Chapter 56: Energy Systems

56.1 What an Energy System Is

An energy system is the complete chain of processes that extracts energy from nature, converts it into useful forms, stores it where needed, transmits it to where it is needed, and delivers it for final use. Energy systems involve primary energy sources (the forms in which energy exists in nature), energy carriers (intermediate forms used to transport or store energy), and end-use technologies (devices that deliver the services users actually want: heat, light, mechanical work, communication). The key physical constraint on all energy systems is the laws of thermodynamics. The first law (energy conservation) requires that energy input equals useful output plus all losses. The second law requires that some energy is always degraded to low-grade heat in any real conversion process; the Carnot efficiency sets the upper bound for thermal machines. No energy system delivers 100% useful output; real systems have cascading losses at every conversion step. Energy is measured in joules (J), but practical energy systems use larger units. 1 kWh

(kilowatt-hour) = 3.6 MJ. 1 toe (tonne of oil equivalent) = 41.87 GJ. 1 quad = 10¹⁵ BTU

1.055 EJ (used in US energy statistics). Global primary energy consumption is approximately 580 EJ/year (as of the mid-2020s), equivalent to a continuous global power use of about 18 TW. Unit Equivalent in Joules Typical Context calorie (nutritional)

4186 J = 4.186 kJ

Food energy 1 kWh

3.6 × 10⁶ J = 3.6 MJ

Household electricity billing toe (tonne oil equivalent)

4.187 × 10¹⁰ J = 41.87 GJ

Oil and gas statistics 1 EJ (exajoule) 10¹⁸ J National/global energy statistics TWh (terawatt- hour)

3.6 × 10¹⁵ J = 3.6 PJ

Grid-scale electricity Mt TNT equivalent

4.184 × 10¹⁵ J

Nuclear weapon yield comparison

Table 56.1

Physics: Foundations of Physical Science Chapter 56 — Energy Systems

56.2 Primary vs Secondary Energy Sources

Primary energy sources are found in nature before any human conversion. They include fossil fuels (coal, oil, natural gas — stored solar energy from ancient photosynthesis), nuclear fuels (uranium, thorium — nuclear binding energy), and renewable flows (solar radiation, wind, falling water, geothermal heat, tidal energy). Primary energy is characterised by energy density (joules per kilogram or joules per litre) and by whether it is a stock (finite quantity that is depleted) or a flow (continuously replenished). Secondary energy carriers are derived from primary sources through conversion: electricity (from generators driven by thermal, hydraulic, wind, or nuclear power), hydrogen (from electrolysis or steam methane reforming), refined petroleum products (from crude oil), and district heat (from co-generation plants). Secondary carriers are more useful than primary sources — they are portable, controllable, and deliverable through infrastructure networks — but conversion always involves losses. Source Type Energy Density CO₂ Intensity (gCO₂/kWh) Key Physics Coal (hard) Fossil stock 29 MJ/kg 820–900 Combustion, Rankine cycle Natural gas Fossil stock 55 MJ/kg 490–550 Combustion, CCGT turbines Crude oil Fossil stock 44 MJ/kg 650–750 Combustion, Otto/diesel cycle Uranium (in LWR) Nuclear stock 45 GJ/kg ≤15 Fission, Rankine cycle Wind Renewable flow Variable (0.5

kW/m² typical)

≤10 Bernoulli, Betz limit Solar PV Renewable flow

W/m²

peak ≤30 Photovoltaic effect Hydro Renewable flow Site-dependent ≤15 Potential energy, Pelton/Kaplan Geothermal Renewable flow Site-dependent 15–100 Heat engines

(low ΔT)

Physics: Foundations of Physical Science Chapter 56 — Energy Systems

Table 56.2

56.3 Mechanical Energy Systems

The simplest energy systems convert gravitational or kinetic energy to mechanical work directly. Waterwheels, windmills, and sailing ships are ancient examples; their modern descendants are hydroturbines, wind turbines, and wave energy converters.

Mechanical energy systems are governed by conservation of energy: E_k + E_p =

constant (no losses), with real systems losing energy to friction, turbulence, and

material vibration. The power available from a mechanical energy flow is P = Fv (linear

motion) or P = τω (rotational motion), where τ is torque and ω is angular velocity.

Flywheels store rotational kinetic energy: E = ½ Iω², where I is the moment of inertia.

Modern flywheel energy storage systems use high-strength composite rotors spinning at 20 000–60 000 rpm in vacuum enclosures, storing tens of kWh with round-trip efficiencies of 90–95%. They are used for frequency regulation in electrical grids and for uninterruptible power supplies.

