a b :

Albedo (reflectivity) of a person’s body surface (0.3)

a o :

Mean albedo of ground-based, solid objects projecting into the sky hemisphere (0.2)

a g :

Albedo of ground (0.3)

A eff :

Effective radiation area (square metres)

A D :

Total body surface area (square metres)

A P :

Projected body surface area (square metres)

e a :

Air vapour pressure (hectopascals)

f eff :

Effective radiation area factor (=A eff/A D)

f p :

Projected area factor per unit of effective radiation body area (=A p /A eff)

\( f_{\text{p}}^{*} \) :

Projected area factor per unit of total body area (=A p /A D  = f p  × f eff)

K b :

Incoming direct beam solar radiation on a horizontal surface (watts per square metre)

\( K_{\text{b}}^{*} \) :

Absorbed direct beam solar radiation on the body surface (watts per square metre)

K d :

Diffuse beam solar radiation from the sky on the horizontal ground surface (watts per square metre)

\( K_{\text{d}}^{*} \) :

Diffuse beam solar radiation from the sky absorbed on the body surface (watts per square metre)

K r :

Total reflected solar radiation by objects and ground (=K ro + K rg) (watts per square metre)

\( K_{\text{r}}^{*} \) :

Total solar radiation reflected by objects and ground absorbed on the body surface (watts per square metre)

K rg :

Solar radiation reflected by the ground (watts per square metre)

K ro :

Solar radiation reflected by objects in the sky hemisphere (buildings, trees and other structures) (watts per square metre)

L :

Net longwave radiation on the body surface (watts per square metre)

L a :

Incoming longwave radiation from the sky to the horizontal ground surface (watts per square metre)

L b :

Longwave radiation emitted from the body surface (watts per square metre)

L g :

Longwave radiation emitted from the ground (watts per square metre)

L o :

Longwave radiation coming from objects in the sky hemisphere to the horizontal ground surface (watts per square metre)

Q :

Net all-wave radiation on the body surface (watts per square metre)

R :

Absorbed solar (shortwave) radiation on the body surface (watts per square metre)

RH:

Relative humidity (decimal)

t :

Transmissivity for direct beam solar radiation of objects obscuring the sun [building = 0, trees vary from 0.15 (spruce) to 0.75 (willow), from Brown and Gillespie (1995)]

T a :

Air temperature (degrees Celsius)

T b :

Skin temperature (degrees Celsius)

Z sl :

The angle between the perpendicular to the object surface and the sun (degrees)

ψ sky :

Sky view factor (decimal)

ψ w_sky :

The view factor of sky seen from the object surface (decimal)

β :

(Solar) altitude angle (degrees)

ε a :

Emissivity of air (0.97 to 0.99)

ε b :

Emissivity of the body surface (0.97)

ε sky :

Sky emissivity

σ :

Stefan–Boltzmann constant (5.67·10−8 W m−2 K−4)

1 Introduction

A number of human thermal exchange models have been developed which include all important modes of energy exchange. Some of those capable of use in outdoor urban environments are the Burt model (Burt 1979 and modified by Tuller 1990), COMFA (COMfort FormulA; Brown and Gillespie 1986, 1995), MENEX (Man-ENvironment heat Exchange; Blazejczyk 1994, 2004, 2005), RayMan (Matzarakis et al. 2000, 2007, 2009) and OUT-SET* (Pickup and de Dear 2000).

Urban outdoor environments have a very important and different climatic variable, solar radiation, compared with indoor environments. Solar radiation can raise the apparent temperature between 7°C and 14°C depending on wind speed (Steadman 1971).

Early versions of different methods for estimating solar radiation absorbed on the human body surface were reviewed by Blazejczyk et al. (1993) using Krys and Brown’s (1990) cylindrical body type equation which is a solar radiation component of the COMFA model. Park and Tuller (2010) analysed human body area factors (effective radiation area and projected area factors) for standing and walking postures and compared these with previous studies, e.g. Underwood and Ward (1966), Fanger (1972), Tanabe et al. (2000) and Kubaha et al. (2004) for standing posture; Ward and Underwood (1967) and Steadman (1979) for walking posture.

Four different approaches have been used to analyse the effects of human body shape on human solar and longwave radiation exchange. The first is experimental results from a photographic method such as used in the Park and Tuller and RayMan models. The second is also experimental results from a standing person’s shadow patterns on the ground surface used in the Burt model or a mannequin as an analogue model of the human body used in the MENEX model. The third assumes the human body is a cylinder used in the COMFA and OUT_SET* models (Kenny et al. 2008; Spagnolo and de Dear 2003). The COMFA model uses a directionless cylinder, and the OUT_SET* model uses Underwood and Ward’s (1966) elliptical cylinder (major axis facing the sun). The last is a cube combined with six-direction measured radiation data (upward, downward, east, west, south and north) (VDI 1998; Huang et al. 2005; Ali-Toudert et al. 2005; Matzarakis et al. 2007; Oliveira and Andrade 2007; Thorsson et al. 2007). Huang et al. (2005) used weighting factors of 0.238 for each of the four cardinal directions and 0.024 for up- and downward. Other researchers adopted the weighting factors from VDI (1998): 0.22 for each cardinal direction and 0.06 for up- and downward based on the directionless mean projected area factors of Fanger (1972).

