Page 36 - Advanced Mine Ventilation

Air Flow in Mine Airways 19

The theory of dimensional analysis will be used to determine the variables that can
be used to predict l. When a viscous fluid flows in a pipe, the frictional stress, s o ,is
dependent on the following variables only:

v; the velocity of the fluid; ft=s L = T
d; the pipe diameter; ft ðLÞ

M
3
r; the density of the fluid; lb=ft 3
L

M
m; the viscosity of the fluid; poise
LT
and e; the pipe roughness; in ft ðLÞ
Mathematically,
s o ¼ F ðv; d; r; m; eÞ (2.2)

Dimensional analysis converts Eq. (2.2) into
c
a d
M L b M M e

¼ ðLÞ ðLÞ
2
T L T L 3 LT
Comparing the power of mass (M), length (L), and time (T) on both sides for
M: 1 ¼ c þ d
L: 1 ¼ a þ b 3c e d þ e
T: 2 ¼ a dor 2 ¼ a þ d

We can eliminate three unknowns by converting a, b, and c into d and e.
a ¼ 2 d; b ¼ (d þ e) and c ¼ 1 d.
Thus Eq. (2.2) can be rewritten as:
2 d ðdþeÞ 1 d d e
s o ¼ cv d r m e
(2.3)
e e
m d 2

or s o ¼ c r$v
vd r d
where c is the constant of proportionality.
Eq. (2.3) shows that frictional losses in a pipe are basically a function of two vari-

vd r
e
ables: is defined as the Reynold’s number, R, and is the roughness factor.
m d
Stanton [2] and Nikuradse [3] have carried out extensive research on measuring l
6
3
e
for Reynold’s number ranging from 10 to 10 and ranging from 1 10 4 to
d
2
1 10 . References can be made to these works for details.
The roughness, e, for various commercial pipes are shown in Table 2.1.
Colebrook [4] studied roughness of many pipes and came up with a single equation
that can be used very conveniently.