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Shortcut Models and Feasibility Considerationsfor Emerging Batch Distillation Columns

Stefan P. Lotter, and Urmila M. DiwekarInd. Eng. Chem. Res., 1997, 36 (3), 760-770• DOI: 10.1021/ie960632n • Publication Date (Web): 03 March 1997

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SEPARATIONS

Shortcut Models and Feasibility Considerations for Emerging BatchDistillation Columns

Stefan P. Lotter and Urmila M. Diwekar*

Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

The transient nature and flexibility of batch distillation allow for configuring the column in anumber of different ways. Some of the new configurations are an inverted column, a middlevessel column, and a multivessel column. These new column configurations have also providednew ways of operation. The preliminary analysis of these emerging columns has shown promisingbehavior, because of the added flexibility. This added flexibility is especially interesting for achemical industry where the quantity and lifetime of the products are uncertain, but it has alsomade the analysis of the system more difficult. Shortcut procedures provide an easy way ofunderstanding the global behavior of complex systems. In this paper we are presenting shortcutprocedures for the newly described batch distillation column configurations. The transientprofiles obtained by the proposed shortcut procedures and rigorous models are compared usingextensive test cases. Global qualitative properties and feasibility criteria are derived for thesenew designs, and a detailed analysis of these configurations is also presented.

1. Introduction

Batch distillation is widely used and is an importantunit operation in the batch processing industry. Theflexibility of batch distillation combined with the inher-ent unsteady nature of the process poses challengingdesign and operation problems. The transient natureof the process allows for configuring of batch columnsin a number of different ways, some of which are shownin Figures 1 and 2. The column at the right in Figure1 is a conventional batch distillation column with areboiler at the bottom and a condenser at the top, whichessentially performs the rectifying operation. In con-trast to a conventional batch distillation column, theinverted column (Figure 1, left) has its storage vesselat the top and the products leave the column at thebottom. Thus, mixtures with a small amount of thelight component can be separated by removing theheavy component as the bottom product. These twocolumns are comparable to the rectifying and strippingparts of a continuous distillation column but withadditional flexibility.Devidyan et al. (1994) presented a batch distillation

column that combines both the rectifying and strippingsections, the middle vessel column (Figure 2, left).Although this column has not been investigated com-pletely, preliminary analysis has demonstrated that itprovides high flexibility and that it is able to removeboth light and heavy impurities by having three productvessels. For example, the composition of the mostvolatile component in a rectifier decreases with time,and the composition of the least volatile component ina stripper also decreases with time. In the middle vesselcolumn, however, these effects can be reversed bysetting the vapor ratios for the top and bottom parts ofthe column appropriately (Diwekar, 1995).

Recently, Skogstestad et al. (1995) described a newcolumn called a multivessel column (Figure 2, right) andshowed that the column can obtain purer products atthe end of a total reflux operation. With this column it

* Author to whom correspondence is addressed. Tele-phone: (412) 268-3003. Fax: (412) 268-3757. email: urmila@cmu.edu.

Figure 1. Inverted batch distillation column (stripper) andconventional batch distillation column (rectifier).

Figure 2. Middle vessel batch distillation column and multivesselbatch distillation column.

760 Ind. Eng. Chem. Res. 1997, 36, 760-770

S0888-5885(96)00632-X CCC: $14.00 © 1997 American Chemical Society

is possible to separate more than three components ata time by installing enough intermediate vessels. Itsdesign, however, is less flexible than the design of aconventional batch distillation column.The two basic modes of operating batch rectifying

columns are (1) constant reflux and variable distillatecomposition and (2) variable reflux and constant distil-late composition of the key component. There is a thirdmode of operation for operating a batch rectifyingcolumn that is neither variable reflux nor constantreflux but instead is in between the two. This type ofoperation is known as optimal reflux or optimal controlpolicy. The optimal reflux policy is essentially a trade-off between the two operating modes of constant refluxand variable reflux and is based on the ability to yieldthe most profitable operation. The calculation of thispolicy is a difficult problem and relies on optimal controltheory (Diwekar, 1995). Although not reported in theliterature, the stripper operating modes, like the batchrectifier, can be defined as (1) the constant reboil ratioand variable bottom composition, (2) the variable reboiland constant bottom composition of the key component,and (3) the optimal reboil policy. Figure 3 shows thefirst two operating modes for the stripper. For themiddle vessel, the combination of the three rectifyingand three reboil policies results in nine possible operat-ing policies. Additionally, one can manipulate (at eachtime step) the parameter q′, or the ratio of vapor boiluprate in the rectifying to the stripping section, which isan important parameter for the middle vessel. Fordifferent kinds of columns combined with differentmodes of operation, the number of possible columnconfigurations tends to be very high, which poses abewildering problem of selecting proper configurationand feasible design in the face of complexities. Althoughcomputers have made it possible to solve large-scalemodels in a reasonable amount of time, shortcut modelsare necessary to derive global properties, such asfeasible regions of operation, which are critical foroptimization, optimal control, and synthesis problems.Even if this information was available in the solutionof a rigorous model, the computational cost of iterativedesign, optimization, optimal control, or synthesis usingrigorous models would be too high. In addition, with ashortcut model it is not necessary to solve systems ofstiff differential equation, which poses an algorithmicproblem. The shortcut method for the batch rectifier(Diwekar, 1988; Diwekar and Madhavan, 1991) is based

