temperature collocation algorithm for fast and robust distillation design

Temperature Collocation Algorithm for Fast and Robust DistillationDesign

Libin Zhang and Andreas A. Linninger*

Laboratory for Product and Process Design, Department of Chemical Engineering,University of Illinois at Chicago, Chicago, Illinois 60607

In this paper, we describe a new minimum bubble-point distance (MIDI) algorithm for assessingthe feasibility of a desired distillation specification. The algorithm computes the rectifying andstripping temperature profiles by temperature collocation on finite elements with orthogonalpolynomials. We discovered the beneficial use of a dimensionless equilibrium tray temperatureas an independent variable. This novel choice is bounded between 0 and 1, improves thenumerical quality of the design problem formulation, and is well-behaved even in the vicinity ofpinch regions. It also employs the fixed points of column sections as collocation points. Adaptive-element boundary placement at saddle temperatures can effectively overcome problems withnumerical instability near pinch regions. Extensions of our MIDI algorithm to the calculationof the minimum and maximum refluxes are also introduced. We show the application of thisalgorithm in the separation of quaternary mixtures and provide an outlook of the methodologyfor optimal column sequencing. Cases studies demonstrate the algorithm’s robustness andreliability.

1. Introduction

Distillative separation is among the least expensivemethods for separating mixtures that exhibit suitablevolatility differences.1 Hence, continuous distillationcolumns, along with their optimal operation and heatintegration, constitute a major engineering activity inrefineries and bulk commodity manufacturing. Thecommon practice for distillation design often involvesnumerous trial-and-error experiments by means ofstate-of-the-art process flowsheet simulators. However,this time-consuming practice does not guarantee theproduction of successful designs; it might not provideany information about the feasibility of a particularspecification in cases when the efforts do not converge.When the design-by-simulation approach is used, in-feasibilities are often discovered only after extensivesimulation studies.2

The classical Underwood method for column profilecomputation is restricted to mixtures of constant rela-tive volatility.3,4 Doherty and co-workers5,6 introducedthe boundary value method (BVM), which examines theintersection of rectifying and stripping profiles graphi-cally. However, the BVM is inconvenient for mixtureswith more than four components because of the lack ofgraphical representations for composition trajectoriesin higher dimensions. Julka and Doherty7,8 extendedthis methodology to multicomponent systems, employ-ing a geometric theory based on topological concepts.They demonstrated that the feed point and C-1 pinchpoints lie in one hyperplane satisfying a zero-volumeformula at minimum reflux. The rectification body (RB)method9 employed pinch points of the top and bottomsections to approximate the space of reachable columnprofiles. If the rectification bodies for the rectifying andstripping section just touch each other, the authorsconjectured that the minimum reflux ratio is obtained.However, these methods are only accurate for high-

purity splits in which the composition profiles approachsaddle pinches. They break down for sloppy separationsin which the intersection of hyperspaces spanned by thepinch point is a necessary but not a sufficient feasibilitycondition. In general (sloppy) specifications, only columnprofiles unambiguously verify the feasibility of a design.

Existing approaches avoid calculating the compositionprofile rigorously because of the huge computationaleffort. This holds particularly true when several col-umns are being designed and optimized simultaneously(i.e., in optimimum column sequencing). Orthogonalcollocation on finite elements (OCFE) has been shownto reduce the problem size substantially, without sig-nificant loss in accuracy compared to the full-order tray-by-tray model. Several researchers have explored col-location techniques for distillation column simulation.11-20

An OCFE method was also deployed successfully forminimum-stage-number designs.21-23 Huss and West-erberg24,25 proposed advantageous variable transforma-tions to ensure high accuracy close to the columns’ endsections when using collocation.

Despite the attention devoted to continuous distilla-tion simulations, less work has been aimed at develop-ing algorithms to determine whether a given specifica-tion, sloppy or sharp, is feasible or not. In this research,we propose a minimum bubble-point distance (MIDI)algorithm for ascertaining the feasibility of any arbi-trary design specification for simple continuous distil-lation columns. Our research objective targets thedevelopment of an efficient and reliable computationalalgorithm for establishing the feasibility or infeasibilityof specifications for sharp or sloppy separations of anynumber of species. The availability of a fast and globallyconvergent feasibility test would introduce an essentialelement for automatic computer-aided distillative sepa-ration synthesis.26 This work constitutes an importantintermediate milestone toward that long-term objective.

Outline. A theoretical basis for the novel robustfeasibility test algorithm is developed in section 2.Fundamental concepts pertinent to the new approach

* To whom correspondence should be addressed. Tel.: (312)996-2581. Fax: (312) 996-0808. E-mail: [email protected].

3163Ind. Eng. Chem. Res. 2004, 43, 3163-3182

10.1021/ie034223k CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 05/15/2004

such as the attainable temperature window (ATW) andthe bubble-point distance (BPD) are introduced. Atemperature finite-element collocation method capableof overcoming stationary points in the compositionprofile is presented. The methodology section alsoincludes numerical results and implementation issues.Section 3 demonstrates the robustness and performancecharacteristics of our approach for constant relativevolatility mixtures, ideal mixtures, and nonideal mix-tures with three or four components. Section 4 extendsour methodology to ascertain the boundaries of feasibleoperation, i.e., it describes the robust computation ofminimum and maximum reflux ratios. Section 5 ad-dresses problematic design tasks such as the separationof mixtures with tangent pinches and provides sugges-tions toward novel computational solution options forstructural design problems such as optimal columnsequencing. The paper closes with conclusions and anoutlook for future research.

2. MethodologyA performance problem predicting expected distillate

and bottoms compositions given the feed and columnparameters for an existing column always has a solutionthat is readily attainable with commercial flowsheetpackages (e.g., AspenPlus, HYSYS, Pro/II, etc.). On theother hand, the inverse design problem seeking thecolumn operation and parameters to achieve desiredseparation targets might not have a solution even for aconsistent set of design specifications. In this paper, weshall, without loss of generality, assume given feedcompositions and column pressure. A simple columnconfiguration has four design degrees of freedom, typi-cally three product-purity specifications and a desiredreflux value, for example.

Unfortunately, no generally applicable, globally con-vergent algorithm exists for determining the feasibilityof a given design specification. The classical Underwoodmethod3,4 converges only for feasible designs; it is oflimited use in the synthesis of distillation trains becauseof its extremely nonlinear behavior. Other design meth-odologies are restricted in either the number of species27

or the properties of the mixture.28 To the best of ourknowledge, no algorithm is robust enough for structuralflowsheet optimizations, which typically lead to large-scale mixed integer nonlinear mathematical program-ming (MINLP) problems.

Critical numerical difficulties in the design problemstem from singularities in the composition profiles nearstationary pinch and saddle points that naturally occureven in ideal mixtures. To overcome the existing short-comings, we propose a combination of two simple butvery effective concepts: (1) transformation of columnprofiles into the space of dimensionless bubble-pointtemperatures; (2) minimization of a bubble-point dis-tance function between stationary profile nodes in thebubble-point temperature space.

