Blanket Wash Solvent Blend Design Using Interval ... - ACS Publications

The search for new solvents is driven by the needs of new applications, new processing requirements, changing environmental regulations, and market ...
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Ind. Eng. Chem. Res. 2003, 42, 516-527

Blanket Wash Solvent Blend Design Using Interval Analysis Manish Sinha† and Luke E. K. Achenie* Department of Chemical Engineering, Unit 3222, University of Connecticut, Storrs, Connecticut 06269

Rafiqul Gani Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark

The search for new solvents is driven by the needs of new applications, new processing requirements, changing environmental regulations, and market demands. Many cleaning solvents used in the lithographic printing industry are on the environmental “hit list” and are to be phased out within the next few years. This paper discusses the systematic design of cleaning solvent blends for lithographic printing (commonly referred to as blanket washes). The design problem consists of a discrete problem involving selection of solvents from a set of pure-component solvents and a continuous problem of finding the blend composition. The simultaneous consideration of associated process constraints, property requirements, and environmental restrictions makes blanket wash design a rather difficult problem. To address this issue, we present a framework for designing cleaning solvent blends that meet thermophysical property requirements and environmental restrictions. The solvent design model is solved using interval analysis. 1. Introduction Solvents are extensively used in industry to clean equipment parts by separating grease and grime, to suspend solids as in inks and paints, to separate solid or liquid components from mixtures prior to purification (liquid-liquid extraction and gas absorption), and for many other purposes. Once media of choice for the processing industry, many organic solvents are being phased out of products and processes for environmental and health reasons.1 One of the most used solvents in a printing press is the “blanket wash”, which is specially formulated to clean ink from lithographic printing presses. There are more than 52 000 lithographic printers in the United States,2 and each one uses an average of 160 gallons per year (for a total of approximately 8 million gallons per year). These solvents are eventually released into the atmosphere, thus posing considerable environmental problems. There is a tremendous need to replace and/or recover these solvents. The search for new solvents is usually based on design heuristics, prior experience, and direct experimental studies. This approach is inherently trial-and-error and therefore costly and time-consuming, and it might not necessarily yield a solvent with the desired performance attributes. For example, the solvent mixture might have a drying time that is too slow for the intended use. Although there is no substitute for experimental study, there is a definite need for a pre-experimental stage that will quickly and cheaply generate new solvents that are promising enough to be considered for the costly and time-consuming experimental stage. In a fairly recent article,3 Zhao et al. discussed different property requirements that should be satisfied in the design of solvents for different applications. In * To whom all correspondence should be addressed. † Current address: Global Alternative Propulsion Center, General Motors, Honeoye Falls, NY 14472 (E-mail: [email protected]).

our discussions with a newspaper company,4 a major performance issue in the selection of a blanket wash solvent is minimization of the effect of a solvent on the surface characteristics of the rubber blanket on which the printing paper is processed. Many solvents swell the rubber blanket. Environmental restrictions and the need to reduce operating costs mandate recycling of the spent solvent in the short term. Previous and existing industrial approaches to solvent selection and substitution have relied on database search and query approaches. For example, SAGE5 and SOLVDB (National Center of Manufacturing Science) have large databases of existing solvents and associated processes. Through query and answer sessions, the user is led to a suggested selection of solvents. Eastman Chemicals and Dow Chemicals have developed a large database of solvents. The database contains solvent properties such as flash point, thermodynamic functions, vapor pressure, and hazard evaluation. Eastman’s solvent alternative strategy aims at matching the solubility constants and evaporation rates.1 Solvents used in industry are often blended together to meet user requirements. These requirements can include limits on boiling point, viscosity and other transport properties, solute-solvent interactions, or solvent power as characterized by solubility parameters. Other implicit requirements on miscibility have to be satisfied to ensure single-phase mixtures. Computeraided product design (CAPD) has emerged as a powerful strategy for identifying promising compounds with prespecified levels of certain thermophysical properties. More formally, CAPD is a reverse-engineering procedure that incorporates desired levels of physicochemical properties directly into the design of products. This approach has been applied to polymer design,6,7 reinforced polymer composite design,8-10 liquid-liquid extractants,11,12 and refrigerants.13-15 CAPD has been applied to solvent design problems as well. These include the design of solvents for liquid-liquid extraction,16,17 gas absorption processes,18 and separation

10.1021/ie020224l CCC: $25.00 © 2003 American Chemical Society Published on Web 01/07/2003

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processes.19-21 Computer-aided mixture design (commonly referred to as the formulation problem) has been applied to solvent blends for paint formulation22 and refrigerant mixture design,23 polymer blend design,9 and coating applications,24 as well as in the paint and ink industry. Generalized approaches for designing mixtures appear in the literature.22,23 Also, many mixing rules for the prediction of mixture properties have been developed and reviewed.25 Even though many water-soluble solvents exist that can be used to make blanket wash formulations, deciding on the composition of the final wash formulation is a trial-and-error procedure. Moreover, mixture property prediction models for aqueous systems are difficult, and the models are highly nonlinear.26,27 Thus, there is tremendous incentive to develop a formulation tool that can design aqueous blanket wash blends in the presence of nonlinear (and probably nonconvex) models. In this study, we employ our interval-arithmetic-based global optimization package LIBRA for the systematic design of optimal waterbased blanket wash systems. This paper is organized as follows. Section 2 describes the use of lithographic blanket washes. The mixture design model is developed in section 3, followed by a description of interval analysis in section 4. Interval analysis forms the basis for the solution algorithm in section 5. A case study is presented in section 6, followed by the study’s conclusions in section 7. Finally, the Appendix provides details of the physical property models employed.

