*3.3.1 Mathematical models*

Mathematical models of varying complexities involving different types of kinetic and film thickness models are used to represent the wastewater treatment process in the biofilm reactor.

#### *3.3.1.1 One dimensional model*

This model accounts the rate of mass transfer to be proportional to the concentration difference between the interface and the bulk fluid. This model assumes no accumulation of the component at the interface under steady state condition. The differential equation governing the fluid phase is given by.

$$-u\frac{dc}{dz} = k\_{\text{g}}a\_{v}\left(c - c\_{s}^{\circ}\right) \tag{27}$$

with the boundary condition:

$$\mathbf{c} = \mathbf{c}\_0 \text{ at } \mathbf{z} = \mathbf{0} \tag{28}$$

In this model, the substrate concentration in the bulk fluid varies only in the axial direction.

The differential equation governing the solid phase is given by.

$$\frac{D\_f}{\xi^{a-1}} \frac{d}{d\xi} \left( \xi^{a-1} \frac{dc\_\sharp}{d\xi} \right) - r\_\sharp(c\_\sharp) = 0 \tag{29}$$

with the boundary conditions:

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$$\frac{dc\_s}{d\xi} = 0 \quad \text{at } \xi = 0$$

$$k\_{\xi} \left(c\_s^\prime - c\right) = -D\_f \frac{\mathbf{dc\_s}}{\mathbf{d}\xi} \quad \text{at } \xi = L\_f \tag{30}$$

The superscript '*a*' in Eq. (29) refers to 1, 2, and 3 for planar, cylindrical and spherical geometries, respectively, and *Df* denotes substrate diffusion coefficient in the biofilm (m<sup>2</sup> /day). The notation for other terms in the above equations is denoted as *kg* is mass transfer coefficient (m/s), *av* is specific surface area of particle (m�<sup>1</sup> ), *c* is substrate concentration in the bulk fluid (kg/m<sup>3</sup> ), *cs* is substrate concentration in the bio-film (kg/m<sup>3</sup> ), *c s <sup>s</sup>* is substrate concentration on the bio-film surface (kg/m<sup>3</sup> ), *ξ* is space coordinate in the bio-film (m), *z* is axial coordinate (*m*) and *Lf* is thickness of the bio-film (*m*).

## *3.3.1.2 Two dimensional model*

This model considers the variation of the substrate concentration in bulk fluid in both the axial and radial directions. The substrate transfer occurring through the biofilm is characterized by physical diffusion and reaction, and this process is described by second order differential equation. Pressure losses along the length of the reactor are neglected. The differential equation governing the bulk fluid phase is given by.

$$
\mu \frac{dc}{dz} = \frac{eD}{r} \left[ \frac{\partial}{\partial r} \left( r \frac{\partial c}{\partial r} \right) \right] - k\_g a\_v \left( c - c\_s' \right) \tag{31}
$$

where *D* is substrate diffusion coefficient in the bulk fluid (*m*<sup>2</sup> /day), *u* is superficial velocity (m/s), *r* is radial coordinate (*m*) and *ε* is porosity of bed. The boundary condition for solving this equation is given in Eq. (28). The solid phase biofilm model for two dimensional model is same as in one dimensional model as in Eq. (29) and its boundary conditions are expressed by Eq. (30).

#### *3.3.1.3 Mass transfer coefficient*

The substrate mass transport occurring from the bulk fluid to the biofilm surface across the diffusion layer is defined by

$$j\_D = \frac{k\_\mathrm{g}(s\varepsilon)^{2/3}}{u} \tag{32}$$

where Sc is Schmidt number (*μ*/*ρD*). The *jD*-factor is calculated using the following correlation:

$$j\_D \varepsilon = \frac{0.765}{\text{Re}^{0.82}} + \frac{0.365}{\text{Re}^{0.386}} \tag{33}$$

$$\text{where} \quad \text{Re} = \frac{\rho u d\_{\text{p}}}{\mu} \\ \text{for} \ 0.01 < \text{Re} < 15,000$$

where *dp* is equivalent particle diameter (m). The mass transfer coefficient, *kg* obtained from Eq. (32) is used in bulk fluid phase equations, Eq. (27) and Eq. (31).

#### *3.3.2 Kinetic models*

Various kinetic models can be used to represent the bioprocess kinetics [28], of which the Haldane and Edward models are given as follows.

### *3.3.2.1 Haldane model*

This rate expression is generally valid when the substrate concentration is high. According to this model, the substrate consumption rate, *rs* in biofilm is given by the following equation:

$$r\_s = (\mu\_{\text{max}} \rho\_s / Y) \xrightarrow[K\_s + C\_s + \frac{C\_s^2}{K\_L}] \tag{34}$$

where *Cs* is substrate concentration, *μmax* is maximum specific growth rate, *ρ<sup>s</sup>* is density of biomass, *Y* is yield coefficient, *Ks* is half velocity constant, and *KI* substrate inhibition constant.

#### *3.3.2.2 Edward model*

This model considers substrate inhibition, according to which the substrate consumption rate, *rs* in biofilm is given by the equation:

$$r\_t = \left(\frac{\mu\_{\text{max}} \rho\_t}{\text{Y}}\right) \left(\exp\left(\frac{-C\_t}{K\_I}\right) - \exp\left(\frac{-C\_t}{K\_s}\right)\right) \tag{35}$$

#### *3.3.2.3 Film thickness*

The biofilm thickness (*Lf*) is has significant influence on substrate conversion. A uniform film thickness with fixed value cannot represent the realistic situation and the film thickness *Lf* is expected to vary along the reactor [35]. The varying thickness of the biofilm is determined by the substrate flux from the bulk fluid into the biofilm as wellas the growth and decay rates of the bacteria. It is assumed that the biofilm is composed of a static portion and a variable portion. The variable portion is supposed to raise with the organic loading rate and lessen with the hydraulic loading rate. The above point of view leads to the following relations for estimating the biofilm thickness:

$$L\_f = a + b \text{ OLR} \tag{36}$$

$$L\_f = a + b\,\text{OLR} - \frac{c}{HLR} \tag{37}$$

where *OLR* is organic loading rate (kg/m<sup>3</sup> /day), *HLR* hydraulic loading rate (m/day), and *a*, *b* and *c* are constants to be determined.

#### **3.4 Procedure for solving the model equations**

To facilitate the solution of mathematical models, the height of the column (1 m) is divided into 50 equal steps, each step representing 0.02 m. In one dimensional model, the fluid phase equation (Eq. (27)) along with its boundary condition (Eq. (28)) is solved for each height step in the axial direction by using Runge–Kutta (RK) 4th order method. The solid phase equation for the biofilm (Eq. (29)) is solved for each segment using orthogonal collocation on finite elements (OCFE) [36]. Two finite elements, each with four internal collocation points are considered for OCFE implementation. In two dimensional model, the fluid phase equation (Eq. (31)) is transformed to ODE by finite difference technique and the resulting equation along with its boundary condition is solved by using 4th order RK method for each height step of the axial direction. The solution for solid phase equation of this model is same as in one dimensional model.
