CONTENTS

1. Editors' preface to the Manchester Physics Series
2. Author's preface
3. 1: Introduction
   1. 1.1 Energy consumption
   2. 1.2 Energy sources
   3. 1.3 Renewable and non-renewable energy sources
   4. 1.4 The form and conversion of energy
   5. Problems 1
4. 2: The atomic nucleus
   1. 2.1 The composition and properties of nuclei
   2. 2.2 Nuclear forces and energies
   3. 2.3 Radioactivity and nuclear stability
   4. Problems 2
   5. Notes
5. 3: Nuclear power
   1. 3.1 How to get energy from the nucleus
   2. 3.2 Nuclear reactions
   3. 3.3 Nuclear fission
   4. 3.4 Controlled fission reactions
   5. 3.5 Nuclear fusion
   6. Problems 3
   7. Note
6. 4: Solar power
   1. 4.1 Stellar fusion
   2. 4.2 Blackbody radiation
   3. 4.3 Solar radiation and its interaction with the Earth
   4. 4.4 Geothermal energy
   5. 4.5 Solar heaters
   6. 4.6 Heat engines: converting heat into work
   7. Problems 4
   8. Notes
7. 5: Semiconductor solar cells
   1. 5.1 Introduction
   2. 5.2 Semiconductors
   3. 5.3 The p–n junction
   4. 5.4 Semiconductor solar cells
   5. Problems 5
   6. Note
8. 6: Wind power
   1. 6.1 A brief history of wind power
   2. 6.2 Origin and directions of the wind
   3. 6.3 The flow of ideal fluids
   4. 6.4 Extraction of wind power by a turbine
   5. 6.5 Wind turbine design and operation
   6. 6.6 Siting of a wind turbine
   7. Problems 6
9. 7: Water power
   1. 7.1 Hydroelectric power
   2. 7.2 Wave power
   3. 7.3 Tidal power
   4. Problems 7
10. 8: Energy storage
    1. 8.1 Types of energy storage
    2. 8.2 Chemical energy storage
    3. 8.3 Thermal energy storage
    4. 8.4 Mechanical energy storage
    5. 8.5 Electrical energy storage
    6. 8.6 Distribution of electrical power
    7. Problems 8
11. Solutions to problems
    1. Problems 1
    2. Problems 2
    3. Problems 3
    4. Problems 4
    5. Problems 5
    6. Problems 6
    7. Problems 7
    8. Problems 8
12. Index
13. EULA

List of Tables

1. Chapter 3
   1. Table 3.1
2. Chapter 5
   1. Table 5.1
3. Chapter 8
   1. Table 8.1

List of Illustrations

1. Chapter 1
   1. Figure 1.1 Illustration of the dramatic rise in annual global energy consumption that occurred between 1820 and 2010.
   2. Figure 1.2 Annual energy consumption for the USA in 2014, by energy source – 81% of energy consumption came from fossil fuels, while nuclear energy and renewable sources provided the remainder.
   3. Figure 1.3 Illustration of the distinction between renewable and non-renewable energy sources, using the examples of solar energy and energy from the fossil fuel oil. Solar energy flows continuously from the Sun, while reserves of oil are finite.
2. Chapter 2
   1. Figure 2.1 Schematic diagram of Rutherford's apparatus for observing the scattering of α particles by a thin gold foil. α particles from the source are collimated into a narrow beam and directed at the foil. The angle θ through which an α particle is deflected at the foil is measured by observing the point at which the particle strikes the fluorescent screen.
   2. Figure 2.2 The scattering of α particles by a gold nucleus. The scattering angle increases as the α particles approach closer to the nucleus, and these particles can be scattered through very large angles, up to 180°.
   3. Figure 2.3 The deflection angle, θ, that an α particle experiences as it passes close to a gold nucleus. The distance b is called the impact parameter. The deflection angle is given by the ratio Δp/p, where p is the momentum of the incident α particle and Δp is the sideways momentum that results from the electrostatic repulsion.
   4. Figure 2.4 The [sin ⁴(θ/2)]^{− 1}angular dependence of , the number of α particles detected per unit area on the fluorescent screen. Note the logarithmic scale for . Although falls rapidly with increasing scattering angle θ, a significant number of α particles are detected at very large angles.
   5. Figure 2.5 Experimental data for the scattering of α particles through a fixed angle of 60° by a lead target shows the breakdown of the Rutherford scattering formula. At an α-particle energy of about 27.5 eV, the data abruptly depart from scattering due to purely electrostatic repulsion. Above this energy the α particles get close enough to the nucleus to come under the influence of the strong but short-range nuclear force.
   6. Figure 2.6 Typical form of the distribution of charge density, ρ(r), for a nucleus as a function of radial distance, r, from the centre of the nucleus. R represents the mean nuclear radius. The skin thickness is the distance over which ρ(r) falls from 90% to 10% of its maximum value. The charge density is roughly uniform over the central region of a nucleus.
   7. Figure 2.7 Schematic diagram of a mass spectrometer. The collimating apertures define a narrow beam of ions from the source. The velocity selector filters the ions according to their velocity and the velocity-selected ions are injected into the momentum selector. The momentum selector disperses the ions according to their mass and a mass spectrum is recorded on the ion detector. The velocity and momentum selectors are immersed in magnetic fields B₁ and B₂, respectively, as indicated by the ⊗ symbols. In the velocity selector there is also an electric field E that is perpendicular to the magnetic field B₁.
   8. Figure 2.8 A mass spectrum of a naturally occurring sample of chlorine. The relative areas of the peaks due to the ³⁵Cl and ³⁷Cl ions are in the ratio 3:1, reflecting the natural abundances of the two isotopes.
   9. Figure 2.9 A plot of the square root of the frequency ν of the K_α characteristic X-ray line of an atom versus its atomic number Z. The linear dependence of on Z is in agreement with Moseley's law.
   10. Figure 2.10 The process of X-ray emission from an atom. (a) An incident electron knocks out an atomic electron from the n = 1 shell, leaving a vacancy in that shell. (b) The resulting vacancy in the n = 1 shell is filled by an atomic electron from the n = 2 shell and this is accompanied by the emission of an X-ray.
   11. Figure 2.11 The disintegration of a deuterium nucleus into a proton (¹H) and a neutron by the absorption of a γ-ray.
   12. Figure 2.12 A plot of binding energy per nucleon B/A versus atomic mass number A for the stable nuclides. Except for the region of low mass number, the binding energy curve has a roughly constant value of ∼8 MeV, indicating that the total binding energy of a nuclide is approximately proportional to A. Peaks occur at ⁴He, ¹²C and ¹⁶O, as these nuclides have particularly high values of binding energy. Also indicated are the nuclides Ni (nickel), which occurs close to the maximum of the curve, and U (uranium), which occurs at the high end of the atomic mass range.
   13. Figure 2.13 The potential energy of two nucleons as a function of their separation, resulting from the strong nuclear force between them. A minimum in the potential energy occurs at an equilibrium separation where the repulsive component of the nuclear force is balanced by the attractive component.
   14. Figure 2.14 (a) The potential energy function, V(r), for neutrons that are bound in a nucleus, where the potential energy V is plotted as function of radial distance r from the centre of the nucleus. In the interior of the nucleus, the potential energy is approximately constant, as depicted by the flat bottom of the potential well. (b) V(r) for protons that are bound in a nucleus, again plotted as function of r. In addition to the nuclear force, protons also experience electrostatic repulsive due to other protons. The addition of this electrostatic repulsion to the nuclear attraction reduces the depth of the potential well for protons. Further away from the nucleus, the protons feel just the long-range electrostatic force, and the combination of the attractive nuclear force and the repulsive electrostatic force results in the formation of a Coulomb barrier.
   15. Figure 2.15 Schematic diagram of the first few energy levels in the ¹²C nucleus, showing the arrangement of the six neutrons and six protons in the levels, which are not drawn to scale. The potential wells for the neutrons and protons are conventionally drawn back to back, as shown. Because ¹²C has relatively few protons, the electrostatic energy due to proton repulsion can be neglected and only the potential energy due to the nuclear force needs to be considered.
   16. Figure 2.16 The one-dimensional infinite potential well. The potential energy V(x) is zero between x = 0 and x = L, but rises abruptly to infinity at x = 0 and x = L, so that any particle in the well will be confined to the region 0 < x < L.
   17. Figure 2.17 (a) The wave functions for a particle confined in an infinite potential well, for quantum number n = 1 to 5. These functions are identical to those for standing waves on a string. (b) The corresponding, quantised energies of the particle in the well, which scale as n².
   18. Figure 2.18 A potential well in which one of the walls is not completely rigid; the potential energy does not become infinite at x = L. Also shown is the wave function that corresponds to the ground state of a particle in the well.
   19. Figure 2.19 Plots of k₂ versus k₁ to determine the possible energies of a particle in a finite well potential such as that shown in Figure 2.18, with a well depth V₀ = ℏ²π²/2mL². In this case, there is just one bound state.
   20. Figure 2.20 A three-dimensional finite potential well to model the deuteron nucleus. This model gives just one bound state of the nucleus with a binding energy of approximately 2 MeV, which is in agreement with the experimentally measured value.
   21. Figure 2.21 A particle of energy E approaches a barrier of height V₀, where E < V₀. Quantum mechanically, the particle can tunnel through the barrier. The wave function of the particle exhibits an oscillatory behaviour on either side of the barrier and has an exponentially decreasing amplitude within the barrier.
