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5345 results in Thermal-fluids engineering

Page 75 of 268

  • 1 - Properties, dimensions, and scales

  • from Part I - Use of experimental data
  • William M. Deen, Massachusetts Institute of Technology
  • Book:
    Introduction to Chemical Engineering Fluid Mechanics
    Published online:
    28 May 2018
    Print publication:
    15 August 2016, pp 3-29

  • Summary

    INTRODUCTION

    Fluids, including both gases and liquids, are materials that deform continuously when subjected to shearing forces. If a flowing fluid contains a dye or other tracer, the labeled region tends to change shape from instant to instant. The viscosity of a fluid is a reflection of its resistance to such deformations. This chapter begins with definitions of viscosity and three other material properties that are important in fluid mechanics, namely density, kinematic viscosity, and surface tension. Representative values of each are presented, and some of the differences between Newtonian and non-Newtonian fluids are described.

    Physical quantities have dimensions, such as mass (M), length (L), and time (T). The concepts of mass, length, and time are more fundamental than the units in a particular system of measurement, such as the kilogram (kg), meter (m), and second (s) that underlie the SI system. To have general validity, a physical relationship must be independent of the observer, including the observer's choice of units. That can be achieved by making each variable or parameter in an equation dimensionless, which means that the physical quantities are grouped in such a way that their dimensions cancel. Dimensionless parameters are ratios, and often the numerator and denominator are each the scale of a variable. A scale is the maximum value of something, such as a velocity or force. Thus, the numerical value of such a group reveals how two things compare, and thereby offers insight into what is important in the process or phenomenon under consideration and what might be negligible. Several dimensionless groups that arise in fluid mechanics are discussed.

    Dimensional analysis, the last major topic of the chapter, provides a systematic way to identify the dimensionless groups that are involved in something of interest. In addition to yielding relationships that are more universal, grouping quantities in this manner minimizes the number of independent variables. This greatly facilitates the design of experiments and interpretation of data, as discussed here and explored further in Chapters 2 and 3.

    FLUID PROPERTIES

    Viscosity

    A definition of viscosity is provided by the idealized experiment in Fig. 1.1. Imagine that a fluid fills a space of thickness H between parallel plates, each of which has an area A. Suppose that the upper plate is pulled to the right (in the x direction) at a constant velocity U, while the lower one is held in place.

  • 4 - Fluid statics: pressure, gravity, and surface tension

  • from Part II - Fundamentals of fluid dynamics
  • William M. Deen, Massachusetts Institute of Technology
  • Book:
    Introduction to Chemical Engineering Fluid Mechanics
    Published online:
    28 May 2018
    Print publication:
    15 August 2016, pp 85-109

  • Summary

    INTRODUCTION

    In a static fluid, gravity causes the pressure to increase with depth. Pressures at fluid–fluid interfaces may be influenced also by surface tension. Both kinds of pressure variation have numerous practical consequences, as will be discussed in this chapter. An advantage of beginning a more detailed analysis of fluid mechanics with statics is that viscous stresses, which are the most difficult conceptually and mathematically, are absent in fluids at rest. Thus, we can concentrate first on pressure, gravity, and surface tension, and return later to a more general and precise description of viscous stresses. Quantifying momentum changes within fluids also can be deferred. Viscous stresses and inertia are considered in detail in Chapter 6.

    This chapter begins the use of vector notation, and it is suggested that the reader review certain parts of the appendix before proceeding. Needed particularly is familiarity with vector representation (Section A.2), vector dot products (Section A.3), the gradient operator (Section A.4), and cylindrical and spherical coordinates (last part of Section A.5).

    PRESSURE IN STATIC FLUIDS

    Properties of pressure

    Pressure is a force per unit area, making it a type of stress. It has three properties, whether or not there is flow.

  • (i) Pressure forces only act normal to (perpendicular to) surfaces.

  • (ii) Positive pressures are compressive (rather than tensile). In other words, pressure pushes on (rather than pulls on) a surface.

