Analysis and Conceptual Design of Ternary ... - ACS Publications

Ind. Eng. Chem. Res. 2002, 41, 3849-3866

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Analysis and Conceptual Design of Ternary Heterogeneous Azeotropic Distillation Processes Rosa Y. Urdaneta, Ju 1 rgen Bausa,† Stefan Bru 1 ggemann, and Wolfgang Marquardt* Lehrstuhl fu¨ r Prozesstechnik, RWTH Aachen, Turmstrasse 46, D-52056 Aachen, Germany

Design phenomena in heterogeneous distillation processes are investigated. Algorithms for trayby-tray calculation are extended for columns with multiple heterogeneous trays in both sections. Minimum-energy calculations and methods for feasibility analysis are investigated for such systems. Based on this research, a new conceptual design method for heterogeneous distillation with one feed and two product streams is developed and tested on some examples showing different residue curve map topology. 1. Introduction The crossing of distillation boundaries using a heterogeneous entrainer is a technique widely used in industry to separate binary and multicomponent azeotropic mixtures. In these processes the immiscibility of two liquid phases is exploited in order to facilitate the separation. In typical designs, a heterogeneous stream is produced at the top of the column which is then split in a decanter into the entrainer-lean distillate product and the entrainer-reach reflux stream. The dehydration of a binary ethanol-water mixture using benzene or cyclohexane as an entrainer is a commonly known example of these types of processes.1-3 In recent years, new approximative methods for the determination of the minimum energy demand4-8 and for the prediction of feasibility9-12 of a distillative separation of ternary and multicomponent nonideal mixtures have been developed. The focus of all of these publications has been largely restricted to homogeneous separations. A design method for ternary heterogeneous separations has been presented by Ryan and Doherty,1 Pham et al.,13 and Pham and Doherty.14 The determination of minimum energy demand and an assessment of the feasibility of the separation is based on the calculation of tray-by-tray profiles from given product compositions at different reflux ratios. Feasibility is evaluated by checking the simple distillation boundaries (SDBs) of the mixture. However, these authors only consider processes with a heterogeneous liquid in the decanter at the top of the column and assume the trays in the column to be operated in the homogeneous region. While this is certainly a very important class of processes, there may be other economically advantageous processes which are excluded by this design method. Wasylkiewicz15 and Wasylkiewicz et al.16,17 also present design methods for heterogeneous separations based on tray-by-tray calculations and Simple Distillation Regions (SDRs). Although it seems that they were able to determine multiple heterogeneous trays in the rectifying section,17 neither an explanation on the procedure of these calculations nor an analysis of the consequences for column design has been given. * Corresponding author. Fax: +49-241-8092326. Phone: +49-241-8094671. E-mail: [email protected]. † Present address: Bayer AG, 51368 Leverkusen, Germany.

Starting from the geometrical analysis of tray-by-tray profiles suggested by Pham et al.,13 this paper will present an extension of the tray-by-tray analysis for multiple heterogeneous trays in both the rectifying and stripping sections of the column. Some unique physical effects of heterogeneous column profiles will be discussed. In particular, we will show that heterogeneous trays in the column complicate minimum energy demand analysis and conceptual column design, because of the resulting multiparametric design problem formulation. Special attention will be paid to the determination of separation feasibility, considering the existence of pinch-point solutions. Further, it will be demonstrated that an analysis based on the widely used SDBs results in an incomplete understanding of the column behavior for heterogeneous separations. It may even lead to qualitatively wrong predictions and hence infeasible designs. Using the insights gained from these findings, a more general design procedure for the determination of feasible operation and minimum energy demand of heterogeneous distillation processes will be presented. Some examples will be shown and analyzed using this new design method. 2. Ethanol Dehydration The dehydration of azeotropic mixtures of alcohols and water is a common industrial application of heterogeneous distillation. In particular, ethanol dehydration has been studied extensively in the literature. It will be used as an illustrative example in the first part of this paper. Ryan and Doherty1 consider benzene as an entrainer and propose distillation sequences with two, three, or four columns. However, cyclohexane is more often preferred as the entrainer because of its low cost and low toxicity.2,3 Hence, it will be used here. However, the residue curve diagrams of both mixtures ethanol/water/cyclohexane (EWC) and ethanol/water/ benzene have similar topologies. Similar designs can, therefore, be expected in both cases. EWC displays a ternary heterogeneous minimum azeotrope and three binary azeotropes. The binary azeotrope between water and cyclohexane (CW) is heterogeneous, whereas the other two azeotropes between ethanol and water (EW) and between ethanol and cyclohexane (EC) are homogeneous (see Figure 1). The EWC system has three SDRs numbered by I-III. These regions are formed by those residue curves that connect

10.1021/ie0107486 CCC: $22.00 © 2002 American Chemical Society Published on Web 05/17/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002

Figure 1. Residue curve map at p ) 1.013 bar for the mixture EWC. The miscibility gap at boiling temperature, some liquidliquid tie lines, and the vaporline are also shown.

Figure 3. Mass balances for the ethanol purification process: (a) overview; (b) detailed enlargement close to the heteroazeotrope.

Figure 2. Flowsheet of a heterogeneous azeotropic distillation process for ethanol purification using cyclohexane as the entrainer.

the ternary heterogeneous azeotrope EWC with the binary azeotropes EC, CW, and EW as well as by the edges of the Gibbs triangle (see Figure 1). Such residue curves are known as SDBs.18 They limit the range of feasible products for open distillation stills or for distillation columns operating at total reflux. Because residue curves often give a good approximation of the column profiles at finite reflux, the SDBs are often used for feasibility analysis of distillation columns operating at finite reflux. The vaporline comprises the locus of vapor concentrations in equilibrium with the heterogeneous liquid concentrations.19 In the EWC system, it is very close to the SDB connecting the EWC and CW azeotropes (see Figure 1). The distillation column sequence considered in this paper is the one proposed by Mu¨ller20 (see Figure 2). The mass balances of the process are illustrated in Figure 3 (for the feed and product specifications of column 2 as given in Table 1). The preconcentrator column (column 1 in Figure 2) operates in the homogeneous region. Its distillate product D1 is fed to the azeotropic column (column 2 in Figure 2), where pure ethanol is obtained as the bottoms product B2. The overhead vapor is condensed, subcooled to 25 °C, and split into two liquid phases in the decanter. The aqueous, entrainer-lean phase (phase I) is partly recycled to the first column (D2). The organic, entrainer-rich phase (phase II) and part of the aqueous entrainer-lean phase (phase I) compose the reflux of column 2. As shown in

Figure 3, the phase split in the decanter of column 2 is used to cross the distillation boundary separating region II from region III to facilitate the production of pure ethanol. Pham et al.13 propose a design method for the determination of minimum reflux for heterogeneous azeotropic columns. The specifications shown in Table 1 are selected to satisfy the first three constraints of their method as follows: 1. The decanter tie line at 25 °C crosses the SDR where the bottoms composition is located (see Figure 3a). 2. B2 is close to the desired pure component (ethanol), and D2 is located on the entrainer-lean side of the heterogeneous region on the tie line specified in step 1. 3. The top vapor composition yn must be selected considering the SDRs. In the EWC system, every composition on the part of the decanter tie line located in distillation region III is a valid choice for yn. This part of the tie line is highlighted in Figure 3b. It can be seen that these first steps limit the applicability of this method to systems showing the same residue curve map topology as that of the EWC mixture, i.e., systems having a minimum boiling ternary azeotrope inside the miscibility gap (e.g., ethanol/water/ benzene used by Ryan and Doherty1 or 2-propanol/ water/benzene by Pham et al.13). The feasibility constraints of steps 1 and 3 are based on the presence of the SDBs. Moreover, the method does not allow the determination of multiple heterogeneous stages and is therefore restricted by the assumption that phase immiscibility only appears in the decanter. Thus, some questions remain unanswered. First, does the assumption of homogeneous operation throughout the column always represent the optimal condition for column

Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3851 Table 1. Specifications for Column 2 in the EWC Example in Figure 3

xE xW xC temp, °C vapor fraction

feed

distillate

bottoms

0.7814 0.1775 0.0411 69.83 0.0

0.5970 0.3272 0.0758 25.0 subcooled

1.0000 8.0 × 10-6 1.0 × 10-10 78.34 0.0

operation? How does the process behavior change if this assumption is relaxed, i.e., when multiple heterogeneous trays exist? What is the best choice for the vapor composition at the top of the column yn, and does every choice of yn within the correct SDR lead to a feasible column? What is the implication of these issues on minimum energy demand and hence process economics? Wasylkiewicz et al.17 suggest and apply a more general design method, which extends the applicability to systems with different topologies and multiple heterogeneous stages in the rectifying section. However, they do not provide a rigorous analysis, which is necessary to properly answer the previous questions. In the following, we will carry out such an analysis. In a first step, the tray-by-tray calculation procedure for the determination of column profiles will be reviewed and extended for multiple heterogeneous trays. 3. Calculation of Column Profiles The sequential tray-by-tray calculation of column profiles has been introduced for homogeneous mixtures by Levy et al.4 Later, Pham et al.13 extended the standard formulation by adding equations for a decanter at the top of the column. They use the following assumptions: constant molar overflow (CMO), saturated liquid feed, theoretical plates, complete separation of the liquid phases in the decanter, and given product compositions. Under these assumptions, the material balances for the rectifying section of the column can be formulated as