   22. Figure 2.22 When light is incident upon the glass/air interface of a prism at an angle greater than the critical angle, it is totally internally reflected. However, because of its wave nature, the light can penetrate an air gap and pass into a second prism when the gap is of the order of the wavelength of the light. The critical angle for glass of refractive index 1.5 is 42°.
   23. Figure 2.23 The Segré chart for the stable nuclides. Each nuclide is indicated by a data point, where its neutron number N is plotted versus its atomic number Z. Also shown is the straight line N = Z.
   24. Figure 2.24 An example of a Poisson distribution where P(n) is the probability of n counts in a particular counting period. In this example the mean number of counts is 3.80.
   25. Figure 2.25 The exponential decay of the number N(t) of radioactive nuclei versus time t. N(t) reduces by a factor of 2 for every half-life t¹/₂.
   26. Figure 2.26 A semi-logarithmic plot of activity versus time t. The slope of the straight line is equal to −λ, where λ is the decay constant of the radioactive nuclide. reduces by a factor of 2 for every half-life t¹/₂.
   27. Figure 2.27 A linear plot of activity versus time t. The shaded area of width dt corresponds to the number of nuclei that have lifetimes between t and (t + dt).
   28. Figure 2.28 The behaviour of α particles, β⁻ particles and γ-rays in a magnetic field, all having the same energy. The relatively light, negatively charged β⁻ particles suffer large deflections in the magnetic field. The much heavier and positively charged α particles are deflected by a much smaller amount and in the opposite direction. The γ-rays are uncharged and are not deflected at all.
   29. Figure 2.29 Potential energy function, V(r), for an α particle trapped inside a daughter nucleus, as a function of radial distance r. The radius r′ can be taken as the sum of the radius of the daughter nucleus and of the α particle. Outside the nucleus, V(r) is the potential due to the 1/r Coulomb repulsion between the α particle and the nucleus. Inside the nucleus, V(r) is represented by a square well, which reflects the potential energy due to the nuclear force. The horizontal line E represents the kinetic energy of the α particle at a large distance from the nucleus. The figure also shows schematically the wave function of the α particle. It is oscillatory inside the nucleus, decreases exponentially in amplitude within the barrier and becomes oscillatory again outside the nucleus. Quantum mechanically, the α particle has a finite probability of tunnelling through the Coulomb barrier.
   30. Figure 2.30 The continuous distribution of kinetic energy for β⁻ particles emitted in the β⁻ decay of a radioactive nuclide. The vertical arrow indicates the maximum kinetic energy that a β⁻ particle can have.
   31. Figure 2.31 A ¹³⁷Cs nucleus may decay to a ¹³⁷Ba nucleus with the emission of a β⁻ particle, leaving the ¹³⁷Ba nucleus in an excited state, labelled ¹³⁷Ba^*. The excited ¹³⁷Ba nucleus then decays to its ground state with the emission of a γ-ray. The energy of the γ-ray is equal to the energy difference (0.66 MeV) between the excited and ground states of the ¹³⁷Ba nucleus.
3. Chapter 3
   1. Figure 3.1 Schematic diagram of the binding energy curve of the nuclides. In a fusion reaction, two light nuclei fuse together to form a heavier nucleus that lies closer to the maximum of the curve at A ∼ 56. In a fission reaction, a heavy nucleus splits into two lighter nuclei that again lie closer to the maximum of the curve.
   2. Figure 3.2 The figure shows a beam of particles that is incident upon a slab of material. The reaction rate depends on the flux, , of the incident beam and also on the number, N, of nuclei that are exposed to the beam.
   3. Figure 3.3 The attenuation of a beam of particles as it passes through three slabs of absorbing material. This picture is consistent with the beam flux reducing exponentially with distance travelled in the slabs.
   4. Figure 3.4 A slab of absorbing material of cross-sectional area S that is uniformly illuminated by a beam of particles. The flux, , of the beam reduces steadily as it passes through the slab. We consider the attenuation of the beam by a thin slice of the material that has width dx.
   5. Figure 3.5 The view that the thin slice of the slab in Figure 3.4 would present to the beam of particles. The slice has a cross-sectional area S and each nucleus has a cross-section σ.
   6. Figure 3.6 The cross-section for the absorption of a neutron by lithium, plotted as a function of incident neutron energy. The cross-section rises steadily as the neutron energy decreases, varying inversely as the neutron velocity v. There is a sharp peak in the cross-section due to a resonance at the neutron energy of about 250 keV. The width of the peak is ΔE.
   7. Figure 3.7 Schematic diagram of a liquid drop undergoing harmonic oscillations.
   8. Figure 3.8 Pictorial representation of the fission of a heavy nucleus. The parameter r is a rough measure of the distortion of the nucleus and the subsequent separation of the two fission fragments.
   9. Figure 3.9 The solid blue curve represents the potential energy, V(r), of a nucleus as a function of the parameter r, where r represents the deformation of the nucleus or its separation into two fragments. The figure shows the presence of a fission barrier, whose height E_b is ∼6 MeV. This barrier inhibits the splitting of the nucleus into two fragments. The dashed blue curve represents V(r) for a nucleus with A ∼ 100. For such a nucleus the height E_b of the fission barrier is ∼50 MeV.
   10. Figure 3.10 The cross-sections for neutron-induced fission of ²³⁵U and ²³⁸U. The cross-section for ²³⁵U steadily increases as the neutron energy reduces while, in contrast, the cross-section for ²³⁸U remains zero until the neutron energy is ∼1 MeV. The sharp structures in the ²³⁵U cross-section are due to resonances that occur at energies of excited states of the compound nucleus ²³⁶U.
   11. Figure 3.11 The form of the mass distribution of fragment nuclei following the fission of a heavy nucleus. The vertical scale corresponding to percentage mass yield is logarithmic. There are maxima in the distribution at A ∼ 95 and A ∼ 135, and fission into fragment nuclei of equal or nearly equal mass is very unlikely.
   12. Figure 3.12 When a fragment nucleus ⁹³₃₇Rb₅₆ β⁻ decays to ⁹³₃₈Sr₅₅, there is a 1.4% probability that the ⁹³₃₈Sr₅₅ nucleus will be left in a highly excited state that has enough energy to decay by the emission of a neutron to ⁹²₃₈Sr₅₄.
   13. Figure 3.13 The process of effusion through a tiny hole. The left-hand chamber contains gas at a relatively high pressure, while the pressure in the right-hand chamber is much lower. The effusion rate of the gas is inversely proportional to the square root of its molecular mass.
   14. Figure 3.14 The gas centrifuge technique to enrich uranium. Uranium hexafluoride gas is passed into a cylinder that rotates at a very high rate, ∼500 000 rpm. The rotation creates a strong centrifugal force that causes the heavier ²³⁸UF₆ molecules to move preferentially towards the outer wall of the cylinder, while the lighter ²³⁵UF₆ molecules collect closer to the centre. The ²³⁵UF₆ molecules are then drawn off the rotating cylinder close to the axis of rotation. In practice, the cylinder is ∼3–5 m tall with a diameter of ∼0.2 m.
   15. Figure 3.15 The principle of a laser ionisation technique for uranium enrichment. Monochromatic laser light with energy hν_exc excites only atoms of the ²³⁵U isotope because the photon energy only matches energy E. A second laser with energy hν_ion then ionises the excited ²³⁵U^* atoms so that a beam of ²³⁵U⁺ ions is obtained that can be separated from the neutral ²³⁸U atoms in the beam by electrostatic deflection.
   16. Figure 3.16 The sharp peaks in the spectrum correspond to resonances in the radiative-capture cross-section of ²³⁸U, in the energy range between 10 and 120 eV. At the resonance energies, the cross-section reaches peak values of 10³–10⁴ b. As fast, prompt neutrons are moderated, they steadily lose energy and have to pass through the energy region containing the resonances. If a neutron encounters a ²³⁸U nucleus when its kinetic energy matches a resonance energy, it is highly likely to initiate a radiative-capture reaction and be lost to the chain reaction.
   17. Figure 3.17 A schematic diagram of a nuclear fission reactor based on thermal fission of uranium, showing its main components. The core of the reactor contains the nuclear fuel, the moderator and the control rods. Thermal energy is extracted by a coolant that circulates through the core and the thermal energy is used to produce steam that drives a turbine, which in turn drives an electrical generator. The core is surrounded by the neutron reflector and housed in a steel pressure vessel, and the reactor is contained within a biological shield.
   18. Figure 3.18 The blue curve is a plot of ⟨σ_fv⟩ against plasma temperature for the D-T reaction, where ⟨σ_fv⟩ denotes the value for σ_fv averaged over all relative speeds; σ_f is the fusion cross-section and v is the relative speed of the participating nuclei. Plots of ⟨σ_fv⟩ for alternative fusion reactions: D-D and D-³He are also shown.
   19. Figure 3.19 The helical path of a charged particle in a magnetic field B.
   20. Figure 3.20 A toroidal magnetic field, in which the field lines are joined. This field is obtained with a coil that is wound in the form of a doughnut.
   21. Figure 3.21 A poloidal field, which is generated by passing a current through the fusion plasma itself.
   22. Figure 3.22 The addition of the poloidal field gives a twist to the toroidal field and the resultant magnetic field lines trace out helical paths.
