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CHAPTER 6

Aseptic processing of liquid/particulate foods

N.G. Stoforos1 & H. Sawada21Department of Chemical Engineering, Aristotle Universityof Thessaloniki, Greece.2Somatech Center, House Food Corporation, Chiba, Japan.

Abstract

A generalized analytical solution to the set of coupled energy balance equationsgoverning heat transfer in a system comprised of uniform radius spherical parti-cles, with nonuniform initial temperature, heated in a well mixed liquid with timevarying temperature, is initially derived. Based on this solution, analytical equationsdescribing the temperature evolution for both fluid and particles, during processingof liquid/particulate food systems in a typical aseptic processing line, consisting ofa heating, a holding, and a cooling section, are explicitly given. Sample calcula-tions, illustrating the effect of the convective heat transfer coefficient at the particlesurface, as well as the overall heat transfer coefficient between the liquid and itssurroundings, are included.

1 Introduction

During conventional in-container thermal processes, the food is first hermeticallyenclosed in an appropriate container (typically metal cans or glass jars) and thenheat treated, according to a specified time–temperature schedule. The applied heattreatment is established in a very precise manner in order to inactivate the targetmicrobial populations (and/or other undesirable agents) that may be present in theproduct, and therefore extend its shelf life to desirable levels, with minimal qual-ity deterioration. An alternative approach is aseptic processing, i.e. a continuousoperation applied for bulk foods processing, in which the product is heat treatedfirst and then aseptically packaged in sterile containers (metal, composite, glass,paperboard laminates and plastic packages) [1]. It is generally accepted that food

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188 Heat Transfer in Food Processing

quality is better preserved in thermal processes characterized by high rates of heattransfer between the heating medium and the product [2], due to the differentialeffect of temperature on the rates of microbial destruction and quality degradationby heat [3]. The high heat transfer rates achieved during aseptic processing can leadto better quality products (e.g. better texture, flavor, or color) compared to tradi-tional canned products. Aseptic processes are commercially attractive because theypossess key advantages such as improved product quality, independent of containersize, ability to operate continuously, reduced operating cost, and convenience ofuse and the plethora of choices in terms of aseptic packages.

As with any thermal process, product safety is of primary importance and itcan be assessed through evaluation of the F value of the process, defined here, inthermobacteriological terms, by eqn (1) [4–6]

FzTref

=∫ tb

ta10(T (t)−Tref)/zdt = DTref (log (Na) − log (Nb)) (1)

The F value summarizes the integrated time–temperature destructive effect of a ther-mal process on a heat labile substance (e.g. microbial spores or a quality attribute)in a single value. It can be defined as the time, at a constant temperature, Tref, thatproduces the same destructive effect as the actual thermal process. Product tem-perature, T , during the actual thermal process can be, and usually is, a function ofprocessing time, as it is revealed in the integral of eqn (1). Comparison of the Fvalue of a process, with the time required to destroy a given percentage of the targetmicrobial population, is the basis for thermal process design.

As suggested by eqn (1), the F value of a process can be calculated either throughthe integral evaluation from time–temperature data (physical-mathematical proce-dures), or the right-hand side of eqn (1) from concentration values at the beginningand the end of the process (in situ and time–temperature integrators [TTI] meth-ods) [7]. Due to inherent difficulties in working with in situ or TTI procedures [8],physical-mathematical methods are considered to be the most appropriate proce-dures when product temperature data can become available [9].

A typical aseptic processing system, i.e. the system used to process the food anddeliver it to the packaging system, consists of a heating, a holding and a coolingsection. The product, with a controlled flow rate, enters the system through theheating section where it is heated to sterilizing temperatures. Then, in the hold-ing section, the product is held for a time sufficient for sterilization, before goingthrough the cooling section, where cooling of the product to filling temperature isachieved. In principle, any pumpable food can be processed aseptically. Neverthe-less, the design of aseptic processes for heterogeneous foods, consisting of a carrierliquid containing discrete solid particles (e.g. a vegetable-based soup containing1-cm sized potato cubes), is far more complicated compared to aseptic processesof homogeneous products. Targeting commercial sterility of the critical region ofthe solid particles has to overcome difficulties associated with temperature mea-surements on moving particles, convective heat transfer coefficient at the particlesurface estimation, particle size and residence time distribution, as well as a numberof critical control points that must be taken into consideration.



