Journal of Food Engineering 71 (2005) 331–340
www.elsevier.com/locate/jfoodeng
Supercritical CO2 extraction of nimbin from
neem seeds––a modelling study
Dayin Mongkholkhajornsilp a, Supaporn Douglas a, Peter L. Douglas
Ali Elkamel b, Wittaya Teppaitoon a, Suwassa Pongamphai a
a
b,*
,
Department of Chemical Engineering, King Mongkut’s University of Technology, Thonburi, Bangkok 10140, Thailand
b
Department of Chemical Engineering, University of Waterloo, Ont., Canada N2L 3G1
Received 20 February 2004; accepted 5 August 2004
Abstract
The extraction of nimbin from neem seeds using supercritical carbon dioxide is investigated in this paper. A model that accounts
for intraparticle diffusion (De) and external mass transfer of nimbin (kf) is presented for the supercritical extraction process. Mass
transfer is based on local equilibrium adsorption between solute (nimbin) and solid (neem solid). The external mass transfer coefficient was determined by fitting the theoretical extraction curve to experimental data. The following range of conditions: 0.24–
1.24 cm3/min of CO2, 10–26 MPa, 308–333 K, 1.0–2.5 g of neem kernel powder and 0.0575–0.185 cm of particle size of neem kernel
powder, were considered. In addition, a new correlation for Sherwood number (Sh) was developed in terms of the dimensionless
groups; Reynolds number (Re) and Schmidt number (Sc) from optimisation results. This correlation was compared to previous correlations and was found to give superior results when compared to experimental data.
2004 Elsevier Ltd. All rights reserved.
Keywords: Nimbin; Neem seeds; Supercritical CO2; Mass transfer; External mass transfer coefficient; Optimisation
1. Introduction
Azadirachta indica, popularly known as ‘‘Neem’’ in
India, is widely found in South Asia, Southeast Asia
and West Africa. In Thailand, the leaves of A. indica
var. siamensis (locally called the Sadao tree) are extensively used as a vegetable, and the leaves and other parts
of the plants are traditionally used for a variety of ailments. The seeds contain approximately 45% oil which
contains loeic acid (50–60%), palmitic acid (13–15%),
stearic acid (14–19%), linoleic acid (8–16%) and arachidic acid (1–3%). The oil is a brownish yellow, non-drying oil with an acrid taste and unpleasant odour. The
quality of the oil differs according to the method of
*
Corresponding author. Tel.: +1 519 888 4567x2913; fax: +1 519
746 4979.
E-mail address: [email protected] (P.L. Douglas).
0260-8774/$ - see front matter 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jfoodeng.2004.08.007
processing. A number of bitter components, viz. nimbin
(0.12%), nimbinin (0.01%), nimbidin (1.4%) and nimbidiol (0.5%) have been identified in neem oil. There are
also pigments, polysaccharides, salts and the proteinaceous material which makes up the cellular matrix of
the seeds (Johnson & Morgan, 1997). In view of the
growing importance of neem oil as a commercial product and the long established uses of neem in indigenous
medicine as a bitter tonic, an anti-malarial and anthelmintic, as a cure for syphilitic conditions and a variety
of skin diseases (Rochanakij, Thebtaranonth, Yenjai,
& Yuthavong, 1985) and using as anti-inflammatory
(ratpaw oedema) and a fairly good antipyretic effect
(pyrogen induced hyperpyrexia in rabbits) (Okpanyi &
Ezeukwu, 1981). Besides these properties, crude neem
oil is reported to have several other biological properties
such as antimicrobial, antifungal and antiviral effects.
A re-investigation of the active principle of the oil was
332
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
Nomenclature
A
ap
b, c
Bi
C
Cp
Cps
Cs
C0
ci
Dax
De
dp
F
F
jD
K
parameter defined in Eq. (A9)
specific surface area (cm1)
parameter defined in Eqs. (A10) and (A11)
Biot number (–)
concentration of solute in the bulk fluid
phase surrounding the particle (g/cm3)
concentration of solute in pore fluid in annulus (g/cm3)
concentration of solute in the pore space at
the surface of the particle (g/cm3)
concentration of solute in solid phase
expressed as mass of solute per unit mass of
solid (g/cm3)
total initial solute concentration (g/cm3)
optimised coefficient (–)
axial dispersion coefficient (cm2/s)
effective diffusivity (cm2/s)
particle diameter (cm)
the cumulative fraction of solute extracted
from model (–)
the cumulative fraction of solute extracted
from experiment (–)
Chilton–Colburn factor of mass transfer (–)
equilibrium constant (–)
considered of interest, as part of a general scheme of research for establishing its industrial uses.
