# LED Thermal Modeling (How to solve heat equation with constant heat source)

I have a mechanical design with LEDs that generate heat. I want to estimate the temperature at the LED junction vs. time, but especially at steady state.

Knowing the LED voltage drop and current, I can estimate the power dissipated via heat (some of the power goes to light, but to be conservative, I thought to estimate heat power as simply I*V).

The LED junction temperature needs to be held below a certain temperature. The thermal system will consist of the LED junction to thermal pad, the circuit board, and the mechanical design. The thermal resistance is given for the junction to thermal pad (about 10 K/W), and the manufacturer provides a couple different circuit board designs, each with their own thermal resistance. The best design can achieve 3-4 K/W, but a less expensive design may be an option. So I need the thermal resistance of the mechanical housing, which is pre-existing, and cannot be changed except for how to attach the LEDs to the housing.

The geometry is somewhat complicated, so I thought I'd start with a simple aluminum block (the circuit board mounts directly to the block), to make sure my modeling is correct, and then move to more complicated geometries. The LEDs are arranged in a line, so I'm going to assume the heat transfer is constant along the LED line (call it the z-axis). I also don't want to consider convection or radiation right now, along the length of the block.

So let's say I attach a heat sink to the end of the block opposite the LEDs, with a low enough thermal resistance to ambient so the heat sink end of the block is at ambient. The thermal resistance of an aluminum block is L/(kA), where L is the length, k is the thermal conductivity (0.25 W/(mm*K)), and A is the cross-sectional area. So if the length is 20 mm (along x-axis), the cross-sectional area is about 300 mm^2 (about 6 mm x 50 mm, with 6 mm along y-axis, and 50 mm along z-axis), the thermal resistance is about 0.27 K/W. The LED power per unit area is 0.0896 W/mm^2 (total power of 26.88 W, spread across the 300 mm^2).

Note: I like to work in mm for this problem.

This problem is 1-D, and from Fourier's conduction law, the temperature drop across the block at steady state should therefore be:

$\Delta T = L \cdot P/(kA) = 7.17 K.$

I want to be able to model this and come up with the same result.

Because the final geometry is more complex, I believe I need to use the Finite Element Method for the final solution, but for starters, it'd be nice to know how to start with the heat equation and end up with the result obtained from Fourier's law (since the former can be derived from the latter).

I found freefem++ (http://www.freefem.org/ff++/index.htm), which can approximate a solution to the heat equation using the Finite Element Method, but variational formulation is beyond my understanding, and none of the examples deal with a constant heat source.

I want to setup the heat equation and boundary conditions. I especially want the steady-state result, but I'm also interested in the temperature evolution vs. time. The (3-D) heat equation is

$${{\partial u} \over {\partial t}} = \alpha \nabla ^2 u + {Q \over {c \rho}},$$ where

$u = u(x,y,z,t)$ = temperature $(K)$

$\alpha = {k \over c \rho}$ = thermal diffusivity $({mm^2 \over s})$

$c$ = specific heat $({J \over {kg \cdot K}})$

$\rho$ = mass density $({kg \over mm^3})$

$Q$ = power per unit volume $({W \over mm^3})$

Since the simple block problem is 1-D, the heat equation becomes:

$${{\partial u} \over {\partial t}} = \alpha {{\partial ^2 u} \over {\partial x^2}} + {Q \over {c \rho}}$$

or

$$u_t = \alpha u_{xx} + {Q \over {c \rho}}$$

Note, Q is a function of $(x, t)$. This paper provides an analytical solution for this 1-D problem: http://people.math.gatech.edu/~xchen/teach/pde/heat/Heat-Duhamel.pdf

In order to use Duhamel's Principle, I first reformulate the problem so the heat source is at $L/2$ the temperature at $x=0$ and $x=L$ is held to zero (whether zero or ambient temperature, the solution is the same). Then the heat source is the Dirac-delta function at $x=L/2$, and the problem becomes:

$$u_t = \alpha u_{xx} + {Q_i \over {c \rho}} \delta(x-{L \over 2}), 0 < x < L, t > 0$$

and the boundary conditions are:

$$u(0,t) = 0, u(L, t) = 0, t > 0$$ $$u(x,0) = 0, 0 \le x \le L$$

Here, $Q_i = 0.896 W/mm^3$ is the heat source. I was able to approximate the solution given in the paper using Matlab (see below), and as $ds \rightarrow 0$, the solution becomes closer to a triangle function centered at $x=L/2$ and increasing asymptotically over time to about 7 K. Note that because the heat source is in the center, I doubled the power and block length, and the solution for the above problem is one half of the block ($x \ge L/2$).

