The document presents a three-level mathematical model of a well-stirred catalytic reactor with bidispersed catalysts. The model describes (1) the completely stirred fluid phase on the first level, (2) diffusion in the macropores on the second level, and (3) diffusion and reaction in the micropores on the third level. The model equations are solved using multilevel collocation to investigate transient phenomena in the reactor via computer simulation.
Quadrupolar structures generated by chiral islands in freely suspended smecti...NunoSilvestre
This document describes a study of quadrupolar defect structures generated by chiral islands in freely suspended smectic C films. The researchers provide a quantitative model to describe interactions between topologically chiral islands that depend on the islands' chirality. Their model uses the Landau theory approach with an order parameter representing the smectic C layers. They find that as the separation between islands increases, the symmetric quadrupolar defect configuration breaks down and the defects begin to follow individual islands in a dipolar structure.
Direct variational calculation of second-order reduced density matrix : appli...Maho Nakata
Presented at GCOE interdisciplinary workshop on numerical methods for many-body correlations, https://sites.google.com/a/cns.s.u-tokyo.ac.jp/shimizu/gcoe
This summary provides the key details about the document in 3 sentences:
The document presents a new simplified analytical model for Chireix outphasing combiners that accounts for impedance mismatch and reflection effects. The model derives simplified expressions for instantaneous efficiency and input/output voltages as functions of the decomposition angle and combiner parameters. Validation of the model is performed through comparisons with experimental and simulation results for different Chireix combiner designs.
Modelling Simul. Mater. Sci. Eng. 8 (2000) 445 C A And C P F E MDierk Raabe
1) The document describes coupling a crystal plasticity finite element model with a probabilistic cellular automaton model to simulate primary static recrystallization in aluminium.
2) The crystal plasticity model accounts for crystallographic slip and rotation during plastic deformation, while the cellular automaton uses a probabilistic rule to model the motion of grain boundaries during recrystallization based on driving forces.
3) The models are coupled by translating variables like texture and dislocation density between the models, mapping the finite element mesh onto the cellular automaton grid, and using output from the finite element model to set parameters like length scale and time step in the cellular automaton.
A novel algorithm for capacitor placement to improve voltage stability of rad...Alexander Decker
This document presents a novel algorithm for determining the optimal location and amount of reactive power injection to improve voltage stability in power systems.
1) It proposes a new "L-Index sensitivity matrix" to identify buses where reactive power should be injected based on how reactive power changes at each bus affect the L-Index values of all buses.
2) The algorithm calculates L-Index values for each bus, derives the L-Index sensitivity matrix, and uses linear programming to determine the location and amount of reactive power injection to minimize L-Index values and improve voltage stability while satisfying constraints.
3) Experimental results on the IEEE 14-bus test system show the effectiveness of the proposed algorithm at improving voltage stability.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Speech recognition using hidden markov model mee 03_19NGUYEN_SPKT
This document describes the implementation of a speech recognition system using hidden Markov models (HMMs) on a DSK-ADSP-BF533 EZ-KIT LITE REV 1.5 digital signal processor board. The key steps taken are:
1. Feature extraction from speech signals by calculating mel frequency cepstrum coefficients (MFCCs). This involves preprocessing the signal, framing and windowing, Fourier transform, mel filtering, and discrete cosine transform.
2. Training HMMs to create models of words using an iterative process that calculates mean, variance, state transition probabilities, and output distributions from multiple utterances of words.
3. Testing utterances against trained models using the Viterbi
Punching Shear Strength of Transversely Prestressed Concrete DecksSana'a Amir
This document discusses punching shear strength in transversely prestressed concrete bridge decks. It introduces compressive membrane action that occurs due to lateral restraint at slab edges, improving punching capacity. Existing research models for predicting capacity are described, including modified Hallgren and engineering methods. Future experiments are proposed to better understand the effects of transverse prestressing level, joint skewness, and loading position on punching strength accounting for compressive membrane action.
Quadrupolar structures generated by chiral islands in freely suspended smecti...NunoSilvestre
This document describes a study of quadrupolar defect structures generated by chiral islands in freely suspended smectic C films. The researchers provide a quantitative model to describe interactions between topologically chiral islands that depend on the islands' chirality. Their model uses the Landau theory approach with an order parameter representing the smectic C layers. They find that as the separation between islands increases, the symmetric quadrupolar defect configuration breaks down and the defects begin to follow individual islands in a dipolar structure.
Direct variational calculation of second-order reduced density matrix : appli...Maho Nakata
Presented at GCOE interdisciplinary workshop on numerical methods for many-body correlations, https://sites.google.com/a/cns.s.u-tokyo.ac.jp/shimizu/gcoe
This summary provides the key details about the document in 3 sentences:
The document presents a new simplified analytical model for Chireix outphasing combiners that accounts for impedance mismatch and reflection effects. The model derives simplified expressions for instantaneous efficiency and input/output voltages as functions of the decomposition angle and combiner parameters. Validation of the model is performed through comparisons with experimental and simulation results for different Chireix combiner designs.
Modelling Simul. Mater. Sci. Eng. 8 (2000) 445 C A And C P F E MDierk Raabe
1) The document describes coupling a crystal plasticity finite element model with a probabilistic cellular automaton model to simulate primary static recrystallization in aluminium.
2) The crystal plasticity model accounts for crystallographic slip and rotation during plastic deformation, while the cellular automaton uses a probabilistic rule to model the motion of grain boundaries during recrystallization based on driving forces.
3) The models are coupled by translating variables like texture and dislocation density between the models, mapping the finite element mesh onto the cellular automaton grid, and using output from the finite element model to set parameters like length scale and time step in the cellular automaton.
A novel algorithm for capacitor placement to improve voltage stability of rad...Alexander Decker
This document presents a novel algorithm for determining the optimal location and amount of reactive power injection to improve voltage stability in power systems.
1) It proposes a new "L-Index sensitivity matrix" to identify buses where reactive power should be injected based on how reactive power changes at each bus affect the L-Index values of all buses.
2) The algorithm calculates L-Index values for each bus, derives the L-Index sensitivity matrix, and uses linear programming to determine the location and amount of reactive power injection to minimize L-Index values and improve voltage stability while satisfying constraints.
3) Experimental results on the IEEE 14-bus test system show the effectiveness of the proposed algorithm at improving voltage stability.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Speech recognition using hidden markov model mee 03_19NGUYEN_SPKT
This document describes the implementation of a speech recognition system using hidden Markov models (HMMs) on a DSK-ADSP-BF533 EZ-KIT LITE REV 1.5 digital signal processor board. The key steps taken are:
1. Feature extraction from speech signals by calculating mel frequency cepstrum coefficients (MFCCs). This involves preprocessing the signal, framing and windowing, Fourier transform, mel filtering, and discrete cosine transform.
2. Training HMMs to create models of words using an iterative process that calculates mean, variance, state transition probabilities, and output distributions from multiple utterances of words.
