4Modelling of drying processes

for food materials

H.T. SabarezCommonwealth Scientific and Industrial Research Organisation–Food and NutritionFlagship, Werribee, Victoria, Australia

4.1 Introduction

Drying is one of the important unit operations in food manufacturing aimed at reduc-

ing the amount of moisture in food products to levels that ensure preservation by inhi-

biting microbial and enzymatic activity and the associated product quality

deterioration. In basic terms, the process of drying (or dehydration) involves the

removal of water from a wet feedstock by inducing phase changes of water from solid

or liquid into a vapour phase via the application of heat (except in the case of osmotic

dehydration during which the water is removed without a change in phase by the dif-

fusion of liquid water from solid foods to osmotic solution through an osmotic pres-

sure difference). Drying usually consumes large amounts of energy and imparts

significant alterations in product quality attributes due to the exposure to longer drying

times or higher temperatures. Today’s increased competition due to globalisation,

together with the growing consumer demand for better quality products, will continue

to drive innovations in the drying process, leading to further efforts in improving the

performance of the existing drying technologies and the development of new drying

concepts crucial for the future sustainability of the food industry. In drying food mate-

rials, there remains a major challenge in removing water from the material in the most

efficient way, with better control of product quality, minimal impact on the environ-

ment, the lowest capital and operating costs of the process. A further challenge arises

from the fact that many food materials with very diverse physical or chemical prop-

erties need to be dried at different scales of production and with very different product

quality specifications (Mujumdar and Wu, 2010).

Mathematical modelling is a useful tool for simulating and testing the performance

of the drying process, which allows one to generate tangible results for a wide range of

scenarios through a virtual laboratory that would be too expensive or time-consuming

to perform at the conceptual stage. It also enables the modeller to predict outcomes for

the optimisation and scaling-up of the improved and new or untested process designs,

accelerating understanding and decreasing the development costs of the process, with-

out the excessive need for labour-intensive trial-and-error experimentation. However,

food materials are extremely complex in their structure and composition, so there is no

universally acceptable way to model their drying behaviour, as exemplified by the vast

volume of literature published on various modelling approaches (Kostoglou et al.,

2013). This also reflects the extreme diversity of the drying mechanisms in food

Modeling Food Processing Operations. http://dx.doi.org/10.1016/B978-1-78242-284-6.00004-0

© 2015 Elsevier Ltd. All rights reserved.

96 Modeling Food Processing Operations

systems. Mujumdar and Huang (2007) stated that most models are applicable for spe-

cific product-equipment combinations. The development of drying models would

therefore continue to improve our attempts to meet these challenges.

The chapter mainly deals with the basic concept of the drying process of food

materials when considering the removal of water from these materials through vapor-

isation of the liquid or solid by supplying different forms of heat. It starts with a brief

overview of the transport phenomena involved in the drying of food materials and the

drying techniques commonly used in the food industry. The latter part of this chapter

focuses on modelling approaches and strategies, providing a case study on the devel-

opment and application of the modelling approach for the optimisation of an industrial

drying process.

4.2 The drying process

4.2.1 Drying mechanisms

The process of drying food materials is extremely complex, involving coupled tran-

sient mechanisms of heat, mass and momentum transfer processes accompanied by

physical, chemical and phase change transformations (Sabarez, 2012). In drying food

materials, two distinct transport mechanisms occur simultaneously, comprising heat

transfer from the drying medium (or heat source) to the food material and water trans-

port from the interior of the solid product to its surface from which the water is even-

tually transported away by a carrier gas (or by the application of vacuum for

nonconvective dryers). A conceptual representation of the transport phenomena

occurring during the drying of a solid food material is illustrated in Figure 4.1.

Hot airTemp

velocityRH

Internal

External

Heating plate

(liquid/vapour)

Shrinkage

Evaporation

Radiation Convection

Heating plate

Conduction

Diffusion

Capillary flow

Figure 4.1 A conceptual representation of the thermal drying process for a solid food material.

Modelling of drying processes for food materials 97

Energy is required to generate a phase change of water from a liquid to a vapour (or

solid to vapour) and to activate molecular movement. Hot air (the most common dry-

ing medium) is employed both to supply heat (by convection) and as the carrier gas to

take away the moisture. Heat may be also supplied by conduction (i.e. from heated

metal surfaces as a heat source) or radiation, or it may be supplied volumetrically

by placing the wet material in a microwave or radio frequency electromagnetic field.

Drying is a highly energy-consuming unit operation due to the high latent heat of

vaporisation of water and the inherent inefficiency of using hot air as the (most com-

mon) drying medium. The three modes of energy transfer (convection, conduction and

radiation) may be used alone or in combination to supply heat from the heat source to

the food materials. According to Mujumdar and Devahastin (2008), over 85% of

industrial dryers are of the convective type with hot air or direct combustion gases

as the drying medium. All modes of heat transfer, except those using electromagnetic

energy (microwave and radio frequency), supply heat at the boundaries of the drying

material so that the heat must diffuse into the solid primarily by conduction. Also, the

different heat transfer modes may be deployed simultaneously or sequentially depend-

ing on individual application in order to achieve improved energy efficiency in the

drying process.

The mass transfer phenomena during drying may be controlled by either the rate of

moisture diffusion (liquid or vapour) within the food matrix (internal transfer) or the

rate of moisture evaporation from the product surface to the drying medium (external

transfer). The internally controlled drying process is mainly influenced by tempera-

ture and predominates once the rate of replenishment of moisture from the interior

to the surface of the product is slower than the external mass transfer rate. The internal

mass transfer process often involves different mechanisms of moisture movement.

Food materials (e.g. fruits, vegetables) are generally porous media containing solid

matrices having void spaces that are filled with gas or liquid (Prakotmak, 2013). In

this case, the food material is treated to undergo deformation (shrinkage) during

the drying process with the transport of water within the material to include in mul-

tiphases. According to Heldman and Hartel (1997), the moisture within the food prod-

uct can migrate in several ways via a number of different mechanisms (liquid or

vapour phase). The liquid transport mechanisms include capillary flow, surface dif-

fusion and liquid diffusion while the vapour transport mechanisms consist of Knudsen

diffusion, mutual diffusion, Stefan diffusion, Poiseuille flow and condensation-

evaporation (Sablani and Rahman, 2007). In liquid diffusion, the rate at which mois-

ture migrates depends on the nature of the food product, temperature and water

concentration difference. In some cases, vaporisation may occur within the product,

and thus water diffuses in the form of vapour through the food matrix with the differ-

ence in vapour pressure as the driving force for moisture transfer. The differences in

pressure between the drying medium and the internal food structure (pressure flow)

and the differences in temperature between the surface and the interior of the product

(thermal flow) may also influence the internal mobility of moisture. Transport of

moisture within the solid may also occur by a combination of the following mecha-

nisms of mass transfer (liquid diffusion, vapour diffusion, pressure differences). The

moisture must travel to the boundary of the material before it is then transported away

98 Modeling Food Processing Operations

by the carrier gas (or by application of vacuum for nonconvective dryers). The mois-

ture transfer from the solid surface to the drying medium is mainly governed by the

properties of the drying air.

The heat and mass transfer phenomena are usually influenced by both temperature

and water concentration differences, as well as the air velocity field, together with the

properties of the material itself. In convective drying, the coupled transfer of heat,

mass and momentum in at least two distinct subdomains (air and food), which simul-

taneously occur both externally and internally to the food matrix during drying, can be

described as follows (Sabarez, 2012): (1) convective and conductive heat transfer in

the air, (2) convective and diffusive water transfer in the air, (3) heat transfer mainly

by conduction within the solid interior, (4) mass transfer in the solid interior by dif-

fusion (liquid or vapour), (5) moisture evaporation at the air–food interface, and (6)

airflows (laminar or turbulent) around the food material. The external transfer rates

for both heat and mass are greatly influenced by the air velocity field (fluid dynamics)

and other drying air properties (i.e. temperature and relative humidity). The internal

heat and mass transfer processes may be also affected by the physical changes

that may occur in the product, including shrinkage, puffing, crystallization and glass

transitions.

4.2.2 Drying techniques

The removal of water from the food materials during drying can be achieved in dif-

ferent ways, and this variety of methods has led to many drying techniques. There are

also tens of thousands of different foodmaterials with very diverse physical and chem-

ical properties that need to be dried for different product specifications and at different

scales of production (Mujumdar and Wu, 2010). The selection of the drying method

for a particular food product is an important step because the drying technique and

operating conditions affect the quality of the dried product as well as its cost. It

depends on various factors, including the type of feed, the amount of moisture, the

drying kinetics, the heat sensitivity, the physical structure of the material to be dried,

the quality requirements of a dried food, and many other factors (Jangam, 2011). Also,

the selection procedures of a drying system include the cost estimation of various

dryers, including capital and operating costs. Table 4.1 summarises a generalised clas-

sification of conventional drying methods applied for drying food materials. There are

numerous criteria used to classify dryers. The classification in Table 4.1 is a rather

coarse representation, and readers may refer to Jangam (2011) and Bansal and

Chung (2007) for additional information on the classification of dryers applicable

for drying food materials.

