Ind. Eng. Chem. Res. 1997, 36, 3727-3738
3727
Process Economics for Commodity Chemicals. 1. The Effect of Fluctuating Costs on Design and Optimization 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 chemicals prices are subject to strong market fluctuations and therefore difficult to forecast with accuracy. This creates difficulties in design and optimization, as the common economic criteria used in formulating design objective functions can be very sensitive to variations in price data. Commodity chemical price data can be described with reasonable accuracy by probability distributions of price ratios. These ratios are time-invariant and arise from the fundamental structure of the petrochemical industry. A new economic criterion, the residual economic function, REF, is derived for commodity chemicals processes, on the basis of price ratio distributions instead of forecasts. Optimization of this new criterion is shown to be equivalent to optimization of both net present value and discounted cash flow rate of return. Unlike other criteria, REF does not depend on any unforeseeable parameters, due to the timeindependent behavior of the cost ratios and the elimination of other economic assumptions in its derivation. REF can be used with any design or optimization method. The new economic criterion is compared with established methods through an example. 1. Introduction The design and optimization of chemical processes depend upon the specification of an overall objective function, which is usually based on profitability or process costs. This objective function should account for the design trade-off between operating costs and initial capital. For example, spending more money on heat exchangers may reduce annual energy costs. We should also consider the time value of money if the results are to be meaningful. There are various measures of profitability; see, for example, Peters and Timmerhaus (1981), Edgar and Himmelblau (1988), or Douglas (1988), which account for the capital-operating cost trade-off and time value of money in different ways. In general, the most favored methods are the calculation of net present value (NPV) or discounted cash flow rate of return (DCFROR). The NPV method tends to favor large projects, while the DCFROR can give misleading results if there are large variations in annual cash flow (Edgar and Himmelblau, 1988). For commodity chemical processes, evaluation of profitability is complicated by the fact that the prices of feedstocks, products, and energy can undergo significant variations over time. For example, Figure 1 shows the variations in product, feed, and energy prices for the hydrodealkylation of toluene to benzene over the years 1991-1995 (in current dollars, i.e., current at the time the prices prevailed) on the basis of U.S. spot market values as reported in Chemical Week, Chemical Marketing Reporter and the Oil & Gas Journal. Most importantly, variations in prices are not synchronous, consequently the margin between product price and costs is subject to interference effects that can cause dramatic changes in cash flow, as illustrated in Figure 2. In the case of commodity chemicals traded on a free market, many unforeseeable factors (e.g., unplanned shutdowns, strikes, natural disasters, etc.) contribute to price variations. These parameters cannot be anticipated; hence the various mathematical methods that can be used to produce forecasts fail to guarantee a * 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)00651-3 CCC: $14.00
satisfactory level of accuracy. Ultimately, the inaccuracies embedded in the price forecasts are transferred to the optimization results. As a consequence, there is no assurance that a design that is optimal for some predicted price fluctuations will be optimal under, or even adaptable to, the actual fluctuating economic environment. So there may be a significant reduction in potential profit if we cannot accurately account for future price changes. It is the purpose of this paper to introduce an objective function for optimization at the design stage, which embeds accurate modeling of price fluctuations in its expression of the trade-off between capital and operating costs. The analysis requires only a knowledge of existing price data in order to describe future price behavior accurately. It enables the engineer to produce single-optimum designs that account for the true average of the future economic conditions without relying on forecasting. The new economic criterion allows for the time value of money without making any assumptions about the plant life or the discount factor. In a future paper we will show how this methodology can also be used to assess process performance in a fluctuating economic environment and develop flexible designs that can adapt to exploit the fluctuating market situation. 2. Background and Previous Work 2.1. Assessment of Profitability under Price Uncertainty. The classical approach to design under price uncertainty is to assume nominal values of the prices of feed, product, etc., and then perform a postdesign sensitivity analysis to examine the effect of oneat-a-time price variation on the overall profitability (Edgar and Himmelblau, 1988; Axtell and Robertson, 1986; Holland et al., 1973). Combined effects of price fluctuations cannot be accommodated using this method; moreover, sensitivity analysis is concerned with the effects of relatively small input variations. This clearly does not apply to commodity chemicals prices, which undergo wide variations. For example, U.S. spot market prices for styrene fluctuated between 19.5 and 63 cts/lb during the period 1991-1995. The modeling and impact of large input fluctuations on design and operation is the subject of flexibility © 1997 American Chemical Society
3728 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
Figure 1. U.S. spot price fluctuation of benzene, toluene, and natural gas over 1991-1995.
Figure 2. Cash flow variation of an HDA process over 1991-1995 (100 000 000 gal capacity).
