Advanced Transport Phenomena (ch. 1)
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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- The resistance of flow is the viscosity.
- Molecular momentum transport between parallel plates.
- Convective momentum transport by bulk fluid motion.
1.1 Newton’s law of viscosity (molecular transport of momentum)
Y V A
F
1.1 Newton’s law of viscosity (molecular transport of momentum)
Y V A
F
dy dv
yx
- Momentum flux (pressure or stress) is proportional to velocity difference (dv) and inversely to distance (dy).
- The proportional coefficient is the viscosity.
- Convective momentum transport by bulk fluid motion.
- Newtonian fluids: the flow resistance of fluids with molecular weight of less t han about 5000 is described by the above equation.
- Non-Newtonian fluids: polymers, pastes, slurries, suspensions … - Flux of x-momentum in the positive y direction
- Downhill from high velocity region to low velocity region.
1.2 Generalization of Newton’s law
zz zy
zx
yz yy
yx
xz xy
xx
zz zy
zx
yz yy
yx
xz xy
xx
p p
p tensor
molecular
- Normal stress and shear stress.
- Viscous stress tensor (
ij) and molecular stress tensor (
ij) - Velocity gradient tensor (v)
k l l
k ijkl
ij
x
tensor v
viscous
1.3 Pressure and temperature dependence of viscosity
- For gases,
1) pressure vs. viscosity?
2) Temperature vs. viscosity?
- For liquids,
1) pressure vs. viscosity?
2) temperature vs. viscosity?
- Consider a pure gas composed of rigid, non-attracting spherical molecules of diameter d and mass m. The number density is taken to be n. The mean molecular velocity:
1.4 Molecular theory of the viscosity of gases at low density
m T du
) u ( f
du ) u ( u uf
8
0 0
T / mu
T e nu m
) u (
f
22 3 2
2
4 2
- Maxwell-Boltzmann distribution of molecular velocity - Z: frequency of molecular bombardment per unit area - is the Boltzamnn constant (=R/N=1.3806610-23 J/K)
u n Z 4
1
- Mean free path (): average distance traveled by a molecule between successive collisions
1.4 Molecular theory of the viscosity of gases at low density
n d
22
1
a y x y a
y x y
yx
Zm v
Zm v
- Maxwell’s viscosity of rigid atom in gas phase.
- is independent of pressure and dependent on square root of temperature
dy v dv
v
x yya
x yy
x3
2
dy u dv
nm
xyx
3
1
3
22 3
1 3
1
d T u m
u
nm
Simple model
- transport properties of intermolecular potential energy - intermolecular force, F(r)=-d/dr
- LJ (Lennard-Jones) 6-12 potential:
- : collision diameter, : characteristic energy
6 12
4 r r
) r (
2 2
16 5 3
2
T m
d
T
m
- ij: flux of j-momentum across a surface perpendicular to the i-direction.
- ij: molecular momentum flux for random molecular motions - Convective momentum flux (vv): momentum by bulk fluid flow
1.5 Molecular theory of the viscosity of liquids
RT G~
V~h e N~ a
0
2
- Viscous momentum flux (ij) and combined momentum flux
1.7 Convective momentum transport
i j
j i i
i
v v
v
v
1. Momentum per unit area per unit time = force per unit are = pressure?
2. Compare the molecular and convective mechanisms for momentum transport?
3. What are the physical meaning of LJ parameters ( and )?
4. Sketch the potential energy function (r) for rigid, non-attracting sphere.
