Modelling of Transport within Porous Solids

Sean Rigby

Chemical Engineering

Problems being addressed

• Intrusion and extrusion of a non-wetting fluid within porous solids– simulation of mercury porosimetry

• Coupled diffusion and reaction with coke deposition in porous heterogeneous catalysts

Mercury porosimetry

• Pore structure characterisation technique based on non-wetting property of mercury.

• Can be used to obtain porosity, pore-size distribution, and pore connectivity descriptors.

• Indirect technique, so interpretation of data is ambiguous, and requires a model.

• More understanding of the physical processes involved should improve interpretation of raw data.

Mercury porosimetry simulations

• Characterise pore structure using MRI.

• Simulate mercury intrusion and retraction in structural models based upon MR images.

• Predict spatial distribution of entrapped mercury (and pressure-volume curves).

• Compare with X-ray tomography images (and other data).

MR data• MR provides 128x128x14 grid-size map of

the spatial distribution of porosity, pore size and tortuosity for ~2-3 mm dia. pellets.

• Image voxel 40x40x250 m

Porosimetry simulations• Predict mercury intrusion and extrusion using an

pseudo-equilibrium, percolation-based model.• Obtain predicted spatial distribution of mercury

for 2D slices.

X-ray tomography• Compare model predictions with X-ray images

using image analysis techniques, e.g. determine typical mercury ganglion size using auto-correlation function.

• Pellet dia. ~ 3 mm; Voxels 14x14x14 m• 2D grid size typically 220x220 pixels

On-going work

• Adapt pseudo-equilibrium model to include kinetic effects during retraction.

• Leading to increase in computing demands.

Computing requirements

• Progress limited by time required for simulations/image analysis to run on a PC.

• Simulations take ~1 day on good desktop PC.

• Image analysis takes ~2-3 days on PC.

Coupled diffusion and reaction processes

• Many diffusion-limited reactions in porous heterogeneous catalysts are accompanied by capillary condensation and/or coking.

• Liquid/solid deposition in pores affects mass transfer rates, and thus catalyst activity.

• How does the pore structure influence these processes?

• What is the best catalyst for a given process?

Diffusion and reaction simulations

• Model porous solid as a three-dimensional cubic lattice of interconnected cylindrical pores – 1000 nodes.

• Use dusty gas model and Kirchoff’s equations to model diffusion and reaction within network.

• For example, a pore network with 1000 nodes, a pore inter-connectivity of 6, 6 finite different points in each pore and 6 components in the vapour results in a system of differential equations with 124,800 unknowns.

Proposed solution of equations

• Rieckmann and Keil (1997) gave the Jacobian matrix for similar systems.

• As the Jacobian matrix is non-symmetric and not diagonally dominant, the system of non-linear equations is very difficult to solve.

• In this work the solution methods of Rieckmann and Keil will be applied. – A Schur complement technique is used to decompose

the Jacobian matrix, in order to decouple the pore equations from the node equations (Rieckmann and Keil, 1997 ).

– A subspace search algorithm is used to ensure that the solution of the model equations converges efficiently (Rieckmann and Keil, 1999)

Implementation• The diffusion/reaction model was coded in the ‘C’

programming language. • The non-linear equations for the single pores were

each solved using the FORTRAN NAG routine C05NBF.

• The decomposition of the Schur complement matrix and solution of the resulting linear system was performed using NAG routines F01BRF, F01BSF and F01AXF.

• The CPU time taken for the diffusion/reaction algorithm to converge on a dedicated DEC-alpha workstation is approximately 1 h.

Further work• Incorporate more complex pore geometry using pore

bond and throat networks?– But equations get more complex too.

Alternate approach• Some catalyst pellets have highly correlated pore

structures which may facilitate a meso-scale approach.

• Meso-scopic model will reduce computing demands and allow larger section of pellet to be modelled than for network models.