2D heat transfer of profiles and heat sinks

Heat flows from where it is generated, in a component, a winding or a conductor, through the part to where it leaves. It leaves by convection to air or a coolant, by radiation, or into a body held at a known temperature, such as a cold plate. The temperatures along the way decide whether a component stays within its limits. Formulas give the conduction through walls, pipes and simple fins. For a real outline with several materials, the temperatures follow from a solution of the heat equation. The tool meshes the part and solves that equation with the finite element method, a steady-state or transient thermal analysis, and shows the temperature field, the hottest point and the heat leaving at every boundary.

Cross-section or flat plate

A cross-section stands for a long part, such as an extruded heat sink or a bus bar, in which the heat flows in the plane of the drawing and not along the part. The Length gives the part's length, and the results hold for that length, with the thermal resistance also per meter. A flat plate stands for a plate in which the heat spreads in its own plane, such as a spreader plate or the base of a housing. The Thickness gives the plate's thickness, and convection and radiation can act on both of its faces as well, with their values in the Part group.

Regions and materials

Every closed region of the drawing gets a number, F1, F2 and so on, and its own material with its thermal conductivity k. A loop inside another on the same layer is a hole, and a loop on a layer of its own is an insert, a region of its own material. Regions that touch exchange heat with perfect contact, unless a contact resistance is set on the edge between them. A region can also generate heat, given in watts for the whole region, such as the losses of a component or the Joule heating of a conductor.

Edge, hole and interface numbers

The outline's edges get E1, E2 and so on from the lowest left corner around the part, every hole one number for its whole boundary, H1, H2 and so on from the left, and every edge between two regions I1, I2 and so on. The boundary conditions refer to these numbers, and the drawing shows them, with each edge's length on hover. The Propagation column carries a row on along the outline, with planex2d's selection propagation.

• Minimal keeps the edge alone.
• Coincident adds the edges joined to it end to end, which on a closed outline is all of it.
• Tangent adds the edges that continue it tangentially, such as the lines on either side of a fillet.
• Curvature adds the edges that continue its curvature as well.

Boundary conditions

• A temperature holds an edge or a hole at a fixed temperature, such as a wall cooled by water.
• A heat input puts a heat flow in watts into an edge or a hole, spread evenly along it.
• Convection gives off q = h (T − T_∞) to a fluid at the ambient temperature T_∞, with the heat transfer coefficient h.
• Radiation gives off q = εσ (T⁴ − T_s⁴) to surroundings at T_s, with the emissivity ε. It is the radiation to open surroundings, without view factors between the part's own faces, so fins facing each other give off less than it says.
• A contact resistance R between two regions makes the temperature jump by q R across the edge between them, such as at a thermal pad or a bolted joint.

Edges and holes not in the table are insulated. Where rows meet on an edge, heat input, convection and radiation add, and a temperature overrides them.

Results

The results are the hottest and the coldest point, the heat entering the part and the heat flow through every boundary condition, which together balance. The thermal resistance is the rise of the hottest point over the lowest reference temperature, an ambient or a held temperature, divided by the heat put in, in K/W. The drawing colors the part by its temperature, or by the heat flux with the list beside Results, and marks the hottest point.

Transient

With Transient on in the Solver group, the part starts at the start temperature and heats up or cools down over the duration, in 150 time steps of the implicit Euler method. Each material then also needs its density and specific heat. The results add the curve of the hottest point over time and the time it takes to cover 63 % of its rise to the steady state, which for a small, well-conducting body is its time constant. The time slider shows the temperature field at moments across the duration, on one color scale.

Simplify outline

Drawings often carry arcs and holes as chains of short lines, which give the outline many edge numbers. With Simplify outline on, planex2d fits arcs to such chains, within a ten-thousandth of the part's size, before the part is numbered and meshed. Turn it off to analyze the outline exactly as drawn. Where the fit misses an arc, the contour checker fits arcs with a tolerance and an edge length of your choice, closes gaps and removes tiny edges, and saves the cleaned drawing for the heat transfer tool.

