Thermodynamic and Heat-Transfer Foundations

Introduction and context

Renewable-energy devices convert naturally replenished resources—solar radiation, wind, flowing water, biomass, geothermal heat, and ocean energy—into useful heat, work, electricity, or fuels. Their design is governed by the same physical principles as conventional energy systems, but renewable resources are often intermittent, diffuse, variable in temperature, and geographically constrained. A solar-thermal collector, wind turbine, heat pump, electrolyser, and thermal-storage unit can therefore be understood only by combining thermodynamics, heat transfer, and fluid mechanics.

Thermodynamics establishes what energy conversion is possible and how much energy is conserved. The second law identifies the direction of real processes and the quality of energy. Heat transfer determines rates of energy movement, while fluid mechanics describes the motion and pressure losses of working fluids. Together these disciplines support energy balances, exergy balances, efficiency calculations, equipment sizing, and identification of avoidable irreversibilities.

Learning objectives

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

1. Apply the first and second laws to closed systems and control volumes.
2. Calculate entropy generation and explain its physical meaning.
3. Distinguish energy quantity from energy quality using exergy.
4. Select appropriate heat-transfer models for conduction, convection, and radiation.
5. Use continuity, momentum, and Bernoulli equations in renewable-energy equipment.
6. Construct energy and exergy balances and evaluate conversion efficiency.
7. Identify major irreversibilities in renewable systems.
8. Interpret these principles in the preliminary design of a renewable-energy plant.

1. System descriptions and properties

A system is the matter or region selected for analysis. Everything outside it is the surroundings. A closed system (control mass) permits energy transfer but not mass transfer across its boundary. An open system (control volume) permits both mass and energy transfer. A system may be stationary or moving, and its boundary may be fixed or deformable.

A property describes the state of a system. Intensive properties, such as temperature T, pressure p, and density ρ, do not depend on system size. Extensive properties, such as mass m, volume V, and total energy E, do depend on size. Specific properties are extensive properties per unit mass, for example v = V/m and u = U/m.

A process is a change of state; a cycle returns a system to its initial state. The working-fluid state is commonly specified by two independent intensive properties when phase and composition are known. In renewable systems, useful properties include enthalpy h, internal energy u, entropy s, and specific heat capacity c_p or c_v.

Enthalpy is defined by

where H is enthalpy, U internal energy, p pressure, and V volume. For steady-flow devices, enthalpy conveniently incorporates flow work. The specific enthalpy is h = u + pv, with units of J kg⁻¹.

2. The first law: conservation of energy

The first law states that energy is conserved. For a closed system,

where ∆E_sys is the change in total system energy, Q is heat transferred to the system, and W is work done by the system. With kinetic and potential contributions,

and ∆KE = m(V₂² − V₁²)/2, ∆PE = mg(z₂ − z₁). The sign convention above treats heat input and work output as positive. Other conventions are valid if used consistently.

For a control volume, the general energy balance is

Here Q̇ and Ẇ are heat-transfer and work rates, ṁ is mass-flow rate, V is fluid speed, z is elevation, and g is gravitational acceleration. The equation assumes one-dimensional average flow at each inlet and outlet and neglects chemical and nuclear energy unless explicitly included.

For steady operation with one inlet and one outlet,

If changes in kinetic and potential energy are negligible, this becomes Q̇ − Ẇ = ṁ(h₂ − h₁). This simplified equation is useful for turbines, compressors, pumps, heat exchangers, and many solar-thermal components.

Energy efficiency

The first-law or energy efficiency compares useful energy output with energy supplied:

For a wind turbine, for example, η_E = P_elec/P_wind, including generator losses if electrical output is used. Efficiency depends on the defined system boundary. A collector efficiency may use incident solar radiation as input, whereas a complete solar-electric plant may include tracking, inverter, cabling, and storage losses.

3. The second law, entropy, and irreversibility

The first law does not distinguish between energy forms or indicate process direction. The second law does. Heat naturally flows from higher to lower temperature, friction converts organized mechanical energy into internal energy, and no heat engine can convert all supplied heat into work.