on the FUG equations (Fenske, Underwood, Gilliland)and has been shown to be successful in identifying thefeasible parameter window for different operation modes,presenting criteria such as minimum reflux and mini-mum number of plates. The method is extremelyefficient and involves only noninteger variables. Thisshortcut model delivers bounds (feasibility window) ondesign variables, especially the number of plates andthe reflux ratio, and is thus useful in design andoptimization.In this paper, we are presenting shortcut procedures

for the newly described batch distillation column con-figurations, namely, the inverted column and the middlevessel column. The models are limited to constantreflux and reboil operation but can be easily extendedto other column configurations and operating modes.For both columns, the shortcut models are based on theassumption that a batch distillation column can beregarded as a continuous distillation column withchanging feed at each time step, modified to suit thedifferent configurations. This study is restricted to idealsystems in order to separate the thermodynamic com-plexity of nonideal and azeotropic mixtures and thecomplexities associated with the flexible, unsteady-statenature of the batch distillation column. The shortcutmodels presented here can be extended to azeotropicsystems based on the variable transformation suggestedin the earlier work (Diwekar, 1991; Kalagnanam andDiwekar, 1993). The transient profiles obtained by theproposed shortcut procedures and rigorous models arecompared using extensive test cases. Global qualitativeproperties and feasibility criteria are derived for thesenew designs, and a detailed analysis of these configura-tions is also presented to help understand the novelbatch distillation column configurations and their op-erating modes.The shortcut procedures neglect the liquid holdups

on the plate which can be a reasonable assumption forcertain applications. However, it should be noted that,similar to the rectifier, holdup affects the batch stripperand the middle vessel in two ways, namely, the dynamicflywheel effect and the steady-state capacitance effect.It is possible to use the compartmental modeling ap-proach to capture the capacitance effect with theshortcut procedures as suggested in Diwekar (1995).However, for the dynamic flywheel effect, the productcomposition profiles predicted by the shortcut methodshow rapid change initially as compared to the profiles

Figure 3. Two operating modes for a batch stripper.

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 761

of the column with large holdup effect, but as both showthe same trends, the shortcut procedure can be used topredict the trends even for columns with large flywheeleffect.

2. Inverted Batch Column (Stripper)

This section presents a shortcut model for the in-verted batch distillation column (stripper), based on theshortcut model for the batch rectifier (Diwekar andMadhavan, 1991). A comparison between the resultsof rigorous simulation and the proposed shortcut methodis made to validate the accuracy of the shortcut method.Using this method, design criteria and the window offeasible operation are provided.2.1. Shortcut Model for the Stripper. Similar to

the shortcut method for the batch distillation rectifier,the main assumption of the shortcut method for thestripper is that the batch distillation column is regardedas a continuous column with changing feed compositionat each time step, as shown in Figure 4. Other assump-tions include constant relative volatility (ideal systems),equal molal overflow, and negligible plate holdups.Thermodynamic equilibrium is assumed between thevapor and liquid leaving each plate (theoretical plate),and the composition of the liquid leaving the still isassumed to be the same as that of the still (well-mixedreservoir). Although the basis of the shortcut methodfor the stripper is similar to the shortcut method forthe rectifier, i.e., based on FUG (Fenske-Underwood-Gilliland) equations, there are differences in the columnbehaviors. The first difference is the definition of thereflux and reboil ratio. Unlike the reflux ratio, thereboil ratio does not vary between zero and infinity butbetween 1 and infinity. This is due to the definition ofthe reboil ratio (eq 1). This difference also reflects in

various equations for the shortcut procedure.For the system shown in Figure 4 we assume that

the distillation is carried out at constant boilup rate Vb.The constant molal overflow assumption leads to thefollowing overall material balance equation:

where F is the feed. A material balance for the keycomponent k over the differential time can be writtenas

where F is the feed to the still, S is the amountremaining in the still, and Rb is the reboil ratio. Thestill composition and bottom composition are given byxs and xbot, respectively. Equation 4 relates the changein the still composition to the bottom product composi-tion. The above two mass balance equations can beconverted to a time-implicit equation similar to theRayleigh equation for simple distillation and for batchrectifier. We will call this equation the modified Ray-leigh equation. The following is the finite differenceapproximation to this modified Rayleigh equation for ncomponents expressed in terms of the key componentk.

As seen in the case of the rectifier, the functionalrelationship between the end compositions is crucial forthe simulation of the complete operation, and it is herethat we need to use the design equations described bythe modified FUG method.Functional Relationship between xbot and xs. At

each instant, there is a change in the still compositionof the key component, resulting in changes in the stillcomposition of all the other components calculated bythe differential material balance equations (eq 5). Forrectifier, Hengestebeck-Geddes’ relation provides therelation between distillate composition and bottomcomposition. For stripper, this equation translates intothe following equation:

where CB is the Hengestebeck-Geddes’ (HG) constant.Fenske (1932) derived an equation to calculate theminimum number of plates for a continuous distillationcolumn with constant relative volatilities in terms of thedistillate composition and feed composition. It can beeasily shown that the Fenske equation for a stripper isgiven by

The minimum number of plates is the number ofequilibrium plates required for a separation at totalreflux conditions and is thus a boundary of the operatingconditions. If Hengestebeck-Geddes’ equation is com-pared with the Fenske equation for the minimumnumber of plates Nbmin for two components, Nbmin hasto be equal to CB. Since the summation of all compo-nents must be equal to 1 (∑i)1

n xbot(i) ) 1), an equation can

be found which relates the new bottom product compo-sition to the current still composition in terms of relativevolatilities and the minimum number of plates (eq 8).

