Minimum Plate Area for Complex Distillation Networks with Fixed

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PROCESS DESIGN AND CONTROL IDEAS Approach to Process Network Synthesis: Minimum Plate Area for Complex Distillation Networks with Fixed Utility Cost† James E. Drake and Vasilios Manousiouthakis* Department of Chemical Engineering, University of California at Los Angeles, Los Angeles, California 90095

In this paper, we employ the infinite dimensional state-space (IDEAS) paradigm to determine the globally minimum plate area for a liquid/vapor equilibrium based separation process (distillation) network. The IDEAS paradigm results in a convex (linear) optimization problem which accounts for all possible design alternatives. The obtained solution is thus guaranteed to be the global optimum over all networks. The power of the IDEAS paradigm is demonstrated in a case study on the separation of a nitrogen/oxygen mixture. The plate area for the IDEAS design is 58% lower than the plate area of the best conventional design. 1. Introduction Engineering is an economically driven discipline whose objective is often the reduction of total process operating costs, utility costs, and capital costs while meeting process quantity and quality objectives. The process objectives often require the separation, and purification, of various stream components by means of distillation. However, distillation is a versatile unit operation which may be operated over a wide range of conditions while achieving process and product specifications. For conventional single input/double output distillation columns, process conditions can range from total reflux (which produces no product) to minimum reflux (which requires an infinite number of plates), neither of which is economically attractive. Aside from these two extremes, there is a myriad of other designs that can be pursued based on engineering intuition, thermodynamic considerations, heuristics, or first principles.1 The resulting networks, however, are not likely to contain advantageous nonintuitive or counterintuitive networks, such as feed bypass in binary distillation,2 or product recycling in ternary distillation.3 These difficulties can be dealt with by the introduction of the state-space process representation,2,4-6 which includes all possible network structures. Nevertheless, the resulting optimization problems are nonconvex, thus providing no guarantee that the objective function value identified is the global minimum. In this paper, the infinite dimensional state-space (IDEAS) process representation is discussed and applied to the problem of determining the minimum plate area for a complex distillation network that separates an oxygen/nitrogen mixture and features a fixed utility † Part of this work was first presented in Session 131 at the 1998 AIChE Annual Meeting, Poster 171z. * To whom correspondence should be addressed. Tel: 310285-9385. Fax: 310-825-2394. E-mail: [email protected].

Figure 1. IDEAS representation of a complex distillation network.

cost. The resulting IDEAS design is then compared with a conventional (McCabe-Thiele) design for the same problem. 2. Infinite Dimensional State-Space (IDEAS) Process Representation The IDEAS process representation addresses the two difficulties associated with other design paradigms: it considers all possible process configurations, and it results in convex (linear) mathematical programs. In the IDEAS paradigm, the process is represented by two interacting networks. The first, the distribution network (DN), contains all mixing and splitting operations. The second, the operator network (OP), contains the unit operation models necessary to meet the process objectives. Figure 1 illustrates an IDEAS representation of a distillation network, consisting of a DN and an OP consisting of a heat exchange network (HEN) and a mass exchange network (MEN). Points (junctions) along

10.1021/ie010735s CCC: $22.00 © 2002 American Chemical Society Published on Web 08/27/2002

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Figure 4. Alternate IDEAS representation of a complex distillation column.

Figure 2. Traditional distillation column.

Figure 3. IDEAS representation of a traditional distillation column.

the edges of the DN represent the quality parameters (composition, pressure, temperature, etc.) associated with the various streams. The sides of the DN represent splitting (entering) junctions [bottom edge and left-hand side (LHS)] and mixing (exiting) junctions [top edge and right-hand side (RHS)]. Process feed stream(s) enter the LHS of the DN and are split into streams, all of which go to all of the DN mixing (exiting) junctions. Those streams going from the LHS to the top of the DN are bypass streams. The streams that go to the RHS are joined with other streams, from the bottom edge of the DN, and exit the DN and enter the OP. Streams exiting the OP reenter the DN on the bottom edge, where they are split, some going to the top edge of the DN, forming process product streams with any feed bypass stream, and others returning to the RHS of the DN for further processing in the OP. Figure 2 is the flowsheet of a simple binary distillation column with saturated liquid feed and bottoms product and a saturated vapor distillate product. The column consists of two multiplate mass exchangers (MEXs), the stripping section (S) and the rectifying section (R), and two heat exhangers, the reboiler (RB) and the condenser (C). Figure 3 is the IDEAS representation of the simple distillation column in Figure 2. The OP consists of two

