Goto

128 The Journal of Supercritical Fluids, 1996, 9, 128-133

Shrinking-Core Leaching Model for Supercritical- Fluid Extraction

Motonobu Goto,* Bhupesh C. Roy, and Tsutomu Hirose

Department of Applied Chemistry, Kumamoto University, Kumamoto 860, Japan

Received August 18, 1995; accepted in revised form December 28, 1995

Extraction or leaching of a solute from a solid material is a process involving mass transfer in the solid matrix. When the solute content in the solid material is sufficiently large as compared to the solubil- ity in fluid phase, the process is similar to that of irreversible desorption. The shrinking-core model was applied to the modeling of the extraction process. The model including axial dispersion in the extraction column was solved numerically. Quasi-steady-state solution without axial dispersion was derived, and the accuracy was discussed in comparison with the numerical solutions. The model calculations gave a good agreement with the experimental extraction curve reported in literature.

Keywords: supercritical extraction, leaching, shrinking core, modeling, mass transfer

INTRODUCTION Supercritical-fluid extraction has been applied to

natural materials to separate the desired components from a solid matrix in several industries such as the food, per- fumery, pharmaceuticals, and the petroleum industry. Most of the work has been devoted to experimental stud- ies where the focus has been on the composition of the extracts or application to new materials.

The extraction process from natural materials such as plants, beans, or seeds involves the release of solutes from the porous or cellular matrix into the solvent. It is a mass-transfer process. The solutes fixed or trapped in a matrix by physical or chemical forces must be released and transferred to the supercritical fluid by dissolving. Then, the dissolved solutes diffuse through the matrix to the surface of a particle. Finally, they move across stag- nant film around a particle to the bulk fluid-phase.

The solutes of interest vary from small to large molecules, or low to high boiling compounds such as flavor components, or essential oil to lipids or carotenoids. Their content in a feed material may be small (less than 1%) or large (40% for seed oil). Thus, rate-limiting process for the extraction of these materials may differ from material to material. The dissolution kinetics involved in a model may depend on the ratio of the solute content to the saturation concentration in the fluid phase, the bound state of the solute on a matrix, and the association with coexisting materials.

It is important to model the extraction process when the extraction operation is to be optimized for commercial application. Unfortunately, the extraction process for

natural materials is not well understood and is not always operated at the optimal conditions.

Several attempts have been devoted for the modeling of the extraction process. 1*2 The extraction of oil from seeds has been most intensively modeled. Since oil content of seeds is relatively high and oil consists of similar compounds, mass-transfer model was often employed. Ficks diffusion model has been applied.2,3 A simpler model has been also applied where mass transfer was ex- pressed by a linear driving-force mode1.4+5 The driving force for the mass transfer may be the difference of the solute concentration between saturated concentration in fluid phase and local concentration in pores in the matrix of a particle. It is evident that natural materials are different from uniformly porous materials such as adsorbents but are cellular structures with partial destruction taking place during the crushing process. Sovova6v7 considered a particle consisting of crushed cells and uncrushed cells. The solutes can be easily extracted from the crushed cell, whereas the extraction from the uncrushed cell is much more difficult.

Desorption models have been developed where local dissolving process is considered to be desorption of solutes adsorbed on a matrix. Recasens et al.* developed a model accounting for local adsorption/desorption kinetics, intraparticle diffusion, and external mass transfer. Goto et a1.9 applied the desorption model to the extraction of essential oils where local adsorption equilibrium was assumed.

When there is a sharp boundary within a particle between extracted part and nonextracted part, the shrinking-

0896-8446/96/0902-0128$7.50/O 0 1996 PRA Press

The Journal of Supercritical Fluids, Vol. 9, No. 2, 1996 Shrinking-Core Leaching Model 129

core leaching model may be useful. The shrinking-core model has been used in solid-fluid reaction,O adsorption, and ion exchange. t2 The shrinking-core leaching model was used for a single particleI and for a packed bed.14 King and Catchpolet4 derived quasi-steady-state analytical solution assuming negligible axial dispersion in a bed.

In this paper we derive a shrinking-core leaching model accounting for intraparticle diffusion, external fluid film mass transfer, and axial dispersion. The model is further simplified to derive an analytical solution. The effect of parameters involved in the model is discussed. The model was applied to the experimental data for oil extraction from seeds.

THEORETICAL DEVELOPMENT Model Including Axial Dispersion. This

model describes the situation of the irreversible desorption followed by diffusion in the porous solid through the pores. When mass transfer rate of the solute in the nonextracted inner part is much slower than that in the outer part where most of the solute has been extracted, or the solute concentration is much higher than the solubil- ity of the solute in the solvent phase, a sharp boundary may exit between the outer and the inner region. A core of inner region shrinks with the progress of the extraction. These situations can be modeled by the shrinking- core model. An example of these situations is the supercritical-fluid extraction of vegetable oils.

