simulation thermal process
TRANSCRIPT
REVIEW ARTICLE
Computational Simulation and Developments Applied to FoodThermal Processing
Roberto A. Lemus-Mondaca • Antonio Vega-Galvez •
Nelson O. Moraga
Received: 4 July 2011 / Accepted: 9 September 2011 / Published online: 3 November 2011
� Springer Science+Business Media, LLC 2011
Abstract New challenges to improve food processing
have created an incentive in the potential use of computer-
aided engineering for simulating thermal processes in foods
as a viable technique to provide effective and efficient
design solutions. Mathematical conjugated and nonconju-
gated models used along suitable numerical methods such
as finite differences, finite elements, and finite volumes in
predicting food freezing, dehydration, and sterilization are
discussed in this review. The application of computational
simulation should be used in combination with conven-
tional techniques such as physical experiments and analyt-
ical solutions in order to enhance the knowledge of fluid
mechanics, heat and mass transfer in foods.
Keywords Food engineering � Convection/diffusion �Dehydration � Freezing � Sterilization �Numerical modeling
List of symbols
C Mass concentration (kg/m3)
Cp Specific heat (J/kg K)
D Mass diffusion coefficient (m2/s)
fpc Liquid phase-change fraction
g Gravitational acceleration (m/s2)
h Heat transfer coefficient (W/m2 K)
hls Latent heat of solidification (J/kg K)
hlv Latent heat of vaporization (J/kg K)
hm Mass transfer coefficient (m/s)
k Thermal conductivity (W/m K)
L Height of the cavity (m)
n Normal direction to the food surface
N Temperature/concentration ratio (dimensionless)
p Pressure (Pa)
q00 Heat flux (W/m2)
R Universal gas constant (J/mol K)
T Temperature (K)
t Time (s)
u–v Velocity components (m/s)
x–y Rectangular coordinates (m)
P Dimensionless pressure
U–V Dimensionless velocity components
X–Y Dimensionless coordinates
Gr Grashof number (=gbTDTL3/m2)
Pr Prandtl number (=m/a)
Ra Rayleigh number (=Gr�Pr)
Re Reynolds number (=Luo/m)
Ri Richardson number (=Gr/Re2)
Sc Schmidt number (=m/D)
Subscripts
ls Liquid to solid
lv Liquid to vapor
max Maximum
min Minimum
o Inlet
pc Phase change
ref Reference
R. A. Lemus-Mondaca (&)
Department of Mechanical Engineering, Universidad de
Santiago de Chile, Santiago, Chile
e-mail: [email protected]; [email protected]
R. A. Lemus-Mondaca � A. Vega-Galvez
Department of Food Engineering, Universidad de La Serena,
La Serena, Chile
e-mail: [email protected]
N. O. Moraga
Department of Mechanical Engineering,
Universidad de La Serena, La Serena, Chile
e-mail: [email protected]
123
Food Eng Rev (2011) 3:121–135
DOI 10.1007/s12393-011-9040-x
Greek symbol
a Thermal diffusivity (m2/s)
bT Thermal expansion coefficient (1/K)
bC Mass expansion coefficient (1/m3)
q Density (kg/m3)
m Kinematic viscosity (m2/s)
s Dimensionless time
r Stress tensor (Pa)
_c Shear rate (1/s)
g Apparent viscosity (Pa s)
h Dimensionless temperature
u Dimensionless concentration
Introduction
Currently, many countries have recorded increasing agro-
industry production of both fresh and products processed.
The three main thermal processes used in the food conser-
vation are dehydration, freezing, and sterilization (canning).
Historically, the research, development, and innovation
regarding new technologies applied to food processes have
been designed to achieve better quality and greater added
value compared to raw materials. Furthermore, the chal-
lenges in the new environmentally friendly food industry
require consistent quality, optimal productivity, process
safety based on energy efficient processes. Adequate
mathematical modeling combined to efficient computational
simulation allow food thermal processes transport phe-
nomena prediction leading to better equipment design and
process control improvement for the food industry [1].
In the last decades, new advances in the computational
simulations have been used to improve the numerical solu-
tion of partial differential equations, especially those of the
convection–diffusion type found in food processes [2].
Finite numerical methods are powerful simulation tools for
analyzing and describing fluid flow processes with complex
geometries in food processing [3, 4]. This is due to the
advantage in using physics-based modeling to make pre-
dictions from the food physical properties combined with
the kinematics and dynamics for each specific process [5].
Finite differences, finite volumes and finite elements meth-
ods are among the most popular techniques for solving fluid
mechanics and heat and mass transfer problems. However,
under the same assumptions it is well recognized that only
the finite volume method offers conservative balances at any
discretization level for a single control volume, a group
control volume, or across the entire solution domain [6, 7].
The FVM has been found to be more accurate than the finite
difference method (FDM) and finite element method (FEM)
in coupled diffusion-convection problems where numerical
tests indicate that it is more stable [6–9].
Computational simulation plays an important role in
food engineering; in particular where the visual simulation
obtained is important in order to enhance thermal processes
optimization and improvement [10]. Results obtained for
velocity vectors, streamlines, temperature, pressure, and
species concentrations in either solid or liquid foods sur-
rounded by heating or cooling fluids can be animated with
visualization tools, thus aiding in the interpretation of the
simulated physical phenomena [11]. The measure of suc-
cess is how well the results of numerical simulation agree
with experimental results in cases where careful laboratory
experiments can be designed and how well the simulations
can predict highly complex phenomena that cannot be
accomplished in the laboratory [12, 13]. The application of
numerical methods in food process engineering can pro-
vide useful answers to complex problems that neither
analytical nor empirical solutions can achieve. Datta [14,
15] performed a wide review of porous media approaches
on food processes with simultaneous heat and mass trans-
fer. The author mentioned that analytical and empirical
solutions are only available for very limited cases such as
the use of the simple heat and mass diffusion equation [16]
with constant thermophysical properties [17], constant
convective coefficients [18], and regular geometries [19].
