Thermodynamic cycle
A thermodynamic cycle is a sequence of thermodynamic processes in which the working substance undergoes a series of changes in state, returning to its initial thermodynamic state at the completion of the cycle, thereby allowing for the periodic conversion of heat energy into mechanical work or the reverse process.[1] This closed-loop nature distinguishes thermodynamic cycles from open processes and forms the basis for analyzing energy transformations in engineering systems.[2] Thermodynamic cycles are fundamental to the operation of heat engines, refrigerators, and heat pumps, where they facilitate efficient energy transfer while adhering to the laws of thermodynamics, particularly the first law (conservation of energy) and the second law (limiting efficiency due to entropy increase).[3] The efficiency of a cycle, often expressed as the ratio of net work output to heat input, is maximized in ideal reversible cycles like the Carnot cycle, which sets the theoretical upper limit for any heat engine operating between two temperatures.[4] Real-world cycles approximate these ideals but account for irreversibilities such as friction and heat losses, influencing design choices in practical applications.[5] Thermodynamic cycles are broadly classified into power cycles, which produce net work from heat (e.g., Rankine for steam turbines, Brayton for gas turbines, Otto and Diesel for internal combustion engines), and refrigeration cycles, which transfer heat from low to high temperatures using work input (e.g., vapor-compression for air conditioning).[6] These cycles underpin diverse technologies, from electric power generation and automotive propulsion to cooling systems and aerospace propulsion, driving advancements in energy efficiency and sustainability.[7]Fundamentals of Thermodynamic Cycles
Definition and Basic Principles
A thermodynamic cycle consists of a series of thermodynamic processes through which a system passes, returning it to its initial thermodynamic state, thereby forming a closed loop in the state space. This cyclical path ensures that the working substance experiences no net change in its properties at the completion of the cycle, allowing for repeatable operation. Prerequisite to understanding cycles are the concepts of thermodynamic systems and equilibrium states: a thermodynamic system is a defined region of matter under study, which may be closed (with fixed mass and no material exchange across boundaries, only energy transfer) or open (permitting mass flow in addition to energy exchange), while an equilibrium state is one in which the system's properties are uniform and do not change spontaneously over time, encompassing mechanical, thermal, and chemical equilibrium.[8][8][8] In engineering applications, thermodynamic cycles are fundamental to devices that convert thermal energy into mechanical work or vice versa, enabling continuous operation in systems such as engines, refrigerators, and power plants without requiring a net alteration in the system's internal state. By facilitating the absorption of heat from a high-temperature source, partial conversion to work, and rejection of the remainder to a low-temperature sink, cycles underpin efficient energy utilization in practical machinery. This repetitive process is essential for sustained performance, as it allows the working fluid to cycle indefinitely, optimizing resource use in thermal systems.[8][8] Thermodynamic cycles are typically visualized on diagrams such as pressure-volume (P-V) or temperature-entropy (T-S) plots, where the processes trace a closed path representing the sequence of state changes. On a P-V diagram, the area enclosed by the cycle corresponds to the net work output (or input) of the cycle, providing a graphical measure of energy conversion efficiency. Similarly, the T-S diagram highlights heat transfers, with the enclosed area relating to the net heat interactions, aiding in the analysis of cycle performance and irreversibilities. These representations leverage the fact that state functions like pressure, volume, temperature, and entropy fully describe equilibrium states along the path.[9][8][8]Heat, Work, and Energy Transfer
