Invariant rectifying-stripping curves for targeting minimum energyand feed location in distillation

Santanu Bandyopadhyay a, Ranjan K. Malik b, Uday V. Shenoy c,*a Energy Systems Engineering, Department of Mechanical Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India

b Computer Aided Design Centre and Department of Chemical Engineering, Indian Institute of Technology, Powai, Bombay 400 076, Indiac Department of Chemical Engineering and Computer Aided Design Centre, Indian Institute of Technology, Powai, Bombay 400 076, India

Abstract

Invariant rectifying-stripping (IRS) curves are proposed that are independent of the feed location and operating reflux of thedistillation column for a given separation problem. IRS curves represent the enthalpy surpluses and deficits in the rectifying andstripping sections, respectively, as a function of temperature for all possible values of reflux and reboil. The IRS curves providea new representation on the temperature-enthalpy diagram to set distillation column targets prior to detailed design for minimumenergy requirement, feed location, feed preconditioning, and side-exchanger loads. The application of the proposed concepts totwo binary distillation examples (one featuring a tangent pinch) and a multicomponent distillation example illustrates theusefulness of the IRS curves in properly locating the feed, determining the minimum utility requirements, and reducing the tediumof repeated simulations. The IRS curves are rigorously invariant and provide the absolute minimum utility requirements forbinary systems (ideal as well as non-ideal); however, they are near-invariant and predict the near-minimum utility requirementsfor multicomponent systems (where the pseudo-binary concept of a light and heavy key is employed).

Keywords: Distillation; Thermodynamic minimum; Temperature–enthalpy diagram; Feed location; Energy targeting; Pinch analysis

1. Introduction

The temperature-enthalpy (T–H) curve for a binarydistillation column at the minimum thermodynamiccondition (MTC) can be generated by solving the cou-pled heat and mass balance equations for the reversibleseparation scheme (Benedict, 1947; Fonyo, 1974; Fitz-morris & Mah, 1980; King, 1980; Naka et al., 1980; Ho& Keller, 1987). The limitations in the sharpness ofreversible multicomponent separations (Fonyo, 1974;Franklin & Wilkinson, 1982) can be overcome by usingthe pseudo-binary concept of a light and heavy keymodel (Fonyo, 1974; Dhole & Linnhoff, 1993).

Dhole and Linnhoff (1993) described a procedure forgenerating a T–H curve (which they called the columngrand composite curve or the CGCC) from a converged

simulation of a distillation column. The calculationprocedure involves determination of the net enthalpydeficit at each stage by generating envelopes from eitherthe condenser end (top-down approach) or the reboilerend (bottom-up approach). However, the values calcu-lated by the two approaches differ for stages with feedsbecause they do not consider the enthalpy balances atthe feed stages. A feed stage correction (FSC) thatrigorously considers the mass and enthalpy balances atfeed stages has been recently proposed by Bandyopad-hyay, Malik and Shenoy (1998) to resolve the dis-crepancy. The invariant rectifying-stripping (IRS)curves proposed here have the FSC built-in, and conse-quently have the advantage of not requiring a separatecorrection procedure to the CGCC.

The CGCC is a T–H curve at the practical near-min-imum thermodynamic condition, which inherently ac-counts for the inevitable feed loss, pressure loss,sharp-separation loss, and loss due to chosen configura-tion. The energy-saving potential for different column

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modifications like reflux reduction, feed conditioning,and scope for side reboiler/condenser can be addressedon such a T–H diagram (Naka et al., 1980; Terranova& Westerberg, 1989; Dhole & Linnhoff, 1993; Ognisty,1995; Hall, Ognisty and Northup, 1995; Trivedi et al.,1996). The horizontal enthalpy gaps at the top andbottom of the CGCC denote the reboiler and condenserloads, respectively. These gaps may be decreased byreducing the reflux (i.e. by increasing the number ofstages). Dhole and Linnhoff (1993) discuss reflux mod-ification and target its scope in terms of the utilityreduction in the reboiler/condenser.

The CGCC depends not only on the operating reflux,but also on the feed location in the column. Dhole andLinnhoff (1993) assumed the feed stage location for thecolumn had been appropriately chosen beforehand. Al-though they indicated that appropriate feed stage loca-tion should be identified before targeting for anycolumn modification, no methodology for locating thefeed was suggested by them. It must be emphasized thatimproper feed location leads to energy penalties in thereboiler and condenser, as well as an erroneous refluxmodification target.

The feed stage location is an important parameter forcolumn optimization, and may be determined throughseveral simulation runs. Different methods for the opti-mal location of the feed stage and their shortcomingsare reviewed by King (1980) and Kister (1992). Thoughthe empirical correlation proposed by Kirkbride (1944)can be utilized to find the approximate feed location, itis not very reliable and satisfactory for asymmetricfeeds (Henley & Seader, 1981). Hengstebeck (1968)proposed a graphical approach to correct the feedlocation from the base case simulation result. Theseparation parameter plot of Hengstebeck used for thispurpose is essentially based on compositions (key ratio)and does not consider enthalpies. A poor feed locationcauses relatively sharp breaks on the separationparameter plot, which may be corrected by relocatingthe feed using a slope criterion on an extrapolatedcurve as discussed by Hengstebeck (1968). The ap-proach proposed here for feed location overcomes thedeficiencies in previous methods and captures composi-tion as well as temperature dependencies. It is reliablefor all types of feeds and accurate as it does not involveextrapolation or slope calculations.

In this paper, targeting procedures for minimumutility consumption and feed location are established.Invariant rectifying-stripping (IRS) curves, that primar-ily depend on the separation problem and not on thecolumn configuration, are proposed for this purpose.These curves are invariant to the operating reflux andthe feed location in the column. They depend only onthe separation and the operating pressure. The invari-ant property of the curves is rigorously proved forbinary and reversible multicomponent separations. The

invariance approximately holds for general multicom-ponent systems, and is demonstrated through a casestudy where the pseudo-binary concept of a light andheavy key is used. The IRS curves provide the feedlocation target in terms of temperature, which may bethen converted to a stage number by a simple method-ology. Thus, a systematic procedure, free from heuris-tics, for locating the feed in a column through pinchanalysis is presented.