Entropy is a property measuring energy dispersal and the restrictions imposed by the second law. For a closed system,

where S is entropy, T_b is the boundary temperature at which heat crosses, and equality applies to a reversible process. The entropy-generation form is

where S_gen ≥ 0 is entropy generated by irreversibility. Units are J K⁻¹, or J kg⁻¹ K⁻¹ for specific entropy.

For a steady control volume,

where T_j is the boundary temperature associated with heat-transfer rate Q̇_j. Thus,

A reversible process has S_gen = 0. Real processes have positive entropy generation due to finite temperature differences, mixing, friction, pressure drops, electrical resistance, turbulence, throttling, and chemical reaction.

The Gouy–Stodola theorem relates lost work to entropy generation:

where I is irreversibility or exergy destruction (W) and T₀ is the ambient reference temperature (K). This relation applies when the surroundings are in a stable reference state and entropy generation is measured consistently.

4. Exergy: the quality of energy

Energy is conserved, but exergy is consumed. Exergy is the maximum theoretical useful work obtainable as a system comes into equilibrium with a specified environment. It depends on the environmental reference state, normally described by T₀ and p₀.

For a flowing stream, neglecting chemical, kinetic, and potential exergy, the physical specific flow exergy is

where ψ is specific flow exergy, and subscript 0 denotes the dead-state environmental condition. If kinetic and potential effects matter,

Chemical exergy must be added for fuels, biomass, and reacting mixtures. For solar radiation, exergy is not simply equal to energy because radiation arrives from an effective source temperature much higher than ambient. A practical solar-exergy analysis must specify the radiation model and environment.

The steady-flow exergy balance is

where İ = T₀Ṡ_gen. The exergy efficiency is

Energy efficiency can be high while exergy efficiency is poor. For example, a heat exchanger may transfer nearly all supplied heat energy, but heat delivered at only slightly above ambient temperature has limited work potential.

5. Heat transfer

Heat transfer occurs because of a temperature difference. The three modes are conduction, convection, and radiation; they may occur simultaneously.

Conduction

Fourier’s law for one-dimensional plane-wall conduction is

where k is thermal conductivity (W m⁻¹ K⁻¹), A area (m²), and dT/dx temperature gradient (K m⁻¹). For a wall of thickness L with constant k and steady one-dimensional conduction,

This assumes no internal heat generation and negligible edge effects. Thermal resistance is R_cond = L/(kA), so Q̇ = ∆T/R_cond. Composite walls use resistances in series.

Convection

Newton’s cooling law is

where h is the convection coefficient (W m⁻² K⁻¹), T_s surface temperature, and T_∞ bulk-fluid temperature. The coefficient is not a material constant; it depends on geometry, velocity, fluid properties, and flow regime. Correlations commonly use dimensionless groups:

where Nu is the Nusselt number, Re Reynolds number, Pr Prandtl number, L_c characteristic length, ρ density, μ dynamic viscosity, and k_f fluid conductivity. A correlation is applicable only within its stated geometry and Reynolds- and Prandtl-number ranges.

Radiation

Thermal radiation requires no material medium. An ideal blackbody emits

where σ = 5.670 × 10⁻⁸ W m⁻² K⁻⁴ is the Stefan–Boltzmann constant. For a small gray surface exchanging radiation with large surroundings,

where ε is emissivity. View factors and surface properties are required for general enclosure problems.

Thermal networks and heat exchangers

For conduction and convection in series,

A simple exchanger energy balance is

provided there is negligible heat loss to the surroundings and no phase change. The log-mean-temperature-difference method uses

where U is the overall heat-transfer coefficient and ∆T_lm = (∆T₁ − ∆T₂)/ln(∆T₁/∆T₂). It applies to steady operation with approximately constant properties and a known flow arrangement.

