Ranz-Marshall :

Validation in 3D:

In order to validate Ranz-Marshall correlation: $$Sh=2+0.552Re^{\frac{1}{2}}Sc^{\frac{1}{3}}$$

Numerous simulations were carried out for solving the Navier-Stockes equation coupled with the convection-diffusion equation around a spherical particle placed in a box, the sphere has radius equal to $2.{10^{-4}}$ and the box dimensions are 15r x 15r where r is radius of the sphere. The boundary conditions are similar to those explained in the introduction.

Geometry and Mesh:

 

             

                                 Geometry                                                                      Mesh

The taken geometry is a sphere of radius $2.{10^{-4}}$ m inside a cubic box of side 15r .The inlet is from down and the outlet is up .

   

 

 

 

                                                      

                          Slice velocity profile                                 Slice concentration profile         

Validation in 2D:

Due to memory constrains we were not able to increase Sherwood number beyond ~6 in 3D as we need to refine the mesh when we increase Péclet number because the thickness of the boundary layer $ \delta$ is inversely proportional to Peclet number:

 

                                                                 $\delta $ ~ $\frac{1}{Pe^{\frac{3}{4}}}$

For this reason we also used 2D axisymetric model as the mesh size stays reasonable for higher values of Pe.

Geometry and Mesh:

 

                                   Geometry                                                                                 Mesh

 

Profiles:

                        Velocity profile                                                    Concentration profile

            

The velocity at the surface is always zero (No slip) but we have  seen that the boundary layer at the particle is getting thinner as we increase $Re$, it’s known that the thickness $\delta$ is inversly proportional to $Re$ and so when Re increases the thickness of the boundary layer decreases, the same applies for the diffusion, the layer thicknes decreases as we increase Péclet number.

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Results:

                                  

In the above figure we compare the Ranz-Marshall correlation to the results obtained by 3D and 2D-axisymetric simulation. We see clearly that the 2D-axisymetric is closer to the correlation, this is because we were able to increase the domain size surrounding the particle in 2D as the mesh size stays finite whereas this was not possible in 3D because the increment of the mesh cells needs higher memory which was limited .

 

                   The variation of Sherwood number as function of Péclet number

We see that 2D simulations are better fitting the Ranz-Marshall  correlation and the range of Péclet is higher , this is because in 2D the mesh can be refined and can be solved by the available memory.