The operation of a diesel engine can be idealized by the cycle shown in Fig. 20–26. Air is drawn into the cylinder during the intake stroke (not part of the idealized cycle). The air is compressed adiabatically, path ab. At point b diesel fuel is injected into the cylinder and immediately burns since the temperature is very high. Combustion is slow, and during the first part of the power stroke, the gas expands at (nearly) constant pressure, path bc. After burning, the rest of the power stroke is adiabatic, path cd. Path da corresponds to the exhaust stroke.
(a) Show that, for a quasistatic reversible engine undergoing this cycle using an ideal gas, the ideal efficiency is


e = 1 - (Vₐ/V𝒸)⁻^γ - (Vₐ/Vᵦ)⁻^γ / γ [(Vₐ/V𝒸)⁻¹ - (Vₐ/Vᵦ)⁻¹] ,


where Vₐ/Vᵦ is the “compression ratio,” Vₐ/V𝒸 is the “expansion ratio,” and γ is defined by Eq. 19–15. (γ = C_P/ C_V)
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Verified step by step guidance

1. Identify the key points in the cycle and the processes between them: Point a (beginning of compression), Point b (end of compression/start of combustion), Point c (end of combustion), and Point d (end of expansion).
2. Understand that the process ab (compression) and cd (expansion after combustion) are adiabatic processes. Recall that for an adiabatic process involving an ideal gas, the relationship PV^γ = constant applies, where P is the pressure, V is the volume, and γ is the heat capacity ratio (C_P/C_V).
3. Recognize that the process bc is an isobaric process (constant pressure). During this process, the volume changes from V_b to V_c, and this can be related to the temperatures at points b and c using the ideal gas law PV = nRT.
4. Apply the efficiency formula for a heat engine, which is e = 1 - Q_c/Q_h, where Q_c is the heat expelled to the cold reservoir and Q_h is the heat absorbed from the hot reservoir. For the cycle described, calculate Q_h during the isobaric process bc and Q_c during the adiabatic process cd.
5. Substitute the expressions for Q_h and Q_c in terms of the temperatures and volumes at points b, c, and d into the efficiency formula. Simplify the expression using the relationships between the volumes (compression ratio V_a/V_b and expansion ratio V_a/V_c) and the adiabatic process equations to derive the final formula for efficiency.

Key Concepts

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In the context of a diesel engine, the compression of air during the intake stroke and the expansion of gases during the power stroke occur adiabatically, meaning that the temperature and pressure change without heat transfer. This principle is crucial for understanding how the engine operates efficiently, as it allows for significant temperature increases during compression.

Compression and Expansion Ratios

The compression ratio (Vₐ/Vᵦ) and expansion ratio (Vₐ/V𝒸) are critical parameters in engine thermodynamics. The compression ratio is the ratio of the cylinder volume at the bottom of the stroke to the volume at the top, while the expansion ratio is the ratio of the volume after combustion to the volume at the end of the power stroke. These ratios influence the efficiency and power output of the engine, as they determine how much the air-fuel mixture is compressed and how much work can be extracted during expansion.

Ideal Gas Law and Specific Heat Ratios

The ideal gas law relates the pressure, volume, and temperature of an ideal gas, and is fundamental in analyzing thermodynamic cycles. The specific heat ratio, γ (C_P/C_V), is the ratio of the specific heat at constant pressure to that at constant volume, and it plays a significant role in determining the efficiency of thermodynamic processes. In the context of the diesel engine cycle, γ affects the calculations of work done during adiabatic processes and is essential for deriving the ideal efficiency formula.