Computation of Conduction and Duct Flow Heat Transfer

Suhas V. Patankar

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9780367450595
• eBook ISBN: 9781351459044
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Mechanical Engineering
• Index: Technology & Engineering

This book describes the computer program CONDUCT in terms of its physical, mathematical, and computational details and its application to heat conduction and duct flow problems. It aims to develop students' problem-solving skills as well as enhance their understanding of these physical processes.

Temperature distribution in a solid with multiple heat sources and sinks

Chapter 1

An Overview

We who live in this computer age are fortunate because we can use computational methods for simulating physical phenomena. In this book, you will be introduced to this exciting activity, which also has great practical utility. An overview of the topics in the book is provided in this chapter.

1.1 Purpose of the Book

Numerical prediction methods play an important role in the analysis and design of practical devices involving heat transfer and fluid flow. The methods, when embodied in suitable computer programs, represent a speedy and economical alternative to experimental measurement. The computational analysis can incorporate the actual details of the geometry, material properties, and boundary conditions and produce complete and detailed information about the fields of temperature, velocity, etc. and about the associated fluxes. In some practical situations, the analysis and design can be wholly done by the use of a computer program. In situations where some experimental testing is desirable, the computational prediction can be used to plan and design the experiments, to significantly reduce the amount of experimentation, and to supplement and enrich the experimental results.

The computer simulation of practical situations involving heat transfer, fluid flow, and related phenomena requires, in general, the solution of a set of nonlinear partial differential equations in three space coordinates and time. Although numerical methods exist for obtaining such a solution, the task of writing and using a sufficiently general computer program for all practical thermofluid processes is quite formidable. Especially to a beginner, such an undertaking may be rather intimidating. A more comfortable entry into the computer simulation activity can be provided by means of a computer program restricted to a subset of the general heat transfer and fluid flow processes. The purpose of this book is to illustrate the construction and application of a general-purpose computer program for a particular subset.

For the computer program presented in this book, two main restrictions are adopted. The first restriction is that of two-dimensionality. There are many physical situations that can be satisfactorily approximated as two-dimensional, and in general, the qualitative features of most practical problems can be studied in a two-dimensional context. Moreover, the framework for a three-dimensional computer program can be conveniently illustrated via a two-dimensional program. The addition of the third dimension to the program makes it more difficult to use by increasing the complexity of the problem and by requiring much greater amounts of computer time and computer memory. Therefore, a two-dimensional program can be considered a more appropriate learning tool. (By the way. a one-dimensional program will be even easier to construct and run. However, for most practical problems, a one-dimensional representation is often too crude to capture, even qualitatively, many interesting features of multidimensional situations.)

The other restriction is that the computer program is designed for conduction-type phenomena. This means that the program solves for the distribution of any scalar variable that is governed by a differential equation similar to the one used for heat conduction. Thus, the program is primarily designed for heat conduction, although it can be used for many other processes that are analogous.

One important class of problems that can be considered as conduction-type is the fully developed flow and heat transfer in ducts. Beyond the entrance region in a duct, there usually exists a region where the longitudinal velocity and the fluid temperature exhibit a certain regular behavior. In this region, the velocity and temperature fields are governed by equations similar to those used for two-dimensional heat conduction.

Other examples of conduction-type situations include: mass diffusion with chemical reaction, flow through porous materials, lubrication flows, heat and moisture transport in soil, potential flows, and electromagnetic fields. Thus, even within the main restrictions of the computer program presented here, a variety of interesting physical phenomena can be simulated and analyzed.

The primary focus in this book is on the analysis of heat conduction and duct flow heat transfer. For that reason, the computer program is given the name CONDUCT, which refers to heat CONduction and DUCT How. The purpose of this book is to describe the computer program CONDUCT in terms of its physical, mathematical, and computational details and to illustrate the application of the program to many problems of engineering interest.

Generally, we look at a computer program as a means of producing quantitative numerical results for a practical problem. However, the computational activity serves an additional, and possibly more important, purpose. Through the computer simulation of physical situations, we can develop a better understanding of, and an insight into, a number of complex physical processes. As you will see. throughout the development of the method and the associated computer program in this book, there is an emphasis on the physical understanding.

