Final answer:
To find the gauge pressure difference required for water to emerge at 12 m/s from a pipe where the small end is 8 m higher than the large end, Bernoulli's equation is used taking into account the conservation of energy within the fluid flow.
Step-by-step explanation:
The student's question is about calculating the gauge pressure difference required at two ends of a tapered pipe for water to emerge at a specified speed from the higher elevation end. To solve this, we need to apply the principle of conservation of energy, specifically Bernoulli's equation, to relate the pressure, velocity, and elevation difference along the pipe.
The equation for Bernoulli's principle is given by:
P1 + 0.5∙ρ∙v1^2 + ρ∙g∙h1 = P2 + 0.5∙ρ∙v2^2 + ρ∙g∙h2
where P is the pressure, ρ is the density of the fluid, v is the velocity, g is the acceleration due to gravity, and h is the elevation. The indices 1 and 2 refer to two different points along the streamline.
Given that the diameter at the large end of the pipe is twice as large as the small end, using the principle of continuity, the velocity at the large end (v1) will be one-fourth of the velocity at the small end (v2), because the cross-sectional area is four times larger. Therefore, v1 = v2/4 = 12 m/s / 4 = 3 m/s.
Assuming the fluid is water, we use a density (ρ) of 1000 kg/m^3 and g as 9.81 m/s^2. The height difference (h2 - h1) is given as 8 m (since the small end is elevated 8 m above the large end).
We simplify the equation to solve for the pressure difference, which is the gauge pressure (P2 - P1), and find it using the provided and derived values.