Solving partial differential equations of gas–solid reactions by orthogonal collocation

In this work the solution of the coupled partial differential equations for noncatalytic gas–solid reactions has been considered by orthogonal collocation. First of all, by an integral transformation and then by applying the orthogonal collocation method, these partial differential equations are converted to the ordinary differential equations. Then the equations are solved and the conversion–time profiles are obtained. The solution of the equations for volume reaction model, grain model and grain model with product layer resistance, modified grain model, random pore model, nucleation model and reaction of two gas with one solid has been presented in this work. The orthogonal collocation is a rapid method for solving of these equations and shows a good accuracy with respect to other solution techniques in the literature.     

A study on deviation of noncatalytic gas-solid reaction models due to heat effects and changing of solid structure

In this study a common type noncatalytic gas-solid reaction is modeled based on some well-known, previously presented mathematical models, including grain, modified grain and additive reaction times models. In order to approach more realistic models, the heat effects and the changing of solid structure effects are considered in the above named mathematical models. The governing equations are developed and solved numerically. Then, the predicted results are compared with available experimental data presented for some important industrial gas-solid reactions. The results reveal shortage of the simplifying assumptions of the referred models to predict solid conversion, as a result of neglecting heat effects and structural changes of solid reactant. In this study, for the first time, the process of the change in the different reaction controlling steps is considered during the reaction time. The results also show that the main rate-limiting resistances convert to each other during the reaction progress. It reveals that the undesirable heat and structural changing effects decrease with decreasing the particle diameter, increasing the convective heat transfer coefficient, and taking appropriate gas temperature. This study shows that considering heat effects and changing of solid structure improve the abilities of previous mathematical models to predict the behavior of noncatalytic gas-solid reactions.     

EFFECTS OF PARTICLE CHARACTERISTICS ON THE PERFORMANCE OF A FAST FLUIDIZED BED REACTOR

A heterogeneous model for the fast fluidized bed reactor which carries out a gas-solid non catalytic reaction is presented. The hydrodynamics of the fast fluidized bed is characterized by the model of Kwauk et al. (1985) which assumes the existence of two phases; a dense phase and a dilute pneumatic transport phase. For a given solid flowrate, the length of the reactor occupied by each phase depends on gas velocity, particle diameter and density and average voidage within the reactor. The gas-solid reaction is assumed to follow the shrinking core model. The solids are assumed to be completely backmixed in the dense phase and move in plug How in the dilute pneumatic transport phase. The gas phase is assumed to be in plug flow in both phases

For given gas and solid flowrates, the transition from the dense phase flow to the fast fluidized bed (containing two regions) as functions of particle size and density is determined using the model of Kwauk et al. (1985). The numerical solution of the governing mass balance equations show that for given solid and gas flowrates, (and average voidage) the gas phase conversion shows an unusual behavior with respect to particle diameter and density. Such behavior is resulted from the effects of particle diameter and density on the reactor volume occupied by each phase and the effect of particle diameter on the apparent reaction rate. The numerical results show that a fast fluidized bed gives the best conversion at large particle density and for the particle diameter which results the fast fluidized bed to be operated near the pure dense phase flow.     

Reactions of solid particles with nonuinform distribution of solid reactant: The volume reaction model

The effect of non-uniform solid reactant distribution on conversion of solid particles in gas-solid reactions is analyzed based on the volume reaction model. Certain special features of such systems are pointed out. The possibility of ash layer formation in the kinetically controlled regime is discussed. Conditions leading to single or double ash layer formation, both at the center and surface of the particle, in the intermediate regime of diffusion with simultaneous reaction are described. Detailed mathematical equations which are useful for calculation of the conversion-time relationship for particles with non-uniform solid reactant distribution are presented. Comparison is made to reaction of uniform particles and differences in required reaction time for desired conversion are outlined.     

Effectiveness factors for solid-catalyzed gas-solid reactions

Effectiveness factors are given for gas-solid reactions catalyzed by a solid catalyst. Since the gasification of the solid reactant takes place at the catalyst-solid reactant interface, the gaseous reactant has to diffuse to the interface. As conversion proceeds, the thickness of the catalyst layer increases causing the effectiveness factor to decrease. Effectiveness factors obtained for the surface reaction show heavier dependence on the order of reaction and the Thiele modulus when compared with the effectiveness factors for the solid-catalyzed gas reactions. For the general case of gaseous reactant consumed by both surface and volume reaction, two Thiele moduli are required and the effectiveness factor decreases exponentially with increasing conversion. The effectiveness factor calculated from literature data on catalytic hydrogasification of coal is given to illustrate this dependence on conversion.     

