Algebraic Techniques for Property Integration via Componentless

3792

Ind. Eng. Chem. Res. 2004, 43, 3792-3798

Algebraic Techniques for Property Integration via Componentless Design X. Qin, F. Gabriel, D. Harell, and M. M. El-Halwagi* Department of Chemical Engineering,Texas A&M University, 3122 TAMU, College Station, Texas 77843-3122

In many cases, design specifications are based on the satisfaction of a set of property constraints. Recently, the novel paradigm “componentless design” was introduced by Shelley and El-Halwagi (Comput. Chem. Eng. 2000, 24, 2081) to identify optimal graphical strategies for property-based problems. The essence of this approach is driven by tracking properties and not chemical components. Notwithstanding the usefulness of this graphical approach, it is limited to the simultaneous tracking of three properties. In this paper, we overcome this limitation by developing an algebraic approach for property integration through componentless design. Process constraints and stream characterization are described using bounds on intensive properties and flows. The problem of allocating streams is mapped into an equidimensional domain of dimensionless property operators. The specific mathematical structure of the set of operator constraints is exploited to develop a constraint-reduction algorithm, which provides rigorous bounds on the feasibility region. The result is an efficient algebraic procedure that identifies optimal allocation in the property-operator domain. The solution is then mapped back to the raw-property domain. To illustrate the applicability of the developed approach, a four-property case study is addressed using the devised algebraic procedure. Introduction Primarily, the tracking of individual species has been the heart of process design and integration. Although the nature and quantity of chemical constituents are important in characterizing key aspects in process design, it is necessary to note that many design problems are not component-dependent or “chemo-centric.” Instead, these problems are driven by properties or functionalities of the streams and not by their chemical constituency. The following are examples of design problems that are based on properties or functionalities: (i) In some cases, the usage of material utilities (e.g., solvents) relies on the material characteristics such as equilibrium distribution coefficients, viscosity, and volatility without the need for chemical characterization of these materials. (ii) Alternatively, constraints on process units that can accept process streams can be based on the properties of the feeds to processing units. For example, the recycle/reuse of waste streams to a papermaking machine is based on properties (e.g., reflectivity, opacity, and density, to name a few). The performance of a heat exchanger is based on the heat capacities and heat-transfer coefficients of the matched streams. The chemical identities of the components are useful only to the extent of determining the values of heat capacities and heat-transfer coefficients. Similar examples can be given for many other units (e.g., vapor pressure in condensers; specific gravity in decantation; relative volatility in distillation; Henry’s coefficient in absorption; density and head in pumps; density, pressure ratio, and heat-capacity ratio in compressors; etc.). (iii) In other problems, the tracking of numerous chemical pollutants can be prohibitively difficult. A common example is the case of continuous mixtures that contain numerous (almost infinite) components such as complex hydrocarbons and lignocellulosic materials. Nonetheless, if one tracks such continuous mixtures on the basis * To whom correspondence should be addressed. Tel.: (979) 845-3484. Fax: (979) 845-6446. E-mail: [email protected]

of their properties, one avoids the need to enumerate the numerous chemical constituents. Recent work by Shelley and El-Halwagi1 has shown that it is possible to tailor conserved quantities, called clusters, that act as surrogate properties and enable the conserved tracking of functionalities instead of components. Clusters are defined as conserved surrogate properties that are tailored to ensure intra- and interstream conservation, which are defined as follows. Intrastream Conservation. For any stream s, the sum of clusters must be conserved, adding up to a constant (e.g., unity), i.e. Nc

Ci,s ) 1 ∑ i)1

s ) 1, 2, ..., Ns

(1)

Interstream Conservation. When two or more streams are mixed, the resulting individual clusters must be conserved via consistent additive rules, preferably in the form of lever arm rules, such as Ns

C hi )

