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Ch E 542 - Intermediate Reactor Analysis & Design
Heat and Mass
Transfer Resistances
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Mass Transfer & Reaction
When convection dominates, the boundary
condition expressing steady state flux continuity
at z=is used;
kcis the convection mass transfer coefficient
AsAcAs CCk W
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Mass Transfer & Reaction
for flow around a sphere (roughly the geometric
shape of a catalyst particle), the convective heat
transfer coefficientcan be found from correlation
such as the following:
t
p
kdhNu
pdvRe
t
Pr
3121 PrRe6.02Nu
AsAcAs CCk W
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Mass Transfer & Reaction
By the heat/mass transfer analogy:
for flow around a sphere, the convective heat transfercoefficient can be found from:
PrSc
NuSh
ABt
c
Dk
kh
AB
pc
DdkSh
pdvRe
ABDSc
3121 ScRe6.02Sh
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Mass Transfer & Reaction
"rW AsAs AsrCk
molar flux to catalyst surface = reaction rate on surface
AsrAsAc CkCCk
AsAc CCk
rc
AcAs
kkCkC
rc
ArcAs
kkCkk"r
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Fast Reaction Kineticsfast reaction kinetics
Ac
r
Arc
rc
ArcAs Ck
k
Ckk
kk
Ckk"r
cr kk
3121
p
ABc ScRe
dD6.0k
21
p
21
61
32
ABc
d
vD6.0k
31
AB
21
p
p
ABc
D
vd
d
D6.0k
Frssling Correlation
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Fast Reaction Kineticsfast reaction kinetics
AcAs Ck"r
cr kk
21
p
21
61
32
ABc
d
vD6.0k
TfD
61
32
AB
DAB
gas
liquid
Tas
Tfd
v21
p
21
to increase kc
v
dp
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kcvdp vdp0.5
: rAs vdp kr kcvdp CAkr kcvdp
:
0 5 10 150
0.02
0.04
0.06
0.08
0.1
rAs vdp
vdp0.5
Reaction and Mass Transfer
reaction
rate limited
mass
transferlimited
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Rate Units for Catalytic Reaction
for single pellets
for packed beds
acsurface area / gramcAA a"r'r
pcc d
6a
ppB
cd
16
d
6
a
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Example Calculation
The irreversible gas-phase reaction AB is carried out in a PBR. The
reaction is first order in A on the surface.
The feed consists of 50%(mol) A (1.0 M) and 50%(mol) inerts and enters
the bed at a temperature of 300K. The entering volumetric flow rate is
10 dm3/s.
The relationship between the Sherwood Number and the ReynoldsNumber for this geometry is
Sh = 100 Re
Neglecting pressure drop, calculate catalyst weight necessary to achieve
60% conversion of A for
isothermal operation
adiabatic operation
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Example Calculation
'rdW
dXF AAo
Mole Balance
Rate Law
AsrAs C'k'r
assume reaction is mass transfer limited
AsAcA CCkW
AAs Wr
'kk
C'kk'r
rc
ArcAs
Re100Sh
21
p
AB
pc vd100
D
dk
gs
cm4242
vd
d
D100k
321
p
p
ABc
Mass Transfer Coefficient
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Example Calculation
'rdW
dXF AAo
Mole Balance
Rate Law
gas-phase, = 0, T = T0, P = P0.Stoichiometry
X1CC AoA
'kk
C'kk'r
rc
ArcAs
gs
cm
4242k
3
c
Energy Balance
Reaction is being carried
out isothermally. Thus,
energy balance not needed
and krf(T)
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Example Calculationgas-phase, = 0, P = P0.Stoichiometry
T
TX1CC oAoA
'kk
C'kk'r
rc
ArcAs
gs
cm4242k
3
c
T1
T
1
R
E
ror
Re'kT'k
'rdW
dXF AAo
Mole Balance
Rate Law Energy Balance
io
piio
r TCF
HXT
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Multicomponent Diffusion
Exact form of the flux equation for multicomponent
mass transport:
A simplified form uses a mean effective binary
diffusivity,
1N,,2,1j,NyyDCN
N
1kkj
1N
1kkjktj
N
1k
kjjjmtj NyyDCN
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Multicomponent Diffusion
The Stefan-Maxwell equations (Bird, Stewart, Lightfoot)
are given for ideal gases:
For binary system:
N
jk 1k
kjjk
jk
jt NyNyD
1
yC
2111121t NNyND
1
yC
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Multicomponent Diffusion
Solved for flux
Simplified for
assumed equimolarcounter-diffusion
2111121t NNyND
1
yC
211112t1 NNyyDCN
112t1 yDCN
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Multicomponent Diffusion
The effective binary diffusivity for species j can then be
defined by equating the driving force terms of the
expression containing Djmand the Stefan-Maxwell
N
jk 1k
kjjk
jk
jt NyNyD
1
yC
N
1k
kjjmjtj NyDyCN
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Multicomponent Diffusion
The effective binary diffusivity for species j can then be
defined by equating the driving force terms of the
expression containing Djmand the Stefan-Maxwell
N
1k
kj
N
jk
1k
kjjk
jk
jmj NyNyNyD
1DN
