Evaluation of models for supercritical fluid extraction

International Journal of Heat and Mass Transfer 72 (2014) 274–287

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Evaluation of models for supercritical fluid extraction Amit Rai a,⇑, Kumargaurao D. Punase b,1, Bikash Mohanty a,2, Ravindra Bhargava a,3 a b

Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India Department of Chemical Engineering, University of Petroleum and Energy Studies, Deharadoon 248007, Uttarakhand, India

a r t i c l e

i n f o

Article history: Received 22 April 2013 Received in revised form 1 December 2013 Accepted 4 January 2014 Available online 31 January 2014 Keywords: Diffusion Mass transfer Numerical analysis Packed bed Simulation Supercritical fluid

a b s t r a c t Various models available in the literature for the modeling of supercritical extraction process are studied and validated using published experimental data. The first model considers internal mass transfer coefficient as the controlling parameter for the extraction process. On the other hand, the second model analyzes the dynamic behavior of the extraction process by considering intra-particle diffusion and external mass transfer. These models have also been studied to understand the effects of various model parameters like intra-particle diffusion, mass transfer coefficients & operating parameters on cumulative extraction yield. The model proposed by Reverchon (1996) [13] predicts a cumulative yield within an error limit of +9% in the MATLAB simulation and +4% to 5% in the FEMLAB simulation. Also, the model proposed by Goto et al. (1993) [8] fits the experimental data of Kim et al. (2007) [19], Skerget and Knez (2001) [20], and Tonthubthimthong et al. (2004) [21] within an error band of +10% to 2%. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the use of supercritical fluid extraction (SFE) process for the removal of organic compounds from different liquid and solid matrices has received much attention. Because, supercritical fluids have several distinctly advantageous properties, such as liquid like density and gas like viscosity and diffusivity, they have high mass-transfer characteristics and their effectiveness can be controlled by small changes in temperature and pressure leading to better fractional separation [1]. The mechanism of supercritical fluid extraction process can be explain by the following steps; (1) Transport of supercritical solvent to the particle surface and then from particle surface to interior of particle by diffusion. (2) Dissolution of the solute with the supercritical solvent. (3) Transport of supercritical solvent with molecules from interior of particle to particle surface. (4) Transport of supercritical solvent and solute molecules from particle surface to bulk solvent. Hence, the possibility of using supercritical solvents at the commercial level has increased in the recent past [2]. To design an extraction plant, it is necessary to have reliable mass-transfer models that will allow the determination of optimum operating condi-

⇑ Corresponding author. Mobile: +91 75790 75744; fax: +91 1332 276535. E-mail addresses: [email protected] (A. Rai), [email protected] (K.D. Punase), [email protected] (B. Mohanty), [email protected] (R. Bhargava). 1 Mobile: +91 9917720952. 2 Tel.: +91 01332 5710. 3 Tel.: +91 01332 285382. 0017-9310/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.01.011

tions [3]. Single stage supercritical extraction and separation produces a quasi-solid extract, which consists of several compound families. However, the yield data and the shape of the extraction curve are influenced by the presence of undesired compounds [4]. Some authors have attempted to describe the evolution of the extraction process by using empirical kinetic equations [5,6]. Heat transfer analogy of a single sphere cooled in a fluid medium was used by Reverchon et al. [7] to describe the extraction process. However, this model describes a highly idealized situation and the performance of the fixed bed of particles used is overestimated. The extraction process was also modeled by integrating the differential mass balances in the solid and fluid phase. Goto et al. [8] described the extraction of peppermint essential oil as a desorption process characterized by the attainment of an instantaneous equilibrium by breaking the peppermint leaves into differential slab elements. Sovova [9,10] modeled the vegetable oil extraction process based on the broken and intact cell model by considering the oil contained as either accessible or inaccessible. The same model was also proposed for pepper extraction [11], where the internal and external mass transfer resistances were taken into account. Goto et al. [12] proposed a shrinking core model and explained the ginger rhizomes extraction considering effective diffusivity and solubility as model parameters. However, the model was unable to describe the experimental results obtained for different particle sizes. Reverchon [13] took into account the shape of the particles (slabs) to obtain a good fit with the experimental data for large particles and found that internal mass transfer controlled the essential

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275

Nomenclature

Reverchon [13] Ap total surface of particles (m2) c extract concentration in the fluid phase (kg/m3) cn fluid-phase concentration in the nth stage (kg/m3) h spatial coordinate along the bed (m) kp volumetric partition coefficient of the extract between the solid and the fluid phase at equilibrium (–) K internal mass-transfer coefficient (m/s) n number of stages deriving from the bed subdivision (–) q extract concentration in the solid phase (kg/m3) qn solid phase concentration in the nth stage (kg/m3) ⁄ q concentration at the solid–fluid interface (kg/m3), t extraction time (s) ti internal diffusion time (s) u superficial velocity (m/s) V extractor volume (m3) W CO2 mass flow rate (kg/s) e bed porosity (–) q solvent density (kg/m3) Goto et al. [8] a1, a2 constants defined by Eq. (3.41) (–) ap specific surface area (1/m) A constant defined by Eq. (3.42) (–) Ab bed cross section area (m2) b, c constant defined by Eq. (3.43) (–) C solute concentration in the solvent (kg/m3) Cp solute concentration within the particle pore (kg/m3) Cp0 solute concentration in the pore phase at t ¼ 0 (kg/m3) Cps solute concentration in the pore space at the particle surface (kg/m3) Cs solute concentration in particle (kg/m3)

oil extraction from sage leaves. Goodarznia and Eikani [14] proposed a model based on differential mass balance on a single particle as well as in the fluid phase and validated the experimental data of Reverchon et al. [7,15] and Sovova [10]. The model also included the effects of internal diffusivity and axial dispersion. The phase equilibrium depends on solute composition, solvent composition, extraction pressure and temperature. It controls the initial extraction period when the fluid phase leaving the extractor is either in equilibrium or is about to attain equilibrium with the solute in the solid phase. When the solute concentration in solid phase is high, like that of vegetable oil in Canola seed, the fluid-phase equilibrium concentration is independent of the matrix and equal to oil solubility. When the initial solute concentration in the plant is low, which is rare for vegetable oils, the equilibrium is usually controlled by solute–solid interaction and the fluid-phase concentration is much lower than the oil solubility. The equilibrium is expressed as a linear relationship between the solid and fluid phase concentrations and the proportionality constant is called the partition coefficient. Goto et al. [16] used the Brunauer–Emmett–Teller adsorption isotherm to simulate a smooth transition between the equilibrium of free solute at high concentrations and the equilibrium of solute–solid interaction at low concentrations. Perrut et al. [17] considered a discontinuous equilibrium to model the sunflower oil extraction process. The fluid-phase concentration is equal to the oil solubility above the discontinuity and is determined by partition coefficient below it. The discontinuous equilibrium curve was successfully applied by Wu and Hou [18] in the simulation of egg yolk oil extraction.

