Chemical Reactor Analysis and Design

Chemical Reactor Analysis and Design

3th Edition

G.F. Froment, K.B. Bischoff†, J. De Wilde

Chapter 3

Transport Processes with Reactions Catalyzed by Solids

Part two Intraparticle Gradient Effects

Introduction

1. Transport of reactants A, B, ... from the main stream to the catalyst pellet surface.

2. Transport of reactants in the catalyst pores.3. Adsorption of reactants on the catalytic site.4. Chemical reaction between adsorbed

atoms or molecules.5. Desorption of products R, S, ....6. Transport of the products in the catalyst

pores back to the particle surface.7. Transport of products from the particle

surface back to the main fluid stream.

Steps 1, 3, 4, 5, and 7: strictly consecutive processes Steps 2 and 6: cannot be entirely separated !

Chapter 2: considers steps 3, 4, and 5Chapter 3: other steps

Molecular-, Knudsen- and surface diffusion in pores

[Adapted from Weisz, 1973]

(mainly encountered in zeolite catalysts)


Molecular diffusion:• Driven by composition gradient• Mixture n components: Stefan-Maxwell:

M

ij ijD

jDiiDj

i D

yyRTp

'

,

,,NN

• Molecular diffusivities (binary):• independent of the composition• inversely proportional to the total pressure (gas)• proportional to T3/2

• Momentum transfer: by collisions between atoms or molecules• Fluxes: expressed per unit external surface of the catalyst particle ijsij DD '

with εs: void fraction of the catalyst particle (m3f / m3

cat)= fraction particle surface taken by pore mouths (Dupuit)


Knudsen diffusion:• Mean free path of the components >>> pore dimensions• Momentum transfer: mainly collisions with the pore walls• Encountered

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

• Knudsen diffusion flux of i : independent of the fluxes of the other components:

l

p

RT

DiiK

iK

,, N

• Knudsen diffusivity:

• function of the pore radius• independent of the total pressure• varies with T1/2

iiK M

RTrD

8

3

2, iKbiK DD ,

', and:

i

j

jK

iK

M

M

D

D

,

,

(Graham’s law)


Simultaneous Molecular and Knudsen diffusion and flux from viscous or laminar flow:

tiK

toi

ijijD

jDiiDj

iK

iK

ip

D

pBy

D

yy

DRTp

'

,

'

,

,,

'

,

,

μ

NNN

Darcy’s permeability constant

Dusty gas model equation (kinetic gas theory)

Viscous flow term:• generally negligible• except when > 10 – 20 (micron-size pores)

iKto DpB ,/

Example: Binary mixture of A and B:

',

',

'

111

1

AKABD

A

BA

A DD

y

D

N

N

For equimolar counterdiffusion:

',

',

'

111

AKABDA DDD Additive resistance relation

(Bosanquet formula)


Simultaneous Molecular and Knudsen diffusion and flux from viscous or laminar flow:Diffusion in a multicomponent mixture:Sometimes Stefan-Maxwell replaced by less complicated equivalent binary mixture equation:

m

k jKj

j

kk

jkjm Dyy

DD 1'

,''

111

N

N


Surface diffusion:• Hopping of molecules from one adsorption site to another• Random walk model

'

2

τ

λkD

s

where:• : the jump length• : the correlation time for the motion• k : a numerical proportionality factor

'

Vary with temperature according to the van ‘t Hoff exponential law

• pre-exponential factor:• energy factor:

020 '/ kD as,

ED = 2Eλ – Eτ ~ 1/number of available sites

known for structured surfaces like zeolites, but much less for amorphous surfaces

• Depends on the surface coverage• More important in micro- than in macroporous material• Driving force: not fluid phase concentration gradient (Fickian law can not be applied)

Diffusion in a catalyst particle

A pseudo-continuum model:

Effective diffusivities:

Catalyst particles: very complicated (3D) pore structure

Model:• Pseudo-continuum• 1D• « Effective » diffusivity

Fick type law:

Pellet surface:

Sphere:

dz

dCDN A

eAA

r

C

rr

CDN AA

eAA

2

²

2


A pseudo-continuum model:

AsA

eA DD

D

'

sm

m

p

f3

in:

« Tortuosity » factor:• Tortuous nature of the pores• Eventual pore constrictions• Typical value: 2 - 3

