Nicholas Ravanelli, PhD

V1.10 (June 12th, 2019)
- Added Strava API integration
- Improved mobile user interface
- Corrected imst definition - Thanks @ Boris Kingma (NTO)
- Auto update activity details when reviewing past thermal audits

V1.01 (May 30th, 2019)
- Updated date/time selector to be more mobile friendly
- Able to fetch historical & 6-days ahead weather data
- Added clothing ensembles and estimate Re,cl

V1.0 (May 25th, 2019)
- Initial Release

The equations used for the "Thermal Audit Simulation" are based on the equations found in:

• The Biophysics of Human Heat Exchange (in Heat Stress in Sport and Exercise, 2019); and
• Human Thermal Environments: The Effects of Hot, Moderate, and Cold Environments on Human Health, Comfort, and Performance, Third Edition

Conceptual Heat Balance Equation:

M - W = ± K ± C ± R ± (E_res + C_res) + E_sk ± S [W/m²]

Heat Production (H_prod):

H_prod = M - W [W/m²]

Metabolic Energy Expenditure (M):
(when not provided)

M = (341 * VO₂) / BSA [W/m²]

Where VO₂ (rate of oxygen consumption) is in L/min, and BSA is body surface area

Rate of oxygen consumption (VO₂ in L/min):
VO₂ can be provided, or estimated using the 2011 Compendium of Physical Activities normative values (in METs) or through selecting modality and providing a parameter. When the later is selected, ACSM equations are used to estimate VO₂.

• Walking:
  • VO₂ = [(0.1 * SPEED [m/min]) + (1.8 * SPEED [m/min] * GRADE[/1]) + 3.5 ] * MASS [L/min]
• Running:
  • VO₂ = [(0.2 * SPEED [m/min]) + (0.9 * SPEED [m/min] * GRADE[/1]) + 3.5 ] * MASS [L/min]
• Cycling:
  • VO₂ = [(10.8 * WATTS / MASS ) + 7] * MASS [L/min]

External Work Rate (W):
(when not provided, i.e. cycling)
If slecting a normative physical activity from the Compendium of Physical Activities

W = M * (0.2 * tanh(0.39 * MET -0.6) ) [W/m²]

Where MET is the metabolic rate; 1 MET = 58.15 W/m²

If selecting the "Walking" or "Running" modality:

W =(10³ * (MASS[kg] * SPEED[m/min] * (%GRADE * 100) )) / (6.12 * 60 * 1000) ) / BSA [W/m²]

Conduction (K):
At present, conduction is assumed negligible.

Convection (C):

C = f_cl * h_c * (T_shell - T_air) * (P_b/760)^0.55 [W/m²]

Where f_cl is the clothing area factor, h_c is the convective heat transfer coefficient, T_shell is the external surface temperature (skin or clothing temperature), T_air is ambient temperature, and P_b is barometric pressure in mmHg

Convective heat transfer coefficient (h_c):
The convective heat transfer coefficient is calculated depending on the activity conducted where v is air velocity (m/s) or loc is locomotion speed (m/s);

• Default (rest or normative activities):
  • if v < 0.2, h_c = 3.1, else h_c = 8.3 * v^0.6 [W/m²/K]
• Stationary cycling:
  • if v < 0.2, h_c = 6, else h_c = 8.3 * v^0.6 [W/m²/K]
• Treadmill:
  • h_c = 8.3 * loc^0.391 [W/m²/K]
• Outdoor Running:
  • h_c = 8.3 * loc^0.531 [W/m²/K]
• Outdoor Cycling:
  • h_c = 8.4 * loc^0.840 [W/m²/K]

Radiation (R):

R = f_cl * h_r * (T_shell - T_air) [W/m²]

Where f_cl is the clothing area factor, h_r is the radiative heat transfer coefficient, T_shell is the external surface temperature (skin or clothing temperature), and T_air is ambient temperature

Radiative heat transfer coefficient (h_r):

h_r = 4 * ɛ * σ * Ar [273.2 + (T_shell + T_r / 2)]³ [W/m²/K]

Where σ is the Stefan–Boltzmann constant, 5.67 × 10^-8 (W/m²/K⁴), ɛ is the area-weighted emissivity of the body surface, and Ar is the effective radiative area, T_shell is the external surface temperature (skin or clothing temperature), and T_r is the mean radiant temperature.

