A Numerical Solution to 2D Flat Plate Problem with Constant Conductivity Heat Transfer

All the engineering devices we use involves the conversion of energy. In this process, a lot of heat released to the surroundings as a part of energy loss. The efficiency of any device can be increased by minimizing loss due to heat transfer. That’s why a lot of research going on around the world in the field of heat transfer. To solve real-life heat transfer problems Numerical methods are preferred, because of less time consumption, ease, and convenience. To get hands-on experience on numerical methods, here you are going to solve the Two-Dimensional flat plate problem with Constant conductivity heat transfer by using Numerical methods.

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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 conductive heat transfer problems using Finite Difference Method (FDM). You need to find the Analytical solutions to, governing differential equations for Two-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:

Two Dimensional Flat Plate Problem: An Aluminium square thin plate having dimension 1m in each side is exposed to the environment. If the temperature at one boundary is maintained at 400 ºC and rest are at 100 ºC, then determine the temperature distribution over it 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 two-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. Remember this is a two-dimensional problem, so you have to discretize both X-axis and Y-axis. Always chose Forward in time and Centre in space discretisation (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: After plotting contours you can observe how temperature distribution occurs along the area of the plate.

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. Minitab; Also, you will be needing Minitab plotting software to plot contours using your result data and compare the solutions.

Programming language: C programming language

• DevC++
• Gnuplot
• Minitab

• CFD
• numerical Methods
• conductive heat transfer
• FDM

A Numerical Solution to 2D Flat Plate Problem with Constant Conductivity Heat Transfer