These natural substances were originally extracted
using organic solvent extraction; quality-aware consumers however preferred cold pressing to avoid consuming
traces of organic solvents. An alternative to cold pressing is supercritical fluid extraction (SFE). Recently,
several studies investigating the supercritical fluid
extraction of natural substance from solid matrices
(seeds, flowers, leaves, etc.) have been published
(Akgün, Akgün, & Dincer, 2000; Ghoreishi & Sharifi,
2001; Tonthubthimthong, 2002). In most of these studies carbon dioxide is the solvent used because of its relatively low critical temperature (304.1 K), non-toxicity,
non-flammability, good solvent power, ease of removal
and low cost. In addition, carbon dioxide has a low surface tension and viscosity and high diffusivity (Roy,
Goto, & Hirose, 1996). Replacing the organic solvent
with supercritical carbon dioxide provides a safety concern associated with solvent extraction, but would result
in an increase in capital and operating costs.
Two different approaches have been proposed for the
mathematical modelling of the supercritical fluid extraction (SFE) of natural substance, those are the empirical
model, it can be useful when information on the mass
transfer mechanisms and on the equilibrium relationships is missing; however, these models are little more
kf
k f
kp
M
m1, m2
N
OBJ
PP
r
Re
Sh
Shop
Sc
SSE
t
Us
z
a
b
s
i
j
external mass transfer coefficient (cm/s)
optimum external mass transfer coefficient
(cm/s)
overall mass transfer coefficient (cm/s)
final extraction time
eigen values defined in Eq. (A8)
total number of experiment = 21
objective function (–)
physical properties
particle radius (cm)
Reynolds number (–)
Sherwood number (–)
optimum Sherwood number (–)
Schmidt number (–)
sum of squared errors (–)
time (s)
superficial velocity (cm/s)
bed height co-ordinate (cm)
bed voidage (–)
particle porosity (–)
total bed volume/volumetric flow rate (s)
experiment number
extraction time for experiment
than an interpolation of the experimental results (Barton, Hugher, & Hussein, 1992; Kandiah & Spiro, 1990;
Naik, Lentz, & Maheshawari, 1989) and differential mass
balance model (Goto, Sato, & Hirose, 1993; Goto, Roy,
Kodama, & Hirose, 1998; Reverchon, 1996; Reverchon
& Poletto, 1996; Roy et al., 1996). To develop a scaleup procedure from laboratory to pilot and industrial
scale, we need mathematical models. Mathematical
models which lack physical properties of materials and
the process have limited viability; although they may fit
some experimental data well they cannot be used to predict the operation under different conditions. However,
rigorous differential mass balance models should be able
to scale-up the process and predict the behaviour of the
process under various conditions.
Several researchers used supercritical CO2 to extract
neem oil from neem seeds. Johnson and Morgan (1997)
studied the selective extraction of nimbin, salanin and
azadirachtin and oil from neem seeds with supercritical
CO2 and supercritical CO2 plus methanol. They found
that pure CO2 could extract nimbin and salanin and
the highest extraction rate of nimbin and salanin occur
at 20.6 MPa and 6% methanol. Recently, Tonthubthimthong (2002) studied nimbin extraction from neem seeds
with supercritical CO2 and supercritical CO2 plus methanol. She found the best extraction conditions to be
308 K, 23 MPa and a flow rate of 1.24 cm3/min for 2 g
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
of neem. She estimated the extraction curve using three
empirical models. A Langmuir-type gas adsorption, a
first order plus dead time model (FOPDT), and a cyclone
model. The empirical models correlate the results the
best because all parameters were fitted using experimental data. Moreover, she used a theoretical model which
has an interaction between solute and solid (nimbin
and neem seeds) based on local equilibrium adsorption
to predict the extraction yield. It could give a good prediction for the effect of CO2 flow rate, pressure, temperature and weight of neem seeds but it showed quite poor
prediction for the effect of particle size of neem seeds.
The purpose of the present work is twofold: first, to
present a theoretical model of nimbin and estimate all
process parameters which are needed in the model. The
second is to develop a new correlation of mass transfer
coefficient specifically tailored for nimbin extraction.
