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Ind. Eng. Chem. Res. 2000, 39, 3912-3919

Graphical Concepts To Orient the Minimum Reflux Ratio Calculation on Ternary Mixtures Distillation J. A. Reyes, A. Go´ mez, and A. Marcilla* Departamento Ingenierı´a Quı´mica, Universidad de Alicante, Apartado n° 99, Alicante 03080, Spain

New methods for orienting the calculation of the minimum reflux ratio for distillation columns have been developed. These methods are based on the geometrical concepts of the Ponchon and Savarit method for binary systems and use the geometrical conditions required to achieve such a situation in ternary systems. These concepts have been applied to ideal and nonideal ternary mixtures. 1. Introduction The minimum reflux ratio is a limiting operation condition that is normally used as a design parameter in distillation problems. Any column operated at a reflux ratio lower than the minimum would never attain the desired separation, even with an infinite number of stages. This limit must be numerically or graphically determined, and the condition can obviously not be proved experimentally. The methods for carrying out such calculations fall into the following categories: (1) Graphical methods. These are strictly applicable only to binary systems, and it is stated that, to achieve the condition of an infinite number of stages, an operating line is required that coincides with a tie line in a given zone of the column, i.e., a pinch point of constant composition is attained. To determine this condition, all sections of the column must be analyzed, and the minimum reflux corresponding to each section compared in order to select the maximum of these characteristic values, which is the minimum limiting reflux ratio searched. This can be expressed by the following equation:

Rmin ) max[Rmin(k)]

(1)

where Rmin(k) is the minimum reflux characteristic of the section k. For binary distillation columns with a single feed, this condition is normally produced when an operating line coincides with the tie line corresponding to the feed condition, except for certain nonideal systems whose y/x equilibrium diagrams show changes in the curvature. (2) Shortcut calculations. These are methods involving the use of one or more simplifying assumptions such as constant relative volatility, constant molar flow across the column, composition and/or temperature in the pinch zone, etc. (3) Strict plate-to-plate calculations, involving stepwise material- and enthalpy-balance calculations until successive stages become identical. This is a tedious iterative procedure, which normally consists of obtaining the function of the number of stages versus the reflux ratio and extrapolating to an infinite number of * Author to whom all correspondence should be addressed. Telephone: (34) 965 903 789. Fax: (34) 965 903 826. E-mail: [email protected].

stages. Obviously, this procedure is very time-consuming, as it involves the calculation of a large number of columns, each one with a large number of stages. Thus, it would be very interesting to develop a procedure to orient and accelerate such a search. Several graphical, shortcut, and rigorous approaches can be found in the literature. Among the graphical methods, those of Ponchon and Savarit and McCabe and Thiele (see King1 and Seader2) are the most frequently used for binary systems. Hengstebeck3 extended the McCabe-Thiele method to multicomponent systems, considering the separation as a binary separation of the key components and recommended a graphical procedure for minimum reflux that allows results that are somewhat lower than true values because this method neglects the presence of light nonkeys just below the feed and heavy nonkeys just above it.4 There are some iterative methods such as the “zero volume” of Julka and Doherty5 that use the points where the operating lines intersect the equilibrium surface to find a close approximation to the minimum reflux in nonideal multicomponent mixtures. There are various classical shortcut methods described in the works cited in the reference list (see, e.g., Sawistowski and Smith6 and Van Winkle7). Among the methods most frequently used are the following: the Underwood8 equation; the Colburn9 method, which is a modification of the Underwood equation; the Underwood method;10 and the Gilliland method.11 All of these methods calculate the minimum reflux by using different equations involving the relative volatilities of the components of the mixture. Generally, the Underwood8 equation is not applicable to a multicomponent mixture, as it involves a considerable error for the minimum reflux ratio. Colburn9 uses two equations to calculate the concentrations of the key components in the rectifying pinch point, which are used in place of that of the feed in the equation of Underwood, leading to a minimum reflux ratio that can be close to the real value, but not always. The method of Underwood10 solves an equation that relates the feed composition, the thermal condition of the feed, and the relative volatility at the average temperature of the column with a factor θ that lies between the relative volatilities of the keys. Gilliland’s method11 yields results similar to those obtained with Underwood’s but has the advantage of using only one equation and not requiring iterative calculations. For complex columns, there are a number of different

10.1021/ie9907021 CCC: $19.00 © 2000 American Chemical Society Published on Web 10/02/2000

