New Approach to the Design and Optimization of Energy Efficient Chemical Processes

Energy use in many process industries is dominated by separation processes. As energy costs are rising rapidly, the.re is a renewed interest in better methodologies for the synthesis, design and/or retrofitting of separation ·processes. In this thesis, a novel method for determinin~ energ~ efficient process designs based on finding the separation with the shortest stripping line distance is proposed. A problem formulation based on mixed integer nonlinear programming (MINLP) is given and a global optimization algorithm is presented for determining energy efficient process designs. A variety of examples of separations involving ideal, non-ideal, azeotropic and reactive mixtures are used to demonstrate the versatility and advantages of the shortest stripping line distance approach over available methods in literature. One of the major advantages of the proposed methodology is that it can be used to identify minimum energy requirement for multi-unit processes such as hybrid separations involving extraction followed by distillation and reaction/separation/recycle processes. The proposed shortest stripping line distance method is extended and a two-level distillation design procedure is developed for finding portfolios of minimum energy designs when specifications are given in terms of key component recoveries. It is shown that the proposed two-level design procedure is flexible and can find minimum energy designs for both zeotropic and azeotropic distillations. It is also shown that the two-level design method encompasses Underwood's solution, when it exists, and can find minimum energy designs when Underwood's method is not applicable. This twolevel design approach also overcomes the well-know · limitation of distillation line methods of sensitivity of column profiles to the product compositions. Non-pinched, mm1mum energy distillation designs are an important and often overlooked class of distillation designs that provided added economic advantages in practice. All current metp.ods for designing distillation columns available in literature are based on the concept of pinch points and are incapable of finding non-pinched, minimum energy solutions. In contrast, it is demonstrated that shortest stripping line distance approach is capable of systematically and reliably finding non-pinched, minimum energy distillation designs as well as providing insights into the reasons for the existence of non-pinched, minimum energy design. These reasons include · trajectories that follow unstable branches of a pinch point curve in azeotropic systems, the inherent looping structure of trajectories in hydrocarbon separations, and the presence of ancillary constraints in multi-unit processes like extraction/distillation. Several examples are studied and many numerical results and geometric illustrations are presented in each section show that the shortest stripping line distance methodology is indeed a powerful and versatile tool for designing energy efficient processes and can be considered as a next generation method for conceptual desigt?. of energy efficient chemical processes.

separation with the shortest stripping line distance is proposed. A problem formulation based on mixed integer nonlinear programming (MINLP) is given and a global optimization algorithm is presented for determining energy efficient process designs.
A variety of examples of separations involving ideal, non-ideal, azeotropic and reactive mixtures are used to demonstrate the versatility and advantages of the shortest stripping line distance approach over available methods in literature. One of the major advantages of the proposed methodology is that it can be used to identify minimum energy requirement for multi-unit processes such as hybrid separations involving extraction followed by distillation and reaction/separation/recycle processes.
The proposed shortest stripping line distance method is extended and a two-level distillation design procedure is developed for finding portfolios of minimum energy -designs when specifications are given in terms of key component recoveries. It is shown that the proposed two-level design procedure is flexible and can find minimum energy designs for both zeotropic and azeotropic distillations. It is also shown that the two-level design method encompasses Underwood's solution, when it exists, and can find minimum energy designs when Underwood's method is not applicable. This two-level design approach also overcomes the well-know · limitation of distillation line methods of sensitivity of column profiles to the product compositions.
Non-pinched, mm1mum energy distillation designs are an important and often overlooked class of distillation designs that provided added economic advantages in practice. All current metp.ods for designing distillation columns available in literature are based on the concept of pinch points and are incapable of finding non-pinched, minimum energy solutions. In contrast, it is demonstrated that shortest stripping line distance approach is capable of systematically and reliably finding non-pinched, minimum energy distillation designs as well as providing insights into the reasons for the existence of non-pinched, minimum energy design.
These reasons include · trajectories that follow unstable branches of a pinch point curve in azeotropic systems, the inherent looping structure of trajectories in hydrocarbon separations, and the presence of ancillary constraints in multi-unit processes like extraction/distillation.
Several examples are studied and many numerical results and geometric illustrations are presented in each section show that the shortest stripping line distance methodology is indeed a powerful and versatile tool for designing energy efficient processes and can be considered as a next generation method for conceptual desigt?. of energy efficient chemical processes.

INTRODUCTION
Separation processes play an important role in chemical, petrochemical, pharmaceutical and related industries. Distillation is the most important and versatile separation process available to date and will continue to be so in the near future. With the rapid increase in global energy costs, it is not only critical to design distillation and other separation processes in energy efficient ways but also to develop newer and .
less energy intensive ways to perform sep_ aration tasks. Hybrid processes such as extraction/distillation appear to be one promising alternative as they have the ability to handle both the throughput and product purities of distillation processes and, at the same time, reduce processing cost significantly.
There are an estimated 40,000 distillation columns in the U.S. Rough estimates put the energy consumed by distillation alone around 6% of the total energy in the U.S.
Many of the existing distillation columns are more than 20-30 years old and were designed using methods that were developed when energy was far less expensive.
Retrofitting these columns to save energy and designing newer replacements will be an on-going task in the process industry as energy costs continue to rise.
Shortcut design methods play an important role in conceptual process design, especially for separation processes. Conceptual or shortcut methods are often used for screening promising alternatives among a larger set of possible designs. They are also used to get quick estimates of capital and operating costs. A good shortcut design 1 method can save time and effort and also lead to more innovative, creative and effective design solutions.
Attempts to design distillation columns in a systematic way started in the early 1900s.
McCabe and Thiele ( 1925) developed a method for the conceptual design of distillation columns for b~nary mixtures. Underwood, on the other hand, developed a shortcut method in 1932 for finding minimum energy requirements for the distillation of multi-component mixtures. As separation processes are energy intensive, the rise in the cost of energy has spawned renewed interest in methods for designing energy efficient chemical processes.

Motivation
As mentioned earlier, methods for conceptual process design were available as early as 1925. Underwood's method, which first appeared in 1932, is perhaps the most famous and widely used shortcut design method and specifically addresses the issue of energy consumption. This method is presently included in most commercially available simulation programs (e.g., Aspen Plus, Proll, etc.). In addition to Underwood's method, there are several other methods that have been developed in recent years that are capable of finding minimum energy requirements for separating a multi-component mixture by distillation. These methods include the boundary value methods of Doherty and co-workers, the rectifying body method, and Vmin diagrams.
A detailed survey of these methods is included in the chapters to follow.
2 Though there are several methods for finding . mimmum energy requirements for separation processes, m9st of them have serious limitations. For example, Underwpod's method is really only applicable when phase behavior of the system can be well approximated by constant relative volatilities; thus it can not be used for azeotropic mixtures. In addition, the performance of a given separation unit often depends on an upstream P.rocess suc. h as a reactor or an extractor . . In these cases, one needs to consider the reactor-separator configuration or the extraction unit simultaneously when designing a process with overall energy efficiency. It is rather surprising that there are no shortcut methods in the literature that allow one to find minimum energy requirements and corresponding operating conditions for these multi-unit processes. Hence, there is a need for a design methodology th.at will both unify all existing methods for finding minimum energy requirement in. chemical processes and overcome the limitations of existing shortcut methods with regard to their inherent reliance on pinch points, their sensitivity to product compositions, and their inability to handle multi-unit processes. The main objective of. this research project was to develop a versatile shortcut method for conceptual design of separation and other chemical processes that will fulfill these needs.

Background
Residue curve maps have long been used during the early stages of synthesis and design to provide insight into feasibility and limitations of separation by distillation, particularly for azeotropic mixtures. Residue curve maps were first proposed by  reflux. This approximation is often good enough for analysis at the conceptual design stage. For azeotropic systems, residue curve maps also help to identify different distillation regions or boundaries for a given mixture.
While the first papers on residue curve maps appeared at the beginning of the 19th century, they have receh:ed renewed attention in the last 20 years or so. There are several recent review papers on residue curve maps -for example, the paper by Pollmann and Blass 7 and the work of Kiva et al. 8 Recently, Lucia and T. aylor 9 have shown that exact distillation boundaries correspond to residue curves with locally longest line integral that connect the unstable and stable node in a particular distillation region. See Fig. 1. This has helped to develop a method that can precisely locate distillation boundaries. More importantly, the work of Lucia and Taylor has clearly shown that this geometrical principle of measuring line integrals readily extends to mixtures with more than three components. For example, Bellows and Lucia 10 have demonstrated that for. four-component mixtures a boundary corresponds to a local maximum in surface area while for more than four components the boundary is a maximum in volume. Taylor et al. 11 have shown that same concept of locally maximum line integrals defines a distillation boundary in reacting mixtures as well.
Since a distillation boundary represents, in some sense, a limiting case in terms of the degree of difficulty of separation,. it is reasonable to draw an intuitive connection 5 between the length of distillation lines and energy consumed. Specifically the locally longest residue curve or distillation line at total reflux obviously corresponds to a separation that uses the most energy. Intuitively then, one might imagine that the shortest residue curve or distillation line should provide some info~ation regarding the easiest or most energy efficient separation. This key observation forms the fundamental idea behind !his dissertation research. In the chapters that follow it will be shown that this connection between distillation line length and energy consumption is in fact true for any kind of mixture. Moreover, it will be quantified and exploited to develop a novel shortcut methodology for designing energy efficient chemical processes.

Layout of the Thesis
This thesis is written in the manuscript format and is organized in following way.
This introduction is followed by chapter 1, in which the basic formulation of the proposed methodology for energy efficient process design is discussed. A detailed literature smyey is given at the start of the chapter to summarize the methods currently available in the literature for finding minimum energy requirement in separation processes, highlighting the advantages and disadvantages of each.
Governing equations used in this entire work, the basic philosophy and principles behind the shortest stripping line distance approach are subsequently discussed.
Problem formulations that take the general form of nonlinear programming (NLP) and Chapter 2 focuses on the use of the shortest stripping distance line approach for designing energy efficient multi-unit processes. In particular, the synthesis ~fa hybrid separation process for purification of acetic acid from an acetic acid solution using extraction with ethyl acetate followed by distillation is discussed. It is shown that the shortest stripping line approach correctly finds the extract target composition that connects . extractor to the distillation column, corresponding to the most energy efficient design for both low and high purity acetic acid. on the analysis of residue. curve maps and have uncovered a fundamental underlying geometric principle that defines distillation boundaries in a precise way. · In the chapters of this dissertation, we use a second fundamental geometric principle -that energy · consumption can be described by the concepts of longest and shortest distillation line -to develop a novel method for designing energy efficient chemical processes.
The new aspects of this work · show that this shortest stripping line approach can find minimum energy requirements for 1) Distillations with feed pinch, saddle pinch, and tangent pinch points.
2) Distillations for which the minimum energy solutions do not correspond to a pinch point.
Other novel features of this work also shows that the shortest stripping line approach 4) Can be used to identify correct processing targets in multi-unit processes.

5) Encompasses longstanding methods for finding minimum energy requirements
including the McCabe-Thiele method and boundary value methods.

11
A back-to-front design approach based . on shortest stripping lines is used so that correct processing targets can be identified so that all tasks in can be synthesized simultaneously in such a way that the most energy efficient designs are achieved.
New problem formulations that take the general form of nonlinear programming (NLP) and mixed integer nonlinear pro~amming (MINLP) problems are given and a novel giobal optimization. algorithm is presented for obtaining energy efficient process designs. A variety of ideal and nonideal distillations, including examples with four or more components, are used to demonstrate the efficacy of the shortest stripping line approach. The examples with more than three components are particularly significant because they clearly illustrate that the proposed approach can be readily used to find minimum energy requirements for distillation problems involving any number of components. Many geometric illustrations are used to highlight the key ideas of the method where appropriate.

