Environmentally benign solvent design by global optimization

Computers and Chemical Engineering 23 (1999) 1381–1394

Environmentally benign solvent design by global optimization

Manish Sinha, Luke E.K. Achenie *, Gennadi M. OstrovskyDepartment of Chemical Engineering, Uni6ersity of Connecticut, Storrs, CT 06269, USA

Accepted 20 September 1999

Abstract

Changes in environmental regulations have often resulted in the need to replace existing solvents with more environmentallybenign substitutes. For example, solvents such as 1,1,1, trichloroethane are being phased out within the next few years. In general,‘designer’ compounds, which have desired properties, can be obtained with the help of computer-aided molecular design (CAMD)approaches. In recent years, the above product design problem has often been posed as a mathematical programming problem.In this framework, the desired attributes of the compound are posed as performance objective and constraint functions. Nonlinearfunctions, such as those associated with solubility parameter models seen in solvent design, can lead to multiple local solutions(i.e. optimization minima). For these cases, the failure to design the globally optimal compound (using a local optimizationapproach) can introduce a source of uncertainty. Thus, there is a need for global optimization techniques in product design. Toaddress this need, we have developed a new global optimization algorithm that exploits the structure of the CAMD formulation.We have used this algorithm to study the design of environmentally benign solvents for surface cleaning applications in theprinting industry. © 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Mixed-integer; Global optimization; Computer-aided molecular design

www.elsevier.com/locate/compchemeng

1. Introduction

Solvents are extensively used as processing materialsin the chemical industry and solvent-based industries(e.g. resin, coating and paint industry). However, manyorganic solvents such as 1,1,1, trichloroethane are beingphased out in response to directives from federal agen-cies such as the EPA and OSHA (Krishner, 1995). Thishas prompted the search for new environmentally be-nign solvent substitutes. Traditionally, the search for asubstitute solvent has been done through experience,database searches and bench scale synthesis of solventalternatives. All these approaches have severe limita-tions and are often unable to effectively find the opti-mal solvent substitute. Computer-aided moleculardesign (CAMD), also referred to as product design, hasemerged as a powerful tool for identifying productalternatives. In this approach, mathematical modelsthat relate different physico-chemical properties to themolecular structure are systematically ‘inverted’ to de-

sign solvent alternatives that have a desired set ofproperties.

Constantinou, Bagherpour, Gani, Klein and Glen(1996) have reviewed the CAMD problem formulationand its applications in detail. Some solvent applicationsfor which the CAMD methodology has been proposedare liquid–liquid extraction (Macchietto, Odele &Omatsone, 1990; Odele & Machietto, 1993), multicom-ponent gas absorption processes (Pistikopoulos & Ste-fanis, 1998), separation processes (Pretel, Lopez, Bottini& Brignole, 1994) and solvent blends for paint formula-tion (Klein, Wu & Gani, 1992). Other applicationsinclude polymer design (Vaidyanathan & El-Halwagi,1994a; Maranas, 1996; Vaidyanathan, Gowayed & El-Halwagi, 1998), and refrigerant design (Joback &Stephanopoulos, 1989; Gani, Nielsen & Fredenslund,1991; Duvedi & Achenie, 1996). A recent issue ofComputers & Chemical Engineering focused on com-puter-aided design of chemical compounds (Mavrovou-niotis, 1998).

Two classes of solution approaches, namely enumer-ation techniques and knowledge based strategies havebeen employed for certain CAMD formulations (Gani

* Corresponding author. Tel.: +1-860-4862756; fax: +1-860-4862959.

E-mail address: [email protected] (L.E.K. Achenie)

0098-1354/99/$ - see front matter © 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 9 8 -1354 (99 )00299 -9

M. Sinha et al. / Computers and Chemical Engineering 23 (1999) 1381–13941382

& Brignole, 1983; Brignole, Bottini & Gani, 1986;Joback & Stephanopoulos, 1989; Pretel et al., 1994).These methods rely significantly on heuristics and userintervention. Another class of solution approaches arestochastic based approaches such as genetic algorithms(Venkatasubramanium, Chan & Caruthers, 1994) andsimulated annealing (Marcoulaki & Kokosis, 1998).

In recent years, the CAMD problem has often beenposed as a mixed integer nonlinear mathematical pro-gramming (MINLP) formulation in which the userspecifications are modeled as constraints and the desir-able property is modeled as an objective function. ACAMD problem may also be formulated as a con-straint satisfaction problem (i.e. feasibility problem).Mathematical programming based solution strategyhave been successfully employed for solving the CAMDproblem (Macchietto et al., 1990; Odele & Machietto,1993; Duvedi & Achenie, 1996; Maranas, 1996; Pistiko-poulos & Stefanis, 1998; Vaidyanathan et al., 1998).

Many methods for solving mathematical program-ming based CAMD problems use outer approximation(OA) algorithm (Viswanathan & Grossman, 1990)which consists of a sequence of iterations consisting ofNLP (nonlinear programming) subproblem and MILP(mixed-integer linear programming) ‘master’ problems.One weakness of such methods is that the NLP sub-problem may get trapped in a local solution for non-convex mathematical programming formulations(nonconvexity in objective function and/or constraints)as shown in Fig. 1.

Another issue in CAMD is the problem of modeluncertainty. The latter is independent of the searchstrategy employed during the design stage. In additionto model uncertainty, another source of uncertainty isthe fact that a local optimization approach may notreach the global solution (i.e. that one cannot find abetter solution), if this solution happens to be the best

even in the presence of other uncertainties (see Fig. 1).Thus, there is a need to develop computationally effi-cient global optimization algorithms for CAMD.

