Using Excel to Implement the Finite Difference Method for 2-D Heat Transfer in a Mechanical Engineering Technology Course

Multi-dimensional heat transfer problems can be approached in a number of ways. Sometimes an analytical approach using the Laplace equation to describe the problem can be used. This involves finding the solution of differential equations, which may be reasonable for Mechanical Engineering Technology (MET) students. However, these students are not always particularly proficient in using this approach. Also, the analysis can get quite complex depending on boundary conditions, often involving advanced mathematics using Bessel functions, Fourier series and other special functions. Graphical methods might be used, but their usefulness extends primarily to discussions about the relationships between isotherms and heat flow paths. Shape factors and other approximations can also be useful in certain instances. None of these seem to provide an especially good approach for MET students. A more practical approach for these students is the use of numerical methods. The finite difference method seems to provide a good approach for MET students. Using this method a student can model fairly complex two-dimensional problems with a variety of boundary conditions using a simple spreadsheet. This paper presents information on how this method is used at Penn State Erie, The Behrend College in a first course in heat transfer for MET students. The method is used to aid in presenting the theory, as well as for a laboratory exercise. The basic equations for a variety of node types are included, as well as equation modifications that are used to account for several thermal loading and boundary conditions. The lectures are reinforced with homework practice problems before the more involved lab exercise. Finally, the lab exercise is included. The exercise is designed to give the students practice using the method. Introduction: The first course in heat transfer for Mechanical Engineering Technology (MET) students at Penn State Erie, The Behrend College focuses primarily on one-dimensional heat transfer with applications. Conduction, convection and radiation are introduced early. Conduction and convection are covered in some detail, including the calculation of convection coefficients using a variety of Nusselt correlations. Only the basics of radiation are included in the course. A section on transient heat transfer is also part of the course, including the lumped mass method, closed solutions for simple shapes and semiinfinite solids. Very little is done with multi-dimensional conduction. Traditionally it was touched on during the treatment of fins and heatsinks. A laboratory exercise was developed to provide a little more coverage of the topic. The exercise requires the students to P ge 24328.2 determine the temperature distribution across a plate with a variety of inputs and boundary conditions. This is done using finite difference formulations with a spreadsheet. Very little time is devoted to this during lectures, so the students are not being asked to develop the formulas that are used on the spreadsheet. Instead, one or two lectures are devoted to a simple treatment of the topic. The basic formula for an interior node is developed, and the others formulas that are needed are simply given to the students. Theory: The finite difference method is a numerical approach to solving differential equations. The fundamental equation for two-dimensional heat conduction is the two-dimensional form of the Fourier equation (Equation 1) 1,2