# How to Solve Heat Transfer and Fluid Flow Problems Using Numerical Techniques by Suhas Patankar

## Numerical Heat Transfer and Fluid Flow: A Review of Suhas V. Patankar's Book

Numerical heat transfer and fluid flow are two important branches of computational science that deal with the simulation and prediction of various physical processes involving heat transfer, fluid flow, chemical reaction, mass transfer, etc. These processes occur in many engineering equipment, natural environment, living organisms, and industrial applications. For example, numerical heat transfer and fluid flow can be used to design heat exchangers, turbines, reactors, boilers, solar collectors, air conditioners, refrigerators, vehicles, buildings, etc. They can also be used to study atmospheric phenomena, ocean currents, climate change, biological systems, combustion, etc.

## Numerical Heat Transfer And Fluid Flow Suhas V.patankar Solution Pdfl

However, numerical heat transfer and fluid flow are also challenging topics that require a good understanding of mathematics, physics, computer science, and engineering. Most physical phenomena are governed by partial differential equations (PDEs) that describe the conservation laws of mass, momentum, energy, species, etc. These PDEs are often nonlinear, coupled, time-dependent, multidimensional, and have complex boundary conditions. Therefore, it is usually impossible to find exact analytical solutions for these PDEs. Instead, numerical methods are needed to approximate these PDEs using discrete algebraic equations that can be solved by computers.

Numerical methods are powerful tools that can provide accurate and efficient solutions for complex problems. However, they also have some limitations and challenges. For example, numerical methods require a proper discretization of the physical domain into a grid or mesh of discrete points or cells. The choice of the grid size, shape, and distribution can affect the accuracy and stability of the solution. Numerical methods also require a proper selection of the discretization scheme that converts the PDEs into algebraic equations. The choice of the discretization scheme can affect the consistency, convergence, and conservation properties of the solution. Numerical methods also require a proper solution technique that solves the algebraic equations iteratively or explicitly. The choice of the solution technique can affect the speed, robustness, and memory requirements of the solution.

Therefore, numerical heat transfer and fluid flow are not only about applying numerical methods to physical problems, but also about understanding the underlying physical aspects of heat transfer and fluid flow, as well as analyzing and interpreting the computed results. This requires a solid foundation of theory and practice, as well as a critical and creative thinking.

One of the books that provides such a foundation is Numerical Heat Transfer and Fluid Flow by Suhas V. Patankar. This book was first published in 1980 and has been widely used as a textbook and reference for students, researchers, and practitioners in this field. The book focuses on the development and application of numerical methods for predicting heat transfer and fluid flow problems mainly based on physical considerations. The book uses simple algebra and elementary calculus to explain the concepts and methods, and provides many examples and exercises to illustrate the applications. The book also provides a deeper understanding of the underlying physical aspects of heat transfer and fluid flow as well as improves the ability to analyze and interpret computed results.

## Overview of the book

The author of the book is Suhas V. Patankar, who is a professor emeritus of mechanical engineering at the University of Minnesota and a pioneer in the field of computational fluid dynamics (CFD). He has made significant contributions to the development and application of numerical methods for heat transfer and fluid flow problems, especially in the areas of pressure correction methods, upwind schemes, body-fitted coordinates, turbulence modeling, etc. He has also authored or co-authored more than 200 publications and several books on this topic.

The main topics and objectives of the book are to introduce the basic concepts and terminology of heat transfer and fluid flow, to develop numerical methods for predicting these processes mainly based on physical considerations, to apply these methods to various practical problems, and to analyze and interpret the computed results. The book also covers some special topics such as inverse problems, multiphysics problems, etc.

The book is organized into nine chapters as follows:

Chapter 1: Introduction

Chapter 2: Discretization Methods

Chapter 3: Heat Conduction

Chapter 4: Convection and Diffusion

Chapter 5: Calculation of the Flow Field

Chapter 6: Finishing Touches

Chapter 7: Special Topics

Chapter 8: Illustrative Applications

Chapter 9: Nomenclature

The book is structured in a logical and progressive way, starting from the basic concepts and terminology, then moving to the discretization methods, then applying these methods to different types of problems, then adding some finishing touches such as complex geometries, turbulence modeling, etc., then exploring some special topics such as inverse problems, multiphysics problems, etc., and finally presenting some illustrative applications that demonstrate the practical use of numerical methods.

## Summary of the main chapters

### Chapter 1: Introduction

This chapter introduces the basic concepts and terminology of heat transfer and fluid flow. It explains what heat transfer and fluid flow are, why they are important for engineering and science applications, what are the main challenges and methods for solving these problems.

The chapter starts by defining heat transfer as the process of energy transfer due to temperature difference between two systems or regions. It then classifies heat transfer into three modes: conduction, convection, and radiation. It explains that conduction is heat transfer due to molecular motion within a medium or between two media in contact; convection is heat transfer due to bulk motion of a fluid over a solid surface or within a fluid; radiation is heat transfer due to electromagnetic waves emitted by a body or received by another body.