analysis, which proposes to embed mathematical models of the uncertainties in the expression of the objective function. The origins of process uncertainty have been classified by Pistikopoulos (1995) into four categories: model-inherent, process-inherent, external, and discrete. Each of these uncertainties can be modeled either by ranges of possible realizations or by probability distributions. There are two approaches to design and operation under uncertainty, namely stochastic and deterministic. Both of these approaches focus on obtaining an optimal design belonging to the process feasible region. Extended reviews of these methods have been presented by Grossmann et al. (1983) and, more recently, by Ierapetritou and Pistikopoulos (1994) and by Pistikopoulos (1995), who presented a model combining the two approaches. In the probabilistic approach, the uncertain parameters are assumed to be described by probability distributions. The objective function maintains the stochastic qualities of the uncertainty representation and is expressed as the expected value of the optimum cost or profit. Mathematically, this expected value is the multiple integral over the parameter ranges of the optimum cost or profit of a design weighted by the joint probability function of the vector of the uncertain parameters (Grossmann and Sargent, 1978). In deterministic approaches, the uncertain parameters are assumed to have realized values, and the objective function is the optimum cost or profit over all the realized parameter values. The deterministic approach considers that each uncertain parameter can vary between fixed lower and upper bounds and proceeds to reduce the problem to a deterministic multiperiod formulation by discretization of the parameter space (Grossmann and Sargent, 1978). The particular case of price uncertainty falls into the category of external uncertainty, which unlike all the
other types of uncertainty does not affect feasibility. The impact of price uncertainty on profitability is evaluated through the calculation of an economic criterion. Therefore, the difficulty of process optimization under price uncertainty involves the modeling of price variations and the assessment of their effect on profitability through the selection of a suitable economic criterion. Price variations can be modeled, with various degrees of accuracy, by bounds, probability distributions, mean values, etc.; however, the modeling of price uncertainty is usually carried out using forecasting methods. Forecasting relies on extrapolation techniques to make use of previous data to predict future trends. The difficulty of forecasting is illustrated by a recent report (Knott, 1995) drafted by executives of major oil and gas companies about the outlook of crude oil prices 15 years on. According to four scenarios, prices were forecasted to range between $10 and $25/bbl (1995 $), i.e., within a (50% bound around the mean price $17.5/bbl. Insights into the petrochemical industry are essential when considering the fluctuation of product prices. The commodity chemicals market constitutes a mature market, which is, therefore, mostly influenced by environmental forces (Clifton et al., 1992; Sedricks, 1994). These are set by the general trends of the economy, i.e., cyclical trends (related to macroeconomic events such as recession or boom) and opportunist effects (related to unpredictable events such as wars, strikes, unplanned shutdowns, etc.). An experimental price forecasting method for petrochemicals is proposed by Sedricks and Carmichael (1993). Their model relies, principally, on estimates of oil prices and of industry capacity utilization. The latter represents the tightness of the supply and is used as an indicator of the industry’s expectation of future prices. The correlation of this indicator with the process profitability of a straggler plant (high production cost) and a pacesetter
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3729
plant (low production cost) is represented by a “hockey stick” curve. They combine these curves with experience curves, supply curves, and predictions of oil prices to obtain medium-term projections of petrochemical prices. The effect of price uncertainty on process profitability is measured by the variation of an economic criterion. The selection of an economic criterion is complicated by the fact that there is no single way to optimize a chemical process, since companies seek various benefits in return for their investments. At the design stage, profitability is usually determined by one of four economic criteria, namely, the total annual cost (TAC), the economic potential (EP), the net present value (NPV), or the discounted cash flow rate of return (DCFROR). (Simple pay-back is not included as it only describes profitability before the initial investment has been recovered.) The TAC and the EP treat the capital cost as an annual charge determined by multiplying the initial capital investment by an annualization factor that accounts for project financing:
TAC ) hQY + AfI0
(1)
EP ) CRY - hQY - AfI0
(2)
where h is the operating cost per unit of production, Q is the production rate of the main product in units per hour, Y is the number of operating hours per year, Af is the annualization factor, I0 is the initial investment, and CR is the revenue from product sales. The annualization factor depends on the plant life and the interest rate at which the management requires projects to pay out, which in turns depends on the debt to equity ratio. Projects with greater expected returns can be financed at a lower debt to equity ratio than projects with low rate of return. The plant life, on the other hand, will remain uncertain until the plant is scrapped; hence, it is difficult to specify Af precisely at the design stage and it is common practice to assume a value (e.g., Douglas (1988) recommends a value of 0.33). The NPV and the DCFROR both reduce all costs to the current year by summing discounted future cash flows: n)L
NPV ) -I0 + DCFROR ) i,
CFnfdn ∑ n)1
s.t.
NPV(i) ) 0
(3) (4)
where L is the plant life, CFn is the cash flow in year n, and i is the interest rate. fd is the discount factor, which is expressed as follows:
fd )
1 1+i
(5)
These criteria are generally recommended as methods of evaluating and ranking project profitabilities; however, Edgar and Himmelblau (1988) reported NPV to give preference to large plant capacities and, in this light, suggested using the modified economic criterion NPV/I0, which we will call the scaled net present value (SNPV):
SNPV )
NPV
1
n)L
I0
n)1
) -1 + I0
∑ CFnfdn
(6)
The evaluation of the economic criteria defined in eqs
1-6 relies upon the assumptions made for the value of the annualization factor, the plant life, the discount factor, and the future cash flows. The discount factor is likely to change as the years go by, the plant life is unpredictable ab initio, and the fluctuations of future prices constitute unforeseeable parameters. Different assumptions will lead to different optimization results, and the values that would be found if perfect information were available, i.e., at the end of the plant life, can only be approximated. Consequently, in the absence of perfect information, current methods of profitability analysis cannot guarantee access to the true optimum design at the design stage. Profitability losses due to inaccurate forecasts can be accounted for by comparison of predicted profits with realized profits. Liu and Sahinidis (1995) looked at how forecast errors affected the solution quality of long-range planning models described as mixed integer linear programs. They simulated random errors ranging from 2 to 20% of actual values and found that profitability was proportionally affected when no plan revision was allowed (i.e., orders and building work commitments have to be honored). However, when plan revision was allowed (i.e., capacity utilization and orders could be modified according to actual demand and prices), they found profitability losses less than 2%. This illustrates the importance of ensuring that all optimum operating points are attainable by the process and that simultaneous adaptation of process operations to reflect changes in market prices occurs. To design a flexible process that will always be adaptable to the market, we require long-term forecasts over the plant lifetime in order to assess the extent of flexibility needed. In the case of commodity chemicals prices, the uncertainty of these long-term forecasts is unlikely to be as low as 2-20%; therefore, the influence of poor forecasts on profitability will be more significant. In the absence of an adequate profitability analysis that accounts accurately for price uncertainty, risk management provides an alternative that aims at the attenuation of profit losses in a fluctuating economic environment. In this context, risk management involves using the market, for instance, by using derivatives (Toal, 1995) or by hedging (Reidy et al., 1977). Hedging on commodities futures market reduces the risk of losing money on a contract, e.g., if the price paid for raw materials by a producer at the time the contract was concluded is higher than the current price at the time of the delivery. However, these risk management methods have themselves proved to be very risky because of the speculative nature of the commodities market, as reported by Thomas (1995). 2.2. Modeling Commodity Price Behavior. Although the spot prices of petrochemicals and fuels fluctuate significantly with time in an unforeseeable way, some observations can be made about the relative variation of energy and product prices on the basis of the fundamental linkage between raw material and product prices. In a free market, variations in energy and feedstock prices are passed on to cause downstream price variations. Sedricks and Carmichael (1993) showed that on a trend-line basis, petrochemical and energy prices (e.g., natural gas) were linear with respect to crude oil prices both in constant U.S. dollars and current U.S. dollars due to the cancellation of inflation by the learning-curve effect. Bacon (1984, 1990) found that the netback (revenue from product sales less marginal costs of refining and transport) is a linear function of the spot price of crude oil, once a time lag (6 days on average) was allowed for product prices to catch up with crude
3730 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
derived from producing one chemical from another and of accounting for the stoichiometry of the main reaction. We write the overall process main reaction in the following general form (Appendix A):
ν1A1 + ν2A2 + ... + νNFANF f δNF+1BNF+1 + δNF+2BNF+2 + ... + δNF+NPBNF+NP (7) where Aj is the feed j, νj is the stoichiometric coefficient of reactant j, Bk is the product k, δk is the stoichiometric coefficient of product k, NF is the total number of feeds, and NP is the total number of products. We can then define the molar price of stoichiometric product, P, as k)NF+NP
Figure 3. Example of the tree structure of the petrochemical industry.
