initialising distillation column models

Computers and Chemical Engineering 23 (2000) 1811–1824

Initialising distillation column models�

Roger Fletcher a, William Morton b,*a Department of Mathematics, Uni6ersity of Dundee, Dundee DDI 4HN, Scotland, UK

b School of Chemical Engineering, Uni6ersity of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, Scotland, UK

Received 9 September 1999; received in revised form 3 January 2000; accepted 3 January 2000

Abstract

Difficulties associated with the optimisation of distillation column models by non-linear programming are considered. Thepaper presents a systematic procedure to enable these difficulties to be overcome and proposes a particular formulation of thedistillation column model. A certain limiting case of the column model is examined, that of infinite reflux or zero feed. Thislimiting case considerably simplifies the model and provides a system of non-linear equations that is readily solved. The solutionof this problem gives useful information about the purity that can be achieved in the general case and the number of plates neededto attain a given level of purity. The limiting problem provides starting values for the solution of the general column and suggestsa homotopy that can be followed if difficulties arise in obtaining convergence. To obtain a stable form of the limiting case requiresthe general column model to be formulated in a certain way, which to our knowledge has not previously been considered. Theideas have been successfully tested on various multi-column flowsheets involving distillation columns with heat integration.© 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Distillation column; Mathematical model; Zero-feed column; Non-linear programming; AMPL

www.elsevier.com/locate/compchemeng

1. Introduction

In this paper we consider problems associated withthe modelling of distillation columns and the optimisa-tion of such models by non-linear programming (NLP).Obtaining convergence of an NLP solver on distillationmodels is an uncertain process. The solver may fail toconverge, or converge to an infeasible solution (that isa solution that does not satisfy the constraints of theproblem) or may converge to a physically unrealisticsolution. This paper aims to present a systematic proce-dure to enable these difficulties to be overcome. It alsoproposes a particular formulation of the distillationcolumn model.

The main idea is to examine a certain limiting case ofthe column model where the feed goes to zero. This‘infinite reflux’ case is shown to considerably simplifythe model and provides a system of non-linear equa-

tions that is readily solved, and whose solution givesuseful information about the purity that can beachieved in the general case and the number of platesneeded to attain a given level of purity. This feature ispotentially useful in cases (e.g. of non-ideal systems)where the analytic solution available from Fenske’sequation which assumes constant relative volatility(King, 1980 p. 426) does not apply.

The limiting problem also provides starting valuesfor the NLP solution of the general column and sug-gests a homotopy that can be followed if difficultiesarise in obtaining convergence. The general columnmodel is formulated in a certain way that provides astable form of the limiting case, and promotes conver-gence from the starting guesses provided by the infinitereflux model.

In Section 2 we describe the formulation which weuse for the mathematical model of a distillationcolumn. This contains some new features devolvingfrom the later part of the paper, although many of theideas are common to other models that have been used.In Section 3 the limiting case is described and devel-oped, based on the solution of a column with zero feed(that is, at infinite reflux). In Section 4 an extra condi-

� A preliminary version of this paper was presented at the DundeeBiennial Conference on Numerical Analysis, June 1999

* Corresponding author. Tel.: +44-131-6504860; fax: +44-131-6506551.

E-mail address: [email protected] (W. Morton)

0098-1354/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 9 8 -1354 (00 )00295 -7

R. Fletcher, W. Morton / Computers and Chemical Engineering 23 (2000) 1811–18241812

tion is determined so that the zero-feed solution be-comes the limiting solution of the general column. Theadvantages of pre-solving this limiting case problem aredescribed and discussed in Sections 4 and 5. Someexperience with these procedures for solving complexmulti-column flowsheets with heat integration is de-scribed, and a case study is presented in Section 6 thatsupports the effectiveness of the technique.

Within the structure imposed by the limiting zero-feed column, there are still various alternative modelformulations that might be used. Section 7 considerssome alternative possibilities and presents evidence sup-porting the particular choice made in Section 2. A tableof notation is provided for the model description inSection 2 onwards.

The experiments with different modelling formula-tions have been carried out using the modelling lan-guage AMPL (Fourer, Gay & Kernighan, 1993). Thisexpedites the modelling very considerably and providesa number of useful structural features for flowsheets.The main advantage however is that it provides auto-matic generation of first and second derivatives of allnon-linear problem functions and hence frees the userfrom this onerous task, whilst ensuring that it is doneaccurately and efficiently (and with due regard to spar-sity). Another advantage is that AMPL is now linked tovarious state of the art NLP solvers, which can beaccessed over the web. An AMPL program that de-scribes our distillation column model is given in theAppendix.

The solver that we have used in our calculations is asequential quadratic programming (SQP) solver, re-ferred to as filterSQP, which users the idea of an NLPfilter (Fletcher & Leyffer, 1997) to promote conver-gence. This method has proved very reliable overall,although some of the QP sub-problems generated bylarge column models have proved to be very ill-condi-tioned, causing difficulties (usually not terminal) for theQP solver.

Before proceeding to describe the column model wedescribe the terminology used in the rest of the paperfor a distillation column. The purpose of a distillationcolumn is to split a given multi-component feed streaminto two separate streams, referred to as the tops andbottoms. The aim is that the tops stream should becomposed of only the more volatile light components,whilst the bottoms stream is composed of only the lessvolatile hea6y components. The so-called light key is theleast volatile light component and the hea6y key is themost volatile heavy component. In practice a totallysharp split between the keys cannot be attained andsome level of impurity in the tops and bottoms streamsmust be accepted (of heavy and light components,respectively). The column consists of a number ofstages or plates, on each of which contact betweenvapour and liquid takes place, promoting the transfer

of light components to the vapour phase towards thetop and heavies to the liquid towards the bottom.

The feed can be separated into its constituent compo-nents by linking a number of distillation columns, andchoosing a different light and heavy key for eachcolumn, as illustrated for example by Fraga and McK-innon (1994).

We only consider the steady state solution of thecolumn, and not the dynamics associated with startingup the column or the effect of fluctuations in the feedcomposition. We assume that equilibrium is reached oneach plate, an assumption which for real columns re-quires modification by using a plate efficiency or a masstransfer based model. The systems studied so far haveassumed ideal vapour–liquid equilibrium (activity co-efficients equal to one) but we expect that the ideaspresented here will also simplify the optimisation ofcolumn models involving non-ideally mixed compo-nents, which we propose to study in future work.

