Minimum Reflux for Batch Distillations of Ideal and Nonideal Mixtures

2732

Ind. Eng. Chem. Res. 1999, 38, 2732-2746

Minimum Reflux for Batch Distillations of Ideal and Nonideal Mixtures at Constant Reflux Jose´ Espinosa and Enrique Salomone* INGARsInstituto de Desarrollo y Disen˜ o, Avellaneda 3657, 3000 Santa Fe, Argentina

This paper presents a short-cut procedure for estimating the instantaneous separation performance in batch columns having an infinite number of stages that it is applicable to nonideal mixtures with any number of components. The model allows the estimation of the instantaneous top composition without resorting to stage-by-stage computation and therefore can be used both as a method for rapid simulation of the batch operation and as a procedure for calculating the minimum-energy demand. A key aspect of the method is the determination of the different regimes that can be found during the operation of batch columns. Each one of the different regimes is controlled by a pinch that determines the geometry of the internal profiles. For systems with constant relative volatilities, the presented approach is compared with the use of the Underwood equation to determine the instantaneous separation column performance. We show that both methods are equivalent. This analysis also gives an interpretation of the application of the Underwood equation in the context of batch rectification. Finally, we analyze the application of the model to highly nonideal mixtures with azeotropes and distillation boundaries and briefly discuss the modifications that have to be introduced to extend the method. Introduction

Table 1. Procedure for the Computation of Rmin and Subprocedure A

The minimum reflux to perform a separation task by batch distillation is, by definition, the reflux required by a hypothetical batch distillation column having an infinite number of stages. Therefore, as is described in a previous work (Salomone et al.1), the computation of the batch minimum reflux requires the prediction of the instantaneous separation performance of a column with infinite stages. For a given initial mixture and once the separation task is specified in terms of two fractional recoveries, the procedure uses simplified dynamic simulations of the operation for different values of the reflux ratio until convergence of both fractional recoveries to the specified values. The simplified dynamic simulation consists of the integration of the material balance for each component assuming no holdups in the column and condenser. Therefore, a simple set of differential equations relating the instantaneous still and distillate compositions were derived. This model has a level of detail suitable for the preliminary design of batch distillation operations, and it is completely general provided there is a suitable method for predicting the instantaneous separation performance of the column. For ideal mixtures, this prediction can be done by using the Underwood equations, and the procedure is summarized in Table 1. A more detailed analysis is presented in the mentioned paper. For nonideal mixtures, the Underwood equations are no longer valid and a method of similar complexity needs to be conceived. Offers et al.2 present a method for computing the minimum instantaneous reflux ratio for a batch rectification column operating at constant distillate composition. On the basis of geometric char* To whom correspondence should be addressed. Tel.: +54 (342) 453-4451. Fax: +54 (342) 455-3439. E-mail: salomone@ arcride.edu.ar.

Step 2

Procedure for the Computation of Rmin define the separation task xi,f Ri,h flc* fhc* guess a value for R

Step 3

integrate

Step 1

dni xi,t ) dnlc xlc,t

tracking flc )

Step 4

up to flc ) flc*

nlc0 - nlc nlc0

C with xi,b ) ni/∑i)1 ni xi,t calculated in subprocedure A compute the heavy key fractional recovery

nhc0 - nhc

fhc )

nhc0

Step 5 Step 6

if fhc 〈〉 fhc*, guess a new value for R and go to step 3 Rmin ) R

Step 1 Step 2

Subprocedure A given xi,b Ri,h R find the C - 1 roots θm of C

Ri,hxi,b

∑R i)C

Step 3

- θm

)0

obtain top compositions xi,t solving the linear system:

Ax ) b Step 4 Step 5

i,h

|

Rj i ) 1, ..., C - 1 aC,j ) 1 Rj - θi bi ) 1 + R i ) 1, ..., C - 1 bC ) 1

ai,j )

if xC,t < 0, then set xC,t ) 0, let C ) C - 1, and go to step 3 return xi,t

acteristics of the compositions profile, the method relates the top and bottom compositions without a stageby-stage calculation. In this paper we develop a simplified method for estimating the separation performance of nonideal mixtures in a batch rectification column with infinite

10.1021/ie990022y CCC: $18.00 © 1999 American Chemical Society Published on Web 05/26/1999

Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999 2733

Figure 1. Residue curve maps for the mixture methanolethanol-2-propanol.

stages. The method extends the ideas of Offers et al.2 to variable distillate composition at a given reflux ratio and therefore is suitable for column simulation. To introduce the concepts behind the method, we will first focus our attention on the geometry of the instantaneous internal profiles in a rectifying column with infinite stages operating at different reflux ratios while inspecting the all-possible distillate compositions from a given instantaneous still composition. Then, we show that the same patterns are found at different times during a batch rectification, and on the basis of these results, we develop an algorithm for ideal ternary and quaternary mixtures. Next, and for the sake of comparison, we show that the Underwood equation applied to the rectification section indirectly reproduces the geometry of the internal profiles and, hence, for systems with constant relative volatilities, the two approaches are equivalent. Afterward, the generalization to nonideal multicomponent systems is developed, and special attention will be given to ternary azeotropic mixtures. For this case, unstable boundaries for the instantaneous distillate composition and stable boundaries for the still compositions are considered. Finally, we explain the use of the new method for the prediction of the minimum reflux ratio needed for a given separation task in batch rectification at constant reflux. Instantaneous Product Composition Regions and Composition Profile Patterns In this section we explore the all-possible distillate compositions that can be achieved from a given still composition in a column with an infinite number of stages by varying the reflux ratio. At different reflux ratios, the internal composition profile is controlled by different pinch locations defining characteristic patterns of shape. For the following analysis, a negligible holdup in the column and condenser is assumed. Consider the ternary mixture methanol, ethanol, and 2-propanol that behaves ideally. Figure 1 shows the

Figure 2. Instantaneous product composition region and controlling pinch points for a given boiler composition (ternary system).

residue curve maps at atmospheric pressure for this system. All of the residue curves originate from the lightest species, methanol (unstable node), and terminate at the heaviest, 2-propanol (stable node). Ethanol acts as a saddle node because any trajectory passing close to it will first approach the node and then will move away after the difference between vapor and liquid states have reached a minimum. Because the thermodynamic behavior of the mixture is ideal, there are no boundaries for the still compositions. Because these curves represent the simple batch distillation for different initial mixture compositions, a great amount of information is contained in these curves and its corresponding distillate compositions. Let us consider the given still composition xB in Figure 2; y*xB represents its vapor in equilibrium and must be located on a tangent to the residue curve that passes through the still composition. From xB, different distillate compositions can be achieved for different values of the reflux ratio. Under total reflux operation, it can be expected that the internal liquid profile resembles the residue curve passing through the still composition. The top product will be the more volatile species methanol, and a pinch will be placed at the top of the column. For small values of the reflux above zero (see Figure 2), the distillates are aligned with the equilibrium vector y*xB-xB starting from y*xB when the reflux equals zero up to the point xD* where the composition of the heaviest component becomes zero. We will refer to R* as the reflux ratio corresponding to this condition. For all of these separations a pinch is located at the end of the column immediately above the boiler. The pinch composition coincides with the composition of the mixture in the still. Offers et al.2 called such instantaneous separations “preferred separations” as an extension of the concept introduced by Stichlmair3 for continuous distillation. As shown by Du¨ssel,4 for a reflux ratio equal to R* a binary saddle pinch xpII between the lightest components methanol and ethanol appears. Because of the saddle, the profile becomes sharp and the internal profile between the instantaneous still composition and

