heat transfer in two-phase solid-liquid food flows- a review 1998

HEAT TRANSFER IN TWO-PHASE SOLID-LIQUID

FOOD FLOWS: A REVIEW

M. BARIGOU, S. MANKAD* (GRADUATE) and P. J. FRYER (FELLOW)

School of Chemical Engineering, University of Birmingham, Birmingham, UK*Department of Chemical Engineering, University of Cambridge, Cambridge, UK

Higher quality food can be produced by continuous aseptic processes rather than by theessentially batch process of in-container sterilization of particulate foods. Incontinuous aseptic processing, a food mixture passes continuously through a heat-

hold-cool system, and is then packaged in presterilized containers. This results in shorterprocessing times, higher production rates, superior product quality, reduced powerrequirements, and improved process control. The design of such a plant requires knowledgeof the rates of heat exchange (both ¯ uid-particle and wall-¯ uid) which take place within theprocess. At present there is a severe lack of understanding of two-phase solid-liquid food ¯ ows,and suitable commercial sterilization schedules must be determined experimentally for everyfood product processed. Until enough knowledge of real ¯ ows has been gained, foodmanufacturers will adhere to conservative approaches in the design of each stage in a heat-hold-cool system. Following the recent review on the ¯ uid mechanics of solid-liquid food¯ ows6 , this paper reviews the current state-of-the-art in the area of heat transfer. The ¯ uid-particle and the wall-¯ uid heat transfer coef® cients are crucial design parameters.Measurement techniques and mathematical models for estimating them are reviewed, theapplication of existing knowledge to the design of continuous aseptic processes is discussed,and research needs are highlighted.

Keywords: aseptic processing; food sterilization; heat transfer coef® cient; mathematicalmodelling; particulate food; solid-liquid ¯ ow

1. INTRODUCTION

Cooking represents a critical step in most food manufactur-ing processes. Thermal processing brings about irreversiblechanges in food textural and sensoric properties, whilst atthe same time achieving the desired level of microbialsterility. It is not possible nor is it necessary to eliminate allviable organisms from the material. Organoleptic andnutritive properties of foods are adversely affected byheat, and the process must only be as severe as necessary toensure commercial sterility. Spore reduction and nutrientloss are governed by different kinetics, the rates of whichare dictated by the processing conditions. A food that isoptimally cooked would be safe but would also havesensoric and nutritious properties that are most acceptable tothe consumer. The optimization of such thermal treatmentsposes a challenging manufacturing problem. The overridingimportance of food safety often results in the food beingexposed to a more severe process than is desirable from aquality aspect, resulting in lower sensoric and nutritionalattributes, especially with sensitive products, than isactually possible. Without con® dence in the design data,processes will always be overdesigned for safety.The problem of optimizing such a process is dif® cult forsingle phase foods; it is even more intractable wheninhomogeneous highly-viscous foodstuffs bearing largesolid food particulates are handled.

The processes which result in quality loss have a loweractivation energy than those which result in sterility1 . Athigh temperatures, sterilization reactions proceed about 100times faster than loss of quality reactions: the time neededfor sterilization is reduced and the amount of quality loss isalso reduced. Hence, there is a clear quality advantage insterilizing at higher temperatures and for shorter times i.e.HTST2 . HTST processes involve heating to temperatures inthe region of 1408 C for about 10 seconds, holding theproduct at the sterilization temperature for long enough toensure suf® cient pathogen spore destruction, and thencooling to packaging temperature. Rather than the essen-tially batch process of canning, the HTST process requiresthe product to pass continuously through the system at high¯ owrates to optimize productivity; this necessitates inde-pendent sterilization of the product and package, and istermed aseptic processing.

The literature on the thermal processing of foods isobviously enormous: here, the focus is on the continuousaseptic processing of food products consisting of bothviscous liquid and (large) solid constituents. This is beingincreasingly considered as a better substitute for traditionalheat treatment in containers3 . Higher production rates,improved product quality because of relatively short heatingand cooling times, reduced energy requirements, and moreamenability to automatic control are all attractive features ofcontinuous aseptic processing. The process in its essential

3

0960±3085/98/$10.00+0.00q Institution of Chemical Engineers

Trans IChemE, Vol 76, Part C, March 1998

form consists of pumping the solid-liquid food mixturethrough a heat exchanger. If this heat exchanger is aconventional one, using convective heat transfer, most ofthe heat goes only into the liquid as the residence time in theheat exchanger is short and the rate of transient-conductionheat transfer inside the particle is low. Therefore, little morethan a steep temperature gradient from the outside to thecentre of the particle will have been established by the timethe particle exits the heat exchanger. The mixture thenpasses into an insulated holding tube where suf® cient time isgiven for the solid particles to be indirectly heated by theliquid to the desired level. From here, the food passes intoone or more heat exchangers where it is cooled prior topackaging: here the particle temperature will often begreater than the surrounding liquid. For example, in theAPV ohmic-heaterprocess4 ,5 , between100and1000kghr- 1 offood mixtures containing 50% solids up to 25 mm indiameter, are pumped through pipes of 0.075-0.15mdiameter, i.e. at average ¯ ow velocities on the order of0.1 ms- 1 .

The technology has been successfully applied to singlephase liquid foods. Its widespread application to foodproducts containing both liquid and solid constituents isseverely limited by the lack of understanding of theprinciples that govern the transfer of heat within suchfoods as they ¯ ow through the processing equipment. Theoften complicated rheology of the foods which usuallyconsist of a highly viscous non-Newtonian liquid carryingalmost neutrally-buoyant solid food pieces, causes the¯ ow through the equipment to be non-uniform and oftenunpredictable6 . The process may thus be subjected to awide distribution of particle concentration, velocities,residence times, and temperatures, thereby causing a wideand still largely unpredictable distribution of thosequality changes that are imparted to the food by theheat treatment. The problem is distinct from that ofprocessing packaged foods, which is discussed in detailby Holdsworth7 .

In conventional heat transfer, the processes which lead tothe heating of particles are:

(i) transfer of heat from some external source to theexchanger surface;(ii) transfer from the exchanger wall to the ¯ uid,characterized by a wall heat transfer coef® cient, hw ;(iii) convective mixing or conduction through the particle-¯ uid mixture to the particle;(iv) transfer from ¯ uid to particle, characterized by aparticle-¯ uid heat transfer coef® cient, hfp ;(v) thermal conduction within the particle.

A successful design of a continuous heat-hold-coolaseptic process for particulate foods couples safetyassurance with quality optimization. The absolute aim isto ensure that the process delivers the necessary microbiallethality to the slowest heating zones within the particleswhilst not over-processing any signi® cant amounts of theproduct. Factors in¯ uencing process design are of threeprincipal categories:

(i) Heat Transfer

In addition to design considerations for processing

liquids, there are a number of extra parameters associatedwith food particle heating which directly affect the design ofequipment used to process liquids with particles1 ,8 ± 1 0 ,

(a) particle sizeÐ the larger the particle the harder it is tosterilize;(b) particle shapeÐ different shapes lead to different ratesof heat transfer to the particle centre;(c) particle thermal propertiesÐ density, speci® c heatcapacity, and thermal conductivity which has the mostimportant effect on particle centre heating;(d) convective heat transfer coef® cient at particle surface(hf p )Ð this is a vital parameter for estimating the steriliza-tion rate at the geometric centre of the particle, but at thesame time it is very dif® cult to measure. As shown inTable 1, many workers have developed dimensionlesscorrelations for predicting hf p of the form: Nu = a+bRecPrd , where, for a single isolated particle, the limitingNusselt number usually corresponds to a = 2. Attentionmust be paid to the de® nition of the Reynolds number asdifferent forms of Re are found; and(e) heat transfer coef® cient between the two-phase mixtureand the pipe wall (hw )Ð which has been much less wellstudied than hfp , but could well be as signi® cant1 1 . Thisparameter has usually been estimated by assuming eitherlaminar or turbulent ¯ uid ¯ ow in the pipe, or assumingan arbitrary high value ( , 1000 Wm- 2K- 1 )1 2 .

Another aspect that should be considered in the design isthe contribution of the cooling stage to product sterilization.The centroids of the particles leaving the holding tube willcool at a ® nite rate and some reaction leading to microbiallethality must occur. Conservative approaches tend toignore this contribution. An optimal process design shouldtake into account all contributions to lethality from theheating, holding and cooling stages.

(ii) Particle and Liquid Flow

In a recent review, Lareo et al.6 discussed outstandingissues in the ¯ uid mechanics of solid-liquid food ¯ ows. Themicrobiological lethality delivered to the particles and thedegree to which quality is maintained is a function of theresidence time of the particles in the process as well as thetemperatures. This must be taken into account whenestablishing a continuous heat preservation process. Thefactors affecting particle residence time are:

(a) particle sizeÐ most food products contain a range ofparticle sizes, which in¯ uences particle ¯ ow behaviour.Particle size together with the properties of the liquid carrieraffect the residence time; over the last ten years someworkers have begun to study particle residence timedistributions1 3 ± 1 5 ;(b) velocity pro® le of the carrier liquid within the ¯ owchannelÐ the residence time of a particle will depend on itsradial location within the conveying liquid;(c) particle concentration pro® le which is usually non-uniform with a minimum at the wall;(d) the heat exchanger con® guration, which will affect the¯ ow ® eld and, thus the particle residence time. For example,the ¯ ow pro® le through the barrel of a scraped surface heatexchanger is far more complex than that through a straighttubular heat exchanger1 6 . On the other hand, the presence of

4 BARIGOU et al.


® ttings and especially the presence of multiple turns inholding tubes is likely to prevent the establishment ofa stable velocity pro® le and in consequence affect theparticle residence time distribution. A good process designshould aim at achieving a narrow particle residence timedistribution.

A continuous heat sterilization process usually has twocritical control points1 0 : (i) monitoring of the largest particlein the product as it has the slowest heating rate at its centre,and on which the design of the process must be based; and(ii) estimation of the shortest particle residence time in theprocess. The latter is a far more complex critical controlpoint as particle residence time is poorly understood. Ingeneral, much remains to be learned about the dynamics offood particles conveyed by a viscous liquid in a continuousaseptic process.

(iii) Quality Optimization

The complexity of designing an aseptic process formaximum product quality is similar to that of optimizingmicrobial lethality; both are limited by the lack of knowl-edge and control of particle ¯ ow and the associatedresidence time distribution. When designing for optimumproduct quality, account must be taken of the temperaturehistory in the carrier liquid as well as within the particles.The degree of product quality expected from the process canonly be estimated by integrating these two temperature-timehistories for the phases together with appropriate kineticparameters for quality: few studies1 7 have examined thisimportant problem.

The type and layout of the process poses a further barrierto quality optimization. In a continuous system, ® nalproduct quality will be a direct consequence of the abilityof the process to achieve truly HTST sterilization. Foroptimum product quality, the time to heat the particlecentroids to the required temperature, to hold them at thistemperature, and then to cool them, must be minimized, andall stages of the process must be optimized. The design ofthe equipment affects both particle residence time and heattransfer rates, and hence the ® nal product quality. Appro-priate equipment must, therefore, be chosen with regard foroptimum quality. Increasing consumer demands for betterquality may well stimulate the development of noveldesigns; for example, some solutions have been suggested,such as the ohmic heater in which heat is generated withinparticles by the passage of electric current, obviating theneed for heat transfer during the heating stage (see forexample Zhang and Fryer1 8 ; Fryer1 9 ). This, however, doesnot eliminate the need for effective coolingÐ indeed, sinceparticles can heat faster than the surrounding ¯ uid duringohmic heating, cooling can be more important to productquality here.

At present, a suitable commercial sterilization schedulemust be established experimentally for every food productprocessed. It is likely that food manufacturers will adhere tothis approach in the design of each stage in a heat-hold-cool system. However, without an understanding of thedesign factors, processes will be set so that products areover-processed to ensure safety. The damage to delicateparticulates resulting from over-processing will restrict therange of food products that could bene® t from continuous

aseptic processing technology; the consumer will not payextra unless products are demonstrably better. The FDAhave recently approved an aseptic process involving par-ticulates1 1 8 ; the use of data and models is to enable processeswhich fully exploit the advantages of continuous processing.

This paper reviews the current state-of-the-art in the areaof heat transfer in two phase solid-liquid ¯ ows. Theapplication of existing knowledge to the design ofcontinuous processes for the heat treatment of particulate-bearing food systems is discussed, and research needshighlighted.

2. FLUID-PARTICLE HEAT TRANSFER

The in¯ uence of the surface heat transfer coef® cient, hf p ,on the thermal response of a particle will vary; for largeparticles internal thermal conduction is the controlling (i.e.slowest) process, so that varying hfp does not affect theprocess time. The particle Biot number, Bi, compares theheat transfer rates resulting from external heat transfer andinternal thermal conduction, and thus gives a measure of themechanism which controls the rate of heating. It is de® nedas

Bi =hfpx

ks

(1)

where ks is the particle thermal conductivity, and x is acharacteristic dimension of the particle. For spheres, forwhich x is the radius, if Bi < 0.1 the conductive resistance inthe solid is small and hf p controls the thermal behaviour,whilst if Bi > 10 the convective resistance is small i.e. hf p islarge and heat transfer is conduction controlled2 0 . Betweenthese two values a combination of mechanisms controls heattransfer. In the limit when Bi ! 0, the particle will beisothermal and heat transfer solely controlled by externalfactors; for this case the problem has an exact analyticalsolution known as Newton’ s law of heating,

mpCp

Ap

lnTp - Tf

Ti - Tf= -hfpt, (2)

which gives the temperature change when a particle of massmp , surface area Ap , and speci® c heat capacity Cp , is heatedfrom temperature Ti to Tp , for time t, in a ¯ uid at constanttemperature Tf.

Published research has adopted two different approachesto determine hf p . Both use a transducer particle with aknown Biot number; the convective heat transfer coef® cientcan be estimated from the temperature history of theparticle. The two approaches are to use particles where: (i)Bi = 0, assuming the entire particle volume is at constanttemperature and then evaluating the heat transfer coef® cientfrom equation (2) or its equivalent for non-uniform ¯ uidtemperature; and (ii) Bi Þ 0, in which the transient heatconduction equation (18a) must be solved to obtain hf p . Thelatter approach can be inaccurate, as particle heating ratebecomes increasingly insensitive to hf p as Bi increases, i.e.as thermal conductivity controls, so data for hf p ® tted fromexperiments with high Bi particles may be worse than fromlow Bi experiments.

Data has been commonly expressed in terms of correla-tions relating Reynolds, Prandtl and Nusselt numbers. Thevelocity that controls the interfacial heat transfer will be the

5HEAT TRANSFER IN TWO-PHASE SOLID-LIQUID FOOD FLOWS


6 BARIGOU et al.


Tabl

e1.

Con

vec

tive

hea

ttr

ansf

erco

ef®

cien

tsfo

rpar

ticl

esin

conti

nuous

¯ow

.

