This chapter introduces scheduling in the simplest production environment, the single-unit environment. The statements of different problems are presented in Section 3.1. Two different types of MIP sequence-based models are presented in Section 3.2. Section 3.3 discusses models based on a continuous time grid, while Section 3.4 presents models based on a discrete time grid. A number of extensions are discussed in Section 3.5, and we close in Section 3.6 with some general remarks.

In this chapter, we discuss scheduling in network environments that are unique in the process industries. We study the basic features of this class of problems, including tasks consuming and producing multiple materials, shared utilities, and time-varying utility cost and capacity. In the next chapter, we will consider additional features, including material consumption and production during batch execution, material transfer activities, and unit deterioration. Problems in sequential environments can be expressed using the traditional representation based on batches (orders), stages (operations), and units (machines). This representation is insufficient for problems in network environments. Thus, we start in Section 7.1 with two frameworks, the so-called state-task network (STN) and resource-task network (RTN), for problem representation and the corresponding problem statements. In Section 7.2, we present two models based on a common discrete time grid, and in Section 7.3 we present a model based on a common continuous time grid.

This chapter provides an overview of scheduling and its role within a manufacturing organization, with special focus on the process industries. Section 1.1 introduces some preliminary concepts, including a definition of scheduling, some simple classes of problems, the role of scheduling in a manufacturing supply chain, and a general problem statement for chemical production scheduling. The different chemical production environments, which play a key role in the definition of the different classes of problems, are introduced in Section 1.2. Section 1.3 presents the different classes of problems, and Section 1.4 outlines the different approaches to scheduling. Finally, an outline of the book is presented in Section 1.5.

Methods to simulate colloid suspension structure, dynamics, and rheology are presented and distinguished by whether the solvent is modeled by a continuum or by meso-scale particles, and whether or not a mesh is used. The unmeshed/continuum solvent methods include various forms of Stokesian Dynamics, while the meshed continuum solvent methods include the Arbitrary Lagrangian–Eulerian method and the Immersed Boundary Method. Mesoscopic particles are used on a mesh in the Lattice–Boltzmann (LB) method and are mesh-free in Dissipative Particle Dynamics (DPD). These and other methods are reviewed and guidance provided on which ones are preferred in various situations. For spherical hard particles at high concentrations, without inertia, when accurate treatment of hydrodynamic interactions is needed, versions of Stokesian dynamics set benchmarks for accuracy and efficiency. Other methods become preferable depending on the speed needed for computation, the numbers of particles to be simulated, how well hydrodynamic interactions need to be modeled, whether Brownian motion or particle inertia is important, the shapes and deformability of the particles, the complexity of the flow domain, and whether the solvent is non-Newtonian. Information in this chapter indicates how well the different methods perform in their present state of development.

Chapter 2 introduces the statistical physics description of the rheology of concentrated colloidal suspensions at low Reynolds number. While the solvent is a Newtonian fluid, the suspension exhibits viscoelasticity and non-Newtonian rheology. After explaining the hydrodynamic interactions between Brownian particles mediated by the intervening solvent flow, two theoretical methods for describing the suspension rheology are discussed. In the Langevin Equation method, stochastic particle trajectories are considered under the influence of direct and hydrodynamic interactions, and solvent-induced fluctuating forces. This method is fundamental to simulation schemes where the macroscopic suspension stress is calculated from time-averaging the microscopic stress over representative trajectories. The main focus is on the second so-called generalized Smoluchowski equation (GSmE) method invoking a many-particles diffusion-advection equation for the configurational probability density of Brownian particles in shear flow. Based on the GSmE, real-space and Fourier-space schemes are discussed for calculating rheological properties including the shear stress relaxation function and steady state and dynamic viscosities. Starting from exact Green-Kubo relations for the shear stress relaxation function, the linear mode coupling theory (MCT) and its non-linear extension, termed Integration Through Transients (ITT), are introduced as versatile Fourier-space schemes. They allow for studying the rheology of concentrated suspensions close to glass and gel transitions.