2734 Ind. Eng. Chem. Res., Vol. 38, No. 7, 1999

Figure 3. Instantaneous product composition region for a given boiler composition (quaternary system).

the saddle lies on a straight line containing both the saddle and the still compositions. For values of the reflux greater than R*, the distillate compositions are located in the binary edge corresponding to the more volatile components between xD* and xDmax. There is no longer a pinch at the end of the column, but the binary saddle pinch remains with its composition xpII (invariant). The internal profiles between the instantaneous still composition and the invariant saddle lie on a straight line. At xDmax the composition of the intermediate component becomes zero; we refer to this reflux condition as R**. Finally, for values of the reflux greater than R**, the profiles resemble the total reflux separation with a pinch at the column top and, hence, there is not a saddle pinch. Thus, the instantaneous product composition region for an infinite stages batch column under both finite and total reflux ratios is defined by the products on the “preferred separation line” (beginning at the vapor in equilibrium with the still composition until the intersection with the edge corresponding to the lower boiler constituents) and those on the binary edge between the lightest species from xD* to xDmax. A global mass balance around the top of the column and the pinch point (the lever arm rule) relates the limiting reflux ratios R* and R** to the pinch compositions as follows:

R* )

R** )

x*D - y*xB y*xB - xB xDmax - ypII ypII - xpII

(1)

(2)

For quaternary mixtures the analysis is quite similar. Let us consider the ideal quaternary mixture methanol, ethanol, 2-propanol, and 1-propanol. Figure 3 shows the results for a given still composition xB. For values of the reflux ratio between 0 and R*, the distillates containing all of the components present in the boiler are located on the corresponding line at “preferred separation”. An invariant quaternary pinch located at the column end having the composition of the still controls the separation. At R* the heaviest component becomes zero in the distillate composition xD*.

When the value of the reflux ratio becomes greater than R*, an invariant ternary saddle pinch containing the more volatile species will appear. The corresponding distillates are restricted to the straight line between xD* and xD**, and no heavy component is present in the distillates. The slope of this line is given by the equilibrium vector at the invariant ternary saddle pinch xpIII. At xD**, the second heaviest component becomes zero and the corresponding reflux ratio is R**. For reflux ratios above R** and until a value of R*** is achieved, the distillate compositions lie on the line between xD** and xD*** and an invariant binary saddle pinch xpII controls the separation. Only methanol and ethanol are present in the distillate. Finally, a distillate containing only methanol can be expected for values of the reflux greater than R***. As for the case of ternary mixtures, the values of the extreme reflux ratios can be calculated through the lever arm rule and the corresponding pinch compositions. From the analysis above, it is clear that the different invariant pinch points are key in the determination of the all-possible distillate compositions for a given instantaneous still composition. They control not only the range of distillate compositions that can be achieved but also the geometry of the internal profiles. In line with this statement, for values of the reflux ratio between R* and R** the profiles will become sharp and will lie on the straight line between the still composition and the invariant ternary pinch when the profiles leave the face of the tetrahedron at the ternary pinch point. When the invariant binary pinch point controls the separation, the distillate profiles leave the edge between methanol and ethanol at the pinch point and approach the still composition on a plane containing the binary pinch, the ternary pinch, and the still composition. Operation Regimes of an Infinite Batch Column at Constant Reflux In the preceding section we discussed the different possible profile patterns and product compositions as the reflux ratio is changed for the same still composition. In a column operating at constant reflux, the still composition is varying all of the time and therefore, during a typical run, different profile patterns may occur for the same reflux ratio. To illustrate the above ideas, let us imagine the operation of an infinite batch column under a finite reflux operation policy for a ternary mixture. At the startup the column will operate at total reflux and, hence, the top composition will be pure volatile component with a pinch at the top of the column. Then, assuming that a high reflux ratio (greater than the R** corresponding to the initial feed) is established, the profiles will behave as in the case of total reflux. The column will be producing a pure distillate during a certain period of time. Because of the depletion of the light component from the still, the column will not be able to sustain a pure top composition and the top will become a binary mixture containing the two lightest components. The situation now is such that the operating reflux ratio is between the R* and R** corresponding to the instantaneous still composition. Therefore, the separation proceeds with a profile having a saddle pinch between the lightest species.


Figure 5. Geometry of the profiles corresponding to total reflux operation (quaternary system). Figure 4. Different regimes of the batch operation for a ternary mixture.

Afterward, the column will operate with a pinch at the end of the column. The pinch composition will coincide with the instantaneous still composition, and the instantaneous value of the operating reflux ratio is between 0 and R*. It is also possible to find the mentioned operating regions for the batch operation of a column with a finite number of stages and holdup. Figure 4 shows these regions for the ideal mixture considered. The simulations were performed with Hysys from Hyprotech Ltd. (1997). The identification of the present operating regime and therefore the knowledge of the controlling pinch point are the bases of the simplified method for estimating the separation performance of a batch rectification column with infinite stages. A key step in this direction is the determination of the instantaneous limiting values of the reflux ratio, R* and R**, together with the invariant binary saddle pinch, xpII. R* can be easily calculated using the lever arm rule, eq 1, computing the composition of the vapor in equilibrium y*xB and xD* as the point aligned with the equilibrium vector y*xB-xB where the composition of the high boiler becomes zero. On the other hand, the determination of R** requires the knowledge of the location of the invariant saddle pinch. As was established by Offers et al.,2 the saddle on the binary edge of the lightest components can be located with the only knowledge of the still composition by imposing the following conditions (Offers et al.2): (i) The liquid profile in the rectifying section between the still composition and the saddle lies on a straight line containing both the saddle and still composition. (ii) The liquid and vapor composition profiles of the rectifying section are parallel lines between the still composition and the saddle. The previous conditions can be mathematically expressed as a characteristic line passing through the still composition and pointing directly to the saddle pinch. This instantaneous characteristic direction of the in-

ternal concentration profile can be calculated as

dx1 C4 - C1 )( dx2 2C3 C1 ) C2 ) C3 ) C4 )

( ( ( (

x(

)

C4 - C1 2C3

) ) ) )

∆y*1 ∆x1

x2

∆y*1 ∆x2

x1

∆y*2 ∆x1

x2

∆y*2 ∆x2

x1

2

+

C2 C3

(3)

where 1 and 2 represent the lightest and intermediate components in the mixture. To determine the saddle pinch location from the two solutions, that of positive slope must be selected. This implies that the two lightest components increase their composition while the composition of the heavy species diminishes along the characteristic line beginning from the instantaneous still composition. For systems with constant relative volatilities, an analytical expression can be deduced, as is shown in Stichlmair et al.5 The determination of R** by means of the lever arm rule, eq 2, is straightforward once the vapor in equilibrium with the saddle pinch composition is obtained. Following a similar analysis for the quaternary ideal mixture, Figures 5-7 show three of the five different regimes that could be present in a batch distillation having an infinite number of stages. As for the ternary case, a finite number of stages and holdup was assumed in the examples. Figure 6 shows the regime under control of an invariant binary saddle pinch. In this case, the liquid and vapor profiles inside the tetrahedron move on parallel planes that intersect the binary edge between the more volatile components at the controlling pinch. On the other hand, Figure 7 shows the regime under control of an invariant ternary saddle pinch. Both


in the first condition, the following quadratic equation can be obtained:

( )]( ) [

[

dx3 dx1

C4 + C6

dx1 dx2

2

+

( )]( )

C5 - C1 - C3

dx3 dx1

dx1 - C2 ) 0 (5) dx2

with the following parameters:

C1 ) C3 ) C5 )

| | |

∂y*1 ∂x1 ∂y*1 ∂x3 ∂y*2 ∂x2

C2 )

x2,x3

C4 )

x1,x2

C6 )

x1,x3

| | |

∂y*1 ∂x2

x1,x3

∂y*2 ∂x1

x2,x3

∂y*2 ∂x3

x1,x2

(6)

The second condition is Figure 6. Geometry of the profiles corresponding to regimes controlled by a binary pinch (quaternary system).

dx3 dy*3 ) dx1 dy*1

(7)

dy*1 )

∂y*1 ∂y*1 ∂y*1 dx1 + dx2 + dx3 ∂x1 ∂x2 ∂x3

dy*3 )

∂y*3 ∂y*3 ∂y*3 dx1 + dx2 + dx3 ∂x1 ∂x2 ∂x3

For this case, the quadratic equation is

[

( )]( ) [ dx2 dx3

C3 + C2

dx3 dx1

2

+

( )]( )

C1 - C9 - C8

dx2 dx3

dx3 - C7 ) 0 (8) dx1

with the three extra parameters defined as

C7 )

|

∂y*3 ∂x1

C8 ) x2,x3

|

∂y*3 ∂x2

x1,x3

C9 )

|

∂y*3 ∂x3

(9)

x1,x2

Figure 7. Geometry of the profiles corresponding to regimes controlled by a ternary pinch (quaternary system).

By considering the following relationship among the differentials:

liquid and vapor profiles departing from the pinch are parallel lines inside the tetrahedron. In this case, the condition of parallelism for the liquid and vapor profiles can be expressed as two mathematical statements. The first condition is

dx2 dx1/dx3 ) dx3 dx1/dx2

dx1 dy*1 ) dx2 dy*2 dy*1 )

∂y*1 ∂y*1 ∂y*1 dx1 + dx2 + dx3 ∂x1 ∂x2 ∂x3

dy*2 )

∂y*2 ∂y*2 ∂y*2 dx1 + dx2 + dx3 ∂x1 ∂x2 ∂x3

(4)

By replacing the differential of the vapor compositions in terms of the differentials of the liquid mole fractions

(10)

Equations 5, 8, and 10 can be solved to compute dx1/ dx2, dx3/dx1, and dx2/dx3, giving thus the characteristic direction. For systems with constant relative volatilities, it is possible to obtain analytical expressions for the parameters in the preceding equations. These expressions are summarized in Appendix A. Estimating the Separation Performance of an Infinite Batch Column The concepts discussed in the preceding sections can be used to develop an algorithm for the prediction of the instantaneous top composition of a batch column having infinite stages, given the still composition and the operating reflux ratio. For ternary mixtures the algorithm is the following:


Step I. Given xB and r. Step II. Calculate y* (vapor in equilibrium with xB) and R* through the lever arm rule, eq 1, with xD* being a distillate without heavy component located at the intersection between the binary edge corresponding to the more volatile components and the line that passes through the equilibrium vector y*-xB. Step III. If r e R*, there exists a pinch at the end of the column and the instantaneous distillate mole fraction is located on the line that passes through the equilibrium vector. Use the lever arm rule to calculate the distillate composition:

xi,D ) y*i + r(y*i - xi,B)

(11)

Step IV. If r > R*, a saddle pinch between the lightest species may control the separation. Follow the next steps to calculate the instantaneous distillate composition. IV.1. Find the direction for the concentration profile as the slope whose value is greater than zero of the two given by eq 3. IV.2. Calculate the equation of the line that passes through the instantaneous boiler composition with the previously selected slope. IV.3. Calculate the binary saddle pinch composition xpII at the intersection of the line that passes through the boiler composition with the characteristic direction and the binary edge corresponding to the more volatile components. IV.4. Calculate the vapor in equilibrium with the saddle composition, ypII. IV.5. Use the lever arm rule to calculate xD:

xi,D ) yi,p + r(yi,p - xi,p)

(12)

Step V. If x2,D < 0, then set x2,D ) 0 and x1,D ) 1, respectively. For this case, a pinch at the top of the column controls the separation. (This condition implies that r > R**.) For quaternary mixtures the algorithm is similar: Step I. Given xB and r. Step II. Assume that a quaternary pinch controls the separation. Then, calculate y* (vapor in equilibrium with xB) and xD through the lever arm rule, eq 11. Step III. If x4,D < 0, then assume that a ternary saddle pinch controls the separation. (This condition implies that r > R*.) Follow the next steps to calculate the instantaneous distillate composition. III.1. Calculate the equation of the characteristic line by solving eqs 5, 8, and 10. III.2. Calculate the composition of the invariant ternary saddle pinch by calculating the intersection between the characteristic line and the plane x1 + x2 + x3 ) 1 (where there is no heavy component). III.3. With the composition of the invariant ternary pinch and its vapor in equilibrium, the composition of the instantaneous distillate can be computed by using the lever arm rule, eq 12. Step IV. If x3,D < 0, then assume that a binary saddle pinch controls the separation. (This condition implies that r > R**.) Then, proceed as in the case of ternary mixtures but considering the composition of the ternary pinch as the still composition. Step V. If x2,D < 0, then set x2,D ) 0 and x1,D ) 1, respectively. A pinch at the column top controls the separation. (This condition implies that r > R***.)

When using the above algorithms for the prediction of the instantaneous top composition within an integration loop for simulating a batch run, the still composition eventually may become a binary mixture. For that situation, only two different regimes could be present. The procedure for binary mixtures is as follows: Step I. Calculate y* by solving the bubble point from the instantaneous boiler composition and the distillate mole fraction using the lever arm rule, eq 11. Step II. Again, if x2,D < 0, then set x2,D ) 0 and x1,D ) 1, respectively. For this case, a pinch at the top of the column controls the separation. Note that the above algorithms are based on reproducing the behavior of the internal profiles by means of the characteristic directions in the neighborhood of the controlling pinch points. An important issue in determining the controlling pinch is the tracking of the distribution of the components. For example, given a quaternary still composition, we begin the algorithm assuming that a quaternary pinch with the composition of the still controls the separation. Hence, the distillate composition is calculated on the line of preferred separation. If the heavy component does not distribute, the equation of the characteristic line is then calculated and consequently the composition of the ternary pinch. Once the vapor in equilibrium with the invariant pinch is obtained, the distillate composition can be calculated for the reflux ratio from the lever arm rule. Again, if the second heaviest species does not distribute, the invariant binary saddle pinch must be calculated to obtain the distillate composition. Finally, if the second lightest species does not distribute, only the lightest species will be present in the distillate and the regime resembles that of total reflux with a pinch at the column top. The algorithms developed above are applicable to ideal or nearly ideal ternary and quaternary systems. They are presented to introduce the reader to the concepts that will be extended to nonideal multicomponent systems in subsequent sections. Interpretation of the Underwood Equations for Batch Distillations In this section, we show that by applying the Underwood equations to calculate the instantaneous distillate composition for a given still composition and reflux ratio it is possible to reproduce the behavior of the column profile in terms of the different regimes and controlling pinch points. Moreover, we show that, for systems with constant relative volatilities, the use of the Underwood equation for the prediction of the separation performance is entirely equivalent to the algorithms of the preceding section. The following discussion also contributes to the understanding of the correct application of the Underwood equations to a batch rectification column, in particular, for those operating conditions when not all of the components in the still distribute along the column. The Underwood equations for a column with both rectifying and stripping sections are