Dryers are commonly classified based on the mode of heat transfer (e.g. convec-

tion, conduction, radiation or a combination thereof). The majority of dryers used in

the food industry are of the convective type; in other words, hot air is used to both

supply heat for the evaporation of water and carry away the evaporated moisture from

the product. These are by far the most common drying method despite their relatively

low thermal efficiency. Hot air produced by indirect heating or direct firing is the most

common drying medium, although for some special applications superheated steam

Table 4.1 A generalised classification of thermal dryers forfood materials

Classification Types of dryers (general characteristics and applications)

Type of feed material l Particlesl Slurry/paste/sludgel Liquid suspension

Processing mode l Batchl Continuous

Mode of heat transfer l Convectionl Conductionl Electromagnetic (RF, ohmic, infrared, microwave)l Combination (hybrid)

Energy sources l Electricityl Gas (natural/LPG)l Solar/windl Biomass

Mode of operation l Cyclicl Intermittentl Continuous

Product temperature l Above freezing pointl Below freezing point

Operating pressure l Atmosphericl Vacuuml High pressure

Modelling of drying processes for food materials 99

has been shown to yield higher efficiency and often higher product quality

(Mujumdar, 1987). In this type of dryer, the drying medium contacts the material

to be dried directly.

Another type of dryer involves supplying heat to the dryingmaterial through heated

metallic or nonmetallic solids (molecular vibration) or stationery fluids (primarily by

molecular collision) by conduction (e.g. drum dryers). In this type, the heat is trans-

ferred from the heat transfer medium (hot gas, steam, thermal fluids, etc) through the

hot metal directly in contact with the drying materials. Because no gas flow is pre-

sented on the wet solid side, the system must either apply a vacuum or use gentle

gas flow to remove the evaporated moisture so that the dryer chamber is not saturated

with vapour. Furthermore, vacuum operation lowers the boiling point of the liquid

being removed; this allows drying of heat-sensitive solids at relatively fast rates. Con-

vection (gas flow) or vacuum operation is still needed to remove the evaporated

moisture.

Dryers are also classified according to the radiant energy supplied in various forms

of electromagnetic waves categorised according to where they lie in the electromag-

netic spectrum (e.g. radio frequency, infrared, microwave). Radiation does not

100 Modeling Food Processing Operations

require a medium and uses the electromagnetic waves emitted by an object for

exchanging heat, and, thus, this is the only form of heat transfer present in vacuum.

The electromagnetic spectrum classifies radiative heat transfer according the wave-

lengths of radiation. In food drying applications, the main types of radiation applied

are infrared (IR), microwave (MW) and radio frequency (RF), which employ a

completely different heating mechanism. This type of heating mode (termed thermalradiation to distinguish it from other forms of radiation such as x-rays and gamma

rays) must be used in conjunction with convection or under vacuum to remove the

evaporated moisture. Radio waves have long wavelengths and therefore have good

penetration. Microwave heating is similar to RF heating because both methods are

quick and consistent, and they target water molecules, resulting in fast volumetric

heating (from the inside out). In most drying operations, energy is transferred from

the surface to the centre of the wet material, with the exception of radiofrequency

and microwave drying, during which the energy supplied generates internal heat

within the solid.

In some cases, the combination of the three modes of heat transfer (convection,

conduction and radiation) is applied for a more energy-efficient drying process.

The refractance window (RW) dehydration method, developed by MCD Technolo-

gies, Inc. (Tacoma, Washington, USA), is a good example of a drying technique that

utilises all the three modes of heat transfer (Figure 4.2). The technology is suitable for

producing dried products from liquid and semiliquid foods (Bolland, 2000). It uses

water as a drying medium to transmit heat into the product to be dried. The product

is evenly applied to the surface of a conveyor belt system (usually an infrared-

transparent plastic) that floats on the surface of heated circulating water. The RW

drying technology utilises the refractive principle of the surface of water, which is

harnessed by creating a window for the passage of infrared energy. In this technology

the three modes of heat transfer occur between the drying medium (water) and

the material to be dried. A number of studies were found to be relevant to the RW

drying process (Ochoa-Martinez et al., 2012; Caparino et al., 2012; Nindo et al.,

2003a; Abonyi et al., 2001; Bolland, 2000; Nindo et al., 2003b, 2004; Clarke,

2004). According to Abonyi et al. (1999), products can be dried in a few minutes with

this technology, unlike hot air or tunnel dryers that can take several hours. Nindo et al.

(2003a) reported that the drying of pumpkin puree from 80% to 5% moisture content

(wet basis) was achieved in less than 5 minutes in both pilot- and commercial-scale

RW dryers with a circulation water temperature of 95 °C, with a 52–70% energy effi-

ciency of the RW drying system.

The other common classification of dryers is based on the type of feed material. For

example, for liquid feed, spray drying is still the most common drying method,

although rotary drum dryers are also popular (Jangam, 2011). Spray drying is a very

expensive technique to use for low-value products, mainly because of its low energy

efficiency (Jangam, 2011). This method has several advantages, including rapid dry-

ing, large throughput and continuous operation (Duffie and Marshall, 1953). How-

ever, due to the relatively high temperatures involved in spray drying processes,

this drying technique (spray drying) may cause losses of certain quality and sensory

attributes, especially vitamin C, b-carotene, flavours and aroma (Dziezak, 1988).

Figure 4.2 Photos of (a) a commercial-scale RW dryer facility (RWD5 Model, MCD

Technologies, USA), (b) the wet-feed end of the dryer, and (c) the dried-product exit end of the

dryer (Sabarez and Chessari, 2006).

Modelling of drying processes for food materials 101

102 Modeling Food Processing Operations

The RW drying technology is also a suitable drying method for producing dried

products from liquid and semi-liquid foods (Bolland, 2000).

Other drying techniques evolved due to the need to produce high-quality dried

products that are ultra-heat-sensitive. Such drying systems include the utilisation of

subfreezing temperature and vacuum for the operating pressure (e.g. freeze drying).

Freeze drying (also known as lyophilisation) is a drying process in which the food is

first frozen then dried by direct sublimation (i.e. phase changes from solid to vapour)

of the ice under reduced pressure (Oetjen and Haseley, 2004; Barbosa-Canovas,

1996). Freeze drying is generally considered to be the best method for the production

of high-quality dried products (Ratti, 2001). But, it suffers from high production costs,

high energy consumption and low throughput (Ratti, 2001; Hsu et al., 2003). The cost

of low-temperature vacuum processing is many times higher than that of other con-

vection dryers, which makes it less attractive for most products (Jangam, 2011).

In recent years, a number of innovative food processing technologies have been

investigated and developed with the aim of improving or replacing conventional pro-

cessing technologies. These novel or emerging technologies take advantage of other

physical phenomena such as sound waves, pressures and electromagnetic fields,

which can be applied for the development of new drying concepts for improving

the quality of food products through gentle processing. In particular, the application

of ultrasonic energy to assist the drying of food materials has been explored for several

decades. It has been known for many years that the energy generated by sound pres-

sure waves could enhance a wide range of processes due to a series of mechanisms

activated by the ultrasonic energy, such as heat, diffusion, mechanical rupture,

chemical effects, and so on (Gallego-Juarez et al., 2007).

Several studies have reported the application of ultrasonic technology in combina-

tion with convective drying processes. A number of investigations have shown the

potential of power ultrasound to improve the drying process of various food materials.

In these studies, the ultrasonic energy was transmitted as either airborne to the surface

of food material (Garcia-Perez et al., 2009; Carcel et al., 2007, 2011a,b; Garcia-Perez

et al., 2007a,b, 2010; Khmelev et al., 2008, 2011; Ozuna et al., 2011; Soria and

Villamiel, 2010) or in direct contact between the product and the vibrating element

(Gallego-Juarez et al., 2007; Gallego-Juarez, 2010; Schossler et al., 2012). In partic-

ular, a promising approach for the application of ultrasound to assist in the convective

food drying of apple slices was developed and tested by Sabarez et al. (2012). This

study was carried out to investigate the effect of ultrasound on drying kinetics and

product quality attributes using the alternative approach for the application of ultra-

sonic energy in the convective drying process. The approach is based on the transmis-

sion of ultrasonic energy using a vibrating stepped-plate ultrasound technique that

relies on combined transmission through air and a series of solid contacts between

the ultrasound element and the product tray as the ultrasonic vibration transmitting

surface (Figure 4.3). The results from this work indicate a significant reduction in dry-

ing time (up to 57%) with the simultaneous application of ultrasound on the convec-

tive drying of apple slices (Figure 4.4). This corresponds to a reduction of energy

consumption by up to 54% with the ultrasound-assisted convective drying process.

The processing variables (i.e. drying temperature, product thickness, ultrasonic

Front viewComputer

Refrigeration

Sensors (x16)

Airf

low

Actuator

Transducer

Vibrating plate

Balance

PLC

Generator

Sample tray(movable)

PLC

Water tank

Cooling coil

Heater

Motor

Exhaust

Fan

Steamgenerator

Inlet

Side view

Figure 4.3 Schematic diagram of a computerised ultrasound-assisted convective drying system

(Sabarez et al., 2012).

Drying time (h)

−US (0 W)

+US (75 W)

+US (90 W)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Moi

stur

e co

nten

t (%

w.b

.)

60

70

80

90

Figure 4.4 Effect of ultrasound on the drying kinetics of apple slices at different ultrasonic

power levels (T¼40 °C; RH¼25%; u¼1.0 m/s; 5 mm thickness) (Sabarez et al., 2012).

Modelling of drying processes for food materials 103

104 Modeling Food Processing Operations

power level) appeared to substantially influence the magnitude of the ultrasonic

energy’s effect in enhancing the drying process, indicating the necessity to establish

the optimum drying conditions for specific product and ultrasonic applications. In par-

ticular, the ability of ultrasound to improve the efficiency of convective drying pro-

cesses seems to be maximised when using low temperature and a high ultrasonic

power level.

