Environ. Sci. Technol. 2010, 44, 5503–5508
Selection of a Multi-Stage System for Biosolids Management Applying Genetic Algorithm Y I T Z H A K S T R A M E R , † A S H E R B R E N N E R , * ,† STUART B. COHEN,‡ AND G I D E O N O R O N §,† Unit of Environmental Engineering, Faculty of Engineering Sciences, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel, Department of Naval Architecture and Marine Engineering, University of Michigan, NA & ME Building, 2600 Draper Drive, Ann Arbor Michigan, 48109-2145, Environment Water Resources, J. Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Kiryat Sde-Boker, 84990, Israel and The Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
Received February 9, 2009. Revised manuscript received May 21, 2010. Accepted June 11, 2010.
An economic analysis and feasibility study of a sequential biosolids management process was developed and tested using Genetic Algorithm (GA). The algorithm was used to identify trends and behaviors of the “Biosolids Process Train”. This heuristic method of analysis is robust in that it will not only simulate different design scenarios, its analysis will also suggest possible solutions which meet predetermined requirements. This concept was adopted because GA’s biggest advantage is the capability to analyze multiple objective functions, design variables, and constraints. The range of “good approximations” provided by the GA solutions could be useful for municipal wastewater planners who need to search for potential alternatives and evaluate new technologies for managing biosolids. The unit processes in the model were arranged sequentially so the effect modifications to thickening and dewatering parameters could easily be observed further along in the process. The model was extended to examine the supernatant return flow quality and the potential impact on the wastewater treatment plant. Results from a sensitivity analysis on operating expenses reveals the impact that fluctuations in fuel, electricity, and labor costs can have on the total biosolids management cost as well as the selection of the appropriate treatment sequence.
Introduction Background to Municipal Sludge Treatment and Biosolids Production. New research and innovations are discovering valuable methods to handle, shrink, and ultimately utilize biosolids (treated sewage sludge). New methods in “mining biosolids” have been introduced to recover useful materials such as organic matter, phosphates, methane, and hydrogen (1). Markets for sludge products include fertilizers and soil * Corresponding author phone: 972-8-647-9029; e-mail: brenner@ bgu.ac.il. † Unit of Environmental Engineering, Faculty of Engineering Sciences, Ben-Gurion University of the Negev. ‡ University of Michigan. § Environment Water Resources, J. Blaustein Institutes for Desert Research, Department of Industrial Engineering and Management, Ben-Gurion University of the Negev. 10.1021/es902981t
2010 American Chemical Society
Published on Web 06/24/2010
additives, ingredients in bricks, cement, and energy (2). Governments and municipalities are constantly battling whether to upgrade or implement new biosolids handling systems. High capital costs and operating expenses, such as energy, labor, and chemicals, can make the cost to treat and dispose of biosolids 50% of the total wastewater utility costs (3). Best practices in biosolids management have shown combining different treatment options to get optimal results. Engineers and planners are now using management models to analyze the flow of biosolids to minimize costs, as well as health and environmental risks, thereby optimizing the end product quality. These models utilize typical sludge characteristics data, treatment parameters, transportation options, and reuse markets. Murray et al. (4) utilized a Life Cycle Inventory (LCI) management model to evaluate biosolids handling options that combine treatment with productive end uses for biosolids, thus treating biosolids as a resource rather than a nuisance. A heuristic method of analysis, such as Genetic Algorithm (GA), can provide decision makers with options for disposal or reuse (5, 6). The algorithm is robust in that it will not only simulate different design scenarios, its analysis will also suggest possible solutions which meet predetermined requirements. Disposing and/or reusing biosolids is a complicated management issue containing multiple stages of pretreatment such as thickening, stabilization, and dewatering (7). Each specific system has design information and operating conditions. Each method of treatment is also dependent on values from previous stages. It is essential that interrelationships between treatment stages are understood to ensure the overall process is feasible, and ultimately, the most economical one. A study from the University of Manitoba in Winnipeg, Canada, explored the biosolids management system, also known as the “Biosolids Process Train” (8). A diagram was provided in this study showing various treatment stages and many individual treatment operations at each stage, which results in many possible treatment combinations. Explained were the key decision