REVIEW ARTICLE

Heat and Mass Transfer Modeling for Microbial Food SafetyApplications in the Meat Industry: A Review

J. F. Cepeda • C. L. Weller • M. Negahban •

J. Subbiah • H. Thippareddi

Received: 21 August 2012 / Accepted: 1 February 2013 / Published online: 26 February 2013

� Springer Science+Business Media New York 2013

Abstract Temperature is an important factor affecting

microbial growth in meat products, and hence the most

controlled and monitored parameter for food safety in the

meat industry. In the last few decades, modeling of heat

and mass transfer in products has gained special attention

in the meat industry as it can be integrated with predictive

microbial models, and eventually with risk assessment

models. Thus, heat and mass transfer models can be used as

practical tools to assess microbial safety of meat products

quantitatively, especially in the event of unexpected pro-

cessing issues such as thermal processing deviations. This

manuscript reviews research efforts related to heat and

mass transfer modeling in meat products that have been

published in recent years. It synthesizes the main ideas

behind modeling of thermal processing in the meat industry

encompassing common considerations and techniques.

This review specially emphasizes in research efforts that

have been oriented to industrial applications, and can be

potentially integrated with food safety tools. Literature

indicates that despite great advances in the field, there are

several challenges that persist and the scientific community

must address them to develop models applicable to the

meat industry.

Keywords Meat processing � Meat cooling � Finite

elements � Numerical analysis � Food safety � Computer

modeling � Predictive microbiology � Meat safety

Introduction

Ready-to-eat meat products are susceptible to rapid

microbial growth unless stored and processed appropri-

ately. Meat processors must control the growth of spoilage

bacteria such as Brochothrix thermosphacta, Pseudomonas

spp., and lactic acid bacteria. Moreover, food safety regu-

latory agencies such as the United States Department of

Agriculture Food Safety and Inspection Service (USDA–

FSIS) issue different policies and regulations that enforce

meat processors to restrict the presence or limit the growth

of foodborne pathogens such as Clostridium perfringens,

Listeria monocytogenes, Salmonella spp., and Escherichia

coli in their products [44].

Product temperature profile during processing and stor-

age is an important factor affecting microbial growth in meat

products. Therefore, it is one of the most controlled and

monitored parameters in the meat industry. Maintaining

product temperature profiles within safe limits that reduce

the risk of potential microbial outgrowth represents a chal-

lenge for some meat processors, especially when processing

products of large mass and volume (e.g., products over 4 kg)

[1]. In the event of a presumptive microbial outgrowth, the

meat industry and regulatory agencies rely on sampling and

microbial testing to determine the safety of the products.

However, sampling and testing may be impractical

because it is time-consuming, generates extra costs, and

J. F. Cepeda � C. L. Weller (&) � J. Subbiah

Department of Biological Systems Engineering, University

of Nebraska-Lincoln, 210, L.W. Chase Hall, East Campus,

Lincoln, NE 68583-0726, USA

e-mail: cweller1@unl.edu

J. F. Cepeda � C. L. Weller � J. Subbiah � H. Thippareddi

Department of Food Science and Technology, University

of Nebraska-Lincoln, Lincoln, NE 68583-0919, USA

M. Negahban

Department of Mechanical and Materials Engineering,

University of Nebraska-Lincoln, Lincoln, NE 68588-0526, USA

123

Food Eng Rev (2013) 5:57–76

DOI 10.1007/s12393-013-9063-6

can be unreliable as microorganisms are not uniformly

distributed in the products. In the last two decades, com-

puter modeling, including heat transfer, predictive micro-

biology, and risk assessment modeling, has gained special

attention in the meat industry as it is a practical resource to

estimate meat safety quantitatively. In fact, regulatory

agencies recommend the use of computer modeling as a

tool to prevent and evaluate the severity of potential

microbial contaminations [77, 78].

The following review focuses on heat transfer modeling

of meat products; emphasizing in practical models with

potential use in food safety industrial applications. It pro-

vides perspective on the evolution of this field, its novel

advances, and its current shortcomings. In addition, it

offers a synopsis of the state-of-the-art in the heat transfer

modeling area, which can be used to get a better under-

standing of future research opportunities.

Thermal Processing in the Meat Industry

Thermal processing of meat products refers to a broad

range of products, processing techniques, and equipment

utilized during cooking and/or cooling. Despite this vast

diversity, most of the thermal processing systems in the

meat industry operate under similar physical principles of

heat and mass transfer [38].

The manufacturing of heat-treated meat products

includes three basic steps: meat-matrix preparation, heat

treatment, and cooling. Although the meat-matrix prepa-

ration may differ between products, the heat treatment and

cooling steps follow similar principles for most products.

During heat treatment (or cooking), products are placed

into industrial meat ovens or smokehouses in which they are

exposed to a hot fluid, usually humid hot air. This step serves

several purposes such as heat stabilization of the meat

matrix, and fixing of meat color. Products are normally

cooked to at least -55 �C (131 F) to ensure proper heat

stabilization of the meat matrix by protein denaturation [5,

60, 90]. This cooking step is also critical for destroying

foodborne pathogens and assuring microbial safety. The

heat treatment must achieve the lethality performance

standard established by the USDA–FSIS (i.e., 7-log reduc-

tion in Salmonella spp. in ready-to-eat poultry, and a 6.5-log

reduction in Salmonella spp. in ready-to-eat beef products).

According to the USDA–FSIS recommended compliance

guidelines, lethality can be accomplished by different tem-

perature and holding time combinations. For instance,

lethality can be accomplished by reaching a minimum

temperature of 60 �C (140 F) and maintaining it across the

whole product for at least 12 min [78].

After the heat treatment, some meat products are exposed

to cold-water showers right before entering the cooling

chamber. This procedure helps drop the surface temperature

of the product, minimizing the weight loss caused by evap-

oration from the surface during the cooling step [1]. Cold-

water showers also help generate a gradient of temperature

within the product that facilitates the heat conduction process

from the core to the surface during the cooling step [12].

Consequently, water-showered products have non-uniform

initial temperatures prior entering the cooler.

Cooling is a critical step for preventing potential growth

of spore-forming bacteria that can survive the heat treat-

ment. Heat-shocked spores of foodborne pathogens may

germinate and grow if the cooling is not rapid and uniform.

In general, processors need to employ fast cooling rates to

minimize the risk of potential microbial outgrowth while

avoiding surface freezing. Ice formation causes micro-

structural changes in meat products that may lead to unde-

sired changes in product properties (e.g., color, viscosity,

pH), cell dehydration, drip loss, and tissue shrinkage [45].

The cooling step must meet the stabilization performance

standard established by USDA–FSIS (i.e., no multiplication

of toxigenic microorganisms such as Clostridium botulinum,

and no more than a 1 log10 multiplication of C. perfringens

within the product). According to the recommended com-

pliance guidelines, stabilization can be accomplished by

following save cooling methods. For instance, the maximum

internal temperature of a non-cured product should drop

from 54.4 to 26.6 �C (130 to 80 F) in less than 1.5 h; and

must reach 4.4 �C (40 F) within the next 5 h.

Cooling can be performed using different methods such

as water immersion, slow air flow, blast air, and vacuum

cooling. Vacuum cooling is a faster method for cooling

compared to water immersion, blast air, and slow air

cooling. Also, vacuum cooling results in a more uniform

temperature distribution of the product during the cooling

process, which is beneficial from the microbial safety

standpoint [72]. Conversely, vacuum cooling may result in

greater weight loss as the heat transfer is mainly governed

by water evaporation from the product surface. Hence, it is

not suitable for various meat products. Sun and Wang [72]

suggested that the excessive weight loss caused by vacuum

cooling could be overcome by using greater rates of water

injection during the meat preparation step. Despite the

potential advantages of vacuum cooling, traditional cooling

methods such as slow air and blast air cooling are more

common in the meat industry.

Modeling Heat and Mass Transfer in Meat Products

Heat and mass transfer in meat products is a complex

phenomenon affected by multiple physics involving energy

transport, mass transport, fluid flow dynamics, and

mechanical deformation (e.g., shrinkage and swelling).

58 Food Eng Rev (2013) 5:57–76

123

In traditional meat processing, products are exposed to a

colder or warmer airflow, which triggers the driving force

for heat transfer. As a result, thermal energy is conducted

across the product domain due to temperature differences

between the core and surface of the product. Heat and mass

are also transferred due to convection between the product

surface and the air. In addition, thermal radiation from the

product surface and from the cooler/over walls can occur.

