## 2 Materials and Methods
### Characterization of Hydraulic Behaviors  
The hydrodynamics of the anaerobic reactor was studied because they significantly influence the rates of biological reactions. They particularly affect the rates of mass transfer and the distribution of reactions along a reactor, both of which determine a reactor's overall performance [20; 36]. The amount of mixing in a reactor also determines the performance of a reactor; therefore, to describe the real behavior of a reactor, the influence of mixing on the mass balance equation must be specified correctly [37]. In this study, the hydrodynamics were characterized by the non-dimensional numbers given by Peclet and Reynolds.  
The mixing intensity of the fluid within a reactor is well described by the axial Peclet number (\(\text{Pe}_{\text{axial}}\)) (see Equation (6)).  
\[\text{Pe}_{\text{axial}}=\frac{\text{V}_{\text{up}}\cdot\text{H}}{\text{D}_{ \text{A}}} \tag{6}\]  
where \(\text{V}_{\text{up}}\), \(\text{H}\), and \(\text{D}_{\text{A}}\) are the up-flow velocity (m/h), bioreactor height (m), and axial dispersion coefficient (m\({}^{2}\)/h), respectively. When \(\text{D}_{\text{A}}\rightarrow\infty\), the value of \(\text{Pe}_{\text{axial}}\) becomes 0 since \(\text{Pe}_{\text{axial}}\) is an inverse function of \(\text{D}_{\text{A}}\). Consequently, the system will operate as a plug-flow reactor since there is no mixing in the axial direction. On the other hand, when \(\rm D_{A}\to 0\), the system will behave as a complete mixture reactor [19]. Various transfer functions have been proposed to estimate the dispersion from either the Reynolds number or a flow velocity [38; 39; 40; 41; 42]. Here, we used an approach described by Equations (7) and (8), which assessed \(\rm D_{A}\) as a function of flow distance  
\[\rm D_{A}=1.03\cdot V_{up}^{1.11}.0.009^{\eta_{j}} \tag{7}\]  
\[\rm n_{j}=\frac{z}{H} \tag{8}\]  
where \(\rm n_{j}\), \(\rm z\), and \(\rm H\) are the values of the normalized height, axial position (\(\rm m\)), and height (\(\rm m\)) of the bioreactor, respectively.  
The amount of turbulence is characterized by the Reynolds number (\(\rm Re\)) and is given by Equation (9) [19; 43]. Where \(\rm V_{up}\), \(\rm d\), \(\rm u_{w}\), \(\rm\rho_{w}\), and \(\rm v_{w}\) are the up-flow velocity (\(\rm m/h\)), bioreactor diameter (\(\rm m\)), dynamic viscosity (\(\rm Pa\cdot s\)), density (\(\rm kg/m^{3}\)), and kinematic viscosity (\(\rm m^{2}/s\)), respectively, Reynolds describes a relationship between inertial to viscous forces [42]. Equation 9 is the most widely used; however, it exits variations of the Reynolds number around noncircular conduits, packed beds, and mixing impellers.  
\[\rm Re=\frac{V_{up}\cdot d}{\
u_{w}}=\frac{\rho_{w}\cdot V_{up}\cdot d}{\mu_{w}} \tag{9}\]  
Turbulence, meanwhile, is the rotational and three-dimensional chaotic movement in all directions of flowing elements, where the resulting net flow is unidirectional. The rapid mixing associated with turbulence enhances the momentum, heat, and mass transfer processes. The intervals of Reynolds include \(\rm Re<2300\), \(2300<\rm Re<4000\), and \(\rm Re>4000\), which correspond to laminar, transient, and turbulent regimen, respectively. However, a typical turbulent regimen truly manifests itself from values of \(\rm Re>10,000\).