Theoretical models for food drying commonly utilize an effective diffusivity solved through curve fitting based on experimental data. This creates models with limited predictive capabilities. Multiscale modeling is one approach which can help transition to a more physics - based model minimizing the empirical information required while improving a model’s predictive capabilities. However, to enable an accurate scaling operation, multiscale models require diffusivity at a fine scale (microscale). Measuring these properties is experimentally costly and time consuming as they are often temperature and/or moisture dependent. This research conducts an inverse analysis on a multiscale homogenization food drying model to deduce the temporal diffusivity of intracellular water. A representation of the real cellular water breakdown was considered and appropria te assumptions to represent its cellular heterogeneity , in relation to time, were investigated. The work uncovered that a linear decrease in intracellular Accepted in Biosystems Engineering 2 water content could be assumed and thus a function for its diffusivity was developed. The proposed function is in terms of sample temperature and intracellular water content opening the possibilities to be applied to various food materials. Keywords: multiscale modeling, cellular diffusivity, homogenization, food drying, temporal Nomenclature Abbreviations FW Free water ICW Intracellular water NMR Nuclear Magnetic resonance Symbol s C Macroscale instantaneous moisture concentration (mol m-3) c Microscale concentration ( mol m-3) cp Specific heat of apple tissue (J kg-1 K-1) cp air Specific heat of air (kJ kg-1 K-1) D Cellular diffusivity (m2 s-1) DAB Binary mass diffusivity (m2 s-1) Deff Effective moisture diffusivity (m2 s-1) effD Average diffusivity (m2 s-1) DFW Diffusivity of free water ( m2 s-1) DICW Diffusivity of intracellular water (m2 s-1) Do Integration constant (m2 s-1) Accepted in Biosystems Engineering 3 Ea Activation energy (J mol-1) hfg Latent heat of evaporation (J kg-1) hm Mass transfer coefficient (m s-1) hT Heat transfer coefficient (W m-2 K-1) k Thermal conductivity of apple tissue (W m-1 K-1) kair Thermal conductivity of air (W m-1 K-1) Lc Characteristic length (m) M0 Initial moisture content (kg kg-1 dry bases) M0wb Initial wet basis moisture content (kg kg-1 wet bases) Mdb Moisture content dry basis (kg kg-1 dry bases) ,expdbM Average experimental moisture content (kg kg-1 dry bases) ,db numM Average predicted moisture content (kg kg-1 dry bases) Me Equilibrium moisture content (kg kg-1 dry bases) Mw Molar mass of water (kg mol-1) n Unit vector normal to the boundary Nu Lc Nusselt number Pr Prandtl number pv air Vapor air pressure of ambient air (Pa) pv eq Equilibrium vapor pressure (Pa) pv sat Saturated vapor pressure of water (Pa) R Universal gas constant (J mol-1 K-1) r, z Macroscale spherical coordinates (m) Re Lc Reynold’s Number Sc Schmidt Number Accepted in Biosystems Engineering 4 ShLc Sherwood number t Time (s) T Macroscale instantaneous Temperature (K) T0 Initial sample temperature (K) Tair Drying air temperature (K) Ts Sample temperature (K) u1 and u2 Corrective factors v Air velocity (m s-1) WC ICW Intracellular water content x, y Microscale cartesian coordinates (m) Greek symbols β Coefficients µair Viscosity of air (kg m-1 s-1) ρ Density of apple tissue (kg m-3) ρair Density of air (kg m-3) ρw Density of water (kg m-3) υair Kinematic viscosity of air (m2 s-1) ω Area of the microscale domain (m2) 1. Introduction Drying is one of the most dominant food preservations and processing techniques. The demand for dried food products has significantly increased over the last decades driving a fast - growing global food drying industry. Although various drying methods, including drying with renewable energy (Bahammou et al., 2019; Bahammou et al., 2020), have been developed, hot Accepted in Biosystems Engineering 5 air or convective drying is still being widely used by industry due to its simplicity. However, convective drying is an energy intensive lengthy proc ess which can cause deterioration of a product’s final quality (Pham et al., 2019). Additionally, drying is a complex multiphysics problem which involves simultaneous heat, mass, and momentum transport processes with continuous phase change (Khan, Joardder, Kumar, & Karim, 2018). In order to address these complex problems and design optimal and effective food drying systems, fundamental understanding of the transport process needs to be developed. Physics- based mathematical modeling can provide deep insight into drying, and as such, many researchers have developed models in an attempt to simulate the complex drying process spanning spatial and temporal dimensions (Khan, Welsh, Gu, Karim, & Bhandari, 2020) . However, food materials are complex and multiscale in nature. Plant -based food materials are structurally heterogeneous with moisture distributed throughout the various cellular environments of the material (Rahman, Gu, & Karim, 2018; Rahman, Joardder, & Karim, 2018). It has been found that the largest proportion of water (around 80%) is located within intracellular spaces (inside cell vacuole), known as intracellular water (ICW), whereas the remainder of the water is located within the intercellular and cell wall environments (Khan, Wellard, Nagy, Joardder, & Karim, 2016). During drying, cellular water can migrate from its cellular compartment to the surrounding environments through various migration pathways. Specifically, ICW migrates from cell to a neighboring cell via diffusion and from cell to intercel lular spaces through periodic cell rupturing (Khan, Wellard, Nagy, Joardder, & Karim, 2017) . Most existing physics -based models have been developed to investigate drying at the macroscale (tissue scale) (Kumar, Joardder, Farrell, Millar, & Karim, 2016) , wh ere the structure of the material is approximated as a representative elementary volume (REV) (Whitaker, 1977), neglecting to consider the material’s dynamic cellular heterogeneity and the migration of ICW (Wang, Chen, & Mujumdar, 2007). The migration of ICW causes