Environ. Sci. Technol. 2005, 39, 2394-2405
Systematic Procedure for Designing Processes with Multiple Environmental Objectives KI-JOO KIM† AND RAYMOND L. SMITH* National Risk Management Research Laboratory, Office of Research and Development, U.S. Environmental Protection Agency, 26 West Martin Luther King Drive, Cincinnati, Ohio 45268
Evaluation of multiple objectives is very important in designing environmentally benign processes. It requires a systematic procedure for solving multiobjective decisionmaking problems due to the complex nature of the problems, the need for complex assessments, and the complicated analysis of multidimensional results. In this paper, a novel systematic procedure is presented for designing processes with multiple environmental objectives. This procedure has four steps: initialization, screening, evaluation, and visualization. The first two steps are used for systematic problem formulation based on mass and energy estimation and order of magnitude analysis. In the third step, an efficient parallel multiobjective steady-state genetic algorithm is applied to design environmentally benign and economically viable processes and to provide more accurate and uniform Pareto optimal solutions. In the last step a new visualization technique for illustrating multiple objectives and their design parameters on the same diagram is developed. Through these integrated steps the decisionmaker can easily determine design alternatives with respect to his or her preferences. Most importantly, this technique is independent of the number of objectives and design parameters. As a case study, acetic acid recovery from aqueous waste mixtures is investigated by minimizing eight potential environmental impacts and maximizing total profit. After applying the systematic procedure, the most preferred design alternatives and their design parameters are easily identified.
1. Introduction Evaluation of environmental, health, and safety indices is very important in designing for the environment due to stringent environmental regulations and high public awareness as well as industry stewardship. Traditional product and process designs have not focused on these indices; however, recent product and process designs have considered these indices from the early design stages for better economic performance and ecological and environmental quality. Life cycle assessment (1) is the most popular method for ecological and environmental evaluations for environmentally benign product design. For environmentally benign process designs, several methodologies have evolved recently. Pistikopoulos and co-workers (2, 3) introduced a methodology for environmental impact minimization that applies life * Corresponding author telephone: (513)569-7161; fax: (513)5697111; e-mail:
[email protected]. † ORISE Post-Doc. 2394
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cycle assessment principles to a process optimization framework. This method defines an environmental impact vector based on short- or long-term effects of impacts and then incorporates the impacts as process design objectives together with economics in a multiobjective optimization problem. This methodology has been applied to a simplified chemical reaction-separation process and a methane chlorination process. Cano-Ruiz and McRae (4) published a review paper about incorporating environmental issues into the design of new processes and manufacturing facilities. This review explains the need to view environmental issues as part of the design objectives rather than as constraints on operations. A method for ecological and economic assessment during process design has been developed by Heinzle et al. (5). On the basis of simple mass balances, they defined three indices: mass loss indices, ecological indices for byproduct formation, and economic indices. Mass loss indices represent a rough measure of the ecological impact of a reaction system and are used to yield ecological and economic indices using weighting factors. The ecological weighting factors are based on three groups of environmental impacts for identifying serious problems of the process. Economic indices rely on raw materials and waste treatment costs and may also include equipment and operating costs. This method was applied to a fine chemical process and provided more cost-effective and environmentally friendlier chemical processes. The Waste Reduction (WAR) algorithm, developed by the U.S. Environmental Protection Agency (6-8), involves potential environmental impact (PEI) balances of mass and energy crossing the system boundary. Through the PEI balance, the generation of potential environmental impacts provides a relative indication of the environmental friendliness or unfriendliness of the system. Eight potential environmental impacts ranging from human and ecological toxicity to global warming potential are used in this algorithm. Applications include the effect of in-process recycle on the potential environmental impacts from a methyl ethyl ketone production plant. Shonnard and Hiew (9) developed an environmental fate and risk assessment tool (EFRAT) to integrate all of the key steps of impact assessment into a single methodology and software tool. After obtaining emission data from mass and energy balances, this tool is applied to evaluate process releases, fate and transport of pollutants based on the Level I multimedia model (10), exposure potentials, and finally relative risks based on nine environmental and human health impact indices. This tool has been applied to designs for solvent recovery from gaseous streams. More recently, Chakraborty and Linninger (11) developed a two-stage combinatorial flow sheet synthesis method for plant-wide waste management. The first stage involves generating candidate process designs guided by regulation limits and property information and then selecting best designs by minimizing cost and a modified global environmental impact vector. The second stage optimizes total cost and a global pollution index for designing environmentally friendlier processes. Even though many useful methodologies for designing environmental processes and systems with multiple objectives are summarized here, they are often limited in their implementation. For example, many of the methods still use binary objectives (e.g., cost and environmental impact) after aggregating all environmental and ecological impacts into a single impact value. This limitation is mainly caused by multiple objectives that increase complexity and difficulty 10.1021/es0490424 CCC: $30.25
2005 American Chemical Society Published on Web 02/24/2005
FIGURE 1. Systematic procedure for multiobjective decision-making. of the design problem. To overcome these obstacles and solve true multiple objectives efficiently, we present a systematic procedure for designing processes with multiple environmental objectives. This procedure has several novel concepts that are applied to analyze the relative importance of environmental impacts, to evaluate true multiple objective problems using a novel multiobjective decision-making tool, and to visualize multiple objectives and multiple design parameters on the same plot. With this procedure, the decision-maker can easily determine more favorable design alternatives based on his or her preferences. The remainder of this paper has been written in four sections. Section 2 explains a basic scheme of the systematic procedure for multiobjective design problems, and the subsequent sections are lined up according to the steps in the procedure. The potential environmental impacts of the WAR algorithm are used as environmental indicators, and the PEI balances based on mass and energy crossing the system boundary are derived in section 3. As actual evaluation of multiobjective design problems is very important, Section 4 describes some of the traditional methods and provides our novel method based on a parallel multiobjective genetic algorithm for fast and accurate evaluations. In section 5, the systematic procedure is applied to design processes for acetic acid recovery from aqueous waste mixtures, and the results including a new visualization method are discussed.
