Modeling Flavor Release from Oil-Containing Gel Particles - American

Chapter 17

Modeling Flavor Release from Oil-Containing Gel Particles Guoping L i a n

Downloaded by AUBURN UNIV on December 5, 2016 | http://pubs.acs.org Publication Date: September 7, 2000 | doi: 10.1021/bk-2000-0763.ch017

Unilever Research Colworth Laboratory, Sharnbrook, Bedford MK44 1LQ, United Kingdom

This paper presents a modelling study of flavor release from oil containing gel particles. A general equation for the diffusion of flavor molecules in gels with micro-inclusion of oil droplets has been formulated. The equation with micro-inclusion is shown in the same format of that without micro-inclusion and degenerates to the latter when the volume fraction of the micro-inclusion approaches zero. The release of a diffusive material from a spherical particle has been previously predicted either by the Crank equation or the Sherwood correlation. The Crank equation considers the diffusion in the particle and ignores the resistance by the bulk fluid. The Sherwood correlation considers the mass transfer from a particle suspended in an infinite fluid. It is proposed that the mass transfer coefficient for the release of flavor from a gel particle should take into account the both effects from the gel particle and the surrounding fluid. The effect of particle size, volume fraction of micro-inclusion and partition coefficient on flavor release rate has been investigated.

Flavor release from oil-containing gel particles involves the diffusion of flavor molecules through a heterogeneous media consisting of distinctively different phases of materials of different diffusion coefficients and partitions. Diffusion in heterogeneous media is also important in a number of areas such as drug delivery (1,5,19), food processing (14), drying and polymer processing (18). Many previous studies have considered the effective diffusion of two-phase systems where one phase is dispersed in the other continuous phase (10,13,18). The diffusive molecules were assumed to be only soluble in the continuous phase and their diffusion coefficient in the dispersed phase was considered to be sufficiently small. The diffusion equation was written as ( l - 0 ) ^ = D Vc e

© 2000 American Chemical Society

Roberts and Taylor; Flavor Release ACS Symposium Series; American Chemical Society: Washington, DC, 2000.

(1)

201

202

where φ is the volume fraction of the dispersed phase and D is the effective diffusion coefficient, c is concentration of flavor and ρ is the gradient operater defined as c

V =

ι — + j — + k— dx dy dz

A number of models have been proposed for the effective diffusion. The commonly used ones include the Maxwell model (12,13,18), which is based on electric conduction theory


Ι+φ and the Mackie-Meares model (11,18), which is based on stochastic obstruction theory 2

(1-0

(1+ΦΫ where D is the diffusion coefficient in the continuous phase. This paper is concerned with the more general case of effective diffusion in twophase materials. The molecules are soluble and diffusive in both phases. The system under consideration is the release of flavor from gel particles with homogeneously distributed micro-inclusion of oil droplets. The size of the dispersed phase is assumed to be sufficiently small such that at the microscopic level the concentrations in the two phases are in instantaneous equilibrium c

c =P c dc

d

(4)

c

where P is the partition coefficient, c and c are the concentrations of flavor in the dispersed phase and continuous phase respectively. In the first part of the paper, the general equation of diffusion in a two-phase dispersion has been formulated. The second part of the paper considers the release of flavor from oil containing gel particles. Many previous studies used the Crank equation based on diffusion for the release of diffusive materials from a spherical particle (1,6,16,19,20). The Crank equation considers the diffusion in the particle and ignores the resistance by the bulk fluid. Mass transfer from a solid phase to a fluid phase has been also a subject of research in fluid mechanics for many years (2,17). For a particle suspended in a fluid, the interfacial mass transfer coefficient is often predicted by Sherwood correlation (17). The Sherwood correlation considers the mass transfer to an infinite fluid from a suspended particle with a constant surface concentration. The penetration may be considered as a special case of interfacial mass transfer theory (3,7,15). The penetration theory assumes that mass transfer between phases is by a short time contact of an element of one phase with another. During the short time contact at the interface, it is assumed that non-steady-state diffusion in a semi-infinite body takes place. After the short time contact, the element is mixed thoroughly in the liquid phase and a new element come into contact. The penetration theory predicts that the mass transfer coefficient is proportional to the square root of diffusion coefficient. dc

d

c



203

Recently, a flavor release model based on two stagnant film theory has been proposed by Hills and Harrison (9). The model assumed that the mass transfer across the food-saliva interface is rate limiting. Harrison and Hills (8) have further extended their interfacial mass transfer model to gelatine-sucrose gels by considering the melting process due to heat transfer. One limitation of their model is that the mass transfer coefficient needs to be obtained by fitting with experimental data. Their model also assumed both the solid food and liquid saliva as homogeneous materials and did not consider micro-inclusions. Thus, it can not be applied to oil-containing gels. In this study, a more general model of mass transfer across oil containing gel particle and liquid interface has been proposed. The newly proposed model includes the effects of both phases (solid and fluid) on the interfacial mass transfer coefficient. It is shown that the Crank equation and Sherwood correlation are special cases of the newly proposed equation. Application of the model has been also demonstrated by predicting the effects of particle size, volume fraction of micro-inclusion (oil) and partition coefficient on flavor release rate.