The four different concepts of the human body shape can create large differences among models in the body area factors used in radiation analysis. The important body area factors are the effective radiation area factor (f eff) and projected area factor (f p). Body shape and posture control the area exposed to direct beam solar radiation (projected area, A p) and the total body surface area exposed to the surrounding radiant environment rather than to other body parts (effective radiation area, A eff). f p (=A p /A eff) and f p * (=A p /A D  = f p  × f eff, where A D is the total body surface area) are used in the calculation of direct beam solar radiation. f p is used in a formula for calculating the mean radiant temperature (T mrt) in predicted mean vote (PMV, Fanger 1972) and RayMan, and f p * is used in the Burt, COMFA, MENEX and OUT_SET* models. f eff (=A eff/A D) is employed to estimate all solar and longwave exchanges.

In this study, results from the radiation components of five existing human thermal exchange models (Burt, COMFA, MENEX, OUT_SET* and RayMan models) along with the six-directional method (VDI 1998) and a new human radiation exchange model, Park and Tuller model, are compared. The latter model employs mean body area factors (f eff and f p) of standing and walking postures determined from a sample of 139 normal- and over-weight Caucasian adults (Park and Tuller 2010). Emphasis is on the effect of differences in human body area factors.

2 Materials and methods

2.1 Radiation models

Net all-wave radiation on the human body surface (Q) is the sum of absorbed total solar (shortwave) radiation (R) and net longwave (terrestrial) radiation (L) on the surface:

$$ \pm Q = R\pm L\left( {\text{watts per square metre}} \right) $$
(1)

2.1.1 Absorbed solar radiation

The units of all radiation streams and used in all models in this study are watts per square metre.

Park and Tuller model

$$ \begin{array}{*{20}{c}} {R = K_{\text{b}}^{*} + K_{\text{d}}^{*} + K_{\text{r}}^{*}} \\ { = \left\{ {f{\text{p}} \cdot f{\text{eff}} \cdot \frac{{K_{\text{b}}}}{{\sin \beta }}t + \frac{1}{2}f{\text{eff}}\left[ {K_{\text{d}} \cdot \psi_{\text{sky}} + \left( {K_{\text{b}} \cdot \cos Z_{\text{sl}} \cdot t + K_{\text{d}} \cdot \psi_{{{\text{w\_sky}}}}} \right)a_{\text{o}} + \left( {K_{\text{b}} \cdot t + K_{\text{d}} \cdot \psi sky + K_{\text{ro}}} \right)a_{\text{g}}} \right]} \right\}\left( {1 - a_{\text{b}}} \right)} \\ { = \left\{ {f_{\text{p}}^{{*}}\frac{{K_{\text{b}}}}{{\sin \beta }}t + \frac{1}{2}f{\text{eff}}\left[ {K_{\text{d}} \cdot \psi_{\text{sky}} + K_{\text{ro}} + K_{\text{rg}}} \right]} \right\}\left( {1 - a_{\text{b}}} \right)} \\ \end{array} $$
(2)

fp* is fp × feff. fp can be calculated with a formula, \( f{\text{p}} = 3.34 \cdot {10^{{ - 7}}}{\beta^3} - 6.60 \cdot {10^{{ - 5}}}{\beta^2} + 8.42 \cdot {10^{{ - 4}}}\beta + 0.297 \), from Park and Tuller (2010) for the mean of directionless standing and walking postures. The formula yields fp within 0.002 of measured fp values. t is transmissivity of objects for direct beam solar radiation between the specific object of interest and the sun. Zsl is the angle between the perpendicular to the object surface and the sun. ψw_sky is the view factor of open sky seen from the object surface. The mean value of feff for standing and walking postures from the sample of Caucasian male and female adults in Canada, 0.836, was used in this model (Park and Tuller 2010).

RayMan model

The equation is assumed to be same as the Park and Tuller model’s except for values of f p and f eff. f p can be calculated with the formula, \( f{\text{p}} = 0.308\cos \left[ {\beta \left( {0.998 - {\beta^2}/50000} \right)} \right] \) (Jendritzky et al. 1990), which was derived from Fanger’s (1972) directionless f p values for standing posture. The formula results are within 0.004 of Fanger’s (1972) measured data. Fanger’s f eff result for standing posture, 0.725, is used in this model (Matzarakis et al. 2009).