Differences in temperature and moisture levels between the

air and the product can cause moisture evaporation from

the product surface. As the product surface dries, internal

moisture transport toward the product surface can occur.

Furthermore, the meat matrix undergoes structural and

physical changes during thermal processing that may also

affect heat transfer rates.

On the other hand, meat-processing environments are

diverse and processing conditions may vary over time.

Hence, heat and mass transfer rates are influenced by

multiple parameters including cooler/oven temperature,

product load, airflow velocity, type of heating/cooling

medium, product arrangement inside cooler/smokehouse,

and type of products (e.g., shape, dimensions, and thermal

properties). Consequently, modeling of heat and mass

transfer of meat products under industrial environments is a

challenging task. During the last two decades, researchers

have proposed different models to describe and simulate

the physics behind heat and mass transfer in meat products

(Table 1). Some of the key modeling principles and

methodologies are described in the following sections.

Governing Equations

Energy and mass transport are important governing equa-

tions for modeling heat transfer at the product level.

Energy Transport

The energy transport equation is commonly described by

transient heat conduction without internal heat generation.

It can be represented in Cartesian (x, y, z) coordinates as:

oq

oxþ oq

oyþ oq

oz¼ qCp

oT

ot

S.T. Boundary conditions and initial conditions

where variables k, q, and Cp represent thermal conductivity,

density, and specific heat, respectively; at a particular time t,

temperature T, and location (x, y, z). The conduction heat

flux q is commonly modeled by Fourier’s law,

q ¼ �k rTð Þ

which states that q is equal to the material’s thermal conduc-

tivity k times the negative local temperature gradient (-rT).

Some researchers have suggested that the nature of heat

conduction in porous materials with non-uniform inner

structure like RTE meats may be better described by non-

Fourier models. It has been shown that heat wave propa-

gation may take a finite time; contrary to the instantaneous

propagation established by Fourier [3, 50, 65]. Moreover, it

is believed that this phenomenon may be further noticeable

at low temperatures such as the ones encountered in meat

cooling, when the energy levels of molecules are highly

reduced [49]. Thus, the traditional Fourier heat conduction

model would include an extra term containing a finite

thermal characteristic time or relaxation time constant (s),

qþ soq

ot¼ �k rTð Þ

resulting in non-Fourier models such as the hyperbolic and

dual phase lag (DPL) models [4, 33, 34].

However, the heat wave propagation velocity in meat

products is high; thus, the finite thermal characteristic time

is minimal and can be neglected [33, 34]. Hence, the

Fourier model provides a practical description of heat

conduction in meat products. Non-Fourier models are yet

to be shown to be of practical value.

Mass Transport

The mass transport governing equation is usually based on

Fick’s law of mass diffusion. It can be represented in

Cartesian (x,y,z) coordinates as:

omw

ot¼ o

oxD

omw

ox

� �þ o

oyD

omw

oy

� �þ o

ozD

omw

oz

� �

where D represents moisture diffusivity in the meat matrix

and mw represents the moisture mass fraction in the meat

product.

Modeling internal water transport is a challenge as it is

not well understood, especially in meat matrices with

complex internal porous structures. Therefore, it has not

been included in multiple models for practical purposes [1,

35, 83]. Such models are built upon the assumption that

internal moisture transport compensates for the moisture

evaporation from the meat surface. As a result, water

activity and moisture concentration on the meat surface are

assumed to remain constant. The validity of those

assumptions is discussed below in the boundary conditions

and thermo-physical properties sections.

Initial Conditions

Considering a uniform initial temperature is a common

assumption when modeling heat transfer in meat prod-

ucts because non-uniform initial temperatures are diffi-

cult to provide. For instance, the final temperature of the

Food Eng Rev (2013) 5:57–76 59

123

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60 Food Eng Rev (2013) 5:57–76

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alh

eat

gen

erat

ion

2D

mas

sd

iffu

sio

n

Hea

tco

nv

ecti

on

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apora

tiv

eh

eat

loss

Ex

per

imen

tal

corr

elat

ions

Rep

ort

edv

alu

esC

on

sid

ered

mult

iple

pro

du

cts

pro

cess

edsi

mu

ltan

eou

sly

Ph

amet

al.

[57]

Tru

jill

oan

dP

ham

[74]

Air

-co

oli

ng

tem

per

atu

rep

rofi

lean

dw

eig

ht

loss

Bee

fca

rcas

s

(3D ap

pro

xim

atio

n,

and

2D

cro

ss-

sect

ion

s)

3D

CF

D

(Flu

ent)

2D

fin

ite

elem

ents

3D

/2D

hea

tco

nd

uct

ion

1D

mas

sd

iffu

sio

n

Hea

tco

nv

ecti

on

Mo

istu

reev

apo

rati

on

Mas

sco

nv

ecti

on

Lo

cal

val

ues

calc

ula

ted

by

regre

ssio

neq

uat

ion

so

bta

ined

from

CF

Dsi

mu

lati

on

s

Reg

ress

ion

equ

atio

ns

Rep

ort

edv

alu

es

Val

idat

edin

ind

ust

rial

env

iro

nm

ents

Food Eng Rev (2013) 5:57–76 61

123

Ta

ble

1co

nti

nu

ed

Ref

eren

ces

Ap

pli

cati

on

Pro

du

ct(g

eom

etry

)T

echniq

ue

(so

ftw

are)

Go

ver

nin

geq

uat

ion

Bo

un

dar

yco

nd

itio

ns

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nsf

erco

effi

cien

tsT

her

mal

pro

per

ties

Ad

dit

ion

ald

etai

ls

Pra

dhan

etal

.[5

8]

Con

vec

tio

nco

ok

ing

tem

per

atu

rep

rofi

lean

dw

eig

ht

loss

Ch

ick

enb

reas

t

(1/2

elli

pse

)

2D

fin

ite

elem

ents

(Mat

lab

)2

Dh

eat

con

duct

ion

2D

mas

sd

iffu

sio

n

Hea

tco

nv

ecti

on

Mo

istu

reev

apo

rati

on

Mas

sco

nv

ecti

on

Rep

ort

edval

ues

Rep

ort

edval

ues

Inte

gra

ted

wit

hfi

rst-

ord

erle

thal

ity

mod

els

for

L.

inn

ocu

a

Rin

ald

iet

al.

[61]

Coo

kin

gte

mp

erat

ure

pro

file

and

wei

ght

loss

Mo

rtad

ella

Bolo

gn

aN

.A.

Hea

tco

nd

uct

ion

Mas

sd

iffu

sio

n

Hea

tco

nv

ecti

on

Mo

istu

reev

apo

rati

on

Mas

sco

nv

ecti

on

Ex

per

imen

tal

Ex

per

imen

tal

Val

idat

edin

anin

du

stri

alp

lant

San

tos

etal

.[6

3,

64]

Wat

er-b

ath

hea

tin

gte

mp

erat

ure

pro

file

Sau

sage

(irr

egu

lar

2D

cross

-sec

tions)

2D

fin

ite

elem

ents

(Mat

lab

)

2D

hea

tco

nd

uct

ion

Hea

tco

nv

ecti

on

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per

imen

tal

val

ues

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ort

edval

ues

Inte

gra

ted

wit

ha

firs

t-ord

erle

thal

ity

mod

elfo

rE

.co

li

Ass

um

edn

on

-tim

e-v

ary

ing

ther

mal

pro

per

ties

and

pro

cess

ing

con

dit

ion

s

Sin

gh

etal

.[6

6]

Ov

en-r

oas

tin

gte

mp

erat

ure

pro

file

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t

(1/4

of

are

ctan

gu

lar

slab

)

Fin

ite

dif

fere

nce

s2

Dh

eat

con

duct

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Hea

tco

nv

ecti

on

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apora

tiv

eh

eat

loss

Arb

itra

ryv

alues

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ort

edv

alu

esN

ov

alid

ated

Con

clud

edth

aten

erg

yre

qu

ired

tom

elt

and

soli

dif

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tsco

uld

be

neg

lect

ed

Sp

ragu

ean

dC

olv

in[6

8]

Ng

adi

and

Hw

ang

[51]

Fry

ing

tem

per

atu

rep

rofi

lean

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ual

ity

Bee

fp

atty

(cy

lin

der

)

2D

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ite

elem

ents

(Fo

rtra

n)