1.2 Capabilities and Limitations of CONDUCT

CONDUCT is designed for the solution of partial differential equations of the heat conduction type. It can calculate the distribution of scalar quantities such as the temperature in heat conduction, the concentration in mass diffusion, the velocity and the temperature in fully developed duct flows, the potential in potential flows, the pressure in porous-material flows, and so on. As we shall see later, such phenomena are governed by the general differential equation expressed as Eq. (3.6) in Chapter 3. Therefore. CONDUCT can be used for the calculation of any variable that is governed by a differential equation of the form specified by Eq. (3.6). Further, we restrict attention to only two-dimensional situations; i.e., the quantities of interest can have significant variations in only two space coordinates. The program can be used for steady or unsteady situations.

CONDUCT is designed to employ three coordinate systems: (1) Cartesian (x, y), (2) axisymmetric (x, r), and (3) polar (θ, r). These are illustrated in Fig. 1.1. For each coordinate system, the program uses a grid network of lines drawn in the two coordinate directions. As a result, the program is ideally suited for shapes of the calculation domain that conveniently fit into one of these coordinate systems. The program can still be used for domain shapes that exhibit a number of geometrical irregularities: but, if the domain is too irregular, the use of CONDUCT is not convenient. Thus, the use of grids in three standard coordinate systems imposes some limitation on the geometrical shapes that can be conveniently handled. We could have removed the limitation, but this would have been at the expense of making the program much more complicated to construct, understand, and use.

For the polar (θ, r) coordinate system, a specific restriction must be mentioned. The circumferential coordinate θ is not allowed to cover the full circle of 360° unless the values of the dependent variables are known at 0 = 0 and θ = 360° or unless these locations represent lines of symmetry. This restriction is not as serious as it appears. For most problems with the full 360 degrees of the circular or annular geometry, there are two or more radial lines of symmetry; so the solution is required only over a sector bounded by two successive symmetry lines. This can be handled by CONDUCT, since the extent of 0 will then be less than 360°. For example, consider the fully developed flow in a circular duct with radial internal fins. If the fins are identical in shape and uniformly spaced over the duct perimeter (which is usually the case in practice), it is sufficient to use CONDUCT to analyze the region between two successive fins (or rather, between the centerline of one fin and the location halfway towards the next fin). If the fins are nonuniformly spaced with no discernable lines of symmetry. CONDUCT cannot be used for such a problem.

Fig 1.1 The three coordinate systems

Within these overall limits, CONDUCT can be applied to a wide variety of problems in heat conduction, fully developed duct flow, and analogous phenomena. The properties like the conductivity or viscosity can be nonuniform; they may depend on position (as in a composite material) and on the temperature and other variables. The duct flow can be laminar or turbulent, Newtonian or nonNewtonian. There can be internal heat generation in a conduction problem: the generation may depend on position and/or on temperature. For all problems, a variety of boundary conditions can be present. Once you fully understand the scope and limitations of the program, you will be able to design a large variety of imaginative applications.

1.3 The Structure of CONDUCT

One purpose in introducing you to the program CONDUCT is to illustrate the construction of a general-purpose computer program, which can be used to solve, within its overall limitations, an endless variety of seemingly different physical problems.

CONDUCT is constructed in two parts: the invariant part and the adaptation part. The invariant part contains the general calculation scheme that is common to all possible applications within the overall restrictions of the program. It is written without any knowledge or assumption about the particular details of the problem to be solved. Normally, there would be no need for you to make any changes in the invariant part of the program. The adaptation part provides the problem specification. It is here that the actual details of the problem such as the geometry, material properties, heat sources, reaction rates, boundary conditions, desired output, etc. are supplied. It then follows that the adaptation part cannot be written “in advance” for the endless variety of practical problems to which the program can be applied. What can be provided is the framework for the adaptation part: but its contents must be written “on demand” to specify the problem at hand. Thus, a general-purpose program of this type consists of the complete invariant part and a skeleton of the adaptation part. The latter is to be completed by the user according to the general instructions provided with the program. Within some overall restrictions, there is considerable freedom in designing the adaptation part. Very complex adaptations can be designed, the possibilities being limited only by the imagination of the user.

