Process Economics for Commodity Chemicals. 2. Design of Flexible

Ind. Eng. Chem. Res. 1997, 36, 3739-3755

3739

Process Economics for Commodity Chemicals. 2. Design of Flexible Processes Karine Boccara and Gavin P. Towler* Department of Process Integration, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, U.K.

Commodity chemical processes are characterized by small profit margins and large, nonsynchronous fluctuations of raw material prices and product prices. Process designs that can adapt their operation to maximize profitability under the prevailing economic conditions can be more profitable than designs optimized under a single set of prices. Price forecasts provide inaccurate information on the future economic environment. When used with current methods of process economics, this leads to poor designs. Instead, the fluctuating economic environment of a process can be described accurately and independently of time by a two-dimensional scatter plot of molar price ratios constructed from existing price data. The economic conditions under which the process will operate most of the time are easily identified on this plot. Processes optimized under these conditions show the highest overall profitability under market fluctuations. Under given economic conditions, the performance of a design can be assessed by comparing its profitability with the maximum profitability that could be achieved under the same economic conditions. By conducting such a comparison across the most likely economic conditions, the designer can carry out sensitivity analysis to select the best design among alternatives. The proposed method of design is illustrated using the example of the hydrodealkylation of toluene to benzene. 1. Introduction Price forecasting methods used in process economics are not able to accurately predict commodity chemical prices over the long-term. In the first paper in this series (Boccara and Towler, 1997) we showed that ratios of the product to feed price (P/F) and energy to feed price (E/F) tend to show time-invariant behavior and thus can be used to develop an alternative economic function that does not require forecasts. Optimization of this function, termed the residual economic function (REF), was shown to be mathematically equivalent to optimization of the ratio of net present value to the initial investment (NPV/I0) and the discounted cash flow rate of return (DCFROR), without requiring assumptions of plant life, interest rate, or price forecasts. The use of REF enables the designer to optimize a commodity chemical process for a single best design under the true average economic conditions, i.e., at the mean of the price ratios. As prices fluctuate around this average economic condition, the single best design operates at suboptimum economic performance. A greater profit might be realized if the process could be adapted to exploit the changing economic environment. This can be achieved by allowing for flexibility at the design stage. A flexible process permits operation over a range of uncertain design parameters around their nominal values and is consequently more capital intensive than a single-optimum process, for which the design parameters are fixed at their nominal values. In the case of price uncertainty, this is compensated by the flexible process achieving greater profit across a wider range of economic conditions. Existing methods of flexibility analysis focus on ensuring process feasibility as uncertain parameters vary. These methods allow the designer to select the most profitable design for a given flexibility requirement; however, they do not address the preceding problem of selecting the process flexibility * Author to whom correspondence should be addressed. E-mail: [email protected]. Telephone: 011 44 161 200 4386. Fax: 011 44 161 236 7439. S0888-5885(96)00652-5 CCC: $14.00

requirement that permits adaptation to maximize profitability as prices fluctuate. Such a selection should be based on the comparison of expected design performance, i.e., the economic penalty of suboptimum operation, of different process alternatives in the real fluctuating economic environment. The process adaptability yielding maximum profit can be found using a threestep approach. First, the most likely future economic conditions must be determined with accuracy. Second, process performance under these conditions must be assessed. Third, flexible and single-optimum process performances must be compared across the range of expected future price changes. The process adaptability that yields maximum profit corresponds to the design with best performance. Once this has been determined, the designer can specify the ranges required for the uncertain parameters and then proceed to design the flexible process using the established methods (e.g., Grossmann et al., 1983). In this paper we will show how the fluctuating economic environment of a process can be mapped accurately by a two-dimensional Economic Conditions (EC) plot of E/F vs P/F. Most likely future conditions are readily found on the EC plot. We will also demonstrate how REF can be used to assess process performance in the fluctuating economic environment. Regions of performance (i.e., 100% corresponds to optimum performance and Z% corresponds to Z% of optimum performance) of single-optimum and flexible process alternatives can be plotted on the EC plot. The comparison of process performance across the range of likely future conditions enables the designer to select at the design stage the process yielding the largest expected profitability in the fluctuating economic environment. This process can then be adapted during operation to exploit the fluctuating market situation. The savings achieved using this new design approach are expected to be significant for commodity chemical processes, which require high operating efficiency to sustain profitable operation in volatile markets. © 1997 American Chemical Society

3740 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997

2. Background and Previous Work It is necessary to consider flexibility in process design to account for uncertainty in the design and operating data. This problem can be divided into two subproblems. The first of these is to assure feasible process operation (i.e., satisfy all process constraints) when design data are not known accurately or process parameters are known to vary. Profitability can then be maximized under operation by optimization of the control variables if sufficient flexibility exists. This subject has been extensively addressed in the literature; for example, see Grossmann et al. (1983) and Pistikopoulos (1995). The second subproblem is the design of processes that are able to adapt to changing circumstances (e.g., economic conditions and feed purity) so as to maximize profitability. Such processes require a degree of overdesign to allow adaptation of operation over a range of design parameters, depending on market circumstances. Formulation of the conditions required for process adaptability can be a preliminary to the design of a flexible process by the methods set out in the work above. A particular case where process adaptability is attractive is that of commodity chemicals, where price fluctuations can cause wide variations in the profit margin. Commodity chemical prices are mostly influenced by environmental forces and opportunist effects (Clifton et al., 1992), neither of which can be satisfactorily accounted for by a probabilistic representation. Consequently, commodity chemical price fluctuations can only be described a priori by forecasts. All commodity chemical prices and energy prices depend on crude oil prices (Sedricks and Carmichael, 1993; Bacon, 1984, 1990); however, the complex interactions of supply and demand patterns for these chemicals yield significant (Boccara and Towler, 1997) and unforeseeable (Knott, 1995) price variations. Hence, neither stochastic nor deterministic approaches provide an adequate representation of single commodity chemical price uncertainty. For this reason, traditional methods based on price modeling cannot guarantee finding the best flexible design that is adaptable to the full range of market fluctuations. Instead, the dependence between commodity chemical prices and energy prices can be accurately described independently of time by the probability distributions of the price ratios P/F and E/F (Boccara and Towler, 1997), which can be determined from existing price data. The expected future economic conditions could therefore theoretically be obtained by computation of the joint probability distribution of the price ratios. This approach would require the designer to select a statistical interval representing the most likely price changes (e.g., the most likely price changes are those occurring 90% of the time). There is no firm grounds on which to base such selections, so this approach is unsatisfactory. The performance of design alternatives under uncertainty is usually compared on the basis of the optimal value at nominal economic conditions of an economic criterion, e.g., net present value (Edgar and Himmelblau, 1988), or residual economic function (Boccara and Towler, 1997), although other criteria are available, e.g., robustness (Rosenhead et al., 1972). The accuracy of these approaches is strongly related to the accuracy of the selected nominal prices in representing fluctuating prices. The means of the price ratios P/F and E/F constitute the only satisfactory representation of the true average economic conditions due to the time-

