Mass Transport
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Transcript Mass Transport
Advanced Transport Phenomena
Module 6 Lecture 26
Mass Transport: Diffusion with Chemical Reaction
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras
1
QUASI-STEADY-STATE (QS) DIFFUSION
OUTSIDE ISOLATED SPHERE
In completely quiescent case, diffusional mass transfer
from/ to sphere occurs at a rate corresponding to Num = 2
If Bm ≡ v wdm/D is not negligible, then:
ln 1 Bm
Num 2
Bm
and
ln 1 Bm
Num
Fm Stefan " blowing "
2
Num ,0
Bm
Results from radial outflow due to net mass-transfer
flux across phase boundary
2
QUASI-STEADY-STATE (QS) DIFFUSION
OUTSIDE ISOLATED SPHERE
vw may be established by physically blowing fluid through
a porous solid sphere of same dia => Bm “blowing”
parameter
vw is negative in condensation problems, so is Bm
Suction enhances Num
3
QUASI-STEADY-STATE (QS) DIFFUSION
OUTSIDE ISOLATED SPHERE
Pew,m alternative blowing parameter, defined by:
and
Pew,m
wd m,0
D
Pew,m
Num
Fm Stefan blowing
Num ,0 exp Pew,m 1
Equivalent to correction factor for “phoretic suction”
4
QUASI-STEADY-STATE (QS) DIFFUSION
OUTSIDE ISOLATED SPHERE
Stefan-flow effect on Num very similar to phoresis effect,
but with one significant difference:
Phoresis affects mass transfer of dilute species, but
not heat transfer
Stefan flow affects both Nuh and Num in identical
fashion, hence not an analogy-breaker
Corresponding blowing parameters:
Bh
vwd h
Pew,h
vwd h,0
5
QUASI-STEADY-STATE (QS) DIFFUSION
OUTSIDE ISOLATED SPHERE
and
Pew,h
Nuh ln(1 Bh )
Fh Stefan blowing
Nuh,0
Bh
exp Pew,h 1
Nuh same function of Bh (or Pew,h) & Pr, as Num is of
Bm (or Pew,m) & Sc
6
QS EVAPORATION RATE OF ISOLATED DROPLET
Droplet of chemical substance A in hot gas
Energy diffusion from hotter gas supplies latent heat
required for vaporization of droplet of size dp
Known:
Gas temperature, T∞
Vapor mass fraction wA,∞
Unknowns:
droplet evaporation rate
m
''
wvw or m p d p2 m''
Vapor/ liquid interface conditions (wA,w, Tw)
7
QS EVAPORATION RATE OF ISOLATED DROPLET
Assumptions:
Vapor-liquid equilibrium (VLE) @ V/L interface
Liquid is pure (wA(l) = 1), surrounding gas insoluble in it
dp >> gas mean free path
Forced & natural convection negligible in gas
Variable thermophysical property effects in gas
negligible
8
QS EVAPORATION RATE OF ISOLATED DROPLET
Assumptions:
Species A diffusion in vapor phase per Fick’s (pseudo-
binary) law
No chemical reaction of species A in vapor phase
Recession velocity of droplet surface negligible
compared to radial vapor velocity, vw, at V/L phase
boundary
9
QS EVAPORATION RATE OF ISOLATED DROPLET
Dimensionless “blowing” (driving force)parameters:
w A, w w A,
Bm
1 w A, w
Bh
c p (T Tw )
LA
where wA,w = wA,eq(Tw; p)
10
QS EVAPORATION RATE OF ISOLATED DROPLET
Species A mass balance:
mA'' ,n continuous, and, in each adjacent phase, given
by:
m wi m i ci Di ,eff gradw
''
i
''
i
Since wA(l) = 1, in the absence of phoresis:
mA'' ,n m'' .w A, w j A'' ,n , w
(applying total mass balance condition mn = 0)
''
11
QS EVAPORATION RATE OF ISOLATED DROPLET
m vw, and
''
Since
j
''
A, n , w
w A, w w A,
w A
DA
DA
dm
n w
We can relate Bm directly to mass fractions of A as:
vwd m wA,w wA,
Bm
D
1 w A, w
12
QS EVAPORATION RATE OF ISOLATED DROPLET
Similarly, energy conservation condition at V/L interface
leads to relation between Bh and T∞-Tw
m'' LA qn'' , w Fourier
(neglecting work done by viscous stresses)
where LA latent heat of vaporization
LA hA Tw hAl Tw ; p
13
QS EVAPORATION RATE OF ISOLATED DROPLET
Heat Flux:
T Tw
T
q Fourier k
k.