This novel approach reduces the dimensionality of thedesign problem; eliminates singularities encountered inthe tray-by-tray approach; extends to any number ofspecies with customary vapor-liquid equilibrium solu-tion models, including constant-relative-volatility, ideal,and nonideal mixtures; and applies to both sharp andsloppy splits. In effect, the column design problembecomes more tractable from a computational point ofview. Before the introduction of our new methodology,a discussion of reachable product compositions is inorder.

2.1. Reachable Compositions and Their Equilib-rium Temperatures. Pinch points delineate the ex-treme points of all possible column profiles for a givendesign specification. For a known distillate compositionxD for a c-component separation, c stationary points arefound without performing tray-by-tray computations byenforcing the pinch equations given by eqs 1 and 2, i.e.,simultaneous compound balances and thermodynamicsequilibrium.29

Figure 1a illustrates that the distillate, d, and stablenode, γ, span all possible rectifying profiles for a givenspecification. Rectifying composition profiles start at thedistillate and terminate in the stationary pinch point.For high-purity separations, the profiles approach asaddle point dividing the composition profile into twobranches that can be reached only after an infinitenumber of equilibrium trays have been traversed.5 Eachpinch point is also associated with a specific bubble-

Figure 1. Fixed points in a ternary mixture and residual errorsof eqs 1 and 2.

Pinch equation for the rectifying section

rxi - (r + 1)xiKi + xD,i ) 0 i ) 1, ..., C (1)

∑xiKi - 1 ) 0 (2)

3164 Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004

point temperature as illustrated in Figure 1b. Conse-quently, realizable rectifying sections exhibit tray equi-librium temperatures within the interval delimited bythe temperatures of the distillate, TD, and the stationarypinch point, TP1. Correspondingly, the bubble points ofthe bottoms, TB, and the stationary stripping pinch, TP2,bracket the temperature range for all possible strippingsections. For a simple column configuration (i.e., withno side trays or multiple feeds), the stripping andrectifying profiles must intersect at the liquid traycomposition of the feed plate. Hence, the temperatureinterval between min(TP1, TB) and max(TP2, TD) consti-tutes the attainable temperature window (ATW) for thegiven column design specification. The ATW marks allpossible equilibrium tray temperatures that can bereached by the column sections, as illustrated in Figure2 (i.e., boiling points of mixtures on an equilibrium tray).If the ATW associated with a design specification isempty (i.e., TP1′ < TP2′,), then one can conclude un-equivocally that the specification is infeasible (e.g.,Figure 2b). By comparing the temperature ranges in the

stripping and rectifying sections, we can already excludemany design specifications without actually performingtray-by-tray computations as expressed in remark 1.This property can also be very valuable in assessing theseparability of mixtures with azerotropes.

Remark 1. Attainable Temperature Windows. Ifthe attainable temperature window (ATW) of the rectify-ing and stripping section is empty, the given designspecification is infeasible.

2.2. Bubble-Point Distance (BPD). In cases ofoverlapping ATWs, more detailed quantitative analysisis required. Numbers of trays or column heights areunsuitable independent variables in numerical compu-tations because of their indefinite value range, namely,]0, ∞[. We propose an affine mapping of the columnheight into a dimensionless bubble-point temperature,Θ. The mapping establishes a bijective relationshipbetween height and bubble-point temperature. We alsoconvert the tray-by-tray difference equations into ordi-nary differential equations as proposed by Doherty.5 Thecomposition trajectories expressed as functions of thenew independent variable, Θ, do not approach infinityhampering the traditional tray-by-tray methods. Usingthe dimensionless temperature instead of the columnheight, we arrive at a system of differential-algebraicequations. Equations 3 determine the composition pro-files, xi(Θ), as a function of the equilibrium constant,Ki; the distillate specification, xD,i; and the reflux, r. Thenew temperature transformation reduces the nonlin-earity of the profile equations significantly, because theuse of temperature as an independent variable makestemperature-dependent expressions explicit. Equation4 represents the stripping profile in term of the bottomscomposition, xB,i, and the reboiler ratio, s. The detailedderivations of continuous rectifying and stripping pro-files as well as the total derivative of the equilibriumconstant, dKj/dΘ, in terms of temperature and composi-tion for a general nonideal mixture model are providedin the appendices.

The bubble-point distance (BPD) is defined as theEuclidean distance between two equilibrium composi-tions belonging to the stripping and rectifying profileswith the same bubble-point temperature. Figure 3depicts two instances of bubble-point distances, dA anddB, for two discrete bubble-point temperatures (TA andTB, respectively), as well as the traces of the bubble-point surfaces in a nonideal ternary mixture. For a

Figure 2. Attainable temperature windows in (top) compositionand (bottom) temperature space.

Continuous rectifying profile equations

dxiR

dΘ) -

(xiR -

r + 1

ryi

R +1

rxD,i)

∑j)1

C [(xjR -

r + 1

ryj

R +1

rxD,j)Kj]

∑j)1

C (dKj

dΘxj

R) )

FR[xR(Θ),Θ,p] i ) 1, ..., C (3)

Continuous stripping profile equations

dxiS

dΘ) -

( s

s + 1yi

S - xiS +

1

s + 1xB,i)

∑j)1

C [( s

s + 1yj

S - xjS +

1

s + 1xB,j)Kj]

∑j)1

C (dKj

dΘxj

S) )

FS[xS(Θ),Θ,p] i ) 1, ..., C (4)

Ind. Eng. Chem. Res., Vol. 43, No. 12, 2004 3165

design specification to be feasible, there must exist apair of compositions with BPD ) 0. If the globallyminimum BPD for a specified separation target isgreater than zero, then this target is guaranteedinfeasible. Figure 4 contrasts a feasible solution and aninfeasible design of a ternary separation problem. Theinfeasible specification has a minimum BPD that isgreater than zero.

Our feasibility test also exploits remark 2 for ternarymixtures:

Remark 2. Shortcut Infeasibility. For a feasibleternary column to exist, it is necessary and sufficient thatthe vectors P1-Xr(TP2) and P2-Xs(TP1) intersect, whereXr(TP2) is the point on the rectifying profile correspondingto the temperature of the stable stripping pinch P2 and

Xs(TP1) is the point on the stripping profile correspondingto the temperature of the stable rectifying pinch P1.

Intuitively, we can then restate remark 2 as follows:for column profiles to be feasible, the linear approxima-tion of the rectifying and stripping profiles takenbetween the two pinch-point temperatures must inter-sect as depicted in Figure 5. The validity of remark 2becomes clearer when one considers strict temperaturemonotonicity of the composition profiles; this propertyis observed in industrially relevant columns. Exceptionsto this rule are possible but require special heat effectssuch as can occur in reactive or extractive distillations.These situations are not the objective of this work.Hence, profiles are assumed not to intersect the samebubble-point surface twice even in highly nonidealsolutions. For mixtures of four or more species, thegeneralization of remark 2 to hypersurfaces is neces-sary, but not sufficient.

