Chapter 3 Steady-State Conduction Multiple Dimensions.ppt

Chapter 3 Steady-State Conduction Multiple Dimensions CHAPER 3 Steady-State Conduction Multiple Dimensions 3-1 Introduction In Chapter 2 steady-state heat transfer was calculated in systems in which the temperature gradient and area could be expressed in terms of one space coordinate. We now wish to analyze the more general case of two-dimensional heat flow. For steady state with no heat generation, the Laplace equation applies. 3-1 Introduction The objective of any heat-transfer analysis is usually to predict heat flow or the temperature that results from a certain heat flow. The solution to Equation (3-1) will give the temperature in a two-dimensional body as a function of the two independent space coordinates x and y. Then the heat flow in the x and y directions may be calculated from the Fourier equations 3-3 Graphical Analysis 3-4 The Conduction Shape Factor Consider a general one dimensional heat conduct- ion problem, from Fourier’s Law: 3-5 Numerical Method of Analysis The most fruitful approach to the heat conduction is one based on ?nite-difference techniques, the basic principles of which we shall outline in this section. 3-5 Numerical Method of Analysis 3-5 Numerical Method of Analysis 3-5 Numerical Method of Analysis Example 3-5 Consider the square shown in the figure. The left face is maintained at 100℃ and the top face at 500℃, while the other two faces are exposed to a environment at 100℃. h=10W/m2·℃ and k=10W/m·℃. The block is 1 m square. Compute the temperature of the various nodes as indicated in the figure and heat flows at the boundaries. Example 3-5 Example 3-5 3-6 Numerical Formulation in Terms of Resistance Elements 3-6 Numerical Formulation in Terms of Resistance Elements 3-7 Gauss-Seidel Iteration 3-7 Gauss-Seidel Iteration Example 3-6 Example 3-6 3-8 Accuracy Consideration Summary Summary Summary Exercises Example 3-2 Example 3-3 Example 3-4 1、Discretization of the solving 2、Discrete equation Taylor series expans