In order to develop and test the nimbin extraction
model, we will use the experimental data obtained by
Tonthubthimthong, Chuaprasert, Douglas, and Luewisutthichat (2001). These experiments were performed
in a 1.5 cm diameter extractor at a wide range of conditions of temperatures, pressures and flow rates. Data for
various weight of neem kernel and different particle size
was also reported.
2. Mathematical modelling
The extraction of natural substances from solid
matrices in packed beds forms the basis of most present
day commercial scale supercritical fluid extraction. Despite this fact, a better understanding of the basic phenomena which controls the process of supercritical
extraction is still required. Extraction phenomena entails a series of sequential steps comprising diffusion of
CO2 into the pores, adsorption of CO2 on the solid surface, formation of an external liquid film around the
solid particles, dissolution and convective transport of
the solute in the bulk fluid phase, desorption of the solute to fluid phase in the pores, diffusion through the
pores, and finally transport to the bulk CO2. Therefore,
there have been many reports measuring and modelling
the rates of mass transfer from beds of organic material.
Most studies have concentrated on the extract quality
and the influence of the operating parameters (pressure
and temperature) on the yield of the extract, its quality
and its solubility in the solvent.
When existing mass transfer rate data are to be extended or scale up calculations have to be carried out
to optimise the process, mathematical models are
needed. The general equations for the process of supercritical fluid extraction (SFE) are similar to mass transport operations involving solids and fluids such as
leaching and adsorption/desorption processes (Erkey,
Guo, Erkey, & Akgernam, 1996; Schueller & Yang,
333
2001). Those models contain two differential solute mass
balances in fluid and solid phase, in addition to a local
equilibrium adsorption that describes the relation between solute and solid.
In what follows we will give a brief overview of the
extraction model on which this study is based on. The
model was first developed by Goto et al. (1993) to predict
concentrations of the solute in both the bulk and solid
phase. Further modifications to the model were made
by Goto, Smith, and McCoy (1990) and Dunford, Goto,
and Temelli (1998). We will then present our estimation
methodology of the model parameters that are specific to
the system under study: nimbin extraction from neem
seeds. The following assumptions are first made:
(1) axial dispersion is negligible,
(2) because of small column diameter, radial dispersion
is also neglected,
(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 neem seeds,
(7) assumed differential bed is gradientless bed in solid
and fluid phase,
(8) physical properties of the supercritical fluid are
constant.
The reactor was assumed to be a fixed bed of the
neem seed particle containing nimbin as the stationary
phase with flowing supercritical carbon dioxide as the
mobile phase. Unsteady state mass conservation was applied from a packed bed of stationary solid particle for
extracting nimbin from neem seeds. The mass balance
for the solute on the mobile (bulk fluid) phase in the
extractor is:
a
oC
oC
o2 C
þ k f ap ð1 aÞðC C ps Þ ¼ U s
þ Dax 2
ot
oz
oz
ð1Þ
where C, Cp and Cps are solute concentration in the bulk
fluid phase, in pores within particle and solute concentration in pore at the surface of particle, respectively.
Dax is an axial dispersion coefficient and kf in an external
mass transfer coefficient of particle. The specific surface
area, ap, is defined by ap ¼ d6p . Fig. 1 shows a schematic
drawing of a particle.
The mass balance for the solute on the stationary
(solid) phase is:
2
b
oC p
oC p
oC s
¼ De 2 ð1 bÞ
ot
or
ot
ð2Þ
where Cs is solute concentration in the solid phase of
particle. The boundary and initial conditions are:
Cðt ¼ 0Þ ¼ 0
ð3Þ
334
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
Fig. 1. Schematic drawing of a porous solid particle.
C p ðt ¼ 0Þ ¼ C p0
ð4Þ
C s ðt ¼ 0Þ ¼ C s0
oC p ¼ k f ½C C ps De
or surface
ð5Þ
C 0 ¼ bC p0 þ ð1 bÞC s0
ð6Þ
ð7Þ
The above model is solved for the unknown concentrations C by further assuming fast adsorption/desorption
(Appendix A). Since in our previous experimental study
(Tonthubthimthong et al., 2001), only the cumulative
nimbin production was measured, the above extraction
model is used to estimate the cumulative fraction of solute extracted up to time t. By definition, this cumulative
quantity is given by:
Z t
1
F ðtÞ ¼
C dt
ð8Þ
ð1 aÞC 0 s 0
Using Eq. (A7) for the value of C gives:
m1 t
k p ap
e 1 em2 t 1
F ðtÞ ¼
m1
m2
sa½b þ ð1 bÞK½m1 m2 ð9Þ
where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2 4c
m1;2 ¼
ð10Þ
2
and with b, c and kp given by Eqs. (A10), (A11) and
(A5), respectively.
and mass transfer properties in order to predict the
extraction yield.
