Physics of Buildings

How it works

How much water grows, and how an air cushion takes it

A 50-gallon heater pushes out about half a gallon every time it heats from cold. Drag the temperatures and see where that number comes from.

9 min read · 3 interactive figures

The density of water against temperature, from NIST’s tables. Drag the blue (cold) and red (hot) points along the curve, or use the sliders, and read how much a heater of the chosen size grows. Default: 50 gallons heated from 50 °F to 120 °F.

Every time a water heater warms a tankful of cold water, the water grows. Not by much: about 1 % between a cold winter supply and a normal thermostat setting. But water is almost incompressible, so in a closed system that 1 % has to be given somewhere to go. An expansion tank gives it a cushion of air to squeeze, and Boyle’s law decides how big that cushion must be.

How much the water grows

Water is densest at about 39 °F (4 °C) and gets lighter, per gallon, as it warms [1]. NIST’s reference tables give its density to six figures: 999.70 kg/m³ at 50 °F (10 °C), 988.53 at 120 °F, 983.20 at 140 °F (60 °C) [2]. A fixed mass of water takes up a volume inversely proportional to its density, so the growth from cold to hot is a ratio of two numbers from the table:

E = Vh · [ ρ(Tcold) / ρ(Thot) − 1 ](1)

For a 50-gallon heater filled at 50 °F and heated to 120 °F, that is 50 × (999.70 / 988.53 − 1) = 0.56 gallons, a little over 1 %. Heated from 40 °F to 140 °F it is 0.85 gallons. The figure at the top of the page is that table drawn as a curve: drag the two points and watch the gap in density, and the gallons, grow as the hot point moves right. The curve steepens as it goes: each degree near 140 °F adds about three times as much volume as a degree near 60 °F.

The heater’s own steel tank grows a little too, by about 35 millionths of its volume per °C [1], which makes room for some of the water. Amtrol’s sizing guide prints expansion factors that match water’s growth less the steel’s, within 0.0003 for every row from 120 °F to 180 °F: 0.0100 at 120 °F and 0.0150 at 140 °F, both from 40 °F [3]. The estimator uses the same correction by default, with a switch to leave it out.

Why a system with no air spikes

Question 1 Prediction: answered by Figure 1 and the reasoning below it.#
Figure 1 A closed system with no air at all, starting at 110 °F and 60 psi (rigid heater and pipes: an upper bound). Drag the red point to warm the water; the pressure is drawn only as far as you have warmed it. The orange line is the T&P valve’s 150 psi setting.#

When the supply is open, the expanding water pushes back into the street main. Close it with a PRV, check valve or backflow preventer and the water can only stay where it is by being squeezed. Water is very stiff. Comparing NIST’s densities at two pressures one megapascal apart gives its compressibility: about 3.1 millionths of its volume per psi in the range a water heater sees [2]. Near 110 °F, one degree of warming grows the water by about 230 millionths of its volume. To hold it in the same space, the pressure has to rise by

Δp ≈ (growth per °F) ÷ (compressibility per psi) ≈ (230 × 10⁻⁶) ÷ (3.1 × 10⁻⁶) ≈ 74 psi per °F(2)

So a degree and a bit takes a 60 psi house to the 150 psi at which the T&P valve opens [5]. Real heaters and pipes stretch under pressure, which softens this (equation (2) is the rigid upper bound), but the rise still takes degrees, not tens of degrees: it is why a closed system without a tank relieves within minutes of the burner starting, every cycle. The heater’s T&P valve then does the job of an expansion tank, which it was never meant to do: Watts states it is for emergency relief and “shall not be used as an operating control” [5]. A. O. Smith says the same thing from the other side: when a backflow preventer seals the system, an expansion tank is required “or the relief valve will discharge on every heating cycle” [6].

Never cap or plug a dripping relief valve. It is the device that stops a heater from turning into a pressure vessel with nowhere to vent. Codes require its drain line to end with no thread and no valve so it cannot be closed off [7]. A drip means something upstream needs attention; a licensed plumber can find what.

