Team:ZJU-China/Modeling

Introduction

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Compartment:The biofilm itself is distinguished from the overlying water and the substratum to which it is attached. A mass-transport boundary layer separates the biofilm from the overlying water.

Within each compartment are components: include different types of biomass ,substrates , products. biomass is often divided into active microbial species, inert cells, and extracellular polymeric substances(EPS).

The components can undergo transformation, transport, and transfer processes. For example, substrate is consumed, and this leads to the synthesis of new active biomass.

All process affecting each component in each compartment are mathematically linked together into a mass balance equation that contains rate terms and parameters for each process.

Model Selection:Many kinds of Mathematics models have been founded to describe a system of biofilm. Models of different dimensions (1d, 2d, 3d) focus on different properties of a biofilm. Since we care most about the oxygen concentration gradients perpendicular to the substratum, numerical 1-dimensional dynamic model(N1) would be a proper choice for us.

Modeling

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An introduction to N1 model:

The core of the N1 model consists of a system of stiff, non-linear partial differential equations. Because of the stiffness of the equation system, integration methods tailored for stiff systems must be used, or the dissolved and the particulate components have to be treated differently. Furthermore, biofilm growth, i.e., the displacement of the interface between biofilm and bulk liquid, creates a so-called moving boundary problem (Kissel et al. 1984; Wanner and Gujer 1986).We performed simulations with the N1 model using the software package AQUASIM (Reichert 1998a, 1998b), since it can handle stiff systems and the moving boundary. The N1 model implemented in AQUASIM is described in next Section.

The processes considered in the model include:

• many transformation processes
• advection and diffusion of attached particulate components in the biofilm solid matrix
• attachment and detachment of particulate components at the biofilm surface and in the biofilm interior
• diffusion of suspended particulate and dissolved components in the biofilm liquid phase and in the liquid boundary layer at the biofilm surface
• complete mixing of suspended particulate and dissolved components in the bulk liquid

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Definitions and equations

As discussed above, biofilms are multiphase systems. Consequently, in the N1 model three different phases are distinguished. The solid attached phase is made up by the particulate components, which form the biofilm solid matrix. In the model of single species of bacteria, the particulate components are assumed to be uniform, defined as

where X_M is the concentration of the attached component, ρ is its density , defined as the mass divided by the volume of the cell or particle, and ε is its volume fraction, defined as volume of the component per unit biofilm volume. The porosity or biofilm pore volume fraction θ is

The pore volume is formed by two phases: the phase of the suspended particulate components with concentrations X_P and the biofilm liquid phase, with the liquid phase volume fraction ε_liquid:

Mass balances in biobilm models

Conservation of mass of a component in a dynamic and open system states that:

The local mass balances are differential equations that express the variation of concentration of a component in time in a point in space as a result of transport and transformation processes. One-dimensional mass balance equations for attached particulate, suspended particulate, and dissolved components can be derived from the general mass balance if gradients are considered only in the direction of z, perpendicular to the substratum

where t is time; z is perpendicular coordinates ; S is the concentration; j is the mass flux along the perpendicular coordinates; and r is the net production rate of the component. For a dissolved substrate S_i, the specific mass flux j_i in the biofilm, can be modeled by Fick’s first law of diffusion as

Where D_i is the molecular diffusivity of substrate in the biofilm. Substituting this equation into the equation (1) yields a mass balance equation that describes the development in time and the spatial profile in the biofilm of the concentration S_i of a dissolved substrate as

where r_i is the net production rate of S_i. Equation (3) has the boundary condition

at the substratum (z=0) and

at the biofilm–bulk liquid interface (z=LF), where LF is the biofilm thickness and S_LF, i is the substrate concentration at the biofilm surface. All terms in equation (6) are given as substrate mass per unit biofilm volume. Equations (6) to (8) also hold for suspended particulate components, i.e., for cells or particles which are suspended in the biofilm pore volume and have the concentration X_P.

For attached particulate components, which form the biofilm solid matrix, transport is assumed to be the result of microbial growth and decay in the biofilm. Growing or shrinking cells lead to a volume expansion or contraction of the biofilm solid matrix, respectively, and to a displacement of neighboring cells (Wanner 1989). This displacement can be interpreted as advective transport and is formally described as a specific mass flux j_M by

where u_F is the distance by which the cells are displaced per unit time. The displacement velocity u_F of a cell at the location z is equal to the added net specific mass production of the cells the biofilm matrix between the substratum and this location:

where r_M is the net production rate of a particulate component in the biofilm matrix. Based on equations (9), with u_F as defined in (10), a mass balance analogous to equation (6) can be derived and used to describe the development in time and the spatial profile in the biofilm of the attached particulate component:

The boundary condition is no-flux condition:

at the substratum. Equation (11) is used to calculate the relative abundance, spatial distribution, and development in time of microbial species and particles in the biofilm.

