Methodology for Early-Stage Technology Assessment and Decision

Ind. Eng. Chem. Res. 2004, 43, 4337-4349

4337

Methodology for Early-Stage Technology Assessment and Decision Making under Uncertainty: Application to the Selection of Chemical Processes Volker H. Hoffmann,*,†,‡ Gregory J. McRae,‡,§ and Konrad Hungerbu 1 hler† Group for Safety and Environmental Technology, ETH Zurich, 8093 Zurich, Switzerland, and Chemical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

In recent years, besides economic questions, environmental problems and subsequent regulatory measures have emerged as a driving force for the chemical industry. It has become essential to identify potential environmental problems as early as possible in the design process in order to avoid costly process changes later. However, in early design stages frequently high uncertainty prevails around possible values that economic, ecological, or technical parameters can assume. In this paper, we propose a new approach on how to select promising process alternatives while taking explicit account of uncertainties. The method is based on approximating flowsheets (as modeled in commercial flowsheet simulators) by response surfaces. Because those response surfaces exhibit a lower complexity than the original flowsheets, much larger design problems can be tackled. This is especially helpful when complex problems, such as multiobjective design problems under uncertainty, have to be solved. While the method we present is generally applicable, it is illustrated with a decision-making case study on selecting a production process for hydrocyanic acid in which more than 400 uncertain variables are treated. The applicability of the method is shown, and optimal process alternatives are identified. 1. Introduction It has long been a subject for researchers and practitioners alike to identify optimal process alternatives when a given chemical substance has to be produced. While significant progress has been achieved regarding the optimization of an economic objective function by varying design variables, the problem has been put into a larger perspective in recent years for three reasons: First, to optimize the allocation of scarce financial resources for research and development, decision makers try to identify and select promising process alternatives as early as possible in order to avoid the development of inferior processes.1-3 At the same time, the emergence of novel technologies has considerably increased the number of potential process alternatives. Consequently, it seems advisable to utilize systematic screening methods to identify promising designs.4 Second, the pressure that stakeholder groups are able to exert on chemical companies has frequently led to the integration of nontraditional objectives into process design efforts, such as the environmental performance or reliabilty. This has fueled the advancement of appropriate indicators and, among others, the development of life cycle assessment (LCA) methodologies to rigorously represent the environmental consequences of production processes.5-7 If nonmonetary goals are in* To whom correspondence should be addressed. E-mail: [email protected]. Tel.: +41-1-6326098. Fax: +41-1-6321189. † ETH Zurich. ‡ Massachusetts Institute of Technology. § Fax: +1-617-2580546. E-mail: [email protected].

cluded in the process optimization, a multiobjective decision situation is created that calls for special mathematical tools as well as for increased problem understanding of the decision maker.8-10 Third, many variables such as future prices of chemicals are uncertain at the time when a process is designed. If the uncertain nature of these variables is not included in the analysis, the results can easily be challenged and it can be misleading to base the selection of alternatives only on “best estimates” for parameters. To cope with this problem, an uncertainty analysis can be employed that enables the decision maker to include stochastic information on risk issues in design and investment decisions. With these three challenges in mind, a decision maker should no longer only identify the most profitable process but rather address more detailed questions, such as what the probability of losses is, what the most important uncertain variables are, and what the probability is that a process A is, economically and environmentally, better than a process B. Accordingly, the following research goal is addressed in this paper: We assume the role of a decision maker who intends to produce a given substance X, for which numerous production alternatives exist. The task is to identify those process alternatives that fulfill the interests of the decision maker in the best possible way. If economic and environmental objectives are pursued, this constitutes a multiobjective optimization problem under uncertainty. This corresponds to the following mathematical representation, in which O$ represents an economic profit-related objective that is to be maximized while OEP describes an environmental damage that should be minimized.

10.1021/ie030243a CCC: $27.50 © 2004 American Chemical Society Published on Web 06/16/2004

4338 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004

Problem 1: An example of a mathematical formulation of a two-objective optimization problem under uncertainty max E[O$(d,c,u)] d,c,u

s.t. E[OEP(d,c,u)] e P(E[h(d,c,u)])0) g 1 - r P(E[g(d,c,u)]e0) g 1 - β To solve the problem with the constraint method,11,12 OEP has been transformed into a constraint. Technical and economic constraints are denoted by the vectors g and h. E[ ] represents the expected value and P( ) a probability, while the vectors r and β denote confidence limits with which the constraints g and h must be met. u characterizes uncertain parameters such as prices, efficiencies, or environmental evaluation factors. d is a vector of design variables that can be fixed before the plant is built but remains constant afterward. The vector of control variables c, such as temperatures of flow rates, can be adjusted during plant operation after some of the uncertainties are resolved. The optimization idea is to choose the design variables d in such a way that the variation of control variables c will yield the best (expected) result given the uncertainty in u. Note that this problem can also be formulated as a two-stage stochastic programming problem.13,14 Also note that the value of flexibility in design decisions (e.g., keeping design options open until uncertainties are resolved) has not been taken into account for simplicity reasons. In the literature, different solution strategies for multiobjective optimization problems have been provided11,12,15 and applied to such objectives as process robustness or flexibility.16,17 Several research teams have tackled optimizations with economic and environmental objectives similar to those of problem 1: Pistikopoulos and co-workers approached the problem of multiobjective optimization under certainty with commercial process simulators18-20 and under uncertainty.21 They also developed stochastic and parametric programming approaches to solve design problems under uncertainty,22,23 while Ciric and co-workers tackled the problem of variable waste treatment costs.24,25 Chakraborty and Linninger developed waste reduction strategies on a plant-wide level and approached decision-making problems under uncertainty.26,27 Characteristic for their method is a two-stage synthesis algorithm that generates all feasible alternatives and rigorously optimizes superior flowsheets. As objectives, they use cost and environmental damage measured by a global pollution index.28 Diwekar and co-workers addressed the problem of how to propagate parameter uncertainties through process models within the framework of commercial process simulators.8,29-31 They implemented several FORTRAN blocks in Aspen Plus that generate samples of each uncertain parameter based on its probability density function (pdf), write these sampled values into the appropriate Aspen blocks, start an optimization, and collect the relevant Aspen outputs. This procedure is repeated for the number of Monte Carlo runs as intended by the user. Recently, Dantus and High proposed to solve multiobjective optimization problems under uncertainty by