Cs0 C0 dp DAB Dax De F h ka kp kf K r R t Us x x0 xp xs y z

a b / h

qs

s IC BC

solute concentration in the solid phase at t ¼ 0 (kg/m3) total solute concentration (kg/m3) particle diameter (m) binary diffusion coefficient (m2/s) axial dispersion coefficient (m2/s) effective intraparticle diffusion coefficient (m2/s) cumulative fraction of solute extracted (–) height of the bed (m) adsorption rate constant (1/s) overall mass transfer coefficient (m/s) external mass transfer coefficient (m/s) equilibrium adsorption coefficient (–) radial position in spherical particle (m) radius of spherical particle (m) time (s) superficial velocity (m/s) dimensionless solute concentration in effluent (–) initial solute mass ratio in the solid phase (–) dimensionless solute concentration in pore (–) dimensionless solute concentration in solid particle (–) solute mass ratio in the fluid phase (kg/kg) bed height coordinate (m) bed void fraction (–) particle porosity (–) dimensionless mass transfer coefficient (–) dimensionless time (–) solid density without void volume of the solid matrix (kg/m3) total bed volume/volumetric flow rate (s) initial condition boundary condition

It can be concluded that there are various models available in the literature that differ not only from a mathematical point of view, but also in terms of mass transfer mechanisms, which control the supercritical extraction process of different matrices. Hence, a single model cannot describe all the experimental results. In all published models, the initial extraction process is governed by the solubility equilibrium between the solute and the fluid phase, which, in most cases, is assumed to be linear as detailed information is not available for complex matrix systems. From a mathematical point of view, all the proposed models are based on differential mass balance integration with some assumptions. Table 1 shows that the published supercritical fluid extraction models differ in the description of phase equilibrium, flow pattern, and solute diffusion in the solid phase. As the experimental data considered in the present study are related to seeds of sage, black pepper, nimbin and caffeine. Amongst these sages, black pepper and nimbin belong to the category of essential oil. The literature shows that for extraction of essential oil as well as caffeine the model developed by Goto et al. [7] and Reverchon [3,12] are suitable. Therefore, in the present paper these models are considered for extraction of oil from seeds of sage, black pepper, nimbin and caffeine. 2. Mathematical modeling The initial distribution of the solute within the solid substrate affects the selection of the possible models. The solute may be free on the surface of the solid material, adsorbed on the outer surface, located within pores or evenly distributed within plant cells. In the

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Table 1 The models based on differential mass balance equations and applied to the extraction of natural products with near-critical CO2. References

Solutea

Equilibrium relationship

Particleb

Solvent flowc

[29] [8] [12] [30] [13] [31] [32] [14] [33] [34] [16] [36] [37] [10] [43] [10] [28] [35] [38] [39] [40] [41] [42]

FO EO EO OR, FO EO EO EO EO EO, FO EO EO, W FO FO FO FO EO FO EO,FO FO EO EO EO EO

Constant Linear Linear Linear Linear Constant Constant Linear Linear Linear BET Combined Combined Constant Constant Approx. linear Combined Combined Linear Linear Polynomial Freundlich isotherm Linear

No internal resistance Porous slab Shrinking core Sphere, cylinder, slab Sphere, cylinder, slab Shrinking core Shrinking core Sphere Sphere No internal resistance Porous particle Porous particle No internal resistance B+I B+I B+I B+I B+I No internal resistance B+I Porous Spherical Porous Spherical Porous Spherical

PF M AD PF PF AD PF AD PF AD M PF PF PF PF PF PF PF PF AD AD AD AD

a b c

C: caffeine; EO: essential oil; FO: fatty oil; OR: oleoresin; W: wax. B + I: broken and intact cells. PF: plug flow; M: mixer; AD: flow with axial dispersion.

present work, an in-depth evaluation of two models [8,4] based on differential mass balance integration have been carried out using the published experimental data of Kim et al. [19], Reverchon et al. [4], Skerget and Knez [20], and Tonthubthimthong et al. [21].

Solute free solvent

2.1. The Reverchon model A model based on the integration of differential mass balances along the extraction bed is proposed with the following assumptions: (1) Plug flow exists in the bed. (2) The axial dispersion in the bed is negligible. (3) The fluid flow rate, temperature, pressure and bed properties are constant. Based on the following assumptions, the model equations (2.1.1) and (2.1.2) is obtained for the situation where the seeds are stationary and oil free solvent is entered at the top of extraction vessel, by taking solute mass balance on the solvent and solid phase over an element of extractor of height dh respectively as described in Fig. 1 [22].

@c @c þ eV þ Ap Kðq  q Þ ¼ 0 @t @t @q ð1  eÞV ¼ Ap Kðq  q Þ @t

uV

c ¼ 0;

q ¼ q0 ;

at t ¼ 0

cð0; tÞ ¼ 0 at h ¼ 0

ð2:1:1Þ

Fig. 1. Schematic diagram of the extraction vessel.

ð2:1:3aÞ ð2:1:3bÞ

Assuming a linear relationship for SFE process between c and q due to lack of experimental phase equilibrium data,

c ¼ kp  q

Solute bearing solvent

ð2:1:2Þ

⁄



δh

ð2:1:4Þ

In Eq. (2.1.2), the fraction ApK/(1  e)V depends on the geometry of particles, though ApK and e are supposed to be constant within the bed. This fraction is dimensionally equal to l/s. Therefore, the internal diffusion time (ti) which, according to the hypothesis, is characteristic of the extraction process and is defined as ti = (1  e)V/ApK and related with the internal diffusion coefficient (Di) by ti = r2/ 15Di, where r is the mean particle radius. Therefore Eq. (2.1.2) becomes Eq. (2.1.5).

@q 1 ¼  ðq  q Þ @t ti

ð2:1:5Þ

Now, the fixed bed is divided into n stages and it is also assumed that the fluid and solid phase concentration is uniform in each stage, which is approximated as a plug-flow extractor through a series of mixed extractors. So model equations (2.1.1) and (2.1.5) are written as a set of 2n ODEs given by Eqs. (2.1.6) and (2.1.7), respectively.

W

q

ðcn  cn1 Þ þ e

v

dcn v dqn þ ð1  eÞ ¼0 n dt n dt

dqn 1 ¼  ðqn  qn Þ ti dt

ð2:1:6Þ ð2:1:7Þ

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Initial conditions are given as:

C p ¼ C P0

cn ¼ 0 and qn ¼ q0

C s ¼ C s0 at t ¼ 0; 0 6 r 6 R   @C p ¼ kf ðC ps  CÞ at r ¼ R for all t  De @r   @C p ¼ 0 at r ¼ 0 for all t @r

at t ¼ 0

Eqs. (2.1.6) and (2.1.7) are solved by using a fourth-order Runge– Kutta method to obtain the solute concentration in solvent and solid phase, respectively with initial conditions.

at t ¼ 0; 0 6 r 6 R

ð2:2:1:3aÞ ð2:2:1:3bÞ ð2:2:1:3cÞ ð2:2:1:3dÞ

C 0 ¼ bC po þ ð1  bÞC s0

2.2. The Goto model In this model, the solid particle is considered as a porous structure and the mechanism of the extraction of solute from porous structures is divided into five steps: (1) Transport of the supercritical solvent molecules from bulk solvent to the particle surface through the boundary layer adjacent to the particle surface. (2) Penetration of the solvent molecules into the pores of the particle (impregnation). (3) Diffusion of the extractable component i.e. solute to the particle surface (internal diffusion). (4) Diffusion of the solute from the particle surface through a stagnant (boundary) layer of the fluid (external diffusion), and (5) Convective transfer of the solute in the intergraded space towards the extractor outlet. The diffusion of the extractable substance through the boundary layer is a relatively slow process. Diffusion in a solid, particularly in a nonporous solid, occurs even more slowly, and it is this step that limits the extraction rate. 2.2.1. General solid phase mass balance The transport of solute (at low concentration) in or out of the particle is assumed to take place by diffusion through a network of pores as described in steps 1–5. As the pore diameter considerably varies, the diffusion is described in terms of an effective pore diffusion coefficient. The solute distribution is assumed to be radially symmetric. The fluid-solute mass balance can be represented for a porous spherical particle as shown in Fig. 2. The differential mass balance equation for solute concentration in the particle pores is written as:

  @C p   2 @C p 1 @ r De @r @C s ¼ 2 b  ð1  bÞ r @t @t @r

ð2:2:1:1Þ

The local extraction rate, which is equivalent to the desorption rate, is assumed to be reversible and linear in terms of adsorption rate constant ka and the adsorption equipment constant K given by Eq. (2.2.1.2)

  @C s Cs ¼ ka C p  @t K

2.2.2. General fluid phase mass balance The concentration C of the solute in the fluid phase at height z in a bed of particulate material depends on the rate of mass transfer from particles at height z, the fluid flow rate and extent of mixing. Eq. (2.2.2.1) is obtained by taking a mass balance around an element Dz of bed height as shown in Fig. 3 [23].

a

@C @C @2C ¼ U s þ Dax 2 þ ap ð1  aÞkf ðC ps  CÞ @z @z @z

ð2:2:2:1Þ

The specific surface area is defined as. ap = 6/dp. The initial and boundary conditions for Eq. (2.2.2.1) are:

C ¼ 0 at t ¼ 0; 0 6 z 6 h

ð2:2:2:2aÞ

C ¼ 0 at z ¼ 0 @C ¼ 0 at z ¼ h @z

ð2:2:2:2bÞ ð2:2:2:2cÞ

2.2.3. Model simplification and analytical solution In order to simplify the model equation and its initial and boundary conditions, the following assumptions are made: (1) axial dispersion is negligible, (2) radial dispersion is also neglected (small column diameter), (3) isothermal process, (4) the packed column is isobaric, (5) no interaction among solutes in the fluid phase or solid phase, (6) local equilibrium adsorption between solute and solid in pore of material, (7) differential bed is gradientless in solid and fluid phase, (8) physical properties of the supercritical fluid are constant. It is assumed that the combined internal and external mass transfer processes are described by a linear driving force approximation, which is derived by assuming a parabolic concentration profile within the particle. Coutlet(t)

ð2:2:1:2Þ

z=h

Eq. (2.2.1.1) can be solved using the following initial and boundary conditions.

r=R

Z+∆Z C(t,r,z) r+ r Particle surface

Z

Solid

Cp(t,r,z)

Fluid Cps(t,R,z) SCF Film Cs(t,r,z)

z=0 Cinlet(t) Fig. 2. Porous solid particle.

Fig. 3. Schematic represent of packed bed element.

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A. Rai et al. / International Journal of Heat and Mass Transfer 72 (2014) 274–287 100 90 80 70 Modelled Yield, (%)

60

Experimental (Reverchon, 1996)

50 40 30 20 10 0 0

50

100

150

200

250

300

350

400

450

Time, (min)

Fig. 4. Comparison between experimental and model predicted data for MATLAB.

kf ap ð1  aÞðC  C ps Þ ¼

15De R2

ð1  aÞðC ps  CÞ

ð2:2:3:1Þ

The overall mass transfer coefficient for a spherical particle is given by Eq. (2.2.3.2)

kp ¼

kf 1 þ Bi =5

ð2:2:3:2Þ

where Bi = kfh/De is Biot number. Average intra-particle and solid concentrations are evaluated using the parabolic profile.

Z R 3 4pr 2 C p ðrÞdr 4pRs 0 Z R 3 Cs ¼ 4pr 2 C s ðrÞdr 4pRs 0

ven by C s ¼ KC p . To further simplify the model equations, some dimensionless variables are defined as: x = C/C0, xp ¼ C p =C 0 , xp ¼ C p =C 0 , xs ¼ C s =C 0 , h = t/s and / = kpaps. In terms of dimensionless variables, Eqs. (2.2.2.1) and (2.2.1.1) can be written as Eqs. (2.2.3.4) and (2.2.3.5):

dx x /ð1  aÞ  xs  þ ¼ x dh a a K  dxs / xs   x ¼ b dh K þ ð1  bÞ K

ð2:2:3:4Þ ð2:2:3:5Þ

Initial conditions are:

Cp ¼

ð2:2:3:3aÞ ð2:2:3:3bÞ

It is assumed that the equilibrium in the pores is established instantaneously for a relatively fast adsorption–desorption rate and is gi-

x ¼ 0 at h ¼ 0 K xs ¼ ½b þ ð1  bÞK

ð2:2:3:6aÞ at h ¼ 0

ð2:2:3:6bÞ

The analytical solution of Eqs. (2.2.3.4) and (2.2.3.5) with its initial conditions is given as:

100 +42%

90

+9%

80

Predicted Yield, (%)

70 60 50 40 30 20 10 0 0

10

20

30

40

50

60

70

80

90

Experimental Yield, (%) Fig. 5. Error between predicted data from Reverchon model and experimental data.

100

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When the mass transfer resistance is negligible, i.e. when / = 1; Eq. (2.2.3.9) reduces to Eq. (2.2.3.10).

100

 FðhÞ ¼ 1  exp 

90 80

ð2:2:3:10Þ

The mass ratio of the solute in the fluid phase as a function of time can be obtained as:

70 Yield, (%)

h ½b þ ð1  bÞKð1  aÞ þ a



60

yðtÞ ¼

50

   

b x0 qs t t  exp a2 þ ð1  bÞ A exp a1 K qCO2 s s

ð2:2:3:11Þ

qCO y

where x ¼ c02 . The mass of extract at the bed outlet can be calculated from Eq. (2.2.3.12):

40 30 20

Modelled

mðtÞ ¼

Experimental (Reverchon, 1996)

10

Z

t

0

yðtÞQ CO2 qCO2 dt

ð2:2:3:12Þ

Using Eq. (2.2.3.8a) in Eq. (2.2.3.8b), we obtain

0 0

100

200

300

  

b 1 t 1 þ ð1  bÞ x0 qs Q CO2 As exp a1 K a1 s   1 t 1  exp a2 þ a2 s

400

mðtÞ ¼

Time, (min) Fig. 6. Comparison between experimental data and Reverchon model predictions with FEMLAB.

xðhÞ ¼ A½expða1 hÞ  expða2 hÞ

ð2:2:3:7Þ

where,

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 1 2 2 b þ b  4c ; a2 ¼ b  b  4c 2 2 ð1  aÞ/ A¼ ½b þ ð1  bÞKaða1  a2 Þ / 1 /ð1  aÞ þ þ b¼ b þ ð1  bÞK a a / c¼ ½b þ ð1  bÞKa

a1 ¼

ð2:2:3:8aÞ ð2:2:3:8bÞ ð2:2:3:8cÞ

FðhÞ ¼

The ordinary differential equations, Eqs. (2.2.3.4) and (2.2.3.5), for the dimensionless solute concentration in the bulk fluid phase and solid phase respectively are used as the model equations with Eqs. (2.2.3.6a) and (2.3.3.6 b) as the initial conditions. The model equations are simplified using Laplace transform to obtain the analytical solution of the model in terms of the dimensionless solute concentration in the bulk fluid phase, given by Eq. (2.2.3.7). Eq. (2.2.3.9), which gives the cumulative fraction of the solute extracted or extraction yield is used for model validation against experimental results with the determination of the constants defined by Eqs. 2.2.3.8a, 2.2.3.8b, 2.2.3.8c, 2.2.3.8d. 2.2.4. Estimation of the physical properties and parameter Identification 2.2.4.1. Estimation of the physical properties. The empirical correlation proposed by Jossi et al. can be used to calculate the viscosity of CO2 as given below [14]: 0:25

ð2:2:3:9Þ

½ðlf  lf Þn þ 104 

¼ 0:10230 þ 0:023364qr

þ 0:058533q2r  0:040758qr3 þ 0:0093324q4r

100 +4

90

-5%

80

Predicted Yield, (%)

70 60 50 40 30 20 10 0 0

10

20

ð2:2:3:13Þ

ð2:2:3:8dÞ

The cumulative fraction of solute extracted up to dimensionless time h is given by:

Z h 1 xdh 1a 0

A expða1 hÞ  1 expða2 hÞ  1  ¼ 1a a1 a2



30

40

50

60

70

80

90

Experimental Yield, (%) Fig. 7. Error between predicted data from Reverchon model and experimental data.

