Exergy Efficiency of Closed and Unsteady-Flow Systems

Limitations of Traditional Efficiency Definitions

The second-law or exergy analysis of energy systems is frequently regarded as a challenging experience by both engineering students and practicing engineers. This difficulty stems primarily from the abstract nature of the topic; unlike energy, which is conserved, entropy is generated and exergy is destroyed. Consequently, there is a recurring tendency in the field to revert to first-law energy balances and energy conversion efficiencies, thereby bypassing the more comprehensive insights offered by exergy balances and exergy efficiency.

Inconsistencies in Current Literature

Exergy efficiency, often referred to as second-law efficiency or effectiveness, lacks a singular, universally applied formulation. While it is fundamentally a measure of thermodynamic perfection—reaching 100 percent in a reversible process where exergy destruction is zero—the literature reveals significant inconsistencies. A review of various studies shows researchers utilizing disparate relations for identical components. For instance, in the analysis of gasoline engines or compression-ignition systems, some authors have been observed using energy terms in the denominator of their efficiency equations instead of exergy terms. Others utilize definitions that are not consistent with one another, such as pairing a ratio of output to input with a ratio of exergy destruction at products to destruction at sources, leading to confusion regarding the true performance of the system.

Failure in Closed and Unsteady Systems

Most traditional exergy efficiency definitions, such as the simple ratio of exergy output to exergy input (η_ex = X_output/X_input), were formulated with steady-flow systems in mind. These ratios assume systems in steady operation, like power plants, where the exergy content remains constant. When these standard definitions are applied to closed systems—such as rigid tanks and piston-cylinder devices—or to unsteady-flow processes, they fail to provide an accurate picture unless the terms are interpreted broadly to account for the change in the system's internal exergy.

Furthermore, traditional ratios comparing actual work to reversible work (W_act/W_rev) possess inherent limitations. In specific piston-cylinder processes, simple ratios can result in zero or even negative values, failing to serve as a meaningful indicator of performance. To address these failures, exergy changes within the system must be meticulously accounted for as either exergy input (expended) or output (recovered). Without this nuanced approach, the degree of approaching reversible operation remains obscured in non-steady applications.

The Exergy Expended-Recovered Framework

The core of the proposed methodology lies in the "expended-recovered" framework, which offers a more inclusive and intuitive basis for evaluating efficiency than traditional "input-output" or "fuel-product" models. The general exergy efficiency relation is defined as:

η_ex = X_recovered / X_expended = 1 − X_destroyed / X_expended

This definition serves as a measure of thermodynamic perfection, where an efficiency of 100 percent represents a reversible process characterized by zero entropy generation and, consequently, zero exergy destruction (X_destroyed = T₀S_gen). By focusing on what is actually consumed versus what is successfully retrieved, this framework encompasses heat, work, and mass transfer interactions simultaneously, making it robust enough for complex thermodynamic cycles.

Accounting for System Exergy Changes

A significant limitation of standard steady-flow efficiency definitions is their failure to account for the internal exergy content of a system. The expended-recovered framework addresses this by treating the system exergy change (ΔX_sys) as an active component of the balance. The treatment of this term depends on whether the system is accumulating or depleting its work potential:

• Positive Exergy Change (ΔX_sys > 0): If the system stores exergy during a process (such as a tank being pressurized or heated), the increase is added to the exergy recovered. In this context, the system acts as a storage medium for the invested exergy resources.
• Negative Exergy Change (ΔX_sys < 0): If the system supplies exergy to drive a process (such as a piston expanding or a battery discharging), the decrease in its internal exergy is treated as part of the exergy expended. Here, the system's own state serves as a primary resource or "fuel."

Mechanisms of Transfer and Limitations

The framework integrates all three mechanisms of exergy transfer: heat, work, and mass. For work, the framework distinguishes between shaft or electrical work (which is the work itself) and moving boundary work, where the atmospheric work (P₀[v₂ − v₁]) must be subtracted to determine the useful exergy transfer. For heat transfer (Q), the exergy is quantified by the Carnot efficiency factor (1 − T₀/T_b). A critical assumption in this framework is the role of the environment temperature (T₀); exergy transfer associated with heat loss from an "extended system" is zero, as the boundary temperature at that limit equals T₀.