2D

hea

tco

nd

uct

ion

2D

mas

sd

iffu

sio

n

Vap

or

gen

erat

ion

Hea

tco

nv

ecti

on

Ev

apora

tiv

eh

eat

loss

Mas

sco

nv

ecti

on

Rep

ort

edv

alues

Ch

oi

and

Ok

os

[13]

Inte

gra

ted

wit

ha

model

top

red

ict

gen

erat

ion

of

het

ero

cycl

icam

ine

Su

nan

dH

u[7

0,

71]

Vac

uu

m-c

oo

lin

gte

mp

erat

ure

pro

file

and

wei

ght

loss

Co

ok

edh

am

(1/8

of

acy

lin

der

)

CF

D

(Fo

rtra

n)

3D

hea

tco

nd

uct

ion

Vap

or

and

mois

ture

dif

fusi

on

(po

rou

sm

ediu

m)

Hea

tco

nv

ecti

on

Ther

mal

radia

tion

Mo

istu

reev

apo

rati

on

Mas

sco

nv

ecti

on

Arb

itra

ryv

alues

Em

pir

ical

val

ues

Rep

ort

edv

alu

esV

alid

ated

inla

bo

rato

ryv

acu

um

coo

ler

Incl

ud

edth

eef

fect

of

po

rous

dia

met

eran

dori

enta

tion

(anis

otr

opy)

toes

tim

ate

mas

str

ansf

erco

effi

cien

t

Incl

ud

edef

fect

of

shri

nk

age

Tru

jill

oan

dP

ham

[74,

75

]

Air

-co

oli

ng

tem

per

atu

rep

rofi

lean

dw

eig

ht

loss

Bee

fca

rcas

s

(3D

reco

nst

ruct

edfr

om

cross

-se

ctio

ns)

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D

(Flu

ent)

Fo

ra

ir:

con

tin

uit

y,

mom

entu

m,

turb

ule

nt

kin

etic

rate

,tu

rbu

lent

dis

sip

atio

nra

te,

hea

tco

nd

uct

ion

and

mois

ture

dif

fusi

on

Fo

rm

eat:

hea

tco

nd

uct

ion

and

mas

sd

iffu

sio

n

Hea

tco

nv

ecti

on

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mal

radia

tion

-Mo

istu

reev

apo

rati

on

Mas

sco

nv

ecti

on

Lo

cal

val

ues

calc

ula

ted

fro

ma

pre

lim

inar

yC

FD

stea

dy

-st

ate

flo

wfi

eld

anal

ysi

s

Fro

mp

rev

iou

sst

ud

ies

Dif

fusi

vit

yfr

om

Tru

jill

oet

al.

[76]

Fo

llo

wed

sim

ula

tio

np

roto

col

pro

pose

db

yH

uan

dS

un

[35]

Ass

um

edco

nst

ant

loca

ltr

ansf

erco

effi

cien

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uri

ng

the

coo

lin

gp

erio

d

Inco

rpora

ted

chan

ges

inw

ater

acti

vit

yo

nth

eca

rcas

ssu

rfac

e

62 Food Eng Rev (2013) 5:57–76

123

Ta

ble

1co

nti

nu

ed

Ref

eren

ces

Ap

pli

cati

on

Pro

du

ct(g

eom

etry

)T

echniq

ue

(so

ftw

are)

Go

ver

nin

geq

uat

ion

Bo

un

dar

yco

nd

itio

ns

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nsf

erco

effi

cien

tsT

her

mal

pro

per

ties

Ad

dit

ion

ald

etai

ls

Wan

get

al.

[83]

Air

-bla

stco

oli

ng

tem

per

atu

rep

rofi

le

Co

ok

edh

am

(1/4

elli

pse

)

2D

fin

ite

elem

ents

(Vis

ual

C?

?)

2D

hea

tco

nd

uct

ion

Co

mbin

edh

eat

con

vec

tio

n

and

ther

mal

radia

tion

Mo

istu

reev

apo

rati

on

Em

pir

ical

corr

elat

ions

for

elli

pso

id

Lew

isre

lati

on

ship

Em

pir

ical

corr

elat

ions

[13

]V

alid

ated

inla

bo

rato

ryai

r-b

last

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ler

Inte

gra

ted

wit

hL

.p

lan

taru

mg

row

thm

od

el

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gan

dS

un

[85–8

7]

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last

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ater

imm

ersi

on

coo

lin

gte

mp

erat

ure

pro

file

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ok

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eat

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ts

(1/4

elli

pse

)

(1/4

bri

ck)

3D

/2D

fin

ite

elem

ents

(Vis

ual

C?

?)

3D

hea

tco

nd

uct

ion

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mbin

edh

eat

con

vec

tio

nan

dth

erm

alra

dia

tio

n

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istu

reev

apo

rati

on

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pir

ical

corr

elat

ions

for

elli

pso

id

Lew

isre

lati

on

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Em

pir

ical

corr

elat

ions

of

mu

ltip

leso

urc

es

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din

terp

ola

tio

nto

acco

un

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rn

on

-un

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itia

lte

mp

erat

ure

s

Val

idat

edin

aco

mm

erci

alai

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ole

r

Stu

die

dth

eef

fect

of

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vel

oci

tyo

nco

oli

ng

tim

e

Wan

gan

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un

[88]

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uu

m-c

oo

lin

gte

mp

erat

ure

pro

file

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wei

ght

loss

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ok

edm

eat

join

ts

(ell

ipso

idan

db

rick

)

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fin

ite

elem

ents

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ual

C?

?)

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hea

tco

nd

uct

ion

wit

hin

ner

hea

tg

ener

atio

n

Vap

or

tran

spo

rtw

ith

inn

erv

apo

rg

ener

atio

n(p

oro

us

med

ium

)

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tra

dia

tio

n

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apora

tiv

eh

eat

loss

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rfac

ep

ress

ure

=v

apo

rp

ress

ure

inv

acu

um

cham

ber

Em

pir

ical

corr

elat

ions

Rep

ort

edv

alu

esV

alid

ated

inla

bo

rato

ryse

ttin

gs

Incl

ud

edth

eef

fect

of

po

rous

dia

met

ero

nm

ass

tran

sfer

coef

fici

ent

Pro

po

sed

empir

ical

corr

elat

ion

sto

rela

tep

rod

uct

geo

met

ric

dim

ensi

on

sw

ith

pro

duct

wei

ght

Food Eng Rev (2013) 5:57–76 63

123

meat-matrix preparation step is used as the uniform initial

temperature to simulate heat transfer during cooking; and

the final cooking temperature is used as the uniform initial

temperature to simulate cooling. However, in real meat-

processing environments, transitory operations and/or

unexpected delays between processing steps are common;

resulting in non-uniform initial temperatures. Such is the

case of water showering prior to air cooling, which drops

the surface temperatures of the product. Hence, initial

temperatures of the product Tinitial should be represented as

a function of location:

Tinitial ¼ f x; y; zð Þ at t ¼ 0

Most proposed models lacked to consider this important

factor, which compromises model performance and limits

model applicability in the meat industry. For instance,

assuming temperatures around the product surface higher

than the actual values may increase the driving force for

convection and evaporation, resulting in overestimation of

cooling rates [12].

Regression models fitting experimental initial tempera-

tures as function of 1D and 2D (x, y) locations have been

used to predict non-uniform initial temperatures [1, 2, 42].

However, this approach is dependant on geometry and

process, which imply that a particular product and process

(e.g., 30 vs. 15 min water showering) would require its

own regression model. A better approach may be the usage

of preliminary analysis to simulate heat and mass trans-

fer taking place during the intermediate processing steps

[12, 74].

Boundary Conditions

Boundary conditions represent a set of physical restraints

that occur at the boundary of the system being analyzed.

When modeling heat transfer at the product level, boundary

conditions would include physics occurring at the product

surface due to interaction with the cooling/heating med-

ium. In traditional meat processing, thermal energy trans-

fers from the product surface to the air and vice versa by

convection and evaporation. In addition, thermal radiation

from the product surface can occur.

Heat Convection

Heat transfer due to convection is an important boundary

condition used for thermal processing modeling of meats.

As it is well known, convection encompasses thermal

energy movement between a fluid (e.g., air, water) and the

meat product surface. This phenomenon is usually modeled

by Newton’s law of cooling in which the heat flow due to

convection is equal to the convective heat transfer coeffi-

cient times the driving force [38]. The convective heat

transfer coefficient (hconv) is a function of the air proper-

ties, airflow conditions, and product shape. The driving

force is the difference between the bulk fluid temperature

(Ta) and the product surface temperature (Ts),

qconv ¼ hconv Ts � Tað Þ

Changes in heat load in the product are notoriously

affected by heat convection. Hu and Sun [35] estimated that

convection can be responsible for removing about 35 % of

the heat load removed during cooling of unwrapped meats.

