A room measures 4.0 m by 3.9 m by 4.5 m. The heating and air-conditioning ducts to and from the room are circular with diameter 0.37 m and all the air in the room is to be exchanged every 14 min.
(a) What is the average flow rate?
m3/s
(b) What is the necessary flow speed in the duct? (Assume that the density of the air is constant.)
(a) (Room volume)/(840 s) = ______ m^3/s
(b) (Flow rate)/(duct area)
To find the average flow rate and the necessary flow speed in the duct, we can use the formula for flow rate:
Flow Rate = Volume / Time
Let's break down the problem step by step:
Step 1: Find the volume of the room.
The volume of the room can be found by multiplying the length, width, and height together.
Volume = length x width x height
Volume = 4.0 m x 3.9 m x 4.5 m
Step 2: Convert the time from minutes to seconds.
Since flow rate is typically measured in cubic meters per second (m^3/s), we need to convert the time from minutes to seconds.
1 min = 60 s
Therefore, the time in seconds is 14 min x 60 s/min.
Step 3: Calculate the average flow rate.
To find the average flow rate, divide the volume of the room by the time in seconds.
Average Flow Rate = Volume / Time
Step 4: Find the cross-sectional area of the duct.
The cross-sectional area of the duct can be found using the formula for the area of a circle.
Area = π * (radius)^2
Since we are given the diameter of the duct (0.37 m), we need to divide it by 2 to get the radius.
Step 5: Calculate the necessary flow speed in the duct.
To find the flow speed, divide the average flow rate by the cross-sectional area of the duct.
Flow Speed = Average Flow Rate / Cross-sectional Area
Let's plug in the values and calculate the answers.
(a) Average flow rate:
Volume = 4.0 m x 3.9 m x 4.5 m
Time = 14 min x 60 s/min
Average Flow Rate = Volume / Time
(b) Necessary flow speed in the duct:
Cross-sectional Area = π * (radius)^2
Flow Speed = Average Flow Rate / Cross-sectional Area