Second-order isothermal reaction in a semi-batch reactor: modeling, exact analytical solution, and experimental verification

2019 ◽  
Vol 4 (11) ◽  
pp. 2011-2020
Author(s):  
Mana Kord ◽  
Ali Nematollahzadeh ◽  
Behruz Mirzayi
Keyword(s):  
Mathematical Model ◽  
Analytical Solution ◽  
Flow Rate ◽  
Experimental Verification ◽  
Exact Analytical Solution ◽  
Batch Reactor ◽  
Second Order ◽  
Reaction Conditions ◽  
Reactor Modeling ◽  
Isothermal Reaction

Mathematical model of a semi-batch reactor (SBR) can be employed for tuning the concentration or flow rate of the external-feed of reactants, to control the reaction conditions and product properties.

The Shakedown Boundary for Parabolic Stress Distribution in NB-3222.5

2013 ◽  
Author(s):  
W. Reinhardt ◽  
A. Asadkarami
Keyword(s):  
Thermal Stress ◽  
Temperature Distribution ◽  
Stress Distribution ◽  
Analytical Solution ◽  
Thermal Loading ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Fe Model ◽  
Membrane Stress ◽  
Shakedown Analysis

The rules for the prevention of thermal stress ratchet in NB-3222.5 address the interaction of general primary membrane stress with two types of cyclic thermal loading. The first is a linear through-wall temperature gradient, for which the shakedown boundary is given by the well-known Bree diagram. The Code provides a second shakedown boundary for the interaction of general primary membrane stress with a “parabolic” temperature distribution. The corresponding ratchet boundary is fully defined in the elastic range, but only three points are given in the elastic-plastic regime. The range of validity of this ratchet boundary in terms of the thermal stress distribution (does “parabolic” mean second-order in the thickness coordinate or any polynomial of degree greater than one? If it is second-order, are there any further restrictions?) is not well defined in NB-3222.5. Using a direct lower bound method of shakedown analysis, the non-cyclic method, an exact analytical solution is derived for the shakedown boundary corresponding to the interaction of general primary membrane stress with a cyclic “parabolic” temperature distribution. By comparison to what is given in NB-3222.5, the thermal condition for which the Code equation is valid is defined and its range of validity is established. To study the transition behavior to the steady state and to confirm the analytical solution, numerical results using an FE model are also obtained.

scholarly journals Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Afrah Sadiq Hasan
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Fractional Order ◽  
Infinite Series ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Integer Order

Numerical solution of the well-known Bagley-Torvik equation is considered. The fractional-order derivative in the equation is converted, approximately, to ordinary-order derivatives up to second order. Approximated Bagley-Torvik equation is obtained using finite number of terms from the infinite series of integer-order derivatives expansion for the Riemann–Liouville fractional derivative. The Bagley-Torvik equation is a second-order differential equation with constant coefficients. The derived equation, by considering only the first three terms from the infinite series to become a second-order ordinary differential equation with variable coefficients, is numerically solved after it is transformed into a system of first-order ordinary differential equations. The approximation of fractional-order derivative and the order of the truncated error are illustrated through some examples. Comparison between our result and exact analytical solution are made by considering an example with known analytical solution to show the preciseness of our proposed approach.

scholarly journals The exact analytical solution for the gas lubricated bearing in the slip and continuum flow regime

2012 ◽  
Vol 91 (105) ◽  
pp. 83-93 ◽  
Author(s):  
Nevena Stevanovic ◽  
Vladan Djordjevic
Keyword(s):  
Analytical Solution ◽  
Flow Rate ◽  
Gas Flow ◽  
Exact Analytical Solution ◽  
The Other ◽  
Two Dimensional ◽  
Lubrication Equation ◽  
Continuum Flow ◽  
Independent Variable ◽  
The Continuum

The exact analytical solution for the compressible two-dimensional gas flow in the microbearing is presented. The general slip-corrected Reynolds lubrication equation is derived and it is shown that it possesses an exact analytical solution. It is obtained by a suitable transformation of the independent variable, and it provides the pressure distribution in the bearing as well as the mass flow rate through it. By neglecting the rarefaction effect, this solution is also applicable to the continuum gas flow in the bearing, which also does not exist in the open literature. The obtained analytical solution can be usefully applied for testing the other, experimental or numerical results.

