Thermopile Spreading-Resistance Lab (2D)

Thermopile Setup

Junction pairs N 12

Hot contact T_h 380 K

Cold plate T_c 300 K

Hot-contact width 30%

Couple material

External load R_L 10.0 Ω

Live Electrical Output

Solved EMF (self-consistent)

0.00 V

Ideal EMF (no spreading)

0.00 V

Spreading loss

Internal R (N·r)

0.0 Ω

Circuit current

0.00 A

Power in load

0.00 W

Power delivered vs load resistance — marker = current R_L

How it works

The 3D version of this simulator hands every junction the same ΔT = T_h − T_c, as if a full-width hot plate touched every leg equally. This 2D cross-section instead solves the actual unsteady 2D heat-diffusion PDE, ∂T/∂t = α∇²T, across the pile's body by explicit finite differences (FTCS) every frame:

T[i,j]ⁿ⁺¹ = T[i,j]ⁿ + α·(T[i+1,j]+T[i-1,j]+T[i,j+1]+T[i,j-1] − 4T[i,j])ⁿ

Boundary conditions:
  bottom edge      → Dirichlet, T = T_c  (full-width cold plate)
  top edge, source → Dirichlet, T = T_h  (hot contact patch only)
  top edge, else    → zero-flux (insulated, mirrored)
  left / right edges → zero-flux (insulated, mirrored)

When the hot contact is narrower than the array (a small heater pad instead of a full-area plate — common when a thermopile is driven by a pipe, a chip, or a point source), heat has to spread laterally through the pile's own body before it reaches junctions far from the patch. Each junction's driving ΔT is read directly off the solved temperature field at its own position, so junctions under the patch run near the full T_h − T_c, while ones near the edges run measurably cooler — a real effect called thermal spreading resistance, well known in TEG module design. The pile's total EMF is then the same series sum as the 3D model, EMF = S·Σ ΔT_i, just built from N genuinely different, self-computed ΔT_i instead of one assumed value.

N slider — resamples the N leg positions across the solved field; more junctions spread further toward the cooler edges.
T_h / T_c — the Dirichlet boundary values that drive the PDE; watch the field visibly re-diffuse toward the new steady state instead of jumping instantly.
Hot-contact width — narrower means more spreading resistance and a bigger gap between the solved and ideal EMF; at 100% the patch covers the full top edge and the two EMFs converge.
R_L slider / Match button — sweeps the load and snaps it to the maximum-power point, exactly as in the 3D circuit model.

Real-world relevance: this is why real thermoelectric generator modules are built no larger than their heat source can actually feed — oversizing the module past the source's footprint wastes area on junctions that never see the full ΔT.

Field: settling…

Hot contact patch (T_h) Cold plate (T_c) Per-junction ΔT bars