Figure 4. Shortcut method for the stripper.

reflux ratio, R ) LdD/dt

reboil ratio, Rb ) VbdBot/dt

(1)

dSdt

) - VbRb

, S0 ) F (2)

xbot(k)VbRb

) -d(xs

(k)S)dt

(3)

dxs(k)

dt) Vb(Rb)S

(xs(k) - xbot

(k) ), xs0(k) ) xF

(k) (4)

xsnew(i) ) xsold

(i) +∆xs

(k) (xbot(i) - xs

(i))old(xbot

(k) - xs(k))old

, i ) 1, 2, ..., n

(5)

xbot(i) ) (Ri

Rk)-CBxs

(i)

xs(k)xbot(k) , i ) 1, 2, ..., n (i * k) (6)

Nbmin )

ln[xs(i)xs(k)

xbot(k)

xbot(i) ]

ln[Ri

Rk]

(7)

762 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997

For a binary mixture, the separation is between onlytwo components; in the case of a multicomponentmixture, however, the separation can be expressed interms of a binary mixture of two key components, LK,the lightest component appearing in the bottom, andHK, the heavy key component defined as the heaviestcomponent in the top. It should be noted that, for mostof the cases, the heavy key component in the stripperis the least volatile component in the mixture, as thestill contains all the feed. Unlike continuous distilla-tion, the key components keep changing for batchdistillation as time progresses. For example, the lightkey component will become lighter than light key, andthe next component in the relative volatility hierarchywill become the light key component.To get a second boundary of the operating condition,

the minimum reboil ratio Rbmin, Underwood’s (1948)equation for continuous distillation columns is appliedto batch distillation columns. At minimum reboil condi-tions, an infinite number of equilibrium stages isrequired to achieve the desired separation.

The q in the above equations represents the feedcondition and is defined as the ratio of heat required tovaporize 1 mol of the feed to the molar latent heat ofthe feed. The φ is the root of the Underwood equationwhich lies between the RLK and RHK. The shortcutmethod assumes that batch distillation can be consid-ered as continuous distillation with changing feed. Inother words, the top product of one time step forms thefeed for the next time step. This is equivalent to havingthe top plate as the feed plate and the feed at its boilingpoint, which means q is unity. Also, the feed composi-tion in the Underwood equations can be replaced by stillcomposition. Therefore, the Underwood equations forthe batch stripper shown in Figure 4 are given by

The Fenske and Underwood equations above providethe limiting boundary conditions for a batch stripperin terms of the still and bottom compositions. Theselimiting conditions should be related to the designvariables of the column such as the reboil ratio Rb andthe number of plates Nb, to complete the analysis. Inbatch rectifier, Gilliland’s correlation (eq 13, Gilliland(1940)) furnishes this information for relating the designvariables N and R with Rmin and Nmin.

For continuous distillation, from where the shortcutprocedure for the rectifier is derived, FUG equationssupply the complete design equations including thestripping section of the column. This is because in thecontinuous distillation the rectifying and strippingsections, and hence reflux and reboil ratios, are con-nected to each other by a steady-state material balanceequation. However, for the batch stripper or for themiddle vessel column, the steady-state balance equationdoes not exist. Therefore, a correlation similar toGilliland’s correlation needs to be obtained for thestripper. To get this correlation, we conducted a largenumber of systematic experiments using the new sam-pling technique (Diwekar and Kalagnanam, 1996). Thesamples are selected from various parameters whichinclude relative volatility (1.0 g R e 5.0), reboil ratio(3.0 g Rb e 12.0), number of plates (4 g Nb e 15), andstill composition (0.05 g xs

(1) e 0.95). At first the X andY factors were defined based on the original Gillilandcorrelation and plotted in Figure 5.

Unlike Gilliland’s original plot for continuous distil-lation, where points line up nicely to form a curve, theresults of the batch stripper in Figure 5 are widelydispersed. The preliminary analysis of the equationssuggested a relative volatility factor to be included inthe stripper correlation. The dependence of relativevolatility is visible in Figure 6, plotted using the newdefinitions of X and Y given below.

A logarithmic curve fit with the following correlationresulted in a value of R2 of 0.92.

