mass transfer in cellular tissues. part i: the mathematical model

Journal ofFood Engineering 13 (1991) 199-220

Mass Transfer in Cellular Tissues. Part I: The Mathematical Model

Michkle Marcotte”

Department of Food Science, University of Alberta, Edmonton, Alherta. Canada T6G 2P5

Christian J. Toupin

Food Research and Development Centre, 3600 Casavant Blvd West, St-Hyacinthe.

Quebec. Canada J2S XE3

Marc Le Maguer

Department of Food Science, University of Guelph, Guelph. Ontario. Canada N 1 G 2 W 1

(Received 8 May 1989; revised version received 25 July 1990: accepted 1 August 1990)

A HS TRA CT

The kinetics of equilibration of a biological structure with osmotic solutions were studied based on the internal cellular structure of the plant material.

The model developed by Toupin (Osmotically induced mass transfer

in biological systems: the single cell and the tissue behavior. PhD thesis, Dept of Food Science, University of Alberta, Edmonton, Alberta, Canada, 1984) was used to describe the mass transfer of sucrose and water in potato material. This model was modified to give a closer thermo- dynamical description of the forces involved in the osmotic process.

The model was used to simulate movement of water and sucrose in potato tubers. The comparison between the simulations and the experimental data is presented in the second part of this paper.

*To whom correspondence should be addressed at: Food Research and Development Centre. 3600 Casavant Blvd West, St-Hyacinthe. Quebec, Canada J2S 8E3.

199 Journal of Food Engineering 0260-8774/91/$03.50 - 0 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

200 M. Marcotte, C. J. Toupin, M. Le Maguer

fi

a,,

ff; d D d I L

Ef Jzr N P 9

4 R Rwm .% T

t: P W

?i

Z

NOTATION

Activity Constant, eqn (26) Constant, eqn (26) Area ( m2) Constant, eqn (9) Standard diffusion coefficient (mz/s) Apparent diffusivity ( m2/s) Length of the ECUC equivalent cylindrical unit cell (m) Phenomenological coefficient (kmo12/Pa s m”) Mass (kg) Molecular weight (kg/kmol) Molality (kmol/lOOO kg water) Flux ( kg/m2 s) Hydrostatic pressure (N/m2) Permeability coefficient (m/s) Constant, eqn (9) Radius (m) Transmembrane transport (kg/s) Universal gas constant (J/mol K) Temperature (K or “C) Barycentric velocity (m/s) Volume ( m3) Partial molar volume (m”/kmol) Weight fraction on a dry weight basis Mole fraction Moisture content on a dry weight basis Distance (m)

Variable, eqn ( 11) Constant, eqn ( 16) Constant, eqn ( 16) Activity coefficient Void fraction Time (s)

Constant, eqns (30) and (31) Chemical potential (J/kmol) Elastic modulus (N) Volumic mass (kg/m”) Tortuosity (-) Matrix potential (N/m2)

Modelling of osmotic treatments ofplants 201

1(, Diffusibility w Weight fraction

Subscripts b

C

ci ~~ fS

i

in

j m 0

Pr s st V

w

Buffer Cellular Refers to the cjcell Cell wall Dry matter Free space Extracellular space Interface Refers to the j th component Membrane Refers to the initial condition Proteins Sucrose Starch Vacuole Water

Superscripts in Inside the cellular volume out In the extracellular space

; Surface Temperature

0 Reference state at the temperature considered and at atmospheric pressure

* Full turgor a3 Infinite dilution

A REVIEW OF EXISTING MODELS

Very little has been published in the area of modelling mass transport in plant material during osmosis. Generally, the assumption of diffusive transfer of solutes from a solid to a surrounding solvent or vice versa is used to model the observed mass transfer in biological structures. Since biological tissues are essentially porous or cellular solids in which a gas and/or a liquid are immobilized, the rate of mass transfer is approxi- mately predicted by appropriate solutions of the simplified unsteady


state second-order Fickian equation provided that the apparent or effective diffusivity is known. Crank (1975) gave a detailed theoretical description of the diffusion process.

