a review of thin layer drying of foods theory
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A Review of Thin Layer Drying of Foods: Theory,Modeling, and Experimental ResultsZafer Erbay a & Filiz Icier ba Graduate School of Natural and Applied Sciences, Food Engineering Branch, Ege University,35100, Izmir, Turkeyb Department of Food Engineering, Faculty of Engineering, Ege University, 35100, Izmir,TurkeyPublished online: 05 Apr 2010.
To cite this article: Zafer Erbay & Filiz Icier (2010) A Review of Thin Layer Drying of Foods: Theory, Modeling, andExperimental 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, LLCISSN: 1040-8398DOI: 10.1080/10408390802437063
A Review of Thin Layer Dryingof Foods: Theory, Modeling,and Experimental Results
ZAFER ERBAY1 and FILIZ ICIER2
1Graduate School of Natural and Applied Sciences, Food Engineering Branch, Ege University, 35100 Izmir, Turkey2Department 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 complexbecause of the differential structure of products. In practice, a food dryer is considerably more complex than a devicethat merely removes moisture, and effective models are necessary for process design, optimization, energy integration, andcontrol. Although modeling studies in food drying are important, there is no theoretical model which neither is practical norcan it unify the calculations. Therefore the experimental studies prevent their importance in drying and thin layer dryingequations 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 appropriatemodel 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 thiscomprehensive study will be beneficial to those involved or interested in modeling, design, optimization, and analysis of fooddrying.
Keywords food drying, thin layer, mathematical modeling, diffusivity, activation energy
INTRODUCTION
Drying is traditionally defined as the unit operation that con-verts a liquid, solid, or semi-solid feed material into a solid prod-uct of significantly lower moisture content. In most cases, dryinginvolves the application of thermal energy, which causes waterto evaporate into the vapor phase. Freeze-drying provides an ex-ception to this definition, since this process is carried out belowthe triple point, and water vapor is formed directly through thesublimation of ice. The requirements of thermal energy, phasechange, and a solid final product distinguish drying from me-chanical dewatering, evaporation, extractive distillation, adsorp-tion, and osmotic dewatering (Keey, 1972; Mujumdar, 1997).
Drying is one of the oldest unit operation, and widespreadin various industries recently. It is used in the food, agricul-tural, ceramic, chemical, pharmaceutical, pulp and paper, min-eral, polymer, and textile industries to gain different utilities.
Address correspondence to: Zafer Erbay, Graduate School of Natural andApplied Sciences, Food Engineering Branch, Ege University, 35100 Izmir,Turkey. Tel:+90 232 388 4000 (ext.3010) Fax: +90 232 3427592. E-mail:[email protected]
The methods of drying are diversified with the purpose of theprocess. There are more than 200 types of dryers (Mujumdar,1997). For every dryer, the process conditions, such as the dry-ing chamber temperature, pressure, air velocity (if the carriergas is air), relative humidity, and the product retention time,have to be determined according to feed, product, purpose, andmethod. On the other hand, drying is an energy-intensive pro-cess and its energy consumption value is 10–15% of the totalenergy consumption in all industries in developed countries(Keey, 1972; Mujumdar, 1997). It is a very important processaccording to the main problems in the whole world such as thedepletion of fossil fuels and environmental pollution. In brief,drying is arguably the oldest, most common, most diverse, andmost energy-intensive unit operation and because of all thesefeatures, the engineering in drying processes gains importance.
In the food industry, foods are dried, starting from their nat-ural form (vegetables, fruits, grains, spices, milk) or after han-dling (e.g. instant coffee, soup mixes, whey). The productionof a processed food may involve more than one drying processat different stages and in some cases, pre-treatment of food isnecessary before drying. In the food industry, the main purpose
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442 Z. ERBAY AND F. ICIER
of drying is to preserve and extend the shelf life of the product.In addition to this, in the food industry, drying is used to obtaina desired physical form (e.g. powder, flakes, granules); to obtainthe desired color, flavor, or texture; to reduce the volume or theweight for transportation; and to produce new products whichwould not otherwise be feasible (Mujumdar, 1997).
Drying is one of the most complex and least understoodprocesses at the microscopic level, because of the difficultiesand deficiencies in mathematical descriptions. It involves si-multaneous and often coupled and multiphase, heat, mass, andmomentum transfer phenomena (Kudra and Mujumdar, 2002;Yilbas et al., 2003). In addition, the drying of food materialsis further complicated by the fact that physical, chemical, andbiochemical transformations may occur during drying, some ofwhich may be desirable. Physical changes such as glass transi-tions or crystallization during drying can result in changes in themechanisms of mass transfer and rates of heat transfer within thematerial, often in an unpredictable manner (Mujumdar, 1997).The underlying chemistry and physics of food drying are highlycomplicated, so in practice, a dryer is considerably more com-plex than a device that merely removes moisture, and effectivemodels are necessary for process design, optimization, energyintegration, and control. Although many research studies havebeen done about mathematical modeling of drying, undoubt-edly, the observed progress has limited empiricism to a largeextent and there is no theoretical model that is practical and canunify the calculations (Marinos-Kouris and Maroulis, 1995).
Thin layer drying equations are important tools in mathemat-ical modeling of drying. They are practical and give sufficientlygood results. To use thin layer drying equations, the drying-ratecurves have to be known. However, the considerable volumeof work devoted to elucidate the better understanding of mois-ture transport in solids is not covered in depth, in practice,drying-rate curves have to be measured experimentally, ratherthan calculated from fundamentals (Baker, 1997). So the ex-perimental studies prevent their importance in drying. There isno review done about the experimental results of the thin layerdrying experiments of foods and mathematical models in thinlayer drying in open literature for more than 10 years. Jayas etal. (1991) have written the last review according to the authors’knowledge. In this study, the fundamentals of thin layer dryingwere explained, and commonly used or newly developed semi-theoretical and empirical models in the literature were shown.In addition, the experimental results gained in the last 10 yearsfor food materials were summarized and discussed.
THE THEORY AND MATHEMATICAL MODELINGOF FOOD DRYING
Mechanisms of Drying
The main mechanisms of drying are surface diffusion orliquid diffusion on the pore surfaces, liquid or vapor diffusiondue to moisture concentration differences, and capillary action
in granular and porous foods due to surface forces. In additionto these, thermal diffusion that is defined as water flow causedby the vaporization-condensation sequence, and hydrodynamicflow that is defined as water flow caused by the shrinkage andthe pressure gradient may also be seen in drying (Strumilloand Kudra, 1986; Ozilgen and Ozdemir, 2001). The dominantdiffusion mechanism is a function of the moisture content andthe structure of the food material and it determines the dryingrate. The dominant mechanism can change during the processand, the determination of the dominant mechanism of drying isimportant in modeling the process.
For hygroscopic products, generally the product dries in con-stant rate and subsequent falling rate periods and it stops whenan equilibrium is established. In the constant rate period of dry-ing, external conditions such as temperature, drying air velocity,direction of air flow, relative humidity of the medium, physicalform of product, the desirability of agitation, and the method ofsupporting the product during drying are essential and the dom-inant diffusion mechanism is the surface diffusion. Toward theend of the constant rate period, moisture has to be transportedfrom the inside of the solid to the surface by capillary forcesand the drying rate may still be constant until the moisture con-tent has reached the critical moisture content and the surfacefilm of the moisture has been so reduced with the appearanceof dry spots on the surface. Then the first falling rate periodor unsaturated surface drying begins. Since, however, the rateis computed with respect to the overall solid surface area, thedrying rate falls even though the rate per unit wet solid sur-face area remains constant (Mujumdar and Menon, 1995). Inthis drying period, the dominant diffusion mechanism is liquiddiffusion due to moisture concentration difference and internalconditions such as the moisture content, the temperature, andthe structure of the product are important. When the surface filmof the liquid is entirely evaporated, the subsequent falling rateperiod begins. In the second falling rate period of drying thedominant diffusion mechanism is vapor diffusion due to mois-ture concentration difference and internal conditions keep ontheir importance (Husain et al., 1972).
Although biological materials such as agricultural productshave a high moisture content, generally no constant rate periodis seen in the drying processes (Bakshi and Singh, 1980). Infact, some agricultural materials such as grains or nuts usuallydry in the second falling rate period (Parry, 1985). Althoughsometimes there is an overall constant rate period at the initialstages of drying, a statement such as the food materials drywithout a constant rate period is generally true.
Mathematical Modeling of Food Drying
Drying processes are modeled with two main models:
(i) Distributed modelsDistributed models consider simultaneous heat and masstransfer. They take into consideration both the internal and
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A REVIEW OF THIN LAYER DRYING OF FOODS 443
external heat and mass transfer, and predict the temperatureand the moisture gradient in the product better. Generally,these models depend on the Luikov equations that comefrom Fick’s second law of diffusion shown as Eq. 1 or theirmodified forms (Luikov, 1975).
∂M
∂t= ∇2K11M + ∇2K12T + ∇2K13P
∂T
∂t= ∇2K21M + ∇2K22T + ∇2K23P
∂P
∂t= ∇2K31M + ∇2K32T + ∇2K33P (1)
where, K11, K22, K33 are the phenomenological coeffi-cients, while K12, K13, K21, K23, K31, K32 are the couplingcoefficients (Brooker et al., 1974).For most of the processes, the pressure effect can be ne-glected compared with the temperature and the moistureeffect, so the Luikov equations become as (Brooker et al.,1974):
∂M
∂t= ∇2K11M + ∇2K12T
∂T
∂t= ∇2K21M + ∇2K22T (2)
Nevertheless, the modified form of the Luikov equations(Eq. 2) may not be solved with analytical methods, be-cause of the difficulties and complexities of real dryingmechanisms. On the other hand, this modified form canbe solved with the finite element method (Ozilgen andOzdemir, 2001).