100

ð2:2:4:1:1Þ

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A. Rai et al. / International Journal of Heat and Mass Transfer 72 (2014) 274–287 100 90 80 70 Model prediction by MATLAB Model prediction by FEMLAB

Yield, (%)

60 50 40 30 20 10 0 0

50

100

150

200

250

300

350

400

450

Time, (min)

Fig. 8. Comparison between Reverchon model prediction with MATLAB and FEMLAB.

0.18 0.5

0.16

0.45 0.4 Cumulative fraction, F (-)

Solute Concntration, X (-)

0.14 0.12 0.1 0.08 0.06 0.04

0.25 0.2 0.15

Modeled Experimental (Kim et al.,2007)

0.05 0

0 0

5

10

Time

15

20

25

The viscosity l⁄ at normal pressure can be calculated as follows:

105 T 0:94 ; r

  fn

0

10

l ¼ 34:0  T r 6 1:50 5   lf n ¼ 17:78  10 ð4:58T r  1:67Þ5=8 ; T r > 1:50

20

30

40

50

Time, θ (-)

(-)

Fig. 9. ODE and analytical solutions for the overallsolute concentration in bulk fluid phase.



0.3

0.1

ODE Solution Analytical Solution

0.02

0.35

Fig. 10. Comparison between experimental data and simulated data.

The axial dispersion coefficient in the fluid phase is calculated using the correlation given by Tan and Liou [26]:

Pe ¼ 1:634 Re0:265 Sc0:919

ð2:2:4:1:3Þ

1=6

Tc

where n ¼ 1=2 2=s . M Pc The particle porosity (b) and the bed void fraction (a) is computed by using the relation b = 1  (qp/qs) and a = 1  (qb/qp), respectively. The effective intra particle diffusion coefficient De is estimated from De = DABb2. The binary diffusion coefficient DAB, can be obtained using Riazi and Whitson correlation [24]. The external mass transfer coefficient, kf, is calculated with the Wakao and Kaquei correlation [8,25].

Sh ¼ 2 þ 1:1Sc1=3 Re0:6

ð2:2:4:1:2Þ

where Sh = 2Rkf/De, Sc ¼ l=ðqCO2 De Þ, and Re ¼ ð2RU s qCO2 Þ=l.

where Peclet number Pe is obtained using the relation Dax(2UsR)/Pe.

2.2.4.2. Model parameters. The experimental data related to extraction of caffeine from green tea [19], sage oil [4], pepper [20], nimbin from neem seed [21] is used for model validation. In the present work, two models are selected and model equations are solved in MATLAB and FEMLAB environment. After solving the model equations, the results are validated with the experimental values using the tuning parameter ‘‘Di’’ and ‘‘K’’ of the model equations.

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1

P redicted cum ulative extraction yield, F (-)

0.9

+15% -2%

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Experimental cumulative extraction yield, F (-) Fig. 11. Error predictions from Goto model and experimental data of [19].

k1 ¼ hf ðxn ; yn Þ   h k1 k2 ¼ hf xn þ ; yn þ 5 5   3h 3k1 9k2 ; yn þ þ k3 ¼ hf xn þ 10 40 40   4h 44k1 56k1 32k3 ; yn þ  þ k4 ¼ hf xn þ 5 45 15 9   8h 19372k1 25360k3 64448k3 212k1 ; yn þ  þ  k5 ¼ hf xn þ 9 6561 2187 6561 729   9017k1 355k1 46732k3 49k5 5103k5  þ þ  k6 ¼ hf xn þ h; yn þ 3168 33 5247 176 18656

Solute concntration in solid phase, Xs (-)

1.4 1.2 1 0.8 0.6 0.4 0.2 Solute Concentration

0 0

5

10

15

20

25

30

Time, θ (-) Fig. 12. Dimensionless average solute concentration profile in solid phase with time.

3.2. The algorithm for solution technique employed for the Reverchon model using MATLAB is as follows

3. Solution technique The solution strategies used for the ordinary differential equations (ODE) and analytical equations are discussed in this section. 3.1. Solution technique for the solution of ODE The Dormand–Prince method of Runge–Kutta family [27] is used for the solution of the model ordinary differential equations. It uses a sampling of slopes through an interval and takes a weighted average to determine the right end point. The method is given as:

 ynþ1 ¼ yn þ

35 500 125 2187 11 k1 þ k3 þ k4 þ k5 þ k6 384 1113 192 6784 84

where k1–k6 is given as:

The 2n differential equations are solved by applying ode45 toolbox of MATLAB 7.0 which works on the principle of Runge–Kutta (Dormand–Prince) method. The advantages of using Dormand–Prince method over other Runge–Kutta methods are that it matches more terms in the Taylor series approximation by taking a weighted average of several derivative approximations.



Step 1: Define all the input parameters like bed void fraction, particle diameter, length of the extractor, solvent flow rate and density of SC-CO2. Step 2: Define the ordinary differential equations, Eqs. (2.1.1) and (2.1.5), for solute concentration in bulk fluid and solid phases depending upon the number of divisions in the bed. Step 3: Give time interval step size for the time span up to which the solution is to be computed. Step 4: Define the initial conditions of the solute concentrations, c0 and q0 in fluid and solid phases, respectively. Step 5: Compute the solution for the solute concentration, c in bulk fluid and solute concentration q in solid phase with respect to time using Runge–Kutta (Dormand–Prince) method. Step 6: Print the calculated values.

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[2.89, 1.84; 2.64]

0.9

10 MPa 18 MPa 20 MPa 26 MPa 10 MPa (Tonthubthimthong et al., 2004) 18 MPa (Tonthubthimthong et al., 2004) 20 MPa (Tonthubthimthong et al., 2004) 26 MPa (Tonthubthimthong et al., 2004)

0.9

[4.16, 2.66; 3.83]

0.8

[2.89, 1.84; 2.64] (Kim et al., 2007) [4.16, 2.66; 3.83] (Kim et al., 2007)

0.7

0.7

[9.39, 6.01; 8.71] (Kim et al., 2007)

Cumulative fraction, F (-)

Cumulative fraction, F (-)

[9.39, 6.01; 8.71]

0.8

0.6 0.5 0.4 0.3 0.2 0.1 0

0.6 0.5 0.4 0.3 0.2

0

10

20

30

40

50

Time, θ (−)

0.1

Fig. 13. Combined effects of mass transfer coefficients and effective diffusivity at P = 40 MPa and T = 313–353 K. Parameters in brackets are. [kf  105 (m/s); kp  105 (m/s); De  109 (m2/s)].

