Microscale modeling of coupled water transport and mechanical deformation of fruit tissue during dehydration

Journal of Food Engineering 124 (2014) 86–96

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Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng

Microscale modeling of coupled water transport and mechanical deformation of fruit tissue during dehydration Solomon Workneh Fanta a, Metadel K. Abera a, Wondwosen A. Aregawi a, Quang Tri Ho a, Pieter Verboven a, Jan Carmeliet c,d, Bart M. Nicolai a,b,⇑ a

BIOSYST-MeBios, KU Leuven, Willem de Croylaan 42, B-3001 Leuven, Belgium Flanders Centre of Postharvest Technology (VCBT), Willem de Croylaan 42, B-3001 Leuven, Belgium Building Physics, Swiss Federal Institute of Technology Zurich (ETHZ), Wolfgang-Pauli-Strasse 15, 8093 Zürich, Switzerland d Laboratory for Building Science and Technology, Swiss Federal Laboratories for Materials Testing and Research (Empa), Überlandstrasse 129, 8600 Dübendorf, Switzerland b c

a r t i c l e

i n f o

Article history: Received 16 August 2013 Received in revised form 30 September 2013 Accepted 2 October 2013 Available online 11 October 2013 Keywords: Structure Turgor Diffusion Mechanics Cell wall

a b s t r a c t Water loss of fruit typically results in fruit tissue deformation and consequent quality loss. To better understand the mechanism of water loss, a model of water transport between cells and intercellular spaces coupled with cell deformation was developed. Pear (Pyrus communis L. cv. Conference) was chosen as a model system as this fruit suffers from shriveling with excessive water loss. A 2D geometric model of cortex tissue was obtained by a virtual fruit tissue generator that is based on cell growth modeling. The transport of water in the intercellular space, the cell wall network and cytoplasm was predicted using transport laws using the chemical potential as the driving force for water exchange between different microstructural compartments. The different water transport properties of the microstructural components were obtained experimentally or from literature. An equivalent microscale model that incorporates the dynamics of mechanical deformation of the cellular structure was implemented. The model predicted the apparent tissue conductivity of pear cortex tissue to be 9.42 ± 0.40  1015 kg m1 s1 Pa1, in the same range as those measured experimentally. The largest gradients in water content were observed across the cell walls and cell membranes. A sensitivity analysis of membrane permeability and elastic modulus of the wall on the water transport properties and deformation showed that the membrane permeability has the largest influence. The model can be improved further by taking into account 3-D connectivity of cells and intercellular pore spaces. It will then become feasible to evaluate measures to reduce water loss of fruit during storage and distribution using the microscale model in a multiscale modeling framework. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Fresh fruits are mostly composed of water, the unique universal solvent that is fundamentally important in all life processes. Water loss equates to loss of saleable weight, and thus means a direct loss in revenue, as well as affects overall fruit quality. Measures that minimize water loss after harvest will usually enhance profitability. Loss in weight of only 5% will cause many perishable commodities to appear wilted or shriveled (Wills et al., 1998). Texture is one of the most important quality attributes of fruit and vegetables. Most plant materials contain a significant amount of water and other liquid-soluble materials surrounded by a semi-permeable membrane and cell wall. The texture of fruits and vegetables is dependent on the turgor pressure, and the composition of indi⇑ Corresponding author at: BIOSYST-MeBios, KU Leuven, Willem de Croylaan 42, B-3001 Leuven, Belgium. Tel.: +32 (0)16 322375; fax. +32 16 322955. E-mail address: [email protected] (B.M. Nicolai). 0260-8774/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfoodeng.2013.10.007

vidual plant cell walls and the middle lamella, which ‘‘glues’’ individual cells together (Barrett et al., 2010). Cell walls are accepted as the main structural component affecting the mechanical properties of fruits and vegetables (Zdunek and Konstankiewicz, 2004; Bourne, 2002; Waldron et al., 2003; Vanstreels et al., 2005). Also the turgor pressure, cell size and shape, volume of vacuole and volume of intercellular spaces, chemical composition have a major influence on tissue strength and macroscopic fruit firmness (Oey et al., 2007). Shrinkage is one of the major physical changes that occur during the dehydration process. It results from the collapse of cells during water evaporation, which has a negative impact on the quality of dehydrated product. At first, shrinkage causes changes in the shape of the product. These changes are due to the stresses developed while water is removed from the material (Rajchert and Rzace, 2009). Shrinkage during dehydration can be classified in three different types (Gekas, 1992): one-dimensional when the volume change follows the direction of diffusion; (2) isotropic or

