# Introduction

Context: .

Most catalytic and petrochemical reactions are operated with fixed bed reactors. In these reactors, catalyst pellets are generally randomly stacked in a large cylindrical vessel and the reactants, usually gas and liquid, are flowing through the bed to react inside the catalyst pellets. Catalyst pellets are made of a porous support onto which are deposited active materials. They are typically from 2 to 5mm in size with spherical, cylindrical or more complex shapes (tri-lobic or quadri-lobic extrudates).  More efficient processes can be designed based on a better comprehension of the physics ruling reactor performances in term of chemical response. A finer representation of the catalytic process allows smaller over-designs which will have fine economical results.

Objectives:

The objectives of this project are to:

■ Validate the Ranz-Marshall correlation

■ Find new correlation for cylindre

■ Study the effect of different stacked configurations on the outlet concentration

Numerical Method:

The used code is Comsol multiphysics version 3.5a through which we solve the  incompressible Navier-Stokes equation coupled with the convection diffusion equation , the calculation is Stationary .

The Incompreisble N-S equation:

$$\frac{ \partial u}{\partial t}+u\cdot{\nabla{u}}-\nu{\nabla^{2}{u}}=-\nabla{w}+g$$

The convection diffusion equation:

$$\frac{ \partial c}{\partial t}=D \nabla^{2}{c} -\vec{v}\cdot{\nabla}{c}$$

For stationary calculation :

$$\frac{\partial{c}}{\partial t}=0$$

The equation becomes:

$$0=\nabla{\cdot}({D\nabla{c}}) -\nabla{\cdot}(\vec{v}{c})+R$$

The dimensionless numbers controlling the problem are :

Reynolds number:

$$Re=\frac{\rho{vL}}{\mu}=\frac{vL}{\nu}$$

Schmidt number:

$$Sc=\frac{\nu}{D}=\frac{\mu}{\rho{D}}$$

Sherwood number:

$$Sh=\frac{KL}{D}$$

Péclet number :

$$Pe=\frac{Lu}{D}={Re}{Sc}$$

Where:

$\bullet \nu$   is the kienamtic viscosity

$\bullet D$   is the Diffusion coefficient

$\bullet \mu$  is the Dymanic viscosity

$\bullet \rho$   is the Density of the fluid

In order to validate our numerical method we have started by 3D simulations of a spherical particle in a flow to study the mass transfer and validate the Ranz-Marshall corellation:

$$Sh=2+0.552Re^{\frac{1}{2}}Sc^{\frac{1}{3}}$$

This relation permits to calculate the mass transfer coefficient by integrating the flux over the particle surface and evaluating Sherwood number .

For the N-S equation:The boundary conditions are:

►No slip at the particle surface and the container walls

►Velocity inlet at the intlet

►Pressure outlet at the outlet

For the convection diffusion equation:The boundary conditions are:

►C=1 at the particle surface

►Insulation symertry at the container walls

►Intlet concentration zero

►outlet :Convective flux