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3913

approaches. Some of these are based on the extension of Underwood’s equations, but the presence of distributed components is a complicating factor.4 Erbar and Maddox12 developed a computer method based on the approach suggested by Bachelor,13 but this method failed if the feed was not in equilibrium. Chien14 suggested a method that does not involve any approximation and is based on the definition of a “recovery” as the ratio of the concentration of a component in the liquid and the vapor cross between two consecutive stages. The general condition for locating the pinch point is to match the relative volatilities of the heavy keys to their recoveries. This method has the limitation that it cannot solve problems involving tangent pinch points in the McCabe-Thiele diagram. Most graphical and shortcut methods give good results, but for systems in which the relative volatility varies widely throughout the column, a more rigorous method may be desired. Rigorous procedures accurately predict minimum reflux but often fail to converge. There are several methods suitable for making this type of calculation by means of a computer.15 On the other hand, several empirical reflux-stage correlations can be found in the literature, whose accuracy is usually satisfactory and which are more or less easy to computerize.4 2. Suggested Criteria for the Calculation of the Minimum Reflux Ratio The main problem involved in the rigorous calculation of the minimum reflux ratio is the great number of iterations involved, each one requiring the calculation of a distillation column. Nevertheless, this search can be oriented and simplified through a consideration of the conditions that must be fulfilled under such operating conditions. This orientation reduces not only the number of iterations but also the number of convergence failures. Thus, the objective of this work is to suggest and test two different methods for making this oriented search, based on geometrical concepts and the design method of distillation columns for multicomponent mixtures proposed by Marcilla et al.16 and by Reyes Labarta.17 The results obtained are compared with those from the shortcut methods and with those from a rigorous method without any orientation, i.e., by running successive calculations, gradually reducing the reflux ratio until the stage number increases to a sufficiently large value. The rigorous calculation of each distillation column was carried out by the procedure developed by Marcilla et al.,16 which consists of a generalization of the Ponchon-Savarit method for binary mixtures to complex columns and the extension of this method to multicomponent mixtures. The procedure solves the problem by employing the same geometrical concepts involved in the Ponchon-Savarit method but using the corresponding analytical equations of equilibrium surfaces and operating lines. Figure 1 shows the graphical representation of this method for a ternary mixture, and Table 1 contains the corresponding algebraic equations (eqs 2-5). This figure shows the location of the net flow streams in the rectifying section ∆(1) and the stripping section ∆(2), the surface corresponding to the equilibrium enthalpy of the saturated vapor H[y(1),y(2)], and the surface corresponding to the saturated liquid, h[x(1),x(2)]. In this graph, as in the Ponchon and Savarit method, the operating lines (eq 4) and the tie lines are

Figure 1. Graphical representation of the Ponchon-Savarit method for a ternary mixture.

alternated until the final residue (R) is reached. The change in the net flow at the origin of the operating line occurs when the criteria for the section change are satisfied (eq 5). In the Ponchon-Savarit method, the minimum reflux condition is attained whenever, in a zone of the column, the composition of the crossing streams between two consecutive stages is constant from one stage to the following. Such a condition implies that the streams crossing between stages must be in equilibrium, and consequently, the minimum reflux condition is equivalent (from a graphical point of view) to a tie line coinciding with an operating line. Therefore, the Ponchon-Savarit method calculates the minimum reflux by searching for the tie line whose prolongation intersects the vertical over its corresponding net flow stream (VONFS) composition farthest from the corresponding equilibrium curve (Figure 2). In this way, it is assured that, when the net flow point is set more separated than this, such a coincidence between tie line and operating line will never occur when the column is operated. It is evident that, in the case of ternary systems (i.e., in three-dimensional space) as opposed to binary systems, not all of the tie-line prolongations intercept the VONFS. Nevertheless, in any single-feed column operating at minimum reflux, there must be at least three tie lines whose projection cross the vertical over one of the net flow points, i.e., those corresponding to the distillate, residue, and the tie line leading to the minimum reflux condition, although other tie lines may exist that satisfy such a condition. The suggested procedures base the search for the minimum reflux on this graphical condition. According to the previous concepts, the methods presented in this work for the calculation of minimum reflux are the following. 2.1. Method 1. The first suggested method consists of calculating the specified column at a reflux ratio higher than the minimum. Obviously, this column must involve a finite number of tie lines lying on a given path in the enthalpy-composition diagram, as well as in the projection on the triangular composition diagram, i.e., the basis of the enthalpy-composition diagram. Therefore, none of these calculated tie lines would yield the minimum reflux ratio. However, we can always find a tie line that will provide a reflux ratio closer to the

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Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000

Table 1. Methods and Equations Used for the Development of the Calculation Scheme Shown in Figure 1 (Marcilla et al.16)a equilibrium calculation of a feed stream

searching of the tie line that passes closest to the feed point (in the enthalpy-composition hyperspace) by the simplex method (Himmelblau)19

net flow point composition and enthalpy

δ(i,k) )

∑P(k-1)z (i,k-1) + Dx (i) - ∑F(k-1)z (i,k-1) P

F

∑P(k-1) + D - ∑F(k-1) ∑P(k-1)H (k-1) + Dh + Q + ∑Q (k-1) - ∑Q (k-1) - ∑F(k-1)H (k-1) P

M(k) )

D

D

D

E

A

(2)

