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Ind. Eng. Chem. Res. 1998, 37, 2250-2270

Multicomponent Distillation Design through Equilibrium Theory Bernardo Neri, Marco Mazzotti,† Giuseppe Storti, and Massimo Morbidelli* Laboratorium fu¨ r Technische Chemie, ETH Zu¨ rich, Universita¨ tstrasse 6, CH-8092 Zu¨ rich, Switzerland

A new tool for the design of multicomponent distillation columns is presented, which is based on analytical solutions of a suitable mathematical model. The assumptions on which it is built are (i) a large number of stages, which makes it possible to apply a continuous description of the column instead of the usual stage-by-stage equations; (ii) constant molar overflow; (iii) constant relative volatility; and (iv) attainment of vapor-liquid equilibrium everywhere along the column. The resulting model equations are solved in the frame of Equilibrium Theory, which was originally developed to describe chromatographic processes. Through this approach explicit results may be obtained and both the steady state and the dynamic behavior of high-purity columns can be analyzed. In particular, this work focuses on the former, thus obtaining the following results: (i) a complete picture of the different separation regions in the operating parameter plane spanned by the flow-rate ratios in the rectifying and stripping section; (ii) the evaluation of the optimal operating conditions corresponding to each different separation regime, together with the proof that these are equivalent to the Underwood minimum reflux conditions; (iii) the explanation of well-known features of multicomponent distillation such as pinch conditions and nonlinear behavior. Numerical examples of binary and multicomponent separations are presented and discussed. 1. Introduction This work is intended to provide a tool for the design and optimization of multicomponent distillation columns, which is particularly suited for application to high-purity columns, i.e., columns with a very large number of stages operated near the minimum reflux limit. This is based on a short-cut approach, which shares not only several assumptions but also simplicity of implementation with other short-cut methods for distillation design. However, the present approach has the unique advantage of being based on a model that allows one to achieve a deep insight on several steadystate and dynamic features of multicomponent distillation columns. In particular, high-purity columns exhibit a highly nonlinear character, which is evidenced by sharp composition and temperature profiles, where large concentration changes occur in small portions of the column, and complex dynamic features, such as high steady-state gains, large response lags, and asymmetric dynamics (Hwang, 1991, 1995). The most popular and easy to use short-cut technique for designing binary distillation columns is the McCabe-Thiele method. This simple method allows one not only to quantitatively predict minimum reflux conditions and minimum number of stages but also to get a clear illustration and a pictorial understanding of these and other concepts, such as that of pinch conditions. Nothing of the kind is available for multicomponent distillation, where the need for a multivariant description of vapor-liquid equilibrium makes it very difficult to visualize the column design procedure. Hence, stage-by-stage models are used, which have a mathematically simple structure but also a serious * To whom correspondence should be addressed. Telephone: ++41-1-6323034. Fax: ++41-1-6321082. E-mail: [email protected]. † Present address: Institut fu ¨ r Verfahrenstechnik, ETH Zu¨rich, Sonneggstrasse 3, CH-8092 Zu¨rich, Switzerland.

drawback. In fact, in the case of high-purity columns, which operate near the minimum reflux or infinite theoretical stage limit, the number of equations is very large (algebraic or ordinary differential equations in the case of steady-state or dynamic calculations, respectively), thus allowing one to reach a limited physical insight into the column behavior. To overcome this limitation, the work by Doherty and co-workers analyzes the classical stage equations through a dynamical systems approach (cf. Julka and Doherty, 1990). In this work a different approach is followed. The distillation column is regarded as a continuous countercurrent contactor with negligible axial dispersion, thus reducing the dynamic model of the column to a set of first-order partial differential equations (Gilles and Retzbach, 1980; Nandakumar and Andres, 1981; Hwang, 1991). These are more complex than stage-by-stage equations, but they are small in number and can be treated using convenient mathematical tools which allow one to reach a deep insight into both steadystate and dynamical column behavior. In this framework, simple distillation columns, i.e., columns with a single feed and two product streams, can be seen as two countercurrent sections, the stripping and rectifying ones, connected by the feed equilibrium stage, together with a condenser and a reboiler (see Figure 1). It is remarkable that following this approach the condenser and reboiler characteristics, i.e., whether they are total or partial, have no influence on the column performances. The column illustrated in Figure 1 may describe not only packed columns but also plate columns when the number of stages is large, i.e., when a fine space description is needed. In practice, this kind of column is used when high-purity separations are designed; hence, an accurate optimal design is required. It is worth noting that a large number of stages is a necessary requirement to achieve high-purity performance; however, as we will see in the following that a wrong choice of operating conditions also leads columns

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Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2251

Figure 1. Schematic representation of a distillation column in terms of two continuous countercurrent sections and one feed stage.

with a large number of stages to poor separation performances. The derivations presented in this work are based on the assumption of constant molar overflow (CMO) and that of negligible mass-transfer resistances, i.e., thermodynamic equilibrium attained everywhere at every time in the column. The latter is equivalent to the assumption of ideal equilibrium stages in stage-by-stage models. Thus, a low-order nonlinear model is obtained that can represent the key nonlinearity of the system, though being simple and having physical significance. In this context, two approaches are possible. On the one hand, a generic equilibrium relationship is considered and the so-called nonlinear wave theory is applied either to study the dynamics of nonlinear distillation columns both theoretically (Hwang, 1991, 1995) and experimentally (Hwang et al., 1996) or to develop nonlinear controllers for high-purity columns (Han and Park, 1993; Balasubramhanya and Doyle, 1997). In this case binary separations are easily handled, but multicomponent ones constitute a rather tough challenge. On the other hand, the further assumption of constant relative volatility (CRV) is enforced; thus, the continuous model of the multicomponent distillation column becomes tractable using powerful mathematical tools and clear analytical results can be obtained. In fact, under these assumptions the model equations constitute a system of hyperbolic first-order partial differential equations (PDEs); these were studied first by Lax (1957) in the general case, then by Amundson and co-workers (Rhee et al., 1970, 1971), and independently by Helfferich and Klein (1970) in the particular case of chromatographic processes characterized by the Langmuir adsorption isotherm. The mathematical structure of the x-y relationship in the case of the vapor-liquid equilibrium under the CRV assumption is the same as that of the Langmuir isotherm in adsorption equilibrium. This last approach has been proposed by Andres and co-workers (Nandakumar and Andres, 1981; Wachter et al., 1988) and is applied here; this is particularly useful for the steady-state analysis of distillation performances. The same approach has been applied by Hwang and Helfferich to the study of the dynamics of

single-section countercurrent separation units (Hwang and Helfferich, 1988, 1989) and by Morari and coworkers to the analysis of batch distillation columns with a middle vessel (Davidyan et al., 1994). It is worth noting that the same assumptions, i.e., CMO and CRV, must be enforced to apply the Underwood criterion, which is the simplest method to determine minimum reflux conditions for multicomponent distillation columns. This was first proposed and demonstrated by Underwood (1948), through an algebraic manipulation of stage-by-stage equations; then it was derived again by Amundson, giving an elegant closed-form solution of the same stage-by-stage equations (Acrivos and Amundson, 1955a,b) and using selfadjoint operators (Ramkrishna and Amundson, 1973); then Julka and Doherty (1990) gave a geometrical interpretation of the Underwood criterion through a dynamic systems approach. It is remarkable and enlightening that the same criterion can be derived using the continuous model applied in this work, as shown by Nandakumar and Andres (1981). The purpose of this work is to apply the continuous description of multicomponent distillation columns under the assumptions discussed above to determine not only the optimal operating conditions for the desired separation but also the regions in the operating parameter space corresponding to the different achievable steady-state separation regimes. This extends the applicability of the continuous model not only to minimum reflux calculations but also to the prediction of the column performances under nonoptimal operating conditions. Moreover, these findings allow one to fully characterize the nonlinear behavior of the column and to clarify well-known features of high-purity columns; in particular, the emergence of pinch conditions and the occurrence of large composition and temperature excursions for small changes of the reflux and reboil ratio are thoroughly discussed. The paper is organized in such a way that the mathematical demonstrations are confined in the appendices, whereas the body of the paper can be followed based on the background material provided in section 3. In section 2 the model equations and the assumptions on which our approach is based are presented with reference to a single section of the column, together with an illustration of the comparison between the description of the column provided by the continuous Equilibrium Theory and that given by a classic stage-by-stage model. In the following section 3 the whole distillation column is considered and the different separation regimes are defined; the role of the operating mode of the condenser and the reboiler is analyzed. In section 4 the main results of the paper are presented; they are summarized in Tables 2 and 3, where the procedure to determine the regions of separation in the operating plane drawn in Figures 9 and 10 is given. Finally, based on these findings in section 5 we discuss the effects of the different parameters on the separation performances and the nonlinear features of the behavior of high-purity distillation columns. 2. Model Equations and Single-Section Analysis 2.1. Stage-by-Stage and Continuous Countercurrent Model. With reference to Figure 1, a simple distillation column is constituted of the reboiler, the condenser, and two sections (namely, the stripping and rectifying sections) connected by the feed plate, where

2252 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998

the mixture to be separated is introduced. Thus, the analysis of a single section provides a basis for the understanding of the column as a whole. This can be carried out by studying the model equations of a single section of the column and their solution. First, let us introduce the classical stage model equations, which will then be recast in continuous form. The following assumptions are enforced: (1) The mass transport resistances are negligible; thus, everywhere in the column the liquid and the vapor phases are locally at equilibrium. (2) Constant molar overflow (CMO) is assumed; thus, the molar flow rates of the liquid and vapor phases are constant along the rectifying and stripping sections, though they can change across the feed plate. These assumptions are usually for the short-cut methods and yield the following time-dependent massbalance equation of the jth component on the nth stage of either the stripping or the rectifying section:

d(hlxnj + hvynj ) ) ls dt L(xn+1 - xnj ) - V(ynj - yn-1 ) (j ) 1, ..., C) (1) j j where ls is the height of the stage, hv and hl are the vapor and liquid holdup per unit length, respectively, t is time, L and V are the molar flow rates of the liquid and vapor phases, respectively, and C is the number of species. The variables xj and yj represent the mole fraction of the jth species in the liquid and vapor phases, respectively, and their superscripts indicate the stage number that increases in the direction of the vapor flow. Our aim is to study high-purity columns, with a large total number of stages N, where the composition change from one plate to the other is small. Therefore, it is reasonable to approximate the discrete composition profiles with continuous functions xj(t,ξ) and yj(t,ξ), where ξ is the axial coordinate defined as ξ ) 1 - n/N:

xj(t,ξn+1) ) xj(t,ξn) -

|

|

∂xj ∂2xj ∆ξ2 - O(∆ξ3) ∆ξ + 2 ∂ξ n 2 n ∂ξ (2)

|

|

∂yj ∂2yj ∆ξ2 ∆ξ + 2 yj(t,ξn-1) ) yj(t,ξn) + + O(∆ξ3) ∂ξ n ∂ξ n 2 (3) where ∆ξ is the distance between two consecutive stages in terms of the ξ coordinate, i.e., ∆ξ ) 1/N, and terms on the order of ∆ξ3 (or higher) are neglected. Substituting eqs 2 and 3 into eq 1 and multiplying it by N yield

lsN

∂[xjhl + yjhv] ∂[xjL - yjV] + ) ∂t ∂ξ ∂2xj ∂2yj 1 L 2 +V 2 2N ∂ξ ∂ξ

[

]

(j ) 1, ..., C) (4)

