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2372 results in Chemical engineering

Page 45 of 119

  • 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.

  • Closure

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 757-757

  • Summary

    The following quote (in italics) at the end of the book by BSL is an appropriate closing remark for this book.

    No engineering project can be conceived, let alone completed, purely through the use of descriptive disciplines, such as transport phenomena. Transport phenomena can, however, prove immensely helpful by providing useful approximations, starting with order-of-magnitude estimates and going on to successively more accurate approximations.

    I have tried to provide the knowledge base for this with scaling concepts, dimensionless arguments, and modeling at three scales, together with some computational snippets.

    Much remains to be done, but the utility of transport phenomena can be expected to increase rather than diminish. Many challenges remain to be met. The quantitative undertones provided by transport phenomena will prove to be an immense help. I hope that you, the students and the readers, will able to participate in this exciting field.

  • 11 - Analysis and solution of transient transport processes

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 377-424

  • Summary

    Mathematical prerequisites

  • • Orthogonal functions; the Sturm–Liouville equation.

  • • Fourier expansion of a function and evaluation of the Fourier coefficients.

  • • The Laplace transform and inverse Laplace transforms.

  • • Complex representation of a sine or cosine function; manipulations of complex variables.

    • In Chapters 8 and 10 we examined a number of steady-state problems in heat and mass transfer. In this chapter we examine unsteady-state problems. Owing to the similar mathematical structure heat and mass transfer problems can be treated in a similar manner. The concentration or temperature is now a function of both time and the spatial coordinates, and therefore the governing equations are now partial differential equations. This adds additional mathematical complexity to the solution. The most general problem in 3D and time and with time-varying boundary conditions needs a numerical solution. However, there are many simpler but nevertheless important problems where analytic solutions are possible and useful. This chapter introduces and solves such problems and illustrates a number of important techniques to obtain the solutions.

    For linear partial differential equations, analytic solutions based on the method of separation of variables are commonly used. It can be applied to problems where the differential equations and the boundary conditions are homogeneous (as will be explained in the text) and applies to problems posed in a finite spatial domain.

  • 9 - Equations of mass transfer

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 296-320

  • Summary

    Mathematical prerequisites

    No new mathematical perquisites are needed for the study of this chapter. The student may wish to revise the Green–Gauss theorem for converting volume integrals to surface integrals for the derivation of macroscopic models. The notion of the substantial derivative must also be revisited.

    Problems involving mass transport are important in many contexts and find applications in chemical, biological, and environmental processes. Some examples were discussed in Chapter 1, and more will follow as we progress further through the study of mass transport processes. We have already solved some illustrative problems involving mass transfer in Chapter 2. The approach demonstrated there employed a problem-to-problem basis. For a specified problem, we studied how the mass conservation principle followed by the constitutive equation can be used for setting up models for problems involving diffusion and in some cases diffusion with reaction. Simple Fick's law was used as the constitutive model for diffusion. The goal of this chapter is to provide a more formal setting and derive general differential equations for mass transfer. The chapter is organized as follows.

    As a preliminary, the concentration and other necessary variables needed to characterize mass transfer are introduced. This is followed by a brief discussion on the conditions prevailing at a phase interface (a gas–liquid boundary, for example).

  • 5 - Equations of motion and the Navier–Stokes equation

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 184-207

  • Summary

    Mathematical prerequisites

    No additional mathematical tools are needed for the study of this chapter. You may wish to review the vector calculus, plus your knowledge of tensors and the various operations on tensor quantities.

    This chapter starts with the development of the differential equation of motion, which is nothing but the statement of the momentum conservation principle. Thus we combine the fluid acceleration calculations derived in Chapter 3 and the representation of forces derived in Chapter 4. This leads to a general equation of motion that is based on the conservation principle alone. This is the basic model for the transport of momentum, which is also called the equation of (fluid) motion. As you may have guessed already, the divergence of the stress tensor will appear as a term, and the model is not in terms of the velocity alone. The model needs to be closed with appropriate constitutive relations between the stress and the strain rate. This requires knowledge of the rheological properties or the flow behavior of the fluid. Hence a discussion of common fluid behaviour is presented, and a classification of the rheological behaviors of fluids is presented next. Fluids obeying a linear relation are referred to as Newtonian fluids.