∑βsCi,s

i ) 1, 2, ..., Nc

(2)

s)1

where C h i is the mean cluster resulting from adding the individual clusters of Ns streams and βs represents the mixing arm of stream s on the ternary cluster diagram. Graphical procedures and some of the key optimization rules can be found in El-Halwagi et al.2 Eden et al.3,4 and Gani and Pistikopoulos5 addressed the problem of simultaneous molecular and process design through the use of property-based representation techniques. Eden et al.2 employed the concept of clustering to reformulate the conventional forward design problem as two reverse problems. First, the design targets (constitutive variables) are identified, and subsequently, the design targets are matched by solving the constitutive equations. For instance, Eden et al.2 used the clustering techniques to obtain optimal prop-

10.1021/ie034183k CCC: $27.50 © 2004 American Chemical Society Published on Web 03/02/2004

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3793

erty profiles for potential fresh resources. Then, by utilizing computer-aided molecular design techniques, candidate fresh resources were generated from the desired property profiles and compared with commercially available chemicals. Gani and Pistikopoulos3 discussed two distinct approaches for property-based modeling and design of products and processes: one is the formulation of a mixed-integer nonlinear programming problem to tackle the modeling and simulation of a process based on properties; the second approach simplifies thermodynamic equations of state by lumping terms to allow for graphical solutions. The obtained graphical solutions provide a basis for more detailed design. Notwithstanding the usefulness and the visual insights of the graphical techniques for clustering-based property integration and componentless design, there are several limitations for such visualization approaches, as described below. Number of Properties. Because of the ternary representation of the graphical clusters, a maximum of three properties can be tackled simultaneously. In many cases, more than three properties must be considered in the analysis to provide a more realistic scenario for the overall problem. Optimal Mixing Points. Shelley and El-Halwagi1 presented an enumeration technique for the identification of the optimal properties of mixtures and the fractional contributions of the contributing streams. Although this approach works for problems of limited size, it becomes tedious for large-scale problems. Scaling Problems. In some applications, the problem data might render a graphical representation with various appropriate scales. In such cases, a single ternary graph cannot capture the right level of details for all of the cluster regions. Instead, the scaling up and down of clusters becomes necessary to provide proper representation. Integration with Other Design Techniques. As a result of its graphical nature, visualization tools cannot be easily integrated with mathematical approaches (for both process and product design). To overcome these limitations, the current paper expands on the ideas and applicability of the approach introduced by Shelley and El-Halwagi1 by incorporating an algebraic procedure for property integration and componentless design. The new approach allows for the expansion of the problem to include any number of properties, which breaks the previous graphical limitation of optimizing at most three properties at one time. Additionally, for mixing problems, the algebraic approach directly determines the optimal fractional contributions of any number of sources. To demonstrate the applicability of this new approach, a modified version of the degreaser case study by Shelley and El-Halwagi1 is solved algebraically. Problem Statement The problem to be addressed in this work can be formally stated as follows: Given is a process with a number, Ns, of streams (sources) that contain a number, Np, of targeted properties. These streams can be utilized in a number, Nj, of process units (sinks) if they satisfy given constraints on the flow rates and properties. Each sink has a set of constraints on functionality (properties) that are described by

lower upper pp,j e pp,j e pp,j p ) 1, 2, ..., Np; j ) 1, 2, ..., Nj (3)

where p is the index of properties and j is the index of sinks. Similarly, there are flow rate constraints on the utilized streams given by

e Fj e Fupper Flower j j

j ) 1, 2, ..., Nj

(4)

Background Equations For the tracking of properties throughout a process, the basic equations from Shelley and El-Halwagi1 are used. The mixing rule for a given property is described by Ns

j p) ) ψp(p

xiψp(pp,i) ∑ i)1

p ) 1, 2, ..., Np

(5)

where xi is the fractional contribution of stream i and ψp(pp,i) is the property operator acting on pp,i. Next, the property operator is made dimensionless by dividing by a reference value of the operator. Therefore

Ωp,i )

ψp(pp,i) ψref p

p ) 1, 2, ..., Np; i ) 1, 2, ..., Ns (6)

where Ωp,i is the dimensionless property operator and ψref p is the chosen reference value of the operator. Next, the augmented property index and the cluster are defined as Np

