The water is supplied at the rate of 30 litres per second from a height of 4 m to a hydraulic ram, which raises 3 litres per second to a height of 18 m from the ram. Determine D’ Aubuisson’s and Rankine’s efficiencies of the hydraulic ram.

Solution

1. D'Aubuisson's Efficiency (\(\eta_D\))

This efficiency is the simple ratio of the energy output (lifting water) to the energy input (supply water).

$$ \eta_D = \frac{\text{Work Output}}{\text{Work Input}} = \frac{q \times h}{Q \times H} $$ $$ \eta_D = \frac{3 \, \text{L/s} \times 18 \, \text{m}}{30 \, \text{L/s} \times 4 \, \text{m}} = \frac{54}{120} $$ $$ \eta_D = 0.45 $$

2. Rankine's Efficiency (\(\eta_R\))

This efficiency considers the net work done. The input energy comes from the "waste" water (\(Q-q\)), and the output work is lifting the delivered water through the net height (\(h-H\)).

$$ \eta_R = \frac{q \times (h - H)}{(Q - q) \times H} $$ $$ \eta_R = \frac{3 \times (18 - 4)}{(30 - 3) \times 4} = \frac{3 \times 14}{27 \times 4} $$ $$ \eta_R = \frac{42}{108} \approx 0.3889 $$

Explanation of a Hydraulic Ram and its Efficiencies

A hydraulic ram is a clever pump that uses the energy of a large amount of water falling a small height to lift a small amount of that water to a much greater height. It works by using the "water hammer" effect. The flowing supply water is suddenly stopped by a valve, creating a high-pressure spike that forces a small portion of the water up the delivery pipe.

• D'Aubuisson's Efficiency: This is a straightforward energy balance. It compares the total potential energy of the water that was lifted to the total potential energy of the water that was supplied.
• Rankine's Efficiency: This is often considered a more practical measure. It recognizes that the water being lifted was already at the supply height, so the useful work is only in lifting it the *additional* height (\(h-H\)). It also correctly identifies that the energy to do this work comes from the water that was *not* lifted, i.e., the "waste" water (\(Q-q\)). This generally results in a lower, but more realistic, efficiency value.