3. Testing utterances against trained models using the Viterbi
Punching Shear Strength of Transversely Prestressed Concrete DecksSana'a Amir
This document discusses punching shear strength in transversely prestressed concrete bridge decks. It introduces compressive membrane action that occurs due to lateral restraint at slab edges, improving punching capacity. Existing research models for predicting capacity are described, including modified Hallgren and engineering methods. Future experiments are proposed to better understand the effects of transverse prestressing level, joint skewness, and loading position on punching strength accounting for compressive membrane action.
Toward overcoming the concept of effective stiffness and damping in the dynam...Alessandro Palmeri
In the current state-of-practice, the time-domain dynamic analysis of structures incorporating viscoelastic members is generally carried out through the Modal Strain Energy (MSE) method, or other procedures somehow based on the quite simplistic idea of substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping.
This crude approximation in civil engineering applications is very often encouraged by manufacturers of the viscoelastic devices themselves, whose interest is to simplify as much as possible the design procedures for structures embedding their products. As an example, elastomeric seismic isolators are generally advertised and sold with a table listing the equivalent values of elastic stiffness and viscous damping ratio for different amplitudes of vibration. Unfortunately, many experimental and analytical studies confirm that the real dynamic behaviour of such devices is much more complicated, and cannot be bend to the interests of manufacturers and designers.
Despite the advances in the field made in the last two decades, two well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structures: first, MSE method and similar procedures are unconditionally assumed to provide good approximations, which are acceptable for design purposes; second, the implementation of more refined approaches is thought to be computationally too expensive, and hence suitable just for a few very important constructions.
In this presentation, as a further contribution to overcome these popular beliefs, a novel time-domain numerical scheme of dynamic analysis is proposed and numerically validated. After a brief review of the LPA (Laguerre’s Polynomial Approximation) technique for one-dimensional viscoelastic members of known relaxation function, the state-space equations of motion for linear structures with viscoelastic components are derived in the modal space. Aimed at making the proposed approach more general, the distribution of the viscoelastic components is allowed to be non-proportional to mass and elastic stiffness, in so removing the most severe limitation of previous formulations. Then, a cascade scheme is derived by decoupling in each time step traditional state variables (i.e. modal displacements and velocities) and additional internal variables. The joint use of modal analysis and improved cascade scheme permits to reduce the size of the problem and to keep low the computational burden. The illustrative application to the small-amplitude vibration of a cable beam made of different viscoelastic materials demonstrates the versatility of the proposed approach. The numerical results confirm a superior accuracy with respect to the classical MSE method, whose underestimate in the low-frequency range can be as large as 75%.
This document summarizes work using molecular dynamics simulations to calculate the viscosity of liquid nickel. The researchers used a modified embedded atom method potential to simulate liquid nickel across a range of temperatures. Preliminary results for the viscosity fell within the range of available experimental data. Future work involves further testing and developing optimized potentials for nickel alloys and calculating other physically relevant parameters for larger scale simulations.
HETEROGENEOUS CLUTTER MODEL FOR HIGH RESOLUTION POLARIMETRIC SAR DATA PROCESSINGgrssieee
This document summarizes research on modeling heterogeneous clutter for high resolution polarimetric synthetic aperture radar (SAR) data processing. It discusses limitations of the conventional Gaussian clutter model and proposes using a spherically invariant random vector (SIRV) model instead. This accounts for heterogeneity and non-Gaussian statistics of SAR images better. It also addresses estimating the texture and speckle covariance matrix parameters for the SIRV model and applying the approach to polarimetric SAR images. Results demonstrate the SIRV model's ability to characterize clutter heterogeneity compared to the Gaussian model.
An algorithm for simulation of achemical transport equation in an aquifer fin...Alexander Decker
This document describes an algorithm developed to simulate chemical transport in an aquifer. The algorithm discretizes and solves the governing chemical transport and groundwater flow equations using an integrated finite difference method. The algorithm is implemented in a computer code to generate predictions about chemical fate and transport processes in the aquifer. Simulation results demonstrate how transport processes, decay, and hydraulic properties influence chemical concentration levels over time.
This document provides an introduction to quantitative x-ray diffraction methods. It discusses that quantitative analysis aims to determine precise structural characteristics and phase proportions from experimental diffraction data. This requires accurate intensity measurements and careful sample preparation to produce uniformly sized, randomly oriented powder. Several quantitative methods are based on comparing intensity ratios between unknown and known internal standard peaks. The most effective methods involve modeling the entire diffraction pattern to duplicate the experimental pattern.
The document analyzes microstrip transmission lines using a quasi-static approach. It presents numerically efficient and accurate formulas to analyze microstrip line structures. The analysis derives formulas for characteristic impedance of microstrip lines based on variables like the normalized strip width, effective permittivity, height of the substrate, and thickness of the microstrip line. It also defines the structure of a microstrip line and formulates the quasi-static analysis by introducing the concept of an effective relative dielectric constant to account for the microstrip being surrounded by different dielectrics like air and the substrate material.
Coupling Maxwell\'s Equations to Particle-Based Simulatorsayubimoak
The document describes a research group's work on simulating semiconductor devices using electromagnetic and particle-based simulation methods. It aims to accurately capture radiated electromagnetic fields from high-frequency nanostructure devices. The group is developing a simulator that couples Maxwell's equations and the Boltzmann transport equation to model carrier transport and electromagnetic wave propagation. It initializes simulations using empirical pseudopotential bandstructure models and calculates particle motion through free flight and scattering.
This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.
This document discusses strategies for parallelizing spectral methods. Spectral methods are global in nature due to their use of global basis functions, making them challenging to parallelize on fine-grained architectures. However, the document finds that spectral methods can be effectively parallelized. The main computational steps in spectral methods are the calculation of differential operators on functions and solving linear systems, both of which can exploit parallelism. Domain decomposition techniques may also help parallelize computations over non-Cartesian domains.
Effective properties of heterogeneous materialsSpringer
1) The document discusses the multipole expansion method (MEM) for analyzing the microstructure and effective properties of composite materials.
2) MEM reduces boundary value problems for heterogeneous materials to systems of linear algebraic equations. It expresses fields like temperature and stress as expansions of basis functions related to inclusion geometry.
3) MEM has been applied to analyze conductivity and elasticity problems in composites with spherical, spheroidal, circular, and elliptic inclusions. It provides analytical solutions for local fields and exact expressions for effective properties involving only dipole moments.
1) The document summarizes research on the ground state of strongly coupled quark matter with a finite isospin chemical potential.
2) A Ginzburg-Landau theory approach is used to qualitatively analyze the phase transition near the critical point in a model-independent way.
3) It is found that at zero isospin chemical potential, the chiral condensation transition becomes first-order at high densities due to the formation of spatial inhomogeneities. At finite isospin chemical potential, charged pion condensation can occur in addition to chiral condensation.
1) A mathematical model was developed to relate the time for a fluid to empty from a capillary viscometer to the fluid's viscosity. Equations were derived based on the Navier-Stokes and continuity equations.
2) Computational fluid dynamics simulations showed that offsetting the capillary tube reduced turbulence and improved laminar flow compared to designs without an offset.
