**1. Introduction**

The phenomenon of anaerobic sludge granulation was first observed in the 1990s. Extensive experimental works have been implemented since then. The culture conditions for forming the anaerobic granular sludge are well understood [1]. A high upflow velocity (usually >1 m/h) is usually required. The diameters of anaerobic granules can be up to 0.15–4 mm, which results in high free sedimentation velocities, that is, 15–50 m/h. The high sedimentation velocities can make a large amount of highly active granular sludge retained in a bioreactor in a highly efficient way. By 2007, the market share for anaerobic granule sludge-based technologies in the field of anaerobic wastewater treatment was 89%. Anaerobic granular sludge-based technologies have been extensively applied to treat wastewater from different industries, including agriculture, food, beverage, alcohol distillery, pulp, and papermaking.

Bioreactors involved in wastewater treatment are complex systems, and many nonlinear biokinetics occur in the bioreactors. A model that can successfully model bioprocesses in the bioreactors is effective in understanding the bioreactors and their manipulation. Versus aerobic wastewater treatment, modeling an anaerobic wastewater treatment is much more difficult. This chapter summarizes different model strategies for a granular sludge bed reactor. These strategies are beneficial for further model development and applications.

## **2. Bioparticle model**

The distribution of microorganisms in an anaerobic granule has big impacts on modeling the bioactivity of this granule. Different microbial structures for granules are identified. A layered and a cluster granular sludge structures are observed [2]. Here, three layers are proposed. The outermost layer includes acidogens and hydrogen-consuming organisms. In the middle layer, hydrogen-producing organisms as well as hydrogen-consuming organisms both exist, while *Methanosaeta* locate in the core layer. In this clustered structure, *Methanosaeta* clusters and zones with syntrophic eubacteria and hydrogenotrophic methanogens scatter in the granule.

A granular sludge bed consists of numeral sludge granules. Modeling substrate degradation in a single sludge granule has other applications. Indeed, understanding bioreactions in a single granule can explain the operation of an entire bioreactor. Two strategies are used to model substrate degradation in a single granule. Modeling strategies are both termed bioparticle models in this study. The bioparticle models are discussed below.

### **2.1 Diffusion-reaction model**

A diffusion-reaction model couples mass transfer and substrate degradation kinetics in a single granule. Some assumptions need to be made to establish a diffusion-reaction model. The shape of real granules in reactors is irregular and nonuniform. In addition, the biogas that results from bioprocesses contributes to the formation of pores in the inner space of a granule. Water and biomass are different materials and constitute a granule. Therefore, substrate diffusion in the inner space of a granule is different at different locations. Nevertheless, some assumptions are adopted for building a typical diffusion-reaction model to simplify the difficulty in modeling and ensure model accuracy. The assumptions are listed here: (1) the granules are spherical and uniform; (2) only radial diffusion transport is considered and is described by Fick's law; (3) the diffusion coefficient is constant; and (4) there are no active biomass gradients in the granules at time zero [3].

A representative granule is assumed in a diffusion-reaction model [3, 4]. A typical diffusion-reaction model is characterized by the following equations:

e is assumed in a diffusion-reaction model [3, 4]. A type del is characterized by the following equations:

$$D\_i \left( \frac{d^2 \mathcal{S}\_i(r)}{dr^2} + \frac{2}{r} \frac{d \mathcal{S}\_i(r)}{dr} \right) + r\_i = 0. \tag{1}$$

with two boundary conditions:

$$\begin{aligned} \text{f. } & \cdots\\ \text{f.cons:}\\ & \frac{dS\_i}{dr} = \mathbf{0}, \quad at \quad r = \mathbf{0} \\ \mathbf{S}\_i = \mathbf{S}\_{i, var}, \quad at \quad r = R \end{aligned} \tag{2}$$

**71**

**Figure 1.**

*biomass growth, and biofilm detachment.*

*Approaches for Modeling Anaerobic Granule-Based Reactors*

there is a pore at the core of the representative granule.

is based on each single individual bacterium.

two-dimensional, or three-dimensional.

isms can be applied in this model strategy.

calibrate the kinetic parameters for each process. In addition, assumption (4) for a diffusion-reaction model may not be true for complex substrates such as carbohydrates. If a carbohydrate is used as the substrate, then a multilayer model could be a better alternative. In the multilayer model, a granule is divided into three layers: H2, producing acetogens; H2, consuming organisms (*Methanothrix*); and acidogens. Furthermore, the boundary conditions (Eq. (2)) should be revised accordingly, while substrate concentrations at the core of a representative granule are not zero or

In the other model, substrate degradation can be coupled with the dynamic growth of a sludge granule. In the dynamic growth process, the sludge granule consists of many bacteria, and the granular surface growth and detachment are involved. The model is called an individual-based model (IBM) because the model

The IBM significantly differs from the above diffusion-reaction model. The size and shape of a single granule are not constant in the IBM. Bacteria grow and can be sheared off in the model, which mimics the natural growth of a single granule. The model has clear and active biomass gradients because the growth of different bacterial species interacts with substrate degradation. The IBM can be one-dimensional,

**Figure 1** shows the model strategy of the IBM model. The IBM model was applied to model the biofilm development in a reverse osmosis module. This data verified the validity of the IBM model [5, 6]. In principal, any kind of microorgan-

*Algorithm steps for the biofilm model including substrate convection, substrate diffusion, substrate reaction,* 

*DOI: http://dx.doi.org/10.5772/intechopen.90201*

**2.2 Individual-based model**

where *Si* is the substrate concentration of component i in the granule, *Si,sur* is the substrate concentration of component *i* in the granule surface, *ri* is the volumetric substrate conversion rate in the granule, and *Di* is the diffusion coefficient of substrate *I*; *r* is the distance from the granule center.

The diffusion-reaction model was successfully applied in an anaerobic ammonium oxidation (ANAMMOX) granule [3]. However, the above diffusion-reaction model must be revised accordingly, while other sludge granules are modeled. The ANAMMOX reaction is a simple and single reaction that involves simple substrates. If a complex substrate is involved in a diffusion-reaction model, then a hydrolysis process as well as other downstream processes are involved, and it is hard to

calibrate the kinetic parameters for each process. In addition, assumption (4) for a diffusion-reaction model may not be true for complex substrates such as carbohydrates. If a carbohydrate is used as the substrate, then a multilayer model could be a better alternative. In the multilayer model, a granule is divided into three layers: H2, producing acetogens; H2, consuming organisms (*Methanothrix*); and acidogens. Furthermore, the boundary conditions (Eq. (2)) should be revised accordingly, while substrate concentrations at the core of a representative granule are not zero or there is a pore at the core of the representative granule.