Thus, both the rectifying and stripping sections of the column can be regarded as countercurrent units described through continuous PDEs. The right-hand side of the last equation is dependent on the second-order space derivatives of the composition profiles and therefore plays the role of an axial mixing term. Let the number of equilibrium stages increase, while the column length and holdups are kept constant. Soon, the right-

hand side of eq 4 becomes negligible and can be dropped. Thus the mass-balance equations can be recast in the following dimensionless form:

∂[xj + ryj] V ∂[mxj - yj] + ) 0 (j ) 1, ..., C) (5) ∂τ Qr ∂ξ where r ) hv/hl, m ) L/V is the molar flow rate ratio, and Qr is a reference flow rate. Finally, the dimensionless time is defined as τ ) Qrt/(Nlshl), where the denominator represents the overall liquid holdup of the column. It is worth noting that the reflux and reboil ratios in the rectifying and stripping sections, respectively, are given in terms of the relevant flow rate ratio as follows:

reflux ratio )

mR LR ) VR - LR 1 - mR

(6)

reboil ratio )

1 VS ) LS - VS mS - 1

(7)

Now, the constant relative volatility assumption model is enforced (CRV), as is typical when short-cut methods are applied. Accordingly, let us consider a C-component system where

yj ) KrRjxj (j ) 1, ..., C)

(8)

where Rj is the relative volatility, defined as the ratio of the partition constant Kj of the jth species to the partition constant Kr of the rth species, taken as the reference component. Partition constants are strongly dependent on temperature, whereas the CRV assumption implies that the relative volatility Rj is regarded as a temperature-independent parameter. The species are numbered by ascending relative volatility; hence, the least volatile species is the first one and the most volatile one is the Cth component. Applying the stoiC yj ) 1 yields chiometric relationship ∑j)1

yj )

Rjxj

)

Rjxj

C

aixi ∑ i)1

δ

(j ) 1, ..., C)

(9)

where δ is defined as C

δ)

1

Rixi ) ∑ K i)1

(10)

r

Hence, it is a function of temperature. 2.2. Equilibrium Theory for Distillation Columns. In a more general frame, the study of countercurrent units where axial dispersion and mass-transport resistances are negligible is referred to as Equilibrium Theory. In this case, PDEs of the type of eq 5 together with an equilibrium model (in this case eq 9) and proper initial and boundary conditions constitute the model equations of the unit. Equilibrium Theory shows that the model equations of the single section of the distillation column have an analytical solution in the case of the Riemann problem (cf. Rhee et al., 1971), i.e., when the initial composition is constant along the section, and streams with constant composition in time are fed at


Figure 2. Schematic diagram of a continuous countercurrent section.

the two ends of the section. With reference to Figure 2, these are given by

xj(0,ξ) ) x0j xj(τ,0) ) xAj yj(τ,1) ) yBj (j ) 1, ..., C)

(11)

Although Equilibrium Theory allows us to obtain the time-dependent solution of the model equations, in this work we are interested only in the steady-state asymptotic behavior. The solution of the steady-state problem for a generic countercurrent unit can be obtained directly, i.e., without considering the transient behavior, as shown in Appendix A. Upon inspection of the structure of the second term of eq 5, it is clear that this solution depends on the value of the parameter m and on the equilibrium relationship (9), together with the composition of the streams fed at the two ends of the unit according to the initial and boundary conditions (eq 11). According to Appendix A, the possible solutions can be classified in three different groups referred to as g-constant state, g-wave, and g-shock, respectively; the occurrence of these is controlled by the operating parameter m. As demonstrated in Appendix B, the structure of conventional distillation columns imposes some limitations on the possible achievable steady state. As a typical situation, at steady state the solution is a g-constant state, where the compositions of the liquid and that of the corresponding equilibrium vapor phase are constant along the section. This composition is, in general, different from those of the inlet and outlet streams fed at the two ends of the section; thus, in a typical case, the solution at steady state is constituted of a constant concentration profile with discontinuities at one or both ends. 2.3. Binary Distillation. Let us consider a binary distillation column operated under the conditions reported in the caption of Figure 3 and let us focus on the rectifying section. In the McCabe-Thiele diagram shown in Figure 3a, we see that the mole fraction of the light component increases along the rectifying section and in this case the composition changes occur mainly near the feed plate. As the top of the column is approached, the rate of concentration increase becomes smaller and smaller, and ultimately the composition change from one stage to the next becomes negligible and the so-called pinch conditions are attained. The corresponding composition profile of the rectifying section of the distillation column is shown in Figure 4a where the results obtained through Equilibrium Theory are compared with those obtained using the stagewise model (this calculation is made using a finite but large number of stages as reported in the caption of the figure). It can be observed that the pinch composition

Figure 3. Binary distillation: McCabe-Thiele diagrams. System and operating parameters: R1 ) 1, R2 ) 2, z1 ) z2 ) 0.5, q ) 1, mS ) 1.37. (a) mR ) 0.75; (b) mR ) 0.62; (c) mR ) 0.67.

and the concentration change near the feed plate are regarded by Equilibrium Theory as a constant state, with the same composition of the state entering the column from the condenser, and a boundary discontinuity, respectively (cf. Appendix A). When described through a stagewise model, high-purity columns, which


given by eq 4 has been studied in detail in chromatographic applications (cf. Mazzotti et al., 1994a). In practice, the solution of eq 5 is a good approximation of the solution of eq 1 and, in addition, pinch conditions are predicted exactly (see Figure 4). The application of these concepts to the dynamics of binary distillation columns has been discussed also by Hwang (1991, 1995). The occurrence of a particular constant state (i.e., pinch conditions) in a section of the column is controlled by the value of the corresponding molar flow rate ratio m. With reference to Figure 3a, if the molar flow rate ratio of the rectifying section mR is decreased (keeping that of the stripping section mS constant), the constant state prevailing in the rectifying section changes, as demonstrated in Figures 3b and 4b. In this case the composition change (a discontinuity in the Equilibrium Theory case or an abrupt continuous transition if the stage model is used) is near the condenser and a constant state with the same composition of the feed plate prevails through the rectifying section; i.e., the pinch conditions occur at the feed plate. It is worth noting that a high-purity distillate is not obtained any more. Comparing parts a and b of Figure 3, it is clear that there exists a value of m, for which a pinch occurs at both ends of the section. This condition is indicated as 1-shock in Appendix A (where the relationship for the corresponding m value is given) and is illustrated in Figures 3c and 4c. According to Equilibrium Theory, for a particular value of mR, a discontinuity is present inside the rectifying section which separates two constant states. In other words, for a particular value of the flow rate ratio a double pinch condition is attained: as can be observed in Figures 3c and 4c, a pinch state corresponds to the composition on the feed plate, whereas the other corresponds to the composition entering the column from the condenser. This particular value of the molar flow rate ratio is critical: since it separates two different operation regimes of the rectifying section: if a flow rate ratio m larger than this is used, the situation illustrated in Figures 3a and 4a is obtained, whereas if a smaller value is adopted, the situation illustrated in Figures 3b and 4b occurs. Thus summarizing, Equilibrium Theory is a useful and powerful tool to analyze pinch conditions in binary and multicomponent distillation columns, as we will see in more detail in the following sections. Pinch conditions are exhibited very often by high-purity columns, with a larger number of plates. Therefore, the whole picture looks consistent; in fact, we develop a tool based on a continuous description of a distillation column, which is proper for a stage column only when the number of stages is large, i.e., a high-purity column, and this tool is especially powerful in describing pinch conditions, which are a well-known feature of highpurity distillation columns. 3. Column Analysis Figure 4. Binary distillation: liquid composition profiles in the rectifying section for the more volatile component (species 2). System and operating parameters as in Figure 3. (s) Equilibrium Theory model; (O) stage model (N ) 92; feed stage, 47).

lead to pinch conditions, exhibit smooth composition changes (shock layers) instead of discontinuous transitions. This is a well-known general result: the relationship between shock discontinuities obtained through Equilibrium Theory models and shock layers calculated using equilibrium dispersive models such as the one

The objective of this section is to study the distillation column as a whole, using the approach and the results obtained in the previous section for a single countercurrent section. To this aim, let us first show that the continuous model provides a realistic description of multicomponent columns by comparing its prediction with that of a multistage model in the case of the separation of a ternary mixture. The composition profiles of the three components along a simple column calculated using the continuous


distillate only. Therefore, the following conditions hold true:

dj ) 0 wj ) zjF j ) 1, ..., p dj, wj > 0 wj + dj ) zjF j ) p + 1, ..., k - 1 wj ) 0 dj ) zjF j ) k, ..., C

(12)

dj and wj are the flow rate of the jth component in the distillate and in the bottoms, respectively. It is clear that the separation regime is fully specified once the numbers p and k are given. According to their definition, p and k obey the following consistency constraints:

2ekeC+1 0 e p e min{k - 1, C - 1}

Figure 5. Ternary distillation: liquid composition profiles. System and operating parameters: R1 ) 1, R2 ) 2, R3 ) 3, z1 ) 0.25, z2 ) 0.30, z3 ) 0.45, q ) 1; mR ) 0.55; mS ) 1.30. Equilibrium Theory model (s). Stage model (N ) 100; feed stage, 51): component 1 (0); component 2 ()); component 3 (O).

model, as presented in Appendix E, are reported in Figure 5, together with the same profiles obtained through the stage model. The system parameters and the operating conditions are given in the caption of the figure. It can be observed that pinch conditions as well as boundary discontinuities occur in both the rectifying and the stripping sections. As far as the first component is concerned, the reflux in the rectifying section is sufficiently high so as to eliminate it from this section. A discontinuity is present across the feed plate, through which this component disappears. Its mole fraction remains constant along the stripping section and increases through a boundary discontinuity at the bottom of the stripping section itself. As far as the third component is concerned, the profile of its mole fraction is symmetric to that of the first component. The second component is distributed between the top and the bottom of the column. Its mole fraction is constant across both sections and changes in a discontinuous fashion across the feed plate, at the top and the bottom of the rectifying and stripping sections, respectively. As well as in the binary case, the continuous model provides a very accurate prediction of the composition of the pinch states, while, on the other hand, concentration changes are viewed as discontinuous transitions. The same conclusions can be reached also in the case of the distillation of a mixture of any number of components. Note that, in general, a constant state prevails in both sections of the column and discontinuities are present at both ends of each section. Moreover, the pinch composition is not necessarily coincident with that of the feed plate or with that of the outlet stream as in the binary case. Let us now proceed to define the different separation regimes of a multicomponent distillation column and the key operating parameters. Afterward, necessary and sufficient conditions for the column operating in a given regime are provided. 3.1. Operating Parameters and (p, k) Separation. Let us consider the separation regime where the least volatile components labeled from 1 to p are present in the bottoms only, the intermediate ones labeled from p + 1 to k - 1 are distributed, and the most volatile species, with indices from k to C, are collected in the

(13)

It is worthwhile noting the following special cases: when k ) C + 1, all components are present in the bottoms (i.e., no component is collected in the distillate only); if p ) 0, all components are present in the distillate; k ) p + 1 means that no component is distributed. In the following we will refer to the separation regime defined by p and k as the (p, k) separation. The above definition corresponds to a socalled sharp separation, where the maximum concentration of the heavy key, p, in the distillate and of the light key, k, in the bottoms is specified to be zero. In the frame of a short-cut design method this is a properly given specification for a high-purity distillation column; a more detailed discussion about the design of sharp separations as compared to nonsharp ones, where maximum nonzero values of the concentrations of the key components are specified, is reported in section 5.4. As shown in section 2, the separation performances of a countercurrent section are determined by its molar flow rate ratio m. Once the m values of both sections, i.e., mR and mS, are given, then the separation regime and the column performances are determined. Thus, the operating parameter space is bidimensional, or, in other words, the system has 2 degrees of freedom. As key operating parameters, it is convenient to use the following ones:

Φ ) δRLR/VR ) δRmR

(14)

Ψ ) δSLS/VS ) δSmS

(15)

The superscripts R and S refer to the rectifying and stripping sections, respectively. Recalling the definition of mR and mS, Φ and Ψ represent the absorption factors (or the reciprocals of the stripping factors) for the reference component in the rectifying and stripping sections, respectively. The effect of these parameters on the behavior of the corresponding section has been fully described in Appendix A. Together, they play a key role in determining the operating conditions of the column. In Appendix A it is shown that there exists a one to one mapping between the set of equilibrium liquid and vapor compositions, given in terms of mole fractions x and y, and the vectors of a (C - 1)-dimensional space. These are referred to as Λ vectors and are defined through eqs 32 and 38 (or alternatively through eqs 33 and 37). This mapping exhibits useful features (see Appendix A), which are particularly helpful when the operating regimes of a countercurrent separation unit, in particular a distillation column, are analyzed.