In thermodynamic cycles, heat (Q) and work (W) represent distinct modes of energy transfer across the system boundary. Heat is the transfer of energy due to a temperature difference between the system and its surroundings, occurring through random molecular motions without macroscopic displacement.[10] In contrast, work involves organized motion, typically arising from the expansion or compression of the system, such as in piston-cylinder arrangements where forces act over distances.[10] Both are path-dependent quantities, meaning their values depend on the specific sequence of states traversed during the cycle, unlike state functions such as internal energy. The first law of thermodynamics, which expresses the conservation of energy, applies directly to thermodynamic cycles. For a complete cycle, the internal energy change (ΔU) returns to zero because the system starts and ends in the same state, leading to the relation Q_net = W_net, where Q_net is the net heat transfer and W_net is the net work.[11] This equality highlights that the net energy input as heat must equal the net energy output as work over the cycle. The work for a process within the cycle is calculated as W = ∫ P dV, integrating the pressure over the volume change.[12] For reversible processes, the heat transfer is given by Q = ∫ T dS, where T is temperature and dS is the infinitesimal entropy change.[13] For heat engines operating on thermodynamic cycles, thermal efficiency (η) quantifies performance as η = W_net / Q_in, where Q_in is the heat absorbed from the high-temperature source. This metric indicates the fraction of input heat converted to useful work, with the remainder rejected as Q_out. Sign conventions in these analyses define positive Q as heat added to the system and positive W as work done by the system on the surroundings, consistent with the first law formulation ΔU = Q - W.[14] These conventions ensure consistent tracking of energy flows in cycle analyses.Common Thermodynamic Processes
Thermodynamic processes represent the fundamental changes in state that a system undergoes, serving as the building blocks for constructing thermodynamic cycles in engines, refrigerators, and other devices. These processes are characterized by constraints on variables such as pressure, volume, temperature, or energy transfer, and they are analyzed using the laws of thermodynamics, particularly the first law relating heat, work, and internal energy changes. For ideal gases, many processes admit simple analytical expressions derived from the ideal gas law and specific heat capacities.[8] An isobaric process occurs at constant pressure, where heat addition or removal leads to changes in volume and temperature while pressure remains fixed. In such a process, the work done by the system is given by W = P \Delta V, where P is the constant pressure and \Delta V is the change in volume; for an ideal gas, this equals P(V_2 - V_1). The heat transfer is Q = \Delta H = m c_p \Delta T, reflecting the enthalpy change, with c_p as the specific heat at constant pressure and m as mass.[8] Isobaric processes are common in open systems like combustion chambers, where expansion at fixed pressure converts heat to work.[8] An isothermal process maintains constant temperature, often achieved through heat exchange with a reservoir, resulting in no change in internal energy for an ideal gas (\Delta U = 0). For an ideal gas, the pressure-volume relation follows PV = constant, and the work done is W = nRT \ln(V_2 / V_1), where n is the number of moles, R is the gas constant, and T is the constant temperature; the heat absorbed equals the work, Q = W.[8] This process exemplifies reversible heat transfer in idealized cycles, balancing compression or expansion without temperature variation.[8] An adiabatic process involves no heat transfer (Q = 0), so changes in internal energy directly equal work done, \Delta U = -W. For a reversible adiabatic process with an ideal gas, the relation is PV^\gamma = constant, where \gamma = c_p / c_v is the ratio of specific heats; equivalently, TV^{\gamma-1} = constant or T P^{1-\gamma}/\gamma = constant. The work is W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} for reversible cases.[8] Adiabatic processes model insulated expansions or compressions in turbines and compressors.[8] An isochoric process, or constant-volume process, features fixed volume (V = constant), resulting in zero work (W = 0) since no displacement occurs. The heat transfer equals the internal energy change, Q = \Delta U = m c_v \Delta T, where c_v is the specific heat at constant volume.[8] Pressure and temperature vary proportionally via the ideal gas law, making this process relevant for heating in closed vessels without mechanical work.[8] A throttling process is an isenthalpic expansion (h_1 = h_2) through a restriction like a valve, with no heat transfer or work, leading to a pressure drop and potential temperature change. For steady flow, the first law simplifies to constant enthalpy, and the Joule-Thomson coefficient \mu = \left( \frac{\partial T}{\partial P} \right)_h determines the temperature shift, which is zero for ideal gases but nonzero for real gases due to intermolecular forces.