MENEX model

$$ R = \left[ {f_p^{*} \cdot K_{\text{b}} + \left( {K_{\text{d}} + K_{\text{r}}} \right)\left( {0.0018 + 0.0462\ln \beta } \right)} \right]\left( {1 - a_{\text{b}}} \right) $$
(3)

This model does not use an f eff. f p * can be calculated from:

$$ \begin{array}{*{20}{c}} {if\beta \leqslant 5^\circ, f_{\text{p}}^{*} = 1.4{e^{{( - 0.51 + 0.368\beta )}}}} \hfill \\ {if\beta > 5^\circ, f_{\text{p}}^{*} = \frac{{26.34}}{\beta } - 0.329} \hfill \\ \end{array} $$
(4)

OUT_SET* model

$$ R = \left\{ {f_{\text{p}}^{*}\frac{{K_{\text{b}}}}{{\sin \beta }} + f{\text{eff}}\left[ {K_{\text{d}} + \left( {K_{\text{b}} + K_{\text{d}}} \right)a_{\text{g}}} \right]} \right\}\left( {1 - a_{\text{b}}} \right) $$
(5)

fp* can be estimated from an equation of Underwood and Ward’s (1966) elliptical cylinder model (orientation: major axis facing the sun):

$$ f_{\text{p}}^{{*}} = \left( {0.42\cos \beta + 0.043\sin \beta } \right) $$
(6)

feff is 0.75 (Jendritzky and Nübler 1981). This model does not have the ½ function in estimating the Kd and Kr components. The ½ function accounts for the fact that a unit area of the vertical body surface is exposed to only ½ of the upper and lower hemispheres. Therefore, the unit area receives only ½ of the measured diffuse beam and reflected radiation on/from a horizontal surface.

Burt model

$$ R = \left\{ {f_{\text{p}}^{{*}} \cdot K_{\text{b}} + \frac{1}{2}f{\text{eff}}\left[ {K_{\text{d}} + \left( {K_{\text{b}} + K_{\text{d}}} \right)a_{\text{g}}} \right]} \right\}\left( {1 - a_{\text{b}}} \right) $$
(7)

feff is 0.725 from Fanger (1972). The fp* equation is originally from Terjung and Louie (1971):

$$ f_{\text{p}}^{{*}} = 4.278\exp \left( { - 0.0512\beta } \right) $$
(8)

COMFA model

$$ R = \left\{ {f_{\text{p}}^{*} \cdot K_{\text{b}} \cdot t + \frac{1}{2}f{\text{eff}}\left[ {K_{\text{d}} + \left( {K_{\text{b}} \cdot \cos Z_{\text{sl}} \cdot t + K_{\text{d}} \cdot \psi_{{{\text{w\_sky}}}}} \right)a_{\text{o}} + \left( {K_{\text{b}} \cdot t + K_{\text{d}} \cdot \psi_{\text{sky}} + K_{\text{ro}}} \right)a_{\text{g}}} \right]} \right\}\left( {1 - a_{\text{b}}} \right) $$
(9)

This model did not originally use half the quantity of diffuse and reflected solar radiation. This was included by Kenny et al. (2008). feff is 0.78 for medium body build erect (standing) posture from Kerslake (1972) (originally from Guibert and Taylor 1952). fp* is estimated from a formula:

$$f^{*}_{p} = \frac{{1/\tan \beta }}{\pi }$$
(10)

2.1.2 Net longwave radiation

All models except the COMFA and Park and Tuller models assume an unobstructed upper (sky) hemisphere, which means ψsky is 1.0 (100%). Therefore, there is no longwave radiation emitted from vertical obstructions such as building and tree components (L o). All models were modified to include this term by incorporating the components of ψsky and 1 – ψsky in the model comparison.

Park and Tuller model

$$ \begin{array}{*{20}{c}} {L = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left[ {\varepsilon_{\text{sky}} \cdot \sigma {{\left( {T_{\text{a}} + 273} \right)}^4}\psi_{\text{sky}} + \varepsilon_{\text{o}} \cdot \sigma {{\left( {T_{\text{o}} + 273} \right)}^4}\left( {1 - \psi_{\text{sky}}} \right) + \varepsilon_{\text{g}} \cdot \sigma {{\left( {T_{\text{g}} + 273} \right)}^4}} \right] - f{\text{eff}} \cdot \varepsilon_{\text{b}} \cdot \sigma {{\left( {T_{\text{b}} + 273} \right)}^4}} \\ { = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left( {L_{\text{a}} + L_{\text{o}} + L_{\text{g}}} \right) - f{\text{eff}} \cdot L_{\text{b}}} \\ \end{array} $$
(11)

feff is 0.836.

