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Numerical simulation of mass transfer during the osmotic dehydration of biological tissues
Computational Materials Science 35 (2006) 75–83 www.elsevier.com/locate/commatsci
Numerical simulation of mass transfer during the osmotic dehydration of biological tissues Long-yuan Li
*
School of Engineering and Applied Science, Aston University, Birmingham B4 7ET, UK Received 1 February 2005; received in revised form 14 March 2005; accepted 15 March 2005
Abstract In this paper a mathematical model based on mass transfer in plant tissues is developed. The model takes into account the diffusion and convection of each constituent within the tissue. The driving force for the convection is assumed to be the gradient of hydrostatic pressure. The mass balance equation for the transport of each constituent is established separately for intracellular and extracellular volumes but taking into account the mass exchange across the cell membrane between the intracellular and extracellular volumes. The mass transfer results in not only the change of intracellular and extracellular volumes but also the shrinkage of whole tissue. The model allows us to quantitatively simulate the time evolution of intracellular and extracellular volumes, which was observed in histological sections under the microscope. 2005 Elsevier B.V. All rights reserved.
1. Introduction Osmotic dehydration is a water removal process in which a tissue consisting of individual cells (such as fruits and vegetables) is immersed in a concentrated osmotic solution containing one or more solutes. During the process, the solute diffuses into the cellular structure. The penetration of the solute creates a chemical potential difference across the cell membrane and draws the water out from the cellular structure. Consequently, there are at least two simultaneous, counter-current flows during the osmotic process: the solute flow from the concentrated solution into the tissue and the water flow out from the tissue into the solution. During the osmotic dehydration, the volume of water flowing out of the system is much higher than the entering volume of solids. Driving forces are maintained due to this difference in flows, and shrinkage of the tissue arises as a consequence. *
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Osmotic dehydration is an important technology that enables both the removal of water from the product and the modification of its functional properties by the impregnation of desired solutes. The process is often applied as a pretreatment process for fruits and vegetables, which reduces the physical, chemical and biological changes during drying at high temperature. Removal of water by osmotic dehydration from biological materials has recently received considerable attention as a potential alternative or supplementary operation to conventional drying, freezing and dehydro-freezing processes. This is because the process is carried out at a low temperature and does not involve the phase change that would be present in the processes of drying and freezing, resulting in high quality products and low operating cost. Moreover, the improvement of textural properties, flavor retention and color stabilization without sulfite addition is also obtained as a result of this form of pretreatment. Osmotic dehydration is a relatively new process applied to food dewatering and as a pre-treatment technology to reduce energy consumption and to improve the
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
quality of fruit products. However, its application in the food industry is still restricted not only due to lack of understanding of the mechanism that controls the simultaneous counter-current mass flows [5,10,11,15] but also, for example, the problem of recycling and/or reuse of diluted solutions. The existing research carried out to model mass transfer in osmotic dehydration is mainly based on FickÕs law using the assumption that the driving force for the process is the solute concentration gradient [1,4,12] or on the mass conservation of solutes and solution together with empirically assumed volume reduction equations at different stages [16,18–20], which can only deal with the problems of one dimension. This paper presents a mathematical model for simulating the mass transfer during osmotic dehydration processes in plant tissues. The model is based on the mechanism of diffusion and convection of any mobile material that can transport in plant tissues. The driving force for the convection is assumed to be the gradient of hydrostatic pressure. The mass balance equation for the transport of each constituent is established separately for intracellular and extracellular volumes but taking into account the mass exchange across the cell membrane between the intracellular and extracellular volumes. The mass transfer results in not only the change of intracellular and extracellular volumes but also the shrinkage of whole tissue. The model allows us to quantitatively simulate the time evolution of intracellular and extracellular volumes.
