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Prediction of water loss and viscoelastic deformation of apple tissue using a multiscale model
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 464111 (http://iopscience.iop.org/0953-8984/26/46/464111) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 29/04/2017 at 12:01 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 464111 (10pp)
doi:10.1088/0953-8984/26/46/464111
Prediction of water loss and viscoelastic deformation of apple tissue using a multiscale model Wondwosen A Aregawi1,3, Metadel K Abera1,3, Solomon W Fanta1, Pieter Verboven1 and Bart Nicolai1,2 1
MeBioS, Department of Biosystems, University of Leuven, 3001 Heverlee, Belgium VCBT, Flanders Centre of Postharvest Technology, 3001 Heverlee, Belgium
2
E-mail: [email protected] Received 24 March 2014, revised 19 May 2014 Accepted for publication 27 May 2014 Published 27 October 2014 Abstract
A two-dimensional multiscale water transport and mechanical model was developed to predict the water loss and deformation of apple tissue (Malus × domestica Borkh. cv. ‘Jonagold’) during dehydration. At the macroscopic level, a continuum approach was used to construct a coupled water transport and mechanical model. Water transport in the tissue was simulated using a phenomenological approach using Fick’s second law of diffusion. Mechanical deformation due to shrinkage was based on a structural mechanics model consisting of two parts: Yeoh strain energy functions to account for non-linearity and Maxwell’s rheological model of visco-elasticity. Apparent parameters of the macroscale model were computed from a microscale model. The latter accounted for water exchange between different microscopic structures of the tissue (intercellular space, the cell wall network and cytoplasm) using transport laws with the water potential as the driving force for water exchange between different compartments of tissue. The microscale deformation mechanics were computed using a model where the cells were represented as a closed thin walled structure. The predicted apparent water transport properties of apple cortex tissue from the microscale model showed good agreement with the experimentally measured values. Deviations between calculated and measured mechanical properties of apple tissue were observed at strains larger than 3%, and were attributed to differences in water transport behavior between the experimental compression tests and the simulated dehydration-deformation behavior. Tissue dehydration and deformation in the high relative humidity range ( > 97% RH) could, however, be accurately predicted by the multiscale model. The multiscale model helped to understand the dynamics of the dehydration process and the importance of the different microstructural compartments (intercellular space, cell wall, membrane and cytoplasm) for water transport and mechanical deformation. Keywords: plant tissue, turgor, nonlinear mechanics, diffusion (Some figures may appear in colour only in the online journal)
1. Introduction Fresh fruit such as apples are mostly composed of water [1]. Water loss of fruit during storage is the main cause for the reduction of fruit quality after harvest. At the microscopic level, fruit tissue undergoes large deformations as a result of hygrostresses 3
associated with dehydration [2–4]. At the microscopic level, deformation of cells occurs as a result of the loss of turgor acting on the cell walls surrounding the cell during the dehydration process [5]. Firmness is an important quality attribute critical to consumer acceptance. Turgor pressure, cell size and shape, presence of intercellular spaces, and chemical composition have
These authors contributed equally to the paper.
0953-8984/14/464111+10$33.00
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© 2014 IOP Publishing Ltd Printed in the UK
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List of symbols b damping factor C10, C01, C11 coefficients of Yeoh hyperelastic potential molar concentration of solutes in Cs water water capacity cψ D diffusion coefficient F force acting upon the node G deformation gradient tensor h thickness of the simulated tissue J water flux k spring constant K water conductivity l cell wall length mi mass of the vertex N unit normal vector ns amount of moles of solutes in the cell Pm permeability of the membrane R universal gas constant (8.314) S second Piola-Kirchhoff stress tensor t time T temperature u net displacement vector of the cell wall v velocity Vw molar volume of water (18 × 10–6) Xdb water content on dry matter basis x position
the volume change due to the shrinkage of the tissue is equal to the volume of water removed from the material [11, 12]. The principle of virtual work has also been applied to hygrostress formation, where the deformation is based on linear elasticity theory [13–16]. Recent work has allowed better understanding and modeling of coupled water transport and large deformation of fruit tissue during dehydration at the macroscale [2]. The disadvantage of continuum models is that they do not allow to investigate the effect of microstructural features of the tissue on macroscopic processes such as water loss and mechanical deformation. In contrast, in microscale models the heterogeneous structure of the tissue is taken into account by representing the complex cellular structure through a geometrical model [17–19]. A microscale water transport model was developed to describe water transport in fruit at the cellular level through the inter-cellular space and cells [20]. Different approaches to model the mechanical behavior of cellular tissues have been reported. A finite element model was developed to simulate the compression of a single suspension-cultured tomato cell [21]. However, the model was limited to mechanical loading and did not account for the deformation due to water loss. Other authors have used smoothed particle hydrodynamics (SPH) for the cytoplasm and a discrete element model (DEM) for the cell wall to simulate mechanical behavior cellular tissue in response to dynamic stimuli [22–24]. The model incorporated the mechanics during and after cell failure and cell wall rupture. However, the model predicts water efflux due to mechanical loading but not water transport as such between the cells. A microscale water transport model coupled with a mechanical deformation model has been developed to describe water transport and mechanical properties of pores, cell walls, cells and cell membranes [5]. Although this model provided detailed predictions of the local microscale water distribution and mechanical deformation, the extension of the microscale modelling techniques to predict the dehydration process of an entire fruit would require far more computational power than is currently available. The multiscale modelling paradigm offers an alternative approach to combine continuum type models defined at the macroscale level with the level of detail models incorporating the microscale features. A multiscale water transport and large deformation model is basically a hierarchy of models which describe the water transport and large deformation at different spatial scales as shown in figure 1. The coupling of the hierarchy of models is via computational analysis i.e. model results relevant to a particular spatial scale are linked to simulations at a different spatial scale. Multiscale modelling has been successfully applied in soft matter systems [25, 26], in material science and geoscience [27–33]. It has also been used as a useful tool for biological applications [34–36]. In this study, a 2D multiscale model is introduced and validated to perform a computational analysis of water transport and large deformation of apple (cv. Jonagold) cortex tissue at different scales (illustrated in figure 2). On the microscale, water transport and deformation of tissue are computed for the actual cell assembly of apple tissue. With these simulations, the apparent water diffusion and mechanical properties of tissues are calculated from a homogenization procedure. These parameters are consequently used to perform simulations of the dehydration and
− Pa mol m − 3 kg kgDM − 1 Pa − 1 m2 s − 1 N − m kg m − 2 s − 1 N m − 1 kg m − 1 Pa − 1 s − 1 m kg − mol m s − 1 J mol − 1 K − 1 Pa S K m m s − 1 m3 mol − 1 kg kg dm − 1 m
Greek symbols ε λ ρw ρdm σ
ψ
strain stretch ratio density of water dry matter density stress water potential
m m − 1 − kg m − 3 kg dm m − 3 Pa Pa
Subscripts a app c d db i m n p s T tn w
air apparent cell damping dry base node number i cell membrane natural pressure solute total tension cell wall
a major influence on tissue strength and macroscopic fruit firmness [6]. To better understand water loss and large deformations during dehydration and how they are affected by microscale features of the tissue, a modelling approach is most appropriate. In the macroscopic, continuum approach, the tissue is considered as homogeneous; lumped parameters are used as apparent material properties, and incorporate the effect of microstructural features such as the ensemble of cell membranes, cell vacuoles pores and cell walls. Macroscopic water transport models in food have been developed by, amongst others [7–9], which all adopted a continuum approach based on Fick’s second law of diffusion. Biological materials such as fruit can be modeled as a nonlinear viscoelastic continuum [10], when viewed at the macroscopic scale. Different approaches to model water transport and shrinkage have been reported. In the simplest approach, 2
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Figure 1. Multiscale hierarchy of apple geometry. (a) Detailed microstructure of the tissue; (b) and (c) tissue and its representative elementary volume; (d) cylindrical apple cortex.
where ρw,c is wet base cell density (kg m − 3), ρc is cell density (kg m − 3), Dc is the water diffusion coefficient inside cells (m2 s − 1), and cw, c is water content on wet base (kg kg − 1). The wet base cell density and dry matter density inside cells are related by the following equation: ρw, c = ρdm, c Xc (2)
where Xc is the dry matter base water content (kg kg dm − 1). Using the relationship between water content on wet and dry matter base the following equation can be derived: Xc . cw, c = (3) 1 + Xc
Figure 2. Schematic outline of the multiscale approach.
The water capacity cψ, c (kg kg dm − 1 Pa − 1) is given by:
shrinkage of apple tissue samples using a macroscale model. The novelty of the current study over our previous work includes that we predict not only the apparent water transport properties but also the non-linear stress-strain relationship from the microscale model, compared them with parameter values obtained experimentally, and used the parameters in a macroscale model.