The physical properties of nimbin needed in the
model were obtained experimentally as in our previous
work (Tonthubthimthong et al., 2001). The particle density, qp was measured by using a pyknometer (Tonthubthimthong, 2002) and found to be 1.1732 g/cm3. Bulk
density, qb was calculated from weight divided by volume of neem kernel powder. The porosity of the neem
kernel powder was estimated from b = 1 (qb/qp). The
bed void fraction, a was obtained from the volume of
neem kernel powder and bulk volume of bed.
The physical properties of supercritical carbon dioxide under various operating conditions are presented
in Table 1. The density of CO2 was calculated by using
the Soave/Redlich/Kwong equation of state. The residual viscosity correlation of Stiel and Thodos (Poling,
Prausnitz, & OÕConnell, 2001) using low-pressure gas
viscosity estimated by the Chung et al. (Poling et al.,
2001) were used to calculate the viscosity of CO2. The
binary diffusion coefficient, DAB of nimbin in CO2 was
obtained using the Riazi and Whitson correlation (Poling et al., 2001). The effective intraparticle diffusion coefficient, De was estimated from De = DABb2. The
equilibrium constant, K of nimbin was estimated separately from experimental measurement of nimbin.
The external mass transfer coefficient, kf can be calculated from Sherwood number (Sh):
kf ¼
DAB Sh
dp
ð11Þ
Different correlations are available in the literature for
the estimation of Sherwood number as a function of
Reynolds number (Re) and Schmidt number (Sc). In this
work, we tested two possible correlations, the Wakao–
Funazkri correlation (Wakao & Funazkri, 1978) and
the Tan correlation (Tan, Liang, Liou, & C, 1988).
These correlations were widely used by various researchers in the past (Goto et al., 1993; Puiggené, Larrayoz, &
Recasens, 1997).
The Wakao and Funazkri (1978) correlation is valid
over the range of Re from 3 to 3000 and Sc from 0.5
to 10,000 and is given by:
b 3. Model parameters
The model presented by Eq. (9) will be used to predict
the nimbin extraction and the results will be compared
with our pervious experimental results (Tonthubthimthong et al., 2001). According to Eq. (9), one needs values of various parameters including physical properties
Table 1
Extraction conditions and estimated transport parameters
P
(MPa)
T
(K)
q
(g/cm3)
l · 104
(g/cm s)
DAB · 105
(cm2/s)
K (–)
10
18
20
20
20
20
20
23
26
328
328
308
313
323
328
333
328
328
0.307
0.631
0.777
0.750
0.694
0.665
0.636
0.706
0.739
2.57
5.01
6.51
6.21
5.62
5.35
5.08
5.78
6.15
14.15
5.10
3.41
3.67
4.30
4.67
5.08
4.20
3.85
1321
79.1
21.5
46.9
36.8
59.6
56.7
59.7
43.7
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
Sh ¼ 2 þ 1:1Re0:6 Sc0:33
ð12Þ
The Tan et al. (1988) correlation is given by:
Sh ¼ 0:38Re0:83 Sc0:33
ð13Þ
Tan et al. (1988) studied measuring extraction rates of
b-naphthol in supercritical CO2 with three sizes of soil
particles. Their correlation can be used over superficial
velocities from 4.4 · 103 to 3.1 · 102 cm/s, Re from 2
to 40 and Sc from 2 to 40.
4. Results and discussion
4.1. Comparison with experimental data
Experimental data from the nimbin extraction from
neem seeds with supercritical carbon dioxide was compared with predictions from the model presented earlier
and with the model parameters estimated as indicated
previously. The cumulative fraction of solute extracted
up to time t, F(t) is compared with experimental measurements, F ðtÞ. Two model predictions using both the
Wakao and Tan correlations were made. The predictions were evaluated by calculating the sum of square
error as given below:
SSE ¼
M
X
ðF t F t Þ
2
ð14Þ
t¼1
The sum is over all data points from 1 to M. Table 2
shows the SSE for different experimental conditions
335
for various mass transfer correlations. The Wakao and
Tan correlations are given in columns two and three in
the table. Columns 4–9 will be discussed later in this
paper. We found that both the Wakao and Tan correlations when used in conjunction with the extraction
model give reasonable results, on average. However, neither correlation could predict the effect of particle size as
evidenced by the high values of SSE falling between
11.76E2 to 113.34E2.
