1998 Mathematical modelling of migration of volatile compounds into packaged food via package free...

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Journcrl of Food Engineer& 36 (1998) 473-4X4

0 1998 Elsevier Science Limited. All rights reserved

Printed in Great Britain

FLSEVIER

PII: SO260.8774(98)00059-4 0260.8774198 $19.00 t 0. 00

Mathematical Modelling of Migration of VolatileCompounds into Packaged Food via Package Free Space.

Part II: Spherical Shaped Food

S. M. Mousavi, S. Desobry* & J. Hardy

Laboratoire de Physico-chimie et Genie Alimentaires, Ecole Nationale Suptrieure

d’Agronomie et des Industries Alimentaires (ENSAIA), Institut National Polytechnique de

Lorraine (INPL), 2, Avenue de la F&et de Haye - BP 172,54500,

Vandoeuvre-l&-Nancy, France

(Received 20 July 1997; accepted 13 April 1998)

ABSTRACT

This st udy deals w i t h t he model l ing of mi grat i on of vol at il e compounds from

w ooden packaging i nto spheri cal foods. Bot h analyt i cal and numeri cal met hodsw ere appli ed t o solv e part i al dif ferent i al equati ons for radi al di fSusi on in a

spheri cal shaped food. A computer program w as developed t o solv e fi ni t e

di fference equat i ons for the di ffusion probl em. A mat hemat i cal dimensionl ess

model w as obtai ned and then val i dat ed experi ment al l y. A cet i c aci d w as chosen

as t he vol at i l e compound for t hi s w ork , as i t appears i n di fberent species of

popl ar t rees w hi ch are w idel y used in w ooden packagi ng i ndustr i es. The

di ffusion coeff i cient of acet i c acid i n simul at ed food w as det ermi ned, as

required experi ment al data for t he model val i dati on. The part it ion coeffi ci ent of

acet i c acid betw een package and ai r and al so betw een ai r and agar gel w as

det erm ined. Head space gas chromat ography w as used t o det erm i ne t he ki net i cs

of acet i c aci d migrat ion t hrough a sphen’cal food. A cet i c aci d concent rat i on w as

measured at di fferent t imes and posit i ons of a spheri cal food anal og and

compared w i t h t he model , 0 1998 Elsevi er Science L imi t ed. A l l ri ghts reserved

NOMENCLATURE

c: Acetic acid concentration in the sphere (puggg ‘)Cv Acetic acid concentration in wood (pg g- ‘)

cs Acetic acid concentration at the surface of spherical simulant (pug g- ‘)

*To whom correspondence should be addressed.

473



474

$,,I

c*;

DS

GCJK

KPPDE

Re

Y

R*

;I5

SC

tt*

S. M. Mousuvi et al.

Concentration at equilibrium (pg g g ‘)Concentrations at time interval (t + At)

Concentrations at initial timeAcetic acid diffusivity in spherical simulant (cm2 s-‘)

Gas chromatographyParameter of space in computer program

Parameter of time in computer program

Partition coefficient

Partial differential equationReynolds number

Variable radius in the sphere (cm)Radius of sphere (cm)

Normalized radius (r/R)Sherwood numberSchmidt number

Time (seconds)Dimensionless time (tD/R’).

INTRODUCTION

In a food packaging system, migration of volatile compounds may occur in two

directions, from food to packaging materials or conversely from packaging intofoodstuffs (Blumenthal, 1997). The first one is economically very important, due to

weight loss and/or loss of nutritional value (Desobry & Hardy, 1994), while the lateris of great importance from a human health point of view (Gilbert, 1976). In this

regard, considerable investigations have been carried out (Chang et al., 1988; Begly& Hollifield, 1990; Mercer et aZ., 1991; Lilja & Piepponnen, 1992) but only fewstudies were related to wooden packaging which is widely used for cheese and fruits.

A wood species, commonly used in this field, is the poplar because of its white colorand low density. Nevertheless, in a wide range of poplar trees, volatile compounds

are numerous (Green Away et al., 1989). Their amount depends upon many factors,such as species, age, season and processing method for the wooden material (Barry,1931; Wise, 1946).