2. Lithographic Blanket Washes Lithographic printing is the most common printing process and is based on the immiscibility of water in oil. Printing ink, which is insoluble in water, comprises resins and pigments suspended in a petroleum-based solvent. The ink is applied on a printing plate, which is pressed onto printing paper to impart the print image. When the plate is dipped in aqueous fountain solution, the ink and fountain solutions repel each other, and the ink is confined to the image area of both the plate and the printed material. Roller blankets are used to carry the print paper. At the end of the print cycle, ink residues on the blanket have to be removed using a solvent-based blanket wash solution. Solvents are extensively used as a major component of ink in the printing industry. The function of a solvent in ink is to act as a vehicle for polymeric resins, pigments, and dyes. The ink solvent also assists in wetting and dispersion of dyes and pigments. In letter press and offset lithographic printing processes, the ink is carried to the plate by means of a train of rubber rollers commonly called blankets, as shown in Figure 1. Thus, a thin film of ink is distributed over a large surface area on the blankets. These ink solvents are volatile and evaporate to leave behind the pigments and resins on the blanket surface. Cleaning is required between print jobs and whenever the residue buildup affects the print quality. Paper fibers, ink residue, paper coating, and dried ink are types of material that must be removed from the rubber blankets. One of the most used solvents for lithographic printing is the “blanket wash”, which is specially formulated to clean ink and other residue from rubber blankets.

Figure 1. Schematic of lithographic printing.

Manual cleaning operations (also termed “rag and bucket”) involve wiping down the blanket cylinder with a cloth wipe dampened with blanket wash solution. The large volume of soiled rags from these operations are routinely sent to industrial launderers who are then faced with the proper disposal of the wastewater resulting from laundering the rags. In addition, the industrial launders are burdened by the inefficiencies of solvent use in the printing industry, as they also have to abide by rigid standards on wastewater pollution levels. Blanket wash solvents are mostly mixtures, as opposed to single-component solvents. As such, next to solvent performance, one of the most pressing concerns of the printing industry with regard to the environment is the volatile organic component (VOC) level of solvents. At present, the VOC levels of solvents used in the printing industry are unusually high, well over 80% and far beyond the industry target of 30%. For example, a commonly used blanket wash, VM&P naphtha, has a 100% VOC content.28 To enhance the cleaning operation, companies sometimes mix solvents from different vendors. However, as noted earlier, this trial-and-error approach is costly and might not necessarily yield a solvent mixture with the desired performance attributes. In addition, the solvent for a cleaning operation might not meet safety, health, and environmental restrictions. Another important issue is minimizing the effect of a solvent, through induced swelling, on the surface characteristics of the rubber blanket. Swelling severely affects the print quality in lithographic processes. 3. Mixture Design Problem Formulation Computer-aided mixture design is composed of three main steps: (i) selection of pure components from a database (for example, the design of a binary mixture from a set of 10 pure components can result in (10 2 ) or 45 combinations), (ii) determination of mixture compositions that satisfy the property targets, and (iii) ranking of the candidate mixtures by some criterion such as overall cost (if a cost model has not already been used in the framework). The first step is a combinatorial problem; the second step is a continuous problem, which could be nonconvex depending on the nature of the property prediction

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techniques employed. We propose to use a mixed-integer nonlinear problem formulation that is general enough to handle several types of mixture estimation techniques. Obviously, if the number of combinations (binary, ternary, etc.) for the pure components is small, one can enumerate them all and solve a series of continuous nonlinear programs. In the proposed formulation, binary variables are used to denote the presence or absence of a purecomponent solvent in the mixture, and a set of continuous variables is used to describe the mole fractions of the components in the mixture. Hence, the formulation is mixed-integer in nature. First, let us introduce the variables

Figure 2. Continuous function and its interval extension.

yi (binary variable) ) 1 if pure component i is present in the mixture 0 otherwise

replace it by the constraint yi < xi < yi(1 - ), where  is a small number (e.g., 0.01).

xi (continuous variable between 0 and 1) ) mole fraction of pure component i in the mixture

4. Interval Analysis Technique for Solving Mixture Problem

{

Other parameters include n, the number of purecomponent solvents (basis set); nmax, the maximum number of pure-component solvents in the blend; and Pij, property j of pure component i. Constraints are imposed to (a) limit the number of pure-component solvents in the blend, (b) ensure that the mole fraction of an absent component is 0, and (c) ensure that the sum of all mole fractions is 1.0. These constraints are by no means exhaustive, and several different ones can be added to achieve a specific solvent mixture design objective. Consider a performance objective function f(x,y) (that might include economics); then, the solvent mixture problem is posed as

Pmix: min f(x,y) x,y

subject to PL e P(x,y) e PU g(x,y) e 0

∑i yi e nmax

(1)

