Critical Reviews in Food Science and Nutrition

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A Review of Thin Layer Drying of Foods: Theory,

Modeling, and Experimental Results

Zafer Erbay & Filiz Icier

To cite this article: Zafer Erbay & Filiz Icier (2010) A Review of Thin Layer Drying of Foods:
Theory, Modeling, and Experimental Results, Critical Reviews in Food Science and Nutrition, 50:5,
441-464, DOI: 10.1080/10408390802437063

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Critical Reviews in Food Science and Nutrition, 50:441–464 (2009)
Copyright
C Taylor and Francis Group, LLC
ISSN: 1040-8398
DOI: 10.1080/10408390802437063

A Review of Thin Layer Drying

of Foods: Theory, Modeling,
and Experimental Results

ZAFER ERBAY1 and FILIZ ICIER2

1
Graduate School of Natural and Applied Sciences, Food Engineering Branch, Ege University, 35100 Izmir, Turkey
2
Department of Food Engineering, Faculty of Engineering, Ege University, 35100 Izmir, Turkey

Drying is a complicated process with simultaneous heat and mass transfer, and food drying is especially very complex
because of the differential structure of products. In practice, a food dryer is considerably more complex than a device
that merely removes moisture, and effective models are necessary for process design, optimization, energy integration, and
control. Although modeling studies in food drying are important, there is no theoretical model which neither is practical nor
can it unify the calculations. Therefore the experimental studies prevent their importance in drying and thin layer drying
equations are important tools in mathematical modeling of food drying. They are practical and give sufficiently good results.
In this study first, the theory of drying was given briefly. Next, general modeling approaches for food drying were explained.
Then, commonly used or newly developed thin layer drying equations were shown, and determination of the appropriate
model was explained. Afterwards, effective moisture diffusivity and activation energy calculations were expressed. Finally,
experimental studies conducted in the last 10 years were reviewed, tabulated, and discussed. It is expected that this
comprehensive study will be beneficial to those involved or interested in modeling, design, optimization, and analysis of food
drying.

Keywords food drying, thin layer, mathematical modeling, diffusivity, activation energy

INTRODUCTION The methods of drying are diversified with the purpose of the
process. There are more than 200 types of dryers (Mujumdar,
Drying is traditionally defined as the unit operation that con- 1997). For every dryer, the process conditions, such as the dry-
verts a liquid, solid, or semi-solid feed material into a solid prod- ing chamber temperature, pressure, air velocity (if the carrier
uct of significantly lower moisture content. In most cases, drying gas is air), relative humidity, and the product retention time,
involves the application of thermal energy, which causes water have to be determined according to feed, product, purpose, and
to evaporate into the vapor phase. Freeze-drying provides an ex- method. On the other hand, drying is an energy-intensive pro-
ception to this definition, since this process is carried out below cess and its energy consumption value is 10–15% of the total
the triple point, and water vapor is formed directly through the energy consumption in all industries in developed countries
sublimation of ice. The requirements of thermal energy, phase (Keey, 1972; Mujumdar, 1997). It is a very important process
change, and a solid final product distinguish drying from me- according to the main problems in the whole world such as the
chanical dewatering, evaporation, extractive distillation, adsorp- depletion of fossil fuels and environmental pollution. In brief,
tion, and osmotic dewatering (Keey, 1972; Mujumdar, 1997). drying is arguably the oldest, most common, most diverse, and
Drying is one of the oldest unit operation, and widespread most energy-intensive unit operation and because of all these
in various industries recently. It is used in the food, agricul- features, the engineering in drying processes gains importance.
tural, ceramic, chemical, pharmaceutical, pulp and paper, min- In the food industry, foods are dried, starting from their nat-
eral, polymer, and textile industries to gain different utilities. ural form (vegetables, fruits, grains, spices, milk) or after han-
dling (e.g. instant coffee, soup mixes, whey). The production
Address correspondence to: Zafer Erbay, Graduate School of Natural and of a processed food may involve more than one drying process
Applied Sciences, Food Engineering Branch, Ege University, 35100 Izmir,
Turkey. Tel:+90 232 388 4000 (ext.3010) Fax: +90 232 3427592. E-mail: at different stages and in some cases, pre-treatment of food is
Zafererbay@yahoo.com necessary before drying. In the food industry, the main purpose
441
442 Z. ERBAY AND F. ICIER

of drying is to preserve and extend the shelf life of the product. in granular and porous foods due to surface forces. In addition
In addition to this, in the food industry, drying is used to obtain to these, thermal diffusion that is defined as water flow caused
a desired physical form (e.g. powder, flakes, granules); to obtain by the vaporization-condensation sequence, and hydrodynamic
the desired color, flavor, or texture; to reduce the volume or the flow that is defined as water flow caused by the shrinkage and
weight for transportation; and to produce new products which the pressure gradient may also be seen in drying (Strumillo
would not otherwise be feasible (Mujumdar, 1997). and Kudra, 1986; Özilgen and Özdemir, 2001). The dominant
Drying is one of the most complex and least understood diffusion mechanism is a function of the moisture content and
processes at the microscopic level, because of the difficulties the structure of the food material and it determines the drying
and deficiencies in mathematical descriptions. It involves si- rate. The dominant mechanism can change during the process
multaneous and often coupled and multiphase, heat, mass, and and, the determination of the dominant mechanism of drying is
momentum transfer phenomena (Kudra and Mujumdar, 2002; important in modeling the process.
Yilbas et al., 2003). In addition, the drying of food materials For hygroscopic products, generally the product dries in con-
is further complicated by the fact that physical, chemical, and stant rate and subsequent falling rate periods and it stops when
biochemical transformations may occur during drying, some of an equilibrium is established. In the constant rate period of dry-
which may be desirable. Physical changes such as glass transi- ing, external conditions such as temperature, drying air velocity,
tions or crystallization during drying can result in changes in the direction of air flow, relative humidity of the medium, physical
mechanisms of mass transfer and rates of heat transfer within the form of product, the desirability of agitation, and the method of
material, often in an unpredictable manner (Mujumdar, 1997). supporting the product during drying are essential and the dom-
The underlying chemistry and physics of food drying are highly inant diffusion mechanism is the surface diffusion. Toward the
complicated, so in practice, a dryer is considerably more com- end of the constant rate period, moisture has to be transported
plex than a device that merely removes moisture, and effective from the inside of the solid to the surface by capillary forces
models are necessary for process design, optimization, energy and the drying rate may still be constant until the moisture con-
integration, and control. Although many research studies have tent has reached the critical moisture content and the surface
been done about mathematical modeling of drying, undoubt- film of the moisture has been so reduced with the appearance
edly, the observed progress has limited empiricism to a large of dry spots on the surface. Then the first falling rate period
extent and there is no theoretical model that is practical and can or unsaturated surface drying begins. Since, however, the rate
unify the calculations (Marinos-Kouris and Maroulis, 1995). is computed with respect to the overall solid surface area, the
Thin layer drying equations are important tools in mathemat- drying rate falls even though the rate per unit wet solid sur-
ical modeling of drying. They are practical and give sufficiently face area remains constant (Mujumdar and Menon, 1995). In
good results. To use thin layer drying equations, the drying-rate this drying period, the dominant diffusion mechanism is liquid
curves have to be known. However, the considerable volume diffusion due to moisture concentration difference and internal
of work devoted to elucidate the better understanding of mois- conditions such as the moisture content, the temperature, and
ture transport in solids is not covered in depth, in practice, the structure of the product are important. When the surface film
drying-rate curves have to be measured experimentally, rather of the liquid is entirely evaporated, the subsequent falling rate
than calculated from fundamentals (Baker, 1997). So the ex- period begins. In the second falling rate period of drying the
perimental studies prevent their importance in drying. There is dominant diffusion mechanism is vapor diffusion due to mois-
no review done about the experimental results of the thin layer ture concentration difference and internal conditions keep on
drying experiments of foods and mathematical models in thin their importance (Husain et al., 1972).
layer drying in open literature for more than 10 years. Jayas et Although biological materials such as agricultural products
al. (1991) have written the last review according to the authors’ have a high moisture content, generally no constant rate period
knowledge. In this study, the fundamentals of thin layer drying is seen in the drying processes (Bakshi and Singh, 1980). In
were explained, and commonly used or newly developed semi- fact, some agricultural materials such as grains or nuts usually
theoretical and empirical models in the literature were shown. dry in the second falling rate period (Parry, 1985). Although
In addition, the experimental results gained in the last 10 years sometimes there is an overall constant rate period at the initial
for food materials were summarized and discussed. stages of drying, a statement such as the food materials dry
without a constant rate period is generally true.

THE THEORY AND MATHEMATICAL MODELING

OF FOOD DRYING Mathematical Modeling of Food Drying

Mechanisms of Drying Drying processes are modeled with two main models:

The main mechanisms of drying are surface diffusion or (i) Distributed models
liquid diffusion on the pore surfaces, liquid or vapor diffusion Distributed models consider simultaneous heat and mass
due to moisture concentration differences, and capillary action transfer. They take into consideration both the internal and
 A REVIEW OF THIN LAYER DRYING OF FOODS 443

2 
external heat and mass transfer, and predict the temperature ∂T ∂ T a1 ∂T
and the moisture gradient in the product better. Generally, =α + (6)
∂t ∂x 2 x ∂x
these models depend on the Luikov equations that come
from Fick’s second law of diffusion shown as Eq. 1 or their where, parameter a1 = 0 for planar geometries, a1 = 1
modified forms (Luikov, 1975). for cylindrical shapes and a1 = 2 for spherical shapes
(Ekechukwu, 1999).
∂M
= ∇ 2 K11 M + ∇ 2 K12 T + ∇ 2 K13 P
∂t The assumptions resembling the uniform temperature distri-
bution and temperature equivalent of the ambient air and product
∂T
= ∇ 2 K21 M + ∇ 2 K22 T + ∇ 2 K23 P cause errors. This error occurs only at the beginning of the pro-
∂t cess and it may be reduced to acceptable values with reducing
the thickness of the product (Henderson and Pabis, 1961). With
∂P
= ∇ 2 K31 M + ∇ 2 K32 T + ∇ 2 K33 P (1) this necessity, thin layer drying gains importance and thin layer
∂t equations are derived.
where, K11 , K22 , K33 are the phenomenological coeffi-
cients, while K12 , K13 , K21 , K23 , K31 , K32 are the coupling Thin Layer Drying Equations
coefficients (Brooker et al., 1974).
For most of the processes, the pressure effect can be ne- Thin layer drying generally means to dry as one layer of
glected compared with the temperature and the moisture sample particles or slices (Akpinar, 2006a). Because of its thin
effect, so the Luikov equations become as (Brooker et al., structure, the temperature distribution can be easily assumed
1974): as uniform and thin layer drying is very suitable for lumped
parameter models.
∂M Recently thin layer drying equations have been found to have
= ∇ 2 K11 M + ∇ 2 K12 T
∂t wide application due to their ease of use and requiring less data
unlike in complex distributed models (such as phenomenologi-
∂T cal and coupling coefficients) (Madamba et al., 1996; Özdemir
= ∇ 2 K21 M + ∇ 2 K22 T (2)
∂t and Devres, 1999).
Thin layer equations may be theoretical, semi-theoretical,
Nevertheless, the modified form of the Luikov equations and empirical models. The former takes into account only the in-
(Eq. 2) may not be solved with analytical methods, be- ternal resistance to moisture transfer (Henderson, 1974; Suarez
cause of the difficulties and complexities of real drying et al., 1980; Bruce, 1985; Parti, 1993), while the others consider
mechanisms. On the other hand, this modified form can only the external resistance to moisture transfer between the
be solved with the finite element method (Özilgen and product and air (Whitaker et al., 1969; Fortes and Okos, 1981;
Özdemir, 2001). Parti, 1993; Özdemir and Devres, 1999). Theoretical models ex-
(ii) Lumped parameter models plain the drying behaviors of the product clearly and can be used
Lumped parameter models do not pay attention to the tem- at all process conditions, while they include many assumptions
perature gradient in the product and they assume a uniform causing considerable errors. The most widely used theoretical
temperature distribution that equals to the drying air tem- models are derived from Fick’s second law of diffusion. Simi-
perature in the product. With this assumption, the Luikov larly, semi-theoretical models are generally derived from Fick’s
equations become as: second law and modifications of its simplified forms (other semi-
theoretical models are derived by analogues with Newton’s law
∂M of cooling). They are easier and need fewer assumptions due
= K11 ∇ 2 M (3)
∂t to using of some experimental data. On the other hand, they
are valid only within the process conditions applied (Fortes and
∂T
= K22 ∇ 2 T (4) Okos, 1981; Parry, 1985). The empirical models have also sim-
∂t ilar characteristics with semi-theoretical models. They strongly
depend on the experimental conditions and give limited infor-
Phenomenological coefficient K11 is known as effective mation about the drying behaviors of the product (Keey, 1972).
moisture diffusivity (Deff ) and K22 is known as thermal
diffusivity (α). For constant values of Deff and α, Equations Theoretical Background
3 and 4 can be rearranged as: Isothermal conditions changing only with time may be as-

2  sumed to prevail within the product, because the heat transfer
∂M ∂ M a1 ∂M rate within the product is two orders of magnitude greater than
= Deff + (5)
∂t ∂x 2 x ∂x the rate of moisture transfer (Özilgen and Özdemir, 2001). It can
444 Z. ERBAY AND F. ICIER

Q Nw Table 1 Values of geometric constants according to the product geometry.