In a further study (Beck et al., 2014), the application of a specially designed ultra-

sonic horn for a completely airborne ultrasound transmission to assist in the convective

drying of a model food system was investigated. This work involved investigations

of the impact of airborne ultrasound at various power levels and different levels of dry-

ing air conditions (i.e. temperature, relative humidity and velocity), using a response

surface methodology (RSM) approach to examine the possible interactions between

these parameters and to find the combination of these factors that yields the best

response. The airborne ultrasound equipment tested in this work was found to enhance

the conventional hot air drying process by significantly reducing the overall drying time

(i.e. by more than 60%). The process parameters (temperature, air humidity, air speed

and ultrasound power level) and their interactions substantially affected the drying pro-

cess, with optimum conditions found using the RSM approach.

In general, the findings from these studies offer a promising alternative to facilitate

the adaptability of the technology in industrial-scale operations because there is no

direct contact between the ultrasound element and the food sample to be dried. Further

research efforts to optimise the technology for application in industrial food drying

and the application to other drying techniques, together with future advancements

in ultrasonic technology, should provide the basis for developing a new ultrasonic dry-

ing technology for adoption in industrial drying practise.

4.3 Modelling approaches

The literature presents different approaches for modelling various drying processes. In

general, the models for the drying of food materials can be categorised into two major

groups: (a) those involving empirical equations and (b) those based on the fundamen-

tal physics of the drying processes. The level of model complexity must be balanced

with the time and cost required to develop and implement it while ensuring that an

adequate accuracy is achieved. In some cases, models are either too simplistic (i.e.

deviating significantly from real processes) or too complex to have any practical

application. It is therefore essential to develop a model that is not only meaningful

and relatively simple to use, but also accurate enough to predict the processes (and

that can be applied in industrial situations beyond those in which experiments were

conducted).

The empirical models (sometimes referred to as characteristic drying rate curves)

are system specific and cannot be generalised, because they do not involve any

physical basis. The most common model under this category is the Page model.

The empirical models, resulting from simple fitting to the experimentally determined

Modelling of drying processes for food materials 105

drying curves (Mujumdar and Huang, 2007; Togrul, 2005), are widespread due to their

simple implementation and adequate description of the specific drying processes.

Although empirical models would produce good results for engineering applications

in the food industry, they frequently do not allow the simulation of experiments car-

ried out under conditions different from those used to identify the model parameters

(Ah-Hen et al., 2013). These models are generally based on simplifying hypotheses

that may not be applicable in some situations (e.g. complex food geometries) and

changing operating conditions during the drying process (i.e. industrial scale). During

the drying process, variation in moisture content and temperature as a function of both

time and space exists inside the material, but this is not included in empirical models,

which may limit their practical application to drying. The dependence of thermophy-

sical and transport properties on product temperature and moisture content is not taken

into account. Also, the models consider only either the internal or external resistance

to mass transfer in an isothermal process (i.e. the analysis of heat transfer is neglected),

and product shrinkage is not considered. Singh et al. (2012), Menges and Ertiken

(2006) and Yaldiz et al., 2001 presented a comprehensive review of the application

of such models in the drying of various food materials.

The second approach is mainly based on models utilising the fundamental laws of

conservation of mass, momentum and energy. These mechanistic (classical) models

couple the fundamental transport equations with the thermodynamically interactive

fluxes and phase equilibrium expressions (Hayakawa and Furuta, 1988). The approach

comprises the physics involved in the drying process, largely represented by coupled

heat and mass transfer equations. For convection-based drying processes, the external

heat and mass transfer rates at the air–food interface could play a significant role in

controlling the drying process, depending on the drying conditions, which are, in turn,

strongly dependent on the drying air velocity field (Sabarez, 2012). It is therefore

important to further account for the momentum transport dynamics to improve the

predictive precision of the model. Besides, the incorporation of fluid dynamics in

the drying process is important for characterising the effect of heat and mass transfer

from the material to the drying medium (i.e. termed as equipment models) for design-

ing a dryer for a particular task. The equipment model describes the transfer process in

the drying system and predicts the instantaneously changing drying conditions of air at

any location in the drying system (i.e. drying chamber). However, there are different

mechanisms proposed under the mechanistic models (e.g. single or multiple phases),

and, generally, these models involve highmathematical complexity and determination

of too many parameters.

A more comprehensive modelling approach involves the numerical computation of

a theoretical model describing the simultaneously coupled transfer of momentum (air

only), heat and mass (both air and food) under transient conditions occurring during

the convective drying of food materials (Sabarez, 2012; Curcio et al., 2008). These

models are based on the fundamental physical principles of the drying process and

take into account the variability of air flows (fluid dynamics) around the food

material. For example, Sabarez (2012) successfully used a similar approach to predict

the moisture content and temperature distributions within prunes during drying.

Aversa et al. (2007) also employed a similar modelling approach to describe the

106 Modeling Food Processing Operations

transport phenomena occurring during the drying of carrot slabs. These models did not

make any distinction between the transport of liquid water and that of vapour within

the food matrix. More recently, Curcio and Aversa (2014) presented and successfully

validated a multiphase transport model for describing the drying process of a porous

material undergoing deformation by considering the conservation of liquid water,

vapour and energy in food, coupled to the conservation of vapour and energy in the dry-

ing air. However, a detailed analysis of the inherent complexity of the coupled transient

phenomena (i.e. heat, mass, momentum and deformation) involved in the drying process

is often regarded as time consuming for practical purposes. Hence, it is necessary to

have a simple, accurate and robust mathematical model with minimum mathematical

complexity to reduce the computational time. With the advent of increasing computing

power it is now possible to undertake simulations of these complex phenomena in less

time. The challenge is to optimise the level of simplification (model complexity) with

the level of accuracy to satisfactorily explain the real system.

4.4 Modelling of the drying process

The development and delivery of a product to market can be a long and expensive

journey. Industrial R&D now extensively employs mathematical modelling

approaches to evaluate new concepts to reduce costs, to minimise time needed and

to intensify innovation. Together with advances in computing capability, the develop-

ment of computational models to accurately simulate complex processes in less time is

one of the great advances in process engineering research. This enables the modeller to

predict outcomes for performance evaluation, optimisation and the scaling-up of new

and untested process designs, without the excessive need for expensive and labour-

intensive trial-and-error experimentation (and the models are not time consuming to test

experimentally). Computational modelling can also be utilized to develop improved

conceptual designs and to optimise operating conditions as a cost-effective route to

intensifying improvements in existing conventional dryers. The extensive characterisa-

tion of drying behaviour using a strictly experimental approach constitutes a formidable

challenge due to the excessively large number of variables that must be considered.

Modelling allows one to conduct a wide range of tests that would be too expensive

or time-consuming to perform, reducing the number of prototypes required. It is also

important to develop a tool that can simulate the product’s drying behaviour and there-

fore allow one to extend understanding beyond the results of experimental drying inves-

tigations. Modelling a drying process (as with modelling any other processes) involves

several steps, includingmodel conceptualisation, mathematical formulation, determina-

tion of model parameters, methods of solution and experimental validation.

4.4.1 Model conceptualisation

Modelling starts by conceptually defining the system and the physics associated with

the process using a geometric representation. The computational domain can be

solved in various dimensional coordinates (i.e. 2D or 3D), depending on the geometric

Modelling of drying processes for food materials 107

complexities of the system, allowing one to satisfactorily depict the real system. At

this stage, it is also important to clarify howmany details and assumptions are required

in order to reach a practical solution and satisfy the need of the application by provid-

ing the level of information required.

A conceptual representation of a convection-based drying process of a solid food sys-

tem is illustrated in Figure 4.1. In this example, themodel food is viewed as a continuum

system with a composite ellipsoidal body comprising two materials with different prop-

erties, representing typical fruit drying. A 2-dimensional (2D) solution is considered to

satisfactorily describe the physics occurring in two distinct subdomains (air and food).

In this example, the drying process is described by the simultaneous transfer of coupled

momentum (air only), heat and mass (air and food) phenomena.

4.4.2 Mathematical formulation

The conceptual model development is followed by the formulation of mathematical

equations that describe the physics of the process. The approach takenwhen formulating

a mathematical model generally depends on the problem being considered. In food dry-

ing processes, two distinct transport mechanisms occur simultaneously, involving heat

transfer from the drying medium to the food material and water transport from the inte-

rior of the solid product to its surface and eventually to the air through evaporation. For

robustness and accuracy, the governing partial differential equations (PDEs) describing

the simultaneous transfer of heat, mass and momentum in two distinct subdomains (air

and food) during the drying of a solid food material are employed.

In convection-based drying processes (e.g. hot air drying, spray drying, etc.), the

heat and mass transfer rates depend on both temperature and concentration differ-

ences, as well as on the air velocity field. The nonisothermal turbulent flow of air

in the drying chamber is described according to the standard k-e model (C.O.M.S.