The objective of this study was to develop and validate a predictive model forvolatile migration from wooden packaging into spherical foods, via the package free

space. Both analytical and numerical methods were applied to solve radial diffusionin a sphere which is not in direct contact with packaging material. In this paper, we

present a mathematical model for migration of wood volatile components into amodel spherical shape food.

MATERIALS AND METHODS

Mathematical modelling

Conceptual modelSome hypotheses and physical conditions which were taken into account (Mousavi etal., 1998), concerning cylindrical food, were also assumed for this paper. Bothpackage material and food were assumed to be isotropic, the diffusion coefficient



Spherical shaped food 475

constant and migration was considered to be a conservation of mass transfer pro-

cess.Surface evaporation of volatile components from wooden packaging occurs during

storage. Since package thickness (x) as compared to its length and/or width is

negligible, volatile concentration can be considered as homogeneous in the packag-ing and we can write:

(1)

From the surface evaporation equation inside the packaging (Crank, 1975). can he

written:

&- n, - = x(L., - c,,)

8x?(2)

then from eqn (1) and eqn (2)c, = c’,,. (3)

Surface evaporation from a sphere results in the following equation:

x(c, - c,,). (4)

Solution of radial diffusion in a sphere

The general equation for radial diffusion in a sphere has been demonstrated byCrank (1975) as follows:

with initial and boundary conditions

t =o, c=O and

r>o. r=O c = finite,

t>O, r=H,

Analytical solution

Introducing a new variable, C = c-c,,, into eqn (5) we obtained a new equation:

(5)



476 S. M. Mousavi et al.

The classical solution of a 2nd order PDE (partial differential equation) by changing

the variable, C = RT, resulted in

linlr co&

C= A- +B- e- lZnr

r r I .

The boundary and initial conditions in the system studied are

t=O, c= -co,

t > 0, r=O, C = finite,

(7)

(84

(8b)

t>O, r=R,

Introducing the initial and boundary conditions into eqn (7) and after several

algebraic manipulations the following mathematical model is obtained:

c-_=l--2 z sin/l,, COSBl,sinlLr*-_- 1

_ jz,,,

co n= I PZ IL --Fe .p,, are non zero positive roots of tar@,, for n = 1,2,3,. . .

/3,,= 1,R; t* = DtlR2; r* = rlR and1R

tanAR =1 -aRID ’

(9)

(10)

By definition, the dimensionless Sherwood number is Sh = 2ctRID.

Numerical solution

For more certainty, a finite difference numerical method was also applied to solve

eqn (5).Introducing the following dimensionless parameters to eqn (5):

Co-C

t * = tDIR2,r* = rlR,c* = - results in

ac* a2c*-=-at* ar*’

with new initial and boundary conditions

t*=o,

CO

2 ac*+J F- ar *

c*=1,

ac*- =o,a t - *

01)

t* > 0, r*=O,



Spherical shaped food 417

t” > 0, r * = 1 [g + $c*]r.*=,=o. (12c)

Equation (11) was discretizedobtain the following equation:

and several algebraic manipulations were utilized to

C*:‘+‘_1

- _ ~

A?’

1+ -+

A?’At”.

Likewise, discretizing boundary conditions leads to:

r*=O, (.*; + 1 ,*” + 1--( I 1

(.*I1+ 1

r*= 1, c*!J+ 1 ~m,wI‘m.,, -

1+ShAr*/2

A computer program was developed (Fig. 1) to solve eqns (9) and (13).

(13)

(14a)

(14bl

Packaging material

Dry rolled wood from poplar trees provided by SIELB (St-Pierre-sur-Dives, France)

was used as packaging material. Volatile compounds in the samples were identifiedby mass spectrometry (FI 8000, Fison, Paris) coupled with gas chromatography.Samples were then cut to the size of 10 cm x 10 cm x one millimeter thick (weight

3.2 g).

Model food

A spherical model food (diameter, 40 mm) was made in a spherical plastic dish and

consisted of 2% agar (BIOKAR, Beauvais, France) and 98% distilled water (w/w).

