zone where the instability effect on mixing appears itself. On the other hand, we tried to fit these data curves (upflow and downflow curves on Figure 8) with our model allowing an external mass transfer to be estimated by a correlation using free convection. The Peclet numbers obtained in the fitting did not change with the experimental conditions and the fraction of cosolvent. We thus conclude that with the present experimental conditions, flow dispersion could not be assessed, due possibly to overlapping external mass transfer. Cosolvent Effects on Mass Transfer Parameters The fitted mass transfer parameters have been evaluated for different pressures, temperatures, flowrates, and fluid mixtures, to assess their effects on mass transfer.

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3515 Table 7. Intraparticule Diffusivity in SC Fluid (CO2 and CO2 + toluene) and Tortuosity Factor in Sintered Porous Pellets P (bar)

T (K)

×108 De (m2/s)

D/Dea

τ

153.66 109.412 32.344 12.68 5.3005

>10 >10 8.09 3.17 1.32

2.327 5.167 4.702

120 150 150 200 200

348.15 358.15 368.15 358.15 368.15

Pure CO2 0.027919 0.024586 0.1011 0.14036 0.39996

150 200 200

368.15 358.15 368.15

CO2 + 6% Toluene 1.4434 0.37159 0.47856

150 150 200 200

358.15 368.15 358.15 368.15

CO2 + 10% Toluene 0.12141 22.074 2.688 1.205 0.41168 4.5423 0.49188 4.4319

a

0.58 1.29 1.17 5.52 0.30 1.13 1.11

D estimated from Catchpole-King correlation.48

Table 7 gives the values of the effective diffusivity and tortuosity factor under different conditions using pure and modified carbon dioxide. This type of dependence has been addressed for pure fluid by several authors.12,13,30,40-42 while for modified carbon dioxide, no report is available, except that of Chou et al.,42 who measured the effect of solute concentration on the diffusion coefficient. Tortuosity factors reported here increase with pressure, just as was also found by Chou et al.42 Tortuosity factor values go down when going from pure carbon dioxide to 6% toluene and then remain about constant in going from 6% to 10%. In the case of pure carbon dioxide, the values of the effective diffusivity increase with temperature, but they change very little in going from 150 to 200 bar. For modified carbon dioxide, the fitted values go down somewhat with pressure (150-200 bar) as we are more away from the critical state. Shown in Figure 10 are the relative changes in effective diffusivity, taking the effective diffusivity in pure carbon dioxide as a reference value. Figure 10a shows that the ratio (De/De0) is insensitive to fluid flowrate, as expected for an internal transport property like the diffusivity. At 150 bar, closer to the critical state of the mixture, the use of cosolvent brings about a quite large increase in the diffusivity (involving a 27-fold increase for 10% toluene), whereas at 200 bar, somewhat far above the critical pressure, the diffusivity is almost constant both for pure carbon dioxide and for modified CO2 (Figure 10b). For constant pressure and flowrate, we see that the ratio of the effective diffusivity varies little with the temperature. Presented in Figure 11 are the variations in the external mass transfer coefficient relative to that without cosolvent. For constant pressure and temperature, the cosolvent considerably enhances the fluid-to-particle mass transfer coefficient. This enhancement increases with fluid flowrate as shown by Figure 11a. At 150 bar, the presence of 6% cosolvent produces a small but significant sizable increase in the mass transfer coefficient (about 12%). The increase is remarkably larger (about 80%) in the presence of 10% toluene (Figure 11b). In the upper pressure setting (200 bar), the enhancement is less. For constant pressure and temperature, the mass transfer coefficient increases very slightly with temperature.

Figure 10. Intraparticle diffusivity enhancement due to cosolvent over that in SC carbon dioxide.

Figure 11. Enhancement in particle-to-fluid mass transfer coefficient due to modifier (toluene).

Tan et al.43 measured and correlated the interface mass transfer coefficient using a simple equation for the system carbon dioxide with β-naphthol as a dissolving

3516 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999

Figure 13. Axial Peclet number data in supercritical fluids (neat and modified CO2) in packed bed.44

Figure 12. (a) Parity plot for Sherwood number in the case of pure and modified CO2. (b) Solid-to-fluid mass transfer correlations for neat and toluene-modified CO2.

