Problem Statement
Find the efficiency of a hydraulic crane, which is supplied 400 litres of water under a pressure of 490.5 N/cm² for lifting a weight of 98.1 kN through a height of 10 m.
Given Data & Constants
- Volume of water supplied, \(V = 400 \, \text{litres} = 0.4 \, \text{m}^3\)
- Supply pressure, \(P = 490.5 \, \text{N/cm}^2\)
- Weight lifted, \(W = 98.1 \, \text{kN} = 98100 \, \text{N}\)
- Height of lift, \(h = 10 \, \text{m}\)
Solution
1. Calculate the Useful Work Output
The work output is the energy used to lift the weight.
2. Calculate the Energy Input
The energy input is the work potential of the pressurized water supplied to the crane. First, convert the pressure to standard units (N/m²).
3. Calculate the Efficiency (\(\eta\))
Efficiency is the ratio of the useful work output to the total energy input.
The efficiency of the hydraulic crane is \(50\%\).
Explanation of Efficiency
The efficiency of any machine is a measure of how well it converts input energy into useful output work. In this hydraulic crane:
- Energy Input: The energy supplied to the system is in the form of pressurized water. This is the total energy available for the crane to use.
- Work Output: The useful work performed is the lifting of the heavy load. This is the primary purpose of the crane.
An efficiency of 50% means that half of the energy supplied by the pressurized water was successfully converted into lifting the weight. The other half was lost, likely due to factors like friction in the hydraulic cylinders and mechanical components, fluid leakage, and heat generation.