Units

Lengths are in millimeters, temperatures in °C, heat flows in watts, the thermal conductivity in W/mK, heat transfer coefficients in W/m²K, the density in g/cm³ and the specific heat in J/kgK. A file is read in the unit it names and converted to millimeters.

1. Load

   The page loads Python and planex2d into this tab, and NGSolve with the first solve, about 23 MB the first time. Your browser keeps them in its cache, so later visits start faster.

2. Read

   planex2d reads the file, joins its closed contours into regions, simplifies the outline and numbers its regions, edges and holes.

3. Solve

   NGSolve meshes the part and solves the heat equation, again whenever a setting changes after the first solve.

4. Save

   The results are saved as a PDF report or a CSV table, made in the tab as well.

The finite element model

The mesh has elements of about a thirtieth of the part's size, finer along arcs, and it is curved to the exact outline. NGSolve solves the heat equation with second-order elements, set from 1 to 5 in the Solver group, and a sparse Cholesky factorization. Radiation makes the problem nonlinear, and Newton's method solves it. Across a contact resistance, each region has a temperature field of its own, coupled along the edge between them. The tests compare the tool with known answers:

• A plane wall and a pipe wall give their heat flows within 0.001 %.
• A fin, and a plate fin cooled over its faces, give the heat flow of the fin formula within 0.1 %.
• Two slabs with a contact resistance give their series resistance, and a slab heating itself its peak temperature q w² / (8 k).
• A radiating wall meets its one-dimensional energy balance, and a cooling body its time constant ρ c V / (h A) within 1 %.

Supported files

Run it in Python

The same analysis takes a few lines with planex2d and NGSolve. The example is the cross-section of an aluminum fin of 50 × 2 mm, its base held at 100 °C and the rest cooled with h = 10 W/m²K at 0 °C. planex2d draws the fin and names its edges, and NGSolve meshes it and solves the heat equation in millimeters, with k = 200 W/mK as 0.2 W/(mm·K) and h as 10⁻⁵ W/(mm²·K). See Meshing with netgen in the docs.

$ pip install "planex2d[occ]" ngsolve

import ngsolve as ngs
import planex as px

fin = px.QuickSketch().Rectangle(50, 2).Face()
for e in fin.edges:
    e.name = "base" if e.midpoint.x == 0 else "air"
mesh = ngs.Mesh(px.OCCGeometry(fin).GenerateMesh(maxh=1))
fes = ngs.H1(mesh, order=2, dirichlet="base")
T, v = fes.TnT()
k, h = 200e-3, 10e-6
a = ngs.BilinearForm(k * ngs.grad(T) * ngs.grad(v) * ngs.dx + h * T * v * ngs.ds("air")).Assemble()
gf = ngs.GridFunction(fes)
gf.Set(100, definedon=mesh.Boundaries("base"))
r = -a.mat * gf.vec
gf.vec.data += a.mat.Inverse(fes.FreeDofs(), inverse="sparsecholesky") * r
print("heat flow", ngs.Integrate(h * gf, mesh, ngs.BND, definedon=mesh.Boundaries("air")) * 1000, "W per meter")

It prints a heat flow of 97.80 W per meter of fin, against 97.80 W from the fin formula √(h P k A) θ tanh(m L_c) with the corrected length L_c = L + t/2, which counts the fin's tip.

Privacy

Limits

• The tool solves heat conduction in solids with constant material values. The flow of the air or the coolant is not computed, and it enters through the heat transfer coefficient h.
• Radiation goes to open surroundings, without view factors between the part's own faces.
• A cross-section assumes no heat flow along the part, and a flat plate a temperature that does not vary through its thickness.
• The material values are typical values. Use the values of your material where they matter.
• Window frames and thermal bridges after ISO 10077-2 and ISO 10211, with their cavity rules, are not part of the tool.
• The tool analyzes one connected part. A file must be planar, and its text, dimensions and hatches are not read.
• A part that fails or a result that looks wrong? Open an issue on GitHub and attach the file if you can.