MathBench

Sensible & Latent Heat Calculator

Split a heating or cooling load into the part that changes temperature and the part that changes moisture. The air-side mode uses the trade's 1.08, 0.68 and 4.5 constants — and derives each one in front of you from air density, specific heat and latent heat, so you can correct them for altitude instead of trusting them blindly. A mass-basis mode handles anything that is not ducted air.

The airstream

Air properties — this is where 1.08, 0.68 and 4.5 come from

Result

—

Showing the work

How this calculator works

Heat that changes temperature is sensible — a thermometer sees it. Heat that changes phase at constant temperature is latent, which means hidden. An air conditioner does both at once, dropping the dry-bulb temperature of the air and condensing water vapour out of it onto a cold coil. Those two loads sum to what the equipment has to carry.

In the field the split is done with three constants — 1.08, 0.68 and 4.5 — that everyone quotes and almost nobody derives. This page builds each one from air density, the specific heat of air and the latent heat of water, because they are not universal constants at all. They are shorthand for standard air at sea level.

That matters the moment you leave sea level. All three are directly proportional to air density, and density follows barometric pressure, so the 1.08 is nearer 0.89 in Denver. Set the site elevation, switch the density basis to computed, and all three move together.

The second mode drops the air assumption and works straight from mass, which covers heating a tank of water, melting ice for thermal storage, or working out why a steam humidifier draws so much power. Pick a material and its properties fill in.

The formula

Air side — start with the mass flow of dry air, not the volume:

    m_dot = 60 × ρ × CFM                      lb of dry air per hour

    Q_sensible = m_dot × c_p × ΔT           = (60 ρ c_p)        × CFM × ΔT
    Q_latent   = m_dot × h_fg × ΔW ÷ 7000   = (60 ρ h_fg/7000)  × CFM × ΔW_grains
    Q_total    = m_dot × Δh                 = (60 ρ)            × CFM × Δh
    SHR        = Q_sensible ÷ Q_total

Where the famous constants come from, for standard air
ρ = 0.075 lb/ft³ (dry air, 14.696 psia, about 70 °F):

    60 × 0.075               = 4.50      the "4.5"
    4.5 × 0.240              = 1.08      the "1.08"    c_p of dry air
    4.5 × 1058 ÷ 7000        = 0.680     the "0.68"    h_fg of water
    4.5 × 1076               = 4842      the "4840"    ΔW in lb/lb

Metric, ρ = 1.201 kg/m³, flow in L/s, c_p in kJ/kg·K, h_fg in kJ/kg:

    Q_sensible = ρ c_p × L/s × ΔT           = 1.21 × L/s × ΔT   W
    Q_latent   = (ρ h_fg ÷ 1000) × L/s × ΔW = 2.96 × L/s × ΔW   W, ΔW in g/kg
    Q_total    = ρ × L/s × Δh               = 1.20 × L/s × Δh   W, Δh in kJ/kg

    1.201 × 1.005            = 1.207    0.240 Btu/lb·°F = 1.005 kJ/kg·K
    1.201 × 2461 ÷ 1000      = 2.956    1058 Btu/lb = 2461 kJ/kg
                                        the ÷1000 turns g/kg into kg/kg

Mass basis — anything that is not ducted air:

    Q_sensible = m × c_p × (T_final − T_start)
    Q_latent   = m × fraction × h        h = h_fg to boil, h_sf to melt

Barometric pressure and moist-air properties:

    p    = 14.696 (1 − 6.8754×10⁻⁶ Z)^5.2559          psia, Z in feet
    W    = 0.621945 p_w ÷ (p − p_w)                   lb water per lb dry air
    v    = 0.370486 (t + 459.67)(1 + 1.607858 W) ÷ p  ft³ per lb dry air
    ρ    = 1 ÷ v
    h    = 0.240 t + W (1061 + 0.444 t)               Btu per lb dry air

The whole derivation is one idea: a fan moves volume, but heat is carried by mass, so these equations convert one to the other first. Sixty minutes per hour times 0.075 pounds per cubic foot is 4.5 pounds of dry air per hour for every CFM, and each constant is that mass flow times one property.

The properties are where the assumptions hide. 0.075 lb/ft³ is ASHRAE standard air: dry air at one standard atmosphere, 14.696 psia, and roughly 70 °F — real room air at 70 °F and 50 percent relative humidity is nearer 0.0740 lb of dry air per cubic foot, because water vapour is lighter than air. 0.240 Btu/lb·°F is dry air, and 1058 Btu/lb is the latent heat of water near 62 °F, the value hiding inside the traditional 0.68.

The entering-and-leaving-conditions option adds the standard psychrometric relations: saturation pressure from the Hyland and Wexler correlation ASHRAE adopts, barometric pressure from the US Standard Atmosphere. The saturation-pressure routine returns 611.66 Pa at 0.01 °C against a true triple-point value of 611.657 Pa.

