Power, Refrigeration, and Energy-Conversion Cycles

Introduction and context

Thermodynamic cycles are recurring sequences of processes that convert energy between heat, work, and internal energy. Power cycles produce net work, refrigeration cycles consume work to transfer heat from a low-temperature region to a high-temperature region, and heat-pump cycles deliver useful heat. Their analysis combines the first and second laws of thermodynamics with heat-transfer principles, fluid mechanics, material limits, and economic and environmental constraints.

The ideal cycle provides a benchmark; the real cycle includes pressure losses, finite-rate heat transfer, mechanical friction, leakage, non-equilibrium effects, and control limitations. At postgraduate level, cycle analysis is therefore not merely a calculation of efficiency. It is an exercise in modelling: selecting the appropriate system boundary, identifying assumptions, determining property states, quantifying irreversibility, and judging whether the result is technically and economically credible.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Formulate energy and entropy balances for cyclic devices.
2. Distinguish power cycles, refrigeration cycles, and heat-pump cycles.
3. Analyse the Rankine, Brayton, Otto, Diesel, and combined cycles.
4. Derive and apply coefficient of performance and efficiency relations.
5. Evaluate regeneration, reheating, intercooling, recuperation, and cogeneration.
6. Interpret real-cycle losses using second-law and exergy concepts.
7. Select and improve a cycle for a specified engineering application.

1. Thermodynamic basis of cycle analysis

For a closed system completing a cycle, the change in stored energy is zero. Consequently, the first law gives

where δQ is differential heat transfer to the system and δW is differential work done by the system. For a steady-flow control volume, neglecting kinetic and potential-energy changes when appropriate,

where q is heat transfer per unit mass, w is work per unit mass, and h is specific enthalpy.

The second law imposes a direction on real processes. For a cycle,

where T_b is the boundary temperature at which heat crosses the system boundary. Equality applies to a reversible cycle. The entropy balance for a steady component is

where s_gen ≥ 0 is entropy generation per unit mass. Entropy generation measures internal irreversibility, although it does not by itself identify the economic importance of a loss.

A useful idealisation is the internally reversible process, for which s_gen = 0. An isentropic process is both adiabatic and internally reversible. It is not correct to call every adiabatic process isentropic: friction, mixing, throttling, and shocks can generate entropy even when q = 0.

1.1 Performance measures

For a heat engine receiving heat Q_H from a high-temperature source and rejecting Q_L to a low-temperature sink,

where W_net = Q_H − Q_L. Thermal efficiency is always less than the Carnot value for the same source and sink temperatures:

with absolute temperatures in kelvin and a reversible heat engine operating between uniform reservoirs at T_H and T_L.

A refrigerator removes Q_L from a low-temperature space while consuming work W_net. Its coefficient of performance is

A heat pump delivers Q_H to a heated space:

For a reversible refrigerator or heat pump between two reservoirs,

and

These expressions show why refrigeration becomes increasingly difficult as the temperature lift increases.

2. Vapour power cycles

2.1 Rankine cycle

The ideal Rankine cycle is the basic model for steam power plants. It consists of:

1. Isentropic pumping of saturated liquid or subcooled liquid from condenser pressure to boiler pressure.
2. Constant-pressure heat addition in the boiler, often including superheating.
3. Isentropic expansion through the turbine.
4. Constant-pressure heat rejection and condensation.

For a steady-flow cycle, neglecting kinetic and potential terms,

and

The component relations are approximately

Here states 1–4 conventionally denote condenser exit, pump exit, boiler exit, and turbine exit. Enthalpies are usually obtained from reliable steam tables or a validated property formulation.

The pump work for an incompressible liquid can be estimated by

where v_f is the liquid specific volume and P is pressure. This approximation is useful because liquid specific volume changes little, but it becomes inadequate near critical conditions or for strongly compressible fluids.

For turbine expansion, the isentropic efficiency is

where h_4s is the exit enthalpy for isentropic expansion to the same exit pressure. Pump isentropic efficiency is

2.2 Rankine-cycle improvements

Superheating increases the average temperature of heat addition and reduces moisture at turbine exit. Reheating divides expansion into two turbine stages: steam expands partly, returns to the boiler for reheating, and expands again. Reheat commonly improves moisture control and may increase efficiency, but it requires additional equipment and pressure losses.

