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Anomalous diffusion of particles in fluids is better described by the fractional diffusion models. A robust hybrid numerical algorithm for a two-dimensional time fractional diffusion equation with the source term is presented. The well-known L1 scheme is considered for semi-discretization of the diffusion equation. To interpolate the semi-discretized equation, orthogonal collocation with bi-quintic Hermite splines as the basis is chosen for the smooth solution. Quintic Hermite splines interpolate the solution as well as its first and second order derivatives. The technique reduces the proposed problem to an algebraic system of equations. Stability analysis of the implicit scheme is studied using H ~ 1 m -norm defined in Sobolev space. The optimal order of convergence is found to be of order O (h 4) in spatial direction and is of order O (Δ t) 2 - α in the temporal direction where h is the step size in space direction and Δ t is the step size in time direction and α is the fractional order of the derivative. Numerical illustrations have been presented to discuss the applicability of the proposed hybrid numerical technique to the problems having fractional order derivative. [ABSTRACT FROM AUTHOR]

The present study proposes a hybrid numerical technique to discuss the solution of non-linear reaction-diffusion equations with variable coefficients. The perturbation parameter was assumed to be time-dependent. The spatial domain was discretized using the cubic Hermite splines collocation method. These splines are smooth enough to interpolate the function as well as its tangent at the node points. The temporal domain was discretized using the Crank-Nicolson scheme, commonly known as the CN scheme. The cubic Hermite splines are convergent of order h4, and the CN scheme is convergent of order Δt². The technique is found to be convergent of order O(h²(γ2εjΔt+γ0(1+α)h²)+Δt²). The step size in the space direction is taken to be h, and the step size in the time direction is Δt. Stability of the proposed scheme was studied using the L2 and L∞ norms. The proposed scheme has been applied to different sets of problems and is found to be more efficient than existing schemes. [ABSTRACT FROM AUTHOR]

In tubular reactors, fouling issues are caused due to two reasons. One is the heating–cooling prerequisite, and the other is the exothermic nature of the low‐density polyethylene (LDPE) polymerization process. These issues must be considered while optimizing LDPE production to provide maximum productivity and a safe operation. However, it is not a simple process because the conversion of the monomer (XM) is generally related to significant profits. This conversion might be performed at high reaction temperatures, resulting in fouling formation. Therefore, in this research, a study of dynamic optimization to find the most efficient production of LDPE in the presence of fouling resistance (Rf) restrictions is conducted. An Rf is employed as a measure of fouling. To establish the highest reactor RfRfmax=50cm2sK/cal$$ \left({R}_{f_{\mathrm{max}}}=50\ \mathrm{cm}2\ \mathrm{s}\ \mathrm{K}/\mathrm{cal}\right) $$, this study employs variations in the heat transfer coefficient (U) calculated from industry data. This dynamic optimization study addresses the optimization challenges using dynopt coded programming based on orthogonal collocation (OC) and sequential quadratic programming methodologies. Beforehand, the LDPE model is validated with industrial data. This study evaluates three possibilities to determine the most optimum reactor performance. The most optimum reactor output is determined from the constrained maximum conversion, which gave 32.15% conversion, while the Rf$$ {R}_f $$was effectively met at 47.37cm2 s K/cal. [ABSTRACT FROM AUTHOR]

An important part of the education of chemical engineers involves the design and assessment of heterogeneous catalytic reactions. They pose fundamental phenomena and constitute invaluable industrial processes. Progressing in this field, especially in graduate‐level courses, entails solving nonlinear differential equations systems, which can only be achieved by using efficient and reliable numerical techniques. In this work, we solve nonisothermal one‐dimensional reaction‐diffusion BVP considering convection in the boundary conditions, by applying orthogonal collocation and arc‐length continuation. Particularly, we predict the effectiveness factor as a function of the Thiele modulus for the main three particle geometries: plane slab, cylinder, and sphere. For chemical engineering students, such problems are at the zenith of the discipline, and the results of such calculations can elucidate basic and advanced concepts of mass and heat transfer as well as chemical reaction engineering. Both Chebyshev and shifted Legendre polynomials were used to transform the boundary value problem into a set of algebraic nonlinear equations. In addition, we also compare our findings against the available analytic solutions whenever possible. Codes in Scilab, Matlab®, and Mathematica© were developed for the solution of such problems and are available in the Supporting Information. There are two possible approaches to compute effectiveness factors: (a) integration as per definition versus (b) differentiation and nonlinear system. It is demonstrable that software performance is dependent on the approach, and that the differentiation approach yields better results. [ABSTRACT FROM AUTHOR]