Hawkes and Flink (1978) tried to quantify the apparent mass transfer coefficient by plotting the normalized solid content of apple slices (percent total solids change based on the initial total solids) versus the square root of time and by calculating the slope at each point of the curve in order to compare different systems (i.e. different osmotic solutions).

More recently, Conway et al. (1983) considered for modelling purposes that only a simple diffusion of water was occurring upon osmotic dehydration of apples in a sucrose solution. The solution given by Crank (1975) was used in order to calculate the theoretical moisture ratio and to compare it with the experimental one. An empirical correlation was found between the diffusivity, the sucrose concentration and the temperature.

The conditions assumed in order to use solutions for the unsteady state Fickian equation do not necessarily simulate an osmosis process, as in osmosis there is usually more than one important flow (Karel, 1975). Lerici et al. ( 1985) pointed out that to characterize an osmotic treatment it is important to take into account not only the weight reduction and the water loss but also the solids gain. In addition to diffusion, it is possible for other mechanisms, such as shrinkage, to be of importance in the overall mass transport (Lenart & Flink, 1984).

Soddu and Gioia ( 1979) tried to model the process of sugar diffusing from a beet immersed in water with an effective diffusion coefficient corrected for the void fraction and the tortuosity. They found that the comparison was unsatisfactory particularly for the experimental data from fresh sugar beets. However, a better agreement was observed for badly damaged beets. Viable cells seemed to play an important role in the overall mass transfer during osmosis.

Another approach was used by Guennegues ( 1986) in order to model the overall mass transfer in the multicomponent system occurring in plant tissue during osmosis. The overall mass transfer was assessed using irreversible thermodynamics, taking into account the fluxes of different components of the system and the interactions between different flows.

The inclusion of a formal description of the tissue structure in the modelling of the drying of plant material was investigated by Roman et al. ( 1983) and Crapiste et al. ( 1984) in their studies on air dehydration of foodstuffs.

There are many models of the fundamental water relations in plant tissue. However, most of these models have been developed to describe the behavior of the tissue under natural growing conditions.

Modelling of osmotic treatments of plants 203

Philip (195&r, b, C) considered the water exchange between a linear aggregation of cells through their membranes. He pointed out that the assumption of the diffusion and osmotic phenomena in whole pieces of tissue being analogous to that in single cells is not justified. The development was made for non-permeating solutes.

Using the irreversible thermodynamic approach (Kedem & Katchalsky, 1958; Dainty, 1963) Molz and Hornberger ( 1973) extended the model developed by Philip (195&r, b, c) to include the effects of a diffusible solute.

Molz and Ikenberry (1974) developed a theory which allowed for a water flux in the cell wall pathway as well as transport from cell to cell through their membranes.

Molz ( 1976) introduced a model which described the water flow through the apoplast and the symplast of the plant tissue for non-diffusing solutes.

It is interesting to note that the model developed by Molz et ul. ( 1979) included some of the discrete cellular geometry and anatomy of the tissue.

The microscopic description of mass transport phenomena in plant storage tissue seemed to offer the greatest potential in modelling the mass transfer in vegetables and fruits during osmotic dehydration. It has been put forward by plant physiologists and used to model the air dehydration of foodstuffs with successful results.

Recently, Toupin (1986) developed a model of the mass transport phenomena in plant material based on the model presented by Molz et

rd. (1979). Diffusion of non-permeating and permeating species in the matrix was considered, as well as the shrinkage of the whole structure. Toupin (1986) showed with sufficient evidence that the transport of matter across biological membranes obeys the laws of thermodynamics of irreversible processes and the permeability data of these membranes can be obtained. The model was compared satisfactorily with experimental measurements in terms of cell volume changes. Most of the prob- lems of this model arose from the description of the tissue behavior. Firstly, some assumptions have been made in order to deal with the global structure changes of the tissue or shrinkage. The critical cell volume was introduced in order to take into account a loss of the integrity of the cells. Secondly, the lack of equilibrium data describing the behavior of the tissue during osmosis, similar to sorption data in drying. led to an improper description of the changes of the different phases of the cell and, in particular, of the cellular volume. The apparent non- osmotic volume was introduced to compensate.