The work extends the energy targeting concepts(Linnhoff, Townsend & Boland, 1982) originally devel-oped for heat exchanger networks (HENs) to distilla-tion columns. The analogs of hot utility load, coldutility load, and DTmin in HENs are reboiler duty,condenser duty, and reflux ratio in distillation. Just asenergy targets for HENs are established in pinch analy-sis (Linnhoff, 1993) ahead of network design based onlyon stream specifications, energy targets for distillationare developed here from IRS curves prior to columndesign based purely on feed/products specifications.The targeting procedures aim at reducing the poten-tially large space of design alternatives to a small set ofpromising designs that merit more detailed attention.Targets provide the direction in which the base-casedesign should be evolved to ensure the optimalsolution.

2. Motivation

Dhole and Linnhoff (1993) suggested that the scopefor reduction in energy requirement by decreasing thereflux ratio may be targeted in terms of the enthalpygap (horizontal distance) of the CGCC pinch from thetemperature axis. The pinch is defined as the point onthe CGCC with the minimum enthalpy value (i.e. clos-est to the temperature axis). The CGCC pinch typicallyoccurs close to the feed stage except for some non-idealbinary systems (where it occurs in either the strippingsection or the rectifying section depending on the va-por–liquid equilibrium). Mathematically speaking, thescope for energy conservation (in terms of reboiler/con-denser loads) by reflux modification as well as theminimum reflux (for a specified separation with a givencolumn configuration) can be estimated from

Qr−Qr,min=Qc−Qc,min=HCGCC,min:Dl(R−Rmin)(1)

2.1. Proper location of feed

Unless the feed is appropriately located in thecolumn, the reflux modification scope predicted by Eq.(1) in terms of HCGCC, min is erroneous. If the feed islocated too high or too low in the column, the CGCCpinch at the feed stage will usually show a reduced

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Fig. 1. Effect of feed location on CGCC: (a) feed is located too low in the column; (b) feed is located too high in the column. For comparison,dashed line shows the CGCC when feed is properly located in the column.

potential for reflux modification in terms of the en-thalpy gap. In such cases, the utility consumption maybe significantly decreased by simply relocating the feedas explained graphically in Fig. 1. Fig. 1a shows atypical CGCC for the case where the feed is located toolow in the column. A sharp change in enthalpy isobserved at the feed stage (at a relatively high tempera-ture closer to the reboiler). This is due to the suddenjump in the driving forces in the column as may be seenon the x–y diagram (King, 1980). If the feed is locatedtoo high in the column, the CGCC (Fig. 1b) shows abehavior similar to that in Fig. 1a but it is flippedvertically in a sense. Fig. 1 includes the CGCC (as adashed line) for the case where the feed is properlylocated maintaining the same number of stages in thecolumn. On comparing the CGCCs for the properly-lo-cated and improperly-located feed, it is observed that asignificant reduction in the utility consumption is possi-ble due to the alteration in the feed location withoutchanging the number of stages. Furthermore, properlocation of the feed enhances the scope for reduction inenergy requirement by increasing the number of stages(often referred to as the scope for reflux modification).It is appropriate to target the reflux modification scopefrom the CGCC only after deciding where the feed is tobe introduced into the column. Thus, the primary goalof this paper is to establish a proper feed location targetthat will minimize the utility consumption for a fixednumber of stages and maximize the scope for energyconservation through reflux modification.

2.2. Reduction in simulation effort

Several simulation runs are usually required to studythe effect of reflux (or number of stages) and feedlocation, as well as determine their optimum values.Furthermore, the generation of a CGCC requires a

simulation run, when the number of stages and/or feedlocation are altered. For illustration, consider a binary(benzene–toluene) distillation column operating at 1.1kg cm−2 pressure with a feed of 100 kg-mol h−1 (atdew point and 1.2 kg cm−2 containing 50% benzene),and 99% product purity desired at both the top and thebottom. Fig. 2 shows three CGCCs for this binaryseparation generated from the following three simula-tions: (a) 20 total stages (including total condenser andreboiler) with the feed at the eighth stage (from the topof the column); (b) 20 total stages with the feed at the18th stage; and (c) 70 total stages with the feed at the25th stage. The three CGCCs are substantially differentin their appearance. However, Fig. 3 shows the datapoints from the three CGCCs (Fig. 2) unified intosimply two curves. In fact, it is possible to coalesce allthe CGCCs for this binary separation problem (corre-sponding to different feed stage locations and totalnumber of stages in the distillation column) into a pairof master curves (which may be called invariant rectify-ing-stripping (IRS) curves). Therefore, an importantmotive of this work is to establish the IRS curves andconsequently reduce the tedium involved in performingrepeated simulations.

3. Invariant rectifying-stripping curves

The invariant rectifying-stripping (IRS) curves aredefined below based on a derivation for a simple distil-lation column (with a single feed and two products) atthe minimum thermodynamic condition (MTC). TheMTC is defined as reversible operation for a columnwith no entropy generation. It corresponds to a columnwith infinite stages having a side exchanger at everystage [as discussed in detail by Bandyopadhyay et al.(1998)]. Furthermore, the operating curve coincideswith the equilibrium curve at MTC (King, 1980).

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Fig. 2. CGCC for benzene–toluene system at different reflux ratios (number of stages) and feed locations: (a) 20-stage column with feed at stage8; (b) 20-stage column with feed at stage 18; (c) 70-stage column with feed at stage 25.