Ta
Product Geometry A1 A∗2
Me
Infinite slab 8/π 2 4L2
Sphere 6/π 2 4r 2
L 3-dimensional finite slab (8/π 2 )3 1/(L1 + L22 + L23 )
2

Mi ∗L is the half thickness of the slice if drying occurs from both sides, or L is the
thickness of the slice if drying occurs from only one side.

∞
 
Me
 1 J02 Deff t
MR = A1 exp − (12)
J2
i=1 0
A2
Q Nw

Figure 1 Schematic view of thin layer drying, if drying occurs from both where, Deff is the effective moisture diffusivity (m2 /s), t is time
sides. (s), MR is the fractional moisture ratio, J0 is the roots of the
be assumed as only Eq. 5 describes the mass transfer (Whitaker Bessel function, and A1 , A2 are geometric constants.
et al., 1969; Young, 1969). Then Eq. 5 can be analytically solved For multidimensional geometries such as 3-dimensional slab
with the above assumptions, and the initial and boundary con- the Newman’s rule can be applied (Treybal, 1968). In brief, the
ditions are (Fig. 1): values of geometric constants are shown in Table 1.
MR can be determined according to the external conditions.
t = 0, −L ≤ x ≤ L, M = Mi (7) If the relative humidity of the drying air is constant during the
drying process, then the moisture equilibrium is constant too. In
t > 0, x = 0, dM/dx = 0 (8) this respect, MR is determined as in Eq. 13. If the relative humid-
ity of the drying air continuously fluctuates, then the moisture
t > 0, x = L, M = Me (9) equilibrium continuously varies so MR is determined as in Eq.
14 (Diamante and Munro, 1993);
t > 0, −L ≤ x ≤ L, T = Ta (10)
(Mt − Me )
MR = (13)
Assumptions: (Mi − Me )

Mt
(i) the particle is homogenous and isotropic; MR = (14)
Mi
(ii) the material characteristics are constant, and the shrinkage
is neglected;
where, Mi is the initial moisture content, Mt is the mean mois-
(iii) the pressure variations are neglected;
ture content at time t, Me is the equilibrium moisture content,
(iv) evaporation occurs only at the surface;
and all these values are in dry basis. If we accept that food ma-
(v) initially moisture distribution is uniform (Eq. 7) and sym-
terials dry without a constant rate period, than Mi is equal to
metrical during process (Eq. 8);
the Mcr which is defined as the moisture content of a material at
(vi) surface diffusion is ended, so the moisture equilibrium
the end of the constant rate period of drying, then Eq. 13 equals
arises on the surface (Eq. 9);
to Eq. 15 and MR can be named as the characteristic moisture
(vii) temperature distribution is uniform and equals to the am-
content (φ).
bient drying air temperature, namely the lumped system
(Eq. 10);
(Mt − Me )
(viii) the heat transfer is done by conduction within the product, φ= (15)
and by convection outside of the product; (Mcr − Me )
(ix) effective moisture diffusivity is constant versus moisture
content during drying. Semi-Theoretical Models
Semi-theoretical models can be classified according to their
Then analytical solutions of Eq. 5 are given below for infinite derivation as:
slab or sphere in Eq. 11, and for infinite cylinder in Eq. 12
(Crank, 1975):
(i) Newton’s law of cooling:
∞
 
 1 (2i − 1) π Deff t
2 2 These are the semi-theoretical models that are derived
MR = A1 exp − (11) by analogues with Newton’s law of cooling. These models
(2i − 1) 2 A2
i=1 can be classified in sub groups as:
 A REVIEW OF THIN LAYER DRYING OF FOODS 445

a. Lewis model known as the Modified Page-I Model:

b. Page model & modified forms
(ii) Fick’s second law of diffusion (Mt − Me )
MR = = exp (−kt)n (19)
The models in this group are the semi-theoretical models (Mi − Me )
that are derived from Fick’s second law of diffusion. These
models can be classified in sub groups as: In addition, White et al. (1978) used another modified form
a. Single term exponential model and modified forms of the Page model to describe the drying of soybeans. This
b. Two term exponential model and modified forms form is generally known as the Modified Page-II Model:
c. Three term exponential model
(Mt − Me )
MR = = exp − (kt)n (20)
(Mi − Me )
The Models Derived From Newton’s Law of Cooling.
Diamente and Munro (1993) used another modified form
a. Lewis (Newton) Model of the Page model to describe the drying of sweet potato
This model is analogous with Newton’s law of cooling so slices. This form is generally known as the Modified Page
many investigators named this model as Newton’s model. equation-II Model:
First, Lewis (1921) suggested that during the drying of
porous hygroscopic materials, the change of moisture con- (Mt − Me )  n
MR = = exp −k t/ l 2 (21)
tent of material in the falling rate period is proportional to (Mi − Me )
the instantaneous difference between the moisture content
and the expected moisture content when it comes into equi- where, l is an empirical constant (dimensionless).
librium with drying air. So this concept assumed that the
material is thin enough, or the air velocity is high, and the The Models Derived From Fick’s Second Law of Diffusion.
drying air conditions such as the temperature and the relative
humidity are kept constant. a. Henderson and Pabis (Single term) Model
Henderson and Pabis (1961) improved a model for drying
dM by using Fick’s second law of diffusion and applied the new
= −K (M − Me ) (16)
dt model on drying of corns. As the derivation was shown in
the previous section, they use Eq. 11. For sufficiently long
where, K is the drying constant (s−1 ). In the thin layer dry- drying times, only the first term (i = 1) of the general series
ing concept, the drying constant is the combination of dry- solution of Eq. 11 can be used with small error. According
ing transport properties such as moisture diffusivity, thermal to this assumption, Eq. 11 can be written as:
conductivity, interface heat, and mass coefficients (Marinos- 
Kouris and Maroulis, 1995). (Mt − Me ) π 2 Deff
MR = = A1 exp − t (22)
If K is independent from M,then Eq. 16 can be rewritten as: (Mi − Me ) A2

(Mt − Me ) If Deff is constant during drying, then Eq. 22 can be rear-

MR = = exp(−kt) (17)
(Mi − Me ) ranged by using the drying constantk as:

(Mt − Me )
where, k is the drying constant (s−1 ) that can be obtained MR = = a exp (−kt) (23)
from the experimental data and Eq. 17 is known as the Lewis (Mi − Me )
(Newton) model
where, a is defined as the indication of shape and generally
b. Page Model
named as model constant (dimensionless). These constants
Page (1949) modified the Lewis model to get a more accurate
are obtained from experimental data. Equation 23 is gener-
model by adding a dimensionless empirical constant (n) and
ally known as the Henderson and Pabis Model.
apply to the mathematical modeling of drying of shelled
b. Logarithmic (Asymptotic) Model
corns:
Chandra and Singh (1995) proposed a new model including
the logarithmic form of Henderson and Pabis model with an
(Mt − Me )
MR = = exp(−kt n ) (18) empirical term addition, and Yagcioglu et al. (1999) applied
(Mi − Me ) this model to the drying of laurel leaves.

Generally, n is named as the model constant (dimensionless). (Mt − Me )

c. Modified Page Models MR = = a exp (−kt) + c (24)
(Mi − Me )
Overhults et al. (1973) modified the Page model to describe
the drying of soybeans. This modified form is generally where, c is an empirical constant (dimensionless).
446 Z. ERBAY AND F. ICIER

c. Midilli Model as:

Midilli et al. (2002) proposed a new model with the addi-
tion of an extra empirical term that includes t to the Hen-
(Mt − Me )
derson and Pabis model. The new model was the com- MR= =a exp (−kt) + (1 − a) exp (−kat) (29)
bination of an exponential term and a linear term. They (Mi − Me )
applied this new model to the drying of pollen, mush-
room, and shelled/unshelled pistachio for different drying Equation 29 is generally known as the Two-Term Exponen-
methods. tial model.
h. Modified Two-Term Exponential Models
(Mt − Me )
MR = = a exp (−kt) + b∗ t (25) Verma et al. (1985) modified the second exponential term
(Mi − Me ) of the Two-Term Exponential model by adding an empirical
constant and applied for the drying of rice.
where, b∗ is an empirical constant (s−1 ).
d. Modified Midilli Model
(Mt − Me )
Ghazanfari et al. (2006) emphasized that the indication of MR = = a exp (−kt) + (1 − a) exp (−gt) (30)
shape term (a) of the Midilli model (Eq. 25) had to be 1.0 at (Mi − Me )
t = 0 and proposed a modification as:
This modified model (Eq. 30) is known as the Verma Model.
(Mt − Me ) Kaseem (1998) rearranged the Verma model by separating
MR = = exp (−kt) + b∗ t (26)
(Mi − Me ) the drying constant term k from g and proposed the renewed
form as:
This model was not applied to a food material, but gave good
results with flax fiber. (Mt − Me )
e. Demir et al. Model MR= =a exp (−kt) + (1 − a) exp (−kbt) (31)
(Mi − Me )
Demir et al. (2007) proposed a new model that was similar
to Henderson and Pabis, Modified Page-I, Logarithmic, and
Midilli models: This modified form (Eq. 31) is known as the Diffusion Ap-
proach model. These two modified models were applied for
(Mt − Me ) some products’ drying at the same time, and gave the same
MR = = a exp [(−kt)]n + b (27)
(Mi − Me ) results as expected (Tořul and Pehlivan, 2003; Akpinar et al.,
2003b; Gunhan et al., 2005; Akpinar, 2006a; Demir et al.,
This model has been just proposed and applied to the drying 2007).
of green table olives and got good results. i. Modified Henderson and Pabis (Three Term Exponen-
f. Two-Term Model tial) Model
Henderson (1974) proposed to use the first two term of the Karathanos (1999) improved the Henderson and Pabis and
general series solution of Fick’s second law of diffusion (Eq. Two-Term models as adding the third term of the general
5) for correcting the shortcomings of the Henderson and series solution of Fick’s second law of diffusion (Eq. 5)
Pabis Model. Then, Glenn (1978) used this proposal in grain for correcting the shortcomings of the Henderson and Pabis
drying. With this argument, the new model derived as: and Two-Term models. Karathanos emphasized that the first
term explains the latest part, the second term explains the
(Mt − Me ) intermediate part, and the third term explains the beginning
MR = = a exp (−k1 t) + b exp (−k2 t) (28) part of the drying curve (MR-t) as:
(Mi − Me )

where, a, b are defined as the indication of shape and gen- (Mt − Me )

erally named as model constants (dimensionless), and k1 , k2 MR = = a exp (−kt)
(Mi − Me )
are the drying constants (s−1 ). These constants are obtained
from experimental data and Eq. 28 is generally known as the + b exp (−gt) + c exp (−ht) (32)
Two-Term Model.
g. Two-Term Exponential Model
Sharaf-Eldeen et al. (1980) modified the Two-Term model where, a, b, and c are defined as the indication of shape and
by reducing the constant number and organizing the second generally named as model constants (dimensionless), and
exponential term’s indication of shape constant (b). They k, g, and h are the drying constants (s−1 ). These constants
emphasized that b of the Two-Term model (Eq. 27) has to be are obtained from experimental data and Eq. 32 is generally
(1 – a) at t = 0 to get MR= 1 and proposed a modification known as the Modified Henderson and Pabis model.
 A REVIEW OF THIN LAYER DRYING OF FOODS 447

Empirical Models 2001; Midilli et al., 2002; Akpinar et al., 2003b; Wang et al.,
2007a). In addition to r, χ 2 and RMSE are used to determine
a. Thompson Model
the best fit. The highest r and the lowest χ 2 and RMSE values
Thompson et al. (1968) developed a model with the experi-
required to evaluate the goodness of fit (Sawhney et al., 1999a;
mental results of drying of shelled corns in the temperature
Yaldiz et al., 2001; Tořul and Pehlivan, 2002; Midilli and Kucuk,
range 60–150◦ C.
2003; Akpinar et al., 2003a; Lahsasni et al., 2004; Ertekin and
Yaldiz, 2004; Wang et al., 2007b). r, χ 2 , and RMSE calculations
t = a ln (MR) + b [ln (MR)]2 (33) can be done by equations below:

N
N
N N i=1 MRpre,i MRexp,i − i=1 MRpre,i i=1 MRexp,i
r = 

 

N 2  (36)
N N 2 N
N i=1 MR2pre,i − i=1 MRpre,i N i= MR2exp,i − i=1 MRexp,i

n
(MRexp,i − MRpre,i )2
where, a and b were dimensionless constants obtained from χ =
2 i=1
(37)
N −n
experimental data. This model was also used to describe the
drying characteristics of sorghum (Paulsen and Thompson,  1/2
1 
N
1973). RMSE = (MRpre,i − MRexp,i )2 (38)
b. Wang and Singh Model N i=1
Wang and Singh (1978) created a model for intermittent
drying of rough rice. where, N is the number of observations, n is the number
of constants, MRpre,i ith predicted moisture ratio values,
MR = 1 + b∗ t + a ∗ t 2 (34) MRexp,i ith experimental moisture ratio values.
Finally, the effect of the variables on model constants can
where, b∗ (s−1 ) and a ∗ (s−2 ) were constants obtained from be investigated by performing multiple regression analysis with
experimental data. multiple combinations of different equations such as the simple
c. Kaleemullah Model linear, logarithmic, exponential, power, and the Arrhenius type
Kaleemullah (2002) created an empirical model that included (Guarte, 1996). These equation types are relatively easy to use in
MR, T , and t. They applied it to the drying of red chillies multiple regression analysis, because they could be linearized.
(Kaleemullah and Kailappan, 2006). The other types of equations must be solved with nonlinear re-
gression techniques and it is too hard to find the solution to such
MR = exp −c∗ T + b∗ t (pT +n) (35) nonlinear equations if there are many parameters. After investi-
gating the effect of experimental variables on model constants,
where, constant c∗ is in ◦ C−1 s−1 , constant b∗ is in s−1 , p is the final model has to be validated by the statistical methods
in ◦ C−1 and n is dimensionless. that are mentioned above.

Determination of Appropriate Model Effective Moisture Diffusivity Calculations

Mathematical modeling of the drying of food products of- Diffusion in solids during drying is a complex process that
ten requires the statistical methods of regression and correlation may involve molecular diffusion, capillary flow, Knudsen flow,
analysis. Linear and nonlinear regression analyses are important hydrodynamic flow, or surface diffusion. With a lumped param-
tools to find the relationship between different variables, espe- eter model concept, all these phenomena are combined in one
cially, for which no established empirical relationship exists. term named as effective moisture diffusivity (Eq. 3). Equations
As mentioned above, thin layer drying equations require MR 22 and 23 are derived for the constant values of Deff (m2 /s) and
variation versus t. Therefore, MR data plotted with t, and re- for sufficiently long drying times. With a simple arrangement,
gression analysis was performed with the selected models to Eq. 39 is obtained:
determine the constant values that supply the best appropriate-
ness of models. The validation of models can be checked with ln (MR) = ln (a) − kt (39)
different statistical methods. The most widely used method in
literature is performing correlation analysis, reduced chi-square and, k is defined as:
(χ 2 ) test and root mean square error (RMSE) analysis, respec- π 2 Deff
tively. Generally, the correlation coefficient (r) is the primary k=− (40)
A2
criterion for selecting the best equation to describe the drying
curve equation and the highest r value is required (O’Callaghan where, A2 is the geometric constant that is shown in Table 1 for
et al., 1971; Verma et al., 1985; Kassem, 1998; Yaldiz et al., main geometries.
448 Z. ERBAY AND F. ICIER

Equation 39 indicates that the variation of ln(MR) values 30.0%

26.8%
versus t is linear and the slope is equal to drying constant
(k). By revealing the drying, the constant effective moisture 25.0%

diffusivity can be calculated easily with different geometries

20.0%
(Eq. 40).

Distribution (%)
As a matter of fact, the drying curves have a concave form 15.5%
15.0%
when the curves of ln(MR)-t are analyzed. The reason for this 12.7%
11.3%
is the assumption of the invariability of the effective moisture 9.9%
8.5% 10.0%
diffusion (independency of Deff from moisture content) during
5.6%
drying while deriving the equations (Bruin and Luyben, 1980). 4.2% 4.2% 5.0%
The concave form of drying curves is caused by variation of 1.4%
the moisture content and Deff during drying. Because of this, 0.0%
the slopes have to be derived from linear regression of ln(MR)-t 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
data. Publishing years
Deff mainly varies with internal conditions such as the prod-
Figure 2 Distribution of the studies according to the publishing years.
uct’s temperature, the moisture content, and the structure. This
is harmonious with the assumptions of the thin layer concept.
But all assumptions cause some errors and Deff is also affected the greater value of Ea means more sensibility of Deff to tem-
from external conditions. These effects are insignificant relative perature (Kaymak-Ertekin, 2002).
to internal conditions while they cannot be disregarded in some To calculate Ea , Eq. 41 is arranged as:
ranges. Drying air velocity is an example of this. Islam and Flink
(1982) explained that the resistance of the external mass transfer Ea 1
ln(Deff ) = ln(D0 ) − 103 × (42)
was important in 2.5 m/s or lower velocities. Mulet et al. (1987) R (T + 273.15)
expressed that drying air velocity affected the diffusion coef-
ficient at an interval of a certain flow velocity. Ece and Cihan Equation 42 indicates that the variation of ln(Deff ) versus
(1993) used a temperature and air velocity dependent Arrhenius [1/(T + 273.15)] is linear and the slope is equal to (−103 .Ea /R),
type diffusivity and Akpinar et al. (2003a) exposed a tempera- so Ea is easily calculated with revealing the slope by deriving
ture and air velocity dependent Arrhenius type diffusivity with from linear regression of ln(Deff )-[1/(T + 273.15)].
experimental data. So, for clarifying the drying characteristics, If the coefficient of the determination value cannot be as
it is important to calculate Deff . high as required, other factors would affect the Deff and they
have to be considered. At this condition, the most appropriate
Activation Energy Calculations method is to reflect these factors to the D0 and perform nonlinear
regression analysis to fit the data. For microwave drying, another
As mentioned above, the factors affecting Deff are significant form was developed to calculate the activation energy by Dadalı
to clarify the drying characteristics of a food product, meanwhile et al. (2007b). They described the Deff as a function of product
the power of the effect is significant. The effect of temperature mass and microwave power level with an Arrhenius equation:
on Deff gains importance at this point. Because temperature has
two critical properties in this matter: 
−Ea m
Deff = D0 exp (43)
Pm
(i) temperature is one of the strongest factor affects on Deff ,
(ii) it is easily calculated or fixed during experiments. where, m is the weight of the raw material (g), Pm is the mi-
crowave output power (W), and Ea is the activation energy for
As a consequence, many researchers studied the effect of
the microwave drying of the product (W/g).
temperature on Deff , and this effect can generally be described
In addition, Dadalı et al. (2007a) used an exponential ex-
by an Arrhenius equation (Henderson, 1974; Mazza and Le
pression based on the Arrhenius equation for prediction of the
Maguer, 1980; Suarez et al., 1980; Steffe and Singh, 1982;
relationship between drying rate constant and effective diffusiv-
Pinaga et al., 1984; Carbonell et al., 1986; Crisp and Woods,
ity as:
1994; Madamba et al., 1996):
 
Ea −Ea m
Deff = D0 exp −10 3
(41) k = k0 exp (44)
R (T + 273.15) Pm

where, D0 is the Arrhenius factor that is generally defined as where, k is the drying rate constant predicted by the appropriate
the reference diffusion coefficient at infinitely high temperature model and k0 is the pre-exponential constant (s−1 ). The acti-
(m2 /s), Ea is the activation energy for diffusion (kJ/mol), R is vation energy values obtained from Eqs. 43 and 44 were quite
the universal gas constant (kJ/kmol.K). The value of Ea shows similar and they showed the linear relationship between the dry-
the sensibility of the diffusivity against temperature. Namely, ing rate constant and effective diffusivity with Eqs. 43 and 44,
 Table 2 Studies conducted on mathematical modeling of sun drying of food products

Product Process conditions # Best model Effects of process conditions on model constants Reference

Apricot T = 27–43◦ C 12 Diffusion a = −116.304 + 5615T – 71.40T 2 + 18567.2RH Toǧrul and Pehlivan, 2004
(Untreated) Approach
b = −4.136 + 0.1924T – 0.00259T 2 + 1.8054RH k = 405.2 – 19.6T + 0.25T 2 – 64RH
T = 27–43◦ C a = −1.3536 – 0.3392T + 0.00548T 2 + 13.64RH b = 0.021 – 0.00371T + 0.000098T 2
(SO2 -sulphured) – 0.00772RH
k = −0.00406 + 0.0239T - 0.000515T 2 –
0.0498RH
T = 27–43◦ C Modified a = 31686.2 – 1537.26T + 18.52T 2 + 86.68RH b = 20632.67 – 993.17T + 11.92T 2
(NaHSO3 - Henderson & – 116.52RH
sulphured) Pabis
c = −9845.92 + 452.37T – 5.304T 2 + 689.51RH k = 0.0783 – 0.00348T – 0.000041T 2
– 0.01064RH
g = 3049.82 – 149.57T + 1.81T 2 + 53.08RH h = 2140.31 – 104.16T + 1.256T 2
+ 14.65RH
Basil — 12 Modified Page-II — Akpinar, 2006b
Bitter leaves — 8 Midilli — Sobukola et al., 2007
Crain-crain leaves
Fever leaves
Figs T = 27–43◦ C 12 Diffusion a = 17947.61 – 899.84T + 10.173T 2 – 15206RH – Toǧrul and Pehlivan, 2004
(Untreated) Approach 18383.1RH2 + 689.56TRH
b = –696.75 + 30.682T – 0.312T 2 + 667.47RH +
826.62RH2 – 24.75TRH
k = –144.51 + 7.257T – 0.0821T 2 + 119.83RH +
152.98RH2 – 5.531TRH
Grape T = 27–43◦ C 12 Modified a = -10403.4 + 440.23T – 4.47T 2 - 764.33RH + Toǧrul and Pehlivan, 2004
(pretreated) Henderson and 10172.7RH2 – 70.584TRH
Pabis
b = 2625.76 – 111.34T + 1.163T 2 + 301.24RH –
1566.3RH2 – 4.752TRH
c = –29575.3 + 1501.73T – 18.9T 2 – 50390.6RH –
7998.7RH2 + 1192.85TRH
k = 181.42 – 6.875T – 0.0673T 2 – 138.64RH +
51.95RH2 + 2.058TRH
g = 318.54 – 12.61T + 0.1305T 2 – 249.37RH +
320.2RH2 + 2.368TRH
h = 16.69 – 0.7479T + 0.000084T 2 + 3.566RH +
1.208RH2 – 0.091TRH
Mint — 12 Modified Page-II — Akpinar, 2006b
(Continued on next page)

449
450
Table 2 Studies conducted on mathematical modeling of sun drying of food products. (Continued)

Product Process conditions # Best model Effects of process conditions on model constants Reference

Mulberry fruits Untreated 2 Henderson and — Doymaz, 2004b

(Morus alba L.) Pabis
Pretreated
Parsley — 12 Verma — Akpinar, 2006b
Peach T = 27–43◦ C 12 Verma a = –4.873 + 0.269T – 0.0000372T 2 + 0.252RH k = –0.5742 + 0.0317T – 0.000449T 2 Toǧrul and Pehlivan, 2004
(Untreated) – 0.0956RH
g = 0.0479 – 0.0000262T + 0.0000361T 2
– 0.0000128RH
Pistachio T = 24–32◦ C 8 Midilli a = 0.9975 + 0.0007 lnT k = 0.1291 + 0.0006 lnT Midilli et al., 2002
(shelled)
n = 0.8828 + 0.0008 lnT b∗ = 0.0490 + 0.0001 lnT
T = 24–32◦ C a = 1.0030 + 0.0003 lnT k = 0.1500 + 0.0002 lnT
(unshelled)
n = 1.1044 + 0.0005 lnT b∗ = 0.0744 + 0.0004 lnT
Plum T = 27–43◦ C 12 Modified a = 3743.05 – 424.11T + 7.65T 2 + 3849.9RH Toǧrul and Pehlivan, 2004
(pretreated) Henderson & + 13477.76RH2 – 147.13TRH
Pabis
b = 4354.1 – 417.01T + 7.379T 2 – 1464.73RH +
21426.01RH2 – 109.47TRH
c = 7273.1 - 829T + 15.042T 2 + 7219.2RH
+ 30018.1RH2 – 314.25TRH
k = -0.0628 + 0.0000905T – 0.000175T 2
– 0.1396RH – 0.5232RH2 + 0.000064TRH
g = 865.08 – 82.384T + 1.427T 2 – 164.32RH
+ 3078.6RH2 – 12.7TRH
h = 758.05 – 72.23T + 1.251T 2 – 141.84RH +
2698.85RH2 – 11.18TRH
 Table 3 Studies conducted on mathematical modeling of food drying performed with convective type batch dryers