The following assumptions have been made to derive the fundamental equations. The solvent flows axially with interstitial velocity, v, through a packed bed in a cylindrical extractor of height, L. Pure solvent enters the bed. The process is isothermal. Considering axial dispersion, the material balance on the bulk fluid-phase in the extractor is

+XD a2C l-&3k, dC at az Laz2 &

---R[C-Ci(R)]. (1)

Time variation of the solid-phase concentration (average oil concentration in a particle) is equated with the rate of mass transfer of solute within external film sur- rounding the particle.

The diffusion in outer region is given by

(2)

Solid phase solute exists within the core, the average value of solid-phase concentration 4 being given by

(4)

Boundary conditions are given as follows. At the core boundary, the concentration in the fluid phase is at its saturation value.

Ci = C,,, at r = r, (5)

Diffusion flux at the outer surface of a particle is equal to the mass transfer through external film.

( 1

Dac, e -& _ = kf[c-cml

r-R (6)

Initial conditions are given as follows:

rc=R at t=O (7)

C=O at t=O. (8)

Danckwerts boundary condition at the inlet of column and the exit condition are given by

vC-D z=O at z=O L az

(9)

&=O at z=L.

Following dimensionless groups are defined to derive dimensionless formulae of the fundamental equations: x = C/C,, 9 Xi = CilCs,, v 5 = t-JR, Z = zlL, a = vR21D,L, 0 = (D,lR2)t, y = qlqo, b = C,,,/g,, Pe = LvlDL, Bi = kfRID,. With these dimensionless groups, eqs l-3 become

3X 2X

-+a5g=KdZ2 ae a fi-53Bi[x-xi(l)]

g=3Bib[x-x,(l)]

+& p$ =o, ( 1

Dimensionless boundary and initial conditions are

xi = 1 at 5 = 5,

=Bi[x-xi(l)]

130 Goto et al. The Journal of Supercritical Fluids, Vol. 9, No. 2, 1996

{,=l at 8=0

x=0 at 8=0

(17)

(18)

x-J-$=0 at Z=O

$=O at Z=l.

(19)

The following equations were derived by rearranging these equations.

ax dx a a2x l--E 3Bi(x - 1) --- %+Z=KdZ2 E l-Bi(l-l/t,) (21)

ag,- bBi( x - 1) &I - cz[l--Bi(l-lit,)]

(22)

These differential equations coupled with boundary and initial conditions were solved numerically by Crank Nicholsons method.

The yield, that is, cumulative amount of extract up to time 8 is given by

9

xde . (23)

Quasi-Steady-State Solution Without Axial Dispersion. When axial dispersion is negligible, a quasi-steady-state solution can be obtained by assuming that changes in & during the residence time of solvent in the column are negligible and that changes in the axial direction of the column at any given time are small. By introducing the dimensionless time B* = 8 - Z/a, integration of eq 2 1 gives

3BiZ

1 l-Bi(l-115,) (24)

By substitution of x into eq 22 and integration

1 4,. (25)

The symbolic integration was done with a computer alge- bra system (MathematicaTM, Wolfram Research Inc.), which gives analytical solution in terms of the exponen- tial integral function. For example computations, numerical values were obtained directly from the analytical solution.

When 5, is further assumed to be constant during integration of eq 22 with eq 24, the folIowing simplified solution can be obtained:

0.6

x

0.4

TABLE I Parameters for Example Calculation

0, L

I;; & % C sat a

b Bi Pe

1.5 x lO-O m2 s- 70x 10.rn 1 X 1O-3 m s-l 1 X 10e3 m 0.4 250 kg m-3 qdlO0 100 (value calculated with above parameters is 95) 0.01 100 1000

b = 0.01

Figure 1. Effect of parameter a on the extracted concentration.

3bBXJ* =l~~~~i(l~~/~,)exp

l--E 3BiZ a& 1-Bi(l-115,) (26) I

This solution is identical to that derived by Catchpole.14 The time required for complete equilibrium extrac-

tion is given by eqs 27 and 28 on an assumption of equilibrium extraction, that is, rate processes do not interfere with the extraction.

1-& e, =- ab&

(27)

(28)

PARAMETRIC ANALYSIS AND ACCURACY OF SIMPLIFIED MODELS

Values of the Parameters Used in the Model. The parameter values used in model calculation were taken from the literature.13 Table I shows the parameters used in this work.

The Effect of Parameter a. One of the important operation parameters is a = (v/L)(R2/D,) which is an inverse of dimensionless residence time. The value of

The Journal of Supercritical Fluids, Vol. 9, No. 2, 1996

0.2

0

b I 0.01 Pe-loo0 - Bi-100

0 1 2 3 4 5

@B.