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Multicomponent Diffusion
use for diffusion of species 1 through stagnant 2, 3, (all
flux ratios are zero for k=2,3,) reduces to
the "Wilke equation"
N
1k
kjj
N
jk 1k
kjjkjk
jm NyN
NyNyD
1
D
1
N
3,2k k1
k
1m1 Dy
y11
D1
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Multicomponent Diffusion
For reacting systems where steady-state flux ratios are
determined by reaction stoichiometry,
N
jk j
kjk
jkjjN
1k k
k
j
N
jk j
kjk
jk
jm
yyD
1
y1
1
y1
yyD1
D
1
constantN
j
j
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Diffusion/Rxn in Porous Catalysts
Effective Diffusivity (De) is a measure of
diffusivity that accounts for the following:
Not all area normal to flux direction is available for
molecules to diffuse in a porous particle (P) Diffusion paths are tortuous ()
Pore cross-sections vary ()
Internal void fraction, s= P
~
DD PAe
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Diffusion/Rxn in Porous Catalysts
Extended Stefan-Maxwell
Solved for binary, steady-state, 1D diffusion
Kj,e
D
jN
1k
D
kj
D
jk
jk,e
jD
NNyNy
D
1p
RT
1
KA,eAB,e0AAB
KA,eAB,eAAB
BA
AB,et
ADDyNN11
DDLyNN11
lnNN1L
DC
N
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Diffusion/Rxn in Porous Catalysts
Define effective binary diffusivity for use in
single reaction multicomponent systems:
dz
dCDN j
jm,ej
Kj,e
N
1k j
kjk
jk,ejm,e D
1yyD
1
D
1
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Quantify De
Random Pore Model
Parallel Cross-linked Pore Model
Pore Network Model of Beeckman & Froment
Tortuosity factor using Wicke-Kallenbach cell
Pore diffusion with
Adsorption Surface Diffusion
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Diffusion/Rxn in Porous Catalysts
steady state mass balance
rate in at r
r
2
Ar r4W rate out at r + r
rr
2
Ar r4W
rate of generation within shell
cmasscatalyst
ratereaction
volumeshell
masscatalyst volumeshell
rr + r
R
rr4
2
m'
Ar
0rr4r
r4Wr4W
2
mC
'
A
rr
2
Arr
2
Ar
0rr
dr
rWd 2C
'
A
2
Ar BA cat
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Diffusion/Rxn in Porous Catalysts
0rSCkrdr
dC
Ddr
d 2Ca
n
An
2A
e
identify boundary conditions
finiteC0rA
symmetry
AsRrA CC
surface
dimensionless
As
A
C
C
R
r
AsA
Cd
dC
R
1
d
dr
R
C
d
d
dr
dC AsA
0CD
Sk
dr
dC
r
2
dr
Cd nA
e
CanA
2
A
2
0D
CRSk
d
d2
d
d n
e
1n
As
2
Can
2
2
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Diffusion/Rxn in Porous Catalysts
define Thiele modulus (n)
0D
CRSk
d
d2
d
d n
e
1n
As
2
Can2
2
e
1n
As
2
Can2
nD
CRSk
0dd2
dd n2
n2
2
understand the Thiele modulus
R0CDRCSk
Ase
nAsCan2
n reaction rate
di f fusion rate
large n- diffusion controlssmall n- kinetics control
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Diffusion/Rxn in Porous Catalystsfirst order
kinetics
(n = 1)
define y =
0d
d2
d
d 212
2
2
e
Can2
1 RD
Sk
322
2
2
2 y2
d
dy2
d
yd1
d
d
2
y
d
dy1
d
d
0yd
yd 212
2
1111 sinhBcoshAy
1B
1
Asinhcosh 11
differential has the solution apply boundary conditions
1,1
finiteis,0
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Diffusion/Rxn in Porous Catalystsfirst order
kinetics
(n = 1)
0d
d2
d
d 212
2
2
e
Can2
1 RD
Sk
0yd
yd 212
2
1111 sinhBcoshAy
1B
1
Asinhcosh 11
differential has the solution apply boundary conditions
1,1
finiteis,0
1
1
sinh
sinh1
As
A
C
C
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Internal Effectiveness Factor ()
The internal effectiveness factor () is a measure
of the relative importance of diffusion to reaction
limitations:
sAsT,Ctoexposedweresurfaceentireifrate
ratereactionoverallactual
As
A"
As
"
A'
As
'
A
As
A
MM
rr
rr
rr
M mol / timer mol / time / mass cat
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Internal Effectiveness Factor ()
Determine MAs(rate if all surface at CAs)
catalystmasscatalystmass
areasurfaceareaunitperrateMAs
'
Asr
aS
CVAsM
x
x
As1Ck
c
3
34
aAs1As RSCkM
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Internal Effectiveness Factor ()
Determine MA(actual rate is equal to reactant
diffusion rate at outer surface)
1
AseA
d
dCRD4M
11
12
1
11
1 sinhsinh1
sinhcosh
dd
1coth 11
1cothCRD4M 11AseA
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Internal Effectiveness Factor ()
Substitute results into definition of
As
A
M
M
c
3
34
aAs1
11Ase
RSCk
1cothCRD4
1cothRSk
D3 11
c
2
a1
e
1coth3 1121
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Revisit and
Thiele modulus -
Derived for spherical particle geometry
Derived for 1storder kinetics
For large , approximately
Internal effectiveness factor -
Assumed =0, correction applied when 0
Assumed isothermal conditions
2
1
21
31n
2
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Non-Isothermal Behavior
For exothermic reactions, can be > 1 as internaltemperature can exceed Ts.
