Page 68 - Bird R.B. Transport phenomena

§2.4 Flow Through an Annulus 53

To check whether the flow is laminar, we calculate the Reynolds number
D(v )p 4(w/p)p
Re = z
TTD/JL
4( 0.00398 -К-)(2.54 ^ X 12 ^ ) ( ^ ^ Y 1.261
\ min/\ m. ft/ \60 s /V

= 2.41 (dimensionless) (2.3-24)
Hence the flow is indeed laminar. Furthermore, the entrance length is
L e = 0.035D Re = (0.035)(0.1/12X2.41) = 0.0007 ft (2.3-25)
Hence, entrance effects are not important, and the viscosity value given above has been calcu-
lated properly.

EXAMPLE 2.3-2 Obtain an expression for the mass rate of flow w for an ideal gas in laminar flow in a long cir-
cular tube. The flow is presumed to be isothermal. Assume that the pressure change through
Compressible Flow in t ^ e ^ ^ е j s n o t v e r v i g e / s o that the viscosity can be regarded a constant throughout.
и
ar
a Horizontal Circular
Tube6
SOLUTION
This problem can be solved approximately by assuming that the Hagen-Poiseuille equation
(Eq. 2.3-21) can be applied over a small length dz of the tube as follows:

( 2 3 2 6 )
-
To eliminate p in favor of p, we use the ideal gas law in the form pip = p /p , where p and p 0
0
o
o
are the pressure and density at z = 0. This gives
( )
The mass rate of flow w is the same for all z. Hence Eq. 2.3-27 can be integrated from z = 0 to
z = L to give

^ (23-28)

Since p\-p\ = (p + p )(p - p ), we get finally
0
L
L
0
«*>-rf** .
(23 29)
where p av g = \(p + p ) is the average density calculated at the average pressure /? avg =
0
L
l(Po + Pi)-
§2.4 FLOW THROUGH AN ANNULUS
We now solve another viscous flow problem in cylindrical coordinates, namely the
steady-state axial flow of an incompressible liquid in an annular region between two
coaxial cylinders of radii KR and R as shown in Fig. 2.4-1. The fluid is flowing upward in

6
L. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, 2nd edition (1987), §17, Problem 6. A
perturbation solution of this problem was obtained by R. K. Prud'homme, T. W. Chapman, and J. R.
Bowen, Appl. Sci. Res, 43, 67-74 (1986).