solute, in terms of the Reynolds and Schmidt numbers only (i.e., RRenScm). In the present work, where the contribution of the Grashof number is unimportant (see Figure 9b), a correlation without a term for freeconvection of the type used by Tan et al.43 is perfectly justified. On the basis of this simplified equation, we studied the change of the leading coefficient R with the fraction of cosolvent, finding that it increases from 0.295 (for pure carbon dioxide) to about 0.589 (for 10% cosolvent). Represented on Figure 12 a is a parity plot for the Sherwood numbers (calcd vs exptl Sh). Most of the data points fall on the diagonal, indicating a sufficiently good fit. In summary, the simplified correlation accounting for a variable amount of cosolvent is as follows:

Sh ) RRe0.8Sc0.33

(41)

Figure 12b shows a comparison between the above correlation and those reported in the literature. Our correlation runs almost parallel to that of Tan et al.43 and to the one we previously published.11 The mass dispersion Peclet number has also been studied in the present work, in the form defined by eq

40. It was observed that the fitted Peclet number is almost insensitive to pressure, temperature, and gravity action. In general, the Peclet number was found insensitive to the presence of cosolvent, except in the case with 10% toluene, where we have observed that the relative Pe/Pe0 changed from 0.83 to 1.15 (at 150 bar and 95 °C). At first, these results were unexpected since it is fairly well-known that the Bodenstein number should vary with the Reynolds number in a packed bed. The results here showed that it was approximately constant instead. Accurate values for Pe cannot be given, as discussed in the parameter sensibility study (Figure 2c). Figure 13 shows a comparison between the Bodenstein numbers fitted in the present study and those corresponding to different Schmidt numbers,44 as well as the Bodenstein numbers from the Gunn45 correlation. It is fairly evident that our values are between those of gases and liquids, hence corresponding to the fluid state (somewhere between a gas and a liquid). In the beginning, we tried to correlate our data with the Catchpole et al.46 correlation, but it was difficult. Undoubtedly this can be due to the fact that their correlation is for small (less than 1 mm) particles. By contrast, the Gunn equation allowed us to correlate our dispersion data very readily. This would confirm Catchpole et al.’s46 recommendation of using other correlation available in the literature for particles larger than 1 mm in diameter. Molecular properties appear as the Schmidt number in Gunn’s correlation. We have used Sc ) 2.1 and 4.5 for the lines of Figure 13. The values used in our study are between these two Sc. As seen in Figure 13, all the measured data points fall between these Schmidt numbers. Finally, as will be apparent from Figure 13 in the central zone (Re ) 10-100) of the plot, the Bodenstein number is very flat with respect to the Reynolds number, for either a gas or a liquid, particularly if some experimental error accompanies the measurements. Conclusions The extraction kinetics of porous metallic pellets loaded with solid solute (β-naphthol) has been studied, with SC carbon dioxide as the solvent, a pure fluid, and a fluid modified with toluene. Prior knowledge about solubility data for β-naphthol in the fluids was available. The extraction kinetics was shown to depend on six

Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 3517

main factors: pressure, temperature, fluid velocity, pellet size, action of gravity, and toluene concentration in the fluid. The most outstanding advantage of using a cosolvent is that it allows for a considerable reduction in extraction pressure (50 bar or more). If one still uses the higher pressure, the cosolvent allows for the reduction of the extraction time by 5-10-fold. In the experimental measurements, an integral fixed bed of particles was employed. Extraction rate data were interpreted with the shrinking-core model, with the assumption of solubility equilibrium at the core boundary. The transport steps involved in extraction are pore diffusion, mass transfer at the solid-fluid interface, and axial dispersive transport in the fluid. The three mass transport coefficients were measured (De, kg, and Pe) as a function of the modifier concentration up to 10% toluene. The resulting correlations indicate that at the higher toluene concentration De can increase as much as 30 times, whereas the value of kg increases only 3 times. On the other hand, the change in the dispersive Peclet number is only 15%. The conclusion would be that a cosolvent strongly accelerates extraction not only because of a driving force enhancement (solubility) but also because of a very important increase in the mass transport coefficients. On the basis of the shrinking-core model, the cosolvent tends to change the extraction controlling resistance from the inner of the particle to its external surface, owing to a 10-fold reduction in the Biot number. For 8 mm pellets, this changes from Bi ) 180 to 5, due to the presence of toluene. Therefore, extraction rate becomes more sensitive to fluid velocity. The results of the present work should allow for the establishment of the ranges of the variables for a process where porous pellets, pulvimetallurgical parts, and similar porous matrixes are cleaned with a supercritical fluid, either pure or modified, as regards to toluene concentration and pellet size. For example, for a pellet loaded with β-naphthol, the best operating conditions are: P ) 150 bar, T ) 95 °C, flowrate ) 4 L/min, pellet size equal or less than 8 mm, and fraction of toluene ) 10%. Under this condition, the extraction yield is 100% in about 2 h. Acknowledgment The authors acknowledge the fellowships received from the Ecole des Mines d’Albi (France), the Spanish Ministry of Foreign Affairs (AECI, Madrid), and to the Generalitat de Catalunya (PIEC Program from CIRIT). Technical assistance and samples of sintered pellets were kindly supplied by the research department of AMES, SA (Barcelona, Spain). Research funds were provided by the Spanish Program for Research and Development through the grants AMB95-0042-C02-02, QUI98-0482- C02-01 and 2FD97-0509-C02-02 from the CICYT (MEC, Madrid) and the European Regional Development funds (FEDER, Brussels). Nomenclature Bi ) Biot number, () kgR/De) Bo ) Bodenstein number, () dpu/Dax) C ) concentration of solute in the fluid phase, kmol/m3 Csat ) concentration of solute at saturation in the fluid phase, kmol/m3