RixD,i

∑i R

i

-Φ

)r+1

(13)


and

RixB,i

∑i R

i

-Ψ

) -s

(14)

where Ri is the constant relative volatility of component i, Φ and Ψ are the roots of the rectifying and stripping sections, respectively, and finally r and s are the reflux and reboil ratios. The physical interpretation of both parameters Φ and Ψ was developed by Ko¨hler et al.6 and can be stated as follows. Let us define the relative volatility of component i with respect to the heaviest species as

Ri ) Ki/Khc

(15)

where the liquid-vapor equilibrium for component i and the heaviest species is given by

Ki ) yi/xi; Khc ) yhc/xhc

(16)

The mass balance for the rectifying section under pinch condition can be written as Figure 8. Interpretation of the minimum reflux given by Franklin and Forsyth.7

Vyi ) Lxi + DxD,i yi ) Kixi ≡ KhcRixi

(17)

Taking into account that V/D ) r + 1 and reordering eq 17, the following can be obtained:

yi xD,i ) (r + 1)xiKhc ) (r + 1) Ri - L/VKhc Ri

(18)

Thus, when eq 18 is summed up for all components, an expression similar to the Underwood equation for the rectifying section (eq 13) is obtained provided that the parameter Φ be defined as

Φ ) L/VKhc

(19)

The parameter Φ defined in terms of eq 19 is known as the stripping factor (Ko¨hler et al.6). An analogous treatment can be done for the stripping section to obtain

Ψ)L h /V h Khc

(20)

Underwood demonstrated that, for a given product composition and reflux ratio (the boilup can be calculated through the overall energy balance), each one of the equations (13) and (14) has as many solutions for the parameters Φ and Ψ as there are components in the mixture. For a multicomponent mixture, the different roots fit into the order of relative volatilities as follows:

0 < Φ1 < RC < Φ2 < RC-1 < ... < R2 < ΦC < R1 RC < Ψ1 < RC-1 < ... < R2 < ΨC-1 < R1 < ΨC

(21)

For each root of eq 13, a pinch can be calculated with eq 18. The same occurs for the stripping section. Thus, for a ternary mixture three pinch points can be calculated for each product with some of these pinch points getting negative values of its concentrations and, hence, having only a theoretical meaning. The polyhedron obtained from the straight lines joining the compositions of the pinch points is called the “distillation space”

(“distillation triangle” in this case). Figure 8 shows the triangles corresponding to the rectifying and stripping sections of a column operating at minimum reflux. The calculations were performed by considering the average relative volatilities between the top and bottom products. Underwood’s main contribution was to provide a method that determines, from the several pinch points, the ones that control the minimum-energy demand. Underwood demonstrated that at minimum reflux some of the parameters Φ and Ψ must coincide, and their values must be between the relative volatilities of the key components. As an example, when the key components are adjacent, there will be only one common root in eqs 13 and 14. A direct consequence of the equality between the parameters Φ and Ψ is that the pinch points Ps>, Prend, Psend, and Prsaddle in Figure 8 are aligned under minimum-energy demand. At operating refluxes below the minimum, the triangles do not touch each other. On the other hand, the triangles penetrate each other for values of the reflux above the minimum. In both cases, there is no more alignment between pinch points. This geometric interpretation was first developed in the paper of Franklin and Forsyth.7 These authors gave also geometric explanations for quaternary mixtures. Note that for the common roots of eqs 13 and 14 it is possible to sum up both expressions to give with the aid of the well-known overall mass balance equation

RixF,i

∑i R

i

-θ

)1-q

(22)

From eq 22, the common roots θ of eqs 13 and 14 can be calculated. The variable q in the above equation is the liquid fraction of the feed. Back to Figure 8, it can be seen that the collinearity between Prend (nonactive under minimum reflux), Psend (active), and Prsaddle (active) and the feed composition is a consequence of the minimum-energy condition. Thus, the rectifying profile below the saddle (we consider the condenser as the column top) lies on a straight line that passes through the saddle pinch and the feed composi-


methodology to multicomponent mixtures, we need to resort to a more abstract mathematical representation. We have learned that the characteristic directions associated with the algorithms are related to linearizations of the profile in the neighborhood of the critical pinch points. This subject was studied in detail by Po¨llmann and Blass8 and the subsequent discussion refers to their analysis. Considering the composition profile as an implicit recurrence Ω(xn,xn+1), for the rectifying section the following expression can be obtained:

0 ) Ω(xn,xn+1) ) xn - yn+1 + ξn(yn+1 - xD) (23) ξn ) - 1/rn

with

Figure 9. Equivalence between the parallelism criterion and the application of the Underwood equations to determine an instantaneous top composition.

tion. Thus, it is clear that the Underwood equations contain implicitly the geometry of the profiles under minimum reflux without having to calculate these profiles. Figure 9 shows the results of the application of the Underwood equations to calculate the instantaneous distillate composition for the still composition of the figure. The common root was calculated by replacing the feed composition of a continuous distillation in eq 22 by the instantaneous still composition. The distillate composition was calculated as explained by Salomone et al.1 for a given reflux ratio. We calculated all of the pinch points in both the liquid and vapor phases by using eqs 13 and 18 and the dew point equations. As the bases of the triangles are parallel and contain the still composition, the direction of the “implicit” profiles below the corresponding saddle coincides with the characteristic directions used in the algorithms of the preceding section. In other words, the saddle pinch composition implicitly contained in the Underwood equations is the same as that predicted from the study of the geometry of the profiles. The advantage for systems with constant relative volatility is that the Underwood method can be applied for multicomponent mixtures while geometrical interpretations can be made only for ternary and quaternary mixtures. On the other hand, the Underwood method cannot be applied to nonideal systems. Generalization for Nonideal Multicomponent Mixtures In the preceding sections we have shown that for ideal systems the Underwood equation can be used to predict the separation performance for any number of components. On the basis of the observation by Du¨ssel4 regarding the geometry of the profiles, we developed a couple of algorithms for ternary and quaternary mixtures that are not restricted to constant relative volatilities. Although geometrical considerations are very useful to gain insight on the problem, to extend the

The value of ξ depends on the actual value of the reflux at stage n. The energy balance around the envelope of mass balance may be used to determine the actual value of the reflux ratio. For constant molar overflow, the internal reflux ratio is constant and equal to the external reflux ratio. The implicit recurrence Ω cannot, in general, be expressed as an explicit recurrence of the form

xn ) Π(xn+1)

(24)

However, it is possible to derive a linear approximation of the explicit recurrence. To do this, the differentials of the implicit recurrence must be calculated (Po¨llmann and Blass8). They derived a vectorial equation that relates the compositions in the neighborhood of pinch points. The linear approximation can be written as

xn - xp ) Π(xn+1) - Π(xp) ≈

( )| ∂Πi ∂xj

(xn+1 - xp)