MENEX model

$$ \begin{array}{*{20}{c}} {L = \frac{1}{2}\varepsilon_{\text{b}}\left[ {\varepsilon_{\text{a}} \cdot \sigma {{\left( {T_{\text{a}} + 273} \right)}^4}\left( {0.82 - 0.25 \cdot {{10}^{{ - 0.094e_{\text{a}}}}}} \right)\psi sky + \varepsilon o \cdot \sigma {{\left( {To + 273} \right)}^4}\left( {1 - \psi_{\text{sky}}} \right) + \varepsilon_{\text{g}} \cdot \sigma {{\left( {T_{\text{g}} + 273} \right)}^4}} \right] - \varepsilon_{\text{b}} \cdot \sigma {{\left( {T_{\text{b}} + 273} \right)}^4}} \\ { = \frac{1}{2}\varepsilon_{\text{b}}\left[ {\varepsilon_{\text{a}} \cdot \sigma {{\left( {T_{\text{a}} + 273} \right)}^4}\left( {0.82 - 0.25 \cdot {{10}^{{ - 0.094e_{\text{a}}}}}} \right)\psi_{\text{sky}} + L_{\text{o}} + L_{\text{g}}} \right] - L_{\text{b}}} \\ \end{array} $$
(12)
$$ e_{\text{a}} = {\text{RH}} \times \exp \left( {18.956 - \frac{{4030.18}}{{T_{\text{a}} + 235}}} \right) $$
(13)

This model adopted the equation of sky emissivity \( \left[ {\varepsilon_{\text{sky}} = \varepsilon_{\text{a}}\left( {0.82 - 0.25 \cdot {{10}^{{ - 0.094e_{\text{a}}}}}} \right)} \right] \) from Geiger (1965). e a is calculated with Antoine’s equation (Parsons 1993). This model does not use an f eff. 0.97 is used for ε a.

RayMan model

This model uses the same Geiger (1965) sky emissivity equation as the MENEX model (Matzarakis et al. 2009). However, to facilitate the comparison of the effects of Fanger’s (1972) f eff value with those used in other models, the longwave radiation formula applied in this paper is assumed to be the same as Park and Tuller’s except the value of f eff, 0.725.

OUT_SET* model

$$ \begin{array}{*{20}{c}} {L = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left\{ {\sigma \left[ {0.7 + 5.95 \cdot {{10}^{{ - 5}}}e_{\text{a}}\exp \left( {\frac{{1500}}{{T_{\text{a}} + 273}}} \right)} \right]{{\left( {T_{\text{a}} + 273} \right)}^4}\psi_{\text{sky}} + \varepsilon_{\text{o}} \cdot \sigma {{\left( {T_{\text{o}} + 273} \right)}^4}\left( {1 - \psi_{\text{sky}}} \right) + \varepsilon_{\text{g}} \cdot \sigma {{\left( {T_{\text{g}} + 273} \right)}^4}} \right\} - f{\text{eff}} \cdot \varepsilon_{\text{b}} \cdot \sigma {{\left( {T_{\text{b}} + 273} \right)}^4}} \\ { = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left\{ {\sigma \left[ {0.7 + 5.95 \cdot {{10}^{{ - 5}}}e_{\text{a}}\exp \left( {\frac{{1500}}{{T_{\text{a}} + 273}}} \right)} \right]{{\left( {T_{\text{a}} + 273} \right)}^4}\psi_{\text{sky}} + L_{\text{o}} + L_{\text{g}}} \right\} - f{\text{eff}} \cdot L_{\text{b}}} \\ \end{array} $$
(14)

The equation for calculating sky emissivity \( \left[ {\varepsilon_{\text{sky}} = 0.7 + 5.95 \cdot {{10}^{{ - 5}}}e_{\text{a}}\exp \left( {\frac{{1500}}{{T_{\text{a}} + 273}}} \right)} \right] \) is from Idso (1981). feff is 0.75.

Burt model

$$ \begin{array}{*{20}{c}} {L = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left[ {\varepsilon_{\text{a}} \cdot \sigma {{\left( {T_{\text{a}} + 273} \right)}^4}\left( {0.55 + 0.065e{{_{\text{a}}}^{{1/2}}}} \right)\psi_{\text{sky}} + \varepsilon_{\text{o}} \cdot \sigma {{\left( {T_{\text{o}} + 273} \right)}^4}\left( {1 - \psi_{\text{sky}}} \right) + \varepsilon_{\text{g}} \cdot \sigma {{\left( {T_{\text{g}} + 273} \right)}^4}} \right] - f{\text{eff}} \cdot \varepsilon_{\text{b}} \cdot \sigma {{\left( {T_{\text{b}} + 273} \right)}^4}} \\ { = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left[ {\varepsilon_{\text{a}} \cdot \sigma {{\left( {T_{\text{a}} + 273} \right)}^4}\left( {0.55 + 0.065e{{_{\text{a}}}^{{1/2}}}} \right)\psi_{\text{sky}} + L_{\text{o}} + L_{\text{g}}} \right] - f{\text{eff}} \cdot L_{\text{b}}} \\ \end{array} $$
(15)

The equation for calculating sky emissivity \( \left[ {\varepsilon_{\text{sky}} = \varepsilon_{\text{a}}\left( {0.55 + 0.065e{{_{\text{a}}}^{{1/2}}}} \right)} \right] \) is originally from Brunt (1932) and evaluated by Iziomon et al. (2003) converting the units of e a from millimetres of mercury to hectopascals. f eff is 0.725. ε a in this model is 0.99.