BPD for More than Four Species. The designspecifications for the separation of a quaternary mixturedo not fully determine the products. In this case, thefeasibility test and determination of the remaining

Figure 3. Bubble-point distance and boiling-point trajectories ina nonideal mixture.

Figure 4. Bubble-point distance function (BPD) for a feasibledesign (+, lower curve) and an infeasible design (*, upper curve)in a ternary separation.

Figure 5. Illustration of shortcut feasibility for a ternary mixture.


compositional degrees of freedom must be performedsimultaneously. Figure 6 depicts a feasible solution fora quaternary problem in a two-dimensional problemspace consisting of the bubble-point temperature andremaining compositional degree of freedom, i.e., distil-late composition, xD,3. Again, the minimum BPD is zerofor a feasible specification. It is worth noting the smoothBPD functions in both cases; this property of the bubble-point distance function is essential for numerical algo-rithms requiring gradient information. Furthermore,the BPD surface, as well as all projections of the BPDfor a given xD,3 value exhibits only one minimum.

2.3. Minimum Bubble-Point Distance (MIDI)Algorithm. Feasible separations targets lead to inter-secting stripping and rectifying profiles, whereas theseprofiles miss each other in infeasible designs. Enforcingthe equality of the feed tray composition in the top andbottom sections is not a good basis for a numericalalgorithm, as it cannot be converged for infeasiblespecifications. To ensure numerical convergence for anysituation, we propose to search for the globally mini-mum BPD instead.

The information flow for the new feasibility test basedon the minimum bubble-point distance (MIDI algo-rithm) is summarized in Table 1. The first step searchesfor the stable pinch points for the stripping and rectify-ing profile. Steps 2 and 3 examines the shortcutinfeasibility (remark 2), which involves the liquid traycompositions corresponding to the bubble points of thestationary nodes X(Θ1) and X(Θ2) of the opposite columnsection. For ternary mixtures, the shortcut infeasibilityof remark 2 can detect many infeasible specificationswithout the need for detailed profile computations. Step4 entails a nonlinear optimization problem seeking theglobal minimum distance between the rectifying and

stripping profiles as formulated in eqs 5-9 below. Thisproblem can also be considered a “dynamic” optimizationproblem in which the dimensionless bubble-point tem-perature, Θ, assumes the role of the independentvariable “time”. The equality constraints of eq 6 cor-respond to the rectifying and stripping profiles of eqs 3and 4. Equality 7 represents the pinch equations yield-ing the dimensionless temperature boundary for thisdynamic optimization problem.30 Inequality constraint8 ensures that the dimensionless temperature, Θ,remains within the bounds of the attainable tempera-ture window (ATW). This mathematical program alwayshas a solution, even for infeasible separation targets.The final output of the feasibility test returns the actualcolumn profile or impossibility of realizing the desiredseparation with simple distillation within the precisionlimits of the algorithm.

2.4. Finite-Element Approximation of ColumnProfiles. As stated above, the mathematical programof eqs 5-9 constitutes a dynamic optimization problemwith differential algebraic (DAE) constraints.31 For itssolution, we approximated the differential equationswith finite temperature elements and collocation onorthogonal polynomials.23,32,33 A detailed discussion oforthogonal collocation on finite elements (OCFE) fordiscretizing differential and partial differential equa-tions can be found in the literature.22,34 This method isalso referred to as spline collocation.35

The continuous differential equations are convertedinto a set of nonlinear algebraic equations in terms ofthe unknown weights, i.e., unknown compositions. TheOCFE discretization renders algebraic expressions ofthe variable derivatives for each collocation node asdescribed in eq 10. With the help of gradient expressionin eq 10, the differential equations become algebraic asshown in eq 11.

The elements of the variable vector xj,c[i], represent

the unknown compositions for each species, i.e., theequilibrium tray composition of each species c at nodej in element i corresponding to temperature, Θj

[i]. Thefinite elements decompose the temperature range intosegments; the nodes of the orthogonal collocation poly-

Figure 6. Bubble-point distance surface for a feasible separationof a quaternary mixture.

Table 1. Pseudocode for the Feasibility Test (MIDIAlgorithm)

1. Compute the stable pinch points in the rectifying andstripping sections, P1 and P2 (eqs 1 and 2)

2. Compute the points Xs(P1) and Xr(P2) corresponding to thepinch point on the opposite profile.

3. IF the shortcut is infeasible, THEN the column specificationsare infeasible. Stop.

4. Perform global minimization of the BPD using orthogonalcollocation on finite elements.

5. IF min ||BPD|| < ε, THEN column specs are feasible. Stop.ELSE design specifications are infeasible. Stop.

Feasibility Test:Mathematical Problem Formulation

minΘ,p

||BPD|| s.t. (5)

dx(Θ)dΘ

) F[x(Θ),Θ,p] (6)

G[x(Θ),Θ,p] ) 0 (7)

Θl e Θ e Θu (8)

0 e x(Θ) e 1 (9)

Discretization of derivatives of node j in element [i]

dxdΘ|Θ)Θ

j[i]

) AΘj[i]x (10)

Transformation of profiles in eqs 6 via point colloca-tion discretization of node j in element [i]

AΘj[i]x ) F(x,Θ,p) (11)


nomials further divide each element according to apredetermined number of roots. Within each element,the variable derivatives are expressed in terms of aderivative matrix, A, of the collocation polynomials andthe unknown liquid tray compositions, x, each one cor-responding to the specific node temperature. Zero-ordercontinuity is ensured from element to element. Thedetailed expression resulting from the OCFE discreti-zation on dimensionless temperature using Lagrangianand Legendre base polynomials for the rectifying andstripping profiles are given in eqs 12-17. The scalarentries, λk(Θj

[i]), of the derivative matrix, A, in eq 12represent the first derivative of the kth collocationpolynomial, lk, evaluated at node j in element [i], i.e.,[∂lk(Θj

[i])]/∂Θ. The elements of the right-hand-side vec-tor, gj,c

[i], involve the expressions and bc, where êj[i]

represents the right-hand sides of the profile equations,i.e., eq 14 for the rectifying section and eq 16 for thestripping profile. The term bc accounts for the productpurity, i.e., eqs 15 and 17. Substitution of the approx-imating functions into eqs 3 and 4 yields a correspond-ing set of residual functions that can be solved to obtainthe desired liquid concentration trajectories in eachcolumn section. It is important to stress that the solv-ability of the resulting design equations is substantiallyenhanced because of the explicit expressions in dimen-sionless temperature as well as the boundedness of alldependent and independent variables between 0 and 1.

The implementation of the MIDI algorithm is dis-cussed in the next subsection.