2. A column model

Mathematical models for distillation columns can beclassified according to whether they consider the detailsof composition, temperature and flow on each plate(the so-called ‘rigorous’ models) or whether theyprovide an overall description of the column usingfewer variables, based on some kind of interpolationbetween behaviours in the limiting cases of total refluxand minimum reflux (King, 1980 p. 428).

In this paper we focus on an idealised, rigorouscolumn model with N+2 equilibrium stages. Theseconsist of N plates together with a condenser andreboiler. Our model assumes the use of a kettle-typereboiler (Sinnott, 1996 p. 686). The plates are numberedas i=1, 2, ... , N with plate 1 being at the bottom of thecolumn and plate N at the top. Quantities associatedwith the reboiler are indexed by i=0 and with thecondenser by i=N+1. The feed to the column isdirected on to plate S and is composed of a mixture ofm components. The components are numbered as j=1, 2, ... , m in order of decreasing volatility, component1 being the most volatile component, and so on. Themodel assumes that the number of stages (N) and thefeed stage (S) are both fixed. The column is shown inFig. 1.

Heat input to the reboiler (at a rate Qr) vapourisessome of the material which passes up the column asvapour. At the top of the column, the vapour is con-densed (with a rate of heat removal Qc). Some of theliquid is returned as reflux to the top of the column.The tops output stream is also taken from the con-denser outflow, and the ratio of the reflux flow to theflow in the tops stream is referred to as the reflux ratio.The bottoms stream is drawn from the liquid in the

R. Fletcher, W. Morton / Computers and Chemical Engineering 23 (2000) 1811–1824 1813

kettle reboiler with a flow rate which preserves overallmaterial balance.

The flow of liquid descending from stage i to i−1 isdenoted by Li (liquid flow) and the flow of vapourupwards from stage i to i+1 by Vi (vapour flow). Themole fraction of component j in the liquid Li is denotedby xij and in the vapour Vi by yij. It is convenient torefer to these quantities by the m-component vectors xi

and yi. The flow of each component of liquid may beexpressed by ljj=Lixij or by li=Lixi. Likewise 6ij=Viyij

or vi=Viyi expresses the component wise flow of va-pour. The temperature of these quantities associatedwith stage i is denoted by Ti. The flow of heat in theliquid from stage i to i−1 is denoted by Hi

L, and theflow of heat in the vapour from stage i to i+1 by H 6i .

Our model assumes that the column operates at auniform pressure P. A more detailed model could allowfor a pressure drop from stage to stage up the columnbut that is not done here. The main effect of a non-con-stant pressure will be to alter slightly the K-values forvapour–liquid equilibrium (Eq. (2.4) below). We do notexpect solutions to he hindered by such a refinement.The liquid flows in the tops and bottoms stream aredenoted respectively by D and B. The fractional com-position of the tops stream is the same as that of therecycled liquid, and is denoted by xN+1. The fractional

composition of the bottoms stream is the same as thatof the reboiler liquid, that is x0. The feed specificationis given and is assumed to be liquid. The total flow inthe feed is denoted by F and its fractional compositionby xF. Componentwise flows for the feed, tops andbottoms streams are denoted by

f=FxF, d=DxN+1, b=Bx0, (2.1)

respectively.The variables are required to satisfy various con-

straints, for example all flows and compositions arenon-negative and the fractional compositions mustsatisfy

%m

j=1

xij=1 (2.2)

and

%m

j=1

yij=1. (2.3)

There are equations for� the material balance of each component at each

stage of the column, and� the overall heat balance on each stage.We give a suitable definition of these constraints below.Since the vapour and liquid on each plate, and also atthe reboiler, are assumed to be in equilibrium, thevapour and liquid compositions are related by

yij=Kijxij (2.4)

where, for ideal mixtures, the Kij are non-linear func-tions of Ti defined by

Kij=1P

exp�

Aj+Bj

Cj+Ti

�. (2.5)

The quantities Aj, Bj and Cj for j=1, ... , m are knownas Antoine constants and are given as data. A relatedcondition is that the condenser outflow must be satu-rated liquid, which can be expressed by the bubblepoint equation:

%m

j=1

KN+1, jxN+1, j=1. (2.6)

We also note that Eq. (2.4) may be expressed as

yi=Kixi (2.7)

where Ki is a diagonal matrix with diagonal elementsKij, j=1, ... , m. The heat flows Hi

L and HiV are also

non-linear functions of the Ti which can be derived byintegrating expressions for the specific heat of the com-ponents, derived from experimental observation. Wehave used relatively simple quadratic expressions forenthalpies, of the form

HiL= %

m

j=1

lij(aj+a %jTi+a¦j T i2) (2.8)

Fig. 1. Distillation column schematic.


and

HiV= %

m

j=1

6ij(bj+b %jTi+b¦j T i2), (2.9)

where the coefficients aj, a %j, a¦j and bj, b %j, b¦j are givenas data. The heat flow in the feed stream is likewisegiven by

HF= %m

j=1

fj(aj+a %jTF+a¦j TF2), (2.10)

where TF is the feed temperature (also given as data).Similar expressions hold for the heat flows HD and HB

in the tops and bottoms streams, namely

HD= %m

j=1

dj(aj+a %jTN+1+a¦j TN+12 ) (2.11)

and

HB= %m

j=1

bj(aj+a %jT0+a¦j T02). (2.12)

In stating the form of the mathematical model to bepresented to the non-linear programming solver, vari-ous considerations must be taken into account. It isusually advisable to avoid redundancy in the constraintformulation, as this can otherwise cause potentialdifficulties due to the loss of linear independence in anylinearised constraints. There is often the option of usingequations to define variables (essentially eliminatingvariables and constraints from the problem), or alterna-tively of leaving the variables and constraints explicitlyin the formulation. The latter gives rise to larger non-linear programming problems, but the equations are ofa less complex nature and there is often more sparsityand structure in the resulting linear systems that can betaken into account. Other issues, such as suitable scal-ing of the variables and constraints, are oftenimportant.

