Heat and Mass Transfer Resistances
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Transcript of Heat and Mass Transfer Resistances
Ch E 542 - Intermediate Reactor Analysis & Design
Heat and MassTransfer Resistances
Mass Transfer & Reaction
• When convection dominates, the boundary condition expressing steady state flux continuity at z= is used;
– kc is the convection mass transfer coefficient
AsAcAs CCk W
Mass Transfer & Reaction
– for flow around a sphere (roughly the geometric shape of a catalyst particle), the convective heat transfer coefficient can be found from correlation such as the following:
t
p
k
dhNu
pdvRe
t
Pr
3121 PrRe6.02Nu
AsAcAs CCk W
Mass Transfer & Reaction• By the heat/mass transfer analogy:
– for flow around a sphere, the convective heat transfer coefficient can be found from:
PrSc
NuSh
ABt
c
Dk
kh
AB
pc
D
dkSh
pdvRe
ABDSc
3121 ScRe6.02Sh
Mass Transfer & Reaction
"rW AsAs AsrCk
molar flux to catalyst surface = reaction rate on surface
AsrAsAc CkCCk
AsAc CCk
rc
AcAs kk
CkC
rc
ArcAs kk
Ckk"r
Fast Reaction Kineticsfast reaction kinetics
Acr
Arc
rc
ArcAs Ck
k
Ckk
kk
Ckk"r
cr kk
3121
p
ABc ScRe
d
D6.0k
21
p
21
61
32AB
c d
vD6.0k
31
AB
21
p
p
ABc D
vd
d
D6.0k
Frössling Correlation
Fast Reaction Kineticsfast reaction kinetics
AcAs Ck"r cr kk
21
p
21
61
32AB
c d
vD6.0k
TfD
61
32AB
DAB
gas
liquid
T as
Tfd
v21
p
21
to increase kc
v
dp
Slow Reaction Kineticsslow reaction kinetics
Arc
Arc
rc
ArcAs Ck
k
Ckk
kk
Ckk"r
cr kk
kr is independent of• fluid velocity• particle size
kc vdp vdp0.5 rAs vdp
k r kc vdp CA
k r kc vdp
0 5 10 150
0.02
0.04
0.06
0.08
0.1
rAs vdp
vd p0.5
Reaction and Mass Transfer
reactionrate limited
masstransferlimited
Rate Units for Catalytic Reaction
for single pellets
for packed beds
ac surface area / gramcAA a"r'r
pcc d
6a
ppB
c d
16
d
6a
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 Reynolds Number for this geometry is
Sh = 100 Re½
• Neglecting pressure drop, calculate catalyst weight necessary to achieve 60% conversion of A for
– isothermal operation– adiabatic operation
Example Calculation
'rdW
dXF AAo
Mole BalanceMole Balance
Rate LawRate Law
AsrAs C'k'r
assume reaction is mass transfer limited
AsAcA CCkW
AAs Wr
'kk
C'kk'r
rc
ArcAs
½Re100Sh
21
p
AB
pc vd100
D
dk
gs
cm4242
vd
d
D100k
321
p
p
ABc
Mass Transfer CoefficientMass Transfer Coefficient
Example Calculation
'rdW
dXF AAo
Mole BalanceMole Balance
Rate LawRate Law
gas-phase, = 0, T = T0, P = P0.StoichiometryStoichiometry
X1CC AoA
'kk
C'kk'r
rc
ArcAs
gs
cm4242k
3
c
Energy BalanceEnergy Balance
Reaction is being carried out isothermally. Thus,
• energy balance not needed• and kr f(T)
Example Calculationgas-phase, = 0, P = P0.StoichiometryStoichiometry
T
TX1CC o
AoA
'kk
C'kk'r
rc
ArcAs
gs
cm4242k
3
c
T
1
T
1
R
E
rorRe'kT'k
'rdW
dXF AAo
Mole BalanceMole Balance
Rate LawRate Law Energy BalanceEnergy Balance
io
piio
r TCF
HXT
Multicomponent Diffusion
• Exact form of the flux equation for multicomponent mass transport:
• A simplified form uses a mean effective binary diffusivity,
1N,,2,1j , NyyDCNN
1kkj
1N
1kkjktj
N
1kkjjjmtj NyyDCN
Multicomponent Diffusion
• The Stefan-Maxwell equations (Bird, Stewart, Lightfoot) are given for ideal gases:
• For binary system:
N
jk 1kkjjk
jkjt NyNy
D1
yC
211112
1t NNyND1
yC
Multicomponent Diffusion
• Solved for flux
• Simplified forassumed equimolarcounter-diffusion
211112
1t NNyND1
yC
211112t1 NNyyDCN
112t1 yDCN
Multicomponent Diffusion
• The effective binary diffusivity for species j can then be defined by equating the driving force terms of the expression containing Djm and the Stefan-Maxwell
N
jk 1kkjjk
jkjt NyNy
D1
yC
N
1kkjjmjtj NyDyCN
Multicomponent Diffusion
• The effective binary diffusivity for species j can then be defined by equating the driving force terms of the expression containing Djm and the Stefan-Maxwell
