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Exergy efficiency

Exergy efficiency, also referred to as second-law efficiency, is a thermodynamic metric that evaluates the quality and effectiveness of energy utilization in a system by quantifying the ratio of the actual useful exergy output to the maximum possible exergy output under reversible conditions, or equivalently, one minus the fraction of exergy destroyed due to irreversibilities.^[1] It represents the potential for work extraction from an energy resource relative to its interaction with a reference environment, typically defined at standard conditions such as 25°C and 1 atm, and is expressed mathematically as \eta_{II} = \frac{X_{\text{out}}}{X_{\text{in}}} = 1 - \frac{I}{X_{\text{in}}}, where I denotes exergy destruction linked to entropy generation.^[1] This concept differs fundamentally from conventional energy efficiency, which adheres to the first law of thermodynamics and treats all forms of energy as equivalent, often yielding misleadingly high values that overlook energy degradation.^[2] Exergy efficiency, grounded in the second law, reveals the true extent of thermodynamic losses—such as those from heat transfer across finite temperature differences or friction—enabling more precise assessments of system performance.^[1] Its importance lies in promoting sustainability: by identifying and minimizing exergy destruction, it extends resource longevity, reduces waste emissions, and lowers environmental impacts, as demonstrated in analyses where improving exergy efficiency in energy systems can decrease pollutant outputs by over 60%.^[3] In engineering practice, exergy efficiency finds broad applications across sectors, including power generation (e.g., coal-fired plants achieving around 36% efficiency), refrigeration cycles, heat exchangers, and industrial drying processes, where it guides optimizations for higher resource utilization and lower operational costs.^[2]^[3] For heat engines specifically, it is calculated as the thermal efficiency divided by the Carnot efficiency, providing a benchmark for reversible ideals.^[1] Overall, exergy efficiency serves as a vital tool for advancing eco-friendly technologies and informing policy decisions aimed at global energy challenges.^[3]

Fundamentals

Definition of Exergy

Exergy is defined as the maximum theoretical useful work that can be obtained from a system as it is brought into complete thermodynamic equilibrium with its reference environment, known as the dead state, through reversible processes interacting only with that environment.^[4] This concept quantifies the quality of energy available for work, distinguishing it from total energy by accounting for the second law of thermodynamics and the inevitability of entropy generation in real processes. The dead state is typically specified by environmental conditions, such as ambient temperature T_0 and pressure P_0, representing the state of minimum energy where no further work can be extracted. The term "exergy" was coined in 1953 by Slovenian engineer Zoran Rant, who introduced it as "Exergie" in German to denote "technical working capacity," building on 19th-century thermodynamic concepts like the availability function developed by J. Willard Gibbs in 1873 and earlier ideas of available work.^[4] For a closed system, exergy Ex is mathematically expressed as

Ex = (U - U_0) - T_0(S - S_0) + P_0(V - V_0),

where U is the internal energy, S the entropy, and V the volume of the system, with subscript 0 denoting values at the dead state; kinetic and potential energies may be added if significant.^[1] Exergy comprises several physical components arising from deviations of the system's state from the dead state. Thermal exergy stems from temperature differences relative to T_0, enabling reversible heat transfer to produce work. Mechanical exergy arises from pressure differences compared to P_0, allowing reversible expansion or compression work. Chemical exergy results from differences in chemical composition with the environment, such as in reactive mixtures, permitting work from diffusion or reaction processes.^[5] Representative examples illustrate these components. Hot water above ambient temperature possesses primarily thermal exergy, as its heat content can theoretically drive a reversible heat engine interacting with the environment until equilibrium at T_0.^[1] Similarly, a compressed gas at pressure exceeding P_0 holds mechanical exergy, extractable via reversible expansion to atmospheric conditions.^[1]