Global optimization based on interval analysis(Vaidyanathan & El-Halwagi, 1994b, 1996) has beenemployed to solve relatively small sized CAMD prob-lems. However realistic CAMD problems have severalinteger variables describing a molecular structure.Moreover, most of the variables participate in nonlin-ear equations. To address this need, we have developeda new global optimization algorithm that exploits thestructure of the CAMD formulation.

Now we need to consider the issue of the validity ofa global optimization approach in the presence ofmodel uncertainty. Let us suppose that the CAMDmodel has integer variables (i.e. structural variables) asthe only design variables. Then exhaustive enumerationwill yield the same global solution as a global optimiza-tion approach. Therefore in this regard enumerationand global optimization have the same deficiency withrespect to model uncertainties. That is, unless modeluncertainties are explicitly accounted for, a global solu-tion may not necessarily be the best solution in thepresence of model uncertainties. In fact the issue ofmodel uncertainty is independent of the search strategyemployed.

The paper is organized as follows, in Section 2, someimportant models for the CAMD approach to designenvironmentally benign solvents are discussed. Next,the development of a global optimization algorithm forthe CAMD problem is presented in Section 3. In Sec-tion 4, we develop the relaxation (underestimator) forthe nonconvex solubility parameter constraint. Finally,in Section 5, the efficacy of the proposed algorithm isdemonstrated via a case study in which the design ofsolvent alternatives for use in a commercial cleaningapplication is considered.

Fig. 1. Schematic of a linear objective function surface with nonlinear constraints. A local optimization approach may not find the optimal pointfrom any of the five initial starting (or initial guess (IG)) points.

M. Sinha et al. / Computers and Chemical Engineering 23 (1999) 1381–1394 1383

Fig. 2. Correlation between environmental measure for bioconcentration factor (log BCF) with octanol–water partition coefficient (log Kow)(Nendza & Hermens, 1995). Printed with kind permission from Kluwer Academic Publishers.

2. Important models for solvent design

2.1. Solubility parameter

The amount of solute that a solvent can dissolve(capacity) and the ability of the solvent to selectivelydissolve a solute (selectivity) are two important proper-ties that determine a solvent’s performance. Theseproperties can be modeled by the equilibrium relation-ships that govern the distribution of a solute in asolvent, which in turn can be modeled by an activitycoefficient model such as UNIFAC (Fredenslund,Gmehling & Rasmussen, 1977). In addition, there areother models for characterizing solvent performancethat are prevalent in the solvent industry. Some of thesemodels are based on the solubility parameter.

The solubility parameter has been successfully usedin solvent-based industry. Applications in which thesolubility parameter approach have been employed in-clude finding compatible solvents for coating resins,finding an effective mass separating agent, predictingthe swelling of cured elastomers by a solvent, andestimating liquid–liquid equilibrium in polymer systems(Grulke, 1989; Archer, 1996). In a recent article, Gess-ner recommends use of solubility parameter for design-ing solvents for surface cleaning and surface coatingapplications (Gessner, 1998). In this work also thesolvent performance is characterized by the Hansensolubility parameter (dT) (Hansen, 1969). This parame-ter in turn is made up of hydrogen bonding interaction(dH), polar interactions (dp), and nonpolar (dispersive)interaction (dD) as shown in Eq. (1).

dT=dD2 +dP

2 +dH2 (1)

Solute and solvents are represented as points in thethree-dimensional Hansen solubility space. The dis-tance, Rij, of a solute (i ) from a solvent ( j ) is ameasure of the solute–solvent interaction (see Fig. 3).Associated with the solute point is a radius of interac-tion, R*. If Rij is less than R*, then the solvent is likelyto dissolve the solute. Rij is given by

Rij=4(dD−dD* )2+ (dP−dP* )2+ (dH−dH* )25 (R*)2

(2)

where

dD=%i

niFDi

Vo+%i

niVi

(2a)

Fig. 3. Solute interaction sphere of phenolic resin and the swellingbox of polyisoprene rubber. The design space is the shaded region(outside the swelling box but inside the interaction sphere).


dP=

'%i

ni(FPi)2

Vo+%i

niVi

(2b)

dH=D%

i

ni(−UHi)

Vo+%i

niVi

(2c)

Here Fdi, FPi and UHi are the group contributionparameters associated with the dispersion, polar andhydrogen bonding solubility parameters, respectivelyfor the ith group. ni is the number of times i th group ispresent in a molecule, Vo is a constant associated withthe liquid molar volume prediction model and Vi is thegroup contribution associated with the ith group forprediction of liquid molar volume.

The group contribution parameters are taken fromvan Krevlen and Hoftyzer’s work on the dispersion andpolar components (van Krevelen & Hoftyzer, 1976) andfrom Hansen and Beerbower for the hydrogen bondingcomponent (Hansen & Beerbower, 1971). For predict-ing the liquid molar volume the model proposed byConstantinou and Gani (1994) is used.

2.2. En6ironmental health and safety requirements

Environmental, health and safety concerns are im-portant issues for solvent selection. Solvent propertiestargeted by environmental regulations include biocon-centration factor (BCF), toxicity and flammability.