Through vectors Λ it is rather straightforward to determine the conditions to operate the column in a specified operating regime, the pinch compositions and location, and the minimum reflux operating conditions. First, let us determine the conditions to operate the distillation column in a given operating regime. In particular, these conditions are expressed in terms of the absorption factors Φ and Ψ, the values of the relative volatility of the species in the feed mixture, Rj, and the vector corresponding to the composition of the feed plate Λ* ) (λ*1, ..., λ*C-1). At this stage of the discussion this vector is still unknown and cannot be given directly in terms of the known feed composition. This difficulty will be overcome in the next section. However, whatever the relationship between the vector Λ* and the feed mixture, it can be stated that this vector identifies univocally the composition of both vapor and liquid streams leaving the feed plate, which is considered an equilibrium stage. The required conditions are given in the following rule, which is demonstrated in Appendix B: Rule 1. Necessary and sufficient conditions for the column operating in the (p, k) separation regime are that the operating parameters Φ and Ψ fulfill the following constraints:

λ* p e Φ < Rp+1, p * 0 0 < Φ < R1, p ) 0

(16)

Rk-1 < Ψ e λ* k-1, k * C + 1 RC < Ψ < ∞, k ) C + 1

(17)

This rule defines the region of the operating parameter plane, constituted of operating points which allow the unit to operate in the (p, k) separation regime; thus, in the following this will be referred to as the (p, k) region. In Appendix B it is also shown that, at steady state, the compositions of the liquid and vapor phases are constant along both the rectifying and stripping sections. The vectors corresponding to the equilibrium composition prevailing inside the rectifying and stripping sections are referred to as ΛR and ΛS, respectively. For a pair of values Φ and Ψ, which according to Rule 1 lead to a (p, k) separation, the corresponding vectors ΛR and ΛS are given according to the following corollary: Corollary 1. When in the (p, k) separation regime, the states prevailing in the rectifying and stripping sections of the column are defined by the vectors:

ΛR ) (R1, ..., Rp, λ* p+1, ..., λ* C-1)

(18)

ΛS ) (λ* 1, ..., λ* k-2, Rk, ..., RC)

(19)

3.2. Partial/Total Condenser and Reboiler. The conditions derived above are very simple but cannot be used directly since they involve the components of the vector Λ*, which, at this stage, are still unknown. However, it is already possible to show that these are independent of how the condenser and the reboiler are operated, i.e., in the total or in the partial mode. This is a consequence of the large number of stages of the high-purity columns under consideration, which allow the adoption of a continuous description of the distillation column itself. In fact, let us refer to Figure 6 and consider the rectifying section and the condenser (simi-

Figure 6. Schematic representation of the condenser and the top of a distillation column at steady state. Operating conditions are such that only components from p + 1 to C are collected in the distillate. The condenser may be partial (D ) DV and DL ) 0) or total (D ) DL and DV ) 0).

lar considerations can be made for the stripping section and the reboiler). When the column is operated in the (p, k) separation regime, according to Appendix B, the vector ΛD associated with the stream entering the column from the top is given by D D ΛD ) (R1, ..., Rp, λp+1 , ..., λC-1 )

(20)

and coincides with that of the distillate. In fact, these two streams either share the same physical state and composition (the case of a total condenser, where D ) DL and DV ) 0) or are equilibrium liquid and vapor phases, respectively (the case of a partial condenser, where D ) DV and DL ) 0). Hence, the values λD i , i ) (p + 1), ..., (C - 1), depend on the condenser features, i.e., on whether it is total or partial. Assuming that the composition of the steady state within the rectifying section, and hence its corresponding ΛR, is known, these λD i values can be calculated first by making material balances through Envelope 2 in Figure 6, thus calculating the values of the distillate flow rates, dj, and the corresponding mole fractions. Then from the knowledge of the mole fractions, the values λD i can be calculated accounting for the operating mode of the condenser: if it is total, then the distillate is liquid and eq 32 applies; if it is partial, then the distillate is vapor and eq 33 is used. In order to complete the analysis, the composition of the vapor stream coming out from the top of the column is calculated through a material balance around the condenser, i.e., through Envelope 1. According to Corollary 1, ΛR is built by using the first p components of ΛD, which are known R values, and the last (C - p + 1) components of Λ*, which are fixed though yet unknown in our analysis. As a consequence, the operating mode of the condenser has no influence on the behavior of the column. The physical reason of this feature is that if a partial condenser is used, this simply represents an additional stage. However, since in this frame the number of stages is very large, actually infinite, the presence of an additional stage is negligible. Accordingly, the effects of the operation of the reboiler on the state prevailing in the stripping section can be easily determined just through the separation requirements. 4. Separation Regions and Optimal Operating Conditions 4.1. Role of the Feed Stage (i.e., Evaluation of Λ*). The conditions on the absorption factors Φ and Ψ


given by Rule 1 are implicit since, according to the material balance around the feed plate, its composition as well as the corresponding vector Λ* is a function of mR and mS, and hence of Φ and Ψ themselves. In this section we will specify the conditions which guarantee the attainment of a (p, k) separation, by determining the relationship between the state of the feed plate and the state of the feed mixture. Let us consider the vector ΛF associated with the feed stream whose components are obtained by solving the following equation:

H(λ) F

C

)

Rjzj

∑ j)1R

j

-λ

- (1 - q) ) 0

(21)

where zj’s are the feed mole fractions, F is the feed molar flow rate, and q is the enthalpic factor, defined as

q)

LS - LR VR - VS )1F F

(22)

Equation 21, which can be viewed simply as a way to define its solutions, is derived in Appendix A through an overall material balance around the whole distillation column; it can be readily proved that it has C solutions. Among these, C - 1 values fulfill the inequalities (34) and define the vector ΛF; hence ΛF ) F ) with R1 < λF1 < R2 < ... < Ri < λFi < Ri+1 < (λF1 , ..., λC-1 F ... < λC-1 < RC. It is worth noting that the mapping between ΛF and the feed composition z is not the same as that defined by eqs 32, 33, and 35 in Appendix A. As proved in Appendix C, Rule 1 and Corollary 1 can be restated, thus obtaining the following rule, which gives the conditions on Φ and Ψ for the attainment of the (p, k) separation in terms of the components of the vector ΛF: Rule 2. Necessary and sufficient conditions for the column operating in the (p, k) separation regime are that the operating parameters Φ and Ψ fulfill the following constraints:

Figure 7. Schematic representation of a multicomponent distillation column operated at steady state in the (p, k) separation regime. The vectors ΛR, Λ*, and ΛS characterize the compositions of the rectifying section, feed plate, and stripping section, respectively.

λFp e Φ < Rp+1, p * 0 0 < Φ < R1, p ) 0

(23)

F , k*C+1 Rk-1 < Ψ e λk-1

RC < Ψ < ∞, k ) C + 1

(24)

It is worth noting that this rule provides explicit conditions, since ΛF depends only upon the composition and the enthalpic factor q of the feed stream, which are known. Through the knowledge of the vector ΛF it is possible to obtain some components of the vector Λ* immediately. In fact, it can be demonstrated (see Appendix C) that the following relationship between Λ* and ΛF holds in a (p, k) region: F λ* i ) λi (i ) p + 1, ..., k - 2)

(25)

while the remaining elements of the Λ* vector are computed through different equations as discussed later. The states prevailing in the sections of a distillation column, when a (p, k) separation is established, are illustrated in Figure 7. These states are obtained

Figure 8. Separation regions in the (Φ, Ψ) plane for a ternary distillation column. System parameters: R1 ) 1, R2 ) 2, R3 ) 3, z1 ) 0.25, z2 ) 0.30, z3 ) 0.45, q ) 0.5.

according to Corollary 1 and eq 25 and are correct provided that Φ and Ψ fulfill the inequalities (23) and (24). The eight (p, k) separation regions in the (Φ, Ψ) plane for the ternary system, whose characteristic parameters are reported in the caption of the figure, are shown in Figure 8. The separation regions are obtained directly from Rule 2 and require the solution of eq 21 only. This equation, though nonlinear, has some particular features that allow an easy numerical solution; in fact, each interval [Ri, Ri+1] contains one and only one of its roots. It is possible to observe that, according to Rule 2, the separation regions are rectangular and not contiguous and, therefore, not all the values of Φ and Ψ are allowed. With reference to Figure 8, if operative conditions belonging to a boundary of one of the separation regions are not allowed, the corresponding boundary is dashed. Thus summarizing, using eq 21 and the known values of the feed composition and the enthalpic factor, the parameters λFi , with i ) 1, ..., C - 1, can be calculated

2258 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 Table 1. Ternary Distillation: Structure of the Λ Vectors Characterizing the Steady-State Compositions in the Rectifying and Stripping Section (p, k) separation

λR1 λR2 λS1 λS2

Table 2. Procedure for the Calculation of the Optimal Operating Point in the (p, k) Region (0 < pe k - 1 < C) F step 1 solve the following equation in the unknowns λFp , ..., λk-1 :

C

(0, 2)

(1, 2)

(0, 3)

(1, 3)

(2, 3)

(0, 4)

(1, 4)

(2, 4)

λ* 1 λ* 2 R2 R3

R1 λ*2 R2 R3

λF1 λ* 2 λF1 R3

R1 λ* 2 λ* 1 R3

R1 R2 λ* 1 R3

λF1 λF2 λF1 λF2

R1 λF2 λ* 1 λF2

R1 R2 λ*1 λ*2

(λF1 ) 1.238, λF2 ) 2.391 in the case illustrated in Figure 8). Then the regions of separation in the (Φ, Ψ) plane are obtained from Rule 2. For each (p, k) region the vectors Λ corresponding to the steady state prevailing in the rectifying and stripping sections, i.e., the pinch compositions, are given by eqs 18 and 19. These are reported in Table 1 for the case illustrated in Figure 8 and could be used to draw a representation of the distillation column such as the one given in Figure 7 for each (p, k) separation. 4.2. Operating Parameter Plane and Optimal Conditions. The result obtained in the previous section is practically useful, because it allows one to calculate all possible steady-state regimes and pinch compositions of a distillation column in a simple way. However, it is not yet fully satisfactory because the composition states in Table 1 and the flow rate ratios mR and mS, which are the manipulated parameters of the unit, depend on the values of Λ*, which are still unknown. This difficulty can be overcome with the help of the result demonstrated in Appendix D, i.e., that when represented in the (mR, mS) plane the regions of separation are indeed contiguous. This result is expected on physical grounds, because mR and mS represent reflux ratios (cf. eqs 6 and 7) and it is not possible that some ranges of values are forbidden. After this observation, a complete analysis of the operating conditions in the (mR, mS) plane can be performed, as reported in the following. For each (p, k) separation, first we determine the optimal operating conditions and then the corresponding separation boundaries in the (mR, mS) plane. The optimal operating conditions for a simple distillation column in a (p, k) separation regime are those making the process satisfy the separation requirements with the minimum operating cost. Accordingly, the operating conditions are optimal when mR is minimum and mS maximum, which implies that the heat exchanged in the reboiler and the condenser is minimum. If mR is reduced below this value, the pth species shows up in the distillate. On the other hand, if mS is increased above this maximum value, the kth species shows up in the bottoms. In Appendix C, the optimal values of Φ and Ψ (i.e., those corresponding to mRmin and mSmax) are derived, thus yielding the following rule: Rule 3. In a (p, k) separation regime, with 0 < p e k - 1 < C, the optimal operating conditions are given by

Φopt ) λFp ) λ* p

(26)

F ) λ* Ψopt ) λk-1 k-1

(27)

It may be noted that the optimal values of Φ and Ψ are on the boundaries of the intervals defined by eqs 23 and 24 for the attainment of the (p, k) separation.

