fluid_mechanics

316 Chapter 6 ■ Differential Analysis of Fluid Flow
v z
r o
r
r i
z
F I G U R E 6.35
The viscous flow through an annulus.
An exact solution
can be obtained for
axial flow in the annular
space between
two fixed,
concentric cylinders.
6.9.4 Steady, Axial, Laminar Flow in an Annulus
r r .
The differential equations 1Eqs. 6.143, 6.144, 6.1452 used in the preceding section for flow in
a tube also apply to the axial flow in the annular space between two fixed, concentric cylinders
1Fig. 6.352. Equation 6.147 for the velocity distribution still applies, but for the stationary
annulus the boundary conditions become v z 0 at r r o and v z 0 for With these
two conditions the constants c 1 and c 2
i
in Eq. 6.147 can be determined and the velocity
distribution becomes
v z 1
4m a 0p
0z b c r 2 r 2 o r 2 i r 2 o
ln1r or i 2 ln r d
r o
(6.155)
The corresponding volume rate of flow is
r o
Q v z 12pr2 dr p 8m a 0p
0z b c r 4 o r 4 i 1r 2 o r 2 i 2 2
d
ln1r or i 2
r i
or in terms of the pressure drop, ¢p, in length / of the annulus
Q p¢p
8m/ c r 4 o r 4 i 1r 2
o r 2 i 2 2
d
ln1r or i 2
(6.156)
The velocity at any radial location within the annular space can be obtained from Eq. 6.155.
The maximum velocity occurs at the radius r r m where 0v z 0r 0. Thus,
r 2 o r 2 12
i
r m c
2 ln1r or i 2 d
(6.157)
An inspection of this result shows that the maximum velocity does not occur at the midpoint of
the annular space, but rather it occurs nearer the inner cylinder. The specific location depends on
r o and r i .
These results for flow through an annulus are valid only if the flow is laminar. A criterion
based on the conventional Reynolds number 1which is defined in terms of the tube diameter2 cannot
be directly applied to the annulus, since there are really “two” diameters involved. For tube cross
sections other than simple circular tubes it is common practice to use an “effective” diameter,
termed the hydraulic diameter, D h , which is defined as
D h
4 cross-sectional area
wetted perimeter
The wetted perimeter is the perimeter in contact with the fluid. For an annulus
D h 4p1r o 2 ri 2 2
21r o r i 2
2p1r o r i 2
In terms of the hydraulic diameter, the Reynolds number is Re rD h Vm 1where V Q
cross-sectional area2, and it is commonly assumed that if this Reynolds number remains below
2100 the flow will be laminar. A further discussion of the concept of the hydraulic diameter as it
applies to other noncircular cross sections is given in Section 8.4.3.