The percentage may be higher in products where

evaporation is negligible. Rates of heat convection usually

increase in proportion with air velocity in the 0–5 m/s

range. This is because higher air velocities would lead to

higher convective heat transfer coefficients. However, rates

of heat convection would likely remain constant for air

velocities above 5 m/s, as the surface temperature would

reach thermal equilibrium faster. Hence, the cooling rate

would be controlled by conduction. As the thermal

conductivity of meats is low, higher air velocities would

not reduce the processing time [85].

Thermal Radiation

Some models include the effect of thermal radiation as a

boundary condition. The net rate at which radiation is

exchanged (qrad) between the surface of the meat (s) and

surrounding radiative surfaces (sur) is,

qrad ¼ er T4s � T4

sur

� �where the emissivity of the meat product (e) expresses

the fraction of incident energy absorbed by the meat, and

the Stefan–Boltzmann constant (r) is equal to 5.676 9

10-8 Wm-2K-4. A common practice to account for

thermal radiation is to consider the surrounding surfaces

to be at the same temperature of the airflow surrounding the

meat products (i.e., Tsur & Ta). In such a case, qrad can be

expressed as:

qrad ¼ er T4s � T4

a

� �¼ hrad Ts � Tað Þ

where hrad represents the radiative heat transfer coefficient.

Therefore, a collective effect of heat convection and

radiation could be calculated by Newton’s cooling law as,

qconv þ qrad ¼ h Ts � Tað Þ

where h represents the combined, also called effective, heat

transfer coefficient [1, 2, 42, 84].

The heat flow due to thermal radiation can be signifi-

cant, especially during heating treatments, as emissivity of

meat products can be as high as 0.9 [59]. However, radi-

ation effects have been neglected in various models [46,

47, 63, 66].

64 Food Eng Rev (2013) 5:57–76

123

Some studies have shown that heat transfer due to radi-

ation should be considered under industrial processing

conditions, as its effect is comparable to the convective heat

transfer phenomenon [42]. Radiation may be responsible for

about 15 % of the heat loss during air cooling [35]. Radia-

tion is theoretically higher during cooking operations; and it

is critical for modeling vacuum cooling [71].

Moisture Evaporation

Another boundary condition frequently used is the heat loss

due to evaporation. Evaporation can be defined as the

process by which available liquid water found at the meat

product surface is converted into vapor. The water phase

change process requires energy (i.e., latent heat of evapo-

ration), which is extracted from the product.

Considering moisture evaporation at the product surface

is particularly important for air cooling of unwrapped meat

products where evaporation may be responsible for

removing about 40 % of the total removed heat load [35].

Evaporation is responsible for about 2–5 % weight loss

during air cooling of unwrapped products [32].

Evaporative heat loss (qevap) is commonly described as,

qevap ¼ kv

omevap

ot

where the latent heat of vaporization (kv) is taken at the

average between the product temperature and the cooling/

heating medium temperature [14]. The evaporative weight

loss rate ðomevap=otÞ can be coupled with the mass transfer

governing equation.

A common strategy is to relate omevap=ot to the con-

vective mass transfer to the surrounding air due to the

vapor pressure difference between the meat surface and the

surrounding air,

omevap

ot¼ k00 Ps � Pað Þ

which can also be written as [14–16],

omevap

ot¼ k00 awps � RHpað Þ

where (k00) represents the mass transfer coefficient. The

following method based on the Antoine equation can be

used to estimate the saturation vapor pressure of water at

surface and cooling/heating medium temperatures, (ps) and

(pa), respectively [14, 70, 76, 84–88]:

p� ¼ exp 23:4795� 3; 990:56

T� þ 233:833

� �

where * can be substituted for the subscripts s or a.

Another similar approach to represent omevap=ot is to

define it as function of the mass fraction gradient of water

(dry basis) between meat surface (mw,s) and the air sur-

rounding the product (mw,a) [57],

omevap

ot¼ k00 mw;s � mw;a

� �

The mass fraction of water (dry basis) at the meat surface

can be calculated from [74]

mw;s ¼aw � ps=pTotal

1� aw � ps=pTotal

18

29

where pTotal represents the total pressure.

Moisture evaporation is a complex process regulated by

several parameters including air relative humidity, air

temperature, air velocity, airflow regime, water activity of

the product, product temperature, meat-matrix structure

(e.g., porosity, skin, fat regions), and type of product cas-

ings (e.g., fibrous casings, collagen casings, natural cas-

ings, nets). High airflows (i.e., 1–3 m/s) are necessary to

increase heat transfer coefficients and reduce processing

times. However, high airflows also increase the mass

transfer coefficients, increasing moisture evaporation and

weight loss rates [7]. Evaporation rates are reduced and can

be negligible when processing in high relative humidity

environments, wrapping the products, and/or lowering

cooking temperatures [66].

The evaporation process can be described in three

stages. During the initial stage, moisture from the meat

surface evaporates at the same rate as from a free water

surface. Then, the moisture evaporation rate from the

product surface decreases as the surface dries. Finally, the

moisture evaporation rate increases due to progressively

rewetting of the product surface by moisture migration

from product interior. Hence, the rate of evaporation is

influenced by the internal moisture transport and transport

properties of the product [32].

When internal moisture transport is not considered, a

practical approach to account for changes in surface

moisture is to consider surface water activity as a function

of time, or to assume a different value of water activity for

each of the three evaporation stages.

Mass Conservation

Models considering internal moisture transport include a

mass conservation boundary condition commonly descri-

bed by,

omevap

ot¼ q

omw

ot

����s

¼ Dqomw

oxnx þ

omw

oyny þ

omw

oznz

� �

which establishes that at the meat surface, the convective

moisture transfer rate to the surrounding air is equal to the

internal mass transport rate by diffusion.

Food Eng Rev (2013) 5:57–76 65

123

Additional Modeling Approaches

The previously discussed governing equations, boundary

conditions, and initial conditions correspond to the most

common methods used to define heat and mass transfer

problems oriented to practical food safety industrial

applications. In recent years, porous-media approaches for

heat and mass transfer modeling in meat products have

been proposed [17, 18, 24, 81]. Meat products may be

considered as capillary porous media as they are solids

having small void spaces filled with air, vapor, and liquid.

Hence, thermal processing of meats can be studied as a

phenomenon involving mass and thermal energy move-

ment through those interconnected void spaces. Porous-

media analysis consider that mass transport through a

porous material may occur due to molecular diffusion (for

gases), capillary diffusion (for liquids), and convection

(pressure driven or Darcy flow) [17, 90].

van der Sman [81] proposed a model based on porous-

media approaches for simulating water transport during meat

cooking. The model is based on the Flory–Rehner theory of

rubber elasticity. The model considers that during cooking

the muscle proteins denature, leading to decrease in their

water holding capacity and to shrinkage of the protein net-

work. The model considers evaporation of water from the

product surface, internal heat conduction, convective heat

transfer by water flow, dripping of water from the surface,

and one-dimensional heat transfer between meat and airflow.

The model was validated with experimental data collected

from roasting processes of rectangular pieces of beef [81].

Dhall et al. [24] proposed a multiphase multicomponent

model for modeling meat cooking based on porous-media

approaches. The model considers the flow of four fluids

through the meat matrix: water phase, liquid fat phase,

water vapor, and air. The definition of the model includes

mass balances equations for each of the fluids, and a global

energy equilibrium equation. The proposed model was

applied to double-sided contact heating of hamburger pat-

ties, considering the simulations from a frozen product.

Thus, phase change was considered in the analysis.

Recent developed models based on porous-medium

approaches are noteworthy. Such models allow for better

understanding of internal moisture transport in meat matri-

ces, including not only different components, but also the

interactions among them (e.g., water release from protein

matrix during heating due to protein denaturation). In

addition, models based on porous-media approaches may

allow for analysis of spatial profiles of not only temperature,

but also moisture content and fat content, which may be used

to determine critical points for microbial food safety.

Nevertheless, models based on porous-media approa-

ches are yet in development phase. Also, there is a lack of

data regarding some of the parameters and transport

properties required for simulations using porous-media-

based models. For instance, diffusivities of liquid water

phase and fat phase due to concentration gradients and

temperature gradients are difficult to estimate, and there is

a lack diffusivity values for meat reported at cooking

temperatures [24].