Experimental determination of effective diffusivities of a component and of the tortuosity

Pulse response technique:• column packed with catalyst (fixed bed)• ideal plug flow pattern ( dt/dp)• tracer pulse injected in carrier gas flow• pulse response measured (reactor outlet)

In Out

Tracer pulse Pulse response

Fixed bed

Pulse widens:• Dispersion in the bed:

• Adsorption on the catalyst surface• Effective diffusion inside the catalyst particle

• Three parameters to be estimated: method of moments


Wicke-Kallenbach cell:• Steady state or transient operation• Single catalyst particle used as membrane• Above membrane: steady flow of carrier gas• Tracer pulse injected into the carrier gas:

• Diffuses through the catalyst membrane• Swept in the compartment underneath by a carrier gas => to detector

• Two parameters to be estimated

Determination tortuosity:• Specific catalyst characterization equipment (mercury porosimetry & nitrogen–sorption and –desorption)


EXAMPLE 3.5.1.2.AExperimental determination of the effective diffusivity of a component and of the catalyst tortuosity by means of the packed column technique

• Pt-Sn-y-alumina catalyst (catalytic reforming of naphtha)• Column internal diameter: 10-2 m• Column length: 0.805 m• Particle radius: 0.975 × 10-3 m• Void fraction of the packing: 0.429 m3

f / m3r

• Catalyst density, ρcat: 1080 kg cat/m3cat

Hg-porosimetry, N2-adsorption and -desorption?,s

Hg porosimetry:• Pore volume as a function of the amount of intruded mercury• Pore radius: calculated from Washburn eq. (cylindrical pores)• At 2000 bar: all pores > 3.3 nm filled with HgNitrogen sorption:• Steep increase of the amount adsorbed at pressure where the macropores are filled by nitrogen through capillary condensation• From Washburn eq.: total volume of adsorbed N2

• The volume of N2 adsorbed until the sharp rise is the meso pore volume


From Van Melkebekeand Froment [1995]


Cumulative pore volume distribution:• Derived from N2 adsorption curve: Broekhoff-De Boer eq. (cylindrical pores)• Inflection point => differential pore volume distribution by a peak at mean pore radiusTracer pulse injected into packed column:• Fitting data: Kubin and Kucera-model => De, KA, and Dax

=>

Remarks:• Performing experiments at various total pressures => possible to distinguish between and • Measurements possible in the absence or presence of reactions

D K


Structure and Network models: (in contrast to Pseudo-continuum model)

• More realistic representation• More accurate

Structure models:• The random pore model• The parallel cross-linked pore model

Network models:1. A Bethe tree model2. Network models for disordered pore media

• Monte Carlo simulation• Effective Medium Approximation (EMA)


Structure models: The random pore model:• Macro- micro pore model [Wakao and Smith, 1962 & 1964]• Application: pellets manufactured by compression small particles• Void fraction- and pore radius distributions: each replaced by two averaged values, for the macro for the micro distribution (often a pore radius of ~100 Å is used as the dividing point between macro and micro)• Micro-pores particles: randomly positioned in pellet space• Macro-pores of the pellet: interstices• Diffusion flux: three parallel contributions:

1. Through the macro-pores2. Through the micro-pores3. Through interconnected macro-micro pores

DDDD

M

MM

M

MMMe 2

2

2

222

1122

11

DDM

MMM

1

3122

KKMABorM DorDDD

111with:


Structure models: The random pore model:

Diffusion areas in random pore model. Adapted from Smith [1970].


Structure models: The parallel cross-linked pore model• Pore size and orientation distribution function:• Pellet flux: integrating flux in single pore with orientation l and accounting for the distribution function:

),( rf

ddrrfN ljlj ),(,N

lwith: : unit vector or direction cosine between l direction and coordinate axes

Example: mean binary diffusivity:dl

dCDN j

jmlj , jljm CD .

