L 29-Heterogeneous Catalysis and Reactor Design Prof. K.K.Pant Department of Chemical Engineering...
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Transcript of L 29-Heterogeneous Catalysis and Reactor Design Prof. K.K.Pant Department of Chemical Engineering...
L 29-Heterogeneous Catalysis and Reactor Design
Prof. K.K.PantDepartment of Chemical Engineering
Diffusion and Reaction in a spherical pellet (Reading assignment: Fogler, Ch 12))
dr
dCD
dr
dycDW A
eA
eAr
0)( 22
rrdr
rWdcA
Ar
0])/([ 22
rrdr
rdrdCDdcA
Ae =r+Δr
Moles = WAr (4 πr2)r
Boundary conditions
-rA=c(-r’A)
r=0, CA finite, r=R, CA=CAS
Molar flux
In – out – disappearance =0 (spherical shell balance)
WAr (4 πr2)r - WAr (4 πr2)r+∆ r - rA’ (4 πr2 c
∆r) =0Dividing by -4 π ∆ r
consider 1st order
0])/([ 22
rrdr
rdrdCDdcA
Ae
0])/([
12
2
AAe Ckr
dr
rdrdCDd
c(-r’A) =-rA
-rA=kCA
What about n-th order ?
0])/([ 22
n
AnAe Ckr
dr
rdrdCDd
Differentiation &Divide by –r2De
Differentiation &Divide by –r2De
Dimensionless eq. – 1st order
About for n-th order ?
02
2
2
n
Ae
nAA CD
k
dr
dC
rdr
Cd
02 1
2
2
A
e
AA CD
k
dr
dC
rdr
Cd 2212
20
d d
d d
22
2
20n
n
d d
d d
Thiele Module
Thiele Module
eD
Rk 21
e
nAsn
D
CRk 12
5
Y = , , =>= y/ d/ d = 1/ (dy/d)- y/ 2
d2/ d2 = 1/ (d2y/d2)- 2/ 2 dy/d + 2y/3 Þd2y/d 2 - ø2y=0Þy= A Cosh ø + B Sinh ø ÞA=0 as φ must be finite at the centre, (B. C. =0, coshø 1; and ÞSinh ø 0.And at =1, =1,=> B= 1/Sinh ø
Thus , = CA/CAs = 1/ [Sinh ø / Sinh ø]
2212
20
d d
d d
Thiele Modulus, n
2 n-1 n2 n As n Asn
e e As
k R C k RC "a" surface reaction rateφ = = =
D D [(C -0)/R] "a" diffusion rate
A 1
As 1
C sinhφ λ1ψ = =
C λ sinhφ
• If n is large – internal diffusion limits the overall rate• If n is small – the surface reaction limits the overall rate
Internal Effectiveness Factor
• Internal effectiveness Factor, is: ranged 0 – 1
• for a first-order reaction in a spherical catalyst pellet
As s
Actual overall rate of reactionη =
Rate of reaction that would result if entire
interior surface were exposed to the external
pellet surface conditions C ,T
' "A A A
' "As As As
-r -r -rη = = =
-r -r -r
1 121
3η = φ cothφ -1
φ
Calculation of Catalytic Effectiveness Factor
)1(3 Coth
η = Actual overall rate(RA /Rate in the absence of diffusion resistance (RAs )
Global rate RA = 4 πR2 De (dCA/dr) at r=R
Or RA = 4 πR De (d /d ) at =1
Þ ((d /d )at =1 = (ø cot h ø-1)
ÞRA = 4 πR De CAS (ø cot h ø-1) Global Rate.ÞThus η = [4 πR De CAS (ø cot h ø-1)] / k’ ρc CAS 4/3 πR 3
Þ η = 3 (ø cot h ø-1)/ k’ ρc R2/De RÞ η = 3 (ø cot h ø-1)/ ø2
for ø> 20, η= 3/ ø strong pore diffusion resistance
Calculation of Catalytic Effectiveness Factor
Catalytic Effectiveness Factor:
wheref- Thiele Modulus
1st order reaction rate:
Spherical Pellet
Cylindrical Pellet
Slab Pellet
)313(1 Coth
DekSaRp /
3
DekSaRp /
2
DekSaL p /
Internal Effectiveness Factor
11
Uses the measured values of the rate of reaction to determine if Internal diffusion controls the rate.
Weisz-Prater Parameter CWP
ηø2 = 3(Ø Coth Ø-1)ηø2 = (observed rate/rate cal. at CAS) x (rate calculated at CAS) / diffusion Rate)
η = (-r’A(obs)/ -r’As
Ø2= -r”AS Sa ρp R2/De CAs = -r’AS ρp R2/De CAs
Weisz – Prater Criterion for internal diffusion
12
ÞCWP= (-r’A(obs)/ -r’As )(-r’AS ρp R2/De CAS)
Þ CWP = (-r’A(obs) (ρp R2/De CAS)
Þ These are measured or known terms.
Þ if CWP << 1, No diffusion limitations and no concentration gradient exists in the pellet.
ÞCWP >> 1, Internal diffusion limits the rate.
13
Non isothermal pellet Energy balance
0])/([ 22
rrdr
rdrdCDdcA
Ae Mass Balance
Energy balance
14
Non isothermal pellet effectiveness factor
Falsified Kinetics
• Measurement of the apparent reaction order and activation energy results primarily when internal diffusion limitations are present.
• This becomes serious if the catalyst pellet shape and size between lab (apparent) and real reactor (true) regime were Too different.
• Smaller catalyst pellet reduces the diffusion limitation higher activation energy more temperature sensitive
• RUNAWAY REACTION CONDITIONS!!!!
Falsified Kinetics•With the same rate of production, reaction order and activation energy to be measured (apparent rate)
17
Overall effectiveness factor (Both internal and external diffusion are important
18
Overall Effectiveness Factor
At Steady state, Moles transported from bulk fluid to the
external surface of the Catalyst (WAr Ap )= Net Rate of
reaction with in and on the pellet,
MA = WAr Ap = -r”A(As+ Ap),=( Molar Flux x Ext. Surface
Area of pellet)
For a single spherical pellet of Radius R,
AP= 4π R2, and As= SA x mass of pellet,(As >>Ap)
AP = (ext. SA/reactor volume) (reactor volume) = ac ∆V
As= (int. SA/.mass of catalyst) (mass cat./vol. cat) (vol
cat/reactor vol.) . Rect vol.) => AS= SA ρ c (1- ø) ∆V