   23. Figure 3.23 A split image showing an interior view of the Joint European Torus vacuum vessel, with a superimposed image of an actual plasma taken with a visible light camera. Only the cold edges of the plasma can be seen, as the centre is so hot that it radiates only in the ultraviolet part of the spectrum. Courtesy of EUROfusion. https://www .euro-fusion.org/2011/08/ the-virtual-vessel-5/ ?view=gallery-428
   24. Figure 3.24 The principle of inertial confinement fusion is to bombard a pellet of fusion fuel from all sides by a short pulse of intense laser radiation. The aim is to compress the pellet to ∼10³ times its normal density and heat it to a temperature of ∼10⁸ K so that thermonuclear fusion occurs.
   25. Figure 3.25 Part of the optics that amplify the energy of laser beams as they make their way towards the target chamber at the National Ignition Facility, USA. Damien Jemison, courtesy of Lawrence Livermore National Laboratory. https://lasers .llnl.gov/media/photo -gallery?id=2012-037864
4. Chapter 4
   1. Figure 4.1 Schematic diagram of the internal structure of the Sun, showing the core, the radiative zone, the convection zone and the photosphere from where the solar radiation is emitted. Also illustrated is the random walk of a photon through the core and radiation zone. In a random walk, each successive step occurs in a random direction with respect to the previous step.
   2. Figure 4.2 A tiny hole in the side of a cavity is a good approximation to a blackbody. Any radiation entering the hole is reflected many times and at each reflection its intensity reduces. If the area of the hole is very small compared with the area of the cavity walls, the radiation has little chance of being reflected out of the cavity and the hole acts as a nearly perfect absorber.
   3. Figure 4.3 Distribution of the radiated power of a blackbody radiator with respect to wavelength λ. The distribution is specified by the spectral radiant exitance W_e(λ), which has units of watts per unit area per unit wavelength (W/m² m). The radiated power due to the wavelength band of width δλ centred at wavelength λ₀ is given by the elemental area W_e(λ₀) × δλ. The total area under the curve is the total power radiated by the blackbody. The figure also defines λ_max, which is the wavelength at which the maximum in the spectral distribution occurs.
   4. Figure 4.4 The spectral power distribution of blackbody radiation for various temperatures. The maxima of the distributions are displaced towards shorter wavelengths as the temperature increases.
   5. Figure 4.5 Spectral power distribution of an incandescent light bulb at a filament temperature of 2500 K. Only about 3% of the emitted radiation falls within the visible region.
   6. Figure 4.6 Electromagnetic standing waves, shown as the blue curves, between two plane mirrors in a laser cavity. The separation of the mirrors is L.
   7. Figure 4.7 Plot of the spectral irradiance against wavelength λ for solar radiation before it enters the Earth's atmosphere, called the extraterrestrial spectrum. Also shown is the corresponding blackbody spectrum for a temperature of 5800 K. The lower horizontal scale is photon wavelength (μm), while the upper horizontal scale is photon energy (eV).
   8. Figure 4.8 Illustration of the various ways in which solar radiation is reflected, scattered or absorbed by the Earth and its atmosphere. About 70% of incoming solar energy is absorbed and the rest is reflected back into space.
   9. Figure 4.9 Comparison of the extraterrestrial spectrum, at the top of the atmosphere, with the solar spectrum at the surface of Earth. The attenuation and dips in the spectrum at Earth's surface are due to various absorption and scattering processes as indicated.
   10. Figure 4.10 Schematic diagrams of (a) vibrational motion of a CO₂ molecule and (b) rotational motion of a CO molecule.
   11. Figure 4.11 Schematic diagram of the photo-dissociation of an O₂ molecule.
   12. Figure 4.12 Solar radiation may come as direct radiation (a) or as diffuse radiation due to scattering of light from atmospheric molecules, called Rayleigh scattering (b), or due to scattering of light from larger particles such as water droplets in a cloud, called Mie scattering (c).
   13. Figure 4.13 When the Sun is vertically above the ground on a clear sunny day, with zenith angle θ_z = 0, each square meter of ground receives ∼1 kW of solar power. This is reduced when the Sun's position makes a non-zero zenith angle, θ_z, with respect to the vertical. This is because the solar radiation has to pass through a greater depth of the atmosphere and because the cross-sectional area A of the radiation is spread over a larger area on the ground.
   14. Figure 4.14 The Earth is titled by the angle of 23.5° with respect to its orbital plane about the Sun. The hour angle, ω, is the angular distance between the meridian of an observer at P and the meridian whose plane contains the Sun. It increases by 15° every hour. The angle of declination, δ, is the angle between Earth's equatorial plane and a line drawn from the centre of Earth to the centre of the Sun. φ is the latitude of point P.
   15. Figure 4.15 Despite the fact that Earth orbits around the Sun, it is simpler to picture the variation in declination angle δ by imagining that the Sun orbits around the Earth. The three orientations of the Sun–Earth system correspond to (in the northern hemisphere): (a) the summer solstice, (b) the autumn solstice, and (c) the winter solstice. At the summer solstice the angle of declination is + 23.5° and at the winter solstice it is − 23.5°.
   16. Figure 4.16 In the summer in the northern hemisphere, the sunlight hits the ground much more directly at point P than at point P ′ at the same latitude in the southern hemisphere. At P ′, a beam of sunlight is spread over a larger area than at P, reducing its irradiance. Furthermore, the sunlight reaching P travels the shorter distance in the atmosphere, and the length of daylight will be much longer in the northern hemisphere. Consequently, the total solar energy deposited per unit per area per day is much greater in the northern hemisphere. The situation is reversed when it is winter in the northern hemisphere.
   17. Figure 4.17 Heat conduction across a slab of material of area A, width L and thermal conductivity κ. The faces of the slab are at temperatures T₁ and T₂, with T₁ > T₂, and H is the heat flux. The direction of heat flow is from high to low temperature.
   18. Figure 4.18 The diffusion of heat along a lagged rod of material. The heat flux into and out of a small slice of the rod between z and z + δz is deduced in the text.
   19. Figure 4.19 (a) A plot of the Earth's surface temperature versus time. (b) A plot of temperature versus time at a distance z = π/β = 9.0 m below the Earth's surface, where the parameter β is defined in the text. Note the time lag between the two plots and the fact that the temperature variation at the given distance below the surface is small.
   20. Figure 4.20 In order to extract shallow geothermal energy, a long length of copper pipe is buried in the ground at a depth of several metres and the pipe is connected to a heat pump. The action of the heat pump is to extract thermal energy from the water in the pipe and deposit it in the building. In the summer months, this process can be reversed so that the air in the building is cooled.
   21. Figure 4.21 The various regions below the surface of Earth. Heat transfer from the mantle maintains a temperature difference across Earth's relatively thin crust of 1000°C, and a temperature gradient of typically 30°C/km. This results in a heat flow through the surface of Earth of ∼0.1 W/m², called deep thermal energy.
   22. Figure 4.22 Schematic diagram showing the principle of operation of a flat plate water heater. The flat plate collects the Sun's radiation and the absorbed heat is transferred to water in copper pipes that are welded to the plate. The area of the flat plate is typically a few square metres.
   23. Figure 4.23 Side view of a flat plate water heater. The flat plate is mounted on a slab of thermally insulating material to reduce heat loss due to conduction. A double-glazed cover shelters the flat plate from the wind and also minimises heat loss due to convection. The heated water is stored in a thermally insulated tank.
   24. Figure 4.24 Blackbody spectra for temperatures of 5800 and 350 K (77°C), corresponding to the temperature of the Sun and a typical heater plate, respectively. The solar spectrum peaks at about 0.5 μm, while the flat plate spectrum peaks at about 10 μm. To fit on the same scale, the solar spectrum has been reduced in height by a factor of approximately 10⁶.
   25. Figure 4.25 Typical absorption and emission coefficients for a metal surface and a semiconductor surface. The vertical axis corresponds to both e_λ and a_λ, as the spectral emissivity of a body, at a given wavelength, is equal to its spectral absorption factor at that wavelength. These surfaces have a high absorption coefficient for solar radiation but a low emission coefficient for the far-infrared radiation emitted by a solar water heater, (∼10 μm).
   26. Figure 4.26 (a) Cross-section of a brick wall that is thermally insulated by a layer of expanded polystyrene. (b) Analogous electrical circuit. The voltage drop V is analogous to the temperature drop across the insulated wall, while the two resistors are analogous to the thermal resistances of the wall and polysytrene lining.
   27. Figure 4.27 (a) Two slabs of insulating material having different values of thermal conductance arranged ‘in parallel’ between two boundaries at different temperatures. (b) Analogous electrical circuit.
   28. Figure 4.28 A hollow cylinder of insulating material with thermal conductivity κ. The radii of the inner and outer surfaces of the cylinder are r₁ and r₂, respectively, and these surfaces are at temperatures T₁ and T₂, respectively, where T₁ > T₂.
   29. Figure 4.29 Vacuum tube collector. The space between the two glass tubes is evacuated, which eliminates thermal conduction and convection. The outer surface of the inner tube is coated with a selective absorption material to maximise absorption of solar radiation. A typical water heater would contain about 20 evacuated tubes.
   30. Figure 4.30 (a) Schematic diagram of a parabolic trough concentrator. Solar radiation is concentrated by parabolic mirrors onto a pipe that runs along the focus of the mirrors. The pipe carries synthetic oil which is heated by the solar radiation. Each mirror can be rotated about the axis containing the pipe to track the position of the Sun. (b) The focusing action of the parabolic mirrors.