The present study proposes a hybrid numerical technique to discuss the solution of non-linear reaction-diffusion equations with variable coefficients. The perturbation parameter was assumed to be time-dependent. The spatial domain was discretized using the cubic Hermite splines collocation method. These splines are smooth enough to interpolate the function as well as its tangent at the node points. The temporal domain was discretized using the Crank-Nicolson scheme, commonly known as the CN scheme. The cubic Hermite splines are convergent of order $ h^4 $, and the CN scheme is convergent of order $ \Delta t^2 $. The technique is found to be convergent of order $ O(h^{2}\big(\gamma_2 \varepsilon_j\Delta t + \gamma_0(1+\bar{\alpha})h^2\big)+\Delta t^2) $. The step size in the space direction is taken to be $ h $, and the step size in the time direction is $ \Delta t $. Stability of the proposed scheme was studied using the $ L_2 $ and $ L_{\infty} $ norms. The proposed scheme has been applied to different sets of problems and is found to be more efficient than existing schemes.

The nonlinear space time dynamics have been discussed in terms of a hyperbolic equation known as a sine-Gordon equation. The proposed equation has been discretized using the Bessel collocation method with Bessel polynomials as base functions. The proposed hyperbolic equation has been transformed into a system of parabolic equations using a continuously differentiable function. The system of equations involves one linear and the other nonlinear diffusion equation. The convergence of the present technique has been discussed through absolute error, L2-norm, and L∞-norm. The numerical values obtained from the Bessel collocation method have been compared with the values already given in the literature. The present technique has been applied to different problems to check its applicability. Numerical values obtained from the Bessel collocation method have been presented in tabular as well as in graphical form. [ABSTRACT FROM AUTHOR]

This paper applies orthogonal collocation based on quadratic B-spline basis functions to solve the Schrödinger equation on uniform and non-uniform grids. The method is well suited for solving different cases of soliton solutions for which the solution behaviour is very complicated. The solutions are found to be consistent with the exact solutions for various non-uniform cases. The overall numerical scheme is proven to be unconditionally stable and utilizes minimal memory storage due to the sparse matrix systems associated with B-spline basis functions. This could prove particularly useful in machine learning applications where repetitive calculations are needed and large amounts of data need to be processed. We found that using non-uniform grids as compared to uniform grids had a profound effect in capturing the correct solution dynamics especially in the 3-soliton case.

In this paper, different geometries and types of catalyst distribution used for fixed bed WGS reactors that can be placed at the outlet of gasifiers have been studied. It has been shown that spherical eggshell catalysts allow significant reductions in catalytic mass compared to uniform pellets. Moreover, thanks to the possibility of using large pellets offered by eggshell catalysts, the pressure drops observed when using these particles remain acceptable (< 5%) even for a long reactor (Z = 2.8 m) and despite a low feed pressure (P0 = 2.5 atm). Simulations also revealed that operating in isothermal or isoperibolic mode at low temperature (T = 630 K) improves significantly the process efficiency. [ABSTRACT FROM AUTHOR]

This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers' equations, including the modified Burgers' equation. The KdV–Burgers' equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space. [ABSTRACT FROM AUTHOR]