2.5. Implementation of the MIDI Algorithm.Pinch Equations. The solution of the pinch equationsfor the computation of all stationary points in step 1can be executed with well-known homotopy continuationmethods.8,36 Determining all pinch points in nonidealmixtures is a multivariable algebraic problem becauseof the dependence of equilibrium constants on composi-tion as well as temperature. Different versions of theMIDI algorithm identify zeros by continuation or inter-val search. The computation of stationary points forideal mixtures simplifies to a one-dimensional zerosearch, because the equilibrium constant depends onlyon the temperature. Bisection and bounded one-dimen-sional Newton methods are successful for ideal mix-tures. For constant-R mixtures, the problem becomes asimple polynomial root finding. This can be addressedby the classical Newton-Horner scheme.37 The MIDIalgorithm also benefits from the fact that unstable nodeslying outside the temperature window spanned by theproduct compositions are not needed.

Choices for Node Placements. The solution ac-curacy benefits from node placement in regions withsteep gradients. It is worthwhile mentioning the ad-vantage of element boundary placement at saddle-pointtemperatures. We observed that the saddle-point tem-perature nearly coincides with the maximum curvatureof the composition profile in most cases. This aspectholds particularly true in sharp-split separations, asdemonstrated in Figure 7. In consequence, it is advis-able to place an element boundary at the saddle-point

Chart 1

gj,c[i] ) êj

[i]xj,c[i] + bcêj

[i] (13)

Right-hand-side expression for the rectifying profile

êj[i] ) -

(1 -r + 1

rKj,c

[i])∑m)1

C [(xj,m[i] -

r + 1

rKj,m

[i] xk,m[i] +

1

rxD,m)Kj,m

[i] ]×

∑m)1

C (dKj,m[i]

dΘxj,m

[i] ) (14)

bc ) 1rxD,c (15)

Right-hand-side expression for the stripping profile

êj[i] ) -

( s

s + 1Kj,c

[i] - 1)∑m)1

C [( s

s + 1Kj,m

[i] xj,m[i] - xj,m

[i] +1

s + 1xB,m)Kj,m

[i] ]×

∑m)1

C (dKj,m[i]

dΘxj,m

[i] ) (16)

bc ) 1s + 1

xB,c (17)


temperature. Numerical problems in the column profilecomputation exhibiting long, flat profile sections can becircumvented by reducing the number of elements.

Initialization. The influence of the initial values onthe convergence and computational effort required tofind an accurate solution was investigated. Three op-tions were explored: (a) Underwood initialization, (b)ODE initialization, and (c) finite-difference initializa-tion. For the Underwood initialization, the equilibriumrelation is approximated by a geometric mean of therelative volatilities at the bottoms and distillate tem-peratures. The initial weights at specified node tem-peratures were extracted by means of the Underwoodmethod using a pseudo-temperature (described in moredetail in section 3). In the ODE initialization, we directlyintegrate the profile with a fourth-order explicit Runge-Kutta method37 under the ideal-mixture assumptionsexpressed in eqs 3 and 4. This initialization techniquealready yields the correct result for ideal mixtures. Thefinite-difference initialization obtains an approximatecomposition at the next node by employing a one-stageexplicit Euler step from the previous node. For allzeotropic mixtures, Underwood initialization is recom-mended, because of its balance between speed andreliability. The ODE initialization proved to be the most

accurate, but it is also most expensive in computationaltime. The finite-difference initialization is fastest, butit often leads to insufficiently accurate profiles, poten-tially causing numerical problems in the subsequentcollocation procedure. This initialization method can beapplied when many elements and nodes are used, sothat the temperature gap is small. The computationalperformance and reliability for these three differentinitialization methods are reported in the Table 2. Wefound the Underwood initialization using the geometric-mean constant relative volatility to exhibit the bestoverall result.

Computation of Minimum Distance. The numer-ical solution of the NLP in eqs 5-9 still poses aformidable challenge because the interesting solutionsare often sought at extreme concentrations. Severalgradient-based and informed-search techniques wereinvestigated for performance and robustness. For thesolution of the nonlinear profile equations, we tested theNewton-Raphson method with Armijo line search,37 theautoscaled Newton solver using dynamic column androw scaling,38 the inexact Newton method based onKrylov subspace iterations,39 the Broyden method37 andthe trust region solver (MINPACK).40 The homotopymethod41 was deployed for the fixed-point equations (eqs1 and 2). For the optimization problem of the minimumBPD, the golden section search, the Newton-Raphsonmethod,37 and the rSQP method,42,43 in combinationwith the Newton-type method for the OCFE, weredeployed in the case studies. We investigated theNewton-type and reduced successive quadratic program-ming (rSQP) methods for the simultaneous solution ofthe dynamic optimization problem as presented in eqs5-9. For ternary mixtures, the direct dynamic program-ming succeeded, but this approach often failed inquaternary problems. The performance results aresummarized in Table 3.

We discovered that the BPD distance function admitsmultiple extrema whenever saddle-point temperaturesfall within the ATW. To locate multiple local minima,MIDI launches multiple searches for each temperaturesubinterval created by saddle-point temperatures withinthe ATW. In the case of multiple local convergence, thesmallest distance is declared to be the “globally” mini-mum BPD. In our experience with design problems internary and quaternary mixtures, MIDI proved reli-able,44 because it warps the design problem into a

Figure 7. Saddle temperature coincident with region of maximumprofile curvature.

Table 2. Initialization Methods Tested for the CaseStudy A

initializationmethod

speed tosolve (s) performance

finite-difference 0.10 occasional convergence failureUnderwood 0.82 always convergentODE 1.26 always convergent

Table 3. Comparison of Performancea of DifferentNumerical Method

example

goldensectionsearch

Newton-Raphsonmethod

homotopycontinuation

method rSQP

ternary mixtureb C C C Cquaternary mixturec C F F Fmin/max reflux ratiod C C C Ccolumn sequencee NA C C C

a C ) converges, F ) fails occasionally. b Ternary mixtureexample from Figure 9. c Quaternary mixture example from Figure12. d Min/max reflux ratio example from Figures 16 and 17.e Column sequence example from Figure 22.


relatively small space in terms of the independentvariables (i.e., Θ, compositional degrees of freedom).Alternatively global search algorithm based on intervalarithmetic could be used.

All solvers were implemented in C or Fortran on aPentium II 450-MHz computer. For industrial users, auser-friendly graphical interface was developed. Userscan input the specifications and select from among allsolution algorithms described above. We also providegraphical interfaces to track the convergence trajectoryof the numerical procedures.45

In a preliminary study, the gradient information inthe optimization problems was obtained by numericaldifferentiation. Gradient information obtained by nu-merical differences can fail in quaternary separationproblems because the gradients are often fairly flatalong the free composition coordinates (e.g., xD,3). Thisfact is quite clear in Figure 6. Inaccurate gradientinformation often leads to near singular matrices,causing the mathematical program to diverge. Theseproblems can be circumvented by using first-ordersensitivity information, which is superior to numericaldifferences, as will be shown in case study E.46

3. Design Applications and Validation

This section demonstrates the application of the newfeasibility test for a wide range of zeotropic mixturesfor both sloppy and sharp splits. For constant-R mix-tures, the MIDI algorithm provides a more robust andfaster alternative to the Underwood method. The MIDIalgorithm extends seamlessly to ideal and nonidealmixtures. The examples described here characterize andvalidate the robustness and performance of the novelfeasibility test.

