---
jupyter:
  description: 'An extruded aluminum heat sink with eight fins in air: a 2D heat
    conduction and convection solve in NGSolve, with one fin compared against fin
    theory.'
---

# Heat sink: temperature in an extruded profile

The example is an extruded aluminum heat sink with a base of `width` = 60 by `base_h` = 6 and `n` = 8 fins, `t` = 2
thick and `H` = 30 high, each with a round bulb on top, in a box of air. A component under the middle of the base puts
20 W into every 100 mm of length, and the fins and the base give it to the air. NGSolve solves the section in 2D, and
one fin is compared with fin theory. Lengths are in mm.

![The temperature in an extruded aluminum heat sink](../_static/gallery/heat-sink.png){.result}

```python
import math
import ngsolve as ngs
from ngsolve.webgui import Draw
import planex as px
```

```python nbsphinx="hidden"
from docs_figures import Row
```

## The section

The base is drawn with its bottom in three parts, the middle one being the `heater`, 20 wide. Every fin is a
rectangle named `fin0` to `fin7`, with a bulb of radius 1.5 on top that is named like its fin. The union keeps the
names on the outline. A box of air, 15 larger on every side, surrounds it, glued to the aluminum so that the two share
their edges, and the edges keep the heat sink's names (Figure 1).

```python
width = 60.0
base_h = 6.0
n = 8
t = 2.0
H = 30.0
heater = 20.0

base = px.QuickSketch((-width / 2, 0))
base.Line((-heater / 2, 0), name="bottom").Line((heater / 2, 0), name="heater").Line((width / 2, 0), name="bottom")
base.Line((width / 2, base_h), name="base").Line((-width / 2, base_h), name="base").Close(name="base")
pitch = (width - t) / (n - 1)
fins = []
for i in range(n):
    x = -width / 2 + i * pitch
    fins.append(px.QuickSketch((x, base_h), dir=(1, 0)).Rectangle(t, H, name=f"fin{i}").Face())
    fins.append(px.QuickSketch().Circle(x + t / 2, base_h + H, 1.5, name=f"fin{i}").Face())
sink = px.Unify([base.Face(), *fins])
sink.name = "aluminum"

box = px.QuickSketch().RectangleC((0, (base_h + H + 1.5) / 2), width + 30, base_h + H + 1.5 + 30, name="far").Face()
box.name = "air"
box.maxh = 5
model = px.Glue([box, sink])
```

```python nbsphinx="hide-input"
Row((model, "The heat sink in its box of air"), caption="The section of the heat sink")
```

## Temperature

NGSolve solves for θ = T − T_air in the aluminum, with the air at `T_air` = 20 °C:

- conduction at `k` = 200 W/(m K), that is 0.2 W/(mm K);
- the heat flux on the heater, 20 W per 100 mm of length, `q` = 0.2 W/mm over its width;
- convection `h_air` = 25 W/(m² K) on the base and the fins;
- the rest of the bottom insulated.

The mesh covers the air as well, and the air acts through the convection on the shared edges. The plot shows the
temperature in °C.

```python
T_air = 20.0
k = 0.2
h_air = 25e-6
q = 0.2
mesh = ngs.Mesh(px.OCCGeometry(model).GenerateMesh(maxh=1))
mesh.Curve(3)
fes = ngs.H1(mesh, order=2, definedon="aluminum")
u, v = fes.TnT()
air = ngs.ds(definedon=mesh.Boundaries("base|fin.*"))
a = ngs.BilinearForm(k * ngs.grad(u) * ngs.grad(v) * ngs.dx + h_air * u * v * air).Assemble()
f = ngs.LinearForm(q / heater * v * ngs.ds("heater")).Assemble()
theta = ngs.GridFunction(fes)
theta.vec.data = a.mat.Inverse(fes.FreeDofs()) * f.vec
temperature = mesh.MaterialCF({"aluminum": T_air + theta}, default=T_air)
print("hottest", round(T_air + max(theta.vec), 2), "°C")
Draw(temperature, mesh)
```

## One fin against fin theory

For a fin with a convective tip and its length corrected by t/2, fin theory gives
q = sqrt(2 h k t) θ_b tanh(m L_c) with m = sqrt(2 h / (k t)), where θ_b is taken at the fin root. The bulb is not part
of the theory and makes the fin slightly more effective. **Check** on fin 3, next to the middle:

```python
i = n // 2 - 1
x = -width / 2 + i * pitch + t / 2
fe = ngs.Integrate(h_air * theta * ngs.ds(f"fin{i}"), mesh)
mf = math.sqrt(2 * h_air / (k * t))
Lc = H + t / 2
theory = math.sqrt(2 * h_air * k * t) * theta(mesh(x, base_h)) * math.tanh(mf * Lc)
print("fin", i, round(fe * 100, 4), "W per 100 mm, fin theory", round(theory * 100, 4), "W")
print("fin efficiency", round(math.tanh(mf * Lc) / (mf * Lc), 3))
```

```python nbsphinx="hidden"
from ngsolve.webgui import Draw as WebguiDraw
import sys
sys.path.insert(0, "..")
from gallery import save_scene, save_figure, HIDDEN, CLEAN
save_scene(WebguiDraw(temperature, mesh, settings={**CLEAN, "Colormap": {"ncolors": 32}}, show=False), "heat-sink")
```