Consider the simplest nonafterburning, single-spool turbojet engine, which is schematically shown in Figure 12.1. Assuming a viable (i.e., stable compressor) operation mode, there are obvious constrains relating the gas-generator components to one another. These generally enforce the uniformity of shaft speed, as well as ensure the mass and energy conservation principles (Figure 12.2).

Utilization of axial-flow compressor stages (Figure 9.1) in gas turbine engines is a relatively recent development. The history of this compressor type began after an era when centrifugal compressors were dominant (Figure 9.2). It was later confirmed, on an experimental basis, that axial-flow compressors can run much more efficiently. Earlier attempts to build multistage axial-flow compressors entailed running multistage axial-flow turbines in the reverse direction. As presented in Chapter 4, a compressor-stage reaction, in this case, will be negative, a situation that has its own performance degradation effect. Today, carefully designed axial-flow compressor stages can very well have efficiencies in excess of 80%. A good part of this advancement is owing to the standardization of thoughtfully devised compressor-cascade blading rules.

In this chapter, a turbomachinery-related nondimensional groupings of geometric dimensions and thermodynamic properties will be derived.

In this chapter, the flow-governing equations (conservation laws) are reviewed, with applications that are purposely turbomachinery related. Particular emphasis is placed on the total (or stagnation) flow properties. A turbomachinery-adapted Mach number definition is also introduced as a compressibility measure of the flow field. A considerable part of the chapter is devoted to the total-relative properties, which, together with the relative velocity, define a legitimate thermophysical state. Different means of gauging the performance of a turbomachine, and the wisdom behind each of them, are discussed. Also explored is the entropy-production principle, as a way of assessing the performance of turbomachinery components. The point is stressed that the calculation of entropy production may indeed be desirable, for it is the only meaningful performance measure that is accumulative (or addable) by its mere definition.

This chapter introduces the reader to the modelling of particle-particle collisions. We assume that two spherical particles collide along the normal axis to plane of contact – that is, we only examine a head-on impact. In the beginning, attention is paid to the contact mechanics. The objective is therefore to prepare the reader for fundamental analysis. First, we investigate a simple case of a single force acting on a surface. This problem gradually extends to a similar contact between two spherical bodies (Hertz theory). Next, these bodies are allowed to move towards each other, and we observe their deformation – i.e., a head-on collision. The collision is also elastic, so there is no mechanical energy loss upon impact. Later, this issue is expanded upon by introducing dissipative forces during the contact in addition to the elastic forces discussed above. These dissipative forces are of different types: both linear and non-linear. Finally, another topic is introduced, which is plastic deformation. Here, the colliding bodies are allowed to deform permanently.

From a historical viewpoint, the centrifugal compressor configuration was developed and used, even in the propulsion field, well before axial-flow compressors were. Due to their large envelope and weight (Figure 11.1), the common belief that such a “bulky” compressor type has no place except in aerospace applications is not exactly accurate. For example, with a typical total-to-total pressure ratio of, for example, 5:1, it would take up to three axial-compressor stages to absorb similar amounts of shaft work that a single centrifugal compressor stage would. In fact, the added engine length, with so many axial stages, would increase the skin friction drag on the engine exterior, almost as much as the profile drag, which is a function of the frontal area.

Interactions between particles in multiphase flow may also involve adhesion – i.e., an attraction between the particles. This issue is the main topic of this chapter. The first sections of the chapter, however, focus on a primary case: forces acting between two solid surfaces close to each other. A typical example is an interaction between two spherical bodies, which mimic two particles in a multiphase flow. This situation is later extended to a more complex case: the bodies change their shape due to these adhesive interactions. For this, two theories were developed in the literature (JKR and DMT), and they are fully described in the chapter. Later, it is shown how these theories can be adopted to investigate particle-particle collisions in a multiphase flow. In other words, this topic constitutes an extension of the previous chapter, where the focus was on purely “mechanical” interactions without considering any adhesive forces. Finally, the last section of the chapter describes rough surfaces. There is a brief description of how this real-life issue influences the adhesion between two bodies in contact.

