Modeling Tools for Environmental Engineers and Scientists

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may have to be included. Examples of fundamental theories and principlesinclude stoichiometry, conservation of mass, reaction theory, reactor theory,and transport mechanisms. A review of theories of environmental processesis included in Chapter 4. A review of engineered environmental systems ispresented in Chapter 5. And, a review of natural environmental systemsis presented in Chapter 6.Task 2: deriving relationships. The next step is to apply and integrate thetheories and principles to derive relationships between the variables of significanceand relevance. This essentially transforms the real system into amathematical representation. Several examples of derivations are included inthe following chapters.Task 3: standardizing relationships. Once the relationships are derived, thenext step is to reduce them to standard mathematical forms to take advantageof existing mathematical analyses for the standard mathematical formulations.This is normally done through standard mathematical manipulations, such assimplifying, transforming, normalizing, or forming dimensionless groups.The advantage of standardizing has been referred to earlier in Chapter 1with Equations (1.1) and (1.2) as examples. Once the calculus that applies tothe system has been identified, the analysis then follows rather routine procedures.(The term calculus is used here in the most classical sense, denotingformal structure of axioms, theorems, and procedures.) Such a calculusallows deductions about any situation that satisfies the axioms. Or, alternatively,if a model fulfills the axioms of a calculus, then the calculus can beused to predict or optimize the performance of the model. Mathematicianshave formalized several calculi, the most commonly used being differentialand integral. A review of the calculi commonly used in environmental systemsis included in Chapter 3, with examples throughout this book.2.2.3 MATHEMATICAL ANALYSISThe next step of analysis involves application of standard mathematicaltechniques and procedures to “solve” the model to obtain the desired results.The convenience of the mathematical representation is that the resultingmodel can be analyzed on its own, completely disregarding the real system,temporarily. The analysis is done according to the rules of mathematics, andthe system has nothing to do with that process. (In fact, any analyst can performthis task—subject matter expertise is not required.)The type of analysis to be used will be dictated by the relationships derivedin the previous step. Generalized analytical techniques can fall into algebraic,differential, or numerical categories. A review of selected analytical techniquescommonly used in modeling of environmental systems is included inChapter 3.© 2002 by CRC Press LLC
2.2.4 INTERPRETATION AND EVALUATION OF RESULTSIt is during this step that the iteration and model refinement process is carriedout. During the iterative process, performance of the model is comparedagainst 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 principlesused to build the model described the system truthfully, its performancemight deviate from the real system because of the inherent assumptionsand simplifications made in Task 3, Section 2.2.1 and the assumptions madein the mathematical analysis. These deviations can be minimized by calibratingthe model to more closely match the real system.In the calibration process, previously observed data from the real systemare used as a “training” set. The model is run repeatedly, adjusting the modelparameters by trial and error (within reasonable ranges) until its predictionsunder similar conditions match the training data set as per the goals and performancecriteria established in Section 2.2.1. If not for computer-based modeling,this process could be laborious and frustrating, especially if the modelincludes several parameters.An efficient way to calibrate a model is to perform preliminary sensitivityanalysis on model outputs to each parameter, one by one. This can identifythe parameters that are most sensitive, so that time and other resources can beallocated to those parameters in the calibration process. Some modern computermodeling software packages have sensitivity analysis as a built-in feature,which can further accelerate this step.If the model cannot be calibrated to be within acceptable limits, the modelershould backtrack and reevaluate the system characterization and/or themodel formulation steps. Fundamental theorems and principles as well as themodel formulation and their applicability to the system may have to be reexamined,assumptions may have to be checked, and variables may have to beevaluated and modified, if necessary. This iterative exercise is critical inestablishing 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 validated,its acceptability will remain limited and questionable. There are nostandard benchmarks for demonstrating the validity of models, because modelshave to be linked to the systems that they are designed to represent.Preliminary, informal validation of model performance can be conductedrelatively easily and cost-effectively. One way of checking overall performanceis to ensure that mass balance is maintained through each of the modelruns. Another approach is to set some of the parameters so that a closed algebraicsolution could be obtained by hand calculation; then, the model outputscan be compared against the hand calculations for consistency. For example,© 2002 by CRC Press LLC
Page 2: © 2002 by CRC Press LLCModeling To
Page 5 and 6: ContentsPrefaceAcknowledgmentsFUNDA
Page 7 and 8: APPLICATIONS8. MODELING OF ENGINEER
Page 9 and 10: The contents of this book are organ
Page 11 and 12: PART IFundamentals© 2002 by CRC Pr
Page 13 and 14: The models resulting from the model
Page 15 and 16: within the grasp of only a few with
Page 17 and 18: 1.2.3 STATIC VS. DYNAMICWhen a syst
Page 19 and 20: Real SystemsMathematical ModelsDete
Page 21 and 22: through several pathways. Consequen
Page 23 and 24: ange of complex environmental appli
Page 25 and 26: APPENDIX 1.1 TYPICAL USES OF MATHEM
Page 27 and 28: APPENDIX 1.2 (continued)Target Audi
Page 29 and 30: characteristics. The system is isol
Page 31 and 32: formation, inferencing, testing, va
Page 33: system-surroundings interactions ca
Page 37 and 38: to fill in where scientific theorie
Page 39 and 40: Figure 2.3 Schematic of real system
Page 41 and 42: Table 2.1 Variables, Symbols, Dimen
Page 43 and 44: ∴Net diffusive inflow = -EA ∂
Page 45 and 46: in terms of the model parameters al
Page 47 and 48: Figure 2.8 Model predictions vs. me
Page 49 and 50: variables. Deterministic systems ca
Page 51 and 52: e possible, and a computational met
Page 53 and 54: or, in matrix forma 11 a 12 a 13a 2
Page 55 and 56: (a)(b)Figure 3.3 (a) Setting up Sol
Page 57 and 58: Figure 3.5 Using MATLAB ® for solv
Page 59 and 60: Another method, known as the Newton
Page 61 and 62: and so on up tof n (x 1 , x 2 , . .
