Modeling Tools for Environmental Engineers and Scientists Episode 3 pot

Before con-tinuing on to the topic of developing mathematical models, it is necessary to formalize certain terminology, definitions, and conventions relating to the modeling process.. 2.

CHAPTER 2

Fundamentals of Mathematical

Modeling

CHAPTER PREVIEW

In this chapter, formal definitions and terminology relating to mathe-matical modeling are presented The key steps involved in developing mathematical models are identified, and the tasks to be completed under each step are detailed While the suggested procedure is not a standard one, it includes the crucial components to be addressed in the process The application of these steps in developing a mathematical model for a typical environmental system is illustrated.

2.1 DEFINITIONS AND TERMINOLOGY IN

MATHEMATICAL MODELING

GENERALbackground information on models was presented in Chapter 1, where certain terms were introduced in a general manner Before con-tinuing on to the topic of developing mathematical models, it is necessary to formalize certain terminology, definitions, and conventions relating to the modeling process Recognition of these formalities can greatly help in the selection of the modeling approach, data needs, theoretical constructs, math-ematical tools, solution procedures, and, hence, the appropriate computer software package(s) to complete the modeling task In the following sections, the language in mathematical modeling is clarified in the context of model-ing of environmental systems

2.1.1 SYSTEM/BOUNDARY

A “system” can be thought of as a collection of one or more related objects, where an “object” can be a physical entity with specific attributes or

characteristics The system is isolated from its surroundings by the “bound-ary,” which can be physical or imaginary (In many books on modeling, the term “environment” is used instead of “surroundings” to indicate everything outside the boundary; the reason for picking the latter is to avoid the confu-sion in the context of this book that focuses on modeling the environment In other words, environment is the system we are interested in modeling, which

is enclosed by the boundary.) The objects within a system may or may not interact with each other and may or may not interact with objects in the sur-roundings, outside the boundary A system is characterized by the fact that the modeler can define its boundaries, its attributes, and its interactions with the surroundings to the extent that the resulting model can satisfy the mod-eler’s goals

The largest possible system of all, of course, is the universe One can, depending on the modeling goals, isolate a part of the universe such as a con-tinent, or a country, or a city, or the city’s wastewater treatment plant, or the aeration tank of the city’s wastewater treatment plant, or the microbial popu-lation in the aeration tank, and define that as a system for modeling purposes Often, the larger the system, the more complex the model However, the effort can be made more manageable by dissecting the system into smaller subsys-tems and including the interactions between them

2.1.2 OPEN/CLOSED, FLOW/NONFLOW SYSTEMS

A system is called a closed system when it does not interact with the sur-roundings If it interacts with the surroundings, it is called an open system In closed systems, therefore, neither mass nor energy will cross the boundary; whereas in open systems, mass and energy can When mass does not cross the boundary (but energy does), an open system may be categorized as a nonflow system If mass crosses the boundary, it is called a flow system

While certain batch processes may be approximated as closed systems, most environmental systems interact with the surroundings in one way or another, with mass flow across the boundary Thus, most environmental sys-tems have to be treated as open, flow syssys-tems

2.1.3 VARIABLES/PARAMETERS/INPUTS/OUTPUTS

The attributes of the system and of the surroundings that have significant impact on the system are termed “variables.” The term variable includes those attributes that change in value during the modeling time span and those that remain constant during that period Variables of the latter type are often referred to as parameters Some parameters may relate to the system, and some may relate to the surroundings

A system may have numerous attributes or variables However, as men-tioned before, the modeler needs to select only those that are significant and relevant to the modeler’s goal in the modeling process For example, in the case of the aeration tank, its attributes can include biomass characteristics, vol-ume of mixed liquor, its color, temperature, viscosity, specific weight, con-ductivity, reflectivity, etc., and the attributes of the surroundings may be flow rate, mass input, wind velocity, solar radiation, etc Even though many of the attributes may be interacting, only a few (e.g., biomass characteristics, vol-ume, flow rate, mass input) are identified as variables of significance and rel-evance based on the modeler’s goals (e.g., the efficiency of the aeration tank) Variables that change in value fall into two categories: those that are gen-erated by the surroundings and influence the behavior of the system, and those that are generated by the system and impact the surroundings The for-mer are called “inputs,” and the latter are called “outputs.” In the case of the aeration tank, the mass inflow can be an input, the concentration leaving the tank, an output, and the volume of the tank, a parameter In mathematical language, inputs are considered independent variables, and outputs are con-sidered dependent variables The inputs and model parameters are often known or defined in advance; they drive the model to produce some output