∂ Xc cψ , c = (4) ∂ ψc
where ψc the water potential of the cell (Pa). Substituting equations (2), (3) and (4) into equation (1) leads to: ⎛ ⎛ ρdm, c cψ , c ⎞ ∂ ρdm, c ⎞ ∂ ψc = ∇⋅ Dc ⎜ ⎟ ∇ψ . ⎜ ρdm, c + xc ⎟ cψ , c (5) ⎝ 1 + xc ⎠ c ⎝ ∂ xc ⎠ ∂t
2. Multiscale model 2.1. Microscale model
For the cell wall, the unsteady-state diffusion model is given by:
2.1.1. Microscale water transport model. The transport of
water in the intercellular space, the cell wall network and cytoplasm were modeled using diffusion laws and irreversible thermodynamics [37]. For the cells, the unsteady-state model of water transport reads:
∂ψ cψ , wρdm, w w = ∇⋅ ρdm, w Dwcψ , w ∇ ψw. (6) ∂t
∂ ρw, c = ∇⋅ ρc Dc ∇ cw, c (1) ∂t
∂ψ cψ , aρdm, a a = ∇⋅ ρdm, a Dacψ , a ∇ ψa. (7) ∂t
For the air, unsteady-state diffusion is modeled by:
3
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Figure 3. Schematic diagram of (a) cell boundary represented as a set of walls (modeled as springs) connected at points called vertices, (b) force at the vertices.
A simple flux law was applied to describe water transport through the cell membrane [37]: Figure 4. Relationship between turgor pressure and water potential at high relative humidity (99/97.7%).
ρ PmVw ( ψc − ψw ). J= w (8) RT
The force contributed by the wall is the resultant of the net turgor pressure force between the two adjacent cells working normal to the wall and the force associated with it:
2.1.2. Microscale mechanics model. A cell micromechanics
model was recently developed [38], and the basic characteristics are summarized below. In this 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 (see figure 3). Newton’s second law was considered to model the shrinkage mechanics. The following system of equations is solved for the velocity and position of the vertices i of the cell wall network:
Fw = Fturgor + Ftn. (13)
The net turgor force on the vertex is calculated by taking the difference in turgor pressure Pc 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 (see figure 3(b)): ⎛1 ⎞ Fturgor = ⎜ ( Pc,1− Pc,2 ) l ⎟ N. (14) ⎝2 ⎠
Hooke’s law was employed to determine the force acts along the wall and its magnitude
dv mi i = FT , i (9) dt
Ftn = −k u (15)
dxi = vi (10) dt
u = l − ln. (16)
To find the positions of each vertex of all cell walls of every single cell and, thus, the shape of the cells with time, a system of differential equations (9) and (10) for the positions and velocities of each vertex were established and solved using a Runge-Kutta fourth and fifth order (ODE45) method.
where m 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 − 1) are the position and velocity of node i, respectively, and FT , i 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, obtained from the water transport model outlined in the previous section, can be converted to turgor pressure using the relationship presented in section 2.1.3. 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
2.1.3. Coupling of water transport and mechanical deformation. The transient water transport model is solved for certain
time steps and the water loss results in loss of water potential in the cells. The change in water potential of the cells induces loss of turgor pressure. This is valid for the high range of equilibrium relative humidity values of the cells during dehydration until turgor drops to zero [5]. The dehydration is thus performed 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 ψs can be obtained from equation (17) ρ ρ n ψs = − CsRT = − w nsRT = − w s RT. (17) mw Xc ms
FT = ∑ Fw + Fd (11) w∈W
Fd = − bv (12) 4
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Table 1. Model parameters for different compartments of apple
(cv. Jonagold) cortex tissue (cell membrane, cell, cell wall and pore space) at 25 °C. Parameter
Value
Dc ρ dm,c Da ρ dm,a Cψ,a Dw ρ dm,w Pm Pc E b
2.2 × 10–9 m2 s − 1 111 kg m − 3 2.42 × 10–5 m2 s − 1 1.184 kg m − 3 1.3 × 10 − 10 Pa − 1 41.9 × 10–11 m2s − 1 58.035 kg m − 3 17 × 10–6 m s − 1 1 MPa 30 MPa 3.5 Ns µm − 1
References [47] [48] [49] [48] [50] [38] [20] [20] [38] [51] [38]
turgor pressure using the relation shown in figure 4. Then this set of turgor pressures was used in the shrinkage mechanics presented in section 2.1.3 to find the new equilibrium configuration of the cells. The 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 eight time steps of 50 s were required to reach equilibrium. The computation time was 2412 s for the unsteady state water transport simulation in each time step on a 8 Gb RAM quad-core PC, and 20 s for the mechanical equilibrium calculations. The numerical values of the water transport and mechanical parameters used in the microscale model are listed in table 1. These parameters were taken from our previous studies and from different literature resources. It should be noted that not all parameters were obtained at exactly the same environmental conditions, which could induce some uncertainty in the simulation results.
Figure 5. Numerical solution of the microscale coupled water transport model with deformation.
The turgor pressure is then equal to ψp = ψc − ψs. (18) The relationship between turgor pressure and water potential is shown in figure 4 for the considered range. 2.1.4. Implementation. Three different tissue samples of
1250 × 1600 μm, which had 80 cells with average cell diameter of 157.39 ± 22 μm and a porosity of 14.97% ± 1%, were generated using random simulation with the microscale mechanics model as described in [37]. Then, the geometric models of apple cortex tissue 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 more than 350 000 quadratic elements with triangular shape for each tissue geometry. A numerical experiment was carried out to calculate apparent material properties to be used in the macroscale model. As it was described in section 2.1.3, a relative humidity of 99% and 97.7% was chosen and applied to the top and bottom of the tissue geometry, respectively, while the other two lateral boundaries were defined to be impermeable. Afterwards, a sequence of time steps was considered for solving the coupled moisture transport and mechanical deformation model (see figure 5). The finite element method was used to discretise non-linear coupled model equations. In every time step the water potential and water content distribution in the tissue samples as well as the water flux through the sample were solved for a given water potential gradient across the sample. The initial water transport calculation was performed on the initial apple cortex tissue geometry that was obtained using a virtual fruit tissue generator. Afterwards, the water potential of each cell was obtained and converted into
2.1.5. Computation of apparent diffusivity and mechanical properties. The microscale model was used for in silico
analysis of water loss and deformation of cells in apple fruit tissue, and for computing the apparent properties of the apple tissue as a whole. The water capacity and dry mass density were combined into an apparent diffusion coefficient Dapp (m2 s − 1) of the tissue from the apparent water conductivity Kapp (kg m − 1 Pa − 1 s − 1) obtained from the microscale simulations using equation (19). Kapp Dapp = . (19) ρdm cψ
The water capacity cψ (kg kgdm − 1 Pa − 1) was calculated from the steady state calculated water content and water potential of the microscale tissues for RH values of 98.5% and 97.5% and temperature of 25 °C using microscale model simulations. The apparent water conductivity was obtained from 5
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Table 2. Apparent water transport and mechanical parameters of apple (cv. Jonagold) cortex tissue at 25 °C.
Parameter
Calculated from microscale model
Experiment
D × 10 − 11 [m2s − 1]
4.25 ± 0.3 RH (99–98%) 3.35 ± 0.2 RH (98–97%) 2.08 ± 0.01 RH (98.5%) 1.45 ± 0.01 RH (97.5%) 0.16 ± 0.1 52.18 ± 6.3 0.99 ± 0.1
10.30 ± 2.3 RH (62–63% [45])
cψ × 10 − 6 [kg kgdm − 1 Pa − 1] C10 [MPa] C01 [MPa] C11 [MPa]
1.06 ± 0.5 RH (97.5% [43]) 0.09 ± 0.03 [2] 44.18 ± 9.6 [2] 0.42 ± 0.1 [2]
the calculated flux for a specified gradient in water potential across the tissue sample:
were calculated from the following mechanical equilibrium, assuming no body and surface forces [42]:
h Kapp = − J (20) Δψ
∇⋅ ( S ⋅ GT ) = 0. (23)
The model equation (23) is based on the theory of compressible hyperelasticity with the decoupled representation of the Helmholtz free energy function with the internal variables [41]. The way in which the second Piola-Kirchhoff stress tensor (S) and the deformation gradient tensor (G) are defined is clearly explained in [2]. The coupling between water transport and the mechanical response (large deformations) is based on [2]. For the purpose of this article, we have selected a 2D geometrical model of the tissue samples, namely a long tissue cylinder (16.6 mm diameter). The model was numerically solved applying the finite element method, using the software Comsol Multiphysics 3.5a (Comsol BV, Sweden). Simulations were performed to compute the dehydration of the cylinder in a cross flow of air of 0.01 m s − 1, 25 °C and 97% RH. The relevant water transport and mechanical properties of apples used in the macroscale model were computed from microscale simulations as described earlier (see table 2).
with J (kg m − 2 s − 1) the total flux through the fruit tissue, Δψ (Pa) the assigned water potential difference between the two opposite sides and h (m) the thickness of the simulated tissue. The apparent mechanical properties of the tissue were calculated from the microscale model by assuming that the change in displacement (deformation) of tissue is due to a change in turgor pressure. This is logical, since turgor pressure generates a stress that leads to the expansion of cell wall [39]. The reverse phenomenon, shrinkage of cells, takes place as a result of turgor loss during dehydration. Thus, in the current study, we assumed that the history of turgor pressure and the resulting expansion or shrinkage of the cell wall network could determine the mechanical properties of the tissue in general. The current approach is plausible from the common type of mechanical tests where external forces are applied to determine tissue properties, because the stresses and the resulting shrinkage can be obtained from the natural dehydration processes. Based on this approach, the average changes in turgor and the corresponding displacement of the tissue in the direction of the applied gradient (see section 2.1.4) were calculated from microscale model simulations. Afterwards, mechanical properties were estimated by fitting these data points into the Yeoh hyperelastic potential with three parameters [2], (see equation (21)).
