A detailed understanding of the biochemical reactions and hydrodynamics inside wastewater treatment plants (WWTPs) can assist designers and engineers in evaluating new plant designs, quantifying management decisions, developing new control schemes and providing safe operator training. In this blog, I will introduce our readers to FLOW-3D’s new Activated Sludge Model (ASM) which dynamically solves massive systems of biochemical reactions.

Aeration tanks

Aeration tanks are where most biochemical reactions happen in the secondary treatment part of a WWTP. Typically, aeration tanks have long paths to allow time for most biochemical reactions to finish. The amount of time a species takes to traverse the entire length of the aeration tank is called the residual time. Oxygen is injected into the aeration tank, triggering the growth of bacteria in the wastewater. The bacteria use the oxygen to break down the waste material in the water, and as they do so, form an aggregate called a floc or sludge blanket. A portion of the activated sludge is then recycled back into the aeration tank to further promote the biochemical treatment of the wastewater.

Schematic of a wastewater treatment plant
Schematic [2] of a WWTP from Zele, Belgium. The green box delineates the secondary treatment process.

Standard systems of bio-chemical reactions

The International Water Association (IWA) has proposed three major mathematical systems of describing biochemical reactions over the last four decades. Each of these systems, ASM-1, ASM-2, and ASM-3, captures the growth and decay dynamics of various species inside an aeration tank with different degrees of detail, ASM-3 being the most comprehensive one. The first system, ASM-1, is shown below in tabular and expanded formats.

ASM-1 system of bio chemical reactions
ASM-1 system of bio-chemical reactions, as an expanded system of coupled partial differential equations.

ASM solver capabilities

Most biochemical reactions are based on the Monod model or a similar model. The Monod model is a mathematical model which predicts the rate of growth and decay of microorganisms and is described by the simple equation,

$latex \displaystyle \frac{\partial C}{\partial t}=k\frac{C}{a+C}$

where a and k are maximum specific growth rate constant and the substrate concentration corresponding to half the maximum specific growth rate. C is the concentration of a microorganism species, which is changing with time, t. The Monod model has the property that it dynamically changes the order of a reaction depending on the concentration of the species.

For C  >> a, the rate of change of C approaches zeroth order.

$latex \displaystyle \frac{\partial C}{\partial t}\sim k$

For C  << a, the rate of change of C approaches first order.

$latex \displaystyle \frac{\partial C}{\partial t}\sim \frac{kC}{a}$

All this is to say that if the concentration of a microorganic species is high, it will decay/grow faster, and, if the quantity of a species is low, it will decay/grow slower. The solution for the Monod equation is given by the Lambert function as follows,

$latex \displaystyle {{C}_{t}}=aW\left( \frac{{{C}_{0}}e\frac{{{C}_{0}}+kt}{a}}{a} \right)$

W→ Lambert W function

$latex \displaystyle W\left( x{{e}^{x}} \right)=x$


Comparison of the analytical solution with the FLOW-3D solution for the simple Monod equation
Comparison of the analytical solution with the FLOW-3D solution for the simple Monod equation

The standard systems describing biochemical reactions contain long chains of Monod terms. FLOW-3D’s ASM model is fully capable of tracking the Monod-based growth and decay of the bacterial species in a WWTP. The ASM model is integrated with FLOW-3D’s fluid dynamics solver so that the bacteria’s movement, based on the velocity and pressure fields, can be coupled with their rate of growth and decay.

In my next blog, I will show the activated sludge model in action using the Zele WWTP case study. 


[1] Henze M., Lossdrecht M.C.M., Ekama G.A., Brdjanovic D., Biological Wastewater Treatment, Principles, Modelling and Design, IWA publishing 2008.

[2] Peterson B., Vanrollenghem P.A., Gernaey K., Henze M. (2002) Evaluation of an ASM-1 model calibration procedure on a municipal–industrial wastewater treatment plant, Journal of Hydroinformatics, 4(1): 15-38.

[3] Henze, M., Grady, C. P. L. Jr., Gujer, W., Marais, G. v. R. & Matsuo, T. (1987) Activated Sludge Model No. 1. IAWPRC Scientific and Technical Reports No. 1. London, UK.

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