3) Experiments were conducted using various fluids to empirically validate the theoretical model. Dimensional analysis provided insights into the sensitivity of design parameters.
Gecco 2011 - Effects of Topology on the diversity of spatially-structured evo...matteodefelice
This document summarizes research on the effect of topology on diversity in spatially-structured evolutionary algorithms (SSEAs). The researchers modeled SSEAs as spreading processes and investigated how network topology influences diversity. They found that lattice networks maintained more diversity than random networks, leading to finding more optima. Rewiring the lattice to have small-world properties showed how network structure can control the spreading of solutions and thus the algorithm's dynamics. This research was a first step toward understanding how to design network topologies to optimize problem-solving in SSEAs.
Narayan Shrestha [Step wise multi-criteria performance evaluation of rainfall...Narayan Shrestha
The first author of the paper is Mr. Shakti PC who is currently in Japan. This paper is published in Journal of Hydrology and Meteorology Nepal (Vol. 7, No. 1) on December 2010.
Conformal Anisotropic Mechanics And Horavas Particleguest9fa195
The document presents a model of classical mechanics that implements anisotropic scaling transformations between space and time consistent with the dispersion relation of Hořava gravity. The resulting action is a generalization of conformal mechanics and reduces to known cases in certain limits. Thermodynamic properties of a gas of free particles under the anisotropic scaling are also derived, matching results from recent literature on anisotropic black branes.
This document discusses smoothed particle hydrodynamics (SPH), a Lagrangian numerical method for simulating fluid flows. SPH uses discrete particles that move with the fluid and sample hydrodynamic properties by averaging values over neighboring particles. The document outlines the basic concepts of SPH, including how it approximates continuous fields with discrete particles and calculates spatial derivatives. It also discusses how SPH can be used to model astrophysical phenomena through numerical simulations since laboratory experiments are often impossible at astrophysical scales.
Study on Reconstruction Accuracy using shapiness index of morphological trans...ijcseit
Basin, lakes, and pore-grain space are important geophysical shapes, which can fit with the several
classical and fractal binary shapes, are processed by employing morphological transformations, and
methods. The decomposition of skeleton network (minimum morphological information) using various
classical structures like square, octagon and rhombus. Then derive the dilated subsets respective degree by
the structures for reconstruct the original image. Through shapiness index of pattern spectrum procedure,
we try test the reconstruction accuracy in a quantitative manner. It gives some general procedure to
characterise the shape-size complexity of surface water body. The reconstruction accuracy is against the
size of water bodies with which we produce the some example of different shapiness index for different
structuring element of shapes. In which quantitative manner approach yields better reconstruction level.
The complexity of water bodies are compared with the surfaces.
Current Research on Quantum Algorithms.pptDefiantTones
This document summarizes research on quantum algorithms being conducted at the Institute for Quantum Information and Matter (IQIM). The research objectives include developing improved methods for fault-tolerant quantum computation, new quantum algorithms beyond the hidden subgroup problem, and simulation methods for quantum many-body systems and local quantum systems. Recent progress includes developing quantum algorithms for simulating particle collisions in fermionic quantum field theories and optimal algorithms for preparing topological quantum error correcting codes. Future work will focus on showing problems in quantum field theory are hard, characterizing logical operations in topological codes, and improving bounds on quantum memory times.
FUZZY LOGIC Control of CONTINUOUS STIRRED TANK REACTOR ProfDrDuraidAhmed
This document describes the use of fuzzy logic control for a continuous stirred tank reactor (CSTR). It begins with an abstract that summarizes modeling the CSTR system using mass and energy balances, and designing a fuzzy logic controller to control the reactor temperature. It then provides more details on mathematical modeling of the CSTR, the basic operations of fuzzy set theory, and the design of the fuzzy logic controller. The controller design involves choosing membership functions to classify the error signal and change in error, then developing fuzzy rules relating the error terms to the control output. Simulation results showed the fuzzy logic controller provided better tracking and regulation than a traditional PID controller.
Conformal Anisotropic Mechanics And The HořAva Dispersion Relationvcuesta
This document summarizes a paper that implements scale anisotropic transformations between space and time in classical mechanics. This results in a system consistent with the dispersion relation proposed in Horava gravity. The paper constructs an action principle for a particle that is invariant under anisotropic scaling transformations between space and time. This action reduces to known systems like conformal mechanics in certain limits of the dynamical exponent z. The paper also analyzes the canonical formalism, equations of motion, symmetries and thermodynamic properties of this anisotropic mechanical system.
Toward overcoming the concept of effective stiffness and damping in the dynam...Alessandro Palmeri
In the current state-of-practice, the time-domain dynamic analysis of structures incorporating viscoelastic members is generally carried out through the Modal Strain Energy (MSE) method, or other procedures somehow based on the quite simplistic idea of substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping.
This crude approximation in civil engineering applications is very often encouraged by manufacturers of the viscoelastic devices themselves, whose interest is to simplify as much as possible the design procedures for structures embedding their products. As an example, elastomeric seismic isolators are generally advertised and sold with a table listing the equivalent values of elastic stiffness and viscous damping ratio for different amplitudes of vibration. Unfortunately, many experimental and analytical studies confirm that the real dynamic behaviour of such devices is much more complicated, and cannot be bend to the interests of manufacturers and designers.
Despite the advances in the field made in the last two decades, two well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structures: first, MSE method and similar procedures are unconditionally assumed to provide good approximations, which are acceptable for design purposes; second, the implementation of more refined approaches is thought to be computationally too expensive, and hence suitable just for a few very important constructions.
In this presentation, as a further contribution to overcome these popular beliefs, a novel time-domain numerical scheme of dynamic analysis is proposed and numerically validated. After a brief review of the LPA (Laguerre’s Polynomial Approximation) technique for one-dimensional viscoelastic members of known relaxation function, the state-space equations of motion for linear structures with viscoelastic components are derived in the modal space. Aimed at making the proposed approach more general, the distribution of the viscoelastic components is allowed to be non-proportional to mass and elastic stiffness, in so removing the most severe limitation of previous formulations. Then, a cascade scheme is derived by decoupling in each time step traditional state variables (i.e. modal displacements and velocities) and additional internal variables. The joint use of modal analysis and improved cascade scheme permits to reduce the size of the problem and to keep low the computational burden. The illustrative application to the small-amplitude vibration of a cable beam made of different viscoelastic materials demonstrates the versatility of the proposed approach. The numerical results confirm a superior accuracy with respect to the classical MSE method, whose underestimate in the low-frequency range can be as large as 75%.
This document summarizes work using molecular dynamics simulations to calculate the viscosity of liquid nickel. The researchers used a modified embedded atom method potential to simulate liquid nickel across a range of temperatures. Preliminary results for the viscosity fell within the range of available experimental data. Future work involves further testing and developing optimized potentials for nickel alloys and calculating other physically relevant parameters for larger scale simulations.