Rjzj

∑R - λ - (1 - q) ) 0 j)1

j

step 2 (a) Φ)λFp F (b) Ψ ) λk-1 (c) wj ) zjF and dj ) 0; j ) 1, ..., p (d) dj ) zjF and wj ) 0; j ) k, ..., C step 3 solve the following linear system of equations in the unknowns VS and wj; with j ) p + 1, ..., k - 1: k-1

Rjwj

∑R -λ

j)p+1 j

p

+ VS ) m

RjFzj

∑R - λ j)1

j

m

F F with λm ) Φ, λp+1 , ..., λk-2 ,Ψ

step 4 (a) dj ) zjF - wj with j ) p + 1, ..., k - 1 k-1 (b) LS ) VS + ∑j)1 wj R S (c) V ) V + (1 - q)F (d) LR ) LS - qF step 5 (a) mR ) LR/VR (b) mS ) LS/VS

The optimal operating point in the (p, k) region, with 0 < p e k - 1 < C, can be calculated through the procedure reported in Table 2. The separation regimes with either p ) 0 or k ) C + 1 do not have any optimal operating point according to the definition given above. In fact, in the (0, k) regime, mR can be reduced to its lower physical limit, i.e., zero, without changing the separation regime, since the rectifying section actually does not have any separation to perform. Similar considerations can be repeated for the (p, C + 1) regime and the possibility of increasing mS to infinity. The demonstration of the procedure reported in Table 2 is straightforward. Step 1 deals with the solution of eq 21. Substeps a and b of step 2 implement eqs 26 and 27, and substeps c and d enforce the separation requirements, which define the (p, k) separation. Step 3 is obtained from eqs 59, by exploiting the results of substep 2c. Step 4 implements different material balances exploiting eq 22 for the definition of the enthalpic factor, q. Step 5 is just the definition of the flow rate ratios. It is worth noting that Table 2 has an algorithmic character. One can observe the presence of indices i and j running from a lower to an upper limit; wherever the former is larger than the latter, the assignment must be ignored or the sum must be set equal to zero. 4.3. Separation Regions in the Operating Parameter Plane. It is remarkable that the optimal operating conditions computed above are equal to those given by the Underwood criterion for a sharp separation and usually referred to as the minimum reflux conditions with an infinite number of ideal stages (Underwood, 1948; Nandakumar and Andres, 1981). Moreover, Equilibrium Theory allows one to calculate not only the optimal operating points but also all the attainable separation regions, which, of course, are not given by the Underwood criterion. According to Appendix D, separation regions are contiguous in the (mR, mS) plane. Therefore, the image of the boundary of the (p, k) region defined by Φ ) Rp+1 coalesces with that of the boundary of the (p + 1, k) F . For the same reason the region defined by Φ ) λp+1 image of the boundary of the (p, k) region defined by Ψ ) Rk-1 coalesces with that of the boundary of the (p, k F . Thus, only one - 1) region defined by Ψ ) λk-2

Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 2259 Table 3. Procedure for the Calculation of the Upper Boundary (Ψ ) Ψopt) of the (p, k) Region (0 < p < k - 1 < C) F step 1 solve the following equation in the unknowns λFp , ..., λk-1 :

C

Rjzj

∑R - λ - (1 - q) ) 0 j)1

j

step 2 (a) choose Φ, with λFp < Φ < Rp+1 F (b) Ψ ) λk-1 (c) wj ) zjF and dj ) 0; j ) 1, ..., p (d) dj ) zjF and wj ) 0; j ) k, ..., C step 3 solve the following linear system of equations in the unknowns VR and dj; with j ) p + 1, ..., k - 1: k-1

Rjdj

∑R -λ

j)p+1 j

C

- VR ) m

RjFzj

∑R - λ j)k

j

m

F F with λm ) Φ, λp+1 , ..., λk-2 ,Ψ

step 4 (a) dj ) zjF - wj with j ) p + 1, ..., k - 1 k-1 (b) LS ) VS + ∑j)1 wj (c) VR ) VS + (1 - q)F (d) LR ) LS - qF step 5 (a) mR ) LR/VR (b) mS ) LS/VS

condition is needed and we will use eqs 26 and 27 to describe the boundaries of the separation regions. As to the boundary defined by eq 26, its image in the (mR, mS) plane can be traced by varying Ψ between Rk-1 and F , while Φ ) Φopt ) λFp , and calculating the correλk-1 sponding point in the (mR, mS) plane. A single point on this line can be calculated using a slightly modified version of the procedure reported in Table 2. In particular, step 2c must be modified as follows: F . Substep 2c: choose Ψ, with Rk-1 < Ψ < λk-1 The procedure must be repeated with different values of Ψ until the boundary is entirely traced. The boundary defined by eq 27 is obtained in a similar way; for the sake of clarity the entire procedure necessary for its calculation is reported in Table 3. When k ) C + 1 or p ) 0 or 0 < p ) k - 1 < C, a simpler approach than the one above allows one to obtain an explicit expression for the boundaries of the corresponding (p, k) region in the (mR, mS) plane. The procedure is described in detail in Appendix D. After repeating the above calculations for all the (p, k) regions of the system under examination, a complete picture of the regions of separation in the operating parameter plane (mR, mS) is obtained. A first example of this is illustrated in Figure 9, where the same regions shown in Figure 8 are drawn. It is worth noting that the (p, p + 1) regions, identifying complete separation conditions where there are no distributed components, are one-dimensional loci given by eq 72. Using Figure 9, it is possible to predict the qualitative performance of the separation for a given pair of values of the reflux ratios mR and mS, by observing the location of the corresponding point in the (mR, mS) plane with respect to the (p, k) regions. The quantitative prediction of the separation performances and the composition profile in the column can be performed following the procedure discussed in Appendix E. On the other hand, for any desired (p, k) separation regime the figure indicates the set of operating conditions which can be chosen to achieve it. As a further and more complex example the separation regimes attainable in the case of the distillation of a five-component mixture are illustrated in Figure 10.

Figure 9. Separation regions in the (mR, mS) plane for the ternary system considered in Figure 8.

Figure 10. Separation regions in the (mR, mS) plane for a fivecomponent distillation. System parameters: R1 ) 1, R2 ) 2, R3 ) 3, R4 ) 4, R5 ) 5, z1 ) 0.1, z2 ) 0.2, z3 ) 0.3, z4 ) 0.25, z5 ) 0.15, q ) 0.5.

Nineteen regions, among which four are one-dimensional loci, are drawn, according to the general rule that for a C-component system (C2 + 3C - 2)/2 regions exist. Out of these C - 1 are one-dimensional complete separation regions, corresponding to the C - 1 possible cuts of a C-component mixture. Although the complete picture is rather complex, its calculation based on the procedures in Tables 2 and 3 is rather simple. This is shown by the examples reported in Tables 4 and 5, which refer to the evaluation of the optimal operating point of the (3, 4) region (components 1, 2, and 3 in the bottoms; components 4 and 5 in the distillate) and of the (2, 5) region (components 1 and 2 in the bottoms; components 3 and 4 distributed; component 5 in the distillate). 5. Discussion 5.1. Role of Reflux and Reboil Ratio. Several remarks can be made about the maps of the separation regions in the (mR, mS) plane illustrated in Figures 9 and 10. First, let us analyze the effect of changing the operating conditions on the separation performances of

2260 Ind. Eng. Chem. Res., Vol. 37, No. 6, 1998 Table 4. Five-Component Distillation: Example of Calculation of the Optimal Operating Conditions in the (3, 4) Region (cf. Figure 10 and Corresponding Parameters) step 1 step 2

step 3

λF3 ) 3.513 (a) Φ ) λF3 ) 3.513 (b) Ψ ) λF3 ) 3.513 (c) w1 ) 0.10F; d1 ) 0 w2 ) 0.20F; d2 ) 0 w3 ) 0.30F; d3 ) 0 (d) w4 ) 0; d4 ) 0.25F w5 ) 0; d5 ) 0.15F 3

VS ) -

j)1

step 4

step 5

Rjzj

∑R - ΦF ) 2.058F j

(a) not applicable 3 (b) LS ) VS + ∑j)1 wj ) 2.658F (c) VR ) VS + (1 - q)F ) 2.558F (d) LR ) LS - qF ) 2.158F (a) mR ) LR/VR ) 0.844 (b) mS ) LS/VS ) 1.292

Table 5. Five-Component Distillation: Example of Calculation of the Optimal Operating Conditions in the (2, 5) Region (cf. Figure 10 and Corresponding Parameters) step 1

step 2

step 3

step 4

step 5

λF2 ) 2.267 λF3 ) 3.513 λF4 ) 4.712 (a) Φ ) λF2 ) 2.267 (b) Ψ ) λF4 ) 4.712 (c) w1 ) 0.10F; d1 ) 0 w2 ) 0.20F; d2 ) 0 (d) w5 ) 0; d5 ) 0.15F VS ) 0.782F w3 ) 0.162F w4 ) 0.057F (a) d3 ) 0.138F d4 ) 0.193F 3 (b) LS ) VS + ∑j)1 wj ) 1.301F (c) VR ) VS + (1 - q)F ) 1.282F (d) LR ) LS - qF ) 0.801F (a) mR ) LR/VR ) 0.625 (b) mS ) LS/VS ) 1.664

Figure 11. Schematic representation of a typical (p, k) region in the (mR, mS) plane, together with the neighboring regions. Its optimal point, Up,k, as well as that of the (p + 1, k - 1) region, Up+1,p-1, are also indicated. Points X, Y, and W represent possible operating points with different characteristics as far as changes of the reflux and reboil ratios are considered.

