Sizing formulas and methodology

The calculators give performance requirements: the First Hour Rating or flow a heater must meet. They do not select equipment, size gas lines, venting or electrical circuits, or check code compliance.

1. Mixed water (hot fraction)

Fixtures deliver water blended to a use temperature. Only part of that flow comes from the heater:

hot fraction = (T_use − T_inlet) ÷ (T_supply − T_inlet)

• T_supply is the tank setpoint or tankless outlet temperature. This is the same energy balance as the F_mix term in ANSI/RESNET/ICC 301.
• Dishwashers and hot-fill clothes washers draw straight from the heater: hot fraction 1.
• A use temperature at or below the incoming temperature is rejected as impossible to blend to.
• A use temperature above the supply temperature is capped at 1 (all hot) with a caution. This is conservative.

2. Tank sizing: First Hour Rating

hot water per use = volume per use × hot fraction (volume per use = flow × duration, or a fixed volume) peak-hour demand = Σ uses × hot water per use required FHR = peak-hour demand × (1 + design margin) minimum FHR shown = required FHR rounded up to a whole gallon

• First Hour Rating is defined in 10 CFR 430 Subpart B Appendix E as “an estimate of the maximum volume of ‘hot’ water that a non-flow activated water heater can supply within an hour that begins with the water heater fully heated”. It is tested with 58 ± 2°F supply water and 125 ± 5°F delivery.
• The design margin defaults to 0%. DOE’s 2001 fact sheet advised choosing an FHR “within about 2 gallons” of peak-hour demand, and no authoritative source recommends a fixed percentage. The iPhone app’s Lifetime Pro lets you add one.
• When incoming water is below 58°F, a note says real delivery will be lower than the published FHR. No authoritative correction factor exists, so none is applied.

3. Tankless sizing: flow at a temperature rise

temperature rise = T_outlet − T_inlet required flow = Σ quantity × fixture flow × hot fraction heat load = 500 × required flow × rise BTU/hr into the water required input = heat load ÷ efficiency BTU/hr electric input = required input ÷ 3,412.14 kW

• Hot water only. A tankless unit heats only the hot share of a blended flow. Counting full fixture flow at the full rise overstates the load. The app offers the conservative all-flow basis in Lifetime Pro; the web calculator shows total fixture flow for reference.
• The 500 constant = 8.33 lb/gal × 60 min/hr × 1 BTU/(lb·°F) ≈ 499.8, rounded. Hot water weighs about 8.24 lb/gal, so 500 overstates the load by about 1%, erring toward adequacy.
• Rated rise. Maximum GPM ratings are measured at a nominal 67°F rise (Appendix E). Above 67°F, a note says delivered flow will be lower.
• Residential gas limit. Consumer gas instantaneous water heaters have a nameplate input below 200,000 BTU/hr (10 CFR 430.2). At or above that estimated input, a caution suggests multiple units or commercial equipment.
• Efficiency presets (thermal efficiency, not UEF): gas non-condensing 82% (federal minimum UEF 0.81 for gas instantaneous heaters; a Rheem 199,900 BTU/hr, 5.0 gpm @ 67°F spec back-calculates to about 84%), gas condensing 95% (ENERGY STAR requires UEF ≥ 0.95), electric 99%.

4. Checking a specific heater

• Tank: meets the requirement when published FHR ≥ required FHR. Tank capacity is informational.
• Tankless flow: published at your rise (within 0.5°F): compared directly. Published at a smaller rise: scaled to yours as flow × rated rise ÷ your rise, labelled an estimate. Published at a larger rise: used as-is. The engine never extrapolates upward.
• Tankless input: meets when maximum input ≥ heat load ÷ efficiency. If efficiency isn’t entered, 82% is assumed for gas and 99% for electric.

5. Fixture defaults (library 2026.1)

Cross-check: AHRI’s peak-hour worksheet allows 20 gallons of hot water per shower and DOE’s 2001 worksheet 15; both say they assume no water conservation measures. The flow-based model gives 20 gallons for a 7.7-minute shower from an older 3.3 gpm head and about 15.7 gallons from today’s 2.5 gpm maximum, at 50°F incoming water and a 120°F tank.

6. Validation limits

Out-of-range values are rejected, not clamped, and every problem is reported at once.

The default incoming temperature, 50°F, is deliberately on the cold side; users are prompted to enter their own. The default setpoint, 120°F, follows DOE (“A setting of 120°F generally meets most household needs”) and CPSC, which urges users to lower water heaters to 120°F to reduce scald risk.

7. Web-only calculations

• Recovery rate: gallons per hour = input × efficiency × 60 ÷ (500 × rise), using the same constant. See the recovery calculator.
• Electric current: amps = kW × 1,000 ÷ volts, total across all circuits. Not a breaker or wire size.
• Equivalent flow at the 67°F rating rise: required flow × your rise ÷ 67 when your rise is above 67°F, otherwise the required flow. This is the smallest rated flow that passes the model check above.
• Mains water temperature estimates (37 cities): NOAA 1991–2020 normals run through the NREL (Burch & Christensen) correlation as implemented in EnergyPlus: mains = (Tavg + 6) + (0.4 + 0.01 × (Tavg − 44)) × (ΔT ÷ 2) × sin(0.986 × (day − 15 − lag) − 90), lag = 35 − 1.0 × (Tavg − 44). The authors report errors around 4°F and ±5°F variation within a water network. Results and caveats.

8. How the web calculator is kept identical to the app

The web engine is a line-by-line TypeScript port of the app’s Swift engine, performing the same arithmetic in the same order. A test harness runs the app’s real Swift engine and the web engine over the same 427 cases: hand-picked worked examples, boundary values, invalid inputs and seeded random inputs across the full range. Every number must match exactly, and every calculation step, warning and validation message must match character for character.

Where the two differ, it is by design: on the web, the full step-by-step breakdown and the model check are free, and some web-only reference tools exist. The app adds saved jobs, PDF reports, custom fixtures, design margin and a side-by-side tank vs. tankless comparison.

Station data for the city estimates

Sources

1. 10 CFR Part 430, Subpart B, Appendix E: Uniform test method for measuring the energy consumption of water heaters, eCFR.
2. 10 CFR 430.2: Definitions (gas and electric instantaneous water heaters), eCFR.
3. 10 CFR 430.32: Energy and water conservation standards (showerheads, faucets, dishwashers), eCFR.
4. WaterSense labeled showerheads, U.S. EPA.
5. Residential water heaters key product criteria, ENERGY STAR.
6. Water Heating: Energy-efficient strategies for supplying hot water in the home (DOE/GO-102001-0785, 2001), U.S. Department of Energy.
7. Tap water scalds (Publication 5098), U.S. Consumer Product Safety Commission.
8. Water heater sizing worksheet, AHRI.
9. Burch & Christensen, Towards Development of an Algorithm for Mains Water Temperature (NREL), ENERGY STAR (hosted).
10. U.S. Climate Normals, 1991–2020, NOAA National Centers for Environmental Information.