Exergy Efficiency

Exergy efficiency, also known as second-law efficiency, serves as a performance metric that evaluates the quality of energy conversion in thermodynamic processes by quantifying the fraction of supplied exergy that is effectively utilized for the intended task. It is formally defined as the ratio of the useful exergy output to the exergy input, expressed as \eta_{ex} = \frac{Ex_{out}}{Ex_{in}} \times 100\%, where Ex_{out} represents the minimum exergy required to accomplish the task and Ex_{in} is the exergy supplied to the system.^[2]^[6] Unlike first-law efficiency, which conserves energy quantity but ignores quality degradation, exergy efficiency incorporates the second law of thermodynamics by accounting for irreversibilities and the reference environment, thereby revealing the true extent of potential work losses in processes. This distinction highlights how exergy efficiency is typically lower than energy efficiency, as it penalizes the dissipation of high-quality energy into lower-quality forms, such as heat rejected to the surroundings.^[2] Exergy efficiency can be categorized into task-based and component-based types. Task-based exergy efficiency focuses on the overall purpose of the system, for instance, in a heating process where \eta_{ex} = \frac{\text{Exergy of [heat](/page/Heat) delivered}}{\text{[Exergy](/page/Exergy) supplied}}, emphasizing the minimum exergy needed to achieve the desired temperature elevation. Component-based efficiency, in contrast, assesses individual elements within a system, such as a compressor or turbine, to pinpoint local irreversibilities.^[7]^[8] For simple work-producing devices, such as turbines, exergy efficiency simplifies to \eta_{ex} = \frac{W_{actual}}{Ex_{available}}, where W_{actual} is the actual work output and Ex_{available} is the maximum reversible work obtainable from the input stream. This formulation underscores the metric's utility in benchmarking real performance against ideal reversible limits.^[2] A primary advantage of exergy efficiency lies in its ability to identify locations and causes of energy degradation, guiding improvements in system design. For example, in throttling processes—where a fluid expands through a restriction without producing work—exergy efficiency approaches zero due to complete destruction of pressure-related exergy potential through irreversibilities.^[9]

Theoretical Aspects

Relation to Carnot Cycle

The Carnot efficiency, \eta_C = 1 - \frac{T_l}{T_h}, where T_h and T_l are the temperatures of the high- and low-temperature reservoirs, respectively, establishes the theoretical maximum efficiency for any heat engine operating between these reservoirs under reversible conditions.^[10] This efficiency reflects the ideal limit dictated by the second law of thermodynamics, where no irreversibilities occur, and all processes—two isothermal heat transfers and two adiabatic expansions/compressions—are perfectly reversible.^[11] In practice, T_l is often taken as the dead-state temperature T_0 of the environment, aligning the cycle with ambient conditions for exergy analysis.^[10] In the context of exergy efficiency, the ideal Carnot heat engine achieves \eta_{ex} = 1, meaning the entire exergy input is converted to useful work without losses due to the absence of irreversibilities.^[11] The exergy input to the cycle comes primarily from the heat Q_h supplied by the hot reservoir at T_h, quantified as Ex_{Q_h} = Q_h \left(1 - \frac{T_0}{T_h}\right).^[10] For the reversible Carnot cycle, the net work output W exactly equals this exergy input, since W = Q_h \left(1 - \frac{T_0}{T_h}\right), resulting in \eta_{ex} = \frac{W}{Ex_{Q_h}} = 1.^[11] This equivalence highlights how exergy efficiency benchmarks real processes against the Carnot ideal, where the cycle's exergy balance shows no destruction: exergy enters via Q_h, is fully utilized to produce W, and the rejected heat Q_l to the environment at T_0 carries zero exergy.^[10] In real heat engines, exergy destruction arises from irreversibilities, leading to \eta_{ex} < 1 and \eta < \eta_C.^[11] The exergy destruction I is given by I = T_0 \sigma, where \sigma is the total entropy generation within the cycle, quantifying the lost work potential due to factors like finite temperature differences during heat transfer or frictional losses in expansions.^[10] For instance, in the exergy flow of a Carnot-like cycle, destruction would occur if the isothermal heat addition involved non-quasistatic processes, reducing the effective work output below the reversible limit and illustrating why practical efficiencies fall short of both \eta_C and full exergy utilization.^[11] This relation underscores exergy efficiency's role in identifying and minimizing such losses to approach the Carnot benchmark.^[10]

