Modeling Tools for Environmental Engineers and Scientists

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suitable. The use of the ithink ® software package is chosen in this example asillustrated in Figure 9.7.The results from this model are presented in Figure 9.8. The slow buildupof the sediment concentration can be seen from this plot. These results are inagreement with those presented by Chapra (1997).9.4 MODELING EXAMPLE: ALGAL GROWTH IN LAKESIn this example, development of a model to describe algal growth in lakesis illustrated, starting from a simple two-component model and graduallyrefining it to make it more realistic. The model assumes the constant volumeof lake, V(L 3 ); complete mixing; nutrient limiting conditions for algaegrowth; and phosphorous recycling to the phosphorous pool after death. Thepreliminary model assumes a constant maximum growth rate, k g,max (T –1 ),and first-order death process of rate constant, k d (T –1 ). The MB equations foralgae, a, and phosphorous, p, are as follows: d pdt d ap =dtk g,maxKs,p + p a – k d a – Q V a = a pa (k d a) + Q V (p in – p) – a pa k g,maxKs,pp+ p awhere Q is the inflow rate (L 3 T –1 ), p in is the influent phosphorous concentration(ML –3 ), K sp is the half saturation constant for p (ML –3 ), and a pa is thestoichiometric ratio of p in a (–). The term (V/Q) can be replaced in terms ofthe hydraulic detention time, HRT = V/Q (T). The above equations are couplednonlinear differential equations and have to be solved numerically. Equationsolver packages such as TK Solver, Mathcad ® , MATLAB ® , or Mathematica ®can be used for solving them; however, if the model parameters such as p inare functions of time, dynamic simulation packages such as ithink ® orExtend would be more efficient in solving them. In this example, the aboveequations are implemented in the ithink ® dynamic simulation package.The model developed using the ithink ® simulation package is illustrated inFigure 9.9. The construction of the flow diagram is relatively straightforward.The model parameters are first assigned numerical values using theContainers. Two Stocks are created to represent a and p; the equations areentered into the Converters, which are automatically compiled by ithink ® andare listed in Table 9.1. The flow diagram visually illustrates the interactionsbetween the algae and phosphorous compartments, and at the same time, itencodes the underlying equations governing the system. The model is set tosolve the differential equations using the Runge-Kutta fourth-order method for© 2002 by CRC Press LLC
Figure 9.9 Algal growth modeled in ithink ® .30 days, with a time step of 0.1 days. Temporal variations of concentrationsof algae and phosphorous, resulting from a step input of p in , are shown inFigure 9.10.The Extend simulation package can also be easily used to model thisproblem as illustrated in Figure 9.11. Two Integrator blocks are used to solvethe coupled differential equations: one for a and the other for p. In this case,the influent concentration of phosphorous, pin, is set as a constant value ofTable 9.1 Model Equations Compiled by ithink ®A(t) = a(t - dt) + (Growth - Death&Washout) * dtINIT a = 0.5INFLOWS: Growth = kgmax*(p/(p+ksp))*aOUTFLOWS: Death&Washout = kd*a + (1/HRT)*ap(t) = p(t - dt) + (Inflow&Release - Upake) * dtINIT p = 1INFLOWS: Inflow&Release = apa*kd*a + (1/HRT)*(pin-p)OUTFLOWS: Uptake = apa*kgmax* (p/(p+ksp))*aapa = 1.5; HRT = 30; kd = 0.1; kgmax = 1; ksp = 2© 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
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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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Page 229 and 230: Figure 8.20 Chemical oxidation proc
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Page 237 and 238: © 2002 by CRC Press LLCFigure 8.27
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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 253 and 254: Figure 8.39 Mathematica ® script f
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Page 259 and 260: Figure 8.44 Concentration in pore g
Page 261 and 262: modeling of a sample problem from T
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
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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 283 and 284: © 2002 by CRC Press LLCFigure 9.20
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 303 and 304: 342Chemical Mass Distribution [%]11
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Page 307 and 308: Hence, Q π tan-1 - 1 - UL - 1Q/
Page 309 and 310: Figure 9.38d Stream lines and veloc
Page 311 and 312: BibliographyBedient P. B., Rifai, H
Page 313: Nirmalakhandan, N., Jang, W., and S
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