oil prices. This time lag implies that although the margin between prices is not constant, the stability of the price relationship causes product prices to be constrained to fluctuate within a certain range. Other variables prevent product price variations from simply copying feedstock price fluctuations. For example, inventories are used by companies as price regulators to smooth out the effects of wide fluctuations in supply and demand patterns (Long, 1990). Investment can also be used as a price regulator; however, the adjustment to demand shifts is much slower compared to stocks usage. Consequently, adjustment to supply and demand fluctuations requires a combination of the long-term policy of investment and the short-term policy of price changes. For commodity chemicals, production costs, and hence margins, are dictated by raw material costs and hence by feedstock and energy prices. The intensity of the energy price dependence relates directly to the energyintensiveness of the process. The interactions between buyers and sellers in the market correspond to massbalance requirements around the network of chemical processes that constitutes the petrochemical industry. Their study, for the aim of long-range planning, has been the basis of extensive work by Rudd et al. on the simulation of this industry (Rudd, 1975; Jimenez and Rudd, 1987; Chavez et al., 1991) and by Grossmann and co-workers on strategic planning of investment and production (Sahinidis et al., 1989; Sahinidis and Grossmann, 1991a,b; Ierapetritou and Pistikopoulos, 1994). From the tree structure of the bulk chemicals industry, partially illustrated in Figure 3, (Rudd, 1975; Jimenez and Rudd, 1987), we can suggest that along a branch (representing a chemical route) two economic observations can be made. First, product and feed prices depend on each other, and second, energy and feedstock prices also depend on each other. Clearly, the strength of this price dependence is subject to some uncertainty, which relates to the fluctuations of supply and demand, the existence of alternative chemical routes, the impact of raw material costs on the total production cost, stock availability, etc. 3. Analysis 3.1. Price Ratio Behavior. In this study, we will use the ratio of the molar price of product formed stoichiometrically to the molar price of feedstock required stoichiometrically, noted P/F, to investigate their price linkage. This stoichiometric molar price ratio provides the double advantage of measuring the benefits
P)
∑
δkΓk
(8)
k)NF+1
and the molar price of stoichiometrically-required feed, F, is j)NF
F)
νjΓj ∑ j)1
(9)
where Γk is the molar price of product Bk and Γj is the molar price of reactant Aj ($‚kmol-1). Example 1. For the hydrodealkylation (HDA) of toluene to benzene the main reaction is
C6H5CH3 + H2 f C6H6 + CH4
(10)
hence, P ) ΓC6H6 (if methane is not salable) and F ) ΓC6H5CH3 + ΓH2. Example 2. For the acetone-phenol process, where cumene is reacted with air the main reaction is
C6H5CH(CH3)2 + (O2 + 3.8N2) f C6H5OH + CH3COCH3 + 3.8N2 (11) hence, P ) ΓC6H5OH + ΓCH3COCH3 and F ) ΓC6H5CH(CH3)2. In addition, we have selected the ratio between the energy price and the molar price of feedstock, noted E/F, to study the dependence of energy and feedstock prices. We can account for the uncertainty of the dependence by using a probabilistic representation of the price ratio variation. The processes selected as examples are the production of benzene by hydrodealkylation (HDA) of toluene, styrene from benzene and ethylene, p-xylene from isomerization of mixed xylenes and ammonia from natural gas. U.S. spot prices of these chemicals and of natural gas were collected on a weekly basis from Chemical Week, Chemical Marketing Reporter, and the Oil & Gas Journal for the period 1991-1994, which corresponds to over 150 data points for each price ratio and hence forms a large enough sample to be submitted to a statistical analysis. Note that the original data for product prices are corrected to the units $‚kmol-1. The results of the statistical analysis of the price ratio variations are presented in Figure 4 in the form of probability distributions of the standardized values (i.e., z-values) of the ratios P/F and E/F. Table 1 summarizes the results of the statistical analysis of price ratios for each process. The ammonia process corresponds to a particular case where the feedstock and energy are both natural gas; consequently, the probability distribution of E/F is a single peak for this process. More generally, it can be observed that the price ratios P/F always present narrower distributions than the ratios E/F, for which
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3731
Figure 4. Probability distributions of the standardized P/F and E/F over 1991-1994 for (i) the HDA process, (ii) the styrene process, (iii) the p-xylene process, and (iv) the ammonia process. Table 1. Statistical Results for the Distributions of P/F and E/F process
feedstock
benzene styrene p-xylene ammonia
toluene ethylene + benzene mixed xylenes natural gas
process
feedstock
benzene styrene p-xylene ammonia