dynamic Accepted in Biosystems Engineering 6 modifications to the materials physical structure and nutritional characteristics effecting the products quality (Pham, Khan, & Karim, 2020) , and therefore needs be considered to accurately predict the drying kinetics and optimize the drying processes. To address such a research problem, multiscale modeling has been introduced to the field of food drying (Welsh, Simpson, Khan, & Karim, 2018). Multiscale modeling is an approach which can help transition towards a more mechanistic model by considering the dynamic water heterogeneity within the material (Welsh, Khan, & Karim, 2021) . A multiscale model is a series of sub -models which investigate a products behavior over multiple spatial scales, modeling an in depth and more realistic descripti on of the underlying physics (Rahman, Joardder, Khan, Nghia, & Karim, 2018) . Recently, researchers have shown their immense interest in multiscale food drying modeling and provided some general background into the topic through comprehensive reviews (Ho et al., 2013; Rahman, Joardder, Khan, et al., 2018; Welsh et al., 2018). Additionally, some researchers have directly attempted to develop multiscale models for food material (Aregawi, Abera, Fanta, Verboven, & Nicolai, 2014; Ho et al., 2011) . However, their attempts did not consider actual drying conditions as they considered natural dehydration and were developed for other applications such as postharvest storage or for some arbitrary moisture removal without conjugal heat and mass transfer (Wijerathne et al., 2019). To overcome this significant research gap, Welsh et al. (2021) developed a basic multiscale food drying model utilizing knowledge of the microstructural evolution. The work investigated convective drying apple tissue at three different temperatu res (45, 60 and 70 °C ) and predicted the experimental data well for drying at low/medium temperatures. However, the model utilized unique spatial and temporal averaged microscale domains where a comprehensive experimental knowledge of the microstructural e volution is required to achieve the averages. This necessity significantly increased the experimental cost of the approach. Moreover, to obtain this type of experimental Accepted in Biosystems Engineering 7 validation data requires very sophisticated experimental facilities such as nuclear ma gnetic resonance (NMR), or X -ray microcomputed tomography which is not always possible or feasible. Moreover, Welsh et al. (2021) model was developed based on the condition- dependent properties, which limits its widespread applicability. The limitations o f the microstructural evaluation via experimental facilities and the condition dependency of the model must be minimized. This can be achieved through the development of a theoretical method to estimate the comprehensive cellular properties of the material. Within published literature, experimental property data is sparse and often unreliable as they vary by 3- 5 orders of magnitude for the same material (Dadmohammadi & Datta, 2019; Saravacos & Maroulis, 2001). One key property of interest in drying of food material is diffusivity. Effective diffusivity is heavily influenced by its moisture content and temperature distribution and therefore numerous models have been constructed to accurately predict the moisture diffusivity of various food materials during drying (Dadmohammadi & Datta, 2019) . Commonly, all mass transport mechanisms are lumped and Fickian diffusion is assumed. The effective diffusivity is then solved through curve fitting techniques to obtain a constant moisture diffusivity. The resulting property is often unable to consider any variations in material, experimental set up and/or psychrometric drying conditions (Dadmohammadi & Datta, 2019). To enhance the predictive capabilities of the moisture diffusivity, the Arrhenius framework is applied. T his allows the property to be suitable for drying at different temperatures, however, the resulting diffusion coefficient is still condition- dependent and is only suitable for that particular material (i.e. carrot or apple) within that drier configuration. Researchers have attempted to overcome this issue by developing new approaches, such as the modified Arrhenius approach (Lentzou, Boudouvis, Karathanos, & Xanthopoulos, 2019) which considers the diffusivity in terms of the materials moisture/deformation. Additionally, Dadmohammadi and Datta (2019) developed a new mechanistic diffusivity function which was Accepted in Biosystems Engineering 8 constructed around porous media modeling and requires unique phase dependent diffusivities for each material. However, these approaches require extensive experimental investigations and therefore generate a dramatic demand in physical and mechanical characterization of each product being dried (Perré, 2007) . Additionally, most studies in literature have investigated bulk level moisture and/or temperature dependent diffusivities (Joardder, Karim, Kumar, & Brown, 2014; Khan, Kumar, Joardder, & Karim, 2017), neglecting the cell level water transport phenomena (such as cell rupture). Therefore, further investigation into cellular properties and their temporal variations are required for multiscale modeling to be widely adopted for food drying. Cellular diffusiv ity properties can be either experimentally investigated or approximated utilizing an inverse problem (optimization). Sibgatullin et al. (2007) investigated the diffusion and relaxation behavior of water in fresh apple parenchyma cells using NMR spectrosco py. Their approach distinguished two components, a slow relaxing component (assigned to the vacuole) for the restrictive diffusion and a fast component (other cell components) for the unrestricted diffusion component. However, this investigation was conducted for fresh apple tissue not food material during drying. Food drying models require temperature and/or moisture dependent properties to describe the complex simultaneous heat and mass transport which occurs. In order to apply Sibgatullin et al. (2007) approach to food drying the technique would need to be