independence of the probability distributions of the price ratios. Boccara and Towler (1997) developed a new economic criterion, named the residual economic function (REF), that is calculated using these ratios instead of prices. This REF is therefore suitable for evaluation of single-optimum design alternatives in a fluctuating economic environment when sufficient data are available to determine average price ratios. When the economic conditions differ from the nominal values chosen, designs optimized under the nominal conditions (i.e., single-optimum designs) operate at suboptimum performance. The assessment of actual process performance against optimal performance (i.e., assessment of operating inefficiencies) can be carried out using internal or external benchmarking. External benchmarking uses the performance of competitors as a target, while internal benchmarking compares a company’s actual performance with its own theoretical best performance. An example of this latter approach is margin opportunity analysis (MOA), which can be carried out on a monthly basis to develop refinery strategic planning (Waguespack, 1995). Grossmann et al. (1983) addressed the problem of selecting the optimum flexibility of a process, which is derived from the optimum trade-off between the expected additional profit derived from flexible operation and the added capital cost of a flexible process. They considered that the probability distributions of the uncertain parameters and the economic penalties incurred during infeasible or suboptimal operation can never be known accurately and consequently proposed to simplify the original stochastic problem by considering two simultaneous objective functions that minimize costs and maximize flexibility. This approach was also followed by Pistikopoulos and Grossmann in their work on retrofit design (1988a,b, 1989). In the particular case of commodity chemical processes, however, neither of these conditions apply. Boccara and Towler (1997) have shown that the probability distributions of the process price ratios can be accurately modeled. In the course of this paper we will also demonstrate that the expected penalties incurred when the economic conditions vary can be accurately predicted at the design stage. This enables us to develop a method for selecting the appropriate flexibility to allow for profitable adaptation to changing markets. 3. Analysis 3.1. Modeling Future Price Variations. 3.1.1. Economic Conditions Plot. Because of the structure of the petrochemical industry, the fluctuating prices of raw materials, products, and energy for a given commodity chemical process are dependent. The uncertainty of this dependence can be expressed independently of time by the probability distributions of the ratio of the product to feed molar prices, P/F, and of the ratio of the energy price to the feed molar price, E/F (Boccara and Towler, 1997). Simultaneous fluctuations of all prices can be stochastically represented by the joint probability function of P/F and E/F. Alternatively, we can construct from the existing price data of a given commodity chemical process the scatter plot of E/F vs P/F, which represents the economic conditions that prevailed through the period covered by the available price data. We call this plot the economic conditions (EC) plot. Figures 1-3 illustrate the EC plots obtained for the production of respectively benzene by hydrodealkylation (HDA) of toluene, styrene from ben-

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3741

Figure 1. EC plot over 1991-1995 for the HDA process.

Figure 2. EC plot over 1991-1995 for the styrene process.

Figure 3. EC plot over 1991-1995 for the p-xylene process.

zene and ethylene, and p-xylene from isomerization of mixed xylenes. These plots were produced using the U.S. weekly spot prices of these chemicals (corrected to $‚kmol-1) and of natural gas, as reported in Chemical Week, Chemical Marketing Reporter, and the Oil & Gas Journal for the period 1991-1995. EC plots possess two significant properties that allow them to quantify price uncertainty effectively. First, all EC plots share the same pattern of dispersion, which reflects the balance between supply and demand common to every commodity. The dispersion of the EC plot is determined by the interactions between prices, which affect the breadth of the price ratio distributions, and the balance between supply and demand, which sets the pattern of dispersion of the data points. Spot prices of commodity chemicals reflect either actual levels of supply and demand for these chemicals or else traders’ appreciation of those levels (Himona, 1987). Reversely, prices can influence levels of supply and demand, the adaptation occurring either instantaneously by changing the level of production or shutting down of plants, or in the long term by using investment to start up new plants or add capacity to existing facilities (Long, 1990). The balance between supply and demand in any market causes the market to exist in one of three states: excess supply, excess demand, or balanced supply and demand. These three distinct states are associated with three types of economic conditions, which, from the viewpoint of the producer, can be thought of as poor, good, and average, respectively. These three economic regions can be distinguished on the EC plots, where each has a different likelihood of occurrence. The most common state of a market is near equilibrium, which corresponds to supply and demand in balance. When this balance is disturbed, the market

Figure 4. Fluctuation of the annual average economic conditions of the HDA process over 1991-1995.

returns to equilibrium as a consequence of the shutting of old plants and the opening of new ones (Pegum, 1995; Garcia et al., 1995a,b, 1996; Walker and Baker, 1996). Under equilibrium conditions, the margin between feed and product prices usually allows for profitable operation. The region of the EC plot corresponding to market equilibrium contains the highest number of points, i.e., highest occurrence of previous economic conditions and hence represents the region of most likely operation. According to Figures 1-3, this region is approximately defined by P/F in the range 1.0-1.2 for the HDA process, 1.2-2.0 for the styrene process, and 1.3-2.3 for the p-xylene process. The poor economic conditions corresponding to product glut are usually adjacent to the conditions of market equilibrium on the EC plot. The distinction between the two regions lies in the much lower probability of occurrence of the region of product glut. According to the EC plot of the HDA process (Figure 1), the economic region of benzene glut corresponds to P/F less than 1.0. In this region, toluene is more expensive than benzene by definition of the price ratio P/F; hence, even at zero operating costs, the HDA process is running at a loss. Note that this region of very adverse economics did not appear on the EC plots of the styrene process (Figure 2) and the p-xylene process (Figure 3) during the period studied. The economic region corresponding to product shortage can be observed in all of the EC plots, defined approximately by the following ranges of P/F: 1.4-1.6 for the HDA process, 2.0-3.4 for the styrene process, and 3.8-5.4 for the p-xylene process. This situation occurs when there is a shortage of product on the market, e.g., following a major producer plant shutdown or the startup of a new consumer plant. The process economics under these conditions are very favorable for producers until market equilibrium is re-established. The second important property of the EC plots is their time-independence. The probability distributions of P/F and E/F being independent of time, the EC plot constitutes a time-independent reference for the price fluctuations of a given process. Consequently, EC plots constructed from existing price data equally reflect future process economic conditions. If we continued to record price ratio data, we would expect the new points to fall within the regions previously observed, and we would not expect the shapes of these regions to be substantially altered unless there was a fundamental change in the structure of the industry, caused, for example, by the widespread adoption of a superior route based upon a different feedstock. This time-invariance is illustrated by the discontinuous line plotted in Figure 4, which is the locus of annual average price ratios for the HDA process over the period 1991-1995. This locus shows that the annual average varies around the overall average with no systematic trend. The time-invariance can also be confirmed by extending the data back over


Figure 5. EC plot over 1976-1996 for the HDA process.

