A new method of locating all pinch points in nonideal distillation systems, and its application to pinch point loci and distillation boundaries

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Abstract

A new method for automatically finding all of the pinch points in a user-specified composition space in nonideal distillations system at any reflux is presented. It does not rely on the solution of ODEs, and neither knowledge of the system topology, nor rigorous simulation is required. Moreover, the method can be applied to any column section, even those within complex configurations. The method works on the principle of a systematic search over an area to find where the conditions for a pinch point are satisfied; this includes nodes outside of the mass balance triangle, which, while physically impossible, do provide useful information. This principle is extended to reflux-parameterised pinch point loci and to finding distillation boundaries accurately. Nonidealities are modelled with the NRTL model, although any model can be used. Only ternary systems have been considered, but the method can extend to higher order systems.

Introduction

Traditionally, residue curve maps (RCMs) have been used as a graphical method to perform shortcut heuristic synthesis of multicomponent distillation systems (Fien & Liu, 1994). Residue curves are constructed mathematically by solving a system of ordinary differential equations, the general case of which is given by Eq. (1), for N  1 components.dxidn=xiyiThe above is an initial value problem, which can be solved using any combination of (xi, …, xN−1) as a starting point. When the system of equations is solved, the result is a trajectory in composition space, known as a residue curve (RC). Performing the integration with a starting point that lies on a specific RC yields that same RC. In a batch distillation system (with changing time t, rather than n, in Eq. (1)), the physical significance of a RC is that it shows how the composition of the liquid changes with time as it is boiled in an open system, if it had the starting composition at a known time t. Since RCs are unique, integrating Eq. (1) backwards from the starting point can tell one what the liquid composition would have been prior to known time t.

All RCs terminate at a stable node, and originate at an unstable one, the compositions of which are at pure components or azeotropes. These compositions are referred to as stationary points. Additionally, in ternary and higher order systems, saddle points can occur, which are simply points in composition space towards which composition trajectories can run past, but at which they can never terminate. The above is also true of any azeotrope, which can either be a stable or unstable node, or a saddle point.

Nodes, or pinch points in distillation systems (used synonymously in this paper1), occur mathematically when dxi/dn = 0 in Eq. (1) for N  1 components (since the mole fractions sum to 1, this condition is automatically met for the Nth component).

By plotting a number of residue curves on the same graph, a RCM is constructed, an example of which is given in Fig. 1 for the acetone, benzene and chloroform (ABC) system. It allows one to consider various trajectories simultaneously, and to get an idea of the system behaviour, known as the topology of the system, which is the structure of how various nodes and saddle points are linked by residue curves.2

With regard to continuous distillation, if a distillation column is operated at infinite reflux, that is, if all of the product is returned to the column, the solution of Eq. (1) yields the composition profile within the column, with a known starting point at a stage n. Residue curves are identical to these column profiles, such that RCMs describe both the batch distillation case and the continuous infinite reflux one.

Using infinite reflux as a limiting case, it is possible to design separation systems, or rather, to get a starting point for column sequencing or for the design of complex columns. Examples of the former can be found in Fien and Liu (1994), as well as in many modern texts on distillation.

While RCMs do have their uses, the assumption of infinite reflux is an unrealistic and impractical one.

To circumvent this shortcoming, Van Dongen and Doherty (1985) introduced an approach to express finite-reflux column profiles in traditional rectifying and stripping sections with an ordinary differential equation analogous to Eq. (1). Tapp, Holland, Hildebrandt, and Glasser (2004) built on the work of Van Dongen and Doherty by extending the latter’s equation for use in generalised column sections—such as those within complex configurations like the Petlyuk Holland, 2005, Holland et al., 2010. Tapp et al. termed their equation the difference point equation (DPE). The DPE is given here as Eq. (2):dxidn=1+1RΔxiyi+1RΔXΔixiwhere Δ = V  L, RΔ = L/Δ, and XΔ=VYTLXT/Δ. The last two are design parameters; RΔ is a ‘reflux ratio’ of the column section (CS).3 Δ is the net flux of material in the CS, and XΔ, the difference point, can be viewed as the pseudo composition of the net flow in the CS. Note that constant molar overflow (CMO) is assumed for Eq. (2) to hold.

When these design parameters are fixed, the solution of Eq. (2) is a column profile, which is entirely analogous to a residue curve, except that it is the column profile at finite reflux.

