Hundredths of a foot
Put the numbers side by side and the scale of a thermosiphon becomes obvious. A circulator in a hydronic system produces something like ten to twenty feet of head. With the defaults here — four feet of height, water at 61.8 lb per cubic foot in the cool leg, an expansion of 0.00035 per °F and a rise around 38 °F — the hot leg comes out about 0.8 lb per cubic foot lighter, and four feet of that difference is roughly 0.05 feet of head. Fifty thousandths of a foot. About 0.02 psi.
Everything else about a natural circulation loop follows from that number being small. A single check valve can exceed it. A trapped air pocket at a high point stops it dead. A bore one nominal size down halves it. The loop does not have any authority to spare, which is why the arrangement matters so much more than the components.
Flow and temperature rise decide each other
This is the reason the page solves rather than calculates. More flow means a smaller rise across the collector; a smaller rise means a smaller density difference; a smaller difference means less head and therefore less flow. Go the other way and it also settles. There is exactly one temperature rise at which the flow the buoyancy produces carries exactly the heat put in, and the page finds it by bisection — the function is strictly decreasing, so the bracket always closes on the single root.
It reports how many steps it took and how far off zero it ended, so you can see it landed rather than gave up. With the defaults it converges to about 0.63 GPM at a 38 °F rise, and reading the heat back out of that answer returns the 12,000 BTU/h that went in.
The bore is the lever
Friction goes as roughly the fifth power of the bore. The table shows what that means in practice: the same loop in half inch and in one inch are not slightly different, they are different systems. This is why thermosiphon installations are plumbed a size or two above what the flow would suggest, and why a run with a lot of tight elbows can fail to circulate at all while the same length of straight tube works fine. The equivalent length field is where fittings enter, and on a short loop they are usually more of it than the pipe is.
Height helps too, but less than people expect. Doubling from four to eight feet does not double the flow, because the friction the extra head buys is also rising faster than linearly. The table on the page shows the actual shape.
Where the arithmetic is weakest
Hazen-Williams was developed for water moving at the speeds distribution pipework runs at, and a thermosiphon at a few hundredths of a foot per second is nowhere near that. In laminar flow the friction goes with the first power of velocity, viscosity matters and roughness essentially does not — which means the C factor is doing work it was never designed for, and a cold glycol mix will do worse than the page suggests. Treat the absolute numbers as indicative and the trends as real.
What the page will not tell you
Whether the loop will start, whether it will keep going, and whether it will run backwards after dark and quietly cool the store overnight. All three are real behaviours of natural circulation systems and none of them is arithmetic. They depend on where the tank sits relative to the collector, on whether there is a high point that collects air, and on what the route offers by way of resistance at almost zero pressure. That is the design, and it belongs to whoever does it.
Questions people ask
How much head does a thermosiphon actually produce?
Very little. With four feet between the collector and the tank, water in the cool leg at 61.8 lb per cubic foot and a 38 degree rise, the hot leg is about 0.8 lb per cubic foot lighter and the driving head is around 0.05 feet — roughly 0.02 psi. A circulator makes ten to twenty feet. That gap is the reason a single check valve or one trapped air pocket can stop a natural circulation loop entirely.
Why does the calculator have to iterate?
Because flow and temperature rise determine each other. More flow gives a smaller rise, a smaller rise gives a smaller density difference, and a smaller difference gives less head and so less flow. Only one rise satisfies both, so the page brackets it and bisects. The function is strictly decreasing, so the bracket always closes, and the page reports the step count and the residual so you can see it landed.
How high above the collector does the tank have to be?
This page states no requirement and gives you the trade instead. The height table runs from two to twelve feet and shows the flow that results at each. Doubling the height does not double the flow, because friction rises faster than linearly with it. What height is workable for a particular arrangement, and whether the loop will reverse overnight without one, is a design question rather than an arithmetic one.
What pipe size does a thermosiphon loop need?
Bigger than the flow alone suggests, because friction goes as roughly the fifth power of the bore. The bore table on the page shows the same loop across six sizes and the spread is dramatic — small bores often have no settling point at all for the heat entered. Fittings matter as much: on a short loop the equivalent length of the elbows and valves is usually more than the pipe itself contributes.
Is this arithmetic accurate?
Indicative rather than exact, and the page says why. Hazen-Williams was developed for water at distribution velocities, and a thermosiphon runs at a few hundredths of a foot per second, which is laminar. In that regime friction scales with the first power of flow, viscosity matters and roughness does not, so the C factor is doing a job it was not designed for. A cold glycol mix will perform worse than the numbers here suggest.