Today, hydrogen has become one of the most promising clean energy. Several processes allow obtaining hydrogen, among them there is the Water Gas Shift (WGS) reaction. On an industrial scale, WGS reaction takes place at high pressure [25–35 bar]. At high pressure, the cost of the process rises due to the energy consumed by compression, and the reduction in the lifetime of the equipment and the catalyst. At low pressures, catalyst lifetime can reach many years and the energy cost is reduced. It is for this reason that we are interested in modelling and simulation of a WGS converter operating at low pressures close to atmospheric pressure. In this work, a numerical study was conducted in order to determine the conditions allowing good rector operating at low pressure. A number of drawbacks of the process were identified. These drawbacks are essentially the non-negligible pressure drops and the strong intraparticle diffusion resistances. The prediction of the concentrations and the reaction rate within the pellet showed that the active zone of the pellet is located near the particle surface. It has also been shown that the resistances to interfacial mass and heat transfer are insignificant. The study of pressure effect showed that the pressure increase reduces the required catalyst mass to achieve equilibrium. Finally, this work revealed that the decrease in temperature and the increase in the concentrations of the reactants by increasing their fluxes, make it possible to increase the effectiveness factor of the catalyst and the conversion of carbon monoxide. Copyright © 2022 by Authors, Published by BCREC Group. This is an open access article under the CC BY-SA License (https://creativecommons.org/licenses/by-sa/4.0).

Fast and robust computation for population balance models is a crucial requirement for simulating particulate systems. Despite all the recent advances in developing relevant computational algorithms, either efficiency or robustness can only be achieved at the expense of the other. However, optimal compromise is possible by having prior estimates of the system's internal length and time scales. Thus, an approximation technique is introduced in this study to extract equilibrium and transient behaviors for a particulate system considering coalescence and breakage phenomena. The approximation method is developed by simplifying assumptions on the available analytical solution for a spatially homogeneous population balance equation with simple kernels. The derived equilibrium and transient equations suggest that the system is governed by a dimensionless group that can describe the equilibrium distribution of the system as well as the rate and the direction that the system is likely to evolve. The method is applied to two different sets of complex breakage and coalescence kernels used for liquid-liquid dispersed systems. The approximated time and length scales were validated with numerical results; thus, the approximation can be used to generate targeted element-based orthogonal collocation grids for fast and robust computation of transient and steady-state particulate systems. This approach can significantly decrease the computation time, typically by 40–70 % for steady-state conditions. • An approximation technique is developed to analyze a particulate system. • The technique suggest that the system is governed by a dimensionless group. • This group describes equilibrium distribution and direction of system evolution. • The estimated time and length scales are in good agreement with numerical results. • The technique was used to generate element-based orthogonal collocation grids. [ABSTRACT FROM AUTHOR]

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The septic Hermite collocation method is used to examine the behavior of time-dependent singularly perturbed convection–diffusion equations in this paper. The solution of such type of equation contains a sharp boundary layer, i.e., the solution changes rapidly for a small region of the domain when the perturbation parameter (ε ) approaches to zero. The scheme over Shishkin mesh is applied for spatial discretization and for time discretization forward differences are used. The ε -uniform convergence of the algorithm is studied and is found up to sixth order of logarithmic factor in space. The method is shown to be parameter-uniform. The method is implemented on singularly perturbed equations for Dirichlet and Robin boundary conditions. The results are given in the form of 2D, 3D graphs, and tabular forms, which indicates that the proposed method is capable of accurately capturing the sharp boundary layers when perturbation parameter tends to zero. The results exhibit that the proposed technique is efficient, reliable, and works even for a very small values of the perturbation parameter. The numerical results corroborate the theoretical results. [ABSTRACT FROM AUTHOR]