repeated in regular intervals in triplicate throughout the drying process. Recently, Dadmohammadi, Kantzas, Yu, and Datta (2020) estimated the permeability and porosity of potato cells using NMR from r aw to processed states. The work uncovered permeability coefficients at varying temperatures which can be used to calculate the capillary diffusion in potato tissue. However, investigating these properties experimentally can become time consuming and costly (Khan, Nagy, & Karim, 2018). Accepted in Biosystems Engineering 9 The alternative approach is to conduct an inverse analysis computationally . If the phase morphology is correctly represented at a cell level, a complete scaling approach can be performed. However, instead of using the multiscale approach to predict macroscopic properties, an inverse analysis can be conducted allowing the local (microscale) properties to be deduced from the macroscopic measurements (Perré, Almeida, Ayouz, & Frank, 2016) . This approach has already been applied to the application of timber drying. Perré et al. (2016) presented a comprehensive strategy to predict different wood properties from anatomical images, particularly focusing on several meshless methods. A key step within the strategy was the representati on of the structure which accounted for the local anisotropy and heterogeneity of the cell walls for the material. Inverse problems have also been utilized on macroscale theoretical models. Weres and Olek (2005) developed a finite element analysis strategy for inverse problems. The proposed algorithm was investigated and validated for two types of timber to identify the thermal conductivity, convective heat transfer coefficient, diffusion coefficient, equilibrium moisture content and convective mass transfe r coefficient. Lentzou et al. (2019) used an inverse problem in a food drying model. The investigation estimated the effective macroscale diffusivity and mass transfer coefficient in a moving boundary model for fruit isothermal drying and shrinkage. However, an inverse study has not been applied in the context of multiscale food drying modeling to uncover the temporal cellular properties. To address this important research gap, this work aims to conducts an inverse analysis for deducing the diffusivity of intracellular water while investigating the temporal water heterogeneity of apple tissue during drying. Utilizing an inverse study to develop cellular properties has the ability to efficiently determine a material’s properties without the complexity and co st of measuring these properties experimentally. The approach will apply multiscale modeling, specifically homogenization, for convective drying at 60°C. A temporal heterogeneity investigation will be performed to compare the real cellular water morphology Accepted in Biosystems Engineering 10 and suitable assumptions which can represent the materials evolution. Finally, the most robust approach uncovered within the heterogeneity investigation will be utilized to construct the property for ICW diffusivity. 2. Materials and methods The cellular str ucture representation is presented first, followed by the multiscale model and solution procedure. 2.1.1 Cellular structure representation The cellular structure of apple consists of a few key components: protoplast (composed of vacuole, tonoplast membrane and cytoplasm), the cell membrane, cell wall and intercellular spaces (Prawiranto et al., 2018). During drying apple tissue deforms and changes due to the various transport mechanisms which occur separately but coexist (Lentzou et al., 2019) . Additionally, dif ferent cellular deformation can occur when drying at different temperatures (Khan, Farrell, Nagy, & Karim, 2018; Mahiuddin, Khan, Duc Pham, & Karim, 2018; Mahiuddin, Khan, Kumar, Rahman, & Karim, 2018) . For this investigation, three distinct stages of cell ular deformation will be referred to, cell intact, cell rupturing and ultimate cell rupture. For convective drying, the drying air temperature generally dictates what cellular deformation occurs. For example, when convective drying at temperatures below 50 °C, cells generally remain intact and the composition of cellular water (water within different cellular environments) remains fairly consistent throughout the entire drying period (Khan, Farrell, et al., 2018). When convective drying at temperatures above 50 °C , cell rupturing occurs, drastically changing the amount of water within different cellular environments (Khan, Farrell, et al., 2018), causing all three stages of cellular deformation to occur . Khan, Farrell, et al. (2018) investigated this phenomenon with NMR and X-ray tomography demonstrating w hen cells collapse, water rushes out of the cells, moving from the intracellular spaces to intercellular Accepted in Biosystems Engineering 11 spaces and thus becomes free water (FW)]. This is visually represented in Figure 1 (c). Once the water becomes free water it can be easily transport therefore changing the transport properties of the material. This current work investigates convective drying granny smith apples at 60 °C , therefore all three stages of cellular deformation occur, and the cellular water heterogeneity dynamically changes during drying. This makes it ideal to construct a versatile property, however adding complexitie s to accurately representing the heterogeneity . With this in mind, the microscale domain considered two subdomains, ICW denoted as ω ICW and FW denoted as ωFW (Khan & Karim, 2017), demonstrated in Figure 1. This groups the effects of the protoplast, cell membrane and cell wall within the ICW domain. Additionally, the intercellular spaces were grouped and represented as a circle located within the center of the microscale, Figure 1 (b). This requires minimal microstructural knowledge but forces the effective diffusivity to be isotropic. These microscale domain approximations are constructed from ICW/FW ratios published in Khan, Farrell, et al. (2018) which were measured using NMR, details outlined in section 2.2.6. Anisotropic diffusivity is outside the scope of the current work, however, it can be introduced into