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2372 results in Chemical engineering

Page 44 of 119

  • 13 - Solutions: advanced and special cases

  • M. Scott Shell, University of California, Santa Barbara
  • Book:
    Thermodynamics and Statistical Mechanics
    Published online:
    05 April 2015
    Print publication:
    16 April 2015, pp 246-279

  • Summary

    In this chapter we extend the fundamental properties of solutions introduced in Chapter 12 to a variety of cases in which a more advanced analysis is required. The uninterested reader may wish to skip these topics. Here we first examine in detail the phenomenology and mathematical description of liquid–vapor equilibrium in multicomponent systems, including nonideal ones. Subsequently, we consider two important classes of systems – polymers and strong electrolytes – for which the ideal solution never provides a reasonable model. We provide the basic models and conceptual underpinnings in these analyses; however, excellent elaborations on these topics can be found in the references noted at the end of the chapter.

    Phenomenology of multicomponent vapor–liquid equilibrium

    Because multicomponent vapor–liquid equilibrium is so important to the chemical process industries, we elaborate on the ideas initially developed in Chapter 12. Before we consider the mathematics of this problem, let us begin by describing some essential phenomenology. The behavior upon heating a pure liquid at constant pressure P is familiar: its temperature increases until the boiling temperature is reached, at which Pvap(Tb) = P. The liquid vaporizes entirely at Tb and then the temperature subsequently increases again as the vapor is heated.

Thermodynamics and Statistical Mechanics
  • An Integrated Approach
  • M. Scott Shell
  • Published online:
    05 April 2015
    Print publication:
    16 April 2015
  • Learn classical thermodynamics alongside statistical mechanics with this fresh approach to the subjects. Molecular and macroscopic principles are explained in an integrated, side-by-side manner to give students a deep, intuitive understanding of thermodynamics and equip them to tackle future research topics that focus on the nanoscale. Entropy is introduced from the get-go, providing a clear explanation of how the classical laws connect to the molecular principles, and closing the gap between the atomic world and thermodynamics. Notation is streamlined throughout, with a focus on general concepts and simple models, for building basic physical intuition and gaining confidence in problem analysis and model development. Well over 400 guided end-of-chapter problems are included, addressing conceptual, fundamental, and applied skill sets. Numerous worked examples are also provided together with handy shaded boxes to emphasize key concepts, making this the complete teaching package for students in chemical engineering and the chemical sciences.

Chromatographic Processes
  • Modeling, Simulation, and Design
  • Roger-Marc Nicoud
  • Published online:
    05 April 2015
    Print publication:
    20 April 2015
  • Addressing all aspects of the design, modeling and simulation of chromatographic processes, this result-oriented primer provides a practical guide to all the necessary approaches, methodologies and tools. Beginning with key definitions and concepts, it builds up from the most simple to the most complex situations, including multicomponent systems, non-uniform velocity profiles, bed instability, particle size distributions, and the influence of complex environments on chromatographic process design. In addition to covering classical approaches, it introduces efficient tools for investigating chromatographic processes, such as the 'Russian-Lego' approach for linear systems, phenomenological models, and specific shortcuts for deriving the key properties of industrial processes. With an emphasis on real-world problems and applications, step-by step modeling design guidelines, and detailed exercises for self-assessment, this is a must-have guide for practitioners and researchers working in chemical, biochemical, food and pharmaceutical engineering.

  • Topical outline

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

  • Summary

    The topical organization of this book is as follows.

    Chapter 1 is the basic introductory material which illustrates the richness of the subject, spanning applications to a wide range of problems in science and engineering. This chapter also provides the introduction to the basics of model building and shows the relationships among models of various levels of hierarchy. The basic vocabulary is introduced, and the physical properties needed in transport problems are discussed. The link between continuum and molecular models is indicated. The chapter concludes with a brief note on the historical development of the subject.

    Chapter 2 illustrates the formulation of model equations for many common transport problems using a basic control-volume-balance type of approach. All three modes of transport are illustrated so that the student can grasp the similarities. Some “standard” problems are illustrated. This chapter is written assuming no significant earlier background knowledge in this field, and is therefore useful to bring such students up to speed.

    The next few chapters, Chapters 3–6, provide the detailed framework for the analysis of momentum transport problems. The kinematics of flow are reviewed in Chapter 3, while the kinetics of flow are discussed in Chapter 4, leading to the derivation of the differential equations for the stress field and the velocity field in Chapter 5.

  • 22 - Mass transport in charged systems

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 725-756

  • Summary

    Transport of ions or charged species is encountered in a wide variety of processes. For example, sodium chloride is ionized in water and exists as sodium and chloride ionpairs. Thus the diffusion of sodium chloride in reality involves the diffusion of positively charged sodium ions and negatively charged chloride ions. Such systems are of importance in electrochemical reaction engineering. Examples can be found, for example, in electro-winning of metals, fuel cells, batteries, etc. Equally important is the transport of charged species in charged membranes or solids carrying a net surface charge. An example of such a system is the ion-exchange membrane, as is widely used in water purification, metal recovery, and selectivity modulation of chemical reactors. Transport of charged species in biological membranes is another example of transport across charged membranes. In addition, a number of techniques to separate large molecules such as DNA and proteins involve application of an electric field. Electrophoresis is such an example. The ionic-transport effects are also important in the study of the surface properties of colloids.

    The key feature to be included in the model for transport in such systems is the charge migration due to the electric field. The constitutive equation for diffusion in such systems is the classical Nernst–Planck equation, which is discussed in Section 22.1.2.

  • 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.

Advanced Transport Phenomena
  • Analysis, Modeling, and Computations
  • P. A. Ramachandran
  • Published online:
    12 September 2018
    Print publication:
    25 September 2014
  • An integrated, modern approach to transport phenomena for graduate students, featuring traditional and contemporary examples to demonstrate the diverse practical applications of the theory. Written in an easy to follow style, the basic principles of transport phenomena, and model building are recapped in Chapters 1 and 2 before progressing logically through more advanced topics including physicochemical principles behind transport models. Treatments of numerical, analytical, and computational solutions are presented side by side, often with sample code in MATLAB, to aid students' understanding and develop their confidence in using computational skills to solve real-world problems. Learning objectives and mathematical prerequisites at the beginning of chapters orient students to what is required in the chapter, and summaries and over 400 end-of-chapter problems help them retain the key points and check their understanding. Online supplementary material including solutions to problems for instructors, supplementary reading material, sample computer codes, and case studies complete the package.

  • 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.

Page 44 of 119