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Nomenclature b cw D E F J Keff k l L mi Pm R T Vw v x

damping factor (–) 1 water capacity (kg kgDM Pa1 ) diffusion coefficient (m2 s1) Young’s modulus of elasticity (Pa) total force acting upon the node (N) water flux (kg m2) water conductivity (kg m1 Pa1 s1) Spring constant (N m1) cell wall length (m) thickness of the simulated tissue (m) mass of the vertex (kg) permeability of the membrane (m s1) Universal gas constant (8.314) (J mol1 K1) temperature (K) molar volume of water (18*106) (m3 mol1) velocity (m s1) position (m)

three-dimensional; and (3) anisotropic or arbitrary. Volume reduction patterns for fruits and vegetables are often of type 3 and in a less extent of type 2. Shrinkage of apple parenchyma, for example, was found to be highly anisotropic (Mavroudis et al., 1998; Moreira et al., 2000). Cellular shrinkage during dehydration has been observed during osmotic dehydration of parenchymatic pumpkin tissue (Mayor et al., 2008), apple (Lewicki and Porzecka-Pawlak, 2005) and convective drying of grapes (Ramos et al., 2004). With respect to modeling mechanical deformation of fruit tissue, most models are based on continuum mechanics. It is often assumed that the biologic material behaves as a nonlinear viscoelastic continuum. Recent work has allowed better understanding and modeling the nonlinear shrinkage of fruit tissue at the macroscale (Aregawi et al., 2013; Defraeye et al., 2013). Most microscopic works on the deformation are based on single cell analysis. Feng and Yang (1973) considered the problem of the deformation and the consequential stresses in an inflated, non-linear elastic, gas-filled spherical membrane compressed between two frictionless rigid plates. Lardner and Pujara (1980) extended this model further by considering the sphere to be filled with an incompressible liquid rather than gas. Their model was able to predict accurately the deformation of sea urchin eggs, as previously reported by Yoneda (1973). Liu et al. (1996) improved the computational algorithm, and applied the model to data on microcapsules. None of these studies allowed for water loss from the sphere. Smith et al. (1998) created a finite element model in which volume loss was included, and applied this to compression data from yeast cells (Smith et al., 2000). Using a finite element method, it was possible in principle to consider any cell wall material constitutive equation, although in practice Smith et al. (2000) only considered the linear elastic case. The more recent work is by Dintwa et al. (2011) who developed a finite element model to simulate the compression of a single suspension-cultured tomato cell, using data from Wang et al. (2004). The model could serve as a basic building block for more complex models for tissue deformation under mechanical loading. The model was limited to mechanical loading and not to the deformation due to water loss and also was applied for single cell and not for a real tissue. We have recently developed a cell growth algorithm that generates representative in silico fruit tissue geometries from increasing cell turgor in and cell wall generation by the individual cells in a tissue (Abera et al., 2013a). Using this algorithm in the reversed sense, it becomes possible to perform simulations of the deformation

Greek symbols e strain (m m1) qw density of water (kg m3) 1 qDM dry matter density (kgDM m3 ) r stress (Pa) w water potential (Pa) Subscripts a air c cell i node m cell membrane s solute T total w cell wall

mechanics of tissue as a result of hydrostatic stress occurring during water loss. We previously also modeled water diffusion in pear fruit tissue samples taking into account the cellular structure of the tissue (Fanta et al., 2013). Shrinkage was, however, taken into account in a static manner by considering different equilibrium states at different water contents, using a global shrinkage coefficient the model lacks to incorporate dynamic deformation due to water loss. Using the cell mechanics algorithm, however, the simulation of the deformation of individual cells in the tissue is possible. The aim of the present work was to combine and apply the microscale transient water transport model with the cell mechanics model for predicting cell and tissue deformation due to water loss in the actual microstructural architecture of the tissue. The model was also used to calculate the apparent water conductivity of the tissue. Pear fruit (Pyrus communis L. cv. conference) was used as a model system. Pears quickly deform resulting in shriveling as a consequence of water loss during low temperature storage (Nguyen et al., 2006).