6. Fluid mechanics for renewable systems

Mass conservation for steady incompressible flow is

For variable-density flow, the general continuity equation must be used; the incompressible form is not appropriate for strongly compressible gases.

The mechanical-energy relation for steady flow between two points is commonly written

Here h_p is pump head, h_t turbine head, h_L losses, and α kinetic-energy correction factor. For uniform turbulent flow, α is often approximated as 1. The loss term represents friction and fittings and is an important source of exergy destruction.

For a hydro turbine, ideal hydraulic power is

where Q is volumetric flow rate and H is available hydraulic head. Actual electrical power is approximately

where η_t and η_g are turbine and generator efficiencies. Wind power available in an undisturbed stream is

with swept area A and wind speed V. The fraction extracted by the rotor is the power coefficient C_p, so P_t = C_p(1/2ρAV³). The Betz limit gives C_p ≤ 16/27 for an ideal actuator-disk rotor under its assumptions; real machines perform below this limit.

7. Energy and exergy analysis of renewable components

A disciplined analysis follows these steps:

1. Define the system boundary and the time basis.
2. State steady or transient assumptions.
3. Identify all mass, heat, work, and radiation interactions.
4. Write the first-law balance before inserting numbers.
5. Write the entropy or exergy balance.
6. Check units, signs, and conservation.
7. Rank losses by exergy destruction and assess improvement options.

Worked example: solar-thermal receiver

A receiver absorbs 100 kW of solar energy. It delivers 72 kW of useful heat to a fluid; 18 kW is lost by convection and radiation, and the remaining loss is due to piping and measurement uncertainty. The fluid enters at 80 °C and leaves at 140 °C. The surroundings are at 25 °C. Determine the energy efficiency and the exergy associated with the useful heat if the heat is supplied reversibly at the mean fluid temperature.

Energy efficiency is

The mean delivery temperature is T_m = (353.15 + 413.15)/2 = 383.15 K. The exergy factor for heat supplied reversibly at constant temperature T_m is 1 − T₀/T_m, with T₀ = 298.15 K. Thus useful-heat exergy is

The exergy efficiency relative to incident solar energy cannot be determined rigorously from this information alone because the exergy of solar radiation is not equal to 100 kW. If, only as a simplified engineering comparison, incident radiation is treated as the input work potential, the ratio is 16.0%. This distinction illustrates why energy and exergy efficiencies must not be confused. The largest improvement opportunities are reducing receiver heat loss, lowering thermal resistance, and matching delivery temperature to the application.

8. Applied case: geothermal binary power plant

Background

This illustrative case concerns a binary geothermal plant using a moderate-temperature geothermal brine. The brine transfers heat through a heat exchanger to an organic working fluid in a closed Rankine cycle. The organic fluid expands through a turbine, is condensed, and is pumped back to the vaporizer.

Problem

The plant must generate electricity while minimizing geothermal-fluid consumption and preventing environmental damage. The resource temperature is insufficient for efficient direct steam generation, and reinjection is required to maintain reservoir pressure and control dissolved-mineral discharge.

Analysis

The plant boundary includes the heat exchanger, turbine, condenser, pump, generator, and auxiliary equipment. The first-law balance tracks geothermal heat input, turbine work, pump work, rejected condenser heat, and net electrical output. The second-law analysis identifies the heat exchanger as a likely major source of exergy destruction because heat is transferred across a finite temperature difference. Additional destruction occurs in the turbine through friction and non-ideal expansion, in the condenser through heat rejection near ambient temperature, and in the pump through hydraulic losses.

A component exergy balance is applied to each device. For the heat exchanger, reducing the temperature difference increases exergy efficiency but also requires greater surface area and cost. For the turbine, improved blade design and careful control of moisture or organic-fluid quality reduce internal losses. For the condenser, lower cooling-water temperature improves cycle performance but may increase water consumption.