2. Mathematical model 2.1. Tissue structure model Consider the simplified structure of a typical plant tissue initially of unit volume. The total volume of the simplified matrix consists of two main components. The first part is the extracellular volume that comprises the total cell wall volume and the total intercellular free space volume. The second part is the intracellular volume that comprises the total cellular volume. The actual tissue thus can be regarded as an aggregation of N cells each of which has an identical volume, that is, N X0 ¼ N ðXc0 þ Xf0 Þ 1
ð1Þ
where N is the total number of cells in the referenced unit volume, X0 is the initial volume of the average cell, Xc0 and Xf0 are the initial intracellular and extracellular volumes of the average cell. To make the mathematical treatment feasible, it is assumed that the actual tissue is represented approximately by a homogeneous and isotropic cubic arrangement of representative spherical cells. Therefore the intracellular and extracellular volumes for the average cell at any time can be expressed as follows:
X ¼ Xc þ Xf ;
Xc ec X0 ;
Xf ef X0
ð2Þ c
f
where X is the volume of the average cell, X and X are the intracellular and extracellular volumes of the average cell, ec and ef are the volume fractions of the intracellular and extracellular volumes in the referenced unit volume. Note that, ec + ef = 1 if the volume change only involves the volume exchange between the intracellular and extracellular volumes and ec + ef 5 1 if the volume change also involves the change of the average cell size. Therefore, the value of ec + ef actually represents the change of the average cell volume during the osmotic process. The diameter of the sphere representing the intracellular volume of the average cell can be calculated based on the spherical arrangement model as follows: c 1=3 c 1=3 c 1=3 6X 6e X0 6e X0 dc ¼ ¼ ; Sc ¼ p ð3Þ p p p where d c is the diameter of the sphere representing the intracellular volume of the average cell and S c is the corresponding surface area of the sphere. The interface surface between the intracellular and extracellular volumes in the referenced unit volume thus is NS c = S c/X0. 2.2. Equations of mass balance When a plant tissue is immersed in a concentrated osmotic solution containing one or more solutes, the solute in the solution will diffuse into the tissue. The penetration of the solute creates concentration gradients within the intracellular and extracellular volumes as well as between the intracellular and extracellular volumes, which drive the water out from the cellular structure. Consequently, there are at least two simultaneous, counter-current flows occurring in three different paths during the osmotic process. The two simultaneous, counter-current flows are the solute flow from the concentrated solution into the tissue and the water flow out from the tissue into the solution. The three paths are the transport within the intracellular volume, namely symplastic transport, the transport within the extracellular volume, namely free-space transport, and the transport across the cell membrane, namely apoplastic transport. In addition, the tissue shrinks because the amount of water flowing out, in general, is more than that of solutes diffusing in. The changes of volumes in the intracellular and extracellular volumes in turn lead to the existence of hydrostatic pressure gradient, which causes the convection flow of solutions in the intracellular and extracellular volumes. The equations of mass balance for each species (solute) within the referenced frame can be expressed as: In the intracellular volume (s = 1,2, . . . , M): o c c Sc ðe C s Þ ¼ rJ cs þ J fc ot X0 s
ð4Þ
L.-y. Li / Computational Materials Science 35 (2006) 75–83
In the extracellular volume (s = 1,2, . . . , M): o f f Sc ðe C s Þ ¼ rJ fs J fc ot X0 s
ð5Þ
where t is the time, C cs and C fs are the concentrations of species s in the intracellular and extracellular volumes, J cs and J fs are the molar fluxes of species s in the intracellular and extracellular volumes, J fc s is the flux of species s from the extracellular volume into the intracellular volume through cell membrane (plasmalemma) and M is the total number of the solutes considered in the solution. The fluxes of species s in the intracellular and extracellular volumes can be expressed as J cs J fs
¼ ¼
Dcs ec rC cs Dfs ef rC fs Dcs
þ V ðe C cs Þ þ V f ðef C fs Þ c
c
ð6Þ ð7Þ
Dfs
where and are the diffusion coefficients of species s in the intracellular and extracellular volumes, V c and V f are the bulk velocities of solutions in the intracellular and extracellular volumes. Note that the gradient operator in Eqs. (4) and (5) is different from that in Eqs. (6) and (7). For the former the derivatives are with respect to the referenced coordinates, while for the latter the derivatives are with respect to the transient coordinates. The relationship between these two gradients can be expressed as follows: rð Þ ¼
ec
1 rð Þ þ ef
ð8Þ
During the mass transport the intracellular and extracellular volumes vary with both time and position. The change of the intracellular and extracellular volumes can be determined by considering the conservation of the total volumes of solutes and water in the intracellular and extracellular volumes. Note that for ideal dilute solutions the volume change due to diffusion can be ignored. Therefore, the following volume conservation equations can be established, oec Sc ¼ rðV c ec Þ þ J fc ot X0 0 f oe Sc ¼ rðV f ef Þ J fc ot X0 0
ð9Þ ð10Þ
where J fc 0 is the volume flux of the water from the extracellular volume into the intercellular volume through cell membrane. The bulk velocities of solutions in the intracellular and extracellular volumes can be determined by employing DarcyÕs law in terms of the pressure gradients in the intracellular and extracellular volumes as follows [6,17]: Vc ¼
Kc rP c ; lc
Vf ¼
Kf rP f lf
ð11Þ
where K c and K f are the medium permeabilities for the solution in the intracellular and extracellular volumes,
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lc and lf are the corresponding dynamic viscosities, Pc and P f are the hydrostatic pressures in the intracellular and extracellular volumes, respectively. The hydrostatic pressures in the intracellular and extracellular volumes are related to their volumes and thus can be assumed as the functions of the intracellular and extracellular volumes as follows: P c P f ¼ Ec P f ¼ Ef ln
Xc Xc0 þ P c0 P f0 Xc0
Xf þ P f0 Xf0
ð12Þ ð13Þ
where E c andE f are the elastic moduli of the intracellular and extracellular volumes, P c0 and P f0 are the initial values of P c and P f, respectively. Eq. (13) is developed on the assumptions made by Dainty [2] and Nobel [9] for biological tissues, which was based on the experimentally found relationship between turgor pressure and cell volume change, whereas Eq. (12) is based on the elastic assumption for cytoplasm. 2.3. Mass exchange across cell membrane The driving force for the apoplastic transport of solutes and water across the cell membrane is the chemical potential difference between the solutions in the intracellular and extracellular volumes, which can be characterised using irreversible thermodynamics with the phenomenological equations of Kedem and Katchalsky [3] as follows (see also [7,8]): ! M X fc J 0 ¼ Lp DP rs RT DC s ð14Þ s¼1
J fc s
¼ ð1 rs ÞC s J fc 0 þ xs RT DC s
ð15Þ
where Lp is the membrane hydraulic conductivity, rs is the membrane reflection coefficient to solute s, xs is the membrane permeability to solute s, DP and DCs are the differences in pressures and concentrations (solute s) between the intracellular and extracellular volumes, C s is the average concentration (solute s) of intracellular and extracellular volumes, R is the universal gas constant, T is the absolute temperature. The sign convention used in Eqs. (14) and (15) is defined as follows. Net fluxes of water and solutes from extracellular volume into intracellular volume are taken as positive. The differences in solute concentrations and in pressures across the membrane are calculated by the extracellular values minus the intracellular values. The two terms in Eq. (14) are called hydraulic flow and osmotic flow; while the two terms related to the solute flux in Eq. (15) are called ultrafiltration and diffusion, respectively [14]. Eqs. (4), (5), (9) and (10) are the governing equations for determining concentrations, C cs and C fs , and volume
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
fractions, ec and ef, in which the mass fluxes in the intracellular and extracellular volumes are defined by Eq. (6) and (7), and the mass exchange between the intracellular and extracellular volumes is defined by Eq. (14) and (15). For given initial and boundary conditions the solute concentrations and volume fractions at any time, at any place can be determined by solving the governing Eqs. (4), (5), (9) and (10), no matter the problem is one-dimensional or two-dimensional.
