Optimisation is one of the most powerful tools in process synthesis. Optimisation involves selecting the best solution from a set of candidate or feasible solutions (El-Halwagi, 2006). Process optimisation problems are formulated as mathematical models, where variables correspond to decisions - e.g., the flowrate of a stream, the amount of
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heat provided by high-pressure steam - ,constraints corresponding to the conceptual model of the system - e.g., mass balance (Klemeš et al, 2010), and the objective function corresponding to the mathematical description of an optimisation criterion in the case of single-objective optimisation, or to the description of two or more criteria in the case of multi-objective optimisation.
Optimisation aims at finding appropriate values the variables, in such a way that (i) Constraints involving these variables are satisfied and
(ii) The objective function of the problem is minimised (or maximised). The constraints define the search space, whilst the objective function is used to determine the most favourable ―point‖ or ―points‖ within this space.
The principles of optimisation theory and algorithms are covered by various books (e.g., Grossmann 1996; Edgar and Himmelbalu 2001; Grossmann and Biegler 2004; Williams, 2005; El-Halwagi, 2006; Ravidran et al., 2006, and Klemeš et al., 2010) and overviews as Friedler (2009, 2010).
In general, a process synthesis problem is defined by specifying the available raw materials, candidate operating units, and desired products. Each of them is given by an individual mathematical model. The models cannot, by themselves, directly constitute the mathematical programming model for the synthesis problem. The mathematical model will be unclear with the risk of failure. The major steps of process synthesis are illustrated in Figure 2.5.
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Mathematical Programming
It is important not to confuse the concept of mathematical programming (MP) with computer programming. Mathematical programming is ―programming‖ in the sense of
―planning‖, as such it has nothing to do with computers (Williams, 2006). However MP becomes involved with computing since practical problems are complex: so large quantities of data and arithmetic can only be solved by the calculating power of a computer.
Model Generation
Cost and constraints for the units and raw materials. Price
and constraints for the products
Mathematic model (LP, MILP, NLP, MINLP)
Solution Procedure
Optimal Network (Flowsheet)
Figure 2.5 Major steps of process synthesis (Klemeš et al., 2010)
The mathematical formulation of an optimisation problem entails the following steps (El-Halwagi, 2006):
29 1. Determining the objective function
2. Developing the game plan to tackling the problem 3. Developing the constraints
4. Improving formulation
In general, modelling begins by accumulating sufficient information about the process in order to develop an understanding of the elements and the relationships between them and to proceed with formulating a mathematical description of the process that as is implemented on a computational platform. A distinctive characteristic of this procedure is its iterative nature (Figure 2.6). The mathematical modelling often leads to changes in the conceptual model; the result is an iterative feedback loop, as shown in the figure. A similar correction loop is also present at the output of the implementation block. The discussion that follows addresses only those activities in Figure 2.6 that involve conceptual and mathematical modelling.
Overview of the MP Development
Mathematical models are classified according to the types of variables (continuous or integer) and constraints (linear or nonlinear). A model itself cannot be a programming.
Programming is an activity of optimisation using an algorithm suitable of solving a given type of model. If a model is linear and described by continuous variables, the corresponding programming is called linear programming (LP). If it is nonlinear, then we are talking about nonlinear programming. In the case of a linear model with mixed continuous and discrete variables, the programming is mixed-integer linear programming (MILP), whist in the presence of non-linearities in the model we are dealing with mixed-integer nonlinear programming (MINLP).
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Figure 2.6 Model creation procedure (Klemeš et al., 2010)
Linear programming (LP) problems appear over a wide range of applications, including transportation, distribution from sources to sinks, and management decisions (see, e.g., Klemeš and Vašek, 1973; Jeżowski, 1990; Jeżowski, Shethna, and Castillo, 2003;
Williams, 2006; El-Halwagi, 2006).
LP problems can conveniently be solved by the simplex method (Dantzig, 1968) and its improvements (see, e.g., Maros, 2003a; Maros, 2003b).
