Basic Mechanical Engineering (3110006)

BE | Semester-1   Summer-2020 | 11-06-2020

Q3) (b)

The efficiency of Otto cycle is a function of compression ratio: Prove it.

 
Figure shows the Otto cycle plotted on p – V diagram. The Otto cycle has following main processes:
 
Adiabatic compression process (1 – 2)
  • At state point 1 cylinder is full of gas with volume V1, pressure p1 and temperaure T1.
  • Piston moves from BDC to TDC and an ideal gas is compressed adiabatially to state point 2 through compression ratio, r = V1 V2  .
  • For adiabatic compression process (1 – 2), p1 V1γ = p2 V2γ
    and T2 T1  = V1 V2 γ - 1 = r γ - 1
    T1 = T2 r γ - 1 
 
Constant volume heat addition (2 – 3)
  • Heat is added at constant volume from an external heat source, both pressure and temperature are increases.
    p2T2 = p3T3
  • Heat supplied to the gas during constant volume process (2 – 3),
    Qs = Q2-3 = mCvT3 - T2
 
Adiabatic expansion process (3 – 4)
  • The increased high pressure exerts a greater amount of force on the piston and pushes it towards the BDC.
  • Expansion of working fluid takes place isentropically and work done by the gas.
  • The volume ratio r = V4 V3  is called expansion ratio.
  • For adiabatic expansion process (3 – 4), p3 V3γ = p4 V4γ
    and T3 T4  = V4 V3 γ - 1 = V1 V2 γ - 1 = r γ - 1
    T4 = T3 r γ - 1 
 
Constant volume heat addition (4 – 1)
  • Heat is rejected to the external sink at constant volume. This process is so controlled that ultimately the working fluid comes to its initial state 1 and the cycle is repeated
    p4T4 = p1T1
  • Heat rejected from the gas during constant volume process (4 – 1),
    Qr = Q4-1 = mCvT4 - T1
 
The expression of efficiency of Otto cycle:
 
η =  Net work done Heat Supplied =  Wnet Qs =  Qs - QrQs
 
η =  mCvT3 - T2 - mCvT4 - T1 mCvT3 - T2 
 
η = 1 -  T4 - T1 T3 - T2 
 
Substitute the value of T1 and T4
 
η = 1 -   T4  r γ-1  - T2r γ-1 T3 - T2 
 
η = 1 -  1  r γ-1  
 
Above equation proved that the efficiency of Otto cycle is a function of compression ratio.