2. Systematic Procedure for Multiobjective Design Problems As described, many decision-making tools for designing environmental processes with multiple objectives have considered only binary objectives such as economic performance and environmental impact. Even though various environmental and ecological impacts are included in the design problem, only a single, combined environmental impact is actually used as a design objective. This is mainly due to the complex nature of multiple objectives, which makes the design problem hard to solve, partly due to complicated topology (e.g., noncontinuous surfaces, deep valleys) of the objective domain and partly due to the prohibitive time scale of conventional solution techniques as the number of objectives increases. To overcome these obstacles and include all aspects of the multiobjective decision-making process, a systematic procedure illustrated in Figure 1 is developed. This procedure composes four steps: initialization, screening, evaluation, and visualization. The first step, “initialization”, involves identification of possible economic and environmental objectives and how to measure these objectives. For the examples described here, environmental objectives are expressed by the potential environmental impacts of the WAR algorithm and are functions of mass and energy crossing the system boundary and the WAR database of normalized impact scores. This step also involves estimation of the amounts of waste and energy emitted to the surrounding environment. This estimation can be done
without any rigorous process simulation, and the estimated values are to be used in the second step to determine relative magnitudes of environmental effects. “Screening”, the second step, performs an order of magnitude analysis of the potential environmental impacts. This analysis determines relative magnitudes of the impacts, and only those having significant magnitudes are used in the final formulation of the environmental design problem. This step also analyzes relative importance of mass and energy contributions to the potential environmental impacts, helping the decision-maker understand the behaviors of the environmental impacts. The third step evaluates the environmental design problem using a multiobjective decision-making tool. Since there are a large number of these tools, section 4.1 summarizes some of the traditional methods such as goal programming and the weighting method, and sections 4.2 and 4.3 explain our novel multiobjective decision-making tool based on a parallel multiobjective genetic algorithm. The last step, “visualization”, is also essential in this procedure because there is no simple way to visualize multiple objectives and design parameters in two- or threedimensional space. A new visualization technique based on a lower triangular set of diagrams and a design alternative diagram shows all of the multiple objectives and illustrates the Pareto optimal solutions and their design parameters on the same two-dimensional space. This technique helps the decision-maker pinpoint the most preferred design alternatives. In addition, this technique is independent of the number of objectives and design parameters.
3. Potential Environmental Impacts The potential environmental impacts of the WAR algorithm (7, 8) are used as the environmental objectives in this paper. A general potential environmental impact balance of mass and energy crossing the system boundary is as follows:
dI in out I energy - ˆI out I energy + ˆIgeneration ) ˆI in mass + ˆ mass - ˆ dt
(1)
where ˆI is the rate of potential environmental impact crossing the system boundary or generated. This PEI balance measures a relative indication of environmental friendliness or unfriendliness of the design under consideration. A design having lower impacts is better in ecological and environmental aspects. The above PEI balance can be applied to batch processes as well as continuous processes. For example, a batch distillation process, one of the common separation techniques for separating valuable components or removing unwanted components in fine chemical and pharmaceutical industries, is illustrated in Figure 2, where a feed is charged to the top still pot and the product is recovered at the bottom. It is known that this column configuration is better for separating a heavy product at the bottom (12). As there is no reaction (Iˆgeneration ) 0), no continuous mass input (Iˆ in mass ) VOL. 39, NO. 7, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. Normalized Impact Scores of Acetic Acid and Coal Energy impact category, j
ψj (kg)
ψjE (MJ)
human toxicity potential by ingestion (HTPI) human toxicity potential by inhalation or dermal exposure (HTPE) terrestrial toxicity potential (TTP) aquatic toxicity potential (ATP) ozone depletion potential (ODP) global warming potential (GWP) photooxidation chemical potential (POCP) acidification potential (AP)
0.1065
7.83E-5
0.0117
1.22E-6
0.1065 0.0107
7.83E-5 2.65E-4 2.03E-9 1.93E-4 7.07E-8 5.98E-3
FIGURE 2. Batch-stripping column for waste solvent recovery. The dotted line is the system boundary. out 0), and no energy output (Iˆ energy ) 0), the potential environmental impact for this batch process becomes:
dI in ) ˆI energy - ˆI out mass dt
(2)
After integration mass I - Io ) I energy - I out in mass I ) (Io - I out ) + I energy in mass ) I residue + I energy in
(3)
where Io is the initial potential environmental impact of the waste solvent. Residue is the amount of mass left in the system after separation. Then, for a given impact category j, the above equation becomes:
FIGURE 3. Basic concepts in multiobjective decision-making. In the past, a combined single value of impacts has been used to evaluate environmental friendliness or unfriendliness of a given system (11, 13, 14) and is defined as: 8
Itotal )
mass energy Ij ) I residue,j + I in,j
)
∑S ψ i
ij
+ QRψEj
(4)
where j is an index for the eight impact categories and i is an index for chemical components. Si is the amount of residue material (in kg) after separation, and QR is energy consumption (in MJ) for that separation task. ψij is a normalized impact score of component i in impact category j, and ψEj is a normalized impact score of energy in impact category j. The underlying basis of the potential environmental impact is that the slope of the database value (impact/release) represents the slope of the traditional dose-effect curve above the threshold value. The impact categories and their normalized scores for a system having acetic acid and coal energy are summarized in Table 1. In this table there is only one component, and thus eq 4 becomes: Ij ) Sψj + QRψEj after dropping the subscript i for simplicity. The impact categories include human, aquatic, terrestrial, and atmospheric environmental impacts. In this analysis ψHTPI and ψTTP have identical scores because they are based on the same indicator for toxicity of chemicals, LD50. The effects of acetic acid on ozone depletion, global warming, photoxidation, and acidification impacts are negligible according to the WAR database, and thus values are not given in this table. However, coal energy definitely affects all of these impact categories. 9
j
j
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i
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where wj is a weighting factor for the impact category j. The weighting factor represents relative or site-specific importance of the impact category. For instance, if the decision-maker were evaluating a process that is located in an urban area having frequent smog alerts, the weighting factor for smog formation (i.e., POCP) would probably receive a high value. If Itotal of process (or system) A is lower than that of process B, then process A would be claimed as an environmentally friendlier process than process B. However, it is not trivial, nor objective, to obtain or guess the proper weighting factors for a combined impact value. Instead of assigning arbitrary weighting factors, the individual impacts Ij are evaluated in this paper in order to analyze all possible behaviors of the potential environmental impacts.
