Operation of Homogeneous Azeotropic Distillation Column Sequences

4512

Ind. Eng. Chem. Res. 2003, 42, 4512-4534

Operation of Homogeneous Azeotropic Distillation Column Sequences Jan Ulrich† and Manfred Morari* Automatic Control Laboratory, ETH-Z, Swiss Federal Institute of Technology, CH-8092 Zu¨ rich, Switzerland

A self-optimizing control concept is provided for sequences of three homogeneous azeotropic distillation columns with two recycles separating a ternary feed of a 020 mixture that has two simple distillation regions that are separated by a residue curve boundary. Studying a case with reduced complexity (columns of infinite length operated at infinite reflux), one of the two recycles (the entrainer recycle) is identified as the key manipulated variable for ensuring optimal operation of the sequence. Different setups of the scheme have different optimal feed regions. The theoretical results are illustrated with rigorous simulations of a 020 mixture: methanol (which acts as an entrainer), 2-propanol, and water. Dynamic simulations confirm that the process operates robustly by using the (L/D)QR scheme for the individual columns and one additional control loop that controls the methanol (entrainer) composition of the distillate by manipulation of the overall feed of one column. Dynamic simulations also confirm that column profiles can cross residue curve boundaries, distillation boundaries, and even azeotropes if the feed composition is changed such that a column profile is only feasible in the other distillation region. 1. Introduction In the chemical industry, distillation is the most common process used to separate multicomponent mixtures. If the mixtures are zeotropic, they can be separated into pure components by a simple distillation sequence. If the mixture is azeotropic, a simple sequence cannot separate the mixture into its pure components. A common technique is to break the azeotrope by a third component, an entrainer, that is added to the mixture. One of the most difficult tasks in designing a separation sequence is the proper choice of entrainers, which has been extensively discussed in the literature1-6 and is also subject of chemical engineering textbooks about separation processes.7,8 Further, the design of azeotropic column sequences for a given mixture has attracted additional research.9-26 The procedures presented in these works are complex optimization problems that can be used to design a sequence, but for operation, physical insight is necessary to overcome the model uncertainties and to propose robust control schemes. The key idea of azeotropic distillation column sequences is to remove one component in the first column and to process the other component, enriched with the entrainer, in a second column. In the second column, the entrainer is recovered and recycled to the first column, while the second component is retrieved. Hence, azeotropic distillation column sequences exhibit recycles. Product recycles change the steady-state and dynamic properties of the system, as extensively discussed by Bill Luyben,27-30 to whom this paper is dedicated. For distillation column sequences with recycles, operation and control is more difficult than for sequences without recycles. * Author to whom correspondence should be addressed. Tel.: +41 1 632 7626. Fax: +41 1 632 1211. E-mail: morari@ aut.ee.ethz.ch. † Present address: BASF Aktiengesellschaft, Conceptual Process Engineering, GIC/P - Q920, 67056 Ludwigshafen, Germany.

In this paper, operation and control of the boundary separation scheme31 (also called a process with border crossing2,6,32 or a light entrainer scheme3,4) is analyzed. The boundary separation scheme is often advantageous. When two components react with each other to form a third component, the third component is often a larger molecule with a higher boiling temperature that can form a minimum-boiling azeotrope with one of the reactants or side products. The best entrainer candidates for breaking possible azeotropes of the products are then either the reactants or a side product of the reaction. An example is the reaction of acetylene with acetone to form methyl isobutanol in the presence of ammonia as a solvent and aqueous KOH as a catalyst.33 After the removal of the acetylene and ammonia, a mixture of acetone, methyl isobutanol, and water is to be separated, which is a 020 mixture. This mixture can be separated with the boundary separation scheme, but the current industrial setup is a heterogeneous sequence.34 Here, the boundary separation scheme consumes less energy for certain feed compositions at the cost of an additional column.35 For the mixture methanol/ 2-propanol/water, homogeneous and heterogeneous sequences were found to have the same setup, but the boundary separation scheme consumes less energy for low 2-propanol contents.35 Although these homogeneous mixtures have the advantage that no additional component has to be introduced to the process (self-entrained mixtures7), they are rarely used in industry. We believe that this process is not more widely adopted because of the more complex process behavior of the boundary separation scheme introduced by the two recycles and the uncertainty in describing the residue curve boundary. The classical heterogeneous sequences with well-documented process behavior are preferred even when these sequences consume more energy for the same equipment. The motivation for this paper is to explain the operation of the boundary separation scheme and to

10.1021/ie020930m CCC: $25.00 © 2003 American Chemical Society Published on Web 06/07/2003

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4513

Figure 1. Column configuration of the boundary separation scheme.

Figure 3. Boundary separation scheme with three different feed locations: setup 1, Fc to column 1; setup 2, Fc to column 2; and setup 3, Fc to column 3.

Figure 2. Qualitative representation of the boundary separation scheme in a [020] residue curve map.

show that a fairly simple control scheme can operate the process reliably. 2. Problem Statement 2.1. Process. The example mixture to be separated is a 020 mixture:36 methanol, 2-propanol, and water. Figure 1 shows the setup of the boundary separation scheme with two columns with 30 stages (including top and bottom) and one column with 50 stages (including top and bottom). Figure 2 shows qualitatively the mass balances and column profiles in a residue curve map. The crude feed, Fc, enters column 1 together with the mixture, M1, of two recycles, R2 and D3, to give F1. Here, water is removed via the bottom, B1. The distillate D1, which consists of methanol, 2-propanol, and water, enters the second column. Here, methanol is removed from the sequence via Dex 2 , and the rest of D2 is recycled to column 1 as R2. In column 3, the remaining two-component mixture of 2-propanol and water, which lies on the left side of the azeotrope, is separated into pure water (B3) and the 2-propanol/water azeotrope (D3), which is recycled to column 1. If the sequence separates only binary 2-propanol/water feeds, it might be better to introduce the crude feed to column 3 for high

2-propanol contents.6,37 For ternary feeds, it might also be desirable to introduce the crude feed into column 2. Figure 3 shows the three different locations of the crude feed Fc. The resulting three configurations are differentiated by the names setup 1, setup 2, and setup 3, respectively. For binary crude feeds, setups 1 and 3 have been discussed in the literature;6,37 setup 2 is new. 2.2. Control Concept. The framework of selfoptimizing control38-41 is applied to the boundary separation scheme to yield a hierarchical control concept with two layers: an optimization layer that computes the optimal values of the chosen reference (controlled) variables and a control layer that ensures that the process operates at the optimal values of the controlled variables. The controlled variables are chosen such that, if they are held constant, they will lead to the optimal adjustments of the manipulated variables and, thereby, to the optimal operating conditions.42 The controlled variables of the optimization layer are identified on the basis of steady-state considerations, and their performance and robustness toward implementation and modeling errors are analyzed. The control layer is realized with simple single-loop PID controllers for the selected controller pairings. For self-optimizing control, a cost function is needed. For a fixed column design, a good cost function is the sum of the reboiler duties QRi , because reboiler duties are the main part of the operating costs

J ) QR1 + QR2 + QR3

(1)

Two main process disturbances are considered for the boundary separation scheme: (1) changes in the crude feed condition (flow rate and composition), and (2) changes in the interconnecting streams caused by rejection of the crude feed disturbances. The main uncertainty is the curvature of the residue curve boundary (distillation line boundary). A control scheme has to be robust against this uncertainty. Because of the complexity of the problem, the process is first analyzed for an asymptotic case: a sequence of ∞/∞ columns, which are columns of infinite length operated at infinite reflux.43,44 Instead of the specific

4514 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003

Figure 4. Eight cases for sequence A. For simplicity, the column profiles are not shown but only the mass balance lines of the columns. The thick dashed, dashed-dotted, and dashed-double-dotted lines denote columns 1, 2, and 3, respectively.

mixture methanol/2-propanol/water, the concept is discussed first for a general 020 mixture of the low boiler L, the intermediate boiler I, and the heavy boiler H. 3. Degrees of Freedom and Constraints of the ∞/∞ Sequence Because of the complexity of the process with two recycles, the analysis starts with three columns in a row (sequence A, section 3.1). Then, one recycle after the other is closed (sequences B and C, sections 3.2 and 3.3, respectively). The recycles change the process properties, especially the feasible feed regions. Therefore, the feasible feed regions are also discussed here. Further, Stichlmair (stated by Laroche et al.4 as a personal communication) suggested to use a third recycle. This option is not considered for the design of the control schemes, as discussed in Appendix A.