toluene ethylene + benzene mixed xylenes natural gas
P/F based on benzene styrene p-xylene ammonia E/F based on natural gas natural gas natural gas natural gas
xj
s
V (%)
1.07 1.621 1.755 1.406
0.0076 0.022 0.017 0.0298
0.71 1.36 0.97 2.12
xj
s
V (%)
0.0808 0.0502 0.067 1
0.0017 0.00113 0.0015 0
2.10 2.25 2.24 0
the coefficient of variation, V, is about twice as large. The dependence between energy and feed prices seems irrespective of the processes and shows a regular pattern (V ) {2.10%, 2.25%, 2.24%}), unlike the dependence between feed and product prices (V ) {0.71%, 1.36%, 0.97%, 2.12%}) which is stronger in some processes than in others. The dependence of product and feed prices is influenced by the impact of the raw material costs on the unit costs of the process. The strength of this dependence is determined by the availability of alternative reaction paths. This is illustrated by the branches of the tree structure of the petrochemical industry. Where few branches arrive at a node (i.e., there are few alternative routes to this chemical), the link between product and feed prices is expected to be strong. Conversely, the dependence will weaken as more reaction paths are available. Using the χ2 criterion (Walpole, 1982), we tested the goodness-of-fit of the observed standardized price-ratio distributions with normal distributions. For each price ratio, we tested the null hypothesis that the collected price data was following a normal distribution. The results of this statistical analysis are summarized in Table 2. The last two columns of Table 2 show that all calculated χ2 are lower than the tabulated χ2, i.e., there is no evidence of real difference between observed and expected values, so we accept the null hypothesis that the ratios P/F and E/F follow normal distributions. The
fourth and fifth columns show the levels of skewness (mean away from 0) and flatness (standard deviation away from 1) of the normal distributions. The important result of this analysis is that a statistical dependence exists between product and feed spot prices, and also between energy and feedstock spot prices, for the commodity chemicals studied. Furthermore, it is possible to model the uncertainty of this dependence by a probability distribution. The uncertainty of the price ratios has a great advantage relative to that of the prices in that it is time-invariant. Indeed, unless the petrochemical industry undergoes fundamental structural changes in the chemical route used to manufacture a product and/or ceases to depend on fossil fuels, the linkage between prices and the associated uncertainty are likely to remain constant. Consequently, the probability distributions of P/F and E/F, which are readily obtained from existing price data, are a very good estimate of future price ratio behavior. In particular, the mean of the ratio P/F or E/F represents the true average, throughout the years, of the changing economic environment. This is illustrated in Figure 5 for the HDA process, where the yearly average value of the price ratios is plotted against time. The average price ratios calculated year-to-year vary around their mean values; however, we observe no systematic trend in their fluctuation. 3.2. New Economic Criterion To Exploit Price Ratio Behavior in Profitability Analysis. The DCFROR (eq 4) and the SNPV (eq 6) are chosen to assess and compare profitability. Both of these criteria are calculated from the process cash flow, i.e., the difference between revenue and production costs. The formation of products from reactants can be expressed through the general eq 7. The revenue derives from the sale of the main product and of salable byproducts. Initially, we will assume that waste products are not salable. The total revenue, CR, can be written in terms of the flowrate, Qk, of any product, Bk, as
3732 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 2. Summary of Statistical Results for the Test of Goodness-of-Fit of the Distributions of P/F and E/F process
feedstock
P/F based on
µa
σ
tabulated χ2 (R,df)
calculated χ2
benzene styrene p-xylene ammonia
toluene ethylene + benzene mixed xylenes natural gas
benzene styrene p-xylene ammonia
-0.1 0 0 0
0.7 0.95 0.95 1.05
3.841 (0.05,1) 3.841 (0.05,1) 5.991 (0.05,2) 3.841 (0.05,1)
1.71 1.73 0.428 1.749
process
feedstock
E/F based on
µa
σ
tabulated χ2 (R,df)
calculated χ2
benzene styrene p-xylene
toluene ethylene + benzene mixed xylenes
natural gas natural gas natural gas
0 -0.05 0.02
1 1.15 1.15
5.991 (0.05,2) 3.841 (0.05,1) 3.841 (0.05,1)
0.88 2.25 0.120
a
Corresponding to the z-values of P/F and E/F calculated from the mean and standard deviations of Table 1.
∀ k ∈ K,
CF )
j)NF Qk j)NF ( νjΓj + wjνjΓj) (15) δk j)1 j)1
∑
∑
where wj represents the ratio of the amount of feed Aj lost to byproduct formation or other waste to the amount of feed Aj used to make useful product. We can define an overall feed loss term, w, as k)NF
w)
wjνjΓj ∑ j)1 j)NF
(16)
νjΓj ∑ j)1
So, on the basis of the main production rate, eq 15 combined with eq 16 becomes:
CF )
Figure 5. Fluctuation of the average value of P/F and E/F with time. Example of the HDA process.