COMFA model

$$ \begin{array}{*{20}{c}} {L = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left\{ {\left[ {1.2\varepsilon_{\text{a}} \cdot \sigma {{\left( {T_{\text{a}} + 273} \right)}^4} - 171} \right]\psi_{\text{sky}} + \varepsilon_{\text{o}} \cdot \sigma {{\left( {T_{\text{o}} + 273} \right)}^4}\left( {1 - \psi_{\text{sky}}} \right) + \varepsilon_{\text{g}} \cdot \sigma {{\left( {T_{\text{g}} + 273} \right)}^4}} \right\} - f{\text{eff}} \cdot \varepsilon_{\text{b}} \cdot \sigma {{\left( {T_{\text{b}} + 273} \right)}^4}} \\ { = \frac{1}{2}f{\text{eff}} \cdot \varepsilon_{\text{b}}\left\{ {\left[ {1.2\varepsilon_{\text{a}} \cdot \sigma {{\left( {T_{\text{a}} + 273} \right)}^4} - 171} \right]\psi_{\text{sky}} + L_{\text{o}} + L_{\text{g}}} \right\} - f{\text{eff}} \cdot L_{\text{b}}} \\ \end{array} $$
(16)

The regression equation for estimating longwave radiation from the clear sky as a function of air temperature is from Swinbank (1963). f eff is 0.78. ε a is 0.98.

2.2 Radiation data

To compare the existing radiation models, clear summer data collected three times on August 10, 2002 were used: in the morning (0730–0900 hours), around noon (1130–1300 hours) and in the afternoon (1530–1700 hours). The research site was Winegard Walk at the University of Guelph, Guelph, Ontario, Canada (Fig. 1). The latitude and longitude of the site are N43°32′, W80°14′. The mean ground elevation is 345 m above sea level. For this study, only the data observed at sunny locations were used as reference data for the radiation analysis: 4 locations (no. 1, 8, 9, 11) in the morning, all 13 locations around noon and 8 locations (no. 1, 3, 5, 8, 9, 10, 11, 13) in the afternoon (Fig. 2 and Table 1).

Fig. 1
figure 1

The view of Winegard Walk and the 13 observation locations

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Fig. 2
figure 2

Observed incoming solar and longwave radiation incident on a horizontal surface

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Table 1 Collected climatic and radiation data on August 10, 2002 in Guelph, Ontario, Canada
Full size table

A Kipp & Zonen CNR1 net radiometer was used to collect both shortwave and longwave radiation. This instrument has two CM 3 pyranometers and two CG 3 pyrgeometers, one of each mounted to face in opposite directions (e.g. up- and downward). Each pyranometer has a 180° field of view, and both of them will cover the entire sphere. Each pyrgeometer’s field of view is only 150°, so both of them will cover 300° not the entire sphere. However, the differences of collected longwave radiation that can be created by the short field of view are negligible, e.g. 5.4 W m−2 when total incoming longwave radiation from an entire hemisphere is 400 W m−2 if the entire hemisphere is assumed as an isothermal environment. Radiation data were collected from six directions (up- and downward and the four cardinal directions) for 1 min for each two directions at the height of 1.2 m and saved every 5 s on a 21x datalogger, made by Campbell Scientific Corporation. To collect air temperature, a Mannix Model CMM880 digital thermo-hygrometer was used. Humidity data were measured at the nearby Guelph Turfgrass Institute. For more detail, see Park (2003).

To simply compare the models, the effect of clothing was not included in the calculation. The albedoes of the body surface (a b), objects in the sky hemisphere (a o) and ground (a g) and emissivity of the body surface (ε b) were set to 0.3, 0.2, 0.3 and 0.97, respectively. These values were frequently used in previous modelling studies. The same values of incoming direct beam (K b), diffuse beam (K d), reflected (K r) by objects (buildings, trees and other structures) in the sky hemisphere (K ro) and by the ground (K rg) solar radiation, and longwave radiation coming from objects in the sky hemisphere (L o) and ground (L g) derived from the measured data were directly used in all radiation models (Table 1 and Fig. 2). The Park and Tuller, RayMan and six-directional models employed measured longwave radiation from the sky (L a). L a values computed from air temperature and humidity via Eqs. 14 through 18 were used in the other four models. Gagge et al. (1969) found that human skin temperature (T b) was between 27°C and 36.5°C in steady-state conditions. Within this range of T b, the greatest difference in longwave radiation emitted by the body surface (L b) between the greatest and lowest f eff compared in this study, 0.836 from the Park and Tuller model and 0.725 from Fanger (1972), was only 6 W m−2. Thus, a T b of 31°C was assumed for all simulations.

