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This numerical study investigates the steady state unimolecular thermal decomposition of a chemical dissolved in water inside a parallel-plate reactor containing four heated circular rods using a penalty Galerkin finite element approach. The reactant fluid enters from the left inlet and exits from the right outlet of the reactor. All solid walls of the reactor are assumed to be thermodynamically isolated. The aim of the investigation is to illustrate the effects of the energy expelled during the reaction, temperature of the heated rods and fluid inlet velocity on the thermal field and concentration of the heat sensitive chemical. The simulation is conducted for different values of inlet velocity and rods temperature taking into consideration and neglecting the reaction energy. From the results, it is concluded that the thermal field and decomposition process of the chemical are significantly influenced by fluid velocity, rods temperature and the reaction type.

A unimolecular reaction is in principle the simplest kind of elementary reaction since it involves the decomposition or isomerization of a single reactant. The next example will illustrate the mechanism of the unimolecular decomposition. The following elementary reaction,

is unimolecular because there is only one molecule reacting, that is, molecule “A” is reacting.

This unimolecular reaction step implies the following rate law,

or, equivalently,

In words, these elementary reactions state that the molecule, A, spontaneously transforms into B at some reaction rate k_{1}. The algebraic sign in front of k_{1} tells whether you are gaining product or losing reactant depending on whether the concentration in the derivative is increasing or decreasing. For example, in Equation (2), “B” is increasing, and in Equation (3), “A” is decreasing.

The study of a unimolecular reaction is of paramount importance in chemical engineering. Recently, various experimental and numerical investigations have been done to examine characteristics and mechanisms of the unimolecular decomposition of many chemicals. In [_{2}OH, are carried out using ab initio electronic molecular structure methods. The thermal unimolecular decomposition of propynal is investigated behind reflected shock waves in [

The present study addresses the unimolecular thermal decomposition of a chemical passing through a parallel plate reactor with four transverse heated rods, which is investigated. The compound to be decomposed is dissolved in water. The water with the dissolved compound first enters the reactor from the left then passes heated circular cylinders before exiting from the right side and hence the reacting fluid is heated as it passes the transverse cylinders. The proposed model is numerically simulated using Galerkin weighted residual finite element method [

The numerical results are presented graphically in terms of contour plots and curves at the longitudinal axis of the reactors for various values of water inlet velocity and rods temperature when the reaction energy is considered and when is neglected.

The three-dimensional (3-D) representation of the reactor geometry is shown in

The computational domain is considered as follows

rods are considered with no slip as walls. The water and the reaction physical properties used in the simulation are listed in

The physical properties of water are assumed to be constant and don’t depend on the thermal field.

In the current problem, we have considered that the water flow is a steady-laminar one. The gravitational force and radiation effect are neglected here. The governing equations that describe the present model are incompressible Navier-Stokes equations, energy balance equation, convection and diffusion equation and equations related to the chemistry of the problem.

Consider the heat sensitive chemical “A” (dissolved in water) undergoes thermal decomposition into fragments “B” according to the unimolecular reaction described by Equation (1). The rate constant

where

The reaction rate

where

In addition, if the decomposition reaction is considered to be exothermic, then the rate of the expelled energy during the reaction Q is given by:

In the present model the fluid flow is described by 2D steady state incompressible Navier-Stokes equations:

Quantity | Water | Quantity | Chemical |
---|---|---|---|

Density | Activation energy | ||

Dynamic viscosity | ^{ } | Heat of reaction | ^{ } |

Specific heat capacity | 4180 | Frequency factor | |

Thermal conductivity | 0.62 | Diffusivity of “A” in water |

where u and v are the velocities in the x and y directions, respectively and p is the pressure.

We used this formulation because the flow regime is laminar and the density is assumed to be constant. It’s known that the Reynolds number indicates whether a flow regime is laminar or turbulent. In the present simulation, we will simulate the decomposition at two inlet normal velocities

where d is characteristic length which is equal to 0.12 for the present model. By calculating the Reynolds numbers in the considered cases, we can conclude that the Reynolds numbers are well within the limits of the lami- nar regime which is Re < 2000. Moreover, assuming that water density is constant and doesn’t depend on the thermal field, the incompressible Navier-Stokes Equations (7)-(9) are appropriate to model the flow.

