Problem:
Consider an incompressible liquid contained in a closed annular region between a tubing
of radius κR and a cylindrical container of inner radius
R
, as shown in the figure. The
liquid in the vertical annulus can be circulated by the following mechanical and thermal
methods.
1.
In the first method
, the tubing is moving upward with a constant velocity u
0
through
the fixed container. As the inner tube is pulled upwards it passes through a tight seal,
and no fluid leaves with it at the top (or enters with it at the bottom). This results in an
isothermal circulating flow of the liquid, which is upward along the moving tubing
and downward along the fixed container wall..It is assumed that the flow is steady. In
petroleum production, flows similar to this occur in annular spaces between tubing
and coil tubing. Another example is the seals of some reciprocating machinery, such
as in the annular space between piston and ring.
a.
Set up the equation of motion for the circulating flow, using the dimensionless
radius
r
r R
=
(
. State clearly how assumptions are used to simplify the flow
equation. (7.5 points)
b.
Define boundary and massconservation conditions that are required to solve the
equation of motion. (7.5 points)
c.
Show that the velocity distribution in the annular region, far from the end
disturbances, has the following form (15 points)
2
1
2
3
0
=C
C ln
C
(1)
z
u
r
r
u
+
+
(
(
2.
In the second method
, the tubing is fixed and heated up by steam. The temperatures
at the outer surface of the tubing wall and the inner surface of the container are
maintained constant at T
1
and T
2
respectively. The liquid density varies to a small
degree with temperature. This results in a similar circulating flow as observed in the
first method.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Peters
 Fluid Dynamics, Equations

Click to edit the document details