the distillation column. To this aim we refer to Figure 11, where a specific (p, k) region with k > p + 2 together with its neighboring regions is considered (such as, for example, the (1, 5) region in Figure 10). A similar but simpler picture applies to regions where p ) 0 or k ) C + 1. Three operating points are indicated: point X in the shadowed zone close to the optimal point of the (p,

k) region (point Up,k, calculated through the procedure reported in Table 2 and corresponding to the Underwood minimum reflux conditions); point Y in the central rectangle of the (p, k) region; point W in the shadowed zone close to the optimal point of the (p + 1, k - 1) region (point Up+1,k-1). Note that when k ) p + 3 (such as the (1,4) and (2, 5) regions in Figure 10), the (p + 1, k - 1) region is, in fact, a complete separation, onedimensional locus. All three operating points, i.e., X, Y, and W, are robust, since small perturbations of the flow rate ratios mR and mS keep the operating point inside the (p, k) region itself and the distillation column in the corresponding (p, k) separation regime. However, finite changes of the operating conditions have different effects in the case of the three points, as discussed in the following. Point Y, as well as all points in the dashed rectangular region, has a normal behavior; in fact, changing either mR or mS yields a change in the set of components collected in the distillate or in the bottoms, respectively. In particular, if mR is increased, the operating point enters the (p + 1, k) region, i.e., through the boundary between the (p, k) and (p + 1, k) regions the (p + 1)th component disappears from the distillate and a cut of the feed mixture with a smaller number of distributed components is obtained. If mR is decreased, the (p - 1, k) region is entered and the pth component appears also in the distillate. By increasing the value of mS, the operating point is driven to the (p, k + 1) region; through the boundary between the (p, k) and (p, k + 1) regions the kth component shows up in the bottoms. On the other hand, if mS is decreased, the (p, k - 1) region is entered and the (k 1)th species disappears from the bottoms. In the cases of points X and W, changing only one parameter at a time gives access only to two neighboring regions and not to all four as in the case of point Y. Let us consider point X, which lies in the zone of the (p, k) region close to its optimal point Up,k. Both an increase of mR and an increase of mS drive the operating point toward the (p, k + 1) region, while, on the other hand, a decrease of either flow rate ratio leads the point into the (p - 1, k) region. It is remarkable that by increasing either mR or mS the same effect is achieved, i.e., the breakthrough of the kth species in the bottoms. Similarly, a decrease of either parameter causes the appearance of the pth component in the distillate. This behavior, which cannot be immediately grasped on physical grounds, is the result of the nonlinear character of the distillation column behavior, which is properly accounted for by the continuous model and is neatly explained by this approach. In all cases discussed above, starting from point X a cut of the feed mixture with a larger number of distributed components is achieved. This deserves a further comment; point Up,k represents optimal, i.e., minimum reflux, operating conditions when it is required that only components (p + 1) to (k - 1) distribute. The point Up,k represents nonrobust operating conditions because even the smallest perturbation may lead the system to an operating region where more components distribute: the kth component in the (p, k + 1) region; the pth component in the (p - 1, k) region; both the pth and the kth components in the (p - 1, k + 1) region. It is worth noting that based on this analysis the (p - 1, k + 1) region will play an important role in the discussion about nonsharp separation specifications in section 5.4.


The situation is somehow symmetric in the case of point W. An increase or a decrease of either operating parameter drives the operating point into the (p + 1, k) or (p, k - 1) regions, respectively. However, in the case of point W these perturbations lead to a smaller number of distributed components. Finally, for operating points within the four triangleshaped zones surrounding the central rectangle of the (p, k) region only three of the neighboring regions are accessible by changing the operating parameters mR and mS one at a time. For (p, p + 2) regions, i.e., where k ) p + 2 and only the (p + 1)th component distributes, the picture is partially different from the one illustrated in Figure 11 (see, for example, the (1, 3) region in Figure 9 and the (1, 3), (2, 4), and (3, 5) regions in Figure 10). The crossing of the boundaries with the (p, k + 1) and (p 1, k) regions shares the same features as in the case illustrated in Figure 11. On the contrary, the (p + 1, k) and (p, k - 1) regions collapse into the onedimensional complete separation (p + 1, p + 2) and (p, p + 1) regions, respectively. Therefore, the neighboring region on the right-hand side of the (p, p + 2) region is the (p + 1, p + 3) region, whereas below it the (p - 1, p + 1) region is found. Therefore, in this case increasing the parameter mR starting from a point such as points Y or W in Figure 11 drives the operating point first onto the boundary of the (p, p + 2) region. This consists of the one-dimensional (p + 1, p + 2) region, where the (p + 1)th species is not present in the distillate any more and the complete separation corresponding to the sharp cut with components 1 to p + 1 in the bottoms and components p + 2 to C in the distillate is achieved. A further increase of mR takes the operating point into the (p + 1, p + 3) region where only the (p + 2)th component distributes. A similar analysis can be performed with reference to the decrease of the parameter mS starting from a point such as Y or W in Figure 11. This discussion clearly indicates that the operating conditions along the complete separation lines, i.e., belonging to one-dimensional (p, p + 1) regions, are not robust at all. In fact, even the slightest perturbation of the operating conditions or the smallest model error would drive the operating point on the complete separation line into a region where at least one species distributes. 5.2. Behavior of Composition Profiles. Let us deepen the analysis of the effect of moving the operating point from one region to a neighboring one by considering not only the separation performances but also the composition profiles. To this aim, any boundary between any pair of regions in Figures 9 and 10 could be considered. Therefore, for the sake of simplicity, we refer to the binary separation used as a case study to draw Figures 3 and 4. The four regions of separation for this binary mixture are drawn with solid boundaries in Figure 12, together with the operating points a, b, and c, corresponding to the operating regimes whose composition profiles are illustrated in Figure 4a-c, respectively. These share the same value of the flow rate ratio in the stripping section, mS, and, in fact, the operation of the stripping section itself does not change qualitatively, as can be readily observed by analyzing the McCabe-Thiele diagrams in Figure 3; in all cases the light component, i.e., component 2, distributes and comes out also with the bottoms.

Figure 12. Binary distillation. (s) Separation regions in the (mR, mS) plane: system parameters and operating conditions in points a, b, and c as in Figure 3. (a) (- -) Separation regions for the same system, but z1 ) 0.5 and q ) 0.82. Point c′: mR ) 0.681. (b) (- ‚ -) Separation regions for the same system, but z1 ) 0.45 and q ) 1. Point c′′: mR ) 0.645.

Therefore, it is interesting to focus on the rectifying section, by first considering point a. This belongs to the (1, 3) region where the heavy component 1 does not distribute, as is apparent from the diagram in Figure 3a. Accordingly, the composition state at the top of the column prevails in the whole rectifying section and a sharp concentration change occurs only at the feed plate. It is worth noting that qualitatively similar profiles are also obtained for all operating points between a and c. On the other hand, point b belongs to the (0, 3) region where both components distribute (see Figure 3b). The composition state of the feed plate prevails in both the stripping and rectifying sections, and in this case the sharp concentration change in the rectifying section is located close to the top of the column. Again, essentially the same profile is obtained for every operating point between b and c. Point c is on the boundary between the two regions. This is reflected by the presence of a concentration front standing somewhere in the middle of the rectifying section (see Figure 4c and also Hwang, 1991). For the given value of the stripping section reflux ratio, mS, point c represents optimal operating conditions of the


rectifying section; in fact, it identifies the minimum flow rate ratio of the rectifying section, mR, which guarantees the achievement of 100% purity in the distillate. This is a nonrobust operating point, since even very small changes of the reflux ratio in the rectifying section drive the column to either the (1, 3) region (with a strong change in the concentration profile, which becomes similar to that in Figure 4a) or the (0, 3) region (with both components distributed). In general terms, if the distillation column is operated under conditions corresponding to a point very close to one of the boundaries of the separation regions, small flow perturbations may lead to a large shift of the composition profiles such as the change from the profile corresponding to a to that corresponding to b. This transition developes in the form of a steep composition front, such as the one shown in Figure 4c, which travels from one end of the column to the other. This corresponds also to the appearance or disappearance of a component from either the distillate or the bottoms; this phenomenon has already been observed (cf. Kovach and Seider, 1987; Rovaglio and Doherty, 1990). Equilibrium Theory provides a framework where this behavior is explained (cf. also Hwang, 1991, 1995; Hwang et al., 1996). Through the calculation of the separation regions in the (mR, mS) plane, a complete picture of the instability boundaries in the operating parameter plane and of the admittable abrupt composition shifts is given. Though based on steady-state considerations, this constitutes an important contribution toward the understanding of the behavior of multicomponent distillation columns and provides a useful tool for their design. Furthermore, it is worth noting that this nonrobust behavior has to be accounted for when the control system for a distillation column is developed. The continuous model used in this work provides also a dynamic description of the distillation column which has been usefully exploited to develop a nonlinear controller (Han and Park, 1993; Balasubramhanya and Doyle, 1997) and to explain peculiar dynamic behaviors. These include the asymmetric dynamics through which distillation columns depart and return to the steady state of maximum separation (Wachter et al., 1988; Hwang, 1991) or the behavior of batch distillation columns with a middle vessel (Davidyan et al., 1994). 5.3. Role of Feed Characteristics. So far, only perturbations in the flow rates have been considered as potential causes of instability. Actually, other sources of large changes in separation performances exist. Since the shape and location of the regions of separation in the (mR, mS) plane depend on the feed properties, i.e., composition and enthalpic factor, any change of these parameters modifies the separation regions boundaries for the system under consideration. In Figure 12a the effect of a decrease of the enthalpic factor from 1 to 0.82 is illustrated (regions with dashed boundaries). It can readily be observed that in this case the optimal point along the straight line between a and b, i.e., point c, falls in the (0, 3) region where both components distribute and a profile such as the one in Figure 4b is established. Its role for the new separation regions is taken by point c′. The opposite effect is illustrated in Figure 12b, where the dashed separation regions correspond to a decrease of the feed concentration of the heavy component, z1, from 0.50 to 0.45. Now, point c falls in the (1, 3) region and the corresponding composi-

tion profile is that of Figure 4a, while its role is now played by point c′′. 5.4. Nonsharp Separation Specifications. The analysis developed in this work is based on sharp separation specifications. The (p, k) region is constituted of operating points achieving a sharp separation between components p and k, which are the heavy and light keys, respectively; in this context sharp means that they are required to be absent from the distillate and the bottoms, respectively. These specifications are consistent with the focus on high-purity distillation columns. The aim of this section is to show that the approach developed in this work allows one to gain a deeper insight also on operations where nonsharp separation specifications are given. In classical applications of the latter kind, an upper bound for the concentrations of the keys in the twoproduct stream is specified and, before applying the Underwood method, the approximate flow rates of each component in the distillate, dj, are estimated using, for example, the Smith-Brinkley or the Fenske method. In this context, the application of the Underwood method to the case where the keys are adjacent, i.e., where in our notation k ) p + 1, requires solution of the following two equations: C

Rjzj

∑ j)1R

j

-λ

- (1 - q) ) 0 C

VR )

Rjdj

∑ j)1R

j

-l

(28)

(29)

