Orthogonal Collocation Technique Research Articles

Geometrically transformed spectral methods to solve partial differential equations in circular geometries, application for multi-phase flow

Circular geometries are ubiquitously encountered in science and technology, and the polar coordinate provides the natural way to analyze them; However, its application is limited to symmetric cases, and it cannot be applied to segments that are formed in multiphase flow problems in pipes. To address that, spectral discretization of circular geometries via orthogonal collocation technique is developed using geometrical mapping. Two analytical mappings between the circle and square geometries, namely, elliptical and horizontally squelched mappings, are employed. Accordingly, numerical algorithms are developed for solving PDEs in circular geometries with different boundary conditions for both steady state and transient problems. Various implementation issues are thoroughly discussed, including vectorization and strategies to avoid solving the differential–algebraic system of equations. Moreover, several case studies for symmetric and asymmetric Poisson equations with different boundary conditions are performed to evaluate several aspects of these techniques, such as error properties, condition number, and computational time. For both steady state and transient solvers, it was revealed that the computation time scales quadratically with respect to the grid size for both mapping and polar discretization techniques. However, due to the presence of the second mixed derivative, mapping techniques are more computationally costly. Finally, the squelched mapping was successfully employed to discretize the two, and three-phase gravity flows in sloped pipes.

Nonlinear programming reformulation of dynamic flux balance analysis models

Dynamic flux balance analysis (dFBA) models are very popular in systems biology because of the many possible applications. dFBA consists of dynamic mass balance equations that describe the concentration of external metabolites and an optimization model to compute the internal flux distribution of the cell. Here, we applied a nonlinear programming formulation for solving dFBA models. The ODE system is discretized using the orthogonal collocation technique and the optimization problem is replaced by its first-order optimality conditions. In addition, an adaptive mesh scheme is applied to improve performance by efficiently handling changes in the active set of constraints. This formulation ensures total differentiability of the model, which can be solved by large-scale solvers and take advantage of automatic differentiation packages. The new approach presents results equivalent to state-of-the-art methods and outperforms them in dynamic optimization problems by avoiding calling an external solver to handle the embedded optimization model.

Some efficient numerical schemes for approximating the nonlinear two-space dimensional extended Fisher-Kolmogorov equation

This paper develops some efficient numerical schemes based on the hybridization of finite difference and orthogonal cubic spline collocation (OCSC) techniques to approximate the nonlinear two-dimensional extended Fisher-Kolmogorov model. This model represents a strong nonlinear fourth-order reaction diffusion evolution model. The proposed strategy is based on second-order backward differentiation formula and θ-scheme (including the Crank-Nicolson and backward Euler methods). It is shown that both the semi-discrete and full discrete approaches are unconditionally stable and convergent by means of the energy analysis procedure in an appropriate Sobolev space. In addition, the optimal convergence rates are obtained including fourth-order and second-order accuracies in space and time directions, respectively. Finally, numerical results assess the accuracy of the proposed methods and verify theoretical estimates.

Numerical Analysis of Alternating Direction Implicit Orthogonal Spline Collocation Scheme for the Hyperbolic Integrodifferential Equation with a Weakly Singular Kernel

This paper studies an alternating direction implicit orthogonal spline collocation (ADIOSC) technique for calculating the numerical solution of the hyperbolic integrodifferential problem with a weakly singular kernel in the two-dimensional domain. The integral term is approximated with the help of the second-order fractional quadrature formula introduced by Lubich. The stability and convergence analysis of the proposed strategy are proven in L2-norm. Numerical results highlight the high accuracy and efficiency of the proposed strategy and clarify the theoretical prediction.

Bypass control of heat exchanger network under uncertainty

AbstractThe dynamic control of the heat exchanger network is important for developing energy‐efficient and safe industrial processes. In such a system, the control is achieved through the bypass stream around the heat exchanger. This work aims to track the setpoint temperature of the mixed stream by manipulating the bypass fraction of the cold stream around the heat exchanger. The implemented control is in a non‐linear model predictive control (NMPC) framework. The first‐principles model of a shell and a tube heat exchanger is used. The orthogonal collocation technique is used to discretize the first‐principles model into the system of algebraic equations. In this work, uncertainty is also considered in the inlet temperature of the hot stream. The uncertain optimal control problem is dealt with by using a scenario tree‐based approximation along with the affine policy‐based method. The results show that, under different scenarios of uncertainty, the controlled variable efficiently tracks the setpoint. In comparison, considering the same scenarios of uncertainty used, the deterministic optimization approach shows significant deviation in the controlled variable from the setpoint as time passes.