The content of this chapter provides a brief overview of the basic concepts of colloid science that will be used in this book. Foundational knowledge is provided by reviewing our understanding of the simplest case of suspensions of hard spheres. First, the characteristic properties of Brownian hard spheres are presented. This includes a discussion of the relevant forces on and between particles in fluids at rest or during flow. On this basis, the microstructure and the phase behavior of the suspensions under consideration are evaluated. The basic rheology of hard sphere suspensions is reviewed in some detail, covering linear and nonlinear shear behavior, oscillatory flow, and also normal stress differences and shear thickening. The rheology is discussed in relation to the effect of flow on microstructure. As a foundation for understanding more complex suspensions, some basic colloidal interaction potentials are introduced along with their resulting, rich phase behavior. A special section of this chapter is dedicated to thixotropy, as this phenomenon occurs in several of the real life systems discussed in subsequent chapters. An appendix reviews the basic rheological concepts as an aid to the reader.

Some industrial products and processes involving colloid rheology are presented within the framework of this book. Paint rheology is presented and analyzed, where effects such as coating defects are avoided by ensuring the right rheology at each stage. Coating formulation is discussed, in particular for waterborne coatings. Carbon black suspensions are also widely used. Their microstructure is considered at different length scales. The resulting rheology includes time effects such as thixotropy. Measurement problems are reviewed as well as their electrical and dielectric behavior. For bitumen and asphalts, thermorheological behavior and aging are important and are controlled by their specific microstructure. A colloidal approach can also be used here. The rheology is discussed on the basis of the Roscoe model. For cement and cement-based products the rheology during shaping and hardening is linked to the underlying physical processes and interactions in the cement. Thixotropy and yield stress are relevant factors here. In the final part large scale processes are tackled. These often involve mixtures of large and small particles. Practical measurements such as the vane and the slump test are discussed, as well as the prediction of suspension behavior in industrial equipment. The compression behavior can be relevant here, including the compressive yield stress.

When the particle concentration and/or the interparticle forces are sufficiently increased, structures with a solid-like response will develop in colloidal systems. This is dealt with in this chapter, mainly for simple systems, comprised of hard or nearly hard spheres with interparticle attractions. Models have been developed for their state diagrams and have been confirmed by a range of experimental techniques. Gel and glass phases can be distinguished. Glasses occur more commonly at sufficiently concentrated suspensions of hard spheres, but also for suspensions of particles with weak attractions. Gelation occurs at lower volume fractions for suspensions with interparticle attractions, which results in either homogeneous (or equilibrium) gels or heterogeneous gels depending on the nature of the forces. The complex rheology of gels and glasses includes nonlinear viscoelasticity, creep, transient start-up shear, yield strain, and stress. Their nonequilibrium nature has significant consequences for their rheology, including time and shear rate effects. Applying shear causes a change in the microstructure, which recovers when the flow is arrested. For glasses this is known as rejuvenation and aging, respectively. Time and shear effects are stronger in gels where more complex microstructures are involved, leading to a more variable, time-dependent, rheological response.

Softness has a great impact on the properties of colloidal suspensions, especially at high concentrations. Particle deformability due to crowding is responsible for elastic interactions strongly affecting the dynamical properties, which therefore differ from those of hard spheres. The universal aspects of the linear and nonlinear rheological response, based on appropriate scaling, are discussed. Different approaches to determine an effective volume fraction and its role on the low frequency plateau modulus in the glassy and jamming regimes are presented. The flow properties often follow Herschel–Bulkley behavior, with the particle microstructure and interactions affecting the yield stress and causing shear banding or wall slip in some cases. Concentrated suspensions exhibit aging and internal stresses with several common but also distinct features compared to hard sphere glasses. The rich state diagrams of mixtures involving soft colloidal glasses and additives (linear polymers, soft or hard particles) suggest the possibility to tailor their flow properties, often in unprecedented ways, by means of osmotic interactions. This wealth of physical properties in relation to particle interactions can be described by different microstructural, statistical, and phenomenological models which offer a valuable predictive toolbox for understanding the complex and tunable rheology of this class of systems.