Aseptic Processing of Liquid Foods 189

An aseptic process for foods containing particulates, and specifically for dicedpotato soup in a carrier of modified food starch, was developed, as a case study,by the National Center for Food Safety and Technology and the Center for AsepticProcessing and Packaging Studies and served as a framework for a validation studyfor filing with FDA [10]. The thermal process was based on the determination of theresidence time distribution of the particles, mathematical modeling of the lethality(F value) for the fastest moving particle, validation of the model with inoculatedpack studies, and the identification of the critical control points for the process. Anumber of articles dealing with the issues involved in designing aseptic processesfor liquid/particulate foods can be found in a collection type of presentation in theliterature [11–21].

Mathematical modeling can be of value for any process. It can give insight to var-ious aspects of the process, preventing unnecessary experimental effort. Referringto aseptic processing of liquid/particulate foods, where there are no conventionalmeans available for particle temperature measurements [18], prediction of producttemperature, and especially, the temperature of the critical region of the criticalparticle, by mathematical modeling, is essential. It enables evaluation of the pro-cess F value, and therefore, assesses the efficacy of the process. The objective ofthis chapter is to present a mathematical model for temperature estimations duringaseptic processing of particulate foods. An analytical solution for processing ofspherical particles in a well-mixed carrier liquid will be given in sufficient details,and sample results, showing the effect of the heat transfer coefficients on producttemperature histories, will be presented.

2 Mathematical analysis

The problem under investigation is predicting the temperature history of both fluidand discrete solid particles during thermal processing of liquid/particulate foodsystems. The liquid portion of the product (herein referred to as liquid) exchangesheat with an external medium (or its surroundings) as well as the contained solidparticles (herein referred to as particles). Due to similarities, as far as heat transfer isconcerned, the following analysis applies equally to both aseptic as well as conven-tional in-container processing in agitating systems. Basic simplifying assumptionsfor the mathematical description of this problem include the following:

1. Heat transfer within particles is purely by conduction.2. The thermophysical properties for both fluid and particles are constant.3. The overall heat transfer coefficient, between the liquid and its surroundings,

as well as the convective heat transfer coefficient at the particle surface are alsoconstant.

4. The temperature of the liquid is uniform. For the case of in-container process-ing, this suggests that the liquid temperature is uniform throughout the con-tainer, although it can vary with processing time. For continuous sterilizationsystems, we assume that the process is under steady-state conditions; every par-ticle travels through the system at the same average longitudinal velocity as that




of the fluid, and the fluid temperature does not vary within each cross-sectionof the processing system (e.g. radially within the tubular heat exchangers andholding tube).

5. Exchange of heat between the external medium and the particles, or among theparticles themselves by direct contact is negligible.

Under the above assumptions, the particle temperature, Tp, is governed by thefollowing heat conduction eqn [22]:

∇2Tp = 1

αp

∂Tp

∂t(2)

subject to a convective boundary condition

kp∇Tp · n = hp(Tps − Tf) (3)

and the appropriate initial condition (allowing for nonuniform initial product tem-perature). (All symbols are defined in the Nomenclature section of this chapter.)

Neglecting the heat accumulation term in the wall separating the product from theexternal medium, an overall energy balance can be written, in one representation,as follows [23]:

ρfCpfεV

dTf

dt= UoAc(Tm − Tf) − ρpCpp

(1 − ε)Vd〈Tp〉

dt(4)

Equation (4) calculates the rate of heat accumulated in the fluid as the differencebetween the rate of heat transferred between external medium and the fluid and therate of heat accumulated in the particles.

For continuous sterilization systems, according to the fourth assumption listedabove, the position variable, z, along the length of the system can be correlatedto the time variable, t, through the average longitudinal velocity of the product,uave, as

t = z

uave(5)

Furthermore, note that according to the same assumption, the initial fluid temper-ature must be constant.