∀ k ∈ K ) {NF + 1, ..., NF + NP}, Qk k)NF+NP CR ) δkΓk (12) δk k)NF+1
∑
In particular, if we choose as a basis the production rate of the main product, k ) NF + 1, the combination of eqs 8 and 12 results in
CR )
QNF+1 P δNF+1
(13)
For the sake of simplicity, the following notation will be used:
Q (1 + w)F δ
(17)
For example, consider the production of benzene from toluene in the HDA process with concurrent formation of byproduct diphenyl through the following reactions:
C6H5CH3 + H2 f C6H5 + CH4 (ν1 ) 1) (ν2 ) 1) (δ ) 1) (δ4 ) 1) 2C6H5 a (C6H5)2 + H2
(18)
which are equivalent to
2C6H5CH3 + H2 f (C6H5)2 + 2CH4 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. At a benzene production rate Q, the total amount of converted toluene is Q/S; hence the stoichiometric feed flowrate and the feed flowrate lost to diphenyl are expressed as follows:
QNF+1 ) Q δNF+1 ) δ
(14)
Feed chemicals are converted into products and byproducts; therefore, the total feed cost, CF, is the cost of the feed required by stoichiometry in order to make the product, together with the feed that is lost to waste production, including amounts of feeds that form byproducts:
C6H5CH3 + H2 f C6H6 + CH4 Q Q Q Q H2 f (C6H5)2 + 2CH4 2C6H5CH3 + Q(1 - S) Q(1 - S) Q(1 - S) Q(1 - S) S 2S S 2S Therefore the total cost of feed is
(19)
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3733
time. Consequently, the expression in brackets can be factorized out of the time summation. Equation 22 then becomes:
F ) ΓC6H5CH3 + ΓH2 w1 )
1-S , S
w2 )
1-S 2S
SNPV )
1-S 1-S ΓC6H5CH3 + Γ S 2S H2 w) ΓC6H5CH3 + ΓH2 )
δ F
( Q ) (Γ S
)
)
1
(CR - CF - CE)nYfdn ∑ n)1
( ∑ ( (( )
n)L
) ) ( ))
n)L
Q P
I0
n)1
δ F
) ()
-1-w -e
(24)
Let xd (d ∈ D ) {1, ..., ND}) be the ND design variables, then n)L
∂(REF ∀ d ∈ D,
∂SNPV
∑ FnYfdn)
n)1
) ∂xd
∂xd ∂REF n)L
FnYfdn + ∑ n)1
) ∂xd
n)L
∂( REF
FnYfdn) ∑ n)1 ∂xd
(25)
The present value of the future feed prices is independent of the design variables; hence, n)L
∂( ∀ d ∈ D,
FnYfdn) ∑ n)1 ∂xd
)0
(26)
So eq 25 becomes
∂REFn)L
∂SNPV
∀ d ∈ D,
) ∂xd
∂xd
FnYfdn ∑ n)1
(27)
Also, the feed price is always strictly positive; therefore, n)L
∑ FnYfdn > 0
(28)
Hence, at optimum profitability we can derive from eq 27
1 ) -1 +
(23)
n)1
Q (P - (1 + w)F) - eE Yfdn δ n
I0
(( )
n)L
∑ n)1
) -1 +
FnYfdn ∑ n)1
(21)
where the energy use, e, is a function of the design variables. A simple estimate of the process energy requirements can be found by applying the method of pinch analysis (Linnhoff et al., 1982). When the cooling medium is air or cooling water, the cost of cooling is assumed to be negligible. For example, the cost of cooling water per kilowatt is typically 1% of the cost of power (Smith, 1995). If the process requires refrigeration, then the cost of cooling is proportional to the cost of electricity. In turn, the price of electricity can be expressed in terms of the energy price (e.g., 4:1 ratio); hence the cost of refrigeration can be accounted for in the process energy cost according to eq 21. Similarly, the cost of electric power and the value of electric power generation can be included in eq 21. Using eqs 13, 14, 17, and 21, we can rewrite eq 6:
I0
n)L
Q P E -1-w -e δ F F REF ) I0
(20)
CE ) eE
SNPV ) -1 +
F
We call the ratio in front of the summation term the residual economic function (REF):
The main costs of waste are the cost of feed loss and, possibly, the disposal cost. This latter cost is proportional to the amount of waste generated (i.e., to the waste flowrate), and since we assume for now that the waste is not salable, the disposal cost is independent of the changing economic environment. In the particular case where waste can be burnt and used as an energy source, it is possible to adapt our costing method by accounting for the potential fuel credit of the waste in the energy cost. Salable byproducts that are not formed through the main reaction can also be accounted for, as described by Boccara (1996). The energy cost, CE, is the product of the process energy use and the energy price, E:
1
E
I0
1-S (2ΓC6H5CH3 + ΓH2) 2S
1+S Γ H2 C6H5CH3 + 2
) ()
-1-w -e
-1 +
1 - S 2ΓC6H5CH3 + ΓH2 2S ΓC6H5CH3 + ΓH2
CF ) Q ΓC6H5CH3 + ΓH2 +
(( )
Q P
E F
n
FnYfdn (22)
At fixed production rate, Q, and for a design based on a single set of optimum conditions, w and e are both functions of the design variables (including the production capacity), all of which are time invariant. In addition, we have shown previously that the distributions of the price ratios P/F and E/F are independent of
∂REF ∂SNPV )0S )0 ∂xd ∂xd
∀ d ∈ D,
(29)
Also,
∀ d ∈ D,
∂2SNPV
∂2REFn)L )
∂xd
2
∂xd
2
FnYfdn + ∑ n)1 n)L
∂( ∂REF ∂xd
∑ FnYfdn)
n)1
∂xd
(30)
3734 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
Using eq 26 the above equation becomes:
SNPV ) -1 + ∀ d ∈ D,
∂2SNPV
∂2REFn)L )
∂xd2
∂xd2
∑ FnYfdn n)1
(31)
1 I0 1