O.L. Multiphysics, 2007). The equations for the momentum transport and continuity

are the following:

r@u

@t�r � + r

Cmk2

ske

� �� ru+ ruð ÞT� �� �

+ ru�ru+rP¼ 0 (4.1)

@r@t

+r ruð Þ¼ 0 (4.2)

turbulence energy equation is given by

The

r@k@t

�r � + rCmk2

e

� �rk

� �+ ru�rk¼ 1

2rCm

k2

eru+ ruð ÞT

� �2

�re (4.3)

the dissipation equation by

and

r@k@t

�r � + rCmk2

e

� �re

� �+ ru�re¼ 1

2rCe1k ru+ ruð ÞT

� �2

�rCe2e2

k(4.4)

108 Modeling Food Processing Operations

The energy balance in the food material for a generalised geometry leads to the tran-

sient heat transfer equation according to Fourier’s law of heat conduction, as shown by

Equation (4.5):

rCp@T

@t

� �+r �krTð Þ¼QM +QU (4.5)

advantage of the numerical approach is that the PDEs can be solved, covering the

The

presence of various mechanisms and source terms. With MW applications, the volu-

metric rate of thermal energy generation (QM) is provided by the dissipated MW

power, which can be evaluated according to the electric field distribution resulting

from solving Maxwell’s equations (Equation (4.6)) (Oliveira and Franca, 2002;

Knoerzer et al., 2008). This field depends on the dielectric properties of the food mate-

rial and, thus, on the temperature and moisture content fields and vice versa (i.e. tem-

perature and moisture content fields depend on the electric field) at any point in the

material (Feng et al., 2012), coupling the governing equations (heat and mass transfer,

electromagnetism). A further challenge is the complicated responses of dielectric

properties to the porosity and compositional changes of the material, in addition to

the travelling microwaves that can also decay, focus and superimpose to further

complicate the calculation (Feng et al., 2012). As a result, the modeller must rely

on significant computational effort and tricky computational strategy to efficiently

solve, in a parallel manner, these coupled equations. Modellers can now overcome

this limitation due to the development of powerful computers, which can handle large

amounts of data, and advanced numerical techniques. Feng et al. (2012) presented

the basics of dielectric heating and drying, examined the heat and mass transfer

models developed for the simulation of microwave drying processes, and discussed

dielectric properties of selected food products as influenced by moisture, temperature,

and porosity. On the other hand, the heat generation rate (QU) imparted to the material

by ultrasonic irradiation dissipated as heat can be estimated according to Equation (4.7)

(Du et al., 1981; Lin, 1995; Zhao and Chen, 2011):

QM ¼ 2pf e0e00r E2 (4.6)

QU ¼ 2I1:1f 1=2

c1grm

� �(4.7)

e same time, the energy balance in the drying air, accounting for both convective

At th

and conductive contributions, is given in Equation (4.8):

raCpa@T2@t

� �+r �karT2ð Þ+ raCpaurT2 ¼ 0 (4.8)

transient moisture transport within the foodmatrix is modelled using the basic law

The

governing the movement of moisture according to Fick’s law of diffusion

Modelling of drying processes for food materials 109

(Equation (4.9)), while the water mass balance in the drying air, taking into account

for both convective and diffusive contributions is given in Equation (4.10):

@c

@t

� �+r �Drcð Þ¼ 0 (4.9)

@c2@t

� �+r �Drc2ð Þ+ urc2 ¼ 0 (4.10)

rticular, the boundary condition at the air–food interface (at t>0) for heat trans-

In pa

fer, considering the mass transfer at the air–food interface, thus coupling the heat and

mass transfer equations simultaneously is given in Equation (4.11). This means that

the heat transported by convection and conduction from the drying air to the food is

partly used to raise the food temperature by conduction and partly for water evapo-

ration at the food surface. To account for the effect of IR, which is mainly taken at

the boundary condition, the last term of Equation (4.11), which is described by the

Stefan-Boltzmann law of thermal radiation, can be added. This allows for the absorp-

tion of the infrared energy (radiative heat flux) across the boundary layer of the prod-

uct (Sabarez and Chessari, 2006). We write this equation as:

�n �krTð Þ¼ lkc c2� csð Þ+ hc T2�Tsð Þ+ e1s T4R�T4

s

� (4.11)

boundary condition at the air–food interface for mass transfer is given in Equa-

The

tion (4.12), which accounts for the balance between the diffusive flux of liquid water

coming from the interior of the product and the flux of vapour from the food surface to

the drying air:

�n Drcð Þ¼ kc c2� csð Þ (4.12)

dition, the development of mathematical models incorporating the associated

In ad

impact on product quality attributes (e.g. colour) is crucial in achieving the optimum

design and operating conditions of a drying system that maximises the retention of the

desired quality attributes of the product. The quality changes (e.g. colour) of the prod-

uct during drying can be modelled using a general kinetic reaction equation:

dC

dt

� �¼�kCn (4.13)

e n is the order of reaction, and k is the reaction rate constant. The positive-or-

wher

negative sign in Equation (4.13) indicates the formation and degradation in the quality

parameter with time, respectively. The kinetic model predicts the development of the

quality parameter in the product on each grid or cell during the drying process, with

the reaction rate depending on the temperature and moisture content of the product,

110 Modeling Food Processing Operations

thus coupling the changes in the product quality parameter with the local heat and

mass transfer calculations (Sabarez, 2014).

4.4.3 Model parameters

The solution of the governing partial differential equations requires knowledge of the

thermophysical and transport properties of the product and air. Many of these param-

eters cannot be assumed constant through the drying process but depend on the tem-

perature or moisture content, if precise drying kinetic predictions are to be achieved

(Sloth et al., 2006). The thermophysical properties of the product (i.e. thermal conduc-

tivity, specific heat capacity and density) are assumed to be dependent on product

composition (i.e. water, protein, fat, carbohydrate and ash) expressed as a function

of the local temperature (ASHRAE, 1995; Choi and Okos, 1987; Sabarez, 2012).

The majority of model parameters for drying processes are shown in Table 4.2. Some

parameters are physically measurable and others are quite difficult and sometimes not

available for food systems.

The transport coefficients for heat and mass required in the boundary condition are

usually estimated from empirical equations involving dimensionless numbers. The

convective heat transfer coefficient required for the boundary condition in the heat

transfer equation is calculated using the Nusselt–Reynolds–Prandtl correlation for

local convective heat transfer for a particular geometry of the food material given

by Heldman and Lund (2007). The mass transfer coefficient, which describes the con-

vective mass transfer at the surface of the product, is obtained using the Sherwood–

Reynolds–Schmidt correlation for average convective mass transfer for a particular

geometry of the food material (Heldman and Lund, 2007). The heat and mass transfer

coefficients can vary significantly depending on the drying parameters (i.e. food size,

air velocity, etc.).

The effective diffusion coefficient is the main parameter for the characterisation of

mass transfer phenomena in solid foods. It is regarded as a lumped property that does

not really distinguish between the transport of water by liquid or vapour diffusion, or

capillary or hydrodynamic flow due to pressure gradient set up in the material during

drying (Mujumdar and Devahastin, 2008). The effective diffusivity depends on geo-

metric shapes and drying conditions, and it is strongly a function of both temperature

and moisture content. In some cases, the well known Arrhenius equation is used to

quantify the influence of temperature on moisture diffusivity (Sablani and Rahman,

2007). However, care should be taken in applying effective diffusivity correlations

obtained experimentally with simple geometric shapes (e.g. slab, cylinder or sphere)

to the more complex shapes because this may lead to incorrect calculated results

(Gong et al., 1997). It is apparent from the relationships reported in the literature that

the effective diffusivity values gradually increased with temperature and decreased

with the decrease in moisture content. In general, the moisture diffusivity values

reported in the literature are in the range of 10�9–10�11 m2/s for drying of food

materials (Sacilik et al., 2006).

Table 4.2 Thermophysical and transport properties usedin modelling drying process.

Parameter Expression/dependence References

Thermophysical properties (food):

Density ¼Sb1 + b2T + b3T

2ð Þi

Xwi

where:

b1,b2,b3 ¼Constants of ith component

Xwi ¼weightfractionof ith component

Choi and Okos

(1987),

Sabarez (2012)

Specific heat

capacity

¼S c1 + c2T + c3T2ð ÞiXw

i

where:

c1,c2,c3 ¼Constantsof ith component

Xwi ¼weightfractionof ith component

Choi and Okos (1987),

Sabarez (2012)

Thermal

conductivity

¼S a1 + a2T + a3T2ð ÞiXv

i

where:

a1,a2,a3 ¼Constantsof ith component

Xvi ¼ volumefractionof ith component

Choi and Okos (1987),

Sabarez (2012)

Transport properties:

Heat transfer

coefficient

¼ f (Nu, Re, Pr, geometry, velocity) Heldman and Lund

(2007)

Mass transfer

coefficient

¼ f (Sh, Re, Sc, geometry, velocity) Heldman and Lund

(2007)

Diffusion

coefficient

¼ f (temperature, moisture content) Sabarez (2012)

Sabarez and Price

(2001)

Thermodynamic properties of moist air:

Density ¼ f (temperature) Pakowski et al. (1991)

Thermal

conductivity

¼ f (temperature) Pakowski et al. (1991)

Specific heat

capacity

¼ f (temperature) Pakowski et al. (1991)

Viscosity ¼ f (temperature) Pakowski et al. (1991)

Note: i¼ food components (protein, fat, carbohydrate, fibre, ash and water).

Modelling of drying processes for food materials 111

4.4.4 Methods of solution

For regularly shaped geometries (infinite cylinder, infinite slab, and sphere) with

proper initial and boundary conditions, together with appropriate simplifications

and assumptions of the mathematical models of a system, it is possible to derive infor-

mation about the system by analytical means, which directly produce general solu-

tions. However, for complex geometries and equations, it is necessary to use

112 Modeling Food Processing Operations

numerical computational methods to provide approximate solutions for the problem

under investigation. The fundamental concept of numerically solving the complex

systems is the discretisation of the geometry of interest to a number of finite elements

or cells, thus reducing the complex governing equations to sets of simple linear or

polynomial equations by employing appropriate approximation techniques. The

numerical methods produce solutions in steps, with each step providing the solution

for one set of conditions and the calculation repeated to expand the range of solutions.