Therefore, porous-media approaches, to date, may be

unfeasible for practical microbial food safety applications.

Simpler models like the ones described in Table 1 seem to

be sufficient for the current needs of the meat industry as it

relates with microbial food safety applications. Advances

in simulation technology and further development on por-

ous-media modeling may provide opportunities of making

those models applicable to practical microbial food safety

applications in the future.

Transfer Coefficients

Heat Transfer Coefficient

The heat transfer coefficient is one of the most difficult

parameters to estimate when building models for industrial

applications. In meat-processing environments, there are

multiple factors that may affect the heat transfer coeffi-

cient. Examples of such factors include irregular product

shapes; variable product arrangement during processing;

turbulent, swirling and non-parallel airflow; variable

boundary layers; local variations of temperature and

thermo-physical properties around the oven/cooler and on

the meat surface; etc. Typical uncertainties of ±10–20 %

in predictions of heat transfer coefficients are frequently

reported in the literature [22].

Different strategies can be used to estimate heat transfer

coefficients. Some of the most common techniques include

the use of experimental values, assumptions from values

reported in the literature, and use of empirical correlations

(Table 2).

Using experimental h values would theoretically

improve model performance as inherent factors of the flow

regimen can be captured. Theoretical h values are com-

monly considered constant during simulations; which may

limit model applicability for industrial use.

The use of empirical correlations is a practical strategy

to incorporate the flow variations like the ones found in

industrial applications [23]. However, empirical correlation

models applicable to irregular geometries can be limited.

When empirical correlations for the specific geometry of

interest are not available, a set of experimental values (e.g.,

by the mass-loss rate method) can be used to select among

empirical correlations available for similar shapes. A

straightforward methodology to select among different

empirical correlations was described by Ryland et al. [62].

66 Food Eng Rev (2013) 5:57–76

123

The convective heat transfer coefficient hconv commonly

includes the effect of forced convection hfc and natural

convection hnc, following the empirical equation proposed

by [8, 21],

hconv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih3

fc þ h3nc

3

q

The radiative heat transfer coefficient is commonly

estimated by [1, 12, 21, 27, 83, 85–87]:

hrad ¼ re TK;s þ TK;a

� �T2

K;s þ T2K;a

�

where TK,s and TK,a correspond to the product surface and

air absolute temperatures in Kelvin, e represents the

emissivity of the product, and r is the Stefan–Boltzmann

constant, 5.676 9 10-8 Wm-2 K-4. Another common

equation used to estimate radiative heat transfer

coefficient is [42]:

hrad ¼ 4reT3a

The heat transfer coefficient is commonly used as a

combined or effective coefficient h representing the

collective effect of convection and radiation. It can be

calculated as [1, 2, 84]:

h ¼ hrad þ hconv

h can also be calculated as [42]:

h ¼ hconv 1þ hrad

hconv

þ kL

hconv

Ps � Pwbl

Ts � Twbl

� � �

Table 2 Strategies to estimate the heat transfer coefficient

Strategy Technique(s) Additional details Reference(s)

Experimental values Back calculations from transient

temperature data

A mathematical model is fitted to experimental data

h is back-calculated

[59, 64]

Measurements with commercial heat

flux/heat transfer sensors

h = heat flux reading/DT [10, 91]

Mass-loss rate and psychrometric methods Considers the constant drying rate period, where the

net heat flux exchange is equal to zero

h = experimental mass loss/latent heat of evaporation

[40, 39]

Empirical correlations

from dimensionless

numbers

Re, Pr, Ra, and Gr dimensionless numbers

calculated for specific flow conditions

Correlations used to estimate Nu for a

particular shape

Nu ¼ f ðRe;Pr;Ra;GrÞBack calculate h from

h ¼ kNuL

RTE meats (ellipsoid)

Nuð Þnc¼ 3:470þ 0:510Ra1=4

Nuð Þfc¼ 2ffiffiffippþ 0:150p1=4Re1=4 þ 0:350Re0:566� �

Pr1=3

Nu ¼ 0:32� 0:22G� �

Re 0:44þ0:23Gð Þ

[1, 2, 12, 92, 93,

67, 62, 85–87,

88]

RTE meats (horizontal ellipsoid)

Nuð Þnc¼ 3:470þ 0:510Ra1=4

Nuð Þfc¼ 2ffiffiffippþ 0:150p1=4Re1=4 þ 0:350Re0:566� �

Pr1=3

[83, 73]

Roast beef (circle)

Nu ¼ 0:683Re0:466Pr1=3

[29, 30]

Beef carcass (vertical surface)

Nuð Þnc¼ 0:555� Pr Grð Þ0:25

Beef carcass (slab)

Nuð Þfc¼ 0:664Re1=2Pr1=3

[20, 21]

Beef carcass (vertical plate)

Nu ¼ 0:228Re0:731Pr1=3

[46]

Beef carcass (fitted)

Nu ¼ C Re m_

Pr1=3TuA

[57, 74]

Cooked meat joints (horizontal cylinder)

Nuð Þnc¼ 1:02� Pr Grð Þ0:148

Nuð Þfc¼ 0:555Re0:633Pr0:333

[85–87]

Reported values Assumptions from values reported in the

literature for similar products and

operating conditions

Typical range

5\h\50 W

m2K

[42, 47, 48]

Preliminary CFD

simulation

Values of steady-state CFD simulation for

the air phase are used to back calculate hh = estimated heat flux from meat surface/DT

Allows estimation of local or average h

[35, 57, 74, 75]

Food Eng Rev (2013) 5:57–76 67

123

where Twbl represents the wet-bulb temperature and Pwbl

corresponds to the water vapor pressure calculated at wet-

bulb temperature.

Davey and Pham [20] accounted for the additional

resistance to heat transfer generated for visible sections of

fat in beef carcasses by estimating an effective heat transfer

coefficient as:

h ¼ 1

1ðhconvþhradÞ þ

Xfat

kfat

�

where Xfat represents fat thickness.

In meat-processing environments, heat transfer coeffi-

cients may vary between locations on the product surface

due to local variations in flow patterns surrounding the

product. Variations in local heat transfer coefficients can be

significant and should be taken into account, especially

when modeling heat transfer of products with complex

shapes. Willix et al. [91] calculated experimental values of

heat transfer coefficients at different locations on the sur-

face of a fiberglass model of a meat product with complex

shape (i.e., side of beef carcass). Authors showed that local

heat transfer variations can be as high as 140 % of the

mean, and they are more notorious as the air velocity and

turbulence intensity increase.

Calculations of local transfer coefficients can be

obtained by studying the fluid flow patterns surrounding the

product by computational fluid dynamics (CFD) analysis.

However, full CFD analysis can be complex as it requires

simultaneous calculation of continuity, momentum, energy

transport, mass transport, and turbulent dissipation rate

equations [74]. Moreover, CFD models may have issues

when modeling turbulent flow [39]. This is because tur-

bulence models require detailed modeling of boundary

layers near the product surface, in addition to several

empirical coefficients that may affect the accuracy of the

prediction [23, 55].

Hu and Sun [35] proposed a three-step methodology to

simulate heat and mass transfer with CFD-generated heat

transfer coefficients (Table 2). The methodology involves

an initial steady-state CFD simulation of the air phase,

followed by back calculation of mean heat transfer coef-

ficients in the second step. In the third step, the calculated

mean heat transfer coefficients can be used for the transient

heat and mass simulation at product level. The Hu and Sun

[35] method was further implemented by Trujillo and

Pham [74, 75] to allow calculation of local heat and mass

transfer coefficients in beef carcass cooling. The main

disadvantage of the three-step methodology is that the

calculated local transfer coefficients have to be assumed

constant during the heat–mass transfer simulation, which

may be unrealistic. Moreover, although the three-step

methodology requires less computational power than a full

CFD simulation, it still requires rigorous analysis; which

may be currently impractical for industrial applications.

Therefore, empirical correlations for estimating heat

transfer coefficients continue to be the best approach to

estimate heat transfer coefficients in models intended for

industrial use. However, CFD modeling can be a valuable

tool to simulate multiple flow conditions and irregular

shapes, and back calculate local heat transfer coefficients

[6, 55]. Then, the multiple CFD-generated heat transfer

coefficients can be fitted into empirical correlations that

allow the prediction of local variations, and inclusion of

turbulence effects. This approach has been successfully

implemented by Pham et al. [57] and Trujillo and Pham

[74].

