# A Multi-Level Approach for Simulation of Storage and Respiration of Produce
## 5 The Porous Continuum Model  
As described in the introduction of this article, and made clear through the approach presented in Sections 2-4, simulation at the shape-level of the produce is computationally expensive and simply not practical for considering large storage regions. The more practical approach is to use information from shape-level simulations as well as experiments to provide closure models that can be used to relegate large regions of stacked produce to a porous-continuum. By the porous-continuum approach, the conservation equations are volume-averaged, which leads to additional terms that can be expressed in terms of known quantities (constitutive models) and closure coefficients. In the present study, the establishment of a representative produce shape, the characterization of a REV (produce-filled box) and the simulation of flow and heat/gas transfers in the REV yields information about pressure drop, heat and mass transfer that is now used to generate closure coefficients. To illustrate the required closure coefficients, the volume-averaged equations of mass, momentum, heat transfer and species (in the absence of external sources) are given, respectively, by [33; 42]:  
\[\
abla.(\left\langle\rho_{f}\right\rangle^{f}\langle\mathbf{v}\rangle)\ =0 \tag{5}\]  
\[\frac{1}{\varepsilon}\
abla.(\left\langle\rho_{f}\right\rangle^{f}\langle \mathbf{v}\rangle\langle\mathbf{v}\rangle)=-\varepsilon\
abla\langle P\rangle^ {f}+\mu_{f}\
abla^{2}\langle\mathbf{v}\rangle-\frac{\varepsilon\mu_{f}}{K} \langle\mathbf{v}\rangle-\frac{\varepsilon\left\langle\rho_{f}\right\rangle^{ f}c_{E}}{\sqrt{K}}|\langle\mathbf{v}\rangle|\langle\mathbf{v}\rangle \tag{6}\]  
\[ \begin{array}{c}\sum\limits_{i}\varepsilon c_{p,i}\frac{\partial(\left\langle \rho_{f}\right\rangle^{f}\left\langle Y_{i,f}\right\rangle^{f}\left\langle T _{f}\right\rangle^{f})}{\partial t}+\sum\limits_{i}c_{pi,i}\
abla.(\left\langle \rho_{f}\right\rangle^{f}\left\langle Y_{i,f}\right\rangle^{f}\left\langle T _{f}\right\rangle^{f}\left\langle\mathbf{v}\right\rangle)\\ =k_{eff,f}\
abla^{2}\left\langle T_{f}\right\rangle^{f}+\sum\limits_{i}\
abla.[\left\langle\rho_{f}\right\rangle^{f}D_{eff,i}\
abla\left\langle Y_{i,f} \right\rangle^{f}(c_{pi,i}\left\langle T_{f}\right\rangle^{f})]\\ +h_{fs}A_{fs}(\left\langle T_{s}\right\rangle^{s}-\left\langle T_{f}\right \rangle^{f})+\ \dot{H}_{fs}^{mass}\\ \varepsilon\frac{\partial(\left\langle\rho_{f}\right\rangle^{f}\left\langle Y _{i,f}\right\rangle^{f})}{\partial t}+\
abla.(\left\langle\rho_{f}\right\rangle ^{f}\left\langle Y_{i,f}\right\rangle^{f}\left\langle\mathbf{v}\right\rangle)= \
abla.(\left\langle\rho_{f}\right\rangle^{f}D_{eff,if}\ \
abla\left\langle Y_{i,f}\right\rangle^{f})+\left\langle\dot{m}_{i_{fs}}\right\rangle \end{array}  \tag{7}\]  
where \(\varepsilon\) represents the volumetric porosity (calculated herein as 0.6687 and 0.622 for tomatoes and avocadoes, respectively) and \(i\) is a species counter, which stands for O\({}_{2}\), CO\({}_{2}\), etc. \(\rho_{f}\) is the fluid density, \(\mathbf{v}=(u,v,w)\) is the velocity vector, \(T\) is local temperature and \(Y_{i}\) is the species mass fraction (note that \(y_{i}\) is the species mole fraction). The \(\left\langle\ \right\rangle\) brackets represent an extrinsic volume averaged quantity that is averaged over a multiphase porous media volume, while \(\left\langle\ \right\rangle^{f}\) represents an intrinsic volume average over only the fluid volume inside the porous region. The hydrodynamic equations (Equations (5) and (6)) are cast in their steady form. It is also noted that the momentum equation herein is cast in its extrinsic form unlike the heat and mass transfer equations (Equations (7) and (8)), which are cast in their intrinsic forms.