In the context of modeling, relationships are sought between inputs and out-puts, with the parameters acting as model coefficients

At this point, a very important factor has to be recognized; in the real sys-tem, not all significant and relevant variables and/or parameters may be accessible for control or manipulation; likewise, not all outputs may be acces-sible for observation or measurement However, in mathematical models, all inputs and parameters are readily available for control or manipulation, and all outputs are accessible It also follows that, in mathematical modeling, modelers can suppress “disturbances” that are unavoidable in the real sys-tems These traits are of significant value in mathematical modeling

However, numerical values for the variables will be needed to execute the model Some values are set by the modeler as inputs Other system parame-ter data can be obtained from many sources, such as the scientific liparame-terature, experimentation on the real system or physical models, or by adapting esti-mation methods Accounts of experimentation techniques and parameter estimation methods for determining such data can be found elsewhere and are beyond the scope of this book

2.2 STEPS IN DEVELOPING MATHEMATICAL MODELS

The craft of mathematical model development is part science and part art

It is a multistep, iterative, trial-and-error process cycling through hypotheses

formation, inferencing, testing, validating, and refining It is common prac-tice to start from a simple model and develop it in steps of increasing com-plexity, until it is capable of replicating the observed or anticipated behavior

of the real system to the extent that the modeler expects It has to be kept in mind that all models need not be perfect replicates of the real system If all the details of the real system are included, the model can become unmanage-able and be of very limited use On the other hand, if significant and relevant details are omitted, the model will be incomplete and again be of limited use While the scientific side of modeling involves the integration of knowledge

to build and solve the model, the artistic side involves the making of a sensi-ble compromise and creating balance between two conflicting features of the model: degree of detail, complexity, and realism on one hand, and the valid-ity and utilvalid-ity value of the final model on the other

The overall approach in mathematical modeling is illustrated in Figure 2.1 Needless to say, each of these steps involves more detailed work and, as men-tioned earlier, will include feedback, iteration, and refinement In the follow-ing sections, a logical approach to the model development process is presented, identifying the various tasks involved in each of the steps It is not the intention here to propose this as the standard procedure for every modeler

to follow in every situation; however, most of the important and crucial tasks are identified and included in the proposed procedure

Figure 2.1 Overall approach to mathematical modeling.

Problem formulation

Mathematical representation

Mathematical analysis

Interpretation and evaluation

of results

2.2.1 PROBLEM FORMULATION

As in any other field of scientific study, formulation of the problem is the first step in the mathematical model development process This step involves the following tasks:

Task 1: establishing the goal of the modeling effort Modeling projects

may be launched for various reasons, such as those pointed out in Chapter 1 The scope of the modeling effort will be dictated by the objective(s) and the expectation(s) Because the premise of the effort is for the model to be sim-pler than the real system and at the same time be similar to it, one of the objectives should be to establish the extent of correlation expected between model predictions and performance of the real system, which is often referred

to as performance criteria This is highly system specific and will also depend

on the available resources such as the current knowledge about the system and the tools available for completing the modeling process

It should also be noted that the same system might require different types

of models depending on the goal(s) For example, consider a lake into which

a pollutant is being discharged, where it undergoes a decay process at a rate estimated from empirical methods If it is desired to determine the long-term concentration of the pollutant in the lake or to do a sensitivity study on the estimated decay rate, a simple static model will suffice On the other hand, if

it is desired to trace the temporal concentration profile due to a partial shut-down of the discharge into the lake, a dynamic model would be required If toxicity of the pollutant is a key issue and, hence, if peak concentrations due

to inflow fluctuations are to be predicted, then a probabilistic approach may have to be adapted

Another consideration at this point would be to evaluate other preexisting

“canned” models relating to the project at hand They are advantageous because many would have been validated and/or accepted by regulators Often, such models may not be applicable to the current problem with or without minor modifications due to the underlying assumptions about the sys-tem, the contaminants, the processes, the interactions, and other concerns However, they can be valuable in guiding the modeler in developing a new model from the basics

Task 2: characterizing the system In terms of the definitions presented

earlier, characterizing the system implies identifying and defining the system, its boundaries, and the significant and relevant variables and parameters The modeler should be able to establish how, when, where, and at what rate the system interacts with its surroundings; namely, provide data about the inflow rates and the outflow rates Processes and reactions occurring inside the sys-tem boundary should also be identified and quantified