3. Numerical results As an example, a one-dimensional transport problem is presented here, which is to simulate the evolution of the osmotic dehydration process of discs of plant tissues. It is assumed that there are two solutes (s = 1,2) in the osmotic solution and therefore there are six coupled partial differential equations that need to be solved. The values used for defining initial conditions (at t = 0), boundary conditions at one side of the disc (x = L = 1 cm) and transport parameters employed in the numerical computations are given in Table 1. The boundary at x = 0 is assumed as the symmetric boundary. The governing equations are solved numerically using Matlab programs. The results are summarised as follows: When the disc is immersed in the concentrated osmotic solution, the solutes in the solution start to diffuse into the disc through both intracellular and extracellular volumes. Fig. 1a shows the concentration distribution profiles of the two solutes at four different times. It can be seen that the concentrations are higher in the extracellular volume than in the intracellular volume. This is because the diffusion coefficients assumed in the extracellular volume are generally higher than those assumed in the intracellular volume. The slight difference in the concentration profiles between the two solutes found in the intracellular and extracellular volumes is also attributed to the difference of diffusion coefficients assumed for the two solutes. The concentration difference between the intracellular and extracellular volumes creates a chemical potential difference across the cell membrane, which drives water to flow from the intracellular volume into the extracellular volume (that is so-called osmotic flow). As a consequence of this osmotic flow, the intracellular volume will decrease, while the extracelluar volume increases. The volume changes in the intracellular and extracellular volumes alter the distributions of hydrostatic pressures, which in turn cause the convective flow of solutions in both the intracellular and extracellular volumes and the hydraulic flow of water through the cell membrane. Fig. 1b shows the distribution profiles of the volume fractions and corresponding hydrostatic pressures in the intracellular and extracellular volumes.
As is to be expected, the extracellular volume increases with time, while the intracellular volume decreases with time. It is noticed from the figure that the distributions of hydrostatic pressures in the intracellular and extracellular volumes are very similar. This is mainly because the elastic modulus of the cell cytoplasm is much smaller than that of the cell wall. Therefore, any significant difference in hydrostatic pressures between the intracellular and extracellular volumes will lead to the change of the intracellular volume and thus the pressure difference will immediately be compensated by the changed intracellular volume. Note that the water flow through the cell membrane changes only the relative values of the intracellular and extracellular volumes. It does not change the whole cell volume. The change of the cell volume is purely due to the convective flows taking place in both the intracellular and extracellular volumes caused by the hydrostatic pressure gradients. Fig. 2 shows the evolution of Table 1 Parametric data used in calculations Parameter
Intracellular
Extracellular
Diffusion coefficients
Dc1 ¼ 0.05Df1 Dc2 ¼ 0.05Df2
Df1 ¼ 10.8 1010 m2/s Df2 ¼ 7.2 1010
Permeability/ dynamic viscosity
Kc lc
Kf lf
Elastic modulus
Ec = 1.0 · 103 Ef = 1.0 · 105
f
¼ 0.05 Klf
Lp = 1.8 · 1015
Membrane hydraulic conductivity
m4/(N s)
N/m2 m3/(N s)
r1 = 0.7 r2 = 0.7
Membrane reflection coefficients Membrane permeabilities
¼ 9.0 1014
Unit
RTx1 = 1.8 · 1011
m/s
RTx2 = 1.8 · 1011 kmol/m3
Initial concentrations
C c1 ¼ 0.001 C c2 ¼ 0.001
C f1 ¼ 0.001 C f2 ¼ 0.001
Initial volume fractions Initial pressures
ec = 0.9
ef = 0.1
P c0 ¼ 1.0 105 P f0 ¼ 1.0 105
N/m2
Concentrations at boundary
C c1 ¼ 0.5 C c2 ¼ 0.5
C f1 ¼ 0.5 C f2 ¼ 0.5
kmol/m3
Volume fractions at boundary
ec = 0.1
ef = 0.1
Pressures at boundary
Pc = 1.0 · 105 Pf = 1.0 · 105
Gas constant and temperature Initial spherical diameter of intracellular volume
RT = 8314 · (273 + 20) d c0 ¼ 1.0 105
N/m2 N m/kmol m
L.-y. Li / Computational Materials Science 35 (2006) 75–83
79
Fig. 1. Distribution profiles of (a) concentrations and (b) volume fractions and hydrostatic pressures in intracellular and extracellular volumes at different times.