Simulated package

In an enclosure of 3.3 liter (Fig. 2), acetic acid migrated from wood samples into a

spherical food which was placed on a plastic stand 12 cm in height. Since theamount of acetic acid in the wood sample was small, 0.1 mmol of acetic acid wasadded into each one to permit more accurate measurements. Acetic acid (/lg g - ’ of

wood) migration into the spherical food analog was measured in triplicate by statichead space gas chromatography (HSGC) (Model 8500 Perkin-Elmer & Co GmbH,

Aberlingen, Germany). Storage temperature was 20°C.



478 S. M. Mousavi et al.

-52ead parameters:

Jmax, Kmax, At* , A r*

, Sh

No

Write: 1 -c* (C/Co)

for all times and

radius

K=K+l

STOP

Fig. 1. Flow diagram of the computer program for numerical solution of radial diffusion in

a sphere.



Spherical shaped food

Spherical model food

Isolated chamber

20°C and 65% humidity

Stand with open sides

T Wooden packaging(1OOcm )

Fig. 2. Assembly to study volatile migration from wooden packaging into a

food. Arrows show migration direction.

1

0.9

0.8

0.7

0.6

C

Y 0.5i,

0.4

0.3

0.2

0.1

O_

spherical model

0.35

0 0. I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I

r*

Fig. 3. Predictive model to calculate the migrant concentration of volatile in the sphere.Numbers on the curves are the values of dimensionless number ‘t*’ (t* = Dt lR ’ ) , and r* = r/R.

Where: D is the volatile diffusivity (cm’ s-l); is the time(s); R is the radius (cm); Sh = 2.



480 S. M Mousavi et al.

TABLE 1

Partition Coefficient of Acetic Acid (Dimensionless) t_ St. dev.

Acetic acid Air/wood Airfgel Gel/wood

KP 0.347 + 0.03 0.061_+ 0.005 5.65-0.5

Measurement of acetic acid concentration in spherical model food

Thin slices of spherical agar gel were withdrawn at r = 1.8 cm and at r = 0.4 cm( f O-1 cm). These samples were weighed and put into a vial. The vial was sealed and

a static desorption was carried out for 30 min at 60°C. After 30 min, it was checkedthat remaining acetic acid in the agar gel was negligible. Injection of the air in thevial was automatically performed twice for each sample.

1

0.8

0.4

0.2

0

r

t

03

b

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

t*

Fig. 4. Normalized concentration versus dimensionless time (t* = Dt/R’) resulting fromexperimental concentration measurement at the surface of the sphere. (r* = 0.9). (0, experi-

mental data and -, best fit curve using theoretical values derived from the model).



Spherical shaped food

Determination of the acetic acid diffusion coefficient in agar gel, Ll,

481

This was carried out by an improved method based on that of Belton and Wilson(1982), described by Mousavi et al. (1998). A long cylinder of 14 cm length and

1.2 cm diameter was held interdiffusing a 0.1 M solution of acetic acid in a ballon of50 ml volume. Acetic acid concentration (pg gg ‘) was measured at different times

and positions in the cylinder by gas chromatography at 20°C. The mean value ofacetic acid diffusivity from 50 tests was 1.1 x lo-‘_tO.l x 1O--5 m2 s -‘.

Determination of the partition coefficient

The partition coefficient of acetic acid in wood/air and also in agar gel/air wasdetermined by the method described and applied by Halek and Levinson (1988).

Different quantities of acetic acid were injected in the vials in which small slices of

0.4

0.2

0

0 0.5 I 1.5 2 2.5 3 3.5 4 4.5 5

t*

Fig. 5. Normalized concentration versus dimensionless time (t* = Dt iR ’ ) resulting fromexperimental concentration measurement in the center of the sphere (r* = 0.2). (0, experi-

mental data and -, best fit curve using theoretical values derived from the model).



482 S. M . Mousavi et al.

wood were held. In a like manner, the tests were repeated for agar gel slices. Thevials were tightly sealed and stored at 20°C for equilibrating (3 days). Concentration

was measured by HSGC.

RESULTS AND DISCUSSION

Predictive model

A computer program was developed to solve eqn (9) and eqn (13). Data from theprogram was introduced to Matlab@ software to plot theoretical curves (Fig. 3). The

models obtained by both numerical and analytical solution were almost identical.This confirms the precision of the methods used to solve the diffusion problem in asphere. A comparison between the two methods shows that the numerical method

was easier for solving diffusion problems,

A dimensionless model was established to predict migrant concentration in thespherical food analog at any time and position in the sphere. The mdoel was

1

0.8

32

.: 0.6

K

4

0.4

0.2

0ec0

I I I I

0.4 0.6 0.8 1Model

Fig. 6. Model validation, confirmed by agreement between theoretical (derived from model)

and experimental data.