The former is eq 21 and only the solution λFp , with Rp < λFp < Rk is required, while, on the other hand, the latter is eq 58 and allows one to calculate VR, by using the solution of the former as λ value and the previously estimated values of the dj quantities (which, in principle, may all be nonzero). To the aim of analyzing the qualitative and quantitative differences between the sharp separation approach followed in this paper and the nonsharp separation approach involving eqs 28 and 29, let us consider the specific example consisting of the minimum reflux calculations reported in Table 4. In this table, the minimum reflux conditions are determined, which allow one to attain a (3, 4) separation in the case of the fivecomponent distillation whose parameters are reported in the caption of Figure 10. This implies that under these conditions only components 1-3 are collected in the bottoms, where component 3 is the heavy key and components 4 (the light key) and 5 are collected in the distillate; no distributed components are present. The calculated flow rate ratios correspond to a minimum reflux ratio of 5.396. Let us now consider the same distillation, with the same key components but nonsharp separation specifications. For the sake of simplicity, we assume that an upper bound for the mole fraction of the heavy key D , and that the correin the distillate is given, xp,max sponding amount of heavy key collected in the distillate is compensated by the same amount of light key collected in the bottoms. This implies that dp ) wk, in general, and d3 ) w4 in our specific example, and hence that the distillate and bottoms overall flow rates do not change with respect to the sharp separation case. In


the frame of Equilibrium Theory, these separation specifications imply that the column operates not in the (3, 4) separation regime any more but in the (2, 5) separation regime, where species 3 and 4 distribute according to the new specifications. Therefore, the nonsharp separation between components 3 and 4 might be seen as a sharp separation between keys 2 and 5, but with a given split of the two distributed components, i.e., species 3 and 4. If, for this last separation, we were interested in calculating the minimum reflux conditions we would follow the procedure reported in Table 5 and end up with the calculated values of the distillate and bottoms flow rates of the distributed components 3 and 4, i.e., d3 ) 0.138F and w4 ) 0.057F as to Table 5. On the contrary, if the split is assigned, as is the case when a nonsharp separation is specified, then d3 and w4 are known and the corresponding flow rate ratios mR and mS must be calculated through eq 29. For the sake of comparison, the vapor flow rate in the rectifying section can be given in terms of the vapor flow rate obtained in the case of the sharp separation, VRsharp, as follows: D VR ) VRsharp + Dxp,max

(

Rp

Rp -

λFp

-

Rk Rk - λFk

)

(30)

where it is worth noting that both terms between brackets are negative. Finally, the minimum reflux ratio for the nonsharp separation can be cast as follows in terms of the same parameter for the sharp separation, Rmin sharp:

(

D xp,max Rk Rp Rmin ) 1 + min min F Rsharp Rsharp Rp - λp Rk - λFk

)

(31)

It can be readily observed that, since the term between brackets on the right-hand side of this equation is always negative, then the minimum reflux ratio obtained for nonsharp specifications is always smaller than Rmin sharp. This is consistent with physical intuition which suggests to us that the larger the key component concentration in the product stream, the smaller the minimum reflux ratio and hence the easier the distillation. Using the above equations, the calculation of the operating point in the (mR, mS) plane for different values D of the upper bound xp,max is straightforward. In the case of the (3, 4) separation above, these are drawn in Figure 13, which is an enlargement of Figure 10 around the minimum reflux point of the (3, 4) region, for values D varying between 0 and 0.01. These points of x3,max belong to the dashed line starting from the optimal point of the (3, 4) region and stretching within the (2, 5) region, as expected. As can be readily observed, the D ) 0.0001 (0) cannot be point corresponding to x3,max distinguished from the optimal point obtained assuming sharp separation; this is justified by the very low upper bound for the heavy key and corresponds to a minimum reflux ratio of 5.394 instead of 5.396, as in the sharp D ) 0.001 (O) is case. The point corresponding to x3,max still very close to the previous one, and the corresponding minimum reflux ratio goes down only to 5.381. D ) 0.01 ()) is Finally, the point corresponding to x3,max further away within the (2, 5) region; however, it is still rather close and, in fact, the corresponding minimum reflux ratio is 5.255, which is only 2.5% less than the

Figure 13. Separation regions in the (mR, mS) plane for a fivecomponent distillation: system parameters as in Figure 10. The dashed line represents minimum reflux operating conditions for nonsharp separations corresponding to different mole fractions of the heavy key in the distillate, xD p , between 0 and 0.01. Symbols: D D 0, xD p ) 0.0001; O, xp ) 0.001; ), xp ) 0.01.

value corresponding to sharp specifications. It follows that the value calculated based on the approach proposed in this paper, i.e., the one corresponding to sharp specifications, is a rather good approximation of those obtained with nonsharp specifications provided that the separation requirements are indeed high-purity specifications, i.e., with upper bounds in mole fractions of the key components of less than 0.01. A few final remarks are worth doing. First, based on eqs 30 and 31, which bear general validity, the results illustrated in Figure 13 could be easily generalized to different separations of the same system or to different systems; quantitative differences would be observed, but similar conclusions would be drawn. It is worth recalling that the importance of sharp specifications was clear also to Underwood; in the first worked-out example of his 1948 paper he chose a 0.0001 upper bound for the mole fraction of the heavy key in the raffinate and calculated the minimum reflux ratio by approximating the distribution of the components in the product streams assuming sharp separation. Second, once the position in the (mR, mS) plane of the minimum reflux point corresponding to nonsharp specifications is known, it is possible to analyze the effect of flow rate disturbances on the separation performances following the considerations illustrated in Figure 11. It may be noted that such nonsharp optimal operating points are located in a position similar to that of point W in Figure 11. This understanding of the consequences of flow rate changes on separation performances in multicomponent distillation is a rather important achievement made possible by the approach presented in this work. Finally, it is worth noting that the results obtained by applying short-cut design methods based on the CRV assumption are affected by this rather simplistic approximation of the real thermodynamic behavior of the system. Therefore, in this context and for high-purity columns the design based on sharp separation specifications does not provide results which are less realistic than those obtained by assuming a finite split of the key components, as in the case of nonsharp specifications.


6. Concluding Remarks After defining the different possible separation regimes of a multicomponent distillation column (through the number of components collected in the two product streams under sharp separation specifications), an exact analytical procedure for the determination of the map of the corresponding separation regions in the operating parameter plane spanned by the flow rate ratios, mR and mS, has been determined. This is based on a continuous model of the column which is suitable to describe not only packed but also plate columns in the limit of a large number of stages, i.e., when high-purity performances are sought. The adopted approach needs some assumptions, namely, that local equilibrium is attained everywhere in the column, axial dispersion is negligible, and CMO and CRV hypotheses hold true. These allow one to obtain a complete picture of the column behavior also in the multicomponent case. These assumptions are typical of short-cut design methods and are consistent with the aim of this work, which is meant to provide a tool for the design of multicomponent distillation columns and a conceptual framework in which to explain the main features of high-purity columns behavior. The developed procedure allows one to calculate optimal operating conditions for each operating regime attainable; these are proven to be the same as those obtained through the Underwood method. Furthermore, also nonoptimal operating conditions are characterized, thus identifying the sets of operating conditions corresponding to the different separation regimes and vice versa. It is shown that the boundaries between adjacent separation regions represent nonrobust operating conditions. Crossing these boundaries yields the appearance or disappearance of one component in or from one product stream and large shifts of composition and temperature profiles, as has been previously observed in experiments. The adopted continuous model allows one to give a thorough explanation of the nonlinear behavior of highpurity columns. Furthermore, also the occurrence of pinch conditions is explained. Despite the several simplifying assumptions on which the adopted model is based, it allows one to calculate the pinch compositions precisely. Through these and through application of proper material balances, also the composition of the product streams can be accurately predicted. Besides the reported results, the developed approach has two further merits. First, it is based on a dynamic model and therefore the analysis of the dynamic behavior and of the steady-state performances of high-purity columns can be done within the same frame and using the same tool (Wachter et al., 1988; Hwang, 1991). Second, a very similar approach has been applied to the analysis of continuous countercurrent chromatographic separation processes of which the practical implementation is the so-called simulated moving bed technology (Storti et al., 1993; Mazzotti et al., 1994b, 1997). Similar results have been obtained in this last case which have proven useful for the interpretation of a large set of experimental data. This strongly supports the conjecture that all countercurrent separation processes may be effectively described within the same Equilibrium Theory framework, in particular for design and control purposes.

Notation C ) overall number of components d ) molar flow rate in the distillate f ) net molar flow rate, defined by eq 46 F ) feed molar flow rate h ) holdup per unit length H ) function, defined by eq 21 K ) partition constant, Kj ) yj/xj ls ) height of the stage L ) molar flow rate of the liquid phase m ) molar flow rate ratio, m ) L/V N ) overall number of stages q ) enthalpic factor of the feed Qr ) reference molar flow rate r ) holdup ratio, r ) hv/hl t ) time V ) molar flow rate of the vapor phase w ) molar flow rate in the bottoms x ) mole fraction in the liquid phase y ) mole fraction in the vapor phase z ) mole fraction in the feed p Z ) feed parameter, Z ) ∑j)1 zj Greek Letters R ) relative volatility, Rj ) Kj/Kr δ ) thermodynamic parameter, defined in eq 10 λ ) component of vector Λ, defined by eq 32 Λ ) Equilibrium Theory vector ξ ) dimensionless axial coordinate, ξ ) 1 - n/N τ ) dimensionless time, τ ) Qrt/(Nlshl) Φ ) absorption factor for the reference component in the rectifying section Ψ ) absorption factor for the reference component in the stripping section Subscripts and Superscripts 0 ) initial state (τ ) 0) a ) constant state at boundary A A ) left-hand side boundary of the countercurrent section (ξ ) 0) b ) constant state at boundary B B ) right-hand side boundary of the countercurrent section (ξ ) 1) c ) constant state D ) distillate F ) feed stream g ) state index i ) vector index j ) component index k ) heavy key component l ) liquid phase n ) stage index p ) light key component r ) reference component R ) rectifying section S ) stripping section v ) vapor phase w ) wave * ) feed plate

Appendix A. Equilibrium Theory Background The main results of Equilibrium Theory in the case of a single countercurrent section, which are relevant to this work, are briefly summarized in this appendix. Note that we adopt the approach based on the ω-transformation (cf. Rhee et al., 1970, 1971; Storti et al., 1989, 1993) rather than the equivalent one based on the h-transformation (cf. Helfferich and Klein, 1970; Hwang and Helfferich, 1989; Helfferich, 1991).


Let us consider a C-component system characterized by the constant relative volatility equilibrium model (9). Through Equilibrium Theory it is shown that there exists a one-to-one mapping between the space of liquid or vapor phase mole fractions and the vectors Λ of a (C - 1)-dimension space. The components of the vector Λ associated with a given composition set, corresponding to liquid and vapor phase mole fractions xj and yj (j ) 1, ..., C), respectively, are obtained as roots of either of the following equations:

Rjxj

C

∑ j)1R

-λ

j

∑ j)1R

-λ

j

(32)

B B Λc ) (λA1 , ..., λAg , λg+1 , ..., λC-1 )

yj

C

)0

)0

(33)

These equations have C - 1 real and positive roots fulfilling the inequalities

R1 e λ1 e R2 e ... e Ri e λi e Ri+1 e ... e RC-1 e λC-1 e RC (34) The special case where one component is absent from a particular stream in the system can be handled by applying the rule that if and only if xj ) yj ) 0, then λ ) Rj. Note that, in this case all the other λ values, with λ * Rj, can be calculated through the same equations (32) and (33) where the missing jth component is obviously not included in the sum. By accounting for the vapor phase stoichiometric relationship, eq 33 can be recast in the following form:

Rjyj

C

∑ j)1R

-λ

j

)1

(35)

It is worth noting that eqs 32, 33, and 35 share the structure C

F(λ) )

bjyj

∑ j)1R

j

-λ

)a

(36)

where yj > 0, bj > 0, Rj > Rj-1 > 0, and a is a generic constant. The left-hand side of this equation is an increasing function of λ, within each interval bounded by Rj and Rj+1, j ) 1, ..., (C - 1). Equation 36 has at least C - 1 solutions fulfilling constraints (34). Moreover, if a * 0 there is another solution λC, with λC < R1 if a > 0 and λC > RC if a < 0. Equations 32, 33, and 35 can be inverted, yielding the mole fractions as functions of the Λ vector’s components: C-1

yj ) (

(40)

where g is an integer number and 0 e g e C - 1. Also the choice of the index g depends on the value of the flow rate ratio m. The gth constant state prevails provided that the following necessary and sufficient conditions on the product mδc are fulfilled, where δc is calculated using eq 39 and the λ values in eq 40 for the particular choice of g that is considered:

0 < mδc e min{λA1 , λB1 } (g ) 0) A B , λg+1 } max{λAg , λBg } e mδc e min{λg+1 (g ) 1, ..., C - 2) (41) A B , λC-1 } e mδc < ∞ (g ) C - 1) max{λC-1

2. g-Wave (g ) 0, ..., C - 1). If, for some g ) 1, ..., C - 1, λBg < λAg , then the homogeneous state may correspond to the vector A B B , λg, λg+1 , ..., λC-1 ) Λw ) (λA1 , ..., λg-1

(42)

This situation occurs if

λBg < mδw ) λg < λAg

(43)

We need not analyze this case, since it never occurs in the units considered in this work. 3. g-Shock (g ) 0, ..., C - 1). If, for some g ) 1, ..., C - 1, λBg > λAg , then two states may prevail in the section. These are associated with the vectors Λa and Λb, respectively: A B B , λAg , λg+1 , ..., λC-1 ) Λa ) (λA1 , ..., λg-1 A B B , λBg , λg+1 , ..., λC-1 ) Λb ) (λA1 , ..., λg-1

(44)

This situation occurs only if

C

(Rj - λi))/( ∏ (Rj - Ri)) ∏ i)1 i)1,i*j C

xj ) yj(

Now let us consider the countercurrent section illustrated in Figure 2 and the two incoming states characterized by the mole fractions xAj corresponding to the vector ΛA (the liquid state) and by the mole fractions yBj corresponding to the vector ΛB (the vapor state). At steady state there are three possible kinds of solution of the Riemann problem defined by eq 5 with boundary and initial conditions (11): 1. g-Constant State (g ) 0, ..., C - 1). A homogeneous state prevails in the whole section, whose specific composition is selected by the value of the net flow rate ratio m. The corresponding Λ vector has the following general structure:

m)

C-1

λi) ∏ Ri)/(∏ i)1,i*j i)1

(38)

It follows that the quantity δ defined by eq 10 is given by C

δ)(

C-1

Rj)/( ∏ λi) ∏ j)1 i)1

g

(37)

(39)

λAg δa

)

λBg δb

C-1

λBj ∏ ∏ j)1 j)g λAj

)

C

(45)

Rj ∏ j)1

In this case the two states are separated by a discontinuity, called a shock transition, which is stationary in the column; i.e., it does not propagate. It is worth noting that when Equilibrium Theory is applied to the time-dependent analysis of a distillation


column, the last equation does not hold, in general, and the propagation velocity of the above-mentioned shock is not zero. It is this velocity that determines the dynamic behavior of the column (Hwang, 1991). Conditions (44) and (45) imply that in the stationary regime the homogeneous state Λa prevails at the side of the section where the liquid enters, whereas Λb prevails at the side where the vapor does. These two states are separated by a standing step transition. Under the assumptions on which the Equilibrium Theory approach is based, its exact position is not independent of the initial state of the column; on the contrary, it is welldefined when column end effects observable under nonequilibrium conditions are accounted for (Hwang and Helfferich, 1988, 1989). In this context we are allowed to disregard it, since our analysis involves only the net flow rate fj of each component; in fact, this quantity depends only on the composition plateaus and not on the position of the concentration transition (cf. also Hwang and Helfferich, 1988; in particular, Figure 8 of their paper where this statement is clearly assessed). The net flow rate fj, which is implicitly defined by the model equation (5), is given by the following relationship:

fj ) Lxj - Vyj ) L(xj - myj) ) Vyj (mδ - Rj) (j ) 1, ..., C) (46) Rj where eqs 9 and 10 have been used. In principle, two different values are obtained on each side of the discontinuity; however, by using eqs 37 and 45, it is readily proved that the net flow rate of each component is the same on either side of the shock. Hence, in this case as well as in the other two cases above, at steady state the net flow rate of each species is uniform along the column. Therefore, although the compositions of the two constant states are different, the relevant quantities used for design purposes, i.e., the net flow rate of each component, can be calculated with reference to either constant state, no matter what the exact position of the step transition is. Based on this observation, both states may be said to be prevailing in the whole section at steady state, since either state can be used to calculate the relevant single-component net flow rates. Thus summarizing, the steady-state solution of the Riemann problem for a single countercurrent column consists of a single homogeneous state, namely, either state Λc given by eq 40 or state Λw of eq 42, or else it consists of the two homogeneous states Λa and Λb given by eq 44 separated by a standing step transition. In all cases the single-component net flow rates are constant through the column and across the discontinuity and are given by eq 46. The results summarized above do not directly determine the states of the outgoing streams, characterized by liquid and vapor mole fractions yAj and xBj , respectively. In general, these are not in equilibrium with the corresponding incoming streams and must be determined through material balances at the column boundaries assuming no accumulation of mass at the boundary itself. The following equations are obtained:

at ξ ) 0:

yAj ) mxAj - fj/V

(47)

at ξ ) 1:

xBj ) yBj /m + fj/L

(48)

These equations imply that there may be a discontinuity at one or both boundaries, which is, in fact, called boundary discontinuity. Appendix B. Proof of Rule 1 and Corollary 1 In this section we will demonstrate Rule 1 and Corollary 1. The results of Appendix A will be used herein. According to the convention adopted to define the (p, k) separation, the molar flow rate of the jth component in the distillate, dj, is equal to zero for all species with index between 1 and p, whereas it is greater than zero for all other species. Accordingly, the molar flow rate of the jth component in the bottoms, wj, is equal to zero for all species with index between k and C, whereas it is greater than zero for all other species. First, let us refer to the rectifying section and to the necessary conditions of Rule 1. By accounting for the equilibrium conditions (9) and eq 46, the jth component material balance for an envelope cutting the rectifying section above the feed plate can be represented by:

dj ) -fRj ) yRj VR - xRj LR )

yRj VR (Rj - mRδR) Rj

(49)

Since dj > 0, if j ) p + 1, ..., C, then it follows that

mRδR ) Φ < Rp+1

(50)

Moreover, yRj ) 0 (j ) 1, ..., p), since for these species dj ) 0. Hence, applying eq 34 and the remark immediately following it, one obtains

λRi ) Ri (i ) 1, ..., p)

(51)

Let us consider the vector ΛD associated with the state entering the column from the condenser. Since the first p components are not present in the distillate, ΛD can be written as follows: D D , ..., λC-1 ) ΛD ) (R1, ..., Rp, λp+1

(52)

Referring to Appendix A, three possible conditions must be analyzed (note that according to the convention adopted therein ΛA ) ΛD and ΛB ) Λ*): 1. g-Constant State. Among the C possible constant states corresponding to the following vectors (see eq 40): D ΛR ) (λD 1 , ..., λg , λ* g+1, ..., λ* C-1) (g ) 0, ..., C - 1) (53)

only the p-constant state fulfills the separation requirements. In fact, all species are present on the feed plate, and therefore λ* i * Rj for all i and j. Combining eqs 51 and 53 leads to the conclusion that g g p. However, according to eq 41 and conditions (34), if g R R > p, then Rp+1 e Rg e max{λD g , λ* g} e m δ ) Φ, which contradicts eq 50. Thus, g must be equal to p, and, therefore, the vector ΛR defined in eq 53 associated with the state prevailing inside the rectifying section has the special form given by eq 18 of Corollary 1. Equation 41 of Appendix A and eq 50, along with the property that max{λ*p, λD p ) Rp} ) λ* p due to eq 34, yield the relationship (16) of Rule 1.


2. g-Wave. This case is not consistent with the separations requirements; hence, it is not acceptable. In fact, in this case it is necessary that g > p, for the same reasons that yield g g p in the previous case. However, this contradicts eq 50, for if q > p, then Rp+1 e Rg < λ*g < λg ) mRδR ) Φ. 3. g-Shock. Two constant states separated by a discontinuity prevail in the column under the conditions reported in Appendix A. The two states are given by eq 44, and it is possible to demonstrate, with consideration similar to those made for the first case, that only a p-shock is possible. According to Appendix A, in the frame of Equilibrium Theory the position of the stationary step transition separating the two states is not defined. For the sake of simplicity, but without loss of generality, we assume that this is close to the feed plate section end. Thus, referring to eq 44, we let ΛR ) Λa, which is exactly eq 18. Note that this condition occurs if Φ ) λ* p, i.e., when Φ attains the lower bound in eq 16. The analysis above demonstrates the necessary conditions of Rule 1. The sufficient conditions are easily proved. In fact, if conditions (16) hold true, then, according to material balances (49), dj g 0 if j ) p + 1, ..., C and dj e 0 if j ) 1, ..., p. Obviously, negative values of dj have no physical meaning, hence, conditions (16) imply null values of yRj (j ) 1, ..., p) so that dj ) 0 for the same species. Accordingly, λRi ) λD i ) Ri for i ) 1, ..., p, since λ*i * Rj for all i and j. Recalling inequalities (34), the following relationship for Φ ) mRδR can be D obtained: max{λ* p, λp ) Rp} ) λ* p e Φ < Rp+1 e D min{λ* p+1, λp+1} (for p ) 0 the lower constraint is Φ > 0). Thus, according to eq 41, the constant state prevailing in the section at steady state is the one given by eq 18. Equations 17 for the stripping section are obtained by following a similar procedure, thus completing the proof of Rule 1. Corollary 1 is easily demonstrated. In fact, the operating conditions belong to a (p, k) region, as is seen in the demonstration of Rule 1; the only states that can prevail inside the rectifying and stripping sections are given by eqs 18 and 19, respectively. Vice versa, if eq 18 holds, then λRi ) Ri; therefore, according to Appendix A, yRi ) 0 (with i ) 1, ..., p). According to material balances (49), dj ) 0 and, therefore, λD i ) Ri for i ) 1, ..., p. Hence, from eq 41 it is D obtained that Φ g max{λ* p, λp ) Rp} ) λ* p (if p ) 0 and Φ > 0). Physical consistency of material balances (49) requires that Φ e Rp+1. If Φ * Rp+1, then the last two conditions on Φ yield eq 16. If Φ ) Rp+1, it is easily shown that conditions for the occurrence of a (p + 1)shock are fulfilled. Thus, recalling Appendix A and the conventions adopted in the demonstration of Rule 1, the state prevailing in the rectifying section is different from that given in eq 18; this excludes Φ ) Rp+1. Repeating the same procedure for the stripping section and eq 19 concludes the demonstration. Appendix C. Proof of Rules 2 and 3 The objective of this appendix is to prove Rules 2 and 3 of section 4. The states in the rectifying and stripping sections are characterized by the vapor phase compositions yRj and ySj , respectively, corresponding, according to Corollary 1, to the vectors

ΛR ) (R1, ..., Rp, λ*p+1, ..., λ* k-2, λ* k-1, ..., λ* C-1) ΛS ) (λ* 1, ..., λ* p, λ* p+1, ..., λ* k-2, Rk, ..., RC)

(54) (55)

Only components labeled from p + 1 to C are present in the rectifying section, whereas only components indexed from 1 to k - 1 are present in the stripping section. The values λ*i (i ) p + 1, ..., C - 1) are defined by the following special case of the mapping relationship (35):

RjyRj

C

∑

j)p+1Rj

-λ

)1

(56)

Multiplying this equation by VR and substituting the material balances (49) yields