(25)

xp

In this equation Π represents the explicit recurrence at a point xn and at the pinch xp, respectively. The above equation can be solved as an eigenvalue problem provided that the differentials of the functional matrix of the unknown explicit recurrence Π, in terms of the known implicit recurrence Ω, can be obtained. Following the hints recommended by Po¨llmann and Blass8 and considering that for constant molar overflow ξ is a constant, we obtain

( )| ∂Πi ∂xj

( )|

) (1 - ξ)

xp

∂yi ∂xj

(26) xp

Given that the functional matrix of the explicit recurrence is proportional to the Jacobian of the equilibrium function, the eigenvalue problem can be stated as

λ‚v )

( )| ∂yi ∂xj

‚v

(27)

xp

Solving this eigenvalue problem will produce C - 1 eigenvectors vj and C - 1 eigenvalues λj. It has been pointed out by Du¨ssel4 that the characteristic directions obtained by imposing the parallelism condition are coincident with the eigenvectors resulting from eq 27. Therefore, the eigenvalue problem can be used for generating the characteristic directions for any number of components. The selection of the proper eigenvector will depend on the controlling pinch where the problem


eq 28 for xD, we need to impose additional conditions for xN that depend on the different profile regime. For small values of the reflux ratio (r < R*), a pinch is located at the end of the column immediately above the boiler. The pinch composition coincides with the composition of the mixture in the still (C components), and therefore xN is equal to xB. The application of eq 28 is straightforward. For higher values of the reflux ratio, a C - 1 saddle pinch xpC-1 will control the separation. The location of this pinch can be estimated by solving the eigenvalue problem in xB and selecting the eigenvector corresponding to the smallest eigenvalue. This selection preserves the nature of the profile in the section going from the bottom to the saddle. (All species but the heaviest augment the composition.) Thus, both xpC-1 and xN are located on the line indicated by the eigenvector vI:

Figure 10. Linearization of the profile in nonideal systems.

xi,B - xi,pC-1 ) vi,I

(29)

xi,B - xi,N ) δvi,I

(30)

and considering that under this regime the heaviest species does not distribute, and extra conditions are imposed:

xC,D ) 0 (31) xC,pC-1 ) 0

Figure 11. Mass balance around the rectifier.

is formulated and will be discussed later in this section when the algorithm is reformulated. At this point we also consider some issues regarding nonideal systems. For these mixtures, the characteristic pinch points are no longer invariant and, in general, the profiles are no longer straight lines connecting them. However, in the neighborhood of the pinch points, the linearization of the profile is still a good approximation. This situation is illustrated in Figure 10 for a ternary system with a controlling binary saddle pinch. The profile leaves the point xB in the direction given by the eigenvector vI (which corresponds to a linerization of the profile). Using this direction to predict the location of the pinch point will produce the point xpLIN, and therefore the application of the lever arm rule, eq 12, in the point xpLIN instead of xp will produce a wrong estimate of the top composition. To overcome this situation, we can resort to the overall mass balance in the column; see Figure 11 (the lever arm rule between the top and bottom) to get / / xi,D ) yi,x + r(yi,x - xi,N) B B

(28)

Because xN is a composition close to xB, then the error introduced by using a linearization of the profile around the still composition is reduced as compared with the application of the lever arm rule in the pinch. To solve

Equations 28-31 form a linear system where the unknowns are the top composition xD, the saddle pinch composition xpC-1, the composition at the end of the column xN, and the scalar values and δ. For an even higher value of the reflux ratio, a C - 2 saddle pinch, xpC-2, will become controlling. Both the top and the pinch will not contain the two heaviest species. The location of this pinch can be estimated by solving the eigenvalue problem in xpC-1 and selecting the eigenvector vII corresponding to the second smallest eigenvalue. Under this regime, xN and xpC-2 are located on a plane defined by the two previous eigenvectors vI and vII. Therefore, eqs 29 and 30 are reformulated as follows:

xi,B - xi,pC-2 ) Ivi,I + IIvi,II

(32)

xi,B - xi,N ) δIvi,I + δIIvi,II

(33)

and the corresponding extra conditions:

xi,D ) 0; xi,pC-2 ) 0

i ) C, C - 1

(34)

When the above ideas are generalized, it can be said that for a given reflux ratio a C - M pinch, xpC-M, will be controlling the separation. Both xN and the controlling pinch xpC-M will be located in the linear Mdimensional space defined by the M eigenvectors obtained solving the eigenvalue problem at the previous pinch points. The system of equations is completed by setting to zero the composition of the last M components for the top and the controlling pinch. Thus, the method requires the recursive solution of a linear system of 3C + 2M equations and variables [3C mole fractions xD, xN and xp; M values for and M values for δ are unknown]. The reflux ratio r, the composition of the liquid in the still xB, its vapor in equilibrium y*B, and the matrix of eigenvectors vi,j are the parameters


of the equations system and, hence, their values are known. We exemplify how the equation system must be solved for the case of a quaternary mixture. Given the instantaneous still composition, the vapor in equilibrium can be calculated by solving a bubblepoint routine. Assume the controlling pinch is in the bottom of the column with the composition of the liquid in the still. Set the value of M equal to zero and solve the equation system. Note that both xN and xp equal xB. If the heavy component does not distribute (detected by the prediction of xC,D < 0), calculate the eigenvector corresponding to the still composition and assign it (that corresponding to the lowest eigenvalue) to vi,1. Set M ) 1, and solve again the equation system. Now, xN will lie on a line through xB, and a ternary pinch will be calculated in addition to a distillate with the three more volatile species. Again, if the second heaviest component does not distribute (xC-1,D < 0), calculate vi,2 at the ternary pinch, set M ) 2, and solve the equation system. As a result, a binary distillate composition, a binary saddle pinch, and a composition of the liquid at the end of the column lying on a plane will be obtained. Finally, if the second lightest component does not distribute, only the most volatile component is obtained at the top for the given reflux ratio. As a test for the methodology, we have performed extensive comparison against the predictions from the Underwood equations for systems with constant relative volatilities and from rigorous simulations for nonideal mixtures. A first set of tests was performed by calculating for a given still composition the distillate compositions at different reflux ratios to determine the allpossible distillate composition that could be achievable at the top of the column. Rigorous simulations were performed using a rectifying column model in Hysys. To simulate an infinite column, we selected columns with 300 trays or more. A second set of tests consisted of the computation of the complete still path at a given reflux ratio and from a given initial still composition. In this case, we compared the results for large reflux ratios with dynamic simulations of a column using a rigorous total reflux model. The maximal recoveries for all of the components of the mixture versus the rectification advance and the predicted sequence of cuts were used as a comparison criterion. In both tests, excellent results were achieved. Highly Nonideal Ternary Mixtures with Azeotropes The preceding method is quite general for nonideal zeotropic mixtures of any number of components. However, highly nonideal and azeotropic mixtures present a series of characteristics that must be considered to correctly predict the instantaneous performance of the separation and, hence, to successfully model the batch distillation operation. There are three main cases for which the previous algorithms have to be modified: (i) mixtures for which the linearization is not always a valid assumption, (ii) mixtures whose still composition path is limited by a stable distillation boundary and, (iii) mixtures with an unstable distillation boundary limiting the top compositions. In this section we analyze several well-known ternary azeotropic mixtures in order to illustrate the mentioned cases, the modifications that have to be introduced into

Figure 12. Internal profiles and instantaneous product composition regions for two mixtures (system acetone-methanol-ethanol).