Nonisothermal control of batch reactor with reaction by digital computer. Experimental verification of the control algorithm based on mathematical model of the system

1983 ◽  
Vol 48 (9) ◽  
pp. 2473-2483
Author(s):  
František Jiráček ◽  
Josef Horák
Keyword(s):  
Mathematical Model ◽  
Heat Carrier ◽  
Reaction Temperature ◽  
Flow Rate ◽  
Control Algorithm ◽  
Operating Temperature ◽  
Batch Reactor ◽  
Time Derivative ◽  
Reactor Model ◽  
Operating Reactor

Two-position feedback temperature control of the mixture in a batch reactor with the strongly exothermic autocatalytic reaction is studied. The control is based on continuous measurement of reaction temperature, evaluation of its time derivative and periodical determination of conversion of the reactant. On basis of numerical solution of equations of the mathematical model of the reactor is then evaluated the highest operating temperature of the mixture at which is still secured a safe operation of the reactor and trajectories along which the given operating temperature could be reached in the shortest time period. The manipulated variable was the flow rate of heat carrier into the cooler. Sampling of the reaction temperature and two-position switching over of flow rate of the heat carrier was performed in real time by use of the digital measuring center Hewlett-Packard 3052A. In the experiments has been verified the effect of changes in reactivity of the mixture, accuracy of the mathematical reactor model and effect of additional noise in the measured reaction temperature on control safety. Results of the experiments have proved that the proposed control algorithm enables safe control of the reaction temperature also in cases when the operating reactor states are unstable at the open control loop and when the cooler has a slow response to changes in the manipulated variable.

scholarly journals Mathematical Model, Simulation and Scale up of Batch Reactor Used in Oxidative Desulfurization of Kerosene

2021 ◽  
Vol 22 (3) ◽  
pp. 11-17
Author(s):  
Ghazwan Ahmed ◽  
Jasim Humadi ◽  
Ahmad Aabid
Keyword(s):  
Mathematical Model ◽  
Kinetic Parameters ◽  
Batch Reactor ◽  
Scale Up ◽  
Oxidative Desulfurization ◽  
Reaction Conditions ◽  
Exponential Factor ◽  
Simulation And Optimization ◽  
The Mathematical Model ◽  
Optimal Reaction

In this paper, a mathematical model for the oxidative desulfurization of kerosene had been developed. The mathematical model and simulation process is a very important process due to it provides a better understanding of a real process. The mathematical model in this study was based on experimental results which were taken from literature to calculate the optimal kinetic parameters where simulation and optimization were conducted using gPROMS software. The optimal kinetic parameters were Activation energy 18.63958 kJ/mol, Pre-exponential factor 2201.34 (wt)-0.76636. min-1 and the reaction order 1.76636. These optimal kinetic parameters were used to find the optimal reaction conditions which used to obtain a high conversion (≥ 99%). These optimal reaction conditions were reaction temperature 379.4 oK and reaction time 160 min. A scale up to batch reactor was conducted using these optimal kinetic parameters and optimal reaction conditions and the results showed the best reactor size that can be used at a diameter of 1.2 m.

scholarly journals An Exact Analytical Solution to the Shallow Water Equations Near Beaches

2017 ◽  
Vol 71 ◽  
pp. 63
Author(s):  
M. Yourdkhani ◽  
Saber Zarrinkamar
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Jacobi Polynomials ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Shallow Waters ◽  
Order Polynomial ◽  
Linear Counterpart

<p>The nonlinear differential equation governing the dynamics of water waves can be well approximated by a linear counterpart in the case of shallow waters near beaches. The linear equation, which is of second order nature, cannot be exactly solved in many apparently simple cases. In our work, we consider the shape of system as a complete second-order polynomial which contains the constant (step-like), linear and quadratic shapes near the beach. We then apply some novel transformations and transform the problem into a form which can be solved in an exact analytical manner via the powerful Nikiforov-Uvarov technique. The eigenfunctions of the problem are obtained in terms of the Jacobi polynomials and the eigenvalue equation is reported for any arbitrary mode. </p>

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