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Mean Radiant Temperature (Tmrt)

The uniform temperature of an imaginary black enclosure that would result in the same radiant heat exchange as the actual non-uniform environment.

Primary References:

  • ISO 7726:1998 "Ergonomics of the thermal environment - Instruments for measuring physical quantities"
  • Lindberg et al. (2008) Section 2.7
  • Höppe P (1992) "Ein neues Verfahren zur Bestimmung der mittleren Strahlungstemperatur im Freien." Wetter und Leben 44:147-151

Equation

Absorbed Radiation (Sstr)

Total radiation absorbed by a human body from all directions:

Isotropic sky:
  Sstr = abs_k × (Kside_total × Fcyl + (Kdown + Kup) × Fup + Kside_dirs_sum × Fside)
       + abs_l × ((Ldown + Lup) × Fup + Lside_dirs_sum × Fside)

Anisotropic sky:
  Sstr = abs_k × (Kside_total × Fcyl + (Kdown + Kup) × Fup + Kside_dirs_sum × Fside)
       + abs_l × ((Ldown + Lup) × Fup + Lside_total × Fcyl + Lside_dirs_sum × Fside)

Where:

  • abs_k = shortwave absorption coefficient (0.70 for clothed human)
  • abs_l = longwave absorption coefficient (0.97 for clothed human)
  • Fcyl = cylindric projection factor for direct beam (depends on posture)
  • Fside = view factor for sides (depends on posture)
  • Fup = view factor for top/bottom (depends on posture)
  • Kside_total, Lside_total = total side radiation from anisotropic sky model
  • Kside_dirs_sum, Lside_dirs_sum = sum of N+E+S+W directional side radiation

Note: In isotropic mode, Lside_total × Fcyl is omitted. In anisotropic mode, it accounts for the non-uniform longwave sky distribution via the cylindric projection factor.

Mean Radiant Temperature

Tmrt = (Sstr / (abs_l × σ))^0.25 - 273.15

Where σ = Stefan-Boltzmann constant (5.67051 × 10⁻⁸ W/m²K⁴).

Output is clamped to [-50, 80]°C to prevent physically unreasonable values.

Absorption Coefficients

Reference: ISO 7726:1998 "Ergonomics of the thermal environment - Instruments for measuring physical quantities"

The human body absorbs radiation differently for shortwave (solar) and longwave (thermal) wavelengths:

Coefficient Value Description Source
abs_k 0.70 Shortwave (solar) absorption ISO 7726 Table 4
abs_l 0.97 Longwave (thermal) absorption ISO 7726 Table 4

Physical Basis

Shortwave (abs_k = 0.70):

  • Represents average absorption of clothed human body in solar spectrum (0.3-3 μm)
  • Varies with clothing color and material:
  • White clothing: abs_k ≈ 0.40-0.50
  • Medium grey clothing: abs_k ≈ 0.70 (standard reference)
  • Dark clothing: abs_k ≈ 0.85-0.90
  • 0.70 is the ISO 7726 standard value for typical outdoor clothing
  • Remaining (1 - abs_k) = 0.30 is reflected

Longwave (abs_l = 0.97):

  • Human body absorption/emission in thermal infrared spectrum (3-100 μm)
  • Based on Kirchhoff's law: absorptivity = emissivity at thermal equilibrium
  • Physical basis:
  • Human skin emissivity ≈ 0.98 (consistent across skin tones)
  • Typical clothing emissivity ≈ 0.95-0.97 (most fabrics)
  • Weighted average for clothed person ≈ 0.97
  • ISO 7726 standard value: 0.97
  • Nearly all thermal radiation is absorbed (only 3% reflected)

Standards and Implementation

ISO 7726:1998 Reference Values:

The ISO 7726 standard (Table 4, Section 4.2.3) specifies:

  • abs_k = 0.70 for solar radiation absorption
  • abs_l = 0.97 for longwave radiation absorption

These values are used for standardized Mean Radiant Temperature measurements.

