This is an author generated postprint of the article: Chan, C.-H., Yusoff, R., & Ngoh, G.-C. (2014). Modeling

and kinetics study of conventional and assisted batch solvent extraction. Chemical Engineering Research and

Design, 92(6), 1169-1186. doi: 10.1016/j.cherd.2013.10.001

*Corresponding author. Tel: +6017 7680611; Fax: +603 79675319; Email address: ch_chan@um.edu.my

The published version is available on http://dx.doi.org/10.1016/j.cherd.2013.10.001

Modeling and kinetics study of conventional and assisted batch

solvent extraction

Chung-Hung Chan a,*, Rozita Yusoff a, Gek-Cheng Ngoh a

a University of Malaya, Department of Chemical Engineering, 50603 Kuala Lumpur, Malaysia.

ABSTRACT:

Batch solvent extraction techniques have been widely explored. On understanding its potential

significance, this article aims to review kinetics and modeling of various extraction techniques

which involve assisted means including microwave-assisted extraction (MAE), ultrasonic-

assisted extraction (UAE), pulse electric field (PEF) and high voltage electrical discharge

(HVED). This review includes a detailed discussion of the instrumental setup, extraction

mechanisms and their distinct advantages and disadvantages. Additionally, the impact of the

operating parameters on the extraction kinetics of the mentioned techniques are highlighted.

The review also covers the mathematical modeling based on Fick’s law, chemical rate law and

empirical models. The established kinetic models of various extractions are also summarized

to facilitate better understanding.

Keywords: kinetic modeling, microwave-assisted extraction, ultrasonic-assisted extraction,

pulse electric field, high voltage electrical discharge

Contents

1. Introduction

2. Batch solvent extraction

3. Assisted solvent extraction techniques

3.1. Microwave-assisted extraction (MAE)

3.2. Ultrasonic-assisted extraction (UAE)

3.3. Electrically-assisted extraction (EAE)

4. Influences of operating parameters on extraction kinetics

4.1. General operating parameters of extraction

4.2. Specific parameters of assisted solvent extraction

5. General Mathematical modeling

5.1. Fick’s law

5.1.1. Mass transfer in solid particles

5.1.2. Mass transfer in solid particles and the extraction solvent

5.1.3. Mass transfer with extraction temperature variation

5.1.4. Mass transfer with degradation of active compounds

5.1.5. Modeling with external mass transfer resistance

5.1.6. Modified Fick’s law

5.2. Rate law

5.3. Empirical equations

6. Extraction models

7. Summary

Acknowledgement

References

Nomenclature

1. Introduction

Batch solvent extraction is commonly used to extract active compounds from plants. The

current food industry adopts this technique and focuses on extracting and recovering valuable

active compounds from different plants and waste residues such as grape pomace (Casazza et

al., 2010), orange peels (Inoue et al., 2010) and olive cake (Cárcel et al., 2010). In conventional

solvent extraction, the solid sample is immersed in the solvent and the extract is collected after

the equilibrium extraction is reached. The efficiency of the solvent extraction can be enhanced

by employing microwaves (Amarni and Kadi, 2010; Gujar et al., 2010; Spigno and De Faveri,

2009; Xiao et al., 2012), ultrasounds (Cárcel et al., 2010; Pan et al., 2012; Stanisavljević et al.,

2007; Velickovic et al., 2008) and electrical fields and charges (Boussetta et al., 2011; El-

belghiti and Vorobiev, 2005; El Belghiti and Vorobiev, 2004; Moubarik et al., 2011) into the

extraction system. These assisted techniques offer unique advantages and features which are

suitable for specific extractions. These non-conventional extraction techniques are presumed

to replace conventional solvent techniques, which makes studying their kinetics mechanisms

and modeling essential. Such studies enable prediction of the extraction behavior which is

considered to be useful for scaling up of the process.

In general, mathematical modeling approach that applies to solvent extraction depends on the

operational mode. Many reviews on the kinetic modeling of continuous solvent extraction such

as supercritical fluid extraction have been reported (Diaz and Brignole, 2009; Oliveira et al.,

2011). Thus this has prompted the review on the kinetic modeling of batch solvent extraction

techniques including the assisted techniques. The fundamental approach to model the

extraction is through derivation of Fick’s law (Chen and Chen, 2011; Cissé et al., 2012; Franco

et al., 2007a; Franco et al., 2007b; Herodež et al., 2003; Hojnik et al., 2008; Rakotondramasy-

Rabesiaka et al., 2010; Tsibranska et al., 2011; Wongkittipong et al., 2004; Xu et al., 2008).

Other mathematical approaches employed include rate law (Pan et al., 2012; Qu et al., 2010;

Rakotondramasy-Rabesiaka et al., 2007; Xiao et al., 2012), Peleg’s empirical model (Boussetta

et al., 2011; Cárcel et al., 2010) and other two-parametric empirical models. To have a better

grasp on the extraction models developed, the derivations of the models together with their

assumptions and applications are elucidated. The information and data presented in this article

are very useful for specific plant extraction processes.

2. Batch solvent extraction

The solvent extraction curve is typically comprised of a fast extraction step (washing stage)

and a slow extraction step (diffusion stage) as shown in Fig. 1 (Franco et al., 2007b; Perez et

al., 2011; So and Macdonald, 1986). The extraction mechanism starts when the solvent

molecules penetrate into the plant matrices, causing the cytoplasm layer to be exposed directly

to the solvent (Crossley and Aguilera, 2001).This facilitates the dissolution of the active

compounds into the solvent. In the beginning of the extraction process, the fast step

corresponds to a constant extraction rate (Rakotondramasy-Rabesiaka et al., 2009). At an

extremely fast rate, the period in this extraction step is difficult to determine (Franco et al.,

2007b). During the slow extraction step, active compounds diffuse from the interior of the plant

matrices and dissolve in the solvent. The extraction yield during this step is greatly dependent

on the cells that remain intact after the washing extraction step (Crossley and Aguilera, 2001).

In fact, the characteristics of washing and diffusion steps in the extraction can be determined

by the proportion of broken and intact cells after sample preparation, e.g. grinding, (So and

Macdonald, 1986).

time

Extr

action y

ield

Washing stage

diffusion stage

Fig. 1: Typical extraction curve of batch solvent extraction of active compounds from plants

Sample grinding and soaking in solvent are commonly applied prior to extraction in order to

reduce the particle size of the sample for better diffusion mechanism (Tsibranska et al., 2011)

and to improve the penetration of the solvent into the plant structure (Gujar et al., 2010).

Improvement of the extraction kinetics can also be achieved using advanced pretreatment such

as steam explosion and instant controlled pressure drop (DIC). These pretreatment methods

fragment the sample forming microspores as it is decompressed through sudden release of high

steam pressure (Ben Amor and Allaf, 2009; Chen and Chen, 2011). This improves the washing

step of the extraction (Chen and Chen, 2011) and enhances the diffusion of the solute into the

solvent (Ben Amor and Allaf, 2009).

Various setups of conventional extraction systems are available. For instance, a basic setup

consists of a stirred vessel with a water bath for temperature control, as shown in Fig. 2. This

setup has been widely applied in the industry to provide convective bulk movement in the

solvent. This reduces the mass transfer barrier and enhances the extraction (Franco et al.,

2007a). In some applications, a condenser is attached to the top of the vessel to prevent

evaporation of solvent due to overheating during the extraction (Xu et al., 2008). The

drawbacks associated with the conventional extraction technique in terms of long extraction

time and high solvent consumption have triggered the development of new solvent extraction

techniques with assisted means to overcome these limitations.