A fourth order orthogonal spline collocation method Interface boundary value problem

Objective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study. Keywords: Helmholtz problem; Orthogonal spline collocation techniques (OSC); Discontinuous data; Super Convergence; Piecewise cubic Hermite basis functions; Almost block diagonal (ABD) structure

Modelling and Dynamic Simulation of One-Dimensional Isothermal Axial Dispersion Tubular Reactors with Power Law and Langmuir-Hinshelwood-Hougen Watson Kinetics

In this paper, the modelling, numerical lumping and simulation of the dynamics of one-dimensional, isothermal axial dispersion tubular reactors for single, irreversible reactions with Power Law (PL) and Langmuir-Hinshelwood-Hougen-Watson (LHHW)-type kinetics are presented. For the PL-type kinetics, first-order and second-order reactions are considered, while Michaelis-Menten and ethylene hydrogenation or enzyme substrate-inhibited reactions are considered for the LHHW-type kinetics. The partial differential equations (PDEs) developed for the one-dimensional, isothermal axial dispersion tubular reactors with both the PL and LHHW-type kinetics are lumped to ordinary differential equations (ODEs) using the global orthogonal collocation technique. For the nominal design/operating parameters considered, using only 3 or 4 collocation points, are found to adequately simulate the dynamic response of the systems. On the other hand, simulations over a range of the design/operating parameters require between 5 to 7 collocations points for better results, especially as the Peclet number for mass transfer is increased from the nominal value to 100. The orthogonal collocation models are used to carry out parametric studies of the dynamic response behaviours of the one-dimensional, isothermal axial dispersion tubular reactors for the four reaction kinetics. For each of the four types of reaction kinetics considered, graphical plots are presented to show the effects of the inlet feed concentration, Peclet number for mass transfer and the Damköhler number on the reactor exit concentration dynamics to step-change in the inlet feed concentration. The internal dynamics of the linear (or linearized) systems are examined by computing the eigenvalues of the linear (or linearized) lumped orthogonal collocation models. The relatively small order of the lumped orthogonal collocation dynamic models make them attractive and useful for dynamic resilience analysis and control system analysis/design studies.

Unit Heads Amidst COVID 19 Pandemic: Work-Life Balance, Talent Management Skills and Institutional Performance

Purpose: The study aims to determine the level of work-life balance, talent management skills, and institutional performance of Northern Iloilo Polytechnic State College (NIPSC) key officials during the COVID 19 pandemic. Methods: This study is descriptive-correlational research using purposive samplings. The 97 key officials from the 7 campuses were selected as respondents of this study. The researcher adopted a questionnaire focused on work balance, talent management skills, and institutional performance. To analyze the data, statistical tools like frequency count, percentage, mean, standard deviation for descriptive, and for inferential were t-test, one-way analysis of variance, and Pearson’s r was utilized. Results: The results revealed that the work-life balance of different units’ heads was ”Highly Balanced”. While the talent management skills were ”Highly Managed”. And, the level of key officials’ institutional performance was ”Outstanding.” Furthermore, the findings also indicate that there are no significant differences in the level of work-life balance, talent management skills, and institutional performance of school officials. The school key officials’ highly balanced lifestyle was able them to cater to the needs of their work and personal life in this catastrophe. Their experiences as key officials improved their management skills and impacted and influenced them to cope with the crisis. Novelty: Higher Education Institutions (HEIs) unit heads should know work-life balance for teachers and management skills to inspire colleagues for institutional performance in this virus crisis. This pandemic teaches key officials to become dynamic and visionary to manage the new educational policies by implementing different modalities and training for teachers. The four mandates such as instruction, research, extension services, and production of the college are prospering despite health protocols restrictions. Keywords: Northern Iloilo Polytechnic State College; key Officials; Pandemic; WorkLife Balance; Management Skills; Institutional Performance 1 Introductionbjective: A higher order numerical scheme for two-point boundary interface problem with Dirichlet and Neumann boundary condition on two different sides is propounded. Methods: Orthogonal cubic spline collocation techniques have been used (OSC) for the two-point interface boundary value problem. To approximate the solution a piecewise Hermite cubic basis functions have been used. Findings: Remarkable features of the OSC are accounted for the numerous applications, theoretical clarity, and convenient execution. The stability and efficiency of orthogonal spline collocation methods over B-splines have made the former more preferable than the latter. As against finite element methods, determining the approximate solution and the coefficients of stiffness matrices and mass is relatively fast as the evaluation of integrals is not a requirement. The systematic incorporation of boundary and interface conditions in OSC adds to the list of advantages of preferring this method. Novelty: As against the existing methodologies it becomes clear from our findings that OSC is dominantly computationally superior. A computational treatment has been implemented on the two-point interface boundary value problem with super-convergent results of derivative at the nodal points, being the noteworthy finding of the study. Keywords: Helmholtz problem; Orthogonal spline collocation techniques (OSC); Discontinuous data; Super Convergence; Piecewise cubic Hermite basis functions; Almost block diagonal (ABD) structure