New challenges in the food industry have created an
incentive to explore the potential of recent advances in
computer-aided engineering. The advanced techniques
with suitable numerical methods are included in novel
software packages for the complete study of main food
preservation techniques: freezing, dehydration, and sterili-
zation [20–22]. Also, mathematical modeling and numeri-
cal simulation can be successfully applied to food
equipment design and development, including new pro-
duction lines [23, 24]. Computer-aided engineering has
been used successfully for the simulation, optimization,
and control in food industry processes [25]. However, in
these processes, the complexity of the nonlinear mathe-
matical models, the use of appropriate initial and boundary
conditions, together with the complex geometries and
variable thermophysical properties make the solution pro-
cedures very complicated [26–28]. Therefore, the devel-
opment of suitable mathematical models along with the use
of efficient numerical methods and object-oriented pro-
gramming is a good approach to enhance the prediction
capabilities for the food industry.
The application of mathematical modeling and compu-
tational simulations can provide useful information that can
effectively and precisely contribute to generate new
knowledge and provide the foundations to achieve better
processes and new developments in the agrofood industry
[29]. To this end, the main motivation of this work is to
present a review including mathematical modeling, com-
puter-based finite numerical methods, fluid mechanics, heat
122 Food Eng Rev (2011) 3:121–135
123
and mass transfer results, discussions, and examples related
to the formulation of two mathematical models: noncon-
jugated cases and conjugated studies applied to food
dehydration, freezing, and sterilization processes. In this
context, the information contained in this overview can be
useful for students, professors, and researchers in the area
of food science and technology.
Physical Layout Analysis
The principles of many thermal processes in solid foods are
based on heat and mass exchanges between a solid food
and the surrounding fluid. Heat transfer in solid foods is
normally modeled by the transient heat diffusion described
by Fourier’s law, while the mass transfer is described by
Fick’s law of mass diffusion [30]. Continuity and Navier–
Stokes equations are used to model Newtonian fluid flow
[6]. A successful approach to describe transport phenom-
enon in the food industry can be achieved by combining the
use of analytical models with numerical methods and
selected physical experiments [1].
The methodology proposed is based on simultaneous
calculation for the fluid mechanics, convective/diffusion
heat and mass transfer in the surrounding fluids and liquid/
solid foods, by using nonconjugated and conjugate models.
The examples shown in this review are related to food
dehydration, freezing, and sterilization processes for 2D
laminar flows of Newtonian and non-Newtonian liquid
foods [28]. Due to the fast advance in hardware [25], the
use of the proposed methodology to other food nonthermal
(high hydrostatic pressure, pulsed electric fields, oscillating
magnetic fields, ultraviolet, and ultrasound) and thermal
(microwave, evaporation, refrigeration, pasteurization, and
distillation) processes [30], either for laminar or turbulent
flows in 2D and 3D, applied to solid, liquid, and mixed
(solid/fluid treated as porous media) foods [14, 20–22] can
be expected in the near future.
Nonconjugated convection/conduction mathematical
models are those in which either solid or liquid foods, with
constant or variable thermophysical properties, are used to
describe heat and mass transfer with prescribed Dirichlet,
Neumann, or Robin boundary conditions on the food sur-
face [31, 32]. Then, a more general physical situation,
where liquid and solid foods are studied along with the
surrounding fluid, is defined as a conjugated convection/
convection or convection/conduction problem [33, 34].
Table 1 provides a list of authors that have carried out
research regarding thermal processes using conjugated or
nonconjugated models.
Table 1 Foods and processes classified according to type of model used
Food Process Solution method Type of model References
Apple Drying Finite differences Nonconjugated Hussain and Dincer [31]
Apple Drying Finite differences Nonconjugated Oztop and Akpinar [76]
Banana Drying Finite volumes Conjugated Lamnatou et al. [36]
Beef Freezing Finite differences Nonconjugated Wang et al. [53]
Beef Freezing Finite elements Nonconjugated Huan et al. [49]
Beef patty Freezing Finite volumes Conjugated Ho et al. [64]
Beef soup Sterilization Finite volumes Nonconjugated Ghani et al. [92]
Carrot Drying Finite elements Conjugated Curcio et al. [21]
Carrot Drying Finite elements Nonconjugated Aversa et al. [74]
Carrot soup Sterilization Finite volumes Nonconjugated Ghani et al. [84]
CMC Sterilization Finite volumes Nonconjugated Varma and Kannan [89]
CMC Sterilization Finite volumes Nonconjugated Farid and Ghani [97]
Eggs Cooling Finite volumes Conjugated Ho et al. [64]
Kiwi Drying Finite volumes Nonconjugated Kaya et al. [33]
Mango Drying Finite elements Nonconjugated Janjai et al. [32]
Meat Freezing Finite volumes Nonconjugated Moraga et al. [59]
Pineapple Sterilization Finite volumes Conjugated Ghani and Farid [40]
Potato Drying Finite differences Nonconjugated Oztop and Akpinar [76]
Potato Drying Finite differences Nonconjugated Hussain and Dincer [31]
Rice Drying Finite differences Nonconjugated Zare et al. [73]
Rice Rehydration Finite elements Nonconjugated Bakalis et al. [75]
Salmon Freezing Finite volumes Conjugated Moraga and Medina [39]
Food Eng Rev (2011) 3:121–135 123
123
Nonconjugated Problems
The nonconjugated models that involve fluid mechanics,
heat and mass transfer have been used in cases where the
food is either a solid or a canned liquid, in which the
mathematical model includes first- (Dirichlet), second-
(Neumann), or third- (Robin) kind boundary conditions at
the food surface (Fig. 1) [35]. Figure 2 shows, in a flow
sheet, the information needed as input: type of food, thermal
properties required, initial and boundary conditions in order
to calculate the dependent relevant variables inside the food,
temperatures and species concentration for solid and liquid
foods, and also velocity and pressure fields for liquid foods.