The purpose of this study was to improve the model developed by Toupin ( 1986). The model was modified to give a closer thermodynamic


description of the forces in the osmotic process. The model was also simplified. Firstly, symplastic transport was not considered. Secondly, the surface area of the plasmalemma of the cells was assumed to be fixed. Finally, sucrose has been chosen as the solute of the osmotic medium. Because of its impermeability to the cell membrane, the exchanges across the plasmalemma were limited to the flux of water. The space available for the movement of sucrose was restricted to the extracellular space between protoplasts. The kinetic behavior of potato tissue during osmosis in sucrose solutions was then quantified.

DEVELOPMENT OF THE MODEL

Qualitative description of the mass transport of sucrose in potato tissue

Nowadays, although the water relation in a single plant cell is adequately described and well understood, the events are complex when osmosis is applied to the whole plant tissue structure (Dainty, 1976).

As indicated in Fig. 1, three accepted pathways that a solute or solvent may follow while traversing a plant tissue have been identified as being responsible for the behavior of the plant material during osmosis (Molz, 1976).

Apoplastic transport, which occurs outside the cell membrane, can be visualized as a diffusion of molecules in the cell wall and the intercellular spaces between cells.

TRANSHEMBRANE TRANSPORT

SYHPLASTIC TRANSPORT I

PLAsnALEnnA PLASnODEsnATA

I

CELL WALL FREE SPACE

Fig. 1. Mechanisms of mass transport inside the biological structure.


The discovery of the plasmodesmata led to another pathway. Symplastic transport is inside the plasmalemma and is characterized by a movement of molecules from one cell to another through small channels (Molz, 1976).

Finally, the transmembrane transport is an exchange between the protoplast and the free space which comprises the intercellular space and the cell wall.

The rate of swelling or shrinking of a plant tissue immersed in an osmotic solution will depend on both extracellular solute diffusion and cell membrane permeation if we consider that there is no particular relation between cells. The behavior of the whole tissue is the same as the behavior of a single cell.

The modelling of the mass transfer of water and sucrose in potato tissue during osmosis requires the definition of a realistic representation of the biological structure in terms of physical dimensions and shape.

Figure 2 represents a typical potato cell. Parenchymatous cells which compose the edible part of the potato are the most common cell type in the tubers. The cells are large, polyhedrical or spherical in shape. The potato tuber possesses a single large vacuole which may occupy over 90% of the total cell volume. The vacuole solution is a relatively diluted homogeneous aqueous phase. Minerals, sucrose, glucose, fructose, organic acids and vitamins are the major solutes. It is known to be a storage reservoir for toxic products and metabolites (Nobel, 1983). The tonoplast separates the vacuole from the cytoplasm. The cytoplasm is a

Tonoplast Cell wall Plasmalemma

Vacuole I Diluted aqueous solution Soluble solids

minerals soluble sugars organic acids vitamins

Intercellular space

>

Cytoplasm

Drganelles

e.g. Starch granules Protein granules for PotatoTissue

Fig. 2. Parenchymatous cell of potato tissue.


more complex phase containing many colloids and membrane-bounded organelles (Nobel, 1983). Reserve materials such as starch and proteins are present in the cytoplasm. In the raw tuber, starch is present as microscopic granules in the leucoplasts lining the interior of the walls of the cells of the parenchyma tissue (Talburt et al., 1975). By contrast, the concentration of soluble solutes is very low. A membrane separates the cell wall from the cytoplasm. The plasmalemma, which is the main barrier, regulates what enters and leaves a plant cell. The selective permeability of the plasmalemma membrane towards sucrose and water determines the osmotic behavior. Because of the cell wall, high hydrostatic pressure inside the cell can exist. It provides the rigidity to allow a build-up of pressure. Cells of potato tubers are closely packed but there is a certain amount of intercellular space. Woolley ( 1962) found the intercellular spaces to be interconnecting and air-filled except at the cut surface, where most spaces were water-filled to a depth of a few microns.

