Modeling the Thin-Layer Drying of Fruits and Vegetables: A Review

Modeling the Thin-Layer Drying of Fruits and Vegetables: A Review

Daniel I. Onwude, Norhashila Hashim, Rimfiel B. Janius, Nazmi Mat Nawi, and Khalina Abdan

Abstract

The drying of fruits and vegetables is a complex operation that demands much energy and time. In practice, the drying of fruits and vegetables increases product shelf-life and reduces the bulk and weight of the product, thus simplifying transport. Occasionally, drying may lead to a great decrease in the volume of the product, leading to a decrease in storage space requirements. Studies have shown that dependence purely on experimental drying practices, without mathematical considerations of the drying kinetics, can significantly affect the efficiency of dryers, increase the cost of production, and reduce the quality of the dried product. Thus, the use of mathematical models in estimating the drying kinetics, the behavior, and the energy needed in the drying of agricultural and food products becomes indispensable. This paper presents a comprehensive review of modeling thin-layer drying of fruits and vegetables with particular focus on thin-layer theories, models, and applications since the year 2005. The thin-layer drying behavior of fruits and vegetables is also highlighted. The most frequently used of the newly developed mathematical models for thinlayer drying of fruits and vegetables in the last 10 years are shown. Subsequently, the equations and various conditions used in the estimation of the effective moisture diffusivity, shrinkage effects, and minimum energy requirement are displayed. The authors hope that this review will be of use for future research in terms of modeling, analysis, design, and the optimization of the drying process of fruits and vegetables.

Keywords: diffusion, drying kinetics, fruits and vegetables, mathematical modeling, thin-layer drying

Introduction

Drying is one of the oldest and a very important unit operation, it involves the application of heat to a material which results in the transfer of moisture within the material to its surface and then water removal from the material to the atmosphere (Ekechukwu 1999; Akpinar and Bicer 2005). It is the most frequent method of food preservation and thereby increases shelf-life and improves product quality. The frequent application of drying in the food, agricultural, manufacturing, paper, polymer, chemical, and pharmaceutical industries for different purposes cannot be overemphasized. In addition to preservation, the reduction in the bulk and weight of dried products reduces handling, packaging, and transportation costs. According to Klemes and others (2008), there are over 200 dryer types which can be used for different purposes. Also, the drying features for pressure, air velocity, relative humidity, and product retention time vary according to the material and method of drying. Furthermore, drying is estimated to consume $10 \%$ to $15 \%$ of the total energy requirements of all the food industries in developed countries (Keey 1972; Klemes and others 2008). Thus, it is energy-intensive. In a nutshell, drying is arguably the

^[1]most long-standing, diverse, and conventional operation. Consequently, the engineering aspects of drying are an essential consideration. According to Kudra and Mujumdar (2002), conventional technologies are still widely preferred industrially as compared to novel technologies. This is for multiple reasons, which include simplicity of dryer construction, ease of operation, as well as the status of familiarity (Araya-Farias and Ratti 2009).

Over time, the models developed have been used in calculations involving the design and construction of new drying systems, optimization of the drying process, and the description of the entire drying behavior including the combined macroscopic and microscopic medium of heat and mass transfer. Thus, it is important to understand the basic idea of modeling the drying kinetics of fruits and vegetables. The drying conditions, type of dryer, and the characteristics of the material to be dried all have an influence on drying kinetics. The drying kinetics models are therefore significant in deciding the ideal drying conditions, which are important parameters in terms of equipment design, optimization, and product quality improvement (Giri and Prasad 2007). So, to analyze the drying behavior of fruits and vegetables it is important to study the kinetics model of each particular product.

Thin-layer drying is a widely used method for determining the drying kinetics of fruits and vegetables (Alves-Filho and others 1997; Chau and others 1997; Kiranoudis and others 1997; Kadam and others 2011). It involves simultaneous heat and mass transfer operations. During these operations, the material is fully exposed to drying conditions of temperature and hot air, thus improving the

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1. MS 20151951 Submitted 24/11/2015, Accepted 11/1/2016. Authors Onwude, Hashim, Janius, Nawi, and Abdan are with Dept. of Biological and Agricultural Engineering, Faculty of Engineering, Univ. Putra, Malaysia, 43400 UPM Serdang, Selangor, Malaysia. Author Onwude is with Dept. of Agricultural and Food Engineering, Faculty of Engineering, Univ. of Uyo, 52021 Uyo, Nigeria. Direct inquiries to author Hashim (E-mail: norhashila@apnn.edu.nm). ↩︎

drying process. The most important aspects of thin-layer drying technology are the mathematical modeling of the drying process and the equipment design which can enable the selection of the most suitable operating conditions. Thus, there is a need to explore the thin-layer modeling approach as an essential tool in estimating the drying kinetics from the experimental data, describing the drying behavior, improving the drying process, and eventually minimizing the total energy requirement.

Fruits and vegetables are highly perishable commodities that need to be preserved to increase shelf-life. The drying process can be predicted using suitable thin-layer models. Several researchers have studied the drying of fruits and vegetables using thin-layer drying models to estimate the drying time of a product (Meisamiasl and Rafiee 2009; Gupta and Alam 2014; Tzempelikos and others 2015). Evidence suggests that these models can further be used to estimate the drying curve and also predict the drying behavior, energy consumption, and heat and mass transfer of the drying process (Murthy and Manohar 2012). However, in practice, there is no single thin-layer model that can be used to effectively generalize the drying kinetics of several fruits and vegetables. This is due to a number of factors including the method of drying, the drying conditions, and the product to be dried. The application of thin-layer drying models to predicting the drying behavior of fruits and vegetables often involves the measurement of the moisture content of the material. This is done after it has been subjected to different drying conditions (temperature, air velocity, and relative humidity) and subsequent correlation with the dominant drying condition to estimate the model parameters. Incorrect collection of experimental data from the thin-layer drying experiments, will affect the drying process and, subsequently, the selection of appropriate thin-layer models. Thus, the selection of the most suitable thin-layer drying model is also a very important tool in describing the drying behavior of fruits and vegetables.

Much research has been carried out over the past few years concerning thin-layer modeling of fruits and vegetables. However, to the best of our knowledge, there has not been a review published on the theories, applications, and comparisons of the existing knowledge within the past 10 y . This gap in knowledge is a serious drawback for future developmental efforts.

Therefore, this article aims to provide a critical literature review of the drying mechanisms, theories, applications, and comparisons of thin-layer drying models for fruits and vegetables since 2005.

Factors affecting drying

The various conditions affecting the drying of fruits and vegetables include air velocity, drying temperature, size and shape of the material, and the relative humidity. Amongst these conditions, the most influential in terms of drying fruits and vegetables are drying temperature and material thickness (Meisami-asl and others 2010; Pandey and others 2010; Kumar and others 2012a). It has been argued that the air velocity rate significantly affects the drying process of food and agricultural products (Yaldiz and others 2001; Krokida and others 2003). However, this is mostly observed for crops such as rice, corn, potatoes, and so on. Studies on the drying of fruits and vegetables indicate that the air velocity has little influence on the drying kinetics of most of them (Tzempelikos and others 2014; Darıc1 and Şen, 2015). Similar results were recorded by Yaldiz and others (2001); Akpinar and others (2003); Krokida and others (2003); Menges and Ertekin (2006); Sacilik (2007); and Meisami-asl and others (2010). These authors have highlighted that the effect of air velocity could depend on the respective heat and mass transfer, which could have either internal or
external resistance. Greater internal resistance exists at a lower air velocity ( $\leq 1.5 \mathrm{~m} / \mathrm{s}$ ) than at a higher flow rate. Generally, this parameter can only have great influence at air velocity above $2.5 \mathrm{~m} / \mathrm{s}$ (El-Beltagy and others 2007; Reyes and others 2007; Perez and Schmalko 2009; Guan and others 2013).