Mass Transfer Coefficient

The mass transfer coefficient (k00) is also a difficult

parameter to estimate when building models for industrial

applications. As in the case of the heat transfer coefficient,

the mass transfer coefficient can be estimated from

experimental values [53], assumptions from values repor-

ted in the literature [58], use of empirical correlations, and

CFD simulations [57, 74, 75].

Nevertheless, the mass transfer coefficient is regularly

estimated from the Lewis relationship [20, 21, 42],

h

k00¼ Cp;aLe2=3

which relates the heat and mass transfer coefficients.

Analogies typically used to represent the proportion

between the heat and mass transfer coefficients based on

the Lewis relationship are presented in Table 3.

Thermo-physical Properties

Air Properties

Air properties such as viscosity, density, thermal conduc-

tivity, and specific heat are frequently calculated using

linear regressions of tabulated values found in the litera-

ture. Usually, they are calculated as a function of air

temperature [1, 37].

Meat Properties

Thermo-physical properties of meats such as specific heat,

thermal conductivity, and density are affected by temper-

ature and composition. It is well known that moisture and

fat contents are the most influencing parameters. Carbo-

hydrate content has been reported to be particularly

important for estimating density of meat emulsions [48].

68 Food Eng Rev (2013) 5:57–76

123

Meat emulsions with high carbohydrate content exhibit

lower densities during cooking due to several chemical and

structural changes take place in the meat matrix (e.g.,

volume expansion due to gelation). Other factors such as

meat fiber orientation have shown to influence the thermal

conductivity of fresh meats [80].

Thermal properties of meat products are fairly under-

stood under processing temperatures observed in traditional

thermal processing (i.e., product temperatures ranging

from 2 to 80 �C). For instance, ground beef exhibits

densities of 1,006–1,033 kg/m3 and thermal conductivities

of 0.35–0.41 W/m �C at 5–75 �C [54]. Thermal conduc-

tivity and specific heat of meat and poultry emulsions range

between 0.26–0.48 W/m �C and 2,850–3,380 J/kg �C,

respectively [25, 47, 48].

The empirical correlations for predicting thermal prop-

erties of food components as functions of composition and

temperature proposed by Choi and Okos [13] are the most

common correlations used when modeling heat and mass

transfer of unfrozen meat products. Studies have shown

that the maximum relative error between the observed and

predicted thermal properties of ready-to-eat meats using

Choi and Okos’s correlations was 5.3 % [62].

Alternative correlations to predict thermal conductivity

of meats have been proposed. For instance, a Kirschner-

based model to predict the thermal conductivity in different

meat emulsions proposed by Marcotte et al. [47, 48], and

linear-regression models to estimate thermal conductivity

of meat as a function of temperature and water content

proposed by Elansari and Hobani [25].

Thermal properties of meats are not fully understood

during freezing stages because the amount of ice formation

as a function of temperature is still under investigation.

Thermal conductivity of frozen meat products may be up to

2–3 times higher than the thermal conductivity of unfrozen

meat products. For instance, thermal conductivity of

fresh chicken and beef rapidly increases from about 0.5 to

1 W/m �C in the 0 to -3 �C range, and it continues increasing

to about 1.5 W/m �C when it reaches -40 �C [80]. van der

Sman [80] proposed a model to predict thermal conductivity

of frozen meats based on composition. The model is based on

a model for ice formation, and a model for predicting water

activity in frozen meats. Additionally, the model takes into

account the fibrous meat structure and the anisotropy of ice

crystals.

Water Activity in Meat

Surface water activity (aw) is another important property to

be estimated when modeling heat transfer. It is affected by

the water concentration on the surface of the product, the

presence of boundary layers and casings, and the rate of

evaporation. aw has been reported to be between 0.95 and

1.0 for most meat products [8, 82]. Hence, it is usually

assumed as a constant value within that range. According

to Daudin et al. [19, 21], aw slowly decreases during meat

cooling operations, but it remains between 0.95 and 1.0

during up to 23 h of cooling even when the drying rates are

very high. Hence, the common practice of considering

constant aw values is a practical strategy that may not have

considerable effects on the overall model performance.

However, there are situations when variations in surface

water activity are significant as the internal water move-

ment rate within the product may not be enough to main-

tain a fully wetted product surface during the entire

processing operation [14–16]. In such cases, models should

include the time-varying effects of aw to avoid incorrect

estimations of heat losses due to evaporation. For instance,

models for cooling of meat products should account for

three different aw values: one to represent the starting

condition, one to represent aw during active cooling, and

one to represent the quasi-equilibrium phase [14–16].

Variation in aw can be also estimated by empirical

models. For instance, van der Sman [80, 82] described an

empirical model to predict water activity in cooked meats

as a function of composition and salt content. The model

stated that aw can be estimated from the individual con-

tribution of added sodium chloride (NaCl) and/or sodium

polyphosphates (NaPP), and remain salts (ash):

aw ¼ aw;NaClaw;NaPPaw;ash

The NaCl contribution aw,NaCl can be estimated by an

approximation of the Pitzer equation [80, 82],

aw;NaCl ¼1

1þMwð1:868þ 0:0582mol1:618Þmol

where the molality mol can be calculated as:

mol ¼ MNaClmNaCl=mwf

The molar mass of water and NaCl are Mw = 18 g/mol and

MNaCl = 58.15 g/mol, respectively. mNaCl corresponds to

mass fraction of NaCl in wet basis (wb), and mwf represents

Table 3 Common strategies to account for mass transfer coefficient

from Lewis relationship

Typical application Equation References

Air cooling k00 ¼ hconv

Cp;a

ScPr

� ��2=3 [20, 21, 42]

Air cooling k00 ¼ hCp;a

Mw

MaP�1

atmðLeÞ�2=3 [10, 91]

Oven roasting, slow air, air

blast, water immersion

cooling

k00 ¼ hconv

64:7kvap

[66, 85–87]

Air cooling k00 ¼ 18h29Cp;aPatm

[14–16, 1,

2, 35, 46,

12]

Food Eng Rev (2013) 5:57–76 69

123

the current mass fraction of free water (wb) which can be

estimated as function of the current mass fraction of water

(mw), proteins (mp), and carbohydrates (mc),

mwf ¼ mw � 0:29mprot þ 0:1mcarb

On the other hand, the NaPP contribution aw,NaPP and the

remaining salts contribution aw,ash can be estimated by

Raoult’s law [80, 82]:

aw;� ¼mwf

mwf þ n�Mwm�=M�

where * can be substituted by the subscripts NaPP or ash.

The molar mass of NaPP and remaining salts are

MNaPP = 376 g/mol and Mash = 72 g/mol; with dissocia-

tion numbers nNaPP = 8 and nash = 2, respectively. In a

posterior study, van der Sman [79] described a method to

predict water activity and water holding capacity of meat

products using the free volume flory–Huggins (FVFH) the-

ory. The method allows for the prediction of water activity

and water holding capacity not only as a function of current

water and salt content, but also as a function of temperature.

For fresh meats having high aw (i.e., aw [ 0.9), aw has

been successfully estimated by the Lewicki equation [74],

mw;db ¼0:0488

ð1� awÞ0:8761� 0:0488

1þ a�34:7794w

where mw,db represents the mass fraction of water in dry

basis (db).

Moisture Diffusivity in Meat

Moisture diffusivity in the meat matrix D is commonly

taken from reported values. However, some empirical

correlations have been proposed. For instance, moisture

diffusivity in lean beef meat as a function of temperature

can be estimated using the following Arrhenius equation

proposed by Trujillo et al. [76],

D ¼ 4:67� 10�5 expð�3757:23=TÞm2s�1

Numerical Analysis

Heat and mass transfer modeling involves complex partial

differential equations that need to be solved by numerical

methods, as analytical solutions do not exist. Finite dif-

ference analysis (FD), finite element analysis (FEA), and

finite volume analysis (FVA) are three common numerical

methods (discretization methods) used in modeling of heat

and mass transfer.

In FDA, the product shape is approximated by a regular

geometry and equations are associated with heat flow

between the nodes. FDA is very simple to implement. It is

a practical method for one-dimensional models, but its

application is limited to products with regular shapes such

as cylinders, slabs, and boxes [22].

FEA is practical for products with irregular shapes. It can

be used to account for non-uniform material composition

and mixed boundary conditions. However, it is a more

complex method than finite differences. FEA states that a

problem involving differential equations over complex

geometries can be simplified by dividing the geometry into

small regions of well-understood behavior, called elements.

