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SolutionThe potential function for the combined flow field is found by superposition: = –ux +R2 x Q – x2 y 2 2 π ln [x 2 y2 ]The contours of constant velocity potentials can be readily constructed withMathematica ® as shown in Figure 6.4. The plot shows that the stagnationpoint is upstream of the pond, implying that water is exiting the upstreamboundary of the pond. This can be verified analytically by determining thevelocity u at (–R,0) and checking if it is less than zero:u (–R,0) = – ∂ ∂x u 1 – 2R 2R 2 2x(–R,0) x2 y 2 (x2 y2 ) Qx 2 2π(x 2 y 2 ) (–R,0)The above reduces to:u (–R,0) = 2u – 2QπR Figure 6.4 Contours of potential function at q = 300.© 2002 by CRC Press LLC
giving the condition Q > 4πRu for flow to occur from the pond through itsupstream boundary. In the above example, this condition is satisfied. The conditionof Q < 4πRu can be readily evaluated by decreasing Q, for example,from 1000π to 600π, and plotting the potential lines as shown in Figure 6.5.The Mathematica ® script can be easily adapted to superimpose the streamfunction on the velocity potential function as shown in Figure 6.6, for the twocases, to illustrate the orthogonality and to describe the flow pattern completely.6.2.3 FLOW OF WATER AND CONTAMINANTS THROUGHTHE SATURATED ZONEPrinciples of groundwater flow and process fundamentals have to be integratedto model the fate and transport of contaminants in the saturated zone.Both advective and dispersive transport of the contaminant have to beincluded in the contaminant transport model, as well as the physical, biological,and chemical reactions.As an initial step in the analysis of contaminant transport, the groundwaterflow velocity components u, v, and w (LT –1 ), and the dispersion coefficients,E i (L 2 T –1 ), in the direction i, can be assumed to be constant withFigure 6.5 Contours of potential function at q = 300.© 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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Page 99 and 100: in the air (M/L -3 ), C a is the co
Page 101 and 102: ackward reactions can expressed in
Page 103 and 104: 4.7.2 ELEMENTARY REACTIONSThe rate
Page 105 and 106: excess energy. The direct photolysi
Page 107 and 108: 4.8 MATERIAL BALANCEAs indicated in
Page 109 and 110: SolutionLet M be the mass of chemic
Page 111 and 112: CpHence, the final result isEXERCIS
Page 113 and 114: APPENDIX 4.1 COMMON PARTITION COEFF
Page 115 and 116: analyzing and modeling them. The ex
Page 117 and 118: \5.2.2 HETEROGENEOUS REACTORSHetero
Page 119 and 120: CC = in -(t t 0 )k C t in0 k- C
Page 121 and 122: Worked Example 5.1A wastewater trea
Page 123 and 124: or,C L = C 0 e -(k /u)L = C 0 e -k
Page 125 and 126: HRT = 1 k (1 - )The overall HRT
Page 127 and 128: The MB equation for the above syste
Page 129 and 130: the particle surface (ML -2 T -1 ),
Page 131 and 132: 0 = QX - Q(X - dX) - K L (aAdz)(X -
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Page 135 and 136: This expression does not provide an
Page 137 and 138: If the WWTP used air instead of pur
Page 139 and 140: and degradation of the natural envi
Page 141 and 142: The solution to the above PDE gives
Page 143 and 144: • river 2: x = 15000.5 dmay ∴u
Page 145 and 146: These functions are valuable tools
Page 147: this problem. Once the basic “syn
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Page 153 and 154: SolutionTo be conservative, it may
Page 155 and 156: zone, because the hydraulic conduct
Page 157 and 158: 6.2.5 FLOW OF AIR AND CONTAMINANTS
Page 159 and 160: time-dependent volumetric flow rate
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 177 and 178: are always expressed in the standar
Page 179 and 180: know the program’s environment. M
Page 181 and 182: typing in the equations in algebrai
Page 183 and 184: accessed in Simulink ® models. The
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 and 192: dependent variable, the dependent v
Page 193 and 194: command is used to keep the plot fr
Page 195 and 196: Figure 7.10 Lake problem solved wit
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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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© 2002 by CRC Press LLCFigure 8.1
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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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