Page 361 - Modelling in Transport Phenomena A Conceptual Approach

9.1. MOMENTUM TRANSPORT 341

The Taylor series expansion of the term ln(1- $) is
1
1
In(l-$) = -$- -$2 - 3+3 - ... (9.1-1 14)
2
Using Eq. (9.1-114) in Eq. (9.1-113) and carrying out the divisions yields

(Po - PL) WB3
&=
8 PL
(9.1-115)
or 7
(Po - PL) WB3 lim (- $- + ...>
2
-
&=
8 PL @-O 3 2
- (Po - PL) WB3
-
12 pL (9.1-116)
Note that Eq. (9.1-116) is equivalent to Eq. (9.1-26).
W Case (ii) K -+ 0

When the ratio of the radius of the inner pipe to that of the outer pipe is close to
zero, i.e., K + 0, a concentric annulus may be considered to be a circular pipe of
radius R. In this case Eq. (9.1-99) becomes

(9.1-117)

Since In0 = -00, Eq. (9.1-117) reduces to

(9.1-118)

which is identical with Eq. (9.1-83).

9.1.5 Physical Significance of the Reynolds Number

The physical significance attributed to the Reynolds number for both laminar and
turbulent flows is that it is the ratio of the inertial forces to the viscous forces.
However, examination of the governing equations for fully developed laminar flow:
(i) between parallel plates, Eq. (9.1-19), (ii) in a circular pipe, Q. (9.1-76),
and (iii) in a concentric annulus, Eq. (9.1-92), indicates that the only forces
present are the pressure and the viscous forces. Inertial forces do not exist in
these problems. Since both pressure and viscous forces are kept in the governing
equation for velocity, they must, more or less, have the same order of magnitude.
Therefore, the ratio of pressure to viscous forces, which is a dimensionless number,
has an order of magnitude of unity.