Lecture 13 – February 2, 2004 Reminder of Quiz in the tutorial session tomorrow We have completed chapters 1-3. Today Chapter 4 -- 2D Conduction There are 3 basic methods of solving a heat transfer problem • Analytical

o Solve the governing Equations to arrive at a mathematical expression for the temperature

o Graphical Method Conduction Shape Factor • Experimental

o Build it, instrument it and measure T • Numerical

o Solve for T at discrete locations by using conservation of energy

o Finite Volume o Finite Difference o Finite Element

Governing Equation Conservation of Energy

• 2D • Steady

To arrive at an analytic solution, also assume that

• k is constant • no generation

This is the famous Laplace equation which applies to many physical systems, and relates to conservation, as we have seen in the case of heat energy. You have also seen this in Fluid Mechanics where conservation of mass leads to the Laplace equation for the stream function.

Graphical Solution Method Build up a diagram of correctly drawn isotherms (temperature contour lines) and adiabats (lines through which there is no heat transfer). Begin by drawing temperature contours with equal increments in temperature. Make sure temperature contours intersect adiabatic boundaries at 90o. Next draw adiabats perpendicular to each contour line. As the area for heat transfer increases, the spacing between the temperature contours should increase to keep a constant heat rate, qi flowing through each heat lane.

If the diagram is constructed such that each qi is equal, then q = Mqi.

Theflux is everywhere given by Fourier’s Law

Since

While this may be quite approximate, the process of drawing the diagram forces you to think of the physics, and to have a good feel for your problem. This is invaluable. Conduction Shape Factor. If we simply write that the heat transfer

we can calculate the flux for a variety of shapes and tabulate representative values of Sk. These are called conduction shape factors.

• Sk can come from numerical solutions o Why not solve your exact problem

then? • Sk can come from Analytic Solutions

o Quite limited in Boundary conditions

2D Conduction – Simple numerical solution Let’s examine a 2D conduction problem with constant conductivity, no generation and use a uniform grid to solve it.

Lets apply conservation of energy to a general control volume inside our domain. With these simplifications, conservation of energy simply says what comes in, goes out.

Now lets evaluate each flux according to the local gradients

If the grid spacing is uniform then this simplifies greatly,

We can easily use a tool like excel to solve this simplified case. When we generalize, and add generation and energy

storage, and more complicated boundary conditions, then we will need to use a more sophisticated package like MATLAB, or even Fluent.