∑i xi ) 1 0 e xi e yi x ) [x1, x2, ..., xn]T, xi ∈ [0, 1] real y ) [y1, y2, ..., ym]T, yi ∈ {0, 1} binary PL and PU are the lower and upper limits, respectively, on a vector of target properties P. These properties can be nonlinear and nonconvex with respect to the search variables xi and yi. The constraints g(x,y) e0 can include phase equilibrium. The last constraint in the above formulation ensures that, if component i is not present in the mixture (i.e., i ) 0), then the corresponding composition xi is also 0. This constraint, however, can lead to cases where the composition of one component is infinitesimally small. To avoid this problem, we

Because many property estimation techniques are generally nonconvex, we have developed an intervalanalysis-based optimization strategy that can design (globally) optimal mixtures. Interval analysis has emerged as a reliable mathematical tool that can automatically generate lower and upper bounds for a function.29 It has been used for solving ordinary differential equations and linear systems, as well as verifying chaos. Interval arithmetic, which is at the heart of interval analysis, was developed by Moore.30 In essence, interval-analysis-based optimization continually deletes portions of the search space with the goal of maintaining a final box of specified width that contains the global solution. A number of interval-based optimization procedures have been developed.8,29,31-33 Most of these procedures are tailored to unconstrained optimization problems. In addition, these techniques have so far handled only continuous variables and not discrete variables. Notwithstanding the attractive features of interval-based global optimization, such approaches are, in general, computationally intensive. To address some of these issues, we have developed new acceleration strategies and extended the capabilities of the algorithm to solve mixed-integer problems. 4.1. Brief Introduction to Interval Analysis. An interval Xi ) [ai, bi] containing a real variable xi is characterized by two real scalars ai and bi such that ai e xi e bi. An interval vector X ) (X1, X2, ..., Xi, ..., Xn)T represents a hyperrectangle (commonly referred to as a hyperbox or simply a box) in an n-dimensional space Rn. The width of a box, denoted by w(X), can be defined as the largest width of the interval X. There are other definitions for the width, for example, the perimeter of the box. An interval extension F(X) of a continuous function f(x) is obtained by replacing all variables with their interval counterparts. The resulting function is an interval function, denoted as F(X) ) [F L, F U], where F L and F U, in general, are loose lower and upper bounds, respectively, on f (x) over the box or domain X (see Figure 2). However, for any single-use expression (one in which each variable occurs only once), the interval extension F(X) (using outward rounding) provides exact bounds (within machine precision) on f(x). In general, the smaller the width of X, the tighter the

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bounds. The true range of f (x) over X is denoted by f (X) [or R(X)] such that f L and f U correspond to the global minimum and maximum, respectively, over X. Moore34 proved that

lim F(X) ) f(X)

w(X)f0

Considerable effort has been expended by interval analysts on producing systematic methods for representing an interval function that gives the sharpest bounds on the range of a real function over an interval.35-37 It can be shown that, for monotonic functions, F(X) provides sharp bounds on f(X). In fact, as one of the reviewers pointed out, for interval extensions in which the bounds are computed in some other way (not necessarily through interval arithmetic), then the upper and lower bounds can be obtained exactly (within machine precision) using function values at the upper and lower bounds. An important property of interval analysis that makes it useful for global optimization is the inclusion of functions. Consider a real-valued function f(x). The interval function F(X) is said to be an inclusion isotone of f(x) if x ∈ X implies f(x) ∈ F(X) and also if Y ⊆ X implies F(Y) ⊆ F(X). Operations such as addition, subtraction, multiplication, and division have been developed that are inclusion isotonic. See refs 29, 30, and 35 for more details on the operations and interval analysis. We summarize some of the notation above as follows: xi ) real scalar number (e.g., 5.0) x ) vector of real scalar numbers xi [e.g., (1.0, 1.4, 2.5)T] Xi ) interval scalar number (e.g., [2.0, 2.2]) X ) vector of interval scalar numbers Xi (e.g., [(1.0, 1.2), (1.4, 1.45), (2.5, 2.6)]T ) f(X) [or R(X)] ) [f L(X), f U(X)] ) range of a function f(x) on X (this is often unknown) F(X) ) [F L(X), F U(X)] ) natural interval extension of a function f(x) on X such that F L(X) e f L(X) e f U(X) e FU(X). 4.2. Current Global Optimization Methods Based on Interval Analysis. Almost all interval-analysisbased global optimization algorithms8,31,38 employ a successive domain reduction approach that eliminates portions of search regions that do not contain the global solution. Consider the continuous optimization model

(globally) min f(x) x

subject to g(x) e 0 h(x) ) 0 x ) [x1, x2, ..., xn]T real x ∈ X0

(2)

Almost all domain reduction algorithms invariably use the following tests to systematically remove portions of X0 that cannot contain the global minimum. (i) Upper Bound Test. If the objective value UPBD ) f(x) corresponding to a point x in the feasible region (i.e., search space that satisfies all constraints) is known,