Worked example

A five-ton air handler on a summer afternoon, which is the calculator's default: 2,000 CFM across the coil, the air dropping 20 °F, and 10 grains of water per pound of dry air condensing out. Standard sea-level air, cp 0.240 Btu/lb·°F, hfg 1058 Btu/lb.

  1. Mass flow of dry air: 60 min/h × 0.075 lb/ft³ × 2,000 CFM = 9,000 lb of dry air per hour
  2. Total constant: 60 × 0.075 = 4.5
  3. Sensible constant: 4.5 × 0.240 = 1.08
  4. Latent constant: 4.5 × 1058 ÷ 7000 = 0.6801, the 0.68
  5. Sensible load: 1.08 × 2,000 × 20 = 43,200 Btu/h
  6. Latent load: 0.6801 × 2,000 × 10 = 13,603 Btu/h
  7. Total: 43,200 + 13,603 = 56,803 Btu/h, which is 4.73 tons at 12,000 Btu/h per ton
  8. Sensible heat ratio: 43,200 ÷ 56,803 = 0.761
  9. Implied enthalpy change: 56,803 ÷ 9,000 = 6.311 Btu/lb; 4.5 × 2,000 × that change returns the same 56,803 Btu/h — the third form of one equation

Rounding that 0.6801 to the book's 0.68 changes the latent load by 0.02 percent. Elevation is a different matter. Move the same job to Denver at 5,280 feet and switch the density basis to computed: pressure falls to 12.10 psia and the density of air at 70 °F and 50 percent relative humidity to 0.06074 lb of dry air per cubic foot. The sensible constant becomes 0.8746, the latent constant 0.5508, and the total load 46,000 Btu/h — same air handler, same airflow, same temperature drop, 19 percent less capacity.

Now the mass-basis mode, at its defaults: 10 lb of water — about 1.2 US gallons — taken from 60 °F to boiling and then boiled away entirely.

  1. Sensible: 10 lb × 1.00 Btu/lb·°F × (212 − 60) = 1,520 Btu
  2. Latent: 10 lb × 100% × 970 Btu/lb = 9,700 Btu
  3. Total: 11,220 Btu, or 3.288 kWh
  4. Latent share: 9,700 ÷ 11,220 = 86.5 percent

Boiling the water takes 6.4 times the energy of heating it there from tap temperature, every bit of that with the thermometer parked at 212 °F.

Assumptions & tips

  • 1.08, 0.68 and 4.5 are sea-level shorthand, not physics. They assume 0.075 lb/ft³. Above roughly 2,000 feet the error is worth correcting, and in Denver or Albuquerque it is large enough to change equipment selection — which is why manufacturers publish altitude derate tables.
  • Grains are the unit that trips people up. There are 7,000 grains in a pound, so a humidity ratio of 0.0093 lb of water per pound of dry air is 65 grains per pound — the unit US psychrometric charts are drawn in.
  • Use the measured airflow, not the nameplate. Every constant is proportional to CFM, so a system delivering 340 CFM per ton instead of the assumed 400 is short on capacity by that same 15 percent. Nameplate airflow assumes clean filters and ductwork built the way it was drawn.
  • A cooling coil only removes moisture if its surface is below the entering air dew point. If it is not, the latent load is zero no matter what the equipment is rated for, the sensible heat ratio is 1.0, and the space gets cold and clammy.
  • Ventilation air is usually the biggest latent load in the building. Outdoor air at 90 °F and 75 percent relative humidity carries about 160 grains per pound against 65 in the space, which is why moisture-transferring energy recovery wheels earn their keep in humid climates.
  • This is a load split, not a load calculation. It tells you how a load divides once you know the airflow and the state change. Finding the load itself is ACCA Manual J for residential work, or the ASHRAE heat balance and radiant time series methods for commercial.
  • Properties drift with temperature, and sometimes it matters. Water is 1.000 Btu/lb·°F at 60 °F and 1.007 near boiling, which is negligible; steam is roughly half that, which is not. Calculate each leg of a process that crosses a phase change separately.

Frequently asked questions

Where does the 1.08 in the sensible heat formula actually come from?

From air density, the specific heat of air, and the sixty minutes in an hour. Airflow in cubic feet per minute becomes a mass flow of dry air when you multiply by 60 minutes per hour and by density: 60 x 0.075 lb/ft3 = 4.5 lb of dry air per hour for every CFM. Multiply that by the specific heat of air, 0.240 Btu per pound per degree Fahrenheit, and you get 4.5 x 0.240 = 1.08 Btu/h per CFM per degree. Nothing else is hidden in it. The same 4.5 multiplied by the latent heat of water divided by 7000 grains per pound gives the 0.68 used with grains, and the bare 4.5 is the constant used with enthalpy in Btu per pound.

Why do some references print 1.10 instead of 1.08, or 0.69 instead of 0.68?

