Steady Flow in a Constricted Hose: Calculating Volume Flow Rate

How can we determine the volume flow rate of olive oil in a constricted hose based on given data? To calculate the volume flow rate of olive oil in a constricted hose, we can use the principles of fluid mechanics and Bernoulli's equation. Given the density of olive oil, diameter of the hose at the inlet and outlet, and the change in pressure between the two sections of the hose, we can determine the volume flow rate by calculating the velocity at the inlet and the area of the hose at the inlet. Let's break down the steps to solve for the volume flow rate.

Calculating Volume Flow Rate of Olive Oil in a Constricted Hose

Given Data:

  • Density of olive oil, ρ = 875 kg/m³
  • Diameter of hose at inlet, d₁ = 3.20 cm = 0.032 m
  • Diameter of hose at outlet, d₂ = 1.25 cm = 0.0125 m
  • Change in pressure between inlet and outlet, ΔP = -4900 Pa (negative as the flow is from higher pressure to lower pressure)

Assumptions:

  • Steady, ideal flow
  • Incompressible fluid

Calculate the area of the hose at the inlet and outlet:

Area at inlet, A₁: π(d₁/2)² = π(0.032/2)² = 8.0425 × 10⁻⁴ m²

Area at outlet, A₂: π(d₂/2)² = π(0.0125/2)² = 1.5708 × 10⁻⁴ m²

Apply Bernoulli's equation to relate pressure, velocity, and density at the inlet and outlet:

Pressure at inlet + (1/2)ρv₁² = Pressure at outlet + (1/2)ρv₂²

Rearrange Bernoulli's equation to solve for velocity at the inlet:

Velocity at inlet, v₁ = √[(2(P₁ - P₂)) / ρ]

Substitute given values to find velocity at the inlet:

Velocity at inlet, v₁ = √[(2(-4900)) / (875)] = 3.90 m/s

Calculate the volume flow rate at the inlet:

Volume Flow Rate, Q = v₁A₁ = (3.90)(8.0425 × 10⁻⁴) = 0.00310 m³/s

Therefore, the volume flow rate of olive oil in the constricted hose is 0.00310 m³/s based on the given data.

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