Autothermal reactors with internal heat exchang—I Numerical and experimental studies in the multiple steady-state region

Detailed numerical and experimental studies in the multiple steady-state region of an autothermal reactor with internal countercurrent heat exchange were conducted. The mildly exothermic water-gas shift reaction was used in the studies. In this distributed-parameter system multiple steady states were found experimentally. Although significant radial and longitudinal temperature gradients exist, one-dimensional (radially-lumped) simulations predict mean behavior quite well. For two-dimensional simulations, Legendre polynomial trial functions give faster convergence than Jacobi polynomials when using orthogonal collocation methods. Inflections in the radial temperature gradients were measured near the hot spot although not predicted by the simulations. Significant temperature jumps at the inner wall upstream of the hot spot were also measured. Production of hydrogen could be conducted at surprisingly mild conditions if the reactor could be controlled in a stable mode essentially identical to the middle unstable steady state.     

Tubular reactor design—I Two dimensional model

Conservation equations for a packed tubular reactor are modelled using a radially varying axial velocity, radial diffusivity and radial thermal conductivity. Radial diffusivity and thermal conductivity profiles are obtained from relations presented in this work. Instead of using a wall heat transfer coefficient, the model accounts for the higher resistance to heat flow near the reactor wall by using a lower value of the thermal conductivity. Axial diffusion is not included in the bulk of the bed, but its effect on the inlet temperature and conversion profiles are accounted for by considering axial diffusion in the pre-reaction zone.The two partial differential equations are converted into ordinary differential equations by using orthogonal collocation in the radial direction. The conventional orthogonal collocation method is modified by adding balance equations at the center of the tube, so as to improve the prediction of hot spot temperature.The model was applied to the sulfur dioxide reactor of Schuler et al. The results agreed quite well with the experimental data without the need for adjustment of parameters or constants.     

Steady‐State Modeling and Simulation of an Axial‐Radial Ammonia Synthesis Reactor

A two‐dimensional model was developed for an axial‐radial ammonia synthesis reactor of the Shiraz petrochemical plant. In this model, momentum and continuity equations as well as mass and energy balance equations are solved simultaneously by orthogonal collocation on the finite element method to obtain pressure, velocity, concentration and temperature profiles in both axial and radial directions. For the catalyst particle, the effectiveness factor is calculated by solving a two‐point boundary value differential equation. The boundary conditions for the Navier‐Stokes and continuity equations are obtained by using equations representing the phenomena of gases splitting or joining in different streams and going through holes in a thin wall. The results of the mathematical model have been compared with the plant data and a good agreement is obtained.     

Modelling of non-catalytic gas—solid reactions—II. Transient simulation of a packed bed reactor

A Transient model for packed bed non-catalytic reactors, which avoids many of the simplifying assumptions of the earlier models, has been developed. The model includes the effects of inter- and intra-pellet transfer resistances and the additional effects of axial dispersion in the bulk fluid. The solution procedure to the system of equations is based on the orthogonal collocation method. The effects of various parameters on the temperature rise encountered in the bed and the possibility of reducing this rise by perturbing the inlet concentration and temperature of the gas are examined.     

Simulation of bulk free radical polymerization of styrene in tubular reactors

A model that takes into consideration radial and axial changes in velocity in a tubular reactor for the thermal polymerization of styrene is used to simulate the effect of changes in inlet tube wall temperature and tube radius. The reactor performance is sensitive to the changes of these parameters. The method of orthogonal collocation is used to discretize the modeling equation in the radial direction and Gear method to solve the resulting stiff differential equations in the axial direction. It is found that reducing the wall temperature and the tube radius along the direction of the flow of the monomer reduces the variation in conversion between the tube center and tube wall and thus are advantageous.     

Solution of diffusion–dispersion models using a computationally efficient technique of orthogonal collocation on finite elements with cubic Hermite as basis

In this paper, linear and non linear diffusion–dispersion models involving fluid flow through porous cylindrical particles are solved using orthogonal collocation on finite elements with cubic Hermite as basis. The technique involves partitioning of axial domain into equal elements and then orthogonal collocation method with cubic Hermite as basis function is applied within each element. Exit concentration profiles are drawn for Peclet numbers ranging from 0 (perfect mixing) to ∞ (perfect displacement). Proposed technique is computationally efficient, stable and yields accurate solution, even for nonlinear stiff problem. Correlation with industrial parameters is also presented.     