The rate internally is thus larger than at the surfaceconditions where is evaluated.
The magnitude of this effect is dependent on Hrxn, Ts, Tmax, and kt(thermal conductivity of the pellet)
and are used to quantify this effect:
can result in mulitple steady states
No multiple steady states exist if Luss criterion is fulfilled
NumberArrheniussRT
E
st
Aserxn
s
smax
Tk
CDH
T
TT
14
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Overall Effectiveness Factor
When both internal AND external diffusion
resistances are important (i.e., the same order of
magnitude), both must be accounted for when
quantifying kinetics. It is desired to express the kinetics in terms of the
bulk conditions, rather than surface conditions:
bulkA,Ctoexposedweresurfaceentireifrateratereactionoverallactual
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Overall Effectiveness Factor
Accounting for reaction both on and within the pellet, the molar ratebecomes:
For most catalyst, internal surface area is significantly higher than the
external surface area:
V1SarM cac"
AA
b
bac"
AcA SaraW
ba
"
AcA SraW
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Overall Effectiveness Factor
ba
"
AcA SraW reaction rate(internal & external surfaces)
VaCCkVaW cAsbulk,AccAr
mass transport rate
internal surfaces not
all exposed to CAs
As1
"
As
"
A Ckrr Relation between CAsand CAdefined by the as:
VSCkVaW baAs1cA
baAs1cAsbulk,Ac SCkaCCk
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Overall Effectiveness Factor
ba
"
AcA SraW reaction rate(internal & external surfaces)
VaCCkVaW cAsbulk,AccAr
mass transport rate
As1
"
As
"
A Ckrr Relation between CAsand CAdefined by the as:
ba1cc
bulk,Acc
As Skka
Cka
C Solving for CAs:
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Overall Effectiveness Factor
ba
"
AcA SraW reaction rate(internal & external surfaces)
VaCCkVaW cAsbulk,AccAr
mass transport rate
ba1cc
bulk,Acc1"
ASkka
Cakkr
Substitution into the rate law:
ba1cc
bulk,Acc
As Skka
Cka
C Solving for C
As:
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Overall Effectiveness Factorsummary of factor relationships:
ba1cc
bulk,Acc1"
ASkka
Cakkr
Rearranging the expression:
bulk,A1
ccba1
CkakSk1
"
bulk,A
"
A rr ccba1 akSk1
"As
"bulk,A
"A rrr
As1
"
As
Ckr
Ab1
"
Ab Ckr
Overall Effectiveness Factor ()
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Weisz-Prater Criterion
Weisz-Prater Criterion is a method of determining if a givenprocess is operating in a diffusion- or reaction-limited regime
CWPis the known as the Weisz-Prater parameter. All
quantities are known or measured.
CWP> 1, severe diffusion limitations
Ase
c
2'
obs,A2
1WPCD
RrC
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Mears Criterion
Mass transfer effects negligible when it is true that
n is the reaction order, and the transfer coefficients kcand h (below) canbe estimated from an appropriate correlation (i.e., Thoenes-Kramers for
packed bed flow)
Heat transfer effects negligible when it is true that
15.0Ck
nRr
Abc
b
'
A
15.0ThR
RErH2
bg
b
'
Arxn
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A li i PBR
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Application to PBRs
Which can be rewritten as:
AbabAb C
UkS
dzdC
Entrance condition:oAb0zAb
CC
Integrating and applying boundary condtion yields:
U
zkS
expCC
ab
AbAb o