# A Multi-Level Approach for Simulation of Storage and Respiration of Produce
## 5 The Porous Continuum Model
Also, as long as \(l<<\) L, where L here is the characteristic macroscopic length taken as the smallest side length of the box, \(\widetilde{\mathbf{v}}\) - \(\langle\mathbf{v}\rangle\), where \(\widetilde{\mathbf{v}}\) is the local microscopic deviation-from-average for velocity, and the average porosity \(\varepsilon\) is nearly fixed throughout the box (i.e., \(\vec{\varepsilon}\ \sim\ O(1),\Delta\varepsilon\ll 1\)), the findings of [PERSON] [43] based on closure mathematics in regards to the permeability and Forchheimer tensors are valid. Relaxing the definition of the Forchheimer tensor to diagonal as well and using a linear correlation to velocity [44], the classic definitions of the coefficients for directional closure--i.e., for \(x\), \(y\) and \(z\)--may be utilised for each Cartesian equation of momentum. Also, one may reduce the flow equation based on translational periodicity on the three directions to essentially the Darcy-Forchheimer equation, given by:  
\[(\frac{\Delta P}{L})_{\text{j}}=\frac{\mu_{f}}{K_{j}}\mathbf{v_{j}}+\frac{ \left\langle\rho_{f}\right\rangle^{f}c_{Ej}}{\sqrt{K_{j}}}\mathbf{v_{j}}^{2} \tag{9}\]  
where \(j\) here may be \(x\), \(y\) or \(z\) and \(\mathbf{v_{j}}\) may be \(u\), \(v\) or \(w\). Now the task turns to using Equation (9) to fit the data from Figure 9, in order to obtain \(K_{j}\) and \(c_{Ej}\) for the three directions. Dropping the \(j\), the data presented in the figure is fitted to an equation of the form:  
\[\frac{\Delta P}{L}=aRe_{d}+b\ Re_{d}^{2} \tag{10}\]  
where \(a=\frac{\mu_{f}{}^{2}}{\left\langle\rho_{f}\right\rangle^{f}d_{p}K_{j}}\) and \(\frac{b}{a}=c_{Ej}\sqrt{K_{j}}/d_{p}\). The average produce diameters are taken as 6.2 cm and 7.75 cm for tomatoes and avocadoes, respectively. Table 1 lists the curve fitting information and the values of \(K_{j}\) and \(c_{Ej}\) for each commodity and direction. The introduction of this anisotropy was necessary as mentioned earlier in Section 4 due to the difference in resistance between the resting direction of the produce (\(y\)) and other directions. It is noted that the resistance to the flow here may be characterized collectively by both coefficients of permeability and Forchheimer as the fit was developed for both the linear range, \(Re<1\) and the inertial range, \(Re\) up to 200. Also, although collectively the tomatoes have generally higher pressure drops as stated earlier, the avocadoes have larger average diameter, and hence this finding does not necessarily reflect on the individual coefficient values. Now we close the heat transfer equations by curve fitting the Nusselt number data in Figure 10. The correlation of choice is given as:  
\[Nu=c\ Re_{d}{}^{m}\Pr^{1/3} \tag{11}\]  
In the case of general averaged Nusselt number, for tomatoes, \(c=0.1861\) and \(m=0.6777\) while for avocadoes, \(c=0.2425\) and \(m=0.5739\) with goodness of fit given by \(\mathrm{R}^{2}\) of 0.9947 and 0.9964, respectively. It is also useful to obtain correlations for purely vertical (i.e., using y-direction data) and purely lateral flow (using an average Nusselt number over \(x\) and \(z\) directions). For purely vertical flow, \(c=0.5644\) and 0.632, \(m=0.6141\) and 0.5383 with \(\mathrm{R}^{2}=0.9963\) and 0.9968 for tomatoes and avocadoes, respectively. As for purely lateral flow, and with much weaker heat transfer as observed, the constants exhibit different order of values. For tomatoes, \(c=0.02342\), \(m=0.906\) and \(\mathrm{R}^{2}=0.9865\) while for avocadoes, \(c=0.05341\), \(m=0.6977\) and \(\mathrm{R}^{2}=0.9855\). It is also worth mentioning that a velocity-weighted representation of the Nusselt number may be used for porous continuum modeling as well (i.e., \(\mathrm{Nu}=\sum\limits_{\mathrm{j}=1}^{3}\frac{\langle\mathbf{v}_{\mathrm{j}} \rangle}{\langle\mathbf{v}\rangle}\mathrm{Nu}_{\mathrm{j}}\) with \(\mathrm{Nu}_{\mathrm{j}}\) evaluated by Equation (11)).