Key"1ords
pinch point and non-pinch solutions, shortest stnpping lines, energy efficiency.

J . . l Introduction
The primary motivation for this work is the current rapidly rising costs of energy. As a result of recent significant increases in global energy demands, and every indication that demand will remain high, it has become increasingly important to consider ways, perhaps unconventional ways, of designing new processes and/or retrofitting ex. isting ones so that they are energy efficient. To do this -to allow engineers to find creative and energy efficient solutions to processing challenges -new methodologies are needed to support synthesis and design efforts. Separation and energy use in many industries is dominated by distillation. There are an estimated 40,000 distillation columns in the U.S. that consume approximately 18% of all of the energy in the manufacturing sector (see the recent DOE workshop study spearheaded by Eldridge et al. 2 ). Because distillation is such a large energy user and because it will continue to be used to address a wide variety of separation needs,. any new synthesis and design methodologies for overall energy efficiency should, in our opinion, include and/or extend techniques for finding minimum energy requirements in distillation. This is the approach we have adopted in this work.
This paper addresses energy efficiency in the design and optimization of separation processes. The particular design and optimization approach proposed in this work 1.s based on the novel concept of shortest separation (stripping) lines, and is a direct outgrowth ofrecent results by Lucia and Taylor 3 , and subsequently Taylor et al. 4 ,that shed new light on residue curves and distillation lines (i.e., that separation boundaries are defined by longest residue curves or distillation lines). Through new global 13 optimization formulations based on shortest separation lines, the proposed methodology l) Encompasses all existing methodologies for finding mm1mum flows and minimum energy requirements in distillation in the presence of feed, saddle or tangent pinch points.
2) Is unaffected by the number of components or the presence of reverse separation.
3) Uses a back-to-front philosophy to identify correct processmg targets for processes with multiple units (e.g., reactors, other separators) such that overal_ l energy consumption is minimized.
4) Can easily find minimum energy solutions that do not correspond to separation pinch points.

5)
Can be readily combined with other synthesis methods such as the attainable regions approach for the simultaneous design of multi-unit processes.
6) Can solve synthesis and design problems other methods cannot solve. 7) Can provide starting values for more detailed process optimization studies. 8) Can be used to establish that longest and shortest paths are unifying geometric principles for the design of energy efficient chemical processes. 9) Provides a new methodology for the teaching and practice of various aspects of energy efficiency in process design that can be easily understood by the general public.
14 The focus of this manuscript is to show that the key synthesis or design idea of the ,nt 0 r shortest stripping lines readily applies to conventional distillation concer 'J processes as well as the synthesis, design or retrofitting of processes such as ctor/senarator/recycle (RSR) processes and hybrid separation schemes. Problem ~a r .
fonnulations that take the general form of nonlinear programming (NLP) and mixed integer nonlinear progra~ming (MINLP) problems . are presented and a global optimization algorithm is presented for obtaining energy ef~cient process designs.

Literature Survey
Many papers on minimum flows and minimum energy use in distillation have been published beginning with the work of Underwood 5 for the case of constant relative volatility. This includes papers on regular columns, columns with side-streams, extractive distillation, azeotropic distillation, reactive distillation, Petlyuk and other multiple column configurations. For single columns, it is well known that minimum energy requirements generally correspond to minimum reflux and/or boil-up ratios and an infinite number of equilibrium stages so that the column just performs the desired .separation (or exhibits one or more pinch points). Most methods for determining minimum energy requirements in this case are based on either methods for. directly finding pinch points or rigorous column simulations. See, for example, Vogelpohl 6 , Hausen 7 , Doherty and co-workers 8 ' 9 • 10 • 11 ,Koehler et al. 12 ,and Urdaneta et al. 13 for methods based on finding pinch points, and Brown and Holcomb 14 , Murdoch and Holland 15 and Acrivos and Amundson 16 ,Shiras et al. 17 ,Bachelor 18 , and Holland and co-workers 19 ' 20 • 21 for methods based on rigorous column simulation. Koehler et al. 22 give a good survey of methods for determining minimum energy requirements for 15 . 1 and multiple column configurations up to 1995 and show that many of the pinch singe point techniques are related to the original method of Underwood -some more strongly than others. They also give an example of a minimum energy column that does not correspond to a pinch point. More recent work by  shows that a simple graphical algorithm based on a maximum in the separation driving force defined as IYLK -XL~I, where the subscript LK denotes the light key component, can be used to determine near minimum (or minimum) energy requirements for conventional distillations. The graphical approach of Vmin diagrams by Halvorsen and Skogestad 24 also is related to the work of Underwood while that of the rectification body method (RBM) of Urdaneta et al. 13 for reactive distillation and Kim 25 for thermally coupled columns are both based on the use pinch points and residue curves.
Finally, the paper by Alstad et al. 26 gives an example of energy savings in complex column configurations using over-fractionation. There are also many other papers on synthesis and design of single and multiple separator configurations. However, these papers do not specifically address minimum energy requirements and therefore they have not been included in this literature survey.

Some Details of Existing Methods for Finding Separation Pinch Points
Current methods for finding pinch points and minimum energy requirements in distillation include boundary value methods, reversible distillation models, eigenvalue meth~ds, separation driving force methods, the rectification body approach, and Vmin diagrams.

I.J.l Boundary Value Method
Over the last twenty years, Doherty and co-workers 8 -11 ' 27 -29 have published several papers and a variety of numerical methods for addressing minimum energy requirement in azeotropic multicomponent distillation, heterogeneous azeotropic distillation · and reactive distillation. Most of these methods are based on finite difference approximations. of column profiles in ordinary differential equation form under the assumption of constant molar overflow (CMO). Conditions such as minimum reflux are determined using a boundary· value method, in which the rectifying profile for the liquid compositions is integrated from top to the feed stage while the stripping profile is integrated from bottom to the feed stage. Thus a feasible column configuration is one in which the rectifying and stripping profiles intersect and the reflux ratio for which these profiles just tom~h each other corresponds to minimum reflux. When only one pinch occurs at minimum reflux it is designated as a feed pinch. This procedure for finding minimum reflux requires calculating column profiles several times.
A second type of pinch point, _ called a saddle pinch, can also appear in a column profile if a saddle point 'attracts' part of the profile. Using-the boundary value approach, Doherty and co-workers show that a saddle pinch is characterized by a colinearity condition -that is, the saddle pinch, feed pinch point, and feed composition are co-linear. The reflux ratio that makes the saddle pinch, feed pinch point, and feed composition co-linear is the minimum reflux ratio, is exact only for ideal mixtures, and is considered a good approximation for non-ideal mixtures. The boundary value C h was initially proposed for homogeneous mixtures, and later extended to approa .
heterogeneous azeotropic distillation by incorporating vapor-liquid-liquid equilibrium in the decanter during the initialization of the rectifying profile calculations. On the other hand, the co-linearity method is not useful for calculating m_ inimum reflux for heterogeneous azeotropic distillations because a saddle pinch may not appear in the case of heterogeneous az~otropic distillation. Barbosa 10 . This zero volume method uses a continuation method to find pinch points of the operating lines in either the rectifying or stripping sections of a column. These fixed points are used to construct a set of special vectors and the value of reflux that makes .the ( ori~nted) volume of these vectors zero corresponds to minimum reflux. For feed pinch points the zero volume method is straightforward. Tangent pinch points, on the other hand, correspond to turning points of the volume with respect to reflux ratio and require a bit more care in computing because of the singularity condition that accompanies any turning point.

l.3.2 Reversible Distillation Models
The method of Koehler et al. 12 is based on a reversible distillation model. This reversibl d. t·11 · · e is 1 ation model assumes that heat can be transferred to and from a column temp erature difference and that no contact of non-equilibrium liquid and vapor at zero streams is allowed. Reversible distillation path equations are derived by rearranging the column material balances as well as the equilibrium relationships for the most and least volatile components. The solution of this reduced set of equations requires that the flow rates of the most and least volatile components be specified at t~e ·feed plate. Koehler et al. show that ~ reversible distillation path is generated by adding . heat continuously along the length of column and consists of exactly all pinch points of an adiabatic (CMO) calculation. The concentration reached in a reversible distillation column section for any given amount of continuously introduced energy exactly corresponds to the stationary concentration that is obtained in an adiabatic (CMO) section, provided the same amount of energy is introduced only at the ends (through condenser or reboiler). This value of energy represents the minimum ·energy requirement for the section. The reversible distillation model approach has also been used to determine tangent pinch points based on a maximum energy criterion. Here a tangent pinch appears if there is a local maximum in the reversible energy profile between the distillate and the computed pinch po!nt composition, provided the energy demand at this maximum exceeds the energy demand at the tangent pinch point.
Numerical methods based on any reversible distillation model require knowledge of the products that can be achieved by the distillation before starting the computations fo~ finding the minimum reflux. The starting point for the novel aspects of this paper is the recent work by · Lucia and Taylor 3 who show that exact separation boundaries for ternary mixtures are given by the set of locally longest residue curves (or distillation lines at infinite reflux) from any given unstable node to any reachable stable node. See Figure I.1.
We then began with the intuitive belief that following the longest residue curve must somehow be related to the highest energy costs associated with performing a given separation. Furthermore, if the longest residue curve is the most costly separation, then the shortest curve should result in the use of the least amount of energy required for the given separation.  boil-up ratios, these equations reduce to the c-1 residue curve equations given by the differential equation x' = y -x.

I.4.2 Remark
In simulating the behavior. of any staged column using the differential equations defined by Equations I.

I.5 Optimization Formulations and Algorithm
In this section we outline a MINLP formulation and suggest a methodology for finding energy efficient process designs. The overall strategy for determining minimum energy requirements proceeds in two stages -an NLP stage in which minimum boil-up ratio is determined followed by an integer programming (IP) stage in which the smallest number of stages at fixed minimum boil-up ratio is determined.
One of the key features of the formulations given in this section is that they apply to mixtures with any number of components and are not restricted to just ternary mixtures. .13) are also discussed.
3) For stages j = 1 to N 8 , calculate x/ using Eq. 8, Xj+l = Xj + x/, and calculate Dj = 29 4 ) If the column has a rectifying section, then calculate r using Eq. 10 and set k =

Nonlinear Programming Algorithm
Step 1 specifies the feed, bottoms, and desired distillate conditions while step 2 simply initializes the boil-up ratio. . profile has not converged. If the rectifying profile leaves the feasible region, then the separation is clearly infeasible for . the current value of reboil ratio. This is step 6a of the algorithm. On the other hand, if the calculated distillate composition has Ve rges to a different distillate product composition, then the separation is also con .
iafeasible. Convergence to~ different distillate composition in step 6b can be easily checked by checking the condition llx'll < £ at the calculated distillate composition.
Convergence is characterized by a very small value of llx'll and some care must be exercised to avoid identifying rectifying saddle pinches as converged distillate compositions. If the current calculated distillate composition is feasible and has not converged, as indicated in step 6c, then the number of rectifying stages is incremented by one and the next rectifying stage liquid composition is calculated by returning to step 5. Optimality with respect to reboil ratio is checked in step 7 of the algorithm. If optimality conditions are satisfied, then the methodology has determi~ed the shortest stripping line from the given bottoms composition to the stripping pinch point curve.
If not, the reboil ratio is reduced using an optimization algorithm and the whole process (i.e. steps 2 to 7 of the algorithm) is repeated. If, on the other hand, the N;LP has reached optimality, then the minimum reboil ratio and minimum stripping line distance are determined and the algorithni goes to step 8, where it begins the integer programming calculations.
•zrer Programming Algorithm .. The integer programming problem has special structure that can be exploited. For Pie once the boil-up ratio that gives the minimum stripping line distance from exam ' · the bottoms composition to the stripping pinch point curve has been determined, we . know that Ns is to be reduced. Remembers remains fixed at Smin (and therefore r is fixed For indirect splits, there is often a feed pinch on the rectifying pinch point curve. In this case, the stripping line does not exhibit a feed pinch and therefore some modification of the algorithm is required. Remember, one must ·still calculate the distance to the stripping pinch point curve to provide a meaningful distance measurement. However, the point (or stage) at which there is a switch from the stripping section to the rectifying section (i.e., the feed tray) is not on the stripping pinch point curve. Therefore, one must determine the feed tray by determining the stripping tray number at which to make the ~witch and, at the same time, ensure that the distillate specifications are met. The most straightforward way to do this is use the feed composition as a target. By this we mean find the stripping profile that passes through the feed point, locate the intersection of this stripping line with the rectifying pinch point curve, identify the corresponding reflux ratio from the rectifying pinch Handling processing targets requires that the ancillary constraints be included in the NLP. We recommend solving this type of NLP using a penalty or barrier function approach by including only the ancillary constraints in the penalty or barrier function tenn. See Lucia et al. 1 for an illustration of this.