HETEROGENEOUS CLUTTER MODEL FOR HIGH RESOLUTION POLARIMETRIC SAR DATA PROCESSINGgrssieee
This document summarizes research on modeling heterogeneous clutter for high resolution polarimetric synthetic aperture radar (SAR) data processing. It discusses limitations of the conventional Gaussian clutter model and proposes using a spherically invariant random vector (SIRV) model instead. This accounts for heterogeneity and non-Gaussian statistics of SAR images better. It also addresses estimating the texture and speckle covariance matrix parameters for the SIRV model and applying the approach to polarimetric SAR images. Results demonstrate the SIRV model's ability to characterize clutter heterogeneity compared to the Gaussian model.
An algorithm for simulation of achemical transport equation in an aquifer fin...Alexander Decker
This document describes an algorithm developed to simulate chemical transport in an aquifer. The algorithm discretizes and solves the governing chemical transport and groundwater flow equations using an integrated finite difference method. The algorithm is implemented in a computer code to generate predictions about chemical fate and transport processes in the aquifer. Simulation results demonstrate how transport processes, decay, and hydraulic properties influence chemical concentration levels over time.
This document provides an introduction to quantitative x-ray diffraction methods. It discusses that quantitative analysis aims to determine precise structural characteristics and phase proportions from experimental diffraction data. This requires accurate intensity measurements and careful sample preparation to produce uniformly sized, randomly oriented powder. Several quantitative methods are based on comparing intensity ratios between unknown and known internal standard peaks. The most effective methods involve modeling the entire diffraction pattern to duplicate the experimental pattern.
The document analyzes microstrip transmission lines using a quasi-static approach. It presents numerically efficient and accurate formulas to analyze microstrip line structures. The analysis derives formulas for characteristic impedance of microstrip lines based on variables like the normalized strip width, effective permittivity, height of the substrate, and thickness of the microstrip line. It also defines the structure of a microstrip line and formulates the quasi-static analysis by introducing the concept of an effective relative dielectric constant to account for the microstrip being surrounded by different dielectrics like air and the substrate material.
Coupling Maxwell\'s Equations to Particle-Based Simulatorsayubimoak
The document describes a research group's work on simulating semiconductor devices using electromagnetic and particle-based simulation methods. It aims to accurately capture radiated electromagnetic fields from high-frequency nanostructure devices. The group is developing a simulator that couples Maxwell's equations and the Boltzmann transport equation to model carrier transport and electromagnetic wave propagation. It initializes simulations using empirical pseudopotential bandstructure models and calculates particle motion through free flight and scattering.
This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.
This document discusses strategies for parallelizing spectral methods. Spectral methods are global in nature due to their use of global basis functions, making them challenging to parallelize on fine-grained architectures. However, the document finds that spectral methods can be effectively parallelized. The main computational steps in spectral methods are the calculation of differential operators on functions and solving linear systems, both of which can exploit parallelism. Domain decomposition techniques may also help parallelize computations over non-Cartesian domains.
Effective properties of heterogeneous materialsSpringer
1) The document discusses the multipole expansion method (MEM) for analyzing the microstructure and effective properties of composite materials.
2) MEM reduces boundary value problems for heterogeneous materials to systems of linear algebraic equations. It expresses fields like temperature and stress as expansions of basis functions related to inclusion geometry.
3) MEM has been applied to analyze conductivity and elasticity problems in composites with spherical, spheroidal, circular, and elliptic inclusions. It provides analytical solutions for local fields and exact expressions for effective properties involving only dipole moments.
1) The document summarizes research on the ground state of strongly coupled quark matter with a finite isospin chemical potential.
2) A Ginzburg-Landau theory approach is used to qualitatively analyze the phase transition near the critical point in a model-independent way.
3) It is found that at zero isospin chemical potential, the chiral condensation transition becomes first-order at high densities due to the formation of spatial inhomogeneities. At finite isospin chemical potential, charged pion condensation can occur in addition to chiral condensation.
1) A mathematical model was developed to relate the time for a fluid to empty from a capillary viscometer to the fluid's viscosity. Equations were derived based on the Navier-Stokes and continuity equations.
2) Computational fluid dynamics simulations showed that offsetting the capillary tube reduced turbulence and improved laminar flow compared to designs without an offset.
3) Experiments were conducted using various fluids to empirically validate the theoretical model. Dimensional analysis provided insights into the sensitivity of design parameters.
Gecco 2011 - Effects of Topology on the diversity of spatially-structured evo...matteodefelice
This document summarizes research on the effect of topology on diversity in spatially-structured evolutionary algorithms (SSEAs). The researchers modeled SSEAs as spreading processes and investigated how network topology influences diversity. They found that lattice networks maintained more diversity than random networks, leading to finding more optima. Rewiring the lattice to have small-world properties showed how network structure can control the spreading of solutions and thus the algorithm's dynamics. This research was a first step toward understanding how to design network topologies to optimize problem-solving in SSEAs.
Narayan Shrestha [Step wise multi-criteria performance evaluation of rainfall...Narayan Shrestha
The first author of the paper is Mr. Shakti PC who is currently in Japan. This paper is published in Journal of Hydrology and Meteorology Nepal (Vol. 7, No. 1) on December 2010.
Conformal Anisotropic Mechanics And Horavas Particleguest9fa195
The document presents a model of classical mechanics that implements anisotropic scaling transformations between space and time consistent with the dispersion relation of Hořava gravity. The resulting action is a generalization of conformal mechanics and reduces to known cases in certain limits. Thermodynamic properties of a gas of free particles under the anisotropic scaling are also derived, matching results from recent literature on anisotropic black branes.
This document discusses smoothed particle hydrodynamics (SPH), a Lagrangian numerical method for simulating fluid flows. SPH uses discrete particles that move with the fluid and sample hydrodynamic properties by averaging values over neighboring particles. The document outlines the basic concepts of SPH, including how it approximates continuous fields with discrete particles and calculates spatial derivatives. It also discusses how SPH can be used to model astrophysical phenomena through numerical simulations since laboratory experiments are often impossible at astrophysical scales.
Study on Reconstruction Accuracy using shapiness index of morphological trans...ijcseit
Basin, lakes, and pore-grain space are important geophysical shapes, which can fit with the several
classical and fractal binary shapes, are processed by employing morphological transformations, and
methods. The decomposition of skeleton network (minimum morphological information) using various
classical structures like square, octagon and rhombus. Then derive the dilated subsets respective degree by
the structures for reconstruct the original image. Through shapiness index of pattern spectrum procedure,
we try test the reconstruction accuracy in a quantitative manner. It gives some general procedure to
characterise the shape-size complexity of surface water body. The reconstruction accuracy is against the
size of water bodies with which we produce the some example of different shapiness index for different
structuring element of shapes. In which quantitative manner approach yields better reconstruction level.
The complexity of water bodies are compared with the surfaces.
