A Numerical solution to One Dimensional Conductive Heat Transfer with Variable Conductivity

Numerical analysis is one of the most researched fields today. Getting an exact solution for physical problems is too difficult and again for most of the physical problems exact solution does not exist at all, so numerical techniques are preferred to obtain a fairly exact solution with ease. In this project, you will write your own codes to simulate temperature distribution over a 1D flat plate and compare its result with the exact solution to check the accuracy of your Numerical solution.

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Heat transfer is classified into various mechanisms, such as conduction, convection, radiation. The simplest mode of heat transfer is by conduction. Conductive heat transfer is the phenomena in which heat is transferred from one body to another due to temperature difference when they came in contact. In this project, you are going to solve a few conductive heat transfer problems using Finite Difference Method (FDM). You need to find the Analytical solutions to, governing differential equations for one-dimensional steady-state conduction, without heat generation and constant thermal conductivity. And after that, you need to find the Numerical solutions for those equations using FDM.

Problem Description:

One Dimensional Flat Plate Problem: A Iron wire of uniform conductivity is exposed to the environment. One side of the wire is in contact with a heat source of 1000°C. The length of the wire is 2 meter and the atmospheric temperature is 30°C. Determine the temperature distribution over a rod during steady state.

Project Description:

1. FDM: This is one of the discretization techniques used in numerical analysis to solve differential equation by approximating the derivatives. Here we discretize both space and time into N no of data points, where we store the data and keep updating as we move forward in time.
2. Taylors Series Expansion: To implement the finite difference method, Taylor series representation of a function should be known. In mathematics, Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. That is, if a function is continuous and we know its derivative at a point, then we can easily determine its value on the next neighboring point in the domain by using Taylor series.
3. Forward in Time Central in Space (FTCS) discretion: Among all the discretization techniques FTCS scheme is more preferred in FDM for producing an accurate and stable result. In this technique, while computing we will march forward in time and central in space.

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Project Implementation:

1. Find Analytical solutions to, governing equations for one-dimensional steady-state conduction, without heat generation and constant thermal conductivity. In this step, you also need to write C programming codes to calculate the temperature distribution over the given plate using the solution you got by solving above differential equations. At the end save all the result data in a text file.
2. Then, you have to find the Numerical solution to those equations using FDM. To do so, write codes to discretize the domain and initialize the boundary condition. Chose Forward in time and Centre in space discretization (FTCS) to get a better result. Interact the solution for a number of times until you get an accuracy of 10e-5. Don’t forget to save the result data in a text file.
3. Plot the result data using Gnuplot or Minitab. For plotting, contours use Miniplot for ease and convenience.
4. Compare your result for an analytical solution and numerical solution by plotting them simultaneously. If both of your solutions are coinciding or near to coincide then your numerical solution is correct.

Project Brief: By doing this project you can observe that, the temperature distribution over the material is linear when the thermal conductivity of the material is assumed to be constant.

Software requirements:

1. Dev-C++: You will be needing Dev-C++ software to write logic and interact the solution for a number of times.
2. Gnuplot: Also, you will be needing plotting software such as Gnuplot to plot the result data and compare the solution.

Programming language: C programming language

• DevC++
• Gnuplot

• Numerical Methods
• Variable conductivity
• FDM

A Numerical solution to One Dimensional Conductive Heat Transfer with Variable Conductivity