Why the water chart does not work here
Hazen-Williams and every friction chart built on it carry the properties of water at ordinary temperatures baked into the coefficient. That is fine for a domestic supply and it is fine for most heating circuits. It is not fine for a ground loop, because the fluid in a ground loop is an antifreeze mix at close to freezing and its kinematic viscosity is several times that of warm water. Viscosity does not appear anywhere in Hazen-Williams, so the chart cannot know it has changed.
The Darcy-Weisbach route used here goes through the Reynolds number, which is velocity times diameter divided by kinematic viscosity, and viscosity is right there in the denominator. At the defaults — 3 GPM through a 1.076 inch bore at 4.5 centistokes — the Reynolds number is 1,959 and the flow is laminar. Warm that same fluid up until the viscosity falls to 1.5 centistokes and the Reynolds number becomes 5,878 and the flow is firmly turbulent. Same pipe, same pump, same gallons a minute, completely different behaviour.
The number that matters is not the head
Head decides what circulator goes on the wall. Whether the flow is turbulent decides whether the loop works, and it is a heat transfer question rather than a pumping one. Heat leaving the fluid has to cross a film against the inside of the pipe wall, and a stirred flow keeps that film thin while a laminar flow lets it sit there as a layer of insulation. A loop that is long enough on paper and laminar in practice does not deliver what the sizing said it would.
This page will not tell you what Reynolds number to design to. There is a figure the industry works to and it is not this page place to state it as a requirement, because it depends on the pipe, the fluid, the temperature and the designer judgement about how much pumping energy the heat transfer is worth. What the page does is take whatever number you were given and report the flow that reaches it at your viscosity — 3.83 GPM on the defaults for a Reynolds number of 2,500, against the 3 GPM entered.
Head does not go as flow squared
Everybody knows friction goes as roughly the square of the flow, and in fully turbulent flow that is close enough. In laminar flow it is wrong: the friction factor is exactly 64 over the Reynolds number, so it falls in inverse proportion to velocity, and it cancels one of the two velocity terms. Head ends up proportional to flow, not to flow squared.
The table on the page walks the flow from 40 percent to double and prints the regime beside each row. On the defaults the circuit is laminar at 3 GPM and takes 5.9 feet of head; at 3.75 GPM the Reynolds number has crossed 2,300 and the head is 13.1 feet. A 25 percent increase in flow has more than doubled the head, because the calculation has switched from one correlation to the other at that boundary. The step in the column is a real discontinuity in the arithmetic and the physics in that region is not smooth either, which is the honest reason not to read too much into a figure sitting right on the edge of it. Where the step lands is set by the viscosity, which moves through the season, which is why the loop that behaved one way in October behaves another in February.
Parallel circuits, and what the header does
Circuits on a common manifold see the same head as each other, so four circuits carrying 3 GPM each is 12 GPM at the head of one circuit, not four times it. That is why the answer barely changes when a fifth circuit is added — the head is set by the longest single path, and the pump simply has to move more gallons at the same pressure.
The header is the exception, because it carries everything at once. Twelve GPM through a 1.917 inch bore over 120 feet is a small contribution here, but it grows fast if the header is undersized, and unlike the buried circuits it can be changed afterwards without a rig. If the header head is a large share of the total, that is normally the cheapest thing on the whole job to fix.
What the wattage is really telling you
Wire to water efficiency on a small wet-rotor circulator is well below what people assume, and the defaults here use 25 percent. At 12 GPM and 14.7 feet of head the hydraulic work is 0.046 horsepower, about 34 watts, and at 25 percent that is 138 watts at the plug. Over 2,500 hours a year it is 344 kWh, 55 dollars at 16 cents. On a system whose entire argument is efficiency, a circulator running most of the year is not a rounding error, and it is worth knowing what it costs before choosing a large pump to cover an uncertain head figure.
Questions people ask
Why does a ground loop need a Reynolds number calculation at all?
Because the fluid is cold antifreeze rather than water, and its viscosity is several times higher. Viscosity is what decides whether flow in the pipe is turbulent or laminar, and that decides how well heat crosses the film against the pipe wall. A laminar loop moves less heat than the sizing assumed, and no amount of extra length compensates cheaply. The friction charts most people reach for have water properties built into a single coefficient and cannot see the difference.
What flow do I need for turbulent flow in the loop?
It depends on the bore, the fluid and its temperature, so there is no single answer and this page states none. The relationship is Reynolds equals 3,162.6 times GPM divided by the bore in inches times the kinematic viscosity in centistokes. At a 1.076 inch bore and 4.5 centistokes, reaching a Reynolds number of 2,500 takes 3.83 GPM. Halve the viscosity, as happens when the loop warms up, and the same Reynolds number arrives at half the flow. What number to design to comes from the designer, not from a calculator.
Is head proportional to the square of the flow?
Only in turbulent flow, and ground loops are often not in turbulent flow. In laminar flow the friction factor is 64 over the Reynolds number, which falls in proportion to velocity and cancels one of the two velocity terms in the head equation, leaving head proportional to flow. The table on this page prints the regime beside each row so the change is visible. Assuming a square law throughout will overstate the head at low flow and understate the jump when the flow trips into turbulence.
Do parallel circuits multiply the head?
No. Circuits fed from a common manifold and returning to a common manifold all see the same pressure difference, so four circuits at 3 GPM each is 12 GPM at the head of one circuit. What multiplies is the flow the pump must deliver, not the head it must produce. The header is different, because it carries every circuit at once, so its own friction is worked at the combined flow through its own bore.
How much does the circulator cost to run?
More than most people expect on a system sold on efficiency. Hydraulic power in horsepower is GPM times head in feet times specific gravity divided by 3,960, and dividing that by the wire to water efficiency of the pump gives the actual draw. At 12 GPM, 14.7 feet of head, a specific gravity of 1.035 and 25 percent efficiency, that is 138 watts, and 2,500 hours a year is 344 kWh. Enter your own efficiency from the pump curve at the duty point rather than the headline figure.