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Trends in Food Science & Technology
journal homepage: www.elsevier.com/locate/tifs
Review
Simulation of food drying processes by Computational Fluid Dynamics
(CFD); recent advances and approaches
Narjes Malekjani, Seid Mahdi Jafari∗
Department of Food Materials and Process Design Engineering, Gorgan University of Agricultural Sciences and Natural Resources, Gorgan, Iran
A R T I C L E I N F O
Keywords:
Computational fluid dynamics (CFD)
Drying
Simulation
Control
Optimization
Food quality
A B S T R A C T
Background: Understanding the mechanisms underlying the drying processes has a critical role in dehydration of
food and agricultural products. Advanced computer modeling and simulation techniques can help in developing
new dryers, modification of current systems, energy saving and process optimization. Also the most important
parameter during the drying food products is food quality (moisture content, crack formation, case hardening,
etc.) which can be enhanced through using appropriate modeling. Computational Fluid Dynamics (CFD) is a
well-known modeling technique which has received more attention in the food industry in the recent years.
Hydrodynamics of fluid flow, heat and mass transfer during drying can be predicted using CFD.
Scope and Approach: This article reviews fundamentals, merits and shortcomings of CFD in the drying process
modeling with a special focus on dehydration of food products. Since the drying is a growing unit operation,
there is an emphasis on investigation of CFD utilization in modeling emerging drying processes of food products
such as microwave assisted drying, infrared and superheated steam drying besides conventional convective
drying systems notably in recent 5 years.
Key Findings and Conclusions: CFD has been considered as a promising method which could help developing the
design of new dryers, enhancing current dryers and the most important aspect of utilization of this method in the
food industry research and development is “food quality” improvement.
1. Introduction
Drying is a multidisciplinary unit operation which can be expressed
as the heart of processing operations in many industries (Dehnad,
Jafari, & Afrasiabi, 2016). This process has versatile utilizations in food,
wood and paper, textile, pharmaceutical, chemical and biological ma-
terial production (Defraeye, 2014). In food processing, drying is also
considered as one of the most crucial practices which is applied to
different types of food products ranging from low moisture agricultural
crops (e.g. rice, corn, and wheat) to intermediate moisture (e.g. pasta,
tea, and coffee) and high moisture (milk, fruits and vegetables) pro-
ducts (Jafari, Azizi, Mirzaei, & Dehnad, 2016; Jafari, Ghanbari, Ganje,
& Dehnad, 2016; Azizi, Jafari Seid, Mirzaei, & Dehnad, 2017). The main
purpose of drying is reducing the water content of the product in order
to reach a safe level to preserve it from deteriorative factors such as
microbial, physical and chemical parameters while retaining its sensory
and nutritional quality and declining the energy and time consumption
and optimization of the process throughput (Bahmani, Jafari, Shahidi,
& Dehnad, 2016; Jafari, Ghalegi Ghalenoei, & Dehnad, 2017). Drying
process has a sophisticated nature. According to Defraeye (2014),
drying is a multicubed operation; multiphase means it involves dif-
ferent phases (solid food products, liquid water and gas phase of drying
air), multiscale ranging from dryer scale to cell walls of the drying
product and multiphysics including simultaneous heat, mass and mo-
mentum transport in addition to many biochemical and chemical pro-
cesses such as nutritional loss, enzymatic reactions, aroma and color
changes, fat oxidation and microbial degradation that are so critical for
ensuring food quality and safety (Defraeye, 2014). Drying is also an
energy intensive process and includes 12–20% of the energy require-
ments of national industries in developed countries (Bardy, Hamdi,
Havet, & Rouaud, 2015), so it has always been a challenging issue in the
food industry (Norton & Sun, 2006; Strumiłło, 2006).
Investigation of the physicochemical changes occurring during
drying in line with developing novel strategies to optimize the energy
consumption, utilization of renewable energy, recovering the heat used
for this process and application of environmentally friendly technolo-
gies are becoming more important in the current world and any
achievement in this area is highly encouraged. These developments
need a lot of time and costly laboratory and experimental efforts.
Mathematical models and computer simulation tools are realistic and
https://doi.org/10.1016/j.tifs.2018.06.006
Received 30 December 2017; Received in revised form 11 June 2018; Accepted 12 June 2018
∗ Corresponding author.
E-mail address: smjafari@gau.ac.ir (S.M. Jafari).
Trends in Food Science & Technology 78 (2018) 206–223
Available online 14 June 2018
0924-2244/ © 2018 Elsevier Ltd. All rights reserved.
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effective alternatives to experimental practices (Hashemi Shahraki,
Jafari, Mashkour, & Esmaeilzadeh, 2014; Jafari, Ganje, Dehnad, &
Ghanbari, 2016; Malekjani, Jafari, Rahmati, Esmaeelzadeh, & Mirzaee,
2013). These approaches can represent an excellent understanding of
the transport phenomena during the drying operation and enhancing
the process control leading to drying optimization and improved food
quality (Jamaleddine & Ray, 2010). The main external parameters
which affect heat and mass transfer are ambient air or any drying fluid
velocity, temperature and relative humidity; while density, porosity,
permeability, specific heat, mass diffusivity and thermal conductivity
are some of the internal parameters affecting the drying process. Si-
mulation can act as virtual sensors of humidity, velocity and tempera-
ture and also internal parameters in inaccessible locations, it is sensitive
to small changes and there are no limitations in testing different and
unusual drying conditions. Also, there is no requirement for excessive
laboratory spaces, skilled operators and additional maintenance. Be-
sides these advantages, there are some shortcomings in utilization of
the computer modeling and simulation methods in the field of food
industry. Multicubed nature of drying process as stated before, lack of
adequate data regarding the material properties, especially physico-
chemical properties of food and agricultural products which are
strongly moisture and temperature dependent, low economic value of
some agricultural products which makes the simulation unworthy, and
shortage of user-friendly computer software packages are some of these
disadvantages (Defraeye, 2014).
Computational fluid dynamics (CFD) is a powerful and advanced
numerical method to solve governing partial differential equations
(PDEs) of mass, momentum and energy conservation in fluid flow and
heat and mass transfer problems (Norton & Sun, 2010). It also could be
noted as a useful tool in food engineering problems. Visualization of the
obtained simulation results for pressure, velocity vectors, temperature
distribution, and species concentration in the form of liquid or solid
during thermal processes using attractive color figures and animations
can help interpreting of the occurring physical phenomena, thus en-
hancing the overall process and product quality (Lemus-Mondaca,
Vega-Gálvez, & Moraga, 2011). CFD was first used in the 1950s and
since then, it has been developed increasingly (Norton, Tiwari, & Sun,
2013). The most important motivation for this study in the first part is
to review fundamentals of CFD including the frameworks, numerical
solution
methods, introducing relevant software and analysis techni-
ques. In the second part of the study, some useful and highlighted ex-
amples of CFD utilization in the field of drying especially novel drying
technologies are presented in order to introduce the capabilities of this
method to researchers and students in the area of food engineering. As
the previous literature of CFD was mainly focused on drying methods
such as spray drying and to some extent fluidized bed drying, in this
study it has been tried to show the ability of CFD in simulation of some
novel drying strategies such as microwave assisted, superheated steam
and infrared drying which are growing quickly nowadays. Un-
fortunately, there is a shortage in literature in application of CFD in
these novel drying technologies but the authors tried to discuss some of
the most important aspects of CFD simulation for the mentioned
methods. Further researches regarding the fundamentals of CFD simu-
lation and its application in drying of food products will be discussed in
the following sections.
2. Principles of CFD in food drying processes
There are some important PDEs which should be solved in order to
determine heat, mass and momentum transfer during drying processes.
Modeling Newtonian fluid flow is performed using Navier- Stokes
equations. Since drying process involves heat transfer and fluid prop-
erties are usually temperature dependent, so energy equation is usually
coupled with Navier- Stokes equation. When a conjugated heat transfer
is being studied, the continuity of thermal energy exchange across the
interface between fluid and solid should be maintained, so, heat
transfer in solid body should be taken into account in CFD simulation.
Continuity equation or conservation law of mass (eq. (1)) im-
plies that there is an axact balance between the incoming flow of mass
in a fluid element with the mass leaving that element:
∂
∂
+ ∂
∂
=
ρ
t x
ρu( ) 0
i
i (1)
where ρ and t are density (kg/m3) and time (s), x is Cartesian co-
ordinates (m), u is velocity component (m/s) and i is Cartesian co-
ordinate index.
Conservation of momentum or Newton's second law (eq. (2))
states that there is a balance between the rate change of linear mo-
mentum and sum of the external forces acting on the fluid element:
⎜ ⎟
∂
∂
+ ∂
∂
= ∂
∂
⎡
⎣
⎢− +
⎛
⎝
∂
∂
+
∂
∂
⎞
⎠
⎤
⎦
⎥ +t
ρu
x
ρu u
x
pδ μ u
x
u
x
ρg( ) ( )i
j
i j
j
ij
i
j
j
i
i
(2)
where j is the Cartesian coordinate index, δ is Kronecker delta, μ is the
dynamic viscosity (kg/ms) and g is acceleration due to gravity (m/s2).
Conservation of energy as the first law of thermodynamics (eq.
(3)) states that there is an equality between the energy changing rate of
a fluid element and added heat or work done on it:
⎜ ⎟
∂
∂
+ ∂
∂
− ∂
∂
⎛
⎝
∂
∂
⎞
⎠
=
t
ρC T
x
ρu C T
x
λ T
x
s( ) ( )a
j
j a
j j
T
(3)
where Ca is specific heat capacity (W/kgK), T is the temperature (K), λ
is thermal conductivity (W/mK) and sT is thermal sink or source (W/
m3) (Norton, Sun, Grant, Fallon, & Dodd, 2007).
Fourier equation (eq. (4)) can be used to determine heat exchange
in an isotropic solid as follows (Norton et al., 2013):
⎜ ⎟
∂
∂
= ∂
∂
⎛
⎝
∂
∂
⎞
⎠
+
t
ρC u
x
λ T
x
s( )a i
j j
T
(4)
Fick's law of mass diffusion (eq. (5)) is analogous to Fourier equa-
tion and is generally solved in drying processes to describe moisture
transfer:
⎜ ⎟
∂
∂
= ∂
∂
⎛
⎝
∂
∂
⎞
⎠
X
t x
D X
x
w
j
eff
w
j (5)
where Deff is the effective diffusion coefficient (m2/s) (Wang & Sun,
2003).
Although direct solving of Navier- Stokes equations is possible for
laminar flow, solving the fluid motion in the Kolmogorov microscales in
turbulent flow is not possible computationally yet, so, Turbulence
models should be solved besides Navier- Stokes equations in the case of
turbulent flow regime (which is generally encountered in food process
modeling because of high flow rates and complex geometry involved).
Selecting an appropriate turbulence model is very critical and has a
direct effect on CFD results. Choosing from available turbulent models
depends on the accuracy and computational time (or cost) (Defraeye,
2014). Generally, turbulence models are categorized based on the
governing equations used in them. There are two distinct groups of
turbulence models: (1) Reynolds averaged Navier- Stokes (RANS)
models, and (2) the models which works on computing fluctuations e.g.
large eddy simulations (LES).
