A cylindrical air duct in an air conditioning system has a length of 3.5 m and a radius of 5.00 X 10^-2 m. A fan forces air (ç = 1.8 X 10^-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.
Your value of ç = 1.8 X 10^-5 Pa·s is called the viscosity.
Get the volume flow rate from
Q = 242 m^3/720 s = 0.336 m^3/s
Get the duct velocity from
V = Q/(pi r^2)
I get 43 m/s, which seems much too high for a well designed A/C system. Pressure drops will be high.
Compute the dimensionless Reynolds number of the flow, which is
(density*diameter*V/ç)
Use pipe flow charts to compute the friction factor, f, which is a function of the Reynolds number.
pressure drop = f*(L/D)*(1/2)(density)*V^2
To determine the difference in pressure between the ends of the air duct, we can use the Hagen-Poiseuille equation, which relates the pressure difference to the flow rate, viscosity, length, and radius of the duct.
The flow rate (Q) is the volume of air passing through the duct per unit time. In this case, the volume of the room (V) is replenished every 12 minutes. So, the flow rate is given by:
Q = V / t
Where:
Q = Flow rate
V = Volume of the room
t = Time taken to replenish the volume
Substituting the given values:
Q = 242 m^3 / 12 min
Note: We need to convert the time into seconds, as the SI unit for viscosity (eta) is in Pa·s.
Substituting the converted values:
Q = (242 m^3 / 12 min) * (60 s / 1 min)
Q = 1210 m^3/s
Next, we can use the Hagen-Poiseuille equation to find the pressure difference. The equation is given by:
ΔP = (8 * η * L * Q) / (π * r^4)
Where:
ΔP = Pressure difference
η = Viscosity
L = Length of the duct
Q = Flow rate
r = Radius of the duct
Substituting the given values:
ΔP = (8 * (1.8 X 10^-5 Pa·s) * 3.5 m * 1210 m^3/s) / (π * (5.00 X 10^-2 m)^4)
Using these values and calculating the equation, we can find the pressure difference between the ends of the air duct.