Because of the relative coherence that exists in plant storage tissue (e.g. potato tissue), one can define a unit cell as a representative microscopic unit of the structure, characterized by the average typical properties of the real cells. Figure 3(a) shows a simplified representation of an

Cellular f- Cytoplasm volume -proteins

4 -starch

Extracellular volume

Vacuole

equeous solution -minersls -soluble

sugars

-cell wall -free space

(4 I

Plasmalemma

UNIT CELL

Fig. 3. (a) Average unit cell; (b) cubic arrangement of average unit cells.


average unit cell of parenchyma potato tissue. Figure 3(b) represents a cubic arrangement of these average unit cells in a tissue structure.

Toupin ( 1986) developed the concept of the equivalent cylindrical unit cell (ECUC) and proposed an arrangement of these ECUC in a hypothetical parenchyma tissue in order to avoid this particular constraint. Figure 4(a) shows the ECUC. Each cell is assimilated as three coaxial cylinders. The inner cylinder (no. 1) acts as a buffer, the middle cylinder (no. 2) represents the extracellular volume which includes the cell wall and the free space, while the outer cylinder (no. 3) comprises the cellular volume with its vacuole and cytoplasm. Figure 4(b) shows that the hypothetical tissue structure is approximated by an arrangement of columns. Each column is formed by a linear assemblage of ECUC.

Toupin (1986) pointed out that this representation allowed for the continuity of the extracellular volume and the discontinuity of the cellular volume of the cells imposed by the tissue structure. This hypothetical tissue exhibits also homogeneous and isotropic properties. The diffusion in the extracellular volume or apoplastic transport is linearized, restricting the analysis of unidirectional bulk diffusion in the tissue. Although the description of mass transport in plant storage tissue is greatly simpli-

w

Buffer

R

Extracellular volume

PRESENTATIVE

Fig. 4. (a) Equivalent cylindrical unit cell; (b) cubic arrangement of ECUCs.


fied by the introduction of the ECUC concept, Toupin et al. (1989) discussed thoroughly the geometrical transpositions needed in order to fully describe the corresponding phenomena.

In order to model the mass transport of sucrose and water in potato tissue during osmosis, a column of ECUC is selected as representing the entire system. The equations describing the isothermal mass transport phenomena are established. The transmembrane transport is modelled by relations based on the theory of irreversible thermodynamics while the overall bulk diffusion occurring in the extracellular space is modelled by relations associated with the extended form of the second-order Fick equation. The changes occurring in the cellular volume in terms of volume and concentration of both species are monitored on a cell-to-cell basis.

Thermodynamic description of the forces involved in exchanges across membranes

The term transmembrane transport (Rwm) indicates the mass transfer of water occurring across the plasmalemma complex. Since the membrane is fully non-permeable to sucrose, only water transfer is possible:

Rwm = L,,A,uW,

where L,, is the phenomenological coefficient describing the water permeability of the plasmalemma membrane. A,u,, is the difference in chemical potential of water across the membrane:

Ay,, = /& -p;; (2)

where ,u&, is the chemical potential of water inside the cellular volume and ,uF$ is the water chemical potential of the extracellular volume with respect to the same reference state.