Kjj

j

kk

N

k jkjm Dy

N

Ny

DD

111

1

with:

ddrrfCD jlljmj ),(.N

l lwith: the tortuosity tensor


Structure models: The parallel cross-linked pore modelLimiting cases:

1) Perfectly communicating pores Cj(z; r, Ω) = Cj(z)

2) Noncommunicating pores:Pure diffusion at steady state, dNj/dz = 0 or Nj = constant

+ no assumption on communication of pores

),(00

rfdCDddrdzN ll

C

C

jjm

L

jz

jL

j

(closest to usual types of catalyst particles)

jsjmj CrdrrD )()()( N

with: κ(r) : a reciprocal tortuosity (results from the integration over Ω)Proper diffusivity: weighted with respect to the measured pore size distribution


Structure models: The parallel cross-linked pore modelLimiting cases:

3) Pore size and orientation effects are uncorrelated

)()(),( frfrf

with:• f(r) : the pore size distribution

• 1)( df

jsjmj CdfrdrD )()(N

Completely random pore orientations=> tortuosity depends only on the vector component cos

3

1cos2 dfdf

= 3


Network models: A Bethe tree model:

branching network of pores:• coordination number of 3• no closed loops

• Higher coordination numbers possible• Pores can have a variable diameter• Main advantage: can yield analytical solutions for the fluxes• Disadvantage: absence of closed loops not entirely realistic


Network models: Disordered pore media:• Amorphous catalysts: no regular or structured morphology• Sometimes structure modified during its application (pore blockage)

• Pore medium description:• Network of channels (preferably 3D)• Size distribution• Disorder to be included: certain fraction of pores blocked

Random number generator(Monte Carlo simulation)

Calculations repeated for same over-all blockage probability & average pore size => calculated set of values of De is averaged

• Effective Medium Approximation (EMA): construct small size network => relation between diffusivity & blockage without considering complete network

Diffusion and reaction in a catalyst particle. A continuum model

First-Order Reactions. The Concept of Effectiveness Factor:

Reaction and diffusion occur simultaneous: Process not strictly consecutive Both phenomena must be considered together

Example: first-order reaction, equimolar counterdiffusion, isothermal conditions, and steady-state: slab of thickness L:

Species continuity equation A: 02

2

sss

eA Ckdy

CdD

with boundary conditions:sss CLC )( at the surface

0)0(

dy

dCs at the center line

LD

k

yD

k

C

yC

eA

s

eA

s

ss

s

cosh

cosh)(

Solution:

' modulus eAs DkL /



with: for a slab of thickness L



Effectiveness factor:

conditionssurfaceatreactionofrate

resistancediffusionporewithreactionofrate

)(

)(1

ssA

csAc

Cr

dWCrW

Observed reaction rate: )( ssAobsA Crr

First-order reaction'

'tanh

Extension to more practical pellet geometries: cylinders or spheres:

e.g., sphere: sAAs

eA rdr

dCr

dr

d

rD

2

2

1

eA

s

D

k

S

V Aris [1957]:



Effectiveness factors for slab (P), cylinder (C), and sphere (S) as functions of the Thiele modulus. Dots represent calculations by Amundson and Luss [1967] and Gunn [1965]. From Aris [1965].


More General Rate Equations. Single rate equation:

Analytical solution not possible

• Generalized modulus (10 – 15% error)

• Numerical solution

2/1

'''

,

)()(2

)(

ss

eqs

C

C

sssAseAs

ssA dCCrCD

Cr

S

V

Coupled multiple reactions:

Numerical solution:• finite difference• orthogonal collocation

Nonane

Dimensionless radial coordinate, r

0.0 0.2 0.4 0.6 0.8 1.0

Pi/P

s

0.0

0.2

0.4

0.6

0.8

1.0

Wilke

Stefan-Maxwell

Catalytic reforming of naphtha on Pt.Sn/alumina. Dimensionless partial pressure profile inside the particle for the reactant nonane. Total pressure: 7 bar, T = 510 °C, molar ratio H2/Hydrocarbons = 5. From Sotelo-Boyas and Froment [2008].

Depends on Cs !