In the case of turbulence, with a statistical probability, it is supposed
that process variables will be in a certain range of values within the
flow regime. So, RANS equations are used. The effect of turbulence on
flow field would be determined through time averaging using these
equations. In other words, these equations are time averaged Navier-
stokes equations of motion for fluid flow (Norton & Sun, 2010). This
averaging method eliminates the stochastic properties of the turbulent
flow regime and generates six additional stresses called Reynolds
stresses which could be modeled by different system of equations (for
more information, please see Norton & Sun, 2010). Some commonly
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
207
used RANS turbulence models are (Table 1):
- Standard −k ε (k represents turbulence kinetic energy and ε for
turbulence dissipation rate)
- Renormalization Group or RNG −k ε
- Realizable −k ε
- −k ω model (k represents turbulence kinetic energy and ω for
specific dissipation rate of kinetic energy)
- Reynolds Stress Model (RSM).
First three models have similar forms and Standard −k ε is the most
commonly used turbulence model due to a better conversion
Table 1
Comparison between Reynolds averaged Navier- Stokes (RANS) models (Farid, 2010; Kuriakose & Anandharamakrishnan, 2010; Norton & Sun, 2006, 2010; Norton
et al., 2013).
Model Description Advantages Disadvantages
Standard k-ε Basics of this model are transport equations accounting
for k , turbulence kinetic energy and ε , turbulence
dissipation rate
Good convergence
Robust
Economical
Reasonably accurate
Satisfactory results in processes
without adverse pressure gradient
Suitable for external flow around
complex geometries
Frequently used in drying processes
with high Reynolds number
Inadequate in some cases due to assumptions and
empiricism which model is based on (the most important
assumption made in this model is assuming an equilibrium
condition for turbulence)
Not highly accurate in complex flow regimes, geometry
and severe pressure gradients
RNG k-ε Similar to Standard k-ε but includes some additional
expressions for development of dissipation rates and some
constants different from Standard k-ε model constants.
Independency from empiricism and
taking anisotropy of complex flows
into account
Better results for recirculating
flows
Some limitations because of assuming eddy viscosity as
being isotropic
Require high power computers
Some convergence difficulties
Realizable k-ε Instead of using constant Empirical turbulence model
constant (like Standard k-ε model), this term is a function
of turbulence properties and mean flow. So, it satisfies
certain mathematical constraints
on the Reynolds stress tensor that are consistent with the
physics of turbulent flows. A new model for the rate of
dissipation is also used.
Same benefits as RNG but works
better in more complex flows such
as jet impingement
Some limitations because of assuming eddy viscosity as
being isotropic
Standard −k ω
model
Basics of this model are transport equations accounting
for k , turbulence kinetic energy and specific dissipation
rate ω (rate of dissipation per unit turbulent kinetic)
Works better for boundary layers
under adverse pressure gradients
Suitable for flows with convoluted
curvatures and internal flows
Frequently used in drying processes
Convergence is more difficult than k-ε
Reynolds Stress
Model (RSM)
This model includes transport equations for the Reynolds
stresses, a transport equation
for the turbulence energy
dissipation rate and three transport equations for the
turbulent fluxes of each scalar property.
Most complete model physically
Suitable in the case of adverse
pressure gradient involved
Suitable for complex flow behavior
Long computational time and powerful memory
Poor convergence was reported in many literature
Fig. 1. Main scheme of CFD analysis.
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
208
(Anandharamakrishnan, 2013). Comparison between RANS models is
presented in Table 1.
The second group of turbulence models, large eddy simulation
(LES), are based on the fact that highly anisotropic large turbulent
eddies are dependent on geometry of flow domain and also mean ve-
locity gradients. The velocity field in these models is separated to a
resolved part representing large eddies and a sub-grid part which re-
present small scales (Norton & Sun, 2006). For more information about
LES models see Norton and Sun (2006).
RANS models yield satisfactory results for simulation complex large
systems with less cost compared to some other methods. Using LES
models is an accurate approach but it requires powerful computers and
is time consuming and consequently expensive. Nowadays advanced
methods like hybrid RANS-LES is utilized which have a higher accuracy
to model drying processes (Kuriakose & Anandharamakrishnan, 2010).
It should be noted that none of the mentioned turbulence models are
complete and the prediction performed by these models is strongly
dependent to the geometry of the system and flow condition.
Solving mentioned PDEs was a difficult problem in the past. The
PDEs had to undergo simplification processes in order to be solved by
hand. Accurate determination of some local phenomena such as fouling,
cleaning, mixing, etc. was not possible using those simplified models.
Modern CFD approaches have overcome this problem (Norton & Sun,
2010). The main scheme of CFD analysis is presented in Fig. 1. Also,
overall procedure of a reliable CFD model has been depicted in Fig. 2.
First of all, it should be noted that simulation procedure may differ
depending on the situation and the information which the researcher
wants to obtain in a special problem. Based on this idea, there are two
distinct modeling approaches named conjugated and non-conjugated
models. If heat and mass transfer in liquid or solid state food surface
with variable or constant thermo-physical properties is studied, it is
called non-conjugated modeling; but, if solid or liquid food material is
modeled along with its surrounding media (gas or liquid), it is named
conjugated modeling (Lemus-Mondaca et al., 2011).
As shown in Fig. 2, a critical determination in CFD analysis for
multiphase systems is choosing between Eulerian- Eulerian (EE), Eu-
lerian- Lagrangian (EL) and volume of fluid (VOF) reference frame-
works to represent the governing equations. There are basic differences
between EE, EL and VOF formulations (Fig. 3). To sum up, EL for-
mulation is generally used for dense systems and individual particles
can be tracked in the flow domain. It is expensive and time consuming
and advanced high speed computers are demanded to solve this for-
mulation. EE is used for both dense and dilute systems and cannot
predict the local behavior of the particles. VOF is applicable to two or
more immiscible fluids by solving a single set of momentum equations
of each fluid and tracking its volume fraction (Anandharamakrishnan,
2013). More information related to EE and EL formulations is provided
by Jamaleddine and Ray (2010).
One of the other most important factors in CFD analysis is mesh
generation. Meshing technology has improved extensively in recent
years and utilization of tetrahedral, hexahedral hybrid and polyhedral
meshes has overcome the limitations of simple meshes (Norton et al.,
2013).
Details about discretization methods (FEM, FVM, and FDM) are not
presented in this article since there are a lot of valuable articles dis-
cussing these methods. For more information about discretization
methods see Mallinson and Norris (2010).
There are many commercial CFD codes which can be used in dif-
ferent processes to model food drying processes. Some of the most
important codes are ANSYS - Fluent, ANSYS - CFX, OpenFOAM,
COMSOL Multiphysics, STAR-CD, FIDAP, ADINA, CFD 2000, PHOEN-
ICS, FLOW 3D, etc. (Bhutta et al., 2012). More details on CFD software
packages and codes are presented by Xia and Sun (2002).
It should be noted that the accuracy of any CFD simulation depends
on many factors such as the assumptions and simplification of the
model which has been made, the empirical correlations used,
simplification of boundary conditions and the geometry in order to
reduce computation time and existence of other physical mechanisms
such as chemical kinetics.
Like any other modeling methods, CFD has some advantages and
disadvantages for drying simulations. Table 2 summarize some of these
aspects (Norton et al., 2013; Xia & Sun, 2002).
Jamaleddine and Ray (2010) reviewed the application of CFD in
some drying systems with a special focus on gas-solid multiphase flows
(Jamaleddine & Ray, 2010). Also, Pragati and Sharma (2012) reviewed
the application of CFD in food processing equipment design e.g.
cleaning of food processing tanks, optimization of hygiene, drying,
pasteurization, sterilization, mixing, baking, refrigeration, heat ex-
changers and crystallization. They explained further research is de-
manded regarding to control of the drying processes and reduction of
energy costs. These authors concluded that CFD can be utilized for
predicting gas flow patterns, particle histories (temperature, velocity,
residence time and impact position), designing drying chambers, scale
up studies and air-particle interactions successfully. They mentioned
lacking spatial homogeneity of air velocity is a major problem resulting
in the instability of CFD data for dryers (Pragati & Sharma, 2012).
In a recent study, Norton et al. (2013) reviewed CFD modeling of
some thermal food processes including sterilization, pasteurization of
canned foods (food in pouches, intact egg, plate heat exchangers for
milk and yoghurt processing), drying (fluidized bed drying, spray
drying and forced convection drying), cooking (natural and forced
convection, commercial baking and microwave and infrared ovens).
They also had a short look on application of CFD in modeling some
emerging technologies such as high pressure thermal processing and
Ohmic heating (Norton et al., 2013). Defraeye (2014) made a com-
prehensive review on computational methods for modeling dehydration
of porous materials specially food products including CFD. He reported
that the number of publications reflecting the application of CFD in
drying technology has increased significantly between 2002 and 2014
(Defraeye, 2014).
Although there are some valuable articles which have reviewed the
studies of CFD modeling, developing drying technologies and in-
creasing the number of publications in this area made the authors
convinced to have a fresh look at novel achievements in this field,
particularly the food drying processes. The main goal of this review
article is investigating the utilization of CFD in the recent publications
and also emerging drying technologies such as microwave assisted,
infrared, and super-steam drying.
3. Classification of food dryers
The drying technologies can be classified based on different aspects.
Some researchers have categorized dryers as direct and indirect, batch
and continuous, in gas or in vacuum, etc. One of the best classification
of dryers has been presented by Mujumdar (2008) who determined four
categories for drying facilities based on the strategy used for drying, the
medium in which the material is being dried, the way that solids are
handled and the heat input mode (Mujumdar, 2008). Fig. 4 demon-
strates briefly some of the dryers in this categorization. Nowadays more
than 85% of dryers in the industry
are convective dryers which utilize
hot air or combustion gases as the drying medium (Moses, Norton,
Alagusundaram, & Tiwari, 2014). The poor quality of the final dried
food products and low yields in these types of dryers is a critical issue.
In order to find more efficient and economical drying methods, scien-
tists have combined some of these different categories and developed
new dryer designs (Moses et al., 2014). For example, microwave as-
sisted convective drying, ultrasound assisted drying or combining flui-
dized bed dryers with vacuum are some of these novel techniques.
4. Application of CFD in convective drying systems
Convective dryers are the most frequent and popular equipment
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
209
used in industrial applications. Hot air or combustion gases might be
used as heat transfer medium in these type of dryers (Moses et al., 2014;
Tzempelikos et al., 2015). The efficiency of convective dryers is de-
pendent to several factors such as drying air speed, temperature, re-
lative humidity and flow uniformity (Chandramohan, 2016). There are
different types of convective dryers like tunnel, tray, fluidized bed,
spray and solar dryers. In the following sections, some of the recent
studies concerning CFD modeling and simulation of convective dryers
in food processing are discussed.
4.1. CFD in spray dryers
Spray drying is a method for converting liquid materials into
powder form. It has the advantages of high drying rates, broad range of
drying temperatures and short drying periods (Jafari et al., 2017). One
of the most important problems in spray dryers is unsteady nature of
flow which may cause wall deposition or overheating of the product.