The extracellular volume is composed of the free space and the cell wall which are usually filled with air. However, any pretreatment, such as placing disks of biological material in distilled water and storing them, ensures not only that the tissue is initially at full turgor but also that the intercellular spaces and cell wall volumes are likely to be water-filled. Sucrose penetrates the extracellular volume as the osmotic treatment progresses and contributes more and more to the osmotic potential of the extracellular space. The interactions between the cellulose matrix and the solution are assumed to be negligible so that the matrix potential in the extracellular space is considered negligible. The pressure potential


term is zero, since the extracellular medium is directly in contact with an osmotic solution open to the atmosphere. Thus, the chemical potential of water in the extracellular space is simply:

,u 0ut _ wm cc::= ST In&

where .Y? T In ciWi reflects the increasing contribution of the sucrose penetrating the extracellular space during the osmotic treatment. Li,,., is calculated by:

4, = Yd, (4) A correlation was made between the activity coefficient, y,, and the

mole fraction of sucrose, x,, using the data from Hougen et nl. (1954).

The following correlation was found:

In yW = - xz( 3.36 + 90*5x, - 623.6.~:)

The temperature correction can be calculated as (Marcotte, 1988):

azi = u’w”, exp(0.000 497 9 .N:t) (6)

The cellular volume includes the vacuole and the cytoplasm which comprises starch and proteins. Restricting the treatment to non-electrolytes with the reference state of pure water at the temperature under con- sideration and at atmospheric pressure, the general expression for the chemical potential of water for a vegetable system is:

(7)

The first term L% T In &,,, the osmotic potential, reflects the contribution of dissolved solutes to the chemical potential of water. The second term v,,,q,,, the matrix potential, arises because of the strong interactions between water and solids of large surface area present in the system. The final term v,(P- P”), the pressure potential, expresses the dependence of the water chemical potential on hydrostatic pressure.

A typical composition of potato tissue is found in the literature (Crapiste & Rotstein, 1982). It is required in order to assign the proper weight of the contribution of the different phases of the potato tissue.

At the vacuole phase, the osmotic potential is predominant (Crapiste & Rotstein, 1982). Since the vacuole aqueous solution is composed of small amounts of minerals and soluble sugars, these solutes contribute to the osmotic potential. For multicomponent systems, a relationship between the partial water activity of each component and the water activity of the mixture was derived by Ross ( 1975) by considering that all


water present in the system forms a solution with each of the components independently of each other. Thus:

On the basis of equivalent binary water-jth component:

~wj= 10 -.V,l I - 1,,1”

X WJ (9)

The water mole fraction

XV X,j=

Xv+ Wjaj

where

WV ai=M,

(10)

(11)

The values of H and q for sugars and minerals are available in the literature (Crapiste & Rotstein, 1982).

At the cytoplasm phase, since starch and the proteins are the major components, the matrix potential is important. Empirical correlations for sorption isotherms of starch and proteins are given by Crapiste & Rotstein ( 1982) and used to estimate the matrix potential:

P~in-P~i$= 2 T ln[ 1 - expj 53*4759X,“;“0’s] (12)

and

#&-$nin= .%5!-[ - 0-0208Xp,“6’*9] (13)

The total water content of the cellular volume is calculated by:

X =X,1 wst + Xrr wpr + XV (14)

The contribution of the cell wall phase was found to be negligible (Crapiste & Rotstein, 1982). In practice, an iteration procedure is used to converge on the chemical potential and the composition of the phases, from the total moisture content, using eqns (8)-( 14). The temperature effect is assumed to be negligible.

The last term to be considered is the pressure potential term of the cell. Although Rotstein and Cornish (19786) found that, for the prediction of the sorptional equilibrium relationship for apples, the pressure potential of the cell (vacuolar phase) could be neglected in the high


moisture content region, according to Dainty (1976) the contribution of the pressure potential term in the water relations of plant cells is very important under normal growing conditions. Rotstein and Cornish ( l978b) defined the high moisture content region of a sorption isotherm as extending from full turgor down to the moisture content at which the pressure inside the cell (mostly from the vacuole and more generally the cellular volume ( V,) which comprises the cytoplasm and the vacuole in the case of potato) is equal to the external pressure. This state is named zero turgor or incipient plasmolysis of the cell.