The numerical methods most commonly used to solve complex equations in drying

processes include finite difference, finite element, and finite volume methods. More

details of these methods in relation to solving complex equations in food processes can

be found elsewhere (Patankar, 1980).

To account for shrinkage (deformation) in the material, the resulting systems of

highly coupled nonlinear PDEs in the space-time domain, together with the set of initial

and boundary conditions, are numerically solved coupled to different spatial discretisa-

tion techniques. The most commonly used spatial discretisations are the Lagrange,

Euler, ALE (Arbitrary Euler Lagrange) and mesh-free methods such as smooth particle

hydrodynamics (SPH) (Quan et al., 2003). In the case of ALE, for example, the bound-

ary conditions control the displacement of the moving mesh with respect to the initial

geometry dependent on the moisture content of the material. The moving boundary dis-

placement is propagated throughout the domain to obtain a mesh deformation every-

where using a Laplace smoothing technique (C.O.M.S.O.L. Multiphysics, 2007).

A number of commercial modelling software packages (e.g. COMSOL Multi-

physics) are currently available for the solution of the resulting systems of linear and

nonlinear equations using the appropriate solver. In some cases, a computer program

written in various programming languages (Delphi, C++, Fortran, etc.) is specifically

developed to iteratively solve the equations used to describe the system (Sabarez,

2014; Sabarez and Chessari, 2006). A user-friendly interface of the computer simulation

tool can also be developed especially in object-oriented programming languages (e.g.

Dephi, C++) for the implementation of the mathematical models. This allows the users

to assess the performance of the drying system as a function of the design and opera-

tional parameters of the drying system, as well as the product properties.

Usually, grid independency tests are carried out to ensure that the solution is inde-

pendent of grid or cell size and to verify whether the numerical solution basically

remains the same with further grid or cell refinements. In numerical solutions, the prob-

lem is approximated by discretising the computational domain into a number of points

(or grids). The associated approximation error depends on the number of discretisation

points, meaning that increasing the number of discretisation points can substantially

reduce the error, but with the expense of additional computer time requirement.

4.4.5 Experimental validation

Experimental validation is an essential step in modelling aimed at ensuring that the

mathematical description of the process captures reality. The outputs generated by

a mathematical model must be comparable to the real world system under investiga-

tion. The quality of representing the real behaviour is confirmed during this validation

phase of model development. Differences in model outputs and those gathered from a

Modelling of drying processes for food materials 113

real system are indicators of the level of simplification of the real world problem. By

optimising the level of simplification (model complexity) with the level of accuracy, a

real world problem can be explained satisfactorily. All models must therefore be val-

idated and verified against good experimental data. This involves the comparison of

predicted values (e.g. temperature, moisture content, etc.) with experimentally

measured data.

The validity of the drying models to represent real systems is usually verified by

determining the mean relative percentage deviation (%P) between experimental and

the predicted values, using the expression described elsewhere (Lomauro et al., 1985;

Madamba et al., 1996; Palipane and Driscoll, 1994). According to Kaymak-Ertekin

and Gedik (2005) andMcLaughlin andMagee (1998), a model is acceptable, or a good

fit, when P<10%. Figure 4.5 shows an example of the comparison of changes in the

average moisture content between the experimental and predicted values during the

finish drying of trellis-dried sultanas performed at three drying air temperature levels,

while keeping the same levels of air velocity (2.0 m/s) and relative humidity (10%).

As can be seen from this figure, the simulated results agreed well with the experimen-

tal data. The%Pwas found to be in the range of 0.3–1.1%, confirming the acceptabil-

ity of the model for describing the finish drying process of sultanas at different

temperature levels. In some cases, the validity of the models is evaluated by compar-

ing the predicted values in any particular drying conditions with the experimental data

(Yaldiz et al., 2001; Togrul and Pehlivan, 2003; Sabarez, 2014). The accuracy of the

model predictions is then evaluated on how closely the measured and predicted values

banded around the straight line, which is indicated by the R2 value. For example,

Figure 4.6 shows the experimental drying curves together with the predicted results

from the model at different levels of relative humidity of the drying air with

R2¼0.9941, indicative of the suitability of the model for describing the drying

Drying time (min)

Moi

stur

e co

nten

t (%

w.b

.)

012.0

12.5

13.0

13.5

14.0

14.5

15.0

15.5

16.0

20 40 60 80 100

T = 60 �C (expt)

T = 70 �C (expt)

T = 80 �C (expt)

T = 60 �C (pred)

T = 70 �C (pred)

T = 80 �C (pred)

120 140

Figure 4.5 Experimental versus predicted drying curves at different air temperatures during

finish drying of trellis-dried sultanas (RH¼10%; u¼2.0 m/s; Mi¼15.2%) (Sabarez, 2014).

Experimental moisture content (%w.b.)12 13 14

Pre

dict

ed m

oist

ure

cont

ent (

%w

.b.)

15

y = 0.8732x + 2.0428; R 2 = 0.9941

RH = 10%RH = 30%RH = 50%Linear fit

16 17 18 19 2012

13

14

15

16

17

18

19

20

Figure 4.6 Experimental versus predicted moisture content at different air relative humidity

levels during finish drying of trellis-dried sultanas (T¼70 °C; u¼2.0 m/s; Mi¼18.8%)

(Sabarez, 2014).

02

4

6

8

10

12

14

60 50 30 20 40 60300

302

304

306

308

310

40x (mm) x (mm)y (mm)

y (mm)40 30 20

5

10

15

z (m

m)

z (�

2 m

m)20

25

30

Simulation MeasurementT ( K)

Figure 4.7 Visual comparison between the simulated (left) and the measured (right) heating of

a model food cylinder (at a discrete time) (Knoerzer et al., 2008).

114 Modeling Food Processing Operations

behaviour of sultanas at different relative humidity levels. In other cases, preliminary

evaluation is carried out by visually comparing the simulated values against the exper-

imental data as illustrated in Figure 4.7.

When a model (i.e. particularly one based on fundamental physics) is properly val-

idated, one can extrapolate the behaviour of a system to a range of parameters not

tested in the experiment. A parametric sensitivity study can be undertaken to further

Modelling of drying processes for food materials 115

investigate the effects of the uncertainties of various input parameters on the model’s

predictions and to demonstrate the usefulness of the predictive tool in identifying crit-

ical operational factors affecting the drying process. The model can then be used to

test a number of scenarios (i.e. different operating conditions and material properties)

to study the interactions between the factors in the system and to examine the critical

parameters affecting the drying process. For instance, Figure 4.8 depicts the effect of

uncertainties in the drying air temperature measurement on the model predictions in

plum drying. It indicates that the uncertainties in the measurement of this parameter

are likely to represent a greater contribution to the accuracy in the model predictions,

and it also demonstrates that accurate measurement of the drying air temperature and

its sensing location are important in plum drying operations.

4.5 Case study

This section presents a characteristic case study for the convective drying of plums to

illustrate the application of the modelling approach in determining the optimal design

and operating conditions in industrial-scale tunnel drying systems. In industrial tunnel

dehydrators (as in any large-scale industrial drying systems), the materials being dried

are typically exposed to the dynamically changing conditions of the drying air (e.g.

temperature) at any time and position, as illustrated in Figure 4.9(a). This requires

a model of the drying system that comprises both material and equipment models,

in which the material model describes the drying kinetics and the equipment model

determines the changes of the condition of the drying medium with time and space

during drying. Together, these models constitute a complete modelling tool capable

of predicting the dynamic behaviour of the drying system. Thus, the prediction of the

drying air stream conditions flowing across the product surface, which would affect

T = 82 �C

T = 80 �C

T = 78 �C

010

20

30

40

50

60

70

80

2 4 6 8 10Drying time (h)

Moi

stur

e co

nten

t (%

w.b

.)

12 14 16 18

Figure 4.8 Predicted effect of air temperature uncertainties on the drying kinetics of prunes

(RH¼15%; u¼5 m/s) (Sabarez, 2012).

1.5

1

Air outlet

Exhaust

(a)

(b)

Fresh air

Burner

Recirculation

Fan

Motor

Fresh fruit

Air inlet

Temp (�C)

86

84

82

80

78

76

74

0.5

10.8

0.60.4

0.2 01

2 Length (m)Width (m)

Hei

ght (

m)

34

5

Figure 4.9 Schematic diagram of a typical tunnel dehydrator for (a) the commercial drying of

prunes and (b) the measured temperature profile across the drying tunnel (Sabarez, 2010).

116 Modeling Food Processing Operations

the drying behaviour of the solid product, at any time and position in the dryer is of

particular importance in simulating the drying process of industrial drying systems for

which a systematic dynamic variation in drying conditions is typical (Sabarez, 2012).