F

∑P(k-1) + D - ∑F(k-1)

mass balance line (operating line)

L(j+1,k) y(i,j,k) - δk(i) H(j,k) - M(k) ) ) V(j,k) x(i,j,k) - δk(i) h(j+1,k) - M(k) y(i,j,k) - δk(i) (there will be a term of the type for each component) x(i,j,k) - δk(i)

equilibrium calculations (tie lines)

by NRTL equation (Renon et al.)22 or by approximate methods

mass and energy balances

intersection between the operating lines and the enthalpy surface by solving the corresponding system of equations

criterion for change of the net flow point

x(1,j,k) x(3,j,k) e1 xC(1,k)

(3)

(4)

(5)

xC(3,k) a

The nomenclature used is shown at the end of this paper.

Figure 2. Minimum reflux condition: Tie line coinciding with an operating line, i.e., the tie-line prolongation intersects the VONFS.

minimum than the previous one. This tie line must satisfy two criteria (see Figure 3): Criterion A: It must pass near the projection of the net flow point. Criterion B: It must intersect or cross over the VONFS farthest away from the equilibrium surface. At this point, it is necessary to note that not all of the calculated tie lines intersect a VONFS. The value of the enthalpy at this intersection or crossing-over point for the VONFS corresponding to a

Figure 3. Graphical representation of the proposed method 1. Prolongation and projection of a tie line.

section of the column allows, by eq 9, the minimum reflux ratio of this section to be calculated. This procedure is applied to all sections of the column in order to estimate the minimum reflux ratio (eq 1). Now, the calculations of distillation columns are repeated, in each case using as the reflux ratio the minimum reflux ratio estimated in the previous iteration. This iterative procedure is repeated until convergence.

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3915

Figure 4 shows a possible situation in a conventional column for the distillation of a ternary mixture. The diagrams presented are the projection over the basis of the enthalpy-composition diagram. The trajectory of the distillation path through the saturated liquid and through the saturated vapor is shown in a discontinuous line, while the tie line leading to the minimum reflux prediction is in bold type. Figure 4a shows the projection, on the composition plane, of the main streams involved in a hypothetical distillation column with a single feed partially vaporized operating at a given reflux ratio higher than the minimum, including the operating lines. Figure 4b shows the prolongations of the tie lines of the column and how these prolongations do not intercept the VONFS. The development of this method involves the statement of the corresponding equations to translate the geometrical concepts to the corresponding algebraic form. For the rectifying section, the specific enthalpy of this point of interception M′(1) is related from the enthalpy balance to the minimum reflux by the following equations:

M′(1) )

DhD + QD QD ) hD + D D

(6)

where the value of QD is obtained from an enthalpy balance around the condenser

QD ) (LD + D)(HD - hD) Figure 4. (a) Location of the different streams, tie lines, operating lines, and trajectories of the liquid and vapor saturated in a ternary distillation column. (b) Prolongations of the tie lines of the column.

This method is the one used in the Ponchon-Savarit method, and here it is directly extended and applied to ternary mixtures. It can be stated that only the intersections from the tie lines corresponding to the calculated trays at each iteration are considered. Thus, the minimum reflux obtained for each section may not be the correct one, but is closer than the previous value, thereby directing the convergence. The reason for operating in this way lies in the fact that the path of the distillation depends on the reflux ratio, i.e., on the position of the net flow points. Thus, a pathway must be set in order to start the calculation, and the tie lines on the different paths are used to assist the search. The number of these tie lines will progressively increase, and one of them will be closest to that corresponding to the minimum reflux ratio. Therefore, the suggested method combines the two criteria because neither of them is a sufficient condition, although both are necessary. Using only criterion A, it is possible that one tie line passes closest to the projection of the net flow point or even intersects it, but the crossing point with the vertical is not the farthest from the equilibrium surface. On the other hand, with criterion B, no difference has been considered between the tie lines crossing over the net flow point and those intercepting it because the method begins with few tie lines. Thus, in this case, it is possible that the tie line selected to provide the operation reflux for the next iteration is a crossing one whose distance from the net flow vertical is high. In these cases, the minimum reflux obtained will be far from the true minimum.

(7)

By combining eqs 6 and 7 and using as M′(1) the value of the enthalpy of the interception point, the value of the minimum reflux ratio is obtained.