4. Multiobjective Decision-Making Tool This section, which corresponds to the third step in the systematic procedure shown in Figure 1, provides a theoretical background for multiobjective optimization problems (MOPs), traditional methods for solving MOPs, and a novel multiobjective decision-making tool developed by the authors. 4.1. Traditional Methods. Figure 3 illustrates several important concepts in multiobjective design problems. If
the design problem is to minimize the objectives, solution a is dominated by or inferior to solutions b and c since solutions b and c have smaller f1 and f2 values than those of solution a. The first design objective (f1) of solution b is superior to that of solution c, but improving the second design objective (f2) of solution b cannot be achieved without sacrificing the first design objective. Thus solutions b and c are called nondominated. The current solutions found by an MOP method are located somewhere in the feasible design space and are used to explore the design space to find these nondominated solutions, often called Pareto optimal solutions, after the French-Italian economist and sociologist V. Pareto (15). The final nondominated solutions are the design alternatives that the decision-maker wants to get and investigate further. A space comprised of an infinite number of Pareto optimal solutions is called a Pareto front, denoted as PF *. Therefore, the goal of multiobjective decision-making tools is to accurately approximate the true Pareto front with a given finite number of solutions. This finite number of solutions is the number of Pareto optimal solutions (Npareto) that the decision-maker wants to see. Due to this approximation with a finite number of solutions, solution accuracy and uniformity in the Pareto front are critical properties of multiobjective decision-making tools. There are many analytical techniques to solve multiobjective design problems. Most of the traditional methods convert multiple objectives to a single objective, which is finally optimized. These methods are generally divided into two basic types: preference-based methods and generating methods. Preference-based methods such as goal programming attempt to quantify the decision-maker’s preference or goal, and with this information, the solution that best satisfies the decision-makers’ preference is then identified. A simple formulation of this method is given below: k
min
∑p | f - G | i
i
i
(6)
i)1
where k is the number of objectives and pi is a priority factor associated with the goal Gi. To find another design alternative, a different set of goals should be provided by the decisionmaker, and the above optimization problem is repeated. Among the literature cited in this paper, Chakraborty and Linninger (11) applied goal programming in their plant-wide waste management. Generating methods, such as the weighting method and the -constraint method, have been developed to find the exact Pareto set or an approximation of it. In the weighting method, a single objective function is formulated by adding up all the objectives with weighting factors and is then optimized to get a Pareto optimal solution. Similarly, a different set of weighting factors is used to find another design alternative. In the -constraint method, which is a pure mathematical method without any adjustable parameters, one objective is arbitrarily selected while the remaining objectives are turned into constraints:
min fi s.t. fj e i where i ) 1, ..., k; i * j
(7)
where i is a randomly chosen value between the minimum and maximum values of fi. Thus, the minimum and maximum values for each design problem should be known a priori. By changing i values, different design alternatives can be obtained (3, 16). However, these preference-based and generating methods have several critical disadvantages in their implementations.
The first disadvantage is the difficulty of quantifying decisionmaker’s preferences in goal programming or assigning sound weighting factors in the weighting method. Often decisionmakers would rather not quantify their preferences; they prefer to see options with various results and choose among them. If the decision-maker does specify preferences, then improper selection of these values results in uneven distribution of the Pareto optimal solutions on the Pareto front or fails to approximate the true Pareto front. The second disadvantage of preference-based and generating methods is that they tend to generate possible solutions for the Pareto front one at a time and then repeat the optimization routine to get another solution. Solution uniformity, which is one of the important properties of multiobjective decision-making tools, is solely governed by how parameters are selected (e.g., goals or weighting factors, or -constraints) to get another Pareto optimal solution. However, even uniform selection of parameters in a multidimensional parameter domain does not guarantee uniformity in the objective domain. In addition, there is no exchange of information like solution crowdedness among independent optimization runs to improve solution uniformity on-the-fly. Another severe problem is that most of these methods are very sensitive to the shape of the Pareto front such as convexity and continuity and cannot handle these problems in many cases. Since nonconvex and/or noncontinuous design problems are common in science and engineering, a new multiobjective decision-making tool is required to handle nonconvex and/or discontinuous systems as well as obtain uniform Pareto optimal solutions. 4.2. Multiobjective Evolutionary Algorithm. The aforementioned problems of the traditional multiobjective decision-making tools can be easily tackled when a multiobjective evolutionary algorithm (MOEA) is applied. Many MOEA implementations, test problems, and applications can be found in a recent book by Coello Coello et al. (17). One of the famous evolutionary algorithms is a genetic algorithm (GA), which is a population-based design technique (18). A group of individuals, called a population, undergoes some evolutionary operations such as selection, crossover, and mutation to find the best-fit individual and eventually the best-fit population. An individual (also called a chromosome or string) is used to represent the design parameters. Then fitness of an individual, an indication of how well an individual can survive under given evolutionary conditions, is calculated based on the objective function value. More fit individuals transmit their genetic properties to next generations through their offspring. Finally, over a large number of generations a population gradually evolves to one filled with best-fit individuals. Due to its population-based nature, an evolutionary algorithm can handle multiple solutions (i.e., individuals) at each iteration, leading to easy implementation of an evolutionary algorithm to multiobjective design problems. Since a MOEA does not rely on derivative information to explore the design surface (which is commonly required in preference-based and generating methods), MOEAs can be used to solve complex nonlinear, nonconvex design problems. MOEAs have been applied in various fields of science and engineering, and some of the MOEA applications in the environmental engineering field include groundwater pollution remediation (19), water quality control (20), and water distribution networks (21). Solution accuracy and uniformity are two important factors for accurate representation of the true Pareto front, and thus MOEA implementations basically differ in two things: how to search toward the Pareto front and how to maintain well-distributed solutions. On the basis of these two important factors, we have developed a new multiobVOL. 39, NO. 7, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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and gets the results back from them. This parallelization model is especially efficient if the evaluation of design alternatives takes up most of the computer time. We have integrated our MSGA on top of PGAPack (28), which is a master-servant genetic algorithm package for single objective optimization problems. The resulting integrated technique, parallel MSGA, is then performed using a Beowulf cluster (29) with 16 Linux servant processors (Red Hat Linux 7.1). The cluster machines each have Athlon XP 2000 CPU and 256 MB RAM. The implemented library for message passing interface (MPI) is MPICH (version 1.2.1) (30).