This comprehensive approach allows for the analysis of systems where there is no apparent external exergy input, such as a compressed-air storage tank discharging through a turbine. In such a case, the decrease in the system's exergy is the expended resource, and the turbine’s work output is the recovered product. By interpreting these terms broadly, the framework eliminates the confusion often found in academic literature regarding the distinction between exergy "loss" and "destruction."

Extended System Boundaries and Exergy Loss

The classification of energy that leaves a system as either exergy loss (X_loss) or internal exergy destruction (X_destroyed) is heavily dependent on the selection of system boundaries. In a standard thermodynamic analysis where the physical device itself constitutes the system, exergy loss represents exergy transferred to the environment—such as thermal energy rejected through heat transfer or the chemical and kinetic exergy of exhaust gases—that is generally unrecoverable. While this energy is eventually destroyed in the surroundings, it is categorized as a loss because the destruction occurs outside the defined physical surface of the device.

To provide a more comprehensive and realistic assessment of thermodynamic perfection, researchers often utilize the concept of an "extended system." This approach expands the boundary to include the immediate surroundings where temperature, pressure, and concentration gradients exist. By extending the boundary to a point where the system reaches the environmental temperature (T₀) and pressure (P₀), the exergy content of waste streams and heat transfers at the boundary becomes zero. Consequently, the term X_loss is eliminated from the exergy efficiency equation.

In these extended systems, all potential waste is accounted for as internal exergy destruction, defined by the relationship:

X_destroyed = T₀S_gen

Under this framework, exergy that would have been labeled as "loss" in a physical system analysis is instead recognized as destruction within the immediate surroundings of the extended system. For example, if a rigid tank experiences heat loss to the environment, the exergy associated with that heat is destroyed in the temperature gradient zone between the tank surface and the ambient air. By including this zone within the system boundary, the analysis faithfully captures the total irreversibility of the process.

This methodology offers significant analytical advantages. It simplifies the evaluation of energy systems by eliminating the need to calculate complex gradients at the physical surface, such as the exact temperature of a tank wall or the specific chemical potential of a purging substance at the moment of release. Instead, engineers can treat the eventual destruction of exergy loss as an actuality rather than a potentiality. This results in a more intuitive measure of efficiency, where the recovered exergy is compared directly against the total exergy expended, providing a clear indicator of how closely the process approximates reversible operation.

Chemical Exergy and Mixing Processes

The total exergy of a system is the sum of various work potentials, including kinetic, potential, and thermal components. However, even when a system reaches thermal and mechanical equilibrium with its environment—a state known as the restricted dead state—it may still possess additional work potential. This potential, termed chemical exergy (x_ch), represents the maximum useful work obtainable as the system undergoes a reversible process from the restricted dead state to a state of total chemical equilibrium with the environment.

Foundations in Mixtures

Chemical exergy is fundamentally rooted in the Gibbs function and chemical potentials. For an ideal gas mixture or ideal solution, where the influence of dissimilar molecules on one another is negligible, the chemical exergy of a component i at the environmental temperature (T₀) is defined by the difference between its chemical potential in the system and its chemical potential in the dead state. Mathematically, for an ideal gas mixture, the chemical exergy per unit mole of the mixture (x_ch,mixture) is derived from the mole fractions (y_i) of the constituents:

x_ch,mixture = ∑ y_i x⁰_ch,i + R_u T₀ ∑ y_i ln y_i

In this expression, x⁰_ch,i represents the standard chemical exergy of the pure substance i, and R_u is the universal gas constant. This formulation is particularly advantageous because it allows for the analysis of mixtures containing substances not present in the reference environment, such as fuels, by utilizing tabulated standard chemical exergy values.

Mechanisms of Mixing and Separation

The physics of mixing processes illustrates the practical significance of these mathematical foundations. Mixing is a spontaneous process that generates entropy and thus destroys exergy. Because exergy destruction is equivalent to lost work (X_destroyed = T₀S_gen), a reversible mixing process must theoretically produce work. Conversely, the separation of a mixture into its pure components requires an input of work. Under reversible conditions, the work potential produced during mixing is exactly equivalent to the minimum work required for separation:

• Reversible Mixing: Produces work as components move toward environmental concentration levels.
• Reversible Separation: Requires work input equal to the exergy change of the mixture components.

Reacting Systems and Gibbs Function

When analyzing reacting systems, such as combustion processes, the chemical exergy calculation must account for changes in molecular structure. This requires the addition of the Gibbs function of formation (g⁰_f) to the chemical exergy terms. The exergy associated with the formation of a substance is the reversible work required or produced when that substance is formed from its stable elements at the standard reference state (T₀, P₀).