Distillation Examples
The next section presents a number of distillation examples that illustrate the use of the shortest stripping lines approach for calculating minimum energy requirements.
These problems include examples of feed, saddle, and tangent pinch points for ideal and non-ideal mixtures as well as problems whose solutions are not pinch points. In the liquid phase is modeled by the UNIQUAC equation, unless otherwise all cases, .fi d All interaction parameters can be found in the appendix. In all case where spect e .
. hed solution is reported, we solved the Nonlinear Programming (NLP) problem a p111C defined by Equations I. 7 to I.12. In all cases where a non-pinched solution is reported we solved the NLP and then the integer programming problem defined by Equations 1.1 4 to I.17 plus Equation~ I.10 to I.12. For the multi-unit process examples an NLP defined by Equations I.7 to I.13 was solved. However, we remark the reader must keep in mind that configuration must be accounted for correctly. That is, a column with stripping and rectifying sections obviously involves a different set of equations · that, for example, a stripping column. All heat duties were determined using energy balance calculations around the reboiler and condenser. Finally, calculations were performed on a Pentium III with a Lahey F77 /EM3~ compiler, a Pentium IV equipped with a Lahey-Fijitsu LF95 compiler, and using Maple.

Binary Mixtures
Binary mixtures can exhibit both feed and tangent pinch points but not saddle pinch points. However, before discussing any ternary examples it is important to define what we mean by feasibility.
Recall the remarks made at the end of section 4 regarding our decisions to integrate both column sections from the bottom up. F~om a mathematical perspective, it is possible to completely specify the bottoms product composition since this simply corresponds to specifying the initial conditions for a nonlinear dynamical system --di s of the number of components in the mixture. As a result, the corresponding regaf es . column trajectory will be unique provided the energy balance is used to define the reflux ratio and the appropriate liquid composition in the strippin~ section is used to initialize the rectifying profile. For each specified bottoms composition, there will be one and only one resulting distillate composition for each choice of boil-up ratio.
Consequently we define . feasibility based on whether the calculated distillate composition satisfies desired distillate compositions constraints and typically use one or more inequalities to define this condition.

Feed Pinch Points
Consider the separation of n-pentane and n-heptane by distillation. The feed, distillate, and bottoms compositions for this example are summarized in Table I. l. The single feed is assumed to be saturated liquid, the liquid and vapor phases are treated as ideal mixtures, and the column is equipped with a total condenser. The distillation is considered feasible ifxn(n-C 5 ) > 0.99.   distance is simply the stripping line measured from the bottoms composition to the pinch point on the equilibrium curve along the x-axis. Reflux ratios that satisfy overall energy balance for the column are also given in Figure. I.2.
The stripping liiie distance of 0.1932 corresponds to a boil-up ratio of s = 0.4750 and represents a case where the reboil ratio is less than the minimum required. This is because the resulting rectifying line has a reflux ratio of r = 0.0266 and does not produce the desired overhead product. Therefore the desired separation is infeasible for s = 0.4750. On the other hand, th~ middle column profile, which is shown in red, corresponds to the minimum boil-up ratio for.:which the desired separation is feasible.
The stripping line distance for a boil-up ratio of s = 0. 7055 in this case is 0.3100, r = 0.5248, and the corresponding rectifying profile a distillate product with a composition of xn = 0.99863 -clearly greater than the specified value of xn. Moreover, for all reboil ratios greater than Smin = 0. 7055, the s'eparation is always feasible and the distance of the stripping line is always greater that 0.3100 -as shown for the case of s = 1.0500, for which the reflux ratio is r = ·i.2693, xn = 0.99943, and the stripping distance is 0.4311. These results are tabulated in Table I

[angent Pinch Points
Consider the distillation of acetone (A) and water (W) at 1 atm . . The equilibrium curve for acetone and water shows an inflection and hence can give rise to a tangent Pinch point that determines the minimum boil.:.up ratio for this distillation. The feed, 39 · ate distillate and bottoms compositions for this distillation are given in tPProxnn ' In this example, the NLP defined by Eqs. 7 to 12 was solved. Figure

,, ent Pinch Points
Tangent pinch points can also determine minimum energy consumption in distillation.
The recovery of acrylic acid from a mixture of acrylic acid (AcA), water (W), and acetic acid (AA) at . atmospheric pressure provides an example of a tangent pinch.
Here the liquid is modeled using the UNIQUAC equation and the vapor is modeled by the Hayden-O'Connell (HQ_C) equation since both acetic acid and acrylic acid show strong vapor phase dimerization. Table I.9 gives the feed, approximate distillate, and bottoms composition. Separation feasibility is defined by the purity of acrylic acid in the overhead product and for this illustration, xn(AcA) :::: 0.99 was used.  This distillation is -an example of a split whose overhead product is close to the ethanol/ethyl acetate/water azeotrope and a bottoms stream is an acetic acid product that t · · con ams small amounts of the other components.  Figure. I. 7 and these boundaries are one-dimensional curves. ·Figure I. 7 also shows three column profiles with corresponding stripping line distances for which two profiles are feasible and one is not.
ample the minimum boil-up ratio that gives the desired separation is Smin = for this ex ' h S tripping line distance is 1.31397, and there is a feed pinch that occurs at the 6.263, t e .
-(0 14850 0.13635, 0.55189). The corresponding reflux ratio is r = pomt XFP -   mn where the feed is saturated liquid and the column is equipped with a total water\ n ,, condenser. Here the primary purpose of the separation is to produce an overhead product that is largely a mi,xture of methanol and ethyl acetate since the low boiling mixture is the methanol-ethyl acetate azeotrope at XAz = (xM, XEAc) = (0.69410, 0.30590). The feed, approximate distillate, and bottoms product compositions are given in Table I.13. Separation feasibility, in this example, is defined by the condition II xo -xo,spec II :S 0.02, where xn,spec is the distillate composition given in Table I.13.
Vapor phase behavior is modeled using the HOC equation.

rifion-Pinched Minimum Energy Solution for a Ternary Mixture
Koehler et ai.2 2 provide an example where minimum energy consumption does not correspond to a pinch point and that it is possible to construct a finite column that uses minimum energy. Consider the separation of a mixture of chloroform, acetone and benzene at atmospheric pressure where the vapor phase is assumed to be ideal. Feed, bottoms, and approximate distillate compositions for this example are shown in Table   1.17. The primary objective of this separation is to produce a chloroform-rich distillate such that x 0 (C) 2: 0.945.
Distillations with minimum energy solutions that do not lie at a pinch point can be solved using a two-step approach based on the concept of shortest stripping line -as described in section 5. First, the shortest stripping line that gives a pinch for the desired separation is determined by solving the NLP defined by Eqs. 7 to 12.
58 . Column Specifications for Chloroform/Acetone/Water Distillation with No Table I

Non-Pinched Minimum Energy Solution for a Six-Component Refinery Mixture
Consider the feed mixture shown in Table I.15 and let the desired separation be a split between the C4's and C 5 's as shown in Table I.18. The liquid and vapor phases for this example are assumed to be ideal solutions where the K-values are given by the method in Wilson 36 . The distillation is considered feasible if the condition II Yn -YD.spec II S 0.03 is satisfied, where YD,spec is given in Table I.18. Surprisingly, this distillation is a more difficult separation than one might imagine because of the relatively volatilities of the components involved. Normal butane distributes more 60 . h expected. Nonetheless, Table I. 19 gives results for two non-pinched readily t an .
for the desired separation. As shown in Table I. 19, the minimum boil-up ratio for this distillation is Smin = 12.669 and corresponds to the shortest stripping line distance of 2.66343 .. However, it is also important to note that this minimum energy solution is not pinched. It is a nonpinched solution that has only 20 stripping stages and 6 rectifying stages and a corresponding minimum reflux ratio ofrmin = 11.669. Moreover, each of the solutions in Table I We explain the non-pinched nature of the minimum energy solution to this problem in the following way. For this indirect split, the overall energy balance for the column dictates that the boil-up ratio cannot go belows= 1 otherwise the corresponding reflux ratio would be less than zero. However, even at slightly greater than one, the stripping feed pinch point is XFP = (0.43039, 0.00312, 0.00001, 0.24100, 0.17500) where the compositions are in the order propane, n-butane, isobutene, iso-pentane, and npentane. At this stripping feed pinch point the composition of propane is already higher than the specified propane composition in the distillate in Table I.18. Since any rectification only increases the propane concentration in the distillate, it is clear that there is not a stripping feed pinch in this column. On the other hand, the rectifying pinch points that are relevant to this separation are severely limited. . For a feed rectifying pinch point to occur, both the composition on some tray for the stripping profile and reflux ratio calculated by o~erall energy balance for a given value of s must match a composition and reflux ratio on the rectifying pinch point curve.
·n this distillation, at relatively low values of reflux ratio the rectifying aowever, 1 .
. t curve moves rapidly to the n-octane comer and we have a similar situation pmch pom to that described for the top of the column. That is, at low values of reflux ratio the rectifying pinch point composition is greater than the specified n-octane concentration in the bottoms is 0.3010. Thus, there is no rectifying feed pinch for this column and the only alternative is a no~-pinched minimum energy solution.
We compared the results in Table I.19 with those predicted by Underwood's method as implemented in the Aspen Plus program DSTWU, which uses constant relative volatility to describe the phase equilibrium. For the Underwood method we assumed that nC 4 and iC 5 were the light and heavy key components respectively, the recoveries for the.light and heavy keys in the distillate were 0.9999 and 0.00025 respectively, and the column was equipped with a partial condenser. Also simple mass balance shows that D/F = 0.5 if the goal is to separate the C/s and C 5 's. The results predicted by Underwood's method differed substantially from those predicted by the shortest stripping line approach when Wilson's method 36 was used to describe the phase equilibrium. DSTWU predicts a minimum reflux ratio of r = 1.3388 and a minimum boil-up ratio of s = 2.3388.
To understand these marked differences we did several things. In the first case, wheres= . 1.3388 was used in our shortest stripping line approach, the propane composition at the stripping pinch point is well above the desired propane composition in the distillate product. Further rectification only makes matters worse and it is not possible to meet the desired specifications shown in Table I.18 at the top and bottom of the column with the minimum boil-up and reflux ratios predicted by Underwood's method. In the second case, we used constant relative volatilities of · 4.9501, 1.9470, 2.4210, 1, 0.8522, and 0.1042 for propane, n-butane, i-butane, ipentane, n-pentane, and n-octane respectively and our shortest stripping line approach.
The minimum boil-up ratio calculated using the shortest stripping line approach and a constant relative volatility model matched the results in Table I. 19 -not those ·given by DSTWU. Finally, we used DSTWU to det~rmine minimum reflux and boil-up ratios for two other problems -the direct split of this six-component mixture whose results are described in Table I. 16 and an indirect split of the ternary mixture described in the next section. Minimum reflux and boil-up ratios predicted by Underwood's method and the shortest stripping line approach agree quite well for the direct split of the six-component refinery mixture. On the other hand, for the indirect split of the ternary mixture described in the next section, DSTWU fails and thus provides no values for the minimum reflux or boil-up ratio. derived directly or ip.directly from pinch points on the stripping line equation (i.e., Eq.