5. Case Study: Waste Solvent Recovery
FIGURE 4. General procedure of multiobjective steady-state genetic algorithm. Nfirst is the number of solutions in the first Pareto front (PF 1) of the current population. Npareto is the number of solutions in the true Pareto front (PF *) after many generations, which corresponds to the number of design alternatives that the decisionmaker wants to consider. jective evolutionary algorithm, called multiobjective steadystate genetic algorithm (MSGA) (14), whose schematic diagram is illustrated in Figure 4. MSGA features nondominated sorting (22) to search toward the Pareto front and fitness sharing (23) and sharing again (14) to maintain welldistributed nondominated solutions. Full descriptions of these three features are explained in the Supporting Information. The innovative ideas of MSGA are (i) a new fitness sharing function, which measures relative crowdedness of design alternatives, for fast and accurate evaluation of survival of fitness; (ii) a “sharing again” procedure for significant improvement of solution uniformity; and (iii) a steady-state genetic algorithm that does not require an external population to archive Pareto optimal solutions. To illustrate solution accuracy and solution distribution of MSGA, it is compared with the two most common MOEA implementations, NSGA-II (24) and SPEA (25). Figure 5 shows typical results of a three-dimensional sphere test problem (14), where three random numbers are drawn in a unit cube. Since the constraint is that the sum of the squares of the numbers is equal to or greater than 1.0, the feasible space is the outer region of the sphere in the cube. All three techniques perform well in approximating the true Pareto front (i.e., the surface of the sphere); however, NSGA-II has definitely failed to evenly distribute Pareto optimal solutions over the Pareto front due to many identical solutions. Spacing (26) measures the variance of the minimum distances between design alternatives and will be zero if all the solutions are equally distributed. The resulting spacing values are 0.503 for NSGA-II, 0.020 for SPEA, and 0.015 for MSGA, which shows a better solution distribution for MSGA. 4.3. Parallel Multiobjective Evolutionary Algorithm. Even though the developed MSGA can generate accurate and uniform Pareto optimal solutions, the main disadvantage of the evolutionary algorithm is computational time. This obstacle can be eased by parallelizing the MSGA. Fortunately, GA can be easily parallelized because it is a population-based technique. The commonly used GA parallelization model is the master-servant model (27), where the master processor assigns unevaluated individuals to the servant processors 2398
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5.1. Initialization. Solvents are widely used as dissolving, separating, drying, and cooling agents and as reaction media in the chemical processing industries. Solvents are essential in making these processes economically feasible; however, waste solvents from these industries are a main source of pollution to the environment if they are not properly controlled. Waste solvents also reduce economic performance due to loss of atom economy. If one has to send these waste solvents to a wastewater treatment facility, then all of their economic value is lost. In addition, wastewater treatment facilities have air emissions as described in AP-42 (31), although this alternative is not considered here. Another commonly practiced method for handling waste solvents is recovery and recycle of useful solvents from waste mixtures (11, 32-34). However, designing a waste solvent recovery process does not guarantee a benefit to the environment nor guarantee economic benefit. Even if a proposed process is economically favorable, it may generate more environmental impacts than the original waste solvents. Or the opposite scenario of environmentally favorable but economically infeasible design can be possible. Therefore, designing environmentally benign processes should incorporate both economic and environmental benefits simultaneously. In this industrial case study, we want to recover acetic acid from aqueous waste mixtures from the pharmaceutical industry. In this industry, batch distillation (as shown in Figure 2) can be a preferred separation technique and is applied to this environmental design problem. The goal of this case study is to design a waste solvent recovery process for maximizing total profit and minimizing all eight potential environmental impacts. Thus the design problem is
min [-profit, IHTPI, IHTPE, ITTP, IATP, IODP, IGWP, IPOCP, IAP]T (8) subject to
g1(x) ) xHOAc g 0.99 g2(x) ) B g 1/6 feed (or 16.7% recovery) where profit is total profit ($/h), Ij is the potential environmental impact j of the WAR algorithm, and gi are design constraints. Design parameters (x) are teq for startup time (h), t for batch time (h), RB for reboil ratio, V for boil-up rate (kmol/h), and N for the number of stages. Note that N is an integer design parameter. The lower and upper bounds for these design parameters are available (14). xHOAc is the mole fraction of acetic acid in the recovered bottom product (B) and is equivalent to technical grade in the market. The constraint for the amount of acetic acid recovery is relaxed in order for parallel MSGA to generate more feasible solutions during the evolutionary process.
