Advance Journal of Food Science and Technology 3(2): 127-131, 2011
ISSN: 2042-4876
© Maxwell Scientific Organization, 2011
Received: January 31, 2011
Accepted: March 03, 2011
Published: April 10, 2011
Mathematical Model on Thin Layer Drying of Finger Millet (Eluesine coracana)
1
G.B. Radhika, 2S.V. Satyanarayana and 3D.G. Rao
B.V. Raju Institute of Technology, Hyderabad, India
2
Jawaharlal Nehru Technological University, Anantapur, India
3
Caledonian College of Engineering, Muscat (Oman)
1
Abstract: Thin layer drying characteristics of the finger millet (Eluesine coracana) samples with an Initial
Moisture Content (IMC) of 38.5%, on dry basis (db) were studied. The drying experiments were carried out
in a conventional tray dryer at temperatures of 50, 70 and 80ºC. The drying data were fitted to nine thin layer
models and a thin layer model for the finger millet was developed by regressing the coefficients of the best fit.
The logarithmic model was found to satisfactorily describe the drying kinetics of the millet. The drying
constants were found to vary linearly with temperature. Also, effective diffusivity was evaluated by using
Fick’s law, which varied from 1.526x10G10 to 2.85x10G10 m2/s. Temperature dependence of diffusivity was
found by Arrhenius type of relationship and the activation energy for the diffusion of the moisture associated
with the millet was found to be 35.37 kJ/gmol.
Key words: Activation energy, drying constants, effective diffusivity, finger millet, mathematical model,
moisture diffusion, thin layer drying characteristics
INTRODUCTION
equation. Activation energy required for moisture
diffusion was also calculated.
Finger millet under study is an annual cereal plant. It
is widely grown in the arid areas of Africa and Asia. It is
the main food grain for many people, especially in the dry
areas of India and Srilanka and is used in the form of
flour. The products made of the finger millet flour are
easily digestible and can be recommended for all age
groups. Malted flour is used as staple food in many parts
of India.
Drying involves simultaneous heat and mass transfer.
The main objective of drying is to reduce the moisture
content and to increase the shelf life. Several
mathematical models have been developed describing the
drying process. However only thin layer drying models
are widely used for the designing of the dryers. Some of
the thin layer models reported were for drying of Toria
seeds (Rangroo and Rao, 1992), Dates (Bakri and
Hobani, 2000), chili pepper (Toyosi and Adeladun, 2010),
two varieties of millet samples (Ojendiran and
Raji, 2010), Srilankan paddy (Syamali et al., 2009),
Sesame seeds (Khazaei and Daneshmandi, 2007),
Amaranth grain (Ronoh et al., 2010), parboiled wheat
(Debabandya and Srinivas, 2005).
The objective of the present study was to develop the
thin layer model and determination of effective diffusion
coefficients for drying of finger millets using tray drier,
which were not reported earlier. Also, the temperature
dependence of diffusivity was assessed by Arrhenius type
MATERIALS AND METHODS
The experimental studies were carried out in
laboratory of department of Chemical Engineering,
B.V.Raju Institute of Technology of Jawaharlal Nehru
Technological University, Hyderabad, India in
August’2010.
Sample preparation: Finger millet samples were
procured from the local market. Initial moisture content of
the samples was determined by the standard oven method
(AOAC, 1960), which was found to be 11.23% (db).
Grain samples of known weight (~ 400 g) were soaked for
3 hrs. at 45ºC in a constant temperature water bath, the
excess water was drained and the surface moisture was
removed. Further, the samples were stored in self sealed
covers for equilibration for 24 h. there by increasing the
moisture content to 38.5% (db)
Experimental apparatus and procedure: Conventional
laboratory tray dryer was used for the study, which
consisted of a heater, digital temperature controller and a
blower. The dimensions of the drying chamber were
0.2*0.2*0.3 m.
