# Conduction and Convection in a Fin of Uniform Cross Section

## Summary

Conduction and convection are two important modes of heat transfer. To understand the mechanisms of these two types of heat transfer processes, we study the temperature distribution in a fin of uniform cross section through analytical and numerical approaches. Various boundary conditions are investigated to demonstrate the effects of conduction and convection on the temperature distribution. Some of the related engineering applications of this study include the design of a heat exchanger with fins or a thermal radiator.

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## Learning Goals

Upon completing this project, students should learn to solve the temperature distribution along a one-dimensional fin analytically and numerically. Specifically, students learn to solve diffusion equations with source using the finite volume method and implement the algorithm via MATLAB programming. Students are expected to build a MATLAB App to perform parametric studies and to visualize the simulation results.

## Context for Use

This activity can be used in junior/senior level engineering courses such as heat transfer and computational fluid dynamics or in a physics course on the concepts of conduction and convection. It requires students to investigate the temperature distribution along a one-dimensional fin through analytical and numerical approaches. Several parameters and boundary conditions are to be investigated, which makes the study comprehensive and challenging. Students are expected to complete the project within one week including modeling and simulation. A written report is required to address all questions and exercises in the activity.

The prerequisites of this project are some familiarity with undergraduate-level heat transfer, ordinary differential equations, and numerical analysis. Students are expected to know some common techniques in solving ordinary differential equations analytically. Also, students need to understand the basic procedures in solving diffusion equations with source, numerically using the finite difference or finite volume method. This activity provides students with the opportunity to develop their computational skills on MATLAB programming and MATLAB App building.

## Description and Teaching Materials

To investigate the temperature distribution in a one-dimensional fin, the fin structure can be modeled by a slender bar with various boundary conditions. First, we will discuss how to model the conduction and convection heat transfer processes based on two physical laws: Fourier's law and Newton's law of cooling. The resulting steady-state, second-order ordinary differential equation can be solved analytically provided with certain boundary conditions. Then we will discuss how to solve the differential equation numerically using the finite volume method. The numerical algorithm will be implemented in MATLAB. A MATLAB App will be built to conduct numerical simulations and to verify the analytical results. Please find more details in the attached file.

## Teaching Notes and Tips

Before assigning this project to students, it is recommended to work through the mathematical model, in particular the various boundary conditions that affect conduction and convection heat transfer of the fin. The instructor can work through an example in the attached teaching material to help students understand the solutions analytically. To implement the finite volume algorithm in MATLAB, it is helpful to provide students with a coding example (attached fv1d_fin.m) which covers discretization, solving equation systems and graphing. Finally, students can follow the MATLAB App Designer tutorial (linked in the references) and the provided app (tempDistrFin.mlappinstall) to build a MATLAB App for the simulations.

## Assessment

The project will be graded on:

1. (15%) Modeling: Students are expected to complete the exercises* in Section 4.2 to understand the solutions analytically.

2. (15%) Numerical Schemes: Develop/complete the finite-volume schemes for the heat transfer problem following the instructions in Sections 5 & 6.

3. (25%) MATLAB Codes: Codes for solving the model equation based on the three prescribed boundary conditions discussed in Sections 4 & 6. The codes produce correct numerical results.

4. (15%) MATLAB App: An interactive MATLAB App can be used to perform parametric studies and to visualize the simulation results.

5. (15%) Numerical Exercises: Students are expected to complete the exercises* in Section 5.1 based on the numerical experiments.

6. (15%) Overall Organization: A final report including description of the physical problem, mathematical modeling, numerical scheme, exercises, and comparison of the numerical and the analytical solutions. The report is organized and easy to follow.

* Hints for selected exercises are provided in the teaching materials.

## References and Resources

[1] Bergman, T. L., Incropera, F. P., DeWitt, D. P., & Lavine, A. S. 2011. Fundamentals of heat and mass transfer. John Wiley & Sons.

[2] Ferziger, J. H., Peri Ìc, M., & Street, R. L. 2002. Computational methods for fluid dynamics. Springer.

[3] MATLAB App Designer. https://www.mathworks.com/products/matlab/app-designer.html

[4] Teaching Fluid Mechanics and Heat Transfer with Interactive MATLAB Apps. https://www.mathworks.com/videos/teaching-fluid-mechanics-and-heat-transfer-with-interactive-matlab-apps-81962.html