Refrigeration Cycles
Refrigeration cycles use work or high-temperature heat to move energy from a colder region to a warmer one. Analysis is reliable only when the desired product, refrigerant state, component boundary, and property source are declared before a coefficient of performance is computed.
Learning objectives
- Distinguish refrigerator and heat-pump products and calculate their coefficients of performance.
- Identify vapor-compression states from absolute pressure, temperature, and phase.
- Apply steady-flow mass, energy, and entropy balances to compressors, valves, and heat exchangers.
- Differentiate isenthalpic throttling, isentropic compression, and actual compressor efficiency.
- Compare vapor, gas, cascade, and absorption architectures within their valid operating domains.
- Audit cycle closure, Carnot limits, entropy generation, exergy destruction, and uncertainty.
Refrigerators and Heat Pumps
Choose the product before the metric
A refrigerator's product is \(Q_L\), heat removed from the cold region; a heat pump's product is \(Q_H\), heat delivered to the warm region. For a cyclic device, \(W_{in}=Q_H-Q_L\), so \(\mathrm{COP}_{HP}=\mathrm{COP}_R+1\). COP may exceed one because it compares moved heat with work; it is not thermal efficiency.
Physical meaning
A COP of four means one unit of work moves four units of the selected heat product. It does not create energy: the condenser rejects the selected cold-side load plus compressor work.
The Reversed Carnot Cycle
For reservoirs at absolute temperatures, reversible limits are \(\mathrm{COP}_{R,rev}=T_L/(T_H-T_L)\) and \(\mathrm{COP}_{HP,rev}=T_H/(T_H-T_L)\). Use kelvins or rankines, never Celsius or Fahrenheit differences mixed with absolute values. Real heat exchangers require refrigerant temperatures beyond the reservoir temperatures, increasing lift and reducing achievable COP. Wet compression and a two-phase expander also make the vapor Carnot arrangement impractical.
The Ideal Vapor-Compression Refrigeration Cycle
State 1 is vapor entering an ideal isentropic compressor; 2 rejects heat in the condenser; 3 is liquid before the valve; 4 follows isenthalpic throttling and enters the evaporator. Each component is a steady open control volume. Use refrigerant tables or validated software, not an ideal-gas equation in the two-phase region.
Problem-solving strategy
- Mark the refrigerated space and decide whether the requested product is cooling or heating.
- Draw component control volumes, flow direction, state numbers, and a heat-in/work-in sign convention.
- Use absolute pressures and temperatures to classify each refrigerant state; quality is valid only inside the saturation dome.
- Read or interpolate properties from one authoritative refrigerant source, recording the fluid version and reference state.
- Close energy around each component and the cycle, then test COP against the reversible limit and entropy generation against zero.
Recognition strategy
A compressor, condenser, valve, and evaporator form the standard vapor-compression family; enthalpy differences expose all requested transfers.
Given information
- Supplied educational property values: \(h_1=398\), \(h_2=438\), and \(h_3=h_4=250\ \mathrm{kJ/kg}\).
- One-kilogram basis, steady operation, absolute state pressures, and temperatures interpreted on an absolute basis where required.
Model and interpretation
Each device is an open steady control volume and the assembly is cyclic. Heat absorbed by the refrigerant and compressor work input are positive. The refrigerant is a real pure substance represented by supplied table values, not an ideal gas; throttling is isenthalpic, compression is ideal isentropic, and kinetic and potential energy are neglected.
Verification
Units: Enthalpy differences produce \(\mathrm{kJ/kg}\), and dividing two specific energies makes COP dimensionless.
Signs and direction: Positive compressor input drives positive cold-side absorption and larger positive warm-side rejection.
Plausibility: The cycle closes because \(q_L+w_c=148+40=188=q_H\ \mathrm{kJ/kg}\). Mass and stored energy return to their initial values. The working-fluid entropy and exergy changes over a cycle are zero; ideal compression has zero generation, while throttling must generate nonnegative entropy.
Engineering interpretation
Each unit of compressor work moves 3.70 units of heat from the cold space, while 4.70 units reach the condenser. Traceable fluid data, pressure drops, superheat, subcooling, and motor losses are needed for equipment selection.
Actual Vapor-Compression Refrigeration Cycle
Actual compressor endpoint
For an adiabatic compressor, \(\eta_c=(h_{2s}-h_1)/(h_{2a}-h_1)\). Because \(\eta_c<1\), actual work and discharge enthalpy exceed isentropic values. Suction superheat protects against liquid ingestion, subcooling can increase refrigerating effect, and line pressure drops alter saturation temperatures. Neither superheat nor subcooling is automatically beneficial after all loads and controls are counted.
Recognition strategy
The isentropic outlet and efficiency define the actual compressor state; the cooling duty then sets mass flow before power is evaluated.
Given information
- Supplied values: \(h_1=410\), \(h_{2s}=445\), \(h_3=h_4=260\ \mathrm{kJ/kg}\), \(\eta_c=0.78\).
- Supplied entropies \(s_1=1.720\), \(s_{2a}=1.755\ \mathrm{kJ/(kg\!\cdot\!K)}\); \(\dot Q_L=60.0\ \mathrm{kW}\).
Model and interpretation
Use steady component control volumes and a refrigerant mass basis. Absolute pressures and temperatures define states in the supplied real-fluid source. Compressor work input and evaporator heat input are positive. Compression is actual adiabatic, the valve is isenthalpic, the model is approximate, and kinetic and potential energy are neglected.