Design of continuous contacting countercurrent unit operations: An approach based on the usage of orthogonal collocation and Matlab

AbstractA counter‐current set‐up is preferred to its co‐current counterpart because substantial improvement in the performance of the unit operation is achieved. Hence, proper design of countercurrent unit operations is a fundamental part of any undergraduate chemical engineering curriculum. The governing equations, derived from material and/or energy balance equations around a continuous‐contact counter‐current unit consist of at least two coupled differential equations where the independent variable is either position or area. Such problems also involve limit conditions at both edges of the unit. In the present paper, we demonstrate that these mathematical problems are readily solved by the orthogonal collocation technique despite their seemingly mathematical complexity especially to undergraduate chemical engineering students. Our approach is applied to following four case studies: (a) countercurrent heat exchanger, (b) gas‐liquid absorption in a packed column, (c) water cooling tower and finally (d) gas permeation using dense membranes. One advantage of the method lays in the fact that both (a) the size of the equipment (eg, height of the gas‐liquid contactor, area of the separation membrane or heat exchanger…) and (b) the compositions and/or temperature profiles are obtained simultaneously in a single fast and easy calculation. These computations always involve a discretization step based on collocation points. This discretization transforms the differential equations into a set of nonlinear algebraic equation, which is readily solved using the ubiquitous command (ie, fsolve) of Matlab®.

Modelling, simulation, and experimental validation for toluene removal from gas phase in a fixed bed adsorption column

ABSTRACTIn this study, an adsorption technique was explored for the removal of toluene present in the gas phase by taking granular activated carbon as an adsorbent in a fixed bed adsorption column. Breakthrough curves were experimentally found out at varying bed height (0.015–0.025 m) and inlet concentration (7000–11500 ppm). Modelling and simulation studies were subsequently carried out to enable the comparison between predicted and experimental breakthrough curves. Here orthogonal collocation technique was employed to solve the model equations by discretising non-linear coupled partial differential equations into set of ordinary differential equations which were solved by using ODE15s solver of Matlab. The discretisation procedure has been described in detail. Finally, the flow rate was found out to be the most sensitive parameter after carrying out sensitivity analysis.

