Diesel Cycle

The ideal Diesel cycle is a four-step thermodynamic model for compression-ignition (diesel) engines. The idealized steps are: 1->2 isentropic (reversible adiabatic) compression of the working fluid (air); 2->3 constant-pressure heat addition (represents fuel injection and combustion at approximately constant pressure); 3->4 isentropic expansion (power stroke); 4->1 constant-volume heat rejection returning the working fluid to the initial state. Key non-dimensional parameters are the compression ratio r = V1/V2 and the cutoff ratio rc = V3/V2. For an ideal gas with constant specific heats, state-property relations for isentropic processes and the energy balance for constant-pressure and constant-volume processes are used to derive temperatures, pressures, heat transfers, and the cycle thermal efficiency. Assumptions commonly used: ideal gas working fluid, constant specific heats (Cp, Cv), reversible (isentropic) compression and expansion, and heat addition at constant pressure.

Governing FormulaKey relations (ideal gas, constant specific heats Cp, Cv, ratio γ = Cp/Cv): - Compression ratio: r = V1 / V2 - Cutoff ratio: rc = V3 / V2 - Isentropic temperature relation (1->2): T2 = T1 * r^(γ-1) - Isentropic pressure relation (1->2): p2/p1 = r^γ - Isentropic expansion (3->4): T4 = T3 * (rc / r)^(γ-1) - Heat added (constant pressure 2->3): Q_in = Cp * (T3 - T2) - Heat rejected (constant volume 4->1): Q_out = Cv * (T4 - T1) - Thermal efficiency (Diesel cycle): η = 1 - Q_out / Q_in which can be written as: η = 1 - (1 / r^(γ-1)) * ( (rc^γ - 1) / (γ * (rc - 1)) ) - Relations between Cp, Cv and gas constant R: Cp - Cv = R, γ = Cp / Cv.

Knowledge Check

10 Questions

1.In the ideal Diesel cycle, which process corresponds to constant-pressure heat addition?

2.Which expression correctly defines the cutoff ratio rc used in describing an ideal Diesel cycle?

3.Which of the following is the correct expression for the thermal efficiency η of an ideal Diesel cycle (constant specific heats)?

4.For an ideal Diesel cycle with γ = 1.4, initial temperature T1 = 300 K and compression ratio r = 18, what is the temperature T2 after isentropic compression? (Use T2 = T1 * r^(γ-1).)

5.Calculate the thermal efficiency of an ideal Diesel cycle with γ = 1.4, compression ratio r = 18 and cutoff ratio rc = 2. Use η = 1 - (1/r^(γ-1)) * ((rc^γ - 1)/(γ( rc - 1))).

6.For an ideal Diesel cycle with T3 = 2000 K, compression ratio r = 16, cutoff ratio rc = 2.5 and γ = 1.4, what is the temperature T4 after isentropic expansion? Use T4 = T3*(rc/r)^(γ-1).

7.Keeping cutoff ratio rc and γ fixed, what is the effect of increasing the compression ratio r on the thermal efficiency of the ideal Diesel cycle?

8.For the same compression ratio and the same heat added per cycle (same Q_in), which ideal cycle is more thermally efficient: Otto or Diesel? Assume idealized cycles with constant specific heats.

9.For an isentropic compression from state 1 to 2 with compression ratio r = 15 and γ = 1.4, what is the pressure ratio p2/p1?

10.Given constant specific heats Cp = 1.005 kJ·kg^-1·K^-1 and Cv = 0.718 kJ·kg^-1·K^-1 (air), and state temperatures T1 = 300 K, T2 = 1 000 K, T3 = 1 800 K and T4 = 800 K in an ideal Diesel cycle, compute the thermal efficiency η = 1 - Q_out/Q_in where Q_in = Cp(T3-T2) and Q_out = Cv(T4-T1).