In the preceding eight chapters we have developed our basic tools and techniques. In this chapter and the next we shall illustrate their use in the derivation and analysis of low-dimensional models of the wall region of a turbulent boundary layer. First, the Navier–Stokes equations are rewritten in a form that highlights the dynamics of the coherent structures (CS) and their interaction with the mean flow. To do this, both the neglected (high) wavenumber modes and the mean flow must be modelled, unlike a large eddy simulation (LES), in which only the neglected high modes are modelled. Second, using physical considerations, we select a family of empirical subspaces upon which to project the equations. Galerkin projection is then carried out. In doing this, we restrict ourselves to a small physical flow domain, and so the response of the (quasi)local mean flow to the coherent structures must also be modelled. This chapter describes each step of the process in some detail, drawing on material presented in Chapters 2, 3, and 4. After deriving the family of low-dimensional models, in the last three sections we discuss in more depth the validity of assumptions used in their derivation. In Chapter 10 we shall describe the use of the dynamical systems ideas, presented in Chapters 5 through 8, in the analysis of these models, and interpret their solutions in terms of the dynamical behavior of the fluid flow.

Introduction

Fluids, that is gases and liquids, are self–evidently prerequisites for normal life. They also play a major role in the production of many artefacts and in the operation of much of the equipment upon which modern life depends. Occasionally, a fluid is the ultimate result of a technological process, such as a liquid or gaseous fuel, so that its existence impinges directly on the public consciousness. More often, fluids are intermediates in processes yielding solid materials or objects, and are then contained within solid objects so that their public image is very much less and their significance not fully appreciated. Nevertheless, every single component of modern life relies upon a fluid at some point and therefore upon our understanding of the fluid state.

The gross behavioral features of a fluid are well understood in the sense that it is easy to grasp that a gas has the property to completely fill any container and that a liquid can be made to flow by the imposition of a very small force. However, beyond these qualitative features lie a wide range of thermophysical and thermochemical properties of fluids that determine their response to external stimuli. This analysis concentrates exclusively on thermophysical properties and will not consider any process that involves a change to the molecular entities that comprise the fluid.

Introduction

The power and versatility of corresponding–states (CS) methods as a prediction tool has been pointed out by Mason and Uribe in Chapter 11 of this volume. Here, however, the strong point of corresponding–states principles is stressed: that methods based on the principle are theoretically based and predictive, rather than empirical and correlative. Thus, while CS cannot always reproduce a set of data within its experimental accuracy, as can an empirical correlation, it should be able to represent data to a reasonable degree but, more important, do what a correlation cannot do – estimate the properties beyond the range of existing data. In this chapter a particular corresponding–states method is reviewed that can predict the viscosity and thermal conductivity of pure fluids and their mixtures over the entire phase range from the dilute gas to the dense liquid with a minimum number of parameters. The method was proposed several years ago by Hanley (1976) and also by Mo & Gubbins (1974, 1976). It led to a computer program known as TRAPP (‘TRAnsport Properties Prediction’ (Ely & Hanley 1981a)). The method is also the basis for two NIST Standard Reference Databases – NIST Standard Reference Database 4 (SUPERTRAPP; Ely & Huber 1990) and NIST Standard Reference Database 14 (DDMIX; Friend 1992). Here, the original TRAPP procedure will be discussed as well as some more recent modifications to it. The performance of the model for viscosity and thermal conductivity prediction will also be examined for selected pure fluids and mixtures.

Introduction

The purpose of this chapter, in a book about transport properties, is to give advice to the reader on the best methods for converting the data, which are usually measured as a function of P and T, to a function of ρ and T, which is the form required for the correlating equations; and, in addition, to provide sources for values of the ideal–gas isobaric heat capacities, which are also required for the transport–property calculations. Both of these purposes can be fulfilled by calculations from a single equation of state which has been fitted to the whole thermodynamic surface. Heat capacities of the real fluid are required only for the calculation of the critical enhancement of the thermal conductivity and viscosity, as described in Chapter 6; discussion of these properties in this chapter will be restricted to Section 8.4.4.