3.1. Case Study A: Constant Relative VolatilityMixtures. For mixtures with constant relative volatili-ties, the Underwood method can be used to assessfeasible column specifications. Attacking design prob-lems with the Underwood method requires the simul-taneous solution of the Underwood roots, an n-dimen-sional root-finding problem, together with the search forthe numbers of trays in the rectifying and strippingsection for profile intersection.3,4 For more than fourspecies with sharp desired product splits, this problemis often numerically unstable and has no solution forinfeasible column specifications. As a consequence,infeasibility, divergence due to sharp split specifications,and insufficiently accurate initialization are numericallyindistinguishable. Hence, the Underwood method doesnot constitute a robust feasibility test for computationalpurposes in separation synthesis.

As a solution, we advocate the modified MIDI algo-rithm. The MIDI algorithm requires the introductionof an artificial bubble-point temperature, θ′, for mixtureswith constant relative volatility by eq 21. For convertingthe constant-R equilibrium condition into the formatrequired by the MIDI algorithm, we define the pure-component vapor pressure in relation to an artificialbubble-point temperature, θ′. The pseudo-vapor pres-sure, Ψi, which is linear in the bubble-point temperatureθ′, as expressed in eq 20, is compatible with theequilibrium condition for constant relative volatility ineq 18, as well as with the ideal solution model in eq 19.In effect, one obtains linear pressure-temperatureexpressions with a temperature scale chosen betweenthe boiling points of the lightest and heaviest com-

pounds as a reference. With this modification, the MIDIfeasibility test is adapted for constant relative volatilitymixtures.

A computation experiment documented in Figure 8applied the feasibility test to 10 000 randomly chosencolumn design problems for a ternary mixture withconstant relative volatility. All design problems con-verged within a few iterations; no convergence failurewas reported. The entire execution time amounted to39.4 CPU s on a Pentium II 450-MHz computer for10 000 design computations! Figure 8 also visualizes allfeasible designs in a composition triangle and lists thenumber of infeasible designs. More design experimentsfor mixture with R ranging from 1 to 1000 are docu-mented elsewhere.44 The algorithm was found to workrobustly and reliably in all constant-R mixtures. Fail-ures were observed only when all species had extremelyclose volatilities (e.g., RA ) 1.1, RB ) 1.05, RC ) 1). Theresults of the ternary case matched the analyticalresults obtained by Doherty and Malone.29 These resultsprovide evidence of the robustness and speed of the newapproach.

3.2. Case Study B: Ideal Mixtures. The MIDIalgorithm for mixtures with ideal vapor-liquid behaviorwas formulated using eqs 11-17. The equilibriumconstant, Ki, is a function of the pure-component vaporpressure only, which are conveniently modeled by theAntoine equation.47 It is recommended that analyticalderivatives available at specialized high-performancephysical property servers48 be used.

The simulation results of Figure 9 depict productcompositions and corresponding column profiles ob-tained by the new feasibility test. The rectifying andstripping profiles in Figure 9a corresponding to sloppyseparation targets run far from the saddle point asexpected. In sharp separations, as depicted in Figure9b, the profiles approach the saddle pinch requiring ahuge number of trays but only a few temperature nodes.These results demonstrate the speed and reliability ofthe feasibility test based on the novel idea of tempera-ture collocation for ideal mixtures in sharp and sloppysplits.

Figure 10 reports typical results of a simulationexperiment with 10 000 randomly chosen column designproblems. Comparison of the feasible designs with thefeasibility boundaries established by Doherty and Ma-lone29 shows good agreement. More computationalresults show stable and rapid performance for a widerange of vapor pressure differences. Numerical failuresoccurred when attempts were made to design separa-tions for extremely narrow boilers such as isomers.

yC )RCxC

∑i)1

C

Rixi

(18)

yC )ψC

PxC (19)

ψC ) PR1

[(R1 - 1)θ′ + 1] (20)

1

R1

[(R1 - 1)θ′ + 1] ∑i)1

C

Rixi - RC ) 0 (21)


However, for narrow boilers, distillation is, in all likeli-hood, not an economical process option.

3.3. Case Study C: Nonideal Zeotropic Mixtures.The MIDI algorithm extends easily to nonideal mix-tures. The vapor-liquid data models employed theAntoine equation for the vapor pressure and the two-parameter Wilson equation for liquid-phase activities.47

Analytical derivatives of the activity coefficients withrespect to temperature were computed by means of ahigh-speed thermodynamic property server.48 The col-location equations for nonideal mixtures are analogousto those presented for ideal solutions, except for expres-sions modeling the equilibrium constant, Ki. Theyintroduce the species activities and their temperaturedependence in the profile equations as quantified in eq22.

When eq 22 is used to include the compositiondependence of the equilibrium constant, all equationspresented for ideal mixtures extend to nonideal mix-tures. Figure 11 reports all feasible specifications uponsolution of 10 000 feasibility tests for the separation ofthe nonideal mixture of acetaldehyde, methanol, andwater. The execution time of 1000 s to perform 10 000random nonideal column design problems is still rea-sonably fast. Extensive design problems for the separa-tion of ideal and nonideal solutions at different feedcompositions and a wide range of volatilities are dis-cussed elsewhere. 44

Whereas feasible regions can be determined analyti-cally for ternary mixture by computations based onpinch regions,27,49 extensions to higher dimensions donot exist. The MIDI algorithm does not suffer from thislimitation.

3.4. Case Study D: Quaternary Mixtures. So far,we have shown the utility of the feasibility test forternary mixtures. In this section, the method is ex-tended to four components, a problem for which analyticmethods for determining feasible regions do not exist.In mixtures with four or more components, the designproblem involves the search for one or more composi-tional degrees of freedom (DOF) of the distillate orbottom product.7 The problem becomes a two-dimen-sional search in the quaternary case, e.g., temperatureand one compositional degree of freedom. The math-ematical formulation is identical to the problem in eq5, when one compositional DOF is considered as anadditional unknown parameter p, e.g., the distillatepurity of a species, xD,3, without loss of generality.

The problem is again a dynamic optimization problemin which temperature plays the role of “time”.50 In near-ideal quaternary mixtures, there are four fixed pointswith two unstable nodes in each column section. Thebubble-point temperatures corresponding to the twosaddles are chosen as element boundaries. Figure 12displays typical results for quaternary separation prob-lems. These diagrams are designed to document theapplicability of the novel approach to quaternary mix-ture of constant R, ideal mixtures and nonideal mix-tures. Other experiments cover a variety of problemsincluding different feeds, direct and indirect splits, anddifferent production specifications. Exhaustive simula-tion results for a variety of nonideal mixtures as wellas a description of the software providing an easy-to-use interface for industrial users are documented else-where.44

The previous sections have demonstrated the suit-ability of the MIDI algorithm as a reliable and fastfeasibility test for ternary and quaternary mixtures. Thenext section extends the proposed method for thecomputation of the feasible operational range, i.e.,minimum and maximum reflux.

