Chebyshev orthogonal collocation technique to solve transport phenomena problems with Matlab® and Mathematica

Mahgoub

et al. 2018

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In the present paper, we apply a new approach, based on mainly two built‐in functions of the computer algebra Mathematica©, in order to estimate the parameters of several models that describe various transient (i.e., time‐dependent) chemical engineering processes. These models usually consist of either a set of ordinary differential equations (ODEs) or partial differential equations (PDEs). Basing on experimental data, it is possible to obtain, both readily and accurately, estimates of the values of the parameters involved in such models by adopting the proposed methodology. We illustrate the approach with three case studies, all related to chemical reactors. Actual experimental data, collected by senior students at King Fahd University of Petroleum & Minerals during their core laboratory course (i.e., Chemical Engineering Lab II or CHE 409) are used in the first two case studies: (1) response to a tracer input of three continuous stirred‐tank reactors in series and (2) saponification reaction of ethyl acetate in a batch reactor. In the first case study, the residence time distribution and the mean residence time in each reactor are deduced from the set of gathered data. Reaction order and rate constant are estimated from the experimental findings in the second case study. A more advanced situation, involving the solution of the advection–diffusion equation applied to a tubular micro‐reactor, is considered in the third case study. Data were collected, as part of a Ph.D. research of S. Mejri, using an experimental set‐up at Energy Intensified Reactor Engineering Lab at the University of Warwick, U.K. Here, we obtained an estimate of the relevant dimensionless number namely, the Péclet number.

y_{j} = cos true(j π / N true), j = 0, \dots, N

and represent the locations of the extrema of the first kind Chebyshev polynomials, T N ( x ).…”

Section: Case Studiesmentioning

confidence: 99%

A model parameter estimation method using Mathematica© applied to transient chemical engineering processes

Mejri

Mahgoub

et al. 2018

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“…When the spectral collocation method is used to solve stationary PDEs, a major challenge faced is the construction of the spatial derivative matrices, D regardless of whether the model is one‐, two‐, or even fully three‐dimensional (1‐, 2‐, or 3‐D). When using the Chebyshev orthogonal collocation, the computation of the one‐dimensional derivative matrix, explicitly defined in Binous et al , is sufficient to treat 1‐D as well as 2‐D and 3‐D problems. Indeed, the derivatives matrices for 2‐D and 3‐D PDEs can be readily derived from the one‐dimensional matrix (i.e., D ) using the Kronecker product [3,4].…”

Section: Theoretical Background On Chebyshev Orthogonal Collocationmentioning

confidence: 99%

“…Several studies using this technique [2,9,14,15], and at least three monographs [7,11,16] presenting the method in detail have been published. A recent set of three pedagogical papers by the authors of this paper described the application of the Chebyshev orthogonal collocation in one‐, two‐, and three‐dimensional domains . In this paper, the orthogonal collocation technique is applied to elucidate six graduate‐level chemical engineering problems.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, the orthogonal collocation technique is applied to elucidate six graduate‐level chemical engineering problems. Table lists these six case studies as well as the challenges and the novelty of the treatment employed in each example (indicated in italic and red in Table ) compared to the problems we have solved in our previous work . All case studies elucidate a chemical engineering problem involving the transfer of mass, heat, and/or momentum.…”

Section: Introductionmentioning

confidence: 99%

“…The approach based on the Chebyshev orthogonal collocation technique is far simpler to implement with mathematical software (e.g., Matlab ® and M ATHEMATICA © ) than with finite differences, finite volumes, and/or finite elements. This approach only requires the calculation of the derivative matrix, D , which is explained briefly in the next section and thoroughly in the published papers and monographs .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Kaddeche

Bellagi

2017

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The 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.

Bellagi

2015

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We present in this pedagogical paper several important aspects of the application of orthogonal collocation for solution of Chemical Engineering problems. These include: the Stefan-Maxwell diffusion; the startup of a Poiseuille Flow in a Newtonian Fluid; the transient heat conduction with temperature-dependent thermal conductivity; and the reaction-diffusion in a drop. The presented method of resolution (i.e., orthogonal collocation) of such problems can be easily introduced in any graduate-level chemical engineering numerical method course. Indeed, this technique has been used for the past four year at King Fahd University of Petroleum & Minerals (KFUPM) in the graduate-level core numerical methods course called CHE 560. ß