Regeneration extracts some turbine steam to heat feedwater before it enters the boiler. Open feedwater heaters mix extracted steam with feedwater; closed heaters transfer heat across a surface. Regeneration raises the feedwater temperature and reduces the external heat required in the boiler. Its effect on net work must be evaluated with mass and energy balances because extraction reduces turbine flow through downstream stages.

The practical design objective is not maximum turbine work alone. It is often minimum levelised cost of electricity subject to limits on boiler temperature, condenser pressure, turbine-exit moisture, material strength, emissions, and cooling-water availability.

3. Gas-turbine and internal-combustion cycles

3.1 Brayton cycle

The ideal simple Brayton cycle consists of isentropic compression, constant-pressure heat addition, isentropic expansion, and constant-pressure heat rejection. It models open gas-turbine systems.

for an ideal gas with constant specific-heat ratio γ and compressor pressure ratio r_p = P₂/P₁. The formula assumes ideal-gas behaviour, constant γ, isentropic compressor and turbine processes, and negligible pressure losses.

For an ideal gas, the isentropic relation is

The net specific work is

where c_p is constant-pressure specific heat. In real equipment, compressor and turbine efficiencies are

A recuperator transfers heat from turbine exhaust to compressed air. Intercooling reduces compressor work, while reheating increases turbine work. Intercooling alone may reduce thermal efficiency because it lowers the temperature entering the combustor; its value depends on pressure ratio and heat-recovery arrangement.

The pressure ratio that maximises ideal Brayton net work is not generally the pressure ratio that maximises efficiency. This distinction is important in gas-turbine optimisation. Real optimum pressure ratios are also constrained by turbine-inlet temperature, component efficiencies, pressure drops, cooling requirements, and off-design operation.

3.2 Otto and Diesel cycles

The air-standard Otto cycle represents spark-ignition engines. Its heat addition is modelled as constant volume. For constant γ,

where r = V_max/V_min is the compression ratio. The air-standard Diesel cycle models compression-ignition engines, with constant-pressure heat addition. Its efficiency is

where ρ = V_3/V_2 is the cut-off ratio. These formulas are ideal comparisons, not direct predictions of brake efficiency. Real engines experience variable specific heats, finite combustion time, heat transfer, pumping losses, friction, incomplete combustion, and emissions constraints.

A dual cycle uses both constant-volume and constant-pressure heat addition and can better approximate some practical combustion processes. Advanced engine analysis may require chemical equilibrium, turbulent combustion, spray dynamics, exhaust-gas recirculation, and after-treatment models.

4. Refrigeration and heat-pump cycles

4.1 Vapour-compression refrigeration

The basic vapour-compression cycle contains an evaporator, compressor, condenser, and expansion valve. The refrigerant evaporates at low pressure, is compressed, condenses at high pressure, and is throttled back to the evaporator pressure.

For the ideal cycle, the compressor is isentropic, the condenser and evaporator operate at approximately constant pressure, and throttling is isenthalpic:

The refrigeration effect per unit mass is

The compressor work is

Therefore,

Subcooling can increase refrigeration effect, while superheating may protect the compressor from liquid entry. Excessive superheat reduces volumetric capacity and may raise discharge temperature. The expansion valve is highly irreversible, but replacing it with an expander can recover work; the added cost and complexity often make this unattractive in small systems.

4.2 Absorption refrigeration

An absorption refrigerator replaces much of the mechanical compressor work with thermal energy. A refrigerant–absorbent pair, such as ammonia–water or water–lithium bromide, circulates through absorber, pump, generator, condenser, expansion device, and evaporator. It is attractive where waste heat, solar heat, or low-cost steam is available. Its COP is generally lower than that of vapour compression, but its electrical demand can be much lower.

The choice of working pair depends on pressure, crystallisation risk, toxicity, corrosion, operating temperature, and separation requirements. Water–lithium bromide systems are unsuitable for sub-zero evaporator temperatures because water is the refrigerant and may freeze.