The 3D nature of the spray dryers makes it impossible to use empirical
models for their describing. So, CFD is a useful tool for modeling this
type of dryers. Eulerian- Lagrangian approach has been used in CFD
modeling of spray dryers in most cases which can predict the motion
path of particles and heat and mass transfer between the particles and
drying air (Lo, 2005; Norton et al., 2013). Spray drying process is
comprised of four major parts including atomization, contact between
the particles and drying air, evaporation of the water and in the last
part, separation of the dried powder from the drying air (cyclone). CFD
Fig. 2. Fundamentals of CFD analysis.
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
210
has been used for modeling all of these parts and it can predict velocity
and temperature distribution, particle size and humidity as well as
several other factors in spray drying processes that effects the final
product quality and dryer design (Anandharamakrishnan, 2013). Al-
though there are a lot of advantages in CFD modeling of spray dryers,
there are also some limitations because of simultaneous existence of
both solids and fluids and difficulty in predicting mass transfer within
the particles and necessity of applying sub-models (e.g. sub-models
accounting for mass transfer, collision between particles, thermal re-
actions, stickiness and agglomeration) in order to take this issue into
account. Also, validation of CFD results in spray drying process is
complicated since local measurement of process parameters is to some
extent impossible and expensive even in a laboratory scale. Most of the
results in literature have been validated using average outlet data. On
the other hand, the results are dependent on empirical models that are
not completely reliable (Mujumdar, Huang, & Chen, 2010).
Kemp and Oakley (2002) stated that the main difficulties in CFD
modeling of spray drying process are:
- There is no similarity in drying hydrodynamics during process
scaling
- Circulation of air inside drying chamber may move particles on
different trajectories
- Heating story may change quality and morphology of the drying
products
- Determination of drying kinetics of material under drying process is
difficult.
There have been many reviews on application of CFD in spray
drying. For instance, Lo (2005) and Kuriakose and
Anandharamakrishnan (2010) have reviewed the application of CFD in
spray drying. Mujumdar et al. (2010) have also studied some advances
in spray drying focusing on CFD modeling. Jamaleddine and Ray
(2010) and Norton et al. (2013) dedicated a part of their studies to
review the application of CFD in spray drying. Most of the CFD studies
in the field of spray drying is concerned to co-current towers so,
Abdullah, Taip, Siti Mazlina, and Abdul Rahman (2017) have recently
reviewed fundamentals of counter current spray dryers. As there are
comprehensive reviews in the field of CFD modeling of spray dryers, we
refer the readers to these articles and only review some of the most
recent studies.
In most papers, CFD simulation has been performed to simulate
hydrodynamics of bed or maximize drying thermal efficiency or low-
ering the overall cost of the spray drying process. Although these factors
are important, the crucial factor in food drying is the product's quality.
Based on this assumption Schmitz-Schug, Kulozik, and Foerst (2016)
studied the impact of spray drying conditions on lysine loss. Reaction
engineering approach (REA) which has been used for modeling drying
kinetics was implemented in CFD analysis by STAR-CCM + Software.
The model representing kinetics of lysine loss took temperature,
moisture and physical state of lactose into account. The residence time
and properties of particles was solved using CFD and coupled with re-
action kinetic model to predict lysine loss. They suggested that this
approach could be used in optimization of spray drying for dairy
powders in order to prevent lysine loss (Schmitz-Schug et al., 2016).
Jaskulski, Atuonwu, Tran, Stapley, and Tsotsas (2017) performed a
similar study and tried to investigate thermal inactivation of whey
proteins during spray drying of skim milk using ANSYS- Fluent CFD
package. Particle moisture, temperature profile and residence time was
predicted by evaporation and particle formation extended models and
were utilized as input data in a quality model of inactivation kinetics.
Then, the resulted quality model was implemented into the CFD code
and simulation was conducted and validated. Also, Differential Scan-
ning Calorimetry (DSC) was used to measure whey protein inactivation
during drying experiments. A good agreement was shown between
experimental and predicted whey activity that represented the suc-
cessful development of the CFD code. So, this model can be suggested to
predict the whey protein loss during spray drying of skim milk
(Jaskulski et al., 2017).
Many CFD models have not been verified in the literature. So, data
Fig. 3. (a) Eulerian–Eulerian, (b) Eulerian–Lagrangian approach illustration.
Table 2
Advantages and disadvantages of CFD modeling in food drying processes.
Advantages Disadvantages
Providing detailed understanding of heat, mass and momentum
transfer in the drying system
Very small time steps are needed in some cases because of different time scale of fluid flow, heat, mass and
scalar transport resulting in long computation times
Declining scale up problems for drying systems Incapability of online controlling the thermal processes
Working as virtual sensors in drying systems to improve final product
quality
Lack of adequate data about physicochemical properties of food materials during drying
Simulating unusual conditions such as hot temperatures or
dangerous environment
Shrinkage during drying which makes mesh generation more complex
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
211
provided by such studies can hardly be used in a large scale. In a recent
work by Lisboa, Duarte, and Cavalcanti-Mata (2018), an industrial scale
spray dryer was modeled. They used simplified basic models to estimate
key parameters for spray drying design. The model was validated for a
wide range of food products
at different temperature and relative hu-
midity levels. The authors suggested that such models could be used
instead of complicated CFD models in many applications (Lisboa et al.,
2018).
The authors think that although velocity and flow pattern during
spray drying can be predicted easily in CFD packages, but it is necessary
to conduct studies concerning drying process control and energy and
optimization in this era. Also, some simplifying assumptions like con-
sidering spatial homogeneity which leads to unrealistic results should
be avoided. Finally, since designing a spray dryer which produces high
quality products is a very sophisticated process, it is thought that be-
sides aspects related to drying cost and efficiency, correlating quality
kinetics of degradation of food products to the transport phenomena
occurring during the process may be the most holistic approach in spray
drying of valuable food and pharmaceutical products which is still
lacking and could be focused with more details.
4.2. CFD in fluidized bed dryers (FBD)
Fluidization is a deep-rooted technique which has been applied
successfully in many industrial applications (Lettieri & Mazzei, 2009).
FBDs can be used for the drying of wet particulate materials that behave
like a fluid in the drying chamber. High heat and mass transfer coef-
ficients between drying materials and drying air medium due to their
high contact and gentle mixing leads to elevated drying rates compared
to some other drying methods. However, there are some shortcomings
in this process e.g. scale-up issues, uneven fluidization and product
quality. Solving these problems depends on gaining a comprehensive
knowledge about the process and reliable mathematical correlations
which can be used to predict, optimize and scale-up the process
(Mortier et al., 2011). CFD has been used extensively to explore the
interactions between multiple phases and prediction of different phe-
nomena in this drying system (Lettieri & Mazzei, 2009; Malekjani,
2017). In regard of existing different phases and chemical reactions
which may be conducted in such a process, the CFD modeling is a
complex procedure. One of the most important difficulties in CFD
modeling of FBDs is modeling the turbulence behavior (because of 3D
and transient nature). Numerical modeling of fluidized beds is per-
formed by coupled solving of mass, momentum and energy equations
with the equations describing interphase interactions and one of the
Fig. 4. Different aspects for classification of food dryers.
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
212
main difficulties in CFD simulation of fluidized bed dryers is modeling
multiphase interactions (Philippsen, Vilela, & Zen, 2015). As it was
mentioned in the previous section, Eulerian- Eulerian and Eulerian-
Lagrangian approaches are used to simulate multiphase flows. Eulerian-
Eulerian approach is widely used for CFD modeling of the fluidized
beds. One of the other important limitations in CFD modeling of flui-
dized bed dryers is very long computation times required for modeling
just a fraction of drying process (Szafran & Kmiec, 2005), although with
the continuous progress in computers and invention of powerful su-
percomputers, these calculations can now be performed in a very short
time.
Verification of the CFD models in FBDs has been conducted using X-
ray imaging, optical fiber and pressure probes, laser and 3D capacitance
imaging methods (Mortier et al., 2011). Some of the applications of CFD
in FBD include designing different parts of the dryers especially gas
distribution section and draft tube and presenting online models in
order to control the drying process (Norton et al., 2013).
Mortier et al. (2011) have reviewed modeling of FBD for wet
granular materials. Lettieri and Mazzei (2009) discussed some of the
challenges on the CFD modeling of FBDs. Pan, Chen, Liang, Zhu and
Luo (2016) reviewed the fundamentals and applications of three phase
fluidized bed reactors. Jamaleddine and Ray (2010) and Norton et al.
(2013) dedicated a part of their works to review the studies concerning
the application of CFD in modeling spouted and fluidized bed drying
processes. In a recent study by Philippsen et al. (2015), they presented
different methods of FBD modeling including CFD.
Spouted bed dryers are used instead of FBDs for drying larger,
irregularly shaped, sticky and heavy particles. In this method, there is a
high speed injection of a gas flow which moves the solids to the center
of the drying chamber until reaching the top of it
(Anandharamakrishnan, 2013).
Nazghelichi, Jafari, Kianmehr, and Aghbashlo (2013) studied the
hydrodynamics and heat transfer in a lab scale FBD for drying carrot
cubes using Fluent. Three levels of bed height, cube size and inlet air
temperature were investigated in order to evaluate the effects of each
factor on energy optimization ratio. Also, Taguchi technique was used
to rank the mentioned factors. The results revealed that the cube size
had the most significant effect among the others. The authors declared
that their method could be applied in energy utilization optimization in
FBD drying process. Their findings showed that as the particle size is
smaller, the bed is deeper and higher and the drying air temperature is
higher, the energy utilization is more (Nazghelichi et al., 2013).
Azmir, Hou, and Yu (2018) coupled CFD and discrete element
method DEM method to describe heat and mass transfer in a fluidized
bed dryer containing spherical corn kernels. DEM method is used to
investigate the granular flow. Each grain is tracked using Newton's laws
of motion while all forces due to gravity, particle-particle, particle-wall
and electrostatic fields are considered. This method helps in gaining
more detailed information in particle scale. In this approach, the on-line
data about particle velocity, temperature, position and moisture con-
tent produced by DEM is used for solving CFD governing equations.
Then the results of CFD analysis would be incorporated into the DEM.
These authors used 2D and 3D geometry for CFD and DEM analysis,
respectively. An in-house developed code was used for CFD-DEM ana-
lysis. They examined the effects of drying air temperature and velocity
on grain and air moisture content. The results showed that as the drying
air velocity and temperature rise, the drying rate increases. They de-
scribed the corn kernel quality by standard deviation of its moisture
distribution and reported that in higher air velocities and lower drying
air temperatures, the quality of drying material is superior (Azmir et al.,
2018).
In Table 3, a brief overview of CFD application in the field of flui-
dized and spouted bed drying of food and agricultural products has
been provided.
Ta
bl
e
3
So
m
e
of
th
e
C
FD
st
ud
ie
s
in
th
e
fi
el
d
of
fl
ui
di
ze
d
an
d
sp
ou
te
d
be
d
dr
yi
ng
of
fo
od
an
d
ag
ri
cu
lt
ur
al
pr
od
uc
ts
.