6
). In this section we show that the shortest stripping line approach can also easily find feed pinch points in the rectifying section as well as multiple pinch points. It is well known that indirect splits often give rise to rectifying feed pinch points and/or  +Feasible if llYn -YD,specll :S 0.01; * Saturated liquid ( q = 1) F this example we solved the NLP problem defined by Eqs. 7 to 12 with the or .
odifications described for finding a rectifying pinch point as given in the algorithm . Figure I.9 shows the results for two separate profiles that make the desired section. separation given in Table 1 Figure 9 for clarity. Second, it again. clearly shows that it is the shortest stripping line that is important in finding minimum energy requirements -not the distance of the stripping line plus rectifying line. Third, this example illustrates that there is a very simple way of deciding whether there is a potential feed pinch in the rectifying or stripping section and h~w to find ~ good approximation of the feed pinch point. Notice that the stripping profiles cross th~ rectifying pinch point curve before they cross _ the stripping pinch point curve. This, ~e believe, clearly suggests that there is a potential feed pinch in the rectifying section and not in the stripping section. Moreover, note that the extended stripping line corresponding to minimum boil-up ratio passes through the f~ point. Thus, the intersection of this actual stripping profile with the rectifying pinch point curve represents a ~seful processing target for the amount of separation that needs to be accomplished (or the number of stages) in the stripping section of the column design that gives a rectifying feed pinch. Thus all one needs to do is find the stripping profile that passes through the feed point, locate the intersection of this stripping line with the rectifying pinch point curve, say Xpp, identify the corresponding reflux ratio from the rectifying pinch point curve, say rpp, and count the number of stri . ppmg stages needed to get from xB to Xpp. If there is a feed pinch point in the rectifying section, then the reflux ratio, r, calculated from overall energy balance (i.e., . 1 10) should match closely with the value of rpp. As in the case of the six-Bquat1on · t refinery example, each non-pinched solution shown in Table I.21 defines a  cmnponen  ' f boil-up ratios that meet the desired separation for the given number of range o .
• m· g and rectifying stages. For example, for all s = [20.28, 23.19], a column stnPP configured with 34 stripping stages and 4 rectifying stages easily makes the desired separation given in Table 1..20. Finally, despite all of these problem characteristics, the shortest stripping line approach easily identifies the minimum boil-up ratio and hence minimum energy requirements for this separation.

Minimum Energy Requirement for Multi-Unit Process
One of the key features of the concept of shortest stripping line is its ability to find minimum energy requirements for multi-unit processes. In ·doing so, it provides correct processing targets so that the overall process uses minimum energy. Two examples of multi-unit processes -a hybrid extraction/distillation process and reaction/separation/recycle system -are given. In each case the NLP problem defined by Equations I.7 to I.13, which include ancillary .constraints, was solved. The reason these additional constraints are required is to define correct processing targets that are constrained to lie on surfaces defined by liquid-liquid equilibrium curves or reaction equilibrium curves. . · additional energy requirement information. The process flow diagram provide some _ d · g acid by extraction and distillation is shown in Figure 10. To correctly for pro ucin . minimum energy use, it is necessary to determine the extent of extraction detennme that results in the subsequent distillation processes using minimum energy such that the acetic acid specifications in the bottoms stream of the acid recovery column are still met. To do this, car~ful attention must be paid to the fact that the feed to the acetic acid recovery column must lie on the binodal curve. Thus there is a correct processing target (i.e., extract composition) that results in minimum energy use.  Table   l.22. The t mos energy efficient · solution to the acid recovery column is a stripping 70 "th 17 stages and is clearly not pinched, has a stripping line distance of Ds = column w1 9 d corresponds to the minimum boil-up ratio of Smin = 9.10. It is also worth t.365 an . th t the reason for the differences in energy requirements for the two feasible nonng a · . . columns is not the difference in the boil-up ratios but rather the difference in stnppmg .
tbfoughput to the acetic acid recovery column, which in tum, is due to the significant difference in the extent of ex,traction.

Reaction/Separation/Recycle Versus Reactive Distillation
In this section we compare two processes for producing Methyl Tertiary Butyl Ether (MTBE) -reaction/separation/recycle (RSR) and reactive distillation -as shown in   (1) and methanol (2)

Reaction/Separation/Recycle System
The objective of this RSR process is to produce pure MTBE. However, producing pure isobutene at the top of the column in the RSR process is not a concern as it is in reactive distillation because the overhead product in the RSR process can be recycled to the reactor. Figure 1.13 shows various stripping lines for the mixture isobutene (I), methanol {M) and MTBE at 8 atm, for the production of high purity MTBE. Also shown in Figure 13 are the chemical equilibrium curve (under the assumption that the Damkohler number is high enough to drive the reaction to equilibrium), the distillation boundary for the case of no reaction, the attainable region for PFR's for a range of isobutene and methanol feeds, and the distances of various stripping lines.

74
"ble stripping columns for the production of pure MTBE that are shown in The feas1 figure 13 assume that the reactor effluent is on the chemical equilibrium line. We th . 1 1 · n Figure 13 as in other figures in this manuscript, the stripping line distance note a ' . ured from the bottoms composition to the stripping pinch point curve and only 1 s meas the stripping line at the very bot~om _ of the triangular region is infeasible. However, it is clear from Figure 13 that the (back-to-front) approach to synthesis and design based on the concept of shortest separation line easily identifies the correct PFR reactor effluent target composition so that the RSR process uses minimum energy. This reactor effluent target, in tuni, identifies the overall feed (fresh feed plus recycle) to the reactor by following the appropriate PFR trajectory in the attainable region toward the hypotenuse.
The net result of this is that if minimum energy is the objective, then the overall feed to the reactor should not consist of a stoichiometric (or 1: 1) ratio of isobutene and methanol but should be a mixture of 58-mol% isobutene and 42-mol% methanol. This ratio of reactants to the column is easily determined by extrapolating the PFR trajectory back to the hypotenuse. On the other hand, the overall feed to the process · is equimolar mixture of isobutene and methanol and is fixed by overall mass balance to . the RSR process. Numerical results for this RSR process are summarized in Table   1.23. 2) Due to the presence of the distillation boundary, little is gained by rectification and therefore separation can be achieved using a stripping column, in which the overhead product is recycled back to the reactor.
3) The energy of any PPR increases as conversion approaches the chemical equilibrium line. However, these energy requirements are insignificant compared to the energy requirements for separation. 76   4 ) As in the case of the hybrid separation scheme, the proposed back-to-front approach based on the concept of shortest separation lines clearly identifies the correct reactor effluent target for the desired MTBE product. Moreover, this effluent target does not lie at a pinch point for the stripping column.
S) The stripping column design that uses minimum energy corresponds to the shortest stripping line distance of 0.5316, Smin = 0.917, and has 37 stages. It is clear from Figure 13 that this design is not pinched.
6) The reactor effluent target determined from the shortest stripping line distance shows that minimum energy consumption requires a reactor feed of 58 mol% isobutene and 42 mol% methanol. composition is specified exactly as in the RSR process while the distillate specification defining feasibility is xn(iC4) ::::_ 0.998.

Governing Equations
The equations used to determine distillation lines for the reactive distillation processes differ in format from Equations I.1 to I.6 an~ are, therefore, summarized here. The total and component material balances are given by where Vi is the stoichiometric coefficient for the ith component and the unsubscripted variable, v, in Equation I.19 is the stoichiometric coefficient for the overall reaction and has the value of -1 for the production of MTBE from isobutene and methanol.
The Variable, ET, in Equation I. 19 and I.20 denotes the extent of reaction for the entire d signifies that the number of degrees of freedom is one higher than that for colU111Il an t . nal distillation. Thus for the three component mixture under consideration 1 ..,ven 10 ecify two mole fractions in both the distillate and bottoms streams. The we can sp 11 and component material balances together with the mole fraction summation overa equations may then be solved for the flow ratios, F /B and D/B, as well as the overall extent of reaction that is n~eded to achieve the specified product purities.
The model equations for the jth stage in an RD column include the overall and component material balances where Ej is the extent of reaction on the jth stage. To these equations we add · the familiar equations of phase equilibrium, mole. fraction summation, and the stage energy balance, which in the assumed absence of heat effects simplifies to Vj = Yj-l · Calculation of the composition profile and associated stripping line distance begins by solving the equations for the reboiler. This is a special case of Equations I.21 and I.22 in which Vo = 0 and V 1 I B = s and provides the composition of the vapor leaving the reboiler, the composition of the liquid entering the reboiler, the temperature, and the unknown flows. Moving from the reboiler to each stage in the stripping section, we 79 . the calculations in a similar manner until we reach the pinch at the end of the continue . . line However, these stage-to-stage calculations in the stripping section stopping .
. that Equations I.21 and I.22 be augmented by the simple equation Ej = 0, which require S th e condition of no reaction in the stripping section. However, not all of these stripping lines are candidates for the stripping section of a reactive distillation column to make MTBE with the specified composition. Other stripping ratios intermediate between those shown given in Table I.23 will lead to profiles that do not have a stage composition on the reaction equilibrium line.
Once we have located a feasible feed stage composition we can continue to solve the model equations for the stages in the reactive rectifying section. The model for the feed stage and all higher stages necessarily includes the reaction equilibrium equation for the composition of the liquid entering the feed stage from the stage above together with the equations of phase equilibrium for the stage above because it is these equations that determine the temperature at which the activity coefficients in Equation