Again, completely analogously to RCMs, the DPE allows for construction of column profile maps (CPMs), which are essentially RCMs, but at finite reflux. These maps trace the liquid composition profile in a generalised column section, defined as a section of the column in which there is no addition or removal of energy or mass. Since there is an XΔ value for each component, XΔ,i, XΔ can be described as a unit vector that has the same number of elements as components in the system, i.e. XΔ = [XΔ,1, XΔ,2, …, XΔ,N]. It is noteworthy that Eq. (2) collapses to Eq. (1) as RΔ  ∞, i.e. infinite reflux, which indicates that RCMs can be considered as a subset of CPMs. Furthermore, the mathematical conditions for a node or pinch point also remain the same as in RCMs: dxi/dn = 0 in Eq. (2) for N  1 components. An example of the CPM of the ABC system, with RΔ = 6 and XΔ = (0.9, 0.05, 0.05) is given in Fig. 2.

How CPMs relate to RCMs has been examined in some detail by Tapp et al. (2004), and Holland, Tapp, Hildebrandt, and Glasser (2004), and Holland, Tapp, Hildebrandt, Glasser, and Hausberger (2004), but even comparing Figs. 1 and 2 will give the reader some insight into this relationship. CPMs are relatively new way of looking at distillation processes; as such, not much literature on the topic is available outside of the authors’ research group yet, although its use is becoming more widespread, e.g. see the work of Linninger and co-workers Kim et al., 2010, Linninger, 2009, and Tian, Sun, and Guo (2009). For the unfamiliar reader’s convenience, some of the more important aspects of CPMs are reiterated below.

Although this paper deals predominantly with nonideal systems, it is convenient to consider a system exhibiting constant relative volatility (CRV) as a start. Such a system might have, for example, relative volatility α = (2, 1, 1.5), such that the components are light, heavy, and intermediate, in that order.

Let us arbitrarily consider a rectifying section terminated by a condenser, such that the interpretation of the design parameters is natural to the reader: RΔ is the reflux ratio, and let us say that the column is producing a distillate of xD = XΔ = (0.8, 0.1, 0.1). Fig. 3 gives the RCM (i.e. CPM with RΔ  ∞) of this system, as well as the CPM at RΔ = 5. What is important to note is how the mass balance triangle has shifted at finite reflux (this phenomenon is the “moving triangles” that forms the basis of Holland, Tapp, Hildebrandt, Glasser, & Hausberger, 2004), and the equivalence of the topology of the two triangles, regardless of their different shapes and positions. The vertices of this finite-reflux transform triangle are the pinch points at finite reflux. With respect to the MBT, when reflux is lowered to a finite value, the topology changes. However, with respect to the shifted or transform triangle, the topology remains unchanged. In effect, all that has changed between the two scenarios is that the nodes have moved. Note that most of the pinch points that make up the vertices of the two transform triangles are outside of the real space. Although the compositions outside of the real space are physically unattainable, they are topologically important, since knowledge of their positions allows us to understand the behaviour of the column section and its structure. For example, even though a node is outside of the MBT, its position influences the behaviour of column profiles within the real space, and, e.g. if it is a stable node, all profiles will run towards it, regardless of the physical impossibility of reaching it.

Bausa et al. (1998) and Tapp et al. (2004) both noted what has just been illustrated, i.e. that when reflux is changed, the nodes of a system move in the composition space away from their infinite-reflux positions.4

Fig. 3 illustrates that CPMs are simply a transform of RCMs in composition space. This particular feature of the CPM method is one of its most powerful; it allows for the synthesis of columns by manipulating these triangles of known topology to achieve the behaviour that the designer requires. It departs from—and in our opinion, improves on—the traditional design methods which consider only one profile at a time, since the column profile map scans all possible profiles and behaviours for the chosen design parameters.

‘Unusual’ behaviour can also occur at low RΔ values. For example, as the nodes move with decreasing reflux, their stability can also change. Past a certain reflux, a stable node may suddenly become an unstable node or a saddle point, thereby dramatically changing the behaviour within the distillation column. Furthermore, as the nodes move, they tend to move towards one another, and can ‘collide’ and merge in the composition space, decreasing the number of unique nodes. For example, compare Figs. 1 and 2: in the former (at infinite reflux) there are seven nodes, while in the latter, only five exist. These phenomena have been examined in some detail by Beneke, Hildebrandt, and Glasser (2010), and mentioned by Bausa et al. (1998) and Holland, Tapp, Hildebrandt, and Glasser (2004).

Finally, we address the concept of pinch point loci (PPL). A pinch point locus (also known as a pinch point curve) is the locus of pinch points that traces the node positions as an independent design parameter is varied. Usually, this is RΔ, but XΔ could be used, depending on what is more convenient for the particular design problem. The allows one to determine easily which values of the parameter will lead to placement of the pinch points that results in the desired behaviour within the column.

An example, which unifies all of the discussed concepts and proposed methods into a practical application for the synthesis of a distributed-feed column, is presented at the end of this paper.