(ii) Lumped parameter modelsLumped parameter models do not pay attention to the tem-perature gradient in the product and they assume a uniformtemperature distribution that equals to the drying air tem-perature in the product. With this assumption, the Luikovequations become as:
∂M
∂t= K11∇2M (3)
∂T
∂t= K22∇2T (4)
Phenomenological coefficient K11 is known as effectivemoisture diffusivity (Deff) and K22 is known as thermaldiffusivity (α). For constant values of Deff and α, Equations3 and 4 can be rearranged as:
∂M
∂t= Deff
[∂2M
∂x2+ a1
x
∂M
∂x
](5)
∂T
∂t= α
[∂2T
∂x2+ a1
x
∂T
∂x
](6)
where, parameter a1 = 0 for planar geometries, a1 = 1for cylindrical shapes and a1 = 2 for spherical shapes(Ekechukwu, 1999).
The assumptions resembling the uniform temperature distri-bution and temperature equivalent of the ambient air and productcause errors. This error occurs only at the beginning of the pro-cess and it may be reduced to acceptable values with reducingthe thickness of the product (Henderson and Pabis, 1961). Withthis necessity, thin layer drying gains importance and thin layerequations are derived.
Thin Layer Drying Equations
Thin layer drying generally means to dry as one layer ofsample particles or slices (Akpinar, 2006a). Because of its thinstructure, the temperature distribution can be easily assumedas uniform and thin layer drying is very suitable for lumpedparameter models.
Recently thin layer drying equations have been found to havewide application due to their ease of use and requiring less dataunlike in complex distributed models (such as phenomenologi-cal and coupling coefficients) (Madamba et al., 1996; Ozdemirand Devres, 1999).
Thin layer equations may be theoretical, semi-theoretical,and empirical models. The former takes into account only the in-ternal resistance to moisture transfer (Henderson, 1974; Suarezet al., 1980; Bruce, 1985; Parti, 1993), while the others consideronly the external resistance to moisture transfer between theproduct and air (Whitaker et al., 1969; Fortes and Okos, 1981;Parti, 1993; Ozdemir and Devres, 1999). Theoretical models ex-plain the drying behaviors of the product clearly and can be usedat all process conditions, while they include many assumptionscausing considerable errors. The most widely used theoreticalmodels are derived from Fick’s second law of diffusion. Simi-larly, semi-theoretical models are generally derived from Fick’ssecond law and modifications of its simplified forms (other semi-theoretical models are derived by analogues with Newton’s lawof cooling). They are easier and need fewer assumptions dueto using of some experimental data. On the other hand, theyare valid only within the process conditions applied (Fortes andOkos, 1981; Parry, 1985). The empirical models have also sim-ilar characteristics with semi-theoretical models. They stronglydepend on the experimental conditions and give limited infor-mation about the drying behaviors of the product (Keey, 1972).
Theoretical Background
Isothermal conditions changing only with time may be as-sumed to prevail within the product, because the heat transferrate within the product is two orders of magnitude greater thanthe rate of moisture transfer (Ozilgen and Ozdemir, 2001). It can
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444 Z. ERBAY AND F. ICIER
L
Nw
Me
Q
Mi
Ta
Q
Me
Nw
Figure 1 Schematic view of thin layer drying, if drying occurs from bothsides.
be assumed as only Eq. 5 describes the mass transfer (Whitakeret al., 1969; Young, 1969). Then Eq. 5 can be analytically solvedwith the above assumptions, and the initial and boundary con-ditions are (Fig. 1):
t = 0, −L ≤ x ≤ L, M = Mi (7)
t > 0, x = 0, dM/dx = 0 (8)
t > 0, x = L, M = Me (9)
t > 0, −L ≤ x ≤ L, T = Ta (10)
Assumptions:
(i) the particle is homogenous and isotropic;(ii) the material characteristics are constant, and the shrinkage
is neglected;(iii) the pressure variations are neglected;(iv) evaporation occurs only at the surface;(v) initially moisture distribution is uniform (Eq. 7) and sym-
metrical during process (Eq. 8);(vi) surface diffusion is ended, so the moisture equilibrium
arises on the surface (Eq. 9);(vii) temperature distribution is uniform and equals to the am-
bient drying air temperature, namely the lumped system(Eq. 10);
(viii) the heat transfer is done by conduction within the product,and by convection outside of the product;
(ix) effective moisture diffusivity is constant versus moisturecontent during drying.
Then analytical solutions of Eq. 5 are given below for infiniteslab or sphere in Eq. 11, and for infinite cylinder in Eq. 12(Crank, 1975):
MR = A1
∞∑i=1
1
(2i − 1)2exp
[−
(2i − 1)2π2Defft
A2
](11)
Table 1 Values of geometric constants according to the product geometry.
Product Geometry A1 A∗2
Infinite slab 8/π2 4L2
Sphere 6/π2 4r2
3-dimensional finite slab (8/π2)3 1/(L21 + L2
2 + L23)
∗L is the half thickness of the slice if drying occurs from both sides, or L is thethickness of the slice if drying occurs from only one side.
MR = A1
∞∑i=1
1
J 20
exp
[−
J 20 Defft
A2
](12)
where, Deff is the effective moisture diffusivity (m2/s), t is time(s), MR is the fractional moisture ratio, J0 is the roots of theBessel function, and A1, A2 are geometric constants.
For multidimensional geometries such as 3-dimensional slabthe Newman’s rule can be applied (Treybal, 1968). In brief, thevalues of geometric constants are shown in Table 1.
MR can be determined according to the external conditions.If the relative humidity of the drying air is constant during thedrying process, then the moisture equilibrium is constant too. Inthis respect, MR is determined as in Eq. 13. If the relative humid-ity of the drying air continuously fluctuates, then the moistureequilibrium continuously varies so MR is determined as in Eq.14 (Diamante and Munro, 1993);
MR = (Mt − Me)
(Mi − Me)(13)
MR = Mt
Mi
(14)
where, Mi is the initial moisture content, Mt is the mean mois-ture content at time t,Me is the equilibrium moisture content,and all these values are in dry basis. If we accept that food ma-terials dry without a constant rate period, than Mi is equal tothe Mcr which is defined as the moisture content of a material atthe end of the constant rate period of drying, then Eq. 13 equalsto Eq. 15 and MR can be named as the characteristic moisturecontent (φ).
φ = (Mt − Me)
(Mcr − Me)(15)
Semi-Theoretical Models
Semi-theoretical models can be classified according to theirderivation as:
(i) Newton’s law of cooling:These are the semi-theoretical models that are derived
by analogues with Newton’s law of cooling. These modelscan be classified in sub groups as:
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A REVIEW OF THIN LAYER DRYING OF FOODS 445
a. Lewis modelb. Page model & modified forms
(ii) Fick’s second law of diffusionThe models in this group are the semi-theoretical modelsthat are derived from Fick’s second law of diffusion. Thesemodels can be classified in sub groups as:a. Single term exponential model and modified formsb. Two term exponential model and modified formsc. Three term exponential model
The Models Derived From Newton’s Law of Cooling.
a. Lewis (Newton) ModelThis model is analogous with Newton’s law of cooling somany investigators named this model as Newton’s model.First, Lewis (1921) suggested that during the drying ofporous hygroscopic materials, the change of moisture con-tent of material in the falling rate period is proportional tothe instantaneous difference between the moisture contentand the expected moisture content when it comes into equi-librium with drying air. So this concept assumed that thematerial is thin enough, or the air velocity is high, and thedrying air conditions such as the temperature and the relativehumidity are kept constant.
dM
dt= −K (M − Me) (16)
where, K is the drying constant (s−1). In the thin layer dry-ing concept, the drying constant is the combination of dry-ing transport properties such as moisture diffusivity, thermalconductivity, interface heat, and mass coefficients (Marinos-Kouris and Maroulis, 1995).If K is independent from M,then Eq. 16 can be rewritten as:
MR = (Mt − Me)
(Mi − Me)= exp(−kt) (17)
where, k is the drying constant (s−1) that can be obtainedfrom the experimental data and Eq. 17 is known as the Lewis(Newton) model
b. Page ModelPage (1949) modified the Lewis model to get a more accuratemodel by adding a dimensionless empirical constant (n) andapply to the mathematical modeling of drying of shelledcorns:
MR = (Mt − Me)
(Mi − Me)= exp(−ktn) (18)
Generally, n is named as the model constant (dimensionless).c. Modified Page Models
Overhults et al. (1973) modified the Page model to describethe drying of soybeans. This modified form is generally
known as the Modified Page-I Model:
MR = (Mt − Me)
(Mi − Me)= exp (−kt)n (19)
In addition, White et al. (1978) used another modified formof the Page model to describe the drying of soybeans. Thisform is generally known as the Modified Page-II Model:
MR = (Mt − Me)
(Mi − Me)= exp − (kt)n (20)
Diamente and Munro (1993) used another modified formof the Page model to describe the drying of sweet potatoslices. This form is generally known as the Modified Pageequation-II Model:
MR = (Mt − Me)
(Mi − Me)= exp −k
(t/ l2
)n(21)
where, l is an empirical constant (dimensionless).
The Models Derived From Fick’s Second Law of Diffusion.
a. Henderson and Pabis (Single term) ModelHenderson and Pabis (1961) improved a model for dryingby using Fick’s second law of diffusion and applied the newmodel on drying of corns. As the derivation was shown inthe previous section, they use Eq. 11. For sufficiently longdrying times, only the first term (i = 1) of the general seriessolution of Eq. 11 can be used with small error. Accordingto this assumption, Eq. 11 can be written as:
MR = (Mt − Me)
(Mi − Me)= A1 exp
(−
π2DeffA2
t
)(22)
If Deff is constant during drying, then Eq. 22 can be rear-ranged by using the drying constantk as:
MR = (Mt − Me)
(Mi − Me)= a exp (−kt) (23)
where, a is defined as the indication of shape and generallynamed as model constant (dimensionless). These constantsare obtained from experimental data. Equation 23 is gener-ally known as the Henderson and Pabis Model.
b. Logarithmic (Asymptotic) ModelChandra and Singh (1995) proposed a new model includingthe logarithmic form of Henderson and Pabis model with anempirical term addition, and Yagcioglu et al. (1999) appliedthis model to the drying of laurel leaves.