Use of orthogonal collocation methods for the modeling of catalyst particles—I. Analysis of the multiplicity of the solutions

Different collocation methods are compared in their ability to predict the effectiveness factor for the general case of a catalyst particle with both internal and external resistance to mass and heat transfer. The accuracy of the methods in determining the bifurcation points which separate regions of uniqueness and multiplicity of the steady states is tested. First order orthogonal collocation and the linearization method, shown to be almost equivalent for all cases, give poor approximations. Even high order orthogonal collocation is sometimes inaccurate in predicting the low concentration steady states. An excellent alternative is a variation of the Paterson—Cresswell technique which is a combination of the low and high reactivity models. This modified technique is able to predict the existence of five steady states first demonstrated by Hatfield and Airs.     

用正交配置法求解血液透析超滤的传质动力学模型

提出了血液透析、血液超滤和血液透析超滤过程传质的通用模型,并利用正交配置技术分析了影响传质速率的各种因素.结果表明,采用正交配置法进行上述传质过程的模拟时,简单的三点配置即与解析解的结果接近.采用正交配置法比有限差分法简单、快速和准确.     

Evaluation of weighted residual methods for the solution of the pellet equations: The orthogonal collocation,Galerkin, tau and least-squares methods

In recent years a number of publications have adopted the least-squares method for chemical reactor engineering problems such as the population balance equation, fixed bed reactors and pellet equations. Evaluation of the performance of the least-squares method compared to other weighted residual methods is therefore of interest. Thus, in the present study, numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau, and least-squares methods, have been adopted to solve a non-linear comprehensive and highly coupled pellet problem. The methanol synthesis and the steam methane reforming process have been adopted for this study.Based on the residual of the governing equations, the orthogonal collocation method obtains the same accuracy as the Galerkin and tau methods. Moreover, the orthogonal collocation method is associated with the simplest algebra theory and thus holds the simplest implementation. Another benefit of the orthogonal collocation method is that the technique is more computational efficient than the other methods evaluated. The least-squares method does not obtain the same accuracy as the other weighted residual methods. In particular, the least-squares method is not suitable for strongly diffusion limited systems that give rise to steep gradients in the pellet. On the other side, considering the spectral–element framework, the least-squares method is less sensitive to the placement of the element boundaries than observed for the orthogonal collocation, Galerkin and tau methods.The present paper outlines the algebra of the weighted residual methods and illustrate the numerical solution techniques through a simplified pellet problem.     

Effect of sinusoidal variation of feed concentration and temperature on the performance of a packed-bed biological reactor – a theoretical study

The performance of a packed-bed biological reactor has been analysed under sinusoidal variations of substrate concentration and temperature for zero-order, first-order and Michaelis-Menten kinetics. The model equations were solved by the method of orthogonal collocation. The results show that the cyclic steady-state conversion is not affected by cyclic variations in the feed concentration. However, cyclic temperature variations with an amplitude of 20°C significantly decrease the mean exit concentration for zero-order and Michaelis-Menten kinetics compared to the constant-temperature case. The approach to cyclic steady-state conditions is estimated to be somewhat flower for zero-order kinetics than for the other kinetics models investigated. We conclude that temperature variations during the day or changes in the performance of upstream plant will not adversely affect the performance of a packed-bed biological reactor.     

A301氨合成催化剂工程参数研究与测定Ⅱ床层传热参数的实验测定

实验测定了Ａ３０１氨合成催化剂填充床内的二维温度分布。采用正交配置法求解拟均相二维传热数学模型,获得了床层的径向有效导热系数和壁给热系数。并将其与雷诺数关联得到了可在实验条件范围内使用的经验关联式。     

Analysis of non-isothermal catalytic reactions. Limiting effect of non-key component

An improved method for predicting effectiveness factors for pseudo first order catalytic reactions with limiting non-key reactant has been derived. The method takes account of intraparticle temperature and concentration profiles in addition to interphase effects and solution of the system equations is obtained using the orthogonal collocation technique which is simple and involves only algebraic equations. The depth of penetration of reactant is also determined by this method. The results are discussed and compared with earliler work.     

多孔催化剂效率因子的多组分扩散模型(Ⅳ)有死区圆柱状中温变换催化剂效率因子的二维模型及其求解

建立了有死区圆柱状WB-2中温变换催化剂效率因子计算的二维数学模型,用正交配置法进行计算,所得结果与实验测定值比较,相对误差绝对值的平均值为11.50%.     