Fig. 2: Schematic diagrams of conventional and assisted extraction systems

3. Assisted solvent extraction techniques

Recent development in solvent extraction techniques focuses on enhancing the conventional

techniques with the assistance of microwave heating, ultrasonic radiation, electrical fields and

charges. These processes can be incorporated into solvent extraction or as sample pretreatment

prior to extraction. Their mechanisms, advantages and drawbacks are discussed in the

following sections.

3.1. Microwave-assisted extraction (MAE)

One of the assisted means used for enhancing the conventional solvent extraction system is

MAE. In this system, the heating efficiency is improved by applying microwaves. The

microwave radiation penetrates into the targeted material and interacts with the polar molecules

through ionic conduction and dipole rotation (Sparr Eskilsson and Björklund, 2000) to generate

heat. The localized heating is based on the dielectric constant of the material (Mandal et al.,

2007). The effectiveness of MAE is attributed to its localized heating which increases the

internal pressure of the cells and consequently ruptures them (Zhou and Liu, 2006). The active

compounds then elute from the cells and get dissolved in the surrounding solvent. The

schematic diagram of MAE instrumental setup is illustrated in Fig. 2. Closed type, opened type

and other modified MAE setup can be found in the literature (Chan et al., 2011).

MAE has been proven to enhance the extraction yields and shorten the extraction time in many

extractions (Chen et al., 2007; Li et al., 2010; Yan et al., 2010). For instance, the concentration

of phenolic compounds extracted from black tea by MAE after 90 s was 43% higher than that

using traditional brewing after 210 s (Spigno and De Faveri, 2009). Besides, microwave

heating can significantly improve the washing step of extraction. As reported in the kinetic

study of MAE of oils from olive cake (Amarni and Kadi, 2010), the rate constant of the washing

step of MAE was 17 times greater than that of the conventional extraction. This is probably

due to the rupture of the plant structure by microwave heating that enhanced the penetration of

the solvent into the interior structure of the plant sample (Gujar et al., 2010). The downside of

MAE can be referred to in the work (Chan et al., 2011).

3.2. Ultrasonic-assisted extraction (UAE)

Ultrasonic means is another technique which is capable of enhancing its mass transfer

mechanism in the extraction process. UAE provides stirring and thermal effects for the

extraction solvent, as well as structural effects for the solid sample. Ultrasound generates the

growth of bubbles inside liquids causing the cavitation phenomenon to occur, where the

cavitation bubbles implode asymmetrically near the solid surface (Leighton, 1998). This

phenomenon generates microjets in the direction of the solid surface and creates micro stirring

effects at the interface. It also reduces the effect of the boundary layers of the extraction and

results in enhanced mass transfer (Floros and Liang, 1994; Leighton, 1998). In addition, the

movement and implosions of the bubbles repeatedly squeeze and release the sample in what is

known as the “Sponge effect” (Juárez et al., 1999), can create micro channels in the sample

which improves the penetration of the solvent and provides larger contact area for mass transfer

(Juárez et al., 1999; Muralidhara et al., 1985). Another effect of ultrasound is attributed to its

thermo-acoustic influence which also has impact on the mass transfer resistance due to heating

(Mason and Lorimer, 2002). The instrumental setup for UAE is shown in Fig. 2. The extraction

temperature of the system is normally controlled through a water bath whereas other more

advanced setups of UAE can be referred to in the literature (Shirsath et al., 2012).

In UAE, the washing stage is enhanced due to cell destruction, improved solvent penetration

and mass transfer intensification (Vinatoru et al., 1999) but the diffusion stage is unffected

(Milić, 2013). As reported in UAE of oil from tobacco seeds (Stanisavljević et al., 2007),

ultrasound improved the washing of oil from native seeds and some degree of milling effects

was observed in the seeds. However, the effect on ground seeds was insignificant

(Stanisavljević et al., 2007). The stirring effect of UAE in reducing external mass transfer

resistance for diffusion was better than that provided by conventional extraction using

mechanical agitation as the former gave better extraction yields (Cárcel et al., 2010). This was

proven by the results obtained from UAE of phenolic compounds from pomegranate peel in

which significant advantages were observed over conventional extraction techniques in terms

of extraction yields and extraction time (Pan et al., 2012). Nevertheless, UAE was less efficient

than conventional techniques in certain cases where oxidation and degradation of active

compounds occurred under prolonged sonication (Stanisavljevic et al., 2008; Stanisavljević et

al., 2007). The interaction between the highly reactive hydroxyl radicals (resulted from

sonication) and the active compounds is responsible for the degradation (Karabegovic et al.,

2011). The key feature related to UAE is short extraction time though at times it has no positive

effect on the extraction yield as compared to conventional techniques (Stanisavljević et al.,

2007; Velickovic et al., 2008).

3.3. Electrically-assisted extraction (EAE)

The kinetics of mass transfer can also be improved by exerting electrical effect during

extraction. There are two types of EAE techniques; high voltage electrical discharge (HVED)

and pulsed electric field (PEF). These two techniques operate in different mechanisms which

require different types of electrodes and applied voltage. As shown in Fig. 2, HVED requires

a high voltage generator and a needle electrode for arc discharge while PEF requires two

parallel plate electrodes for the induction of electric field.

The effectiveness of HVED on the extraction depends on the electrical breakdown between the

electrodes when high voltage is applied (Klimkin, 1990). This introduces energy directly into

the solvent-solid mixture through a plasma channel, which it is formed by a high voltage

electrical discharge generated between two submerged electrodes (Bogomaz et al., 1991). The

electrical breakdown creates high pressure shock waves and bubble cavitations which can

damage the cell structure and results in particle fragmentation (Gros et al., 2003; Touya et al.,

2006). Consequently, a better penetration of solvent into the solid particle enhances the mass

transfer mechanism. The performance of HVED in the extraction of polyphenols from wine

by-products was reported (Boussetta et al., 2011; Boussetta et al., 2012; Liu et al., 2011), and

the advantages of which were credited to its short treatment time (few ms) and low energy

consumption (10-50 kJ/kg) (Gros et al., 2003). With regards to the operational aspect, the

electrode gap distance is very critical for discharge formation and the optimum distance is very

much dependent on the nature of the extraction. An optimum electrode gap distance reduces

the amount of energy needed for plasma channel formation (Lang et al., 1998) and strengthens

the electric field, thus optimizes the discharge intensity (Sun et al., 1998). In addition, the pH

of the extraction system also affects the performance of HVED. The sample was found to be

relatively stable in acidic solution as compared to alkaline solutions as the latter can degrade

polyphenols when treated by HVED (Boussetta et al., 2011). This might have been caused by

the hydroxyl radicals which could have damaged the extracted compounds through oxidative

chemical reactions (Bogomaz et al., 1991; Chen et al., 2004; Ershov and Morozov, 2008).

Another electrically assisted technique is the pulsed electric field (PEF) which is based on the

electroporation phenomenon where it changes the permeability of cell membrane through a

potential difference across the membrane (Morales-Cid et al., 2010; Zimmermann et al., 1974).

During electroporation, molecular orientation takes place where the polar molecules align

themselves with the electric field and migrate to the membrane induced by the electric field

(Morales-Cid et al., 2010). The electrocompression exerted on the membrane ruptures the

membrane and creates pores on it (Soliva-Fortuny et al., 2009; Zimmermann et al., 1974). This

can result in a temporary (reversible) or permanent (irreversible) loss of membrane

permeability without having thermal alteration of the membrane (Morales-Cid et al., 2010;

Zimmermann et al., 1974). The permeability loss and the pore formation depend on the

induction of critical electric field strength and cell sizes in a range of 1-2 kV/cm for plant cells

sizes of 40-200 µm (Heinz et al., 2001). PEF can substantially enhance the mass transfer in

biological tissues of different food plants with very low heating as well as improving both the

extraction yields during the washing step and the diffusion step (El-belghiti and Vorobiev,

2005).