In the rest of this section we describe a formulationwhich is reasonably compact and well-scaled, andwhich has worked well in practice. It is based mainly onthe standard MESH equations for rigorous columnmodelling (Sinnott, 1996 p. 452). The important featureof our approach is a particular selection of independentvariables. We then use some of the MESH equations tosolve for the dependent variables, so as to assist conver-gence of the solution of the column and to avoidunphysical variable values, e.g. negative componentflows within the column.

We start by introducing a new independent variableu, 05u51, which measures the overall recovery in thetops stream, or the proportion of the feed liquid flowthat is directed to the tops stream. This is somewhatdifferent to other models that have been suggested, andis motivated by the ideas pursued in Sections 3 and 4.Thus we can express the total tops and bottoms flowsas

D=uF and B= (1−u)F. (2.13)

The overall component-wise material balance for thecolumn is f=d+b, or alternatively, after using Eqs.(2.1) and (2.13) and dividing through by F, as

xF=uxN+1+ (1−u)x0. (2.14)

This equation is well-scaled and has a useful propertywhich we exploit in Section 4.

We next consider the liquid and vapour flows Li andVi. These must satisfy the equations

Vi=Li+1+D, i=S, ... N (2.15)

and

Li=Vi−1+B, i=1, ... S. (2.16)

This is a consequence of equating the net flow betweenstages with the output flow at the tops and bottoms. Inour model we select V0, ... , VS−1, and LS+1, ... , LN+1

independent non-negative variables, and use Eqs. (2.15)and (2.16) to determine the remaining values of Vi andLi. The reason for this is that the non-negativity of theremaining flows is automatically satisfied and does nothave to be enforced as an inequality constraint in themodel.

There are various ways in which the xi and yi vari-ables and the equilibrium constraints might be handled,and this is discussed in more detail later in this section.We have chosen to have the xi, i=0, ... , N as indepen-dent variables, and we use Eq. (2.4) to determine yi,i=0, ... , N, as dependent variables. The physical condi-tion that the condenser does not affect the materialcomposition may be stated as

xN+1=yN (2.17)

and enables us to express xN+1 as a dependent variable.As the total flows Li and Vi are available, we can nowdetermine the componentwise flows by using li=Lixi

and vi=Viyi.We are now in a position to state componentwise

material balance equations for within the column. Byequating the net flow between stages with the outputflow, we deduce the equations

vi= li+1+d i=S, ... N−1 (2.18)

and

li=vi−1+b i=1, ... S. (2.19)

In fact, Eq. (2.18) also holds for i=N but this isalready implied by Eqs. (2.15) and (2.17).

In a similar way we can also state the heat balanceequations. We choose Qr to be an independent variablein the model. We can calculate the feed and outputflows f, d and b as in Eq. (2.1), and hence the corre-sponding heat flows can be calculated from Eqs. (2.10),(2.11) and (2.12). An overall heat balance is achievedsimply by defining the condenser heat duty Qc by


Qc=Qr+HF−HD−HB. (2.20)

Heat balance within the column is obtained by equatingthe net heat flow between stages with the heat output atthe top (or heat input, at the bottom) of the column.This gives the equations

HiV−Hi+1

L =Qc+HD i=S, ... , N (2.21)

and

HiV−Hi+1

L =Qr−HB i=0, ... , S−1. (2.22)

We have already remarked that the variables xi and yi

need to satisfy the normalization conditions (Eqs. (2.2)and (2.3)). In fact it is necessary to enforce only one setof these conditions. This can be seen by summing overthe components in Eqs. (2.18) and (2.19), and usingEqs. (2.15) and (2.16). Imposing Eq. (2.2) as the nor-malisation provides equations that are linear, whichmight be used to advantage by the non-linear program-ming solver. In practice however, imposing Eq. (2.3)explicitly seems to provide somewhat better perfor-mance. Thus the conditions

%m

j=1

yij=1 i=0, ... , N. (2.23)

are included in the model.This completes the definition of the non-linear equa-

tions used by the model. To these we add lower andpossibly upper bounds on the independent variables ina suitable way. An AMPL model is given in the ap-pendix which precisely describes the definition of theindependent and dependent (defined) variables, the con-straints of the model, and the lower and upper boundsthat are specified.. There are no other inequality con-straints that need be added to determine the solution.However we have found it useful to add the monotonictemperature condition

Ti]Ti+1 i=0, ... , N (2.24)

which helps the non-linear programming solver toavoid temperature estimates which are not physicallyacceptable. Much of the subsequent discussion in thepaper proceeds on the assumption that none of thebounds or inequality constraints are active at a solutionto the column model, which is usually the case inpractice.

It is a useful exercise to count the number of degreesof freedom inherent in the column model. We shallassume that the column pressure P and the variables Nand S, describing the column configuration, are fixed.Of course the effect of varying these quantities is also ofinterest, but here we concentrate on the more simplecase. Indeed, N and S are integer variables and cannotbe varied unless we extend the formulation to onerequiring solution by mixed integer non-linear pro-gramming. The independent variables of the model are

the split parameter u, the reboiler heat duty Qr, thetemperatures T0, ... , TN+1, the flows V0, ... , VS−1 andLS+1, ... , LN+1, and the fractional compositionsx0, ... , xN. This gives a total of mN+m+2N+5 vari-ables. Any other quantities that arise can be defined asfunctions of these independent variables. The non-lin-ear equations which remain, and which we impose onthe model, are the N+1 normalization conditions (Eq.(2.23)), the mN material balance equations (Eqs. (2.18)and (2.19)), the N+1 heat balance equations (Eqs.(2.21) and (2.22)), the m equations (Eq. (2.14)) arisingfrom overall material balance, and the single equation(Eq. (2.6)) relating to the condenser outflow, giving atotal of mN+m+2N+3 independent equations. Thusthe distillation column model has two degrees offreedom.

This freedom may be taken up in various ways: forexample it is usually desirable to minimise the operatingcost of the column, of which the reboiler heat duty Qr

is a major item, whilst imposing constraints on theamount of impurity in the tops and bottoms streams.