AUPi )

∑ Ωp,i

(7)

Ωp,i AUPi

(8)

p)1

Cp,i )

where AUPi is the augmented property index of stream i and Cp,i is the cluster of property p in stream i. Constraint Reduction Approach Given a process with Ns sources, each containing a finite number of properties, the optimal mixing and allocation of these streams to a given sink containing certain property constraints can be obtained using the above-described equations. First, we exploit the special mathematical structure of the property operator constraints to find rigorous bounds on the feasibility domain. In particular, we identify the feasible intervals of fractional contributions of streams. Then, by incorporating the source cost function, the first technique can be expanded to determine the optimal mixture that provides the minimum resource cost. In the development of the constraint-reduction algorithm, the special features of the property constraints are examined. Each sink has property constraints with upper and lower boundaries as described in eq 3. Using the mixing rule operators and appropriately chosen reference values, these upper and lower property constraints can be transformed into the corresponding Ωmin and Ωmax values for each property using eq 6. Therefore, the following inequality expression can be obtained

eΩ h p e Ωmax Ωmin p p

p ) 1, 2, ..., Np

(9)

3794 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004

Similarly, any feasible combination of sources can be expressed in terms of their mixture properties, which can be translated into an equal number of bilinear expressions using the mixing rule operators and appropriately chosen reference values. The general dimensionless mixing rule for a given property p and Ns sources can be written as Ns

Ω hp )

xiΩp,i ∑ i)1

p ) 1, 2, ..., Np

(10)

Additionally, the summation of the fractional contributions of the sources leads to the following equation Ns

xi ) 1 ∑ i)1

(11)

Combining eqs 10 and 11, the following expression can be generated Ns-1

xi(Ωp,i - Ωp,N ) ∑ i)1

Ω h p ) Ωp,Ns +

p ) 1, 2, ..., Np

s

(12)

Using eqs 9 and 12, the following inequality expression can be generated Ns-1

xi(Ωp,i - Ωp,N ) e Ωmax ∑ p i)1

e Ωp,Ns + Ωmin p

s

p ) 1, 2, ..., Np (13)

Now, eq 13 can be rearranged and separated to form the following two sets of inequalities Ns-1

xi(Ωp,i - Ωp,N ) ∑ i)1

p ) 1, 2, ..., Np

xi(Ωp,i - Ωp,N ) e Ωmax - Ωp,N ∑ p i)1

p ) 1, 2, ..., Np

Ωmin - Ωp,Ns e p

s

(14a)

Ns-1

s

s

(14b)

Inequality sets 14a and 14b provide the basis for the constraint-reduction algorithm, i.e., the algebraic approach. However, several additional physical inequality constraints must be considered as well. The expressions in 15a and 15b constitute the additional physical inequality constraints

xi g 0

i ) {1, ..., Ns}

(15a)

property, there are two inequalities that have the same Ns-1 xi(Ωp,i - Ωp,Ns), on opsummation term, namely, ∑i)1 posite sides of the inequality. Therefore, only one of these sums can be used to eliminate an xi term from the pair of inequalities 14a and 14b for another property. The same step is repeated for the Np properties to identify the most compact feasibility region in the fractional-contribution domain. Indeed, during the first elimination, the inequalities are combined such that 2(NpC2 + Np) + 1 inequalities are generated for each sink, where NpC2 is defined as the number of combinations of two properties that can be generated from Np properties. This is because each property contains two inequalities; therefore, each property can be combined with another property in two ways. Furthermore, only one inequality from set 15a will participate in the firstround elimination. The active inequality in 15a will be the nonnegativity constraint for the xi term being eliminated. The inactive inequalities from 15a will carry over into the next elimination round, where again, only one inequality will participate in the elimination. Finally, inequality 15b can be combined with each pair of property inequalities once and also with the active nonnegativity constraint. This systematic elimination can then be repeated to further eliminate an additional xi term. The process is repeated until only one xi term remains, at which time, the inequalities are solved (all Ω values are numbers calculated from the source and sink information) to provide a range for the remaining xi term. This process can then be repeated to provide a range for all of the various xi terms. The result is a rigorous bound on the feasible range for each fractional contribution. This constraint-reduction algorithm is illustrated in the flowchart presented in Figure 1. In the case of a process with multiple process sources and a fresh resource, the algorithm can be used to calculate the minimum fractional contribution of the fresh resource. In some cases, there might be multiple fresh resources. For such cases, the developed algorithm can be modified to address cost (instead of flow) minimization. By utilizing the resource cost function, the optimal composition of the mixture and the minimum resource cost can be determined. The resource cost function for systems in which there is a set price for each source can be generically represented by the equation Ns

