An engine pumps water from a river 10m below its own level and discharges it through a nozzle of diameter 10cm with a speed of 50ms^-1.find the power required assuming:(a)no losses.(b)70% efficiency.water weighs 10^3kgm^-3.(g=10ms^-2)
To calculate the power required by the engine to pump water, we need to use the equation:
Power = Force × Velocity
We can start by finding the force required to pump the water.
(a) Assuming no losses:
1. Calculate the cross-sectional area of the nozzle:
The nozzle has a diameter of 10 cm, which is equal to 0.1 m.
The cross-sectional area is given by A = πr², where r is the radius.
So, A = π(0.1/2)² = 0.00785 m².
2. Calculate the force exerted by the water:
The force is given by F = pressure × area.
The pressure is the weight of the water column above the nozzle, which is equal to ρgh, where ρ is the density of water, g is the acceleration due to gravity, and h is the height difference.
ρ = 1000 kg/m³ (density of water)
g = 10 m/s² (acceleration due to gravity)
h = 10 m (height difference)
F = 1000 kg/m³ × 10 m/s² × 10 m × 0.00785 m² = 785 N.
3. Calculate the power required:
The power is given by Power = Force × Velocity.
Velocity = 50 m/s (given)
Power = 785 N × 50 m/s = 39,250 W (or 39.25 kW).
Therefore, the power required, assuming no losses, is 39,250 W (or 39.25 kW).
(b) Assuming 70% efficiency:
To calculate the power required with 70% efficiency, we need to adjust the power value obtained in step 3.
1. Calculate the mechanical power output:
Mechanical Power = Power / Efficiency = 39,250 W / 0.70 = 56,071.43 W.
Therefore, the power required, assuming 70% efficiency, is approximately 56,071.43 W.