A flywheel has I = 50 kg·m² and spins at 30

000 rpm = 3142 rad/s. Energy stored: E = ½ × 50 × 3142² = ½ × 50 × 9.87 × 10⁶ =

2.47 × 10⁸ J = 247 MJ

68.5 kWh. This is enough to power an average home for about 2–3 days. Physics: Foundations of Physical Science Chapter 56 — Energy Systems

56.4 Thermal Power Generation

The majority of the world's electricity is generated by thermal power plants, which convert heat from combustion (fossil fuels, biomass) or nuclear fission into electrical energy via a heat engine driving a generator. The working fluid in most thermal power plants is steam (water vapour), cycled through the Rankine cycle: heating at constant pressure, expansion through a turbine (work output), condensation back to liquid, and pumping back to high pressure.

η_Carnot = 1 −T_C / T_H

A modern supercritical coal power plant operates at T_H

873 K (600°C) and T_C

308 K (35°C). Carnot efficiency: 1

308/873 = 64.7%. Actual efficiency:

43–46% (due to friction, heat losses, pump and turbine irreversibilities). A combined cycle gas turbine (CCGT) plant uses a gas turbine (Brayton cycle) at high temperature and recovers waste heat in a steam turbine (Rankine cycle), achieving efficiencies of 55–60%. All thermal power plants must reject waste heat to a cold reservoir — typically a river, lake, ocean, or cooling tower. This waste heat is not a design failure but a thermodynamic necessity. A 1 GW electrical plant at 40% efficiency must reject 1.5 GW

of waste heat (since P_in = P_out + P_waste; P_waste/P_out = (1 η)/η = 1.5 at 40%).

Cooling towers are concrete hyperbolic structures that transfer this heat to the atmosphere by evaporation. The white plume from a cooling tower is water vapour, not pollution.

A coal plant burns 1 kg of coal (heating value

29 MJ/kg) per second. Electrical output = 29 MW × 0.40 (efficiency) = 11.6 MW. Waste

heat rejected = 29

11.6 = 17.4 MW. CO₂ emission: coal is approximately 85% carbon;

0.85 kg C per second produces 0.85 × 44/12 = 3.12 kg CO₂/s = 3.12/(11.6 × 10 ³) =

269 g CO₂/kWh. Physics: Foundations of Physical Science Chapter 56 — Energy Systems

56.5 Hydroelectric and Fluid Power Systems

Hydroelectric power converts the gravitational potential energy of elevated water into

electrical energy. Water falling through height h releases potential energy E = mgh per

unit mass. A turbine converts this kinetic energy of the water flow into rotational mechanical energy, which drives a generator.

P = η ρ g Q h

Hydroelectric plants range from small run-of-river systems (no significant reservoir, head 2–10 m, power tens of kW) to large impoundment facilities (reservoir stores months of flow, head hundreds of metres, power gigawatts). The Three Gorges Dam in China (22.5 GW capacity) is the world's largest hydroelectric facility, generating about 100 TWh/year — enough to supply roughly 15 million average Chinese households. Pumped-storage hydroelectricity (PSH) is the dominant form of large-scale electrical energy storage worldwide. During periods of excess generation (low demand, high renewable output), electricity pumps water uphill to a reservoir. During peak demand, water is released to generate electricity. Round-trip efficiency is approximately 75–85%. PSH accounts for about 94% of the world's installed grid energy storage capacity. Tidal energy exploits the potential and kinetic energy of ocean tides, which arise from the gravitational forces of the Moon and Sun on the oceans. Tidal barrages (La Rance, France, 240 MW) capture tidal flow through sluice gates and turbines. Tidal stream generators (analogous to underwater wind turbines) extract kinetic energy from tidal currents where flow is fast (e.g., the Pentland Firth, UK, with currents up to 4 m/s).

A river flows at Q = 250 m³/s through a turbine

with hydraulic head h = 80 m. Efficiency = 0.88. Power = 0.88 × 1000 × 9.81 × 250 ×

80 = 172.7 × 10⁶ W

173 MW. This is a medium-sized hydroelectric plant. Physics: Foundations of Physical Science Chapter 56 — Energy Systems

56.6 Solar Energy Systems

Solar energy reaches Earth's surface at a mean irradiance of about 170 W/m² globally

(averaged over day/night, latitude, and weather). Peak irradiance on a horizontal

surface can reach 1000 W/m² under full sun in the tropics. Two main technologies

convert solar radiation to useful energy: photovoltaics (PV, converting light directly to electricity) and solar thermal (concentrating heat for hot water, space heating, or electricity generation). Photovoltaic cells use the photoelectric effect in semiconductor p-n junctions. Incident

photons with energy E = hf ≥E_gap excite electrons across the band gap, creating

electron-hole pairs that are separated by the built-in electric field of the junction and collected as direct current. Silicon single-crystal cells have band gaps of 1.12 eV; this matches the solar spectrum well, but photons with energy less than 1.12 eV (infrared) are not absorbed, and photons with energy much greater than 1.12 eV (UV) deliver most of their energy as waste heat. These losses account for the theoretical Shockley- Queisser efficiency limit of about 33% for a single-junction silicon cell.