components: a MEN containing the stripping section (S) and rectifying section (R) and a HEN containing the reboiler (RB) and condenser (C). Stream numbers correspond to the stream numbers in Figure 2. Streamflows are clearly indicated. Utilities to the reboiler and condenser are not shown for clarity. An alternative IDEAS representation for an isobaric distillation process network is shown in Figure 4. In this representation, all DN streams are considered to be saturated, a characteristic of distillation systems. Additionally, splitting and mixing of streams also occur in the OP, in the form of a splitting operator (SO) and a mixing operator (MO), respectively. Each stream exiting the DN has a distinct combination of quality variables (composition, temperature, etc.) and, upon entering the SO, is split into streams which enter the various process elements of the OP (i.e., MEN, HEN). In the MO, those streams exiting the OP with identical quality variables are combined, before entering the DN. To allow process feed streams and product streams to be at any thermal state (subcooled, saturated, or superheated), these feed (product) streams are processed through the HEN before (after) entering (leaving) the DN. With reference to Figure 4, streams entering and leaving the DN, SO, and MO are represented by sequence doublets of the form (v, ζ) ) {v(j), ζ(j)}∞1 , where the scalar v(j) contains the quantity information (molar flow rate) of the jth stream. The n-dimensional vector ζ(j) contains the quality information (composition, temperature, etc.) of the jth stream. Because the process is isobaric, then ζ(j) } [ζ1(j), ζ2(j), ..., ζn-1(j), ζn(j)]T ) [x1(j), x2(j), ..., xn-1(j) hˆ (j)]T, where xi(j) is the mole fraction of component i in stream j, and hˆ (j) is the specific molar enthalpy of stream j. Because all flows, v(j), are nonnegative, considering their infinite sum to be bounded yields v ∈ l1+. Additionally, mole fractions and enthalpies are finite, and thus ζ ∈ l∞n. We further assume, without loss of generality, that ζ(j) ∈ Qn ⊂ Rn. One unique feature of the IDEAS representation is the complete interconnection of the DN inlet and outlet streams, allowing all possible process networks to be considered. These intra-DN connections are specified by sequence triplets, (zsw, ξ, θ), where the flow rate to quality ξ(k) from quality θ(j) is given by zsw(k,j). Because all flows zsw(k,j) are nonnegative and their sum is finite, zsw ∈ l1+(2). A complete description of the properties of these sequences may be found in work by Wilson and Manousiouthakis.7

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The resulting IDEAS model then consists of the following set of equations:

q(i) g 0, i ) 1, ∞ w12(j)

∞

T

min x,w,wji,y,wl,zyu,zwu,zyx,zwx,r

A(j) ) U

AI )

A(j) ∑ j)1

(1)

P(j) w22(j)

(2)

V SF G

H

C

C ) c q(1) + c q(∞) ∞

u(i) )

∑ zwu(k,i) + k)1 ∑ zyu(k,i), k)1

y(k) )

∑ i)1 ∑ i)1

zyu(k,i) +

i ) 1, ∞

(5)

∑ i)1

zyx(k,i), k ) 1, ∞

zwu(k,i) +

∑ i)1

zwx(k,i), k ) 1, ∞

(7) π2k(j) ) [

∞

R′(i) zwu(k,i) + ∑σ(i) zyx(k,i) + ∑ i)1 i)1 rδ(k) [0, 0, ..., δn(k)]T, k ) 1, ∞ (9)

∑ w12(j) + ∑ w22(j) + m∈S ∑ w1(m),

j∈S1i

j∈S2i

i ) 1, ∞

i

(10)

) {j:δ (j) ) δ(i)}, k ) 1, 2; i ) 1, ∞ Si ) {m:δ(m) ) δ(i)}, m ) 1, ∞

x(i) )

k

∑ w13(j) + ∑ w23(j) + m∈Z ∑ w1(m),

j∈Zi1

j∈Zi2

i ) 1, ∞

i

(11)

