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The above equation is implemented as a spreadsheet model as shown inFigure 5.10. This model calculates the stripping efficiency for K a–w valuesranging from 0.05 to 0.8, at G/Q values of 1, 5, 10, 15, 20, and 30. The contourplot generated from this model shows the required relationship betweenthe three process parameters. Such a plot provides additional insight and aidsin rapid evaluation of the overall process.EXERCISE PROBLEMS5.1. Develop unsteady state MB equations for biomass and substrate concentrations,X and S (M/L –3 ), respectively, in a batch bioreactor employingthe following variables: maximum growth rate, µ max (T –1 ); half saturationconstant, K s (ML –3 ); first-order biomass death rate, k d (T –1 ); firstorderbiomass respiration rate, k r (T –1 ); and yield coefficient, Y [M cells(M substrate) –1 ]. Note that the biomass death process releases organiccarbon back into the substrate pool, whereas the respiration processdoes not.5.2. Using the same notation as in problem 5.1, develop MB equations forbiomass and substrate in a CMFR with the following additional variables:the flow rate, Q (L 3 T –1 ); influent substrate concentration, C in(M/L –3 ); and volume of the reactor, V (L 3 ). Assume negligible biomassconcentration in the influent.5.3. The aeration tank in a wastewater treatment plant is based on a CMFRdesign, using pure oxygen with bubble diffusers. A chemical plantwishes to discharge a waste stream containing 140 mg/L of toluene intothe sewer system. The permit granted to the wastewater treatment plant(WWTP) limits the maximum concentration of toluene in its effluent to1 mg/L. You are required to estimate the allowable discharge rate fromthe chemical plant, assuming that the WWTP influent previously carriedno traces of toluene.Oxygen transfer efficiency of aerators = 20%Waste flow rate through aeration basin = 1 MGDSurface area of aeration basin = 4000 sq ftHydraulic residence time= 12 hrsOxygen requirement= 200 mg/L of reactor volumeHenry’s Constant of oxygen = 32 –in wastewaterHenry’s Constant of toluene = 0.45 –in wastewaterDO level maintained in aeration basin = 3.0 mg/LK L for toluene/K L for oxygen = 0.65 –Average temperature = 30ºC© 2002 by CRC Press LLC
If the WWTP used air instead of pure oxygen, do you think the chemicalplant would be able to discharge at a higher rate than the one calculatedabove? Explain without any calculations.5.4. Refer to Equation (5.35) where the overall mass transfer coefficient withreference to the liquid-side film, K L , is dimensioned as (ML –2 T –1 ),while in Equation (4.28), it is defined and dimensioned as (LT –1 ).Reconcile these two forms of K L .5.5. Sparged tanks have been proposed for the removal of synthetic organicchemicals (SOCs) from water. Here, SOCs can be removed by twomechanisms—volatilization and oxidation. The oxidation process canbe approximated by a first-order process of rate constant k O3 . Followingthe approach and the notation used in developing Equation (5.49), andassuming the liquid phase to be completely mixed, develop a model todescribe the effluent concentration of the SOC from the tank. The modelshould be in terms of the parameters Q, G, V, K a–w , and K G a defined inEquation (5.49); and τ, the hydraulic detention time; and C O3 , the dissolvedconcentration of ozone in the reactor.5.6. Continuing the above problem 5.4, construct MB equations for ozone inthe gas and liquid phases. The rate of loss of ozone in the gas phaseshould equal the rate of consumption by the reaction in the liquid phaseplus the rate of outflow in the effluent. Hence, derive an expression forC O3 that can be used in the result derived in the above problem 5.4.© 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
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
Page 105 and 106: excess energy. The direct photolysi
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
Page 119 and 120: CC = in -(t t 0 )k C t in0 k- C
Page 121 and 122: Worked Example 5.1A wastewater trea
Page 123 and 124: or,C L = C 0 e -(k /u)L = C 0 e -k
Page 125 and 126: HRT = 1 k (1 - )The overall HRT
Page 127 and 128: The MB equation for the above syste
Page 129 and 130: the particle surface (ML -2 T -1 ),
Page 131 and 132: 0 = QX - Q(X - dX) - K L (aAdz)(X -
Page 133 and 134: chemical engineering applications.
Page 135: This expression does not provide an
Page 139 and 140: and degradation of the natural envi
Page 141 and 142: The solution to the above PDE gives
Page 143 and 144: • river 2: x = 15000.5 dmay ∴u
Page 145 and 146: These functions are valuable tools
Page 147 and 148: this problem. Once the basic “syn
Page 149 and 150: giving the condition Q > 4πRu for
Page 151 and 152: directions, and the last term repre
Page 153 and 154: SolutionTo be conservative, it may
Page 155 and 156: zone, because the hydraulic conduct
Page 157 and 158: 6.2.5 FLOW OF AIR AND CONTAMINANTS
Page 159 and 160: time-dependent volumetric flow rate
Page 161 and 162: Solution(1) The steady state concen
Page 163 and 164: As a first step in modeling a river
Page 165 and 166: where a 1,4 is the conversion facto
Page 167 and 168: © 2002 by CRC Press LLCFigure 6.8
Page 169 and 170: Figure 6.106.5 Consider the stream
Page 171 and 172: Case 4: Pulse Source in Two Dimensi
Page 173 and 174: End users often adapted these model
Page 175 and 176: to further enhance the capabilities
Page 177 and 178: are always expressed in the standar
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Page 181 and 182: typing in the equations in algebrai
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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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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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