(

yn+1 ) 1 -

)

D D x + x Vn+1 n Vn+1 D

(1)

This equation relates the concentration of the vapor phase yn+1 on tray n + 1 to the concentration of the liquid phase xn on tray n, which is above tray n + 1. The composition of the liquid concentration xn+1 on tray n + 1 can then be obtained from the equilibrium expressions:

yn+1 ) K(xn+1,Tn+1)xn+1

(2)

∑ yn+1,k - 1

GE-model; vapor phase (param source) decanter, NRTL; column, ext. NRTL; ideal (Keil et al.21)

0.542 1.013

) 3) only 1 degree of freedom is available. Hence, the vapor compositions in equilibrium with heterogeneous liquid mixtures are situated on a curve known as the vaporline. Any other vapor composition is in equilibrium with a homogeneous liquid mixture. The algorithm of Pham et al.13 does not restrict the location of yn+1; therefore, yn+1 is usually not positioned on the vaporline, and xn+1 is not heterogeneous. On the other hand, if yn+1 were determined exactly on the vaporline, the temperature and composition of both liquid phases on the tray I II , and xn+1 ) result from a VLLE dewn + 1 (Tn+1, xn+1 point calculation, but the phase ratio cannot be determined.22 Therefore, the composition of the heterogeneous liquid mixture (xn+1), which is necessary for the next tray-by-tray step, is not known. For these reasons, it is always assumed by Ryan and Doherty,1 Pham et al.,13 and Pham and Doherty14 that two liquid phases only occur in the decanter. 3.1. Downward Calculation. We develop a system of equations that allows the recursive calculation of the state variables on tray n + 1 in the rectifying section if the state variables on tray n above, the product stream and composition (D and xD), and the condenser duty QD are known. First heterogeneous trays and next homogeneous trays are considered. We will drop the CMO assumption and replace it by adding the energy balance to the set of tray-by-tray equations of Pham et al.13 because the CMO assumption is obviously not valid for the highly nonideal mixtures encountered in heteroazeotropic distillation. 3.1.1. Heterogeneous Trays. Assuming two liquid phases and a vapor phase in equilibrium on tray n, the following balances can be formulated to calculate tray n + 1 for the downward tray-by-tray step (see Figure 4):

0 ) LIn + LII n - Vn+1 - D

(4)

I II + LII 0 ) LIn xn,i n xn,i - Vn+1yn+1,i - DxD,i, i ) 1, ..., C (5) L II 0 ) LInhL(xIn,Tn,p) + LII n h (xn ,Tn,p) -

Vn+1hV(yn+1,Tn+1,p) - D(hD - QD/D) (6)

C

0)

D/F pressure (bar)

(3)

k)1

The column profile for the rectifying section can be calculated downward recursively starting from the overall liquid composition of the reflux leaving the decanter at the top of the column. The profile for the stripping section is determined analogously by calculating upward starting from the vapor composition leaving the reboiler (see work by Pham et al.13 for details). According to the Gibbs phase rule, C - 1 degrees of freedom are available in vapor-liquid equilibrium (VLE) calculations at fixed pressure, whereas only C 2 degrees of freedom exist in the vapor-liquid-liquid equilibrium (VLLE). Therefore, in a ternary system (C

Considering the new dimensionless variables v ) V/D and φ ) LII/(LI + LII) and eliminating the total balance of the column (eq 4), we obtain a system of 3C + 3 equations I II 0 ) (1 - vn+1)[(1 - φ0n)xn,i + φ0nxn,i ] + vn+1yn+1,i - xD,i, i ) 1, ..., C (7)

0 ) (1 - vn+1)[(1 - φ0n)hL(xIn,Tn,p) + φ0nhL(xII n ,Tn,p)] + vn+1hV(yn+1,Tn+1,p) - (hD - QD/D) (8) C

0)

I - yn+1,k) ∑ (xn+1,k k)1

(9)

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Figure 4. Balance envelope for the rectifying section of a heterogeneous azeotropic distillation column.

Figure 5. Balance envelope for the decanter.

C

0)

II (xn+1,k - yn+1,k) ∑ k)1

(10)

I I ,yn+1,Tn+1,p) xn+1,i , i ) 1, ..., C 0 ) yn+1,i - Ki(xn+1 (11) II II 0 ) yn+1,i - Ki(xn+1 ,yn+1,Tn+1,p) xn+1,i , i ) 1, ..., C (12) I II in 3C + 3 unknown variables xn+1 , xn+1 , yn+1, Tn+1, φ0n, 0 and vn+1. If a feasible solution φn with 0 e φ0n e 1 does not exist, the assumption of a heterogeneous liquid phase does not hold. In this case, tray n + 1 is homogeneous and we have to switch to the equations for a homogeneous tray (see section 3.1.2) and specify a value of φn because it is no longer a variable that can be determined from the tray-by-tray calculation step. 3.1.2. Homogeneous Trays. Using the mass, component, and energy balances between trays n and n + 1 as shown in Figure 4 and defining v ) V/D as before, we define the balance equations for homogeneous trays as

0 ) (1 - vn+1)xn,i + vn+1yn+1,i - xD,i, i ) 1, ..., C

(13)

0 ) (1 - vn+1)hL(xn,Tn,p) + vn+1hV(yn+1,Tn+1,p) (hD - QD/D) (14) C

0)

∑ (xn+1,k - yn+1,k)

(15)

k)1

0 ) yn+1,i - Ki(xn+1,yn+1,Tn+1,p) xn+1,i, i ) 1, ..., C (16) This system has 2C + 2 equations in the unknown variables xn+1, yn+1, Tn+1, and vn+1. As soon as xn+1 is calculated, its stability can be verified using some phase stability test. We suggest to use the test developed by Bausa and Marquardt,22 which presents a good compromise between reliability and computational effort. If xn+1 is unstable, the vapor composition yn+1 on tray n + 1 is located exactly (subject to some tolerance) on the vaporline of the miscibility gap, and therefore, the set of equations (7)-(12) for a heterogeneous tray n + 1 must be solved. 3.1.3. Calculation of Rectifying Section Profiles. Obviously, it is a nontrivial problem to properly switch from the heterogeneous region to the homogeneous

Figure 6. Decanter balances for the EWC system (p ) 1.013 bar) at boiling temperature.

region. This issue will further be explored by studying the situation at the top of the column. Figure 5 shows the balance envelope for the decanter at the top of the column. Such a balance envelope can be considered the first tray of the column. Hence, it can be described with the same set of equations (7)-(12) as the other trays. The top tray of the column is then tray 2. For convenience, it is now assumed that the decanter operates at boiling temperature. If a lower temperature is specified for the decanter, the product composition must be situated on the binodal at that temperature. According to the decanter material balances (see eqs 4, 5, and 7), the compositions xD, x1, and y2 must lie on a straight line. Additionally, liquid-liquid equilibrium in the decanter demands that xD and x1 have to lie on the same tie line. Hence, the tie line and the material balance line for tray 2 have to coincide (see Figure 6). Equations 7-12 can be solved if the decanter tie line intersects the vaporline. The phase ratio φ1 is then fixed to the solution φ1 ) φ01. Hence, φ1 is not a degree of freedom anymore (as considered in the method of Pham et al.13) if the top tray in the rectifying section (n ) 2) is specified to be heterogeneous. In general terms, the system of equations (7)-(12) can be solved if the material balance line for tray n + 1 intersects the vaporline. The resulting values are only valid if 0 e φ0n e 1, in which case φn can be fixed to φn ) φ0n. Before proceeding with the downward recursion, we have to decide whether we want to switch from a heterogeneous stage n to a homogeneous stage n + 1.

Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3853

If tray n + 1 is supposed to be homogeneous, eqs 7-12 have to be replaced by the homogeneous tray equations (13)-(16). Their solution requires the overall liquid concentration xn on tray n. This is not available because a solution of eqs 7-12 for the heterogeneous tray n only provides φn-1, but not φn. Hence, φn has to be specified in the sense of a degree of freedom, either if a heterogeneous state is not possible anymore or not desired on tray n + 1. In such a case, the tray-by-tray calculation is continued from tray n + 1 using the homogeneous set of equations (13)-(16) presented above using the following overall liquid compositions and enthalpy:

xn ) (1 - φn)xIn + φnxII n L

h (xn,Tn,p) ) (1 - φn)h

L

(xIn,Tn,p)

(17) L

+ φ nh

(xII n ,Tn,p)

Figure 7. Schematic representation of the treelike structure of the column profiles within the miscibility gap on the rectifying section.

