Modeling Tools for Environmental Engineers and Scientists

esponses are additive in their effects, i.e., the output is directly proportionalto the input, and outputs satisfy the principle of superpositioning. Forinstance, if an input I 1 to a system produces an output O 1 , and another inputI 2 produces an output of O 2 , then a combined input of (αI 1 + βI 2 ) will producean output of (αO 1 + βO 2 ). Superpositioning cannot be applied in nonlinearmodels.In the lake example, if the reactions undergone by the pollutant in the lakeare assumed to be of first order, for instance, then the linearity of the resultingmodel allows superpositioning to be applied. Suppose the input to the lakeis changed from a steady state condition, then the response of the lake can befound by adding the response following the general solution (due to the initialconditions) to the response following the particular solution (due tothe input change) of the differential equation governing the system.1.2.6 ANALYTICAL VS. NUMERICALWhen all the equations in a model can be solved algebraically to yield asolution in a closed form, the model can be classified as analytical. If that isnot possible, and a numerical procedure is required to solve one or more ofthe model equations, the model is classified as numerical.In the above example of the lake, if the entire volume of the lake isassumed to be completely mixed, a simple analytical model may be developedto model its steady state condition. However, if such an assumptionis unacceptable, and if the lake has to be compartmentalized into severallayers and segments for detailed study, a numerical modeling approach hasto be followed.A comparison of the above classifications is summarized in Figure 1.1.Indicated at the bottom section of this figure are the common mathematicalanalytical methods appropriate for each type of model. These classificationsare presented here to stress the necessity of understanding input data requirements,model formulation, and solution procedures, and to guide in the selectionof the appropriate computer software tool in modeling the system. Mostenvironmental systems can be approximated in a satisfactory manner by linearand time variant descriptions in a lumped or distributed manner, at leastfor specified and restricted conditions. Analytical solutions are possible forlimited types of systems, while solutions may be elaborate or not currentlyavailable for many others. Computer-based mathematical modeling usingnumerical solutions can provide valuable insight in such cases.The goal of this book is to illustrate, with examples, the application of avariety of software packages in developing computer-based mathematicalmodels in the environmental field. The examples included in the book fallinto the following categories: static, dynamic, continuous, deterministic(probabilistic, at times), analytical, numerical, and linear.© 2002 by CRC Press LLC