Tutorial/HW Week #7 WRF Chapters 22-23 ; WWWR Chapters 24-25 ID Chapter 14 Tutorial #7 WWWR# 24.1,...
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Transcript of Tutorial/HW Week #7 WRF Chapters 22-23 ; WWWR Chapters 24-25 ID Chapter 14 Tutorial #7 WWWR# 24.1,...
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 during the week 9-13 March, 2015.
• By either volunteer or class list.
• Homework #7
• (self practice)
• WWWR #24.2
• ID # 14.25.
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 AAA
n
iii
n
ii
n
iii
1
1
1
vvv
c
cn
iii
1
vV
– 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 relationdz
d
RT
D
dz
duVv cABc
AzzA
,
dz
d
RT
DcVvcJ cABAzzAAzA
)( ,,
Diffusion Coefficient
• Fick’s law proportionality/constant
• Similar to kinematic viscosity, , and thermal diffusivity,
t
L
LLMtL
M
dzdc
JD
A
zAAB
2
32
, )1
1)((
• Gas mass diffusivity– Based on Kinetic Gas Theory
– = mean free path length, u = mean speed
– Hirschfelder’s equation:
uDAA 3
1*
2/13
22/3
2/3
* )(3
2
AAAA M
N
P
TD
DAB
BAAB P
MMT
D
2
2/1
2/3 11001858.0
– Lennard-Jones parameters and from tables, or from empirical relations
– for binary systems, (non-polar,non-reacting)
– Extrapolation of diffusivity up to 25 atmospheres
2BA
AB
BAAB
2
1
1,12,2
2/3
1
2
2
1
TD
TD
ABAB T
T
P
PDD
PTPT
Binary gas-phase Lennard-Jones “collisional integral”
– With no reliable or , we can use the Fuller correlation,
– For binary gas with polar compounds, we calculate by
23/13/1
2/1
75.13 1110
BA
BAAB
vvP
MMT
D
*
2196.00 T
ABD
where
bb
PBAAB TV
232/1 1094.1,
ABTT /* 2/1
BAAB
bT23.1118.1/
)exp()exp()exp( ****0 HT
G
FT
E
DT
C
T
ABD
and
– For gas mixtures with several components,
– with
2/1BAAB
3/1
23.11
585.1
bV
nn DyDyDyD
1
'31
'321
'2
mixture1 /...//
1
nyyy
yy
...32
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.
BAB r
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
3/1A
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
• 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
AAA M
NTuD
8
33*
• But if Kn >1, then
• If both Knudsen and molecular diffusion exist, then
• with
• For non-cylindrical pores, we estimate
Apore
A
poreporeKA M
Td
M
NTdu
dD 4850
8
33
KAAB
A
Ae DD
y
D
111
A
B
N
N1
AeAe DD 2'
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 oABAe
pore
s
d
d
22
1 2
( )( ) (1 )pore s
pore
d dF
d
• F2 is the hydrodynamic hindrance factor, one equation is by Renkin,
532 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
....
0vcsc
dVt
dA nv
• For in – out,– in x-dir,
– in y-dir,
– in z-dir,
• For accumulation,
xxAxxxA zynzyn ,,
yyAyyyA zxnzxn ,,
zzAzzzA yxnyxn ,,
zyxtA
• 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 BA B A Br r
t
vvvnn BBAABA
BA rr
• So by conservation of mass,
• Written as substantial derivative,
– For species A,
0
t
v
0 vDt
D
0 AAA r
Dt
Dj
• In molar terms,
– For the mixture,
– And for stoichiometric reaction,
0
A
AA R
t
cN
0)(
BABA
BA RRt
ccNN
0)(
BA RRt
ccV