MULTIPLICITY AND STABILITY ANALYSIS OF AGGLOMERATION CONTROLLED PRECIPITATION

The possibility of multiplicity in continuous isothermal MSMPR precipitators has been explored for agglomeration controlled conditions and general criteria, independent of nucleation kinetics, are developed for stability and multiplicity of the steady states. For the Volmer model of primary nucleation and the magma dependent power law model of secondary nucleation, parameter regions are determined in which multiple steady states exist, and their linear stability is analyzed. The analysis holds in general for all types of agglomeration kernel. For the Volmer nucleation kinetics three steady states exist in the region of multiplicity with the “middle” one always being unstable. The analysis for magma dependent power law model showed multiplicity regions having as many as four steady states, the number depending on magma and kinetic order. Unlike the case of molecular growth control, limit cycle behaviour is not possible, and the approach to the steady state is always asymptotic     

等温CMSMPR结晶器定态及瞬态特性分析(Ⅰ)存在多定态及发生持续振荡的条件

运用分岔理论 ,对同时发生初级成核和二次成核的等温连续混合悬浮混合排料 (CMSMPR)结晶器的定态特性进行了分析 ,推导了系统存在多定态及发生持续振荡的参数条件 .以结晶过程的物性参数、动力学参数以及操作参数为基础 ,应用非线性方程组的延拓方法分别求解极限点条件方程组及Hopf分岔点条件方程组 ,确定了系统存在多定态及发生持续振荡的参数边界     

Multiplicity in Continuous Adiabatic MSMPR Reactive Precipitators

The possibility of both concentration and temperature multiplicities has bcen studied for the case of acontinuous adiabatic mixed suspension mixed product removal(MSMPR)reactive precipitaior.A Process in-volving homogeneous chemical reaction in first order reaction kinetics with respect to each of the reactive compo-nents and subsequent crystallization described by conventional power law growth and power law magma depen-dent nucleation models is considered.The temperature dependency of each of these kinetics is described by Ar-rhenius relations.Parameter regions are determined in which multiple steady states exist.The linear stability ofthese steady states is analyzed by using the Routh criterion approach.     

MULTIPLICITY IN CONTINUOUS CRYSTALLIZERS: ADIABATIC REACTIVE PRECIPITATION

The application of the theory of both concentration and temperature multiplicities is extended to the case of a reactive precipitation system in a continuous mixed suspension mixed product removal (MSMPR) crystallizer. A process involving elementary homogeneous chemical reaction with first order reaction kinetics with respect to each of the reactive components and subsequent crystallization described by conventional power law growth and power law magma dependent nucleation models is considered; the temperature dependency of each of these kinetics is described by Arrhenius relations. Exact uniqueness and multiplicity (i.e. multiple steady states) criteria are developed. The stability of these steady states is analysed using small perturbations around the steady state     

同时发生初次和二次成核且为聚结控制的沉淀过程多定态及稳定性分析

The possibility of multiplicity in an isothermal continuous mixed suspension-mixed product removal crystallizer is explored using the bifurcation theory.A process involving agglomeration controlled precipitation is considered in which secondary nucleation occurs simultaneously with primary nucleation.The determinant equations for the existence of multiple steady states are developed and the multiplicity boundaries dependent on the physical and kinetic properties and operational parameters of the process are obtained by resolving these determinant equations.The number of steady states in the precipitator for various multiplicity regions is determined and the linear stability of these steady states is analyzed by using the Routh criterion.     

等温CMSMPR结晶器定态及瞬态特性分析(Ⅱ)分岔行为及瞬态特性

应用非线性方程组的延拓解法 ,对同时发生初级成核和二次成核的等温连续混合悬浮混合排料(CMSMPR)结晶器的定态方程组进行了求解 ,考察了结晶过程的定态在不同参数区域之间的分岔行为 ,确定了各参数区域内结晶过程的定态数目 ,并应用Routh -Hurwitz准则对各定态的稳定性进行了分析 .采用四阶Runge -Kutta法求解CMSMPR结晶器的动态方程组 ,分析了在多定态区域以及持续振荡区域中结晶过程的瞬态特性     

Bifurcation analysis of two continuous bioreactors operated in series with recycle

In this paper, bifurcation analysis has been carried out for two continuous bioreactors operated in series with recycle from the second reactor. The existence of multiplicity of steady states is analyzed by considering Contois growth kinetics in the process model. It was observed that there exist two possible steady states of which one is trivial (wash out condition). Stability analysis is carried out to determine the stability of these steady states and it was observed that both these steady states are unstable in nature. Bifurcation analysis has been carried out for substrate and biomass concentration with dilution rate as the bifurcation parameter. Effect of recycle ratio, substrate separation factor and biomass separation factor is studied and analyzed. It was observed that Hopf bifurcation occurs at a dilution rate of 1.0208 with purely imaginary Eigen values which showed that sustained oscillatory behavior exists in the substrate concentration of the second reactor. The significance of different bifurcation points and the operating conditions by considering biomass and substrate concentrations in each reactor is studied and it was observed that the bioreactors need to be operated at intermediate dilution rates to obtain improved conversion and yield.     

