Calculate design power and efficiency of the system

Question

Calculate the design power and efficiency of the system if output is 3kilowatts, of the following parameters: Diameter of solid cylinder = 100mm Length of the cylinder = 750mm Density = 8900kgm^-3 Time to be taken = 1, 8 seconds Initial rpm = 600 Final rpm = 2200

Answer 1

3kW=3000 watts;

1 watt = 1 Joule/sec = Newton*metres/sec = mass*acceleration*metres/sec = volume*density*acceleration*metres/sec.

Volume of cylinder=0.750*(0.05)^2π=0.00589 cu m. Mass=8900*0.00589=267π/16=52.4253kg approx.

Change in revolution=2200-600=1600rpm=1600/60rps. Rate of change=80/(30*1.8)=400/27 revs/sec/sec.

1 rev = 2πr=0.1π=0.3142m, so rate of change (acceleration)=(400/27)*0.1π=40π/27=4.6542m/s/s.

Mass*acceleration=(267π/16)(40π/27) Newtons. Final rps=2200/60rps=110/3rps=110*0.1π m/s=11π m/s.

Energy in watts=(267π/16)(40π/27)11π=8432 watts approx = 8.432kW.

Output is 3kW, so efficiency is 3/8.432=0.36 approx or 36%.