The above correlation completes the shortcut modelfor the batch stripper. The following procedure il-lustrates the shortcut method for a single time step indetail for constant reboil mode of operation. Note thatchoosing the proper light key and heavy key componentsis crucial for the success of this method.2.2. Solution Procedure. At any time step the still

composition, xs(i), i ) 1, 2, ..., n, can be found from the

previous time steps using eq 5. Then the followingprocedure is used to obtain the bottom compositions.1. Assume the initial value of CB (0 < CB e Nb).2. Calculate the bottom product composition of the

key component k using the HG equation and summationof all composition.

xbot(k) )

1

∑i)1

n (Ri

Rk)-CBxs

(i)

xs(k)

(8)

∑i)1

n RixF(i)

Ri - φ) 1 - q (9)

-Rbmin ) ∑i)1

n Rixbot(i)

Ri - φ(10)

∑i)1

n Rixs(i)

Ri - φ) 0 (11)

-Rbmin ) ∑i)1

n Rixbot(i)

Ri - φ(12)

X )R - Rmin

R + 1; Y )

N - Nmin

N + 1

Y ) 1 - exp[(1 + 54.4X)(X - 1)

(11 + 117.2X)xX ] (13)

X )Rb - Rbmin

Rb; Y )

Nb - NbminNb + 1

X )Rb - Rbmin

Rbln(RLK/RHK); Y )

Nb - NbminNb + 1

Y ) 0.2478 - 0.0965 ln(3.784X) (14)

xbot(k) )

1

∑i)1

n (Ri

Rk)-CBxs

(i)

xs(k)

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 763

3. Substitute the value in Hengestebeck-Geddes’equation and find the bottom composition, xbot

(i) , i ) 1, 2,..., n.

4. Solve Underwood equations for φ and obtain thevalue of Underwood’s minimum reboil ratio, Rbminu.

5. Calculate Y using Nbmin equal to CB and solve themodified Gilliland correlation for the stripper for X toobtain Gilliland’s minimum reboil ratio, Rbming.

6. For the correct value of CB, Rbminu should be equalto Rbming. Therefore, the value of the quantity Gbcshould be zero (within a tolerance).

7. Calculate Gbc and find whether it is zero within atolerance. If Gbc is approximately zero, the solution isconverged for this time step; otherwise, use the New-ton-Raphson method to calculate the new CB, andrepeat all the steps from step 2.2.3. Model Validation. To verify the accuracy of

the shortcut model, several systematic experiments areconducted using the rigorous model (zero holdup) for thestripper implemented in MultiBatchDS (Diwekar,1996) and the shortcut model presented here. At firstbinary mixtures of relative volatility varying from 4.0to 1.15, reboil ratio from 3 to 12, and bottom plates from4 to 15 are used. Table 1 presents the 10 representativecases from this study. Table 1 also presents the averageerror in the prediction of the bottom and still composi-tions. From the 10 cases in Table 1, details of the twocases (case 1 and case 9) are presented in Figures 7 and8 which plot the composition profiles of the least volatilecomponent. These two cases represent an easy (highR) and a difficult (low R) separation, respectively. Table1 also reports four cases of the ternary mixture. It canbe seen that the shortcut model predicted reasonablyaccurate results for binary as well as multicomponentsystems. The accuracy of the shortcut method dependson the accuracy of the modified Gilliland correlation;therefore, for specific applicability one can improve thisempirical correlation further.Figures 7 and 8 also show that the bottom composition

of the least volatile component decreases as the distil-lation progresses, a condition used for deriving one ofthe feasibility criteria for the stripper.2.4. Feasibility of the Stripper. The determina-

tion of feasibility regions is crucial for optimization,synthesis, design, or control of a column. It is possibleto easily identify the feasibility region with the help ofthe shortcut method and impose bounds. This sectionpresents the feasibility criteria for the inverted columnoperating under constant reboil ratio mode and il-lustrates it with an example.For specified average bottom product composition, the

inverted batch distillation column has a lower boundRbMIN on the reboil ratio and the Fenske minimumnumber of plates provides the lower boundNbmin on thenumber of plates. For the stripper, the initial composi-tion of the least volatile component is the highest anddecreases as the distillation progresses (as observed inFigures 7 and 8). If the initial value of Rb is such thatthe bottom composition of the least volatile componentis less than the specified average, then the goal ofattaining the specified average purity is impossible tomeet for the given number of plates. This criterionprovides the lower limit RbMIN on the value of Rb, whereRbMIN is defined as the value of Rb required to obtainthe bottom composition of the least volatile componentequal to the specified average composition, for the givenNb. The reboil ratio at constant reboil policy has to beat least RbMIN to reach the desired product compositionat initial conditions even if in this case the amount ofproduct is almost zero. RbMIN is calculated by obtainingNbmin and Rbmin from Fenske’s and Underwood’s equa-tions by equating the bottom composition at the initialcondition to the average specified composition and thenusing the modified Gilliland correlation to calculate the

Figure 5. Gilliland’s plot for a batch distillation stripper.

