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Air at 20°C flows through a circular duct. The absolute pressure is 100.8 kPa at point A and 101.6 kPa at point B. Determine the volumetric discharge through the duct.

Answer :

To determine the volumetric discharge through the circular duct, we first need to calculate the air velocity using Bernoulli's equation:

P₁ + ½ρv₁² = P₂ + ½ρv₂²

Where:
P₁ = absolute pressure at point A = 100.8 kPa
P₂ = absolute pressure at point B = 101.6 kPa
ρ = density of air at 20°C = 1.204 kg/m³
v₁ = velocity of air at point A
v₂ = velocity of air at point B

We know that the temperature of the air is constant at 20°C, so we can assume that the density is constant throughout the duct. Rearranging the equation and solving for v₁, we get:

v₁ = √[(2(P₂ - P₁))/ρ]

v₁ = √[(2(101.6 - 100.8))/1.204]

v₁ = 24.9 m/s

Now that we have the air velocity, we can calculate the volumetric flow rate using the formula:

Q = A × v

Where:
Q = volumetric flow rate
A = cross-sectional area of the duct
v = air velocity

Since the duct is circular, the cross-sectional area can be calculated using the formula:

A = πr²

Where:
r = radius of the duct

We don't have the radius of the duct, but we can use the hydraulic diameter as an approximation, which is defined as:

Dh = (4A) / P

Where:
Dh = hydraulic diameter
A = cross-sectional area of the duct
P = perimeter of the duct

For a circular duct, the perimeter is equal to the circumference, so we can write:

P = 2πr

Substituting this into the hydraulic diameter equation, we get:

Dh = (4πr²) / (2πr)

Dh = 2r

Now we can approximate the cross-sectional area of the duct as:

A ≈ π(Dh/2)² = πr²

Substituting the values we have, we get:

A ≈ π(0.1 m)² = 0.0314 m²

Finally, we can calculate the volumetric flow rate as:

Q = A × v₁

Q = 0.0314 m² × 24.9 m/s

Q = 0.7818 m³/s

Therefore, the volumetric discharge through the circular duct is approximately 0.7818 m³/s.

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