variables such as total solids content, metal concentrations, and availability of land for disposal that would be used to help determine the appropriate “process train” to be implemented. An analysis was suggested to compare the total cost per dry metric ton [US $/t] required to achieve desired quality for a selected mode of disposal or reuse. The study identified a need for a method that can comparatively analyze management options and can select Biosolid Process Train solutions based on predefined requirements (8). Genetic Algorithm: Background and Theory. Genetic Algorithm is a heuristic optimization method that searches for feasible solutions based on a set of design criteria (9). The algorithm first creates a population of individuals, each called chromosomes, where “genes” on the chromosome represent design variables. The algorithm tests each chromosome to determine how well it solves the problem. The algorithm assigns a “fitness” score and those that pass a predetermined set of requirements “live,” while those that do not “die” (Figure 1). The first step of the GA is to represent random solutions in a binary form. New potential solutions (or chromosomes) are created by the process of reproduction called “selection.” Two parent chromosomes are selected for “mating” thereby recombining to form a “child.“ This child chromosome is added to the population to undergo the fitness test which VOL. 44, NO. 14, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Decision makers need to understand relationships of important aspects such as fixed capital costs, operation and maintenance expenses, potential benefits, solids quality, and environmental impact. Also, the wastewater treatment plant (WWTP) planners need to be aware of the effects of supernatant return flow quality from solid/liquid separation processes.
Methodology and Approach
FIGURE 1. Diagram of the GA process with chromosomes representing potential solutions and genes representing design variables in binary form. determines whether the solutions are suitable. The algorithm can also include a process of “mutation” where individual genes can be altered to create and test new chromosomes. Conditions for stopping are predefined by modifying the maximum generations or number of generations resulting in no change in the objective value. Genetic Algorithm requires the user to define an objective function such as minimizing cost or maximizing profits. The user then defines all the design variables and constraints that impact the solution. The major difference between the GA and common optimization methods such as Linear Programming is that the GA cannot guarantee its solution is the global optimum. The GA is heuristic in nature in that it provides multiple feasible (local) solutions to a problem. Linear Programming, on the other hand, will guarantee that its solution is the most optimal (global) one. The GA is generally utilized in large search areas where the application of other optimization techniques would be difficult. It is also applied to decision making situations where the goal is to find multiple, feasible solutions (5). Subject to the GA’s heuristic nature, the solutions provided by this algorithm can be defined as “good approximations” which have solved the objective defined in the model. This could be useful to municipal wastewater treatment planners who need to search for potential alternatives for treating and disposing or reusing biosolids. Scientists at Cranfield University in England applied a GA to a multiobjective, multistage metal forming process (10). A framework was proposed for applying GA to processes with sequential relationships between two stages. Each stage has its own objectives and requirements which affect the performance of the overall system. In the current work, a GA for optimizing a multistage biosolids management process was formulated and tested. Problem Statement. Lack of management research has shown a demand for modeling software to help decision makers choose the appropriate and best Biosolids Process Train. Decision makers need to know the potential treatment options available, understand their impact, and adjust parameters to meet requirements and government policy. 5504
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Modeling Approach. The following were conducted to accomplish the goals and targets of the work: (i) research the application of GA to similar processes such as Cranfield University’s application of GA to a multiobjective, multistage manufacturing process (9); (ii) research, define, and select different biosolids treatment systems to be modeled (generic modeling); (iii) complete a full mass and flow balance for each system selected; (iv) collect all relevant costs and quality data for the model; (v) test and screen different algorithm techniques for building the model architecture; (vi) build and test a prototype model for review based on various combinations of treatment stages; (vii) construct a model using feedback from the prototype; (viii) analyze the completed model’s constraints and their effect on results; and (ix) conduct a sensitivity analysis of model parameters