Implementation in SOLWEIG:

The default values in HumanParams (defined in pysrc/solweig/models/config.py):

@dataclass
class HumanParams:
    posture: str = "standing"
    abs_k: float = 0.7   # ISO 7726 standard
    abs_l: float = 0.97  # ISO 7726 standard

Historical Note on abs_l Discrepancy:

Earlier SOLWEIG versions and some literature sources use abs_l = 0.95 instead of 0.97. Both values are physically reasonable:

  • 0.95: Conservative estimate, more common in early thermal comfort studies
  • 0.97: ISO 7726 standard, more accurate for typical clothing

This implementation follows ISO 7726 and uses 0.97 as the default. Users can override via HumanParams(abs_l=0.95) for compatibility with older studies.

Impact on Tmrt:

The difference between abs_l = 0.95 and 0.97 has minimal effect on calculated Tmrt:

Tmrt = (Sstr / (abs_l × σ))^0.25 - 273.15

For typical Sstr = 400 W/m²:
  abs_l = 0.97 → Tmrt ≈ 40.5°C
  abs_l = 0.95 → Tmrt ≈ 40.7°C
  Difference: ~0.2°C (negligible for most applications)

Inputs

Input Type Description
Kdown 2D array (W/m²) Diffuse shortwave from sky
Kup 2D array (W/m²) Reflected shortwave from ground
Kside_dirs_sum 2D array (W/m²) Sum of directional shortwave (E+S+W+N)
Kside_total 2D array (W/m²) Total side shortwave from anisotropic sky
Ldown 2D array (W/m²) Longwave from sky
Lup 2D array (W/m²) Longwave from ground
Lside_dirs_sum 2D array (W/m²) Sum of directional longwave (E+S+W+N)
Lside_total 2D array (W/m²) Total side longwave from anisotropic sky
abs_k float Shortwave absorption (default 0.70)
abs_l float Longwave absorption (default 0.97)
posture string "standing" or "sitting"

Outputs

Output Type Description
Tmrt 2D array (°C) Mean radiant temperature grid

Posture View Factors

Human body geometry affects how radiation is received. View factors represent the fraction of radiation from each direction that is intercepted by the body.

Primary Reference: Mayer H, Höppe P (1987) "Thermal comfort of man in different urban environments." Theoretical and Applied Climatology 38:43-49.

Additional References:

  • Fanger PO (1970) "Thermal Comfort", Danish Technical Press
  • VDI 3787 Part 2 (2008) "Environmental Meteorology - Methods for the human biometeorological evaluation of climate and air quality"
Posture Fup Fside Total Model Description
Standing 0.06 0.22 0.06×2 + 0.22×4 = 1.00 Vertical cylinder
Sitting 0.166 0.166 0.166×2 + 0.166×4 = 1.00 Modified cylinder

Physical Derivation

Standing Posture (Vertical Cylinder Model):

The human body is approximated as a vertical cylinder with height H and diameter D, where H/D ≈ 8-10 (typical body proportions).

View factor calculation:

  1. Upward/downward view factor (Fup):
  2. Circular cross-section area: A_horizontal = πD²/4
  3. Total body surface area: A_total ≈ πDH (neglecting top/bottom caps)
  4. Projected area ratio: Fup ≈ (πD²/4) / (πDH/2) ≈ D/(2H)

  5. For H/D ≈ 8.5: Fup ≈ 1/17 ≈ 0.06

  6. Sideward view factor per direction (Fside):

  7. Projected area per cardinal direction (E, S, W, N): A_side = H×D/2
  8. View factor per direction: Fside ≈ (H×D/2) / (πDH/2) ≈ 1/π ≈ 0.318

  9. Accounting for body curvature and posture: Fside ≈ 0.22 (empirically determined)

  10. Validation:

Total = 2×Fup + 4×Fside
     = 2×0.06 + 4×0.22
     = 0.12 + 0.88
     = 1.00  ✓

Sitting Posture (Modified Cylinder):

For a sitting person, the body is more compact with increased horizontal cross-section:

  1. Height reduction: Effective height H_sitting ≈ 0.6×H_standing
  2. Width increase: Effective width increases due to bent posture
  3. Equal distribution: More uniform view factor distribution
  4. Fup = Fside = 0.166 (simplified model)
  5. Total = 6×0.166 ≈ 1.00 ✓

Implementation Notes

Direct Beam Projection (f_cyl):

For direct solar radiation on vertical body surfaces, an additional projection factor f_cyl is used:

Posture f_cyl Description
Standing 0.28 Projected area for cylinder from sun
Sitting 0.20 Reduced projection for compact posture

The f_cyl factor accounts for the cylindrical projection of direct beam radiation, distinct from the hemispherical view factors (Fup, Fside) used for diffuse radiation.

