Solving singular control problems using uniform trigonometrization method

This study investigates the application of a recently developed construct, the uniform trigonometrization method (UTM), to the singular control problems in chemical engineering. The UTM involves minimal modifications to the original problem, thereby generating near-singular control solutions that can be used for conceptual design and serve as an alternate to direct techniques like nested and simultaneous approaches. Eight classical singular control problems with known analytical solutions and three complex singular control problems from chemical engineering domain are solved in this study. The results obtained using the UTM for these problems are found to match well with the literature and are of higher resolution as compared to the results obtained using a direct pseudospectral-based solver. The ability of the UTM to handle complex chemical engineering problems with both singular controls and state path constraints has also been demonstrated in this study.     

A bilevel NLP sensitivity‐based decomposition for dynamic optimization with moving finite elements

Optimal control has guided numerous applications in chemical engineering, and exact determination of optimal profiles is essential for operation of separation and reactive processes, and operating strategies and recipe generation for batch processes. Here, a simultaneous collocation formulation based on moving finite elements is developed for the solution of a class of optimal control problems. Novel features of the algorithm include the direct location of breakpoints for control profiles and a termination criterion based on a constant Hamiltonian profile. The algorithm is stabilized and performance is significantly improved by decomposing the overall nonlinear programming (NLP) formulation into an inner problem, which solves a fixed element simultaneous collocation problem, and an outer problem, which adjusts the finite elements based on several error criteria. This bilevel formulation is aided by a NLP solver (the interior point optimizer) for both problems as well as an NLP sensitivity component, which provides derivative information from the inner problem to the outer problem. This approach is demonstrated on 11 dynamic optimization problems drawn from the optimal control and chemical engineering literature. © 2014 American Institute of Chemical Engineers AIChE J, 60: 966–979, 2014     

A simultaneous approach for singular optimal control based on partial moving grid

Singular optimal control plays an important role in process engineering, including optimal operation of batch and semi-batch reactions. However, for many practical applications, accurate solution of singular optimal control profiles is still an open issue. In particular, numerical optimization must deal with an ill-conditioned problem that often leads to very slow convergence or failure. Starting from the nested approach in our previous work in 2016, this study develops a more efficient strategy for singular control through a heuristic approach for the outer problem. The approach includes three stages. Starting from a coarse distribution of finite elements, sufficiently many finite elements are inserted where control profiles are steep and fixed gridpoints are inserted on the basis of error estimation of state profiles. Then, moving gridpoints are inserted where the modified switching function is violated. Initial junctions are obtained by moving the latest inserted gridpoints. Moreover, further mesh refinements are considered based on switching point detection and a moving grid point update strategy, until modified switching conditions are satisfied over the whole-time span. A key feature of this approach is that only a subset of finite elements needs to move during optimization. Complexity of the optimization formulation is considerably decreased compared to our previous work. This approach is demonstrated on eight classical singular control problems with known solutions, as well as six complex singular control problems drawn from the chemical engineering literature.     

Development of a computer algorithm based on a conjugate gradient approach for optimization of fed-batch fermentations

The problem of optimization of fed-batch fermentations using the substrate feed rate as the control variable is singular in nature. Previous approaches, including the boundary condition iteration method and transformation to a nonsingular problem using a different control variable, do not work well for solving optimization of systems governed by more than four differential equations. The applicability of a first-order conjugate gradient algorithm for optimizing fed-batch fermentations was tested for systems of varing complexity. This approach does not need any variable transformation ora priori knowledge of the control arc sequence. Constraints on the feed rate are handled in a simple and direct manner. The algorithm worked very well for three, four, and five-dimensional singular systems. The correctness of the optimal profile was judged by observing the variation in the sign of the gradient of the Hamiltonian. The gradient was found to be zero during the singular period and had the appropriate sign on the boundary arcs. The optimization method based on conjugated gradient approach can be complementary to the boundary condition iteration method for determination of the exact optimum profile.     

