Modeling Tools for Environmental Engineers and Scientists

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In this example, a mathematical model is developed to describe thisprocess to gain a better understanding of the sensitivity of the process to thevarious parameters. The following assumptions are made in model development:biological growth is single-substrate limited; biokinetic rates are substrateinhibited according to the Haldane rate expression; the OBR process iscompletely mixed; and equilibrium conditions exist at the carbon particle surface,according to the Freundlich isotherm. Model development follows thework of Goeddertz et al. (1988). The MB equation for substrate concentration,C (ML –3 ), can be stated as follows:where d(V C) = (biodegradation rate) + (desorption rate)dt d C µ = –dt Y X dasmk V (C s – C)Cµ= µ maxK s C C 2KV is the volume of the bulk liquid phase (L 3 ), X is the biomass concentration(ML –3 ), Y is the biomass yield (–), k d is the desorption rate constant (LT –1 ),a s is the specific surface area of the carbon (L 2 M –1 ), m is the mass of carbon(M), and C s is the substrate concentration at the carbon-liquid interface(ML –3 ). It is convenient to lump the group of variables (k d a s m/V) for measurementand computation purposes. Using the Freundlich isotherm relationship,C s can be expressed as follows:1C s = k mq nwhere q is the mass of substrate (M) adsorbed on the carbon at any time. Anexpression for q is as follows:fq = q o – V (C – C o) ∆ GY where q o is the initial mass of substrate adsorbed on the carbon (M), C o is theinitial concentration of the substrate in the bulk liquid (ML –3 ), and ∆G is thebiomass grown during regeneration (ML –3 ). To determine ∆G, MB on biomasscan be written as follows: d G = µ Xdtl© 2002 by CRC Press LLC
and d X = (µ – b)Xdtwhere b is the biomass death rate (T –1 ). The regeneration efficiency, R (%),can be determined from:R = 100 q o – qq The model equations contain coupled algebraic equations and ODEs withnonlinear terms. They can be solved readily using dynamic simulation-basedpackages such as ithink ® , Extend , or Simulink ® . In this example, the modeldeveloped with ithink ® is illustrated in Figure 8.31. The three ODEs for C,X, and G are represented by Stocks and are interconnected according to theequations. The fourth-order Runge-Kutta method is chosen to solve themodel over a time span of 96 hours and a time step of 0.05 hours. The equationsunderlying the model and the model parameters are automatically compiledby ithink ® into a list as shown in Table 8.4.oFigure 8.31 Bioregeneration process modeled in ithink ® .© 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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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 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 241: Figure 8.29 Results of activated ca
Page 245 and 246: Figure 8.32 Results of bioregenerat
Page 247 and 248: under a set of known data, the stat
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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 and 258: To generalize the approach, the gov
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
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
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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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