Arteries carrying warm blood outward and veins carrying cooled blood back run side by side along the limb, exchanging heat directly instead of losing it all to the air. Along the vessel length x (0 = core, 1 = extremity), for artery temperature Ta and vein temperature Tv:
∂T_a/∂t = -v·∂T_a/∂x - k·(T_a - T_v)
∂T_v/∂t = +v·∂T_v/∂x + k·(T_a - T_v) - L(x)·(T_v - T_amb)
L(x) = loss coefficient, largest near the extremity (x→1)
k is the countercurrent coupling — how tightly the two vessels are thermally bonded (the anatomical "rete mirabile" seen in arctic mammal legs, penguin feet, and tuna/whale flippers). v is proportional to blood flow rate.
- Ambient temperature — colder air pulls more heat out at the extremity, cooling the returning vein.
- Core body temperature — sets the fixed temperature blood enters the limb at.
- Blood flow rate — faster flow (v) advects heat down the limb faster, giving the exchanger less time per unit length to trade heat.
- Countercurrent coupling — the exchange strength k; drag to 0% to see the "no adaptation" case where the whole limb runs near core temperature and loses far more heat.
- Countercurrent ON/OFF — toggles whether the vein runs backward (real countercurrent) or forward alongside the artery (co-current, the much less effective arrangement); compare "core heat retained" between the two.
Core heat retained is measured against a single-pass baseline (blood entering and leaving at core temperature with no exchange at all): the fraction of that maximum possible loss actually avoided thanks to the exchanger. This is why an arctic fox can stand on snow with a foot only a few degrees above freezing while its core stays at 38 °C — almost all the heat that would otherwise leave with the returning blood is recaptured before it ever reaches the paw.