E_{photon} = hf = hc / λ

(minimum for PV absorption:

E_{photon} ≥E_{gap})

⁹ Commercial silicon PV cells achieve efficiencies of 20–23%. Multi-junction concentrator cells (used in spacecraft and concentrated solar power) achieve over 40%. The capacity factor of solar PV (fraction of time at peak output) is 10–25% in most locations, determined by solar resource and local weather. The cost of solar PV has fallen by over 90% since 2010, making it the cheapest source of new electricity generation in much of the world as of the mid-2020s. Concentrated solar power (CSP) uses mirrors or lenses to concentrate sunlight onto a receiver, generating high-temperature heat that drives a conventional steam turbine. CSP plants can incorporate thermal storage (molten salt at 390–560°C) that allows electricity generation for hours after sunset. Parabolic trough, power tower (heliostat field), and linear Fresnel designs are the main CSP configurations. Physics: Foundations of Physical Science Chapter 56 — Energy Systems

56.7 Wind Energy Systems

Wind turbines extract kinetic energy from moving air. The power available in a wind

stream passing through area A at speed v is P_wind = ½ρAv³ (kinetic energy flux). No

turbine can extract all of this: the Betz limit (derived from conservation of momentum applied to the actuator disc model) states that the maximum extractable fraction is 16/27 59.3%.

P_{extracted} = ½ ρ A v³ × C_p

(C_p ≤16/27 ≈0.593)

Modern onshore wind turbines have rotor diameters of 100–175 m and rated capacities of 3–6 MW. Offshore wind turbines reach 15–22 MW (rotor diameters up to 280 m). Capacity factors for onshore wind are typically 25–40%; offshore 40–60%. Wind turbines operate between cut-in speed ( 3 m/s, below which the blades do not generate useful power) and cut-out speed ( 25 m/s, above which the turbine is shut down to prevent damage).

m/s wind. Swept area A = π × 60² = 11 310 m². Wind power available = ½ × 1.2 × 11

310 × 10³ = 6.786 × 10⁶ W = 6.79 MW. At C_p = 0.48: output = 6.79 × 0.48 = 3.26

MW. At 15 m/s (50% more wind): power ∝v³, so output = 3.26 × (15/10)³ = 3.26 ×

3.375 = 11.0 MW.

The tip-speed ratio (TSR) λ = ωR/v is the ratio of blade tip speed to wind speed.

Optimal TSR for maximum C_p is about 6–8 for modern three-bladed turbines. At TSR

< optimal, the rotor turns too slowly to sweep all the wind efficiently; at TSR > optimal, turbulence from one blade interferes with the next. Physics: Foundations of Physical Science Chapter 56 — Energy Systems

56.8 Nuclear Energy Systems

Nuclear fission releases energy when heavy nuclei (uranium-235, plutonium-239) absorb a neutron and split into two smaller nuclei plus 2–3 fast neutrons and energy.

The energy per fission event is about 200 MeV = 3.2 × 10 ¹¹ J. This is about 50 million

times more energy per reaction than chemical combustion; 1 kg of U-235 undergoing complete fission releases the equivalent energy of about 2700 tonnes of coal.

E = Δm c²

⁶ In a nuclear reactor, a controlled chain reaction is maintained. Each fission event

produces 2–3 neutrons; exactly one must on average cause another fission (k = 1,

criticality) for a sustained reaction. Moderators (water, heavy water, graphite) slow fast neutrons to thermal energies where U-235 fission cross-section is much higher. Control rods (boron, hafnium, cadmium) absorb neutrons and allow the reaction rate to be regulated. The heat produced heats coolant (water, heavy water, sodium, helium), which drives a steam turbine through a Rankine cycle. Nuclear power plants produce no CO₂ during operation. Life-cycle emissions (including construction, fuel mining, and decommissioning) are 5–15 gCO₂/kWh, comparable to wind and lower than solar PV. The main challenges are: high capital cost and long construction time, management of spent fuel (high-level radioactive waste with long- lived isotopes), and public concern about safety (following accidents at Three Mile Island 1979, Chernobyl 1986, and Fukushima 2011). Nuclear fusion (Chapter 53) remains a long-term research goal. Reactor Type Moderator Coolant Key Characteristics Pressurised Water Reactor (PWR) Light water Light water (pressurised) Most common globally; ~70% of fleet Boiling Water Reactor (BWR) Light water Light water (boils in core) Simpler; steam goes directly to turbine CANDU (PHWR) Heavy water (D₂O) Heavy water Uses natural (unenriched) uranium High Temperature Gas Reactor Graphite Helium Higher efficiency; inherently safe Fast Breeder None (fast Liquid sodium Breeds Pu-239 from U- Physics: Foundations of Physical Science Chapter 56 — Energy Systems Reactor neutrons) 238; uses more fuel