An important property possessed by any solution tothe constraints of the model is that it is scalable in thesense that if the feed flow F is changed by some factor,then an equivalent solution may be obtained by chang-ing the flows Li, Vi and Qr by the same factor, andleaving the remaining independent variables u, Ti and xi

unchanged.A major difficulty with the solution of distillation

column models is that of inducing convergence in thenon-linear programming solver. Particularly it is valu-able to be able to obtain estimates that satisfy theconstraints of the model to good accuracy. If poorestimates are generated during the solution process thenthe linearised model used by the solver can be unsatis-factory, which can lead to failure to converge, orconvergence to a point at which the constraint lineari-sations are inconsistent. It is not too difficult to providereasonable estimates for some of the independent vari-ables, for example the Ti and the xi, but the amount offlow within the column (Li and Vi) is not easy toestimate, if a particular purity is required. For a fixedcolumn configuration, if the flows are increased withoutlimit, the product purities asymptotically approach lim-its that depend on the number of plates above andbelow the feed and the relative volatilities of the keycomponents. In the rest of the paper we look at meanswhereby good initial estimates for all independent vari-ables might be obtained.

Before passing on to this topic, we note that ourchoice of independent variables is but one of a numberof alternative possibilities for defining the model. Theapproach that we have used is to some extent motivatedby the ideas described in Sections 3 and 4, but is alsoguided by practical experience with a variety of possiblechoices. We describe some of the alternative possibili-ties that we have investigated in Section 7.


3. The zero-feed column

The new ideas for column initialization in this paperhave come about by considering what happens to thecolumn as Qr��, the case of infinite reflux ratio. Byscalability, taking this limit is essentially equivalent tofixing Qr and decreasing the feed flow rate F to zero.Fixing Qr removes one degree of freedom, and we shallremove the other degree of freedom by adding a singleimpurity constraint. The form we have chosen for theimpurity constraint is

x0,lk

x0,hk

=xN+1,hk

xN+1,lk

(3.1)

where lk and hk index the light key and heavy keycomponents, respectively. Since the heavy key compo-nent, is the major source of impurity in the tops stream,and likewise for the light key component and thebottom stream, this constraints essentially the fractionalimpurity in the light and heavy components. Of courseone might equally argue the case for a different mea-sure of impurity. However, the main point is that theexact level of impurity is not specified and this con-straint achieves a symmetry between the relative keycompositions in the tops and bottoms products. In-creasing the heat rate tends to increase the sharpness ofthe split, but as Fenske’s equation makes clear there isa limiting product impurity at infinite reflux that pre-vents a sharp split. This limiting impurity can be re-duced by increasing the number of plates, N.

Adding the impurity constraint and fixing Qr, trans-forms the general column model of Section 2 into awell-determined system of nM+m+2N+4 variablesand constraints, with zero degrees of freedom.

We now consider the special case of this system inwhich F=0 (the Zero-Feed Column). This causes con-siderable simplifications in the non-linear system. Firstof all the overall material balance for the column istrivially satisfied by f=d=b=0 and so we need notinclude the variable u. Also the vapour and liquidmaterial flow between the plates must balance, so itfollows that

Vi=Li+1 and yi=xi+1, i=0, ... N. (3.2)

We use these equations to eliminate the Vi and yi

variables. In regard to heat flow, there is a constant netheat flow Qr upwards between each stage. Thus theheat balance equations may be expressed as

Viqi=Qr i=0, ... , N (3.3)

where

qi= %m

j=1

xij((bj+b %jTi+b¦j T i2)

− (aj+a %jTi+1+a¦j T i+12 )). (3.4)

Thus, once the Ti and xi are known. Eqs. (3.4) and (3.3)enable the flows Vi and hence Li to be determined. Thuswe are left with a problem in which only the variablesTi and xi are present.

The equilibrium conditions (Eq. (2.7)) become

xi+1=Kixi i=0, ... , N. (3.5)

Given the Ti and hence Ki, these are simply recurrencerelations from which the vectors xi may be calculated.Thus we can use Eq. (3.5) to eliminate all but m of thexii variables. However there is an aspect of numericalstability to consider here, and it is important to run therecurrence relations up the column for the heavy com-ponents (as xi, j=Ki−1,jxi−1,j for increasing i ), anddown the column (as xi, j=xi+1,j/Ki+1,j for decreasingi ) for light components. This ensures that the compo-nents are exponentially decreasing and avoids anydifficulty due to exponential growth. (Such ideas arefamiliar from the old Lewis–Matheson stage-by-stagedesign procedure for multi-component columns (seeKing, 1980 p. 450)).

The resulting problem has N+2 variables Ti and mvariables xij (these are the x0,j for the heavy componentsand xN+1,j for the light components with which therecurrences are initiated). The constraints remaining arethe N+2 normalization equations Eq. (2.2) for i=0, ..., N+1, the saturation condition (Eq. (2.6)) and theimpurity condition (Eq. (3.1)).

Counting variables and equations we see that thissystem has m−2 degrees of freedom. Yet we startedout with a general system with zero degrees of freedom.When this issue first arose, the explanation for thismismatch was by no means obvious, and the means ofresolving it even less so, especially with the modellingequations that were in use at the time. The resolution ofthis issue is described in the next section.

4. The limiting zero-feed column (LZFC)

The answer to the questions posed above is that thezero-feed column does indeed have m−2 degrees offreedom. However it does not quite answer the questionposed in Section 2 as to what is the limiting solution ofthe general column equations as F�0. In this case,although the overall material balance equation f=d+bis trivial in the limit, with zero on both sides, theequation

xF=uxN+1+ (1−u)x0 (4.1)

obtained by dividing out F is still valid in the limit. Thisis obvious in retrospect, having seen this equation inSection 2, but it was certainly not obvious when thisstudy began. In fact it was the resolution of this issuethat led us to model the general column in the waydescribed in Section 2. The equation has good numeri-


cal properties for use in the general column model. It iswell scaled (with terms of order unity), and it also enablesthe general column model to be solved reliably for verysmall values of F, thus permitting the effective use ofhomotopies for the general case, starting from F=0.Furthermore we see why the reflux ratio would be anunsatisfactory choice of independent variable, since itgoes to infinity as F�0.

As hoped, Eq. (4.1) solves the problem of the extradegrees of freedom. It adds an extra variable 0 to theproblem and m extra constraints. However one of theproduct stream normalisation constraints becomes re-dundant, as it is implied by summing over the compo-nents in Eq. (4.1). It probably does not matter which ofthese normalisations is dropped. To preserve symmetry,we have dropped both the product stream normalisa-tions. Instead we retain Eq. (2.2) for i=1, ... , N, andimpose the extra condition

%m

j=1

(xN+1, j−x0, j)=0. (4.2)

This gives a well-determined non-linear system withm+N+3 equations and variables.