Y)

Ns-1

xi e 1 ∑ i)1

cixi ∑ i)1

(16)

(15b)

The constraints in 15a characterize the nonnegativity condition for the fractional contributions, where as those in 15b ensure that the fractional contributions for Ns 1 streams do not exceed unity. With inequality sets 14 and 15, the feasible range of fractional contributions can be determined. Because of the special structure of inequalities 14a and 14b, these expressions can be combined with constraints 15a and 15b to eliminate a fractional contribution, xi, term. Upon inspection of the structure of inequalities 14a and 14b, it is noticed that, for each

where Y is the total cost and ci is the cost per unit of source i. By combining eqs 16 and 11, an xi term (denoted by xs) can be eliminated, thereby generating the expression Ns-1

Y ) CNs +

(ci - cN )xi ∑ i)1 s

(17)

Next, eq 17 can be rearranged and solved for the fractional contribution of the (Ns - 1)th source, which is given by

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3795

Figure 1. Constraint-reduction algorithm to optimize fractional contributions. Ns-2

YxNs-1 )

s

i ) {1, ..., Ns - 2}

xi g 0

(ci - cN )xi - cN ∑ i)1

Ns-2

s

Y-

(18)

cNs-1 - cNs

(ci - cN )xi - cN ∑ i)1 s

cNs-1 - cNs Furthermore, eq 18 can be substituted into inequality sets 14a, 14b, 15a, and 15b to create the following constraints Ns-2

Y-

Ns-2

xi(Ωp,i - Ωp,N ) + ∑ i)1


s

cNs-1 - cNs

s

(Ωp,Ns-1 - Ωp,Ns) g Ωmin - Ωp,Ns p Ns-2

Y-

Ns-2

∑ i)1

xi(Ωp,i - Ωp,Ns) +


s

cNs-1 - cNs (Ωp,Ns-1 - Ωp,Ns) e Ωmax - Ωp,Ns p l

(19)

Ns-2

Y-

Ns-2

∑ i)1

xi(Ωz,i - Ωz,Ns) )


s

cNs-1 - cNs (Ωz,Ns-1 - Ωz,Ns) g Ωmin - Ωz,Ns z Ns-2

Y-

Ns-2

xi(Ωz,i - Ωz,N ) + ∑ i)1 s


s

cNs-1 - cNs (Ωz,Ns-1 - Ωz,Ns) e Ωmax - Ωz,Ns z

s

g0

(20)

Ns-2

YN -2 s

∑ i)1


cNs-1 - cNs

s

e1

By inspection, inequalities 19 and 20 can be systematically combined to eliminate an xi term, similarly to how inequalities 14a, 14b, 15a, and 15b were combined. During the first elimination, 2(NpC2) + 3Np + 2 inequalities will be generated. The number of inequalities generated is determined in a fashion similar to that described in the previous section regarding the combination of inequalities 14a, 14b, 15a, and 15b. Again, several nonnegativity constraints will be inactive and, therefore, carried on to the next elimination. The systematic elimination can then be repeated to remove an additional xi term. The constraint-reduction procedure is repeated until only Y remains, at which time, the inequalities are solved, and the minimum and maximum values of Y are determined. Next, the minimum Y value (corresponding to the minimum-cost solution) is substituted into the set of inequalities containing only Y and one xi term to determine the optimal mixture. Solving the inequality set for the xi term results in a range of values for the xi term. However, because the substituted value of Y will occur at a vertex point, there will be only one corresponding xi term, thereby establishing a one-to-one correspondence between cost and fractional contribution. The only exception occurs in the rare occasion when the cost function coincides with one of the peripheral lines for the feasibility region, resulting in a degenerate solution. This process can then be repeated with the inequality set containing Y and two xi terms and so on, until all xi


