Modeling Tools for Environmental Engineers and Scientists Episode 9 ppsx

PA RT II Applications Chapter 08 11/9/01 9:33 AM Page 195 © 2002 by CRC Press LLC CHAPTER 8 Modeling of Engineered Environmental Systems CHAPTER PREVIEW In this chapter, 12 examples of engineered systems are illustrated. The selected examples include steady and unsteady state analyses using algebraic and differential equations, solved by analytical, trial and error, and numerical methods. Computer implementation of the math- ematical models for the above are presented. The rationale for select- ing appropriate software packages for modeling the different problems and their merits and demerits are discussed. 8.1 INTRODUCTION T HIS chapter will serve as the “capstone” chapter for engineered systems in that the principles and philosophies covered in the previous chapters are integrated and applied in the modeling of engineered environmental sys- tems. Engineered systems involving steady and unsteady conditions are mod- eled applying the general theories presented in Chapter 4 to various reactor configurations discussed in Chapter 5 utilizing the software packages identi- fied in Chapter 7. The first example illustrates the entire modeling process from model development to computer implementation to calibration to vali- dation. The rest of the examples illustrate the model development and com- puter implementation procedures. 8.2 MODELING EXAMPLE: TRANSIENTS IN SEQUENCING BATCH REACTORS This example is based on a treatability study on a high-strength waste from a soft drink bottling facility, reported by Laughlin et al. (1999). Due to the Chapter 08 11/9/01 9:33 AM Page 197 © 2002 by CRC Press LLC high variation in waste flow rate and its constituents, a sequencing batch reac- tor (SBR) configuration was chosen to pretreat this waste prior to discharge into the city sewers. This example illustrates the modeling of the SBR process to predict the temporal variation of substrate, dissolved oxygen, and biomass growth during the fill and react phases of the process. Specifically, the objective of the original study was to investigate whether the bioprocess would be limited by dissolved oxygen levels due to the high substrate concentrations at the beginning of the react phase and to test the hypothesis that an oxygen saturation model can be used to model such limi- tation. The modeling goal is to achieve a correlation of r 2 > 0.8 between pre- dictions and observations of substrate and biomass concentrations at the end of the react phase. The study consisted of laboratory testing and mathematical modeling. Two bench-scale reactors (“Blue” and “Green”) we re-operated in parallel under identical conditions to test reproducibility. These reactors were fed with the high substrate waste from the bottling plant at various concentrations, but the concentrations were maintained constant during each test run. One set of experimental data was used to calibrate the mathematical model, and several other sets were used to validate the model under various flow rates, initial COD concentrations, and operating conditions. Details of the experimental studies can be found in Laughlin et al. (1999). 8.2.1 MODEL DEVELOPMENT Following the classifications in Section 5.22 in Chapter 5, SBRs can be categorized as flow, unsteady during the fill phase, and nonflow, unsteady during the react phase. The significant processes occurring during the fill period are dilution and endogenous decay. The processes occurring during the react period are substrate utilization, microbial growth, endogenous decay, oxygen uptake, and oxygen transfer. Therefore, two sets of material balance (MB) equations have to be developed—one for the fill phase and one for the react phase, with MB equations for biomass, substrate, and dissolved oxygen (DO) for each set. During the fill phase, the only processes occurring are dilu- tion and decay of biomass; hence, the MB equations are as follows: MB on biomass: ᎏ d d C t b ᎏ = – ᎏ V 0 Q + C Q b T ᎏ – k d C b MB on substrate: ᎏ d d C t s ᎏ = ᎏ V Q 0 C + s, Q in t ᎏ – ᎏ V 0 Q + C s Qt ᎏ Chapter 08 11/9/01 9:33 AM Page 198 © 2002 by CRC Press LLC MB on dissolved oxygen: ᎏ dC d o t xy ᎏ = ᎏ Q V C 0 o + xy Q ,i t n ᎏ – ᎏ V Q 0 C + ox Q y t ᎏ – bЈf d C b During the react phase, the initial high concentration of the substrate can cause oxygen limiting conditions. To simulate such conditions, a dual Monod’s kinetic function is included to modify the biomass growth rate and the sub- strate utilization rate under low DO levels. MB on biomass: ᎏ d d C t b ᎏ = ΂ ᎏ µ K m S ax ϩ C s C C s b ᎏ ΃ × ΂ ᎏ K O C ϩ ox C y oxy ᎏ ΃ – k d C b MB on substrate: ᎏ d d C t s ᎏ = – ΂ ᎏ Y µ (K ma S x C ϩ s C C b s ) ᎏ ΃ × ΂ ᎏ K O C ϩ ox C y oxy ᎏ ΃ MB on dissolved oxygen: ᎏ dC d o t xy ᎏ = K L a(C * oxy – C oxy ) – aЈ ΂ ᎏ d d C t s ᎏ ΃ – bЈf d C b The model parameters were determined from independent experiments, except for the half saturation constant for oxygen K o , which was established during the model calibration process. The variables are defined in Table 8.1. Table 8.1 Parameters for SBR Model Symbol Definition Value Units a′ Oxygen-substrate stoichiometric coefficient 0.2 mg/mg b′f d Respiration rate constant 0.0075 1/hr C b Biomass (MLSS) concentration Variable mg/L C oxy Dissolved oxygen concentration Variable mg/L C* oxy Saturated dissolved oxygen concentration 7.7 mg/L k d Biomass death rate 0.0004 1/hr K L a Overall mass transfer coefficient for aeration 12.8 1/hr K o Half saturation constant for oxygen uptake 90 mg/L K s Half saturation constant for substrate 800 mg/L Q Waste flow rate Variable L/hr t Time Variable hr V 0 Volume remaining in tank at start of fill phase 1.8 L µ max Maximum specific growth rate 0.2 1/hr Chapter 08 11/9/01 9:33 AM Page 199 © 2002 by CRC Press LLC Because the ODEs are coupled, a numerical method has to be used. Further, the inputs are constants; hence, mathematical packages as well as dynamic simulation packages can be used in this example. Models developed with Mathcad ® and ithink ® are illustrated in the following sections. 8.2.1.1 Mathcad ® Model The Mathcad ® model is shown in Figure 8.1. The model parameters are first declared in the top section of the sheet. Several parameters are entered using logical statements to switch their values depending on whether the cal- culations are in the fill phase (t < tf) or the react phase (t > tf). The syntax of the logic statements in Mathcad ® is very similar to that in Excel ® . The differential equations governing the system are entered as a matrix in D, which is then fed to the built-in routine, rkfixed. The solution is returned as a four-column matrix in Z, containing the time, biomass, substrate, and DO concentrations as a function of time. Finally, the fourth column is plotted to show the DO variation. Notice that in this model, the aeration is switched off during the fill period by the following statements: Kla(t) = if(t<tf, 0, 12.8) and µm(t) = if(t<tf, 0, 5.0). 8.2.1.2 ithink ® Model The model flow diagram for the SBR developed with ithink ® is shown in Figure 8.2. It shows the three separate, but interconnected, model segments, each describing the fate of substrate, biomass, and DO. The ghosting feature of ithink ® is used here to minimize the complexity of the flow diagram. For example, instead of feeding inputs directly from the substrate stock directly to the converters where it is used, ghosts of the substrate stock (indicated by dashed lines) are used. Model parameters, inputs, and intermediate calculations are contained within the containers indicated by circles. Components of the MB equations are embedded into the converters. All the parameters, inputs, and equations compiled by ithink ® are shown in Table 8.2. The initial values for the three stocks are set for each stock individually. The Runge-Kutta fourth-order method is selected for the three stocks. A user-friendly graphic user interface (GUI) for this model is presented in Figure 8.3. Using the built-in features of ithink ® , a GUI is constructed that enables users to adjust several model parameters such as fill time, waste flow rate, and influent COD, interactively. The users can also evaluate the effect of turning on aeration during the fill phase. Chapter 08 11/9/01 9:33 AM Page 200 © 2002 by CRC Press LLC Figure 8.1 SBR model in Mathcad ® . Chapter 08 11/9/01 9:33 AM Page 201 © 2002 by CRC