Equations (2) and (4), subject to the appropriate initial and boundary conditions,describe both the heating and the cooling cycle of an in-container sterilizationprocess, as well as the heating, the holding, and the cooling section of an asepticprocess. Differences though in the values of several constants, for example, in theoverall or the convective heat transfer coefficient at the particle surface betweendifferent cycles can and do exist. Concerning the initial condition for the producttemperature, we should notice here that the particle temperature profile at the end ofeach section represents the initial particle temperature distribution during the nextprocessing section, thus making the problem rather complicated. Furthermore, thesolution to the above system of differential equations, a coupled problem, repre-sents a difficult task, since particle and fluid temperatures cannot be obtained inde-pendently. Several solutions, for given particle geometries, with or not additional




assumptions, targeting in-container or continuous sterilization of liquid/particulatefood products, have been presented in the literature.

In general, four approaches to the solution of the heat transfer equations havebeen identified in the published models:

(a) Use of prescribed liquid temperature profiles to solve for particle temperature[13, 24–27]

(b) Use of numerical methods to solve the conduction equation and the overallenergy balance alternately [15, 28–39]

(c) Use of a combination of analytical and numerical techniques for satisfying theconduction and the overall energy balance equation [40–42]

(d) Use of analytical methods to obtain a closed form solution that satisfies boththe conduction and the energy equation simultaneously [43–48].

As can be expected, a variety of different assumptions, simplifications, and targetingsystems (including particle geometry) are associated with the existing publishedmodels. In the remaining of this chapter, an analytical solution developed and usedby Sawada [47–49] will be presented.

2.1 Generalized solution for spherical particles with nonuniform initialtemperature heated in a variable temperature medium

The solution refers to isotropic spherical particles of uniform radius, heated in awell-mixed fluid of possibly time varying temperature, and allows for nonuniforminitial particle temperature.

2.1.1 Governing equationsEquating the rate of heat accumulation in the particles to the rate of heat enteringthe particles through their surface, i.e.

ρpCpp(1 − ε)V

d〈Tp〉dt

= hpApt(Tf − Tps) (6)

and substituting the last term of eqn (4) for eqn (6), while noting that, for equal diam-eter spherical particles, Apt = 3(1 − ε)V/Rp, the system of the governing equationswith the appropriate initial and boundary conditions can be written as

∂2(rpTp)

∂r2p

= 1

αp

∂(rpTp)

∂t(7)

and

ρfCpfεV

dTf

dt= UoAc(Tm − Tf) − 3hp(1 − ε)V

Rp(Tf − Tps) (8)

with the following initial

Tp(rp, t = 0) = Tpi(rp) (9)

Tf(t = 0) = Tfi (10)




and boundary conditions

−kp∂Tp

∂rp

∣∣∣∣rp=Rp

= hp(Tp(rp = Rp) − Tf) (11)

Tp(rp = 0, t) = finite (12)

Through the use of dimensionless variables and temperature differences,

r = rp

Rp(13)

θ = αpt

R2p

(14)

�f = Tf − Tfi (15)

�p = Tp − Tfi (16)

�m = Tm − Tfi (17)

F = Tpi − Tfi (18)

and by defining the following parameters,

Bi = hpRp

kp(19)

m = 3hp(1 − ε)Rp

αpερfCpf

(20)

and

n = UoAcR2p

αpερfCpf

(21)

eqns (7)–(12) are reduced to eqns (22)–(27), respectively:

∂2(r�p)

∂r2 = ∂(r�p)

∂θ(22)

d�f

dθ= n(�m − �f) − m(�f − �p(r = 1, θ )) (23)

�p(r, θ = 0) = F(r) (24)

�f(θ = 0) = 0 (25)

∂�p

∂r

∣∣∣∣r=1

= −Bi(�p(r = 1) − �f) (26)




and

�p(r = 0, θ ) = finite (27)

2.1.2 SolutionFollowing a procedure described by Carslaw and Jaeger [22], we first seek a solu-tion �∗

p, the sum of two functions,

�∗p = �1 + �2 (28)

�1 being the solution for the unit instantaneous spherical surface temperature sourceof radius r′ at time zero in an infinite medium, and �2 being the solution of theequation

∂2(r�2)

∂r2 = ∂(r�2)

