Coefficient of Performance (COP) in Thermal Machines

• Coefficient of Performance is a dimensionless metric that quantifies the efficiency of refrigeration and heat-pump systems by relating useful heat transfer to the work input.
• The analysis shows that COP is bounded by the Carnot limit and further limited by irreversibilities, finite-time operations, and internal dissipation effects.
• Extending across classical, quantum, and nanoscale systems, COP optimization integrates material properties, operational conditions, and design trade-offs to enhance overall performance.

The coefficient of performance (COP) is a dimensionless metric that quantifies the efficiency of cooling and heat-pump systems by relating useful heat transfer to the required work or electrical input. COP is central to the engineering and physical analysis of thermal machines and serves as an upper-bounding or target value in theoretical, experimental, and optimization studies of both classical and quantum refrigeration devices.

1. Definition and Physical Principles

The COP of a refrigeration or heat-pump system is defined as the ratio of useful heat transfer to the input work:

• For refrigeration (extraction of heat $Q_c$ from a cold reservoir using work $W$):

$\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$

where $Q_c$ is the heat absorbed from the cold bath and $W$ is the work input per cycle or per unit time (Ferrantelli et al., 2012).

• For heat-pump operation (delivery of $Q_h$ to a hot reservoir):

$\mathrm{COP_{hp}} = \frac{Q_h}{W}$

The theoretical maximum for COP is set by the reversible Carnot limit:

$\varepsilon_{\mathrm{C}} = \frac{T_c}{T_h - T_c}$

where $T_c$ and $T_h$ are the absolute temperatures of the cold and hot reservoirs, respectively (Mansoori, 2013).

COP can also be formulated at the system level to include all subsystem consumptions:

$W$0

with $W$1 the net heat removed, $W$2 the secondary-fluid pump consumption, and $W$3 the remainder (compressors, fans, etc.) (Ferrantelli et al., 2012).

2. COP in Classical and Nonlinear Irreversible Thermodynamics

In the context of irreversible processes, COP acquires bounds stricter than the Carnot value due to heat leaks, finite-time dissipation, and internal irreversibilities. For example, the minimally nonlinear irreversible thermodynamics (MNLIT) model gives (Izumida et al., 2014):

$W$4

with optimization over fluxes yielding explicit upper and lower bounds dependent on internal dissipation parameters. Maximum achievable COP is always strictly less than the Carnot limit except in reversible operation.

Finite-time models based on low-dissipation assumptions or endoreversible thermodynamics often optimize COP not directly, but under various figures of merit that balance efficiency and cooling power, such as $W$5- or $W$6-criteria (Hu et al., 2013, Nilavarasi et al., 2020). Universal bounds emerge under these criteria, including:

• Curzon–Ahlborn-type bound at maximum $W$7: $W$8
• Upper bound at maximum $W$9: $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$0
• For the $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$1-criterion: $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$2 (Nilavarasi et al., 2020, Hu et al., 2013)

3. COP in Quantum and Information-Theoretic Refrigerators

Quantum machines extend the concept of COP to situations where energy quantization, coherence, and information flow are fundamental (Fu et al., 16 Jul 2025, Singh et al., 2019, Yuan et al., 2014, Joseph et al., 2020). For quantum refrigerators (e.g., Otto or three-level absorption cycles), the COP typically preserves the same formal structure,

$\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$3

but with $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$4 and $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$5 computed as expectation values over quantum processes (e.g., master equations, population transfers).

Notable quantum results include:

• The Otto bound: $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$6 for frequency gaps $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$7, $\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$8 (Fu et al., 16 Jul 2025, Singh et al., 2019).
• In catalyzed quantum refrigerators, the COP can surpass the Otto limit and approach the Carnot bound by exploiting a catalyst space, with

$\mathrm{COP} \equiv \varepsilon = \frac{Q_c}{W}$9

where $Q_c$0 is the catalyst dimension (Fu et al., 16 Jul 2025).

Quantum stochastic models also analyze the distribution (not just the mean) of COP over finite times, revealing nontrivial time dependence and fluctuation bounds linked to entropy production and full counting statistics (Jiao et al., 2020, Okada et al., 2016).