N
1kkj
N
jk 1kkjjk
jkjmj NyNyNy
D1
DN
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
1kkjj
N
jk 1kkjjk
jk
jm NyN
NyNyD1
D1
N
3,2k k1
k
1m1 Dy
y11
D1
Multicomponent Diffusion• For reacting systems where steady-state flux ratios are
determined by reaction stoichiometry,
N
jk j
kjk
jkjjN
1k k
kj
N
jk j
kjk
jk
jm
yyD1
y11
y1
yyD1
D1
constantN
j
j
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 PA
e
Diffusion/Rxn in Porous Catalysts
• Extended Stefan-Maxwell
• Solved for binary, steady-state, 1D diffusion
Kj,e
Dj
N
1k
Dkj
Djk
jk,ej D
NNyNy
D1
pRT
1
KA,eAB,e0AAB
KA,eAB,eAAB
BA
AB,etA DDyNN11
DDLyNN11ln
NN1LDC
N
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 D1
yyD
1D
1
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
Diffusion/Rxn in Porous Catalysts
steady state mass balance
rate in at r
r
2Ar r4W
rate out at r + r
rr
2Ar r4W
rate of generation within shell
c
mass catalyst
rate reaction
volume shell
mass catalyst volume shell
rr + r
R
rr4 2m'
Ar
0rr4r
r4Wr4W
2mC
'A
rr
2Arr
2Ar
0rr
dr
rWd 2C
'A
2Ar
BA cat
Diffusion/Rxn in Porous Catalysts
0rr
dr
rWd 2C
'A
2Ar
dr
dCDW A
eAr
0rrrdr
dCD
dr
d 2C
'A
2Ae
0rSrrdr
dCD
dr
d 2Ca
"A
2Ae
2AnaCA
2Ana
'A
2An
"A
CkSr
CkSr
Ckr
0rSCk 2Ca
nAn
rate equationdefinitions
substitute Fick’s Law
Diffusion/Rxn in Porous Catalysts
0rSCkrdr
dCD
dr
d 2Ca
nAn
2Ae
identify boundary conditions
finiteC0rA
symmetry
AsRrA CC
surface
dimensionless
As
A
C
C
R
r
AsA C
d
dC
R
1
d
dr
R
C
d
d
dr
dC AsA
0CD
Sk
dr
dC
r
2
dr
Cd nA
e
CanA2A
2
0D
CRSk
d
d2
d
d n
e
1nAs
2Can
2
2
Diffusion/Rxn in Porous Catalysts
define Thiele modulus (n)
0D
CRSk
d
d2
d
d n
e
1nAs
2Can
2
2
e
1nAs
2Can2
n D
CRSk
0d
d2
d
d n2n2
2
understand the Thiele modulus
R0CD
RCSk
Ase
nAsCan2
n
reaction rate
diffusion rate
large n - diffusion controls
small n - kinetics control
Diffusion/Rxn in Porous Catalystsfirst orderkinetics(n = 1)
define y =
0d
d2
d
d 212
2
2
e
Can21 R
D
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
1A sinhcosh 11
differential has the solution apply boundary conditions
1 ,1
finite is ,0
Diffusion/Rxn in Porous Catalystsfirst orderkinetics(n = 1)
0d
d2
d
d 212
2
2
e
Can21 R
D
Sk
0yd
yd 212
2
1111 sinhBcoshAy
1B
1A sinhcosh 11
differential has the solution apply boundary conditions
1 ,1
finite is ,0
1
1
sinh
sinh1
As
A
C
C
Thiele Modulus
As
A
C
C
Internal Effectiveness Factor ()
• The internal effectiveness factor () is a measure of the relative importance of diffusion to reaction limitations:
sAs T ,C to exposed weresurface entire if rate
rate reaction overall actual
As
A"As
"A
'As
'A
As
A
M
M
r
r
r
r
r
r
M mol / timer mol / time / mass cat
Internal Effectiveness Factor ()
• Determine MAs (rate if all surface at CAs) catalyst mass
catalyst mass
area surfacearea unit per rateMAs
'Asr
aS
CVAsM
x
x
As1Ck
c3
34
aAs1As RSCkM
Internal Effectiveness Factor ()
• Determine MA (actual rate is equal to reactant diffusion rate at outer surface)
1AseA d
dCRD4M
11
12
1
11
1 sinh
sinh1
sinh
cosh
d
d
1coth 11
1cothCRD4M 11AseA
Internal Effectiveness Factor ()
• Substitute results into definition of
As
A
M
M
c
334
aAs1
11Ase
RSCk
1cothCRD4
1cothRSk
D3 11
c2
a1
e
1coth3
1121
1coth3
1121
Internal Effectiveness Factor ()
1coth3
1121
12
131
21
101
ac1
e21
SkD
R33
1 20
small dp
Internal Effectiveness Factor ()
1coth3
1121
12
131
21
101
ac1
e21
SkD
R33
1 20
reactionrate
limited
internaldiffusionlimited
Revisit and
• Thiele modulus - – Derived for spherical particle geometry– Derived for 1st order kinetics
• For large , approximately
• Internal effectiveness factor - – Assumed =0, correction applied when 0– Assumed isothermal conditions
21
213
1n
2
Non-Isothermal Behavior
• For exothermic reactions, can be > 1 as internal temperature can exceed Ts.
