First Principles - Sustainable Design: The Science of Sustainability and Green Engineering - page 72

number of moles for the entire fluid:

n A

x A = (2.18)

n A + n B + n C +···

The mole fraction value is always between 0 and 1. The mole fraction may be
converted to mole percentage as

= ×

x A% x A 100 (2.19)

For gases, the mole fraction is the same as the volumetric fraction of each gas in
a mixture of more than one gas.
The amount of resistance to flow when it is acted on by an external force,
especially a pressure differential or gravity, is the fluid's viscosity. This is a crucial
fluid property used in numerous green engineering applications, particularly in
air pollution plume characterization, sludge management, and wastewater and
drinking water treatment and distribution systems.
Bernoulli's equation states that when fluid is flowing in a long, horizontal pipe
with constant cross-sectional area, the pressure along the pipe must be constant.
However, as the fluid moves in the pipe, there will be a pressure drop. A pressure
difference is needed to push the fluid through the pipe to overcome the drag
force exerted by the pipe walls on the layer of fluid that is making contact with
the walls. Since the drag force exerted by each successive layer of the fluid on each
adjacent layer is moving at its own velocity, a pressure difference is needed (see
Fig. 2.10). The drag forces are known as viscous forces . Thus, the fluid velocity is
not constant across the pipe's diameter, owing to the viscous forces. The greatest
velocity is at the center (farthest away from the walls), and the lowest velocity is
found at the walls. In fact, at the point of contact with walls, the fluid velocity is
zero.
So if P 1 is the pressure at point 1 and P 2 is the pressure at point 2, with the
two points separated by a distance L , the pressure drop P is proportional to the
flow rate:

P = P 1 − P 2 (2.20)

Figure 2.10 Viscous flow
through a horizontal pipe. The
highest velocity is at the center of
the pipe. As the fluid approaches
the pipe wall, the velocity
approaches zero.

1 2

L

P 1 P 2

υ