Two-dimensional heat conduction

The method of weighted residuals can be used to reduce the dimensionality of a problem. To illustrate this, let us consider a two-dimensional heat conduction problem  

We will get an approximate solution over the x domain and then solve the resulting ordinary differential equation in y. The problem is symmetric about x = 1/2 and vanishes at a = 0 and x = 1. We want to choose a trial function that has these properties. As discussed by Finlayson (1972), the quadratic form 

If we use the collocation method and choose the collocation point as the midpoint a = 1/2, then the weighted residual of equation (8.11.9) is 

This ordinary differential equation has the analytical solution 

As discussed by Finlayson (1972), a good way to check the accuracy of the approximate solution is to compare the average heat flux at the boundary using the MWR solution method to an analytical solution. The average heat flux is given by 

---

Green s identities for a 2D Laplace s equation (heat conduction) Here, we will demonstrate how to develop Green s identities for a two-dimensional heat conduction problem, which for a material with constant properties is described by the Laplace equation for the temperature, i.e.,... 

Assume that the following equation describes the two-dimensional heat conduction in an insulated pipe... 

C Whal is an inilial condition How many initial conditions do we need to specify for a two-dimensional heat conduction problem ... 

The finite difference formulation above can easily be extended to two- nr threc-dimen.sinnal heat transfer problems by replacing each second derivative by a difference equation in that direction. For example, the finite difference fomiulalion for steady two-dimensional heat conduction in a region with... 

EXAMPLE 5-3 Steady Two-Dimensional Heat Conduction in L-Bars... 

C 1 he explicit finite difference formulation of a general interior node for transient two-dimensional heat conduction is given by... 

C Consider transient two-dimensional heat conduction in a rectangular region that is to be solved by (he explicit method. If all boundaries of the region are either insulated or at specified temperatures, express the stability criterion for this problem in its simplest form. 

For the special case of a stationary fluid, u = v = Q and the energy equation reduces, as expected, to the steady two-dimensional heat conduction equation,... 

The Finite Element Method is a very powerful and convenient tool to obtain temperature fields accounting for the variable material properties in the analysis. Figure 19 shows a two-dimensional model for FEM analysis. For calculation, the temperature on the rake face of a diamond tool is calculated. Two-dimensional heat conduction can be expressed by the equation ... 

Figure 4.4-1. Graphical curvilinear-square method for two-dimensional heat conduction in a rectangular flue.

This shape factor S has units of m and is used in two-dimensional heat conduction where only two temperatures are involved. The shape factors for a number of geometries have been obtained and some are given in Table 4.4-1. 

In Section 4.4 we discussed methods for solving two-dimensional heat-conduction problems using grap]iical procedures and shape factors. In this section we consider analytical and numerical methods. 

Derivation of the method. Since the advent of the fast digital computers, solutions to many complex two-dimensional heat-conduction problems by numerical methods are readily possible. In deriving the equations we can start with the partial differential equation (4.15-5). Setting up the finite difference of d T/dx, ... 

Two-Dimensional Heat Conduction and Different Boundary Conditions. A very... 

The biharmonic and Laplace s equations are the governing differential equations for many two dimensional problems in linear elasticity and heat transfer. Among these problems are those which arise in the theory of thin plates, two dimensional thermal stresses, torsion and bending of prismatic bars, and two dimensional heat conduction. 

Solve the following two-dimensional heat conduction equation ... 

---