combining a process simulator and an external optimization by stochastic annealing.9 In this approach, first an external optimization block selects values for process parameters. Then a sampling block draws values from the distributions of uncertain parameters, the values are entered in the process simulator, and the flowsheet is computed. This is repeated for each Monte Carlo run. Then a new set of process parameters is selected by the (external) optimization block, and the whole procedure is repeated. Consequently, for each set of optimization values a Monte Carlo simulation with N runs has to be carried out until a solution is considered optimal. While the approaches cited above have contributed significantly to the development of the field, two principal problems seem to remain: On the one hand, it has to be noted that the use of process simulators in the context of multiobjective problems under uncertainty causes considerable computational requirements either when an optimization routine within the process simulator is required for each Monte Carlo run8 or when each optimization step of an external optimization routine requires N Monte Carlo runs.9 The computational demands make it difficult to systematically analyze large-scale problems with many alternatives and many uncertain parameters. On the other hand, more discussion seems necessary to illustrate the conclusions that a decision maker can draw from the uncertainty analysis, and in order to choose between investments in process A or B, stochastic information should be used more rigorously. In this paper, we address both shortcomings. First, a novel solution strategy for multiobjective optimization problems under uncertainty is proposed that avoids the limitations cited above by utilizing response surface techniques (section 2). Second, we demonstrate with a detailed case study for the production of hydrocyanic acid (HCN) how a decision maker can exploit the probabilistic information to arrive at less subjective investment decisions (section 3). The paper concludes with a summary of the most important findings (section 4). 2. Response-Surface-Based Multiobjective Optimization under Uncertainty To overcome the potentially prohibitive computational requirements mentioned above, the method presented in this paper relies on the use of response surface methods for solving complex process optimization problems under uncertainty. A response surface function uses a limited number of model evaluations to approximate the underlying full model with a simplified mathematical representation.32-38 By construction of response surface functions of process models, it becomes possible to represent the (implicit) process model by a number of (explicit) polynomials. These polynomials can be substituted for the process simulator model in all further calculations. Thus, the expensive optimization within the process simulator can be avoided and is replaced by the inexpensive optimization of polynomials. This leads to significant computational savings, and much larger problems can be tackled. Hence, a more systematic evaluation of process alternatives is possible that explicitly includes probabilistic information in the decision-making process. The solution evolves in five steps, as illustrated in Figure 1 and explained in the following subchapters, First, a reduced process model is set up (step 1). Next,

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4339

Figure 1. Response-surface-based procedure for process optimization under uncertainty.

uncertainty information is integrated into the model (step 2) before in a third step the actual response surfaces are constructed (step 3). Then a Monte Carlo simulation is run with the response surface model, and the alternatives are evaluated and optimized according to economic and ecological indicators (step 4). Finally, results can be analyzed and conclusions can be drawn for the design problem (step 5). Subsequent sections describe these steps in more detail. Note that methods exist to quickly screen alternatives so that response surfaces are typically only constructed for the most promising alternatives.4 2.1. Step 1: Creation of a Reduced Process Model. The first step to solving the optimization problem consists of building the underlying process models. While all calculations (unit operations and process evaluations) could be carried out within a process simulator environment,8 it is not always necessary to build such complex flowsheets at the early stages of process evaluation. When analyzing the different types of unit operation models, one finds that only complex equilibrium models such as reactors or distillation columns have to be modeled in detail. Other unit operations that a process model frequently contains do not necessarily require a simulator if the models can be approximated by simple input-output relationships. This applies, for example, to operations such as the processing of solids or to operations that are run with efficiencies close to 100%, such as absorbing a strong base in a strong acid. If no significant loss of information occurs, then these unit operations can be described with linear relationships outside the process simulator environment. The construction of these reduced models helps to limit computational efforts and allows one to approach larger problems. For the same purpose, it is useful to carry out the evaluation of the alternatives

outside the simulator. Bearing these considerations in mind, we propose to structure the problem as a series of embedded models as shown in Figure 2. At the heart of all calculations are equilibrium models within a process simulator that will later be replaced by the polynomial models. Around that, technical inputoutput models are built that are, in turn, surrounded by evaluation models. The authors are well aware of the need to define ranges of validity of response surfaces and to implement appropriate convergence tests. Our experience shows, however, that in early stages of process evaluation the errors in approximation are significantly smaller than the uncertainties in other parts of the problem. 2.2. Step 2: Integration of Uncertainty. Parameter uncertainties can be described with pdf’s that can be propagated through the model. The most important issues in representing uncertainty in response surface models are which probability distributions to use for describing uncertain parameters (a) and how to screen the uncertain inputs to find the ones that have to be modeled explicitly (b). (a) Retrieving Uncertainty Distributions. When trying to obtain probability distributions, uncertainties in technical, economic, and environmental parameters can be distinguished. While the distribution of technical parameters such as heat-transfer coefficients or physical property data is usually based on experiments or on literature data, the distribution of economic parameters is usually based on Markov-type models that rely on historic data.10,39,40 Alternatively, price predictions can be based on the subjective belief of experts, e.g., marketing or purchasing managers, etc. Finding distribution functions for environmental evaluation factors such as the impact of air emissions on ecotoxicity is more difficult because the underlying


Figure 2. Relation between different types of models.