   31. Figure 4.31 Schematic diagram of a solar power plant consisting of a central receiver that is surrounded by plane mirrors (heliostats). The heliostats concentrate solar radiation onto the receiving tower, which contains molten salt. This molten salt is pumped to a storage tank and can then be used to heat water for a steam turbine. The storage tank enables the molten salt to be used when the Sun is not shining.
   32. Figure 4.32 A photograph of the Gemasolar Solar Power Facility, which is situated in Fuentes de Andalucía, Spain. It has 2650 heliostats, which have a total reflective area of 304 750 m² and produces an electrical power of ∼20 MW. The inclusion of the molten salt heat storage system enables independent electrical generation for up to 15 hours without any solar radiation. Photo: Sener Group. Taken from http://www.torresolenergy .com/TORRESOL/image -library/en
   33. Figure 4.33 Illustration of the work done when a gas expands at constant pressure. (a) Gas at pressure p is contained in a cylinder with a moveable and frictionless piston. At equilibrium, the downward force due to the weight of the piston and atmospheric pressure is balanced by the upward force due the pressure of the gas. (b) If the piston moves an infinitesimal distance, dx, the work, dW, done by the gas is equal to pdV, where dV = Adx is the infinitesimal change in volume of the gas.
   34. Figure 4.34 A p–V diagram for a gas undergoing an expansion where the pressure p varies with volume V. The elemental area is the work, dW, done when the gas volume changes by dV. The total work performed by the gas is represented by the area under the p–V curve between V₁ and V₂.
   35. Figure 4.35 Pictorial representation of a heat engine that operates between two thermal reservoirs at temperatures of T_H and T_C, where T_H > T_C. A quantity of heat, Q_H, is absorbed from the hot reservoir and a quantity of heat Q_C is discarded to the cold reservoir. The work output of the engine is W = (Q_H − Q_C).
   36. Figure 4.36 Illustration of a gas in a cylinder undergoing a Carnot cycle between hot and cold reservoirs, and the corresponding p–V curves. Step 1: isothermal expansion of gas from a to b while in contact with the hot reservoir at temperature T_H; step 2: adiabatic expansion of gas from b to c; step 3: isothermal compression of gas from c to d while in contact with the cold reservoir at temperature T_C; and step 4: adiabatic compression of gas from d to a. Work is done by the gas during steps 1 and 2. Work is done on the gas during steps 3 and 4. The net work W produced by the gas during the full cycle is represented by the grey area enclosed by the solid curves.
   37. Figure 4.37 The efficiency of an ideal heat engine as a function of the temperature T_H of the hot reservoir. The temperature of the cold reservoir is constant and equal to 10°C. This graph demonstrates the need for high temperatures of the energy source for the efficient conversion of heat into work.
   38. Figure 4.38 The principle of operation of an impulse steam turbine. The nozzle, with a reducing bore, produces a jet of steam with high velocity. This jet is directed at bucket-shaped blades attached to a rotor. The steam jet impinging on the blades exerts a tangential force on the rotor, causing it to rotate. In this way, the kinetic energy of the steam is converted into mechanical energy of the rotor.
   39. Figure 4.39 Schematic diagram of a steam turbine. The turbine has a series of moving rotors, which are mounted on a central axle. The rotors are separated from each other by a set of stationary nozzles, which are attached to the casing of the turbine. Superheated steam from the boiler passes through the nozzles which accelerate the steam to high velocity, producing jets of steam that are directed at the rotor blades. This produces a tangential force on the rotors, causing them to rotate. The rotary motion of a steam turbine about an axle is an important feature because it is ideally suited to turning the spindle of an electrical generator. The steam is converted back to water in the condenser.
   40. Figure 4.40 Successive rotors in a steam turbine are separated from each other by a set of stationary nozzles which are attached to the casing of the turbine. This figure illustrates the arrangement of nozzles and rotors. The nozzles change the direction of the steam jets between successive rotors and optimise the angle at which the steam jets strike the blades.
   41. Figure 4.41 Schematic representation of a refrigerator and also of a heat pump. By putting work into the engine, heat can be transferred from a cold reservoir to a hot reservoir.
5. Chapter 5
   1. Figure 5.1 The photoelectric effect in which UV light is incident upon a metal surface. An electrode is held at a positive potential with respect to the metal surface and collects emitted photoelectrons, which then flow in the external circuit, producing a current i. (Note that conventional current flows in the opposite direction to electron flow.)
   2. Figure 5.2 Schematic diagram of a solar cell, which is formed from a p–n junction in a semiconductor material. Incident photons cause the generation of electron–hole pairs in or close to the depletion layer that exists at the junction. There is an inbuilt electrical field across the depletion layer that causes the electrons to move into the n-type material and the holes to move into the p-type material. Electrons generated by the incident photons flow around the external circuit from the n-type to the p-type, where they combine with the holes. In this way, electrical power is delivered to an external load.
   3. Figure 5.3 Schematic representation of how energy levels of closely spaced atoms evolve into energy bands. The energy gaps between the allowed bands are called forbidden energy gaps because an electron cannot have an energy that lies within these gaps.
   4. Figure 5.4 Electronic band structures for: (a) a typical conductor like sodium in which the valence band is partially full; (b) a conductor like magnesium in which the allowed energy bands overlap; (c) a typical insulator, where there is a large forbidden energy gap between the valence and conduction bands; and (d) a semiconductor where the forbidden energy gap is small.
   5. Figure 5.5 Generation of electron–hole pairs by thermal excitation of electrons from the valence band to the conduction band of a semiconductor. The electrons that are excited to the conduction band leave behind holes in the valence band.
   6. Figure 5.6 If a potential difference is applied across a semiconductor, the free electrons move toward the positive end, while holes appear to move in the opposite direction towards the negative end. Because the holes are positively charged, the flow of holes in one direction is equivalent to a flow of electrons in the opposite direction. Hence, the net current in the semiconductor is the sum of the electron and hole currents.
   7. Figure 5.7 A two-dimensional representation of the crystal structure of silicon. Each silicon atom is surrounded by four other silicon atoms and shares one of its electrons with one of its neighbours, which also contributes an electron to this sharing process.
   8. Figure 5.8 Creation of a spare electron in the crystal lattice of silicon by a donor impurity atom of phosphorus, which has one more electron than silicon.
   9. Figure 5.9 (a) In an n-type semiconductor at absolute zero (T = 0 K), all the electrons that are bound to the donor atoms are in donor levels that lie just below the bottom of the conduction band. (b) At temperatures above absolute zero, these weakly bound electrons can be thermally excited into the conduction band and, at room temperature (300 K), essentially all these electrons are promoted into the conduction band.
   10. Figure 5.10 Creation of a hole in the crystal lattice of silicon by an acceptor impurity atom of aluminium, which has one less electron than silicon. The aluminium atom accepts an electron from a neighbouring silicon atom to complete the valence bond, thus creating the hole at the site of that silicon atom.
   11. Figure 5.11 (a) In a p-type semiconductor at absolute zero (T = 0 K), the electrons remain in the valence band. (b) At temperatures above absolute zero, the electrons can be thermally excited into acceptor levels that lie just above the top of the valence band. This creates holes in the valence band.
   12. Figure 5.12 The figure shows isolated pieces of p-type and n-type semiconductor. For the sake of clarity, only the acceptor and donor ions and the respective majority carriers are shown. The acceptor and donor ions are denoted by the symbols and respectively. In the p-type, there is an abundance of holes, which form the majority carriers. These are denoted by the ‘+’ symbol. Similarly in the n-type, there is an abundance of electrons, which are the majority carriers, denoted by the ‘−’ symbol. In general, the concentration of holes in the p-type will be different from the concentration of electrons in the n-type.
   13. Figure 5.13 The p–n junction in equilibrium. Diffusion of electrons and holes across the junction and their subsequent recombination produce a depletion layer that is devoid of mobile charge carriers. When electrons diffuse away from the n-type region, they leave behind positively charged donor ions. Similarly, when holes diffuse away from the p-type region they leave behind negatively charged acceptor ions. This double layer of charge causes an electric field to be set up across the junction, producing a difference in potential energy between the n and p regions. This produces a potential barrier of height, V₀, that inhibits holes from the p region diffusing into the n region and similarly inhibits electrons diffusing into the p region. At equilibrium, there must be no net current flow across a p–n junction. Hence, the drift current, i_drift, is exactly counterbalanced by the diffusion current i_diff, as indicated by their respective arrows on the figure.
   14. Figure 5.14 A reverse-biased p–n junction. The effect of the bias voltage V is to push the majority charge carriers away from the junction, increasing the width of the depletion region. The bias voltage V is dropped across the high-resistance depletion layer, increasing the height of the potential barrier from V₀ to (V₀ + V) and reducing the diffusion current to essentially zero. The drift current due to thermally generated electron–hole pairs is essentially insensitive to the bias voltage and remains the same as for the p–n junction in equilibrium. Thus the arrow representing the diffusion current is much reduced in length but the length of the drift arrow remains the same.
   15. Figure 5.15 A forward-biased p–n junction. The effect of the bias voltage V is to push the majority carriers toward the junction, reducing the width of the depletion region. The bias voltage is again dropped across the depletion layer and has the effect of reducing the height of the potential barrier to (V₀ − V), so that more majority carriers can surmount the reduced barrier. This increases the diffusion current, which becomes larger than the drift current, as indicated by the length of the respective arrows.
