Simulation of chemical reactors using the least-squares spectral element method

A numerical comparison of the pseudo-homogeneous, conventional heterogeneous, and simplified heterogeneous reactor models is performed for the steam methane reforming process. The pseudo-homogeneous reactor model consists of a set of partial differential equations, where the diffusional limitations are accounted for by considering the effectiveness factor. The heterogeneous model is divided into two categories: conventional and simplified heterogeneous models. In the conventional heterogeneous model, there are separate equations for the fluid phase species mass balance, the fluid inside the catalyst pores. This is type of reactor model is needed when there is a considerable amount of interphase mass transfer resistance present in the process. However, in the simplified heterogeneous reactor model the mass transport phenomenon is accounted for by accounting the efficiency factors. The model is validated against literature data. Several closures for the intra-particle mass diffusion fluxes, the Maxwell-Stefan, Wilke, dusty gas, and Wilke-Bosanquet models, have been compared on the level of the catalyst pellet and the impacts of the different particle flux closures on the reactor performance are investigated.The simulations show that the conventional heterogeneous reactor model is necessary for the SMR process, because the effectiveness factor values of different reactions of the SMR process vary along the reactor axis. The maximum deviation between the pseudo-homogeneous and the conventional heterogeneous reactor model is less than 38 %, whereas between the conventional and simplified heterogeneous reactor models it is less than 21 %. A parametric study of the transport phenomena on the pellet level is recommended prior to any large-scale reactor simulation to determine what the rate-determining transport mechanisms are.

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“…Reactor specification for the SMR process [12,61] extended Fick diffusion flux model originally designed for binary mixtures. [18] 4.…”

Section: Boundary Conditionsmentioning

confidence: 99%

“…The time dependent governing equations for the different fixed bed reactor models are solved using the leastsquares spectral element methods (LS-SEM). [7,10,61] The basic idea in the LSM is to minimize the integral of the square of the residual over the computational domain.…”

Section: Model Solutionmentioning

confidence: 99%

A numerical study of fixed bed reactor modelling for steam methane reforming process

Rout

Jakobsen

2015

Can J Chem Eng

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“…In the third modeling approach the generic population balance equation is derived by Maxwellian averaging a statistical Boltzmann-type of equation in terms of number probability densities, in line with the work of Reyes [107], Lafi and Reyes [59], Carrica et al [11], Lasheras et al [60], and Jakobsen et al [19,91,96,[123][124][125]137]. In this framework, a Boltzmann type equation is Maxwellian averaged over the entire particle velocity space only thus the resulting dispersed phase governing equations are expressed explicitly in terms of the inner coordinates, the external physical space coordinates and time so that further details of the dispersed flow and chemical process behavior can be resolved and described directly and not only the moments of the dispersed phase properties.…”

Section: Four Alternative Population Balance Frameworkmentioning

confidence: 99%

“…They were able to derive Tambour's sectional equations from the kinetic level. This type of modeling framework has also been investigated by Jakobsen et al [19,91,96,[123][124][125]137] in order to derive a general set of balance equations for the dispersed phase populations. Starting out from a Boltzmann-type equation, a set of consistent PBE, mass, momentum, species mass and enthalpy (temperature) equations were derived for the dispersed phase and expressed in terms of a probability density function.…”

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confidence: 99%

The Population Balance Equation

Jakobsen

2014

Chemical Reactor Modeling

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The chemical engineering community began the first efforts that can be associated with the concepts of the population balance in the early 1960s. The familiar application of population balance principles to the modeling of flow and mixing characteristics in vessels was formally organized by Danckwerts [18]. Certain distribution functions were then defined for the residence times of fluid elements in a process vessel. The residence time distribution function give information about the fraction of the fluid that spends a certain time in a process vessel. Himmelblau and Bishoff [37] describe how the residence time and other age distributions are defined, how they can be measured, and how they can be interpreted. This chapter focuses on the derivation and use of the general population balance equation (PBE) to describe the evolution of the fluid particle distribution due to the advection, growth, coalescence and breakage processes.The historical derivation of the general population balance equation for countable entities is outlined. Two fundamental modeling frameworks emerge formulating the early population balances, in quite the same way as the kinetic theory of dilute gases and the continuum mechanical theory were proposed deriving the governing conservation equations in fluid mechanics. A third less rigorous approach is also used formulating the population balance directly on the macroscopic averaging scales, an analogue to the multiphase mixture models. A fourth less computationally demanding approach is to formulate a moment form of the population balance equation.Considering dispersed two-phase flows, a few research groups adopted a statistical Boltzmann-type equation determining the rate of change of a suitable defined distribution function. Performing the Maxwellian averaging integrating all terms over the whole velocity space to eliminate the velocity dependence, one obtains the generic population balance equation. Rigorous closures are required for the unknown terms resulting from the averaging process. However, by employing the conventional Chapman-Enskog approximate expansion method solving the Boltzmann-type equation this theory provides rational means of understanding for the one way coupling between microscopic particle physics and the average macroscopic continuum properties. Moreover, to represent complex problem physics an optimized choice of H. A. Jakobsen, Chemical Reactor Modeling,

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“…However, in recent years the least-squares method has been adopted for the solution of chemical reactor problems, e.g. the work of Dorao [23], Zhu [24], Patruno [25], Sporleder [26], and Rout [27]. In the field of chemical reactor engineering, the least-squares method has most frequently been applied to PB problems, but the least-squares method has also been applied for the solution of the pellet equations and fixed packed bed reactor models [28,29].…”

Section: Introductionmentioning

confidence: 99%

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

Solsvik

Jakobsen

2016

Applied Mathematical Modelling

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The behavior of dispersed systems such as gas-liquids or liquid-liquid systems depends on the characteristics of the dispersed phase. The population balance (PB) equation is encountered in numerous engineering disciplines in order to describe complex processes where the accurate prediction of the dispersed phase plays a major role for the overall behavior of the system. In the present study, the orthogonal collocation, Galerkin and least-squares methods have been adopted to solve a non-linear PB equation which consists of both breakage and coalescence terms. The performance of the methods is demonstrated by comparing the numerical solution results with the (manufactured) analytical solution of the problem. For the least-squares method, the choice of linearization technique influences the numerical performance, whereas the Galerkin and orthogonal collocation methods obtain the same numerical accuracy for both the Picard and Newton iteration techniques. The least-squares method suffers from lack of convergency using the Picard method. On the other hand, if the Newton method is employed, the least-squares method obtains the same accuracy as the Galerkin and orthogonal collocation methods.

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Simulation of chemical reactors using the least-squares spectral element method

Cited by 30 publications

References 17 publications

A numerical study of fixed bed reactor modelling for steam methane reforming process

A numerical study of fixed bed reactor modelling for steam methane reforming process

The Population Balance Equation

Spectral solution of the breakage–coalescence population balance equation Picard and Newton iteration methods

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