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Tutorial/HW Week #7
WWWR Chapters 24-25
ID Chapter 14
Tutorial #7
WWWR# 24.1, 24.12, 24.13,24.15(d), 24.22.
To be discussed on March
12, 2013. By either volunteer or
class list.
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Molecular Mass Transfer
Molecular diffusion
Mass transfer law components:
Molecular concentration:
Mole fraction:
(liquids,solids) , (gases)
c
cy
c
cx AA
AA
RT
p
V
n
Mc AA
A
AA
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For gases,
Velocity:mass average velocity,
molar average velocity,
velocity of a particular species relative to mass/molar average is
the diffusion velocity.
P
p
RTP
RTpy AAA
n
i
ii
n
i
i
n
i
ii
1
1
1
vv
v
c
cn
i
ii 1
v
V
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mol
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Flux:A vector quantity denoting amount of a particular species that
passes per given time through a unit area normal to the vector,
given by Ficks First Law, for basic molecular diffusion
or, in the z-direction,
For a general relation in a non-isothermal, isobaric system,
AABA cD J
dz
dcDJ AABzA ,
dz
dycDJ AABzA ,
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Since mass is transferred by two means:
concentration differences
and convection differences from density differences
For binary system with constant Vz,
Thus,
Rearranging to
)( ,, zzAAzA VvcJ
dz
dycDVvcJ AABzzAAzA )( ,,
zAA
ABzAA Vcdz
dycDvc ,
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As the total velocity,
Or
Which substituted, becomes
)(1 ,, zBBzAAz vcvcc
V
)( ,, zBBzAAAzA vcvcyVc
)( ,,, zBBzAAAA
ABzAA vcvcydz
dycDvc
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Defining molar flux, N as flux relative to a fixed z,
And finally,
Or generalized,
AAA c vN
)( ,,, zBzAAA
ABzA NNydz
dycDN
)( BAAAABA yycD NNN
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Related molecular mass transferDefined in terms of chemical potential:
Nernst-Einstein relation dz
d
RT
D
dz
duVv cABcAzzA
,
dz
d
RT
D
cVvcJ
cAB
AzzAAzA
)( ,,
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Diffusion Coefficient
Ficks law proportionality/constant
Similar to kinematic viscosity, , and
thermal diffusivity, a
t
L
LLMtL
M
dzdc
J
D A
zA
AB
2
32
,
)1
1
)((
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Gas mass diffusivity
Based on Kinetic Gas Theory
= mean free path length, u = mean speed
Hirschfelders equation:
uDAA 3
1*
2/13
22/3
2/3
* )(3
2
AA
AAM
N
P
TD
DAB
BA
ABP
MMT
D
2
2/1
2/3 11001858.0
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Lennard-Jones parameters and e from tables,
or from empirical relations
for binary systems, (non-polar,non-reacting)
Extrapolation of diffusivity up to 25
atmospheres
2
BAAB
BAAB eee
2
1
1,12,2
2/3
1
2
2
1
TD
TD
ABABT
T
P
PDD
PTPT
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Binary gas-phase Lennard-Jones
collisional integral
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With no reliable ore, we can use the Fuller
correlation,
For binary gas with polar compounds, we
calculate by
23/13/1
2/1
75.1311
10
BA
BA
AB
vvP
MMT
D
*
2
196.00 T
ABD
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where
bb
PBAAB
TV
232/1 1094.1,
ABTT e /*
2/1
e
e
e BAAB
bT23.1118.1/ e
)exp()exp()exp( ****0 HTG
FT
E
DT
C
T
ABD
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and
For gas mixtures with several components,
with
2/1BAAB 3/1
23.11585.1
bV
nn DyDyDyD
1
'
31
'
321
'
2
mixture1/...//
1
nyyy
yy
...32
2'
2
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2
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Liquid mass diffusivity
No rigorous theoriesDiffusion as molecules or ions
Eyring theory
Hydrodynamic theory Stokes-Einstein equation
Equating both theories, we get Wilke-Chang eq.B
ABr
TD
6
6.0
2/18104.7
A
BBBAB
V
M
T
D
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For infinite dilution of non-electrolytes in
water, W-C is simplified to Hayduk-Laudie eq.
Scheibels equation eliminates B,
589.014.151026.13 ABAB VD
3/1
A
BAB
V
K
T
D
3/2
8 31)102.8(A
B
VVK
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As diffusivity changes with temperature,
extrapolation of DAB
is by
For diffusion of univalent salt in dilute solution,
we use the Nernst equation
n
c
c
ABT
ABT
TT
TT
D
D
1
2
)(
)(
2
1
F
RTDAB )/1/1(
200
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Pore diffusivity
Diffusion of molecules within pores of poroussolids
Knudsen diffusion for gases in cylindrical pores
Pore diameter smaller than mean free path, and
density of gas is low
Knudsen number
From Kinetic Theory of Gases,
poredKn
AAA M
NTuD
8
33*
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But if Kn >1, then
If both Knudsen and molecular diffusion exist, then
with
For non-cylindrical pores, we estimate
A
pore
A
porepore
KAM
TdMNTdudD 48508
33
KAAB
A
Ae DDy
D111 a
A
B
N
N1a
AeAe DD2' e
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Example 6
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Types of porous diffusion. Shaded areas represent nonporous solids
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Hindered diffusion for solute in solvent-filled
pores
A general model is
F1 and F2 are correction factors, function of porediameter,
F1 is the stearic partition coefficient
)()( 21 FFDDo
ABAe
pore
s
d
d
2
2
1 2
( )( ) (1 )
pore s
pore
d dF
d
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F2 is the hydrodynamic hindrance factor, one
equation is by Renkin,
53
2 95.009.2104.21)( F
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Example 7
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Convective Mass Transfer
Mass transfer between moving fluid with
surface or another fluid
Forced convection
Free/natural convection
Rate equation analogy to Newtons cooling
equation
AcA ckN
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Example 8
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Differential Equations
Conservation of mass in a control volume:
Or,
inout + accumulationreaction = 0
.... 0vcsc dVtdA nv
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For inout,
in x-dir,
in y-dir,
in z-dir,
For accumulation,
xxAxxxAzynzyn
,,
yyAyyyA zxnzxn ,,
zzAzzzA yxnyxn ,,
zyxt
A
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For reaction at rate rA
,
Summing the terms and divide by xyz,
with control volume approaching 0,
zyxrA
, , , , , , 0A x x x A x x A y y y A y y A z z z A z z A An n n n n n
rx y z t
, , ,0AA x A y A z An n n r
x y z t
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We have the continuity equation for
component A, written as general form:
For binary system,
but and
0
A
AA r
t
n
n n 0A BA B A Br rt
vvvnn BBAABA
BA rr
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So by conservation of mass,
Written as substantial derivative,
For species A,
0
t
v
0 vDtD
0 AAA r
Dt
D j
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In molar terms,
For the mixture,
And for stoichiometric reaction,
0
A
AA Rt
cN
0)( BABA
BA RRt
ccNN
0)( BA RR
t
ccV