Set a showerhead's flow against a tub and the model marks the minute the shower passes it. Does the heating energy cross at that minute too?
Ask whether a shower or a bath uses more water and "the bath" is the easy answer, one that needs no counting. That answer treats the two as fixed amounts, one larger than the other, like two jugs standing side by side.
Only one of them behaves like a jug. In the model a tub is filled once and then stays put, while a showerhead adds the same litres for every minute it runs.
That makes the comparison a question about minutes. At some minute the shower catches the tub, and the sliders below move that minute along with a second one, where the heating energy catches up.
Of the five sliders, “Showerhead flow rate” sets how fast the head pours and “Water drawn for the bath” sets how much goes into the tub. “How long the shower runs” moves a marker along the time axis. The last two, “Temperature rise for the shower” and “Temperature rise for the bath”, set how far the heater lifts each lot of water. Below them, the “Energy price per kWh (optional)” box takes a price you type in; left empty, it shows no cost.
The page opens at 9.50 L/min against 80 L, and the model puts the water crossover at 8.4 minutes. The “2.0 gpm (7.57) — WaterSense” preset stretches that to 10.6 minutes for the same tub, and “2.5 gpm (9.46) — EPCA max” makes it 8.5. Across the whole flow slider the minute runs from 20.0 down to 4.0.
At the opening setting the model lifts the shower's water 27 K and the tub's 25 K, so each shower litre carries more heat than each tub litre. That brings the energy crossover forward to 7.8 minutes, 37 seconds before the water one. Between the two marks the model has the shower using less water than the tub but more energy. Press “Same temperature both” and the gap closes to nothing.
In this model only unequal rises can part the two marks, and the ratio of the rises makes the energy minute a fixed multiple of the water one; flow and tub size move both marks by the same factor.
Both panels share one horizontal axis, minutes of shower. On each, the sloping line is the shower and the flat line is the tub. The tub's line is level because how long anyone showers is not one of its inputs. It stands for the tub's own total, in litres on one panel and kilowatt-hours on the other.
The dashed vertical line is the crossover. To its left the model has the shower using less than the tub, and to its right more. It moves in inverse proportion to flow, so doubling the flow halves the crossover minute. The dotted vertical, with a dot where it meets the rising line, sits wherever “How long the shower runs” is set.
The lower panel is the one to watch as the two rises go from equal to unequal. While they match, its dashed line sits directly under the upper one and there is only one mark to see. Once they differ the two lines separate. The shaded band between them is the stretch of minutes in which the model has the shower ahead on one panel and behind on the other. Every axis rescales to fit, so check the axis labels before comparing two settings.
The shower's volume is flow times time; the tub's is a single number.
Setting the two volumes equal gives the first crossover:
Energy is the same water carried through a temperature rise, E = ρ·V·c·ΔT, so the shower's energy grows as ρ·Q·t·c·ΔTs while the tub's stays at ρ·V·c·ΔTb. Equating those two gives the second crossover:
Two things follow from the shape of that fraction alone, whatever numbers go into it. First, ρ and c appear on both sides and cancel. They set how tall the energy lines are, but they cannot move the minute where the two lines meet.
Second, equal rises make the ratio exactly one, and the energy crossover then lands on the water one. In that case the lower panel says nothing the upper one had not already said.
Start from the opening setting, 9.50 L/min against an 80 L tub, and press “Same temperature both” so both rises sit at 25 K. Count the water first: 8 minutes of shower is 9.50 × 8 = 76.0 L, which leaves the shower 4.0 L short of the tub's 80.0 L. At 9.50 litres a minute that gap takes 0.42 of a minute, so the shower draws level at 8 minutes and 25 seconds.
Now count the heat, without using the answer the water count gave. At ρ = 1 kg/L, 76.0 L of water weighs 76.0 kg, and lifting it 25 K takes 76.0 × 4.186 × 25 ≈ 7 953 kJ. The tub takes 8 372 kJ, so the shower is 419 kJ short, and each further minute adds 994 kJ. Dividing 419 by 994 gives the same 0.42 of a minute, and the same 8 minutes and 25 seconds.
The closed form agrees: 80 ÷ 9.50 = 8.4 minutes, which is what the “Water crossover” card reads.
Every figure above is worked out under five things the model holds fixed.
The two measured constants. c is 4.186 kJ/(kg·K) and ρ is 1 kg/L. Both drift with temperature, and both are held still here. The conversions, by contrast, are definitions. One US gallon is 3.785411784 L, and one kilowatt-hour is 3 600 kJ because an hour holds that many seconds. The gallon figure only turns the two US ceilings into litres per minute. The kilowatt-hour figure only restates energy in another unit, and neither appears in the crossover formula.
The flow presets are ceilings. They are the most a product may pour under each rule or label, not a measurement of any installed head. The tub volume is not sourced. None of the sources below sets how much water a bath holds, so that slider's opening value is a choice, as are the opening flow of 9.50 L/min, the 8 minutes and the two rises.
The plumbing. The model lifts a mass of water by ΔT and stops there. It has no heater efficiency, no standby loss from a tank and no heat left behind in the pipe run. Nor does it count the cold water run off while the hot arrives, which belongs on the water panel. Because flow is held constant, a tank running dry mid-shower is outside the model as well.
The bath counts as one number at one temperature, so the model has no cooling during the soak, no hot water added partway through, and no allowance for the volume a body pushes aside, which lowers how much has to be drawn.
The tariff. Energy price enters as one multiplication at the end. Standing charges, block rates, time-of-use pricing and separate water and wastewater billing are all absent, so the output prices the energy alone and stops short of a bill.
Both crossovers are checked by a second path. A loop pours one minute of water at a time until it passes the tub, and its answer is held against the closed form above. The energy is worked out twice as well, once through kilojoules and once with a kilowatt-hour constant from the start. What I could not check is listed on the About page, under How I check things.
Sources.
Chance
Move the door count and the number of doors the host opens. The model puts switching ahead of staying by the doors you did not pick, divided by the doors you could still swap to — two to one at three doors. Opening more doors raises that edge; adding doors without opening more lowers it.
0.667 the model's switch win rate at three doors
Chance
Two groups where one option leads in each group, and trails once the model pools them.
19.4 pp the model's pooled gap in the Berkeley preset, against a lead in both departments
Chance
A single queue against one queue per counter, on average wait and on a bad visit (the 95th percentile).
5.4× the model's cut in mean wait from one line, at four counters 80% busy