3. Results 3.1. Microscale model 3.1.1. Water potential and water content profile of apple tissue during dehydration. Water potential and content profiles of
one of the apple cortex tissues are shown in figure 6. Gradients of water potential can be observed from one cell to another in the direction of the applied gradient (see section 2.1.4). Cells at the same horizontal position, which is perpendicular to the applied water potential gradient, tend to have similar and uniform water potential, which is due to the high water conductivity within the cells. The bottom side of the sample is dehydrated more severely while the cell turgor decreases. The water content of the cells remains relatively uniform and constant during the course of dehydration. This is plausible, since the water content of cells do not change abruptly, as a result of shrinkage during water loss [5].
⎛ ⎛ ⎞ 1 ⎞⎛ 2 σYeoh ( λ ) = 2 ⎜ λ2 − ⎟ ⎜ C10 + 2C01⎜ λ2 + − 3 ⎟ ⎝ ⎝ ⎠ λ ⎠⎝ λ ⎛ ⎞2 ⎞ 2 + 3C11⎜ λ2 + − 3 ⎟ ⎟ . ⎝ ⎠ ⎠ λ
(21)
2.2. Macroscale model of water transport coupled with deformation
A macroscopic phenomenological approach was used to model water transport in fruit tissue, where water moves in the tissue as a consequence of a gradient in water potential [40]:
3.2. Apparent apple tissue properties
∂ψ ρdm cψ = ∇⋅ K ∇ ψ . (22) ∂t
The apparent water capacity of tissue was calculated from microscale model simulations for the RH of 98.5% and 97.5% and compared well with the water capacity of apple (cv. Jonagold) cortex tissue, which was calculated from desorption isotherm that we measured previously [43] at
Materials undergoing large deformations are best described by nonlinear elasticity theory according to [41], instead of by linear elasticity theory. The stresses in the fruit tissue 6
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Figure 6. Water potential (Pa) (left) and water content (kg kgdm − 1) (right) of an apple (cv. Jonagold) cortex tissue sample (1.25 mm
thickness) dehydrated for 500 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 = 200 s, (c) t = 500 s.
97.5% RH (see table 2). In general, the water capacity will decrease with RH. The apparent diffusivity of the tissue samples was calculated from the obtained fluxes for the RH levels of 99–98% and 98–97%, using the microscale model. Compared with the diffusivity value obtained from measurements, the apparent diffusivity calculated from the microscale simulations was found to be lower (see table 2). The deviation of predicted apparent value could be attributed to the fact that the model in this study did not account for cell to cell water transport through plasmodesmata. The measured apparent diffusivity and water capacity showed large variations (see table 2), which could be due to microstructural variations of the tissue. Microscale simulations with three different tissue structures showed that differences in microstructure affected the water transport in the tissue, but the resulting variation was smaller as compared with the experiments. Figure 7 displays the calculated average stress-strain response of tissue during dehydration. The apparent mechanical properties were calculated by fitting the nonlinear mechanical model (with Yeoh hyperelastic potential) to the data points obtained from microscale simulations. The agreement between measured and predicted values of mechanical parameters at small strains is excellent, but above 3% strain clear discrepancies can be observed. This
Figure 7. Stress–strain response of apple cortex tissue from microscale simulations for three random tissue geometries denoted by different symbols (□, ○, and ◊). The solid, dot and dashed lines are the corresponding constitutive model fits. The (Δ) and the dash-dot line represent an actual compression experiment and the constitutive model fit.
7
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Figure 8. Simulation results for drying experiment at 97% RH, 25 °C. Contour plots of the water potential (Pa) after 4 h using experimental
(a) and apparent parameters obtained from mircoscale simulations (b). Evaluation of the average water content (c) and areal shrinkage (d) with dehydration time; the dashed and solid line represent macroscale simulations using experimental and apparent parameters, respectively.
measured diffusivity was relatively higher than the one calculated from microscale simulations. From figure 8(d) it can observed that the rate of areal shrinkage of the cylindrical sample was non-linear. Fast deformation occurred at the beginning of dehydration, which gradually decreases with time. The percentage of areal shrinkage as simulated using measured material properties was higher than using computed ones, again due to the discrepancy between measured and computed material properties. In general, a good agreement was found between macroscale simulations.
could be due to the fact that in reality compression tests break cells so that more water is lost to the environment than mild dehydration process, thus resulting in larger deformation in the compression experiment. Another reason for the deviation between predicted and measured hygromechanical properties is the 2D cross-sectional simplification of the tissue that is essentially 3D, consisting of ellipsoid cells and a 3D network of pores. The estimated parameters from microscale simulations are close but not equal to the experimentally obtained values from compression tests (see table 2).
4. Discussion 3.3. Macroscale model
In this study, a microscale model of coupled water transport and large deformation was used to predict apparent tissue properties such as tissue diffusivity and viscoelastic properties. These properties were then used in a macroscale model to predict shrinkage of apple cortex cylinder during dehydration. While we validated the macroscale model previously [2, 44], in this paper we validated the microscale model and the overall multiscale model. The microscale coupled water transport and large deformation model was validated by
The 2D macroscale coupled water transport and mechanical model was solved using calculated apparent water transport and mechanical properties. In figure 8 this simulation is compared to one in which the measured parameters were used. The water loss percentage calculated from the macroscale simulation using measured apparent material properties was higher than using apparent properties computed from microscale simulations (see figure 8(c)). This is due to the fact that the 8
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comparing macroscopic apparent water transport and mechanical properties computed from microscale model with measured ones [2, 43, 45]. The effect of random variation in tissue structure on transport and mechanical properties was investigated by taking three different geometrical models of apple cortex tissue and was found to be insignificant. However, if anisotropy of cell shape is present [46], the variation of the microstructure would likely become more significant e.g. for hypodermal and epidermal tissue of fruit. Shrinkage adversely affects the quality of fruit during storage at too low relative humidity. Experimental and simulation results of our previous studies using a macroscale model showed that the apple tissue undergoes large deformations during dehydration [2, 44]. This is confirmed by the current study using microscale model simulations, and even at high RH. The Yeoh hyperelastic model successfully estimated the nonlinear curve characteristic of the tissue obtained from microscale model simulations. The apparent mechanical properties obtained from the microscale simulations were in the range of measured ones [2]. Macroscale model simulations were performed using the apparent water transport and mechanical properties of the tissue obtained from the microscale model simulations and measurements. The predicted water loss and areal shrinkage using computed or measured apparent material properties corresponded well. The 2D microscale model of water transport coupled with mechanical deformation does not take into account the connectivity of cell walls and air spaces that can affect the transport phenomena and mechanics of a tissue considerably [34]. 3D modeling of water transport and cell mechanics at the microscale is required to investigate this. The aim of the current paper, was to present a multiscale methodology for hygromechanical analysis of fruit, and thus it was opted to use a simplified 2D approach before elaborating the more cumbersome 3D model. An additional comment can be made here. Moisture transport is different from gas transport as we have shown that the cell membrane is a more determining factor than the pore space connectivity [20], which is opposite to the case of gas diffusion in tissues. In gas diffusion analysis, we found significant differences between 2D predicted and measured gas diffusion properties and thus a 3D analysis was required [34]. Here, for moisture transport we already find reasonable agreement of the 2D predictions with measurements. To be conclusive, however, indeed a 3D analysis will be required.
