Optimization of water network integrated with process models

In this paper, a novel approach for the synthesis of water network incorporated with process models is introduced. The process models are utilized to relate the variables (i.e., flow rate and concentration) of process output (typically defined as internal water source) with those of process input (i.e., water sink). A generalized water network superstructure is developed to embed all possible process units and all the connections among resources, interceptors, process units, and wastes. The problem is formulated as four optimization problems (minimum freshwater flow rate, intercepted flow rate, intercepted mass load, and number of connections), and the four models are solved in sequence to locate the targets. A literature case is used to validate the proposed approach. Moreover, a sour water network of a practical refinery plant is presented to illustrate the applicability and effectiveness of the proposed approach.     

Reduction method for the non-linear analysis of symmetric anisotropic panels

Reduction method and computational procedures are presented for reducing the size of the analysis model and the number of degrees of freedom used in predicting the non-linear response of symmetric anisotropic panels. The two key elements of the method are (a) operator splitting, or decomposition of the characteristic arrays of the finite element model into sums of orthotropic and non-orthotropic contributions, (b) application of a reduction method through the successive use of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate a small number of global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the classical Rayleigh-Ritz technique. The global approximation vectors are selected to be those commonly used in single (or multiple) parameter perturbation techniques, namely a non-linear solution corresponding to zero non-orthotropic arrays and a number of its derivatives with respect to an anisotropic tracing parameter (and possibly, to a load or arc-length parameter in the solution space). The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding orthotropic structure. The effectiveness of the proposed reduction method is demonstrated by means of a numerical example, and its potential for solving quasi-symmetric non-linear problems of anisotropic panels is discussed.     

A generalised guideline for process changes for resource conservation networks

In the past one and a half decades, resource conservation network (RCN) synthesis has been well accepted by both academics and industrial practitioners in enhancing sustainability aspect for the process industry. Various insight-based pinch analysis and mathematical optimisation techniques have been proposed to synthesise optimum RCN. For the former, the developments to date have been focusing on various targeting and design tools, with very little emphasis on the systematic assessment of process changes. The latter involves the modification of process operating conditions (e.g., flowrate, concentration, temperature, etc.), which leads to further reduction of minimum fresh resource flowrates. In this study, the plus-minus principle in heat exchanger network synthesis is extended for use with graphical targeting tool in assessing opportunities for process changes in the RCN of fixed-flowrate problems, aiming to further reduce its minimum fresh resource flowrate. Literature examples are used for illustration.     

A direct boundary element approach for unsteady non-linear heat transfer problems

Unsteady non-linear problems are usually solved based on the time marching scheme together with iterative methods. In this paper, we employed an example to illustrate a new direct boundary element technique, which can provide accurate solutions of unsteady non-linear problems without the need for iterations. The principle and validity of the technique is demonstrated by considering an unsteady non-linear hyperbolic heat transfer problem, together with a fully implicit difference scheme in the time domain. It illustrates that the proposed non-iterative boundary element approach is numerically rather efficient for unsteady non-linear problems. Thus, it provides an alternative to traditional iterative methods for unsteady non-linear problems.     

Combined use of BEM and genetic algorithms in groundwater flow and mass transport problems

The boundary element method (BEM) can be used very efficiently, in solving groundwater flow problems. Genetic algorithms (GAs) constitute a very efficient optimization tool. In this paper, BEM and GAs have been combined to find optimal solutions in three classes of commonly encountered groundwater flow and mass transport problems: (a) determination of transmissivities in zoned aquifers (inverse problem), based on a restricted number of field measurements; (b) minimization of pumping cost from any number of wells under various constraints; and (c) hydrodynamic control of a contaminant plume, by means of pumping and injection wells. Application examples show that the proposed combination is very efficient in optimizing development and protection of groundwater resources.     