There are many advantages of using the solution of anLZFC problem as a preliminary to solving a generalcolumn problem. We itemise these as follows and expandon the points in the subsequent text.� The LZFC problem is much smaller than the general

column problem.� The LZFC problem is much easier to initialize and

converge.� The solution of the LZFC provides estimates of the

minimum impurity that can be obtained for anygiven number of plates.

� The LZFC solution provides good estimates of Ti, xi

and the flows Li and Vi, with which to initialize thegeneral case.

� The LZFC solution may be used to initialize ahomotopy approach by which to solve the generalcase.

� Good results have been obtained on a variety offlowsheets involving linked distillation columns, withand without heat integration.

The most obvious advantage of solving the LZFCproblem is that it is much smaller than the generalcolumn problem, having m+N+3 variables as againstmN+m+2N+5 variables. The LZFC problem doesnot involve the non-linear heat balance constraints, andthe flow variables Li and Vi, whose values are not easilyestimated. This avoids some of the difficulty inherent inthe general problem. Moreover good estimates of Ti

and the required xi variables are readily provided forthe LZFC.

Very good estimates of u and the required xi vari-ables are readily obtained by assuming that the split is

sharp. Thus we set

u=%lk

1

xjF, (4.3)

that is the sum of light component mole fractions in thefeed, (or the fractional tops stream recovery assuming aperfectly sharp split). Then for a light component weset

xN+1, j=xjF/u (4.4)

and likewise for a heavy component

x0, j=xjF/(1−u) (4.5)

These settings are close to being normalized (Eq. (4.2)),once the remaining xN+1,j, and x0,j variables are deter-mined from the recurrence relations. The extreme tem-peratures T0 and TN+1 can now be accurately estimatedby solving the non-linear equations

%m

j=hk

K0, jx0, j=1 and %lk

j=1

KN+1, jxN+1, j=1 (4.6)

For T0, it is sufficient just to tabulate S K0,jx0,j for alikely range of temperature and then use linear interpo-lation, and similarly for TN+1. The effect on T0 andTN+1 of varying the column pressure P can also beinvestigated at this Juncture. Intermediate values of Ti

can be estimated by some interpolation procedure. Wehave found that linear interpolation is quite adequate.The resulting LZFC problems are typically solved fromthis initial approximation in about six or seven itera-tions of our NLP solver (effectively implementing someversion of the Newton–Raphson method).

As in Eq. (3.1), we have measured the impurity of theLZFC solution by looking at the ratio of the fractionsof the light key and heavy key components in the x0

and xN+1 variables. The function of the impurity con-straint has been to equate the relative impurity of thetops and bottoms streams. The actual level of impuritygiven by the LZFC solution is a lower bound on theimpurity that can be obtained in the general columnwith finite reflux. If we are interested in obtaining aparticular level of impurity (say 1%) in the solution ofthe general column, then we might try solving theLZFC for increasing values of N until this level ofimpurity is attained. In fact it is important to look fora lower level of impurity in the LZFC, so that the valueof Qr in the general case is not too large. (Recall thatthe LZFC solution essentially corresponds to Qr=�when F\0).

Of course the variation of N can also be investigatedat a later stage of the process but this procedureprovides a useful way of obtaining reasonable esti-mates. In particular it provides estimates for which theresulting general NLP problem has a feasible solution.This is very important in that many NLP solvers per-form badly when faced with an infeasible problem.


5. Solution of columns with finite reflux

One main purpose of the LZFC solution is toprovide good estimates of the variables for the generalcolumn problem, with non-zero feed and finite refluxratio. In this respect, we shall consider the form of themodel having two degrees of freedom (or three degreesof freedom if P is included — see Section 2). Thus F isfixed and Qr becomes an independent variable. Becauseof scalability, it suffices to take F=1 and this alsoensures that the material balance constraints are wellscaled. Exactly what objective function is selected is amatter of choice. So far we have considered simplyminimising Qr which is related to both column operat-ing and capital costs (through steam requirements andcolumn diameter needed to accommodate the vapourflow without flooding, respectively). This ignores theeffect of column height (number of plates) on capitalcost but then in our model N is fixed in any case. Wehave fixed a particular level of impurity, typically 1%,and imposed the inequality constraints

x0,lk50.01 x0,hk and xN+1,hk50.01 xN+1,lk (5.1)

on the problem. These are usually both active at thesolution as would be expected.

It has been observed that the Ti and xi from theLZFC problem are usually close to the values for thegeneral problem. This is because even quite large rela-tive changes in small impurity levels have little effect onthe mole fractions of abundant components (lightabove and heavy below the feed) and hence on bubblepoint temperatures. An initial value of Qr has to beprovided, and it is important to choose units of heatflow so that the numerical value of Qr (and hence of theHi

L and HiV) is around unity. This ensures that the heat

balance equations are well-scaled. Given Qr, we can usethe LZFC solution, together with Eqs. (3.4), (3.3) and(3.2)), to initialise the flows Li and Vi that are requiredin the general case. A feature of this procedure is thatthe initial point for the general column is usually rea-sonably close to feasibility. The main issue is then thatof how to provide a good estimate of Qr. It may bepossible to do this on the basis of a priori knowledgeabout the column and where possible we aim to do this.If the effective relative volatilities of the componentsare known then the Underwood equations (King, 1980p. 420) can be solved to obtain the minimum refluxratio, Rm. If we suppose the actual reflux ratio in ouroptimised column is somewhat greater than this (say bya factor 1.25 as in a well-known rule of thumb (Peters& Timmerhaus, 1991 p. 371), and the feed is saturatedliquid, then the vapour boilup can be estimated as

V0= (1.25 Rm+1)uF

This value should be compared with the solution of theLZFC problem, and Qr and all internal column flows

scaled accordingly. As an alternative to direct use of thethermodynamic package incorporated in the model, theeffective relative volatility of the keys (aK) can becalculated from the Fenske equation rearranged as

aK=�x0,lkxN+1,hk

x0,hkxN+1,lk

�1/NZF

where NZF is the number of plates in the LZFCproblem.

Alternatively, if a good estimate of Qr cannot bemade (e.g. if the liquid mixture were non-ideal), then ahomotopy approach to finding the solution can be veryeffective and is outlined in the next paragraph.

