Wet-Bulb Temperature, Heat Index and WBGT Calculator

The main result follows the 2017 ASHRAE Handbook—Fundamentals, chapter 1, as openly implemented and documented by PsychroLib 2.5. Saturation pressure uses equations 5 and 6 in kelvin, with the ice/water switch at the water triple point (0.01 °C) to avoid an artificial inversion discontinuity. Vapor pressure is $p_v=RH\,p_{ref}/100$ and humidity ratio is $W=0.621945p_v/(p-p_v)$.

The liquid-water and ice-bulb balances are bracketed separately: equation 33 on 0 °C to the dry bulb when the dry bulb is nonnegative, and equation 35 on −100 °C to min(0 °C, dry bulb). Each admissible bracket is solved independently for 32 bisections. A bracket is admissible when the target humidity ratio lies between its endpoint balances (with a 1e−12 kg/kg numerical allowance). In the liquid branch, $W=[(2501-2.326T_w)W_s-1.006(T-T_w)]/[2501+1.86T-4.186T_w]$, using °C and kg/kg. The reported main result uses the liquid-water root whenever it exists, otherwise the ice-bulb root. The reported phase and both admissible candidate roots are listed in the calculation. A warning appears whenever both phase brackets admit solutions, including coincident roots at freezing; the solver does not choose a phase by its initial guess. The closure residual checks the selected phase's balance only and cannot establish uniqueness. Dew/frost point independently inverts saturation pressure between −100 °C and the dry bulb.

The Stull 2011 empirical fit uses °C, relative humidity as percent and arctangents in radians; see Practical Meteorology, chapter 4, equation 4.19, attached page 9 (printed page 95). It is a sea-level comparison, not a pressure correction.

The NWS heat-index algorithm first computes the simple Steadman-consistent expression. Its average with air temperature is used only to screen whether to apply the regression. Below 80 °F for that average, return the unaveraged simple expression (Pasche et al., 2025, Appendix B, equation B5); at 80 °F or higher, use the Rothfusz regression and the low-/high-humidity adjustments. The published formula uses °F and RH in percent. NWS 2001 wind chill uses $35.74+0.6215T-35.75v^{0.16}+0.4275Tv^{0.16}$, with $T$ in °F and $v$ in mph. All reported readings are converted to the displayed units. These two indices remain separate.

The chart is drawn from the same saturation-pressure and mixing-ratio equations at the selected pressure. Its temperature bounds follow the dew/frost point and dry bulb with padding; the humidity scale comes from saturation at the upper displayed temperature. Tick intervals are rounded for readability and may be fractional g/kg for cold air. This is an original simplified psychrometric chart, not a reproduced standards figure.

WBGT from wind and sunlight

Liljegren et al. (2008), DOI: 10.1080/15459620802310770, supplies the globe and natural wet-wick heat balances and heat/mass-transfer correlations. The openly published Argonne reference equations are expressed here as energy residuals, each bracketed and solved with 32 bisections. Sun elevation and direct-horizontal beam fraction are supplied explicitly, rather than inferred from a location and timestamp. Globe diameter is 0.150 m, following the Japan MoE description of its approximately 15 cm black globe; the reference C code’s 0.0508 m diameter is replaced explicitly in both this solver and the independently compiled comparison. Other sensor and surface reference values remain: wick diameter 0.007 m, wick length 0.0254 m, globe/wick emissivity 0.95, globe albedo 0.05, wick albedo 0.4, ground albedo 0.45 and ground emissivity 0.999.

For either sensor, incoming absorbed shortwave and longwave radiation balance convection; the wick also loses latent heat by evaporation. Convective coefficients use Reynolds and Nusselt correlations, with properties at the mean sensor/air temperature. The wet-wick mass-transfer balance uses the Schmidt number and the reference heat/mass-transfer relation. The WBGT model retains its own Buck liquid-water vapor-pressure relation, including the 1.004 enhancement factor. This is separate from the ASHRAE psychrometric saturation relation. Maximum absolute residual must be below 0.001 W/m²; this numerical tolerance is not a statement of prediction accuracy.

With solar load, $WBGT=0.7T_{nwb}+0.2T_g+0.1T_a$. Without solar load, $WBGT=0.7T_{nwb}+0.3T_g$. The no-solar choice removes shortwave power while retaining the open-sky longwave balance. Natural wet bulb is the temperature of the exposed, naturally ventilated wick; it is not substituted with the thermodynamic ASHRAE or Stull result. The original C reference uses a 0.02 K iteration threshold; its separately executed globe outputs agree with the bisection values within 0.025 K. Natural wet-bulb and WBGT comparisons allow 0.08 K because of the documented latent-heat correction below as well as stopping error. This is a comparison of slightly different model implementations, not a 0.08 K prediction-accuracy claim. Those reference cases validate implementation behavior, not environmental accuracy.

