catalytic reaction engineering...reaction engineering professor of industrial chemistry department...
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CatalyticReaction Engineering
Professor of Industrial ChemistryDepartment of Chemical and Metallurgical EngineeringSchool of Chemical TechnologyAalto UniversityEmail: [email protected] 1, E404
Yongdan Li
Nov-Dec, 2018
8. Overall Diffusion
8.1 Evaluation of Diffusion Limitations
A quick and dirty estimate of diffusion limitations
2. Mears criterion for external diffusion
Observed reaction rates are used
1. Weisz-Prater criterion for internal diffusion
3
8.1 Weisz-Prater Criterion for Internal Diffusion
1)coth(3
112
1
η
(1st order, spherical particles)
)1coth(3 11
2
1 η
2
1WPC η
Weisz-Prater criterion is written as Cwp:
Rearranging internal effectiveness factor:
(8-2)
(8-1)
4
η = Effectiveness factor of internal diffusion
Ф1= Thiele modulus of first order reaction
Cwp = Weisz Prater parameter
8.1 Weisz-Prater Criterion for Internal Diffusion
Internal effectiveness factor and Thiele modulus have already
been determined earlier
A
As
( )'
'
r obsη
r
2 22 As As
1
As As
'' '
a c c
e e
r S ρ R r ρ R
D C D C
2 22 As AA
1
As As As
( )( )' ''
c cWP '
e e
r ρ R r obs ρ Rr obsC
r D C D C
Weisz-Prater parameter (1st order reaction):
R=catalyst particle radius, m
ρc=solid density of catalyst pallet, kg/m3
CAs=surface reactant concentration, mol/m3
Sa=internal surface area/mass of catalyst, m2/g
De=Effective diffusivity, m2/s
(8-3) (8-4)
(8-5)
5
8.1 Weisz-Prater Criterion for Internal Diffusion
All terms are measured or known, if limited by internal
diffusion
22 A
1
As
( )'
cWP
e
r obs ρ RC
D C
If CWP << 1 => no internal diffusion resistance
If CWP >> 1 => internal diffusion limits the reaction
(8-5)
6
8.1 Mears Criterion for External Diffusion
A b
Ab
( )'
c
r obs ρ RnMR
k C
External diffusion is evaluated with MR
ρb=bulk density of catalyst bed, kg/m3
=(1-ɛ)ρc (ɛ=voidage)
n=reaction order
R=catalyst particle radius, m
ρc=solid density of catalyst pallet, kg/m3
CAb=bulk reactant concentration, mol/dm3
kc=mass transfer coefficient, m/s
kc can be calculated e.g. by Thoenes-
Kramers correlation (for flow throu-
gh a packed bed)
Others can be measured
MR<0.15
External diffusion is neglected(8-6)
7
8.2 Overall Effectiveness Factor
What happens if both external AND internaldiffusion exist simultaneously…?
8
8.2 Overall Effectiveness Factor
Useful for first order reactions
External and internal diffusion in packed catalyst beds
In steady state, molar flow of reactant A to the catalyst surface can be expressed
MA= molar flow of reactant A to the catalyst surface
ac = external surface area of catalyst particles per unit volume of the reactor
V = the volume
A A Δr cM W a V (8-7)
9
8.2 Overall Effectiveness Factor
Because often Sab >> ac (porous particle)
Molar rate of A transfers to the surface, MA, is equal to the net (total) rate
of reaction on and within the pellet
A A (external area internal area)''M r
A A b( Δ Δ )''
c aM r a V S ρ V
ρb=bulk density of catalyst bed, kg/m3
=(1-ɛ)ρc (ɛ=voidage)Sa=internal surface area/mass of catalyst, m2/g
A A b A A b( )'' ''
A r c c a r c aM W a V r a S ρ V W a r S ρ
(8-8)
(8-9)
10
8.2 Overall Effectiveness Factor
For external mass transfer
WAr ac = kc(CAb- CAs) ac
Since internal resistance affects the system, CAs cannot be neglected in this case
(as earlier)
''