Mean Radiant Temperature (T_r):

If the activity is conducted indoors or at night, T_r is assumed to be equal to air temperature.

If air velocity (v) is below 0.15 m/s:

T_r = [ (t_g + 273)⁴ + ( 0.25 * 10⁸ / ɛ ) * [(|t_g - t_a|)/d]^1/4 * ( t_g - t_a) ]^0.25 -273 [°C]

If air velocity (v) is > 0.15 m/s:

T_r = [ (t_g + 273)⁴ + ( 1.1 * 10⁸ * v^0.6 / ɛ * d^0.4 )* ( t_g - t_a) ]^0.25 -273 [°C]

Where t_g is black globe temperature (°C), ɛ is the emissivity of a standard black globe (0.95), and d is the diameter of the globe.

Solar Radiation (R_solar):

Total clear sky solar radiation, used to estimate black globe temperature (t_g) is estimated using equations from Campbell and Norman (1998):

R_solar,cs = K_b + K_d + K_r [W/m²]

Where K_b (direct beam radiation);

K_b = K_p * cos(ψ) [W/m²]

Where ψ is the solar zenith angle estimated using time of day and day of year using a validated Solar Positioning Algorithm, and K_p is the direct irradiance received on the Earth surface perpendicular to the beam;

K_p = K_o * τ^m [W/m²]

Where K_o is the solar constant (~1367 W/m²), τ is the atmospheric transmittance which can be given or default to 0.6, and m is the optical mass number derived as:

m = P_b / ( 101.3 * cos(ψ) ) [ND]

Where P_b is the atmospheric pressure in kPa, and ψ is the solar zenith angle.

K_d (diffuse radiation) is calculated as;

K_d = 0.3 * (1 - / τ^m) * K_o * cos(ψ) [W/m²]

Where τ is the atmospheric transmittance which can be given or default to 0.6, and m is the optical mass number, and ψ is the solar zenith angle.

K_r (reflected solar radiation) is calculated as:

K_r = α_gr * K_t [W/m²]

Where α_gr is the albedo of the ground surface (default is 0.3) and K_t is the sum of K_b and K_d.

Lastly, cloud cover alters solar radiation and thus is accounted for using the following equation from NASA:

R_solar = R_solar,cs *(1 - (0.75*n^3.4)) [W/m²]

Where n is the percent (0-1) of cloud coverage.

Black globe temperature (T_g):

When not provided, black globe temperature (T_g) is estimated using the equation from Hajizadeh et al. 2017.

T_g = (0.01498 * $solarrad) + (1.184 * $Tair) - (0.0789 * $relative_humidity) - 2.739;

Respiratory heat loss (E_res + C_res):

E_res + C_res = 0.0014 * (M * (34 - T_a) ) + 0.0173 * (M * (5.87 - P_a) ) [W/m²]

Where M is metabolic rate in W/m², T_a is ambient temperature in °C, and P_a is the ambient vapour pressure in kPa.

Evaporative requirements for heat balance (E_req):

E_req is calculated by rearranging the conceptual heat balance equation:

E_req = (M - W) ± K ± C ± R ± (E_res + C_res) [W/m²]

Maximum evaporative capacity of the environment (E_max):

Assuming negligible clothing or nude:

E_max = (P_sk,s - P_a) / (1/h_e)[W/m²]

Where the P_sk,s - P_a is the ambient vapor pressure difference between skin and air in kPa, and h_e is the evaporative heat transfer coefficient.

If clothing is worn and locomotion or air velocity exceed 0.2 m/s, E_max is derived using dynamic evaporative resistance clothing equations found in Chapter 9 which are also used in the PHS model.