The environmental and health impact of a solventhave been correlated to the octanol–water partitioncoefficient (Kow) by many researchers (Lyman, Reehl &Rosenblatt, 1981; Horvath, 1992; van Leeuwen & Her-mens, 1995). Kow is a measure of hydrophobicity andrelates the equilibrium partition of solvent betweenwater and a water-immiscible liquid phase. For mostchemicals log Kow ranges from −3 to 7 and may beestimated by Hansch and Leo’s approach (Hansch &Leo, 1979), as follows

Log Kow=sum of fragments (x0)+ factors (x1) (3)

Here empirically derived atomic or group fragmentconstants (x0) and structural factors (x1) are used.

Several environmental and health metrics such as(BCF and LC50) depend on the Kow. A solvent with ahigh bioconcetration factor has a greater likelihood ofaccumulating in and harming living tissue. One ofseveral correlations for BCF shown in Eq. (4) (Veith &Konasewich, 1975), also see Fig. 2. (Nendza & Her-mens, 1995)

log BCF=0.76 (log Kow)−0.23 (4)

LC50 measures the lethal concentration of a chemicaland can be used to quantify the solvent toxicity. The

LC50 value has been correlated by the following rela-tion (Konemann, 1981)

log LC50= −0.87(log Kow)−0.11 (5)

Although the theoretical basis of these models is lim-ited, they provide practical tools for risk assessment. Inthis case study, the environmental impact of the solventis considered only by constraining Kow value. Thisimplicitly constrains BCF and LC50 for the solvent.

A solvent that can form potentially explosive vapormixtures with air is an occupational hazard and oftenresults in higher insurance costs. Safety requirementscan be quantified by the solvent’s flash point. Most ofthe predictive techniques relate the flash point to theboiling point. For example, Butler, Cooke, Lukk andJameson (1956) proposed a correlation for paraffins,aromatics and cyloparaffins

Tf=0.683TB−119 (6)

Other equations showing direct relationship betweenthe flash point and the boiling point have been pro-posed and are reviewed by Lyman et al. (1981). In thiscase study we implicitly account for safety by constrain-ing the boiling point. We note that all correlationsincluding the ones discussed above have a limited re-gion of validity.

2.3. Other considerations

One important consideration in solvent substitutionis the desire to use existing equipment and processingtechnology. This requires that both the old and thesubstitute solvent should have similar transport andother properties such as viscosity, diffusivity, density,surface tension, heat capacity (Cp), and vapor pressure(Pvp). Zhao and Cabezas (1988) have reviewed molecu-lar property considerations for design of substitutesolvents.

One important property in many solvent designproblems is the heat of vaporization (DH6) which is agood measure of the energy efficiency associated withthe drying operation. The trivial observation that thesolvent has to be a liquid at the operating conditionleads to constraints on the boiling and melting point ofthe solvent molecules. Additional constraints areneeded to design structurally feasible molecular struc-tures (see for example Churi & Achenie, 1996). Con-straints can also be imposed to eliminate unstablemolecules. For example, a small molecule made up ofthree ether group may not exist. In the next section weconsider the structure of the CAMD problem for sol-vent design and develop a global optimization al-gorithm that exploits this structure.


3. Solvent design: a reduced space branch and boundapproach

3.1. The CAMD problem formulation

A typical molecular design problem may bemodeledas

minimizeU

(maximize) pobj(U, u) (PMD)

subject to

pkL5pk(U, u)5pk

U Ö k=1, 2, 3, ... , m1

hj(U, u)=0.0 Ö j=1, 2, 3, ... , m2

where U is a vector of variables that define the molecu-lar structure, u is a vector of group contributionparameters, pobj is the objective function (some undesir-able property to be minimized during design), pk is thespecified property requirements constrained between anupper (pk

U) and a lower value (pkU) and hj are the

equality constraints generally associated with structuralfeasibility requirements. The number of inequality con-straints is 2m1 and the number of equality constraints ism2.

Most of the property constraints are of the form

pk=��j nju j

1/�j nju j2na

where ‘1’ and ‘2’ correspond to

group contribution parameters associated with a prop-erty of type ‘1’ and another molecular property of type‘2’; a is an exponent (generally equal to 1). For exam-ple, molar density is the ratio of molar weight (‘1’) andmolar volume (‘2’). Other examples of such modelsinclude glass transition temperature and specific heatcapacity (Maranas, 1996). Transformation of such con-straints into a linear form is straightforward. Howeversome property constraints are of the form

pk= fNL1 �%

j

nju ja�/fNL

2 �%j

nju jb�

where f 1NL and f 2

NL are nonlinear functions. Groupcontribution models for solubility parameter are of thisform (Eqs. (2b) and (2c)). Other examples of thesemodels include Reidels method for prediction of heat ofvaporization and vapor pressure by correspondingstates correlations (Reid, Prausnitz & Poling, 1987). Itis not always possible to reformulate these constraintsinto linear or convex forms.

The nonlinear mathematical programming model forthe CAMD problem (PMD) has the following features:1. It is a mixed integer nonlinear problem (MINLP)

problem involving a large number of variables.2. The number of linear constrains are much greater

than the number of nonlinear constraints.3. Most of the components of the design vector (U)

participate in the nonlinear terms.

We now present the development of a branch andbound global optimization algorithm that exploits thisproblem structure. At first we lay out the steps of abranch and bound algorithm, discuss its limitation forthe CAMD problem, and then propose modifications.

3.2. Branch and bound

Consider a general MINLP problem

f=minimizex,y

f(x, y, u) (PORG)

subject to

gi(x, y, u)50 i=1,2, ... , n

hj(x, y, u)=0 j=1,2, ... , m

xLBxBxU, y� (0, 1)

where x is a vector of continuous variables, y is a vectorof integer (binary) variables and u is a vector ofparameters. There are n inequalities and m equalityconstraints.