this model but ungrouping the intercellular spaces, considering the real cellular structure of the material or considering an off -centered circle. It should be noted, this would significantly increase the cost of the model. [Figure 1 can be placed here] 2.2 Multiscale model The macroscale transport model used an axisymmetric coordinate system and assumed the following: (1) internal convective flow and heat generation is neglected, (2) drying air Accepted in Biosystems Engineering 12 properties are constant at 60°C , (3) axisymmetric heating occurs, (4) moisture is only evaporated from the surface and (5) thermal equilibrium exists betw een all phases . Although food materials are multiphase in nature, contribution gas transport is insignificant in drying. The addition of gas transport in a physics -based model increases its computational cost and does not provide significant benefits for predicting the drying kinetics, particularly in convective drying at 60 °C . Therefore, in this research, it was assumed that the impact of gas transport can be neglected and hence was not included in the model. To achieve the multiscale model, homogenization was applied. Homogenization allows a heterogeneous material to be approximated as a homogeneous medium, replacing a spatially varying property with an effective or equivalent counterpart, visually represented in Figur e 1 (a). Homogenization introduces a second scale into the computational model, a microscale, representing the heterogeneous structure of the material, Figure 1 (b) to (d). Homogenization is focused around a limit, where as the scale parameter (the ratio between the length of each scale, L /ℓ) tends to zero, the heterogeneities vanish, leading to a hom ogeneous material (Pavliotis & Stuart, 2008) . To achieve this goal, an arrangement of periodic microscales can be introduced within the macroscale domain, Figure 1 (a). When assuming a periodic domain, the effective diffusivity can be derived utilising the standard method of homogenization by asymptotic expansion. This routine derivation is well documented in literature (Auriault, Boutin, & Geindreau, 2010; Carr & Turner, 2014) . Therefore, the microscale mass transport equation is , 0cc cDDtx x y y ∂∂ ∂ ∂ ∂+− +− =  ∂∂ ∂ ∂ ∂  , (1) where c is the microscale concentration (mol m-3) and D is the cellular diffusivity (m2 s-1) equaling D ICW if x,y ϵ ωICW and DFW if x,y ϵω FW. The microscale is coupled to the macroscale mass conservation (Golestani, Raisi, & Aroujalian, 2013), Accepted in Biosystems Engineering 13 10eff effC CCDr Dt rr r z z∂ ∂ ∂∂ ∂   + − +− =  ∂ ∂ ∂∂ ∂   , (2) where C is the macroscale instantaneous moisture concentration ( mol m-3), t is the time (s) and Deff is the effective moisture diffusivity (m2 s-1). In drying food material, the Soret effect (mass flux due to a temperature gradient) is considered to be small (Datta, 2007; Itaya, Kobayashi, & Hayakawa, 1995) and therefore was not considered. Also, property data on the Soret effect is generally unavailable. As multiscale modeling advanc es in the field, these effects can be investigated in depth in the future. The 2D homogenized effective diffusivity is given by, 12 1211(, ) ( 1 ) (, ) ,11(, ) (, ) ( 1 )effuuDxy d Dxy dxx DuuDxy d Dxy dyyωω ωωωωωω ωωωω∂∂ +∂∂= ∂∂+ ∂∂∫∫ ∫∫ (3) and ω is the area of the m icroscale domain, u1 and u2 are the corrective factors and the solution of the periodic cell problem: ( ( ) 0 1, 2j Du j∇⋅ ∇ + = =je , (4) where, 10,01 = =  12ee , (5) ju is periodicω− , (6) 10j Audω ω=∫ . (7) The solution of u j is unique only up to an additive constant, Equation (4) (Auriault et al., 2010). Therefore, requiring a zero mean constraint to achieve a single unique soluti on, Accepted in Biosystems Engineering 14 Equation (7). However, the effective homogenized diffusivity only requires the gradient of u j, therefore any of these unique solutions are suitable (Carr & Turner, 2014). The macroscale energy balance solves for the instantaneous temperature T (K) based on Fourier’s law of heat transfer, given by (Golestani et al., 2013), 10pT TTc kr kt rr r z zρ∂ ∂ ∂∂∂   + − +− =  ∂∂ ∂∂ ∂    , (8) where ρ is the density of apple tissue (kg m-3), cp is the specific heat of apple tissue (J kg-1 K- 1) and k is the thermal conductivity of apple tissue (W m-1 K-1). 2.2.1 Boundary conditions The mass flux at the exterior boundaries is given by, ()v eq v air eff mpp DCh RT−⋅− ∇ =n , (9) where hm is the mass transfer coefficient (m s-1), n is the unit vector normal to the boundary, p v eq is the equilibrium vapor pressure (Pa), pv air is the vapor air pressure of ambient air (Pa) and R is the universal gas constant (J mol-1 K-1). The sorption isotherm of apple tissue was used to derive the equilibrium vapor pressure, p v eq (Pa), given as (Ratti, Crapiste, & Rotstein, 1989), 43.949 0.696 0.0411( )exp( 0.182 0.232 ln ( ) )dbM v eq v sat db db v sat p pT M e M pT− − =−+, (10) where pv sat is the saturated vapor pressure of water (Pa) and M db is the moisture content (kg kg- 1 dry basis). The saturated vapor pressure is (Vega- Mercado, Marcela Góngora -Nieto, & Barbosa-Cánovas, 2001), ( )42 735800.22061.3915 0.0486exp 0.4176 10 0.1445 10 6.546lnv satTT p T TT−−− +− +=×−×+ . (11) Accepted in Biosystems Engineering 15 Heat lost due to evaporation and c onvective heat transfer were incorporated within t he heat transfer boundary condition, defined as, [ ]() ()v eq v air T air m fg wpp k T hT T h hM RT− ⋅−∇ = − −n (12) where hT is the heat transfer coefficient (W m-2 K-1), Tair is the drying air temperature (K), M w molar mass of water ( kg mol-1) and hfg is the latent heat of evaporation (J kg-1). Finally, by symmetry the remaining boundary conditions is, 0effDC⋅− ∇ =n (13) [ ] 0 kT⋅−∇ =n (14) 2.2.2 Macroscale diffusivity The macroscale diffusivity is calculated by one of two techniques, technique 1 through an Arrhenius relationship ( D1,eff) and technique 2 considering an average ( D2,eff), 1, 0aE RT effD De−= , (15) 2,eff effDD= , (16) where Do is an integration constant (m2 s-1), Ea is the activation energy (J mol-1) and effD is the average diffusi vity (m2 s-1). Do, Ea and effD will be uncovered through the optimization outlined in Section 2.2.5. As the model was developed based on single -phase moisture transport, Arrhenius law is applicable and commonly used (Kumar, Millar, & Karim, 2015). 