Figure 8. Feasible designs (right) filtered from 10 000 random specifications (left) (R1 ) 6.35, R2 ) 2.47, R3 ) 1).

dKi

dΘ)

1

P(dpis

dΘγi + pi

s(∑j)1

C ∂γi

∂xj

‚dxj

dΘ+

∂γi

∂Θ)) (22)


4. Feasible Range of Column Operation:Minimum and Maximum Reflux

During the early stages of distillation design, it isoften useful to ascertain the range of refluxes capableof realizing desired product purities. Existing graphicalmethods for determining the minimum reflux arelimited to binary and ternary mixtures.51-53 Shortcutmethods apply only to constant relative volatility mix-tures3,4,54,55 and/or sharp split assumptions.10 In general,there exist definite lower and upper reflux bounds forfeasible column operation. It is possible to relate theproblem of feasible refluxes to the feasibility testdiscussed previously, when considering three productspecifications as given and the reflux ratio as an openvariable. Then, the problem can be restated as thesearch for the interval [rmin, rmax] for which the feasibil-ity test is positive.

4.1. Minimum Reflux Ratio. The compact math-ematical formulation of our new feasibility test allowsfor the accurate computation of the minimum feasiblereflux ratios for desired product purities. A new methodbased on the feasibility test for minimum reflux ratiosmakes use of remark 3:

Remark 3. Minimum Reflux Ratio, rmin. A refluxspecification leading to a zero bubble-point distance(BPD) between the rectifying and stripping profiles atthe stationary pinch point is the minimum.

Figure 9. Feasible composition profiles of an ideal mixture: (a)sloppy split specification, (b) sharp split specification.

Figure 10. Experiments with 10 000 random designs for assess-ing the feasibility of separating an ideal mixture of pentane,hexane, and heptane.

Figure 11. Experiments with 10 000 random designs for assess-ing the feasibility of separating a nonideal mixture of acetaldehyde,methanol, and water.


For a ternary mixture, three possible pinch scenariosare depicted in Figure 13. The desired value correspondsto the smallest of the three possible cases.

We propose a two-level optimization algorithm forcalculating the minimum reflux ratio, as shown inFigure 14. The inner loop executes the feasibility testusing the MIDI algorithm as discussed in the previoussections. The outer loop modifies the reflux ratio until

the limit of feasible operation is identified within desiredaccuracy. A modified golden section method was em-ployed to endow the algorithm with robustness.37 Figure15 depicts the information flowsheet of the outer-loopoptimizer for updating the reflux ratio. We bracket thereflux ratio between 0 and 1 using the dimensionlessreflux F ) r/(r + 1). The initial guesses are typicallygiven at 0.01, 0.44, 0.66, and 0.999 as the starting

Figure 12. Feasible specification of quaternary mixtures. (a) Constant-R mixture [A(1), B(2), C(3), D(4)] (RA ) 8, RB ) 4, RC ) 2, RD )1. (b) Ideal mixture [pentane (1), hexane (2), heptane (3), octane (4)]. (c) Nonideal mixture [methanol (1), ethanol (2), n-propanol (3),acetic acid (4)].


interval. The search produces a series of contractingbounds until rmin is found. The MIDI algorithm enforces

the feasibility test for each specification in the innerloop. Two branches can be met depending on whethera feasible specification was found in the interval. Branch1 determines whether none of the four values in thecurrent interval is feasible. It compares the signs of thegradients of the minimum distances to determine thenext step according to two possible scenarios. (1a) Whenthe signs of the gradients are the same, the procedureterminates and concludes that the specifications areinfeasible. (1b) If the signs differ, a new search islaunched within the subinterval corresponding to re-fluxes with sign changes as the new upper and lowerboundaries. The range they span is divided consistentwith the golden section rule. Branch 2 sets the smallestfeasible reflux as the upper bound and subdivides theadjacent interval according to the golden cut. Finally,the minimum reflux ratio is obtained when all bound-aries are within a small interval of length ε. rmin is equalto the last upper bound, which is a feasible reflux.

Typical results for two ideal ternary mixtures ofpentane, hexane, and heptane and methanol, ethanol,and n-propanol for sloppy and sharp separations aredepicted in Figure 16. Table 4 contrasts the outcomesof other minimum-reflux algorithms for a variety ofseparation problems to those of our temperature col-location approach. The numerical results of all methodsare almost identical for the sharp-split specification, asexpected. Figure 16a and b shows the correct minimum-reflux profiles obtained with the MIDI algorithm in allcircumstances. These diagrams also explain the devia-tions befalling pinch alignment methods, as the pinchpoints and the feed points for sloppy specifications are

Figure 13. Three different pinch scenarios at minimum refluxof a ternary constant-R mixture.

Figure 14. Overview of the extended MIDI algorithm for comput-ing minimum and maximum reflux ratios.


not collinear at minimum reflux. This shortcoming alsooccurs for pinch alignment in nonideal mixtures, forwhich MIDI is still accurate. We found previous meth-ods to exhibit errors amounting to as much as 20%compared to the correct minimum reflux obtained withthe MIDI algorithm.

4.2. Maximum Reflux Ratio. The maximum refluxis the largest reflux ratio for which a simple columnspecification can be realized. Unless the distillate andbottoms happen to lie on the same residue curve, whichwould be an unusual coincidence, a given separationtarget is realizable only up to a particular reflux limit,rmax. This explanation also makes clear why total refluxdoes not always render the best product splits. A refluxr is at its feasible maximum for a given separationtarget if the rectifying profile goes through the bottomsproduct or if the stripping profile penetrates the oppositeliquid equilibrium point of a tie line through thedistillate composition. This is expressed in remark 4.

Remark 4. Maximum Reflux Ratio, rmax. A refluxleading to a zero bubble-point distance (BPD) betweenthe rectifying and stripping profiles at the bottomsproduct temperature (bubble point) or distillation prod-uct temperature (dew point) is the maximum.

Figure 17 shows the solution to the maximum refluxproblem for a direct or indirect split of the ideal mixturepentane, hexane, and pentane obtained by the MIDIalgorithm. The temperature collocation algorithm of thispaper provides the first computational procedure forcomputing the maximum reflux ratio without resortingto graphical means. We should also note that thedistillate composition is in phase equilibrium with theliquid composition on the top stage. The stripping profilepasses through a liquid composition point in equilibrium

with the desired distillate composition xD, as depictedin Figure 17b.

Figures 18 and 19 document the successful extensionof the new approach to the calculation of the minimumand maximum reflux ratios for a nonideal quaternaryseparation of methanol (1), ethanol (2), n-propanol (3),and acetic acid (4). The activities were modeled with atwo-parameter Wilson solution model with Wohl’s ex-pansion.47 The minimum reflux ratio is reached whenthe minimum BPD equals zero at the stationary node,i.e., the stable node pinch on the opposite column profile(Figures 18a and 19a). For the maximum reflux, thecomposition profile passes through the product (Figures18b and 19b). Although graphical representations arenot available for mixtures with more than four compo-nents, the computational approaches described in thispaper extend to multicomponent problems.

