Page 118 - Bird R.B. Transport phenomena

§3.7 Dimensional Analysis of the Equations of Change 103

in which the kinematic viscosity v = /л/р is used. Normally both tanks will operate in the
same gravitational field g, = g , so that Eq. 3.7-41 requires
n
(
N, \Dj (3.7-42)
Substitution of this into Eq. 3.7-40 gives the requirement

(3.7-43)
This is an important result—namely, that the smaller tank (II) requires a fluid of smaller kine-
matic viscosity to maintain dynamic similarity. For example, if we use a scale model with
=
^
£*II iDu then we need to use a fluid with kinematic viscosity v xl = /V8 in the scaled-down
experiment. Evidently the requirements for dynamic similarity are more stringent here than
in the previous example, because of the additional dimensionless group Fr.
In many practical cases, Eq. 3.7-43 calls for unattainably low values of v u. Exact scale-up
from a single model experiment is then not possible. In some circumstances, however, the ef-
fect of one or more dimensionless groups may be known to be small, or may be predictable
from experience with similar systems; in such situations, approximate scale-up from a single
9
experiment is still feasible.
This example shows the importance of including the boundary conditions in a dimen-
sional analysis. Here the Froude number appeared only in the free-surface boundary condi-
tion Eq. 3.7-36. Failure to use this condition would result in the omission of the restriction in
Eq. 3.7-42, and one might improperly choose v l{ = v v If one did this, with Re n = Rei, the
Froude number in the smaller tank would be too large, and the vortex would be too deep, as
shown by the dotted line in Fig. 3.7-3.

EXAMPLE 3.7-3 Show that the mean axial gradient of the modified pressure & for creeping flow of a fluid of
constant p and /JL through a tube of radius R, uniformly packed for a length L > D p with
>
Pressure Drop for solid particles of characteristic size D p « R, is
Creeping Flow in a
Packed Tube K(geom) (3.7-44)

Here (• • •) denotes an average over a tube cross section within the packed length L, and the
function K(geom) is a constant for a given bed geometry (i.e., a given shape and arrangement
of the particles).

SOLUTION We choose D p as the characteristic length and (v z) as the characteristic velocity. Then the inter-
stitial fluid motion is determined by Eqs. 3.7-8 and 3.7-9b, with v = \/{v z) and 9> =
3 0)D I,/IJL{V-), along with no-slip conditions on the solid
(9? - < ) surfaces and the modified pres-
sure difference A(2P) = (0> o ) - ($>,). The solutions for v and Ф in creeping flow (D (v )p/fi
p
z
—> 0) accordingly depend only on г, 0, and z for a given particle arrangement and shape. Then
the mean axial gradient
CUD D
D p p e
\dz = -j- (% - & ) (3.7-45)
J I L
L Jo
depends only on the bed^geometry as long as R and L are large relative to D . Inserting the
p
foregoing expression for Ф, we immediately obtain Eq. 3.7-44.

For an introduction to methods for scale-up with incomplete dynamic similarity, see R. W. Powell,
4
An Elementary Text in Hydraulics and Fluid Mechanics, Macmillan, New York (1951).