Today, hydrogen has become one of the most promising clean energy. Several processes allow obtaining hydrogen, among them there is the Water Gas Shift (WGS) reaction. On an industrial scale, WGS reaction takes place at high pressure [25-35 bar]. At high pressure, the cost of the process rises due to the energy consumed by compression, and the reduction in the lifetime of the equipment and the catalyst. At low pressures, catalyst lifetime can reach many years and the energy cost is reduced. It is for this reason that we are interested in modelling and simulation of a WGS converter operating at low pressures close to atmospheric pressure. In this work, a numerical study was conducted in order to determine the conditions allowing good rector operating at low pressure. A number of drawbacks of the process were identified. These drawbacks are essentially the non-negligible pressure drops and the strong intraparticle diffusion resistances. The prediction of the concentrations and the reaction rate within the pellet showed that the active zone of the pellet is located near the particle surface. It has also been shown that the resistances to interfacial mass and heat transfer are insignificant. The study of pressure effect showed that the pressure increase reduces the required catalyst mass to achieve equilibrium. Finally, this work revealed that the decrease in temperature and the increase in the concentrations of the reactants by increasing their fluxes, make it possible to increase the effectiveness factor of the catalyst and the conversion of carbon monoxide. [ABSTRACT FROM AUTHOR]

This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space.

ABSTRACT: In this study a historical perspective of the mass continuity equation for fully developed laminar flow in the lumen side of a hollow-fiber membrane contactor is presented with respect to the lumen-wall boundary condition (BC). It is shown that the constant wall concentration case (Dirichlet boundary condition) imposed by the Graetz-Lévêque postulations is a sub-case of the mixed Neumann-Dirichlet linear BC largely overestimating the performance of such contactors. For the linear BC the analytical solution derived by the separation of variables method is revisited proving that it is very accurate and practical even in the region very close to the entrance of the computational domain. The analysis is extended by incorporating and solving nonlinear lumen-wall BCs with the method-of-lines approach by discretizing the radial domain using the Gauss-Jacobi orthogonal collocation and integrating the resulting initial-value differential-algebraic system. The analytical solution, the derivation of the collocation matrices and the numerical solution are presented with the aid of the open-source SageMath and the commercial package Maple.

This study proposes a mathematical model constructed in spherical coordinates to reveal the mass and temperature changes that occur during the freeze-drying of apricot fruit. A kind of polynomial approach known as the orthogonal collocation method was applied to find the solution of governing equations. The physical properties and some transport parameters that describe the freeze-drying of apricot fruit were estimated by using the Levenberg-Marquardt algorithm based on the nonlinear least-square method. The effect of product radius, drying temperature, and product shape on the length of the primary drying stage was investigated by the determined transport parameters in the model. Furthermore, the effect of the freezedrying process on product color and total phenolic content (TPC) was examined, and it was observed that the TPC value of apricot fruit increased from 295.60 ± 16.55 to 602.63 ± 17.74 mg GAE (Gallic acid equivalent) / 100 g DM. [ABSTRACT FROM AUTHOR]

• Experimental TGA data sets usually contain noise. • Reaction rates obtained by finite difference method are highly fluctuated. • Smooth reaction rate can be obtained by orthogonal collocation method. • Less than 20 data points are enough for single step reactions. The orthogonal collocation method offers a solution to the challenge of noise in thermogravimetric analysis data, improving upon the differential methods like the Friedman isoconversional method. This new technique requires only a few data points for interpolation to recreate conversion-time or conversion-temperature curves, from which a smooth reaction rate curve can be derived by differentiating polynomials. Remarkably, this method demands fewer than 20 data points for a precise estimation of reaction rates in single step reactions. The method's validity is supported by both simulated and actual reactions. Users can implement this technique using provided Gnu Octave (a free alternative to MATLAB) codes. Furthermore, the activation energy results of experimental data obtained from the Freidman and orthogonal collocation methods were in good agreement with those evaluated from the integral Vyazovkin and Ortega methods. [Display omitted] [ABSTRACT FROM AUTHOR]

• Feed is distributed in the modern ammonia synthesis reactors in radial- axial direction. • The reactor has three adiabatic beds and two coolers after the first and the second beds. • Two dimensional mathematical modeling predicts the process well. • Results are compared with industrial data and CFD analysis. • Orthogonal collocation on finite element method is used for mathematical modeling. A three-bed ammonia reactor with the radial-axial flow and two heat exchangers in the rector's middle has been studied. According to the fluid regime type with two-dimensional axial-radial flow and intercooler, this reactor is different from the previous generation. Two-dimensional mathematical modelling is utilized to predict fluid flow, concentration, and temperature profiles along the reactor. The Navier-Stokes, mass, and heat partial differential equations are solved simultaneously via orthogonal collocation on the finite element. The results have been compared with industrial data, and good accordance has been observed. There are industrial data in the bed outlet, and results along the reactor cannot be validated. Therefore, CFD analysis of three beds is performed, and results are compared with results of two-dimensional mathematical modeling, and good agreement is observed. [Display omitted] [ABSTRACT FROM AUTHOR]