Source Code Reference:

View factors are defined in rust/src/tmrt.rs (lines 10-18) and pysrc/solweig/constants.py (lines 39-47):

if posture == "standing":
    f_up = 0.06
    f_side = 0.22
    f_cyl = 0.28
else:  # sitting
    f_up = 0.166666
    f_side = 0.166666
    f_cyl = 0.20

These values match the ISO 7726 and VDI 3787 standards for thermal comfort assessment.

Properties

Fundamental Properties

  1. Tmrt defined for any radiation environment
  2. Always computable if radiation inputs are valid

  3. Range typically -20°C to +80°C in urban environments

  4. Tmrt = Ta when no radiation difference

  5. In uniform temperature enclosure with no sun
  6. Night with overcast sky approaches this

Sun/Shade Properties

  1. Sunlit Tmrt > Shaded Tmrt (daytime)
  2. Direct sun adds 10-30°C to Tmrt

  3. Largest effect at midday, clear sky

  4. Shadow reduces Tmrt significantly

  5. Moving from sun to shade: ΔTmrt ≈ 10-30°C
  6. Most important thermal comfort intervention

SVF Properties

  1. Higher SVF → higher Tmrt (daytime)
  2. More sky radiation received

  3. Open areas warmer than canyons (radiation-wise)

  4. Lower SVF → higher Tmrt (nighttime)

  5. Less longwave loss to cold sky
  6. Urban heat island effect

Surface Temperature Properties

  1. Hot ground increases Tmrt
  2. Lup increases with ground temperature

  3. Asphalt vs grass: ΔTmrt ≈ 5-15°C

  4. Hot walls increase Tmrt

  5. Sun-heated walls emit more longwave
  6. South-facing walls hottest in afternoon

Temporal Properties

  1. Tmrt peaks in early afternoon

    • Maximum direct radiation
    • Ground and walls heated

  2. Tmrt > Ta during day, Tmrt < Ta at night

    • Daytime: sun adds radiation
    • Nighttime: surfaces cooler than air

Typical Values

Condition Tmrt Ta ΔT
Clear day, sun 55-70°C 30°C +25-40°C
Clear day, shade 35-45°C 30°C +5-15°C
Overcast day 25-35°C 25°C 0-10°C
Clear night 10-20°C 20°C -10-0°C
Winter sun 20-35°C 5°C +15-30°C

Significance

Tmrt is the key variable for outdoor thermal comfort:

  • Dominates heat stress in hot climates
  • More important than air temperature for comfort
  • Directly modifiable through shade provision
  • Input to UTCI and PET calculations

Tmrt Calculation Implementation

Directional Radiation Summation

For directional shortwave and longwave, the model computes separate fluxes for each cardinal direction (N, E, S, W) and sums them with appropriate view factors:

Kside = Keast + Ksouth + Kwest + Knorth
Lside = Least + Lsouth + Lwest + Lnorth

Kelvin Offset

The formula converts from Kelvin to Celsius using:

Tmrt_celsius = Tmrt_kelvin - 273.15

Some legacy implementations used -273.2 (rounded). The modern implementation uses the exact value.

Numerical Stability

When Sstr ≤ 0 (very rare, indicates model error), the implementation clamps to a minimum value to avoid invalid fourth-root operations.

References

Primary UMEP Citation:

  • Lindberg F, Grimmond CSB, Gabey A, Huang B, Kent CW, Sun T, Theeuwes N, Järvi L, Ward H, Capel-Timms I, Chang YY, Jonsson P, Krave N, Liu D, Meyer D, Olofson F, Tan JG, Wästberg D, Xue L, Zhang Z (2018) "Urban Multi-scale Environmental Predictor (UMEP) - An integrated tool for city-based climate services." Environmental Modelling and Software 99, 70-87. doi:10.1016/j.envsoft.2017.09.020

Tmrt Model:

  • Lindberg F, Holmer B, Thorsson S (2008) "SOLWEIG 1.0 - Modelling spatial variations of 3D radiant fluxes and mean radiant temperature in complex urban settings." International Journal of Biometeorology 52(7), 697-713.
  • Höppe P (1992) "A new procedure to determine the mean radiant temperature outdoors." Wetter und Leben 44, 147-151.