Tutorial/HW Week #7courses.nus.edu.sg/course/chewch/CN2125E/lectures/Week7.pdf · 2020. 3. 10. ·...
Transcript of Tutorial/HW Week #7courses.nus.edu.sg/course/chewch/CN2125E/lectures/Week7.pdf · 2020. 3. 10. ·...
Tutorial/HW Week #7
WRF Chapters 22-23; 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
10, 2020.
• By either volunteer or
class list.
Molecular Mass Transfer
• Molecular diffusion
• Mass transfer law components:
– Molecular concentration:
– Mole fraction:
(liquids,solids) , (gases)
c
cy
c
cx A
AA
A
RT
p
V
n
Mc AA
A
AA
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 AA
A
n
i
ii
n
i
i
n
i
ii
1
1
1
vv
v
c
cn
i
ii 1
v
V
mol
– 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 Fick’s 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 A
ABzA ,
dz
dycDJ A
ABzA ,
– 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 A
ABzzAAzA )( ,,
zAA
ABzAA Vcdz
dycDvc ,
• As the total velocity,
• Or
• Which substituted, becomes
)(1
,, zBBzAAz vcvcc
V
)( ,, zBBzAAAzA vcvcyVc
)( ,,, zBBzAAAA
ABzAA vcvcydz
dycDvc
• 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
• Related molecular mass transfer
– Defined in terms of chemical potential:
– Nernst-Einstein relation
dz
d
RT
D
dz
duVv cABc
AzzA
,
dz
d
RT
DcVvcJ cAB
AzzAAzA
)( ,,
Diffusion Coefficient
• Fick’s law proportionality/constant
• Similar to kinematic viscosity, n, and
thermal diffusivity, a
t
L
LLMtL
M
dzdc
JD
A
zA
AB
2
32
,)
1
1)((
• Gas mass diffusivity
– Based on Kinetic Gas Theory
– l = mean free path length, u = mean speed
– Hirschfelder’s equation:
uDAA l3
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
– 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
Binary gas-phase Lennard-Jones
“collisional integral”
– With no reliable or e, we can use the Fuller
correlation,
– For binary gas with polar compounds, we
calculate by
23/13/1
2/1
75.13 1110
BA
BA
AB
vvP
MMT
D
*
2196.00 T
ABD
where
bb
PBAAB
TV
232/1 1094.1,
ABTT e /* 2/1
e
e
e BAAB
bT23.1118.1/ e
)exp()exp()exp( ****0 HT
G
FT
E
DT
C
T
ABD
and
– For gas mixtures with several components,
– with
2/1
BAAB
3/1
23.11
585.1
bV
nn DyDyDyD
1
'
31
'
321
'
2
mixture1/...//
1
nyyy
yy
...32
2'
2
2
• Liquid mass diffusivity
– No rigorous theories
– Diffusion 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
– For infinite dilution of non-electrolytes in
water, W-C is simplified to Hayduk-Laudie eq.
– Scheibel’s 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
V
VK
– 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
ll2
• Pore diffusivity
– Diffusion of molecules within pores of porous
solids
– 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
l
AAA M
NTuD
ll 8
33*
• 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
Td
M
NTdu
dD 4850
8
33
KAAB
A
Ae DD
y
D
111
a
A
B
N
N1a
AeAe DD 2' e
Example 6
Types of porous diffusion. Shaded areas represent nonporous solids
– Hindered diffusion for solute in solvent-filled
pores
• A general model is
• F1 and F2 are correction factors, function of pore
diameter,
• F1 is the stearic partition coefficient
)()( 21 FFDD o
ABAe
pore
s
d
d
2
2
1 2
( )( ) (1 )
pore s
pore
d dF
d
• F2 is the hydrodynamic hindrance factor, one
equation is by Renkin,
53
2 95.009.2104.21)( F
Example 7
Convective Mass Transfer
• Mass transfer between moving fluid with
surface or another fluid
• Forced convection
• Free/natural convection
• Rate equation analogy to Newton’s cooling
equation
AcA ckN
Example 8
Differential Equations
• Conservation of mass in a control volume:
• Or,
in – out + accumulation – reaction = 0
....0
vcscdV
tdA nv
• For in – out,
– in x-dir,
– in y-dir,
– in z-dir,
• For accumulation,
xxAxxxA zynzyn ,,
yyAyyyA zxnzxn ,,
zzAzzzA yxnyxn ,,
zyxt
A
• For reaction at rate rA,
• Summing the terms and divide by xyz,
– with control volume approaching 0,
zyxrA
, , , , , ,0
A x x x A x x A y y y A y y A z z z A z z AA
n n n n n nr
x y z t
, , , 0AA x A y A z An n n r
x y z t
• 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 B
A B A Br rt
vvvnn BBAABA
BA rr
• So by conservation of mass,
• Written as substantial derivative,
– For species A,
0
t
v
0 v
Dt
D
0 AAA r
Dt
Dj
• In molar terms,
– For the mixture,
– And for stoichiometric reaction,
0
A
AA R
t
cN
0)(
BA
BABA RR
t
ccNN
0)(
BA RR
t
ccV