4.3 Refrigerant selection and environmental constraints

A refrigerant must provide suitable thermodynamic performance, chemical stability, material compatibility, safety, and acceptable environmental impact. Important indicators include ozone-depletion potential, global-warming potential, flammability, toxicity, and atmospheric persistence. Natural refrigerants such as ammonia, carbon dioxide, hydrocarbons, water, and air may offer low environmental impact but introduce pressure, safety, or efficiency challenges.

Regulatory compliance must be assessed over the entire life cycle, including leakage, recovery, servicing, end-of-life destruction, and indirect emissions from electricity consumption.

5. Combined cycles, cogeneration, and exergy

A combined gas–steam cycle uses a Brayton topping cycle and a Rankine bottoming cycle. The gas-turbine exhaust supplies heat to a heat-recovery steam generator. This arrangement exploits the high-temperature exhaust that would otherwise be rejected and can achieve substantially higher efficiency than either cycle alone.

Cogeneration, or combined heat and power, supplies electricity and useful process heat. A simple electrical efficiency does not fully represent its benefit. Depending on the objective, one may report electrical efficiency, total-utilisation efficiency, or an exergy-based performance measure. Heat delivered at a high temperature has greater ability to perform work than low-temperature heat.

For a stream or energy transfer, the physical exergy relative to an environment at T_0 and P_0 represents its maximum useful work potential. Exergy destruction is related to entropy generation by the Gouy–Stodola relation:

where E_D is exergy destruction, T_0 is ambient absolute temperature, and S_gen is total entropy generation. Units are consistent: kelvin × joules per kelvin = joules. This relation applies when the reference environment is defined and the system is evaluated under the assumptions of classical thermodynamics.

Exergy analysis identifies where energy quality is destroyed. A condenser may reject a large quantity of energy but destroy less exergy per unit heat than combustion, throttling, or a large temperature-difference heat exchanger. Energy and exergy analyses should therefore be used together rather than treated as substitutes.

6. Worked example: ideal Rankine-cycle estimate

An ideal Rankine cycle receives steam at 8 MPa and 480 °C and condenses at 10 kPa. For illustration, take representative property values from a consistent steam-property table: h_1 = 191.8 kJ/kg for saturated liquid at 10 kPa, v_f = 0.00101 m³/kg, h_3 = 3348 kJ/kg, s_3 = 6.66 kJ/(kg·K). At 10 kPa, take h_f = 191.8 kJ/kg, h_fg = 2392 kJ/kg, s_f = 0.649 kJ/(kg·K), and s_fg = 7.50 kJ/(kg·K).

Pump work is

Thus h_2 ≈ 199.9 kJ/kg. At the turbine exit, s_4s = s_3. The quality is

and

The turbine work, heat input, and net work are approximately

Therefore,

The result is illustrative because property values have been rounded. The turbine exit quality of about 0.80 indicates approximately 20% liquid by mass, which may be excessive for turbine-blade durability. Reheating or a higher-quality exhaust condition should therefore be investigated rather than accepting efficiency alone as the design criterion.

7. Applied case: waste-heat recovery in a manufacturing plant

Background

An illustrative food-processing plant operates a natural-gas-fired thermal process. Its exhaust gas leaves at a high temperature, while the plant purchases electricity for refrigeration and low-pressure steam for cleaning.

Problem

The existing arrangement rejects exhaust heat and uses separate electric refrigeration and steam generation. Energy bills are high, and the plant must reduce carbon emissions without compromising hygiene or production reliability.

Analysis

A preliminary audit establishes exhaust temperature and flow rate, steam demand profiles, refrigeration load, electricity tariff, gas price, and operating hours. The engineering team then evaluates three alternatives: a recuperated gas-turbine combined heat-and-power system, a waste-heat boiler supplying process steam, and an absorption chiller driven by recovered heat.

The assessment uses first-law energy balances, second-law exergy accounting, pinch analysis for heat-exchanger temperature matching, and a life-cycle cost model. Safety review addresses combustible gas, pressure vessels, refrigerant exposure, combustion products, and emergency shutdown.

Approach and solution

The selected configuration uses a waste-heat boiler for the stable steam load and a small absorption chiller for part of the summer refrigeration demand. Electric refrigeration remains for peak and low-temperature loads. Heat exchangers are designed with allowable minimum temperature differences, fouling margins, condensate drainage, and cleanability requirements.