n w
dh
Mass Flux:
DA
''
m 2
.ln 1 Bm
dp
Relating the two:
''
n,w
''
k / cp
q
n Fourier
''
m
2
.ln 1 Bh
LA
dp
14
QS EVAPORATION RATE OF ISOLATED DROPLET
Driving forces are related by:
1 Bh 1 Bm
Le
where Le = DA/ [k/cp] = Lewis number
Yields equation for Tw
Solution yields Bh, Bm
15
QS EVAPORATION RATE OF ISOLATED DROPLET
Single droplet evaporation rate
k
m p 2 d p
c
p
Equating this to
3
d
d
4 p
l
dt 3 2
c p T Tw
.ln 1
LA
(overall mass balance on the entire droplet )
We find that dp2 decreases linearly with time:
d d
2
p
2
p ,o
g
8
l
c p T Tw
t
ln 1
LA
16
QS EVAPORATION RATE OF ISOLATED DROPLET
Setting dp = 0 yields characteristic droplet lifetime:
tlife,vap
d p2,0
g
8
l
c p T Tw
ln 1
LA
17
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Catalyst impregnated throughout with porous pellets
To avoid having to separate catalyst from reaction
product
Pellets are packed into “fixed bed” through which
reactant is passed
Volume requirement of bed set by ability of reactants to
diffuse in & products to escape
Core accessibility determined by pellet diameter,
porosity & catalytic activity
18
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Assumptions in continuum model of catalytic pellet:
SS diffusion & chemical reaction
Spherical symmetry
Perimeter-mean reactant A number density nA,w at R = Rp
Radially-uniform properties (DA,eff, k”’eff, , …)
First-order irreversible pseudo-homogeneous reaction
within pellet
19
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
SS nA(r) profile within pellet satisfies local species A
mass-balance:
0 div( j ) r
''
A
since
'''
A,eff
j''A DA,eff gradwA
and
r
'''
A,eff
k wA
'''
eff
20
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Then, nA(r) satisfies:
dnA
1 d 2
'''
. r DA,eff
keff nA
2
r dr
dr
Relevant boundary conditions:
n A R p n A, w
and
dnA / dr r 0 0
21
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Applying species A mass balance to a “microsphere” of
radius e, and taking the limit as e 0:
lim
e 0
j A'' ,r lim
e 0
1 e '''
2
. r 4 r dr
2 eff
4e 0
which, for finite rA''' (o) leads to:
dnA / dr r 0 0
22
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Once nA(r) is found, catalyst utilization (or effectiveness)
factor can be calculated as:
cat
actual reaction rate within pellet
reaction rate within pellet if nA r nA,w
or
cat
Rp
0
dn
2
A
'''
2
D
4
R
keff nA r 4 r dr
A, eff
p
dr r R p
3
3
4
R
4
R
p
'''
p
'''
keff nA, w .
keff nA, w .
3
3
23
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
By similitude analysis:
r
nA
c
fct , f fct ; f
R
n A, w
p
and, therefore:
1 dc
cat 2
cat f
f d 1
where the Thiele modulus, f, is defined by:
DA,eff
f Rp . '''
k
eff
1/2
f 2 relevant Damkohler number; ratio of characteristic
diffusion time (Rp2/DA,eff) to characteristic reaction time,
24
(k”’ )-1
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
c (; f) normalized reactant-concentration variable,
satisfies 2nd-order linear ODE:
1 d
d
2 dc
2
f
c
d
subject to split bc’s:
c (1) 1
dc / d 0 0
25
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Solution to this two-point BVP:
sinh f
c
sinh(f )
or, explicitly:
nA r sinh f r / Rp
n A, w
r / Rp sinh(f )
Catalyst-effectiveness factor is explicitly given by:
cat
3 1
1
.
