## 2 Simulation Methodology
### Governing Equations  
Reynolds-Averaged Navier-Stokes (RANS) Equations [22]--Equations (1) and (2) are RANS equations commonly used in commercial CFD codes [18]:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho}{\partial x_{i}}(\rho u_{i })=0 \tag{1}\]  
\[\frac{\partial}{\partial_{t}}(\rho u_{i})+\frac{\partial\rho}{\partial x_{j}} \big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p}{\partial x_{i}}+\frac{ \partial}{\partial x_{j}}\left[\mu\left(\frac{\partial u_{i}}{\partial x_{j}} +\frac{\partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u _{l}}{\partial x_{l}}\right)\right]+\frac{\partial}{\partial x_{j}}\left(- \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\right) \tag{2}\]  
Here, \(t\) is a basic mathematical promise representing the partial derivative with respect to the time variable. \(\delta_{ij}\) is the \"Kronecker delta\" function for two variables \(i\) and \(j\). If the \(i\) and \(j\) have the same value, it becomes 1, then it becomes 0. \(\rho\) is the density of the slurry. \(p\) represents the slurry pressure. \(\mu\) represents the dynamic viscosity of the slurry, and \(u_{i}\) represents the velocity of the slurry. \(x_{i}\), \(x_{j}\) are Cartesian coordinates in index notation. \(u_{i}\) is the vector of averaged velocity field. \(u_{i}\), \(u_{j}\) are Cartesian velocity in index notation. \(\overline{u_{i}^{\prime}}\), \(\overline{u_{j}^{\prime}}\) are mean and fluctuating velocity components. \(-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\) is Reynolds stresses tensor.  
Figure 2: Top views of the R0 pad and R32 pad.  
Species Transport Equation [22]--Equation (3) is a conservation equation for calculating the local mass fraction of slurry (\(Y_{i}\)) by convection and diffusion [22].  
\[\frac{\partial}{\partial t}(\rho Y_{i})+\
abla\cdot\left(\rho\overrightarrow{v} Y_{i}\right)=-\
abla\cdot\overrightarrow{J}_{i}+R_{i}+S_{i} \tag{3}\]  
where \(\overrightarrow{v}\) is mass averaged velocity vector; \(R_{i}\) is the net rate of production of the species \(i\), which is zero in this simulation; \(S_{i}\) is the volume of the supplied slurry, which is generated every time step by a volume flow rate of 160 mL/min; and \(J_{i}\) is the diffusion flux for the species \(i\), which is determined by the mass diffusion equations below.  
Mass Diffusion Equations [22]--There are significant differences in mass diffusion flux between the laminar and turbulent regions. In this study, the laminar region and turbulent region were modeled simultaneously.  
\[\overrightarrow{J}_{i}=-\rho D_{i,m}\
abla Y_{i}-D_{T,i}\frac{\
abla T}{T} \tag{4}\]  
\[\overrightarrow{J}_{i}=-\left(\rho D_{i,m}+\frac{\mu_{t}}{Sc_{t}}\right) \
abla Y_{i}-D_{T,i}\frac{\
abla T}{T}\,\ \ \ \left(Sc_{t}=\frac{\mu_{t}}{\rho D_{t}}\right) \tag{5}\]  
\(D_{i,m}\) and \(D_{T,i}\) represent the mass diffusion coefficient for species \(i\) and thermal diffusion coefficient, respectively. Only \(D_{i,m}\) was considered in the laminar region, while a stronger diffusion effect was applied in the turbulent region because \(S_{ct}=0.7\). \(\mu_{t}\) is turbulent viscosity. \(D_{t}\) is turbulent diffusion coefficient. \(S_{ct}\) is turbulent Schmidt number. \(\frac{\
abla T}{T}\) is temperature gradient. \(D_{T,i}\) was set to zero so that the effect of temperature could be ignored.