3.1. Sequence A: Three Columns in a Row. Figure 4 shows, in the center, three columns in a row without any recycles. A distillation column separating a c-component mixture has c + 2 degrees of freedom. Hence, a three-column sequence has 3c + 6 degrees of freedom. For a given external crude feed Fc, c degrees of freedom are fixed: c - 1 for the feed composition and one for the feed flow rate. In addition, 2c degrees of freedom are fixed by the selected connections (D1 ) F2 and B2 ) F3). For the ∞/∞ analysis, the reflux for each column is infinite, thereby fixing three more degrees of freedom. Hence, 3c + 3 degrees of freedom are fixed, leaving three degrees of freedom for the sequence. By specifying B1, D2, and D3, all flow rates and product compositions are determined. In general, each column profile can be of type I, II, or III,44 which gives 33 ) 27 combinations. However, not all of them are feasible or make sense. If the profile of

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4515 Table 1. Possible Combinations of Profile Types for Sequence A: Three Columns in a Row case column

1

2

3

4

5

6

7

8

9

10

11

12

1 2 3 feasible? degrees of freedom

II I I no -

II I II yes 3

II I III yes 3

II III I no -

II III II yes 3

II III III yes 3

III I I no -

III I II yes 3

III I III yes 3

III III I no -

III III II yes 3

III III III yes 3

column 1 is a type I profile, the distillate D1 will consist of pure L (unstable node). The subsequent columns cannot perform any further, useful separation. Therefore, these cases will be disregarded, reducing the possible cases to 18. If the profile of column 2 is of type II, the bottoms B2 will consist of pure H (a stable node), thus allowing no further separation in column 3. This reduces the number of useful combinations to 12 (there are 9 cases of column 2 profiles of type II, but three of them include a column 1 of type I). These 12 cases are listed in Table 1 [for a type III profile, the I/H azeotrope (saddle) is located in the middle of the column]. For cases 1 and 7, column 3 does not perform any separation that column 2 cannot do. For cases 4 and 10, type I for column 3 is infeasible for distillate flows D3 greater than 0. Hence, eight cases remain for a setup that allows for the separation of a three-component feed into products on both sides of the separatrix. These cases are also shown in Figure 4. Here, a II-I-II profile means that column 1 has a profile of type II, column 2 of type I, and column 3 of type II. The feasible feed region of the sequence with the bottom composition of column 1 (xB1) in the convex set is the convex set of the 020 mixture. Multiple steady states are not possible. 3.2. Sequence B: Three Columns with One Recycle. Figure 5 shows the three-column sequence with D3 recycled to the feed of column 1. In general, closing a recycle does not change the number of degrees of freedom, but some specific profile combinations change the number of degrees of freedom because the recycle imposes constraints so that just one specific combination of flow rates is feasible. Here, only the II-I-II profile is discussed (see Ulrich35 for the other profiles). Figure 6a shows the mass balances for a II-I-II profile. The mixing of D3 with Fc is indicated by the thin dashed line. The external flow rates (B1, D2, and B3) are given by the composition of the ternary crude feed Fc because of the compositions of the pure products xB1, xD2, and xB3. This fixes two of the three degrees of freedom (the third flow rate, e.g., B3 results from the overall mass balance). Hence, sequence B has one degree of freedom, for example D3. The recycle imposes further constraints, indicated by the two additional thin dashed lines. Following the idea of Doherty and Caldarola,1 the mass balance around columns 2 and 3 has to be fulfilled for a feasible setup

xD1D1 ) xD3D3 + (xD2D2 + xB3B3)

(2)

With

xD2B3 )

(xD2D2 + xB3B3) D2 + B3

(3)

it follows from eq 2 that xD2B3 has to be collinear with xD1 and xD3 (thin dashed line in the residue curve map

Figure 5. Sequence B: three columns with one recycle (D3).

in Figure 6a). xD2B3 follows from the overall mass balance

xFcFc ) xB1B1 + xD2B3(D2 + B3)

(4)

In the residue curve map (Figure 6a), xD2B3 can be determined by the intersection of the line connecting xFc and xB1 with the line connecting xD2 and xB3. The feasible feed region is constructed as follows. For the locations of xD2B3 and xD3 shown in Figure 6a, the mass balance line (eq 2) is a secant to the residue curve boundary, i.e., it crosses the boundary twice. Increasing the recycle D3 forces xD1 to move toward the boundary, and at the point where xD1 is at the boundary (Figure 6b), increasing D3 is possible only if xD3 moves toward the I/H azeotrope. In the limiting case of infinite D3, the compositions xF1, xD1, xB2, and xD3 all coincide in the I/H azeotrope, as shown in Figure 6d. Figure 6c shows that the amount of L in Fc, which equals the amount of L in D2B3, can be decreased at constant amount of H until the mass balance line around columns 2 and 3 is tangent to the boundary. Then, xD1 lies on the boundary. The amount of L in D2B3 can be further decreased only if xD3 moves toward the binary azeotrope (Figure 6d), which increases D3. In the limiting case of minimum L content at constant H content, D3 is again infinite, and xD1 again lies on the binary azeotrope, as do xF1, xB2, and xD3. The tangent to the boundary at that point determines the xD2B3 with a minimum amount of L. This gives the feasible feed region, as indicated by the shaded region in Figure 6d. This construction of the feasible feed region is consistent with the abstract statements of Serafimov et al.9,10 For sequence B, multiple steady states are possible: 1. Input Multiplicities. For the case shown in Figure 6a, the flow D3 can be continuously varied for a constant xD3. This gives a set of feasible compositions xD1, which all lie on the line connecting xD2B3 with xD3 given by the mass balance line of the subsystem around columns 2 and 3, eq 2. The same external flow rates and compositions can be reached for a continuum of different D3’s. 2. State Multiplicities. For a constant D3, it is possible to find a feasible set of compositions xD1, xB2, and xD3 for a continuous set of compositions xD1. These sets are illustrated in Figure 7a. (These kind of multiplicities were also observed for a heterogeneous sequence.45) 3.3. Sequence C: Three Columns with Two Recycles. Figure 3 shows the three different setups for sequence C. Setup 1 has the same configuration as sequence B but a second recycle R2, giving two degrees of freedom for a II-I-II profile. The effect of R2 on the process is illustrated in Figure 8 for a II-I-II profile. With recycle R2, the mass balance around columns 2


Figure 6. Derivation of the feasible feed region for the II-I-II profile of sequence B.

and 3 changes to become B3 xD1D1 ) (xD2R2 + xD3D3) + (xD2Dex 2 + x B3) ex

xD1D1 ) xM1(R2 +D3) + xD2 B3(Dex 2 + B3)

(5)

ex

From eq 5, it follows that xD1, xD2 B3, and xM1 are collinear. The composition xM1 is determined by the mixing of the two recycle streams R2 and D3. As Figure ex

8 shows, the line connecting xD2 B3 with xD3 (xM1 for R2 ) 0) is completely in the nonconvex set for R2 ) 0. Because xD1 has to lie in the convex set (the residue curve boundary also belongs to the convex set), the feed is infeasible for R2 ) 0. For increasing R2, xM1 moves toward xD2. At some point, xM1 enters the convex set. Now, xD1 might also lie in the convex set, so that the sequence would be feasible. Because R2 can be varied between 0 and ∞, xM1 can move between xD2 and xD3. ex

Thus, it is possible to find an R2 for each possible xD2 B3 such that xD1 lies in the convex set and xD1 is collinear ex

with xM1 and xD2 B3. Hence, the region of feasible feeds of the three-column sequence is now the triangle ex

spanned by xB1, xD2 , and xB3. For a II-I-II profile, this corresponds to the whole ternary composition space, including the nonconvex set. For sequence C, the input and state multiplicities that were observed for sequence B are also possible. For any fixed R2, D3 can be fixed in a certain range to give the same external outputs. For each fixed D3, a set of solutions is possible (Figure 9).

Figure 7. Sequence B: II-I-II profile showing the solution space for constant D3.

Figure 8 also shows the mass balances for a II-I-II profile of setups 2 and 3. As for setup 1, the feasible feed regions of these two setups are the whole composition triangle for a II-I-II profile. Different from setup 1, setups 2 and 3 are infeasible without the recycle R2. The reason is that xD3 lies in the nonconvex set. Therefore, mixing of D3 with R2 > 0 is required to place xF1 in the convex set for a feasible column 1 profile. Even though all three setups have the same structure, the different positions of the crude feed change the performance of the setups. 4. Optimization of the ∞/∞ Sequence For an ∞/∞ column, the reboiler duty is infinite, and an approximation is needed to find an objective function


Figure 9. Sequence C, setup 1: II-I-II profile with fixed R2 showing the solution space for constant D3.

can be solved for the weighted cost function (eq 7). For a sequence of ∞/∞ columns, which will be investigated here using the residue curve boundary, a further approximation is made: k1 ) k2 ) k3 ) k. In this case, J is minimal if the sum of the distillate flow rates is minimal

JD ) D1 + D2 + D3

(8)

The distillate flows can be calculated by the mass balances around the subsystems that are indicated by the dashed lines in Figure 3 for sequence C. The mass balances can be formulated in the following form

[

][ ]

D2 D3 1 D1 xD H -xH -xH D1 D2 3 D2 ) b(xFc, Fc) xL -xL -xD L D3 1 -1 -1

(9)

The vector b depends on the setup and can be related to the external flows via the overall mass balance B3 xFcFc ) xB1B1 + xD2Dex 2 + x B3

Figure 8. Sequence C: mass balances for a II-I-II profile of setups 1, 2, and 3.

for the ∞/∞ analysis. The reboiler duty QRi of a finite column operated with a fixed reflux-to-distillate ratio ri ) Li/Di is proportional to the distillate flow rate Di7,46

QRi ) pi(ri + 1)Di ) kiDi

(6)

where pi is a proportionality factor related to the enthalpy of vaporization of the mixture. pi will be constant if the enthalpy of vaporization does not depend on the composition of the vapor flow. Using eq 6, the cost function (eq 1) can be written as

(10)

(see eq 20 in section 4.1 as an example). The optimization problem can be formulated as follows for the II-I-II profile (xD2 at pure L)

JD min ) min (D1 + D2 + D3) y

subject to

(1)

[

][ ]

1 3 D1 -xD xD H 0 H Fc D1 D3 D 2 ) b(x , Fc) xL -1 -xL D 3 1 -1 -1

(11)

(12)

(2)

D1, D2, and D3 are nonnegative.