II
. Because the stripping section of the RD column cames out exactly the same separation as the stripping column in the RSR process it follows that the RD column with the lowest energy demand is the same as that in the RSR process and corresponds to the shortest stripping line distance. Numerical results for this RD process are shown in Table I  It is also important to note that the feed composition plays an important role in design and operation of MTBE reactive distillation columns. For example, if the feed composition approaches equal parts isobutene and methanol, then the reflux ratio must increase significantly in order to ensure that there is sufficient liquid to return to the .
column and it actually operates as a (reactive) distillation column. In the example of Lee and Westerberg 40 , the feed is 60% isobutene, the reflux ratio is 14 and there are 16 stripping stages and 3 reactive stages in the rectifying section including the feed stage. there may also be other compounds present in the feed, which among other things, gllarantees that there is sufficient liquid to return to the column as reflux.
ct it is not surPrising that the minimum energy stripping section is the same 1nretrospe th R SR and RD processes given the vessel configurations and recycle stream for e shown in Figure L 12. The volatilities of the compoun~s involved in this process are such that the reaction should take place above the feed; thus, the same non-reactive stripping line suffices for }?oth processes. However, one should be cautious about generalizing this result since there is a wide variety of possible reactive distillation column gonfigurations 39 . We also note here that for columns that attain reaction equilibrium on each stage the material balances can be written in form of transformed composition variabl~s 27 ' 39 and it is easy to show that the lowest energy design corresponds to that with the shortest stripping line when expressed in terms of these transformed composition variables.
Finally we remark that if distillate product is actually drawn from the RD process, this will alter the overall mass balance for the RD process and change, perhaps significantly,_ the design that uses minimum energy. Nevertheless, our back-to-front approach based on shortest stripping line can be used to find minimum energy designs for this case as well. shortest stripping line approach finds correct processing targets in multi-unit processes so that the overall process consumes minimum energy. Results for two examples of multi-unit proce~ses -an extraction/distillation process for the separation of acetic acid and water ·and MTBE production using reactive distillation and a reactorseparation-recycle process -were presented to support these claims.
Finally, we close with the remark that the concept of shortest stripping line is a fundamental principle in separations that encompasses many approaches to minimum energy consumption in distillation processes.  Chem. Eng. Sci. 1961, 16, 143.  ·n the petroleum, chemical, pharmaceutical, and other industries for many aiauons d this is unlikely to change. These unit operations, as well as others, will years, an ~ the primary means of separation in many industries for the foreseeable future.

remain .
Other separation techniques like chromatography and membrane separation simply cannot provide the purity and volume to be competitive. However, distillation consumes significant amounts of energy. While some believe that these unit operations are mature technologies and that there is little to be gained from research in ICParations like distillation and crystallization, we disagree with this viewpoint for two reasons. First, with the recent significant increase in global energy demands and every indication that demand will remain high, it is important to consider ways of designing new separation processes and retrofitting existing ones so they are energy efficient.
Hybrid separations such as extraction followed by distillation and reactive distillation can often be used to reduce the energy costs of conventional distillation alone.
Second, the approach taken in. this work is a direct outgrowth of recent results that shed new light on residue curves and distillation lines and it is unlikely that we would have uncovered the proposed characterization ·of energy efficient separations without '.isorption/distillation, and so on) represent one way of reducing the energy costs of . distillation alone. The motivation for this work comes from our fundamental belief that there is a connection between the length of residue curves (or distillation lines) and the energy needed to perform a given separation. In particular, we began with the intuitive belief that following the longest residue curve must somehow be related to the highest energy costs associated with performing a given separation. Furthermore, if the longest residue curve is the most costly separation; -then the shortest curve lhould result in the use of the least amount of energy required for the given separation  Pennit energy balances to be included in the determination of minimum reflux. d co-workers 16 • 17 • 1 8 also present a method for computing minimum reflux oUand an ventional and c_ omplex column configurations that is based on their 8-method. 9-th od is also a tray-by-tray approach closely related to the Thiele-Geddes e roe eth d While these methods are considered more-rigorous than techniques that solve JD 0 .
. h oint equations because they can incorporate energy tray effects, they are also pmc P ' laborious since they involve energy as-well as mass balance equations. Finally, ore .
;mch points can also be determined in a reliable manner by integrating the differential can be related to minimum reflux and boil-up ratios, and thus minimum energy use.

U.4 A Methodology for Energy Efficient Hybrid Separations
The remainder of this paper presents a methodology for determining minimum energy irements for a given separation based on the concept of shortest separation lines.
To make the methodology clear, the separation of acetic acid and water is used. · Acetic acid is ari important chemical commodity because many intermediates (e.g., vinyl acetate monomer, terphthalic acid, acid anhydride, and various solvents) are manufactured from low water-content acetic acid. The separation of acetic acid and water by conventional distillation is known to be energy intensive and does not represent best industrial practice. For dilute solutions of acetic acid in water (i.e., at or below 30 wt% = 11.5 mol% acetic acid) hybrid separation using liquid-liquid extraction followed by distillation is often used. ounts of water to be condensed overhead, only to be re-vaporized internally in wgeam .
lumn Conventional distillation of acetic acid and water also requires high boil- 2) What is the number of stages for the subsequent distillations?
3) How much extraction sI:iould be performed so that the subsequent distillations use a minimum amount of energy and still produce the desired acetic acid composition?
In reality these questions are strongly interrelated. More<?ver, the synthesis and design of the distillations require comparisons of columns that have different feeds because they depend on the separation performed by the extraction column. This is more ehallenging than the problems studied by Fidkowski 7 or those presented in the review paper of Koehler et al. 9 where the feed under consideration remains fixed. In our case, Dlus t be car eful to make meaningful comparisons of all of the separations involved.

Acetic Acid No Solvent Rec cle
Consider the use of a stripping column to recover acetic acid from a feed that is the extract from a liquid-liquid extraction column. Let ethyl acetate be the solvent used to extract acetic acid from a water solution of 11.5-mol% acetic acid. Let the desired bottoms composition be high purity acetic acid with XB = (0.9999, 5 x 10-5 , 5 x 10-5), Where the components are ordered acetic acid (l ), ethyl acetate (2) and water (3). We emphasize that no solvent recycle is considered in the analysi~ at first; solvent recycle is addressed later in the paper. Note also that there are no separation boundaries   where the liquid is sub-cooled to 298.15 K. the so ven · Extraction column stream molar flow rates and compositions are summarized in Table   Table II Fig. II3, under the condition of no recycle.      where D represents a distance function along a discrete trajectory, II . II denotes the two-norm, Xj and YJ represent the liquid and vapor compositions on stage j, XB is the bottoms composition, and c(xK) is some constraint function that defines any auxiliary conditions that must be met to make the des~gn feasible. For example, for the illustrative example, c(xK) = 0 can be viewed as a constraint that forces the liquid composition on tray K, XK, for the acetic acid recovery column to lie at some point, xbr.., on the binodal curve. Note here that the unknown optimization variable is the boil-up ratio and the optimal trajectory is ac~ually a sequence of liquid compositions denoted by {xj}* that is assumed to be piece-wise linear. Also remember, for discrete stages the integration step size is h = 1; thus the upper limit on the summation in Equation Il.6 represents some large number of stages. We typically use Ns = 300 as an approximation for the number of stripping stages. Because the ancillary constraints can give rise to a feasible region that is a disjoint set of distillation lines, as illustrated in the hybrid separation of acetic acid and water, enforcing auxiliary constraint satisfaction from one optimization iteration to the next would require the optimizer to jump from one feasible distillation line to another. This where Pis some penalty parameter and c(x) is a shorthand notation for any auxiliary constraints. Note that the modified objective function is still a function of boil-up ratio b t · · u is now differentiable. However, the use of penalty or barrier functions can ltiple minima in the modified objective function as shown in

D.S.3 Generalizations and Other Formulations
The nonlinear programming problem defined by Equations II.6 through II.9 can be further generalized by using any appropriate set of constraint functions that define feasibility. Other formulations for more conventional separation problems are also possible within the theory of shortest separation lines.
. . ur opinion straightforward to imagine conditions similar to Eq. 9 for other It is, m o ' hybrid separation system configurations. The primary requirement of these more general auxiliary constraints is that they define a feasible region in some meaningful way. Given that, the auxiliary constraints can be written in the general form c(x1, x2, ••• , XN, yi, y 2 , ... , YN) = 0, where now c is some vector function of the liquid and vapor compositions throughout the separator. Since phase equilibrium implies that Yj = yj(Xj}, these constraints can actually be written in the compact form as simply c(x) = 0.

Conventional Separator Designs
Here we shift focus by considering single column designs and showing how the concept of shortest separation lines readily extends to more traditional settings in which the feed is specified, a prescribed separation is demanded, and conventional separators with rectifying and stripping sections are considered as design alternatives.
One important difference betwee~ this type of synthesis problem and the synthesis of hybrid separation schemes is that the feed to the primary recovery column is not fixed in the latter.
For the purpose of illustration, the use of both rectification and stripping in the acetic acid recovery column to achieve essentially the same desired high purity acetic acid separation is considered. Consider However, this is simply a necessary synthesis tactic. When stages are actually stepped off starting from XB, the end point of the top stage is unlikely to occur exactly at YD· The same is true for ternary and other multi-component mixtures.
The extension of the concept of shortest separation lines is quite straightforward, even for conventional separators. In particular, we still use the distance of the stripping line from the desired bottoms comp~sition to the stripping pinch point curve as the correct measure of energy requirements -even though the separator has a rectifying section.
To see why this is correct, consider an alternative separator with both a rectifying and stripping section for making the same separation that the optimal stripping column for high purity acetic acid recovery does. See Table II In this figure, actual stage compositions are indicated by the filled squares and for clarity, no tie lines have been shown in the liquid-liquid region. Of course it is rather . that these alternate column designs will not result in a lower energy obvious . ent than the stripping column determined previously because the boil-up ratio requ1rem . . "ficantly higher than minimum boil-up. See also Table II subsets, was presented for directly finding the most energy efficient hybrid separation designs. The approach of shortest separation lines was generalized to conventional separators and was shown to represent a unifying principle for generating separation process designs that are energy efficient.

Other Applications
We have also applied the concept of shortest separation lines to a variety of single distillation columns that exhibit feed, saddle point, or tangent pinch points, reactive distillation columns, and columns that have minimum energy requirements that do not 132 occur at a pinch point. These results are the subject of a separate paper on the . . 1 of shortest separation lines. In addition, we have applied the concept of pnnc1p e shortest separation lines to multi-unit reaction/separation/recycle (RSR) processes such as the production of MTBE from isobutene and methanol. In all cases, we have been able to illustrate that minimum energy requirements correspond to the shortest separation line.  Recently, Lucia and co-workers have used a distillation line approach to develop the concept of shortest stripping line distance approach to minimum energy designs of distillation columns and multi-unit processes. It is well known that distillation line methods can be very sensitive to specified product compositions. A two-level distillation design procedure is proposed for finding portfolios of minimum energy designs when specifications are given in terms of key component recoveries. Thus product compositions are not specified but calculated. It is shown that the proposed two-level design procedure is flexible and can find minimum energy designs for both zeotropic and azeotropic distillations. It is also shown that the two-level design method encompasses Underwoo~' s solution but can find minimum energy designs components that are heavier than the heavy key and components lighter than the light key respectively. t ly Lucia et al. 6 have developed a novel and comprehensive approach to More recen ' JDinitnUlll energy requirements in distillations as well as multi-unit processes based on P t of shortest stripping line distance. This work clearly shows that minimum the conce · eq uirements for all types of processes, distillations, hybrid separations like energy r extraetion/distillation, and reaction, separation, recycle processes, can be determined in 8 straightforward geometric and intuitive manner by finding the shortest stripping line distance for the problem at hand. This new approach is quite general, encompasses many existing methods for finding minimum energy requirements, and is also capable of finding minimum energy requirements that do not correspond to pinch points -something the other methods cannot do.
It is well known that any methodology based on distillation lines can be very sensitive to specified product compositions. Small variations in product compositions can result in very large changes in minimum energy demands! Moreover, there are cases in which numerical difficulties arise in generating stripping and/or rectifying profiles that meet product specifications -even though these profiles are in theory possible.
These numerical difficulties are often due to rounding and truncation errors. The main purpose of this paper is to present a two-level distillation design methodology that addresses the sensitivity of distillation line methods to specified product compositions and design feasibility. The inner loop of our two-level design method is comprised of the shortest stripping line approach, which determines minimum energy requirements for fixed bottoms composition. The outer loop, on the other hand, is a Gauss-Newton th t is used to adjust the bottoms composition. In addition, the numerical strategy a . that comes from the outer loop provides a straightforward way of analysts d .
the sensitivity of distillation line trajectories to bottoms product understan mg composition.
We also show that our two-level methodology encompasses Underwood's method as a special case of the shortest stripping line approa~h (Lucia et al.6) by demonstrating that the minimum boil-up ratio determined by Underwood's method with vapor-liquid equilibrium given by constant relative volatilities corresponds to the minimum of all shortest stripping line distances for a given set of key component recovery fractions. Finally, we show that Underw~od's method often fails to find even a feasible design for problems involving mixtures that form azeotropes but that the proposed two-level design approach easily finds a portfolio of minimum energy designs in these cases.
Accordingly this paper is organized in the following way. First a very brief summary of Underwood's method is presented. This is followed by a description of a two-level algorithm for design and optimi~ation based on processing target. The description of the inner loop, which is the shortest stripping line approach of Lucia et al. 6