Product Process conditions (o C; m/s; g water/kg da; mm) # Best model Effects of process conditions on model constants Reference

Apple (slice) T = 60–80 υ = 1.0–1.5 13 Midilli a = 1.004084 – 0.000073T – 0.001960υ+ k = –0.006391 + 0.000065T Akpinar, 2006a
3.944759ω + 0.009775υ+ 1.576723ω
ω = 8 × 8 × 18 – 12.5 × n = 1.187734 + 0.002467T b∗ = 0.000082 – 0.000002T –
12.5 × 25 – 0.128878υ – 202.536ω 0.000041υ+ 0.041667ω
Apple (Golden) T = 60–80 υ = 1.0–3.0 14 Midilli a = 1.4678 − −0.0067T k = Menges and
1.0835υ 0.1316 n = 0.8867b∗ = 0.0030 Ertekin, 2006a
Apple pomace T = 75–105 10 Logarithmic a = 271.15 – 8.91T + 0.097T 2 – 3.52T 3 k = –0.61 + 0.02T – 0.0002T 2 + Wang et al., 2007a
0.0000008T 3
c = –267.45 + 8.82T – 0.096T 2 + 0.0004T 3
Apricot T = 47.3–61.74 υ = 0.707–2.3 14 Midilli a = 1.069931 – 0.001297T – 0.004534υ+ Akpinar et al.,
0.005478RSC 2004
RSC = 0–2.25 rpm k = –0.086272 + 0.001775T + 0.035643υ+
(SO2 -sulphured) 0.009545RSC
n = 1.705840 – 0.013076T – 0.167507υ –
0.020810RSC
b∗ = 0.010122 – 0.000162T – 0.001439υ –
0.000240RSC
T = 50–80 υ = 0.2–1.5 14 Logarithmic a = 1.13481exp(0.018352υ) k = 0.001269 + 0.000018T Toǧrul and
(SO2 -sulphured) x+ 0.00105υ Pehlivan, 2003
c = –1.16416 + exp(1.6982/T ) – 0.0138υ
Bagasse T = 80–120 υ = 0.5–2.0 12 Page k = 0.49123557038 + 0.0031094667H – Vijayaraj et al.,
0.0031183596869T – 0.03947507753υ+ 2007
0.113762212L
H = 9–24 L = 20–60 n = –0.86990405 + 0.238750462logt –
1.175456904k
Bay leaves T = 40–60 RH = 5–25% 15 Page k = exp(-4.4647 + 0.07455T – 0.00714RH) n = 1.14325 Gunhan et al.,
2005
Black Tea T = 80–120 υ = 0.25–0.65 5 Lewis k = 0.12563υ 1.15202 exp(−209.12341/Tabs ) Panchariya et al.,
2002
Carrot (slice) T = 60–90 υ = 0.5–1.5 4 Modified Page-II k = 42.66υ 0.3123 (2L)−0.8437 exp(–2386.6/T ) Erenturk and
Erenturk, 2007
L = 2.5–5 n = 5.48υ −0.0846 (2L)−0.1066 exp(–452.5/T )
Citrus aurantium leaves T = 50–60 RH = 41–53% 13 Midilli a = –49.079 + 1.838T – 0.0167T 2 k = –13.604 + 0.498T – Mohamed et al.,
0.004518T 2 2005
.
V = n = 37.447 – 1.346T + 0.01231T 2 b∗ = –0.451 + 0.01576T –
0.0277 − −0.0833m3 /s 0.00014T 2
Coconut (Young) T = 50–70 (Osmotically L = 2.5–4 3 Page k = 21.8exp(–2136.9/Tabs ) Madamba, 2003
pre-dried)
n = 0.098 – 0.082L
Dates T = 70–80 (Sakie var.) 3 Page k = –2.463 + 0.0613T – 0.00035T 2 n = –1.228 + 0.0524T – Hassan and
0.00032T 2 Hobani, 2000
T = 70–80 (Sukkari var.) k = 0.00000027T 3.0511 n = –4.437 + 0.1353T –
0.00085T 2
Echinacea angustifolia T = 15–45 υ = 0.3–1.1 4 Modified Page-II k = 0.07υ 0.1793 (2r)−1.2349 exp(-20.66/T ) Erenturk et al.,
2004
(Continued on next page)

451
452
Table 3 Studies conducted on mathematical modeling of food drying performed with convective type batch dryers. (Continued)

Product Process conditions (◦ C; m/s; g water/kg da; mm) # Best model Effects of process conditions on model constants Reference

r = root size (mm) n= 0.96υ −0.0139 (2r)−0.0433 exp(-1.73/T )

Eggplant T = 30–70 υ = 0.5–2.0 14 Midilli a = 0.98979 − 0.08071 ln υk = Ertekin and
0.00160T 1.55945 n = Yaldiz, 2004
1.09877 + 0.29745 ln υb∗ = 0.00062
Figs (whole) T = 46.1–60 υ = 1.0–5.0 7 Logarithmic a = 1.12998 + 0.0006324T - 0.0368791υ - Xanthopoulos et
0.00410299H al., 2007
H = 8.14–13.32 k = −0.0898261 + 0.00244127T +
0.00445721υ −0.0000864371H
c = −0.161594 − 0.000764116T +
0.0347936υ+ 0.00720103H
Grape (Sultana) T = 32.4–40.3 υ = 0.5–1.5 8 Two-term a = 0.336 - 0.004T k1 = 7.703 – 8.717 lnυ Yaldiz et al., 2001
b = 0.806υ −0.039 k2 = -0.141 + 0.048 lnT
Grape (Thompson seedless) T = 50–80 υ = 0.25–1.0 3 Page k = 2.91 × 106 υ 0.22 exp(5749.05/T ) Sawhney et al.,
(pretreated) 1999a
n = 1.14
T = 50–70 υ = 0.25–1.0 - k = 3720000υ 0.19 H −0.13 exp(-6032/Tabs ) Pangavhane et al.,
2000
RH = 13–23% n = 1.107
Green bean T = 50–80 υ = 0.25–1.0 12 Page k = 0.3560 – 0.1407υ n = 0.7832 + 0.0892lnυ Yaldiz and
Ertekin, 2001
Green chilli T = 40–65 RH = 10–60% 2 Page k = 0.008759 – 0.00027T + Hossain and Bala,
0.000000282T 2 + 0.00166υ – 0.01058RH 2002
+ 0.009057RH2
υ = 0.1–1.0 (Over/underflow) n = 0.563021 + 0.006435T + 0.088298υ +
0.63696RH
T = 40–65 RH = 10–60% k = −0.02184 + 0.000781T –
0.0000068T 2 + 0.004522υ +
0.004437RH – 0.01335RH2
υ = 0.1–1.0 (Through n = 0.580425 + 0.00465T + 1.7177υ –
flow) 1.2991υ 2 – 1.2421RH + 1.3845RH2
Green pepper T = 50–80 υ = 0.25–1.0 12 Diffusion Approach a = −1.6626 + 1.7015υ b = 0.5868 – 0.0172υ Yaldiz and
Ertekin, 2001
k = 0.3549 – 0.1489υ
Hazelnut T = 100–160 8 Thompson a = −116.05 + 0.656T b = −19.89 + 0.122T Özdemir and
Devres, 1999
T = 100–160 Mi = 12.3 % 3 Two-term a = 0.535 - 0.00058T k1 = 0.465 Özdemir et al.,
(moisturized) 2000
b = 0.00058 + 236248.7T k2 = 4.52
T = 100–160 Mi = 6.14 % Two-term a = 0.434 - 0.00304T k1 = 0.566
(untreated)
b = 0.00304 + 236248.7T k2 = 5.29
T = 100–160 Mi = 2.41 % Two-term a = 0.714 − 0.00356T k1 = 0.286
(pre-dried)
b = 0.00356 + 236248.7T k2 = 2.89
 Kale T = 30–60 L = 10–50 4 Mod. Page-I k = exp(8.0487 – 3836.1/Tabs ) n = 0.894653 Mwithiga and
Olwal, 2005
Kurut T = 35–65 11 Two-term - Karabulut et al.,
2007
Onion T = 50–80 υ = 0.25–1.0 12 Two-term a = 0.4866 + 0.6424 lnυ k1 = 0.1557 + 0.1995 lnυ Yaldiz and
Ertekin, 2001
b = 0.5143 – 0.6424 lnυ k2 = 0.1117 – 0.0992 lnυ
T = 50–80 υ = 0.25–1.0 - Henderson and Pabis a = 1.01 Sawhney et al.,
1999b
H = 6.5–10.5 k = 122.34υ 0.31 exp(-3020/Tabs )
(pretreated)
Paddy (parboiled) T = 70–150 υ = 0.5–2.0 - Lewis k = 0.02υ 0.473 L−0.699
d T 0.478 Rao et al., 2007
Ld = 50—200
Parsley T = 56–93 9 Page k = 0.000012T 0.706263 n = 0.293914T 0.299815 Akpinar et al.,
2006
Peach slice T = 55–65 6 Logarithmic - Kingsley et al.,
2007
Blanched with %1 KMS
or AA
Pistachio nuts T = 25–70 6 Page k = −0.00209 + 0.000208T + 0.00502υ 2 n = 0.844 + 0.00262T – 0.106υ Kashaninejad
et al., 2007
Pistachio T = 40–60 υ = 0.5–1.5 8 Midilli a = 0.9968 + 0.0007 lnT k = 0.1493 + 0.0006 lnT Midilli et al., 2002
RH = 5–20% (shelled) n = 0.9178 + 0.0008 lnT b∗ = 0.0501 + 0.0001 lnT
T = 40–60 υ = 0.5–1.5 a = 0.9968 + 0.0003 lnT k = 0.1545 + 0.0002 lnT
RH = 5–20% (unshelled) n = 0.9247 + 0.0005 lnT b∗ = 0.0486 + 0.0004 lnT
Plum (Stanley) T = 60–80 υ = 1.0–3.0 14 Midilli a = 2.5729 − 0.3726 lnT k = 0.2643υ 0.3665 Menges and
(pretreated) Ertekin, 2006b
n = 0.00011T 2.1554 b∗ = −0.0044
T = 60–80 υ = 1.0–3.0 a = 3.2180 − 0.5255 lnT k = 0.2288υ 0.2994
(untreated)
n = 0.000057T 2.3144 b∗ = −0.0028
Pollen T = 45 8 Midilli a = 0.9987 + 0.0003 lnT k = 0.2616 + 0.0002 lnT Midilli et al., 2002
n = 0.5869 + 0.0005 lnT b∗ = 0.0609 + 0.0004 lnT
Potato (slice) T = 60–80 υ = 1.0–1.5 13 Midilli a = 0.986173 + 0.000069T + 0.005702υ+ Akpinar, 2006a
0.098206ωk = -0.015582 + 0.000156T +
0.013467υ+ 0.266761ω
ω = 8 × 8 × 18 − 12.5 × n = 1.218379 + 0.000802T – 0.162776υ –
12.5 × 25 138.528ω
b∗ = 0.0000085 + 0.00000029T –
0.0000393υ – 0.0203022ω
Prickly pear fruit T = 50–60 8 Two-term a = −2.9205 + 0.1117T – 0.0011T 2 k1 = 1.1619 – 0.0439T + Lahsasni et al.,
0.0004T 2 2004
b = 2.3099 – 0.0547T + 0.0005T 2 k2 = -0.0764 + 0.0027T
– 0.000021658T 2
(Continued on next page)

453
454
Table 3 Studies conducted on mathematical modeling of food drying performed with convective type batch dryers. (Continued)

Product Process conditions (◦ C; m/s; g water/kg da; mm) # Best model Effects of process conditions on model constants Reference