Table 56.3

56.9 Batteries and Energy Storage

Energy storage is essential for balancing supply and demand in electrical grids (especially with variable renewables), for mobile applications (electric vehicles), and for backup power. Batteries convert chemical energy to electrical energy through electrochemical reactions at two electrodes separated by an electrolyte. The voltage of a battery cell is determined by the electrochemical potential difference between the anode and cathode materials. Energy density (Wh/kg or Wh/L) determines how much energy can be stored per unit mass or volume. Power density (W/kg) determines how fast energy can be delivered. Cycle life is the number of charge- discharge cycles before capacity degrades significantly. Battery Type Voltage (V) Energy Density (Wh/kg) Cycle Life Applications Lead-acid 2.0/cell 30–50 200–500 Car starter, UPS, low-cost storage Nickel-metal hydride (NiMH) 1.2/cell 60–90 500–1000 Hybrid vehicles (Prius), consumer electronics Lithium-ion (LFP) 3.2/cell 100–160 2000–6000 Grid storage, EVs (long-range focus) Lithium-ion (NMC) 3.7/cell 150–220 1000–2000 EVs (energy density focus), portable electronics Lithium-ion (NCA) 3.6/cell 200–260 500–1500 High-performan ce EVs (Tesla 18650) Sodium-ion 3.1–3.3/cel 100–150 1000–3000 Stationary Physics: Foundations of Physical Science Chapter 56 — Energy Systems l storage, lower cost alternative Flow battery (vanadium) Variable 15–30 (system) >10 000 Grid-scale, long duration storage

Table 56.4

The specific energy of a battery cell is limited by the electrochemistry: E = Q × V,

where Q is the charge capacity (coulombs) and V is the cell voltage. For lithium-ion, the theoretical specific energy of the active materials alone is about 400–600 Wh/kg; actual cells deliver 150–260 Wh/kg because they include inactive mass (current collectors, casing, electrolyte, separators). Improving specific energy requires either finding electrode materials with higher capacity (Ah/kg) or higher voltage, or reducing inactive mass.

56.10 Electrical Transmission and Distribution

Electrical power is transmitted from generating stations to consumers through a network of transmission lines (high voltage, long distance) and distribution lines (lower

voltage, local). Power loss in a transmission line is P_loss = I²R, where R is the line

resistance and I is the current. For fixed power P = IV, increasing voltage reduces

current in proportion, reducing losses by I².

P_{loss} = I² R = P^2 R / V^2

Transformers (Chapter 48) step voltage up at the generating station (e.g., from 11 kV to 400 kV for long-distance transmission) and step it down progressively for distribution and final use. A typical UK domestic supply is 230 V (single-phase); US is 120/240 V. Industrial supplies are 11 kV or 33 kV. The efficiency of modern power transformers is very high (99.5–99.7%), making them one of the most efficient energy conversion devices.

A power station generates 500 MW and

transmits it through a 400 kV, 300 km line with resistance 0.06 Ω/km (total R = 18 Ω).

Current I = P/V = 500 × 10⁶ / 400 × 10³ = 1250 A. Power loss = I²R = 1250² × 18 =

28.1 MW = 5.6% of transmitted power. If transmitted at 11 kV (36 times lower): I = 45

455 A; loss = 45455² × 18 = 37.2 GW — vastly exceeding the generated power, which

Physics: Foundations of Physical Science Chapter 56 — Energy Systems shows why high voltage transmission is essential. High-voltage direct current (HVDC) transmission uses power electronics (rectifiers and inverters) to convert AC to DC for long-distance transmission and back to AC for distribution. HVDC has lower line losses than HVAC for distances over 600 km (overhead) or 100 km (underground/submarine cable), because capacitive charging losses in AC cables are eliminated. HVDC is preferred for long submarine cables (e.g., interconnectors between countries) and for integrating distant renewable resources.

