Inside the tank: why hot water runs out, and how the anode saves the steel

A 40-gallon water heater holds about 23,000 Btu of heat above the temperature of the water coming in, and its burner adds about 30,800 Btu an hour. A shower at 2.5 gallons a minute uses heat faster than that, so it lives off the store, and the store runs out in about 18 minutes. This essay follows that heat: how the tank keeps hot water separate from cold, why the burner’s heat arrives too late, and how a rod of magnesium or aluminum keeps the steel tank from rusting through.

Hot on top: stratification and the dip tube§

Water expands as it warms. At 60 °F a cubic centimetre of water weighs 0.99907 grams; at 120 °F, 0.98870 grams ^[1]. One percent is enough: hot water floats on cold the way oil floats on water, and a tank left alone settles into layers, hottest at the top. Engineers call this stratification, and a water heater is built around it. Hot water leaves from the top. Cold water enters through the dip tube, a plastic tube that runs from the cold inlet almost to the bottom of the tank, so that incoming water arrives below the hot water instead of in the middle of it ^[2][3].

Stratification is what lets a tank deliver water at full temperature until nearly all of it is gone. During a shower the cold water does not dilute the hot: it lifts it. A boundary between hot and cold, the thermocline, climbs from the bottom while the outlet stays at the set temperature. When the boundary reaches the top, the outlet temperature falls within a few minutes. Anything that stirs the tank blurs the boundary and wastes hot water: a jet of cold water that churns the bottom, or a dip tube broken off near the top, which drops cold water straight into the hot layer, where it sinks and mixes. In an experimental study, shielding the cold inlet with a curved cover so that water entered more gently gave about 17 percent more usable hot water ^[4].

The figure at the top of the page draws the tank as 30 horizontal layers. Each second, the shower’s share of tank water leaves the top layer, every layer moves up, and cold water enters the layer where the dip tube ends. Heat goes into the layer at the burner or the element, and wherever a layer is warmer than the one above it, the two mix, as warm water would rise. That one rule makes the thermocline, the lukewarm water below it, and the broken dip tube’s mixing all come out on their own.

Why a tank runs out: an energy balance§

The unit of heat on every American water heater is the British thermal unit, roughly the heat that warms one pound of water by 1 °F. Water’s specific heat is 4.184 joules per gram per degree Celsius ^[5]; with the NIST conversion factors ^[6] that is 0.9993 Btu per pound per °F, and a gallon of 60 °F water weighs 8.34 pounds ^[1]. So it takes 8.33 Btu to warm a gallon by 1 °F:

A 40-gallon tank at 120 °F filled from a 50 °F supply holds 8.33 × 40 × 70 ≈ 23,300 Btu more than cold water would. A shower does not use 120 °F water directly. The valve mixes hot and cold to the temperature you want, and the share that comes from the tank is fixed by the same heat bookkeeping:

For a 105 °F shower that is 55/70 = 0.79, so a 2.5 gpm shower head takes 1.96 gpm from the tank. The heater’s side of the ledger is its recovery rate: the gallons per hour it can warm through a given rise. A gas burner turns part of its fuel into heat in the water, the part federal test rules call its recovery efficiency ^[7]; the rest leaves up the flue.

A. O. Smith’s 40-gallon GCBL-40 burns 40,000 Btu an hour and lists a recovery of 41 gallons per hour ^[8]. If that is for the 90 °F rise that spec sheets usually quote, equation (3) says about 77 percent of the gas ends up in the water, the figure the simulation uses. At our 70 °F rise the same burner heats 53 gallons an hour, 0.88 gallons a minute. A 4.5 kW electric element puts all of its 15,355 Btu an hour into the water and recovers about 20 gallons an hour at a 90 °F rise.

Here is the catch. While the shower runs, cold water flows into the bottom of the tank at 1.96 gallons a minute, right where the burner is. The burner’s 30,800 Btu an hour, spread over that stream, warms it by only

to about 81 °F: lukewarm water that settles below the thermocline and is no use for a 105 °F shower. The burner is working the whole time, and its heat is in the tank, but at the wrong temperature. So the shower lasts about as long as the stored hot water does, roughly 18 minutes in the simulation against 17 with the burner off, and when the boundary reaches the top the shower turns lukewarm instead of cold, which is what people describe as “it runs out, then it’s just warm”. After the shower ends, the burner heats that lukewarm water back to 120 °F and the tank recovers.