Model based control of a liquid swelling constrained batch reactor subject to recipe uncertainties

This work presents the application of nonlinear model predictive control (NMPC) to a simulated industrial batch reactor subject to safety constraint due to reactor level swelling, which can occur with relatively fast dynamics. Uncertainties in the implementation of recipes in batch process operation are of significant industrial relevance. The paper describes a novel control-relevant formulation of the excessive liquid rise problem for a two-phase batch reactor subject to recipe uncertainties. The control simulations are carried out using a dedicated NMPC and optimization software toolbox OptCon which implements efficient numerical algorithms. The open-loop optimal control problem is computed using the multiple-shooting technique and the arising nonlinear programming problem is solved using a sequential quadratic programming (SQP) algorithm tailored for large-scale problems, based on the freeware optimization environment HQP. The fast response of the NMPC controller is guaranteed by the initial value embedding and real-time iteration technologies. It is concluded that the OptCon implementation allows small sampling times and the controller is able to maintain safe and optimal operation conditions, with good control performance despite significant uncertainties in the implementation of the batch recipe.     

Simultaneous design and NMPC control under uncertainty and structural decisions: A discrete-steepest descent algorithm

In this article, we address the integration of design and nonlinear model-based control under uncertainty and structural decisions for naturally ordered structures. We propose an algorithmic framework to determine the optimal location of process units or streams over an ordered discrete set that operates in closed-loop with a model-based controller. The formulation corresponds to a mixed-integer bilevel problem (MIBLP) that is transformed into a single-level mixed-integer nonlinear problem (MINLP) using a KKT transformation strategy. In our methodology, the integer decisions are partitioned into subsets called external variables, such that the MINLP is decomposed into an integer-based master problem and primal subproblems with fixed discrete variables. The master and primal problems are solved using a Discrete-Steepest Descent Algorithm (D-SDA). We illustrate the discrete-based methodology in a case study for a binary distillation column. The D-SDA showed an improved performance compared to a benchmark continuous-based formulation using differentiable distribution functions (DDFs).     

Boundary value problems in transport with mixed or oblique derivative boundary conditions—I: Formulation of equivalent integral equations

When an internally heated body is cooled along its boundary by a peripherally flowing fluid that is continually replenished from an external source, a differential energy balance on the boundary leads to unfamiliar boundary conditions. Such boundary conditions involve mixed second derivatives with respect to spatial variables, which under additional assumptions (such as an infinite heat transfer coefficient) lead to oblique derivative boundary conditions; i.e. at the boundary an oblique derivative of the temperature is specified. Classical attempts at solution of elliptic partial differential equations with oblique derivative boundary conditions have been through the establishment of equivalent singular integral equations, using complex analytic continuation. The theory of singular integral equations is complicated, however.Using appropriate Green's functions, the boundary value problems of interest have been reduced to equivalent integral equations in this work. While oblique derivative boundary value problems are shown to lead to singular integral equations, the mixed derivative boundary value problem is shown to yield Fredholm integral equations directly. This surprising finding is mathematically significant, because Fredholm integral equations are solved more easily, and physically significant because the mixed derivative boundary condition is the more realistic condition in the present context. A method of solution of Fredholm integral equations is discussed.More complicated boundary conditions in which axial conduction in the coolant fluid is important have also been shown to lead to Fredholm integral equations. Finally a transient problem has been formulated.     

Exact solutions for a class of heat and mass transfer problems

A number of heat and mass transfer problems of chemical engineering interest involve the convective diffusion equation of the form where θ = θ(X₁, X₂). Exact solutions for such problems are developed in terms of well-known functions which have been thoroughly studied in recent years. Several problems which have appeared in the literature, solved by completely numerical methods, are re-examined and new problems are discussed and solved. The results of the present analysis are compared with those obtained by other methods where possible. The problem of axial diffusion of heat or mass is solved in terms of known functions. The present formulation is shown to be particularly useful in the analysis of conjugated boundary value problems, i. e. for problems involving heat or mass transfer across an interface where the interfacial boundary condition is not known a priori but is related to the temperature or concentration fields in the adjacent phases.     

Sensor network design for maximizing process efficiency: An algorithm and its application

Sensor network design (SND) is a constrained optimization problem requiring systematic and effective solution algorithms for determining where best to locate sensors. A SND algorithm is developed for maximizing plant efficiency for an estimator‐based control system while simultaneously satisfying accuracy requirements for the desired process measurements. The SND problem formulation leads to a mixed integer nonlinear programming (MINLP) optimization that is difficult to solve for large‐scale system applications. Therefore, a sequential approach is developed to solve the MINLP problem, where the integer problem for sensor selection is solved using the genetic algorithm while the nonlinear programming problem including convergence of the “tear stream” in the estimator‐based control system is solved using the direct substitution method. The SND algorithm is then successfully applied to a large scale, highly integrated chemical process. © 2014 American Institute of Chemical Engineers AIChE J, 61: 464–476, 2015     