Rmin )

( ) LD D

)

min

M′(1) - hD -1 HD - hD

(8)

In a similar way, according to Marcilla et al.,18 for any another section k of the column, it can be stated that Rmin(k) ) 1

{

M′(k) [

∑P(k) + D - ∑F(k)] - ∑P(k) H (k) - ∑Q (k) p

E

(HD - hD)

D

∑Q (k) + ∑F(k) H (k) - H F

F

(HD - hD)

]

+

D

(9)

where M′(k) represents the enthalpy of the interception point in each section k. Once the minimum refluxes for all sections of the column are obtained, eq 1 yields the minimum reflux for the column. This procedure is repeated until the reflux considered and that calculated in the next iteration are identical. Evidently, for ternary and higher-order systems, the graphical resolution is very difficult or even impossible, and the problem must be solved analytically. The distance between a tie line corresponding to stage j and the projection of the net flow point ∆(k), in any of the diagrams in Figure 4, can be readily calculated

d(j,k) )

|Aδ(1,k) + Bδ(2,k) + C|

xA2 + B2

(10)

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Table 2. Different Cases Used to Illustrate the Proposed Procedures for the Calculation of the Minimum Reflux Ratioa case

mixture

mole fraction of 1, zF(1)

mole fraction of 3, zF(3)

enthalpy hF (kcal/mol)b

distillate flow rate D (kmol/h)

recovery % of 1

recovery % of 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

a a a a a a a a a a a b b b b a a a a a a a a b b b b b c c c c c c c c

0.60 0.50 0.18 0.70 0.30 0.394 0.10 0.50 0.40 0.40 0.50 0.55 0.30 0.25 0.55 0.60 0.60 0.60 0.60 0.50 0.50 0.50 0.50 0.55 0.55 0.55 0.55 0.40 0.55 0.30 0.60 0.40 0.70 0.65 0.35 0.45

0.394 0.30 0.50 0.12 0.30 0.60 0.30 0.40 0.40 0.40 0.30 0.30 0.50 0.35 0.15 0.394 0.394 0.394 0.394 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.40 0.30 0.65 0.30 0.50 0.20 0.15 0.50 0.40

3244 (sl) 3255 (sl) 3661 (sl) 2971 (sl) 3383 (sl) 3593 (sl) 3480 (sl) 3337 (sl) 3415 (sl) 3415 (sl) 3255 (sl) 1544 (sl) 1497 (sl) 1595 (sl) 1577 (sl) 2000 (ul) 7000 (lv) 11017 (sv) 12000 (ov) 2900 (ul) 7000 (lv) 10846 (sv) 12000 (ov) 1000 (ul) 7000 (lv) 10387 (sv) 11000 (ov) 1785 (sl) 1727 (sl) 1520 (sl) 1653 (sl) 1614 (sl) 1668 (sl) 1825 (sl) 1683 (sl) 1708 (sl)

60.0 69.0 50.5 86.0 69.4 40.0 69.6 59.5 59.5 58.6 69.0 69.0 50.0 65.0 69.0 60.0 60.0 60.0 60.0 69.0 69.0 69.0 69.0 69.0 69.0 69.0 69.0 59.8 68.5 35.5 69.0 50.0 78.5 83.5 50.0 59.5

97.3 97.4 97.3 97.3 97.3 90.0 97.3 97.3 97.3 90.0 97.0 96.9 97.0 96.9 96.9 97.3 97.3 97.3 97.3 97.4 97.4 97.4 97.4 97.4 97.4 97.4 97.4 90.0 93.0 90.0 96.0 97.3 97.3 97.3 97.3 96.0

97.4 98.2 97.4 97.4 97.4 90.0 97.4 97.4 97.4 90.0 97.0 97.6 94.0 97.6 97.6 97.4 97.4 97.4 97.4 98.2 98.2 98.2 98.2 98.2 98.2 98.2 98.2 90.0 91.0 90.0 96.0 97.4 97.4 97.4 97.4 96.0

a Cases 1-11 and 16-23 correspond to mixture a, cases 12-15 and 24-27 correspond to mixture b, and cases 28-36 correspond to mixture c. b lv, liquid-vapor mixture; ov, overheated vapor; sl, saturated liquid; sv, saturated vapor; and ul, undercooled liquid).

where δ(i,k) are the compositions of ∆(k) and A, B, and C are the coefficients of the tie line (j) equation

A ) [y(2,j,k) - x(2,j,k)]

(11)

B ) -[y(1,j,k) - x(i,j,k)]

(12)

C ) x(2,j,k) [y(1,j,k) - x(1,j,k)) - x(1,j,k) (y(2,j,k) x(2,j,k)] (13) The interception point between the vertical over ∆(k) and a tie line can be obtained by solving the system formed by their corresponding straight line equations

M′(i,k) - h(j,k) z(i) - x(i,j,k) ) y(i,j,k) - x(i,j,k) H(j,k) - h(j,k)

(14)

z(i) ) δ(i,k)

