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The dual cycle is sometimes used to approximate actual cycles as the time taken for
heat transfer in the engine is not zero for the Otto cycle (so not constant volume). In the
Diesel cycle, due to the nature of the combustion process, the heat input does not occur
at constant pressure.
The dual cycle consists of following operations:
1-2 Adiabatic compression
2-3 Addition of heat at constant volume.
3-4 Addition of heat at constant pressure.
4-5 Adiabatic expansion.
5-1 Rejection of heat at constant volume.

4.5 Analysis of Power Cycles
Let us derive the expression of the thermal efficiency of the dual cycle. The actual
Otto, Diesel and Trinkler cycles can be approximated more closely by the ideal dual
cycle by ε, λ and ρ are defined as
V
  1 - the compression ratio (4.4)
V
2
P
  3 - the pressure ratio (4.5)
P
2
V
  4 - the cutoff ratio, as the ratio of the cylinder volumes after and before the (4.6)
V
3
combustion process.

For the adiabatic processes 1-2
k-1
T 2 = T 1ε .
For the constant volume processes 2-3
k-1
T 3 = λT 2 = T 1ε λ.
For the constant pressure processes 3-4
k-1
T 4 = ρT 3 =T 1ε λρ.
For the adiabatic processes expansion 4-5

k 1  k 1 
T V  V V  1
5 4 4 2 k 1 
       ,
T  V   V V   k 1 
4  5   3 1 
from that
1
k
T  T  k 1  T 
5 4 k 1 1


An amount of the Addition of heat and Rejection of heat is in the dual cycle

I
II
q 1 = q + q = C vm ( T 3 - T 2 ) + C pm ( T 4 – T 3) = C vm[T 1ε k-1 (λ-1) + kλ(ρ-1)];
k
q 2 = C v ( T 4 – T 1 ) = C vmT 1(λρ -1).

Then the thermal efficiency of the dual cycle

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