) -1 + From the combination of eqs 28 and 31, we conclude that the second derivatives of SNPV and REF have the same sign. In addition, we have shown in eq 29 that SNPV and REF are equivalent at optimum, therefore maximizing SNPV, i.e., maximizing profitability, is equivalent to maximizing REF. A demonstration of the equivalence between optimum DCFROR and REF is presented in Appendix B. The main advantage of using REF over NPV, SNPV, or DCFROR is that its evaluation avoids making assumptions embedded in these three criteria. These assumptions are the plant life, the discount factor, and the forecast of future cash flows. None of these unforeseeable parameters are present in the expression of REF, which relies only on known parameters, namely, the production rate, the feed flowrate, the functions w and e, and the price ratios P/F and E/F. The production rate is assumed to be fixed at this stage; however, we address varying production rate below. The value of the feed flowrate and the functions w and e are controlled by the designer and by the operator if some flexibility is allowed for. Furthermore, the price ratios P/F and E/F can be evaluated using existing price data. Consequently, there are no unforeseeable variables in the expression of REF, which makes it a powerful mathematical tool expressing profitability under price uncertainty. Use of REF enables the designer to carry out profitability analysis at the design stage, without requiring forecasts of future prices, a prediction of the process lifetime, or any assumption regarding interest rates. It is also important to note that the use of REF does not entail that the designer use either sophisticated software or simple short-cut calculations to carry out design optimization. The use of REF as an economic criterion is not restricted to any particular optimization method. 3.3. Fluctuating Production Rate. If we assume that process-operating efficiency is not affected by changes in production levels, then costs and revenue should be affected in the same proportion by a change in production rate. We would also obtain the same result if we assumed that when the production rate varies all process units suffer the same reduction in efficiency. Each year of the plant life can be separated into NTn time periods, where each time period corresponds to a constant level of production. The cash flow in a particular time period is then represented as a weighted cash flow where the weight is a function of the throughput level. The NPV can therefore be expressed as
I0
n)L t)NTn
( ∑ (CR - CF - CE)tωt)nYfdn ∑ n)1 t)1
( (
n)L t)NTn
∑ ∑ n)1 t)1
Q δ
(P -
))
(1 + w)F) - eE ωt Yfdn t
1 ) -1 + I0
( ( (( ) ) ( )) )
n)L t)NTn
∑ ∑ n)1 t)1
Q
P
δ
F
-
1-w -e
Q δ
(( ) P
F
n
E F
) ()
-1-w -e
Ftωt Yfdn
t
n
E F
) -1 + I0
×
n)L t)NTn
( ∑ Ftωt)nYfdn ∑ n)1 t)1
n)L t)NTn
) -1 + REF
( ∑ Ftωt)Yfdn ∑ n)1 t)1
(33)
where Q is now the maximum production rate. There are two possible modes of operation common to every commodity process. In the first mode, the product is sold through contracts, and consequently, levels of production are set by delivery constraints. Alternatively, the commodity can be traded on the spot market in which case the global levels of supply and demand set the production levels through pricing. The actual operation over the process life is most likely to be a combination of the two above scenarios, due to the nature of the commodity markets. It is therefore reasonable to assume that ωt is set by external constraints over which the operator has no control. To this extent, ωt is no different than Ft and eq 32 enables us to conclude that a design based on the optimization of REF will yield maximum profitability independent of the values of both ωt and Ft. The above equation allows the extension of the conclusions we have drawn for the optimization of commodity chemical processes at a given level of production to permit optimization with a variable level of production, where the exact production level at any future time is not specified at the design stage. Again, the suitability of REF as an optimization criterion is manifest. 3.4. Example. To illustrate the use of REF as an economic criterion and compare its benefits with other economic criteria, we propose to solve the following problem. We wish to produce a commodity chemical formed through the following reactions:
feed f product h W2 n)L
NPV ) -I0 +
feed f W1
∑ CFnfdn n)1
n)L t)NTn
) -I0 +
∑(∑ n)1 t)1
CFtωt)nfdn
(32)
where t is a time period and ωt is a weighting factor. In terms of SNPV:
(34)
The conversion of product into W2 is Xeq and the selectivity of feed to W1 is (1 - S1). Prices of feed, product, and energy are uncertain and wastes W1 and W2 are not salable. The available design alternatives (Figure 6) are either (1) to separate the product from W1 and W2 and dispose of both wastes or (2) to proceed to the further separation of W1 from W2 and recycle W2 to the reactor. The separation of the product from
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3735 Table 3. Data for Designs 1 and 2 assumptions (designs 1 and 2) S1
Xeq
L
Q
0.96
0.30
5
100
operating costs ($‚h-1) design (1) (2)
feed
energy 150*E 365*E
MF,1ΓF MF,2ΓF
capital cost ($‚h-1) reactor
column 1
column 2
3 × 105MF,10.6Af/Y 3 × 105MR,20.6Af/Y
(M1,1 + 20)/Af/2.26 (M1,2 + 20)/Af/2.26
N/A (M2,2 + 35)/Af/2.26
Table 4. Summary of Price Data previous year
xj
s
ΓF ΓP E
9.58 21.3 1.81
0.417 1.33 0.0834
previous 5 years
xj
s
P/F E/F
2.21 0.190
0.0263 0.00263
and energy is expensive whereas design 2 is favored by the reverse situation. In this example, P and F are expressed as
P ) ΓP F ) ΓF
Figure 6. Alternative designs: (1) no recycle of W2 and (2) recycle of W2.