models are very complex and the potential range of parameter values as judged by (often disagreeing) experts is large. Another difficulty is inherent to life cycle approaches: While it is comparatively easy to evaluate the effects of emissions of a single process, all environmental effects of upstream processes (so-called “background processes”, such as production of raw materials and utilities) also have to be included in the uncertainty analysis and have to be propagated through the LCA model. This can be done by carrying out all matrix inversions during LCA with distribution functions instead of using deterministic values.10 (b) Identifying the Most Important Parameters. Process models frequently contain hundreds of uncertain parameters, and in most cases not all of them can or need to be modeled. Hence, the influence that each parameter has on the results has to be elucidated in order to decide whether it should be included in the analysis. While technical parameters in input-output models (e.g., catalyst losses) and evaluation parameters (e.g., prices or environmental damage factors) do not enter the response surface (compare Figure 2), it is undesirable to include a large number of technical parameters in the “equilibrium sphere” because computational costs can increase, especially for complex response surfaces with high-order approximations and when cross-terms between the inputs are included. Literature suggests several approaches to how to identify the relevant parameters.32,41,42 On the basis of a thorough understanding of the underlying model, a first set of parameters might be selected. If there are only a small number of parameters, the further analysis should consist of a sensitivity study that perturbs the full model sequentially around the most likely parameter values. Should the number of parameters be large, a fractional factorial design can be applied in which several parameters are perturbed simultaneously.43 An additional advantage of this method is to detect crosseffects between different parameters. 2.3. Step 3: Construction of Response Surfaces. With the deterministic equivalent modeling method (DEMM) developed by Tatang, Wang, and McRae, a tool exists that can represent large black-box models as a set of polynomials.44,45 The general idea on which this method is built is to transform a stochastic black-box model with uncertain input parameters into a deterministically equivalent model that consists of a series

of polynomial functions. To achieve this, the method relies on polynomial chaos expansion46 and the probabilistic collocation approach.47 While polynomial chaos expansion approximates random variables as a series of polynomials, the collocation approach fits these polynomials to the response surface in such a way that the residual error is minimized. Thus, the black-box model can be substituted by a small number of polynomials for all further calculations. Consequently, uncertainty analysis can be performed on the deterministically equivalent model, for which computation times are usually orders of magnitude lower than those for the original model. The mathematical description of DEMM including a revision of polynomial chaos expansion and a discussion on the propagation of uncertainties can be found elsewhere.44,45 The procedure to retrieve a polynomial representation of a process model can be summarized as follows. Given a model with uncertain parameters, the order of approximation of the response surface has to be chosen. Next, collocation points at which the process model will be evaluated have to be determined. The response of the original model serves to find the coefficients of the polynomials. The accuracy of the approximation can be estimated by comparing the model results with the results of higher-order response surfaces, and the variance contribution of each input parameter can be calculated. Additionally, the response surface representation has to be compared to that of the original model in order to estimate the magnitude of error. If the error is acceptable, the polynomials can serve as a substitute for the original process model. If, on the other hand, the error is unacceptable, a higher-order approximation has to be chosen. As this increases the dimensionality of the calculations, a reduction of the number of uncertain parameters may be necessary in the selection of which can be based on the variance contributions computed earlier. 2.4. Step 4: Monte Carlo Simulation and Optimization. Once the polynomial model has been constructed, it can be substituted for the technical equilibrium model. Hence, a fast and reliable model is available that is suited for all further calculations, especially Monte Carlo simulations and optimizations. Depending on the desired accuracy, the sample size of the Monte Carlo simulation is chosen, typically 100010 000 runs. Then, the model can be optimized with


commercially available genetic algorithms by adjusting the control variables. In the following paragraphs, assessment methods are provided that can serve to evaluate process alternatives. While these indicators should be understood as examples, it is important to realize that the success of process selection efforts depends to a large extent on how well those indicators correspond to the objectives of the decision maker. Accordingly, great care should be taken in choosing indicators. (a) Economic Evaluation with the Total Annualized Profit per Service Unit (TAPPS). In the literature on chemical engineering economics, a large number of methods are offered on how to compare the profitability of processes.48,49 One of these measures is the TAPPS.4 As shown in eq 1, the TAPPS is calculated per functional unit S of main product and comprises the prices Pin and Pout of input and output material streams Min and Mout as well as the annuity AN of the initial investment C0 > 0. The annuity can be derived according to eq 2, with r as the discount rate and T as the lifetime of the plant.50

TAPPSProduct )

MoutPout - MinPin - AN S

AN ) C0/

[

1 1 r r(1 + r)T

]

(1) (2)

TAPPS, as illustrated in eq 1, comprises all cash-flow effects that stem from buying or selling materials or utilities and contains the annuity of the initial investment. Hence, TAPPS can be understood as the potential maximum profit per unit of product, while the real profit will be smaller because costs for labor and overhead are not included in the calculation. (b) Environmental Evaluation with the Ecoindicator 99. While a variety of indicators that rely on the principles of life cycle analysis can be defined,51-53 here the application of the Eco-indicator 99 is suggested.54 The Eco-indicator 99 is a damage-oriented method for assessing adverse environmental effects on human health, ecosystems, and natural resources in Europe. The evaluation proceeds in four steps. First, starting from the input-output data of a process (inventory data), the fate of emissions and the extraction of resources are analyzed. Second, the exposure that the emissions lead to and the effects of today’s resource extraction on future harvesting are considered. This is done for different impact categories such as climate change or respiratory effects. Third, the damages that the effects in each impact category impose on the three safeguard subjectsshuman health, ecosystems, and resourcessare estimated. Last, all damages are normalized, and the three damage categories are evaluated relative to each other. The Eco-indicator 99 (EI99) can be calculated according to eq 3. The damage factor of a substance Dij characterizes the contribution that the emission/extraction of one unit of a substance i has on impact category j. mi is the corresponding mass flow, wk the weight that is attributed to damage category k, and Nk the normalization factor for damage category k.