[15] This irreversible process is essential in refrigeration cycles for expanding refrigerants.[16] Thermodynamic processes are classified as reversible or irreversible based on entropy generation. A reversible process is quasi-static, maintaining equilibrium at every stage with no friction or dissipation, resulting in zero entropy change for the system and surroundings (\Delta S = 0); it represents the theoretical maximum efficiency, as all changes can be undone without net effects.[8] In contrast, an irreversible process involves nonequilibrium phenomena like friction or unrestrained expansion, generating entropy (\Delta S > 0) and reducing available work, as seen in real devices where losses degrade performance.[8] The distinction implies that cycles composed of reversible processes achieve ideal limits, while irreversibilities in actual systems lower efficiency.[8]| Process | Constraint | Key Relation (Ideal Gas) | Work W | Heat Q |
|---|---|---|---|---|
| Isobaric | Constant P | P = const | P \Delta V | m c_p \Delta T |
| Isothermal | Constant T | PV = const | nRT \ln(V_2 / V_1) | = W (since \Delta U = 0) |
| Adiabatic | Q = 0 | PV^\gamma = const | \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} (rev.) | 0 |
| Isochoric | Constant V | V = const | 0 | m c_v \Delta T |
| Throttling | Constant h | h_1 = h_2 | 0 | 0 |
Classification of Thermodynamic Cycles
Power Cycles
Power cycles are thermodynamic cycles engineered to convert thermal energy into mechanical work, characterized by a net positive work output that exceeds any required input work, and they form the operational basis of heat engines. These cycles operate by absorbing heat from a high-temperature source and rejecting a portion to a low-temperature sink, with the difference enabling useful work production.[17][18] The general structure of power cycles encompasses sequential processes: heat addition at elevated temperatures to increase the internal energy of the working fluid, followed by expansion that extracts mechanical work; subsequent heat rejection dissipates excess thermal energy, and compression prepares the fluid for the next cycle by requiring work input. This framework ensures the system returns to its initial state after one complete cycle, allowing continuous operation. Applications of power cycles span diverse technologies, including internal combustion engines for vehicular propulsion, steam turbines in electricity generation, and gas turbines for both power production and aviation.[19][20][21] Efficiency in power cycles is constrained by fundamental thermodynamic principles, with the Carnot efficiency serving as the theoretical upper limit determined by the temperature ratio of the heat source and sink, beyond which no real engine can operate. Power cycles are further distinguished as open or closed based on working fluid management: closed cycles recirculate a sealed fluid, such as vapor in steam systems, maintaining constant mass; open cycles, like those in gas turbines, involve continuous intake and exhaust of the fluid, often approximated using air-standard assumptions for analysis.[22][23][24] Historically, power cycles evolved significantly in the 19th century amid the Industrial Revolution, with early steam engines driving mechanization; the Rankine cycle, a cornerstone for vapor power systems, was systematically described by Scottish engineer William John Macquorn Rankine in his 1859 Manual of the Steam Engine and Other Prime Movers, providing the thermodynamic foundation for efficient steam turbine operations.[25]Refrigeration and Heat Pump Cycles
Refrigeration and heat pump cycles are thermodynamic cycles that operate as reversed heat engines, using external work input to transfer heat from a low-temperature reservoir to a higher-temperature one, thereby achieving a cooling effect in the low-temperature space.[26] These cycles violate the natural direction of heat flow dictated by the second law of thermodynamics without work input, making them essential for cooling and heating applications.[27] The performance of a refrigeration cycle is quantified by its coefficient of performance (COP), defined as the ratio of heat absorbed from the cold reservoir (Q_c) to the work input (W): \text{COP}_R = \frac{Q_c}{W}.[26] The vapor-compression cycle is the most common type of refrigeration cycle, consisting of four main components: a compressor, a condenser, an expansion valve (or throttle), and an evaporator.[28] In the cycle, the refrigerant enters the compressor as a low-pressure vapor and is compressed to high pressure and temperature (process 1-2), increasing its ability to release heat.