(7)

(3)

xD1 lies in the convex set.

The ki values of finite columns depend on the mixture and on the column design. A result of the mass balances is that D1 is a linear function of D2 and D3. Hence, the optimum depends on two relations among the ki’s, for example, k2/k1 and k3/k1. If the ki’s and data on the distillation lines of column 1 are taken from rigorous simulations or experiments, the optimization problem

(4)

xD3 liesDin the nonconvex set. x 1D1 - xD2D2 B2 lies in the nonconvex set, x ) D1 - D2

J ) k1D1 + k2D2 + k3D3

(5)

where the vector of unknowns is D1 D3 D3 1 y ) [D1, D2, D3, xD H , xL , x H , xL ]

(13)


Figure 11. Sequence C, setup 1: II-I-II profile with a fixed R2 and minimal D3. Figure 10. Residue curve map of a 020 mixture with the residue curve boundary as defined by eq 14.

First, the results are illustrated with a residue curve boundary (RCB) that is given by a simple quadratic expression RCB 2 xRCB ) 2(xRCB +1 L H ) - 3xH

(15)

For a II-I-II profile, sequence C, setup 1, has two degrees of freedom, for example, R2 and D3. JD is minimized over these two degrees of freedom. Figure 9 showed a set of solutions for fixed D3. This set reduces to a single point if D3 is minimized for any given R2. At the minimum, xD3 is at the saddle I/H azeotrope (Figure 11). As mentioned in section 3.3, D3 is minimal if xD3 is at the I/H azeotrope. The variation of R2 changes the location of xD1, which is given by eq 5. From the lever rule, it follows that M1 ) R2 + D3 is at a minimum for ex

a given xD2 B3 when xD1 is farthest from xM1. This is the case when xD1 lies on the residue curve boundary. Then, the lever rule gives

R2 + D3 Dex 2

+ B3

)

d2 - d1 d2 ) -1 d1 d1

(16)

As shown in Figure 11, the scalars d1 and d2 represent the distances between the two compositions xD1 and ex

xD2 B3, respectively, and the line R that connects xD2 with xD3. The two-dimensional vector n, defined as

[ ]([ ] [ ]) 0 -1 1 0

2 3 xD xD H H 2 3 xD xD L L

(17)

is perpendicular to the line R. Using this vector, the scalar distances d1 and d2 can be calculated as

(14)

Figure 10 shows the residue curve boundary (RCB) in these coordinates. The location of the I/H azeotrope is az xaz ) [xaz H , xL ] ) [0.5, 0]. To illustrate the curvature of the boundary, a straight line connects L with the I/H azeotrope. The tangent in the azeotrope has the slope 2B3 -1, which gives xD > 0.5 as the condition for feasible L feeds for R2 ) 0 (compare Figure 6d). This specific example does not affect the generality of the results, but it enables a good illustration. 4.1. Sequence C, Setup 1. First, the optimization problem (eq 11) is further simplified. The mass balance around column 1 is D1 ) Fc + R2 + D3 - B1. Together D with D2 ) R2 + Dex 2 , this gives the cost function J (eq 8) as

JD ) 2(R2 + D3) + Fc - B1 + Dex 2

n)

([ ] [ ]) ([ ] [ ])

d1 ) nT

1 3 xD xD H H 1 3 xD xD L L ex

T

d2 ) n

2 B3 xD H ex

2 B3 xD L

-

3 xD H 3 xD L

(18)

(19)

For a given feed Fc, the flow rates of Dex 2 and B3, as ex

well as xD2 B3, are constant. Because xM1 can only move along the line R, d2 is constant. Figure 11 shows that d2 > d1. Hence, eq 16 indicates that R2 + D3 is minimized if d1 is maximized. In other words, the optimal location of xD1 on the boundary is where the distance d1 is maximal. Choosing xD1,opt such that d1 is maximal determines xM1 and the recycle flows R2 and D3. Therefore, the optimal operating point of the sequence can be determined by knowledge of the residue curve map alone. Note that xD1,opt is determined solely by the maximum of d1, which does not depend on d2. d2 only changes as a function of the crude feed composition xFc. Thus, the recycle flow rates depend on xD1 and xFc, but the value of xD1,opt that gives the minimum of R2 + D3 is independent of xFc. This property makes xD1 a good candidate for a self-optimizing controlled variable. This possibility is discussed in section 5. The above optimality condition was also derived for binary I/H feeds.31 Because binary I/H feeds are special cases of ternary feeds, the results for ternary feeds can be easily reduced to the binary case: For binary I/H feeds, no entrainer L leaves the process. Hence, Dex 2 is ex

0, and xD2 B3 is at the origin (I). Nevertheless, d2 is independent of the position of xM1, and R2 + D3 is minimal at the maximum of d1 (similar results are obtained for a nonmonotonic boundary35). Another important property of d1 is that it determines the feasibility of a sequence. Specifically, if d1 > 0, the boundary separation scheme is feasible. If d1 e 0, the boundary separation scheme is infeasible. For d1 ) 0, xD1 lies on line R in Figure 11, and xB2 is at the I/H azeotrope. This would disable any further separation in columns 2 and 3. For d1 < 0, xD1 lies on


xD3. Because this result has implications on the controllability of the process, it is discussed in detail in section 5. (2) For xFc in region 2, sequence C is feasible for xD1,opt (R2 > 0) and for R2 ) 0 (sequence B). (3) For xFc in region 3, R2 has to be greater than 0 for a feasible setup. Sequence C with xD1,opt gives the minimum value of JD. For the methanol/ethanol/water system, two different column configurations were found by Bauer and Stichlmair with MINLP optimization.16 Depending on the feed composition, one optimal setup had one recycle, and the other had two recycles. Using the residue curve map, it can be shown that the feed composition for the first configuration lies in region 1 whereas the feed for the second configuration lies in region 2. Hence, the conclusions of this framework are consistent with MINLP optimization. Reformulation of the Optimization Problem. The above analysis simplifies the original optimization problem (eq 11). The vector of unknowns (eq 13) is reduced from seven elements to five (xD3 is fixed at the azeotropic composition). Constraint 3 is now an equality constraint (xD1 ∈ residue curve boundary), and constraints 4 and 5 are obsolete. For setup 1, vector b in eq 9 depends on B3, which can be expressed in terms the crude feed using eq 10

[ ][

xBH3B3 0 b ) xBL 3B3 ) 0 (1 - xFHc - xFLc)Fc B3 Figure 12. (a) Derivation of the feasible feed region for a given xD1; the dashed line represents eq 5. (b) Three different feed regions.

the right side of line R, so xB2 lies in the convex set. Hence, xB3 is at pure H, and not at pure I as desired. Feasibility of the Optimal Operating Point. For any given residue curve, xD1 can be varied along the residue curve boundary, and the maximum d1 can be identified. A small problem remains. xD1,opt is not feasible for all crude feed compositions xFc. This difficulty is explained in Figure 12, which shows the products of a II-I-II profile with xD3 being at the I/H azeotrope. The mass balance around columns 2 and 3 (eq 5) gives xM1 xD1

ex

D2 B3

xFc.

xM1

as a function of and x , which is given by is obtained by the mixing of R2 and D3 and lies on the line connecting xR2 with xD3. This determines R2 + D3 for xD1,opt (the optimal location in this case). If the L content in Fc increases as indicated by the arrow in ex

Figure 12a, xD2 B3 has to move toward L. This causes xM1 to move toward xD3. At a certain L content in Fc, xM1 coincides with xD3, which implies that R2 is 0. Nevertheless, D3 still has a finite value (Figure 12b). At this point, sequences B and C are identical because the feasible composition xD1 for sequence B is the same as xD1,opt. If the L content in Fc is increased further, xD1 has to move away from the optimal point toward L. Hence, the feeds with higher L contents (region 1 in Figure 12b) are infeasible for xD1,opt. The following points summarize this discussion: (1) For xFc in region 1, the minimum of the recycles is always at R2 ) 0 (which corresponds to sequence B). Nevertheless, there always exists a feasible xD1 for every R2 > 0 that places xM1 on the line connecting xD2 and

]

(20)

For crude feeds in region 1 (Figure 12b), the optimum is at R2 ) 0, and xD1,opt depends on xFc. In this case, xD1,opt can be calculated directly from xFc.35 For crude feeds in regions 2 and 3 in Figure 12b, xD1 and xD3 are constant at the optimal point. From eqs 9 and 20, it follows that all distillate flows are proportional to the quantity (1 xFHc - xFLc) in this case. Hence, the optimal JD is also proportional to (1 - xFHc - xFLc). In particular, for the example residue curve (eq 14), xD1,opt ) [0.25, 0.375], xD3,opt ) [0.5, 0], and Fc Fc JD 1 ) 15(1 - xH - xL )Fc

(21)

Fc Figure 13 shows JD 1 as a function of x , including the optimal cost function for xFc in region 1, as well as for setups 2 and 3, as derived next. 4.2. Sequence C, Setup 2. Following the graphical arguments of setup 1, xD3,opt is at the azeotrope, and xD1,opt is the same as for setup 1. Unlike for setup 1, xD1,opt is feasible for all crude feed compositions for setup 2. The vector b in eq 9 depends on Fc and B3 via

[

][

-xFHcFc xBH3B3 - xFHcFc b ) xBL 3B3 - xFLcFc ) -xFLcFc B3 - Fc (-xFHc - xFLc)Fc

]

(22)