A 2D axis-symmetric model was developed to describe the simultaneous transfer of

momentum (air only), heat and mass (air and food) occurring in convective air drying

of fruits (e.g. plums). The governing PDEs describing the simultaneous transfer of

heat, mass and momentum in two distinct subdomains (air and food) during the drying

of plums were presented in previous studies (Sabarez, 2010, 2012). The nonisothermal

Modelling of drying processes for food materials 117

turbulent flow of air in the drying chamber is described according to the standard k-emodel (C.O.M.S.O.L. Multiphysics, 2007). The resulting systems of highly coupled

nonlinear PDEs in the space-time domain, together with the set of initial and boundary

conditions, were numerically solved by the finite element method (FEM) coupled to

the ALE procedure to account for the shrinkage phenomenon, using a commercial

software package (C.O.M.S.O.L. Multiphysics, 2007). The details of the numerical

solution are presented in previous studies (Sabarez, 2010, 2012). Also, the solution

of the governing PDEs requires knowledge of the thermophysical and transport prop-

erties of the product and air. The model parameters used in this work are given in pre-

vious studies (Sabarez, 2012, 2014).

A computer-controlled experimental drying system (Figure 4.3) was specifically

developed to study the drying kinetics of various food materials under controlled con-

ditions over a wide range of operating conditions for use inmodel validation. The exper-

imental set-up was designed to allow simulation of a typical commercial dehydration

system. The purpose-built test drying facility incorporated a number of special features,

including a fully programmable cyclic control of process conditions (i.e. temperature,

humidity, and air flow), ultrasonic unit and a dedicated weighing system. It was

equipped with controllers to control the process variables. A number of additional

sensors (i.e. thermocouples, infrared noncontact temperature sensors, air velocity

sensors, relative humidity probes, etc.) were interfaced to a computer-based data

acquisition and control system for further online monitoring and recording of the var-

ious processing conditions. Further details of the experimental drying setup can be

found in previous studies (Sabarez, 2012, 2014; Sabarez et al., 2012; Beck et al., 2014).

Figure 4.10 shows the measured surface and centre temperatures of the product

together with the predicted values. It can be seen from this figure that there is a good

agreement between the experimental data and predicted values. These results confirm

the suitability of the model for describing the heat transfer process during the drying of

Drying time (h)

Surface (expt)

Centre (expt)

Surface (pred)

Centre (pred)

020

25

30

35

40

Frui

t tem

pera

ture

(�C

)

45

50

55

60

65

70

75

2 4 6 8 10 12 14 16 18 20

Figure 4.10 Predicted versus experimental fruit temperature profile at different locations in the

fruit (T¼70 °C; RH¼35%; u¼5.7 m/s) (Sabarez, 2012).

118 Modeling Food Processing Operations

plums, and they demonstrate that the thermophysical parameters used in the model are

reasonable. Similar trends were also found for other drying conditions investigated

(Sabarez, 2010). This validates the dependency of the product’s thermal properties

on both temperature and moisture content.

Figure 4.11 shows the drying curves of plums predicted by the model for the two

experimental drying tests performed at different air temperatures and relative humid-

ity levels under the same air velocity (5.7 m/s). In the moderate drying conditions, the

air temperature was maintained at 70 °Cwith a relative humidity of 35%, while, in the

more intense conditions, the air temperature was 80 °C with a relative humidity of

15%. These conditions were selected to simulate the extreme drying conditions

typically found in industrial tunnel drying operations. As can be seen from this figure,

the simulated results agree well with the experimental data. Also, Sabarez (2012)

presented further validations to verify the predictive capability of the model over a

range of conditions. The results confirm the validity of the model and demonstrate

that the parameters used in the model are reasonable, indicating the suitability of

the model for describing the drying process of plums under various conditions.

A number of numerical scenarios were also tested for different operating condi-

tions andmaterial properties to study the interactions between the factors in the system

and to identify critical operational factors that offer significant and measurable oppor-

tunities for improvement in the drying process. The conditions of the drying air (i.e.

airflow, temperature and relative humidity) are considered to be the main factors

influencing the drying performance in tunnel dehydrators. In particular, the effect

of different air velocity levels was taken as an example to demonstrate the impact

of this parameter on the drying kinetics (Figure 4.12). In this instance, an increase

in air velocity leads to a pronounced decrease of the drying time but only to a certain

level. Beyond this level the air velocity plays a proportionally decreasing role in

Expt (T = 70 �C; RH = 35%)

Expt (T = 80 �C; RH = 15%)

Pred (T = 70 �C; RH = 35%)

Pred (T = 80 �C; RH = 15%)

10

20

30

40

50

60

70

80

0 2 4 6 8 10Drying time (h)

Moi

stur

e co

nten

t(%

w.b

.)

12 14 16 18 22 2420

Figure 4.11 Predicted versus experimental drying kinetics of plums (u¼5.7 m/s) (Sabarez,

2012).

Drying time (h)

Moi

stur

e co

nten

t (%

w.b

.)

0 2 4 6 8 10 12

u = 1.0 m/s

u = 3.0 m/s

u = 5.0 m/s

u = 7.0 m/s

14 16 18 2010

20

30

40

50

60

70

80

Figure 4.12 Predicted effect of air velocity on the drying kinetics of prunes (T¼80 °C;RH¼15%) (Sabarez, 2012).

Modelling of drying processes for food materials 119

reducing the drying time. This has significant implications for the energy consumption

of the drying process, as shown later.

The advantage of the proposed numerical model is that the temperature and

moisture distributions across the solid food domain, as well the changes of the con-

dition of the drying air with location, can be established at any time during drying

(Figure 4.13(b)). This is important for simulating the drying process that will take into

account the dynamic changes in the drying conditions, allowing simulations that

mimic the industrial-scale tunnel drying of plums in both parallel-flow and

counter-flow modes of operation (Figure 4.14). It should be noted that the mode of

operation in a tunnel drying system is one of the key factors that significantly influ-

enced the drying performance (Sabarez, 2010).

A number of computer simulations were further carried out to study the drying per-

formance under various drying conditions in an industrial tunnel drying operation. It is

well known that the air velocity field greatly influences the heat and mass transfer

rates at the food–air interfaces. Therefore, the temperature and concentration of mois-

ture in the product and the drying air are basically controlled by the level of air veloc-

ity and its distribution. So, the effect of different air velocity levels in the drying tunnel

operated in parallel-flow mode was used to demonstrate the impact of this parameter

on the drying kinetics and energy consumption. The other selected conditions are rep-

resentative for the current commercial tunnel drying operation for plums (Sabarez,

2010). Figure 4.15 depicts the simulated effect of different levels of air velocity on

both drying time and energy consumption in an industrial tunnel drying operation.

As can be observed from this plot, there appears to be an optimum level of air velocity

required to achieve better drying performance, which can be found at the intersection

of the plots. Under these conditions, the optimum air velocity level appears to be

around 4–5 m/s. As the air velocity increases the energy consumption also appears

to increase. This is obvious because increases in air volume would result in increased

energy requirements for heating the large volume of air to the desired temperature

0(a) (b)

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.01 0.02

x (m)

y (m

)

y (m

)

x (m)

0.03 0−0.01

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.01 0.02 0.03 0.04 0.0435 63

80

100

120

140

160

180

200

220

240

260

278

0.0436

0.0438

0.044

0.0449Air Food

Time=64300[s]; Food: concentration, c [kg/m^3]; Air: concentration, c2 [kg/m^3]; Arrow: Velocity field [m/s]

0.0448

0.0446

0.0444

0.0442

Figure 4.13 Predicted product moisture concentration, moisture concentration and velocity

profiles of the drying air during the drying of plums (T¼80 °C; RH¼15%; u¼5.7 m/s)

(Sabarez, 2012).

Drying time (h)0 2 4 6 8 10 12

Parallel-flow (predicted)

Counter-flow (predicted)

Parallel-flow (measured)

Counter-flow (measured)

14 16 18 2010

20

30

40

Moi

stur

e co

nten

t (%

w.b

.)

50

60

70

80

Figure 4.14 Simulated and measured drying kinetics of prunes in industrial-scale drying for

both modes of operation (u¼5 m/s) (Sabarez, 2010).

120 Modeling Food Processing Operations

15

14

13

12

11

10

9

80

01

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9 10Velocity (m/s)

Dry

ing

time

(h)

Drying time Energy use

Ene

rgy

use

(GJ)

Figure 4.15 Effect of air velocity on drying time and energy consumption in an industrial-scale

drying operation (inlet: T¼85 °C, RH¼15%; recycle: T¼70 °C, RH¼30%, ratio¼90%;

ambient: T¼25 °C, RH¼65%) (Sabarez, 2010).

Modelling of drying processes for food materials 121

level. On the other hand, the drying time seems to significantly decrease as the air

velocity increases but only to a certain point. Beyond this point, the air velocity plays

a proportionally decreasing role in reducing the drying time.

Furthermore, the predictions in the distributions of moisture content and temper-

ature across the food materials are important for characterising the quality changes

during drying. The development of mathematical models for improved understanding

of the underpinning heat and mass transfer mechanisms controlling the drying process

and the associated impact on product quality attributes is crucially important for

achieving the optimum design and operating conditions of a drying system that max-

imises the retention of the desired quality attributes of the product. One of the impor-

tant quality attributes that usually accompanies dehydration of food products,

particularly for fruits (e.g. grapes), is the change of product colour due to browning

reactions (i.e. enzymatic and nonenzymatic). The ability to predict changes in product

colour during drying would be useful for optimising the drying process in order to

produce the desired premium colour attributes. For example, a kinetic model was

coupled to the heat and mass transfer calculations to describe the drying kinetics

and the evolution of product colour during the finish drying of trellis-dried sultanas

(Sabarez, 2014). This allows simultaneous predictions of the moisture content, tem-

perature and colour profiles of the product in a space-time domain during the drying

process as a function of various operating conditions, establishing the optimal drying

conditions for producing the desired premium colour attributes of the product. This

approach could be extended to other food products, and it could incorporate other

product quality attributes.