developments could focus on 3D modeling to elaborate the effect of interconnectivity on the macroscopic water transport and mechanical property of a tissue, simulating compression test and extending the model formulations to predict more severe dehydration (at lower RH) as would be expected to occur during drying processes. Acknowledgements The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 226783 and 245288. The authors would also like to acknowledge the Fund for Scientific Research—Flanders (Grant no. FWO G.0603.08) and the KU Leuven (project OT 08/023 and 12/055) for financial support. References [1] Wills R, McGlasson B, Graham D and Joyce D 1998 Postharvest: An Introduction to the Physiology and Handling of Fruit, Vegetables and Ornamentals 4th edn (Willingford: Cab International) [2] Aregawi W A, Defraeye T, Verboven P, Herremans E, De Roeck G and Nicolai B 2013 Modeling of coupled water transport and large deformation during dehydration of apple tissue Food Bioprocess Technol. 6 1936–78 [3] Pieczywek P M and Zdunek A 2014 Finite element modelling of the mechanical behaviour of onion epidermis with incorporation of nonlinear properties of cell walls and real tissue geometry J. Food Eng. 123 50–9 [4] Ghysels P, Samaey G, Tijskens B, Van Liedekerke P, Ramon H and Roose D 2009 Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics Phys. Biol. 6 016009 [5] Fanta S W, Abera M K, Aregawi W, Ho Q T, Verboven P, Carmeliet J and Nicolai B M 2014 Microscale modeling of coupled water transport and mechanical deformation of fruit tissue during dehydration J. Food Eng. 124 86–96 [6] Konstankiewicz K and Zdunek A 2001 Influence of turgor and cell size on the cracking of potato tissue Int. Agrophysics 15 27–30 [7] Dhall A and Datta A K 2011 Transport in deformable food materials: a poromechanics approach Chem. Eng. Sci. 66 6482–97 [8] Rakesh V and Datta A K 2011 Microwave puffing: determination of optimal conditions using a coupled multiphase porous media—large deformation model J. Food Eng. 107 152–63 [9] Van der Sman R G M 2013 Modeling cooking of chicken meat in industrial tunnel ovens with the Flory–Rehner theory. Meat Sci. 95 940–57 [10] Wineman A 2009 Nonlinear viscoelastic solids—a review Math. Mech. Solids 14 300–66 [11] Pioletti D P, Rakotomanana L R, Benvenuti J F and Leyvraz P F 1998 Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons J. Biomech. 31 753–7 [12] Moreira R, Figueiredo A and Sereno A 2000 Shrinkage of apple disks during drying by warm air convection and freeze drying Dry. Technol. 18 279–94 [13] Niamnuy C, Devahastin S, Soponronnarit S and Vijaya Raghavan G S 2008 Modeling coupled transport phenomena and mechanical deformation of shrimp
5. Conclusion A multiscale model of water transport with mechanical deformation in apple cortex tissue was developed to study water loss and shrinkage of fruit cylinders. Using in silico experiments with the coupled water transport and mechanical model at the microscale, we predicted apparent diffusivity and nonlinear mechanical properties that compared favourably with experimental measurements. A multiscale approach provides insight how microscale features of the tissue affects the dynamics of dehydration in apple and in fruit tissue in general. Future 9
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[33] Makov G, Gattinoni C and De Vita A 2009 Ab initio based multiscale modelling for materials science Model. Simul. Mater. Sci. Eng. 17 1–17 [34] Ho Q T, Verboven P, Verlinden B E, Herremans E, Wevers M, Carmeliet J and Nicolaï B M 2011 A three-dimensional multiscale model for gas exchange in fruit Plant Physiol. 155 1158–68 [35] Seymour R S 2001 Diffusion pathway for oxygen into highly thermogenic florets of the arum lily Philodendron selloum J. Exp. Bot. 52 1465–72 [36] Wood B D, Quintard M and Whitaker S 2002 Calculation of effective diffusivities for biofilms and tissues Biotechnol. Bioeng. 77 496–516 [37] Nobel P S 1991 Physicochemical and Environmental Plant Physiology (San Diego: Academic) [38] Abera M K, Fanta S W, Verboven P, Ho Q T, Carmeliet J and Nicolai B M 2012 Virtual fruit tissue generation based on cell growth modelling Food Bioprocess Technol. 6 859–69 [39] Kroeger J H, Zerzour R and Geitmann A 2011 Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth PLoS One 6 e18549 [40] Nguyen T A, Verboven P, Schenk A and Nicolaï B M 2007 Prediction of water loss from pears (Pyrus communis cv. Conference) during controlled atmosphere storage as affected by relative humidity J. Food Eng. 83 149–55 [41] Holzapfel G A 2000 Nonlinear Solid Mechanics. A Continuum Approach for Engineering (Chichester: Wiley) [42] Bonet J and Wood R D 1997 Nonlinear Continuum Mechanics for Finite Element Analysis (Cambridge: Cambridge University Press) [43] Veraverbeke E A, Verboven P, Scheerlinck N, Lan Hoang M and Nicolaı̈ B M 2003 Determination of the diffusion coefficient of tissue, cuticle, cutin and wax of apple J. Food Eng. 58 285–94 [44] Aregawi W, Defraeye T, Saneinejad S, Vontobel P, Lehmann E, Carmeliet J, Derome D, Verboven P and Nicolaï B 2013 Dehydration of apple tissue: intercomparison of neutron tomography with numerical modelling Int. J. Heat Mass Transf. 67 173–82 [45] Verstreken E, Van Hecke P, Scheerlinck N, De Baerdemaeker J and Nicolaï B 1998 Parameter estimation for moisture transport in apples with the aid of NMR imaging Magn. Reson. Chem. 36 196–204 [46] Gibson L J and Ashby M F 1998 Cellular Solids: Structure and Properties (Oxford: Pergamon) [47] Holz M, Heil R, Sacco A, Chemie P, Karlsruhe U, Karlsruhe D, Foggia U, Napoli V and August A 2000 Temperature-dependent self-diffusion coefficients of water and six selected molecular liquids for calibration in accurate 1 H NMR PFG measurements Phys. Chem. Chem. Phys. 4740–2 [48] Fanta S W, Vanderlinden W, Abera M K, Verboven P, Karki R, Ho Q T, De Feyter S, Carmeliet J and Nicolaï B M 2012 Water transport properties of artificial cell walls J. Food Eng. 108 393–402 [49] Lide R L and Frederikse H P R 1994 Hand Book of Chemistry and Physics (Boca Raton, FL: CRC Press) [50] Coulson J M and Richardson J F 2004 Fluid Flow, Heat Transfer and Mass Transfer (Oxford: Butterworth-Heinemann) [51] Wu N and Pitts M J 1999 Development and validation of a finite element model of an apple fruit cell Postharvest Biol. Technol. 16 1–8
during drying in a jet spouted bed dryer Chem. Eng. Sci. 63 5503–12 [14] Kowalski S J and Rajewska K 2002 Drying-induced stresses in elastic and viscoelastic saturated materials Chem. Eng. Sci. 57 3883–92 [15] Inazu T, Iwasaki K and Furuta T 2005 Stress and crack prediction during drying of Japanese noodle (udon) Int. J. Food Sci. Technol. 40 621–30 [16] Shim V P W, Yang L M, Lim C T and Law P H 2003 A viscohyperelastic constitutive model to characterize both tensile and compressive behavior of rubber Polymer 42 523–31 [17] Yao Z and Le Maguer M 1996 Mathematical modelling and simulation of mass transfer in osmotic dehydration processes. Part I: conceptual and mathematical models J. Food Eng. 29 349–60 [18] Toupin C J and Le Maguer M 1989 Osmotically-induced mass transfer in plant storage tissues: a mathematical model. Part II J. Food Eng. 10 97–121 [19] Marcotte M, Toupin C J and Le Maguer M 1991 Mass transfer in cellular tissues. Part I: the mathematical model J. Food Eng. 13 199–220 [20] Fanta S W, Abera M K, Ho Q T, Verboven P, Carmeliet J and Nicolai B M 2013 Microscale modeling of water transport in fruit tissue J. Food Eng. 118 229–37 [21] Dintwa E, Jancsók P, Mebatsion H K, Verlinden B, Verboven P, Wang C X, Thomas C R, Tijskens E, Ramon H and Nicolaï B 2011 A finite element model for mechanical deformation of single tomato suspension cells J. Food Eng. 103 265–72 [22] Van Liedekerke P, Tijskens E, Ramon H, Ghysels P, Samaey G and Roose D 2010 Particle-based model to simulate the micromechanics of biological cells Phys. Rev. E 81 061906 [23] Van Liedekerke P, Ghysels P, Tijskens E, Samaey G, Smeedts B, Roose D and Ramon H 2010 A particle-based model to simulate the micromechanics of single-plant parenchyma cells and aggregates Phys. Biol. 7 026006 [24] Van Liedekerke P, Ghysels P, Tijskens E, Samaey G, Roose D and Ramon H 2011 Mechanisms of soft cellular tissue bruising. A particle based simulation approach Soft Matter 7 3580 [25] Praprotnik M, Site L D and Kremer K 2008 Multiscale simulation of soft matter: from scale bridging to adaptive resolution Annu. Rev. Phys. Chem. 59 545–71 [26] Peter C and Kremer K 2009 Multiscale simulation of soft matter systems—from the atomistic to the coarse-grained level and back Soft Matter 5 4357 [27] Ricard J, Guigné F and Laiarinandrasana L 2014 Damage and fracture mechanisms of polyoxymethylene: multiscale experimental study and finite element modeling Eng. Fract. Mech. 115 270–83 [28] Regenauer-Lieb K and Yuen D A 2003 Modeling shear zones in geological and planetary sciences: solid- and fluidthermal–mechanical approaches Earth-Sci. Rev. 63 295–349 [29] Fish J and Yu Q 2001 Multiscale damage modeling for composite materials : theory and computational framework 2. 0 mathematical homogenization for damaged composites Int. J. Numer. Methods Eng. 52 161–91 [30] Gitman I M, Askes H and Sluys L J 2008 Coupled-volume multi-scale modelling of quasi-brittle material Eur. J. Mech. A 27 302–27 [31] Goldenfeld N, Athreya B P and Dantzig J A 2006 Renormalization group approach to multiscale modelling in materials science J. Stat. Phys. 125 1015–23 [32] Elliott J A 2011 Novel approaches to multiscale modelling in materials science Int. Mater. Rev. 56 207–25
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Prediction of water loss and viscoelastic deformation of apple tissue using a multiscale model
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 464111 (http://iopscience.iop.org/0953-8984/26/46/464111) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 29/04/2017 at 12:01 Please note that terms and conditions apply.