FIGURE 5. Pareto optimal solutions of the three-dimensional sphere problem.
TABLE 2. Order of Magnitudes for Analyzing Potential Environmental Impacts category, j
O (ψj)
O (ψjE)
O (ψj/ψEj )
O (S/QR)
O (Imass/Ienergy)
O (Sψj/QR)
O (Ij /QR)
HTPI, TTP HTPE ATP ODP GWP POCP AP
-1 -2 -2 -∞ -∞ -∞ -∞
-4 -6 -3 -9 -4 -7 -2
3 4 1 -∞ -∞ -∞ -∞
-1 -1 -1 -1 -1 -1 -1
2 (1.88) 3 (2.73) 0 (0.36) -∞ -∞ -∞ -∞
-2 -3 -3 -∞ -∞ -∞ -∞
-2 -3 -3 -9 -4 -7 -2
The total profit is defined as:
profit )
B × PriceHOAc total annualized cost (9) total batch time yearly operating hours
where PriceHOAc is the purchase price of technical grade acetic acid. The first term represents recovery profit, and the second term indicates total annualized cost. The total annualized cost includes costs of the distillation column, heat exchangers, and utilities. The required parameters for evaluating the total annualized cost have been described (14). The expression for the potential environmental impacts is already given in eq 4. If the waste solvent mixture of 100 kmol has 30 mol % of acetic acid, then the initial potential environmental impacts are as given below: IHTPI
IHTPE
ITTP
IATP
IODP
IGWP
IPOCP
IAP
191.7
21.1
191.7
19.3
0
0
0
0
Note that there is no impact caused by energy consumption yet. The Pareto optimal solutions should have lower IHTPI, IHTPE, ITTP, and IATP values than these initial values even if energy is utilized for waste solvent recovery. In addition, IODP, IGWP, IPOCP, and IAP, which are impact generations due to energy consumption, should be minimized. 5.2. Screening. The multiobjective environmental design problem shown in eq 8 can be solved using the developed parallel MSGA. However, careful observation of the normalized impact scores in Table 1 and an “order of magnitude” analysis for the potential environmental impacts can eliminate some of the environmental objectives. If we know the approximated values of the mass left in the still pot S and the reboiler heat duty QR in eq 4 a priori, we can estimate the relative importance of the potential environmental impacts and their mass and energy contributions to the impacts. Suppose that the waste solvent of 100 kmol contains 30 mol % of acetic acid and half of it is recovered. Then B (acetic acid recovery) and S (i.e., amount of residue) are equal to 15 kmol (900 kg). The heat supplied to the reboiler is essentially used for providing reboil reflux
for the column, and thus the total heat consumption to the reboiler can be estimated using the following equation (35)
QR ≈ ∆Hv,avgRBB
(10)
where ∆Hv,avg is the average heat of vaporization of the mixture. From a calculation described in the Supporting Information, the approximated heat duty is 12400 MJ. On the basis of these estimated values of S and QR and the normalized impact scores in Table 1, the order of magnitude analysis is summarized in Table 2. Since HTPI and TTP have the same normalized impact scores, they are listed on the same row in this table. The order of magnitudes, O(‚), of ψj, ψEj , and ψj/ψEj are calculated from the values given in Table 1. As ODP, GWP, POCP, and AP have zero normalized impact scores for acetic acid, O(ψj) and O(ψj/ψEj ) of these impact categories become negative infinity. The O(Imass/Ienergy) in the sixth column represents the relative importance of mass and energy contributions to the environmental impacts and, from eq 4, is defined as:
Imass energy
I
∑Sψ i i
)
QRψEj
ψj
ij
w
ψEj
×
S QR
for i ) 1
(11)
where O(ψj/ψEj ) is already known. Since the assumed S is 900 kg and the estimated QR is 12 400 MJ, O (S/QR) is -1 (more precisely, -1.14) for all impact categories. Note that O(S/QR) from the experimental results is -1.08 ( 0.35, which agrees quite well with the estimated order of magnitude. Thus O(Imass/Ienergy) can be estimated by just adding the fourth (O(ψj/ψEj )) and fifth (O (S/QR)) columns, as shown in the sixth column with experimental values in parentheses. Higher O(Imass/Ienergy) means higher mass contribution to the potential environmental impact. For example, O(Imass/Ienergy) of IHTPI (and also ITTP) is two, indicating that there is a greater mass contribution than energy contribution to human toxicity by ingestion and terrestrial toxicity. This mass contribution is even greater in IHTPE whose O(Imass/Ienergy) is three. However, ATP has zero order of magnitude, and this means that both VOL. 39, NO. 7, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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mass and energy contributions are important in measuring ATP. As explained, the O(Imass/Ienergy) of ODP, GWP, POCP, and AP are negative infinity since there is no mass contribution. Besides the relative importance of mass and energy contributions, the relative magnitude of the impact categories is also crucial in analyzing the multiobjective design problems. The last column O(Ij/QR) represents the relative magnitude of the potential environmental impact categories and, again from eq 4, is defined as:
Ij Sψj ) + ψEj QR QR
(12)
The O(Sψj/QR) in the seventh column can be easily calculated by adding the second and fifth columns, and O(ψEj ) is already given in the third column. Thus, the order of magnitude of the left side of eq 12 (i.e., the last column of Table 2) is the dominant term between the third and seventh columns. The orders of magnitude of ODP and POCP are relatively very small as compared to the other values in the last column of Table 2. For POCP to be considered, the weighting factor of POCP, wPOCP (introduced in eq 5), would have to be O(4) times higher than that of HTPE or ATP, and this magnitude of a weighting factor is generally unrealistic. Thus, ODP and POCP impacts can be removed from the environmental design problem. Two additional potential environmental impacts can be removed from the original design problem. GWP is also relatively small but not negligible. However, its behavior is exactly the same as AP since they are only a function of energy use. This can lead to removal of GWP from the design problem. HTPE can also be eliminated because HTPE has smaller relative importance than HTPI (based on the last column) and has similar trend parallel to that of HTPI (based on the sixth column, which shows that both are predominantly a function of mass of residue). 5.3. Evaluation. On the basis of the order of magnitude analysis in the previous section, the original multiobjective design problem expressed in eq 8 can be reduced to the following form:
{
f1(x) ) - profit f2(x) ) IHTPI min f3(x) ) IATP f4(x) ) IAP
(13)
where IHTPI is predominantly a function of mass of residue, IAP is a function of energy use, and IATP depends on both mass and energy. Other impact categories have been eliminated because they have similar trends (e.g., TTP and HTPE are similar to HTPI; ODP, GWP, and POCP are similar to AP) and because their relative impacts are so small (e.g., ODP and POCP). Simulation of batch distillation (shown in Figure 2) for acetic acid recovery is conducted using a professional batch distillation simulation package, MultiBatchDS (35). The batch distillation model has 80 ordinary differential equations and 202 algebraic equations when the maximum number of stages is 40 (including a top still and a reboiler), which is the upper bound of the number of stages, and the number of components is two (water and acetic acid). The differential equations are solved using LSODE (36), not Runge-Kutta method, because of the stiffness of the differential equations. The population size is 400, and this large population size is caused by the tight acetic acid purity and the constraint handling method (37) implemented in our parallel MSGA. The constraint handling method requires at least one feasible 2400