EI99 )

wk

∑k N ∑j ∑i Dijmi k

(3)

Regarding uncertainty analysis, the Eco-indicator 99 provides most damage factors as a mean value with information about the distribution function and the standard deviation. These are estimated based on the individual uncertainties that enter the assessment during the fate and effect analyses. Providing uncertainty data in this way allows one to engage in a life cycle uncertainty propagation that includes all background processes in the analysis, as explained above. 2.5. Step 5: Analysis of the Results. To address the questions of a decision maker before a potential investment that were raised in the Introduction, we propose to analyze the results of the optimization in four steps. 1. During an absolute analysis of one process alternative under certainty (i.e., if “best guesses” are chosen for parameters), the flowsheet is examined, contributions of the most important mass and energy streams are presented, and ideas for process improvements can be offered. 2. An absolute analysis of one process alternative under uncertainty can elucidate the probability of losses or extreme environmental consequences and can reveal the most important contributors to this uncertainty in the objective functions. 3. A relative analysis of several process alternatives under certainty can illustrate how key decisions such as the process configuration or the values of design variables can influence the emergence of superior processes. 4. Finally, a relative analysis of several process alternatives under uncertainty should be employed in order to find the probability that a process A is better than a process B. This is the question a decision maker should be interested in the most because it best describes the decision situation he or she frequently encounters. Each of these types of analyses will be illustrated in more detail with the case study in section 3.5. 3. Application to the Decision-Making Case Study on HCN The response-surface-based method for multiobjective process optimization under uncertainty was applied to a case study in order to identify the best among several process alternatives to produce HCN. HCN can be generated in a variety of ways,4,55-58 and earlier studies showed that by combining different unit operation technologies about 1200 process alternatives are technically feasible.4 Here, we examine nine process alternatives that look particularly promising (compare Figure 3): For the production of HCN, a BMA reactor is employed that consists of a large number of tubes coated with a layer of Pt catalyst. Inside the tubes, HCN is produced according to the endothermic reaction (R1) while heat is provided by the combustion of fuel gas outside of the tubes. Side products of this reaction are

CH4 + NH3 ) HCN + 3H2 (∆HR ) 252 kJ/mol) (R1) acetonitrile and nitrogen. Unreacted ammonia can be removed from the gas mixture by absorption in sulfuric acid and sale of the resulting ammonium sulfate in liquid or solid form. Alternatively, if the ammonia is absorbed in phosphoric acid, the resulting ammonium phosphate solution can be regenerated and the ammonia


Figure 3. Reduced BMA flowsheet with input-output unit operations (example).

can be recycled to the reactor. HCN is isolated from the gas mixture by aqueous absorption and distillation, yielding HCN with a purity of 99.5%. HCN that remains in the gas stream leaving the aqueous absorber is captured in an NaOH absorber, in which HCN is converted into sellable NaCN. The remaining gas stream can serve to produce pure hydrogen in a pressure-swing adsorption, it can be fed into a methanation unit to yield hydrogen of lower quality, or it can be combusted to recover the heating value. Finally, wastewater is treated with hydrogen peroxide. The plant capacity was set to 50 kt/a HCN, while a byproduction of up to 4 kt/a NaCN was permitted. Because there are three unit operations for removing unreacted ammonia and three different uses for hydrogen, nine overall process alternatives were analyzed. The discount rate of the investment was set to 15%, while a triangular distribution between 10 and 25 years was assumed for the lifetime of the plant. In this situation, a decision maker is typically interested in several questions before committing to an investment: Do any of the above process alternatives generate positive economic cash flows and, if so, which of the alternatives is the most promising? Additionally, it will be questionable to which values the technical control variables should be set in order to ensure a Pareto-optimal operation of the plant and which out of the Pareto-optimal operation points should be realized. If the decision maker additionally considers uncertainty, it should be elucidated what the probability is that a process A is better than a process B and what the most important uncertain parameters are. The solution procedure for the case study follows the outline as discussed in section 2: a reduced flowsheet representation is created (section 3.1), uncertain parameters are integrated into the model representation (section 3.2), and response surfaces are constructed (section 3.3). The polynomial model is employed in a Monte Carlo simulation and optimization (section 3.4), and finally the results are analyzed (section 3.5). 3.1. Building the Process Models. In principle, all unit operations of the process under consideration could be represented in a process simulator and a detailed simulation could be performed.1 However, the increased accuracy due to detailed modeling was traded off against increased computational complexity, and rigorous models were avoided as far as possible. Hence, a reduced

flowsheet was built, as shown in Figure 3. For all equilibrium calculations, Aspen Plus was used while input-output models were set up in Excel. During the course of the uncertainty analysis, the assumption that input-output models are sufficient was validated because variations in the characteristic parameters such as the reaction yield in the absorbers did not have a significant influence on the objective functions. Hence, the construction of reduced process models decreased computational complexity with only minor reductions of technical accuracy. A more detailed description of the complexity reduction and the structure of the reduced models can be found elsewhere.1 An additional benefit of this proceeding was that the number of variables that had to be treated during the generation of response surfaces was greatly reduced. 3.2. Integration of Uncertainty. (a) Retrieving Uncertainty Distributions. The HCN optimization problem described above contained about 500 variables including about 450 uncertain parameters and about 50 design and control variables. For each of the uncertain variables, pdf’s were derived. Technical parameters were assessed based on the literature data and on discussions with plant managers and engineers. Information about economic parameters was deduced from historic prices provided by industrial partners. Last, the pdf’s of environmental evaluation factors were computed with the so called EnvEvalTool developed by Cano.10 While more information on how to retrieve uncertainty distribution is provided elsewhere,10 Table 1 exemplifies the probability distribution of some of the uncertain evaluation parameters. (b) Selecting Uncertain Parameters. As a next step, the variables that were included in the construction of response surfaces for the equilibrium models were selected. Figure 4 gives a detailed view of the equilibrium model that had to be substituted with polynomials: the stream “GAS” contains the HCN-rich gas mixture that is absorbed in the aqueous “ABSORBER”. “OFFGAS” with traces of HCN is led to the NaOH absorber, while the aqueous HCN solution is distilled in “DIST”. The wastewater of the distillation column is recycled to the absorber, while HCN with a desired purity is produced. The boxes in Figure 4 show uncertain parameters as well as design and control variables that are explained in more detail in Tables 2

Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 4343 Table 1. Probability Distributions of Important Evaluation Parameters parameter name

meaning

distribution

1

P(NH3) P(NatGas) P(HCN) P(NaCN) P(H2-HQ) EP(CO2) EP(CH4) EP(NatGas)

price of ammonia [$/kg] price of natural gas [$/kg] HCN price [$/kg] price of NaCN [$/kg] price of high-quality H2 [$/kg] damage factor for CO2 emissions to air [EP/kg] damage factor for CH4 consumption [EP/kg] damage factor for natural gas consumption [EP/kg]

triangular triangular triangular triangular triangular log-normal 2 log-normal 2 log-normal 2

0.17 0.24 0.99 0.92 1 -5.2151 -2.227 -2.065

characteristic valuesa 2 3 0.21 0.28 1.2 1.1 1.6 0.439 0.261 0.184

0.25 0.31 1.29 1.25 1.7 4.8 × 10-6 (shift) 0.028 (shift) -7.3 × 10-3 (shift)

a For triangular distributions, characteristic value (CV) 1 is the lower bound, CV 2 the most probable value, and CV 3 the upper bound. log-normal 2 refers to a distribution that is generated by taking the exponential values of the corresponding normal distribution. For log-normal 2 distributions, CV 1 is the mean of the corresponding normal distribution, CV 2 the standard deviation of the corresponding normal distribution, and CV 3 a constant by which the distribution has to be shifted.

Figure 4. HCN purification system: Aspen Plus flowsheet to be substituted by polynomials (with parameters under consideration; detail of Figure 3).

and 3. Note that not only parameters within the purification section but also those variables that determine the composition of all streams that enter the purification section had to be included in the response surface generation. This especially applies to the composition of the feed gas and the yield in the reactor. Also note that a distinction between discrete and continuous variables is helpful in order to simplify the generation of response surfaces. While it is possible to create response surfaces that contain discrete variables with DEMM,45 in this analysis changes in discrete variables were modeled as different cases in order to improve the accuracy of the response surfaces. The continuous variables were subjected to a sensitivity analysis in which the objective functions were evaluated for the mean and two extreme values of each input parameter. As a first criterion, a minimum sensitivity, defined as the difference in the objective functions when the parameters were set at their extreme values compared to the base case, of (2% in either the economic or the environmental dimension was required to include a variable in the further analysis. Surprisingly, only about half of the examined variables,

as again listed in Tables 2 and 3, exhibited this sensitivity and were hence included in the polynomial model. 3.3. Construction of Response Surfaces. The DEMM was used to generate response surfaces based on the selection of uncertain input parameters as explained above. As output variables, two types of variables were chosen: those variables that connect the equilibrium-based purification system with subsequent unit operations (i.e., mass flows) and those that are direct inputs from or emissions to the environment [i.e., mass flows M( ) or energy flows Q( )]. For each of these variables (i.e., response surfaces), the order of approximation was increased until the difference between a response surface and the response surface of the next higher order was below 0.5%. More information on the structure of the polynomials can be found elsewhere.1 Once the polynomial representation was retrieved, the quality of the fit was examined by comparing the results of the polynomial representation with the ones from the original Aspen simulations. It was acknowledged that response functions usually show the best approximation in the center of the sampled region but a less adequate

4344 Ind. Eng. Chem. Res., Vol. 43, No. 15, 2004 Table 2. Probability Distributions of Uncertain Parameters in the Equilibrium Models parameter name

meaning

ethene Y(HCN) Eff(Dist) NRTL-R C(AN) Y(N2) R(CO) Eff(Abs) NRTL-B1 NRTL-B2 U(HX)

ethene content in the reactor feed [%] HCN yield in the reactor [%] efficiency of distillation column [%] NRTL interaction parameter for the HCN-water system acetonitrile conversion in the reactor [%] N2 yield in the reactor [%] HCN/CO molar ratio in reactor [-] efficiency of the absorber [%] NRTL interaction parameter for the HCN-water system NRTL interaction parameter for the HCN-water system heat-transfer coefficient in the heat exchangerb [W/m2‚K]

characteristic valuesa 1 2 3

distribution uniform triangular uniform triangular uniform uniform uniform uniform triangular triangular triangular

0 0.85 0.7 0.345 0.95 0.35 1500 0.5 588.3 244.5 227

0.03 0.9 0.9 0.386 1 0.5 3000 0.7 653.7 271.7 284

in polynomial model?

0.95 0.421

719.0 298.9 341

yes yes yes yes no no no no no no no

a For triangular distributions, characteristic value (CV) 1 is the lower bound, CV 2 the most probable value, and CV 3 the upper bound. For uniform distributions, CV 1 is the lower bound and CV 2 the upper bound. b Refers to heat exchanger HX in Figure 3.

Table 3. Design and Control Parameters in the Equilibrium Model variable name

meaning

type of variable

range (bounds) lower upper

included in the polynomial model?

N(Abs) N(Dist) Surplus(NH3) A(HX) DS(Bottom) T(CH2O-A) F(CH2O-A) split to WW DS(Top) T(CGAS-A) T(GAS IN)

number of stages in the absorber number of stages in distillation ratio between molar flows of NH3 and CH4 into the reactor size of the heat exchanger [m2] wastewater design specification for HCN [ppm] temperature of water flow into the absorber [°C] flow rate of water into the absorber [m3] ratio of wastewater flow/recycle flow HCN design specification [%]a temperature of gas flow into the absorber [°C] temperature of gas flow into the purification system [°C]

design design control design control control control control control control control

10 10 1.01 1000 1 15 90 0.05 98 20 80

discrete parameter discrete parameter yes yes yes yes yes yes yes no no

25 25 1.1 3000 1000 35 270 0.5 99.5 40 120

a The product quality used was included as a variable in order to find the potential effects that would result from a higher or lower product specification.