In nonconjugated models the convective heat and mass
transfer coefficients are required as a needed input to the
model [36]. The accurate values for these local coefficients
may not always be easily found in literature for nonlinear
transient models since they change in time and space. Food
freezing, drying, and non-Newtonian fluid sterilization are
three examples of food processes that have been studied
using the nonconjugated approach [2]. The physical, math-
ematical, and computational aspects of these processes have
been examined by using numerical methods such as finite
differences, finite volumes, and finite elements [37].
The alternative boundary conditions applied along the
food surface that can be used to assess heat transfer are:
First kind ðDirichlet): T ¼ Tref ð1Þ
Second kind ðNeumann) : �koT
on¼ q00 ð2Þ
Third kind ðRobin) : �koT
on¼ h ðTwall � TfluidÞ þ hmhlv
� ðCwall � CfluidÞ ð3Þ
The symbols used are: k, thermal conductivity (W/m K);
Tref, reference temperature (K); q00, heat flux (W/m2); n,
normal direction to the food surface; Twall, food wall
temperature (K); Tfluid, fluid temperature (K); Cwall, mass
concentration at the wall (kg/m3); Cfluid, fluid mass con-
centration (kg/m3); h, heat transfer coefficient (W/m2 K);
hm, mass transfer coefficient (m/s); and hlv, latent heat of
vaporization (J/kg K). The evaporation at the food surface
included in the last term of Eq. 3 couples heat and mass
transfer unsteady diffusion equations.
Conjugated Problems
The predictions of heat transfer in foods using numerical
methods have been accomplished in the past mainly based
on the use of a mathematical model that includes the heat
diffusion equation inside the food with external heat con-
vection incorporated in the boundary conditions by means of
a heat transfer convective coefficient, which is neither
always available nor easily extrapolated to physical prob-
lems of interest [38]. The accurate quantification of conju-
gate fluid mechanics and heat transfer can lead to
improvements in the characterization and description of
drying, sterilization, and freezing processes. The successful
use of this type of models applied to food industry can
contribute to reduce energy consumption, experimental cost,
and working time. In this approach the complexity of the
mathematical model increases but the introduction of heat
transfer convective coefficients (global and/or locals) that
affect the uncertainty in the calculations is not required [39].
Furthermore, recent advances in modern computing power
allows the use of finite numerical methods (differences,
xa
Equipment: freezer, drier or sterilizer
Unknown variables:Solid food: T(x,y,t)=T; C(x,y,t)=C Liquid food: V(x,y,t)=V
Know boundary condition: Surrounding fluid: ( ) ( )fmf C,h;T,h
( )( )( )ft,by,xH
n
t,by,xk φφφ −=
∂=∂− =
mhorhH
CorT
==φ
External surface
y
b
Fig. 1 Physical situation of food and surrounding fluid with
nonconjugated boundary conditions
Non conjugated model
Step 1. Known food characteristics Given: Geometry and dimension
Step 2. Known initial conditions Given: Velocity, pressure, temperature,
concentration in food
Step 3. Known thermophysical properties Solid food: ρ, Cp, k, D
Liquid food: ρ, Cp, k, D, μ, σ=σ (γ ), β T, βC
Step 4. Known boundary conditions Heat transfer: h, fT
Mass transfer: hm, fC
Step 5. Solve PDEs inside the food - Unsteady heat conduction eqn. - Unsteady coupled mass diffusion eqn.
Step 6. Find dependent variables Inside the food: V(x,y,t); P(x,y,t);
T(x,y,t); C(x,y,t)
Fig. 2 Flow diagram of the nonconjugated model
124 Food Eng Rev (2011) 3:121–135
123
volumes, elements) to attempt a simultaneous solution for
the transport phenomena inside the food and in the sur-
rounding fluid [30] using the conjugated model approach.
The conjugate heat and mass transfer mathematical
model can be described by two coupled systems of
equations, one for the surrounding air and the other for
the food to freeze, dry, or sterilize (Fig. 3) [21, 34, 39,
40]. In addition, Lamnatou et al. [36] established that for
modeling and simulating the thermal process one should
take into account the interaction between momentum, heat
and mass transfer within the solid and liquid food and the
transfer to the surrounding fluid. These authors explained
that conjugated models do not require prior knowledge of
convective heat and mass transfer coefficients on the
surface of the solid and liquid foods, where these coeffi-
cients can be evaluated as a part of the computational
simulation. Thus, a more accurate description about the
fluid mechanics, heat and mass transport phenomena
occurred during the food conservation processes can be
obtained in any domain where both food and surrounding
fluid are interacting [28, 36].
Also, conjugated models could automatically exclude
the need to use surface transfer coefficients in processes
simulation; however, these local surface transfer coeffi-
cients can be obtained as post-processing after temperature
and concentration distributions have been obtained for both
the surrounding liquid and for the solid or liquid food. For
convenience, Fig. 4 shows a flow sheet with the 6 steps
needed when solving a food process with the conjugated
model.
Examples of conjugate boundary conditions for heat
transfer between food and fluid in normal direction to the
wall are indicated in Eq. 4. In addition, local convection
heat transfer coefficients can be calculated from the tem-
perature fields in the food and in the surrounding fluid as
indicated in Eq. 5:
ðTfoodÞwall ¼ ðTfluidÞwall;
kfood
oTfood
on
� �wall
¼ kfluid
oTfluid
on
� �wall
ð4Þ
q00 ¼ kfoodðTfood � TwallÞDn
; h ¼ q00
Twall � Tfluid
ð5Þ
where Tfood is the food temperature (K); kfood is food
thermal conductivity (W/m K); kfluid is surrounding fluid
thermal conductivity (W/m K); n is the normal direction to
the food surface; q00, heat flux (W/m2); and Twall, food wall
temperature (K).