MSMPR连续结晶器的多定态

对混合悬浮混合排料(MSMPR)连续结晶器的定态方程组进行了剖析,应用非线性方程组的延拓求解法,求得了定态方程组的分岔图,即MSMPR连续结晶器状态参数与操作参数的依赖关系.利用分岔理论对MSMPR连续结晶过程的定态数目及相应的参数范围进行了预测,导出了连续结晶过程存在多定态或唯一定态的判据。该判据仅与过程的动力学参数和操作参数有关,而与过程的状态参数无关。最后应用Routh-Hurwitz准则,确定了连续结晶过程各定态的稳定性。     

Modelling of chemical reactors-XXXII: Nonisothermal Nonadiabatic Tubular Reactor. One Dimensional Model-detailed Analysis

A detailed analysis of the region of multiple steady states in nonisothermal-nonadiabatic systems is performed. It is shown that in a certain region of Damköhler numbers five steady states are possible. These steady states are referred to as the kinetic, diffusion and intermediate regimes. Numerical procedures for calculation of regions of multiplicity are described. Numerical analysis of the problem indicates that with higher value of the cooling parameter β multiplicity is suppressed. For higher values of the Peclet number multiple steady states disappear. The analysis has shown that in nonadiabatic systems Frank-Kameneckij approximation of the temperature dependence of the reaction rate must be used with some care.     

A basic population balance model for fluid bed spray granulation

A basic population balance approach is developed for a granulation process in a fluid bed spray granulator. The particle size distribution predicted by the model is confirmed by plant data. Hence this model is considered to be useful to describe and optimize an industrial process. The model depends on a limited number of parameters (most of these factors can be measured or are known): the spray volume flux, the nucleation fraction (the fraction of the spray volume flux which leads to new particles formed), the nucleation particle diameter, the product withdrawal threshold diameter, and the product withdrawal rate. Analysis of the model reveals a steady-state constraint; a steady state does not exist if the nucleation fraction is too large. For cases where the steady state does exist, the steady-state particle size distribution is solved analytically. A numerical implementation of the model is used to illustrate the transient evolution of the process. The steady-state solution appears to be stable for a constant nucleation fraction. However, if the nucleation fraction depends on the bed height the steady state can be unstable. Such a situation may occur if the spray inlet is near the height of the bed surface. Instead of convergence towards a steady state, the transient solution displays ongoing oscillatory behavior with an oscillation period of a number of hours. A linear stability analysis is performed to confirm the findings on the stability of the steady state.     

Steady-state multiplicity in a solid oxide fuel cell: Practical considerations

Steady-state multiplicity in a solid oxide fuel cell (SOFC) in three modes of operation, constant ohmic external load, potentiostatic and galvanostatic, is studied using a detailed first-principles lumped model. The SOFC model is derived by accounting for heat and mass transfer as well as electrochemical processes taking place inside the fuel cell. Conditions under which the fuel cell exhibits steady state multiplicity are determined. The effects of operating conditions such as convection heat transfer coefficient and inlet fuel and air temperatures and velocities on the steady state multiplicity regions are studied. Depending on the operating conditions, the cell exhibits one or three steady states. For example, it has three steady states: (a) at low external load resistance values in constant ohmic external load operation and (b) at low cell voltage in potentiostatic operation.     

Analysis of multiple steady states of complex biochemical reactions

A general theory for the identification of criteria within which an enzymic reaction can exhibit multiplicity (i.e. multiple steady states) is reviewed. Application of the theory to four types of kinetic models encountered in biochemical reactions is illustrated and figures are presented to delineate the region of multiplicity. The stability of the steady states is analysed for small perturbation about the steady state.     

The influence of reactant adsorption on the multiplicity and stability of the steady state of a catalyst particle

The influence of a resistance to reactant adsorption on the multiplicity and stability of the steady state of a catalyst particle, promoting an exothermic reaction, is considered. For activated adsorption new regions of multiplicity are uncovered in the parameter space. It is shown that multiple solutions can arise for a surface reaction having zero activation energy. No such behaviour is observed for non-activated adsorption. The effect of non-activated adsorption is largely to diminish the regions of multiplicity.The influence of reactant adsorption resistance on the stability of the steady state is most complex and no general conclusions are drawn. A few specific cases are considered. The presence of an additional heat generation term due to adsorption has a great stabilising influence and, in addition, tends to dramatically reduce “hot-spot” temperatures which may arise during transient behaviour.     