Pumpkin (slice) T = 60–80 υ = 1.0–1.5 13 Midilli a = 0.966467 + 0.000184T + 0.007014υ k = 0.005645 - 0.000095T Akpinar, 2006a
+ 0.003791υ
n = 0.572175 + 0.009074T b∗ = 0.000050 - 0.000001T –
– 0.064652υ 0.000024υ
Red chillies T = 50–65 4 Kaleemullah c∗ = 0.0084766 b∗ = -0.34775 Kaleemullah and
Kailappan,
2006
m = 0.00004934 n = 1.1912
T = 40–65 υ = 0.12–1.02 2 Lewis k = 0.003484 – 0.000222T + 0.00000366T 2 Hossain et al.,
– 0.007085RH + 0.00572RH 0.002738υ – 2007
0.001235υ 2
RH = 10–60
Red pepper T = 55–70 11 Diffusion Approach a = 1844.324 – 493.320 lnT b = 1.033970exp(-12.2945/Tabs ) Akpinar et al.,
2003c
k = 63319.52exp(-4973.88/Tabs )
Rice (rough) T = 22.3–34.9 RH = — Page k = -0.00209 + 0.000208T + 0.00502υ 2 n = Basunia and Abe,
34.5–57.9% 0.844 + 0.00262T – 0.106υ 2001
T = 5–35 υ = 0.75–2.5 4 Henderson and Pabis a = 18.1578 – 1.49019υ -0.027191T – Iguaz et al., 2003
0.263827RH +0.00453363T υ+
0.000966809TRH + 0.00304256RHυ
RH = 30–70% k = 0.00301414 – 0.000021593T +
0.0000000389067T 2 + 0.00000478υ
Stuffed Pepper T = 50–80 υ = 0.25–1.0 12 Two-term a = 0.6315 – 0.2957υ k1 = 0.0224exp(4.7396υ) Yaldiz and
Ertekin, 2001
b = 0.3679 + 0.2962υ k2 = 0.0677 – 0.0117 lnυ
Wheat (parboiled) T = 40–60 6 Two-term a = 0.03197T – 1.009 k1 = −0.034 Mohapatra and
Rao, 2005
b = -0.032T + 1.9918 k2 = −0.009
Yoghurt (strained) T = 40–50 υ = 1.0–2.0 9 Midilli a=1 k = −0.0005569 + Hayaloglu et al.,
0.00001205T + 0.0002047υ 2007
n = 1.7 b∗ = −0.00003489 -
0.00000038T – 0.00000542υ
 A REVIEW OF THIN LAYER DRYING OF FOODS 455

Table 4 Studies conducted on mathematical modeling of food drying conducted by natural convection in a drying cupboard

Product Process conditions # Best model Effects of process conditions on model constants Reference

Mushroom T = 45◦ C 8 Midilli a = 0.9937 + 0.0003 lnT k = 0.7039 + 0.0002 lnT Midilli et al., 2002
n = 0.8506 + 0.0005 lnT b∗ = –0.0064 – 0.0004 lnT
Pollen a = 0.9975 + 0.0007 lnT k = 1.0638 + 0.0006 lnT
n = 0.5658 + 0.0008 lnT b∗ = –0.0432 – 0.0001 lnT

and described as: widely because of low technology and energy requirements such
that modeling studies conducted on sun drying have preserved
kth = λDeff (45) its importance as shown in Table 2.
th
The most popular thin layer drying method in literature and
industrial applications is hot air drying using convection as the
where, kth is the theoretical value of drying rate constant ob-
main heat transfer mechanism. Generally, heated air is blown
tained from Eq. 44 (s−1 ), (Deff )th is the theoretical effective
to the product and the drying rate is increased with the help of
diffusivity value obtained from Eq. 43 (m2 /s) and λ is the em- the forced convection. The main modeling studies executed with
pirical constant (m−2 ). this method within the last 10 years were compiled and shown in
Table 3. Furthermore, the modeling in a drying cupboard without
STUDIES CONDUCTED ON MODELING OF FOOD the effect of airflow, done for some products, was summarized
DRYING WITH THIN LAYER CONCEPT in Table 4.
The improving effect of electrical heating methods on drying
The considerable volume of work devoted to elucidating a processes, especially microwave and infrared, is strong. These
better understanding of moisture transport in solids is not cov- methods can shorten the drying time, and many modeling studies
ered in depth, and the reason for this is that, in practice, drying- for these processes were performed with the thin layer concept
rate curves have to be measured experimentally, rather than cal- (Table 5).
culated from fundamentals (Baker, 1997). So the experimental Furthermore, various pre-treatments are done to the raw food
studies prevent their importance in drying, especially for food products to facilitate the drying and to improve the product
products, and there have been many studies done in the last 10 quality. These processes affect the drying kinetics directly and
years in literature. The distribution of the studies according to many investigators used the thin layer concept to explain the
the publishing years was summarized in Fig. 2. This graph shows effects of various pre-treatments, especially in fruit drying. The
the increasing interest to the thin layer drying investigations in studies conducted on the effects of pre-treatments to the drying
recent years. kinetics are shown in Table 6.
Process conditions, the product, and the drying method are As mentioned above, the effective moisture diffusivity is
important variables in thin layer drying modeling. The main a useful tool in explaining the drying kinetics, and activation
parameter in this article was chosen as the drying method for
the categorization of the reviewed studies.
MD; 6.9% SD; 8.3%
The oldest method of drying is sun drying. Due to requiring
extensive drying area and long drying time, microbial risks can ICD; 6.9% DC; 1.4%
appear in many products. On the contrary, it has been used

Others; 8.0% ID; 4.2%

Fruits; 36.8%
Medical & FBD; 1.4%
aromatic
plants; 20.7%

Grains; 12.6%

Vegetables;
21.8% CBD; 70.8%

Figure 3 Distribution of the product types used in studies. Figure 4 Distribution of the drying methods used in studies.
456
Table 5 Studies conducted on mathematical modeling of food drying with thin layer concept and performed by electrical methods.

Product DM Process conditions # Best model Effects of process conditions on model constants Reference

Apple (slice) ID T = 50–80◦ C 10 Modified Page eq-II k = –9.08244 + 1.580765 lnT n = 11.49544 – 1.74016 lnT Toǧrul, 2005
l = –0.628792 + 0.574354 lnT
Apple Pomace MD Pm = 150–600 W Untreated 10 Page k = –0.01783 + 0.0001303Pm n = 1.6747 – 0.00728Pm Wang et al., 2007b
Pm = 180–900 W Hot air pre-dried k = 0.02484 + 0.000479Pm n = 0.8704 – 0.00104Pm
ICD T = 55–75◦ C Untreated 10 Logarithmic a = –20.71196 + 0.72489T – 0.00567T 2 c = 21.80075 – 0.72728T + Sun et al., 2007
0.00569T 2
k = 0.16955 – 0.00485T + 0.00003485T 2
T = 55–75◦ C Hot air pre-dried Page k = 0.11269 – 0.0034T + 0.00002615T 2 n = –8.6026 + 0.30111T –
0.00221T 2
Barley ICD I = 0.167–0.5 W/cm2 υ = 0.3–0.7 m/s — Page k = 0.80495 + 7.2839I 2 + 1.4943RH – Afzal and Abe,
1.6662υ – 1.3368Mi 2000
RH = 36–60% Mi = 25–40% n = 0.97857 + 0.7309I + 0.4604RH –
0.41773υ
Carrot ID T = 50–80◦ C 5 Midilli a = 64T −0.716565 n = 0.117979exp(0.006983T ) Toǧrul, 2006
k = 111T −1.67037 b∗ = –0.000051exp(0.004993T )
Olive husk ICD T = 80–140◦ C — Midilli a = 0.96656exp(0.00032696T ) n = 1.87693 – 0.01393T + Celma et al., 2007
0.00004891T 2
k = –0.00234 + 0.00054676lnT b∗ = [–564428.48 + 9055.14T –
37.28T 2 ]−1
Onion ICD I1 = 0.5–1.0 kW/kg υ = 0.1–0.35 m/s 3 Page k = 0.058exp(2.5681I1 + 1.841υ – 0.022L2 Wang, 2002.
– 0.0608RH2
RH = 28.6–43.1% L = 2–6 mm n = 1.3658
I = 2.65–4.42 W/cm2 T = 35–45◦ C 9 Logarithmic a = 0.725 + 0.0415I + 0.00331T + 0.054υ Jain and Pathare,
k = 1.573 – 0.357I – 0.0339T + 0.0555υ 2004
υ = 1.0–1.5 m/s c = 0.00651 – 0.00121I + 0.000223T –
0.00584υ
 A REVIEW OF THIN LAYER DRYING OF FOODS 457

Table 6 Studies conducted on the effect of pretreatment applications on the drying behaviors

Process Best Deff

Product DM conditions Pretreatments # model (m2 /s) Reference

Banana CBD T = 50◦ C Untreated 3 Two-term 4.3E-10 - 13.2E-10 Dandamrongrak et al.,

υ = 3.1 m/s 2002
Blanched
Chilled
Frozen
Blanched & Frozen
Mulberry fruits CBD T = 50◦ C Untreated 6 Logarithmic 2.23E-10 – 6.91E-10 Doymaz, 2004c
(Morus alba L.) υ = 1.0 m/s
Dipped in HW
Dipped in AEEO
Dipped in AA, then
AEEO
Dipped in CA, then
AEEO
Dipped in HW, then
AEEO
Mulberry fruits SD — Untreated 2 Henderson and Pabis 4.26E-11 Doymaz, 2004b
(Morus alba L.)
Dipped in AEEO 4.69E-10

energy is important in describing the sensibility of Deff with The distribution of the drying methods used in the studies
temperature. The values of Deff and Ea calculated by the thin is shown in Fig. 4. This graph displays that the interest of the
layer concept were collected in Table 7. Furthermore, Ea val- investigators to the convective type batch dryers in food drying
ues for microwave drying calculated by the Dadalı model were processes. 70.8% of the studies reviewed have used convec-
shown in Table 8. tive type batch dryers in their experiments. At the same time,
Approximately a hundred articles on the thin layer drying this graph shows the increasing interest of the electrical drying
modeling have been published in the last 10 years. Replicated methods, especially infrared drying. 18% of the reviewed stud-
studies on the same product and method have not been reviewed ies conducted on electrical drying methods and 11.1% of all
in this article, only represented articles were chosen. The results the studies were used in various types of infrared dryers. The
of the representing studies were interpreted and discussed to intensity of the infrared dryers can be explained as the harmony
attain some general approaches in the thin layer drying of foods. of infrared theory and thin layer concept.
Figure 3 shows the distribution of the product types used in Marinos-Kouris and Maroulis (1995) compiled the 37 dif-
the studies. The most widely studied product types are fruits ferent effective moisture diffusivity value intervals that were
(36.8%) and vegetables (21.8%). But the intensity of medical calculated by the experiments. They expressed that the diffusiv-
and aromatic plants is very interesting (20.7%) because they are ities in foods had values in the range 10−13 to 10−6 m2 /s, and
very suitable for thin layer drying. most of them (82%) were accumulated in the region 10−11 to

Number of Products
Number of Products 1 29
1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 1.00E-05
1.00E-05

1.00E-06
1.00E-06

1.00E-07
1.00E-07
Deff (m2/s)

1.00E-08
Deff (m2/s)

1.00E-08

1.00E-09
1.00E-09

1.00E-10
1.00E-10

1.00E-11 1.00E-11

1.00E-12 1.00E-12

1.00E-13 1.00E-13

Figure 5 Distribution of effective moisture diffusivity values compiled from Figure 6 Distribution of effective moisture diffusivity values compiled from
studies. studies in which the experiments were done with convective type batch dryer.
458 Z. ERBAY AND F. ICIER

Table 7 Effective moisture diffusivity and activation energy values calculated by thin layer concept in literature