Mathematical Models
Fluid mechanics and heat and mass transfer in liquid foods
and the surrounding cooling or heating fluids are predicted
and described by transport equations based on conservation of
mass (continuity), linear momentum, energy and mass
transfer. They are completed by adding two algebraic equa-
tions: the state equation and the constitutive equation [41].
The nonconjugated and conjugated problems are studied
under the assumptions of laminar flow, incompressible fluids,
with negligible volume change, absence of heat generation
inside the food, and negligible thermal radiation around the
food. In general, laminar flows are assumed, since turbulent
flow modeling would require to add one to five additional
equations depending on the turbulence model used [42, 43].
xa
Equipment: freezer, drier or sterilizer
Unknown variables:Solid food: T(x,y,t)=T; C(x,y,t)=C Liquid food: V(x,y,t)=V
Unknown variables:
Surrounding fluid: ( ) ( )( ) ( )⎩
⎨⎧
t,y,xC;t,y,xT
t,y,xP;t,y,xV
y
b
Fig. 3 Physical situation of food and surrounding fluid with conju-
gate boundary conditions
Conjugated model
Step 1. Known food and equipment characteristics Given: Geometry and dimension
Step 2. Known initial conditions Given: Velocity, pressure, temperature,
concentration, etc. in food and surrounding fluid
Step 3. Known thermophysical properties Solid food: ρ, Cp, k, D
Liquid food: ρ, Cp, k, D, μ, σ (γ ), βT, βC
Step 4. Known external thermophysical properties Surrounding fluid: ρ, Cp, k, D, μ, σ (γ ), βT, βC
Step 5. Solve PDEs in food and surrounding fluid Inside the food: - Unsteady heat conduction eqns.
- Unsteady coupled mass diffusion eqns.
Surrounding fluid: - Continuity eqn. - Linear momentum eqn. in each direction
- Energy eqn. - Unsteady convective/diffusion mass eqn.
Step 6. Find dependent variables Inside the food: V(x,y,t); T(x,y,t); C(x,y,t)
Surrounding fluid: V(x,y,t); P(x,y,t) T(x,y,t); C(x,y,t)
Fig. 4 Flow diagram of the conjugated model
Food Eng Rev (2011) 3:121–135 125
123
Solid and non-Newtonian liquid foods are nonporous. In these
foods, water transport is considered only due to the relatively
simple phenomena of molecular diffusion [14]. Only density
is allowed to change linearly with temperature according to
the Boussinesq approximation.
Dimensional Mathematical Model
In cases where properties change with temperature, such as
freezing in solid foods, the unsteady 2D mathematical
model for natural convection is:
Continuity equation:
oqotþ oqu
oxþ oqv
oy¼ 0 ð6Þ
X linear momentum equation:
oqu
otþ u
oqu
oxþ v
oqu
oy¼ � op
oxþ orxx
oxþ oryx
oy
� �ð7Þ
Y linear momentum equation:
oqv
otþ u
oqv
oxþ v
oqv
oy¼ � op
oyþ orxy
oxþ oryy
oy
� �
þ qgbT T � Trefð Þþ qgbC C � Crefð Þ ð8Þ
Heat transfer equation, including solid–liquid phase change
of water content inside food:
1þ hls
qCp
ofpc
oT
� �oðqCpTÞ
otþ u
oðqCpTÞox
þ voðqCpTÞ
oy
¼ o
oxkoT
ox
� �þ o
oykoT
oy
� �ð9Þ
Mass transfer equation:
oC
otþ u
oC
oxþ v
oC
oy¼ o
oxD
oC
ox
� �þ o
oyD
oC
oy
� �ð10Þ
In the above equations the symbols used are: C, mass con-
centration (kg/m3); Cp, constant pressure-specific heat (J/
kg K); D, mass diffusion coefficient (m2/s); fpc, liquid phase-
change fraction; hls, latent heat of solidification (J/kg K); g,
gravitational acceleration (m/s2); k, thermal conductivity (W/
m K); r, stress tensor (Pa); q, density (kg/m3); bT, thermal
expansion coefficient (1/K); bC, mass expansion coefficient
(1/m3); p, pressure (Pa); t, time (s); u–v, velocity components
(m/s); T, temperature (K); and x–y, coordinates (m).
Dimensionless Mathematical Model
In cases where the change of physical properties can be
assumed to be negligible, such as in simplified models for
solid drying and sterilization and non-Newtonian liquid
foods, respectively, a dimensionless mathematical model
can be used. The dimensionless dependent (v, T, C) and
independent (x, y, t) variables are defined in the usual way
for mixed convection [44]:
X ¼ x
LY ¼ y
Ls ¼ tuo
LU ¼ u
uo
V ¼ v
uo
P ¼ p
qu2o
ð11Þ
h ¼ T � Tmin
Tmax � Tmin
u ¼ C � Cmin
Cmax � Cmin
ð12Þ
and hence the dimensionless numbers involved are:
Re ¼ Luo
mPr ¼ m
aSc ¼ m
DRi ¼ Gr
Re2ð13Þ
GrT ¼gbTL3 Tmax � Tminð Þ
m2GrC ¼
gbCL3 Cmax � Cminð Þm2
ð14Þ
In natural-convection-controlled processes, the velocity
scale, dimensionless variables, and commonly used
parameters are:
uo ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigbRðTmax � TminÞ
pg� ¼ g
gref
_c� ¼ L
uo
_cP ¼ pL
grefu2o
ð15Þ
where the dimensionless numbers used are:
Pr ¼ grefCp
kRa ¼ qgbL3ðTmax � TminÞ
gref � að16Þ
and the governing equations written in dimensionless form
becomes:Continuity equation:
oU
oXþ oV
oY¼ 0 ð17Þ
X linear momentum equation for mixed convection:
oU
osþ U
oU
oXþ V
oU
oY¼ � oP
oXþ 1
Re
o2U
oX2þ o2U
oY2
� �ð18Þ
X linear momentum equation for natural convection:ffiffiffiffiffiffiRa
Pr
roU
osþ U
oU
oXþ V
oU
oY
� �¼ � oP
oXþ g�
o2U
oX2þ o2U
oY2
� �
ð19Þ
Y linear momentum equation for mixed convection:
oV
osþ U
oV
oXþ V
oV
oY¼ � oP
oYþ 1
Re
o2V
oX2þ o2V
oY2
� �
þ Riðhþ NuÞ ð20Þ
Y linear momentum equation for natural convection:ffiffiffiffiffiffiRa
Pr
roV
osþ U
oV
oXþ V
oV
oY
� �¼ � oP
oYþ g�
o2V
oX2þ o2V
oY2
� �
þffiffiffiffiffiffiRa
Pr
rh ð21Þ
126 Food Eng Rev (2011) 3:121–135
123
Heat transfer equation for mixed convection:
ohosþ U
ohoXþ V
ohoY
� �¼ 1
Re Pr
o2hoX2þ o2h
oY2
� �ð22Þ
Heat transfer equation for natural convection:
ffiffiffiffiffiffiffiffiffiffiffiffiRa Prp oh
osþ U
ohoXþ V
ohoY
� �¼ o2h
oX2þ o2h
oY2
� �ð23Þ
Mass transfer equation for mixed convection:
ouosþ U
ouoXþ V
ouoY
� �¼ 1
Re Sc
o2uoX2þ o2u
oY2
� �ð24Þ
Symbols included in previous equations are: a, thermal
diffusivity (m2/s); m, kinematic viscosity (m2/s); _c, shear
rate (1/s); s, dimensionless time; g, apparent viscosity
(Pa s); gref, reference apparent viscosity (Pa s); g*,
dimensionless apparent viscosity; h, dimensionless tem-
perature; u, dimensionless concentration; N, temperature/
concentration ratio (dimensionless); uo, inlet velocity (m/
s); L, height of the cavity (m); P, dimensionless pressure;
R, universal gas constant (J/mol K); U–V, dimensionless
velocity components; X–Y, dimensionless coordinates; GrT,
thermal Grashof number (=gbTDTL3/m2); GrC, mass Gras-
hof number (=gbCDCL3/m2); Pr, Prandtl number (=m/a); Sc,
Schmidt number (=m/D); Ra, Rayleigh number (=Gr�Pr);
Re, Reynolds number (=Luo/m); and Ri, Richardson number
(=Gr/Re2).
Overview and Development
Food Freezing
Food freezing is a widely used preservation method
because frozen products can be stored for long periods, due
to the inhibition of the microbial growth and the reduction
of biochemical and enzyme reaction rates. As a result, the
food can be stored for long periods without practical
alteration of the initial characteristics [45]. However, the
food freezing may alter quality characteristics such as fla-
vor and texture, which in turn can affect their acceptance
[46]. In the food industry, the most common way of
freezing and thawing liquid and solid foods is to use either
cold/hot air or cold/hot water. Inside the solid foods, heat
transfer is by conduction whereas in liquid foods it is by the
combined convective/conduction heat transfer mechanisms
[47]. Therefore, a precise knowledge of the fluid dynamic,
heat and mass transfer by diffusion and convection is
important to adjust the freezing–thawing process variables
in order to better preserve and retain the quality of the
product [48].
Nonconjugated Cases
The freezing process is difficult to describe due to the
nonlinear heat diffusion equation in which the thermo-
physical properties such as density, specific heat, enthalpy,
and thermal conductivity vary with temperature [49].
However, freezing as a food preservation process must
achieve optimum quality of the finished product as a pri-
mary objective and hence a good prediction of freezing
time and time-evolution of temperature distribution are
important. The common practice to construct a mathe-
matical diffusion model has been based on the solution of
the energy and mass equations with a third-kind boundary
condition on the surface. Spatial and time variations of the
local convective heat transfer coefficients caused errors in
the range of 5–15% compared to analytical models and
with respect to the experimental data [50–52]. In industrial
practice a simplified fast predictive method is preferred and
the development of simple and available software is
desired [53]. Wang and Sun [54, 55] studied the two- and
three-dimensional transient cooling processes of roasted
and cooked meat using the FEM with variations in the food
physical properties. They calculated the moisture loss rate
(weight loss) during the cooling process. The convective
coefficients were obtained by using an analytical equation.
These coefficients were incorporated in the heat conduction
model. The numerical values were validated with experi-
mental results showing a low deviation between them.
Campanone et al. [56, 57] developed a generalized math-
ematical model to simulate the coupled heat and mass
transfer during food refrigeration in air. The developed
model considers food geometry, surface water evaporation,
variable physical properties, and variable external tem-
perature and humidity. The numerical technique used was
the FDM with a Crank–Nicolson scheme and results were
compared with those obtained by analytical solutions as
well as with experimental data. Wang et al. [53] carried out
a study on the unsteady one-dimensional freezing in
spherical and cylindrical foods. The FDM with the Crank–
Nicolson scheme was used in the numerical simulation.
The water phase-change problem in the freezing process
was solved with the apparent heat capacity approach and
the physical property change was described by a quadratic
curve. The predicted values were validated with a set of
actual experimental values, with high correlation coeffi-
cients (r2 [ 0.99), which means that the model could be
used to predict the freezing time and the temperature his-
tory of different food geometries at different cooling air
temperatures. Huan et al. [49] analyzed freezing and
thawing processes for food by using FEM. The authors
evaluated the effect of different freezing parameters (food
Food Eng Rev (2011) 3:121–135 127
123
shapes, freezing temperature, and air velocity) on the
freezing time. The heat transfer coefficients varied with the
freezing time and temperature. The final results showed
that freezing temperature and air velocity were the
important factors affecting food freezing rate. Other
authors such as Califano and Zaritzky [26] and Zhao et al.
[58] have also found that accurate predictions can be made
with the FEM for elliptical cylinders of minced beef and
albacore tuna, respectively.
