Modeling Tools for Environmental Engineers and Scientists

where K is the sum of the first-order rate constants for all the consumptivereactions. The solution to the above ordinary differential equation (ODE)can yield the concentration of the pollutant in the lake, C, as a function oftime under predefined initial conditions, C 0 , and forcing waste load functions,W (t) . The forcing functions can be some function of time, continuousor discontinuous. Obviously, the nature of W(t) should be known beforeattempting to solve this ODE. A general solution to the above can be foundfrom the following:where tC = e– W (t) e t dt C 0 e – t (7.2)V = K Q V K + 1 τ (7.3)Obviously, the result will depend on the nature of the forcing function, W (t) .The simplest case is the steady state solution, with C 0 = 0, under a constantforcing function, W(t) = W 0 , when the result can be found to be the following:W0WC ss = 0 (7.4)V • V • K Qwhere C ss is the concentration of the pollutant in the lake at steady state.Because the explicit solution for the problem as described by the generalequation [Equation (7.1)] will depend on the initial conditions and the forcingfunction, it is desirable to develop simulation models that can solve thegoverning equations in a general manner rather than in a problem-specificmanner. Some common types of W (t) functions in this problem are as follows:• constant loading:W (t) = W 0• linearly increasing or decreasing load:W (t) = W 0 ± at• exponentially increasing or decreasing load:W (t) = W 0 e ±λt• step increase or decrease from a background load of W 0 :W (t) = W 0 for t < 0 and W (t) = W 0 ± W for t < 0• impulse load of mass, m:W (t) = mδ(t), where δ(t) = 0 for t ≠ 0 and ∞ –∞ δ(t)dt = 1• sinusoidal loading:W (t) = W ave + W 0 sin (ωt – θ)© 2002 by CRC Press LLC