Numerical Heat Transfer and Fluid Flow

Suhas Patankar

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9781351991513
• Edition: 1
• Topic: Scienze fisiche
• Subtopic: Termodinamica

This book focuses on heat and mass transfer, fluid flow, chemical reaction, and other related processes that occur in engineering equipment, the natural environment, and living organisms. Using simple algebra and elementary calculus, the author develops numerical methods for predicting these processes mainly based on physical considerations. Through this approach, readers will develop a deeper understanding of the underlying physical aspects of heat transfer and fluid flow as well as improve their ability to analyze and interpret computed results.

CHAPTER
ONE

INTRODUCTION

1.1 SCOPE OF THE BOOK

Importance of heat transfer and fluid flow. This book is concerned with heat and mass transfer, fluid flow, chemical reaction, and other related processes that occur in engineering equipment, in the natural environment, and in living organisms. That these processes play a vital role can be observed in a great variety of practical situations. Nearly all methods of power production involve fluid flow and heat transfer as essential processes. The same processes govern the heating and air conditioning of buildings. Major segments of the chemical and metallurgical industries use components such as furnaces, heat exchangers, condensers, and reactors, where thermofluid processes are at work. Aircraft and rockets owe their functioning to fluid flow, heat transfer, and chemical reaction. In the design of electrical machinery and electronic circuits, heat transfer is often the limiting factor. The pollution of the natural environment is largely caused by heat and mass transfer, and so are storms, floods, and fires. In the face of changing weather conditions, the human body resorts to heat and mass transfer for its temperature control. The processes of heat transfer and fluid flow seem to pervade all aspects of our life.

Need for understanding and prediction. Since the processes under consideration have such an overwhelming impact on human life, we should be able to deal with them effectively. This ability can result from an understanding of the nature of the processes and from methodology with which to predict them quantitatively. Armed with this expertise, the designer of an engineering device can ensure the desired performance–the designer is able to choose the optimum design from among a number of alternative possibilities. The power of prediction enables us to operate existing equipment more safely and efficiently. Predictions of the relevant processes help us in forecasting, and even controlling, potential dangers such as floods, tides, and fires. In all these cases, predictions offer economic benefits and contribute to human well-being.

Nature of prediction. The prediction of behavior in a given physical situation consists of the values of the relevant variables governing the processes of interest. Let us consider a particular example. In a combustion chamber of a certain description, a complete prediction should give us the values of velocity, pressure, temperature, concentrations of the relevant chemical species, etc., throughout the domain of interest; it should also provide the shear stresses, heat fluxes, and mass flow rates at the confining walls of the combustion chamber. The prediction should state how any of these quantities would change in response to proposed changes in geometry, flow rates, fluid properties, etc.

Purpose of the book. This book is primarily aimed at developing a general method of prediction for heat and mass transfer, fluid flow, and related processes. As we shall shortly see, among the different methods of prediction, the numerical solution offers great promise. In this book, we shall construct a numerical method for predicting the processes of interest.

As far as possible, our aim will be to design a numerical method having complete generality. We shall, therefore, refrain from accepting any final restrictions such as two-dimensionality, boundary-layer approximations, and constant-density flow. If any restrictions are temporarily adopted, it will be for ease of presentation and understanding and not because of any intrinsic limitation. We shall begin the subject at a very elementary level and, from there, travel nearly to the frontier of the subject.

This ambitious task cannot, of course, be accomplished in a modest-sized book without leaving out a number of important topics. Therefore, the mathematical formulation of the equations that govern the processes of interest will be discussed only briefly in this book. For the complete derivation of the required equations, the reader must turn to standard textbooks on the subject. The mathematical models for complex processes like turbulence, combustion, and radiation will be assumed to be known or available to the reader. Even in the subject of numerical solution, we shall not survey all available methods and discuss their merits and demerits. Rather, we shall focus attention on a particular family of methods that the author has used, developed, or contributed to. Reference to other methods will be made only when this serves to highlight a certain issue. While a general formulation will be attempted, no special attention will be given to supersonic flows, free-surface flows, or two-phase flows.

An important characteristic of the numerical methods to be developed in this book is that they are strongly based on physical considerations, not just on mathematical manipulations. Indeed, nothing more sophisticated than simple algebra and elementary calculus is used. A significant advantage of this strategy is that the reader, while learning about the numerical methods, develops a deeper understanding of, and insight into, the underlying physical processes. This appreciation for physical significance is very helpful in analyzing and interpreting computed results. But, even if the reader never performs numerical computations, this study of the numerical methods will provide–it is interesting to note–a greater feel for the physical aspects of heat transfer and fluid flow. Further, the physical approach will equip the reader with general criteria with which to judge other existing and future numerical methods.

1.2 METHODS OF PREDICTION

Prediction of heat transfer and fluid-flow processes can be obtained by two main methods: experimental investigation and theoretical calculation. We shall briefly consider each and then compare the two.

1.2-1 Experimental Investigation

The most reliable information about a physical process is often given by actual measurement. An experimental investigation involving full-scale equipment can be used to predict how identical copies of the equipment would perform under the same conditions. Such full-scale tests are, in most cases, prohibitively expensive and often impossible. The alternative then is to perform experiments on small-scale models. The resulting information, however, must be extrapolated to full scale, and general rules for doing this are often unavailable. Further, the small-scale models do not always simulate all the features of the full-scale equipment; frequently, important features such as combustion or boiling are omitted from the model tests. This further reduces the usefulness of the test results. Finally, it must be remembered that there are serious difficulties of measurement in many situations, and that the measuring instruments are not free from errors.