A hybrid artificial neural network method with uniform design for structural optimization

This paper presents a new hybrid artificial neural network (ANN) method for structural optimization. The method involves the selection of training datasets for establishing an ANN model by uniform design method, approximation of the objective or constraint functions by the trained ANN model and yields solutions of structural optimization problems using the sequential quadratic programming method (SQP). In the proposed method, the use of the uniform design method can improve the quality of the selected training datasets, leading to a better performance of the ANN model. As a result, the ANN dramatically reduces the number of required trained datasets, and shows a good ability to approximate the objective or constraint functions and then provides an accurate estimation of the optimum solution. It is shown through three numerical examples that the proposed method provides accurate and computationally efficient estimates of the solutions of structural optimization problems.     

An integral formulation applied to the diffusion and Boussinesq equations

A new integral method is proposed here to solve the diffusion equation (confined flow) and the Boussinesq equation (unconfined flow) in a two-dimensional porous medium. The method, based on Green's theorem, derives its integral representation from the portion of the original differential equation with the highest space derivatives so that the resulting kernel of the integral representation is not time dependent. Compared to an earlier integral formulation, namely the direct Green function, based on the same theorem, the kernel is simpler so that the present theory provides a more efficient numerical model without compromising accuracy. An iterative scheme is employed along with the theory to achieve solutions to the non-linear Boussinesq equation. Concepts used in the finite difference and finite element methods enable simplification of the temporal derivative. The method is tested with success on a number of numerical examples from groundwater flow.     

A THIRD-ORDER NUMERICAL SCHEME WITH UPWIND WEIGHTING FOR SOLVING THE SOLUTE TRANSPORT EQUATION

Solute transport in the subsurface is generally described quantitatively with the convection–dispersion transport equation. Accurate numerical solutions of this equation are important to ensure physically realistic predictions of contaminant transport in a variety of applications. An accurate third-order in time numerical approximation of the solute transport equation was derived. The approach leads to corrections for both the dispersion coefficient and the convective velocity when used in numerical solutions of the transport equation. The developed algorithm is an extension of previous work to solute transport conditions involving transient variably saturated fluid flow and non-linear adsorption. The third-order algorithm is shown to yield very accurately solutions near sharp concentration fronts, thereby showing its ability to eliminate numerical dispersion. However, the scheme does suffer from numerical oscillations. The oscillations could be avoided by employing upwind weighting techniques in the numerical scheme. Solutions obtained with the proposed method were free of numerical oscillations and exhibited negligible numerical dispersion. Results for several examples, including those involving highly non-linear sorption and infiltration into initially dry soils, were found to be very accurate when compared to other solutions. © 1997 by John Wiley & Sons, Ltd.     

Efficient Multilevel MINLP Strategies for Solving Large Combinatorial Problems in Engineering

This paper reports on the experience gained in solving large combinatorial problems by using the Outer-Approximation /Equality-Relaxation (OA/ER) algorithm in two multilevel MINLP strategies. The first one is a Linked Multilevel Hierarchical Strategy (LMHS) and the second one is a Reduced Integer Space (RIS) strategy. Both strategies are used to decompose the original MINLP problem in a hierarchical manner into several MINLP levels that are much easier to solve than the original one. While the first LMHS strategy can be applied to problems that contain only simple mixed-integer constraints, e.g. standard dimensions, the RIS strategy can be used to solve problems with more complex mixed-integer constraints, e.g. different design equations for alternative units. The LMHS strategy is rigorous and can solve convex problems to global optimal solutions. On the other hand, when the RIS strategy is applied for the solution of large combinatorial problems, the global optimality cannot be guaranteed, but very good solutions can be obtained. The synthesis problem of a roller steel gate for a hydroelectric power station with 19623 binary variables is presented to illustrate the LMHS strategy, whilst the synthesis problem of a heat exchanger network comprising different types of exchangers with 1782 binary variables is presented to present the RIS strategy.     