2.2.3 Other input parameters Accepted in Biosystems Engineering 16 The input parameters used in the model are calculated from the experimental conditions and presented in Table 1. More details on their calculation can be found in appendix A. [Table 1 can be placed here] 2.2.4 Solution Procedure Due to the inverse nature of the investigation, the solution procedure contained various steps to construct the property. The full solution procedure is summarized in Figure 2 and outlined in detail below. 2.2.5 Solving the macroscale model The first major step in the solution procedure was to solve the macroscale drying model (denoted in the green subsection in Figure 2). The drying model solves for the heat and mass transport, while calculating the optimal effective diffusivity for the heterogeneity investigation. The effective macroscale diffusivity ( D eff) is unknown and predicted by mini mizing the error between the experimental and predicted moisture content curves over the entire drying period. To implement this, a Levenberg- Marquardt optimization algorithm coupled with the finite element method was applied . The objective function is, 2 ,exp ,, ( ) () (, ) db db numeff i eff OF D M t M t D dt = −∫ (17) where ,expdbM is the average experimental moisture content ( dry bases ) and ,db numM is the average predicted moisture content ( dry bases ) from the model. The Levenberg-Marquardt algorithm was used for it higher convergence rate in least-square- type problems (Lentzou et al., 2019) . The optimization and macroscale heat and mass t ransport model was solved in COMSOL Multiphysics 5.3a. Accepted in Biosystems Engineering 17 2.2.6 Temporal heterogeneity investigation Once the optimal effective diffusivity is uncovered the temporal heterogeneity investigation can be conducted (denoted in the orange subsection Figure 2 ). The investigation consists of using the uncovered optimal diffusivity to inversely calculate the diffusivity of ICW, with respect to time, by varying the microscale domains according to the ICW/FW ratio associated with each approach. The investigation considered four approaches: a base case and three approaches to approximate the material’s dynamic heterogeneity. Base case The base case for the heterogeneity investigation utilizes published microscale experimental data to consider the ICW/FW ratio in apple tissue drying at 60 °C . This ratio is then used to construct the microscale domain required for the homogenization, allowing a representation of the real cellular water breakdown to be considered. Examples of these microscale domains can be seen in Figure 1 (b) to (d). The specific work which is utilized is Khan, Farrell, et al. (2018) . Khan, Farrell, et al. (2018) used NMR to investigate the cellular transport phenomena of apple tissue during convective drying. Their results tracked the ICW and FW breakdown with respect to time. Also, th e work demonstrates that ICW migrates from intracellular spaces to intercellular spaces by cell walls progressively rupturing while drying at higher temperatures. Their results are further validated with X-ray tomography images. Khan, Farrell, et al. (201 8) findings are summarized in Figure 3, with examples of the microscale domains constructed from the ratio. It should be noted, within the data there are many fluctuatio ns in the proportions of ICW and FW throughout the drying period. According to the arguments of Khan, Farrell, et al. (2018), these fluctuations are mainly caused by cells collapsing at different stages of drying. The peaks in the ICW content curve indicate the points Accepted in Biosystems Engineering 18 where the cell membranes collapse. Within Figure 3, the total amount of water (ICW + FW) is considered 100% at each point of the drying process. [Figure 3 can be placed here] Approaches to approximate the temporal heterogeneity The other approaches in the heterogeneity investigation utilize common assumptions to approximate the temporal heterogeneity. These assumptions are important to investigate as consi dering the real microscale experimental data will not always be possible/feasible due to its cost. Three approaches, each corresponding to an assumption, were investigated to approximate the temporal heterogeneity, approach 2: the average cellular structur e, approach 3: ICW content linearly decreases with time and approach 4: diffusivity by stages. All approaches and their associated ICW/FW ratio in relation to time are summarized in Figure 3. Approach 2 considers the average cellular water ratio with respect to time. This creates a single microscale domain and a constant diffusivity coefficient for the entire drying period. Considering constant diffusivity is very common in literature (Rahman, Joardder, Khan, et al., 2018) . Approach 3 assumes the temporal heterogeneity caused by the cell rupturing can be approximated by the ICW content linearly decreasing as drying occurs. This is a basic assumption and aligns with previously published finding (Halder, Datta, & Spanswick, 2011) . Approach 4 introduces a new concept and considers diffusivity in three stages . Each stage corresponds to different cellular deformation stage ; stage 1: cells intact, stage 2: cell rupturing occurs and stage 3: ultimate cell rupturing. This approach assumes cell rupturing during stage 2 significantly influences the results and is a key interest of the model. Each stage is defined by the magnitude and fluctuations of ICW in the base case. Stage 1, when the cells are intact, occurs first and is when the ICW content experiences minimal deviation from its initial Accepted in Biosystems Engineering 19 percentage. Stage 2, when cell rupturing occurs, exists in the middle stages of drying when the majority of fluctuations in ICW occur. Stage 3, ultimate cell rupturing, occurs in the latter stages of drying when the percentage of ICW drops off significantly. The stages are represente d in Figure 3. Each approach will be investigated and evaluated on their ability to represent the base case, relative experimental cost and applicability to represent ICW diffusivity ( D ICW) as a function. T he homogenization and inverse calculation of D ICW was solved using MATLAB 2018a coupled with Gmsh (Geuzaine & Remacle, 2009) to construct the microscale domains. 