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VolatilizationHydrolysisPhotolysisBiodegradationDissolved formof toxicantDiffusionAdsorptionDesorptionBiouptakeDepurationSedimentsSuspendedsolidsScourSettlingFigure 2.4 Environmental processes acting upon dissolved toxicant.• The river flow and waste input are the only inflows into the system,and the river flow is the only outflow from the system.• The volumetric flow of the waste stream is negligible compared to theriver flow.With these assumptions, the system is reduced to a simplified form as shownin Figure 2.5. The variables of significance and relevance are listed in Table2.1 along with their respective symbols, dimensions, and common units.Figure 2.5 Schematic of the simplified system.© 2002 by CRC Press LLC
Table 2.1 Variables, Symbols, Dimensions, and UnitsVariable Symbol Dimension UnitRiver flow rate Q L 3 T 3 cfsConcentration of toxicant C ML –3 mg/LFirst-order reaction rate constant k T –1 1/dayArea of flow A L 2 sq ftVelocity of flow U LT –1 ft/sDispersion coefficient E L 2 T –1 sq miles/dayLength in direction of flow x L milesTime t T dayStep 2: Mathematical RepresentationTask 1: identifying fundamental theories. Having simplified the system,the appropriate fundamental theorems, laws, and principles necessary to buildthe model to achieve the goal can be identified as follows:• Toxicant mass balance based on conservation of mass:Rate of changeof mass oftoxicant insidesystemNet rate of Net rate of Net rate ofadvective diffusive loss of= + – (2.1)inflow of inflow of toxicant duetoxicant toxicant to reactions• Advective mass flow rate based on continuity:= [river flow rate, Q] × [concentration of toxicant, C] (2.2)• Dispersive mass flow rate based on Fick’s Law:= [dispersion coefficient, E] × [area of flow, A]× [concentration gradient, ∂C/∂x] (2.3)• Mass loss due to reaction based on first-order reaction kinetics:= [reaction rate constant, k] × [volume, V] × [concentration, C] (2.4)If other processes, such as interactions with suspended solids, bioaccumulation,volatilization, etc., are to be included, then appropriate principles andrelationships have to be included in this listing.Task 2: deriving relationships. The fundamentals identified in Task 2 cannow be combined to derive an expression for the output(s). This involves© 2002 by CRC Press LLC
Page 2: © 2002 by CRC Press LLCModeling To
Page 5 and 6: ContentsPrefaceAcknowledgmentsFUNDA
Page 7 and 8: APPLICATIONS8. MODELING OF ENGINEER
Page 9 and 10: The contents of this book are organ
Page 11 and 12: PART IFundamentals© 2002 by CRC Pr
Page 13 and 14: The models resulting from the model
Page 15 and 16: 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
Page 23 and 24: ange of complex environmental appli
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: Figure 2.3 Schematic of real system
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 and 68: over the entire step, which may be
Page 69 and 70: Hence, the original problem is now
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
Page 77 and 78: contaminants that do not undergo an
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
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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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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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