MR = (Mt − Me)
(Mi − Me)= a exp (−kt) + c (24)
where, c is an empirical constant (dimensionless).
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446 Z. ERBAY AND F. ICIER
c. Midilli ModelMidilli et al. (2002) proposed a new model with the addi-tion of an extra empirical term that includes t to the Hen-derson and Pabis model. The new model was the com-bination of an exponential term and a linear term. Theyapplied this new model to the drying of pollen, mush-room, and shelled/unshelled pistachio for different dryingmethods.
MR = (Mt − Me)
(Mi − Me)= a exp (−kt) + b∗t (25)
where, b∗ is an empirical constant (s−1).d. Modified Midilli Model
Ghazanfari et al. (2006) emphasized that the indication ofshape term (a) of the Midilli model (Eq. 25) had to be 1.0 att = 0 and proposed a modification as:
MR = (Mt − Me)
(Mi − Me)= exp (−kt) + b∗t (26)
This model was not applied to a food material, but gave goodresults with flax fiber.
e. Demir et al. ModelDemir et al. (2007) proposed a new model that was similarto Henderson and Pabis, Modified Page-I, Logarithmic, andMidilli models:
MR = (Mt − Me)
(Mi − Me)= a exp [(−kt)]n + b (27)
This model has been just proposed and applied to the dryingof green table olives and got good results.
f. Two-Term ModelHenderson (1974) proposed to use the first two term of thegeneral series solution of Fick’s second law of diffusion (Eq.5) for correcting the shortcomings of the Henderson andPabis Model. Then, Glenn (1978) used this proposal in graindrying. With this argument, the new model derived as:
MR = (Mt − Me)
(Mi − Me)= a exp (−k1t) + b exp (−k2t) (28)
where, a, b are defined as the indication of shape and gen-erally named as model constants (dimensionless), and k1, k2
are the drying constants (s−1). These constants are obtainedfrom experimental data and Eq. 28 is generally known as theTwo-Term Model.
g. Two-Term Exponential ModelSharaf-Eldeen et al. (1980) modified the Two-Term modelby reducing the constant number and organizing the secondexponential term’s indication of shape constant (b). Theyemphasized that b of the Two-Term model (Eq. 27) has to be(1 – a) at t = 0 to get MR= 1 and proposed a modification
as:
MR= (Mt − Me)
(Mi − Me)=a exp (−kt) + (1 − a) exp (−kat) (29)
Equation 29 is generally known as the Two-Term Exponen-tial model.
h. Modified Two-Term Exponential ModelsVerma et al. (1985) modified the second exponential termof the Two-Term Exponential model by adding an empiricalconstant and applied for the drying of rice.
MR = (Mt − Me)
(Mi − Me)= a exp (−kt) + (1 − a) exp (−gt) (30)
This modified model (Eq. 30) is known as the Verma Model.Kaseem (1998) rearranged the Verma model by separatingthe drying constant term k from g and proposed the renewedform as:
MR= (Mt − Me)
(Mi − Me)=a exp (−kt) + (1 − a) exp (−kbt) (31)
This modified form (Eq. 31) is known as the Diffusion Ap-proach model. These two modified models were applied forsome products’ drying at the same time, and gave the sameresults as expected (Torul and Pehlivan, 2003; Akpinar et al.,2003b; Gunhan et al., 2005; Akpinar, 2006a; Demir et al.,2007).
i. Modified Henderson and Pabis (Three Term Exponen-tial) ModelKarathanos (1999) improved the Henderson and Pabis andTwo-Term models as adding the third term of the generalseries solution of Fick’s second law of diffusion (Eq. 5)for correcting the shortcomings of the Henderson and Pabisand Two-Term models. Karathanos emphasized that the firstterm explains the latest part, the second term explains theintermediate part, and the third term explains the beginningpart of the drying curve (MR-t) as:
MR = (Mt − Me)
(Mi − Me)= a exp (−kt)
+ b exp (−gt) + c exp (−ht) (32)
where, a, b, and c are defined as the indication of shape andgenerally named as model constants (dimensionless), andk, g, and h are the drying constants (s−1). These constantsare obtained from experimental data and Eq. 32 is generallyknown as the Modified Henderson and Pabis model.
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A REVIEW OF THIN LAYER DRYING OF FOODS 447
Empirical Models
a. Thompson ModelThompson et al. (1968) developed a model with the experi-mental results of drying of shelled corns in the temperaturerange 60–150◦C.
t = a ln (MR) + b [ln (MR)]2 (33)
r = N∑N
i=1 MRpre,iMRexp,i − ∑Ni=1 MRpre,i
∑Ni=1 MRexp,i√(
N∑N
i=1 MR2pre,i − (∑N
i=1 MRpre,i
)2)(N
∑Ni= MR2
exp,i − (∑Ni=1 MRexp,i
)2) (36)
where, a and b were dimensionless constants obtained fromexperimental data. This model was also used to describe thedrying characteristics of sorghum (Paulsen and Thompson,1973).
b. Wang and Singh ModelWang and Singh (1978) created a model for intermittentdrying of rough rice.
MR = 1 + b∗t + a∗t2 (34)
where, b∗ (s−1) and a∗ (s−2) were constants obtained fromexperimental data.
c. Kaleemullah ModelKaleemullah (2002) created an empirical model that includedMR, T , and t. They applied it to the drying of red chillies(Kaleemullah and Kailappan, 2006).
MR = exp −c∗T + b∗t (pT +n) (35)
where, constant c∗ is in ◦C−1s−1, constant b∗ is in s−1, p isin ◦C−1 and n is dimensionless.
Determination of Appropriate Model
Mathematical modeling of the drying of food products of-ten requires the statistical methods of regression and correlationanalysis. Linear and nonlinear regression analyses are importanttools to find the relationship between different variables, espe-cially, for which no established empirical relationship exists.
As mentioned above, thin layer drying equations require MRvariation versus t . Therefore, MR data plotted with t , and re-gression analysis was performed with the selected models todetermine the constant values that supply the best appropriate-ness of models. The validation of models can be checked withdifferent statistical methods. The most widely used method inliterature is performing correlation analysis, reduced chi-square(χ2) test and root mean square error (RMSE) analysis, respec-tively. Generally, the correlation coefficient (r) is the primarycriterion for selecting the best equation to describe the dryingcurve equation and the highest r value is required (O’Callaghanet al., 1971; Verma et al., 1985; Kassem, 1998; Yaldiz et al.,
2001; Midilli et al., 2002; Akpinar et al., 2003b; Wang et al.,2007a). In addition to r , χ2 and RMSE are used to determinethe best fit. The highest r and the lowest χ2 and RMSE valuesrequired to evaluate the goodness of fit (Sawhney et al., 1999a;Yaldiz et al., 2001; Torul and Pehlivan, 2002; Midilli and Kucuk,2003; Akpinar et al., 2003a; Lahsasni et al., 2004; Ertekin andYaldiz, 2004; Wang et al., 2007b). r, χ2, and RMSE calculationscan be done by equations below:
χ2 =∑n
i=1 (MRexp,i − MRpre,i)2
N − n(37)
RMSE =[
1
N
N∑i=1
(MRpre,i − MRexp,i)2
]1/2
(38)
where, N is the number of observations, n is the numberof constants, MRpre,i ith predicted moisture ratio values,MRexp,i ith experimental moisture ratio values.
Finally, the effect of the variables on model constants canbe investigated by performing multiple regression analysis withmultiple combinations of different equations such as the simplelinear, logarithmic, exponential, power, and the Arrhenius type(Guarte, 1996). These equation types are relatively easy to use inmultiple regression analysis, because they could be linearized.The other types of equations must be solved with nonlinear re-gression techniques and it is too hard to find the solution to suchnonlinear equations if there are many parameters. After investi-gating the effect of experimental variables on model constants,the final model has to be validated by the statistical methodsthat are mentioned above.
Effective Moisture Diffusivity Calculations
Diffusion in solids during drying is a complex process thatmay involve molecular diffusion, capillary flow, Knudsen flow,hydrodynamic flow, or surface diffusion. With a lumped param-eter model concept, all these phenomena are combined in oneterm named as effective moisture diffusivity (Eq. 3). Equations22 and 23 are derived for the constant values of Deff (m2/s) andfor sufficiently long drying times. With a simple arrangement,Eq. 39 is obtained:
ln (MR) = ln (a) − kt (39)
and, k is defined as:
k = −π2Deff
A2(40)
where, A2 is the geometric constant that is shown in Table 1 formain geometries.
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448 Z. ERBAY AND F. ICIER
Equation 39 indicates that the variation of ln(MR) valuesversus t is linear and the slope is equal to drying constant(k). By revealing the drying, the constant effective moisturediffusivity can be calculated easily with different geometries(Eq. 40).
As a matter of fact, the drying curves have a concave formwhen the curves of ln(MR)-t are analyzed. The reason for thisis the assumption of the invariability of the effective moisturediffusion (independency of Deff from moisture content) duringdrying while deriving the equations (Bruin and Luyben, 1980).The concave form of drying curves is caused by variation ofthe moisture content and Deff during drying. Because of this,the slopes have to be derived from linear regression of ln(MR)-tdata.
Deff mainly varies with internal conditions such as the prod-uct’s temperature, the moisture content, and the structure. Thisis harmonious with the assumptions of the thin layer concept.But all assumptions cause some errors and Deff is also affectedfrom external conditions. These effects are insignificant relativeto internal conditions while they cannot be disregarded in someranges. Drying air velocity is an example of this. Islam and Flink(1982) explained that the resistance of the external mass transferwas important in 2.5 m/s or lower velocities. Mulet et al. (1987)expressed that drying air velocity affected the diffusion coef-ficient at an interval of a certain flow velocity. Ece and Cihan(1993) used a temperature and air velocity dependent Arrheniustype diffusivity and Akpinar et al. (2003a) exposed a tempera-ture and air velocity dependent Arrhenius type diffusivity withexperimental data. So, for clarifying the drying characteristics,it is important to calculate Deff.