• The rate internally is thus larger than at the surface conditions 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
Number ArrheniussRT
E
st
Aserxn
s
smax
Tk
CDH
T
TT
14
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,C to exposed weresurface entire if rate
rate reaction overall actual
Overall Effectiveness Factor
• Accounting for reaction both on and within the pellet, the molar rate becomes:
• For most catalyst, internal surface area is significantly higher than the external surface area:
V1SarM cac"AA
b
bac"AcA SaraW
ba"AcA SraW
Overall Effectiveness Factorba
"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 CAs and CA
defined by the as:
VSCkVaW baAs1cA
baAs1cAsbulk,Ac SCkaCCk
Overall Effectiveness Factorba
"AcA SraW reaction rate
(internal & external surfaces)
VaCCkVaW cAsbulk,AccAr mass transport rate
As1"As
"A Ckrr Relation between CAs and CA
defined by the as:
ba1cc
bulk,AccAs Skka
CkaC
Solving for CAs:
Overall Effectiveness Factorba
"AcA SraW reaction rate
(internal & external surfaces)
VaCCkVaW cAsbulk,AccAr mass transport rate
ba1cc
bulk,Acc1"A Skka
Cakkr
Substitution into the rate law:
ba1cc
bulk,AccAs Skka
CkaC
Solving for CAs:
Overall Effectiveness Factorsummary of factor relationships:
ba1cc
bulk,Acc1"A Skka
Cakkr
Rearranging the expression:
bulk,A1ccba1
CkakSk1
"bulk,A
"A rr ccba1 akSk1
ccba1 akSk1
"As
"bulk,A
"A rrr
As1"As Ckr
Ab1"Ab Ckr
Overall Effectiveness Factor ()
Weisz-Prater Criterion• Weisz-Prater Criterion is a method of determining if a given
process is operating in a diffusion- or reaction-limited regime – CWP is the known as the Weisz-Prater parameter. All
quantities are known or measured.
– CWP << 1, no C in the pellet (kinetically limited)
– CWP >> 1, severe diffusion limitations
Ase
c2'
obs,A21WP CD
RrC
Mears’ Criterion
• Mass transfer effects negligible when it is true that
– n is the reaction order, and the transfer coefficients kc and h (below) can be 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
15.0
ThR
RE rH2bg
b'Arxn
0rdz
dCU
dz
CdD A
'A
Ab2Ab
2
AB
Application to PBRs• Shell balance on
volume element Az
• Mole flux of A
• First order reaction
0rdz
dWb
'A
Az
UCdz
dCDW Ab
AABAz
Aba'Ab
'A CkSrr
0CkS Abab
0CkSdz
dCU
dz
CdD Abab
Ab2Ab
2
AB
Application to PBRs
• Axial dispersion negligible (relative to forced axial convection) when…– dp is the particle diameter
– Uo is the superficial velocity of the gas
– Da is the effective axial dispersion coefficient
a
po
Abo
pb'A
D
dU
CU
dr
Which can be rewritten as:
Application to PBRs
Which can be rewritten as:
AbabAb C
U
kS
dz
dC
Entrance condition:oAb0zAb CC
Integrating and applying boundary condtion yields:
U
zkSexpCC ab
AbAb o
U
zkSexpCC ab
AbAb o