Final answer:
To calculate the required gauge pressure for water to emerge from the small end of a tapered pipe with a specified speed and elevation difference, Bernoulli's principle is applied, considering fluid density, gravity, velocities, and elevations at both ends of the pipe.
Step-by-step explanation:
To determine the required gauge pressure difference between the large and small ends of the pipe, we can apply the principles of fluid dynamics. Specifically, the Bernoulli's principle comes into play, which relates the speed, pressure, and elevation of a fluid in steady flow.
According to Bernoulli's equation, for an incompressible and non-viscous fluid:
P1 + 0.5ρv1² + ρgh1 = P2 + 0.5ρv2² + ρgh2
Where:
- P1 and P2 are the pressures at the large and small ends of the pipe respectively,
- ρ is the density of the fluid (for water, typically ρ = 1000 kg/m³),
- v1 and v2 are the velocities at the large and small ends of the pipe respectively,
- σ is the acceleration due to gravity (approximately 9.81 m/s²),
- h1 and h2 are the elevations at the large and small ends of the pipe respectively.
Assuming that the large end is at h1 = 0 m and that the water emerges at the small end (h2 = 8 m) with velocity v2 = 12 m/s, and that the velocity at the large end (v1) is negligible due to its much larger cross-sectional area (based on the continuity equation ρA1v1 = ρA2v2), we can calculate the gauge pressure P1 - P2 required to produce this flow. Take into account that the elevation difference adds a potential energy component to the equation.