The advection–dispersion phenomenon can be observed in various fields of science. Mathematically, this process can be studied by considering empirical models, high-order differential equations, and fractional differential equations. In this paper, a fractional model considered to represent the transport of passive tracers carried out by fluid flow in a porous media is studied both in the direct and inverse contexts. The studied mathematical model considers a one-dimensional fractional advection–dispersion equation with fractional derivative boundary conditions. The solutions of both direct and inverse problems are obtained by using the orthogonal collocation method and the differential evolution optimization algorithm approaches, respectively. In this case, the source term along the spatial and time coordinates is taken as a design variable. The obtained results with the solution of the direct problem are compared with those determined by using an implicit finite difference scheme. The results indicate that the proposed approach characterizes a promising methodology to solve the direct and inverse fractional advection–dispersion problems. [ABSTRACT FROM AUTHOR]

A comprehensive diffusion dispersion model has been presented for displacement washing of porous particles. Model equations have been divided into two phases, namely bulk fluid phase and particle phase. Both the phases have been characterized by particle geometry and pore radius of particles. Inter-pore and intra-pore solute concentrations have been related to Langmuir adsorption isotherm. Nonlinear set of model equations has been solved by using the technique of orthogonal collocation on finite elements with Lagrangian basis. Effect of different parameters such as Péclet number, bed porosity and distribution ratio has been shown graphically via breakthrough curves and surface plots. Stability of the numerical technique has been checked by L2 and L∞ norms for different values of parameters. Validity of the model on the laboratory-scale washer has been verified by comparing experimental and model-predicted values. Applicability of the model has also been discussed through industrial parameters. [ABSTRACT FROM AUTHOR]

A 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®. [ABSTRACT FROM AUTHOR]

For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hp-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory. [ABSTRACT FROM AUTHOR]

Highlights • Simultaneous heat and mass transfer considerations of frost formation. • Orthogonal collocation method to spatially discretize differential equations. • Front-fixing method to account for the air-frost moving boundary. • Supersaturation at the air-frost interface an important adjustable parameter. • Frost properties are predicted as a function of position as well as time. Abstract The analysis of the process of frost formation on a cold surface exposed to humid air yields a system of differential equations. In this study, the orthogonal collocation method is used to help solve the system of equations resulting from the flow of humid air over heat exchanger fins maintained at a temperature below both the dew-point temperature of water vapor in air and the freezing point. The temporal dependence of the frost growth creates a moving boundary that is addressed using a front-fixing method. This proposed method allows solving for important variables such as frost thickness, temperature distribution, and heat flux through the frost layer. Model results were found to agree closely with available experimental data. In all data sets modeled, it was found that super saturation at the frost-air interface must be included in the model to accurately predict frost growth. [ABSTRACT FROM AUTHOR]

Abstract 2018 marked the 80th anniversary of Lanczos (1938) and 2017 was the 50th anniversary of Villadsen and Stewart (1967), both seminal works in development of the Orthogonal Collocation Method, also known as the Pseudospectral Method and Differential Quadrature Method. Due to these milestones, a retrospective on development of the method seems appropriate. The method's development as a Method of Weighted Residuals (MWR), with numerical integration, is reviewed. It is placed into historical perspective and developments in disparate disciplines are reconciled. We show the equivalence of methods that were thought by many to be different. Guidance in navigating the minefield of terminology is also provided. In addition to reviewing the development of the method, we show some refinements which are not known to most practitioners. Chief amongst these are the correct treatment of boundary conditions involving fluxes. We demonstrate that a proper treatment of flux boundary conditions can sometimes reduce errors by several orders of magnitude. Highlights • Moments, tau and collocation at Gauss points are equivalent. • Weak formulation correct for flux boundary conditions — Lobatto, Chebyshev. • Symmetric matrix formulation described — Gauss, Lobatto, Radau. • Simple extension to finite element collocation demonstrated — Gauss, Lobatto. • Hybrid-Collocation-Galerkin, Lobatto-Galerkin and hp Spectral Elements equivalent. [ABSTRACT FROM AUTHOR]