Crystallization of aluminium hydroxide from the reactive NaAl(OH)4–NaHCO3 solution: Experiment and modeling

The crystallization kinetics of aluminium hydroxide from the sodium aluminate solution reacted with sodium bicarbonate were systematically investigated in a steady-state MSMPR (mixed-suspension mixed-product removal) crystallizer for the first time, and the expressions of the nucleation rate, growth rate and the agglomeration kernel of aluminium hydroxide were successfully regressed. The aluminium hydroxide particles precipitated from the reactive system are identified as gibbsite by XRD and SEM examinations. The volume growth rate order of gibbsite with respect to the relative supersaturation of the solution is above the linear growth rate order, and the spiral growth mechanism for the growth of the basal face of gibbsite in the reactive system was further identified by the growth rate, morphology analysis as well as the calculated surface entropy factor. The secondary nucleation rate of gibbsite from the reactive system is three to four orders of magnitude larger than that from seeded process reported in the only available literature reference. The agglomeration kernel of gibbsite in the reactive system increases linearly with growth rate and residence time, and the positive order about 0.55 of magma density is thoroughly different from the negative order of magma density for gibbsite agglomeration in seeded process presented in the literature.     

Steady‐State Multiplicity in High‐Pressure Bubble Column Reactors

The effect of pressure on the possible existence of multiple steady states in bubble column reactors is investigated. A mathematical model involving fast pseudo first‐order kinetics is employed to describe the performance of non‐isobaric, non‐isothermal reactors. The numerical analysis reveals that the existence of multiplicity is sensitive to pressure variation, yet high‐pressure operating conditions do not necessarily lead to a higher likelihood of multiplicity in these contactors.     

Heat and mass transfer in monolithic honeycomb catalysts—I.

The body of information presented in this paper is directed to engineers and scientists concerned with control of automobile emissions and exhaust gases from some industrial processes. The differential equations describing heat and mass transfer in a monolithic honeycomb catalyst are developed. Following transport mechanism is considered: convective heat and mass transfer in the holes of the structure, longitudinal thermal conductivity of the honeycomb support and gas-to-solid heat and mass transfer. The magnitudes of governing parameters for monolithic modules in use are discussed. Two methods for the numerical solution of a system of coupled, nonlinear ordinary differential equations with split boundary conditions are proposed. The first method-shooting procedure can be used only for problems with low values of Peclet number. For high values of Peclet number finite-difference approach along with the Newton-Raphson algorithm is suggested. It is shown that two stable steady states exist in certain regions of operation of a particular monolithic structure. The all-metal monolithic supports are more prone to multiplicity of steady states than the ceramic ones. For ceramic supports the two-phase piston-flow model is sufficiently accurate.     

在铂催化剂上-氧化碳氧化反应的多定态特性

对铂催化剂上一氧化碳氧化反应的多定态特性进行了实验研究,借助于奇异理论和突变理论,对实验结果进行了分析,找出了系统可能存在的定态数目,并预测出在定态数目不同的操作参数区域内,描述系统状态变量与操作参数的分岔图型式,最后,尝试利用系统的多定态特性识别反应的动力学模型,取得了满意的效果.     

在铂催化剂上-氧化碳氧化反应的多定态特性

对铂催化剂上一氧化碳氧化反应的多定态特性进行了实验研究,借助于奇异理论和突变理论,对实验结果进行了分析,找出了系统可能存在的定态数目,并预测出在定态数目不同的操作参数区域内,描述系统状态变量与操作参数的分岔图型式,最后,尝试利用系统的多定态特性识别反应的动力学模型,取得了满意的效果.     

A Plug Flow Model for Chemical Reactions and Aerosol Nucleation and Growth in an Alkali-Containing Flue Gas

The paper presents a numerical model for the simulation of gas to particle conversion and the chemical changes during cooling of a flue gas from the combustion of fuels rich in volatile alkali species. For the homogeneous nucleation of alkali species the model uses the classical theory modified by the Tolman coefficient for the size dependent surface tension. A special adaptation of the theory to the nucleation of seeds of chloride salt from an equilibrium mixture of monomer and dimer chloride vapor molecules is also made. The growth of particles occurs by condensation and agglomeration. Different multicomponent growth models are treated. The local gas phase composition is determined from a gas phase chemical equilibrium calculation combined with finite reaction rate kinetics for slower reactions. The model is useful in the analysis of boiler operation with respect to the formation of particles, HCl, SO₂, and deposits.