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 464111 (10pp)
doi:10.1088/0953-8984/26/46/464111
Prediction of water loss and viscoelastic deformation of apple tissue using a multiscale model Wondwosen A Aregawi1,3, Metadel K Abera1,3, Solomon W Fanta1, Pieter Verboven1 and Bart Nicolai1,2 1
MeBioS, Department of Biosystems, University of Leuven, 3001 Heverlee, Belgium VCBT, Flanders Centre of Postharvest Technology, 3001 Heverlee, Belgium
2
E-mail: [email protected] Received 24 March 2014, revised 19 May 2014 Accepted for publication 27 May 2014 Published 27 October 2014 Abstract
A two-dimensional multiscale water transport and mechanical model was developed to predict the water loss and deformation of apple tissue (Malus × domestica Borkh. cv. ‘Jonagold’) during dehydration. At the macroscopic level, a continuum approach was used to construct a coupled water transport and mechanical model. Water transport in the tissue was simulated using a phenomenological approach using Fick’s second law of diffusion. Mechanical deformation due to shrinkage was based on a structural mechanics model consisting of two parts: Yeoh strain energy functions to account for non-linearity and Maxwell’s rheological model of visco-elasticity. Apparent parameters of the macroscale model were computed from a microscale model. The latter accounted for water exchange between different microscopic structures of the tissue (intercellular space, the cell wall network and cytoplasm) using transport laws with the water potential as the driving force for water exchange between different compartments of tissue. The microscale deformation mechanics were computed using a model where the cells were represented as a closed thin walled structure. The predicted apparent water transport properties of apple cortex tissue from the microscale model showed good agreement with the experimentally measured values. Deviations between calculated and measured mechanical properties of apple tissue were observed at strains larger than 3%, and were attributed to differences in water transport behavior between the experimental compression tests and the simulated dehydration-deformation behavior. Tissue dehydration and deformation in the high relative humidity range ( > 97% RH) could, however, be accurately predicted by the multiscale model. The multiscale model helped to understand the dynamics of the dehydration process and the importance of the different microstructural compartments (intercellular space, cell wall, membrane and cytoplasm) for water transport and mechanical deformation. Keywords: plant tissue, turgor, nonlinear mechanics, diffusion (Some figures may appear in colour only in the online journal)
1. Introduction Fresh fruit such as apples are mostly composed of water [1]. Water loss of fruit during storage is the main cause for the reduction of fruit quality after harvest. At the microscopic level, fruit tissue undergoes large deformations as a result of hygrostresses 3
associated with dehydration [2–4]. At the microscopic level, deformation of cells occurs as a result of the loss of turgor acting on the cell walls surrounding the cell during the dehydration process [5]. Firmness is an important quality attribute critical to consumer acceptance. Turgor pressure, cell size and shape, presence of intercellular spaces, and chemical composition have
These authors contributed equally to the paper.
0953-8984/14/464111+10$33.00
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© 2014 IOP Publishing Ltd Printed in the UK
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J. Phys.: Condens. Matter 26 (2014) 464111
List of symbols b damping factor C10, C01, C11 coefficients of Yeoh hyperelastic potential molar concentration of solutes in Cs water water capacity cψ D diffusion coefficient F force acting upon the node G deformation gradient tensor h thickness of the simulated tissue J water flux k spring constant K water conductivity l cell wall length mi mass of the vertex N unit normal vector ns amount of moles of solutes in the cell Pm permeability of the membrane R universal gas constant (8.314) S second Piola-Kirchhoff stress tensor t time T temperature u net displacement vector of the cell wall v velocity Vw molar volume of water (18 × 10–6) Xdb water content on dry matter basis x position
the volume change due to the shrinkage of the tissue is equal to the volume of water removed from the material [11, 12]. The principle of virtual work has also been applied to hygrostress formation, where the deformation is based on linear elasticity theory [13–16]. Recent work has allowed better understanding and modeling of coupled water transport and large deformation of fruit tissue during dehydration at the macroscale [2]. The disadvantage of continuum models is that they do not allow to investigate the effect of microstructural features of the tissue on macroscopic processes such as water loss and mechanical deformation. In contrast, in microscale models the heterogeneous structure of the tissue is taken into account by representing the complex cellular structure through a geometrical model [17–19]. A microscale water transport model was developed to describe water transport in fruit at the cellular level through the inter-cellular space and cells [20]. Different approaches to model the mechanical behavior of cellular tissues have been reported. A finite element model was developed to simulate the compression of a single suspension-cultured tomato cell [21]. However, the model was limited to mechanical loading and did not account for the deformation due to water loss. Other authors have used smoothed particle hydrodynamics (SPH) for the cytoplasm and a discrete element model (DEM) for the cell wall to simulate mechanical behavior cellular tissue in response to dynamic stimuli [22–24]. The model incorporated the mechanics during and after cell failure and cell wall rupture. However, the model predicts water efflux due to mechanical loading but not water transport as such between the cells. A microscale water transport model coupled with a mechanical deformation model has been developed to describe water transport and mechanical properties of pores, cell walls, cells and cell membranes [5]. Although this model provided detailed predictions of the local microscale water distribution and mechanical deformation, the extension of the microscale modelling techniques to predict the dehydration process of an entire fruit would require far more computational power than is currently available. The multiscale modelling paradigm offers an alternative approach to combine continuum type models defined at the macroscale level with the level of detail models incorporating the microscale features. A multiscale water transport and large deformation model is basically a hierarchy of models which describe the water transport and large deformation at different spatial scales as shown in figure 1. The coupling of the hierarchy of models is via computational analysis i.e. model results relevant to a particular spatial scale are linked to simulations at a different spatial scale. Multiscale modelling has been successfully applied in soft matter systems [25, 26], in material science and geoscience [27–33]. It has also been used as a useful tool for biological applications [34–36]. In this study, a 2D multiscale model is introduced and validated to perform a computational analysis of water transport and large deformation of apple (cv. Jonagold) cortex tissue at different scales (illustrated in figure 2). On the microscale, water transport and deformation of tissue are computed for the actual cell assembly of apple tissue. With these simulations, the apparent water diffusion and mechanical properties of tissues are calculated from a homogenization procedure. These parameters are consequently used to perform simulations of the dehydration and
− Pa mol m − 3 kg kgDM − 1 Pa − 1 m2 s − 1 N − m kg m − 2 s − 1 N m − 1 kg m − 1 Pa − 1 s − 1 m kg − mol m s − 1 J mol − 1 K − 1 Pa S K m m s − 1 m3 mol − 1 kg kg dm − 1 m
Greek symbols ε λ ρw ρdm σ
ψ
strain stretch ratio density of water dry matter density stress water potential
m m − 1 − kg m − 3 kg dm m − 3 Pa Pa
Subscripts a app c d db i m n p s T tn w
air apparent cell damping dry base node number i cell membrane natural pressure solute total tension cell wall
a major influence on tissue strength and macroscopic fruit firmness [6]. To better understand water loss and large deformations during dehydration and how they are affected by microscale features of the tissue, a modelling approach is most appropriate. In the macroscopic, continuum approach, the tissue is considered as homogeneous; lumped parameters are used as apparent material properties, and incorporate the effect of microstructural features such as the ensemble of cell membranes, cell vacuoles pores and cell walls. Macroscopic water transport models in food have been developed by, amongst others [7–9], which all adopted a continuum approach based on Fick’s second law of diffusion. Biological materials such as fruit can be modeled as a nonlinear viscoelastic continuum [10], when viewed at the macroscopic scale. Different approaches to model water transport and shrinkage have been reported. In the simplest approach, 2
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J. Phys.: Condens. Matter 26 (2014) 464111
Figure 1. Multiscale hierarchy of apple geometry. (a) Detailed microstructure of the tissue; (b) and (c) tissue and its representative elementary volume; (d) cylindrical apple cortex.
where ρw,c is wet base cell density (kg m − 3), ρc is cell density (kg m − 3), Dc is the water diffusion coefficient inside cells (m2 s − 1), and cw, c is water content on wet base (kg kg − 1). The wet base cell density and dry matter density inside cells are related by the following equation: ρw, c = ρdm, c Xc (2)
where Xc is the dry matter base water content (kg kg dm − 1). Using the relationship between water content on wet and dry matter base the following equation can be derived: Xc . cw, c = (3) 1 + Xc
Figure 2. Schematic outline of the multiscale approach.
The water capacity cψ, c (kg kg dm − 1 Pa − 1) is given by:
shrinkage of apple tissue samples using a macroscale model. The novelty of the current study over our previous work includes that we predict not only the apparent water transport properties but also the non-linear stress-strain relationship from the microscale model, compared them with parameter values obtained experimentally, and used the parameters in a macroscale model.
