1. Multiscale Packed Bed Reactor with Extra Dimension
An extra dimension approach to model the mass and reaction distribution along the 3D reactor and within each catalyst pellet along the reactor length. The pellet radial dimension constitutes the extra 4th dimension.
A 𝑘 𝐵
𝑅 𝐴 =−𝑘∙ 𝐶 𝐴
𝑅 𝐵 =𝑘∙ 𝐶 𝐵
A is reactant and B is product. First order irreversible
catalytic reaction:
Raction rates follow
moles/(m3·s)
Macroscale
transport in reactor
Microscale
transport and reaction in pellet
Inflow
CA=1 mol/m3, CB=0 mol/m3

2. Equations In bed volume, transport by convection and diffusion
In bed volume, velocity determined by Darcy’s law
In pellet, reaction and transport by diffusion, spherical formulation
At pellet boundary, flux continuity and concentration continuity is assured

3. Results, macroscale Bed pressure distribution, gauge pressure.
Concentration distribution of reactant A in the macro space. A is consumed by the pellet reactions along the flow path.

4. Results, microscale Example plots of the pellet concentrations
at center of column, at 1 m height position

5. Results, microscale Concetration distibution in one catlytic pellet:
The product B has higher concetration at the center and diffues to the surface leaves to the fluid
3D representation of the product
concentrations

6. Step-by-step instructions are available in model documentation
Supporting slides follow, use only if needed.

7. Preferences Dialog Box
Preferences Dialog Box. Click Model Builder, select the Enable technology preview functionality check box, and then click OK.

8. Define Parameters & Variables
Load .txt files

9. Add Extra Dimension and Attach Dimensions
Right Click to Component 1 to add
Extra dimension, choose 1D for pellet
Right Click Component1>Definitions
And choose Extra Dimensions>Attached Dimensions

10. Create geometry of packed bed reactor

11. SCreate geometry for extra dimension which is 1D, for normalized pellet radius
Note that x=0 represent the center of pellet and x=1 is the surface of the pellet

12. Meshing Extra dimension geometry
Mesh the extra dimension geometry
Using the above values for size

13. Meshing the reactor geometry
Use structured meshes by combining triangular with swept meshes
There are 5 elements in the swept direction

14. Assign Material Property and choose water from add materials node

15. Define Physics
Darcy’s Law for fluid flow
Assign porosity and permeability
in matrix properties for packed bed

16. Add Transport of Diluted Species with number of dependent variable as 2 (A & B)
Assign Diffusion coefficient of A & B inside the packed bed column

17. Activate Advanced Physics Options

18. Right Click chds interface and choose weak contribution

19. Step 14: Within weak contribution
Select all domains in both Domain Selection and Extra Dimension 1
Make sure to choose “Attached Dimensions 1” in “Extra Dimension attachment”
4*pi*N*r_pe*r^2*(-DAp*pellet_CAr*test(pellet_CAr)-DBp*pellet_CBr*test(pellet_CBr)+r_pe^2*(react*test(pellet_CB)-react*test(pellet_CA)))

20. Make sure the first spatial coordinates is “r” for the Extra dimension this is because
The weak expression is has gradient term that is in r-direction

21. Define Auxiliary Variable, right click Weak Contribution 1
Field Variable name is “pellet_CA” this is the concentration of A inside the pellet

22. Define Auxiliary Variable, right click Weak Contribution 1
Field Variable name is “pellet_CB” this is the concentration of B inside the pellet

23. Add Inflow boundary condition, Only specie A comes in
A is converted to B from reaction inside the pellet
A > B

24. Add Outflow boundary condition

25. Results: Plotting concentration inside the pellet along the 3D reactor
Make a duplicate of solution 1
Under Data Sets
Change the Component to
Extra Dimension 1 in the setting
window

26. Add 1d Plot group and make sure the Data set is selected as
Solution 3

27. Add Line Plot and Type the following expression
“comp1.atxd3(0,0,0.1,pellet_CA)”
This gives concentration of A inside pellet at position z=0.1 in the column
Similar line plot is added for concentration of B

28. Results: Volume plot of concentration of A in the entire reactor