(18) The situation is illustrated in Figure 6 for the EWC example where the decanter tie line has been chosen such that it intersects the vaporline. This figure illustrates the variation of the concentrations x2 for different choices of φ1. If φ1 ) φ01, the concentrations 0 xI2(φ01) and xII 2 (φ1) are calculated and x2 will lie somewhere on the tie line of tray 2. For φ1 * φ01, x2 lies in the homogeneous region. It is close to the EW edge for φ1 < φ01 and close to the EC edge for φ1 > φ01. Based on this reasoning, any profile in the rectifying section is determined not only by specification of the product (D and xD), condenser duty (QD), and column pressure as in the case of homogeneous separations but in addition by the number of heterogeneous trays k and by the phase ratio φk on the last heterogeneous tray in the rectifying section. Whereas for homogeneous profiles all design variables on tray n are already known before the variables on tray n + 1 are calculated, as has been shown in section 3.1.2, for heterogeneous profiles one state variable of tray n, namely, the phase ratio φn, can only be calculated together with the other state variables on tray n + 1. In some cases (see section 7), we may reenter a heterogeneous region while recursively stepping down the column from a homogeneous tray. Such a behavior can be detected by a phase stability test, as indicated in section 3.1.2. The suggested procedure allows the calculation of the complete set of column profiles in contrast to the method of Pham et al.13 that will only permit calculation of a subset of these profiles, which is characterized by homogeneous trays in the rectifying section only. The set of physically plausible column profiles forms a treelike structure as exemplarily shown in Figure 7. Starting from the decanter tray 1, which is heterogeneous, an infinite number of possible states on tray 2, and therefore of possible column profiles, can be constructed by variation of φ1. However, there is only one solution for tray 2 (with φ2 ) φ02), giving rise to a heterogeneous liquid phase on tray 3 (see Figure 7). All other profiles leave the miscibility gap in the next step (tray 3). 3.2. Upward Calculation. During upward recursion, the states on tray n - 1 can be calculated if the states on tray n below, the bottoms product (B and xB), and the reboiler duty QB are known (see Figure 8). Condenser and reboiler duty can be related with the energy

Figure 8. Balances for an upward tray-by-tray step.

balance of the column

FhF ) DhD - DQD + BhB - BQB

(19)

3.2.1. Heterogeneous Trays. Assuming two liquid phases on tray n - 1 (Figure 8) and using the dimensionless variable v ) -V/B, the mass, component, and energy balances can be written as a system of 3C + 3 I II , xn-1 , yn-1, equations in the unknown variables xn-1 Tn-1, φn-1, and vn as follows: 0 I 0 II )xn-1,i + φn-1 xn-1,i ] + vnyn,i 0 ) (1 - vn)[(1 - φn-1 xB,i, i ) 1, ..., C (20) 0 I 0 )hL(xn-1 ,Tn-1,p) + φn-1 hL 0 ) (1 - vn)[(1 - φn-1 II (xn-1 ,Tn-1,p)] + vnhV(yn,Tn,p) - (hB - QB/B) (21) C

0)

I (xn-1,k - yn-1,k) ∑ k)1

(22)

C

0)

II - yn-1,k) ∑ (xn-1,k

(23)

k)1

I I 0 ) yn-1,i - Ki(xn-1 ,yn-1,Tn-1,p) xn-1,i , i ) 1, ..., C (24) II II 0 ) yn-1,i - Ki(xn-1 ,yn-1,Tn-1,p) xn-1,i , i ) 1, ..., C (25)

This set of equations can be solved directly for the unknowns on tray n - 1. If no physically meaningful

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0 solution with φn-1 ∈ [0, 1] exists, the liquid on tray n - 1 is homogeneous and the equations for a homogeneous tray (section 3.2.2) have to be solved. 3.2.2. Homogeneous Trays. Mass, component, and energy balances are written as a system of 2C + 2 equations in the unknown variables xn-1, yn-1, Tn-1, and vn:

0 ) (1 - vn)xn-1,i + vnyn,i - xB,i, i ) 1, ..., C (26) L

L/B in the stripping section. With these abbreviations, the homogeneous tray balances for both the rectifying and stripping sections, eqs 13-16 and 26-29, respectively, can be reformulated in order to calculate the pinch-point solutions to result in

0 ) lxi + vyi - zi, i ) 1, ..., C C

0)1-

∑ (xn-1,k - yn-1,k)

(31)

C

0)1-

C

0)

∑ xk

k)1

V

0 ) (1 - vn)h (xn-1,Tn-1,p) + vnh (yn,Tn,p) (hB - QB/B) (27)

(30)

∑ yk

(32)

k)1

(28)

k)1

0 ) yn-1,i - Ki(xn-1,yn-1,Tn-1,p) xn-1,i, i ) 1, ..., C (29) This set of equations can readily be solved if the state on tray n is known. 3.2.3. Calculation of Stripping Section Profiles. It can be seen that the heterogeneous balances in the stripping section (20)-(25) allow to calculate not only the compositions of the vapor and liquid phases (yn-1, I II , and xn-1 ) but also the phase ratio φn-1 on the xn-1 same tray n - 1. Therefore, in contrast to the rectifying section, the phase ratio cannot represent any degree of freedom here. Because the bottoms product in the EWC process is homogeneous, the set of equations (26)-(29) is solved first. After this set of equations is solved, the stability of the liquid phase with overall composition xn-1 is checked on the basis of a phase stability test.22 If it is found to be unstable, then two liquid phases with I II and x˜ n-1 result from the phase test concentrations x˜ n-1 on xn-1. However, these concentrations do not correspond to the heterogeneous solution, because there are also modifications of boiling temperature and enthalpy due to the phase split. However, the values can be used as initial guesses for the subsequent solution of the heterogeneous set of equations (20)-(25). If two liquid phases exist on tray n, eqs 20-25 are first solved for tray n - 1. If no physically meaningful solution is obtained, then the two phase solutions have to be determined through eqs 26-29 and the resulting homogeneous liquid phase xn-1 is checked with the stability test. 3.3. Calculation of Pinch-Point Solutions. As pointed out by Bausa et al.,8 the analysis of tray-bytray profiles leads to the fact that the column profiles touch one or more pinch points under minimum reflux. Authors such as Julka and Doherty,5 Ko¨hler et al.,6 Poellmann et al.,7 and Bausa et al.,8 among others, have used pinch-based approximate methods for the determination of minimum energy demand, especially for homogeneous systems. In the following, we will show that the behavior of heterogeneous distillation processes is also governed by the pinch points of both column sections. Therefore, we will briefly introduce pinch-point equations next, which, in principle, follow from the tray balances if the states on adjacent trays are equated. To formulate a general set of pinch-point equations, we introduce the variables z ) xD, hN ) hD - QD/D, v ) V/D, and l ) -L/D in the rectifying section and the variables z ) xB, hN ) hB - QB/B, v ) -V/B, and l )

0 ) yi - Ki(x,y,T,p) xi, i ) 1, ..., C

(33)

This set of equations allows one to calculate the homogeneous pinch branches of the system. Then, the energy parameter hN is determined from

0 ) lhL(x,T,p) + vhV(y,T,p) - hN

(34)

For the heterogeneous case, the pinch-point equations can be formulated for both column sections as

0 ) l[(1 - φ)xIi + φxII i ] + vyi - zi, i ) 1, ..., C (35) C

0)1-

xIk ∑ k)1

0)1-

∑ xIIk k)1

0)1-

∑ yk k)1

(36)

C

(37)

C

(38)

0 ) yi - Ki(xI,y,T,p) xIi , i ) 1, ..., C

(39)

0 ) yi - Ki(xII,y,T,p) xII i , i ) 1, ..., C

(40)

The heterogeneous pinch branches are solved first, and then the energy parameter hN is determined from

0 ) l(1 - φ)hL(xI,T,p) + lφhL(xII,T,p) + vhV(y,T,p) - hN (41) Because pure components and azeotropes can also be pinch-point solutions, in which case the vapor fraction v is infinite, some modifications have to be applied to the set of pinch-point equations (30)-(33) and (35)-(40), respectively, to avoid numerical problems. More details on the solution of the pinch-point equations for the homogeneous as well as the heterogeneous case can be found in work by Bausa.23 4. Minimum Energy Demand The minimum energy demand of the separation is of major concern for the design of distillation columns not only in the homogeneous but also in the heterogeneous case. Therefore, we attempt to transfer well-known methods for the estimation of the minimum energy demand in homogeneous distillation to the heteroge-

Ind. Eng. Chem. Res., Vol. 41, No. 16, 2002 3855 Table 2. Range of O1 as a Function of Reboiler Duty QB QB/F r s (MJ/kmol) (L1/D) (Vn/B) 77.141 74.855 73.569 72.127