Figure 6. Modified gilliland plot for a batch distillation stripper.

xbot(i) ) (Ri

Rk)-CBxbot

(k)

xs(k)xs(i), i ) 1, 2, ..., n (i * k)

∑i)1

n Rixs(i)

Ri - φ) 0

-Rbminu ) ∑i)1

n Rixbot(i)

Ri - φ

X )Rb - Rbming

Rbln(RLK/RHK); Y )

Nb - CB

Nb + 1

Y ) 0.2478 - 0.0965 ln(3.784X)

Gbc )Rbminu - Rbming

Rb

764 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997

Rb from the givenNb and the values of Rbmin andNbmin.It should be remembered that this Rb which is equal toRbMIN is different from Underwood’s Rbmin, whichstands for the minimum reboil ratio at an infinitenumber of plates and is also lower than RbMIN. Theminimum reboil ratio Rbmin increases with time, thusshowing that the separation becomes more difficult.Since the actual reboil ratio has to be higher than RbMIN,

the minimum reboil ratio serves as an indicator for thedifficulty of a separation and gives information for thequestion if a column can be realized economically. Table2 shows the feasibility region for the stripper.The example presented in Table 3 serves as an

illustration of the feasibility concept. An equimolarbinary mixture is fed into an inverted batch distillationcolumn and stripped off of the less volatile component.The purity of the bottom product was specified to bexbot-ave(2) ) 0.75. At first consider case I, the compositionprofile of the bottom product for this case is shown inFigure 9. One can easily see that the compositionprofile cannot achieve the product purity specification.The figure also shows the results of case II, where thereflux ratio is 10 times higher, and it is still not possibleto achieve the given specification. The reason can befound easily by calculating the lower bound ofNb, given

Table 1. Test Cases for the Validation of the Shortcut Method

test R1 R2 Nb Rb xF(1) average error for xs (%) average error for xbot (%)

1 3.85 1.0 7 9.75 0.59 0.003 0.012 3.55 1.0 10 10.875 0.23 0.002 0.013 2.95 1.0 5 11.437 0.374 0.054 0.114 2.8 1.0 14 6.937 0.194 0.003 0.115 2.35 1.0 13 10.312 0.518 0.004 0.016 2.05 1.0 6 8.062 0.158 0.065 0.1067 1.75 1.0 8 11.718 0.662 0.447 0.9518 1.6 1.0 4 7.218 0.482 0.558 1.1049 1.45 1.0 14 9.468 0.302 0.078 0.55210 1.15 1.0 6 10.593 0.806 2.845 6.269

test R1 R2 R3 Nb Rb average error for xs (%) average error for xbot (%)

1 1.7 1.2 1.0 3 8 0.257 0.0772 2.5 1.3 1.0 7 6 6.00 1.6003 4.5 2.3 1.0 5 6 1.277 0.5714 5.6 3.3 1.0 5 4 0.183 0.741

Vb ) 100 mol/h F ) 100 mol/h T ) 3 h

Figure 7. Validation of the shortcut method for a batch stripper.

Figure 8. Validation of the shortcut method for a batch stripper.

Figure 9. Concept of Nbmin.

Table 2. Feasible Region for a Batch Stripper (ConstantReboil Mode)

final still composition 0 e xs∞

(HK) e xbot(HK)

bottom composition xbot(HK) e xbot

(HK) e 1reboil ratio 1 e Rb / RbMIN e ∞number of plates Nbmin e Nb

Table 3. Input for Illustration of the Nbmin and RbMINConcept

parameter case I case II case III case IV

Rb 5 50 3.0 2.0Nb 1 1 6 6Vb 100 100 100 100.0R1 1.5 1.5 1.5 1.5R2 1.0 1.0 1.0 1.0

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 765

by Nbmin, and it was found to be 2.7 (including thereboiler). Since Nb should be greater than Nbmin(actually Nbmin - 1 when reboiler is excluded), it wasimpossible to attain a given separation. Now considercases III and IV where Nb was increased from 1 to 6.The lower bound for the reboil ratio, RbMIN, was foundto be 2.668 for the given value of Nb, and rigoroussimulations with R g RbMIN (case III) and R e RbMIN(case IV) are computed. Figure 10 shows that, in caseIV, the specified product conditions could not be reachedbecause the reboil ratio was too small, whereas in caseIII, the desired composition could be reached at thebeginning of the operation. To obtain a higher amountof product, the actual reboil ratio has to be increasedappropriately.

3. Middle Vessel Column

The middle vessel column was proposed by Devidyanet al. (1994). This column consists of a rectifier and aninverted column connected by a still vessel (Figure 2).Hasebe et al. (1995) describe this connection as a heatintegration for the rectifier and stripper. This config-uration enables the simultaneous separation of light andheavy impurities and therefore offers more flexibility.In previous works, the directions a separation can takehave been examined and the existence of equilibriumplateaus has been proved (Devidyan et al., 1994).Rigorous simulations of this column with

MultiBatchDS (Diwekar, 1996) show a promising be-havior, as it can reach higher purities than conventionalbatch distillation columns and offers greater operationflexibility. If, for example, a binary mixture specifiedin Table 4 with an equimolar feed is distilled for 3 h toachieve 30 mol of distillate and bottom product, theresults will differ from similar operations with a rectifieror a stripper. The average compositions of distillate andbottom product for the middle vessel are about 4%higher, even if none of the other parameters werechanged and the same amount of distillate and bottomproduct was gathered, compared to the compositions ofsimilar operations with a rectifier and a stripper.The middle vessel column makes it possible to copy

the behavior of a column operated at variable reflux

without having to change the reflux or reboil ratio. Thegoal of variable reflux operation is to maintain aspecified product composition by changing the refluxratio. This can be achieved in a middle vessel columnby keeping the composition of the middle vessel con-stant. This means that the vapor boilup rates for thetop and bottom part of the column and reflux and reboilratios have to be designed in a way to make up thedifference in mass balance equation (e.g., Figures 11 and12 for q′ ) Vt/Vb ) 1). Since this column consists of arectifier and an inverted column both connected by thereservoir in the middle, its behavior becomes lesspredictable than a conventional column. For example,at constant reflux and reboil operation, the compositionof the heavy key in the bottom product can increase withtime (Figure 12, q′ ) 10), whereas in an inverted batchdistillation column, it would decrease. A similar effectcan be shown for the distillate (Figure 11, q′ ) 0.1). Thisbehavior results from an increase of the composition ofthe light key component in the middle vessel due to