Page 63 and 64: c. Second-order equation with equid
Page 65 and 66: Figure 3.9 Solution of ODE by Mathe
Page 67 and 68: over the entire step, which may be
Page 69 and 70: Hence, the original problem is now
Page 71 and 72: 2∂ f f i1,j1 - f i-1,j1 - f i1,j-
Page 73 and 74: modeling, in diagnosis and troubles
Page 75 and 76: For the reach x ≥ a: C S D K
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Page 79 and 80: known as an intensive property. Oth
Page 81 and 82: Dissolved mass in sample = dissolve
Page 83 and 84: Figure 4.2 Illustration of steady s
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4.3.2.2 Dalton’s LawDalton’s La
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Table 4.2 Different Forms of Quanti
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M olesoxygenHence, K a-w = = 8 - 3
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SolutionThe flux, N, which is the d
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Figure 4.3 Illustration of the Two-
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where C s is the concentration of t
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Fraction in dissolved form = f d =
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in the air (M/L -3 ), C a is the co
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ackward reactions can expressed in
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4.7.2 ELEMENTARY REACTIONSThe rate
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excess energy. The direct photolysi
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4.8 MATERIAL BALANCEAs indicated in
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SolutionLet M be the mass of chemic
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CpHence, the final result isEXERCIS
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APPENDIX 4.1 COMMON PARTITION COEFF
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analyzing and modeling them. The ex
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\5.2.2 HETEROGENEOUS REACTORSHetero
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CC = in -(t t 0 )k C t in0 k- C
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Worked Example 5.1A wastewater trea
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or,C L = C 0 e -(k /u)L = C 0 e -k
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HRT = 1 k (1 - )The overall HRT
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The MB equation for the above syste
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the particle surface (ML -2 T -1 ),
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0 = QX - Q(X - dX) - K L (aAdz)(X -
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chemical engineering applications.
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This expression does not provide an
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If the WWTP used air instead of pur
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and degradation of the natural envi
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The solution to the above PDE gives
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• river 2: x = 15000.5 dmay ∴u
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These functions are valuable tools
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this problem. Once the basic “syn
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giving the condition Q > 4πRu for
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directions, and the last term repre
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SolutionTo be conservative, it may
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zone, because the hydraulic conduct
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6.2.5 FLOW OF AIR AND CONTAMINANTS
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time-dependent volumetric flow rate
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Solution(1) The steady state concen
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As a first step in modeling a river
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where a 1,4 is the conversion facto
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© 2002 by CRC Press LLCFigure 6.8
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Figure 6.106.5 Consider the stream
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Case 4: Pulse Source in Two Dimensi
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End users often adapted these model
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to further enhance the capabilities
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are always expressed in the standar
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know the program’s environment. M
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typing in the equations in algebrai
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accessed in Simulink ® models. The
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By combining the above simple funct
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Concentration [mg/L]Figure 7.3 Lake
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saving considerable model building
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dependent variable, the dependent v
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command is used to keep the plot fr
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Figure 7.10 Lake problem solved wit
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simulation. This enables, for examp
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Figure 7.13 Lake problem modeled in
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It is hoped that the above overview
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APPENDIX 7.2 EXAMPLES OF TYPES OF E
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CHAPTER 8Modeling of EngineeredEnvi
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MB on dissolved oxygen: dC od xyt=
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Table 8.2 SBR Model Equations Gener
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Figure 8.4 Predicted vs. measured C
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mathematical model. First, a proces
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Figure 8.9 Pretreatment system—ef
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Figure 8.11 CMFRs in series: optimi
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Other improvements can include alte
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Figure 8.15 Model of wastewater tre
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8.5 MODELING EXAMPLE: CHEMICAL OXID
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When Mathematica ® cannot find ana
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Figure 8.20 Chemical oxidation proc
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fairly large and may not be suitabl
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where q is the mass of color adsorb
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Figure 8.29 Results of activated ca
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and d X = (µ - b)Xdtwhere b is the
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Figure 8.32 Results of bioregenerat
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under a set of known data, the stat
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MB across element on oxygen in gas
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Figure 8.38 Typical results of oxyg
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Figure 8.39 Mathematica ® script f
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entered as a function of x and y, u
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To generalize the approach, the gov
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Figure 8.44 Concentration in pore g
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modeling of a sample problem from T
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Figure 9.2 Lakes in series modeled
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Figure 9.4 Two lakes in series mode
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In this example, a two-compartment
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in the sediments, v b is the burial
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Figure 9.9 Algal growth modeled in
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Figure 9.11 Algal growth modeled in
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Once the basic model is constructed
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Figure 9.15 Graphical user interfac
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Figure 9.17 Contaminant transport v
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Figure 9.19 Contaminant transport v
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Figure 9.21 Chemical equilibrium mo
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MB on liver:MB on bones: d P2 = k 2
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Table 9.3 Model Equations in ithink
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Figure 9.25 Problem definition for
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Figure 9.28 Groundwater flow visual
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W and the second to V, in line 6. T
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Figure 9.30 Three-dimensional conto
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VerticalDispersion CoefficientsHori
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configuration of the unit world pro
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342Chemical Mass Distribution [%]11
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water quality standard of 500 mg/L
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Hence, Q π tan-1 - 1 - UL - 1Q/
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Figure 9.38d Stream lines and veloc
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BibliographyBedient P. B., Rifai, H
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Nirmalakhandan, N., Jang, W., and S
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