Figure 2. Constraint-reduction algorithm to minimize cost. Table 1. Property Values for Each Source property

fresh

sulfur (wt %) 0.1 density (kg/m3) 610 RVP (atm) 2.1 viscosity (cP) 0.178

240 K condensate 250 K condensate 0.7 580 5.2 0.256

1.5 650 2.5 0.22

values are known. The algorithmic flowchart for this procedure can be seen in Figure 2. At first glance, the proposed approach appears to apply only to well-mixed systems; however, the pathindependent nature of the mixing rules allows the proposed technique to be applied equally well to distributed systems. Additionally, the proposed approach can also be applied to problems with multiple objective functions. The explicit nature of the developed procedure allows standard multiobjective optimization techniques to be applied, e.g., noninferior curves. Case Study To implement this new algebraic procedure of property integration, a revised case study from the literature was analyzed. Shelley and El-Halwagi1 proposed a VOC recovery system for a degreasing plant that utilizes a fresh solvent in two units (absorber and degreaser) with their property and flow rate constraints. The main task in the case study was to recover a process resource and recycle it back into the system to lower the consumption of fresh resource at minimum cost. For this case study, rather than letting the temperature range vary, two condensate temperatures were chosen (240 and 250 K), and each condensate temperature was made into a source. Additionally, the absorber was neglected because it is not feasible to recycle condensate at any temperature level. In the original case study, three properties were considered: sulfur content, density, and Reid vapor pressure (RVP). We revise the case study by adding a fourth property, viscosity. The graphical approach of

Shelley and El-Halwagi1 is limited to a maximum of three properties. Therefore, there is an incentive to use the developed algebraic procedure, which is applicable to higher-dimension problems. The source properties are listed in Table 1. Furthermore, the degreaser has a range of acceptable properties and flow rates given by

0.0 e Sdegreaser (wt %) e 1.0 555 e Fdegreaser (kg/m3) e 615 2.1 e RVPdegreaser (atm) e 4 0.171 e µdegreaser (cP) e 0.202 36.6 e Fdegreaser (kg/min) e 36.8

(21)

In addition to the property values and constraints of the sources and sink, the mixing rules for those properties are needed. The mixing rules for the four targeted properties are given by

1 Fh

Ns

)

1

xi ∑ i)1 F

i

Ns

RVP1.44 )

xiRVP1.44 ∑ i i)1 Ns

S h)

xiSi ∑ i)1 Ns

log(µ j) )

xi log(µi) ∑ i)1

(22)

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3797 Table 2. Calculated Ω Values for Each Source Ωi Ωj Ωk Ωl

fresh

240 K condensate

250 K condensate

0.200 1.639 2.911 0.909

1.400 1.724 10.741 0.718

3.000 1.538 3.741 0.797

Table 3. Calculated Ωmin and Ωmax Values for the Degreaser

Ωi Ωj Ωk Ωl

Ωmin

Ωmax

0.000 1.626 2.910 0.844

2.000 1.802 7.360 0.931

combinations, it turns out that the 11 inequalities above can be transformed into 21 new inequalities with x1 as the only variable (inequality 31 cannot be combined with any other inequality, but it is carried through to bound x1). For example, inequalities 25 and 29 can be combined to form the following inequality

0.367x1 g 0.199

(34)

