## 2 TLC Description and Mathematical Modeling
### Hydrodynamic Modeling  
The open channel flow behavior is evaluated based on the Reynolds number and hydraulic diameter \((D_{h})\), which are given below:  
\[D_{h}=\frac{4 dW}{2d+W} \tag{1}\]  
\[Re=\frac{\rho D_{h}U}{\mu} \tag{2}\]  
where \(W\) represents the channel width (m), d is the channel or water depth (m), and U denotes the average water velocity (m/s). Water was taken as the working fluid with a density \((\rho)\) 1000 (kg/m\({}^{3}\)) and viscosity (\(\mu\)) of 0.001 (Pa. s). The flow behavior in the TLC reactor was turbulent with a \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Slope angle & \(1^{\circ},2^{\circ};3^{\circ}\) \\ Depth ratio & 5.6 mm; 7.5 mm; 10 mm \\ Aspect ratio & 180; 260; 340 \\ Mass flow rate & 1.6–3.6 kg/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of important parameters of the thin-layer cascade reactor used in this study. Figure 1: Computational model of thin-layer cascade (TLC) used in this study (**a**) xy-view and (**b**) xyz-view. Reynolds number, ranging from 6300 to 14,200 [23; 24; 25]. The geometrical effects were studied by using a dimensionless number, depth ratio, and aspect ratio (AR), as given below:  
\[AR=\frac{Channel\ width}{Channel\ depth}.\]  
The hydraulic power required for mixing is the product of the flow rate Q (m\({}^{3}\)/s), cultural density \(\rho\) (kg/m\({}^{3}\)), gravity constant g (m/s\({}^{2}\)), and head loss. For the case of the cascade reactor with an inclined surface of length L (m), the hydraulic power \(E_{D}\) (W/m\({}^{2}\)) is given by:  
\[E_{D}=\rho gLQI \tag{3}\]  
where \(I\) represent the inclination angle of the surface. Q is the product of liquid velocity U (m/s), thickness of algal suspension d (m) and width of the reactor W (m). Thus, the hydraulic power per unit area (W/m\({}^{2}\)) is given by [14; 26]. \[E_{D}/WL=\rho gUdI \tag{4}\]  
The k-\(\omega\) turbulence model was adopted to model the turbulence in an open channel. The simulation of turbulent flow with a k-\(\omega\) model requires an initial velocity \(U_{o}\), initial turbulent intensity \(I_{T}\) and turbulent length scale \(L_{T}\) as a boundary condition. \[U_{O}=AU \tag{5}\]  
\[I_{T}=0.16(Re)^{\frac{-1}{8}} \tag{6}\]  
\[L_{T}=0.07D_{h} \tag{7}\]  
where \(U_{O}\) represent the initial velocity (m/s), A is the cross-sectional area (m\({}^{2}\)), and U denotes the average velocity (m/s). The continuity equation for k-\(\omega\) turbulence model is given as follows:  
\[\rho\
abla.(\mathbf{u})=0. \tag{8}\]  
The momentum equation derived from the Bernoulli equation is given as follows:  
\[\rho\frac{d\mathbf{u}}{dt}+\rho(\mathbf{u}.\
abla)\mathbf{u}=\
abla.\big{[}- \rho\mathbf{l}+(\mathbf{u}+\mu_{T})\big{(}\
abla_{u}+\big{(}\
abla\mathbf{u}^ {T}\big{)}\big{)}\big{]}+\mathbf{F} \tag{9}\]  
where \(\mathbf{F}\) represents the body force (N/m\({}^{3}\)), \(\mathbf{u}\) is the velocity vector (m/s), and \(\mathbf{I}\) represents the identity matrix. The k-\(\omega\) approach uses two equations to represent the turbulence properties of the flow. The first equation begins with the turbulent kinetic energy k, which determines the energy in the turbulence, and the second equation begins with turbulent dissipation rate \(\omega\), which determines the scale of turbulence. The governing equations of the turbulent kinetic energy and turbulent dissipation rate for k-\(\omega\) turbulence model are as follows:  
Figure 1:0  
Figure 1:1  
where \(\sigma_{\omega}\) and \(\sigma_{k}\) represent the Prandtl numbers for the dissipation rate and kinetic energy, respectively. The typical values of the model constant used in the present study are given as \(\alpha=0.55\), \(\sigma_{k}^{*}=0.5\), \(\sigma_{\omega}=0.5\), and \(\beta^{*}=0.09\).