Modeling Tools for Environmental Engineers and Scientists

orC e – t W oV1 – µ e(–µ)t b where = [(1/τ) + k] = 1.23 yr –1µ = 0.05 yr –1W o = 1080 lbs/day; and V = 3.5 × 10 9 ft 3Hence, C e –1.23t [1.70e 1.18t b].The integration constant b can be found by substituting the initial condition:Therefore, b = –0.07.C = 1.63 mg/L at t = 0,Hence, the concentration in the lake, C (mg/L), after the introduction ofthe ban can be described by the following equation:C = e –1.23t [1.70e 1.18t – 0.07]In Chapter 7, several variations of this problem will be modeled with differenttypes of software packages.6.3.2 RIVER SYSTEMSIn a simple analysis of the fate of a substance discharged into swiftly flowingrivers and streams, river systems can be characterized as plug flow systemswith negligible dispersion. In a more realistic analysis, dispersion might haveto be included as described in the next section. Under the assumption of idealplug flow conditions, the water flows without any longitudinal mixing in thedirection of flow but with instantaneous mixing in all directions normal to theflow. Thus, the concentration of substances in the water column can vary withdistance along the direction of flow as well as with time, but it is uniform atany cross-section. Such a condition can be analyzed by setting up a differentialmass balance, assuming constant flow rate and first-order reactions: d(A xdxC) = Q t C – (Q t dQ)(C dC) ± kA x dxC ± S D A x dx (6.40)dtwhere A x is the area of flow at a distance X from the origin (L 2 ), x is the distancemeasured from origin along direction of flow (L), and, S d is the strengthof a distributed source or a sink (ML –3 T –1 ).© 2002 by CRC Press LLC