In most cases, NLP is difficult to solve, certain limitations on the constraints and objective function may be necessary for it to be practically solvable by specific methods (Seidler, Badach, and Molisz, 1980; Banerjee and Lerapetritou, 2003; Sieniutycz and Jeżowski, 2009). General technique for solving mixed-integer programming problems is the branch and bound framework (Land and Doig, 1960) where the original problem is solved via the systematic generation and solution of a set of relaxed subproblems.
BEGIN
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Process synthesis is an inventive step; in fact, it is one of the earlier actions to be taken by the process designer to create the structure, network or flowsheet of a process, satisfying the given requirements in terms of constraints and specifications and attaining the prescribed objectives. In this thesis, P-graph approach is used for network synthesis which will be discussed later in Chapter 5.
The relationships among the mathematical models, the process being modelled, and the solver being deployed are usually complicated, which makes it difficult to establish the most effective and valid model. There has only been a limited discussion of generating mathematical models in the literature, and the topic is covered in only a few publications (see, e.g., Grossmann, 1990; Kovacs et al., 2000) concerning specific areas.
Tools for mathematical programming optimisation and process synthesis
Process optimisation problems in chemical engineering are generally complex tasks of a considerable scale and comprehensive interactions. The application of information technology and computer software tools are essential for providing fast and as much as possible accurate solutions with a user-friendly interface. General purpose optimisation and modelling tools overviews have been available throughout the years of development, from dedicated conferences and publications (Klemeš and Vašek, 1973;
Grossmann and Daichendt, 1996; Casavant and Côté, 2004; Gani, 2008; Friedler, 2009, 2010). A number of computer based systems have been developed to support process engineers in energy and mass balance calculations. However, due to the substantial on-going funding needed for continuous development, only a limited number have remained on the market. They have only been secured by a substantial number of continuous sales.
In this research work, GAMS (General Algebraic Modelling System), a high-level modelling language and efficient interface for mathematical programming is used.
GAMS is designed for modelling linear, nonlinear and mixed-integer optimisation
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problems. This system is tailored for complex, large-scale modelling applications and allows the user to build large maintainable models that can be adapted to new situations (GAMS, 2010).
GAMS is especially useful for handling large, complex, one-of-a-kind problems which may require several revisions in order to establish an accurate model. The user can change the formulation quickly and easily, can specify different models, and can convert from linear to nonlinear models. GAMS was the first algebraic modelling language (AML), and is similar in form to similar programming languages. Models are described as algebraic statements which are easy to read for both for humans and machines.
GAMS has the ability to formulate models within many different types of problem classes. The same data, variables, and equations within different types of models can be used at the same time. GAMS provides a range of different types of solvers for different types of models such as (i) BARON (Branch-And-Reduce Optimisation Navigator) for solving non-convex optimisation problems to global optimality, (ii) CPLEX for Linear Programming (LP), Mixed-Integer Programming (MIP), Quadratically Constraint Programming (QCP) and second order cone programs, and Mixed-Integer Quadratically Constraint Programming (MIQCP) based on the Cplex Callable Library (iii) DICOPT for solving MINLP models (iv) OQNLP for global optimisation of smooth constrained problems with either all continuous variables or a mixture of discrete and continuous variables.
Various processes have been modelled by using GAMS e.g. synthesis of (i) Mass exchange networks (Chen and Ciou, 2007) , (ii) Water networks (Chew et al., 2008;
Tokos and Novak Pintaric, 2009), (iii) Biogas production (Drobez et al., 2009) .
33 2.5 P-graph Framework
The P-graph framework is robust; its algorithms have been validated as mathematically rigorous in that they are based on a set of axioms (Friedler et al., 1992). These axioms express the necessary structural properties for feasible process networks. The algorithms are able to guarantee the resultant mathematical model’s validity, reduce the search space, and generate the optimal solution.