f tanh f f
26
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Catalyst effectiveness factor for first-order chemical reaction in a porous solid sphere
(adapted from Weisz and Hicks (1962))
27
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
cat 1 for f 0
cat 3 / f for f 1
Reaction only in a thin shell near outer perimeter of
pellet
Alternative presentation of cat : based on dependence
2
f
on
4 3
r
/ MA /
Rp
3
3 . rA,obs / M A
4 DA,eff nA,w Rp
DA,eff nA,w / Rp2
A,obs
Independent of (unknown)
keff'''
28
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Catalyst effectiveness factor vs experimentally observable (modified) Thiele
modulus (adapted from Weisz and Hicks (1962))
29
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Following additional parameters influence cat
E
Arr
RTe
DA,eff nA,e H
keff Te
n reaction order
DA, fluid Nu m
Bim
.
D
A,eff 2
k fluid Nu h
Bih
.
k 2
eff
30
STEADY MASS DIFFUSION WITH SIMULTANEOUS
CHEMICAL REACTION: CATALYST PELLET
Representative Parameter Values for Some Heterogeneous Catalytic Reactions
(after Hlavacek et al (1969))
31
TRANSIENT MASS DIFFUSION: MASS
TRANSFER (CONCENTRATION) BOUNDARY
LAYER
Discussion for thermal BL applies here as well
d m 4 Dt
1/2
Thermal BL “outruns” the MTBL:
dh
dm D
1/ 2
Le
1/ 2
D << (Le << 1) for most solutes in condensed phases
(especially metals)
Ratio holds for time-averaged penetration depth in
periodic BC case as well
32
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Analogies to Energy Transfer:
“analogy conditions” apply, heat-transfer
equations can be applied to mass-transfer by
substituting:
When
D,
Pr Sc,
Nu h Nu m
Sth Stm ,
Rah Ram
33
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Analogies to Energy Transfer:
Mass transfer of dilute species A in straight empty tube
flow (by analogy):
wA,w wA,b ( z )
exp 4m .Nu m .F (entrance)
wA,w wA,b (0)
where
1
z
m
. Sc v / DAmix
Re.Sc d w
34
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Nu m 0.023 Re0.8 Sc1/3
(turbulent )
and
1 7.60 8/3 1/8 (laminar)
m
F (entrance)
2/3
1 z / dw
(turbulent)
35
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Analogies to Energy Transfer:
Packed duct (by analogy):
Num,bed
dj A / (a ''' A0 dz )
DA w A, w w A,b / e / 1 e . d p ,eff
Since, in the absence of significant axial dispersion:
djA m.dwA,b ,
36
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Analogies to Energy Transfer:
Packed duct (by analogy):
We find:
w A, w w A,b ( z )
Num,bed Rebed , Sc z
exp 6(1 e ).
.
w A, w w A,b (0)
e Rebed .Sc
d p ,eff
where
2/3
Num,bed Sc0.4 0.4Re1/2
0.2Re
bed
bed
if 3 ≤ Rebed ≤ 104, 0.6 ≤ Sc, 0.48 ≤ e ≤ 0.74
Quantity in square bracket = Bed Stanton number for
mass transfer, Stm,bed
37
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Analogies to Energy Transfer:
Packed duct (by analogy):
In terms of Stm,bed
wA,w wA,b ( z )
exp Stm,bed a''' z
wA,w wA,b (0)
'''
where, as defined earlier, a (= 6(1-e)/dp) interfacial
area per unit volume of bed
38
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Analogies to Energy Transfer:
Packed duct (by analogy):
Height of a transfer unit (HTU) is defined by:
wA,w wA,b ( z)
z
exp
wA,w wA,b (0)
( HTU )
HTU bed depth characterizing exponential approach
to mass-transfer equilibrium
39
CONVECTIVE MASS TRANSFER IN LAMINARAND TURBULENT-FLOW SYSTEMS
Analogies to Energy Transfer:
Packed duct (by analogy):
In the case of single-phase fluid flow through a packed bed,
HTU = (a”’Stm,bed)-1
Widely used in design of heterogeneous catalytic-flow
reactors and physical separators
No chemical reaction within fluid
Also to predict performance of fluidized-bed contactors,
using e(Rebed) correlations
40