For a reacting system, the total chemical exergy is expressed as:

X_ch = ∑ N_i g⁰_f,i + ∑ N_i x⁰_ch,i + R_u T₀ ∑ N_i ln y_i

By comparing the total chemical exergy of the reactants to that of the products, researchers can determine the maximum work output of a steady-flow chemical reaction. This value, W_{rev,out,reaction}, serves as the ultimate benchmark for evaluating the second-law efficiency of chemical energy conversion technologies.

Case Study: Piston-Cylinder and Sub-Environmental States

The application of the proposed exergy efficiency methodology to specific thermal processes reveals significant disparities between first-law energy conservation and second-law thermodynamic perfection. By evaluating processes through the lens of expended versus recovered exergy, the methodology accounts for the quality of energy across varying environmental conditions and system configurations.

Piston-Cylinder Water Heating

In a controlled heating process involving a piston-cylinder device, saturated liquid water is heated at a constant pressure of 200 kPa. Under these specific conditions, the system yields an exergy efficiency of 39.5%. This value represents the ratio of the increase in the system's non-flow exergy to the exergy expended via heat and work. Unlike thermal efficiency, which might only track the quantity of heat added, the 39.5% figure provides a measure of how much of the high-quality energy input was successfully stored as potential work within the water’s state change, accounting for the inherent irreversibilities of the heating process.

Sub-Environmental Cooling and Refrigeration

Systems operating below the environmental temperature, such as those utilizing R-134a at -15.6 °C, require a nuanced interpretation of exergy flow. According to the methodology, when a system's temperature is lower than the environment (T_b < T₀), heat transfer to the system actually decreases its exergy. In these cases, the exergy decrease must be identified as an expenditure. This is because restoring the cold system to its original state would require a work input, typically modeled as a reversible refrigerator. Identifying these exergy decreases as expenditures ensures that the efficiency calculation accurately reflects the cost of maintaining sub-environmental states.

Electric Resistance Heating

The methodology highlights the extreme thermodynamic inefficiency of electric resistance heating. When an electric heater operates at 25 °C in a 10 °C environment, the resulting exergy efficiency is remarkably low—only 5%. While a first-law analysis might suggest 100% energy efficiency (as all electricity is converted to heat), the exergy analysis reveals a massive destruction of work potential. Using high-grade electrical exergy to produce low-grade thermal energy at a temperature only slightly above the environment represents a near-total loss of the resource's capacity to do useful work.

Unsteady-Flow in Rigid Tanks

The evaluation of unsteady-flow processes, such as the filling of rigid tanks, is managed by balancing the exergy supplied via the line against the accumulation within the vessel. In these scenarios:

• Supply Line Exergy: The flow exergy of the mass entering the tank is treated as the primary expenditure.
• Tank Accumulation: The net change in the non-flow exergy of the mass within the tank serves as the recovered exergy.
• Mechanism: By comparing the exergy of the fluid in the supply line to the resulting exergy increase in the tank's contents, the methodology captures the irreversibilities inherent in the filling process, such as throttling and mixing.

Quantitative Review of Results

The application of the expanded exergy efficiency definitions to closed and unsteady-flow systems reveals specific numerical benchmarks for thermodynamic performance. These results quantify the degree of perfection achieved in practical processes relative to the theoretical reversible limit, where exergy destruction would be zero. By calculating the ratio of recovered exergy to expended exergy, research at the institute has identified efficiency ranges that highlight both the potential of modern systems and the inherent limitations imposed by temperature gradients.

Efficiency Benchmarks in Closed Systems

In closed-system applications, the interpretation of exergy change as either an input or an output is critical for determining efficiency. Specific experimental and numerical cases provide the following results:

• Rigid Tank Work-Input: In a process involving a rigid tank where work is the primary input to increase the internal exergy of the fluid, the system converted 80.2% of the consumed exergy into stored exergy. The remaining 19.8% represents exergy destruction occurring within the system due to internal irreversibilities.
• Steam-Filling Process: For an unsteady-flow application involving the filling of a 0.5 m³ tank with steam, the exergy efficiency was calculated at 77.2%. This illustrates the effectiveness of exergy accumulation relative to the flow exergy supplied by the steam stream.