∂θ(29)

which vanishes at time zero. �∗p satisfies the energy balance equation (eqn (23))

and the boundary conditions (eqns (26) and (27)).The Laplace transforms of eqns (23), (24), and (29) are, respectively,

s�̄f = n(�̄m − �̄f) − m(�̄f − �̄p(r = 1, s)) (30)

∂�̄p

∂r

∣∣∣∣∣r=1

= −Bi(�̄p(r = 1) − �̄f) (31)

and

∂2(r�̄2)

∂r2 = sr�̄2 (32)

The subsidiary solution is of the form [22]

�̄∗p = 1

8πrr′√s

{e−√

s|r−r′| − e−√s|r+r′|} + A sinh (r

√s)

r(33)

Determining A such that eqn (33) satisfies eqns (30) and (31) simultaneously, weobtain,

for 0 < r′ < r,

�̄∗p = sinh (r′√s)[(m + n + s)

√s cosh (

√s(1 − s)) + {Bi(n + s) − (m + n + s)} sinh (

√s(1 − s))]

4πrr′√s[(m + n + s)√

s cosh (√

s) + {Bi(n + s) − (m + n + s)} sinh (√

s)]

+ �̄mnBi sinh (r√

s)

r[(m + n + s)√

s cosh (√

s) + {Bi(n + s) − (m + n + s)} sinh (√

s)](34)

For r < r′, r and r′ have to be interchanged in the first part of eqn (34).




Using the inversion and the convolution theorem [50], the inversion is foundto be

�∗p =

∞∑j=1

2C2(λj) sin (λjr) sin (λjr′)e−λ2j θ

4πrr′C1(λj)

+∫ θ

0

2nBi

r

∞∑j=1

λj sin (λjr)

C1(λj)�m(τ )e−λ2

j (θ−τ )dτ (35)

The parameters C1(λj) and C2(λj) are given by

C1(λj) = (2(Bi − 1) + (m + n − λ2j ))λj sin (λj) − (Bi(n − λ2

j ) − 2λ2j ) cos (λj)

(36)

and

C2(λj) = (m + n − λ2j )λj sin (λj) − (Bi(n − λ2

j ) − (m + n − λ2j )) cos (λj) (37)

where λj (j = 1, 2, . . .) are the positive roots of

(m + n − λ2)λ cos (λ) + (Bi(n − λ2) − (m + n − λ2)) sin (λ) = 0 (38)

Equation (35) holds for either 0 < r′ < r, or r < r′.Taking a source of strength 4π(r′)2F(r′)dr′ and integrating the first part of eqn

(35) with respect to r′ from 0 to 1 we obtain the solution for �p as

�p = 2

r

∞∑j=1

C2(λj) sin (λjr)e−λ2j θ

C1(λj)

∫ 1

0r′F(r′) sin (λjr

′)dr′

+ 2nBi

r

∞∑j=1

λj sin (λjr)

C1(λj)

∫ θ

0�m(τ )e−λ2

j (θ−τ )dτ (39)

where we have interchanged the order of summation and integration assuming thatthe infinite series is uniformly convergent.

Similarly, for the fluid temperature, the subsidiary solution due to the unit instan-taneous spherical surface temperature source of radius r′ is

�̄∗f = m sinh (r′√s)

4πr′[(m + n + s)√

s cosh (√

s) + {Bi(n + s) − (m + n + s)} sinh (√

s)]

+ �̄mnBi sinh (r√

s)

(m + n + s)√

s cosh (√

s) + {Bi(n + s) − (m + n + s)} sinh (√

s)(40)




The inversion is found as before, ending with the fluid temperature solution�f as

�f = 2m∞∑

j=1

λje−λ2

j θ

C1(λj)

∫ 1

0r′F(r′) sin (λjr

′)dr′

+ 2n∞∑

j=1

λj(λj cos (λj) + (Bi − 1) sin (λj))

C1(λj)

∫ θ

0�m(τ )e−λ2

j (θ−τ )dτ (41)

2.1.3 Roots of the characteristic equationA short discussion should be made concerning the roots λj (j = 1, 2, . . .) of eqn (38),which can be written as

λ cot(λ) = 1 − Bi + m Bi

(m + n − λ2)(42)