then any subregion of X0 (namely, X) satisfying F L(X) > UPBD does not contain the global solution and can be deleted from the search space. (ii) Infeasibility Test. For a subregion X, if G L(X) > 0, then X does not contain any feasible region and can be deleted from the search space (domain). (iii) Monotonicity Test. For an unconstrained problem, at the optimal point, the gradient f ′(x) ) 0. Let F ′(X) be the interval extension of f ′(x). Now, if 0 ∉ F ′(X), then the subregion cannot contain an optimal point, so only the edge of the subregion is retained, and the rest deleted. Note that this test is not appropriate for constrained systems; it can only be applied to a subregion X for which G U(X) e 0 and H(X) ) [H L(X), H U(X)] ) [0, 0]. (iv) Nonconvexity Test. This test is based on the principle that, at the optimal point, the curvature of the surface defined by the objective function is positive, in other words, the Hessian of f(x) is positive semidefinite. Let H ˆ (X) be the interval Hessian of the function. To apply this test, one checks whether H ˆ (X)is positive semidefinite, usually true when the diagonal elements H ˆU ii (X) < 0 for all values of i ) 1, 2, ..., n. If not, then the subregion is deleted. Again, this test is not appropriate for constrained systems; it can only be applied to a subregion X for which GU(X) e 0 and H(X) ) [H L(X), H U(X)] ) [0, 0]. (v) Distrust Region Test. In the distrust region method8 (for constrained problems), the idea is that, once an infeasible point, x, is found, a box X is constructed around it such that GL(X) > 0. Then, the boxed region X is deleted from the search space. Vaidyanathan et al. did not directly deal with equality constraints. 4.3. Modifications Employed in this Study. An interval-based global optimization algorithm can be constructed on the basis of the above tests. However, in our experience, such an algorithm is computationally slow, especially for problems with a large number of constraints. Additional domain reduction tests are proposed next. (i) Upper Bound via SQP local optimization. For the initial search space defined by X0, a good upper bound on the global solution, f UPBD, is found using the locally optimal sequential or successive quadratic programming (SQP) approach.39 SQP has proven to be a very powerful algorithm for gradient-based local optimization. It often requires fewer function evaluations than other competing gradient-based algorithms.40 Subsequently, at any iteration k, the upper bound (UPBDk) for a subregion Xk is found via SQP. If this upper bound is lower than the overall upper bound UPBD, then UPBDk replaces UPBD. In our experience, in many cases, SQP finds the global solution in the first few iterations (of the global optimization algorithm), and the remaining iterations are merely used to verify global optimality. (ii) Local Feasibility Test. Here, the idea is to relax the optimization model and consider only the convex constraints and determine whether this relaxed search space contains a feasible solution. This requires the prior specification of which constraints are convex and which are not. This specification is not always straightforward. However, linear equality and linear inequality constraints are simple convex constraints. Using this reduced set of constraints, the feasibility of a subregion Xk is checked by solving the following feasi-

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Figure 3. Branching strategy on continuous and binary variables.

bility problem

and the second box is represented by

{

[xi, xi] ∀ i * k

min µ µ,x

gconvex(x) e µ hlinear(x) ) 0 x ∈ Xk

(3)

The above problem is solved via a local optimization algorithm (SQP). Note that, for this problem, the local and global solutions are identical. If the problem is infeasible (i.e., µ > 0), then Xk cannot have any feasible point and can be deleted from the search space. (iii) Extension to MINLP Problems. Mixture design problems have relatively small dimensions. For a design with a basis set of m pure components, the interval dimension is 2m. As indicated earlier, current interval-based global optimization algorithms only solve continuous optimization problems. An extension of the algorithm is required to solve mixed-integer nonlinear programming problems (MINLP) such as the mixture design problem discussed earlier. Let a hyperbox be represented by [xi, xi], ∀ i ) 1, 2, ..., n. This box can be partitioned into two subboxes at a branch point xk*. The first box is represented by

{

[xi, xi] ∀ i * k [xk, x/k] otherwise

[x/k, xi] otherwise

We have employed a modified partitioning strategy for binary variables. If a variable k is a binary variable then the partitioning results in two points 0 and 1 on the kth dimension. For the above case, a binary partition (see Figure 3) will result in two boxes, namely,

first box )

{

and

second box )

[xi, xi] ∀ i * k [0, 0] otherwise

{

[xi, xi] ∀ i * k [1, 1] otherwise

The branching strategy at each iteration plays an important role in the efficiency of this algorithm. The accuracy of the bounds of functions and constraints using interval analysis also depends on the width of X. To address both of these issues, branching is performed along the dimension that corresponds to the maximum width. It is known that this is not necessarily an optimal branching strategy; however, we have had satisfactory experience with it. 5. LIBRA: An Interval-Based Domain Reduction Algorithm Here, the implementation of a global optimization algorithm that utilizes the domain reduction strategy