56.11 Efficiency and Energy Losses in Infrastructure

Every stage of an energy system loses some energy, typically as heat. The overall efficiency of a system is the product of the efficiencies of each stage. A typical electricity-to-heat

chain in an old electric resistance heater: power plant (40%) ×

transmission (95%) × distribution (97%) × heater (100%) = 36.9% primary energy to

useful heat. Replacing the resistance heater with a heat pump (COP 3.0) gives: 40% ×

95% × 97% × 300% = 110.7% — more useful heat than the primary energy input to

the power plant, because the heat pump also draws heat from the environment.

η_{system} = η_1 × η_2 × η_3 × …

The coefficient of performance (COP) of a heat pump is the ratio of useful heat output

to electrical work input: COP = Q_H / W. For a Carnot heat pump: COP_max = T_H /

(T_H

T_C). A ground-source heat pump with T_C = 7°C (280 K) and T_H = 35°C (308

K) has COP_max = 308/28 = 11.0; practical COPs are 3–5.

Electric vehicles (EVs) are substantially more energy-efficient than internal combustion engine (ICE) vehicles. An ICE vehicle converts about 20–25% of fuel energy to wheel motion (the rest is heat). An EV converts about 85–90% of battery energy to wheel motion. Even accounting for power-plant losses in generating the electricity, EVs typically deliver 2–3 times more useful mobility per unit of primary energy than ICE vehicles, and 4–6 times more when powered by renewable electricity. Energy System Primary-to-Service Efficiency Notes Incandescent bulb (electricity from coal) 4%

Plant 40% × grid 95% ×

bulb 10% Physics: Foundations of Physical Science Chapter 56 — Energy Systems LED bulb (electricity from coal) 36%

Plant 40% × grid 95% ×

LED 95% LED bulb (solar PV, direct) 18%

PV 20% × LED 95%

ICE vehicle (petrol) 15–25% Tank-to-wheel; city driving lower EV (grid electricity from coal) 36%

Plant 40% × grid 95% ×

EV 90% EV (solar PV) 17%

PV 20% × EV 90%

Natural gas condensing boiler (space heat) 93% Latent heat recovery Heat pump (ground- source, from coal plant) 120–150% COP 3–5

×

plant/grid efficiency

Table 56.5

56.12 Grid Stability and Energy Demand

An electrical grid must balance supply and demand instantaneously — electricity cannot easily be stored in large quantities, and if supply exceeds demand, frequency Physics: Foundations of Physical Science Chapter 56 — Energy Systems rises; if demand exceeds supply, frequency falls. Grid frequency (50 Hz in Europe, 60 Hz in North America) is the real-time indicator of this balance. Generators slow down if they are overloaded (demand > supply); they speed up if underloaded (supply > demand). Grid stability is maintained by a hierarchy of responses: primary frequency response (automatic, within seconds, from generator governor systems); secondary frequency control (automatic generation control, minutes timescale); and tertiary control (manual dispatch of reserve generation, tens of minutes). Large inertia (from spinning generator masses) provides the system with time to respond before frequency deviates dangerously. As fossil-fuel synchronous generators are replaced by variable-speed inverter-based renewable generators, traditional rotational inertia is reduced — requiring new approaches such as virtual inertia from grid-scale batteries. Demand variability follows predictable daily patterns (morning and evening peaks), seasonal patterns (winter heating demand, summer cooling demand), and unpredictable short-term fluctuations (e.g., large simultaneous kettle use at the end of a popular TV show — a real phenomenon that UK grid operators must plan for). Demand response programmes incentivise large electricity consumers to shift or curtail usage during peak demand periods. Integrating large amounts of variable renewable generation (wind and solar) increases the challenge of balancing because their output depends on weather. Solutions include: flexible backup generation (gas peakers, hydro), demand response, energy storage (batteries, pumped hydro, hydrogen), and interconnection with neighbouring grids so surplus renewable generation in one region can serve demand in another. Long- duration energy storage (8–100 hours) is an active area of technology development.

— becomes less meaningful in a grid with high renewable penetration. What matters is flexibility: the ability to ramp generation up and down quickly in response to changes in renewable output and demand. Physics determines the ramp rates: hydro and gas can ramp in seconds to minutes; coal plants take tens of minutes to hours.