Effective Diffusion The mass transfer rate in the continuous phase is given by Fick's first law of diffusion

j =-D Wc c

c

(5)

c

By analogue, the mass transfer in the dispersed phase is given by

J =-D Vc d

d

(6)

d

where D and Dj are the diffusion coefficients in the continuous phase and dispersed phase respectively. Recalling eq 4, the mass transfer in the dispersed phase may be written as c

j =-D P Vc d

d

(7)

d

Thus, the mass transfer equation in the dispersed phase may be treated as the same of the continuous phase but with a different diffusion coefficient of DJP^. From eq 5 and eq 7, the overall mass transfer through a composite material may be deduced as follows

j = -D

Vc

(8)

where D is effective diffusion coefficient, min(D , DjPj^KD^maxiDc, DjP ). The effective diffusion coefficient depends on the microstructure of the dispersed phase and may be estimated by empirical correlation, e.g. the general Maxwell equation (12), by replacing Dj by P^Dj e

c

where ξ is a shape factor (ξ=2 for dispersed spheres).


dc

204 Particularly, if D »D , eq 9 reduces to eq 2. Introduce concentration of flavor in the two-phase structured particle defined as c

d

an effective

c = c (1-φ) + φο = c (1 - φ + φΡ ) p

c

ά

c

ά0

(10)

It may be derived that the mass conservation equation of diffusion may be written as follows

-JL = V{D Vc ) ot p

(11)

p

D

where D„ep = Downloaded by AUBURN UNIV on December 5, 2016 | http://pubs.acs.org Publication Date: September 7, 2000 | doi: 10.1021/bk-2000-0763.ch017

n

1-Φ + ΦΡ*

Note that eq 11 is in the same format of diffusion equation in homogeneous continua. At the special case P = 0 and D «D , eq 11 degenerates to eq 1, a special case studied by many previous researchers. On the other hand, when the volume fraction of the dispersed phase is reduced zero, φ = 0, eq 11 degenerates to the diffusion equation in homogeneous one-phase media. dc

d

c

Flavor Release Crank Model Consider flavor release from an oil-containing gel particle suspended in an aqueous phase. If the gel particle is spherical in shape, the diffusion eq 11 of flavor molecules in the particle may be written in terms of spherical polar coordinates as follows

oc

D

d

ot

r

or

n

n

ο

oc

n

or

Eq 12 may be solved analytically under the boundary conditions of c„(0 = 0 if r = R

dc

(13)

n

—£. = 0

6t

if

r=0

where R is the radius of the sphere. From the analytical solution, the amount of diffusing substance leaving the sphere is expressed by (4)

M

η=οπ n

0

1

R

where M is the initial total amount of the diffusive substance in the sphere and M is the amount of the substance diffused out of the sphere. The underlying assumption of the boundary condition of eq 13 is that substance that diffused out of the sphere is instantaneously removed and there is no external 0

t


205 resistance in the surrounding aqueous phase to the mass transfer. It also implies that the surrounding aqueous phase is treated as an infinite sink. Eq 14 contains an infinite number of terms and is very difficult to apply. In practice, the early stage of the equation can be approximated by the following equation to a good accuracy (7)

A, t

M,

= 6 j - ^ r - 3 - £ r2 -

(15)

R

2

Alternatively, noticing that the leading term in eq 14 is exp(-D7i?t/R ),

we may


also approximate the Crank equation by the leading term. The best fit to the Crank solution using the first-term is given by f M l

= 1 - exp -1.2-

r\ ~2 D π* p

2

V

R

\

t

(16)

A comparison o f the above approximate expressions with the exact Crank solution is shown in Figure 1. It can be seen that the approximation of eq 15 agrees very well with the exact solution in the early stage but under-predicts the exact solution significantly as the time increases. The leading term approximation underpredicts the flavor release rate initially, but this under prediction is reduced as the time is increased.