Table 4. Comparison of the Middle Vessel, Rectifier, and Stripper

column R1 R2 R Rb Nt Nb Vt Vb xDav(1) xs(1) xbot-ave

(2)

rectifier 1.7 1.0 4.0 8 50 0.9194 0.3202stripper 1.7 1.0 5.0 8 50 0.6690 0.8942middle vessel 1.7 1.0 4.0 5.0 8 8 50 50 0.9522 0.4941 0.9427

Figure 10. Concept of RbMIN.

Figure 11. Distillate composition profiles for constant reflux andconstant reboil operating mode.

Figure 12. Bottom product composition profiles for constantreflux and constant reboil operating mode.

766 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997

removal of the heavy components through the strippingpart of the middle vessel column. The average productcomposition is higher than the composition resultingfrom a calculation where the product compositiondecreases with time.3.1. Shortcut Model for the Middle Vessel Col-

umn. The shortcut model for the middle vessel columncan be easily derived from the rectifier and the strippershortcut model and is described below.Since the mass balance around the middle vessel

constitutes the main difference between a middle vesselcolumn, an inverted column, and a rectifier, it isresponsible for the change in column behavior and hastherefore to be examined thoroughly.

where Vt is the vapor boilup rate for the top part of thecolumn and xD

(k) is the distillate composition of the keycomponent. The combination of the overall and com-ponent mass balance leads to an equation that updatesthe still composition in terms of the distillate and thebottom product composition.

Once the composition of the distillate and the bottomproduct is known, the new still composition can becalculated. These two compositions result from Hengest-ebeck-Geddes’ equation, with CB and CT standing forthe minimum number of plates for the top, Ntmin, andthe bottom part, Nbmin, of the column, respectively.

The bottom composition of the key component and thedistillate composition of the key component can beexpressed in terms of the still composition.

Similarly, Underwood’s equations can predict Rbmin andRmin.

The relationships between R, Rmin, Nt, and Ntmin forthe top and between Rb, Rbmin, Nb, and Nbmin for thebottom are given by empirical relations such as theGilliland correlation (shown in Figure 13). Since, in thecase of middle vessel, we have to solve the equationsiteratively for both the stripping section and the rectify-ing section of the column, we simplified this empiricalcorrelation further by assuming linear correlations givenbelow. These correlations are obtained by fitting thedata for several columns as shown in Figures 14 and15. From the figures it can be seen that the linearcorrelation is only valid between Yt e 0.6 for the rectifierand Yb e 0.55. For wider applicability one can alwaysuse eq 13 for the rectifying section and eq 14 for thestripping section.

Thus, eqs 15-26 form the shortcut model for themiddle vessel column. At each time step the modelneeds to solve three differential equations (eqs 15, 16,and 18) and the algebraic equations consisting of eq 19,HG equations, and FUG equations iteratively. Thefollowing subsection describes this procedure for con-stant reflux/constant reboil mode of operation.3.2. Solution Procedure. At any time step the still

composition, xs(i), i ) 1, 2, ..., n, can be found from the

previous time steps using eq 19. Then the followingprocedure is used to obtain the bottom and top composi-tions.1. Assume the initial values of CT and CB. Remember

that CT and CB should be greater than zero but less thanthe number of plates in the respective sections.2. Calculate the distillate and bottom product com-

positions of the key component k by using the corre-sponding Hengestebeck-Geddes equation and summa-tion of all compositions.

dSdt

) -VbRb

- VtR + 1

, S0 ) F (15)

dBotdt

) VbRb

, Bot0 ) 0.0 (16)

xbot(k)VbRb

+ xD(k) VtR + 1

) -d(xs

(k)S)dt

, xs(k) ) xF(k) (17)

dxs(k)

dt) Vb(Rb)S

(xs(k) - xbot

(k) ) + Vt(R + 1)S

(xs(k) - xD

(k)),

xs0(k) ) xF

(k) (18)

xsnew(i) ) xsold

(i) +∆xs

(k)(xbot(i) - xs

(i))old(xbot

(k) - xs(k))old

+

∆xs(k)(xD

(i) - xs(i))old

(xD(k) - xs

(k))old, i ) 1, 2, ..., i * k, n (19)

xD(i) ) (Ri

Rk)CTxs

(i)

xs(k)xD(k), xbot

(i) ) (Ri

Rk)-CBxs

(i)

xs(k)xbot(k) , i )

1, 2, ..., n (20)

Ntmin )

ln[xD(i)xD(k)

xs(k)

xs(i)]

ln[Ri

Rk]

