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37586 results in Cambridge Textbooks

Page 1489 of 1880

  • 6 - Illustrative flow problems

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 208-250

  • Summary

    Mathematical prerequisites

    The ability to find general solutions to second-order differential equations is needed, as well as the procedure to find the integration constants and find solutions to specific problems.

    The chapter aims essentially to illustrate the solution to some common and well-studied fluid-flow problems in laminar flow. The problems analyzed are mainly those where the Navier–Stokes (N–S) equations can be simplified and analytical solutions are possible. Steady-state flow problems are discussed. The types of problems examined here and a brief classification of flow problems are presented below.

    Flows can be classified into external flows and internal flows. External flow is in a semi-infinite domain and the flow is always developing and of boundary-layer nature. Thus there is a region near the solid surface called a boundary layer where the major changes in velocity can be anticipated. The thickness of this boundary layer increases as one moves along the flow direction. Further the flow is 2D (or even 3D) and both components of the velocity are needed to describe the flow correctly. The solution of the N–S equations are therefore more complex than those for internal flows, and for boundary-layer problems they can be classified as parabolic partial differential equations. Details of external flow are postponed until a later chapter. However, some key results that are also useful for study of heat and mass transfer in boundary layers are summarized.

  • 1 - Introduction

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 1-55

  • Summary

    Mathematical prerequisites

  • • The notion of scalars and vectors.

  • • Common coordinate systems (rectangular, cylindrical, and spherical) and the notion of components of vectors in various coordinates.

  • • The gradient of a scalar field and its physical significance.

  • • The dot product of two vectors.

  • • The definition and meaning of the directional derivative.

  • • The expression for gradient in common coordinate systems.

    • The goal of this chapter is to introduce the scope of the subject matter of transport phenomena, which essentially deals with flow of fluids, flow of heat, and flow of mass. This subject provides us with the basic tools to build models for systems or processes. These tools find application in a wide range of areas drawn from different disciplines. The analysis of transport processes which you will learn from this book can be effectively used in these areas. At the end of this chapter we will provide a run-down of some of these important areas of applications.

    The chapter starts off with an introduction of the basic methodology of analysis and modeling of transport processes. Some important definitions and terminology that will reappear throughout the text are presented. Since transport phenomena are closely related to model development, we then provide a general discussion on the (philosophy of) modeling of processes and systems.

  • 19 - Radiation heat transfer

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 656-677

  • Summary

    Radiation can be viewed as the transfer of energy due to electromagnetic wave propagation. Heat transfer at high temperatures is always accompanied by radiation. This chapter provides the theory needed to analyze these processes and illustrates the theory with sample problems. First we review some basics of electromagnetic waves and introduce several laws to describe the energy distribution in radiating systems. In particular, the Stefan–Boltzmann law is widely used in radiation heat transfer calculations.

    The media involved in radiation heat transfer can be either non-participating or participating. The distinction and how this affects the type of modeling effort is described. An important simplification in non-participating media is that only the heat exchange between surfaces present in the system needs to be considered. These types of models are often referred to as surface-to-surface models. How the surfaces are oriented towards each other and how they see each other becomes an important consideration in the calculation of the rate of heat exchange between the surfaces. This is characterized by a view-factor parameter. Illustrative examples of calculation of this parameter are presented. If the surfaces absorb the radiation completely, they are called black bodies. The calculation of the radiation heat exchange in such cases is rather simple. Knowledge of the view factor and the Stefan–Boltzmann law is sufficient to calculate the heat transfer rate in such systems.

  • Preface

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp xvii-xx

  • Summary

    The analysis, modeling, and computation of processes involving the transport of heat, mass, and momentum (transport phenomena) play a central role in engineering education and practice. The study of this subject originated in the field of chemical engineering but is now an integral part of most engineering curricula, for example, in biological, biomedical, chemical, environmental, mechanical, and metallurgical engineering both at undergraduate and at graduate level. There are many textbooks in this area, with varying levels of treatment from introductory to advanced, all of which are useful to students at various levels. However, my teaching experience over thirty years has convinced me that there is a need for a book that develops the subject of transport phenomena in an integrated manner with an easy-to-follow style of presentation. A book of this nature should ideally combine theory and problem formulation with mathematical and computational tools. It should illustrate the usefulness of the field with regard to practical problems and model development. This is the primary motivation for writing this book. This comprehensive textbook is intended mainly as a graduate-level text in a modern engineering curriculum, but parts of it are also useful for an advanced senior undergraduate class. Students studying this book will understand the methodology of modeling transport processes, along with the fundamentals and governing differential equations. They will develop an ability to think through a given physical problem and cast an appropriate model for the system.

