Modeling Tools for Environmental Engineers and Scientists

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One of the unknown concentrations, C r , can be eliminated between the twoequations to yield the following:d V r C pp –QC pdtNote that V r is a variable, and therefore, another relationship is needed tosolve the above differential equation. The left-hand side of the above equationcan be expanded using the chain rule:d V r C pp = 1 p V r d Cpd t C d V p r d dt t Recognizing that the rate of decrease of volume inside the vessel, (–dV r /dt) Q, the flow rate out of the vessel, and the volume remaining at any time,V r = V 0 – Q t , the MB equation reduces to: d Cd pt= Q ( 1 – p)V C P– Qt0By separating the variables, the solution to the above can be developedas follows: d Cp = Q ( 1 – p) dtC V – QtIntegrating between the following initial conditions:andFigure 4.5 A membrane separation system.P0t = 0, C p0 = pC r 0© 2002 by CRC Press LLC
CpHence, the final result isEXERCISE PROBLEMSt = t, C p = C pd Cp = Q(1 – p) t pCr 0C 1 dt0 V 0 – QtpCln ppC= (1 – p) ln V 0 – QtV r 0C p = pC r 0 V 0 – QtV (1 – p)4.1. A mixture of benzene and toluene exists as a gas-liquid system in equilibriumin a closed system at 80ºC. The gas phase contains 65% molebenzene and 35% mole toluene. Assume that the vapor pressure of benzene= 756 and that of toluene = 287 mm Hg at 80ºC.a. Find the total pressure of the system.b. What is the composition of the liquid phase?4.2. In a gas-liquid binary system containing a solute A, the bulk mole fractionof A in the gas phase, Y 1 , is 0.00035, and that in the liquid phase,X 1 , is 0.03. Assume X 1 * = 0.02, Y i = 0.0004, and k L = 0.01.a. Determine the Henry’s Constant of the solute.b. Construct the equilibrium diagram for the system.c. Determine Y 1 * and X i .d. Determine the overall mass transfer coefficients, K L and K G .e. In what direction will the solute tend to flow?f. List four different equations to calculate the mass transfer flux andverify that all four equations yield the same result.4.3. At an industrial facility, an open holding tank is used to equalize andstore wastewaters for weekly discharge. Water samples taken from thistank indicate 2 mg dissolved oxygen in a 100-g water sample.a. Determine the direction of oxygen flux under this condition.b. When equilibrium is reached, what would be the dissolved oxygenconcentration in the water (in mg/L)?c. To contain air emissions, this tank is now being covered, and the headspaceis pressurized to 1.1 atm. Also, the headspace is enriched with oxygen.What should be the oxygen content (in %) in the headspace toensure 20 mg/L of dissolved oxygen when the tank is discharged?Assume Henry’s Constant of oxygen = 2.28 × 10 7 mm Hg/mole fraction.4.4. A continuous flow of an aqueous waste stream containing toluene ispretreated by holding it in a closed tank and scavenging the headspacewith a continuous flow of air.00© 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
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
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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
Page 105 and 106: excess energy. The direct photolysi
Page 107 and 108: 4.8 MATERIAL BALANCEAs indicated in
Page 109: SolutionLet M be the mass of chemic
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
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Page 131 and 132: 0 = QX - Q(X - dX) - K L (aAdz)(X -
Page 133 and 134: chemical engineering applications.
Page 135 and 136: This expression does not provide an
Page 137 and 138: If the WWTP used air instead of pur
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
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Page 157 and 158: 6.2.5 FLOW OF AIR AND CONTAMINANTS
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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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