In a P-graph, one class of nodes is assigned to operating units or activities and the other to their inputs and outputs. Raw materials, resources (precursors), and preconditions (activating entities) are inputs to the operating units; products, effects (resulting entities), and targets are outputs from the operating units. Table 2.3 shows the P-graph representation of process structure elements.
Table 2.3 P-graph symbols that represent process elements (Klemeš et al., 2010) Process element P-graph representation
Raw material or precursor Final product or final target
Intermediate material or entity
By-product
Operating unit
In a process network, functional units that perform operations (e.g., mixing, reacting, separating) are termed operating units’. These operating units, which correspond to the blocks in a process flowsheet, alter the physical and/or chemical states of materials being processed or transported. Such transformations are carried out by one or more unit operations, and the overall process converts raw materials into the desired
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product(s). A process may also generate by-products, which are either recoverable for further use or treated as waste.
In process network synthesis, a material is uniquely defined by its components and their concentrations - in other words, by its composition, which is identified by a symbol used to mark the material. Two classes of materials (or material streams) are associated with any operating unit: Input materials and output materials. For example, operating unit O2
in Figure 2.7 consumes raw materials E and F while producing intermediate material C and by-product B. Note that a material may consist of more than one component.
Figure 2.7 P-graphs representing the process structure of three operating units (Klemeš et al., 2010)
The P-graph provides not only a formal description of the process but also an unambiguous representation of the possibilities for structural decisions. If an operating unit requires multiple inputs, each provided by a single operating unit, then structural alternatives cannot be defined. In contrast, if multiple operating units are capable of providing a particular input, then any combination of these units can, in theory, be used.
In Figure 2.7 (a), for example, materials C and D are necessary inputs to operating unit O1. Material C can only be produced by operating unit O2, and material D can only be
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produced by operating unit O3. Thus, for unit O1 to operate it is necessary that units O2
and O3 are both included in the process structure. In Figure 2.7 (b), however, material C can be produced by unit O2, unit O3, or both. In addition to unambiguous structural representation, the P-graph framework also provides a set of rigorous and effective algorithms for the synthesis and optimisation of process networks.
The extreme complexity of process network synthesis is mainly due to the problem’s combinatorial nature. This complexity grows exponentially with the number n of candidate operating units, because the optimal network must be found among 2n possible combinations of the units (i.e., alternative networks) unless some possibilities can be eliminated (e.g., by heuristics) in advance. The factor 2n is derived by simple induction. The first observe action is that a single additional decision (regarding the inclusion or exclusion of an operating unit) doubles the number of potential design alternatives: 2n×2= 2n+1. This means that a designer contemplating a system, e.g. with a total of 35 operating units, is faced with more than 34 billion (235 = 3.436 × 1010) alternative arrangements.
Reducing such large a number of alternatives requires robust decision-making tools that are mathematically rigorous (preferably axiomatic) and effectively implementable on computers. These ends have been met largely by employing the well-established mathematics of graph theory, which can be regarded as a branch of combinatorics.
Thus was developed the graph-theoretic, algorithmic method was developed, as described in this section. This method is based on using P-graphs to extract those universal combinatorial features (properties) inherent in feasible processes. Such properties can be expressed mathematically as a set of axioms that characterize the combinatorial feasibility of processing networks.
A given process network is said to be combinatorially feasible (or to be a solution structure) if it satisfies the following five structural axioms (Friedler et al., 1995).
(S1) Every final product and target is represented in the structure.
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(S2) An entity represented in the structure has no input if and only if it represents a raw material or precursor.
(S3) Every operating unit represented in the structure is defined in the problem.
(S4) Any operating unit represented in the structure has at least one path leading to a final product or a final target.
(S5) An entity belongs to the structure if and only if it is either an input entity to or an output entity from at least one operating unit already represented in the structure.