A As A 1 As
'' '' ''r ηr r ηk C (8-11)
Consumption rate for a first order reaction
It is not possible to measure CAs=> elimination
(8-10)
11
k1’’= Rate constant per unit area
CAb ≠ CAs ≠ CA and only CAb can be measured
By combining equations (8-9),(8-10) and (8-11):
''
Ab As 1 b As(C )c c ak a C ηk S ρ C
For CAs
AbAs ''
1 b
c c
c c a
k a CC
k a ηk S ρ
8.2 Overall Effectiveness Factor
(8-12)
(8-13)
12
8.2 Overall Effectiveness Factor
By substituting CAs in first order rate equation:
'' ''Ab1''
1 b
c cA
c c a
k a Cr ηk
k a ηk S ρ
''
1 Ab''
1 b1 ( )a c c
ηk C
η k S ρ / k a
Where is internal effectiveness factor
As s
Actual overall (observed) reaction rate
reaction rate if entire interior surface were exposed to
the conditions of external surface ,
η
C T
(8-14)
(8-15)
13
8.2 Overall Effectiveness Factor
Overall effectiveness factor is defined as
Actual overall (observed) reaction rate
reaction rate if whole particle were exposed to the conditions of bulk , Ab bC T (8-16)
Based on the definitions of Ω and equation (8-14):
A Ab( )'' ''r Ω r ''
1 Ab''
1 b1 ( )a c c
ηk C
η k S ρ / k a
''
1 b1 ( )a c c
ηΩ
η k S ρ / k a
A
''r
So the overall effectiveness factor is
''
1 b1 ( )a c c
η
η k S ρ / k a
Ab( )''r
(8-19)
(8-17) (8-18)
14
8.2 Overall Effectiveness Factor
''
1 b1 ( )a c c
ηΩ
η k S ρ / k a
Large flow rate results in large external transfer coefficient kc
Neglect
Therefore, in the case without external diffusion
Ω
Overall effectiveness factor approaches internal effectiveness factor
(8-20)
(8-21)
15
8.3 Mass Transfer and Reaction in a Packed Bed
Isomerization reaction in a packed bed reactor
z z+z z = L
FAb
Ac
z = 0
V
Plug flow assumed
Constant volumetric flow rate 0 (U = 0/Ac)
Ac=cro-sectional area of the tube, dm2
CAb=bulk gas concentration of A, mol/dm3
0=volumetric flow rate, dm3/s
U=superficial velocity, dm/s
FAb=molar flow, mol/s
(8-22)
16
Z = the length of the packed bed, m
8.3 Mass Transfer and Reaction in a Packed Bed
AAb Ab Δ( Δ ) 0'
b cz z zF F r A z
10'Ab
A b
C
dFr ρ
A dz
Molar balance for A in steady state:
Dividing by Acz and taking the limit by z→0
in - out + generated = 0
(8-24)
(8-23)
17
8.3 Mass Transfer and Reaction in a Packed Bed
A Ab
' 'r r Ω
Ab Ab Ab''' ''
a ar r S k C S
A Ab'''
ar Ωk C S
-r’A = real reaction rate
Assuming first order isomerization reaction A → B
Substituting into Equation (25)
(8-25)
(8-26)
(8-27)
18
k’’ = Rate constant per unit area
8.3 Mass Transfer and Reaction in a Packed Bed
Rate equation is substituted in molar balance equation
Ab b
1''Ab
a
C
dFΩk C S ρ
A dz
Constant volumetric flow rate
→ concentrations can be used in the balance equation
U = 0/Ac 1/Ac = U/0
Ab b Ab b
0
'' ''Ab Aba a
dF dCUΩk C S ρ U Ωk C S ρ
dz dz
(8-28)
(8-30)
(8-22) (8-29)
19
8.3 Mass Transfer and Reaction in a Packed Bed
Ab b''Aba
dCU Ωk C S ρ
dz
The final form of the balance equation
Ab bAb Ab
b
''
''
a
a
dC Ω ρ k SC a C
dz U
Ω ρ k Sa
U
Initial conditions (integration limits): CAb = CAb0 when z = 0
Ab
Ab0
AbAb Ab0
Ab 0
ln ln
C z
C
dCadz C C a z
C
(8-30) (8-31)
(8-32)
20CAb0 = Initial phase concentration, mol/dm3
8.3 Mass Transfer and Reaction in a Packed Bed
For bulk concentration as a function of reactor length
Ab Ab0ln lnC C a z Ab Ab0
azC C e
b ''
Ab Ab0
aΩ ρ k S z
UC C e
Conversion at the reactor outlet, z = L
b( '' )Ab
Ab0
1 1 aΩ k S L /UCX e
C
(8-32) (8-33)
(8-34)
(8-35)
21
8.4 Reaction Rate Dependency