Two numbers decide which side wins. If the heater can warm the whole hot share of the shower as fast as it is used, the shower never runs cold:

For this burner that is 1.1 gpm, so a 1-gallon-a-minute trickle runs indefinitely and a 2.5 gpm shower does not. Double the burner and the keep-up flow doubles to 2.2 gpm, still short of 2.5; but now equation (4) warms the incoming water to about 113 °F, above the shower temperature, and that water extends the shower to about 35 to 36 minutes. A bigger tank only adds its own gallons: 50 gallons last about 22 minutes. Cold winter supply water works the other way: at 40 °F the tank holds more heat but each gallon of shower needs more of it, and the same shower ends sooner.

This is also why the federal first-hour rating, not tank size, is the number to compare. It is the most hot water a heater can deliver in an hour starting fully heated, measured by drawing at 3 gallons a minute from a 58 °F supply, stopping each draw when the outlet falls 15 °F below its peak, and starting the next when the thermostat is satisfied ^[7]. It counts both the store and the recovery: the 40-gallon GCBL-40 is rated at 68 gallons ^[8]. The shower-time calculator runs the layered model for your own heater.

Electric tanks: where the element sits§

An electric heater has two elements, one low and one high in the tank, each with its own thermostat, and the control energizes only one at a time, with the upper element first ^[9][7]. Stratification explains the arrangement. An element can heat only the water at and above it, because heated water rises. When a big draw brings cold water up to the top thermostat, the upper element heats the top of the tank first, so some hot water is back within minutes; then the lower element heats the rest. Most of the time the lower element does the work, which is why it usually fails first, and why a dead lower element leaves exactly the symptom A. O. Smith describes: “some hot water but not as much as before” ^[9]. In the figure, choose “Electric, lower element dead” and the hot water ends a little above the upper element. Element and thermostat checks are 240-volt work for a licensed electrician or plumber.

The anode: a battery that protects the tank§

The tank is steel with a thin glass lining, and the lining is never perfect: at its gaps, iron meets water. Iron in water with dissolved oxygen is a small galvanic cell, and the iron is its anode, the electrode that oxidizes and dissolves ^[10]. Connect a more active metal to it, and the roles change: the active metal becomes the anode and corrodes, and the iron becomes the cathode, where only oxygen is reduced. This is cathodic protection, and the textbook example is the water heater ^[10]. The rod screwed into the top of the tank is there to be eaten.

Which metal will do it is set by the ladder of standard electrode potentials. The more negative a metal’s potential, the more readily it gives up electrons: magnesium −2.372 V, aluminum −1.662 V, zinc −0.762 V, iron −0.447 V ^[11]. Any metal below iron on that ladder protects it, and the gap is the driving voltage of the cell: about 1.9 V for magnesium against iron, 1.2 V for aluminum and 0.3 V for zinc.

How long a rod lasts is Faraday’s law. Every atom that dissolves releases a fixed number of electrons, so the charge a rod can deliver is its mass divided by its molar mass, times the electrons per atom, times the Faraday constant, 96,485 coulombs per mole ^[12][13]. With atomic weights of 24.3 for magnesium, 27.0 for aluminum and 65.4 for zinc ^[14]:

and the rod’s life is that charge divided by the current it supplies. Figure 2 does the arithmetic for a rod you choose.

The three metals trade strength for life. Magnesium drives hardest; aluminum carries more charge per kilogram at a lower voltage. The anode’s protective action also produces hydrogen, and where the water carries sulfate and sulfate-reducing bacteria and little oxygen, that hydrogen ends up as hydrogen sulfide, the rotten-egg smell: A. O. Smith’s remedy is a less active anode plus chlorination by a service agency, never removing the anode ^[3]. A powered anode replaces chemistry with a power supply: an inert titanium rod drives a small direct current into the tank, so nothing is consumed ^[15]. How fast a sacrificial rod wears depends on the water; softened water wears it faster ^[16], and one manufacturer asks for an inspection after six months and then at least yearly ^[9].