This type of program structure may be somewhat unfamiliar to you. Many of you are probably accustomed to programs that are complete and require only the numerical input data. Such programs, even when they are highly complicated, cannot match the virtually unlimited flexibility and the open-ended applicability of a program like CONDUCT, in which you can provide the problem description and design the desired output by writing a Fortran subroutine.

Of course, you do not need this flexibility when you use the program for routine computations for a small class of problems. For such applications, you can design the adaptation part such that it works in an automatic manner, simply asking for the values of a few input parameters. Thus, you treat CONDUCT as a toolkit for constructing specialized programs that, although of limited applicability, are easy to use.

It may appear that the use of a general-purpose computer program for a simple problem is rather cumbersome, because the program asks too many questions. It needs to know: the number of dependent variables; variations of viscosity, conductivity, and diffusion coefficients: the distributions of the source and sink terms for all the variables: the details of the boundary conditions for the relevant equations; and so on. However, certain facilities in CONDUCT have been designed to overcome this difficulty: they make the program delightfully easy to apply to simple problems.

This desirable characteristic of the program has been achieved by a judicious use of “default values” for many parameters and variables. In other words, certain quantities are assumed to have the most commonly encountered values, unless they are overwritten by you in the adaptation part. As a result, the adaptation part for a simple problem can be very short. The length and the complexity of the adaptation part increases only in proportion to the complexity of the problem at hand.

Your responsibility of designing the adaptation part is further lightened by the provision of a number of general utilities in the invariant part of the program. These utilities are not an essential part of the program, but are provided for your convenience. Examples of such utilities are: routines for the construction of uniform and nonuniform grids, and a routine for the printout of the grid and the field variables.

Because successful utilization of CONDUCT depends on how correctly you prepare the adaptation part, a complete appreciation of the entire program is essential. This book provides all the necessary details of the invariant part and gives instructions and guidance for the design of the adaptation part. Illustrative adaptations of the program (given in Chapters 8, 10, and 11) serve as examples of the actual implementation of these ideas.

A computer program such as CONDUCT can be endlessly improved and streamlined. In this sense, there is no “perfect” or “best” program. What is presented in this book is the version that I think is adequate for our purposes. But it can certainly be improved. At first, it will be better for you to understand the program as it is and to learn to use it well. Later on, you may wish to design improved versions of the program for your own purposes.

CONDUCT is written in Fortran 77 and can be used on nearly all computers without modification. The results presented in this book have been obtained by using the Microsoft FORTRAN Compiler Version 4.1 on an IBM PC. Depending on the word length used by your computer, your results may differ slightly. This is especially true of certain quantities that should be theoretically zero, but acquire a small value because of computer roundoff. Unless the discrepancies are very large, there is no need to worry about them.

1.4 Outline of the Book

The general concept of numerical methods is described in Chapter 2, where our particular numerical technique is developed in detail with reference to the one-dimensional heat conduction situation. Although limited to one dimension, this chapter explains nearly all important ideas needed for our later work. Therefore, a good understanding of the material in this chapter is highly desirable.

The mathematical formulation for the general physical phenomena of interest is presented in Chapter 3. There we discuss the heat conduction equation and generalize it to represent other analogous processes. The computer program CONDUCT provides a calculation scheme for the solution of this general equation. Please note that Chapter 3 is not intended for providing complete information about the derivation of the heat conduction equation, the formulations for other proces...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• Copyright Notice
• Chapter 1 An Overview
• Chapter 2 Introduction to Numerical Methods
• Chapter 3 General Mathematical Framework
• Chapter 4 Structure of the Computer Program
• Chapter 5 Numerical Scheme and Its Implementation
• Chapter 6 Invariant Part the Computer Program
• Chapter 7 Adaptation Part of the Computer Program
• Chapter 8 Adaptation Examples: Heat Conduction
• Chapter 9 Flow and Heat Transfer in Ducts
• Chapter 10 Duct Associated Adaptation Examples: Flow and Heat Transfer
• Chapter 11 Additional Adaptations of CONDUCT
• Chapter 12 Closing Remarks
• Appendix A Listing of the Invariant Part of CONDUCT
• Appendix B List of Fortran Names
• Appendix C Default Values
• Appendix D User’s Checklist for ADAPT
• Nomenclature
• References
• Index