The amounts of solar and longwave radiation mentioned above were calculated from radiation data collected on a horizontal surface. They were not the quantities of radiation on a body surface perpendicular to the sun’s rays. The Burt and COMFA models used f p * values developed on the basic concept of shadow areas of a standing person on the horizontal ground surface (Terjung and Louie 1971, Fig. 3a). In contrast, the OUT_SET*, RayMan and Park and Tuller models took the perpendicular to the sun’s rays surface area concept (Fig. 3b). The perpendicular concept models should be multiplied by 1/cos(90 − β) = 1/sinβ to facilitate direct application of direct beam solar radiation measured on a horizontal surface. β is solar altitude (elevation).

Fig. 3
figure 3

Two different approaches to obtain projected areas: a shadow area on the horizontal ground surface (Burt and COMFA models), b shadow area on the surface perpendicular to sun’s rays (RayMan, OUT_SET* and Park and Tuller model) (created using VectorWorks 2008)

Full size image

All models except the Park and Tuller and COMFA models are for unobstructed horizons. They do not have components for the open sky view factor (ψsky) and reflected solar radiation from objects in the sky hemisphere (K ro). Therefore, fixed values of diffuse and reflected solar radiation with sky view factors were used in the solar radiation calculation of all models:

$$ K_{\text{d}} \to K_{\text{d}} \cdot \psi_{\text{sky}} $$
(17)
$$ \left( {K_{\text{b}} + K_{\text{d}}} \right)a_{\text{g}} \to \left( {K_{\text{b}} \cdot \cos Z_{\text{sl}} \cdot t + K_{\text{d}} \cdot \psi_{{{\text{w\_sky}}}}} \right)a_{\text{o}} + \left( {K_{\text{b}} \cdot t + K_{\text{d}} \cdot \psi_{\text{sky}} + K_{\text{ro}}} \right)a_{\text{g}} = K_{\text{ro}} + K_{\text{rg}} = K_{\text{r}} $$
(18)

The quantities of absorbed solar and net longwave radiation on the body surface area were compared among the models. The differences among them would be caused by their different adopted concepts of body shapes which resulted in various body area factors. They also use different methods of determining longwave radiation from the clear sky.

3 Results and discussion

3.1 Comparison of projected area factors (\( f_{\text{p}}^{*} \))

Variations in the key factor in the analysis of direct beam solar radiation among models, f p *, are presented in Fig. 4. The horizontal shadow areas used in the Burt, COMFA and MENEX models were multiplied by sinβ to convert them to surface areas perpendicular to the sun’s rays, equivalent to the projected area factors used in the other models.

Fig. 4
figure 4

Various comparisons of projected area factors (f p *). a Comparison of projected area factors (f p *), b comparison of f p * of Park and Tuller, RayMan and COMFA models and c comparison of f p * of Park and Tuller, RayMan and OUT_SET*

Full size image

The differences in f p * between the Park and Tuller model and Fanger’s results used in the RayMan model are small, around 0.03 when β is less than 65°, and decrease as β approaches 90° (Fig. 4a). Next closest results to the Park and Tuller model were from the COMFA model which used a cylindrical body model. The maximum difference between them was 0.07 at β = 0°. The difference decreased continually until β = 63° and increased again up to 0.068 at β = 90° (Fig. 4b). The OUT_SET* model used the results of Underwood and Ward’s (1966) photographic method, but only considered the anterior/posterior of the body facing the sun, not the body sides (see Kerslake 1972). This overestimated f p * compared with a body exposed to the sun from a variety of directions. This creates more differences with the other photographic methods such as the Park and Tuller and RayMan models (Fig. 4c). The MENEX and Burt models had the highest values at β = 5° and 20°, respectively, and crossing points with the Park and Tuller model around β = 42° and 65°. Differences between f p * are greatest at low solar altitudes (Fig. 4a).

3.2 Absorbed radiation comparison

All five radiation estimating models and one experimental model, the six-directional method, were compared with the Park and Tuller model in absorbed solar, net longwave and net all-wave radiation because the Park and Tuller model was recently developed from a larger sample of people than previous studies (Park and Tuller 2010), and its results were close to the median of a group of models whose results were relatively close, i.e. six-directional method, COMFA, RayMan and Park and Tuller (Fig. 6).

3.2.1 Absorbed direct beam solar radiation (\( K_{\text{b}}^{*} \))

In this study, β was 23.0–35.9° in the morning (0734–0845 hours), 60.1–62.4° around noon (1133–1244 hours) and 41.7–29.7° in the afternoon (1533–1642 hours).