2.3 Property development for the diffusivity of intracellular water Once the ideal approach is uncovered t he relationship development for diffusivity of ICW ( DICW) can be conducted (denoted in yellow subsection Figure 2). The property development was considered in terms of two variables, sample temperature ( Ts) and ICW content ( WC ICW). Multiple regression with a logarithmic transformation was utilized and solved in MATLAB. After a brief investigation to determine the nature of the data, the initial function for D ICW was assumed as, 22 12 3 4 exp ( )ICW o s ICW s ICW D T WC T WCββ β β β= ++ + + (18) 3. Results: The results consist of three main parts, each corresponding to a subsection in the solution procedure : the inverse analysis of the food drying model, the temporal heterogeneity investigation and the estimation of ICW diffusivity. The predicted results were compared to Khan, Farrell, et al. (2018) experimental data to validate the model. Within Khan, Farrell, et al. (2018) the apple samples were dried up to 430 min to reach the products’ final equivalent moisture content, around 10-15% wet basis, for proper preservation. Accepted in Biosystems Engineering 20 3.1 Inverse analysis on the macroscale effective diffusivity The results of the optimization investigatio n are presented in Figure 4 and Figure 5. The investigation considered two optimal scenarios: assuming an Arrhenius approach to diffusivity [ Equation (15)] and assuming an average effective diffusivity (effD) is sufficient [ Equation (16)]. Assuming an Arrhenius approach to diffusivity assumes temperature is the dominant influencer on diffusivity. This resulted in a D o of 8.77×10-7 m2 s-1 and Ea of 20.871 kJ mol-1. This approach predicted moisture transport accurately with a low mean relative error (MRE) of 5.96% ( Figure 4 ). The average temperature profile also described the experimental data well with a MRE of 1.97%. The value for the integration constant ( Ea) agrees with the existing literature falling within the expected range of 12.7- 110 kJ mol-1 for various food and agricultural material (Zogzas, Maroulis, & Marinos -Kouris, 1996). In particular, E a for apple has been reported at 24.51 kJ mol-1 for drying at 40-60 °C (Wang, Sun, et al., 2007) and 29.65 kJ mol-1 for drying at 50-80 °C (Kara & Doymaz, 2015). The alternative approach (optimal scenario 2) found an optimal average effective diffusivity ( Deff) of 3.30 ×10-10 m2 s-1 with MREs of 5% and 1.77% for the average moisture and temperature profiles respectively. This effective diffusivity fell within the expected diffusivity range for most food material , 10-9-10-11 m2 s-1 (Panagiotou, Krokida, Maroulis, & Saravacos, 2004; Saravacos & Maroulis, 2001). Additionally, as expected the calculated diffusivities were less than the self -diffusion of water at 60 °C (4.748×10-9 m2 s-1) (Lentzou et al., 2019) . The effective diffusivity can also be compared to 0.713×10-11 – 7.66×10-10 m2 s-1 for Red Delicious apples at a temperature range of 35 -55°C (Beigi, 2016). [Figure 4 can be placed here] [Figure 5 can be placed here] Accepted in Biosystems Engineering 21 3.2 Temporal heterogeneity investigation Investigating the temporal heterogeneity utilizing an inverse study can be complex. Due to the nature of the approach, the diffusivity of ICW is inversely calculated. This means all approaches essentially achieve the same macroscale diffusivity result. This can make it hard to evaluate the performance of each approach. However, each approach creates unique relationships for the diffusivity of ICW, and not all these relationships can be easily described by functions for future models. The end goal of applying multiscale modeling is to move towards a more mechanistic modeling approach. This cannot be achieved without suitable property relationships at a microscale. Therefore, each approach is evaluated on their ability to represent the real cellular water breakdown (base case), relative experimental cost and applicability to represent ICW diffusivity ( D ICW) as a function. The results of the temporal heterogeneity investigation can be seen within Figure 6 to Figure 9. Approach 1 considered previously published experimental results to create a base case of achieving an optimal Arrhenius relationship. The approach assumed optimal scenario 1 was ideal and the results can be seen in Figure 6. Within the experimental data, the FW content fluctuates, and when considering two subdomains at a microscale this trend is reflected in the diffusivity of ICW. The diffusivity initially increases, however, once cell rupturing occurs, the diffusivity oscillates. Towards the end, ultimate cell rupturing occurs causing the diffusivity of ICW to drastically drop to achieve the optimal Arrhenius trend. It should be noted, it is likely there is a few mechanisms occurring at the latter stage of drying contributing to the low diffusivity of the ICW subdomain. The ICW subdomain encompasses the protoplast, cell membrane and cell wall which are known to deform/change due to the drying process. Joardder, Brown, Kumar, and Karim (2015) demonstrated that the cell wall thickness halved during convective drying of Granny Smith apples. Prawiranto et al. (2018) modeled the Accepted in Biosystems Engineering 22 dehydration of cellular food tissue (at an air temperature of 25 °C). The work theorized and demonstrated that the dried material forms an additional barrier restricting the moisture within the sample in the later stages of dehydration. The presence of cell rupturing likely causes the additional restricting barrier to have a smaller effect on the diffusivity in the later stages of drying. However, it could still be contributing to the drastic drop off. [Figure 6 can be placed here] Approach 2 considered