) {j:π (j) ) γ(i)}, k ) 1, 2; i ) 1, ∞ Zi ) {m:ω(m) ) γ(i)}, m ) 1, ∞ k

Zki

where ∞

q(i) +

H C (λw (j,i) + λw (j,i))[hˆ (δ(j),T(i)) ∑ j)1 1

1

∞

hˆ (ω(j),T(i+1))]w1(j) +

(λH ∑ u (j,i) + j)1

λCu (j,i))[hˆ (R(j),T(i)) - hˆ (R′(j),T(i+1))]u(j) + ∞

C (λH ˆ (β′(j),T(i)) ∑ y (j,i) + λy (j,i))[h j)1 ∞

hˆ (β(j),T(i+1))]y(j) +

H (νw (j,i) + ∑ j)1 3

∞

C (j,i))hˆ [π1(j),Tlsat(π1(j))]rπ(j) + νw 3 ∞

νCy (j,i))β′n(j) rβ′(j) +

(13) j ) 1, ∞

w12(j)

w12(j)

w22(j)

w22(j)

+ Rk

w12(j)

w22(j)

[( ) ( ) [( ) w12(j)

w22(j)

+

w12(j)

δ1k(j) + δ2k(j)

(14)

]

]

,

(δ11(j) - π11(j)) + δ21(j)

j ) 1, ∞ (15)

δ1k(j) + 2 w2(j) w12(j) 1 Rk (δ1(j) w22(j)

]

δ2k(j) -

π11(j))

+

]

,

δ21(j)

k ) 1, n - 1; j ) 1, ∞ (16)

n-1

∞

∞

where

)

R1π11(j)

R′(i) zyu(k,i) + ∑σ(i) zyx(k,i) + ∑ i)1 i)1

Ski

π1k(j)

(6)

rβ′(k) [0, 0, ..., β′n(k)]T, k ) 1, ∞ (8)

w(i) )

R1π11(j)

)

w23(j),

[( ) ( ) [( ) R1π11(j)

∞

∞

δ(k) w(k) )

∑

k)2

∞

∞

w(k) )

n-1

(4)

∞

∑ zwx(k,i) + k)1 ∑ zyx(k,i), k)1 ∞

β′(k) y(k) )

i ) 1, ∞

w22(j)

R1π11(j)

∞

∞

x(i) )

(3)

1 ) π11(j) +

)

w13(j),

(νH ∑ y (j,i) + j)1

C (νH ∑ w (j,i) + νw(j,i))δn(j) rδ(j) ) j)1

q(i+1), i ) 1, ∞ (12)

Rkπ1k(j)]-1Rkπ1k(j), ∑ k)1

k ) 1, n - 1; j ) 1, ∞ (17)

Equation 1 is the objective function of the optimization, which is the total plate area of the MEXs in the IDEAS design, AIT. Equation 2 is used to calculate the plate area, A(j), of the jth MEX, where P(j) is the number of plates in the jth MEX, w22(j) is the vapor molar flow rate in the jth MEX, VS is the superficial velocity, and FG is the vapor molar density. The superficial velocity can be determined using Carey’s relationship8 for a given tray spacing and liquid seal height, and component data are from Timmerhaus and Flynn.9 The requirement of a fixed utility cost is expressed in eq 3. Equations 4-7 represent mass balances at junctions on the left, bottom, top, and right edges, respectively, of the DN. Equations 8 and 9 are the component and enthalpy balances at junctions on the top and right edges, respectively, of the DN, where rβ′(k) and rδ(j) are the “flows” of heat added to, or removed from, junctions β′(k) and δ(k), respectively, to ensure saturated conditions. The SO mass balances are represented in eq 10, while the MO mass balances are expressed in eq 11. The HEN operator is based on the pinch formulation in Papoulias and Grossmann.10 Temperature interval boundaries consist of the various saturated liquid and vapor temperatures corresponding to the sequences ζ(j) and utility temperatures. Equation 12 is the enthalpy balance within the ith HEN temperature intervals, where q(1) is the hot utility usage (QH) available at T(1) and q(∞) is the hot utility usage (QC) available at T(∞). Equation 13 states the second law of thermodynamics, requiring that heat flow from higher temperatures to lower temperatures. In this work, and without loss of generality, we consider ∆Tmin ) 0. Equations 14 and 15 comprise the single-plate, equilibrium, equimolar overflow MEX model. The equimolar overflow requirement is stated in eq 14. From eq 15, candidate ratios of w12(j)/w22(j) are determined, given δ1k(j), δ2k(j), j ) 1, ∞, k ) 1, n - 1, and π11(j), j ) 1, ∞, where w12(j) and w22(j) are the molar flow rates of the

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Figure 5. IDEAS design convergence characteristic.