Finger millet samples of around 100g were dried in
a laboratory tray drier at temperatures of 50, 70 and
Corresponding Author: S.V. Satyanarayana, Jawaharlal Nehru Technological University, Anantapur, India. Tel: 91-9849509167;
Fax: 91-8554-273007
127
Adv. J. Food Sci. Technol., 3(2): 127-131, 2011
Table 1: Thin layer mathematical models
S.No. Model name
1.
Page
2.
Modified page
3.
Modified page II
4.
Lewis
5.
Henderson and Pabis
6.
Logarithmic
7.
Two term model
8.
Simplifed Fick’s diffusion
9.
Wang and Singh
Model equation
MR = exp(-ktn)
MR = exp(-kt)n
MR = exp(-c(t/L2)n)
MR = exp(-kt)
MR = a exp(-kt)
MR = a exp(-kt)+c
MR = aexp(-kot)+bexp(-k1t)
MR = aexp(-c (t/L2))
MR = 1+at+bt2
can be used to evaluate effective diffusivity of spherical
particles
80ºC.The air velocity was 2.2 -2.4 m/s, which was
measured using the hot air anemometer. Moisture loss
was recorded using a digital balance with an accuracy of
±0.01g. Drying studies were continued till the equilibrium
was reached. The data was collected in triplicates.
MR =
Mathematical modeling: Drying characteristics can be
investigated by effectively modeling the drying behavior.
Hence, the experimental data of the finger millet at three
different temperatures were fitted to nine thin layer drying
models, presented in Table 1. The dimensionless moisture
ratio in these models was given by equation MR = (MMe)/(Mo-Me) where M is the moisture content at any
time, Mo is the initial moisture content and Me is the
equilibrium moisture content. The values of Me may be
relatively small compared to M and Mo, so the equation
can be simplified to MR = M/Mo (Akgun and Doymaz,
2005; Togrul and Pehlivan, 2002; Thakor et al., 1999).
The non linear regression analysis in the present
study was performed using the software MATLAB 7.0.
The goodness of the fit of the tested mathematical models
to the experimental data was evaluated with the
correlation coefficient (R2), the reduced chi square (P2)
and the root mean square error (RMSE). The best fit was
that which results in higher R2 and the lowest P2 and
RMSE. (Yaldiz and Ertekin, 2004; Gunhan et al., 2005;
Ozdemir and Devres, 1999). The reduced P2 and RMSE
were evaluated as:
∑ ( MR
=
n
χ2
exp,i − MR pre.i
i =1
∑ ( MR
i=1
pre ,i
N
− MRexp,i
6
Π2
∞
∑n
n= 1
1
2
⎛ − n 2 DΠ 2t ⎞
⎟
exp⎜
r2
⎝
⎠
(3)
where, D is the effective diffusivity and R is the radius of
the grain. For long drying times Eq. (3) can be further
simplified to first term of the series (Tutuncu and Labuza,
1996). Thus Eq. (3) can be written in the logarithmic form
as Eq. (4):
InMR = ln
6
Π 2 Dt
2 −
r2
Π
(4)
Diffusion coefficients were determined by plotting ln
MR verses drying time, t. The plot yields a straight line
with the slope of B2D/r2 from which the effective
diffusivity was evaluated.
Temperature dependence of diffusivity and calculation
of activation energy: Temperature dependence of
diffusivity was studied by the Arrhenius type relationship,
since temperature plays a significant role in the process of
drying (Ozdemir and Derves, 1999). The relation is given
by equation
D = Do exp (-Ea/RT), where Ea is the activation energy
2
required for moisture diffusion (kJ/gmol), R is the
universal gas constant.