A plot of the functions φ1(λ) and φ2(λ) defined through eqns (43) and (44) as

φ1(λ) = λ cot(λ) (43)

and

φ2(λ) = 1 − Bi + m Bi

(m + n − λ2)(44)

gives a graphical approach in approximating the roots of eqn (38) as the intersectionof the curves of the two functions. Such a plot is shown in Fig. 1 for Bi = 5,

4π3π2ππ

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12 φ1(λ)φ2(λ)

λn

√m + n

Figure 1: The first five roots of the characteristic equation (eqn (38) for Bi = 5,m = 10, and n = 10).




m = 10, and n = 10. One can observe that within each interval from (k − 1)π tokπ (k = 1, 2, . . .) there lies always a root, while one additional root exists in theinterval where the value of

√m + n is located (Fig. 1).

3 Study of aseptic processing of liquid/particulate foods

Equations (39) and (41) can be used to predict the temperature of the liquid and theparticles during a particular process, based on the initial temperature distributionwithin the solid particles, F(r′), as well as the medium temperature evolution,�m(τ ). For some special cases, depending on the exact form of the F(r′) and�m(τ ) functions, eqns (39) and (41) can be integrated analytically and give closedform solutions.

In the remaining of the chapter we apply the generalized equations described sofar to study the aseptic processing of liquid/particulate foods. The system we referto is a typical aseptic line consisting of three sections: the heating, the holding, andthe cooling section. The product is introduced to the system through the heatingsection (typically a scraped-surface heat exchanger using steam as heating medium)where the product temperature rises from its initial to a predetermined value. Next,the product travels through the holding section (holding tube), where, based on itslongitudinal velocity, it spends the required time for thermal sterilization. In theholding tube product temperature might change due to thermal losses to the envi-ronment (air). After exiting the holding tube, the product goes through the coolingsection of the system (a scraped-surface heat exchanger using water as coolingmedium) where its temperature falls to a value according to the specifications. Allprocessing conditions should be based on exact calculations, targeting a particularsterilization value.

The system to be analyzed operates under steady-state conditions and the producttravels through each section with the same average longitudinal velocity, such thateqn (5) can be used to correlate processing time, t, with the position (length) withinthe system, z. In addition to the assumptions listed earlier, we further assume (inorder to simplify the evaluation of the integrals) that

1. the external medium temperature within each section (heating, holding, andcooling) is constant;

2. the initial product temperature (before the product enters the heating section) isconstant. This suggests that initial particle and fluid temperatures are constantand equal.

3.1 Heating section

Under the earlier assumptions, the equations that describe the temperature evolutionduring the heating of the product are obtained from eqns (39) and (41) for Tpi = Tfi,and constant heating medium (steam) temperature, i.e.

F = Tpi − Tfi = 0 (45)




and

Tm = Tst (46)

Evaluating the integrals in eqns (39) and (41) we end up with

Tp = Tst − (Tst − Tfi )2nhBih

r

∞∑j=1

sin (λhj r)

λhj Ch

1(λhj )

e−(λhj )2θ (47)

and

Tf = Tst − (Tst − Tfi )2nh∞∑

j=1

λhj cos (λh

j ) + (Bih − 1) sin (λhj )

λhj Ch

1(λhj )

e−(λhj )2θ (48)

As usual, application of L’Hospital’s rule gives the equation applicable at the centerof the sphere. Note that in eqns (47) and (48) above, we explicitly used particleand fluid temperatures instead of temperature differences (�). Furthermore, weintroduced a superscript, ‘h,’ to several parameters (i.e. nh, Bih, λh

j , and Ch1(λh

j )) toindicate that these particular parameters refer to the heating section. We shouldmention here that system and process characteristics can be different in the varioussections of the aseptic unit (e.g. the value of Uo, the overall heat transfer coefficientbetween the liquid and its surroundings) and therefore several parameters appearingin the equations might correspond to different values. We use superscripts, ‘H’and ‘c,’ for parameters referring to the Holding and the cooling section of thesystem.

3.2 Holding section

Mathematical description of product temperature during the holding section be-comes more complicated, compared to the heating section, since particles enter theholding section, possibly, with an initial temperature distribution. Product temper-ature at the end of the heating section i.e. at time θh, will serve as the initial producttemperature for the holding section. Thus, as far as the initial particle temperaturein the holding section is concerned, this will be the one calculated from eqn (47)for θ = θh, different for each radial position within the particles. Concerning initialfluid temperature for the holding section, this will be constant, termed TH

fi , obtainedfrom eqn (48) for θ = θh.