The following section illustrates the influences of the important operating parameters on the

extraction kinetics of the various assisted solvent extraction techniques discussed above.

4. Influences of operating parameters on extraction kinetics

The effect of the general operating parameters such as the extraction solvent, solvent to feed

ratio, sample particle size and temperature are considered crucial for both conventional and

non conventional techniques. These parameters as well as some specific parameters for certain

assisted solvent extraction techniques such as microwave power, intensity of ultrasonic,

electrical field intensity and number of charges are discussed hereinafter to provide a better

insight of the extraction kinetics.

4.1. General operating parameters of extraction

The extraction solvent and its concentration play an important role in the extraction of active

compounds from plants. Different extraction solvents have different abilities to overcome the

energy barrier, which is also known as the activation energy of extraction (Rakotondramasy-

Rabesiaka et al., 2007; Spigno and De Faveri, 2009). This energy is required for the solvent to

penetrate into the interior of the plant cells. Aqueous organic solvents are usually employed in

most of the extraction. The mixture of ethanol and water is frequently used as an extraction

solvent in the extraction of active compounds from Fumaria officinalis L. (Rakotondramasy-

Rabesiaka et al., 2010), chestnut tree wood (Gironi and Piemonte, 2011) and grape seeds

(Bucic-Kojic et al., 2007). A suitable extraction solvent can enhance the washing step and

shorten the extraction time (Rakotondramasy-Rabesiaka et al., 2007). It can also improve the

diffusivity of the solute in the solvent and subsequently maximize the equilibrium extraction

yield (Xu et al., 2008).

Solvent to feed ratio is another important parameter which when applied correctly can decrease

the mass transfer barrier during the diffusion of active compounds and subsequently enhance

the extraction yield (Franco et al., 2007a; Qu et al., 2010). However, if the ratio is beyond the

optimum, the excess solvent does not have a significant effect on the equilibrium extraction

yields resulting in solvent wastage. It is worthy to note that the initial extraction rate during

the washing period is not significantly affected by the solvent to feed ratio (Herodež et al.,

2003). On the other hand, extraction which is carried out at low solvent to feed ratio tends to

reach equilibrium much faster than those carried out at high solvent to feed ratio due to lower

equilibrium yields (Stanisavljević et al., 2007).

The extraction kinetics is also greatly affected by the particle size of the plant sample. Slight

changes in particle size can significantly affect the extraction result. The extraction yield during

the washing step can be improved by smaller particle size of the sample (Qu et al., 2010) but it

becomes unfavorable when a small particle size sample leads to high extraction yields of

undesirable compounds (Cissé et al., 2012). In terms of extraction kinetics, smaller particle size

increases the diffusivity and enhances the mass transfer mechanism in the diffusion step. This

is due to larger contact surface area with the solvent and shorter average diffusion path of the

active compounds from the solid to the solvent (Cissé et al., 2012; Herodež et al., 2003; Hojnik

et al., 2008). As a result, shorter extraction time is required. The classic example is the

extraction of antioxidants from pomegranate marc (Qu et al., 2010) whereby a remarkable

reduction of extraction time from 90 min to 2 min was achieved when the particle size was

reduced from 3.5 mm to 0.2 mm. One worth-noting point is that the particle size does not affect

the initial extraction rate in the washing step provided that the internal diffusion of the active

compounds is rate limiting (Herodež et al., 2003). Even though the effect of particle size on

the extraction process is obvious, it depends on the geometry of the extraction sample. For a

sample with plate geometry such as leaves, the effect of the particle size is not significant since

the relevant dimension for the diffusion of active compounds is the thickness of the leaves

(Wongkittipong et al., 2004). The effect might turn significant only when the particle size of

the leaves are reduced below its thickness such as in powder form.

Conventionally, the extraction temperature is monitored as it affects the stability of the active

compounds and the extraction performance. For extracting thermally stable compounds, better

extraction yield at elevated temperatures was achieved in shorter extraction time (Cissé et al.,

2012). This can be explained by the high temperature that increases the diffusivity of the solute

and decreases the energy barrier of the extraction (Cissé et al., 2012; Rakotondramasy-

Rabesiaka et al., 2007). The solvation power of the extraction solvent is also enhanced allowing

more active compounds to be dissolved in the solvent (Cissé et al., 2012; Rakotondramasy-

Rabesiaka et al., 2007; Xu et al., 2008). Furthermore, the extraction rate at the washing step

was reported to increase with temperature which could probably be due to increasing solvation

power of the solvent (Rakotondramasy-Rabesiaka et al., 2007, 2010). On the contrary, for

thermal sensitive compounds, elevated extraction temperature exerts negative effect on

extraction and has to be avoided at all costs. The general trend exhibited by these compounds

is that the stability decreases with increasing extraction temperature (Cissé et al., 2012). This

also signifies that the rate of degradation or decomposition of thermal sensitive compounds

depends on the extraction temperature (Xiao et al., 2012). Thus, the selection criteria of a

suitable extraction temperature should be based on both the extraction efficiency and thermal

stability of the active compounds.

4.2. Specific parameters of assisted solvent extraction

Having elucidated the pertinent operating parameters for solvent extraction, specific

parameters of certain assisted solvent extraction techniques will be discussed. These

parameters have equally important influence on the extraction efficiency. For instance,

microwave power controls the rate of heating during MAE extraction. As reported, higher

microwave powers decrease the extraction time but increase the extraction yields (Chemat et

al., 2005; Mandal and Mandal, 2010; Xiao et al., 2008). In the stability study of flavonoids

during microwave radiation (Biesaga, 2011), it was found that increasing the microwave power

amplifies the degradation as higher microwave heating causes sudden rise in temperature.

Elevated temperatures may result in overheating and undesired solvent evaporation, leading to

poor yields especially for thermally sensitive extracts. Similarly, the selection of microwave

power and extraction time is crucial to prevent thermal degradation of heat sensitive active

compounds.

For UAE, the effect of the intensity of ultrasounds on the extraction yields is significant (Ma

et al., 2009). In general, increased intensity level enhances the extraction yields and speeds up

the extraction regardless of the mode of radiation (Pan et al., 2012). However, this phenomenon

is not applicable to all cases as ultrasonic extraction does not impose any significant

enhancement on the extraction in some applications (Mircea, 2001). The evidence can be

observed from the poor ultrasound effect on both the yield and the kinetics in the extraction of

oil from woad seeds (Isatis tinetoria), independent of sonication conditions (Romdhane and

Gourdon, 2002). The poor effects imposed might be due to the geometrical shape of the treated

sample. While ultrasound enhances the extraction of olive leaves as the laminar shaped leaves

probably had received the ultrasound homogeneously but this was not the case for the grape

stalks which are of irregular shape (Cárcel et al., 2010).

The high voltage electrical discharge (HVED) in the context of electrical assisted extraction is

the key factor accountable for the extraction yields. The treatment time of HVED is based on

the number of discharges where the typical peak pulse voltage is at 40 kV (Boussetta et al.,

2011; Liu et al., 2011; Moubarik et al., 2011). Increasing the number of discharges accelerates

the EAE extraction and thus enhances the equilibrium yields. Both the extraction yields during

washing and diffusion stages is increased by increasing the number of discharges until it

reaches an optimum point (Boussetta et al., 2011; Moubarik et al., 2011). The washing and

diffusion coefficient are also improved by increasing the number of discharges.

Analogous to the previous observation on HVED, the equilibrium extraction yields of EAE at

the washing and diffusion stages, increases with increasing intensity of pulse electrical field

(PEF) regardless of the extraction rate that remain unchanged (El-belghiti and Vorobiev, 2005;

El Belghiti and Vorobiev, 2004; Moubarik et al., 2011). Similar effects were also reported

when the number of pulses were increased in the extraction of sugar from beets (El Belghiti

and Vorobiev, 2004). In this study, the yield of solute increased from 15 to 30% which

corresponds to an increase in the number of pulses from 100 to 250.