A new mathematical solution for predicting char activation reactions

The differential conservation equations that describe typical gas-solid reactions, such as activation of coal chars, yield a set of coupled second-order partial differential equations. The solution of these coupled equations by exact analytical methods is impossible. In addition, an approximate or exact solution only provides predictions for either reaction- or diffusion-controlling cases. A new mathematical solution, the quantize method (QM), was applied to predict the gasification rates of coal char when both chemical reaction and diffusion through the porous char are present. Carbon conversion rates predicted by the QM were in closer agreement with the experimental data than those predicted by the random pore model and the simple particle model.     

MODELLING GAS-SOLID FLOW IN A PNEUMATIC-FLASH DRYER

Investigations of aerodynamics of gas-solid flow in a pneumatic-flash dryer in semiindustrial scale have been carried out. Apparatus was composed of three elements with varying cross-sectional area connected together, i.e. expanding cone, decreasing cone and a vertical pipe with constant diameter. A mathematical model of the dryer is based on the continuity equations for both gas and solid phase and on differential equations for momentum balance of the gas-solid mixture and momentum balance of the solid phase. The model has been solved by means of Gear's numerical method. The effect of various empirical correlations for the solid-wall friction factor has been shown. Distributions, resulting from the model, for pressure, gas velocity, panicle velocity, voidage and residence time of panicle along the axis of apparatus have been presented. The results of numerical calculations have been verified on the basis of measurements in pan.     

A correlation to the solution of the two-phase dispersion model

A correlation to the solution of the two-phase dispersion model has been developed for gas-solid fluidized bed reactors operating in the bubbling regime. An analytical solution was obtained for fractional gas conversion by using an exponential function to characterize the dense phase gas concentration profile. The coefficient of the exponential function was found to depend on gas axial dispersion and, in order to determine this parameter, a Peclet number correlation was developed. Model predicted gas conversions were in excellent agreement with experimental conversions for a variety of fluidized bed reaction data over a conversion range from 2.5 to 99%.     

MODELLING GAS-SOLID FLOW IN A PNEUMATIC-FLASH DRYER

ABSTRACT

Investigations of aerodynamics of gas-solid flow in a pneumatic-flash dryer in semiindustrial scale have been carried out. Apparatus was composed of three elements with varying cross-sectional area connected together, i.e. expanding cone, decreasing cone and a vertical pipe with constant diameter. A mathematical model of the dryer is based on the continuity equations for both gas and solid phase and on differential equations for momentum balance of the gas-solid mixture and momentum balance of the solid phase. The model has been solved by means of Gear's numerical method. The effect of various empirical correlations for the solid-wall friction factor has been shown. Distributions, resulting from the model, for pressure, gas velocity, panicle velocity, voidage and residence time of panicle along the axis of apparatus have been presented. The results of numerical calculations have been verified on the basis of measurements in pan.     

Modeling of drag with the Eulerian approach and EMMS theory for heterogeneous dense gas-solid two-phase flow

Combined with the Eulerian approach, energy minimization multi-scale (EMMS) theory was used to develop a new theoretical model for the drag between the gas and solid phases in dense fluidized systems. The energy minimization was used in the solution procedure as an additional stability condition to close the conservation equations. The model was derived without introducing any empirical factors, so it can be used for more flow conditions in circulating fluidized beds (CFBs) than empirical models, especially for heterogeneous gas-solid two-phase flows that include cluster formation. Non-uniform particle distribution in computational cells, which is usually not described by the differential equations, is also considered in the new drag model. Both the drag values given by the model and simulation results for real systems agree well with experimental data. The results show that the model reasonably describes the interactions between the gas and particle phases in dense flows.     

Simulation of cyclic dynamics in isothermal gas-solid reaction systems effects of the frequency of sinusoidal perturbation in the bulk gas concentration

The dynamic responses of a gas-solid reaction system to a sinusoidal perturbation in the bulk gas concentration are analyzed. Effects of the frequency variations on the gas and solid concentration profiles and the solid conversion are comprehensively examined based on the volume reaction model under the isothermal condition. The results show that the solid conversion can be accelerated as much as 50% and retarded as much as 50% depending on the frequency and the amplitude of the perturbation. The relationships between the reactant concentrations at the surface and the center are elucidated by means of phase-plane plots.     

Analytical prediction of conversion-time behaviour of gas-solid noncatalytic reactions

A procedure has been suggested for analytical prediction of conversion-time behaviour of gas-solid non-catalytic reactions which are first order with respect to the gaseous reactant but of any general form with respect to the solid species. Based on this method analytical solutions have been presente for some of the commonly encountered rate forms including the grain model. These solutions predict the behaviour of the system extremely well upto abou 50–70% conversion of the solid.     

Noncatalytic gas-solid reactions in a vertical pneumatic transport reactor

A heterogeneous model is developed to account for noncatalytic gas-solid reactions in a vertical pneumatic transport reactor. The model takes into consideration both the positive and negative variations of the solid porosity and the variation of the gas diffusivity with the reaction. The method of lines utilizing the second order centered finite difference scheme for the spatial discretization is employed to obtain the model solution. Experiments utilizing a vertical pneumatic transport reactor of a laboratory scale are performed to study the reaction between limestone and sulfur dioxide generated from coal combustion. The reactor is of 16.2 cm ID and 610 cm in length. The experimental data for sulfur retention are reported for various superficial gas velocities and calcium-sulfur molar ratios. Verification of the model with experimental data is conducted. The agreement between the model prediction and experimental data is satisfactory.     