Figs. 2 and 3 show comparison plots of experimental
F(t) vs t for various particle sizes with the predictions
that used the Wakao and Tan correlations. Both correlations show little sensitivity to particle size as illustrated
by the narrow spread in the curves. The reason for the
poor fit may lie in the fact that both Wakao and TanÕs
correlations were based on experimental studies outside
the range of conditions used in this study. In our study
the Re varied from 0.16172 to 1.189 and the Sc varied
from 10 to 45. WakaoÕs correlation was generated using
liquid–gas experiments at ambient conditions and over a
range of Re from 3 to 3000 and Sc 0.5–10,000 whereas
TanÕs correlation was generated from experimental results over a smaller range of Re and Sc, 2–40 and 2–
40, respectively. In addition, Tan et al. (1988) found that
the effect of pressure and temperature on mass transfer
rates was not significant, but that particle size has an effect. They studied the relationship between mass transfer
rates and Re at three particle sizes. They found that the
larger particle sizes resulted in larger errors between
their experimental data and their model; we observed
the same tendency as shown in Fig. 3.
Table 2
Results of statistical analysis of different correlations
SSE (·102)a
Effect
3
Flow rate (cm /min)
0.24
0.62
1.24
10
18
20
23
26
308
313
323
328
333
1.0
1.5
2.0
2.5
0.0575
0.1015
0.1440
0.1850
Pressure (MPa)
Temperature (K)
Weight of neem (g)
Particle size (cm)
a
SSE ¼
PM
t¼1 ðF t
F t Þ2 .
Wakao
Tan
M1
M2
M3
M4
M5
M6
1.11
4.03
3.43
1.37
1.19
3.99
2.34
5.98
9.46
2.15
13.32
4.00
2.44
13.17
2.00
4.00
2.29
16.92
38.88
66.19
113.34
1.02
2.09
1.60
1.47
1.14
2.07
3.29
2.75
5.35
0.71
8.22
2.07
1.47
5.85
0.38
2.07
3.05
11.76
26.02
43.09
77.21
0.95
1.53
0.82
1.41
1.29
1.52
5.15
0.88
0.97
3.25
5.11
1.52
1.38
3.41
0.09
1.52
3.72
4.01
1.72
1.04
0.55
0.95
1.52
0.83
1.41
1.30
1.51
5.18
0.89
1.01
3.15
5.06
1.51
1.38
3.34
0.09
1.51
3.74
4.23
1.82
1.02
0.55
0.95
1.50
1.76
1.45
1.43
1.49
4.55
1.42
2.94
1.15
5.61
1.49
1.37
3.22
0.08
1.49
3.79
8.57
4.59
1.63
6.66
0.95
1.43
1.05
1.41
1.43
1.42
5.16
1.03
1.84
1.02
4.89
1.42
1.37
2.84
0.07
1.42
3.94
6.51
4.23
1.34
1.39
0.95
1.37
1.70
1.54
1.70
1.37
4.78
1.42
3.26
1.09
5.14
1.37
1.44
2.49
0.07
1.37
4.11
9.04
7.81
3.01
2.67
1.35
1.26
0.87
1.59
2.11
1.27
5.81
1.01
2.33
1.83
3.89
1.26
1.63
1.47
0.16
1.26
4.75
7.27
11.03
13.37
24.52
336
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
1
0.0575 cm
F [-]
0.8
0.0575 cm
0.1015 cm
0.6
0.1015 cm
0.144 cm
0.4
0.144 cm
0.185 cm
0.2
0.185 cm
0
0
50
100
150
200
250
300
Time (min)
Fig. 2. Comparison of extraction yield predicted from WakaoÕs
correlation with experimental data for various particle sizes and at
308 K and 20 MPa.
of the squares of the difference between the experimental
extraction yield, F t , and the predicted extraction yield,
Ft, using GotoÕs model (Eq. (9)) as shown in Eq. (15);
X
2
Objective
function
¼
ðF t F t Þ
ð15Þ
min
kf
t
The algorithm to calculate the optimum external mass
transfer coefficients, k f , is shown in Fig. 4. The unknown
parameter is the external mass transfer coefficient, kf
at each experiment i. The model was fit to the experimental data (function of time j) with k f as the optimised
parameter. The optimum external mass transfer coefficients, k f , determined from the solution of Eq. (15) are
presented in Table 3. From Table 3 we can see that the
optimum external mass transfer coefficient varies from
3.18E05 cm/s to 1.64E03 cm/s; these values are reasonable when compared with earlier experiments (Tan
et al., 1988).
