Modeling Tools for Environmental Engineers and Scientists

y a continuous, constant input load of W. The steady state condition isfound by setting the first term in the left-hand side of Equation (6.22) tozero, yielding:W W QCC ss = = = in Cin = (6.38)V (1 kτ)V 1 V 1 τ k τ k 6.3.1.2 General SolutionA more general situation, where the application of a new step load to a lakewith an initial concentration of C i causes a transient response, can be simulatedby Equation (6.34), giving the solution as follows (Schnoor, 1996):C = C o exp – 1 τ + k t Cin 1 1 – exp – 1 τ + k t (6.39)+ kτIt is apparent that, by setting t to infinity, the first term in the right-hand sideof Equation (6.39) dies off to zero, and the second term approaches the steadystate value given by Equation (6.38).The above equations can be applied to model the fate and transport of severalwater quality parameters such as pathogens, BOD, dissolved oxygen,nutrients, organic chemicals, metals, etc. Once the water column concentrationsof these are established from the above equations, their impacts on othernatural compartments of the lake systems such as suspended solids, biota,fish, sediments, etc., can also be analyzed. The above results can also beapplied to simulate lakes in series, such as the Great Lakes, and compartmentalizedlakes. Examples of such modeling are presented in the followingchapters in this book.Worked Example 6.6A lake of volume V of 3.15 × 10 9 ft 3 is receiving a flow, Q, of 100 cfs. Afertilizer has been applied to the drainage basin of this lake, resulting in aload, W, of 1080 lbs/day to the lake. The first-order decay rate K for this fertilizeris 0.23 yr –1 .(1) Determine the steady state concentration of the chemical in the lake.(2) If a ban is now applied on the application of the fertilizer, resulting in anexponential decline of its inflow to the lake, this decay can be assumedto be according to the equation W = 1080e –µt , where µ = 0.05 yr –1 and tis the time (yrs) after the ban. Develop a model to describe the concentrationchanges in the lake.© 2002 by CRC Press LLC