For industrial drying, higher drying rates can be achieved with a minimum drying time when drying at higher velocities and temperatures (Erbay and Icer 2010). However, drying at a very high temperature (above $80^{\circ} \mathrm{C}$ ) (Shi and others 2008; Chen and others 2013) and higher velocity (above $2.5 \mathrm{~m} / \mathrm{s}$ ) could adversely affect the final quality of the material and increase the total energy demand (Sturm and others 2012). The higher air velocity increases heat transfer and total energy requirement during constant drying rate period. Consequently, it is not advisable to dry at extremely high temperature and air velocity.

The size and shape are also important parameters in the drying of fruits and vegetables. It is safe to note that most fruits and vegetables are dried using the thin-layer concept which means that the size of the material is reduced to dimensions that will enable uniform distribution of the drying air and temperature over the material. The shape factor is integrated into the kinetics models of drying to reduce the effect of product shape on the drying process (Pandey and others 2010). Furthermore, during the drying of fruits and vegetables, the relative humidity of the drying chamber often fluctuates due to the conditions of the ambient temperature and relative humidity of the environment, hence this has less influence on the entire drying process (Aghbashlo and others 2009; Sturm and others 2012; Misha and others 2013). In summary, during the drying process, the air velocity and relative humidity are the least significant factors that affect the drying kinetics of fruits and vegetables.

Thin-Layer Drying Theories and Modeling Drying mechanism

According to the American Natl. Standards Inst. and the American Society of Assoc. Executives (ANSI/ASAE 2014) a thin-layer is a layer of material fully exposed to an airstream during drying. Figure 1 shows a schematic of a drying chamber and product layers along the drying tray. The thickness of the layer should be uniform and should not exceed 3 layers of particles. It is assumed that the temperature distribution of a thin-layer material is uniform. This is due to the thin-layer characteristics, thus making use of lumped parameter models suitable for thin-layer drying. It is imperative to note that the this concept can be applied to (1) a single material freely exposed to the drying air or one layer of the material and (2) a multilayer of different slice thicknesses, provided the drying temperature and the relative humidity of the drying air are in the same thermodynamic condition at any time of the drying process, which thus can be applied to the mathematical estimations of the drying kinetics. However, Kucuk and others (2014) reported that the thickness of a thin layer can be increased provided there is an increase in the drying air velocity and also if the simultaneous heat and mass transfers of the material are in equilibrium with the thermodynamic state of the drying air.

Erbay and Icer (2010) reported that the mechanisms of drying all kinds of foods include surface diffusion, liquid/vapor diffusion, and capillary action within the porous region of foods. However, it has been widely reported that the dominant mechanism of moisture removal from fruits and vegetables is diffusion (Akpinar 2006a; Doymaz 2007; Raquel 2007; Duc and others 2011; Hashim and others 2014). Further, the rate of diffusion depends on the moisture content and the nature of the material. Diffusion determines

Figure 1-Schematic diagram of a drying chamber and product layers along the drying tray.
the drying rate, which can be expressed as the moisture content changes ( g of water/g of solid). However, during drying, the dominant mechanism can change due to a change in the physical structure of the drying solid after a long period of time (Jangam and Mujumdar 2010). Thus, determining the dominant mechanism can be very useful information in regards to modeling the drying process of fruits and vegetables.

Figure 2 describes the drying rate and temperature as a function of time. This rate curve can also be used in identifying the dominant mechanism of a product during drying. In the initial drying period, the equilibrium air temperature $\left(T_{\mathrm{air}}\right)$ is usually greater than the temperature of the product (Carrin and Crapiste 2008). Therefore, the drying rate between A and B increases with an increase in temperature of the product until the surface temperature attains equilibrium (Corresponding to line B to C). Under constant conditions, the drying process of agricultural and biological products has been described as a number of steps consisting also of an initial constant rate period ( B to C ) during which drying occurs as if pure water is being evaporated, and one or several falling rate periods where the moisture movement is controlled by combined external-internal resistances or by either external or internal resistance to heat and mass transfer (Araya-Farias and Ratti 2009). Mostly, many fruits and vegetables dry during the falling rate periods because the drying process is controlled by a diffusion mechanism. Drying usually stops when steady state equilibrium is reached (Erbay and Icer 2010). During the constant rate period the physical form of the product is affected, especially the surface of the product. This period is largely controlled by capillary and gravity forces. The conditions of the drying process, like the temperature, drying air velocity, and relative humidity, also affect the product during this stage. The first falling rate period (C to D) begins when the surface film of the product appears to be dry, and the moisture content has decreased to its critical moisture content $\left(M R_{c}\right)$. As drying continues, the material will then experience a change from the first falling rate period to a phenomenon known as the second falling rate period (D to E).

Tzempelikos and others (2015) reported that the drying curve of quinces shows only the presence of the falling rate period. Daricı and Şen (2015) reported that the drying process of kiwi takes in both the constant and falling rate period, stating that
the resistance to the moisture diffusion within kiwi is negligibly small as otherwise a reasonably long constant rate period would be expected. Darvishi and others (2014) reported that the entire drying process of lemon fruit occurs during the falling rate period, stating that diffusion was the dominant physical mechanism for moisture movement. In addition, for vegetables, Saeed and others (2008) reported that the diffusion mechanism also controlled the moisture movement of the drying process of Roselle (Hibiscus sabdariffa L.) which totally took place during the falling rate period. Ayadi and others (2014) found that the drying of spearmint leaves occurs during the falling rate period which is enormously influenced by the drying temperature. Similar results regarding the diffusion mechanism as the dominant controlling mechanism of the drying process of fruits and vegetables have been reported extensively in the literature. This dominant mechanism, which results in the falling drying rate period, is further exemplified in studies of the thin-layer drying of mango (Akoy 2014), pumpkin (C.moschata) (Hashim and others 2014), starfruit (Dash and others 2013), carrot (Aghbashlo and others 2009), kastamonu garlic (Sacilik and Unal 2005), and beetroot (Kaur and Singh 2014).

However, Seremet (Ceclu) and others (2015) reported an initial slight constant time period ( 5 to 10 min ) and subsequent falling rate period during the drying of pumpkin at drying temperature range of 50 to $70^{\circ} \mathrm{C}$. They stated that due to the high initial moisture content ( $90.85 \%$ w.b.) of the product, in order to remove the unbound water from the surface, an initial constant rate period was observed. Reyes and others (2007) also reported the presence of both an initial constant rate period and a falling rate period during the drying of carrots. The presence of the constant rate period can be attributed to the method and conditions of drying. Diamante and others (2010a) also attributed the presence of the constant rate period in the drying of green and gold kiwi fruits to the lower drying air velocity used.

In summary, an all-inclusive drying profile for fruits and vegetables may consist of 3 drying stages: an initial slight constant rate period (products with high moisture content), a first falling rate period, and a second falling rate period. In practice, recent evidence suggests that the drying of fruits and vegetables occurs only during the falling rate period with the initial slight constant rate period said to be negligible.

Figure 2-A typical drying curve of agricultural products showing constant rate and falling rate periods. (Adapted from Carrin and Crapiste 2008).

Thin-layer drying models

Some selected thin-layer drying models of fruits and vegetables are shown in Table 1. These models are often employed to describe drying fruits and vegetables and may be classified into 3 groups based on their comparative advantages and disadvantages and also their derivation. These are theoretical, semitheoretical, and empirical models. The most widely applied categories of thin-layer models are the semitheoretical and empirical models (Ozdemir and Devres 2000; Panchariya and others 2002; Akpinar 2006a; Doymaz 2007; Raquel and others 2011). These categories of models take into account the external resistance to moisture transport process between the material and atmospheric air, provide a greater extent of accurate results, give a better prediction of drying process behaviors, and make less assumptions due to their
reliance on experimental data. Thus, these models have proved to be the most useful for dryer engineers and designers (Brooker and others 1992). However, they are only valid within the applied drying conditions. On the other hand, the theoretical models make too many assumptions leading to a considerable number of errors (Henderson 1974; Bruce 1985), thus limiting their utilization in the design of dryers.