5. Mathematical modeling of batch solvent extraction

Many mathematical approaches can be used to model the extraction process. The modeling

equations are either theoretically derived or empirically formulated. The widely employed

Fick’s law of diffusion, chemical kinetic equations and other two-parametric empirical

equations are applicable for most extraction curves. Therefore, this section concentrates on the

derivation and applications of the mathematical modeling of batch solvent extraction.

5.1. Fick’s Law

The diffusion step in batch type extraction depends on two extraction mechanisms; internal

diffusion and external diffusion. The internal diffusion of active compounds, as explained in

Fick’s law, is driven by the difference in concentration between the plant matrix and the bulk

solvent (Bird et al., 2006) as follows:

dx

dCDN (1)

where N is the mass flux of the solute, C is the concentration of the solute in the solid particle,

D is known as the diffusivity or diffusion coefficient for the solute in the solvent, and x is the

distance in the direction of the transfer. For external diffusion, the active compounds diffuse

from the external surface of the solid to the bulk liquid. The determination of the rate limiting

mechanism in the diffusion stage is important in kinetic modeling as it determines a suitable

mathematical approach to model the extraction. To ensure efficient extraction, the external

mass transfer resistance has to be minimized so that the rate of extraction is dependent only on

the internal diffusion of the active compounds. Diffusivity in Fick’s law in Eq. (1) is an

important property that indicates the rate of mass transfer and it is useful for equipment design

(Perez et al., 2011). Most of the kinetic modeling in the literature investigates the diffusivity

or other mass transfer coefficients in solvent extraction.

Characterization of extraction can be performed via derivation of Fick’s law with initial and

boundary conditions. The solution of the mass transfer problem can be obtained analytically or

numerically depending on the complexity of the equations involved. Some basic assumptions

(Crank, 1975) which can be used to simplify the mass transfer problem are as follows:

i. Symmetrical and porous sample particles. The geometry of solid particles is assumed

to be spherical with radius of R or thin plate with half thickness of L.

ii. The solid particle is assumed to be of a pseudo-homogeneous medium. The

concentration of the active compounds in the solid particle depends on time and radius,

r or thickness, x.

iii. Uniform distribution of active compounds in the sample matrix.

iv. Homogeneous mixing between solvent and plant sample particles. The concentration

of the solute in the solvent only depends on time.

v. The mass transfer of active compounds from the solid is a diffusion phenomenon in

which the diffusion coefficient is independent of time.

vi. Diffusion of the solute and other compounds are in parallel and no interaction between

them.

vii. External mass transfer resistance is negligible. The concentration of the solute in the

solvent at the interior of the solid particle is equal to the concentration of the solute in

the bulk solvent.

5.1.1. Mass transfer in solid particles

One of the assumptions of the mass transfer is to treat the external mass transfer resistance as

negligible, which is crucial and it depends on the nature of the extraction. This assumption

simplifies most of the extraction problems. The extraction process model can thus be developed

by considering only the mass balance in a spherical solid particle as shown:

)( CDt

C

(2)

Where, t is the extraction time. Considering solely the spherical geometry of particles with

radius r, the respective initial and boundary conditions can be written as follows:

rCCt 0,0 (3)

RrCCt i 0,0 (4)

00,0

r

r

Ct (5)

where, C0 is the initial concentration of solute in the sample particle, Ci is the concentration of

solute at the interface of sample particle. With the assumption of negligible external mass

transfer resistance, the concentration at the particle interface will become zero as described in

Eq. (4). The ordinary differential equation (ODE) for both spherical and plate geometry of

sample can then be expressed as in Eq. (6) and Eq. (7) respectively (Crank, 1975):

Spherical:

2 2

0

210

2 ( 1)1 sin exp

n

ni

C C R nr Dn t

C C r n R R

(6)

Plate:

2 2

0

200

4 ( 1) (2 1) (2 1)1 cos exp

2 1 2 4

n

ni

C C n x n Dt

C C n L L

(7)

The mass of solute transferred from the sample particle at any time, M can be calculated by

integrating the concentration of solute over the radius or thickness of the particles in Eq. (6)

and Eq. (7) to obtain Eq. (8) and Eq. (9) respectively.

Spherical:

12

22

22exp

161

n R

tDn

nM

M

(8)

Plate:

02

22

22 4

)12(exp

)12(

181

n L

Dtn

nM

M

(9)

where, M∞ is the total amount of solute transferred after infinite time. After a certain time lapse

or usually after the washing stage, only the first term of the series remains significant (Spiro,

1988). Both Eq. (8) and Eq. (9) can then be reduced to the following form (Perez et al., 2011):

( )1 BtM t

AeM

(10)

where A is the model constant, and B is the diffusion rate constant. Theoretically, B = π2D/r2

is for spherical particles and B = π2D/4L2 is for plate particles. The constant B might not

applicable under certain conditions when its value has to depend on the geometry of plant

samples. The expression in Eq. (10) can be further modified based on non-extracted fraction of

the solute in the sample particle, E. Rearrangement of which has the simplified version as

shown below (Chen and Chen, 2011):

BtAeM

tME

)(

1 (11)

BtAE lnln (12)

Alternatively, the concentration of solute in the extraction solvent at any time, c can be

expressed by considering only the first term of Eq. (8) and (9) (Spiro, 1988) with the simplified

forms shown as follows:

Spherical:

2

87.9498.0ln

R

Dt

cc

c

(13)

Plate:

24

87.921.0ln

L

Dt

cc

c

(14)

where c∞ is the concentration of solute in extraction solvent after infinite time. By plotting Eq.

(13) or Eq. (14) using experimental extraction curve, two intersecting straight lines can be

drawn and the slope of the first line is steeper than the second. The intersection between the

lines is the transition point. This denotes the point where the extraction changes its phase from

the washing step to the diffusion step (Kandiah and Spiro, 1990; Spiro et al., 1989). To achieve

better modeling results, Osburn and Katz (1944) suggested that the modeling of the extraction

process should consider both the washing and diffusion steps in Eq. (13) or Eq. (14) to yield

the following equations:

Spherical:

2

22

22

12

12expexp

6

R

tDf

R

tDf

c

cc

(15)

Plate:

2

22

22

12

12 4exp

4exp

8

L

tDf

L

tDf

c

cc

(16)

where f1 and f2 are fractions of the solute extracted from the washing and diffusion stages with

diffusion coefficient of D1 and D2, respectively. The parameters D2 and f2 can be determined

from the slope and the intersection points of Eq. (13) or Eq. (14) as the second exponential

term is significant for the second stage of the extraction. In the early stage of extraction, the

second exponential term is close to unity thus D1 and f1 can be determined.

5.1.2. Mass transfer in solid particles and the extraction solvent

The models developed in the previous section can be theoretically obtained by solving the mass

transfer in solid particles. To make the models more realistic, mass balance in the solvent

should be taken into account. From the mass balance in solid particles shown previously in Eq

(2), the mass balance in the solvent can be expressed as shown in Eq (17):

)(tJdt

dcVL (17)

where, VL is the volume of solvent used in the extraction. The initial extraction conditions and

the boundary conditions are presented in Eq. (18 &19) and Eq. (20-22) respectively:

rCCt 0,0 (18)

00 ct (19)

At center, r = 0

0,0

r

Cr (20)

At interface, r = R

Diffusive flux from the solid particle (Cissé et al., 2012; Wongkittipong et al., 2004; Xu et al.,

2008):

r

CDAtJ s

)( (21)

Incoming flux in solvent (Tsibranska et al., 2011; Wongkittipong et al., 2004):

dt

dCVtJ s)( (22)

where As is the specific area of the solid particle and Vs is the volume of plant sample. There

are two different boundary conditions at the solid interface which need to be considered;

namely the diffusive flux for the binary mixture in Eq. (21) and the incoming flux of the solvent

in Eq. (22). The same assumption of negligible external mass transfer resistance applies for

both Eq. (21) and Eq. (22). These equations can be used interchangeably or together depending

on the number of independent variables that need to be solved (Wongkittipong et al., 2004).