0 0

2

4

6

8

10

12

14

Time, θ (-)

1 [29.66, 19.00; 27.50] [6.31, 4.03; 5.80] [4.16, 2.66; 3.83] [4.16, 2.66; 3.83] (Kim et al., 2007) [6.31, 4.03; 5.80] (Kim et al., 2007) [29.66, 19.00; 27.50] (Kim et al., 2007)

0.9

Cumulative fraction, F (-)

0.8

Fig. 16. Effect of pressure on extraction yield at constant temperature 328 K and flow rate 0.62 cm3/min.

3.3. The solution technique and algorithm employed for the Reverchon model using FEMLAB is as follows

0.7

The partial differential equations, Eqs. (2.1.1) and (2.1.5), are solved by FEMLAB using Eqs. (2.1.3a) and (2.1.3b) as initial and boundary conditions, respectively. Firstly, Eqs. (2.1.1) and (2.1.5) are converted into a dimensionless form by introducing some variable such as T ¼ Ks t, H ¼ Ku h. Then Eq. (2.1.1) is written as:

0.6 0.5 0.4 0.3

   @c @c Ap c q ¼0 þ þ @T @H kp V     @q Ap e  c ¼ q @T kp V 1e

0.2 0.1 0 0

10

20

30

40

ð3:3:1Þ ð3:3:2Þ

50

Time, θ (−) Fig. 14. Effect of mass transfer coefficients and effective diffusivity at T = 323 K and P = 10–40 MPa. Parameters in brackets are [kf  105 (m/s); kp  105 (m/s); De  109 (m2/s)].

Eq. (2.1.5) was solved in FEMLAB after adding a diffusion term to it, which is a function of the Peclet number, defined in terms of an effective diffusion coefficient. Eq. (3.3.2) becomes:

1

3

1.24 cm /min 3 0.62 cm /min

0.9

1

3

0.24 cm /min 3

0.9

0.24 cm /min (Tonthubthimthong et al., 2004)

0.8

3

0.62 cm /min (Tonthubthimthong et al., 2004) 3

Cumulative fraction, F (-)

Cumulative fraction, F (-)

0.8 0.7 0.6 0.5 0.4 333 K

0.3

323 K

0.7 0.6 0.5 0.4 0.3 0.2

308 K

0.2

1.24 cm /min (Tonthubthimthong et al., 2004)

308 K (Tonthubthimthong et al., 2004)

0.1

323 K (Tonthubthimthong et al., 2004)

0.1

333 K (Tonthubthimthong et al., 2004)

0 0

2

4

6

8

10

12

Time, θ (-)

Fig. 15. Effect of temperature on extraction yield (F) at pressure = 20 MPa, solvent flow rate = 0.62 m3/min.

0 0

2

4

6

8

10

12

Time, θ (-) Fig. 17. Effect of solvent flow rate on extraction yield at P = 20 MPa and T = 328 K.

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1 0.9

Cumulative fraction, F (-)

0.8 0.7 0.6 0.5 0.4

1.85 mm 1.445 mm

0.3

1.015 mm 0.575 mm

0.2

1.850 mm (Tonthubthimthong et al., 2004) 1.445 mm (Tonthubthimthong et al., 2004)

0.1

1.015 mm (Tonthubthimthong et al., 2004) 0.575 mm (Tonthubthimthong et al., 2004)

0 0

2

4

6

8

10

12

14

16

18

Time,θ θ (-) Fig. 18. Effect of particle size on extraction yield at P = 20 MPa and T = 308 K.

Table A1 Experimental conditions and process parameters for sage oil extraction [13]. S. No.

Parameter

Magnitude

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

P T Q e

9 MPa 323 K 8.83 g/min 0.4 285 kg/m3 2.31  105 Pa s 0.75  103 m 0.24  105 m2/s 8.48  1012 m2/s 6  1013 0.455  103 m/s 1.91  105 m/s 0.06 m 0.17 m 0.160 kg 0.2

q l dp DL De Di u kf dE He W kp

Table A2 Experimental conditions and process parameters for pepper extraction [20]. S. No.

Parameter

Magnitude

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

a

0.26 0.3 0.25  103 m 24,000 m1 60 mg/g of pepper 475 bar 80 °C 32.6 1/h 110.43 s 2.652 0.6213  105 m/s 1.987 0.4446  105 m/s 39.08  109 m2/s 11.78 5.70

b dp ap Co P T Qv

s Re kf Bi kp De / K

   @c @c Ap c 1 @2c q ¼ þ þ @T @H kp Pe @H2 V

Eq. (3.3.3) can be rearranged to Eq. (3.3.4) so that it resembles the standard form of convection diffusion equation in FEMLAB with transient analysis.

@c 1 @2c  @T Pe @H2

! ¼

   Ap c @c q  kp @H V

ð3:3:4Þ

Eqs. (3.3.2) and (3.3.4) are solved using FEMLAB. The algorithm used for the solution of the above partial differential equations is given as: Step 1: Open FEMLAB and choose Chemical Engineering Module, Mass Balance, Convection and Diffusion, and Transient Analysis. The default variable name c is used for Eq. (3.3.4). Step 2: For the solute concentration in the solid phase, use the Multiphysics menu, and choose the same equation. This time, however, change the name of the variable to q. Add this equation to the problem. Step 3: Draw a line from x = 0 to x = 1 and set the mesh to have 60 elements, or 121 nodes. For the two problems, then, there are 242 degrees of freedom. Step 4: Ensure that the concentration equation for c is selected (under the Multiphysics menu). Under the Physics/Subdomain settings, the following equation is displayed:

dts

@c þ r  ðDrcÞ ¼ R  u  rc @t

Rearrange this equation in such a way that Eq. (3.3.4) is obtained. This is done by setting the following parameters:

dts ¼ 1; D ¼

1 ; Pe

R¼

Ap  rate; V

u¼u

Also, select the artificial diffusion option and choose the Petrov– Galerkin method to consider the effect of additional diffusion in the problem. Choose the Init tab and set the initial concentration to zero. Step 5: Under Options/Expressions/Subdomain Expressions, define the following term for the equilibrium equation.

rate ¼ ðq  c=kp Þ ð3:3:3Þ

Step 6: Under the Multiphysics menu, select the second equation for q. The equation is of a similar nature and uses the following parameters:

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Table A3 Experimental conditions and process parameters in the model at various conditions for caffeine extraction [19].

q (kg/m3)

Q  106 (kg/h)

s (s)

Re

kf  105 (m/s)

Bi

kp  105 (m/s)

De  109 (m2/s)

/

K

Effect of pressure 10 323 20 323 40 323

390 790 920

28.08 28.08 28.08

227.64 462.95 542.56

0.32 0.23 0.19

29.66 6.31 4.16

2.81 2.83 2.83

1.84 2.66 6.01

27.50 5.80 3.83

4.32 1.87 1.44

3181 784 229

Effect of temperature 40 313 40 323 40 353

960 920 820

28.08 28.08 28.08

566.15 542.56 843.59

0.18 0.19 0.22

2.89 4.16 9.39

2.80 2.83 2.85

1.84 2.66 6.01

2.64 3.83 8.71

1.04 1.44 2.91

564 229 210

P (MPa)

T (K)

b = 0.61, a = 0.84, ap = 11,538 m1, dp = 0.520 mm and C0 = 34.832 mg/g of green tea.