Four of the six models (Park and Tuller, RayMan, COMFA and MENEX) form a group where values of absorbed direct beam solar radiation (K b *) are all relatively close, within 50 W m−2 (Fig. 5). The Burt and OUT_SET* model results were above those of this model group. The Burt model deviated in the morning and afternoon and the OUT_SET* model at all times of the day. This mirrors the patterns of the projected area factor f p *. The Terjung and Louie (1971) method used in the Burt model had the highest f p * values of all models. This occurred at solar altitudes around 20°. Thus, the Burt model deviates from other models in f p * and K b * at times and observation sites with solar altitudes near this value (Figs. 4 and 5). Its estimated K b * was up to 151 W m−2 greater than that of the Park and Tuller model in the morning. The OUT_SET* model had a relatively high f p * value compared with most other models at all solar altitudes less than 80°. The MENEX model was within the group in the morning and afternoon except for station 1 in the morning (Morn_1) and had a difference of only around 30 W m−2 with the group’s median around noon (Fig. 5).

Fig. 5
figure 5

Comparison of absorbed direct beam solar radiation on the body surface in the morning, around noon and in the afternoon

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The closest results with the Park and Tuller model’s were those of the RayMan model which had values that were 15–20 W m−2 lower in the morning and afternoon and 16 W m−2 lower around noon (Fig. 5). COMFA model results were very close to those of the Park and Tuller model around noon, within 1–4 W m−2.

3.2.2 Absorbed total solar radiation (R)

The OUT_SET* model had the greatest quantity of absorbed total solar radiation among all models for the day as a whole (Fig. 6a). It has a relatively high projected area factor and direct beam solar radiation as noted above. It is one of the models where diffuse beam and reflected solar radiation measured on a horizontal surface were not reduced by one half, producing greater quantities of these two radiation streams than most of the other models.

Fig. 6
figure 6

Comparison of a absorbed total solar radiation, b net longwave radiation, c net all-wave radiation and d differences in absorbed and emitted radiation between the Park and Tuller model and the other models

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The MENEX model also had relatively high values of absorbed total solar radiation throughout the day (Fig. 6a). Its estimated direct beam solar radiation is close to that of most other models during the morning and afternoon and the lowest of all models at noon (Fig. 5). However, it differs from most other models in the lack of the ½ function correction for diffuse sources of radiation on/from a horizontal surface used as input in this study. It also does not apply an effective radiation area factor (f eff) to these diffuse sources of radiation. These two factors combine to give the MENEX model the greatest diffuse beam and reflected solar radiation absorbed on the human body surface (Fig. 6d). The current version of MENEX has reduced the importance of diffuse radiation (http://www.igipz.pan.pl/geoekoklimat/blaz/MENEX_2005.pdf).

The COMFA model had the closest values to the Park and Tuller model around noon, about 4 W m−2 differences. It had 14–27 W m−2 more absorbed total solar radiation in the morning and afternoon. The Burt model also had close values to those of the Park and Tuller model around noon but had high values in the morning and afternoon close to those of the MENEX model because of its high f p * values. The six-directional method continually yielded 8–44 W m−2 more than the Park and Tuller model’s values. The RayMan model has a similar f p (=A p /A eff) as the Park and Tuller model, but f eff of the Park and Tuller model is 0.836 which is higher than RayMan’s 0.725. Therefore, the RayMan f p * (=A p /A D = f p × f eff) values were around 0.03 lower all day than the Park and Tuller model’s. The lower RayMan f p * and f eff values created somewhat lower absorbed direct beam, diffuse beam and reflected solar radiation than the Park and Tuller model’s (Fig. 6d). Therefore, its absorbed total solar radiation was 23–30 W m−2 lower.

3.2.3 Net longwave radiation (L)

The range between model estimates of net longwave radiation was not great, 31 W m−2 between the OUT_SET* and Park and Tuller models in the morning and 23 W m−2 between the MENEX and Park and Tuller models around noon and in the afternoon (Fig. 6b). The effective radiation area factor (f eff) values and methods determining incoming longwave radiation from the open sky were the variables creating differences between models in this study. Differences in employed f eff affect both the gain of absorbed and the loss of emitted longwave radiation. One offsets the other. The difference in net longwave radiation between models with unequal f eff values will depend on the difference between incoming and emitted longwave radiation. The differences between these two radiation streams were not great in the very warm, sunny and summer conditions during our observations. The maximum daily mean difference in open sky longwave radiation (L a) between models was only 19 W m−2 (Fig. 7). These two factors combined to limit the magnitude of net longwave radiation differences between models.