   16. Figure 5.16 A summary of the currents that flow across a p–n junction for: (i) p–n junction in equilibrium, (ii) reverse-biased junction and (iii) forward-biased junction. The drift current is relatively insensitive to bias voltage and is essentially the same for all three conditions. At equilibrium, when there is no bias, the diffusion and drift currents are equal and opposite. When the p–n junction is reverse-biased, the diffusion current is essentially zero. When the p–n junction is forward-biased, the diffusion current is much greater than the drift current.
   17. Figure 5.17 The current–voltage characteristic of a p–n junction, where V is the applied voltage and i is the current flowing through the junction. As the voltage V is increased in the forward direction, corresponding to forward bias, the current i increases rapidly. However, when the voltage V is increased in the reverse direction, corresponding to reverse bias, the reverse saturation current i₀ remains essentially constant. The magnitude of i₀ is exaggerated for the sake of clarity.
   18. Figure 5.18 The density of states function Z(E) for electrons in the conduction band of a semiconductor.
   19. Figure 5.19 The Fermi–Dirac distribution F(E), which gives the probability of a particular state at energy E being occupied by an electron, which is a fermion. The distribution is shown for T = 0 K and for increasing temperatures. At any finite temperature, the probability of occupation of an energy level at E = E_F is 0.5, where E_F is defined as the Fermi energy.
   20. Figure 5.20 (a) The density of state functions Z(E) for the conduction and valence bands of a semiconductor. (b) The Fermi–Dirac distribution in an intrinsic semiconductor, in which the Fermi energy is centred on the middle of the band gap. At finite temperatures this Fermi–Dirac distribution has tails that extend into the valence and conduction bands. (c) The concentration of electrons in the conduction band is obtained by summing the product Z(E)F(E), and is equal to the shaded area as indicated. The concentration of holes in the valence bans is obtained by summing the product Z(E)[1 – F(E)] and is equal to the other shaded area, as indicated. The two shaded areas are the same, reflecting the fact that the number of electrons in the conduction band is equal to the number of holes in the valence band in an intrinsic semiconductor.
   21. Figure 5.21 (a) The Fermi energy in a p-type semiconductor lies just above the top of the valence band. (b) In an n-type semiconductor, the Fermi energy lies just below the bottom of the conduction band.
   22. Figure 5.22 The energy levels in an unbiased p–n junction. Note that the Fermi energy is constant throughout the semiconductor. There is a step change, eV₀, in energy between the n and p regions. This inhibits electrons from the n region diffusing into the p region. Similarly, holes are inhibited from diffusing into the n region.
   23. Figure 5.23 The energy levels in a forward-biased p–n junction. The energy bands in the n-type are raised relative to the bands in the p-type, decreasing the energy difference between them by eV. It then becomes easier for the electrons in the n region to diffuse into the p region and for the holes in the p region to diffuse into the n region. The Fermi level is no longer constant throughout the semiconductor; the Fermi levels on the two sides are displaced by the magnitude of the bias voltage V.
   24. Figure 5.24 The energy levels in a reverse-biased p–n junction. The energy bands in the n-type are lowered relative to the p-type. Electrons from the n region and holes from the p region have an even higher barrier to climb and relatively few of these majority carriers diffuse across the junction. Again, the Fermi level is no longer constant throughout the semiconductor.
   25. Figure 5.25 When a photon is incident upon a semiconductor material, an electron may be promoted from the valence band to the conduction band if the photon has an energy hν that is greater than the band gap E_g.
   26. Figure 5.26 When a photon is incident upon the depletion region of a p–n junction, the photon may produce an electron–hole pair. The inbuilt electric field across the junction causes the electron and hole to separate and drift across the depletion layer. The electron moves into the n region and the hole moves into the p region. In terms of the energy level diagram shown, an electron is promoted to the conduction band and a hole is left in the valence band. The promoted electron then ‘rolls down the hill’ into the n-type, as the conduction band of the n-type lies energetically below the conduction band of the p-type. Similarly, the holes, of opposite charge, move into the valence band of the p-type, which is at a lower energy for them. This movement of electrons and holes across the junction results in a flow of electron current in an external circuit; electrons flow from the n region around the external circuit and enter the p region where they recombine with holes. Electron–hole pairs that are generated within a diffusion length or so of the depletion layer may also contribute to the flow of charge carriers across the junction.
   27. Figure 5.27 The attenuation coefficient curves for silicon (Si) and gallium arsenide (GaAs) as a function of photon energy. Each of the curves shows a sharp onset at the respective band gap energies: 1.11 eV for Si and 1.43 eV for GaAs. Photons of energy less than the band gap of a semiconductor are transmitted with zero or very little absorption. The attenuation coefficient is much lower for the indirect band gap semiconductor Si than the direct band gap semiconductor GaAs over most of the solar wavelength range.
   28. Figure 5.28 (a) A solar cell in the short-circuit mode with short-circuit current i_sc. (b) A solar cell in the open-circuit mode with open-circuit voltage V_oc. (c) A solar cell connected to a load of finite resistance R_L. The current flowing through the load is less than i_oc and the voltage across the load is less than V_oc.
   29. Figure 5.29 A solar cell is conventionally considered to be a battery that delivers a positive current i_cell to an external load. V is the voltage output of the cell.
   30. Figure 5.30 The solid curve is a plot of the output current i_cell of a solar cell against its output voltage V. The shaded area of the graph is the quadrant in which the cell generates electrical power. The figure also indicates how the i_cell–V curve moves upward as the illumination of the cell is increased, as shown by the dashed curves.
   31. Figure 5.31 A solar cell can be represented by a current source, delivering current i_photo, that is connected in parallel with a diode and with the load. The diode represents the p–n junction of the cell. For a given illumination, i_photo is fixed. As the load draws more current, i.e. i_cell increases, the current, i, flowing through the diode must decrease according to i_cell = i_photo − i and this also reduces the voltage V in accordance with the diode equation, i = i₀(e^eV/kT − 1).
   32. Figure 5.32 (a) The i_cell–V curve for a solar cell in the power generation quadrant. Maximum power P_mp occurs for V = V_mp, i_cell = i_mp and is given by the area of the shaded rectangle. The figure also illustrates that, in practice, V_mp ≈ V_oc, and i_mp ≈ i_sc. The ratio of the maximum power, P_mp, and the product, i_scV_oc, is the fill factor. (b) For V = 0 (closed circuit), i_cell = i_sc, and the output power P is zero. For V = V_oc (open circuit), i_cell = 0 and P is also zero.
   33. Figure 5.33 A blackbody spectrum for a temperature of 5800 K. Also indicated is the critical wavelength (1.12 μm) for silicon. Photons lying in region A do not have enough energy to cause promotion of an electron across the band gap. For photon energies greater than the band gap, a fraction of their energy is dissipated as heat in the crystal lattice. The area of region C represents the power that is lost in this way. The area of region B represents the only part of the solar power that can, in principle, be obtained from a solar cell. The area of region B compared with the total area under the blackbody curve is about 45%.
   34. Figure 5.34 For photon energies greater than the band gap, the excited electron goes into an energy level that lies above the bottom of the conduction band. This electron very quickly drops down to the bottom of the conduction band and its excess energy (hc/λ − E_g) is dissipated as heat in the crystal lattice.
   35. Figure 5.35 The process of radiative recombination. An electron recombines with a hole and the result is that a photon is emitted with an energy that is roughly equal to the band gap.
   36. Figure 5.36 The efficiency of a solar cell as a function of band gap energy, E_g, when the cell is illuminated by solar radiation and atmospheric effects are taken into account. This curve is deduced from the theoretical considerations of Shockley and Queisser. The maximum efficiency is 34% and this is achieved for a band gap of 1.34 eV.
   37. Figure 5.37 The construction of a typical solar cell. The solar cell is based on a thin single crystal of silicon, the kind used in microelectronic devices. A p–n junction is formed within the silicon layer. The thickness (∼1 μm) of the n-type region is much less than the thickness (∼300 μm) of the p-type region, but the n-type is more heavily doped. This allows most of the absorption of light to occur in the p region. Typically a cell may be ∼10 cm × 10 cm in size. A grid of wires is placed between an anti-reflectance coating and the semiconductor surface to provide electrical contact to the cell. There is also an electrode at the back of the cell to complete the circuit. Light collection can be enhanced by texturing the front surface of the solar cell by a chemical etching process.
   38. Figure 5.38 An array of silicon solar cells at the Nellis Solar Power Plant located within the Nellis Air Force Base in Clark County, Nevada, USA. The plant has nearly 6000 solar panels and these can be rotated about a single axis to track the Sun. The peak power capacity of the plant is approximately 14 MW. Courtesy of United States Air Force. http://www.nellis.af.mil/photos/media_search.asp?q=solar&btnG.x=0&btnG.y=0
   39. Figure 5.39 (a) The principle of operation of the multi-junction solar cell, which consists of a stack of three sub-cells. The topmost sub-cell is made from a semiconductor that has a large band gap and absorbs photons at the blue end of the solar spectrum. The transmitted photons encounter the middle sub-cell that is made from a semiconductor that has a medium band gap and absorbs lower-energy, green photons. The bottom sub-cell is made from a semiconductor with a low band gap that absorbs the lowest-energy photons, at the red end of the spectrum. The tunnel junctions provide low-loss electrical and optical connections between adjacent sub-cells. (b) The three regions of the solar spectrum that are absorbed by the three different semiconductor materials, each contributing to the photocurrent. The shaded areas represent the electrical power obtained.