Table A4 Experimental conditions and process parameters in the model at various conditions for nimbin extraction [21].

q (kg/m3)

Q  106 (m3/min)

s (s)

Re

kf  105 (m/s)

Bi

kp  105 (m/s)

De  109 (m2/s)

/

Effect of flow rate 20 328 20 328 20 328

759 759 759

1.24 0.62 0.24

213.8 427.7 1105

0.84 0.42 0.16

2.40 1.91 1.49

6.63 5.28 4.14

1.03 0.93 0.82

1.09 1.09 1.09

34.9 15.3 8.5

59.56 59.56 59.56

Effect of pressure 10 328 18 328 20 328 26 328

330 727 794 825

0.62 0.62 0.62 0.62

427.7 427.7 427.7 427.7

0.43 0.42 0.42 0.39

4.35 2.09 1.91 1.53

4.57 5.20 5.28 5.47

2.27 1.02 0.93 0.73

2.86 1.21 1.09 0.83

97.4 43.8 39.7 31.1

1321 79.06 59.56 43.67

Effect of temperature 20 308 875 20 323 791 20 333 729

0.62 0.62 0.62

427.7 427.7 427.7

0.38 0.41 0.42

1.41 1.86 1.96

5.53 5.29 5.26

0.67 0.90 0.95

0.77 1.06 1.12

11.1 14.9 15.7

21.51 36.76 56.67

P (MPa)

T (K)

K

b = 0.6143, a = 0.6142, ap = 3857 m1, dp = 0.6 mm and C0 = 0.2646 mg/g of neem kernel powder.

Table A5 Experimental conditions and process parameters for particle size in the model nimbin extraction [21]. P (MPa)

T (K)

dp (mm)

qb (kg/m3)

s (s)

Re

kf  105 (m/s)

Bi

kp  105 (m/s)

/

K

ap

a

b

20 20 20 20

308 308 308 308

1.840 1.445 1.015 0.575

435 452 458 458

444.8 427.7 422.6 422.6

1.18 0.92 0.65 0.37

0.69 0.80 0.99 1.45

7.93 7.52 6.70 5.54

0.26 0.32 0.43 0.69

1.43 2.20 4.16 11.9

21.51 21.51 21.51 21.51

1203 1607 2308 4074

0.629 0.614 0.610 0.610

0.629 0.614 0.610 0.610

dts ¼ 1; D ¼ 0;

R¼

    Ap v oidage   rate 1  v oidage V

Step 7: Set the boundary conditions for the first equation to have concentration equal to 1.0 at the left-hand side and equal to the convective flux at the right-hand side. Since the solid material cannot flow in or out when it is in the extractor, the boundary conditions for the second equation are Insulation/Symmetry (i.e. no flux) at both ends. Step 8: Define the parameters used in both Eqs. (3.3.2) and (3.3.4) under the Options/Constants. Step 9: The time dependent analysis is obtained using Solve/Solver parameters. Define the time interval for which the solution is to be obtained and then press ‘=’ to solve the problem. Step 10: The extract concentration in the fluid phase and the extract concentration in the solid phase can be plotted by choosing Plot Parameters. In the Line tab, select either c or q as required. The variation of extract concentration with respect to time can be obtained under Post-processing/Domain Plot Parameters in Line tab.

3.4. The algorithm for solution technique of ODE employed for the Goto model using MATLAB is as follows Step 1: Define all the input parameters like bed void fraction, particle diameter and porosity, overall mass transfer coefficient, solvent flow rate and volume of the bed. Step 2: Define the ordinary differential equations, Eqs. (2.2.3.4) and (2.2.3.5), for solute concentration in bulk fluid and solid phases respectively. Step 3: Give time interval step size for the time span up to which solution is to be computed. Step 4: Define the initial conditions, Eqs. (2.2.3.6a) and (2.2.3.6b), of the solute concentrations in both the phases. Step 5: Compute the solution for the solute concentration in bulk fluid and solid phases with respect to time using Runge– Kutta (Dormand–Prince) method. Step 6: Print the calculated values. 3.5. The algorithm for analytical solution technique employed for the Goto model using MATLAB is as follows The analytical solution of the model is obtained from Eq. (2.2.3.9) with equilibrium adsorption constant, K, as a fitting

A. Rai et al. / International Journal of Heat and Mass Transfer 72 (2014) 274–287

parameter. The minimum average absolute relative deviation (AARD) between the experimental extraction yield and predicted extraction yield is chosen to determine the best fit value of K. Average absolute relative deviation (AARD) is given as:



100

yieldcalc:  yieldexp: AARDð%Þ ¼

N i¼1 yieldexp: N X

The algorithm for the solution technique employed is as follows: Step 1: Input the parameters like bed void fraction, particle diameter, porosity, overall mass transfer coefficient, solvent flow rate and volume of the bed. Step 2: Define the equilibrium adsorption constant, K. Step 3: Compute the constants used for the solution of Eq. (2.2.3.9) Step 4: Solve the model equation (2.2.3.9) to compute the extraction yield. Step 5: Compare the predicted extraction yield with experimental yield using AARD. Step 6: If AARD is minimum, stop computing otherwise go to step 2. 4. Results and discussion The properties of SFE and experimental data related to the extraction of sage oil, pepper, caffeine from green tea, nimbin from neem seed, used for model validation are given in Appendix A. The results of the available models along with their inherent weaknesses are discussed in this section. 4.1. Results obtained from model proposed by Reverchon Fig. 4 shows the variation of percentage normalized yield with time for operating conditions of T = 323 K, P = 9 MPa, solvent flow rate of 8.83 g/min and the particle size of 0.75 mm. The experimental data points are superimposed on the simulated results and it is observed that the predicted results bear a close semblance to the experimental results. Fig. 5 shows a parity plot to compute the extent of error. From the parity plot, it can be seen that the model represents 86% experimental data within +9% errors. It can be concluded that the model over predicts the experimental values slightly. Fig. 6 shows the predicted percentage yield and experimental data with respect to time for the model of Reverchon [4] on FEMLAB. From Fig. 6, it can be seen that the predicted results match with the experimental data within an acceptable error limit. Fig. 7 shows that the deviation of the calculated results from the experimental data for the Reverchon [4] model on FEMLAB and it can be seen that the results match with the experimental values within an error band of +4% to 5%. The results obtained by solving ODE and PDE are compared with the experimental data of Reverchon [4] as shown in Fig. 8. The maximum deviation between these two methods of solution is found to be about 28%. It should be noted that the solution obtained by solving linear and non-linear coupled partial differential equations using FEMLAB which uses finite element method are expected to be more accurate. The results clearly show that the computer programme developed in the present work under MATLAB environment for ODE is moderately good as far as accuracy is concerned.

285

concentration within the bulk fluid phase (x) and dimensionless time (h) for numerical and analytical ODE solutions. 4.2.1. Validation of model proposed by Goto with experimental data of Kim et al. To examine the reliability of the model of Goto et al. [8], the results obtained are validated with the experimental data of Kim et al. [19]. To validate the model, presented by Eqs. (2.2.3.4) and (2.2.3.5) along with initial and boundary conditions given by Eqs. (2.2.3.6a) and (2.2.3.6b), respectively, The cumulative extraction yield computed using Eq. (2.2.3.9) and Fig. 10, is graphically represented by plotting the cumulative fraction of the solute extracted (F) on the vertical axis and the dimensionless time (h) on the horizontal axis along with experimental data of [19]. From Fig. 10, it is evident that model predictions match excellently with that of experimental data. To compute the extent of error, a parity plot is drawn in Fig. 11 and it is clear that the model predictions are within +15% to 2% of the experimental values. It can be concluded that the model slightly over predicts the experimental values. The model proposed by Goto et al. [8] considers the transfer of solute from solid matrix to bulk liquid phases through different seamless steps. The solute in the solid phase is transferred to the supercritical fluid present in the pores of the solid matrix, which is then transported to the liquid bulk. Various curves are plotted, as given below, to effectively understand the depletion of the solute from the solid phase and depict how it is transferred to the supercritical fluid in the pores. 4.2.2. Variation of dimensionless average solute concentration profile in solid phase with time The solute concentration profile (xs) in the solid phase of the particle with respect to the dimensionless time (h) is shown in Fig. 12. It can be seen that the solute concentration is decreasing with respect to time due to the diffusion of solute from solid phase to the pores and then to bulk fluid by convective mass transfer. 4.2.3. Effect of model parameters on cumulative extraction yield Once the model has been validated using the experimental data of Kim et al. [19] for caffeine extraction, it can now be used to study the effect of the operating parameters on the extraction yield, as shown below. The effect of various system parameters such as effective intra-particle diffusion coefficient (De), external mass transfer coefficients (kf), and overall mass transfer coefficients (kp) on the cumulative extraction yield were studied using the model of Goto et al. [8] and then validated using the experimental data for caffeine extraction by Kim et al. [19].