Spherical shaped ,food 483

obtained with Sh = 2, corresponding to natural convection at the sphere boundary,

as demonstrated by the Chilton-Colbern analogy (Saatjian, 1993) when the Rey-nolds number is nearly zero:

Sh = 2 + ().6~e0~s~c0~‘3~ (15)

It was experimentally verified that this kind of transfer may be well defined by the

Sherwood number at the boundaries.

Model validation

Acetic acid concentrations at r” = 0.9 and r* = 0.2 were measured at different times.

Predicted and measured concentrations of acetic acid at Y* = 0.2 were compared and

demonstrated a good agreement between experimental and theoretical values (Fig.4). Nearly the same agreement was observed for migrant concentrations at r* = 0.9

(Fig. 5) with slight overprediction.

In these figures, C,, is the concentration at equilibrium which is related to theinitaial volatile concentration in the wood packaging material: C, = C, x 5.6 (Table

1). For example, a spherical food with a diameter of 4 cm (such as apricot) which isstored in wooden packaging will be at equilibrium with the acetic acid concentration

in the package, after 12 days. In isothermal conditions and identical initial concen-trations, theoretical and experimental results were very close (Fig. 6).

CONCLUSION

The present model permitted the prediction of acetic acid migration from wooden

packaging into spherical foods. Foodstuff may be protected from the risk of con-tamination by either simple isolation or use of a barrier between it and the wooden

packaging. The theoretical approach was generalized for the cases where volatileconcentration is independent of initial concentration and time. This gave a dimen-

sionless model which is probably usable for more general purposes, i.e. otherpackaging materials and other volatiles.

Computer programs developed in this study can be used to solve similar diffusion

problems in food processing. The model can be applied to real food and packagingsystems taking into account the hypotheses described in the mathematical approach.

ACKNOWLEDGEMENTS

Authors acknowledge the SIELB for financial support.

REFERENCES

Barry, J. (1931). La Chimie du Bois. Dunod, Paris.Begly, T. H. & Hollifield. H. C. (1990). Evaluation of polyethylene tetraphtalate cyclic

migration from microwave food packaging using temperature-time profiles. Food Addit.

Cont ami n., 23, 1575-1579.Belton, P. S. & Wilson, R. H. (1982). An experimentally simple method for measuring

diffusion in food gels. J. Food Technol., 17, 531-539.



484 S. M Mousuvi t al.

Blumenthal, M. M. (1997). How food packaging affects food flavor. 1. Food Technol., 51(l),71-74.

Chang, S., Guttman, C. M., Sanchez, I. C. & Smith, L. E. (1988). Theoretical and computa-

tional aspects of migration of package components to food. In Food and Packaging

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Crank, J. (1975). The M athemati cs of Di fi sion, 2nd edn, Oxford University Press, Oxford.

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Gilbert, S. G. (1976). Migration of minor constituents from food packaging materials. J. Food

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Green Away, W., Scaysbrook, T. & Whatley, F. R. (1989). Head space volatile from propolis.

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Halek, G. W. & Levinson, J. J. (1988). Partitioning behavior and off-flavor thresholds in

cookies from plastic packaging film printing ink compounds. J. Food Chem., 53(6),

1806-1808.Lilja, H. L. & Piepponen, S. (1992). Leaching of aluminium dishes and packages. Food Addit.

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Mercer, A., Castle, L. & Comyn, J. (1991). Evaluation of a predictive mathematical model of

di-(Zethylhexyl) adipate plasticiser migration from PVC film into foods. Food Addit.

Contamin., 7, 497-507.Mousavi, S. M., Desobry, S. & Hardy, J. (1998). Mathematical modeling of migration of

volatile compounds into packaged food via package free space. Part I: Cylindrical shaped

food. J. Food Engng., 36,453-472.

Saatdjian, S. (1993) Phtnom& es de Transport et leurs R& solut ions Numi ri ques. Polytechnica,

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