the change of the average cell volume during the process. It is noticed that the volume change starts from the place close to the side exposed to the osmotic solution and gradually moves in as time goes on. Fig. 3 provides a comparison between the present numerical
simulation and the experimental results reported by Stuart [13] for a potato disk of 14 mm in diameter and 1 mm in thickness, subjected to an osmotic dehydration. It can be seen that the present numerical results are well consistent with the experimental data if the parametric
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
Fig. 3. Evolution of the cell volume during the process of osmotic dehydration (Dc1 ¼ 5.4 1010 m2 =s, Df1 ¼ 0.95 1010 m2 =s, Dc2 ¼ 0.67Dc1 , Df2 ¼ 0.67Df1 , ecb ¼ efb ¼ 0.2, other values are the same as those given in Table 1).
Fig. 2. Distribution profiles of the cell volume changes at different times.
values are chosen correctly. From the figure it is also seen that the maximum volume reduction can reach to 50% for a 40 min treatment. The performance of the present model is highly dependent upon the values employed for the parameters describing diffusion and convection in the intracellular and extracellular volumes, and the membrane constants describing the fluxes of solutes and solution flow over
the cell membrane. The influence of specific variables on the overall behaviour of the model can be illustrated effectively by using the model to predict the response of the system studied to changes in values of these variables. Figs. 4 and 5 show the distribution profiles of solute concentrations and volume fractions after the 38 min
(a) Intracellular K/µ/5 K/µ 5K/µ
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4 0.6 Distance, cm
0.8
1
Fig. 4. Influence of permeability/dynamic viscosity on the process (t = 38 min).
L.-y. Li / Computational Materials Science 35 (2006) 75–83
(a) Intracellular
C1, mole/l
0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
D/5 D 5D
0.4
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
81
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 5. Influence of diffusion coefficient on the process (t = 38 min).
(a) Intracellular Lp/2 Lp 2Lp
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 6. Influence of membrane hydraulic conductivity on the process (t = 38 min).
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
(a) Intracellular ω/20 ω 20ω
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 7. Influence of membrane permeability on the process (t = 38 min).
(a) Intracellular σ =0.3 σ =0.7 σ =1
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 8. Influence of membrane reflection coefficients on the process (t = 38 min).
L.-y. Li / Computational Materials Science 35 (2006) 75–83
treatment, when the ratios of permeability-to-dynamic viscosity and diffusion coefficients in both the intracellular and extracellular volumes are altered by a factor of 5. It is seen from Fig. 4 that the increase in the ratios of permeability-to-dynamic viscosity can significantly increase the convective flow of the water in the extracellular volumes although it slightly reduces the transport speed of solutes in both the intracellular and extracellular volumes. On the other hand, the increase in diffusion coefficients speeds up the transport of solutes in both the intracellular and extracellular volumes and also increases the exchange flows of solutes and solution across the cell membrane (see Fig. 5). Figs. 6–8 show the influence of the membrane constants on the transport of solutes and solution. It is seen from Fig. 6 that the increase of the membrane hydraulic conductivity has had a little influence on the transport of solutes but has significantly increased the water flow between the intracellular and extracellular volumes. In comparing to the membrane hydraulic conductivity, the influence of the membrane permeability (see Fig. 7) and membrane reflection coefficients (see Fig. 8) on the water flow between the intracellular and extracellular volumes seems not very significant, although their influence on the transport of solutes in both the intracellular and extracellular volumes are visible. 4. Conclusions A mathematical model based on mass transfer in plant tissues has been developed. The mass balance equation for the transport of each constituent in the tissue was established separately for intracellular and extracellular volumes but taking into account the mass exchange across the cell membrane between the intracellular and extracellular volumes. The mass transport in intracellular and extracellular volumes includes the diffusion due to concentration gradient and the convection due to the hydrostatic pressure gradient, whereas across the cell membrane it includes the osmotic and ultrafiltrational flow. The proposed mass transfer equations were used to determine the concentrations and volume fractions in the intracellular and extracellular volumes in which the hydrostatic pressures are assumed as functions of the intracellular and extracellular volumes. Numerical examples have been provided, which showed that the mass transfer taking place in the tissue results in not only the change of intracellular and extracellular volumes but also the shrinkage of whole tissue. The model allows us to predict the time evolution of intracellular and extracellular volumes as well as the time history of the whole tissue change.