Product DM Process conditions Deff (m2 /s) Ea (kJ/mol) Reference

Apple (slice) CBD T = 60–80◦ C υ = 1.0–1.5 m/s 8.41E-10 – 20.60E-10 — Akpinar et al., 2003b
ω = 8 × 8 × 18–12.5
× 12.5 × 25 mm
Apple pomace CBD T = 75–105◦ C 2.03E-9 – 3.93E-9 24.51 Wang et al., 2007a
MD Pm = 150–600 W Untreated 1.05E-8 – 3.69E-8 — Wang et al., 2007b
Pm = 180–900 W Hot air pre-dried 2.99E-8 – 9.15E-8
ICD T = 55–75◦ C Untreated 3.48E-9 – 6.48E-9 31.42 Sun et al., 2007
T = 55–75◦ C Hot air pre-dried 4.55E-9 – 8.81E-9 29.76
Apricot CBD T = 50–80◦ C υ = 0.2–1.5 m/s 4.76E-9–8.32E-9 — Toǧrul and Pehlivan,
(SO2 -sulphured) 2003
Bagasse CBD T = 80–120◦ C υ = 0.5–2.0 m/s 1.63E-10 – 3.2E-10 19.47 Vijayaraj et al., 2007
H = 9–24 g/kg L = 20–60 mm
Basil SD — 6.44E-12 — Akpinar, 2006b
Bitter leaves SD — 43.42E-10 — Sobukola et al., 2007
Black Tea CBD T = 80–120◦ C υ = 0.25–0.65 m/s 1.14E-11 – 2.98E-11 406.02 Panchariya et al.,
2002
Carrot (slice) CBD T = 50–70◦ C υ = 0.5–1.0 m/s 7.76E-10 – 93.35E-10 28.36 Doymaz, 2004a
ω = 10 × 10 × 10–20
× 20 × 20 mm
(pretreated)
ID T = 50–80◦ C 7.30E-11 – 15.01E-11 22.43 Toǧrul, 2006
Coconut (Young) CBD T = 50–70◦ C L = 2.5 – 4 mm 1.71E-10 – 5.51E-10 81.11 Madamba, 2003
(Osmotically
pre-dried)
Crain-crain leaves SD — 52.91E–10 — Sobukola et al., 2007
Fever leaves SD — 48.72E–10 —
Grape (Chasselas) CBD T = 50–70◦ C (1 ) 49 Azzouz et al., 2002
Grape (Sultanin) CBD T = 50–70◦ C (2 ) 54
Green bean CBD T = 50–70◦ C 2.64E-9 – 5.71E-9 35.43 Doymaz, 2005
FBD T = 30–50◦ C υ = 0.25 − 1.0m/s — 29.57 – 39.47 Senadeera et al., 2003
RH = 15% LD = 1:1, 2:1, 3:1
Hazelnut CBD T = 100–160◦ C 2.30E-7 – 11.76E-7 34.09 Özdemir and Devres,
1999
T = 100–160◦ C Mi = 12.3 % 3.14E-7 – 30.95E-7 48.70 Özdemir et al., 2000
(moisturized)
T = 100–160◦ C Mi = 6.14 % 3.61E-7 – 21.10E-7 41.25
(untreated)
T = 100–160◦ C Mi = 2.41 % 2.80E-7 – 15.65E-7 36.59
(pre-dried)
Kale CBD T = 30–60◦ C L = 10–50 mm 1.49E-9 – 5.59E-9 36.12 Mwithiga and Olwal,
2005
Kurut CBD T = 35–65◦ C 2.44E-9 – 3.60E-9 19.88 Karabulut et al., 2007
Mint SD - 7.04E-12 - Akpinar, 2006b
CBD T = 30–50◦ C υ = 0.5 − 1.0m/s 9.28E-13 – 11.25E-13 61.91 – 82.93 Park et al., 2002
T = 35–60◦ C υ = 4.1m/s 3.07E-9 – 19.41E-9 62.96 Doymaz, 2006
Mulberry fruits CBD T = 60–80◦ C υ = 1.2m/s 2.32E-10 – 27.60E-10 21.2 Maskan and Göüþ,
(Morus alba L.) 1998
Okra MD Pm = 180–900 W m = 25–100 g 2.05E-9 – 11.91E-9 - Dadalı et al., 2007b
Olive cake CBD T = 50–110◦ C 3.38E-9 - 11.34E-9 17.97 Akgun and Doymaz,
2005
Olive husk ICD T = 80–140◦ C 5.96E-9 – 15.89E-9 21.30 Celma et al., 2007
Paddy (parboiled) CBD T = 70–150◦ C 6.08E-11 - 34.40E-11 21.90 - 23.88 Rao et al., 2007
υ = 0.5–2.0 m/s (3 )
Ld = 50–200 mm
Parsley SD - 4.53E-12 - Akpinar, 2006b
Peach slice CBD T = 55–65◦ C 3.04E-10– 4.41E-10 - Kingsley et al., 2007
(Blanched with %1
KMS or AA)
Peas FBD T = 30–50◦ C - 42.35 – 58.15 Senadeera et al., 2003
υ = 0.25–1.0 m/s
RH = 15%
(Continued on next page)
 A REVIEW OF THIN LAYER DRYING OF FOODS 459

Table 7 Effective moisture diffusivity and activation energy values calculated by thin layer concept in literature (Continued)

Product DM Process conditions Deff (m2 /s) Ea (kJ/mol) Reference

Pestil SD L = 0.71–2.86 mm 1.93E-11 – 9.16E-11 - Maskan et al., 2002

CBD T = 55–75◦ C L = 0.71–2.86 mm 3.00E-11 – 37.6E-11 10.3 – 21.7
Pistachio nuts CBD T = 25–70◦ C 5.42E-11 – 92.9E-11 30.79 Kashaninejad et al.,
2007
Plum (variety: Sutlej CBD T = 55–65◦ C (Untreated) 3.04E-10 – 4.41E-10 - Goyal et al., 2007
purple)
T = 55–65◦ C (Blanched)
T = 55–65◦ C (Blanched with KMS)
Plum (Stanley) CBD T = 60–80◦ C υ = 1.0 − 3.0m/s 1.20E-7 – 4.55E-7 - Menges and Ertekin,
(pretreated) 2006b
T = 60–80◦ C υ = 1.0 − 3.0m/s 1.18E-9 – 6.67E-9
(untreated)
T = 65◦ C υ = 1.2m/s (Dipped 2.40E-10 - Doymaz, 2004d
in AEEO)
T = 65◦ C υ = 1.2m/s 2.17E-10
(untreated)
Potato (slice) FBD T = 30–50◦ C υ = 0.25 − 1.0m/s - 12.32 – 24.27 Senadeera et al., 2003
RH = 15% AR = 1:1, 2:1, 3:1
Red chillies CBD T = 50–65◦ C 3.78E-9 – 7.10E-9 37.76 Kaleemullah and
Kailappan, 2006
Rice (rough) CBD T = 5–35◦ C 5.79E-11 – 17.15E-11 18.50 – 21.04 Iguaz et al., 2003
υ = 0.75–2.5 m/s
RH = 30–70%
Spinach MD Pm = 180–900 W 7.6E-11 – 52.4E-11 - Dadali et al., 2007c
m = 25–100 g .
Tarhana Dough ID T = 60–80◦ C L = 1–6 mm 4.1E-11 – 50.0E-11 41.6 – 49.5 Ibanoǧlu and Maskan,
Untreated 2002
T = 60–80◦ C L = 1–6 mm Cooked 7.7E-11 – 67.0E-11 20.5 – 24.9
Wheat (parboiled) CBD T = 40–60◦ C 1.23E-10 -2.86E-10 37.01 Mohapatra and Rao,
2005
Yoghurt (strained) CBD T = 40–50◦ C υ = 1.0 − 2.0m/s 9.5E-10 – 1.3E-9 26.07 Hayaloglu et al., 2007

eff = D0 exp(-Ea /RTabs )exp(-(dTabs + e)M) Deff = 0.0016exp(-Ea /RTabs )exp(-(0.0012Tabs + 0.309)M)
(1) D

eff = D0 exp(-Ea /RTabs )exp(-(dTabs + e)M) Deff = 0.522exp(-Ea /RTabs )exp(-(0.0075Tabs + 1.829)M)
(2) D

eff = (67.37 + 110.8υ – 14.64Ld + 0.5946T – 4.706υLd + 0.696Ld – 0.0369Ld T )×10–

(3) D 2 12

10−8 m2 /s. In this study, 52 different diffusivity intervals were expressed. The accumulation of the values is in the region 10−10
compiled and shown in Fig. 5. The biggest Deff values were to 10−8 m2 /s (75%).
between 10−5 and 10−6 (product number 23 to 26). The biggest On the other hand, the distribution of Deff values according
4 values gained in hazelnut drying and the drying temperatures to the drying method was plotted. Figure 6 showed the distribu-
of these experiments were between 100–160◦ C. These temper- tion of Deff values collected from the studies reviewed, in which
ature values are too high for food drying, so these values were the experiments were conducted with a convective type batch
not taken into consideration for creating general and appropriate dryer. Disregarding the hazelnut values as mentioned above, the
statistics. Except these values, the effective moisture diffusivity accumulation of Deff values of the foods that were dried in a
values in foods are in the range 10−12 to 10−6 m2 /s and this convective type batch dryer is in the region 10−10 to 10−8 m2 /s
range is more narrow than what Marinos-Kouris and Maroulis (86,2%).
Figure 7 is arranged according to the Deff values obtained
by electrical methods. All values of infrared drying without the
Table 8 Activation energy values calculated by Dadalı model airflow were in the region 10−10 to 10−9 m2 /s (ID). Deff values
for infrared drying systems that contain airflow mechanisms
Product Process conditions Ea (W/g) Reference (ICD) appeared approximately in 10−8 m2 /s level. This showed
Mint Pm = 180–900 W 11.05(2) – 12.28 (1) Özbek and Dadali, 2007 that the drying rate for ICD were faster as expected, because of
Okra m = 25–100 g 5.54(1) Dadalı et al., 2007a the enhancing effect of the airflow. In addition, the microwave
5.70(2) Dadalı et al., 2007b dryer (MD) values were higher than the convective type batch
Spinach 9.62 (2) – 10.84 (1) Dadali et al., 2007c
dryers, and this was harmonious with the theory.
(1) k = k0 exp(-Ea .m/Pm ) During the sun drying experiments (Fig. 8), the ambient tem-
(2) D
eff = D0 exp(-Ea .m/Pm ) perature in Nigeria increased up to 44◦ C, while in Turkey the
460 Z. ERBAY AND F. ICIER

Number of Products maximum temperature value was measured as 36◦ C. Because of

1 2 3 4 5 6 7 8 9 10
1.00E-05 the temperature difference, the values gained in Nigeria (prod-
uct number 3, 4 and 5) were higher than the others, and this
1.00E-06
showed the critical effect of the temperature on Deff .
1.00E-07 Finally, the activation energy values in literature were com-
MD
MD piled and graphed in Fig. 9. In this graph, the black tea value
Deff (m2/s)

1.00E-08
ICD ICD ICD MD was disregarded. Ea of black tea was 406.02 kJ/mol and this
1.00E-09 value is too high according to others. As shown in Fig. 9, all
ID ID
ID
MD other values (41 different products) are in the range of 12.32 to
1.00E-10
82.93 kJ/mol. The accumulation of the values was in the range
1.00E-11 of 18 to 49.5 kJ/mol (80.5%).
1.00E-12

1.00E-13 CONCLUSIONS
Figure 7 Distribution of effective moisture diffusivity values compiled from
studies in which the experiments were done by electrical methods. In this study, the most commonly used or newly developed
thin layer drying models were shown, the determination meth-
ods of the appropriate model were explained, Deff and Ea cal-
culations were expressed, and experimental studies performed
within the last 10 years were reviewed and discussed.
Number of Products
1 2 3 4 5 6 7 8 9
The main conclusions, which may be drawn from the results of
1.00E-05 the present study, were listed below:
1.00E-06

a. Although there are lots of studies conducted on fruits, veg-

1.00E-07
etables, and grains, there is insufficient data in drying of
Deff (m2/s)

1.00E-08 other types of foods, for example meat and fish drying.
b. The effective moisture diffusivity values in foods were in
1.00E-09
the range of 10−12 to 10−6 m2 /s and the accumulation of
1.00E-10 the values was in the region 10−10 to 10−8 m2 /s (75%).
1.00E-11
In addition, 86.2% of Deff values of the foods dried in a
convective type batch dryer were in the region 10−10 to 10−8
1.00E-12
m2 /s.
1.00E-13 c. The studies showed that electrical drying methods were faster
than the others.
Figure 8 Distribution of effective moisture diffusivity values compiled from
studies in which the experiments were done by sun drying. d. The effect of temperature on Deff was critical.
e. The activation energy values of foods were in the range of
12.32 to 82.93 kJ/mol and 80.5% of the values were in the
region 18 to 49.5 kJ/mol.
90.00

80.00
ACKNOWLEDGEMENT
70.00

60.00 This study is a part of the MSc. Thesis titled “The investiga-
Ea (kJ/mol)

50.00 tion of modeling, optimization, and exergetic analysis of drying

40.00
of olive leaves,” and supported by Ege University Scientific
Research Project no. of 2007/MÜH/30.
30.00

20.00

10.00 NOMENCLATURE
0.00
0 7 14 21 28 35 42
a empirical model constant (dimensionless)
a∗ empirical constant (s−2 )
Number of Products

Figure 9 Distribution of activation energy values compiled from studies. a1 geometric parameter in Eqs. 5, 6
 A REVIEW OF THIN LAYER DRYING OF FOODS 461

A1 , A2 geometric constants R universal gas constant (kJ/kmol.K)

