Carnot Otto Diesel Joule Cycles

Overview

Engineering reference data for Carnot Otto Diesel Joule Cycles in steam condensate.

Key Formulas

Steam Quality

$x = \frac{m_v}{m_{total}}$

Mass fraction of vapor in two-phase mixture.

Enthalpy of Wet Steam

$h = h_f + x \cdot h_{fg}$

Specific enthalpy of wet steam.

Flash Steam

$m_{flash} = m_{liquid} \frac{h_f - h_{f2}}{h_{fg2}}$

Steam generated when condensate flashes to lower pressure.

Condensate Load

$m_c = \frac{Q}{h_{fg}}$

Condensate generated by heat transfer.

Variables

Cycle Descriptions

Carnot Cycle

The Carnot cycle is an ideal thermodynamic cycle representing the maximum possible efficiency for any heat engine operating between two temperature reservoirs. Heat addition and rejection occur isothermally, while the working fluid undergoes isentropic compression and expansion.

The efficiency is given by:

$$\eta_{\text{Carnot}} = 1 - \frac{T_{\min}}{T_{\max}}$$

where $T_{\max}$ is the absolute temperature of the hot reservoir (K) and $T_{\min}$ is the absolute temperature of the cold reservoir (K).

Otto Cycle

The Otto cycle models the ideal spark-ignition (e.g., gasoline) internal combustion engine. Heat addition (combustion) and heat rejection occur isochorically (at constant volume).

The efficiency depends on the compression ratio $\varepsilon$ and the isentropic exponent $\kappa$:

$$\eta_{\text{Otto}} = 1 - \varepsilon^{1 - \kappa}$$

where $\varepsilon = \frac{V_{\max}}{V_{\min}}$ is the compression ratio and $\kappa = \frac{c_p}{c_v}$ is the ratio of specific heats.

Diesel Cycle

The Diesel cycle models the ideal compression-ignition (diesel) internal combustion engine. Heat addition (combustion) occurs isobarically (at constant pressure), while heat rejection is isochoric.

The efficiency is calculated using the compression ratio $\varepsilon$, isentropic exponent $\kappa$, and the cutoff ratio $\Phi$:

$$\eta_{\text{Diesel}} = 1 - \frac{1}{\varepsilon^{\kappa - 1}} \left( \frac{\Phi^{\kappa} - 1}{\kappa (\Phi - 1)} \right)$$

where $\Phi = \frac{V_3}{V_2}$ is the cutoff ratio (ratio of volume after and before combustion).

Joule Cycle (Brayton Cycle)

The Joule cycle is the ideal thermodynamic cycle for gas turbines. It consists of isentropic compression and expansion, with isobaric heat addition and rejection in the combustor and heat exchanger.

The thermal efficiency depends on the pressure ratio:

$$\eta_{\text{Joule}} = 1 - \frac{1}{r_p^{(\kappa-1)/\kappa}}$$

where $r_p = \frac{P_2}{P_1}$ is the pressure ratio across the compressor. Alternatively, it can be expressed in terms of the temperatures at the compressor inlet ($T_1$) and outlet ($T_2$):

$$\eta_{\text{Joule}} = 1 - \frac{T_1}{T_2}$$

Cycle Process Comparison

The four cycles differ fundamentally in how heat addition and rejection occur:

Cycle Type Summary

The Carnot cycle establishes the theoretical maximum efficiency between two temperature limits, with both heat addition and rejection occurring isothermally. In practice, this is not achievable in real engines.

The Otto cycle models spark-ignition engines where combustion occurs rapidly at nearly constant volume, making it applicable to gasoline engines.

The Diesel cycle models compression-ignition engines where fuel injection and combustion occur at constant pressure, applicable to diesel engines.

The Joule cycle (also known as the Brayton cycle) serves as the comparison cycle for gas turbines with continuous flow, where compression and expansion occur adiabatically.

Key Variables Reference

Efficiency Formulas and Definitions

Carnot Cycle Efficiency

The Carnot cycle represents the maximum theoretical efficiency between two thermal reservoirs.

$\eta_{\text{Carnot}} = 1 - \frac{T_{\text{min}}}{T_{\text{max}}}$

• Process: Heat addition and rejection occur isothermally (constant temperature).
• Significance: It is the most efficient cycle possible for a given temperature range, setting an upper limit for all heat engines.

Otto Cycle Efficiency

The Otto cycle models spark-ignition (gasoline) engines where heat addition is instantaneous (constant volume).

$\eta_{\text{Otto}} = 1 - \varepsilon^{1 - \kappa}$

• Process: Combustion (heat addition) is isochoric (constant volume).
• Variable $\varepsilon$: Compression ratio ($V_{\text{max}} / V_{\text{min}}$). Higher ratios generally increase efficiency.

Diesel Cycle Efficiency

The Diesel cycle models compression-ignition (diesel) engines where heat addition occurs at constant pressure.

$\eta_{\text{Diesel}} = 1 - \frac{\Phi^\kappa - 1}{\varepsilon^{\kappa - 1} \kappa (\Phi - 1)}$

• Process: Combustion (heat addition) is isobaric (constant pressure).
• Variable $\Phi$: Injection ratio or cutoff ratio ($V_3 / V_2$). It represents the volume expansion during fuel injection.

Joule (Brayton) Cycle Efficiency

The Joule cycle is the ideal cycle for continuous-flow gas turbines.

$\eta_{\text{Joule}} = 1 - \frac{T_1}{T_2}$

• Process: Heat addition and rejection occur at constant pressure.
• Note: The formula given in the source ($T_1/T_2$) uses the temperature ratio across the compressor, which is related to the pressure ratio. The standard form is $\eta = 1 - r_p^{(1-\kappa)/\kappa}$ where $r_p$ is the pressure ratio.

Key Definitions from Cycle Processes

• Isothermal: Constant temperature ($T$).
• Isochoric: Constant volume ($V$).
• Isobaric: Constant pressure ($P$).
• Isentropic: Constant entropy ($s$), which is an ideal, reversible adiabatic process. The isentropic exponent $\kappa$ (ratio of specific heats, $c_p/c_v$) is critical in these efficiency formulas.

Efficiency Formulas (LaTeX Notation)

For precise mathematical representation, the efficiency formulas for each cycle are given below:

$$\mu_{\text{Carnot}} = 1 - \frac{T_{\text{min}}}{T_{\text{max}}}$$ $$\mu_{\text{Otto}} = 1 - \varepsilon^{1 - \kappa}$$ $$\mu_{\text{Diesel}} = 1 - \frac{\Phi^\kappa - 1}{\varepsilon^{\kappa - 1} \kappa (\Phi - 1)}$$ $$\eta_{\text{Joule}} = 1 - \frac{T_1}{T_2}$$

Where:

• $\varepsilon$ is the compression ratio,
• $\kappa$ is the isentropic exponent,
• $\Phi$ is the injection ratio (Diesel cycle),
• $T_{\text{min}}$, $T_{\text{max}}$, $T_1$, and $T_2$ are temperatures in Kelvin (K).

These formulas are essential for evaluating cycle performance and are derived from thermodynamic principles for each cycle type.

References

• Original Source