For the example residue curve (eq 14), the optimal operating point is at xD1 ) [0.25, 0.375] and xD3 ) [0.5, 0]. From eqs 9 and 22, it follows that Fc Fc JD 2 ) (17xH + xL )Fc

(23)

4.3. Sequence C, Setup 3. For setup 3, the optimization problem (eq 11) cannot be simplified in the same


D3 1 Figure 14. Sequence C, setup 3: optimal xD L and xL as a function of the crude feed composition.

ex JD 3 ) 2(R2 + B2) + Fc + 2D2 - B3

Figure 13. Sequence C: cost functions for setups 1, 2, and 3 as functions of the crude feed composition.

manner as for setups 1 and 2 because of the mass balance around columns 1 and 2

xD3D3 ) xB2B2 + (xB1B1 + xD2Dex 2 ) ex B1D2

xD3D3 ) xB2B2 + x

(B1 + Dex 2 )

(24) (25)

ex

xD3 has to be collinear with xB2 and xD2 B1. Therefore, setup 3 is infeasible for every feed Fc that contains L when xD3 is at the I/H azeotrope (compare Figure 8). Consequently, xD1 and xD3 are both functions of xFc at the optimal operating point. To simplify the optimization, the cost function (eq 8) is formulated as a function of the two recycles R2 and B2 ex ex JD 3 ) (R2 + B2 + D2 ) + (R2 + D2 ) + (Fc + B2 - B3)

(26)

Hence, the optimal point is where the sum of the two recycles R2 and B2 is minimal (eq 26). Using graphical arguments (Figure 8), the lever rule shows that B2 will always have the smallest possible value (independent of R2) if xB2 lies on the binary I/H edge. To identify the optimal operating point, xD1 is varied along the boundary. For each xD1, xD3 is determined that gives xB2 on the I/H edge using the mass balance for column 2. The flow rates are determined using eq 9 with the vector b that depends on B1 and Dex 2

[

][

ex 2 -xBH1B1 - xD -xFHcFc H D2 B1 D2 ex b ) xL B1 - xL D2 ) -xFLcFc -B1 - Dex (-xFHc - xFLc)Fc 2

]

(27)

Figure 13 shows JD 3 as a function of the feed composition at the optimal point. JD 3 is not a linear function of xFc. Optimal operation of this setup will be far more difficult than that of the other two setups because the optimal compositions xD1 and xD3 depend on the feed composition xFc (Figure 14). 4.4. Comparison. Figure 15 shows the three cost functions, JD i , at the optimal operating points for each crude feed composition as a function of the crude feed composition. Setup 2 has the lowest energy consumption


5. Control of the ∞/∞ Sequence

Figure 15. Comparison of all three setups of sequence C as shown in Figure 13.

for high I contents and the best operability, because xD1,opt is feasible for all compositions and xD1 lies such that d1 > 0 by definition. Figure 8 shows that, for setup 2, xD1 will lie to the left of the line connecting xD2 and xD3 if xB1 is maintained at pure H because, by definition, xF1 lies on the line connecting xD2 and xD3. In contrast, Figure 8 also shows that, for setup 1, xF1 will lie on the right side of the line connecting xD2 and xD3 if xFc lies on the right side of that line. Hence, it is possible that xD1 lies such that d1 < 0 is also possible, which makes setup 1 a bit more difficult to operate. In addition, xD1,opt is not feasible for setup 1 for low I contents. However setup 1 has the best performance for low to medium I contents. The optimal cost function of setup 3 is always greater than that of setup 2, except for binary I/H feeds, in which case they are equal. For binary I/H feeds, Dex 2 is 0, and xD3, being at the I/H azeotrope, gives a feasible setup according to the mass balance around columns 1 and 2 (eq 24). In this case, the column products and distillate flow rates of setup 3 are equal to those of setup 2. For finite columns, the sum of the reboiler duties will be lower when a binary I/H feed is introduced into column 3 instead of column 2 because the I content of the crude feed does not pass through the reboiler of column 2 if Fc goes directly into column 3. However, for ternary feeds, a finite sequence with setup 3 will have the highest operating costs and the most difficult operating behavior, as the optimal products xD1 and xD3 both depend on xFc. Therefore, setup 3 will not be considered further. These results are independent of the shape of the residue curve boundary as long as the boundary lies to the left of the line connecting L and the I/H azeotrope. The region in which setup 1 is better than setup 2 can be well predicted using just the residue curve map information with ki values obtained by rigorous simulations and eq 7, rather than eq 8, as the cost function for the optimization problem (eq 11). With some simplifying assumptions, this method can even be extended to heterogeneous sequences.35

5.1. Selection of Controller Pairings. The ∞/∞ sequence has four degrees of freedom and four controlled variables are required for setups 1 and 2. Possible candidate controlled variables are only the distillate and bottom flow rates of the columns and the distillate and bottom compositions. Distillate and bottom flow rates should not be controlled variables because they lock the material balance. Hence, only the distillate and bottom compositions remain as candidate controlled variables. For an ∞/∞ column, not all values of the top and bottom compositions give a unique D/F ratio. For example, specifying the bottom purity xBH1 ) 1 will give a type II profile, and the ratio D/F will not be uniquely defined. In contrast, specifying xBH1 ) 1 - , with arbitrarily small but greater than 0, will give a type III profile with a unique D/F ratio. As a result, xD1 lies on the boundary. This gives the following controller pairings: (1) Column 1. D1 is manipulated to control xBH1 ) 1 - . The column profile is of type III, and xD1 lies on the residue curve boundary. 2 (2) Column 2. D2 is manipulated to control xD L ) 1 - . The column profile is of type III, and xB2 lies on the binary I/H edge. (3) Column 3. D3 is manipulated to control xBI 3 ) 1 - . The column profile is of type III, and xD3 is at the binary I/H azeotrope. This fixes three degrees of freedom. The fourth degree 1 of freedom is fixed by controlling xD L . This gives the fourth controller pairing, which is the central controller pairing of the sequence: D1,opt 1 (4) Sequence. R2 ensures that xD (because L ) xL of the first controller pairing, xD1 lies on the residue curve boundary). The key point of this controller pairing is that R2 changes the overall feed composition to 1 column 1 such that xD L is at the optimal position. A similar impact of the entrainer recycle was also observed for a heterogeneous sequence by Rovaglio et al.47 These results are also consistent with the operation concept for a heterogeneous column analyzed by Ulrich and Morari.34 The self-optimizing properties of the controlled 1 variable xD L are evaluated next (1) for different crude feed compositions xFc and (2) for xD3 also being in the 3 nonconvex set (in this case, xD H is the controlled variB3 able for column 3 instead of xI ). In this evaluation, perfect control of the other three control loops is assumed. Hence, xD1 always lies on the residue curve boundary, and the L content of xD1 uniquely identifies the location of xD1 for a residue curve boundary monotonic in L. Manipulation of R2 can adjust xD1 but not always reach xD1,opt. Therefore, the influence of R2 on the sum of the distillate flows JD is discussed next for fixed but different crude feeds to analyze the robustness toward disturbance 1. Concerning disturbance 2, the analysis is first done for xD3 at the I/H azeotrope and then repeated if column 3 does not reach the I/H azeotrope in the distillate. A fold bifurcation will occur for the fourth controller pairing if xD3 is in the nonconvex set. For finite columns, xD3 will always be in the nonconvex set. Without loss of generality, the calculations are illustrated with the example residue curve boundary (eq 14). 5.2. Controlled Variable xD1 for xD3 at the I/H Azeotrope. Figure 16 shows the pairwise dependence of different operating parameters for xD3 at the I/H


Figure 16. Sequence C, setup 1: pairwise dependence of different operating parameters for xD3 at the I/H azeotrope for five different feed compositions.

azeotrope for five different crude feed compositions. If JD is plotted as a function of R2, it can be seen that feed i lies in the infeasible region for xD1,opt. As stated in section 4.1, the minimum of JD occurs at R2 ) 0. The slope of the curve is always greater than 0. For feed ii, xD1,opt is just feasible, and the minimum is also at R2 ) 0, with a slope of 0 in this point. Feed iii lies in the region where xD1,opt is feasible for sequence C and the minimum of JD is for R2 > 0, but R2 ) 0 is also feasible.