Par

ticl

esh

ape

Rey

nold

san

dsi

zeh

fpnum

ber

Exper

imen

tal

Ref

eren

ce

(mm

)C

arri

er¯

uid

(Wm

-2K

-1)

range

Pr

Nu

Corr

elat

ion

met

hod

Bap

tist

aet

al.1

05sp

her

eC

MC

Ther

moco

uple

1.2

7±

2.2

3(0

.10±0.6

0%

)(a

lum

iniu

m)

w/w

)st

atio

nar

y56

±2612

4.1

<R

e p<

636

69

±1810

9.0

±88.3

Nu

s=

2+

(0.0

25

60.0

04)P

r1 /3

sG

r1/2

;2.8

<G

r<

4840

Nu=

Nu

s+

(0.2

06

0.0

6)R

e(0.6

76

0.0

3)

pP

r(0.3

8 60.0

3 )s

rota

ting

67

±1782

0.1

<R

e p<

801

71

±5340

2.5

±60.7

Nu=

Nu

s+

(0.0

81

60.0

4)R

e(0.7

06

0 .04)

pP

r(0.4

26

0.0

6)

s

Nu=

Nu

s+

(0.1

76

0.0

6)R

e(0.7

16

0.0

3)

pP

r(0.4

2 60.0

4 )s

´d

p d t

(0.2

86

0.0

5)

`N

us:

Nuss

elt

num

ber

for

nat

ura

lco

nvec

tion

and

non-N

ewto

nia

n¯

uid

Pr s

:gen

eral

ized

Pra

ndtl

num

ber

for

zero

¯uid

vel

oci

ty,

bas

edon

intr

insi

cvis

cosi

ty.

Man

kad

etal.

36

spher

ew

ater

±292

<R

e p<

2433

±20

±60

Nu=

2+

1.4

1R

e0.4

7p

Ther

moco

uple

15

(copper

)poly

acry

lam

ide

±114

<R

e p<

893

±20

±48

Nu=

2+

3.2

0R

e0.3

8p

(0.5

%w

/w)

gly

cero

l:67%

w/w

±18

<R

e p<

140

±16

±36

Nu=

2+

4.9

8R

e0.3

9p

83%

w/w

±5

<R

e p<

97

±9±

34

Nu=

2+

5.6

7R

e0.4

9p

for

all

¯uid

suse

d:

Nu=

2+

0.9

7R

e0 .45

pP

r0.2

5

Gad

onna

etal

.75sp

her

ew

ater

290±

466

1400

±2300

±4.8

7±

13.4

2N

u=

2+

4.9

8R

e0.2

34

d p d t

1.4

41

Liq

uid

cryst

al9.1

±24.0

(poly

pro

pyle

ne)

346±

1587

2300

±5000

±5.3

3±

45.7

2N

u=

2+

19.3

1R

e0 .279

dp dt

2 .217

290±

1587

2300

±5000

±4.8

7±

45.7

2N

u=

2+

0.0

933R

e0.7

27p

;84

<R

e p<

2300

Nu=

2+

0.2

28R

e0 .639

p

dp dt

0 .514

;84

<R

e p<

2300

Nu=

2+

14.4

4F

r0.7

77p

;0.1

77

<F

r p<

1.7

14

Nu=

2+

129.3

5F

r0.7

12

p

d p dt

1.9

9

;0.1

77

<F

r p<

1.0

42

Kel

lyet

al.

27cu

be:

10.0

±32.0

wat

er±

239

<R

e p<

5012

2.3

±4.3

14

±76

Nu=

0.2

98R

e0 .62

pP

r0.3

6T

her

moco

uple

spher

e:9.5

±30.0

Nu=

2.0

+0.2

06R

e0 .66

pP

r0.3

9(o

pen

chan

nel

cyli

nder

:d

p=

12.7

±¯

ow

)32.5

;l=

12.8

±31.9

(alu

min

ium

)C

MC

±4.5

<R

e p<

177

0.0

2±

0.1

417

±53

Nu=

21.1

3R

e0 .12

pP

r-0 .

076

(0.8

±1.5

%w

/w)

Nu=

2.0

+19.3

6R

e0 .132

pP

r-0.0

8



Tabl

e1.(C

ont

inued

)

Par

ticl

esh

ape

Rey

nold

san

dsi

zeh f

pnum

ber

Exper

imen

tal

Ref

eren

ce(m

m)

Car

rier

¯uid

(Wm

-2K

-1)

range

Pr

Nu

Corr

elat

ion

met

hod

AÊst

roÈm

and

spher

e,cu

be

sili

cone

oil

,110±

450

±±

±N

u=

8+

2.5

3R

e0.5

4;

par

ticl

eat

50

mm

from

axis

Ther

moco

uple

Bar

k48

8±16

star

ch(6

%w

/w)

of

rota

tion.

(sta

tionar

y(l

ead)

Nu=

8+

2.2

5R

e0.6

1;

par

ticl

eat

11

mm

from

axis

par

ticl

ein

aof

rota

tion.

rota

ting

¯uid

)

Bal

asubra

nia

mS

pher

eC

MC

±±

Ther

moco

uple

and

Sas

try

34

12.8

±22.3

(0.2

,0.5

,363

±2010

0.0

058±

798

104.5

±2443

Liq

uid

cryst

al0.8

%w

/w)

1<

Re p

<193

Rel

ativ

evel

oci

ty

Aw

uah

etal.

115

cyli

nder

:C

MC

Ther

moco

uple

d p=

16.0

±23.0

;(0

.5,1%

w/w

)l=

22.3

-40.0

carr

ot

80

±456

±1.2

3±

7.0

2±

Nu=

2.4

5(P

rGr )

0.1

08;

0.1

´10

4<

Gr

<10.9

6´

10

4

pota

to100±

556

±1.2

3±

7.0

2±

Nu=

2.0

2(P

rGr )

0.1

13;

0.1

1´

10

4<

Gr

<11.1

´10

4

Bham

idip

ati

and

cylinder

CM

C±

0.3

10

±1.6

87

1.3

2±

5.1

2±9.2

6N

u=

0.2

7R

e0 .2

pP

r0.3

3R

emote

Sin

gh

106

dp=

10.7

;0.5

±1.2

%w

/w4.7

3´1

04

tem

per

ature

l=

22.6

senso

r

Dam

ayan

dP

ain

76

spher

e19.4

±26.3

aw

ater

990

±2260

8950±

22000

±±

Nu=

2+

0.0

92R

e0 .76

dp d t

1 .221

Mel

ting

poin

tin

dic

ator

19.0

±26.0

b1460±

3346

8000±

24000

±±

Nu=

2+

0.2

25R

e0 .626

Nu=

2+

1.2

21R

e0 .432

Fr0

.163

p

Mat

eria

lw

ith

chan

ge

of

phas

eat

Nu=

2+

Re0

.52

p

d p dt

0.8

8

-128C

aan

d08C

b.

Mw

angi

etal

.35

Spher

egly

cero

l±

563±

6000

3.6

8±

5.5

2±

Lam

inar

:N

u=

0.1

0R

e0 .58P

r0 .33

Mel

ting

poin

t8

±12.7

(19,

21,23%

Turb

ule

nt:

Nu=

0.0

336R

e0.8

0P

r0.3

2in

dic

ator

w/w

)

Zuri

tzet

al.2

8m

ush

room

CM

CT

her

moco

uple

20.6

±28.2

548

±1175

0.1

23

±1.9

41.1

4±

7.2

217.2

±50.2

9N

u=

2+

28.3

7R

e0 .233

Pr0

.143

dp d t

1 .787

(alu

min

ium

)d p

:dia

met

erof

pro

ject

edca

pper

imet

er

8 BARIGOU et al.


Tabl

e1.(C

ont

inue

d)

Par

ticl

esh

ape

Rey

nold

san

dsi

zeh

fpnum

ber

Exper

imen

tal

Ref

eren

ce(m

m)

Car

rier

¯uid

(Wm

-2K

-1)

range

Pr

Nu

Cor

rela

tion

met

hod

Cha

ndar

ana

etal

.69cu

be

wat

er64.7

±107.1

1287.3

±880.7

±±

Nu=

2.0

+0.0

333R

e1.0

8T

her

moco

uple

(silic

one)

star

ch55.6

3±

89.5

1.2

3±27.3

89.5

±376.2

±N

u=

2.0

+0.0

282R

e1.6

Pr0 .

89

(2±

3%

w/w

)

Sas

try

etal

.46

spher

ew

ater

688±

3005

3600

±27300

±±

Nu=

26.8

1+

0.0

0455R

e (d

p/d

t)M

ovin

g

13.3

±23.9

Nu=

6.0

23x1

0-6

Re1 .

79(d

p/d

t)1.7

1F

r-0.6

4th

erm

oco

uple

(alu

min

ium

)N

u=

0.0

46R

e+

41.5

4(d

p/d

t)-

35.6

5F

r-

5.4

2N

u=

0.1

25R

e+

85.6

7(d

p/d

t)-

342.5

Fr

-0.0

144R

e (d

p/d

t)+

0.0

113R

eFr

+500.3

Fr (

d p/d

t)-

0.0

173R

eFr (

d p/d

t)-2

6.5

6.

Fr=

0.3

9-

14.8

3;

dp/d

t=

0.2

618

-0.6

273

Inco

rper

aan

d¯

atpla

te±

±±

±±

Nu=

0.3

32R

e0.5

Pr0

.33

±D

eW

itt10

9

Cha

ndar

ana

etal

.26cu

be

wat

er65.6

7±

107.1

1761

±2144

±±

Nu=

0.8

5R

e0.4

34

Ther

moco

uple

(silic

one)

star

ch55.6

3±

89.5

5.6

±141.8

40.8

±1564

±N

u=

0.5

5R

e0.4

38P

r0 .349

(2±

3%

w/w

)

Zuri

tzet

al.

68m

ush

room

CM

C548±

1175

0.1

1±

2.0

01.0

5±

8.3

6±

Nu=

2+

7.9

464R

e0 .207

95P

r0.1

4413

Ther

moco

uple

(alu

min

ium

)N

u=

22.7

+0.0

36R

e0.4

5P

r0.5

3

Hep

pel

l25

Spher

ew

ater

2180

±7870

5250

±50000

±110.8

±400.2

±B

iolo

gic

al3

star

ch930

30

±±

±5%

w/w

Chu

chott

aworn

107

±±

±±

±±

Nu=

2.0

+0.3

7R

e0.6

1P

r0.5

1±

Whitak

er67

spher

eai

r,oil,w

ater

±3.5

´10

4<

Re p

0.7

1±380

<7 .

6´1

04

Nu=

2.0

+(0

.4R

e1/2

p+

0.0

6R

e2/3

p)P

r0.4

l lp

0 .25

±

Noord

sij

and

spher

ew

ater

±400-1

350

500±

2000

±N

u=

10

+0.6

08R

e1 /2

wP

r1 /3

±R

otte

112

(nic

kel

)in

itia

lly

obta

ined

for

mas

str

ansf

er



Tabl

e1.(C

ont

inued

)

Par

ticl

esh

ape

Rey

nold

san

dsi

zeh f

pnum

ber

Exper

imen

tal

Ref

eren

ce(m

m)

Car

rier

¯uid

(Wm

-2K

-1)

range

Pr

Nu

Corr

elat

ion

met

hod

Ran

zan

dli

quid

dro

pai

r±

1<

Re p

<10

40.6

±400

±N

u=

2.0

+0.6

Re0 .

5p

Pr0 .

33

Evap

ora

tion

Mar

shal

l660.6

±1.1

from

pure

liquid

dro

ps

Kra

mer

s65sp

her

eai

r,oil

,w

ater

±R

e p<

10

50.7

±400

10±

400

Nu=

2.0

+1.3

Pr0

.15+

0.6

6P

r0 .31R

e0 .5

p±

(ste

el)

Willi

ams11

4sp

her

e±

±>

200

±±

Nu=

0.3

7R

e0.6

Pr0

.33

±

Johnst

one

etal.

110

spher

e±

±>

500

±±

Nu=

0.7

14R

e0 .5P

r0 .5

±

Lja

chow

ski1

11

spher

e(m

etal

)ai

r±

200±

30000

±±

Nu=

0.6

1R

e1/2

±sp

her

e(s

teel

,m

ercu

ry)

wat

er±

3500±

15000

±±

Nu=

0.0

85R

e0 .78

±

Vyro

ubow

113

spher

eai

r±

200±

3000

±±

Nu=

0.5

8R

e1/2

Pr1

/3±

(met

al)

FroÈ

szli

ng

108

spher

eai

r±

2±

800

0.6

±2.7

±N

u=

2.0

+0.5

5R

e1/2

Pr1

/3±

(wat

er,

nit

roben

zene,

anil

ine,

nap

hth

alen

e)

relative velocity between particle and ¯ uid (i.e. the slipvelocity); some correlations do not include this but use theoverall velocity instead. However, the slip velocity in realmixtures is often not known and may be dif® cult to de® ne.To de® ne slip as the difference between the mean ¯ uidvelocity and particle velocity is not complete, because of thecomplexity of the ¯ ow and the liquid velocity gradient. Forexample, several workers including Lareo1 5 have found thatparticles in a viscous carrier ¯ uid travel faster than the mean¯ uid velocityÐ this arises because particles congregateinto the centre of the ¯ ow where the liquid is fastest.Experiments in a well-de® ned ¯ ow ® eld may thus givecorrelations in terms of a slip Reynolds number that cannotbe estimated for a real system. Conversely, whilst experi-ments in real ¯ ows give actual hfp data, the ¯ ow ® eld whichproduces them is not known. Figure 1 shows some existingresults for forced convection heat transfer. The data werecalculated from cases where correlations were given interms of the particle-¯ uid slip velocity; hence, this plotexcludes data such as that of Sastry et al.2 1 where thecorrelations were not derived in terms of slip velocity. Allcorrelations are plotted for water at 15 8 C (Pr = 7.425) sothat the correlations may be compared on the same basis.This graph shows that data are scattered over a wide rangeof Nusselt numbers, and there is no clear associationbetween the different sets of data. This may be due to anumber of effects including: (i) experimental inaccuracies,such as the presence of natural convection at low Re; (ii)discrepancies in the real velocity ® eld experienced by theparticle, where the slip quoted is not representative of the¯ ow ® eld around the particle; and (iii) different measure-ment techniques.

2.1. Factors Governing Fluid-Particle Heat Transfer

Results for some of the main investigations on ¯ uid-particle heat transfer are summarized in Table 1, in terms ofcorrelations for Nusselt number. The correlations re¯ ect thefact that the convective heat transfer coef® cient isin¯ uenced by several factors (see for example, Hendrickxet al.2 2 and Stoforos2 3 ):

2.1.1. Fluid viscosityThe effect of viscosity is shown by the difference in hf p

values obtained for low viscous ¯ uids, e.g. water, and moreviscous ¯ uids, e.g. aqueous CMC solution. Lenz and Lund2 4

obtained lower hf p values in a 60% aqueous sucrose solutionthan in water. Heppell2 5 determined hfp by measuring thedestruction of Bacillus stearothermophilus in calciumalginate particles (3.4 mm dia.) ¯ owing through a holdingtube. For a 5% starch solution, heat transfer was so low thattwice the residence time (and thus holding tube length) wasneeded to achieve the sterilization effect predicted for thecase of in® nite hf p , i.e. zero resistance to interfacial heattransfer. Chandarana et al.2 6 measured greater hf p values forwater (65.67±107.11Wm- 2 K- 1 ) ¯ owing past siliconecubes than for starch solutions (55.63±89.5 Wm- 2 K- 1 )over the same temperature range (133±1458 C). Kelly etal.2 7 measured hf p using aluminium cubes (10.0±32.0mm),spheres (9.5±30.0mm dia.) and cylinders (12.7±32.5 mmdia.; 12.8±31.9 mm length) in open channel ¯ ow, usingwater and aqueous CMC solutions (0.8, 1.0, 1.2, 1.5% w/w)at 558 C, ¯ owing at velocities 8.7-41.2mm/s. The particleswere stationary and had a thermocouple imbedded at theirgeometric centre to monitor temperature. An increase in¯ uid viscosity caused a signi® cant decrease in hf p for all¯ ow velocities. Other researchers2 8 ± 3 1 also reported that hf p

increases with decreasing CMC concentration.Results can be interpreted in terms of the particle

Reynolds number, preferably slip Reynolds number, ratherthan just viscosity. Viscosity has a number of effects: it willin¯ uence the ¯ uid velocity pro® le and ¯ ow regime withinthe tube as well as the drag force on the particle, and thus theparticle-liquid slip.