Often, creating a schematic, graphic, or pictographic model of the system (a two-dimensional model) to visualize and identify the boundary and the

system-surroundings interactions can be a valuable aid in developing the mathematical model These aids may be called conceptual models and can include the model variables, such as the directions and the rates of flows crossing the boundary, and parameters such as reaction and process rates inside the system Jorgensen (1994) presented a comprehensive summary of

10 different types of such tools, giving examples and summarizing their char-acteristics, advantages, and disadvantages

Some of the recent software packages (to be illustrated later) have taken this idea to new heights by devising the diagrams to be “live.” For example,

in a simple block diagram, the boxes with interconnecting arrows can be encoded to act as reservoirs, with built-in mass balance equations With the passage of time, these blocks can “execute” the mass balance equation and can even animate the amount of material inside the box as a function of time Examples of such diagrams can be found throughout this book

Another very useful and important part of this task is to prepare a list of

all of the variables along with their fundamental dimensions (i.e., M, L, T )

and the corresponding system of units to be used in the project This can help

in checking the consistency among variables and among equations, in trou-bleshooting, and in determining the appropriateness of the results

Task 3: simplifying and idealizing the system Based on the goals of the

modeling effort, the system characteristics, and available resources, appropri-ate assumptions and approximations have to be made to simplify the system, making it amenable to modeling within the available resources Again, the primary goal is to be able to replicate or reproduce significant behaviors of the real system This involves much experience and professional judgement and an overall appreciation of the efforts involved in modeling from start to finish

For example, if the processes taking place in the system can be approxi-mated as first-order processes, the resulting equations and the solution proce-dures can be considerably simpler Similar benefits can be gained by making assumptions: using average values instead of time-dependent values, using estimated values rather than measured ones, using analytical approaches rather than numerical or probability-based analysis, considering equilibrium vs non-equilibrium conditions, and using linear vs nonlinear processes

2.2.2 MATHEMATICAL REPRESENTATION

This is the most crucial step in the process, requiring in-depth subject mat-ter expertise This step involves the following tasks:

Task 1: identifying fundamental theories Fundamental theories and

princi-ples that are known to be applicable to the system and that can help achieve the goal have to be identified If they are lacking, ad hoc or empirical relationships

may have to be included Examples of fundamental theories and principles include stoichiometry, conservation of mass, reaction theory, reactor theory, and transport mechanisms A review of theories of environmental processes

is included in Chapter 4 A review of engineered environmental systems is presented in Chapter 5 And, a review of natural environmental systems

is presented in Chapter 6

Task 2: deriving relationships The next step is to apply and integrate the

theories and principles to derive relationships between the variables of sig-nificance and relevance This essentially transforms the real system into a mathematical representation Several examples of derivations are included in the following chapters

Task 3: standardizing relationships Once the relationships are derived, the

next step is to reduce them to standard mathematical forms to take advantage

of existing mathematical analyses for the standard mathematical formulations This is normally done through standard mathematical manipulations, such as simplifying, transforming, normalizing, or forming dimensionless groups The advantage of standardizing has been referred to earlier in Chapter 1 with Equations (1.1) and (1.2) as examples Once the calculus that applies to the system has been identified, the analysis then follows rather routine pro-cedures (The term calculus is used here in the most classical sense, denoting formal structure of axioms, theorems, and procedures.) Such a calculus allows deductions about any situation that satisfies the axioms Or, alterna-tively, if a model fulfills the axioms of a calculus, then the calculus can be used to predict or optimize the performance of the model Mathematicians have formalized several calculi, the most commonly used being differential and integral A review of the calculi commonly used in environmental sys-tems is included in Chapter 3, with examples throughout this book

2.2.3 MATHEMATICAL ANALYSIS

The next step of analysis involves application of standard mathematical techniques and procedures to “solve” the model to obtain the desired results The convenience of the mathematical representation is that the resulting model can be analyzed on its own, completely disregarding the real system, temporarily The analysis is done according to the rules of mathematics, and the system has nothing to do with that process (In fact, any analyst can per-form this task—subject matter expertise is not required.)

