10
1 Fundamental Concepts of Fluid Mechanics for Mine Ventilation
As the fluid is incompressible, the control volume crossing 1 and 2 is the same
size, and therefore has the same mass. Taking out the common factor, one arrives at:
1
2
ρV
v
2
2 − v
2
1
+ ρV g(h 2 − h 1 ) = (P 2 − P 1 )V
Which simplifies to:
1
2
ρ
v
2
2 − v
2
1
+ ρg(h 2 − h 1 ) = (P 2 − P 1 )
Then, regrouping terms leads to Bernoulli’s equation (Eq. 1.12):
1
2
ρv
2
1 + ρgh 1 + P 1 =
1
2
ρv
2
2 + ρgh 2 + P 2
(1.12)
This expression is stated as:
“The sum of kinetic per unit volume
1
2
ρv
2
i
, potential per unit volume
ρgh i
and static energy per unit volume (P i ) remains constant for any two points along a
streamline”.
Sometimes the Bernoulli equation is regrouped as follows (Eq. 1.13):
P i
ρg
+
v
2
i
2g
+ z i = H i
(1.13)
where
• P i
ρ g
: Pressure head or static head (m),
•
v
2
i
2 g
: Velocity head (m),
• z i : Elevation head (m), and
• H i : Total head (m).
Keep in mind that the units are meters of the circulating fluid and not the fluid
present in the gauge, which will generally be different from the circulating fluid.
Exercise 1.2 Demonstrate that the units in the terms
P
ρ
g and
v
2
2 g
of Bernoulli’s
equation are meters.
Solution
P
ρ g
N
m 2
K g
m 3
N
K g
= m
v
2
2 g
m
s
2
m
s 2
= m