4.2. Results obtained from the model proposed by Goto

4.2.3.1. Effect of temperature on mass transfer coefficient and effective diffusivity. Fig. 13 shows the combined effects of mass transfer coefficients coupled with effective intra-particle diffusion coefficient on cumulative extraction yield (defined in bulk liquid) for a pressure of 40 MPa and temperature ranging from 313 to 353 K. It is noted that for a given dimensionless time, the cumulative extraction yield (F) increases with increase in temperature. The model predictions match excellently with experimental data of Kim et al. [19]. The observation can be explained by the fact that the mass transfer coefficients increases with rise in temperature as a result of high diffusivity of caffeine in SC-CO2 at higher temperatures. Thus the rate of mass transfer of solute to bulk liquid phase increases with increase in temperature.

The model proposed by Goto et al. [8] is solved numerically and analytically. The analytical solution proposed by Goto et al. [8] for the above model is represented by Eq. (2.2.3.7). Fig. 9 is drawn to show the comparison between dimensionless overall solute

4.2.3.2. Effect of pressure on mass transfer coefficient and effective diffusivity. Fig. 14 shows that for a given dimensionless time, the cumulative extraction yield (F) increases with increase in pressure. The observation can be explained by the fact that the solubility of

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caffeine in SC-CO2 increases at higher pressures despite the fact that mass transfer coefficients decreases with rise in pressure. Consequently, the mass transfer resistance increases resulting in the decrease of the extraction rate. 4.2.4. Effect of operating parameters on cumulative extraction yield In this section, the effects of temperature, pressure, solvent flow rate and particle size on the cumulative extraction yield are investigated using the data obtained for nimbin extraction from neem seed. 4.2.4.1. Effect of temperature on cumulative extraction yield. Fig. 15 shows the effect of temperature on cumulative extraction yield at pressure of 20 MPa and a solvent flow rate of 0.62 cm3/min. Fig. 15 shows that for a given dimensionless time, the cumulative extraction yield (F) increases with increase in temperature. However, in this respect a reversal trend is observed from experimental investigation due to Tonthubthimthong et al. [21]. 4.2.4.2. Effect of pressure on cumulative extraction yield. Fig. 16 shows the influence of pressure on the extraction yield at constant temperature of 328 K and solvent flow rate of 0.62 cm3/min. For a given dimensionless time (h), the cumulative extraction yield (F) increases with increase in pressure. The observation can be explained by the fact that with the increase in pressure from 10 to 26 MPa, the solubility increases rapidly. It has a positive effect on extraction process. However, the effective diffusivity and mass transfer coefficients decreased with higher pressure. The mass transfer resistance increased resulting in a decrease in the extraction rate showing negative effect of pressure on the extraction process. At low pressures, the positive effects on the extraction process overcome the negative effects and hence the cumulative extraction yield (F) increases with increasing pressure. 4.2.4.3. Effect of solvent flow rate on cumulative extraction yield. Fig. 17 shows the effect of flow rate on extraction yield at P = 20 MPa and T = 328 K. For a given dimensionless time, the cumulative extraction yield, F increases with increase in solvent flow rate. This observation can be explained by the fact that with the increase in SC-CO2 flow rate from 0.24 to 1.24 cm3/min, the mass transfer coefficient increases and thus the solute transported per unit time to bulk liquid increases leading to higher ‘F’ values on increasing the flow rate for a given time. 4.2.4.4. Effect of particle size on cumulative extraction yield. Fig. 18 shows the effect of particle size on nimbin extraction at 308 K, 20 MPa and a flow rate of 0.62 cm3/min. For a given dimensionless time (h), the cumulative extraction yield (F) increases with decrease in particle size. The observation can be explained by the fact that smaller particles offer higher interfacial area for mass transfer. When the particle size decreases, the interface area increases leading to higher mass transfer. This, in turn, the ‘F’ value increases with time.

5. Conclusions 5.1. Supercritical fluid extraction is a green and effective extraction process which produces an extract with little or no residual solvent, superior purity and high yield under lower operating cost as compared to solvent-based methods. A large number of useful species can be extracted using SFE technique. 5.2. The models proposed by Goodarznia et al. [14], Goto et al. [8,12], Marrone et al. [28], Reverchon [13], and Sovova [9,10] are studied in the present investigation to compute the extraction yield. Out of these models, the model based

5.3.

5.4.

5.5.

5.6.

on differential mass balances proposed by Reverchon [13] and the local adsorption equilibrium model proposed by Goto et al. [8] are found to be best as these address the internal diffusion and the dynamic behavior of the extraction process incorporating intra-particle diffusion as well as external mass transfer of SFE respectively. The model of Reverchon [13] fits 86% experimental data within +9% error when MATLAB environment was used for solving the model equations and +4% to 5% error when it was modeled using in FEMLAB environment. The numerical solution provided by FEMLAB is found to be better than the solution obtained by MATLAB environment showing a maximum deviation of 28%. The model of Goto et al. [8] fits the experimental data given by Kim et al. [19], Skerget and Knez [20], and Tonthubthimthong et al. [21] within an error band of +10% to 2%. At high temperature, pressure and solvent flow rate for smaller particle size, the highest yield of the product is obtained.

