ChE 400 - Reactive Process Engineering
ChE
What happened so far…
1
9 Heterogeneous Catalysis requires more than just reaction:
(external) mass transfer – (internal) pore diffusion – reaction !
9 Concentration profiles in and around a catalyst pellet for different ratelimiting steps
9 Mass-transfer and (1st order) reaction: keff = km×kr /(km+kr)
9 Calculate km from Sherwood number!
Be careful to pick the right correlation! Make sure your set-up is not
mass-transfer limited when measuring heterogeneous reaction kinetics!
…what about Pore Diffusion Limitations??
ChE 400 - Reactive Process Engineering
ChE
Catalyst Surface Area
2
Rate of catalytic surface reactions is proportional to the catalyst surface area.
However, accessibility of that surface area is typically limiting, as most of it
is inner surface area inside the porous catalyst structure.
Typical catalyst specific surface areas:
Scat = 1 - 500 m2/g
Assume a pellet with 1mm diameter. Total surface: Atot =
Sphere volume: V =
Density (Al2O3): ρ = 3.9 × 106 g/m3
Total mass: M =
Specific external surface area: Acat = Atot/M =
Essentially
ysts isis
Essentiallyall
allthe
thesurface
surfacearea
areaof
ofmost
mosttypical
typicaltechnical
technicalcatal
catalysts
internal
internalsurface
surfacearea!
area!
Hence, a catalyst can only be used at its maximum potential, if diffusion inside
the pore structure is not limiting…
ChE 400 - Reactive Process Engineering
ChE
Internal Mass Transfer Limitations
3
To be able to calculate the reaction rate inside a catalyst pore, we need to know the
concentration profile inside the pore. How can get this information??
Mass balance over (pore) volume element:
CAs
flow in – flow out + production by reaction = 0
CA(z)
d 2C A ⎛ 4 kr′′ ⎞
−⎜
⎟C = 0
⎜ d P DA ⎟ A
dz 2
⎝
⎠
This is a 2nd order ODE which can be solved with the boundary conditions CA(0) = CAs
and (dCA/dz)z=l = 0.
e − λ (l − x ) + eλ (l − x )
cosh(λ (l − x))
C A ( x) = C As
=
C
As
e − λl + eλl
cosh(λl )
(check the detailed math in the textbook)
ChE 400 - Reactive Process Engineering
ChE
Effectiveness Factor
4
We don’t really care about the details of this concentration profile, but we do care
about how it influences the effective reaction rate!
For this purpose, we define:
(integral mean of ) real reaction rate at local concentrations
“effectiveness
η =
factor”
ideal reaction rate at (external ) surface concentrations
l
∫ k ′′ C ( z ) dz
r
…which gives…
..where we defined for convenience: λ =
=
A
z =0
kr′′ C As l
1 cosh(λ (l − z ))
tanh(λl )
= ∫
dz =
l 0 cosh(λl )
λl
l
4 kr′′
d P DA
The dimensionless group (λ l) which appears in the effectiveness factor is also called
Thiele Modulus
φ= λl= l
4 kr′′
d P DA
Instead of the Thiele modulus, you will also
often find the Damköhler number
DaII = (φ)2
5
How does the effectiveness factor η change with the Thiele modulus φ?
So, the plot of η vs φ looks like
η=
1
tanh φ
φ
η
ChE 400 - Reactive Process Engineering
ChE
Effectiveness Factor & Thiele Modulus
0
0.1
Limiting cases:
φφ small,
small, ηη →
→ 11
φφ large,
φ
large, ηη →
→ 1/
1/φ
1
φ
10
ChE 400 - Reactive Process Engineering
ChE
Effectiveness Factor: Example
6
Fischer-Tropsch synthesis:
synthesis
synthesis of higher hydrocarbons from synthesis gas.
Important large-scale processes in South Africa (Sasol). Outside South Africa, only one plant
operating (Shell, in Malaysia). FT is expected to gain importance in near future due to an
anticipated change in the petro-chemical industry from oil-based to natural gas- and coal-based
processes.
Post et al., AIChE J. 35 (1989) 1107
ChE 400 - Reactive Process Engineering
ChE
Exercise: MTL in a Catalytic Reactor
7
You are designing a catalytic reactor (VR= 10 l) for a reaction A -> B, for which a
pseudo-homogeneous rate coefficient of kR= 0.01 s-1 has been measured. The
reactor is filled with 10,000 catalyst pellets. Each pellet has a total surface area
of about 1000 cm2. The pellets are spherical with a diameter of dcat= 1 mm and
have pores with a typical pore diameter of dP = 10-4 mm and a characteristic
length l= 0.3 mm.