[6] The hot vapor then flows to the condenser, where it rejects heat to the surroundings and condenses into a high-pressure liquid (process 2-3).[28] The liquid refrigerant passes through the expansion valve, undergoing a throttling process that reduces its pressure and temperature (process 3-4), before entering the evaporator.[27] In the evaporator, the low-pressure liquid absorbs heat from the cooled space and evaporates back into vapor (process 4-1), completing the cycle.[28] Absorption cycles differ from vapor-compression by using heat as the primary energy input rather than mechanical work, employing a binary mixture of a refrigerant and an absorbent.[29] A typical pair is ammonia as the refrigerant and water as the absorbent, where the process involves absorption and desorption driven by temperature differences.[29] In the absorber, the refrigerant vapor dissolves into the absorbent, releasing heat and forming a strong solution; this solution is then heated in a generator to desorb the refrigerant vapor, which is condensed and evaporated similarly to vapor-compression but without a compressor.[30] The weak solution returns to the absorber after heat exchange, enabling heat-driven operation suitable for waste heat or solar energy sources.[29] Heat pump cycles utilize the same fundamental processes as refrigeration cycles but emphasize heat delivery to the high-temperature reservoir for heating purposes, such as space or water heating.[26] The COP for a heat pump is defined as the ratio of heat rejected to the hot reservoir (Q_h) to the work input: \text{COP}_{HP} = \frac{Q_h}{W}.[26] Since Q_h = Q_c + W, the heating COP is always greater than the refrigeration COP by unity for the same cycle.[6] These cycles find widespread applications, including household refrigerators and freezers for food preservation, air conditioners for building cooling, and industrial chilling systems for processes like food processing or chemical manufacturing.[31] Heat pumps are commonly used for residential and commercial space heating, leveraging ambient air, ground, or water sources.[32] The maximum possible COP for reversible refrigeration and heat pump cycles is given by the Carnot limits: for refrigeration, \text{COP}_{R,\text{Carnot}} = \frac{T_c}{T_h - T_c}, and for heating, \text{COP}_{HP,\text{Carnot}} = \frac{T_h}{T_h - T_c}, where T_c and T_h are the absolute temperatures of the cold and hot reservoirs, respectively.[26] Real vapor-compression systems achieve COP values of 3-5 for typical air conditioning conditions (e.g., T_c = 5^\circC, T_h = 35^\circC), which is approximately 30-55% of the Carnot COP due to irreversibilities like compressor inefficiencies and pressure drops.[27] Absorption cycles typically have lower COPs, around 0.5-0.7, reflecting their reliance on lower-grade heat inputs.[29]Ideal Versus Real Cycles
Assumptions in Ideal Cycles
In the analysis of ideal thermodynamic cycles, several simplifying assumptions are employed to facilitate theoretical calculations of performance metrics such as efficiency and net work output. These include the treatment of all processes as quasi-static and reversible, meaning they occur infinitely slowly with no dissipative effects like friction or throttling, allowing the system to remain in thermodynamic equilibrium at every stage. Additionally, no heat losses to the surroundings are assumed, so heat transfer occurs only between the working fluid and designated thermal reservoirs during specified processes. For ideal gas power cycles (e.g., air-standard models of Otto, Diesel, and Brayton cycles), the working fluid is modeled as an ideal gas obeying the equation of state PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is absolute temperature; specific heats at constant pressure (c_p) and constant volume (c_v) are taken as constant, independent of temperature, which simplifies enthalpy and internal energy changes to linear functions of temperature. In contrast, for vapor power cycles like the ideal Rankine cycle, real fluid properties from tables (e.g., steam tables) are used, accounting for phase changes and variable properties.[33][34] A particularly common framework is the air-standard cycle, where air is assumed to be the working fluid with constant composition and properties matching those of dry air at standard conditions (e.g., molecular weight of 29 g/mol, c_p = 1.005 kJ/kg·K, c_v = 0.718 kJ/kg·K). Under these assumptions, combustion is idealized as external heat addition at constant volume or pressure, without chemical reactions altering the fluid's composition, and the cycle is closed with the same mass of air recirculating. This model applies broadly to gas power cycles like Otto and Diesel, replacing complex fuel-air interactions with heat transfer from an infinite-capacity source.