For example, in the modeling of extraction process by Wongkittipong et al. (2004), Eq. (21)

and Eq. (22) were used to solve the mass transfer problem that involved solid particles with

both cylindrical and plate geometry. This modeling approach is also applicable for other

geometry when the geometry shape factor is known (Wongkittipong et al., 2004).

5.1.3. Mass transfer with extraction temperature variation

The modeling approach presented so far is confined to specific extraction temperature. To

investigate the influence of temperature on the extraction process, Arrhenius equation shown

in Eq. (23) can be used to describe the effect of temperature on the diffusivity of the system.

TR

EAD aexp'

(23)

where, A’ is the pre-exponential factor and Ea is the activation energy of the Arrhenius model.

By comparing the reference diffusivity value, Dref at the reference temperature, Tref, the

diffusivity D at the extraction temperature T can be expressed as follows:

TTR

E

TREA

TREA

D

D

ref

a

refa

a

ref

11exp

exp'

exp'

(24)

By substituting this temperature related expression into the modeling system previously

discussed, the influence of temperature on the extraction process can be determined (Xu et al.,

2008).

5.1.4. Mass transfer with degradation of active compounds

When the extraction involves thermally sensitive compounds such as vitamins, the extraction

profile is driven by two processes namely; the diffusion of active compounds from plant

samples and the thermal degradation of active compounds in the extraction solvent (Xiao et al.,

2012):

ionDecompositSolventplantkDdeg

To improve the accuracy of the modeling, degradation terms can be added as first order rate

equations in the mass balance expressions for solid particles and solvent as shown in Eq. (25)

and Eq. (26) respectively (Cissé et al., 2012).

Mass balance in solid particle with degradation term:

CkCDt

Cdeg)(

(25)

Mass balance in solvent with degradation term:

LL cVktJdt

dcV deg)(

(26)

where, kdeg describes the degradation constant of the extraction which can be related to

Arrhenius equations as shown in Eq.(27):

TR

Ekk aexpdeg

(27)

where, k∞ is the degradation rate constant for the active compound. The parameters in Eq. (27),

i.e. k∞ and Ea, and the initial and boundary conditions in Eq. (18-22) are required to solve the

thermal degradation associated with the mass transfer problem.

5.1.5. Mass transfer with external mass transfer resistance

When the external mass transfer resistance in the extraction system becomes significant, the

convective mass transfer coefficient should be considered. A dimensionless form of Eq. (2) is

required to model this particular condition (Franco et al., 2007b).

t

Y

r

Y2

2

(28)

where the dimensionless groups for radius, extraction time and the yield can be defined as in

Eq. (29 – 31):

Dimensionless radius

R

rr

(29)

Dimensionless extraction time

2R

tDt

(30)

Dimensionless yield

)(

)(

0 e

e

CC

CCY

(31)

where Ce is defined as the concentration of solute which remains in the sample particle after

infinite extraction time. The initial and boundary conditions are similar to those in Eq. (3) and

Eq. (5), only the boundary conditions at the interface are different and defined as follows:

c

cc

D

RkY

D

Rk

r

Yt cc,0

(32)

where, kc is the convective mass transfer coefficient. This mass transfer problem was reported

by Walas (1991) and the suggested analytical solution is shown below:

1

_2

__

)sin(),(

n

nn t

r

BtrY

(33)

while Bn can be determined from Eq. (34):

2))(sin(2

)cos()sin(4

nn

nnnn

nnB

(34)

where λn is the eigenvalues of the function given in Eq. (35):

1)(cot

D

Rkg c

(35)

The average non-extracted fraction of the solute in the sample particle can be obtained from

Eq. (33) as follows:

1

0

____

),.( rdtrYY

(36)

1

2_

)exp()8415.0()(

n

nnn tBtY

(37)

These series can be truncated to the first five terms to estimate the non-extracted oil fraction

with minor error.

5.1.6. Modified Fick’s law

The models presented earlier are derived fundamentally and tre suitable to be used in scaling

up study and equipment design as the parameters involved in the equations have real physical

meanings. More simplified models like the film theory (Stankovic ́et al., 1994; Veljkovic ́and

Milenovic ,́ 2002) and unsteady state diffusion through plant material (Ponomaryov, 1976;

Velickovic et al., 2006) can be adopted to describe the washing step and the diffusion step in

the extraction process. These two-parametric equations are derived from Fick’s law and are

expressed in Eq. (38) and (39) respectively:

Film theory:

1 (1 ) ktcb e

c

(38)

Unsteady state diffusion theory:

'

0

(1 ') k tCb e

C

(39)

where, b and b’ denote the coefficients for extraction kinetics in the washing step while k and

k’ are the coefficients for the diffusion step. To express Eq. (39) on the basis of the amount of

solute extracted in the extraction solvent, the equation can be modified into Eq. (40):

'0

0

(1 ') k tC Cb e

C

(40)

where (C0-C) denotes the amount of solute dissolved in the extraction solvent. The modeling

equations involving Fick’s law are commonly used in the modeling of solvent extraction as

they represent the fundamental theory for mass transfer.

In the next section, adaptation of another theory in the modeling of solvent extraction will be

discussed.

5.2. Rate law

Besides the expressions derived from Fick’s Law, rate law had also been adapted in the

modeling of solvent extraction of active compounds from various plants in addition to being

employed to investigate the degradation rate of active compounds in the solution as discussed

previously (Xiao et al., 2012). Extraction models based on a second-order rate law are normally

applied in conventional and non-conventional extractions (Pan et al., 2012; Qu et al., 2010;

Rakotondramasy-Rabesiaka et al., 2007; Rakotondramasy-Rabesiaka et al., 2009). The rate of

dissolution of active compounds of a plant into the extraction solvent is given as follows:

21 cckdt

dc (41)

where k1 is the second order extraction rate constant. Taking the initial and boundary conditions

as t = 0 to t and c = 0 to c, the integrated rate law can be obtained:

tkc

tkcc

1

12

1

(42)

By linear transformation of Eq. (42), the rate constant k1 can be determined by fitting Eq. (43)

with experimental data. Subsequently Eq. (44) can be obtained from Eq. (43).

c

t

ckc

t2

1

1 (43)

ctckt

c2

11

1 (44)

c/t in Eq. (44) indicates the initial extraction rate, which can also be denoted by h, and can be

defined by Eq. (45) when the extraction time t approaches zero.

21 ckh

(45)

The concentration of solute in the extraction solvent at any time can then be described as:

cth

tc

1 (46)

5.3. Empirical equations

In the modeling of solvent extraction, various empirical models have been employed, either

developed from the fundamental models or adapted models as discussed above. The empirical

models are more suitable for extraction processes involving assisted means such as microwave,

ultrasound and electrical as they cannot be adequately described theoretically. The most

commonly used empirical model was proposed by So and Mcdonald (1986) and Patricelli et

al. (1979) which has the form shown in Eq. (47).

)exp(1)exp(1 tkctkcc ddww (47)

where, cw and cd are the amounts of solute extracted in the solvent during the washing step and

the diffusion step, respectively. The amount of solute extracted can be expressed per mass of

sample used, or expressed in fraction by comparing with the equilibrium yield. kw and kd

represents the coefficients of extraction kinetics during the washing step and the diffusion step,

respectively. The empirical Eq. (47) resembles the model proposed by Osburn and Katz (1944)

previously shown in Eq. (15) and Eq. (16).