   40. Figure 5.40 Various ways to concentrate solar radiation onto a solar cell: (a) a Fresnel lens, (b) a parabolic mirror, and (c) a combination of the two.
   41. Figure 5.41 Schematic diagram of a quantum dot solar cell. The cell consists of quantum dots made from cadmium selenide (CdSe). These are attached to nanoparticles of titanium oxide (TiO₂), which in turn are attached to a transparent but conducting electrode. The quantum dots and nanoparticles are contained in an electrolyte that typically contains polysulphide ions. An incident photon excites an electron into the conduction band of a quantum dot, leaving behind a hole in the valence band. This excited electron rapidly moves into an TiO₂ nanoparticle. From there the electron passes into the transparent electrode and moves around the external circuit to the platinum electrode. The polysulphide ions serve to transport electrons from the platinum electrode back to the quantum dots.
6. Chapter 6
   1. Figure 6.1 Circulation of the Earth's atmosphere in the absence of the Coriolis force. The air rises at the equator, producing low pressure there, and sinks at the colder poles, giving high pressure there.
   2. Figure 6.2 (a) The trajectory of a rocket in the northern hemisphere if the Earth did not rotate. (b) The trajectory of the rocket as the Earth does rotate. For an observer at point A, the rocket is deflected to the right of its direction of travel because of Earth's rotation.
   3. Figure 6.3 (a) The velocity components of a rocket that is fired in a northerly direction. (b) The rocket is deflected to right of its direction of travel according to an observer at point A because of the Earth's rotation.
   4. Figure 6.4 The cells of circulation in the atmosphere caused by the Earth's rotation, and the directions of the resulting global winds.
   5. Figure 6.5 Schematic for the flow of a fluid past a cylinder. (a) Steady or laminar flow, where the overall pattern of streamlines does not change with time; the streamlines curve around the cylinder becoming closer together at the top and bottom of the cylinder. (b) Turbulent flow where the flow pattern changes continuously, making it irregular and chaotic.
   6. Figure 6.6 The figure shows a fluid that flows from left to right down a pipe of decreasing cross-section. The shaded volume on the left depicts the volume of fluid that flows through area A₁ in time Δt. The volume that flows through area A₂ in time Δt is depicted by the shaded area on the right. These volumes must be equal according to the continuity equation.
   7. Figure 6.7 A very small cube of fluid moving along a particular streamline with velocity u = ds/dt. The inset shows an enlarged view of the front face of this cube, which has length l. The pressure p will, in general, vary along the streamline, as also will the velocity of the cube.
   8. Figure 6.8 The principle of operation of a Venturi meter, which is used to measure fluid velocity in a pipe.
   9. Figure 6.9 A container of water with a small hole in its side that produces a jet of water. H is the depth of the container and h is the distance of the hole below the surface of the water. The jet of water strikes the floor at a distance x from the side of the container.
   10. Figure 6.10 (a) When the air flows past an aeroplane wing, the streamlines crowd together above the wing, producing a lower pressure in that region. The upward force on the wing is then greater than the downward force on the wing and there is a net upward or lift force, F_L. The wing is also subject to a drag force F_D, which is due to frictional forces between the wing and the flow of air. (b) F_L can be increased by tilting the blade away from the incident wind direction within a certain range; the angle through which it is tilted is called the angle of attack, α. However, this also increases F_D.
   11. Figure 6.11 A section of a hosepipe with a uniform cross-sectional area A that carries water at velocity u₀ from left to right. In time Δt the volume of water that passes through area A is Au₀Δt.
   12. Figure 6.12 A column of air that is incident upon a wind turbine. The wind speed steadily decreases as the air approaches and passes through the turbine blades, which explains the evolution of the cross-sectional area of the air column. The area A_T represents the area swept out by the blades of the turbine. The wind exerts a force on the rotor blades and power is extracted from the wind.
   13. Figure 6.13 A plot of the ratio P_T/P₀ against a. This ratio is a measure of the fraction of wind power extracted by a turbine and is called the power coefficient. The parameter a is the fractional decrease in wind speed between the initial wind speed and the wind speed at the turbine blades. The maximum value of the power coefficient occurs when a =1/3, when it has the value 0.59.
   14. Figure 6.14 (a) One of the blades of a three-blade rotor as viewed from the front of the rotor. The blades rotate at angular velocity Ω in the plane that is perpendicular to the wind direction. An element of the blade at radial distance r from the rotor hub is indicated by the shaded area. It has linear velocity u_lin = rΩ. The blade can be twisted about its long axis to control the angle of attack. (b) The cross-section of the shaded element as viewed from the blade tip. When a wind flows across a turbine blade, a pressure difference develops between the two surfaces of the blade and a lifting force is produced. As the blade is attached to the hub of the rotor, the lifting force causes the blade to rotate about the axis of the rotor.
   15. Figure 6.15 A turbine blade experiences the incident wind with velocity u₀ coming directly towards it. As the blade is rotating, it also experiences air moving towards it in its plane of rotation with velocity u_rot = u_lin = rΩ. These two winds combine vectorially to produce a resultant wind with velocity u_rel.
   16. Figure 6.16 (a) The forces acting on a turbine blade are the lift force F_L and the drag force F_D. The lift force is perpendicular to the direction of the incoming air flow and is the consequence of the unequal pressure on the upper and lower surfaces of the blade. The drag force is parallel to the direction of incoming air flow, and is due to frictional forces between the wing and the flowing air. The lift force, F_lift, and drag force, F_drag, combine to produce the total force, F_tot, acting on the element. This total force has a component F_pow in the plane of the rotation of the blades and it is this component that produces a torque on the rotor and hence the production of power by the turbine. The magnitude of F_pow, and hence the amount of power absorbed from the wind, is controlled by adjusting the angle of attack α.
   17. Figure 6.17 (a) The horizontal axis wind turbine is the most common type today and typically has three blades. (b) The Darrieus wind turbine exemplifies those types that rotate about a vertical axis.
   18. Figure 6.18 The main components of a horizontal axis wind turbine. These include the rotor, the gearbox and the electrical generator. The rotor consists of the turbine blades and a supporting hub that connects the blades to the main shaft of the turbine. The housing that contains the mechanical and electrical components of the turbine and protects them from the weather is called the nacelle. The whole assembly is mounted on a support tower. The yaw mechanism consists of a large bearing that connects the body of the turbine to its support tower and is used to turn the turbine into the wind. The purpose of the gearbox is to match the rate of rotation of the rotor (5–20 rpm) to that of the generator (∼1000 rpm). Wind turbines also incorporate a brake that can be engaged to close down the turbine for maintenance or for when the wind speed is so high that it could damage the turbine.
   19. Figure 6.19 The operating regions of a wind turbine. At very low wind speeds, there is insufficient torque exerted on the turbine blades to make them rotate. However, as the speed increases, the turbine begins to rotate and to generate electrical power. The minimum speed at which the turbine blades rotate is called the cut-in speed. As the wind speed rises, the output power rises rapidly; the blue shaded area in the figure is where the turbine produces output power. When the wind speed becomes greater than ∼ 14 m/s, the absorbed wind power is limited to a constant value to avoid exceeding safe electrical and mechanical loading limits. This limit is called the rated power output and the speed at which it is reached is called the rated output speed. At wind speeds above the cut-out speed, the turbine rotor is stopped from turning to avoid excessive operating loads.
   20. Figure 6.20 A possible electrical arrangement for a variable-speed wind turbine that is not connected to a national grid. This arrangement provides electrical power for applications, such as resistive elements to heat water that do not need constant AC frequency or supply voltage. It also provides a separate 240 V, 50 Hz AC voltage line for applications that are sensitive to frequency and amplitude.
   21. Figure 6.21 (a) A simple version of an alternator that converts mechanical energy to electrical energy in the form of alternating current. The wire loop of area A rotates with constant angular frequency ω about the axis shown. The magnetic field B is uniform and constant. (b) Side view of the alternator.
   22. Figure 6.22 (a) The variation of the induced emf, ϵ, with time for the alternator in Figure 6.22; (b) the corresponding orientations of the loop.
   23. Figure 6.23 Variation of wind speed, u(z), with vertical height, z. Within the height of local obstacles, wind direction changes erratically and there may be large-scale fluctuations in wind speed. Above this erratic region, u(z) is found to vary logarithmically with vertical height.
   24. Figure 6.24 The London Array offshore wind farm located 20 km off the UK's Kent coast. The London Array has 175 turbines and delivers 630 MW of power. It is the largest wind farm in Europe by megawatt capacity. Courtesy of London Array Limited. http://www .londonarray.com/offshore-2/
   25. Figure 6.25 A schematic diagram of adjacent turbines in a wind farm that are separated by 5D, where D is the rotor diameter.
   26. Figure 6.26 A particularly effective site for a wind turbine is on top of a hill overlooking the surrounding countryside. This gives a wide view of the prevailing wind. In addition, the wind speed will increase towards the top of the hill. The wind becomes compressed on the hillside facing the prevailing wind direction as it reaches the top of the hill. This in turn increases the wind speed.
7. Chapter 7
   1. Figure 7.1 A schematic diagram of a hydroelectric plant. The water reservoir is connected to a turbine by a large pipe called a penstock. The filter keeps debris from entering the turbine, while the valve controls the water flow and hence the rotational speed of the turbine and the electrical generator. The vertical distance h between the surface of the water in the reservoir and the turbine is called the head. The turbine drives a generator that produces electrical power at an output voltage of typically 400 VAC.