Press LLC 8.2.2 MODEL CALIBRATION One set of experimental data was used to establish a value for K o . Starting from literature values, a trial-and-error approach was used to find the optimal value of K o = 90 mg/L. The criterion was to achieve a correlation of r 2 > 0.8 between calculated and measured COD and biomass values at the end of the react phase. Some of the biokinetic parameters were also adjusted to be within ±10% of the measured values to improve the degree of fit between cal- culated and measured COD and MLSS results. Figure 8.2 SBR model in ithink ® . Chapter 08 11/9/01 9:33 AM Page 202 © 2002 by CRC Press LLC 8.2.2.1 Model Validation The model is validated using the following measures: (1) Predicted vs. measured concentrations of substrate (COD) and biomass (MLSS) at the end of the react phase (2) Predicted vs. measured temporal concentration profiles of substrate (COD) and biomass (MLSS) during the fill and react phases Table 8.2 SBR Model Equations Generated by ithink ® STOCK EQUATIONS : Biomass(t) = Biomass(t – dt) + (Input2 – Death) * dt INIT Biomass = 5000 INFLOWS : Input2 = if(TIME < FillTime) then (-(FlowIn/Vol@t)*Biomass) else (Monod*OxyControl*Biomass) OUTFLOWS : Death = Biomass*kd Oxygen(t) = Oxygen(t – dt) + (Aeration – Endo – Dilution) * dt INIT Oxygen = .1 INFLOWS : Aeration = if (TIME < FillTime) then 0 else (Kla*(Csat–Oxygen)–OxyUptake) OUTFLOWS : Endo = Biomass*b Dilution = if (TIME < FillTime) then (-FlowIn*Oxygen/Vol@t) else 0 Substrate(t) = Substrate(t – dt) + (lput1 – Biouptake) • dt INIT Substrate = 100 I NFLOWS : Iput1 = if(TIME < FillTime) then (FlowIn*SubstrateIn/Vol@t)– (FlowIn*Substrate/Vol@t) else 0 OUTFLOWS : Biouptake = if (TIME < FillTime) then 0 else (Monod*Biomass/Yield)*OxyControl CONSTANTS : b = 0.0075 Yield = 0.53 Csat = 7.7 QDesign = 3.2 SubstrateIn = 338 FillTime = 1 InitVol = 1.8 kd = 0.01/24 KLa = 12 Ko = 9 Ks = 800 MuMax = 5/24 Monod = if (TIME < FillTime) then 0 else MuMax*Substrate/(Ks+Substrate) OxyControl = Oxygen/(Oxygen+Ko) OxyUptake = –a*DERIVN(Substrate,1) FlowIn = IF(TIME < FillTime) then QDesign else 0 Vol@t = InitVol+ (if(TIME < FillTime) then TIME*QDesign else FillTime*Qdesign) a = if (TIME < FillTime) then 0 else 0.2 Chapter 08 11/9/01 9:33 AM Page 203 © 2002 by CRC Press LLC (3) Predicted vs. measured temporal DO profiles during the fill and react phases (4) Predicted vs. measured long-term substrate (COD) removal efficiencies (5) MB closure at the end of the react phase to check if the increase in bio- mass equaled the decrease in substrate times yield Results of these comparisons are presented in Figures 8.4 to 8.7, which indi- cate that the model performance is within the expected goals. 8.3 MODELING EXAMPLE: CMFRs IN SERIES FOR TOXICITY MANAGEMENT This example is a modified version adapted from Weber and DiGiano (1996). An industry is considering equipping an existing tank (of volume V) Figure 8.3 Graphical user interface for sequencing batch reactor model in ithink ® . Chapter 08 11/9/01 9:33 AM Page 204 © 2002 by CRC Press LLC Figure 8.4 Predicted vs. measured COD and MLSS after the react phase. Figure 8.5 Predicted vs. measured COD and MLSS during the fill and react phases. as an activated sludge pretreatment system for treating their waste stream to meet the sewer discharge permit. This pretreatment system is expected to receive BOD and a nonbiodegradable, toxic chemical according to the sched- ule shown below, every two days: 0:00 9:00 10:00 11:00 12:00 13:00 14:00 16:00 to to to to to to to to 9:00 10:00 11:00 12:00 13:00 14:00 15:00 0:00 BOD conc., 50 60 75 100 125 142 150 50 C (mg/L) Toxicant conc., 0 30 45 50 50 50 40 0 C T (mg/L) Chapter 08 11/9/01 9:33 AM Page 205 © 2002 by CRC Press LLC [...]