The type of analysis to be used will be dictated by the relationships derived

in the previous step Generalized analytical techniques can fall into algebraic, differential, or numerical categories A review of selected analytical tech-niques commonly used in modeling of environmental systems is included in Chapter 3

2.2.4 INTERPRETATION AND EVALUATION OF RESULTS

It is during this step that the iteration and model refinement process is car-ried out During the iterative process, performance of the model is compared against the real system to ensure that the objectives are satisfactorily met This process consists of two main tasks—calibration and validation

Task 1: calibrating the model Even if the fundamental theorems and

prin-ciples used to build the model described the system truthfully, its perform-ance might deviate from the real system because of the inherent assumptions and simplifications made in Task 3, Section 2.2.1 and the assumptions made

in the mathematical analysis These deviations can be minimized by calibrat-ing the model to more closely match the real system

In the calibration process, previously observed data from the real system are used as a “training” set The model is run repeatedly, adjusting the model parameters by trial and error (within reasonable ranges) until its predictions under similar conditions match the training data set as per the goals and per-formance criteria established in Section 2.2.1 If not for computer-based mod-eling, this process could be laborious and frustrating, especially if the model includes several parameters

An efficient way to calibrate a model is to perform preliminary sensitivity analysis on model outputs to each parameter, one by one This can identify the parameters that are most sensitive, so that time and other resources can be allocated to those parameters in the calibration process Some modern com-puter modeling software packages have sensitivity analysis as a built-in fea-ture, which can further accelerate this step

If the model cannot be calibrated to be within acceptable limits, the mod-eler should backtrack and reevaluate the system characterization and/or the model formulation steps Fundamental theorems and principles as well as the model formulation and their applicability to the system may have to be reex-amined, assumptions may have to be checked, and variables may have to be evaluated and modified, if necessary This iterative exercise is critical in establishing the utility value of the model and the validity of its applications, such as in making predictions for the future

Task 2: validating the model Unless a model is well calibrated and

vali-dated, its acceptability will remain limited and questionable There are no standard benchmarks for demonstrating the validity of models, because mod-els have to be linked to the systems that they are designed to represent Preliminary, informal validation of model performance can be conducted relatively easily and cost-effectively One way of checking overall perform-ance is to ensure that mass balperform-ance is maintained through each of the model runs Another approach is to set some of the parameters so that a closed alge-braic solution could be obtained by hand calculation; then, the model outputs can be compared against the hand calculations for consistency For example,

by setting the reaction rate constant of a contaminant to zero, the model may be easier to solve algebraically and the output may be more easily compared with the case of a conservative substance, which may be readily obtained Other informal validation tests can include running the model under a wide range of parameters, input variables, boundary conditions, and initial values and then plotting the model outputs as a function of space or time for visual interpre-tation and comparison with intuition, expecinterpre-tations, or similar case studies For formal validation, a “testing” data set from the real system, either his-toric or generated expressly for validating the model, can be used as a bench-mark The calibrated model is run under conditions similar to those of the testing set, and the results are compared against the testing set A model can

be considered valid if the agreement between the two under various condi-tions meets the goal and performance criteria set forth in Section 2.2.1 An important point to note is that the testing set should be completely independ-ent of, and differindepend-ent from, the training set

A common practice used to demonstrate validity is to generate a parity plot

of predicted vs observed data with associated statistics such as goodness of fit Another method is to compare the plots of predicted values and observed data as a function of distance (in spatially varying systems) or of time (in tem-porally varying systems) and analyze the deviations For example, the num-ber of turning points in the plots and maxima and/or minima of the plots and the locations or times at which they occur in the two plots can be used as com-parison criteria Or, overall estimates of absolute error or relative error over a range of distance or time may be quantified and used as validation criterion

Murthy et al (1990) have suggested an index J to quantify overall error in

dynamic, deterministic models relative to the real system under the same

input u(t) over a period of time T They suggest using the absolute error or the relative error to determine J, calculated as follows:

J = T

o

e(t) T e(t)dt or J = T

o

˜e(t) T ˜e(t)dt

where e(t) = y s (t) – y m (t) or ˜e(t) = 

y

e

s

( (

t t

) )



y s (t) = output observed from the real system as a function of time, t

y m (t) = output predicted by the model as a function of time, t

2.2.5 SUMMARY OF THE MATHEMATICAL MODEL

DEVELOPMENT PROCESS

In Chapter 1, physical modeling, empirical modeling, and mathematical modeling were alluded to as three approaches to modeling However, as could

be gathered from the above, they complement each other and are applied together in practice to complete the modeling task Empirical models are used

to fill in where scientific theories are nonexistent or too complex (e.g., non-linear) Experimental or physical model results are used to develop empirical models and calibrate and validate mathematical models

The steps and tasks described above are summarized schematically in Figure 2.2 This scheme illustrates the feedback and iterative nature of the process as described earlier It also shows how the real system and the

“abstract” mathematical system interact and how experimentation with the real system and/or physical models is integrated with the modeling process It is hoped that the above sections accented the science as well as the art in the craft of mathematical model building

Figure 2.2 Steps in mathematical modeling.