the algorithm, and we show the kind of analysis that can be done by the application of the corrected algorithm. Mixtures for Which the Linearization Is Not Always a Valid Assumption. Figure 12 shows the allpossible distillate compositions for a still composition composed for acetone-methanol-ethanol [0.3, 0.4, 0.3]. This Figure also shows the internal profiles for several reflux ratios. At reflux ratios below R*, the distillate compositions lie on the preferred separation line. The corresponding profiles present a pinch at the column end that equals the composition of the mixture in the still. For values of the operation reflux above R*, a saddle controls the separation and the distillates are located on the edge of the triangle corresponding to the two lightest components. As can be seen for two different cases (a, b), the composition of the saddle remains no more as invariant (Du¨ssel4). However, the profiles for different reflux ratios approach the still composition following a straight line and, hence, the linearization of the profiles around the still composition remains a good approximation. Errors below 1.5% in the lightest component were obtained for the still composition of the figure with respect to simulations performed with Hysys. Figure 12 also shows another example corresponding to a mixture of composition [0.4, 0.1, 0.5]. Again, for this case the controlling pinch may be either the one at the composition corresponding to the still or the saddle. However, in this region, a linearization of the profiles around the still composition is not longer a good approximation for the liquid profiles. In this case, the linearization of the profile predicts a pinch point located between the composition xD* (xD2a in Figure 12) corresponding to the preferred separation and the unstable node (the azeotrope). Therefore, the predicted characteristic line produces an estimation of xN located between the vapor in equilibrium with the still composition, yB*, and the distillate composition xD. It can be deducted by inspecting Figure 11 that this arrangement violates the mass balance because yB* must ever be amidst the liquid compositions xN and xD. We have detected that this behavior is associated with regions where the residue curve map presents inflec-


Figure 13. Still paths at different reflux ratios for two starting mixtures (system acetone-methanol-ethanol).

tions. For these cases, it is a better strategy to predict the saddle point through a stage-by-stage calculation from the already known distillate condition and the reflux ratio at the preferred separation (i.e., xD* and R*). Thus, starting at the distillate composition, a downward stage-by-stage calculation is performed until a pinch condition is reached. The new characteristic line is the one defined by the still composition xB and the computed saddle pinch. Note that even though this alternative requires some recursive computation it is still much simpler than solving the whole column profile iterating on a distillate composition. Also, this approach is more robust than solving the pinch equations with the given xD* and R* as parameters. It is well-known that for highly nonideal systems the pinch equation system may have multiple solutions, and therefore the selection of the active pinch becomes a problem (Aguirre and Espinosa9). In terms of our algorithm, that means that validity proof of the linearization must be done for each still composition, and according to this, the linearization is done by the eigenvector method or the pinch computation alternative. By using the corrected algorithm, the maximum error in predicting the top composition of the lightest species was about 6%. In addition, we used the corrected algorithm within an integration loop to study the dynamic evolution of a batch rectification. Figure 13 shows different still paths for different values of the reflux ratio for two different initial still mole fractions. For the mixture xB1, with composition [0.5, 0.3, 0.2] and a very high value of the reflux ratio, the still path moves on a straight line, away from the minimum-temperature azeotrope between acetone and methanol. During this time, the azeotrope is collected in the distillate drum. As was demonstrated by Van Dongen and Doherty,10 the distillate at each moment of time must be located anywhere on a tangent to the still path. At large reflux ratios and large numbers of trays, the still path follows a straight line from the unstable node (the azeotrope) through the initial composition in the still. Once the liquid in the still does not contain more of the volatile species acetone and considering that the reflux ratio is great enough to produce methanol at the top of the column, the still

moves away from the intermediate (saddle node) until the heavy component (stable node) remains as the unique component in the still. It must be noted that in this example the azeotrope acts as a limitation for the possible distillates and, hence, it must be introduced in our algorithm as a boundary for the distillates. We will explain how to introduce boundaries for the top compositions at the end of this section. On the other hand, no boundaries influence the still path. The results obtained are in agreement with those obtained at operation near total reflux conditions. For the given initial mixture, the azeotrope is collected at the beginning of the operation, followed by a cut of pure methanol, and finally the heavy species ethanol is obtained at the top of the column. Figure 13 also shows several still paths for the case of a mixture xB2, with the composition [0.7, 0.1, 0.2]. In this case, operation at a high reflux ratio produces an initial cut containing the azeotrope, a second cut where pure acetone is collected at the top of the column, and finally pure ethanol which can be obtained at the top. The two different behaviors of the mixtures considered at operation near total reflux have led several authors to define two different distillation regions in the composition simplex (Bernot et al.11,12 and Safrit and Westerberg13). Note, however, that for purposes of a short cut only the azeotrope must be considered as a limiting factor for the distillate compositions. Moreover, a still path from xB2 (Figure 13) corresponding to low reflux ratios can traverse the line between the azeotrope and the pure ethanol, which in turn is considered to divide the simplex into two different distillation regions. The concept developed by the mentioned authors is useful for determining the different cuts that could be expected operating near total reflux depending on the initial feed composition according to the definition given by Ewell and Welch.14 We follow the conventional definition of a simple distillation region, which is defined in terms of the origin and destination of residue curves. Still Compositions Limited by a Stable Distillation Boundary. Another important aspect can be analyzed from the mixture acetone-chloroform-benzene that presents two distillation regions. As is shown in Figure 14, a distillation separatrix runs from the maximum-temperature azeotrope between chloroform and acetone (saddle node) to the heaviest benzene (stable node). Each one of the distillation regions has a different unstable node. Pure acetone is the unstable node for initial compositions in the upper region. On the other hand, pure chloroform acts as an unstable node for starting mixtures in the lower distillation field. If the initial mixture lies in the upper region and for high reflux ratios, pure acetone is obtained as the top product during the first step of the batch distillation. The still path moves along a straight line away from the acetone vertex. At the intersection of the still path with the distillation separatrix, the concentration of acetone in the distillate sharply decreases to a value that lies at the tangent to the distillation separatrix at the point of intersection. During the second step of the batch distillation, the still path moves along the distillation separatrix, with the corresponding top compositions lying on tangents to the still path. Figure 15 shows the all-possible distillate compositions for an instantaneous still composition on the distillation border (xB2). Only products on the preferred separation line are


Figure 14. Still paths at different reflux ratios for two starting mixtures of a system showing a stable separatrix (system acetonechloroform-benzene).

Figure 16. Top compositions versus the advance of the rectification for a starting mixture in the upper distillation field (system acetone-chloroform-benzene).

