Modelling and performance analysis for cumene production process in a four-layer packed bed reactor

A mathematical model is developed and designed for the cumene reactor in cumene production process in Hindustan Organic Chemicals Limited (HOCL), Kochi with improved operating conditions. High purity cumene is produced by the alkylation of benzene with propylene in this catalytic condensation process where solid phosphoric acid (SPA) is used as the catalyst. The mathematical model has been derived from mass and energy balance equations considering the reactor as fixed packed bed reactor and two different numerical methods are presented here to solve the modelling equations. The explicit finite difference method (FDM) involves the approximation of derivatives into finite differences, and in the other one, orthogonal collocation (OC), Ordinary Diffeential Equations (ODEs) are formed at the collocation points and are solved using Runge–Kutta fourth order numerical scheme. Here the analysis shows that the predictions from the model are in good alignment with the plant data. The combined feed has the optimum value of 1:2:8 for propylene, propane and benzene and the profiles of temperature and concentration can be obtained along the reactor. The model has been implemented in COMSOL Multiphysics as a packed bed reactor using the same parameters collected from the plant of study. It has been found that the reaction occurs at a satisfactory level even with a low temperature than the reactor temperature at the plant by changing the catalytic particle size. The reaction performance is also analysed for the physical properties like porosity and catalyst size.

Solution of mass-spring-damper fractional systems using Caputo derivative and orthogonal collocation

PurposeIn this contribution, the solution of Mass-Spring-Damper Systems in the fractional context by using Caputo derivative and Orthogonal Collocation Method is investigated. For this purpose, different case studies considering constant and periodic sources are evaluated. The dimensional consistency of the model is guaranteed by introducing an auxiliary parameter. The obtained results are compared with those found by using both the analytical solution and the predictor-corrector method of Adams–Bashforth–Moulton type. The influence of the fractional order on the mechanical system is evaluated.Design/methodology/approachIn the present contribution, an extension of the Orthogonal Collocation Method to solve fractional differential equations is proposed.FindingsIn general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics.Originality/valueThe development of a new numerical method to solve fractional differential equations is the major contribution.

Modelling and Dynamic Simulation of One-Dimensional Isothermal Axial Dispersion Tubular Reactors with Power Law and Langmuir-Hinshelwood-Hougen Watson Kinetics

In this paper, the modelling, numerical lumping and simulation of the dynamics of one-dimensional, isothermal axial dispersion tubular reactors for single, irreversible reactions with Power Law (PL) and Langmuir-Hinshelwood-Hougen-Watson (LHHW)-type kinetics are presented. For the PL-type kinetics, first-order and second-order reactions are considered, while Michaelis-Menten and ethylene hydrogenation or enzyme substrate-inhibited reactions are considered for the LHHW-type kinetics. The partial differential equations (PDEs) developed for the one-dimensional, isothermal axial dispersion tubular reactors with both the PL and LHHW-type kinetics are lumped to ordinary differential equations (ODEs) using the global orthogonal collocation technique. For the nominal design/operating parameters considered, using only 3 or 4 collocation points, are found to adequately simulate the dynamic response of the systems. On the other hand, simulations over a range of the design/operating parameters require between 5 to 7 collocations points for better results, especially as the Peclet number for mass transfer is increased from the nominal value to 100. The orthogonal collocation models are used to carry out parametric studies of the dynamic response behaviours of the one-dimensional, isothermal axial dispersion tubular reactors for the four reaction kinetics. For each of the four types of reaction kinetics considered, graphical plots are presented to show the effects of the inlet feed concentration, Peclet number for mass transfer and the Damköhler number on the reactor exit concentration dynamics to step-change in the inlet feed concentration. The internal dynamics of the linear (or linearized) systems are examined by computing the eigenvalues of the linear (or linearized) lumped orthogonal collocation models. The relatively small order of the lumped orthogonal collocation dynamic models make them attractive and useful for dynamic resilience analysis and control system analysis/design studies.

A computationally efficient technique for the solution of pulp washing models

An efficient numerical technique for the solution of the pulp washing model is proposed in this study. Two linear and one nonlinear model are explained with quintic Hermite collocation method. In this technique, quintic Hermite polynomials (C2 continuous) are used as a basis function and orthogonal collocation method is applied within each element of the partitioned domain. For accuracy and applicability of the method, a comparison of the numerical results with analytic ones is made. The method is found to be stable using stability analysis and convergence criteria. The effect of Peclet number on exit solute concentration and other parameters is presented in the form of breakthrough curves. The results are derived for a broad range of parameters and the present method is found to be more useful and refined for solving the two-point boundary value problems.

Single and Multiobjective optimal control of epidemic models involving vaccination and treatment

Introduction: A rigorous multiobjective optimal control strategy (that does not require the use of weighting functions) of the epidemic models that consider vaccination and treatment strategies is presented. Modifications of the standard susceptible-infectious-removed, susceptible-exposed-infectious-removed, and the modified susceptible-infectious-removed models are dynamically optimized to minimize the number of infected individuals while, controlling the rate at which the individuals are vaccinated and treated. Method:The optimization program, Pyomo , where the differential equations are automatically converted to a Nonlinear Program using the orthogonal collocation method is used for performing the dynamic optimization calculations. The Lagrange-Radau quadrature with three collocation points and 10 finite elements are chosen. The resulting nonlinear optimization problem was solved using the solver BARON 19.3, accessed through the Pyomo-GAMS27.2 interface. Results: The computational results how that the multiobjective optimal control profiles generated by this strategy are very similar to those produced when weighting functions are used. Conclusion: The main conclusion of this work is to demonstrate that one can perform a rigorous dynamic optimization of epidemic models without the use of weighting functions that have the potential to produce some uncertainty and doubt in the results, in addition to dealing with unnecessary additional variables.

Effect of an axial-radial plate reactor modifications on a mega methanol plant production

Axial-radial flow plate reactors have been recently considered as efficient and practical types of reactors for methanol synthesis. Generally, an axial–radial reactor (AR) consists of two main parts namely the axial section and the radial section and the vast majority of the feed enters the radial section. Moreover, the structure of AR has a space above the axial part, which can add an adiabatic bed in the system. In this study, the performance of two novels AR configurations is investigated to improve the effectiveness of the axial–radial plate reactor. In the first configuration, the optimum length of the adiabatic bed is calculated and the adiabatic bed is located above the axial section inside the AR and is named IAAR. Therefore, in IAAR the feed of the axial section just enters the adiabatic bed and warms up. On the other configuration, the adiabatic bed with the optimum length is placed outside the reactor and is named OAAR. Therefore, in OAAR the total feed passes through the adiabatic bed, highly warms up, then cools to the optimum temperature in a heat exchanger, and finally enters AR. Two-dimensional mathematical modeling via orthogonal collocation on the finite element method is developed to compare the performance of two configurations. The results show that the maximum proportion of methanol produces in IAAR, which is approximately 3.8% higher than that produced in conventional AR due to utilizing an adiabatic bed inside the AR and superior gas distribution in the process. Momentum, mass, and heat equations are calculated and molar flow rates, mole fractions and temperatures are depicted along the radius and the length of the three configurations.