The position and stability of pinch points in a system determine the topology of CPMs of the various column sections that make up a column. It can be used to the designer’s advantage with the CPM method, not only to synthesise a feasible column, but also to gain a better understanding of the choices that design parameters have on the final design.

Pinch points in CRV systems are straightforward to find; in fact, they can be found by software with symbolic calculation capabilities.

However, for highly nonideal systems that can only be modelled accurately by more complicated models such as NRTL or Wilson, this problem becomes substantially more difficult: not only is it mathematically far more difficult to solve, but, as discussed earlier, nodes have highly complex behaviour at low reflux ratios, and can merge with one another, or ‘disappear’ altogether.5 Prior to this work, unless the full CPM was plotted, it was impossible to know how many nodes actually exist, what their locations are and what the structure of the CPM is.

The algorithms presented here are intended as a tool to aid work with column profile maps (especially for automation thereof), but not as a standalone synthesis/design technique. Tapp et al. (2004), Holland et al., 2004a, Holland et al., 2004b and Holland et al. (2010) have comprehensively covered their novel design method—most of which requires pinch points—but not a way of finding pinch points or of constructing pinch point loci, which is what this paper aims to do. Naturally, the method can be used for any other application that requires pinch points. One non-CPM example of such an application is for use with the rectification body method Bausa et al., 1996, Bausa et al., 1998, which relies on pinch points.

This algorithm for finding nodes is a ‘brute-force’ one, which is to say that it populates the given region with potential starting points in a systematic manner (although only the most promising are used). While this is not computationally efficient—in fact, it is only feasible due to the computing power available nowadays—it is a requirement at this stage, as will become apparent when the algorithm is described later. We make no claims that it is the most efficient way to find a specific node, but, to the best of our knowledge, it is the only method which can find all of the nodes in a CPM automatically, given minimal information (as mentioned above, only RΔ, XΔ, and the bounds of where to search in the composition space).

The example provided later on serves not only to show how the concepts and methods presented here tie in with each other for the design of distillation columns, but it also provides justification for why one might want to find all of the nodes on a CPM automatically.

Section snippets

Literature survey

A number of methods exist for finding the pinch points in a distillation system, usually as a step towards finding the minimum reflux for a two-section column, based on the fact that the minimum reflux is observed when the end pinch of the stripping section just touches the rectifying profile, or vice versa, depending on the type of split (Doherty & Malone, 2001). Of course, it is immediately apparent that these approaches are then limited to column configurations for which they are

Locating nodes

A necessary and sufficient condition for the existence of a stationary point (node) is that the derivative in Eq. (2) is zero for N  1 components, as has already been mentioned. In other words, in order to find a pinch point on a CPM, we must find a liquid composition vector that satisfies the system of equations:dx1dn=0=1+1RΔx1y1+1RΔXΔ1x1dxN1dn=0=1+1RΔxN1yN1+1RΔXΔN1xN1On RCMs, there are at least three pinch points, meaning that there are three or more different compositions which

Pinch point loci

When keeping XΔ at a fixed composition (or unfixed, but with some known relationship to RΔ) and varying the RΔ parameter, the pinch points move in the composition space. It is possible to trace a locus of all these pinch point compositions, parameterised by RΔ, thus providing information which is invaluable for choosing parameters in the design of columns, as is demonstrated with an example at the end of this paper.

The same approach as for finding pinch points is applied, except that the full

Finite-reflux distillation boundaries

Distillation boundaries are of great practical interest in designing distillation columns, since they represent a limit to what paths a column profile or residue curve can follow. With the knowledge of pinch point compositions, it is possible to find the distillation boundaries (at infinite or, more importantly, finite reflux) in a ternary system with a high degree of accuracy—one that is more than sufficient for design purposes. Emphasis has been placed here on finite reflux, because the

Unifying example: distributed-feed column

We present an example here which unifies the concepts and methods discussed in this paper, i.e. column profile maps, pinch points, pinch point loci, XΔ points outside of the real space, and finite-reflux distillation boundaries, with application to a practical problem.

The CPM method does not have any particular advantages over traditional methods for the design of simple columns, i.e. those with one feed stream, a distillate and a bottoms stream, a condenser and a reboiler, and two column

Conclusion

A new algorithm for locating nodes in generalised column sections has been proposed. It has been demonstrated to locate all of the pinch points in highly nonideal systems successfully. An extension of the method was made to construct pinch point loci, which can give valuable insight into what nodes exist in a system, even outside of the mass balance triangle, and how they can be manipulated under finite reflux conditions, and moved into or out of the real space.

The approach is versatile in that

Acknowledgements

We thank Ronald Abbas for helpful discussions regarding the distributed feed example. The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the NRF.

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