Figure 17. Top compositions versus the advance of the rectification for a starting mixture in the lower distillation field (system acetone-chloroform-benzene). Figure 15. Internal profiles and instantaneous product composition regions for two mixtures (system acetone-chloroformbenzene).

permitted for this case. As demonstrated by Van Dongen and Doherty,10 the distillate composition has to lie on a straight line that forms a tangent to the still path. The distillation border acts as a boundary for the still composition and, hence, the top products must lie on tangents to this curve. The mentioned authors also showed why the separatrix acts as a limit for the still compositions. They termed this boundary a stable separatrix. Note, however, that the distillation border joins a saddle with a stable node (benzene). From the analysis above, it is clear that the distillation border does not act as a boundary for the distillate compositions. The reason is that the boundary presents a curvature and, hence, distillate can be obtained with compositions pertaining to the lower distillation region (Du¨ssel4 and Stichlmair and Fair15). Figure 16 shows the distillate composition vs the rectification advance for a starting mixture with composition [0.5, 0.4, 0.1] that lies on the upper distillation

field. This figure qualitatively resembles Figure 6.14 presented in Stichlmair and Fair.15 Figure 14 also shows the change in the state of the boiler for different reflux ratios for an initial mixture belonging to the lower distillation region [0.1, 0.7, 0.2]. Because of the curvature of the distillation separatrix, the composition of chloroform must present a minimum, as is shown in Figure 17. From this example it is clear that the algorithm must consider the influence of stable separatrixes on the still path behavior. Moreover, a nonlinear mathematical representation of the boundary should be considered to correctly predict the behavior of systems with curved stable boundaries, and the location of the still composition with respect of the boundary must be verified. If the still composition pertains to the stable boundary, only a preferred separation is permitted and therefore the algorithm should include this constraint. By using a corrected algorithm, we obtained a very good agreement between predicted and simulated values using Hysys. As an example, errors in predicting the lightest top composition below 1% were found for the fixed composition xB1 [0.3, 0.3, 0.4] in Figure 15.


Figure 19. Component recoveries versus the advance of the rectification for a starting mixture in the left distillation field (system methanol-1-propanol-water). Figure 18. Still paths at different reflux ratios for two starting mixtures of a system showing an unstable separatrix (system methanol-1-propanol-water).

Top Compositions Limited by an Unstable Distillation Boundary. Let us consider the system methanol-1-propanol-water that shows a minimum-temperature azeotrope between the heaviest components. The system presents a distillation boundary that runs from the lightest species methanol to the azeotrope. Figure 18 shows still paths for two starting mixtures at different reflux ratios. In this case, the distillation separatrix acts as a boundary for the distillate compositions. Hence, this separatrix is termed an unstable separatrix. Note that in this case the separatrix joins an unstable node (methanol) with a saddle (the azeotrope). To reproduce the behavior in both distillation regions, a linear approximation of the boundary can be included in our procedure. We use the linear approximation to calculate the limiting value x*D that in this case contains all of the components of the mixture. For reflux ratios above R*, eq 31 must be replaced for a linear representation of the unstable boundary in terms of the two lightest components. By running simulation using the corrected algorithm, we obtained the results in Figures19 and 20. Here we show the recoveries of the components near total reflux operation for the two cases under consideration. While for xB1 [0.3, 0.2, 0.5] the sequence of cuts is (i) methanol, (ii) the azeotrope, and (iii) water, a change in the cuts can be expected for xB2 [0.3, 0.5, 0.2]. Although the first two cuts remain the same, 1-propanol instead of water can be collected at the end of the operation for the last case. From the analysis above, it is clear that both stable and unstable separatrixes should be considered in the short-cut procedure in order to reproduce the behavior of highly nonideal mixtures. There exist several methods to obtain these boundaries both qualitatively (Bernot et al.,11,12 Po¨llmann et al.,16 Rooks et al.,17 and Safrit and Westerberg13) and rigorously (Po¨llmann et al.16). In this work we used a linear approximation for the unstable separatrixes and nonlinear approximations for stable separatrixes. So far, we have summarized the three main cases where our algorithm has to be modified in order to keep properly representing the behavior of highly nonideal

Figure 20. Component recoveries versus the advance of the rectification for a starting mixture in the right distillation field (system methanol-1-propanol-water).

or azeotropic mixtures. The cases and the proposed solutions were analyzed as isolated effects for ease of presentation. However, in an arbitrary mixture, it is likely that more than one of these problems may appear concurrently. In fact, this is the case with several of the examples we have presented here. We have succeeded in articulating all of the needed corrections and checking in a general purpose algorithm applicable to any ternary mixture. Without entering into the details of the procedure, it can be said that the overall strategy is as follows: (i) Whenever a mixture presents stable separatrixes, the algorithm should check for each new still composition if the separatrix has been reached. When this occurs, possible solutions for the distillate composition are constrained to be in the direction of the preferred separation. From now on, the still path evolution is enforced to follow the separatrix until a stable node is reached. (ii) Also for each new still composition the possible distillate composition has to be checked for violations of the distillate boundaries. For mixtures without unstable separatrixes, these boundaries will be the


natural boundaries of the composition space (for ternary mixtures, they are the binary axis); otherwise, linear representations of the boundaries are introduced. Note that this check has to be done after the check for stable separatrixes and that it is applicable indistinctly of the direction of search of the one constrained to the preferred separation or freely computed by solving the eigenvalue problem. (iii) Finally, each time the eigenvalue problem is solved the obtained eigenvector has to be validated for its suitability to predict the pinch location, and eventually the alternative method is used. Method for the Calculation of the Minimum Reflux In a previous work (Salomone et al.1), a method for the computation of the batch minimum reflux was developed for mixtures with constant relative volatilities. For a given initial mixture and once the separation task is specified in terms of two fractional recoveries, the procedure uses simplified dynamic simulations of the operation for different values of the reflux ratio until convergence of both fractional recoveries to the specified values. A summary of the method is reproduced here in Table 1. As is noted in the first part of Table 1, the prediction of the instantaneous separation performance of an infinite stage column is required in step 3. For systems with constant relative volatility, the Underwood equation was proposed for such purposes in the subprocedure A of Table 1. Here we analyzed the implications of using the Underwood equation. The extension of the method to nonideal multicomponent mixtures is straightforward. The prediction of the instantaneous top composition needed in step 3 is done by using the new developed algorithm. Conclusions We have developed a short-cut procedure for estimating the instantaneous separation performance for batch columns having an infinite number of stages that is applicable to nonideal mixtures. The model assumes a negligible holdup for the column and condenser and allows the estimation of the instantaneous top composition without resorting to stage-bystage computation. The key aspect of the method is the determination of the operating regimes at each instant of time. The regime is characterized by a controlling pinch, which in turn controls the geometry of the internal profiles. With this model, both a method for rapid simulation of the batch operation and a procedure for calculating the minimum-energy demand can be developed. We first analyzed ideal systems, and for introducing the main concepts behind the method, we developed algorithms for ideal ternary and quaternary mixtures using the parallelism condition developed by Du¨ssel.4 Then, the presented approach was compared with the use of the Underwood equation to determine the instantaneous separation column performance. We showed that for a system with constant relative volatilities, both methods are equivalent. This analysis also gives an interpretation of the application of the Underwood equation in the context of batch rectification.

Afterward, we generalized the method to multicomponent nonideal zeotropic mixtures. The parallelism condition developed by Du¨ssel4 is replaced by a linearization of the liquid composition profile around pinch points and presented a mathematical formulation allowing extension to systems with more than four components. The model has a level of detail enough to preserve the nonideal thermodynamic behavior and still simple enough to allow rapid simulation of dynamic runs. Therefore, it should be considered in the scope of the conceptual design. Finally, we analyze the application of the developed model to highly nonideal mixtures with azeotropes and distillation boundaries and briefly discuss the modifications that have to be introduced to extend the method. Both cases of stable and unstable separatrixes were taken into consideration. To validate the model, we have compared the all-possible top compositions for a given boiler composition at different reflux ratios with the results from rigorous simulations of columns with 300 or more stages. Also, we compare the predicted top cuts from our model with those from rigorous simulations of columns operating near total reflux. For these cases, the limiting recoveries were also considered. Given the latest significant advances in the thermodynamic analysis of azeotropic mixtures allowing the prediction of azeotropes, distillation boundaries together with feasible sequences of distillate cuts, and maximum attainable separations, the method in this paper incorporates this knowledge and extends the analysis to the operational issues in batch distillation. Therefore, an immediate application is the computation of the minimum reflux ratio for a batch rectification. Acknowledgment We are grateful to CONICET (Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas) for supporting this research. Appendix The following are analytical expressions for the parameters used in the computation of the characteristic line in ternary systems, eq 3. They are systems with constant relative volatilities.