We employ a branch and bound method for solvingthe problem. In a conventional branch and boundmethod a region is partitioned into subregions Tk,which has the form

Tk={x, y : xL(k)5x5xU(k); yL(k)5y5yU(k)}

i.e. a subregion (Tk) is defined by bounds on the designvariables (x, y). We will designate the optimizationproblem for subregion Tk as P (k)

ORG.In order to estimate the lower bound (LBk) of the

objective function f(x, y, u) in the subregion Tk, weobtain a relaxation of the problem (P (k)

ORG) by replacingthe nonlinear constraints and objective function (ifnonlinear) by their linear underestimators. In effect weobtain a mixed integer linear programming (MILP)problem which can be solved by any standard MILPsolver such as lp–solve (Schwab, 1997) or OSL (IBM,1992).

If the solution point (x*, y*) of the lower boundproblem satisfies all constraints in problem P (k)

ORG, thenit is a valid solution for the calculation of the upperbound (UBk= f(x*, y*, u)). In the case of a linearobjective function, LBk=UBk and the solution point(x*, y*) is a global minimum for f(x*, y*, u) in Tk. Ifthe solution for the calculation of the lower boundproblem does not satisfy all constraints of P (k)

ORG thenthe upper bound (UBk) for subregion Tk is computedby replacing the nonlinear constraints and objective (ifnonlinear) by their respective linear overestimators. Ageneric branch and bound algorithm for global opti-mization (Horst & Tuy, 1990) have the following steps.

Step 0: Specify a tolerance (o). Compute the lowerbound (LBo) and upper bound (UBo) for the originalregion To and insert it in a list L. Set the overall lowerbound LWBD to−� and overall upper bound UPBDto −�.


Fig. 4. Flowchart for global optimization via reduced space branchand bound.

If there is a large number of search variables partici-pating in the nonlinear constraints or objective func-tion, step 2 may generate a large number of subregions(or nodes). The above is true for the CAMD problem.To circumvent this dimensionality problem, we havedeveloped a branch and bound algorithm that performsbranching based on certain splitting functions, whichare linear functions of x and y.

3.3. Using splitting functions in the branch and boundalgorithm

In our approach, a region is not defined by boundson the variables but by bounds (al and bl) on thesplitting functions (cl). The problem formulation forsubregion Tk becomes

f=minimizex,y

f(x, y, u)

gi(x, y, u)50 i=1,2, ... , n

hj(x, y, u)50 i=1,2, ... , m (P. ORGK )

al5cl(x, y, u)5bl l�SF

xLBxBxU, y� (0, 1)

where cl (x, y, u) are linear splitting functions and SF isa set of splitting functions.

In the proposed algorithm, partitioning of a regioninto two subregions is performed on one of the splittingfunctions as described below.

Modified step 2: If LBk\LWBD set LWBD=LBk.If UBkBUPBD set UPBD=UBk. Partition Tk intotwo subregions (Tk

P and TkQ) by splitting on one split-

ting function. For example the rth function, such thatTk

P={x, y : ar5cr{x, y)Bcr(x*, y*); al5cr(x, y)5bl

Ö(1�SF and l"r)}. Similarly TkQ={x, y : cr(x*, y*)5

cr(x, y)5br ; al5cr(x, y)5bl Ö(1�SF and l"r)},where x*y* is the solution point for the calculation ofthe lower bound.

The proposed algorithm is also shown as a flowchartin Fig. 4. To construct underestimators and overestima-tors, we represent a nonlinear function as a tree struc-ture (see Fig. 6) with the design variables at the lowestlevel. The underestimators are constructed in a stepwiseprocedure starting from the lowest level going up to thefirst level for computation of bounds. This is followedby a reverse sweep from the first level to the lowest levelfor construction of the underestimators. In our ap-proach, the dimensionality of the lower and upperbound problem is no larger than the dimensionality ofthe original problem PORG. It is interesting, and in factfortuitous, that there are instances where the number ofbranching variables actually decreases. This translatesto a smaller number of partitions and therefore asmaller number of iterations. The details of the al-gorithm for constructing the underestimators and over-estimators are presented in Appendix A.

Step 1: Remove a region Tk from the list L that hasthe lowest LBk (lower bound). If the list L is emptythen the problem is infeasible and has no solution.

Step 2: If LBk\LWBD set LWBD=LBk and ifUBkBUPBD set UPBD=UBk. Partition the region Tk

into s subregions (Tk1, Tk

2, … , Tks ) by splitting one or

more of the variables (x, y). In many algorithms onlyone of the design variables is bisected.

Step 3: Compute lower and upper bounds (LBki , UBk

i

Öi=1, 2, … , s) for each subregion generated in step 2.A subregion Ti

k is inserted in the list L if it satisfies (i)the problem of calculation of lower bound resulted in afeasible solution, and (ii) LBk

iBUPBD.Step 4: If �UPBD–LWBD�\o then go to step 2, else

the algorithm has converged to the global solution.Note that in order to estimate the lower and upper

bound we need to solve an MILP. As a result, thetermination criteria stated in STEP 4 guarantees thatthe final solution is the global solution for the originalproblem PORG. We note that the final solution point tothe original problem (PORG) corresponds to the solutionof the upper bound problem for the region j such thatUBj=UPBD.