C
as
e
de
sc
ri
pt
io
n
M
od
el
in
g
m
et
ho
ds
So
ft
w
ar
e
Fi
nd
in
gs
A
ut
ho
rs
Sp
ou
te
d
be
d
dr
yi
ng
of
gr
ai
ns
at
di
ff
er
en
tg
as
in
le
tv
el
oc
it
ie
s,
gr
ai
n
de
ns
it
ie
s
an
d
di
am
et
er
s
Eu
le
ri
an
m
ul
ti
ph
as
e
m
od
el
w
as
us
ed
to
de
sc
ri
be
th
e
hy
dr
od
yn
am
ic
s
of
th
e
be
d
an
d
gr
ai
n
he
ig
ht
in
th
e
be
d
w
it
h
di
ff
er
en
t
m
od
el
pa
ra
m
et
er
s
(t
ur
bu
le
nt
m
od
el
s,
in
it
ia
l
an
d
m
ax
im
um
co
effi
ci
en
t
of
pa
ck
in
g)
Fl
ue
nt
-
Im
po
rt
an
ce
of
in
le
t
ai
r
ve
lo
ci
ty
-
Im
po
rt
an
ce
of
se
le
ct
in
g
su
it
ab
le
tu
rb
ul
en
ce
m
od
el
s
-
G
oo
d
ag
re
em
en
t
be
tw
ee
n
ex
pe
ri
m
en
ta
l
an
d
si
m
ul
at
io
n
re
su
lt
s
(S
ob
ie
sk
i,
20
08
)
Sp
ou
te
d
be
d
dr
yi
ng
V
ar
io
us
im
po
rt
an
t
fe
at
ur
es
of
si
m
ul
at
io
n
su
ch
as
se
le
ct
in
g
be
tw
ee
n
2D
an
d
3D
sp
ac
e,
ge
om
et
ry
,s
tr
uc
tu
re
d
or
un
st
ru
ct
ur
ed
gr
id
,c
on
fi
gu
ra
ti
on
of
m
ul
ti
ph
as
e
m
od
el
,t
he
tu
rb
ul
en
ce
m
od
el
an
d
th
e
se
ns
it
iv
it
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of
th
e
m
od
el
to
ph
as
e
an
d
fl
ow
pa
ra
m
et
er
ch
an
ge
s.
Fl
ue
nt
-
Ev
er
y
fa
ct
or
ha
s
it
s
ow
n
eff
ec
t
on
si
m
ul
at
io
n
re
su
lt
s
so
,
th
e
fu
nc
ti
on
an
d
pr
ac
ti
ca
l
im
pl
ic
at
io
n
sh
ou
ld
be
ex
am
in
ed
ca
re
fu
lly
to
im
pr
ov
e
co
m
pu
ta
ti
on
al
re
su
lt
s
(S
ob
ie
sk
i,
20
10
)
Fl
ui
di
ze
d
be
d
dr
yi
ng
of
so
yb
ea
n
m
ea
l
at
di
ff
er
en
t
te
m
pe
ra
tu
re
s,
ve
lo
ci
ti
es
an
d
be
d
he
ig
ht
s
Eu
le
ri
an
-E
ul
er
ia
n
ap
pr
oa
ch
w
it
h
in
co
rp
or
at
io
n
of
m
as
s
tr
an
sf
er
m
od
el
Sy
m
al
al
O
'B
ri
en
dr
ag
m
od
el
w
as
se
le
ct
ed
as
th
e
be
st
m
od
el
M
FI
X
-
D
ry
in
g
co
nd
uc
te
d
in
fa
lli
ng
ra
te
pe
ri
od
an
d
di
ff
us
io
n
is
th
e
m
os
t
im
po
rt
an
t
co
nt
ro
lli
ng
pa
ra
m
et
er
.
-
Te
m
pe
ra
tu
re
ha
d
a
si
gn
ifi
ca
nt
eff
ec
t
on
dr
yi
ng
bu
t,
th
e
eff
ec
ts
of
ve
lo
ci
ty
an
d
be
d
he
ig
ht
w
er
e
no
t
si
gn
ifi
ca
nt
(d
a
Si
lv
a,
de
So
uz
a,
da
C
os
ta
,
de
M
at
os
Jo
rg
e,
&
Pa
ra
ís
o,
20
12
)
C
ir
cu
la
ti
ng
fl
ui
di
ze
d
be
d
dr
yi
ng
of
so
m
e
se
ed
s
w
er
e
st
ud
ie
d
fo
r
ev
al
ua
ti
on
of
th
e
in
fl
ue
nc
e
of
ax
ia
l
pr
es
su
re
va
ri
at
io
ns
w
it
h
ga
s
fl
ow
ra
te
an
d
so
lid
ci
rc
ul
at
io
n
ra
te
.
A
n
Eu
le
ri
an
m
od
el
w
it
h
ki
ne
ti
c
th
eo
ry
of
gr
an
ul
ar
fl
ow
A
ns
ys
-F
lu
en
t
-
Pr
es
su
re
dr
op
de
cl
in
ed
al
on
g
th
e
hi
gh
er
le
ng
th
-
D
ev
ia
ti
on
be
tw
ee
n
th
e
ex
pe
ri
m
en
ta
l
an
d
pr
ed
ic
te
d
da
ta
w
er
e
sa
ti
sf
ac
to
ry
an
d
in
th
e
ra
ng
e
of
7–
9.
5%
.
(P
ri
ya
,P
ra
de
ep
,&
Sa
ra
va
na
n,
20
17
)
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
213
4.3. CFD in solar dryers
Sun drying is one of the oldest and cheapest methods of drying food
and agricultural products but, it is labor-intensive, long and has the risk
of pollution of food with insects, molds, etc. So, solar dryers have been
developed because they are easier to control and protection of the
product from pollution is possible (Sanghi, Ambrose, & Maier, 2017).
Solar dryers are categorized into four major types: direct, indirect,
mixed, and hybrid. CFD can be used to predict velocity and temperature
distribution profiles in these dryers (Chauhan, Kumar, & Tekasakul,
2015). Chauhan et al. (2015) have reviewed the application of com-
puter modeling in solar dryers. In two other recent studies, Milczarek
and Alleyne (2017) and Prakash and Kumar (2017) reviewed mathe-
matical modeling of solar dryers.
In a study by Romero, Cerezo, Garcia, and Sanchez (2014), they
investigated an indirect solar dryer for vanilla drying. Variation of
temperature inside the dryer was predicted using ANSYS- Fluent CFD
package. Their results showed a good agreement between experimental
and predicted data at collector outlet while inside the chamber, some
deviations were found. The authors ascribed these variations to ap-
proximation of heat transfer coefficient and suggested using a variable
heat transfer coefficient as a function of time during day for solar dryers
(Romero et al., 2014).
In another study, an indirect solar cabinet dryer was simulated using
ANSYS- Fluent for drying sliced tomatoes. The ambient local weather
was used as the boundary condition. Temperature and velocity dis-
tribution were predicted. Thermal efficiency of the dryer without any
load, under half load and full load was measured. The maximum and
minimum temperatures in the dryer were predicted to be 70 and 27 °C,
respectively. Thermal efficiency of the dryer was 21.5% (Tesfamariam,
Bayray, Tesfay, & Hagos, 2015).
Solar cabinet drying process of corn has also been studied using
CFD. Temperature, humidity and velocity profiles of the drying air was
simulated and validated with experimental data. The major objective of
this study was developing a predictive model based on local changes of
the weather in order to enhance drying efficiency. A 3D model was
developed by solving continuity, Navier- Stokes and energy equations
using ANSYS- Fluent package. Moisture transfer equation was solved
independently. Thin layer drying models were also used to model the
drying process of corn kernels. The porosity of the flow fields was
considered in all of the governing equations. Radiative transfer equa-
tion (RTE) was added as a source term to energy equation to represent
the radiative effects. The effect of turbulence was also evaluated using
RNG −k ε model. The local weather data was used as boundary con-
dition. The results showed that the model over-predicted both tem-
perature and humidity to some extent. A stagnation which was seen in
experimental humidity data was predicted by model accurately. This
model was able to simulate the drying process under overcast condi-
tions and amount of removing the moisture was 32% less than fair
weather conditions (Sanghi et al., 2017).
Recently, Orbegoso, Saavedra, Marcelo, and La Madrid (2017)
performed numerical simulation of one-step and three-step solar col-
lectors for solar drying of cocoa bean. The aim of their study was si-
mulating heat transfer due to convection and radiation in order to de-
termine the best configuration of the collectors regarding outlet drying
air temperature and drying thermal efficiency. Mass, momentum and
energy equations were solved coupled with discrete ordinate model
(DOM) for discretizing RTE and standard −k ε turbulence model in an
Eulerian framework using ANSYS- Fluent package. The results showed
that the three-step collector located between the second and third
channel in dryer was the most thermally efficient configuration (67%
more efficient than one-step). These authors showed the importance of
CFD modeling in design and optimization of energy in such type of
dryers (Orbegoso et al., 2017).
Some limitations in CFD modeling of solar dryers include transient
nature of mixed and natural convection of airflow within the dryer,
variations and complexity of the boundary conditions which is the local
weather, shrinkage in drying material which necessities applying
moving meshes, existing solar radiation besides long wave radiation,
more sophisticated drying kinetics than forced convection systems be-
cause of partial rehydration (Prakash & Kumar, 2017).
5. CFD in freeze drying
Lyophilization or freeze drying is an excellent method for retaining
original quality attributes of the dried products. This method preserves
the biological activity of food components, flavors, colors, and aroma
but it is a complicated, costly and time-consuming process. In this
method, the water inside the food material is first frozen; then the
created ice is sublimated (Nakagawa & Ochiai, 2015). This process can
be implemented at atmospheric pressure or under vacuum (Li,
Stawczyk, & Zbicinski, 2007).
There have been some studies on the modeling of freeze drying
process for food products. Some researchers have focused on mathe-
matical modeling of this process (Bubnovich, Quijada, & Reyes, 2009;
Bubnovich, Reyes, Quijada, & Mahn, 2012; Nakagawa & Ochiai, 2015;
Nam & Song, 2007), and some others have emphasized on modeling the
freeze dryer parts such as drying shelf (Cheng & Tsai, 2012), condensers
(Petitti, Barresi, & Marchisio, 2013) or heat exchangers (Cheng, Tsai,
Cheng, & Chen, 2014). There are a few studies regarding CFD simula-
tion of freeze drying of food products (Coletto, 2015; Li et al., 2007).
Li et al. (2007) developed a two-dimensional model based on film
sublimation and uniformly reaching ice front (URIF) model using
FLUENT 6.1 software. The model took into account phase changes and
water vapor diffusion inside the porous media. Species Transport model
in FLUENT was used to simulate the transport of non-condensable
species and to predict the rate of sublimation. The drying samples
(apple cubes) were divided into four zones including the frozen front
(pure ice plate), an ice-vapor interface,
the dry zone (porous zone) and
a gas phase. The frozen front zone was treated as a wall where ice
sublimation starts in and a UDF was used to determine the sublimation
rate. A source term was added to the adjacent walls in order to consider
the effect of water vapor generation on the gas phase species distribu-
tion. Flow through the dry zone (porous zone) was simulated using
inbuilt CFD model for porous media. The results were validated with
the experimental data of apple cubes and showed a reasonable agree-
ment. Results showed the internal resistance of porous medium is
dominant. Absorption and desorption of the samples enhanced moisture
transport (Li et al., 2007).
