A room has a volume of 164 m3. An air-conditioning system is to replace the air in this room every 26.8 minutes, using ducts that have a square cross section. Assuming that air can be treated as an incompressible fluid, find the length of a side of the square if the air speed within the ducts is (a) 3.00 m/s and (b) 8.00 m/s.
Required volume flow rate:
164m^3/(26.8*60 s) = 0.102 m^3/s
That equals (Area)*(Velocity).
For a square duct, Area = a^2.
a is the side length of the square.
a = sqrt(0.102/V)
Plug in the numbers.
Solve for a.
To find the length of the side of a square duct in an air-conditioning system, we can use the formula Q = Av, where Q is the volume flow rate, A is the cross-sectional area of the duct, and v is the air speed.
Step 1: Calculate the volume flow rate Q
The volume of the room is given as 164 m3, and the air-conditioning system is designed to replace the air every 26.8 minutes. Therefore, we need to calculate the volume flow rate Q, which is the volume of air passing through the duct per unit time.
To find Q, we divide the volume of the room by the time required to replace the air:
Q = 164 m3 / 26.8 min
Step 2: Convert the time to seconds
Since air speed is commonly provided in meters per second, we need to convert the time to seconds, as follows:
26.8 min x 60 sec/min
Step 3: Calculate the volume flow rate Q (continued)
Now that we have the time in seconds, we can calculate Q:
Q = 164 m3 / (26.8 min x 60 sec/min)
Step 4: Use the formula Q = Av to find the cross-sectional area A
We are given the air speed v in meters per second. Rearranging the formula, we get:
A = Q / v
Step 5: Calculate the length of a side of the square duct
Since the duct has a square cross-section, and the area of a square is given by A = side^2, we can find the length of the side of the square duct:
side = √(A)
(a) Given air speed v = 3.00 m/s:
Calculate Q using the above steps, then substitute the values into the formula A = Q / v to find A. Finally, calculate the square root of A to find the length of a side.
(b) Given air speed v = 8.00 m/s:
Perform the same calculations using the new air speed value to find the length of a side.