3.1. In6ariant rectifying cur6e (T 6s. HR)

For the envelope in Fig. 4a, the overall mass balanceand component balance are

Vmin=Lmin+D (2)

Vminy*=Lminx*+DxD (3)

Eqs. (2) and (3) yield the minimum flows for liquid(Lmin) and vapor (Vmin) to be

Lmin=D(xD−y*)/(y*−x*) (4)

Vmin=D(xD−x*)/(y*−x*) (5)

The enthalpy balance for the envelope is used to evalu-ate the enthalpy surplus (HR) from

VminHV=LminHL+DHD+HR (6)

Eqs. (4)–(6) may be combined to yield the following:

HR=D [HV(xD−x*)/(y*−x*)

−HL(xD−y*)/(y*−x*)−HD] (7)

Eq. (7) may be rewritten in terms of the slope of the(rectifying) line joining a point on the operating curvewith the distillate point (xD, xD) as shown in Fig. 4b.Thus,

HR=D [(HV−HLSR)/(1−SR)−HD] (8)

where SR=slope of the rectifying line= (xD−y*)/(xD−x*)=Lmin/Vmin.

The quantity HR signifies the minimum condenserload required to carry out a separation from x* to xD.By rotating the rectifying line from y*=xD to x*=xB

with the distillate point (xD, xD) as the pivot, its slope iscontinuously varied and the enthalpy surplus (HR) cal-culated from Eq. (7) or Eq. (8) (for all possible valuesof reflux). This enthalpy surplus is then plotted as afunction of temperature to give a T versus HR curvewhich may be termed the invariant rectifying curve. Atypical invariant rectifying curve is shown in Fig. 4c.

3.2. In6ariant stripping cur6e (T 6s. HS)

For the envelope in Fig. 5a, the analogs for Eqs.(2)–(5) are

Lmin=Vmin+B (9)

Lminx*=Vminy*+BxB (10)

Lmin=B(y*−xB)/(y*−x*) (11)

Vmin=B(x*−xB)/(y*−x*) (12)

The enthalpy deficit (HS) for the envelope in Fig. 5amay be determined from

LminHL+HS=VminHV+BHB (13)

On combining Eqs. (11)–(13), the following expressionfor HS is obtained.

HS=B [HV(x*−xB)/(y*−x*)−HL(y*−xB)/(y*−x*)

+HB] (14)

An alternative form of Eq. (14) may be obtained interms of the slope of the (stripping) line joining a point

Fig. 3. Invariant rectifying-stripping (IRS) curves based on coalescingthe CGCCs from Fig. 2.

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Fig. 4. Generation of invariant rectifying curve: (a) rectifying section of a column for determination of enthalpy surplus; (b) rectifying line on x–ydiagram; (c) typical invariant rectifying curve.

on the operating curve with the bottoms point (xB, xB)as shown in Fig. 5b. On denoting the slope of thestripping line by SS= (y*−xB)/(x*−xB)=Lmin/Vmin,

HS=B [(HV−HLSS)/(SS−1)+HB] (15)

The quantity HS signifies the minimum reboiler loadnecessary to carry out a separation from x* to xB. Asbefore, the stripping line may be rotated from x*=xB

to y*=xD with the bottoms point (xB, xB) as the pivotand the enthalpy deficit (HS) continuously computedfrom Eq. (14) or Eq. (15) (for all possible values ofreboil). This enthalpy deficit is then plotted as a func-tion of temperature to yield a T versus HS curve whichmay be termed the invariant stripping curve. Fig. 5cshows a typical invariant stripping curve.

3.3. IRS cur6es

When the invariant rectifying curve (Fig. 4c) and theinvariant stripping curve (Fig. 5c) are plotted on thesame T–H axis, the invariant rectifying-stripping (IRS)curves (T–HR–HS) are obtained (see Fig. 3). Details ofthe procedure for actually generating such IRS curvesare provided in the application examples discussedlater.

Physically, the IRS curves correspond to the enthalpysurpluses and deficits for the rectifying and strippingsections, respectively, for all possible values of refluxand reboil. It must be emphasized that the enthalpysurpluses and deficits are calculated on the basis of theminimum flows by neglecting the effect of the feed. Thecurves extend from TB to TD on the temperature scale.They correspond to the MTC (rather than merely theminimum reflux or infinite stages) and consequentlyrepresent heat cascades based on an infinite number ofside exchangers.

The invariance property of these curves is discussed

next. A binary two-phase system has exactly two de-grees of freedom as per Gibb’s phase rule. On specify-ing the operating pressure and the separation, thedistillation problem becomes deterministic. Therefore,HR and HS are functions of temperature only. In otherwords, the IRS curves are invariant to the feed locationand the operating reflux for a distillation system whoseoperating pressure and separation are specified.

The IRS curves may be used to target the feedlocation and the minimum energy requirement for thedistillation system as described next.

4. Feed location target

The material, component, and enthalpy balances forthe overall column (Fig. 6) are

F=D+B (16)

FzF=DxD+BxB (17)

DHD+BHB−FHF=Qr−Qc D (18)

Eq. (18) shows that the parameter D (which is theconstant enthalpy difference for the utility requirementsof the column from the first law of thermodynamics)really depends on the separation problem (i.e. D, HD,B, HB, F and HF) and not on the column operation (i.e.Qr and Qc).

Eqs. (16)–(18) may be combined with Eqs. (7) and(14) to determine the following relationship between HR

and HS:

HS=HR+F [HL(zF−y*)/(y*−x*)

−HV(zF−x*)/(y*−x*)+HF]+D (19)

The above relation may be employed to target the feedlocation. For this purpose, a fundamental analysisneeds to be performed at the feed stage.

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Fig. 5. Generation of invariant stripping curve: (a) stripping section of a column for determination of enthalpy deficit; (b) stripping line on x–ydiagram; (c) typical invariant stripping curve.

4.1. Feed location criterion

The material balance, component balance and en-thalpy balance at a feed stage (Fig. 6) are as follows:

Lin+Vin+F=Lout+Vout (20)

LinxF* +VinyF* +FzF=LoutxF* +VoutyF* (21)

LinHL+VinHV+FHF=LoutHL+VoutHV (22)

Eqs. (21) and (22) assume the composition and molarenthalpy changes of the saturated liquids and vaporsover the feed stage are negligible. This assumptionholds when the feed stage is pinched or the column isoperating at the MTC. After some algebraic manipula-tions, Eqs. (20)–(22) give

HL(zF−yF* )/(yF* −xF* )−HV(zF−xF* )/(yF* −xF* )+HF

=0 (23)

On substituting Eq. (23) into Eq. (19), the followingrelation is obtained:

HS=HR+D at the feed stage (24)

Eq. (24) defines the criterion for the proper location ofthe feed at MTC. The next step is to work out theimplications of this criterion on the IRS curves.

