How to Calculate Gauge Pressure for Water Exiting a Tapered Pipe?
To find the gauge pressure required for water to emerge from the small end of a tapered pipe with a speed of 12 m/s when the small end is elevated 8 m above the large end of the pipe, we can use Bernoulli's equation. This equation relates the pressure, velocity, and elevation of an incompressible fluid and is stated as:
P + (1/2)ρv^2 + ρgh = constant
Where:
P is the pressure within the fluid,
ρ (rho) is the density of the fluid,
v is the velocity of the fluid,
g is the acceleration due to gravity, and
h is the elevation above a reference point.
At the large end, assuming the water is essentially static (v = 0), the equation becomes:
P_large + ρgH = constant
At the small end, where the water is moving at 12 m/s, and the end is 8 meters above the large end, the equation is:
P_small + (1/2)ρv^2 + ρgh = constant
To find the gauge pressure, we subtract the pressure at the large end from the pressure at the small end:
Gauge Pressure = P_small - P_large = (1/2)ρv^2 + ρgh
Substituting ρ (the density of water) as 1000 kg/m^3, g (gravity) as 9.81 m/s^2, v as 12 m/s, and h as 8 m, we get:
Gauge Pressure = (1/2) (1000 kg/m^3)(12 m/s)^2 + (1000 kg/m^3)(9.81 m/s^2)(8 m)
This calculation will give us the needed gauge pressure to achieve the specified conditions.