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Mass transport and reaction in catalyst pellets

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Abstract

When heterogeneous chemical reactions take place in porous catalysts, mass transport can occur by bulk diffusion, Knudsen diffusion, and convective transport. Previous studies of these phenomena have been largely based on Maxwell's ‘dusty gas’ model with the convective transport or ‘Darcy flow’ added to the diffusive transport. This is done in order to satisfy one of the limiting conditions encountered in the study of flow in porous media. A more fundamental approach consists of the use of the method of volume averaging and the general form of the species momentum equation. For an N-component system, this leads to N independent flux relations to be used in conjunction with the volume-averaged species continuity equations.

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Abbreviations

Å A (t):

surface area of a species body, m2

a v :

interfacial area per unit volume, m-1

A γe :

area of entrances and exits for the γ-phase contained within the averaging volume, m2

A γK :

area of the γ-κ surface contained within the averaging volume, m2

b A :

species A body force, N/kg

b :

mass average body force, N/kg

B :

inverse tortuosity tensor for bulk diffusion

c :

total molar concentration, moles/m3

c A :

species A molar concentration, moles/m3

ĉ A :

surface concentration of species A, moles/m2

〈CA〉2 :

intrinsic phase average molar concentration, moles/m3

\(\tilde c_A\) :

c A − 〈CA〉2, spatial deviation concentration, moles/m3

c A :

mean molecular speed for species A, m/s

\({\text{D}}_{{\text{A,eff}}}^{\text{K}}\) :

binary diffusion coefficient, m2/s

D K, eff A :

Knudsen diffusion coefficient for species A, m2/s

f :

vector that maps ▽〈P A 〉γ into ∼P A , m

g :

gravitational vector, m/s2

G :

second order tensor that maps 〈N A〉γ into N A for free molecule flow conditions

H :

inverse tortuosity tensor for Knudsen diffusion

I :

unit tensor

j A :

c A u * A , molar diffusive flux, moles/m2s

K :

Darcy's Law permeability tensor, m2

L :

macroscopic length scale, m

L D :

diffusive length, m

l γ :

characteristic length for the γ-phase, m

l A :

mean free path for species A, m

M A :

molecular weight of species A, kg/mole

n :

outwardly directed unit normal vector

n γK :

unit normal vector directed from the γ-phase toward the κ-phase

n γ :

outwardly directed unit normal vector at the entrances and exits of the γ-phase contained within the averaging volume

N A c A v A :

molar flux of species A, moles/m2s

〈N A〉γ:

intrinsic phase average of the species A molar flux, moles/m2s

\~N A :

spatial deviation of the molar flux of species A, moles/m2s

p :

total pressure, N/m2

P :

p + ϱφ, total pressure over and above the hydrostatic pressure, N/m2

P A :

partial pressure of species A, N/m2

〈p A〉γ:

intrinsic phase average partial pressure, N/m2

\(\tilde P_A\) :

PA − 〈p A〉γ, spatial deviation partial pressure, N/m2

P A pA + ϱAφA :

partial pressure of species A over and above the hydrostatic pressure of species A, N/m2

p ab :

diffusive force exerted by species B on species A, N/m3

ℛ:

universal gas constant, N m/moles K

R A :

molar rate of production of species A owing to homogeneous chemical reaction, moles/m3s

\(\hat R_A\) :

molar rate of production of species A owing to heterogeneous chemical reaction, moles/m2s

r A :

mass rate of production of species A owing to homogeneous chemical reaction, kg/m3s

r 0 :

radius of the averaging volume, m

r :

position vector, m

t :

time, s

t A :

species stress vector, N/m2

T A :

species stress tensor, N/m2

T :

total stress tensor, N/m2

T :

temperature, K

〈T〉:

spatial average temperature, K

u A :

v A − v, mass diffusion velocity, m/su *A vA − v*, molar diffusion velocity, m/s

u o :

velocity of the rigid, solid phase relative to some inertial frame, m/s

v A :

species velocity, m/s

v :

mass average velocity, m/s

v * :

molar average velocity, m/s

v *A :

species velocity of those molecules of species A generated by chemical reaction, m/s

∨ A (t):

volume of a species A body, m3

∨:

averaging volume, m3

V γ :

volume of the γ-phase contained within the averaging volume, m3

V κ :

volume of the κ-phase contained within the averaging volume, m3

〈v〉:

phase average, mass average velocity, m/s

w :

arbitrary velocity vector, m/s

x A c A /c :

mole fraction of species A

〈X A 〉:

intrinsic phase average mole fraction

\(\tilde \chi _A\) :

X A − 〈X A 〉γ, spatial deviation mole fraction

∈γ :

Vγ/V volume fraction of the γ-phase

εA :

sum of all terms in the species A momentum equation that are small compared to the diffusive force, N/m3

μ:

viscosity of the γ-phase, Ns/m2

ϱA :

mass density of species A, kg/m3

ϱ:

total mass density, kg/m3

τ a :

species viscous stress tensor, N/m2

τ:

total viscous stress tensor, N/m2

τ:

tortuosity factor

φ:

total body force potential function, Nm/kg

φ a :

species body force potential function, Nm/kg

π:

3.1416

ω a :

ϱ a /ϱ mass fraction of species A

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Author notes

  1. Stephen Whitaker

    Present address: Department of Chemical Engineering, University of California, 95616, Davis, CA, U.S.A.

Authors and Affiliations

  1. Department of Chemical Engineering, University of Houston, 77004, Houston, TX, U.S.A.

    Stephen Whitaker

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  1. Stephen Whitaker
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Whitaker, S. Mass transport and reaction in catalyst pellets. Transp Porous Med 2, 269–299 (1987). https://doi.org/10.1007/BF00165785

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  • DOI: https://doi.org/10.1007/BF00165785

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