The development of the biofilm thickness in time is the result of the net production of biomass in the biofilm, as described by equation (10), of the attachment at the biofilm surface of microbial cells and particles suspended in the bulk liquid, and of the detachment of microbial cells and particles from the biofilm surface to the bulk liquid. It is modeled as

where u_de and u_at are the detachment velocity and the attachment velocity, respectively. The velocity u_de yields a phenomenological description of the decrease of the biofilm thickness per unit time as result of the detachment process, i.e., erosion or sloughing.

Attachment refers to the adsorption of microbial cells suspended in the bulk liquid to the biofilm surface and is modeled by an attachment velocity as

where k is the attachment rate coefficient and X_L is the concentration at the bulk liquid side of the biofilm surface of the suspended particulate component.

The environment of the biofilm is modeled as a completely mixed volume of water, termed bulk liquid. Conversion processes in the bulk liquid can be equally important as those that take place in the biofilm. For each dissolved and particulate component considered, an additional mass balance equation of the form

is needed, where C_in,i and C_B, i are the influent and bulk liquid concentrations, respectively, of the dissolved or suspended particulate component i, V_B is the bulk liquid volume, Q is the rate of flow through the bulk liquid, A_F is the biofilm surface area, j_F,i is the mass flux across the biofilm surface , and r_B,i is the production rate of the component.

Transformation processes

In modeling, the transformation processes are represented mathematically with rate expressions, or equations that tell how fast the component is produced or consumed. The rate is proportional to the concentrations of one or more of the components and one or more kinetic parameters. We describe the synthesis of microbial biomass with the Monod equation for one limiting substrate:

where μ is the specific growth rate, μ_max the maximum specific growth rate, S is the concentration of the rate-limiting substrate, and K_S is the concentration giving one-half the maximum rate.

When multiple substrates are rate limiting, the Monod equation typically is extended to include the effects of each substrate influencing the rate of microbial synthesis. For instance, is the rate of microbial synthesis may be limited by the concentrations of the electron donor (S_1, e.g. COD) and the electron acceptor (S_2, e.g. Oxygen) then the specific growth rate can be described with a multiplicative-Monod expression (Bae and Rittmann 1996):

Synthesis is not the only transformation process relevant to microorganisms in a biofilm. Indeed, the microorganisms undergo a number of loss processes, including endogenous decay (or inactivation), predation, and detachment. The inactivation process is common in almost any biofilm and serves as a model for how to express a biomass-loss process.

Synthesis is not the only transformation process relevant to microorganisms in a biofilm. Indeed, the microorganisms undergo a number of loss processes, including endogenous decay (or inactivation), predation, and detachment. The inactivation process is common in almost any biofilm and serves as a model for how to express a biomass-loss process.A simple and common means to represent inactivation/decay is with a first-order loss rate

where r_in is the inactivation rate and b is the first-order inactivation rate constant. Because utilization of a substrate is what brings about biomass synthesis, the rate of substrate utilization is proportional to the growth rate of X:

where Y is the true-yield coefficient, or the ratio of biomass produced per unit of substrate consumed. Similar equations could be founded to describe rates in other processes. Following table summarizes the basic rate expressions for heterotrophic bacteria in a convenient and commonly used matrix format. The Monod half saturation constant for oxygen K in processes of growth and respiration are assumed to be equal.

Simulation

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Calculate growth and oxygen concentration gradient of a biofilm which consists of Heterotrophic bacteria. The water inflow at a constant rate contains substrates and Oxygen. Growth occurs with Monod-type rate laws. Respiration occurs with specific rate. A program variable LF referring to Biofilm Thickness and 3 dynamic volume state variables O_2, S, X referring to oxygen, substrates and the Heterotrophic bacteria are defined. All necessary constants and initial conditions are defined in following table. Some of their values are based on average results provided by IWA.

Definition of Process:

Define a dynamic process for growth, a dynamic process for respiration, and a dynamic process for inactivation. The microbial stoichiometric numbers of the processes of S (referring to substrates) and O_2 (referring to Oxygen) and rate of the processes are shown in following table.

Simulation results

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Definition of simulation

Define an active calculation with proper steps of 1 day.

Fit above graph into polynomial, we have:

Where the equation of the polynomial is:

Reference

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[1]	O Wanner, H Eberl, E Morgenroth, D Noguera, C Picioreanu, B Rittmann, M Loosdrecht. ”Mathematical Modeling of Biofilms, IWA Task Group on Biofilm Modeling.”, Scientific and Technical Report, 2006.
[2]	P Reichert. “AQUASIM 2.0-Tutorial” Swiss Federal Institute for Environmental Science and Technology, 1998.
[3]	P Reichert. “AQUASIM 2.0-User Manual” Swiss Federal Institute for Environmental Science and Technology, 1998.
[4]	LI Tian-cheng, LI Xin-gang, ZHU Shen-lin.”Two-Dimension Dynamic Simulations on Biofilm Forming and Developing”. Acta Scientiarum Naturalium Universitatis Sunyatseni, 2005.