(15)

where eq 14 is the equation of the straight line defined by the tie line passing through the equilibrium points [x(i,j,k), h(j,k)] and [y(i,j,k), H(j,k)]. Obviously, there exists one composition-dependent term for each component. Equation 15 is the equation of the vertical over the net flow point where δ(i,k) represents the composition of this net flow point. [z(i), M′(i,k)] represents the coordinates of the intersection point between both straight lines. When this system has a solution, the enthalpy of interception point M′(k) can been obtained. In other cases, when the tie line considered does not intercept but crosses over the vertical at ∆(k), the enthalpy of the crossing point is obtained from the point

at which the distance between the two straight lines is minimum. This point can be calculated by the following procedure: First, it is necessary to determine the equation of the plane π that contains the vertical straight line and that is also perpendicular to the tie line. Then, the intersection of this plane with the tie line determines the crossing point searched. This intersection point is calculated by solving the corresponding system of equations. When more than one tie line intercepts or crosses over near the vertical over one net flow point, those leading to the greater reflux ratio will be selected, because each tie line crossing over the VONFS predicts a minimum reflux situation, i.e., a different pinch zone, and the values selected will be those that satisfy eq 1. 2.2. Method 2. In this second method, no distillation columns are calculated, but the distance between any straight line defined by a tie line and the projection of the net flow point of the corresponding section in the projection of the enthalpy-composition diagram is minimized. At the minimum reflux ratio condition, this distance must obviously be zero. The simplex flexible method19 has been applied to carry out these calculations, and obviously, the objective function is the distance previously mentioned (eq 10), with the composition of the saturated liquid x(i) as a parameter. For each saturated liquid composition, the calculation of the vapor in equilibrium can be made by any thermodynamic model, such as the NRTL equation, or by the approximate methods suggested by Marcilla et al.16 The

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3917 Table 3. Minimum Reflux Ratio Calculated by Several Methods: Shortcut Methods (I, Underwood Equation; II, Colburn Method; III, Underwood Method; and IV, Gilliland Method), Proposed Procedures [V, Suggested Method 1 Using the Two Criteria Together (A + B) and VI, Method 2] and Rigorous Method (VII)a (LD/D)min by several methods shortcut

suggested

rigorous

case

I

II

III

IV

V (A + B)

VI

VII

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

1.102 [-5.5] 0.887 [-4.4] 1.358 [-5.0] 0.639 [-0.9] 0.882 [-3.7] 1.174 [-5.5] 0.971 [-9.4] 1.107 [-2.8] 1.105 [-1.2] 0.763 [-6.0] 0.866 [-4.2] 0.756 [14] 0.676 [33] 0.894 [65] 0.845 [38] 1.095 [-0.1] 1.120 [-23] 1.130 [-42] 1.130 [-46] 0.884 [-4.9] 0.932 [-20] 0.977 [-36] 0.987 [-41] 0.777 [6.4] 0.712 [-4.9] 0.570 [-37] 0.546 [-43] 0.724 [40] 0.664 [19] 0.870 [9.4] 0.901 [25] 0.866 [17] 0.762 [8.1] 0.848 [16] 0.975 [33] 0.899 [23]

1.113 [-4.5] 1.108 [20] 1.447 [1.3] 0.796 [23] 0.969 [5.8] 1.185 [-4.6] 0.926 [-14] 1.136 [-0.3] 1.164 [4.1] 0.877 [8.0] 0.966 [6.9] 0.642 [-2.9] 0.541 [6.3] 0.537 [-0.7] 0.592 [-3.6] 1.088 [-0.7] 1.127 [-23] 1.134 [-41] 1.134 [-45] 1.023 [10] 1.081 [-7.1] 1.048 [-32] 1.037 [-38] 0.746 [2.2] 0.691 [-7.7] 0.654 [-28] 0.613 [-36] 0.579 [12] 0.564 [0.7] 0.826 [3.9] 0.811 [12] 0.780 [5.5] 0.686 [-2.7] 0.710 [-2.6] 0.829 [13] 0.764 [4.5]

1.102 [-5.5] 0.859 [-7.4] 1.270 [-11] 0.619 [-4.0] 0.894 [-2.4] 1.181 [-4.9] 1.038 [-3.2] 1.051 [-7.7] 1.025 [-8.3] 0.789 [-2.8] 0.842 [-6.9] 0.648 [-2.0] 0.472 [-7.3] 0.592 [9.4] 0.668 [8.8] 1.095 [-0.1] 1.121 [-23] 1.132 [-42] 1.132 [-46] 0.856 [-8.0] 0.904 [-22] 0.948 [-38] 0.958 [-43] 0.667 [-8.6] 0.442 [-41] 0.288 [-68] 0.266 [-72] 0.593 [15] 0.579 [3.4] 0.786 [-1.1] 0.756 [4.7] 0.762 [3.1] 0.715 [1.4] 0.781 [7.1] 0.814 [11] 0.784 [7.3]

