Water flows at a rate of 0.040 m3/s in a 0.15 m diameter pipe that contains a
sudden contraction section to a 0.06 m diameter pipe. Determine the pressure
drop across the contraction section. Determine also the pressure difference due to
the losses.
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To determine the pressure drop across the contraction section, you can use the principle of continuity and Bernoulli's equation. Here's how:
1. Apply the principle of continuity: According to the principle of continuity, the mass flow rate of an incompressible fluid remains constant in a closed system. Therefore, you can equate the mass flow rates before and after the contraction:
A1 * V1 = A2 * V2,
where A1 and A2 are the cross-sectional areas of the pipe before and after the contraction, respectively, and V1 and V2 are the velocities of the water at those sections.
2. Convert the given diameter into the corresponding areas:
A1 = π * (0.15/2)^2 = π * 0.0225 m^2,
A2 = π * (0.06/2)^2 = π * 0.009 m^2.
3. Rearrange the equation from step 1 to solve for V2:
V2 = (A1 * V1) / A2.
4. Substitute the given flow rate into the equation:
0.040 m^3/s = (π * 0.0225 m^2 * V1) / (π * 0.009 m^2).
5. Simplify the equation:
V1 = (0.040 m^3/s * 0.009 m^2) / 0.0225 m^2 = 0.016 m/s.
6. Apply Bernoulli's equation: Bernoulli's equation relates the pressure, velocity, and height of a fluid. The equation can be written as:
P1 + 0.5 * ρ * V1^2 + ρ * g * h1 = P2 + 0.5 * ρ * V2^2 + ρ * g * h2,
where P1 and P2 are the pressures at sections 1 and 2, ρ is the density of the fluid, g is the acceleration due to gravity, and h1 and h2 are the elevations of sections 1 and 2, respectively.
7. Assume that the elevation difference is negligible (h1 = h2 = 0). Thus, the equation simplifies to:
P1 + 0.5 * ρ * V1^2 = P2 + 0.5 * ρ * V2^2.
8. Since the pressure drop across the contraction is the difference between P1 and P2, rearrange the equation to solve for the pressure drop:
(P1 - P2) = 0.5 * ρ * (V2^2 - V1^2).
9. Substitute the known values into the equation:
(P1 - P2) = 0.5 * ρ * ((0.016 m/s)^2 - (0.040 m/s)^2).
10. Finally, calculate the pressure drop:
(P1 - P2) = 0.5 * ρ * (-0.001504 m^2/s^2).
The pressure difference due to losses can be calculated using empirical equations such as Darcy-Weisbach or Hazen-Williams. These equations take into account factors such as pipe roughness, friction losses, and fittings. To provide a more accurate answer, specific information about the pipe, such as its material and length, is needed.