Falsification of rate coefficients and activation energy by diffusion limitations

Consider nth order reaction:

nssobsA Ckr

nssCk

1

~

2/1

1

2

nsseAobsA CkD

nV

Sr

Introduce generalized modulus:

observed rate: order (n+1)/2

only correct for 1st order reaction

Also: kkobs

2/1/0

/

1

2 RTERTED eAeA

nV

SD

22/1

ln EEE

Td

kdE Dobs

obs

effective diffusion

(strong pore diffusion limitations)


ln

ln

2

11

d

d

E

Eobs

ln

ln

2

1

d

dEEEE Dobs

Weisz and Prater [1954]:

or:

Languasco, Cunningham, and Calvelo [1972]:

lnd

lnd

2

1obs

nnn

EXAMPLE 3.7.A EFFECTIVENESS FACTORS FOR SUCROSE INVERSION IN ION EXCHANGE RESINS

First-order reaction: 612661262112212 OHCOHCOHOHC H

(sucrose) (glucose) (fructose)

Studied in particles with different size [Gilliland, Bixler, and O’Connell, 1971]


EXAMPLE 3.7.A EFFECTIVENESS FACTORS FOR SUCROSE INVERSION IN ION EXCHANGE RESINS

dp (mm) as sk )( 1 04.0/ kk

eAs DkR /

0.04 0.0193 1.0 0.53 1.0 0.27 0.0110 0.570 3.60 0.600 0.55 0.00664 0.344 7.35 0.352 0.77 0.00487 0.252 10.3 0.263

a Calculated on the basis of approximate normality of acid resin = 3N.DeA = 2.69 × 10-7 cm2/s.Separately measured:

dp (mm) E (kJ/mol) 0.04 105 0.27 84 0.55 75 0.77 75

Homogeneous acid solution 105 From data at 60 and 70°C. ED = 34 kJ/mol obsE =

2

34105 = 70 kJ/mol theor.

Compare: Diffusional resistance decreases selectivity !

Influence of diffusion limitations on the selectivities of coupled reactions

[Wheeler, 1951]Parallel, independent reactions:

RA 1 , with order 1a

SB 2 , with order 2a

Absence of diffusion limitations:

2

1

2

1a

Bs

aAs

S

R

C

C

k

k

r

r

With pore diffusion limitations:

2

1

22

11a

Bs

aAs

obsS

R

C

C

k

k

r

r

Strong pore diffusion limitations: i ~ i/1

2/1

1

1

2

1

1

2

2

1

1

1a~

asBs

asAs

eB

eA

obsS

R

C

C

D

D

k

k

ar

r

First-order reactions & eBeA DD

sBs

sAs

obss

R

C

C

k

k

r

r

2

1


[Wheeler, 1951]Consecutive first-order reactions:

SRA 21

Absence of diffusion limitations:

As

Rs

A

R

C

C

k

k

r

r

1

21


Species continuity equations A and R, for slab geometry:

AssAs

eA Ckdz

CdD 12

2

RssAssRs

eR CkCkdz

CdD 212

2

Selectivity

L

As

L

RsAs

L

A

L

R

obsA

R

dzCk

dzCkCk

dzr

dzr

r

r

0

1

0

21

0

0

Selectivity:


[Wheeler, 1951]Consecutive first-order reactions:


sAs

sRs

C

C

k

k

1

2

1

212

1

/1

with:i

ii

tanh eisii DkL /

eR

eA

Dk

Dk

1

2

2

1

2

with i = 1, 2

Strong pore diffusion limitation and for DeA = DeR:

obsA

R

r

r

~ s

As

sRs

C

C

k

k

1

21

1

1

sAs

sRs

C

C

k

k

kk 1

2

12 /1

1

Compare: Diffusional resistance decreases selectivity !

Criteria for the importance of intraparticle diffusion limitations

Determining kinetic parameters from experimental data:• kρs not available yet• Criteria importance pore diffusion not explicitly containing kρs also needed !1) Experiments with two different sizes of catalyst:

Assume kρs and DeA same for both sizes

2

1

2

1

L

L

S

VL with:

2

1

2

1

obs

obs

r

rand:

No intraparticle diffusion limitations: 21 = 1

Strong intraparticle diffusion limitations: = 1/

1

2

1

2

2

1

L

L

r

r

obs

obs

:

2) Weisz-Prater criterion [1954]:

2

2obs

sseA

sA

CD

LrFirst-orderreaction:

No pore diffusion limitation: << 1, η = 1 => 2 << 1; Strong pore diffusion limitation: >> 1, η = 1/ => 2 >> 1.

Extendable via generalized modulus

Combination of external and internal diffusion limitations

s

seA

ssg dz

dCDCCk

Nonisothermal particles

Practical situations:• Internal temperature gradients unlikely• Internal gradients unlikely to cause particle instability

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