on the total biosolids management cost. Case Study. This model was developed for sludge produced from a traditional activated sludge WWTP, treating 14,000 m3/day and servicing a population of 70,000 inhabitants. The treatment plant produces 3.2 t/d primary sludge at a solids concentration of 4%, of which 90% is volatile solids. The secondary clarifier produces 2.2 t/d secondary excess sludge at a solids concentration of 0.8%, of which 85% is volatile solids. Genetic Algorithm Model. The GA management model was developed using Microsoft Excel with OptWorks add-in software from Pi Blue Software Inc (11). OptWorks was compared to a number of potential programs and was selected for its low cost, friendly use, and flexibility. Figure 2 shows the biosolids treatment layout with the various options that were chosen for this model. Data variables are represented by the subscripts i (treatment stage) and j (treatment alternative at each stage i). The search space for this model offers 640 potential treatment options, however each alternative method of treatment has its design parameters which make the search space much larger. The model is flexible in that treatment alternatives within each treatment stage can be substituted or easily removed. For this analysis a thickening stage was only applied to the secondary excess sludge. Thickened sludge was then combined with the primary sludge prior to the stabilization stage. A mass and flow balance was conducted for each treatment alternative in the model. Some systems require effluent such as wash water during treatment, while others recycle filtered water back to the WWTP. Mathematical expressions for each treatment option were separated into four main categories (inputs, process variables, process calculations, and outputs). Input variables were determined as a result of calculations from the previous stage (i ) i - 1). Process variables include system data such as solids capture rate, schedule of operation, and solids loading rate. Process calculations utilized process variables to solve capacity requirements, energy requirements, and associated costs. Output variables consisted of the sludge conditions to be treated by the next stage, biogas produced, and supernatant recycle flow quantity and quality. Objective Function. For this model, the objective is to minimize the total biosolids treatment fixed capital costs and operation and maintenance (O&M) expenses. The annual fixed costs were calculated for annual payback payments by utilizing the Capital Recovery Factor (CRF) based on the
FIGURE 2. Biosolids management options with i representing the treatment stage and j representing the treatment alternative at each treatment stage. interest rate of 6% and lifespan of the equipment. The lifespan for mechanical and nonmechanical equipment was determined to be 8 years and 20 years, respectively. Annual treatment cost for the combined sludge was calculated by adding the annual fixed costs and annual O&M expenses for each stage of treatment after the thickened and primary sludge were combined (US $/yr for total amount treated). This allows estimating the cost of dry weight [US $/t] represented in eq 1. To solve total cost, this value was added to the unit cost for the thickening treatment stage [US $/t] (eq 2) 7
Total Cost ) COSTTS1,j +
∑ (ACC
i,j
+ COSTOMi,j)
i)2
TScombined × 365 (1)
where COSTTS1,j is treatment cost for the thickening at stage i ) 1, thickening alternative j [US $/t]; COSTOMi,j is operation and maintenance (O&M) cost for treatment stage i, alternative j [US $/yr]; ACCi,j is annual capital cost for treatment stage i, alternative j [US $/yr]; and TScombined is mass of combined dry solids of thickened and raw primary sludge [t/day]. COSTTS1,j )
ACC1,j + COSTOM1,j TSxv × 365
(2)
where COSTTS1,j is treatment cost for the thickening at stage i ) 1, thickening alternative j [US $/t]; and TSxv is mass of excess sludge intended for thickening [t/day]. Definition of the Decision Variables. The model both prompts the user to input decision variables and calculates them based on a set of design equations. The model uses these values to populate and test potential solutions. Decision variables are separated into two categories: design and operation variables. A detailed list of operation variables is available (20). The operating values were used to calculate design variables such as system capacities and land requirements. Definition of the Model Constraints. The algorithm is designed to give the objective value a penalty for solutions that do not meet constraints defined in the model (11). The first constraints refer to heavy metal regulations defined by environmental agencies (12, 14, 15). These agencies defined the allowable heavy metal concentration that can be applied
