Water flows at a rate of 0.040 m3 / s in a horizontal pipe whose diameter is reduced from 15 cm to 8 cm by a reducer. If the pressure at the centerline is measured to be 480 kPa and 440 kPa before and after the reducer, respectively, determine the irreversible head loss in the reducer. Take the kinetic energy correction factors to be 1.05. Answer: 0.963 m

hL = 0.9627 m

Explanation:

Given

Q = 0.040 m³/s (constant value)

D₁ = 15 cm = 0.15 m ⇒ R₁ = D₁/2 = 0.15 m/2 = 0.075 m

D₂ = 8 cm = 0.08 m ⇒ R₂ = D₂/2 = 0.08 m/2 = 0.04 m

P₁ = 480 kPa = 480*10³Pa

P₂ = 440 kPa = 440*10³Pa

α = 1.05

ρ = 1000 Kg/m³

g = 9.81 m/s²

h₁ = h₂

hL = ? (the irreversible head loss in the reducer)

Using the formula Q = v*A ⇒ v = Q/A

we can find the velocities v₁ and v₂ as follows

v₁ = Q/A₁ = Q / (π*R₁²) = (0.040 m³/s) / (π * (0.075 m) ²) = 2.2635 m/s

v₂ = Q/A₂ = Q / (π*R₂²) = (0.040 m³/s) / (π * (0.04 m) ²) = 7.9577 m/s

Then we apply the Bernoulli law (for an incompressible flow)

(P₂ / (ρ*g)) + (α*v₂² / (2*g)) + h₂ = (P₁ / (ρ*g)) + (α*v₁² / (2*g)) + h₁ - hL

Since h₁ = h₂ we obtain

(P₂ / (ρ*g)) + (α*v₂² / (2*g)) = (P₁ / (ρ*g)) + (α*v₁² / (2*g)) - hL

⇒ hL = ((P₁-P₂) / (ρ*g)) + (α / (2*g)) * (v₁²-v₂²)

⇒ hL = ((480*10³Pa-440*10³Pa) / (1000 Kg/m³*9.81 m/s²)) + (1.05 / (2*9.81 m/s²)) * ((2.2635 m/s) ² - (7.9577 m/s) ²)

⇒ hL = 0.9627 m