Considering the heat transfer is done through conduction and convection, the next energy balance equation used to model the energy transport in the reactor

where Q is a sink or source term, which here is the energy expelled during the reaction. So, if the reaction isn’t exothermic then Q = 0, whenever if the reaction is exothermic then

The mass transfer of the compound “A” in the reactor is governed by the convection and diffusion equation:

where

The proper boundary conditions of the considered problem are listed below:

The numerical technique has been used to solve the Navier-Stokes, energy balance and mass transport Equations (7)-(11) subject to the given boundary conditions is the finite element formulation based on the Galerkin weighted residual method. The applications of this method are well described in [

As illustrated in [

where

Using Equation (12), the momentum balance equations (8) and (9) reduce to

Expanding the velocity components (u,v), temperature (T) and concentration (

where N is the number of nodes for each biquadratic element.

Based on the Galerkin weighted residual finite element method, the weight functions are identical to the elements shape functions

Biquadratic shape functions with three point Gaussian quadrature is used to calculate the integrals in the residual Equations (16)-(19). In Equations (16) and (17), the second integral containing the penalty parameter

The non-linear residual equations are solved using Newton-Raphson method to determine the coefficients of the expansions in Equation (15). The boundary conditions are incorporated into the assembled global system of nonlinear equations to make it determinate.

Eight node biquadratic elements have been used with each element. Before evaluating Gauss integration, the coordinate x − y must be mapped into of the natural coordinate

where

Consequently, the domain integrals in the residual equations are approximated using eight node biquadratic basis functions in

In the present study, the model validation is justified by studying the number of iterations needed for convergence and error analysis of the finite element method.

In order to verify the numerical validation of the numerical scheme, the convergence history for calculations is presented in

lated within about 50 iterations. The calculations were performed on a Pentium IV computer and a typical run for a solution required about 2 CPU minutes.

Equations (10), (11), (13) and (14) with the given boundary conditions have been solved numerically using a penalty finite element method based on Galerkin weighted residuals. The computational domain in

The results have been presented graphically in both contour plots and one-dimensional representation in order to illustrate the effects of the reaction energy, heated rods temperature and fluid inlet velocity on the thermal field and decomposition of heat sensitive chemical “A”. The numerical results are displayed as contour plots in

As shown in

Thus, the decomposition of the compound “A” is greatly affected with the fluid inlet velocity. Also, in the absence of the reaction energy i.e. H = 0, the decomposition of “A” will be influenced by the temperature of the transverse heated rods as illustrated in

A comparison between

All the previous discussed results can be deduced by examining the results in the 1-D representation shown in

The influences of the inlet velocity, rods temperature and energy expelled during the reaction on the thermal field can be summarized form

From

of the rods temperature on concentration of “A”.

Comparing

The main objective of the current investigation is to study the influences of reactant fluid inlet velocity, rods temperature and reaction type on the unimolecular thermal decomposition of a chemical dissolved in water inside a parallel-plate reactor having four transverse heated rods. The penalty Galerkin weighted residuals finite element method is used to obtain smooth and reliable solutions for the considered model. Summarizing all the results discussed above will lead to the following conclusions:

1. The decomposition of the dissolved chemical is greatly affected by the fluid inlet velocity.

2. Decreasing the fluid inlet velocity will increase the amount of the chemical being decomposed.

3. The decomposition of the chemical is significantly influenced by the reaction type from the thermal sight.

4. The decomposition process is slightly influenced by the temperature of circular rods.

5. The increasing in the fluid velocity and temperature along the reactor is of an oscillating type due to the consequently heated rods.

6. The obtained results are largely consistent with physical and chemical thermodynamic.

The author would like to thank Deanship of Scientific Research at Majmaah University, Saudi Arabia for the financial grant received in conducting this research.

Mohamed M.Mousa,11, (2016) Finite Element Simulation of an Unimolecular Thermal Decomposition inside a Reactor. Journal of Applied Mathematics and Physics,04,328-340. doi: 10.4236/jamp.2016.42040