56.13 Common Mistakes in Energy Systems Analysis

• Confusing power and energy. Power (watts, W) is the rate of energy flow; energy (joules, J, or kilowatt-hours, kWh) is the total amount. A 3 GW nuclear plant

produces 3 GW of power; in one year it produces 3 GW × 8760 h × 0.90

Physics: Foundations of Physical Science Chapter 56 — Energy Systems

(capacity factor) = 23 652 GWh = 23.7 TWh of energy. Capacity and annual

generation are different quantities. • Applying efficiency to the wrong quantity. Carnot efficiency limits the fraction of heat that can be converted to work; it does not apply to energy flows that do not involve heat-to-work conversion (e.g., PV panels, fuel cells, batteries). Applying the Carnot limit to a battery is a category error. •

Forgetting the v³ dependence of wind power. A 10% increase in average wind

speed increases available wind power by (1.10)³

1.33 — a 33% increase in power, not 10%. Wind resource assessment errors compound rapidly. • Thinking renewable generation capacity equals renewable generation. A 1 GW

solar farm with a 15% capacity factor generates 1 GW × 8760 h × 0.15 = 1314

GWh/year. A 1 GW nuclear plant with 90% capacity factor generates 7884 GWh/year — six times more energy, from the same installed capacity. • Confusing CO₂ per kWh (generation emissions intensity) with total CO₂ per year (total emissions). Total emissions depend on both intensity and how much energy is generated. A very low-emissions source (nuclear) that generates a lot can have a larger impact on decarbonisation than a zero-emissions source (offshore wind) that generates less in a specific context. • Assuming 100% renewable grids are straightforward. Matching variable supply to demand at all hours requires storage, interconnection, flexible demand, or dispatchable backup. The physics of energy storage (battery energy density, pumped hydro geography) and transmission (line capacity, losses) constrain what is achievable and at what cost.

Never use efficiency without specifying the input and output energy forms. Thermal efficiency, electrical efficiency, exergy efficiency, and round-trip efficiency all measure different things and can give very different numbers for the same system. Always state: efficiency of converting what, to what, under what conditions.

Chapter Summary

Energy systems chain together primary sources, conversion technologies, transmission infrastructure, storage, and end-use devices. At every stage, thermodynamics, electromagnetism, fluid mechanics, and nuclear physics determine efficiency and loss. Physics: Foundations of Physical Science Chapter 56 — Energy Systems •

Energy and power: E (joules, kWh) vs P (watts); system efficiency = product of all

stage efficiencies; second law limits all thermal conversions. •

Thermal power: Rankine cycle; Carnot limit η = 1

T_C/T_H; CCGT achieves

55–60%; all plants reject waste heat to cold reservoir. •

Hydro: P = ηρgQh; pumped storage for grid-scale electricity storage; tidal

barrages and tidal stream. •

Solar PV: photovoltaic effect; threshold wavelength from E_gap; Shockley-

Queisser limit 33%; cost has fallen 90%+ since 2010. •

Wind: P = ½ρAv³ × C_p; Betz limit C_p ≤16/27; cubic wind speed dependence;

capacity factor 25–60%. •

Nuclear: E = Δmc²; 200 MeV per fission; 50 million times more energy-dense

than coal; controlled by k = 1 criticality; low CO₂ lifecycle.

• Batteries: electrochemical energy storage; energy density limited by electrode chemistry; lithium-ion dominant; flow batteries for long-duration grid storage. •

Transmission: P_loss = I²R = P²R/V²; high voltage reduces losses; transformers

step voltage; HVDC for long submarine cables. • Grid stability: frequency as real-time supply-demand indicator; inertia buffers response time; variable renewables reduce inertia; storage and flexibility needed.

Key Terms

primary energy

Energy in its natural form before human conversion: fossil fuels,

nuclear fuels, and renewable flows (solar, wind, hydro, geothermal, tidal).

energy carrier

An intermediate form of energy derived from primary sources and used

to transport or store energy

electricity, hydrogen, heat, petroleum products.

Physics

Foundations of Physical Science

Chapter 56 — Energy Systems

Rankine cycle

The thermodynamic

cycle of a steam power plant

heat input at

constant pressure, adiabatic expansion through a turbine, condensation, and pumping back to high pressure.

Carnot efficiency

η_Carnot = 1

T_C/T_H; the maximum possible efficiency of a heat

engine; depends only on the temperatures of the hot and cold reservoirs; always less than 1.

hydraulic head (h)

The effective

vertical height through which water falls in a hydroelectric system; determines the potential energy per unit mass (gh) available for conversion.

photovoltaic effect

The generation of electric current when photons with energy ≥

E_gap excite electrons across a semiconductor band gap, producing electron-hole

pairs that are separated by a p-n junction.

Shockley-Queisser limit

The theoretical maximum efficiency of a single-junction solar

cell ( 33% for silicon); set by the band-gap mismatch with the solar spectrum.