Optimization by lumped control of reactors with langmuir-hinshelwood catalyst deactivation

A method is proposed for solving the problem of temperature optimal control in tubular fixed-bed reactors with reaction systems described by Langmuir-Hinshelwood-Hougen-Watson kinetic equations. The optimization problem is formulated by N state differential equations corresponding to the N differential fixed-bed reactors in which the integral reactor is divided. It is solved using the control vector parameterization computational technique. The proposed method when applied to a simple reaction system reported previously in the literature gives analogous results, and thus validates the theory. This theory is applied to the dehydrogenation of benzyl alcohol to benzaldehyde. An analysis of optimality problem shows a strong influence of the temperature dependence of the ratio of reaction rate to deactivation reaction rate on the optimal policy.     

Simultaneous solution approach to model‐based experimental design

A model‐based experimental design is formulated and solved as a large‐scale NLP problem. The key idea of the proposed approach is the extension of model equations with sensitivity equations forming an extended sensitivities‐state equation system. The resulting system is then totally discretized and simultaneously solved as constraints of the NLP problem. Thereby, higher derivatives of the parameter sensitivities with respect to the control variables are directly calculated and exact. This is an advantage in comparison with proposed sequential approaches to model‐based experimental design so far, where these derivatives have to be additionally integrated throughout the optimization steps. This can end up in a high‐computational load especially for systems with many control variables. Furthermore, an advanced sampling strategy is proposed which combines the strength of the optimal sampling design and the variation of the collocation element lengths while fully using the entire optimization space of the simultaneous formulation. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4169–4183, 2013     

Adaptation strategies for real-time optimization

Challenges in real-time process optimization mainly arise from the inability to build and adapt accurate models for complex physico-chemical processes. This paper surveys different ways of using measurements to compensate for model uncertainty in the context of process optimization. Three approaches can be distinguished according to the quantities that are adapted: model-parameter adaptation updates the parameters of the process model and repeats the optimization, modifier adaptation modifies the constraints and gradients of the optimization problem and repeats the optimization, while direct input adaptation turns the optimization problem into a feedback control problem and implements optimality via tracking of appropriate controlled variables. This paper argues in favor of modifier adaptation, since it uses a model parameterization and an update criterion that are well tailored to meeting the KKT conditions of optimality. These considerations are illustrated with the real-time optimization of a semi-batch reactor system.     

A Numerical and Experimental Study of Natural Convective Heat and Mass Transfer from a Vertical Porous Plate

ABSTRACT

A finite difference scheme for solving the problem of natural transport of heat, mass, momentum and species concentration along vertical porous plates is presented. Several drying related problems are numerically solved, by including a gas-injection boundary condition directly into the governing equations. The effect of variable physical properties is investigated by means of direct comparison against experimental data obtained through holographic interferometry. The relative importance of wall diffusive and convective fluxes is examined. Sherwood and Nusselt numbers can be accurately obtained by means of the proposed techniques.     

The effect of adhesion on axisymmetric contact between a spherical punch and FGM coating

Abstract

This paper investigated the axisymmetric contact problem of a functionally graded material (FGM) coating by considering the adhesion effect. The Maugis–Dugdale adhesion theory is adopted to describe the adhesion effect between the contact bodies. The Hankel integral transform is applied to obtain the singular integral equations which represent the axisymmetric adhesive contact problem of FGM coating with exponential varying modulus. The impact of the adhesion parameter, gradient index of FGM coating and thickness of coating on the contact behaviour is presented by numerically solving the singular integral equations. The results provide a method to improve the contact deformation and damage by adjusting the gradient of the FGM coating.     

Experimental methods in chemical engineering: Computational fluid dynamics/finite volume method—CFD/FVM

Computational fluid dynamics (CFD) applies numerical methods to solve transport phenomena problems. These include, for example, problems related to fluid flow comprising the Navier–Stokes transport equations for either compressible or incompressible fluids, together with turbulence models and continuity equations for single and multi-component (reacting and inert) systems. The design space is first segmented into discrete volume elements (meshing). The finite volume method, the subject of this article, discretizes the equations in time and space to produce a set of non-linear algebraic expressions that are assigned to each volume element—cell. The system of equations is solved iteratively with algorithms like the semi-implicit method for pressure-linked equations (SIMPLE) and the pressure implicit splitting of operators (PISO). CFD is especially useful for testing multiple design elements because it is often faster and cheaper than experiments. The downside is that this numerical method is based on models that require validation to check their accuracy. According to a bibliometric analysis, the broad research domains in chemical engineering include: (1) dynamics and CFD-DEM, (2) fluid flow, heat transfer, and turbulence, (3) mass transfer and combustion, (4) ventilation and the environment, and (5) design and optimization. Here, we review the basic theoretical concepts of CFD and illustrate how to set up a problem in the open-source software OpenFOAM to isomerize n-butane to i-butane in a notched reactor under turbulent conditions. We simulated the problem with 1000, 4000, and 16 000 cells. According to the Richardson extrapolation, the simulation underestimates the adiabatic temperature rise by 7% with 16 000 cells.     