4.2. Translated IRS cur6es

Given the fact that enthalpies are relative (i.e. theenthalpy difference is important rather than the abso-lute enthalpy), the IRS curves can be horizontallyflipped. Moreover, the invariant rectifying curve and/orthe invariant stripping curve may be translated horizon-tally. The following convention may be adopted totranslate the IRS curves in accordance with Eq. (24).Depending on the sign of D (as defined in Eq. (18)), thetranslations may be conveniently classified into threecases:

(a) If D=0 (i.e. Qr=Qc), then the invariant rectifyingas well as stripping curve need not be translated(Fig. 7a).

(b) If D\0 (i.e. Qr\Qc), then the invariant rectifyingcurve is translated to the right by D, with no shiftin the invariant stripping curve (Fig. 7b).

(c) If DB0 (i.e. QrBQc), then the invariant strippingcurve is translated to the right by �D� with no shift inthe invariant rectifying curve (Fig. 7c).

Fig. 6. Feed stage analysis of an MTC column.

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Fig. 7. Translated IRS curves: (a) D=0 (Qr=Qc); (b) D\0 (Qr\Qc); (c) DB0 (QrBQc).

Note that the condenser and reboiler loads are ap-proximately equal [case (a)] when the feed and productsare saturated liquids (Terranova & Westerberg, 1989),which is not an uncommon situation. The other twocases occur depending on the thermal condition of thefeed and products, e.g. case (b) can occur when the feedis a subcooled liquid and case (c) when the feed is asaturated vapor with the products being saturatedliquids.

Mathematically, the horizontal translations of theIRS curves may be represented as:

HRT=HR+D/2+ �D/2� (25)

HST=HS−D/2+ �D/2� (26)

Eqs. (24)–(26) may be combined to obtain HST=HRT

at the feed stage. Thus, the important conclusion is thatthe point of intersection of the translated IRS curves(as shown in Fig. 7) defines the target temperature forlocating the feed (TF).

4.3. Feed stage location methodology

Eq. (24) defines the feed location target in terms oftemperature. Note that this feed location target is inde-pendent of the operating reflux (in contrast to thegraphical McCabe–Thiele and Ponchon–Savarit meth-ods). The target may be converted from temperature tostage number using the following methodology. Thefirst step is to calculate HRT and HST from Eqs. (7),(14), (25) and (26) using stage-by-stage values outputtedby a converged simulation of a distillation column.Having generated HRT and HST as functions of bothstage number N and temperature T, the second step isto superpose the data points on the IRS curves (asshown in Fig. 8, where each open circle corresponds toa stage in the column). The feed stage in the columnsimulation is shown by a filled-in circle. If the feed isproperly located (Fig. 8a), then the feed stage will bevery close to the intersection point of the translatedIRS curves. Furthermore, the rectifying and stripping

sections of the simulated column will correspond to theportion of the invariant rectifying curve below TF andthe portion of the invariant stripping curve above TF,respectively.

When the feed is not properly located (Fig. 8b), thenthe stage number corresponding to the intersectionpoint of the translated IRS curves (denoted by NI) willnot coincide with the feed stage number in the columnsimulation (denoted by NF). Then, the feed stage relo-cation may be performed using N %F=mean(NF, NI).Here, N %F denotes the stage to which the feed needs tobe relocated in the next simulation and therefore re-quires to be rounded off to the nearest integer. Clearly,convergence to the proper feed stage location isachieved when N %F=NF. Experience shows that thecorrection for the feed location based on the geometricmean leads to fast convergence (typically, within aboutthree iterations).

Conceptually, the above methodology attempts tosystematically redistribute the open circles (correspond-ing to the column stages) over the IRS curves such thatthe filled-in circle (representing the feed stage) is eitherjust above or just below the intersection point of thetranslated IRS curves. It must be noted that themethodology may be implemented directly with thesimulation output (without plotting the circles on theIRS curves) by ensuring that the feed stage temperaturefrom the simulation matches the target temperature TF.

5. Minimum energy target

If the feed is properly located at TF, then the abso-lute minimum energy requirements for a binary distilla-tion process may be established as follows. The portionof the invariant rectifying curve below TF and theportion of the invariant stripping curve above TF maybe circumscribed by a right-angled trapezium. Then, thepinch on the IRS curves is defined as the point touchingthe vertical side of the trapezium. The widths of theparallel sides of the trapezium at the top and bottom

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Fig. 8. Locating the feed stage on translated IRS curves: (a) feed is located appropriately; (b) feed is located too high in the column.

define the minimum energy targets for the reboiler andcondenser, respectively (see Fig. 7). These minimumenergy targets are related to and, in a sense, define theminimum reflux target. As an aside, it may be notedthat the operating reflux target (for grassroots cases)and the reflux modification target (for retrofit cases)involve a cost optimization where the tradeoff betweenutility cost (based on the reboiler and condenser loads)and capital cost (based on the column diameter andnumber of stages) needs to be explored.

Fig. 7 illustrates the case where the intersection pointof the translated IRS curves determines the pinch. Thisis often the case; mathematically, it requires the IRScurves to be monotonic in nature. However, exceptionsexist as demonstrated in the example later (on thenon-ideal acetic acid–water system).