2.634 2.530 2.472 2.406

4.348 4.219 4.146 4.065

φ1,min

φ1,max

φ01

∆φk

0.771 39 0.780 67 0.786 06 0.792 46

0.771 81 0.780 75 0.786 12 0.792 46

0.769 78 0.778 51 0.783 74 0.789 92

0.000 42 0.000 08 0.000 06 0.000 00

neous case. We propose first some modifications to the boundary value method (BVM) initially introduced by Levy et al.,4 considering the balances presented in section 3 for the calculation of heterogeneous column profiles. Then, the results will be compared to two pinchbased methods, the zero-volume criterion (ZVC) of Julka and Doherty5 and the rectification body method (RBM) of Bausa et al.8 4.1. BVM. Given the product and the feed streams, a proposed separation will only be feasible if the rectifying and stripping profiles intersect. According to the BVM, the smallest energy resulting in an intersection of the profiles is the minimum energy demand of the process. Therefore, for homogeneous separation processes at pressure p, the minimum energy demand QBmin is a function of the product specifications (B, xB, D, and xD) only. QB must be iteratively modified until the minimum energy demand QBmin is reached. A more complicated situation holds for the heterogeneous region. The concentration profiles are determined not only by the product specifications and the energy QB but also by the number of heterogeneous trays k and the phase ratio φk on the last heterogeneous tray. This situation does not occur in the stripping section as explained in section 3.2.3. As a consequence, the minimum energy demand QBmin of heterogeneous separations at given pressure p depends on k and φk in addition to the product specifications (B, xB, D, and xD). These additional degrees of freedom increase the complexity of the determination of QBmin significantly. To determine the minimum energy demand of a ternary separation with a decanter at the top of the column by means of the BVM, the following procedure can be used: 1. Select k. For example, in the case of the EWC separation, only the decanter is considered to be in a heterogeneous state, i.e., k ) 1. 2. Determine QB such that the rectifying and stripping section profiles intersect. 3. Determine those values of φk with 0 e φk,min e φk e φk,max e 1 that allow profile intersection. If the range ∆φk ) φk,max - φk,min is sufficiently small, the minimum energy demand QB(k) has been calculated; if not, reduce QB until ∆φk < with a small tolerance . 4. Increase k by one and repeat from step 2 until QB cannot be improved. This procedure is applied to column 2 of the EWC process starting with k ) 1 and QB/F ) 77.141 MJ/ kmol.20 This reboiler duty corresponds to a reflux ratio r ) 2.634 (r ) L1/D) and to a reboil ratio s ) 4.348 (s ) Vn/B). The range ∆φk is determined, for which the intersection of the profiles can be achieved (step 3 above). Then, the reboiler duty is reduced iteratively until ∆φk < . In Table 2, the values for φ1,min, φ1,max, φ01, and ∆φk are shown for decreasing reboiler duties QB/F. Using ) 10-5, the minimum reboiler duty is 72.127 MJ/kmol (r ) 2.406 and s ) 4.065). The column profiles for this design are shown in Figure 9. As can be seen from Table 2, φ1 * φ01. Thus, only one heterogeneous tray (the decanter) is needed in the rectifying

Figure 9. Tray-by-tray profiles for the rectifying and stripping sections at QBmin/F ) 72.127 MJ/kmol (r ) 2.406 and s ) 4.065) and φ1 ) 0.792 46.

Figure 10. Tray-by-tray profiles for the rectifying and stripping sections at QB/F ) 72.108 MJ/kmol (r ) 2.406 and s ) 4.065) and two different values of φ1.

section. Generally, only choices of φ1 within the range given by φ1,min and φ1,max yield feasible designs. For choices of φ1 outside this range, two different phenomena occur (see Figure 10). If φ1 < φ1,min, the rectifying profile runs from x1 toward the water vertex as if φ1,min < φ1 < φ1,max, but profile intersection cannot be achieved. If φ1 > φ1,max, the rectifying profile abruptly changes direction and now ends close to the cyclohexane vertex. This behavior can be ascribed to the appearance of a saddle pinch, as will be shown below. The feasible range [φ1,min, φ1,max] can be manipulated by raising the reboiler duty above the minimum energy. For values of φ1 outside the range spanned by φ1,min and φ1,max but corresponding to a top vapor composition y2 within the same SDR as xB, feasible designs cannot be attained regardless of the choice for QB. Thus, not all choices of y2 which obey the design rules of Pham et al.13 guarantee feasible operation. This phenomenon has already been observed by Pham et al.13 (p 1590) and severely limits the general applicability of their design method. It is possible to obtain a profile with one additional heterogeneous tray by choosing k ) 2 which results in φ1 ) φ01 and thus placing y2 on the vaporline. However,

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Figure 11. Profile trajectories for φ1 ) φ01 and φ2 ) 1, conditions at which the minimum energy for profile intersection is QB/F ) 96.762 MJ/kmol (r ) 3.529 and s ) 5.454).

for φ2 ) 1 a reboiler duty of QB/F ) 96.762 MJ/kmol (r ) 3.529 and s ) 5.454) is necessary to widen ∆φ1 such that it includes φ01 and still makes the profiles intersect (see Figure 11). This reboiler duty is 34% higher than the minimum energy calculated with only one heterogeneous tray. Choosing some φ2 < 1 further increases the energy demand. A third heterogeneous tray cannot be achieved in the EWC system, because there is no valid solution for φ02. As we can see in Figure 11, the balance line between D and x2 does not intersect the vaporline, and thus y3, which has to lie on the same balance line (according to eq 7), cannot be located on the vaporline. Therefore, for the given product specifications, the EWC column cannot have more than two heterogeneous stages (decanter and top tray). Although for EWC the minimum energy demand of the process corresponds to a liquid-phase split only in the decanter, such a pattern cannot be generalized to all heterogeneous systems, as has been assumed by Pham et al.13 A different ternary mixture with a residue curve map topology similar to the one of EWC will be shown in section 7. Multiple heterogeneous trays are required in the rectifying section in order to reach the minimum energy demand in that case. 4.2. ZVC. On the basis of the BVM, Levy et al.4 developed an algebraic criterion to determine minimum reflux which has been extended by Julka and Doherty5 to the ZVC for homogeneous distillation. Here, the reflux ratio is chosen such that the feed composition, a saddle pinch point, and a stable node are located on a straight line. For indirect splits, the saddle belongs to the rectifying section; it is a binary pinch located between the two lighter components. The stable node of the stripping section (see Figure 2 of Levy et al.4) represents the intersection of the profiles (feed tray). For the heterogeneous case, Pham et al.13 could not use the ZVC because they did not find any binary saddle for the system 2-propanol/water/benzene. However, using the homotopy continuation method for the calculation of all branches of the pinch solutions (see also section 3.3) developed by Bausa,23 it is possible to determine the solution for this saddle in the EWC example, which corresponds to the ternary saddle r2a in Figure 12. In this figure and later, the pinch-point solutions are classified as proposed by Julka and

Figure 12. Minimum energy demand of the EWC process according to ZVC, QB/F ) 66.284 MJ/kmol (r ) 2.14 and s ) 3.736).

Doherty.5 The pinch points of the rectifying section are identified by r, and the ones for stripping section are labeled by s. Having obtained all pinch-point solutions, the ZVC can now be applied to heterogeneous splits. When r2a, s1, and the concentration of the feed mixture xF are placed on the same line (see Figure 12), the minimum energy demand can now be estimated to QB/F ) 66.284 MJ/kmol (r ) 2.14 and s ) 3.736), which is 8% lower than the value obtained from the BVM in section 4.1. At this level of energy, the feasible range for φ1 is extremely small and φ1 has to be chosen very accurately (∆φ1 ) 2 × 10-15 and φ1,min ) 0.822 259 022 025 940) in order to establish a feasible profile. As before, choosing φ1 outside such range does not allow profile intersection. Application of the heterogeneous BVM introduced in section 4.1 (iteratively decreasing ∆φ1 to zero) would have made it extremely time-consuming to calculate φ1 with this accuracy. This result shows the advantage of using pinch points for targeting minimum energy demand of multicomponent separations, especially for heterogeneous mixtures, where column profiles show a high sensitivity to the choice of the phase ratio φk and the number of heterogeneous trays k. However, the applicability of the ZVC method for heterogeneous columns suffers from the disadvantage that the topology of the split and the pinch point determining minimum reflux must be known a priori, as observed by Bausa et al.8 for homogeneous mixtures. 4.3. RBM. Bausa et al.8 have recently developed the RBM to estimate the minimum energy demand in nonideal multicomponent distillation processes. This new method uses a geometrical analysis of the tray-bytray profiles, where the set of attainable trajectories is approximated by means of so-called rectification bodies (RBs). The method may be interpreted as a generalization of the geometrical interpretation of Underwood’s equation presented by Franklin and Forsyth,24 but it is not limited to ideal mixtures. The RBs are constructed from the pinch-point solutions with the general rule that the number of stable eigenvectors in the pinch points visited by a tray-by-tray profile must increase monotonously. Thus, every profile, running from the product composition toward a stable node, can run through a saddle but never through an unstable node.