Second Law Efficiency Under Constraints

Finite-time thermodynamics extends the analysis of heat engines beyond the idealized reversible Carnot cycle, which assumes infinite time for isothermal heat transfer processes, to account for real-world operations where finite time is required to achieve practical power outputs. In practical systems, prioritizing maximum power over perfect reversibility introduces irreversibilities primarily in the heat exchange with finite temperature gradients, leading to reduced efficiencies compared to the reversible limit.^[12] A key result in this framework is the Curzon-Ahlborn efficiency, derived for an endoreversible Carnot-like heat engine operating at maximum power, given by

\eta_{\text{CA}} = 1 - \sqrt{\frac{T_{\text{L}}}{T_{\text{H}}}},

where T_{\text{H}} and T_{\text{L}} are the temperatures of the hot and cold reservoirs, respectively; this expression provides an upper bound on the efficiency achievable under finite-rate heat transfer constraints. The second law efficiency, or exergy efficiency, under these maximum power conditions is then \eta_{\text{ex,MP}} = \eta_{\text{CA}} / \eta_{\text{C}}, where \eta_{\text{C}} = 1 - T_{\text{L}}/T_{\text{H}} is the Carnot efficiency, yielding a value less than the reversible exergy efficiency of 1.^[13] This exergy efficiency arises from the endoreversible engine model, in which internal processes are reversible but external heat transfers occur irreversibly across finite temperature differences; optimization involves maximizing power P = W / t, where W is the work output and t is the cycle time, by balancing exergy flows with the trade-off between efficiency and finite conduction rates modeled via Newton's law of cooling.^[12] For instance, in Stirling engines, which approximate the Carnot cycle through regeneration, operation at maximum power results in a 20-30% reduction in exergy efficiency relative to the reversible case due to these finite-time irreversibilities.^[14] Recent developments in the 2020s have focused on exergy optimization under maximum power constraints for renewable energy systems facing variable loads, such as hybrid solar-wind setups, where finite-time models help balance fluctuating inputs to achieve higher overall exergy utilization without excessive storage demands.^[15]

Practical Applications

In Thermodynamic Systems

Exergy analysis provides a second-law perspective on the performance of thermodynamic systems, revealing inefficiencies beyond those captured by first-law energy balances. In power cycles such as the Rankine cycle, which is widely used in steam power plants, the boiler typically accounts for the largest share of exergy destruction, often exceeding 40-50% of the total, primarily due to irreversibilities in the combustion process, including chemical reactions, heat transfer across finite temperature differences, and mixing of fuel and air.^[16] These losses highlight how much of the fuel's exergy potential is dissipated as unusable heat, even though first-law efficiencies may appear high. For instance, in a typical coal-fired Rankine cycle, combustion-related exergy destruction in the boiler can reach around 48% of the inlet exergy, underscoring the need for advanced combustion technologies to mitigate these irreversibilities.^[17] The exergy efficiency of individual components in thermodynamic systems, such as turbines, compressors, or heat exchangers, is quantified using the formula:

\eta_{ex,comp} = 1 - \frac{Ex_{destroyed}}{Ex_{in}}

where Ex_{destroyed} represents the exergy destroyed within the component, calculated as Ex_{destroyed} = T_0 \Delta S_{gen}, with T_0 being the reference environment temperature and \Delta S_{gen} the entropy generation.^[18] This metric directly measures how closely a component approaches reversible operation, with values below 80-90% indicating significant potential for improvement through reduced friction, optimized pressure ratios, or minimized temperature gradients. In turbines and compressors, for example, exergy destruction arises from non-ideal expansions or compressions, leading to efficiencies that pinpoint mechanical and fluid dynamic losses.^[18] A practical case study is the gas turbine combined cycle (GTCC), where exergy analysis reveals an overall exergy efficiency of approximately 40-50%, compared to a first-law thermal efficiency of around 60%, with stack gas losses contributing substantially to the discrepancy due to high-temperature exhaust carrying away unused exergy.^[19] In GTCC systems, the combustion chamber and heat recovery steam generator dominate exergy destruction, often accounting for over 60% of total losses, while stack emissions represent an exergy loss of 10-20% from unrecovered heat potential.^[19] This contrast emphasizes how exergy efficiency better captures the quality degradation of energy flows, guiding optimizations like intercooling or advanced blade designs to recover more work potential. To visualize these inefficiencies, exergy flow diagrams—adapted Sankey diagrams that track exergy streams rather than energy—illustrate the distribution of destructions and losses across components, enabling engineers to identify and target high-loss areas such as the combustor or exhaust.^[20] In refrigeration systems, which operate as reversed thermodynamic cycles, exergy efficiency is typically low, often below 30%, with the compressor being the primary source of losses due to irreversible compression and heat generation from friction and non-ideal gas behavior.^[21] For vapor-compression refrigeration units, compressor exergy destruction can constitute up to 50-70% of the total system losses, driven by polytropic inefficiencies and elevated discharge temperatures that reduce the coefficient of performance relative to the Carnot ideal.^[22] This low efficiency underscores the challenges in achieving reversible heat pumping, prompting design improvements like variable-speed compressors or alternative refrigerants to enhance exergy utilization in cooling applications.^[21]

In Energy Conversion Processes

In energy conversion processes, exergy efficiency plays a crucial role in evaluating the performance of fuel processing and conversion systems, such as combustion and electrochemical devices, by quantifying the useful work potential preserved from chemical energy inputs. Chemical exergy represents the maximum work obtainable from a fuel as it reacts to environmental conditions, often approximated for hydrocarbons like methane as the lower heating value (LHV) multiplied by a factor of approximately 1.04, yielding about 831 kJ/mol for methane given its LHV of 802 kJ/mol.^[23] In combustion processes, exergy efficiency is defined as the ratio of the exergy content in the products to that in the reactants, typically ranging from 94% to 97% for the oxidation reaction itself, highlighting the near-reversible nature of fuel combustion despite temperature gradients that contribute to irreversibilities.^[24]^[25] For steady-flow processes common in energy conversion, the exergy balance equation governs the analysis:

\sum \dot{\text{Ex}}_{\text{in}} = \sum \dot{\text{Ex}}_{\text{out}} + \dot{W} + \dot{I}

where \sum \dot{\text{Ex}}_{\text{in}} is the total exergy inflow rate, \sum \dot{\text{Ex}}_{\text{out}} is the outflow rate, \dot{W} is the useful work output rate, and \dot{I} is the irreversibility rate representing destroyed exergy. This balance reveals opportunities to minimize losses in systems like fuel cells, where proton exchange membrane fuel cells (PEMFCs) achieve exergy efficiencies up to 80% through direct electrochemical conversion of hydrogen, avoiding the high irreversibilities of combustion.^[26] In contrast, internal combustion engines typically exhibit around 40% exergy efficiency due to thermal and mechanical losses in indirect heat-to-work conversion.^[26] A representative case is biomass gasification, where exergy efficiency hovers around 60% in direct gasification configurations, primarily limited by char formation that retains unreacted carbon and reduces syngas yield.^[27] Post-2010 advancements, such as incorporating char or activated char catalysts, have improved efficiencies by enhancing gas heating value and reducing tar, thereby targeting exergy losses in the reaction zone.^[28] Environmentally, exergy losses in integrated gasification combined cycle (IGCC) plants correlate with emissions, as higher irreversibilities in gasification and combustion stages lead to greater fuel consumption and CO₂ release; carbon capture integration can mitigate this by recovering up to 90% of emissions while preserving overall plant exergy efficiency near 40%.^[29]