5. Limitations and Future Research Directions

This section points out difficult situations related totangent pinches and describes methods for overcomingthem. Furthermore, we suggest an extension of thenovel methodology for separation synthesis problems.

5.1. Validation against Flowsheet Simulator. Toassess the accuracy of the approach against a com-mercial flowsheet simulator, we validated the resultsof temperature collocation against HYSYS. Figure 20compares the column profiles obtained by the MIDIalgorithm and with the commercial flowsheet simulatorHYSYS. Even though we used identical vapor-liquiddata sets, as expected, differences remain for tworeasons: (a) The OCFE formulation solves continuousprofile equations. It is well-known that this formulation

Figure 15. Detailed information flow diagram for computing the minimum reflux ratio based on the MIDI algorithm.


derived from a first-order profile approximation deviatesslightly from the tray-by-tray computations.5 (b) TheOCFE model does not account for the effects of heat (i.e.,the assumption of constant molar overflow). In light ofthe inherent model differences, the level of disagreementbetween the continuous collocation method and the tray-by-tray model is judged to be acceptable for practicaldesign purposes. More comparison and validation stud-ies are recorded elsewhere.44

5.2. Tangent Pinch. Tangent pinches can occur innonideal mixtures. In a separation problem with tan-gent pinches, the composition profiles exhibit additionalstationary nodes, i.e., tangent pinch nodes. The exist-ence of a tangent pinch can be detected by bifurcationanalysis, a detailed discussion of which is beyond thescope of this paper.56,57 In a separation controlled by atangent pinch, the composition profile terminates at the

tangent pinch point. Such a situation is depicted inFigure 21a, for which the MIDI algorithm accuratelyrenders the shorter profile between the points xD andT. In near-tangent-pinch situations, as depicted inFigure 21b, the MIDI algorithm stably traverses theproblematic column region. Standard tray-by-tray meth-ods would fail because of the requirement for infinitelymany trays, or they would simply produce inaccurateprofiles because of the accumulation of round-off errors.The MIDI algorithm behaves well in actual tangentpinch control. In near-tangent-pinch situations, theMIDI algorithm passes the region with similar robust-ness as shown in the vicinity of saddle points.

It should be noted that the feasibility test only aimsto establish whether a design is realizable. Even whenfeasible, a column operation might be found to beuneconomical. Such questions require economic tradeoffs;

Figure 16. Minimum reflux ratio of an ideal mixture: (a) sloppysplit, (b) sharp split.

Figure 17. Maximum reflux ratio of an ideal mixture: (a) passthe bottom production, (b) pass the distillation liquid composition.


a simple example for the advantageous use of MIDI fora column design optimization problems is presented inthe next subsection.

5.3. Column Sequencing. The speed of the algo-rithm holds high promise for its use in computer-aided

separation synthesis and optimization design problems.With rapid iterations and reliable detections of infea-sibility, it is tempting to solve the problem of optimalcolumn sequences for the achievement of desired prod-ucts in closed form. The reduction in problem size

Table 4. Comparison of Minimum Reflux Ratios (rmin) and Relative Errors (er) Obtained by Different Methods

Underwoodmethod

zero-volumemethod RBM9 BVM6 MIDIsplit

type rmin er (%) rmin er (%) rmin er (%) rmin er (%) rmin er (%)

Constant Relative Volatility Mixturea

sloppy 1.63 2.97 1.22 27.4 1.22 27.4 1.68 1 1.663 0sharp 2.844 0.2 2.851 0.03 2.851 0.03 2.85 0 2.85 0

Ideal Mixtureb

sloppy 0.99 25.6 0.99 25.6 1.33 2.2 1.36 0sharp 1.54 1.9 1.54 1.9 1.54 1.9 1.51 0

Nonideal Mixturec

sloppy 0.64 31 0.64 31 0.93 8.6 1.01 0sharp 0.43 2.3 0.43 2.3 0.43 2.3 0.42 0a Constant-relative-volatility mixture’s specifications from Figure 13a and b. b Ideal mixture’s specifications from Figure 16a and b.

c Nonideal mixture’s specificatiosn from Figure 11.

Figure 18. Limiting refluxes for a direct split of a quaternarymixture.

Figure 19. Limiting refluxes for an indirect split of a nonidealquaternary mixture.


afforded by temperature collocation on finite elementsspeeds convergence; its compactness and scaled dimen-sionless variable space enhance the stability of theoptimization procedure.18

The performance of the new approach within thecolumn optimization problem also benefits from theready availability of sensitivity information offered bythe BPD formulation. First-order sensitivity informationrequires only a simple differentiation of the OCFEmodel equations with respect to desired design vari-ables, e.g., reflux. This is easily accomplished thanksto the compact expressions in terms of the orthogonalpolynomials. Hence, the gradients required for theHessians can be computed with little extra effort andsuperior precision over expensive and inaccurate nu-merical differentiations. In our preliminary studies ofoptimum column sequencing problems, we found a 40%

reduction in CPU time when using sensitivity informa-tion rather than numerical differentiation. Even largergains can be expected in more complex synthesisproblems, i.e., more species and longer separationtrains. MIDI plus sensitivity information also convergedfrom initial values where regular SQP failed.

Case Study E: Column Sequence for MinimumOperating Cost. We studied the optimal sequencingof a pentane-heptane-hexane separation task with theproposed feasibility test. The objective of this optimiza-

Figure 20. Composition profile for an OCFE model and flowsheetsimulator (HYSYS).

Figure 21. Composition profiles showing the presence of atangent pinch.


tion approach was to identify a minimum operating costseparation train for predefined purification targets (i.e.,pentane in P1 ) 99.5%, heptane in P2 ) 96.0%, andhexane in P3 ) 84.5%) such as are often required forhigh-purity pharmaceutical products or solvent recoverytasks. A very simple performance model according tothe marginal cost approach was adopted.58 Accordingly,operating cost were specified as functions of only thevapor rate. The optimum sequencing task consists offinding the configuration with the smallest total vaporrate, Q1 + Q2. Two configurations are possible, namely,a direct sequence and an indirect sequence, as depictedin Figure 22. The solution of the optimization problemoutlined in eqs 23-27 is equivalent to finding theminimum reflux for each of the columns, where BPD1and BPD2 are the bubble-point distances of the first andsecond columns, respectively, between the rectifying andstripping profiles at the corresponding temperatures Θ1and Θ2.

