고급전달공정
Advanced Transport Phenomena (ch. 17)
Major: Interdisciplinary program of the integrated biotechnology
Graduate school of bio- & information technology Young-il Lim (N110), Lab. FACS
Young-il Lim (N110), Lab. FACS
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Announcement
- KIChE spring meeting, 2010 2010. 04. 21-23 (Wed. - Fri.)
2010. 04. 22 (Thu) 4. 29. PM 6-9 (communication
& discussion)
- FOA (fundamentals of Adsorption) conference, Japan 2010. 05. 22 – 28 (Sun. – Fri.)
2010. 05. 27 (Thu) 5. 20. PM 6-9
- Mid-term exam: April 29
th, 2010 PM 2-5 (open book)
Ch. 17 Diffusivity and the mechanisms of mass transport
- Newton’s law of viscosity, Fourier’s law of heat conduction, Fick’s law of diffusion.
- Movement of chemical species A through a binary mixture of A and B because of a co ncentration gradient of A.
- Maxwell-Stefan equation for multicomponent gases at low density.
17.1 Fick’s law of binary diffusion (molecular mass transport) D Y A
AB A
Ay
0
Y V A
F
- Mass flux is proportional to mass fraction difference (dw) and inversely to dist ance (dy).
- The proportional coefficient is the diffusivity.
- jAy + jBy = 0
- Mass flux in the positive y direction
- Downhill from high concentration region to low concentration region.
- Concentration gradient is thought of as a driving force - Diffusivity has units of [m2/s]
17.1 Fick’s law of diffusion (molecular mass transport)
dy D d
j
Ay
AB
AD Y A
AB A
Ay
0
) v v
(
j
Ay
A Ay
yj
A D
AB
AB BA
B
D
j
Ch. 17 Diffusivity and the mechanisms of mass transport
Ch. 17 Diffusivity and the mechanisms of mass transport
- Gas diffusivity at low density: independent of wA, increase with T, decrease with P - Liquid/Solid diffusivity: strongly concentration dependent, increase with T
- Concentration diffusion, thermal diffusion (from temperature gradient), pressure d iffusion (from pressure gradient), forced diffusion (from unequal external force) - diffusivity tensor (AB): isotropic, & anisotropic fluid
17.1 Fick’s law
] [
j
A
AB
ACh. 17 Diffusivity and the mechanisms of mass transport
17.2 Pressure and temperature dependence of diffusivity
- For gases,
1) pressure vs. diffusivity?
2) Temperature vs. diffusivity?
3) Self diffusion? See Fig. 17.2-1.
- For liquids,
1) pressure vs. diffusivity?
2) temperature vs. diffusivity?
Ch. 17 Diffusivity and the mechanisms of mass transport
- Consider a pure gas composed of rigid, non-polar spherical molecules of diameter dA and m ass mA. The number density is taken to be n. From kinetic theory, mean molecular speed (u), wall collision frequency per unit area (Z), mean free path (), and last collision dista nce (a) are:
m T du
) u ( f
du ) u ( u uf
8
0 0
17.3 Kinetic theory of diffusion in gases at low density
u n Z 4
1
21d2n
3 a 2
Ch. 17 Diffusivity and the mechanisms of mass transport
17.3 Kinetic theory of diffusion in gases at low density
- For accurate results, Chapman-Enskog kinetic theory should be used (see Eq. 17.3-1).
dy w dw
w
A yya
A yy
A3
2
dy D dw
v w
dy u dw
v w n
A y AA
A
A y
A Ay
*
3
1
1
3 2 3
1
2 A A
AA
d
T u m
D
*Simple model
Ch. 17 Diffusivity and the mechanisms of mass transport
- Kinetic theory vs. hydrodynamics theory (Nernst-Einstein equation).
- Nernst-Einstein equation: diffusivity of a single particle or solute molecule A through a stationary medium B is given by
17.4 Theory of diffusion in binary liquids
) F / u ( kT
DAB A A
In which B is the viscosity of the pure solvent, RA is the radius of the solute particle, and AB is the coefficient of sliding friction.
- no slip condition (Stokes’s law): AB = - complete slip condition: AB = 0
A B AB
A B
AB A B
A A
R R
R F
u
6 1 2
3
A B AB
A B AB
R D kT
R D kT
4 6
In which uA/FA is the mobility (the steady-state velocity attained by the particle und er the action of unit force)
Ch. 17 Diffusivity and the mechanisms of mass transport
17.6 Theory of diffusion in polymers
DAB 1M
- Mass concentration (), mass flux, & mass average velocity
- Molar concentration (c = /M), molar flux, & molar average velocity - Mass fraction ( = /), and mole fraction (x = c/c)
- Mass average velocity and molar average velocity
N v v
1
17.5 Theory of diffusion in colloidal suspensions
2
1 DAA* M
17.7 Mass and molar transport by convection
N
* x v
v
1
Ch. 17 Diffusivity and the mechanisms of mass transport
- Molecular mass flux and molar flux
A AB
A A
A (v v) D
j
17.7 Mass and molar transport by convection
A AB
* A A
*
A c (v v ) cD x
j
- Diffusion velocities: (vA-v), (vA-v*).
- Convective mass flux and molar flux (by bulk motion of fluid) v
v v
vx y z
c v*x c v*y c vz* c v*
- Molecular and convective mass flux = combined mass flux - Molecular and convective molar flux = combined molar flux
j v (v v) v v n
j c v c (v v )c v c v
N * * * *
Ch. 17 Diffusivity and the mechanisms of mass transport
- Maxwell-Stefan equation
N ..., , , , ),
N x N
x cD (
) v v
D ( x
x N x N 1 12 3
1 1
17.9 Maxwell-Stefan equations for multi-component
- The equation is derived from kinetic theory and Chapman-Enskog theory
- Important difference between binary diffusion and multi-component diffusion
- binary diffusion: movement of A is always proportional to the concentration gradient of species A.
- multi-component diffusion: reverse diffusion, osmotic diffusion, diffusion barrier
17.10 Questions for discussion
