# A Multi-Level Approach for Simulation of Storage and Respiration of Produce
## 3 The Inner-Produce Gas Respiration Model
[6; 38] for O\({}_{2}\), CO\({}_{2}\) and C\({}_{2}\)H\({}_{4}\):  
\[r_{O_{2}}=-\frac{r_{\mathrm{O_{2}}max}y_{\mathrm{O_{2}}}}{K_{m\mathrm{O_{2}}}+ y_{\mathrm{O_{2}}}(1+\frac{y_{\mathrm{CO_{2}}}}{K_{\mathrm{m\mathrm{CO_{2}}}}})} \tag{1}\]  
\[r_{\mathrm{CO_{2}}}=\frac{r_{\mathrm{CO_{2}}max}y_{\mathrm{O_{2}}}}{K_{m \mathrm{CO_{2}}}+y_{\mathrm{O_{2}}}(1+\frac{y_{\mathrm{CO_{2}}}}{K_{\mathrm{ m\mathrm{CO_{2}}}}})} \tag{2}\]  
\[r_{\mathrm{C_{2}}H_{4}}=\frac{r_{\mathrm{C_{2}}H_{4}max}(y_{\mathrm{O_{2}}}^{ h})}{\left(K_{m\mathrm{C_{2}}H_{4}}^{h}\right)+\left(y_{\mathrm{O_{2}}}^{h} \right)} \tag{3}\]  
The equations provide volume sources with units of (cm\({}^{3}\) of gas constituent \(i\)/kg of fruit/day) with concentrations of gases in terms of mole fractions. The different constants present in the expressions are Arrhenius constants taking the form \(A\times exp\left(-\frac{E_{\mathrm{d}}}{RT}\right)\). The constants are available from [PERSON] et al. [6; 38]. Equations (1)-(3) were changed to the form:  
\[\dot{m}_{i}^{\prime\prime}=(1.392E-15)\cdot r_{i}\cdot\rho_{fruit}(\frac{V_{ fruit}}{A_{surface}})\frac{MW_{i}\ P}{T} \tag{4}\]  
where \(\dot{m}_{i}^{\prime\prime}\) is in (kg of \(i\)/m\({}^{2}\)s). To validate the boundary source implementation in a discretized CFD framework, the technique was utilized to reproduce the 0-D data obtained by [PERSON] et al. [6; 38] for experimental data on the respiration of a tomato (of 130 g) and an avocado (of 200 g) inside a 2020 cm\({}^{3}\) closed container. Based on these weights and the volumes of the produce utilized in [6; 38], produce densities were estimated as 1400 kg/m\({}^{3}\) and 790 kg/m\({}^{3}\) for tomatoes and avocadoes, respectively, while the volume-to-surface area ratio is calculated from SolidWorks\({}^{\mathrm{TM}}\)[29] as 0.00945 m and 0.01248 m, respectively. Figure 7 shows the CFD validation setup for an individual tomato and an individual avocado produced using the digital generation approach described in Section 2. In each case, the produce is centered on the base of a cylindrical container of diameter 137 mm and height 137 mm (i.e., total volume of 2020 cm\({}^{3}\)) and the space between the produce and the container is discretized using 432,911 and 386,183 tetrahedral elements for the case of tomato and avocado, respectively. The container contains still air and the change in composition was assumed to not affect the pressure inside the container, which is compatible with the work of [PERSON] et al. [6; 38] since only 10 cm\({}^{3}\) was withdrawn for measurements from the container every hour for the six days of simulation time. In terms of boundary conditions, the walls of the container and the surface of the fruit were all considered walls. The CFD simulations were carried out using ANSYS CFX\({}^{\mathrm{TM}}\)[39], wherein the conservation of mass, momentum and species were solved to simulate the transient gas exchanges and the laminar, incompressible, Newtonian motion of air in the space between the container walls and the fruit. The expressions given in Equations (1)-(4) were implemented using CFX expression language (CEL) to create boundary sources at the fruit surfaces. Solutions were obtained in less than 30 minutes of real time on a Core\({}^{\mathrm{TM}}\) i7 processor and 16 GB of RAM, using a time-step size of 30 minutes to simulate six days of storage time. Smaller time steps did not have a significant impact on the results. Convergence was obtained to a maximum level below \(10^{-4}\) for all transport equations. Figure 8 shows the results of the validation for different produce temperatures for tomatoes and avocadoes, respectively. In both cases, the transient CFD predictions are in excellent agreement with the experimental results of [PERSON] et al. [6; 38] indicating that the boundary approach to modelling inner-produce respiration is sufficient to obtain results of the air-side species concentrations and, inherently, volume-averaged results for the fluid-surrounding of produce.