to land per year. The algorithm provides results ensuring that heavy metal concentrations do not exceed government standards when applied as fertilizer. Another constraint defined for the model was that biosolids must undergo stabilization prior to reuse. This is a minimum standard set by the U.S. EPA for class B biosolids (16). The last constraint defined was that solids concentration must be at least 20% prior to windrow composting (13). Genetic Algorithm Run Parameter Selection. Parameters that influence the algorithm’s performance are “selection type”, “crossover type”, “crossover probability”, “mutation type”, and “mutation probability” (11). The selection type parameter represents how potential solutions are selected and compared, and the speed it takes for the algorithm to converge to a near optimal result. Crossover type and crossover probability represent the method and frequency where solutions are crossed to create new solutions. Mutation type and mutation probability correspond to the method and frequency of changes in a single design variable (11). The model was tested using different combinations of the run parameters to examine how the GA converges to a near optimal result. The parameters chosen for this work were a tournament style selection process with four tournament participants, two point crossover (70% probability), and design wise mutation (20% probability). These values coincided with the software manufacturer’s suggestion for operating the model to encompass the entire search space and provide steady convergence to a near optimal result (11). OptWorks GA software prompts the user to input run parameters which influence the conditions for stopping as well as conditions for the algorithm’s performance (11). The “population size” was set to 25, which means that for each generation, the algorithm provided 25 potential solutions. When the “maximum generation” of 100 is reached, the algorithm will stop. If no change in the best objective value occurs after 30 generations (convergence generations), the algorithm is designed to stop. Output/Input and the Macro Procedure. The algorithm was designed to populate potential solutions where the output of a treatment stage becomes the input for the subsequent one. This was accomplished with Visual Basic Macro software. The code for the Macro consists of a series of “If-Then” statements. The Macro is designed to choose the output data for each selected process for each treatment stage. The subsequent stage then calls on that data, essentially VOL. 44, NO. 14, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 4. Annual O&M expenses and capital cost breakdown for the lowest cost treatment scenario calculated from 30 model runs. FIGURE 3. Histogram displaying algorithm results for generations 1, 5, 10, and 25. The population size for each generation is 25 resulting in 25 solutions for each generation. The objective value [$/t] includes penalties for results violating constraints. populating it with the previous stage’s results. The Macro is run at each iteration of the algorithm, which ensures the algorithm is following a sequential process. Model Verification and Results. Figure 3 was used to verify that the GA results correspond with the objective function to minimize total costs. This graph was used to examine the convergence of the algorithm’s solutions. Figure 3 is a histogram displaying the results of generations 1, 5, 10, and 25. Each generation produced a population size of 25 solutions. The x-axis displays the objective value, which was total treatment cost per dry metric ton [US $/t] and penalties due to violating constraints. The algorithm is designed to penalize the objective function for solutions that violate constraints that are defined in the model. Figure 3 shows a shift in data from right to left which means that as the generations increase, the objective values decrease. This verifies the algorithm is searching for results that correspond with the objective function to minimize total costs.
Results Demonstrating Results. Results from 30 runs of the model with the original objective function provided 15 different treatment scenarios. The model does not define the most optimal solution. However, the model considers these 15 different treatment scenarios as potentially good solutions which have all passed a predetermined set of requirements defined in the model. Statistical results from 30 runs of the
model show an average treatment cost of 350 US $/t and a standard deviation of 60 US $/t. The lowest-cost treatment scenario from the 30 model runs was the following (see Table 1): gravity thickener (1,2) f combined sludge f aerobic stabilization (2,2) f without conditioning (3,3) f drying bed (4,3) f without drying/ advanced stabilization (5,5) f truck transport (6,1) f land application/fertilizer (7,2) [Numbers next to each treatment alternative are represented as (i, j) where i is the treatment stage and j is the treatment alternative]. This scenario’s minimal total treatment cost is 254 US $/t. A cost analysis of the minimum treatment solution calculated from 30 runs of the model shows that approximately 58.6% of treatment cost is fixed costs, and 41.4% is allocated to operation and maintenance expenses (Figure 4). The model provided other potential solutions that also passed the fitness test. These are feasible solutions that could be an alternative to the previous result (see Table 2). Sensitivity Analysis. A sensitivity analysis was conducted to study the effect of model parameters on total biosolids management cost. Detailed predefined model parameters that served this purpose, such as unit electricity costs, labor wage, land costs, construction costs, chemicals costs, and transportation distances, are available elsewhere (20). A Student t test was used to determine the sensitivity of O&M on the total biosolids management cost for 30 model runs for each O&M parameter change. The alpha value was set to 5% significance level with 29 degrees of freedom (df). The results were significantly different for fuel, labor, and electricity costs (P-value less than 0.05). The P-value (Table 3) achieved for fuel, labor, and electricity is less than 0.05 which reveals that the data are statistically different when