2.1.2. Particle slip velocityDifferences in solid-¯ uid translational velocities will

increase interphase heat transfer rates; the relative linear androtational ¯ uid motion over the particle gives increasedforced convection, and thus enhanced heat transfer. Typicalslip velocities occurring in food ¯ ows are in the region1 5

of 0.1 to 1 cms- 1 , and can affect heat transfer signi® cantly.For example, a single food particle of diameter 10 mm witha slip velocity of 1 cms- 1 would, in water, have a Nusseltnumber of 45, as estimated from the Ranz-Marshallcorrelation, equation (7). This is a considerable enhance-ment over the value Nu = 2, i.e. no slip, used in manyexisting theoretical models. Particle slip velocity will besimilar to the sedimentation velocity.

The information reported in the literature suggests thatstationary particles in a ¯ ow usually have higher surfaceheat transfer coef® cients than particles moving with the¯ uid. If, however, the relative velocity between particle and¯ uid is the same, the heat transfer coef® cient for a stationaryparticle in a moving ¯ uid should be similar to that for aparticle moving within its carrier ¯ uid. Chang and Toledo3 3

reported an increase in hf p from 239 to 303 Wm- 2K- 1 whenthe slip velocity increased from 0.38 to 0.86cms- 1 , forstationary potato cubes in water, and a non-Newtoniansolution with 35% sucrose. The same trend was observed byBalasubramaniam and Sastry3 4 for moving aluminiumspheres in continuous tube ¯ ow using CMC solutions.They used three different techniques to measure hf p : movingthermocouple, relative velocity, and liquid crystal; these aredescribed later. Other workers have reported similar

10 BARIGOU et al.


Figure 1. Variation of Nusselt number with slip Reynolds number forexisting experimental data.

experimental ® ndings (see for example, Sastry et al.2 1 ;Zuritz et al.2 8 ; Mwangi et al.3 5 ; Mankad et al.3 6 ). Datacorrelations are thus often expressed in terms of the slipReynolds number, although this may be dif® cult todetermine in practice.

The importance of slip velocity was shown by Mankadet al.3 7 , and Mankad and Fryer3 8 , who used a one-dimensional theoretical model to compute sterilizer lengthsfor various solid-liquid slip velocities and delivered solidsconcentration for a ® xed ¯ ow rate. Changing the slipvelocity between solid and liquid, and thus hf p , affected therequired sterilizer length greatly, highlighting the need toincorporate slip effects in process design. The heating tubelength was found to be a strong function of slip velocity.

2.1.3. Particle concentration and particle-particleinteractions

Particle concentration and particle-particle interactionsare the factors with the least understood effects on hf p .Existing results for the canning process are often irreconcil-able; for example, the experiments of Lenz and Lund2 4

showed a decrease in hf p with an increase in particle loading,whereas Hassan3 9 observed the opposite.

The in¯ uence of particle-particle interactions on ¯ uid-particle heat transfer has not been studied in detail by eitherfood scientists or process engineers. Two effects may occur:

(i) localÐ particle-particle interactions can disturb the ¯ uidstreamlines locally, which will affect the ¯ ow ® eld aroundsuccessive particles and, depending on the Reynolds number,may enhance the convective heat transfer coef® cient;(ii) overallÐ where particle concentration is high, ¯ uidchannelling through particles can occur and can signi® -cantly affect the heat transfer coef® cient in sections of themixture; in places higher, in places lower (see, for example,Agarwal4 0 and Paterson et al.4 1 ).

Dutta and Sastry3 2 studied the velocity distribution ofpolystyrene spheres (9.5 mm dia., 1044.5kgm- 3 ) in CMCsolutions. Particle motion was videotaped; it was seen thatas well as particle collision, particle-particle interactionsalso occur via a hydrodynamic attraction-repulsion due topressure changes in the inter-particle gap. This phenomenonwas previously observed by Davis et al.4 2 who showed theexistence of a signi® cant pressure pulse between approach-ing particles in a liquid, which may cause particles todeform before collision. Dutta and Sastry3 2 also observedthat particle concentration affected the interactions asparticle motion became restricted at high solids fractions,and that particles can change velocity as a result ofinteraction with others. They did not, however, study theeffect of particle interactions on hfp .

Mwangi et al.3 5 measured hf p in a holding tube usinga melting point indicator technique with polymethyl-methacrylate transducer particles (dp = 8.0-12.7mm; q p =1021.36±1100.74kgm- 3 ) in aqueous glycerine (19, 21,23% w/w; q f =1020-1100kgm- 3 ). The value of hf p

increased by 80 to 200% when the solids fraction wasincreased from a single particle to 3.2% w/w. This increasewas attributed to the disturbance of the ¯ ow ® eld around thetransducer particle caused by the presence of other particles,especially at high Reynolds numbers, which led to thethinning of the boundary layer around the particle.

The effect of wake formation behind a particle on hf p was

studied by Kelly et al.2 7 and Mankad1 1 . Kelly et al.2 7

studied the effect of a sphere placed at a distance l upstreamfrom a spherical transducer particle in water (T = 58 8 C;uf =0.0275ms- 1 ). The effect of varying the separationdistance l and the sizes of the two particles on hf p wasinvestigated. When both particles were of the same size(11.9 mm dia.), or the particle upstream of the transducerparticle was smaller (18.8mm upstream of 25.1mm dia.particle or 25.1 mm upstream of 30.0 mm dia. particle), as lincreased from zero to 4 mm hf p increased from that for theisolated particle to a maximum, falling off gradually back toits initial value at about l =100mm. When the particleupstream was larger (25.1mm upstream of 11.9mm dia.particle or 25.1 mm upstream of 9.5 mm dia. particle), hf p

decreased slightly as l increased to 4 mm then peaked to amaximum at l =50 mm, falling back to its initial value atl =100mm. Particle wake effects are thus complex anddepend strongly on the relative sizes of the interactingparticles.

Little information exists on how much heat transfer canbe enhanced through particle-particle interactions in con-tinuous ¯ ows. Multiple particle systems, however, havebeen widely studied in the ¯ uidization literature (see, forexample, Agarwal4 0 ). Three main con® gurations are ofimportance: particulate ¯ uidization where particles are notin contact with each other and their distribution is due tosystem hydrodynamics; packed beds where particles are inmutual contact and their orientation, random or regular, isset by the process; and distented beds where the particles are® xed in a regular orientation but are not necessarily incontact. For constant ¯ ow rate, heat transfer has been foundto increase in distented beds as voidage decreases4 0 . At allbut very low Reynolds numbers, heat transfer coef® cients inpacked beds, which will have the most inter-particleinteractions, are considerably higher (approximately afactor of 2.5-3) than heat transfer coef® cients for singleparticles1 1 ,4 3 . Particle interactions will thus contributesigni® cantly to heating and cooling rates and the effect ofsolids fractions should be taken into consideration inprocess design.

2.1.4. Particle rotationThe slip velocity between particle and ¯ uid will not be

due to translational ¯ ow alone, but will also result from anyparticle rotation. An estimate of the possible enhancementof heat transfer can be made by assuming that the rate ofrotation about the z axis, x , of food particles ¯ owing in apipe is approximately equal to the ¯ uid vorticity (assuggested in experiments by Stockman4 4 ), i.e.,

x = 1

2

¶ufy

¶x - ¶ufx

¶y, (3)

where uf x and uf y are the ¯ uid velocities in the x and ydirections, respectively. For fully developed laminar ¯ ow

¶ufy / ¶x is zero, so the ¯ uid velocity distribution in the xyplane is given by:

ufx = ufmax1 -

r2t

R2t

, (4)

where rt represents radial position within the tube, Rt is the



tube radius, and ufm a x is the maximum ¯ uid velocity on thecentreline. Combining equations (3) and (4) leads to:

x = -ufmax

rt

R2t

, (5)

where the negative sign indicates that the rotation isclockwise.

In food processing, a typical mixture in a tube of radius0.05 m would have a maximum centreline velocity of about0.1 ms- 1 . Therefore, the maximum particle angular velocity(i.e. at rt =Rt) will be 2 s- 1 . Using the expression derivedby Krieth et al.4 5 (equation (10)) the maximum Nusseltnumber around a typical food particle (e.g. a carrot inwater), of diameter 10 mm, would be equal to 13.

The magnitude of this Nusselt number shows that particlerotation can appreciably enhance particulate heat transfer.However, the analysis of Stockman4 4 was for a singleparticle ¯ owing in a tube. The observations of Lareo1 5 showthat, for solids concentrations above 5%, particles near thetube wall where maximum rotational velocities are expectedto occur, appear not to rotate; due to the ¯ ow deformationcaused by high particle concentrations. This suggests that itis not safe to assume heat transfer is enhanced by particlerotation in food ¯ ows; more work is needed before theeffects can be con® dently incorporated in process design.

2.1.5. Particle size: wall-particle effectsParticle size is an important parameter which in¯ uences

hf p : as the particle-tube diameter ratio increases, the effectof the wall will become increasingly important. Particle-¯ uid heat transfer coef® cients have normally been measuredin a uniform ¯ ow ® eld: the velocity gradient near a wall willresult in distribution of relative velocity across the particle.Con¯ icting results for the effect of this have been reportedin the literature. Sastry et al.4 6 , using a moving thermo-couple attached to a metal sphere in water, obtained resultswhich showed an increasing hf p with increasing particle-tube diameter ratio. More recently, however, Balasubrama-niam and Sastry3 4 detected a much less signi® cant trend inCMC solutions. The discrepancy between the ® ndings wasattributed to the difference in ¯ ow regimes studied(turbulent for water, Re =7300±43600; and laminar forCMC, Re= 0.0058±798) and the associated effects ofparticle radial location. Particle radial location was notcontrolled in either study but would be expected to mattermore in laminar ¯ ow, where the velocity distribution ishighly non-uniform. This data is dif® cult to reconcile withthat of (i) Mwangi et al.3 5 who, working with freely movingparticles containing a melting point indicator, reportedconsiderable increases in hf p (58.3-1301.3Wm- 2 K- 1 ) withincreasing particle size (8.0±12.7 mm) in glycerine solu-tions (Rep =73.1-369.4) within a holding tube of 50.8 mmdia.; and (ii) Kelly et al.2 7 who detected a decrease in hf p

with increasing particle size in open channel ¯ ow.One problem with interpreting the data is the dif® culty in

determining relative velocities. This may explain observa-tions such as those of Zuritz et al.2 8 who used three differentsizes (20.6±28.2 mm) of mushroom-shaped aluminiumparticles in a CMC solution, and reported an increase inhf p (548-700Wm- 2K- 1 ) with particle size. However,Chandarana et al.2 6 using various sizes of diced carrotparticles and silicone cubes in water, found that hf p

decreased with increasing cube size. However, whilst

Zuritz et al.2 8 experimented on a narrow tube, Chandaranaet al.2 6 used a pipe of very large section. The in¯ uence ofparticle size on hf p will be due in part to the rise in therelative velocity between ¯ uid and particle, due to areduction in the effective ¯ ow area in the presence oflarger particles.

2.2. Experimental Studies of Food Flows

The above discussion shows that the literature is contra-dictory in places. The discrepancies and inconsistenciescould be due to experimental error, but it is more likely thatthey result from the different experimental techniques usedand the dif® culty in de® ning and relating the experimentalconditions under which they were used. Research usingstandardized experimental approaches is required to pro-duce data from well-characterized situations.

It must be noted that in this paper data has been quoted asreported in the original references. The accuracy of some ofthe results is doubtful (too many signi® cant ® gures) giventhe dif® culties associated with the experimental techniques.

Determination of the particle-¯ uid heat transfer coef® -cient requires information on particle temperature historyunder realistic conditions: correlating the data requiresinformation on the ¯ ow ® eld around the particles. Thedif® culty of monitoring, either directly or indirectly, thetemperature of a moving particle without interfering with itsmotion, has led a number of investigators to resort to the useof stationary particles in determining hf p (such as Chang andToledo3 3 ; Chandarana et al.4 7 , 6 9 ; Zuritz et al.2 8 ; AÊstroÈ m andBark4 8 and Mankad et al.3 6 ). Because of the dif® culty inmeasuring the temperature inside a food particle ¯ owing ina continuous process, some food manufacturers also resortto the use of ¯ uid-particle heat transfer simulation systemswhich measure temperature transients inside a stationaryparticle. For example, Dignan et al.4 9 cite a manufacturer ofaseptic processing equipment that offers such a simulationsystem (the Continuous Process Simulator, Cherry-BurrellCorp, Cedar Rapids, Iowa). In this type of simulator, theparticle is ® xed on the end of a temperature sensor, allowedto equilibrate to a uniform initial temperature, and thenheated by a continuous hot stream of the product liquidphase. The temperature transient is then used to calculatethe F-value for the product. In the test, care is taken to avoidthermal channelling effects, i.e. allowing heat to traveldown the temperature probe to the centre of the particle. Therate of heating will be a function of hf p , and thus the ¯ uidvelocity; any simulator must, therefore, allow accuratemeasurement and control of this velocity.

The true relative velocity between particle and ¯ uid in thereal processing system must be used in the test. This isdif® cult, both because particles will travel at differentvelocities in a real system, and because the contribution ofparticle rotational motion is not usually known. Conse-quently, when applying the information gathered from asimulation test to the design of a process, a conservativeapproach, i.e. an unrealistically low hfp , is usually adopted.Simulators with stationary particles can provide valuabledata that can be used to validate mathematical models orcheck measurements from other experimental techniques;however, the experimental set-up does not take into accountthe particle dynamics of a real solid-liquid ¯ ow. A numberof different experimental approaches have been developed

12 BARIGOU et al.


for determining hfp between a moving particle and its carrier¯ uid; these are discussed below.

2.2.1. Biological or biochemical methodsBiological or biochemical methods of assessing the

effectiveness of a continuous heat treatment process usebacterial spores or biochemicals which are encapsulatedinside a carrier. The inoculated carrier is introduced in the¯ ow, and after the heat treatment, the remaining concentra-tion is determined. The principle has been exploited indifferent ways to determine the heat transfer coef® cient,albeit indirectly.

2.2.1.1. Particles inoculated with sporesThe `biological thermocouple’ system has been used for

the last 30 years. Bacterial spores are used to monitor thelethality of the process. These are encapsulated in a carrier,so the spores do not come into direct contact with the foodproduct. Measurements are thus not affected by factors suchas pH, oxidation/reduction, and food nutrient level. Bymeasuring the number of surviving spores at the end of aprocess, the sterilizing ef® ciency of a commercial heatpreservation food process can be determined.

This concept has been used successfully to measure thelethality delivered to cans by P¯ ug and coworkers5 0 ± 5 6 ; herethe biological indicator units were rods of 2 in length and0.25 in diameter. The technique has been extended tocontinuous ¯ ows by Hersom and Shore5 7 , who used 5 mmdiameter glass bulbs, carrying a known number of spores,inserted inside food particles to measure the heat treatmentdelivered to the particle centres in a continuous system.These type of units have a number of advantages: (i) thelocation of the spores in the particle is precisely known, i.e.they are located near the centre of the particle; (ii) when abulb is recovered it is clear that all the spores have beenrecovered; (iii) the calibration procedure gives an accurateexperimental relationship between the number of survivingspores and the F-value, thus obviating the need to measurethe exact inoculum level (N0 ) and thermal destruction rates(D-values). When a large number of samples are assessed anaccurate estimate of the F-value delivered to the centre ofthe particles can be obtained.