Figure 13. EWC process at QB/F ) 66.284 MJ/kmol (ZVC): (a) RBs, tray-by-tray profiles, and pinch solutions; (b) eigenvectors and pinch-point solutions.

The RBM does not have the disadvantage of the ZVC that the topology of the split and the active pinch points have to be known in advance. However, because the RBs are only a linear approximation of the curved concentration profiles, the accuracy of this method can be low in cases where the profiles show strong curvature (Bausa et al.8). While such behavior is rarely observed for homogeneous separations, it is much more common for heterogeneous systems. The EWC separation is a good example for this behavior. Figure 13a illustrates the construction of the RBs for the minimum reboiler heat duty determined with the ZVC. The column profiles are also shown in order to illustrate the quality of the approximation of the RB. It can be seen that the approximation of the RB (s3, s2b, s1) is satisfactory in the stripping section. In the rectifying section, four bodies have to be constructed. The RB enclosed by the reflux composition x1, the saddle pinch r2a, and the stable node r1b approximates the tray-by-tray rectifying profile as shown in Figure 13a. Obviously, the approximation of the rectifying profile by this composite rectification body is poor. Therefore, the bodies do not intersect, and the application of the RBM yields a result for the minimum energy demand that is approximately 300% higher than the results obtained with ZVC and BVM. For this reason, it is necessary to refine the RBM in those cases where profiles with high curvature occur. As proposed by Bausa et al.,8 information on the eigenvectors in the pinch points (see Figure 13b) could be used for this purpose. A refinement of the RBM will be presented in a forthcoming publication. 5. Identification of Distillation Regions The regions within the composition space where all of the possible column profiles exist are called distillation regions. Obviously, their knowledge is crucial for the assessment of feasible product specifications. Two types of such regions have been identified in recent years. We have already talked about the SDRs, surrounded by SDBs, which are considered as limits of feasible products at total reflux. On the other hand, it has been demonstrated first by Vogelpohl,25 and later by Nikolaev et al.26 and Fidkowski et al.,12 that the

location of the distillation boundaries, and therefore the extension of the distillation regions, depends on the reflux ratio as well as on the product compositions. The boundaries at operational reflux conditions have been named continuous distillation boundaries (CDBs) by Fidkowski et al.12 Often, the continuous distillation regions (CDRs) surrounded by the CDBs only differ slightly from the SDRs, but not qualitatively. For this reason the SDRs are usually used as an approximation of the CDRs in process synthesis. However, there are many distillation systems where the SDRs are qualitatively different from the CDRs. Hence, the approximation of the CDR by the SDR does not hold. The EWC separation represents such a case. Superposing the rectifying profiles for the EWC system shown, e.g., in Figures 9, 11, 12, and 13a on Figure 1, where the SDBs of the EWC system are shown, we can clearly observe that the actual column profiles cross two of the three SDBs. Therefore, the SDRs cannot be employed for the analysis of feasibility in the EWC system, and the CDRs must be used instead. 5.1. Determination of Distillation Regions. Bausa et al.8 point out that the solution of the pinch-point equations (see section 3.3) coincides at total reflux with all pure components and azeotropes for each column section. Thus, the rectifying section of the EWC system has seven pinch-point solutions at total reflux: one stable node on each vertex of the composition space, one saddle on each binary azeotrope, and one unstable node on the heterogeneous ternary azeotrope. These rectifying pinch points represent the sequence of boiling temperatures in increasing order. Although the pinchpoint solutions for the stripping section are identical to those of the rectifying section at total reflux, the stability of the pinch points behaves contrary. Thus, in the stripping section there is one unstable node on each vertex of the composition space, one stable node on the heterogeneous ternary azeotrope, and finally one saddle for each binary azeotrope, where the stability of their eigenvectors is opposite to the one observed in the rectifying section. In the stripping section, the pinchpoint solutions represent the sequence of boiling temperatures in decreasing order.

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Figure 14. EWC process at QB/F ) 66.284 MJ/kmol (ZVC): (a) CDR I and II of the rectifying section, CDR IV of the stripping section (two rectifying profiles are also shown for φ1 > φ1,max and φ1 < φ1,max); (b) relative position of the top tray liquid composition x2 with respect to the CDB between the unstable node r3 and the saddle r2a.

Because the pinch points are identical to azeotropes and pure components at total reflux, the CDRs coincide with the SDRs of Figure 3. Furthermore, the distillation regions are identical for the rectifying and stripping sections. When the reflux is decreased, some of the pinch points move outside the composition space on their solution branch, while others move inside, as pointed out by Bausa et al.8 in Appendix B. Consequently, the location of the pinch points depends on the product concentrations. Thus, two different sets of pinch-point solutions and CDRs exist, one for the rectifying and another one for the stripping section. Because the vertices of the CDRs are the pinch points, whose location depends on the reboiler heat duty (or reflux ratio), the shape and size of the CDBs is also a function of the reboiler duty. The CDBs for each column section can be exactly determined by calculating tray-by-tray profiles starting at each pinch point. It is worthwhile noting that only pinch-point solutions within the composition space will be considered to determine physically meaningful CDRs. Using this procedure, two CDRs can be identified for the rectifying section (see Figure 14). They are named (r3, r2a, r1a, r2b) or I and (r3, r2a, r1b, r2b) or II according to the pinches needed for their construction. The stripping section has only one CDR named (s3, s2b,

s1, s2a) or IV. While the bottoms composition and bottoms profile are confined to CDR IV at all times, the rectifying profile may be located in either of the two CDRs of the rectifying section I or II depending on where the starting point of the profile x1 is located on the decanter tie line. This observation easily allows one to understand the reason for the sensitivity of the course of the rectifying profile with respect to the selection of φ1 for the EWC example (see Figure 10). A modification of φ1 directly influences the position x1 of the reflux stream on the decanter tie line (see eq 17 and Figure 6). Whereas in homogeneous systems the rectifying and stripping profiles will always belong to the CDR where the distillate or bottoms composition lie, respectively, this can be different in the case of a heterogeneous distillate product because the rectifying profile depends on the selection of φ1. Because a subcooled decanter has been chosen in the EWC process, the position of x1 cannot be directly associated with a CDR. Instead, the liquid composition x2 of tray 2, which depends on φ1, is at boiling temperature and can be compared with the CDB between r3 and r2a (see Figure 14b). As can be seen in Figure 14, the rectifying profile of the EWC system will belong to the CDR where the top tray composition x2(φ1) is located. For φ1 ) φ1,max ) φ1,min and QB ) QBmin, an accumulation of trays occurs near saddle r2a and the profile exactly matches the CDB of intersection located between the pinch points r3, r2a, and r1b. If, for small positive , φ1 ) φ1,max - , the composition x2 and the rest of the rectifying profile will lie inside CDR II. For a small change of φ1 to φ1 ) φ1,max + , the composition x2 moves into CDR I and the inflection of the profile occurs. Because the CDR depends on the heat duty, this inflection can also occur when φ1 is kept constant but QB is changed to higher levels. This sensitivity increases the difficulty of finding the minimum energy demand by the BVM. This is not the case for pinch-based methods. It can be seen from Figure 14 that the minimum energy demand obtained with the ZVC (see Figure 12) coincides with the energy identified to make the CDR of both sections just touch each other (see Figure 14). Hence, a minimum-energy situation is characterized by intersection of one CDR of the rectifying section and one CDR of the stripping section with minimal overlap. Obviously, this criterion is similar to that of the RBM. In fact, the RB constitutes a polytope approximation of the CDR. Depending on the true curvature, the RBM may or may not result in a good estimate for the minimum energy demand. In the multicomponent case, CDRs are expensive to compute and the RBM is an attractive alternative. 5.2. Comparing SDRs and CDRs. As already stated CDR and SDR coincide at total reflux. However, when Figures 3 and 14 are compared, it can be seen that the CDRs at operational reflux differ not only quantitatively but also qualitatively from the SDRs. Obviously, SDR III (see Figure 3) has vanished with decreasing reflux while SDR I and SDR II have somewhat moved with the respective pinch points to form CDR I and CDR II (see Figure 14) at minimum energy demand. This change of CDR topology in the EWC process can be explained by a close inspection of the pinch branches of the rectifying section. These pinch branches are shown in Figure 15. Here, the pinch-point solutions for an


Figure 15. Pinch lines and pinch-point solutions for the rectifying section at QB/F ) 1070 MJ/kmol (r ) 48.938 and s ) 60.385). Right inset: saddle and stable node pinch lines of the rectifying section on the EW edge around the EW binary azeotrope. Axis x: ethanol composition. Axis y: reboiler duty QB/F.