[35][33] These assumptions enable closed-form analytical solutions for cycle performance. For instance, the thermal efficiency of an ideal reversible cycle operating between high-temperature reservoir T_H and low-temperature reservoir T_L is derived as \eta = 1 - \frac{T_L}{T_H}, representing the maximum possible efficiency bounded by the second law of thermodynamics. Such derivations rely on the reversibility assumption to equate heat input and output via entropy balances, yielding expressions for net work as W_{net} = Q_H (1 - \frac{T_L}{T_H}), where Q_H is the heat absorbed at T_H. For air-standard cycles with constant specific heats, efficiency further simplifies to functions of compression ratio or pressure ratio, such as \eta = 1 - \frac{1}{r^{k-1}} for the Otto cycle, where r is the compression ratio and k = c_p / c_v.[33] While these idealizations provide valuable benchmarks for understanding fundamental limits, they inherently overestimate real-world performance by neglecting irreversibilities like fluid friction, heat leaks, and variable specific heats, which reduce actual efficiencies below theoretical values (e.g., real Rankine cycles achieve 30-42%, Brayton 30-40%, Otto 20-35%).[33]Deviations and Modeling in Real Systems
In real thermodynamic cycles, deviations from ideal models arise primarily from irreversibilities such as friction in moving components, heat transfer losses across finite temperature differences, behavior of non-ideal gases under high pressures or temperatures, pressure drops due to fluid flow resistances, and the inherent limitations of finite-time processes that prevent quasi-static conditions.[36][37] These factors introduce entropy generation, reducing the overall reversibility and performance compared to idealized assumptions of isentropic compression/expansion and isothermal heat transfer.[21] To model these real-world effects, engineers employ approaches like the mean effective pressure (MEP), which quantifies the average pressure exerted on the piston during a cycle to assess net work output relative to displacement volume, aiding comparisons across engine designs.[38] Polytropic processes, described by the relation PV^n = \constant, where n is the polytropic index (typically between 1 and \gamma for gases, accounting for heat transfer and friction), provide a more accurate representation of compression and expansion than purely isentropic paths.[6] Exergy analysis further evaluates losses by tracking the available work potential degraded by irreversibilities, highlighting inefficiencies in heat and work transfers.[39] These deviations significantly lower cycle efficiency compared to ideal predictions; for instance, in Rankine cycles, pump work, which is typically about 1% of turbine work (leading to a net efficiency reduction of around 0.3-1%), plus additional losses from friction and pressure drops, consumes a portion of the turbine output.[40] Similarly, in Otto cycles, various irreversibilities, including combustion inefficiencies from incomplete fuel burning and heat losses to cylinder walls, contribute to real thermal efficiencies of typically 20-35%, compared to ideal air-standard values of 52-60% for common compression ratios of 8-12, representing an overall reduction of 25-40%.[41] Correction methods mitigate these losses through regenerative cycles, which recover waste heat via internal heat exchangers to preheat working fluids, boosting efficiency by 5-20% in gas turbine applications.[42] Multi-staging, such as intercooling in compression or reheat in expansion, reduces work input and temperature extremes, improving overall performance in large-scale power systems.[43] Advanced simulations using computational fluid dynamics (CFD) model complex fluid behaviors, pressure gradients, and heat flows to optimize component designs and predict real efficiencies with high fidelity.[44] In general, these real-system deviations typically reduce power output by 20-60% and increase fuel consumption by 30-150% compared to ideal cycles, with the exact figures varying by cycle type (e.g., smaller for Rankine, larger for Otto), underscoring the need for precise modeling to enhance energy utilization in engineering applications.[42][45]Key Examples of Thermodynamic Cycles
Carnot Cycle