Figure 2.8 depicts two process structures that are not combinatorially feasible. The P-graph in Figure 5.2 (a) shows a process structure in which material F is consumed as an input. Yet because material F is not a raw material and was never produced, the structure is not combinatorially feasible according to Axiom (S2). In the P-graph of Figure 2.8 (b), operating unit O3 produces only by-product B. Here O3 does not output any final product or material that is later used to yield a final product, so the process structure violates Axiom (S4). In short, the structural properties expressed by Axioms (S1)–(S5) are necessary conditions for process structures to be feasible. This means that reducing the search space to combinatorially feasible structures does not result in the loss of any practically feasible or optimal processes.
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Figure 2.8 P-graphs representing process structures that violate (a) Axiom (S2) or (b) Axiom (S4) (Klemeš et al., 2010).
The P-graph’s mathematical engine: MSG, SSG, and ABB
When combined with the structural axioms, P-graph representation makes it possible to implement effective algorithms for the structural analysis, synthesis, and optimisation of process structures. The Maximal Structure Generation (MSG) algorithm (Friedler et al., 1992) generates a superstructure that can be rigorously proven to incorporate each combinatorially feasible process structure. Then the Solution Structure Generator (SSG) algorithm (Friedler, Varga, and Fan, 1995) is used to enumerate all the combinatorially feasible process structures that satisfy Axioms (S1) – (S5) or the Accelerated Branch and Bound (ABB) algorithm (Friedler et al., 1996) is used to generate the optimal process structure together with a ranked, finite list of near-optimal structures.
Figure 2.9 illustrates the connections among the three algorithms. Algorithm MSG generates the maximal structure, and can be followed either by algorithm SSG to generate the combinatorially feasible process structures or by algorithm ABB to generate the optimal and near-optimal processes. Algorithms MSG and SSG require the
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list of candidate operating units as input, each defined by the set of its input materials - preconditions - and products (effects). Algorithm ABB requires, in addition to these data, quantitative information relevant to assessing network optimality e.g., prices of raw materials, costs and capacity constraints of operating units).
Software tools for P-graph: PNS Editor
PNS Editor (2010) is a software package solving Process Network Synthesis (PNS) based on P-graph approach. The aim of the PNS problem is to examine the feasible structures and select the optimum from among them. The optimal structure can be assessed in terms of cost, profit, etc. Both structural information (which functional units are connected and how) and sizing information (how much is produced from a given material) are needed in order to define the optimal structure,.
Figure 2.9 Inputs to and outputs from the three P-graph algorithms (Klemeš et al., 2010).
39 The issues addressed by the PNS Editor are:
(i) How to represent the building blocks of a process network?
(ii) What are the solution structures of the problem?
(iii) What is the maximal structure (which includes all solution structure)?
(iv) What is the optimal structure?
The maximal structure comprises all the combinatorially feasible structures capable of yielding the specified products from the specified raw materials. Certainly, the optimal network or structure is among these feasible structures. During the composition phase, the nodes are linked, again step wise and layer by layer, starting from the shallowest end, i.e., final-product end, of the remaining input structure by assessing if any of the linked nodes violates one or more of the axioms, as described in Friedler et al. (1996).
MSG is performed transparently, as the maximum structure is the input for the Solution Structure Generation (SSG) algorithm. SSG gives all of the combinatorially feasible solution structures of a given problem. Often, the number of feasible structures is still too large to use for explicit enumeration. The ABB algorithm determines the optimal structure without generating all the solutions. It needs the structural relationships between materials, operating units and some additional information as to the costs of each raw material, fixed and proportional costs of the operating units, and constraints on quantity of materials or capacities of the operating units. This method has been implemented in several different process synthesis case studies e.g.:
(i) Azeotropic distillation system (Feng et al., 2000), (ii) Heat exchanger network (Nagy et al., 2001),
(iii) Reduction of carbon emissions involving fuel cell combined cycles (Varbanov and Friedler, 2008).
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2.6 Graphical targeting approaches for heat recovery systems, hydrogen recovery