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FIGURE 6. Pareto optimal solutions for xF ) 0.30. solution from the initial population. The replacement size, which is the number of new individuals created at each generation, is 200. The difference between the population and replacement sizes is equal to the size of the Pareto optimal solutions (Npareto), which is the number of design alternatives that the decision-maker wants to consider. Parallel MSGA is performed with 16 servant processors, and the evolutionary process is terminated after 1000 generations. The Linux cluster completed this job in 16.7 h. Without parallelization the total computing time could be approximately 10 days (16.7 h × 16 machines × 90% parallel computing efficiency due to communications). This reduction in computation time shows the benefit of using the parallel MSGA and is an indication of greater time savings for more complex problems. 5.4. Visualization of Pareto Optimal Solutions. 5.4.1. Traditional Representation. Figure 6 is a traditional way of illustrating multiple objectives, in which three potential environmental impacts are shown with respect to the total profit for xF (mole fraction of acetic acid in the feed) of 0.30. Each plot in this figure has the same 200 Pareto optimal solutions but has a different y-axis. Several interesting results can be found from this figure. First of all, the impact values of HTPI and AP are very large compared to the ATP values due to their high normalized impact scores. The variation is even larger in AP since AP is initially zero. Second, the IATP trend is somewhat between the behaviors of the massdominated impact (i.e., IHTPI) and the energy-dominated impact (i.e., IAP). This is caused by the zero order of magnitude of Imass/Ienergy of ATP, which means that both mass and energy contributions to the environmental impact are important. Another interesting result is that there are some economically unfavorable design alternatives (i.e., negative total profit) in the Pareto optimal solutions, and the percentage of these alternatives is roughly 7%. In this region, IHTPI is almost constant while IATP and IAP are increasing. Only the
FIGURE 9. Three-dimensional representation of the four objectives for xF ) 0.30. FIGURE 7. IHTPI and IAP changes with respect to the reboil ratio.
FIGURE 8. Traditional diagram for two objectives: IHTPI and IAP. reboil ratio RB among the design parameters can explain this behavior of the potential environmental impacts. Figure 7 shows IHTPI, IAP, and the bottom production rate (B) with respect to the reboil ratio. As the reboil ratio is increased, energy consumption (implicitly expressed as IAP) is almost linearly increased while the bottom production is increased at first and then has little change later. The constant values of IHTPI in this economically unfavorable region are mainly governed by the constant bottom production (i.e., recovery rate) even at high reboil ratio and high energy consumption. This supports the assertion that Imass has a greater contribution to IHTPI than Ienergy does, as expected according to the “order of magnitude” analysis. Also supporting the analysis, IAP keeps increasing in this region because IAP is only a function of energy, which is strongly governed by the reboil ratio. The order of magnitude analysis also illustrates an interesting relationship between HTPI and AP. The inverse proportionality shown in Figure 8 is caused by the dominant contribution of Imass to IHTPI and dominant contribution of Ienergy to IAP. A sharp increase of IAP at IHTPI of around 92 means that energy consumption is rapidly increased due to a high reboil ratio and a constant bottom production rate. A three-dimensional plot, another traditional way of visualization, can also be used in this case because the original design problem has been reduced to a four-objective design
problem. Figure 9 is a three-dimensional representation of the four objectives, where the total profit, IHTPI, and IAP are on the three-dimensional axes while IATP is the gray color scale on the Pareto optimal surface. Darker gray color is for lower IATP values while lighter color is for higher IATP values. Since both mass and energy contributions are important to the aquatic toxicity potential (ATP), lower IATP values are scattered on the middle region for the Pareto optimal surface. From this Pareto front, the decision-maker can find his or her preferred region of design alternatives. For example, the circled region in this figure is good for high total profit and relatively low IATP, IHTPI, and IAP. 5.4.2. New Visualization Technique. Even though the traditional visualization methods as shown in Figures 6 and 9 are useful in analyzing the general behaviors of Pareto optimal solutions and determining preferred design alternatives, there is no simple way to provide the corresponding design parameters. The decision-maker should open the original data (generally in text or spreadsheet format), look up the objective function values, and finally read the matching design parameters. In addition, the traditional visualization methods can only be used for problems having less than five objectives. For these reasons a new visualization technique is needed and embedded as the last step in the systematic procedure shown in Figure 1. This new visualization technique gives all of the results of the traditional visualization methods plus more features such as an integrated view of objectives and design parameters, easy assessment for preferred design alternatives, and easy evaluation of best and worst scenarios. Moreover, this technique is independent of the number of objectives, the number of design parameters, and the number of designs. The new visualization technique has two steps: one for obtaining a lower triangular set of diagrams (or design planes) (Figure 10) and the other for visualizing both objectives and design parameters in a design alternative diagram (Figure 11). If there are k objectives, (k - 1)2 binary pairs of the design k-1 objectives are possible. As only ∑i)1 i pairs are unique among all the pairs, a lower triangular set of diagrams can be constructed systematically. This construction scheme borrows several procedures from our multiobjective decisionmaking tool, parallel MSGA. For each binary pair, all 200 Pareto optimal solutions enter a “nondominated sorting” module where a new set of Pareto optimal solutions is found in terms of the given binary design objectives. The number of new Pareto optimal solutions for the binary pairs varies. In this work these sets range in size from 13 to 77 solutions for each binary pair. The resulting set goes through the sharing again” module where the most crowded design alternative VOL. 39, NO. 7, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 10. Lower triangular set of diagrams (or design planes) for the Pareto fronts of the given binary pairs. Each design plane is designated by a Roman numeral. Open symbols show the same designs in multiple design planes.