liquid and vapor streams, respectively, entering the jth MEX. Equation 16 expresses the composition of the liquid stream exiting the jth MEX, π1i (j), i ) 2, n - 1, while eq 17 is the equilibrium condition that allows evaluation of the composition of the vapor stream exiting the jth MEX, π2i (j), i ) 1, n - 1. An MEX is n-1 i πk(j) ) 1, i ) 1, 2, feasible if w12(j)/w22(j) > 0, and ∑k)1 i ∀ j ) 1, ∞ and 0 e πk(j) e 1, i ) 1, 2, k ) 1, n - 1, ∀ j ) 1, ∞. For a feasible MEX, the input flow rate to output flow rate map is linear. Thermodynamic properties are calculated assuming constant specific heats; a common, constant heat of vaporization; constant relative volatility; and that the relationship between the temperature and vapor pressure of oxygen is described by the Antoine equation log PV ) A1 + B1/(T + C1). These assumptions are for ease of calculation. They in no way restrict the general applicability of the IDEAS paradigm. 2.1. Solution Method. The aforementioned objective function (eq 1) and utility cost, DN, SO, MO, HEN, and MEN constraints (eqs 2-16) are all linear, thus giving rise to a linear programming formulation. This guarantees the global optimality of the identified solution. This is a unique feature of the IDEAS formulation. A series of finite dimensional subproblems are then solved, whose objective function value converges to the objective function value of the infinite dimensional problem.11,12 This convergence property is illustrated in Figure 5 for the considered case study. 2.2. Example of a Counterintuitive Process Element. The IDEAS process paradigm, free from heuristics regarding the choice of distillation sequence constituents, allows for the consideration of counterintuitive “reverse” exchangers in the design process, similar to the zones of reverse fractionation observed in multicomponent distillation.13 Intuition suggests that, when separating components based on their relative volatility, the more volatile component should migrate to the more volatile (vapor) phase. However, the opposite effect occurs in reverse exchangers: the more volatile component migrates to the less volatile phase (liquid). While normal exchangers operate below the equilibrium surface, reverse exchangers operate above it. The ability of reverse exchangers to produce low volatility vapors (like a conventional reboiler) and high volatility liquids (like a conventional condenser) without the use of thermal utilities can prove useful in minimum utility cost designs. 2.3. Representation of IDEAS Designs. A binary, isobaric distillation system with saturated liquid and vapor streams allows another, reduced, IDEAS representation. Using the symbols listed in the IDEAS Sequences Notation section, the distillation column in

Figure 6. Reduced IDEAS representation of a traditional distillation column.

Figure 2 is shown in this reduced form in Figure 6, for the special case that the rectifying section consists of two plates (plates 1 and 2), while the stripping section consists of three plates (plates 3-5). Vapor junctions are represented by squares in the left-hand column, while liquid junctions are represented in the right-hand column. Volatility and temperature increase as the junctions are higher in the diagram, as indicated by the arrow to the left of the column. A traditional column consists of normal plates. The change in vapor stream composition through a plate is shown by an arc connecting the two saturated vapor junctions corresponding to the vapor inlet and outlet compositions, while the change in the liquid stream composition is indicated in a like manner. A line connecting the two arcs indicates that the streams are brought into contact. It may be seen from the IDEAS Sequences Notation section, because the arcs are inside the columns of junctions, that this is a “normal” MEX. Therefore, the vapor flow is upward, becoming more volatile, while the liquid flow is downward, becoming less volatile. The condenser (C) is indicated by the horizontal line above the plates, with the arrow indicating a change in state from a saturated vapor to a saturated liquid of the same composition. In a similar manner, the reboiler (RB) is represented by a horizontal line with an arrow pointing to a saturated vapor junction of the same composition. The L/V ratio for a specific plate is equal to the vertical distance between the inlet and outlet vapor junctions divided by the vertical distance between the inlet and outlet liquid junctions. Thus, one can clearly

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Table 1. Process Conditions

feed stream product 1 product 2

xN 2

xO2

T, K

P, kPa

flow, kg‚mol/s

0.70 0.90 0.10

0.30 0.10 0.90

83.4 80.4 87.5

101.3 101.3 101.3

10.0 7.5 2.5

Table 2. Design Comparison Data converged converged IDEAS IDEAS conventional without reverse with reverse design exchangers exchangers area (m2) total exchangers reverse exchangers percent savings

434.7

183.5 63 0 58.