1.2-2 Theoretical Calculation

A theoretical prediction works out the consequences of a mathematical model, rather than those of an actual physical model. For the physical processes of interest here, the mathematical model mainly consists of a set of differential equations. If the methods of classical mathematics were to be used for solving these equations, there would be little hope of predicting many phenomena of practical interest. A look at a classical text on heat conduction or fluid mechanics leads to the conclusion that only a tiny fraction of the range of practical problems can be solved in closed form. Further, these solutions often contain infinite series, special functions, transcendental equations for eigenvalues, etc., so that their numerical evaluation may present a formidable task.*

Figure 1.1 Grid layout for a numerical solution for the temperature field.

Fortunately, the development of numerical methods and the availability of large digital computers hold the promise that the implications of a mathematical model can be worked out for almost any practical problem. A preliminary idea of the numerical approach to problem solving can be obtained by reference to Fig. 1.1. Suppose that we wish to obtain the temperature field in the domain shown. It may be sufficient to know the values of temperature at discrete points of the domain. One possible method is to imagine a grid that fills the domain, and to seek the values of temperature at the grid points. We then construct and solve algebraic equations for these unknown temperatures. The simplification inherent in the use of algebraic equations rather than differential equations is what makes numerical methods so powerful and widely applicable.

1.2-3 Advantages of a Theoretical Calculation

We shall now list the advantages that a theoretical calculation offers over a corresponding experimental investigation.

Low cost. The most important advantage of a computational prediction is its low cost. In most applications, the cost of a computer run is many orders of magnitude lower than the cost of a corresponding experimental investigation. This factor assumes increasing importance as the physical situation to be studied becomes larger and more complicated. Further, whereas the prices of most items are increasing, computing costs are likely to be even lower in the future.

Speed. A computational investigation can be performed with remarkable speed. A designer can study the implications of hundreds of different configurations in less than a day and choose the optimum design. On the other hand, a corresponding experimental investigation, it is easy to imagine, would take a very long time.

Complete information. A computer solution of a problem gives detailed and complete information. It can provide the values of all the relevant variables (such as velocity, pressure, temperature, concentration, turbulence intensity) throughout the domain of interest. Unlike the situation in an experiment, there are few inaccessible locations in a computation, and there is no counterpart to the flow disturbance caused by the probes. Obviously, no experimental study can be expected to measure the distributions of all variables over the entire domain. For this reason, even when an experiment is performed, there is great value in obtaining a companion computer solution to supplement the experimental information.

Ability to simulate realistic conditions. In a theoretical calculation, realistic conditions can be easily simulated. There is no need to resort to small-scale or cold-flow models. For a computer program, there is little difficulty in having very large or very small dimensions, in treating very low or very high temperatures, in handling toxic or flammable substances, or in following very fast or very slow processes.

Ability to simulate ideal conditions. A prediction method is sometimes used to study a basic phenomenon, rather than a complex engineering application. In the study of a phenomenon, one wants to focus attention on a few essential parameters and eliminate all irrelevant features. Thus, many idealizations are desirable–for example, two-dimensionality, constant density, an adiabatic surface, or infinite reaction rate. In a computation, such conditions can be easily and exactly set up. On the other hand, even a very careful experiment can barely approximate the idealization.

1.2-4 Disadvantages of a Theoretical Calculation

The foregoing advantages are sufficiently impressive to stimulate enthusiasm about computer analysis. A blind enthusiasm for any cause is, however, undesirable. It is useful to be aware of the drawbacks and limitations.

As mentioned earlier, a computer analysis works out the implications of a mathematical model. The experimental investigation, by contrast, observes the reality itself. The validity of the mathematical model, therefore, limits the usefulness of a computation. In this book, we shall be concerned only with computational methods and not with mathematical models. Yet, we must note that the user of the computer analysis receives an end product that depends on both the mathematical model and the numerical method. A perfectly satisfactory numerical technique can produce worthless results if an inadequate mathematical model is employed.

For the purpose of discussing the disadvantages of a theoretical calculation, it is, therefore, useful to divide all practical problems into two groups:

Group A: Problems for which an adequate mathematical description can be written. (Examples: heat conduction, laminar flows, simple turbulent boundary layers.)

Group B: Problems for which an adequate mathematical description has not yet been worked out. (Examples: complex turbulent flows, certain non-Newtonian flows, formation of nitric oxides in turbulent combustion, some two-phase flows.)

Of course, the group into which a given problem falls will be determined by what we are prepared to consider as an “adequate” description.

Disadvantages for Group A. It may be stated that, for most problems of Group A, the theoretical calculation suffers from no disadvantages. The computer solution then represents an alternative that is highly superior to an experimental study. Occasionally, however, one encounters some disadvantages. If the prediction has a very limited objective (such as finding the overall pressure drop for a complicated apparatus), the computation may not be less expensive than an experiment. For difficult problems involving complex geometry, strong nonlinearities, sensitive fluid-property variations, etc., a numerical solution may be hard to obtain and would be excessively expensive if at all possible. Extremely fast and small-scale phenomena such as turbulence, if they are to be computed in all their time-dependent detail by solving the unsteady Navier-Stokes equations, are still beyond the practical reach of computational methods. Finally, when the mathematical problem occasionally admits more than one solution, it is not easy to determine whether the computed solution corresponds to reality.

Research in computational methods is aimed at making them more reliable, accurate, and efficient. The disadvantages mentioned here will diminish as this research progresses.

Disadvantages for Group B. The problems of Group B share all the disadvantages of Group A; in addition, there is the uncertainty about the extent to which the computed results would agree with reality. In such cases, some experimental backup is highly desirable.

Research in mathematical models causes a transfer of problems from Group B into Group A. This research consists of proposing a model, working out its implications by computer analysis, and comparing the results with experimental data. Thus, computational methods play a key role in this research. A striking example of this role can be found in the recent development ...