ID.1.1 A Brief Summary of Underwood's Method
Underwood's original method Underwood 1 for finding minimum reflux ratios is well known and several modi(lcations and extensions (e.g., Shiras,et al. 7 ; Barnes et al. 8 ) have been developed over the ye':lrs. and Sea e ' separations correspond to a double feed pinch point and the resulting l ·on for minimum reflux ratio, rrn1n, is given by express (III.1) where it is assumed that the feed is saturated liquid with a composition of XF, Xp is a pinch point, xn is the distillate composition, a is the relative volatility, and where the subscripts LK and HK denote the light and heavy key components respectively. For class 1 separations Xp = XF and Eq. III.1 is easily applied.
For class 2 separations Eq. III.1 still applies. However, there is either a rectifying or stripping pinch but not both. Thus Xp is not known and iteration is required. Different cases must be considered depending on which components are suspected of distributing. Class 2 separations require root finding to determine the root or roots, 8, that satisfy (III.2) where q is the thermal quality of the feed and where the subscript r denotes a reference component such as the heavy key.

One of the great appeals of Underwood's method is that is simple to program and easy
It also finds pinch points without regard for column composition profiles. to use.
Thus the convergence difficulties experienced by, for example, boundary value methods (i.e., trajectories that do not meet) are irrelevant in Underwood's method.
However, it does have some disadvantages. Underwood's method is based constant relative volatility and on recovery fractions of key components in the product streams, which can be satisfied by a range of product compositions. Product compositions cannot be specified directly in Underwood's method. Consequently if certain product compositions are required, something in addition to Underwood's method is needed.
Moreover, Underwood's method can fail on problems involving azeotropic mixturesas we demonstrate in the Numerical Examples section of this article.

ID.2 A Design & Optimization Methodology for Hitting Processing Targets
In this section, we describe a two-level design and optimization algorithm for finding or getting as close as possible t~ specific processing target compositions. The inner loop of this algorithm is the shortest stripping line approach, in which minimum energy requirements are determined for fixed bottoms composition.
In most distillation design problems the bottoms composition is often not known a priori.

Sensitivity Information for the Inner Problem
In order to actually compute the minimum stripping line distance with respect to boilup ratio using a Newton-based optimization method, sensitivity or partial derivative infonnation quantifying the change in trajectory with respect to boil-up ratio is required. This information can be computed efficiently using the implicit function recurs10n formulae for the partial derivatives and actually . this partial derivative information during the process of generating a calculating . .
Here the goal is to find expressions for the changes in Xj and yj with respect trajectory .

Partial Derivatives of the Distance Function
To use any Newton-based optimization method like the terrain method of Lucia and co-workers (Lucia and Feng 10 ), first and second derivatives of distance with respect to boil-up ratio are required. These derivatives depend on the sensitivities Jj-l for j = 1, ... , Ns. To begin, note that the distance along any stripping line trajectory in going from tray j to tray j+ 1 is given by (III.14) By the implicit function theorem

fil.3 A Two-Level Algorithm for Energy Efficient Design and Optimization
The overall algorithm is very simple.

1)
Given key component recovery fractions and a ~arget composition, XT, guess 2) Solve the inner problem for Smin(xB).

)
Using Smin(XB) from step 2, use the outer algorithm to calculate XB,new(Smin).
S) Set Xs = XB,new(Smin) and return to step 2. ·on fonnulae for calculating the changes in trajectory with respect to boil-up recurs1 · ratio (i.e., Eqs. III.12 and III.13) and the recursion formulae for determining the partial derivatives of distance function with respect to boil-up ratio (i.e., Eqs. III.26 and m.28) are needed.
Step 3 defines a simple measure of closeness to the desired target.
Step 4 is the outer sub-problem, which updates the recovery fractions of the non-key components and is solved by the Gauss-Newton strategy (i.e., Eq. III.35). The necessary partial derivatives for .solving the outer sub-problem by a Gauss-Newton method are given by Eq. III.36 and ,Eqs. III.39 to III.41. In our opinion, a Gauss-Newton method is appropriate for solving the outer problem because we are not necessarily interested in fast convergence. Rather, we are interested in a methodology that is robust, generates a number (or portfolio) of different minimum energy designs, and shows how these minimum energy designs are related to Underwood's method for a variety of situations.

ID.3.1 Advantages of the Proposed Two-Level Approach
The proposed two-level approach has several advantages because it 1) Permits many minimum energy designs to be investigated in one sweep.

)
Allows for the investigation of direct, indirect and transitions splits in one sweep.
157 I Can handle bounds on lighter than light and heavier than heavy key recovery fractions to be included. . Depending on the problem specifications, one or both initializations will split.
to the transition split -if it exists. Note that if the target composition is converge selected as the feed composition (i.e., XT = xF), then the two-level algorithm asymptotically approaches a transition split (or double feed pinch point) for class 1 separations. Also note that the primary difference between direct and indirect splits in the context of Underwood's method is the choice of light and heavy key components. Thus the proposed two-level algorithm is readily applied to either case by simply varying the choice of light and heavy key components. This process of spanning direct and indirect splits provides a convenient way to -µnderstand the effect of the recovery fractions of non-key components. b U nds are easily included in the two-level design algorithm (i.e., Eqs. III.31 to These o

IJ.Determines Feasible Designs that Underwood's Method Cannot Find
For mixtures that form azeotropes, it is well known that Underwood's method can have difficulties and fail to find a feasible design regardless of whether one of the distillation product compositions is azeotropic or not. Difficulties arise because the concept of light and heavy key component can be skewed for azeotropic mixtures, making the Underwood equations ill-defined. In contrast, the two-level design approach has no difficulties whatsoever in finding feasible minimum energy designs for distillations involving mixtures that form azeotropes.

ID.4 Numerical Examples
In this section, we illustrate two:--level design and optimization methodology for a number of multicomponent mixtures and consider direct, indirect, and transition splits.
In all cases, the calculations were performed in .double precision arithmetic using a Pentium IV personal computer with the Lahey-Fujitsu compiler (LF95).

ID.4.1 Example 1
The primary purpose of this first example is to present the details of the two-level design method for a very simple case. This example was adapted from Doherty and s (p 124 2001) and involves the separation of a mixture of methanol (1) distillation lines generally fix the bottoms and top compositions in the problem definition and are not easily compared to Underwood's method. Therefore, the ·column specifications were changed slightly, as shown in Table III.1, and given in tenns of recoveries so a more direct comparison between the two-level design methodology proposed in this paper and the work of Underwood can be made. when the processing target is set to the feed composition (i.e. XT = Xp), the separations 1 design methodology converges to the Underwood's solution, which in this iwo-leve .
double-feed pinch (or transition split). We also discuss other advantages case 1s a offered by our two-level design methodology.

Evolution o(Direct Splits
jli:DS One way to initialize our two-level design methodology is to set the ethanol recovery fraction, ffa in the top product such that the separation is a direct split (e.g., rE = 0.96).
This choice of recovery fraction is arbitrary and other appropriate initial guesses are equally useful and will result in convergence to Underwood's solution. Ideally, the initial guess should be away from the transition split so that the recovery fraction (or composition) iterates sample an appropriate range of the feasible range. Once the recoveries of all components are specified, the composition of the bottom and top products can be easily calculated. From this, the two-level design methodology alternates between the shortest stripping line approach to find the corresponding minimum energy requirement for tp.e column and the outer loop to update values of the recovery fractions, as described by the equations from the previous section. nl 2 . Two-Level Iterations Initialized Using a Direct Split The solution for Underwood's method is also shown in Table III  The results in Table III thus the glob 1 · · · · a mm1mum energy design for fixed key component recovery fract10ns -

164
. d ·t is understood that the composition of the resulting product streams is not a proV1de 1 .d ti· on in deciding what is optimal. cons1 era Table III Note that the norm of the targeting function decreases monotonically as the two-level design procedure approaches the Underwood solution, and that fast convergence of the outer loop is not necessarily desirable if the goal of the engineering investigation is to generate a portfolio of minimum energy designs.

Evolution of Indirect Splits
It is important to note that any physically meaningful value of rE is possible but it is often easiest to initialize the two-level method and find an initial feasible design with either an approximate direct or indirect split. Here we initialize the proposed twolevel algorithm with a starting guess for the recovery of the non-key component that corresponds to an indirect split. Tc:> explore various designs starting from the indirect split, the ethanol recovery fraction in the bottom product was initialized to rE = 0.04. Table III.4 shows the iterations given by the two-level approach starting from the indirect split. Here we use a line search parameter value of~ = rrn1n/2srnin· Note again that the two-level approach converges to the solution given by Underwood's method -this time from the indirect split -and provides a portfolio of minimum energy designs. Moreover, the same shortest stripping line interpretation of d's method is valid here. That is, Underwood's solution for class 1 Underwoo . s corresponds to the minimum shortest stripping line distance (or the global separation JDinirnUlll stripping line distance) and thus the global minimum energy design.

• 11
note that the norm of the targeting function decreases monotonically as the Fma y, two-level design procedure converges to Underwood's solution.
Ill 3 summarizes all of the calculations given in Tables III.2, III It is also interesting to note that the new estimate predicted by solving the outer loop problem often lands very close to the minimum of each curve for the case of the direct split but that the minima for the curves corresponding to indirect splits can be outside the feasible region -except specifically for the curve that gives Underwood's solution.
. hy it is often a good idea to use a line search parameter ~ < 1 for the two-'fhis IS W d ·gn procedure when it is initialized using the indirect split. The biggest advantage of our two-level design methodology is that it offers a systematic way of using distillation line methods to explore a portfolio of feasible minimum energy designs that encompass Underwood's solution.  Tables III.2 to III.5 satisfy the recovery  and reco  · constraints for the key components and each solution is a minimum energy design for a particular non-key recovery fraction. The resulting designs span the entire range of non-key component recoveries and converge to Underwood's solution. Moreover, each of these minimum energy designs is obtained by using the shortest stripping line method for the corresponding inner loop problem.

ID.4.2 Example 2
The second example involves the separation of the quaternary hydrocarbon mixture at 400 psia. The specific feed composition and recovery fractions are shown in Table   m.6.