a longer period. Figure 5 shows the EC plot for the HDA process over the entire period 1976-1996, containing over 900 data points. Even though the data extends to before the existence of a true hydrocarbon commodities market and three oil price shocks occurred in this period, the pattern of the data is similar to that in Figure 1. All the points for which P/F is greater than 1.6 occurred during the spring of 1987, due to a conjunction of circumstances (shutdowns, etc.) that caused a severe benzene shortage and which has not occurred at any other time before or since (Figure 6). The data in Figure 5 were taken from the same sources as Figures 1-3 and from the Historical Monthly Energy Review. 3.1.2. Importance of Accurate Selection of Representative Prices. To obtain single-optimum designs (i.e., designs that are optimum under a single set of prices) that suffer minimal profitability losses in a volatile market environment, it is necessary to optimize under a set of representative values of each price. The selection of representative commodity chemical prices partly depends on the choice of economic criterion, due to the different treatment of the time value of money in the various economic methods. The total annual cost (TAC) and the economic potential (EP) account for the time value of money using a constant annual charge factor; consequently, these measures should be optimized at the mean of the forecasted prices over the process life. The net present value (NPV) and the discounted cash flow rate of return (DCFROR) discount each year’s cash flow differently; therefore the forecasted mean of the prices over a long period of time is a poor measure of the prices for these economic criteria. Instead, the full forecast or the expected average value of each year’s prices should be chosen. Flexible designs are required to provide sufficient flexibility to adapt their operation to future price fluctuations so as to ensure minimum profitability losses; hence, they should be designed to perform best under a range of representative prices given either by a full price forecast or by a few likely future prices. From the above, we conclude that when forecasting future prices the most relevant issue to process design is the ability to accurately select the representative set of prices appropriate to the economic criterion chosen. The additional information contained in a detailed forecast does not provide any more information for the design of the process but instead contributes to a more accurate assessment of the value of future profit, which, naturally, is highly relevant to the investor. 3.1.3. Forecasting and the EC Plot. Because of the properties described above, the EC plot provides the designer with a map of future relative price variations as opposed to the information on future single prices that is provided by forecasts. A price forecast is a representation of the expected behavior of future prices and is characterized by prices set to occur at given future times. Each of these sets of prices can be

translated uniquely into the price ratios P/F and E/F; hence, a forecast over a specific time interval can be plotted on the EC plot as a vector defined by the forecasted set of economic conditions. The start point of a forecast can be taken as the current set of prices. Although these values are known accurately, their location on the EC plot is arbitrary and without an understanding of price ratio behavior this method can lead to the forecast originating in a region of the plot that is not typical of normal operation. A “better” selection corresponds to the average economic conditions; however, if these are calculated on the basis of all the price data, then their values may be distorted by data corresponding to regions of product shortage or oversupply, again leading to poor representation of normal operating conditions. Moreover, individual prices have a moving average; consequently, the calculated value of the average prices depends on the period covered. The behavioral trend of future prices can be forecasted by the regression of past price data. In this method, the trend is determined either by the most recent points or by the widest scattered, which causes the analysis to be inconsistent. In particular, if the starting point is in the region of product glut or shortage, then the predicted trend will always lead away from equilibrium, which is extremely unlikely to occur in practice. Also, the underlying assumption that the price behavior is a trend is oversimplifying and would only apply to shortterm forecasts; however, these forecasts are not useful at the design stage, where we require long-term forecasts over the project life. Actual price fluctuations for a commodity chemical are similar to an irregular oscillation as illustrated by the price data for the HDA process after 1986, shown in Figure 6. The cyclical behavior can be accounted for by setting discontinuous trends. Continuous and discontinuous trends can be plotted on the EC plot as continuous and discontinuous lines, Figures 7a and 7b, respectively. Clearly, neither of these gives an adequate representation of the full range of price variation, and both are subject to gross errors from a poor choice of the starting point or data set for trend regression. An alternative approach is to consider two scenarios, corresponding to a “best case” and “worst case”. With this approach we still encounter difficulty in selecting a starting point; however, by allowing vectors in opposite directions, the coverage of economic conditions is somewhat improved. The forecasts are still confined to lines on the EC plot, as shown in Figure 7c. Finally, the most conservative forecast gives a range between pessimistic and optimistic estimates. Such a forecast forms half a bow tie on the EC plot as illustrated in Figure 7d. This gives the best range of all forecasts described but still has a poor coverage of the total space of the EC plot. Better coverage of the economic conditions space can be obtained by using multiple forecasts; however, no single function will adequately describe the full range of economic conditions. In conclusion, forecasting methods do not give rigorous coverage of operating conditions, and they do not provide consistent rules for the selection of their starting point and trend. Consequently, commodity chemical forecasts are unreliable because of their unknown uncertainty. In particular, traditional forecasting methods cannot guarantee the accurate selection of repre-


Figure 6. U.S. spot prices of benzene, toluene, and natural gas over 1976-1996.

Figure 7. Price forecasts plotted on the EC plot show inconsistencies and poor coverage of economic conditions.

sentative prices. Unlike forecasts, the EC plot has a known time-independent uncertainty and therefore constitutes a reliable tool for identification of future price ratios.

3.1.4. Selection of Representative Prices. The EC plot offers both a quantitative and a qualitative assessment of future economic conditions. Each of the three regions that can be distinguished on the EC plot

3744 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 1. Price Basis for Example 1 design

region

average P/F

average E/F

Xopt

A B

usual operation product shortage

1.074 1.466

0.080 0.075

0.647 0.798

Table 2. NPV of Designs A and B over 1991-1995 for Different Values of the Interest Rate interest rate (%)

NPV,A (million $)

NPV,B (million $)

∆NPV (million $)

5 15 25 60

88.1 66.2 51.3 24.9

31.5 21.9 15.6 5.0

56.6 44.3 35.7 19.9

has a different probability of occurrence and also corresponds to economic conditions at which a process will achieve different levels of profitability. According to the scatter of the EC plot obtained from existing price data (Figures 1-3), only a few points constitute the most profitable region, which demonstrates that this region has a low probability of occurrence. Hence, if we design a process so that it can be run at maximum profitability solely in the most profitable region, it is not likely to be close to optimum performance during most of the process life time, which will mainly be constituted of economic conditions worse than those it was designed for. When the economic conditions fall into the region of product glut, the plant minimizes its losses by reducing production or shutting down; hence, the region of product glut is not representative of plant operation either. Instead, the price changes that are most significant in profitability analysis are those most likely to occur. We should ensure adaptability to these most probable market fluctuations, under which the process will normally be operated. The following example justifies this approach. 3.1.5. Example 1: HDA Process To Produce Benzene from Toluene. In this example we consider two single-optimum design alternatives of the HDA process. Design A is chosen as the optimum process under economic conditions belonging to the region of most likely operation, and design B is chosen as the optimum process when the prevailing economic conditions fall into the region of benzene shortage. For the purpose of this example, we assume that the HDA plant shuts down its operation when the economic conditions correspond to benzene glut. The aim of this example is to compare the profitability achieved by the two designs over the period 1991-1995. The selected measure of profitability is the net present value (NPV). For the purpose of this example, we have chosen the HDA flowsheet reviewed in Section 4, which features a hydrogen-recovery process and heat integration. The conversion of toluene is the only optimization variable and was optimized using the residual economic function, REF (Boccara and Towler, 1997). Table 1 lists the average price ratios in the region of usual operation and in the region of product outages over the period covered. The single-optimum conversion of designs A and B is also indicated in this table. Table 2 shows the value of NPV calculated for each design alternative at different values of the interest rate using the price data for 19911995. According to Table 2, design A achieves a much higher overall profitability than design B regardless of the interest rate, demonstrating that the HDA process is best designed under the most likely economic conditions, rather than under economic conditions that give highest profit.