RjxRj

C

R

R

V )L

∑ j)p+1R

-λ

j

C

+

Rjdj

∑ j)p+1R

j

-λ

(57)

Equations 56 and 57 are obviously fulfilled by the roots of eq 56 λ ) λ*i (i ) p + 1, ..., C - 1), for which the first term of the right-hand side of the last equation equals zero (cf. eq 32). It follows that these λ values are solutions also of the following equation:

Rjdj

C

R(λ) )

∑

j)p+1Rj

-λ

- VR ) 0

(58)

It is worth noting that this equation shares the same structure of eq 36. Accounting for material balance equations (49) shows that this is fulfilled also by λ ) mRδR ) Φ. Therefore, it has C - p solutions, which according to eqs 34 and 50 fulfill the inequalities Φ < Rp+1 < λ* p+1 < ... < λ* k-2 < Rk-1 < λ* k-1 < ... < RC-1 < λ* C-1 < RC. Along the same lines, the following equation is obtained for the stripping section: k-1

S(λ) )

Rjwj

∑ j)1 R

j - λ

+ VS ) 0

(59)

whose k - 1 solutions are λ ) λi (i ) 1, ..., k - 2) and λ ) mSδS ) Ψ, fulfilling the inequalities R1 < λ* 1 < R2 < ... < λ* p < Rp+1 < λ* p+1 < ... < λ* k-2 < Rk-1 < Ψ. Now, it is necessary to establish the proper relationships between the results obtained so far about eqs 58 and 59 and the roots of eq 21, which may be calculated readily from known quantities (namely, the composition and enthalpic factor of the feed). Substituting the overall material balances, i.e., Fzj ) wj + dj, and eq 22, i.e., the definition of the enthalpic factor, q, as well as accounting for the separation requirements defining the


this case not only S(Ψ) ) 0 but also H(Ψ) ) R(Ψ) ) 0. Thus summarizing:

Φmin ) λFp ) λ* p

(64)

F ) λ* Ψmax ) λk-1 k-1

(65)

Now it is possible to prove that eq 64 defines the minimum reflux conditions for the rectifying section, while, on the other hand, eq 65 defines the maximum reflux condition for the stripping section. In other words, the two equations define the optimal operating conditions for both sections. In fact, substituting eq 10 in the definition of Φ yields

Φ ) mRδR ) mR/Kr

Figure 14. Graphical representation of functions R(λ) (cf. eq 58), S(λ) (cf. eq 59), and H(λ) (cf. eq 60) in the range Rp < λ < Rp+1.

The reference volatility Kr in the rectifying section is an increasing function of temperature; if mR ) LR/VR increases, while mS is kept constant, the state in the rectifying section will, of course, be enriched in the more volatile components and accordingly the corresponding equilibrium temperature will decrease. Therefore, it turns out that

(p, k) regime, eq 21 can be recast as

[

H(λ) ) F

C

)

∑ j)1

j

-λ

Rj - λ Rjwj

∑ j)1 R

j

-λ

]

- (1 - q)

Rj(wj + dj)

k-1

)

Rjzj

C

∑ j)1R

- VR + VS C

+

∑

Rjdj

j)p+1Rj

R

S

-V +V ) -λ R(λ) + S(λ) ) 0 (60)

Let us refer to Figure 14 for a graphical representation of the above-defined functions, namely, H(λ), R(λ), and S(λ). It can be readily noted that they exhibit the same structure as eq 36 of Appendix A. First, let us consider the operating parameter of the rectifying section, Φ. The following properties hold true: R(Φ) ) 0; S(λ*p) ) 0; R(λ) is strictly increasing in the interval between Rp and Rp+1, and so is S(λ); according to eq 16 of Rule 1, Rp < λ* p e Φ < Rp+1. It follows that S(Φ) g 0 and R(λ* p) e 0. Therefore, H(Φ) ) S(Φ) g 0 and H(λ*p) ) R(λ*p) e 0. Thus, as illustrated in Figure 14, due to the increasing character of H(λ), its zero λFp fulfills the constraint λ*p e λFp e Φ. When combined with eq 16, this leads to the following inequality:

λFp e Φ < Rp+1

(61)

Similar considerations apply to the stripping section, thus yielding the following inequality:

|

Kr - mR(∂Kr/∂mR) ∂Φ ) >0 ∂mR mS Kr2

F λk-1

Appendix D. Geometrical Features of the Separation Regions in the (mR, mS) Plane In this appendix the geometric features of the separation regions in the (mR, mS) plane are discussed. Equations 61 and 62 indicate that the separation regions are not contiguous in the (Φ, Ψ) plane. This feature disappears if the same regions in the (mR, mS) plane are considered. To demonstrate this, let us consider the (p, k) region and refer to the minimum value of Φ, i.e., Φ ) λFp . From eqs 14, 25, and 39 the minimum value of mR can be calculated as

λ*i ) λFi (i ) p + 1, ..., k - 2)

(63)

Let us now prove Rule 3, starting with eq 26. According to eq 61, the minimum value of Φ attainable within the (p, k) region considered equals λFp . In this case Φ is a solution not only of eq 58 but also of eq 21, i.e., not only R(Φ) ) 0 but also H(Φ) ) 0; it follows that S(Φ) ) 0, F , and in too. Similarly, the maximum value of Ψ is λk-1

C-1

∏ ∏ λ*i i)p i)k-1 λFi

(62)

This completes the proof of Rule 2. It is worth noting that when k - 1 > p + 1, then eq 60 shares the common solutions of eqs 58 and 59 and therefore eq 25 holds true:

(67)

and therefore for a given constant value of mS, the minimum value of Φ (eq 64) corresponds to the minimum value of mR, mRmin, i.e., to minimum reflux conditions. Similar considerations demonstrate that, if mR is kept constant, eq 65 defines mSmax. Thus, if both eq 64 and eq 65 are fulfilled, the operating conditions are optimal with respect to the reflux and reboil ratios for the prescribed (p, k) separation.

k-2

Rk-1 < Ψ e

(66)

mRmin )

C

(68)

∏ Rj

j)p+1

It is worth noting that since λD p ) Rp < λ* p, the condition (45) of Appendix A is fulfilled; hence, a p -shock prevails in the rectifying section. According to Appendix A, in this case a discontinuity separates two constant states. One of these states corresponds to ΛR given by eq 54, whereas the other is associated with the vector Λb ) (R1, ..., Rp-1, λ*p, ... λ*C-1) (see eq 44). This is exactly the constant state which would prevail in the rectifying section if the (p - 1, k) regime were established. If one


looks at this situation from the point of view of the state Λb and of the (p - 1, k) regime, the corresponding value of Φ would be Φ ) Rp, i.e., the upper bound given by eq 23 where p is substituted by p - 1. Therefore, it is demonstrated that two different Φ values can be calculated corresponding to the two states across the standing discontinuity. These correspond to the minimum and maximum values for the (p, k) and (p - 1, k) regimes, respectively. However, it has been shown that they yield a single value of mR, given by eq 68, thus proving that the (p, k) and (p - 1, k) regions in the (mR, mS) plane are indeed contiguous. The same proof can be repeated with reference to mS, whose maximum value is given by k-1

p

λ* ∏ ∏ i i)1 λFi

mSmax

)

i)p+1

k-1

(69)

Rj ∏ j)1 Now let us consider the regions where k ) p + 1. In this case there is no distributed component and the separation regions can be defined in a more direct way. In fact, from the definition of the enthalpic factor (eq 22) and from the overall material balance of the column, the following general relationship can be obtained:

F mS - mR ) VR q + (1 - q)mS

(70)

Moreover, the overall material balance of the rectifying section can be written as C

VR - LR ) F

∑ zj ) F(1 - Z)

(71)

j)p+1

p zj. Combining the last two equations, where Z ) ∑j)1 one obtains

mS )

q + (1 - q)mR - ZmR q + (1 - q)mR - Z

(72)

which proves that the (p, p + 1) region is a onedimensional locus in the (mR, mS) plane. These lines have the point (1, 1) in common and terminate in the point defined by Φ ) Ψ ) λFp , which represent the optimal operating conditions for the regime considered. It can be readily noticed that eq 72 defines a straight line only in the case where q ) 1, i.e., the case of a saturated-liquid feed. In the (p, C + 1) regions, all components in the distillate are present also in the bottoms and, according to condition (17) in Rule 1, mS has no upper value. Only eqs 61 and 64 must be considered. Accounting for eq 25 yields the following explicit expression for mRmin: C-1

mRmin )

λFi ∏ i)p C

(73)

∏ Rj

j)p+1

In the (0, k) regions all components in the bottoms are

present also in the distillate. According to condition (16) of Rule 1, the lower bound on mR is zero. Only eqs 62 and 65 must be considered and the following equation is obtained: k-1

mSmax )

λFi ∏ i)1 k-1

(74)

Rj ∏ j)1 Appendix E. Calculation of Concentration Profiles and Separation Performances The purpose of this appendix is to describe the procedure which is followed in the frame of Equilibrium Theory to determine the separation performances achieved for given values of the operating conditions mR and mS. For a given feed composition and enthalpic factor, the separation regions are calculated through the procedure described in section 4. Let us select mR and mS within a given (p, k) region. The quantities λFi , i ) p + 1, ..., k - 2, together with the distillate and bottoms flow rates of the species from 1 to p and from k to C are readily determined through step 1 and substeps a, c, and d of step 2 in Table 2. Moreover, VR, LR, VS, and LS are calculated through eq 70 and the definition of the enthalpic factor q, i.e., eq 22. In order to fully characterize the column regime, an overall number of 3(C + k - p) + 1 unknowns must be determined. These are the parameters λ* i, i ) 1, ..., p, k - 1, ..., C - 1; the distillate and the bottoms flow rates dj and wj, with j ) p + 1, ..., k - 1; the mole fractions in the rectifying and stripping sections, i.e., xRj and ySj , j ) p + 1, ..., C, and xSj and ySj , j ) 1, ..., k - 1; and finally δR, δS, Φ, and Ψ. A system constituted of the same number of nonlinear equations must be solved: eqs 37 and 38 to determine the unknown mole fractions in the two sections of the column; eq 10 to determine δR and δS; eqs 14 and 15 to define Φ and Ψ; the material balance equations (49) for the species collected in the distillate, i.e., those with j ) p + 1, ..., C, and the corresponding ones for the species collected in the bottoms with j ) 1, ..., k - 1. An efficient iterative procedure for the solution of this system can be easily devised. Literature Cited Acrivos, A.; Amundson, N. R. On the steady state fractionation of multicomponent and complex mixtures in an ideal cascade. Part I. Chem. Eng. Sci. 1955a, 4, 29. Acrivos, A.; Amundson, N. R. On the steady state fractionation of multicomponent and complex mixtures in an ideal cascade. Part II. Chem. Eng. Sci. 1955b, 4, 68. Balasubramhanya, L. S.; Doyle, F. J., III. Nonlinear control of a high-purity distillation column using a travelling-wave model. AIChE J. 1997, 43, 703. Davidyan, A. G.; Kiva, V. N.; Meski, G. A.; Morari, M. Batch distillation in a column with a middle vessel. Chem. Eng. Sci. 1994, 49, 3033. Gilles, E. D.; Retzbach, B. Reduced models and control of distillation columns with sharp temperature profiles. Proc. 19th IEEE Conf. Decision Control 1980, 2, 860. Han, M.; Park, S. Control of high-purity distillation column using a nonlinear wave theory. AIChE J. 1993, 39, 787.

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Received for review October 6, 1997 Revised manuscript received March 17, 1998 Accepted March 18, 1998 IE970708V

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