Ci ) concentration of solute within the fluid-filled pores, kmol/m3 D ) molecular diffusivity, m2/s D1 ) binary molecular diffusivity (CO2, toluene), m2/s D2 ) binary molecular diffusivity (CO2, β-naphthol), m2/s Dax ) axial dispersion coefficient, m2/s De ) effective diffusivity, m2/s De0 ) effective diffusivity in the case of pure CO2, m2/s dp ) particle diameter, m Fo ) Fourier number, () (De/R2)t) Gr ) Grashof number, [ ) (d3pg/ν2)(∆F/F*)) k12 ) interaction parameter for mixing rule. kg ) fluid overall mass transfer coefficient, m/s kg0 ) fluid overall mass transfer coefficient in the case of pure CO2, m/s l ) half length, m L ) half length of cylinder (or particle), m lc ) length of cylindrical core, m Le ) length of extractor, m N ) amount of solute, kmol n, m ) constants in the equation of Tan et al.43 P ) pressure, bar L eu Pe ) Peclet number, ) Dax q0 ) initial Solute concentration based on solid volume, kmol/m3 q ) solute concentration based on solid volume, kmol/m3 r ) radial coordinate within body, m R ) Pellet radius, m rc ) core radius in the shrinking-core model, m Re ) Reynolds number, () FUdp/µ) s ) dimensionless Laplace transform variable Sc ) Schmidt number, () ν/D) Sh ) Sherwood number, () Kgdp/De) t ) time, s T ) temperature, K u ) interstitial velocity, m/s U ) superficial velocity, m/s x ) dimensionless concentration in bulk fluid phase, () C/Csat) xi ) dimensionless concentration in pores, () Ci/Csat) q y ) dimensionless solid-phase concentration, ) q0 z ) axial coordinate in bed, m

( )

( )

Greek Letters R ) constant in eq 41 and the equation of Tan et al.43 b ) bed void fraction p ) particle porosity ∆F ) absolute difference, ()|F* - F|), kg/m3 F ) fluid density, kg/m3 F* ) fluid density when it is saturated, kg/m3 µ ) dynamic viscosity, Pa s ν ) kinematic viscosity, m2/s pD τ ) tortuosity factor, ) De ξ ) dimensionless radial coordinate

( )

Subscripts and Superscripts ax ) axial b ) bed c ) core e ) effective p ) particle sat ) saturation

Literature Cited (1) Ting, S. S. T.; Macnaughton, S. J.; Tomasko, D. L.; Foster, N. R. Solubility of Naproxen in Supercritical Carbon Dioxide and with Cosolvents. Ind. Eng. Chem. Res. 1993, 32, 1471.