Outcome

The hybrid arrangement reduces purchased steam and peak electrical demand while preserving redundancy. Its success depends not only on nominal COP or heat-recovery efficiency, but also on seasonal load matching, maintenance access, controls, and the economic value of recovered heat.

Learning points

1. Cycle selection must follow the load profile rather than a single peak condition.
2. Waste heat is useful only when temperature, timing, and location match a demand.
3. Exergy analysis helps distinguish valuable high-temperature recovery from low-grade heat.
4. Environmental and safety performance must be evaluated with thermodynamic performance.

8. Limitations, design practice, and contemporary development

Ideal-cycle results should be treated as reference points. Reliable design requires validated fluid properties, component maps, uncertainty estimates, off-design simulation, and experimental or field data. Common modelling errors include mixing gauge and absolute pressure, using Celsius in Carnot relations, ignoring pressure losses, assuming constant specific heats over wide temperature ranges, and applying saturated-property values to superheated or compressed states.

Best practice is to state the system boundary, sign convention, reference state, property source, steady-state assumption, and neglected terms. A model should be subjected to sensitivity analysis: compressor or turbine efficiency, condenser temperature, ambient temperature, fouling, and pressure ratio often dominate performance.

Digital twins and machine-learning tools can predict degradation, detect refrigerant leakage, optimise dispatch, and estimate remaining useful life. They do not replace conservation laws or engineering judgement. Training data may omit abnormal conditions, and opaque models can conceal physically impossible predictions. Hybrid models combining thermodynamic constraints with data-driven estimation are generally more defensible for safety-critical systems.

Emerging directions include supercritical carbon-dioxide Brayton cycles, organic Rankine cycles for low-grade heat, transcritical CO₂ refrigeration, high-temperature heat pumps, thermal energy storage, hydrogen and synthetic fuels, and integrated energy systems. Their assessment requires simultaneous attention to efficiency, transient behaviour, materials, toxicity or flammability, supply chains, and life-cycle emissions.

Chapter Summary

Power cycles convert heat into net work; refrigeration and heat-pump cycles use work or high-temperature heat to move energy against a temperature gradient. The first law determines energy quantities, while the second law limits performance and identifies irreversibility. Rankine cycles dominate steam power generation, Brayton cycles model gas turbines, Otto and Diesel cycles provide ideal internal-combustion benchmarks, and vapour-compression cycles support most mechanical refrigeration. Regeneration, reheating, intercooling, recuperation, combined cycles, and cogeneration improve performance when their additional cost and complexity are justified. Exergy analysis complements energy analysis by quantifying the destruction of work potential. Practical cycle design must integrate properties, component efficiencies, heat transfer, controls, environmental regulation, safety, economics, and real operating profiles.

Key Terms

Thermodynamic cycle: A sequence of processes returning a system to its initial state.

Heat engine: A device that produces net work while receiving heat.

Refrigerator: A device that removes heat from a low-temperature region.

Heat pump: A device that delivers heat to a high-temperature region.

Thermal efficiency: Net work divided by heat supplied.

Coefficient of performance: Useful refrigeration or heating effect divided by work input.

Carnot cycle: A reversible benchmark cycle operating between two temperature limits.

Rankine cycle: A vapour power cycle using pumping, boiling, expansion, and condensation.

Brayton cycle: A gas-turbine cycle involving compression, combustion or heating, expansion, and exhaust.

Regeneration: Feedwater heating using extracted turbine steam.

Reheating: Heating the working fluid between stages of expansion.

Recuperation: Internal heat recovery from hot exhaust to compressed fluid.

Isentropic efficiency: Actual performance relative to ideal isentropic performance.

Throttling: Pressure reduction with approximately constant enthalpy.

Exergy: Maximum useful work relative to a specified environment.

Entropy generation: Measure of irreversibility in a process.

Cogeneration: Simultaneous production of power and useful heat.

Pressure ratio: Ratio of high-side to low-side pressure, or compressor exit to inlet pressure as defined.

Turbine-exit quality: Vapour mass fraction in a wet steam mixture.