increasing reboiler heat duty (QB/F ) 1070 MJ/kmol, r ) 48.938, and s ) 60.385) are also shown. It can be noted that at this energy level, which is well above operational energy, the saddle pinch solution r2c (originated from the binary EW azeotrope) and the stable node pinch r1c (originated from the ethanol vertex) have moved toward each other. The cutout in Figure 15 shows the correlation of reboiler heat duty and location of the pinch points on this pinch branch. For high energies, two pinch points, namely, r1c and r2c, exist on this pinch branch. For decreasing QB, the saddle (dashed line) and the node (solid line) sections of the branch approach each other until they finally coincide in a saddle-node bifurcation for QB/F ) 975.194 MJ/kmol. Below this QB, no pinch point exists on this branch. Consequently, r1c and r2c are no longer available to form the CDR corresponding to SDR III at total reflux in Figure 3. Figure 16 exhibits the change of CDR topology from three CDRs at high reboiler heat duties above QB/F ) 975.194 MJ/kmol to two CDRs at reboiler heat duties below QB/F ) 975.194 MJ/kmol. This change of topology is not grasped when feasibility

analysis is solely based on the SDR, and therefore feasible regions for column profiles at operational energy cannot be predicted using the SDR. The same phenomenon also occurs in the separation of 2-propanol/water/cyclohexane,23 in the separation of 2-propanol/water/benzene,13 and in the separation of ethanol/water/benzene.1 All of these mixtures have very similar topology (two homogeneous binary azeotropes, one heterogeneous binary azeotrope, one heterogeneous ternary azeotrope, and a similar sequence of boiling temperatures). Therefore, the vanishing of a CDR at lower reflux is characteristic for heterogeneous mixtures with a residue curve map topology similar to the one in Figure 1 for EWC. These examples have shown cases where a distillation boundary vanishes at finite reflux. There is, however, nothing to prevent that nonexistent boundaries at total reflux appear at finite reflux. 5.3. Conditions for a Feasible Design. The analysis of the CDB allows us to establish that the proposed EWC separation is feasible if the concentration of the first tray in the rectifying section under boiling temperature conditions (in our case x2 on tray 2) is located in the rectifying CDR II, from where it is possible to reach the stripping CDR IV in Figure 14. As discussed in section 4.1, this requirement can be achieved by a proper choice of φ1 e φ1,max. Because it is not desirable to backmix a more aqueous-rich phase than necessary in the decanter, the economic optimum is φ1 ) φ1,max at minimum energy demand. This analysis completely replaces the design guideline of Pham and Doherty,14 stating that y2, x2, xF, and xf (feed tray composition) must belong to the SDR where the bottoms product lies. The EWC separation is feasible despite x2 and xF lying outside the stripping SDR (compare, e.g., Figure 11, where x2 is identified, with the stripping SDR in Figure 3). A very important requirement for a feasible separation is that the decanter tie line must be on the same side of the ternary heterogeneous azeotrope as the bottoms concentration. In the EWC system, it has to lie above the EWC azeotrope. This is equivalent to the condition of Pham et al.,13 who state that at least part of the decanter tie line should be located in the appropriate distillation region. Such an appropriate distillation region can be identified with the next rule: The

Figure 16. Topology of CDRs for the rectifying section of the EWC process at different finite reflux ratios.

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6. New Design Method for Heterogeneous Columns

Figure 17. Pinch lines of the rectifying section for an infeasible position of the distillate composition (intersection of the BD balance line and tie line on the ternary azeotrope). Pinch-point solutions at finite reflux ratio are also shown.

material balance line of the column must not intersect the tie line of the heterogeneous ternary azeotrope extended within the Gibbs triangle as shown in Figure 17. If that were the case, it would not be possible to find a rectifying profile that runs from the concentration x1 toward the saddle r2a and then toward the pure water edge (stable node r1b), but instead all profiles would run through the saddle r2b toward node r1b (see Figure 17) or node r1a, depending on φ1 selection. This behavior can easily be deduced from the presence of the unstable node r3. In Figure 17, the pinch line of this unstable pinch is represented. It originates from the ternary heterogeneous azeotrope. When the decanter tie line lies below this azeotrope, the unstable branch, and thus any solution for r3, lies above this tie line and saddle r2a (in Figure 17) is not attainable by a tray-by-tray concentration profile from x1 for any energy supply QB or phase ratio φ1, because of the direction of the unstable eigenvectors shown in Figure 17. Therefore, the tie line which lies exactly on the ternary heterogeneous azeotrope limits the obtainable distillate composition. This fact also shows the importance of operating the decanter at 25 °C. In this way, the decanter tie line is moved away from the heterogeneous azeotrope EWC to locate it in an adequate position. This limitation always needs to be considered when a minimum ternary heterogeneous azeotrope in the mixture exists, such as those for the 2-propanol/water/cyclohexane (Bausa23), 2-propanol/water/benzene (Pham et al.13), and ethanol/water/benzene (Ryan and Doherty1) separations.

In section 4, three techniques for an estimation of the minimum energy demand were presented. A heterogeneous version of the BVM shows the dependency of the profiles in the rectifying section on three parameters, namely, QB, k, and φk. A strong sensitivity not only to product specifications but also to the phase ratio φk can be observed. The ZVC requires previous knowledge of the split topology and the pinch point determining minimum reflux, which is typically not available. The RBM, though a powerful method in many cases, shows a poor approximation of the typically strongly curved CDRs of heteroazeotropic columns. Hence, the CDRs are used directly instead of the RBM approximation in this paper. The CDBs are here obtained from tray-by-tray calculations starting at the pinch point. To avoid excessive numerical calculations, the starting point of the tray-by-tray calculation is perturbed along the eigenvectors of the pinch. In section 5, some assessment of feasible column specifications has been presented. It has been established that one CDR of the stripping section and one CDR of the rectifying section just touch each other at minimum energy conditions. When more than one CDR exists in one column section (like in the rectifying section of the EWC system), the CDR that allows intersection must be identified. For a heterogeneous system like EWC, the product composition or a part of its decanter tie line must lie within this CDR for feasibility. For homogeneous products (like the bottoms product in the EWC system), the tray-by-tray profile of the stripping section will belong to the CDR where the product composition is located. Based on these observations, a new design method for single-feed heterogeneous columns has been developed, which can also be applied to homogeneous systems: 1. Choose feed and product compositions (xF, xB, and xD) according to the material balance of the column. If a minimum ternary heterogeneous azeotrope exists, verify that the overall material balance line from xD to xB does not intersect the VLL tie line of this azeotrope. 2. Choose some reboiler duty QB. 3. Calculate the condenser duty QD using the energy balance of the column, eq 19. 4. Calculate the pinch-point solutions of both column sections (section 3.3). 5. Determine the CDBs for both column sections from tray-by-tray calculations (sections 3.1 and 3.2) starting from the pinch points, and identify the CDRs surrounded by these CDBs (section 5). Then, identify the CDRs which comprise the product compositions. These regions are named CDRD and CDRB if they contain the distillate or bottoms product concentrations or their associated tie lines if a decanter is used, respectively. 6. Check the intersection between CDRD and CDRB for QB chosen in step 2. If CDRD and CDRB intersect, decrease QB and go back to step 3. If CDRD and CDRB do not intersect, increase QB and go back to step 3. If CDRD and CDRB intersect each other with minimal overlap, go to step 7. 7. If the distillate product is heterogeneous and there is a decanter at the top of the column, select φ1 such that the liquid composition of the top tray x2 lies on any one of the CDBs that surround the CDRD of intersection. Such a CDB is here identified as CDBD. If φ1 ) φ01, the


existence of multiple heterogeneous trays in the rectifying section must be considered. If φ1 * φ01 must be selected, phase split will only occur in the decanter. The composition of the reflux stream x1 is not used here because, in cases where the decanter does not operate at boiling temperatures, as in EWC example, the position of x1 cannot be compared with the CDBD. 8. If the distillate product is homogeneous, verify that its composition xD lies on the CDBD. If this condition is not fulfilled, choose a different xD and repeat from step 1. 9. Finally, when φ1 ) φ01, the total number of heterogeneous trays k of the rectifying profile can be obtained from the correspondent CDBD. In the heterogeneous region of a rectifying profile, the condition φn ) φ0n is fulfilled for the first n ) k - 1 heterogeneous trays. For the last heterogeneous tray k, φk must be selected to guarantee profile intersection. For the stripping section, the total number of heterogeneous trays can be obtained from the heterogeneous region of the CDBB. Contrary to the design method of Pham et al.,13 where φ1 must be previously calculated to locate x2 and y2 in the stripping SDR, this procedure does not require a priori knowledge of φ1 because the minimum energy does not depend on φ1 but only on the intersection of CDRD and CDRB. Note, however, that feasibility at minimum energy demand does not necessarily guarantee feasibility at higher energy levels because the CDRs change both qualitatively and quantitatively with varying reflux. As pointed out by Levy et al.,4 an operational reflux ratio is usually selected between 1.05 and 1.5 times the minimum reflux. Once the minimum energy demand has been determined with the design method presented above and an operational reflux ratio has subsequently been chosen, the feasibility of the separation must be rechecked following steps 3-6 and 8. From steps 7 and 9 a recommended value for φ1 will be obtained. However, for operational reflux a feasible range (φ1,min to φ1,max) will exist where φ1 can be located. 7. Examples Using the new design method for heterogeneous columns, three different ternary separations are studied: two of them with a heterogeneous distillate product but different topology and the other one with a homogeneous distillate product and a heterogeneous bottoms product. Some of these separations have proven to be difficult to design. However, even in these cases, the new design method generates feasible and economically attractive designs. 7.1. 2-Propanol/Water/Cyclohexane. The dehydration of 2-propanol in a heterogeneous column using cyclohexane as the entrainer23,27 is topologically similar to that of the EWC process presented throughout the last sections. Nevertheless, it is presented here because, in contrast to the EWC process, multiple heterogeneous trays are required in the rectifying section in order to reach the true minimum energy demand of the process. Figure 18 shows the azeotropes and SDRs for this separation (to compare with EWC, see Figure 3). The mixture exhibits one ternary heteroazeotrope (PWC) and three binary azeotropes (PW, CW, and PC), one for each binary subsystem. Feed and product specifications are given in Table 3. The distillate product and its corresponding tie line have been chosen using step 1 of the new design algorithm.