Evaporative heat transfer coefficient (h_e):

h_e = 16.5 * h_c [W/m²/K]

Where h_c is the convective heat transfer coefficient in W/m²/K and 16.5 is the Lewis Relationship (in K/kPa) .

Sweating efficiency (n):

ω_req = E_req / E_max

if ω_req < 1, n = 1 - (ω_req² / 2); else n = (2 - ω_req)² / 2

Evaporation from the skin (E_sk):

Where ω_max is the maximum attainable skin wettedness and is assumed to be 0.72 for unacclimated, 0.84 for partially acclimated (Ravanelli et al. 2018), and 1.0 for fully acclimated (Candas et al. 1979);

When E_req < (E_max * ω_max);

E_sk = E_req [W/m²]

When E_req > (E_max * ω_max):

E_sk = E_max [W/m²]

When E_sk > maximum whole body sweat rate:

E_sk = maximum whole body sweat rate [W/m²]

Sweat rate required (S_req):

S_req = [ ( E_req / n ) * BSA ] / HL_vap,sw * 60 [g/min]

Where BSA is body surface area (calculated using the Dubois & Dubois equation) HL_vap,sw is the heat liberated when evaporating one gram of sweat (default: 2426 J).

If S_req > maximum sweat rate, then S_req = maximum sweat rate.

Skin Temperature (T_sk):

Skin temperature (T_sk) is estimated using equations from Mehnert et al. (2000) for nude and clothed individuals:

Nude: T_sk = 7.19 + 0.064 * T_a + 0.061 * T_r + 0.198 * P_a - 0.348 * V_a + 0.616 * T_re [°C]

Clothed: T_sk = 12.17 + 0.020 * T_a + 0.044 * T_r + 0.194 * P_a - 0.253 * V_a + 0.0029 * M + 0.513 * T_re [°C]

Rectal Temperature (T_re):

As rectal temperature (T_re) is required to satisfy the equations for T_sk, it is loosely approximated as:

T_re = 0.0036 * M [W/m²] + 36.6

if T_re > 40°C, T_re = 40°C

Clothing area factor (f_cl)

During rest and still air (< 0.2 m/s):

f_cl = 1 + [ ( 0.31 * I_cl ) / 0.155 ] [ND]

Where I_cl is the static clothing insulation in clo units.

Evaporative Resistance of the clothing (R_e,cl)

When not provided, and during rest and still air (< 0.2 m/s):

R_e,cl = 0.18 * (I_cl / 0.155) [m²/K/W]

Where I_cl is the static clothing insulation in clo units.

If locomotion or air velocity is > 0.2 m/s, dynamic clothing insulation and evaporative resistance is calculated using equations presented in Chapter 9

Clothing surface temperature (T_cl):

h_r = 4 * ɛ * σ * Ar/Ad * (273.2 + ((T_cl + T_r)/2)³

And;

T_cl = (( ( 1/ I_cl ) * T_sk ) + (f_cl * ((h_c * T_a) + (h_r * T_r)))) / ((1/$I_cl) + (f_cl * (h_r + h_c)))

Where ɛ is emissivity, σ is the Stefan-Boltzmann constant, Ar/Ad is the effective radiative surface area, T_cl is clothing temperature, T_r is radiant temperature, T_r is air temperature, I_cl is the clothing thermal insulation, f_cl is the clothing area factor, and h_c is the convective heat transfer coefficient.

T_cl Is derived using iterative techniques starting at T_cl = 0, until the difference between successive T_cl is < 0.01

Brief Summary

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In order to serve the best possible experience with the Human Thermal Audit Simulation Application, all data stored in this database will be kept indefinitely. This is so we can re-calculate any results based on improvements within the application to ensure revisiting past thermal audit simulations are up-to-date.

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At any time, you can request to receive an exported file of any input data you have provided to us. You can also request that we erase any data we hold about you. This does not include any data we are obliged to keep for administrative, legal, or security purposes. As this information does not contain any a users personally identifiable information (e.g. name, email, civic address), we may require additional information from you to accurately identify the data associated to your inputs (e.g. the TA_ID or hyperlink to the Thermal Audit Results).

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