5.1. Relation between IRS cur6es and CGCC

The invariant property of the IRS curves allows theCGCC to be readily constructed from a knowledge ofthe stage temperatures. Fig. 9 shows how the CGCCs inFig. 2 may be derived from the IRS curves in Fig. 3after appropriate translation as per Eqs. (25) and (26).The CGCCs (horizontally flipped with respect to thosein Fig. 2) are shown by heavy lines and the IRS curvesby dashed lines in Fig. 9. To visualize the CGCCs asshown in Fig. 2, it is convenient to reflect the heavylines on Fig. 9 in mirrors represented by vertical lines ata distance (on the abscissa) corresponding to the actualcondenser or reboiler load (whichever is larger). Thisensures consistency with the convention (Dhole &Linnhoff, 1993; Shenoy, 1995) where the enthalpy gapsat the top and bottom on the CGCCs denote the actualheat duties of the reboiler and condenser, respectively.The mathematical relation to obtain the CGCCs (with-out additional enthalpy calculations) by exploiting theinvariant property of the IRS curves is given by

HCGCC=QC+Hdef (27)

where Hdef= −HR in the rectifying section, and Hdef=−HS+D in the stripping section of the column.

The maximum scope for decrease in utility consump-tion by reflux reduction (i.e. by increasing the numberof stages) is given by the enthalpy gap (horizontaldistance) of the pinch point from the vertical mirror.The pinch is defined as the point closest to the verticalmirror. For the IRS curves, only the portion of theinvariant rectifying curve below TF and the portion ofthe invariant stripping curve above TF must be consid-ered while determining this enthalpy gap. As expected,Fig. 10 (which is a magnified view of Fig. 9a) shows thescope for reflux reduction according to the pinch basedon the IRS curves (on appropriately locating the feed)to be higher than that based on the CGCC. Accord-ingly, Eq. (1) may be written more accurately as

Qr−Qr,min=Qc−Qc,min=Qr−HS,TF=Qc−HR,TF

:Dl(R−Rmin) (28)

where HR,TF and HS,TF are the enthalpy values at TF onthe (not translated) invariant rectifying and strippingcurves, respectively.

6. Application to binary distillation

The enthalpy surplus (HR) and the enthalpy deficit(HS) may be directly evaluated from Eqs. (7) and (14)provided the enthalpies and compositions on the stageas well as for the feed and products are known. Forbinary systems, thermodynamic models for enthalpyand vapor–liquid equilibrium may be readily used forthis purpose. In general, the right-hand-sides of Eqs. (7)and (14) may be conveniently calculated for each stageof a distillation column from the output of a converged

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Fig. 9. CGCCs and their relation to IRS curves for benzene–toluene system: (a) 20-stage column with feed at stage 8; (b) 20-stage column withfeed at stage 18; (c) 70-stage column with feed at stage 25.

simulation. In the examples that follow, the simulationsare performed using the PRO/II (1994–95) softwarebased on the problem data given in Table 1. Stagenumbering starts from the top of the column with 1denoting the condenser. The IRS curves are generatedusing the DISTARG (1998) software, which is primar-ily based on Eqs. (7) and (14). The results for all theexamples are summarized in Table 2 for readycomparison.

6.1. Benzene– toluene example

Fig. 11a shows the CGCC for this binary distillationproblem based on a simulation of a column comprising20 total stages (including total condenser and reboiler)with the feed at the 15th stage. The simulated columnhas an operating reflux ratio of 5.2384 with reboiler andcondenser duties of 1.5152×106 and 2.2976×106 kcalh−1, respectively. The CGCC pinch shows the energysavings potential by reflux reduction to be 0.0608×106

kcal h−1. Consequently, the minimum reflux ratio fromEq. (1) is 5.0734.

Fig. 11b shows the translated IRS curves (on notingthat D= −0.7824×106 kcal h−1). Their point of inter-section provides the target temperature for locating thefeed as 101.23°C. The parallel sides of the right-angledtrapezium (that appropriately circumscribes the curves)establish the minimum load targets for the reboiler andcondenser to be 0.5805×106 and 1.3629×106 kcalh−1, respectively. The corresponding minimum refluxratio is 2.7005. As the IRS curves are monotonic, theirintersection point also denotes the pinch.

On superposing the circles corresponding to the stagetemperatures (not shown), it is observed that the feed atthe 15th stage is located too low in the column. Then,on using the feed stage location methodology (discussedearlier) and relocating the feed based on the geometricmean correction, the circles redistribute themselves suchthat the feed stage (filled-in circle) approximately coin-cides with the intersection point of the translated IRS

curves (Fig. 11b) when the column is re-simulated withthe feed on the tenth stage. The feed at the tenth stageyields reboiler and condenser loads of 0.8411×106 and1.6235×106 kcal h−1, respectively. These are the mini-mum duties for a 20-staged column as may be verifiedthrough a simulator by varying the feed location.

Simulation of a 100-stage column with the feed at the50th stage yields an operating reflux ratio of 2.7024with a feed stage temperature of 101.3°C and therebyvalidates the targets (as shown in Table 2).

6.2. Acetic acid–water example

Fig. 12a shows the CGCC for this non-ideal binarydistillation problem generated from a simulation of acolumn consisting of 30 total stages (including totalcondenser and reboiler) with the feed at the 29th stage.The column simulation gives an operating reflux ratioof 12.1404 with reboiler and condenser duties of10.3121×106 and 10.3078×106 kcal h−1, respectively.The CGCC pinch shows the scope for energy savingsby reflux reduction to be 0.4259×106 kcal h−1 and thecorresponding minimum reflux ratio to be 11.5974.

The translated IRS curves (Fig. 12b) show an inter-section point at 99°C, which defines the target tempera-ture for locating the feed in this problem. Thetrapezium (which appears more of a rectangle becauseD of 0.0043×106 kcal h−1 is practically insignificant)appropriately circumscribes only the portion of theinvariant rectifying curve below 99°C and the portionof the invariant stripping curve above 99°C. The invari-ant rectifying curve is not monotonic and consequently,the pinch does not coincide with the intersection pointof the IRS curves. Rather, there is a tangent pinch inthe rectifying section. The trapezium shown in Fig. 12bhas its vertical side just touching this pinch with theparallel sides defining the targets for the minimumreboiler load and minimum condenser load to be2.2951×106 and 2.2908×106 kcal h−1, respectively.The corresponding minimum reflux ratio as per Eq. (1)is 1.9203.