∂ Xc cψ , c = (4) ∂ ψc
where ψc the water potential of the cell (Pa). Substituting equations (2), (3) and (4) into equation (1) leads to: ⎛ ⎛ ρdm, c cψ , c ⎞ ∂ ρdm, c ⎞ ∂ ψc = ∇⋅ Dc ⎜ ⎟ ∇ψ . ⎜ ρdm, c + xc ⎟ cψ , c (5) ⎝ 1 + xc ⎠ c ⎝ ∂ xc ⎠ ∂t
2. Multiscale model 2.1. Microscale model
For the cell wall, the unsteady-state diffusion model is given by:
2.1.1. Microscale water transport model. The transport of
water in the intercellular space, the cell wall network and cytoplasm were modeled using diffusion laws and irreversible thermodynamics [37]. For the cells, the unsteady-state model of water transport reads:
∂ψ cψ , wρdm, w w = ∇⋅ ρdm, w Dwcψ , w ∇ ψw. (6) ∂t
∂ ρw, c = ∇⋅ ρc Dc ∇ cw, c (1) ∂t
∂ψ cψ , aρdm, a a = ∇⋅ ρdm, a Dacψ , a ∇ ψa. (7) ∂t
For the air, unsteady-state diffusion is modeled by:
3
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Figure 3. Schematic diagram of (a) cell boundary represented as a set of walls (modeled as springs) connected at points called vertices, (b) force at the vertices.
A simple flux law was applied to describe water transport through the cell membrane [37]: Figure 4. Relationship between turgor pressure and water potential at high relative humidity (99/97.7%).
ρ PmVw ( ψc − ψw ). J= w (8) RT
The force contributed by the wall is the resultant of the net turgor pressure force between the two adjacent cells working normal to the wall and the force associated with it:
2.1.2. Microscale mechanics model. A cell micromechanics
model was recently developed [38], and the basic characteristics are summarized below. In this 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 (see figure 3). Newton’s second law was considered to model the shrinkage mechanics. The following system of equations is solved for the velocity and position of the vertices i of the cell wall network:
Fw = Fturgor + Ftn. (13)
The net turgor force on the vertex is calculated by taking the difference in turgor pressure Pc 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 (see figure 3(b)): ⎛1 ⎞ Fturgor = ⎜ ( Pc,1− Pc,2 ) l ⎟ N. (14) ⎝2 ⎠
Hooke’s law was employed to determine the force acts along the wall and its magnitude
dv mi i = FT , i (9) dt
Ftn = −k u (15)
dxi = vi (10) dt
u = l − ln. (16)
To find the positions of each vertex of all cell walls of every single cell and, thus, the shape of the cells with time, a system of differential equations (9) and (10) for the positions and velocities of each vertex were established and solved using a Runge-Kutta fourth and fifth order (ODE45) method.
where m 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 − 1) are the position and velocity of node i, respectively, and FT , i 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, obtained from the water transport model outlined in the previous section, can be converted to turgor pressure using the relationship presented in section 2.1.3. 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
2.1.3. Coupling of water transport and mechanical deformation. The transient water transport model is solved for certain
time steps and the water loss results in loss of water potential in the cells. The change in water potential of the cells induces loss of turgor pressure. This is valid for the high range of equilibrium relative humidity values of the cells during dehydration until turgor drops to zero [5]. The dehydration is thus performed 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 ψs can be obtained from equation (17) ρ ρ n ψs = − CsRT = − w nsRT = − w s RT. (17) mw Xc ms
FT = ∑ Fw + Fd (11) w∈W
Fd = − bv (12) 4
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Table 1. Model parameters for different compartments of apple
(cv. Jonagold) cortex tissue (cell membrane, cell, cell wall and pore space) at 25 °C. Parameter
Value
Dc ρ dm,c Da ρ dm,a Cψ,a Dw ρ dm,w Pm Pc E b
2.2 × 10–9 m2 s − 1 111 kg m − 3 2.42 × 10–5 m2 s − 1 1.184 kg m − 3 1.3 × 10 − 10 Pa − 1 41.9 × 10–11 m2s − 1 58.035 kg m − 3 17 × 10–6 m s − 1 1 MPa 30 MPa 3.5 Ns µm − 1
References [47] [48] [49] [48] [50] [38] [20] [20] [38] [51] [38]
turgor pressure using the relation shown in figure 4. Then this set of turgor pressures was used in the shrinkage mechanics presented in section 2.1.3 to find the new equilibrium configuration of the cells. The 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 eight time steps of 50 s were required to reach equilibrium. The computation time was 2412 s for the unsteady state water transport simulation in each time step on a 8 Gb RAM quad-core PC, and 20 s for the mechanical equilibrium calculations. The numerical values of the water transport and mechanical parameters used in the microscale model are listed in table 1. These parameters were taken from our previous studies and from different literature resources. It should be noted that not all parameters were obtained at exactly the same environmental conditions, which could induce some uncertainty in the simulation results.
Figure 5. Numerical solution of the microscale coupled water transport model with deformation.
The turgor pressure is then equal to ψp = ψc − ψs. (18) The relationship between turgor pressure and water potential is shown in figure 4 for the considered range. 2.1.4. Implementation. Three different tissue samples of
1250 × 1600 μm, which had 80 cells with average cell diameter of 157.39 ± 22 μm and a porosity of 14.97% ± 1%, were generated using random simulation with the microscale mechanics model as described in [37]. Then, the geometric models of apple cortex tissue 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 more than 350 000 quadratic elements with triangular shape for each tissue geometry. A numerical experiment was carried out to calculate apparent material properties to be used in the macroscale model. As it was described in section 2.1.3, a relative humidity of 99% and 97.7% was chosen and applied to the top and bottom of the tissue geometry, respectively, while the other two lateral boundaries were defined to be impermeable. Afterwards, a sequence of time steps was considered for solving the coupled moisture transport and mechanical deformation model (see figure 5). The finite element method was used to discretise non-linear coupled model equations. In every time step the water potential and water content distribution in the tissue samples as well as the water flux through the sample were solved for a given water potential gradient across the sample. The initial water transport calculation was performed on the initial apple cortex tissue geometry that was obtained using a virtual fruit tissue generator. Afterwards, the water potential of each cell was obtained and converted into
2.1.5. Computation of apparent diffusivity and mechanical properties. The microscale model was used for in silico
analysis of water loss and deformation of cells in apple fruit tissue, and for computing the apparent properties of the apple tissue as a whole. The water capacity and dry mass density were combined into an apparent diffusion coefficient Dapp (m2 s − 1) of the tissue from the apparent water conductivity Kapp (kg m − 1 Pa − 1 s − 1) obtained from the microscale simulations using equation (19). Kapp Dapp = . (19) ρdm cψ
The water capacity cψ (kg kgdm − 1 Pa − 1) was calculated from the steady state calculated water content and water potential of the microscale tissues for RH values of 98.5% and 97.5% and temperature of 25 °C using microscale model simulations. The apparent water conductivity was obtained from 5
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Table 2. Apparent water transport and mechanical parameters of apple (cv. Jonagold) cortex tissue at 25 °C.
Parameter
Calculated from microscale model
Experiment
D × 10 − 11 [m2s − 1]
4.25 ± 0.3 RH (99–98%) 3.35 ± 0.2 RH (98–97%) 2.08 ± 0.01 RH (98.5%) 1.45 ± 0.01 RH (97.5%) 0.16 ± 0.1 52.18 ± 6.3 0.99 ± 0.1
10.30 ± 2.3 RH (62–63% [45])
cψ × 10 − 6 [kg kgdm − 1 Pa − 1] C10 [MPa] C01 [MPa] C11 [MPa]
1.06 ± 0.5 RH (97.5% [43]) 0.09 ± 0.03 [2] 44.18 ± 9.6 [2] 0.42 ± 0.1 [2]
the calculated flux for a specified gradient in water potential across the tissue sample:
were calculated from the following mechanical equilibrium, assuming no body and surface forces [42]:
h Kapp = − J (20) Δψ
∇⋅ ( S ⋅ GT ) = 0. (23)
The model equation (23) is based on the theory of compressible hyperelasticity with the decoupled representation of the Helmholtz free energy function with the internal variables [41]. The way in which the second Piola-Kirchhoff stress tensor (S) and the deformation gradient tensor (G) are defined is clearly explained in [2]. The coupling between water transport and the mechanical response (large deformations) is based on [2]. For the purpose of this article, we have selected a 2D geometrical model of the tissue samples, namely a long tissue cylinder (16.6 mm diameter). The model was numerically solved applying the finite element method, using the software Comsol Multiphysics 3.5a (Comsol BV, Sweden). Simulations were performed to compute the dehydration of the cylinder in a cross flow of air of 0.01 m s − 1, 25 °C and 97% RH. The relevant water transport and mechanical properties of apples used in the macroscale model were computed from microscale simulations as described earlier (see table 2).
with J (kg m − 2 s − 1) the total flux through the fruit tissue, Δψ (Pa) the assigned water potential difference between the two opposite sides and h (m) the thickness of the simulated tissue. The apparent mechanical properties of the tissue were calculated from the microscale model by assuming that the change in displacement (deformation) of tissue is due to a change in turgor pressure. This is logical, since turgor pressure generates a stress that leads to the expansion of cell wall [39]. The reverse phenomenon, shrinkage of cells, takes place as a result of turgor loss during dehydration. Thus, in the current study, we assumed that the history of turgor pressure and the resulting expansion or shrinkage of the cell wall network could determine the mechanical properties of the tissue in general. The current approach is plausible from the common type of mechanical tests where external forces are applied to determine tissue properties, because the stresses and the resulting shrinkage can be obtained from the natural dehydration processes. Based on this approach, the average changes in turgor and the corresponding displacement of the tissue in the direction of the applied gradient (see section 2.1.4) were calculated from microscale model simulations. Afterwards, mechanical properties were estimated by fitting these data points into the Yeoh hyperelastic potential with three parameters [2], (see equation (21)).