2. Model formulation 2.1. Microscale model of water transport coupled with deformation 2.1.1. Microscale water transport model Cortex tissue of pear consists of an agglomerate of cells and intercellular spaces of different shapes and sizes (Verboven et al., 2008). To take into account this microstructure, we have introduced the microscopic layout into the modeling as the computational geometry of the model. The transport of water in the intercellular space, the cell wall network and cytoplasm were modeled using diffusion laws and irreversible thermodynamics (Noble, 1991). The full derivation of the diffusion equation for the tissue compartment (cell, cell wall and intercellular space) can be found in our previous work (Fanta et al., 2013). For the cells, the unsteady-state model of water transport reads:



qDM;c þ xc

   @ qDM;c qDM;c cw;c @w rwc cw;c c ¼ r  Dc @xc @t 1 þ xc

ð1Þ 1

where qDM,c is the dry matter density of the cell (kgDM m3 ), xc the 1 dry matter base water content (kg kgDM ), Dc the water diffusion

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coefficient inside cells (m2 s1), cw,c the water capacity 1 (kg kgDM Pa1 ), and wc the water potential of the cell (Pa). For the cell wall, the unsteady-state diffusion model is given by:

cw;w qDM;w

@ww ¼ r  qDM;w Dw cw;w rww @t

ð2Þ 1

where cw,w is the water capacity of the wall (kg kgDM Pa1 ), qDM,w 1 the dry matter density (kgDM m3 ), ww the water potential (Pa), and Dw the water diffusion coefficient (m2 s1). Unsteady diffusion in the air phase is described by:

cw;a qDM;a

@wa ¼ r  qDM;a Da cw;a rwa @t

ð3Þ

Fd ¼ bm

is a damping force, expressed as a product of a damping factor b and the vertex velocity v. The force Fw is the resultant of the net turgor pressure force Fturgor between the two adjacent cells working normal to the wall and the force associated with it.

Fw ¼ Fturgor þ Fs

J¼

qw Pm V w RT

ðwc  ww Þ

ð4Þ 2

3

with J the water flux (kg m ), qw density of water (kg m ), Pm the membrane permeability (m s1), Vw the molar volume of water (18106 m3 mol1), R the universal gas constant (8.314 J mol1 K1), T the temperature (K), wc the water potential at the cytoplasma (interior) side of the membrane (Pa), and ww the water potential at the (exterior) cell wall side of the membrane (Pa). 2.1.2. Microscale mechanics model A cell micromechanics model was used to calculate the deformation as a result of turgor loss of the individual cells in the tissue as a result of water transport (Abera et al., 2013a). In the model, the cell is represented as a closed thin walled structure, maintained in tension by turgor pressure. The cell boundary is represented as a set of walls (modeled as springs) connected at points called vertices. The cell walls of adjacent cells are modeled here as parallel and linearly elastic elements which obey Hooke’s law, an approach similar to that taken in other plant tissue models (Rudge and Haseloff, 2005; Dupuy et al., 2008, 2010). The shrinkage mechanics is modeled by considering Newton’s law. The following system of equations is solved for the velocity and position of the vertices i of the cell wall network:

mi

dmi ¼ FT;i dt

dxi ¼ mi dt

ð6Þ

where mi is the mass of the vertex (kg) which is assumed to be unity in order to simplify the model, which makes the rate of change of velocity (acceleration) of the vertices equal to the net force acting on the vertex; xi (m) and vi (m s) are the position and velocity of node i, respectively, and FT,t is the total force acting upon this node (N). Cell shrinkage or growth is then the result from the action of forces caused by a decrease or increase, respectively, of turgor pressure acting on the cell wall. The water potential of each cell from water transport simulation can be converted to turgor pressure using the relationship outlined below. The resultant force on each vertex, the position of each vertex, and, thus, the shape of the cells is then computed as follows. The total force acting on a vertex is given by the formula:

FT ¼

X

Fw þ Fd

Fturgor ¼

ð7Þ

w2W

where Fw are forces contributed by the set of walls w incident to the vertex, and

   1 P cell;1  Pcell;2 l n 2

ð10Þ

with n the unit normal vector to the cell wall, while l is the actual cell wall length at the current time. The force Fs acts along the wall and its magnitude is determined by Hooke’s law:

Fs ¼ ku

ð11Þ

where u the net vector of the cell wall,

kuk ¼ l  ln

ð12Þ

with ln the natural length of the unpressurised cell wall (m) and k the spring constant (N/m). The latter is calculated from the Young’s modulus of elasticity E (Pa) by dividing the tensile stress r (Pa) by the tensile strain e (m m1) in the elastic (initial, linear) portion of the stress–strain curve.