Computational Considerations

Methods for Exergy Analysis

Exergy analysis begins with establishing an exergy balance for a thermodynamic system, which quantifies the useful work potential of energy streams relative to a reference environment, known as the dead state. The dead state is typically defined by environmental conditions such as T_0 = 25^\circ \text{C} and P_0 = 1 \, \text{atm}, representing the state of thermodynamic equilibrium where no further work can be extracted.^[1] To perform the balance, the process is subdivided into components or control volumes, and mass, energy, and entropy balances are first solved to determine stream properties like temperature, pressure, enthalpy, and entropy.^[30] Exergy for each stream is then calculated, distinguishing between physical exergy (due to temperature and pressure deviations from the dead state) and chemical exergy (due to composition differences). For flow streams, the specific flow exergy \psi is given by

\psi = (h - h_0) - T_0 (s - s_0) + \frac{V^2}{2} + gz + \text{chemical exergy},

where h and s are enthalpy and entropy, subscript 0 denotes dead-state values, and kinetic, potential, and chemical terms are included as applicable.^[1] Exergy destruction, representing irreversibilities, is computed from the balance equation for steady-state systems:

\sum \dot{\psi}_{\text{in}} - \sum \dot{\psi}_{\text{out}} + \sum \dot{Q} \left(1 - \frac{T_0}{T}\right) - \dot{W} = \dot{X}_{\text{destroyed}},

where \dot{X}_{\text{destroyed}} = T_0 \dot{S}_{\text{gen}} and \dot{S}_{\text{gen}} is entropy generation.^[1] An advanced extension is exergoeconomic analysis, which integrates exergy balances with economic costing to allocate expenses based on exergy flows and identify cost-ineffective components. This method combines thermodynamic inefficiencies with financial metrics, such as levelized cost of energy, to optimize system design by quantifying the cost of exergy destruction.^[31] Cost allocation follows the principle that costs are apportioned proportionally to exergy contributions, using the cost balance for each component:

\dot{C}_F + \dot{Z} = \dot{C}_P + \dot{C}_L,

where \dot{C}_F is the cost rate of fuel exergy input, \dot{Z} is the capital and operating cost rate, \dot{C}_P is the cost rate of product exergy output, and \dot{C}_L accounts for losses. The unit exergy cost c_k = \dot{C}_k / \dot{Ex}_k is solved iteratively across the system.^[32] Several software tools facilitate exergy analysis by automating balances and calculations. The Engineering Equation Solver (EES) solves coupled nonlinear equations for thermodynamic properties and exergy terms, supporting custom scripts for physical and chemical exergy in complex cycles.^[33] Aspen Plus enables exergy evaluation through process simulation, where users define the dead state and compute stream exergies via built-in enthalpy/entropy functions or custom blocks for balances.^[30] For open-source options, ExerPy, a Python library updated post-2020 (version 0.0.3 in 2025), performs component-level exergy analysis by importing data from simulators like Aspen Plus or TESPy, calculating physical and chemical exergies to pinpoint inefficiencies.^[34] A specific procedure for evaluating exergy efficiency in a power plant involves tabulating exergy rates for all inputs (e.g., fuel, heat), outputs (e.g., work, heat rejection), and internal destructions across components. The overall exergy efficiency is then

\eta_{\text{ex}} = 1 - \frac{\dot{X}_{\text{destroyed, total}}}{\dot{X}_{\text{in, total}}},

where total destruction is summed from component balances, and input exergy includes fuel chemical exergy plus any auxiliary work. This approach, applied to steam or gas turbine plants, reveals major loss sites like combustion chambers, guiding retrofits.^[35] Exergy analysis integrates with life cycle assessment (LCA) by using cumulative exergy demand (CExD) as a sustainability metric, quantifying total exergy extracted from natural resources across a product's lifecycle, including extraction, production, use, and disposal. CExD extends traditional energy indicators by accounting for resource quality (e.g., fossil fuels, metals, water) in exergy equivalents (MJ-eq), enabling comparison of environmental impacts without subjective weighting. For instance, in building materials, CExD highlights non-energetic resource demands like metals, which can constitute up to 38% of total exergy in metallic products.^[36]