According to Dainty (1976), for small changes in cell volumes, one can write:

Nilsson et al. (1958) have found that the elastic modulus, g, is a function of the turgor pressure for potatoes:

/I, and p2 are found in the Literature for potato tissue. p, is defined as the variation of 6 with respect to (P, - Pz), the hydrostatic pressure in the cellular volume. p2 is the value of 6 at incipient plasmolysis. (PC - P:!) is the excess of the turgor pressure over the atmospheric pressure.

The equation relating pressure potential to the cellular volume is integrated:

(17)

where (P: - P;) is defined as the excess of pressure above the atmospheric pressure at full turgor. The term V,/ L’z represents the ratio of the actual cellular volume to the cellular volume at full turgor.

The chemical potential inside the cellular volume, ,uz,,, - ,H z:, can be estimated either from the vacuole phase contribution or from the cytoplasm phase contribution, since we assume that, at any time, there is an equality between the chemical potential of both phases. The chemical potential inside the cellular volume is calculated from the vacuole phase contribution. Since p $m = ,~4 :A’:

A/G,, = Vw(Pc- PF)+ ,SU ltl Ciwv- ST In ~wi

Thus the flux of water across the membrane is expressed by:

(18)


(19)

From the literature, the permeability coefficient of the plasmalemma membrane, YW,, can be found. Rotstein and Cornish (1978~) observed a certain degree of confusion in their survey of the literature values of the permeability constant which mostly depends on the driving force and the mathematical expressions for the water permeability. The phenomenological coefficient is defined as:

(20)

It is important to notice that ,M:; - ,u$’ varies with the position in the extracellular space whereas ,U t - ,u Eb varies on a cell-to-cell basis.

Extracellular equations of continuity

In developing the equations describing the conditions associated with the mass transport in the extracellular space, the following assumptions are added to the list of assumptions made by Toupin (1986) in order to model the mass transfer in the extracellular cylinder:

(1) isothermal mass transport in a semi-infinite medium is considered;

(2) the space available for the sucrose to move is restricted to the extracellular volume (i.e. sucrose does not permeate the plasmalemma membrane);

(3) the plasmalemma membrane is permeable to water.

Although p varies with the concentration of sucrose and water with respect to time and distance, the variation effect of the density with the concentration of sucrose and water in the interstitium is assumed to be negligible in comparison to the contribution of the transmembrane flux and the convective and diffusional fluxes.

A differential volume element of cross-sectional area A i is defined in the extracellular volume. Figure 5 shows the contribution of the different fluxes in the differential volume element. Applying the law of conservation of mass for each species, one can write various mass balances. Since it is a binary system, two of the three possible equations are independent. The equation for sucrose and the total equation of continuity are used. The convective diffusion equation in the extracellular space for sucrose reads:


Osmatic Solution

>

AR Rc

CellUl8r Volume

Rwm

T

\/ Ri

f’w/z+sz +- NWIZ lnterstitium

NsIz*6z -+ NSIZ

Buffer 11‘1

Fig. 5. Mass balance in a differential volume clement.

The second term on the right-hand side of eqn (2 1) is of particular relevance as it characterizes the changes in the bulk movement of the sucrose solution with respect to z. It is found from the total equation of continuity which reads:

a21 2nRi 1 1 aA. -=Rwm az Aip A, a0

The apparent diffusivity is defined as:

(22)

(23)

The diffusibility ( I+!J) is introduced in order to take into account the effect of the microstructural properties of the extracellular volume which can hinder the process of diffusion along the pathway. The hindrance effect of the interstitium is a function of the combined resistances of the intercellular space (ly,) and the cell wall (q!~,,,) (Toupin, 1986). Since there is a negligible resistance to be expected in the free space volume, the diffusibility $_+, = 1, whereas the diffusibility of the cell wall is defined as:

$!I,, = : ‘ W

(24)

214 h4. Marcotte, C. J. Toupin, h4. Le Maguer

where E,, is the void fraction of the cell wall and rcw the cellulosic matrix. Finally:

V= ’ -[(l - V,,)V~,lVil

is the tortuosity of

(25)

where V,, is the total volume of the cell wall comprising the voids and spaces as well as the cellulosic fibers. It is assumed to remain constant.

The system water-sucrose solution is well documented in the literature in terms of volume fixed-frame-of-reference diffusion coefficients (D,). The description of the concentration dependence of the diffusion coefficient of sucrose in water can be found experimentally. Toupin (1986) fitted a polynomial to measurements obtained at different concentrations (Landolt-Bornstein tables, 1969):

In ++ =a,,+a,p, ( i \

(26)

where D, is the volume fixed-frame-of-reference diffusion coefficient of sucrose in water at a concentration p,, and D$ is the volume fixed- frame-of-reference diffusivity at infinite dilution. Henrion ( 1964) has shown that the volume fixed-frame-of-reference diffusivity is a unique function of concentration. The diffusivity at infinite dilution for other temperatures can be estimated from the relationship D,p/ T= 0.0157.

Yao ( 198 1) pointed out that, when dealing with diffusion, the diffusional flux with respect to a fixed volume frame-of-reference is usually used and the diffusion coefficients are measured accordingly. In this particular case, both diffusion coefficients are equal. Consequently, the volume fixed-frame-of-reference solution diffusion coefficient (D,) was corrected using the partial molar volume of water (VW), the molecular weight of water (M,) and the average density (p), because of the choice of the frame of reference (i.e. fixed mass frame-of-reference). Partial molar volumes and molecular weights are obtained from standard tables (Weast etal., 1983-1984).

Intracellular equations of change

According to Toupin (1986), it is reasonable to assume a perfect mixing in the cellular volume. In contrast, here the plasmalemma surface area is assumed to be fixed and the symplastic transport is not considered. The equations are established on a cell-to-cell basis.

The changes in the mass concentration of species present in the cellular volume are functions of the water loss from the cellular volume through the membrane. Since the symplastic transport is assumed


negligible, the cells behave independently of each other. For each cellular volume, one can write specifically a water mass balance. The concentration of water in the cellular volume is expressed in terms of total moisture content (X) on a dry weight basis. This is convenient since a dehydration process takes place in the cellular volume:

dX ’ 2nRi -=

I Rwm dz

d6’ mdm (27)

The mass of dry matter of the cellular volume is estimated from the full turgor material:

m dm G I/cPwWdm

where ti()d,,, is the proportion of dry matter in the cellular volume of the potato tissue at full turgor.

The cellular volume variation is a function of the volume of water loss through the membrane on a cell-to-cell basis. Neglecting the change of volume on mixing:

d V, ’ 2JcRi -= de IL!,, I

v,,, Rwm dz i29)

Geometrical time derivative relationships

Toupin ( 1986) introduced the shrinkage of the biological structure in the model describing the osmotic treatment of the plant material. In the study on the internal structural changes of the potato tissue during osmotic dehydration, Marcotte (1988) confirmed the three stages of dehydration assumed by Toupin (1986). However, it was found experimentally that in the third stage of dehydration, the changes in total volume are the result of the combined effects of the reduction in extracellular and cellular volume so that:

dv d V, -CK- d8 d0

(30)

It has to be pointed out that volume changes are also included in mass conservation equations through the variation of the surface area of extracellular space (A i) which is related to the extracellular volume changes of eqn (30). Finally, the equation for total volume changes reads:


s=(l+K)$ (31)

where the compliance factor ( FC) describes how cellular volume changes affect the extracellular volume of the cell and the total volume of the cells. From full turgor to incipient plasmolysis K = 0. As soon as the pressure potential is zero, 1c = - 1. The equilibrium study has shown that K = 1 when a loss of the integrity of the cells is observed (Marcotte, 1988). The geometrical time derivative relationships of the ECUC are listed in Table 1.

