Microchip heat flow - 2D thermal map

What is modelled

A silicon die (12 x 12 x 0.5 mm) with four cores and a cache/uncore area is pressed against a heatsink through a 50 micrometre layer of thermal paste. Heat is produced in the cores, spreads sideways through the silicon, crosses the paste into the heatsink block and is carried into the room air by the fins and the fan. The temperature field is stepped explicitly on a 24 x 24 grid; the heatsink is one lumped thermal mass.

Equations

• Die cells: C dT/dt = P + k d ∑(Tnb − T) − Gv (T − Ts) with k = 130 W/m/K, ρc = 1.63 MJ/m³/K, edges insulated. The explicit step is 0.5 ms, below the stability limit C / (4 k d + Gv).
• Paste conductance per cell: Gv = A / (tpaste/kpaste + d/2k) with 50 micrometres of paste; dry or aged paste (0.5 W/m/K) gives a far larger temperature step than good paste (4-12 W/m/K).
• Heatsink: Cs dTs/dt = ∑ Gv(T − Ts) − Gsa (Ts − Troom) with Cs = 240 J/K and Gsa = 1.2 + 2.8 f0.8 W/K. The 0.8 exponent follows the forced-convection scaling Nu ∝ Re0.8 for turbulent flow over fins; the numbers are chosen to resemble a small tower cooler, not measured from one.
• Power: core power rises linearly with load from 2 to 22 W, the uncore adds up to 10 W, and leakage adds 8 W at 60 C doubling every 30 K (a simple exponential fit to subthreshold leakage).
• Throttling: above 97 C the dynamic power is scaled down by a first-order controller (floor 30 %), the way thermal management lowers the clock and voltage.

Simplifications

• Constant silicon conductivity (it actually falls about 30 % between 25 and 100 C), a single heatsink node (real sinks have spreading and fin gradients) and a uniform paste layer.
• No package lid, no heat leaving through the board, no radiation, and no transient air heating inside the case. Core size, power and sink numbers are illustrative of a desktop processor.
• The leakage curve and the throttle controller are generic models, not a particular chip's firmware.

Model checks

• Total energy is conserved to rounding error: energy produced equals energy stored plus energy carried away.
• Uniform power with a fixed-temperature sink reproduces the single-time-constant exponential and the exact steady value P/G.
• A cosine temperature pattern decays at the analytic rate that combines lateral diffusion and vertical loss.
• With the whole stack the steady die temperature equals P times the series sum of the paste and heatsink resistances.