Chapter 8 focuses on rotor blade technology, covering design, materials, manufacture, and testing. The role of fibre-reinforced composites is discussed, examining their superior mechanical and manufacturing properties. Their property of anisotropy enables composites to be tailored to match the direction of principal stresses in the most material-efficient way. Blade structural design is illustrated using bending theory for a cantilever beam, with stress and strain equations developed for a composite structure. The importance of section thickness and cross-sectional geometry is illustrated using the SERI/NREL blade profiles. An overview of blade attachment methods considers adhesive bonded root studs, T-bolts, and fibre-embedded studs that are integrated during the blade-moulding process. Most large blades are nowadays manufactured by vacuum resin infusion moulding (VRIM), and the chapter includes a description of this technique. There is a section on wood-laminate blades, which are still used in some applications, and comments on blade balancing and testing. The chapter concludes with a review of blade weight and technology trends based on some historic commmercial blade designs.

This chapter summarises the key aerodynamic theory of horizontal-axis wind turbine rotors. The actuator disc concept leads to the relationships between induced velocity, axial thrust, and power extraction. The theory is extended to multiple streamtubes, which, combined with 2D wing theory, establish the basis of blade-element momentum (BEM) theory. A straightforward mathematical treatment of BEM theory is included, with an iterative procedure suitable for coding. Measurements from a full-scale rotor illustrate the applicability of BEM theory but also its fundamental limitations: the latter are described, and measures are outlined to compensate for them in practical BEM codes. Simple relationships are given for the axial and tangential load distributions on an optimal HAWT blade. The structure of the rotor wake is described, leading into a description of vortex-wake theory, which provides a more physically realistic description of the airflow. Vortex wake codes are described in non-mathematical terms. The chapter includes wake measurements from full-scale wind turbines and small models. Vorticity maps from the latter verify the underlying mathematical model of a helical vortex wake.

The final chapter is a largely non-technical overview of economic and political aspects of wind energy policy. The cost of wind energy is assessed in terms of Levelised Cost of Energy (LCoE), with equations given in full and simplified forms. Using a large database, historic installed costs for UK wind both onshore and offshore are given, from the earliest projects to the present day. The observed trends are discussed. Operational and balancing costs are outlined, the latter reflecting the intermittency of wind power. LCoE estimates are made for a range of installed costs and output capacity factors at typical discount rates and compared with current generation prices. The chapter considers the economics of onsite generation with the example of a private business using wind energy to offset demand; the energy displacement and export statistics are extrapolated to compare with a national scenario for 100% renewable electricity generation. The topic of ownership is introduced and examined in the context of the UK’s first community-owned windfarm. The chapter concludes with a brief review of UK renewable energy policy, which originated with legislation to protect the nuclear power industry.

Chapter 7 considers structural loading and response of horizontal-axis machines, with some theoretical background and illustrative measurements from different wind turbine types. The chapter begins with a recap of the dynamics of a single degree of freedom system, leading into a discussion of multi-DOF systems and modal analysis. The cyclic loads affecting a wind turbine structure are described, including wind shear, tower shadow, and rotationally sampled turbulence. The concepts of stochastic and deterministic loading are explained and the principle of aerodynamic damping illustrated. Qualitative descriptions are given of gyroscopic, centrifugal, and electromechanical loading. The phenomenon of blade edgewise stall vibration is explained, with discussion of mechanical damper solutions. The last part of the chapter draws on an early experimental campaign in which the dynamic loading on a full-scale wind turbine was measured and compared with the results of software simulation. Results from the same trials also demonstrate the difference in rotor thrust loading arising from positive and negative pitch control. The chapter concludes with a brief summary of fatigue prediction methods.

If you’ve worked through Chapters 1 and 2, you’ve already seen several references to the Schrödinger equation and its solutions. As you’ll learn in this chapter, the Schrödinger equation describes how a quantum state evolves over time, and understanding the physical meaning of the terms of this powerful equation will prepare you to understand the behavior of quantum wavefunctions. So this chapter is all about the Schrödinger equation, and you can read about the solutions to the Schrödinger equation in Chapters 4 and 5.

In the first section of this chapter, you’ll see a “derivation” of several forms of the Schrödinger equation, and you’ll learn why the word “derivation” is in quotes. Then, in Section 3.2, you’ll find a description of the meaning of each term in the Schrödinger equation as well as an explanation of exactly what the Schrödinger equation tells you about the behavior of quantum wavefunctions. The subject of Section 3.3 is a time-independent version of the Schrödinger equation that you’re sure to encounter if you read more advanced quantum books or take a course in quantum mechanics.

To help you focus on the physics of the situation without getting too bogged down in mathematical notation, the Schrödinger equation discussed in most of this chapter is a function of only one spatial variable (x). As you’ll see in later chapters, even this one-dimensional treatment will let you solve several interesting problems in quantum mechanics, but for certain situations you’re going to need the three-dimensional version of the Schrödinger equation. So that's the subject of the final section of this chapter (Section 3.4).

Origin of the Schrödinger Equation

If you look at the introduction of the Schrödinger equation in popular quantum texts, you’ll find that there are several ways to “derive” the Schrödinger equation. But as the authors of those texts invariably point out, none of those methods are rigorous derivations from first principles (hence the quotation marks). As the brilliant and always-entertaining physicist Richard Feynman said, “It's not possible to derive it from anything you know. It came out of the mind of Schrödinger.”

So if Erwin Schrödinger didn't arrive at this equation from first principles, how exactly did he get there? The answer is that although his approach evolved over several papers, from the start Schrödinger clearly recognized the need for a wave equation from the work of French physicist Louis de Broglie.

The conclusions reached in the previous chapter concerning quantum wavefunctions and their general behavior are based on the form of the Schrödinger equation, which relates the changes in a particle or system's wavefunction over space and time to the energy of that particle or system. Those conclusions tell you a great deal about how matter and energy behave at the quantum level, but if you want to make specific predictions about the outcome of measurements of observables such as position, momentum, and energy, you need to know the exact form of the potential energy in the region of interest. In this chapter, you’ll see how to apply the concepts and mathematical formalism described in earlier chapters to quantum systems with three specific potentials: the infinite rectangular well, the finite rectangular well, and the harmonic oscillator.

Of course, you can find a great deal more information about each of these topics in comprehensive quantum texts and online. So the purpose of this chapter is not to provide one more telling of the same story; instead, these example potentials are meant to show why techniques such as taking the inner product between functions, finding eigenfunctions and eigenvalues of an operator, and using the Fourier transform between position and momentum space are so important in solving problems in quantum mechanics. As in previous chapters, the focus will be on the relationship between the mathematics of the solutions to the Schrödinger equation and the physical meaning of those solutions. And although we live in a universe with (at least) three spatial dimensions in which the potential energy may vary over time as well as space, most of the essential physics of quantum potential wells can be understood by examining the one-dimensional case with time-independent potential energy. So in this chapter, the Schrödinger equation is written with position represented by x and potential energy by V(x).

Infinite Rectangular Potential Well

The infinite rectangular well is a potential configuration in which a quantum particle is confined to a specified region of space (called the “potential well”) by infinitely strong forces at the edges of that region. Within the well, no force acts on the particle. Of course, this configuration is not physically realizable, since infinite forces do not occur in nature. But as you’ll see in this section, the infinite rectangular potential well has several features that make this a highly instructive configuration.