CBEN 518: Reaction Kinetics and Catalysis

Heterogeneous Catalysis

bulk fluid stream

Porous Catalyst 𝑪𝑨𝑺

𝑺

𝑪𝑨

1

𝑪𝑨𝑺

2𝑪𝑨𝒍 3

𝑪𝑹𝒍

4

𝑪𝑹𝒔

5

𝑪𝑹𝑺

𝑺

6

𝑪𝑹

7

Adsorption Isotherms

Mass Transfer

Intraparticle Transport

POROUS CATALYST

N.R. BoyleCBEN 518

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Surface Area to Mass Ratio

•Why use porous catalysts?•Want high surface area to expose more of the active (expensive) catalytic material and avoid sintering

•Typically surface area ~10-1000 m2/g• This requires lots of pores!

•Example of why porous structures are necessary•Non-porous alumina sphere 1 mm in diameter:

2 -4 2

3 -4 33

2 23

4

Surface Area 4πr 4π(5x10 m)4 4 kgMass πr ρ π(5x10 m) 40003 3 m

Surface Area 3 m 1kg m1.5x10Mass 5x10 *4000 kg 1000g g

Put this In Perspective

6.6 kg to 666.6 kg of alumina is required to achieve 10-1000 m2 of surface area!

Given 1 sphere is 2.1x10-6 kg that equates to 3.17 million to 317 million alumina spheres

Why use porous catalysts?

This value is orders of magnitude too small!!!

Characterization of Porous Catalysts•Some definitions:

•Porosity: es = pore volume/geometric volume•Surface Area: Ap (typically B.E.T. measurement)•Particle density: p (mass catalyst/geometric volume)•Skeletal density: s (mass catalyst/solid volume--no pores)•Bulk density: b (mass catalyst/reactor volume)•Pore Volume: Vp (pore volume/mass of catalyst)

•Average pore diameter•Assume all n pores are same length L and radius r:

2p

p

p

p

Total pore volume n πr L V

Total pore surface area n 2πr L A

V rA 2

Completely uniform (may be a bad approx..)

𝜺𝒔=𝑽 𝑷 𝝆𝑷=[ 𝟏𝝆𝑷− 𝟏𝝆𝒔 ]𝝆𝑷=𝟏−

𝝆 𝑷

𝝆 𝒔

• Range of pore sizes in which different types of diffusion occur– Molecular diffusion (regular)

+ Molecule – molecule Þ Important when mean free path <<

pore radius – Knudsen diffusion

+ Molecule – pore Þ Important for small pore radius

– Configurational diffusion+ Mainly encountered with zeolite

catalysts

• Molecular (bulk) diffusion dominates if r >> 50 nm (at 1 atm)

• Always dominates in liquids where mean-free-path comparable to molecular diameter

Transport in Pores

Mean Free Path (MFP)

•The mean free path is defined as:

• It represents the average distance travelled by a particle between collisions

•Example MFP for air at different air pressures:

Vacuum range Pressure in hPa (mbar) Molecules / cm3 Molecules / m3 Mean free path

Ambient pressure 1013 2.7 × 1019 2.7 × 1025 68 nm[4]

Low vacuum 300 – 1 1019 – 1016 1025 – 1022 0.1 – 100 μm

Medium vacuum 1 – 10−3 1016 – 1013 1022 – 1019 0.1 – 100 mm

High vacuum 10−3 – 10−7 1013 – 109 1019 – 1015 10 cm – 1 km

Ultra high vacuum 10−7 – 10−12 109 – 104 1015 – 1010 1 km – 105 km

Extremely high vacuum <10−12 <104 <1010 >105 km

Molecular Diffusion

•Driven by a composition gradient • In a mixture of n components, the partial pressure gradient is given by the Stefan-Maxwell equation:

•Fluxes are expressed per unit external surface area of the catalyst particle, so diffusivity had to be reduced by a factor of εs (void fraction of the catalyst particle)

Knudsen Diffusion

• When the MFP is much larger than the pore dimensions, the momentum transfer mainly results from collisions with the pore walls

• Typically encountered at pressures below 5 bar and pore sizes between 3 and 200 nm

• Flux is then written as

Where l is a vacant active site in the catalyst

• Knudsen diffusive flux is independent of the fluxes of other components

• The diffusivity is given by

• The ratio of diffusivity for components i and j is given by Graham’s Law

Knudsen Diffusion•Like molecular diffusion, Knudsen diffusion flux is related to the total particle surface area:

•When both types of diffusion occur and there is flux form viscous or laminar flow, the partial pressure gradient is given by:

Where Bo is Darcy’s permeability constant The viscous flow term is general negligible, except when

Micron size pores

Surface Diffusion•Proceeds by hopping of the molecules from one adsorption site to another

•Diffusivity is given by

Where λ is the jump length, τ’ is the correlation time for the motion and k is a numerical proportionality factor

•Surface diffusivity has been shown to depend on the surface coverage and to be more important in micro- than in macroporous material

Effective Diffusivities

•The internal pore structure of catalyst particles is very complicated, therefore describing diffusion from external surface to the active sites is not a simple task

•Practically, the catalyst particle is generally considered as a continuum through which the molecules move by “effective” diffusion

•Or spherical coordinates,

Where τ is tortuosity of the catalyst pore (ranges from 1 for straight pores to 3-7 for crooked pores)

Saturday, April 22, 2023 C Mark MaupinChEN 518

Other ways to define diffusivity in pores (not going to cover in class but you need to read about them in your text)

•The continuum model w/ global characteristics (void fraction, tortuosity) is convenient but it is not accurate

• May lead to inaccurate prediction of catalyst performance • With increased computational power, other models have been

developed which are more realistic

•Random Pore Model •Parallel Cross-linked Pore Model •Network Models

Estimating Temperature Differences

• Plug in relationship for kg and hf

• For gases flowing in packed beds, the values of the groups are such that :

• The maximum possible temperature different would occur for complete conversions and very rapid reaction and heat release so

N.R. BoyleCBEN 518

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