3.4. Illustrati6e example: MINLP test problem

The following MINLP problem (Kocis & Grossman,1988) will help illustrate the steps of the proposedalgorithm. The problem has three binary variables andtwo continuous variables. Nonconvexitities are presentin the two equality constraints. The three inequalitiesare linear.

minimize f=2x1+3x2+1.5y1+2y2−0.5y3

subject to:

h1(x, y)=x12+y1−1.25=0.0

h2(x, y)=x21.5+1.5y2−3.00=0.0

x1+y151.60

1.33x2+y250.0

−y1−y2+y350

x]0, y�{0, 1}3

The equality constraints are broken into two inequali-ties. Equality hl(x, y) is decomposed to g1(x, y)andg2(x, y). Similarly h2(x, y) is decomposed to g3(x, y)andg4(x, y). Here

g1(x, y)=x12+y1−1.2550.0,

g2(x, y)=x12+y1−1.25]0.0

g3(x, y)=x21.25+1.5y2−3.0050.0

g4(x, y)=x21.25+1.5y2−3.00]0.0.

The nonlinearities are only associated with the twoterms, x1

2 and x21.5, involving variables x1 and x2. For

this problem, the linear splitting functions are c1,=x1

and c2=x2. The problem is solved with the initialbounds of {05c15100} and {05c25100}. The al-gorithm took 39 iterations to converge. At the globalsolution f *=7.667, x*= [1.118, 1.310] and y*=[0, 1, 1]. The reduction in the number of branchingdimensions translates to smaller number of nodes vis-ited during the branch and bound operation. To verifythis claim, the same problem was again solved with allthe five variables as branching variables. This con-verged to the same global solution, but after 227 itera-tions. This illustrative example demonstrates thecomputational advantage of the proposed strategy, es-pecially when many variables participate in the nonlin-ear terms.

4. Construction of linear estimator for the solubilityparameter design constraint

The solvent performance constraint illustrated in Eq.(2) makes the solvent design problem nonconvex andresults in multiextremality. We now present an ap-

proach for the construction of linear underestimator forthis solvent design constraint. Let us introduce thefollowing functions

cD=%j

njFDj

cP=%j

nj(FPj)2

cH=%j

nj(−UHj)

cV=Vo+%j

njVj (7)

The solvent design constraint (Eq. (2)) can then beexpressed as

4�cD

cV

−dD*�2

+�cP

cV

−dP*�2

+�cH

cV

−dH*�2

5 (R*)2

Multiply both sides by cV2 to obtain

f10=4(cD−dD* cV)2+ (cP−dp*cV)2

+ (cHcV−dH* cV)2−R2cV250.

fo is made up of four separable terms. The first and thefourth terms are squares of linear equations. The sec-ond and the third terms are relatively more compli-cated. Next, we will describe a tree representation of thethird term. This representation will then be used toconstruct its underestimator. The third term is

g1o= (cHcV−dH* cV)2

Let us introduce a function such that g11= (cHcV−

dH* cV). Then we obtain g1o,= (g1

1)2 and

g11=

14

[(cH+cV)2− (cH−cV)2−4dH* cV].

Next, we introduce the second level functions (g12 and

g22) such that g1

2=cH+cV and g22=cH−cV

and third level function such that g13=cH and g2

3=cV.Note that the third level functions are linear withrespect to the design variables nj. Now g1

1= (1/4)[(g1

2)2− (g22)2−4dH* cV] or g0={(1/4)[(g1

2)2− (g22)2−

4dH* cV]}2 or g10={(1/4)[(cH+cV)2−(cH−cV)2

−4dH* cV]}2. The functions g12 and g1

2 are themselvesrelated to g1

3 and g23 by the square root. The third level

functions in turn are linear functions with respect to thedesign variables (nj). The bounds on cH and cV areused to construct bounds on the second level functions,g1

2, and g22. This is further used to construct the bounds

on the first level function, g11.

The linear underestimators are constructed in a re-verse sweep that starts at the first level and goes down.First, the linear underestimator of g1

o is constructed interms of g1

1 such that L [go, S1]=m1(g11)+m2. The sign


of m1 (which depends on the bounds on g11, determines

whether the function m1(g11) is convex or concave with

respect to g12 and g2

2. The underestimator now has thefollowing form

L [go,S2]=m3(g12)+m4(g2

2)+m5(CV)+m6

or

L [go, S2]= (m3−m4)(cH)+ (m3+m4)(cV)+m5(cV)

+m6.

The signs of (m3−m4) and (m3+m4) are subsequentlyused to construct the underestimator with respect to cH

and cV. After rearranging the terms the linear underes-timator is represented as

L [go, S2]=m8CH+m9(CV)+m10.

Thus, the underestimator will be linear with respect toCD, CP, CH and CV; therefore, it will be linear in termsof the search variables nj. At any iteration in a branchand bound strategy, the subregion is represented by thebounds on the splitting functions. Based on this infor-mation the coefficients m1, (m13−m4) and (m3+m4) arecalculated and a decision about construction of theunderestimator is taken at two levels as shown in Fig.5. Similarly, the overestimators can be constructed in asimilar fashion. Further details can be found in Ap-pendix A.

In the next section, we apply the global optimizationalgorithm and underestimator for the solubilityparameter model in a case study to design solventmolecules in a surface cleaning application.

5. Case study: single component blanket wash design

In a lithographic printing process, ink is carried tothe impression plate by means of a train of rubberrollers commonly called ‘blankets’. The cleanliness ofthe blanket is of primary concern for production ofhigh-quality images. One of the most used solvents in

the printing industry is called the ‘blanket wash’ whichis specially formulated to clean residue ink from litho-graphic printing presses. There are more than 52 000lithographic printers in the United States (Adrian,1991). Many solvents used in commercial blanketwashes are to be phased out in near future due toenvironmental regulations. The ‘Printing Industry ofAmerica’ (PIA) and EPA have taken a major initiativeto find environmentally benign solvent substitutes forthis application (Design for the Environment Program,1997).