In the homotopy approach, we fix the value ofQr=1 and impose the impurity constraint Eq. (3.1) inplace of Eq. (5.1). We have seen in Section 2 that thisgives rise to a well-determined system of non-linearequations. The LZFC solution is the solution of thisproblem for F=0. We now solve the non-linear systema number of times, for a sequence of values of F,starting from as close to F=0 as is necessary toconverge the system. At each stage we monitor theimpurity in the output streams and adjust the value ofF with the aim of converging to the required level ofimpurity (1% say). Each stage of the homotopy issolved using the best previous solution as an initialguess. Our experience is that each system solves veryquickly, typically in as few as two or three iterations.Once the desired level of impurity is obtained, the flows(including Qr) are rescaled to correspond to the actualfeed flow F that is given. This approach can be auto-mated using a path following technique. However wehave found that the informal approach works very welland the AMPL language readily permits the manualadjustment and repeated solution of the non-linearsystem in the manner described.

We have successfully solved a variety of columnsusing these ideas, not only as individual columns, butalso within a flowsheet of linked columns for the pur-pose of separation. We have also tackled more compli-cated optimisation problems involving heat integrationbetween the columns. We have successfully solved allthe problems described by Fraga and McKinnon (1994)(see the flowsheets in Fig. 3, (a)–(d)), both for the fixedpressures given there, and also with the optimum varia-tion of pressures. These are quite large problems havingfour columns and about 1000–2000 independent vari-ables and constraints, depending on the numbers ofplates used. Our experience is that an NLP solver caneasily run into difficulty with such problems and eitherfail to converge, or converge to an infeasible solution.Our method of initialisation provides estimates that areclose to feasibility, and we have found it advantageousto maintain near-feasibility in the estimates generatedby the NLP solver. Our filter-type SQP solver permitsthis to be done by imposing an upper bound on the


amount of constraint infeasibility. The next sectiondescribes a case study showing how a particular columnmodel was converged.

6. A case study

A five-component problem was modelled, with

C3H8, i−C4H10, n−C4H10, i−C5H12, n−C5H12.

The heavy key was i-C5H12. Physical property data forthese components is as follows.

Liquid enthalpy Vapour enthalpyAntoine coefficients

Bj CjAj aj a %j a¦j bj b %j b¦j

−1872.46 247.99 0 74.096C3 0.9403613.7110 20014 68.253 0.10695i-C4 13.5231 −2032.73 240.00 0 141.71 0.20279 20719 90.516 0.13728

−2154.90 238.73 0n-C4 141.7113.6632 0.20279 22385 91.657 0.12940−2348.67 233.10 0 156.8013.6188 0.19927i-C5 26034 109.658 0.17415−2477.07 233.21 0n-C5 156.8013.8183 0.19927 27499 111.322 0.16868

The feed composition was xF= [0.05, 0.15, 0.25,0.20, 0.35] with the components in the order above.A column was modelled with N=20 trays, thefeed on stage S=12 and at a pressure of P=725kPa.

End product compositions were initialised as xN+1=[1/9, 3/9, 5/9, 0, 0] and x0= [0, 0. 0. 4/11, 7/11], inproportion to the feed and assuming a sharp splitbetween the key components. The tops recovery u wasinitialised as 0.45. Bubble point calculations on theguessed bottoms and distillate compositions gave thecorresponding saturated liquid temperatures as106.11°C and 52.36°C, respectively. The column tem-perature profile was initialised by linear interpolationbetween the end values.

The infinite reflux column converged in six SQPiterations, with the following history of constraintinfeasibilities:

Iteration Sum of equation infeasibilities

0 5.4831 1.3592 0.728

0.36034 0.023

3.79×10−556 1.67×10−10

At the infinite reflux solution, u=0.450017, the reboilerand condenser temperatures were T0=106.11 and TN+

1=52.37°C. The temperature and inter-tray flows areshown below. Vapour and liquid are equal at infinitereflux. These flows correspond to a heat input of 105

kJ/unit time.

TStage Flow

0 4.596106.111 4.605105.74

4.616105.3623 4.626104.94

4.635104.4744.6425 103.914.645103.126

7 4.643101.884.63799.738

96.039 4.64590.2410 4.707

4.84882.811175.6012 5.026

5.17270.34135.26214 67.225.30865.5115

64.4716 5.3315.33963.63175.33018 62.665.28361.16195.16420 58.27–52.3721

The product compositions were solved as (tops)

xN+1= [0.111, 0.333, 0.555, 1.65×10−4, 4.12×10−6]

and (bottoms)


Figure 2. Cumulative composition profiles for case study at finitereflux.

0.451018, was similar to the infinite reflux value. This isunsurprising since the component recoveries shouldchange only modestly. Reboiler and condenser temper-atures were also little changed at T0=105.91 and TN+1

=52.52°C. Stage temperatures were generally ratherlower below the feed than for infinite reflux and higherabove, the biggest difference being on stage 14, wherethe temperature was 71.99 as against 67.22°C. Thereflux ratio for the column was

0.735406/0.451018=1.63

The product compositions were now (tops)

xN+1= [0.111, 0.332, 0.550, 5.50×10−3, 1.23×10−3]

and (bottoms)

x0= [7.02×10−10, 7.62×10−5, 3.60

×10−3, 0.360, 0.637]

The largest relative changes from infinite reflux were inthe non-key compositions, which again is to beexpected.

A plot of cumulative composition profiles againststage number is shown in Fig. 2. In this graph, thelowest curve is the mole fraction of the heaviest compo-nent (n-pentane) and the next up is the sum of the twoheaviest (n-pentane and iso-pentane). This curve is theboundary between the heavy and light key, in which theabsence of much change in slope near the feed on stage12 suggests that the column might be more appropri-ately designed with more plates and less reflux, to meetthe specified separation. Compositions for the light key(n-butane) and lighter components can be inferred fromthe higher curves.