Review Questions

Short Answer Questions

1. Why is an adiabatic process not necessarily isentropic?
2. Define thermal efficiency and coefficient of performance.
3. State the four ideal processes of the Rankine cycle.
4. What is the purpose of regeneration in a steam power plant?
5. Why does a throttling valve have constant enthalpy under the usual steady-flow model?

Long Answer Questions

1. Explain the ideal and actual Rankine cycles, including turbine and pump isentropic efficiencies.
2. Compare the ideal Brayton, Otto, and Diesel cycles and discuss the limitations of air-standard analysis.
3. Explain the operation and performance calculation of a vapour-compression refrigeration cycle.
4. Discuss how reheating, intercooling, recuperation, and regeneration affect cycle performance.
5. Explain the role of exergy analysis in evaluating combined heat-and-power systems.

Analytical / Application Questions

1. A proposed gas-turbine system has a high pressure ratio but poor compressor efficiency. Analyse why its net work and efficiency may both be unsatisfactory, and recommend additional data required for redesign.
2. A refrigeration plant experiences a rise in condensing temperature during summer. Predict the qualitative effects on compressor work, refrigeration capacity, and COP, and explain the thermodynamic basis.
3. For a manufacturing plant with seasonal steam and cooling loads, develop a method for deciding between waste-heat steam generation, absorption cooling, and continued electric refrigeration. Include energy, exergy, economic, safety, and environmental criteria.

Multiple Choice Questions with Answers

Question 1

For a heat engine operating between reservoirs at T_H and T_L, the maximum possible thermal efficiency is:

A. 1 − T_H/T_L

B. 1 − T_L/T_H

C. T_L/T_H

D. T_H/T_L

Correct Answer: B. 1 − T_L/T_H

Question 2

In an ideal throttling process, which property remains approximately constant?

A. Temperature

B. Entropy

C. Enthalpy

D. Pressure

Correct Answer: C. Enthalpy

Question 3

The coefficient of performance of a refrigerator is defined as:

A. W_net/Q_L

B. Q_H/W_net

C. Q_L/W_net

D. Q_H/Q_L

Correct Answer: C. Q_L/W_net

Question 4

The principal function of a condenser in a vapour-compression refrigerator is to:

A. Absorb heat from the refrigerated space

B. Reject heat and condense the refrigerant

C. Increase refrigerant entropy by throttling

D. Compress saturated liquid

Correct Answer: B. Reject heat and condense the refrigerant

Question 5

For an ideal gas Brayton cycle with constant γ, increasing pressure ratio generally changes the ideal efficiency according to:

A. η = 1 − 1/r_p^((γ−1)/γ)

B. η = 1 − r_p^((γ−1)/γ)

C. η = r_p/(γ−1)

D. η = γ/r_p

Correct Answer: A. η = 1 − 1/r_p^((γ−1)/γ)

Question 6

In a Rankine cycle, regeneration primarily:

A. Lowers the average temperature of heat addition

B. Heats feedwater using extracted turbine steam

C. Eliminates condenser heat rejection

D. Makes turbine expansion isothermal

Correct Answer: B. Heats feedwater using extracted turbine steam

Question 7

The Gouy–Stodola relation connects exergy destruction with:

A. Enthalpy and pressure

B. Ambient temperature and entropy generation

C. Specific volume and heat capacity

D. Turbine pressure ratio and moisture

Correct Answer: B. Ambient temperature and entropy generation

Question 8

A heat pump’s COP is related to the refrigerator COP for the same cycle by:

A. COP_HP = COP_R − 1

B. COP_HP = COP_R

C. COP_HP = COP_R + 1

D. COP_HP = 1/COP_R

Correct Answer: C. COP_HP = COP_R + 1

Question 9

A major reason for superheating steam before turbine expansion is to:

A. Increase condenser pressure

B. Reduce moisture at turbine exit

C. Eliminate boiler heat transfer

D. Make pump work dominant

Correct Answer: B. Reduce moisture at turbine exit

Question 10

A combined gas–steam cycle improves performance mainly by:

A. Rejecting all exhaust heat directly to the environment

B. Using gas-turbine exhaust heat to generate steam

C. Replacing the turbine with a throttling valve

D. Operating the condenser at atmospheric pressure only

Correct Answer: B. Using gas-turbine exhaust heat to generate steam