This example illustrates that the most profitable commodity chemical processes are not opportunist designs that perform best under conditions of high profitability that occur infrequently. Rather, they are designs that yield moderate profits with highly efficient performance most of the time. 3.2. Optimum Designs. 3.2.1. Process Performance in the Fluctuating Economic Environment. Boccara and Towler (1997) introduced the economic function REF:

Q P E -1-w -e δ F F REF ) I0

(( )

) ()

(1)

where Q is the main product flow rate, δ is the stoichiometric coefficient of the main product in the main reaction, w is an overall feed loss term, e is the process energy use, and I0 is the initial investment. This function was derived from the net present value, and the first paper of this series showed that optimization of REF is equivalent to optimization of both NPV/I0 and DCFROR. Under given economic conditions, optimal profitability is characterized by maximum REF; hence, plotting loci of maximum REF on the EC plot shows how the optimal profitability is affected by the fluctuating economic environment. In particular, the locus of REFmax equal to zero gives the break-even locus at zero cash flow. This locus separates the EC plot into regions of profit-making (to the right of REFmax equal to zero) and loss-making (to the left of REFmax equal to zero). Process optimization in the loss-making region is treated separately in Appendix A. Process alternatives can be compared on the basis of their profitability. Under given economic conditions, the flowsheet with the highest profitability, i.e., highest REF, is the local best process. More generally, the EC plot can be separated into regions with the same local best flowsheet (Figure 8). The best single-optimum design alternative has the greatest profitability at average economic conditions, which can be taken as the mean of the price ratios. The latter represents the true average of the fluctuating prices because of its time-independence; however, it may not be representative of the most likely economic conditions due to the bias toward extreme economic conditions mentioned earlier. Instead, a more accurate set of representative economic conditions is the average of the most likely economic conditions, or better still, the entire region of usual operation. If one process flowsheet is the best design across the whole of this region, then it is clearly the best design overall. For example, in Figure 8, flowsheet 1 clearly gives the best coverage of normal operating conditions. 3.2.1.1. Example 2. To illustrate how the EC plot and REF allow the assessment of profitability changes as economic conditions vary, we propose to solve the following problem. We wish to produce a commodity chemical formed through the following reactions:

feed f product h W2 feed f W1

(2)

The available flowsheet alternatives (1) and (2) are illustrated in Figure 9. The conversion of product into W2 is X and the selectivity of feed to product is S1.


Figure 8. Identification of the local best flowsheet using the break-even loci between topologies.

Figure 10. EC plot for examples 2 and 3.

materials of construction. If the temperature and pressure of reaction are constrained, then conversion and selectivity can both be expressed as a function of the residence time, which can be eliminated to give the selectivity as a function of the conversion. It follows that in many cases we can simplify the problem by taking conversion as the only design variable to be optimized. For the purpose of the example, we use the following expression: Figure 9. Alternative flowsheet: (1) no recycle of W2 and (2) recycle of W2.

In this study, we concentrate on commodity chemical processes that possess the dual characteristics of wide price variations and low profit margins. One consequence of these conditions is that raw material price increases cannot be entirely absorbed by a reduction of profit margins and hence must be transferred into an increase in product price. Another consequence is that the price of the materials flowing through the process is the greatest contributor to profitability and controls its value. Therefore, in commodity chemical processes the variables affecting the mass flows have the most significant impact on process profitability, and it follows that their correct optimization is essential. Whatever the process involved, these variables are essentially the reactor conversion and selectivity. These two variables are functions of the reactor temperature, pressure, and residence time. It has been found (Grossmann and Halemane, 1982) that process variables are often set at constraint limits in the optimum process design. In particular, the temperature and pressure of reaction are likely to be set at maximum or minimum values, subject to constraints imposed by reaction yield and reactor

S1 ) 1.96 - exp(Xeq - X)

(3)

where the equilibrium conversion, Xeq, was taken as 0.30. The problem cost data are given by Boccara and Towler (1997). The energy cost was taken as:

CE )

100S1(1 - X) E (1 - S1)Q

(4)

for flowsheet 1, and

CE )

200S1

E

(1 - S1)Q

(5)

for flowsheet 2. Figure 10 represents the process EC plot, where the usual three economic regions can be distinguished. The expression of REF for flowsheets 1 and 2 can be derived from the process cost data. The break-even loci of flowsheets 1 and 2 correspond to REFmax,(1) and REFmax,(2) equal to zero, respectively. The EC plot can then be divided into three regions: where neither process is profitable, where flowsheet 1 yields the greatest profitability (REFmax,(1) > REFmax,(2)) and where


Figure 11. Separation of the EC plot into three profitability regions.

flowsheet 2 yields the greatest profitability (REFmax,(2) > REFmax,(1)). The last two regions are separated by the locus:

REFmax(1) ) REFmax(2)

(6)

The above equation and the break-even loci of both flowsheets are represented on the EC plot (Figure 11). The EC plot now conveys enough information to make a design decision at average economic conditions, i.e., at the mean of the price ratios. Under these conditions, we conclude from Figure 11 that flowsheet 2 yields a higher profitability than flowsheet (1). The single best design for flowsheet 1, referred to as design 1, corresponds to a conversion of X(1) ) 0.20 and the single best design for flowsheet 2, referred to as design 2, corresponds to a conversion X(2) ) 0.19. 3.2.2. Performance Index. The economic performance of a process design under given economic conditions can be measured by comparison with the performance of the locally-optimum design under the same conditions. This can be measured by a performance index, PI, defined as

PI )

(

REF REFopt

)

× 100

(7)

P/F,E/F

where REFopt is the greatest value of REF that can be obtained from the most profitable process alternative at the given economic conditions. Consequently, REFopt changes with the values of the price ratios and may refer to different process flowsheets in different regions of the EC plot. Expression of the performance index in the loss-making region is treated as a separate issue in Appendix C. For any design, contours of this performance index can be plotted on the EC plot to show how close the design is to achieving the local optimum REF (e.g., 9599%, etc.). As we move away from the specific set of economic conditions at which the process was optimized, the performance departs from its maximum value. The contours of suboptimal performance show how sensitive design profitability is to the changing economic environment. Narrow contours describe designs that are very sensitive to changing economic conditions while wide contours characterize designs with little sensitivity to price fluctuation. The values of the performance index across the region of most likely operation are particularly relevant to the expected behavior of the profitability; hence, design alternatives can be compared on the basis of this sensitivity analysis across the region of most likely operation. 3.2.2.1. Example 3. To illustrate how the performance index can be used to carry out sensitivity analysis and make design decisions, we will pursue the study of the problem introduced in Section 3.2.1.1. In

Figure 12. Performance of designs 1 and 2 in the region of usual operation.

the previous example, we identified design 2 as the best design at average economic conditions. If, however, we wish to account for the most likely price fluctuations, we should compare the performance of the singleoptimum designs on the basis of flowsheets 1 and 2 over the entire region of usual operation. The expression of the performance index of each of these designs under given economic conditions requires the knowledge of the local optimum profitability, REFopt, for the local best flowsheet. In this example, the analysis of the most profitable flowsheets across the EC plot defines the three regions illustrated in Figure 11. The performance of designs 1 and 2 is illustrated in Figure 12, parts a and b, respectively. These figures show that across the region of usual operation, design 1 performs in the range 90-100% of local optimum profitability whereas design 2 performs in the range 94100%; hence, we should select design 2 which is more profitable than design 1 across the region of most likely price changes. The performances of the two designs are very similar, and we may want to check our decision against the design performance across the entire EC plot. Figure 12a shows that under all previous economic conditions design (1) performs in the range 80-100%, whereas Figure 12b illustrates that, under the same conditions, design 2 achieves a higher profitability in the range of performance 94-100%. In this particular example, design 2 performs better than design 1 both at average conditions and during usual operation. Note that the optimum design of flowsheet 1 is not very sensitive to changes in the economic conditions, which is why the performance of design 1 is maximum, i.e., 100%, throughout the region where flowsheet 1 is the most profitable alternative. 3.3. Flexible Designs. In the previous approach we concentrated on single-optimum designs that were optimized under a unique set of economic conditions and


hence achieve optimum performance only when these conditions take place. We will now consider designs that can be adapted to exploit price changes. 3.3.1. Flexible Design Performance. If we consider the reaction conversion as the only optimization variable, we can characterize a flexible design by its conversion range. This conversion range for flexibility, ∆X, can be defined as follows:

∆X ) [XL,XU]

(8) Figure 13. Flexible region and flexible design operation.