Activation Energy Calculations
As mentioned above, the factors affecting Deff are significantto clarify the drying characteristics of a food product, meanwhilethe power of the effect is significant. The effect of temperatureon Deff gains importance at this point. Because temperature hastwo critical properties in this matter:
(i) temperature is one of the strongest factor affects on Deff,(ii) it is easily calculated or fixed during experiments.
As a consequence, many researchers studied the effect oftemperature on Deff, and this effect can generally be describedby an Arrhenius equation (Henderson, 1974; Mazza and LeMaguer, 1980; Suarez et al., 1980; Steffe and Singh, 1982;Pinaga et al., 1984; Carbonell et al., 1986; Crisp and Woods,1994; Madamba et al., 1996):
Deff = D0 exp
(−103 Ea
R (T + 273.15)
)(41)
where, D0 is the Arrhenius factor that is generally defined asthe reference diffusion coefficient at infinitely high temperature(m2/s), Ea is the activation energy for diffusion (kJ/mol), R isthe universal gas constant (kJ/kmol.K). The value of Ea showsthe sensibility of the diffusivity against temperature. Namely,
26.8%
11.3%9.9%
15.5%
8.5%
12.7%
4.2%5.6%
4.2%
1.4%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
2007200620052004200320022001200019991998
Publishing years
Dis
trib
utio
n (%
)
Figure 2 Distribution of the studies according to the publishing years.
the greater value of Ea means more sensibility of Deff to tem-perature (Kaymak-Ertekin, 2002).
To calculate Ea , Eq. 41 is arranged as:
ln(Deff) = ln(D0) − 103 Ea
R× 1
(T + 273.15)(42)
Equation 42 indicates that the variation of ln(Deff) versus
[1/(T + 273.15)] is linear and the slope is equal to (−103.Ea/R),so Ea is easily calculated with revealing the slope by derivingfrom linear regression of ln(Deff)-[1/(T + 273.15)].
If the coefficient of the determination value cannot be ashigh as required, other factors would affect the Deff and theyhave to be considered. At this condition, the most appropriatemethod is to reflect these factors to the D0 and perform nonlinearregression analysis to fit the data. For microwave drying, anotherform was developed to calculate the activation energy by Dadalıet al. (2007b). They described the Deff as a function of productmass and microwave power level with an Arrhenius equation:
Deff = D0 exp
(−Eam
Pm
)(43)
where, m is the weight of the raw material (g), Pm is the mi-crowave output power (W), and Ea is the activation energy forthe microwave drying of the product (W/g).
In addition, Dadalı et al. (2007a) used an exponential ex-pression based on the Arrhenius equation for prediction of therelationship between drying rate constant and effective diffusiv-ity as:
k = k0 exp
(−Eam
Pm
)(44)
where, k is the drying rate constant predicted by the appropriatemodel and k0 is the pre-exponential constant (s−1). The acti-vation energy values obtained from Eqs. 43 and 44 were quitesimilar and they showed the linear relationship between the dry-ing rate constant and effective diffusivity with Eqs. 43 and 44,
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Tabl
e2
Stud
ies
cond
ucte
don
mat
hem
atic
alm
odel
ing
ofsu
ndr
ying
offo
odpr
oduc
ts
Prod
uct
Proc
ess
cond
ition
s#
Bes
tmod
elE
ffec
tsof
proc
ess
cond
ition
son
mod
elco
nsta
nts
Ref
eren
ce
Apr
icot
T=
27–4
3◦C
(Unt
reat
ed)
12D
iffu
sion
App
roac
ha
=−1
16.3
04+
5615
T–
71.4
0T2+
1856
7.2R
HTo
grul
and
Pehl
ivan
,200
4
b=
−4.1
36+
0.19
24T
–0.
0025
9T2+
1.80
54R
Hk
=40
5.2
–19
.6T
+0.
25T
2–
64R
HT
=27
–43◦
C(S
O2-s
ulph
ured
)a
=−1
.353
6–
0.33
92T
+0.
0054
8T2+
13.6
4RH
b=
0.02
1–
0.00
371T
+0.
0000
98T
2
–0.
0077
2RH
k=
−0.0
0406
+0.
0239
T-
0.00
0515
T2
–0.
0498
RH
T=
27–4
3◦C
(NaH
SO3-
sulp
hure
d)
Mod
ified
Hen
ders
on&
Pabi
s
a=
3168
6.2
–15
37.2
6T+
18.5
2T2+
86.6
8RH
b=
2063
2.67
–99
3.17
T+
11.9
2T2
–11
6.52
RH
c=
−984
5.92
+45
2.37
T–
5.30
4T2+
689.
51R
Hk
=0.
0783
–0.
0034
8T–
0.00
0041
T2
–0.
0106
4RH
g=
3049
.82
–14
9.57
T+
1.81
T2+
53.0
8RH
h=
2140
.31
–10
4.16
T+
1.25
6T2
+14
.65R
HB
asil
—12
Mod
ified
Page
-II
—A
kpin
ar,2
006b
Bitt
erle
aves
—8
Mid
illi
—So
buko
laet
al.,
2007
Cra
in-c
rain
leav
esFe
ver
leav
esFi
gsT
=27
–43◦
C(U
ntre
ated
)12
Dif
fusi
onA
ppro
ach
a=
1794
7.61
–89
9.84
T+
10.1
73T
2–
1520
6RH
–18
383.
1RH
2+
689.
56T
RH
Togr
ulan
dPe
hliv
an,2
004
b=
–696
.75
+30
.682
T–
0.31
2T2+
667.
47R
H+
826.
62R
H2
–24
.75T
RH
k=
–144
.51
+7.
257T
–0.
0821
T2+
119.
83R
H+
152.
98R
H2
–5.
531T
RH
Gra
peT
=27
–43◦
C(p
retr
eate
d)12
Mod
ified
Hen
ders
onan
dPa
bis
a=
-104
03.4
+44
0.23
T–
4.47
T2
-76
4.33
RH
+10
172.
7RH
2–
70.5
84T
RH
Togr
ulan
dPe
hliv
an,2
004
b=
2625
.76
–11
1.34
T+
1.16
3T2+
301.
24R
H–
1566
.3R
H2
–4.
752T
RH
c=
–295
75.3
+15
01.7
3T–
18.9
T2
–50
390.
6RH
–79
98.7
RH
2+
1192
.85T
RH
k=
181.
42–
6.87
5T–
0.06
73T
2–
138.
64R
H+
51.9
5RH
2+
2.05
8TR
Hg
=31
8.54
–12
.61T
+0.
1305
T2
–24
9.37
RH
+32
0.2R
H2+
2.36
8TR
Hh
=16
.69
–0.
7479
T+
0.00
0084
T2+
3.56
6RH
+1.
208R
H2
–0.
091T
RH
Min
t—
12M
odifi
edPa
ge-I
I—
Akp
inar
,200
6b(C
onti
nued
onne
xtpa
ge)
449
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embe
r 20
13
Tabl
e2
Stud
ies
cond
ucte
don
mat
hem
atic
alm
odel
ing
ofsu
ndr
ying
offo
odpr
oduc
ts.(
Con
tinu
ed)
Prod
uct
Proc
ess
cond
ition
s#
Bes
tmod
elE
ffec
tsof
proc
ess
cond
ition
son
mod
elco
nsta
nts
Ref
eren
ce
Mul
berr
yfr
uits
(Mor
usal
baL
.)U
ntre
ated
2H
ende
rson
and
Pabi
s—
Doy
maz
,200
4b
Pret
reat
edPa
rsle
y—
12V
erm
a—
Akp
inar
,200
6bPe
ach
T=
27–4
3◦C
(Unt
reat
ed)
12V
erm
aa
=–4
.873
+0.
269T
–0.
0000
372T
2+
0.25
2RH
k=
–0.5
742
+0.
0317
T–
0.00
0449
T2
–0.
0956
RH
Togr
ulan
dPe
hliv
an,2
004
g=
0.04
79–
0.00
0026
2T+
0.00
0036
1T2
–0.
0000
128R
HPi
stac
hio
T=
24–3
2◦C
(she
lled)
8M
idill
ia
=0.
9975
+0.
0007
lnT
k=
0.12
91+
0.00
06ln
TM
idill
ieta
l.,20
02
n=
0.88
28+
0.00
08ln
Tb
∗=
0.04
90+
0.00
01ln
T
T=
24–3
2◦C
(uns
helle
d)a
=1.
0030
+0.
0003
lnT
k=
0.15
00+
0.00
02ln
T
n=
1.10
44+
0.00
05ln
Tb
∗=
0.07
44+
0.00
04ln
T
Plum
T=
27–4
3◦C
(pre
trea
ted)
12M
odifi
edH
ende
rson
&Pa
bis
a=
3743
.05
–42
4.11
T+
7.65
T2+
3849
.9R
H+
1347
7.76
RH
2–
147.
13T
RH
Togr
ulan
dPe
hliv
an,2
004
b=
4354
.1–
417.
01T
+7.
379T
2–
1464
.73R
H+
2142
6.01
RH
2–
109.
47T
RH
c=
7273
.1-
829T
+15
.042
T2+
7219
.2R
H+
3001
8.1R
H2
–31
4.25
TR
Hk
=-0
.062
8+
0.00
0090
5T–
0.00
0175
T2
–0.
1396
RH
–0.
5232
RH
2+
0.00
0064
TR
Hg
=86
5.08
–82
.384
T+
1.42
7T2
–16
4.32
RH
+30
78.6
RH
2–
12.7
TR
Hh
=75
8.05
–72
.23T
+1.
251T
2–
141.
84R
H+
2698
.85R
H2
–11
.18T
RH
450
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nloa
ded
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e U
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embe
r 20
13
Tabl
e3
Stud
ies
cond
ucte
don
mat
hem
atic
alm
odel
ing
offo
oddr
ying
perf
orm
edw
ithco
nvec
tive
type
batc
hdr
yers
Prod
uct
Proc
ess
cond
ition
s(o
C;m
/s;g
wat
er/k
gda
;mm
)#
Bes
tmod
elE
ffec
tsof
proc
ess
cond
ition
son
mod
elco
nsta
nts
Ref
eren
ce
App
le(s
lice)
T=
60–8
0υ
=1.