Abstract Weakly-supported electrolyses are frequently encountered in organic electrosynthesis in small scale flow reactors. Sometimes "self-supported" electrolyses are reported where no supporting electrolyte is deliberately added at all. To interpret the steady-state and transient behavior of these electrochemical micro- and millireactors, an orthogonal collocation method based on Chebyshev polynomials was developed for calculating the concentration and current distributions in parallel plate and annular geometries. This method takes into account diffusion, migration, and convection in unidirectional flow for an arbitrary number of dilute ionic species. In particular, the method is flexible with regard to the electrode and homogenous reactions so that intricate reaction schemes are conveniently simulated. As a case study, the effect of the supporting electrolyte concentration and the stability of the generated reactive intermediate ions on a generic paired electrosynthesis is investigated. The numerically calculated concentration profiles and current responses are then compared with asymptotic approximations. This reveals that fundamentally different operating regimes exist for a self-supported paired electrolysis compared to more conventional electrolyses where an excess of supporting electrolyte is added. The Matlab code and a detailed user guide is freely available via the Supplementary Information. Graphical abstract Image 1 Highlights • An orthogonal collocation method for the simulation of electrochemical microreactors. • Enables fast transient simulations, such as voltammetry, in flow reactors. • The method is flexible w.r.t. user-specified electrode and homogeneous reactions. • Case study of self-supported paired electrosynthesis discussed in detail. [ABSTRACT FROM AUTHOR]

This numerical study is aimed at investigating the convective heat transfer and flow fluid inside a horizontal circular tube in the fully-developed laminar flow regime under the constant wall temperature boundary condition,, is commonly called the Graetz Problem that our goal is to get the steady temperature distribution in the fluid. The complexity of the partial differential equation that describes the temperature field with the associated linear or nonlinear boundary conditions is simplified by means of numerical methods using current computational tools. The simplified energy equation is solved numerically by the orthogonal collocation method followed by the finite difference method (Crank–Nicholson method).The calculations were effected through a FORTRAN computer program and the results show that orthogonal collocation method giving better results than Crank–Nicholson method.

This study develops the energy balance based on Jacobi collocation method for accurate prediction of conservative nonlinear oscillator models with a single collocation point. The node points are taken as the roots of Jacobi orthogonal polynomials. Several examples are included to demonstrate the applicability and accuracy of the proposed algorithm, and some comparisons are made with the existing results. The method is suitable and the approximate frequencies are valid for small as well as large amplitudes of oscillation. Excellent agreement with exact ones is presented for the first order approximation.

This is a theoretical study about the influence of turbulence on momentum and heat transport in a packed-bed with low tube to particle diameter ratio. The hydrodynamics is given here by the time-averaged Navier-Stokes equations including Darcy and Forchheimer terms, plus a κ-ε two-equation model to describe a 2D pseudo-homogeneous medium. For comparison, an equivalent conventional flow model has also been tested. Both models are coupled to a heat transport equation and they are solved using spatial discretization with orthogonal collocation, while the time derivative is discretized by an implicit Euler scheme. We compared the prediction of radial and axial temperature observations from a packed-bed at particle Reynolds numbers (Rep) of 630, 767, and 1000. The conventional flow model uses effective heat transport parameters: wall heat transfer coefficient (hw) and thermal conductivity (keff), whereas the turbulent flow model includes a turbulent thermal conductivity (kt), estimating hw via least-squares with Levenberg-Marquardt method. Although predictions of axial and radial measured temperature profiles with both models show small differences, the calculated radial profiles of the axial velocity component are very different. We demonstrate that the model that includes turbulence compares well with mass flux measurements at the packed-bed inlet, yielding an error of 0.77 % in mass flux balance at Rep = 630. We suggest that this approach can be used efficiently for the hydrodynamics characterization and design and scale-up of packed beds with low tube to particle diameter ratio in several industrial applications. [ABSTRACT FROM AUTHOR]