E¼

r Fs =A0 Fs l0 ¼ ¼ e u=l0 A0 u

ð13Þ

where Fs is the force exerted on an object; A0 is the original crosssectional area through which the force is applied (m2); and l0 is the original length of the object (m). Hooke’s law can then be derived from this formula, which describes the stiffness of an ideal spring:

Fs ¼

EA0 u ¼ ku l0

ð14Þ

so that

k¼ ð5Þ

ð9Þ

The net turgor force on the vertex is calculated by taking the difference in turgor pressure Pcell of the two adjacent cells multiplied by half the length of the wall as it is divided by the two incident vertices defining the wall:

1

where cw,a is the water capacity of the air (kg kgDM Pa1 ), qDM,a the 1 dry matter density (kgDM m3 ), wa the water potential (Pa), and Da the water diffusion coefficient (m2 s1). A simple flux law was applied at the cell membrane (Noble, 1991):

ð8Þ

EA0 l0

ð15Þ

To find the positions of each vertex of all cell walls of every single cells and, thus, the shape of the cells with time, a system of differential equations for the positions and velocities of each vertex were established and solved using a Runge–Kutta fourth and fifth order (ODE45) method. 2.1.3. Model coupling The transient water transport model is solved for certain time steps and the water loss results in loss of water potential in the cells. This change in water potential of the cells results in loss of turgor pressure. This is only true for the high range of equilibrium relative humidity values of the cells during dehydration until turgor drops to zero. When tissue drying proceeds at lower values of relative humidity, cell protoplasts will detach from the cell walls creating air spaces inside the cell wall and collapse of the cell wall, effectively changing the water transport and deformation mechanisms. This process is not calculated here. The dehydration is thus carried out in the relative humidity range of 99–97.7%. Below this value of relative humidity the turgor pressure is zero and the osmotic potential will be equal to the water potential. The osmotic potential ws can be obtained from the following equation:

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S.W. Fanta et al. / Journal of Food Engineering 124 (2014) 86–96 Table 1 Tissue properties and model parameters (25 °C). Parameter Dc

qDM,c Da

qDM,a Cwa Dw

qDM,w Pm

Value

Source 9

2

1

2.2  10 m s 111 kg m3 2.42  105 m2 s1 1.184 kg m3 1.3  1010 Pa1 41.9  1011 m2 s1 58.035 kg m3 17  106 m s1

Holz et al. (2000) Experimental result, clear fruit juice Lide and Frederikse (1994) Coulson and Richardson (2004) Fanta et al. (2012) Fanta et al. (2012) Estimate based on literature data range (Ramahaleo et al., 1999; Ferrando et al., 2002; Suga et al., 2003; Moshelion et al., 2004; Murai-Hatano and Kuwagata, 2007; Volkov et al., 2007)

Table 2 Mechanical properties. Fig. 1. Relationship between turgor pressure and water potential at high relative humidity.

ws ¼ C s RT ¼ 

qw mw

ns RT ¼ 

qw ns xc ms

RT

ð16Þ

The turgor pressure is then equal to

wp ¼ wc  ws

Parameter

Value

Source

Pcell E b C a

1 MPa 26.4–52.8 MPa 3.5 N s/lm 3.8 0.72

Abera et al. (2013a,b) Wu and Pitts (1999) Abera et al. (2013a,b) Abera et al. (2013a,b) Abera et al. (2013a,b)

ð17Þ

The relationship between turgor pressure and water potential is shown in Fig. 1 for the range considered. 2.2. Initial tissue geometry The initial geometry of the tissue was generated in silico using the microscale mechanics model outlined in Section 2.1.2. A Voronoi tessellation was used to generate a start topology of the cells. Anisotropic cell expansion then resulted from turgor pressure acting on the yielding cell wall material until full turgor (1 MPa) was reached. The size of the initial geometry was 200  200 lm and contained 60 Voronoi cells. The resulting geometry was 750  550 lm and had an average cell area of 8.28 ± 0.97 lm2 and a porosity of 6.42%. More details can be found in Abera et al. (2013a) who showed that this procedure yields tissue geometries that correspond well to actual ones visualized by synchrotron tomography.The cell wall was defined by shrinking the cell geometry until the desired cell wall thickness was obtained. The cells, the pores and the cell walls were then exported as separate bodies so that different material properties could be specified. 2.3. Model parameters The water transport model parameters were obtained from our previous work (Fanta et al., 2012, 2013) and are listed in Table 1. The water content of the cell wall and cytoplasm was calculated from their moisture isotherms (Fanta et al., 2012, 2013). The mechanical properties were determined by Abera et al. (2013a) and are listed in Table 2. We have assumed the cross sectional area of the cell wall to be 1 lm2, and the average of the initial resting length of the cell walls obtained from the Voronoi tessellation was used to calculate the k value. The nominal values in the tables were used for the simulations, while also a sensitivity analysis was conducted with respect to the main parameters, as explained below. 2.4. Implementation details The geometric models of pear cortex tissue constructed by Abera et al. (2013a) were imported into Comsol Multiphysics