183.5 60 0 58.

see that in the rectifying section (plates 1 and 2) L/V < 1, while L/V > 1 in the stripping section (plates 3-5). 3. Case Study: Distillation of a Nitrogen/ Oxygen Mixture Nitrogen and oxygen are among the largest volume chemicals produced in the United States. Nitrogen and oxygen production in the U.S. during 1998 was 29.9 × 109 and 18.0 × 109 kg, respectively, surpassed only by sulfuric acid production.14 Because of its economic importance, distillation of a nitrogen/oxygen mixture is a suitable process for the demonstration of the power and flexibility of the IDEAS. The case study determines the minimum plate area for a fixed utility cost of $50 000/s for the separation of a saturated vapor feed into one saturated liquid and one saturated vapor product at 1.013 × 105 Pa. Table 1 contains the process conditions for this case study. Designs derived from the IDEAS paradigm and a conventional design method are compared. Without loss of generality, mixing is only permitted between an adjacent liquid or an adjacent vapor junction (local mixing), with the exception of streams forming process product streams where generalized, or global, mixing is permitted. One of the features of this problem is that each design has not only the same fixed utility cost but also, because the feed and product specifications are identical, the same hot and cold utility usage rates: 30.36 and 45.45 MJ/s, respectively. Their difference, 15.09 MJ/s, represents the difference in the enthalpies between products and the feed. A comparison of the results of two converged IDEAS designs, one of which allows for reverse exchangers, and a conventional design are shown in Table 2. The IDEAS designs show a plate area 58% less than that of the conventional design minimum. It should be noted that both IDEAS designs feature the same minimum area values and that no reverse exchangers are employed on the converged IDEAS design allowing for their presence. The minimum plate area conventional design is shown in Figure 7. Saturated vapor at 83.4 K is fed into the column. The bottoms product is removed as a saturated liquid, before the total reboiler, containing 90 mol % oxygen at 87.5 K, while the distillate product, a saturated vapor, contains 90 mol % nitrogen and is removed before the total condenser at 80.4 K. The column is operated at a reflux ratio of 0.98, 67% higher than the minimum reflux ratio of 0.58. No heat integration is possible. The total plate area for the best

Figure 7. Conventional design.

Figure 8. Reduced representation of IDEAS design with 35 junctions.

conventional design in 434.7 m2. The design equations for the conventional design are presented in the appendix. An IDEAS design with 35 junctions allowing reverse exchangers is shown in Figure 8. The area of this design is 196.2 m2, 54.7% less than the area of the conventional design and 6.9% more than the area of the converged IDEAS design (183.5 m2). The design consists of 60 MEXs, none of which are reverse exchangers. (The lack of reverse exchangers can be attributed to the requirement that the utility cost to be greater than the minimum utility cost.) There are also 7 condensers, 18 evaporators, and 4 heat exchangers adjusting final

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Figure 9. Mass-transfer composite diagram, conventional design.

Figure 10. Mass-transfer composite diagram, converged IDEAS design.

product constituent stream enthalpies prior to mixing. The feed stream is introduced into the IDEAS representation on the left (vapor) side of the diagram. Bottoms product is withdrawn from one liquid junction without the use of a heat exchanger to adjust the enthalpy. The distillate product, on the other hand, is formed by combining four vapor streams at different compositions, one of which is identical in composition and enthalpy with the distillate product specifications. The enthalpy of the other three streams is adjusted to the specified product enthalpy. Four streams are then mixed to produce the specified distillate product. There are no exchangers at the extremes of the IDEAS diagram because in these regions saturated liquid and saturated vapor approach each other in composition, thereby requiring MEXs with large plate areas. The reason for the differences in plate area may be ascertained by examining mass-transfer composite diagrams, introduced by El-Halwagi and Manousiouthakis,15,16 for both the IDEAS and conventional designs. Comparing the mass-transfer composite diagram for the conventional design (Figure 9) with the mass-transfer composite diagram for the converged IDEAS design (Figure 10), we notice that in the IDEAS design about 4.0 kg‚mol/s is transferred between the rich and lean streams, as compared to about 5.8 kg‚mol/s transferred in the conventional design. This fact, in itself, results in a smaller plate area in the IDEAS design. In addition, a comparison of the IDEAS design representation (Figure 8) with a similar representation of the conventional design (Figure 11) is enlightening. As expected, the conventional design above the feed point shows larger changes in liquid composition than in vapor composition on a given plate, consistent with L/V < 1. The opposite is true in the stripping section where L/V > 1. An examination of the IDEAS design shows that, for a given plate, the vapor stream composition changes are larger than the liquid stream composition changes, implying that L/V > 1 for all plates. The average L/V for the conventional design is 0.82, com-

Figure 11. Reduced representation of conventional design.

