## 2 Materials and Methods
### Numerical Simulation Models and Methods  
#### 2.3.1 Fluid Transport Equations  
Generally, the grain piles were considered as porous media when conducting numerical simulation research on mechanical ventilation [31]. The control equations for internal flow and coupled heat and moisture transfer in grain piles will now be given.  
The continuity equation is obtained as follows [32]:  
\[\frac{\partial(\varepsilon\rho_{a})}{\partial t}+\
abla\cdot(\varepsilon\rho_ {a}\overrightarrow{v})=0 \tag{1}\]  
where \(\varepsilon\) is the porosity; \(\rho_{a}\) is the air density; \(t\) is the time; \(\overrightarrow{v}\) is the vector velocity of air; and \(\
abla\) is a vector symbol representing divergence.  
The moisture transfer equation is expressed as [33]:  
\[\frac{\partial(\varepsilon\rho_{a}w)}{\partial t}+\
abla\cdot(\varepsilon\rho_ {a}\overrightarrow{v}w)=\
abla\cdot(\varepsilon\rho_{a}D_{eff}\
abla w)+S_{w} \tag{2}\]  
where \(w\) is the absolute moisture content in the air between grains; \(D_{eff}\) is the effective diffusion coefficient of water vapor; and \(S_{w}\) represents the moisture source terms for the moisture absorption and desorption of grain particles in the porous medium domain of grain piles, which can be expressed as:  
\[S_{w}=(1-\varepsilon)\rho_{t}\frac{\partial W}{\partial t} \tag{3}\]  
where \(W\) is the moisture in wheat piles, and \(\rho_{t}\) is the density of the grain kernels on a dry basis.  
The momentum conservation equation was expressed as [34]:  
\[\frac{\partial(\varepsilon\rho_{a}\overrightarrow{v})}{\partial t}+\
abla \cdot(\varepsilon\rho_{a}\overrightarrow{v})=-\varepsilon\
abla p+\
abla \cdot(\varepsilon\mu\
abla\overrightarrow{v})+S_{i} \tag{4}\]  
Figure 2: Schematic diagram of measurement points: (a) vertical section measurement point position; (b) horizontal section measurement point position.  
where \(\
abla p\) is the pressure gradient; \(\mu\) is the air viscosity; and \(S_{i}\) is the source term of the momentum equation in the porous media of grain piles, which can be expressed as:  
\[S_{i}=-(\frac{\mu}{\alpha}\overrightarrow{v}+\frac{C_{2}\rho_{a}}{2}\Big{|} \overrightarrow{v}\Big{|}\overrightarrow{v}) \tag{5}\]  
where \(\mu\) is the air viscosity, \(\alpha\) is the viscous resistance coefficient, and \(C_{2}\) is the inertial drag coefficient. The viscous resistance coefficient and the inertial drag coefficient can be expressed as follows:  
\[\frac{1}{\alpha}=\frac{150\mu(1-\varepsilon)^{2}}{dp^{2}\varepsilon^{3}}\
u \tag{6}\]  
\[C_{2}=\frac{150\mu(1-\varepsilon)^{2}}{dp^{2}\varepsilon^{3}}\
u+\frac{1.75 \rho_{a}(1-\varepsilon)}{dpe}\
u^{2} \tag{7}\]  
The energy transfer equation based on the local equilibrium heat model of porous media (air temperature equal to grain temperature) is expressed as [34]:  
\[\frac{\partial(\varepsilon\rho_{a}C_{pu}T_{a}+(1-\varepsilon)\rho_{g}C_{pg}T_ {a})}{\partial t}+\
abla\cdot(\varepsilon\rho_{a}\overrightarrow{v}C_{pu}T_ {a})=\
abla\cdot(\varepsilon k_{eff}\
abla T_{a})+S_{h} \tag{8}\]  
where \(\rho_{g}\) is the volume density of the grain piles; \(T_{a}\) is the temperature; \(K_{eff}\) is the effective thermal conductivity of the grain pile; and \(S_{h}\) is the energy source term for the grain pile domain, which can be expressed as:  
\[S_{h}=(1-\varepsilon)\rho_{t}\frac{\partial W}{\partial t}h_{v} \tag{9}\]  
where \(h_{v}\) is the latent heat of evaporation of water.