Question

For modeling and calculation purposes, architects treat air as an incompressible fluid. As an architect's intern, you are doing the specs on a dorm air conditioning system that is designed to replace the air in each room every twenty-nine minutes. If the rooms each have a volume of 175 m3 and they are supplied by ducts with a square cross section, determine the following. (a) the length of each side of a duct if the air speed in the duct is to be 3. 2 m/s m (b) the length of each side of a duct if the air speed at the duct is to be a value twice this speed. M

Answer 1

(a) To determine the length of each side of a duct if the air speed in the duct is to be 3.2 m/s, we can use the equation:

Volume flow rate = Area x Air speed

The volume flow rate is the volume of air that needs to be supplied to each room every 29 minutes, which is:

Volume flow rate = 175 m^3 / 29 min = 6.03 m^3/s

The area of the duct can be found by rearranging the equation:

Area = Volume flow rate / Air speed

Substituting the given values, we get:

Area = 6.03 m^3/s / 3.2 m/s = 1.885 m^2

Since the duct is square, each side of the duct will have the same length, which is:

Side length = sqrt(Area) = sqrt(1.885 m^2) = 1.373 m

Therefore, the length of each side of a duct if the air speed in the duct is to be 3.2 m/s is 1.373 m.

(b) To determine the length of each side of a duct if the air speed at the duct is to be twice the previous speed, we can use the same equation:

Volume flow rate = Area x Air speed

The volume flow rate is still the same, but the air speed is now 2 x 3.2 m/s = 6.4 m/s. Substituting the values, we get:

Area = 6.03 m^3/s / 6.4 m/s = 0.941 m^2

The length of each side of the duct is:

Side length = sqrt(Area) = sqrt(0.941 m^2) = 0.970 m

Therefore, the length of each side of a duct if the air speed at the duct is to be twice the previous speed is 0.970 m.