2015_10_12_Calculating Pressure Drop Across Film
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Transcript of 2015_10_12_Calculating Pressure Drop Across Film
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CBEN 518: Reaction Kinetics and Catalysis
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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
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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!!!
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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
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• 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
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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
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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)
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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
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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
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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
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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)
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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
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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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