Diesel cycle

The Diesel cycle is assumed to have constant pressure during the initial part of the combustion phase (${\displaystyle V_2}$ to ${\displaystyle V_3}$ in the diagram, below). This is an idealized mathematical model: real physical diesels do have an increase in pressure during this period, but it is less pronounced than in the Otto cycle. In contrast, the idealized Otto cycle of a gasoline engine approximates a constant volume process during that phase.

Idealized Diesel cycle

The image shows a p–V diagram for the ideal Diesel cycle; where ${\displaystyle p}$ is pressure and V the volume or ${\displaystyle v}$ the specific volume if the process is placed on a unit mass basis. The idealized Diesel cycle assumes an ideal gas and ignores combustion chemistry, exhaust- and recharge procedures and simply follows four distinct processes:

• 1→2 : isentropic compression of the fluid (blue)
• 2→3 : constant pressure heating (red)
• 3→4 : isentropic expansion (yellow)
• 4→1 : constant volume cooling (green)

The Diesel engine is a heat engine: it converts heat into work. During the bottom isentropic processes (blue), energy is transferred into the system in the form of work ${\displaystyle W_{in}}$, but by definition (isentropic) no energy is transferred into or out of the system in the form of heat. During the constant pressure (red, isobaric) process, energy enters the system as heat ${\displaystyle Q_{in}}$. During the top isentropic processes (yellow), energy is transferred out of the system in the form of ${\displaystyle W_{out}}$, but by definition (isentropic) no energy is transferred into or out of the system in the form of heat. During the constant volume (green, isochoric) process, some of the energy flows out of the system as heat through the right depressurizing process ${\displaystyle Q_{out}}$. The work that leaves the system is equal to the work that enters the system plus the difference between the heat added to the system and the heat that leaves the system; in other words, net gain of work is equal to the difference between the heat added to the system and the heat that leaves the system.

• Work in (${\displaystyle W_{in}}$) is done by the piston compressing the air (system)
• Heat in (${\displaystyle Q_{in}}$) is done by the combustion of the fuel
• Work out (${\displaystyle W_{out}}$) is done by the working fluid expanding and pushing a piston (this produces usable work)
• Heat out (${\displaystyle Q_{out}}$) is done by venting the air
• Net work produced = ${\displaystyle Q_{in}}$ - ${\displaystyle Q_{out}}$

The net work produced is also represented by the area enclosed by the cycle on the p–V diagram. The net work is produced per cycle and is also called the useful work, as it can be turned to other useful types of energy and propel a vehicle (kinetic energy) or produce electrical energy. The summation of many such cycles per unit of time is called the developed power. The ${\displaystyle W_{out}}$ is also called the gross work, some of which is used in the next cycle of the engine to compress the next charge of air.

Maximum thermal efficiency

The maximum thermal efficiency of a Diesel cycle is dependent on the compression ratio and the cut-off ratio. It has the following formula under cold air standard analysis:

${\displaystyle {\eta }_{th}=1-\frac{1}{r^{\gamma -1}}\left(\frac{{\alpha }^{\gamma }-1}{\gamma (\alpha -1)}\right)}$

where

${\displaystyle {\eta }_{th}}$ is thermal efficiency

${\displaystyle \alpha }$ is the cut-off ratio ${\displaystyle \frac{V_3}{V_2}}$ (ratio between the end and start volume for the combustion phase)

r is the compression ratio ${\displaystyle \frac{V_1}{V_2}}$

${\displaystyle \gamma }$ is ratio of specific heats (C_p/C_v)

The cut-off ratio can be expressed in terms of temperature as shown below:

${\displaystyle \frac{T_2}{T_1}={\left(\frac{V_1}{V_2}\right)}^{\gamma -1}=r^{\gamma -1}}$

${\displaystyle {\displaystyle T_2=T_1{r}^{\gamma -1}}}$

${\displaystyle \frac{V_3}{V_2}=\frac{T_3}{T_2}}$

${\displaystyle \alpha =\left(\frac{T_3}{T_1}\right)\left(\frac{1}{r^{\gamma -1}}\right)}$

${\displaystyle T_3}$ can be approximated to the flame temperature of the fuel used. The flame temperature can be approximated to the adiabatic flame temperature of the fuel with corresponding air-to-fuel ratio and compression pressure, ${\displaystyle p_3}$. ${\displaystyle T_1}$ can be approximated to the inlet air temperature.

This formula only gives the ideal thermal efficiency. The actual thermal efficiency will be significantly lower due to heat and friction losses. The formula is more complex than the Otto cycle (petrol/gasoline engine) relation that has the following formula:

${\displaystyle {\eta }_{otto,th}=1-\frac{1}{r^{\gamma -1}}}$

The additional complexity for the Diesel formula comes around since the heat addition is at constant pressure and the heat rejection is at constant volume. The Otto cycle by comparison has both the heat addition and rejection at constant volume.

Comparing efficiency to Otto cycle

Comparing the two formulae it can be seen that for a given compression ratio (r), the ideal Otto cycle will be more efficient. However, a real diesel engine will be more efficient overall since it will have the ability to operate at higher compression ratios. If a petrol engine were to have the same compression ratio, then knocking (self-ignition) would occur and this would severely reduce the efficiency, whereas in a diesel engine, the self ignition is the desired behavior. Additionally, both of these cycles are only idealizations, and the actual behavior does not divide as clearly or sharply. Furthermore, the ideal Otto cycle formula stated above does not include throttling losses, which do not apply to diesel engines.