Current Research on Quantum Algorithms.pptDefiantTones
This document summarizes research on quantum algorithms being conducted at the Institute for Quantum Information and Matter (IQIM). The research objectives include developing improved methods for fault-tolerant quantum computation, new quantum algorithms beyond the hidden subgroup problem, and simulation methods for quantum many-body systems and local quantum systems. Recent progress includes developing quantum algorithms for simulating particle collisions in fermionic quantum field theories and optimal algorithms for preparing topological quantum error correcting codes. Future work will focus on showing problems in quantum field theory are hard, characterizing logical operations in topological codes, and improving bounds on quantum memory times.
FUZZY LOGIC Control of CONTINUOUS STIRRED TANK REACTOR ProfDrDuraidAhmed
This document describes the use of fuzzy logic control for a continuous stirred tank reactor (CSTR). It begins with an abstract that summarizes modeling the CSTR system using mass and energy balances, and designing a fuzzy logic controller to control the reactor temperature. It then provides more details on mathematical modeling of the CSTR, the basic operations of fuzzy set theory, and the design of the fuzzy logic controller. The controller design involves choosing membership functions to classify the error signal and change in error, then developing fuzzy rules relating the error terms to the control output. Simulation results showed the fuzzy logic controller provided better tracking and regulation than a traditional PID controller.
Conformal Anisotropic Mechanics And The HořAva Dispersion Relationvcuesta
This document summarizes a paper that implements scale anisotropic transformations between space and time in classical mechanics. This results in a system consistent with the dispersion relation proposed in Horava gravity. The paper constructs an action principle for a particle that is invariant under anisotropic scaling transformations between space and time. This action reduces to known systems like conformal mechanics in certain limits of the dynamical exponent z. The paper also analyzes the canonical formalism, equations of motion, symmetries and thermodynamic properties of this anisotropic mechanical system.
This presentation was provided by Rebecca Benner, Ph.D., of the American Society of Anesthesiologists, for the second session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session Two: 'Expanding Pathways to Publishing Careers,' was held June 13, 2024.
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
How Barcodes Can Be Leveraged Within Odoo 17Celine George
In this presentation, we will explore how barcodes can be leveraged within Odoo 17 to streamline our manufacturing processes. We will cover the configuration steps, how to utilize barcodes in different manufacturing scenarios, and the overall benefits of implementing this technology.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
1. Chemical Engineering Science 56 (2001) 659}666
Multilevel modelling of heterogeneous catalytic reactors
B. G. Lakatos*
& &
Department of Process Engineering, University of Veszprem, H-8200 Veszprem, Hungary
Abstract
The paper presents the fundamental elements of the multilevel mathematical models of heterogeneous catalytic reactors. These
models are derived by means of the volume-averaging method, appropriately modi"ed for the dispersed systems, and provide
a theoretical basis for modelling heterogeneous systems across scales. The properties of the models are shown and analysed.
A three-level model of a well-stirred catalytic reactor with bidispersed catalysts is presented. The model equations are solved by means
of multilevel collocation. The transient behaviour of the reactor is investigated by means of computer simulation. 2001 Elsevier
Science Ltd. All rights reserved.
Keywords: Multilevel model; Volume averaging; Well-stirred catalytic reactor; Bidispersed catalyst; Multilevel collocation
1. Introduction hierarchy of the system we can reformulate and general-
ise these models about the so-called multilevel dynamical
In reaction engineering, pseudo-homogeneous models models, providing in this way a suitable theoretical basis
are often used for modelling heterogeneous catalytic re- for modelling and computing the heterogeneous systems
actors, taking into consideration the e!ects of intrapellet across scales.
processes only by means of e!ectiveness factors. When, In the paper, some fundamental aspects of the multi-
however, the time scales of processes inside the catalyst level dynamical models are presented. The hierarchy of
pellets and those of the continuous #uid phase are of the modelling levels is derived and analysed. A three-level
similar order of magnitude then this approach can lead model is derived for well-stirred catalytic reactors with
to signi"cant errors, especially under dynamic condi- bidispersed catalysts and some transient phenomena are
tions, as was revealed by Ramkrishna and Arce (1989). studied by simulation.
The two-phase heterogeneous models represent a more
rational basis for modelling heterogeneous systems since
these models handle all processes occurring simulta- 2. Multilevel dynamical models
neously in the #uid phase and inside the catalysts. These
models were used mainly for kinetic studies (Dogu 2.1. Volume averaging for dispersed phase
& Smith, 1975; Datar, Kulkarni & Doraiswamy, 1987;
Burghardt, Rogut & Gotkowska, 1988; Keil, 1996; Mol- K Consider the two-phase heterogeneous systems shown
ler & O'Connor, 1996) and parameters estimation schematically in Fig. 1 in which the interfacial surfaces
(Imison & Rice, 1975; Do & Rice, 1982; Kim & Chang, between the phases are complex and may vary in time.
1994), while Arce and Ramkrishna (1991), Trinh and Let be a scalar quantity which in the - and -phases is
Ramkrishna (1994) and recently Vernikovskaya, Zagor- denoted, respectively, by and . Then the variation of
? @
nikov, Chumakova and Noskov (1999) applied them for inside the phases is described by the balance equations
studying dynamic phenomena. In all cases, the hetero- *
geneous models were determined on phenomenological G #
( j )" , i" , , (1)
G *t G G
basis. However, taking into consideration the discrete
nature of the catalyst phase and the characteristic where j is the #ow density and is the volumetric source
G G
density of quantity . The transport across the -
G
interface is described by the boundary conditions
* Correspondence address: Department of Process Engineering, Uni-
versity of Veszprem, P.O. Box 158, H-8201, Veszprem, Hungary. n (j ! w)#n ( j ! w)" , (2)
? ? ? ? @ @ @ @ ?@
0009-2509/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 2 7 3 - 6
2. 660 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666
volume < for the -phase that
?
< "constant ¸ (5)
? ?
for which the conditions
;¸ ; , (6)
? ? ?
J (7)
@ ?
are valid then the single-level mathematical model can be
converted into a two-level mathematical model by means
of the modied volume averaging technique.
In this case, the phase average of quantity in the
-phase is dened in the usual way (Whitaker, 1967;
Slattery, 1967; Gray, 1975)
1 1
1 2 d d, (8)
? ? ?