83
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Numerical simulation of mass transfer during the osmotic dehydration of biological tissues Long-yuan Li
*
School of Engineering and Applied Science, Aston University, Birmingham B4 7ET, UK Received 1 February 2005; received in revised form 14 March 2005; accepted 15 March 2005
Abstract In this paper a mathematical model based on mass transfer in plant tissues is developed. The model takes into account the diffusion and convection of each constituent within the tissue. The driving force for the convection is assumed to be the gradient of hydrostatic pressure. The mass balance equation for the transport of each constituent is established separately for intracellular and extracellular volumes but taking into account the mass exchange across the cell membrane between the intracellular and extracellular volumes. The mass transfer results in not only the change of intracellular and extracellular volumes but also the shrinkage of whole tissue. The model allows us to quantitatively simulate the time evolution of intracellular and extracellular volumes, which was observed in histological sections under the microscope. 2005 Elsevier B.V. All rights reserved.
1. Introduction Osmotic dehydration is a water removal process in which a tissue consisting of individual cells (such as fruits and vegetables) is immersed in a concentrated osmotic solution containing one or more solutes. During the process, the solute diffuses into the cellular structure. The penetration of the solute creates a chemical potential difference across the cell membrane and draws the water out from the cellular structure. Consequently, there are at least two simultaneous, counter-current flows during the osmotic process: the solute flow from the concentrated solution into the tissue and the water flow out from the tissue into the solution. During the osmotic dehydration, the volume of water flowing out of the system is much higher than the entering volume of solids. Driving forces are maintained due to this difference in flows, and shrinkage of the tissue arises as a consequence. *
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Osmotic dehydration is an important technology that enables both the removal of water from the product and the modification of its functional properties by the impregnation of desired solutes. The process is often applied as a pretreatment process for fruits and vegetables, which reduces the physical, chemical and biological changes during drying at high temperature. Removal of water by osmotic dehydration from biological materials has recently received considerable attention as a potential alternative or supplementary operation to conventional drying, freezing and dehydro-freezing processes. This is because the process is carried out at a low temperature and does not involve the phase change that would be present in the processes of drying and freezing, resulting in high quality products and low operating cost. Moreover, the improvement of textural properties, flavor retention and color stabilization without sulfite addition is also obtained as a result of this form of pretreatment. Osmotic dehydration is a relatively new process applied to food dewatering and as a pre-treatment technology to reduce energy consumption and to improve the
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
quality of fruit products. However, its application in the food industry is still restricted not only due to lack of understanding of the mechanism that controls the simultaneous counter-current mass flows [5,10,11,15] but also, for example, the problem of recycling and/or reuse of diluted solutions. The existing research carried out to model mass transfer in osmotic dehydration is mainly based on FickÕs law using the assumption that the driving force for the process is the solute concentration gradient [1,4,12] or on the mass conservation of solutes and solution together with empirically assumed volume reduction equations at different stages [16,18–20], which can only deal with the problems of one dimension. This paper presents a mathematical model for simulating the mass transfer during osmotic dehydration processes in plant tissues. The model is based on the mechanism of diffusion and convection of any mobile material that can transport in plant tissues. The driving force for the convection is assumed to be the gradient of hydrostatic pressure. The mass balance equation for the transport of each constituent is established separately for intracellular and extracellular volumes but taking into account the mass exchange across the cell membrane between the intracellular and extracellular volumes. The mass transfer results in not only the change of intracellular and extracellular volumes but also the shrinkage of whole tissue. The model allows us to quantitatively simulate the time evolution of intracellular and extracellular volumes.
2. Mathematical model 2.1. Tissue structure model Consider the simplified structure of a typical plant tissue initially of unit volume. The total volume of the simplified matrix consists of two main components. The first part is the extracellular volume that comprises the total cell wall volume and the total intercellular free space volume. The second part is the intracellular volume that comprises the total cellular volume. The actual tissue thus can be regarded as an aggregation of N cells each of which has an identical volume, that is, N X0 ¼ N ðXc0 þ Xf0 Þ 1
ð1Þ
where N is the total number of cells in the referenced unit volume, X0 is the initial volume of the average cell, Xc0 and Xf0 are the initial intracellular and extracellular volumes of the average cell. To make the mathematical treatment feasible, it is assumed that the actual tissue is represented approximately by a homogeneous and isotropic cubic arrangement of representative spherical cells. Therefore the intracellular and extracellular volumes for the average cell at any time can be expressed as follows:
X ¼ Xc þ Xf ;
Xc ec X0 ;
Xf ef X0
ð2Þ c
f