   2. Figure 7.2 The figure represents the flow of a viscous fluid in a pipe of internal diameter D. The horizontal arrows represent the magnitude of the velocity of the fluid at various values of radial distance r. The velocity of the fluid is zero at the walls of the pipe due to frictional forces and reaches a maximum at the centre of the pipe. The motion of the fluid is rather like a set of concentric tubes sliding relative to each other; the tube at the centre of the pipe moves fastest and the outermost tube is at rest. The viscous forces oppose the sliding of the tubes. The result is that the velocity profile of the fluid is parabolic in shape.
   3. Figure 7.3 Principle of operation of a Pelton impulse turbine. The turbine has a series of cups attached to a revolving wheel. The jet of water impinges on the cups and is deflected so that the water suffers a change in momentum. This produces a tangential force on the wheel that causes it to rotate.
   4. Figure 7.4 The power delivered to an impulse turbine can be increased by the use of additional water jets. Of course, the total flow of water from the jets must be less than the flow of water out of the water reservoir.
   5. Figure 7.5 Plots of the functions y = f(x) and y = f(x − b). The shapes of the two functions are the same, but y = f(x − b) is displaced by distance b along the positive x-axis with respect to y = f(x).
   6. Figure 7.6 The function y = f(x − vt) at three successive instants of time, separated by time interval δt. The rate at which the function moves to the right is the velocity v.
   7. Figure 7.7 The figure illustrates the travelling sinusoidal wave y = Asin 2π[(x − vt)/λ], where λ and A are the wavelength and amplitude of the wave, respectively. The wave propagates along the x-axis and the displacement is along the y-axis, at right angles to the propagation direction. The wave travels at velocity v in the positive x-direction. The dotted parts of the curves indicate that the wave extends to large distance in both directions.
   8. Figure 7.8 Segment of a taut string between x₀ and x₀ + δx that carries a wave. The directions of the tension T acting on each end of the segment are indicated.
   9. Figure 7.9 (a) Snapshot of a portion of a taut string carrying a travelling sinusoidal wave, over one complete wavelength λ. (b) Variation of the instantaneous velocity dy/dt of the wave. (c) Variation of the instantaneous kinetic energy, K. (d) Variation of the instantaneous potential energy, U.
   10. Figure 7.10 Part of a sinusoidal wave travelling on a taut string at velocity v towards the right. (a) Displacement of the wave. (b) Energy distribution in the wave. The energy is transported with the wave at velocity v.
   11. Figure 7.11 The propagation of the modulated wave in a dispersive medium. The wave is shown at several successive instants of time, separated by time interval δt. The wave is shown as the solid line and is contained within the envelope of the modulation, which is represented by the dashed black lines. The vertical arrows indicate a particular crest of the wave, which travels at the phase velocity v_p. The black dots indicate a particular maximum of the envelope, which travels at group velocity v_g. In this example, v_p > v_g and so the wave crest moves forward through the envelope as the wave propagates. This can be seen from the changing relative positions of the arrows and the bold dots.
   12. Figure 7.12 The figure shows a wave group that results from summing together a large number of sinusoidal waves that cover a narrow range of frequencies centred about a mean frequency. The figure shows the wave group at three successive instants of time, separated by time interval δt for the case where the phase velocity v_p is larger than the group velocity v_g. A crest of the wave, indicated by the arrows, travels a distance 2v_p × δt, while the envelope travels a distance 2v_g × δt. We can imagine crests of the wave appearing at the back (left-hand) side of the envelope, passing through the envelope and disappearing at the front of the envelope. As the amplitude is only non-zero within the spatial extent of the group, it follows that the wave energy is transported at the group velocity v_g.
   13. Figure 7.13 The figure illustrates the motion of a small ball floating on a water surface at nine equally spaced instants of time as a wave passes by. As the wave moves forward, so too does the ball. However, as the ball reaches the wave trough, it starts to move backwards and eventually ends up at its initial position, which coincides with the position of the following wave crest. The ball moves in a uniform circular motion in a clockwise direction as shown in the inset of the figure and there is no net forward movement of the ball as the wave propagates. Notice that the ball never leaves the surface of the water, which is depicted as the blue lines.
   14. Figure 7.14 Motion of a number of water particles that are located on the water surface, which is shown in blue. The figure shows the particles at three equally spaced instants of time. All the particles move clockwise in circular orbits, but there is a phase difference between their respective motions. As a water particle falls when a wave crest passes it by, so the next particle rises to take its place on the crest.
   15. Figure 7.15 The circulating motion of particles in a water wave results in a wave profile that is described as a trochoid. A trochoidal curve can be generated as shown in this figure. Here, the circle of radius R rolls anti-clockwise beneath the black horizontal line. The point A, at a distance H from the centre of the circle, traces out a trochoid as indicated by the blue curve. As can be seen, the shape of the trochoid resembles a sinusoidal wave of wavelength 2πR and amplitude H.
   16. Figure 7.16 In a deep-water wave, the water particles that lie below the water surface also move in a uniform circular motion, but the orbital radius decreases exponentially with vertical distance below the surface.
   17. Figure 7.17 As the radius of orbiting water particles in deep water falls of exponentially, the water at depths below about λ/2 is not affected by the motion of the waves. Hence, buoyancy tanks used to support, say, an offshore oil rig in deep water are not appreciably affected by waves if their depth is greater than about half the wavelength of the waves.
   18. Figure 7.18 In shallow water, the water particles in a wave are affected by the presence of the sea bed. This results in the particles moving in elliptical orbits. As the depth of a water particle increases, its vertical component of velocity decreases but its horizontal component remains the same, and so the ellipses become flatter towards the sea bed.
   19. Figure 7.19 In considering wave motion, we imagine shallow water to be composed of thin slices of water, each contained between two vertical sides. These slices have width δx along the x-axis, length l along the wavefront, and height h₀, equal to the depth of the still water. The width δx of the slices is exaggerated for the sake of clarity. When the surface of the water is perturbed by a wave, the slices change shape to follow the profile of the wave. However, the volume of each slice remains constant.
   20. Figure 7.20 The figure illustrates a wave propagating along the surface of shallow water. It shows how a slice of the water is perturbed to follow the wave profile. The height of the slice increases to h and the width changes to δx + (ξ₂ − ξ₁). Any variation in the height of the water is small compared with the depth of the water and the wavelength of the wave. Hence, we neglect any variation in the height of the perturbed slice across its width.
   21. Figure 7.21 This figure illustrates the difference in hydrostatic pressure across a perturbed slice of water resulting from the variation in height of the water surface as a wave passes by.
   22. Figure 7.22 A perturbed slice of shallow water gains potential energy because its centre of mass rises from h₀/2 to h/2. It also has kinetic energy because it has horizontal velocity v_x.
   23. Figure 7.23 The water particles in a wave on deep water follow circular paths with velocity v = rω, where r is the radius and ω is the angular frequency. The figure shows an elemental volume of water in the wave at the instant of time when its instantaneous velocity v is pointing upwards.
   24. Figure 7.24 In an overtopping energy converter, waves break over the sea wall whose height is above the mean sea level and the water is collected in a reservoir. The collected water is returned to the sea via a turbine, which drives an electrical generator. In order to maximise the amount of collected water, a tapered channel funnels the incoming waves into the reservoir.
   25. Figure 7.25 Operating principle of an oscillating water column energy converter. The water level in the chamber rises and falls in synchronism with the incoming waves and sets up an oscillating column of air. As a result, the air in the chamber is forced forwards and backwards through a Well's turbine and this causes the turbine to rotate. The turbine is connected to a conventional generator that produces the electricity.
   26. Figure 7.26 The Edinburgh duck, also known as Salter's Duck. It floats in the water and as waves pass beneath it, the duck rocks back and forth. This rocking motion absorbs the mechanical energy in the wave. The absorbed energy is converted into electrical energy by a combination of hydraulic rams and an electrical generator that are contained within the duck.
   27. Figure 7.27 Illustration of the Pelamis energy converter, which is a semi-submerged, articulated structure. It consists of long cylindrical sections that are linked to shorter sections by hinged joints. The short sections contain the machinery that generates the electricity. Pelamis is tethered to the seabed by an anchor chain and undulates up and down with the motion of the waves, conforming to the local shape of the wave. The undulating motion of the device is converted into electrical energy by a combination of hydraulic rams and an electrical generator.
   28. Figure 7.28 Schematic diagram of a hydraulic motor. As the sections of the Pelamis undulate with the action of the waves, high-pressure fluid is forced through pumps that are connected to a hydraulic motor. These hydraulic components are located in the short sections of the Pelamis.
   29. Figure 7.29 Action of a hydraulic motor. High-pressure fluid passes through the input port and travels between the gear teeth and the motor housing, before leaving via the output port. This flow of fluid causes the gears to rotate. The shaft of one of the gears is connected to an electrical generator.
   30. Figure 7.30 The Pelamis wave energy converter at sea. Courtesy of Umanji Solar. http://buildipedia.com/aec -pros/public-infrastructure/ pelamis-wave-energy-converter-renewable-energy-from-ocean-waves
   31. Figure 7.31 The Earth and the Moon mutually attract each other as described by the universal law of gravity. The gravitational pull on the water on the side facing the Moon is greater than the gravitational pull on the Earth because it is closer to the Moon. As the water is fluid, it flows to produce a bulge of water on the side facing the Moon. The gravitational pull on the water on the far side of the Earth is less than the gravitational pull on the Earth and so water flows to produce a bulge of water on that side too. Then as the Earth revolves about its axis, a given geographical location on Earth, say point P, experiences two high tides and two low tides each day.