, Nbmin )

ln[xs(i)xs(k)

xbot(k)

xbot(i) ]

ln[Ri

Rk]

(21)

xD(k) )

1

∑i)1

n (Ri

Rk)CTxs

(i)

xs(k)

, xbot(k) )

1

∑i)1

n (Ri

Rk)-CBxs

(i)

xs(k)

(22)

∑i)1

n Rixs(i)

Ri - φ) 0 (23)

Rmin + 1 ) ∑i)1

n RixD(i)

Ri - φ, - Rbmin ) ∑

i)1

n Rixbot(i)

Ri - φ(24)

Yt ) 0.5515 - 0.5948Xt (25)

Yb ) 0.6187 - 0.5655Xb (26)

Xt )R - Rmin

R + 1; Yt )

Nt - NtminNt + 1

Xb )Rb - Rbmin

Rb; Yb )

Nb - NbminNb + 1

xD(k) )

1

∑i)1

n (Ri

Rk)CTxs

(i)

xs(k)

, xbot(k) )

1

∑i)1

n (Ri

Rk)-CBxs

(i)

xs(k)

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 767

3. Find all other distillate and bottom compositionsby using HG equations.

4. Solve the Underwood equations for φ and obtainthe value of Underwood’s minimum reflux ratio, Rminu,and minimum reboil ratio, Rbminu.

5. Calculate Yt and Yb usingNtmin (equivalent to CT)and Nbmin (equivalent to CB) and solve the modifiedGilliland correlations for the Xt and Xb, respectively.

6. For the correct values of CT, Rminu should be equalto Rming; hence, Gtc should be equal to zero. Similarly,Rbminu should be equal to Rbming. Therefore, the valueof the quantity Gbc should be zero.

7. For the top column iterate on CT till Gtc is zerowithin a tolerance, and for the bottom portion of thecolumn iterate on CB for Gbc to be negligible.3.3. Model Validation. The middle vessel column

offers greater flexibility to batch distillation, and thereare a large number of various design variables whichcan be changed to achieve the desired separations. Inthe validation of the shortcut model for the middlevessel column, a large number of parameters are thusvaried. First, a binary mixture is tested and thentesting is extended to multicomponent mixtures. Therelative volatility of the light component (R1), the refluxratio (R), the reboil ratio (Rb), the number of equilibriumplates for the top and bottom section (Nt, Nb), and theratio between the vapor rates (q′ ) Vt/Vb) are changed(Table 5). To cover as many variations as possible, asystematic sampling across all the variables is carriedout using the HSS sampling technique (Diwekar andKalagnanam, 1996). For the examples in Table 5, thefeed mixture is equimolar and the second relativevolatility is set at a constant.The differences between the results of the rigorous

simulation and the shortcut model are negligible for thebinary feed mixture. The average product compositionsvary 2% at most, whereas the compositions at each timestep differ by less than 1% on average. Thus theshortcut procedure for the middle vessel simulates thedistillation with satisfying accuracy for a binary mixture

Figure 13. Gilliland’s plot for the middle vessel column.

Figure 14. Linear curve fit for the stripping section in the middlevessel column.

Figure 15. Linear curve fit for the rectifying section in the middlevessel column.

xD(i) ) (Ri

Rk)CTxD

(k)

xs(k)xs(i), xbot

(i) ) (Ri

Rk)-CBxbot

(k)

xs(k)xs(i), i )

1, 2, ..., n (i * k)

∑i)1

n Rixs(i)

Ri - φ) 0

Rminu + 1 ) ∑i)1

n RixD(i)

Ri - φ, -Rbminu ) ∑

i)1

n Rixbot(i)

Ri - φ

Xt )R - Rming

R + 1; Yt )

Nt - CT

Nt + 1

Yt ) 0.5515 - 0.5948Xt

Xb )Rb - Rbming

Rb; Yb )

Nb - CB

Nb + 1

Yb ) 0.6187 - 0.5655Xb

Gtc )Rminu - Rming

R + 1

Gbc )Rbminu - Rbming

Rb

768 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997

at constant reboil and constant reflux policy (Figures16 and 17). Note that, because of the limitation of thelinear form of the Gilliland correlations for the top andbottom sections of the column, the test examples do notinclude extreme values of Xt or Xb.For the shortcut procedure for multicomponent mix-

tures light and heavy key components have to be chosenproperly and can also change during the simulation. Tovalidate the shortcut model for the middle vesselcolumn, several ideal ternary mixtures have been tried.For these test cases the average error is found to be lessthan 5%.3.4. Feasibility Regions for the Middle Vessel

