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.
• Homework #7
• (self practice)
• WWWR #24.2
• To be discussed during the • ID # 14.25.
week 7-11 March, 2016.
• By either volunteer or
class list.
Molecular Mass Transfer
• Molecular diffusion
• Mass transfer law components:
– Molecular concentration:
A
nA p A
cA 


MA V
RT
– Mole fraction:
cA
xA 
c
(liquids,solids) ,
cA
yA 
c
(gases)
p A RT p A

For gases, y A 
P RT
P
n
– Velocity:
mass average velocity,
v
n
 v 
i 1
n
i
i
 i

i 1
i
vi

i 1
n
molar average velocity,
V
c v
i
i 1
i
c
velocity of a particular species relative to mass/molar average is
the diffusion velocity.
– 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
J A  DABcA
or, in the z-direction,
J A, z   DAB
dc A
dz
For a general relation in a non-isothermal, isobaric system,
J A, z  cDAB
dy A
dz
– Since mass is transferred by two means:
• concentration differences
• and convection differences from density differences
• For binary system with constant Vz,
• Thus,
J A , z  c A (v A , z  Vz )
J A, z  c A (v A, z  Vz )  cDAB
dy A
dz
• Rearranging to
c Av A, z  cDAB
dy A
 c AVz
dz
• As the total velocity,
1
Vz  (c A v A , z  c B v B , z )
c
• Or
c AVz  y A (c Av A, z  cB vB , z )
• Which substituted, becomes
c Av A, z  cDAB
dy A
 y A (c A v A, z  c B v B , z )
dz
• Defining molar flux, N as flux relative to a fixed z,
N A c A v A
• And finally,
N A, z  cDAB
dy A
 y A ( N A, z  N B , z )
dz
• Or generalized,
N A  cDABy A  y A (N A  N B )
• Related molecular mass transfer
– Defined in terms of chemical potential:
d c
DAB d c
v A , z  Vz  u A

dz
RT dz
– Nernst-Einstein relation
J A, z
DAB d c
 c A ( v A , z  Vz )   c A
RT dz
Diffusion Coefficient
• Fick’s law proportionality/constant
DAB
 J A, z
M
1
L2

 ( 2 )(
)
3
dc A dz
L t M L 1 L
t
• Similar to kinematic viscosity, n, and
thermal diffusivity, a
• Gas mass diffusivity
– Based on Kinetic Gas Theory
DAA*
1
 lu
3
– l = mean free path length, u = mean speed
DAA* 
2T 3 / 2
3
(
 3N
 P MA
3/ 2
2
A
)1/ 2
– Hirschfelder’s equation:
 1
1 
0.001858T 


M
M
A
B


2
P AB
D
3/ 2
DAB
1/ 2
– Lennard-Jones parameters  and e from tables,
or from empirical relations
– for binary systems, (non-polar,non-reacting)
 AB 
 A  B
2
e AB  e Ae B
– Extrapolation of diffusivity up to 25
atmospheres
DABT2 ,P2
 P1  T2 
 DABT1 ,P1   
 P2  T1 
3/ 2
 D T1
 D T2
Binary gas-phase Lennard-Jones
“collisional integral”
– With no reliable  or e, we can use the Fuller
1/ 2
correlation,
3
10 T
DAB 

1.75
 1
1 



 MA MB 
P  v A   v B

1/ 3 2
1/ 3
– For binary gas with polar compounds, we
calculate  by
   D0
0.196

*
T
2
AB
where
 AB
3 2
1
.
94

10
P
1/ 2
  A B  ,  
VbTb
T  T / e AB
*
e AB  e A e B 



  
1/ 2
e /   1.181  1.3 Tb
2
 D0 
A
T 
* B
C
E
G



*
*
*
exp( DT ) exp( FT ) exp( HT )
and
 AB   A B 
1/ 2
1/ 3
 1.585Vb 
 
2 
 1  1.3 
– For gas mixtures with several components,
D1 mixture
– with
1
 '
y2 / D1 2  y3' / D13  ...  yn' / D1 n
y2
y 
y2  y3  ...  yn
'
2
• Liquid mass diffusivity
–
–
–
–
No rigorous theories
Diffusion as molecules or ions
Eyring theory
Hydrodynamic theory
• Stokes-Einstein equation
DAB
T

6r B
– Equating both theories, we get Wilke-Chang eq.
DAB  B 7.4 10  B M B 

T
VA0.6
8
1/ 2
– For infinite dilution of non-electrolytes in
water, W-C is simplified to Hayduk-Laudie eq.
DAB  13.26 10 
5
1.14 0.589
B
A
V
– Scheibel’s equation eliminates B,
D AB  B
K
 1/ 3
T
VA
  3V  2 / 3 
K  (8.2 108 ) 1   B  
  VA  
– As diffusivity changes with temperature,
extrapolation of DAB is by
 Tc  T2 

 
( DABT2 )  Tc  T1 
( DABT1 )
n
– For diffusion of univalent salt in dilute solution,
we use the Nernst equation
DAB
2 RT

(1 / l0  1 / l0 ) F
• 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
l
• Knudsen number
Kn 
d pore
• From Kinetic Theory of Gases,
DAA*
lu
l 8NT


3 3 M A
• But if Kn >1, then
DKA
d pore 8NT
T

u
 4850d pore
3
3
M A
MA
d pore
• If both Knudsen and molecular diffusion exist, then
• with
1
1  ay A
1


DAe
DAB
DKA
NB
a  1
NA
• For non-cylindrical pores, we estimate
D
'
Ae
 e D Ae
2
Example 6
Types of porous diffusion. Shaded areas represent nonporous solids
– Hindered diffusion for solute in solvent-filled
pores
• A general model is
DAe  D F ( ) F2 ( )
o
AB 1
• F1 and F2 are correction factors, function of pore
diameter,
d

s
d pore
• F1 is the stearic partition coefficient
 (d pore  d s )2
2
F1 ( ) 

(1


)
2
 d pore
• F2 is the hydrodynamic hindrance factor, one
equation is by Renkin,
F2 ( )  1  2.104  2.09 3  0.95 5
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
N A  kc c A
Example 8
Differential Equations
• Conservation of mass in a control volume:

c.s.  v  n dA  t c.v. dV  0
• Or,
in – out + accumulation – reaction = 0
• For in – out,
– in x-dir, nA, x yz
x  x
– in y-dir, n A, y xz
y  y
 n A, y xz
– in z-dir, nA, z xy
z  z
 nA, z xy z
 nA, x yz
• For accumulation,
 A
xyz
t
x
y
• For reaction at rate rA,
rAxyz
• Summing the terms and divide by xyz,
nA, x x x  nA, x x
x

nA, y y y  nA, y y
y

nA, z z z  nA, z z
z
 A

 rA  0
t
– with control volume approaching 0,
 A



n A, x  n A, y  n A, z 
 rA  0
x
y
z
t
• We have the continuity equation for
component A, written as general form:
 A
 nA 
 rA  0
t
• For binary system,
 n A  nB  
   A  B 
t
  rA  rB   0
• but n A  n B   A v A   B v B  v
• and rA  rB
• So by conservation of mass,

  v 
0
t
• Written as substantial derivative,
D
   v  0
Dt
– For species A,
D A
Dt
   j A  rA  0
• In molar terms,
c A
NA 
 RA  0
t
– For the mixture,
 c A  cB 
  N A  N B  
 ( RA  RB )  0
t
– And for stoichiometric reaction,
c
  cV   ( RA  RB )  0
t