where X is the volume of the average cell, X and X are the intracellular and extracellular volumes of the average cell, ec and ef are the volume fractions of the intracellular and extracellular volumes in the referenced unit volume. Note that, ec + ef = 1 if the volume change only involves the volume exchange between the intracellular and extracellular volumes and ec + ef 5 1 if the volume change also involves the change of the average cell size. Therefore, the value of ec + ef actually represents the change of the average cell volume during the osmotic process. The diameter of the sphere representing the intracellular volume of the average cell can be calculated based on the spherical arrangement model as follows: c 1=3 c 1=3 c 1=3 6X 6e X0 6e X0 dc ¼ ¼ ; Sc ¼ p ð3Þ p p p where d c is the diameter of the sphere representing the intracellular volume of the average cell and S c is the corresponding surface area of the sphere. The interface surface between the intracellular and extracellular volumes in the referenced unit volume thus is NS c = S c/X0. 2.2. Equations of mass balance When a plant tissue is immersed in a concentrated osmotic solution containing one or more solutes, the solute in the solution will diffuse into the tissue. The penetration of the solute creates concentration gradients within the intracellular and extracellular volumes as well as between the intracellular and extracellular volumes, which drive the water out from the cellular structure. Consequently, there are at least two simultaneous, counter-current flows occurring in three different paths during the osmotic process. The two simultaneous, counter-current flows are the solute flow from the concentrated solution into the tissue and the water flow out from the tissue into the solution. The three paths are the transport within the intracellular volume, namely symplastic transport, the transport within the extracellular volume, namely free-space transport, and the transport across the cell membrane, namely apoplastic transport. In addition, the tissue shrinks because the amount of water flowing out, in general, is more than that of solutes diffusing in. The changes of volumes in the intracellular and extracellular volumes in turn lead to the existence of hydrostatic pressure gradient, which causes the convection flow of solutions in the intracellular and extracellular volumes. The equations of mass balance for each species (solute) within the referenced frame can be expressed as: In the intracellular volume (s = 1,2, . . . , M): o c c Sc ðe C s Þ ¼ rJ cs þ J fc ot X0 s
ð4Þ
L.-y. Li / Computational Materials Science 35 (2006) 75–83
In the extracellular volume (s = 1,2, . . . , M): o f f Sc ðe C s Þ ¼ rJ fs J fc ot X0 s
ð5Þ
where t is the time, C cs and C fs are the concentrations of species s in the intracellular and extracellular volumes, J cs and J fs are the molar fluxes of species s in the intracellular and extracellular volumes, J fc s is the flux of species s from the extracellular volume into the intracellular volume through cell membrane (plasmalemma) and M is the total number of the solutes considered in the solution. The fluxes of species s in the intracellular and extracellular volumes can be expressed as J cs J fs
¼ ¼
Dcs ec rC cs Dfs ef rC fs Dcs
þ V ðe C cs Þ þ V f ðef C fs Þ c
c
ð6Þ ð7Þ
Dfs
where and are the diffusion coefficients of species s in the intracellular and extracellular volumes, V c and V f are the bulk velocities of solutions in the intracellular and extracellular volumes. Note that the gradient operator in Eqs. (4) and (5) is different from that in Eqs. (6) and (7). For the former the derivatives are with respect to the referenced coordinates, while for the latter the derivatives are with respect to the transient coordinates. The relationship between these two gradients can be expressed as follows: rð Þ ¼
ec
1 rð Þ þ ef
ð8Þ
During the mass transport the intracellular and extracellular volumes vary with both time and position. The change of the intracellular and extracellular volumes can be determined by considering the conservation of the total volumes of solutes and water in the intracellular and extracellular volumes. Note that for ideal dilute solutions the volume change due to diffusion can be ignored. Therefore, the following volume conservation equations can be established, oec Sc ¼ rðV c ec Þ þ J fc ot X0 0 f oe Sc ¼ rðV f ef Þ J fc ot X0 0
ð9Þ ð10Þ
where J fc 0 is the volume flux of the water from the extracellular volume into the intercellular volume through cell membrane. The bulk velocities of solutions in the intracellular and extracellular volumes can be determined by employing DarcyÕs law in terms of the pressure gradients in the intracellular and extracellular volumes as follows [6,17]: Vc ¼
Kc rP c ; lc
Vf ¼
Kf rP f lf
ð11Þ
where K c and K f are the medium permeabilities for the solution in the intracellular and extracellular volumes,
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lc and lf are the corresponding dynamic viscosities, Pc and P f are the hydrostatic pressures in the intracellular and extracellular volumes, respectively. The hydrostatic pressures in the intracellular and extracellular volumes are related to their volumes and thus can be assumed as the functions of the intracellular and extracellular volumes as follows: P c P f ¼ Ec P f ¼ Ef ln
Xc Xc0 þ P c0 P f0 Xc0
Xf þ P f0 Xf0
ð12Þ ð13Þ
where E c andE f are the elastic moduli of the intracellular and extracellular volumes, P c0 and P f0 are the initial values of P c and P f, respectively. Eq. (13) is developed on the assumptions made by Dainty [2] and Nobel [9] for biological tissues, which was based on the experimentally found relationship between turgor pressure and cell volume change, whereas Eq. (12) is based on the elastic assumption for cytoplasm. 2.3. Mass exchange across cell membrane The driving force for the apoplastic transport of solutes and water across the cell membrane is the chemical potential difference between the solutions in the intracellular and extracellular volumes, which can be characterised using irreversible thermodynamics with the phenomenological equations of Kedem and Katchalsky [3] as follows (see also [7,8]): ! M X fc J 0 ¼ Lp DP rs RT DC s ð14Þ s¼1
J fc s
¼ ð1 rs ÞC s J fc 0 þ xs RT DC s
ð15Þ
where Lp is the membrane hydraulic conductivity, rs is the membrane reflection coefficient to solute s, xs is the membrane permeability to solute s, DP and DCs are the differences in pressures and concentrations (solute s) between the intracellular and extracellular volumes, C s is the average concentration (solute s) of intracellular and extracellular volumes, R is the universal gas constant, T is the absolute temperature. The sign convention used in Eqs. (14) and (15) is defined as follows. Net fluxes of water and solutes from extracellular volume into intracellular volume are taken as positive. The differences in solute concentrations and in pressures across the membrane are calculated by the extracellular values minus the intracellular values. The two terms in Eq. (14) are called hydraulic flow and osmotic flow; while the two terms related to the solute flux in Eq. (15) are called ultrafiltration and diffusion, respectively [14]. Eqs. (4), (5), (9) and (10) are the governing equations for determining concentrations, C cs and C fs , and volume
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
fractions, ec and ef, in which the mass fluxes in the intracellular and extracellular volumes are defined by Eq. (6) and (7), and the mass exchange between the intracellular and extracellular volumes is defined by Eq. (14) and (15). For given initial and boundary conditions the solute concentrations and volume fractions at any time, at any place can be determined by solving the governing Eqs. (4), (5), (9) and (10), no matter the problem is one-dimensional or two-dimensional.