A related approach has been used for many years to assessbiologically the sterilization of retorted low-acid cannedfood products. This involves inoculating the geometriccentre of a number of actual food particles with heatresistant bacterial spores. It has been used in continuous¯ ows; examples include Bacillus stearothermophilus incalcium alginate particles2 5 , and PA3679 in turkey cubes5 8 .The inoculated particles are recovered at the end of theprocess, the number of surviving spores is evaluated througha count-reductionprocedure, and the sterilization of the heattreatment is determined. In canning, the concept is simpleand can be implemented readily. It is much more dif® cult todo in continuously ¯ owing systems, the major obstaclesbeing: (i) a large number of particles is required if theprocess is to be reasonably assessed; (ii) the particles mustmaintain their integrity throughout the system; (iii) ideally,the inoculated particles should be recovered at exit from theholding tube and instantly cooled to mimic the possible caseof a particle breaking up at the beginning of the coolingphase. Some data for heat transfer coef® cients was obtainedby Heppell2 5 using this method. These results are probably

not accurate, due to problems in getting an accurate count,but it represents a very valuable attempt.

This method needs calibration data for the number ofviable spores in the food product as a function of F, againstwhich the actual measurements can be evaluated. These canonly be obtained through quantitative studies of sporesin the product. Dignan et al.4 9 have discussed the prob-lems encountered in direct inoculation of food particles tostudy continuous ¯ ows; the accuracy of the method dependson a number of key factors including: (i) calibration must beconducted under identical conditions of micro-organismdestruction as for the tests of the actual process; (ii) theexact inoculum level for the particles (N0 ) must bedetermined as some spores will leak out of the particles;(iii) the residence times of the inoculated particles must berepresentative of the fastest particle in the system; (iv) thesize of the spore carrier particles must be representative ofthe largest particle size in the process, and they mustmaintain their full size during the test programme. Thespatial distribution of the bacterial micro-organism withinthe particles must also be known; if spores are distributedthroughout the particle the result will be an integratedsterilization value rather than the sterilization value for theslowest heating point.

2.2.1.2. Simulated food particles inoculated with bacterialspores

The problems associated with the use of biologicallyinoculated food particles to validate continuous sterilizationprocesses has led some researchers to use simulatedparticles as biological indicators. For example, Hunter5 9

used Bacillus anthracis inside polymethyl-methacrylatespheres. Spores in this material, however, are subjected todry heat destruction, so that the death rate becomes afunction of the moisture content of the plastic. Dallyn et al.6 0

who used inoculated alginate-gel particles found that sporesin alginate systems are stable for long periods of time andyield reproducible results.

Most of the dif® culties associated with the use ofinoculated food particles are also present when usingsimulated particles; the advantages of using simulatedparticles being that particle size is uniform and controlled,and the inoculum level can be uniform throughout theparticle. Such systems have been widely used by a numberof groups. Kim et al.6 1 ,6 2 describe the use of chemically andmicrobiologically inoculated particles to examine the APVohmic heating process for sterilization of high-solidsfraction foods. They demonstrated the success of thetechnique but did not attempt to obtain heat transfercoef® cients, since, for validating an individual process, itis necessary only to show that the design F-value is realized.

2.2.1.3. Time-temperature integratorsA time-temperature indicator (TTI) is a small device that

contains a microbiological or enzymic agent which willundergo an irreversible time-temperature dependent changewhen exposed to a heat treatment (for example Hendrickxet al.6 3 ; van Loey et al.6 4 ). All TTIs require propercalibration and testing under conditions representative ofthe real process before experimentation2 2 . They are noteasily adaptable to real processing systems as extensiveexperimentation is required to calibrate them under differentprocessing conditions.



Biological methods are simple in principle but ratherintricate in practice. They do not require transparent carriersand can, therefore, be used under opaque ¯ ow conditions.Successful implementation, however, requires a correcttechnological approach, a well-controlled environment, andaccurate quantitative microbiology. These techniques canbe successfully used to prove that a process works,but unless the whole temperature-time history of the particleis recorded, they will not be suitable for measurements ofhf p .

2.2.2. Relative velocity methodIn this method, the relative velocity between a transducer

particle and the carrier ¯ uid is measured from thevideotaped motion of ¯ uid tracers and the particle. Thetechnique has recently been employed by Balasubramaniamand Sastry3 4 . Fine polystyrene particles ( , 1 mm) were usedto trace the ¯ ow ® eld of CMC solutions (0.2, 0.5, 0.8%conc) around a moving single aluminum particle (12.8, 16.0,22.3 mm) in a glass tube (50.8 mm dia, 1.12 m length)inclined upwards at 2.08%. The motion of the particle wasvideotaped, and particle and tracer positions located usingsquare grids drawn over the tube. The slip velocity atselected times was determined from the videotaped imagesby following the passage of a selected tracer particle overthe test particle. Estimates of the surface heat transfercoef® cient can then be obtained from standard correlationsavailable in the literature (see Table 1), such as:

Kramers65:

Nu = 2.0 + 1.3Pr0.15 + 0.66Pr0.31Re0.5p , (6)

applicable over the range: Rep < 105; 0.7 < Pr < 400;

Ranz and Marshall66:

Nu = 2.0 + 0.6Re0.5p Pr0.33, (7)

applicable over the range: 1 < Rep < 104; 0.6 < Pr < 400;

Whitaker67:

Nu = 2.0 + 0.4Re1 / 2p + 0.06Re2 / 3

p Pr0.4 l

l p

0.25

, (8)

applicable over the range: 3.5 ´104 < Rep < 7.6 ´104;0.71 < Pr < 380; 1.0 < l / l p < 3.2.

where l and l p are the ¯ uid viscosities evaluated at themean bulk temperature, and at the particle-¯ uid interfacetemperature, respectively. The limiting Nusselt number, forthermal conduction at zero particle Reynolds number, canbe shown theoretically to be 2.0.

Balasubramaniam and Sastry34 found that, even thoughthe mean ¯ uid and particle velocities were similar,signi® cant local slip velocities in the range 0.02±0.19 ms-1 were measured. Observation of the ¯ ow ® eldaround the particle also revealed the existence of thefollowing patterns: (a) translation, (b) rotation, (c) forma-tion of vortices, and (d) particle tumbling in a radialdirection. Equations (6)±(8) were used to calculate hfp usingthe measured slip values. The Ranz-Marshall correlation,equation (7), yielded the lowest hfp values and the Whitakercorrelation, equation (8) yielded the highest. The resultsshowed hfp values considerably higher than those predictedfrom zero slip velocity.

The following techniques rely on the direct determinationof the particle temperature history.

2.2.3 Thermocouple methodHere, a thermocouple embedded inside a transducer

particle records the temperature history at its centre. Thethermocouple method was ® rst used with stationaryparticles (such as Zuritz et al.68; Chang and Toledo33;Chandarana et al.69; Zuritz et al.28; AÊstroÈ m and Bark48;Awuah et al.70 and Mankad et al.36). Zuritz et al.68 used astatic mushroom-shaped metal particle inside a tube throughwhich a non-Newtonian liquid was pumped at low rates.Estimates of hfp in the range 548±1175 Wm-2K-1 weredetermined from temperature transients within the particle,and the data were correlated in terms of Nusselt, Reynoldsand Prandtl numbers. These results, however, are notrepresentativeof particledynamicsunder real ¯ ow conditions.

To investigate the effects of the ¯ ow ® eld around aparticle, AÊstroÈ m and Bark48 immersed spherical/cubic (8±17 mm) test particles of lead, polyacrylamide, calciumalginate, and turnip inside a 200 mm diameter ¯ askcontaining 10 litres of either silicone oil or a 6% starchsolution. The temperature of the transducer particle and ofthe ¯ uid 50 mm away from the particle was monitored.From these measurements, hfp was determined (i) directlyfrom Newton’ s law of heating for particles for whichBi < 0.1, or (ii) for particles with low thermal diffusivities,by matching the theoretical solution of the heat conductionequation with experiment using different hfp values.Experiments were carried out with the ¯ uid at rest as wellas in motion; ¯ uid motion was created either by recirculat-ing the test liquid through a pump or by rotating the ¯ ask atdifferent speeds with the particle position at different radiiin the liquid bath. The effects of the ¯ ow ® eld around theparticle, particle size and shape, particle and ¯ uid thermalproperties, and ¯ uid viscosity were investigated.

Both natural and forced convective ¯ uid motion wasfound to affect hfp strongly. The free convective ¯ ow due tothe introduction of a cold particle into a static liquid gave hfp

values in the initial phase of the temperature transient, 2 to 5times higher than during the rest of the experiment. Heattransfer coef® cients found from the middle and the endperiods of the transient were very similar indicating that theeffect of free convection became small. In contrast, whenliquid was circulated around the particle the maximum hfp

due to free convection was experienced at the end of theheating curve. Initially, heat transfer between the coldparticle and the hot liquid is retarded by a cool interfacialliquid layer of high viscosity, leading to a lower hfp. As theparticle heats, this surface layer heats, leading to themagnitude of hfp being determined by the ability of thesurrounding ¯ uid to continuously supply the particle surfacewith hot ¯ uid. When a lead sphere or cube was located at thecentre of the rotating vessel, i.e. exposed to ¯ uid in purerotation, hfp was similar to that obtained in stagnant liquid( , 100 Wm-2K-1). As the particle was moved 11 mm awayfrom the centre, where the ¯ ow ® eld could be described aseccentric rotation, hfp increased in the range 100±170 Wm-2K-1, as the rotational velocity was increasedfrom 0 to 60 rpm; at 60 rpm the slip velocity was0.069 ms-1. At a position 50 mm from the centre, where¯ uid motion is mainly translational, the ability to renew hot¯ uid at the particle surface was considerably higher which

14 BARIGOU et al.


led to enhanced hfp values in the range 110±450 Wm-2K-1,as rotation increased from 0 to 60 rpm; at 60 rpm, the slipvelocity was 0.31ms-1 which is representative of ¯ uid ¯ owpast a sedimented particle in a process plant.

Extensive static-particle experiments were conducted byMankad1 1 and Mankad et al.3 6 using a 15 mm static hollowcopper sphere to measure the ¯ uid-particle heat transfercoef® cient. The sphere consisted of a thin copper shell(0.5 mm) and contained a germanium resistance heater. TwoK-type thermocouples measured the surface temperature;one at the front stagnationpoint, and one at 90 degrees to thefront stagnation point, with the aim of monitoring the spheresurface temperature to detect any variations in heat transfercoef® cient with radial position. The particle was used in a¯ ow loop measuring 2.5 m in total length, and 0.1 m indiameter; to ensure a smooth steady ¯ ow, a ¯ ow calmingsection consisting of a bundle of narrow capillaries was usedupstream to give a ¯ at ¯ uid velocity pro® le. This simpli® esestimation of the ¯ uid velocity at the particle face.

Experiments were conducted with water (Rep =292±2433), PAA (Rep =114±893), and Glycerol solutions(Rep =5±140), over a ¯ uid temperature range 13-18 8 C.The solid surface temperature varied between 23-35 8 C.The work consisted of: single particle experiments wherethe particle was positioned in the centre of the tube; two-particle experiments where the heat transfer coef® cient ofthe transducer particle was measured at varying distancesbehind another similar particle in the tube to investigateparticle wake effects; packed bed experiments where theheat transfer coef® cient was measured for a sphere in a closepacked arrangement of spheres; and variable voidage bedexperiments to simulate real continuously ¯ owing foodmixtures.

For a single isolated particle the results were in goodagreement with the predictions of the Ranz-Marshallcorrelation (equation (7)). In the two-particle experimentNu values were only slightly enhanced ( , 10%) when theinter-particle distance was increased from 10 to 50 mm. Thisenhancement was attributed to the reduction in boundarylayer thickness around the transducer particle; eddiesgenerated by the upstream particle, superimposed on the¯ ow ® eld around the transducer, give mixing that reducesthe ¯ uid boundary layer thickness and thus resistance to heattransfer. The Ranz-Marshall correlation was able to predicthf p for the two-particle system, but only outside the laminar-turbulent transitional ¯ ow regime (Rep < 500 6 250). Whenthe transducer particle was located within a randomlypacked bed of hollow polypropylene spheres of the samediameter, the Nu values, for similar Rep values, were onaverage 2.3 times higher than for an isolated particle. Theenhancement was small at low Re; satisfactory agreementwith the Ranz-Marshall correlation was obtained for Re lessthan about 100. At high Re, eddy formation and recircula-tion caused by adjacent particles may have been responsiblefor the greater increase in heat transfer observed. Signi® cantchannelling can also occur in randomly packed beds wherethere are large variations in local voidage, and may increaseor decrease local heat transfer rates. Channelling effects alsoincrease with tube Reynolds number. Experiments withvariable voidage beds involved locating the transducerparticle in the centre of a partially packed arrangement of15 mm polypropylene spheres. The sphere spacing wasvaried to give spatially uniform packing voidages of 70, 80,

and 90%. Heat transfer to the particle was found to increasewith decreasing voidage. This increase was dependent onthe Reynolds numbers (tube or particle), with a highdependence for water (high Re’ s) and a low dependence forglycerol (low Re’ s). Again, for any bed voidage, the Ranz-Marshall correlation was only found to be suitable at lowReynolds numbers ( , Re < 100).

The thermocouplemethod has also been used with movingparticles2 1 ,3 4 ,4 6 . The approach was introduced by Sastry etal.2 1 ,4 6 ; a thermocouple was attached to a hollow aluminiumsphere (13.3±23.9mm dia), with a wall thickness which gaveit a density similar to that of a food particle (1006kgm- 3 ).The design was chosen to allow Newton’ s law of heating(equation (2)) to be used whilst also allowing a more realisticsimulation of the dynamics of a real food particle.

The temperature history of the particle was determinedwhilst moving it in the ¯ uid with the same speed as that ofa free particle. In the studies of Sastry et al.2 1 ,4 6 , themean velocity of the particle without the thermocoupleattached was measured using two photoelectric sensorslocated at either end of the ¯ ow section. When thethermocouple was attached to the particle, the thermocouplewire was pulled from the downstream end at the samevelocity as that of a free particle. The initial stage where theparticle undergoes an acceleration from zero velocity uponintroduction into the ¯ ow to the desired velocity wasomitted from the analysis, as the slip velocity during thisstage is very high and is not truly representative of real ¯ owconditions in holding tubes. Tests were conducted in ¯ owsections 0.4-1.3 m long and for ¯ ows in the range 2.69-6.68 ´10-4 m3s-1. The method was modi® ed by Balasu-bramaniam and Sastry3 4 so that the transducer particle wasnot pulled from the downstream end, but instead wasintroduced from upstream. This reduced the time required tomatch the velocity of the thermocouple-attached particle tothat of a free particle.

The stationary thermocouple method has the advantage ofbeing relatively simple and can be used with opaque ¯ uids.However, the technique will underestimate hfp because the¯ ow ® eld around the transducer particle is not fullyrepresentative of the true ¯ ow ® eld surrounding a freelymoving particle. One interesting recent modi® cation isClement et al.7 1 who use a stationary particle in which a heatpipe is inserted; the total heat ¯ ow to the particle can then bemeasured. The ingenious moving thermocouple method asused by Sastry and coworkers is more complicated than thestationary particle method but allows the effects of ¯ ow tobe estimated. One problem is that the thermocouple wire islikely to interfere with the particle dynamics and, inparticular, restrain its rotational motion: presumably themethod will still underestimate the true value. Neithermethod gives the particle its total freedom: the next sectionsconsider possible alternatives in which the particle ¯ owsfreely through the tube.

