The Capacity at N2 = 1200 rpm: 1.394 m^3/s
Static pressure at N2 = 1200 rpm: 1339.54 Pa
Total pressure at N2 = 1200 rpm: 6852.
The solution to the problem:
Given :Fan diameter (D) = 0.3 m, Initial fan speed (N1) = 1000 rpm ,New fan speed (N2) = 1200 rpm ,Air density (ρ) = 1.225 kg/m^3 ,Fan efficiency (η) = 0.8
Solution:
1. Calculate the tip speed ratio:
The tip speed ratio (λ) is a dimensionless parameter that relates the fan's tip speed to its diameter and rotational speed. It is defined as:
λ = N * D / (60 * π)
where:
N is the fan speed in rpm
D is the fan diameter in m
π is the mathematical constant pi (approximately 3.14159)
At the initial speed (N1 = 1000 rpm), the tip speed ratio is:
λ1 = 1000 * 0.3 / (60 * π) = 5.236
At the new speed (N2 = 1200 rpm), the tip speed ratio is:
λ2 = 1200 * 0.3 / (60 * π) = 6.283
2. Calculate the flow coefficient:
The flow coefficient (ϕ) is a dimensionless parameter that relates the fan's volume flow rate to its tip speed and diameter. It is defined as:
ϕ = Q / (π * D^2 * N / (60 * 2))
where: Q is the volume flow rate in m^3/s
From Problem 12-1, we know that the fan has an outlet area of 0.84 ft² (0.08 m²) and is delivering 2000 cfm (0.94 m³/s) of air.
We can use this information to calculate the flow coefficient at the initial speed:
ϕ1 = 0.94 / (π * 0.3^2 * 1000 / (60 * 2)) = 0.45
Assuming that the fan's flow coefficient remains constant as the speed increases, we can calculate the flow coefficient at the new speed:
ϕ2 = 0.45 * (λ2 / λ1)^0.5 = 0.51
3. Calculate the volume flow rate:
The volume flow rate (Q) is the rate at which air is flowing through the fan. It is measured in m^3/s.
At the new speed, the volume flow rate can be calculated using the flow coefficient and tip speed ratio:
Q2 = ϕ2 * π * D^2 * N2 / (60 * 2) = 1.394 m^3/s
4. Calculate the static pressure:
The static pressure (Ps) is the pressure difference between the inlet and outlet of the fan when there is no velocity at either location. It is measured in Pa.
At the new speed, the static pressure can be calculated using the flow coefficient, tip speed ratio, and air density:
Ps2 = 0.5 * ρ * (N2 * D / (60 * 2))^2 * ϕ2^2 = 1339.54 Pa
5. Calculate the total pressure:
The total pressure (Pt) is the sum of the static pressure and the velocity pressure. It is measured in Pa.
At the new speed, the total pressure can be calculated using the static pressure and the velocity pressure:
Pt2 = Ps2 + 0.5 * ρ * (N2 * D / (60 * 2))^2 = 6852.04 Pa
6. Calculate the shaft power:
The shaft power (Psh) is the power required to drive the fan. It is measured in W.
At the new speed, the shaft power can be calculated using the volume flow rate, total pressure, and fan efficiency:
Psh2 = Q2 * Pt2 / η = 11937.83 W