7. Alternative column models

In this section, we describe some alternative possibil-ities to that in Section 2 for defining the general columnmodel, and we compare their performance on somelarge flowsheets with linked columns. An obvious possi-bility is to include both the xi and yi as independentvariables, and to impose the equilibrium constraints(Eq. (2.4)) explicitly. This increases the dominant termin the variable count from mN to 2mN and so approx-imately doubles the size of problem. Although theremay be some advantage to be gained from having theextra variables present, it does put up some of thehousekeeping costs associated with the non-linearsolver. Another interesting possibility is to designatev0, ... , vS−1 and lS+1, ... , lN+1 as the independent vari-ables, in place of the Li, Vi and xi. The informationcontent of this representation is essentially the same.This cuts down the number of independent variables byabout N, and can be thought of as a null-space parame-trisation of the material balance equations (Eqs. (2.18)

x0= [2.80×10−5, 2.60×10−7, 1.08

×10−4, 0.364, 0.636]

It may be noted that the impurity condition (Eq. (3.1))was satisfied, since

0.000108/0.364:0.000165/0.555

For the finite reflux case, the feed rate was set to be 1kmol on the same time basis as before. The column wassolved by NLP using the infinite reflux solution asinitial guess. This includes compositions on all stages(not shown above). The objective was to minimise Qr,the reboiler heat duty, subject to the ratio of keycomponents in the product being 100:1 (heavy: light inthe bottoms and light: heavy in the tops).

Convergence was obtained in 9 SQP iterations asfollows:

Sum of equationIteration Reboiler duty (kJ/infeasibilities kmol feed)

6.237 0.100000.6751 0.1192

2 0.804 0.07230.715 0.04893

0.04230.21440.03670.16750.03190.2266

0.012 0.033278 0.03318.28×10−5

1.76×10−99 0.0331

At, the solution, heat duties were Qr=0.0331 andQc=0.0230 kJ/kmol feed. The tops recovery, at u=


and (2.19)). In this case the equilibrium constraints (Eq.(2.4)) need to be explicitly included. In practice nosignificant gain for this version has been obtained, nodoubt because the dominant term in the number ofvariables is still mN.

In Section 3 it is pointed out that for the infinitereflux column the dominant term mN can be removedby using recurrence relations to generate all but m ofthe variables xij. The same idea can be used for thegeneral column, following the approach behind theMcCabe–Thiele construction for binary column design,although the recurrence relations are considerably morecomplicated. The material balance equations (Eqs.(2.18) and (2.19)) and the equilibrium equations (Eq.(2.4)) are rearranged in a suitable way. As before, for alight component, we carry out the recurrence down thecolumn, in which case the appropriate expressions are

xij= (Li+1xi+1, j+dj)/(KijVi) i=N, ... , S (7.1)

and

xij= (Li+1xi+1, j−bj)/(KijVi) i=S−1, ... , 1. (7.2)

For a heavy component, the recurrence is carried outup the column, and the expressions are

xij= (Vi−1Ki−1, jxi−1, j+bj)/Li i=1, ... , S (7.3)

and

xij= (Vi−1Ki−1, jxi−1, j−dj)/Li i=S+1, ... , N.(7.4)

There is also a complication that is not present inSection 3. The recurrences involve d and b, and theseare defined by d=DxN+1 and b=Bx0. Thus all thecomponents of xN+1 and x0 are needed to initiate theserecurrences. (In Section 3, only the light components ofxN+1 and heavy components of x0 are needed.) Thusboth xN+1 and x0 become independent variables in themodel, and there are m extra constraints which equatethe variables x0,j (for a light component) or xN+1,j (fora heavy component) to the values obtained on the laststage of the recurrences. Thus the independent variablesappropriate to this formulation are the split parameteru, the reboiler heat duty Qr, the temperatures

T0, ... , TN+1, the flows V0, ... , VS−1 and LS+

1, ... , LN+1, and the fractional compositions x0 andxN+1. This gives a total of 2m+2N+5 independentvariables.

In Table 1 we show some numerical experience withthese different formulations when applied to two in-stances of a linked column problem, starting from thesame initial point. The heading 0mN relates to theformulation just described, in which the dominant mNterm is absent, whilst 1mN relates to the formulation ofSection 2, and 2mN relates to the extended formulationin which both xij and yij are independent variables. Thefirst instance is that of column sequences (b), (c) and (d)in Fraga and McKinnon (1994) but with no heat inte-gration and the pressures fixed as in their solution. Thesecond instance is similar, but allows the column pres-sures to vary subject to minimum temperature con-straints on tops products (to allow water or air cooling)and a minimum pressure of one atmosphere.

The different formulations all find the same solutionand the number of iterations taken does not varygreatly within each problem instance. However theformulation of Section 2 is clearly superior on time periteration, and this supports our preference for thisapproach. The reason for the poor performance of the0mN formulation seems likely to be that the consider-able increase in complexity of (Eq. 7.1–7.4) causes adecrease in the sparsity of the Jacobian matrix, andhence an increase in the time taken to solve the QPproblems in the SQP method. On the other hand, theextended formulation 2mN does not seem to provideany useful gains in sparsity to offset the increased sizeof the NLP problem.

Acknowledgements

We gratefully acknowledge the valuable con-tributions of Drs David Gay and Sven Leyfler inregard to integrating the AMPL system and our NLPsolver.

Appendix A. Notation

UnitsSymbol Description

Aj, Bj, Cj Antoine coefficients forvapour pressure (Eq.(2.5))

aj, a %j, a¦j liquid enthalpy coeffi-cients (Eq. (2.8))

B kmol/unit bottoms flowtime

Table 1Comparison of different column formulations

c Vari-Formula- c Itera- Time (s) s/iterationtionstion (mN) ables

438 46.526 120908931 29 7.4215

18.31488 54830251 23970 47.0442

483 7.91 897 6117.78122 461492


bottoms flowrate ofkmol/unitbj

time component jkmol/unitD tops (distillate) flow

timekmol/unitdj tops flowrate of com-

ponent jtimeF feed flowkmol/unit

timekmol/unitfj feed flowrate of compo-

nent jtimeHi

L liquid enthalpy flowkJ/unit timefrom stage i

HiV vapour enthalpy flowkJ/unit time

from stage iKij K value of component–

j on stage idiagonal matrix of K-–Ki

values on stage Ikmol/unitLi total liquid flow from

stage itimekmol/unitlij liquid flow of compo-

nent j from stage itimem number of chemical–

components–N number of stages

column pressurekPaPQ heat dutykJ/unit time

net molar heat flow upkJ/unit timeqi

column below stage iRm minimum reflux ratio–

feed stage–S°CTi temperature on stage i

total lapour flow fromkmol/unitVitime stage i

kmol/unit6ij vapour flow of compo-nent j from stage itime

xij liquid mole fraction of–component j on stagei

xi liquid mole fraction–vector on stage i

xF – feed (liquid) mole frac-tion vector

yij – vapour mole fraction ofcomponent j on stagei

bj, b %j, b¦j vapour enthalpy coeffi-–cients (Eq. (2.9))

u overall recovery in tops–stream

Superscriptr reboiler

condensercF feed stream

tops streamD

B bottomsstream

Appendix B

c column.modc Original AMPL coding by Roger Fletcher

c Program to find a solution of the distil-lation column equations.