  • 14 - Scaling and perturbation analysis

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 484-522

  • Summary

    Mathematical prerequisites

  • • Linear algebra and the notion of the echelon form of a matrix.

  • • The concept of expansion in series in terms of a given (usually small) parameter or in terms of gauge functions.

  • • The concept of a singular perturbation problem.

  • • Matching and patching between the inner and outer solutions.

    • Some of these mathematical topics will be explained in the chapter. For additional reading on asymptotic and perturbation methods the book by Bender and Orszag (1978) is a good starting reference.

    We have presented by now a detailed introduction to the analysis and modeling of transport phenomena. Basic differential equations were developed and examples were shown for all three modes of transport and some coupled problems in transport. We also studied how models at higher levels can be developed by either cross-sectional averaging or volume averaging. The goal of this chapter is to look into the details of some mathematical underpinning of the transport equations as well as some novel methods for solving transport equations. The chapter examines a number of important techniques for the analysis of transport models. Three tools are examined here.

  • 15 - More flow analysis

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 523-565

  • Summary

    Mathematical prerequisites

  • • Basic eigenfunctions of PDEs.

  • • Fundamental solutions of PDEs.

  • • Legendre functions.

  • • Harmonic functions and general solutions to the Laplace equation.

    • Chapter 6 provided some illustration of the analysis of flow problems. The problems studied were usually 1D flow with the velocity changing in the cross-flow direction. The focus of this chapter is on studying somewhat more complex flow problems in further detail. The flow problems studied here are mathematically more involved and involve non-zero velocity components in two or more directions. The problem looks formidable at first sight, but can be broken down into a number of common types, each of which can be solved by a different set of simplifications. Hence it is useful to classify the types of problems which will be studied in this chapter and look at the main features of each of these types of flow. The mathematical method for solving these problems will also be briefly summarized, together with some computational procedures.

  • Low-Reynolds-number flows or Stokes flow. This applies for flow of a highly viscous liquid at relatively low velocity. This has a wide range of applications in particle settling, colloidal suspension flow, and other fields. The viscous effects dominate, and one can reduce the Navier–Stokes (N–S) equation simply to the Stokes equation as discussed in Section 5.4.2 by setting the Reynolds number to zero.

  • […]

  • 12 - Convective heat and mass transfer

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 425-452

  • Summary

    Mathematical prerequisites

    The chapter needs the use of two functions in mathematics:

  • • hypergeometric functions

  • • incomplete gamma functions

    • No additional prerequisites are needed.

    The method of separation of variables and series solution of PDEs should be revised at this stage. Numerical solution of PDEs is another useful mathematical tool for the type of problems examined here.

    In this chapter we examine problems in convective heat and mass transfer mainly for laminar internal flows. Owing to the similar mathematical structure of the problems both heat and mass transfer problems can be treated in a similar manner. The concentration or temperature is now a function of both coordinate directions, namely the axial distance and the radial (or cross-flow) direction, and therefore the governing equations are now partial differential equations (PDEs). Mathematically speaking, the PDEs are similar to that for transient heat transfer studied in the previous chapter, and hence the method of separation of variables can be used. But in the context of convective transport these PDEs, although often still linear, have variable (position-dependent) coefficients. This adds additional mathematical complexity to the solution, mainly in the form of the eigenfunctions and the calculations of the eigenvalues. These details will be examined in this chapter. The structure of this chapter is as follows.

  • 7 - The energy balance equation

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 251-268

  • Summary

    Mathematical prerequisites

    No new mathematical prerequisites are needed for this chapter. However, the student may wish to refresh his or her knowledge of the following topics:

  • • tensor dot and double dot products

  • • index notation and the summation convention

  • • the Green–Gauss theorem

    • The objective of this chapter is to derive differential equations governing the flow of energy. The first goal is to derive the general energy balance which includes all the modes of energy. The internal and kinetic energy will appear as the important terms here.

    A special equation known as the kinetic-energy equation can be obtained without the energy balance simply by starting with the equation of motion (just take the dot product). If this equation is subtracted from the general energy balance, an equation for transport of internal energy alone can be obtained. This equation can be “closed” by applying the thermodynamic relation which relates internal energy to temperature. Thus a differential equation for the temperature field can be obtained. This equation is the starting point in heat-transfer analysis.

    The dimensionless form of the heat equation reveals the characteristic parameters which are important in heat transfer. We also state the various types of boundary conditions which are commonly used in heat transfer. The general dimensionless form together with these (problem-specific) boundary conditions will form the backbone for engineering analysis of heat transport problems.

