Essay 4 Numerical Solutions of the Equations of Heat Transfer and Fluid Flow

4.1 Introduction

In Essay 3, it was shown that heat conduction is governed by a partial differential equation. It will also be demonstrated later that fluid flow and convective heat transfer are also described by partial differential equations. The capability of solving partial differential equations in their unapproximated form is confined to analog devices, a prominent example of which is the differential analyzer. That device, which was prominent between 1930 and 1945, was not able to provide solutions of sufficient accuracy to meet the exacting needs of critical engineering projects. Digital computers are not capable of solving unapproximated differential equations. To enable differential equations to be solved digitally, it is necessary that they be approximated. There are three prominent strategies for approximating partial differential equations that are used at present. They are: (a) Finite Element Method (FEM) (b) Finite Difference Method (FDM) (c) Finite Volume Method (FVM) All of these approaches transform the solution task to that of solving a set of simultaneous algebraic equations in a solution space which is subdivided into a number of small elements. The process of transforming a solution space in which the variables are continuous to a space which there is an assemblage of small elements is called discretization. An alternative descriptions of this process is termed meshing. In the discretized space, the variables are not continuous. It is useful to demonstrate, at least for one of the discretization approaches, how the governing differential equations are discretized. In that regard, it is important to recognize that different degrees of mathematical sophistication are needed for the discretization depending on the specific approach. Among the three listed strategies, the finite element method is based on the most sophisticated mathematics. The degree of mathematical...

4.1 Introduction

In Essay 3, it was shown that heat conduction is governed by a partial differential equation. It will also be demonstrated later that fluid flow and convective heat transfer are also described by partial differential equations. The capability of solving partial differential equations in their unapproximated form is confined to analog devices, a prominent example of which is the differential analyzer. That device, which was prominent between 1930 and 1945, was not able to provide solutions of sufficient accuracy to meet the exacting needs of critical engineering projects. Digital computers are not capable of solving unapproximated differential equations. To enable differential equations to be solved digitally, it is necessary that they be approximated. There are three prominent strategies for approximating partial differential equations that are used at present. They are: (a) Finite Element Method (FEM) (b) Finite Difference Method (FDM) (c) Finite Volume Method (FVM) All of these approaches transform the solution task to that of solving a set of simultaneous algebraic equations in a solution space which is subdivided into a number of small elements. The process of transforming a solution space in which the variables are continuous to a space which there is an assemblage of small elements is called discretization. An alternative descriptions of this process is termed meshing. In the discretized space, the variables are not continuous. It is useful to demonstrate, at least for one of the discretization approaches, how the governing differential equations are discretized. In that regard, it is important to recognize that different degrees of mathematical sophistication are needed for the discretization depending on the specific approach. Among the three listed strategies, the finite element method is based on the most sophisticated mathematics. The degree of mathematical...