Moraga et al. [59] used a mathematical model which
shows unsteady 2D temperature distributions for freezing a
cylindrical ground beef piece, with dependent temperature
thermal properties and variable convective boundary con-
ditions. These local heat transfer coefficients are a function
of freezing time and space. Figure 5 shows the predicted
temperature distributions calculated by the FVM, at dif-
ferent time intervals in ground meat cylinder. This
numerical prediction of freezing curves was found to have
deviations of 2.5% with respect to the experimental data
due to the experimental determination of heat transfer
coefficients and freezing air temperature. The accuracy
obtained may allow this approach to be used as a guideline
for freezing experiments, freezing equipment design, and
frozen food production [39]. Similar results, as well as
freezing time and temperature distributions, have been
reported by Zhao et al. [58], Ohnishi et al. [60], Haiying
et al. [61], Delgado and Rubiolo [48] and Li et al. [62] for
fish, vegetables, and beef samples, by applying different
analytical and numerical methods to predict freezing times,
freezing temperatures, and final product quality.
Conjugated Studies
Several techniques based on discrete equations, such as
FDM and FEM, have been used most frequently in freezing
and melting problems for irregularly shaped foods. On the
other hand, advances made in computational fluid dynamics,
mainly throughout the FVM, have made possible the con-
jugate analysis of fluid dynamics and heat transfer [63]. The
freezing process is difficult to predict due to the nonlinear-
ities caused by the phase change of the water content in the
food and those in the heat diffusion equation, in which the
food thermophysical properties such as density, specific
heat, enthalpy, and thermal conductivity vary continuously
with temperature [49]. Physical, mathematical, and com-
putational aspects of freezing and thawing processes have
been examined using different numerical methods such as
FDM, FVM, and FEM [37]. Moraga and Medina [39] using
the FVM have achieved a good accuracy with experimental
temperature data during salmon meat freezing by forced
convection. The food physical properties were temperature
dependent and the air temperature inside a freezing chamber
varied with time. The researchers found errors in freezing
time prediction between experimental and numerical data
Solid food:
Find: T(r, ,t)
External boundary condition
Know: ( ) ( )t,h;,tT f θθ
Fig. 5 Temperature
distributions during the freezing
of a cylindrical ground beef
piece [59]
128 Food Eng Rev (2011) 3:121–135
123
from 2.0 to 10%. In addition, the local convective heat
transfer coefficients predicted from the numerical simula-
tion were found to reach values between 15 and 30 W/m2 K,
for different food surfaces and freezing times. Ho [64] pre-
sented a 3D conjugated heat transfer model for the analysis
of food freezing, using a conjugated heat transfer method
and the enthalpy method to solve the energy equation across
the fluid–solid interface. The results predicted by the model
were compared with the experimental data available in the
literature. Good overall agreement was obtained.
Moraga and Barraza [28] presented a numerical simu-
lation of fluid flow and heat transfer during natural con-
vection between air and a food in a freezer. Figures 6a, b
show temperature and moisture distributions in the air and
in the food for 2 time instants. Weight loss by mass transfer
(water vapor) from the surface of the food toward the
surrounding air was calculated. After air temperature
reached a temperature of -30 �C in all the freezer area at a
time of 10,200 s, the liquid water had changed to ice in the
lower half portion of the food. Finally, the temperature in
the food was below -15 �C, with a quick decreasing in the
values near the surface. Food weight loss, calculated from
the amount of water lost during the freezing process, was
with a 1.5% of the food original weight.
Food Dehydration
Dehydration is useful to preserve food quality and stability,
reducing water activity by decreasing the water content, and
avoiding potential deterioration and contamination during
long storage periods at ambient temperature [19, 65]. Also,
food quality is preserved, hygienic conditions are improved,
and product loss is diminished [66]. Other important
objectives of food dehydration are weight and volume
reduction, intended to decrease transportation and storage
costs [67]. However, the sensorial and nutritional quality of
a conventionally dried product (hot air) can be drastically
reduced compared to that of the original product [65].
Several methods or combinations of dehydration meth-
ods can be used, including solar drying, hot-air drying,
freeze-drying, osmotic dehydration, spray-drying, and
vacuum-impregnation, among others [68]. In addition, the
consumption of dehydrated food has been increasing due to
the development of new products because of the easy
incorporation of dried food in prepared dishes, yogurt, and
bakery and pastry products. For this reason and considering
that dehydrated foods are an important source of vitamins,
minerals, and fiber, dried food can be also considered a
component or an ingredient of functional foods [69].
Nonconjugated Cases
Drying is a simultaneous heat and mass transfer process with
physical, chemical, and nutritional changes, over times
which are affected by parameters related to internal and
external heat and mass transfer processes [70]. The param-
eters involved include external temperature, velocity, and
relative humidity relative to ambient air, while internal
parameters may include density, permeability, porosity,
Equipment: freezer
Solid food: Find: T(x,y,t);C(x,y,t) Surrounding fluid:
Find: T(x,y,t); C(x,y,t) Find: V(x,y,t)
(a) (b)
Fig. 6 Unsteady a temperature
and b moisture content
distribution in solid food and
surrounding fluid [28]
Food Eng Rev (2011) 3:121–135 129
123
mass diffusivity, specific heat, and thermal conductivity
[33]. Therefore, adequate mathematical models and efficient
solution procedures for heat and mass transfer processes are
required to improve drying conditions [71]. Hussain and
Dincer [72] investigated the drying of rectangular pieces of
apple and potato heated by hot air. Results computed by the
FDM described the simultaneous heat and mass transfer
occurring under the same drying conditions where the
mathematical model used to predict the drying process
considered unsteady 2D heat conduction and mass diffusion.
Comparison with the experimental results shows that a good
numerical prediction of the temperature and moisture at the
center food was achieved with the numerical method, with
deviations of the order of 2.0% with respect to the experi-
mental values (Fig. 7). Zare et al. [73] and Aversa et al. [74]
have found very accurate predictions using FDM and FEM
for rectangular pieces of rough rice and carrots, respectively.