Peleg’s model (Peleg, 1988), which was used to describe the sorption curves, was adapted for

the modeling of solvent extraction process as shown in Eq. (48):

tKK

tcc

210

(48)

where K1 is the model rate constant, K2 is the model capacity constant and C0 is usually equal

to zero. Both the Peleg’s model and the second-order integrated rate law in Eq. (42) are

hyperbolic equations. Other empirical models proposed for solvent extraction include

Ponomaryov equation (Ponomaryov, 1976; Velickovic et al., 2006) and other two-parametric

empirical models such as parabolic diffusion model, power law and etc (Kitanovic et al., 2008).

6. Extraction models in batch solvent extraction

The kinetic models for various batch extraction techniques are tabulated in Table 1. This table

summarizes the model parameters with their associated operating conditions for the extraction.

There are two types of model parameters in the extraction namely; the rate of extraction either

for the washing or the diffusion steps, and the extraction capacity or equilibrium extraction

yield. The model parameters presented in Table 1 can be varied according to the plant sample

used, as the content of the active compounds may differ due to geographical location, weather

variation and soil conditions of the plantation. As a result, the kinetics as well as the extraction

capacity of the techniques at their designated operating conditions can be affected. As shown

in Table 1, the yield of the total extractive substances, total phenolic compounds, oil content

or the individual active compounds can be modeled by the corresponding equations given either

in terms of the concentration in the extraction solvent or the amount of extractives per sample

used.

From Table 1, it can be observed that the conventional extraction techniques are usually

modeled using Fick’s law derivatives models as the model parameters contain physical

meanings which can be used for further interpretations. For example; Biot number (Bi) that

expresses the relative significance of internal and external mass transfer resistance (Seikova et

al., 2004). The theoretical based kinetic models are also suitable for scaling up purposes. On

the other hand, in the modeling of assisted solvent extraction techniques related to microwave,

ultrasonic, electrical field and charges; simplified models from modified Fick’s law, empirical

models and rate law are preferred as these techniques implicate much more complicated mass

transfer problems.

Considering the sensitivity of the models’ coefficients with the change in the extraction

conditions, the increase or the decrease of the models’ coefficients would indicate certain

kinetic behavior of the extraction and the trend is dependent on their mathematical equations.

Most of the models’ coefficients increase to indicate enhanced kinetic (faster rate). However,

some coefficients, e.g. K1 and K2 of Peleg model, decreases to indicate a similar behavior.

Based on the tabulated data listed in Table 1, the effects of conventional operating parameters

on the extraction kinetics are in descending order of significance such as particle size of sample,

solvent to feed ratio and extraction temperature. Decrease in the particle size of sample and

increase in the solvent to feed ratio will enhance the diffusivity of Fick’s law derivatives models

(D), the washing coefficient (b’) and the diffusion coefficient (k’) of modified Fick’s law

models, and also the initial extraction rate (h) and extraction rate constant (k1) of rate law

models, predominantly. Once the parameters reached their optimum value, further increase in

the coefficients will not be significant. On the other hand, increase in the extraction temperature

also enhances the diffusivity (D). However, it may have opposite effect beyond certain

optimum temperature. This could probably be due to thermal degradation of active compounds.

Furthermore, the effects of specific parameters of assisted extraction on the models’

coefficients as presented in Table 1 indicates that as microwave power (MAE) increases, the

washing and diffusion stages in terms of extraction yields (cw and cd) and kinetics coefficients

(kw and kd) generally will be enhanced provided there is no thermal degradation of active

compounds during the extraction. Furthermore, increase in the intensity of ultrasounds (UAE)

enhances the washing stage by increasing the initial extraction rate (h), and also the extraction

rate constant (k1) in the rate law models. In EAE, increase in the number of discharge (HVED)

improves both the washing and diffusion stages in terms of extraction yields (cw and cd) and

kinetics coefficients (kw and kd). However, increase in the intensity of electric field (PEF), can

only enhance the extraction yields (cw and cd).

Table 1: Kinetic models of batch solvent extraction

Authors Extraction Techniques Operating

conditions Equations Model parameters

Perez et al.

(2011)

oil from

confectionery,

oilseed

(Helianthus

annuus) and

wild

(Helianthus

petiolaris)

sunflower

seeds.

conventional

extraction

n-hexane,10 ml/g, 5

g sample, particle

sizes of 667 μm

(confectionery), 624

μm (oilseed) and 586

μm (wild), stirring,

40-60 oC.

10

(spherical)

Confectionery:

A = 0.1361-0.1516 (50 oC) a

B = (1.19-1.66) x10-4 s-1 (50 oC) a

D = (1.34-1.87) x10-12 m2s-1

Oilseed:

A = 0.1148-0.1446 (40 oC) a

B = (2.09-5.10) x10-4 s-1 (60 oC) a

D = (2.06-5.03) x10-12 m2s-1

Wild:

A = 0.1798-0.21 (40 oC) a

B = (1.04-1.35) x10-4 s-1 (60 oC) a

D = (0.96-1.18) x10-12 m2s-1

Ben Amor and

Allaf (2009)

anthocyanins

from Malaysian

Roselle

(Hibiscus

sabdariffa)

DIC

pretreatment

followed by

conventional

extraction

water,100 ml/g, 2 g

DIC treated and

untreated sample,

particle sizes of 135

μm, stirring, 100 oC

11 (plate) DIC treated sample:

D = (4.62-6.11) x10-11 m2s-1

Untreated sample:

D = 4.19 x10-11 m2s-1

Hojnik et al.

(2008)

lutein from

Marigold flower

petals

conventional

extraction

hexane, 10 ml/g

(total 500 ml

solvent), particle

sizes < 315 μm,

stirring, 20-60 oC,

simultaneous

hydrolysis by adding

10% (w/v) alkali

solution at 7.5 ml/g

of sample mass into

extraction system to

obtain free lutein

15 20 oC:

D1= 6.128 x10-12 m2s-1

D2 = 0.011 x10-12 m2s-1

40 oC:

D1 = 2.175 x10-12 m2s-1

D2 = 0.018 x10-12 m2s-1

60 oC:

D1 = 1.503 x10-12 m2s-1

D2 = 0.011 x10-12 m2s-1

Franco et al.

(2007a)

oil from Rosa

rubiginosa

conventional

extraction

ethanol,15-50 ml/g,

particle sizes < 600

μm, stirring, 50 oC

13 15 ml/g:

B = 0.0004 s-1

D = 0.61 x10-11 m2s-1

25 ml/g:

B = 0.0021 s-1

D = 3.19 x10-11 m2s-1

50 ml/g

B = 0.0046 s-1

D = 6.99 x10-11 m2s-1

Herodež et al.

(2003)

carnosic acid

(CA), ursolic

acid (UA) and

oleanolic acid

(OA) from

Balm (Melissa

officinalis L.)

leaves

conventional

extraction

ethanol, 4-10 ml/g

(total 500 ml

solvent), particle

sizes of 200-400 μm,

0-80 oC

16 CA, 20 oC, 4-10 ml/g:

D1 = (0.42-3.07) x10-11 m2s-1 (10 ml/g) a

D2 = (0.039-0.061) x10-11 m2s-1 (10

ml/g) a

UA, 20 oC, 4-10 ml/g:

D1 = (0.48-4.29) x10-11 m2s-1 (10 ml/g) a

D2 = (0.03-0.106) x10-11 m2s-1 (10 ml/g)

a

OA, 20 oC, 4-10 ml/g:

D1 = (0.28-2.59) x10-11 m2s-1 (8 ml/g) a

D2 = (0.045-0.119) x10-11 m2s-1 (6 ml/g)

a

CA, 4 ml/g, 0-80 oC:

D1 = (0.29-0.52) x10-11 m2s-1 (40 oC) a

D2 = (0.013-0.039) x10-11 m2s-1(40oC) a

UA, 4 ml/g, 0-80 oC:

D1 = (0.45-0.63) x10-11 m2s-1 b

D2 = (0.027-0.041) x10-11 m2s-1 (0 oC) a

OA, 4 ml/g, 0-80 oC:

D1 = (0.40-1.72) x10-11 m2s-1 (0 oC) a

D2 = (0.044-0.084) x10-11 m2s-1 (20oC) a

Tsibranska et al.