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is presented. In our implementation, we employ the interval arithmetic C++ class definitions, data structures, and basic linear algebra subroutines BLAs from Iwaarden.33 The BLAs routines use the IEEE standard for directed roundings; this means that the global solution is not missed. The reader should consult Iwaarden’s thesis for further details.33 Alternatively, the reader can request a copy of the original web download of the routines from the corresponding author. In the local optimization solver (a C++ implementation of Biegler and Cuthrell’s SQP algorithm41), the derivative and Hessian information for all functions and constraints are computed via the automatic derivative package ADOL-C.42 For these computations, the function and constraints have to be supplied in a defined format. Automatic derivative is also used for the interval functions and constraints. The algorithm is implemented in the following steps. Step 1. Prepare the input data. The problem is specified including the dimension size, numbers of inequality and equality constraints, and variable indices corresponding to binary variables (if MINLP). Bounds on the variable (in other words the original box entered as an interval vector X0) are specified and define the original search space. Two lists are initialized, box_list (containing the search space) and good_list (containing the candidates for global solution). A tolerance () is specified that defines the maximum width of the interval corresponding to the final solution box that is acceptable. Finally, the original region X0 is inserted as the only box in box_list. Step 2. Check the feasibility of the original region X0. If feasible, continue to step 3. Otherwise, terminate. Step 3. Initialize the lower bound (LWBD) to -∞ and the upper bound (UPBD) to the objective value corresponding to the feasible local minimization solution (from SQP). Note that feusibility is again checked in the interval analysis sense after a call to SQP. If a local solution is not found, then set UPBD to +∞. Step 4. Check the content of box_list. If empty, go to step 8; otherwise, go to step 5. Step 5. Remove the box Xk from the top of box_list. Find an upper bound UPBDk ) f(x/k) to this box via SQP. If UPBDk < UPBD, then UPBD ) UPBDk. If F L(Xk) > UPBD, then delete Xk and go to step 4. Next, determine whether montonicity and nonconvexity tests are applicable. Finally, apply the local feasibility tests. If any of these tests deletes Xk then go the step 4. Step 6. Check the maximum width of Xk. If less than the specified tolerance, add the box to good_list and go to step 4. Otherwise, go to step 7. Step 7. Apply the back-boxing technique (the process of identifying a box that surrounds a given point such that the objective function is convex on that box33) to box the maximum convex region around the upper bound solution point x/k. Add the solution to good_list and partition the remaining search space of Xk. If backboxing is not applicable, then partition the box about its maximum width as described earlier. Add the partitioned boxes to box_list. Here, the boxes are inserted in an ordered list with the box with the largest maximum width on top. Note that back-boxing is defined only for an unconstrained problem. For a constrained optimization problem, we proceed in one of two ways: (a) apply back-boxing to the objective function plus constraints, which have been weighted with a penalty parameter, or (b) apply back-boxing to the

objective function and check whether the constraints are satisfied in the resulting box. Go to step 4. Step 8. Rank order the boxes in good_list in terms of their objective function values. Delete boxes for which F L is greater than UPBD. Terminate and return the list good_list containing the globally optimal solution or set of solutions. The attractive features of this algorithm are summarized as follows: (1) Analytical expressions for gradients and Hessians for objective functions and constraints are not required. They are not computed by finite differences; rather their analytical forms are automatically constructed via the use of the automatic derivative package. (2) If a problem has more than one global solution, then the algorithm finds all of the globally optimal solution points. (3) This algorithm can be used for design under uncertain parameters. In effect, if a parameter is not exactly known, rather its nominal point and corresponding confidence interval or some error band is known, then it can be directly used as an interval parameter. Because parameter correlations are not taken into account in the use of the hyperbox defined by the parameter intervals, parametric uncertainty addressed in this way will result in rather conservative solutions. This algorithm has been tested for many benchmark NLP and MINLP problems. We have shown that, for MINLP problems, the splitting strategy does not result in a total enumeration of all possible solutions.

6. Case Study: Design of Environmentally Acceptable Blanket Wash Blends Printers have traditionally had little alternative to using products that contain volatile organic compounds (VOCs) and toxic materials in their printing operations. Recently, the U.S. EPA (U.S. Environmental Protection Agency) performed a study on lithographic blanket wash solvents and published a detailed report on various solvents used as components in blanket wash formulations and their risk levels.43 This report found that many commercial blanket wash solvents are formulated from inexpensive petroleum distillates that are byproducts of petroleum refining. Thus, these solvents are not systematically designed and pose considerable environmental health and safety risks when used in lithographic printing. 6.1. Motivation for Aqueous Blends. Nearly all conventional blanket washes contain VOCs.2 In Connecticut alone, about 13.5 tons/year of spent solvent is disposed of by the printing industry.44 Of this, only 0.69 ton is aqueous solutions, 10.57 tons is nonhalogenated solvents, and 2.26 tons is halogenated solvents. Weltman45 discussed the benefits of replacing halogenated degreasers by aqueous substitutes. The Printing Industry of America (PIA) and the U.S. EPA started a major initiative in the early 1990s to search for alternative water-based blanket wash solvents.28 Fairly recently, the Toxic Research Institute46 developed a water-based solvent called Printwise that can be used as a blanket wash. The AG Environmental Products company recently commercialized a waterbased blanket wash product called Soygold. This solvent is a methyl ester of soybean oil and is completely miscible in water. All of these water-based blanket

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Table 1. Two Aromatic Hydrocarbons Used in Many Commercial Blanket Washes

Table 2. Solvents Used in the Case Study to Design Blendsa

a Sources: Ref 28 and SOLVDB, Solvent Database at http://solvdb.ncms.org/solvdb.htm, National Center of Manufacturing Sciences, Ann Arbor, MI.

