A three-phase nonequilibrium model for catalytic distillation

2009 ◽  
Vol 63 (2) ◽  
Author(s):  
Marcel Kotora ◽  
Zuzana Švandová ◽  
Jozef Markoš
Keyword(s):  
Effective Diffusivity ◽  
Solid Catalyst ◽  
Algebraic Equations ◽  
Catalytic Distillation ◽  
Nonequilibrium Model ◽  
Three Phase ◽  
Nonlinear Algebraic Equations ◽  
Steady State Simulation ◽  
Heat Transfers ◽  
Fortran Programming Language

AbstractNonequilibrium model for steady state simulation of catalytic distillation is presented. Mathematical model takes into account both mass and heat transfers across the gas liquid interface and through the liquid-solid (catalyst) interface. Equations describing the mentioned phenomena are based on the effective diffusivity approach. The resulting system of nonlinear algebraic equations was implemented in the FORTRAN programming language and solved by the BUNLSI (Ferraris & Tronconi, 1986) solver. The described model was verified using the experimental data obtained from a continuous distillation column equipped with catalytic packing. As an experimental model system, synthesis of propyl propionate from propan-1-ol and propionic acid was chosen. Comparison of experimental and simulation data is presented, and appropriateness of the developed model for other types of catalytic distillation processes is discussed.

Modeling of the Catalytic Distillation Process for the Synthesis of Ethyl Cellosolve Using A Three-Phase Nonequilibrium Model

2002 ◽  
Vol 1 (1) ◽  
Author(s):  
Yuxiang Zheng ◽  
Flora T. T. Ng ◽  
Garry L. Rempel
Keyword(s):  
Theoretical Plate ◽  
Solid Catalyst ◽  
Catalytic Distillation ◽  
Nonequilibrium Model ◽  
Liquid Phases ◽  
Three Phase ◽  
Model Predictions ◽  
Plant Data ◽  
Model Equations ◽  
Good Agreement

The catalytic distillation (CD) process for the synthesis of ethyl cellosolve from ethanol and ethylene oxide on molecular sieve catalyst NKC-01 in a 200 mm CD pilot column was simulated using the three-phase nonequilibrium model which was developed in our laboratory. The main feature of this model is that the actual rates for transport and reaction are used and the stage efficiency or HETP (height equivalent to a theoretical plate) is not required. The effect of multicomponent mass and heat transfer between vapor and liquid phases as well as between liquid and solid (catalyst) phases was taken into account according to the Maxwell-Stefan equations. The Newton-Raphson method was used to solve the model equations. The simulation profiles of the temperature and composition along the column are in good agreement with the pilot CD plant data obtained for the synthesis of ethyl cellosolve. Good agreement between model predictions and experimental data is also obtained for the yield and selectivity of the ethyl cellosolve. This model could be extended to simulate other CD processes and commercial scale CD plants.

Use of Different Numerical Solution Approaches for a Three-Phase Slurry Catalytic Reactor Model

2003 ◽  
Vol 1 (1) ◽  
Author(s):  
Mardonny Nazareno Barreira ◽  
Eduardo Coselli Vasco de Toledo ◽  
Rubens Maciel Filho ◽  
Maria das Graças Enrique
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Orthogonal Collocation ◽  
Reactor Model ◽  
Three Phase ◽  
Steady State Model ◽  
Nonlinear Algebraic Equations ◽  
Volume Method

In this paper two different numerical approaches are used to find out the steady-state solution for multiphase chemical reactors. Such systems are distributed parameter systems and the steady-state formulation problem leads to a system of nonlinear algebraic equations. The problem is to solve the equations in a robust way so that reliable predictions can be made. Bearing this in mind in this work two methods are implement and their performances compared, to know the Orthogonal Collocation Method e the Finite Volume Method. The results show that in spite of good qualitative agreement in the predictions the latter approaches in more suitable to have a direct solution of the steady-state model of the reactor.

scholarly journals Modeling of multiphase flow at solid particles hydropelling

2019 ◽  
pp. 67-71
Author(s):  
A.Yu. Dreus ◽  
A.V. Haminich ◽  
N.V. Koval ◽  
S.V. Dziuba
Keyword(s):  
Multiphase Flow ◽  
Gas Flow ◽  
Solid Phase ◽  
Flow Simulation ◽  
Solid Particles ◽  
Hydraulic Lift ◽  
Algebraic Equations ◽  
Flow Parameters ◽  
Three Phase ◽  
Nonlinear Algebraic Equations