With the above remarks, and for

Tm = Ta (49)




the equations for particle and fluid temperature in the holding tube are explicitlygiven as

Tp = Ta + (Tst − Ta)2nHBiH

r

∞∑i=1

sin (λHi r)

λHi CH

1 (λHi )

e−(λHi )2(θ−θh)

− (Tst − THfi

)2BiH

r

∞∑i=1

λHi sin (λH

i r)

CH1 (λH

i )e−(λH

i )2(θ−θh)

− (Tst − Tfi )4nhBih

r

∞∑i=1

∞∑j=1

C2(λHi ) sin (λH

i r)I(λHi , λh

j )

CH1 (λH

i )λhj Ch

1(λhj )

e−(λHi )2(θ−θh)e−(λh

j )2θh

(50)

and

Tf = Ta − (Tst − THfi

)2mH∞∑

i=1

λHi cos (λH

i ) − sin (λHi )

λHi CH

1 (λHi )


− (Ta − THfi

)2nH∞∑

i=1

λHi cos (λH

i ) + (BiH − 1) sin (λHi )

λHi CH

1 (λHi )


− (Tst − Tfi )4mHnhBih∞∑

j=1

∞∑i=1

λHi I(λH

i , λhj )

CH1 (λH

i )λhj Ch

1(λhj )

e−(λHi )2(θ−θh)e−(λh

j )2θh

(51)

where, for λHi �= λh

j ,

I(λHi , λh

j ) = 1

(λHi )2 − (λh

j )2

(λH

i sin (λhj ) cos (λH

i ) − λhj sin (λH

i ) cos (λhj ))

(52)

while, for λHi = λh

j ,

I(λHi , λh

j ) = 1

2λHi

(λH

i − sin (λHi ) cos (λH

i ))

(53)

3.3 Cooling section

Similar to the holding section, in the cooling section, i.e. for times θ ≥ θH, forwhich

Tm = Tc (54)




particle and fluid temperature are given by the following equations:

Tp = Tc − (Tc − T cfi

)2ncBic

r

∞∑k=1

sin (λckr)

λckCc

1(λck)

e−(λck )2(θ−θH)

+ (Ta − T cfi

)2Bic

r

∞∑k=1

(nc − (λck)2) sin (λc

kr)

λckCc

1(λck)


+ (Tst − Ta)4nHBiH

r

∞∑k=1

∞∑i=1

Cc2(λc

k) sin (λckr)I(λc

k , λHi )

Cc1(λc

k)λHi CH

1 (λHi )

· e−(λck )2(θ−θH)e−(λH

i )2(θH−θh)

− (Tst − THfi

)4BiH

r

∞∑k=1

∞∑i=1

λHi Cc

2(λck) sin (λc

kr)I(λck , λH

i )

Cc1(λc

k)CH1 (λH

i )


i )2(θH−θh)

− (Tst − Tfi )8nhBih

r

∞∑k=1

∞∑i=1

∞∑j=1

Cc2(λc

k)CH2 (λH

i ) sin (λckr)I(λc

k , λHi )I(λH

i , λhj )

Cc1(λc

k)CH1 (λH

i )λhj Ch

1(λhj )


i )2(θH−θh)e−(λhj )2θh (55)

and

Tf = Tc − (Ta − T cfi

)2mc∞∑

k=1

λck cos (λc

k) − sin (λck)

λckCc

1(λck)


− (Tc − T cfi

)2nc∞∑

k=1

λck cos (λc

k) − (Bic − 1) sin (λck)

λckCc

1(λck)


+ (Tst − Ta)4mcnHBiH∞∑

k=1

∞∑i=1

λckI(λc

k , λHi )

Cc1(λc

k)λHi CH

1 (λHi )

e−(λck )2(θ−θH)e−(λH

i )2(θH−θh)

− (Tst − THfi

)4mcBiH∞∑

k=1

∞∑i=1

λckλ

Hi I(λc

k , λHi )

Cc1(λc

k)CH1 (λH

i )e−(λc

k )2(θ−θH)e−(λHi )2(θH−θh)

− (Tst − Tfi )8mcnhBih∞∑

k=1

∞∑i=1

∞∑j=1

λckCH

2 (λHi )I(λc

k , λHi )I(λH

i , λhj )

Cc1(λc

k)CH1 (λH

i )λhj Ch

1(λhj )


i )2(θH−θh)e−(λhj )2θh

(56)

where, the I(λck , λH

i ) function has the same definition as the I(λHi , λh

j ) one (see eqns(52) and (53)).




