Modeling Tools for Environmental Engineers and Scientists

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Figure 3.11 Runge-Kutta method in Excel ® .3.3.3 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONSSystems of coupled ODEs are rather common in many environmental systemsand several examples will be considered in detail in the following chapters.The Runge-Kutta method can be extended to systems of ODEs as well as tohigher-order differential equations. Higher-order equations have to be reduced tofirst-order equations by introducing new variables before applying the Runge-Kutta method. For example, consider the second-order differential equation:2 yd d x2 = g x,y, d ydx(3.27)The order of the above can be reduced from 2 to 1 by introducingz = d ydzd whereby = dxd x d x2 (3.28)2 y© 2002 by CRC Press LLC
Hence, the original problem is now equivalent to one with two coupled ODEs:d zd x= g(x,y,z) and d yd = f(x,y,z) = z (3.29)xThe Runge-Kutta method can now be applied to each of the above two ODEs:(Ky n1 = y n 0 2K 1 2K 2 K 3 )(3.30)6where(Lz n1 = z n 0 2L 1 2L 2 L 3 )(3.31)6K 0 = hf(x n ,y n ,z n )L 0 = hg(x n ,y n ,z n )K 1 = hf(x n + 0.5h,y n + 0.5K 0 ,z n + 0.5L 0 )L 1 = hg(x n + 0.5h,y n + 0.5K 0 ,z n + 0.5L 0 )K 2 = hf(x n + 0.5h,y n + 0.5K 1 ,z n + 0.5L 1 )L 2 = hg(x n + 0.5h,y n + 0.5K 1 ,z n + 0.5L 1 )K 3 = hf(x n + h,y n + K 2 ,z n + L 2 )L 3 = hg(x n + h,y n + K 2 ,z n + L 2 )The above algorithms are available in many software packages as built-inroutines for easy application. Several examples of such applications to a widerange of problems will be presented in the following chapters.3.3.4 PARTIAL DIFFERENTIAL EQUATIONSFor a large variety of environmental systems, the dependent variable isexpressed in more than one independent variable. For example, the concentrationof a pollutant in a river may be a function of time and river miles, whereasthat in an aquifer may be a function of the three spatial dimensions as well asof time. Such systems are modeled using partial differential equations.Partial differential equations may be classified in terms of their mathematicalform or the type of problem to which they apply. The general form of asecond-order PDE in two independent variables is as follows:2 f2∂∂ f ∂ f ∂f∂fA(x,y) ∂ x2 B(x,y) C(x,y) E∂ x∂y∂2y x,y,, ∂ x ∂ y (3.32)If B 2 – 4AC < 0, the equation is classified as elliptic; if B 2 – 4AC = 0, parabolic;and if B 2 – 4AC < 0, hyperbolic. To solve such PDEs, initial conditions,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
Page 17 and 18: 1.2.3 STATIC VS. DYNAMICWhen a syst
Page 19 and 20: Real SystemsMathematical ModelsDete
Page 21 and 22: through several pathways. Consequen
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Page 25 and 26: APPENDIX 1.1 TYPICAL USES OF MATHEM
Page 27 and 28: APPENDIX 1.2 (continued)Target Audi
Page 29 and 30: characteristics. The system is isol
Page 31 and 32: formation, inferencing, testing, va
Page 33 and 34: system-surroundings interactions ca
Page 35 and 36: 2.2.4 INTERPRETATION AND EVALUATION
Page 37 and 38: to fill in where scientific theorie
Page 39 and 40: Figure 2.3 Schematic of real system
Page 41 and 42: Table 2.1 Variables, Symbols, Dimen
Page 43 and 44: ∴Net diffusive inflow = -EA ∂
Page 45 and 46: in terms of the model parameters al
Page 47 and 48: Figure 2.8 Model predictions vs. me
Page 49 and 50: variables. Deterministic systems ca
Page 51 and 52: e possible, and a computational met
Page 53 and 54: or, in matrix forma 11 a 12 a 13a 2
Page 55 and 56: (a)(b)Figure 3.3 (a) Setting up Sol
Page 57 and 58: Figure 3.5 Using MATLAB ® for solv
Page 59 and 60: Another method, known as the Newton
Page 61 and 62: and so on up tof n (x 1 , x 2 , . .
Page 63 and 64: c. Second-order equation with equid
Page 65 and 66: Figure 3.9 Solution of ODE by Mathe
Page 67: over the entire step, which may be
Page 71 and 72: 2∂ f f i1,j1 - f i-1,j1 - f i1,j-
Page 73 and 74: modeling, in diagnosis and troubles
Page 75 and 76: For the reach x ≥ a: C S D K
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Page 79 and 80: known as an intensive property. Oth
Page 81 and 82: Dissolved mass in sample = dissolve
Page 83 and 84: Figure 4.2 Illustration of steady s
Page 85 and 86: 4.3.2.2 Dalton’s LawDalton’s La
Page 87 and 88: Table 4.2 Different Forms of Quanti
Page 89 and 90: M olesoxygenHence, K a-w = = 8 - 3
Page 91 and 92: SolutionThe flux, N, which is the d
Page 93 and 94: Figure 4.3 Illustration of the Two-
Page 95 and 96: where C s is the concentration of t
Page 97 and 98: Fraction in dissolved form = f d =
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
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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
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
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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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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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Modeling Tools for Environmental Engineers and Scientists

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