Purpose. Development of methods for calculating the parameters of the solid particles lifting from the bottom of reservoirs based on the three-phase (gas-liquid-solid particles) flow simulation in an air-lift pipe. Methodology. Mathematical modeling of multiphase flows based on hydraulic ratios. In doing so, heights of up to 10 m (short airlift) are considered with corresponding physical assumptions. Findings. A methodological support has been developed for calculating rational parameters of modes of transporting solid particles with a short airlift. The solution of the problem of the solid particles hydraulic lift by a short airlift is reduced to solving a system of nonlinear algebraic equations. Parametric calculations were carried out and the dependences of the gas flow rate required to ensure a given airlift performance were determined, and the influence of the geometric characteristics of the pipelines on the flow of the solid phase was estimated. Originality. The proposed method allows one to determine the parameters of the solid particles hydraulic lift in a short airlift by solving a system of nonlinear algebraic equations, without using the apparatus of mathematical physics methods. Practical value. It consists in the development of a method for calculating and determining rational parameters of pulp transportation processes in short airlifts. These methods make it possible to determine the dependence of the air flow on the solid particles flow and substantiate the corresponding geometric and flow parameters of the hydraulic lift system. Key words: short airlift, multiphase flow, solid particles hydraulic lift, technological calculatio.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

scholarly journals A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
E. H. Doha ◽  
D. Baleanu ◽  
A. H. Bhrawy ◽  
R. M. Hafez
Keyword(s):  
Numerical Solutions ◽  
Rational Functions ◽  
Absolute Error ◽  
Delay Systems ◽  
Infinite Interval ◽  
Algebraic Equations ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Linear And Nonlinear ◽  
Pseudospectral Scheme

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small.

scholarly journals Modal Perturbation Method for the Dynamic Characteristics of Timoshenko Beams

2005 ◽  
Vol 12 (6) ◽  
pp. 425-434 ◽  
Author(s):  
Menglin Lou ◽  
Qiuhua Duan ◽  
Genda Chen
Keyword(s):  
Perturbation Method ◽  
Dynamic Characteristics ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Solution Process ◽  
Equation Of Motion ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Shear Distortion

Timoshenko beams have been widely used in structural and mechanical systems. Under dynamic loading, the analytical solution of a Timoshenko beam is often difficult to obtain due to the complexity involved in the equation of motion. In this paper, a modal perturbation method is introduced to approximately determine the dynamic characteristics of a Timoshenko beam. In this approach, the differential equation of motion describing the dynamic behavior of the Timoshenko beam can be transformed into a set of nonlinear algebraic equations. Therefore, the solution process can be simplified significantly for the Timoshenko beam with arbitrary boundaries. Several examples are given to illustrate the application of the proposed method. Numerical results have shown that the modal perturbation method is effective in determining the modal characteristics of Timoshenko beams with high accuracy. The effects of shear distortion and moment of inertia on the natural frequencies of Timoshenko beams are discussed in detail.

Periodic Response of a Link Coupling With Clearance

1989 ◽  
Vol 111 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Y. S. Choi ◽  
S. T. Noah
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Solution Procedure ◽  
Contact Forces ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Contact Points ◽  
Integration Schemes ◽  
Nonlinear Algebraic Equations

The nonlinear, steady-state response of a displacement-forced link coupling with clearance with finite stiffness is determined. The solution procedure is derived from satisfying the boundary conditions at the contact points and then solving the resulting nonlinear algebraic equations by setting the duration of contact as a parameter. This direct approach to determining periodic solutions for systems with clearances with finite stiffness is substantially more efficient than numerical integration schemes. Results in terms of contact forces and durations of contact are pertinent to fatigue and wear studies. Parametric relations are presented for effects of the variation of damping, stiffness, exciting displacement, and gap length on the dynamic behavior of the link pair.

Periodic Solutions in Rotor Dynamic Systems With Nonlinear Supports: A General Approach

1989 ◽  
Vol 111 (2) ◽  
pp. 187-193 ◽  
Author(s):  
C. Nataraj ◽  
H. D. Nelson
Keyword(s):  
Dynamic Systems ◽  
Original Problem ◽  
Squeeze Film ◽  
Algebraic Equations ◽  
Trigonometric Collocation ◽  
Nonlinear Algebraic Equations ◽  
The Stability ◽  
Future Work ◽  
Rotor Dynamic ◽  
Large Rotor

A new quantitative method of estimating steady state periodic behavior in nonlinear systems, based on the trigonometric collocation method, is outlined. A procedure is developed to analyze large rotor dynamic systems with nonlinear supports by the use of the above method in conjunction with Component Mode Synthesis. The algorithm discussed is seen to reduce the original problem to solving nonlinear algebraic equations in terms of only the coordinates associated with the nonlinear supports and is a big improvement over commonly used integration methods. The feasibility and advantages of the procedure so developed are illustrated with the help of an example of a typical rotor dynamic system with an uncentered squeeze film damper. Future work on the investigation of the stability of the periodic response so obtained is outlined.

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