Sewage at a certain pumping station is raised vertically by 5.27 m at the rate of 1 930 000 liters each day. The sewage, of density 1 050 kg/m3, enters and leaves the pump at atmospheric pressure and through pipes of equal diameter.
(a) Find the output mechanical power of the lift station.
(b) Assume an electric motor continuously operating with average power 4.35 kW runs the pump. Find its efficiency.
To answer these questions, we need to use the following formulas:
1. Output mechanical power (P) = Work done (W) / time (t)
2. Work done (W) = Force (F) x distance (d)
3. Force (F) = Mass (m) x acceleration (a)
4. Power (P) = Work done (W) / time (t)
5. Efficiency (η) = (Output power / Input power) x 100
Let's solve these step by step:
(a) Find the output mechanical power of the lift station:
First, we need to find the force exerted by the sewage. We can calculate the force using the formula:
F = m x a
Given:
Density of sewage (ρ) = 1050 kg/m³
Volume of sewage each day (V) = 1930000 liters = 1930000 dm³ (1 liter = 1 dm³)
Acceleration due to gravity (g) = 9.8 m/s²
To find the mass (m) of the sewage, we can use:
m = ρ x V
Convert the liters to cubic meters:
V = 1930000 dm³ = 1930 m³
Substitute the values into the equation:
m = 1050 kg/m³ x 1930 m³ = 2026500 kg
Now, we can calculate the force:
F = m x a = 2026500 kg x 9.8 m/s² = 19874700 N
The work done (W) is given by:
W = F x d
The distance (d) is given as the height the sewage is raised, which is 5.27 m. So:
W = 19874700 N x 5.27 m = 104676369 Nm or Joules (J)
Finally, we can calculate the output mechanical power (P):
P = W / t
Since the volume is given for a day, we need to convert it to seconds:
t = 24 hours x 60 minutes x 60 seconds = 86400 seconds
Substituting the values:
P = 104676369 J / 86400 s = 1210.76 W
The output mechanical power of the lift station is approximately 1210.76 Watts.
(b) Find the efficiency:
Given:
Average electric power (input power) = 4.35 kW = 4350 W
Efficiency (η) = (Output power / Input power) x 100
Substitute the values into the equation:
η = (1210.76 W / 4350 W) x 100 = 27.89%
The efficiency of the electric motor operating the pump is approximately 27.89%.