Thermal efficiency
Thermal efficiency is a dimensionless performance metric in thermodynamics that quantifies the effectiveness of a device or process in converting thermal energy into useful work, defined as the ratio of the net work output to the total heat input supplied to the system.[1][2] It is typically expressed as a percentage and applies to heat engines, which operate by absorbing heat from a high-temperature source, performing work, and rejecting waste heat to a low-temperature sink.[3] The fundamental formula for thermal efficiency, denoted as η, is η = W / Q_H, where W is the useful work done and Q_H is the heat absorbed from the hot reservoir; equivalently, it can be written as η = 1 - Q_C / Q_H, with Q_C representing the heat rejected to the cold reservoir.[1][2] In practice, thermal efficiency is constrained by the second law of thermodynamics, which prohibits 100% conversion of heat to work due to inevitable entropy increases and waste heat production.[1] The theoretical maximum efficiency for any heat engine operating between two temperatures is given by the Carnot efficiency: η_Carnot = 1 - T_C / T_H, where T_C and T_H are the absolute temperatures (in Kelvin) of the cold and hot reservoirs, respectively.[3][2] Real-world systems fall short of this limit due to irreversibilities such as friction, heat losses, and non-ideal processes, resulting in typical efficiencies of 25–35% for gasoline engines, 30–35% for diesel engines, and around 33% for nuclear power plants.[2] Thermal efficiency is central to the analysis of various thermodynamic cycles that model practical engines.[3] For instance, the Otto cycle, which approximates spark-ignition internal combustion engines, has an efficiency of η_Otto = 1 - (1 / r)^{γ-1}, where r is the compression ratio and γ is the specific heat ratio of the working fluid (approximately 1.4 for air).[3] The Diesel cycle, used in compression-ignition engines, achieves slightly higher efficiency through higher compression ratios but involves constant-pressure heat addition, modifying the formula to account for a cutoff ratio.[3] These cycles, along with others like the Rankine cycle in steam turbines, underscore efforts to optimize energy conversion in power generation, transportation, and industrial processes, where improving efficiency reduces fuel consumption and environmental impact.[1][2]Fundamentals
Definition and Principles
Thermal efficiency is a dimensionless measure of the performance of a thermodynamic system that converts thermal energy into useful work or desired heat transfer, defined as the ratio of the useful energy output to the total energy input supplied to the system.[1] This metric quantifies how effectively a device, such as a heat engine or boiler, utilizes input energy while accounting for losses due to irreversibilities like friction or heat dissipation.[4] For heat engines, thermal efficiency is commonly expressed by the formula \eta = \frac{W_\text{net}}{Q_\text{in}}, where W_\text{net} is the net work output and Q_\text{in} is the heat input from the high-temperature source.[1] In systems focused on heat transfer, such as boilers, it is calculated as the ratio of heat delivered to the working fluid to the energy content of the fuel input.[5] As a ratio, thermal efficiency is unitless and ranges from 0 to 1, often presented as a percentage from 0% to 100%; it cannot exceed 100% because the useful output cannot surpass the total input, in accordance with the conservation of energy principle.[4] The Second Law of Thermodynamics imposes fundamental limits on achievable efficiency by prohibiting complete conversion of heat to work.[1] The concept originated with Sadi Carnot's 1824 publication Reflections on the Motive Power of Fire, which analyzed the efficiency of heat engines and laid foundational principles for thermodynamics by examining the conversion of heat into mechanical work.[6] A basic example is a boiler's efficiency, given by \eta = \left( \frac{\text{[heat](/page/Heat) exported to [fluid](/page/Fluid)}}{\text{[heat](/page/Heat) provided by [fuel](/page/Fuel)}} \right) \times 100\%, which typically ranges from 80% to 98.5% in modern high-efficiency systems (such as condensing boilers) depending on design and operating conditions as of 2025.[7] For a furnace, similar calculations assess how much of the fuel's combustion energy successfully heats the target medium versus being lost to the environment.[4]Thermodynamic Foundations