Challenges in Exergy Efficiency Calculations

One of the primary challenges in exergy efficiency calculations stems from the selection of the dead state, which represents the reference environment against which exergy is measured.^[37] The standard dead state often assumes ambient conditions at 298 K and 1 atm, but real-world variations in temperature, pressure, and humidity—such as those in different climatic regions—can significantly alter results.^[38] For instance, studies on thermal power plants and absorption chillers have shown that shifting dead state temperatures between 10°C and 30°C can change exergy efficiency values by up to 6% relative, depending on the system components, highlighting the sensitivity of analyses to site-specific environmental data.^[39]^[40] This variability complicates comparisons across global applications and underscores the need for standardized yet adaptable reference states. Data uncertainties in thermodynamic properties, particularly for complex mixtures, further complicate exergy efficiency computations through error propagation in the underlying equations. Properties such as enthalpy, entropy, and specific heat for multi-component systems often rely on equations of state like PC-SAFT, which introduce uncertainties from model parameters and experimental data scatter.^[41] In exergy calculations, these propagate via partial derivatives in expressions for physical and chemical exergy, potentially amplifying errors in efficiency metrics by several percent through propagation in thermodynamic properties.^[42] Accurate property databases mitigate some issues, but incomplete data for novel fuels or high-pressure conditions remains a persistent limitation, demanding rigorous sensitivity analyses in practice. Scale-dependent issues arise when applying exergy analysis across micro- and macro-level systems, often leading to oversimplifications like neglecting kinetic exergy contributions. In microscopic or component-level analyses, such as in microelectronics cooling or nanoscale devices, kinetic and potential exergies must be explicitly accounted for due to high velocities and gradients.^[43] Conversely, in large-scale industrial plants, these terms are frequently omitted as they represent negligible fractions of total exergy (typically <1% in steady-flow processes), but this approximation can distort efficiency estimates if transient flows or turbulence are present.^[44] Such inconsistencies hinder seamless scaling from design prototypes to full operations, requiring hybrid modeling approaches to bridge micro-macro discrepancies. In renewable energy systems, an emerging challenge post-2020 involves handling variable exergy inputs from sources like solar and wind, which disrupt traditional steady-state assumptions in efficiency calculations. Unlike constant-fuel conventional systems, intermittent renewables exhibit fluctuating exergy availability due to weather dependencies, necessitating dynamic modeling that incorporates time-varying dead states and storage effects.^[45] For example, solar thermal plants can experience notable drops in exergy efficiency under partial load compared to steady-state predictions, while wind turbines require transient simulations to capture gust-induced losses.^[46] This shift demands advanced computational tools for real-time exergy tracking, as static analyses overestimate performance in variable environments. Critiques of exergy efficiency highlight its overemphasis on thermodynamic ideals at the expense of economic and practical constraints, prompting calls for hybrid metrics in recent studies. Pure exergy analyses often prioritize irreversibility minimization but overlook capital costs, lifecycle emissions, or operational feasibility, leading to theoretically optimal designs that are economically unviable.^[46] In response, 2023 research advocates integrating exergy with exergoeconomic factors—such as levelized cost of exergy—to balance efficiency with affordability, particularly in hybrid renewable setups where storage adds complexity.^[47] These hybrid approaches reveal that standalone exergy metrics can undervalue resilient, lower-efficiency systems that align better with real-world sustainability goals.

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(PDF) Exergy and Exergy-Economic Approach to Evaluate Hybrid ...
Jan 11, 2023 · The aim of this paper is to present a review of exergy and exergy-economic approaches to evaluate hybrid renewable energy systems in buildings.