These simulation models were validated by comparing the
predicted results with experimental data in each case and
they found that the numerical methods were reliable in
predicting the moisture and temperature at the center of
rough rice and carrots during the drying process. An inter-
esting study of rice rehydration was carried out by Bakalis
et al. [75] where a nonlinear dependence of effective dif-
fusivity with respect to moisture content was found to be a
critical issue to estimate cooking times.
Oztop and Kavak [76] studied heat and moisture trans-
port during apple and potato slice drying. Numerical pre-
diction with the FVM was found to be in good agreement
with the experimental results. This comparison also
described that moisture showed a symmetrical distribution
inside the food due to the use of a constant heat transfer
coefficient on the food surface. De Lima et al. [77] pre-
sented a 2D diffusional model to predict simultaneous mass
transfer and shrinkage using the FVM for banana drying.
They concluded that numerical simulation provided an
accurate prediction of heat and mass diffusion inside
spherical foods with variable properties that were almost
impossible to obtain with analytical solutions.
Da Silva et al. [78] proposed the use of the FVM to
investigate the two-dimensional heat and mass diffusion for
cowpea grain during drying. The diffusion equations were
discretized with a fully implicit formulation, generalized
coordinates, and boundary condition of the first kind. The
numerical solutions obtained were found to be in fairly good
agreement with known analytical solutions. Wu et al. [79]
developed a 3D theoretical model to describe the coupled
heat and mass transfer by the FVM inside a single rice kernel
during drying. A Fortran-90-based computer code was used
to simulate the transient moisture content distributions
inside a rice kernel. The authors found a very good agree-
ment between simulated and experimental results.
Conjugated Studies
Heat and mass transfer in food depends on both temperature
and concentration differences, but also on the surrounding
air temperature, velocity, and water content which strongly
influence heat and mass transfer rates at the food–air inter-
faces [21]. Air temperature and velocity are difficult to
measure during industrial operations because several sen-
sors must be placed at various positions and locations of the
incoming air flow. In some drying tests for several fruits it
has been found that the degree of fruit dryness depended on
the location within the drier, because the drying rate
depended mainly on air flow (air velocity) in the drying
chamber [13]. Also, air velocity gradients within the driers
have been found to cause variations in the drying rates and in
moisture content. Therefore, computer simulation can be
used as a time-saving method to control the dynamics of the
drying process with reduced costs [36, 73]. Curcio et al. [21]
presented a numerical simulation using the FEM to describe
the simultaneous momentum, heat and mass transfer
occurring in a convective drying process under turbulent
conditions around a vegetable sample, without the specifi-
cation of interfacial heat and mass transfer coefficients.
They also showed experimental results which were in
agreement with respect to the prediction models. Figure 8
(b)(a)
300
305
310
315
320
325
330
Tem
pera
ture
(K
)Time (s)
Experimental
Numerical
0.70
0.75
0.80
0.85
0.90
0.95
1.00
0 300 600 900 1200 1500 1800 0 600 1200 1800 2400 3000 3600 4200
Moi
stur
e co
nten
t (d
imen
sion
less
)
Time (s)
Experimental
Numerical
Fig. 7 Measured and predicted
center a temperature and
b moisture content distribution
in a rectangular food [72]
130 Food Eng Rev (2011) 3:121–135
123
shows velocity distribution on a rectangular food sample
with the following characteristics: time = 60 min, food
temperature = 289 K, food moisture content = 0.85 kg
water/kg wet basis, velocity inlet = 1.5 m/s, air tempera-
ture = 318 K, and relative humidity of air = 20%.
Lamnatou et al. [36] proposed a conjugated model using
the FVM to evaluate heat and mass transfer for agricultural
products drying. This particular case shows an application
of the drying process in porous products. The results
showed that the adequate combination of parameters such
as material aspect ratio, fluid flow rate, blockage type, and
contact surfaces variation can lead to higher heat and mass
transfer coefficients resulting in better product quality. The
authors concluded that this methodology could be used to
analyze the transport phenomena in any type of convective
dryer, including those utilizing solar energy. In addition, it
may be valuable in the optimization of drying chamber
design in order to achieve a more uniform drying and
higher heat/mass transfer rates.
Nowadays, an efficient way to find the local convective
heat and mass transfer coefficients can be achieved by the
internal/external heat and mass coupling using CFD pack-
ages (Fluent�, CFX�, Phoenix�, CFD Design�, Blue Ridge
Numeric’s, Inc.) [1, 13, 33, 71, 80]. Other researchers, for
example Mathioulakis et al. [81] and Mirade and Daudin
[82], have focused mainly on providing information on air
circulation inside the driers in order to improve the drying
efficiency.
Food Sterilization
Sterilization has been the most widely used thermal process
for food preservation during the twentieth century. During
solid and liquid food sterilization, rapid and uniform
heating are desirable to achieve a predetermined level of
sterility with low energy consumption, minimum destruc-
tion of nutrients, preserving the organoleptic characteristics
of the food being processed [83]. Moreover, the high heat
resistance of bacterial spores has a great importance in the
sterilization process for low-acid foods [84]. Liquid foods
are non-Newtonian and hence model fluids such as ben-
tonite suspensions and sodium carboxy methylcellulose
(CMC) solutions that exhibit non-Newtonian behavior have
been extensively used in heat transfer studies [27, 83].
Sterilization of food in cans has been well studied, both
experimentally and theoretically [40, 85]. The effect of the
heat sterilization process of canned foods on their quality
and nutrient retention has been a major concern in thermal
processing of food since the beginning of the canning
industry [86].
