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)
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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
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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
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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)
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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
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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   hAl  Tw ; p 
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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
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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
 


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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
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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
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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
 4e 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 
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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,
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(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
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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 
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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'''
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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 4m .Nu m    .F (entrance)
wA,w  wA,b (0)


where
1
z
m 
.  Sc  v / DAmix 
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 Sc0.4  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