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MBW:Dynamic Biodegradation model with Flocculation

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MBW: Flocculated Growth of Microbes

Article Review by Robert Griffin Hale Brandt, Bernd W., and Sebastiaan A. L. M. Kooijman. "Two Parameters Account for the Flocculated Growth of Microbes in Biodegradation Assays." Biotechnology and Bioengineering 70.6 (2000): 677-84.


Brandt and Kooijan focus on modeling the dynamics of biodegradation of organic compounds by microbes. Previous models and data assume individual cell suspension and neglect the formation of flocs or aggregated microbes. Flocs can greatly reduce biodegradation because cells near the center of the floc can either be considered dead biomass or considered to have a reduced consumption. Brandt and Kooijan use a simplistic model to include the effects of flocs. It is assumed that flocs have two phases and are modeled as spheres. Initially small flocs increase biomass at an exponentially rate. At a certain volume, the floc transition to a linear growth rate when a dead ‘kernel’ forms in the center of the biomass. It is also assumed that the substrate is held at a constant concentration exterior to the floc. The transition is a result of diffusion-limited substrate, which dictates the radius of living microbes. Beyond the maximum radius, cells no longer receive adequate nutrients and die. Eventually the flocs will grow to a critical radius and rupture. The dead biomass is released and the surface of critical floc is then assumed to separate into smaller flocs and the cycle repeats.

Context/Biological phenomenon under consideration

Biodegradation is the main method for the processing of household chemicals in wastewater treatment plants. Commonly, the chemicals are added to a large tank of ‘sludge’ that contains live microbial organisms. These microbes feed on the chemicals, metabolizing them to a stable or less harmful state. Many studies of the exact processes within the plants’ tanks have been done. The researchers try to model exact growth rates of the microbes and subsequent breakdown of the substrate being controlled. Often however, the models used assume that the microbes are dispersed throughout the tank evenly; as in an idealized suspension. By assuming true suspension, these models are ignoring the fact that microbes often form coagulates or floc. The process of these clusters forming and growing is known as flocculation.

Flocculation has obvious, and so-far unaccounted, effects on the dynamics of a wastewater treatment facility.

Illinois wastewater treatment facility

When a floc forms, it at first grows exponentially in volume. The cells all have sufficient access to the substrate being processed, and thus are limited only by the external concentration of this substrate, and its diffusion rate. However, as the floc size increases past a certain point, cells in the interior of the floc can no longer receive an adequate supply of nutrients. Because the floc is not highly diffusive to the substrate, and the fact the cells consume substrate as it passes to the center, a size is reached past which a ‘kernel’ is formed. The kernel is found below a layer of living cells and can be thought of as the core of the floc. This kernel is composed of dead or dormant cells, which no longer actively grow, and therefore, no longer consume substrate. Thus, large flocs act to slow the wastewater treatment process, as the consumption rate compared to biomass is decreased. The critical point at which a kernel forms depends on the surrounding substrate concentration, the consumption rate of individual cells, the minimum substrate concentration required for survival and the diffusivity of the floc itself.

Once the kernel has formed, the increase in volume is determined solely by the growth of the living exterior layer. Thus, growth now slows to a roughly linear rate, similar to the tapering of a logistic growth model. The spherical floc will continue to grow, albeit at a reduced speed, until a second critical size is reached. At this point, the floc is no longer physically stable and will fragment. The dead core is released into the environment, and the thin living layer breaks up into multiple small pieces, each of which form its own new floc. At this point the process begins again for the small, kernel-less flocs. The critical size for self-destruction is determined by many factors: the adhesiveness of neighboring cells and their density, and most importantly the sheer forces felt from perturbations in the water, due to stirring and fluid convection.


In today's hyper-engineered and critically designed world, the applications for fluid dynamic models are huge. Applications range from sewage water treatment and large scale water distribution to raw water collection and large-scale reservoirs. Each of these areas contain large lists of variables that affect the flow and circulation of the water. Often, the systems are so large and complex that any type of controlled experiment would be far too expensive to construct. Thus, to save construction costs and valuable resources, computer simulated models are used instead, to best investigate the often hard-to-observe phenomena. Using the computer models, engineers can work together with the mathematicians to best increase efficiency in cost management and also control risks to individuals in the plants and those in the general public. Before any large-scale system is installed, rigorous tests can be applied via simulations to determine whether the design is worth constructing. In addition, an engineer can also simulation the effect of minor alterations to an already existing infrastructure. This will prove invaluable in the coming years as we struggle to increase efficiency to save valuable resources and energy. Using computer models, we will be able to systematically identify the weak points of a system and update them in the most effective ways.