This paper presents the design of PID-type controllers for a one-dimensional, distributed heat conduction system starting from the parabolic partial differential equation modelling the system. The partial differential equation modelling the system was lumped using the orthogonal collocation method resulting in third-order lumped model. This lumped model was then used for the controller design based on a method of PID controller design previously developed for higher-order systems by the authors. The results of closed-loop simulations demonstrate the superior performance of the PID-type controllers so designed. The approach presented is quite general and may be used to carry out PID-type controller designs for other single-input, single-output distributed parameter systems. [ABSTRACT FROM AUTHOR]

We present a new method for the synthesis of mass exchanger networks (MENs) involving packed columns. Simultaneous synthesis of MENs is typically done through the use of mixed-integer nonlinear program (MINLP) optimization, with simplifications made in the mathematical representations of the exchangers due to computational difficulty in solving large non-convex mixed-integer problems. The methodology proposed in this study makes use of the stage-wise based superstructure MINLP formulation for the network synthesis. This stage-wise superstructure model incorporates fixed mass transfer coefficients, fixed column diameters, no pressure drops, and unequal compositional mixing for models. In this paper, the simplified MINLP model is further improved by including a detailed individual packed column design in a non-linear programming (NLP) sub-optimization step, where orthogonal collocation is utilized for the partial differential equations, and optimal packing size, column diameter, column height, pressure drops, and fluid velocities. Detailed designs are then used to determine correction factors that update the simplified stage-wise superstructure models to more accurately portray the chosen design. Once the MINLP is updated with these correction factors, the model is re-run, with new correction factors obtained. This iterative procedure is repeated until convergence between the objective function of the MINLP and that of the NLP sub-optimization is achieved, or until a maximum number of iterations is reached. The methodology is applied to two examples and is shown to be robust and effective in generating new topologies, and in finding superior networks that are physically realizable. [ABSTRACT FROM AUTHOR]

A nonlinear state observer is designed for a thermal energy storage with solid/liquid phase change material (PCM). Using a physical 2D dynamic model, the observer reconstructs transient spatial temperature fields inside the storage and estimates the stored energy and the state of charge. The observer has been successfully tested with a lab-scale latent heat storage with a single pass tube bundle and the phase change material located in a shell around each tube. It turns out that the observer robustly tracks the real process data with as few as four internal PCM temperature sensors. [ABSTRACT FROM AUTHOR]

Methanol is usually produced in large amounts in fixed-bed reactors under steady-state conditions using Cu/ZnO/Al 2 O 3 -catalyst. In this paper, possible improvements through forced periodic operation compared to optimal steady-state solution are investigated. The numerical study is done with a simplified model which describes a hypothetical 7-meter-long isothermal and isobaric fixed-bed reactor. The forced periodic operation is implemented using harmonic forcing and square-waves. Multi-objective optimization is performed for different operating conditions quantifying the achievable molar flow rate of methanol at the reactor outlet and the yield of methanol based on total carbon. The resulting PDE-constrained optimization problems are solved using a simultaneous approach, with orthogonal collocation as discretization. The results show most significant improvements through forced periodic operation with square-waves. The periodic operation can provide theoretical improvements of up to 50% for the methanol mean outlet molar flow rate and improvements of up to 10% for the yield based on total carbon. • Forced periodic operation theoretically improves the performance of methanol synthesis in fixed-bed reactors • Square-wave modulation provides more significant improvement than harmonic modulation • Less hydrogen is needed with forced periodic operation compared to steady-state operation • Orthogonal collocation is used to discretize the partial differential equation. [ABSTRACT FROM AUTHOR]