Figure 12. Heat-Transfer composite diagram, IDEAS minimum plate area design.

pared with 7.71 for the IDEAS design in Figure 6. This is the primary factor resulting in a lower plate area for the IDEAS design. It is also instructive to examine the heat-transfer composite diagram for the converged IDEAS minimum plate area design allowing reverse exchangers (Figure 12) with that for the corresponding IDEAS minimum utility design18 (Figure 13). The higher utility cost of the minimum plate area design implies that its heat transfer is less efficient than that of the minimum utility design. This is manifested in two ways. First, about 8 times more heat is exchanged in the minimum utility design, 10 000 versus 1200 MJ/s, allowing more heat integration opportunities. Second, the hot and cold streamlines are superimposed for the minimum utility design, indicating efficient reversible heat exchange. These lines for the minimum plate area are separated

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and the fixed utility cost, CU, are known, the following model may be used to determine the total plate area of the design:

FhF + QH ) DhD + BhB + QC

(A1)

CU ) cHQH + cCQC

(A2)

C

Figure 13. Heat-transfer composite diagram, IDEAS minimum utility design.

by several degrees, especially below 82 K, indicating a lower efficiency transfer of heat. 4. Conclusions The IDEAS paradigm considers all possible process networks and results in a linear optimization problem. As such, it is guaranteed that the objective function value identified is the global optimum. The IDEAS method is utilized to determine the minimum plate area for a fixed utility cost for the separation of a nitrogen/ oxygen mixture into nitrogen-rich and oxygen-rich streams. The obtained IDEAS design is compared to a conventional design, and its minimum area is found to be 58% lower than that of the conventional design. IDEAS may be used either directly for process design or indirectly for design targeting. In the latter case, the design engineer would use the IDEAS design as a target against which to compare alternative designs. The application of the IDEAS paradigm to other process network synthesis problems is limited only by the designer’s imagination and will be the subject of future work. 5. Physical Parameters Antoine equation constants for oxygen: A1 ) 8.82082; B1 ) 319.013; C1 ) -6.45. Relative volatilities: R1 ) RO2/O2 ) 1.000; R2 ) RN2/O2 ) 4.173. Reference temperature: Tref ) 70 K. Latent heat: λ ) 6.2 × 106 J/kg‚mol. Pressure: P ) 1.0113 × 105 Pa. MEN plate spacing: 0.59 m. MEN liquid seal: 0.07 m. Heat capacities:

CLp , J/(kg‚mol‚K) CVp , J/(kg‚mol‚K)

N2

O2

5.743 × 104 2.926 × 104

5.312 × 104 2.966 × 104

Acknowledgment The support of the National Science Foundation is gratefully acknowledged under Grants CTS 9528553, GER 9554570, and CTS 9876489. Appendix: Determining Conventional Column Design Parameters for a Fixed Utility Cost The conventional design was developed using the McCabe-Thiele design paradigm. Because the feed stream molar flow rate and enthalpy, F and hF, distillate product molar flow rate and enthalpy, D and hD, bottoms product molar flow rate and enthalpy, B and hB, latent heat, λ, hot and cold utility prices, cH and cC,

y) y)

L ) Q /λ

(A3)

V ) QH/λ

(A4)

L L ) V L+D

(A5)

Lo B + Vo ) Vo Vo

(A6)

(VL)x + z [1 - (VL)]

(A7)

Lo Lo x + zB 1 Vo Vo

(A8)

D

( ) [ ( )] y)

R1 x 1 + (R1 - 1)x

AC )

VPr + VoPs V SF G

(A9) (A10)