0–1.
513
Mid
illi
a=
1.00
4084
–0.
0000
73T
–0.
0019
60υ+
3.94
4759
ω
k=
–0.0
0639
1+
0.00
0065
T
+0.
0097
75υ+
1.57
6723
ω
Akp
inar
,200
6a
ω=
8×
8×
18–
12.5
×12
.5×
25n
=1.
1877
34+
0.00
2467
T
–0.
1288
78υ
–20
2.53
6ωb
∗=
0.00
0082
–0.
0000
02T
–0.
0000
41υ+
0.04
1667
ω
App
le(G
olde
n)T
=60
–80
υ=
1.0–
3.0
14M
idill
ia
=1.
4678
−−0
.006
7Tk
=1.
0835
υ0.
1316
n=
0.88
67b
∗=
0.00
30M
enge
san
dE
rtek
in,2
006a
App
lepo
mac
eT
=75
–105
10L
ogar
ithm
ica
=27
1.15
–8.
91T
+0.
097T
2–
3.52
T3
k=
–0.6
1+
0.02
T–
0.00
02T
2+
0.00
0000
8T3
Wan
get
al.,
2007
a
c=
–267
.45
+8.
82T
–0.
096T
2+
0.00
04T
3
Apr
icot
T=
47.3
–61.
74υ
=0.
707–
2.3
14M
idill
ia
=1.
0699
31–
0.00
1297
T–
0.00
4534
υ+
0.00
5478
RSC
Akp
inar
etal
.,20
04R
SC=
0–2.
25rp
m(S
O2-s
ulph
ured
)k
=–0
.086
272
+0.
0017
75T
+0.
0356
43υ+
0.00
9545
RSC
n=
1.70
5840
–0.
0130
76T
–0.
1675
07υ
–0.
0208
10R
SCb
∗=
0.01
0122
–0.
0001
62T
–0.
0014
39υ
–0.
0002
40R
SCT
=50
–80
υ=
0.2–
1.5
(SO
2-s
ulph
ured
)14
Log
arith
mic
a=
1.13
481e
xp(0
.018
352υ
)k
=0.
0012
69+
0.00
0018
T
x+
0.00
105υ
Togr
ulan
dPe
hliv
an,2
003
c=
–1.1
6416
+ex
p(1.
6982
/T)
–0.
0138
υ
Bag
asse
T=
80–1
20υ
=0.
5–2.
012
Page
k=
0.49
1235
5703
8+
0.00
3109
4667
H–
0.00
3118
3596
869T
–0.
0394
7507
753υ
+0.
1137
6221
2L
Vija
yara
jeta
l.,20
07
H=
9–24
L=
20–6
0n
=–0
.869
9040
5+
0.23
8750
462l
ogt
–1.
1754
5690
4kB
ayle
aves
T=
40–6
0R
H=
5–25
%15
Page
k=
exp(
-4.4
647
+0.
0745
5T–
0.00
714R
H)
n=
1.14
325
Gun
han
etal
.,20
05B
lack
Tea
T=
80–1
20υ
=0.
25–0
.65
5L
ewis
k=
0.12
563υ
1.15
202ex
p(−2
09.1
2341
/Tabs)
Panc
hari
yaet
al.,
2002
Car
rot(
slic
e)T
=60
–90
υ=
0.5–
1.5
4M
odifi
edPa
ge-I
Ik
=42
.66υ
0.31
23(2
L)−
0.84
37ex
p(–2
386.
6/T
)E
rent
urk
and
Ere
ntur
k,20
07L
=2.
5–5
n=
5.48
υ−0
.084
6(2
L)−
0.10
66ex
p(–4
52.5
/T)
Cit
rus
aura
ntiu
mle
aves
T=
50–6
0R
H=
41–5
3%13
Mid
illi
a=
–49.
079
+1.
838T
–0.
0167
T2
k=
–13.
604
+0.
498T
–0.
0045
18T
2M
oham
edet
al.,
2005
. V= 0.
0277
−−0
.083
3m3/s
n=
37.4
47–
1.34
6T+
0.01
231T
2b
∗=
–0.4
51+
0.01
576T
–0.
0001
4T2
Coc
onut
(You
ng)
T=
50–7
0(O
smot
ical
lypr
e-dr
ied)
L=
2.5–
43
Page
k=
21.8
exp(
–213
6.9/
Tabs)
Mad
amba
,200
3
n=
0.09
8–
0.08
2LD
ates
T=
70–8
0(S
akie
var.)
3Pa
gek
=–2
.463
+0.
0613
T–
0.00
035T
2n
=–1
.228
+0.
0524
T–
0.00
032T
2H
assa
nan
dH
oban
i,20
00T
=70
–80
(Suk
kari
var.)
k=
0.00
0000
27T
3.05
11n
=–4
.437
+0.
1353
T–
0.00
085T
2
Ech
inac
eaan
gust
ifol
iaT
=15
–45
υ=
0.3–
1.1
4M
odifi
edPa
ge-I
Ik
=0.
07υ
0.17
93(2
r)−
1.23
49ex
p(-2
0.66
/T)
Ere
ntur
ket
al.,
2004
(Con
tinu
edon
next
page
)
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17
Sept
embe
r 20
13
Tabl
e3
Stud
ies
cond
ucte
don
mat
hem
atic
alm
odel
ing
offo
oddr
ying
perf
orm
edw
ithco
nvec
tive
type
batc
hdr
yers
.(C
onti
nued
)
Prod
uct
Proc
ess
cond
ition
s(◦
C;m
/s;g
wat
er/k
gda
;mm
)#
Bes
tmod
elE
ffec
tsof
proc
ess
cond
ition
son
mod
elco
nsta
nts
Ref
eren
ce
r=
root
size
(mm
)n
=0.
96υ
−0.0
139(2
r)−
0.04
33ex
p(-1
.73/
T)
Egg
plan
tT
=30
–70
υ=
0.5–
2.0
14M
idill
ia
=0.
9897
9−
0.08
071
lnυk
=0.
0016
0T1.
5594
5n
=1.
0987
7+
0.29
745
lnυb
∗=
0.00
062
Ert
ekin
and
Yal
diz,
2004
Figs
(who
le)
T=
46.1
–60
υ=
1.0–
5.0
7L
ogar
ithm
ica
=1.
1299
8+
0.00
0632
4T-
0.03
6879
1υ-
0.00
4102
99H
Xan
thop
oulo
set
al.,
2007
H=
8.14
–13.
32k
=−0
.089
8261
+0.
0024
4127
T+
0.00
4457
21υ
−0.0
0008
6437
1Hc
=−0
.161
594
−0.
0007
6411
6T+
0.03
4793
6υ+
0.00
7201
03H
Gra
pe(S
ulta
na)
T=
32.4
–40.
3υ
=0.
5–1.
58
Two-
term
a=
0.33
6-
0.00
4Tk
1=
7.70
3–
8.71
7ln
υY
aldi
zet
al.,
2001
b=
0.80
6υ−0
.039
k2
=-0
.141
+0.
048
lnT
Gra
pe(T
hom
pson
seed
less
)T
=50
–80
υ=
0.25
–1.0
(pre
trea
ted)
3Pa
gek
=2.
91×
106υ
0.22
exp(
5749
.05/
T)
Saw
hney
etal
.,19
99a
n=
1.14
T=
50–7
0υ
=0.
25–1
.0-
k=
3720
000υ
0.19
H−0
.13ex
p(-6
032/
Tabs)
Pang
avha
neet
al.,
2000
RH
=13
–23%
n=
1.10
7G
reen
bean
T=
50–8
0υ
=0.
25–1
.012
Page
k=
0.35
60–
0.14
07υ
n=
0.78
32+
0.08
92ln
υY
aldi
zan
dE
rtek
in,2
001
Gre
ench
illi
T=
40–6
5R
H=
10–6
0%2
Page
k=
0.00
8759
–0.
0002
7T+
0.00
0000
282T
2+
0.00
166υ
–0.
0105
8RH
+0.
0090
57R
H2
Hos
sain
and
Bal
a,20
02
υ=
0.1–
1.0
(Ove
r/un
derfl
ow)
n=
0.56
3021
+0.
0064
35T
+0.
0882
98υ
+0.
6369
6RH
T=
40–6
5R
H=
10–6
0%k
=−0
.021
84+
0.00
0781
T–
0.00
0006
8T2+
0.00
4522
υ+
0.00
4437
RH
–0.
0133
5RH
2
υ=
0.1–
1.0
(Thr
ough
flow
)n
=0.
5804
25+
0.00
465T
+1.
7177
υ–
1.29
91υ
2–
1.24
21R
H+
1.38
45R
H2
Gre
enpe
pper
T=
50–8
0υ
=0.
25–1
.012
Dif
fusi
onA
ppro
ach
a=
−1.6
626
+1.
7015
υb
=0.
5868
–0.
0172
υY
aldi
zan
dE
rtek
in,2
001
k=
0.35
49–
0.14
89υ
Haz
elnu
tT
=10
0–16
08
Tho
mps
ona
=−1
16.0
5+
0.65
6Tb
=−1
9.89
+0.
122T
Ozd
emir
and
Dev
res,
1999
T=
100–
160
Mi=
12.3
%(m
oist
uriz
ed)
3Tw
o-te
rma
=0.
535
-0.
0005
8Tk
1=
0.46
5O
zdem
iret
al.,
2000
b=
0.00
058
+23
6248
.7T
k2
=4.
52T
=10
0–16
0M
i=
6.14
%(u
ntre
ated
)Tw
o-te
rma
=0.
434
-0.
0030
4Tk
1=
0.56
6
b=
0.00
304
+23
6248
.7T
k2
=5.