Nonconjugated Cases
The accurate knowledge of the convective heat transfer
coefficient is essential to predict the sterilization process
[87]. However, in industrial practice the measurement of
heat transfer coefficient in an operating food plant can be
quite difficult due to time restrictions and the cost involved
[88]. The numerical solution describing convective flow
inside canned food has been developed by Varma and
Kannan [89]. Natural convection induced by thermal
buoyancy effects in a gravitational force field has been
observed in many applications [90]. Jung and Fryer [91]
reported a potential optimization approach to be used for
food quality and safety by means of the computational
modeling of a continuous sterilization process. Kurian et al.
[88] determined the effect of the inclination angle (ranging
from 0� to 180�) and Rayleigh number on an inclined
cylinder on thermal internal natural convective heat
transfer under buoyancy-induced flows, using simulation
with a commercial CFD code. Varma and Kannan [85, 90]
investigated enhancing natural convective heat transfer in
canned food sterilization through container shape and ori-
entation modification, using a CFX� commercial software
to solve the governing continuity, momentum, and energy
equations. They used CMC as the food simulator to study
the laminar flow behavior. They also determined the
slowest heating zone (SHZ) temperature for three geome-
tries. Ghani et al. [92–94] studied and simulated 3D
unsteady, SHZ, container shape and orientation, effects of
rotation and nutrient loss (vitamin C) for canned liquid
food sterilization by using the finite volume methods with
the Phoenics� software. Siriwattanayotin et al. [95] pre-
dicted the natural convection and changes of sugar con-
centration during the sterilization of canned liquid food
using the CFX� software. The results showed a good fit
between the calculated temperatures with respect to
Surrounding fluid: Find: T(x,y,t); C(x,y,t) Find: V(x,y,t)
Solid food: Find: T(x,y,t);C(x,y,t)
r [m
]
z [m]
0.1
0.0
80.
060.
040.
020.
0
0.06 0.08 0.1 0.12 0.14 0.16 0.18
velocity [m/s]
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Equipment: dryer
Fig. 8 Velocity field developed close to the rectangular food [21]
Food Eng Rev (2011) 3:121–135 131
123
experimental values. Taking into account all these con-
siderations, the use of this computational software is highly
recommended to simulate the liquid food sterilization
process using low CPU time for calculations.
Moraga et al. [96] studied a thin cylindrical can con-
taining a non-Newtonian liquid food: CMC suspended in
water (0.85% w/w) as aqueous food simulator (Fig. 9). The
can was half submerged in a fluid at a constant temperature
of 394 K. The liquid inside the container was initially
heated by conduction where the temperature increase
generates density gradients near the walls. Figure 9 shows
the physical situation, dimensionless temperature distribu-
tion, velocity vectors, and streamlines during sterilization
in a cylindrical can with an aspect ratio 1.56 (height/
diameter). Food sterilization, including the viscosity tem-
perature variation, was predicted using a noncommercial
computational program based on the FVM. The results
show that the time required for sterilization is strongly
dependent on the liquid food rheological behavior and a
recirculation flow pattern was found inside the cylindrical
container for the aqueous food simulator.
Conjugated Studies
Most mathematical models used in the past have consid-
ered food heating by conduction with prescribed convec-
tive boundary conditions [34]. The required processing
time is generally determined by using either an analytical
or a numerical solution for the unsteady state heat con-
duction equation [40]. Therefore, it is necessary to include
natural convection for liquid foods, which occurs due to
density gradients within the fluid caused by the temperature
gradients, to find the slowest heating point (SHZ) and thus
correctly predict this critical zone [95, 97]. Ghani and Farid
[40] calculated flow patterns, temperature distribution, and
shapes of the SHZ during heating of solid–liquid food
mixtures (pineapple slices with its moisture) in a cylin-
drical can heated by condensing steam. The authors eval-
uated two configurations: (1) pineapple slices floating in
the juice and (2) pineapple slices located at the base of the
can. The partial differential equations describing the con-
servation of mass, momentum, and energy were solved
numerically using a commercial software (Phoenics�),
based on the FVM. Saturated steam at 121 �C was used as
the heating medium. The liquid was assumed to have
constant properties, except viscosity (temperature depen-
dent) and density (Boussinesq approximation). The results
described the action of natural convection on the heating,
liquid flow patterns, and the shape and space evolution of
the slowest heating zone (SHZ), which eventually was
located in a region that was about 30–35% of the can height
from the bottom (Fig. 10). In addition, the simulations
showed that the location of the solid (pineapple slices) in
the can influence significantly the rate of heating as well as
the natural convection.
Conclusions and Future Trends
This review covers the application and development of
computational simulation based on finite numerical meth-
ods in the food process engineering. Examples of food
freezing, drying, and sterilization processes have been
described by using different numerical methods, with and
External boundary condition
Liquid food: Find: T(r,z,t) Find: V(r,z,t)
Know: fT;h
Fig. 9 Dimensionless
temperature distribution,
velocity vectors, and
streamlines for CMC during
sterilization [96]
132 Food Eng Rev (2011) 3:121–135
123
without the direct use of convective coefficients as an
external boundary condition input for the mathematical
model. A new procedure defined as conjugated model,
without the use of convective coefficients in the mathe-
matical model that includes the external environment sur-
rounding the food, has been reviewed through different
investigations which were analyzed and discussed. How-
ever, these mathematical models should be validated by
physical experiments because these models use many
approximations as well as a few assumptions that should be
based on food science knowledge.
In the following years, a considerable growth in the
development and application of computational simulation in
the food industry can be expected. It is noteworthy that
computer simulations can reduce costs, processing time, and
equipment optimization, together with allowing a more
detailed physical visualization of fluid dynamics and heat
and mass transfer during thermal processing. All these
applications and developments will contribute to enhance
computational simulations to be used as a powerful engi-
neering tool in the food processing industry in a near future.
Acknowledgments The authors acknowledge the financial support
of CONICYT–Chile through FONDECYT PROJECT–1111067.
Roberto A. Lemus-Mondaca acknowledges the financial support
given by the Doctoral National Fellowship of the Advanced Human
Capital Program CONICYT-Chile and DIGEGRA-USACH.
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