1.102 [-5.5] 0.856 [-7.8] 1.308 [-8.5] 0.614 [-4.8] 0.860 [-6.1] 1.179 [-5.1] 0.998 [-6.9] 1.055 [-7.4] 1.028 [-8.1] 0.753 [-7.3] 0.841 [-7.0] 0.670 [1.4] 0.560 [10] 0.677 [25] 0.684 [11] 1.096 [0.0] 1.121 [-23] 1.132 [-42] 1.132 [-46] 0.856 [-8.0] 0.904 [-22] 0.948 [-38] 0.958 [-43] 0.690 [-5.5] 0.459 [-39] 0.301 [-67] 0.278 [-71] 0.618 [20] 0.603 [7.7] 0.799 [0.5] 0.811 [12.3] 0.789 [6.8] 0.716 [1.6] 0.787 [8.0] 0.856 [16.9] 0.802 [9.7]

1.167 (4) [0.1] 0.905 (5) [-2.5] 1.430 (4) [0.1] 0.645 (5) [0.0] 0.916 (5) [0.0] 1.242 (5) [0.0] 1.045 (4) [-2.5] 1.136 (3) [-0.3] 1.118 (4) [0.0] 0.813 (5) [0.1] 0.904 (5) [0.0] 0.661 (5) [0.0] 0.510 (7) [0.2] 0.542 (8) [0.2] 0.597 (11) [-2.8] 1.099 (5) [0.3] 1.460 (6) [0.0] 1.920 (2) [-0.8] 2.078 (7) [0.0] 0.930 (4) [0.0] 1.160 (4) [-0.3] 1.502 (2) [-2.0] 1.672 (5) [-0.1] 0.730 (4) [0.0] 0.745 (3) [-0.5] 0.904 (6) [0.0] 0.957 (3) [-0.2] 0.516 (4) [0.0] 0.560 (5) [0.0] 0.796 (4) [0.1] 0.723 (4) [0.1] 0.740 (3) [0.1] 0.705 (13) [0.0] 0.729 (14) [0.0] 0.733 (4) [0.1] 0.731 (3) [0.0]

1.171 [0.4] 0.922 [-0.6] 1.430 [0.1] 0.639 [-0.9] 0.913 [-0.3] 1.242 [0.0] 1.062 [-0.9] 1.139 [0.0] 1.114 [-0.4] 0.812 [0.0] 0.904 [0.0] 0.661 [0.0] 0.509 [0.0] 0.541 [0.0] 0.614 [0.0] 1.099 [0.3] 1.464 [0.3] 1.935 [0.0] 2.077 [0.0] 0.913 [-1.8] 1.152 [-1.0] 1.526 [-0.5] 1.667 [-0.4] 0.646 [-11.5] 0.671 [-10.4] 0.871 [-3.7] 0.947 [-1.3] 0.516 [0.0] 0.560 [0.0] 0.795 [0.0] 0.723 [0.1] 0.740 [0.1] 0.705 [0.0] 0.730 [0.1] 0.732 [0.0] 0.731 [0.0]

1.166 (22) 0.928 (17) 1.429 (20) 0.645 (21) 0.916 (20) 1.242 (27) 1.072 (26) 1.139 (22) 1.118 (25) 0.812 (25) 0.904 (23) 0.661 (23) 0.509 (22) 0.541 (25) 0.614 (25) 1.096 (20) 1.460 (25) 1.935 (18) 2.078 (19) 0.930 (24) 1.164 (24) 1.533 (24) 1.674 (18) 0.730 (26) 0.749 (16) 0.904 (22) 0.959 (13) 0.516 (24) 0.560 (25) 0.795 (15) 0.722 (25) 0.739 (17) 0.705 (24) 0.729 (18) 0.732 (24) 0.731 (25)

a Cases 1-36 are those of Table 2. The numbers in parentheses show the number of iterations required for convergence, and the numbers in square brackets correspond to the deviation of minimum reflux ratio for each case from the real value obtained from strict method. (A value lower that 0 indicates a minimum reflux lower than the true value.)

procedure must be applied in each section of the column, selecting the maximum of all of them (eq 1). It is important to note that the proposed methods are used in the same way with ideal and nonideal mixtures. The difference will only be in the calculation time as the equilibrium calculations in the nonideal case are slower than those in the ideal case. The suggested methods calculate the minimum reflux of distillation columns working with a given feed and a specified percentage of separation between key components. It is evident that the problem can only be solved if the specifications of the column are compatible with the possible presence of different boundary zones in the system.20,21 In other cases, the algorithm will not converge in the first trial, thus advising about that inconsistency. Results and Discussion The methods suggested have been applied to three different ternary mixtures: (a) benzene (1)-cyclohexane (2)-toluene (3), (b) methanol (1)-acetone (2)-water (3), and (c) methanol (1)-ethyl acetate (2)-water (3). Each of the mixtures was in a distillation column having a single feed stream, and the composition and thermal state of the feed mixtures were varied.