Direct Split
The two-level design methodology is initialized to a direct split by setting the non-key rec · · ovenes of 1so-pentane (r 1 p) and n-pentane (rNP) to 0.98. Using these non-key . and the key component recoveries in Table III * Recovery fraction of non-key components (i-pentane, n-pentane) in bottom product **St. · · · nppmg hne distance measured from XB to stripping pinch point curve . h ase of example 1, the two-level design methodology generates a portfolio of As1nt ec nUnimum energy designs as it alternates between inner and outer loops. This portfolio . ·mum energy designs is summarized in Table III.7, along with Underwood's of min• solution obtained by using relative volatilities calculated at the feed comp9sition given in Table III.6.
It can be seen that the outer loop converges monotonically to a solution very close to the Underwood solution. Also, for all inner loop (or shortest stripping line) problems, the solution is considered feasible if the distillate product satisfies the condition llYnyo,specll :=: 0.05. It is important to remember that in this example the K-Wilson model (i.e., Eq. III.45) was used to describe vapor-liquid equilibrium instead of assuming constant relative volatilities. Hence the final solution shown in Table III. 7, as expected, differs to greater extent from Underwood's solution than the results for example 1. However, this example illustrates two important aspects regarding the proposed methodology.
1) It is independent of the num~er of non-key components and thus is applicable to mixtures with any number of components.
2) Any thermodynamic model can be used to describe vapor-liquid equilibrium, provided the necessary derivative information is obtained properly.  Table III. 7. Table III

Indirect Splits
As in the first example, it is possible to initialize the two-level design algorithm with a starting guess for the recoveries of the non-key component that corresponds to an indirect split. Thus to explore various designs starting from the indirect split, the reco · · venes of 1so-pentane (rIP) and n-pentane (rNP) in bottom product were set to 0.02 176 II t ·vely the condition llYn -YD,specll :S 0.05. Table III.10 provides additional information regarding the two-level design portfolio shown in Table III Similar to the direct split, when initialized from an indirect split, the two-level design approach converges to a solution close to that given by Underwood's method and a portfolio of minimum energy designs. Thus the same shortest stripping generates .

. t rpretation that Underwood's solution corresponds to the global minimum in
Jine in e . . g line distances, requires minimum reboil and reflux ratio and thus represent a stnpptn global minimum energy design is valid here.
Also note that the norm of the targeting function decreases monotonically as the twolevel design procedure converges to Underwood's solution. Figure Table III We remark that the final solution to which the two-level design methodology converges from indirect split is not as close as the one reached from the direct split.

178
. d e to the numerical difficulties associated with finding a design with a double This IS U feed pinch.
While this difficulty will vary depending on the specific example, it is always possible fi nd a solution which is close enough to Underwood's solution for engineering use. to 1 . Figure

Direct Split
As in the earlier examples, the processing target for this example is set to the feed composition. To initialize the two-level design methodology, the non-key recovery was first set to a value (rB = 0.98) that makes the separation a direct split. . this initialization, the two-level design methodology alternates between the using d outer loops producing several minimum energy designs. III 13 on the other hand, gives additional information regarding this portfolio of JninimUDl energy designs. Figure  To re-emphasize, this example demonstrates that the two-level design methodology can be applied to any non-ideal vapor-liquid mixtures using suitable phase models, simple or complicated. Moreover, this flexibility is useful when volatilities change over a wide range due to the non-ideal nature of mixture under consideration and where Underwood's method, which is based on assumption of constant relative volatilities, is expected to have greater error in calculating minimum energy requirements. However, what is advantageous is that for the proposed two-level design methodology, the design problem can be specified in a way that is analogous to Underwood's method using only two key component recoveries. Finally, this example illustrates that for the specific set of key component recoveries used here, it is not 'bl to initialize the two-level design methodology by setting the non-key p<>SSl e n ent (benzene) recovery to a value which will make the split close to an indirect comP 0 · l .t This is due to the fact that it is not possible to find a feasible minimum energy sp 1. d ·gn with a rectifying pinch that satisfies the constraints for the given key est component recoveries. Thus the design portfolio for this example covers designs from direct split to the approximate transition split.

Example 4
This fourth example involves the separation of a four-component azeotropic mixture at atmospheric pressure, where the liquid phase is modeled by the UNIQUAC equation and the vapor phase is ideal. The purpose of this example is to show that Underwood's method fails while illustrating the applicability of the two-level design methodology to azeotropic systems. Table III.14 shows the feed composition, the heavy and light key components, and the desired recoveries for this separation.
This particular mixture has two .binary azeotropes at atmospheric pressure -a methanol/acetone azeotrope, (xM, xA) = (0.2343, 0. 7657), and an ethanol/water azeotrope (XE, xw) = (0.8874, 0.1126). The methanol/acetone azeotrope is minimum boiling and is the only stable node for this system. Bellows and Lucia 13  As expected, the design portfolio spans a smaller range of non-key component recoveries than designs in earlier examples. Also, note that the norm of the targeting function in Table III Table III.17 and is considered feasible if llYn -YD,specll :S 0.065. . . t distillation regions, as shown in Figure III.9. Note that the specifications distinc given in Table IIl.17 correspond to a distillation in the left hand side of Fig. 9, where the distillate product is a cleaner water stream (i.e., cleaner than the feed) and where fonnic acid is designated as the light key component and acetic acid is the heavy key component. Water is the non-key component in this illustration.   shortest stripping line interpretation and that the proposed two-level design procedure converges to that solution when the feed composition is used as the processing target.
On the other hand, for azeotropic mixtures, it was shown that Underwood's method fails to find a feasible design whereas the two-level design procedure provides a correct interpretation of minimum energy requirements in terms of a non-zero valued, global minimum in. the no~ of the targeting function. Finally, the mathematical machinery needed to implement the two-level design methodology was presented in detail.

Appendix 111.1
This appendix pr<?vides an implicit theorem analysis of phase equilibrium equations.
The main result is the definition of the ( c-1) x ( c-1) matrix of partial derivatives of y with respect to x, J yx, which accounts for summation equations for both liquid and vapor phases as well as the implicit dependence of temperature.

IV.1 Introduction
In their review paper, Koehler et al. (19_ 95) summarize the state of the art as it relates to finding minimum energy designs for distillation columns. They give a very good overview of methods available for finding minimum energy designs that correspond to itlP@ Q_oints and clearly point to the need for a systematic methodology for finding non-pinched, minimum energy designs with the following quote on page 1016 of their paper: "This special case of a minimum energy throughput without a pinch will not be handled by any of the published approximation procedures. Exact column simulations are here unavoidable." Unfortunately, there was no progress in finding a systematic methodology for finding non-pinched, minimum energy distillation designs until the work of .
In a recent paper,  give a comprehensive treatment of a new unifying principle in energy efficient process design -the shortest stripping line distance approach. This new approach states that the most energy efficient designs for processes in which distillation is involved correspond to the shortest stripping line distance for the distillation(s). Of course, the implicit assumption in this approach is that distillation is the largest energy consumer in many multi-unit processes and, for the most part, this is a very good assumption.   The purpose of this paper is to provide a more detailed description of how the shortest l stripping line distance methodology can be used to systematically and intelligently find non-pinched, minimum energy process designs and to address the broader question -what give rise to non-pinched designs? Accordingly, this paper is organized in the following way. A motivating example is presented first. Next the shortest stripping line distance approach of  is summarized. This is followed by an analysis of the conditions that give rise to non-pinched designs for single columns and multi-unit processes that involve distillation. Next a number of example problems are presented to support our analysis. This article ends with some conclusions regarding the findings of this work.

IV.2 Motivating Example
In this section, we present a non-pinched, minimum energy distillation design taken from the open literature that gives a modest savings in capital investment costs. 204

The Non-Pinched Distillation Exam le o Koehler et al. Revisited
Consider a column design given in  that was studied by Lucia et al. (Z007). The specifications for this distillation are shown in Table IV   used to generate the results in this paper. Nonetheless, the behavior is clearly qualitatively the same as that given in . The important thing to notice is that the pinched solution for Smin  provides for ancillary constraints such as requiring a liquid stage composition to lie on a binodal curve, where the integer K denotes the stage index for which the ancillary constraint is satisfied. See  for other distillate specifications and a discussion of ancillary constraints.
Note that the unknown optimization variable for the problem defined by Eqs. IV.1 to IV.7 is the boil-up ratio, s, and the optimal trajectory is actually a sequence of liquid compositions denoted by {xj}* that is assumed to be piece-wise linear. We typically use Ns = 300 in Eq. IV.1 to approximate an infinite number of stages in the stripping section, which are numbered from bottom to top.

IV.3.2 Integer Programming
To look for solutions that do not correspond to pinch points, we use a simple integer programming strategy to determine if it is possible to reduce the number of stripping stages from infinity to some reasonable finite number without increasing the boil-up · and reflux ratios by solving the following problem (IV.14) c(xK) = 0 for some KE [1, N] (IV.15) Note that the only the unknown optimization variable in this IP problem formulation is the number of stages, Ns. Moreover, the boil-up from the NLP problem is used as a constraint to fix the boil-up ratio in the integer program.
Alternation between the NLP and IP can be performed as many times as needed. For example, suppose the initial NLP with 300 stages yields a solution, Smin = 2, and then the IP results in a reduction in the number of stripping and rectifying stages to Ns = 75 and Nr = 10. One could then retm;n to the NLP with Ns = 75 and Nr = 10 and attempt to reduce the boil-up ratio below the initial calculated value of Smin = 2. If no further reduction in boil-up ratio is determined, then the algorithm terminates. If, on the other hand, the boil-up can be reduced, then the algorithm would return to the IP to try and further reduce the number of stages. This procedure, as stated, can be repeated as many times as needed until no further reduction in either boil-up ratio or number of stages occurs at which point the algorithm terminates.

JV. 3 .3 Optimization Algorithm & Implementation
Lucia et al. also give an optimization algorithm that alternates between solving the NLP and IP sub-problems, where the NLP problem is solved using the terrain method and integer bisection is used to solve the IP sub-problem. Alternation between the NLP and IP sub-problems can be repeated as many times as necessary. We refer the reader to the paper by  for the details of the optimization algorithm.

IV.4 What Gives Rise to Low Energy, Non-Pinched Solutions?
In our opinion, this is a very important question simply because there is absolutely no understanding of non-pinched, minimum energy designs described in the open literature. We gave some indication of what can cause the existence of non-pinched, minimum energy designs for a single distillation column involving an azeotropic mixture in the introduction. However, from our experience, we know the situation is more complicated than this. There are several ways in which non-pinched, minimum energy process designs can occur. Specifically, non-pinched, mimmum energy solutions can exist when 1) Certain product specifications and ancillary conditions, as in hybrid separation processes such as reactive distillation and extraction-distillation, must be satisfied.
2) The separation under consideration contains a) A maximum boiling azeotrope. b) A stripping pinch point curve with unstable branches. c) A product composition that lies near a distillation boundary.