c Components are given in order of de-creasing volatility.

set COMPONENTS ordered; param H – KEY sym-bolic in COMPONENTS; c heavy key

c thermopack coefficients

param Aant {COMPONENTS}; param Bant{COMPONENTS}; param Cant {COMPONENTS};param alp0 {COMPONENTS}; param alp1{COMPONENTS}; param alp2 {COMPONENTS};param bet0 {COMPONENTS}; param bet1{COMPONENTS}; param. bet2 {COMPONENTS};

c fixed column parameters

integerNParam c numberof trays

\2;Param S integer c feed

tray\1, BN;

P\0;Param c columnpressure

xFParam c feedmole{COMPONENTfractionsS}\=0;

F\=0;Param c totalfeed flow

Param c feedTF;temperature

T– lb; param T –Param c stagetempera-ub\T– 1b;turebounds

param f {j in COMPONENTS}:=F*xF[j]; cfeed flows

c independent variables


c heat duty atvar Qr \=0;reboiler

var theta \=0, B=1; c tops/bottomssplitc stagevar T

{0...N+1} \=T– 1b, temperaturesB=T– ub;

c total flowvar z {0...N} \=0;towards the feedc liquidvar x {0...N,

COMPONENTS}\=0; compositions

c NB: for i BS, z[i] is a vapour flow, andfor i \ =S, z[i] is a liquid flow

c defined (dependent) variables

var K {I in 0...N+1, j inCOMPONENTS}=

c equilibriumexp(Aant[j]+Bant[j]/(Cant[j] constants+T[i]))/P;

var y {i in 0...N, j inCOMPONENTS}=K[i,j]*x[i,j];

c vapourcompositions

varD=theta*F;var d {j in c tops flows

COMPONENTS}=D*y[N,j];var B=(1−theta)*F;var b {j in c bottoms flows

COMPONENTS}=B*x[0,j];var L {i in 1...N+1}=if c total liquid

flowsi B=S then z[i−1]+Belse z[i−1];

var V {I in 0...N}=ifi BS then z[i] elsez[i]+D;

c total vapourflowscvar 1{i in 1.A+l, j in

COMPONENTS}= componentwiseliquid flows

if i \N thenL[N+1]*y[N,j] elseL[i]*x[i,j];

var v{i in 0...N, j in ccomponentwiseCOMPONENTS}vapour flows=V[i]*y[i,j];

var HL{i in1...N+1}=1e−6*sum {jin COMPONENTS}

c liquid heat1[i,j]*(alp0[j] flows+T[i]*(alp1[j]+alp2[j]*T[i]));

var HV{i in0...N}=1e−6*sum {j inCOMPONENTS}

v[i,j] c vapour heat*(bet0[j] flows+T[i]*(bet1[j]+bet2[j]*T[i]));

param HF:=1e−6*sum {jin COMPONENTS}

c feed heatf [j]*(alp0[j] flow+TF*(alp1[j]+alp2[j]*TF));

var HD=1e−6*sum {j inCOMPONENTS}

d [j]*(alp0 [j]+T[N+1] c tops heat*(alp1[j]+alp2[j]*T flow[N+1]));

var HB=1e−6*sum {j inCOMPONENTS}

b [j]*(alp0 [j]+T [0] c bottoms heat*(alp1[j]+alp2[j]*T flow[0]));

c cooler dutyvar Qc=Qr+HF−HD−HB;

c normalization equationssubject to nm1 {i in 0..N}: sum {j inCOMPONENTS} y[i,j]=1;

c material balance equations

subject to mb1 {i in 0..S−1, j inCOMPONENTS}: l[i+1,j]=v[i,j]+b[j];subject to mb2 {i in S..N−1, j inCOMPONENTS}: v[i,j]= l[i+ l,j]+d[j];

c heat balance constraintssubject to hb1 {i in 0..S−1}:

HV[i]+HB=HL[i+1]+Qr;subject to hb2 {i in S..N}:

HV[i]=HL[i+1]+Qc+HD;

c condenser recycle is saturated liquid

subject to crs: sum {j in COMPONENTS}y[N,j]*K(N+l,jl= l;

c consistency of feed and tops/bottomsoutflow


subject to ctb {j in COMPONENTS}:xF[j]=theta*y[N,j]+(1−theta)*x[0,j];

c monotonic temperatures

subject to mt1 {i in 0..N}:T[i+1] B =T[i];

c purity condition

subject to pr1: x[0,prev(H – KEY,COMPONENTS)]B =0.01*x[0,H – KEY];

subject to pr2: y[N,H – KEY]B=0.01*y[N,prev(H-KEY,COMPONENTS)];

c objective function

minimize cost: Qr;

References

Fletcher, R., & Leyffer, S. (1997) Nonlinear programming without apenalty function. Dundee numerical analysis report NA/171.Mathematical Programming (submitted for publication).

Fourer, R., Gay, D. M., & Kernighan, B. W. (1993). AMPL, Amodelling language for mathematical programming. Danvers, MA:Boyd and Fraser.

Fraga, E. S., & McKinnon, K. I. M. (1994). CHiPS: a processsynthesis package. Chemical Engineering Research and Design,72(A3), 389–394.

King, C. J. (1980). Separation processes (2nd ed.). New York: Mc-Graw-Hill.

Peters, M. S, & Timmerhaus, K. D. (1991). Plant design and econom-ics for chemical engineers (4th ed.). New York: McGraw-Hill.

Sinnott, R. K. (1996). Chemical engineering, 6olume 6: design (2nd ed.revised). Oxford: Pergamon Press.

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initialising distillation column models

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