where C0 is the total solute concentration in the particle (kg/m3), the particle porosity is represented by eP, K is the partition coefficient of the solute in the solvent, x0 is the initial solute mass ratio in the particle (kg solute/kg particle), and the real solid density pS is measured as kg particle/m3 particle volume.

The solute mass balance equations within the interstitial space and the porous particle can be written in terms of dimensionless variables. The use of dimensionless variables requires the introduction of dimensionless time, 0, which is the real time t divided by the CO2 residence time, t (in seconds). The parameter t is the total bed volume divided by the volumetric flow rate of supercritical fluid at the extractor vessel conditions:

where: x is equal to C/C0 (dimensionless solute concentration in the fluid phase); dimensionless time, 0, is found as t/t; e is the void fraction of the bed; the dimensionless mass transfer coefficient, f, is equal to kpapt with kpap being the combined mass transfer coefficient; lastly, xs (Cs/C0) is the dimensionless solute concentration in the solid. The initial conditions are:

This model has been successful in describing the CO2 extraction of fatty acids and carotenoids from the microalgae Spirulina maxima (Canela et al., 2002) and volatile oils from various plant materials: Croton zehntneri Pax et Hoff (Sousa et al., 2005), rosemary (Rosmarimus officinalis) (Carvalho et al., 2006) and fennel (Foeniculum vulgare) (Moura et al., 2005). This model was applied to the extraction of lipids from P. irregulare because of the similarity of experimental materials, extraction procedures and experimental variables (Cantrell, 2006).

Logistic model

The logistic model (LM) may be applied to the solute transfer to the fluid phase while neglecting accumulation and dispersion of the solute in the fluid phase since these phenomena have no significant influence on the process when compared with the effect of convection (Martinez et al., 2003; Campos et al., 2005). This model observes that when the extraction time approaches infinity, the mass of the extracted material tends to a fixed value asymptotically. This value can be considered the total extractable mass at the given process conditions. The OEC can be represented by the following equation with only two adjustable parameters (b, tm)

with mext as the mass of the extract (kg), bed length, H (m), extraction time, t (s), mt as the total initial mass of extract in the solid (kg), and the adjustable parameters b (s-1) and tm (s).

The physical meaning of the adjusted parameter tm corresponds to the instant in which the extraction rate of the solute reaches its maximum. The parameter b is more than likely a function of mass flow rate and density of the supercritical fluid (Martinez et al., 2003). This model can be applied to OECs that consider the solute as a single pseudo-compound or group of compounds. The fungal oil in this study is considered to be a single pseudo-compound despite the variety of glycerides and fatty acids the oil contains.

The first part of the extraction process is limited by the solubility equilibrium between the oil and CO2 solvent. This CER period is linear with the partition coefficient usually being a part of the constant of proportionality. Once the easily accessible surface oil diffuses into the flowing supercritical CO2, it is necessary for the lipids to diffuse through the biomass and appear on the substrate surface. The oil extraction rate starts to decrease since diffusion is slower than the convective mass transfer and a FER period is seen. Once diffusion completely controls the mass transfer process, the extraction is said to be in a DC regime (Reverchon et al., 2000; Ferreira et al., 2002; Sovova, 2005).

OECs during the SFE process are plots of the cumulated extract yield versus the extraction time. Figure 10.5 shows an example OEC fitted with the Goto and logistic models for extraction of fungal oil. The first part of the OEC is governed by the solubility equilibrium between the solute and fluid phase. Once the easily accessible solute is depleted, then DC mass transfer occurs (Reverchon et al., 2000; Ferreira et al., 2002; Sovova, 2005). Mathematical models have been proposed to correlate OECs during the SFE process (Goto et al., 1993; Reis-Vasco et al., 2000; Reverchon et al., 2000; Martinez et al., 2003; Sovova, 2005).

Other models

Marrone et al. (1998) modeled supercritical extraction of almond oil from crushed almond seeds of three different sizes at 35 MPa and 40 °C. The following assumptions were made: the oil was considered a single pseudo-component; the solute concentration was dependent only on time and the axial coordinates; uniform temperature, pressure, flow conditions existed along the extraction vessel; negligible axial dispersion and constant solid mass existed in the vessel during the extraction process.

Their model was based on physical evidence of broken and intact oil cells and considered two different phases of the extraction process. The initial phase contained freely available oil and was contained within the broken cavities on the surface of the crushed particles and an oil phase was contained inside the particles or internal surfaces. A good fit was observed for experimental data with an internal mass transfer coefficient of 7.5 x 10-9 m/s.

Sovova (1994) and Sovova et al. (1994b) modeled grape oil extraction at 28 MPa and 40 °C with grape seed of different particle size, flow rates and flow directions. Plug flow was observed for downward flow of compressed

Extraction time (min)

Fig. 10.5 Experimental and modeled OEC for SFE of Pythium irreguläre oil at 20.7 MPa, 40 °C and a CO2 flow rate of 3.94 x 10-6 kg/s. Total oil in biomass is from SFE and Hexane: isopropanol (3:2) extractions (0.168 g) (Cantrell, 2006).

Extraction time (min)

Fig. 10.5 Experimental and modeled OEC for SFE of Pythium irreguläre oil at 20.7 MPa, 40 °C and a CO2 flow rate of 3.94 x 10-6 kg/s. Total oil in biomass is from SFE and Hexane: isopropanol (3:2) extractions (0.168 g) (Cantrell, 2006).

gas, whereas extraction was retarded by natural convection in the case of up-flow. The up-flow model with parallel plug flow more closely represented the extraction process. Roy et al. (1996) modeled oil extraction from freeze-dried ginger root as a function of flow rate of CO2, pressure, temperature and particle size. The extraction process was controlled by intra-particle diffusion within the root. The rate of extraction increased with small particle size due to a decrease in the diffusion path. A crossover effect was observed with temperature and pressure. High temperatures increased extraction rates at 24.5 MPa, but low temperatures increased extraction at 10.8 MPa. When applied to experimental results, a shrinking-core model with effective diffusivity and solubility as fitting parameters fitted the data for large particle sizes. Goodarznia and Eikani (1998) developed a two-phase model composed of solid and supercritical phases, which when tested for essential oil extraction showed a dependence on particle size and shape.

Most of the mathematical models presented in the literature describe the OECs based on the mass balance for the extraction bed and the trans port phenomena that occur inside it. The extraction bed can be divided into two phases: (1) a solid phase that contains the solute and (2) a fluid phase, composed of the supercritical solvent with the extract dissolved in it. Many of these models assume: constant temperature and pressure; solute-free entering solvent; a linear equilibrium relation between solid and fluid phases - and the intra-particle transport is described by a diffusional process (Goto et al., 1993; Martinez et al., 2003; Campos et al., 2005; Sovova, 2005).

The concept of broken and intact cells has been introduced to account for the sudden change between the two phases of extraction: the linear constant equilibrium phase and the DC extraction phase. This concept takes into account the easily accessible solute from broken cells and the more difficult to extract part of the intact cells located at the particle core. Many studies have successfully applied these types of models to the extraction of essential oils from seeds and plant material (Goto et al., 1993; Reverchon et al., 2000; Ferreira et al., 2002; Martinez et al., 2003; Campos et al., 2005; Sovova, 2005).

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