The Carnot cycle, proposed by French engineer Sadi Carnot in his 1824 treatise Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance (Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power), represents an idealized, reversible thermodynamic cycle that serves as the theoretical benchmark for the maximum efficiency of heat engines and refrigerators operating between two thermal reservoirs.[46] Carnot's analysis focused on the fundamental limits of converting heat into mechanical work, assuming no dissipative losses such as friction or heat transfer across finite temperature differences, thereby laying the groundwork for the second law of thermodynamics.[47] The cycle comprises four reversible processes executed with an ideal working fluid, typically modeled as a perfect gas: (1) isothermal expansion at the high temperature T_h, during which heat Q_h is absorbed from the hot reservoir while the gas expands and performs work; (2) adiabatic expansion, where the gas continues to expand without heat transfer, cooling to the low temperature T_c; (3) isothermal compression at T_c, rejecting heat Q_c to the cold reservoir as the gas is compressed; and (4) adiabatic compression, returning the gas to the initial state at T_h without heat exchange.[48] These processes ensure the cycle is fully reversible, with the net work output equal to the difference between heat absorbed and rejected. On the pressure-volume (P-V) diagram, the Carnot cycle forms a closed loop consisting of two hyperbolic isotherms—where pressure decreases as volume increases at constant temperature—and two steeper adiabatic curves connecting them, with the enclosed area quantifying the net work done by the cycle.[49] In contrast, the temperature-entropy (T-S) diagram depicts the cycle as a rectangle: horizontal lines represent the isothermal processes, with entropy increasing during heat addition at T_h and decreasing by the same amount during rejection at T_c, while vertical lines indicate the adiabatic processes where entropy remains constant; the rectangular area corresponds to the net work, interpreted as T_h \Delta S - T_c \Delta S, where \Delta S is the entropy change magnitude. The efficiency \eta of the Carnot cycle, defined as the ratio of net work output to heat input (\eta = W / Q_h), derives from the first and second laws of thermodynamics. By the first law, W = Q_h - |Q_c| for a cycle, so \eta = 1 - |Q_c| / Q_h. The second law, applied to the reversible cycle, requires zero net entropy change (\Delta S = 0), implying Q_h / T_h = |Q_c| / T_c and thus |Q_c| / Q_h = T_c / T_h, yielding \eta = 1 - \frac{T_c}{T_h}, where temperatures are absolute (in kelvin); this expression depends solely on the reservoir temperatures, independent of the working fluid.[49] For refrigerators operating in reverse, the Carnot cycle provides the maximum coefficient of performance, T_c / (T_h - T_c), establishing the reversible limit for heat extraction from a cold reservoir.[48] This efficiency formula underscores profound implications: the Carnot cycle achieves the highest possible conversion of heat to work between given temperatures, with all real cycles exhibiting lower efficiency due to inherent irreversibilities like friction and non-equilibrium heat transfer.[50] Carnot's theorem formalizes this boundary, stating that no heat engine operating between two specified temperatures can surpass the efficiency of a reversible (Carnot) engine between the same reservoirs, a principle that prohibits perpetual motion machines of the second kind and enforces the directional flow of energy dictated by the second law.[47]Otto Cycle
The Otto cycle serves as the ideal thermodynamic model for spark-ignition internal combustion engines, such as those used in gasoline-powered automobiles. It was developed based on the four-stroke engine patented by Nikolaus August Otto in 1876, which marked the first practical implementation of a cycle featuring intake, compression, power, and exhaust strokes.[51][52] This cycle approximates the behavior of the engine by simplifying combustion as heat addition and exhaust as heat rejection, focusing on the closed-system thermodynamics of the working fluid during the compression and expansion phases. The Otto cycle comprises four distinct processes in a closed system:- Isentropic compression (1-2): The piston compresses the air-fuel mixture adiabatically and reversibly, increasing pressure and temperature without heat transfer or friction.
- Constant-volume heat addition (2-3): Combustion occurs at fixed volume near top dead center, modeled as instantaneous heat input from an external source, raising temperature and pressure sharply.
- Isentropic expansion (3-4): The hot gases expand adiabatically and reversibly, performing work on the piston as volume increases.
- Constant-volume heat rejection (4-1): Residual heat is expelled at fixed volume, cooling the gases back to the initial state.
These processes repeat in a cyclic manner, converting thermal energy into mechanical work.[53][54]