The semitheoretical models are usually obtained from solutions of Fick’s second law and variations of its simplified forms. The semitheoretical and some empirical models provide an understanding of the transport processes and demonstrate a better fit to the experimental data (Janjai and others 2011). Both the empirical and semitheoretical models have similar characteristics. The main challenges faced by the empirical models are that they depend largely on experimental data and provide limited information about the heat and mass transfer during the drying process (Erbay and Icier 2010). Due to the characteristics of the semitheoretical and empirical models and the high moisture content property of many fruits and vegetables, these models are widely applied in estimating drying kinetics.

Therefore, the pages that follow will attempt to discuss in detail the semitheoretical and empirical models used in the different literature sources as applied to thin-layer drying of fruits and vegetables.

Model classification

Drying processes are usually modeled using 2 main models, the distributed element model and the lumped element model (Erbay and Icer 2010). These are now described individually.

Distributed element model. This model or system is based on the interaction between time and one or more spatial variables for all of its dependent variables. The distributed element model considers the simultaneous mass and heat transfer for the drying processes. It is important to note that the pressure effect is negligible compared to the temperature and moisture effect as reported by Brooker and others (1974).

Lumped element model. This model or system considers the effect of time alone on the dependent variables. The lumped element model does not consider the change in temperature of a product and assumes a uniform distribution of drying air temperature within the product. The model includes assumptions from the Luikov equations, that is, the pressure variable is negligible and the temperature is constant (Luikov 1975). The model is presented in Eq. 1 and 2 below:

$$\begin{aligned} & \frac{\delta M}{\delta t}=\nabla^{2} K_{1} \\ & \frac{\delta T}{\delta t}=\nabla^{2} K_{12} \end{aligned}$$

where $K_{1}$ is the effective diffusivity $(D)$ and $K_{12}$ is known as the thermal diffusivity $(\alpha)$. For the constant values of $\alpha$ and $D$, Eq. 1 and 2 can further be presented as

$$\begin{aligned} & \frac{\delta M}{\delta t}=D\left[\frac{\delta^{2} M}{\delta x^{2}}+\frac{b_{1}}{x} \frac{\delta M}{\delta x}\right] \\ & \frac{\delta T}{\delta t}=\alpha\left[\frac{\delta^{2} T}{\delta x^{2}}+\frac{b_{1}}{x} \frac{\delta T}{\delta x}\right] \end{aligned}$$

In the view of Ekechukwu (1999), the assumptions for parameter $b_{1}$ are reported as $b_{1}=0$ (plate geometries), $b_{1}=1$ (cylindrical

Table 1-Thin-layer models for the drying of fruits and vegetables.

S. No	Model name	Model	Reference
1.	Newton model	$M R=\exp (-k t)$	El-Beltagy and others (2007)
2.	Page model	$M R=\exp \left(-k t^{n}\right)$	Akoy (2014); Tzempelikos and others (2014)
3.	Modified page (II)	$M R=\exp \left[-\left(K t\right)^{n}\right]$	Vega and others (2007)
4	Modified page (III)	$M R=k \exp \left(-t / d^{2}\right)^{n}$	Kumar and others (2006)
5.	Henderson and Pabis model	$M R=a \exp \left(-k t^{n}\right)$	Meisami-asl and others (2010); Hashim and others (2014)
6.	Modified Henderson and Pabis model	$M R=a \exp (-k t)+b \exp (-g t)+c \exp (-h t)$	Zenoozian and others (2008)
7.	Midilli and others model	$M R=a \exp (-k t)+b t$	Darvishi and Hazbavi (2012); Ayadi and others (2014)
8.	Logarithmic model	$M R=a \exp (-k t)+c$	Rayaguru and Routray (2012); Kaur and Singh (2014)
9.	Two-term model	$M R=a \exp \left(-K_{1} t\right)+b \exp \left(-K_{2} t\right)$	Sacilik (2007)
10.	Two-term exponential model	$M R=a \exp \left(-k_{0} t\right)+(1-a) \exp (-k_{1} a t)$	Dash and others (2013)
11.	Hii and others model	$M R=a \exp \left(-K_{1} t^{n}\right)+b \exp \left(-K_{2} t^{n}\right)$	Kumar and others (2012b)
12.	Demir and others model	$M R=a \exp \left(-K t\right)^{n}+b$	Demir and others (2007)
13.	Verma and others model	$M R=a \exp (-k t)+(1-a) \exp (-g t)$	Akpinar (2006)
14.	Approximation of diffusion	$M R=a \exp (-k t)+(1-a) \exp (-k b t)$	Yaldiz and Ertekịn (2007)
15.	Modified Midilli and others	$M R=a \exp (-k t)+b$	Gan and Poh (2014)
16.	Aghbashlo and others model	$M R=\exp \left(\frac{K}{1-k_{0}^{2} t}\right)$	Aghbashlo and others (2009)
17.	Wang and Singh	$M R=1+a t+b t^{2}$	Omolola and others (2014)
18.	Diamante and others model	$\ln (-\ln M R)=a+b(\ln t)+c(\ln t)^{2}$	Diamante and others (2010)
19.	Weibull model	$M R=\alpha-b \exp \left(-k_{0} t^{n}\right)$	Tzempelikos and others (2015)
20.	Thompson	$t=a \ln (M R)+b[\ln (M R)]^{2}$	Pardeshi (2009)
21.	Silva and others model	$M R=\exp \left(-a t-b_{\mathrm{w}} \bar{t}\right)$	Pereira and others (2014)
22.	Peleg model	$M R=1-t /(a+b t)$	Da Silva and others (2015)

shapes), and $b_{1}=2$ (spherical shapes). However, these assumptions result in error for the temperature reading at the beginning of the drying process (Erbay and Icier 2010).

Theoretical models

The theoretical models consider both the external and internal resistance to moisture transfer. They involve the geometry of the material, its mass diffusivity, and the conductivity of the material (Cihan and Ece 2001). Thus the resistances can be estimated from Eq. 3 and 4 because these equations describe the mass transfer (Erbay and Icer 2010). Subsequently, the solution of Fick’s second law of diffusion is widely applied as a theoretical model in the thin-layer drying of food products (Kucuk and others 2014).

Semitheoretical models

The semitheoretical models are derived from the theoretical model (Fick’s second law of diffusion) or its simplified variation (Newton’s law of cooling). The Lewis, Page, and Modified Page semitheoretical models are derived from Newton’s law of cooling. The (i) exponential model and simplified form, (ii) 2-term exponential model and modified form, and (iii) 3-term exponential model and simplified form are all derived from Fick’s second law of diffusion (Erbay and Icier 2010). Factors that could determine the application of these models include the drying temperature, drying air velocity, material thickness, initial moisture content, and relative humidity (Panchariya and others 2002; Erbay and Icer 2010). Furthermore, under these conditions it can be noted that the complexity of the models can be attributed to the number of constants. In respect to the scientific literature, the number of constants varies between 1 (Newton model), 5 (Hii and others model), and 6 (Modified Henderson and Pabis model) (see Table 1). More so, Table 2 shows the relationship between some thin-layer model constants and drying conditions for various fruits and vegetables. Taking the number of constants into consideration, both the Hii and others model and the Modified Henderson and Pabis model can be said to be complex, while the Newton model is the simplest. However, the selection of the most appropriate model for describing the drying behavior of fruits and vegetables does not depend on the number of constants. Rather, it depends on various statistical indicators. The
statistical indicators that have often been used to successfully select the most appropriate drying models as reported in the literature (Akpinar 2006b; Babalis and others 2006; Menges and Ertekin 2006; Doymaz 2007; Vega and others 2007; Saeed and others 2008; Erbay and Icier 2010; Fadhel and others 2011; Kadam and others 2011; Rasouli and others 2011; Akoy 2014; Gan and Poh 2014; Tzempelikos and others 2014; Darcu and Şen 2015; Onwude and others 2015a; Tzempelikos and others 2015) include $R, R^{2}\left(t^{2}\right), x^{2}, S S E, R M S E, R R M S, E F, M P E$, and $M B E$. The higher the values of $R$ and $R^{2}$ of a particular model the better the model is in predicting the drying behavior of fruits and vegetables. Similarly, the lower the values of $x^{2}, S S E, R M S E, R R M S, E F, M P E$, and $M B E$ of a particular model the more suitable the model is in predicting the drying kinetics of the particular product (Kucuk and others 2014). The semitheoretical models made available in the literature over the past 10 y are discussed below. These models have been widely used in expressing the thin-layer-drying kinetics of fruits and vegetables as shown in Table 2.