3. Results 3.1. Microscale model 3.1.1. Water potential and water content profile of apple tissue during dehydration. Water potential and content profiles of
one of the apple cortex tissues are shown in figure 6. Gradients of water potential can be observed from one cell to another in the direction of the applied gradient (see section 2.1.4). Cells at the same horizontal position, which is perpendicular to the applied water potential gradient, tend to have similar and uniform water potential, which is due to the high water conductivity within the cells. The bottom side of the sample is dehydrated more severely while the cell turgor decreases. The water content of the cells remains relatively uniform and constant during the course of dehydration. This is plausible, since the water content of cells do not change abruptly, as a result of shrinkage during water loss [5].
⎛ ⎛ ⎞ 1 ⎞⎛ 2 σYeoh ( λ ) = 2 ⎜ λ2 − ⎟ ⎜ C10 + 2C01⎜ λ2 + − 3 ⎟ ⎝ ⎝ ⎠ λ ⎠⎝ λ ⎛ ⎞2 ⎞ 2 + 3C11⎜ λ2 + − 3 ⎟ ⎟ . ⎝ ⎠ ⎠ λ
(21)
2.2. Macroscale model of water transport coupled with deformation
A macroscopic phenomenological approach was used to model water transport in fruit tissue, where water moves in the tissue as a consequence of a gradient in water potential [40]:
3.2. Apparent apple tissue properties
∂ψ ρdm cψ = ∇⋅ K ∇ ψ . (22) ∂t
The apparent water capacity of tissue was calculated from microscale model simulations for the RH of 98.5% and 97.5% and compared well with the water capacity of apple (cv. Jonagold) cortex tissue, which was calculated from desorption isotherm that we measured previously [43] at
Materials undergoing large deformations are best described by nonlinear elasticity theory according to [41], instead of by linear elasticity theory. The stresses in the fruit tissue 6
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Figure 6. Water potential (Pa) (left) and water content (kg kgdm − 1) (right) of an apple (cv. Jonagold) cortex tissue sample (1.25 mm
thickness) dehydrated for 500 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 = 200 s, (c) t = 500 s.
97.5% RH (see table 2). In general, the water capacity will decrease with RH. The apparent diffusivity of the tissue samples was calculated from the obtained fluxes for the RH levels of 99–98% and 98–97%, using the microscale model. Compared with the diffusivity value obtained from measurements, the apparent diffusivity calculated from the microscale simulations was found to be lower (see table 2). The deviation of predicted apparent value could be attributed to the fact that the model in this study did not account for cell to cell water transport through plasmodesmata. The measured apparent diffusivity and water capacity showed large variations (see table 2), which could be due to microstructural variations of the tissue. Microscale simulations with three different tissue structures showed that differences in microstructure affected the water transport in the tissue, but the resulting variation was smaller as compared with the experiments. Figure 7 displays the calculated average stress-strain response of tissue during dehydration. The apparent mechanical properties were calculated by fitting the nonlinear mechanical model (with Yeoh hyperelastic potential) to the data points obtained from microscale simulations. The agreement between measured and predicted values of mechanical parameters at small strains is excellent, but above 3% strain clear discrepancies can be observed. This
Figure 7. Stress–strain response of apple cortex tissue from microscale simulations for three random tissue geometries denoted by different symbols (□, ○, and ◊). The solid, dot and dashed lines are the corresponding constitutive model fits. The (Δ) and the dash-dot line represent an actual compression experiment and the constitutive model fit.
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Figure 8. Simulation results for drying experiment at 97% RH, 25 °C. Contour plots of the water potential (Pa) after 4 h using experimental
(a) and apparent parameters obtained from mircoscale simulations (b). Evaluation of the average water content (c) and areal shrinkage (d) with dehydration time; the dashed and solid line represent macroscale simulations using experimental and apparent parameters, respectively.
measured diffusivity was relatively higher than the one calculated from microscale simulations. From figure 8(d) it can observed that the rate of areal shrinkage of the cylindrical sample was non-linear. Fast deformation occurred at the beginning of dehydration, which gradually decreases with time. The percentage of areal shrinkage as simulated using measured material properties was higher than using computed ones, again due to the discrepancy between measured and computed material properties. In general, a good agreement was found between macroscale simulations.
could be due to the fact that in reality compression tests break cells so that more water is lost to the environment than mild dehydration process, thus resulting in larger deformation in the compression experiment. Another reason for the deviation between predicted and measured hygromechanical properties is the 2D cross-sectional simplification of the tissue that is essentially 3D, consisting of ellipsoid cells and a 3D network of pores. The estimated parameters from microscale simulations are close but not equal to the experimentally obtained values from compression tests (see table 2).
4. Discussion 3.3. Macroscale model
In this study, a microscale model of coupled water transport and large deformation was used to predict apparent tissue properties such as tissue diffusivity and viscoelastic properties. These properties were then used in a macroscale model to predict shrinkage of apple cortex cylinder during dehydration. While we validated the macroscale model previously [2, 44], in this paper we validated the microscale model and the overall multiscale model. The microscale coupled water transport and large deformation model was validated by
The 2D macroscale coupled water transport and mechanical model was solved using calculated apparent water transport and mechanical properties. In figure 8 this simulation is compared to one in which the measured parameters were used. The water loss percentage calculated from the macroscale simulation using measured apparent material properties was higher than using apparent properties computed from microscale simulations (see figure 8(c)). This is due to the fact that the 8
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comparing macroscopic apparent water transport and mechanical properties computed from microscale model with measured ones [2, 43, 45]. The effect of random variation in tissue structure on transport and mechanical properties was investigated by taking three different geometrical models of apple cortex tissue and was found to be insignificant. However, if anisotropy of cell shape is present [46], the variation of the microstructure would likely become more significant e.g. for hypodermal and epidermal tissue of fruit. Shrinkage adversely affects the quality of fruit during storage at too low relative humidity. Experimental and simulation results of our previous studies using a macroscale model showed that the apple tissue undergoes large deformations during dehydration [2, 44]. This is confirmed by the current study using microscale model simulations, and even at high RH. The Yeoh hyperelastic model successfully estimated the nonlinear curve characteristic of the tissue obtained from microscale model simulations. The apparent mechanical properties obtained from the microscale simulations were in the range of measured ones [2]. Macroscale model simulations were performed using the apparent water transport and mechanical properties of the tissue obtained from the microscale model simulations and measurements. The predicted water loss and areal shrinkage using computed or measured apparent material properties corresponded well. The 2D microscale model of water transport coupled with mechanical deformation does not take into account the connectivity of cell walls and air spaces that can affect the transport phenomena and mechanics of a tissue considerably [34]. 3D modeling of water transport and cell mechanics at the microscale is required to investigate this. The aim of the current paper, was to present a multiscale methodology for hygromechanical analysis of fruit, and thus it was opted to use a simplified 2D approach before elaborating the more cumbersome 3D model. An additional comment can be made here. Moisture transport is different from gas transport as we have shown that the cell membrane is a more determining factor than the pore space connectivity [20], which is opposite to the case of gas diffusion in tissues. In gas diffusion analysis, we found significant differences between 2D predicted and measured gas diffusion properties and thus a 3D analysis was required [34]. Here, for moisture transport we already find reasonable agreement of the 2D predictions with measurements. To be conclusive, however, indeed a 3D analysis will be required.
