**2.2 Individual-based model**

*Bacterial Biofilms*

the granule.

ticle models are discussed below.

**2.1 Diffusion-reaction model**

**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

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 biopar-

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].

cal diffusion-reaction model is characterized by the following equations:

\_ *dSi*

*Di* ( *d*2 \_ *Si*(*r*) *dr*<sup>2</sup> + \_2 *r* \_ *dSi*(*r*)

of substrate *I*; *r* is the distance from the granule center.

with two boundary conditions:

A representative granule is assumed in a diffusion-reaction model [3, 4]. A typi-

*dr* <sup>=</sup> 0, *at <sup>r</sup>* <sup>=</sup> <sup>0</sup>

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

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

*Si* = *Si*,*sur*, *at r* = *R*

*dr* ) <sup>+</sup>*ri* = 0. (1)

(2)

**70**

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 is based on each single individual bacterium.

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, two-dimensional, or three-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 microorganisms can be applied in this model strategy.

### **Figure 1.**

*Algorithm steps for the biofilm model including substrate convection, substrate diffusion, substrate reaction, biomass growth, and biofilm detachment.*

The implementation of an IBM model requires a big computational workload because a modeling domain must be divided into numerous micro grids. Hence, the implementation of an IBM model at a reactor scale would require huge computational workload and appear to be impossible.