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Fig. 10. Minimum energy targets and scope for reflux modification from (a) CGCC and (b) IRS curves.

On superposing the circles corresponding to thestages on the IRS curves (not shown), it is observedthat the feed at the 29th stage is located too low in thecolumn. Use of the feed stage location methodologyand re-simulation of the column with the feed on the25th stage results in the filled-in circle (feed stage)relocating itself in the immediate neighborhood of theintersection point on the translated IRS curves (Fig.12b). As in the benzene–toluene example, it can beproven through simulations that this feed stage locationyields the minimum reboiler and condenser duties forthe given number of stages.

Simulation of a 300-stage column with the feed ap-propriately located at the 276th stage results in anoperating reflux ratio of 1.960 with a feed stage temper-ature of 99.1°C thereby validating the targets (Table 2).Note that the ratio of the feed stage number to the totalnumber of stages is not constant for a non-ideal system.

Finally, it may be noted that the minimum energytargets in Table 2 based on the IRS curves differconsiderably from those based on the CGCC becausethe feeds are grossly mislocated. The difference wouldnot be dramatic had the feed been properly locatedduring the CGCC generation. Importantly, the IRScurves (being independent of feed location) not onlyprovide the true minimum energy target even with agrossly mislocated feed, but also provide a target forproperly locating the feed.

7. Application to multicomponent distillation

For multicomponent distillation, many methods existto approximately predict the minimum energy require-ments through plate-to-plate calculations (i.e. simula-tion). Koehler, Aguirre and Blass (1995) have reviewedthe most important methods for calculating minimumenergy requirement for ideal and nonideal distillation

including the work of Doherty and co-workers (Levy,VanDongen and Doherty, 1985; Knight & Doherty,1986; Julka & Doherty, 1993). Koehler et al. (1995)observed that most methods are based on the simula-tion of a portion or of an entire column, and a ‘univer-sal method which finds minimum energy consumptionfor nonideal and multicomponent distillation, at thetouch of a button, has not yet been developed.’

The IRS-based method, discussed below, is a simula-tion-based approximate method to predict minimumenergy requirement for general multicomponent distilla-tion. However, in addition to minimum energy, targetsfor feed location, feed conditioning and side-exchangerscan be simultaneously set through the IRS curves asdiscussed in the following sections.

For reversible multicomponent distillation, the de-grees of freedom are still two (Koehler, Aguirre &Blass, 1991). By arguments analogous to those for thebinary case, the system becomes deterministic. The IRScurves are invariant to the feed location and the operat-ing reflux on specifying the operating pressure and theseparation. But, the sharpness of separation is generallylimited in reversible multicomponent distillation(Fonyo, 1974; Franklin & Wilkinson, 1982). However,this limitation can be overcome during the generationof IRS curves using the pseudo-binary concept of alight and heavy key (Fonyo, 1974) that defines a practi-cal near-minimum thermodynamic condition (Dhole &Linnhoff, 1993).

For illustration, consider a multicomponent (hep-tane-octane-nonane-decane-C15 example in Table 1)distillation column operating at 200 kPa pressure with afeed of 1000 kg-mol h−1 (at 100°C and 200 kPacontaining 20% of each component), and 0.9% octanedesired at the bottom and 0.9% nonane at the top. Forthis multicomponent separation (more details of whichare available in the next sub-section), Fig. 13 showsdata for rectifying (T vs. HR) and stripping (T vs. HS)

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Table 1Data for various examples

System Acetic acid–waterBenzene–toluene Heptane-octane-nonane-decane-C15NRTL with enthalpy from LKThermodynamic method SRKSRK1 kg cm−21.1 kg cm−2 200 kPaOperating pressure

100 kg mol h−1 100 kg mol h−1Feed data 1000 kg mol h−1Flow-rate50% each 69% waterComposition 20% each

Pressure 1.2 kg cm−2 1 kg cm−2 200 kPaDew point Bubble pointTemperature 100°C99% benzene 86.1% waterSpecifications 0.9% nonaneTop99% tolueneBottom 0.9% octane99% acetic-acid

Table 2Comparison of targets from IRS curves with CGCC targets and simulation resultsa

Benzene–toluene Acetic acid–water Multicomponent

SIMULATIONSIMULATION CGCCIRSCGCCIRSSIMULATIONCGCCIRSNt=300, NF=276 Nt=180, NF=80Nt=100, NF=

50

1.3629 1.3635 2.2908Qc,min 9.88192.2368 2.3218 14.8490 19.4265 14.8539

1.4544 57.65420.5810 2.2951 9.8861 2.3261 57.6494Qr,min 0.5805 62.2256

2.7024 1.2727 1.66505.0734 1.17342.7005Rmin 1.96011.59741.9203

N.A.101.23TF (°C) 159.599.00101.3 N.A. N.A.159.4199.198.36 115.2 98.3 151.1 and 167.6 158.1106.1 101.3101.23TP (°C) 151.2 and 167.8

a Qc,min and Qr,min in 106 kcal h−1 for benzene–toluene and acetic acid–water systems. Qc,min and Qr,min in MMBtu h−1 for multicomponentsystem.

curves generated from five different simulations onvarying the feed stage (NF) and the total number ofstages (Nt). The three data sets corresponding to rela-tively low reflux ratios (R=1.462, 1.315, and 1.305, i.e.close to the minimum reflux Rmin:1.173) practicallydefine unique rectifying and stripping curves; however,as the reflux ratio increases, the data sets (for R=3.404and 5.190) show a certain degree of deviation. Thus, thecurves show near-invariance, to the total number ofstages and feed location, close to the minimum reflux.