Figure 2. Effect of parameter a on the solid-phase concentration.

a-10 b- 0.01 Bi-100 -

x

0 0.5 1 1.5 2 2.5

wee

Figure 3. Effect of Peclet number (Pe) on the extracted concentration.

1 a-70 :

0.8

0.8 X

0.4

0.2

ot- - 1 1

2 e/e, 3 4 5

Shrinking-Core Leaching Model 131

a increases with flow rate and particle size, and decreases with extractor length and intraparticle diffusivity. Figure 1 shows variations of effluent concentration, x, at the exit of the extractor as a function of dimensionless time, e/e,. Figure 2 shows average solid-phase concentration, 7, which is equivalent to

132 Goto et al. The Journal of Supercritical Fluids, Vol. 9, No. 2, 1996

0.8

0.6 X

0.4

6

CO, flow rate: 0.2 m3/h (S

0.8

0.6 X

0.4

0.2

1 2 3 4 5 1;

0.8 Bi=l

0.6 : X

0.4

0.2 :

o--- --- .--- -- ---..- - 1

2 O/8, 3 4 5

Figure 5. Comparison of models at various Biot number (a = 100, b = 0.01).

Accuracy of Quasi-Steady-State Solu- tions. Figure 4 shows the comparisons of dimensionless effluent concentration, x at the extractor exit among the numerical solution, quasi-steady-state (QSS) solution, eq 24 and simplified quasi-steady-state (SQSS) solution, eq 26. Comparison is made for four different values of a in Figures 4 (a), (b), (c), and (d). The SQSS solution gives always larger value of x. For larger value of a (a > 200), QSS solution agreed well with the numerical solution for the entire range of dimensionless time, 0/e,. However, for smaller value of a, the QSS solution devi- ates from the numerical solution at smaller time. In these cases accuracy of the QSS solution is worse than the SQSS solution at smaller time.

The Effect of Biot Number. Relative im- portance of intraparticle diffusion to external fluid-solid mass transfer is represented by Biot number, Bi = kfR/D,. Figure 5 shows the effect of Bi on the accuracy of the solutions. When Bi is small, external mass transfer signifi- cantly affects to the extraction behavior. Thus, the exit

35 MPa, 316 K I

- Model (eqs 21-23)

0.2 0.3 0.4 0.5 Gas flow [m3 (STP)]

Figure 6. Comparison of model calculation and experimental extraction curve of oil from rape seeds with carbon dioxide.

31 I I I I I

0 - I 35 MPa, 316 K q 0 Experimental data (Brunner. 19841 ~2~+&\~ - Model (eqs 21-22) %~ 1

0 0.1 0.2 0.3

Gas flow [m3 (STP)] 0.4 0.5

Figure 7. Comparison of model calculation and experimental extraction rate of oil from rape seeds with carbon dioxide.

concentration at the initial time is smaller than unity for smaller Bi. The accuracy of quasi-steady-state solutions increases with decrease in Bi. When intraparticle diffusion controls the extraction process for large Bi, sudden change of the concentration at the surface of a particle at the initial stage of the extraction results in inadequate situation for quasi-steady-state assumption.

COMPARISON WITH EXPERIMENTAL DATA

The shrinking-core model was applied to the experimental data. The extraction data of oil from rape seed with supercritical carbon dioxide measured by Brunner2 was used to compare with the model calculation. Radius of the crashed rape seed was 0.25 mm and the extractor was 17 mm dia. and 220 mm long. Figure 6 shows the extraction curve, that is, cumulative amount extracted ver- sus carbon dioxide flowed at two operating conditions. Figure 7 shows the extraction rate, which corresponds to

The Journal of Supercritical Fluids, Vol. 9, No. 2, 1996 Shrinking-Core Leaching Model 133

the differentials of the cumulative extraction curve. All the parameters necessary for model calculation were obtained from the literature* except for effective intraparticle diffusivity, D,, and axial dispersion coefficient, DL. External mass-transfer coefficient estimated by Wakao and Kagueis correlation15 was adopted as used by Brunner.*

NOTATION

Estimation method of axial dispersion in supercriti- Cal-fluid extractor is not available in literature. Erkey and Akgermani6 measured axial dispersion by the chromato- graphic technique. They obtained particle Peclet number (=2Rv/DL) in range from 0.058 to 0.244 for naphthalene- alumina system. When these values are converted to Peclet number (=Lv/DL), one obtains Pe = 25-107 for the experimental condition. Thus, we adopted the value of 50 for Peclet number. Axial dispersion did not influence to the extraction curve in this case.

Bi c ci c Sal

$

L Pe 9 ?

