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

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

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.

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

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

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

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

Define Parameters & Variables Load .txt files

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

Create geometry of packed bed reactor

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

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

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

Assign Material Property and choose water from add materials node

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

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

Activate Advanced Physics Options

Right Click chds interface and choose weak contribution

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)))

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

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

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

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

Add Outflow boundary condition

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

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

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

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