  • 20 - More convective mass transfer

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 678-699

  • Summary

    This chapter continues the theme of convective transfer from Chapter 12, which discussed mainly problems with internal laminar flow. This chapter extends the study to other flow problems (e.g., external flow) and parallels Chapter 18 on convective heat transfer. The main goal is to examine how mass transfer rates and transfer coefficients can be calculated for illustrative problems for a prescribed or computed velocity profile. First we start with a well-defined flow problem, viz., laminar external flow over a flat plate with a low-flux model for mass transport taking place from (or to) the plate surface. This problem of mass transfer in boundary-layer flow is analyzed by four techniques, namely (i) dimensionless analysis, (ii) scaling analysis, (iii) integral analysis, and (iv) the similarity-transformation method. As discussed earlier, these are some of the widely used tools for solution of transport problems. Applying them to the same problem will give the student a good understanding of these tools. The similarity to heat transfer is also pointed out. One important difference is that mass transfer is often accompanied by chemical reaction. Hence we also examine how the chemical reaction modifies the rate of mass transfer.

    The next problem examined in this chapter is the high-flux case where convection induced by mass transfer cannot be neglected. Various engineering models useful for such systems are presented.

  • 17 - Turbulent-flow analysis

  • P. A. Ramachandran, Washington University, St Louis
  • Book:
    Advanced Transport Phenomena
    Published online:
    12 September 2018
    Print publication:
    25 September 2014, pp 592-618

  • Summary

    Mathematical prerequisites

    No additional prerequisites are needed. Some idea of the Fourier transform is, however, useful to understand the last section of this chapter.

    Turbulence is the result of flow instability. Any small disturbance could lead to a chaotic type of flow with velocity fluctuating on a small time scale around a mean value. This happens (usually) when the viscous forces are much smaller than the inertial forces. The viscosity effect stabilizes the flow and dampens any disturbance. In the absence of significant viscous forces, any disturbance persists and leads to a continuous fluctuation of velocity, leading to turbulent flow. Thus turbulent flows are characterized by small random fluctuations around a mean value and essentially are chaotic unsteady-state phenomena.

    The goal of this chapter is to provide an introductory treatment of turbulent flow. The first goal is look at the time averaging of the Navier–Stokes (N–S) equations. The main idea is that, although the velocity field may be random in nature, the statistical (average) properties of flow are quite reproducible. This suggests that some time-averaged form of equation of motion can be used for practical applications. The problem associated with time averaging is the closure problem. As we have mentioned many times before, the averaging leads to a loss of information.

  • 3 - Phases, interfaces, dispersions, and the first three principles of thermodynamics

  • Elias I. Franses, Purdue University, Indiana
  • Book:
    Thermodynamics with Chemical Engineering Applications
    Published online:
    28 May 2018
    Print publication:
    25 August 2014, pp 47-69

  • Summary

    INTRODUCTION

    Despite the historical tradition of calling the generalized energy principle the “First Law” of thermodynamics (see Chapter 4), there are three other principles that logically precede the First Law. These principles are “axioms” or postulates. They cannot be proven from more elementary principles. In fact, the “First Law” is also an axiom, not a law. These three principles are based on common-sense empirical (or experimental) observations and idealizations of such observations. Some of these principles are covered, in various forms, in certain advanced books on thermodynamics, such as that of Modell and Reid (1983) and the book of Denbigh (1981). In order to describe and explain these principles, the concepts of equilibrium and non-equilibrium phases, in gravity or in the absence of gravity effects, the types of phases, and the extensive and intensive, or specific, properties of a phase will be first defined and explained. These concepts are essential also for discussing the subsequent principles, and for applying thermodynamics. The phase boundaries, and interfaces or surfaces, between different phases will be defined and explained, followed by the definitions of dispersions. Finally, the “empirical temperature,” or the commonly used temperature concept, and equations of state are defined and explained. This is a chapter with a small number of equations.

    I suggest that some readers study first Section 3.2, then Sections 3.8–3.11, and then go back to Sections 3.3–3.7, in order to understand first the importance of the phase concept in the first two new thermodynamics principles, which involve the existence and the properties of the equilibrium phase.

Page 45 of 119