For the process to be capable of achieving operation at any conversion in the specified range, it will be necessary to incorporate some overdesign into all process equipment. Thus, each process operating parameter will in turn be required to have a range of realizable values. Determination of the appropriate conversion range therefore specifies the flexible design problem, which can then be tackled using methods already described, such as that of Grossmann et al. (1983). More generally, a flexible process may be able to operate over a range of several design parameters:

∆xd ) [xd,L,xd,U]

∀d∈D

(9)

where D is the set of flexible design parameters. This case provides a greater number of design options; however, for commodity chemicals it is likely that process conversion and selectivity will dominate the process economics for reasons already described. We can therefore proceed by first examining optimization of conversion only and then introduce other flexible design parameters at a later stage. The performance of a flexible design can be assessed by the performance index defined by eq 7. Flexible designs are assumed to adapt their operation to the price fluctuations, i.e., the values of the variables e and w may be altered in the expression of REF. Consequently, the expression of REF is not unique for a flexible design. It is assumed that adaptation to current market prices is instantaneous; therefore, the profitability of the flexible design under given economic conditions is the maximum profitability that could be achieved by this design at the prevailing prices. Under fluctuating economic conditions, a given flexible design is operated most profitably at maximum cash flow. We can demonstrate mathematically (see Appendix B) that the maximum cash flow is a function of the conversion and the price ratio E/F; however, it is independent of the value of P/F. Consequently, to a given value of E/F there corresponds a single-optimum value of the conversion across the whole range of P/F, i.e., the loci of optimum conversion for maximum cash flow are parallel to the abscissa. A flexible design can therefore adapt its operation to maximize cash flow over a band of economic conditions bounded by (E/F)L and (E/F)U, which correspond respectively to the loci of optimum conversion XL and XU. We call this band the flexible region, which is illustrated in Figure 13. When the prevailing economic conditions belong to the flexible region, the flexible design can be operated at optimum cash flow at the optimum conversion, Xopt. When the economic conditions correspond to a value of E/F greater than (E/F)U, the feed is cheap relative to energy costs. If process energy costs are dominated by separation and heating of recycle streams, then high conversion is necessary to reduce recycle costs, even at the cost of higher selectivity losses. In this case, the

flexible design maximizes its profitability when run at the conversion XU. Reversely, when the value of E/F is smaller than (E/F)L, the feed is expensive relative to energy costs and selectivity losses should be kept to a minimum; therefore profitability is maximized for a conversion XL. The best flexible design operations are represented in Figure 13. Depending on the prevailing market conditions, the performance of the flexible design, PIflex, is given by the equations below. The expression of the performance index of flexible designs in the loss-making region is treated as a separate issue in Appendix C. We define R as the cash flow contribution to the economic criterion REF:

R)

Q P E -1-w -e δ F F

(( )

) ()

(10)

Therefore REF is expressed as

(11)

REF ) R/I0 Across the flexible region,

PIflex )

Rmax 1 × 100 I0,flex REFopt

(12)

At lower E/F,

REFflex,X)X PIflex )

L

× 100

(13)

U

× 100

(14)

REFopt

At higher E/F,

REFflex,X)X PIflex )

REFopt

Note that according to the above strategy for flexible design operation, the flexible design may yield a higher cash flow than the local optimum design but will require a comparatively higher investment to do so. A flexible process never achieves local optimum profitability because of its greater investment compared with a singleoptimum design; hence, flexible process performance is always suboptimal. However, suitably operated flexible designs have the advantage of being less sensitive to the fluctuating economic environment than singleoptimum designs across the flexible region. At the same level of performance, flexible designs have wider suboptimal regions than single-optimum designs on the EC plot, although their highest level of performance is always lower than that of the single-optimum design, i.e., lower than 100%. 3.3.2. Appropriate Flexibility. The appropriate flexibility is the design flexibility that allows process


C6H5CH3 + H2 F C6H6 + CH4 (15) 2C6H6 h (C6H5)2 + H2

Figure 14. Lower and upper bounds on the appropriate flexibility.

adaptation under the most likely price changes in the most profitable way, i.e., that yields the highest value of the performance index across the region of usual operation. The appropriate flexibility is associated with a conversion range. As highlighted in the previous section, the advantage of a flexible design over single-optimum designs is that it can be operated at higher cash flow over the flexible region; hence, the appropriate flexibility has a conversion range included in but not necessarily equal to the range of optimum conversions bounding the region of usual operation. Figure 14 illustrates the selection of the lower bound, XAF,L, and the upper bound, XAF,U, of the appropriate flexibility. The performance of a flexible design alternative is given by the value of its performance index across the region of usual operation. The variation of this index can be illustrated by a map of performance, which represents contours of levels of performance superimposed on the EC plot. A region of performance of Z% is an area of the EC plot where design profitability is in excess of Z% of local optimum profitability, i.e., REFopt. The regions of performance of design alternatives differ by their location on the EC plot and by their levels. Design alternatives can be evaluated by comparison of their map of performance with the scatter of the EC plot. The EC plot enables us to assess the relevance of both the position and the level of the regions of performance. This allows us to select the most profitable flexible design for each process alternative. For example, a design achieving 99% performance in a region of the EC plot where economic conditions are unlikely to prevail will not be as profitable as a design achieving the same performance in an economic region of higher probability. We can then compare the best designs of each process alternative to determine the best design overall. 4. Case Study: HDA Process To illustrate the proposed method for design of commodity chemical processes, we have chosen the welldocumented example of the design of the HDA (hydrodealkylation) process for making benzene from toluene. In this example we will derive the expression of REF for the HDA process. We will also represent the HDA economic environment and compare the performance of different HDA process alternatives. Ultimately, we wish to identify the design that can be operated with the highest profitability under the most likely economic conditions. 4.1. Expression of REF. In the HDA process, toluene is reacted with hydrogen to form benzene, which is partly lost to diphenyl according to the following reactions:

4.1.1. Assumptions for Stream Prices and Values. The process feeds are pure toluene and hydrogen containing 5 mol % methane. The molar price of toluene is ΓC6H5CH3, and in this example, we assume the price of fresh hydrogen, ΓH2/CH4, to be 2.4 times its fuel value (on the basis of the natural gas price in $/Mscf). The main product of the HDA process is benzene, with a molar price of ΓC6H6. A purge is required to prevent the accumulation of methane in the recycle of unreacted hydrogen. This is assumed to contain only hydrogen and methane. The value of the purge stream, ΓH2/CH4, is given by its fuel value. The diphenyl byproduct is separated from toluene and leaves the process in a waste stream, which is approximated to be pure diphenyl. For the purpose of this example the value of the diphenyl waste stream, Γ(C6H5)2 is assumed to be its fuel value, although we could assign a higher value if we wished to treat diphenyl as a recoverable byproduct. 4.1.2. Breakdown of the Cash Flow. The definition of REF (eq 1) relies on the expression of the cash flow, which is the difference between the revenue and the total cost of feed and energy. The revenue and the total feed cost depend on the expression of the overall main process reaction and of the side reactions. Because of the loss of hydrogen in the purge, extra hydrogen is fed to the process, and consequently the feed is nonstoichiometric. Hence, the overall main process reaction is expressed as follows:

C6H5CH3 + (1 + β2)H2 f C6H6 + CH4 + β2H2 (16) where β2 accounts for the nonstoichiometric hydrogen and is the ratio of the hydrogen lost in the purge to the production rate of benzene. The side reaction, which should not feature the main product benzene (Boccara, 1996), can be expressed as follows:

2C6H5CH3 + H2 f (C6H5)2 + CH4

(17)

The production rate of benzene, Q, is set to 120 kmol‚h-1 and the selectivity for benzene, S, is defined as the ratio of the number of moles of benzene produced to the total number of moles of toluene converted. According to these definitions, the mass balances corresponding to the overall main reaction and the side reaction are as follows:

C6H5CH3 + (1 + β2)H2 f C6H6 + CH4 + β2H2 β2 Q (1 + β2)Q Q Q Q

(18)

2C6H5CH3 + H2 f (C6H5)2 + 2CH4 Q(1 - S) Q(1 - S) Q(1 - S) Q(1 - S) S 2S S 2S Using the above equation, we can express each term of the cash flow. The total cost of feed, CF, is as follows:


(

CF ) Q ΓC6H5CH3 + ΓH2/CH4 + β2ΓH2/CH4 + 1-S 1-S ΓC6H5CH3 + Γ S 2S H2/CH4 1 1+S + β2 ΓH2/CH4 ) Q ΓC6H5CH3 + S 2S

(

(

)

)

)

(19)

The process revenue, CR, is given by

(

CR ) Q ΓC6H6 +

1-S Γ + β′2Γ′H2/CH4 2S (C6H5)2

)

(20)

where β′2 is the ratio of the purge flow rate and the benzene production rate. The process energy source is taken as natural gas and the energy cost, CE, is

CE ) eE

(21)

where e is the process energy use and E is the energy price. Using the above breakdown of the cash flow and the general expression of REF, we derive the following expression of REF for the HDA process:

[(

REF ) Q

(( Q

ΓC6H6

ΓC6H5CH3

(

-

)

1 + S

)

) )

]/

1-S 1+S E k′′ + β2 k + β′2k′ - e I 2S 2S ΓC6H5CH3 0 (22)

where I0 is the initial capital investment, k′ and k′′ are the molar heat values of the purge stream and the diphenyl waste stream, respectively, and k is the ratio of the molar value of hydrogen in the feed to the energy price. 4.2. Process Alternatives and Design Assumptions. In this case study we consider the following HDA process alternatives: (1) no heat integration and no H2 recovery unit, (2) heat integration and no H2 recovery unit, (3) no heat integration and H2 recovery unit, and (4) heat integration and H2 recovery unit. The simplified flowsheet of the process alternatives 1-4 is illustrated in Figure 15. The selected hydrogenrecovery process is a pressure-swing-adsorption (PSA) unit. The PSA unit performance and cost were modeled using correlations presented by Towler et al. (1996). In the absence of a PSA unit, it is assumed that the fraction of hydrogen in the recycle stream of hydrogen/ methane is 0.4, which is a typical value used in industry. This value could be optimized, but it is found that the overall process performance is only weakly sensitive to the hydrogen to methane ratio over the range 0.3-0.5. The process heat recovery is calculated using the pinch analysis method (Linnhoff et al., 1982). Designs with heat integration used the target hot utility load with a minimum approach temperature of 10 °C. Again, this variable can be optimized, but the optimum is very broad and the overall process costs are much more sensitive to the reactor conversion. Designs without heat integration are assumed to have all process heating and cooling provided by hot and cold utilities. The HDA process reaction conditions and the capital costs were taken from Douglas (1988). Under a given set of economic conditions, each of the HDA process alternatives can be optimized using REF

Figure 15. Simplified HDA process.

as the objective function, in combination with any method of optimization. For the purpose of this example, we consider the conversion of toluene as the only design variable to be optimized, according to the arguments developed in Section 3.2.1. and above. The analytical relationship between the selectivity for benzene and the conversion of toluene was taken from Douglas (1988). The optimization problem was solved on a spreadsheet, using the optimization solver available with Microsoft Excel 5.0. A more thorough optimization, including other parameters such as the hydrogen to methane ratio and the minimum approach temperature, could be attempted once the higher level economic analysis had reduced the options to a few of the more attractive process flowsheets. 4.3. Economic Environment. 4.3.1. EC Plot. According to eq 2 and by definition of the economic measure REF, the simultaneous fluctuations of all HDA process prices are embedded in the variations of the price ratios ΓC6H6/ΓC6H5CH3 and E/ΓC6H5CH3. The scatter plot of E/ΓC6H5CH3 vs ΓC6H6/ΓC6H5CH3 (Figure 1) provides the complete picture of the fluctuating economic environment and constitutes the EC plot of the HDA process on the basis of spot market prices. The three economic regions of the HDA EC plot that were discussed in Section 3.1.1. are clearly visible in Figure 1. 4.3.2. Selection of the Economic Conditions Requiring Highest Performance. The break-even loci of the four process alternatives are illustrated in Figure 16. The very poor economics of the HDA process are clearly stressed on this figure where, once the process costs are taken into account, at least half of the economic conditions fall into the region where all process alternatives are uneconomic. It should be noted that this EC plot is based on spot market prices but that margins on contract prices are not significantly higher. The poor economics of the HDA process, combined with the high cost of shutdowns and the possible integration of the HDA process with more profitable downstream processes, may constrain the HDA process to operate at a loss from time to time. These losses can be minimized by designing the HDA process so that it can be operated at high process performance under the conditions yielding losses. In the particular case of the HDA process, losses are registered under economic conditions belonging not only to the region of product glut but also to the region of most likely operation (Figure 16). Under the extremely poor conditions of benzene glut, when benzene becomes cheaper than toluene, it is reasonable to assume that HDA operators either halt


Figure 18. Grid superimposed on the region of usual operation. Figure 16. Break-even loci of the four HDA process alternatives.