29T
=10
0–16
0M
i=
2.41
%(p
re-d
ried
)Tw
o-te
rma
=0.
714
−0.
0035
6Tk
1=
0.28
6
b=
0.00
356
+23
6248
.7T
k2
=2.
89
452
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ded
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e U
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17
Sept
embe
r 20
13
Kal
eT
=30
–60
L=
10–5
04
Mod
.Pag
e-I
k=
exp(
8.04
87–
3836
.1/T
abs)
n=
0.89
4653
Mw
ithig
aan
dO
lwal
,200
5K
urut
T=
35–6
511
Two-
term
-K
arab
ulut
etal
.,20
07O
nion
T=
50–8
0υ
=0.
25–1
.012
Two-
term
a=
0.48
66+
0.64
24ln
υk
1=
0.15
57+
0.19
95ln
υY
aldi
zan
dE
rtek
in,2
001
b=
0.51
43–
0.64
24ln
υk
2=
0.11
17–
0.09
92ln
υ
T=
50–8
0υ
=0.
25–1
.0-
Hen
ders
onan
dPa
bis
a=
1.01
Saw
hney
etal
.,19
99b
H=
6.5–
10.5
(pre
trea
ted)
k=
122.
34υ
0.31
exp(
-302
0/T
abs)
Padd
y(p
arbo
iled)
T=
70–1
50υ
=0.
5–2.
0-
Lew
isk
=0.
02υ
0.47
3L
−0.6
99d
T0.
478
Rao
etal
.,20
07L
d=
50—
200
Pars
ley
T=
56–9
39
Page
k=
0.00
0012
T0.
7062
63n
=0.
2939
14T
0.29
9815
Akp
inar
etal
.,20
06Pe
ach
slic
eT
=55
–65
6L
ogar
ithm
ic-
Kin
gsle
yet
al.,
2007
Bla
nche
dw
ith%
1K
MS
orA
APi
stac
hio
nuts
T=
25–7
06
Page
k=
−0.0
0209
+0.
0002
08T
+0.
0050
2υ2
n=
0.84
4+
0.00
262T
–0.
106υ
Kas
hani
neja
det
al.,
2007
Pist
achi
oT
=40
–60
υ=
0.5–
1.5
8M
idill
ia
=0.
9968
+0.
0007
lnT
k=
0.14
93+
0.00
06ln
TM
idill
ieta
l.,20
02R
H=
5–20
%(s
helle
d)n
=0.
9178
+0.
0008
lnT
b∗
=0.
0501
+0.
0001
lnT
T=
40–6
0υ
=0.
5–1.
5a
=0.
9968
+0.
0003
lnT
k=
0.15
45+
0.00
02ln
T
RH
=5–
20%
(uns
helle
d)n
=0.
9247
+0.
0005
lnT
b∗
=0.
0486
+0.
0004
lnT
Plum
(Sta
nley
)T
=60
–80
υ=
1.0–
3.0
(pre
trea
ted)
14M
idill
ia
=2.
5729
−0.
3726
lnT
k=
0.26
43υ
0.36
65M
enge
san
dE
rtek
in,2
006b
n=
0.00
011T
2.15
54b
∗=
−0.0
044
T=
60–8
0υ
=1.
0–3.
0(u
ntre
ated
)a
=3.
2180
−0.
5255
lnT
k=
0.22
88υ
0.29
94
n=
0.00
0057
T2.
3144
b∗
=−0
.002
8Po
llen
T=
458
Mid
illi
a=
0.99
87+
0.00
03ln
Tk
=0.
2616
+0.
0002
lnT
Mid
illie
tal.,
2002
n=
0.58
69+
0.00
05ln
Tb
∗=
0.06
09+
0.00
04ln
T
Pota
to(s
lice)
T=
60–8
0υ
=1.
0–1.
513
Mid
illi
a=
0.98
6173
+0.
0000
69T
+0.
0057
02υ+
0.09
8206
ωk
=-0
.015
582
+0.
0001
56T
+0.
0134
67υ+
0.26
6761
ω
Akp
inar
,200
6a
ω=
8×
8×
18−
12.5
×12
.5×
25n
=1.
2183
79+
0.00
0802
T–
0.16
2776
υ–
138.
528ω
b∗
=0.
0000
085
+0.
0000
0029
T–
0.00
0039
3υ–
0.02
0302
2ωPr
ickl
ype
arfr
uit
T=
50–6
08
Two-
term
a=
−2.9
205
+0.
1117
T–
0.00
11T
2k
1=
1.16
19–
0.04
39T
+0.
0004
T2
Lah
sasn
ieta
l.,20
04b
=2.
3099
–0.
0547
T+
0.00
05T
2k
2=
-0.0
764
+0.
0027
T
–0.
0000
2165
8T2
(Con
tinu
edon
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page
)
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Sept
embe
r 20
13
Tabl
e3
Stud
ies
cond
ucte
don
mat
hem
atic
alm
odel
ing
offo
oddr
ying
perf
orm
edw
ithco
nvec
tive
type
batc
hdr
yers
.(C
onti
nued
)
Prod
uct
Proc
ess
cond
ition
s(◦
C;m
/s;g
wat
er/k
gda
;mm
)#
Bes
tmod
elE
ffec
tsof
proc
ess
cond
ition
son
mod
elco
nsta
nts
Ref
eren
ce
Pum
pkin
(slic
e)T
=60
–80
υ=
1.0–
1.5
13M
idill
ia
=0.
9664
67+
0.00
0184
T+
0.00
7014
υk
=0.
0056
45-
0.00
0095
T
+0.
0037
91υ
Akp
inar
,200
6a
n=
0.57
2175
+0.
0090
74T
–0.
0646
52υ
b∗
=0.
0000
50-
0.00
0001
T–
0.00
0024
υ
Red
chill
ies
T=
50–6
54
Kal
eem
ulla
hc∗
=0.
0084
766
b∗
=-0
.347
75K
alee
mul
lah
and
Kai
lapp
an,
2006
m=
0.00
0049
34n
=1.
1912
T=
40–6
5υ
=0.
12–1
.02
2L
ewis
k=
0.00
3484
–0.
0002
22T
+0.
0000
0366
T2
–0.
0070
85R
H+
0.00
572R
H0.
0027
38υ
–0.
0012
35υ
2
Hos
sain
etal
.,20
07
RH
=10
–60
Red
pepp
erT
=55
–70
11D
iffu
sion
App
roac
ha
=18
44.3
24–
493.
320
lnT
b=
1.03
3970
exp(
-12.
2945
/Tabs)
Akp
inar
etal
.,20
03c
k=
6331
9.52
exp(
-497
3.88
/Tabs)
Ric
e(r
ough
)T
=22
.3–3
4.9
RH
=34
.5–5
7.9%
—Pa
gek
=-0
.002
09+
0.00
0208
T+
0.00
502υ
2n
=0.
844
+0.
0026
2T–
0.10
6υB
asun
iaan
dA
be,
2001
T=
5–35
υ=
0.75
–2.5
4H
ende
rson
and
Pabi
sa
=18
.157
8–
1.49
019υ
-0.0
2719
1T–
0.26
3827
RH
+0.0
0453
363T
υ+
0.00
0966
809T
RH
+0.
0030
4256
RH
υ
Igua
zet
al.,
2003
RH
=30
–70%
k=
0.00
3014
14–
0.00
0021
593T
+0.
0000
0003
8906
7T2+
0.00
0004
78υ
Stuf
fed
Pepp
erT
=50
–80
υ=
0.25
–1.0
12Tw
o-te
rma
=0.
6315
–0.
2957
υk
1=
0.02
24ex
p(4.
7396
υ)
Yal
diz
and
Ert
ekin
,200
1b
=0.
3679
+0.
2962
υk
2=
0.06
77–
0.01
17ln
υ
Whe
at(p
arbo
iled)
T=
40–6
06
Two-
term
a=
0.03
197T
–1.
009
k1
=−0
.034
Moh
apat
raan
dR
ao,2
005
b=
-0.0
32T
+1.
9918
k2
=−0
.009
Yog
hurt
(str
aine
d)T
=40
–50
υ=
1.0–
2.0
9M
idill
ia
=1
k=
−0.0
0055
69+
0.00
0012
05T
+0.
0002
047υ
Hay
alog
luet
al.,
2007
n=
1.7
b∗
=−0
.000
0348
9-
0.00
0000
38T
–0.
0000
0542
υ
454
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embe
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13
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., 2002n = 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:
kth = λDeffth (45)
where, kth is the theoretical value of drying rate constant ob-tained from Eq. 44 (s−1), (Deff)th is the theoretical effective
diffusivity value obtained from Eq. 43 (m2/s) and λ is the em-pirical constant (m−2).
STUDIES CONDUCTED ON MODELING OF FOODDRYING WITH THIN LAYER CONCEPT
The considerable volume of work devoted to elucidating abetter understanding of moisture transport in solids is not cov-ered in depth, and the reason for this is that, in practice, drying-rate curves have to be measured experimentally, rather than cal-culated from fundamentals (Baker, 1997). So the experimentalstudies prevent their importance in drying, especially for foodproducts, and there have been many studies done in the last 10years in literature. The distribution of the studies according tothe publishing years was summarized in Fig. 2. This graph showsthe increasing interest to the thin layer drying investigations inrecent years.
Process conditions, the product, and the drying method areimportant variables in thin layer drying modeling. The mainparameter in this article was chosen as the drying method forthe categorization of the reviewed studies.
The oldest method of drying is sun drying. Due to requiringextensive drying area and long drying time, microbial risks canappear in many products. On the contrary, it has been used
Vegetables; 21.8%
Fruits; 36.8%
Grains; 12.6%
Medical & aromatic
plants; 20.7%
Others; 8.0%
Figure 3 Distribution of the product types used in studies.
widely because of low technology and energy requirements suchthat modeling studies conducted on sun drying have preservedits importance as shown in Table 2.