2 r r, t

V

x xi Y L

z &

E c

Biot number = k,RID, concentration in bulk fluid-phase, mol mm3 concentration in pores, mol mm3 saturation concentration, mol rnA3 effective intraparticle diffusivity, m2 s-r axial dispersion coefficient, m* s-r external mass-transfer coefficient, m s-l length of extractor, m Peclet number, LvlD, solid-phase concentration, mol mm3 average value of q, mol me3 initial solid-phase concentration, mol mm3 particle radius, m radial coordinate

Intraparticle effective diffusivity is most important parameter for the extraction process from solid materials. However, accurate value of the intraparticle effective diffusivity can not be estimated, because the value changes de- pending on the material and environment. We adopted 1.5 x IO-i0 m* s-i as an initial value for intraparticle effective diffusivity as given in Table I. By fitting the calculated results with the experimental data, we optimized the value to be 1.5 x lo-i0 m* s-i (20.5 MPa, 324.7 K) and 0.75 x lo-i0 m* s-i (35 MPa, 316 K). Wilke and Lees correla- tion17 gives the dependence of temperature and pressure on diffusivity. Diffusivity is proportional to T[K13*lP[bar] according to the correlation. From this relation the ratio of diffusivity at 20.5 MPa and 324.7 K to diffusivity at 35 MPa and 316 K is calculated to be 1.8. This value is in good agreement with the ratio of the optimized intraparticle effective diffusivity, 1.5 x lo-lo/O.75 x lo-i0 = 2.0.

radius of unleached core, m time, s interstitial fluid velocity, m s-r dimensionless concentration in bulk fluid-phase, C/C,,, dimensionless concentration in pores, CilC,, dimensionless solid-phase concentration, q/q0 average value of y axial coordinate in extractor bed voidage dimensionless time, (D,lR*)t dimensionless radial coordinate, r/R dimensionless radius of unleached core, r,lR

REFERENCES (1)

As shown in Figures 6 and 7, the results calculated by the model (eqs 21-23) agreed well. Diffusion within a particle was rate-determining step in the extraction process. In these cases, parameters, a and b, were as follows:

Akgerman, A.; Madras, G. Supercritical Fluids Fundamentals for Application; Kiran, E.; Sengers, J. M. H. L., Eds.; Kluwer Academic Publishers: Dordrecht, 1994; p 669. Brunner, G. Ber. Bunsenges. Phys. Chem. 1984,88, 887. Reverchon, E.; Donsi, G.; Osseo, L. S. Ind. Eng. Chem. Res. 1993,32, 2721. Lee, A. K. K.; Bulley, N. R.; Fattori, M.; Meisen, A. .I. Am. Oil Chem. Sot. 1986.63, 921. Schaeffer, S. T.; Zalkow, L. H.; Teja, A. S. .I. Supercrit. Fluids 1989, 2, 15. Sovova, H. Chem. Eng. Sci. 1994,49, 409. Sovova H.; Kucera, J.; Jez, J. Chem. Eng. Sci. 1994, 49, 415. Recasens, F.; McCoy, B. J.; Smith, J. M. AIChE .I. 1989, 35, 951.

a = 7.1, b = 0.014 (20.5 MPa, 324.7 K) Goto, M.; Sato, M.; Hirose, T. J. Chem. Eng. Jpn. 1993, 26, 401.

a = 4.9, b = 0.054 (35 MPa, 316 K)

Since parameter a is relatively small, simplified analytical solutions (eqs 25 or 26) can not be used accurately as in- dicated in Figure 4.

(2)

(3)

(4)

(5)

(6) (7)

(8)

(9)

(10)

(11)

(12) (13)

CONCLUSIONS Shrinking-core model was developed for the extrac-

tion from solid materials. Numerical solution of the model equations simulated extraction behavior. Quasi- steady-state solution gave a good approximation for larger value of parameter a. However, the QSS solution cannot be utilized when parameter a is small and Bi is large. Although example calculation was demonstrated for supercritical-fluid extraction, the model can be applied to the other extraction or leaching processes.

(14)

(15)

(16) (17)

Levenspiel, 0. Chemical Reaction Engineering; John Wiley & Sons: New York, 1972; p 361. Hall, K. R.; Eagleton, L. C.; Acrivos, A.; Vermeulen, T. Ind. Eng. Chem. Fundam. 1966,5, 212. Guria, C.; Chanda, M. Trans I Chem. E 1994, 72, 503. Jones, M. C. Supercritical Fluid Technology - Reviews in Modern Theory and Applications; Bruno, T. J.; Ely, J. F., Eds.; CRC Press: Boca Raton, FL, 1991; p 365. King, M. B.; Catchpole 0. Extraction of Natural Products Using Near-Critical Solvents; King, M. B.; Bott, T. R., Eds.; Blackie Academic & Professional: Glasgow, 1993; p 184. Wakao, N.; Kaguei, S. Heat and Mass Transfer in Packed Beds; Gordon and Breach: New York, 1982; p 156. Erkey, C.; Akgerman, A. AIChE J. 1990,36, 17 15. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw Hill: New York, 1987; p 587.

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