To find the width of the ducts, we use the formula for volume flow rate and solve for the cross-sectional area. The width of the ducts is 9.49 m if the air flows at 5.0 m/s and 21.21 m if the air flows at 1.0 m/s.
Step-by-step explanation:
To find the width of the ducts, we first need to calculate the volume flow rate of air that needs to be replaced in the house every hour. The volume of the house is given as 450 m³. Since the air in the ducts flows at 5.0 m/s, we can calculate the volume flow rate using the formula Q = A × v, where Q is the volume flow rate, A is the cross-sectional area of the ducts, and v is the velocity of the air. Rearranging the formula to solve for A, we have A = Q / v. Substituting the values, we get A = (450 m³/h) / (5.0 m/s) = 90 m². Since the ducts have a square cross section, the width is equal to the length, so the width of the ducts is √A = √90 m² = 9.49 m.
To determine the size of the ducts if the air is flowing at 1.0 m/s, we can use the same formula. The volume flow rate remains the same (450 m³/h), so substituting the values in the formula A = Q / v, we get A = (450 m³/h) / (1.0 m/s) = 450 m². Since the ducts have a square cross section, the width is equal to the length, so the width of the ducts is √A = √450 m² = 21.21 m.
The probable question can be: A house has a volume of 450 m3. The air-conditioning system is designed to replace the air in the house every hour, using ducts that have a square cross section. Assume that we can treat air as an incompressible fluid. (a) The air in the ducts flows at 5.0 m/s, find the width of the ducts. (b) How big would the ducts need to be if the owner wanted the air to only flow at 1.0 m/s?