the wastes is achieved by Column 1; Column 2 separates waste W1 from waste W2. We are to select the most profitable design by carrying out short-cut calculations. Profitability is assessed by each of the following economic criteria: TAC, EP, NPV, and REF. Mass balances around the process lead to the following flowrate expressions for design 1:
MF,1 ) M1,1 )
Q S1(1 - Xeq)
(35)
(37)
where ΓP is the product price and ΓF is the feed price. The price data are shown in Table 4. Table 5 gives the price models selected for the assessment of the various profitability criteria. The results of the profitability analysis are given in Table 6. This table illustrates the implications of various assumptions on the optimization result. In particular, we have investigated the influence of the interest rate, i.e., the annualization factor, and the forecasted prices. Using a single economic criterion, different values of these variables result in the selection of different designs. The results are, therefore, not conclusive for TAC, EP, and NPV. However, because REF has the advantage of being independent of these assumptions, we can draw significant conclusions from its optimization results. In this example, using REF we select design 2 as the single best optimum design under the prevailing fluctuating economic environment. 4. Conclusions
where MF,1 is the fresh feed flowrate and M1,1 is the flowrate to Column 1. For design 2:
MF,2 )
Q S1
(1 + XeqS1) MR,2 ) M1,2 ) Q S1 M2,2 ) Q
(1 - (1 - Xeq)S1) S1
(36)
where MF,2 is the fresh feed flowrate, MR,2 is the inlet flowrate to the reactor, M1,2 is the flowrate to Column 1, and M2,2 is the flowrate to Column 2. The problem assumptions, specifications, and cost data are given in Table 3 for designs 1 and 2. It is assumed that the main cost of waste is the cost of feed material lost to producing waste products. Table 3 shows that design 1 is favored when the feed is cheap
In this paper we have introduced a new economic criterion, REF, which can be used in optimizing the design of commodity chemical processes. This criterion was derived on the basis of the fundamental dependence between petrochemical prices and crude oil prices. In particular, future prices will reflect this dependence, which is inherent to the petrochemical industry and will prevail as long as no major restructuring of that industry occurs. In order to represent this dependence, we have introduced the stoichiometric molar price ratios P/F and E/F. These price ratios are not the result of any forecast but rather derive from existing price data. Spot price data for some existing commodity processes show that these ratios can be represented by normal distributions. By using REF as an objective function in combination with any optimization method, knowledge of the mean of the distributions of the price ratios P/F and E/F enables the designer to find the best singleoptimum design under the fluctuating economic environment. Although the use of REF as an economic criterion is not restricted to processes involving chemi-
3736 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 Table 5. Price Models criterion
data
model
ΓF
ΓP
TAC, EP previous year previous year NPV previous year REF
previous year optimistic scenario pessimistic scenario previous year pessimistic forecasta previous 5 years
mean xj - s xj + s optimistic forecasta xjn (n ) 1, ..., 5) mean ratios
xj xj + s xj - s xjn (n ) 1, ..., 5) xjn (n ) 1, ..., 5) xj
xj xj - s xj + s xjn (n ) 1, ..., 5) xjn (n ) 1, ..., 5) xj
a
E xj xjn (n ) 1, ..., 5) xj
Details of the forecasts are given in Appendix C.
Table 6. Results
(1)
(2)
most profitable design
17.67 17.88 18.48 17.27 -0.31 -0.52 -0.04 -0.99 16.90 -3.64 4.77
17.82 17.85 17.25 18.45 -0.46 -0.49 1.19 -2.17 15.34 -3.08 4.84
(1) at i ) 5% (2) at i ) 10% (2) (1) (1) at i ) 5% (2) at i ) 10% (2) (1) (1) (2) (2)
design criterion TAC (million $‚year-1)
model
mean mean optimistic scenario pessimistic scenario EP (million mean mean $‚year-1) optimistic scenario pessimistic scenario NPV (million $) optimistic forecast pessimistic forecast REF (kmol‚h-1‚ mean ratio -1 million $ )
cals subject to fluctuating prices, it is particularly useful for petrochemical processes in which raw material prices undergo significant variations. Unlike TAC, EP, NPV, and DCFROR, this new criterion does not involve the calculation of profit, which is, in essence, why it is so powerful. Indeed, the calculation of profit is directly related to the value of future prices, which are unforeseeable. Hence, any economic criterion requiring price forecasts is inherently inaccurate and consequently leads to inaccurate optimization results. This result was illustrated by an example where TAC, EP, NPV, and REF were compared. The main restriction on the use of REF is that it requires the maximum production rate to be specified. For this reason, it can be used following the results of long-range planning, i.e., once expansions have been scheduled, e.g., following the method of Sahinidis et al. (1989). Nomenclature Af ) annualization factor (year-1) (in the example 3.3.4., the expression of Af(i,L) is taken from Douglas (1988)) Aj ) feed j Bk ) product k BNF+1 ) main product CE ) energy cost ($‚h-1) CF ) total feed cost ($‚h-1) CI ) intermediate product I cI,1 ) stoichiometric coefficient of intermediate product CI in the first intermediate reaction cI,m ) stoichiometric coefficient of intermediate product CI in intermediate reaction m CR ) revenue ($‚h-1) CFn ) cash flow in year n ($) CFt ) cash flow in period t ($) D ) {1, ..., ND} set of design variables df ) degree of freedom DCFROR ) discounted cash flow rate of return, eq 4 e ) energy use (MMBtu‚h-1) E ) price of energy ($‚MMBtu-1) EP ) economic potential ($‚year-1), eq 2 fd ) discount factor, eq 5 F ) molar price of feedstock required stoichiometrically, eq 9 ($‚kmol-1) Fn ) value of F in year n ($‚kmol-1)