2.2.4. Liquid crystal methodStoforos and Merson7 2 , and subsequently Balasubrama-

niam and Sastry3 4 ,7 3 , and Zitoun and Sastry3 0 , have used amethod whereby a free moving particle is coated with aliquid crystal. The liquid crystal, being heat sensitive,changes colour gradually and reversibly with temperature asa result of molecular structure rearrangement. The colour-temperature response of the liquid crystal can be calibrated,



and the particle temperature history is inferred either fromvisual colour determination, or (more accurately) fromimage analysis of the videotaped colour changes. Balasu-bramaniam and Sastry7 4 have reviewed the use of thermo-chromic liquid crystals in heat transfer studies and theirapplications to food processing research.

Stoforos and Merson7 2 used the method to measure hf p forspherical particles in axially rotating cylinders. Theyreported a value of 2326 Wm- 2 K- 1 for 25.4 mm Te¯ onparticles in water in a cylinder rotating at 102 rpm. Usingthis technique, Balasubramaniam and Sastry3 4 measuredvalues of 857-2010Wm- 2 K- 1 for particles in CMCsolutions in tube ¯ ow (see Table 1), and in scraped surfaceheat exchangers7 3 . The same technique was employed byZitoun and Sastry3 0 to determine hfp values in the range 551±887Wm- 2K- 1 for aluminium cubes in CMC under variousexperimental conditions. Effects of ¯ owrate, viscosity,particle size, and radial location were investigated.

Gadonna et al.7 5 have further developed the technique byusing a transducer particle made up of two concentric spheres,the core of the particle was a polypropylene sphere precoatedwith a black paint and then coated with a liquid-crystal; thiswas subsequently coated with a layer of transparent epoxyresin of known thermal diffusivity. The coating protects theliquid crystal from the ¯ uid, frictional wear and ultravioletlight; in addition,as the crystal is not in direct contact with the¯ uid in the pipe, it changed colour progressively and muchmore slowly, thus enabling easy detection of such colourchanges. Thermal equilibration of the test particle wasachieved in a thermo¯ ask before it was introduced in the¯ ow. The procedure was based on visual determination ofchanges in the liquid-crystal colour with temperature.

Tap water was pumped through a horizontal tube (3.10mlength and 0.042m dia) under the conditions: 1225 <Re < 5434;0.216 < dp / dt < 0.571;1.016 < s < 1.050.Experi-mentaldata for single particles were used with an inverse heatconduction model to evaluate hfp. The results showed clearlythat particle ¯ ow behaviour, and in particular radial location,slip velocity, and rotational motion had a predominant role indetermining hfp. Particle-tube diameter ratio was also foundto be an important parameter. Correlations for predicting hfp

under the various ¯ ow regimes investigated were proposed(see Table 1).

The liquid crystal technique can provide useful informa-tion, if properly used, but does have limitations. Colourcalibration is dif® cult; and it must be emphasized that eachcalibration curve between temperature and Hue value isunique as colour determination depends both on theequipment and the environment in which it is used (differentlightening chambers, light bulbs, dust formation on lightbulbs, image processing system etc.). Furthermore, themethod can only be used with optically transparent liquids,and cannot be used under UHT conditions since liquidcrystals cannot be used above about 1158 C.

2.2.5. Melting point indicator methodMwangi et al.35 used an approach inspired by the liquid-

crystal method. A polymeric material that changes colour ata speci® c temperature was located inside hollow transparentparticles, polymethyl-methacrylate spheres 8, 9.6, and12.7 mm diameter. Colour indicators of different meltingpoints were placed inside the particles. Aqueous glycerinesolutions were used as the suspending ¯ uid. The time for the

surface of the indicator inside the particle to reach itsmelting point and a colour change to be visually observedwas recorded. The particles containing the indicator couldnot be reused as the colour change was irreversible.

Convective heat transfer coef® cients in the range 58.3±1301.3Wm-2K-1 were obtained for nearly-neutrally buoy-ant particles in shear ¯ ows inside a holding tube for particleReynolds numbers of 73.1±369.4. The heat transfercoef® cients were found to increase signi® cantly withincreasing ¯ ow rate and particle-tube diameter ratio. Heattransfer was enhanced by 80 to 200% when the particleconcentration was increased from a single particle to 3.22%.

Being a visual technique, the melting point indicatormethod is limited to transparent ¯ uids and, therefore, canonly be used with model systems. Damay and Pain76 used asimilar phase change technique in which a particle isintroduced in a horizontal pipe and the change in its mass ismeasured. The heat transfer coef® cient is then deduced froma thermal balance on the particle equating the energysupplied to the particle to the energy of phase change.

2.2.6. Remote temperature sensor methodAll of the previous experimental techniques are limited

by the need to measure temperature either by thermocoupleor by an optical method. An ideal measurement techniquewould be one in which the temperature of a particle could befollowed freely-moving non-invasively. Some progress hasbeen made with `temperature pills’ which use a particle witha quartz crystal inside acting as the temperature sensingelement. The element resonates at a temperature-dependentfrequency and invokes a coil circuit and thus generates amagnetic signal, as described by Balasubramaniam andSastry7 7 . An external receiver converts the magnetic signalinto a temperature reading. The temperature history of theparticle is monitored as it moves through the tube bymoving an external antenna along with the particle. The datacollected are then used in conjunction with a suitablemathematical model to estimate hf p .

Bhamidipati and Singh3 1 ,7 8 were the ® rst to exploit thistechnique. They used a cylindrical particle (22.6 mm length,10.7 mm dia) of known density and speci® c heat capacity,which broadcasts its temperature to an antenna coil, which ismoved along the outer surface of the tube as the particlesensor moves inside the tube, allowing the temperature-timepro® le of the particle to be obtained. The test ¯ uids wereaqueousCMC solutions (0.5±1.2% conc) heated to 82.228 C.They reported hf p values for a single particle ranging from108.08 to 195.83Wm- 2K- 1 .

Being non-invasive, the technique does not interferewith particle trajectory as temperature is being measured,and has the added advantage of yielding local hf p values.However, the large size and density of the broadcastingparticle is severely limiting: it is crucial that the sensingparticle behaves the same way as the food. In addition,quartz crystals used in the temperature sensor cannotwithstand high temperatures (>1218 C) which makes thetechnique currently unsuitable for UHT processing. If theparticle were made smaller it would be a very attractivetechnique.

A novel non-invasive technique is described by Ghironand Litch® eld7 9 who determine temperature using magneticresonance techniques: the change in temperature of amagnetic particle is detected by its change in magnetization.

16 BARIGOU et al.


This type of technique offers the scope for real processmeasurement, but is again not fully developed.

2.3. Experimental Studies in Process Engineering

Within process engineering the variation of Nusseltnumber with particle-¯ uid relative velocity has been studiedby many workers over the last ® fty years, for single particle,packed bed, and ¯ uidized systems. Few of the results arerelevant to heat transfer problems in food ¯ ows, but some ofthe techniques used may be adapted to food applications.

The most fundamental work, and the basis of muchsubsequent research, was that of Ranz and Marshall6 6 , whoexamined evaporation of pure liquid drops (0.6 to 1.1 mmdia) in air at 2208 C, for particle Reynolds numbers between0 to 200, and developed the well-known Ranz-Marshallcorrelation, equation (7).

Shallcross and Wood8 0 used published data to develop ageneral correlation for heat transfer from spheres to air inthe particle Reynolds number range 100-105 ,

Nu = 2 + 0.363Re0.547p . (9)

These authors noted two important requirements in design-ing experiments to determine heat transfer coef® cientsaround a stationary particle: (i) the particle should besupported at the rear stagnation point, and (ii) for laminar¯ ow the inlet ¯ uid stream should have negligible turbu-lence. The authors suggested that results from experimentsthat do not observe these criteria can give Nusselt numbersup to 10% too high. The use of cross¯ ow supports andunsteady ¯ ow can thus lead to erroneous readings.

Particle rotation has been observed to occur within foodprocessing, particularly at low particle concentrations and¯ owrates (for example by Liu et al.8 1 ). As discussed insection 2.1.4, this rotational motion is expected to affect theconvective heat transfer coef® cient. Quantitative studies onparticle rotation in food ¯ ows are, however, very limited.Some data is available in the general literature; for exampleKrieth et al.4 5 studied the behaviour of rotating spheres in aquiescent ¯ uid using 2 in and 6 in diameter metallic spheresin water, air, oil, or mercury. Experiments were conductedfor Reynolds numbers based on angular velocity(Rex = 2 q f x R2

p / l ) in the range 0 to 9 ´105 . The resultsshowed, as expected, that the convective heat transfercoef® cient increased with increasing rotational velocity. Itwas observed that at Reynolds numbers below 5 ´105 , a¯ uid boundary layer separation zone was con® ned to 2degrees either side of the equator (measured usingphotographs); above this Reynolds number the separationzone increased beyond 2 degrees, in proportion to theangular velocity, thus enhancing the heat transfer. Situationsof interest to food processing usually involve low Reynoldsnumbers of the order of 100 or less, since the carrier ¯ uid isusually a viscous solution (e.g. sauce, syrup). Twocorrelations were derived from these results; the low Relimit is:

Nu = 0.43Re0.5x Pr0.4 (10)

for Gr < 0.1Re2x ; Rex < 5 ´105; 0.7 < Pr < 217.

The effect of particle concentration on heat transfer wasconsidered in detail by Agarwal4 0 who produced acorrelation for the change in heat transfer coef® cient due

to the presence of other particles,

Nupb = e -1.23Nusp, (11)

where Nusp is the Nusselt number for a single isolatedparticle at the same super® cial ¯ uid velocity as the packedor expanded bed of voidage e , and Nup b is the Nusseltnumber for a particle in an expanded bed. This is anattractively simple correlation which has been used byNixon et al.8 2 as the basis for correlating data for food-type¯ uids and shown to work well: such an approach may beuseful for other workers.

3. OTHER FACTORS AFFECTING HEATTRANSFER

3.1. Wall-Fluid Heat Transfer

3.1.1. Estimations of wall-¯ uid heat transfer coef® cient infood processing

The convective heat transfer coef® cient, hw , between the¯ uid-particle mixture and the processing tube surface is notwell characterized in food processing, in particular withrespect to the effect of particle loading. This parameter iscrucial in determining the heating tube length, and henceindirectly affects the holding tube length (as particulateheating is related to ¯ uid heating rates/residence time).Mathematical models (such as Mankad1 1 ) suggest that thewall-¯ uid heat transfer coef® cient is as important forprocess heat transfer as is the ¯ uid-particle heat transfercoef® cient. Existing research in this area is sparse and thereare no sound studies on the effect of hw on sterilizer design.This is re¯ ected in the wide variety of conventions used byvarious authors who have compiled mathematical modelsfor sterilizers. The wall heat transfer coef® cient haspreviously been estimated using equations for liquidlaminar ¯ ow assuming constant wall temperature4 7 , turbu-lent ¯ ow using the Dittus-BoÈ elter equation8 3 , or entered asan arbitrary value depending on the type of heater/cooler;for example, in their computational models, SkjoÈ ldebrandand Ohlsson8 4 ,8 5 adopted a value of 1200Wm- 2K- 1 whilstMankad and Fryer3 8 used a value of 225Wm- 2K- 1 . For agiven system, these different approaches give widelydifferent estimates for hw . In addition, they are all basedon conservative design assumptions rather than hardexperimental data, and do not take into account particleeffects which are likely to be signi® cant. It is, therefore,evident more data on wall-¯ uid heat transfer in food ¯ ows isneeded.

One recent study by Sannervik et al.1 2 determined wallheat transfer coef® cients in a tubular heat exchanger(35.6 mm dia, 6 m length) by measuring inlet and outlettemperatures and performing a heat balance on the system.The ¯ owing mixture consisted of sodium alginate particlesof average diameter 5.6 mm suspended in a 4% pseudo-plastic starch solution. Particle concentrations of 0, 10, 20,and 30% were used. Experiments showed that (i) thepresence of particles had a strong in¯ uence on heat transferrates from the wall, and (ii) wall heat transfer coef® cients inthe heating section were always higher than in the coolingsection because of lower liquid viscosity in the thermalboundary layer. An increase in particle concentration from 0to 10% approximately doubled hw in both sections.Increasing the solids loading from 10 to 20% and from 20



to 30% led to further heat transfer enhancements of , 25%in the heating section and , 35% in the cooling section.

A plausible explanation is that particles interact with thewall boundary layer, causing ¯ uid mixing and enthalpyexchange with the ¯ uid inside the pipe, a mechanism similarto heat transfer by turbulent eddy convection. The authorssuggested the use of an equation originally proposed bySingh8 6 for a single-phase power-law ¯ uid with a fullydeveloped velocity pro® le:

Nu = 1.753n + 1

4n

1 / 3

Gzm, (12)

where n is the power law index, Gz=MCp /kl is the Graetznumber (M is mass ¯ ow rate, l is heater/cooler length, k isthermal conductivity,and Cp is speci® c heat capacity). For asingle phase liquid the equation has a theoretically derivedexponent m= 1/3. For solid-liquid mixtures, Sannervik etal.1 2 found that overall m increased with particle concentra-tion in the range 0.297-0.519, and had slightly higher valuesin the cooling section than in the heating section.

One reason for this lack of progress has been the lack ofadequate techniques for measuring heat ¯ ow across a pipewall to a two-phase mixture. Heat ¯ ux sensors may providean attractive solution to this problem. These have been usedin a wide range of applications such as determining thethermal properties of insulation systems, heat loss determi-nation in building structures, monitoring solar heat collectorperformance, and aerodynamic wind tunnel studies (seefor example, Ortolano and Hines8 7 and Mohan et al.8 8 ). Thefood industry, however, does not seem to have exploitedsuch transducers, which consist of a differential thermo-couple with a thin foil type thermopile bonded to both sidesof a known thermal barrier; the difference in temperatureacross the thermal barrier is proportional to the heat ¯ owthrough the sensor, which is attached to the surface of thepipe. Nixon et al.8 2 have shown the feasibility of thetechnique.

3.1.2. Estimations of wall-¯ uid heat transfer coef® cient inprocess engineering

Heat transfer between a surface and a ¯ uid is ubiquitouswithin process engineering through the study of heatexchangers. A considerable volume of literature exists onthe prediction of heat transfer; however, the majority of thisliterature is not relevant to estimating the wall heat transfercoef® cient in food ¯ ow, because the packed and ¯ uid bedheat transfer literature is largely concerned with gas-solidsystems.

SchluÈ nder8 9 reviewed transport processes in packed beds,and highlighted that void fractions can be greater at the wallthan in the bulk. For a randomly packed bed of uniformspheres the bulk voidage is about 0.4, whilst the wallvoidage can reach 0.5. A two channel model, representing¯ ow at the wall and in the bulk, was developed to evaluatethe velocity ratio between ¯ uid at the wall and ¯ uid in thebulk, using the Ergun equation,

D P

l =150l uf (1 - e )2

e 3d2p

+1.75q f u

2f (1 - e )

e 3dp, (13)

where l is the length of the bed and D P is the pressure dropthrough the bed, to obtain expressions for the ¯ uid velocity

in each channel, which were then combined to give anequation for the velocity ratio. For a randomly packed bedof spheres, it was shown that the velocity of the ¯ uid at thewall was 2.81 times greater than that of the ¯ uid in the bulkof the bed. This shows that where variations in cross-sectional voidage occur, appreciable ¯ uid bypassing canarise. In a food ¯ ow, this might give signi® cant channellingaround localized zones of high solids concentration leadingto variations in particle heat transfer coef® cients withinthe ¯ ow. Furthermore, high ¯ uid velocities near the wallwould invalidate the often-used assumptions of fullydeveloped laminar or turbulent ¯ ow in estimating the wallheat transfer coef® cient. Experimental work on packed bedshas con® rmed that the presence of particles in a tube mayresult in ¯ ow channelling at the wall4 1 . This can lead tosigni® cantly higher heat transfer coef® cients than thoseoccurring in a tube conveying a single phase ¯ uid. Thiswork is important in providing insight into particle effectson heat transfer taking place at the wall.