AR aspect ratio (dimensionless) RH relative humidity (%)
b empirical model constant (dimensionless) RMSE root mean square error
b∗ empirical constant (s−1 ) RSC rotary speed column (rpm)
c empirical model constant (dimensionless) T temperature (o C)
c∗ empirical constant (o C−1 s−1 ) Tabs absolute temperature (K)
d empirical constant (K−1 ) t time (s)
e empirical constant (dimensionless) x diffusion path (m)
Deff effective moisture diffusivity (m2 /s) χ2 reduced chi-square
(Deff )th theoretical value of effective moisture diffusiv- υ. velocity (m/s)
ity (m2 /s) V volumetric flow rate (m3 /s)
D0 Arrhenius factor (m2 /s) ω dimensions (mm)
Ea activation energy for diffusion (kJ/mol) or (W/g) α thermal diffusivity (m2 s)
in Eqs. 43,44 λ empirical constant defines relationship between
g drying constant obtained from experimental Deff and Ea (m−2 )
data (s−1 ) φ characteristic moisture content (dimensionless)
h drying constant obtained from experimental # number of models tested
data (s−1 )
H humidity (g water / kg dry air) Abbreviations
i number of terms of the infinite series
I radiation intensity (W/cm2 ) AA ascorbic acid solution
J0 roots of Bessel function AEEO alkali emulsion of ethyl oleate
k, k1 , k2 drying constants obtained from experimental CA citric acid solution
data (s−1 ) CBD convective type batch dryer
k0 pre-exponential constant (s−1 ) DC drying cupboard
kth theoretical value of drying constant (s−1 ) DM drying method
K drying constant (s−1 ) FBD fluid bed dryer
K11 , K22 , K33 phenomenological coefficients in Eqs. 1–4 HW hot water
K12 , K13 , K21 , coupling coefficients in Eqs. 1, 2 ICD infrared convective dryer (with airflow)
K23 , K31 , K32 ID infrared dryer (without airflow)
l empirical constant (dimensionless) MD microwave dryer
L thickness of the diffusion path (m); slice thick- SD sun drying
ness (mm) in Tables 3,5,7
L1 , L2 , L3 dimensions of finite slab (m)
Ld grain depth (mm)
LD length per diameter (dimensionless) REFERENCES
m sample amount (g)
Afzal, T.M. and Abe, T. (2000). Simulation of moisture changes in barley dur-
M local moisture content (kg water/kg dry matter) ing far infrared radiation drying. Computers and Electronics in Agriculture.
or (% dry basis) 26:137–145.
Mcr critical moisture content (% dry basis) Akgun, N.A. and Doymaz, I. (2005). Modelling of olive cake thin-layer drying
Me equilibrium moisture content (% dry basis) process. Journal of Food Engineering. 68:455–461.
Mi initial moisture content (% dry basis) Akpinar, E.K. (2006a). Determination of suitable thin layer drying curve
model for some vegetables and fruits. Journal of Food Engineering. 73:75–
Mt mean moisture content at time t (% dry basis) 84.
MR fractional moisture ratio (dimensionless) Akpinar, E.K. (2006b). Mathematical modelling of thin layer drying process un-
MRexp,i ith experimental moisture ratio (dimensionless) der open sun of some aromatic plants. Journal of Food Engineering. 77:864–
MRpre,i ith predicted moisture ratio (dimensionless) 870.
n empirical model constant (dimensionless); Akpinar, E., Midilli, A. and Bicer, Y. (2003a). Single layer drying behaviour
of potato slices in a convective cyclone dryer and mathematical modeling.
number of constants in Eq. 37 Energy Conversion and Management. 44:1689–1705.
N number of observations Akpinar, E.K., Bicer, Y., and Midilli, A. (2003b). Modeling and experimental
Nw drying rate (kg/m2 s) study on drying of apple slices in a convective cyclone dryer. Journal of Food
p empirical constant (o C−1 ) Process Engineering. 26:515–541.
P pressure (kPa) Akpinar, E.K., Bicer, Y. and Yildiz, C. (2003c). Research note: Thin layer drying
of red pepper. Journal of Food Engineering. 59:99–104.
Pm microwave output power (W) Akpinar, E.K., Sarsilmaz, C., and Yildiz, C. (2004). Mathematical modelling of
Q heat transfer rate (W) a thin layer drying of apricots in a solar energized rotary dryer. International
r correlation coefficient; radius (m) in Table 1 Journal of Energy Research. 28:739–752.
462 Z. ERBAY AND F. ICIER

Akpinar, E.K., Bicer, Y., and Cetinkaya, F. (2006). Modelling of thin layer Erenturk, S., and Erenturk, K. (2007). Comparison of genetic algorithm and
drying of parsley leaves in a convective dryer and under open sun. Journal of neural network approaches for the drying process of carrot. Journal of Food
Food Engineering. 75:308–315. Engineering. 78:905–912.
Azzouz, S., Guizani, A., Jomaa, W., and Belghith, A. (2002). Moisture diffu- Erenturk, K., Erenturk, S. and Tabil, L.G. (2004). A comparative study for the
sivity and drying kinetic equation of convective drying of grapes. Journal of estimation of dynamical drying behavior of Echinacea angustifolia: regres-
Food Engineering. 55:323–330. sion analysis and neural network. Computers and Electronics in Agriculture.
Baker, C.G.J. (1997). Preface. In: Industrial Drying of Foods. Baker, C.G.J. 45:71–90.
Eds., Chapman & Hall, London. Ertekin, C., and Yaldiz, O. (2004). Drying of eggplant and selection of a
Bakshi, A.S., and Singh, R.P. (1980). Drying Characteristics of parboiled rice. suitable thin layer drying model. Journal of Food Engineering. 63:349–
In: Drying’80, Mujumdar, A.S. Eds., Hemisphere Publishing Company, 359.
Washington DC. Fortes, M., and Okos, M.R. (1981). Non-equilibrium thermodynamics approach
Basunia, M.A., and Abe, T. (2001). Thin-layer solar drying characteristics of to heat and mass transfer in corn kernels. Trans. ASAE. 22:761–769.
rough rice under natural convection. Journal of Food Engineering. 47:295– Ghazanfari, A., Emami, S., Tabil, L.G., and Panigrahi, S. (2006). Thin-layer
301. drying of flax fiber: II.Modeling drying process using semi-theoretical and
Brooker, D.B., Bakker-Arkema, F.W., and Hall, C.W. (1974). Drying Cereal empirical models. Drying Technology. 24:1637–1642.
Grains. The AVI Publishing Company Inc., Westport, Connecticut. Glenn, T.L. (1978). Dynamic analysis of grain drying system. Ph.D. Thesis,
Bruce, D.M. (1985). Exposed-layer barley drying, three models fitted to new Ohio State University, Ann Arbor, MI (unpublished).
data up to 150◦ C. Journal of Agricultural Engineering Research. 32:337–347. Goyal, R.K., Kingsly, A.R.P., Manikantan, M.R., and Ilyas, S.M. (2007). Math-
Bruin, S., and Luyben, K. (1980). Drying of Food Materials. In: Advances in ematical modelling of thin layer drying kinetics of plum in a tunnel dryer.
Drying. pp. 155–215, Mujumdar, A.S. Eds., McGraw-Hill Book Co., New Journal of Food Engineering. 79:176–180.
York. Guarte, R.C. (1996). Modelling the drying behaviour of copra and development
Carbonell, J.V., Pinaga, F., Yusa, V., and Pena, J.L. (1986). Dehydration of of a natural convection dryer for production of high quality copra in the
paprika and kinetics of color degradation. Journal of Food Engineering. Philippines. Ph.D.Dissertation, 287, Hohenheim University, Stuttgart, Ger-
5:179–193. many.
Celma, A.R., Rojas, S., and Lopez-Rodriguez, F. (2007). Mathematical mod- Gunhan, T., Demir, V., Hancioglu, E., and Hepbasli, A. (2005). Mathematical
elling of thin-layer infrared drying of wet olive husk. Chemical Engineering modelling of drying of bay leaves. Energy Conversion and Management.
and Processing. (article in press). 46:1667–1679.
Chandra, P.K. and Singh, R.P. (1995). Applied Numerical Methods for Food Hassan, B.H., and Hobani, A.I. (2000). Thin-layer drying of dates. Journal of
and Agricultural Engineers. pp. 163–167. CRC Press, Boca Raton, FL. Food Process Engineering. 23:177–189.
Crank, J. (1975). The Mathematics of Diffusion. 2nd Edition, Oxford University Hayaloglu, A.A., Karabulut, I., Alpaslan, M., and Kelbaliyev, G. (2007). Mathe-
Press, England. matical modeling of drying characteristics of strained yoghurt in a convective
Crisp, J. and Woods, J.L. (1994). The drying properties of rapeseed. Journal of type tray-dryer. Journal of Food Engineering. 78:109–117.
Agricultural Engineering Research. 57:89–97. Henderson, S.M., and Pabis, S. (1961). Grain drying theory I: Temperature
Dadalı, G., Kılıç, D., and Özbek, B. (2007a). Microwave drying kinetics of effect on drying coefficient. Journal of Agricultural Engineering Research.
okra. Drying Technology. 25:917–924. 6:169–174.
Dadalı, G., Kılıç Apar, D., and Özbek, B. (2007b). Estimation of effective mois- Henderson, S.M. (1974). Progress in developing the thin layer drying equation.
ture diffusivity of okra for microwave drying. Drying Technology. 25:1445– Trans. ASAE. 17:1167–1172.
1450. Hossain, M.A., and Bala, B.K. (2002). Thin-layer drying characteristics for
Dadali, G., Demirhan, E., and Özbek, B. (2007c). Microwave heat treatment of green chilli. Drying Technology. 20:489–502.
spinach: drying kinetics and effective moisture diffusivity. Drying Technol- Hossain, M.A., Woods, J.L., and Bala, B.K. (2007). Single-layer drying char-
ogy. 25:1703–1712. acteristics and colour kinetics of red chilli. International Journal of Food
Dandamrongrak, R., Young, G., and Mason, R. (2002). Evaluation of various Science and Technology. 42:1367–1375.
pre-treatments for the dehydration of banana and selection of suitable drying Husain, A., Chen, C.S., Clayton, J.T., and Whitney, L.F. (1972). Mathematical
models. Journal of Food Engineering. 55:139–146. simulation of mass and heat transfer in high moisture foods. Trans. ASAE.
Demir, V., Gunhan, T. and Yagcioglu, A.K. (2007). Mathematical modelling 15:732–736.
of convection drying of green table olives. Biosystems Engineering. 98:47– Iguaz, A., San Martin, M.B., Mate, J.I., Fernandez, T., and Virseda, P. (2003).
53. Modelling effective moisture diffusivity of rough rice (Lido cultivar) at low
Diamante, L.M., and Munro, P.A. (1993). Mathematical modelling of the thin drying temperatures. Journal of Food Engineering. 59:253–258.
layer solar
. drying of sweet potato slices. Solar Energy. 51:271–276. Islam M.N. and Flink J.M. (1982). Dehydration of potato: I.Air and solar drying
Doymaz, I. (2004a). Convective air drying characteristics of thin layer carrots. at low air velocities. J.Food Technol. 17:373–385.
Journal. of Food Engineering. 61:359–364. Ýbanolu, Þ., and Maskan, M. (2002). Effect of cooking on the drying behaviour
Doymaz, I. (2004b). Pretreatment effect on sun drying of mulberry fruits (Morus of tarhana dough, a wheat flour-yoghurt mixture. Journal of Food Engineer-
alba L.). Journal of Food Engineering. 65:205–209. ing. 54:119–123.
.
Doymaz, I. (2004c). Drying kinetics of white mulberry. Journal of Food Engi- Jain, D., and Pathare, B. (2004). Selection and evaluation of thin layer drying
neering. 61:341–346. models for infrared radiative and convective drying of onion slices. Biosystems
.
Doymaz, I. (2004d). Effect of dipping treatment on air drying of plums. Journal Engineering. 89:289–296.
of Food Engineering. 64:465–470. Jayas, D.S. Cenkowski, S., Pabis, S., and Muir, W.E. (1991). Review of thin-
.
Doymaz, I. (2005). Drying behaviour of green beans. Journal of Food Engi- layer drying and wetting equations. Drying Technology. 9:551–588.
neering. 69:161–165. Kaleemullah, S. (2002). Studies on engineering properties and drying kinetics
.
Doymaz, I. (2006). Thin-layer drying behaviour of mint leaves. Journal of Food of chillies. Ph.D.Thesis, Department of Agricultural Processing, Tamil Nadu
Engineering. 74:370–375. Agricultural University: Coimbatore, India.
Ece M.C. and Cihan A. (1993). A liquid diffusion model for drying rough rice. Kaleemullah, S., and Kailappan, R. (2006). Modelling of thin-layer drying
Trans. ASAE. 36:837–840. kinetics of red chillies. Journal of Food Engineering. 76:531–537.
Ekechukwu, O.V. (1999). Review of solar-energy drying systems I: an overview Karabulut, I., Hayaloglu, A.A., and Yildirim, H. (2007). Thin-layer drying char-
of drying principles and theory. Energy Conversion & Management. 40:593– acteristics of kurut, a Turkish dried dairy by-product. International Journal
613. of Food Science and Technology. 42:1080–1086.
 A REVIEW OF THIN LAYER DRYING OF FOODS 463