The First Law of Thermodynamics, which embodies the principle of conservation of energy, forms the foundational energy balance for calculating thermal efficiency in thermodynamic systems. It states that the change in internal energy of a closed system, denoted as ΔU, equals the heat added to the system Q minus the work done by the system W: ΔU = Q - W (where W is the work done by the system).[8] This equation ensures that energy is neither created nor destroyed, allowing efficiency to be expressed as the ratio of useful work output to heat input, such as η = W / Q_in for heat engines. The Second Law of Thermodynamics introduces fundamental limits on thermal efficiency through the concept of entropy and the inevitability of irreversibilities. It asserts that in any energy conversion process, the total entropy of an isolated system cannot decrease and typically increases, reflecting the natural tendency toward disorder. This entropy increase arises from irreversibilities like friction, heat transfer across finite temperature differences, and mixing, which dissipate useful energy as waste heat, preventing complete conversion of thermal energy into work. For reversible processes, the change in entropy ΔS is given by the integral of reversible heat transfer δQ_rev divided by temperature T: \Delta S = \int \frac{\delta Q_\text{rev}}{T} This relation highlights that even in ideal cases, entropy balances impose constraints, as not all heat input can yield work without some rejection at lower temperatures.[9] Two equivalent statements of the Second Law underscore these efficiency bounds. The Clausius statement declares that heat cannot spontaneously flow from a colder body to a hotter one without external work, implying that thermal processes require temperature gradients and cannot achieve perfect efficiency without auxiliary input.[10] Complementing this, the Kelvin-Planck statement posits that no heat engine operating in a cycle can absorb heat from a single reservoir and convert it entirely into work; some heat must always be rejected to a colder reservoir as waste.[11] Together, these laws establish why real-world thermal efficiencies fall below theoretical maxima, as seen in idealized reversible cycles like the Carnot cycle, serving as prerequisites for analyzing limitations in all thermodynamic devices.Heat Engines
Carnot Efficiency
The Carnot cycle, introduced by Sadi Carnot in his 1824 work Réflexions sur la puissance motrice du feu, represents an idealized reversible thermodynamic cycle for a heat engine operating between two thermal reservoirs at constant temperatures.[12] It consists of four reversible processes: isothermal expansion of the working fluid, where heat is absorbed from the hot reservoir; adiabatic expansion, where no heat is exchanged and the fluid does further work; isothermal compression, where heat is rejected to the cold reservoir; and adiabatic compression, returning the fluid to its initial state./Thermodynamics/Thermodynamic_Cycles/Carnot_Cycle) This cycle assumes perfect reversibility, with no frictional losses or other irreversibilities, making it a theoretical benchmark for maximum efficiency./Thermodynamics/Thermodynamic_Cycles/Carnot_Cycle) The efficiency of a Carnot engine, denoted as \eta_{Carnot}, is given by the formula \eta_{Carnot} = 1 - \frac{T_c}{T_h}, where T_h is the absolute temperature of the hot reservoir and T_c is the absolute temperature of the cold reservoir, both measured in Kelvin./15:_Thermodynamics/15.04:_Carnots_Perfect_Heat_Engine-_The_Second_Law_of_Thermodynamics_Restated) This expression derives from the entropy balance in a reversible cycle, where the total change in entropy \Delta S over the complete cycle must be zero for the system to return to its initial state without net entropy production. To outline the derivation, consider the heat transfers: during isothermal expansion at T_h, the entropy change is \Delta S_h = Q_h / T_h, where Q_h > 0 is the heat absorbed; during isothermal compression at T_c, \Delta S_c = -Q_c / T_c, where Q_c > 0 is the heat rejected. The adiabatic processes contribute no entropy change due to reversibility and zero heat transfer. For the cycle, \Delta S = \Delta S_h + \Delta S_c = 0, yielding Q_h / T_h = Q_c / T_c, or Q_c / Q_h = T_c / T_h. The efficiency, defined as the net work output divided by heat input (\eta = W / Q_h = (Q_h - Q_c) / Q_h), then simplifies to \eta_{Carnot} = 1 - T_c / T_h. This result was rigorously established as the upper limit for any heat engine by William Thomson (Lord Kelvin) in 1851, demonstrating that no real engine can exceed it without violating the second law of thermodynamics.