Figure 18. Phase diagram with SDB of the PWC system (p ) 1.013 bar); material balance of the heteroazeotropic column.

When steps 2-7 of the design procedure of section 6 are applied, a minimum energy demand of QBmin/F ) 40.77 MJ/kmol (r ) 2.167 and s ) 1.608) corresponding to CDRD and CDRB just touching is determined for the PWC separation. Figure 19 shows the CDRs of both column sections as well as the column profiles. In the stripping section, only one CDR exists. This is CDRB because B is contained in it. In the rectifying section, two CDRs exist (I and II). As in the EWC example, the decanter tie line occupies two CDRs; therefore, both are identified as CDRD from step 5 in the rectifying section. From step 7, we find φ1 ) φ01 and locate x2 on the CDBD between the pinch points r2a and r1b (see Figure 19). In total, eight heterogeneous trays for the PWC example are determined by step 9 of the design algorithm, despite the similarity with the EWC mixture, where only one heterogeneous tray was necessary (i.e., the decanter; section 4). Different designs can be obtained by arbitrarily choosing a smaller or larger number of heterogeneous trays. However, further analysis of such columns using the heterogeneous BVM shows that in this case the energy demand must be increased significantly in order to guarantee feasibility. A rather dramatic increase of energy demand is observed when the design method of Pham et al.13 is used to generate a design for a column with only one heterogeneous tray. Figure 20 shows the tray-by-tray profiles with only one heterogeneous tray in the rectifying section for this column, at the minimum energy calculated for eight heterogeneous trays with the new design algorithm. It can clearly be seen that at this energy level the rectifying and stripping profiles do not intersect. In fact, intersection can only be achieved when the stable node s1 coincides with the ternary azeotrope, which only occurs at total reflux conditions. Because multiple heterogeneous trays are not calculated in the rectifying section, the saddle r2a cannot be reached and instead the rectifying profile runs on the decanter tie line toward the homogeneous region, as Figure 20 shows. This results from the unstable pinch-point solution in the rectifying section r3 coinciding almost with the reflux composition x1 and therefore with the decanter tie line. Thus, the PWC example nicely demonstrates that multiple heterogeneous trays in the rectifying section

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Table 3. Column Specifications for the PWC and AWV Examples figure 18

22

xP xW xC temp, °C vapor fraction xA xW xV temp, °C vapor fraction

feed

distillate

bottoms

0.7820 0.2070 0.0100 78.18 0.0 0.5130 0.2810 0.2060 42.5 subcooled

0.4018 0.5706 0.0276 63.86 0.0 0.0004 0.5768 0.4228 66.04 0.0

1.0000 1.0 × 10-6 1.0 × 10-5 82.24 0.0 1.0000 1.0 × 10-10 1.0 × 10-10 118.13 0.0

Figure 19. CDR and tray-by-tray profiles of the PWC separation at QBmin/F ) 40.77 MJ/kmol (r ) 2.167 and s ) 1.608).

Figure 20. Rectifying profile of the PWC system with only one heterogeneous tray at QBmin/F ) 40.77 MJ/kmol.

can lead to greatly improved designs and that the optimal number of heterogeneous trays cannot be determined a priori even if good designs for the separation of similar mixtures are already known. 7.2. Acetic Acid/Water/Vinyl Acetate. The separation of acetic acid from water and vinyl acetate (AWV) corresponds to the last stage of the vinyl acetate monomer process presented by Luyben and Tyreús.29 According to this process, the ternary feed is separated into pure acetic acid as the bottoms product and a binary heterogeneous mixture of water and vinyl acetate. This heterogeneous mixture is split into an organic phase, which is rich in vinyl acetate, and an aqueous phase in a decanter at the top of the column. Both phases are

D/F pressure (bar)

GE model; vapor phase (param source) NRTL; ideal (Wang et al.27)

0.364 1.013

0.487 1.013

UNIQUAC; association effect30 (Aspen Plus28)

Figure 21. Residue curve map and miscibility gap at boiling temperatures at p ) 1.013 bar for the AWV system.

withdrawn as individual distillate products. For simplicity, the term distillate product will denote the mixture of these two individual product streams (see Table 3) for the remainder of this example. The residue curve map of the system is shown in Figure 21. This system is characterized by association in the vapor phase and dimerization of the associating component, which is considered using the model presented by Nothnagel et al.30 Using the new design method, the minimum energy demand of this process is determined as QBmin/F ) 49.238 MJ/kmol (r ) 1.361 and s ) 4.014). Figure 22a shows the CDRs at this energy. Only one CDR exists in both the rectifying and stripping sections. Although the decanter tie line intersects the vaporline, φ01 > 1 results at this energy level. This indicates that it is impossible to choose a physically valid liquid composition such that the corresponding vapor phase is located on the vaporline. Therefore, it is not possible to choose more than k ) 1 heterogeneous trays. The phase ratio φ1 can be chosen arbitrarily. For all choices of φ1, the profile of the rectifying section coincides with the CDB between the pinch points r2 and r1. It can be seen in Figure 22a that the CDRs of the rectifying and stripping sections largely overlap. However, operation of the column at lower energy is impossible because of the appearance of a tangent pinch in the rectifying section near the water vertex at lower energy supply. Figure 22b shows the CDRs at QB/F ) 49.003 MJ/kmol < QBmin/F (φ01 > 1). Two new homogeneous pinch-point solutions (r2a and r1b in Figure 22b) have appeared. The saddle r2a and the stable node r1b give rise to two new CDBs in the rectifying section. A


Figure 22. CDR of the rectifying and stripping sections for the AWV process: (a) at minimum energy demand (QBmin/F ) 49.238 MJ/ kmol); (b) at a lower energy demand (QB/F ) 49.003 MJ/kmol).

total of three CDRs are present in this section, identified as I-III in Figure 22b. Any tray-by-tray profile starting at the distillate product or at any liquid composition on the decanter tie line lies in CDR III and thus stops at the stable node r1b. Therefore, the profiles of the rectifying and stripping sections do not intersect. Although CDR I intersects the CDR of the stripping section at this energy level, the CDRD corresponds to CDR III (step 5 of the method). Therefore, no intersection between CDRD and CDRB occurs, and consequently the energy supply must be increased according to step 6. In this way, the identification of the attainable CDRs, as proposed in the design method, allows one to identify tangent pinch points and thus to avoid wrong predictions for minimum energy demand. As already stated, phase split occurs only in the decanter at minimum energy demand. However, a small increase in the energy supply (i.e., QB/F g 1.0035QBmin/ F) allows one to obtain up to three heterogeneous stages (φ1 ) φ01, φ2 ) φ02, and φ3 ) φk). Consequently, multiple heterogeneous trays are likely at operational reflux. This fact offers the advantage to select an adequate φn in order to reduce the number of theoretical stages in the rectifying section, shifting the rectifying profile trajectory away from the water vertex (saddle r2 in Figure 22a), where a high number of trays are needed to cross over the well-known binary tangent point on the acetic acid/water edge. All examples presented so far had a heterogeneous distillate product and a decanter separating at the top of the column. One of the nice properties of this class of processes is that the phase ratio in the decanter φ1 is available as a design degree of freedom. This decouples the computation of pinch points and the calculation of column profiles. The pinch equations only depend on xD and QB, whereas φ1 is used to place the liquid composition of the reflux stream x1 (which is the starting point of the rectifying section profile) on the corresponding CDB. The next example will deal with a process having a homogeneous distillate product and will discuss the resulting implications on process design. 7.3. Acetone/Water/Toluene. Siegert et al.31 present a heterogeneous process for the dehydration of acetone using toluene as the entrainer. The acetone/water/ toluene (AWT) system has one binary heterogeneous minimum azeotrope (WT) on the water/toluene edge (see