  • (iii) Pressure is isotropic. That is, the pressure P has a single value at any point in a fluid, and tends to act equally in all directions.

    • Properties (i) and (ii) provide a mechanical definition of pressure and establish the direction of pressure forces when P > 0. Property (iii) is a consequence of (i), as shown at the end of this section. If (i) and (iii) seem contradictory, keep in mind that the pressure itself is a scalar, whereas the pressure force acting on a surface is a vector. The orientation of the surface determines the direction of the force vector, but not the value of P.

    Static pressure equation

    A differential equation that describes pressure variations in a static fluid will be derived using a force balance on the small fluid cube in Fig. 4.1.

  • 12 - Pipe flow: entrance effects, fittings, and compressibility

  • from Part IV - Macroscopic analysis
  • William M. Deen, Massachusetts Institute of Technology
  • Book:
    Introduction to Chemical Engineering Fluid Mechanics
    Published online:
    28 May 2018
    Print publication:
    15 August 2016, pp 322-361

  • Summary

    INTRODUCTION

    The calculation of pressure drops and flow rates for fully developed flow in various kinds of conduits was discussed in Chapter 2. The resistance to such flows is due to a wall shear stress τw that is independent of conduit length, and it was seen that information on τw could be correlated in terms of the friction factor f. All that is needed for design calculations involving incompressible flow in long pipes is fluid property data and plots or formulas for f. However, any system that distributes a fluid to multiple locations, whether in an oil refinery, a microfluidic device, or the body, consists of conduits of finite length that are connected to ones of different size and to tanks or other dissimilar elements. Additional flow resistances arise from sudden changes in diameter or the presence of valves and fittings. Although the additional resistances are negligible in some systems, in others their total exceeds that from plain sections of pipe. Tube inlets coincide with abrupt reductions in diameter and create entrance regions where the flow adjusts to the new cross-section. The resistance there exceeds that for fully developed flow, as detailed in Section 12.2. Section 12.3 extends the discussion of “other” resistances to pipe fittings and valves and includes basic information on pumps, thereby providing a foundation for practical designs.

    Our default assumption for all flow regimes and geometries has been that the fluid density is constant. Assuming a gas to be incompressible is satisfactory if, as is often the case, the change in pressure is a small fraction of the absolute pressure. Gas flows in very long pipes or at velocities comparable to the speed of sound are exceptions, and are the focus of Sections 12.4 and 12.5, respectively. Introduced there are some of the distinctive features of compressible flow, including how “choking” limits flow rates through pipes or nozzles.

    ENTRANCE EFFECTS

    Entrance length

    Something we have not yet accounted for in calculations is that the velocity profile in a tube or other conduit becomes fully developed only after a certain distance from the inlet. An inlet typically involves a sudden reduction in cross-sectional area, as at position 1 in Fig. 12.1. The radius ratio will be denoted as β = R/R0. For a pipe connected to a large tank, β is practically zero.

  • 10 - Turbulent flow

  • from Part III - Microscopic analysis
  • William M. Deen, Massachusetts Institute of Technology
  • Book:
    Introduction to Chemical Engineering Fluid Mechanics
    Published online:
    28 May 2018
    Print publication:
    15 August 2016, pp 261-288

  • Summary

    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.

  • 11 - Macroscopic balances for mass, momentum, and energy

  • from Part IV - Macroscopic analysis
  • William M. Deen, Massachusetts Institute of Technology
  • Book:
    Introduction to Chemical Engineering Fluid Mechanics
    Published online:
    28 May 2018
    Print publication:
    15 August 2016, pp 291-321

  • Summary

    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.

  • Preface

  • William M. Deen, Massachusetts Institute of Technology
  • Book:
    Introduction to Chemical Engineering Fluid Mechanics
    Published online:
    28 May 2018
    Print publication:
    15 August 2016, pp xxi-xxiii

  • Summary

    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.

Page 75 of 268