3.5a (Comsol AB, Stockholm, SE) for numerical computation of the water exchange using the model equations outlined above. Meshing was performed automatically by the Comsol mesh generator and produced 303,824 quadratic elements with triangular shape by the automatic Comsol mesh generator. The procedure is outlined in Fig. 2. A sequence of time steps was considered for solving the coupled moisture transport and mechanical deformation model. In every time step the transient moisture potential field was solved. The non-linear coupled model equations were discretised over the discretisation mesh using the finite element method. A direct solver was applied for solving the resulting set of ordinary differential equations with an accuracy threshold less than 106. The results of the simulations were the water potential and water content distribution in the tissue samples as well as the water flux through the sample for a given water potential gradient across the sample. The sample was the entire tissue geometry used in the computations. Subsequently the corresponding mechanical equilibrium was calculated using a dedicated Matlab code (Matlab 7.6.0, The Mathworks, Natick, MA). The whole system of equations was numerically solved using a Runge–Kutta method of order 4 and 5. The simulation was iterated until a mechanical equilibrium state was reached. This equilibrium was assumed once the velocity of all points was below a given threshold, as the velocities would go to zero only when the system would be at a steady state. The resulting tissue geometry was then introduced and meshed again in Comsol and the next time step was initiated. In total four time steps of 50 s or eight time steps of 25 s were required to reach equilibrium. The initial water transport calculation was done on the initial pear cortex tissue geometry that was obtained using virtual fruit tissue generator with simulation time of 50 s. From this simulation, the water potential of each cells was obtained and converted into turgor pressure using the relation shown in Fig. 1. Then these set of turgor pressures are used in the shrinkage mechanics (presented in Section 2.2.2) to find the new equilibrium configuration of the cells. Computation time was 2 h for the unsteady state water transport simulation in each time step on a 8 GB of RAM quad-core PC, and 25 s for the mechanical equilibrium calculations.

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2.5. Apparent water conductivity of pear cortex tissue In silico analysis was carried out to study microscale water exchange in pear fruit tissue. A difference in relative humidity of 99/97.7% was applied to the top and bottom of the tissue geometry, respectively, while the other two lateral boundaries were defined to be insulated. The macroscopic water conductivity Keff (kg m1 Pa1 s1) of the tissue sample was computed at steady state from:

K eff ¼ J

L Dw

ð18Þ

with J (kg m2 s1) the total flux through the fruit tissue, Dw (Pa) the assigned water potential difference between the two opposite sides and L (m) the thickness of the simulated tissue. The minus sign indicates that the water diffuses from high to low potential. 2.6. Sensitivity analysis A sensitivity analysis was performed to study how sensitive a particular predicted model output was with respect to changes in model parameters. First, the effect of tissue geometry was evaluated. Hereto different tissues were randomly generated using the cell growth algorithm (Abera et al., 2013a). Then, effects of variability in tissue properties were quantified. The membrane permeability (Pm) was expected to have the highest influence on the tissue conductivity (Fanta et al., 2013), while the elastic modulus of the cell wall (E) had the largest influence on the mechanics of the tissue (Abera et al., 2013a). Here, we thus investigated the effects of both parameters on the dynamic dehydration process. The perturbation of the parameters was taken to be 10 times of the nominal value of Pm which has been shown to vary in a large range (2.5–3000 lm s1, Fanta et al., 2013) and 10% of the nominal value of E. 3. Results 3.1. Water potential and water content profile of pear tissue during dehydration Model simulations that incorporate the dynamics of mechanical deformation of the cellular structure were performed on the fruit cortex tissue samples, where a difference of RH (99–97.7%) across the tissue samples was applied (Fig. 3) such that the bottom of the sample is dehydrated to completely remove the cell turgor (1 MPa). Cells at the same position in the gradient tend to have similar and uniform water potential, which is logical due to the high water conductivity inside the cells. Gradients mainly exist from one cell to another in the direction of the applied gradient. The water content of the cells remains relatively uniform and constant during the dehydration process. This is logical as the cells shrink upon water loss, but do not change drastically their water content. 3.2. Deformation of pear tissue during dehydration During the dehydration process, there is a relatively large shrinkage at the early time and no significant change after 200 s of simulation which indicates that steady state has been achieved. This could be indeed the case, considering the small sample size considered (0.6 mm thickness). The high shrinkage at the early time was also observed on dehydration of grape tissue by Ramos et al. (2004). The deformation of the individual cells in the tissue can be seen in Fig. 3. Deformation occurs in all dimensions as the turgor loss in the cells relaxes the entire cell wall, but locally depends on the

