Bond Graph Modeling of an Integrated Biological Wastewater Treatment System

Compared with abstract pure mathematical representation of a system and rough engineering drawings, the concept of bond graphs provide us a unique graphical view based on energy interactions among different domains. Bond graphs sit in the middle between equation sets and schematic block graphs, combining most of the advantages from both. The bond graph approach has already been shown to be effective for modeling, analysis, and design of engineering systems with applications in mechanical translation and rotation, electrical circuits, thermal, hydraulic, magnetic, chemical and other physical domains [1, 2]. In this study, we extend the concepts of bond graphs to the biological domain by defining the basic effort and flow variables, as free energy of formation, gi (KJ/mol) and molar flowrate, ṅi (mol/s), respectively of each species i involved in a reaction. Power/energy (KJ/s) is the product of effort and flow, which are the basic variables that allow systems with diverse energy domain to be treated in a unified manner. Further, since the chemical/biochemical reactions involve stoichiometric relationships, based on the stoichiometry coefficient, υi, for species i, the biochemical affinity, υigi, and electron equivalent flow, ṅi/υi [3], are introduced as a conversion of the free energy and molar flowrate. For each species, a transformer with the stoichiometry coefficient as the modulus for that reactant/product represents the conversion of bio-chemical energy between an inactive state (the product of free energy and molar flowrate) and an active state (biochemical affinity and electron equivalent flow). As shown in the lower panel of Figure 1, the difference between the forward (Af ) and reverse (Ar) affinities constitute the true driving force for the chemical reaction to occur. Applying the newly defined protocols, it is shown how to represent enzyme catalyzed microbe growth, present in wastewater treatment systems and described by MichaelisMenten, two-step reversible kinetics and Monod kinetics [4, 5] using bond graphs. Figure 2 is a bond graph model of a general Michaelis-Menten equation given by