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: roberto.lemusm@usach.cl; rlemus@userena.cl

R. A. Lemus-Mondaca � A. Vega-Galvez

Department of Food Engineering, Universidad de La Serena,

La Serena, Chile

e-mail: avegag@userena.cl

N. O. Moraga

Department of Mechanical Engineering,

Universidad de La Serena, La Serena, Chile

e-mail: nmoraga@userena.cl

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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