Current models take into account many factors on all scales; ranging from the micro to the macro. For example, effective water treatment requires a strong grasp of the biological systems developing, the chemical reactions and mass diffusion, as well as the hydraulic conditions caused by the treatment system being used. A true insight into how the hydrodynamics and chemical equations effect each-other can lead to novel treatment standards and practices that can help to save time, money, and lives.Today, the chemical, mechanical and manufacturing industries use computational fluid dynamic models to test their products and systems. The water treatment sector has lagged behind slightly, but has recently seen a large movement towards improved models and applications.

In the United States and abroad, fluid dynamics models have begun to lead to increased understanding, and innovation. However, most of these models glaze over or ignore completely the effects of flocculation on the results they are looking for. Often flocculation is removed from the equations because of it is difficult to accurately describe mathematically and measure in experiments. The inherent fractal nature of floc formation makes accurate spatial models of the phenomena computationally taxing. To adapt these models and make them temporally dynamic would require more computing power than is available to the standard engineer, or even larger corporations. The size of the data sets, and dimensions of the variables would be staggering to consider using current models. Such variables are floc size distribution, shape, composition, density, and structural stability. In addition, there are various behaviors and processes related to flocculation including aggregation, sedimentation, fragmentation, deposition and suspension.

For this reason, a novel approach to describing flocculation is required.

Parameter Definitions

D diffusion coefficient

[E] energy density

[{E_{G}}] maximum energy density

f(L) scaled functional response {\frac  {X(L)}{X_{K}+X(L)}}

f_{\dagger } scaled functional response L=L_{M}

{\bar  {f}} average scaled response

g energy investment ratio: {\frac  {[E_{G}]}{[E_{M}]}}

k_{E} specific energy conductance: {\frac  {[P_{A}m]}{[E_{M}]}}

k_{M} maintenance rate coefficient: {\frac  {[P_{M}]}{[E_{G}]}}

j_{{Xfm}} mass-specific maximum uptake rate

j_{{XM}} mass-specific maintenance rate

{l^{*}}_{d} scaled radius at division {\frac  {L_{d}}{L_{M}}}

l_{M} scaled maximum thickness of living layer: {\frac  {L_{M}}{L_{D}}}

l_{T} scaled total radius of floc: {\frac  {L_{T}}{L_{D}}}

L_{D} diffusion length {\sqrt  {{\frac  {DX_{K}}{j_{{Xfm}}X_{F}}}}}

L_{B} radius at birth

L_{d} radius at division

L_{M} maximum thickness of living layer of floc

L_{T} total radius of floc


L_{\dagger } radius of dead kernel of floc

M_{E} amount of reserves in living biomass

M_{{E\dagger }} amount of reserves in dead biomass

M_{V} amount of living biomass in floc

{M_{V}\dagger } amount of dead biomass in floc

n number of daughters from a floc

[p_{{Am}}] volume-specific maximum assimilation rate

[p_{M}] volume-specific maintenance rate

r specific growth rate of cells

r_{F} specific growth rate of flocks

t time

t_{b} time at birth

t_{d} interdivision time

V_{d} volume at division

V_{M} volume of living biomass

V_{T} total volume of biomass

V_{\dagger } volume of dead biomass

x scaled substrate concentration: {\frac  {X}{X_{K}}}

X(L) substrate concentration at L

X_{K} saturation coefficient of scaled functional response

X_{F} amount of biomass / volume floc

X_{V} structural biomass / volume

X_{{V\dagger }} dead biomass / volume

X_{\dagger } minimum substrate concentration for support

y_{{XV}} substrate (X) needed per biomass(V) formed

y_{{XE}} substrate (X) needed per reserve (E) formed


The purpose of the model is to simulate the dynamics of Flocs, and their effects on reactor dynamics. It is therefore first necessary to approximate the growth of these flocs.

Floc Growth

Normal flocs are irregular in shape but for the purpose of this model they will be estimated as spherical. It will also be assumed that the growth of the floc is slow compared to diffusion such that steady states can be achieved. The volume-specfic biomass will additionally be held constant. There are two types of flocs: Small flocs were all microbes receive adequate nutrition via diffusion and large flocs where the core consists of dead biomass.

If L_{T}\leq L_{M} the floc grows exponentially as seen in the following equation. This represents small flocs where all cells receive adequate nutrition.