Fig. 1. Two-phase systems with di!erent -phase scales. 4 4?
where # and is the characteristic function
? @ ?
of the set of points of the -phase, but the phase average
of quantity in the -phase takes the form
where w is the velocity of the interface, while denotes
?@ 4@
the surface source density of quantity . 1 2 1 2 n d , (9)
G @ @ . @ @
The set of equations (1) and boundary conditions (2)
form a microscopic mathematical model of the two- where 1.2 denotes the average of over a particle:
phase systems under investigation. In fact, this is .
a single-level model since all variables are represented on 1
1 2 d. (10)
the same, microscopic, level. However, such mathemat- @ . @
@ 4@
ical description is very complicated and impracticable. In Eq. (9) n is the population density function of the
One procedure for working around this problem is the elements of the dispersed -phase, termed next particles,
use of local averaging method (Whitaker, 1967; Slattery, dened as n : R;R;RPR by means of which
1967; Gray, 1975) by means of which Eqs. (1) are aver-
n(x, t, ) d expresses the number of particles having
aged over a representative local volume to obtain @ @
volume ( , #d ) at moment of time t in the aver-
a simplied, well-structured, but usually not closed set of @ @ @
aging volume associated with coordinate x.
averaged equations. Applying some closure scheme re- A second average that is important in this method is
sults in an averaged formulation of the dynamic behav- the intrinsic phase average
iour of multiphase systems. This formulation is also
a single-level mathematical model since it is expressed 1
1 2@ 1 2, (11)
purely in terms of the averaged, macroscopic, quantities. @ @
@
This procedure, however, leads to meaningful results
where is the volume fraction of the -phase
only if (1) the characteristic lengths and of both @
phases are of similar order of magnitude, where is the 4@K
n d (12)
distance over which varies signicantly and is the @ @ @
characteristic length of the phases; (2) there exists such
averaging volume while the partial volume of the -phase is given as
constant ¸ (3) 4@K
n d . (13)
@ @ @
that the three characteristic lengths satisfy the inequali-
ties On the basis of Eqs. (9)}(13), the volume-averaging
theorem and the general transport theorem can also be
;¸; . (4)
written in terms of the population density function of the
These conditions may be satised for systems in Fig. 1a -phase.
and b, but not for system shown in Fig. 1c in which the
-phase is, in essence, a dispersed phase. Accordingly, 2.2. Two-level model for heterogeneous systems
next the -phase will be treated as a continuous one.
Here, the characteristic lengths of the phases may di!er Applying the volume-averaging operator (8) to
signicantly, but if it is possible to dene such averaging Eqs. (1), making use of the appropriate forms of the
3. B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 661
volume-averaging theorem and of the general transport
theorem, and considering the boundary conditions (2),
for the system in Fig. 1c, the following set of equations
are obtained.
For the continuous -phase we obtain one equation:
* 1 2?
1 2? ? ? #1 2?
( 1 2?1 2?)
? *t ? ? ? ?
!
( 1D 2?
1 2?)! 1 2?
? ? ? ? ?
4@L d
! 1 2 @ n d
@ . dt @
4@L Fig. 2. Splitting a single modelling level into two levels of a two-phase
# n j n dA d
@ @ @ system.
@ 4@
4@L
! n n dA d . (14)
?@ @ @ The set of equations (14)}(17) is termed two-level
@ 4@
model, since the dependent variables and system para-
The left-hand-side terms of Eq. (14) describe the motion meters that are involved in it can be divided into two
of quantity in the continuous phase, while the right- equivalent classes, termed here modelling levels. Vari-
hand-side terms describe the changes of due to the ables 1 2? and n, for instance, are dened as variables of
variation of volume of the particles, and to the transfer of ?
the macroscopic level, while remained a variable of
through the surfaces of particles, and production of @
the microscopic level. The former is termed higher mod-
due to some surface source, respectively. On the left- elling level, whilst the latter is the lower modelling level.
hand side, the conductive term was closed by means of Therefore, a single-level model was converted into
the usual dispersion-type closure model (Carbonell a two-level one as it is shown in Fig. 2. The particles on
Whitaker, 1983; Quintard Whitaker, 1993). the higher modelling level are represented as point sinks
For the dispersed -phase we obtain two equations. immersed and moving in the continuous phase, while on
The population balance equation the lower level are described as material bodies of nite
*n * d volume which have their own inner world, often with
#
(1 2 n)# @ n 1 2 n (15) complex processes.
*t @ . * dt @ .
@ In the two-level model (14)}(17), a number of processes
describes the behaviour of particles immersed and mov- occurring on two di!erent scales is collected and related.
ing in the continuous phase, while the equation For instance, it describes three-dimensional motions in
the physical space on two scales, particles are assumed to
*
@ #
( #q ) (16) be of general form, or both surface and volumetric source
@ *t @ @ @ @ @ terms are present. This model, being in fact cognition-
oriented one, provides a framework for modelling hetero-
describes the motion of quantity inside the particles.
geneous systems across scales, as well as for their multi-
Here, q denotes some non-convective component of
@ level computations. A further advantage of this model is
the #ow density that may be of complex nature, depend-
that during the course of developing it reliable estimates
ing on the structure of particles. Eqs. (14)}(16) are com-
can be obtained for the model errors, so that purpose-
pleted with the constraint for the volume fractions of
oriented models, useful for engineering computations,
phases
can be derived by its successive reduction allowing at the
4@L same time exact error analysis.
# n d 1 (17)
? @ @
2.3. Generalisation: multilevel dynamical models
as well with the appropriate boundary and initial condi-
tions. The boundary conditions for Eq. (14) describe the Generalisation of the two-level models leads to the
connection of the system with the environment while the concept of multilevel dynamical models. In fact, when the
boundary conditions for Eq. (16) describe the connection particles are of complex nature, composed, from instance,
between the interior of a particle and its continuous from a great number of particles of much smaller size,
phase environment. The details of the derivation of Eqs. then the volume averaging process on this second, small-
(14)}(17), as well as of the derivation of equations of er, scale may also lead to meaningful and much simpler
motion of the system will be published elsewhere. models. In this way a three-level model can be dened.
4. 662 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666
We note here that it is often useful to describe even the
continuous -phase by means of a two-level model, vis-
ualising it as consisting of a large number of #uid ele-
ments that simultaneously interact with each other and
with the particles of the dispersed -phase. Such model
was derived and presented recently by Lakatos, Ulbert
and Blickle (1999) for describing micromixing in continu-
ous crystallizers.
3. Three-level models of catalytic reactors with bidisper-
sed catalysts
Consider a continuous catalytic reactor in which an
APProduct-type reaction occurs and both the #uid and
Fig. 3. Coupling the modelling levels in a three-level mathematical solid phases can be considered perfectly mixed. Let the
representation. catalyst (macro-) particles be agglomerates of small por-
ous (micro-) particles what means that the catalyst is of
biporous nature. Let us assume that there exists some
Here, the modelling levels are coupled with each other as resistance to the mass transfer at the #uid phase}macro-
it is shown schematically in Fig. 3. particles interface, adsorption of the reagent A occurs at
Based on the extension of a two-level model into the the surface of the microparticles with Freundlich-iso-
three-level representation, a denition of the n-level mod- therm, and the rst-order reaction occurs only in the
els can be formulated as follows: microparticles.