(1)
N−z
n
RMSE =
)
Reference
Page (1949)
White et al. (1981)
Diamante and Munro (1991)
Bruce (1985)
Henderson and Pabis (1961)
Togrul and Pehlivan (2002)
Henderson (1974)
Diamante and Munro (1991)
Wang and Singh (1978)
)
2
RESULTS AND DISCUSSION
(2)
The effect of drying temperature at a particular air
velocity (2.2-2.4 m/s) on drying curves was presented in
Fig. 1, which indicated that the moisture ratio decreased
with the increased drying time. Further it can be observed
from the drying curves that the temperature significantly
affects the rate of drying and the total drying process was
found to be occurred in the falling rate period only. This
indicated that the process describing the drying behavior
of the finger millet was diffusion governed (Singh and
Sodhi, 2000).
where, MRexp,i is the ith experimentally observed moisture
ratio, MRpre,i is the ith predicted moisture ratio, N is the
number of observations and Z, the number of constants in
models (Akpinar, 2006).
Evaluation of effective diffusivities: The drying
characteristics in the falling rate period can be described
by using Fick’s diffusion equation (Crank, 1975), Eq. (3)
128
Adv. J. Food Sci. Technol., 3(2): 127-131, 2011
Table 2: Statistical results of different thin layer models
Model constants
-------------------------------------------------------------------Model
Temperature (ºC)
Lewis
50
k = 0.00927
70
k = 0.01749
80
k = 0.02281
Page
50
k = 0.008799
n = 1.011
70
k = 0.03243
n = 0.8531
80
k = 0.03849
n = 0.8667
Modified page
50
k = -0.01421
n = -0.6547
70
k = 0.04349
n = 0.4022
80
k = 0.1249
N = 0.1826
Henderson and Pabis
50
a = 1.033
k = 0.009648
70
a = 0.9541
k = 0.01657
80
a = 0.9706
n = 0.1826
Logarithmic
50
a = 1.011
c = 0.04089
k = 0.0108
70
a = 0.9416
c = 0.07519
k = 0.0222
80
a = 0.9031
c = 0.077
k = 0.0302
Two term model
50
a = 1.033
b = -0.08725
k1 = 0.0097
k2 = 1.911
70
a = 0.5648
b = 0.5027
k1 = 0.0404
k2 = 0.0101
-5
80
a = -0.01622
b = 6.86e
k1 = 0.0734
k2 = 0.0145
Wang and Singh
50
a = -0.003137
b = -0.02947
t = 2.613
70
a = -0.01271
b = 4.26e-5
t = 1.456
-5
80
a = -0.01622
b = 6.86e
t = 1.2463
Simplified Fick’s diffusion
50
L = 13.02
a = 1.033
c = 1.634
70
L = 9.516
a = 0.9535
c = 1.499
80
L = 10.58
a = 0.9706
c = 2.464
Modified page equation – II 50
L = 0.731
c = 0.01773
n = 0.2803
70
L = 5.63
c = 1.862
n = 0.2977
80
L = 10.57
c = -2.181
n = -1.169
The moisture ratio calculated from the drying data at
different temperatures was fitted to the thin layer models
given in Table 1. The statistical regression results of
different models, including the drying model coefficients
were listed in Table 2. The R2 values were greater than
0.97 for the different models except for Wang and Singh
model. Further, it is assumed that the model which has
highest R2 and the lowest P2 and RMSE could be
considered as the best fit. Consequently logarithmic
model was selected to predict the drying characteristics in
the present study.
Hence, the effect of temperature on the drying
constants of the Logarithmic model was taken into
account by developing the relation between these
constants and the drying temperature. The regression
equations relating the constants of the selected model and
the drying temperature are the following:
P2
0.008328
0.02508
0.02472
0.008222
0.01054
0.01495
0.007489
0.02508
0.02472
0.004412
0.02175
0.02369
0.001683
0.005159
0.001278
0.004412
0.9988
0.009432
0.001512
0.9985
0.01119
0.001629
0.8789
0.9287
0.9236
0.9978
0.9833
0.9782
0.9963
0.9786
0.9773
0.09915
0.06984
0.07441
0.01329
0.03476
0.04113
0.01731
0.03733
0.04202
0.2458
0.09267
0.08306
0.004414
0.02175
0.02369
0.007489
0.02508
0.02472
80°C
70°C
50°C
Moisture ratio
0.8
0.6
0.4
0.3
0.0
0
50
100
150
200
250
Drying time(min)
300
350
Fig. 1: Drying curves of finger millet at different temperatures
Figure 2 gives the experimental MR verses t curve as
compared to that of predicted by the logarithmic model at
the three temperatures and it may be observed from the
figure that there is good matching between experimental
values and predicted values.