Figure 17. Local best flowsheets across the EC plot.

production and sell their stocks of toluene or else build inventory of benzene to sell when prices improve. Consequently, a flexible HDA process need not be required to adapt to these extreme conditions; however, it should be designed to operate at high performance across the region of usual operation, including that part of this region when losses are incurred. This latter region is illustrated for process alternative 4 by the shaded area in Figure 16. 4.4. Calculation of the Local Optimum Profitability. 4.4.1. Local Best Flowsheet. The local best flowsheets are obtained by identifying the economic conditions that correspond to the break-even loci between topologies. The result of this calculation is plotted in Figure 17, which shows that throughout the EC plot the most profitable process alternative is either process 4 or process 2. Since processes 1 and 3 are designs that have no heat integration, we also know that they will be more sensitive to E/F than processes 2 and 4. Since they have lower profitability and greater sensitivity, we can discard these alternatives from the remainder of the analysis. The local best flowsheet in the region of usual operation can correspond to either process 4 or process 2, so neither of these processes emerges as a clear favorite. 4.4.2. Local Best Design. As discussed earlier, design alternatives should be compared on the basis of their performance in the region of usual operation. This screening relies on comparison of the contours of the performance for each design alternative. A region of performance of Z%, i.e., PI ) Z%, corresponds to an area of the EC plot where the selected design performs in excess of Z% of the local optimum profitability when Z is positive. When Z is negative (Appendix B), a region of performance Z% refers to a region of the EC plot where the performance index is equal to or lower than Z%. The placement of these performance regions requires prior knowledge of the local optimum profitability throughout the region of interest. A satisfactory approximation of the performance map can be obtained

Figure 19. Map of performance at PI ) 90% for different designs of each flowsheet. Table 3. Single-Optimum Designs under the Economic Conditions Set by the Grid P/F

E/F

best process

Xopt

P/F

E/F

best process

Xopt

1 1.05 1.1 1.15 1.2 1 1.05 1.1 1.15 1.2 1 1.05

0.04 0.04 0.04 0.04 0.04 0.063 0.063 0.063 0.063 0.063 0.086 0.086

(4) (2) (2) (2) (2) (4) (4) (2) (2) (2) (4) (4)

0.539 0.486 0.633 0.687 0.718 0.591 0.591 0.618 0.682 0.716 0.627 0.627

1.1 1.15 1.2 1 1.05 1.1 1.15 1.2 1 1.05 1.1 1.15 1.2

0.086 0.086 0.086 0.109 0.109 0.109 0.109 0.109 0.132 0.132 0.132 0.132 0.132

(2) (2) (2) (4) (4) (4) (2) (2) (4) (4) (4) (2) (2)

0.598 0.677 0.715 0.655 0.655 0.688 0.669 0.713 0.677 0.677 0.693 0.660 0.710

by calculating PI for a grid of points across the region of usual operation (Figure 18). Table 3 gives the optimum flowsheet and conversion at each of these grid points. The optimum conversion varies from 0.49 to


Figure 20. Maps of performance for designs 2D, 2E, 4F, and 4G. Table 4. Design Conversions Considered for the HDA Process letter

A

B

C

D

E

F

G

H

I

conversion 0.49 0.52 0.54 0.55 0.60 0.62 0.64 0.69 0.71

0.72 for flowsheet 2 and from 0.54 to 0.69 for flowsheet 4 across the range of interest. 4.5. Selection of the Most Profitable Design. To simplify the following section, we introduce a notation to label different designs. Table 4 gives a set of conversions, each labeled by a letter. A design noted 2A corresponds to a single-optimum design of the process with flowsheet 2, having a conversion given by column A of Table 4. A design noted 4BF corresponds to a process of flowsheet 4, with flexibility to operate over a range of conversion between the values given by columns B and F of Table 4. 4.5.1. Single-Optimum Designs. The comparison of single-optimum design alternatives is based on their coverage of the EC plot at similar levels of performance; however, in a first screening, we can restrict this comparison to the observation of the evolution with conversion (which defines a single-optimum design) of a particular contour corresponding to a given level of performance. If we assume that the distribution of the economic ratio data in the EC plot will remain similar in future years, then we can compare the coverage of

the region of performance against the existing scatter of the EC plot. Once we have identified promising designs, we can then proceed to a more detailed examination of the coverage of their regions of performance at different levels in order to select from among them the best alternative(s). Figure 19 represents the evolution of the contours of 90% PI with conversion for designs based on flowsheets 2 and 4. Figure 19a shows the 90% PI contours for flowsheet 2, from which we can readily see that there is a performance trade-off across the EC plot. Designs at low conversion perform well in a small area of the EC plot when the economic conditions fall in the region of low E/F near the locus of REFmax,(2) ) 0, which is a high-probability region according to the existing scatter. As the conversion increases, poorer performance is observed in that region of the EC plot; however, the size of the region of greater than 90% PI increases rapidly. A compromise between these two extremes favors designs 2D and 2E, which both cover wide and highly probable regions of the EC plot at high performance. Similarly, Figure 19b shows the 90% PI contours for designs of flowsheet 4. As the conversion increases, the size of the region of 90% performance increases; however, as shown previously, the coverage of the region at low E/F near the locus of REFmax,(4) ) 0 worsens.


designer is therefore faced with a choice between two good designs and requires additional criteria to decide which is best, e.g., least investment, lowest maintenance, etc. 5. Conclusions

Figure 21. Map of performance for design 4FG.

According to Figure 19b, 4G and 4H are the most promising designs. Figure 20 illustrates the contours of performance of designs 2D, 2E, 4G, and 4H, superimposed onto the cost data of the region of normal operation for 1991-1995. The comparison of the contours with the scatter of the EC plot results in the rejection of 2E as it only offers high performance in a part of the EC plot that is sparsely populated. Moreover, this design performs very poorly in the loss-yielding region where we expect to operate for a significant fraction of the time. Designs 4G and 4H have similar contours; however, 4G has a coverage of the EC plot at a higher performance in the very likely region corresponding to low E/F near the locus of REFmax,(4) ) 0; consequently we can reject 4H. We retain 2D and 4G for further analysis. 4.5.2. Flexible Designs. If we follow the procedure of design selection used in the previous section, then we conclude that no flexible design of flowsheet 2 performs better than the best single-optimum design 2D. This can be explained by the very close profitability (or performance) of different single-optimum designs of topology 2; consequently, the flexible designs cannot provide enough additional profit to offset their greater investment. Unlike flowsheet 2, flowsheet 4 has promising flexible designs. The contours of performance of the best among them, 4FG, are shown in Figure 21. Figure 21 illustrates that a flexible design has wider suboptimal regions than a single-optimum design, e.g., compare the contours of performance at 80% of designs 4G and 4FG. The regions of performance of designs 4G and 4FG are very similar, which is expected due to the narrow flexibility range of design 4FG; however, as the flexibility becomes larger, the regions of performance of the flexible designs are expected to differ substantially from those of single-optimum designs due to the increased difference between the investments and cash flows of the two designs. The comparison of the regions of performance of 4G and 4FG favors the latter, which has a wider coverage of the region of usual operation at 80% of performance and a slightly better performance both in the loss-making region and in the region of the EC plot at low E/F near the locus of REFmax,(4) ) 0. 4.6. Conclusions for the HDA Process Design. Two designs, 2D and 4FG, remain after this initial analysis of the regions of performance. 2D performs better than 4FG under low values of E/F, whereas 4FG performs better than 2D at high values of E/F. In the loss-making region 4FG clearly has a higher performance than 2D. 4FG requires an initial investment of $8.9 million whereas 2D requires only $3.9 million. The