Figure 5. Comparison of the response surface with the original results (expected value and uncertainty ranges).

fit toward the ends of the input variable range. Hence, to test the quality of the fit, the output functions were evaluated at the extreme values of each input variable in the polynomial model. This was assumed to provide an approximated lower bound on the quality of the fit. The final quality of the polynomial approximation is shown in Figure 5. The right part of the graph gives ranges for the relative error in the fitted variables, defined as the difference between the “true” (i.e., simulated) and the approximated solutions, relative to the simulated solution. These errors were as low as (0.5% for some variables but could also amount to (3% or not cover the true value at all (e.g., heat duty for the water cooler). Nevertheless, the error in the objective functions, displayed in the left part of the graph, was about -1% to +1% for the environmental objective and -1% to +3% for the economic objective. This error,

induced by substituting the true model with response surfaces, was considered to be acceptable compared to the much higher uncertainty induced by varying the uncertain parameters. To derive a polynomial representation, only about 50100 evaluations of the full model were necessary, depending on the order of approximation and the number of cross-terms. These response surfaces could then be used for all nine process alternatives. Besides the considerable reduction of the number of simulations, this effect also contributed to the fact that the DEMMbased proceeding was also significantly faster per simulation than running a Monte Carlo simulation with Aspen Plus models because input values were not varied at random but according to preselected collocation points. A detailed analysis of computational efforts can be found elsewhere.1 3.4. Monte Carlo Simulation and Evaluation. Next, the original process model was substituted with the polynomial response surface, and Monte Carlo simulations with Latin-Hypercube sampling were performed. The model was then optimized with a commercial genetic algorithm. More precisely, Problem 1 was solved in the following way: For each set of process configuration variables (ammonia removal techniques, hydrogen usages, etc.) and for a given expected environmental impact ( constraint) measured by the Ecoindicator 99, the expected TAPPS was maximized by varying those control variables that were chosen during the sensitivity analysis (temperature and flow rate of water into the absorber, wastewater design specification, and wastewater split ratio). Subsequently, other Eco-indicator values (i.e., constraints) were chosen until the whole solution space was sampled. This proceeding was performed for each of the nine process


alternatives. Accordingly, the results that are offered in the following subchapter are comprised of one Pareto curve, generated by the variation of the continuous variables, for each unique combination of process configuration variables. The different Pareto curves are then combined in one Pareto plot that displays all optimized alternatives. The alternatives were evaluated according to the economic and environmental indicators introduced in section 2.4, TAPPS and EI99. As a functional unit, 1 kg of HCN with a purity of 99.5% was chosen, while the balance region was comprised of all unit operations mentioned before, from reactor to waste stream treatments. The environmental damage of producing HCN was computed by allocating credits for the production of byproducts. Full credits were assigned for the production of pure hydrogen and for the generation of steam from the flue gases of the reactor, while ammonium sulfate (solid or in solution) was regarded as a waste product and earned no credits. In those alternatives in which the H2 off-gases were combusted, credits for conserving an amount of natural gas with an equal heating value were given (augmented by credits for avoiding the emission of CO2). The credit for NaCN production was based on the combined damages of HCN and NaOH production. To study the damages that are caused by the consumption of upstream materials and utilities, the UCPTE electricity mix for Western Europe was used, and also for chlorine production, Western European market shares and production technologies were analyzed. For the generation of pure hydrogen, on the other hand, only market shares for the United States were available. Inventory data for the production of ammonia, phosphoric acid, platinum, sulfuric acid, and zeolites were obtained from the ESU database,59 while damages due to methane production were extracted from the SimaPro database.60 Inventory data on hydrogen peroxide were acquired from EMPA,61 whereas all other modules, including steam and natural gas, were contained in the EnvEvalTool.10 3.5. Analysis of the Results. The analysis of the results is presented in those four steps that were introduced in section 2.5. First, a traditional process analysis is performed in which the contributions of the most important mass and energy streams are presented and ideas for process improvements are offered (section 3.5.1). Second, the absolute decision of whether to install one particular process alternative or not is examined in detail, with uncertainty information being considered (section 3.5.2). Third, the decision situation is complicated by introducing other process alternatives that are compared on an absolute basis (section 3.5.3). Finally, a relative analysis of alternatives is performed in order to find the optimal process when uncertainty is included (section 3.5.4). 3.5.1. Traditional Process Analysis with Expected Values. As an illustrative case, the BMA process with phosphoric acid absorption of ammonia and the combustion of hydrogen-rich off-gases was analyzed (compare Figure 3). The Pareto plot of the expected economic and environmental performance of this process is displayed in Figure 6. Interestingly, there is not one set of continuous variables that dominates all other combinations of parameters, but a Pareto curve is formed when the continuous parameters are varied. Note that all points on the curve are optimal points; i.e.,

Figure 6. Pareto plot for the BMA process with H3PO4 absorption and combustion of H2.