3518 Ind. Eng. Chem. Res., Vol. 38, No. 9, 1999 (2) Sunol, A. K.; Hagh, B.; Chen, S. Entrainer Selection in Supercritical Extraction. In Supercritical Fluid Technology; Penninger, J. M. L., Radosz, M., McHugh, M. A., Eds.; Elsevier: Amsterdam, 1985; p 451. (3) Brunner, G. Gas Extraction; Springer: New York, 1994. (4) Tan, C. S.; Weng, J. Y. Solubility Measurement of Naphthol Isomers in Supercritical CO2 by a Recycle Technique. Fluid Phase Equilib. 1987, 34, 37. (5) Maillet, C.; Abaroudi, K.; Trabelsi, F.; Recasens, F. UPC Barcelona, Spain. Unpublished results, 1997. (6) Abaroudi, K.; Maillet, C.; Trabelsi, F.; Recasens, F. The Use of Cosolvents on Supercritical Fluid Cleaning of Sintered Metallic Pellets. Proceeding of the 5th Meeting on Supercritical Fluids Materials and Natural Products Processing, Nice, France, 1998; p 181. (7) Fink, S. D.; Hershey, H. C. Modeling the Vapor-Liquid Equilibria of 1,1,1-Trichloroethane + Carbon Dioxide and Toluene + Carbon Dioxide at 308, 328, and 358K. Ind. Eng. Chem. Res. 1990, 29, 295. (8) Bennecke, J. F.; Eckert, C. A. Phase Equilibria for Supercritical Fluid Process Design. AIChE J. 1989, 35, 1409. (9) Jones, M. C. Mass Transfer in Supercritical Extraction from Solid Matrixes. In Supercritical Fluid Technology: Reviews in Modern Theory and Applications; Bruno, T. J., Ely, J. F., Eds.; CRC Press: Boca Raton, FL, 1991. (10) Goto, M.; Roy, B. C.; Hirose, T. Shrinking-Core Leaching Model for Supercritical Fluid Extraction. J. Supercrit. Fluids 1996, 9, 128. (11) Stu¨ber, F.; Va´zquez, A. M.; Larrayoz, M. A.; Recasens, F. Supercritical Fluid Extraction of Packed Beds: External Mass Transfer in Upflow and Downflow Operation. Ind. Eng. Chem. Res. 1996, 35, 3618. (12) Stu¨ber, F.; Julien, S.; Recasens, F. Internal Mass Transfer in Sintered Metallic Pellets Filled with Supercritical Fluid. Chem. Eng. Sci. 1997, 35, 3527. (13) Knaff, G.; Schlu¨nder, E. U. Mass Transfer for Dissolving Solids in Supercritical Carbon Dioxide, Part II: Resistance in the Porous Layer. Chem. Eng. Process. 1987, 21, 193. (14) King, M. B.; Catchpole, O. J. Physico-Chemical Data Required for the Design of Near-Critical Fluid Extraction Process. In Extraction of Natural Products using Near-Critical Solvents, King, M. B., Bott, T. R., Eds.; Blackie Academic & Professional: Glasgow, 1993. (15) Puiggene´, J.; Larrayoz, M. A.; Recasens, F. Free Liquidto-Supercritical Fluid Mass Transfer in Packed Beds. Chem. Eng. Sci. 1997, 52, 195. (16) Abaroudi, K. Ph.D. Thesis (in progress), UPC Barcelona, 1999. (17) AMES; AMES Impregnating Methodologies; Sant Vicenc¸ dels Horts; Barcelona, Spain, 1992. (18) Spencer, C. F.; Daubert, T. E.; Danner, R. P. A Critical Review of Correlations for the Critical Properties of Defined Mixtures. AIChE J. 1973, 19, 522. (19) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987. (20) King, M. B.; Bott, T. R.; Barr, M. J.; Mahmud, R. S. Equilibrium and Rate Data for the Extraction of Lipids Using Compressed Carbon Dioxide. Sep. Sci. Technol. 1987, 22 (2), 1103. (21) Sovova´, H. Rate of the Vegetable Oil Extraction with Supercritical CO2-I. Modelling of Extraction Curves. Chem. Eng. Sci. 1994, 49, 409. (22) Sovova´, H.; Kucera, J.; Jez, J. Rate of the Vegetable Oil Extraction with Supercritical CO2-II. Extraction of Grape Oil. Chem. Eng. Sci. 1994, 49, 415. (23) Lim, G. B.; Holder, G. D.; Shah, Y. T. Solid Fluid Mass Transfer in a Packed Bed under Supercritical Conditions. In Supercritical Fluid Science and Technology. Johnston, K. P., Penninger, J. M. L., Eds.; ACS Symposium Series 406; American Chemical Society: Washington, DC, 1989; Chapter 24. (24) Tan, C. S.; Liou, D. C. Axial Dispersion of Supercritical Carbon Dioxide in Packed Beds. Ind. Eng. Chem. Res. 1989, 28, 1246. (25) IMSL Math Library Fortran Subroutines for Mathematical Applications; IMSL Inc: Houston, 1989. (26) Riggs, J. B. An Introduction to Numerical Methods for Chemical Engineers; Texas Tech University Press: Lubbock, Texas, 1988.

