Modeling Tools for Environmental Engineers and Scientists

4.7.2 ELEMENTARY REACTIONSThe rate or kinetics of these reactions can be quantified using the Law ofMass Action, which states that the rate is proportional to the concentrationof the reactants. For example, consider the reaction: A + B → C. The rate ofconsumption of species A by this reaction is given by the Law of Mass Actionas follows: d CA = –kf(C A ,C B ) (4.48)dtwhere k is a temperature-dependent reaction rate constant, and the functionf(C A ,C B ) has to be determined experimentally. A common general form ofthis function is d CA = –kC A,C β B(4.49)dtwhere the powers to which the concentrations are raised are referred to as thereaction order. In the above example, the reaction is of order α with respectto species A, and order β with respect to B, and the overall order of the reactionis n = (α + β). Because most modeling efforts build upon submodels ofsingle chemicals, the following sections will deal with the equation: d C = –kC n (4.50)dtwhere n = 0 for zero order, n = 1 for first order, or n = 2 for second order. Themass removal rate by such a reaction can be found from V(dC/dt), where V isthe volume in which the reaction is occurring. The change in concentration asa function of time and the time to reduce the initial concentration by 50%(half-life) can be found by solving the above equation for the appropriatevalue of n and initial concentration of C 0 . The results are summarized in Table4.3. The order of the reaction and the rate constant have to be determinedexperimentally. The procedures are well established and can be found in severaltextbooks on reaction kinetics.4.7.3 ENZYME-MEDIATED REACTIONSThe Michaelis-Menten model of microbial growth is a classic example ofan enzyme-mediated two-step reaction where the substrate S is converted tothe product P in the presence of a catalytic enzyme, E:E S I [ES] r E P© 2002 by CRC Press LLC