Chemical engineering

Summary

Introduction

From a mechanistic point of view, there are two basic forms of heat transfer: conduction and radiation. Other heat transfer processes such as convection, boiling, and condensation involve conduction aided by fluid motion (and sometimes radiation). Our discussion of heat transfer modeling starts with the case of steady-state conduction without flow.

Although steady conduction is normally regarded as a straightforward case, in numerous instances modeling is required and can be helpful. In particular, the electrical analogy between one-dimensional heat conduction and electric current flow allows an easier visualization and can lead to more straightforward solutions. Often this involves simplifying the geometry of a problem. Some two-dimensional cases can be closely modeled using the fin solution. Several means to model two- and three-dimensional conduction are described in this chapter, ranging from circuit models, to graphical approximations, to the use of known solutions for specific geometries. Beyond the geometry, convective and radiative boundary conditions have a substantial influence on overall heat transfer while bringing in another level of uncertainty. These ambiguities may be heightened by less than complete knowledge of material composition and the resulting thermophysical properties.

Molecular View

From a molecular point of view, conduction occurs when a temperature difference is present across a solid, liquid, or gas. By molecular motion, energy is transferred from the warm, more energetic molecules or atoms to the cooler, less energetic molecules. In solids, the energy is transferred by lattice vibrations called phonons and free electron motion. In gases, the energy is transferred by the motion of relatively widely spaced molecules. The energetic molecules collide with less energetic ones and transfer energy to them. In liquids, the molecular spacing is much closer but again the energy transfer takes place by virtue of molecular motion. Note that heat transfer is energy transfer by virtue of a temperature difference. The flow of a warm fluid out of a volume of space represents an energy transfer but not necessarily a heat transfer.

The molecular view of conduction heat transfer is useful to gain a physical understanding of the process. As a good rule of thumb, a material that is a good electrical conductor is also a good conductor of heat.

Summary

Introduction

Heat exchangers transfer heat from one fluid stream to another. They are used in power plants, vehicles, air conditioners, computer data centers, and a wealth of other applications. The fluids may both be liquids, both be gases, or be a combination of the two. The exchangers increase the overall energy conversion efficiency of power and propulsion applications. In some instances, heat exchangers involve simultaneous heat and mass transfer, as in evaporative cooling towers and so-called swamp coolers used to chill air in hot, dry climates. Heat exchanger problems usually involve either the performance prediction of an existing device (sometimes called “rating”) or the design of a new heat exchanger for a given set of performance goals. Calculations of heat exchanger performance bring in principles of conduction for extended or finned surfaces and of convection either inside or outside tubing. For high-temperature applications, thermal radiation can also be important.

In this chapter, we focus on general principles of heat exchanger operation, the influence of geometry, and limiting performance levels. These ideas will be applied to make basic estimates of actual or limiting performance.

Heat Exchanger Geometry

Heat exchangers usually consist of an array of tubes, plates, or other solid surfaces that physically separate the hot stream from the cold one while allowing heat transfer between the two. A heat exchanger is shown schematically in Figure 5.1. Hot fluid enters at THin and leaves at THout . The cold stream enters at TCin and exits at TCout . The internal geometry of the heat exchanger is usually arranged to assure uniform flow distribution of both streams over their respective cross-sectional areas.

The flow arrangement may have the two fluid streams flowing parallel in the same direction or in opposite directions (counterflow). In other designs the two fluids streams may flow at right angles (crossflow), or they may have a combination of these, such as found in a shell and tube heat exchanger. Figure 5.2 illustrates several examples: a compact exchanger in crossflow, a shell and tube exchanger in counterflow, and a plate and frame exchanger.

Energy Balance, Limiting Cases

Usually, conditions in a heat exchanger are in steady state, although devices such as a rotary heat exchanger have periodic conditions in which the solid heat exchanger material alternatively contacts the cold and then the hot fluid streams.

Summary

Convection is heat conduction into a medium that moves. The motion is generally nonuniform, and we are often most concerned with heat flow through a wall perpendicular to the direction of the flow.

The modeling of convection heat transfer always involves two elements. The first is to determine the local rate of heat transfer from wall to fluid. The second is to determine the streamwise energy balance on the fluid. Usually, these two elements are coupled because streamwise temperature variations affect the local temperature differences that drive heat transfer. Only occasionally will the rate of heat transfer be set independently, such as by electrical heating of the wall.

The determination of convection heat transfer rates always begins with the identification of the flow field, and, in particular, it requires one to distinguish between laminar and turbulent flow. Laminar flows often admit relatively simple modeling, in which an estimate of the local boundary layer thickness may be used to determine local convection thermal resistance by analogy to heat conduction. In turbulent flow, thermal resistance has a less obvious relationship to boundary layer thickness, and empirical data must usually be applied. The local laminar thermal resistance is relatively sensitive to the upstream thermal history and wall boundary conditions, whereas the turbulent thermal resistance is not.

For internal flows, bulk flow energy conservation provides a powerful tool for assessing streamwise variations in temperature and other properties. For external flows, the streamwise growth of the boundary layer is of primary importance, requiring consideration of the changing velocity field as well as the temperature history. Such problems can sometimes be approached using the integral energy conservation equation.

In addition to situations that have a mean flow past a wall, some situations of technical importance involve stirred tanks or free surfaces above turbulent pools in which heat transfer to the bulk is driven by turbulent mixing with negligible mean flow. Surface renewal models can be applied to estimate heat transfer coefficients in these cases if certain information about the mixing process is available.

Buoyancy-driven flows add a further level of complexity, in that the fluid velocity depends entirely on the temperature differences that are present during convection. Here the velocity and temperature fields must usually be analyzed simultaneously.

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.

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.

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 τ w that is independent of conduit length, and it was seen that information on τ w 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.

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.

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Dedication

APPENDIX B - Configuration Factors Between Two Bodies of Finite Size

2 - Steady-State Conduction Heat Transfer

Summary

5 - Heat Exchangers

Summary

4 - Modeling Convection

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Author index

8 - Approximations for viscous flows

1 - Properties, dimensions, and scales

Summary

Part IV - Macroscopic analysis

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

Summary

Appendix - Vectors, tensors, and coordinate systems

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

Summary

Subject index

2 - Pipeflow

9 - Laminar flow with inertia

Contents

7 - Unidirectional flow

10 - Turbulent flow

Summary

5 - Fluid kinematics

Part III - Microscopic analysis

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