1,opt is at R2 ) 0. For feeds ii-v, xD can be reached. The L control law that always reaches the optimal operating point is:

D1,opt 1 1,opt If xD , decrease R2 until xD is reached L > xL L or R2 is 0. D1,opt 1 , increase R2. If xD L < xL

ex

For feed iv, xD2 B3 lies on the tangent of the residue curve in the I/H azeotrope. For R2 approaching 0, the sum of the distillates goes to infinity; for R2 ) 0, the sequence is infeasible. This applies also for feed v: sequence C is infeasible without any recycle R2, and JD goes to infinity as R2 approaches 0. In section 4.1, it was derived that xD1 opt is independent of xFc and that minimizing JD (eq 8) and maximizing d1 (eq 18) are equivalent if all ki are equal. The plot 1 with JD as a function of the controlled variable xD L (Figure 16) shows a constraint minimum for feed i. For all other feeds, the minimum of JD is at the same 1 D composition xD L , but the values of J depend on the feed composition xFc, as expected. In contrast, the optimal value of d1 is independent of the crude feed 1 composition, but the maximum is at the same xD L D1 (Figure 16). This shows that xL is a good controlled variable (optimal value ) 0.375 for the example residue curve). For the implementation of the controller, the connection between the manipulated and controlled variables 1 has to be analyzed. Therefore, Figure 16 also shows xD L 1 as a function of R2. xD is a monotonic function of R 2. L 1,opt D For feed i, xD is not feasible, but the minimum of J L

d1 also combines the information of xD2 and xD3 with xD1 and is an alternative controlled variable. The dependence of d1 on R2 is also shown in Figure 16. Here, the controller would have to maximize d1 by manipulating R2. Therefore, d1 should not be used as a controlled variable, but it helps to identify the optimal value of 1 xD L : d1 is a measure for the performance of the sequence that is independent of the crude feed composition xFc. Qualitatively, all of the statements made in this section hold when xD3 is not at the I/H azeotrope but on the residue curve boundary. The same also holds when xD2 is not at pure L but on the residue curve boundary. In contrast, there is a qualitative change for xD2 or xD3 in the nonconvex set. If xD2 lies in the nonconvex set, the profile of column 2 has to be of type II. For the ∞/∞ sequence, this is not possible, as discussed in section 3.1. It is possible, however, for finite columns. If xD3 lies in the nonconvex set, the profile of column 3 will still be a type II profile, which is a feasible ∞/∞ profile. Therefore, the influence of xD3 lying in the nonconvex set on xD1 is discussed next. 5.3. Controlled Variable xD1 for xD3 in the Nonconvex Set. Figure 17 shows the same as Figure 16


Figure 17. Sequence C, setup 1: pairwise dependence of different operating parameters for xD3 ) [0.48, 0] for six different feed compositions.

but with xD3 ) [0.48, 0], i.e., xD3 lies in the nonconvex set. For feeds I-III, the same statements hold as for the case in which xD3 is at the I/H azeotrope. For feed IV, the tangent condition for identifying the feasible region changes. The mass balance line of columns 2 and 3 is tangent to the residue curve boundary, but it touches the boundary not at the I/H azeotrope but somewhere else. Therefore, JD is finite as R2 goes to 0. For feeds V and VI, a minimum amount of R2 is required for a feasible operating point of the sequence. Figure 17 also shows JD as a function of the controlled variable 1 D3 in the nonconvex set. The self-optimizing xD L for x 1 properties of xD L are maintained, but the location of the optimum is shifted, which is also reflected in d1. For all feeds, two solutions exist for a given R2 and xD3 in the nonconvex set. For the implementation of the controller, the change between xD3 lying at the I/H azeotrope and in the nonconvex set is drastic and is caused by the occurrence of the second solution: a fold 1 bifurcation appears. Figure 17 also shows xD L as a D1 D function of R2. For x 3 in the nonconvex set, xL is a monotonic function of R2 over an extensive range, which can be used for the control algorithm. At some point, 1 however, the gain changes sign for R2 as input and xD L as output. This is a fold bifurcation where the stability of the system changes because a pole moves from the left half-plane to the right half-plane. Because the individual columns do not exhibit multiple steady states and are all stable, this instability is introduced by the recycle. The effect is analogous to a multivariable system with partial feedback control.48,49 For the cases indicated with the dashed line, the reason for the second solution is that xM1 can lie in the nonconvex set, so that the mass balance line around

columns 2 and 3 crosses the boundary twice (compare Figure 6a for R2 ) 0). xM1 is given by

xM1(R2 + D3) ) xD2R2 + xD3D3

(28)

For R2 > 0 and xM1 in the nonconvex set, R2 + D3 has to go to infinity if xM1 approaches the residue curve boundary because xD1 approaches xM1. For this case, xM1 is given by the overall feed and is constant. Equation 28 indicates that D3/R2 is constant for a given xM1. Hence, for R2 going to infinity, R2 + D3 also must go to infinity with the slope (1 + D3/R2). If R2 is reduced on the lower branch, xM1 will move toward xD3. As a result, R2 + D3 will go through a minimum. If xM1 goes further toward the boundary, R2 must increase again, together with R2 + D3, because xD1 and xM1 move closer to each other. This does not depend on the fact that the mass balance line of columns 2 and 3 might cross the boundary twice at some point, as it does for case V. For case VI, the L content in Fc is too low, so the mass balance of columns 2 and 3 does not cross the residue curve boundary twice. The properties of the controlled and manipulated variables change slightly for setup 2. In contrast to setup 1, the optimal composition of xD1 is feasible for 1 all feeds. As for setup 1, xD L is a monotonic function of R2 for xD3 at the I/H azeotrope, whereas a second solution appears for the same reason as for setup 1 for xD3 in the nonconvex set (see Ulrich35 for details). 6. Self-Optimizing Control of the Finite Sequence 6.1. Optimization. The boundary separation scheme shown in Figure 1 has seven degrees of freedom for fixed

4524 Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 Table 2. Optimal Parameters of Sequence C, Setup 1 (NRTL-IG Parameter Set and Feed 3 of Table 3) specified variable

value

manipulated variable

B1 xwater L1 R2 D2 xmethanol B2 xmethanol B3 x2-propanol D3 xwater

0.9998 kg/kg 2.7 kg/h 1.004 kg/h 0.9998 kg/kg 0.0005 kg/kg 0.9998 kg/kg 0.125 kg/kg

D1 L1 R2 D2

value 2.167 kg/h 2.7 kg/h 1.004 kg/h 1.337 kg/h 2.053 kW 0.333 kg/h 0.720 kW

QR2 B3 QR3

Table 3. Three Different Crude Feeds (p ) 1.1 bar, T ) 20 °C) feed

flow rate (kg/h)

xmethanol (kg/kg)

x2-propanol (kg/kg)

xwater (kg/kg)

1 2 3

1 1 1

0.6 0.5 0.333

0.066 0.166 0.333

0.334 0.334 0.334

column top pressures (in this example, p1 ) 1.1 bar with a pressure drop/column of ∆pcol 1 ) 0.015 bar, p2 ) 1 bar col with ∆pcol 2 ) 0.025 bar, and p3 ) 1 bar with ∆p3 ) 0.015 bar for columns 1-3). The constraints for the 1 product purities fix three degrees of freedom: (1) xBwater D2 ) 0.9998 kg/kg, (2) xmethanol ) 0.9998 kg/kg, and (3) B3 ) 0.9998 kg/kg. x2-propanol Four degrees of freedom remain to minimize the sum of the reboiler duties (eq 1) to determine the optimal operating point. The optimization of the ∞/∞ sequence (section 4.1) gives four additional conditions: (4) xD3 at the binary 2-propanol/water azeotrope, (5) xB2 on binary 2-propanol/ water edge, (6) xD1 on the residue curve boundary, and (7) xD1 such that the sum of the distillate flows is minimal. Conditions 4 and 5 can be used to initialize the 3 optimizer close to the optimal point: xD water close to the B2 azeotrope and xmethanol very small. For condition 6, the internal flows of column 1 (L1) are varied such that xD1 lies close to the residue curve boundary or even beyond it, as tray columns can cross residue curve boundaries.4 The location of xD1 on the residue curve defines the sum of the distillate flows (condition 7). By varying R2, the feed composition of column 1 is changed and because of condition 6, xD1 moves along the boundary for a fixed xB1 (condition 1). To minimize the sum of the reboiler B2 3 duties, all four variables (xD water, xmethanol, L1, and R2) were varied. Table 2 lists the optimal parameters for a base feed (feed 3 in Table 3); the specified variables are the specifications in AspenPlus (four of them were varied to minimize the sum of the reboiler duties), and the manipulated variables are the variables that the steady-state solver of AspenPlus varies to find the solution for the given specifications. 6.2. Selection of Controller Pairings. The results of the ∞/∞ analysis are used to derive the controller pairings for the finite sequence. An ∞/∞ column has one degree of freedom, and specifying, for example, the bottom composition at 1 - gives one particular distillate composition. A distillation column of finite length and finite reflux has two degrees of freedom for a fixed length. This adds one degree of freedom per column to the sequence. Hence, seven controlled variables are needed for the finite sequence. The first five of the seven requirements of the ∞/∞ sequence (section

Figure 18. Sum of reboiler duties for three different feeds (methanol, 2-propanol, water) as a function of reflux L1 and reflux ratio L1/D1.