(2011)

total phenolic

compounds

from Sideritis

ssp. L.

conventional

extraction

80% Ethanol, 15

ml/g, particle size of

40 μm, stirring, room

temperature

2, 17-20,

22

D = 1.5 x10−12 m2s-1

Gujar et al.

(2010)

thymol from

seeds of

Trachyspermum

ammi

MAE methanol, 30 ml/g, 1

g sample, 35-45 oC

(regulated by

microwave power at

0-300 W)

2, 17-20,

22

35 oC:

D = 1.835 x10-13 m2s-1

40 oC:

D = 2.46 x10-13 m2s-1

45 oC:

D = 3.24 x10-13 m2s-1

Xu et al. (2008) isoflavones

from stem of

Pueraria lobata

(Willd.)

conventional

extraction

50% n-butanol, 50

ml/g, 4 g sample,

particle size of 400-

800 μm, stirring, 25 oC

2, 17-21 D = 1.7 x10-11 m2s-1

Wongkittipong

et al. (2004)

andrographolide

from

Andrographis

paniculata

conventional

extraction

(Soxhlet)

60% ethanol, 50

ml/g, particle size of

600-800 μm, 22-60 oC

2, 17-22 D = (8.43-52.1) x10-14 m2s-1 (60 oC) a

Cissé et al.

(2012)

anthocyanins

from Hibiscus

sabdariffa

conventional

extraction

water, 25 ml/g,

particle sizes of 150

μm, stirring, 25-90 oC

Ea = 61 kJ mol-1 c

k∞ = 44200 s-1 c

18-21, 25-

27

D = (3.9-13.5) x10-11 m2 s-1 (90 oC) a

Franco et al.

(2007b)

oil from Rosa

rubiginosa

conventional

extraction

92% ethanol, 50

ml/g, particle sizes of

250, 350 and 750

μm, stirring, 50 oC

28-37 250 μm:

D = 5.2 x10-11 m2s-1

kR/D = 0.153

350 μm:

D = 7.4 x10-11 m2s-1

kR/D = 0.131

750 μm:

D = 30.0 x10-11 m2s-1

kR/D = 0.127

Velicˇkovic´ et

al. (2006)

extractive

substances from

Salvia

officinalis L.

(SO) and Salvia

glutinosa L.

(SG) sage

UAE petroleum ether, 70%

ethanol and water as

solvent, 10 ml/g, 10 g

sample, total nominal

power of 3x50 W (40

kHz), 40 oC

38, 39 SO, petroleum ether:

b = 0.143

k = 1.14 x10-3 min-1

b' = 0.358

k' = 4.50 x10-3 min-1

SO, 70% ethanol:

b = 0.299

k = 2.21 x10-3 min-1

b' = 0.325

k' = 2.54 x10-3 min-1

SO, Water:

b = 0.251

k =1.86 x10-3 min-1

b' = 0.454

k' = 5.47 x10-3 min-1

SG, petroleum ether:

b = 0.144

k = 0.69 x10-3 min-1

b' = 0.50

k' = 5.09 x10-3 min-1

SG, 70% ethanol:

b = 0.254

k = 2.67 x10-3 min-1

b' = 0.276

k' = 3.05 x10-3 min-1

SG, water:

b = 0.233

k =1.51 x10-3 min-1

b' = 0.422

k' = 4.06 x10-3 min-1

Stanisavljević et

al. (2007)

oil from

tobacco

(Nicotiana

tabacum L.)

seeds

UAE n-hexane and

petroleum ether as

solvent, 3-10 ml/g, 5

g sample, total

nominal power 3x 50

W (40 kHz), 40 C

39 n-hexane:

b' = 0.5-0.6 (10 ml/g) a

k' = 4-7.8 x10-3 min-1 (3 ml/g) a

petroleum ether:

b' = 0.4-0.7 (10 ml/g) a

k' = 4-14 x10-3 min-1 (3 ml/g) a

Karabegovic et

al. (2011)

flavonoids from

Artemisia

vulgaris (AV)

and Artemisia

campestris

(AC)

UAE methanol, 10 ml/g,

10 g sample, total

nominal power 3x 50

W (40 kHz), 25 C

40 AV:

b'= 0.45

k'= 3.0 x10-3 min-1

AC:

b' = 0.4

k' = 3.8 x10-3 min-1

Pan et al. (2012) total phenolic

compounds

from

pomegranate

peel

UAE water, 50ml/g,

operational modes:

1. Continuous mode,

intensity of 2.4-59.2

W/cm2 (20 kHz)

2. Pulse mode,

intensity of 59.2

W/cm2 (20 kHz),

pulse duration (s)

/interval (s) of 2/2,

5/5 and 5,15

41-46 Continuous mode:

h = 0.031-1.398 gL-1min-1 (59.2 W/cm2)

a

k1 = 0.012-0.185 Lg-1min-1 (59.2

W/cm2) a

Pulse mode, 2/2

h =1.128 gL-1min-1

k1 = 0.158 Lg-1min-1

Pulse mode, 5/5

h =1.456 gL-1min-1

k1 = 0.199 Lg-1min-1

Pulse mode, 5/15

h =0.498 gL-1min-1

k1 = 0.065 Lg-1min-1

Qu et al. (2010) total phenolic

compounds

from

pomegranate

peel

conventional

extraction

water, 10-50 ml/g, 2

g sample, particle

sizes of 0.2-3.5 mm,

stirring, 25-90 oC

.

41-46 50 ml/g, 0.2-3.5 mm, 25 oC:

h = 0.375-44.229 gL-1min-1 (0.2 mm) a

k1 = 0.115-7.685 Lg-1min-1 (0.2 mm) a

10-50 ml/g, 0.2 mm, 25 oC:

h = 12.967-50.905 gL-1min-1 (10 ml/g) a

k1 = 0.603-2.575 Lg-1min-1 (50 ml/g) a

50 ml/g, 0.2 mm, 25-90 oC:

h = 10.512-100.721 gL-1min-1 (95 oC) a

k1 = 1.948-6.314 Lg-1min-1 (95 oC) a

Rakotondramas

y-Rabesiaka et

al. (2009)

protopine from

Fumaria

officinalis

L.

conventional

extraction

0-94% aqueous,

6.25-100 ml/g (total

500 g solvent),

particle size of 0.4-

0.5 mm, pH of 5.8,

stirring, 30 oC

41-46 0-94% ethanol, 25 ml/g:

h = 3.8-15.9 mgL-1min-1 (44 % ethanol) a

k1 = 2.3-6.2 Lg-1min-1 (0% ethanol) a

water, 6.25-100 ml/g:

h = 5.69-20.4 mgL-1min-1 (6.25 ml/g) a

k1 = 1.3-36.4 Lg-1min-1 (100 ml/g) a

44% ethanol, 6.25-100 ml/g:

h =7.28-30.3 mgL-1min-1 (6.25 ml/g) a

k1 = 0.5-17.5 Lg-1min-1 (100 ml/g) a

Moubarik et al.