Verification
Units: \(\mathrm{kg/s}\) times \(\mathrm{kJ/kg}\) gives \(\mathrm{kW}\), and mass flow times entropy difference gives \(\mathrm{kW/K}\).
Signs and direction: Actual discharge enthalpy exceeds the isentropic value, required power is positive, and entropy generation is positive.
Plausibility: Condenser rejection is \(0.400(454.87-260)=77.95\ \mathrm{kW}\), equal to \(60.0+17.95\ \mathrm{kW}\). Cycle mass, energy storage, entropy, and exergy state changes close; at \(T_0=298\ \mathrm{K}\), compressor exergy destruction is \(4.17\ \mathrm{kW}\), nonnegative as required.
Engineering interpretation
Compressor irreversibility raises discharge temperature and electrical demand. The stated efficiency and properties are supplied study inputs, not vendor guarantees; off-design maps, motor efficiency, oil management, and uncertainty require review.
Second-Law Analysis of Refrigeration Cycles
Minimum work is set by reservoir temperatures, while actual excess work is associated with exergy destruction. For a declared environment, \(\dot X_d=T_0\dot S_{gen}\). Component entropy balances expose compression, throttling, mixing, and finite-temperature heat-transfer losses. A COP below Carnot is necessary but not sufficient: every component must also satisfy nonnegative entropy generation.
Selecting the Right Refrigerant
Selection balances saturation pressures, critical temperature, volumetric capacity, transport properties, material and lubricant compatibility, toxicity, flammability, ozone depletion, global-warming impact, availability, and regulation. A fluid name alone is insufficient: mixtures may glide in temperature and require composition control. Use current safety codes and authoritative property databases; never infer release properties from a classroom table.
Heat Pump Systems
Air, water, and ground sources impose different temperatures and fouling risks. Defrost and auxiliary heat can dominate seasonal performance. Capacity and COP must be evaluated together at design conditions, not extrapolated from one rating point.
Recognition strategy
Two reservoir temperatures establish a reversible heat-pump COP; a supplied fraction of that limit converts the benchmark to screening performance.
Given information
- Heating product \(\dot Q_H=120\ \mathrm{kW}\); \(T_H=318\ \mathrm{K}\), \(T_L=273\ \mathrm{K}\), all absolute.
- Supplied second-law performance factor \(\mathrm{COP}_{actual}/\mathrm{COP}_{rev}=0.46\); environment \(T_0=293\ \mathrm{K}\), \(P_0=101.3\ \mathrm{kPa}\) absolute.
Model and interpretation
The boundary encloses a steady cyclic heat pump; heating delivered and work input are positive. Reservoirs are isothermal, property detail is embedded in the supplied approximate factor, and refrigerant mass returns each cycle. Kinetic and potential energy are neglected; no ideal-gas or phase-quality calculation is attempted.
Verification
Units: COP is dimensionless; kilowatts divided by COP remain kilowatts, and \(\mathrm{kW}/\mathrm{K}\) follows from exergy divided by kelvins.
Signs and direction: Actual input exceeds reversible input, making destroyed exergy and entropy generation positive rather than impossible negative values.
Plausibility: Cold-side absorption is \(120-36.91=83.09\ \mathrm{kW}\), so the cycle energy balance closes. The independent reservoir audit gives \(120/318-83.09/273=0.0730\ \mathrm{kW/K}\ge0\), and \(293(0.0730)=21.4\ \mathrm{kW}\) confirms nonnegative exergy destruction. Actual COP remains below 7.07.
Engineering interpretation
The exergy loss follows the entropy balance evaluated with the actual hot- and cold-side heat rates. Subtracting reversible from actual work would be inconsistent here because those cases have different cold-side loads. Certified maps, load uncertainty, auxiliary power, refrigerant safety, and code review remain necessary.
Innovative Vapor-Compression Systems
Match architecture to lift
Multistage compression with intercooling reduces work; flash economizers separate phases; cascade systems couple distinct refrigerant loops through an intermediate heat exchanger. The cascade exchanger imposes equal and opposite heat rates only after mass bases and heat loss are defined. Intermediate temperature is iterated to balance compressor ratios, approach temperatures, fluid limits, and cost.
Gas Refrigeration Cycles
A reversed Brayton loop keeps the working fluid gaseous: compression, warm-side cooling, expansion, and cold-side heating. For an ideal gas, authoritative variable-temperature \(h\) data or justified \(c_p\) approximations replace saturation tables; \(c_p\) belongs with enthalpy changes and \(c_v\) with internal-energy changes. Turbine output offsets compressor input. This branch is distinct from vapor compression and involves neither quality nor throttling unless hardware explicitly includes a valve. Choking and shocks require separate compressible-flow analysis, not cycle formulas.
Absorption Refrigeration Systems
Absorption systems replace most compressor work with a generator, absorber, solution pump, and internal heat exchange. COP commonly compares cooling with generator heat, so it cannot be compared directly with an electrically driven COP without fuel and exergy boundaries. Pair equilibrium, crystallization risk, corrosion, pressure, and safety constrain operation.
Chapter recap
- Choose cooling or heating product, then define steady component boundaries and signs.
- Identify phases at absolute states and use traceable refrigerant properties; quality belongs only in a saturated mixture.
- Distinguish isentropic compression from actual efficiency and isenthalpic throttling from reversible expansion.
- Before moving on, be able to close mass and energy ledgers, enforce Carnot and entropy limits, and state the evidence still needed for a professional selection.