So now what happens when I go to 2-D? As I said, the real geometry is complex (but can be modeled as a 2-D problem, since the geometry along the z-axis is essentially constant), so I don't expect an analytical solution. Perhaps my simple 1-D example is not sufficient to demonstrate how to solve the problem with the Finite Element Method.

What if the heat source came in from the left and the bottom of the block was held at constant temperature?

Does the Duhamel Principle extend to 2-D? If so, and I approximate the auxiliary homogeneous problem with FEM, then how do I transform that approximation into the nonhomogenous solution?

Alternately, how would I formulate the nonhomogenous problem using variational formulation, in order to be able to use do FEM analysis directly?

## Matlab approximation to 1-D analytical solution

Here is my Matlab code that approximates the solution using Duhamel's Principle. I did the approximation using both Fourier series and Green's function.

% Approximate the analytical solution of the heat equation with a heat
% source in the center of a block.

% System parameters.
H = 6;               % the block height (mm)
L = 40;              % the block length (mm)
W = 50;              % the block width (mm)
kAl = 0.25;          % Aluminum thermal conductivity (W/(mm*K))
c = 897;             % Aluminum specific heat capacity (J/(kg * K)).
rho = 2.7E-6;        % Density (kg/mm^3).
alpha = kAl / (c * rho);   % Thermal diffusivity (mm^2/s).
Qi = 2 * 27 / 300;          % Input power per unit volume length (?).

dx = 0.2;
dt = .2;
x = 0:dx:L;
tmax = 10;
t = 0:dt:tmax;

% Approximate heat equation using Fourier series and Duhamel's Principle.
ds = 0.1;
N = 200;
n = 1:N;
b = 2*Qi*sin(n*pi/2)/(c*rho*L);
% As N goes to infinity, the solution
% approximates a triangle function centered on L/2.  Because we can't go to
% infinity, there will always be a sharp spike at x = L/2.
u = zeros(length(x), length(t));
for xi = 1:length(x)
for ti = 1:length(t)
tc = t(ti);
for ni = 1:length(n)
s = 0:ds:tc;
sint = 0;
for si = 1:length(s)
sint = sint + b(ni)*exp(-alpha*(n(ni)*pi/L)^2*(tc-s(si)))*ds;
end
u(xi, ti) = u(xi, ti) + sin(n(ni)*pi*x(xi)/L) * sint;
end
end
end

figure;
mesh(t, x, u);
ylabel('x (mm)');
xlabel('t (s)');
zlabel('Temperature (deg C)');
title('Approximation to heat equation solution with constant heat source at L/2, using Fourier series');

% Approximate solution using Green's function.  Note that as ds -> zero,
% the solution approximates a triangle function centered at L/2, and
% increasing asymptotically over time.
u = zeros(length(x), length(t));
N = 40;
n = -N:N;
ds = 0.01;
for xi = 1:length(x)
for ti = 1:length(t)
tc = t(ti);
if tc == 0
continue;
end
s = 0:ds:(tc-ds);
for si = 1:length(s)
nint = 0;
for ni = 1:length(n)
nint = nint + exp(-(x(xi)-2*n(ni)*L-L/2)^2/(4*alpha*(tc-s(si)))) - ...
exp(-(x(xi)-2*n(ni)*L+L/2)^2/(4*alpha*(tc-s(si))));
end
u(xi, ti) = u(xi, ti) + ...
(Qi/(c*rho)) * nint * ds / sqrt(4*pi*alpha*(tc-s(si)));
end
end
end

figure;
mesh(t, x, u);
ylabel('x (mm)');
xlabel('t (s)');
zlabel('Temperature (deg C)');
title('Approximation to heat equation solution with constant heat source at L/2, using Green''s function');

