Modeling Tools for Environmental Engineers and Scientists

5.3.1 COMPLETELY MIXED BATCH REACTORSIn completely mixed batch reactors (CMBRs), the reactor is first chargedwith the reactants, and the products are discharged after completion of thereactions. During the reaction, inflow and outflow are zero, and the volume,V (L 3 ), remains constant, but the concentration of the material undergoing thereaction changes with time, starting at an initial value of C 0 . The MB equationfor a CMBR during the reaction is as follows: d(V C) = –rV = –kCV (5.1)dtwhere r is the rate of removal of the material by reactions (ML –3 T –1 ), and kis the first-order reaction rate constant (T –1 ). The solution to the MB equationis as follows:orC = C 0 e –kt (5.2)t = – 1 k Cln C (5.3)where C is the concentration of the material at any time, t, during the reaction.5.3.2 SEQUENCING BATCH REACTORIn sequencing batch reactors (SBRs), a sequence of processes can takeplace in the same reactor in a cyclic manner, typically starting with a fillphase. Reaction can occur during the fill phase of the cycle as the volumeincreases and can continue at constant volume after completion of the fillphase. On completion of the reaction, another process can take place, or thecontents can be decanted to complete the cycle. The volume, V t , at any time,t, during the fill phase = V 0 + Qt, where V 0 is the volume remaining in thereactor at the beginning of the fill phase (i.e., t = 0), and Q is the volumetricfill rate (L 3 T –1 ). The MB equation during the fill phase, with reaction, forexample, is as follows:which can be expanded to:0 d(V tC) = rV t QC in = –kCV t QC in (5.4)dtd[(V 0 Qt)C] = –kC(V 0 Qt) QC in (5.5)d tThe solution to the above MB equation is:© 2002 by CRC Press LLC