Figure 23a). The heterogeneous feed mixture (F) is separated into a ternary acetone-rich distillate product (D) and a binary heterogeneous mixture of water and toluene (B). The product specifications of the separation are given in Table 4. It will be shown that, unlike the other example processes presented above, it is not possible to fix all of these specifications before applying the design method. In particular, the distillate composition xD needs to be adjusted during the design procedure (see step 8 in section 6). In the AWT separation the complete stripping section and the feed tray are heterogeneous while the distillate composition is homogeneous. Therefore, the profile of the rectifying section must enter the miscibility gap on some tray in order to get an intersection of the profiles. Section 3 describes under which constraints such behavior of the rectifying profile is possible: Starting from a homogeneous tray n, the next tray n + 1 can only be heterogeneous when its vapor composition is located exactly on the vaporline. However, under this constraint, 1 degree of freedom in choosing xD is lost because the vaporline only has C - 2 degrees of freedom. In terms of the design method for heterogeneous processes, this characteristic property of the AWT separation can be interpreted as follows: The CDB of the rectifying section that runs from the homogeneous region to the heterogeneous saddle r2 (Figure 23b) is the only possible column profile that allows transition from the homogeneous to the heterogeneous region because only the liquid compositions xn of the heterogeneous section of this boundary are in equilibrium with vapor compositions lying on the vaporline and only on this boundary the condition φn ) φ0n is fulfilled. Thus, xD must be located on this CDB in order to achieve a feasible separation. This design rule is enforced by step 8 of the design procedure. When the composition of acetone in the distillate product is fixed to xD,acetone ) 0.8, a minimum heat duty of QBmin/F ) 11.85 MJ/kmol (r ) 1.732 and s ) 0.358) and the corresponding distillate product xD ) (0.800, 0.1516, 0.0484) are determined by iteratively applying the design procedure. At this level of energy, the saddle of the rectifying section r2 and the stable node of the stripping section s1 lie on the same tie line. Figure 23a shows the column profile, whereas the pinch lines and the CDBs in the rectifying section for this design are shown in Figure

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Figure 23. (a) Tray-by-tray profiles and pinch-point solutions of both column sections at QBmin/F ) 11.85 MJ/kmol (r ) 1.732 and s ) 0.358). (b) Pinch lines and CDB in the rectifying section of the AWT process. Table 4. Column Specifications for AWT Examples figure 23

24a

24c

xA xW xT temp, °C vapor fraction xA xW xT temp, °C vapor fraction xA xW xT temp, °C vapor fraction

feed

distillate

bottoms

0.1100 0.4460 0.4440 73.34 0.0 0.1100 0.4460 0.4440 73.34 0.0 0.1100 0.4460 0.4440 73.34 0.0

0.8000 0.1516 0.0484 58.66 0.0 0.8000 0.1700 0.0300 58.42 0.0 0.8000 0.1300 0.0700 58.90 0.0

1.0 × 10-10 0.4923 0.5077 84.39 0.0 1.0 × 10-10 0.4900 0.5100 84.39 0.0 1.0 × 10-10 0.4964 0.5036 84.39 0.0

23b. Thus, the location of both pinches r2 and s1 on the same tie line is identical to intersection of the CDRs of both column sections in this special case. Starting from the distillate composition, the column profile approaches the saddle pinch r2 and the bulk liquid compositions xn remain close to this pinch point as long as φn ) φ0n. If φn * φ0n is selected before reaching saddle r2, the rectifying profile shifts to the homogeneous region at tray n + 1 and r2 is not attainable. Consequently, the energy supply must be increased in order to achieve intersection of the profiles. The rectifying profiles in Figures 23a and 24a,c only differ in slight perturbations of distillate product composition and reboiler heat duty (see Table 4) and show the sensitivity of the profiles with respect to this composition. The reboiler duties in Figure 24a,c correspond to the situation when the saddle of rectifying section r2 and the stable node of stripping section s1 lie on the same tie line, as in Figure 23. At such energy levels, the stripping section CDRs and the rectifying section CDRs just touch. However, step 8 of the design procedure identifies that xD does not lie on CDBD. Therefore, the split is not feasible, and a different xD must be chosen. Note that in order to simultaneously find the minimum heat duty and the corresponding distillate composition two intertwined iterations had to be carried out (steps 6 and 8 of the design procedure). This is necessary because the location of the pinch

D/F pressure (bar)

GE model; vapor phase (param source) UNIQUAC; ideal (Aspen Plus28)

0.1375 1.013 UNIQUAC; ideal (Aspen Plus28) 0.1375 1.013 UNIQUAC; ideal (Aspen Plus28) 0.1375 1.013

points and the course of the CDB are functions of both distillate product composition and energy. The solution presented above was found using a multiple shooting technique.23 8. Conclusions In this paper, a comprehensive study of design phenomena in ternary heterogeneous distillation columns has been presented. An extension to the algorithm of Pham et al.13 for the calculation of tray-by-tray profiles in heterogeneous systems has been developed, which allows the determination of multiple heterogeneous stages. Furthermore, the algorithm is not restricted by the CMO assumption. It has been shown that both the number of heterogeneous trays k and the bulk liquid concentration on the last heterogeneous column tray determined by φk can be viewed as design decisions. This results in an unlimited number of mathematically consistent column profiles. However, only a limited number of these profiles correspond to a feasible separation. The appearance of these additional degrees of freedom for the calculation of tray-by-tray profiles complicates the application of tray-by-tray-based design methods such as the BVM significantly. On the other hand, the application of pinch-based methods for the estimation of the minimum energy demand such as RBM or ZVC eliminates the dependency on k and φk. However, the applicability of ZVC


Figure 24. Infeasible splits for the AWT process. Tray-by-tray profiles in both column sections (a and c) and pinch lines and CDBs of the rectifying section (b and d) are shown: (a and b) QB/F ) 11.88 MJ/kmol (r ) 1.729 and s ) 0.359); (c and d) QB/F ) 11.78 MJ/kmol (r ) 1.717 and s ) 0.356).

suffers greatly from the disadvantage that the topology of the split and the set of active pinch points must be known a priori.8 The RBM is not subject to such limitations and can thus be used as an entirely general method for the calculation of the minimum energy demand. It can be extended to analyze heterogeneous columns and is shown to successfully predict qualitatively correct pinch paths for all examples given in this paper. However, the approximation of highly nonlinear CDRs by RBs with linear boundaries leads to significant errors in the minimum energy calculation. Therefore, a better method for construction of the RBs which is capable of covering the curvature properly has to be developed in future work. The examples presented here demonstrate as well that the application of feasibility analysis based on residue curve maps and SDBs often leads to substantial errors for heterogeneous separations. Considering the resulting limitations of existent design methods for heterogeneous distillation processes, a new procedure for this task has been proposed in this paper. This new design method makes use of pinch-point calculation and CDBs in order to determine the feasible regions for column profiles in both sections of the column as a function of the reboiler heat duty. The minimum energy demand is obtained by checking the CDBs of both

column sections controlling the separation for intersection while adjusting the heat duty. In this paper, tray-by-tray calculations were employed for the determination of the CDBs. The computational effort for the construction of the CDR is relatively high. However, it is still manageable for ternary separations. In a forthcoming publication, the CDBs will be replaced by curved RBs in order to extend the method to quaternary and higher dimensional systems. In addition, a significant reduction of computational effort is expected. Future work will address a refinement of the design procedure and extensions to heterogeneous reactive distillation, complex column arrangements, and multicomponent separations. Acknowledgment The authors thank DFG (Deutsche Forschungsgemeinschaft) for the financial support of this research. R.Y.U. is grateful for financial support provided by FUNDAYACUCHO (Fundacioń Gran Mariscal de Ayacucho, Venezuela) and for administrative support in Germany by DAAD (Deutscher Akademischer Austausch Dienst). The comments of the reviewers have improved the presentation of the results and are therefore appreciated.

3866


Notation Symbols B ) bottoms product flow rate [kmol/s] C ) number of components D ) distillate flow rate [kmol/s] F ) feed flow rate [kmol/s] h ) molar enthalpy [kJ/kmol] K ) equilibrium constant L ) liquid flow rate [kmol/s] p ) pressure [bar] Q ) energy [MW] ri ) pinch i in the rectifying section r ) reflux ratio si ) pinch i in the stripping section s ) reboiler ratio T ) temperature [°C] V ) vapor flow rate [kmol/s] x ) liquid-phase composition x ) vector of liquid-phase composition y ) vapor-phase composition y ) vector of vapor-phase composition z ) net product composition z ) vector of net product composition Greek Letter φ ) liquid-phase ratio Subscripts B ) bottom D ) distillate F ) feed f ) feed tray i ) component k ) last heterogeneous tray n ) tray number N ) net product Superscripts L ) liquid phase V ) vapor phase I ) liquid phase I II ) liquid phase II 0 ) on vaporline

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Received for review September 10, 2001 Revised manuscript received March 1, 2002 Accepted March 4, 2002 IE0107486

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