Models derived from Newton’s law of cooling.

Newton model. This model is sometimes referred to in the literature as the Lewis model or the Exponential model, Single exponential model. It is said to be the simplest model because of the single model constant. In the past, this model has been widely applied in describing the drying behavior of several food and agricultural products. Recently, it has occasionally been found suitable for describing the drying behavior of some fruits and vegetables:

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=\exp \left(-k^{\prime}\right)$$

where $k$ is the drying constant $\left(\mathrm{s}^{-1}\right), M R$ is the moisture ratio, $M$ is the dry basis moisture content at any time $t, M_{0}$ is the initial dry basis moisture content of the sample, and $M_{c}$ is the equilibrium moisture content. Furthermore, the Newton model has been found to be suitable in describing the drying behavior of strawberry and red chili as shown in Table 2.

Page model. The Page model or the Modified Lewis model is an empirical modification of the Newton model, whereby the errors associated with using the Newton model are greatly minimized

Table 2-Studies conducted on thin-layer drying modeling of fruits and vegetables in the past 10 years.

Material	Drying method	Process conditions	Eqn. No#	Best model	Relationship between model constants and process condition	Reference
Apple	TD	$w=8 \mathrm{~mm} ; h=8 \mathrm{~mm} ; L=18 \mathrm{~mm} ; A=w \times h$ $\times L ; T=60$ to $80^{\circ} \mathrm{C} ; V=1$ to $1.5 \mathrm{~m} / \mathrm{s}$	11	Midilli and others	$\begin{aligned} & \mathrm{a}=1.004084-0.000013 T-0.001960 V+ \\ & 3.944759 A \\ & \mathrm{k}=-0.006391+0.000065 T+0.009775 V+ \\ & 1.576723 A \\ & \mathrm{n}=1.187734+0.002467 T-0.128878 V- \\ & 202.536 A \\ & \mathrm{~b}=0.000082-0.000002 T-0.000041 V+ \\ & 0.041667 A \end{aligned}$	Akpinar (2006)
Apple slices	LHCD	$h=3$ to $7 \mathrm{~mm} ; T=50$ to $95^{\circ} \mathrm{C} ; V=1 \mathrm{~m} / \mathrm{s}$	11	Midilli and others	$-$	Zarein and others (2013)
Apple slices (Golab)	OD	$V=0.5 \mathrm{~m} / \mathrm{s} ; T=40$ to $80^{\circ} \mathrm{C} ; h=2$ to 6 mm	11 and 9	Midilli and others; Henderson and Pabis	Henderson and Pabis: $a=0.971449897+0.002099298 T-$ $0.10565552 h$ $k=-1.061883481+T^{0.022639541}-$ $0.004633918 h$	Meisami-asl and others (2010)
Apple slices (McIntosh)	LHCD	$T=50$ to $70^{\circ} \mathrm{C} ; V=0.6$ to $1.4 \mathrm{~m} / \mathrm{s} ; h=4$ to 12 mm	12	Logarithmic	$-$	Kaleta and Górnicki (2010)
Apricot	LHCD	$T=60$ to $100^{\circ} \mathrm{C} ; h=5 \mathrm{~mm} ; V=0.2 \mathrm{~m} / \mathrm{s}$	22	Diamante and others	$-$	Diamante and others (2010)
Banana (Musa acuminata)	LHCD	$T=40$ to $70^{\circ} \mathrm{C}$	6 and 25	Page; Silva and others	$-$	Pereira and others (2014)
Banana (Musa spp)	LHCD	Intermittent time $=0.5,1.0$ and $2.0 \mathrm{~h} ; T=$ $70^{\circ} \mathrm{C} ; V=0.55 \mathrm{~m} / \mathrm{s} ; R H=51 \%$ to $56 \%$	26	Peleg	$-$	Da Silva and others (2015)
Banana (Musa sapientum, shum)	OMD	$d=1.8 \mathrm{~cm} ; L=10 \mathrm{~cm} ; T=25$ to $55^{\circ} \mathrm{C} ; C_{4}=$ $30 \%$ to $60 \% ; \mathrm{NaCl}=0 \%$ to $10 \%$	26	Peleg	$-$	Mercali and others (2010)
Banana (Mabonde) slices	DMO	$h=5 \mathrm{~mm} ; P=100$ to 300 W	21	Wang and Singh	$-$	Omolola and others (2014)
Banana slices	ISD	$h=5 \mathrm{~mm} ; V=4.23 \mathrm{~m} / \mathrm{min}$	21	Wang and Singh	$-$	Fadhel and others (2011)
Banana slices	LHCD	$T=50$ to $80^{\circ} \mathrm{C} ; V=2.4 \mathrm{~m} / \mathrm{s} ; R H=4 \%$ to $25 \%$	6 and 12	Page; Logarithmic	$-$	Doymaz (2010)
Basil leaves	TTD	$T=55$ to $65^{\circ} \mathrm{C} ; h=0.33 \pm 0.08 \mathrm{~mm}$	12	Logarithmic	$-$	Kadam and others (2011)
Basil leaves	OSD	-	7	Modified page (II)	$-$	Akpinar (2006)
Beetroot	TD and MD	$w=10 \mathrm{~mm} ; h=10 \mathrm{~mm} ; L=3 \mathrm{~mm} ; T=55$ to $75^{\circ} \mathrm{C} ; P=540$ to 1080 W	13 and 12	Two term; Logarithmic	$-$	Kaur and Singh (2014)
Bitter melon (Momordica charantia) slices	LHCD	$h=0.5$ to $1.0 \mathrm{~cm} ; T=50$ to $80^{\circ} \mathrm{C} ; V=$ $1.2 \mathrm{~m} / \mathrm{s}$	6	Page	$-$	Chen and others (2013)
Blueberries	IR	Treatment $=$ Infusion; $h=10 \mathrm{~mm} ; T=60$ to $90^{\circ} \mathrm{C} ; I=4000 \mathrm{~W} / \mathrm{m}^{2}$	24	Thompson	$-$	Shi and others (2008)
Carrot	LHCD	$T=50$ to $70^{\circ} \mathrm{C} ; V=1 \mathrm{~m} / \mathrm{s} ; R H=35 \%$ to $42 \%$	20	Aghbashlo and others	$k_{1}=54.18 e^{\left(\frac{2951.2}{1000}\right)}$ $k_{1}=-4.2710^{-10} e^{\left(\frac{-4746.3}{1000}\right)}$ $g=0.005636^{T}-0.184143$ $a=0.37742$	Aghbashlo (2009) Botelho and others (2011)

(Continued)

Table 2-Continued.