FIGURE 11. Pareto optimal solutions and their optimal design parameters in this design alternative diagram. Results are segmented according to the design planes of Figure 10. t ) batch time (h); RB ) reboil ratio; V ) vapor flow rate (kmol/h); N ) number of stages. is removed, and then the fitness sharing module assigns new fitness values to the remaining members. These sharing again and fitness sharing modules are repeated until the number of uniformly spaced design alternatives is equal to a certain number of solutions that the decision-maker wants to see (six in this example). Thus, these procedures offer six representative design alternatives in each binary pair and in total 36 design alternatives. The resulting lower triangular set of diagrams (or design planes) is illustrated in Figure 10. Note that some of the Pareto optimal solutions are identical 2402
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across the different pairs of binary design objectives. For example, design numbers 1, 7, and 13 are identical solutions. Generally many of the end points of the Pareto fronts are identical to another end point. The profit-IHTPI pair (design plane I) has a long Pareto front that expands to a negative total profit. This long front is mainly caused by the strong influence of the amount of residue on both design objectives. Many design and operating conditions are possible to change the amount of recovered acetic acid and offer a very wide scope of design and operating
TABLE 3. Best and Worst Solutions in the Pareto Optimal Solutions profit best profit worst IHTPI best IHTPI worst IATP best IATP worst IAP best IAP worst
teq (h)
t (h)
RB
V (kmol/h)
N
B (%)
profit ($/h)
IHTPI
IATP
IAP
0.51 1.95 1.96 0.63 1.17 1.95 0.53 1.95
1.14 5.74 5.13 1.06 3.63 5.74 1.29 5.74
11.33 38.09 32.15 8.26 12.47 38.09 9.12 38.09
111.60 107.14 100.89 39.14 44.38 107.14 35.76 107.14
14 38 35 18 38 38 38 38
37.3 53.8 54.7 16.7 26.4 53.8 16.9 53.8
81.95 -38.16 -22.93 22.98 27.34 -38.16 4.04 -38.16
121.50 93.11 92.88 160.30 110.88 93.11 159.87 93.11
14.83 21.10 19.53 17.04 14.17 21.10 16.98 21.10
60.71 273.11 237.24 21.74 70.43 273.11 21.39 273.11
conditions. The profit-IATP pair (design plane II) has a relatively short Pareto front, but all the Pareto optimal solutions are economically favorable. The profit-IAP pair (design plane III) also has economically favorable solutions, although the right end point (i.e., design number 18) is quite close to zero profit. The design alternatives for maximum profit (i.e., design alternatives 1, 7, 13) have relatively low IATP and IAP values, which can be seen by looking at these categories horizontally across the design planes of Figure 10. The IHTPI-profit (design plane I) and IHTPI-IATP (design plane IV) look very similar after rearranging and imposing one plane onto the other. Both Pareto fronts have very steep slopes at low IHTPI values (e.g., design alternatives 6 and 19), which correspond to designs with high acetic acid recovery and energy consumption due to severe design and operating conditions. Due to different contributions of mass and energy to the IHTPI and IAP, they are inversely proportional to each other (also shown in Figure 8), and design plane V has a very long Pareto front. The “IATP-” pairs (design planes II, IV, and VI) have small changes of the objectives partly due to the very small normalized impact scores for ATP. As O(IATP) is zero, both mass and energy contributions are important. However, the opposite trends of mass and energy contributions also make IATP small. Using these design planes and their corresponding design alternatives, the overall behaviors of binary Pareto fronts and design parameters can be easily visualized. Figure 11, a design alternative diagram, shows these Pareto fronts in closed symbols and the corresponding design parameters in open symbols. Note that the design alternative number is based on the Pareto optimal solutions of the lower triangular set of diagrams, not based on arbitrary designs nor sensitivityor scenario-based numbers. Again it is clearly observed that IATP has little changes in this figure. However, it cannot be removed from formulating the design problem even though its value is small because its weighting factor (wATP) is not known. The preferred region found in the traditional visualization method as shown in Figure 9 can also be identified in this new visualization technique. The preferred region corresponds to the design alternatives 1-3, 7-9, and 13-14 in Figure 11. In addition, the decision-maker can see the matching design parameters, which cannot be easily illustrated in the traditional visualization method. Now the decision-maker can determine the design alternatives that satisfy his or her preference. For example, if the decision-maker wants low acidification potential and positive total profit, he/she can find several promising candidates such as design alternatives 15-18, 29-30, and 34-36. Note that design alternatives 18, 30, and 36 are identical designs. Then the decision-maker can compare design parameters. If mild design parameters are preferred, then design alternatives 15 and 16 can be chosen for detailed investigation.