External mass transfer-limited reactions in packed beds
Robert the Worrier (Example 14-4)
(8-36)
(8-37)
(8-38)(8-39)
(8-40)
6(1 ) / c pa d
22ɛ = Void of the bed dp = Particle diameter, m
8.4 Reaction Rate Dependency
Internal mass transfer-limited reactions
Large Thiele modulus
Surface-reaction-limited reactions
(8-41)
(8-42)
(8-43)
(8-44)(8-45)23
8.4 Reaction Rate Dependency
Limiting
parameter
Flow
rate
Catalyst
particle size
Temperature
External
diffusion
U1/2 (dp)-3/2 linear
Internal
diffusion
independent (dp)-1 exponential
Surface
reaction
independent independent exponential
Surface reaction
stronger dependent
24
8.4 Reaction Rate Dependency
How to determine diffusion limitations experimentally (in packed bed)?
Internal (pore)
diffusion
• Keep catalyst mass, mass flow rate
and reactant inlet concentration
constant
• Change catalyst particle size and
observe reaction rate/conversion
External (film)
diffusion
• Keep reactant inlet concentration and
catalyst mass to mass flow rate ratio
constant
• Change linear flow velocity and
observe reaction rate/conversion
25
8.4 Reaction Rate Dependency
How to determine diffusion limitations experimentally (in packed bed)?
Exp. Particle size Superfacial flow
rate
Reaction rate
I 1 high 3
II 3 low 1
III 3 high 1
• Experiments II and III: no external diffusion with large particles no external diffusion with small particles
• Experiments I and III:• strong pore diffusion
26
Summary
27
Evaluation of Diffusion LimitationsWeisz-Prater Criterion for Internal Diffusion
Mears Criterion for External Diffusion
Both external and
internal diffusion exist
internal effectiveness factor
Overall effectiveness factor Ω
Mass Transfer and Reaction in a Packed Bed
Reaction Rate Dependency
External mass transfer-limited reactions in packed beds
Internal mass transfer-limited reactions
Surface-reaction-limited reactions
How to determine diffusion limitations
28
8.5 Warming-up
EXAMPLE
The catalytic reaction
takes place within a fixed bed containing spherical porous catalyst X22. Figure E1 shows
the overall rates of reaction at a point in the reactor as a function of temperature for
various entering total molar flow rates, FT0 .
Figure E1:
Reaction rates in a catalyst bed.
Gas properties:
Diffusivity: 0.1 cm2/s
Density: 0.001 g/cm3
Viscosity: 0.0001 g/cm·s
Bed properties:
Tortuosity of pellet: 1.414
voidage=0.3
8.5 Warming-up
EXAMPLE
(a) Is the reaction limited by external diffusion?
(b) If your answer to part (a) was “yes,” under what conditions of those shown
(i.e., T, FT0) is the reaction limited by external diffusion?
(c) Is the reaction “reaction-rate-limited”?
(d) If your answer to part (c) was “yes,” under what conditions of those shown
(i.e., T, FT0) is the reaction limited by the rate of the surface reactions?
(e) Is the reaction limited by internal diffusion?
(f) If your answer to part (e) was “yes,” under what conditions of those shown
(i.e., T, FT0) is the reaction limited by the rate of internal diffusion?
(g) For a flow rate of 10 mol/h, determine (if possible) the overall effectiveness factor,
Ω, at 362 K.
(h) Estimate (if possible) the internal effectiveness factor, η, at 367 K
29
8.5 Warming-up
EXAMPLE
(i) If the concentration at the external catalyst surface is 0.01 mol/dm3, calculate (if
possible) the concentration at r = R/2 inside the porous catalyst at 367 K. (Assume a
first-order reaction.)