Material	Drying method	Process conditions	Eap. No#	Best model	Relationship between model constants and process condition	Reference
Carrot (pomace)	LHCD	$T=60$ to $75^{\circ} \mathrm{C} ; V=0.7 \mathrm{~m} / \mathrm{s} ; h=10 \mathrm{~mm}$	15	Hi and others	$\begin{aligned} & a=13905.60529-619.95731 T+ \\ & 9.18210 T^{2}-0.04516 T^{3} \\ & k=-0.21037+0.01238 T-0.00022 T^{2} \\ & n=16.45430-0.69205 T+0.01039 T^{2}- \\ & 0.00005 T^{3} \\ & c=13908.27+620.14037 T- \\ & 9.18481 T^{2}+0.4517 T^{3} \\ & g=0.05894+0.00082 T-0.00005 T^{2} \end{aligned}$	Kumar and others (2012b)
Chili (pickino)	OD and FBD	$T=45$ to $65^{\circ} \mathrm{C} ; V=2.4 \mathrm{~m}^{3} / \mathrm{min}$ (FBD)	11	Midilli and others	$-$	Mihindukulasuriya and others (2013)
Date palm fruit	MD	$P_{d}=4.0$ to $9.5 \mathrm{w} / \mathrm{g}$	6	Page	$\begin{aligned} & k=0.8094 P_{d}^{-1.2282} \\ & n=-0.0189 P_{d}^{3}+0.374 P_{d}^{2}-2.3423 P_{d}+ \\ & 6.8142 \end{aligned}$	Darvishi and Hazbavi (2012)
Fig (Ficus carica)	TTD	$T=55$ to $85^{\circ} \mathrm{C} ; V=0.5$ to $3.0 \mathrm{~m} / \mathrm{s}$	13	Two term	$-$	Babalis and others (2006)
Garlic (Allium sativum L)	LHCD	$T=50$ to $70^{\circ} \mathrm{C} ; h=2$ to 4 mm	23	Weibull	$\begin{aligned} & a=5.994251 \times h^{-0.164} \exp \left(\frac{-516.322}{h a h}\right) \\ & b=6.02554 \times 10^{-6} \times h^{1.065} \exp \left(\frac{2426.964}{h a h}\right) \end{aligned}$	Rasouli and others (2011)
Golden apples	LHCD	$T=60$ to $80^{\circ} \mathrm{C} ; V=1$ to $3 \mathrm{~m} / \mathrm{s}$	11	Midilli and others	$\begin{aligned} & a=1.4678-0.0067 T \\ & k=1.0835 V^{0.1316} \\ & n=0.8867 \end{aligned}$	Menges and Ertekin (2006)
Green bean	OSD	$\begin{aligned} & V=0.5 \text { to } 1.5 \mathrm{~m} / \mathrm{s} ; T=39.09 \text { to } 43.81^{\circ} \mathrm{C} ; \\ & R H=49.91 \% \text { to } 65.06 \%; S_{R}=752.10 \\ & \mathrm{~W} / \mathrm{m}^{2} \end{aligned}$	6	Page	$\begin{aligned} & b=0.0030 \\ & k=0.3560-0.1407 V \\ & n=0.7832+0.0892 \ln (V) \end{aligned}$	Yaldiz and Ertekyn (2007)
Green bean (Phaseolus vulgaris L)	LHCD	$L=4 \pm 0.1 \mathrm{~cm} ; T=50$ to $70^{\circ} \mathrm{C} ; R H=8 \%$ to $25 \%$	6	Page	$-$	Doymaz (2005)
Green peas	LHCD	$T=55$ to $75^{\circ} \mathrm{C} ; V=1.67 \mathrm{~m} / \mathrm{s}$	24	Thompson	$\begin{aligned} & a=-0.013046 T^{3}+2.54139 T^{2}- \\ & 162.3588 T+3362.0605 \\ & b=-0.001602 T^{3}+0.29470 T^{2}- \\ & 17.78930 T+359.30170 \end{aligned}$	Pardeshi (2009)
Green pepper	OSD	$\begin{aligned} & V=0.5 \text { to } 1.5 \mathrm{~m} / \mathrm{s} ; T=38.56 \text { to } 42.52^{\circ} \mathrm{C} ; \\ & R H=42.20 \% \text { to } 59.36 \%; S_{R}=738.54 \\ & \mathrm{~W} / \mathrm{m}^{2} \end{aligned}$	18	Approximation of diffusion	$\begin{aligned} & a=-1.6626+1.7015 V \\ & k=0.3549-0.1489 V \\ & b=0.5868-0.0172 V \end{aligned}$	Yaldiz and Ertekyn (2007)
Green table olives	LHCD	$L=24$ to $26 \mathrm{~mm} ; d=19$ to $21 \mathrm{~mm} ; T=40$ to $70^{\circ} \mathrm{C} ; V=1.0 \mathrm{~m} / \mathrm{s} ; R H=15 \% \pm 2 \%$	16	Demir and others	$-$	Demir and others (2007)
Hawthorn	LHCD	$V=0.8 \mathrm{~m} / \mathrm{s} ; T=50$ to $70^{\circ} \mathrm{C}$	11	Midilli and others	$-$	Unal and Sacilik (2011)
Jackfruit	TD	$h=3 \mathrm{~mm} ; T=50$ to $80^{\circ} \mathrm{C}$	11	Midilli and others	$\begin{aligned} & k=-1 \times 10^{-5} T^{2}+0.001 T-0.058 \\ & n=0.013 T \\ & b=-1 \times 10^{-6} T^{2}+0.12 \\ & a=1.00 \end{aligned}$	Saxena and Dash (2015)
Jackfruit	LHCD	$h=5 \pm 1 \mathrm{~mm} ; T=40$ to $70^{\circ} \mathrm{C}$; triangle: $w=$ $5.5 \mathrm{~cm} ;$ height $=6.5 \mathrm{~cm} ;$ rectangle: $w=$ $3.0 \mathrm{~cm} ; L=5.5 \mathrm{~cm} ;$ square: $w=4 \mathrm{~cm} ; L=$ 4 cm	19	Modified Midilli and others	$-$	Gan and Poh (2014)
Kiwifruit	LHCD	$T=50$ to $80^{\circ} \mathrm{C} ; V=0.5$ to $2.0 \mathrm{~m} / \mathrm{s} ; h=4$ and $6 \mathrm{~mm} ; R H=5 \%$ to $20 \%$	11	Midilli and others	$-$	Danco and Sen (2015)
Kiwifruit	LHCD	$T=60$ to $100^{\circ} \mathrm{C} ; h=5 \mathrm{~mm} ; V=0.2 \mathrm{~m} / \mathrm{s}$	22	Diamante and others	$-$	Diamante and others (2010a)
Kiwifruit	LHCD	$h=0.006$ to $0.4 \mathrm{~m} ; T=30$ to $90^{\circ} \mathrm{C}$	6	Page	$\begin{aligned} & n=0.796 \\ & k=4.756 \times 10^{-5} T-5.54 \times 10^{-4} \end{aligned}$	Simal (2005)

Table 2-Continued.