   32. Figure 7.32 A particle on the side of the Earth closest to the Moon experiences a larger gravitational force of attraction from the Moon than an identical particle on the far-side. The difference in these gravitational forces depends on the cube of the distance L.
   33. Figure 7.33 The figure shows the positions of three identical particles, one at the Earth's centre and the others at points A and B on opposite sides of the Earth. The Moon's gravitational forces, F₀, F_A and F_B, that act on these particles are represented by black arrows. The tidal force acting on a particle at a given point is equal to the vector difference between the gravitational force exerted on the particle and the gravitational force, F₀, exerted on an identical particle at the Earth's centre. The resulting tidal forces at points A and B are indicated by the blue arrows and point away from Earth's centre.
   34. Figure 7.34 The figure shows the construction to obtain the tidal force acting on a particle at an arbitrary point P on the Earth's surface. F_P is the gravitational force acting on the particle, while F_O is the gravitational force acting on a particle at the Earth's centre. The blue arrow represents the resultant tidal force F_T.
   35. Figure 7.35 The blue arrows represent the direction and relative strength of the tidal force at various places on the Earth's surface.
   36. Figure 7.36 This figure shows the view of the Earth–Moon system from Polaris, the north star. From this direction the Earth rotates about its axis in an anti-clockwise direction, while the Moon orbits the Earth also in an anti-clockwise direction. Initially, point P on earth is directly below the Moon, which is at position A. After one full revolution of the Earth about its axis, the Moon has moved from A to B. So the Earth must then rotate a further amount until point P is once again directly ‘under’ the Moon. It takes the Earth about 50 minutes to catch up with the Moon.
   37. Figure 7.37 This figure shows the Earth–Moon system when the moon's angle of declination δ has a positive value and when the Moon is directly above the equator, with δ = 0. The figure also compares the orientations of the tidal bulges for these two cases; the tidal bulges have been greatly exaggerated for the sake of clarity. As δ increases, the tidal bulges move to follow the Moon and no longer align with the Earth's equator. From the figure we can see that the two high tides at latitude P will have different heights. Furthermore, at high latitudes such as at Q, there will be only one tide rather than two.
   38. Figure 7.38 When the Moon is in line with the Sun, at positions A and B, the gravitational pull of the Sun enhances the gravitational pull of the Moon. At these times, the tidal range is greatest and the tides are called spring tides. When the Moon is on a line that is perpendicular to the direction of the Sun, at positions C and D, the gravitational pull of the Sun and of the Moon act in different directions. The effect of this is to reduce the tidal range and such tides are called neap tides. The figure shows the corresponding phases of the Moon.
   39. Figure 7.39 In reality, the tidal bulges are not aligned with the Earth–Moon axis. There is a phase lag between the peak of the bulge and the ‘overhead’ position of the Moon because the tidal bulge cannot move quickly enough to keep up with the motion of the Moon.
   40. Figure 7.40 The figure shows a simple model of a bay. The rising and falling tide at the entrance to the bay acts as periodic driving force that causes the body of water within the bay to rise and fall. The body of water also has a resonant frequency. If the frequency of the driving tide (∼12.5 hr) is close to this resonant frequency, a standing wave of water is set up in the bay and this amplifies the tidal range. The first resonant mode in the bay occurs for L = λ/4, where λ is the wavelength of the standing wave. The blue dashed curves represents the elevation of the water in the bay at resonance. Maximum elevation is obtained at the closed end of the bay.
   41. Figure 7.41 An example of the variation of tidal range over the period of a month, which was recorded at Bridgeport, Connecticut, USA. The sea level rises and falls in a sinusoidal-like fashion with a period of ∼12.5 hr, due to the Earth's rotation. The tidal range varies over the course of the month due to the relative motions of the Moon and Sun, giving rise to two spring tides and two neap tides per month.
   42. Figure 7.42 A schematic diagram of a tidal range power plant. By suitable manipulation of the sluice gates, the turbine is driven by water that flows into the reservoir as well as by water that flows out as the tide rises and falls.
   43. Figure 7.43 A photograph of the Rance Tidal Power Station which is located on the estuary of the Rance River in Brittany, France, where the average tidal range is a massive 8 m. The Rance tidal basin has a total area of 22.5 km², and the power station has 24 turbines which produce a peak output power of 240 MW and an average power of 62 MW. Source: http://theearthproject.com/ wp-content/uploads/2016/01/La-Rance-Tidal-Power-Plant.jpg
   44. Figure 7.44 A schematic diagram of the SeaGen tidal current plant at Strangford Lough in Northern Ireland. The rotor diameter of each turbine is 16 m and each turbine drives a generator. The turbines have a feature that allows them to operate for water flowing into or out of the lough. The crossbeam is attached to a collar which slides over a central column. This allows the turbines to be lifted above the water line for routine maintenance.
8. Chapter 8
   1. Figure 8.1 (a) A one-dimensional model of heat diffusion in a cylindrical block of granite rock. A quantity of heat is deposited at the centre of the rock, x = 0, at time t = 0. (b) This heat then diffuses along the rock, setting up a variation in temperature along the rock given by the function T(x, t).
   2. Figure 8.2 Schematic diagram of a pumped hydroelectric storage plant. During periods of low electricity demand, water is pumped to the upper reservoir. Then during periods of high demand the water in the upper reservoir is used to drive a turbine. Often the pump and turbine are combined into a single unit, as shown.
   3. Figure 8.3 Llyn Stwlan, the upper reservoir of the Ffestiniog Pumped Power Station in North Wales. Courtesy of Arpingstone. https://commons.wikimedia .org/wiki/File:Stwlan.dam.jpg
   4. Figure 8.4 Schematic diagram of a compressed air storage plant. During periods of low electricity demand, air is compressed and stored in an underground salt mine or cavern. Heat generated in this compression stage is dissipated in the surrounding rock. During periods of high demand, the compressed air is released to turbines to generate electricity. The compressed air is heated by natural gas before it enters the turbines to compensate for the heat that it lost during the compression stage.
   5. Figure 8.5 Schematic diagram of an advanced adiabatic compressed air energy storage plant. In this system, the heat produced in the compression stage is used to heat oil or molten salt that is stored in thermally insulated containers. This stored thermal energy is then used to heat the compressed air before it enters the turbines.
   6. Figure 8.6 A solid uniform disc has a moment of inertia equal to , where M is its mass and R is its radius.
   7. Figure 8.7 Schematic diagram of a flywheel energy storage system. The flywheel is mounted on the same shaft as a combination electric motor/generator. The electric motor accelerates the flywheel to high angular velocity. The generator produces electrical power on demand by decelerating the flywheel. Energy losses due to frictional forces are minimised by mounting the flywheel assembly on magnetic bearings in an evacuated chamber.
   8. Figure 8.8 Schematic diagram of a supercapacitor. The capacitor plates are formed from porous carbon with an exceptionally high effective surface area. The plates are separated by an electrolyte containing a mix of positive and negative ions. These form layers of electric charge of opposite polarity to the electrode's polarity, and these layers are at an extremely small distance from the electrodes, ∼0.5 nm. These two features, together with the high relative permittivity of the electrolyte, lead to extremely high values of capacitance.
   9. Figure 8.9 Circuit diagram for a DC power supply. The diodes rectify the AC output voltage of the transformer. The capacitor acts as a store of electrical energy that serves to smooth the rectified voltage. The upper part of the figure shows the voltage waveform at various points in the circuit.
   10. Figure 8.10 Schematic diagram of a superconducting magnetic energy storage system. The superconducting coil of the inductor is typically made of an alloy of niobium and titanium and is bathed in liquid helium in a cryostat. A DC current of ∼100 A circulates in the coil, producing a magnetic field of ∼20 T. The stored energy is released by discharging the coil to an external circuit.
   11. Figure 8.11 Schematic diagram of a single cell of a lead-acid battery. The cell has two plates that are made in the form of grids of typically lead-calcium alloy that support the active materials: spongy lead in the cathode and lead oxide in the anode. These plates are immersed in an electrolyte of dilute sulphuric acid, which contains hydrogen ions (H⁺) and sulphate ions (SO₄⁻). The ions migrate through the electrolyte, while electrons travel around the external circuit. The arrows indicate the directions of the ions and electrons during the discharge of the battery.
   12. Figure 8.12 Schematic diagram of a fuel cell using hydrogen as fuel and oxygen as the oxidising agent. It shows the anode and cathode electrodes and the electrolyte that separates them. Each of these electrodes is coated with a thin layer of catalytic material, which facilitates the chemical reactions occurring at each electrode. Hydrogen gas is fed into one side of the cell and oxygen gas into the other. The action of the cell is that hydrogen ions migrate from the anode to the cathode where they combine with oxygen molecules and electrons that have passed through the external circuit to form molecules of water. A voltage of ∼0.7 V is developed between the anode and cathode and a DC current is delivered to the load.
   13. Figure 8.13 Arrangements for sending electrical power via a national grid. The AC voltage from a power station is transformed to much higher voltages to minimise power losses in the transmission line. At the other end of the line, it is transformed to lower voltages to suit industrial, commercial or domestic consumers. The transmission lines are usually made from aluminium, which has relatively low electrical resistivity and density.

Guide

1. Cover
2. Table of Contents
3. 1