TABLE 1. Breakdown of Values Calculated by the Model for Lowest-Cost Treatment Alternative Selecteda annual annual fixed O&M cost capital cost [$/yr] [$/yr]
equipment capacity/area required
treatment stage
treatment alternative
design variables
thickening stabilization conditioning dewatering drying/advanced stabilization transport
gravity thickener (1, 2) aerobic stabilization (2, 2) without conditioning (3, 2) drying bed (4, 3) without drying/advanced stabilization (5, 5) truck transport (6, 1)
6,661 136,670 67,800 -
8,125 62,919 45,284 -
area: 97 m2 volume: 2932 m3 land area: 0.4 ha -
38,635
43,703
disposal/Reuse
land application/ fertilizer (7, 2)
18,036
28,911
vehicles required: 1 truck @ 138 trips/yr 15 m3 capacity land area required: 5.2 ha/yr; AR: 14.7 t/ha/yr land area + safety factor: 7.2 ha/yr; application vehicles: 1 tractor @ 6 m3 capacity
SLR: 22.6 kg/m2-day VSD: 47.5%; SRT: 20 days SC: 70.7% -
a Results show equipment capacity, design variables, and associated costs calculated by the model. Numbers next to each treatment alternative are represented as (i, j) where i is the treatment stage and j is the treatment alternative. SLR, Solids loading rate; VSD, volatile solids destroyed; SRT, solids retention time; SC, solids capture rate; AR, application rate for fertilizer use.
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without stabilization (2,4)
anaerobic digestion (2,1)
386 24
gravity belt thickener (1,1) gravity thickener (1,1) 383 14
a
drying bed (4,3)
centrifuge dewatering (4,3) centrifuge dewatering (4,3)
chemical conditioning with polymer (3,1) chemical conditioning with polymer (3,1) without conditioning (3,2) aerobic stabilization (2,2) 317 18
Numbers next to each treatment alternative are represented as (i, j) where i is the treatment stage and j is the treatment alternative.
truck transport (6,1)
truck transport (6,1)
truck transport (6,1)
land application/ fertilizer (7,2) land application/ fertilizer (7,2) land application/ fertilizer (7,2) landfill (7,1) truck transport (6,1)
without drying/advanced stabilization (5,5) without drying/advanced stabilization (5,5) without drying/advanced stabilization (5,5) fluidized bed reactor (5,1) drying bed (4,3) without conditioning (3,2) aerobic stabilization (2,2) 281 22
gravity belt thickener (1,1) gravity thickener (1,2)
disposal reuse transport drying/advanced stabilization dewatering conditioning stabilization thickening total treatment cost [$/t] model run
TABLE 2. Feasible Treatment Scenarios Calculated by the Modela
adjusting those O&M parameters. There was no significant change in data when lime and polymer cost were adjusted. The largest change in total treatment cost occurred when labor costs were adjusted from 15 to 22 US $/h. A 47% raise in labor cost increases the average total treatment costs by 15%. A 180% rise in the unit electricity cost results in an 18% average increase in total treatment cost. When the electricity cost reached 0.14 US $/kWh, the total treatment cost dropped, resulting from or “in a” a switch in process selection. The model made a shift in selecting anaerobic digestion when the unit electricity costs reached 0.14 US $/kWh. The aerobic stabilization process requires a tremendous amount of electricity to operate blowers that supply oxygen to the aeration tank. Anaerobic digestion, on the other hand, requires much less electricity than aerobic stabilization to break down organic material. Supernatant Recycle Flow Analysis. A critical parameter that will influence the selection of the appropriate biosolids treatment scheme is the quality of supernatant recycle flow back to the WWTP. Supernatant flow depends on the biosolids treatment processes, which separate water from solids. Certain processes are known to have direct effects on quality which can influence the operating conditions of the WWTP. For example, drying beds are known to have a solids capture rate range of 70%-80% (17). A low solids capture rate from drying beds results in a higher solids concentration in the recycle flow as compared to centrifuge dewatering (85-90% solids capture rate) (17). Another issue is the high nutrient loads in the recycle flow as a result of anaerobic digestion. Wastewater treatment plants have reported problems with increases of nitrogen and phosphorus in the supernatant recycle flow. This can account for 10%-30% of the influent load (8). The result is an