The invariant property of the IRS curves does nothold rigorously for multicomponent systems becausethe distribution of the mole fractions of the compo-nents depends on the operating reflux of the column.Stupin and Lockhart (1968) noted that this distributionbears a non-linear relationship for any finite operatingreflux. The separation of the extreme components im-proves with decreasing reflux and the other way aroundfor intermediate components. From the Gilliland (1940)stage-reflux correlation, it may be noted that the changein reflux with stages is insignificant for a high numberof stages (typically, if N\3Nmin). The temperature vs.composition (T–x–y) and the enthalpy vs. composition(H–x–y) behaviors of pseudo-binary systems do not

change significantly (Johnson & Morgan, 1985; Cam-pagne, 1993) near the minimum reflux for the column.Therefore, the IRS curves for any pseudo-binary systemcan be taken to be invariant of the number of stagesand the feed location for targeting purposes. Essen-tially, IRS curves for establishing targets in multicom-ponent systems must be generated through a simulationwith a high number of stages (i.e. at a low reflux ratio).The targets need to be finally verified through rigorouscolumn simulation.

7.1. Fi6e components example

The feed and product specifications for this exampleproblem are described by Dhole and Linnhoff (1993).The feed stage location, the thermodynamic methodand the two specifications used in the column simula-tion are however not explicitly reported. The 5-compo-nent distillation problem is simulated using the PRO/II(1994–95) software for a column with 18 total stages(including partial condenser and reboiler) and feed atthe ninth stage. The mole fractions of nonane in the topproduct and octane in the bottom product are bothspecified to be 0.009. The thermodynamic method is

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Fig. 11. Minimum energy targets for benzene–toluene system from (a) CGCC and (b) IRS curves.

Fig. 12. Minimum energy targets for acetic acid–water system from (a) CGCC and (b) IRS curves.

based on the SRK (Soave–Redlich–Kwong) equationof state. The condenser and reboiler duties (in MMBtuh−1) from the simulation are 39.72 and 82.52, respec-tively. These compare reasonably well with the valuesof 39.6 and 83.3 reported by Dhole and Linnhoff(1993). The simulation shows the condenser and re-boiler temperatures to be 140.4 and 207.4°C, respec-tively. Although the condenser temperature compareswell with the value of 140.3°C given by Dhole andLinnhoff (1993), the reboiler temperature is about3.6°C higher than their reported value of 203.8°C.

For the purpose of generating the CGCC and theIRS curves (Fig. 14), heptane and octane are groupedas the light keys whereas the rest are grouped as theheavy keys in a manner similar to that adopted byDhole and Linnhoff (1993). The distance of the CGCCpinch (which occurs at the eighth stage on Fig. 14a)from the temperature axis represents the scope forenergy conservation and is observed to be 20.29MMBtu h−1 (corresponding to a minimum reflux ratioof 1.6650 in contrast to an operating reflux ratio of3.4044). Dhole and Linnhoff (1993) observed that theCGCC shows scope for energy savings by about 20–22MMBtu h−1 through reduction in the reflux ratio.

Fig. 14b shows the translated IRS curves (D=42.8MMBtu h−1) based on a simulation of a 50-stagecolumn with the feed at the 20th stage. The point ofintersection of these curves provides the target tempera-ture for locating the feed as 159.41°C. The invariantrectifying and stripping curves are both not monotonicand consequently, two pinches are observed at 151.1and 167.6°C. The parallel sides of the circumscribingtrapezium establish the minimum reboiler and con-denser load targets to be 57.6494 and 14.8490 MMBtuh−1, respectively. The corresponding minimum refluxratio is 1.2727. On superposing the circles correspond-ing to the stage temperatures (not shown on Fig. 14b,but can be visualized from Fig. 14a), the feed at theeighth stage is observed to be the appropriate location.

Simulation of a 180-stage column with the feed be-tween the 80th and 90th stage yields the feed stagetemperature to be 159.5°C with reboiler and condenserduties of 57.6542 and 14.8539 MMBtu h−1, respec-tively. The pinch zones from the simulation are foundto be at 151.2 and 167.8°C. These values compare wellwith the targets. The reflux ratio target (1.2727) differsfrom the operating reflux ratio (1.1734) based on the

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Fig. 13. Near-invariant property of (a) rectifying and (b) stripping curves for multicomponent system at different operating conditions.

Fig. 14. Minimum energy targets for multicomponent system from (a) CGCC and (b) IRS curves.

simulation because of the approximate value of l usedin Eq. (28).

8. IRS curves for feed preconditioning and side-ex-changer targets

Although the focus in this paper has been on mini-mum energy and feed location targets, the IRS curveshave the potential to provide targets for feed precondi-tioning and side-exchangers.

Feed preconditioning causes a change in the feedenthalpy. If QF is the amount of heat exchanged withthe feed, then Eq. (18) shows that the parameter D willbe changed by the same amount. For the sake ofconcreteness, consider preheating the feed by QF for thecase (DB0) depicted by the translated IRS curves inFig. 7c. Then, the invariant stripping curve will furthermove to the right by an amount QF according to Eq.(26) as shown by the dashed curve in Fig. 15a. The newintersection point defines the target temperature for thefeed stage (denoted by TFP). The important conclusionfrom Fig. 15a is that the difference between the trans-lated IRS curves at a certain temperature TFP specifiesthe amount of heat required to change the feed stage

temperature from TF to TFP. Mathematically, (HRT−HST)�TFP=QF. Since either QF or TFP may be arbitrar-ily varied, the equation defines the feed preconditioningtarget through a continuous mapping between QF andTFP. Clearly, QF may be computed if TFP is specifiedand vice versa.