One of the interesting results is in design planes II, III, and VI. These design planes have economically favorable solutions along the entire Pareto front. The corresponding design parameters reveal that the reboil ratio should be small in order to achieve positive total profits. Since the total profit is the difference between the recovery profit, which is the value of recovered acetic acid divided by the total batch time, and equipment and utility costs, the small reboil ratio significantly decreases batch time and equipment and utility costs even though the value of the recovered acetic acid is reduced. 5.5. Best and Worst Design Alternatives. The new visualization technique, based on a lower triangular set of diagrams and a design alternative diagram, can also be used to analyze the best and worst scenarios. The best and worst design alternatives for each design objective are also valuable in analyzing the economic, ecological, and environmental aspects, and are summarized in Table 3. Note that the negative sign of the objective profit is dropped in this table, so the positive values in this column mean positive total profit. The best profit ($81.95/h or $650 000/yr) can be obtained at mild design conditions, especially at a small stage number. The worst profit is obtained at high acetic acid recovery rate. As expected, pursuing high recovery rate may not be economically and environmentally sound because a high recovery rate requires severe design conditions such as long operation time, high reboil ratio, high boil-up rate, and a large number of stages. Even though a high recovery rate increases the value of the recovered product, severe design conditions also increase the costs of labor (in terms of time), distillation column, heat exchangers, and utilities. Further, these severe design conditions significantly increase nonmass-dominated environmental impacts such as IATP and IAP due to high energy consumption. Thus, this worst profit design alternative also gives the worst aquatic toxicity potential and acidification potential values. In this case study, the three design alternatives, worst profit, worst IATP, and worst IAP, are the same design alternative. The best IHTPI is 92.88, which is a 51.5% reduction of the initial impact value. This design alternative is achieved by recovering more acetic acid since IHTPI is a strong function of the amount of the residual waste (S). As explained in the previous paragraph, this high recovery rate mandates severe design conditions, resulting in low profit and high IATP and IAP. The worst IHTPI value is 160.30, which corresponds to 16.4% reduction of the impact. Ironically, the mild design conditions for this worst IHTPI offer positive total profit and very small IAP generation. As the amount of acetic acid recovery is very small, both recovery profit and cost are relatively small, resulting in low total profit. The best IATP solution reduces 26.5% of the environmental impact from its initial value. Because both mass and energy contribute to the aquatic toxicity potential, the design conditions are rather gentle, except the number of stages. A large stage number requires high energy input, but a relatively small reboil ratio and low boil-up rate reduce energy VOL. 39, NO. 7, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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consumption, resulting in good performance of the other design objectives. The best IAP, 21.39, is obtained by recovering a very small amount of acetic acid (16.9% recovery yield) from the initial waste mixture within a short batch time. This operating condition minimizes energy consumption, resulting in the smallest acidification potential and a medium aquatic toxicity potential. However, only a small amount of acetic acid is recovered, and thus human toxicity potential by ingestion is very close to its worst value. In general, the best IAP design alternative is very close to the worst IHTPI design alternative since they have opposite contributions of mass and energy to the environmental impacts. The worst design alternative for IAP is identical to the design alternatives of worst profit and worst IATP, and they are very similar to the best design for IHTPI. This means that recovering more acetic acid from waste solvent requires more energy input and higher design costs, resulting in the lowest total profit and the highest environmental impacts. Even worse, the aquatic toxicity potential is higher than the initial value, 19.26. Thus, depending on the relative importance of various objectives, designing a process just to recover more valuable material from waste mixtures may not be a good design alternative for the environment as well as total profit.
Acknowledgments We thank the Oak Ridge Institute for Science and Education (ORISE) for their financial support for this project. We also thank Dr. Paul Harten, Mr. Jerry Waterman, and Mr. Greg Tucker for building and maintaining the Beowulf cluster. We also thank Dr. Urmila Diwekar (University of Illinois, Chicago) for providing the batch distillation simulation package, MultiBatchDS.
Supporting Information Available Detailed explanations of the multiobjective steady-state genetic algorithm (MSGA) and the reboiler heat duty calculation are available free of charge via the Internet at http://pubs.acs.org.
Notation AP
acidification potential
ATP
aquatic toxicity potential
B
acetic acid recovery (kmol)
GWP
global warming potential
HTPE
human toxicity potential by inhalation or dermal exposure
HTPI
human toxicity potential by ingestion
Ij
potential environmental impact of category j
MPI
message passing interface
MSGA
multiobjective steady-state genetic algorithm
N
number of stages
ODP
ozone depletion potential
PEI
potential environmental impact
PF i
ith Pareto front
PF *
true Pareto front (PF * = PF 1 after many evolutions)
PGAPack
parallel genetic algorithm package
POCP
photooxidation chemical potential
QR
energy consumption in the reboiler (MJ)
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RB
reboil ratio
Si
amount of residue material in the reboiler (kg)
t
batch distillation time (h)
teq
start-up time (h)
TTP
terrestrial toxicity potential
V
vapor flow rate (kmol/h)
WAR
waste reduction algorithm developed by U.S. EPA
∆Hv
heat of vaporization (kJ/mol)
ψij
normalized impact score of component i in impact category j
ψEj
normalized impact score of energy in impact category j
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Received for review June 23, 2004. Revised manuscript received January 7, 2005. Accepted January 10, 2005. ES0490424
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