In a more recent study, Coletto (2015) aimed to simulate the entire
process of freeze drying instead of a piece of product. He focused on
atmospheric freeze drying (AFD) using wheat bran as an adsorbent in a
fluidized bed dryer to enhance the AFD process efficiency. ANSYS-
Fluent was used to simulate the process. Eulerian- Eulerian aproach for
fluidized bed dryer and hydrodynamics of the dryer was successfully
simulated (Coletto, 2015).
6. CFD in novel drying technologies
In order to achieve better results of energy efficiency and product
quality, new dryer designs are emerging from the results of scientific
research. Modern technologies such as microwave and ultrasound are
used to enhance existing drying systems for food products. Some other
technologies have been developed recently, e.g. superheated steam
drying. Most of the researches in the field of drying are conducted on
novel drying technologies so that the widespread and commercial use of
these systems are very limited due to lack of sufficient understanding of
the process costs and commercial design and large-scale applications
(Moses et al., 2014). Therefore, CFD models can provide new insights in
developing and designing these systems in the industrial scale. In the
following subsections, some of the less common drying systems along
with application of CFD in modeling and simulation of these processes
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
214
Ta
bl
e
4
R
ec
en
t
C
FD
st
ud
ie
s
in
th
e
fi
el
d
of
no
ve
l
dr
yi
ng
te
ch
no
lo
gi
es
fo
r
th
e
fo
od
an
d
ag
ri
cu
lt
ur
al
pr
od
uc
ts
.
C
as
e
de
sc
ri
pt
io
n
M
od
el
in
g
m
et
ho
ds
Ty
pe
of
m
od
el
G
eo
m
et
ry
an
d
di
m
en
si
on
s
D
is
cr
et
iz
at
i-
on
m
et
ho
d
Ph
ys
ic
al
pr
op
er
ti
es
M
ic
ro
w
av
e
as
si
st
ed
dr
yi
ng
C
FD
si
m
ul
at
io
n
of
sp
he
ri
ca
l
m
oi
st
pa
rt
ic
le
s
(s
oy
be
an
s)
in
a
m
ic
ro
w
av
e
as
si
st
ed
fl
ui
di
ze
d
be
d
dr
ye
r
in
or
de
r
to
st
ud
y
th
e
be
ha
vi
or
of
ga
s-
so
lid
fl
ow
•T
w
o-
fl
ui
d
Eu
le
ri
an
-E
ul
er
ia
n
m
od
el
an
d
ki
ne
ti
c
th
eo
ry
of
gr
an
ul
ar
fl
ow
w
as
us
ed
•U
D
F
fo
r
ca
lc
ul
at
io
n
of
m
ic
ro
w
av
e
po
w
er
•C
al
or
im
et
ri
c
m
et
ho
d
fo
r
in
it
ia
l
m
ic
ro
w
av
e
po
w
er
de
ns
it
y
de
te
rm
in
at
io
n
an
d
a
m
od
ifi
ed
eq
ua
ti
on
w
er
e
us
ed
•s
ta
nd
ar
d
−
k
ε
m
od
el
w
it
h
so
m
e
m
od
ifi
ca
ti
on
s
w
as
us
ed
to
m
od
el
tu
rb
ul
en
ce
•C
on
ju
ga
-
te
d
•2
D
ax
is
-s
ym
m
et
ry
ge
om
et
ry
(h
al
f
of
a
cy
lin
dr
ic
al
co
lu
m
n
re
pr
es
en
ti
ng
th
e
dr
yi
ng
ch
am
be
r
w
as
se
le
ct
ed
as
co
m
pu
ta
ti
on
al
do
m
ai
n)
•F
V
M
•D
ie
le
ct
ri
c
co
ns
ta
nt
an
d
di
el
ec
tr
ic
lo
ss
fa
ct
or
as
a
fu
nc
ti
on
of
te
m
pe
ra
tu
re
an
d
m
oi
st
ur
e
co
nt
en
t;
•t
he
rm
al
co
nd
uc
ti
vi
ty
of
dr
yi
ng
m
at
er
ia
l
as
a
fu
nc
ti
on
of
m
oi
st
ur
e
co
nt
en
t;
•t
he
rm
od
yn
am
ic
pr
op
er
ti
es
of
ai
r
as
a
fu
nc
ti
on
of
te
m
pe
ra
tu
re
M
ic
ro
w
av
e
dr
yi
ng
of
sp
he
ri
ca
lly
sh
ap
ed
fo
od
pr
od
uc
ts
•M
ax
w
el
l's
el
ec
tr
om
ag
ne
ti
c
fi
el
d
eq
ua
ti
on
s
an
d
po
ro
us
m
ed
ia
m
od
el
s
w
er
e
so
lv
ed
•A
so
ur
ce
te
rm
w
as
ad
de
d
to
th
e
en
er
gy
eq
ua
ti
on
in
or
de
r
to
re
pr
es
en
t
th
e
eff
ec
t
of
th
e
m
ic
ro
w
av
e
fr
om
th
e
el
ec
tr
om
ag
ne
ti
c
m
od
el
.
•D
ar
cy
's
la
w
w
as
us
ed
as
su
m
in
g
fl
ow
in
a
po
ro
us
m
ed
ia
co
nt
ai
ni
ng
lo
w
er
pe
rm
ea
bi
lit
y
in
st
ea
d
of
so
lv
in
g
st
an
da
rd
N
av
ie
r-
St
oc
ks
eq
ua
ti
on
s
•C
on
ju
ga
-
te
d
•3
D
ge
om
et
ry
(m
ic
ro
w
av
e
ca
vi
ty
an
d
sp
he
ri
ca
l
sa
m
pl
e
w
it
h
di
am
et
er
s
of
0.
06
×
0.
03
×
0.
01
2
m
)
•F
EM
•D
ie
le
ct
ri
c
co
ns
ta
nt
,
di
el
ec
tr
ic
lo
ss
fa
ct
or
,
sp
ec
ifi
c
he
at
an
d
th
er
m
al
co
nd
uc
ti
vi
ty
as
a
fu
nc
ti
on
of
te
m
pe
ra
tu
re
In
te
rm
it
te
nt
m
ic
ro
w
av
e
co
nv
ec
ti
ve
dr
yi
ng
co
ns
id
er
in
g
sh
ri
nk
ag
e
an
d
po
ro
si
ty
w
as
co
m
pa
re
d
w
it
h
th
e
m
od
el
w
it
ho
ut
sh
ri
nk
ag
e
fo
r
ap
pl
e
sl
ic
e
dr
yi
ng
•L
am
be
rt
's
la
w
w
as
us
ed
to
ca
lc
ul
at
e
m
ic
ro
w
av
e
en
er
gy
ge
ne
ra
ti
on
w
it
hi
n
th
e
dr
yi
ng
pr
od
uc
ts
.
•N
on
-
co
nj
u-
ga
te
d
•2
D
ax
is
-s
ym
m
et
ry
ge
om
et
ry
fo
r
a
cy
lin
dr
ic
al
sa
m
pl
e
w
it
h
di
am
et
er
of
40
m
m
an
d
th
ic
kn
es
s
of
10
m
m
•F
EM
•T
he
di
al
ec
ti
c
pr
op
er
ti
es
w
er
e
co
ns
id
er
ed
as
a
fu
nc
ti
on
of
m
oi
st
ur
e
co
nt
en
t.
•T
he
rm
o-
ph
ys
ic
al
pr
op
er
ti
es
as
a
fu
nc
ti
on
of
liq
ui
d
an
d
ga
s
vo
lu
m
e
fr
ac
ti
on
in
th
e
sa
m
pl
e
In
fr
ar
ed
dr
yi
ng
Th
e
in
fl
ue
nc
e
of
pe
ne
tr
at
io
n
de
pt
h
of
in
fr
ar
ed
ra
di
at
io
n
on
te
m
pe
ra
tu
re
an
d
m
oi
st
ur
e
co
nt
en
to
fr
ic
e
ke
rn
el
s
du
ri
ng
dr
yi
ng
•F
ic
k'
s
se
co
nd
la
w
of
di
ff
us
io
n
an
d
he
at
tr
an
sf
er
eq
ua
ti
on
w
as
so
lv
ed
.
•T
he
ra
di
at
io
n
so
ur
ce
te
rm
in
he
at
tr
an
sf
er
eq
ua
ti
on
w
as
as
su
m
ed
to
ex
is
t
in
th
e
m
od
el
w
it
h
th
e
pe
ne
tr
at
io
n
de
pt
h
of
in
fi
ni
ty
an
d
w
as
ne
gl
ec
te
d
in
th
e
m
od
el
w
it
h
ze
ro
pe
ne
tr
at
io
n
de
pt
h
an
d
th
e
ra
di
at
io
n
w
as
tr
ea
te
d
as
a
bo
un
da
ry
co
nd
it
io
n.
•N
on
-
co
nj
u-
ga
te
d
•O
ne
-e
ig
ht
h
of
th
e
ke
rn
el
ge
om
et
ry
(e
lli
ps
oi
d)
w
as
se
le
ct
ed
•F
EM
•T
he
rm
al
co
nd
uc
ti
vi
ty
,
de
ns
it
y
an
d
sp
ec
ifi
c
he
at
as
a
fu
nc
ti
on
of
m
oi
st
ur
e
co
nt
en
t
A
no
ve
l
ty
pe
of
in
fr
ar
ed
dr
ye
r
co
m
bi
ne
d
w
it
h
a
so
la
r
co
lle
ct
or
an
d
a
he
at
re
co
ve
ry
un
it
in
or
de
r
to
de
cr
ea
se
th
e
hi
gh
am
ou
nt
of
en
er
gy
re
qu
ir
ed
fo
r
in
fr
ar
ed
dr
yi
ng
of
m
el
on
sl
ic
e
•D
ry
in
g
si
m
ul
at
io
n
w
as
co
nd
uc
te
d
at
su
rf
ac
e
te
m
pe
ra
tu
re
s
of
50
an
d
60
°C
an
d
th
e
ai
r
ve
lo
ci
ty
of
0.
5
m
/s
.
•S
te
ad
y
st
at
e
co
nd
it
io
n
•D
ry
in
g
ai
r
w
as
co
ns
id
er
ed
as
an
id
ea
l
ga
s
m
ix
tu
re
.
•E
ne
rg
y
eq
ua
ti
on
,
−
k
ε
st
an
da
rd
w
al
l
fu
nc
ti
on
s
tu
rb
ul
en
ce
m
od
el
an
d
M
ix
tu
re
m
ul
ti
ph
as
e
m
od
el
w
as
se
le
ct
ed
an
d
bo
th
ai
r
th
e
so
la
r
co
lle
ct
or
an
d
dr
yi
ng
ch
am
be
r
w
as
si
m
ul
at
ed
•F
lu
id
s
in
th
e
sy
st
em
w
er
e
th
e
re
m
ov
ed
va
po
r
fr
om
th
e
m
el
on
sl
ic
es
an
d
dr
yi
ng
ai
r.