Here is the continuation of the article with HTML formatting: laminar flow and turbulent flow. It explains that laminar flow is a smooth and orderly flow in which fluid particles move in parallel layers or streamlines; turbulent flow is a chaotic and irregular flow in which fluid particles move in random directions and cross streamlines.

The chapter then introduces the mathematical description of physical phenomena using partial differential equations. It explains that PDEs are equations that involve partial derivatives of a dependent variable with respect to two or more independent variables. It shows how to derive the PDEs for heat conduction, convection-diffusion, and fluid flow using the conservation laws of mass, momentum, and energy. It also shows how to apply the boundary conditions and initial conditions to these PDEs.

The chapter then explains the need for numerical methods and their advantages and limitations. It explains that numerical methods are methods that use discrete approximations to solve PDEs using computers. It lists some of the advantages of numerical methods such as flexibility, generality, accuracy, efficiency, etc. It also lists some of the limitations of numerical methods such as discretization errors, truncation errors, round-off errors, stability issues, convergence issues, etc.

### Chapter 2: Discretization Methods

This chapter introduces the general procedure for discretizing a partial differential equation. It explains what discretization is, why it is necessary, and how it is done.

The chapter starts by defining discretization as the process of dividing the physical domain into a finite number of discrete points or cells and replacing the PDE by a set of algebraic equations at these points or cells. It explains that discretization is necessary because computers can only handle finite and discrete data and operations.

The chapter then describes the general procedure for discretizing a PDE as follows:

Choose a grid or mesh that covers the physical domain and satisfies the boundary conditions.

Choose a control volume around each grid point or cell that encloses the physical phenomena.

Choose a discretization scheme that converts the PDE into an algebraic equation for each control volume using interpolation or integration techniques.

Choose a solution technique that solves the algebraic equations iteratively or explicitly using linear or nonlinear solvers.

The chapter then explains the different types of discretization methods and their properties. It classifies discretization methods into two categories: finite difference methods and finite volume methods. It explains that finite difference methods are based on replacing the derivatives in the PDE by finite differences using Taylor series expansion; finite volume methods are based on integrating the PDE over each control volume using divergence theorem. It compares the advantages and disadvantages of these two methods in terms of accuracy, conservation, flexibility, etc.

The chapter then explains the criteria for selecting a suitable discretization method for a given problem. It lists some of the factors that affect the choice of discretization method such as grid type, grid size, grid distribution, grid quality, problem type, problem complexity, problem geometry, etc. It also gives some examples of common discretization methods for different types of problems such as heat conduction, convection-diffusion, fluid flow, etc.

### Chapter 3: Heat Conduction

This chapter applies different discretization methods to one-dimensional and two-dimensional heat conduction problems. It shows how to obtain numerical solutions for these problems and how to analyze and interpret the results.

The chapter starts by reviewing the governing equation and boundary conditions for heat conduction problems. It explains that heat conduction is governed by Fourier's law of heat conduction that states that the heat flux is proportional to the temperature gradient; this law leads to a second-order PDE called the heat equation that describes the temperature distribution in a medium; this equation is subject to various types of boundary conditions such as Dirichlet (specified temperature), Neumann (specified heat flux), Robin (specified heat transfer coefficient), etc.

The chapter then applies different discretization methods to one-dimensional heat conduction problems with different boundary conditions. It shows how to use finite difference methods such as forward difference, backward difference, central difference, etc., to obtain algebraic equations for each grid point; it also shows how to use finite volume methods such as power-law scheme, exponential scheme, etc., to obtain algebraic equations for each control volume; it then shows how to solve these algebraic equations using solution techniques such as Gauss elimination, Gauss-Seidel, etc. It also shows how to calculate the heat flux and the heat transfer rate using the numerical solutions.

The chapter then extends the discretization methods to two-dimensional heat conduction problems with different boundary conditions. It shows how to use finite difference methods such as five-point scheme, nine-point scheme, etc., to obtain algebraic equations for each grid point; it also shows how to use finite volume methods such as hybrid scheme, QUICK scheme, etc., to obtain algebraic equations for each control volume; it then shows how to solve these algebraic equations using solution techniques such as ADI, SOR, etc. It also shows how to calculate the heat flux and the heat transfer rate using the numerical solutions.

The chapter then analyzes and interprets the numerical results for heat conduction problems. It explains how to evaluate the accuracy and stability of the numerical solutions using error analysis, convergence analysis, grid independence test, etc. It also explains how to compare the numerical solutions with analytical solutions or experimental data where available. It also explains how to visualize and present the numerical results using contour plots, surface plots, vector plots, etc.

### Chapter 4: Convection and Diffusion

This chapter applies different discretization methods to one-dimensional and two-dimensional convection-diffusion problems. It shows how to obtain numerical solutions for these problems and how to analyze and interpret the results.

The chapter starts by reviewing the governing equation and boundary conditions for convection-diffusion problems. It explains that convection-diffusion is governed by a first-order PDE that describes the transport of a scalar quantity (such as temperature, concentration, etc.) by a fluid flow; this equation consists of two terms: a convection term that represents the advection of the scalar quantity by the fluid velocity; and a diffusion term that represents the diffusion of the scalar quantity by molecular motion; this equation is subject to various types of boundary conditions such as Dirichlet (specified scalar value), Neumann (specified scalar flux), Robin (specified scalar transfer coefficient), etc.

