Modeling Tools for Environmental Engineers and Scientists

The MB equation for the above system under steady state conditions isas follows:0 = QC in – Q w C w – (Q – Q w )C out – raV (5.26)where r is the substrate uptake rate per unit reactive area (ML –2 T –1 ), a is thereactive area per unit volume of the reactor (L 2 L –3 ), and V is the volume ofthe reactor (L 3 ). The Monod’s expression for reaction rate is:r = r max CK s C (5.27)where r max is the maximum surface substrate utilization rate (ML –2 T –1 ) andK s is the half-saturation constant (ML –3 ). The reactive surface area can beexpressed as follows:a = a p C p p (5.28)where a p is the reactive surface area per unit volume of the flocs (L 2 L –3 ), C pis the concentration of flocs in the reactor (ML –3 ), and ρ p is the density of theflocs (ML –3 ).The following assumptions are made to simplify the analysis: the dissolvedsubstrate does not undergo any reaction in the settling tank; concentrationof the dissolved substrate, C w , and the water flow rate, Q w , in thesolids wasting line are negligible when compared to the corresponding valuesin the influent and effluent; thus, C = C out and Q – Q w ~ Q. Hence, combiningthe above Equations (5.26), (5.27), and (5.28) gives the following:0 = QC in – QC – r max CK s C a C p p p V (5.29)Even though the above expression is algebraic, it is somewhat difficult tosolve for C in the above form. If the reactor concentration, C, is small comparedto K s (i.e., K s >> C), the reaction can be approximated by a first-orderreaction with a rate constant = (r max /k s ), and Equation (5.29) can be readilysolved to give the solution as follows:C = C in 1 + –1 r maxK a p C p VQ = Cin 1 + r maxK a p C pτ –1 (5.30)5.4.1.2 Packed Bed ReactorsPacked bed reactors for contacting liquids (and gases) with solids aretypically based on the PFR configuration. In essence, the fluid carrying thes© 2002 by CRC Press LLC