What is the energy availability equation?
The energy availability equation, also known as exergy, quantifies the maximum amount of useful work that can be extracted from a system as it undergoes a reversible process to reach equilibrium with its environment, a state referred to as the 'dead state'. Unlike energy, which is conserved, exergy can be destroyed by irreversibilities such as friction or heat transfer across a finite temperature difference. The specific form of the availability equation depends on the type of system being analyzed—whether it is a closed system (non-flow) or an open system (steady-flow).
The energy availability equation for a closed system
For a closed system, which exchanges energy but not mass with its surroundings, the change in availability ($\Delta A$) between an initial state 1 and a final state 2 is defined by the following equation:
$\Delta A = (U_1 - U_2) - T_0(S_1 - S_2) + P_0(V_1 - V_2)$
- $U_1$ and $U_2$ are the internal energies of the system at the initial and final states.
- $T_0$ is the absolute temperature of the environment (the dead state).
- $S_1$ and $S_2$ are the entropies of the system at the initial and final states.
- $P_0$ is the absolute pressure of the environment.
- $V_1$ and $V_2$ are the volumes of the system at the initial and final states.
This equation reveals that the useful work potential is influenced by changes in internal energy, the heat exchanged with the environment due to entropy changes, and the boundary work done by the system against the environment's pressure. The term $-T_0(S_1 - S_2)$ represents the energy that becomes unavailable for work due to entropy changes, which is a direct consequence of the second law of thermodynamics.
The Gibbs free energy as a special case
The Gibbs free energy equation, $\Delta G = \Delta H - T\Delta S$, is a specific instance of the availability equation, relevant for processes occurring at a constant temperature and pressure. It represents the maximum non-expansion work that can be extracted from a system under these specific conditions. The Gibbs free energy is therefore a measure of chemical availability, determining the spontaneity of chemical reactions.
The energy availability equation for a steady-flow system
In an open system, or steady-flow process, both mass and energy can cross the control volume boundaries. The maximum useful work (or shaft work, $W_{sh, max}$) is described by the following equation, which accounts for changes in enthalpy, entropy, kinetic energy, and potential energy between the inlet and outlet:
$W_{sh, max} = (H_1 - H_0) - T_0(S_1 - S_0) + \frac{C_1^2 - C_0^2}{2} + g(Z_1 - Z_0)$
This is often simplified by the steady-flow availability function, $B = H - T_0S$, where the maximum useful work is the change in B ($B_1 - B_0$) between the initial and dead states. The additional terms for kinetic ($C^2/2$) and potential ($gZ$) energy are also considered when significant.
Factors influencing energy availability
Several factors affect a system's total energy availability, often leading to a reduction in potential useful work:
- Irreversibilities: All real-world processes are irreversible, meaning they involve friction, heat transfer over finite temperature differences, and mixing. These processes generate entropy, which in turn destroys exergy.
- Temperature difference: The maximum efficiency of a heat engine is determined by the Carnot efficiency, which depends on the temperature difference between the heat source and sink. A larger temperature gradient translates to higher thermal exergy.
- Chemical potential: For systems involving chemical reactions, the difference in chemical potential between substances and their reference state environment determines the chemical exergy. This is a crucial factor in fuel cell and combustion analysis.
- System properties: The internal energy, entropy, pressure, and volume of a system all affect its availability. Exergy is a property that depends on both the system's state and the defined reference environment.
- Kinetic and potential energy: While often negligible in thermodynamic analysis, significant changes in a system's velocity or elevation relative to the dead state can contribute to its total availability. For instance, the exergy of wind is its kinetic energy relative to the calm atmosphere.
Comparison of Energy (First Law) vs. Exergy (Second Law) Analysis
| Feature | First Law (Energy) Analysis | Second Law (Exergy) Analysis |
|---|---|---|
| Core Concept | Conservation of energy: Energy can be neither created nor destroyed. | Destruction of available work: Exergy is always destroyed in real processes. |
| Efficiency Measure | Thermal efficiency ($\eta$): Measures energy output relative to energy input. Can exceed 100% for heat pumps. | Exergy efficiency ($\epsilon$): Measures exergy output relative to exergy input. Always less than or equal to 100%. |
| Energy Quality | Does not differentiate between energy quality. All forms of energy are treated as equivalent. | Distinguishes between energy quality. High-grade energy (electricity) has higher exergy than low-grade energy (heat). |
| Reference State | Not required for calculations. | Depends on a reference state (the dead state) for the environment. |
| Identifies Losses | Identifies energy losses from the system, like heat transfer to surroundings. | Identifies both exergy losses (to surroundings) and exergy destruction (internal irreversibilities). |
| Primary Insight | Total energy balance for a system. | The potential for process improvement by minimizing irreversibilities. |
Conclusion
The energy availability equation, or exergy, is a powerful tool derived from the second law of thermodynamics that allows for a deeper understanding of energy systems beyond the simple first law energy balance. By quantifying the maximum useful work potential relative to a reference environment, exergy analysis highlights inefficiencies and the true 'quality' of energy. Whether for a closed system using the availability function $\phi = U + P_0V - T_0S$ or a steady-flow system using the function $B = H - T_0S$, the calculation of exergy provides a critical metric for optimizing energy conversion and usage. Understanding and minimizing the destruction of exergy is key to improving the efficiency and sustainability of any energy-dependent process.
For further reading, the thermodynamic concepts of exergy and availability are comprehensively covered in the Wikipedia article on Exergy.