In this section we present a case study CAMD prob-lem for to design blanket wash substitute solvents. Thedetailed model development has been reported in anearlier work (Sinha & Achenie, 1998). In short, an idealblanket wash solvent should have the following at-tributes: (a) ability to rapidly dissolve dried ink; (b)have minimal drying time; (c) should not swell therubber blanket; (d) be nonflammable; and (e) be envi-ronmentally benign.

Drying time is correlated with the heat of vaporiza-tion (DHV) of the solvent, which is minimized duringdesign stage. Low DHV translates to higher evaporationrate and hence shorter drying time. It also translates tolower utility costs associated with the drying operation.

Ink residue is assumed to consist of phenolic resin(Super Bakacite® 1001, (Cunningham, 1998)) with dd=23.3, dp=6.6 and dh=8.3 MPa1/2 and R*=19.8MPa1/2. Solvents that can effectively dissolve the inkresidue have the following solute–solvent interactionconstraint

Rij= (4(idD−23.3)2+ (idP−6.6)2+ (idH−8.3)2)1/2

B19.8

A lithographic blanket is generally made of polyiso-prene rubber. Natural rubber (polyisoprene) has 25%swelling range between polar solubility parameter (dp)values 0−6.3 MPa1/2 (i.e. a solvent that does not havedp in this range will not significantly swell the blanket).To ensure minimal blanket swelling we add a constraint

Fig. 5. Decision tree for construction of underestimator for the solubility parameter constraint.


on dp to be greater than 6.3. Thus, the ability todissolve dried ink but at the same time not swell therubber blanket is ensured by constraining the solventdesign to a specified region in the solubility parameterspace as shown in Fig. 3.

We note that the correlations for both log BCF andlog LC50 depend linearly on log Kow. Therefore con-straining log Kow is sufficient in the sense that we neednot explicitly constrain the log BCF and the log LC50.In a nutshell, a solvent’s environmental impact is in-ferred from its Kow value. Here we constrain the log Kow

of the solvent to a small value (i.e. less than 4.0). Toensure that the solvent is a liquid at ambient tempera-ture, the boiling point (Tb) should be larger than 323 Kand the melting point (Tm) should be less than −223K. The presence or absence of structural group in amolecule is represented by a binary matrix U (Churi &Achenie, 1996) with elements Uij such that

Uij=!1

if ith position in a molecule has jth structural group

0 otherwiseNote that nj=�i Uij. The size of the molecule is limited

to a maximum of five structural groups��i �j Uij55

�in this case study. The octet rule for acyclic moleculesas proposed by Odele et al. is used (Odele & Machietto,1993).

%i

%j

Uij(2−6j)=2

where 6j is the valency of jth structural group.The basis set consists of twelve groups: CH3-, CH2-,

Ar-, Ar= , -OH, CH3CO-, -CH2CO-, -COOH,CH3COO-, -CH2COO-, -CH3O, -CH2O-. Ar- refers tobenzene ring with one less hydrogen (valency of one)and Ar= refers to a benzene ring with two less hydro-gen atoms (valency of two). Many chlorinated com-pounds (e.g. 1,1,1, trichloroethane) are being phasedout for their ozone depleting potential (ODP). We haveaddressed this by not including any structural groupthat has chlorine, in our basis set. This basis set can beused to design straight chain hydrocarbons, aromatics,alcohols, ketones, esters, ethers and acids. Heat ofvaporization, boiling point and melting point are calcu-lated by the Constantinou and Gani (1994) method.The resulting CAMD formulation for blanket washsolvent design is shown below.

minimizeUij

%i

%j

UijhVj (P3)

subject to:

%i

%j

Uij55 (Molecular size constraint)

%i

%j

Uij(2−6j)=2 (Octet rule)

exp((%i

%j

Uijtmj)/102.425)5223

(Melting point constraint)

exp((%i

%j

Uijtbj)/204.4)]323

(Boiling point constraint)

%i

%j

Uij(xo)j+%i

%j

Uij(xo)j54.0

(Octanol-water partition constraint)

4ÃÃ

Ã

Á

Ä

%i

%j

Uij*FDj

Vo+%i

%j

UijVj

−23.3ÃÃ

Ã

Â

Å

2

+ÃÃ

Ã

Á

Ä

'%i

%j

Uij*(FPj)2

Vo+%i

%j

UijVj

−6.6ÃÃ

Ã

Â

Å

2

+ÃÃ

Ã

Á

Ä

'%i

%j

Uij*(−UHj)'Vo+%

i

%j

UijVj

−8.3ÃÃ

Ã

Â

Å

2

5 (19.8)2

(Solubility parameter constraint)'�%i

%j

Uij(FPj)2�Vo+%

i

%j

UijVj

]6.3 (Swelling constraint)

cD5%i

%j

Uij*FDj5cD (Splitting function 1)

cP5%i

%j

Uij*FPj5cP (Splitting function 2)

cH5%i

%j

Uij*(−UHj)5cH (Splitting function 3)

cV5Vo+%i

%j

UijVj5cV (Splitting function 4)

where h6j, tbj and tmj are group contribution parametersassociated with DHv, Tb, and Tm. Solubility parameterand swelling constraints are nonlinear and nonconvex.The splitting functions are cD, cP, cH and cV. Thesubregion is thus represented by four pairs of addi-tional linear constraints that confine the search betweenthe maximum and minimum values of these splittingfunctions.

