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CFO has now become an integral part of the engineering design and analysis. Engineers can make use of the CFO tools to simulate fluid flow and heat transfer phenomena in a design and predict the system performance before manufacturing. The advantages of CFO are numerous, namely, fewer iterations to the final design, shorter time to launch the product, fewer expensive prototypes and so on. Furthermore, CPO provides a cost-efficient means of testing new designs and concepts that would otherwise be too expensive and hazardous to investigate.
Much of the material in this textbook has been used in a post -graduate course at the Indian Institute of Technology , Kanpur for over a decade. The book is suitable as a text for a one-semester course at the post-graduate or advanced undergraduate level. It can also be used for self-study by practising engineers. The book primarily follows finite difference method of discretization.
However, in the Appendix A, other important schemes such as finite element and finite volume are also discussed. An emphasis has been laid on the physical understanding of the problems. Most of the methods have been illustrated with detailed example problems and the solution procedure.
Several exercise problems are given at the end of various chapters. Readers are encouraged to solve these problems, to get a better understanding of various numerical techniques discussed in the book. Chapter 7 gives details of two new numerical methods and their applications. Chapter 8 illustrates the application of CFO in solving industrial problems. The softcover version of this book also contains a floppy diskette.
The diskette contains 21 files comprising 10 programs, I subroutine and 10 output files. I would like to acknowledge the interaction with the students, both in and outside the class, which has greatly contributed towards the shaping of this book. Their suggestions and comments have been useful in writing this text.
I have also been benefitted by the lively discussions with some of my colleagues. Special thanks are due to the post-graduate students Suresh Singh, Vipin Kumar, R Mahesh Kumar and Kali Sanjay who have assisted me in developing some of the computer programs in the floppy diskette. I am also grate- ful to the anonymous reviewer whose valuable comments and suggestions for Preface vii improvement have gone a long way in the formation of the final version of this book. The typing was carried out with great care and patience by U S Mishra.
The 1. Gratitude not expressible in words is due to my parents for their blessings 1. I0 Summary 2 7 9 6. Reference 8. The simulation of an industrial system on computer involves mathematical representation of the physical processes undergone by the various components of the system, by a set of equations usually differential equations transformed to difference equations which are in turn solved as a set of simultaneous algebraic equations. At this stage, a reader uninitiated into the numerical methods may ask the question "What is the role of computer here?
The aforesaid query is a valid one. Many seem to forget that some of the numerical schemes e. Finite-Difference that are extensively used today for solution of problems on computer were devel- oped when computer was not even invented. Now, to return to the original ques- tion, the answer is that with the aid of the algorithm of the solution method trans- lated into a programming language like FORTRAN fed into a computer which does the arithmetic operations at a tremendous speed one can obtain the solution of mathematical equations in seconds or even in fraction of a second.
A simple example will clarify this point. One can very easily solve a set of three linear simultaneous algebraic equations. Typically in this method, for a system o f n equations the total number of multiplications and divisions is roughly.!. A mainframe computer e. A personal computer e. A floating point operation is an arithmetic operation addition, subtraction, multiplication and division on operands which are real numbers with fractional parts.
NormaUy multiplications and divisions are counted asinajor arithmetic op- erations as compared to addition and subtraction on a computer. It is no wonder that J. A large variety of problems with made this remark in "Heat, like gravity, penetrates every substance of the different levels of complexity can be simulated on a computer. Lord Kelvin, in allows models and hence physical understanding of the problem to be obtained a rough estimate of the age of earth based on an idea proposed by Fourier improved.
It is similar to conducting experiments. Modem dating methods have example, modelling loss of coolant accident LOCA in nuclear reactors, revealed the age of earth to be approximately 4. So Kelvin's result numerical simulation of spread of fire in a building and modelling of was not really too far off the mark considering the fact that the data for the meas- incineration of hazardous waste.
Experiments are still required to get an insight into the molten earth when cooling began available with him at that time were not very phenomena that are not well understood and hence cannot be translated into the ,ccurate. The aforesaid example is probably the first known application of heat language of mathematics and also to check the validity of the results of computer transfer simulation.
TRANSFER Ifope looks at a classical textbook on fluid dynamics and heat transfer, one would Fluid flow and heat transfer playa very important role in nature, living organisms find only a handful of analytical or exact solutions.
In actual situations, prob- and in a variety of practical situations. In many applications flow and heat transfer are accompanied cal solutions to be obtained. The various applications of fluid flow niques for most problems of practical interest. However, necessary experimentation must still be done in checking and solar power plants.
We imagine that the domain is filled by a grid, and seek the values of temperatures at the grid points. The Thus, even with a fairly coarse grid, the number of operations is quite large. If each solution has to be done in about an hour, then the equations to solve per variable. The peak megaflop rating of modem supercom- powerful and widely applicable. From the aforesaid exam- x Grid points ple, it is clear why we need supercomputers with speeds in thousands of megaflops known temperatures range to solve extremely complex problems.
It may involve full scale, small scale or blown-up scale models. The major disadvantages of experimental investigations are high cost, measurement difficulties and probe errors.
Often, small scale models do not al- ways simulate all the features of the full scale set-up. The advantage is that it is most realistic. Only a handful of analytical solutions are available in heat trans- F ig. Furthermore, the analytical solutions often contain infinite series, 1. In general, the problem complexity is described by the formula c Numerical Method As explained in Sec. They are in the speed of arithmetic operations of computers. The great popularity of finite-difference like two-dimensionality, insulated or isothermal boundary, infinite methods is mainly due to their straight-forwardness and relative simplicity by reaction rate.
On the other hand, it is extremely difficult, if not impossible which a newcomer in the field is able to obtain solutions of simple problems. As to set up the same in experiments. However, several shortcomings and limitations of finite-difference method came to light when researchers tried to solve problems with increasing Name of the Method Advantages Disadvantages degree of physical complexity such as, for example, flows at higher Reynolds, 1.
Fasel, It is based on integral minimization principle and provides piecewise or regional approxima- usually in and physics tions to the governing equations. How- to compute 3.
Hence, variational principle-based FEM is limited to solutions of and complicated problems physics creeping flow and heat conduction problems. Much current research is in progress in the use of this of a problem by numerical method is then given with the aid of a figure along with method. A comparison of advantages and disadvan- tages of numerical methods vis-a-vis analytical and experimental methods is then c Spectral Method Spectral methods are generally much more accurate than provided in detail.
Finally, the chapter concludes with a brief description of the simple first or second order finite-difference schemes. In spectral methods, in various methods of discretization and the justifications for the choice of the finite contrast with the discretization as in finite-difference methods, the approximation difference method as the discretization scheme used in this book is based on expansions of independent variables into finite truncated series of smooth mostly orthogonal functions.
A disadvantage of spectral methods is their relative complexity in compari- son with standard finite-difference methods. Also the implementation of complex 1. Clarendon Press, Oxford, The differential equation is integrated 3. Field, George B, Gerrit L. Verschuur, and Cyril Ponnamperuma, Cosmic over each control volume. Piecewise profiles expressing the variation of the un- Evolution: An Introduction to Astronomy, Houghton Mifflin Company, Boston, known between the grid points are used to evaluate the required integrals.
The Jaluria, Yogesh and Kenneth E. Torrance, Computational Heat Transfer, group of grid points Patankar, Hemisphere, Washington, D. The major advantage of this method is its physical soundness. The disadvan- 5. Orszag, S A and M.
Computer Simulation of Flow and Heat Transfer (Without Diskette)
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Computer simulation of flow and heat transfer
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