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Thermo & Stat Mech - Spring 2006 Class 15
1
Thermodynamics and Statistical Mechanics
Transport Processes
Thermo & Stat Mech - Spring 2006 Class 15
2
Mean Free Path
The average distance that a particles travels between collisions is called the mean free path, . In order to have a collision, the centers of two particles of radius R must approach to within a distance of 2R = d. The picture on the next slide illustrates this point.
Thermo & Stat Mech - Spring 2006 Class 15
3
Collision Cross Section
d2
Thermo & Stat Mech - Spring 2006 Class 15
4
Mean Free Path
)/(
1
1 so ,1
vvnv
vnvn
rel
relrel
Thermo & Stat Mech - Spring 2006 Class 15
5
Mean Free Paths(?)
nn
nn
nn
171.0
1
2
1
177.0
1
4
3
175.0
1
4
3
Thermo & Stat Mech - Spring 2006 Class 15
6
Mean Free Path
mc
mmrelm
relrel
vf
nn
vvnvvnv
vnvn
1163.0
8
1
)/2(
1
)/(
1
1 so ,1
Thermo & Stat Mech - Spring 2006 Class 15
7
Some Molecular Speeds
ProbableMost 2
3
3)(
8)(
max
2
0
22
0
m
kTv
m
kTvv
m
kTdvvfvv
m
kTdvvvfv
rms
Thermo & Stat Mech - Spring 2006 Class 15
8
Effusion
If there is a hole in a container of gas, molecules will escape through it. The rate at which they pass through the hole is equal to the flux of particles hitting the hole times the area of the hole, A. Then the rate of change of the number of molecules in the container is:
Adt
dN
Thermo & Stat Mech - Spring 2006 Class 15
9
Effusion Limit
AvnAdt
dNvn
DD
DD
41
41
22
2222
and Then,
4.0or 16.0 i.e.
16or 44
If
If the hole is small equilibrium conditions prevail, and is known.
Thermo & Stat Mech - Spring 2006 Class 15
10
Transport Processes
1. Molecular diffusion. The movement of the particles is of interest. This is particle transport.
2. Thermal conductivity. Particles carry energy from a high temperature region to a lower temperature region. This is energy transport.
3. Viscosity. Drag is created when particles move to a region which is moving at a different speed. This is momentum transport.
Thermo & Stat Mech - Spring 2006 Class 15
11
Transport Processes
dy
dv
A
F
dy
dT
dt
dQ
A
dy
ndD
xx
y
.3
1.2
.1
Thermo & Stat Mech - Spring 2006 Class 15
12
Viscosity
Consider a plate moving parallel to another plate with a layer of gas between them. The gas in contact with either plate is at rest relative to the plate, so there is a velocity gradient in the gas. A force F is applied to each plate to maintain the motion, and the area of the upper plate is A.
Thermo & Stat Mech - Spring 2006 Class 15
13
Viscosity
Thermo & Stat Mech - Spring 2006 Class 15
14
Viscosity
Gas diffuses up and down, transferring x momentum from layer to layer. The gas that moves in the + y direction is going slower than the layer it moves into, and the gas that moves in the – y direction is going faster than the layer it moves into.
Thermo & Stat Mech - Spring 2006 Class 15
15
Viscosity
Thermo & Stat Mech - Spring 2006 Class 15
16
Viscosity
First we need to know how far particles travel between collisions in the y direction. We must find the average value of y = cos , averaged over the flux of molecules.
dddvvvfn
d cossin])([4
Thermo & Stat Mech - Spring 2006 Class 15
17
Viscosity
3
2)2(
3
1)(
41
cossin)(4
cos
2
0
2
0
2
0
vv
y
vn
dddvvvfn
y
dy
Thermo & Stat Mech - Spring 2006 Class 15
18
Viscosity
(poise) 31
31
31
34
41
32
41
32
41
mvndy
dvmvn
A
FG
dy
dvmvn
dy
dvmvnGGG
dy
dvvmvnG
dy
dvvmvnG
xx
xx
xx
xx
Thermo & Stat Mech - Spring 2006 Class 15
19
Viscosity
2
31
31
33
8
18
8
18
d
mkTmkT
n
mm
kTn
nm
kTvmvn
Thermo & Stat Mech - Spring 2006 Class 15
20
Diffusion
vD
dy
ndv
dy
ndvGG
dy
ndynvG
dy
ndynvG
31
31
34
41
32
41
32
41
)(
)(
Thermo & Stat Mech - Spring 2006 Class 15
21
Thermal Conductivity
A
v
N
Tc
dy
dvn
dy
dvnGG
dt
dQ
A
dy
dyvnG
dy
dyvnG
31
34
41
32
41
32
41
1
)(
)(
Thermo & Stat Mech - Spring 2006 Class 15
22
Thermal Conductivity
A
v
A
v
A
v
A
v
N
cvn
dy
dT
N
cvn
dy
dvn
dt
dQ
A
dy
dT
N
c
dy
dT
dT
d
dy
d
N
Tc
31
31
311
Thermo & Stat Mech - Spring 2006 Class 15
23
Transport Processes
mvndy
dv
A
F
N
cvn
dy
dT
dt
dQ
A
vDdy
ndD
xx
A
v
y
31
31
31
.3
1.2
.1