5.1. Results and discussion

The problem has 60 binary variables and 15 con-straints. The algorithm took 31 iterations. Since maxi-mums of two MILPs are solved per iteration, the totalnumber of MILPs solved is no more than 62. Theglobally optimal solution found is methyl-ethyl ketone(MEK) with molecular structure CH3-CH2-CO-CH3.


Table 1Result of solvent design case study, optimal molecules and their properties

Diethyl ketone (DEK) Ethylene glycol mono methyl etherMethyl ethyl ketone (MEK) Comments

Structure

40.12DH6 (KJ/mol) 47.635.47 Computed(objective function) 34.5 38.37 – ExperimentalTB (°C) 81 112.5 114.2 Computed

101.0 124.679.6 ExperimentalTM (°C) −80.0 −66.4 −68.9 Computed

−39.0 −85.1−86.0 Experimental15.13dD (Mpa1/2) 15.33 15.25 Computed16.0 15.8 – Experimental

7.34 7.768.64 ComputeddP (Mpa1/2)9 0 7.6 – Experimental

4.28 17.084.65 ComputeddH (Mpa1/2)4.7 – Experimental5.18.9 11.916.86 Ra=19.2Rij

log Kow 1.59 2.37 −0.65 Computed0.91 −0.770.29 Experimental

0.97log BCF 1.556 −0.724 Computed0.0 0.46 −6.152 Computedb

0.46 −6.80.0 ExperimentalTf (°C) 4.9−16.4 6.2 Computed

13.0 43.0−9.0 Experimental2.3 0.53 ExperimentalEvaporation ratec 3.8

Ketone Glycol etherKetoneSolvent type

a Computed via correlation from experimental log Kow value.b Computed via correlation from predicted log Kow value.c Experimental value of the evaporation rates relative to evaporation rate of butyl actate. Note that this value has a good correlation with the

objective function (DHv).

The DHv (also the objective function) associated withMEK is predicted as 35.471 KJ/ mol. This moleculewas found at the 10th iteration with a overall upperbound (UPBD) of 35.471 and an overall lower bound(LWBD) of 33.99. The algorithm finally converged withMEK as the global solution after 21 more iterations.To find the second best molecule, additional constraintwas added to eliminate MEK from the design space.The next best molecule designed is diethyl ketone(DEK). This process was repeated to find the third bestsolvent namely ethylene glycol monomethyl ether(EGME). The results are presented in Table 1. Thistable also lists the predicted properties along with theexperimental values. The model error associated withDHv, Tb, Tm and solubility parameters is relatively low.The predicted value of log Kow shows a considerablemodel error and consistently predicts a larger valuecompared to the experimental value. This error alsoresults in an error in log BCF. However, it is interestingto note that if the experimental value of log Kow is usedin Eq. (4), then the model prediction for log BCF isvery accurate for the three molecules designed. Table 1also lists the evaporation rate of each solvent. There is

an inverse relationship between DHv and the evapora-tion rate. Hence in effect, minimization of DHv (objec-tive function) results in the maximization of theevaporation rate. Thus the molecules designed havehigh evaporation rate and consequently will have mini-mal drying time.

6. Conclusions

This paper presents a framework that incorporatesenvironmental and health considerations in solvent de-sign. Solvent design models based on solubility parame-ters are nonlinear and often result in a nonconvexCAMD formulation. We note that many nonlinearcorrelations for CAMD have a structure similar to thatof solubility parameter models and are a source ofnonconvexity. To solve these CAMD problems, wehave developed a global optimization algorithm thatexploits the problem structure. We have also presenteda strategy for the construction of linear underestimatorsfor nonlinear models often seen in solvent design. Theunderestimators are used in the reduced space branch


and bound strategy to solve the case study solventdesign problem. We have introduced the idea of split-ting functions that result in a smaller number of branch-ing nodes. For example in the solvent design case study,instead of branching on all 60 binary variables (thatparticipate in nonlinear terms), we only need to branchon four splitting functions.

We are continuing to address other issues. In thisregard, more complete models for molecular structurethat account for connectivity information (i.e. howdifferent structures are connected to each other, Churi& Achenie, 1996) are being considered. In additionother property requirements on viscosity, surface ten-sion, ozone depletion potential, etc. are beingconsidered.

In the proposed global optimization algorithm weonly need to solve a sequence of MILPs. Therefore weare investigating methods for solving MILPs more effi-ciently. It is interesting to note that, in our limitedexperience with the proposed algorithm, the computa-tional effort, measured in terms of the number ofiterations, grows linearly as the problem size increases.However, as the problem size increases Ip–solve is notvery robust. As a result, we are currently replacingIp–solve by more computationally efficient MILPsolvers such as OSL. This modification is not expectedto reduce the number of iterations. Instead, it is ex-pected to reduce the computational effort per iteration.

In this paper, only pure component product design isconsidered. Mixture design problems are often posed asMINLP problems. This algorithm can be used forsolving nonlinear and nonconvex mixture design prob-lem as well. Incorporation of model uncertainties is alsobeing considered.

Appendix A. Multilevel approach to construction ofunderestimator

McCormick (1976) suggested a method for construc-tion of convex underestimators and concave overesti-mators for functions that are the product of twounivariate functions, f(x)g(y). Extension for productsof univariate functions was developed by Maranas andFloudas (1995). In this paper, we discuss another ap-proach for construction of underestimator andoverestimator.