washes have very low VOC levels and low environmental impacts, are less toxic, are easy to recover, and are amenable to safe disposal. Surfactants and dispersants give solvents better cleaning characteristics and have found increasing use in many blanket wash formulations.43 Many blanket wash formulations use alkyl benzene sulfonates (ABSs) as surfactants. These are water-soluble surfactants and cannot be used in hydrocarbon-based blanket washes. Thus, there is a strong incentive to design water-based blanket wash solvent blends. This case study explores the systematic development of aqueous blends for use as blanket wash solvents. 6.2. Case Study Objective. Even though many water-soluble solvents exist that can be used to make blanket wash formulations, determining the composition of the final wash formulation is a trial-and-error procedure. Moreover, mixture property prediction models for aqueous systems are difficult and highly nonlinear.26,27 Thus, there is a tremendous incentive to develop a formulation tool that can design aqueous blanket wash blends in the presence of nonlinear (and probably nonconvex) models. In this study, we employ our interval-arithmetic-based global optimization package LIBRA for the systematic design of optimal water-based blanket wash systems. 6.3. Basis Set. The EPA report on blanket wash risk assessment43 lists 40 different formulations (or solvent blends) used as blanket washes by different printing facilities throughout the United States. However, be-

cause of propriety concerns, the compositions of these formulations are not reported. Of these, 21 formulations contain petroleum distillates (hydrocarbons and/or aromatic hydrocarbons), which pose considerable environmental health and safety risks. Two common aromatic hydrocarbons used in blanket washes are 1,2,4-trimethyl benzene (C9H12) and isomers of xylene (C8H10). Trimethyl benzene has a flash point of 54.4 °C and a log KOW of 3.78. Isomers of xylene have flash point as low as 17 °C and a log KOW of 3.15. Thus, both are flammable and have high bioaccumulation and toxicity levels, as shown in Table 1. Because these solvents are not miscible in water, they are not used as pure components for the design of aqueous blanket wash solvent blends. The pure-component solvents employed in this case study are nonhalogenated and nonaromatic watersoluble compounds. Also, only those solvents that have relatively small environmental and health impacts are selected. These solvents are listed in Table 2. The desired attributes for optimal blanket wash formulation are defined in Table 3. These attributes target the solvent power, flow characteristics, surface contacting, and environmental impact. Note that, by constraining both the density and the viscosity, we have constrained the kinematic viscosity (µ/F) of the blends. The purecomponent properties of the basis set are presented in Table 4. Also, note that these properties are experimental properties.

Ind. Eng. Chem. Res., Vol. 42, No. 3, 2003 523 Table 3. Desired Attributes of an Optimal Blanket Wash Blend attribut

range

solvent powera density, F (as specific gravity) viscosity, µ (cP) surface tension, σ (dyn/cm2) vapor pressure, Psat (mmHg) inhalation exposure, IE (mg/day) permissible exposure limit, PEL (mg/m3)

Rij < R* 0.9-1.1 0.8-1.4 30.0-45.0 0-2 0-2 0-100

Vc(min ) )

Mmix )

Vc(ij) )

σmix1/4 ) ψwσw1/4 + ψoσo1/4

The mathematical formulation of the problem is

log10

Pblend: minimize R ) [4(δD -

+ (δP -

δ/P)2

+ (δH -

δ/H)2]1/2

[

(ψw)q

(1 - ψw)

δP )

∑i ΦiδDi

]

ψo ) 1 - ψw ) log10

[

∑ xiPsat e Psat_max

∑i ΦiδPi

[

]

(xwVw)q (xwVw + xoVo)1-q + xoVo 2/3

]

q σoVo - σwVw2/3 0.441 T q (surface tension constraint)

subject to δD )

(viscosity constraint)

σL eσmix eσU

Based on solubility interaction radius of blend and polymeric resin. The resin is a phenolic resin, Phenodur 373 U, (Barton, 1985). Its solubility parameters are δD ) 19.7, δP ) 11.6, and δH ) 14.6, and its interaction radius is R* ) 12.7 (all in MPa1/2).

ij

∑i xiMi

(Vci1/3 + Vcj1/3)3 8

a

δ/D)2

∑i ∑j xixjVc(ij)

(vapor pressure constraint)

118.8xIFi e PELI (OSHA constraint on permissible exposure limit)

δH )

∑i ΦiδHi

(0.48At)Gi e IEmax xw > 0.3

Φi )

FL e

xiVi

∑i xiVi

∑ xiMi e FU ∑xiVi

(mole fraction constraint)

More details of these models are described in the Appendix. An MINLP model was formulated and solved two ways. In case 1, the model (PBLEND) was solved by fixing the binary variables, resulting in an NLP model. Specifically, each binary mixture was constructed by fixing the binary variables for water and one of the other pure component solvents to 1; the remaining binary variables were set to 0. The surface tension model requires the calculation of the water fraction Ψw, which becomes an additional search variable. In case 2 (PBLEND_MINLP), the MINLP model was solved rigorously, i.e., without fixing the binary variables. This is a relatively more difficult problem, with 17 variables and 8 binary variables. Also, in this case, we only considered two-component blends (i.e., binary blends, not to be confused with binary variables). Thus, the solution approach not only picks the best combination of two-component solvents (combinatorial problem) but also finds the optimal composition (continuous problem).

(density constraint)

ηL e ηmix e ηU ln(ηmixmix) ) x1 ln(η11) + x2 ln(ηww)

)

(water fraction greater than 30%)

∑xi ) 1

(solvent power)

(inhalation exposure limit)

Vc2/3 (TcM)1/2

Table 4. Pure-Component Properties of Solvents in the Basis Set component

µ (cP)

σ dyn/cm2)

Psat (mmHg)

δD (MPa1/2)

δP (MPa1/2)

δH (MPa1/2)

Fa

methyl ethyl ketone GBL NMP propylene glycol DGME DGEE R-terpinol water

0.378 1.700 1.660 19.000 3.480 3.850 36.500 1.000

24.600 40.430 40.700 36.510 28.190 29.530 31.600 70.000

95.300 3.200 0.334 0.200 0.180 0.126 0.490 50.000

14.100 18.600 16.500 11.800 16.200 16.200 13.900 26.500

9.300 12.200 10.400 13.300 9.200 9.200 7.900 23.300

9.500 14.000 13.500 25.000 14.300 12.300 10.200 14.800

0.801 1.120 1.001 1.034 1.229 1.025 0.819 1.000

a

As specific gravity.