Material	Drying method	Process conditions	Eqn. No#	Best model	Relationship between model constants and process condition	Reference
Kiwifruit (Hayward)	CTD	$T=40$ to $80^{\circ} \mathrm{C} ; V=0.5$ to $1.5 \mathrm{~m} / \mathrm{s} ; h=2$ to 6 mm	6	Page	$\begin{aligned} & K=0.0042517-0.00012228 T+ \\ & 0.0000014281 T^{2} \\ & n=0.60498+0.0218756 T-0.0001634 T^{2} \end{aligned}$	Mohammadi and others (2008)
Mango slices	LHCD	$T=60$ to $80^{\circ} \mathrm{C} ; V=0.5 \mathrm{~m} / \mathrm{s} ; h=3 \mathrm{~mm}$	6	Page	-	Akoy (2014)
Mango ginger (Curcuma amanda roxb)	MD	$h=1.77 \pm 0.02 \mathrm{~mm} ; P=315$ to 800 W	11	Midilli and others	-	Murthy and Manohar (2012)
Mango slices (Mangifera indica L.)	LHCD	$L=0.45 \mathrm{~m} ; w=0.34 \mathrm{~m} ; h=0.03 \mathrm{~m} ; V=$ 1.76 to $1.91 \mathrm{~m} / \mathrm{s} ; T=50$ to $80^{\circ} \mathrm{C}$	11	Midilli and others	-	Corzo and others (2011)
Mint leaves	OSD	-	7	Modified page (II)	-	Akpinar (2006)
Onion slices	LHCD	$T=50$ to $70^{\circ} \mathrm{C} ; V=0.5$ to $2.0 \mathrm{~m} / \mathrm{s} ; h=5 \pm$ 0.1 mm	6	Page	Vertical: $k=-0.0099+2.7 \times 10^{-4} T+3.3 \times 10^{-4} V$ $n=1.24-0.0036 T+0.037 V$ Horizontal: $k=-0.022+5.5 \times 10^{-4} T+2 \times 10^{-4} V$ $n=1.23-0.0045 T+0.034 V$	El-mesery and Mwithiga (2012)
Onion slices	OSD	$h=12.5 \mathrm{~mm} ; V=0.5$ to $1.5 \mathrm{~m} / \mathrm{s} ; T=40.52$ to $44.32^{\circ} \mathrm{C} ; R H=48.91 \%$ to $60.28 \%$; $S_{R}=752.10 \mathrm{~W} / \mathrm{m}^{2}$	13	Two term	$\begin{aligned} & a=0.4866+0.6424 \ln (V) \\ & K_{0}=0.1557+0.1995 \ln (V) \\ & b=0.5143-0.6424 \ln (V) \\ & K_{1}=0.1117-0.0992 \ln (V) \end{aligned}$	Yaldjiz and Ertekÿn (2007)
Onion slices	IR + LHCD	$h=2$ to $6 \mathrm{~mm} ; T=60$ to $80^{\circ} \mathrm{C} ; T_{f}=30$ to $50^{\circ} \mathrm{C} ; V=0.8$ to $2.0 \mathrm{~m} / \mathrm{s}$	8	Modified page (III)	$\begin{aligned} & K=-0.04127+0.00055 T- \\ & 0.00027 h+1.6704 \times 10^{-06} T_{f}+ \\ & 0.00158 V+3.086 \times 10^{-05} t^{2} \end{aligned}$	Kumar and others (2006)
Parsley	OSD	-	17	Verma and others	-	Akpinar (2006)
Pepper	MD	$d=0.7 \pm 0.1 \mathrm{~cm} ; L=6 \pm 1 \mathrm{~cm} ; P=180$ to 540 W	11	Midilli and others	$\begin{aligned} & k=0.0847 \exp (0.0031 P) \\ & n=3 \times 10^{-8} P^{3}-3 \times 10^{-5} P^{2}+ \\ & 0.0136 P-0.2157 \\ & b=8 \times 10^{-8} P^{3}-0.0019 P+0.0218 \\ & a=3 \times 10^{-7} P^{2}-0.0003 P+1.0474 \end{aligned}$	Darvishi and others (2014)
Persimmon slices (Diospyros kakil)	CTD	$T=50$ to $70^{\circ} \mathrm{C} ; h=3$ to $8 \mathrm{~mm} ; V=2 \pm$ $0.1 \mathrm{~m} / \mathrm{s}$	11, 6 and 23	Midilli and others; Page; Weibull		Doymaz (2012)
Pineapple	FIR + HCD	$h=15 \mathrm{~mm} ; l=1$ to $5 \mathrm{~kW} / \mathrm{m}^{2} ; T=40$ to $60^{\circ} \mathrm{C} ; V=0.5$ to $1.5 \mathrm{~m} / \mathrm{s}$	11	Midilli and others		Ponkham and others (2012)
Plum	PSD	Blanching $=20 \mathrm{~s} ; T=85^{\circ} \mathrm{C} ; v=0.8 \mathrm{~m} / \mathrm{s}$	13	Two term	-	Jazini and Hatamipour (2010)
Pumpkin	LHCD	Osmotic treatment: $h=2 \mathrm{~cm} ; w=2 \mathrm{~cm} ; L=$ $2 \mathrm{~cm} ; T=50$ to $60^{\circ} \mathrm{C}$	13 and 10	Two term (sucrose treated) Modified Henderson and Pabis (pretreated)	-	Zenoozian and others (2008)
Pumpkin slices	OSD	$\begin{aligned} & h=12.5 \mathrm{~mm} ; V=0.5 \text { to } 1.5 \mathrm{~m} / \mathrm{s} ; R H= \\ & 45.00 \% \text { to } 57.70 \% ; T=39.10 \text { to } \\ & 43.63^{\circ} \mathrm{C} ; S_{R}=738.54 \mathrm{~W} / \mathrm{m}^{2} \end{aligned}$	18	Approximation of diffusion	$\begin{aligned} & a=0.8095 \exp (-0.0794 V) \\ & k=0.2082 V^{-0.3032} \\ & b=0.6857 \exp (0.4860 V) \end{aligned}$	Yaldjiz and Ertekÿn (2007)
Pumpkin slices	TD	$h=5 \mathrm{~mm} ; d=35 \mathrm{~mm} ; T=60$ to $80^{\circ} \mathrm{C} ; V=$ 1 to $1.5 \mathrm{~m} / \mathrm{s}$	11	Midilli and others	$\begin{aligned} & a=0.966567+0.000184 T+0.007014 V \\ & k=0.005645-0.000095 T+0.003791 V \\ & n=0.572175+0.009074 T-0.064652 V \\ & b=0.000050-0.000001 T-0.000024 V \end{aligned}$	Akpinar (2006)
Pumpkin (C. maxima)	LHCD	$T=30$ to $70^{\circ} \mathrm{C}$	6	Page	-	Guiné and others (2011)
Pumpkin (C. maxima) slices	PPCD	$T=50$ to $70^{\circ} \mathrm{C} ; h=25 \mathrm{~mm} ; d=20 \mathrm{~mm} ; V$ $=2.5 \mathrm{~m} / \mathrm{s}$	11	Midilli and others	-	Perez and Schmalko (2009)

Table 2-Continued.