{\frac  {dV_{T}}{dt}}=rV_{t}

However if the floc has a living mantel and a core consisting of a dead biomass the volume changes as follows

{\frac  {d}{dt}}V_{T}(t)={\frac  {d}{dt}}[V_{M}(t)+V_{\dagger }(t)]=rV_{M}(t)

The dead biomass is a result of cells not receiving enough nutrients via diffusion. Using the two previous equations one can solve for the rate of change of the radius, which can be seen below.

{\frac  {d}{dt}}L_{t}={\begin{cases}{\frac  {r}{3}}L_{T}&{\textrm  {for}}\,\,L_{T}\leq L_{M}\\rL_{M}(1-{\frac  {L_{M}}{L_{T}}}+{\frac  {{L}_{M}^{2}}{3{L}_{T}^{2}}})&{\textrm  {for}}\,\,L_{T}>L_{M}\end{cases}}


Now that the growth rate of a floc has been modeled it is necessary to describe the life cycle of a floc. A floc will continue to grow until it reaches a critical volume and then the floc will rupture. This rupture is due to mechanical instability. The critical size is a function of the shear forces and floc porosity, age of floc. For this model a single variable V_{d} accounts for these parameters denoting the maximum volume. It is assumed that the living mantle dissociates into daughter flocs with out changing thickness. The number of daughter flocs is a result of the total volume and the living mantle. The living mantle is dependent on the concentration of the substrate.

A figure of this cycle can be seen below. 
Floc life.png

It is now possible to equate the time it takes for a floc to reach the critical volume or the interdivision time t_{d}

rt_{d}={\begin{cases}3\,\,ln2;&{\textrm  {for}}\,\,{l}_{d}^{*}\leq 1,\\3\,ln2-1+{l}_{d}^{*}-{\frac  {\pi }{3{\sqrt  {3}}}}+{\frac  {1}{{\sqrt  {3}}}}\arctan({\sqrt  {3}}(2{l}_{d}^{*}-1))+{\frac  {1}{2}}ln(1-3{l}_{d}^{*}+3{l}_{d}^{{*2}});&{\textrm  {for}}\,\,{l}_{d}^{*}>1\end{cases}} 

The relationship between the specific floc growth rate and that of free cells, non-dimensionalized by interdivision time can be seen below.

r_{F}(X)={\frac  {ln\,\,n(X)}{t_{d}(X)}}

The critical size of the floc depends on the concentration of substrate as well as the growth model. Below the equation is divided by the specific growth rate. The growth rate {r} is further investigated in the #Cellular Growth section.

{\frac  {r_{F}(X)}{r}}={\frac  {ln\,\,n(X)}{rt_{d}(X)}}

A plot showing the ratio of r_{F} to r as a function of critical size can be seen below.

RF to r as a function of the size at division l*d.png

Substrate Concentration and Reactor Dynamics

Now that the growth and life of a floc are defined the model explores the steady states of the floc. The steady states will help determine the maximum thickness for the living mantle, and thus the amount a metabolic activity occurring in a floc. Cells require substrate that is transported via diffusion. The substrate concentration profile in a sphere as a result of diffusion is as follows.

{\frac  {\delta }{\delta t}}X(L,t)={\frac  {D}{(L_{T}-L)^{2})}}{\frac  {\delta }{\delta L}}{\Bigg \{}(L_{T}-L)^{2}{\frac  {\delta }{\delta L}}X(L,t){\Bigg \}}-f(L,t)j_{{Xfm}}X_{F}

It is important to assume that the concentration of the substrate is constant and the growth is slow compared the substrate profile.

At steady state conditions the concentration of the substrate is not changing. The cells at the edge of the dead core receive exactly the nutrients needed in order to survive. This boundary condition can be seen in the below equation.

0={\frac  {1}{(l_{T}-l)^{2}}}{\frac  {d}{dl}}{\Bigg \{}(l_{T}-l)^{2}{\frac  {dx}{dl}}{\Bigg \}}-f(l)

With the steady states and boundary conditions defined and assuming that the curvature of the floc is small the maximum living thickness of the floc can be solved for as seen below.

l_{M}(x_{\dagger })=2^{{-1/2}}\int \limits _{{x_{\dagger }}}^{{x_{0}}}{\Bigg \{}y-x_{\dagger }+ln\,{\frac  {1+x_{\dagger }}{1+y}}{\Bigg \}}^{{-1/2}}dy

The effect of concentration on the floc radius and maximum living radius can be seen in the following two figures respectfully.

X vs L.png
LM vs X0.png

If a Marr-Pirt model for cellular growth is used: the biodegradation of a batch reactor is as follows.