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Table 5. Computational Results of Blend Design Case Studya component

Rij

MEK-water GBL-water NMP-water propylene glycol-water DGME-water DGEE-water R-terpineol-water

10.34 4.40 6.54 12.94 10.02 8.58 10.16

any component-water

4.40

x1

xw

case 1 (PBLEND) problem formulation 0.14 0.86 0.45 0.55 0.46 0.54 0.06 0.94 0.07 0.93 0.10 0.90 0.08 0.92 case 2 (PBLEND_MINLP) problem formulation 0.45 0.55 (GBL) (water)

Ψw

iter.b

0.31 0.10 0.07 0.47 0.25 0.18 0.16

31 17 33 29 23 25 21

3.4 2.5 4.1 5.9 13.3 8.7 5.1

0.10

1013

242.0

CPU (s)

a Note from Table 3 that Rij for the ink residue is 12.7. Thus, a blend’s Rij value has to be less than 12.7 for it to effectively dissolve and clean the ink residue. b Number of iterations.

Table 6. Blend Properties component

Fa

Psat (mmHg)

IEb (mg/day)

PELc (mg/m3)

δD (MPa1/2)

δP (MPa1/2)

δH (MPa1/2)

σmix (dyn/cm2)

µmix (cp)

MEK-water butyrolactone-water NMP-water propylene glycol-water DGME-water DGEE-water R-terpineol-water

0.910 1.093 1.001 1.007 1.067 1.011 0.922

13.628 1.440 0.154 0.012 0.013 0.013 0.037

116.825 15.235 1.828 0.115 0.125 0.195 0.563

15.348 58.045 54.339 7.130 8.917 12.173 8.237

20.860 20.359 18.259 23.457 23.484 21.841 21.075

16.932 14.672 12.669 21.230 19.171 16.922 16.669

12.389 14.178 13.729 16.911 14.654 13.669 12.819

35.000 42.862 42.380 38.350 30.260 31.572 33.592

0.830 1.363 1.322 1.218 1.181 1.236 1.361

a

As specific gravity. b Inhalation exposure. c Permissible exposure limit.

6.4. Results and Discussion. The models PBLEND and PBLEND_MINLP were solved (on a Pentium II 450-MHz Xeon computer running the Windows NT operating system) to obtain seven different binary mixtures, as shown Table 5. From Table 5, consider, for example, the propylene glycol and water blend, for which the globally minimal objective function, Rij, is 12.94. Thus, any propylene glycol and water blend will not fall within the interaction radius (12.7) of phenolic resin. Therefore, it is expected that no propylene glycol and water blend will be effective in dissolving phenolic resin. Among all solutions, the lowest objective value is achieved by a γ-butyrolactone and water blend with an interaction radius of 4.4. The attributes of the solvent blends in Table 5 are listed in Table 6. Water (with pure σ ) 70) is the major component in all of the blends shown in Table 6. We note that the surface tension is highly nonlinear in that a small organic fraction in the aqueous blends results in a very large change in surface tension. For example, 6% of propylene glycol (with σ ) 36.51) reduced the aqueous blend’s surface tension from 70 to 38.35. This behavior is also true in practice, as verified by many experimental results.47 Case 2 (PBLEND_MINLP) was solved next. The solution found was identical to that obtained in the first case (aqueous blend with γ-butyrolactone at a mole fraction 0.450). However, for this problem, the number of iterations (1013) is much higher. Consequently, the CPU time is relatively large (242.3 s). Figure 4 clearly shows that water is ineffective as a blanket wash solvent because it is very polar (large δP), has a high hydrogen solubility parameter (δH), and is located far from ink residue in the solubility parameter space. However, the designed aqueous blends fall within the interaction radius of ink residue and hold promise as effective blanket wash solvents. 7. Conclusion Computer-aided blend design is a highly complicated problem. The general understanding of variations in

blend properties with variations in composition has improved substantially over the past few decades, and many companies have developed computer models to predict blend properties. These models are being used successfully by companies such as DuPont.27 However, using such models directly to discover optimal blends is nontrivial. Moreover, the identification of binary solvent blends from a set of single-component solvents involves a combinatorial search for an optimal pair. To address this task, we have developed an optimization framework for mathematical representation of the solvent blend design problem. The mathematical framework is an MINLP problem. To solve such design problems, we have also developed an interval-based global optimization tool called LIBRA. This framework and solution approach has been used to solve an industrially relevant problem of designing optimal blends for blanket wash applications in the printing industry taking into account solvent power, viscosity, and surface tension. Two binary mixture design problem case studies were solved, for which we were able to identify a globally optimal blend composition. The issue of economics is very important and quite difficult as pricing information fluctuates from time to time and from vendor to vendor. We recently published a paper on single-component blanket wash design that also considered solvent recovery costs.48 Here, the costs were associated only with the recovery process and not with regard to solvent unit volume cost. If a cost model is available, it can easily be incorporated into the suggested framework either as a performance objective or as a constraint. At the time we performed the simulations, we had no access to reliable unit costs for the single-component solvents. One should bear in mind that this paper is a proof of concept, and in that regard, it has served its purpose. As the article title says, here, we are considering solvent mixture or blend design. To try to mirror actual indutrial practice, the blend design consists of mixing existing single-component solvents. The more general