If the variables x , x ,2 and equations F , F ,2 Considering the system in terms of the multilevel mod-
with the corresponding boundary conditions B , B ,2 els, here the rst modelling level is formed by the com-
of a distributed parameter model can be divided into pletely stirred #uid-phase. The second-level model
n equivalent classes that can be grouped into a hierarchy describes the di!usion-type motion in the macropores,
in such a way that and the third modelling level is in fact the mathematical
M :F (x , f [x ]), B (u), description of the di!usion and reaction in the micro-
pores. Denoting the third phase as -phase, the rst ( )
M :F (x , f [x ]), B (x ), level is described by the averaged equations (14) and (15),
M :F (x , f [x ]), B (x ), the second ( ) level is also described by similar averaged
equations obtained by replacing the indices by and
$ by , and the third level is described by Eq. (16)
M :F (x ), B (x ), (18) replacing the index by . Under such conditions, the
L L L L L purpose-oriented model of the system, obtained through
then the model is called n-level (dynamical) model with a number of simplifying steps, and with the further
the notion assumptions:
x Yx Yx Y2Yx , E the reactor is isotherm;
L
E the macroparticles can be modelled as spheres, cylin-
M YM YM Y2YM , (19)
L ders or plates, while the microparticles can be
where Y denotes the relation higher level } lower level, modelled as spheres;
F are the equations of the ith level, f are functionals of E both the macro and microparticles are monodispersed
G G
their arguments transmitting the in#uence of the (i!1)th with constant characteristic sizes R and R , respec-
@ A
level to the ith level, while B are the boundary conditions tively;
G
transmitting the in#uence of the ith level to the (i!1)th E the catalytic activity is constant,
level. consists of the following equations and boundary
The multilevel dynamical models are characterised by conditions:
a number of special properties because of the character-
istic hierarchy of the levels expressed by Eqs. (18). For Fluid-phase * rst ( ) level:
instance, they exhibit some memory features and a char-
dc m(1! ) D *c
acteristic structure of eigenvalues due to the e!ects the F: ? q(c !c )! ? @ @ @ .
? ? dt ?GL ? R *r @ @
lower levels, often leading to complex dynamic phe- @ @ P 0
nomena. (20)
5. B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 663
Macropores * second ( ) level: and dimensionless parameters
*c D * *c D m(1! )
@ @ @ rK @ p ? @, p ? @p ,
@ *t rK *r @ *r qR
@ @ @ @ ?
F : (21)
@ 3(1! ) D R c
3(1! ) D *c p @ A $ A @, p @K p ,
! @ A A A , c D R 3(1! )
R *r A A @K @ A @ $
A A P 0
k R kR
subject to the boundary conditions p P @ , p R @ , p cI .
D D @K
A @ A @
*c *c k
B : @ 0, @ R [c !c @ @ ]. Eqs. (20)}(25) are written in dimensionless form as
@ *r @ @ *r @ @ D ? @P 0
@ P 0 @ P 0
d *
(22) p ? ! !p @ , (28)
d ?GL ? *
Micropores * third ( ) level: E
* 1 * * *
*c D * *c @ @ !p A , (29)
F: A A A r A !k c (23) * * * *
A A *t r *r A *r P A K
A A A
* p *
subject to the boundary conditions A A !p (30)
* * A
*c
B: A 0, c A A a cI . (24)
A *r A H A P 0 $ @ subject to the boundary conditions
H P 0
In general, the initial conditions are * *
@ 0, @ p [ ! ], (31)
* * ? @ E
c (0)c , c (r ,0)c (r ), E E
? ? @ @ @ @
* *
c (r , r ,0)c (r , r ).
A A @ A A @
(25) A 0, A p I (32)
* * @
Naturally, the rst-level boundary conditions are in- K K
cluded into the equation F since it describes a com- and to the initial conditions
?
pletely stirred #uid}solid suspension. Equations F and
? (0) , ( ,0) ( ), ( , ,0) ( , ). (33)
F have been simplied signicantly because of the
@ ? ? @ @ A A
monodisperse nature of particles, since in this case the The problem (28)}(33) was solved numerically by de-
population density functions have the delta function veloping a two-level orthogonal collocation scheme with
forms n N (r !R ), i , . By the same reasons,
G G G G the aid of which the partial di!erential equations might
the functionals f and f transmitting the e!ects of the
? @ be converted into a system of ordinary di!erential ones.
and levels to the and levels, respectively, have According to the method of orthogonal collocation (Vil-
taken the forms ladsen Stewart, 1967), approximate solutions were
*c (t) constructed for and in terms of orthogonal poly-
@ c (r , t) (r !R ) dr @ A
*r @ @ @ @ @ @ @
(26) nomials of degree N and M in and , respectively,
@ P 0 written in the forms
and
,
I ( , ) d ( ) L
*c (r , t) @ L
(34)
A @ c (r , r , t) (r !R ) dr . (27)
*r A @ A A A A L
A PA 0A and
+ ,
4. Solution to the model equations I ( , , ) g ( ) L K, (35)
A KL
K L
Before solving numerically the problem (20)}(25) we where the coe$cients d , n1, 2,2, N#1 and
rewrite it in dimensionless form by introducing the di- L
g , m1, 2,2, M#1, n1, 2,2, N#1 are un-
mensionless variables KL
known functions of the dimensionless time . Note that
r r D polynomials (32) and (33) satisfy the boundary conditions
@ , A , t @ , at the values 0 and 0 due to their appropriate
R R R
@ A @ forms.
c The N and M interior collocation points were selected
A , c c
A ?K K to be the roots of the shifted Legendre polynomials of
$
6. 664 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666
order N and M, dened as
r¸ Q (r)¸ Q (r) dr , n, m0,1, 2,2, (36)
L K LK
where is the Kronecker delta. The (N#1)-th and
LK
(M#1)-th collocation points were selected 1
,
and 1.
+
Substituting functions (34) and (35) into Eqs. (29) and
(30), respectively, and requiring those to be satised at all
collocation points, as well as taking into consideration
the boundary conditions (31) and (32), together with the
Eq. (28), a set of (N#1) (M#1) ordinary di!erential
equations was obtained.
The resulting set of ODE was solved by means of the
semi-implicit Runge}Kutta method (Villadsen Michel-
sen, 1978). The program was written in C/C# imple-#, Fig. 4. E!ects of the shape of macroparticles and of the rate of the
mented on PC, and was used for investigating di!erent #uid-particle mass transfer on the transient responses of the reactor.
transient phenomena of the catalytic reactors with bidis-
persed catalysts by computer simulation. In the simula-
tions, the following base-case values of parameters were
used: p 1.65, p 0.2 m, p 2, p 1.2, p 15,
p R, p 1, k0.5.
5. Simulation results and discussion
First we consider the system without reaction, i.e. the
vessel is assumed to be operated as an adsorber with
bidispersed adsorbents. Here the transient responses of
the system to the step changes of the adsorbate were
studied.
The numerical experiences have shown that in the
interval 10(¹ /¹ D R/D R(10 of the ratio
@ A A @ @ A Fig. 5. Transient processes in the #uid, in the macroparticles and in the
of the time scales of macro- and microparticles, the tran-
microparticles.
sient behaviour of the adsorber was a!ected either by the
macro- and micropore mass transport. At larger values of
this ratio the micropore transport, while for smaller
tration proles in the macro- and microparticles are
values the macropore transport was the dominant factor
shown for two moments of dimensionless time .
a!ecting the transient responses.
Consider now the system with chemical reaction, i.e.