Phase conditions for stability of multi-loop control systems

Utilizing the concepts of the internal model control and the structured singular value, Grosdidier and Morari (1986) have proposed the μ-interaction measure for multiloop control system designs. Under the μ-interaction measure plot that constrains the magnitude of each closed-loop transfer function, a stable multiloop control system can be designed. However, for some processes, the plot fails to provide a stable multiloop control system with integral action and a dynamically better control system is discarded. Here we introduce a phase stability condition that reduces the above possibility when used together with the μ-interaction measure. The small gain theorem with linear fractional transformations is used to calculate the phase condition. As in the gain margin and phase margin design for single-input single-output processes, a reliable multiloop control system can be designed with the gain constraints of the μ-interaction measure and the proposed phase constraints.     

Analysis of stresses in adhesive joints applicable to IC chips using symbolic manipulation and the numerical method

In this study, a concentrated force is applied to both adherends bonded by an adhesive under pin–pin boundary conditions. First a mathematical model is derived with governing equations and boundary conditions. These complicated, and analytically problematic, coupled equations are solved numerically using symbolic manipulation and singular value decomposition (SVD). Also discussed are the effects of major factors, including the relative thickness of adherends, joint length and the action point of the concentrated force on the peel and shear stresses in the adhesive layer. This study identifies the conditions under which the upper adherend without breakage can be fully separated from the lower adherend. Particularly, it is found that the thickness of the lower adherend should be greater than ten times that of the adhesive layer but less than one-third that of the upper adherend, the adhesive layer should be relatively thin (h _a ≤ 0.01 mm), and the adhesive joint should be relatively short (thickness to length ratio γ ₁ ≥ 0.08).     

Moving horizon approach of integrating scheduling and control for sequential batch processes

Online integration of scheduling and control is crucial to cope with process uncertainties. We propose a new online integrated method for sequential batch processes, where the integrated problem is solved to determine controller references rather than process inputs. Under a two‐level feedback loop structure, the integrated problem is solved in a frequency lower than that of the control loops. To achieve the goal of computational efficiency and rescheduling stability, a moving horizon approach is developed. A reduced integrated problem in a resolving horizon is formulated, which can be solved efficiently online. Solving the reduced problem only changes a small part of the initial solution, guaranteeing rescheduling stability. The integrated method is demonstrated in a simulated case study. Under uncertainties of the control system disruption and the processing unit breakdown, the integrated method prevents a large loss in the production profit compared with the simple shifted rescheduling solution. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1654–1671, 2014     

Wolfsmilchgewächse (Euphorbiaceen) als nachwachsende Ressourcen der Oleochemie – Aktuelle bioorganische Analytik und prognostische Präventivtoxikologie

Species of Spurge (Euphorbiaceae) as Renewable Resources of Oleochemistry – Current Bioorganic Analytics and Prognostic Preventive Toxicology The botanical family of spurge (Euphorbiaceae) consists of about 8000 species. This reservoir of wild-type plants, an considerable number of which have economically interesting properties, may obtain significant importance in the search of renewable resources to replace currently used fossile resources. Therefore, for example, Euphorbia lathyris L., for instance, the herbaceous gopher plant or caper spurge, can be expected to became valuable because of the high oleic acid content in its seed. Plenty of problems, however, have to be solved on the way from a wild to a domesticated type, which can be cultivated in quantities as high as those of rape or sunflower. Trials to solve these problems as well as toxicological control are part of a multidisciplinary research program concerning Euphorbia lathyris L. (EULA), with the project “Renewable Raw Materials Program”, supported by the Federal Ministry of Agriculture (BMELF). Encouraging results have been achieved during the first years of research already, offering hope that mass cultivation of E. lathyris may become a realistic alternative for the production of oleic acid, but numerous problems still have to be solved.