the average ICW content from the base case to construct the microscale domain. The approach considered optimal scenario 2 and the results can be seen in Figure 7. Approach 2 resulted in a single microscale domain and a single D ICW for the duration of drying. Considering the average diffusivity rather than the temperature dependent approach (Arrhenius relationship) resulted in slightly lower MREs, however the approach required comprehensive knowledge of the microstructural evolution of the sample (comprehensive experimental data). Specifically, approach 2 considers the average ICW content over the entire drying period to construct its microscale domain. Recording the ratio of cellular water content is not trivial and requires an NMR or similar approach (Khan et al., 2016). Approach 2 resulted in an average ICW diffusivity of 1.5×10 -10 m2 s-1. Most literature on the diffusivity of food material during drying has only investigated the tissue level (macroscale) (Welsh et al., 2018) . This adds complexity to comparing the average ICW diffusivity uncovered. Research has been published on the ICW diffusivity of fresh Granny Smith apples (high moisture content) with a value of 3.8×10 -10 m2 s-1 (Sibgatullin et al., 2007). Convective drying significantly effects the material’s diffusion process due to the continuous moisture loss and cell ruptures. Additionally, it is known that the effective diffusivity decreases with the decrease in the moisture content in the sample (Dadmohammadi & Datta, 2019). [Figure 7 can be placed here] Accepted in Biosystems Engineering 23 Approach 3 assumes that the fluctuating experimental results can be approxi mated through assuming a linear decrease in the ICW content at the microscale. Assuming a linear decrease in ICW content (representing cellular deformation/rupturing) is consistent with Halder et al. (2011). The results for this approach can be seen in Figure 8. The diffusivity initially increases similar to base case. After the initial increase, the ICW diffusivity progressively decreases to achieve the Arrhenius relationship. The approach resulted in an ICW diffusivity range of 1.92×10 -10 to 1.12×10-11 m2 s-1. [Figure 8 can be placed here] Approach 4 considers diffusivity in individual stages. The stages are defined on the cellular deformation which occurs throughout the drying process. For convective drying at 60 °C, three stages were considered: stage 1 cell membrane are intact, stage 2 pr ogressive cell rupturing occurs and stage 3 ultimate cell rupturing. This approach results in separate diffusivities and microscale domains for each stage, Figure 9. The microscale domains were constructed through utilizing the same experiment data as the base approach to create averages for each stage. [Figure 9 can be placed here] 3.3 Comparison of temporal heterogeneity approaches Approach 2 (average), approach 3 (linear decrease) and approach 4 (diffusivity by stages) all approximate the cellular deformation of approach 1 (base case) through different assumptions. Considering the average or diffusivity by stages requires comprehens ive knowledge of the microstructural evolution which generally results in high experimental cost. These approaches do describe the temporal heterogeneity, however multiscale models excel in optimization scenarios where modeling is utilized to calculate the optimal drying conditions and dryer design taking advantage of generalized properties. As comprehensive knowledge of Accepted in Biosystems Engineering 24 the microstructure evolution is required, considering the average or diffusivity by stages would limit the model’s advantages in these situations, restricting the applications of any property constructed. Additionally, approach 4 (diffusivity by stages) is an interesting concept but shows no significant advantage in this application. The approach could show potential at higher drying temperatures where significant changes to a materials microstructure occur within a particular stage, i.e. case hardening. Approach 3 (linear decrease) is interesting. The approach requires some microstructural knowledge (i.e. the initial and final ICW content) however significantly less microstructural knowledge compared to the other approaches. Additionally, literature already exists for the initial ICW content for a variety of fruits and vegetables, which minimizes the experimental cost (Halder et al., 2011; Khan et al., 2016). Approach 3 also demonstrated the most practicality. The approach was able the capture the initial increase in ICW diffusivity which occurred within the base case, before progressively decreasing to achieve the Arrhenius relationship. The refore, approach 3 (linear decrease) was selected for the property development. 3.4 Diffusivity estimation for ICW Approach 3 (linear decrease) was utilized for the ICW diffusivity relationship development. The property was developed in terms of sample tempe rature ( T s) and intracellular water content ( WC ICW). The finalized function was, 22exp 110.32 0.55536 0.095148 0.00090738 0.0007235ICW s ICW s ICW D T WC T WC = −+ + − − (19) Accepted in Biosystems Engineering 25 A visualization of the ICW diffusivity function can be found in Figure 10 . Within this contour plot, the ICW diffusivity for any given sample temperature and ICW content can be seen within the given range. All coefficients have P -values under 0.05 and thus were cons idered significant. The function has a R2 of 0.999, though, it should be noted the property is developed off the assumption (linear decrease) with a small data set. Some interesting observations can be made by investigating and comparing this function to published literature. The ICW content trend for apple tissue has been reported for convective drying at 45, 60 and 70 °C (Khan, Farrell, et al., 2018). The work reported that at low drying temperatures (45 °C) the ICW content remains fairly constant (88%-79%) whereas at medium to high temperatures (above 50 °C) the ICW fluctuates before dropping off in the latter stages of drying. Additionally, other authors have also demonstrated this trend for other food materials (Halder