[13] The implications of Carnot efficiency are profound in thermodynamics: it depends solely on the reservoir temperatures, independent of the working fluid or engine design, highlighting the fundamental role of temperature differentials in energy conversion./15:_Thermodynamics/15.04:_Carnots_Perfect_Heat_Engine-_The_Second_Law_of_Thermodynamics_Restated) Greater efficiency is achieved with a larger temperature difference (T_h - T_c), but practical constraints limit T_h by material properties and T_c by environmental conditions, underscoring why real systems operate below this ideal./15:_Thermodynamics/15.04:_Carnots_Perfect_Heat_Engine-_The_Second_Law_of_Thermodynamics_Restated)Practical Cycle Efficiencies
Practical heat engine cycles approximate the ideal Carnot cycle but incorporate irreversible processes such as constant-volume or constant-pressure heat addition and rejection, resulting in lower thermal efficiencies. These cycles form the basis for common devices like internal combustion engines, gas turbines, and steam power plants, where efficiency depends on parameters like compression or pressure ratios. While the Carnot efficiency represents the theoretical maximum for given temperatures, practical cycles achieve 20-50% efficiency under typical operating conditions due to these simplifications.[14] The Otto cycle models spark-ignition engines, such as those in gasoline automobiles, featuring isentropic compression, constant-volume heat addition, isentropic expansion, and constant-volume heat rejection. Its thermal efficiency is given by \eta_\text{Otto} = 1 - \frac{1}{r^{\gamma-1}} where r is the compression ratio and \gamma is the specific heat ratio (approximately 1.4 for air). Higher compression ratios increase efficiency, but practical limits due to knocking constrain r to 8-12, yielding automotive Otto cycle efficiencies of 20-30%.[14][15] The Diesel cycle describes compression-ignition engines used in trucks and generators, with isentropic compression, constant-pressure heat addition, isentropic expansion, and constant-volume heat rejection. The efficiency formula is \eta_\text{Diesel} = 1 - \frac{1}{r^{\gamma-1}} \cdot \frac{\rho^\gamma - 1}{\gamma (\rho - 1)} where \rho is the cutoff ratio (volume ratio during heat addition). Diesel cycles allow higher compression ratios (14-25) without pre-ignition risk, achieving practical efficiencies of 30-40%.[16][17] In the Brayton cycle, which powers gas turbines for aircraft and power generation, the processes involve isentropic compression, constant-pressure heat addition, isentropic expansion, and constant-pressure heat rejection. The thermal efficiency is \eta_\text{Brayton} = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}} where r_p is the pressure ratio. Simple-cycle gas turbines operate at pressure ratios of 10-20, resulting in efficiencies of 30-40%, though combined cycles can exceed this by recovering exhaust heat.[18][15] The Rankine cycle underpins steam power plants, consisting of a pump, boiler (constant-pressure heat addition), turbine (isentropic expansion), and condenser (constant-pressure heat rejection). Its efficiency is calculated as \eta_\text{Rankine} = \frac{(h_3 - h_4) - (h_2 - h_1)}{h_3 - h_2} where h denotes specific enthalpy at the respective states (1: pump inlet, 2: boiler inlet, 3: turbine inlet, 4: condenser inlet). Practical Rankine cycles in coal or nuclear plants achieve 30-40% efficiency, limited by condenser temperatures and material constraints on boiler pressures.[19][15]| Cycle | Typical Application | Practical Efficiency Range |
|---|---|---|
| Otto | Spark-ignition engines | 20-30% |
| Diesel | Compression-ignition engines | 30-40% |
| Brayton | Gas turbines | 30-40% |
| Rankine | Steam power plants | 30-40% |