Fig. 2. Procedure to couple water transport model with deformation.

turgor difference between adjacent cells Shrinkage of the cells is more severe on the drying bottom side of the sample than on the top equilibrium side. As the tissue is a dynamic assembly of pressurised cells that are in mechanical equilibrium, deformation of one cell will, however, affect the shape and size of other cells as well. Therefore, all cells in the tissue will deform to some extent. The reduction in the overall cell size distribution is evident from Fig. 4. The relative deformation resulting from simulations with time steps of 25 s and 50 s is shown in Fig. 5. The relative error is less for the final time and higher for initial time this is because moisture loss is higher at early times and at the end of 200 s steady state will be achieved. The relative average cell area in steady state is around 60%, and identical for the two simulations with the different time steps. 3.3. Validation of the overall shrinkage of the sample As a validation we calculated the actual water loss of pear from the desorption isotherm that we measured previously (Nguyen et al., 2004) at 99% and 98.25% (average RH for the simulation) and found it equal to 35.62%. By assuming that shrinkage is proportional to water loss this thus corresponds to a volume loss of 35.62%. This value is comparable with the proportional volume loss of 38.47% that was calculated from the simulation. 3.4. Average profile of the water potential and water content The average water potential and water content is calculated based on the potential and water content distributions across the tissue. Fig. 6a shows how the water potential decreases with time and reaches an equilibrium at 200 s. The drastic decrease in water potential is in line with Fig. 3. Fig. 6b displays the average water content change, which in relative terms is much smaller than the change in water potential. This clearly indicates that early stages of slow dehydration

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1

Fig. 3. Water potential (Pa) (left) and water content (kg kgDM ) (right) of a pear cortex tissue sample (0.6 mm thickness) dehydrated for 200 s, applying a difference of 1.3% RH (corresponding to a drop of 1 MPa turgor) across the tissue (at 25 °C). (a) t = 0 s, (b) t = 50 s, (c) t = 100 s, (d) t = 150 s and (e) t = 200 s.

processes are not necessarily associated with large changes in water content, but rather will result in deformation due to loss of turgor. Fig. 6c shows the average water content change of the wall with time. 3.5. Deformation of different tissue structures The macroscopically observed variation in water conductivity was linked to variability in the microstructure of the tissue.

Simulations for three different pear cortex tissue structures randomly generated in this study showed that differences in porosity (6.42%, 5.94%, 5.39%), connectivity and cell distribution affected the water transport in the tissue to a very minor extent (Fig. 7). The difference in conductivity between the different tissue samples is small (9.44  1015, 9.35  1015 and 9.19  1015 kg m1 s1 Pa1). It also had a minor effect on the average water potential and water content (Fig. 6). A similar result has been obtained for a microscale model of gas exchange in fruit tissue (Ho et al., 2009).

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Fig. 4. Cell size distribution as it changes over time for a pear tissue sample (0.6 mm thickness) dehydrated for 200 s, applying a difference of 1.3% RH (corresponding to a drop of 1 MPa turgor) across the tissue (at 25 °C).

Fig. 5. Time step dependence of the relative average cell area changes versus time for a pear tissue sample (0.6 mm thickness) dehydrated for 200 s, applying a difference of 1.3% RH (corresponding to a drop of 1 MPa turgor) across the tissue (at 25 °C).

3.6. Sensitivity analysis 3.6.1. Effect of membrane permeability The sensitivity of the tissue deformation, water potential and water content to the membrane permeability is shown in Fig. 8. In previous work, this parameter was identified to have a major effect on tissue conductivity. It is seen that the membrane permeability during dynamic water loss also has a large influence on the relative cell average area, water potential and water content. Reported values of membrane permeability cover several orders of magnitude. Seguí et al. (2006) reported values higher to 3000 lm s1, while other authors obtained values that were much lower, from 2.5 to 500 lm s1 on different plant tissues (Ramahaleo et al., 1999; Ferrando and Spiess, 2002; Suga et al., 2003; Moshelion et al., 2004; Murai-Hatano and Kuwagata, 2007; Volkov et al., 2007). Dedicated experiments will be needed to better understand the effect of membrane permeability of fruit cortex cells. 3.6.2. Effect of elastic modulus of the cell wall The sensitivity of water loss and deformation to the elastic modulus of the wall for simulation coupled with dynamic deformation is shown in Fig. 9. It is seen that the elastic modulus

Fig. 6. Average change in water potential (a) and water content (b) of three different tissues and (c) typical average drop in water content of the cell wall for a pear tissue sample (0.6 mm thickness) dehydrated for 200 s, applying a difference of 1.3% RH (corresponding to a drop of 1 MPa turgor) across the tissue (at 25 °C).

of the wall has less influence on the water potential and water content, but a more significant influence on the deformation.