Betz limit

16/27

59.3%; the maximum fraction of wind kinetic energy that any wind turbine can extract; derived from conservation of momentum applied to the rotor disc.

power coefficient (C_p)

The ratio of extracted wind power to available wind power;

C_p ≤16/27 (Betz limit); modern turbines achieve C_p

0.45–0.50.

mass defect (Δm)

The difference in mass between reactants and products in a nuclear

reaction; the lost mass appears as energy E = Δmc²; positive for exothermic

reactions (fission, fusion).

criticality (k = 1)

The condition in a nuclear reactor where exactly one neutron from

each fission event causes another fission, sustaining a steady chain reaction.

energy density

Energy stored or released per unit mass (Wh/kg, MJ/kg) or per unit

volume (Wh/L, MJ/L); determines the mass and volume of fuel or battery required for a given energy storage.

capacity factor

The ratio of actual energy generated over a period to the maximum

possible energy if the plant ran at full rated capacity continuously; accounts for downtime, variable resource, and part-load operation.

grid frequency

The frequency of alternating current on an electrical grid (50 Hz in

Europe, 60 Hz in North America); maintained at the reference value by continuous matching of generation to demand.

coefficient of performance (COP)

For a heat pump: COP = Q_H/W (heat delivered

divided by electrical work input); can exceed 1 because heat is moved from a cold source, not created from scratch.

Physics

Foundations of Physical Science

Chapter 56 — Energy Systems

Concept Review Questions

Answer in complete sentences. No calculations required unless stated.

1. A 1 GW coal power plant has a thermal efficiency of 40% and a 90% capacity

factor. How much coal (heating value 29 MJ/kg) does it consume per year? How much waste heat does it reject per year?

2. Explain why high-voltage transmission (400 kV) reduces power losses compared

to lower-voltage transmission (11 kV) for the same power level. Use the formula

P_loss = I²R to support your explanation.

Physics: Foundations of Physical Science Chapter 56 — Energy Systems

3. Explain the Betz limit. Why can't a wind turbine extract all the kinetic energy

from the wind passing through it? What would happen to the air behind the turbine if all kinetic energy were extracted?

4. Compare the energy density of uranium in a light water reactor to coal (coal: 29

MJ/kg; U-235 complete fission: 80 TJ/kg). By what factor is nuclear more energy- dense? What practical implication does this have?

5. A pumped-storage hydroelectric plant stores energy by pumping 10 million

cubic metres of water uphill by 400 m. How much energy does it store in GWh?

(Round-trip efficiency = 80%.) How long could this supply a 1 GW demand?

6. Explain the difference between a battery's energy density (Wh/kg) and power

density (W/kg). Why might a battery with high energy density but low power density be unsuitable for a vehicle acceleration application?

7. Why does increasing variable renewable penetration on an electrical grid create

challenges for frequency stability, even if total annual generation is sufficient? What physical quantity, previously provided by fossil-fuel generators, is reduced?

8. A household installs a ground-source heat pump with COP = 4.0 instead of an

electric resistance heater. The electricity is generated at a 40% efficient gas power plant. Compare the primary gas energy consumption per MJ of heat delivered for each option.

Power Plant Analysis

9. A coal power plant (efficiency 38%) burns coal with heating value 28.5 MJ/kg. It

generates 850 MW of electricity. Find: (a) the primary thermal power input; (b) the mass of coal burned per second; (c) the waste heat rejected per second; (d) the CO₂ emission rate (coal is 80% carbon by mass).

10. A CCGT plant operates with a gas turbine at T_H = 1400°C and exhaust at 550°C

(this heat goes to a steam turbine with cold reservoir at 30°C). Find the Carnot efficiency of each stage and estimate the maximum overall efficiency of the combined cycle. Physics: Foundations of Physical Science Chapter 56 — Energy Systems

11. A nuclear power station (PWR) uses fuel with an effective energy release of 45

GJ per kg of uranium. It operates at 33% thermal efficiency with capacity factor 90% and rated electrical output 1.2 GW. Find: (a) annual electrical generation (TWh); (b) annual thermal energy input (PJ); (c) annual uranium fuel consumption (tonnes).

12. A hydro plant has a head of 120 m, flow rate of 400 m³/s, and efficiency 91%. (a)

Find the electrical output in MW. (b) How many tonnes of CO₂ per year would an equivalent coal plant (38% efficiency, 0.9 kg CO₂/kWh electrical) emit?

Efficiency Problems

13. A household uses 3500 kWh/year for heating. Compare the annual primary

energy input (coal at power plant, 40% efficiency; grid transmission 95%; distribution 97%) for: (a) electric resistance heaters; (b) a heat pump with COP = 3.5.

