Modeling Tools for Environmental Engineers and Scientists

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On substituting for g and j from the above, C 0 = W/αQ, where W is the massrate of discharge into the river. The final model for this problem is, therefore:Step 4: Interpretation of ResultsC = αWQ e gx for x ≤ 0 (2.26)C = αWQ e jx for x ≥ 0 (2.27)The calibration and validation of the model will be highly problemspecific.Initial interpretations can include simulations, sensitivity analysis,and comparison with other similar systems. As a first step in this case, themodel can be run with typical parameters and known inputs. The output canbe used to corroborate the performance of the model against intuition, pastexperience, or literature results to verify that the model outputs generally followthe observed profiles and are within reasonable ranges. Because the resultin this case is a function of x rather than a numerical value, it may be usefulto plot the variation of C as a function of x to check if the model is reflectingthe spatial concentration profile.As an example of calibration and validation, let us use an artificial data set(adapted from Thomann and Mueller, 1987):Distance (mi) –15 –10 –5 0 5 10 15Concentration (mg/L) 0.43 1.22 3.50 10.00 4.49 2.02 0.91This data set had been collected on an estuary flowing at 100 cfs with an averagecross-sectional area of 10 × 10 5 sq ft, receiving a waste input of 372,000lbs/day at 0 miles. The decay rate of the waste material was measured to be0.1 day –1 . The dispersion for this estuary was estimated to be in the range of2–5 sq miles/day.As a first step, an appropriate value for the dispersion coefficient, E, hasto be established. The data from the upstream portion of the estuary are usedto calibrate the model to fit the four data points by running the model withvarious values of E ranging from 2 to 5 sq miles/day. Given the value of E,the remaining three data points may be used to validate the model. These considerationsare illustrated in Figure 2.8, from which the value of E can be estimatedto be 3 sq miles/day. With this value for E, the model is capable offitting the upstream data points as well as predicting the downstream datapoints. While acknowledging that the validation and calibration exercise inthis example is somewhat academic, the very same procedure is used in practiceto calibrate and validate mathematical models.© 2002 by CRC Press LLC
Figure 2.8 Model predictions vs. measured data.The value of the simplifying assumptions made at the beginning of theanalysis is in the reduction of the governing equation to a form amenable toa closed solution. Such an analysis provides highly useful insight and acts asa stepping stone for the modeler to gradually add pertinent details andrefinements to build more realistic models. Needless to say, as the modelsbecome more detailed and realistic, analytical methods of solving the governingequations can be extremely complicated or impossible. In such cases,using computer-based, numerical approaches is the only way to complete themodeling process. In the next chapter, some of the common analytical andnumerical approaches used in environmental modeling efforts will be reviewed.© 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 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: in terms of the model parameters al
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
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
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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
Page 313:
Nirmalakhandan, N., Jang, W., and S
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