A
ir
m
as
s
fl
ow
in
dr
ye
r
in
le
t,
th
e
ra
te
of
he
at
ra
di
at
io
n
an
d
co
nv
ec
ti
on
he
at
tr
an
sf
er
w
er
e
in
tr
od
uc
ed
as
bo
un
da
ry
co
nd
it
io
ns
.
•V
el
oc
it
y
co
nt
ou
rs
an
d
te
m
pe
ra
tu
re
di
st
ri
bu
ti
on
pr
ofi
le
s
w
er
e
de
te
rm
in
ed
us
in
g
C
FD
re
su
lt
s
•C
on
ju
ga
-
te
d
•3
D
ge
om
et
ry
of
so
la
r
co
lle
ct
or
(8
0×
50
×
11
cm
)
an
d
dr
yi
ng
ch
am
be
r
(1
14
×
64
×
37
cm
)
•F
V
M
•E
ff
ec
ti
ve
m
oi
st
ur
e
di
ff
us
iv
it
y
an
d
m
as
s
tr
an
sf
er
co
effi
ci
en
t
of
fo
od
sa
m
pl
e
w
er
e
ca
lc
ul
at
ed
us
in
g
C
FD
(c
on
tin
ue
d
on
ne
xt
pa
ge
)
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
215
Ta
bl
e
4
(c
on
tin
ue
d)
C
as
e
de
sc
ri
pt
io
n
M
od
el
in
g
m
et
ho
ds
Ty
pe
of
m
od
el
G
eo
m
et
ry
an
d
di
m
en
si
on
s
D
is
cr
et
iz
at
i-
on
m
et
ho
d
Ph
ys
ic
al
pr
op
er
ti
es
V
el
oc
it
y
an
d
te
m
pe
ra
tu
re
di
st
ri
bu
ti
on
an
d
w
at
er
vo
lu
m
e
fr
ac
ti
on
in
th
e
ch
am
be
r
an
d
ne
ar
ap
ri
co
ts
w
er
e
de
te
rm
in
ed
in
an
in
fr
ar
ed
dr
ye
r
fo
r
ap
ri
co
ts
•V
al
id
at
io
n
ex
pe
ri
m
en
ts
w
er
e
co
nd
uc
te
d
at
ai
r
ve
lo
ci
ty
le
ve
ls
of
0.
5
an
d
0.
25
m
/s
an
d
fi
xe
d
ap
ri
co
t
su
rf
ac
e
te
m
pe
ra
tu
re
s
of
60
an
d
65
°C
.
•C
al
cu
la
ti
ng
R
ey
no
ld
's
N
um
be
r
re
ve
al
ed
la
m
in
ar
fl
ow
in
si
de
th
e
ch
am
be
r
so
th
e
N
us
se
lt
N
um
be
r
an
d
he
at
tr
an
sf
er
co
effi
ci
en
t
w
er
e
ea
si
ly
ca
lc
ul
at
ed
us
in
g
re
le
va
nt
re
la
ti
on
sh
ip
s.
D
iff
us
iv
it
y
of
va
po
r
in
to
ai
r
w
as
al
so
ca
lc
ul
at
ed
us
in
g
th
e
an
al
og
y
be
tw
ee
n
he
at
an
d
m
as
s
tr
an
sf
er
.
•
−
k
ε
tu
rb
ul
en
ce
m
od
el
w
as
us
ed
•C
on
ju
ga
-
te
d
•3
D
ge
om
et
ry
•S
ix
in
fr
ar
ed
la
m
ps
an
d
a
tr
ay
(4
0×
80
cm
)
co
nt
ai
ni
ng
ha
lf
ap
ri
co
ts
w
er
e
m
od
el
ed
.
•D
ry
in
g
ch
am
be
r
(1
14
×
64
×
64
cm
)
•F
V
M
•H
ea
t
an
d
m
as
s
tr
an
sf
er
co
effi
ci
en
ts
w
er
e
ca
lc
ul
at
ed
ba
se
d
on
th
e
re
su
lt
s
Su
pe
rh
ea
te
d
st
ea
m
dr
yi
ng
Su
pe
rh
ea
te
d
st
ea
m
fl
ui
di
ze
d
be
d
dr
yi
ng
pr
oc
es
s
fo
r
ra
pe
se
ed
s
•T
w
o-
ph
as
e
fl
ow
eq
ua
ti
on
s
fo
r
an
ax
is
-s
ym
m
et
ri
ca
l
cy
lin
dr
ic
al
dr
yi
ng
ch
am
be
r
w
as
so
lv
ed
•H
ea
ta
nd
m
as
s
tr
an
sf
er
an
d
dr
ag
m
od
el
s
w
er
e
in
co
rp
or
at
ed
in
to
th
e
so
ft
w
ar
e
us
in
g
a
U
D
F.
•T
he
in
le
t
ve
lo
ci
ty
an
d
te
m
pe
ra
tu
re
of
su
pe
rh
ea
te
d
st
ea
m
w
er
e
fi
xe
d.
•C
on
ju
ga
-
te
d
•A
xi
sy
m
m
et
ri
c
cy
lin
dr
ic
al
dr
yi
ng
ch
am
be
r
(1
20
×
25
0
m
m
)
•F
V
M
–
Su
pe
rh
ea
te
d
st
ea
m
dr
yi
ng
to
st
ud
y
te
m
po
ra
l
an
d
sp
at
ia
l
te
m
pe
ra
tu
re
an
d
m
oi
st
ur
e
ch
an
ge
s
of
a
si
ng
le
pe
lle
t
•R
ey
no
ld
s-
A
ve
ra
ge
d
N
av
ie
r-
St
oc
ks
(R
A
N
S)
eq
ua
ti
on
s
w
er
e
us
ed
m
od
el
•
−
k
ω
tu
rb
ul
en
ce
m
od
el
w
as
us
ed
•C
on
ju
ga
-
te
d
•3
D
ge
om
et
ry
(a
qu
ar
te
r
of
a
pe
lle
t
(1
2.
7×
35
.5
m
m
)
an
d
dr
yi
ng
ch
am
be
r
(2
72
×
35
6
m
m
))
•F
V
M
•T
he
rm
o-
ph
ys
ic
al
pr
op
er
ti
es
an
d
m
oi
st
ur
e
di
ff
us
iv
it
y
as
a
fu
nc
ti
on
of
te
m
pe
ra
tu
re
an
d
m
oi
st
ur
e
co
nt
en
t
C
as
e
de
sc
ri
pt
io
n
Bo
un
da
ry
co
nd
it
io
n
M
es
h
ty
pe
O
th
er
co
ns
id
er
at
io
ns
So
ft
w
ar
e
Fi
nd
in
gs
A
ut
ho
rs
M
ic
ro
w
av
e
as
si
st
ed
dr
yi
ng
C
FD
si
m
ul
at
io
n
of
sp
he
ri
ca
l
m
oi
st
pa
rt
ic
le
s
(s
oy
be
an
s)
in
a
m
ic
ro
w
av
e
as
si
st
ed
fl
ui
di
ze
d
be
d
dr
ye
r
in
or
de
r
to
st
ud
y
th
e
be
ha
vi
or
of
ga
s-
so
lid
fl
ow
•V
el
oc
it
y
in
le
t
•P
re
ss
ur
e
ou
tl
et
•S
tr
uc
tu
re
d
te
tr
ah
ed
ra
l
m
es
h
•N
eg
le
ct
in
g
te
m
pe
ra
tu
re
gr
ad
ie
nt
in
si
de
th
e
pa
rt
ic
le
s
an
d
sh
ri
nk
ag
e
•G
A
M
BI
T
6.
3.
26
•F
LU
EN
T
6.
3.
26
•S
m
al
l
de
vi
at
io
ns
of
si
m
ul
at
io
n
re
su
lt
s
fr
om
th
e
ex
pe
ri
m
en
ta
l
da
ta
co
ul
d
be
re
la
te
d
to
la
bo
ra
to
ry
m
ea
su
re
m
en
t
er
ro
rs
oc
cu
rr
ed
w
hi
le
de
te
rm
in
in
g
th
e
in
it
ia
lp
ow
er
de
ns
it
y
of
m
ic
ro
w
av
e,
ai
r
te
m
pe
ra
tu
re
,v
el
oc
it
y
an
d
re
la
ti
ve
hu
m
id
it
y
(R
an
jb
ar
an
&
Za
re
,2
01
2)
M
ic
ro
w
av
e
dr
yi
ng
of
sp
he
ri
ca
lly
sh
ap
ed
fo
od
pr
od
uc
ts
•S
ur
ro
un
di
ng
ai
r
te
m
pe
ra
tu
re
an
d
re
la
ti
ve
hu
m
id
it
y
w
as
us
ed
as
bo
un
da
ry
co
nd
it
io
n
fo
r
tr
an
sp
or
t
m
od
el
s.
•F
or
ce
d
co
nv
ec
ti
on
w
as
co
ns
id
er
ed
as
bo
un
da
ry
co
nd
it
io
n
fo
r
en
er
gy
eq
ua
ti
on
•T
et
ra
he
dr
al
m
es
h
el
em
en
ts
Sh
ri
nk
ag
e
w
as
no
t
co
ns
id
er
ed
•C
O
M
SO
L
M
ul
ti
ph
ys
ic
s
M
ax
w
el
l's
eq
ua
ti
on
s
w
er
e
so
lv
ed
by
th
e
R
F
m
od
ul
e,
m
as
s
tr
an
sf
er
by
Tr
an
sp
or
to
fD
ilu
te
d
Sp
ec
ie
s
an
d
Tr
an
sp
or
t
of
C
on
ce
nt
ra
te
d
m
od
ul
e,
he
at
tr
an
sf
er
by
H
ea
t
tr
an
sf
er
in
Fl
ui
ds
m
od
ul
e
an
d
m
om
en
tu
m
eq
ua
ti
on
by
D
ar
cy
's
la
w
•I
nt
er
m
ed
ia
te
si
ze
d
sp
he
re
s
sh
ow
ed
m
or
e
m
ic
ro
w
av
e
ab
so
rp
ti
on
an
d
ex
pl
od
ed
du
ri
ng
dr
yi
ng
•U
ni
fo
rm
an
d
lo
w
te
m
pe
ra
tu
re
dr
yi
ng
in
sm
al
le
r
si
ze
pa
rt
ic
le
s
•C
ap
ill
ar
y
di
ff
us
io
n
do
m
in
at
es
in
sm
al
le
r
si
ze
d
sa
m
pl
es
(Z
hu
et
al
.,
20
15
)
In
te
rm
it
te
nt
m
ic
ro
w
av
e
co
nv
ec
ti
ve
dr
yi
ng
co
ns
id
er
in
g
sh
ri
nk
ag
e
an
d
po
ro
si
ty
w
as
co
m
pa
re
d
w
it
h
th
e
m
od
el
w
it
ho
ut
sh
ri
nk
ag
e
fo
r
ap
pl
e
sl
ic
e
dr
yi
ng
•F
or
ce
d
co
nv
ec
ti
on
an
d
ev
ap
or
at
io
n
at
sa
m
pl
e
su
rf
ac
e
•T
ri
an
gu
la
r
m
es
h
w
it
h
a
ho
m
og
en
ou
s
el
em
en
t
si
ze
–
•A
rb
it
ra
ry
La
gr
an
gi
an
-
Eu
le
ri
an
fr
am
ew
or
k
fo
r
m
ov
in
g
m
es
h
an
d
N
on
-li
ne
ar
So
lid
M
ec
ha
ni
cs
m
od
ul
e
fo
r
la
rg
e
de
fo
rm
at
io
n
w
er
e
us
ed
in
.C
O
M
SO
L
•M
as
s,
en
er
gy
an
d
m
om
en
tu
m
go
ve
rn
in
g
eq
ua
ti
on
s
fo
r
liq
ui
d
w
at
er
an
d
va
po
r
an
d
ai
r
in
si
de
th
e
sa
m
pl
e
w
er
e
so
lv
ed
us
in
g
Tr
an
sp
or
t
of
D
ilu
te
d
Sp
ec
ie
s,
H
ea
tT
ra
ns
fe
r
in
Fl
ui
ds
an
d
D
ar
cy
's
La
w
m
od
ul
es
,
re
sp
ec
ti
ve
ly
.
