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© 2002 by CRC Press LLCFigure 8.43 Flow through porous media modeled in ithink ® .
Figure 8.44 Concentration in pore gases in the layers as a function of time.EXERCISE PROBLEMS8.1. Refer to Exercise Problem 5.1. Solve the coupled differential equationsderived in that problem using the following biokinetic data: µ max = 0.3hr –1 , k d = k r = 0.01 hr –1 , Y = 0.45 gr cells/gr C in substrate, and K s = 150mg/L. Hence, plot the temporal variation of biomass and substrate.Assume that the initial concentrations of biomass and substrate are5 mg/L as C and 1000 mg/L as C, respectively.8.2. Refer to Exercise Problem 5.2. Solve the coupled differential equationsderived in that problem using the same biokinetic data in ExerciseProblem 8.1 for three different hydraulic retention times (HRT) of 5 hr,10 hr, and 20 hr. Hence, plot the temporal variation of biomass and substratefor each HRT.Use the following additional data: V = 10 L; S in = 1000 mg/L as C; andinitial concentrations of biomass and substrate in the reactor are5 mg/L and 0 mg/L as C, respectively.© 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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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
Page 207 and 208: MB on dissolved oxygen: dC od xyt=
Page 209 and 210: © 2002 by CRC Press LLCFigure 8.1
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 239 and 240: where q is the mass of color adsorb
Page 241 and 242: Figure 8.29 Results of activated ca
Page 243 and 244: and d X = (µ - b)Xdtwhere b is the
Page 245 and 246: Figure 8.32 Results of bioregenerat
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Page 251 and 252: Figure 8.38 Typical results of oxyg
Page 253 and 254: Figure 8.39 Mathematica ® script f
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Page 257: To generalize the approach, the gov
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Page 263 and 264: Figure 9.2 Lakes in series modeled
Page 265 and 266: Figure 9.4 Two lakes in series mode
Page 267 and 268: In this example, a two-compartment
Page 269 and 270: in the sediments, v b is the burial
Page 271 and 272: Figure 9.9 Algal growth modeled in
Page 273 and 274: Figure 9.11 Algal growth modeled in
Page 275 and 276: Once the basic model is constructed
Page 277 and 278: Figure 9.15 Graphical user interfac
Page 279 and 280: Figure 9.17 Contaminant transport v
Page 281 and 282: Figure 9.19 Contaminant transport v
Page 285 and 286: Figure 9.21 Chemical equilibrium mo
Page 287 and 288: MB on liver:MB on bones: d P2 = k 2
Page 289 and 290: Table 9.3 Model Equations in ithink
Page 291 and 292: Figure 9.25 Problem definition for
Page 293 and 294: Figure 9.28 Groundwater flow visual
Page 295 and 296: W and the second to V, in line 6. T
Page 297 and 298: Figure 9.30 Three-dimensional conto
Page 299 and 300: VerticalDispersion CoefficientsHori
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Page 307 and 308: 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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