14. An electric vehicle has battery capacity 75 kWh, drivetrain efficiency 88%, and

energy consumption 15 kWh/100 km. The charging efficiency is 92%. If electricity is generated from natural gas at 50% efficiency, find the primary energy (MJ) per Physics: Foundations of Physical Science Chapter 56 — Energy Systems 100 km of driving. Compare to a 6 L/100 km petrol car (petrol heating value 32 MJ/L, ICE efficiency 22%).

15. LED lighting converts 95% of electrical energy to light; incandescent bulbs

convert 10%. If electricity is generated from solar PV at 20% efficiency, find the

primary solar energy (W/m² of panel) needed to produce 800 lumens of light

from each bulb type. (Assume 100 lm/W for LED, 15 lm/W for incandescent, and

standard irradiance of 170 W/m² annual average.)

16. A steam turbine takes in steam at 500°C (773 K) and exhausts at 40°C (313 K).

The Carnot efficiency is 59.5%, but the actual isentropic efficiency of the turbine blades is 88% and mechanical/generator losses are 3%. Find the actual electrical output efficiency of the turbine-generator set.

Energy Conversion Exercises

17. A wind turbine has rotor diameter 150 m and operates in mean wind speed 9

m/s with C_p = 0.47. (a) Find the mean electrical output. (b) Find annual energy

generation if capacity factor = 35%. (c) How many households (each using 4500

kWh/year) does this turbine serve?

18. A solar farm covers 200 hectares with PV panels of efficiency 21% and packing

factor 0.70 (70% of area covered). Mean irradiance is 180 W/m². Find the mean

electrical output and annual generation. (1 hectare = 10 000 m²)

Physics: Foundations of Physical Science Chapter 56 — Energy Systems

19. A uranium-235 fission releases 200 MeV. Find: (a) this energy in joules; (b) the

energy per kg of U-235 if complete fission occurs; (c) the mass defect per fission

event as a fraction of the U-235 mass (235 u = 235 × 1.66 × 10 ²⁷ kg).

20. A Li-ion battery pack (NMC chemistry) has energy density 200 Wh/kg and

powers a vehicle requiring 180 Wh/km. How heavy is the battery needed for a 500 km range? Compare to a petrol tank for the same range at 8 L/100 km (petrol density 0.74 kg/L). Grid and Storage Applications

21. A 500 MW power station transmits electricity through a 220 kV, 200 km line

(resistance 0.08 Ω/km). Find: (a) the current in the line; (b) the power loss; (c) the

fraction of power lost; (d) the voltage drop across the line.

22. Grid frequency in the UK is 50 Hz. A sudden loss of a 1.8 GW generator causes

frequency to begin falling. The rate of change of frequency (RoCoF) is estimated

at 0.6 Hz/s initially. If the system inertia is H = 3 seconds (stored kinetic energy /

Physics: Foundations of Physical Science Chapter 56 — Energy Systems rated power), find the total synchronous generation connected to the grid. (Use:

RoCoF = (P_imbalance) / (2H × P_system); solve for P_system.)

23. A pumped-storage plant has upper reservoir volume 8 × 10⁶ m³, head 550 m,

turbine-generator efficiency 88%, and pump efficiency 87%. (a) Find the gross stored energy (GWh). (b) Find the electrical energy recoverable on discharge (GWh). (c) Find the round-trip efficiency (electricity in to electricity out). (d) If the plant discharges at full power of 1600 MW, how many hours of storage does it provide?

24. A community installs 20 MW of solar PV (capacity factor 13%) and 5 MW of

wind (capacity factor 35%) to meet an average demand of 8 MW. (a) Find mean solar and wind generation. (b) Is average generation sufficient? (c) Identify the physical problem and propose two storage or backup solutions, explaining the physics of each.

Chapter 56 Checkpoint

Before moving on to Chapter 57, confirm you can do each of the following. I can… See Section Explain the role of thermodynamics and the second law in all energy systems 56.1–56.2 Physics: Foundations of Physical Science Chapter 56 — Energy Systems Calculate Carnot efficiency and actual plant efficiency from operating temperatures 56.4

Apply P = ηρgQh to hydroelectric power output

56.5

Apply the photovoltaic threshold condition E_{photon} ≥E_gap

to solar cells 56.6

Calculate wind turbine output using P = ½ρAv³ × C_p and the

Betz limit 56.7

Explain nuclear fission energy release using E = Δmc² and the

chain reaction 56.8 Compare battery chemistries by energy density, cycle life, and application 56.9

Calculate transmission line losses using P_loss = I²R = P²R/V²

56.10 Calculate and cascade system efficiencies through multiple conversion stages 56.11 Explain grid frequency as a supply-demand indicator and describe stability challenges 56.12 Identify common mistakes in energy calculations, especially power vs energy 56.13