(2011)

extractive from

Foeniculum

vulgare

Pretreatments

by PEF, ED

and UAE

followed by

conventional

extraction

pretreatment:

1. PEF, water, 2

ml/g, 6.5g sample

(moisture 88-92%),

field intensity 0-430

V/cm (1000 Hz),

1000 pulses, pulse

repetition time of

10ms

2. ED, water, 2ml/g,

400 g sample, 40 kV,

pulse repetition time

of 10ms, 0-90 pulses

3. UAE, water, 2

ml/g, 30 g sample,

400 W/cm2 (40 kHz),

0-100 min

extraction:

Water, 2 ml/g,

stirring, 20 oC

47 PEF:

cw = 0.43-0.54 (430 V/cm) a, d

cd = 0.31-0.43 (430 V/cm) a, d

kw= 0.17-0.22 b

kd = 0.011-0.015 b

ED:

cw = 0.43-0.55 (90 pulses) a, d

cd = 0.31-0.43 (90 pulses) a, d

kw= 0.214-0.286 (90 pulses) a

kd = 0.0144-0.0187 (90 pulses) a

UAE:

cw = 0.47-0.58 (100 min) a, d

cd = 0.27-0.39 (100 min) a, d

kw= 0.17-0.20(100 min) a

kd = 0.010-0.013 (100 min) a

Amarni and

Kadi (2010)

oil from olive

cake

MAE hexane, 3 ml/g, 50 g

sample, stirring,

microwave power of

180-720 W

47 cw = 4.02-4.22 g oil/ 100 g sample (720

W) a

cd = 0.58-1.07 g oil/ 100 g sample (720

W) a

kw = 7.54-250.14 (720 W) a

kd = 0.54-0.90 (540 W) a

Bucic-Kojic et

al. (2007)

total

polyphenols

from

grape seeds

conventional

extraction

50% ethanol, 40

ml/g, 0.5 g sample,

particle sizes

of >0.63 mm, 0.63-

0.4 mm, 0.4-0.16

mm, extraction

temperature at 25 oC, 50 oC and 80 oC

48 >0.63 mm:

K1= 0.71-1.154 min (mg GAE/gdb)-1 (80 oC) e

K2 = 0.0393-0.0651 (mg GAE/gdb)-1 (80 oC) e

0.63-0.4 mm:

K1= 0.3056-0.4256min (mg GAE/gdb)-1

(80 C) e

K2 = 0.0375-0.0448 (mg GAE/gdb)-1 (80

C) e

0.4-0.16 mm:

K1= 0.0645-0.0869min/mg (50 C) e

K2 = 0.0175-0.0193 mg-1(50 C) e

a The bracketed operating condition refers to the upper limit of the model parameter. b The effect of operating

condition on the model parameter is not significant. c the parameters used for computational obtained from other

studies. d Coefficient based on c/c∞. e The bracketed operating condition refers to lower limit of the model

parameter.

7. Summary

Various solvent extractions inclusive of kinetic modeling details have been compiled and

summarized in Table 2. This review gives an in-depth discussion on the extraction kinetics of

conventional and assisted solvent extraction techniques together with their pertinent operating

parameters. Mathematical modeling equations and kinetic models of extraction are also

presented. This review provides useful information pertaining to mathematical modeling of

various batch solvent extractions. The model parameters presented can also be utilized for the

comparative and scaling up studies.

Table 2: Summary of kinetic modeling of batch solvent extraction

Batch solvent extraction

Conventional Microwave-

assisted Ultrasonic-

assisted Electrically-assisted

Extraction

Technique

batch extraction

with stirring and

temperature

control

microwave-

assisted

extraction

(MAE)

ultrasonic-

assisted

extraction (UAE)

high voltage electrical

discharges (HVED);

pulsed electric field (PEF)

Features

temperature

control, agitation

microwave

heating

ultrasonic

radiation

electric field, electrical

discharges

Extraction

mechanism

Washing and

diffusion of active

compounds

Localized

heating of

microwave

builds internal

pressure to

rupture plant

cells

Cavitation

phenomenon

which provides

stirring and

structure effects

on plant structure

Electrical breakdown

creates high pressure

shock waves to damage

the cell structure (HVED);

Electroporation

phenomenon creates pore

on the membrane of plants cells (PEF)

Improvements on extraction

kinetic

/

improve

extraction rate in

washing and diffusion step

resulted in short

extraction time

improve rate in

diffusion step, enhance

equilibrium

extraction yield

enhance extraction yields

during washing and diffusion steps (PEF),

accelerate extraction and

improve yields (HVED)

Important

parameters

solvent nature,

particle size of

sample and its

structure, solvent

to feed ratio,

temperature

microwave

power

ultrasound

intensity

intensity of Electric field

(PEF), Number of pulses

(PEF), Numbers of

discharge (HVED)

Modeling

equations

employed

Fick's law

derivations,

chemical rate law

and Peleg's model

Fick's law

derivations,

Patricelli's model

modified Fick's

law, chemical rate

law,

Patricelli's model

ACKNOWLEDGEMENTS

This work was carried out under the Centre for Separation Science and Technology (CSST),

University of Malaya and financially supported through UMRG (RP002A-13AET) and PPP

(PV062/2011B) grants.

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NOMENCLATURE

A model constant (dimensionless)

A’ pre-exponential factor (dimensionless)

As specific area of the solid particle (m2)

B diffusion rate constant (s-1)

b coefficient of extraction kinetic in the washing step (dimensionless)

b’ coefficient of extraction kinetic in the washing step (dimensionless)

C concentration of solute in the solid particle (g m-3)

c concentration of solute in the extraction solvent at any time (g m-3)

C0 initial concentration of the solute in the sample particle (g m-3)

c∞ concentration of solute in the extraction solvent after infinite time (g m-3)

cd amount of solute extracted in the solvent during diffusion step (dimensionless)

Ce concentration of solute remains in the sample particle after infinite time (g m-3)

Ci concentration of solute at the interface of the sample particle (g m-3)

cw amount of solute extracted in the solvent during the washing step (dimensionless)

D diffusion coefficient (m2 s-1)

D1 diffusion coefficient during the washing stage (m2 s-1)

D2 diffusion coefficient during the diffusion stage (m2 s-1)

Dref diffusion coefficient at the reference temperature (m2 s-1)

E non extracted fraction of solute in the sample particle (dimensionless)

Ea activation energy

f1 fraction of solute extracted from the washing stage (dimensionless)

f2 fraction of solute extracted from the diffusion stage (dimensionless)

h initial extraction rate (g L-1 min-1)

k coefficient of extraction kinetic in the diffusion step (min-1)

k’ coefficient of extraction kinetic in the diffusion step (min-1)

k1 second order extraction rate constant (L g-1 min-1)

K1 model rate constant (min mg-1)

K2 model capacity constant (mg-1)

k∞ degradation rate constant (s-1)

kc convective mass transfer coefficient (m s-1)

kd coefficient of extraction kinetics during the diffusion step (min-1)

kdeg degradation constant of the extraction (s-1)

kw coefficient of extraction kinetics during the washing step (min-1)

L half of the thickness of the solid particle (m)

M mass of solute transferred from the sample particle at any time (g)

M∞ total amount of solute transferred after infinite time (g)

N mass flux of the solute (g m-2 s-1)

r radial distance in the diffusion direction (m)

R radius of the solid particle (m)

r dimensionless radius (dimensionless)

t time (s)

T temperature (oC)

t dimensionless time (dimensionless)

Tref reference temperature (oC)

VL volume of the solvent used in the extraction (m3)

Vs volume of the plant sample (m3)

x distance in the diffusion direction (m)

Y dimensionless yield (dimensionless)

Y average non-extracted fraction of solute in the sample particle (dimensionless)