The cheapest sequence for the separation of a 100kmol/h feed mixture employs a direct split in the firstcolumn for the removal of pentane with a minimumreflux ratio of r ) 1.53. In the second binary column,hexane and heptane are separated, leading to a totalvapor rate of 63.71 kmol/h. The best indirect sequencerequires a total vapor rate of 79.3 kmol/h. The optimalcolumn configuration operates at minimum refluxes ascan be inferred from Figure 22. Compared to the

marginal cost method, which is limited to constantrelative volatilities, our method is able to produceresults that extend to ideal and nonideal mixtures.

Conclusions and Significance

In this paper, we propose a novel fast and reliablealgorithm for testing the feasibility of a distillativeseparation target. The nucleus of the new approach isthe dimensionless bubble-point temperature as an inde-pendent variable, instead of the commonly used columnheights or number of trays. The novel MIDI algorithmwas shown herein to be effective for a wide variety ofseparation problems involving both sharp and sloppyspecifications. The temperature collocation techniqueoffers critical advantages over the regular tray-by-traymodels, which can lock up in singular points (i.e., saddleand pinch points) requiring an infinite number of traysto be overcome. The temperature collocation method alsodramatically reduces the problem size, thereby provid-ing a computationally efficient feasibility test. Anotheradvantage of temperature collocation lies in the im-proved numerical scale. Even independent variables aredimensionless, bounded within known limits. Betterproblem scaling was found to lead to superior numericalstability. The advantages of the MIDI algorithm werefully exploited in minimum- and maximum reflux ratiocomputations. Promising opportunities in column se-quencing in which an inexpensive feasibility tests wasneeded in each iteration were briefly demonstrated.

In the future, we will extend the new methodology toazeotropic mixtures and apply the MIDI algorithm tocomplex column configurations. We will explore thepotential of the MIDI algorithm in conjunction withstructural optimization techniques for the synthesis ofdistillative networks with desired performances.

Acknowledgment

Acknowledgment is made to the donors of the Petro-leum Research Fund, administered by the ACS, forpartial support of this research (ACS-PRF 35702-G9).

Figure 22. Direct and indirect sequences and profiles for the minimum-cost sequence for the desired product recovery of pentane, hexane,and heptane.

minΘ1,Θ2,r1,r2

(Q1 + Q2) (23)

s.t.

||BPD1|| e ε (24)

||BPD2|| e ε (25)

Θ1l e Θ1 e Θ1

u (26)

Θ2l e Θ2 e Θ2

u (27)


The authors gratefully acknowledge Professor LorenzT. Biegler from Carnegie Mellon University for provid-ing the source code of his rSQP algorithm.

Nomenclature

C ) number of componentsC1, C2 ) operating costs of cooling waterBPD ) bubble-point distanceK ) number of nodes in each elementKi ) equilibrium constant of component in ) order of polynomialNB ) tray number of the rectifying profileNT ) tray number of the stripping profileNS ) number of elements in each profilepi

s ) vapor pressure of component i, kPaP ) total pressure, kPap ) parameterQ1, Q2 ) vapor rates, kmol/hr ) reflux ratios ) reboiler ratioT ) temperature, Kxi ) liquid mole fraction of component iyi ) vapor mole fraction of component i

Greek Letters

Ri ) relative volatility of component iΘ ) dimensionless temperature, Θ ) (T - Tmin)/(Tmax -

Tmin)θ′ ) pseudo temperatureγi ) liquid activity coefficient of component iψi ) pseudo-vapor pressure of component i, kPa

Subscripts

B ) bottoms productc ) component indexD ) distillation producti, j ) indices for nodes and elementsp ) pinch point or fixed point

Superscripts

l ) lower boundR ) rectifying sectionS ) stripping sectionu ) upper bound

Appendix I

Differential Equation Model with Temperatureas the Independent Variable. The continuous dif-ferential equation models for both the rectifying sectionand the stripping section of the column with columnheight as the independent variable are given in eqs A-1and A-2.5

Equation A-3 is the bubble-point equation with equi-librium constants Kj, and compositions xj.

Total differentiation of eq A-3 with respect to x1 gives

Equation A-4 can be expanded further to yield

Rearrangement and collection of items in eq A-6 pro-duces eq A-7 and an expression for dx1/dT in eq A-8.

The derivatives dxj/dx1 can be eliminated with thecolumn height h to give the desired relationship for dx1/dT as shown in eq A-9.

Inserting eq A-1 into A-9 provides the temperaturedependence of the liquid composition profiles, xi, for allspecies, as desired.

The derivation can be repeated in analogy for thestripping profile. In eq A-10, the differential of theequilibrium constant with respect to temperature, dKi/dT is refined further as explained in Appendix II.

Appendix II

The temperature differential of the equilibrium con-stant, Ki, with respect to T includes the differential

d(∑j)1

C

{Kj[T,x(T)]xj})

dx1

) 0 (A-4)

∑j)1

C {dKj[T,x(T)]

dx1

xj + Kj

dxj

dx1} ) 0 (A-5)

∑j)1

C {dKj[T,x(T)]

dT

dT

dx1

xj + Kj

dxj

dx1} ) 0 (A-6)

∑j)1

C {dKj[T,x(T)]

dT

dT

dx1

xj} ) - ∑j)1

C

Kj

dxj

dx1

(A-7)

dx1

dT) -

∑j)1

C {dKj[T,x(T)]

dTxj}

∑j)1

C

Kj

dxj

dx1

(A-8)

dx1

dT)

-

∑j)1

C {dKj[T,x(T)]

dTxj}

∑j)1

C

Kj

dxj

dh

dh

dx1

) -dx1

dh

∑j)1

C [dKj[T,x(T)]

dTxj}

∑j)1

C

Kj

dxj

dh(A-9)

dxi

dT) - (xi -

r + 1

ryi +

1

rxi,D) ×

∑j)1

C {dKj[T,x(T)]

dTxj}

∑j)1

C [(xj -r + 1

ryj +

1

rxj,D)Kj]

(A-10)

dxi

dh) xi - r + 1

ryi + 1

rxi,D (A-1)

dxi

dh) s

s + 1yi - xi + 1

s + 1xi,B (A-2)

∑j)1

C

{Kj[T,x(T)]xj} ) 1 (A-3)


pressure and differentials of the activity coefficients, asin eq A-12.

The differential of the liquid activity is a function of boththe temperature and the composition for nonidealmixtures. The differential of the activity coefficient withrespect to temperature follows from the chain rule.

Hence, the desired expression for Ki, dependent on bothtemperature, T, and composition, x, follows in eq A-14.

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Ki )pi

sγi[T,x(T)]P

(A-11)

dKi

dT) 1

P(dpis

dTγi + pi

s dγi[T,x(T)]dT ) (A-12)

dγi

dT) ∑

j)1

C ∂γi

∂xj

dxj

dT+

∂γi

∂T(A-13)

dKi

dT)

1

P[dpis

dTγi + pi

s(∑j)1

C ∂γi

∂xj

dxj

dT+

∂γi

∂T)] (A-14)


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Received for review October 30, 2003Revised manuscript received March 18, 2004

Accepted March 22, 2004

IE034223K


temperature collocation algorithm for fast and robust distillation design

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