6.1) can be directly transferred to a set of controlled D2 B3 B2 1 3 , xmethanol , x2-propanol , xmethanol , and xD variables: xBwater water. D 1 Only the conditions for x are a bit more difficult to transfer. If xD1 lies on the residue curve boundary, the position of xD1 is uniquely identified by the methanol content of xD1, giving the sixth controlled variable, D1 xmethanol . To fulfill the seventh condition, the reflux of the column must be high enough to keep the column profile close to the boundary. For these seven controlled variables, seven manipulated variables are needed. For a steady-state analysis, there is no difference between specifying the distillate flow rate and reboiler duty, specifying the reflux and bottom flow rates, or specifying any other pairing of the four variables D, B, L, and QR as manipulated variables. All of these alternatives have the same steady-state properties when special cases such as multiplicities caused by the nonlinear mass-to-molar transformation50-52 are excluded. For the following steady-state analysis, the seven manipulated variables of Table 2 were chosen for convenience. For a distillation column that separates a binary feed, a constant reflux-to-feed ratio is slightly superior to a constant reflux-to-distillate ratio.38 However, a result of the recycles in the system is that D1 and F1 ) Fc + R2 + D3 are closely related. Hence, a constant refluxto-distillate ratio is a good choice in this case for a selfoptimizing controlled variable, as is actually confirmed by the following case study. The reflux L1 is varied while the six controlled variables are kept constant (five at D1 ) 0.617 kg/kg). the values given in Table 2 and xmethanol Figure 18 shows the sum of reboiler duties as a function

Ind. Eng. Chem. Res., Vol. 42, No. 20, 2003 4525 Table 4. Sets of Controlled Variables for Sequence C, Setup 1 (NRTL-IG Parameter Set) set 1

set 2 (and 2a)

set 3 (and 3a)

variable

value

variable

value

variable

value

B1 xwater D1 xmethanol L1/D1 D2 xmethanol

0.9998 kg/kg 0.617 kg/kg 1.3 0.9998 kg/kg


0.9998 kg/kg 0.617 kg/kg 1.3 (1.56) 0.9998 kg/kg


0.9998 kg/kg 0.617 kg/kg 2.8 (3.36) kg/h 0.9998 kg/kg

B2 xmethanol B3 x2-propanol D3 xwater

0.0005 kg/kg 0.9998 kg/kg 0.125 kg/kg

L2/D2

4.05 (4.86) 0.9998 kg/kg 5.10 (6.12)

L2

5.4 (6.48) kg/h 0.9998 kg/kg 2.5 (3) kg/h

B3 x2-propanol L3/D3

B3 x2-propanol L3

Table 5. Determination of the ki of Eq 6 for the NRTL-IG Parameter Set and Feed 3 (Table 3) column

QRi (kW)

Di (kg/h)

ki [kW/(kg/h)]

1 2 3

1.403 2.047 0.719

2.160 1.333 0.494

0.650 1.536 1.455

of the reflux L1 for different feed compositions. The optimal reflux L1 is a function of the water content in the crude feed, but, as expected, the optimal reflux ratio L1/D1 does not depend on the crude feed composition. The optimal reflux-to-distillate ratio is 1.25 for an NRTL-IG parameter set. For more robust convergence behavior of the sequence, 1.3 was chosen in the case studies. Table 4 shows the resulting set of controlled variables (set 1). Table 4 also lists two more sets of controlled variables. The idea behind set 2 is that the composition of the feed to column 2 (xD1) is constant at the optimal point and only the flow rate varies. For specified purities of the products of column 2, the split is constant, and the internal flows are proportional to the feed flow rate for a fixed feed composition. If xD1 is constant, the refluxto-distillate ratios of columns 2 and 3 will not depend on the crude feed composition. Another option is to keep the reflux at a high but constant value to allow simple one-point control. Therefore, set 3 is also discussed. In practice, distillation columns are operated with a higher reflux to increase robustness. Therefore, sets 2 and 3 are also analyzed for a reflux increased by 20% (sets 2a and 3a, respectively). 6.3. Evaluation of Set 1 of Controlled Variables. As for the ∞/∞ sequence, set 1 of the controlled variables is evaluated with respect to the key controller pairing, D1 R2/xmethanol . For three different feeds (Table 3), each of which lies in a region that give a qualitatively different behavior (this was derived in section 4.1), rigorous simulations are compared to the predictions obtained from the ∞/∞ analysis. Figure 19 shows J and d1 as D1 D1 functions of xmethanol and xmethanol as a function of R2 (the ∞/∞ predictions were obtained with ki values determined from the optimal solution of feed 3, Table 5). The reason for the large quantitative deviation is that the residue curve boundary and the distillation line boundary differ. Nevertheless, the ∞/∞ predictions compare well with the qualitative picture of the operating parameters of the rigorous simulations, confirming D1 that xmethanol is a good controlled variable. Figure 19 further confirms that the optimal value of J depends on the crude feed composition whereas d1 is independent of the feed composition if it is in region 2 or region 3. The symbol indicates the location of the minimum of J. The maximum of d1 is close to that point. This confirms that d1 is a good and independent measure of the performance of the process.

Figure 19. Sequence C, setup 1 set 1: J and d1 as functions of D1 D1 the controlled variable xmethanol , and xmethanol as a function of R2 (solid lines, rigorous simulation; dashed lines, ∞/∞ predictions). D1 and R2 is further The relation between xmethanol D investigated here. x 3 lies in the nonconvex set, and a second solution is expected and shown in Figure 19.


Figure 20. Sequence C, setup 1: comparison of the performance of five different sets of controlled variables for different crude feed compositions xFc (disturbance).

Whereas R2 ) 0 was reached for the rigorous simulation for feeds 1 and 2, it was not possible to converge the D1 < 0.38 kg/kg by varying R2 (see sequence for xmethanol Ulrich35 for a detailed discussion). 6.4. Comparison of the Sets of Controlled Variables. Figure 20 shows the sum of the reboiler duties for the three sets of controlled variables as a function c of the crude feed composition: xFwater is constant at Fc 0.334 kg/kg while xmethanol is varied between 0.003 and 0.653 kg/kg. Set 1 is the best set. The dependence of J Fc is roughly piecewise linear for set 1. At the on xmethanol Fc where the optimum of J is not at R2 > value of xmethanol 0 but at R2 ) 0, the slope changes. As long as xD1,opt is feasible, set 2 is identical to set 1. Set 3b is for L1 ) 5.6 kg/h, L2 ) 10.8 kg/h, and L3 ) 5 kg/h, which is twice the nominal values given in Table 4. It has the highest energy consumption of set 3 but covers the same feed compositions as sets 1 and 2a. To summarize, set 1 is the best option, but it requires seven composition measurements. Set 2 has the same performance as set 1 but is not robust against changes in the crude feed composition: the reflux might go below the minimum reflux for columns 2 and 3 if xD1 changes with the crude feed composition. Set 2a is the alternative for this sequence because it requires fewer composition measurements, it has an acceptable loss for different crude feed compositions, and it has the necessary robustness toward minimum reflux problems. Sets 3, 3a, and 3b are unacceptable alternatives. Therefore, set 2a will be implemented. 6.5. Temperature Controllers for Implementation of Set 2a. All columns are operated with a constant reflux-to-distillate ratio. Three of the four composition controllers are realized as cascaded controllers, as is often done in practice; a stage temperature is controlled by manipulating, for example, the reboiler duty. The setpoint of the temperature controller is adjusted such that the product purity is as desired. Before this concept is implemented, the temperatures are selected and evaluated. Figure 21 shows the temperature profiles of the three columns for set 2a of the controlled variables for feed 3 (Table 3). As the controlled temperature, the temperature of the stage where the front is steepest is chosen. For column 1, this is the temperature on stage 28

Figure 21. Temperature profiles for columns 1, 2, and 3 operated in setup 1 with set 2a of controlled variables. 1 (Tcolumn ) 91.78 °C); for column 2, it is also stage 28 28 column 2 (T28 ) 73.28 °C); and for column 3, it is stage 16 column 3 ) 81.49 °C). The resulting set of controlled (T16 variables is called set 4a. Figure 22a shows the controlled temperatures for the Fc c three columns as a function of xmethanol for xFwater ) 0.334 kg/kg for the specifications of set 2a (Table 4) as illustrated in Figure 20. The temperature front in column 1 is very sensitive to the changes in the crude feed composition and hence varies with xFc. For the 1 , will be higher chosen setpoint, the bottom purity, xBwater Fc than 0.9998 kg/kg for xmethanol < 0.333 kg/kg and lower Fc > 0.333 kg/kg. For than 0.9998 kg/kg for xmethanol Fc xmethanol < 0.528 kg/kg, the controlled temperatures of columns 2 and 3 are constant. The reason is that xD1 is constant, giving a constant T28 for column 2 and a constant xB2. This gives a constant T16 for column 3. For Fc >0.528 kg/kg, the optimal operating point is xmethanol where R2 ) 0 and xD1 changes. This also changes the temperature fronts in columns 2 and 3, as reflected in the change of the two temperatures in this steady-state analysis. In a dynamic simulation, these temperatures might also change because of the pressure change caused by the different loads of the columns for varying crude feed compositions.


(

K ) kc 1 +

τds 1 + τis 0.1τds +1

)

(29)

The sensor dynamics are modeled with a first-order lag

f)

Figure 22. Sequence C, setup 1: (a) three controlled temperatures for set 2a (over-refluxed columns, fixed product compositions) Fc Fc for different xmethanol at xwater ) 0.334, (b) three product purities for set 4a (over-refluxed columns, fixed temperatures).

