Environmental Engineering Reference
In-Depth Information
r
= R
Integrating Eq. (4.6) between r to R at the boundary conditions
:
= v , we obtain a formula to calculate the velocity of the liquid layers located
at a distance r from the axis of the tube:
2
2
2
P QR
r
����
( )
vr
=+
v
*
 
-
€‚ ƒ„
€‚
(4.18)
*
2
ƒ„
π
RR
R
 
-
Let's insert Eq. (4.18) in Eq. (4.11), integrate over the radius of all sections from 0
to
. Then we get a formula for the flow of liquid from the nanotube:
R
4
8
lR
h ��
2
*
R
QQv pR
=
+
*
 
-
ƒ„
P
*
4
(4.19)
D
R
 
-
or
2
4
8 lv R
h ����
R
e=
*
*
+
*
€‚€‚
ƒ„ƒ„
4
D
pR
R
R
v :
from which we candetermine theunknown
2
4
��
Q
R
R
��
v
=
P
e−
*
 
-
€‚ €‚
*
ƒ„
(4.20)
π
R
2
RR
ƒ„
 
-
*
Figure 4.17 shows the profile of the radial velocity of water particles in a
carbon nanotube with a diameter of 2.77 nm, calculated using the Eq. (4.18) at
19
, e = 114. The velocity at the border v is equal to 11.55
m/s. It is seen that the velocity profile is similar to a parabolic shape, and at the
same time agrees with the calculations of Thomas John and McGaughey Alan
[73], obtained by using the model of molecular dynamics.
The calculations suggest the following conclusions. Flow of liquid in a nano-
tube was investigated using synthesis of the methods of the continuum theory and
molecular dynamics. Two models are considered. The first is based on the fact
that fluid in the nanotube behaves like a viscoplastic. A method of calculating the
value of limiting shear stress is proposed, which was dependent on the nanotube
radius. A simplified model agrees quite well with the results of the molecular
simulations of fluid flow in carbon nanotubes. The second model assumes the ex-
istence of an empty interlayer between the liquid molecules and wall of the nano-
tube. This formulation of the task is based on the results of experimental works
known from the literature. The velocity profile of fluid flowing in the nanotube is
practically identical to the profile determined by molecular modeling.
Q -
2
4, 75
10
n / m
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