213
3) The separation under consideration has stripping and rectifying trajectories that exhibit looping and intersect on their way to their respective pinch point curves and the trajectory in at least one section of the column shows reverse separation of one of the key components.
In all cases, the common thread seems to be the complicated interplay between the distillation line profiles and the pinch point curves.  have clearly demonstrated that hybrid separations like extraction-distillation systems and reactive distillations often have non-pinched, minimum energy solutions because the feed to the distillation must lie on a binodal or chemical equilibrium curve and because low· energy distillation line trajectories, which have pinch points outside the liquid-liquid region, fortuitously have a liquid stage composition that lands on the binodal or equilibrium curve! For example, in the hybrid separation of acetic acid and water using ethyl acetate as the solvent that was studied by  the extract from the liquid-liquid extraction column, which is the feed to the primary acid recovery column, must lie on the binodal curve.
Thus the feed to the primary column is constrained. For a relatively low purity acetic acid bottoms product (xAA :S 0.995), several low energy distillation line trajectories have a liquid stage co~position that lands on the binodal curve well before reaching their corresponding pinch points, which lie in the single liquid region. Thus the path of the trajectories combined with a feed that ~ust lie on the liquid-liquid equilibrium curve forces the design to be non-pinched. Similarly, in the reactive distillation example, the stream leaving the bottom stage of the reactive rectifying section of the reactive column lies on the chemical equilibrium curve. Here again there is a liquid stage composition for some of the low energy distillation lines that lands on the chemical equilibrium curve long before ever reaching the stripping pinch point curve.
Again, see . Thus in both hybrid separation examples, problem specifications and ancillary constraints that restrict the feed to the distillation help define minimum energy demands that are not pinched.
2) In the case of single distillation columns, there can be a variety of reasons for the existence of non-pinched, minimum energy designs. For zeotropic mixtures, the pinch point curves generally show no bifurcation behavior. However, the existence of a n~n-pinched design is tightly tied to the relationship between the actual · distillation line trajectory (i.e., the liquid composition profile and the corresponding boil-up and reflux ratios), the pinch points, and the boil-up ratios associated with the pinch points.
Mixtures that can form azeotropes, on the other hand, can have pinch point curves that exhibit bifurcations. Non-pinched. designs for columns separating azeotropic mixtures exist whenever part of the liquid composition profile follows an unstable branch of the pinch point curve so that tray compositions correspond to unstable pinch point compositions. In addition, the boil-up ratio for the actual column must be greater than the boil-up ratio for any given unstable pinch point.
3) There are situations that we have encountered where the stripping and/or rectifying line trajectory passes near each other well away from any pinch point curves. More 215 specifically, for the illustrative example that we provide for this situation, rectifying lines (including the one for minimum reflux ratio) loop around and pass very near stripping lines before converging to their respective pinch points. It is the looping structure of the rectifying and stripping line trajectories that gives rise to non-pinched designs and one in particular that uses minimum energy!

IV.5 Results and Analyses
In this section, we present a number of examples that have non-pinched minimum energy solutions and describe in detail how we use the concept of shortest stripping line distance to find these non-pinched, minimum energy designs. All numerical calculations were done in double precision arithmetic using a Pentium IV computer with the Lahey-Fujitsu (LF95) compiler.

IV.5.1 Example 1
This first example involves the separation of acetic acid, formic acid and water at atmospheric pressure, where the .UNIQUAC equation of  was used to-model the liquid phase and the vapor phase was modeled by the Hayden-O'Connell (HOC) equation to account for hydrogen bonding (i.e., vapor phase ~ dimerization df acetic acid and formic acid). The purpose of this example is twofold.
1) To show that the non-pinched design example of  is not an isolated case but there appears to be a well defined set of characteristics which give rise to this behavior. Z) To show that pinched, mm1mum energy distillation profiles that follow unstable branches of a pinch point curve give rise to non-pinched designs for the same boil-up ratio.
Two different separations are discussed. Feed, distillate, and bottoms compositions for each separation are given in Table IV     where the liquid and vapor are modeled using a correlation given by  where Pc,i, T c,i, and roi are the critical pressure, critical temperature and acentric factor for the ith component. For this example, we have used critical properties given in .

224
The feed to the column is a mixture of n-butane (n-C4), iso-pentane (~-C 5 ), n-pentane (n-Cs) and n-hexane (n-C6) and is saturated liquid. The column specifications are given in Table IV.5. In our approach, the feed and bottoms compositions are fixed and the distillation is considered feasible if llYn -YD ( calc )I :S 0.06. Note that the light and heavy key components for this separation are i-pentane and n-pentane respectively and that both the bottoms and distillate products lie on different faces of the tetrahedral feasible region.     give rise to the potential for the stripping and rectifying trajectories to intersectprovided the boil-up ratio and corresponding reflux ratio are large enough. Table IV.7 shows the liquid compositions for the top of the stripping section and bottom of the rectifying section for the minimum energy distillation design, which has 81 total stages, and clearly shows there is no feed pinch. Note that the iso-pentane composition increases, as it should, then decreases -which indicates that iso-pentane is not being stripped from the liquid.

Remarks
There are several additional remarks that are relevant to this hydrocarbon distillation example.
1) The reasons for the existence of non-pinched, minimum energy designs for the hydrocarbon distillation given here also explain the results for the sixcomponent non-pinched example presented in . In that case, n-butane is the light key component and goes through a maximum in composition and thus forms a loop in the stripping section.
2) In this example there is no pinched design that uses minimum energy from which to find a non-pinched minimum energy design. However, this present no computational difficulties for the shortest stripping line distance approach.
This type of non-pinched design can be determined in exactly the same manner that designs with rectifying pinch points are determined (see, . Starting from the bottoms composition, one simply determines the transition (or feed) stage that gives a feasible design and then continues by reducing the boil-up ratio and determining the number of stages in each section of the column needed to maintain feasibility.
3) Using both our own in-house version of Underwood's method, we calculated values of minimum reflux ratio and minimum boil-up ratio of rmin = 4.17611 and Srnin = 7 .29344 for this example for relative volatilities calculated at feed conditions. DSTWU in Aspen Plus also gives very similar results. However, the values of reflux ratio and boil-up ratio given by Underwood's method do not yield a feasible column design! 4) Rigorous simulations with RADFRAC also re-verified the validity of the design given by the shortest stripping line distance approach and the failure of Underwood's method to yield anything useful in this case. That is, for the column specifications given in Table IV  The last example in this article involves the separation of chloroform, acetone, carbon tetrachloride, and benzene at atmospheric pressure. The UNIQUAC equation of  was used to model the liquid phase and the vapor phase was assumed to be ideal. The purpose of this example is to show that non-pinched designs can arise in mixtures with any number of components and that the proposed shortest 231 stripping line methodology for finding these non-pinched designs is unaffected by the number of components. Table IV.8 shows the column specifications for this example.     Since the composition for stage 136 corresponds to unstable pinch point but occurs at a higher value of boil-up ratio (and reflux ratio), it is possible to reduce the number of stripping stages in the pinched design and find many non-pinched, minimum energy designs for the desired separation. The best design, in our opinion, is the one that we report.
Note that this example demonstrates that non-pinched, minimum energy solutions can occur in a mixture with any number of components and is independent of the nature of

IV.5.4 Example 4
Here we briefly re-visit the non-pinched, mm1mum energy design for the six component hydrocarbon separation recently studied by . The purpose of this discussion is to show that the non-pinched, minimum energy design for this six component hydrocarbon example has trajectories that exhibit the same loping structure described in Example 2 -even though the desired separation is closer to a direct split. In this example, the phase equilibrium is modeled using the K-value correlation of , where the critical properties have been taken from . Table IV .10 gives the column specifications for this separation.  report a non-pinched, minimum energy design for this separation that has a minimum boil-up ratio of Smin = 12.669 that corresponds to the shortest 236 stripping line distance of 2.66343. This minimum energy design has 20 stripping stages, 6 rectifying stages, and a corresponding minimum reflux ratio ofrmin = 11.669.
Underwood's method, on the other hand, predicts a minimum reflux ratio of r = 1.3388 and a minimum boil-up ratio of s = 2.3388 and does not provide a feasible . _ solution for this separation.    Note that n-C4, which was designated as the light key component in this example, goes through a maximum in composition on tray 15 in the stripping section and that i-C 4 exhibits a maximum in composition on stage 26, which is at the top of the rectifying section. These composition maxima are characteristic signatures of the looping structure of the trajectories in non-pinched designs.

IV.6 Comparisons with Rigorous Column Simulations
In order to provide ~ome assessment of the quality of the non-pinched, minimum energy designs computed using the shortest stripping line distance approach, we  procedures. Nonetheless, we think the reader will agree that the liquid composition profiles, minimum boil-up ratio, and required number of stripping and rectifying

IV. 7 Discussions and Conclusions
In this paper, it was shown that the shortest stripping line distance approach represents a rigorous and systematic procedure for determining non-pinched, minimum energy distillation designs. In addition, several reasons that underlie the existence of nonpinched, minimum energy distillation designs were identified and discussed. These reasons include 1) The combination of certain product specifications and ancillary conditions, as in hybrid separation processes such as reactive distillation and extraction-distillation.
b) A stripping pinch point curve with stable and unstable branches.

.
c) A product composition that lies near a distillation boundary.
3) Separations that have stripping and rectifying trajectories that exhibit looping and intersect on their way to their respective pinch point curves, where the trajectory in at least one section of the column shows reverse separation of one of the key components.
We close this article with a discussion of two issues associated with non-pinched, minimum energy designs that we believe are important. First, in cases where pinched, minimum energy designs exist alongside non-pinched, minimum energy designs, the non-pinched designs offer the advantage of not having to necessarily use conventional rules of thumb to determine the rough size of a column necessary to make the desired separation. Typical design protocols often find pinched designs and then use rules of thumb to estimate the number of actual stages (or packing height) required to make the desired separation at modest energy consumption. It is common, for example, to take the minimum boil-up (or reflux) ratio, multiply it by a factor between 1.1 and 1.5 (see  to give an operating boil-up o_ r reflux ratio, and then determine the number of stages required by trial and error. When non-pinched, minimum energy designs exist, there is no need to increase the minimum boil-up ratio, if the number of stages needed for the separation is small enough to represent a column that can be built -since it would only result in a column that unnecessarily uses more energy than needed. In addition, note that the existence of non-pinched, minimum energy designs also show that increasing the number of stages beyond that predicted by the non-243 pinched solutions does not necessarily result in any better separation. In fact, this practice could lead to wasted capital investment costs.
Second, and perhaps more important, are cases where there is no pinched, minimum energy distillation design. In these cases, it is clear that the shortest stripping line distance approach provides design solutions that no other methodology can. More specifically, if one treats the problem at hand in a manner similar to the rectifying pinch case described in , then it is clear that the shortest stripping line distance approach can reliably and systematically find non-pinched, minimum energy designs. to use different techniques depending on the nature of mixture or number of components and represents a general purpose shortcut method for providing a good conceptual design of any kind of separation process.

Nomenclature
Moreover, it was shown that the shortest stripping line approach finds correct processing targets m multi-unit processes so that the overall process consumes minimum energy. Hybrid separation of high and lower purity acetic acid by extraction with ethyl acetate followed by distillation, the production of MTBE using reactive distillation and a reactor-separation-recycle process were used as examples to illustrate key concepts and identify important numerical characteristics of this class of synthesis problems. At the present time, there is no shortcut method other than the shortest stripping line distance approach that can handle this type of multi-unit synthesis problem.
In manuscript III, the shortest stripping line approach was further extended to develop a novel two-level distillation design methodology for generating portfolios of In summary, the shortest stripping line distance approach is a fundamental geometric design concept that provides a unified, versatile, and rigorous shortcut methodology for the design, synthesis and retrofitting of energy efficient chemical processes. It overcomes many of the limitations of available design methods and readily extends to situations where no other method is applicable. Thus the shortest stripping line methodology represents a powerful next generation shortcut technique for energy efficient process design.