# Settling velocity in the NanoFASE model

If the particle density of suspended particulate matter (SPM) for a given size class, $$\rho _{spm,n}$$, (kg/m3) (which includes the inflow of "new" SPM from erosion or other tributaries) is greater than the density of water $$\rho _{w}$$ (kg/m3), then the settling of that sediment size class is triggered. The settling velocity $$W _{spm,n}$$ for an individual SPM size class is calculated as:

 $$w_{\text{SPM},n} = \begin{cases} \frac{\eta}{d_n} d_{*,n}^3 \left( 38.1 + 0.93d_{*,n}^{12/7} \right)^{-7/8} & \mbox{if } \rho_{\text{SPM},n} > \rho_{\text{w}} \\ 0 & \mbox{if } \rho_{\text{SPM},n} \leq \rho_{\text{w}} \end{cases}$$ where $$d_{*,n} = (\Delta g/\eta^2)^{1/3} d_n$$ is an effective particle diameter (m), $$g$$ is the gravitational acceleration (m/s2), $$d_n$$is the diameter (m) of the SPM particle in the size class, $$\eta$$ is the kinematic viscosity (kg/m.s) of water, and $$\Delta = (\rho _{SPM,n/}\rho_{w} )-1$$

## Execution

The calculated settling velocity is used to compute a settling rate constant $$k_{\text{settle}} = w_{\text{SPM},n} / D$$(/s), where $$D$$ is the depth of the water column (m). On a timestep of length $$\delta t$$, the mass of SPM in the size class lost to the bed sediment due to settling (kg) is thus

$$\mathbf{j}_{\text{SPM,dep},n} = \textbf{m}_{\text{SPM},n} k_{\text{settle},n} \delta t$$

where $$\mathbf{m}_{\text{SPM},n}$$(kg) is the mass of suspended sediment in the size class within the river reach being simulated. The deposited mass per unit area of the bed sediment (kg/m2) is given by

$$\mathbf{M}_{\text{dep},n} = \mathbf{j}_{\text{SPM,dep},n} / l f_{\text{m}} W$$

where $$l$$ is the linear reach length (m), $$W$$ is the reach width (m) and$$f_(m)$$ is a factor to account for meandering.