1
0.0575 cm
0.0575 cm
0.1015 cm
0.1015 cm
0.144 cm
0.144 cm
0.185 cm
0.185 cm
F [-]
0.8
0.6
0.4
0.2
START
i =0
i = i+1
Initial kfi
PPi
0
0
50
100
150
200
250
300
Time (min)
Goto’s Model
Eqs. (9)
Fig. 3. Comparison of extraction yield predicted from TanÕs correlation with experimental data for various particle sizes and at 308 K and
20 MPa.
Fi,j
NO
From the above discussion it is clear that neither the
Wakao nor Tan mass transfer correlations are adequate
for nimbin extraction. An attempt will be made in the
next section to develop a correlation more representative
of the nimbin extraction system under study.
Choose
new kf
j= M
YES
OBJ = Σ (Fi , j − Fi , j )
M
2
Fi , j
j =1
4.2. New correlation development
NO
Our objective here is twofold. First we want to determine the best or optimal external mass transfer coefficient, kf, to fit GotoÕs model using experimental data.
Then with these new values of kf, we would like to
develop a new external mass transfer coefficient correlation which is similar in form to the Wakao or Tan
correlations.
4.2.1. Determination of optimal k f
We determined the optimum external mass transfer
coefficient, k f , at each experiment (from 21 experiments)
by choosing the best or optimal k f to minimise the sum
min OBJ
YES
k*fi
NO
i=N
YES
STOP
Fig. 4. External mass transfer coefficient optimisation procedure.
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
Table 3
Optimum external mass transfer coefficients of nimbin extraction
k f (cm/s)
Effect
Flow rate (cm3/min)
Pressure (MPa)
Temperature (K)
Weight of neem (g)
Particle size (cm)
0.24
0.62
1.24
10
18
20
23
26
308
313
323
328
333
1.0
1.5
2.0
2.5
0.0575
0.1015
0.1440
0.1850
2.66E04
1.72E04
3.06E04
1.64E03
6.62E04
1.72E04
–
1.23E04
7.86E05
2.84E04
6.91E05
1.72E04
2.87E04
1.09E04
2.23E04
1.72E04
–
5.89E05
4.76E05
4.36E05
3.18E05
4.2.2. Development of external mass transfer coefficient
models
The next step was to use the optimum external mass
transfer coefficients, k f , determined from the previous
part to determine the most appropriate mass transfer
correlation. Mass transfer is normally quantified by
the Sherwood number, Sh = kfdp/DAB, which represents
the mass transfer between a surface and a fluid. The
Sherwood number is normally written as a function of
the Reynolds number, Re = dpUsq/l, and the Schmidt
number, Sc = l/qDAB. The generalised relationship between Sh, Re and Sc is shown by Eq. (16):
Sh ¼ c1 þ c2 Rec3 Scc4
ð16Þ
Eq. (16) is often written as the Frössling equation (17)
(Frössling, 1938):
Objective function ¼
min
c ;c
1
X
2
337
ðShop;i ðc1 þ c2 Reci 3 Scci 4 ÞÞ
2
i
c3 ;c4
ð18Þ
The optimum Sherwood number, Shop,i, was calculated
from the optimum external mass transfer coefficient,
k f;i as Eq. (19) and was presented in Table 3:
Shop;i ¼
k f;i d p;i
DAB;i
ð19Þ
We used the optimum external mass transfer coefficient,
k f , to calculate the optimum Sh for all experiments,
Shop,i, and then developed to be the function of Re
and Sc for each experiment i. Finally, a minimisation
method was used to determine all coefficients. By applying the Chilton–Colburn jD factor of mass transfer as
Eq. (20) and then Eq. (16) can be replaced to Eq. (21)
which was used in mass transfer researches (Ranz &
Marshall, 1952; Wakao & Funazkri, 1978).