— - Crank solution Early time approximation First order approximation

0.0 Time, s Figure 1. Dimensionless flavor release rate for a sphere predicted early time approximation and first order approximation.

by Crank


solution,

206 Sherwood Correlation Mass transfer from particles suspended in fluid has been a subject in fluid mechanics and chemical engineering for many years (2,17). Prediction of mass transfer from a suspended particle is analogous to the prediction of heat transfer and dimensionless correlations have been applied. The Nusselt and Prandtl numbers have been substituted by the Sherwood and Schmidt numbers. For mass transfer between a fluid and solid spherical particle, the following equation has been suggested (17) ί λ

dM = iKRShD dt


t

(Π)

-c

4

Pf

where D is the diffusion coefficient of the fluid, P (=1-φ+φΡ^) is the effective partition coefficient between the particle and the fluid, c is the concentration in the bulk fluid and Sh is the Sherwood number. The Sherwood number is a dimensionless parameter measuring the mass transfer between two phases. The following correlation has been suggested f

pf

f

Sh = 2 + 0.6Re

1/2

m

Sc

(18)

where Re and Sc are the Reynolts and Schmidt numbers defined as 2

Re =

RU

P P

c ν Sc = • 2pR

η

where u is the relative velocity of the particle in relation to the fluid, ρ is the density, p

and η is the viscosity of the fluid. The effective concentration, c , is related to the amount of substance diffused out, M„ by p

3

M,=M -^-R c 0

(19)

p

Under the condition of an infinite sink, Cf= 0. Substitution of this condition and eq 19 into eq 17 leads to the following expression f

dt

-(M -M) 2R P,pf

(20)

0

Z

Eq 20 may be integrated to give the amount of diffusing substance leaving the sphere expressed by

M_

- = 1 - exp V

3ShDf

(21)

2

2R P.pf

J

which is also in the same format of eq 16.


207


Interfacial Mass Transfer With the Sherwood correlation, the diffusion resistance in the suspended particle is assumed to be sufficiently small and the mass transfer in the fluid phase is rate limiting. On the other hand, the Crank equation applies to the other extreme, i.e. where diffusion in the particle becomes rate limiting. While both the Crank equation and Sherwood correlation are two special cases, the interfacial mass transfer theory may be applied to the mass transfer between two phases under more general conditions. The interfacial mass transfer theory may be visualized by the two-film theory and the rate of mass transfer from a sphere to the surrounding fluid may be expressed as (75) λ dMt (22) = AKR k - c 2

f

dt

Pf

J

where k is the mass transfer coefficient across the interface The interfacial mass transfer coefficient is related to the mass transfer coefficients of the particle k and fluid kf by c

1

1

1

- = k

+ P

k

(23) k

pf p

f

By analogue, with the continuous phase as an infinite sink (i.e. c = 0), eq 22 can be integrated to give the following expression f

M,

- 1 - exp

3k k, p

R(k +P k ) f

pf

-t

(24)

p

Again, it is a first-order kinetics equation in the same form of eq 16 and eq 21. Specially, if the mass transfer in the particle becomes rate limiting, we have P jk «kf. Comparing eq 24 with eq 16 leads to P

p

2

n D cp *„=1·23R

(25)

By analogue, the mass transfer coefficient in the fluid may be derived as

ShD

f

*/="

(26)

2R

Thus, a more general equation for flavor release based on interfacial mass transfer theory is derived.

Results and Discussion Eq 22 combined with eq 23, 25 and 26 forms the basic model for describing the flavor release from oil containing gel particles to liquid (e.g. saliva). The equation may be solved analytically under few special conditions. For instance, if the volume


208 fraction of gel particles in the fluid is φ , the finite concentration of flavor in the fluid may be expressed as ρ

(27)

AnB}

f

By substitution of eqs 19 and 27 to eq 22, the following expression is derived d M

3 k

dt'

RP,pf-[Μ -(\ 0

φ Ρ )Μ,]

+

ρ

(28)

ρ/


This equation can be integrated to give the amount of flavor released to the fluid expressed as

3(l + *,JV)*

l-exp

M, =• ^ΦΡΡ,

(29)

RP.Pf

pf

The system reaches equilibrium when the amount of flavor released is given by

M, =

2

(30)

Analytical solutions to eq 22 can be also obtained, as shown by eq 24, if the fluid is assumed under the condition of an infinite sink. The time required to release half of the total flavor, tVi, may be obtained from eq 24 as follows

IP,Pf ^1/2

~

2

\.2π Ό„

η

2

R \n2

ShD,

(31)

cp V Particularly, when the diffusion in the particle becomes rate limiting, the half-life is given by the following equation 2

^1/2

R \n2 „ , •(Ι-φ + φΡ*)

~

(32)

On the other hand, if the mass transfer in the surrounding fluid is rate limiting, the half-life is then obtained as 2

2i? ln2 ^1/2

"~ '

3ShD

-(Χ-Φ

+

ΦΡ*)

(33)

Examples of predicted half-life time of oil-containing gel particles are shown in Figures 2 and 3, using eq 32. Here, a constant effective diffusion coefficient of l.Oe10 m /s is used. In Figure 2, the half-life time is predicted as functions of partition coefficient and particle radius. The volume fraction of oil droplets is assumed to be 10%. It can be seen that as the partition coefficient from the dispersed phase of oil droplets to the continuous phase of gel is increased, the half-life time required to release half of the flavor also increased. For flavor compounds of low partition coefficients, particle size needs to be increased in order to achieve same release time of higher partition coefficients. 2



209

Figure 2. Predicted half-life time as a function ofpartition coefficient for a spherical particle containing 10% of oil.