3.4 Sample calculations

A FORTRAN program was written, based on the equations presented in the earlierparagraphs (i.e. eqns (47), (48), (50), (51), (55), and (56)), for time–temperaturedata generation. Sample calculations were performed for the operational and prod-uct parameters presented in Tables 1 and 2. Liquid and particle temperatures, atthe surface and the center, were calculated for each section (heating, holding, andcooling) of an aseptic unit. Various values for the overall and the convective heattransfer coefficient at the particle surface at the different sections were assumed.Based on finite difference approximations of the differential equations, two numer-ical models (a fully explicit and a fully implicit scheme) were used in a comparisonstudy with very good agreement with analytical results [48].

Figures 2–4 show the effect of Biot number on the temperature profiles for givenvalues of Uo, at each processing section, whereas Fig. 5 presents data to illustrate –in comparison to Fig. 2 – the effect of Uo. As can be understood from the boundarycondition, eqn (11), and the energy balance equation, eqn (8), knowledge of heattransfer coefficients Uo, and hp is essential in predicting time–temperature profilesin liquid/particulate systems. Nevertheless, a discussion on heat transfer coefficientdetermination is beyond the scope of this chapter.

Table 1: System characteristics used for the simulation.

Heating Holding Cooling

Ac (m2) 0.4775840 0.1492450 0.4775840V (m3) 0.1361110 0.0017723 0.1361110External medium 140 30 20

temperature (◦C)

Table 2: Product properties and parametersused for the simulation.

ρf (kg/m3) 1063Cpf

(J/kgK) 3680ρp (kg/m3) 1063Cpp

(J/kgK) 3517kp (W/mK) 0.62Rp (m) 0.00955ε 0.6Tfi (◦C) 30Tpi (◦C) 30




0

20

40

60

80

100

120

140

0 200 400 600 800 1000 1200 1400Time (s)

Tem

pera

ture

(°C

)

MediumLiquidSurfaceCenter


Figure 2: Liquid and particle (surface and center) temperature profiles during asep-tic processing of liquid/particulate foods for Bih = BiH = Bic = 1.048and Uh

o = 4000 W/(m2K), UHo = 2 W/(m2K), and Uc

o = 2000 W/(m2K).

0

20

40

60

80

100

120

140

0 200 400 600Time (s)

Tem

pera

ture

(°C

)



Figure 3: Liquid and particle (surface and center) temperature profiles during asep-tic processing of liquid/particulate foods for Bih = BiH = Bic = 10 andUh


o = 2000 W/(m2K).

0

20

40

60

80

100

120

140

0 200 400 600Time (s)

Tem

pera

ture

(°C

)



Figure 4: Liquid and particle (surface and center) temperature profiles during asep-tic processing of liquid/particulate foods for Bih = BiH = Bic = 40 andUh


o = 2000 W/(m2K).




0

20

40

60

80

100

120

140

0 400 800 1200 1600 2000Time (s)

Tem

pera

ture

(°C

)



Figure 5: Liquid and particle (surface and center) temperature profiles during asep-tic processing of liquid/particulate foods for Bih = BiH = Bic = 1.048and Uh


o = 500 W/(m2K).

4 Conclusions

A generalized analytical solution for fluid and particle temperature predictions ina system comprised of uniform-sized spherical particles, heated in a well-mixedliquid with time varying temperature, was presented. The solution was used tosimulate the heating, holding, and cooling sections of an aseptic processing unit.The presented model has potential application in the study and the design of bothconventional in-container processes as well as aseptic processes of liquid/particulatefoods.