The specifications for the examples selected to illustrate the proposed methods are provided in Table 2, with the recovery percentages of component 1 in the top product and component 3 in the bottom product specified. For all examples, the feed flow rate is 100 mol kg/ h. The distillate flow rate and the reflux ratio are the other variables needed to solve the problem. The reflux ratio is the variable modified in each iteration to reach the minimum, and the values for the other variables are shown in Table 2. Cases 1-11 and 16-23 correspond to the first mixture (a), cases 12-15 and 2427 correspond to the second mixture (b), and cases 2836 correspond to the third mixture (c). The pressure in the top of the column is 1 atm, and the pressure drop considered is 0 in all cases. The minimum reflux values obtained by the proposed procedures were compared to those obtained by the application of a rigorous method, consisting of the extrapolation of the reflux to an infinite number of stages, and to those obtained by the application of the Underwood equation and the shortcut methods of Colburn, Underwood, and Gilliland. Table 3 shows the results obtained for minimum reflux in the 36 cases studied using four conventional shortcut methods (I-IV), those obtained by applying the procedures proposed in the present work (V and VI),

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and also those obtained with the rigorous method (VIII), used for the comparison of all the applied methods. The numbers in brackets in Table 3 show the number of iterations required for convergence. Table 3 also shows the percentage deviation of the minimum reflux ratio calculated (square brackets) for each case and by each method with respect to the value obtained by the rigorous method. These deviations are coincident with those normally reported in the works cited in the reference list for the conventional shortcut methods. As an example, Sawistowski and Smith7 show that the Underwood equation considering the pinch point composition to be the feed composition differs from the real value by 243%, the approximate method of Colburn differs by 5.4%, the Underwood method differs by 13.5%, and the Gilliland method differs by 44.6%. The Underwood equation is the approach yielding the poorest results, because in multicomponent mixtures, the pinch points are usually far from the conditions at equilibrium with the feed. The Colburn method occasionally provides good results, but not always; consequently, it must be considered with care, especially for nonideal systems. The Underwood method yields good results in most of the cases analyzed. The Gilliland method has the advantage of using a simple equation, but in some instances yields poor results. The first method suggested (V in Table 3), combining the two criteria A and B, is the one yielding the best results in all cases. Although the reduction in the number of iterations is lower in the nonideal casesthan in the ideal cases, the proposed method involves a remarkably lower number of iterations, indicating that these criteria are a good way of orienting the iterations. The suggested method 2 (VI in Table 3), as well as the method 1, yields very good results in all cases, but involves a lower calculation time than the first, both in the ideal and nonideal cases. 3. Conclusions An analysis and extension of the geometrical concepts of Ponchon and Savarit has been developed and applied for the calculation of the minimum reflux ratio in ternary systems. The two methods, which have been suggested to orient the iteration to calculate the minimum reflux ratio in ternary distillation columns, provide excellent results very close to those of the rigorous solution. The first method suggested, which combines the two criteria, is the one leading to better results, as it uses the two necessary conditions to be fulfilled by a tie line leading to the minimum reflux ratio. Both methods suggested are very simple, can be readily implemented on a computer, and are reliable for calculating the minimum reflux ratio for both ideal and nonideal mixtures. Symbols A, B, C: coefficients of the equation of a straight line. component 1: light key component. component 3: heavy key component. D: distillate flow or point representative of the distillate in the diagrams. d(j,k): distance between the projection of the tie line corresponding to stage j and the projection of ∆(k). F: feed flow or point representative of the feed in the diagrams.

H: enthalpy of a vapor stream. H: enthalpy of a point of enthalpy-composition diagram, in any thermal condition. H(j,k): enthalpy of a vapor stream leaving tray j in section k. HD: enthalpy of the vapor leaving the top of the column. h: enthalpy of a liquid stream. h(j,k): enthalpy of a liquid stream leaving tray j in section k. hD: enthalpy of the distillate (liquid). HF(k): enthalpy of the feed between sections k and k+1. HP(k): enthalpy of the side product between sections k and k+1. i: subindex referring to a component. j: subindex referring to a tray. k: subindex for a section of a rectifier. LD: reflux flow. LF(k): saturated vapor in equilibrium with the feed F(k). L(j,k): liquid stream leaving a tray j in the section k. (LD/D): reflux ratio. (LD/D)min: minimum reflux ratio. lv: liquid-vapor mixture M(k): enthalpy of the net flow point. M′(k): enthalpy of the net flow point at minimum reflux condition. ov: overheated vapor QD: heat duty in the condenser. R: residue flow or point representative of the residue in the diagrams. Rmin: minimum reflux ratio Rmin(k): minimum reflux ratio calculated for section k. sl: saturated liquid sv: saturated vapor ul: under cooled liquid VF(k-1): saturated vapor in equilibrium with the feed F(k-1). V(j,k): vapor stream leaving a tray j in the section k. VONFS: vertical over the corresponding net flow stream composition. x(i,j,k): composition of component i in the liquid stream leaving stage j of section k. x′(i,j,k): projection of x(i,j,k) point, on the triangular composition diagram, i.e., the basis of the enthalpycomposition diagram. xC(i,k): composition of component i in the saturated liquid in the intersection between saturated liquid surface and the straight-line ∆(k)-∆C. xD(i): composition of component i in the distillate. xR(i): composition of component i in the residue. y(i,j,k): composition of component i in the vapor stream leaving stage j of section k. y′(i,j,k): projection of y(i,j,k) point on the triangular composition diagram. z(i): composition of a point of enthalpy-composition diagram, in any thermal condition. zF(k): composition of the feed between sections k and k+1. zP(k): composition of the side product between sections k and k+1. ∆(k): net flow point corresponding to section k. ∆′(k): projection of ∆(k) on the triangular composition diagram. ∆C: net flow point corresponding to the feed zone. δ(i,k): composition of the net flow point of section k. ΣF(k-1): sum of all feeds entering above section k in a complex column. ΣP(k-1): sum of all products leaving the column above section k.