Irreversibility and Temperature Gradients

The review of results demonstrates that significant exergy destruction occurs during heat transfer processes involving large temperature differences. For example, the transfer of heat from a reservoir maintained at 500 °C to water resulted in substantial irreversibility. This destruction is fundamentally tied to the temperature gradients existing between the source and the working fluid, as exergy transfer associated with heat is proportional to the Carnot efficiency factor (1 - T₀/T_b). As the boundary temperature T_b deviates from the environment temperature T₀, the potential for exergy destruction increases if the heat is not transferred through a reversible mechanism.

Components of Exergy Destruction

A consistent finding across the analyzed applications is that total exergy destruction is not localized solely within the device. To achieve a comprehensive efficiency review, the total exergy destruction must be calculated as the sum of two distinct parts:

By accounting for both internal and external effects, the exergy efficiency serves as a faithful indicator of how well a system performs compared to the best possible performance under the second law of thermodynamics.

Key findings

• Superiority of the Recovered-Expended Model — The ηex = Xrecovered / Xexpended definition provides consistent results where the Wact/Wrev ratio fails, such as constant-pressure heating at 100 kPa where actual useful work is zero.
• Boundary Impact on Efficiency — Using an extended system boundary captures unavoidable external irreversibilities, resulting in more conservative and realistic efficiency values.
• Resistance Heating Inefficiency — While 100% energy efficient, electric resistance heaters for space heating are only about 5% exergy efficient due to the high-grade electricity being degraded to low-temperature heat.
• Exergy Destruction Relationship — Exergy destruction (Xdestroyed) is always equal to the product of environmental temperature (T0) and entropy generation (Sgen).

Method and assumptions

The researchers developed general exergy efficiency relations using a first-principles thermodynamic approach. The methodology distinguishes between 'physical systems' and 'extended systems' to clarify the treatment of exergy loss. System boundaries were modeled to include immediate surroundings where gradients occur, ensuring that boundary temperatures match the environment (T0). Calculations for fluid properties (Water, R-134a) were performed using Engineering Equation Solver (EES) Version 10. The analysis assumes the environment is a large, simple compressible system at a fixed dead state (typically 25 °C, 100 kPa) with fixed chemical composition. Chemical exergy was modeled using ideal gas and ideal solution assumptions, incorporating standard Gibbs functions of formation for reacting systems.

Where it applies

• Compressed Air Energy Storage (CAES) — Evaluating the efficiency of discharging air through turbines where exergy is expended from system storage.
• Transient HVAC Analysis — Determining the exergetic performance of electric heaters and heat pumps during start-up or variable load conditions.
• Chemical Process Engineering — Calculating the minimum work required for gas separation or the maximum work available from mixing and chemical reactions.

Terms used

• Dead State — A state where a system is in thermal, mechanical, and chemical equilibrium with its environment, possessing zero exergy.
• Exergy Destruction — The loss of work potential caused by internal irreversibilities such as friction or heat transfer across finite temperature differences.
• Chemical Exergy — The maximum useful work obtainable as a system reaches chemical equilibrium with the environment.
• Reversible Work — The maximum theoretical useful work output or minimum work input for a process between two specified states.
• Restricted Dead State — A condition where a system is in thermal and mechanical equilibrium with the environment but not necessarily chemical equilibrium.

Questions and answers

Why is the standard efficiency ratio unsuitable for a piston-cylinder device?

If a piston expands at atmospheric pressure, the actual useful work is zero, making the ratio Wact/Wrev zero regardless of the heat input. The expended-recovered framework avoids this by accounting for the increase in the fluid's internal exergy as a recovered quantity.

How should heat loss to the environment be treated in exergy calculations?

It can be treated as exergy loss (Xloss) if analyzing the physical system boundary, but it is better captured as exergy destruction (Xdestroyed) by using an extended system boundary that reaches the environmental temperature.

What is the difference between exergy efficiency and thermal efficiency?

Thermal efficiency compares work output to heat input, while exergy efficiency compares work output to the work potential (exergy) of that heat input, providing a measure of how closely the system approaches a reversible Carnot cycle.

Does exergy efficiency apply to systems below the temperature of the surroundings?

Yes; however, in such cases, transferring heat to the system may actually decrease its exergy (work potential), meaning the exergy change is treated as an expenditure rather than a recovery.

How to cite

Çengel, Y.A.; Kanoğlu, M. Exergy Efficiency of Closed and Unsteady-Flow Systems. Entropy 2025, 27, 943. https://doi.org/10.3390/e27090943