# Compression and Fungal Heat Production in Maize Bulk Considering Kernel Breakage
## 3 Results and Discussion
### Physical Modeling of Heat Transfer and Verification  
To investigate heat transfer due to physical factors during the testing process, finite element (FE) software COMSOL Multiphysics was used to establish numerical modeling. For simplicity, the maize kernels were assumed as porous media with isotropic properties and in local thermodynamic equilibrium with their surrounding air. Additionally, the laminar air in the void between kernels was incompressible under vertical load. However, the buoyant force and natural convection due to temperature gradient can be considered. The flow rate is assumed to be unchanged, and the 3D flow field can be expressed as  
\[\frac{\partial u_{j}}{\partial x_{j}}=0 \tag{5}\]  
where \(u_{j}\) (\(j\) = 1,2,3) is the air flow rate in the direction \(x_{j}\), \(u_{1}\) = \(u\), \(u_{2}\) = \(u_{3}\) = \(v\). In rectangular Cartesian coordinate system, \(x_{1}\) = \(x\), \(x_{2}\) = \(y\), \(x_{3}\) = \(z\). The Brinkman-Darcy formulation was incorporated with the maize domain to represent the air flow. Additionally, the momentum satisfies the Boussinesq's approximation, then  
\[\rho_{a}\frac{\partial u_{i}}{\partial t}+\frac{\rho_{a}u_{j}}{\phi}\frac{ \partial u_{i}}{\partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{ \partial}{\partial x_{j}}\left[\mu\frac{\partial u_{i}}{\partial x_{j}} \right]+\rho_{0}g\beta(T-T_{0})-\frac{\phi\mu u_{i}}{K} \tag{6}\]  
where \(\rho_{a}\) is the density of air, \(u_{i}\) is the speed of airflow, \(t\) is the storage time, \(p\) is the air pressure, \(\phi\) is the porosity of maize bulk, \(K\) is the permeability of maize bulk, \(T\) is the temperature of the air and maize kernels, \(\rho_{0}\) is the density of air at the reference temperature (\(T_{0}\)), \(g\) is the gravity vector, \(\beta\) is the coefficient of volumetric expansion of the air, and \(\mu\) is the viscosity of air. As there is a local thermodynamic equilibrium between maize kernels and the surrounding air, the governing conservation equation of thermal energy is  
\[\rho_{b}c_{b}\frac{\partial T}{\partial t}+(\rho_{a}c_{a})u_{j}\frac{\partial T }{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[k_{b}\frac{\partial T }{\partial x_{j}}\right]+\rho_{b}h_{s}\frac{\partial W}{\partial t} \tag{7}\]  
where \(c_{a}\) is the specific heat of air; \(\rho_{b}\), \(c_{b}\), and \(k_{b}\) are the dry density, specific heat, and the effective thermal conductivity of maize bulk, respectively; \(h_{s}\) is the heat sorption of water on maize, and \(W\) is the moisture content of maize in dry basis. Maize is a living organism with water absorption and desorption feature. Thus, the moisture balance formula is written as  
\[\frac{\partial}{\partial t}[(\varepsilon\rho_{a}w)+(\rho_{b}W)]+\rho_{a}u_{j} \frac{\partial w}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[ \frac{D_{\varepsilon}\varepsilon}{\tau}\frac{\partial}{\partial x_{j}}(\rho _{a}w)\right] \tag{8}\]  
where \(\tau\) is the tortuosity factor of maize bulk, \(D_{v}\) is a dimensionless rate coefficient for moisture exchange between air and maize kernels, and \(w\) is the moisture content of water vapor in the air on a dry basis. To verify the effectiveness of the numerical model, physical heat transfer for maize was investigated with both testing and FE methods. The test was conducted by placing the sample within the model box (including the top plate) in a freezer. Before testing, the model box, top plate, and spread-open kernels were placed in the freezer, with the door open for hours to reach a uniform initial temperature. Then, the whole maize kernels were put into the box and covered by the top plate. The tests were begun after closing the door, and the temperature of the freezer was set at 6 \({}^{\circ}\)C. The test lasted for 350 min only. Therefore, biochemical heat production could be ignored, and only physical heat transfer occurred during the test. Two T-type thermocouples were set at the central point of the sample and in the freezer to monitor the temperature change.