In other words, dividing the domain into simple small ele-

ments (e.g., triangles, tetrahedrons, or cubes) allows com-

plicated differential equations to be easily solved [52].

Hence, the behavior of a complex domain can be approxi-

mated by studying and accounting for the behavior of its

simple elements [94]. A step-by-step methodology to solve

heat transfer models for meat products by 3D FEA was

described in full detail by Cepeda et al. [11].

In FVA, the object is divided into small elements as in

finite elements; but the equations are imposed on the

control volumes rather than on the mesh nodes.

Various commercial/proprietary software packages

include general implementations of FDA, FEA, and FVA.

They can be used to solve heat and mass problems by

defining the specific model and inputting parameters

through user-friendly interfaces. Some common commer-

cial/proprietary packages include COMSOL Multiphysics,

ANSYS FLUENT, Abaqus SIMULIA, and NEi Nastran. It

is also common to build custom-made algorithms in differ-

ent computer programming languages/platforms (e.g.,

Matlab, C??, Visual Basic, and Java) to solve the models.

However, custom-made algorithms are usually restricted to

1D or 2D analysis due to programming complexity

(Table 2), although there are few exceptions including

Cepeda et al. [12], Santos [63], and Wang and Sun [84]. It is

important to validate custom-made algorithms by compar-

ing predicted temperatures against temperatures predicted

with analytical solutions (e.g., spheres and cylinders), or by

comparing simulation results against results obtained from

commercial/proprietary software packages.

Models for Food Safety Industrial Applications

Predictive microbiology models are useful tools for food

safety management and decision making in the meat indus-

try. They can be used to support hazard analysis, determi-

nation of critical limits, estimation of potential impact of

processing deviations, and simulation of multiple processing

scenarios for quantitative microbial risk assessment. A

thorough review on model definition of predictive microbial

models has been provided by Huang and Sheen [36].

Temperature is an important intrinsic factor affect-

ing microbial growth and inactivation. As a result, most

70 Food Eng Rev (2013) 5:57–76

123

predictive microbial models require the temperature pro-

files of the product as an input. Such temperature profiles

can be: (1) provided from recorded data, (2) estimated by

lumped-capacitance or analytical methods, or (3) estimated

by heat and mass transfer models like the ones discussed in

this review (Table 1).

Recording temperature profiles is a common practice in

the meat industry. However, the accuracy of the recorded

data may be compromised due to issues related with selec-

tion of adequate temperature probes (e.g., thick metal probes

that allow conduction of the heat through the metal sheath to

the tip of the probe), probe calibration, and probe placement

in irregular-shaped products [12]. Furthermore, temperature

is monitored usually at the core of a single product; which

does not reflect the actual temperature distribution of the

whole product, and may not capture differences among

products being processed simultaneously.

Simplified methods to estimate temperature profiles

involve the use of basic analytical solutions of heat transfer

problems, or the use of lumped-capacitance methods which

assume that the temperature of the product changes expo-

nentially with time,

T � Ta

T0 � Ta

¼ exp � hAs

qVCp

t

�

where h is the effective heat transfer coefficient (W/

m2 �C), Ta is the ambient temperature (�C); and T0 is the

initial temperature (�C), q is density (kg/ml), V is the vol-

ume (m3), and Cp is the specific heat (J/kg �C) of the meat

product.

However, simplified methods to estimate temperature

profiles may be inaccurate and only work under strict

conditions (e.g., uniform initial temperatures, Biot number

\0.1, and qVCp/hA ratio equal to total process time) [36].

Heat and mass transfer models like the ones discussed in

this review (Table 1), allow estimation of more accurate

temperature profiles. Each of the models included in this

review has weak and strong aspects that may affect its

applicability in industrial food safety applications. Con-

siderations involving definition of physics, product shape,

thermal properties, transfer coefficients, processing condi-

tions, in addition to validation procedures and model

availability, should be taken into account when selecting a

model for industrial use; and when developing future

models oriented toward food safety industrial applications.

Important considerations and future research opportunities

based on the strong and weak points of the current avail-

able models are discussed in the following sections.

Physics Considerations

The majority of the heat and mass transfer models that have

been developed in recent years consider physics occurring in

two dimensions. Such simplification is used to reduce model

complexity and computational time. However, two-dimen-

sional analysis may hold for different products where sym-

metry assumptions are valid (e.g., regular-shaped sausages)

[47, 48, 63], but it can be a source of errors for simulating heat

and mass transfer in products with complex shapes (e.g., bone-

in products, irregular-shaped ready-to-eat meats) [21, 42, 58].

On the other hand, heat and mass transfer is a multi-

physics phenomenon involving coupled energy transport,

internal mass transport, fluid flow variations, mechanical

deformation, and more. Internal mass transport and

mechanical deformation are yet poorly understood [26, 69],

and therefore neglected in many models for practical rea-

sons [1, 35, 83]. Research efforts by Datta [17, 18] and van

der Sman [79, 81] have provided better understanding of

internal moisture transport in meat products. Internal mass

transport in meat products should be further studied and

incorporated into future developed models.

It is also important to find a good balance between

inclusion of sophisticated physics and practicability for

industrial use. Including sophisticated physics may cause a

significant increase in model complexity and computational

time, which may not be reflected in significant improvement

of model accuracy. A perfect example was described by

Trujillo and Pham [74] in which a complete 3D CFD anal-

ysis of a beef carcass cooling proved to be impractical due to

technology limitations and excessive computational time

requirements. A simplified model integrating steady CFD

analysis with unsteady FE analysis was sufficient to provide

accurate temperature predictions in a reasonable time.

Advances in technology and commercial simulation

software may now provide the opportunity to incorporate

multiphysics analysis occurring in three dimensions [74,

75]; Le [53]. Also, different open-source algorithms to

solve three-dimensional problems such as the one devel-

oped by Cepeda et al. [12] are available.

Product Shape Considerations

Heat and mass transfer is shape dependent. Thus, adequate

shape of a product should be considered, especially in

products with complex shapes when shape simplifications

(e.g., assuming the shape of a chicken breast as an ellipse)

may lead to notorious deviations in the predictions. Current

computer-aided engineering (CAE) softwares such as

SolidWorks, CATIA, and Pro/ENGINEER provide tools to

build irregular product geometries, but yet very few models

have taken shape irregularity into account. Some examples

of models accounting for irregular-shaped product include

Trujillo and Pham [74] for cooling of beef carcasses,

Santos et al. [64] for heating of sausages, Goni and Sal-

vadori [29, 30] for roasting of beef, and Cepeda et al. [12]

for cooling of ready-to-eat meats.

Food Eng Rev (2013) 5:57–76 71

123

Irregular shapes can be generated by rough interpolation

of cross-sectional images, and by 3D scanning. Rough

shape interpolations from cross-sectional images can be

carried out by commercial CAE software [29, 30]. In

addition, there are commercial 3D scanners available in the

market that would allow a 3D representation of the surface

of an irregular-shaped product [41].

When dealing with meat products having a complex

internal composition (e.g., bone-in products), computer

tomography (CT) scanning, and medical imaging software

(e.g., materialize mimics) would allow for the inclusion of

the multiple materials and components found in the prod-

ucts (e.g., bone, fat, meat, skin). CT scans allow identifi-

cation of the different components of a product based on

the gray value of CT images (Fig. 1). A methodology to

build 3D representations from CT scans has been described

by Cepeda et al. [11, 12].

The effect of shape as a function of product weight may

be determined by interpolating and scaling 3D shapes of

known product weights.

Thermo-physical Properties Considerations

The majority of the heat and mass transfer models consider

meat products as materials with uniform composition. This

is a fair approximation for products such as cooked ready-

to-eat meats, but may be invalid for products exhibiting

complex compositions that may lead to significant varia-

tions in thermal properties such as bone-in products.

Geometry definition via CT scanning allows for the

incorporation of non-uniform product composition, without

adding major complexity to a heat and mass transfer

model. CT scanning can be used for identification and

selection of different materials (e.g., bone, fat, meat, skin)

present in a product, as each material would exhibit dif-

ferent image gray values. Then, geometries for each

material present in a product can be reconstructed from the

CT images, and proximate compositions can be assigned to

each material. Traditional empirical correlations may be

used to estimate thermo-physical properties of each mate-

rial as a function of temperature.