Solution
(a) Is the reaction limited
by external diffusion?
30
8.5 Warming-up
EXAMPLE
Limiting
parameter
Flow
rate
Catalyst
particle size
Temperature
External
diffusion
U1/2 (dp)-3/2 linear
Internal
diffusion
independent (dp)-1 exponential
Surface
reaction
independent independent exponential
Reaction rate dependency
31
8.5 Warming-up
EXAMPLE
(a) Is the reaction limited
by external diffusion?
YES
(b) what conditions?
All temperautres, FT0=10
mol/h. The rate of reaction
changes with Flow rate and
increases linearly with
temperature.
32
8.5 Warming-up
EXAMPLE
(c) Is the reaction “reaction-rate-limited”?
(e) Is the reaction limited by internal diffusion?
33
8.5 Warming-up
EXAMPLE
Limiting
parameter
Flow
rate
Catalyst
particle size
Temperature
External
diffusion
U1/2 (dp)-3/2 linear
Internal
diffusion
independent (dp)-1 exponential
Surface
reaction
independent independent exponential
Reaction rate dependency
Surface reaction
stronger dependent
34
8.5 Warming-up
EXAMPLE
(c) Is the reaction limited
by surface reaction?
YES
(e) Is the reaction limited
by internal diffusion?
YES
(f) T>367K, 1000, 5000 mol/h
(d) T<367K, 1000, 5000 mol/hT<362K, 100 mol/h
35
8.5 Warming-up
EXAMPLE
(g) For a flow rate of 10 mol/h, determine (if possible) the overall effectiveness factor,
Ω, at 362 K.
Actual overall (observed) reaction rate
reaction rate if whole particle were exposed to the conditions of bulk , Ab bC T
External and internal diffusion is eliminated
Reaction is “reaction-rate-limited”
Actual overall (observed) reaction rate, 10 mol/h, 362K
Rate of surface-reaction limited reaction, 362K
(8-16)
(8-46)
36
8.5 Warming-up
EXAMPLE
(g) For a flow rate of 10 mol/h, determine (if possible) the overall effectiveness factor,
Ω, at 362 K.
Ω =−𝑟𝐴(362𝐾, 10 𝑚𝑜𝑙/ℎ)
−𝑟𝐴(362𝐾, 5000 𝑚𝑜𝑙/ℎ)
Ω =0.26
0.70= 0.37
(d) T<367K, 1000, 5000 mol/hT<362K, 100 mol/hReaction limited
(8-47)
(8-48)
37
8.5 Warming-up
EXAMPLE
(h) Estimate (if possible) the internal effectiveness factor, η, at 367 K
As s
Actual overall (observed) reaction rate
reaction rate if entire interior surface were exposed to
the conditions of external surface ,
η
C T
Focus on 5000 mol/hAt 5000 mol/h, no external diffusion
No internal diffusion
Rate of surface-reaction limited reaction,
5000 mol/h, 367 K
(8-15)
38
8.5 Warming-up
EXAMPLE
Focus on 5000 mol/h, 367 K
T<367K, 5000 mol/hReaction limited
η =−𝑟𝐴(𝑎𝑐𝑡𝑢𝑎𝑙)
−𝑟𝐴(𝑒𝑥𝑡𝑟𝑎𝑝𝑜𝑙𝑎𝑡𝑒𝑑)
η =1.2
1.4= 0.86
(8-49)
(8-50)
39
8.5 Warming-up
EXAMPLE
(i) If the concentration at the external catalyst surface is 0.01 mol/dm3, calculate (if
possible) the concentration at r = R/2 inside the porous catalyst at 367 K. (Assume a
first-order reaction.)
1
1
sinh λ1ψ
λ sinh
A
As
C
C
1λ =
2
r
R
1 12
1
3( coth 1)η
0.86η
1=1.60
CA
(8-1)
(8-51) (8-52)
40
CatalyticReaction Engineering
Professor of Industrial ChemistryDepartment of Chemical and Metallurgical EngineeringSchool of Chemical TechnologyAalto UniversityEmail: [email protected] 1, E404
Yongdan Li
Nov-Dec, 2018