Dual Cycle

The dual (mixed) cycle is an air‑standard internal combustion cycle that models heat addition as a combination of a constant‑volume portion followed by a constant‑pressure portion. The idealized five processes (1→2→3→4→5→1) are: (1→2) reversible adiabatic (isentropic) compression, (2→3) constant‑volume heat addition, (3→4) constant‑pressure heat addition, (4→5) reversible adiabatic (isentropic) expansion, and (5→1) constant‑volume heat rejection. The dual cycle reduces to the Otto cycle if the constant‑pressure heat addition vanishes, and to the Diesel cycle if the constant‑volume heat addition vanishes. The analysis usually assumes an ideal gas (air) with constant specific heats (cv, cp) and ratio of specific heats γ = cp/cv.

Governing FormulaAssumptions: ideal gas (air), constant specific heats cv and cp, γ = cp/cv, R = cp - cv. Key relations (state temperatures): T2 = T1 * r^(γ-1) where r = V1/V2 (compression ratio) Let τ = T3/T2 (temperature rise across constant‑volume heat addition) T3 = τ * T2 T4 = rc * T3 where rc = V4/V3 (cutoff ratio for the constant‑pressure addition) T5 = T4 * (V4/V1)^(γ-1) = T4 * (rc/r)^(γ-1) (isentropic expansion). Equivalent algebraic simplification gives T5 = T1 * τ * rc^γ. Heat added per unit mass: q_in = cv (T3 - T2) + cp (T4 - T3). Heat rejected per unit mass: q_out = cv (T5 - T1). Thermal efficiency: η = 1 - q_out / q_in = 1 - cv (T5 - T1) / [ cv (T3 - T2) + cp (T4 - T3) ]. Net work per unit mass: w_net = q_in - q_out. Pressure relation for isentropic steps: p2/p1 = (T2/T1)^(γ/(γ-1)). Specific volume from ideal gas: v = RT/p. Mean effective pressure (MEP) (indicated) = w_net / (v1 - v2). (Use SI units; when numerical values are required, state cv, cp, γ explicitly.)

Knowledge Check

10 Questions

1.Which sequence of processes correctly describes the ideal air‑standard dual cycle (states 1→2→3→4→5→1)? State types use: 'isentropic compression', 'constant volume heat addition', 'constant pressure heat addition', 'isentropic expansion', 'constant volume heat rejection'.

2.Under what limiting conditions does the dual cycle reduce exactly to the Otto cycle and to the Diesel cycle, respectively?

3.Which expression is the correct general thermal efficiency η (air‑standard) for the dual cycle using temperatures at states 1–5?

4.Numerical: For an ideal dual cycle with air (γ = 1.40, cv = 718 J·kg^-1·K^-1, cp = 1005 J·kg^-1·K^-1), compression ratio r = 8, cutoff ratio rc = 2, intake temperature T1 = 300 K, and τ = T3/T2 = 3, what is the thermal efficiency η (percent)? (Use p1 = 100 kPa if needed.)

5.Using the same cycle data as the previous question (r = 8, rc = 2, τ = 3, T1 = 300 K, same cv and cp), what is the net work output w_net per unit mass (J·kg^-1)?

6.Numerical: For the same case with p1 = 100 kPa, what is the pressure at the end of constant‑volume heat addition p3 (in MPa)? Use γ = 1.40 and the same T values (T1 = 300 K, T2 ≈ 688.8 K, T3 ≈ 2066.4 K).

7.Qualitative: How does an increase in the cutoff ratio rc (with r and τ fixed) generally affect the thermal efficiency of a dual cycle?

8.Conceptual/numerical: If the dual cycle has τ = 1 (no constant‑volume heat addition), which of the following statements about the q_in and resulting cycle type is correct?

9.Using the same numerical case (r = 8, rc = 2, τ = 3, T1 = 300 K, p1 = 100 kPa), compute the indicated mean effective pressure (IMEP) in MPa. Use R = 287 J·kg^-1·K^-1 to get specific volumes.

10.In the dual cycle, at which state (or states) is the peak pressure attained, and why?