1 Module 4:
Multicomponent Transport
Lecture 35:
Thermal, Forced and Pressure Diffusion
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering 2 (I) Thermal Diffusion
Figure showing steady-state binary thermal diffusion in a bulb apparatus. The mixture
of gases A and B tends to separate under the influence of the thermal gradient.
The temperature gradient will cause a mass flux given by (see Eq. 3.7.16)
j
(T )
Az
=−
C2
ρ
M A M B DAB
kT dT
T dz
(3.7.26)
This mass flux will tend to develop a concentration gradient which in turn will yield a
(ordinary) diffusion flux equal to:
j
( x)
Az
=−
C2
ρ
M A M B DAB
dxA
dz
(3.7.27)
At steady state the net mass flux will be zero.
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering 3 j
(T )
Az
+j
( x)
Az
dx A
kT dT
=0⇒
=−
dz
T dz
T2
kT
T2
⇒ x A 2 − x A1 = − ∫ dT ≈ − kT ln
T
T1
T1
Or,
T2
x A 2 − x A1 = − kT ln ,
T1
(3.7.28)
Where kT is evaluated at a “mean” temperature Tm
Recommended mean temperature:
T1T2
T2
ln
Tm =
T2 − T1 T1
Equation (3.7.28) is useful for estimating the order of magnitude of thermal diffusion (Soret) effects.
(II) Pressure Diffusion
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering 4 Figure showing steady-state pressure diffusion in a centrifuge. The mixture of A and B tends to
separate by virtue of the pressure gradient produced in the centrifuge.
Pressure gradient in centrifuge
dp
= − ρ g Ω = − ρΩ 2 R0 ,
dz
Where Ω is the angular velocity.
At steady-state the net mass flux
j Az
is zero.
Equation (3.7.16) yields
d
⎛g ⎞
ln x A = − ⎜ Ω ⎟ ( M A − ρVA ) .
dz
⎝ RT ⎠
Integration of the above equation yields
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering 5 VB
VA
⎛ X A ⎞ ⎛ X B0 ⎞
gΩ z ⎤
⎡
⎜
⎟ ⎜
⎟ = exp ⎢(VA M B − VB M A )
RT ⎥⎦
⎣
⎝ X A0 ⎠ ⎝ X B ⎠
Where VA , VB are assumed to be constants independent of composition
(III) Forced Diffusion
Electrolytic Cell
Figure showing Concentration Polarization
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering (3.7.29)
6 Assumptions:
a) Ions M+ (i.e., Ag+) and X- (i.e., Cl-) diffuse in water independently of one another (dilute solution)
b) Activity coefficient of ions is unity
Because of dilute and quiescent solution assumption:
N i ≈ J i* (i.e., v* = 0)
Cell current is given by
I = N M + = J M*( x+ ) (ordinary diffusion) + J M*( g+ ) (forced diffusion)
(3.7.30)
0 = N X − = J X*(−x ) + J X*(−g )
Assumption (b) implies that
giz =
Also we have:
So we have:
(3.7.31)
dGi = RT ( d ln xi )
ε i ⎛ dφ ⎞
⎜− ⎟,
mi ⎝ dz ⎠
where ε i is the charge and φ is the potential.
⎛ dx x ε dφ ⎞
J lz* = −cDiw ⎜ i + i i
⎟
⎝ dz kT dz ⎠
(3.7.32)
Combining, we get
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering 7 ⎛ dx + x M + ε i dφ ⎞
I z = −cDM + w ⎜ M +
⎟
⎜ dz
kT dz ⎟⎠ ,
⎝
⎛ dxX − xX − ε i dφ ⎞
0 = −cDX − w ⎜⎜
−
⎟⎟ .
kT dz ⎠
⎝ dz
Summing up and taking into account that
I z = −2CDM + w
dxM +
dz
xM + = x X −
(electro-neutrality), we get
,
Which on integration yields
xM + − x0
z
=−
Iz
,
2CDM + w
(3.7.33)
where x 0 is the mole fraction at the cathode.
This shows a linear concentration gradient and that the maximum current that can be drawn is found
by setting x0=0.
I max = −
2CDM + w xM +
L
.
(3.7.34)
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering 8 This equation shows that there is a limit to the rate at which metal can be deposited on the cathode.
This is set by mass transfer.
NPTEL , IIT Kharagpur, Prof. Saikat Chakraborty, Department of Chemical Engineering