Karathanos, V.T. (1999). Determination of water content of dried fruits by drying Overhults, D.G., White, G.M., Hamilton, H.E., and Ross, I.J. (1973). Drying
kinetics. Journal of Food Engineering. 39:337–344. soybeans with heated air. Trans. ASAE. 16:112–113.
Kashaninejad, M., Mortazavi, A., Safekordi, A., and Tabil, L.G. (2007). Thin- Özbek, B., and Dadali, G. (2007). Thin-layer drying characteristics and mod-
layer drying characteristics and modeling of pistachio nuts. Journal of Food elling of mint leaves undergoing microwave treatment. Journal of Food En-
Engineering. 78:98–108. gineering. 83:541–549.
Kaseem, A.S. (1998). Comparative studies on thin layer drying models for Özdemir, M., and Devres, Y.O. (1999). The thin layer drying characteristics of
wheat. In 13th International Congress on Agricultural Engineering, Vol. 6:2– hazelnuts during roasting. Journal of Food Engineering. 42:225–233.
6. February, Morocco. Özdemir, M., Seyhan, F.G., Bodur, A.Ö., and Devres, O. (2000). Effect of initial
Kaymak-Ertekin, F. (2002). Drying and rehydrating kinetics of green and red moisture content on the thin layer drying characteristics of hazelnuts during
peppers. Journal of Food Science. 67:168–175. roasting. Drying Technology. 18:1465–1479.
Keey, R.B. (1972). Introduction. In: Drying Principles and Practice. pp. 1–18. Özilgen, M., and Özdemir, M. (2001). A review on grain and nut deterioration
Keey, R.B. Eds., Pergamon Press, Oxford. and design of the dryers for safe storage with special reference to Turkish
Kingsley, R.P., Goyal, R.K., Manikantan, M.R., and Ilyas, S.M. (2007). Effects hazelnuts. Critical Reviews in Food Science and Nutrition. 41:95–132.
of pretreatments and drying air temperature on drying behaviour of peach Page, G.E. (1949). Factors ınfluencing the maximum rate of air drying shelled
slice. International Journal of Food Science and Technology. 42:65–69. corn in thin-layers. M.S.Thesis, Purdue University, West Lafayette, Indiana.
Kudra, T., and Mujumdar, A.S. (2002). Part I. General Discussion: Conventional Panchariya, P.C., Popovic, D., and Sharma, A.L. (2002). Thin-layer modelling
and Novel Drying Concepts. In: Advanced Drying Technologies. pp. 1–26. of black tea drying process. Journal of Food Engineering. 52:349–357.
Kudra, T., and Mujumdar, A.S. Eds., Marcel Dekker Inc., New York. Pangavhane, D.R., Sawhney, R.L., and Sarsavadia, P.N. (2000). Drying kinetic
Lahsasni, S., Kouhila, M., Mahrouz, M., and Jaouhari, J.T. (2004). Drying studies on single layer Thompson seedless grapes under controlled heated air
kinetics of prickly pear fruit. Journal of Food Engineering. 61:173–179. conditions. Journal of Food Processing and Preservation. 24:335–352.
Lewis, W.K. (1921). The rate of drying of solid materials. I&EC-Symposium of Park, K.J., Vohnikova, Z., and Brod, F.P.R. (2002). Evaluation of drying pa-
Drying, 3(5):42. rameters and desorption isotherms of garden mint leaves (Mentha crispa L.).
Luikov, A.V. (1975). Systems of differential equations of heat and mass transfer Journal of Food Engineering. 51:193–199.
in capillary-porous bodies (review). International Journal of Heat and Mass Parry, J.L. (1985). Mathematical modeling and computer simulation of heat and
Transfer. 18:1–14. mass transfer in agricultural grain drying. Journal of Agricultural Engineering
Madamba, P.S. (2003). Thin layer drying models for osmotically pre-dried Research. 54:339–352.
young coconut. Drying Technology. 21:1759–1780. Parti, M. (1993). Selection of mathematical models for drying in thin layers.
Madamba, P.S., Driscoll, R.H., and Buckle, K.A. (1996). Thin-layer drying Journal of Agricultural Engineering Research. 54:339–352.
characteristics of garlic slices. Journal of Food Engineering. 29:75–97. Paulsen, M.R., and Thompson, T.L. (1973). Drying endysus of grain sorghum.
Marinos-Kouris, D., and Maroulis, Z.B. (1995). Transport Properties in the Trans.ASAE. 16:537–540.
Drying of Solids. In: Handbook of Industrial Drying. pp. 113–160. Mujumdar, Pinaga, F., Carbonell, J.V., Pena J.L., and Miguel, I.J. (1984). Experimental
A.S. Eds., 2nd Edition, Marcel Dekker Inc., New York. simulation of solar drying of garlic slices using adsorbent energy storage bed.
Maskan, M., and Göüþ, F. (1998). Sorption isotherms and drying characteristics Journal of Food Engineering. 3:187–203.
of mulberry (Morus alba). Journal of Food Engineering. 37:437–449. Rao, P.S., Bal, S., and Goswami, T.K. (2007). Modelling and optimization of
Maskan, A., Kaya, S., and Maskan, M. (2002). Hot air and sun drying of grape drying variables in thin layer drying of parboiled paddy. Journal of Food
leather (pestil). Journal of Food Engineering. 54:81–88. Engineering. 78:480–487.
Mazza, G., and Le Maguer, M. (1980). Dehydration of onion: Some theoretical Sawhney, R.L., Pangavhane, D.R., and Sarsavadia, P.N. (1999a). Drying kinetics
and practical considerations. Journal of Food Technology. 15:181–194. of single layer Thompson seedless grapes under heated ambient air conditions.
Menges, H.O., and Ertekin, C. (2006a). Mathematical modeling of thin layer Drying Technology. 17:215–236.
drying of Golden apples. Journal of Food Engineering. 77:119–125. Sawhney, R.L., Sarsavadia, P.N., Pangavhane, D.R., and Singh, S.P. (1999b).
Menges, H.O., and Ertekin, C. (2006b). Thin layer drying model for treated and Determination of drying constants and their dependence on drying air param-
untreated Stanley plums. Energy Conversion and Management. 47:2337– eters for thin layer onion drying. Drying Technology. 17:299–315.
2348. Senadeera, W., Bhandari, B.R., Young, G., and Wijesinghe, B. (2003). Influence
Midilli, A., and Kucuk, H. (2003). Mathematical modelling of thin layer drying of shapes of selected vegetable materials on drying kinetics during fluidized
of pistachio by using solar energy. Energy Conversion and Management. bed drying. Journal of Food Engineering. 58:277–283.
44:1111–1122. Sharaf-Eldeen, Y. I., Blaisdell, J.L., and Hamdy, M.Y. (1980). A model for ear
Midilli, A., Kucuk, H., and Yapar, Z. (2002). A new model for single-layer corn drying. Transaction of the ASAE. 23:1261–1271.
drying. Drying Technology. 20:1503–1513. Sobukola, O.P., Dairo, O.U., Sanni, L.O., Odunewu, A.V. and Fafiolu, B.O.
Mohamed, L.A., Kouhila, M., Jamali, A., Lahsasni, S., Kechaou, N., and (2007). Thin layer drying process of some leafy vegetables under open sun.
Mahrouz, M. (2005). Single layer solar drying behaviour of Citrus auran- Food Science and Technology International. 13:35–40.
tium leaves under forced convection. Energy Conversion and Management. Steffe, J.F., and Singh, R.P. (1982). Diffusion coefficients for predicting rice
46:1473–1483. drying behavior. Journal of Agricultural Engineering Research. 27:189–193.
Mohapatra, D. and Rao, P.S. (2005). A thin layer drying model of parboiled Strumillo, C., and Kudra, T. (1986). Heat and Mass Transfer in Drying Processes.
wheat. Journal of Food Engineering. 66:513–518. In: Drying: Principles, Applications and Design. Gordon and Breach Science
Mujumdar, A.S. (1997). Drying Fundamentals. In: Industrial Drying of Foods. Publishers, Montreux.
pp. 7–30. Baker, C.G.J. Eds., Chapman & Hall, London. Suarez, C., Viollaz, P., and Chirife, J. (1980). Kinetics of Soybean Drying.
Mujumdar, A.S., and Menon, A.S. (1995). Drying of Solids: Principles, Clas- In: Drying’80. pp. 251–255. Mujumdar, A.S. Eds. Hemisphere Publishing
sification, and Selection of Dryers. In: Handbook of Industrial Drying. pp. Company, Washington DC.
1–40. Mujumdar, A.S. Eds., 2nd Edition, Marcel Dekker Inc., New York. Sun, J., Hu, X., Zhao, G., Wu, J., Wang, Z., Chen, F., and Liao, X. (2007).
Mulet, A., Berna, A., Borras, M., and Pinaga, F. (1987). Effect of air flow rate Characteristics of thin-layer infrared drying of apple pomace with and without
on carrot drying. Drying Technology, 5(2):245–258. hot air pre-drying. Food Sci Tech Int. 13:91–97.
Mwithiga, G., and Olwal, J.O. (2005). The drying kinetics of kale (Brassica Thompson, T.L., Peart, P.M., and Foster, G.H. (1968). Mathematical simulation
oleracea) in a convective hot air dryer. Journal of Food Engineering. 71:373– of corn drying: A new model. Trans. ASAE. 11:582–586.
378. Toǧrul, H. (2005). Simple modeling of infrared drying of fresh apple slices.
O’Callaghan, J.R., Menzies, D.J., and Bailey, P.H. (1971). Digital simulation of Journal of Food Engineering. 71:311–323.
agricultural dryer performance. Journal of Agricultural Engineering Researh. Toǧrul, H. (2006). Suitable drying model for infrared drying of carrot. Journal
16:223–244. of Food Engineering. 77:610–619.
464 Z. ERBAY AND F. ICIER

Toǧrul, Ý.T., and Pehivan, D. (2002). Modelling of drying kinetics of single and without hot air pre-drying. Journal of Food Engineering. 80:536–
apricot. Journal of Food Engineering. 58:23–32. 544.
Toǧrul, Ý.T., and Pehlivan, D. (2003). Modelling of drying kinetics of single Whitaker, T., Barre, H.J., and Hamdy, M.Y. (1969). Theoretical and experimental
apricot. Journal of Food Engineering. 58:23–32. studies of diffusion in spherical bodies with variable diffusion coefficient.
Toǧrul, Ý.T., and Pehlivan, D. (2004). Modelling of thin layer drying kinetics of Trans. ASAE. 11:668–672.
some fruits under open-air sun drying process. Journal of Food Engineering. White, G.M., Bridges, T.C., Loewer, O.J., and Ross, I.J. (1978). Seed coat
65:413–425. damage in thin layer drying of soybeans as affected by drying conditions.
Treybal, R.E. (1968). Mass Transfer Operations, 2nd Edition, McGraw Hill, ASAE paper no. 3052.
New York. Xanthopoulos, G., Oikonomou, N., and Lambrinos, G. (2007). Applicability of
Verma, L.R., Bucklin, R.A, Ednan, J.B., and Wratten, F.T. (1985). Effects of dry- a single-layer drying model to predict the drying rate of whole figs. Journal
ing air parameters on rice drying models. Transaction of the ASAE. 28:296– of Food Engineering. 81:553–559.
301. Yagcioglu, A., Degirmencioglu, A., and Cagatay, F. (1999). Drying charac-
Vijayaraj, B., Saravanan, R., and Renganarayanan, S. (2007). Studies on thin teristics of laurel leaves under different conditions. Proceedings of the 7th
layer drying of bagasse. International Journal of Energy Research. 31:422– international congress on agricultural mechanization and energy, ICAME’99,
437. pp. 565–569, Adana, Turkey.
Wang, J. (2002). A single-layer model for far-infrared radiation drying of onion Yaldiz, O., and Ertekin, C. (2001). Thin layer solar drying of some vegetables.
slices. Drying Technology. 20:1941–1953. Drying Technology. 19:583–597.
Wang, C.Y., and Singh, R.P. (1978). A single layer drying equation for rough Yaldız, O., Ertekin, C., and Uzun, H.I. (2001). Mathematical modelling of thin
rice. ASAE Paper No. 3001. layer solar drying of sultana grapes. Energy. 26:457–465.
Wang, Z., Sun, J., Liao, X., Chen, F., Zhao, G., Wu, J., and Hu, X. (2007a). Yilbas, B.S., Hussain, M.M., and Dincer, I. (2003). Heat and moisture diffusion
Mathematical modeling on hot air drying of thin layer apple pomace. Food in slab products to convective boundary conditions. Heat and Mass Transfer.
Research International. 40:39–46. 39:471–476.
Wang, Z., Sun, J., Chan, F., Liao, X., and Hu, X. (2007b). Mathemat- Young, J.H. (1969). Simultaneous heat and mass transfer in a porous solid
ical modelling on thin layer microwave drying of apple pomace with hygroscopic solids. Trans. ASAE. 11:720–725.