developments could focus on 3D modeling to elaborate the effect of interconnectivity on the macroscopic water transport and mechanical property of a tissue, simulating compression test and extending the model formulations to predict more severe dehydration (at lower RH) as would be expected to occur during drying processes. Acknowledgements The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 226783 and 245288. The authors would also like to acknowledge the Fund for Scientific Research—Flanders (Grant no. FWO G.0603.08) and the KU Leuven (project OT 08/023 and 12/055) for financial support. References [1] Wills R, McGlasson B, Graham D and Joyce D 1998 Postharvest: An Introduction to the Physiology and Handling of Fruit, Vegetables and Ornamentals 4th edn (Willingford: Cab International) [2] Aregawi W A, Defraeye T, Verboven P, Herremans E, De Roeck G and Nicolai B 2013 Modeling of coupled water transport and large deformation during dehydration of apple tissue Food Bioprocess Technol. 6 1936–78 [3] Pieczywek P M and Zdunek A 2014 Finite element modelling of the mechanical behaviour of onion epidermis with incorporation of nonlinear properties of cell walls and real tissue geometry J. Food Eng. 123 50–9 [4] Ghysels P, Samaey G, Tijskens B, Van Liedekerke P, Ramon H and Roose D 2009 Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics Phys. Biol. 6 016009 [5] Fanta S W, Abera M K, Aregawi W, Ho Q T, Verboven P, Carmeliet J and Nicolai B M 2014 Microscale modeling of coupled water transport and mechanical deformation of fruit tissue during dehydration J. Food Eng. 124 86–96 [6] Konstankiewicz K and Zdunek A 2001 Influence of turgor and cell size on the cracking of potato tissue Int. Agrophysics 15 27–30 [7] Dhall A and Datta A K 2011 Transport in deformable food materials: a poromechanics approach Chem. Eng. Sci. 66 6482–97 [8] Rakesh V and Datta A K 2011 Microwave puffing: determination of optimal conditions using a coupled multiphase porous media—large deformation model J. Food Eng. 107 152–63 [9] Van der Sman R G M 2013 Modeling cooking of chicken meat in industrial tunnel ovens with the Flory–Rehner theory. Meat Sci. 95 940–57 [10] Wineman A 2009 Nonlinear viscoelastic solids—a review Math. Mech. Solids 14 300–66 [11] Pioletti D P, Rakotomanana L R, Benvenuti J F and Leyvraz P F 1998 Viscoelastic constitutive law in large deformations: application to human knee ligaments and tendons J. Biomech. 31 753–7 [12] Moreira R, Figueiredo A and Sereno A 2000 Shrinkage of apple disks during drying by warm air convection and freeze drying Dry. Technol. 18 279–94 [13] Niamnuy C, Devahastin S, Soponronnarit S and Vijaya Raghavan G S 2008 Modeling coupled transport phenomena and mechanical deformation of shrimp
5. Conclusion A multiscale model of water transport with mechanical deformation in apple cortex tissue was developed to study water loss and shrinkage of fruit cylinders. Using in silico experiments with the coupled water transport and mechanical model at the microscale, we predicted apparent diffusivity and nonlinear mechanical properties that compared favourably with experimental measurements. A multiscale approach provides insight how microscale features of the tissue affects the dynamics of dehydration in apple and in fruit tissue in general. Future 9
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[33] Makov G, Gattinoni C and De Vita A 2009 Ab initio based multiscale modelling for materials science Model. Simul. Mater. Sci. Eng. 17 1–17 [34] Ho Q T, Verboven P, Verlinden B E, Herremans E, Wevers M, Carmeliet J and Nicolaï B M 2011 A three-dimensional multiscale model for gas exchange in fruit Plant Physiol. 155 1158–68 [35] Seymour R S 2001 Diffusion pathway for oxygen into highly thermogenic florets of the arum lily Philodendron selloum J. Exp. Bot. 52 1465–72 [36] Wood B D, Quintard M and Whitaker S 2002 Calculation of effective diffusivities for biofilms and tissues Biotechnol. Bioeng. 77 496–516 [37] Nobel P S 1991 Physicochemical and Environmental Plant Physiology (San Diego: Academic) [38] Abera M K, Fanta S W, Verboven P, Ho Q T, Carmeliet J and Nicolai B M 2012 Virtual fruit tissue generation based on cell growth modelling Food Bioprocess Technol. 6 859–69 [39] Kroeger J H, Zerzour R and Geitmann A 2011 Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth PLoS One 6 e18549 [40] Nguyen T A, Verboven P, Schenk A and Nicolaï B M 2007 Prediction of water loss from pears (Pyrus communis cv. Conference) during controlled atmosphere storage as affected by relative humidity J. Food Eng. 83 149–55 [41] Holzapfel G A 2000 Nonlinear Solid Mechanics. A Continuum Approach for Engineering (Chichester: Wiley) [42] Bonet J and Wood R D 1997 Nonlinear Continuum Mechanics for Finite Element Analysis (Cambridge: Cambridge University Press) [43] Veraverbeke E A, Verboven P, Scheerlinck N, Lan Hoang M and Nicolaı̈ B M 2003 Determination of the diffusion coefficient of tissue, cuticle, cutin and wax of apple J. Food Eng. 58 285–94 [44] Aregawi W, Defraeye T, Saneinejad S, Vontobel P, Lehmann E, Carmeliet J, Derome D, Verboven P and Nicolaï B 2013 Dehydration of apple tissue: intercomparison of neutron tomography with numerical modelling Int. J. Heat Mass Transf. 67 173–82 [45] Verstreken E, Van Hecke P, Scheerlinck N, De Baerdemaeker J and Nicolaï B 1998 Parameter estimation for moisture transport in apples with the aid of NMR imaging Magn. Reson. Chem. 36 196–204 [46] Gibson L J and Ashby M F 1998 Cellular Solids: Structure and Properties (Oxford: Pergamon) [47] Holz M, Heil R, Sacco A, Chemie P, Karlsruhe U, Karlsruhe D, Foggia U, Napoli V and August A 2000 Temperature-dependent self-diffusion coefficients of water and six selected molecular liquids for calibration in accurate 1 H NMR PFG measurements Phys. Chem. Chem. Phys. 4740–2 [48] Fanta S W, Vanderlinden W, Abera M K, Verboven P, Karki R, Ho Q T, De Feyter S, Carmeliet J and Nicolaï B M 2012 Water transport properties of artificial cell walls J. Food Eng. 108 393–402 [49] Lide R L and Frederikse H P R 1994 Hand Book of Chemistry and Physics (Boca Raton, FL: CRC Press) [50] Coulson J M and Richardson J F 2004 Fluid Flow, Heat Transfer and Mass Transfer (Oxford: Butterworth-Heinemann) [51] Wu N and Pitts M J 1999 Development and validation of a finite element model of an apple fruit cell Postharvest Biol. Technol. 16 1–8
during drying in a jet spouted bed dryer Chem. Eng. Sci. 63 5503–12 [14] Kowalski S J and Rajewska K 2002 Drying-induced stresses in elastic and viscoelastic saturated materials Chem. Eng. Sci. 57 3883–92 [15] Inazu T, Iwasaki K and Furuta T 2005 Stress and crack prediction during drying of Japanese noodle (udon) Int. J. Food Sci. Technol. 40 621–30 [16] Shim V P W, Yang L M, Lim C T and Law P H 2003 A viscohyperelastic constitutive model to characterize both tensile and compressive behavior of rubber Polymer 42 523–31 [17] Yao Z and Le Maguer M 1996 Mathematical modelling and simulation of mass transfer in osmotic dehydration processes. Part I: conceptual and mathematical models J. Food Eng. 29 349–60 [18] Toupin C J and Le Maguer M 1989 Osmotically-induced mass transfer in plant storage tissues: a mathematical model. Part II J. Food Eng. 10 97–121 [19] Marcotte M, Toupin C J and Le Maguer M 1991 Mass transfer in cellular tissues. Part I: the mathematical model J. Food Eng. 13 199–220 [20] Fanta S W, Abera M K, Ho Q T, Verboven P, Carmeliet J and Nicolai B M 2013 Microscale modeling of water transport in fruit tissue J. Food Eng. 118 229–37 [21] Dintwa E, Jancsók P, Mebatsion H K, Verlinden B, Verboven P, Wang C X, Thomas C R, Tijskens E, Ramon H and Nicolaï B 2011 A finite element model for mechanical deformation of single tomato suspension cells J. Food Eng. 103 265–72 [22] Van Liedekerke P, Tijskens E, Ramon H, Ghysels P, Samaey G and Roose D 2010 Particle-based model to simulate the micromechanics of biological cells Phys. Rev. E 81 061906 [23] Van Liedekerke P, Ghysels P, Tijskens E, Samaey G, Smeedts B, Roose D and Ramon H 2010 A particle-based model to simulate the micromechanics of single-plant parenchyma cells and aggregates Phys. Biol. 7 026006 [24] Van Liedekerke P, Ghysels P, Tijskens E, Samaey G, Roose D and Ramon H 2011 Mechanisms of soft cellular tissue bruising. A particle based simulation approach Soft Matter 7 3580 [25] Praprotnik M, Site L D and Kremer K 2008 Multiscale simulation of soft matter: from scale bridging to adaptive resolution Annu. Rev. Phys. Chem. 59 545–71 [26] Peter C and Kremer K 2009 Multiscale simulation of soft matter systems—from the atomistic to the coarse-grained level and back Soft Matter 5 4357 [27] Ricard J, Guigné F and Laiarinandrasana L 2014 Damage and fracture mechanisms of polyoxymethylene: multiscale experimental study and finite element modeling Eng. Fract. Mech. 115 270–83 [28] Regenauer-Lieb K and Yuen D A 2003 Modeling shear zones in geological and planetary sciences: solid- and fluidthermal–mechanical approaches Earth-Sci. Rev. 63 295–349 [29] Fish J and Yu Q 2001 Multiscale damage modeling for composite materials : theory and computational framework 2. 0 mathematical homogenization for damaged composites Int. J. Numer. Methods Eng. 52 161–91 [30] Gitman I M, Askes H and Sluys L J 2008 Coupled-volume multi-scale modelling of quasi-brittle material Eur. J. Mech. A 27 302–27 [31] Goldenfeld N, Athreya B P and Dantzig J A 2006 Renormalization group approach to multiscale modelling in materials science J. Stat. Phys. 125 1015–23 [32] Elliott J A 2011 Novel approaches to multiscale modelling in materials science Int. Mater. Rev. 56 207–25
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