Efficient Simulation of Novel Electrode Architectures

The use of high-capacity, thick electrodes for energy-dense lithium ion cells is limited by poor material utilization, especially in high-rate scenarios. This is primarily due to high resistances to Li-ion transport arising from the length scales and tortuosity in thick porous electrodes.1,2 Novel electrode architectures can be engineered to reduce overall transport limitations by creating additional pathways for lithium ion diffusion and migration (Fig.1). These architectures consist of the active material distributed in such a way as to enable higher overall ion fluxes and intercalation rates, ensuring higher active material utilization than would be possible with the same amount of active material in a conventional homogeneous distribution.1,2 Since the design of these architectures is essentially a problem of improving ion transport, the macro-homogeneous electrochemical models of Newman and coworkers (pseudo-3D models) are widely used to study the relationship between electrode architectures and electrochemical performance .4 Studies have mainly focused on two-dimensional (2D) simulations of the effect of specific geometric parameters such as pillar widths, heights, and aspect ratios.1,3,5,6 While these studies provide useful insights, they are limited to a given set of battery chemistry parameters. Any change in electrode chemistry or material properties will entail repeating the entire analysis, often using detailed finite-element simulations, which can be extremely computationally expensive. 3 The purpose of this work is therefore to develop a 2D Li-ion cell model capable of simulating novel electrode architectures in a computationally efficient manner. We use the governing equations of the pseudo-3D model. Orthogonal collocation techniques used in past work will be adapted to enable fast solution of the model equations.7 Comparisons with finite-element simulations will be presented to demonstrate the accuracy and computational performance of the new 2D cell model. The fast simulation is then used to evaluate certain representative architectures by iterating through a parameter space that includes electrode and electrolyte properties in addition to geometric parameters. The performance of these architectures will be reported in terms of metrics relating to discharge performance, ion concentration profiles, and material utilization trends. The computationally fast simulation can potentially provide greater flexibility by enabling access to a larger parameter space, and by permitting the evaluation of a larger number of designs than would be possible by conventional simulations of parametric variations. The fast simulation is also envisaged to be integrated into optimization algorithms in future work. Acknowledgements The authors are thankful for financial support by the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium). References C-J. Bae, C.K. Erdonmez, J.W. Halloran, and Y-M. Chiang, Mater., 25, 1254-1258 (2012).P. Smyrek, J. Proll, H.J. Seifert, and W. Pfleging, J. Electrochem. Soc., 163, A19-A26 (2016).C.L. Cobb and M. Blanco, J. Power Sources, 249, 357-366 (2014).M. Doyle, T. F. Fuller, and J. Newman, J. Electrochem. Soc., 140, 1526–1533 (1993).V. Zadin, H. Kasemagi, A. Aabloo, and D. Brandell, J. Power Sources, 195, 6218-6224 (2010).P. Priimagi, D. Brandell, S. Srivastav, A. Aabloo, H. Kasemagi, and V. Zadin, Elect. Acta, 209, 138-148 (2016).P.W.C. Northrop, M. Pathak, D. Rife, S. De, S. Santhanagopalan, and V.R. Subramanian, J. Electrochem. Soc., 162, A940-A951 (2015). Figure 1

Pomodoro: A Novel Toolkit for Dynamic (MultiObjective) Optimization, and Model Based Control and Estimation

This paper presents the software package Pomodoro which includes a collection of algorithms and tools for dynamic optimization. It also introduces the framework for multiobjective problems, model predictive control and state estimation. Pomodoro is implemented in Python and utilizes CasADi as a backbone to formulate the problem. It uses orthogonal collocation technique to solve the dynamic optimization problems and efficient third-party solvers are employed to solve the resulting nonlinear programs. The design of the software and its main modules are discussed and the user-friendliness of the software is demonstrated with the help of tutorial examples for each problem class. Lastly, the advantages and limitations of this software are discussed.

Orthogonal spline collocation scheme for multiterm fractional convection‐diffusion equation with variable coefficients

The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two‐dimensional multiterm fractional convection‐diffusion reaction equation with variable coefficients. The optimal estimates in Hj (j = 0, 1, 2) norms at each time step are derived. Also, estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.

Solution of six chemical engineering problems using the Chebyshev orthogonal collocation technique

AbstractThe Chebyshev orthogonal collocation is a powerful and accurate numerical technique, which is also relatively easy to implement using modern mathematical software (e.g., MATHEMATICA © and MATLAB ®). This technique allows the elucidation of engineering problems involving partial differentials equations or boundary value problems. Six chemical engineering problems concerning the transfer of momentum, heat, and mass, with varying degree of complexity, are elucidated. The results obtained using the subject numerical technique were systematically bench‐marked against the results derived from alternative approaches. Benchmarking indicates good agreement between the results, despite the simplicity of this technique. Thus, this technique has a great potential for the elucidation of problems, for which no known solutions exist.