Equation A1 is the enthalpy balance around the column. In conjunction with eq A2, the fixed utility cost, it is used to solve for the reboiler and condenser loads. Then the liquid flow rate in the rectifying section and the vapor flow rate in the stripping section may be determined from eqs A3 and A4. After calculation of the slopes of the rectifying and stripping sections from eqs A5 and A6, the equations for the rectifying and stripping operating lines are generated by eqs A7 and A8. The numbers of plates in the rectifying section, Pr, and stripping section, Ps, were calculated by Martin’s method,17 using eqs A7 and A8, together with eq A9 for the equilibrium line. Individual plate areas are calculated by the method previously described. The total plate area for the conventional design, AC, is calculated from eq A10. The rectifying section, with a vapor throughput of 14.83 kg‚mol/s, contains 1.9 plates with a total area of 297.2 m2, while the stripping section has a vapor throughput of 4.83 kg‚mol/s, contains 2.67 plates, and has a total area of 137.5 m2. The total plate area of the conventional design is 434.7 m2. Notation A(j) ) area of the jth MEX AT ) total plate area of the IDEAS design AC ) total plate area of the conventional design A1 ) Antoine equation coefficient B ) molar flow rate of bottoms product of a traditional distillation column B1 ) Antoine equation coefficient C1 ) Antoine equation coefficient CLp ) molar liquid heat capacity at constant pressure CU ) total fixed utility cost CVp ) molar vapor heat capacity at constant pressure cC ) cold utility unit cost cH ) hot utility unit cost

Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002 4991 D ) molar flow rate of distillate product for a traditional distillation column F ) molar flow rate of feed stream for atraditional distillation column hˆ (j) ) specific molar enthalpy of stream j hB ) specific molar enthalpy of the bottoms product hD ) specific molar enthalpy of the distillate product hF ) specific molar enthalpy of the feed stream L ) molar liquid flow through an MEX L ) molar liquid flow in the rectifying section of a traditional distillation column Lo ) molar liquid flow in the stripping section of a traditional distillation column P ) pressure P(j) ) number of plates in the jth MEX PV ) vapor pressure Pr ) number of plates in the rectifying section based on Martin’s method Ps ) number of plates in the stripping section based on Martin’s method QC ) cold utility usage QH ) hot utility usage q(1) ) cold utility usage q(∞) ) hot utility usage rβ′(k) ) heat added to the kth junction on the top of the DN to achieve saturation of the mixed stream formed there rδ(k) ) heat added to the kth junction on the RHS of the DN to achieve saturation of the mixed stream formed there Ski ) {j: δk(j) ) δ(i)}, k ) 1, 2, i ) 1, ∞ Si ) {m: δ(m) ) δ(i)}, m ) 1, ∞ T ) temperature T(i) ) ith temperature in the HEN temperature interval diagram Tref ) reference temperature V ) molar vapor flow rate through an MEX V ) molar vapor flow rate in the rectifying section of a traditional distillation column Vo ) molar vapor flow rate in the stripping section of a traditional distillation column VS ) superficial velocity w12(1) ) molar liquid flow rate through the jth MEX w22(1) ) molar vapor flow rate through the jth MEX x ) mole fraction of the most volatile component in liquid phase y ) mole fraction of the most volatile component in vapor phase xB ) mole fraction of the most volatile component in bottoms product xD ) mole fraction of the most volatile component in distillate product xF ) mole fraction of the most volatile component in feed stream Zki ) {j: πk(j) ) γ(i)}, k ) 1, 2, i ) 1, ∞ Zi ) {m: ω(m) ) γ(i)}, m ) 1, ∞

the the

IDEAS Notation

the

the the

Ri ) relative volatility of component i relative to the least volatile system component ∆Tmin ) minimum temperature difference required for heat transfer λ ) molar latent heat FG ) molar vapor-phase density I+ ) space of nonnegative integers Qn ) space of rational numbers of dimension n Rn ) space of all real numbers of dimension n

ln∞ ) n-dimensional vector with the absolute value of any ∞ element finite, supi)1,2,...,n∑j)1 |aj(i)| < ∞ ) all elements nonnegative and sum of elements finite, l+ ∞ ∞ ∑i)1 a(i) < ∞, infi)1,...,∞a(i) g 0 n l1 (N) ) nonnegative N-tuples, f(k1,...,kN), with the sum of elements finite, ∑k∞1)1...∑k∞Nf(k1,...,kN) < ∞, infk1,...,kN∈I+f(k1,...,kN) IDEAS Sequences (v, ζ) ) infinite sequences consisting of the elements {v(j), ζ(j)}∞1 v(j) ) single element subsequence containing the molar flow rate of stream j ζ(j) ) n-element subsequence containing the quality parameters of stream j: composition, pressure, temperature, etc. (u, R) ) process feed streams before passing through the HEN (u, R′) ) process feed streams entering the DN after passing through the HEN (w, δ) ) streams exiting the DN and entering the splitting operator (SO) (w1, δ) ) streams exiting the SO and entering the HEN (w2, δ) ) streams exiting the SO and entering the MEN (w1, ω) ) streams exiting the HEN and entering the mixing operator (MO) (w2, π) ) streams exiting the MEN and entering the mixing operator (MO) (x, γ) ) streams exiting the MO and entering the DN (y, β′) ) streams exiting the DN forming product streams before passing through the HEN (y, β) ) streams exiting the DN forming product streams after passing through the HEN zsw(k,j) ) molar flow from quality junction k on side s of the DN to quality junction j on the w side of the DN zyu(k,j) ) molar flow to quality junction β′(k) on the top edge of the DN from quality junction R′(j) on the LHS of the DN zwu(k,j) ) molar flow to quality junction δ(k) on the RHS of the DN from quality junction R′(j) on the LHS of the DN zyx(k,j) ) molar flow to quality junction β′(k) on the top edge of the DN from quality junction γ(j) on the bottom of the DN zwx(k,j) ) molar flow to quality junction δ(k) on the RHS of the DN from quality junction γ(j) on the bottom edge of the DN