(27) Brunner, G. Mass Transfer in Gas Extraction. In Supercritical Fluid Technology; Penninger, J. M. L., Radosz, M., McHugh, M. A., Eds.; Elsevier: Amsterdam, 1985. (28) Lim, G. B.; Holder, G. D.; Shah, Y. T. Mass Transfer in Gas-Solid Systems at Supercritical Conditions. J. Supercrit. Fluids 1990, 3, 186. (29) Zehnder, B.; Trepp, Ch. Mass-Transfer Coefficients and Equilibrium Solubilities for Fluid-Supercritical-Solvent Systems by Online Near-IR Spectroscopy. J. Supecrit. Fluids 1993, 6, 131. (30) Madras, G.; Thibau, C.; Erkey, C.; Akgerman, A. Modeling of Supercritical Extraction of Organics from Solids Matrixes. AIChE J. 1994, 40, 777. (31) Sunol, A. K.; Akman, U. cited in ref 15. (32) Hafner, K. P.; Pouillet, F. L. L.; Liotta, C. L.; Eckert, C. A. Solvatochromic Study of Basic Cosolvents in Supercritical Ethane. AIChE J. 1997, 43, 847. (33) Macnaughton, S. J.; Alessi, P.; Cortesi, A.; Kikic, I.; Foster, N. R. Predictive and Experimental Methods for the Choice of Cosolvents in the Supercritical Fluid Extraction of Pesticides. Hutchenson, K. W., Foster, N. R, Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 1995; p 126. (34) Van Alsten, J. G.; Eckert, A. E. Effect of Entrainers and of Solute Size and Polarity in Supercritical Fluid Solutions. J. Chem. Eng. Data 1993, 38, 605. (35) Benneker, A. H.; Kronberg, A. E.; Post, J. W.; Van Der Ham, A. G. J.; Westerterp, K. R. Axial Dispersion in Gases Flowing Through a Packed Bed at Elevated Pressures. Chem. Eng. Sci. 1996, 51, 2099. (36) Benneker, A. H.; Kronberg, A. E.; Westerterp, K. R. Influence of Buoyancy Forces on the Flow of Gases through Packed Beds at Elevated Pressures. AIChE J. 1998, 44, 263. (37) Debenedetti, P. G.; Reid, R. C. Diffusion and Mass Transfer in Supercritical Fluids. AIChE J. 1986, 32, 2034. (38) Lim, G. B.; Shin, H. Y.; Noh, M. J.; Yoo, K. P.; Lee, H. Subcritical to Supercritical Mass Transfer in Gas-Solid System. Proceeding of the International Symposium Supercritical Fluids; Strasbourg, France, 1994; p 141. (39) Lee, C. H.; Holder, G. D. Use of Supercritical Fluid Chromatography for Obtaining Mass Transfer Coefficients in Fluid-Solid Systems at Supercritical Conditions. Ind. Eng. Chem. Res. 1995, 34, 906. (40) Recasens, F.; McCoy, B. J.; Smith, J. M. Desorption Processes-Supercritical Regeneration of Activated Carbon. AIChE J. 1989, 35, 951. (41) Erkey, C.; Akgerman, A. Chromatography Theory: Application to Supercritical Extraction. AIChE J. 1990, 36, 1715. (42) Chou, S. H.; Wong, D. S. H.; Tan, C. S. Adsorption and Diffusion of Benzene in Activated Carbon Dioxide at High Pressures. Ind. Eng. Chem. Res. 1997, 36, 5501. (43) Tan, C. S.; Liang, S. K.; Liou, D. C. Fluid-Solid Mass Transfer in a Supercritical Fluid Extractor. Chem. Eng. J. 1988, 38, 17. (44) Levenspiel, O. Chemical Reaction Engineering; Wiley & Sons: New York, 1972. (45) Gunn, D. J. Axial and Radial Dispersion in Fixed Beds. Chem. Eng. Sci. 1987, 42, 363 (46) Catchpole, O. J.; Bernig, R.; King, M. B. Measurement and Correlation of Packed-Bed Axial Dispersion Coefficients in Supercritical Carbon Dioxide. Ind. Eng. Chem. Res. 1996, 35, 824. (47) Peng, D.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (48) Catchpole, O.; King, M. B. Measurement and Correlation of Binary Diffusion Coefficients in Near Critical Fluids. Ind. Eng. Chem. Res. 1994, 33, 1828. (49) Wakao, N.; Kaguei, S. Heat and Mass Transfer in Packed Beds; Gordon and Breach: New York, 1982. (50) Reisenberg, D. New Methods for the Estimation of the Viscosity Coefficients of Pure Gases. AIChE J. 1975, 21, 181.

Received for review February 10, 1999 Revised manuscript received June 11, 1999 Accepted June 16, 1999 IE990105E