Study of the countercurrent–concurrent gas–liquid flow configuration impact on ethylene hydrogenation within structured catalyst bed: Experiment and modeling

Abstract In this paper catalytic hydrogenation of ethylene was studied using liquid propane with 1.5% of ethylene content and hydrogen gas in 30 barg trickle bed reactor under two flow configurations: countercurrent and concurrent. Trickle bed experiments were then performed using an experimental package comprising high pressure pilot reactor, gas chromatograph apparatus, dosing pump and required instrumentation system to compare countercurrent and concurrent sampled data furthermore structured catalyst of alumina–nickel was used as catalytic bed. Trickle bed modeling was achieved using a package of 12 dimensionless two-dimensional homogeneous mass and momentum transfer partial differential equations, the model parameters and kinetic equations. Orthogonal collocation technique was employed to solve developed PDEs. Predicted performance of the trickle bed by mathematical modeling was compared by experimental results and there was good consistency between them. The experimental results showed zero concentration of ethylene through countercurrent flow. Finally, the experimental analysis reports emphasize on zero concentration of ethylene in output of countercurrent bed whilst increasing the Reynolds number did not conclude to nil concentration of ethylene in output of concurrent bed.

Collocation Technique for Numerical Solution of Integral Equations with Certain Orthogonal Basis Function in Interval [0, 1

This paper is concerned with the construction of a class of polynomial orthogonal with respect to the weight function w(x)=1-x2 over the interval [0,1]. The zeros of these polynomials were employed as points of collocation for the orthogonal collocation technique in the solution of integral equations. The method is illustrated with some numerical examples and the results obtained show that the method is effective.

Numerical elucidation of three‐dimensional problems in the chemical engineering graduate curriculum

ABSTRACTWe illustrate in the present paper how steady‐state and unsteady‐steady three‐dimensional transport phenomena problems can be solved using the Chebyshev orthogonal collocation technique. All problems are elucidated using the ubiquitous software: MATLAB®. The treated case studies include: (1) unsteady‐state heat conduction in a parallelepiped body, (2) quenching of a brick, (3) transient diffusion of a colorant in a liquid, and (4) steady‐state natural convection in an enclosure. Whenever possible, the results obtained with the spectral method are bench‐marked against the analytical solution in order to validate the proposed numerical technique. © 2016 Wiley Periodicals, Inc. Comput Appl Eng Educ 24:866–875, 2016; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21756

Numerical solution of heat conduction problems using orthogonal collocation on finite elements

Technique of orthogonal collocation along with finite elements has been presented to solve the linear and non linear heat conduction problems numerically. Choice of Lagrangian interpolation polynomials as base function has been opted to discretize the trial function. Error analysis has been discussed in terms of element size for both the linear and non linear problems. Proposed technique has been applied on different types of linear and non linear heat conduction problems and the numerical values are plotted using 2D and 3D graphs.

Solving two‐dimensional chemical engineering problems using the chebyshev orthogonal collocation technique

ABSTRACTThe present paper describes how to apply the Chebyshev orthogonal collocation technique to solve steady‐state and unsteady‐steady two‐dimensional problems. All problems are solved using one single computer algebra, Mathematica©. The problems include: (1) steady‐state heat transfer in a rectangular bar, (2) steady‐state flow in a rectangular duct, (2) steady‐state heat transfer in a cooling cylindrical pin fin, (4) steady‐state heat conduction in an annulus, (5) unsteady‐state heat transfer in a rectangular bar, and finally (6) unsteady‐state diffusion reaction system. Whenever possible the results obtained with orthogonal collocation are compared to the analytical solution in order to validate the applied numerical technique. © 2015 Wiley Periodicals, Inc. Comput. Appl. Eng. Educ. 24:144–155, 2016; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21680

An ADI extrapolated Crank–Nicolson orthogonal spline collocation method for nonlinear reaction–diffusion systems on evolving domains

We consider the approximate solution of nonlinear reaction–diffusion systems on evolving domains that arise in a variety of areas including biology, chemistry, ecology and physics. By mapping a fixed domain onto the evolving domain at each time level, we generalize to evolving domains the ADI extrapolated Crank–Nicolson orthogonal spline collocation technique developed in [8,9] for fixed domains. The new method is tested on the Schnakenberg model and we demonstrate numerically that it preserves the second-order accuracy in time and optimal accuracy in space for piecewise Hermite cubics in various norms. Moreover, the efficacy of the method is demonstrated on several test problems from the literature which involve various types of domain evolution but for which exact solutions are not known.