Column. Due to the nature of the middle vessel

column, finding feasibility regions is more difficult thanfor the batch distillation stripper. Three possible trendsin composition profiles are observed when extensive testcases of a binary mixture are conducted. These threetrends are also observed for ternary and multicompo-nent systems with particular combinations of param-eters.1. The distillate and bottom product compositions are

constant throughout operation.2. The distillate composition of the light key compo-

nent increases, and the bottom product composition ofthe heavy key component decreases.3. The bottom product composition of the heavy key

component increases, and the distillate composition ofthe light key component decreases.In the middle vessel column the dynamic behavior of

the two sections is governed by the interactions of thetwo columns through the differential mass balanceequation which is dependent on the design parametersNt, Nb, R, and Rb as well as q′. Therefore, it is difficultto derive the feasibility regions for the middle vesselwithout detailed study of the parameter interactions.However, a preliminary judgment can be made for thefeasibility windows for the three types of behavior(trends) given above.The first case is similar to the continuous distillation

where compositions are not changing significantly overthe time period; the feasibility window can be specifiedin terms of the minimum reflux, minimum reboil ratio,and minimum number of plates given by Nt e Ntmin,Nb e Nbmin, R e Rmin, and Rb e Rbmin. In the secondcase the bottom product composition for the heavy keycomponent is decreasing, a behavior similar to that ofthe batch stripper. However, the top product composi-tion of the light key component is increasing, a behaviordifferent from that of a rectifier but desirable. There-fore, qualitatively one can argue that the limitingconditions in this case are due to the bottom column,and the stripper feasibility window for the bottom partof the column holds well. Similarly, for the third casethe rectifying part of the column is limiting and arectifier feasibility criterion is applicable.

4. Conclusions

In this paper shortcut models for the batch distillationstripper and middle vessel column are presented. Theseshortcut models provide a fast and reasonably accuratedesign tool for the emerging batch distillation columns.The models are based on the Fenske-Underwood-Gilliland method for continuous columns and hence areapplicable to nearly ideal systems and columns withnegligible holdups. Additionally, with these models itis possible to identify the feasibility window for opera-tion of these new configurations. This paper alsopresented the feasibility criteria based on the shortcutmodel equations for these columns. This information

Table 5. Test Cases for the Validation of the Shortcut Method for the Middle Vessel Column

test R1 R2 Nt Nb R Rb q′ average % error in composition max % error in composition

1 3.0175 1.0 8 6 6.0 6.57 1.64 0.085 0.182 2.835 1.0 6 9 5.0 5.86 1.46 0.047 0.123 2.6525 1.0 9 7 4.0 5.14 1.28 0.115 0.34 2.2875 1.0 8 8 6.8 3.71 0.92 1.139 4.455 2.105 1.0 5 7 5.8 7.90 0.74 0.437 0.676 1.9225 1.0 9 5 4.8 7.18 0.56 0.856 1.987 1.74 1.0 7 9 3.8 6.47 0.38 0.803 2.378 1.5575 1.0 8 8 7.6 5.76 0.20 1.141 2.539 1.375 1.0 5 6 6.6 5.04 1.98 0.904 1.7410 1.1925 1.0 9 9 5.6 4.33 1.80 0.193 0.36

Figure 16. Validation of shortcut method for the middle vesselcolumn.

Figure 17. Validation of shortcut method for the middle vesselcolumn.

Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997 769

is crucial for iterative design, optimization, synthesis,and control problems.

Nomenclature

Bot ) amount in the reboiler (mol)CB ) constant in the Hengestebeck-Geddes equation forthe stripper and stripping (bottom) part of the middlevessel

CT ) constant in the Hengestebeck-Geddes equation forthe rectifying (top) part of the middle vessel

D ) total distillate (mol)F ) total feed (mol)HK ) heavy key componentLK ) light key componentn ) number of componentsNb ) number of theoretical plates in the stripper or bottompart of the middle vessel

Nt ) number of theoretical plates in the top part of themiddle vessel

Nbmin ) minimum number of plates in the stripper orbottom part of the middle vessel

Ntmin ) minimum number of plates in the top part of themiddle vessel

q ) feed condition defined as the ratio of heat required tovaporize 1 mol of the feed to the molar latent heat ofthe feed

q′ ) ratio of the vapor rate in the top section of the middlevessel to the vapor rate in the bottom section [Vt/Vb]

R ) reflux ratioRmin ) minimum reflux ratioRMIN ) minimum feasible reflux ratio for the constantreflux operations

Rming ) minimum reflux ratio given by the Gillilandcorrelation

Rminu ) minimum reflux ratio given by the Underwoodequations

Rb ) reboil ratioRbmin ) minimum reboil ratioRbMIN ) minimum feasible reboil ratio for the constantreboil operations

Rbming ) minimum reboil ratio given by the Gillilandcorrelation

Rbminu ) minimum reboil ratio given by the Underwoodequations

S ) amount remaining in the still (mol)S0 ) initial amount in the still (mol)t ) integration time (h)xbot-ave(i) ) average bottom product composition of compo-nent i (mole fraction)

xbot(i) ) liquid composition of component i in the bottomproduct (mole fraction)

xD(i) ) distillate composition of component i (mole fraction)xDav(i) ) average distillate composition of component i (molefraction)

xF(i) ) feed composition of component i (mole fraction)xs(i) ) liquid composition of component i in the still (molefraction)

Vb ) vapor rate in the bottom section of the column (mol/h)

Vt ) vapor rate in the top section of the column (mol/h)

Greek Symbols

Ri ) relative volatility of component iφ ) Underwood constant

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Received for review October 8, 1996Revised manuscript received December 9, 1996

Accepted December 12, 1996X

IE960632N

X Abstract published in Advance ACS Abstracts, February1, 1997.

770 Ind. Eng. Chem. Res., Vol. 36, No. 3, 1997