Temperature Distribution in a Cross Section
Overview
This project combined elements of heat transfer with elements of computing techniques to create a problem that was super fun to solve. First, it required a fundamental knowledge of heat transfer to solve the most basic version of the taken approach. Once the heat transfer was complete, the coding infrastructure needed to solve the problem at scale was built around it.
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Project Parameters
Below are the known variables for the project. The total width and height of the cross-section were 50 mm and 200 mm, respectively, but due to the assumption of symmetry, the cross-section was simplified into 4 similar sections as depicted by the red rectangle below. The inner sides of the 25x100 mm simplified cross-section were assumed as an insulated surface due to the symmetry of the cross-section. The top and bottom sides were cooled with the same convection coefficient, while the left and right sides were cooled with the same convection coefficients. The left and right sides were cooled with multiple convection coefficients ranging from 10 to 100 [W/m^2-K]. There is also heat being generated at a volumetric rate of 150,000 [W/m^3].
How to Approach the Problem
The fundamental problem with this project is the temperature values are not known at any location. To solve for one temperature you have to know the next and so on. This leads to the process of nodal analysis by the finite difference method. This process allows the temperatures at each node to be solved for simultaneously by performing an energy balance at each nodal point and construction a system of linear equations. An example of an energy balance on one of the inner nodes is shown below. The system of equations will have as many unknowns and equations as there are nodes in the cross-section, so as the nodes increase and the temperature precision becomes greater, the system becomes too large to handle by hand and must be solved with the help of some software.
Constructing a Nodal Network
The nodal network in theory must follow a few guidelines for the nodal analysis to work. The nodes must be evenly spaced in the horizontal and vertical directions, and they must line up in a grid like manner. To simplify the analysis, the nodal network used for this project was chosen to have an equal number of nodes in the vertical and horizontal directions. It was also noticed that the nodes in certain subsections can be treated with the same energy balance equations. The nine different subsections (show in blue), as well as a basic nodal network (shown as the dark dots) was placed on top of the original cross section, all shown below.
How Does MATLAB Understand the Nodal Network?
Once the nodal network was constructed in theory, it was then converted into the language of MATLAB. There were many ways to represent a 2-D matrix in MATLAB, for instance, by making a 2-D matrix... but it was more interesting to use a 1-D vector to represent the 2-D matrix. An energy balance was performed at each of the nine subsections to generate a set of equations unique to each subsection. Once the equations were made, MATLAB had to know when to apply each set of equations. The nine test cases listed below were used for the nine subsections mentioned earlier. Note that m is the size of the square matrix.
index = 1
index > 1 & index < m
index = m
mod(index-1,m) = 0
else
mod(index,m) = 0
index = m^2-m+1
index > m^2-m+1 & index < m^2
index = m^2
Solving the System of Linear Equations
Once the nodal network was translated into MATLAB, a system of linear equations was created and solved. All nine energy balance solutions associated with the nine different subsections were solved by hand and paired with the correct test case from the list above. Each time the program looped through an iteration it added a vector of coefficients to a matrix of coefficients, and a value representing the right side of the linear equation to the vector of knowns. After the iteration was complete there was an (m^2) x (m^2) matrix of coefficients and a vector of size m holding the known values in the equations. The temperatures were then solved for and the contour plots created.
Solutions
The goal of this project was to find the temperature distributions within a rectangular cross section. The results of the analysis from above can be seen below in the 2 heat maps. The plot on the left shows the temperature distribution with a convection coefficient of 10 [W/m^2-K] being applied to the right side of the cross section. The plot on the right shows the temperature distribution with a convection coefficient of 100 [W/m^2-K] being applied to the right side of the cross section. The increased cooling effectiveness can be seen clearly as the right plot is slightly darker all the way down the right edge, compared to the left plot.
When the above heat maps are examined further it can be found that the color map for the left plot is associated with a different temperature range than that of the right plot. To more clearly depict the effectiveness of the increased convection coefficient, the temperature ranges associated with the heat map were adjusted to span the overall temperature range of both plots. The results of this can be seen below.
