Biomedical Engineering Reference
In-Depth Information
Table 2.1
Lennard-Jones characteristic energies and characteristic diameters of common
gases
[1]
Gas
Characteristic Energy (3/
k
B
)
Characteristic Diameter s (nm)
Air
97.0
0.362
N
2
91.5
0.368
CO
2
190.0
0.400
O
2
113.0
0.343
Ar
124.0
0.342
10
23
J/K,d
ij
¼ c
ij
¼
Boltzmann constant: k
B
¼
1.38
1.
and often assumed to be 1.
Table 2.1
lists the parameters of some common gases. With the Lennard-
Jones potential, the force between the molecules can be derived as:
c
ij
s
r
7
d
ij
s
r
13
d
f
ij
ð
Þ
d
r
¼
r
48
3
s
F
ij
¼
:
(2.2)
The Lennard-Jones model results in the characteristic time:
s
¼ s
p
M=3
(2.3)
where
M
is the molecular mass. This characteristic time corresponds to the oscillation period between
repulsion and attraction. Furthermore, the model allows the determination of the dynamic viscosity of
a pure monatomic gas
[2]
:
10
26
p
2
:
68
MT
m
¼
(2.4)
s
2
U
where the collision integral
is a function of the dimensionless temperature
k
B
T
/
3
describing the
deviation from rigid sphere behavior, with
k
B
being the Boltzmann constant.
Fig. 2.1
depicts the
function of
U
is on the order of 1. The above equation allows
the determination of Lennard-Jones parameters
s
and
3
from the measurement of viscosity
m
,
a macroscopic continuum property.
Example 2.1
(
Estimation of gas viscosity using kinetic theory
). Estimate the viscosity of pure
nitrogen at 25
C.
Solution.
Using the Lennard-Jones parameters of nitrogen listed in
Table 2.1
, the dimensionless
temperature is:
U
. The value of the collision integral
U
k
B
T
3
¼
25
þ
273
¼
3
:
26
:
91
:
5
According to the diagram in
Fig. 2.1
, the collision integral of N
2
at 25
C is 0.95. The estimated
viscosity is then:
10
26
p
p
28
10
26
2
:
68
MT
2
:
68
ð
25
þ
273
Þ
10
5
Pa
m
¼
¼
¼
1
:
90
$
sec
:
0
10
9
2
s
2
U
:
368
0
:
95
Search WWH ::
Custom Search