The most popular thin layer drying method in literature andindustrial applications is hot air drying using convection as themain heat transfer mechanism. Generally, heated air is blownto the product and the drying rate is increased with the help ofthe forced convection. The main modeling studies executed withthis method within the last 10 years were compiled and shown inTable 3. Furthermore, the modeling in a drying cupboard withoutthe effect of airflow, done for some products, was summarizedin Table 4.
The improving effect of electrical heating methods on dryingprocesses, especially microwave and infrared, is strong. Thesemethods can shorten the drying time, and many modeling studiesfor these processes were performed with the thin layer concept(Table 5).
Furthermore, various pre-treatments are done to the raw foodproducts to facilitate the drying and to improve the productquality. These processes affect the drying kinetics directly andmany investigators used the thin layer concept to explain theeffects of various pre-treatments, especially in fruit drying. Thestudies conducted on the effects of pre-treatments to the dryingkinetics are shown in Table 6.
As mentioned above, the effective moisture diffusivity isa useful tool in explaining the drying kinetics, and activation
DC; 1.4%
SD; 8.3%MD; 6.9%
ICD; 6.9%
ID; 4.2%
FBD; 1.4%
CBD; 70.8%
Figure 4 Distribution of the drying methods used in studies.
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13
Tabl
e5
Stud
ies
cond
ucte
don
mat
hem
atic
alm
odel
ing
offo
oddr
ying
with
thin
laye
rco
ncep
tand
perf
orm
edby
elec
tric
alm
etho
ds.
Prod
uct
DM
Proc
ess
cond
ition
s#
Bes
tmod
elE
ffec
tsof
proc
ess
cond
ition
son
mod
elco
nsta
nts
Ref
eren
ce
App
le(s
lice)
IDT
=50
–80◦
C10
Mod
ified
Page
eq-I
Ik
=–9
.082
44+
1.58
0765
lnT
n=
11.4
9544
–1.
7401
6ln
TTo
grul
,200
5l=
–0.6
2879
2+
0.57
4354
lnT
App
lePo
mac
eM
DP
m=
150–
600
WU
ntre
ated
10Pa
gek
=–0
.017
83+
0.00
0130
3Pm
n=
1.67
47–
0.00
728P
mW
ang
etal
.,20
07b
Pm
=18
0–90
0W
Hot
air
pre-
drie
dk
=0.
0248
4+
0.00
0479
Pm
n=
0.87
04–
0.00
104P
m
ICD
T=
55–7
5◦C
Unt
reat
ed10
Log
arith
mic
a=
–20.
7119
6+
0.72
489T
–0.
0056
7T2
c=
21.8
0075
–0.
7272
8T+
0.00
569T
2Su
net
al.,
2007
k=
0.16
955
–0.
0048
5T+
0.00
0034
85T
2
T=
55–7
5◦C
Hot
air
pre-
drie
dPa
gek
=0.
1126
9–
0.00
34T
+0.
0000
2615
T2
n=
–8.6
026
+0.
3011
1T–
0.00
221T
2
Bar
ley
ICD
I=
0.16
7–0.
5W
/cm
2υ
=0.
3–0.
7m
/s—
Page
k=
0.80
495
+7.
2839
I2+
1.49
43R
H–
1.66
62υ
–1.
3368
Mi
Afz
alan
dA
be,
2000
RH
=36
–60%
Mi=
25–4
0%n
=0.
9785
7+
0.73
09I+
0.46
04R
H–
0.41
773υ
Car
rot
IDT
=50
–80◦
C5
Mid
illi
a=
64T
−0.7
1656
5n
=0.
1179
79ex
p(0.
0069
83T
)To
grul
,200
6k
=11
1T−1
.670
37b
∗=
–0.0
0005
1exp
(0.0
0499
3T)
Oliv
ehu
skIC
DT
=80
–140
◦ C—
Mid
illi
a=
0.96
656e
xp(0
.000
3269
6T)
n=
1.87
693
–0.
0139
3T+
0.00
0048
91T
2C
elm
aet
al.,
2007
k=
–0.0
0234
+0.
0005
4676
lnT
b∗
=[–
5644
28.4
8+
9055
.14T
–37
.28T
2]−
1
Oni
onIC
DI 1
=0.
5–1.
0kW
/kg
υ=
0.1–
0.35
m/s
3Pa
gek
=0.
058e
xp(2
.568
1I1+
1.84
1υ–
0.02
2L2
–0.
0608
RH
2W
ang,
2002
.
RH
=28
.6–4
3.1%
L=
2–6
mm
n=
1.36
58I
=2.
65–4
.42
W/c
m2
T=
35–4
5◦C
9L
ogar
ithm
ica
=0.
725
+0.
0415
I+
0.00
331T
+0.
054υ
k=
1.57
3–
0.35
7I–
0.03
39T
+0.
0555
υ
Jain
and
Path
are,
2004
υ=
1.0–
1.5
m/s
c=
0.00
651
–0.
0012
1I+
0.00
0223
T–
0.00
584υ
456
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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◦Cυ = 3.1 m/s
Untreated 3 Two-term 4.3E-10 - 13.2E-10 Dandamrongrak et al.,2002
BlanchedChilledFrozenBlanched & Frozen
Mulberry fruits(Morus alba L.)
CBD T = 50◦Cυ = 1.0 m/s
Untreated 6 Logarithmic 2.23E-10 – 6.91E-10 Doymaz, 2004c
Dipped in HWDipped in AEEODipped in AA, then
AEEODipped in CA, then
AEEODipped in HW, then
AEEOMulberry fruits
(Morus alba L.)SD — Untreated 2 Henderson and Pabis 4.26E-11 Doymaz, 2004b
Dipped in AEEO 4.69E-10
energy is important in describing the sensibility of Deff withtemperature. The values of Deff and Ea calculated by the thinlayer concept were collected in Table 7. Furthermore, Ea val-ues for microwave drying calculated by the Dadalı model wereshown in Table 8.
Approximately a hundred articles on the thin layer dryingmodeling have been published in the last 10 years. Replicatedstudies on the same product and method have not been reviewedin this article, only represented articles were chosen. The resultsof the representing studies were interpreted and discussed toattain some general approaches in the thin layer drying of foods.
Figure 3 shows the distribution of the product types used inthe studies. The most widely studied product types are fruits(36.8%) and vegetables (21.8%). But the intensity of medicaland aromatic plants is very interesting (20.7%) because they arevery suitable for thin layer drying.
1.00E-13
1.00E-12
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-051 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52
Number of Products
Def
f (m
2/s)
Figure 5 Distribution of effective moisture diffusivity values compiled fromstudies.
The distribution of the drying methods used in the studiesis shown in Fig. 4. This graph displays that the interest of theinvestigators to the convective type batch dryers in food dryingprocesses. 70.8% of the studies reviewed have used convec-tive type batch dryers in their experiments. At the same time,this graph shows the increasing interest of the electrical dryingmethods, especially infrared drying. 18% of the reviewed stud-ies conducted on electrical drying methods and 11.1% of allthe studies were used in various types of infrared dryers. Theintensity of the infrared dryers can be explained as the harmonyof infrared theory and thin layer concept.
Marinos-Kouris and Maroulis (1995) compiled the 37 dif-ferent effective moisture diffusivity value intervals that werecalculated by the experiments. They expressed that the diffusiv-ities in foods had values in the range 10−13 to 10−6 m2/s, andmost of them (82%) were accumulated in the region 10−11 to
1.00E-13
1.00E-12
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-051 29
Number of Products
Def
f (m
2/s)
Figure 6 Distribution of effective moisture diffusivity values compiled fromstudies in which the experiments were done with convective type batch dryer.
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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., 2003bω = 8 × 8 × 18–12.5
× 12.5 × 25 mmApple 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., 2007bPm = 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., 2007T = 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(SO2-sulphured)
4.76E-9–8.32E-9 — Togrul and Pehlivan,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., 2007H = 9–24 g/kg L = 20–60 mm
Basil SD — 6.44E-12 — Akpinar, 2006bBitter leaves SD — 43.42E-10 — Sobukola et al., 2007Black Tea CBD T = 80–120◦C υ = 0.25–0.65 m/s 1.14E-11 – 2.98E-11 406.02 Panchariya et al.,
2002Carrot (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 Togrul, 2006Coconut (Young) CBD T = 50–70◦C L = 2.5 – 4 mm 1.71E-10 – 5.51E-10 81.11 Madamba, 2003
(Osmoticallypre-dried)
Crain-crain leaves SD — 52.91E–10 — Sobukola et al., 2007Fever leaves SD — 48.72E–10 —Grape (Chasselas) CBD T = 50–70◦C (1) 49 Azzouz et al., 2002Grape (Sultanin) CBD T = 50–70◦C (2) 54Green 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., 2003RH = 15% LD = 1:1, 2:1, 3:1
Hazelnut CBD T = 100–160◦C 2.30E-7 – 11.76E-7 34.09 Ozdemir and Devres,1999
T = 100–160◦C Mi = 12.3 %(moisturized)
3.14E-7 – 30.95E-7 48.70 Ozdemir et al., 2000
T = 100–160◦C Mi = 6.14 %(untreated)
3.61E-7 – 21.10E-7 41.25
T = 100–160◦C Mi = 2.41 %(pre-dried)
2.80E-7 – 15.65E-7 36.59
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., 2007Mint 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., 2002T = 35–60◦C υ = 4.1m/s 3.07E-9 – 19.41E-9 62.96 Doymaz, 2006
Mulberry fruits(Morus alba L.)