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3919 ΣQE(k-1): complex ΣQF(k-1): complex

sum of all heat removed above section k in a column. sum of all heat inputs above section k in a column.

Literature Cited (1) King, C. J. Separation Processes; McGraw-Hill: New York, 1980. (2) Seader, J. D.; Henley, E. J. Separation Process Principles; John Wiley & Sons, Inc.: New York, 1998. (3) Hengstebeck, R. J. Distillation; Reinhold: New York, 1961. (4) Kister, H. Z. Distillation Design; McGraw-Hill: New York, 1992. (5) Julka, V.; Doherty M. F. Geometric Behavior and Minimum Flows for Nonideal Multicomponent Distillation. Chem. Eng. Sci. 1990, 45, 1801. (6) Sawistowski, H.; Smith, W. Me´ todos de Ca´ lculo en los Procesos de Transferencia de Materia; Alhambra: Madrid, 1967. (7) Van Winkle, M. Distillation; McGraw-Hill Chemical Engineering Series; McGraw-Hill: New York, 1967. (8) Underwood, A. J. V. The Theory and Practice of Testing Stills. Trans. Am. Ins. Chem. Eng. 1932, 10, 102. (9) Colburn, A. P. The Calculation of Minimum Reflux Ratio in the Distillation of Multicomponent Mixtures. Trans. Am. Inst. Chem. Eng. 1941, 37, 805. (10) Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Calculation of Minimum Reflux Ratio. J. Inst. Pet. 1946, 32, 614. (11) Gilliland, E. R. Estimate of the Number of Theoretical Plates as a Function of the Reflux Ratio. Ind. Eng. Chem. 1940, 32, 1220. (12) Erbar, R. C.; Maddox R. N. Minimum Reflux Rate for Multicomponent Distillation Systems by Rigorous Plate Calculations. Can. J. Chem. Eng. 1962, 40, 25.

(13) Bachelor, J. B. How to Calculate Minimum Reflux. Pet. Refiner 1957, 36 (6), 161. (14) Chien, H. Y.A Rigorous Method for Calculating Minimum Reflux Rates in Distillation. AIChE J. 1978, 24, 4 (4), 606-613. (15) Holland, C. D. Multicomponent Distillation; Prentice Hall: New York, 1963. (16) Marcilla, A.; Go´mez, A.; Reyes, J. A. New Method for Designing Distillation Columns of Multicomponent Mixtures. Lat. Am. Appl. Res. 1997, 27 (1-2), 51-60. (17) Reyes Labarta, J. A. Disen˜ o de Columnas de Rectificacio´ n y Extraccio´ n Multicomponente. Ca´ lculo del Reflujo Mı´nimo. Ph.D. Dissertation, University of Alicante, Alicante, Spain, 1998. (18) Marcilla, A.; Ruiz, F.; Go´mez, A. Graphically Found Trays and Minimum Reflux for Complex Binary Distillation for Real Systems. Lat. Am. Appl. Res. 1995, 25, 87-96. (19) Himmelblau, D. M. Process Analysis Statistical Methods; John Willey & Sons: New York, 1968. (20) Foucher, E. R.; Doherty, M. F.; Malone M. F. Automatic Screening of Entrainers for Homogeneous Azeotropic Distillation. Ind. Eng. Chem. Res. 1991, 30, 760-773. (21) Safrit B. T.; Westerberg A. W. Algorithm for Generating the Distillation Regions for Azeotropic Multicomponent Mixtures. Ind. Eng. Chem. Res. 1997, 36, 1827-1840. (22) Renon, H.; Asselineau, L.; Cohen, C.; Raimbault, C. Calcul sur ordinateur des e´ quilibres liquide vapeur et liquide-liquide; Collection “Science et Technique du Pe´trole”; Institut Franc¸ ais du Pe´trole: Rueil-Malmaison, France, 1971; Number 17.

Received for review September 17, 1999 Revised manuscript received March 13, 2000 Accepted July 22, 2000 IE9907021