An equation of state for a pure fluid relates the various equilibrium thermodynamic properties to one another and will usually be largely an empirical function, although at the limits it will approach theoretical values. In general, entirely theoretical equations of state are unable to represent measured data to within their experimental accuracy. If the accuracy of the calculated transport properties is to be as high as possible, it is important that the most accurate equation of state be used for calculating the appropriate densities.

Accurate knowledge of transport properties of pure gases and liquids, and of their mixtures, is essential for the optimum design of the different items of chemical process plants, for determination of intermolecular potential energy functions and for development of accurate theories of transport in dense fluids. A previous IUPAC volume, edited by Wakeham et al. (1991), also produced by Commission I.2 through its Subcommittee on Transport Properties, has described experimental methods for the accurate determination of transport properties. However, it is impossible to measure these properties for all industrially important fluids, and their mixtures, at all the thermodynamic states of interest. Measurements therefore need to be supplemented by theoretical calculations.

This present volume, which is complementary to the previous publication, discusses the present state of theory with regard to the dilute–gas state, the initial density dependence, the critical region and the very dense gas and liquid states for pure components and mixtures. In all cases, the intention is to present the theory in usable form and examples are given of its application to nonelectrolyte systems. This will be of particular use to chemical and mechanical engineers. The subtitle of this volume ‘Their correlation, prediction and estimation’ reflects the preferred order of application to obtain accurate values of transport properties. Careful correlation of accurate experimental data gives reliable values at interpolated temperatures and pressures (densities), and at different compositions when the measurements are for mixtures. Unfortunately, there are only a limited number of systems where data of such accuracy are available.

The Commission on Thermodynamics of the Physical Chemistry Division of the International Union of Pure and Applied Chemistry is charged by the Union with the duty to define and maintain standards in the general field of thermodynamics. This duty encompasses matters such as the establishment and monitoring of international pressure and temperature scales, recommendations for calorimetric procedures, the selection and evaluation of reference standards for thermodynamic measurements of all types and the standardization of nomenclature and symbols in chemical thermodynamics. One particular aspect of the commission's work from among this set is carried forward by two subcommittees: one on thermodynamic data and the other on transport properties. These two subcommittees are responsible for the critical evaluation of experimental data for the properties of fluids that lie in their respective areas and for the subsequent preparation and dissemination of internationally approved thermodynamic tables of the fluid state and representations of transport properties.

The Subcommittee on Transport Properties has discharged its responsibilities through the work of groups of research workers active in the field drawn from all over the world. These groups have collaborated in the preparation of representations of the viscosity, thermal conductivity and diffusion coefficients of pure fluids and their mixtures over wide ranges of thermodynamic states. The representations have almost always been based upon an extensive body of experimental data for the property in question accumulated over many years by the efforts of laboratories worldwide.

Introduction

It is now estimated that there are some 50 million pure chemicals known of which some 20,000 are listed as high–volume, major chemicals by the European Economic Community (Forcheri & de Rijk 1981) some of which may be transported across national borders. For each pure fluid there are approximately 30 properties which are of technological significance of which twelve are functions of temperature and pressure. If just these twelve properties are considered and it is assumed that measurements at only ten pressures and ten temperatures are required then to provide the necessary information for only one pure fluid requires 1200 measurements. If all the pure species and all possible mixtures from among the set of bulk chemicals are included and composition is allowed as a variable then it is rather easy to estimate that, even for a generous estimate of the rate of experimental data acquisition in the world, the total effort required to fulfill the needs identified in Chapter 2 would exceed 100 billion man–years. This figure makes it immediately obvious that industry's needs for physical property data can never be met by measurement alone. It is therefore necessary to replace a complete program of measurements by an alternative strategy designed to meet the same objective. The philosophy and methods for the establishment of such a strategy have been discussed by many authors and have been updated regularly and most recently by Nieto de Castro & Wakeham (1992).