with initial condition C = C0 at t = 0 and boundary condition dC/dr = 0

where

C = the concentration of solute in the sphere at time t C = the final concentration of solute at the surface D = the diffusion coefficient r = distance from the center of the sphere R = the radius of the sphere

The analytical solution takes the form of an infinite series with 'n' terms, for the total amount of a species diffusing from the sphere (Crank, 1975):

where Mt and M„ represent the total amount of solute leaving the sphere at time t and total amount of solute extracted over infinite time, respectively. D represents the effective diffusivity for porous media with void volume, and tortuosity factor (Walker, 1997).

The mass transfer rate for SFE for lipids from natural materials involving high initial concentration of extract (such as in oil seeds) in a fixed bed typically remains constant and then declines. SFE involves the control of solubility by manipulating temperature and pressure. Natural materials contain multiple components whose solubility and extractabilities are difficult to predict (Lira, 1996). Mathematical aspects related to SFE of lipids were discussed by King and List (1996); topics covered included solubility (Maxwell, 1996), phase equilibria (King and List, 1996), mass transfer

(Eagger, 1996), fractionation (Peter, 1996) and modeling of SFE of lipids (Goodrum et al., 1996).

Reverchon (1996) modeled supercritical extraction of sage oil from leaves at 9 MPa and 50 °C, for four different particle sizes. A model was proposed based on a mass balance along the extraction bed. Diffusivity of solute was the only adjustable parameter in the model. In the model a mass balance over an element of the extractor of height 'dh' was written as:

with initial conditions t = 0, C = 0 and C = C0 and boundary conditions h = 0, C(0,t) = 0 where e is the bed porosity, V is the extractor volume, C is the extract concentration in fluid phase, C is the extract concentration in solid phase, C* is the concentration at the solid-fluid interface, u is the superficial solvent velocity, m is the coefficient dependent on particle geometry, Di is diffusion coefficient, h is bed height and l = Vp/Ap (particle volume/particle surface) is a characteristic dimension. Testing of the model with experimental results suggested internal mass transfer to be the controlling step for the extraction process. The particle's shape was important for modeling experimental data, with a spherical shape giving a good fit.

Reverchon et al. (1999) modeled the fractional extraction of fennel seed oil and essential oil in two stages. In the first step, conducted at 9 MPa and 50 °C, essential oils were selectively extracted and then, at 20 MPa and 40 °C, the remaining vegetable oil was extracted. The flow rates tested were 8.33, 16.67 and 25 g/min. The model described vegetable oil extraction and was based on differential mass balances around the concept of broken and intact cells, with the internal mass transfer coefficient as an adjustable parameter. Essential oil extraction was modeled as desorption from vegetable matter with a low resistance to mass transfer, having the same internal mass transfer coefficient value as that of seed oil extraction. Both models represented good fits to experimental data.

An extraction model for pennyroyal essential oil by Vasco et al. (2000), with extraction at 10 MPa and 50 °C for different particle sizes (0.3, 05 and 0.7 mm) and different CO2 flow rates (18.6, 25.8 and 37.2 g/min), utilized

T. dC dC . , dC uV — + eV— + (1 -e)V— = 0 dh dt dt

where l2

axial dispersion effects based on the desorption of oil near the leaf surface and mass transfer resistance in the internal part of the vegetable structure. They divided the extraction process into two parts for the purpose of modeling; the first part of the extraction described adsorption equilibrium with superimposed axial dispersion, whereas in the second part of the extraction process internal mass transfer was assumed to be the controlling factor. Yield curves for all particle sizes and flow rates of CO2 fitted fairly well with an internal mass transfer coefficient K as an adjustable parameter. Akgun et al. (2000) described the extraction and modeling of lavender flower essential oil with supercritical CO2 in a semi-continuous system at 8-14 MPa pressure, 35-50 °C temperature and 1.092-2.184 g/min flow rate ranges of CO2. They used a quasi-steady-state model as a function of extraction time, flow rate, pressure and temperature, with inter-particle diffusion coefficient as an adjustable parameter. The model was satisfactorily correlated with experimental data with best fitted value of effective diffusivity (1.2 x 10-11 m2/s).

Reverchon et al. (2000) conducted experiments for SFE extraction of hiprose seed oil at different pressures (10.34, 20.68, 41.37 and 68.94 MPa), temperatures (40, 50 and 70 °C) and flow rates (1, 2, 4 and 6 g CO2/min) with different particle sizes of seeds (0.42, 0.79 and 1.03 mm); experiments were validated with a mathematical model based on the structure of hiprose seed particles. For modeling purposes they assumed oil to be a single pseudo-component, the extraction bed was assumed to be continuous, and the pressure and temperature gradients along the column were neglected. The volume of the solid was assumed constant and the solute concentration in the fluid phase was assumed to be dependent only on time t and axial coordinates. Axial dispersion was neglected. Their model was given as

at where q* = KeqC, C = C0 at time t = 0 for each z, q = q0 at time t = 0 for each z, C = 0 at z = 0 for each t. Here u is superficial velocity of supercritical CO2, e is void fraction of extractor, ps is density of hiprose seed, pf is density of supercritical CO2, Y is the yield of oil seed, a is the specific surface of the vegetable matter, Keq is the linear equilibrium constant, C and q are the concentrations expressed as a mass ratio of oil in fluid phase and solid phase, respectively, and q* is the concentration of oil in solid phase at the solid-fluid interface. z is the axial coordinate and Ki is the internal mass transfer coefficient. The model assumed internal mass transfer coefficient as linearly variable and fitted well to the experimental data.

Goto et al. (1993) extracted peppermint oil with supercritical CO 2 at varying conditions (313-353 °K, 8.83-19.6 MPa) and studied extraction curves and extraction rates of major components (L-menthol and menthone). A mathematical model was also developed based on local adsorption equilibrium of essential oil lipid in leaves as well as mass transfer (Goto et al., 1993, 1998). Their model was based on the following assumptions: (1) leaves are porous solids with essential oil and lipid; (2) essential oils are extracted from leaves as if desorbed from solid biological tissue where lipids are associated with essential oils; (3) essential oils dissolved in supercritical fluid diffuse to the external surface and through the external film to be carried away by bulk flow. The adsorption equilibrium constant, determined by fitting the theoretical extraction curve to experimental data, increased with temperature and decreased with pressure.

Canela et al. (2002) applied the Goto model to supercritical fluid extraction of fatty acids and carotenoids from microalgae (Spirulina maxima) to describe the extraction process. In their study to determine the kinetic parameters, extraction experiments were conducted at varying pressures (15, 16.5 and 18 MPa) and temperatures (20, 25 and 30 °C), and the yield and composition were determined at a constant solvent flow rate (3.33 x 10-5 kg/s). They applied the Goto et al. (1993) model assuming the substrate to be a porous matrix with diffusion occurring through the inside of the particle pores and with mass transfer resistance being offered by the film around the particle.

The Goto model (Goto et al., 1993) treats the solid particle as a solid substrate with a porous matrix. The solute diffuses within the pores and through a film surrounding the particles, then is extracted into the supercritical fluid and carried away by the bulk flow. At equilibrium, the total initial solute concentration existing in both the solid phase and at the surface is given by a linear relationship

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