Approach and outcome

A practical design uses pinch analysis to select heat-exchanger temperature approaches, variable-speed pumps to accommodate resource variation, and reinjection monitoring to protect the reservoir. Energy analysis determines net power; exergy analysis ranks improvements. A design that raises gross turbine output but greatly increases pumping power may reduce net power and should not be accepted without a complete plant balance.

Learning points

The case demonstrates that renewable status does not eliminate thermodynamic losses. Resource temperature, heat-exchanger matching, auxiliary loads, environmental reference conditions, and reinjection strategy all influence sustainability. Exergy analysis helps distinguish unavoidable resource limitations from avoidable engineering losses.

9. Limitations, sustainability, and professional practice

Thermodynamic models are idealizations. Constant-property assumptions may fail over large temperature ranges; steady-state models may hide storage dynamics; one-dimensional heat-transfer models may neglect edge effects; and empirical fluid correlations have limited validity. Engineers should state assumptions, report uncertainty, and perform sensitivity analysis.

Renewable projects also require life-cycle and environmental assessment. A heat pump with high coefficient of performance may still have climate impacts from refrigerant leakage or carbon-intensive electricity. Biomass combustion requires sustainable feedstock accounting, air-pollution control, and attention to land-use change. Hydropower design must address fish passage, sediment transport, downstream flow, and community impacts. Geothermal and mineral extraction require groundwater and induced-seismicity monitoring.

Digital sensors and artificial intelligence can estimate fouling, predict wind or solar availability, and optimize dispatch. However, data-driven models do not replace conservation laws. Sensor calibration, cybersecurity, explainability, physical constraints, and safe fallback control are essential. Professional practice requires compliance with applicable pressure-vessel, electrical, building, environmental, occupational-safety, and grid-interconnection standards.

Chapter Summary

The first law provides energy conservation for closed systems and control volumes. Enthalpy, kinetic energy, and potential energy are particularly useful in steady-flow analysis. The second law establishes process direction and requires nonnegative entropy generation. Exergy measures the maximum useful work relative to an environment; exergy destruction equals T₀S_gen and reveals where improvement is most valuable.

Conduction follows Fourier’s law, convection is represented by Newton’s law of cooling, and radiation follows the Stefan–Boltzmann relation. Continuity and mechanical-energy equations describe flow through pumps, turbines, pipes, hydro systems, and wind machines. Energy efficiency measures quantity conversion, whereas exergy efficiency measures quality conversion. Renewable systems must therefore be designed using both balances, with explicit attention to irreversibility, auxiliary power, resource constraints, environmental impacts, and uncertainty.

Key Terms

System: Matter or region selected for thermodynamic analysis.

Control volume: Open-system boundary through which mass and energy may pass.

Enthalpy: Property defined by H = U + pV.

First law: Principle that energy is conserved.

Second law: Principle governing process direction and entropy generation.

Entropy: Thermodynamic property used to quantify energy dispersal and irreversibility.

Entropy generation: Entropy produced by an irreversible process.

Exergy: Maximum theoretical useful work relative to a reference environment.

Dead state: Condition in which a system is in equilibrium with its environment.

Irreversibility: Lost work caused by entropy generation.

Conduction: Heat transfer through a material due to a temperature gradient.

Convection: Heat transfer between a surface and a moving fluid.

Radiation: Heat transfer by electromagnetic emission.

Thermal resistance: Opposition to heat transfer in a thermal network.

Nusselt number: Dimensionless ratio relating convective to conductive heat transfer.

Reynolds number: Dimensionless indicator of flow regime.

Entropy efficiency: Not a standard substitute for exergy efficiency; exergy efficiency compares useful and supplied exergy.

Power coefficient: Ratio of power extracted by a wind rotor to incident wind power.

Irreversibility: Exergy destroyed, equal to T₀S_gen.

Pinch analysis: Method for matching hot and cold streams to reduce heat-transfer driving-force losses.

Review Questions

Short Answer Questions

1. Distinguish a closed system from a control volume.
2. Why can energy efficiency be high while exergy efficiency is low?
3. State the physical meaning of entropy generation.
4. List the three fundamental modes of heat transfer.
5. What assumptions are required for using P = ρgQH for hydraulic power?