A BEST-FIRST SEARCH STRATEGY FOR ENERGY RELAXATION IN MER HEAT EXCHANGER NETWORKS

The pinch design method efficiently generates a maximum energy recovery (MER) network which meets the utility targets for a given value of the minimum approach temperature difference. However, this MER design usually contains a significantly greater number of heat exchanger units than the theoretical minimum. Loop breaking and energy relaxation may be used to eliminate these additional units. At each stage of loop breaking, it has been recommended that the unit with the smallest heat load should be removed. However, this study shows that this heuristic can lead to suboptimal designs with respect to energy consumption. An alternative systematic method is presented which reduces the number of units such that the energy penalty is a minimum. A mixed integer non-linear program (MINLP) is formulated with the objective of minimizing the energy consumption for a given number of units. Subsequent loop-network interaction analysis helps in identifying the exchanger units which are good candidates for removal. A lower bound on the consequent energy penalty is also evaluated. These bounds are employed in a ‘best-first’ search strategy to solve the proposed model. On removal of the candidate unit/units, the resulting topology transforms the MINLP to a non-linear/linear program (NLP/LP) which is solved by conventional algorithms. A loop and path identification algorithm (LAPIT), based on graph theory, has been developed as an aid to these computations.     

Reduction methods for nonlinear steady-state thermal analysis

Hybrid analysis techniques based on the combined use of finite elements and the classical Bubnov–Galerkin approximation are presented for predicting nonlinear steady-state temperature distributions in structures and solids. In these hybrid techniques the modelling versatility of the finite element method is preserved and a substantial reduction in the number of degrees-of-freedom is achieved by expressing the vector of nodal temperatures as a linear combination of a small number of global-temperature modes, or basis vectors. The Bubnov–Galerkin technique is then used to compute the coefficients of the linear combination (i.e. the amplitudes of the global–temperature modes). The basis vectors chosen are the path derivatives commonly used in perturbation techniques, namely, the derivatives of the nodal–temperature vector with respect to a preselected control (or path) parameter(s). The vectors are generated by using the finite element model of the initial discretization. Also, the performance of alternate sets of basis vectors is investigated. In the alternate sets, only a few path derivatives are generated, and they are augmented by a constant vector representing a uniform temperature rise (or drop), and by reciprocal vectors with nonzero components equal to the reciprocals of the nonzero components of the path derivatives. A problem-adaptive computational algorithm is presented for efficient evaluation of global approximation vectors and generation of the reduced system of equations and for monitoring the accuracy of the reduced system of equations. The potential of the proposed reduction methods for the solution of large-scale, nonlinear steady-state thermal problems is also discussed. The effectiveness of these methods is demonstrated by means of four numerical examples, including conduction, convection and radiation modes of heat transfer. This study shows that the use of the uniform-temperature mode and the path derivatives as global approximation vectors significantly increases the accuracy of the solutions obtained by reduction methods, thereby enhancing the effectiveness of these methods for the solution of large-scale, nonlinear thermal problems.     

A diversity-enriched variant of discrete PSO applied to the design of water distribution networks

The design of water distribution networks (WDNs) is addressed by using a variant of the particle swarm optimization (PSO) algorithm. This variant, which makes use of a discrete version of PSO already considered by the authors, overcomes one of the PSO's main drawbacks, namely its difficulty in maintaining acceptable levels of population diversity and in balancing local and global searches. The performance of the variant proposed here is investigated by applying the model to solve two standard benchmark problems: the Hanoi new water distribution network and the New York Tunnel water supply system. The results obtained show considerable improvements in both convergence characteristics and the quality of the final solutions, and near-optimal results are consistently achieved at reduced computational cost.     

Synthesis of near-optimal topologically constrained property-based water network using swarm intelligence

The sustainability of water resources is one of the major concerns of the world. Industries are finding ways to minimize water costs and to reduce water pollution through efficient use of supplies. One of effective solutions is the use of water pinch technology or process integration techniques. Property integration emerged as the third trend of process integration technique (aside from energy and mass integration) to solve the problem of water network synthesis by considering physical stream properties such as pH or resistivity. Various techniques have been done to solve this property-based problem. In this paper, mixed integer non-linear programming (MINLP) problems with topological constraints is studied. Nowadays, various evolutionary algorithms are also used to solve such MINLP problems. In this study, particle swarm optimization is used to solve the property integration problem. A comparison is made between this algorithm and commercial genetic algorithm (GA) applications. Particle swarm optimization is observed to perform favorably compared to GA.     