用正交配置技术估计固定床内的传热参数

用正交配置技术与非线性最小二乘估计固定床内的传热参数.用多通道巡回检测装置同时测定沿床层径向分布的40个正交配置点上的温度.系统地考察了床高、Re_p、d_t/d_p、气体入口温度、壁温对K_(er)和h_w的影响,并提出了以下的关联式k_(er)=0.199+0.015d_t/d_p+0.0020/1+14.15(d_p/d_t)~2Re_ph_w=33.4+4.23d_t/d_p+0.331/1+8.45(d_p/d_t)~2Re_p(5.5相似文献   

Modeling of continuous-flow bioreactors with a biofilm with the use of orthogonal collocation on finite elements

The paper discusses mathematical modeling of two types of continuous bioreactors, a continuous stirred tank bioreactor (CSTBR) and a tubular bioreactor with biofilm on inner walls. The method of orthogonal collocation on finite elements (OCFE) was used for the modeling of a microbiological process occurring in the biofilm. Single-substrate Monod and Haldane kinetic models, and double-substrate Haldane–Monod kinetic model were used. The accuracy of the OCFE method was evaluated by comparison with the solution obtained using shooting method (SM). Profiles of substrate concentration, biofilm effectiveness factors, steady state-branches of CSTBR and distribution of substrate concentrations and biofilm thickness along the length of the tubular bioreactor were determined. Steady-state branches of the CSTBR were compared with experimental results from the literature. It was shown that the mathematical model gives good agreement. The efficiency of the OCFE method was demonstrated even for very steep gradients of concentrations in the biofilm.     

Use of orthogonal collocation on finite elements with moving boundaries for fixed bed catalytic reactor simulation

A numerical procedure for the solution of transient models for the simulation of fixed bed catalytic reactors is developed. The most general model examined includes axial dispersion in the external fluid phase, interphase mass and heat transfer resistances,intraphase mass resistance and any given kinetic scheme with complex reaction rate expressions. The solution technique is based on the method of lines, in which the space variables are discretized using the orthogonal collocation method on finite elements, with elemination of the node unknown functions, coupled with an integration method for stiff ordinary differential equations. An efficient procedure for updating during the integration the position of the finite element boundaries, in order to follow the movement of steep concentration and temperature profiles along the space variable during the reactor transient, is proposed. Application examples of the developed computer program are also given.     

KINETIC CHARACTERISTICS OF A BATCH ADSORBER OR AN ION EXCHANGE DEVICE OPERATED UNDER NONISOTHERMAL CONDITIONS

A model was formulated for a batch adsorber or ion exchange device with heat generation inside the bulk liquid due to mixing or electrical heating and due to heat of adsorption. Internal and external particle mass and heat transfer gradients and heat transfer through the vessel wall were included. The effective diffusion coefficient was taken to be temperature dependent. Numerical calculations (by orthogonal collocation) give conditions for the existence of intra particle nonisothermity and show the effect of mixing and process temperature on adsorption kinetics.     

Mathematical Modeling of Coke Formation and Deposition due to Thermal Cracking of Petroleum Fluids

A two‐dimensional mathematical dynamics model is presented to predict coke formation due to thermal cracking inside the tubes of fired heaters on two types of petroleum fluid. The laminar and turbulent flows are analyzed for both petroleum fluids. The second‐order k‐? standard model is adopted to make this mathematical model more accurate than previous models of coke formation. The radial and axial variations for temperature, velocity, and concentration due to the high temperature gradients inside the tubes are considered in the model equations. The finite volume method is the numerical model used to discretize the conservation equations. The proposed model is suitable to predict coke formation inside heater tubes since it indicates operational conditions where coke formation is minimized.     

非恒温非绝热固定床反应器非均相模型及其求解

非恒温非绝热固定床反应器广泛应用于强放热的反应体系，由于传热、传质阻力的存在，使得流体中的温度、浓度和催化剂表面处不同，按拟均相二维模型求得的结果和实际情况存在较大的偏差。为此利用分层次的反应器建模思想，建立了非恒温非绝热固定床反应器的非均相二维模型，并用正交配置法对该模型进行了求解，示例给出了萘氧化成邻苯二甲酸酐的模拟结果。     

KINETIC CHARACTERISTICS OF A BATCH ADSORBER OR AN ION EXCHANGE DEVICE OPERATED UNDER NONISOTHERMAL CONDITIONS

Abstract

A model was formulated for a batch adsorber or ion exchange device with heat generation inside the bulk liquid due to mixing or electrical heating and due to heat of adsorption. Internal and external particle mass and heat transfer gradients and heat transfer through the vessel wall were included. The effective diffusion coefficient was taken to be temperature dependent. Numerical calculations (by orthogonal collocation) give conditions for the existence of intra particle nonisothermity and show the effect of mixing and process temperature on adsorption kinetics.