Ft ) value of F in period t ($‚kmol-1) h ) operating cost per unit of production ($‚kmol-1) i ) interest rate I0 ) initial investment ($) K ) {NF + 1, ..., NF + NP} set of reaction products L ) plant life (year) Mj,i ) flowrate to Column j for design i (kmol‚h-1) MF,i ) feed flowrate for design i (kmol‚h-1) MR,i ) flowrate to reactor for design i (kmol‚h-1) ND ) total number of design variables NF ) total number of feeds NP ) total number of products formed through the overall process main reaction NPV ) net present value ($), eq 3 NTn ) number of time periods with constant production rate in year n P ) molar price of product formed stoichiometrically, eq 8 ($‚kmol-1) Q ) production rate of main product BNF+1 (kmol‚h-1) Qk ) production rate of Bk (kmol‚h-1) REF ) residual economic function (kmol‚h-1‚$-1), eq 24 S ) selectivity S1 ) selectivity of feed to product in Section 3.4. s ) sample standard deviation SNPV ) scaled net present value ( ), eq 6 TAC ) total annualized cost ($‚year-1), eq 1 V ) s/xj × 100 ) coefficient of variation (%) w ) ratio of price of feed lost to waste production to price of feed required stoichiometrically to make the product, eq 16 xj ) sample mean X ) conversion Xi ) conversion of single best design of flowsheet (i) Xeq ) equilibrium conversion of product to W2, Section 3.4. xd ) design variable d xjn ) average value in year n Y ) operating hours per year (h‚year-1) (Y ) 8150 h.year-1 in all numerical examples) z ) (x - µ)/σ ) standardized value of the observation x Indices d ) design variable j ) feed k ) products formed through the overall process main reaction, eq 3.3 n ) year m ) intermediate reaction t ) time period (e.g., month) Subscripts eq ) reaction equilibrium i ) flowsheet i Greek Symbols R ) level of confidence Γi ) molar price of compound i ($‚kmol-1) Γj ) molar price of feed Aj ($‚kmol-1) Γk ) molar price of product Bk ($‚kmol-1) δ ) stoichiometric coefficient of main product BNF+1 in the overall process main reaction δk ) stoichiometric coefficient of product Bk δ′k,m ) stoichiometric coefficient of product Bk in intermediate reaction m
Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997 3737 µ ) mean νj ) stoichiometric coefficient of reactant Aj ν′j,m ) stoichiometric coefficient of reactant Aj in intermediate reaction m σ ) standard deviation Φ ) present value of the future prices of feed, eq 42 f)n [(of - ef)2]/ef, where of and ef are the observed and χ2 ) ∑f)1 expected frequencies, respectively, of the fth cell (Walpole, 1982) ωt ) weight of cash flow CFt in period t
∀ d ∈ D, ∂fdn
n)L
∂Φ
∑
) ∂xd
F nY
∂xd
n)1 n)L
∑
)
∂((1 + DCFROR)-n) FnY
∂xd
n)1 n)L
∂DCFROR
FnY(-n)(1 + DCFROR)-(n+1) ∑ n)1
)
∂xd
(44)
Appendix A Stoichiometry of the Overall Process Main Reaction. The overall process main reaction is obtained by combining the intermediate reactions yielding the main product. The first intermediate reaction is of the following form:
ν′1,1A1 + ... + ν′NF,1ANF f c1,1C1 + ... + cNI,1CNI
Also,
(38)
where CI is the intermediate product I, cI,1 is the stoichiometric coefficient of intermediate product I in the first intermediate reaction, and NI is the total number of intermediate components. The following intermediate reactions can be written as
ν′1,mA1 + ... + ν′NF,mANF + c′1,mC1 + ... + c′NI,mCNI f δ′NF+1,mBNF+1 + ... + δ′NF+NP,mBNF+NP (39) where BNF+1 is the main product. The intermediate reactions can always be combined to give an overall process main reaction of the same form as eq 7. As a result of this combination, the reactants of the main reaction are the process feeds and the products only include those formed in stoichiometric amount with the main product, i.e., the main reaction does not include byproducts formed by selectivity losses. Further discussion of cases such as nonstoichiometric feed and salable byproducts is given by Boccara (1996).
∀ n,
n)L
1 ) REF
∑ FnYfdn n)1
(40)
Fn > 0
∀ n,
n>0
∀ n,
Y>0
(1 + DCFROR)-(n+1) > 0
∀ n,
-(F)nYn(1 + DCFROR)-(n+1) < 0 (46)
Hence, at optimum profitability,
∂Φ ∂DCFROR )0S )0 ∂xd ∂xd
∀ d ∈ D,
1 (REF ) ) 0 S ∂DCFROR )
∂ ∀ d ∈ D,
∂xd
∂xd -1 ∂REF 0S ) 0 (48) REF2 ∂xd
And REF is always a finite number; hence we can conclude from the above equation
∂REF δDCFROR )0S )0 δxd ∂xd
∀ d ∈ D,
∀ d ∈ D,
∂2Φ ) ∂xd2
n)L
(41)
(49)
Taking second derivatives with respect to xd of eq 44 gives
FnYn(n + 1)(1 + DCFROR)-(n+2) ∑ n)1
fd ) (1 + DCFROR)-1
(47)
Which, using eq 43, is equivalent to
n)L
where
(45)
So,
Appendix B Demonstration of the Equivalence between DCFROR and REF at Optimum. At DCFROR,
∀ n,
∑ FnY(-n)(1 + DCFROR)
(
∂xd
∂2DCFROR
-(n+1)
n)1
We define Φ as
)
∂DCFROR
∂xd2
2
+
(50)
Hence, at optimum, using eq 49, eq 50 simplifies to n)L
Φ)
FnYfdn ∑ n)1
(42)
Hence eq 40 can be rewritten as
1 ) REFΦ So we can write
∀ d ∈ D,
∂2Φ ) ∂xd2
n)L
(43)
∑
n)1
And,
FnY(-n)(1 + DCFROR)-(n+1)
∂2DCFROR ∂xd2
(51)
3738 Ind. Eng. Chem. Res., Vol. 36, No. 9, 1997
(
)
∂2Φ ∂ -1 ∂REF ) ) 2 2 ∂x ∂x ∂xd d REF d
∀ d ∈ D,
( )
2 ∂REF REF3 ∂xd
2
-
1 ∂2REF (52) REF2 ∂xd2
Using eq 49 this becomes
∀ d ∈ D,
-1 ∂2REF ∂2Φ ) 2 ∂xd REF2 ∂xd2
(53)
The combination of eqs 51 and 53 gives the following at optimum:
∀ d ∈ D,
-1 ∂2REF ) REF2 ∂xd2 ∂2DCFROR
n)L
∑ FnY(-n)(1 + DCFROR)
-(n+1)
∂xd2
n)1
(54)
Also,
-1