An optimization approach for the synthesis of recycle and reuse water integration networks

This paper presents a convex mathematical programming model for the global optimization of recycle/reuse water networks. The model is based on a general superstructure that includes the major configurations of interest such as segregation, mixing, recycle, bypass, and treatment of streams needed to satisfy the process and environmental constraints. The basic idea of the model formulation is to consider component balances, treating as optimization variables the individual flowrates. This formulation avoids the bilinear terms that appear when the compositions and total flowrates are considered as optimization variables. The objective function consists in the minimization of the total annual cost including the fresh sources costs, the treatment units costs (which are reformulated as convex functions) and the piping costs. Four examples problems are solved to show the applicability of the proposed model.     

Synthesis of resource conservation network with sink–source interaction

This article presents a mathematical model for the synthesis of resource conservation networks with interception placement. A comprehensive superstructure that incorporates all possible network configurations is used to facilitate the formulation. The synthesis task involves the allocation and interception of process sources to satisfy process sinks and environmental constraints. In particular, the interaction between the sinks and sources is addressed as the subject of the present study. Two literature case studies are solved to illustrate the proposed approach.     

A SIMULATED ANNEALING ALGORITHM FOR MULTIOBJECTIVE OPTIMIZATION

This paper describes a novel implementation of the Simulated Annealing algorithm designed to explore the trade-off between multiple objectives in optimization problems. During search, the algorithm maintains and updates an archive of non-dominated solutions between each of the competing objectives. At the end of search, the final archive corresponds to a number of optimal solutions from which the designer may choose a particular configuration. A new acceptance probability formulation based on an annealing schedule with multiple temperatures (one for each objective) is proposed along with a novel restart strategy. The performance of the algorithm is demonstrated on three examples. It is concluded that the proposed algorithm offers an effective and easily implemented method for exploring the trade-off in multiobjective optimization problems.     

Approximate solutions for non-linear random vibration problems

Two methods are described for obtaining approximate solutions to non-linear random vibration problems. The first method approximates the non-linear system with the linear system whose corresponding probability density function best solves the Fokker-Planck equation associated with the non-linear system. In the second method, a class of non-linear systems with known solutions to the Fokker-Planck equation is used to best approximate the non-linear system of interest. Two illustrative examples are presented and the results are compared with existing methods.     

A hybrid regularization method for inverse heat conduction problems

This paper presents a hybrid regularization method for solving inverse heat conduction problems. The method uses future temperatures and past fluxes to reduce the sensitivity to temperature noise. A straightforward comparison technique is suggested to find the optimal number of the future temperatures. Also, an eigenvalue reduction technique is used to further improve the accuracy of the inverse solution. The method provides a physical insight into the inverse problems under study. The insight indicates that the inverse algorithm is a general purpose algorithm and applicable to various numerical methods (although our development was based on FEM), and that the inverse solutions can be obtained by directly extending Stolz's equation in the least‐squares error (LSE) sense. Direct extension of the present method to the inverse internal heat generation problems is made. Four numerical examples are given to validate the method. The effects of the future temperatures, the past fluxes, the eigenvalue reduction, the varying number of future temperatures and local iterations for non‐linear problems are studied. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of boundary-value problems describing the non-isothermal processes of elastoplastic deformation taking into account the loading history

We discuss the theory and approximate methods for solving boundary-value problems of thermoplasticity in a quasi-static formulation when the process of non-isothermal elastoplastic deformation of a body is a sequence of equilibrium states. In this case, the stress-strain state depends on the loading history, and the process of inelastic deformation is to be observed over the whole time interval being studied. The boundary-value problem is stated as a non-linear operator equation in the Hilbertian space. The conditions that provide the existence, uniqueness and continuous dependence of the generalized solution on the applied loads and initial strains are defined. A convergence of the methods of elastic solutions and variable elastic parameters is studied to solve the boundary-value problems describing the non-isothermal processes of active loading taking into account the initial strains dependent on the deformation history and heating. __________ Translated from Problemy Prochnosti, No. 1, pp. 69–99, January–February, 2006.