Gunn and Khalid9 0 investigated wall heat transfercoef® cients in packed beds, using an externally heatedpacked bed (diameter 3.75 in, length 12 in), ® lled with eitherlead shot, glass ballotini, or nickel particles, in the range1-7 mm. Compressed air was passed through the bed andthe particle Reynolds number varied between 1 and 400.The air temperature was measured at the entrance of the bedand over the cross section of the exit by means of severalannular banks of thermocouples. This information was usedtogether with the ¯ uid ¯ owrate and wall temperature toestimate the wall heat transfer coef® cient. The wall Nusseltnumber was found to vary linearly with particle Reynoldsnumber, the values ranging between 1.5 and 150; nocorrelation could be derived due to the high amount ofscatter in the results.

Dixon et al.9 1 determined the diffusive mass transfercoef® cient between the inner surface of a tube coated witha thin layer of water soluble material (2-naphthol,cinnamic acid, or benzoic acid) and a packed bed oflarge particles. Water was pumped through the packedcolumn to allow the soluble material to dissolve from thewalls, and the exit ¯ uid stream was sampled at differentradial positions for solute content. Different particlediameters were used so that the tube to particle diameterratio varied between 3 and 12 (in food processes this ratiotypically falls between 4 and 8); the particle Reynoldsnumber was in the range 50-500. Differential mass balanceswere used to estimate the wall mass transfer coef® cient, anddata were correlated by

Shw = 1.0 - 1.5dp

dt

1.5Sc1/ 3Re0.59

p , (14)

where Shw is the wall Sherwood number and Sc is theSchmidt number.

In a more recent paper, Jamialahmadi et al.9 2 measuredheat transfer coef® cients to a solid-liquid ¯ uidized bed in acylindrical tube (D =23.8 mm; l =160 mm) with water asliquid phase and cylindrical steel particles (dp (mm) ´l(mm) =1.0 ´1.0; 1.6 ´1.6; 2.5 ´2.5), glass (2, 4.5 mm dia),nickel (4.5 mm dia), copper (4.5 mm dia) and lead (4.5 mmdia) spheres as solid phase. The suspended solids gave riseto enhanced heat transfer, but no further improvement wasobserved once the behaviour of the bed changed from

18 BARIGOU et al.


particulate to aggregate. The results were modelled usinganalogy with nucleate-boiling heat transfer; this predictedthe experimental results together with extensive data fromthe literature with good accuracy. The work resulted in thefollowing correlations:

for particulate ¯ uidization,

hw = hc + 1.5dp

D

0.256

(1 - e )0.5075 hp - hc); (15a)

and for aggregated ¯ uidization,

hw = hc + 8.64dp

D

0.358

(1 - e SB)0.353

(1 - e )1.077(hp - hc); (15b)

where, dp is particle diameter, D is ¯ uidized bed diameter,hw is the wall heat transfer coef® cient, hc and hp areconvective heat transfer coef® cients for regions of the wallunaffected and affected by particles (correlation data forthese coef® cients are given in the paper), e is ¯ uidized bedvoidage, and e S B is packed bed voidage. The correlationswere each compared to twenty six of the most frequentlyrecommended correlations in the literature, and were foundto have much smaller average relative errors of prediction.The data suggests dependence on both particle size andsolids fraction; although it is not known whether any of thistype of correlation applies in foods, the data suggests waysof treating food ¯ ow information. No set of results isdirectly applicable to food ¯ ow situations where the solidsare entirely mobile; research is still needed to quantify thewall heat transfer coef® cient in such continuous processes.

3.2. Mixing of the Liquid Phase

This review has shown that, of the data required fordesign, most information is available on ¯ uid-particle heattransfer and little is available on wall-¯ uid heat transfer.However, the temperature distribution within the liquid willalso be very important: for example, it is possible that, for ahigh viscosity carrier ¯ uid in which little mixing takesplace, transport of heat through the ¯ uid may be controlledessentially by thermal conduction. It is commonly assumedthe liquid is stirred suf® ciently well that radial temperaturegradients can be neglected (see section 4); but there is verylittle data available on the actual temperature gradientswithin real food systems. Lefebvre and Leuliet9 3 show thethermal ® eld within viscous liquids and show that very largetemperature gradients can arise in high viscosity solutions:the controlling parameter is the ratio of the Grashof andReynolds numbers. High viscosity ¯ uids are commonlyused to minimize particle sedimentation; however, if theyresult in low heat transfer rates product quality might belowered.

4. HEAT TRANSFER MODELLING

4.1. Modelling Studies in Food Processing

Ideally, to ensure the commercial sterility of continuouslyprocessed solid-liquid foods without too much loss inquality, thermal process calculations should be based onaccurate temperature-time data taken within particles asthey ¯ ow through the heating, holding and cooling sections

of the sterilizer. However, as discussed above it is notpossible at present to measure the temperature of foodparticles in continuous sterilizers. Mathematical techniquesusing accurate experimental data (i.e. derived from systemsanalogous to those occurring in industrial food processes)are needed to estimate the temperature-time history in theparticle and hence ensure the correct level of sterility isattained within the product without compromising itsquality.

A number of different modelling studies have beencarried out in an attempt to predict the temperaturevariations within a particle moving through a continuousaseptic process. Most of the models proposed to dateincorporate two fundamental assumptions: (i) there is zeroslip velocity between liquid and particle, hence it is oftenassumed that Nu =2 for particle-liquid heat transfer, i.e. thatthe mode of heat transfer from liquid to particle is purelyconductive8 3 ; and (ii) the ¯ uid phase is a radially-mixedplug ¯ ow with zero axial dispersion, in which all theparticulates travel at the same velocity4 7 . As noted already,neither of these approximations is likely to be correct. Thesection below describes how this problem has beenmodelled; a model needs to re¯ ect experimental data topredict ¯ uid and particle temperature pro® les across theheating, holding and cooling sections.

4.1.1. Thermal balancesThe simplest modelling technique is to carry out energy

balances for ¯ uid and particles in each section of theprocess. This leads to the estimation ® rst of the particlesurface temperature, and then to the calculation of theparticle centre temperature. Once temperature pro® les at thecoldest points, usually the geometric centres of the particles,are known, lethality and cooking levels can be calculatedusing the standard equations (16a) and (16b), respectively,

F =t

0

10(T-Tfref )/ zf dt, (16a)

where, F which is the F-value, is an equivalent heating timefor which the product could be held at a constant referencetemperature, Tf r e f, to give the same ® nal concentration ofmicrobial pathogens as a processing time, t, for which thetemperature, T, changes. Tfr e f depends on the organismbeing inactivated or the indicator organism used (e.g.121.18 C for C. Botulinum), and zf is the temperature changewhich produces a 10-fold change in reaction rate fromthe rate at the reference temperature (e.g. 108 C for C.Botulinum)1 ,2 .

Product quality loss is estimated using the cook value, C,also known as the C-value, a parameter de® ned in a similarway to the F-value, which gives a measure of the extent ofnutrient loss in units of time,

C =t

0

10(T-Tcref )/ zcdt, (16b)

where Tc r e f is a reference temperature dependant on thenutrient under consideration (typically 1008 C), and zc is thetemperature change which produces a 10-fold change inreaction rate from the rate at the reference temperature(typically 25 8 C)1 ,2 .



Models are required to calculate temperature pro® les;those for calculating processing duties for a two-phasemixture generally ® rst consider an energy balance on the¯ uid over some incremental heat exchanger length8 3 ,

Outlet enthalpy of liquid =inlet enthalpy of liquid + heat transfer from walls

+ heat transfer from particles

(Cpw)f (Tf + D Tf )= (Cpw)f Tf + UhxAhx(Th - Tf )

+ hfpApnp(Tps - Tf ), (17)

where, Cp is speci® c heat capacity, w is the ¯ uid mass¯ owrate, Tf is the ¯ uid temperature, Th the heat exchangerwall temperature, Tp s the particle surface temperature, np isthe number of particles present in the section, each havingindividual surface area Ap , Uh x is the overall heat transfercoef® cient at the heating/cooling surface, and Ah x is theheater/cooler surface area available for heat transfer. Thisequation assumes radially uniform ¯ uid temperature.

The thermal response of the particle can then be predictedby solving the transient heat conduction equation,

= .(kp = Tp)= (Cp q )p¶Tp

¶t. (18a)

The two phase thermal balances are coupledby the interfacialconvective heat transfer term between ¯ uid and particle,

= .(kp = Tp)= hfp(Tf - Tps). (18b)

These energy balances can be used in all sections of the ¯ owsystem. If the holding tube is adiabatic then the termUhxAhx(Th - Tf ) in equation (17) can be neglected. Once thetemperature pro® les for the two phases are known, i.e. bysolving equations (17)±(18), then the level of sterility andquality loss can be estimated from equations (16).

Many workers have adopted the above techniques toanalyse the sterilization process. The major dif® culties inthis approach are that (i) radial uniformity is assumed, and(ii) in order to solve the above set of equations, the twoconvective heat transfer coef® cients, hf p and hw , arerequired. A too high or too low estimate can result ineither quality loss or an insuf® ciently sterile product.

4.1.2. Analytical solutions to the heat conduction equationIn a few idealized cases the solution to the heat transfer

equation is algebraic; many such solutions are given in textssuch as Carslaw and Jaeger9 4 . For a sphere, equation (18a)can be rewritten in spherical co-ordinates;

¶Tp

¶t = a s

R2p

¶¶rp

r2p¶Tp

¶rp, (19)

and series solutions to equation (19) can be used if (i) theparticle is heated or cooled by contact with a constanttemperature environment, and (ii) the physical propertiesare constant. Equation (20) shows the thermal response of asphere as a function of its radius, Rp , and time, t, expressedin terms of the Fourier number (Fo= a s t/R

2p) when a sphere,

initially at Ti, is brought in contact with a ¯ uid at Tf

with in® nite heat transfer coef® cient at the ¯ uid particle

interface,

Tp - Tf

Ti - Tf=

¥

j=1

2(-1)j+1 sin(j p rp / Rp)

jp rp

e(-j2 p 2Fo). (20)

Most of the early work on solids sterilization used this typeof equation, ® rst for canned foods, and then for particles incontinuous processing. One of the ® rst models was that ofBall around 1920, developed by Ball and Olson9 5 forsterilization during canning. At high t, all but the j= 1solution of equation (20) (or its equivalent for other particlegeometries) are negligible, so the temperature transientbecomes a single exponential, and a semi-logarithmic plotof temperature against time is linear. The difference intemperature between the two phases at any instant is knownas the thermal lag; in the exponential region, this lag isconstant. Some modi® cations are needed to approximate thenon-logarithmic behaviour at the start and end of thesterilisation process. Ball and Olsen9 5 derived an equationfor the accumulation of sterility in a process of heatingtemperature, Th , and cooling temperature, Tc . The equationsare not fully accurate, but were put to effective use beforecomputers were available to solve equation (18a). They giveadequate results for heating but poor results for the coolingstage.

More recently, Dail9 6 modi® ed Balls method to calculateholding times for different particle shapes. Series analyticalsolutions for heat transfer into an in® nite slab and in® nitecylinder were obtained to give equations similar to equation(20) which were then simpli® ed, using only the ® rst terms ofthe series. This was developed further by Larkin9 7 whoadapted the method to calculate holding times for foodprocessing systems where the semi-logarithmic tempera-ture-time pro® le is not linear (i.e. the thermal response is notpurely exponential). This method uses a ® nite differencesolution of equation (18a) to calculate the centre tempera-ture of a conduction-heated spherical particle movingthrough the heat-hold-cool sections of an aseptic processingsystem. The method was only approximate; it assumes boththat the heat transfer coef® cient is in® nite and that the ¯ uidtemperature pro® le is known.

4.1.3. Computational modelsMore rigorous models for continuous aseptic processing

have been developed in recent years, based on solid andliquid energy balances, such as equations (17)±(18).Formulation of the models requires assumptions aboutheat transfer and ¯ ow behaviour, and the degree ofcomplexity of the constitutive equations depends on thesimplifying assumptions made.

Initial work did not attempt to solve the temperaturebalance for the ¯ uid as well as the solid. de Ruyter andBrunet8 assumed a linear ¯ uid temperature pro® le with timethrough each component of the sterilizer. Using these preset¯ uid temperatures the temperature of the solid phase wascalculated at each point in the sterilizer. The model wasused to estimate holding times for solids of differentdiameters; as expected, longer holding times were necessaryto sterilize large particles compared to small ones. Theauthors assumed that the convectiveheat transfer coef® cientbetween solid and liquid was in® nite, and that both phasesmoved at identical velocities through the system. Theassumption that the ¯ uid temperature pro® le is ® xed implies

20 BARIGOU et al.


that solids heating has a negligible effect on ¯ uidtemperature. Strictly, this assumption violates the law ofconservationof energy and in some cases can lead to seriousinaccuracies; for example, in the presence of a highconcentration of large particles, the effect of the solidsheat capacity would alter the ¯ uid temperature pro® lesigni® cantly. In such a situation, the supposition of a ® xed¯ uid temperature pro® le becomes questionable.

Similar work was carried out by Manson and Cullen9 whoinvestigated the effects of particle residence times on theoverall lethality level of cylindrical solids moving through aheat-hold-cool sterilizer. The model also used a ® xed ¯ uidtemperature pro® le, and solids temperatures were calculatedfrom these preset ¯ uid temperatures through the sterilizer.The ¯ uid-particle interfacial heat transfer coef® cient wasassumed in® nite. This model represents the ® rst attempt toconsider residence time distributions. The holding tube wasmodelled as eleven radial shells. Equal particle concentra-tions were assigned to each shell, and the particle velocitywas then set equal to the maximum ¯ uid velocity, based onfully developed laminar ¯ ow in the shell. This enabled arange of particle velocities to be assigned, thus incorporatingthe effects of a residence time distribution. No modi® cationswere made to the mass balance to account for differences insolids velocity.

The results showed that a parabolic ¯ ow gave longerheating and holding tube lengths than a plug ¯ ow particleresidence time distribution. Thus, assuming that all particlesremain in the system for equal times could result ininsuf® cient solids sterilization.No experimental validation ofthe assumptions was given, but the model was a signi® canttheoretical advance.Although it is questionablewhether solidparticles would adopt a parabolic velocity pro® le akin to thatof the ¯ uid6 , however, the ideas represented in the model areclearly correct.

The ® rst model to calculate the liquid temperature pro® leas well as the solid was that of Sastry8 3 who modelledprocess schedules for mushroom-shaped particles of varioussizes at a constant concentration of 44%. The three stages ofthe continuous aseptic process were modelled using energybalances for each stage. Instead of modelling the effect ofparticle size distribution,heat transfer to a group of average-sized particles was evaluated. The ¯ uid-particle heattransfer coef® cient was based on the limiting value for asphere in a stagnant ¯ uid, i.e. Nu = 2 which is the case forzero slip velocity. The energy balances for the two phases(equations (17) and (18a)) were solved simultaneously usingan iterative procedure. The ¯ uid temperature predictionswere then used as a time-dependant convective boundarycondition at the particle surface; this data was used todetermine particle temperatures.