Because they pick slightly different property values. Using the specific heat of dry air, 0.240 Btu/lb.F, gives 1.08. Using the specific heat of moist air, which is 0.240 + 0.444 times the humidity ratio and works out near 0.244 at ordinary indoor humidity, gives 1.10. On the latent side, a latent heat of 1058 Btu/lb gives 0.680, while the 1076 Btu/lb figure that ASHRAE uses for the pound-per-pound form gives 0.69 and the familiar constant 4840. All of these are the same equation with a different property value substituted, and the spread between them is 1.7 percent, which is far smaller than the error you make by ignoring altitude in Denver.

What is a normal sensible heat ratio?

For a typical office or home cooling coil, roughly 0.75 to 0.85. Restaurants, gyms, locker rooms, natatoriums and spaces with heavy outside air run lower, sometimes 0.55 to 0.70, because people and ventilation air bring in moisture without much heat. Dry climates run higher, and a pure heating or reheat load has a sensible heat ratio of 1.0 because nothing changes the moisture content. The number matters because equipment is rated at a particular sensible heat ratio: a coil selected for 0.80 that meets a 0.65 load will hold temperature and lose the humidity battle.

Do I really have to correct 1.08 for altitude?

Yes, above about 2000 feet. The constants are proportional to air density, and density falls with barometric pressure. At 5000 feet the standard atmosphere is 12.23 psia against 14.696 at sea level, so the air is about 17 percent thinner and 1.08 becomes 0.90. Using the sea-level constant in Denver overstates the capacity of a given airflow by more than twenty percent, which is enough to undersize a coil or oversize a fan. Set the elevation on this page and the derived constants update. Very hot air matters too: at 120 F the density is nearly 9 percent below standard, which is why the same correction is applied to combustion and dryer exhaust work.

What is the difference between sensible heat and latent heat?

Sensible heat changes temperature and a thermometer sees it. Latent heat changes phase at constant temperature and a thermometer sees nothing at all. Warming a pound of water from 60 F to 212 F takes 152 Btu; boiling that same pound away at 212 F takes another 970 Btu, more than six times as much, with no change in temperature while it happens. In an air conditioner the latent part is the energy taken out of the air when water vapour condenses on the cold coil and runs down the drain, which is why a unit that is short-cycling can hold the thermostat setpoint and still leave the room clammy.

Sources

  1. 2021 ASHRAE Handbook — Fundamentals — American Society of Heating, Refrigerating and Air-Conditioning Engineers. ashrae.orgThe Psychrometrics chapter is the source of the standard-air definition of 0.075 lb/ft³, the barometric-pressure-versus-elevation relation, the humidity ratio W = 0.621945 p_w/(p − p_w), the moist-air specific volume and enthalpy relations used here, and the Hyland and Wexler saturation-pressure correlation. The nonresidential cooling and heating load calculations chapter is where the air-side equations appear in the 60·ρ·CFM form from which 1.08, 0.68, 4.5 and 4840 are derived on this page.
  2. NIST Chemistry WebBook — Thermophysical Properties of Fluid Systems — National Institute of Standards and Technology, Standard Reference Database 69. webbook.nist.govSaturation properties of water behind the mass-basis figures: an enthalpy of vaporisation of 2,256 kJ/kg (970 Btu/lb) at 100 °C and one atmosphere, and a liquid specific heat of 4.187 kJ/kg·K at 15.6 °C, which is 1.000 Btu/lb·°F and rises only to about 1.007 near boiling. Also the saturation pressures the psychrometric routine is checked against.
  3. Fundamentals of Heat and Mass Transfer, 8th edition — T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, Wiley. wiley.comThe thermophysical property tables behind the mass-basis material list — the specific heats of aluminium, copper, carbon and stainless steel, concrete, brick, glass, oak, ethanol and ice, together with the latent heats of fusion quoted for aluminium and copper.
  4. U.S. Standard Atmosphere, 1976 — NOAA, NASA and the United States Air Force; NASA-TM-X-74335. ntrs.nasa.govThe atmosphere model behind the pressure-versus-elevation relation used to correct air density, and the source of the sea-level reference of 14.696 psia and the 12.1 psia figure quoted for Denver.
  5. NIST Special Publication 811 — Guide for the Use of the International System of Units (SI) — National Institute of Standards and Technology, 2008 edition. nist.govThe exact conversion factors used by the metric toggle: 1 Btu(IT) = 1,055.056 J, 1 Btu/lb = 2.326 kJ/kg exactly, 1 lb = 0.45359237 kg exactly, 1 ft = 0.3048 m exactly, and the 7,000 grains that make up a pound.
  6. ANSI/ASHRAE Standard 55 — Thermal Environmental Conditions for Human Occupancy — American Society of Heating, Refrigerating and Air-Conditioning Engineers. ashrae.orgThe comfort standard behind the indoor design conditions used in the examples, and the reason the latent side of a load matters at all — comfort depends on humidity as well as temperature.
  7. ANSI/ACCA 2 Manual J — Residential Load Calculation, 8th edition — Air Conditioning Contractors of America. acca.orgThe full residential procedure referred to in the closing tip. This page splits a load you already know into its sensible and latent parts; Manual J is how you establish the load in the first place.

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