One of the parameters that describe particle-particle collision is a coefficient of restitution. This can be simply defined as a ratio of the post-collisional and pre-collisional relative velocity. This chapter is devoted to this topic. As it is straightforward to measure this parameter experimentally, different practical techniques have been used by the researchers, and they are depicted here. Factors such as material properties and pre-collisional conditions are discussed, and it is shown how they influence the value of the coefficient of restitution. It is worth noting that the coefficient of restitution can also be found theoretically by exploiting the relationships previously discussed in the book, especially in Chapter 3. This is described in detail in this chapter. The chapter therefore returns to the previously considered mathematical models. Finally, the chapter concludes with two additional sections focusing on special cases: collisions of granules and nanoparticles., respectively. These particular types of particles have unique features that greatly influence the collision process and restitution coefficient.

This chapter summarises the topics previously discussed in Chapters 2-8. The objective is to illustrate how to create a computer code that simulates a flow of solid particles in a fluid. First, a model is shown that accounts for the motion of particles due to various particle-fluid forces introduced in Chapter 2. Later, it is emphasised that the particles may collide, and this can be described using the techniques mentioned in Chapters 3-8. Finally, a new problem is introduced (not considered in the previous chapters) – collision detection. This issue is crucial for deciding which particles flowing in a system could potentially collide during a time step. The chapter also unveils an algorithm in which the collision detection mode is implemented.

This chapter explains the hard-sphere model of particle-particle collision. This model exploits impulse equations that directly relate the pre-collisional and post-collisional velocities of the particles. Thus, this model does not track the deformation history that was done in the prior chapters. As a result, we obtain ready analytical solutions so that the computational time is short. First, the chapter shows a standard hard-sphere model for a “mechanical” collision of two bodies. Different strategies are presented, such as the so-called two- and three-parameter hard-sphere model. Later, an extension of these models is shown that also accounts for adhesive interactions. Although, due to its simplicity, the hard-sphere model may not account for various physical phenomena between colliding particles, it may still be used in many applications. In this chapter, the reader is again provided with a computer code.

A brief introduction to gas turbine engines was presented in Chapter 1. Review of the different engines included in this chapter reveals that most of these engine components are composed of “lifting” bodies, termed airfoil “cascades,” some of which are rotating, while others are stationary. These are all, by necessity, bound by the hub surface and the engine casing (or housing), as shown in Figures 2.1–2.5. As a result, the problem becomes one of the internal-aerodynamics type, as opposed to such traditional external-aerodynamics topics as “wing theory” and others. Referring, in particular, to the turbofan engines in Chapter 1 (e.g., Figure 1.3), these components may come in the form of ducted fans. These, as well as compressors and turbines, can be categorically summed up under the term “turbomachines.” Being unbound, however, the propeller of a turboprop engine (Figure 1.2) does not belong to the turbomachinery category.

The book’s final chapter pays attention to various issues that can be encountered when investigating multiphase flows. This chapter can be read independently, although on a few occasions it refers to some selected problems from the prior topics. First, this chapter treats a multiphase flow as a system of spherical particles with some given concentration and with some average distance between the particles. Later, the chapter looks into the particle reaction as immersed in a fluid (discussion so-called response times), and it is shown how the presence of the particles influences the fluid flow by discussing the concept of phase coupling and suspension viscosity. Next, we consider the issue of the dispersion of particles as they are subject to turbulent flows, and how the particles may gather in some selected flow zones (preferential concentration). The fact that the particles may be of different sizes is later analysed by investigating the particle size distribution. The final sections of the chapter are dedicated to collision frequency and a particular case of a flow through a particle bed.

Historically, the first axial turbine utilizing a compressible fluid was a steam turbine. Gas turbines were later developed for engineering applications where compactness is as important as performance. However, the successful use of this turbine type had to wait for advances in the area of compressor performance. The viability of gas turbines was demonstrated upon developing special alloys that possess high strength capabilities at exceedingly high turbine inlet temperatures.