Figure 7. Comparison of relative economic costs and environmental damages (BMA process with phosphoric acid absorption and combustion of hydrogen and maximization of TAPPS).

given the set of process configuration and design variables, processes to the upper left of the curve were infeasible, while processes on the lower right were inferior. While the choice of process configuration variables and design variables determines the position of the process in the Pareto plot, the change of continuous parameters causes a variation of about 10% in both objective functions. An important part of process analysis covers the question of how a process can be improved. For this purpose, mass and energy balances of the process were examined and the contributions to the objective functions of each chemical substance were computed. Figure 7 illustrates that about three-quarters of the economic costs and environmental damages are due to the consumption of raw materials for the reaction, while the contribution of gaseous emissions is comparatively small. This effect is inherent to the Eco-indicator 99 in which the consumption of raw materials has a high importance compared to impacts from emissions. Similarly, the absence of damages from the emission of HCN to the environment is due to two reasons. On the one hand, it is current industry practice to design HCN production plants in such a way that emissions are almost completely avoided. On the other hand, though, HCN emissions constitute a local environmental effect that is not captured by a transnational indicator such as the Eco-indicator 99. Also note that in most cases the relative contribution of substances to economic costs and environmental damages is similar. Ammonia, however, is environmentally about twice as costly as it is economically, while emissions after waste treatment do not bear economic costs at all. Investment costs and brine, on the other hand, do not show environmental damages that correspond to its high economic costs. The process designer can use this information of relative contributions to costs and damages to target


Figure 8. Pareto curve with uncertainties for the BMA process (BMA process, H3PO4 absorption, combustion of H2; mean values with 90% intervals as uncertainty bars; same processes as those in Figure 6 but on a different scale).

his efforts for improving the process. For example, one suggestion of Figure 7 is that the reactor should be the prime target for process improvements because both economic and environmental performances would benefit from reducing the consumption of raw materials or natural gas. This has partly been realized by a Degussa invention to use more reaction tubes per combustion chamber in order to decrease the energy losses.62 3.5.2. Absolute Analysis of One Process Alternative under Uncertainty. While the analysis so far has been based on mean values, additional insights can be gained when uncertainty information is included. Figure 8 displays the Pareto curve of the example process and the two-dimensional uncertainty bars (90% intervals). While the position of the curve is identical with the one shown in Figure 6, a number of conclusions concerning the size of the uncertainty bars can be drawn. First, one finds that the uncertainty bars overlap almost completely. This could suggest that the different points on the Pareto curve cannot be distinguished and hence that it does not matter which point on the curve is chosen. This argument is challenged extensively in section 3.5.4. Second, the size of the bars can be analyzed. The economic uncertainty is about (20%, which seems to be slightly above values in the open literature.8,9,48 Considering that HCN is a bulk chemical with presumably low margins, this uncertainty can be crucial for the question of whether the process can be operated profitably. At the same time, the environmental uncertainty amounts to about -40% and +75%, which is equivalent to a variation factor of about 3 around the mean. This value is clearly higher than the results of other studies,9 but it is important to realize that uncertainties always depend on the problem context and on the indicator that is employed and that in this research project, as opposed to other studies, all uncertainties in background processes have been included. Hence, an increased uncertainty had to be expected. On the basis of these results, the most urgent question for the decision maker is whether to go ahead with the development of the process despite uncertain results. The probability of losses can be identified with a cumulative distribution function (cdf), as shown in Figure 9. A loss occurs if the TAPPS of the process minus all cost that are not covered by the TAPPS (for example, operating labor, overhead costs, etc.) is negative. Hence, if the economic uncertainty bars do not cross, a certain threshold value for additional costs, for example, $0.55/kg of HCN, the decision maker can have a confidence of 95% that the process will be profitable. Should the uncertainty bars cross that threshold value, then there is a chance that a process might be unprofitable, although the mean value of the TAPPS indicates

Figure 9. Cumulative distribution of economic results (BMA process with phosphoric acid absorption and combustion of hydrogen and maximization of TAPPS).

Figure 10. Pareto curves of different alternatives and mean values (BMA processes with hydrogen combustion, purification, or methanation and with phosphoric or sulfuric acid absorption).

profitability. Hence, examining the cdf of the results can offer the decision maker valuable insights into how the likelihood of losses is distributed. Corresponding considerations can be made for the environmental objective. If the probability of losses seems unacceptable, the decision maker could decide not to install the process at all or could postpone his decision until more accurate data are available and uncertainty has been reduced. This leads to the question of which uncertain parameters contribute most to the overall uncertainty in the results because if a reduction of uncertainty is necessary, these parameters should be tackled first. An analysis of the Spearman rank correlation coefficients63 between the objective functions and each uncertain parameter reveals that the most important economic uncertainties are the prices of HCN and NaCN, while other prices are significantly less relevant. The technical parameters with the highest correlation to TAPPS are the combustion efficiency and the heat recovery of the reactor.1 The most important uncertain parameters for the environmental objective are the damage factors of NOX, SO2, and particulate matter to respiratory effects, but in general the observed correlation is not very high. However, if the decision maker wanted to reduce the uncertainty in the environmental objective function, more information about the damages that result from the air emission of NOX, SO2, and particulate matter should be collected in order to decrease the standard deviation in the corresponding pdf’s. 3.5.3. Absolute Analysis of Several Process Alternatives under Certainty. Figure 10 elucidates the absolute analysis of process alternatives and shows the Pareto plot of the expected performance of the nine different process alternatives that were introduced above. While all distribution functions and constraints were held constant, process configuration variables were


Figure 11. Pareto curves of different alternatives and 90% confidence intervals (same processes as those in Figure 10 but on a different scale).