The first step involves representation of a complexfunction in terms of simpler one-dimension functions.The only arithmetic operations relating these functionsto the original function are addition, subtraction,square, and square root. Such a representation has atree structure as shown in Fig. 6. The original functionrepresented as 8o

1, corresponds to the root node (zerothlevel). The child nodes emanating from the root nodecorrespond to the first level functions, represented as8 i

(1)(i th function at the first level). The next level corre-sponds to the functions at the second level (8 i

(2)) and soon. As shown in Fig. 6, the number of nodes at kthlevel is nk. The functions at each level are repeatedlydecomposed till the functions at the lowest level childnodes are linear function of the search variables. Hencethis multilevel scheme attempts to represent the originalfunction as a tree function. A mathematical representa-tion of tree function is shown in Eq. (A1).

8 ik= %

nk+1

j=1

mj(8 jk+1)tj (A1)

where mj is the coefficient for the jth child function8 (K+1), and there are m such child functions. Weconsider the case when the power tj for the jth functiontakes the value 0.5, 1, or 2. For a linear group contribu-tion model, at the lowest level in the tree structure, mj

corresponds to the group contribution parameter uj andtj is 1.

We can obtain linear estimators of the kth levelfunction 8 (k) in terms of the function at the level(k+1), 8 i

(k+1). There are standard techniques for con-structing linear underestimators for separable functionsthat can be represented as Eq. (A1). Only the functionsfor which tj equals 2 (square) or 0.5 (square root) needto be linearized. Depending on the sign of the coeffi-cient, mj corresponding to the jth function at the (k+1)th level, the function 8 i

(k) can be concave or convexFig. 6. A multilevel representation of a nonlinear function as a treefunction.


Fig. 7. A multilevel representation of the function f1o=x0x1x2x3, as a tree function.

with respect to the functions, 8 i(k+1). For concave

functions the linear underestimator for region Sik has

the form

L [ f(8 ik);Si

k]= f(8 ik)+

[ f(8 ik)− f(8 i

k)]

8 ik−8 i

k(8 i

k−8 ik) (A2)

where Sik={c i

k : c ik5c i

k5c ik}. c( and c(( correspond

to the minimum and maximum values for ci.For convex functions, the linear underestimator in

region Sik has the form

L [ f(8 ik);Sk]= f %(8̂ i

k)(8 ik−8̂ i

k)+ f(8̂ ik) (A3)

where 8ki is the mid point of 8k

i in the region Sk, f %(8ki )

is the derivative of function f(8ki ) at the midpoint.

If the bounds on the search variables are known,then the bounds or range on 8 i

o, 8 i1, 8 i

2 … 8 iN can be

computed. This results in a bound propagation atdifferent levels of the functions beginning from thelowest level. For example, let us consider the followingfunction

f10=x0x1x2x3

Let us introduce two functions f 11=x0x1+x2x3 and

f 12=x0x1−x2x3. Now, using the following identity

ab=14

(a+b)2−14

(a−b)2

we can represent the original function as f10=1/

4( f11)2−1/4( f2

1)2. At this level let us introduce thefunctions corresponding to the second level f1

2=x0+x1,f 2

2=x0−x1, f32=x2+x3 and f4

2=x2−x3. Usingthese second level functions, we can represent the firstlevel functions f1

1 and f21 as

f11=

14

( f12)2+

14

( f22)2+

14

( f32)2+

14

( f42)2

and

f21=

14

( f12)2+

14

( f22)2−

14

( f32)2−

14

( f42)2.

Here in three levels (see Fig. 7) we have reduced theoriginal function and represented it as a superpositionof univariate functions. The only operations are plus,minus and the square. Given the bounds on x0, x1, x2

and x3, bounds on the second level functions f12, f2

2, f32,

and f42 can be computed via interval arithmetic opera-

tors (Moore, 1966). Similarly, the bounds on the firstlevel functions f1

1 and f21 can be computed. Thus the

estimation of bounds in the tree function starts fromthe bottom level and goes up. Care must be taken tocompute the true range, as the interval operations mayresult in overestimation of the bounds.

The underestimators are constructed in the reverseorder. First the original function ( f0) is linearized withrespect to the set of first level function (here f1

1 end f21).

Next, the resulting linear expression of f11 and f2

1 isconsidered. Underestimators are constructed for eachterm of this expression with respect to the second levelfunctions f1

2, f22, f3

2, and f42. At this stage the expression

for the underestimator is a linear function of f12, f2

2, f32

and f42. In this way we obtain the linear relations which

connect linear underestimators of the kth level with thelinear underestimator for the (k+1)th level (k=0, 1, … , N−1). These second level functions are them-selves linear functions of the original variable x0 x1, x2,and x3. At the next stage, we can obtain underestimatorof all the levels as linear functions of x0 x1, x2, and x3.


In turn we have computed the linear underestimator ofthe original function f1

0. This operation must be per-formed from the bottom level to the top level.

In a similar fashion linear overestimators can beconstructed. For this only Eqs. (A2) and (A3) has to bemodified. If the function is concave then the right handside of Eq. (A3) describes the functional form of theoverestimator. If the function is convex then the righthand side of Eq. (A2) describes the functional form ofthe overestimator. The remaining steps are same asthose for construction of underestimators. Thismethodology has also been demonstrated via construc-tion of linear underestimator for the solubility parame-ter constraint in Section 4.

In contrast to the McCormick’s approach, in ourapproach the dimensionality of the lower and upperbound problem is no larger than the dimensionality ofthe original problem PORG. However, there are in-stances where the number of branching variables actu-ally decreases. This translates to smaller number ofpartitions and therefore a smaller number of iterations.

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