•I
nt
er
m
it
te
nc
y
of
m
ic
ro
w
av
e
re
su
lt
in
un
if
or
m
m
oi
st
ur
e
an
d
te
m
pe
ra
tu
re
di
st
ri
bu
ti
on
•A
go
od
ag
re
em
en
t
re
po
rt
ed
be
tw
ee
n
th
e
ex
pe
ri
m
en
ta
l
an
d
si
m
ul
at
io
n
da
ta
•T
ak
in
g
th
e
sh
ri
nk
ag
e
in
to
ac
co
un
th
el
pe
d
to
ac
hi
ev
e
m
or
e
re
al
is
ti
c
ex
pl
an
at
io
n
of
he
at
an
d
m
as
s
tr
an
sf
er
(J
oa
rd
de
r
et
al
.,
20
17
)
In
fr
ar
ed
dr
yi
ng
•C
O
M
SO
L
M
ul
ti
ph
ys
ic
s
(c
on
tin
ue
d
on
ne
xt
pa
ge
)
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
216
Ta
bl
e
4
(c
on
tin
ue
d)
C
as
e
de
sc
ri
pt
io
n
Bo
un
da
ry
co
nd
it
io
n
M
es
h
ty
pe
O
th
er
co
ns
id
er
at
io
ns
So
ft
w
ar
e
Fi
nd
in
gs
A
ut
ho
rs
Th
e
in
fl
ue
nc
e
of
pe
ne
tr
at
io
n
de
pt
h
of
in
fr
ar
ed
ra
di
at
io
n
on
te
m
pe
ra
tu
re
an
d
m
oi
st
ur
e
co
nt
en
to
fr
ic
e
ke
rn
el
s
du
ri
ng
dr
yi
ng
•C
on
ve
ct
io
n
an
d
ev
ap
or
at
io
n
at
th
e
su
rf
ac
e
as
bo
un
da
ry
co
nd
it
io
n
fo
r
en
er
gy
eq
ua
ti
on
an
d
eq
ui
lib
ri
um
be
tw
ee
n
m
oi
st
ur
e
tr
an
sf
er
re
d
to
th
e
su
rf
ac
e
an
d
th
e
w
at
er
w
hi
ch
le
av
es
th
e
sa
m
pl
es
th
ro
ug
h
su
rr
ou
nd
in
g
dr
yi
ng
ai
r
•T
et
ra
he
dr
al
m
es
h
el
em
en
ts
•R
ic
e
ke
rn
el
s
w
er
e
as
su
m
ed
to
be
co
m
po
se
d
of
th
re
e
di
ff
er
en
t
pa
rt
s;
hu
sk
,
br
an
an
d
th
e
en
do
sp
er
m
.
•B
ot
h
m
od
el
s
co
ul
d
ac
cu
ra
te
ly
pr
ed
ic
t
dr
yi
ng
cu
rv
es
an
d
te
m
pe
ra
tu
re
du
ri
ng
dr
yi
ng
•m
ax
im
um
pr
ed
ic
te
d
te
m
pe
ra
tu
re
di
ff
er
en
ce
w
as
1.
5
°C
be
tw
ee
n
tw
o
m
od
el
s
(W
u
et
al
.,
20
17
)
A
no
ve
l
ty
pe
of
in
fr
ar
ed
dr
ye
r
co
m
bi
ne
d
w
it
h
a
so
la
r
co
lle
ct
or
an
d
a
he
at
re
co
ve
ry
un
it
in
or
de
r
to
de
cr
ea
se
th
e
hi
gh
am
ou
nt
of
en
er
gy
re
qu
ir
ed
fo
r
in
fr
ar
ed
dr
yi
ng
of
m
el
on
sl
ic
e
•H
ea
t
fl
ux
to
th
e
ho
ri
zo
nt
al
pa
rt
of
th
e
co
lle
ct
or
an
d
co
nv
ec
ti
on
he
at
in
th
e
co
nt
ac
t
re
gi
on
of
am
bi
en
t
ai
r
an
d
in
su
la
te
d
w
al
ls
•T
et
ra
he
dr
al
m
es
h
el
em
en
ts
•T
he
fl
ow
w
as
as
su
m
ed
st
ea
dy
st
at
e
•A
N
SY
S
-
Fl
ue
nt
,
D
ES
IG
N
M
O
D
EL
ER
an
d
M
ES
H
pr
og
ra
m
s
w
er
e
us
ed
fo
r
so
lv
in
g
th
e
m
od
el
s,
ge
ne
ra
ti
ng
th
e
ge
om
et
ry
an
d
th
e
m
es
h,
re
sp
ec
ti
ve
ly
.
•T
he
pe
rf
or
m
an
ce
of
so
la
r-
in
fr
ar
ed
dr
ye
r
ha
s
be
en
be
tt
er
w
it
h
th
e
ut
ili
za
ti
on
of
th
e
so
la
r
en
er
gy
an
d
he
at
re
co
ve
ry
un
it
s
•T
he
ex
pe
ri
m
en
ta
l
an
d
th
eo
re
ti
ca
l
fi
nd
in
gs
w
er
e
in
a
go
od
ag
re
em
en
t
(A
kt
aş
;Ş
ev
ik
,
A
m
in
i,
&
K
ha
nl
ar
i,
20
16
)
V
el
oc
it
y
an
d
te
m
pe
ra
tu
re
di
st
ri
bu
ti
on
an
d
w
at
er
vo
lu
m
e
fr
ac
ti
on
in
th
e
ch
am
be
r
an
d
ne
ar
ap
ri
co
ts
w
er
e
de
te
rm
in
ed
in
an
in
fr
ar
ed
dr
ye
r
fo
r
ap
ri
co
ts
•V
ap
or
in
je
ct
io
n
ra
te
w
as
ap
pl
ie
d
to
dr
yi
ng
sa
m
pl
e
su
rf
ac
es
as
a
bo
un
da
ry
co
nd
it
io
n
to
av
oi
d
si
m
ul
at
in
g
th
e
la
te
nt
he
at
ca
lc
ul
at
io
ns
du
ri
ng
di
ff
us
io
n
pr
oc
es
s
•T
et
ra
he
dr
al
m
es
h
(m
es
h
de
ns
it
y
ne
ar
th
e
IR
la
m
ps
an
d
ap
ri
co
t
w
as
hi
gh
er
th
an
ot
he
r
va
ca
nt
po
si
ti
on
s
in
dr
yi
ng
ch
am
be
r)
•O
ne
of
th
e
m
ai
n
as
su
m
pt
io
ns
in
th
is
st
ud
y
w
as
ne
gl
ec
ti
ng
th
e
sa
m
pl
e
sh
ri
nk
ag
e.
•S
te
ad
y
st
at
e
fl
ow
•k
-ɛ
tu
rb
ul
en
t
m
od
el
w
as
ap
pl
ie
d
•A
N
SY
S-
Fl
ue
nt
(V
O
F
m
ul
ti
ph
as
e
m
od
el
),
D
ES
IG
N
M
O
D
EL
ER
an
d
M
ES
H
so
ft
w
ar
e
w
er
e
us
ed
to
so
lv
e
th
e
eq
ua
ti
on
s,
cr
ea
ti
ng
ge
om
et
ry
an
d
m
es
h.
•C
FD
he
lp
ed
ob
ta
in
in
g
un
if
or
m
ai
r
ve
lo
ci
ty
an
d
op
ti
m
iz
at
io
n
of
dr
yi
ng
effi
ci
en
cy
(A
kt
aş
et
al
.,
20
17
)
Su
pe
rh
ea
te
d
st
ea
m
dr
yi
ng
Su
pe
rh
ea
te
d
st
ea
m
fl
ui
di
ze
d
be
d
dr
yi
ng
pr
oc
es
s
fo
r
ra
pe
se
ed
s
–
•N
ot
sp
ec
ifi
ed
•D
ry
in
g
pa
rt
ic
le
si
ze
w
as
co
ns
id
er
ed
co
ns
ta
nt
du
ri
ng
dr
yi
ng
•F
LU
EN
T
•T
he
se
au
th
or
s
di
d
no
t
ta
ke
in
to
ac
co
un
t
ev
ap
or
at
io
n
fr
om
th
e
m
at
er
ia
l
an
d
on
ly
co
ns
id
er
ed
co
nd
en
sa
ti
on
of
st
ea
m
;s
o,
th
er
e
w
er
e
so
m
e
va
ri
at
io
ns
be
tw
ee
n
ex
pe
ri
m
en
ta
l
an
d
C
FD
re
su
lt
s
in
th
e
co
ns
ta
nt
ra
te
pe
ri
od
in
th
ei
r
st
ud
y
(X
ia
o
et
al
.,
20
12
)
Su
pe
rh
ea
te
d
st
ea
m
dr
yi
ng
to
st
ud
y
te
m
po
ra
l
an
d
sp
at
ia
l
te
m
pe
ra
tu
re
an
d
m
oi
st
ur
e
ch
an
ge
s
of
a
si
ng
le
pe
lle
t
•V
ar
yi
ng
bo
un
da
ry
co
nd
it
io
n
ac
co
rd
in
g
to
th
e
st
ag
es
of
dr
yi
ng
•T
et
ra
he
dr
al
m
es
h
w
it
h
in
cr
ea
se
d
de
ns
it
y
ne
ar
th
e
w
al
ls
–
•S
IM
PL
EC
al
go
ri
th
m
in
A
N
SY
S
-
C
FX
pr
og
ra
m
•G
oo
d
ag
re
em
en
t
w
it
h
a
m
ea
n
re
la
ti
ve
pe
rc
en
ta
ge
er
ro
r
le
ss
th
an
or
eq
ua
l
to
10
%
(R
am
ac
ha
nd
ra
n
et
al
.,
20
17
)
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
217
are discussed. More details are presented in Table 4.
6.1. CFD in microwave-assisted drying processes
Microwave drying refers to the utilization of electromagnetic ra-
diations in order to produce a volumetric heating pattern. This process
causes a meaningful time and energy reduction and improved product
quality compared to conventional drying systems (Schubert & Regie,
2006).