Material	Drying method	Process conditions	Eap. No#	Best model	Relationship between model constants and process condition	Reference
Pumpkin (C. pepo)	CTD	$T=60$ to $80^{\circ} \mathrm{C} ; h=10$ to $30 \mathrm{~mm} ; V=$ $1.5 \mathrm{~m} / \mathrm{s}$	12	Logarithmic	$-$	Olurin and others (2012)
Pumpkin slices (C. pepo L.)	CTD	$T=50$ to $60^{\circ} \mathrm{C} ; V=1.0 \mathrm{~m} / \mathrm{s} ; R H=15 \%$ to $25 \%$	12 and 17	Logarithmic; Verma and others	$-$	Doymaz (2007)
Pumpkin (C. pepo L)	LHCD	$T=40$ to $60^{\circ} \mathrm{C} ; V=0.8 \mathrm{~m} / \mathrm{s}$	13 and 12	Two term; Logarithmic	$-$	Sacilik (2007)
Pumpkin (C. moschata)	LHCD	$h=3$ to $7 \mathrm{~mm} ; V=1.16 \mathrm{~m} / \mathrm{s} ; T=50$ to $80^{\circ} \mathrm{C}$	15	Hii and others	$\begin{aligned} & a=0.761+0.0369 h-0.00530 T \\ & b=0.297+0.00454 T-0.0405 h \\ & K_{1}=0.1528 \exp ^{2.6765 h} K_{2}= \\ & -0.00501+0.000393 T- \\ & \left(8.84 E-006 T^{2}\right)+\left(615 E-008 T^{3}\right) n= \\ & -10.581+0.427 T-0.00344 T^{2} \end{aligned}$	Onwude and others (2015b)
Quercus	LHCD	$T=50$ to $70^{\circ} \mathrm{C} ; V=0.5$ to $1 \mathrm{~m} / \mathrm{s}$; Optimum: $T=70^{\circ} \mathrm{C}$ and $V=1 \mathrm{~m} / \mathrm{s}$	6	Page	$\begin{aligned} & k=-0.7 V^{0.64} e^{1 / 100} \\ & n=0.067 V^{0.051} e^{1 / 100} \end{aligned}$	Tahmasebi and others (2011)
Quince slices	LHCD	$T=40$ to $60^{\circ} \mathrm{C} ; R H=10 \% ; h=12 \mathrm{~mm} ; V=$ $2 \mathrm{~m} / \mathrm{s}$	23	Weibull	$\begin{aligned} & \alpha=-0.150540+0.004924 T-0.000045 T^{2} \\ & b=-1.231270+0.011399 T-0.000120 T^{2} \\ & k_{0}=0.264050-0.009648 T+0.000150 T^{2} \\ & n=1.504030-0.026073 T+0.000320 T^{2} \end{aligned}$	Tzempelikos and others (2015)
Quince slices	LTCD	$T=40$ to $60^{\circ} \mathrm{C} ; V=1$ to $3 \mathrm{~m} / \mathrm{s} ; R H=10 \%$	6	Page	$-$	Tzempelikos and others (2014)
Red chili	SDA	Pretreatment: $T=40$ to $65^{\circ} \mathrm{C} ; R H=10 \%$ to $60 \% ; V=0.12$ to $1.02 \mathrm{~m} / \mathrm{s}$	5	Newton	$\begin{aligned} & k=0.003484-0.000222 T+3.66 \times \\ & 10^{-6} T^{2}-0.007085 R H+ \\ & 0.00572(R H)^{2}+0.002738 V- \\ & 0.001235 V^{2} \end{aligned}$	Hossain and others (2007)
Saffron (Crocus sativus L)	ID	$T=60$ to $110^{\circ} \mathrm{C} ; h=0.7 \mathrm{~mm}$	11	Midilli and others	$-$	Akhondi and others (2011)
Spearmint	ATB	$T=40$ to $55^{\circ} \mathrm{C}$	11	Midilli and others	$\begin{aligned} & a=6.40163-0.31813 T+0.00616 T^{2}- \\ & 3.921010^{-5} T^{3} \\ & k=-1.49365+0.12232 T-0.00309 T^{2}+ \\ & 2.4973310^{-5} T^{3} \\ & b=-0.11604+0.00739 T- \\ & 1.55610^{-4} T^{2}+1.0810^{-6} T^{3} \\ & n=25.89022-1.79971 T+0.04203 T^{2}- \\ & 3.2032710^{-4} T^{3} \end{aligned}$	Ayadi and others (2014)
Starfruit slices	LHCD	$T=60$ to $80^{\circ} \mathrm{C} ; h=5 \mathrm{~mm}$	6	Page	$-$	Hii and Ogugo (2014)
Starfruit slices	TD	$h=4 \mathrm{~mm} ; T=50$ to $80^{\circ} \mathrm{C}$	14	Two term exponential	$-$	Dash and others (2013)
Stone apple	LHCD	$T=40$ to $70^{\circ} \mathrm{C} ; h=8 \mathrm{~mm} ; V=1.1 \pm$ $0.2 \mathrm{~m} / \mathrm{s}$	12	Logarithmic	$\begin{aligned} & a=0.001 T+0.945 \\ & k=1.49 E-06 T^{2}-0.0001 T+0.005 \\ & c=8 E-05 T^{2}-0.01 T+0.256 \end{aligned}$	Rayaguru and Routray (2012)
Strawberry	IFSD	Pretreatment: $I=600 \mathrm{~W} / \mathrm{m}^{2}$	5	Newton	$k=0.0042 E A / M+0.0342 T$	El-Beltagy and others (2007)
Stuffed pepper	OSD	$\begin{aligned} & V=0.5 \text { to } 1.5 \mathrm{~m} / \mathrm{s} ; T=39.57 \text { to } 45.82^{\circ} \mathrm{C} ; \\ & R H=45.03 \% \text { to } 63.40 \% ; S_{R}=752.10 \\ & \mathrm{~W} / \mathrm{m}^{2} \end{aligned}$	13	Two term	$\begin{aligned} & a=0.6315-0.2957 V \\ & K_{D}=0.0224 \exp (4.7396 V) \\ & b=0.3679+0.2962 V \\ & K_{1}=0.0677-0.0117 \ln (V) \\ & a=0.1285+0.0206 T+0.1299 R H \\ & k=0.0373+0.000014 T+0.00075 R H \\ & b=0.1478+0.0329 T+0.1402 R H \end{aligned}$	Yaldiz and Ertekjin (2007)
Tomato (cv. Milen) slices	STD; OSD	$T=22.4$ to $35.6^{\circ} \mathrm{C} ; R H=14.5 \%$ to $50.9 \%$; $S_{R}=202.3$ to $767.4 \mathrm{~W} / \mathrm{m}^{2}$	18	Approximation of diffusion	$\begin{aligned} & a=0.1285+0.0206 T+0.1299 R H \\ & k=0.0373+0.000014 T+0.00075 R H \\ & b=0.1478+0.0329 T+0.1402 R H \end{aligned}$	Sacilik and others (2006)

by the addition of a dimensionless empirical constant $(n)$ :

$$M R=\frac{\left(M-M_{e}\right)}{\left(M_{0}-M_{e}\right)}=\exp \left(-k t^{n}\right)$$

where $n$ is known as the model constant (dimensionless).
This model has 2 constants and is widely used as the basis for most semitheoretical thin-layer models. Moreover, the Page (1949) model has been adopted as an American Standard in thin-layer modeling of agricultural and biological products (ANSI/ASAE 2014). From Table 2, it can be seen that the Page model, along with the Midilli and others model are the most suitable in describing the drying behavior of various fruits and vegetables. This model was found to be the most appropriate in describing the drying behavior of banana, date palm, green bean, kiwifruit, mango, onion, bitter melon (Momordica charantia), persimmon, pumpkin, quercus, quince, and star fruit as shown in Table 2.

Modified Page model. As the name implies, this is a modification of the Page model. Erbay and Icier (2010) reported 3 forms of the Modified Page model (I, II, and III). For the purpose of this literature review, the following Modified Page models (Eq. 7 and 8) have been found to be the most suitable in describing the drying behavior of different fruits and vegetables. They include

$$\mathrm{MR}=\frac{\left(M-M_{e}\right)}{\left(M_{0}-M_{e}\right)}=\exp \left(-(k t)^{n}\right)$$

Equation 7 is widely regarded as the Modified Page model (II). This model has 2 constants and has been applied in predicting the drying kinetics of mint leaves and basil leaves as shown in Table 2.

$$\mathrm{MR}=\frac{\left(M-M_{e}\right)}{\left(M_{0}-M_{e}\right)}=k \exp \left(-t / d^{2}\right)^{n}$$

where $d$ is an empirical constant (dimensionless)
Equation 8 can be called the Modified Page model (III). This model has 3 constants and can successfully describe the drying behavior of onion (see Table 2).

Models derived from Fick’s second law of diffusion.

Henderson and Pabis or single-term model. This model is the first term of the general solution of the Fick’s second law of diffusion (Eq. 27). This can also be regarded as a simple model with only 2 model constants. The Henderson and Pabis (1961) model has been effectively applied in the drying of crops such as corn and millet. However, it has not been quite so successful in describing the drying behavior of fruits and vegetables, since the model has been found applicable only to apple (See Table 2):

$$M R=\frac{\left(M-M_{e}\right)}{\left(M_{0}-M_{e}\right)}=a \exp (-k t)$$

where $a$ represents the shape of the materials used (dimensionless).
Modified Henderson and Pabis model. The modified Henderson and Pabis model is a third term general solution of the Fick’s second law of diffusion (Eq. 27) for correction of the shortcomings of the Henderson and Pabis model. It has been reported that the first term explains the last part of the drying process of food and agricultural products, which occurs largely in the falling late period, the second term describes the midway part, and the third term explains the initial moisture loss of the drying process (Erbay and Icier 2010). The model contains 6 constants (Eq. 10), just like the Hii and others model (modified 2-term model), and based on this the model has been referred to as complex thin-layer model.