CBD T = 60–80◦C υ = 1.2m/s 2.32E-10 – 27.60E-10 21.2 Maskan and Gouþ,1998
Okra MD Pm = 180–900 W m = 25–100 g 2.05E-9 – 11.91E-9 - Dadalı et al., 2007bOlive cake CBD T = 50–110◦C 3.38E-9 - 11.34E-9 17.97 Akgun and Doymaz,
2005Olive husk ICD T = 80–140◦C 5.96E-9 – 15.89E-9 21.30 Celma et al., 2007Paddy (parboiled) CBD T = 70–150◦C
υ = 0.5–2.0 m/sLd = 50–200 mm
6.08E-11 - 34.40E-11(3)
21.90 - 23.88 Rao et al., 2007
Parsley SD - 4.53E-12 - Akpinar, 2006bPeach slice CBD T = 55–65◦C
(Blanched with %1KMS or AA)
3.04E-10– 4.41E-10 - Kingsley et al., 2007
Peas FBD T = 30–50◦Cυ = 0.25–1.0 m/sRH = 15%
- 42.35 – 58.15 Senadeera et al., 2003
(Continued on next page)
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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., 2002CBD 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: Sutlejpurple)
CBD T = 55–65◦C (Untreated) 3.04E-10 – 4.41E-10 - Goyal et al., 2007
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(pretreated)
1.20E-7 – 4.55E-7 - Menges and Ertekin,2006b
T = 60–80◦C υ = 1.0 − 3.0m/s(untreated)
1.18E-9 – 6.67E-9
T = 65◦C υ = 1.2m/s (Dippedin AEEO)
2.40E-10 - Doymaz, 2004d
T = 65◦C υ = 1.2m/s(untreated)
2.17E-10
Potato (slice) FBD T = 30–50◦C υ = 0.25 − 1.0m/s - 12.32 – 24.27 Senadeera et al., 2003RH = 15% AR = 1:1, 2:1, 3:1
Red chillies CBD T = 50–65◦C 3.78E-9 – 7.10E-9 37.76 Kaleemullah andKailappan, 2006
Rice (rough) CBD T = 5–35◦Cυ = 0.75–2.5 m/sRH = 30–70%
5.79E-11 – 17.15E-11 18.50 – 21.04 Iguaz et al., 2003
Spinach MD Pm = 180–900 Wm = 25–100 g
7.6E-11 – 52.4E-11 - Dadali et al., 2007c
Tarhana Dough ID T = 60–80◦C L = 1–6 mmUntreated
4.1E-11 – 50.0E-11 41.6 – 49.5.Ibanoglu and Maskan,
2002T = 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
(1)Deff = D0exp(-Ea /RTabs )exp(-(dTabs + e)M) Deff = 0.0016exp(-Ea /RTabs )exp(-(0.0012Tabs+ 0.309)M)(2)Deff = D0exp(-Ea /RTabs )exp(-(dTabs + e)M) Deff = 0.522exp(-Ea /RTabs )exp(-(0.0075Tabs+ 1.829)M)(3)Deff = (67.37 + 110.8υ – 14.64Ld+ 0.5946T – 4.706υLd+ 0.696L2
d – 0.0369LdT )×10–12
10−8 m2/s. In this study, 52 different diffusivity intervals werecompiled and shown in Fig. 5. The biggest Deff values were
between 10−5 and 10−6 (product number 23 to 26). The biggest4 values gained in hazelnut drying and the drying temperaturesof these experiments were between 100–160◦C. These temper-ature values are too high for food drying, so these values werenot taken into consideration for creating general and appropriatestatistics. Except these values, the effective moisture diffusivityvalues in foods are in the range 10−12 to 10−6 m2/s and thisrange is more narrow than what Marinos-Kouris and Maroulis
Table 8 Activation energy values calculated by Dadalı model
Product Process conditions Ea (W/g) Reference
Mint Pm = 180–900 W 11.05(2) – 12.28 (1) Ozbek and Dadali, 2007Okra m = 25–100 g 5.54(1) Dadalı et al., 2007a
5.70(2) Dadalı et al., 2007bSpinach 9.62 (2) – 10.84 (1) Dadali et al., 2007c
(1)k = k0exp(-Ea.m/Pm)(2)Deff = D0exp(-Ea.m/Pm)
expressed. The accumulation of the values is in the region 10−10
to 10−8 m2/s (75%).On the other hand, the distribution of Deff values according
to the drying method was plotted. Figure 6 showed the distribu-tion of Deff values collected from the studies reviewed, in whichthe experiments were conducted with a convective type batchdryer. Disregarding the hazelnut values as mentioned above, theaccumulation of Deff values of the foods that were dried in a
convective type batch dryer is in the region 10−10 to 10−8 m2/s(86,2%).
Figure 7 is arranged according to the Deff values obtainedby electrical methods. All values of infrared drying without theairflow were in the region 10−10 to 10−9 m2/s (ID). Deff valuesfor infrared drying systems that contain airflow mechanisms(ICD) appeared approximately in 10−8 m2/s level. This showedthat the drying rate for ICD were faster as expected, because ofthe enhancing effect of the airflow. In addition, the microwavedryer (MD) values were higher than the convective type batchdryers, and this was harmonious with the theory.
During the sun drying experiments (Fig. 8), the ambient tem-perature in Nigeria increased up to 44◦C, while in Turkey the
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460 Z. ERBAY AND F. ICIER
MDMD
MD
MD
ICDICDICD
IDIDID
1.00E-13
1.00E-12
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-051 2 3 4 5 6 7 8 9 10
Number of Products
Def
f (m
2/s)
Figure 7 Distribution of effective moisture diffusivity values compiled fromstudies in which the experiments were done by electrical methods.
1.00E-13
1.00E-12
1.00E-11
1.00E-10
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-051 2 3 4 5 6 7 8 9
Number of Products
Def
f (m
2/s)
Figure 8 Distribution of effective moisture diffusivity values compiled fromstudies in which the experiments were done by sun drying.
0.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
80.00
90.00
0 7 14 21 28 35 42
Number of Products
Ea
(kJ/
mo
l)
Figure 9 Distribution of activation energy values compiled from studies.
maximum temperature value was measured as 36◦C. Because ofthe temperature difference, the values gained in Nigeria (prod-uct number 3, 4 and 5) were higher than the others, and thisshowed the critical effect of the temperature on Deff.
Finally, the activation energy values in literature were com-piled and graphed in Fig. 9. In this graph, the black tea valuewas disregarded. Ea of black tea was 406.02 kJ/mol and thisvalue is too high according to others. As shown in Fig. 9, allother values (41 different products) are in the range of 12.32 to82.93 kJ/mol. The accumulation of the values was in the rangeof 18 to 49.5 kJ/mol (80.5%).
CONCLUSIONS
In this study, the most commonly used or newly developedthin 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 performedwithin the last 10 years were reviewed and discussed.The main conclusions, which may be drawn from the results ofthe present study, were listed below:
a. Although there are lots of studies conducted on fruits, veg-etables, and grains, there is insufficient data in drying ofother types of foods, for example meat and fish drying.
b. The effective moisture diffusivity values in foods were inthe range of 10−12 to 10−6 m2/s and the accumulation ofthe values was in the region 10−10 to 10−8 m2/s (75%).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
m2/s.c. The studies showed that electrical drying methods were faster
than the others.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 theregion 18 to 49.5 kJ/mol.
ACKNOWLEDGEMENT
This study is a part of the MSc. Thesis titled “The investiga-tion of modeling, optimization, and exergetic analysis of dryingof olive leaves,” and supported by Ege University ScientificResearch Project no. of 2007/MUH/30.
NOMENCLATURE
a empirical model constant (dimensionless)a∗ empirical constant (s−2)a1 geometric parameter in Eqs. 5, 6
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A REVIEW OF THIN LAYER DRYING OF FOODS 461
A1, A2 geometric constantsAR aspect ratio (dimensionless)b empirical model constant (dimensionless)b∗ empirical constant (s−1)c empirical model constant (dimensionless)c∗ empirical constant (oC−1s−1)d empirical constant (K−1)e empirical constant (dimensionless)Deff effective moisture diffusivity (m2/s)(Deff)th theoretical value of effective moisture diffusiv-
ity (m2/s)D0 Arrhenius factor (m2/s)Ea activation energy for diffusion (kJ/mol) or (W/g)
in Eqs. 43,44g drying constant obtained from experimental
data (s−1)h drying constant obtained from experimental
data (s−1)H humidity (g water / kg dry air)i number of terms of the infinite seriesI radiation intensity (W/cm2)J0 roots of Bessel functionk, k1, k2 drying constants obtained from experimental
data (s−1)k0 pre-exponential constant (s−1)kth theoretical value of drying constant (s−1)K drying constant (s−1)K11, K22, K33 phenomenological coefficients in Eqs. 1–4K12, K13, K21, coupling coefficients in Eqs. 1, 2K23, K31, K32
l empirical constant (dimensionless)L thickness of the diffusion path (m); slice thick-
ness (mm) in Tables 3,5,7L1, L2, L3 dimensions of finite slab (m)Ld grain depth (mm)LD length per diameter (dimensionless)m sample amount (g)M local moisture content (kg water/kg dry matter)
or (% dry basis)Mcr critical moisture content (% dry basis)Me equilibrium moisture content (% dry basis)Mi initial moisture content (% dry basis)Mt mean moisture content at time t (% dry basis)MR fractional moisture ratio (dimensionless)MRexp,i ith experimental moisture ratio (dimensionless)MRpre,i ith predicted moisture ratio (dimensionless)n empirical model constant (dimensionless);
number of constants in Eq. 37N number of observationsNw drying rate (kg/m2s)p empirical constant (oC−1)P pressure (kPa)Pm microwave output power (W)Q heat transfer rate (W)r correlation coefficient; radius (m) in Table 1
R universal gas constant (kJ/kmol.K)RH relative humidity (%)RMSE root mean square errorRSC rotary speed column (rpm)T temperature (oC)Tabs absolute temperature (K)t time (s)x diffusion path (m)χ2 reduced chi-squareυ velocity (m/s).V volumetric flow rate (m3/s)ω dimensions (mm)α thermal diffusivity (m2s)λ empirical constant defines relationship between
Deff and Ea (m−2)φ characteristic moisture content (dimensionless)# number of models tested
Abbreviations
AA ascorbic acid solutionAEEO alkali emulsion of ethyl oleateCA citric acid solutionCBD convective type batch dryerDC drying cupboardDM drying methodFBD fluid bed dryerHW hot waterICD infrared convective dryer (with airflow)ID infrared dryer (without airflow)MD microwave dryerSD sun drying
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