A cylindrical air duct in an air conditioning system has a length of 3.5 m and a radius of 5.00 10-2 m. A fan forces air (ç = 1.8 1 x0-5 Pa·s) through the duct, such that the air in a room (volume = 242 m3) is replenished every 12 minutes. Determine the difference in pressure between the ends of the air duct.
To determine the difference in pressure between the ends of the air duct, we can use the equation for the flow rate of a fluid through a pipe:
Q = πr^4ΔP / 8ηL
Where:
Q = Flow rate (m^3/s)
r = Radius of the pipe (m)
ΔP = Pressure difference between the ends of the pipe (Pa)
η = Viscosity of the fluid (Pa·s)
L = Length of the pipe (m)
First, let's find the flow rate of the air through the duct. We know that the air in the room (volume = 242 m^3) is replenished every 12 minutes. Therefore, the flow rate can be calculated as:
Q = Volume / Time
Q = 242 m^3 / (12 * 60 s)
Q = 0.336 m^3/s
Next, we can rearrange the equation to solve for the pressure difference:
ΔP = 8ηLQ / πr^4
Substituting the given values:
η = 1.8 x 10^-5 Pa·s
L = 3.5 m
r = 5.00 x 10^-2 m
Q = 0.336 m^3/s
π = 3.14159
ΔP = 8 * (1.8 x 10^-5 Pa·s) * (3.5 m) * (0.336 m^3/s) / (3.14159 * (5.00 x 10^-2 m)^4)
Evaluating this expression will give us the difference in pressure between the ends of the air duct.
To determine the difference in pressure between the ends of the air duct, we can use the Poiseuille's Law for flow through a cylindrical pipe:
∆P = (8ηLQ) / (πr^4)
Where:
∆P = Pressure difference
η = Viscosity of air (given as 1.8 1 x 10^-5 Pa·s)
L = Length of the air duct (given as 3.5 m)
Q = Volumetric flow rate (needed to calculate)
r = Radius of the air duct (given as 5.00 x 10^-2 m)
To calculate the volumetric flow rate, we need to first find the time it takes to replenish the room with air (12 minutes) and then calculate the average air velocity:
V = V_room / t_replenish
Where:
V = Average air velocity
V_room = Volume of the room (given as 242 m^3)
t_replenish = Time to replenish the room (given as 12 minutes)
Once we have the average air velocity, we can calculate the volumetric flow rate using the formula:
Q = A * V
Where:
Q = Volumetric flow rate
A = Cross-sectional area of the air duct (given as πr^2)
V = Average air velocity
Now we have all the necessary values to calculate the pressure difference:
1. Calculate the average air velocity:
V = V_room / t_replenish
V = 242 m^3 / (12 minutes * 60 seconds/minute)
V = 0.337 m/s
2. Calculate the volumetric flow rate:
Q = A * V
Q = (πr^2) * V
Q = (π * (5.00 x 10^-2 m)^2) * 0.337 m/s
Q = 2.642 x 10^-3 m^3/s
3. Calculate the pressure difference:
∆P = (8ηLQ) / (πr^4)
∆P = (8 * (1.8 x 10^-5 Pa·s) * 3.5 m * 2.642 x 10^-3 m^3/s) / (π * (5.00 x 10^-2 m)^4)
After calculating the above expression, you will find the difference in pressure between the ends of the air duct in pascals (Pa).