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.