Heat pump versus electric water heater: the annual saving

Why the sign is settled and only the margin is in play

The engine prices the identical hot-water year twice — once straight through the tariff, once through the coefficient first — and subtracts. Load and tariff appear on both sides, so both cancel out of the question of who wins: with the machines on one meter there is no second fuel market, no calorific value, no spark spread. The verdict is decided by a single input, the coefficient, and any value above unity decides it the same way.

What load and tariff do instead is set the stakes. The saving is the resistance bill times the share the coefficient forgives — in words, one minus the reciprocal of the coefficient — so a bigger household or a dearer tariff scales the same forgiveness into more money without touching the verdict. That is the structural difference from the gas contest next door: there the prices choose the winner; here they only choose the prize.

The forgiven share grows with the coefficient, but by ever-thinner slices: the step up from the floor to a middling rating forgives a large fraction of the bill, while each further step forgives only a sliver of what remains. That curvature matters at the shop — a premium unit’s rating gap overstates its bill gap, and the fair comparison between two candidate units is their savings, never their coefficients.

The declared reverse workflow turns the relation into a pricing question: fix the annual saving that would make the swap worthwhile — the figure that clears a payback horizon, or matches a rebate case — and let the engine solve for the tariff that delivers it on this load at this rating. A bill priced above the solved tariff has already made the case; one priced below it states the shortfall in the bill’s own units.

This page is the water-heating twin of the heat-pump-versus-gas comparison, and the differences are as instructive as the likeness. That contest can genuinely flip sign, so it ends in a verdict; this one cannot, so it ends in a magnitude — and the magnitude’s real job is downstream, where the payback sibling sets it against the unit’s price premium and turns forgiveness per year into years to break even.

When this calculation is used

• Sizing the operating case for a heat-pump water heater before buying: the resistance bill, the compressor bill and the annual gap on one screen.
• Feeding the payback question: this saving is the denominator the retrofit payback sibling divides the unit’s price premium by.
• Solving backwards — the declared reverse workflow — from a target annual saving to the tariff that would deliver it, to see whether the local price makes the case.
• Checking a claimed saving from a label or a rebate programme against the household’s own load, rating and tariff.

Worked example

Run the pack’s declared anchor contest: the trio’s shared hot-water year, a typical heat-pump water heater at its uniform-energy-factor-type rating against the bare resistance element, both billed at the same mid-range residential tariff.

The engine returns both annual costs and the saving between them. Unlike the space-heating contest against gas, this difference cannot come back negative while the machine deserves its name: any coefficient above unity leaves something over, so the reading is not a verdict but a magnitude — the annual sum the coefficient forgives.

Every figure in this example is produced by the certified engine when the calculator loads, checked against the signed pack’s declared test vectors; nothing on this page stores an answer. Slide the coefficient upward and watch the saving grow by ever-thinner slices — the diminishing return that decides how much a premium rating is actually worth.

What each input represents

Assumptions and limits

• Both machines serve the identical hot-water load at the identical tariff, so the comparison isolates the coefficient; a swap that also changes habits or the rate plan must be priced through the two cost pages separately.
• One year-average coefficient stands in for the heat-pump side; the cost pages own the account of what a cool room, a tall temperature lift and backup-element hours take from it.
• The saving is operating cost only: the unit’s purchase premium, fitting and any electrical work belong to the payback sibling this saving feeds.
• The tariff is a snapshot, and the saving scales with it one for one — a dearer future strengthens the case and a cheaper one weakens it, without ever reversing the sign.

What the guards protect against

• The coefficient may not fall below unity, and the pack ships the refusal as a declared test vector: below it the challenger would lose to the incumbent by physics, and the page would no longer be comparing a heat pump. At exactly unity the machines tie and the saving vanishes by construction.
• The coefficient is also capped above anything a tank-mounted compressor achieves, so a rating misread — an energy factor scaled a hundredfold — is refused rather than silently manufacturing an impossible saving.
• The load must be positive and is bounded far above any household’s annual demand, catching watt-hours entered where kilowatt-hours belong before they inflate both bills and the gap alike.
• The tariff is held to a plausible residential band on both sides at once: a minor-unit slip would scale the saving in proportion, so it is refused rather than allowed to resize the prize silently.

Provenance

Heating-fuel cost-comparison method: each machine’s cost of delivered heat as the tariff over its efficiency, compared on an identical hot-water load

The same annual hot-water load is priced once directly at the tariff and once through the coefficient of performance, and the difference is returned as the annual saving; the declared reverse workflow solves for the tariff that yields a chosen saving.

Screening and reference material, to be checked against the governing standard and a qualified engineer; not a design determination. The signed pack carries its own citation, which displays from the verified leaf when the calculator loads.