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I know what Edison would do. Just build one and measure. You've got some assembly that's hot on one end and cold on the other, transmitting a certain amount of heat energy per second, linear in the temperature difference. I can think of several ways to measure that. There's a story of how he asked some prospective engineers the volume of a light bulb. Some got all mathy, but one just poured in some water & poured it into a measuring cup. Who do you think got the job? –  Mike Dunlavey Nov 22 '11 at 19:07
Patrick, whenever you're around next, could you check on the last edit made to this question and verify whether it's correct or not? –  David Z Jun 20 '12 at 3:29
@David It looks like someone commented out the function declaration without removing the return statement. I tend to put all my Matlab scripts in functions as a habit -- this way I don't depend on variables that might exist in the workspace, but which I forgot about after removing their initialization in my script. If I need to debug, I'll just set a breakpoint at a suitable place in the script. But it's true, the function line isn't needed, and your fix should make it run again. –  Patrick Jun 25 '12 at 19:04
Patrick: Yes, I made the edit to the script file, but forgot to scroll down to the bottom and check it there was a return to comment out. I seen nowhere in the script that the function was being called. Thanks for catching that. –  night owl Jun 28 '12 at 6:24
@night owl The entire script is the function. I "call" it by hitting F5 in Matlab, or call it from the command prompt. As I said in my other message, this is just my habit in order to avoid mixing workspace variables with those used in the script. For the purposes of this post, a function is not necessary. –  Patrick Jun 28 '12 at 16:42

I solved this type of problem before both experimentally and numerically. These problems are not easily solved analytically. I do not recommend an analytical approach other than to provide some intuition.

To take a numerical approach, consider http://www.amazon.com/Transfer-Mcgraw-Hill-Series-Mechanical-Engineering/dp/0073529362 . This text has methodologies for finite difference that are quite easy to understand. The idea is to set up the finite difference equations, solve them with a matrix inversion. For the time evolution problem, you merely increment in time and solve the matrix again and again.

With specialized problems like these, you have a few choices. One is to build a simple finite difference model and debug it with known solutions (again go to Hollman). The other solution is a more exotic finite element solution. Be wary, it is easy to go wrong and to waste a lot of time meshing geometry.

Let me know if you need specific help. You can see some of my papers at:

http://spie.org/Publications/Proceedings/Paper/10.1117/12.842043

http://spie.org/Publications/Proceedings/Paper/10.1117/12.842881

The real fun is in transient problems! These models were done in Nastran and match the experimental data extremely well. To get a good match with experiment, pay close attention to boundary conditions (completely insulating and perfect temperature baths).

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If your goal is practical, and you're looking for a numerical answer, why not use some numerical method to solve the equation? Mathematica, Maple, and Matlab (I think) all have tools for doing this.

As far as the steady state solution, you can simplify the problem greatly since as t goes to infinity, the derivative of u with respect to t goes to zero, giving you a simple Poisson Equation.

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Last time I had a 2D thermal problem, I wanted to match Greens functions on the various boundaries, but that is tricky (algebra and programming wise), i.e. the chance of a coding blunder was too great. Since modern computers are amazingly fast brute force machines successive relaxation on a grid is so trivial to program you almost can't mess it up, and solution time is quite minimal. –  Omega Centauri May 26 '11 at 4:31
I edited my question to (hopefully) clarify what I'm asking. I expect to need to solve the problem numerically, but I'm not sure how to formulate the problem to do so. I realize the time term vanishes as t goes to infinity, but I would like to include it if possible. –  Patrick May 26 '11 at 14:06
@Pat. Many time independent problems are solved by running a time simulation until equilibrium is reached, so it is perfectly acceptable to formulate/solve it as a time dependent problem. –  Omega Centauri May 26 '11 at 18:01