This results in the following x1 constraint

x1 g 0.54

There are two objectives for the case study: (1) Identify the optimal allocation of process and external sources that will minimize the flow rate of the fresh resource while satisfying all property constraints. (2) Identify the optimal allocation of process and external sources that will minimize the total cost while satisfying all property constraints Solution. With the information provided in Table 1, as well as eqs 21 and 22 and the arbitrary reference values, the property values and constraints for sulfur content, density, RVP, and viscosity can be converted into the Ωi, Ωj, Ωk, and Ωl domains, respectively, using ref ref eq 6. The arbitrary reference values for ψref i , ψj , ψk , ref and ψl were taken to be 0.5, 0.001, 1, and 0.15, respectively. The reference values were chosen to keep the relative magnitudes of the Ω values comparable. The Ω values that resulted for the three sources can be seen in Table 2. For the remainder of this study, the fresh solvent, the 240 K condensate, and the 250 K condensate will be indicated by subscripts 1, 2, and 3, respectively. Moreover, the Ωmin and Ωmax parameters described in eq 9 can be seen in Table 3 for the degreaser. For comparison with the graphical approach, the AUP for the fresh resource in Table 2 would be 5.659, and the resulting clusters for the fresh resource would be as follows: Ci ) 0.035, Cj ) 0.29, Ck ) 0.514, and Cl ) 0.161. Now, utilizing inequality sets 14a, 14b, 15a, and 15b, coupled with the data provided in Tables 2 and 3, the following inequalities can be obtained for this case study

(35)

By performing the constraint-reduction algorithm on pairs of inequalities, the following range for x1 can be determined

0.54 e x1 e 1.0

(36)

Therefore, the minimum fractional contribution of the fresh resource is 0.54. Using the total flow rate to the sink (36.6 kg/min), this fractional contribution corresponds to a minimum flow rate of the fresh resource of 21.2 kg/min. This is the optimum solution to part 1 of the case study. Similarly, by performing the algebraic algorithm, the range for x2 can be determined to be

0 e x2 e 0.344

(37)

Additionally, inequality sets 14 and 15 can be expressed in terms of either x1 and x3 or x2 and x3 and then combined and solved for the range of x3, which is as follows

0 e x3 e 0.279

(38)

-2.8x1 - 1.6x2 g -3.0

(23)

Now that the solution to part 1 of the case study has been identified, the second part can be solved. To determine the optimal recycle strategy, the cost function must be incorporated into the analysis using the algorithm summarized by Figure 2. For this case study, the pricing of each source is known and is as follows: fresh resource, $0.08/kg; 240 K condensate, $0.038/kg; and 250 K condensate, $0.022/ kg. This results in the following cost function

-2.8x1 - 1.6x2 e -1

(24)

Y ) 0.08x1 + 0.038x2 + 0.022x3

0.101x1 + 0.186x2 g 0.088

(25)

0.101x1 + 0.186x2 e 0.264

(26)

By transforming eq 39 into the form presented in eq 17, the following expression can be obtained

-0.831x1 + 6.999x2 g -0.831

(27)

Y ) 0.058x1 + 0.016x2 + 0.022

-0.831x1 + 6.999x2 e 3.619

(28)

0.113x1 - 0.079x2 g 0.047

(29)

0.113x1 - 0.079x2 e 0.134

(30)

x1 g 0

(31)

x2 g 0

(32)

x1 + x2 e 1

(33)

By inspection, it is possible to combine certain pairs of inequalities to remove x2. By finding all possible

(39)

(40)

Then, by solving for x2 and substituting the expression into inequality sets 14 and 15, inequality sets of the form of 19 and 20 can be obtained, as shown below

3x1 - 100Y g -5.2

(41)

3x1 - 100Y e -3.2

(42)

-0.573x1 + 11.625Y g 0.344

(43)


-0.573x1 + 11.625Y e 0.52

(44)

-26.02x1 + 437.438Y g 8.793

(45)

-26.02x1 + 437.438Y e 13.243

(46)

0.399x1 - 4.938Y g -0.062

(47)

0.399x1 - 4.938Y e 0.025

(48)

x1 g 0

(49)

-3.625x1 + 62.5Y g 1.375

(50)

-2.625x1 + 62.5Y e 2.375

(51)