C1 )

Rac[1 + (Rbc - 1)xb,B] N2

C2 ) C3 ) C4 )

(1 - Rbc)xa,BRac N2 (1 - Rac)xb,BRbc N2

Rbc[1 + (Rac - 1)xa,B] N2

N ) 1 + (Rac - 1)xa,B + (Rbc - 1)xb,B The following are analytical expressions for the parameters used in the computation of the characteristic line in quaternary systems, eqs 5 and 8. They are systems with constant relative volatilities.


C1 )

C2 )

|

∂y*1 ∂x1

|

∂y*1 ∂x2

)

R1[1 + (R2 - 1)x2 + (R3 - 1)x3] N2 ∂y*3 C7 ) ∂x1

x2,x3

)

|

N2

x1,x3

)

)

(1 - R1)x3R3

|

)

(1 - R2)x3R3

N2

x2,x3

(1 - R2)x1R1

C8 ) ∂y*1 C3 ) ∂x3

|

∂y*3 ∂x2

x1,x3

N2

(1 - R3)x1R1

N2 ∂y*3 R3[1 + (R1 - 1)x1 + (R2 - 1)x2] C9 ) ) ∂x3 x1,x2 N2 x1,x2

|

C4 ) C5 )

|

∂y*2 ∂x2

|

∂y*2 ∂x1

)

)

x2,x3

N2

R2[1 + (R1 - 1)x1 + (R3 - 1)x3] N2

x1,x3

C6 )

(1 - R1)x2R2

|

∂y*2 ∂x3

x1,x2

)

(1 - R3)x2R2 N2

N ) 1 + (R1 - 1)x1 + (R2 - 1)x2 + (R3 - 1)x3 Nomenclature B ) bottom flow rate C ) number of components D ) distillate flow rate, amount of distillate d ) differential F ) starting mixture fi ) fractional recovery of component i Ki ) yi/xi L ) internal liquid flow rate M ) number of nondistributing components ni ) amount of component i in the still P ) pinch point q ) liquid fraction of the feed r, R ) reflux ratio s ) boilup v ) eigenvector V ) internal vapor flow rate xB ) still composition xD ) distillate composition xi ) liquid mole fraction of component i xp ) controlling pinch yi ) vapor mole fraction of component i y*xB ) vapor in equilibrium with xB Greek Symbols Ri ) relative volatility of component i Φ ) roots of the Underwood equation (eq 13) Ψ ) roots of the Underwood equation (eq 14) θ ) roots of the Underwood equation (eq 19) Ω ) implicit recurrence representing the profile (eq 23) Π ) explicit recurrence representing the profile (eq 24) ξ ) parameter in eq 23 λ ) eigenvalue Subscripts B ) still D ) distillate

f ) initial feed hc ) heavy component i ) generic component lc ) light component n ) stage number in a rectifying section m ) stage number in a stripping section max ) maximum min ) minimum p ) pinch t ) top Superscripts * ) equilibrium or special condition max ) maximum r ) rectifying s ) stripping II ) binary III ) ternary

Literature Cited (1) Salomone, H. E.; Chiotti, O. J.; Iribarren, O. A. Short-Cut Design Procedure for Batch Distillations. Ind. Eng. Chem. Res. 1997, 36 (1), 130-136. (2) Offers, H.; Du¨ssel, R.; Stichlmair, J. Minimum Energy Requirement of Distillation Processes. Comput. Chem. Eng. 1995, 19, Supplement, S247-S252. (3) Stichlmair, J. Zerlegung von Dreistoffgemischen durch Rektifikation. Chem.-Ing.-Tech. 1998, 60 (10), 747-754. (4) Du¨ssel, R. Zerlegung azeotroper Gemische durch BatchRektifikation. Ph.D. Dissertation, Technische Universita¨t Mu¨nchen, Mu¨nchen, Germany, 1996. (5) Stichlmair, J.; Offers, H.; Potthoff, R. W. Minimum Reflux and Minimum Reboil in Ternary Distillation. Ind. Eng. Chem. Res. 1993, 32, 2438-2445. (6) Ko¨hler, J.; Po¨llmann, P.; Blass, E. A Review on Minimum Energy Calculations for Ideal and Nonideal Distillations. Ind. Eng. Chem. Res. 1995, 34, 1003-1020. (7) Franklin, N. L.; Forsyth, J. S. The Interpretation of Minimum Reflux Conditions in Multi-Component Distillation. Trans. Inst. Chem. Eng. 1953, 31, 363-388. Also reprinted in Trans. Inst. Chem. Eng. 1997, 75 (Jubilee Supplement), S56-S81. (8) Po¨llmann, P.; Blass, E. Best Products of Homogeneous Azeotropic Distillations. Gas Sep. Purif. 1994, 8 (4), 193-227. (9) Aguirre, P.; Espinosa, J. A Robust Method to Solve Mass Balances in Reversible Column Sections. Ind. Eng. Chem. Res. 1996, 35, 559-572. (10) van Dongen, D. B.; Doherty, M. F. On the Dynamics of Distillation ProcessessVI. Batch Distillation. Chem. Eng. Sci. 1985, 40 (11), 2087-2093. (11) Bernot, C.; Doherty, M. F.; Malone, M. F. Patterns of Composition Change in Multicomponent Batch Distillation. Chem. Eng. Sci. 1990, 45 (5), 1207-1221. (12) Bernot, C.; Doherty, M. F.; Malone, M. F. Feasibility and Separation Sequences in Multicomponent Batch Distillation. Chem. Eng. Sci. 1991, 46 (5/6), 1311-1326. (13) Safrit, B. T.; Westerberg, A. W. Algorithm for Generating the Distillation Regions for Azeotropic Multicomponent Mixtures. Ind. Eng. Chem. Res. 1997, 36, 1827-1840. (14) Ewell, R. H.; Welch, L. M. Rectification in Ternary Systems Containing Binary Azeotropes. Ind. Eng. Chem. 1945, 37 (12), 1224-1231. (15) Stichlmair, J.; Fair, J. R. Distillation: Principles and Practice; Wiley-VCH: New York, 1998; Chapter 6. (16) Po¨llmann, P.; Bauer, M. H.; Blass, E. Investigation of vapor-liquid equilibrium of nonideal multicomponent systems. Gas. Sep. Purif. 1996, 10 (4), 225-241. (17) Rooks, R. E.; Julka, V.; Doherty, M. F.; Malone M. F. Structure of Distillation Regions for Multicomponent Azeotropic Mixtures. AIChE J. 1998, 44 (6), 1382-1391.

Received for review January 12, 1999 Revised manuscript received March 31, 1999 Accepted April 1, 1999 IE990022Y

Minimum Reflux for Batch Distillations of Ideal and Nonideal Mixtures

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