  • 10 - Illustrative mass transfer problems

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 321-376

  • Summary

    Mathematical prerequisites

    No new mathematical tools are needed for this chapter. Students should refresh their knowledge of solution methods for ordinary second-order differential equations.

    This chapter demonstrates the use of equations of mass transfer for illustrative problems in mass transfer. This chapter is organized in a manner somewhat parallel to Chapter 8, so that the student can appreciate the commonalities in the analysis of heat and mass transfer problems. It would be appropriate at this point to classify the types of problems analyzed in this chapter so that the reader can get an overall perspective.

    The chapter starts with steady-state problems with no superimposed external flow in the system. Such problems are often referred to as diffusion in stagnant media. The terminology stagnant can be misleading in many cases, which will become clear during the detailed discussion. Steady-state problems with no reaction are analyzed first, and problems posed in one spatial dimensions are considered. Such problems have applications, for instance, (i) in diffusivity measurements (the Arnold cell), (ii) in developing mass transfer models, and (iii) in membrane transport. Examples of such applications are presented.

    We then proceed to analyze systems with chemical reactions. Two cases will be considered, namely (i) heterogeneous reactions and (ii) homogeneous reactions, and the difference in the model formulation will be clarified.

  • 21 - Mass transfer: multicomponent systems

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 700-724

  • Summary

    Mathematical prerequisites

  • • Matrix representation for the solution of a set of first-order differential equations (see Chapter 2, Section 2.2.2).

  • • Solution methods for boundary-value problems (see Section 10.9).

    • The main focus of this chapter is on models for multicomponent (systems with more than two components) diffusion and applications of these to common problems in mass transfer. The basic constitutive equation for modeling mass transport is usually based on Fick's law. The key postulate of Fick's law is that the flux of each component is proportional to its own concentration gradient. In other words, the diffusion rate for each component is treated independently by invoking its own diffusivity, and interaction effects are ignored. Strictly speaking, this assumption is valid only for binary systems for ideal gases. However, in view of its simplicity it is also used as an approximation for multicomponent systems. Here some value of a pseudo-binary diffusion coefficient for each of the species in the mixture is assumed, and Fick's law is used as the transport law for each of the species. This is a good approximation under several conditions, e.g., (i) if the components of interest are present in small quantities in a mixture with a large excess of one component (usually referred to as the solvent), (ii) for mixtures where the components are of similar size and chemical type.

  • 16 - Bifurcation and stability analysis

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 566-591

  • Summary

    Mathematical prerequisites

  • • Taylor-series expansion and the notion of the Jacobian.

  • • Differential equations for dynamic systems and their matrix representation.

  • • Complex-variable representation of trigonometric functions.

  • • The concept of an eigenvalue of a differential operator and eigenfuncitons.

    • Non–linear differential equations exhibit a wide range of complexities such as multiple steady states, bifurcations, and chaos. The problem is particularly important since the Navier-Stokes (N–S) equations have the non-linear term (υ • ∇)υ which lies at the root of chaos and turbulent flow behavior. The first goal of this chapter is to introduce mathematical tools to analyze the stability of dynamical systems. A common method is linear stability analysis. Here the base flow or a base steady state is perturbed by a small amount and modeled as a dynamical system. Then we can examine whether the flow returns to the original steady state (i.e., whether the perturbations decay with time) or whether the flow remains transient. When this method is applied to the N–S equations, we can derive a widely used differential equation, viz., the Orr–Sommerfeld equation, which is useful for analysis of the stability of the flow. This chapter provides an introductory treatment of these topics.

  • 13 - Coupled transport problems

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 453-483

  • Summary

    Mathematical prerequisites

    No additional prerequisites are needed for this chapter.

    The study of all of the three transport mechanisms individually was presented in the earlier chapters. The goal of this chapter is to analyze problems where two or more modes of transport have to be analyzed together. Such problems are generally referred to as coupled problems. In industrial practice coupled problems are more common, and hence it is important to get a good understanding of the modeling, solutions, and computations of such problems. Therefore I have devoted a whole chapter to this discussion, and will illustrate the methodology with a number of examples.

    It is useful to distinguish between one-way coupling and two-way coupling. In many problems, the information from one transport problem is needed for the solution of the other transport problem, but the solution for the first transport problem can be done separately without consideration of the second problem. Such problems are referred to as one-way coupling. In a sense, we have already seen one example of this, viz., convective heat and mass transfer. Information on the velocity profile is obtained from momentum transfer considerations only and then used in heat and mass transfer.

Page 1489 of 1880