Figure 22b shows the product purities for set 4a. As Fc expected, xB1 > 0.9998 kg/kg for xmethanol < 0.333 kg/kg and lower for higher methanol contents. In region 1 (R2 ) 0 is optimal solution), the purity specifications for the bottoms’ purities of columns 1 and 3 cannot be met. If the sequence is operated with a crude feed close to region 1, the temperature set points will have to be adjusted. 6.6. Control Scheme. Figure 23 shows the control scheme of the process shown in Figure 1 for set 4a of controller pairings. The level controllers and the refluxto-distillate ratio controllers are implemented with high gain, simulating perfect control. The three controllers TC1, TC2, and TC3 control the selected temperatures by manipulating the reboiler duties. The fourth controller R2C is a composition controller. An inspection of Figure 23 suggests that D1, B2, and D3 are uncontrolled flows. This can cause so-called snowball effects.29 If all flows in a recycle loop are uncontrolled, an infinite number of values are able to fulfill the mass balances. Hence, a disturbance can lead to a large increase of the recycle streams. To prevent this, one stream in the loop has to be flow-controlled.29 For the boundary separation scheme, all distillate flows are indirectly controlled via the reboiler duties as a result of the constant reflux-to-distillate ratios. Hence, snowball effects cannot take place. All four controllers are linear PID controllers (proportional amplification of the controller error with integral and derivative action) of the following form

1 τfs + 1

(30)

The time constant τf of the temperature sensor dynamics is assumed to be 1 min; that of the composition measurement dynamics is 5 min. All columns are modeled as simple packings with an HETP of 0.05 m and an initial liquid volume fraction of 0.05; the column diameter is 0.1 m (the equipment is not optimized to a specific packing, nor is the diameter of the column optimized for the expected throughput). The pressure drop is 15 mbar/column for the 30-stage column (including top and bottom) and 25 mbar/column for the 50stage column (including top and bottom). The hold-up of the reflux drum and bottom is 3.56 × 10-3 m3, and the hold-up per stage is 0.02 × 10-3 m3. The controllers are tuned for each column separately using the Ziegler-Nichols method for a constant column feed: the gain of a proportional controller is increased until a sustained oscillation is reached, giving the ultimate gain (ku) and the ultimate period (Pu) for the selection of the controller parameters. Controller tuning with a step answer is difficult because of the strong nonlinearities of the columns: gain and time constants depend strongly on the chosen step size. Therefore, the oscillatory experiment was chosen. The key aspect of the following simulations is to show that the scheme works in general with simple controllers. Any particular tuning for a real plant setup is beyond the scope of this work. Table 6 lists the parameter values obtained. TC1 is much faster than R2C, and detuning is not necessary. For column 2, the tuning was easy, but for column 3, it was not possible to increase the gain such that a stable oscillation was reached. The parameters of column 1 also work for column 3 and are therefore taken to illustrate the dynamics of the scheme. Figures 24 and 25 show the reaction of the full threecolumn sequence to changes in the crude feed composition xFc. The controllers can adjust the process well. All reboiler duties scale linearly with the 2-propanol content in the crude feed, as expected. The only drawback is that the purity of xB1 cannot be kept within specifications. As discussed above, the setpoint of TC1 has to be adjusted. 7. Further Analysis 7.1. Influence of Model Uncertainties. The main D1 problem of the controlled variable xmethanol is that the optimal location depends on the curvature of the boundary, which is unknown in reality. The uncertainty is illustrated in Figure 26, which shows the residue curve boundary calculated by four different activity coefficient models: NRTL, Wilson, UNIQUAC, and UNIFAC. (For each model, there are two different data sets in the AspenPlus database: VLE-IG and VLE-Lit. To differentiate these data sets, the extension IG or Lit is added to the abbreviation of the activity coefficient model. For example, NRTL-IG corresponds to the activity coefficient model NRTL with the VLE-IG parameter set; NRTL-Lit corresponds to the VLE-Lit parameter


Figure 23. Control scheme for the boundary separation scheme. Table 6. Parameters of the PID Controllers (Eq 29) controller

ku

Pu (min)

TC1 TC2 TC3 R2C

0.278 kW/K 2.222 kW/K 8.3 kg/h/(kg/kg)

2 1.5 32

τi (min)

τd (min)

0.167 kW/K 1 1.333 kW/K 0.75 0.167 kW/K 1 5 kg/h/(kg/kg) 16

0.25 0.1875 0.25 4

kc

set.) The performance of the boundary separation scheme depends on the approximation of the residue curve boundary by the activity coefficient model. Using the optimization of the ∞/∞ sequence, this can be illustrated just with calculations of the residue curve boundary. Choosing k1 ) k2 ) k3 ) 1 kW/(kg/h) for simplicity (clearly, ki depends on the activity coefficient model) gives

kW ) kg/h min Fc c - xmethanol )Fc (31) Jp (1 - xFwater

Jmin ) (D1 + D2 + D3)min

Jmin is a proportionality factor and depends on xD1,opt, p which is also a function of the ki, the weighting factors. Table 7 lists the different optimal locations of xD1, Jmin p , as functions of the activity coefficient models and dmax 1 and the different parameter sets. These results reflect an uncertainty concerning the economic performance of the scheme. Just recently, experimental data were published for the methanol/2-propanol/water mixture53-55 that confirmed the curvature of the residue curve boundary (see Ulrich35 for a detailed discussion). Calculations with Wilson-IG, NRTL-IG, UNIQUACIG, and UNIQUAC-Lit give essentially the same performance, as reflected in Jp. Calculations with WilsonLit and NRTL-Lit predict extremely high values of Jp, and the performance of the scheme is best if UNIFAC is used as the activity coefficient model. Figure 27 shows the comparison of different activity coefficient models for feed 3 (Table 3) using the controlled variables of set 2a (Table 4) that were obtained

Figure 24. Dynamic simulation of the three-column sequence for changes in the feed composition xFc (L ) methanol, I ) 2-propanol, and H ) water). Series 1: from region 3 to region 1 and back.

for NRTL-IG. As predicted by the ∞/∞ system (Table 7), the boundary separation scheme will have the best performance if UNIFAC is used as the activity coef-


Figure 25. Dynamic simulation of the three-column sequence for changes in the feed composition xFc (L ) methanol, I ) 2-propanol, and H ) water). Series 2: from the convex set to the nonconvex set and back. Table 7. Optimal Location of xD1 for Different Activity Coefficient Models [ki ) 1 kW/(kg/h), p ) 1 bar] model

azeotrope xwater

D1,opt xwater

D1,opt xmethanol

Jmin p [kW/(kg/h)]

NRTL-IG Wilson-IG UNIQUAC-IG UNIFAC UNIQUAC-Lit Wilson-Lit NRTL-Lit NRTL-Krishna55

0.1304 0.1187 0.1270 0.1201 0.1265 0.1188 0.1187 0.1304

0.04032 0.03528 0.03690 0.03831 0.05985 0.00294 0.01768 0.06387

0.5526 0.5824 0.5916 0.3944 0.4051 0.9618 0.8325 0.4656

13.5 15.6 16.0 5.98 15.4 148 107 43.7

dmax 1 0.0180 0.0143 0.0150 0.0344 0.0154 0.0016 0.0022 0.0058

ficient model in the rigorous calculations. The rigorous simulations with NRTL-IG, Wilson-IG, UNIQUAC-IG, and UNIQUAC-Lit are very similar to each other. The D1 optimal value for xmethanol is around 0.6 kg/kg, and it depends only weakly on the chosen parameter set. D1 Hence, xmethanol is a good self-optimizing controlled variable that is also robust against these model uncertainties. 7.2. Operation of Column 1. The composition xD1 is a critical operating parameter. If the specified value of xD1 is not reached correctly, the sequence might become infeasible. The condition for feasibility is that d1 (eq 18) is greater than 0. The sensitivity of this condition is illustrated with a case study using WilsonIG as the parameter set (R2 and D3 enter column 1 above stage 15 and Fc above stage 20, as indicated by

Figure 26. Residue curve boundary and the resulting xD1,opt values for VLE-IG and VLE-Lit data sets.

the notation F-Stages ) 15/15/20). Figure 28a shows 1 d1 and xBwater as functions of the distillate flow rate D1 (ranging from 0 to F1) for a feed of setup 1 (F1 ) 4.696 F1 1 ) 0.4737 kg/kg, xFwater ) 0.1435 kg/kg). kg/h, xmethanol The critical point here is illustrated with Figure 28b, which shows an enlargement of Figure 28a. For a feasible sequence (d1 > 0), D1 has to be smaller than 4.26 kg/h (upper bound). For a purity larger than 0.9998 kg/kg, D1 has to be larger than 4.188 kg/h. In this region, the pinches of the column change. The pinches can be identified if the total column hold-up (TCH) composition is calculated (eq 32), as introduced by Dorn and Morari56 n

∑ xk Mk

xTCH )

k)1

n

(32)

∑ Mk

k)1

xk is the composition vector at stage k, and Mk is the mass hold-up of stage k. For the calculations, the holdup of the reflux drum and the bottoms is 1 kg, and the hold-up per stage is 0.285 kg. Figure 29 shows the total column hold-up composition (eq 32) as a function of D1. Three different pinches can be identified: (1) pinch in column top (pure methanol) for 0 < D1 < 2.24 kg/h, (2)


Figure 28. Sequence C, setup 1, column 1 (Wilson-IG): (a) d1 B1 B1 and bottom composition xwater and (b) d1 and 1 - xwater as functions of D1. Figure 27. Sequence C, setup 1, set 2a: J as a function of D1 D1 and xmethanol as a function of R2 using different activity xmethanol coefficient models.

pinch in the column middle (2-propanol/water azeotrope) for 2.24 kg/h < D1 < 4.188 kg/h, and (3) pinch in the column bottom (pure water) for 4.188 < D1 < 4.696 kg/ h. Figure 29 confirms that the desired operating range (4.188 kg/h < D1 < 4.26 kg/h) lies exactly at the border between two pinches. If the column is operated such that the pinch is in the bottom, the bottom purity will be very robust toward small fluctuations in D1 and B1 because a large amount of water would have to be removed from the column. This gives a large transition time that makes the process robust toward small fluctuations (see Ulrich35 for a detailed discussion of the transient behavior of the column). This transition time also has to be considered in the startup of the column. In contrast, xD1 is very sensitive to increasing distillate flow rates. For setup 1 of sequence C, this can lead to the possibility of d1 being smaller than 0 (compare Figure 28). This condition must be avoided, and column 1 should be operated with the pinch close to the 2-propanol/water azeotrope in the middle of the column because, then, xD1 will always lie on the boundary for a sufficiently high reflux. Note that, for the constraint 1 xBwater ) 0.9998 kg/kg that was used in all previous studies, the pinch is always in the middle of column 1. For tray columns, xD1 can actually cross the residue curve boundary for lower refluxes, which improves the

Figure 29. Total column hold-up composition of column 1 (setup 1) as a function of D1.

performance of the scheme. (This was one of the reasons for the difference between the ∞/∞ column predictions and the rigorous simulations shown in Figure 19.) As postulated in section 4.4, setup 2 has the best operability because the feasibility of the sequence is given by definition if column 1 is operated such that xB1 is pure water and D1 is smaller than F1. The reason is that d1 > 0 in this case. This can be illustrated with a parameter study, as shown in Figure 30 for a feed to F1 column 1 for setup 2 (F1 ) 4.696 kg/h, xmethanol ) F1 0.5217 kg/kg, xwater ) 0.0567 kg/kg). The optimal operating point is the same as for the feed of setup 1.