The air cushion: Boyle’s law in the tank

At house pressures, air is thousands of times easier to squeeze than water, which is the whole point of an expansion tank. Inside it, a diaphragm separates a sealed charge of air from the water side, which is connected to the cold pipe. The air is charged, with the tank empty of water, to the house pressure [8]. As the heater pushes its expansion in, the air is squeezed, and Boyle’s law says that for a fixed amount of air at a fixed temperature, pressure times volume stays constant [9]. The law needs absolute pressure, the gauge reading plus the atmosphere, 14.7 psi at sea level [10]:

(Phouse + 14.7) · Vair, before = (Ppeak + 14.7) · Vair, after(3)

A 2-gallon tank charged to 60 psi on a 60 psi house starts with 2 gallons of air at 74.7 psia. Push 0.28 gallons of expansion in (one recovery after a long shower, in the home-page simulation) and the air shrinks to 1.72 gallons, so its pressure rises to 74.7 × 2 / 1.72 = 86.9 psia, or 72 psi on the gauge. That is the gentle rise you see in step 3 of the home page’s tour. This is the same physics, and the same engine, as the well pressure tank’s drawdown; there the air pushes water to the taps, here it soaks up the heater’s growth.

Question 2 Prediction: answered by Figure 2.#

Figure 2 draws equation (3) as a curve: pressure against the water in the tank. The amber line is the house pressure; the small amber point is the tank before heating; the red point is after, and you can drag it to push in more or less expanded water. The curve is flat at first and steep at the end, because each extra gallon squeezes a smaller and smaller pocket of air.

Figure 2 Pressure in a precharged expansion tank against the water in it (Boyle’s law). Defaults: a 2-gallon tank charged to 60 psi on a 60 psi house, with 0.28 gallons of expansion pushed in, which brings it to 72 psi. Drag the red point; change the precharge, house pressure and tank size.#

Why the precharge equals the house pressure

Turn the precharge in Figure 2 down to 30 psi with the house at 60. The amber “before” point slides to the right: the house pressure has already pushed 0.8 gallons into the tank before any heating, because the air only stops giving way when it reaches 60 psi. That water takes up room the expansion needed, and the same 0.28 gallons now drives the pressure to 83 psi. Amtrol illustrates exactly this with a 40 psi tank on a 70 psi supply and concludes that the precharge must be matched to the incoming pressure before installation [3]. A. O. Smith lists an unmatched air charge among the causes of a weeping T&P valve [6].

Turn it up to 90 instead, and the tank takes no water at all until the pressure reaches 90 psi: the system has to climb there with no cushion, almost instantly by equation (2), before the air begins to help. Matched, the tank is empty of water but at the house pressure the moment heating starts, so every bit of its air is available. Precharge adjustment is done with the tank empty and the system cold; makers reserve it for licensed installers [8].

From Boyle’s law to a tank size

Sizing reverses the question: given the expansion E, how big must the tank be so that the pressure stays below a chosen maximum Pmax? With the precharge equal to the house pressure PL, equation (3) rearranges to a factor that multiplies the expanded water:

Vtank = E × (Pmax + 14.7) / (Pmax − PL)(4)

Amtrol calls the fraction the design pressure factor and prints it as a table; equation (4) reproduces every entry to the printed tenth, from 1.5 (150 psi maximum, 40 psi line) to 5.7 (100 psi maximum, 80 psi line) [3]. Its worked example: a 240-gallon heater at 150 °F, 60 psi line pressure, 125 psi maximum. The expansion factor is 0.0179, so E = 4.3 gallons; the factor is 139.7 / 65 = 2.1; the tank needs 4.3 × 2.1 = 9.0 gallons [3]. Unrounded, the same numbers give 9.3.

The inverse of the factor is the share of a tank that can take water, and makers list it as an “acceptance volume”. Watts’ PLT-5, a 2.1-gallon tank, is listed as accepting 1.26 gallons [4]; equation (4) gives 1.28 at a 40 psi supply with a 125 psi maximum, and about 1.0 at 60 psi. The higher the house pressure, the less of any tank is useful, because the air starts out already squeezed.

The answer depends on the maximum you choose, and makers choose differently. Amtrol’s quick-sizing chart uses the 150 psi T&P setting itself [11]; Watts’ calculator says the final pressure should never exceed 85 % of the relief valve’s rating [4]; Watts’ relief valve data sheet asks that normal operating pressure stay below 75 % of its setting [5]. For a 50-gallon heater heated from 40 °F to 140 °F on a 60 psi house, the minimum tank is 1.4 gallons at a 150 psi maximum, 1.6 at 127 psi (85 %), 1.8 at 112 psi (75 %) and 2.2 at 100 psi. That spread is why the estimator asks for the maximum and shows its working: it is an estimate for understanding and comparison, not a code determination or a sizing guarantee. The installer, the tank maker’s instructions and the local code decide.

Who does the work. Expansion tanks, PRVs and gas water heaters are licensed-professional work in most places, and Amtrol requires its tanks to be installed by one [8]. This page explains the physics so you can follow the numbers and describe a problem; it is not an installation procedure.