Thermal balance equations were ® rst solved to establish a¯ uid temperature pro® le along the tube, assuming particlesand ¯ uid travelled at the same velocity. The effect of thisliquid temperature pro® le on different particles and processconditions was then studied. The ¯ uid temperature was set,and the thermal response of the particle calculatedaccordingly; no account was taken of any effect that theparticle had on the liquid temperature. This could causeerrors; for example when a large cold particle is heated bythe liquid, the liquid temperature will be reduced. Data onthe effect of velocity and particle size must, therefore, betreated with caution. Subsequent simulations studied the

effects of both particle size, and the two convective heattransfer coef® cients (¯ uid-particle and wall-¯ uid). Datawere also presented for the processing of particles whichtravelled at different speeds or which were of different sizesto the uniformly sized particles used to establish the basecase.

The simulations showed that (i) the length of the holdingtube was sensitive to the heat transfer coef® cient only forBi <10, when external heat transfer is signi® cant2 0 ; (ii) asthe heat transfer coef® cient increased, the holding tubelength decreased to an asymptotic value corresponding tointernal-conduction-controlled heat transfer (i.e. Bi >10);and (iii) the holding tube length decreased with increasingaverage particle residence time, and increased with particlesize.

An investigation into different possible methods oflicensing sterilization procedures was carried out byChandarana et al.4 7 and Chandarana and Gavin9 8 for theNFPA (National Food Processors Association) in Washing-ton. When a product is licensed in the USA, it must beshown to be sterile using an approved technique. Amathematical model using three-dimensional transientheat conduction equations and energy balances wasformulated to estimate heat transfer into foods passingthrough a heat-hold-cool system. The model consisted of thesolution of: (i) equation (18a) for a cube, (ii) equation (17)for the three cases of heating, holding and cooling usingdifferent temperatures and heat transfer coef® cients whichwere computed from literature correlations, and (iii)coupling of the two phase thermal balances via solution ofequation (18b).

Three possible approaches to scheduling and licensingcommercial processes were considered, based on variousconservative assumptions:

(i) particles increase in both temperature and sterility in theheat exchanger and holding tube;(ii) increase in particle temperature occurs solely in theholding tube, i.e. only the liquid is heated in the heatingsection, and no sterility accumulates in the heating section;(iii) both phases increase in temperature in the heatexchanger, but sterility is assumed to accumulate only inthe holding tube.

Heat transfer between the particles and ¯ uid wasestimated for the conservative case of pure thermalconduction between the solids and ¯ uid i.e. Nu =2. Theintegrated lethality of the particulate product was calculatedand the three models compared. Unsurprisingly, it wasfound that case (i) was closest to an actual system, whereboth temperature and lethality accumulate in both heaterand holding tube; this produced the most realistic results(i.e. most similar to the results from an actual steriliser).This work has formed the basis of the mathematical modelused to validate an aseptic process1 1 8 .

The model of Chandarana and Gavin9 8 was adopted byLee et al.9 9 to examine the in¯ uence of changes in inputparameters (i.e. product and process variables) on theholding tube length required for sterility, peroxidaseretention and thiamine retention, the latter two related toproduct quality and cook value. The study aimed to provideinsight into the possible margins of error in modelpredictions due to inaccurate estimation of the inputparameters, such as particle and ¯ uid physical properties.



A summary of the main variables considered, and theirassociated sensitivities is presented in Table 2. Over theranges investigated, particle size, particle density, particlespeci® c heat capacity, and product ¯ owrate had the greatesteffect on holding tube length and thiamine retention;changes in peroxidase retention were minimal for allparameters considered. The most sensitive factors indetermining the process were particle size and product¯ owrate. Changes in ¯ uid properties such as thermalconductivity and viscosity showed little effect on holdingtube length or cook value parameters. The study did notconsider parameters such as heat transfer coef® cients; theeffects of these may be more important than those examined.

McKenna and Tucker1 0 0 used a ® nite difference model topredict the centre temperatures of cylindrical, spherical, andcubic particles subjected to a constant ¯ uid temperature of1358 C. The computed temperature history pro® les were thenused to estimate the F-value and C-value for the processedproduct. The code was validated by comparing thecomputed temperature-time pro® les with the pro® lesdetermined experimentally from a single transducer particlewith a thermocouple at its centre, placed in a ¯ ow loopwhere the ¯ uid heated the particle; theory and experimentwere in good agreement. The convective heat transfercoef® cient was an input variable in the model, and wasestimated using the Ranz-Marshall correlation (equation(7)).

The model of SkjoÈ ldebrand and Ohlsson8 4 ,8 5 extends thatof Sastry8 3 by allowing a greater parameter variation to beimposed on the sterilizer design. The liquid energy balancewas as used in the original model, but the solids energybalance equations were modi® ed to allow calculations forspheres or cylinders. The programme describes a systemcomprising a chosen number of heating units, a holdingsection, and a chosen number of cooling units. The product¯ ows through the process at a selected mass ¯ owrate.Conduction heat transfer is assumed within the particles,although a convective boundary condition at the particlesurface is used to determine the rate of thermal interchangewith the ¯ uid. Other assumptions made are identical to thosein other models, namely that the liquid is perfectly mixed inall sections of the sterilizer, that each particle is totallysurrounded by liquid, and that all particles have the samesize and geometry.

The model calculated processing duties, equipment sizes,and product sterilization and quality measures. In each case,all particles were considered spherical. Particle residencetimes were simulated by inputting a ® nite value for the

velocity ratio of an average moving particle to the fastestmoving particle. Both phases were assumed to move atidentical velocities, hence heat transfer enhancements dueto slip were ignored. The heat transfer coef® cient betweenthe ¯ uid and the heater/cooler tube walls was arbitrarily setat 1200 Wm- 2 K- 1 . The effects of various parameters on theholding tube length were studied as follows:

(i) Particle-¯ uid heat transfer coef® cientÐ the holding tubelength generally decreased with increasing heat transfercoef® cient, as shown by Sastry8 3 . At low heat transfercoef® cients (<100Wm- 2 K- 1 ), the tube length was highlysensitive to even small changes in hf p because of the lowparticle Biot number. At higher values (>200Wm- 2 K- 1 )even a large change in hfp had little impact on the length ofthe holding tube.(ii) Particle diameterÐ for small particles the desiredsterility was achieved in the heating unit; hence no holdingtube was required. Above some diameter, a function of thesterilization temperature, thermal conduction within theparticle became an important factor in reaching the desiredsterilization temperature and thus in¯ uenced the holdingtube length required to achieve the desired level of sterility.Above this critical diameter, further increases in particlediameter generated a linear increase in holding tube length.(iii) Particle concentrationÐ particle loading emerged as akey factor in determining the holding tube length in that itaffected the heat capacity of the system; higher concentra-tions required longer holding tubes. For the cases studied,the ¯ uid had a greater thermal capacity than the solids.(iv) Flow conditions in the holding tubeÐ the simulationinvestigated the effects of varying the ratio of the averageliquid velocity to that of the fastest particle in the range 0.1±1. A slow, approximately linear decrease was observed inthe holding tube length as the velocity ratio was increased.

It is not clear how the liquid temperature pro® le wascalculated for the cases where different conditions (such asparticle diameter) were simulated; it is not, therefore,possible to tell whether this model represents a signi® cantimprovement on that of Sastry8 3 .

A recent model by Cacace et al.1 0 1 simulated particleheating and holding for a 20% delivered solids concentra-tion of 10 mm potato cubes in a sodium chloride solution,using a ® xed ¯ uid temperature pro® le recorded from anexisting sterilizer and entered into the model as an inputvariable. The particle temperature pro® le was estimated bysolving equation (18a) using the ¯ uid temperature as aboundary condition. The value of hf p was assumed to be

22 BARIGOU et al.


Table 2. Sensitivity parameters; results of Lee et al.99.

% change % change in % change in % change inParameter in parameter hold tube length peroxidase retention thiamine retention

Particle size -50% to +50% -79% to +142% +4% to -6% +209% to -90%Particle loading -50% to +50% +10% to -7% no change +10% to -8%Particle thermal conductivity -10% to +10% +10% to -8% -0.5% to +0.5% -15% to +13%Particle density -2% to +2% -4% to +4% +0.1% to -0.1% +4% to -4%Particle speci® c heat capacity -2% to +2% -3% to +3% +0.1% to -0.1% +3.5% to -3.5%Fluid thermal conductivity -6% to +6% +1.5% to -1.5% no change -1% to +1%Fluid viscosity -70% to +70% -10% to +8% no change -10% to +8%Product ¯ owrate -25% to +25% -87% to +208% -1.2% to +1.8% +10% to -75%

1000Wm- 2K- 1 . The code calculated the degree ofmicrobial spore destruction (Saccharomyces Cerevisiae)for a particle cube as it moved through a heating/holdingarrangement. Unlike other models reported in the literature,this was experimentally validated. The number of microbialspores of Saccharomyces Cerevisiae was determined beforeand after processing, and the results were compared to thosegenerated by the model; the theoretical and empirical resultswere found to be in good agreement.

Additional experimental work was conducted to estimatehf p by holding a potato cube stationary in a stream of ¯ uidand recording the temperatures of the ¯ uid, particle surface,and particle centre. The heat transfer coef® cient was thenrelated to slip velocity, although no correlation was derived.The measured values of heat transfer coef® cient rangedbetween 300 Wm- 2 K- 1 and 2000 Wm- 2 K- 1 for slipvelocities between 20 cms- 1 and 29 cms- 1 , respectively.The average particle and ¯ uid velocities in the two phasemixture were measured, and the slip velocities for this caseranged between 0 and 5 cms- 1 ; these are signi® cantly higherslips than those observed by Lareo1 5 .

Mankad et al.3 7 proposed a one-dimensional model tostudy the effect of particle-liquid slip on a homogeneousfood ¯ ow, in which sterilizer lengths for various solid-liquidslip velocities and delivered solids concentration werecomputed at a ® xed ¯ ow rate. For low slip velocities, theenhancement of interfacial heat transfer coef® cient reducedthe process length. The model was subsequently extendedby Mankad and Fryer3 8 to include non-homogeneousstrati® ed ¯ ows which can occur both in horizontal andvertical situations6 . The model was used to study phase andslip velocity effects that a heterogeneous ¯ ow might have ina practical situation, and to demonstrate how a full modelmight be used if ¯ ow patterns and particle residence timedistribution were known. The strati® ed ¯ ow was assumed toconsist of two interconnected one-dimensional ¯ ow zones.A sedimented bed lying underneath a low particleconcentration region was modelled; (each with its ownsolids fraction, phase velocities, and particle-liquid slip).Slip velocity was low within the bed and high in the regionabove it, to suspend the particles, so, the solids heating ratein each zone was different. All particles within each regionwere assumed spherical, and travelled at the same velocitywith no axial inter-particle conduction. The liquid in eachregion was assumed well mixed with no radial temperaturegradient, and travelled at a uniform velocity. There wasno interchange of particles between the two regions butinter-¯ uid heat transfer could occur.

Mass and energy balances were written for each zone; thetwo zones were assumed to ® ll half the tube volume, and formass balance purposes all of the particles were assumed tobe contained in the bed. The energy balance included heataccumulation, thermal diffusion, convective ¯ uid ¯ ow, andheat exchange with the surroundings (wall-¯ uid, particle-¯ uid, and ¯ uid-¯ uid heat transfer between the two zones).

The wall-¯ uid heat transfer coef® cient was arbitrarily setat 225 Wm- 2K- 1 whereas values of hf p were estimatedfrom the Ranz-Marshall correlation (equation (7)). The¯ uid-¯ uid heat transfer coef® cient between the two zoneswas also arbitrarily ® xed at 500 Wm- 2 K- 1 , to represent anintermediate case of incomplete ¯ uid mixing, for moststudies, but the effects of setting the value to 0 and105 Wm- 2K- 1 were also investigated. Studies were

conducted for a typical food processing Biot numberrange of 0.1-10. The range of process parameters modelledare summarized in Table 3.

The results showed clear differences in heating ratesbetween parts of the ¯ ow, and demonstrate the effects ofphase velocity and slip velocity. Phase velocity controlledthe length of the heating tube; here the process is governedby the ¯ uid temperature, whilst slip velocity controlled thelength of the hold tube required to sterilize particles whoseheating rate was strongly dependent on hf p . Phasetemperature was also crucial; hold tube length was stronglyin¯ uenced by the temperatures at the exit of the heatingsection. Equality of phase velocities was found to give thebest results but this, however, is not usually achievable in anindustrial context. The details of the results re¯ ect the casestudied, but the model does highlight the factors thatin¯ uence the process following the models of Manson andCullen9 and Sastry8 3 . It shows that it is not suf® cient toconsider parts of the process in isolation but rather the wholeprocess needs to be considered and an integrated approachshould be followed in design.

Although the model includes heat transfer coef® cientdata, it uses a number of simplifying assumptions whichmay not re¯ ect the conditions of industrial processes. Forinstance, the assumption of a strati® ed ¯ ow with twoseparate zones each characterized by uniform velocity andtemperature pro® les is unlikely to be true in practice6 . Toextend models requires more information on realisticvelocity distributions, and realistic wall-¯ uid and ¯ uid-¯ uid heat transfer coef® cients. This information is crucial indetermining the solids heating rate, and hence the hold tubeand the overall process length. More work is needed in thisarea.

Although a number of mathematical models for hetero-geneous aseptic food processing have been produced, theyare often incomplete. Oversimpli® cations regarding bothheat transfer and residence times mean that most models areonly suitable for very speci® c cases, i.e. where the Biotnumber is high, or for plug ¯ ow. Any model should solve forboth ¯ uid and solid phases temperatures iteratively, andincorporate different phase velocities and heat transfercoef® cients. As accurate data is lacking for factors such asparticle concentration, the complex solid-liquid ¯ owpattern, and residence time distributions, most modelsmay be unacceptable for accurate use. The ¯ uid-particleheat transfer coef® cient used in the models are either basedon the theoretical limit of Nu = 2, or exists in the form of aconstant value (i.e. assuming particle conduction controlsheat transfer). Similarly the wall heat transfer coef® cient isderived from assuming either pure laminar ¯ ow in a pipe, orentered as an arbitrary constant.

Whilst the basic approach of heat balances and conduc-tion equations outlined is correct, accurate heat transfercoef® cients and two-phase ¯ ow data are needed to predicttemperature distributions. The assumptions of many modelsfor the wall-¯ uid and ¯ uid-particle heat transfer coef® cients,are not based on food ¯ ow observation or experimentationand so will give model results that can only be consideredqualitative.

4.2. Modelling Studies in Process Engineering

One of the primary limitations of the work described



above is the lack of knowledge of heat transfer coef® cients.Most models have used experimental data obtained forisolated single particles which, as already noted, does notfully represent the heat transfer processes in such ¯ ows. Onepossible better approach, is to consider the ¯ owingparticulate system as a packed bed, and to produce amodel based on data and correlations for these types ofsystems. This is not fully accurate, since in a packed bed theparticles are stationary. However, considerable work inprocess engineering has studied heat transfer and ¯ owbehaviour for high concentrations of solids in ¯ uids, albeitat particle sizes, concentrations, and Reynolds numbers notcharacteristic of food ¯ ows. These models may still provideconsiderable insight into heat transfer and ¯ ow behaviour,and some of the data and techniques used may also beadopted for food ¯ ow systems. To develop a model for food¯ ows, it might be possible to use existing data for heat ormass transfer in packed beds.

