Modeling Tools for Environmental Engineers and Scientists

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Figure 7.8 Lake example modeled in MATLAB ® for various exponentially decaying loads.supplied in the following lines. A For loop is set up to solve the governingODE for µ ranging from 0 to 0.5, in steps of 0.1. A built-in function, ode45,is used here to solve that ODE. The custom function Lake defined in theM-File is used as one of the arguments for ode45, which identifies the ODEto be solved. The range of the independent variable over which the solutionis sought and the initial value of the dependent variable are also specified bythe other two arguments of this call. The result from ode45 is returned in theform of a matrix [t, C], which is then fed to the built-in plot function to generatethe graph of C vs. t. The text command is used to add annotations tothe plot by specifying the locations and the text to be inserted. The hold on© 2002 by CRC Press LLC
command is used to keep the plot frame the same until all the µ values havebeen evaluated.As can be seen from this example, several lines of commands have to betyped in the Command window every time the program is run. A major partof the commands shown in this example can be moved to the Lake M-File orto another M-File to make the program less tedious to maintain and run.MATLAB ® also contains a set of tools to construct a graphic user interface toimprove the interactivity.Plotting graphs in MATLAB ® also has to be scripted with specificationsfor all the plot objects, such as axis labels, annotations, etc., as in the case ofMathematica ® . In addition, unit conversion factors have to be provided by themodeler/user as MATLAB ® does not maintain dimensional homogeneity. Ascan be seen from this example, the MATLAB ® environment demands somedegree of programming skills, as does the Mathematica ® environment.7.5.5 LAKE PROBLEM MODELED IN TK SolverFigure 7.9 shows the TK Solver model for the steady state solution for thelake problem, illustrating the backsolving feature. The equation, as entered inthe Rule Sheet, is in the same form as Equation (7.1), but the C ss is entered asa known value in the Variable Sheet (= 1.63 mg/L) so as to determine theallowable waste load, W 0 . When run, TK Solver inverts the equation specifiedin the rule sheet to calculate the unknown W 0 and returns the result(= 1081) in the output column. Also of interest to note, in TK Solver, all theparameters in the right-hand side of the equation need not be known to solvethe model—this is not possible with the other packages.TK Solver also has the ability to maintain dimensional homogeneity andto perform calculations in a consistent set of units to provide the result in thecorrect unit; however, this feature is not fully automatic as in the case ofMathcad ® . The developer has to incorporate a unit conversion sheet specifyingthe conversion factors between the units in which the calculations aredone and the units in which the variables are displayed in the output sheet.This feature is also illustrated in Figure 7.9, where the reaction rate constantis entered in day –1 instead of in yr –1 ; TK Solver automatically calculates anddisplays the same numerical value as before for the waste load in the unitsspecified in the output sheet.In contrast to Mathcad ® , in TK Solver, the various components of themodel, such as inputs, equations, and plots, etc., are displayed in separatewindows. Also, the model is not live as in the other packages. After makingany changes in the model parameters, the user has to use the menu commandSolve to recalculate the results or update the plots. TK Solver includes aMathLook window, which reformats the line form of the equation entered intothe Rule sheet, into a standard mathematical form for checking complexequations or for presentation purposes.© 2002 by CRC Press LLC
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© 2002 by CRC Press LLCModeling To
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ContentsPrefaceAcknowledgmentsFUNDA
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APPLICATIONS8. MODELING OF ENGINEER
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The contents of this book are organ
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PART IFundamentals© 2002 by CRC Pr
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The models resulting from the model
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within the grasp of only a few with
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1.2.3 STATIC VS. DYNAMICWhen a syst
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Real SystemsMathematical ModelsDete
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through several pathways. Consequen
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ange of complex environmental appli
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APPENDIX 1.1 TYPICAL USES OF MATHEM
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APPENDIX 1.2 (continued)Target Audi
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characteristics. The system is isol
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formation, inferencing, testing, va
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system-surroundings interactions ca
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2.2.4 INTERPRETATION AND EVALUATION
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to fill in where scientific theorie
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Figure 2.3 Schematic of real system
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Table 2.1 Variables, Symbols, Dimen
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∴Net diffusive inflow = -EA ∂
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in terms of the model parameters al
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Figure 2.8 Model predictions vs. me
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variables. Deterministic systems ca
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e possible, and a computational met
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or, in matrix forma 11 a 12 a 13a 2
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(a)(b)Figure 3.3 (a) Setting up Sol
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Figure 3.5 Using MATLAB ® for solv
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Another method, known as the Newton
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and so on up tof n (x 1 , x 2 , . .
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c. Second-order equation with equid
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Figure 3.9 Solution of ODE by Mathe
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over the entire step, which may be
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Hence, the original problem is now
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2∂ f f i1,j1 - f i-1,j1 - f i1,j-
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modeling, in diagnosis and troubles
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For the reach x ≥ a: C S D K
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contaminants that do not undergo an
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known as an intensive property. Oth
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Dissolved mass in sample = dissolve
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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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Page 145 and 146: These functions are valuable tools
Page 147 and 148: this problem. Once the basic “syn
Page 149 and 150: giving the condition Q > 4πRu for
Page 151 and 152: directions, and the last term repre
Page 153 and 154: SolutionTo be conservative, it may
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Page 157 and 158: 6.2.5 FLOW OF AIR AND CONTAMINANTS
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Page 161 and 162: Solution(1) The steady state concen
Page 163 and 164: As a first step in modeling a river
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Page 167 and 168: © 2002 by CRC Press LLCFigure 6.8
Page 169 and 170: Figure 6.106.5 Consider the stream
Page 171 and 172: Case 4: Pulse Source in Two Dimensi
Page 173 and 174: End users often adapted these model
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Page 185 and 186: By combining the above simple funct
Page 187 and 188: Concentration [mg/L]Figure 7.3 Lake
Page 189 and 190: saving considerable model building
Page 191: dependent variable, the dependent v
Page 195 and 196: Figure 7.10 Lake problem solved wit
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Page 199 and 200: Figure 7.13 Lake problem modeled in
Page 201 and 202: It is hoped that the above overview
Page 203 and 204: APPENDIX 7.2 EXAMPLES OF TYPES OF E
Page 205 and 206: CHAPTER 8Modeling of EngineeredEnvi
Page 207 and 208: MB on dissolved oxygen: dC od xyt=
Page 211 and 212: Table 8.2 SBR Model Equations Gener
Page 213 and 214: Figure 8.4 Predicted vs. measured C
Page 215 and 216: mathematical model. First, a proces
Page 217 and 218: Figure 8.9 Pretreatment system—ef
Page 219 and 220: Figure 8.11 CMFRs in series: optimi
Page 221 and 222: Other improvements can include alte
Page 223 and 224: Figure 8.15 Model of wastewater tre
Page 225 and 226: 8.5 MODELING EXAMPLE: CHEMICAL OXID
Page 227 and 228: When Mathematica ® cannot find ana
Page 229 and 230: Figure 8.20 Chemical oxidation proc
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Page 241 and 242: 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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© 2002 by CRC Press LLCFigure 9.20
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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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