3.7. Prediction of tissue conductivity with deformation The effective conductivity of the entire tissue in steady state was calculated from the obtained fluxes for the RH levels of 99– 97.7% and temperature of 25 °C. The values correspond well with the range of measured values of tissue conductivity at 25 °C as can be seen in Table 3. From the previous work of (Fanta et al., 2013), the effective conductivity is increasing towards the high range of experimental effective conductivity, probably due to the larger deformation than estimated in our previous work. The smaller deviation of predicted effective conductivity values may be attributed to the fact that the 2D model is unable to take into ac-

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1

Fig. 7. Effect of random variation in original tissue structure on (a) water potential (Pa) and (b) water content (kg kgDM ) for three pear tissue samples (0.6 mm thickness) dehydrated for 200 s, applying a difference of 1.3% RH (corresponding to a drop of 1 MPa turgor) across the tissue (at 25 °C).

count the 3D connectivity of the pore space (Ho et al., 2009). Measured values of the effective water conductivity of pear tissue showed a large variation (Table 3).

4. Discussion Microscale water transport in fruit tissue was coupled with dynamic deformation using mechanistic models, and used to calculate cell deformation as well as effective tissue properties such as tissue conductivity as a basis for a multiscale model of water transport in fruit (Ho et al., 2011). Compared to previous work on deformation of single cells (Feng and Yang, 1973; Lardner and Pujara, 1980; Smith et al., 2000; Dintwa et al., 2011) the present work allows to compute deformation of cells in a tissue under water loss. The tissue context is relevant because loss of turgor of one cell will affect neighboring cells as well. Indeed, cell wall tension and extension is a result of the balance of pressures in the two neighboring cells of the cell wall. As a consequence, when turgor is lost, the cell wall relaxes deforming all touching cells. The deformation also occurs in all dimensions because the turgor acts on all cell walls surrounding the cell. Therefore, even if the water transport is essentially one directional (in the case studied here), deformation of the tissue will also occur in directions perpendicular to that of the global water flux through the tissue. This will be a main reason why fruit shrivel upon water loss (Nguyen et al., 2006; Veraverbeke et al., 2003): the cuticle can be regarded as a relatively inelastic material that will maintain its surface area during water loss, while the underlying cell layers tend to shrink in every dimension. The present model predicts this behavior.

The water transport modeling approach presented here is similar to that presented by Esveld et al. (2012a,b), who used a microscale model to investigate effective water transport properties of complex porous foods. While the work of Esveld and coworkers considered vapor transport and sorption into the solid matrix, here we considered dehydration from plant cells under turgor confined by a cell wall matrix. In addition, deformation was included. In our approach, the changing structure of the tissue directly affects also the water loss, which has not been achieved in any previous work to our knowledge. The value of the effective tissue conductivity obtained from the simulation was closer to the experimental average of late picked pears compared to those obtained from the previous work (Fanta et al., 2013) that incorporated deformation in a static manner. The values from static and dynamic simulations are significantly different indicating the difference in results of the static approach and more physically correct dynamic approach that includes a mechanic deformation model. The water content of the wall decreased faster compared to the average water content of the tissue. The dry weight basis equilibrium water content of the cell is initially high as observed in sorption isotherm experiments (Fanta et al., 2013). It appears that the cell wall is an important component to quickly equilibrate the tissue system to a new water status. The consequences of changes in water content on the cell wall mechanics were not included in this work. However, for the high relative humidity range considered in this work this should not present a major issue, as shown in the sensitivity analysis. Understanding cell wall structure and mechanics remains a major area of plant research as evidenced in recent literature (Yi and Puri, 2012; Dyson et al., 2012; Park and Cosgrove, 2012).

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Fig. 8. Sensitivity to membrane permeability of (a) relative average cell area, (b) water potential and (c) water content for a pear tissue sample (0.6 mm thickness) dehydrated for 200 s, applying a difference of 1.3% RH (corresponding to a drop of 1 MPa turgor) across the tissue (at 25 °C).

The sensitivity of the simulation results to cell membrane permeability showed that it has a large influence on the water transport and deformation. Correct understanding and measurement of cell membrane permeability is still an active area of research (Ramahaleo et al., 1999; Ferrando and Spiess, 2002; Suga et al., 2003; Moshelion et al., 2004; Seguí et al., 2006; Murai-Hatano and Kuwagata, 2007; Volkov et al., 2007).