... subscript 1 for the first reactor and subscript 2 for the second reactor: MB equations for toxicant: dCT,1 1 ᎏᎏ = ᎏᎏ (CT,in,1 – CT,1) dt τ1 and dCT,2 1 ᎏᎏ = ᎏᎏ (CT,1 – CT,2 ) dt τ2 MB equations for BOD: dC1 1 ᎏᎏ = ᎏᎏ Cin,1 – ␣1C1 dt τ1 and dC2 1 ᎏᎏ = ᎏᎏ C1 – ␣ 2C2 dt τ2 © 2002 by CRC Press LLC Chapter 08 11 /9/ 01 9: 33 AM Page 210 Now, the equations are coupled ODEs with arbitrary inputs, and again, a... The second and third arguments specify the lists for the dependent variable and the independent variable, C and t, respectively, in this example Following the Rule sheet, the Functions sheet automatically lists the builtin procedure, RK4-se, and the custom procedure, “Equation.” By opening the procedure sheet for “Equation,” the governing ODE for this problem is entered in the bottom section, and the... carbon dose, and W(t) is the cumulative mass of carbon added Setting W(t) = DQt, the MB equation reduces to the following (Weber and DiGiano, 199 6): ΂ ΃ dC Cin C ᎏᎏ ϭ ᎏᎏ – ᎏᎏ [1 – ηkat] dt τ τ Even though the final result appears to be simple, it is a nonhomogeneous ODE, and its solution is not straightforward Procedures to derive the analytical solution to the above can be complicated, and as such,... solutions cannot be found for the governing equations Therefore, it is necessary to resort to dynamic simulation of the entire system The plant was originally designed based on conventional design equations for the trickling filter and the activated sludge processes Similar equations for those processes will be adapted in this model Traditionally, designs of equalization tanks for environmental systems... using standard mathematical notations that are automatically converted to the two-dimensional form by Mathcad® The left-hand side of this equation indicates the dependent and independent variables in the equation, t and C, respectively The built-in Runge-Kutta numerical method is evoked by the call rkfixed, with the arguments specifying the dependent variable (C), the initial and final values for the... different syntax for the different routines in MATLAB® and Mathematica®, in ithink®, the procedure for assembling the flow diagram is more or less the same for all types of problems The model developed with the dynamic simulation package, Extend™, is shown in Figure 8.24 The model parameters and the initial concentration, Co, are stored in Constant blocks The right-hand side of the ODE is formulated in... Webber and DiGiano ( 199 6) recommended a graphical procedure for solving this problem Further, it would be preferable to be able to solve the equations repeatedly under ranges of input values and under different inputs Hence, a computer-based model can be of significant benefit Spreadsheet programs can be used in this case; however, they can become © 2002 by CRC Press LLC Chapter 08 11 /9/ 01 9: 34 AM... arbitrary toxicant and BOD input profiles and modify them to simulate other scenarios Figure 8.10 Pretreatment system—ithink® model of modified system © 2002 by CRC Press LLC Chapter 08 11 /9/ 01 9: 33 AM Page 211 Figure 8.11 CMFRs in series: optimization study Figure 8.12 Pretreatment system: concentration profiles under optimized conditions © 2002 by CRC Press LLC Chapter 08 11 /9/ 01 9: 33 AM Page 212 Figure... equation, the initial condition, the dependent variable, and the independent variable At this step, the equation is entered in symbolic form, and Mathematica® instantly returns the solution in the output line, Out(1), also in symbolic form, evaluating the integration constant automatically based on the initial © 2002 by CRC Press LLC Chapter 08 11 /9/ 01 9: 33 AM Page 218 Figure 8.17 Chemical oxidation process... In(2), the model parameters are first assigned numeric values (Cin = 40; HRT = 8; and η.k.a = 2.50 ), and Dsolve is called again in symbolic form Mathematica® returns the solution in the output line Out(3), in numeric form, substituting the given parameter values Finding the solution in numeric form is a prerequisite for plotting the results Finally, a call to Plot the results is made in the input . facility, reported by Laughlin et al. ( 199 9). Due to the Chapter 08 11 /9/ 01 9: 33 AM Page 197 © 2002 by CRC Press LLC high variation in waste flow rate and its constituents, a sequencing batch. balance (MB) equations have to be developed—one for the fill phase and one for the react phase, with MB equations for biomass, substrate, and dissolved oxygen (DO) for each set. During the fill phase, the. model performance is within the expected goals. 8.3 MODELING EXAMPLE: CMFRs IN SERIES FOR TOXICITY MANAGEMENT This example is a modified version adapted from Weber and DiGiano ( 199 6). An industry