One of the main advantages of microwave and microwave-assisted
drying is energy efficiency. Energy saving in this type of dryers is
mainly due to shorter drying times and higher dying rates. Another
reason for energy efficiency of microwave-assisted drying is that, mi-
crowave energy is absorbed only by dielectric materials, so, energy loss
through air or transfer to oven walls, conveyors and other parts of the
equipment is negligible which can result in significant energy savings.
For instance, 25–90% drying time reduction and 400–800% increment
in drying rate and 32–71% energy saving has been reported during
microwave-assisted drying in comparison with convective drying
techniques. This higher energy efficiency is more dominant specially in
falling rate drying period (Moses et al., 2014).
Application of microwave radiation combined with other drying
techniques especially convective drying helps to overcome some lim-
itations such as non-homogenous heat distribution, unwanted food
quality changes (textural deterioration, color, nutritional loss, etc.) and
also, low penetration depth of microwave into the food products (Feng,
Yin, & Tang, 2012; Malekjani, Emam-Djomeh, Hashemabadi, & Askari,
2018; Moses et al., 2014). Microwave-assisted hot air drying is usually
employed at the final stages of the drying process. The most common
applications of this technology include microwave-assisted vacuum
drying, microwave-assisted fluidized bed drying applied to granular
food and agricultural products (notably spouted bed drying for heat
sensitive products), and microwave assisted vacuum drying
(Rattanadecho & Makul, 2016).
Prediction of heat and mass transfer in microwave-assisted tech-
nologies is a challenging task due to high dependency of dielectric
constant ( ′ε ) and dielectric loss factor (ε") to the physical properties of
drying material such as temperature, moisture content, porosity and
changes in its chemical composition (Feng et al., 2012).
There are various methods for the modeling of microwave-assisted
drying process. It involves simultaneous solving of the models which
describe microwave heat generation and the models which can predict
temperature distribution inside the food product. Solving Maxwell's
equations (eqs. (6)–(9)) which are four fundamental governing equa-
tions can describe microwave energy absorption and pattern accurately.
Although modeling microwave heating with Maxwell's equations is
highly accurate, it is sophisticated and additional data about dielectric
properties of the food material is required.
∇ × = − ∂
∂
E B
t (6)
∇ × = + ∂
∂
H J D
t (7)
∇⋅ =D q (8)
∇⋅ =B 0 (9)
where E and H are electric (V/m) and magnetic field intensity (A/m), D
and B are electric displacement (N/V m) and magnetic induction (T)
respectively. J is current flux (A/m2) and q is electric charge density.
Lambert's law (eq. (10)) is used for the determination of microwave
energy and it is a straightforward formulation; however, it doesn't en-
sure determining a comprehensive scheme of electromagnetic field
distribution (Liu, Wang, & Sakai, 2005; Yanniotis, 2007). So, it is pos-
sible to model microwave drying by solving governing heat and mass
transfer equations while adding the microwave volumetric heating as a
source term with exponential dissipation (Lambert's law) as an alter-
native approach for solving complicated Maxwell's equations. Many
authors have applied Lambert's law in modeling of microwave drying
(Arballo, Campañone, & Mascheroni, 2012; Hemis, Choudhary, &
Watson, 2012; Joardder, Kumar, & Karim, 2017; Kumar, Joardder,
Farrell, Millar, & Karim, 2016; Souraki & Mowla, 2008). Lambert‘s law
has some limitations. First, the size of the sample should be semi-in-
finite. Second, effects of standing wave are not considered, and finally,
penetration of microwave energy is one-dimensional. In Lambert‘s law
it is supposed that incident energy is normal to the surface and energy
dissipation has exponential manner.
= −P x P e( ) βx0 2 (10)
where P(x) is the amount of power dissipation at depth x (W), x is the
depth from the samples surface (m), P0 is the power at the surface (W),
and β is the attenuation constant (m−1) which is dependent to velocity
of radiation (m/s), frequency (Hz) and loss tangent (Pitchai, 2011).
Ranjbaran and Zare (2012) performed a comprehensive CFD simu-
lation of soybean drying in a microwave assisted fluidized bed dryer.
They used a calorimetric method (Yang & Gunasekaran, 2004; Zare &
Ranjbaran, 2012) in order to determine the initial microwave power
density and proposed a modified equation to calculate microwave heat
source. The model could predict the effects of different levels of mi-
crowave power densities and initial drying air temperature on product
moisture content, drying air temperature and absolute humidity with
an appropriate agreement between simulation and experimental data.
They claimed that the small deviations of simulation results from the
experimental data could be related to laboratory measurement errors
occurred while determining the initial power density of microwave, air
temperature, velocity and relative humidity (Ranjbaran & Zare, 2012).
Radio frequency (RF) module in Finite element software COMSOL
Multiphysics is a useful and user-friendly tool in CFD modeling of mi-
crowave drying. Zhu, Gulati, Datta, and Huang (2015) performed a
more basic study on microwave drying of spherically shaped food
products (potato spheres). Instead of using the experimental calori-
metric method to determine the microwave power density, these au-
thors coupled the solution of electromagnetic governing models with
multiphase porous media models. Maxwell's electromagnetic field
equations were solved to identify a more actual explanation of electric
field distribution inside the samples and microwave oven. They re-
ported the use of less sophisticated Lambert's law in most cases can only
present a qualitative explanation of microwave penetration within the
material and moisture content profiles so, they selected Maxwell's
equations. Temperature distribution, point temperatures, moisture
content and pressure changes in three different sizes of potato spheres
and the oven were tracked during the simulation process (Zhu et al.,
2015). They found that different size samples had different responses to
microwave heating at the same drying conditions. The intermediate size
spheres showed more heat absorption in the microwave oven which
caused their explosion during drying process while smaller size parti-
cles experienced uniform temperature and had better quality. They
also, reported that the model was very sensitive to mass transfer coef-
ficients of surrounding air within the oven. The model data was in a
reasonable agreement with the experimental data (Gulati, Zhu, Datta, &
Huang, 2015).
In another study, Joardder et al. (2017) used Lambert's law instead
of solving Maxwell's equations so that intermittent microwave con-
vective drying with a multiphase porous media model was simulated.
The model was solved for liquid, gas and solid phases and considered
shrinkage and porosity during drying; it was compared with the model
without shrinkage. Temperature, moisture content, porosity, and den-
sity was determined using CFD and apple slice was used for validating
the simulation process. A good agreement reported between the ex-
perimental and simulation data. Taking the shrinkage into account
helped to achieve more realistic explanation of heat and mass transfer.
The model provided a good insight of drying process and these
N. Malekjani, S.M. Jafari Trends in Food Science & Technology 78 (2018) 206–223
218
researchers suggested it could be applied in process quality enhance-
ment and optimization. The authors reported that some parameters e.g.
the rate of evaporation, effective thermal conductivity, capillary dif-
fusion and gas pressure which cannot be predicted by simpler models,
could be predicted using their developed model. These parameters can
help in some practical conditions instead of real time calculations
(Joardder et al., 2017).
Although some efforts have been made in CFD modeling of micro-
wave assisted drying but there are still important challenges in this
area. Variation of dielectric and thermal properties of food products
with temperature and frequency changes, lack of adequate data about
dielectric properties of food and agricultural products, scarcity of ap-
propriate CFD packages including physics of microwave process com-
pletely for solving the related equations in real situation, considering
actual conditions in microwave ovens including the complex oven
geometry are some of these issues (Pitchai, 2011; Sosa-Morales,
Valerio-Junco, López-Malo, & García, 2010).
6.2. CFD in infrared drying processes
Wavelengths between 0.75 and 1000 μm in the electromagnetic
spectrum is called infrared which is used in modern food processing
technologies (Moses et al., 2014; Riadh, Ahmad, Marhaban, & Soh,
2015). Infrared drying is a novel drying method which benefits from a
higher rate of dehydration, efficient conversion of electrical energy into
heat, the selectivity of heating and higher speed of shutting down and
starting up the drying processes. It also requires a small space, the
control of the processing unit is easy and it has few installation and
capital costs (Tsotsas & Mujumdar, 2011).
One of the most critical aspects of infrared dryer utilization is en-
ergy saving which could be due to supplying the energy directly to the
drying material without dispersing it into other objects. The energy
efficiency of infrared dryers is related to the absorption properties of
drying material that determines the characteristics of the dryer (Pawar
& Pratape, 2015).
CFD modeling of infrared dryers has recently been conducted in
some researches and it is still a new concept. Wu, Zhang, and Li (2017)
explained that there are two assumptions which can be made in mod-
eling infrared drying based on infinity and zero penetration depth of
infrared radiation. As the penetration depth of the infrared waves is
about a few millimeters, for the products with a larger size than several
millimeters such as peanut, soybean, etc. it can be assumed as a
boundary condition for heat transfer. In the products with limited size
like wheat, rice, etc. it can be considered as an internal heat source or a
heat transfer boundary condition. These authors used both concepts in
order to study the influence of penetration depth of infrared radiation
on drying rice kernels. The radiation source term in heat transfer
equation was assumed to exist in the model with the penetration depth
of infinity and was neglected in the model with zero penetration depth
and the radiation was treated as a boundary condition. The results
showed that both models could accurately predict drying curves and
temperature during drying and the maximum predicted temperature
difference was 1.5 °C between two models so both assumptions can be
applied in CFD modeling of infrared drying for small sized particles (Wu
et al., 2017).
In order to calculate infrared power source term, the penetration
depth of IR waves should be defined, which is the depth at which the
absorbed infrared power declines to 1/e of its original value at the
surface of the material. It can be considered that 63.21% [(1-(1/e)] of
the power at the surface is the effective fraction of power in order to
produce volumetric heat. So, the volumetric heat generated per unit
volume of the layer at any the particle temperature T( )p which it pe-
netrates can be calculated using eq. (11):
=
−
−
( )
P T
P T
πρ R R
( )
( ) 1
( )
IR p
IR sur p e
p p pen
,
1
4
3
3 3
(11)
where PIR is the infrared power density (W/kg), PIR sur, is the initial in-
frared power (W) absorbed at the surface of the particle, Rp is particle
radius (m) and Rpen is internal penetration layer radius (m).
PIR sur, is a function of particle surface temperature and can be cal-
culated by using eq. (12).
=
+ − +
+ +− −
−
P T
σ T T
( )
[( 273.15) ( 273.15) ]
IR sur p
e p
ε
ε A F
ε
ε A
,
4 4
1 1 1e
e e e p
p
p p (12)
where εp and εe are emissivity of particle and emitter, A is surface area
and −Fe p is the view factor (part of radiation which strikes the particle
surface from the emitter surface) (Nejadi & Nikbakht, 2017).
Aktaş, Şevik, Amini, and Khanlari (2016) investigated a novel type
of infrared dryer combined with a solar collector and a heat recovery
unit in order to decrease the high amount of energy required for in-
frared drying. They studied heat and mass transport with a three di-
mensional CFD model to investigate drying kinetics of melon slice and
flow behavior in the drying system. Velocity contours and temperature
distribution profiles were determined using CFD results. They reported
that the performance of solar-infrared dryer has been better with the
utilization