3. Numerical results As an example, a one-dimensional transport problem is presented here, which is to simulate the evolution of the osmotic dehydration process of discs of plant tissues. It is assumed that there are two solutes (s = 1,2) in the osmotic solution and therefore there are six coupled partial differential equations that need to be solved. The values used for defining initial conditions (at t = 0), boundary conditions at one side of the disc (x = L = 1 cm) and transport parameters employed in the numerical computations are given in Table 1. The boundary at x = 0 is assumed as the symmetric boundary. The governing equations are solved numerically using Matlab programs. The results are summarised as follows: When the disc is immersed in the concentrated osmotic solution, the solutes in the solution start to diffuse into the disc through both intracellular and extracellular volumes. Fig. 1a shows the concentration distribution profiles of the two solutes at four different times. It can be seen that the concentrations are higher in the extracellular volume than in the intracellular volume. This is because the diffusion coefficients assumed in the extracellular volume are generally higher than those assumed in the intracellular volume. The slight difference in the concentration profiles between the two solutes found in the intracellular and extracellular volumes is also attributed to the difference of diffusion coefficients assumed for the two solutes. The concentration difference between the intracellular and extracellular volumes creates a chemical potential difference across the cell membrane, which drives water to flow from the intracellular volume into the extracellular volume (that is so-called osmotic flow). As a consequence of this osmotic flow, the intracellular volume will decrease, while the extracelluar volume increases. The volume changes in the intracellular and extracellular volumes alter the distributions of hydrostatic pressures, which in turn cause the convective flow of solutions in both the intracellular and extracellular volumes and the hydraulic flow of water through the cell membrane. Fig. 1b shows the distribution profiles of the volume fractions and corresponding hydrostatic pressures in the intracellular and extracellular volumes.
As is to be expected, the extracellular volume increases with time, while the intracellular volume decreases with time. It is noticed from the figure that the distributions of hydrostatic pressures in the intracellular and extracellular volumes are very similar. This is mainly because the elastic modulus of the cell cytoplasm is much smaller than that of the cell wall. Therefore, any significant difference in hydrostatic pressures between the intracellular and extracellular volumes will lead to the change of the intracellular volume and thus the pressure difference will immediately be compensated by the changed intracellular volume. Note that the water flow through the cell membrane changes only the relative values of the intracellular and extracellular volumes. It does not change the whole cell volume. The change of the cell volume is purely due to the convective flows taking place in both the intracellular and extracellular volumes caused by the hydrostatic pressure gradients. Fig. 2 shows the evolution of Table 1 Parametric data used in calculations Parameter
Intracellular
Extracellular
Diffusion coefficients
Dc1 ¼ 0.05Df1 Dc2 ¼ 0.05Df2
Df1 ¼ 10.8 1010 m2/s Df2 ¼ 7.2 1010
Permeability/ dynamic viscosity
Kc lc
Kf lf
Elastic modulus
Ec = 1.0 · 103 Ef = 1.0 · 105
f
¼ 0.05 Klf
Lp = 1.8 · 1015
Membrane hydraulic conductivity
m4/(N s)
N/m2 m3/(N s)
r1 = 0.7 r2 = 0.7
Membrane reflection coefficients Membrane permeabilities
¼ 9.0 1014
Unit
RTx1 = 1.8 · 1011
m/s
RTx2 = 1.8 · 1011 kmol/m3
Initial concentrations
C c1 ¼ 0.001 C c2 ¼ 0.001
C f1 ¼ 0.001 C f2 ¼ 0.001
Initial volume fractions Initial pressures
ec = 0.9
ef = 0.1
P c0 ¼ 1.0 105 P f0 ¼ 1.0 105
N/m2
Concentrations at boundary
C c1 ¼ 0.5 C c2 ¼ 0.5
C f1 ¼ 0.5 C f2 ¼ 0.5
kmol/m3
Volume fractions at boundary
ec = 0.1
ef = 0.1
Pressures at boundary
Pc = 1.0 · 105 Pf = 1.0 · 105
Gas constant and temperature Initial spherical diameter of intracellular volume
RT = 8314 · (273 + 20) d c0 ¼ 1.0 105
N/m2 N m/kmol m
L.-y. Li / Computational Materials Science 35 (2006) 75–83
79
Fig. 1. Distribution profiles of (a) concentrations and (b) volume fractions and hydrostatic pressures in intracellular and extracellular volumes at different times.