Appendix A The data used for the model validation and discussion of parametric effects is given in this section (see Tables A.1–A.5). References [1] M. Mukhopadhyay, Natural Extracts using Supercritical Carbon Dioxide, CRC Press, Boca Raton, FL, 2000. pp. 1–9. [2] P. Munshi, S. Bhaduri, Supercritical CO2: a twenty-first century solvent for the chemical industry, Curr. Sci. 97 (2009) 63–72. [3] D. Ghonasgi, S. Gupta, K.M. Dooley, F.C. Knop, Measurement and modeling of supercritical carbon dioxide extraction of phenol from water, J. Supercrit. Fluids 4 (1991) 53–59. [4] E. Reverchon, M. Poletto, Mathematical modelling of supercritical CO2 fractionation of flower concretes, Chem. Eng. Sci. 51 (1996) 3741–3753. [5] S.N. Naik, H. Lentz, R.C. Maheshawari, Extraction of perfumes and flavours from plant materials with liquid carbon dioxide under liquid–vapor equilibrium conditions, Fluid Phase Equilib. 49 (1989) 115–126. [6] M. Spiro, M. Kandiah, Extraction of ginger rhizome: partition constants and other equilibrium properties in organic solvents and in supercritical carbon dioxide, Int. J. Food Sci. Technol. 25 (1990) 566–575. [7] E. Reverchon, G. Donsi, L.S. Osseo, Modeling of supercritical fluid extraction from herbaceous matrices, Ind. Eng. Chem. Res. 32 (1993) 2721–2726. [8] M. Goto, M. Sato, T. Hirose, Extraction of peppermint oil by supercritical carbon dioxide, J. Chem. Eng. Jpn. 26 (1993) 401–407. [9] H. Sovova, Rate of the vegetable oil extraction with supercritical CO2—I. Modelling of extraction curves, Chem. Eng. Sci. 49 (1994) 409–414. [10] H. Sovova, J. Kucera, J. Jez, Rate of the vegetable oil extraction with supercritical CO2—II. Extraction of grape oil, Chem. Eng. Sci. 49 (1994) 415– 420. [11] H. Sovova, J. Jez, M. Bartlova, J. Stastova, Supercritical carbon dioxide extraction of black pepper, J. Supercrit. Fluid 8 (1995) 295–301. [12] M. Goto, B.C. Roy, T. Hirose, Shrinking-core leaching model for supercriticalfluid extraction, J. Supercrit. Fluids 9 (1996) 128–133. [13] E. Reverchon, Mathematical modelling of supercritical extraction of sage oil, AIChE J. 42 (1996) 1765–1771. [14] I. Goodarznia, M.H. Eikani, Supercritical carbon dioxide extraction of essential oils: modeling and simulation, Chem. Eng. Sci. 53 (1998) 1387–1395. [15] E. Reverchon, S.L. Osseo, Supercritical CO2 extraction of basil oil: characterization of products and process modeling, J. Supercrit. Fluids 7 (1994) 185–190. [16] M. Goto, B.C. Roy, A. Kodama, T. Hirose, Modeling of supercritical fluid extraction process involving solute–solid interaction, J. Chem. Eng. Jpn. 31 (1998) 171–177. [17] M. Perrut, J.Y. Clavier, M. Poletto, E. Reverchon, Mathematical modelling of sunflower seed extraction by supercritical CO2, Ind. Eng. Chem. Res. 36 (1997) 430–435. [18] W. Wu, Y. Hou, Mathematical modeling of extraction of egg yolk oil with supercritical CO2, J. Supercrit. Fluid 19 (2001) 149–159. [19] W. Kim, J. Kim, S. Oh, Supercritical carbon dioxide extraction of caffeine from Korean green tea, Seper. Sci. Technol. 42 (2007) 3229–3242. [20] M. Skerget, Z. Knez, Modelling high pressure extraction processes, Comput. Chem. Eng. 25 (2001) 879–886. [21] P. Tonthubthimthong, P.L. Douglas, S. Douglas, W. Luewisutthichat, W. Teppaitoon, L. Pengsopa, Extraction of nimbin from neem seeds using

A. Rai et al. / International Journal of Heat and Mass Transfer 72 (2014) 274–287

[22] [23]

[24]

[25] [26] [27] [28]

[29]

[30] [31]

[32]

[33]

supercritical CO2 and a supercritical CO2–methanol mixture, J. Supercrit. Fluid 30 (2004) 287–301. N.R. Bulley, M. Fattori, Supercritical fluid extraction of vegetable oil seeds, J. Am. Oil Chem. Soc. 61 (1984) 1362–1365. M.A.A. Melreles, G. Zahedi, T. Hatami, Mathematical modeling of supercritical fluid extraction for obtaining extracts from Vetiver root, J. Supercrit. Fluids 49 (2009) 23–31. D. Mongkholkhajornsilp, S. Douglas, P.L. Douglas, A. Elkamel, W. Teppaitoon, S. Pongamphai, Supercritical CO2 extraction of nimbin from neem seeds – a modelling study, J. Food Eng. 71 (2005) 331–340. H. Peker, M.P. Srinivasan, J.M. Smith, B.J. McCoy, Caffeine extraction rates from supercritical carbon coffee beans with dioxide, AIChE J. 38 (1992) 761–770. C. Tan, D. Liou, Modeling of desorption at supercritical conditions, AIChE J. 35 (1989) 1029–1031. J.R. Dormand, P.J. Prince, A family of embedded Runge–Kutta formulae, J. Comput. Appl. Math. 6 (1980) 19–26. C. Marrone, M. Poletto, E. Reverchon, A. Stassi, Almond oil extraction by supercritical CO2: experiments and modeling, Chem. Eng. Sci. 53 (1998) 3711– 3718. A.K.K. Lee, N.R. Bulley, M. Fattori, A. Meisen, Modelling of supercritical carbon dioxide extraction of canola oilseed in fixed beds, Journal of American Oil Chemist Society 63 (1986) 921–925. O.J. Catchpole, J.B. Gray, B.M. Smallfield, Near-critical extraction of sage, celery, and coriander seed, Journal of Supercritical Fluids 9 (1996) 273–279. B.C. Roy, M. Goto, T. Hirose, Extraction of ginger oil with supercritical carbon dioxide: experiments and modeling, Industrial and Engineering Chemistry Research 35 (1996) 607–612. M. Akgun, N.A. Akgun, S. Dincer, Extraction and Modeling of Lavender Flower Essential Oil Using Supercritical Carbon Dioxide, Industrial and Engineering Chemistry Research 39 (2000) 473–477. J.M. del Valle, J.M. Aguilera, An Improved Equation for predicting the solubility of vegetable oils in supercritical CO2 Industrial and Engineering Chemistry Research 27 (1988) 1551–1553.

287

[34] E. Reverchon, C. Marrone, Supercritical extraction of clove bud essential oil: isolation and mathematical modeling, Chemical Engineering Science 52 (1997) 3421. [35] E. Reverchon, G. Della Porta, G. Lamberti, Modelling of orange flower concrete fractionation by supercritical CO2, Journal of Supercritical Fluids 14 (1999) 115–121. [36] M. Perrut, J.Y. Clavier, M. Poletto, E. Reverchon, Mathematical modelling of sunflower seed extraction by supercritical CO2, Industrial and Engineering Chemistry Research 36 (1997) 430–436. [37] W. Wu, Y. Hou, Mathematical modeling of extraction of egg yolk oil with supercritical CO2, Journal of Supercritical Fluids 19 (2001) 149–159. [38] M.J. Cocero, J. Garcia, Mathematical model of supercritical extraction applied to oil seed extraction by CO2 + saturated alcohol- I, Desoption Model. Journal of Supercritical Fluid 20 (2001) 229–243. [39] E.M.C. Reis-Vasco, J.A.P. Coelho, A.M.F. Palavra, C. Marrone, E. Reverchon, Mathematical modelling and simulation of pennyroyal essential oil supercritical extraction, Chemical Engineering Science 55 (2000) 2917– 2922. [40] I. Zizovic, M. Stamenic, A. Orlovic, D. Skala, Supercritical carbon dioxide extraction of essential oils from plants with secretory ducts: Mathematical modelling on the micro-scale, Journal of Supercritical Fluids 39 (2007) 338– 346. [41] S.M. Ghoreishi, S. Sharifi, Modeling of supercritical extraction of mannitol from plane tree leaf, Journal of Pharmaceutical and Biomedical Analysis 24 (2001) 1037–1048. [42] M.A.A. Melreles, G. Zahedi, T. Hatami, Mathematical modeling of supercritical fluid extraction for obtaining extracts from vetiver root, Journal of Supercritical Fluids 49 (2009) 23–31. [43] J. Stastova, J. Jez, M. Bartlova, H. Sovova, Rate of the vegetable oil extraction with supercritical CO2— III. Extraction from sea buckthorn, Chemical Engineering Science. 51 (1996) 4347–4352.