122 Modeling Food Processing Operations

4.6 Future directions

The drying processes that are currently used in the food industry will continue to play a

significant role in food manufacturing as long as they are still viable and have not

reached their limit of performance. As with many other food processes, further rede-

sign and optimisation of these existing drying technologies are crucial in order for the

industry to become more productive and sustainable. In addition, new challenges are

continually emerging as new innovative drying technologies appear to overcome the

limitations of the conventional drying processes. Modelling strategies will be needed

to further assist in developing such incremental improvements of the existing drying

processes and in intensifying innovation of new drying concepts for effective and effi-

cient implementation at an industrial scale.

With advances in computing capabilities, further progress can be made in the devel-

opment of advanced and realistic multiscale drying models that couple the transport

phenomena (heat and mass transfer, fluid dynamics), physical or structural changes,

chemical reactions, phase changes, complex food compositions and other physical phe-

nomena (e.g. acoustic, electric, electromagnetic fields). This is important in understand-

ing the length- and time-scale interactions involved in simulating the drying processes

for scaling-up and optimisation, without excessive trial-and-error and the associated

costs of physical experimentations. In addition, modelling the drying process in a

way that incorporates the prediction of food qualities (i.e. sensorial, functional and nutri-

tional) will also be important in the development of new drying technologies allowing

the manipulation and control of food quality to achieve the desired attributes.

These developments, together with advances in sensing and visualisation tech-

niques, will enable the development of real-time model-based control systems for dry-

ing processes, ensuring an efficient process, safe operation and a higher-quality

product. The development of advanced sensing and instrumentation capabilities

should also improve the availability of experimental data that would otherwise be very

difficult to obtain in some drying systems due to the challenges associated with

measurement-taking (e.g. hostile environment of high temperatures and/or pressures).

Appendix: Nomenclature

c

water concentration in food (mol/m3)

c1

ultrasonic propagation velocity in capillary (m/s)

c2

water concentration in air (mol/m3)

C

colour parameters (L*, a*, b*) (–) Cp specific heat (J/kgK)

Cm

model parameter (–)

Ce1

model parameter (–)

Ce2

model parameter (–)

D

effective water diffusivity in food (m2/s)

E

electric field (V/m)

f

ultrasonic frequency (kHz)

g

acceleration of gravity (m/s2)

Modelling of drying processes for food materials 123

I

ultrasonic intensity (W/m2)

hc

heat transfer coefficient (W/m2K)

k

thermal conductivity (W/mK)

kc

mass transfer coefficient (m/s)

M

moisture content (% wet basis)

n

direction normal to surface (–)

P

pressure (Pa)

rm

average radii of capillary (m)

RH

relative humidity (%)

T

food temperature (°C) T2 air temperature (°C) TR temperature of the radiator (°C) t time (s)

Subscripts

a

air (–)

i

initial (–)

s

food surface (–)

Greek letters

e

dissipation rate (m2/s3)

e1

emissivity (–)

e0

dielectric constant of vacuum (A s/V m)

er00

real part of complex permittivity (–)

l

latent heat of evaporation (J/kg)

�

dynamic viscosity (N.s/m2)

k

turbulence energy (m2/s2)

r

density (kg/m3)

sk

model parameter (–)

s

Stefan-Boltzmann constant (–)

u

velocity (m/s)

References

Abonyi, B.I., Tang, J., Edwards, C.G. (1999). Evaluation of energy efficiency and quality reten-

tion for the Refractance Window™ drying system. Research Report, Washington State

University, Pullman.

Abonyi, B.I., Fenh, H., Tang, J., Edwards, C.G., Chew, B.P., Mattison, D.S., Fellman, J.K.,

2001. Quality retention in strawberry and carrot purees dried with Refractance Window

system. J. Food Sci. 67 (2), 1051–1056.

Ah-Hen, K., Zambra, C.E., Aguero, J.E., Vega-Galvez, A., Lemus-Mondaca, R., 2013. Mois-

ture diffusivity coefficient and convective drying modeling of murta (Ugni molinae

Turcz): influence of temperature and vacuum on drying kinetics. Food Bioprocess Tech.

6, 919–930.

ASHRAE, 1995. Handbook of Fundamentals. American Association of Heating, Refrigeration

and Air Conditioning Engineers, Inc., New York.

124 Modeling Food Processing Operations

Aversa, M., Curcio, S., Calabro, V., Iorio, G., 2007. An analysis of the transport phenomena

occurring during food drying process. J. Food Eng. 78, 922–932.

Bansal, P.R., Chung, K.Y., 2007. Food drying equipment and design. In: Hui, Y.H., Clary, C.,

Farid, M.M., Fasina, O.O., Noomhorm, A., Welti-Chenis, J. (Eds.), Food Drying Science

and Technology: Microbiology, Chemistry, Applications. DESTech Publications, Inc,

Pennsylvania, USA, Chapter 15, pp. 359-402.

Barbosa-Canovas, G.V., 1996. Dehydration of Foods. Chapman and Hall, New York.

Beck, S.M., Sabarez, H.T., Gaukel, V., Knoerzer, K., 2014. Enhancement of convective drying

by application of airborne ultrasound: a response surface approach. Ultrason. Sonochem.

21, 2144–2150.

Bolland, K.M., 2000. A new low-temperature/short-time drying process. Cereal Foods World.

45, 293–296.

Multiphysics, C.O.M.S.O.L., 2007. Chemical Engineering Module Model Library. COMSOL

AB, Stockholm, Sweden.

Caparino, O.A., Tang, J., Nindo, C.I., Sablani, S.S., Powers, J.R., Fellman, J.K., 2012. Effect of

drying methods on the physical properties and microstructures of mango (Philippine

“Carabao” var.) powder. J. Food Eng. 111, 135–148.

Carcel, J.A., Garcia-Perez, J.V., Riera, E., Mulet, A., 2007. Influence of high intensity ultra-

sound on drying kinetics of persimmon. Dry. Technol. 25, 185–193.

Carcel, J.A., Garcia-Perez, J.V., Riera, E., Mulet, A., 2011a. Improvement of convective drying

of carrot by applying power ultrasound – influence of mass load density. Dry. Technol.

29, 174–182.

Carcel, J.A., Garcia-Perez, J.V., Benedito, J., Mulet, A., 2011b. Food process innovation

through new technologies: use of ultrasound. J. Food Eng. 110 (2), 200–207.

Choi, Y., Okos, M.R., 1987. Effects of temperature and composition on thermal properties

of foods. In: Le Maguer, M., Jelen, P. (Eds.), Food Engineering and Process Applications,

vol. 1. Elsevier Applied Science Publishers, New York, pp. 269–312.

Clarke, P.T., 2004. Refractance Window – DownUnder. In: Proceedings of the 14th

International Drying Symposium (IDS 2004), Sao Paulo, Brazil, August 2004,

pp. 813–820, vol B.

Curcio, S., Aversa, M., 2014. Influence of shrinkage on convective drying of fresh vegetables: a

theoretical model. J. Food Eng. 123, 36–49.

Curcio, S., Aversa, M., Calabro, V., Iorio, G., 2008. Simulation of food drying: FEM analysis

and experimental validation. J. Food Eng. 87, 541–553.

Du, G.H., Zhu, Z.M., Gong, X.F., 1981. Acoustics Foundation, vol. 2. Shanghai Science and

Technology Press, Shanghai.

Duffie, J., Marshall, W., 1953. Factors influencing the properties of spray-dried materials.

Chem. Eng. Process. 49, 417–423.

Dziezak, J.D., 1988. Microencapsulation and encapsulated ingredients. Food Technol. 42 (4),

136–151.

Feng, H., Yin, Y., Tang, J., 2012. Microwave drying of food and agricultural materials: basics

and heat and mass transfer modeling. Food Eng. Rev. 4, 89–106.

Gallego-Juarez, J.A., 2010. High-power ultrasonic processing: recent developments and pro-

spective advances. Phys. Procedia 2, 35–47.

Gallego-Juarez, J.A., Riera, E., de la Fuente Balanco, S., Rodriguez-Corral, G., Acosta-

Aparicio, V.M., Blanco, A., 2007. Application of high-power ultrasound for dehydration

of vegetables: processes and Devices. Dry. Technol. 25 (11), 1893–1901.

Garcia-Perez, J.V., Carcel, J.A., Benedito, J., Mulet, A., 2007a. Power ultrasound mass transfer

enhancement in food drying. Food Bioprod. Process 85, 247–254.

Modelling of drying processes for food materials 125

Garcia-Perez, J.V., Carcel, J.A., Benedito, J., Riera, E., Mulet, A., 2007b. Influence of process

variables on hot air drying assisted by power ultrasound. In: Proceedings of the 19th Inter-

national Congress on Acoustics, Madrid, Spain.

Garcia-Perez, J.V., Carcel, J.A., Riera, E., Mulet, A., 2009. Influence of the applied acoustic

energy on the drying of carrots and lemon. Dry. Technol. 27 (2), 281–287.

Garcia-Perez, J.V., Puig, A., Perez-Munuera, I., Carcel, J.A., Riera, E., 2010. Kinetic andmicro-

structural changes induced by power ultrasound application on convective drying of egg-

plant. In: Proceedings of the 20th International Congress on Acoustics (ICA 2010),

Sydney, Australia.

Gong, Z.-X., Devahastin, S., Mujumdar, A.S., 1997. A two-dimensional finite element

model for wheat drying in a novel rotating jet spouted bed. Drying Technol. Int. J.

15, 575–592.