the change of the average cell volume during the process. It is noticed that the volume change starts from the place close to the side exposed to the osmotic solution and gradually moves in as time goes on. Fig. 3 provides a comparison between the present numerical
simulation and the experimental results reported by Stuart [13] for a potato disk of 14 mm in diameter and 1 mm in thickness, subjected to an osmotic dehydration. It can be seen that the present numerical results are well consistent with the experimental data if the parametric
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
Fig. 3. Evolution of the cell volume during the process of osmotic dehydration (Dc1 ¼ 5.4 1010 m2 =s, Df1 ¼ 0.95 1010 m2 =s, Dc2 ¼ 0.67Dc1 , Df2 ¼ 0.67Df1 , ecb ¼ efb ¼ 0.2, other values are the same as those given in Table 1).
Fig. 2. Distribution profiles of the cell volume changes at different times.
values are chosen correctly. From the figure it is also seen that the maximum volume reduction can reach to 50% for a 40 min treatment. The performance of the present model is highly dependent upon the values employed for the parameters describing diffusion and convection in the intracellular and extracellular volumes, and the membrane constants describing the fluxes of solutes and solution flow over
the cell membrane. The influence of specific variables on the overall behaviour of the model can be illustrated effectively by using the model to predict the response of the system studied to changes in values of these variables. Figs. 4 and 5 show the distribution profiles of solute concentrations and volume fractions after the 38 min
(a) Intracellular K/µ/5 K/µ 5K/µ
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4 0.6 Distance, cm
0.8
1
Fig. 4. Influence of permeability/dynamic viscosity on the process (t = 38 min).
L.-y. Li / Computational Materials Science 35 (2006) 75–83
(a) Intracellular
C1, mole/l
0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
D/5 D 5D
0.4
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
81
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 5. Influence of diffusion coefficient on the process (t = 38 min).
(a) Intracellular Lp/2 Lp 2Lp
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 6. Influence of membrane hydraulic conductivity on the process (t = 38 min).
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L.-y. Li / Computational Materials Science 35 (2006) 75–83
(a) Intracellular ω/20 ω 20ω
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 7. Influence of membrane permeability on the process (t = 38 min).
(a) Intracellular σ =0.3 σ =0.7 σ =1
C1, mole/l
0.4 0.3
C2, mole/l
0.4 0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
1
0
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
ε
(b) Extracellular
0
0.2
0.4
0.6
0.8
1
0
1
1
0.5
0.5
0
0
0.2
0.4 0.6 Distance, cm
0.8
1
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.8
1
0.4 0.6 Distance, cm
Fig. 8. Influence of membrane reflection coefficients on the process (t = 38 min).
L.-y. Li / Computational Materials Science 35 (2006) 75–83
treatment, when the ratios of permeability-to-dynamic viscosity and diffusion coefficients in both the intracellular and extracellular volumes are altered by a factor of 5. It is seen from Fig. 4 that the increase in the ratios of permeability-to-dynamic viscosity can significantly increase the convective flow of the water in the extracellular volumes although it slightly reduces the transport speed of solutes in both the intracellular and extracellular volumes. On the other hand, the increase in diffusion coefficients speeds up the transport of solutes in both the intracellular and extracellular volumes and also increases the exchange flows of solutes and solution across the cell membrane (see Fig. 5). Figs. 6–8 show the influence of the membrane constants on the transport of solutes and solution. It is seen from Fig. 6 that the increase of the membrane hydraulic conductivity has had a little influence on the transport of solutes but has significantly increased the water flow between the intracellular and extracellular volumes. In comparing to the membrane hydraulic conductivity, the influence of the membrane permeability (see Fig. 7) and membrane reflection coefficients (see Fig. 8) on the water flow between the intracellular and extracellular volumes seems not very significant, although their influence on the transport of solutes in both the intracellular and extracellular volumes are visible. 4. Conclusions A mathematical model based on mass transfer in plant tissues has been developed. The mass balance equation for the transport of each constituent in the tissue was established separately for intracellular and extracellular volumes but taking into account the mass exchange across the cell membrane between the intracellular and extracellular volumes. The mass transport in intracellular and extracellular volumes includes the diffusion due to concentration gradient and the convection due to the hydrostatic pressure gradient, whereas across the cell membrane it includes the osmotic and ultrafiltrational flow. The proposed mass transfer equations were used to determine the concentrations and volume fractions in the intracellular and extracellular volumes in which the hydrostatic pressures are assumed as functions of the intracellular and extracellular volumes. Numerical examples have been provided, which showed that the mass transfer taking place in the tissue results in not only the change of intracellular and extracellular volumes but also the shrinkage of whole tissue. The model allows us to predict the time evolution of intracellular and extracellular volumes as well as the time history of the whole tissue change.
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