Figure 30. Sequence C, setup 2, column 1 (Wilson-IG): (a) d1 B1 B1 and bottom composition xwater and (b) d1 and 1 - xwater as functions of D1.

7.3. Crossing of the Residue Curve Boundaries. A disturbance in column 1 might result in xD1 lying such that d1 < 0. This gives a different steady-state profile. Figure 31 shows two steady-state profiles. The profile of steady state a, which is for d1 > 0, lies completely in the nonconvex set, whereas the profile of steady state b, which is for d1 < 0, lies mainly in the convex set but crosses the residue curve boundary. Figure 32 shows the dynamics of column 2 with the (L/D)QR scheme for a change in xD1 from steady state a to steady state b and back to steady state a. These results confirm that a column profile can cross a distillation boundary if the mass balance forces the column profile to do so. During these transitions, the simple controller can keep the desired purity for xD2 while xB2 crosses the azeotrope. Figure 33 shows the dynamics of column 3 with the (L/D)QR scheme for a step change in the feed composition xB2, which can be caused by the above-described transient of column 2 that results in xB2 lying on the other side of the azeotrope. As a result, the mass balance forces the column profile to cross the azeotrope. After the step change at 20 h, the bottom flow goes to 0. The reboiler duty increases to keep T16 at the set point. Hence, the bottom level decreases. In this transition, the top composition stays at the azeotrope, and the bottom composition crosses the azeotrope. In that process, there is a point at which the whole column profile has an azeotropic composition, giving a lower temperature at the same pressure. The set point of the temperature controller is just reached because of the pressure increase during the transition. Approximately 12 h after the step change, the bottom consists of pure

Figure 31. Two steady-state profiles of column 2. Steady state D1 D1 a: xwater ) 0.0282 kg/kg, xmethanol ) 0.617 kg/kg. Steady state b: D1 D1 ) 0.07 kg/kg, xmethanol ) 0.617 kg/kg. xwater

water (all of the 2-propanol was squeezed out via the distillate). Now, the reboiler duty decreases, the bottom level increases, and at hour 80, the bottom flow is greater than 0 again. At hour 100, the composition is again changed to a value at the other side of the azeotrope, and the column returns to the steady state with pure 2-propanol in the bottom. 8. Conclusions The operation of a sequence of three homogeneous azeotropic distillation columns with two recycles (boundary separation scheme) separating a three-component 020 mixture into pure components was analyzed. The process has seven degrees of freedom. There are three different setups of the boundary separation scheme. Studying a case with reduced complexity (columns of infinite length operated at infinite reflux) showed that the third setup should not be used. Arguments for which of the other two setups should be used were derived. Further, one of the two recycles (the recycle of the entrainer) was identified as the key manipulated variable to ensure the feasibility and optimality of the process. The key aspect is that the mass balance of


Figure 32. Transients between steady state a and steady state b (Figure 31), showing that the mass balance forces xB2 to cross the azeotrope (indicated by the thin solid lines in the upper right plot).

Among the analyzed control schemes, the recommended control scheme uses the (L/D)QR scheme with constant (L/D) and QR controlling a temperature front for each column, as well as an additional control loop that controls the entrainer content in the sequence, in particular the entrainer fraction of the distillate of column 1, by manipulating the overall feed of column 1, in particular the entrainer recycle flow. The advantage of this scheme is that it requires only one online composition controller, while the other compositions can be controlled by adjusting the setpoints of the temperature controllers. The self-optimizing properties are maintained. The key aspect of the sequence, the entrainer hold-up, will become even more pronounced if the boundary separation scheme is operated with a binary external feed of an azeotropic mixture because then, the entrainer can neither leave nor enter the system. A buffer tank would be needed to allow the entrainer hold-up to change. Dynamic simulations showed that the boundary separation scheme operates well using single-loop linear PID controllers. Dynamic simulations showed further that column profiles can cross residue curve boundaries, distillation boundaries, and even azeotropes if the feed composition is changed such that a column profile is feasible only in the other distillation region. The operation concept of the boundary separation scheme can be transferred to heterogeneous distillation column sequences of heterogeneous 222-m mixtures: the entrainer recycle has to be manipulated such that the distillate of the heterogeneous column lies on the tie line that lies close to or at the ternary heterogeneous azeotrope. This is consistent with the results of Rovaglio et al.,47 who also recognized the importance of the entrainer hold-up in the system. Appendix A. Boundary Separation Scheme with Three Recycles

Figure 33. Changes in xB2 such that the mass balance forces xB3 to cross the azeotrope (indicated by the thin solid lines in the upper two plots).

column 1 has to be manipulated such that the distillate composition is at the optimal point. A sensitivity analysis shows that a control scheme that keeps the column compositions at the specified values is selfoptimizing with respect to energy consumption and robust toward model uncertainties.

Figure 34a shows the boundary separation scheme with three recycles suggested by Stichlmair (stated by Laroche et al.4 as a personal communication). The idea is to use a product recycle, R3, that places F2 away from the boundary into the nonconvex set. It was postulated that such a setting might lead to improved economic operation.4 However, this configuration does not reduce the sum of the distillate flows JD. In section 4.1, it was shown that JD is minimized if d1 is maximized. Figure 34b shows that d1 does not depend on R3 because R3 lies inside the subsystem shown in Figure 34a. Hence, the three distillate flows D1, D2, and D3 also do not depend on R3, and R3 has no influence on the cost function of the ∞/∞ column. If the columns were operated with constant reflux-to-distillate ratios, recycle R3 would change the values of the ki such that the sum of the reboiler duties would be increased. The minimum of the reboiler duties would then be at R3 ) 0. Moreover, the third recycle suggests that the boundary separation scheme might also be feasible for a straight boundary because the feed to column 2 is placed into the nonconvex set even for a straight boundary. (For a straight boundary, the term nonconvex set is no longer exact; the nonconvex set in this case is defined as the area spanned by L, I, and the I/H azeotrope.) However, the mass balance around the dashed subsystem (Figure 34a) shows that this sequence is also infeasible for a straight boundary: Even though R3 places F2 in the nonconvex set, the net flux through the


Figure 34. Extension: boundary separation scheme with three recycles.

boundary is 0 for a straight boundary because xD1 and xM1 have to lie at the same point in that case. Then, D1 ex ) R2 + D3. Hence, Dex 2 and B3 are 0 independent of R3. In operation of the sequence, R3 might be useful. If xD1 moves such that d1 < 0 (caused, for example, by a disturbance), R3 might be used to place the feed to column 2 such that the disturbance in column 1 is not propagated to columns 2 and 3. This can be implemented with an override controller that ensures that xF2 is at the desired point until xD1 returns to that desired point. Following the idea of Bausa and Tsatsaronis,57 cyclic operation of D1 or R3 might be an option to reduce the recycle flows by changing the mass balances. However, these changes are symmetrical, and hence, the average flux over the boundary remains constant. Literature Cited (1) Doherty, M.; Caldarola, G. Design and Synthesis of Homogeneous Azeotropic Distillations 3: The Sequencing of Columns for Azeotropic and Extractive Distillations. Ind. Eng. Chem. Fundam. 1985, 24 (4), 474-485. (2) Stichlmair, J.; Fair, J.; Bravo, J. Separation of Azeotropic Mixtures via Enhanced Distillation. Chem. Eng. Prog. 1989, 85 (1), 63-69. (3) Laroche, L.; Bekiaris, N.; Andersen, H.; Morari, M. Homogeneous Azeotropic Distillation: Comparing Entrainers. Can. J. Chem. Eng. 1991, 69 (12), 1302-1319. (4) Laroche, L.; Bekiaris, N.; Andersen, H.; Morari, M. The Curious Behavior of Homogeneous Azeotropic Distillations Implications for Entrainer Selection. AIChE J. 1992, 38 (9), 13091328. (5) Foucher, E.; Doherty, M.; Malone, M. Automatic Screening of Entrainers for Homogeneous Azeotropic Distillation. Ind. Eng. Chem. Res. 1991, 30 760-772. (6) Stichlmair, J.; Herguijuela, J.-R. Separation Regions and Processes of Zeotropic and Azeotropic Distillation. AIChE J. 1992, 38 (10), 1523-1535. (7) Doherty, M.; Malone, M. Conceptual Design of Distillation Systems; McGraw-Hill: New York, 2001.

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Received for review November 20, 2002 Revised manuscript received March 11, 2003 Accepted March 17, 2003 IE020930M

Operation of Homogeneous Azeotropic Distillation Column Sequences

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