Effective diffusivities of the finger millet at three
different temperatures was evaluated by plotting ln MR vs
t (Fig. 3) and the data was presented in Table 3. The
values varied from 1.526 x 10G10 to 2.851 x 10G10 m2/s,
and as expected the diffusion coefficient has increased
R2 = 0.9991
R2 = 0.9923
R2 = 0.9343
Thus, the thin layer model for the finger millet was
MR=
RMSE
0.01613
0.03541
0.0393
0.01629
0.02355
0.03157
0.01697
0.03633
0.04059
0.01303
0.03383
0.03974
0.008204
0.01693
0.01116
0.01356
1.0
MR = a exp(-kt)+c where a, k, c are constants
a = 1.188 – 0.003546T,
k = -0.0213 + 0.00064T,
c = -0.01847 + 0.001162T,
R2
0.9966
0.9807
0.9773
0.9965
0.9919
0.9862
0.9963
0.9807
0.9773
0.9978
0.9833
0.9782
0.9992
0.9989
0.9987
0.9978
(1.188 – 0.003546T) exp[(0.0213 - 0.00064T)t]+
( -0.01847 + 0.001162T)
129
Adv. J. Food Sci. Technol., 3(2): 127-131, 2011
Table 3: Effective diffusivities of finger millet at different temperatures
S. No.
Temperature (0C)
Diffusivity (m2/s)
1
50
1.526x10-10
2
70
2.259x10-10
3
80
2.851x10-10
1.0
Exp 80
Pre 80
Exp 70
Pre 70
Moisture ratio
0.8
Pre 50
the range reported in literature, 10G9 to10G11 m2/s
(Madamba et al., 1996).
Further, a plot (Fig. 4.) was made between 1/T verses
lnD and it may be observed from Fig. 4 that a linear
relationship is obtained. Therefore, the diffusion kinetics
follows Arrhenius type relationship. The slope of this line
gives the value of Ea/R, from which the activation energy
was evaluated as 35.37 kJ/gmol.
0.6
0.4
0.3
0.0
0
20
40
60
80
100 120
Drying time(min)
140
160
180
CONCLUSION
Fig. 2: Drying curves for the experimental data and that
predicted based on the logarithmic model
-0.8
The conclusions drawn from the present study were
that the Logarithmic model gave an excellent fit for the
drying data of the finger millet. The effective diffusivities
increased with the drying temperature and varied from
1.526 x 10G10 to 2.851 x 10G10 m2/s. The temperature
dependence of diffusivity follows Arrhenius type of
relationship and the activation energy for the diffusion of
moisture was found to be 35.37 kJ/gmol.
-0.9
ACKNOWLEDGMENT
-1.0
One of the authors (G.B. Radhika) acknowledge the
Management, Principal and the Head, Department of
Chemical Engineering, B.V. Raju Institute of Technology
for providing the necessary facilities to carry out the
work.
-0.5
50°C
70°C
80°C
-0.6
InMR
-0.7
-1.1
0
20
40
80
60
100
Drying time(min)
120
140
160
Fig. 3: lnMR vs drying time (min)
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AUTHOR’S CONTRIBUTION
G.B. Radhika: Main worker for the paper as a part of
her Ph.D. work, collected the experimental data, and
processed the data, prepared the manuscript. S.V.
Satyanarayana: Co-supervisor for the Ph.D. work, assisted
in data processing and drafting the manuscript. D.G. Rao:
Supervisor for the Ph.D. work, mainly conceived the
project, assisted in data processing, redrafted the
manuscript with editorial corrections.
131