Fig. 4 shows the in#uence of the shape of macropar-
the vessel is assumed to be operated as a catalytic reactor
ticles on the relative uptake given as
with bidispersed catalysts, but with di!erent boundary
O /F@ EQ ds conditions. Namely, we assume that the reagent is #ow-
u( ) /E E , (37) ing through the vessel without any catalyst under sta-
/F@ EQ ds
/E E tionary conditions, when suddenly a given amount of
for two di!erent values of parameter p expressing the empty catalyst is introduced into the vessel. If we can
in#uence of the rate of mass transfer between the #uid assume that the disturbance of the hydrodynamics of the
phase and macroparticles. In the case of negligible mass vessel is negligible, then the initial conditions of the
transfer resistance, only small in#uence of the shape of problem (28)}(32) can be written in the form
macroparticles is observed while for small rate of mass
(0) , ( ,0)0, ( , ,0)0. (38)
transfer the macroparticle shape becomes an important ? ? @ A
factor of the transients. Numerical experiences have shown that minima might be
The whole transient process occurring simultaneously detected in the #uid-phase concentration of the reactor in
at the three modelling levels of the system, computed for a wide range of parameters, similarly to that observed for
the empty vessel initial conditions and step change of the monoporous catalysts (Imison Rice, 1975; Do Rice,
adsorbate at the inlet, is presented in Fig. 5. The concen- 1982). The place of minima on the time axis and their
7. B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666 665
6. Conclusions
The multilevel mathematical models, presented in
the paper in general terms, were derived by means
of the volume-averaging method, appropriately modied
for the dispersed systems. It might be applied for de-
scribing the heterogeneous catalytic reactors because
of the discrete nature of the catalyst phase and of the
corresponding hierarchy of the system. The multilevel
dynamical models, in their original cognition-oriented
form provide a suitable theoretical basis for modelling
and computing the heterogeneous systems across
scales.
A three-level dynamical model of a well-stirred cata-
Fig. 6. The in#uence of the voidage of the reactor on the minima of
#uid concentration. lytic reactor with bidispersed catalysts was presented and
analysed in terms of the multilevel models. The model
equations were solved by means of a two-level ortho-
gonal collocation scheme. This method has proved suit-
able for computing the concentration distributions in
both the macropores and micropores in transient states
of the reactor. By means of computer simulation several
transient phenomena have been considered, showing the
in#uence of the shape of macroparticles, of the mass
transfer resistance at the #uid}macroparticle interface,
and of the voidage of the reactor on the transient behav-
iour of the reactor.
Notation
a parameter of the Freundlich isotherm
$
Fig. 7. Macropore and micropore concentration distributions at di- A surface, m
mensionless time 10. C concentration, kg m
d coe$cients of the polynomial (Eq. (34))
L
D di!usion coe$cient, m s
values are strongly a!ected by the relative amount of the D matrix of the di!usion coe$cients
catalyst as it is seen in Fig. 6. This gure shows the g coe$cients of the polynomial (Eq. (35))
KL
history of the #uid concentration for di!erent values of j #ow density, quantity m s
the reactor voidage . The capacity of catalyst increases k exponent of Freundlich-isotherm
?
with decreasing voidage and the minima arise at smaller k reaction rate coe$cient, s
P
moments of time. Fig. 7 presents the concentration distri- k mass transfer rate coe$cient, s
R
butions of the macropores and micropores for two di!er- ¸Q shifted Legendre-polynomial of order m
K
ent ratios of the time scales of macro- and microparticles ¸ characteristic length scale of averaging, m
at the dimensionless time 10. This gure illustrates n population density function, no m
well that for ¹ /¹ +10 the micropores are lled with n normal unit vector
@ A
reactant continuously as compared to the macropore p dimensionless parameter
H
concentration distribution, while in the case of q volumetric #ow rate, m s
¹ /¹ +10, this concentration is practically zero q #ow density, quantity m s
@ A
since the reactant di!used into the micropores is com- r radial coordinate, m
pletely consumed by the reaction. R radius, m
The minima presented in the reactor responses arise t time, s
because of the memory e!ects of the lower levels repre- u relative uptake
sented by the macro- and microparticles. The di!erences ¹ time scale of di!usion, s
in the #uid concentration seem to be characteristic for v velocity, m s
applying it into some preliminary estimation of the sys- volume, averaging volume, m
tem parameters. w velocity of the -interface, m s
8. 666 B. G. Lakatos / Chemical Engineering Science 56 (2001) 659}666
Greek letters Datar, A., Kulkarni, B. D., Doraiswamy, L. K. (1987). E!ectiveness
factors in bidispersed catalysts: The e!ect of di!usivity variations.
volume fraction Chemical Engineering Science, 42, 1233}1238.
Do, D. D., Rice, R. G. (1982). The transient response of CSTR
dimensionless macropore radius containing porous catalyst pellets. Chemical Engineering Science, 37,
dimensionless concentration 1471}1475.
characteristic microscopic length scale, m Dogu, G., Smith, J. M. (1975). A dynamic method for catalyst
characteristic macroscopic length scale, m di!usivities. AIChE Journal, 21, 58}69.
dimensionless micropore radius Gray, W. G. (1975). A derivation of the equations for multiphase
transport. Chemical Engineering Science, 30, 229}233.
volume source density Imison, B. W., Rice, R. G. (1975). The transient response of CSTR to
density, kg m a spherical particle with mass transfer resistance. Chemical Engin-
surface source density, quantity m s eering Science, 30, 1421}1423.
intensive (scalar) quantity Keil, F. J. (1996). Modelling of phenomena within catalyst particles.
characteristic function of a set Chemical Engineering Science, 51, 1543}1568.
Kim, D. H., Chang, K. S. (1994). Time-domain solutions of un-
steady-state mass transfer models for biporous catalyst in a continu-
Indices ous-#ow gradientless reactor. Chemical Engineering Science, 49,
2173}2185.
m maximal value Lakatos, B. G., Ulbert, Zs., Blickle, T. (1999). Population bal-
-phase, #uid phase ance model for micromixing in continuous crystallizers. Pro-
ceedings of the 14th Symposium on Industrial Crystallization,
-phase, macropore Cambridge.
-phase, micropore Moller, K. P., O'Connor, C. T. (1996). The measurement of di!usion
K
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1.2 volume-averaging operator 3403}3408.
scalar product Quintard, M., Whitaker, S. (1993). Transport in ordered and dis-
ordered porous media: volume-averaged equations, closure prob-
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Slattery, J. C. (1967). Flow of viscoelastic #uids through porous media.
The author would like to thank the Hungarian Re- AIChE Journal, 13, 1066}1071.
search Foundation and the Ministry of Education of Trinh, S., Ramkrishna, D. (1994). Feasibility of pattern formation
Hungary for supporting this work under Grants in catalytic reactors. Chemical Engineering Science, 49,
T 023183 and FKFP 0520/1999, respectively. 1585}1599.
Vernikovskaya, N. V., Zagornikov, A. N., Chumakova, N. A.,
Noskov, A. S. (1999). Mathematical modeling of unsteady-state
operation taking into account adsorption and chemisorption pro-
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