Modeling Tools for Environmental Engineers and Scientists

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contain references to other cells, which in turn, contain the known inputparameters. These cells are referred to by the column-row notation.Alternatively, cells can be assigned appropriate names through the Insert >Name > Define . . . menu sequence and can be referred to by their namesinstead of by the generic column-row notation. This feature may be convenientwhen building and troubleshooting large models in Excel ® .Several simple yet important scenarios may be simulated by this model.For example, one can vary the Q and V values to understand the impact ofthe detention time on the in-lake steady state concentration or compare theimpact of conservative substances (by setting K = 0) against reactive ones(K ≠ 0). This same lake problem can now be posed in different ways: forexample, one way would be to determine the allowable pollutant load, W, thatwould not violate a given in-lake concentration as stipulated, for example, bythe water quality standards for the lake. In such a case, this basic spreadsheetmodel will not be able to back-calculate the answer instantly unless a trialand-errorapproach or a built-in function such as goal seeking is adapted.This form of the spreadsheet model cannot be generalized for different W (t)functions, because a separate model has to be formulated depending on theforcing function. The corresponding algebraic solution must be known inexplicit form before constructing the model. To model this lake problem in ageneral manner, to simulate, for example, different types of waste load inputs,numerical approaches have to be adapted.For instance, consider the case where the lake is initially under pristineconditions, and a step input of W = 1080 lbs/day is input to the lake at timet = 0. After five years, this load is reduced instantaneously to W = 500 lbs/day.To solve the governing ODE under the above conditions, Euler’s numericalmethod introduced in Chapter 3 has to be used. Figure 7.3 shows such animplementation. The model parameters are defined first. The time step hasbeen chosen as h = 0.25 years. (A more accurate solution and a smoother plotcan be obtained by using a smaller step such as h = 0.05 years.) The wasteload input column “W” is filled with the logical expression, if (t < 5, 1080,500), describing the partial step shutdown. By filling the “W” column in thismanner, one can easily change the time step h in cell D13 to observe its effecton the final result, without having to adjust the “W” column manually. Thelast column contains dc/dt, which, in essence, is the right-hand side of theODE to be solved. As can be seen from this implementation, spreadsheetsenable simple ODEs to be solved relatively easily through numerical methods.7.5.2 LAKE PROBLEM MODELED IN Mathcad ®The Mathcad ® model of the lake problem is illustrated in Figure 7.4. Inthis model, a case in which W(t) is a time-dependent function is illustrated.Because the particular solutions of the governing differential equations take© 2002 by CRC Press LLC
Concentration [mg/L]Figure 7.3 Lake example modeled in Excel ® under unsteady state conditions.different forms for different initial conditions and forcing functions, the completesolution must be known in advance to build models in Excel ® . Theequation processing engines in mathematical packages such as Mathcad ® cansolve differential equations in a general manner without the need to modifythe basic model. The forcing function in this example is defined to be a partialexponentially declining shutdown from 1080 to 500 lbs/day after sixyears. In Mathcad ® , this partial step shutdown is encoded in a simple statementas follows:W(t) = if(t > 6 * 365 * 86400, (1080 * e –0.005t + 500), 1080)which is very similar to the format used in the Excel ® spreadsheet packagefor logic statements.One of Mathcad ® ’s built-in Runge-Kutta procedures, rkfixed, is called tomodel the problem using the basic form of Equation (7.1). This procedure is© 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
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4.3.2.2 Dalton’s LawDalton’s La
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Table 4.2 Different Forms of Quanti
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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.
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
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
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Page 181 and 182: typing in the equations in algebrai
Page 183 and 184: accessed in Simulink ® models. The
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Page 189 and 190: saving considerable model building
Page 191 and 192: dependent variable, the dependent v
Page 193 and 194: command is used to keep the plot fr
Page 195 and 196: Figure 7.10 Lake problem solved wit
Page 197 and 198: simulation. This enables, for examp
Page 199 and 200: Figure 7.13 Lake problem modeled in
Page 201 and 202: It is hoped that the above overview
Page 203 and 204: APPENDIX 7.2 EXAMPLES OF TYPES OF E
Page 205 and 206: CHAPTER 8Modeling of EngineeredEnvi
Page 207 and 208: MB on dissolved oxygen: dC od xyt=
Page 209 and 210: © 2002 by CRC Press LLCFigure 8.1
Page 211 and 212: Table 8.2 SBR Model Equations Gener
Page 213 and 214: Figure 8.4 Predicted vs. measured C
Page 215 and 216: mathematical model. First, a proces
Page 217 and 218: Figure 8.9 Pretreatment system—ef
Page 219 and 220: Figure 8.11 CMFRs in series: optimi
Page 221 and 222: Other improvements can include alte
Page 223 and 224: Figure 8.15 Model of wastewater tre
Page 225 and 226: 8.5 MODELING EXAMPLE: CHEMICAL OXID
Page 227 and 228: When Mathematica ® cannot find ana
Page 229 and 230: Figure 8.20 Chemical oxidation proc
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© 2002 by CRC Press LLCFigure 8.27
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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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© 2002 by CRC Press LLCFigure 9.20
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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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