increase in oxygen required to remove ammonia during waste activated sludge treatment. Moreover, optional modifications of plant size and layout are required. Two different treatment scenarios were compared while examining the impact of stabilization processes on supernatant nutrient load. It was assumed that nitrogen content in the influent is 70 mg N/L, phosphorus concentration is around 15 mg P/L, and the centrifuge for dewatering operates 8 h per day, 5 days per week. A condition was set in the model where nitrogen and phosphorus concentrations in the supernatant flow for sludge undergoing anaerobic digestion was 600 mg N/L and 200 mg P/L, respectively. The first supernatant data analyzed came from the following treatment scenario: gravity belt thickening (1,1) f combined aerobic stabilization (2,2) f polymer conditioning (3,1) f centrifuge dewatering (4,2). The increase of nitrogen and phosphorus loads on the WWTP from this scenario was found to be negligible. However, changing the stabilization stage to anaerobic digestion increased nitrogen loads on the WWTP by 11% and phosphorus loads by 17%. Sensitivity Analysis in Reference to the Constraints. An analysis was performed to determine the sensitivity of model constraints on the total treatment cost. One constraint that was defined in the model was that for biosolids to be reused as fertilizer it must be stabilized. The model was rerun 30 times with the original parameters with this constraint removed from the model. The lowest cost scenario selected by the model was exactly the same scenario as the first results, however, without the stabilization stage. The model calculated an annual treatment cost of 371,081 US $/yr which is 206 US $/t. A comparison was made to the previous result from the model prior to removing the stabilization constraint. The previous result calculated an annual treatment cost of 458,330 US $/year which is 254 $/t. This shows that the impact the stabilization constraint has on the treatment scheme is 84,264 US $/year or 19% of the annual treatment cost. Figure 5 compares how this constraint influences each treatment VOL. 44, NO. 14, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 3. Comparative Sensitivity Results of O&M Parameters on the Total Biosolids Management Cost fuel cost 1 $/L mean [$/t] SD minimum [$/t] P-value
4 $/L
349 415 62 79 254 305 3.32 × 10-4
polymer cost
labor cost
lime cost
electricity cost
1 $/kg
5 $/kg
15 $/h
22 $/h
0.10 $/kg
0.90 $/kg
368 110 258
349 62 254
349 62 254
402 110 279
369 74 256
349 62 254
0.23
process. The graph illustrates that there are increased dewatering and transportation costs when the biosolids are not stabilized. The greatest opportunity to improve the overall treatment process based on results from this model would be to implement a cheaper stabilization process that fulfills government regulations and does not increase transportation and dewatering costs.
0.02
0.147
0.05 $/kWh
0.14 $/kWh
349 412 62 97 254 287 1.27 × 10-3
may show whether GA is truly the best modeling method for biosolids management.
Supporting Information Available Complete list of model parameters and constraints, model design equations,anddetailedexplanationofunitprocesses.Thismaterial is available free of charge via the Internet at http://pubs.acs.org.
Discussion Genetic Algorithm’s major advantage is the capability to analyze multiple objective functions, design variables, and constraints. It is recommended that the model should be populated with multiple requirements defined by the decision makers. For instance, WWTP managers may define a maximum return flow solid concentration or maximum nitrogen load acceptable. These can be defined in the model to provide feasible solutions which meet demands defined by the wastewater treatment plant. The model can be used to evaluate and compare new technologies to existing technologies on the market. Potential revenues from biosolids products may show profit from implementing a biosolids treatment scenario. When the model was developed, the cost data were adjusted to 2007 currency, using the Marshall and Swift Equipment Cost Index and the Engineering News Record Construction Cost Index (18, 19). It is recommended to use site-specific cost data relevant for the problem to be solved. This paper tested the method of using Genetic Algorithm solely for selecting the appropriate Biosolid Treatment Train. Improvements to the graphical-user-interface may be required to provide an efficient, user-friendly tool for WWTP planners. Market research and analysis may also be required to provide a product that fully integrates the different needs of the industry. Future work may consist of comparing results from this model with different search and modeling techniques. This
FIGURE 5. Sensitivity analysis on the constraint that sludge must be stabilized prior to reuse as fertilizer. Results show that scenarios without stabilization can increase dewatering and transport costs. Numbers next to each treatment alternative are represented as (i, j) where i is the treatment stage and j is the treatment alternative.
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