Figure 3. Predicted half life time as a function of volume fraction of oil for a spherical particle with an oil-gel partition coefficient of400.


210

Figure 3 shows the effects of the volume fraction of oil droplets and particle size on the half-life time, where the partition coefficient of dispersed oil droplets to continuous phase is set to 400. Apparently, as the volume fraction of the dispersed oil droplets is increased, the effective partition coefficient P f ( = l - φ + φ Ρ ^ ) is also increased. As a result, the half-life time increases. p


Conclusions The diffusion of molecules in two phases of distinctively different materials has been discussed. The system under consideration is the release of flavor from gel particles with homogeneously dispersed micro-inclusion of oil droplets. Assuming that the size of the dispersed oil droplets is sufficiently small such that at the microscopic level the concentrations in the two phases are in instantaneous equilibrium, a general diffusion equation has been derived in terms of effective diffusion coefficient and effective concentration. The general diffusion equation is shown to be in the same format of the ordinary diffusion equation of one phase continuous media. When the volume fraction of the dispersed phase approaches zero, the equation degenerates to the latter. The release of a diffusive material from a spherical particle has been previously predicted either by the Crank equation or the Sherwood correlation. The Crank equation considers the diffusion in the particle and ignores the resistance by the bulk fluid. The Sherwood correlation considers the mass transfer from a particle suspended in an infinite fluid. It is proposed that the mass transfer coefficient for the release of flavor from a gel particle should take into account of the both effects from the gel particle and the surrounding fluid. The newly proposed flavor release equation degenerates to the Crank equation if the diffusion in the particle is rate-limiting, or the Sherwood correlation when the mass transfer in the surrounding fluid becomes ratelimiting. Application of the newly proposed model has been demonstrated by examining the effect of particle size, volume fraction of micro-inclusion and partition coefficient on flavor release half-life time.

References 1. 2. 3. 4. 5. 6.

Baker, R. Controlled Release of Biologically Active Agents. John Wiley & Sons, New York, 1987. Batchler, G. K., J. Fluid Mech., 1979, 95, 369-400. Coulson, J. M. and Richardson, J. F. Chemical Engineering, Pergamon Press, Oxford, 1993. Crank, J. The Mathematics of Diffusion, Oxford University Press, London, 1956. Guy, R. H., Hadgraft, J., Kellaway, I. W. and Taylor, M. J., Int. J. Pharmaceutics, 1982, 11, 199-207. Kurnik, R. T., and Potts, R. O., J Controlled Release, 1997, 45, 257-264.


211

7. 8. 9. 10.


11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Harrison, M., Brian, P. H., Bakker, J. and Clothier, T. J. Food Sci., 1997, 62, 653-664. Harrison, M. and Hills, B. P. Int. J. Food Sci. & Tech., 1996, 31, 167-176. Hills, B. P. and Harrison, M. Int. J. Food Sci. & Tech., 1995, 30, 425-436. Jonsson, B. Wennerstrom, H. Nilsson, P. G. and Linse, P., Colloid & Polymer Sci., 1986, 264, 77-88. Mackie, J. S. and Meares, P. , Proc. R. Soc. London, A, 1955, 232, 498-509. Maxwell, J. C. Α., Trteatise on Electricity and Magnetism, 2 ed.; Clarendon Press, Oxford, 1881; Vol. 1., p435. Muhr, A. H., Blanshard, J. M. V., Polymer., 1982, 23, 1012-1026. Stapley, A. G. F., Fryer, P. J. and Gladden, L. F., AIChE J., 1998, 44, 1777-1789. Saravacos, G. D. In Engineering Properties of Foods; Rao, M. A. and Rizvi, S. S. H., Eds, Marcel Dekker, New York. 1986. Schwartzberg, H. G. J. Food Sci., 1975, 40, 211-213. Sherwood, T. K., Pigford, R. L., and Wilke, C. R. Mass Transfer, McGraw-Hill, New York, 1975. Waggoner, R. Α., Blum, F. D. and MacElroy, J. M. D., Macromolecules, 1993, 26, 6841-6848 Washington, C. Int. J. Pharmaceutics, 1990, 58, 1-12. Wedzicha, B. L. and Couet, C., Food Chemistry, 1996, 55, 1-6 nd


Modeling Flavor Release from Oil-Containing Gel Particles - American

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