A solar-input condition uses $S\le0.85(1367\,\mathrm{W/m^2})\sin\alpha$, the reference model’s normalized-solar ceiling with Earth–Sun distance fixed at 1 AU. It asks for corrected inputs instead of silently clipping measured irradiance. The entered direct fraction is the direct contribution divided by global horizontal irradiance, not direct normal irradiance.

Japanese risk bands

The Ministry of the Environment’s English WBGT Guideline, crediting the Japan Sports Association’s 2019 guidebook, supplies the exercise categories: below 21 °C, Almost safe; 21 to below 25 °C, Caution; 25 to below 28 °C, Warning; 28 to below 31 °C, Severe warning; 31 °C or higher, Danger. Messages paraphrase the guidance. Thresholds use the unrounded WBGT estimate and include the lower edge in each higher band. The Ministry’s daily-life table groups values below 25 °C together; this tool explicitly labels the finer exercise classification. Guidance checked 25 October 2026. This local estimate does not issue an official alert: Japan’s ordinary Heat Stroke Alert uses a forecast WBGT of at least 33 °C at a reporting location within its regional forecast system. Follow the official service and current guidance.

Correction to the open reference’s latent-heat term

The pinned Argonne C function evap has an increasing latent heat with temperature: 2336.2 kJ/kg at 10 °C and 2407.3 kJ/kg at 40 °C. This conflicts with the decreasing physical trend and NISTIR 5078, Table 1, attached page 1: 2477.2 kJ/kg at 10 °C and 2406.0 kJ/kg at 40 °C. The present wick balance corrects the arithmetic sign while retaining the reference constants: $L_v=2.4073\times10^6+2370(313.15-T_{film})$ J/kg with $T_{film}$ in kelvin. This gives 2478.4 and 2407.3 kJ/kg respectively, within 1.5 kJ/kg of the independently published values. Latent heat therefore falls with temperature, as required. The original uncorrected C outputs are retained as cross-method references with the discrepancy explicitly identified in each case’s source; they are not treated as exact physical truth.

RH reference below freezing

The input defaults to relative humidity over liquid water, including below freezing, as reported by common weather sensors (Campbell Scientific HygroVue 5 documentation). Select the explicit ice reference only when the source data use it. Above 0.01 °C both choices use the existing ASHRAE water saturation pressure. Below 0.01 °C the water reference uses Murphy and Koop (2005), equation 10, attached page 14, with temperature in kelvin and pressure in pascals. It is used only within its published 123–332 K range. The ASHRAE ice saturation pressure and the separate ice/liquid bulb balances remain unchanged.

Vapor pressure follows the selected reference, $p_v=(RH/100)p_{ref}$. The solver receives the equivalent stable-phase RH, $RH_{phase}=100p_v/p_{ws}$, where the stable phase is ice below 0.01 °C and water above. Inputs with $RH_{phase}>100$ are rejected as unsupported ice supersaturation; no result is presented as a supercooled-cloud or supersaturated equilibrium calculation. A bounded internal RH keeps invalid-input evaluation finite while the error blocks results; it does not replace an accepted sensor reading. The displayed RH limit is $100p_{ws}/p_{ref}$, capped at 100%. At −20 °C it is about 82.28% for water-referenced observations. The source's rounded 82% example therefore gives a frost point slightly below −20 °C. Humidity sweeps stop at that limit. Psychrometric chart curves retain the explicitly labeled stable-phase RH convention, while the entered point uses the converted vapor pressure.

Longwave-correlation guard

The inherited clear-sky fit is $\epsilon_{sky}=0.575 e_a^{0.143}$ with vapor pressure $e_a$ in hPa from the WBGT Buck relation. WBGT is supported here only when $0<\epsilon_{sky}\le1$, in addition to the nominal input ranges. The raw fitted value is shown in How it’s calculated and is not clipped. For example, saturated 40 °C gives about 1.06435 and is rejected for both solar and no-solar exposure. The sensor solve is skipped; globe temperature, natural wet bulb, WBGT, its risk category and its closure residual are withheld, and the wind curve is blank. Other supported psychrometric results remain available. A warning identifies the radiation-model limitation. A small numerical balance residual cannot validate this empirical fit's extrapolation. Saturated no-solar checks cover the supported side near 32 °C, the excluded side at 33 °C, and the reviewer's 40 °C example.

Ideal moist air at uniform absolute pressure with no supersaturation relative to the stable phase. Entered RH is relative to liquid water by default, even below freezing; an explicit ice-reference choice is available below 0.01 °C. Vapor pressure is converted to the solver's stable-phase reference before the wet-bulb and dew/frost-point solves. Ice-supersaturated observations are rejected rather than interpreted as a different humidity convention. The ice-bulb branch represents an ice-covered bulb at or below 0 °C; the liquid branch represents a water-covered bulb at or above 0 °C. If both roots exist, liquid water is the stated reporting convention, not a prediction of the actual wick's phase. No supercooled liquid-water bulb or transient freezing is modeled.