{\frac  {d}{dt}}X_{{V\dagger }}={\Bigg (}{\frac  {({\frac  {L_{d}}{L_{M}}})^{3}-2^{{-3}}}{({\frac  {L_{d}}{L_{M}}})^{3}}}-1{\Bigg )}^{{-1}}X_{V}r_{F}(X)

This final equation uses the assumptions previously stated and the Marr-Pitt cellular growth model to simulate the dynamics of flocs in biodegradation.

Cellular Growth

Thus far, the growth of a floc has been defined as a scaled function of a known cellular growth rate: the Marr-Pitt model However, the Marr-Pitt is overly simple. In a more accurate model for growth rate of the flocs, the specific growth rate {r} is found using the Dynamic Energy Budget (DEB) Theory (Kooijman,2000). According to DEB, cells can only grow while reserves are sufficient. Both the energy required of cellular maintenance [p_{M}] and growth [{E_{G}}] detract from the reserves. Also, a parameter known as the 'investment ratio' {g} is introduced to account for the varying resources devoted to either growth or maintenance. When taking this into account, the growth rate of the living cells now becomes:

r={\frac  {k_{E}{\bar  {f}}-k_{M}g}{{\bar  {f}}+g}}{\Bigg (}1-ln{\frac  {f_{0}+g}{f_{\dagger }+g}}{\Bigg )}^{{-1}}

Using this new rate definition, the change in floc radius can be recomputed as follows:

{\frac  {d}{dt}}L_{T}=\int \limits _{0}^{{L_{M}}}{\frac  {p_{{Am}}f(L)-[E_{m}]({\frac  {d}{dL}}f)({\frac  {d}{dt}}L_{t})-[p_{M}]}{[E_{m}]f(L)+[E_{G}]}}

Which is obviously more complex than the original time rate of the radius of the floc. This is due to the variation of parameters previously held constant. To allow easier computation, though, we can use the same simplifications as in the previous sections, and the above equation becomes:

{\frac  {d}{dt}}L_{T}\simeq rL_{m}{\Bigg \{}1-{\frac  {L_{M}}{L_{T}}}+{\frac  {{L}_{M}^{2}}{3{L}_{T}^{2}}}{\Bigg \}}. The only difference being that {r} is now the DEB specific growth rate as defined above.


Taking into account the effects of flocculation on reactor dynamics greatly improves the accuracy of predictions being made. This accuracy can deepen our understanding of the requirements and restrictions made on manufacturing and safety standards. The limitations placed on the growth of actively reacting biomass are based on observations made by many different laboratories. This slowed growth rate helps to improve the predictions made of reactors. Instead of assuming continued exponential growth of cells proportional to the nutrients available, the model smartly rectifies the growth of flocs as their size grows beyond sustainability.

Other improvements over current models are made in the interaction between cells and the substrate being consumed. Older models, assuming cell suspension, have used various functional responses based on the concentration of substrate and other parameters. However, few accounted for the need for substrates to diffuse through layers of the flocs to reach internal cells. In this way, the rate of reaction is reduced significantly, and the new model accounts for this (fig. 5).

The model does have some shortcomings, and variables that are ignored or held to be constant. Just like any model, steady-state assumptions lead to vastly simpler answers, but at a cost to accuracy. The shape of the flocs is assumed to be spherical; the irregularities ignored. This assumption affects all further aspects of the model, including growth rate, size at division and consumption rate. Another assumption made is that the volume at division is held constant, based on hydrodynamic conditions and other characteristics of the flocs. In reality, division volume will vary with respect to many characteristics of the reactor. Also, it will vary in space, as shear forces will be much greater in some locations.

Overall, the assumptions made are not implausible. By considering the effects of flocculation, the model greatly improves upon its contemporaries. Brandt and Kooijman did not run any experiments to test their model, but used existing data sets to verify their model.

Further Study

Further research could be done on flocculation. The effects of the shape and size of the containing vessel should be studied, as hydrodynamic forces vary greatly in differing vehicles. Figuring out the metabolic activity of the kernel could also prove helpful. The paper assumed a steady state distribution of substrate such that the kernel had absolutely no metabolic activity. In reality, this may not be the truth, as the interior cells may realize their reduced supply of nutrients and slow growth, thus reducing the expenses of survival. This would allow the kernel to have a continued effect on chemical decomposition.

Currently, it would prove very difficult, if not impossible, to explicitly measure some of the states of a floc, as they are often on the micro-scale. But, as nanotechnology improves, perhaps soon we will be able to answer some of the unknowns left in the understanding of reactor dynamics.

External Links

Computational fluid dynamics with flocculation

Illinois water treatment plant

Flocculation (wikipedia)

Flocculants general info