jD ¼
Sh
ReSc0:33
Sh ¼ c1 þ c2 Rec3 Sc1=3
ð20Þ
ð21Þ
Furthermore, others used equations that are similar to
Eq. (21) but with c1 = 0, the mass transfer coefficient
can then be predicted from the Frössling type equation
(Tan et al., 1988; Tai, You, & Chen, 2000; Baudot,
Floury, & Smorenburg, 2001). The value of c3 depends
on the type of equipment and system. It was reported
to vary between 0.5 for packed beds, (Cussler, 1997)
and 0.80 by others (Tan et al., 1988; Puiggené et al.,
1997) as a result c3 was constrained between 0.5 and
0.8 as follows:
min Objective function
c1 ;c2
c3 ¼0:50:8
c4 ¼0:33
¼
X
ðShop;i ðc1 þ c2 Reci 3 Scci 4 ÞÞ2
ð22Þ
i
Sh ¼ 2 þ 0:6Re0:5 Sc0:33
ð17Þ
However, we have chosen to use Eq. (16) as the basis for
the development of various mass transfer correlations
thereby allowing coefficients c1, c2, c3 and c4 to take
on values that best fit the data. We modified the external
mass transfer coefficient correlations for nimbin extraction with supercritical carbon dioxide, in the range of
Reynolds number from 0.1689 to 1.2918 and Schmidt
number from 6 to 25 using the GAMS optimisation system with the NLP (non linear programme) solver to obtain the parameters (ci). The modified external mass
transfer coefficient correlation can be developed from
the minimization of an objective function which is the
sum of the squares of an error between an optimum
Sherwood number (Shop,i) and modified correlation as
Eq. (18);
Table 4 shows the various correlations that were considered. One can see that the Wakao–Funazkri model is
similar to the Frössling equation with slight modifications in c2 and c3 and TanÕs correlation is also a Frössling
equation with c1 = 0. For the sake of completeness we
have considered a wide variety of forms. M1, relaxes
all constraints, ci, and results in the best fit of all models
with the lowest value of the objective function, 0.3191.
The value of the objective function in M1 is a lower limit
or target, against which, to compare all other models.
M1 is a modified Wakao model and the fit is very good
but the values of the parameters are far from what might
be considered reasonable. Therefore, based on our previous discussion concerning the general form of external
mass transfer coefficient correlation, we constrained various ci. We started by constraining c4 = 0.33 resulting in
338
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
Table 4
Results of objective function of different external mass transfer
correlation (Set of 21 experimental points was used)
External mass transfer
coefficient correlation
Wakao–Funazkri
Tan
M1
M2
M3
M4
M5
M6
OBJ
Sh = 2 + 1.1 (Re0.6Sc0.33)
Sh = 0.38 (Re0.83Sc0.33)
Sh = 1.309 0.258 (Re0.023Sc0.463)
Sh = 1.661 0.517 (Re0.014Sc0.33)
Sh = 0.541 0.133 (Re0.5Sc0.33)
Sh = 3.173 (Re0.06Sc0.85)
Sh = 0.085 (Re0.298Sc0.33)
Sh = 0.135 (Re0.5Sc0.33)
262.617
4.100
0.3191
0.3195
0.5267
0.3536
0.7287
0.9185
objective function than the Wakao and Tan models;
especially in its ability to predict the effect of particle
size. Figs. 5–8 show the extraction curve from the model
with kf from the Wakao, Tan, M1 and M6 correlations.
Table 2 presents the SSE results for all models considered in this work. The extraction yield of the effect of
particle size using correlation M6 is shown in Fig. 9.
1
0.8
Tan
M6
0.4
M1
Exp
0.2
0
0
50
100
150
200
250
300
Time (min)
Fig. 6. Comparison of extraction yield predicted from different
correlations with experimental data for a particle size of 0.1015 cm.
1
0.8
Wakao
F [-]
M2 which is the case in all reported mass transfer correlations. The objective function increased somewhat to
0.3195, however, the negative value of c3 on Re is worrisome since it is always found that Re has a positive influence on the mass transfer coefficient. Therefore, in model
M3 the value of c3 was constrained to be between 0.5 and
0.8, similar to values that are reported in the literature.
The objective function increased further to 0.5265. The
negative value of c2 is also unacceptable since it indicates
that an increase in Re and Sc will diminish the Sherwood
number which we know is not reasonable. Therefore, we
further constrained c2 to be positive for models M4, M5
and M6 with various constraints on c3 and c4.