The chapter then applies different discretization methods to one-dimensional convection-diffusion problems with different boundary conditions. It shows how to use finite difference methods such as upwind scheme, central difference scheme, etc., to obtain algebraic equations for each grid point; it also shows how to use finite volume methods such as power-law scheme, exponential scheme, etc., to obtain algebraic equations for each control volume; it then shows how to solve these algebraic equations using solution techniques such as Gauss elimination, Gauss-Seidel, etc. It also shows how to calculate the scalar flux and the scalar transfer rate using the numerical solutions.

The chapter then extends the discretization methods to two-dimensional convection-diffusion problems with different boundary conditions. It shows how to use finite difference methods such as five-point scheme, nine-point scheme, etc., to obtain algebraic equations for each grid point; it also shows how to use finite volume methods such as hybrid scheme, QUICK scheme, etc., to obtain algebraic equations for each control volume; it then shows how to solve these algebraic equations using solution techniques such as ADI, SOR, etc. It also shows how to calculate the scalar flux and the scalar transfer rate using the numerical solutions.

The chapter then analyzes and interprets the numerical results for convection-diffusion problems. It explains how to evaluate the accuracy and stability of the numerical solutions using error analysis, convergence analysis, grid independence test, etc. It also explains how to compare the numerical solutions with analytical solutions or experimental data where available. It also explains how to visualize and present the numerical results using contour plots, surface plots, vector plots, etc.

### Chapter 5: Calculation of the Flow Field

This chapter applies different discretization methods to one-dimensional and two-dimensional fluid flow problems. It shows how to obtain numerical solutions for these problems and how to analyze and interpret the results.

The chapter starts by reviewing the governing equations and boundary conditions for fluid flow problems. It explains that fluid flow is governed by a set of PDEs that describe the conservation laws of mass (continuity equation), momentum (Navier-Stokes equations), and energy (energy equation) for a fluid; these equations are nonlinear, coupled, time-dependent, and multidimensional; they are subject to various types of boundary conditions such as no-slip condition (zero velocity at solid walls), free-slip condition (zero shear stress at solid walls), inlet condition (specified velocity or pressure at fluid entry), outlet condition (specified velocity or pressure at fluid exit), symmetry condition (zero normal velocity and zero normal gradient at symmetry planes), periodic condition (equal values at periodic boundaries), etc.

Here is the continuation of the article with HTML formatting: It explains that elliptic problems are those that involve a balance between diffusion and reaction terms; parabolic problems are those that involve a balance between diffusion and convection terms; hyperbolic problems are those that involve a balance between convection and reaction terms. It also explains how to identify the type of problem based on the characteristics of the PDE such as the order, linearity, coefficients, etc.

The chapter then applies different discretization methods to one-dimensional and two-dimensional fluid flow problems of different types. It shows how to use finite difference methods such as explicit method, implicit method, Crank-Nicolson method, etc., to obtain algebraic equations for each grid point; it also shows how to use finite volume methods such as SIMPLE algorithm, SIMPLER algorithm, SIMPLEC algorithm, etc., to obtain algebraic equations for each control volume; it then shows how to solve these algebraic equations using solution techniques such as TDMA, SOR, GMRES, etc. It also shows how to calculate the pressure and the velocity fields using the numerical solutions.

The chapter then analyzes and interprets the numerical results for fluid flow problems. It explains how to evaluate the accuracy and stability of the numerical solutions using error analysis, convergence analysis, grid independence test, etc. It also explains how to compare the numerical solutions with analytical solutions or experimental data where available. It also explains how to visualize and present the numerical results using contour plots, surface plots, vector plots, etc.

### Chapter 6: Finishing Touches

This chapter adds some finishing touches to the discretization methods for heat transfer and fluid flow problems. It shows how to deal with complex geometries and irregular grids, how to model turbulence effects, and how to couple heat transfer and fluid flow.

The chapter starts by explaining how to treat complex geometries and irregular grids using body-fitted coordinates and nonorthogonal grids. It explains that body-fitted coordinates are a transformation of the physical coordinates into a computational domain that conforms to the geometry of the problem; nonorthogonal grids are grids that do not have orthogonal angles between adjacent grid lines or faces. It shows how to apply these techniques to discretize the governing equations using finite difference methods or finite volume methods. It also shows how to improve the accuracy and stability of these techniques using deferred correction, Rhie-Chow interpolation, etc.

The chapter then explains how to model turbulence effects using eddy viscosity models and Reynolds-averaged equations. It explains that turbulence is a phenomenon that involves random fluctuations of velocity, pressure, temperature, etc., at various scales; eddy viscosity models are models that assume that the effect of turbulence can be represented by an effective viscosity that depends on the local flow characteristics; Reynolds-averaged equations are equations that are obtained by averaging the governing equations over time or space to separate the mean and fluctuating components of the flow variables. It shows how to apply t