et al., 2011) . The current study investigates drying at 60 °C. To further investigate the proposed function, a case study for drying apples at 45 °C can be considered. When convective drying at 45 °C, it is known minimal fluctuations in the WC ICW occur (78- 88%) in apple tissue (Khan, Farrell, et al., 2018) . Additionally, the product will only experience temperatures between 25 °C (its initial temperature) to 45 °C (its possible maximum). If these are considered within the developed function, the resulting ICW diffusivity would be between 1.41×10-10 to 1.8×10-10 m2 s-1. When this potential minimum and maximum of the material’s ICW diffusivity is upscaled through homogenization, the effective macroscale diffusivity of the material would be 1.99×10 -10 to 2.569×10-10 m2 s-1, which agrees with published literature (Beigi, 2016; Kara & Doymaz, 2015). [Figure 10 can be placed here] Accepted in Biosystems Engineering 26 Though the proposed function [Equation (19) ] is constructed for apple, the ICW content (WC ICW) and its trend are different for each food material during drying. This opens the possibility to appl y the function to other food materials. Literature has reported the initial ICW content for various fruits and vegetables i.e. apricot, pear, potato and eggplant (Halder et al., 2011; Khan et al., 2016), however much more work is still required. Therefore, the robustness of the proposed equation and its applicability to other food materials must be investigated in future works with a larger dataset. The proposed function does have the possibility of transitioning towards a generalized diffusivity for food material during drying. Note that the residuals for the proposed function can be found in Appendix A. 4. Discussion and outlook The developed relationship for ICW diffusivity considers two subdomains at the microscale, ICW and FW. This allows the water heterogeneity to be considered while maintaining a reasonable computational cost. The magnitude of the effective diffusivity uncovered within the inverse analysis is on the mid to lower end of the expected range, 10 -9- 10-11 m2 s-1 (Panagiotou et al., 2004; Saravacos & Maroulis, 2001), for food material. This is important to remember when evaluating the proposed equation for property prediction. The existing macroscale effective diffusivity data within literature should be treated cautiously. Many of the published values are larger than the reported coefficients for self -diffusion and/or diffusion of species in the solvent with no porous structure (Lentzou et al., 2019). Also, many consider a n idealized non-deforming geometry, often causing large error betwee n the predictive model and experimental data towards the end of drying (Tzempelikos et al., 2015) . The effective macroscale moisture diffusivity could be 0.1–50% smaller than free diffusion as the food matrices are porous media (Khaled & Vafai, 2003) . Defo rmation is an important factor Accepted in Biosystems Engineering 27 to consider (Joardder, Kumar, Brown, & Karim, 2015). Whether it’s to model the physical process (Mayor & Sereno, 2004) or to incorporate its effects within the appropriate properties. This research incorporated its effects th rough the inverse study and heterogeneity consideration. Also, alternatively a basic method of incorporating some shrinkage is to consider the average radius within Fick’s equation rather than the initial radius. The developed relationship for ICW diffusi vity encompasses the effects of the vacuole, tonoplast membrane, cytoplasm, cell membrane and cell wall. This is because only two subdomains are considered at the microscale. This lumping could have a negative effect on the proposed functions applicability to other food materials. The developed ICW diffusivity function is in terms of temperature and the moisture content of the intracellular sub-domain (ICW moisture). This allows the developed property to be theoretically suitable for a wide range of psychrometric drying conditions. Additionally, as the property was developed through a temporal heterogeneity investigation while drying at 60 °C cell rupturing was considered, improving the developed property’s predictive capabilities when significant cell rupturing occurs. Theoretical models nowadays have become more complex and more physics -based which allows the generalization of properties to become a real possibility in the near future. 5. Conclusion The development of the property relationship for ICW is a step towards a more mechanistic approach to diffusivity. The inverse analysis and temporal heterogeneity investigation allowed the real temporal cellular morphology to be investigated and appropriate assumptions to be compared. The property relationship was developed in terms of two variables, material temperature and intracellular water content. This opens the possibility of using the function for other food materials, however the proposed function is yet to be validated Accepted in Biosystems Engineering 28 for food material other than apple. Additionally, though the ICW diffusivity relationship was constructed from approach 3 (linear decrease), the temporal heterogeneity inv estigation provides insight into different options available to consider the temporal heterogeneity of the material. The other approaches investigated can be used in different scenarios where they are suitable. For example, the average ICW diffusivity can be used to cut down on computational cost when knowledge of the microstructural evolution is well known and considering diffusivity by stage can be used when the effect of cell rupturing is of key interest. The proposed function for ICW diffusivity will al low future models to consider a more realistic description of the physics during food drying. Additionally, the inverse method developed can be applied to other properties (i.e. thermal conductivity) to further improve the accuracy and predictive capabilit ies of mathematical models for food drying. Acknowledgements This study was funded by the Department of Education and Training (AUS) through a Research Training Program (Stipend) Domestic (RTP) scholarship.