A number of parameters must be further taken into consideration for a completemathematical description of liquid/particulate aseptic systems. Among them, theselection of appropriate values for the overall and especially the convective heattransfer coefficient at the particle surface is of particular importance in processdesign. Their effect on fluid and particle temperature evolution has been illustrated.Furthermore, the closed-form solution presented here can serve for estimation ofheat transfer coefficients from experimental data. Another key issue refers to par-ticle residence time distribution during the process. Time, a critical, but usuallywell-defined and controlled factor in traditional heat treatments of foods, becomesnow a parameter with an inherited variability and high uncertainty. Residence timedistribution during aseptic processing could make calculations rather complex andshould be properly addressed.

Despite its generalized nature, the analytical solution presented in this chaptercarries with it all the limitations originated from the assumptions imposed by thegoverning equations. Limitations, for example, arise from the assumed particlespherical geometry, or the constant values assumed for the various thermal prop-erties of the product. Numerical solutions are more versatile and such approaches




might be more realistic in describing the actual process. Nevertheless, the analyticalsolution presented here can be used as a limiting case solution to test the accuracyof various numerical models intended for use in designing thermal processes forliquid/particulate foods.

Nomenclature

Latin symbols

A constant in eqn (33)Ac system heat transfer surface area, m2

Apt total surface area of particles, m2

Bi Biot number (Bi = hpRp/kp), dimensionlessCp specific heat, J/(kg◦C)C1(λj) constant defined by eqn (36)C2(λj) constant defined by eqn (37)DT (noted also as D) decimal reduction time or death rate constant –

time at a constant temperature, T , required to reduce by 90% theinitial spore load (or, in general, time required for 90% reduction ofa heat labile substance), usually in min

FzT (or simply F) time at a constant temperature, T , required to destroy

a given percentage of microorganisms whose thermal resistance ischaracterized by z, or, the equivalent processing time of a hypotheti-cal thermal process at a constant temperature that produces the sameeffect (in terms of spore destruction) as the actual thermal process,usually in min

F(r) initial particle temperature difference, defined by eqn (18), ◦Chp convective heat transfer coefficient at the particle surface, W/(m2◦C)I(λi, λj) function defined by eqns (52) and (53)kp particle thermal conductivity, W/(m◦C)m constant parameter defined by eqn (20), dimensionlessN spore load (or concentration of a heat labile substance), number of

sporesn constant parameter defined by eqn (21), dimensionlessn unit outward normal vector at the particle surfaceRp radius of particle, mr radial position within the particle, defined by eqn (13), dimensionlessrp radial position within the particle, mr′ radius of unit instantaneous spherical surface temperature source, ms Laplace transform variableT temperature, ◦CTa ambient (medium) temperature, ◦CTc cooling medium temperature, ◦CTfi initial temperature of liquid phase, ◦C




Tpi initial particle temperature, ◦CTps particle surface temperature, ◦C〈Tp〉 volume averaged particle temperature, ◦CTst steam (medium) temperature, ◦Ct time, suave average longitudinal velocity of the product, m/sUo overall heat transfer coefficient between the liquid and its surround-

ings, W/(m2◦C)V system volume, m3

z longitudinal distance, m; or, temperature difference required toachieve a decimal change of the DT value, ◦C

Greek letters

αp particle thermal diffusivity, αp = kp/(ρpCpp), m2/s

ε volume fraction of liquid phase, dimensionless�f liquid phase temperature difference, defined by eqn (15), ◦C�m external medium temperature difference, defined by eqn (17), ◦C�p particle temperature difference, defined by eqn (16), ◦C�1 solution for the unit instantaneous spherical surface source of radius

r′ at time zero in an infinite medium�2 solution of eqn (29) with zero initial conditionθ time, defined by eqn (14), dimensionlessθh time at the end of the heating section, dimensionlessθH time at the end of the holding section, dimensionlessλj jth positive root of the characteristic equation, eqn (38)ρ density, kg/m3

φ1(λ) function defined by eqn (43)φ2(λ) function defined by eqn (44)

Subscripts

a initial conditionb final conditionm external mediumf fluidp particle (solid)ref reference value

Superscripts

c cooling phaseH holding phaseh heating phase




Symbols

¯ Laplace transform∗ solution due to the unit instantaneous spherical surface temperature source

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