Long Answer Questions

1. Derive the steady-flow energy equation from the general control-volume balance and explain the meaning of each term.
2. Explain the Gouy–Stodola theorem and discuss how it guides engineering improvement.
3. Compare conduction, convection, and radiation in a solar-thermal collector.
4. Develop an energy and exergy analysis procedure for a heat pump.
5. Explain the roles of heat-exchanger temperature differences, pressure losses, and auxiliary power in a geothermal plant.

Analytical / Application Questions

1. A hydro site has Q = 12 m³ s⁻¹, H = 18 m, turbine efficiency 0.91, and generator efficiency 0.96. Calculate net electrical power, neglecting pipe losses.
2. A wall has area 20 m², thickness 0.10 m, conductivity 0.8 W m⁻¹ K⁻¹, and face temperatures 90 °C and 30 °C. Calculate the steady conductive heat-transfer rate and identify the assumptions used.
3. A renewable plant has a measured entropy-generation rate of 4 kW K⁻¹ at an ambient temperature of 298 K. Calculate exergy destruction and explain which additional measurements would be needed to locate the dominant component losses.

Multiple Choice Questions with Answers

Question 1

For a steady, one-inlet one-outlet control volume with negligible kinetic and potential-energy changes, which relation is correct?

A. Q̇ − Ẇ = ṁ(h₂ − h₁)

B. Q̇ + Ẇ = ṁ(h₂ + h₁)

C. Q̇ − Ẇ = ṁ(h₁ − h₂)²

D. Q̇ = Ẇ + ṁ(h₁ + h₂)

Correct Answer: A. Q̇ − Ẇ = ṁ(h₂ − h₁)

Question 2

The entropy generation of a real thermodynamic process is:

A. Always negative

B. Always zero

C. Nonnegative

D. Equal to the heat-transfer rate

Correct Answer: C. Nonnegative

Question 3

The Gouy–Stodola relation for exergy destruction is:

A. I = S_gen/T₀

B. I = T₀S_gen

C. I = T₀/S_gen

D. I = Q̇T₀

Correct Answer: B. I = T₀S_gen

Question 4

Which heat-transfer mode can occur through a vacuum?

A. Conduction only

B. Convection only

C. Radiation

D. Conduction and convection only

Correct Answer: C. Radiation

Question 5

A hydro turbine receives 10 m³ s⁻¹ of water through a head of 20 m. Taking ρ = 1000 kg m⁻³ and g = 9.81 m s⁻², the ideal hydraulic power is approximately:

A. 19.6 kW

B. 196 kW

C. 1.96 MW

D. 19.6 MW

Correct Answer: C. 1.96 MW

Question 6

The Reynolds number primarily helps characterize:

A. Radiative emissivity

B. Flow regime

C. Thermal resistance of a wall

D. Dead-state pressure

Correct Answer: B. Flow regime

Question 7

For a wind turbine, the power available in an undisturbed wind varies with wind speed approximately as:

A. V

B. V²

C. V³

D. 1/V

Correct Answer: C. V³

Question 8

Which statement about exergy is correct?

A. It is always conserved in real processes

B. It depends on the reference environment

C. It is identical to internal energy

D. It can be negative for every useful product

Correct Answer: B. It depends on the reference environment

Question 9

For a plane wall with constant conductivity and no internal heat generation, increasing wall thickness while holding temperatures fixed generally:

A. Increases heat transfer

B. Decreases heat transfer

C. Leaves heat transfer unchanged

D. Reverses heat-flow direction

Correct Answer: B. Decreases heat transfer

Question 10

A major source of exergy destruction in a heat exchanger is:

A. Heat transfer across a finite temperature difference

B. Mass conservation

C. The definition of enthalpy

D. A reversible pressure measurement

Correct Answer: A. Heat transfer across a finite temperature difference