基于变尺度格子气方法的气固相间曳力模型

基于变尺度格子气方法,建立了气固相间曳力模型,对气固两相之间的耦合机制进行了研究,从微观角度出发,得到了气固两相之间相互作用的宏观行为。文章用此模型对气固两相流系统进行了模拟,模拟结果与四种经典曳力模型的模拟结果一致,验证了模型的正确性和有效性。     

CFD analysis of hydrodynamic, heat transfer and reaction of three phase riser reactor

Catalytic cracking reaction and vaporization of gas oil droplets have significant effects on the gas solid mixture hydrodynamic and heat transfer phenomena in a fluid catalytic cracking (FCC) riser reactor. A three-dimensional computational fluid dynamic (CFD) model of the reactor has been developed considering three phase hydrodynamics, cracking reactions, heat and mass transfer as well as evaporation of the feed droplets into a gas solid flow. A hybrid Eulerian-Lagrangian method was applied to numerically simulate the vaporization of gas oil droplets and catalytic reactions in the gas-solid fluidized bed. The distributions of volume fraction of each phase, gas and catalyst velocities, gas and particle temperatures as well as gas oil vapor species were computed assuming six lump kinetic reactions in the gas phase. The developed model is capable of predicting coke formation and its effect on catalyst activity reduction. In this research, the catalyst deactivation coefficient was modeled as a function of catalyst particle residence time, in order to investigate the effects of catalyst deactivation on gas oil and gasoline concentrations along the reactor length. The simulation results showed that droplet vaporization and catalytic cracking reactions drastically impact riser hydrodynamics and heat transfer.     

液体与多孔固体颗粒间反应的模型研究

为探讨流休一多孔固体颗粒非催化反应模型中液固反应与气固反应的差别,本文用合成的一系列具有不同孔结构参数的多孔颗粒,考察了液固界面间的浸润现象(毛细现象)作为一种质量传递过程,对液体-多孔颗粒间的反应率～时间关系的影响,根据液体反应物在反应初始时刻进入颗粒的不同方式,定义了三类浸润条件,并对其中的两类提出了质量平衡方程和解析解.     

Gas-solid reaction model for a shrinking spherical particle with unreacted shrinking core

A heterogeneous reaction model is developed to represent the gas-solid reaction consisting of two-step reactions: formation of a solid intermediate on the core of unreacted solid and consumption of the solid intermediate. The presented model takes into account the shrinkage of both the unreacted core and the solid particle itself. The model successfully represents the fluorination of uranium dioxide where uranium hexafluoride gas is produced through uranyl fluoride as a solid intermediate. The rates of the shrinkage are discussed in detail for general cases by solving an ordinary differential equation for two radii of the particle and the unreacted core. Two parameters, the ratio of the reaction rates between the two steps and the non-dimensional diffusion rate of a reactant gas into the intermediate, control the rates of the shrinkage. Effective range of the model is specified in terms of the two parameters.     

PSEUDO TWO-DIMENSIONAL MODEL FOR A PNEUMATIC DRYER

Pneumatic drying of chemical products has been frequently used in chemical industries. The increase in the use of this unit operation requires the knowledge of the dynamic of the gas-solid flow in tubes. The mathematical models of vertical pneumatic conveying found in the literature mostly consider the flow steady and one dimensional. However, experimental evidences suggest that radial profiles of the basic variables of the flow exist. In this work a model is proposed for vertical pneumatic conveying considering axial and radial profiles for gas and solids velocities, porosity and pressure. The conservation equations for energy and mass of water were written to extend the model to a pneumatic dryer. The equations of the model were solved using finite difference method and the results show the axial and radial variations of gas and solid temperatures, gas humidity and particle moisture content in the dryer.     

PSEUDO TWO-DIMENSIONAL MODEL FOR A PNEUMATIC DRYER

ABSTRACT

Pneumatic drying of chemical products has been frequently used in chemical industries. The increase in the use of this unit operation requires the knowledge of the dynamic of the gas-solid flow in tubes. The mathematical models of vertical pneumatic conveying found in the literature mostly consider the flow steady and one dimensional. However, experimental evidences suggest that radial profiles of the basic variables of the flow exist. In this work a model is proposed for vertical pneumatic conveying considering axial and radial profiles for gas and solids velocities, porosity and pressure. The conservation equations for energy and mass of water were written to extend the model to a pneumatic dryer. The equations of the model were solved using finite difference method and the results show the axial and radial variations of gas and solid temperatures, gas humidity and particle moisture content in the dryer.