changed so that the absorption of ammonia with phosphoric and sulfuric acid and different treatments for the hydrogen-rich off-gases were analyzed. It can be seen that each alternative forms similar Pareto curves as discussed before, but the process configuration and design variables determine the position of the curves in the two-dimensional Pareto space. In each case, the phosphoric acid absorption dominates the absorption with sulfuric acid. On the other hand, production of hydrogen is always, independent of the ammonia absorption technology, superior to combustion and methanation. This analysis indicates that the best process should comprise the phosphoric acid absorption as well as the production of pure hydrogen. However, it should be noted that the differences between some of the alternatives are not very big. For example, once a hydrogen treatment is chosen, the question of whether to use phosphoric or sulfuric acid for the absorption is of minor economic importance. This, on the one hand, suggests that for the economic selection other factors besides the TAPPS might have to be considered as well, such as experience with the technology, the licensing situation, or other more strategic aspects.1 On the other hand, though, this finding underlines the importance of integrating uncertainty in the decision-making process, because otherwise one could argue that “the alternatives are so similar to each other that it is irrelevant which one to choose”. A more detailed analysis shows that this is not the case (see section 3.5.4). 3.5.4. Relative Analysis of Several Process Alternatives under Uncertainty. When uncertainty information is included, the clear decision to install the phosphoric acid/hydrogen production process that was derived from Figure 10 might be questioned because uncertainty bars overlap almost completely. This is displayed in Figure 11 (expected performance of each process with 90% intervals). Hence, one cannot clearly say that a process A is better than a process B. Instead, the probability has to be found that A is better than B. The key to solving this problem is to realize that different process alternatives have many uncertainties in common. For example, all processes are subject to uncertain HCN prices, and as long as the processes produce similar amounts of HCN, the price uncertainty will affect all processes similarly and hence does not matter for the relative decision between the two processes. The reduced influence of shared uncertainties can be elucidated by a pointwise computation of the performance ratio of a process and a base case process.10 Assume that the decision maker was particularly interested in the process that promised the highest economic profit, the BMA process with phosphoric acid

Figure 12. Evaluation of all process alternatives relative to a base case. Mean values with 90% uncertainty intervals.

absorption, H2 production, and maximized TAPPS. Then this process would be chosen as a base case, and the performances of all other alternatives that are shown in Figure 11 were computed relative to that base case. The results of this calculation are displayed in Figure 12. In this graph, a relative TAPPS below 1 and a relative EI99 greater than 1 indicate that a process is worse than the base case. This opens the door for a number of conclusions. One can note that the relative location of the curves is still the same as that in Figures 10 and 11. However, because of the effect that common uncertainties were removed by the pointwise relative analysis, the uncertainty bars are significantly shorter than before. It is now possible to distinguish the processes relative to the base case with a statement such as “process A has a probability of X% to be better than the base case”. This statement becomes possible by examining to which extent the uncertainty bars cross the vertical or the horizontal axes that intersect in the base case. For example, consider the BMA process with sulfuric acid absorption, H2 production, and maximized TAPPS (process A in Figure 12). Because none of the error bars crosses the 1-1 axes, the probability that the sulfuric acid process is economically and environmentally worse than the base case is 100%. Opposed to that, consider the BMA process with sulfuric acid absorption, H2 production, and minimized EI99 (process B in Figure 12). The environmental confidence interval (90% interval) crosses the performance of the base case at about 35%. Hence, there is a 35% chance that process B is environmentally preferable to the base case, although the expected environmental performance of process B is worse than the expected environmental performance of the base case. With this information the decision maker can (a) clearly distinguish the different processes and (b) base his decision of which process to implement on the corresponding probabilities that one alternative is better than the other. Additionally, it is possible to compare the shape of the pdf’s of two processes and to define additional measures of interest such as the variance or the downside risk in order to come to a more conscious decision on which process to choose.1 4. Conclusions and Future Research In summary, it can be stated that the presented method for process optimization under uncertainty based on response surfaces is well suited for the identification of Pareto-optimal process alternatives under uncertainty. The response-surface-based method that has been introduced here is considerably faster and allows one to approach bigger design problems com-


pared to performing Monte Carlo simulations with process simulators. These improvements come at the expense of an increase in the uncertainty of the objective functions because the response surfaces are only an approximation of the true model response. However, in the example studied here, this induced uncertainty turned out to be negligible compared to the uncertainty contribution of the input parameters. Although a good polynomial approximation was found in this case study, it has to be noted that this might not always be the case and the problem-specific nature of this approach has to be stressed. Especially, problems with many discrete parameters and nonsmooth response surfaces can be cumbersome to fit, but more cases have to be analyzed in order to find guidelines on when to apply this method. When the presented method for performing uncertainty analyses is applied even to complex decisionmaking problems, a number of benefits can be reaped: An explicit analysis of uncertainty adds far more information to a decision situation than “just” answering the question of what the probability is that process A is better than B. The cdf of the results can guide the decision maker on such important questions as whether to go ahead with a project or what the probability of losses is. When uncertainty has to be decreased before an investment decision can be made, then the contributions of each parameter to the overall uncertainty can suggest where to spend financial resources in order to improve the information content of the model. Furthermore, performing an uncertainty analysis requires the decision maker to structure the problem more carefully and to acknowledge his or her uncertainty about the influence of parameter variations. Especially, thinking about parameter correlations and how to include them in the model can provide an improved understanding of the problem at hand. Despite the advantages of a formal assessment of uncertainty, one has to be aware that the conclusions drawn from the analysis can always be only as good as the underlying assumptions. This is especially important when deriving probability distributions for parameters and when analyzing the correlation structure between them, notably in the environmental domain. Similar considerations apply to the choice of (environmental) indicators. To a large extent, the results of the analysis depend on the characteristics of the indicator; hence, it is important to guarantee that the decision maker agrees with the value statements and assumptions that form an integral part of the indicator. Regarding the Eco-indicator 99, obvious assumptions concern its geographic scope, the selection of damages that are evaluated, and the weighting between these damages, but more subtle assumptions may be hidden in the way in which environmental effects are modeled. The applicability of the response-surface-based method has been shown with a case study on HCN production. Regarding this case study, the relative analysis of uncertainties illustrated that the BMA process with phosphoric acid absorption and the production of pure hydrogen is, apart from parameter variations that lead to tradeoffs on the Pareto curve, the most preferable among the alternatives that were considered. Before this process is implemented, the decision maker should analyze which other effects besides those already incorporated in the uncertainty assessment could be relevant. For example, what would the consequences be if a competitor produced a main product with a higher

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Received for review March 17, 2003 Revised manuscript received December 19, 2003 Accepted February 20, 2004 IE030243A

Methodology for Early-Stage Technology Assessment and Decision

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