The psychrometer drawing is schematic, with illustrative column heights. The main result is thermodynamic wet bulb; a real ventilated sensor can differ because of radiation, ventilation, wick condition and instrument coefficients. WBGT uses its own local wind at 2 m; NWS wind chill uses separate station wind at about 10 m. The thermodynamic wet bulb, Stull comparison and heat index do not depend on either wind input. There is no evolving time state, so animation would not describe this calculation.

WBGT assumes steady outdoor conditions, liquid water on the wick, an unobstructed sky and uniform ground at air temperature. Ground reflectance is fixed at the reference value 0.45, which represents a reflective surface and can overestimate solar loading over darker ground. Sky longwave emission follows the reference clear-sky empirical humidity relation only while its raw fitted emissivity is at most one; otherwise WBGT is unavailable. This guard is a necessary physical bound, not independent validation of the fit for every accepted state. Cloud longwave effects are not separately supplied. Globe size is 150 mm, distinct from the smaller reference-code globe. Wind below 0.13 m/s uses the reference correlation’s 0.13 m/s floor. Sun elevation and beam fraction are user estimates unless measured. Local hot walls, indoor radiant sources, body heat, clothing and activity are not modeled. The drawings are original schematics and the risk-band diagram is an original presentation of public guidance.

Supported inputs: −40 to 60 °C dry bulb, 1–100% relative humidity on the selected reference and 60–110 kPa absolute pressure, subject to the stable-phase saturation limit. Zero humidity is excluded because dew point would not be finite. The ASHRAE numerical solve is more precise than the source correlations and measurement inputs; extra digits are not extra accuracy. Near freezing, both ice and liquid phase solutions can exist. A warning identifies this nonunique phase choice even when each root has a tiny residual. The main result prefers liquid water whenever its nonnegative root exists; otherwise it uses the nonpositive ice-bulb root. Use the phase of the actual sensor when interpreting the candidate results. A separate closure check detects a failed numerical balance, not multiple roots.

Stull's published range is −20 to 50 °C and 5–99% RH, with an exception for cold, dry conditions; this tool conservatively shows it only at 0–50 °C and 5–99% RH. The reported mean absolute fit error is below 0.3 °C at standard sea level, with larger errors possible. It has no pressure dependence.

Heat index is shown only for air at 80–112 °F and a calculated heat index at or below 137 °F, a conservative subset of the NWS charts. Cooler conditions and extreme extrapolations show “Outside range.” It assumes shade and light wind and does not include solar load or an entered-wind correction. Wind chill requires air at or below 50 °F and wind above 3 mph; this tool additionally limits it to 60 mph, the published chart extent. Elsewhere it shows “Not applicable.”

The 35 °C wet-bulb figure is a theoretical heat-balance benchmark, not a safe/unsafe boundary. Vecellio et al. (2022) measured uncompensable heat stress below it in young healthy adults. This tool supplies environmental quantities and an estimated WBGT, not medical advice, safe exposure durations or occupational work/rest limits. Check applicable workplace guidance separately.

WBGT is deliberately displayed only for 20–50 °C air, at least 5% RH and 80–110 kPa pressure. Local 2 m wind is 0–20 m/s (correlation floor 0.13 m/s), solar irradiance 0–1200 W/m², sun elevation 5–90° and direct-horizontal fraction 0–0.9; irradiance must also satisfy the angle-dependent ceiling. Outside the nominal warm-weather domain, or whenever the raw clear-sky emissivity fit exceeds 1, the WBGT cards state that it is unavailable while supported psychrometric results remain available. The wind curve is deliberately blank there. Use the no-solar choice at night; its open-sky assumptions do not represent indoors or a shaded street canyon. Uncertainty in radiation, ground reflectance, local wind and sensor exposure can move an estimate across a Japanese risk threshold. Use an appropriate on-site WBGT instrument for operational decisions. The listed Japanese exercise bands are guidance as accessed on 25 October 2026, not a claim that workplace, military or sports rules are interchangeable. Check the rules in force for the intended activity.

Below freezing, common liquid-water-referenced observations must not be entered as ice RH. The adjacent input help and reference selector specify this convention. The admissible maximum water RH is temperature-dependent (about 82.28% at −20 °C for these saturation correlations). Higher readings may describe real metastable ice supersaturation, but that condition and supercooled liquid-water bulbs are outside this equilibrium tool. Check the sensor reference and reading; do not switch the reference merely to make an out-of-domain state pass. The displayed humidity sweep is restricted to the admissible maximum. The WBGT sky-fit guard applies in sunlight as well as without solar load; numerical closure is checked only after that model-domain test.