INTRODUCTION

As discussed in Chapters 2 and 3, laminar flows tend to be unstable at large Reynolds number. A transition to turbulence occurs when Re exceeds a critical value, which depends on the system but is often in the thousands to hundreds of thousands. Turbulence is ubiquitous in everyday life, the natural environment, and large-scale, continuous processes. One of its hallmarks is a marked increase in wall shear stresses. That is indicated by much higher friction factors for flow in pipes (Fig. 2.2) or past flat plates (Fig. 3.5), and is a reflection of the more rapid cross-stream transfer of momentum in turbulent than in laminar flow. In laminar flow, momentum transfer across streamlines is due only to the molecular-level friction that underlies viscosity. In turbulent flow, transient eddies or vortices of varying size are superimposed on the average motion. The eddies, which vary randomly from instant to instant, transport momentum at rates that greatly exceed those from the molecular mechanism.

The random velocity fluctuations associated with the eddies ensure that no two turbulent flows are ever exactly the same. Deterministic modeling of flow details must be abandoned and statistical descriptions used instead. In engineering, the momentary fluctuations are of secondary interest and it is desired mainly to predict time-smoothed or time-averaged quantities. Even so, all available tools must be brought to bear: dimensional analysis, experimentation, analytical theory, and computation. With the present state of knowledge, experimental findings are particularly crucial. Whereas the details of a laminar flow often can be predicted from first principles, turbulence calculations nearly always contain an element of empiricism.

This chapter is organized as follows. After several aspects of turbulence are described qualitatively, there is a discussion of insights into velocity and length scales that can be gained by combining key observations with dimensional analysis. How to reformulate the continuity and Navier–Stokes equations in terms of time-smoothed quantities, a procedure called Reynolds averaging, is then described. The concept of an eddy diffusivity is introduced, which is the classical approach for making turbulence calculations practical. Several examples then illustrate the application of this concept to shear flows in conduits and boundary layers.

INTRODUCTION

Microscopic analysis, as discussed in Part III, is sometimes impractical. Analytical or even numerical solutions of the differential equations can be very hard to obtain for systems with complex shapes or multiple length scales. The difficulty in predicting velocity and pressure fields tends to increase with the Reynolds number and be greatest with turbulent flow. Macroscopic analysis, which employs integral (control volume) rather than differential (point) balances, provides a way to approach such difficult problems. When using only integral balances, much detail and some precision are sacrificed to obtain useful results. Such analysis is valuable even if only the form of a relationship can be predicted, leaving a coefficient to be evaluated experimentally. The benefits of adopting a more macroscopic viewpoint have been illustrated already by the integral analysis of laminar boundary layers Section 9.4). Conserving momentum only for the boundary layer as a whole, and not necessarily at each point, greatly reduced the computational effort but still yielded acceptable accuracy.

The most direct way to derive a macroscopic balance is to state the volume and surface integrals that represent, respectively, the rate of accumulation of the quantity of interest in a control volume and the net rate at which it enters across the control surface. Any internal formation or loss of the quantity is represented by another volume integral. An alternate method, and sometimes the only option, is to integrate a differential conservation equation over the volume of interest. In this chapter these complementary approaches are used to derive integral balances for mass (Section 11.2), momentum (Section 11.3), and mechanical energy (Section 11.4). The momentum equation, which initially considers only fluid–solid interfaces, is extended then to systems with free surfaces (Section 11.5); no modification of the other integral balances is needed. The use of each kind of macroscopic balance is illustrated by examples.

CONSERVATION OF MASS

General control volume

The continuity equation was derived in Section 5.2 by imagining a small, cubic volume of fluid and equating the rate of increase of its mass with the net rate of mass entry across its surface. Letting the volume approach zero led to the differential equation that expresses conservation of mass at a point. The objective now is to derive a mass balance for a control volume of any size or shape.

WHAT IS CHEMICAL ENGINEERING FLUID MECHANICS?

Quantitative experimentation with fluids began in antiquity, and the foundations for the mathematical analysis of fluid flow were well established by the mid 1800s. Although a mature subject, fluid mechanics remains a very active area of research in engineering, applied mathematics, and physics. As befits a field that is both fascinating and useful, it has been the subject of innumerable introductory textbooks. However, only a few have focused on the aspects of fluid mechanics that are most vital in chemical engineering.

Certain results that stem from conservation of mass and momentum in fluids cut across all fields. However, the kinds of flow that are of greatest interest differ considerably among the various branches of engineering. One thing that distinguishes fluid mechanics in chemical engineering from that in, say, aeronautical or civil engineering, is the central importance of viscosity. Viscous stresses are at the heart of predicting flow rates in pipes, which has always been the main application of fluid mechanics in process design. Moreover, chemical engineering encompasses many technologies that involve bubbles, drops, particles, porous media, or liquid films, where small length scales amplify the effects of viscosity. Surface tension, usually not a concern in other engineering disciplines, also can be important at such length scales. In addition, in chemical engineering applications even gases usually can be idealized as incompressible. Another feature of chemical engineering fluid mechanics is an emphasis on microscopic analysis to calculate velocity fields. Determining velocities and pressures, and finding the resulting forces or torques, is often not an end in itself. Detailed velocity fields are needed to predict concentration and temperature distributions, which in turn are essential for the analysis and design of reactors and separation devices. Of lesser concern than in some other disciplines are the fluid dynamics of rotating machinery, flow in open channels, and flow at near-sonic velocities (where gas compressibility is important). Thus, chemical engineering fluid mechanics is characterized by a heightened interest in the microscopic analysis of incompressible viscous flows. Biomedical and mechanical engineers share some of the same concerns.