Is the overvall reaction mass transfer limited, pore diffusion limited or reaction
limited?
(Assume DA,eff = 10-5 cm2/s and Sh = 2.0).
(external) mass transfer:
km = Sh DA/dcat =
k” =
Hence…
pore diffusion:
Thiele modulus:
φ=
Hence…
ChE 400 - Reactive Process Engineering
ChE
Recap: Where are we…?
8
effectiveness
factor
Thiele Modulus
η=
real reaction rate at local concentrations
ideal reaction rate at external surface concentrations
φ = λ l = l
4 kr′′
dP DA
l2
4 kr
4
φ =l
kr ⋅
=
d P DA d P
DA
2
2
(1)
We know that pore diffusion can substantially alter the catalytic reaction rate.
(2)
We know that the “real reaction rate” is: r = η rideal = η kr” Cas
(3)
We can calculate η from φ : η = tanh(φ) / φ
(4)
We can calculate φ from the above equations.
However, the Thiele Modulus was derived for a 1st order reaction in a single
straight, cylindrical pore, or for irregular pores in a porous catalyst slab!
We need to generalize this for other catalyst shapes, other reaction orders, and
less idealized configurations…!
ChE 400 - Reactive Process Engineering
ChE
η & φ : Catalyst Geometries
9
What if we do not have a 1st order reaction on a spherical catalyst…?
Different catalyst geometries:[See Eqns. (7.60), (7.63)]
Asymptotic solutions for different catalyst geometries converge, if l is replaced by
a characteristic length (Vcat/Acat), where Acat is the external surface area of the
pellet!
Some common
catalyst shapes:
sphere
cylinder
slab
1
1/Φ
ChE 400 - Reactive Process Engineering
ChE
η & φ : Reaction Orders
10
What if we do not have a 1st order reaction on a spherical catalyst…?
Different reaction orders:
Again, the result is (almost) invariant against changes, if a ‘modified’ Thiele modulus
is used:
3rd
2nd 1st 0 order
n−1
Vcat n +1 4 kr′′ CAs
⋅
φ=
Acat
2
dP DA
(valid for n > 0!)
ChE 400 - Reactive Process Engineering
ChE
η & φ : Pore Diameter
11
What if we do not have regular, straight cylindrical pores…?
The pore diameter dP can be replaced by an effective pore diameter based on
(easily) observable quantities:
For an ideal, infinitely long cylinder with diameter d, the surface-to-volume ratio is 4/d.
For any catalyst, we can measure (or often are given as information by the manufacturer) the
surface area per unit mass Sg.
The pore volume per unit mass is the porosity divided by the particle density: εP/ρcat.
The average surface area per pore volume is thus: Sg ρcat/εP.
Thus, we can now define an effective pore diameter deff to replace the ‘ideal’ pore diameter in
our derivation:
4 Sg ⋅ρcat
deff
=
ε
Hence, the effective pore diameter deff is:
Substituting into the expression for the Thiele modulus yields:
ChE 400 - Reactive Process Engineering
ChE
η & φ : More on Pores…
12
We are still assuming idealized pores...
To account for the non-ideal shape of the pores, an effective diffusion coefficient
Deff must be introduced:
porosity
when nothing known, assume 0.1
ChE 400 - Reactive Process Engineering
ChE
Pore Diffusion: Mechanisms
13
Dominant in
large pores:
T3
DAB ∝
Ptot
Typical values at RT:
10-3 – 10-5 cm2/s
Eact ≈ 0.5 . ΔHads
Dominant in
small pores:
T
Mi
(rP in cm, Mi in g/mol,
T in K!
Yields DKi in cm2/s)
(also called
‘configurational’
diffusion)
DKi = 9700 rP
Note the different dependencies!
Intermediate
))-1-1== (D
))-1-1++(D
))-1-1
AB
K
Intermediatepore
poresizes:
sizes: (D
(Deff
(D
(D
eff
AB
K
(‘Bosanquet
-equation’)
(‘Bosanquet-equation’)
ChE 400 - Reactive Process Engineering
ChE
Diffusion Coefficients
14