Initial conditions

The tissue is initially at full turgor. It has been equilibrated with pure water. On average, the dimensions and contents of the cells are equal. In the interstitium at any point, there is no sucrose: ~$1 i = 0. Finally, there is no transmembrane transport (Rwm = 0) or diffusive transfer (NW, NJ i.e. t, = 0, dv/dz = 0.

TABLE 1 Geometrical Time Derivative Relations of the ECUC

(32)

(33)

(34)

(35)

(36)

(37)


Boundary conditions

Three locations are considered: the surface of the tissue, the center. and the interface between two adjoining cells.

Sur$ace of the tissue

Assuming that there is no resistance to mass transfer at the surface of the biological structure, the concentration of sucrose in the interstitium at the surface is equal to the concentration of sucrose of the bath solution or osmotic solution. For a system under a vigorous agitation, the assumption is justified so that ,obl ‘i = p,,,.

Center of the tissue

Considering a semi-infinite medium, the center is defined as a plane parallel to the surface situated at a distance equal to the thickness of the tissue. For any point located on that plane and beyond, the extracellular and cellular conditions remain at any time equal to the initial conditions: psi,, = 0. Since the intracellular equations of change are defined with respect to the mass of dry matter, X1(, =X0. For all species, no diffusive transfer occurs (i.e. dv/dzl, and v I,, equal zero).

Inte$ace between two adjoining cells

Toupin et al. (1989) have made a detailed analysis of the problem occurring at the surface between adjoining cells. As pointed out by Toupin (1986), the major consequence of describing the behavior of the representative column on a cell-to-cell basis is that it creates discon- tinuities at the interface between two adjoining cells. A special analysis of mass transfer is required to link the transport phenomena occurring in the interstitium of each cell at the interface. Assuming that at any interface, the interstitium sucrose concentrations on both sides are the same (,o,,.,. = p,,,_, ,,), it then follows that the convective transfer for sucrose is given by:

[A iv Ic;,in = [A iu lc;+ 1.i” and for diffusive transfer:

(38)

(39)

218 M. Marcotte. C. J. Toupin, M. Le Maguer

TABLE 2 Summary of the Model Equations

Description Equation number.s

Transmembrane transport (Rwm) out _ Out*

-~wm iuWl! ,nO

- ~u:“,-.4V, - Pressure potential

- J&n - Rwm Extracellular equation of continuity - sucrose and total Intracellular equations of change - Total moisture content - Cellular volume Geometrical time derivative relations Interface between two cells

(3)-(b)

(W (20) (19)

(21)-(26)

(27)~(28) (29)

(30)~(37) (38)-(39)

Equations (38) and (39) state that there is conservation of mass for both sucrose and water in the interstitium.

CONCLUSION

Table 2 summarizes the analysis by presenting the list of the model equations.

In the second part of this paper, the results of the simulations will be presented and compared with experimental data on the osmotic treatment of potato tubers in sucrose solutions.

ACKNOWLEDGEMENTS

The financial support offered by the Fends pour La Formation des Chercheurs et /‘Aide h la Recherche from the Government of Quebec is gratefully acknowledged by the first author (M.M.).

Acknowledgement is also made to the Natural Sciences and Engineer- ing Research Council of Canada for financially supporting this work.

REFERENCES

Conway, J., Castaigne, F., Picard, G. & Vovan, X. (1983). Mass transfer considerations in the osmotic dehydration of apples. Can. Znst. Food Sci. Technol. .I., 16(l), 25-9.

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mass transfer in cellular tissues. part i: the mathematical model

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