Within process engineering, heat transfer in solid-liquid¯ ows is very important in catalytic systems and absorbers.Many of the models and theories for predicting equipmentsizes from heat transfer data have proved reliable. In recentyears, most processes have tended to be designed usingcomputational solutions of the fundamental governingequations, unlike the mainly traditional practices of thefood industry.

One of the ® rst relevant analytical models was developedby Amundson1 0 2 , to predict phase temperatures in ® xed andmoving beds. A two dimensional energy balance, based on¯ ow in a horizontal cylinder, was derived for the ¯ uid phase.The solid particles were assumed spherical and their energybalance was based on one-dimensional transient heatconduction into a sphere. Internal heat generation wasincluded within the solids energy balance to account for heatgeneration during catalytic reaction. The solids phaseenergy balance was identical to equation (18a) with anadditional heat generation term, and subject to boundarycondition (18b).

The ¯ uid phase energy balance was two-dimensional,estimating the ¯ uid temperature at a radial, and axial

position at time t,

3(wk)s

Rp(q u)s

¶Tp

¶rp rp=Rp

= (wCp)f¶Tf

¶x

+ e kf¶2Tf

¶r2t

+1

rt

¶Tf

¶rt, (21a)

with boundary condition,

-kf¶Tf

¶rt rt=Rt

= Uw(Ta - Tf ), (21b)

where; w is the mass ¯ owrate, Tp is the particle temperature,Tf is the ¯ uid temperature, Ta is the ¯ uid temperature atinlet, k is thermal conductivity, Cp is speci® c heat capacity,Uw is the overall heat transfer coef® cient at the tube wall, us

is the solids phase velocity, Rp is the particle radius, Rt is thetube radius, rp and rt are particle and tube radial positionsrespectively, x is axial position, and e is the bed voidage.

The two equations were written in dimensionless formand then solved by Laplace transforms and Bessel functions.A number of solutions for different cases were presented(such as small particles, no heat generation, etc.). Thesolution presented for a ® xed bed could be used without anyiterative numerical procedure, however, for moving bedsystems an iterative solution was required.

Three models describing unsteady state heat transfer inpacked beds were given by Wakao and Kaguei1 0 3 , usingenergy balances for the two phases;

(i) The Schumann model, the least complex of the three,assumes ideal plug ¯ ow in the ¯ uid, and no thermalresistance in the particle (i.e. Bi=0). The ¯ uid energybalance may then be written as,

¶Tf

¶t = -uf¶Tf

¶x -hfpas

e (Cp q )f

(Tf - Ts). (22a)

The solids phase temperature is calculated using Newton’ s

24 BARIGOU et al.


Table 3. Range of process parameters used in the simulation of Mankad and Fryer38.

Manipulated parameter Simulation range

Slip velocity above bed (ms- 1) 10- 4-10- 2

Slip velocity within bed (ms- 1) 0.1, 0.5, 1 times the value above bedSolids fraction above bed negligibleSolids fraction within bed 30±50%Fluid-¯ uid heat transfer coef® cient (Wm- 2K- 1) 0, 500, ¥ (in practice 10,000)

Property Fluid Solid

Density (kg m- 3) 1000 1000Thermal conductivity (Wm- 1K- 1) 0.645 0.645Speci® c heat capacity (J kg- 1 K- 1) 4184 4184Viscosity (Pa s) 0.025 ±Initial temperature ( 8 C) 15 15

Tube diameter (m) 0.075 (heat and hold sections)Particle diameter (m) 0.01Total mixture ¯ owrate (m3 s-1) 0.001Total delivered solids concentration 20%Heater temperature ( 8 C) 150

law of heating, using a pre-determined heat transfercoef® cient, hf p , thus,

(1 - e )¶Ts

¶t =hfpas

(Cp q )s(Tf - Ts). (22b)

These two equations are effectively simpli® ed versions ofequations (18a) and (17).

(ii) The C-S model (continuous solid) develops theSchumann model slightly further. The ¯ uid energy balanceis identical to that derived in the Schumann model butincludes a thermal diffusion term; this assumption iscommon in packed beds. The solid particles are stillassumed to have in® nite conductivity (i.e. Bi=0), so thateach individual particle is isothermal, but a conductivityterm (i.e. a ¶2Ts / ¶x2 term) is included to calculate inter-particle heat transfer between particles in contact:

for the ¯ uid,

¶Tf

¶t =kf

e (Cp q )f

¶2Tf

¶x2 - uf¶Tf

¶x -hfpas

e (Cp q )f

(Tf - Ts);

(23a)

for the solid,

(1 - e )¶Ts

¶t = ks

(Cp q )s

¶2Ts

¶x2 +hfpas

(Cp q )s

(Tf - Ts). (23b)

(iii) The D-C model (dispersed-continuous) is now the mostwidely used. The model is based on the ¯ uid havingdispersed plug ¯ ow and the intra-particle temperaturehaving radial symmetry. The solids energy balance isthe one dimensional transient heat conduction equation(equation (19)),

for the ¯ uid,

¶Tf

¶t = a f¶2Tf

¶x2 - uf¶Tf

¶x -hfpas

e (Cp q )f

Tf - Ts | rp=Rt;

(24a)

for the solid,

¶Tp

¶t =a s

R2p

¶¶rp

r2p¶Tp

¶rp, (24b)

with boundary condition,

ks¶Ts

¶rp rp=Rp

= hfp Tf - Ts | rp=Rp . (24c)

Apart from the inclusion of a thermal diffusion term, and theabsence of an external heat transfer term in the ¯ uid energybalance, the D-C model is identical to equations (17)±(18),which are commonly used in food processing to estimatesolid-liquid temperature pro® les.

These models constitute the basis for much of currentequipment design. Due to the close similarity to an actualfood sterilizer, the work of Amundson1 0 2 has strong appli-cability to food ¯ ow situations; modern computers can nowsolve the iterative equations fully. Computational solution offundamental equations, which may need to be speci® callyderived, such as those presented by Wakao and Kaguei1 0 3 ,provides the best way of obtaining accurate models, as theinitial assumptions serve to de® ne the model.

5. CONCLUSIONSÐ THE NEED FOR FUTUREWORK

Continuous aseptic processing of solid-liquid foodmixtures is an elegant alternative to canning technology.However, the dif® culty in predicting the complex two-phase¯ ow and heat transfer rates, and hence in ensuringcommercial sterility and optimum quality, makes it anintricate industrial problem. The process is often subjectedto a wide distribution of particle concentration, velocities,residence times, and temperatures, thereby causing a wideand still largely unpredictable distribution of those qualitychanges that are imparted to the food by the heat treatment.As a result, the design of such a process remains to a greatextent empirical.

The FDA have recently validated an aseptic process forparticulates, using a protocol which incorporates (i) RTDdetermination using the Hall effect sensor techniquedeveloped by CCFRA (for details see Lareo et al.6 ), (ii) amodel for the thermal behaviour of the system developedfrom that of Chandarana and Gavin9 8 developed by Leeet al.9 9 . Calculations were con® rmed by an inoculated packstudy. Recent US work reported through a series ofworkshops1 1 9 ± 1 2 3 describes problems in developing andvalidating an aseptic process, and con® rms both (i) theimportance of accurate knowledge of the interfacial heattransfer coef® cient1 2 3 , and (ii) the consequences of theincorrect choice of hf p in any model1 2 0 .

Particle heat-transfer coef® cients, like particle tempera-tures, cannot at present be measured directly and reliably ina continuous heat-hold-cool system: the main controltechnique remains to be the measurement of liquidtemperature. Although considerable experimental andtheoretical work has been done on the particle surfaceheat transfer coef® cient, applications to real food ¯ ows areseverely limited. These limitations arise because theexperimental equipment and techniques adopted are onthe main not representative of commercial food sterilizers.The degree of relevance of the results to real foodprocessing situations vary from one study to another, asdemonstrated in Figure 1 and Table 1. The key problem isthat of knowing the ¯ ow ® eld round the particle; the simplede® nition of slip, the difference between mean ¯ ow andparticle velocities, is not appropriate. Extensive experi-mental data is available for stationary particles, but it is notclear how this relates to the industrial case of translating androtating particles. Some progress has been madeÐ viamoving thermocouples, liquid crystals, and temperaturepillsÐ to analyse the real problem: these experiments havebeen invaluable, but more work is needed to de® ne theinterfacial heat transfer coef® cient to a level where it can beused with con® dence.

Very little is known about the wall-¯ uid heat transfercoef® cient in food processing, and in particular the effect ofparticle loading. This parameter is crucial for designing thelength of the heating tube, and hence also in¯ uences thelength of the holding tube. The process engineeringliterature is rich on this subject but, unfortunately, most ofit is not relevant to the ¯ ow of food mixtures. Progress inthis area has been hampered by the lack of appropriatemeasurement techniques.

Hard experimental data and theoretical models are stillneeded to predict the fundamental parameters that in¯ uence



process design. Two important areas can be identi® ed wherethere is a clear de® ciency of experimental data:

(i) Particle concentration effectsÐ the process engineeringliterature has mainly concerned itself with packed beds, butthe effect of bed voidage on the particle heat transfer doesnot seem to have been properly investigated. On the otherhand, food science/technology research has mainly dealtwith single, and often static, particles. The practical realityin terms of solids concentration lies between these twoextremes and needs to be rigorously addressed.(ii) Fluid ¯ owrate effectsÐ solid-liquid ¯ ows in indus-trial sterilizers6 usually correspond to tube Reynoldsnumbers below 500, and particle slip Reynolds numbersof less than 100. Most of the existing research, particularlythat reported in process engineering literature, has beenconducted at Reynolds numbers far greater. Future researchneeds to be targeted at realistic food ¯ ow regimes.

Although a number of modelling studies have beenreported they have not had a major industrial impact.Mathematical models for sterilizers have to be built onoversimplifying assumptions which make them eitherunrealistic or suitable for only very limited cases ofpractical food processing. For instance, most models adoptheat transfer coef® cients (¯ uid-particle and wall-¯ uid) thatare largely based on inappropriate empirical relationshipssuch as single particle expressions or turbulent ¯ owconditions. Factors such as particle concentration, the two-phase ¯ ow pattern, and residence time distribution have notbeen considered due to the poor current understandingof thecomplex physics of such ¯ ows. The often used assumptionof no slip between particle and ¯ uid (i.e. Nu = 2) is clearlytoo conservative, but does at least represent a minimum safevalue for the processing of the particle phase, although itmay by reducing the heat transfer between particle andliquid, lead to an overestimate of liquid temperature1 2 0 .Particle slip is an important characteristic of solid-liquidfood ¯ ows and will enhance ¯ uid-particle heat transferconsiderably above the Nu = 2 limit. Particle rotation willalso enhance heat transfer; it is not known how importantthis will be under conditions of moderate to high solidsloadings. At this stage it might be reasonable to assume thatheat transfer effects due to particle rotation in food ¯ ows arenot signi® cant; again this will result in too low a heattransfer coef® cient.

The effects of particle-particle interactions are not wellunderstood and require further extensive research. Theseinteractions can give rise to ¯ uid streamline disturbanceswhich may be superimposed on the ¯ ow ® eld aroundsuccessive particles and, depending on the Reynoldsnumber, can give rise to enhanced particle heat transfercoef® cients. Furthermore, where particle concentration ishigh, ¯ uid channelling can occur and this can signi® cantlyaffect the heat transfer coef® cient.

A combination of experimental and theoretical workrelevant to real solid-liquid ¯ ow situations is needed toidentify the optimal conditions for the design and operationof continuous aseptic processes. Future developments ininstrumentation will probably provide the necessaryinformation on the important heat transfer parameters.For instance, improved remote temperature sensor methodswill give much better data on the particle heat transfercoef® cient (for example, using the methodology of Sastry’ s

`moving thermocouple’ technique without having theconstraining thermocouple), whilst heat ¯ ux sensors mayprovide the solution to the measurement of the wall-¯ uidheat transfer coef® cient. On the theoretical front, computa-tional solutions are required for the fundamental equationsgoverning the ¯ ow and heat transfer in solid-liquid ¯ ows.This is a much more complex problem and progress in thisarea is likely to be slow because of the great computationaldif® culty in handling multiple particle systems of thistype1 0 4 .

NOMENCLATURE

as solids speci® c surface area, m2 m- 3

Ahx heat exchanger heat transfer surface area, m2

Ap particle surface area, m2

Aw wall heat transfer surface area, m2

Bi Biot number (hfpx k-1s ), -

C cook value, minCp speci® c heat capacity, J kg-1 K-1

dp particle diameter, mdt tube diameter, mF sterility value, minFo Fourier Number ( a st x- 2), -g acceleration due to gravity, m s- 2

Gr Grashof number (gx3 j D T q 2f l

- 2), -hfp particle convective heat transfer coef® cient, W m- 2K-1

hw wall convective heat transfer coef® cient, W m- 2K- 1

k thermal conductivity, W m-1K-1

l length, mmp particle mass, kgnp number of particles, -Nu Nusselt number (hfpx k-1

f ), -P pressure, PaPe PeÂclet Number (uf x a -1), -Pr Prandtl number (Cpl k- 1

f ), -Q volumetric ¯ owrate, m- 3 s-1

r radius, mRe tube Reynolds number ( q fumdt l -1), -Rep particle Reynolds number ( q fufdp l - 1), -Reslip particle slip Reynolds number ( q fuslipdp/l -1), -Rex rotational Reynolds number (Rex = 2 q f x R2

p l -1), -Rp particle radius, mrp radial position in particle, mrt radial position in tube, mRt tube radius, ms relative density, -T temperature, Kt time, min or sTcref cook reference temperature, 8 CTfo original ¯ uid temperature, 8 CTfref sterility reference temperature, 8 CTh heating temperature, 8 CTi initial particle temperature, 8 CTps particle surface temperature, 8 CTw wall temperature, 8 Cu velocity, m s-1

ufmax maximum ¯ uid velocity, m s-1

ufx ¯ uid velocity in x-axis vector, m s- 1

ufy ¯ uid velocity in y-axis vector, m s- 1

Uhx heat exchanger overall heat transfer coef® cient, W m- 2K- 1

um mean mixture ¯ ow velocity, m s-1

uslip slip velocity, m s-1

Uw overall heat transfer coef® cient at tube wall, W m- 2K-1

w mass ¯ ow rate, kg s-1

x characteristic length, mzc cook thermal death time constant, 8 Czf sterility thermal death time constant, 8 C

Subscriptsf ¯ uidp particles solids

26 BARIGOU et al.


Greek symbolsa thermal diffusivity, m2 s-1

e bed voidage, -j coef® cient of cubical expansion, K- 1

l dynamic viscosity, Pa sl p interfacial viscosity, Pa sm kinematic viscosity, m2 s-1

q density, kg m- 3

x angular velocity, s- 1

AbbreviationsFDA Food and Drug AdministrationRTD Residence Time DistributionCCFRA Chipping Campden Food Research Association

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ACKNOWLEDGEMENTS

SMs PhD work was funded as part of a DTI Link scheme on foodprocessing, in collaboration with Campden Food and Drink ResearchAssociation and sponsored by Unilever UK Central Resources Ltd, APVBaker Ltd, Alfa Laval Pumps Ltd, HJ Heinz Co Ltd, Master Foods, andNestle UK Ltd.

ADDRESS

Correspondence about this paper should be addressed to Dr M. Barigou,School of Chemical Engineering, University of Birmingham, Edgbaston,Birmingham B15 2TT, UK.

The manuscript was received 25 July 1997 and accepted for publicationafter revision 5 January 1998



heat transfer in two-phase solid-liquid food flows- a review 1998

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