Considerations Regarding Processing Conditions

Heat and mass transfer models should be able to capture

the effect of time-varying processing conditions (e.g., air

relative humidity, air temperature, air velocity). Moreover,

the models should be able to include the effect of inter-

mediate processing steps (e.g., water showering) and

delays occurring in real processing environments that may

generate non-uniform initial temperatures.

Authors such as Cepeda et al. [12], Amezquita [1, 2],

Kuitche and Daudin [42], and Trujillo and Pham [74] have

incorporated the effect of non-uniform initial temperatures

in the models by regressions of experimental initial data, or

by preliminary heat transfer simulations. The use of pre-

liminary heat and mass transfer simulations is a practical

solution to account for typical intermediate processing steps

or delays occurring in meat processing. Further research and

model validations should be aimed at this area.

On the other hand, research efforts in recent years have

been concentrated on modeling heat and mass transfer of a

single product, as opposed to multiple products being

processed simultaneously. The model developed by Le

Page et al. [53] is one of the few that incorporate simula-

tions of stacked products, but it does not consider irregular-

shaped products. The lack of models involving irregular

products being processed simultaneously is due primarily

to issues related with modeling complexity and technology

limitations. However, recent advances in technology and

simulation software may overcome these issues. A good

example is the CFD model for simulating air cooling of

multiple beef carcasses proposed by Kuffi et al. [41].

Considerations Regarding Transfer Coefficients

The majority of the heat and mass transfer models devel-

oped in recent years use empirical correlations to estimate

Fig. 1 3D modeling of complex geometries by computer tomography CT scanning

72 Food Eng Rev (2013) 5:57–76

123

transfer coefficients. This approach has been valid for

multiple applications such as air cooling, vacuum cooling,

roasting and convection cooking of meat products of reg-

ular shapes. However, it is a fact that highly irregular

product geometries and challenging processing conditions

encountered in the meat industry may generate significant

variations in local heat transfer coefficients that cannot be

captured by traditional empirical correlations. CFD analy-

sis of airflow surrounding the products has been success-

fully used to estimate local transfer coefficients [35, 75].

However, CFD modeling, to date, seems to be impractical

as a tool for creating models that are easily applicable to

the meat industry. CFD models may be complex and

demand high computational power. Thus, they may not be

suitable for various practical applications in the meat

industry due to computational power limitations of current

technology. Nevertheless, simplified CFD models can be

used to run multiple simulations and generate data to

develop empirical correlations that allow estimation of

local transfer coefficients [74].

Validation Procedures

Model validation is critical to ensure model applicability in

the meat industry. However, most heat and mass transfer

models are validated under controlled laboratory settings;

thus, model performance under challenging meat-process-

ing facility conditions is uncertain. A limited number of

models have been validated in commercial meat facilities.

Examples of such validated models include the air-cooling

model for cooked boneless ham developed by Amezquita

et al. [1, 2], the air-cooling model for ready-to-eat meat

products developed by Cepeda et al. [12], and the models

for air cooling of beef carcasses developed by Trujillo and

Pham [74]. Further research efforts are needed to take the

validation process to the next level and prove that the

models can perform well in real industrial environments.

Another aspect to keep in mind is that meat processors

may not be able to provide all the input parameters

required for the simulation (e.g., product geometry, non-

uniform initial temperatures, time-varying processing

conditions). Hence, the model validation should include a

sensitivity analysis to determine how well the model can

perform with limited or uncertain input parameters. A

method to perform a sensitivity analysis has been described

by Kuitche et al. [42]. An example of model adaptation to

limited input parameters has been described by Cepeda

et al. [12].

Integration with Predictive Microbial Models

A limited number of heat and mass transfer models have

been integrated with predictive microbial models. A reason

for the scarcity of integrated models may be due to the fact

that experts in predictive microbiology modeling may not

have enough background in heat and mass transfer mod-

eling; and vice versa. Hence, further multidisciplinary

research and collaborations are needed to develop inte-

grated models that can be used to support food safety in the

meat industry.

Examples of integrated heat–mass transfer and predic-

tive microbial models include a model for air cooling of

ready-to-eat meats integrated with various dynamic

microbial growth models through a website [12]; a model

for air cooling of cooked boneless ham integrated with

microbial growth model of C. perfringens [2]; a model for

air-blast cooling of cooked ham integrated with a Lacto-

bacillus plantarum growth model [83]; a model for water-

bath heating of sausages integrated with a lethality model

for E. coli 0157:H7 [64]; a convection cooking model for

chicken breast integrated with lethality models for Listeria

innocua [58]; a model for inactivation kinetics of Salmo-

nella spp., L. monocytogenes, and other microorganisms

during pasteurization treatments of different food matrices

[28]; a model to predict E. coli growth in cartoned meat

undergoing thawing [56]; a model to estimate the effect of

drying and heating on the growth and inactivation of Lis-

teria [95]; and a cooking–cooling model for sausages

integrated with process lethality calculations for Salmo-

nella senftenberg, E. coli, L. monocytogenes, and Entero-

coccus faecalis [47, 48]. Additional integrated models not

included above have been discussed in a review by Lebert

and Lebert [43].

Model Availability

The main shortcoming of current heat and mass transfer

models is that most of them are not be readily available to

meat processors. There is a need for validated heat and

mass transfer models that can be easily accessed by meat

processors. Examples of research efforts focused on mak-

ing models available to meat processors include a general-

purpose food safety software developed by Halder et al.

[31], and a food safety website developed by Cepeda et al.

[12].

Halder et al. (2011) proposed a general-purpose pre-

dictive software package for food safety. The software

allows custom selection of product geometry, product

composition, processing conditions, and initial conditions

to simulate heat and mass transfer. The software allows

simulations using traditional heat and mass models, and

sophisticated models involving porous-media approaches.

The disadvantage of this software is that it is mainly a user-

friendly interface to COMSOL multiphysics, a commercial

simulation software. All the numerical and post-processing

analysis is performed in COMSOL multiphysics. Hence,

Food Eng Rev (2013) 5:57–76 73

123

meat processors must have a license for commercial use of

COMSOL multiphysics, in addition to basic knowledge/

training on post-processing methods in COMSOL; which

may be unattainable for some meat processors.

On the other hand, Cepeda et al. [12] developed a pro-

totype of an online tool for food safety (i.e., http://food

safety.unl.edu). The website allows online simulations

using various dynamic predictive microbial models, as well

as heat and mass transfer models validated in industrial

settings. However, the website is, to date, in development

phase and it currently has very limited models available.

Summary

In the last few decades, modeling has gained special

attention in the meat industry as it is considered as a

practical tool to assess meat safety quantitatively. Multiple

heat and mass transfer models for meat products have been

developed in recent years. Models range from traditional

energy and mass transport models to elaborated models

involving internal moisture transport and porous-media

approaches. However, despite the great advances in the

field, heat and mass transfer modeling of meat products

remains largely a research tool. Some of the currently

available models were built upon assumptions (e.g., sim-

plified physics, regular-shaped products, uniform product

compositions, average transfer coefficients, use of pro-

prietary software) that may limit applicability for food

safety industrial use.

Advances in technology and simulation software may

now allow further development of models that can capture

the complexity of heat–mass transfer physics that takes

place in industrial meat processing. Research efforts ori-

ented toward development of practical heat and mass

transfer models that can consider irregular-shaped prod-

ucts, products with non-uniform compositions (e.g., bone-

in products), products being processed simultaneously,

local transfer coefficients, and non-uniform initial condi-

tions, and that can be easily accessed by meat processors

are needed. Adequate estimation of transfer coefficients, as

well as of thermo-physical properties affecting microbial

growth such as water activity, is critical for accuracy of the

models. Hence, there is a need for validated methods to

accurately estimate transfer coefficients and thermo-phys-

ical properties, especially for irregular-shaped products,

and products with complex surfaces (e.g., products with

skin and/or significant sections of fat). Special consider-

ations regarding validations in commercial meat-process-

ing facilities, and integration with predictive microbial

models and/or risk assessment models are encouraged for

future models.

Acknowledgments This study is a contribution of the University of

Nebraska Agricultural Research Division, supported in part by funds

provided through the Hatch Act, USDA. Additional support was

provided by the USDA-IREE-CGP National Integrated Food Safety

Initiative Competitive Grants Program (Grant Contract Number:

2004-51110-01889). Mention of a trade name, proprietary products,

or company name is for presentation clarity and does not imply

endorsement by the authors or the University of Nebraska.

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