However, it should be emphasized also that with 6 parameters, many more than 6 data points are required to compute the model. The model is not that complex with the advent of computers, but, statistically, a good degree of freedom is required for confidence, and this will require many data points.

$$M R=\frac{\left(M-M_{e}\right)}{\left(M_{0}-M_{e}\right)}=a \exp (-k t)+b \exp (-g t)+c \exp (-h t)$$

where $a, b$, and $c$ are dimensionless model constants, and $g$ and $h$ are the drying constants $\left(\mathrm{s}^{-1}\right)$.

From Table 2 it can be said that this model does not effectively describe the drying process of most fruits and vegetables. This model has been found to only successfully describe the drying kinetics of pretreated pumpkin.

Midilli and others model. Midilli and others (2002) proposed a new model by a modification of the Henderson and Pabis model by the addition of an extra $t$ with a coefficient. The new model, which is a combination of an exponential term and a linear term, has been validated by testing the model on mushroom, pollen, and pistachio.

$$M R=a \exp (-k t)+b t$$

where $a$ and $b$ are the model constants and $k$ is the drying constant $\left(\mathrm{s}^{-1}\right)$ to be estimated from the experimental data.

This model is sometimes called the Midilli Kucuk model or the Midilli model. It contains 3 constants and has been found to be the best in describing the drying behavior of different fruits and vegetables. From Table 2 it can be seen that this model is noted as the most suitable in over $24 \%$ of the literature sources reviewed. Thus, it has been found to be suitable in describing the drying kinetics of fruits and vegetables such as apple, chili, golden apples, hawthorn, jackfruit, kiwifruit, mango, ginger, pepper, persimmon, pineapple, pumpkin, saffron, and spearmint.

Logarithmic model. This model is also known as an asymptotic model and is another modified form of the Henderson and Pabis model. It is actually a logarithmic form of the Henderson and Pabis model with the addition of an empirical term. The model contains 3 constants and can be expressed as

$$M R=\frac{\left(M-M_{e}\right)}{\left(M_{0}-M_{e}\right)}=a \exp (-k t)+c$$

where $c$ is a dimensionless empirical constant.
Table 2 shows that this model has been found to be the fourth best thin-layer model in describing the drying kinetics of various fruits and vegetables. Consequently, the model has produced the best fit in predicting the drying kinetics of apple, basil leaves, beetroot, pumpkin, and stone apple.

Two-term model. The 2 -term model is a second term general solution of the Fick’s second law of diffusion. The model contains 2 dimensionless empirical constants and 2 model constants which can be derived from experimental data. The first term describes the last part of the drying process, while the second term describes the beginning of the drying process. For most fruits and vegetables with high moisture content, this model can well be suitable as it assumes a constant product temperature and diffusivity throughout the drying process. This model well describes the moisture transfer of the drying process, with the constants representing the physical

properties of the drying process:

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=a \exp \left(-K_{1} t\right)+b \exp \left(-K_{2} t\right)$$

where $a$ and $b$ are dimensionless empirical constants, and $K_{1}$ and $K_{2}$ are the drying constants $\left(\mathrm{s}^{-1}\right)$.

From Table 2 it can be seen that this model is the best in describing the drying behavior of beetroot, fig, onion, plum, pumpkin, and stuffed pepper.

Two-term exponential model. The 2-term exponential model is a modification of the 2-term model by reducing the number of constants and modifying the indication of shape constant (b) of the second exponential term. Erbay and Icier (2010) emphasized that constant “b” of the 2-term model (Eq. 13) has to be $(1-\mathrm{a})$ at $t=0$ in order to obtain a moisture ratio of $M R=1$. The model has 3 constants and can be expressed as

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=a \exp (-k t)+(1-a) \exp (-k a t)$$

This model has been found successful in describing the drying kinetics of only star fruit as presented in Table 2.

Hii and others (modified 2-term model). The Hii and others model can also be referred to as a Modified Page model or, more appropriately, a Modified 2-term model. The model involves a combination of the Page and the 2-term model. The first part of the model is exactly as the Page model. However, it more theoretically describes the model as a modified 2-term model with the inclusion of a dimensionless empirical constant " $n$." The model contains 5 constants and can be referred to as a complex model in this regard. Hii and others (2009) proposed this model for the drying of cocoa beans. However, it has been found appropriate in describing the drying kinetics of some fruits:

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=a \exp \left(-K_{1} t^{n}\right)+b \exp \left(-K_{2} t^{n}\right)$$

The Hii and others model has been successfully applied to the drying of carrot pomace and pumpkin as presented in Table 2.

Demir and others model. A modification of the Henderson and Pabis model and the Logarithmic model was proposed by Demir and others (2007) for drying of green olives (Table 2). This model contains 4 constants with 3 dimensionless empirical constants. This model can be expressed as

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=a \exp \left((-k t)^{n}\right)+b$$

Verma and others model. This model is another modification of the 2-term model with 4 model constants. The Verma and others (1985) model has been applied successfully in describing the drying kinetics of parsley and pumpkin as shown in Table 2.

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=a \exp (-k t)+(1-a) \exp (-g t)$$

where $g$ is also a drying constant $\left(\mathrm{s}^{-1}\right)$.
Approximate diffusion model. The Approximate Diffusion model is another modification of the 2-term exponential model with the separation of the drying constant " $k$ " and $t$ with a new dimen-
sionless constant " $b$ " in the second part of the model:

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=a \exp (-k t)+(1-a) \exp (-k b t)$$

where $b$ is also a dimensionless model constant.
This model has been applied with great success in the determining the drying kinetics of green pepper, pumpkin, and tomato (Table 2).

Modified Midilli and others model. As the name implies, this model is a modification of the Midilli and others model. The model has been found to successfully predict only the drying kinetics of jackfruit as can be seen in Table 2. The model is expressed as

$$M R=a \exp (-k t)+b$$

Empirical models

Empirical models give a direct relationship between the average moisture content and the drying time. The major limitation to the application of empirical models in thin-layer drying is that they do not follow the theoretical fundamentals of drying processes in the form of a kinetic relationship between the rate constant and the moisture concentration, thus giving inaccurate parameter values. More so, these models do not have a physical interpretation and are wholly derived from experimental data. The 3 most widely applied empirical models for the drying kinetics of fruits and vegetables as reported in the literature, and shown in Table 2, are (i) the Weibull Model (Eq. 23), (ii) Wang and Singh (Eq. 21), (iii) the Diamante and others Model (Eq. 22), and (iv) the Thompson Model (Eq. 23). The following are the most suitable models found to adequately describe the drying kinetics of some fruits and vegetables:

Aghbashlo and others model. Aghbashlo and others (2009) proposed a model that effectively described the thin-layer drying kinetics of biological materials. The model was tested on carrot and compared with other available thin-layer drying models in the literature. It was found that the model best described the drying behavior of carrot. However, this model has not been successful in describing several other fruits and vegetables. The model contains 2 dimensionless constants which are dependent on the absolute temperature of the drying system (Table 2). However, there is no theoretical basis for this model:

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=\exp \left(\frac{K_{1} t}{1+K_{2} t}\right)$$

where $K_{1}$ and $K_{2}$ are drying constants $\left(\min ^{-1}\right)$.
Wang and Singh model. This model was developed for the intermittent drying of rough rice (Wang and Singh 1978). The model gives a good fit to the experimental data. However, this model has no physical or theoretical interpretation, hence its limitation.

$$M R=\frac{\left(M-M_{c}\right)}{\left(M_{0}-M_{c}\right)}=1+a t+b t^{2}$$

where $a\left(\mathrm{~s}^{-1}\right)$ and $b\left(\mathrm{~s}^{-1}\right)$ are dimensionless model constants gotten from the experimental data.

This model has been found to successfully explain the drying behavior of banana as shown in Table 2.

Diamante and others model. Diamante and others (2010b) proposed a new empirical model for the drying of fruits. The experimental data used in developing the model were obtained from the hot air drying of kiwi fruit and apricot and used polynomial regression analysis to determine the values for the model constants.