Diesel

Diesel cycle: thermal efficiency of compression-ignition engines. Theory, formulas, worked examples and interactive graphs.

Conceptual context

The Diesel cycle is the idealized thermodynamic model describing the operation of compression-ignition engines. Unlike the Otto cycle, where combustion is modeled at constant volume, in the Diesel cycle heat addition occurs at constant pressure. This difference has profound consequences: it allows much higher compression ratios and enables the use of denser fuels, but it introduces an efficiency penalty factor that depends on the duration of combustion.

This leaf sits within the thermodynamic cycles block because it extends the Otto cycle analysis framework by incorporating an additional parameter, the cutoff ratio, which quantifies the fraction of volume during which heat is injected. Understanding the Diesel cycle is essential for evaluating truck engines, locomotives, ships, and industrial generators.

🟢 Essential level

A Diesel engine works by compressing air until it reaches a temperature so high that the injected fuel ignites spontaneously, without the need for a spark plug. The idealized cycle consists of four processes: adiabatic compression, heat addition at constant pressure, adiabatic expansion, and heat rejection at constant volume. The thermal efficiency measures what fraction of the energy injected as heat is converted into useful mechanical work.

The key difference from the Otto cycle is that combustion occurs while the piston moves, at constant pressure, which increases the gas volume during heat injection. This introduces the cutoff ratio $r_c$, which measures how much the gas expands during combustion. The higher $r_c$, the more heat is injected but the more efficiency degrades. Efficiency also increases with the compression ratio $r$, but with diminishing marginal returns just as in the Otto cycle.

This trade-off between $r$ and $r_c$ is why Diesel engine design is never just "compress as much as possible": the fuel injection strategy that sets the cutoff ratio matters just as much as the mechanical compression ratio.

🔵 Formal level

Basic formulas

The ideal Diesel cycle consists of four reversible processes: an adiabatic compression from 1 to 2 that raises the pressure and temperature of the air, an isobaric expansion from 2 to 3 where $Q_H$ is absorbed at constant pressure, an adiabatic expansion from 3 to 4 that produces most of the work, and an isochoric process from 4 to 1 where $Q_C$ is rejected to the cold reservoir.

The central efficiency formula is:

Diesel cycle thermal efficiency

This expression shows that $\eta_D$ depends on three parameters: $r$, $r_c$, and $\gamma$. The bracketed factor in the formula is always greater than one for $r_c$ greater than one, meaning that Diesel efficiency is always lower than Otto for the same $r$. However, Diesel engines can operate with much higher $r$ (14 to 25 versus 8 to 12 in Otto), which more than compensates for this penalty.

Real student questions

Question one: if Diesel has worse efficiency than Otto at the same compression ratio, why is it used? Because Diesel engines can use much higher compression ratios (18 to 25 versus 8 to 12) without detonation, which more than compensates for the cutoff factor penalty.

Question two: how is $r_c$ approaching one interpreted? It means combustion would be instantaneous, without isobaric expansion, turning the Diesel into an Otto. It is the minimum fuel limit.

Question three: can a Diesel engine exceed Carnot efficiency? Never. Carnot between the extreme cycle temperatures will always be higher. The difference measures the irreversibilities of the isobaric combustion mode.

Final synthesis

The Diesel cycle establishes the maximum theoretical efficiency of compression-ignition engines as a function of three parameters: the compression ratio, the cutoff ratio, and the adiabatic index. Its efficiency is always lower than Otto for the same compression ratio, but the ability to operate with much higher compression ratios makes Diesel engines globally more efficient in practice. The cutoff ratio quantifies the combustion duration and constitutes the exclusive penalty factor of the Diesel cycle.