M4 is a modified Tan model in that c1 = 0. Naturally
the value of the objective function is somewhat higher
(0.3536) than the result obtained for M1 because of
the additional constrain imposed on c1 (c1 = 0). Based
on the statistical results and the practical arguments discussed above, we have constrained c4 = 0.33 in M5 and
M6 and c3 = 0.5–0.8 in M6. It was found that M6 is the
best realistic modified correlation even though it resulted in a higher objective function (0.9185) than M4
and M5, 0.3536 and 0.7287, respectively because it is
consistent with mass transfer theory as the original correlations of Wakao and Tan.
M6 is a mathematically appropriate correlation to fit
the experimental data because it has a significantly lower
F [-]
Wakao
0.6
Tan
0.6
M6
0.4
M1
Exp
0.2
0
0
50
100
150
200
250
300
Time (min)
Fig. 7. Comparison of extraction yield predicted from different
correlations with experimental data for a particle size of 0.1440 cm.
1
1
0.8
0.8
Wakao
Tan
0.6
M6
0.4
M1
Tan
0.6
F [-]
F [-]
Wakao
M6
M1
0.4
Exp
Exp
0.2
0.2
0
0
0
50
100
150
200
250
300
Time (min)
Fig. 5. Comparison of extraction yield predicted from different
correlations with experimental data for a particle size of 0.0575 cm.
0
50
100
150
200
250
300
Time (min)
Fig. 8. Comparison of extraction yield predicted from different
correlations with experimental data for a particle size of 0.1850 cm.
D. Mongkholkhajornsilp et al. / Journal of Food Engineering 71 (2005) 331–340
1
0.0575 cm
F [-]
0.8
0.0575 cm
0.1015 cm
0.6
0.1015 cm
0.144 cm
0.4
0.144 cm
0.185 cm
0.2
0.185 cm
0
0
50
100
150
200
250
300
Time (min)
Fig. 9. Comparison of extraction yield predicted using the M6
correlation with experimental data for a various particle sizes and at
308 K and 20 MPa.
The linear driving-force approximation was used to
combine internal and external mass transfer processes
is defined by:
15De
ð1 aÞðC ps C p Þ
ðA2Þ
R2
Eqs. (1) and (2) can be reduced to Eqs. (A3) and (A4),
respectively:
dC C
Cs
þ ¼ k p ap ð1 aÞ
C
ðA3Þ
a
dt
s
K
b
dC s
Cs
þ ð1 bÞ
¼ k p ap C ðA4Þ
K
dt
K
k f ap ð1 aÞðC C ps Þ ¼
where kp is the overall mass-transfer coefficient for a
spherical particle, given by:
kp ¼
5. Conclusions
In this paper, the extraction of nimbin from neem
seeds is considered. A theoretical model is presented
and parameters needed in the model were estimated.
Two existing correlations (Tan et al., 1988; Wakao &
Funazkri, 1978) were used in conjunction with the model. The predictions did not compare well to experimental
results. In order to improve the performance of the
model, a new correlation of mass transfer coefficient
was considered. This correlation was obtained by using
non-linear regression analysis. Optimum external mass
transfer coefficient data was based on 21 sets of experiments. Considering the statistical results, M6 (Sh =
0.135Re0.5Sc0.33) was selected to be an appropriate correlation for the prediction of nimbin extraction with
Reynolds number from 0.1689 to 1.2918 and Schmidt
number from 6 to 25.
339
kf
1 þ Bi5
ðA5Þ
and Bi is Biot number, defined by:
Bi ¼
kf r
De
ðA6Þ
The exact solution of Eqs. (A3) and (A4), is:
C ¼ A½em1 t em2 t ðA7Þ
where
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2 4c
2
ðA8Þ
k p ap ð1 aÞC 0
a½b þ ð1 bÞK½m1 m2 ðA9Þ
m1;2 ¼
A¼
b and with
b¼
Acknowledgements
and
The financial support provided by the Royal Golden
Jubilee Ph.D. Program, the Thailand Research Fund
(TRF) is gratefully acknowledged.
c¼
1 k p ap ð1 aÞ
k p ap
þ
þ
as
a
½b þ ð1 bÞK
ðA10Þ
k p ap
as½b þ ð1 bÞK
ðA11Þ
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and solid phase is fast adsorption/desorption equilibrium as given by:
C s ¼ KC p
ðA1Þ
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