4. COP Optimization and System-Level Multivariable Dependence

COP optimization entails maximizing device performance with respect to technological and thermophysical parameters under practical constraints. In large-scale systems, particularly with nontrivial working fluids (e.g., brine in ice rinks), COP depends on a wide range of variables:

$Q_c$1

where:

• $Q_c$2: brine flow rates
• $Q_c$3: temperature lifts
• $Q_c$4: brine density
• $Q_c$5: kinematic viscosity
• $Q_c$6: specific heat
• $Q_c$7, $Q_c$8, $Q_c$9, $W$0: header/pipe geometry
• $W$1: remaining electrical draw (Ferrantelli et al., 2012)

First-order optimality searches reveal dominant dependence on brine specific heat and density, with weak sensitivity to kinematic viscosity and flow. For ice rink applications (Ferrantelli et al., 2012):

• Optimal brine density: $W$2 g/cm³
• Ammonia brines yield COP enhancements of 9–18% versus ethylene glycol due to higher $W$3.

Practical design must account for technical constraints (material costs, bacterial control, slab thickness) and life-cycle economics, not merely the thermodynamic maxima.

5. COP in Thermoelectric, Nonconventional, and Nanoscale Systems

Thermoelectric devices (TEs), relying on the Peltier effect, possess device-specific COP expressions that incorporate electrical, geometric, and material properties:

$W$4

where $W$5 is the number of p-n pairs, $W$6 is the Seebeck coefficient, $W$7 electrical resistance, $W$8 thermal conductance, $W$9 junction temperatures (Saini et al., 2020).

Efficiency optimization requires minimizing leg thickness and fill factor, employing high $Q_h$0 materials, and maximizing convective coefficients, with COPs exceeding 4 possible under optimal current and flow conditions (Bahk et al., 2024). Real-time exergy-based controllers that minimize the dimensionless loss parameter $Q_h$1 have demonstrated further operational COP enhancements (Amiri-Margavi et al., 2024).

At the nanoscale, COP becomes a stochastic quantity that fluctuates on finite timescales, requiring a full counting statistics framework. Mean COP approaches the macroscopic value in the long-time limit, with corrections governed by the Skellam distribution at all times (Okada et al., 2016).

Nonconventional systems utilizing the Dufour effect realize COP as a function of cross-coefficient material properties (e.g., thermal-diffusion ratio), device geometry, and drive field, with molecular dynamics simulations confirming the linear-response analytic theory (Hoshina et al., 2014).

6. COP Bounds, Experimental Validation, and Practical Implications

Upper and lower bounds for COP, tighter than the Carnot value, can be derived from entropy production constraints and are typically given in terms of measurable properties of the working fluid and the cycle (Mansoori, 2013):

• Upper bound: $Q_h$2
• Lower bound: $Q_h$3

In large-scale and practical systems (e.g., refrigeration cycles, ice rink cooling), field measurements and advanced simulation (e.g., COMSOL Multiphysics) validate the optimized theoretical expressions, confirming that theoretical maxima are robust under real-world constraints (Ferrantelli et al., 2012).

Exceeding the classical COP bounds generally requires non-classical resources:

• Quantum catalysis, strong coupling, or reservoir engineering (e.g., squeezing) can elevate the optimizable COP above classical thresholds (Fu et al., 16 Jul 2025, Zhang, 2019).
• Information-processing refrigerators (Maxwell-demon-type) reach the Carnot COP in the quasi-static limit, but operational trade-offs lower performance at finite rates (Joseph et al., 2020).

7. Figures of Merit and Role of COP in Optimization

COP alone can be misleading as an optimization goal because the practical objective is often to maximize either the cooling rate or a trade-off function between efficiency and cooling power. Generalized figures of merit include:

• The $Q_h$4-criterion: $Q_h$5
• The $Q_h$6-criterion: $Q_h$7 These criteria yield universal COP bounds that encompass a wide class of systems, including power-law-dissipative, low-dissipation, minimally nonlinear irreversible, and quantum models (Hu et al., 2013, Nilavarasi et al., 2020, Singh et al., 2019).

In summary, COP remains the central metric for refrigeration and heat-pump efficiency across the classical, quantum, nanoscale, and information-thermodynamic domains. Its maximal attainable values and bounds are shaped by the fundamental thermodynamic laws and detailed trade-offs imposed by technological, material, and operational constraints. Systematic optimization of COP must integrate these multivariable dependencies, the structure of irreversibilities, and (in quantum or mesoscale regimes) the fluctuation properties and additional thermodynamic resources.