Acknowledgment

As in the technique for finding the ranges of source fractional contribution, the above inequalities can be systematically combined to eliminate x1. Then, with only Y remaining in the new inequalities, the range of cost can be obtained. For instance, when inequalities 43 and 47 are combined, the resulting inequality is

4.534Y g 0.255

(52)

Y g 0.056

(53)

Hence

By performing the proposed procedure, the range of Y can be determined. The narrowest range of Y found for the aforementioned set is

0.056 e Y e 0.080

property domain to the property-operator domain. The mathematical structure of the problem was exploited to develop an efficient constraint-reduction procedure. This procedure can be used to identify minimum fractional contributions as well as minimum cost. Once the solution is identified, it is mapped again to the rawproperty domain. The applicability of this methodology was verified through a case study that could not be solved using the graphical approach.

(54)

Therefore, to minimize the cost, the lowest value of Y is selected. Then, the value of Y ) $0.056/min is substituted into the original set of inequalities 41-51 and a range for x1 is determined, which is confined to the single point, 0.54. To determine the value of x2, the values of Y and x1 can be substituted into the cost function (eq 40), which yields x2 ) 0.180. Then, x3 is obtained from eq 11 and found to be 0.279. This is the minimum-cost solution. It is interesting to note that the minimum-cost solution coincides with the minimum-fresh-fractionalcontribution solution. Nonetheless, this is not always the case. For instance, if the price of the 250 K condensate were to change to $0.06/kg, then the optimal solution would not correspond with the minimum usage of the fresh resource. For this case, the optimal Y value is 0.066, which corresponds to optimal values for x1, x2, and x3 of 0.656, 0.344, and 0.000, respectively. Conclusions An algebraic algorithm has been developed to address the problem of property-based integration. This new algorithm broadly expands the range of problems that can be addressed by the visualization technique to include any number of properties and any number of sources. The problem was mapped from the raw-

The financial support of the Gulf Coast Hazardous Substances Research Center and the U.S. Environmental Protection Agency (Grant CR-831276-01-0) is gratefully acknowledged. This work is dedicated to Professor Art Westerberg, whose exceptional work is a constant source of inspiration for all of us. Nomenclature AUP ) augmented property index as defined by eq 7 ci ) cost per unit mass of the ith source, $/kg Cp,i ) cluster of property p in source i as defined by eq 8 Fi ) flow rate of source i, kg/min i ) index for sources j ) index for sinks Nc ) number clusters Nj ) number of sinks Np ) number of properties Ns ) number of sources p ) index for properties xi ) fractional contribution of the ith stream to the total flow rate of the mixture Y ) cost objective function Greek Letters βs ) fractional contribution of stream s to the cluster domain ψp ) operator used in the mixing formula for the pth property as defined by eq 3 Ωp,i ) normalized, dimensionless operator for the pth property of the ith source as defined by eq 6

Literature Cited (1) Shelley, M. D.; El-Halwagi, M. M. Componentless Design of Recovery and Allocation Systems: A Functionality-Based Clustering Approach. Comput. Chem. Eng. 2000, 24, 2081. (2) El-Halwagi, M. M.; Glasgow, I. Em.; Eden, M. R.; Qin, X. Property Integration: Componentless Design Techniques and Visualization Tools. AIChE J., in press. (3) Eden, M. R.; Jørgensen, S. B.; Gani, R.; El-Halwagi, M. M. A Novel Framework for Simultaneous Process and Product Design. Chem. Eng. Process. 2004, 443 (5), 595-608. (4) Eden, M. R; Jørgensen, S. B.; Gani, R.; El-Halwagi, M. M. Property IntegrationsA New Approach for Simultaneous Solution of Process and Molecular Design Problems. Comput.-Aided Chem. Eng. 2002, 10, 79. (5) Gani, R.; Pistikopoulos, E. Property Modeling and Simulation for Product and Process Design. Fluid Phase Equilib. 2002, 43-53, 194.

Received for review October 14, 2003 Revised manuscript received December 8, 2003 Accepted December 12, 2003 IE034183K

Algebraic Techniques for Property Integration via Componentless

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