In this paper we have introduced the EC plot, a new representation of the fluctuating economic environment of commodity chemical processes. This scatter plot represents E/F vs P/F as constructed from existing price data for a given process. The scatter of the data is independent of time. We have shown that a region containing the most likely economic conditions, corresponding to market equilibrium, can be easily identified on this plot. This region contains the economic conditions under which a process will normally operate. We have also demonstrated that the expected profitability losses incurred as the market fluctuates can be assessed at the design stage for both single-optimum designs and flexible designs using a performance index based on the ratio of process profitability to the maximum profitability achievable under local economic conditions. This performance index enables the designer to carry out sensitivity analysis, compare design alternatives, and select among them the process yielding the largest expected profitability in the fluctuating economic environment. Nomenclature CE ) energy cost ($‚h-1) CF ) total feed cost ($‚h-1) CR ) revenue ($‚h-1) DCFROR ) discounted cash flow rate of return e ) energy use (MMBtu‚h-1) E ) price of energy ($‚MMBtu-1) F ) molar price of feedstock required stoichiometrically ($‚kmol-1) I0 ) initial investment ($) I0,flex ) initial investment of a flexible design ($) K ) {NF + 1, ..., NF + NP} set of reaction products k ) ratio of the molar value of hydrogen in the feed to the energy price, HDA case study k′, k′′ ) molar heat values (MMBtu‚kmol-1) Mj,i ) flow rate to Column j for design i (kmol‚h-1) MF,i ) feed flow rate for design i (kmol‚h-1) MR,i ) flow rate to reactor for design i (kmol‚h-1) NPV ) net present value ($) P ) molar price of product formed stoichiometrically ($‚kmol-1) PI ) performance index PI2 ) performance index in the second region PI3 ) performance index in the third region PIflex ) performance index of a flexible design PIflex2 ) performance index of flexible designs in the second region PIflex3 ) performance index of flexible designs in the third region Q ) production rate of main product (kmol‚h-1) R ) cash flow contribution to REF (kmol‚h-1), defined by eq 10 REF ) residual economic function (kmol‚h-1‚$-1), eq 1 REFAB ) REF of design AB S ) selectivity S1 ) selectivity of feed to product in examples 2 and 3 w ) ratio of price of feed lost to waste production to price of feed required stoichiometrically to make the product X ) conversion XI0 ) conversion at minimum I0

Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3753 XR ) conversion at maximum R Xi ) conversion of single best design of flowsheet i Xeq ) equilibrium conversion of Product to W2 in Examples 2 and 3 xd ) design variable d Y ) operating hours per year (h‚year-1) (Y ) 8150 h‚year-1 in all numerical examples) Indices n ) year Subscripts AF ) appropriate flexibility eq ) reaction equilibrium flex ) for flexible design i ) flowsheet i L ) lower bound for flexibility max ) maximum achievable by the design under the local economic conditions opt ) optimum under the local economic conditions U ) upper bound for flexibility Greek Symbols β2 ) parameter accounting for nonstoichiometric hydrogen feed (HDA case study) β′2 ) ratio of purge flow rate and benzene production rate (HDA case study) Γi ) molar price of compound i ($‚kmol-1) δ ) stoichiometric coefficient of main product in the overall process main reaction ∆X ) conversion range for flexibility ∆xd ) design parameter range for flexibility

Appendix A According to the argument developed in Section 3.2.1.1., we consider the conversion, X, to be our only optimization variable. So we can express the conditions of optimum REF as follows:

dI0 dR I -R dREF dX 0 dX )0S )0 2 dX I

Figure 22. The optimum conversion lies between XI0 and XR.

selected according to the following economic criteria:

(23)

R > 0: REFmax

0

R e 0: Rmax

which is equivalent to the condition

dI0 dR ) REFmax dX dX

(26)

Appendix B

(24)

From eq 25, the condition of optimum REF at REF equal to zero is

The initial investment, I0, is a function of the conversion only, whereas R (eq 10) is a function of both the conversion and the price ratios. Maximum R corresponds to

dR )0 dX

Q dw de E dR )0S )0 dX δ dX dX F

(25)

So at REF equal to zero the optimum conversion corresponds to maximum cash flow. As the economic conditions worsen, REF becomes negative and we are no longer maximizing profit but minimizing losses. When REF is negative, trading-off I0 and R makes no economic sense as we would never invest initially in a loss-making venture; hence, REF is constrained to positive values as a design criterion and negative REF is an operating constraint that may occur after an investment has been made. When REF is negative, operation at minimum losses occurs when the cash flow is maximized. Over the entire range of economic conditions, the most profitable design is therefore

()

(27)

We conclude from the above eq that the optimum R is a function of the conversion and the price ratio E/F and that it is independent of the value of P/F. Let us define XR, i.e., XR ) f(E/F), as the conversion obtained at maximum R, and XI0 as the conversion obtained at minimum I0. The variation of I0 and R can be plotted against conversion for a given value of E/F giving a plot that looks like Figure 22a or 22b. Figure 22a shows that as we move to conversions lower than XR or greater than XI0, the investment is increasing while the cash flow is decreasing. Alternatively, Figure 22b shows that as we move to conversions lower than XI0 or greater than XR, the investment is increasing


When REFopt < 0, the performance index in the flexible region, PIflex3, at a given value of E/F is defined as

PIflex3 )

while the cash flow is decreasing. Hence, in both representations the optimum conversion always lies between XR and XI0. Appendix C C.1. Performance Index of Single-Optimum Designs. The criterion for the selection of single-optimum designs of given topology across the EC plot relies on the division of the EC plot into two regions separated by the locus of REFmax equal to zero (cf Appendix A). This dichotomy is important in defining the performance index. In the loss-making region, the local best performance is defined by Ropt (cf Appendix A), and we define the performance index, PI3, for a given value of E/F, as follows:

(

)

Ropt 1 2R 2

(28)

Note that when R e 0, a greater R has a lower absolute value than a lower R, so the definition of the performance index in the loss-making region is inverted compared to its expression in the profit-making region (eq 7). For any design, there is an economic region of the EC plot where REF < 0 and REFopt > 0. In that region, we approximate the variation of the performance index, PI2, with the price ratio P/F by the following linear function:

P PI2 ) a + b × 100 F

(

)

)

Ropt 1 × 100 2Rmax 2

(30)

According to the above equation, the design with maximum flexibility and best topology always yields optimum profitability in the loss-making region. For this reason, the comparison of flexible design performances should be assessed on the grounds of their comparative performances in the profit-making region, i.e., when REFopt > 0. Under poorer economic conditions than those belonging to the flexible region, i.e., at lower E/F, the performance index of a flexible design at a given value of E/F is

Figure 23. PI vs P/F at constant E/F.

PI3 )

(

(29)

where a and b are constants fixed by continuity constraints. Figure 23 illustrates the behavior of the performance index of two designs against P/F at a constant value of E/F. One design is based on the best topology and the other on a topology not as profitable. As illustrated in this figure, the performance index is a continuous but not smooth function. Optimum performance corresponds to a performance index of value 1 when REFopt > 0 and of value 0 when REFopt < 0 (Figure 23). C.2. Performance Index of Flexible Designs. When REF < 0 and REFopt > 0, the performance index at a given value of E/F is expressed by eq 29 inside and outside the flexible region.

PIflex3 )

(

)

Ropt 1 × 100 2Rflex,X)X 2 L

(31)

Similarly, under better economic conditions than those of the flexible region, i.e., at greater E/F, the performance index of a flexible design at a given value of E/F is expressed as follows:

PIflex3 )

(

Ropt

2Rflex,X)X

U

-

)

1 × 100 2

(32)

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Received for review October 14, 1996 Revised manuscript received June 10, 1997 Accepted June 10, 1997X IE960652P

X Abstract published in Advance ACS Abstracts, August 15, 1997.

Process Economics for Commodity Chemicals. 2. Design of Flexible

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