Ind. Eng. Chem. Res., Vol. 42, No. 3, 2003 525

Acknowledgment This material is based upon work partially supported by the National Science Foundation under Grant No. CTS - 9630917. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Nomenclature yi ) binary variable xi ) continuous variable or mole fraction xw ) bulk mole fraction of pure water xo ) bulk mole fraction of pure organic component n ) number of pure-component solvents nmax ) maximum number of pure-component solvents in the blend Pij ) property j of pure component i Xi ) interval X ) interval vector or box w(X) ) width of X F(X) ) interval extension of a continuous function f(x) F L, F U ) lower and upper bounds defining F(X) f(X) ) range of f(x) over X [xi, xi] ) ith dimension of a box in several dimensions δD, δP, δH ) components of solubility parameter Φi ) volume fraction Rj ) radius of interaction Rij ) solute-solvent interaction ηi ) pure-component viscosity T ) temperature Tc ) critical temperature Vi ) volume Vc ) critical volume Vw ) molar volume of pure water Vo ) molar volume of pure organic component M ) molecular weight σmix ) surface tension of mixture σw ) surface tension of pure water σo ) surface tension of pure organic compound

Appendix The mixture property models used in the case study are outlined here. Solvent Power. For a solvent mixture, the components of the solubility parameter can also be computed according to

Figure 4. Location of designed solvent blends in the threesolubility-parameter space. Also shown in the figure are the purecomponent solvents and the ink residue that the blend has to dissolve to be an effective solvent. All of the blends fall within the interaction radius (12.7 MPa1/2) of the ink residue.

problem of first designing single components and simultaneously or sequentially forming blends from the single components is certainly attractive. However, many issues come into play: for example, the problem size grows exponentially, and property models might not be valid/available for all single-component solvents. For this general formulation, from a problem-size point of view, the interval analysis approach to global optimization is not appropriate at this point in time.

δD )

∑i ΦiδDi

δP )

∑i ΦiδPi

δH )

∑i ΦiδHi

where Φi is volume fraction of component i, expressed as

Φi )

xiVi

∑i xiVi

Phenolic resins are commonly used in printing inks. The dried ink (solute) is assumed to consist of phenolic resins, specifically Super Bakacite 1001, Reichhold. The

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Ind. Eng. Chem. Res., Vol. 42, No. 3, 2003

components of the solubility parameter of the resin are nonpolar (δD) ) 23.3, polar (δP) ) 6.6, and hydrogen bonding (δH) ) 8.3 MPa1/2.49 The radius of interaction (Rj) is 19.8 MPa1/2. Thus, a given solvent i that can effectively dissolve the ink residue has the following solute-solvent interaction constraint

the surface tension of pure organic compound, dyn/cm; ψo ) 1 - ψw; and ψw is defined by the relation

log10

[

(ψw)q

(1 - ψw)

]

) log10

Rij ) [4(δDi - 23.3)2 + (δPi - 6.6)2 + (δHi - 8.3)2]1/2 < 19.8 Flow Characteristics (Viscosity). Most liquid mixture viscosity models assume that pure-component models are available. Reid et al.26 discuss two different models for mixture viscosity, namely, (a) the Grunberg and Nissan model and (b) the method of Teja and Rice. Whereas the first model works well for organic liquids, the Teja and Rice model is specifically recommended for aqueous blends. The equation for the binary mixture viscosity is

ln(ηmixmix) ) x1 ln(η11) + x2 ln(η22) with

) Vc(min ) )

Vc2/3 (TcM)1/2

∑i ∑j xixjVc(ij)

M(mix) )

Vc(ij) )

∑i xiMi

(Vci1/3 + Vcj1/3)3 8

where ηi is the pure-component viscosity evaluated at T(Tci/Tcm); Vc and Tc are the critical volume and critical temperature, respectively; and M is the molecular weight. Flow characteristic requirements are posed as

ηL e ηproduct e ηU Surface Contacting. Surface contacting determines how effective the solvent is in wetting the blanket surface and, in turn, characterizes the solvent’s cleaning ability. High surface tension also translates into greater energy utilization, especially if the cleaning is performed via a wiping operation. The surface tension of aqueous solutions is more difficult to predict than that of nonaqueous models because of the nonlinear dependence on mole fraction. Small concentrations of organic material can significantly affect the surface tension value. For many binary organic-aqueous mixtures, the method of Tamura, Kurata, and Odani26,47 is recommended

σmix1/4 ) ψwσw1/4 + ψoσo1/4 where σmix is the surface tension of mixture, dyn/cm; σw is the surface tension of pure water, dyn/cm; σo is

[

]

(xwVw)q (xwVw + xoVo)1-q + xoVo

(

2/3 q σ oV o 0.441 - σwVw2/3 T q

)

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Received for review March 22, 2002 Revised manuscript received September 23, 2002 Accepted October 10, 2002 IE020224L