Cooling Fin

Introduction

Computers consume power and all electronic components have a measurable resistance, so some of the power is converted into heat. To prevent system failure due to overheating of components, computers and other devices require a means of dissipating the heat into the environment. In this project, we created a model of a cooling fin acting as a heat sink for a power source. After representing the heat distribution of the fin with the elliptical partial differential equations describing the steady-state solution, we solved for the temperature distribution of the fin using the thermal convection boundary conditions and other parameters.

The conduction of heat along the side of the fin with the power input obeys Fourier's Law,

\[ q = -K A \nabla u \]

where \(q\) is the heat per unit time, \(A\) is the cross-sectional area of the fin, \(K\) is the thermal conductivity of the material, and \(\nabla u\) is the gradient of the temperature.

The convection of heat as it is dissipated into the surrounding air follows Newton's law of cooling,

\[ q = -H A (u - u_{b}) \]

where \(H\) is the convective heat transfer coefficient, and \(u_{b}\) is the ambient temperature of the air. Let \(\delta\) represent the length in cm of the fin along its shortest edge (the thickness of the fin), and let \(\Delta x\) and \(\Delta y\) represent the lengths of the two longer edges.

For the system to be in thermal equilibrium, the heat entering from the power source through conduction must equal the heat leaving the system through convection, so

\[ -K \Delta y \delta u_{x}(x,y) + \\ K \Delta y \delta u_{x}(x + \Delta x,y) - \\ K \Delta x \delta u_{y}(x,y) + \\ K \Delta x \delta u_{y}(x,y + \Delta y) - \\ 2 H \Delta x \Delta y u(x,y) = 0 \]

Note that we have set the air's bulk temperature, \(u_{b}\) to zero, so our solution u will represent the difference in temperature between the fin and its surroundings.

After dividing both sides by \(\Delta x\) and \(\Delta y\), and re-expressing finite differences \(\delta\) as infintesimal differences, we obtain our elliptical partial differential equation

\[ u_{xx} + u_{yy} = \frac{2 H}{K \delta} u \]

As our solution will be constructed using a matrix with step sizes \(h\) and \(k\), we can use the finite difference approximation to express our PDE as

\[ \frac{ u_{i+1,j} - 2 u_{i,j} + u_{i-1,j} }{ h^{2} } + \\ \frac{ u_{i,j+1} - 2 u_{i,j} + u_{i,j-1} }{ k^{2} } = \frac{2 H}{K \delta} u_{i,j} \]

On each of the fin's edge's, we apply the Robin boundary conditions using the first derivative approximation. We defined our left edge as the one recieving power from another heat source.