The feed preconditioning target based on the IRScurves is precise, in contrast to the fuzzy target (Colum-nTarget, 1994) based on the CGCC. The target fromthe CGCC is approximate because visual inspection isrequired to estimate the extent of the sharp enthalpychange in the CGCC profile near the feed location(Dhole & Linnhoff, 1993; Ognisty, 1995). In fact, suchsharp enthalpy changes in the CGCC may be caused byan inappropriate feed condition or position as observedby Dhole and Linnhoff (1993). Importantly, the IRScurves simultaneously target feed condition and posi-tion, allowing the feed to be properly located after feedpreconditioning using the methodology discussed inSection 4.3

Side exchangers provide increased opportunities forheat integration and reduction in utility costs. As theIRS curves are fundamentally based on the MTC, theydefine the maximum heat load that can be placed onside exchangers at specified temperature levels. Fig. 15b

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Fig. 15. Application of invariant rectifying-stripping curves for: (a) feed preconditioning targets; (b) side-exchanger targets.

shows corners truncated out of the trapezium (in Fig.7c). The upper corner depicts the maximum scope tosupply a portion (Qsr,max) of the required heat througha side reboiler at a temperature (Tsr) below that of themain reboiler. The lower corner shows the maximumpotential to remove a portion (Qsc,max) of the excessheat through a side condenser at a temperature (Tsc)above that of the main condenser. The side exchangertargets from the IRS curves are conceptually equivalentto those defined by earlier workers (Naka et al., 1980;Ho & Keller, 1987).

9. Conclusions

This work provides targets for distillation in terms ofthe minimum reboiler/condenser duties, feed locationand pinch. These targets (with the exception of feedlocation) are analogous to those discussed by Linnhoffet al. (1982) for heat exchanger networks (HENs). Theenergy targets for distillation from the IRS curves arebased purely on feed/products specifications and areestablished prior to column design. In a similar fashion,the energy targets for HENs from pinch analysis (com-posite and grand composite curves) are based purely onstreams specifications and are established prior to net-work design. The analogs of hot utility load, cold utilityload, and DTmin in HENs are reboiler duty, condenserduty, and reflux ratio in distillation. The problem ofdeciding loads amongst multiple utilities in HENs isequivalent to the case of distributing duties betweenside-reboilers and side-condensers in distillation. TheIRS curves have the potential of providing targets forside exchangers on recognizing that the portion of theIRS curves within the circumscribed trapezium corre-sponds to the grand composite curve in HENs atDTmin=0°. The significance of the pinch, in the contextof distillation, may be stated as follows: no (side-)reboiling below the pinch and no (side-)condensingabove the pinch. This is consistent with the observa-

tions of Naka et al. (1980) and Agrawal and Fidkowski(1996).

It may be noted that the minimum energy targetfrom the IRS curves is exact for any binary system,irrespective of its chemical nature. This is superior tothe prediction from the minimum reflux equation ofUnderwood (1948), which assumes constant relativevolatility and constant molar overflow. However, IRScurves for multicomponent systems are based on apseudo-binary approach and this limitation does notexist in Underwood’s method. As discussed earlier, thelimitation may be overcome by generating IRS curvesfrom a simulation of a column with a large number ofstages (i.e. close to the minimum reflux).

The key representation proposed in this work is theIRS curves. Earlier studies had generated only portionsof such T–H curves and, consequently, failed to recog-nize their invariance to feed location. Thus, IRS curvesallow targets to be established for feed location. In fact,IRS curves simultaneously target feed location andminimum energy (which is equivalent to the scope forreflux modification) ahead of configuring the column. Itis inappropriate to locate feed and reduce reflux se-quentially as recommended by earlier works.

Multiple simulations are not required to generate theIRS curves and establish targets from them. On theother hand, multiple simulations are required to settargets based on the CGCC because a new CGCCneeds to be generated after each column modification.Thus, the simulation work is reduced (i.e. no simulationis necessary for binary systems and a single simulationis required for multicomponent systems) during target-ing by IRS curves. Multiple simulations are necessaryonly for configuring the column (i.e. to locate the feedstage appropriately in a column with finite number ofstages).

A preliminary design procedure (Kister, 1992) in-volves four typical steps to determine: (1) minimumnumber of stages (e.g. by Fenske equation); (2) mini-mum reflux (e.g. by Underwood’s equation); (3) actual

1123

number of stages for a given reflux (e.g. by Gilliland’splot or a stage-reflux correlation); and (4) feed location(e.g. by Kirkbride’s equation). This work provides animproved methodology for steps (2) and (4). In otherwords, steps (1) and (3) are done as before. Theoptimum reflux could be established by minimizing thetotal annual cost target starting with the IRS curves(Bandyopadhyay, 1999). Thus, the IRS-based methodprovides a systematic energy analysis tool to generatea thermodynamically efficient configuration of a distil-lation column while using some of the existing methodsof preliminary design. Further, the possibility for inte-gration (Ho & Keller, 1987) of the distillation columnwith the background process can be conveniently ex-plored on a temperature–enthalpy diagram since thebehavior of the column is depicted by a set of invariantcurves.

It may be noted that for retrofit cases, feed relocationis easier to implement and has no major cost implicationwhen compared to reflux modification. Appropriatelyrelocating the feed may also help in debottlenecking.Furthermore, IRS curves are independent of the en-thalpy of the feed and allow targeting for feed precon-ditioning (preheating/cooling). Current work is directedtowards targeting the efficiency of feed preconditioningin terms of impact on reboiler/condenser loads.

10. Notation

B bottom product molar flowCGCC column grand composite curve

distillate molar flowDF feed molar flow

enthalpyHIRS invariant rectifying-stripping (curves)

liquid molar flowLLee–KeslerLKminimum thermodynamic conditionMTCstage numberN

NRTL non-random two-liquidheat dutyQreflux ratio (L/D)RSoave–Redlich–KwongSRKslope of rectifying/stripping lineStemperatureTvapor molar flowVmole fraction in liquidxmole fraction in vaporymole fraction in feedzenthalpy difference defined in Eq. (18)Dheat of vaporizationl

Subscriptbottom productB

condenserccolumn grand composite curveCGCCdistillateD

def deficitfeedFpreconditioned feedFP

I intersection point on translated IRScurves

in in to (entering) a stageL liquid

maximummaxmin minimum

out of (leaving) a stageoutpinchpreboilerrrectifying curveRrectifying curve (translated)RTside condensersc

sr side reboilerstripping curveSstripping curve (translated)STtotal (stages)tat temperature TFTFvaporV

Superscriptequilibrium condition*

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