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Figure 6.6 Contours of potential and stream functions at q = 500 and q = 300.space and time. A generalized three-dimensional (3-D) material balanceequation can then be formulated for an element to yield: ∂ C = –∂t u ∂ C v ∂ C w ∂ C∂x∂y∂z2 E x ∂ C E ∂ C2 y E ∂ C2 z2∂x∂y∂z22 ± r (6.10)where the left-hand side of the equation represents the rate of change in concentrationof the contaminant within the element, the first three terms withinthe square brackets on the right-hand side represent the advective transportacross the element in the three directions, the next three terms within the nextsquare brackets represent dispersive transport across the element in the three© 2002 by CRC Press LLC
directions, and the last term represents the sum of all physical, chemical, andbiological processes acting on the contaminant within the element.An important physical process that most organic chemicals and metalsundergo in the subsurface is adsorption onto soil, resulting in their retardationrelative to the groundwater flow. For low concentrations of contaminants, thisphenomenon can be modeled assuming a linear adsorption coefficient andcan be quantified by a retardation factor, R (–), defined as follows: bR = 1 K d (6.11)nwhere K d is the soil-water distribution coefficient (L 3 M –1 ) = S/C, n is theeffective porosity (–), ρ b is the bulk density of soil (ML –3 ) = ρ s (1 – n), S isthe sorbed concentration (MM –1 ), and ρ s is the density of soil particles(ML –3 ). Introducing the retardation factor, R, into Equation (6.10) and usingretarded velocities, u, v,and w (LT –1 ), and retarded dispersion coefficients,E i (L 2 T –1 ), results in the following: ∂ C = –∂tu ∂ C v ∂ C w ∂ C∂x∂y∂z2 E x ∂ C E y ∂ C Ez ∂ C222∂x∂y∂z22 ± R1 r (6.12)The solution of the above PDE involves numerical procedures. However,by invoking simplifying assumptions, some special cases can be simulatedusing analytical solutions. These cases can be of benefit in preliminary evaluations,in screening alternative management or treatment options, in evaluatingand determining model parameters in laboratory studies, and in gaininginsights into the effects of the various parameters. They can also be used toevaluate the performance of numerical models. With that note, two simplifiedcases of one-dimensional flow (1-D) are given next.Impulse input, 1-D flow with first-order consumptive reactionA simple 1-D flow with first-order degradation reaction of the dissolvedconcentration, C, such as a biological process, with a rate constant, k (T –1 ),can be described adequately by the simplified form of Equation (6.12):2 ∂ C u ∂C = – E ∂tR ∂xR ∂ C k∂x2 – C (6.13)RFor example, an accidental spill of a biodegradable chemical into the aquifer canbe simulated by treating the spill as an impulse load. For such an impulse input© 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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Page 103 and 104: 4.7.2 ELEMENTARY REACTIONSThe rate
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Page 107 and 108: 4.8 MATERIAL BALANCEAs indicated in
Page 109 and 110: SolutionLet M be the mass of chemic
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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
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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 and 148: this problem. Once the basic “syn
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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
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
Page 175 and 176: to further enhance the capabilities
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
Page 197 and 198: simulation. This enables, for examp
Page 199 and 200: 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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