Answer :
Final answer:
Doubling the cross-sectional area of a circular duct while halving the velocity of air results in no change to the overall rate of ventilation, as the increase in area exactly compensates for the reduction in velocity.
Explanation:
If the cross-sectional area of a circular duct is doubled and the velocity of air is reduced to half, the effect on the rate of ventilation must be considered through the lens of the continuity equation and basic principles of fluid dynamics. The rate of ventilation, or flow rate, in such a situation can be represented by the product of the cross-sectional area of the duct and the velocity of the air flowing through it. Doubling the area effectively doubles the capacity for air to flow through the duct per unit time, whereas halving the velocity reduces this capacity by half.
Mathematically, if the initial flow rate is given by A1 × V1, where A is the cross-sectional area and V is the velocity, and knowing that the new area is 2A1 and the new velocity is V1/2, the new flow rate becomes 2A1 × (V1/2) = A1 × V1, which is equal to the original flow rate. Thus, despite the changes in both cross-sectional area and velocity, the overall rate of ventilation remains unchanged.