Fig. 9. Sensitivity to cell wall elastic modulus of (a) relative average cell area, (b) water potential and (c) water content for a pear tissue sample (0.6 mm thickness) dehydrated for 200 s, applying a difference of 1.3% RH (corresponding to a drop of 1 MPa turgor) across the tissue (at 25 °C).

The 2D microscale model coupled with deformation lacks to incorporate connectivity of cell walls and air spaces that can affect considerably the transport phenomena (Ho et al., 2009, 2011). Further advances require that 3D modeling of water transport and cell mechanics in the microstructure of the tissue is targeted to explain the effect of interconnectivity on the macroscopic water transport of a tissue. Thereto, 3D imaging of the cellular tissue with advanced

S.W. Fanta et al. / Journal of Food Engineering 124 (2014) 86–96 Table 3 Apparent cortex tissue (K  1015 kg m1 s1 Pa1). Experiment (Nguyen et al., 2004)a 3.92 ± 0.76 to 11.6 ± 2.4

water

conductivity

of

Model Steady state model (Fanta et al., 2013) 6.10 ± 0.14

‘Conference’

pear

Transient model 9.42 ± 0.39

a

Values are given for optimal and late picking dates of the pears ± standard deviation for each.

image analysis or 3D tissue modeling will be required to provide representative 3D cellular models. The latter is currently under progress based on the 2D cell growth algorithm (Abera et al., 2013b). 3D imaging of fruit tissue is possible using X-ray microtomography; however, the image processing to obtain 3D models is still cumbersome (Verboven et al., 2008; Ho et al., 2011; Herremans et al., 2013). 5. Conclusions A microscale water transport model coupled with mechanical model in pear cortex tissue was presented and incorporated water transport and mechanical properties of the pores, cell walls, cell and cell membranes and the representative tissue morphology. The model predicted the effective tissue conductivity of pear cortex tissue in the same range as those measured experimentally but was significantly different from other modeling approaches that ignored modeling the deformation mechanics. The model presents a major step towards better understanding of the changes in tissues during dehydration that result in complex phenomena such as shriveling during storage or large scale deformations during drying. Next steps include elaborating the model in 3D, application to different tissue types and extending the model formulation to cope with more severe dehydration relevant to drying processes. Acknowledgements Financial support by the Flanders Fund for Scientific Research (project FWO G.0603.08), KU Leuven (project OT 08/023, project OT/12/055) and the EC (project InsideFood FP7-226783) is gratefully acknowledged. The opinions expressed in this document do by no means reflect their official opinion or that of its representatives. References Abera, M.K., Fanta, S.W., Verboven, P., Ho, Q.T., Carmeliet, J., Nicolaï, B.M., 2013a. Virtual fruit tissue generation based on cell growth modeling. Food and Bioprocess Technology 6, 859–869. Abera, M.K., Verboven, P., Herremans, E., Defraeye, T., Fanta, S.W., Ho, Q.T., Carmeliet, J., Nicolai, B.M., 2013b. 3D virtual pome fruit tissue generation based on cell growth modeling. Food and Bioprocess Technology. http:// dx.doi.org/10.1007/s11947-013-1127-3. Aregawi, W.A., Defraeye, T., Verboven, P., Herremans, E., De Roeck, G., Nicolai, B.M., 2013. Modeling of coupled water transport and large deformation during dehydration of apple tissue. Food and Bioprocess Technology 6, 1963–1978. Barrett, D.M., Beaulieu, J.C., Shewfelt, R., 2010. Color, flavor, texture, and nutritional quality of fresh-cut fruits and vegetables: desirable levels, instrumental and sensory measurement, and the effects of processing. Critical Reviews in Food Science and Nutrition. 50, 369–389. Bourne, M., 2002. Food Texture and Viscosity: Concept and Measurement, second ed. Academic Press, London. Coulson, J.M., Richardson, J.F., 2004. Fluid Flow Heat Transfer and Mass Transfer. sixth ed. Elsevier, India. Defraeye, T., Aregawi, W.A., Saneinejad, S., Vontobel, P., Lehmann, E., Carmeliet, J., Verboven, P., Derome, D., Nicolai, B.M., 2013. Novel application of neutron radiography to forced convective drying of fruit tissue. Food and Bioprocess Technology: An International Journal. doi: 10.1007/s11947-012-0999-y. Dintwa, E., Jancsok, P., Mebatsion, H.K., Verlinden, B., Verboven, P., Wang, C.X., Thomas, C.R., Tijskens, E., Ramon, H., Nicolai, B., 2011. A finite element model

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