Hayakawa, K.I., Furuta, T., 1988. Thermodynamically interactive heat and mass transfer

coupled with shrinkage and chemical reaction. In: Singh, R.P., Medina, A.G. (Eds.), Food

Properties and Computer-Aided Engineering of Fodd-processing System. Kluwer Aca-

demic Pub, Dordrecht, Boston & London.

Heldman, D.R., Hartel, R.W., 1997. Principles of Food Processing. Chapman andHall, NewYork.

Heldman, D.R., Lund, D.B., 2007. Handbook of Food Engineering. Marcel Dekker, Inc.,

New York, pp. 409–479.

Hsu, C.L., Chen, W., Weng, Y.M., Tseng, C.Y., 2003. Chemical composition, physical prop-

erties, and antioxidant activities of yam flours as affected by different drying methods.

Food Chem. 83, 85–92.

Jangam, S.V., 2011. An overview of recent developments and some R&D challenges related to

drying of foods. Drying Technol. Int. J. 29 (12), 1343–1357.

Kaymak-Ertekin, F., Gedik, A., 2005. Kinetic modeling of quality deterioration in onions during

drying and storage. J. Food Eng. 68, 443–453.

Khmelev, V.N., Barsukov, R.V., Abramenko, B.S., Genne, D.V., 2008. Research and develop-

ment of ultrasonic device prototype for intensification of drying process. In: In 9th Inter-

national Workshop and Tutorials EDM’2008.

Khmelev, V.N., Shalunov, A.V., Barsukov, R.V., Abramenko, D.S., Lebedev, A.N., 2011. Stud-

ies of ultrasonic dehydration efficiency. J. Zhejiang. Univ. Sci. A 12 (4), 247–254.

Knoerzer, K., Regier, M., Schubert, H., 2008. A computational model for calculating temper-

ature distributions in microwave food applications. Innovat. Food. Sci. Emerg. Tech.

9, 374–384.

Kostoglou, M., Chrysafis, N., Andritsos, N., 2013. Modeling tomato dehydration in a tunnel

dryer using geothermal energy. Drying Technol. Int. J. 31, 5–15.

Lin, R.T., 1995. Introduction of Porous Media Heat and Mass Transfer. Science Press, Beijing.

Lomauro, C.J., Bakshi, A.S., Labuza, T.P., 1985. Moisture transfer properties of dry and semi

moist foods. J. Food Sci. 50, 397–400.

Madamba, P.S., Driscoll, R.H., Buckle, K.A., 1996. Thin-layer drying characteristics of garlic

slices. J. Food Eng. 29, 75–97.

McLaughlin, C.P., Magee, T.R.A., 1998. The determination of sorption isotherm and the isos-

teric heats of sorption for potatoes. J. Food Eng. 35, 267–280.

Menges, H.O., Ertiken, C., 2006. Mathematical modeling of thin layer drying of golden apples.

J. Food Eng. 77, 119–125.

Mujumdar, A.S., 1987. Handbook of Industrial Drying. Marcel Dekker, Inc., New York.

Mujumdar, A.S., Devahastin, S., 2008. Fundamental principles of drying. In: Mujumdar, A.S.

(Ed.), Guide to Industrial Drying—Principles, Equipments and New Developments. Three

S Colors Publications, Mumbai, India, pp. 1–21.

126 Modeling Food Processing Operations

Mujumdar, A.S., Huang, L.X., 2007. Global R&D needs in drying. Dry. Technol. 25, 647–658.

Mujumdar, A.S., Wu, Z.H., 2010. Thermal drying technologies: new developments and future

R&D potential. In: Jangam, S.V., Thorat, B.N. (Eds.), R&DNeeds, Challenges and Oppor-

tunities for Innovation in Drying Technology’, e-Book.

Nindo, C.I., Feng, H., Shen, H.Q., Tang, J., Kang, D.H., 2003a. Energy utilisation and microbial

reduction in a new film drying system. J. Food Process Pres. 27, 117–136.

Nindo, C.I., Sun, T., Wang, S.W., Tang, J., Powers, J.R., 2003b. Evaluation of drying technol-

ogies for retention of physical quality and antioxidants in asparagus (Asparagus officinalis,

L.). Lebensm. Wiss u. Technol. 36, 507–516.

Nindo, C.I., Tang, J., Powers, J.R., Bolland, K., 2004. Energy consumption during Refractance

Window evaporation of selected berry juices. Int. J. Energy Res. 28, 1089–1100.

Ochoa-Martinez, C.I., Quintero, P.T., Ayala, A.A., Ortiz, M.J., 2012. Drying characteristics of

mango slices using the Refractance Window™ technique. J. Food Eng. 109, 69–75.

Oetjen, G.W., Haseley, P., 2004. Freeze-Drying, second ed. Wiley-VCH GmbH& Coo. KGaA,

Germany.

Oliveira, M.E.C., Franca, A.S., 2002. Microwave heating of foods. J. Food Eng. 53, 347–359.

Ozuna, C., Carcel, J.A., Garcia-Perez, J.V., Mulet, A., 2011. Improvement of water transport

mechanisms during potato drying by applying ultrasound. J. Sci. Food Agric. 2511–2517.

Pakowski, Z., Bartczak, Z., Strumillo, C., Stenstrom, S., 1991. Evaluation of equations approx-

imating thermodynamic and transport properties of water, steam and air for use in CAD of

drying processes. Drying Technol. Int. J. 9, 753–773.

Palipane, K.B., Driscoll, R.H., 1994. Thin-layer drying characteristics of macadamia in-shell

nuts and kernels. J. Food Eng. 23, 129–144.

Patankar, S., 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corpora-

tion, New York.

Prakotmak, P., 2013. Finite element modeling of heat and mass transfer in food materials during

microwave heating. J. Appl. Sci. Res. 9 (12), 6115–6121.

Quan, I.X., Birnbaum, N.K., Cowler, M.S., Gerber, B.I., Clegg, R.A., Hayhurst, C.J., 2003.

Numerical simulation of structural deformation under shock and impact loads using a

coupled multi-solver. In: Proceedings of the 5th Asia-Pacific Conference on Shock and

Impact Load on Structures, Hunan, China.

Ratti, C., 2001. Hot air and freeze-drying of high-value foods: a review. J. Food Eng.

49, 311–319.

Sabarez, H.T. (2010). Improving prune dehydration cost efficiency. Unpublished report pre-

pared by CSIRO Food and Nutritional Sciences for Horticulture Australia Limited

(HAL), Sydney, Australia.

Sabarez, H.T., 2012. Computational modeling of the transport phenomena occurring during

convective drying of prunes. J. Food Eng. 111 (2), 279–288.

Sabarez, H.T., 2014. Mathematical modeling of the coupled transport phenomena and color

development: finish drying of trellis-dried sultanas. Dry. Technol. 32, 578–589.

Sabarez, H.T. and Chessari, C. (2006). High quality fruit and vegetable ingredients for func-

tional foods: Modeling and optimisation of Refractance Window drying technology.

Unpublished report prepared by CSIRO for Nutradry Pty Ltd, Brisbane, Australia.

Sabarez, H.T., Price, W.E., 2001. Modeling simultaneous heat and mass transfer during drying

of prunes. Acta Horticult. 566, 421–428.

Sabarez, H.T., Gallego-Juarez, J.A., Riera, E., 2012. Ultrasonic-assisted convective drying of

apple slices. Drying Technol. Int. J. 30 (9), 989–997.

Sablani, S.S., Rahman, M.S., 2007. Fundamentals of food dehydration. In: Hui, Y.H., Clary, C.,

Farid, M.M., Fasina, O.O., Noomhorm, A., Welti-Chenis, J. (Eds.), Food Drying Science

Modelling of drying processes for food materials 127

and Technology: Microbiology, Chemistry, Applications. DESTech Publications, Inc.,

Pennsylvania, USA, Chapter 1, pp. 1–42.

Sacilik, K., Elicin, A.K., Unal, G., 2006. Drying kinetics of uryani plum in a convective hot-air

dryer. J. Food Eng. 76, 362–368.

Schossler, K., Jager, H. And, Knorr, D., 2012. Effect of continuous and intermittent ultrasound

on drying time and effective diffusivity during convective drying of apple and red bell pep-

per. J. Food Eng. 108, 103–110.

Singh, S.P., Jairaj, K.S., Srikant, K., 2012. Universal drying rate constant of seedless grapes: a

review. Renew. Sust. Energ. Rev. 16, 6295–6302.

Sloth, J., Kiil, S., Jensen, A.D., Andersen, S.K., Jorgensen, K., Schiffter, H., Lee, G., 2006.

Model based analysis of the drying of a single solution droplet in an ultrasonic levitator.

Chem. Eng. Sci. 61, 2701–2709.

Soria, A.C., Villamiel, M., 2010. Effect of ultrasound of the technological properties and bio-

activity of food: a review. Trends Food Sci. Technol. 21, 323–331.

Togrul, H., 2005. Simple modeling of infrared drying of fresh apple slices. J. Food Eng.

71, 311–323.

Togrul, I.T., Pehlivan, D., 2003. Modeling of drying kinetics of single apricot. J. Food Eng.

58, 23–32.

Yaldiz, O., Ertekin, C., Uzun, H.I., 2001. Mathematical modeling of thin layer solar drying of

sultana grapes. Energy 26, 457–465.

Zhao, F., Chen, Z., 2011. Numerical study on moisture transfer in ultrasound-assisted

convective drying process of sludge. Dry. Technol. 29, 1404–1415.