Greek Letters

Spaces

Infinite Sequences

4992

Ind. Eng. Chem. Res., Vol. 41, No. 20, 2002

Literature Cited (1) Malone, M. F.; Glinos, K.; Marquez, F. E.; Douglas, J. M. Simple, Analytical Criteria for the Sequencing of Distillation Columns. AIChE J. 1985, 31, 683. (2) Bagajewicz, M. J.; Manousiouthakis, V. Mass/Heat Exchange Network Representation of Distillation Networks. AIChE J. 1992, 38, 1769. (3) Kovacs, Z.; Friedler, F.; Fan, L. T. Recycling in a Separation Process Structure. AIChE J. 1993, 39, 1087. (4) Roxenby, S.; Manousiouthakis, V. Non-Isothermal Separable Mass Exchange Networks: Minimum Utility Cost through the State Space Approach; Internal Report; UCLA Chemical Engineering Department: Los Angeles, CA, 1993. (5) Roxenby, S. Nonisothermal Separable Mass Exchange NetworkssMinimum Utility Cost through the State Space Approach; Report LUTKDH/(TKKA-5004)/1-133; Department of Chemical Engineering I, Lund University: Lund, Sweden, 1994. (6) Gupta, A.; Manousiouthakis, V. Waste Reduction through Multicomponent Mass Exchange Network Synthesis. Comput. Chem. Eng. 1994, 18, S585. (7) Wilson, S. D.; Manousiouthakis, V. IDEAS Approach to Process Network Synthesis: Application to Multicomponent MEN. AIChE J. 2000, 46, 2408. (8) Treybal, R. E. Mass Transfer Operations; McGraw-Hill: New York, 1956. (9) Timmerhaus, K. D.; Flynn, T. M. Cryogenic Process Engineering; Plenum Press: New York, 1989. (10) Papoulias, S. A.; Grossmann, I. E. A Structural Optimiza-

tion Approach to Process Synthesis. Comput. Chem. Eng. 1983, 7, 707. (11) Sourlas, D.; Manousiouthakis, V. Best Achievable Decentralized Performance. IEEE Trans. Autom. Control 1995, 41, 1858. (12) Sourlas, D. D.; Manousiouthakis, V. Best Achievable Performance: Non-Switching Compensation for Multiple Models. Int. J. Robust Nonlinear Control 1999, 9, 521. (13) King, C. J. Separation Processes, 2nd ed.; McGraw-Hill: New York, 1980. (14) Drake, J. E. Globally Optimum Process Network Synthesis Using the Infinite DimEnsionAl State-space (IDEAS) Representation. Ph.D. Dissertation, University of California, Los Angeles, Los Angeles, CA, 2000. (15) Production: Modest Gains. Chem. Eng. News 1999, 77, 34. (16) El-Halwagi, M. M.; Manousiouthakis, V. Synthesis of Mass Exchange Networks. AIChE J. 1989, 17, 1233. (17) Martin, J. J. Analytical Solution of Equilibrium Stage OperationssApplication to Rectification with Varying Saturated Enthalpies and to Liquid-Liquid Extraction. AIChE J. 1963, 9, 646. (18) Drake, J. E.; Manousiouthakis, V. IDEAS Approach to Process Network Synthesis: Minimum Utility Cost for Complex Distillation Networks. Chem. Eng. Sci. 2002, in press.

Received for review September 4, 2001 Accepted June 27, 2002 IE010735S