Fig. 7
figure 7

Comparison between measured incoming longwave radiation from the open sky to the horizontal ground surface (L a) and computed L a results from the Burt, COMFA, MENEX and OUT_SET* models

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Computed open sky incoming longwave radiation exceeded the measured value for all models that employed the computed values, OUT_SET*, COMFA, Burt and MENEX (Fig. 6b). The OUT_SET* model’s method gave the greatest values. This combined with its relatively low f eff (0.75) which limited the magnitude of the negative net longwave radiation gave it the greatest values during the morning. A high f eff (1.0) and relatively high computed incoming sky longwave radiation gave the MENEX model the highest values during mid-day and the afternoon. The Park and Tuller model used the lower measured open sky longwave radiation values. Its relatively high f eff accentuated the magnitude of its negative net longwave radiation, and it had the lowest net longwave radiation throughout the day (Fig. 6b). The values of the Park and Tuller and RayMan models were nearly identical near noon and in the afternoon.

3.2.4 Net all-wave radiation (Q)

Most of the variations in net all-wave radiation between models resulted from absorbed solar radiation. The greatest difference overall between the Park and Tuller model and the other models in net all-wave radiation was with the OUT_SET* model (Fig. 6c). The mean difference over the entire day was +146 W m−2, composed of the OUT_SET* model’s 128 and 18 W m−2 greater absorbed total solar and net longwave radiation, respectively (Fig. 6a, b). The second greatest difference was a 122 W m−2 greater mean for the MENEX model. MENEX and OUT_SET* were the two models with the greatest absorbed total solar radiation for the day as a whole. The MENEX model also had the greatest net longwave radiation. The six-directional method had 36 W m−2 greater net all-wave radiation than the Park and Tuller model, 29 and 7 W m−2 more absorbed total solar and net longwave radiation, respectively.

The COMFA model had close results with the Park and Tuller model around noon. This model had larger differences at the lower solar altitudes (39, 10 and 27 W m−2 in the morning, noon and afternoon). The RayMan model had the opposite phenomenon to the COMFA model. Its net all-wave radiation difference with the Park and Tuller model was relatively constant throughout the day, 15, 27 and 23 W m−2 lower in the morning, noon and afternoon with the greater differences coming at the higher solar altitudes. The Burt model had the closest results with the Park and Tuller model around noon, but much larger differences in the morning (+147 W m−2) and afternoon (+94 W m−2) due to the greater gaps in absorbed direct beam solar radiation. Mean differences by time presented above between the Park and Tuller model and the other models are shown in Fig. 6d.

3.3 Correlation among the models

Pearson’s product–moment correlation coefficients between models were determined for absorbed total solar, net longwave and net all-wave radiation (Tables 2 and 3). The inputs were the 25 time–site values from the models.

Table 2 The correlation among the models for absorbed total shortwave (top half) and net longwave (bottom half) radiation
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Table 3 The correlation among the models for net all-wave radiation
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Time and site differences in longwave radiation were determined by the variations in measured quantities of longwave radiation, air temperature and humidity depending on the model. The major control of differences in the computed open sky incoming longwave radiation was air temperature. The time–site correlation between air temperature and the measured amount of incoming longwave radiation from the upper hemisphere was 0.963. Hence, the time–site variations in longwave radiation were similar for all models producing very significant correlations in human body net longwave radiation, all over 0.95 and many over 0.99 (Table 2).

Correlations for absorbed total solar radiation were determined mainly by variation patterns of the projected area factor (f p *) with solar altitude. Models whose graphs of f p * with solar altitude have similar shape through the range of solar altitudes encountered in the field observations (23° to 63°) will have high correlation coefficients (Fig. 4, Table 2). Those whose shapes differ will have low correlations. Correlation coefficients between the Park and Tuller, RayMan and OUT_SET* models were all very significant (over 0.994). The Burt and MENEX models whose curves have unique shapes have the lowest correlations with other models.

The correlations for net all-wave radiation were also very significant among the Park and Tuller, RayMan and OUT_SET* models even though all models except the Burt model had correlations over 0.90 (Table 3). Only these three models had high correlations for both absorbed total solar and net longwave radiation. The strongest correlation was between the Park and Tuller and RayMan models, r = 0.999, as expected. The correlations between the Park and Tuller and OUT_SET* models and between the RayMan and OUT_SET* models were r = 0.953 and r = 0.947, respectively (Table 3).

4 Conclusions

The radiation components of five existing human thermal exchange models (Burt, COMFA, MENEX, OUT_SET* and RayMan models), one experimental model [six-directional method from VDI (1998)] and the new Park and Tuller model were compared.

Results of all models were referenced against those of the Park and Tuller model. This is not to suggest this model is the most appropriate for all applications. It is the one that employs the most recently derived human body radiation area factors. It puts the comparisons in the time perspective. Direct comparison with this model eases the identification of differences that might arise when using older, widely employed or more recent methods. The Park and Tuller radiation area factors were developed from a larger sample of actual adults and used up-to-date video and computer technology compared with most of the older methods. However, it has not undergone the testing of extensive applications experienced by the older methods.

Absolute net longwave radiation differences between models were not great (maximum difference, 31 W m−2). Although effective radiation area factors (f eff) had a wide range from 0.725 to 1.0, the magnitude of the differences between human body absorbed and emitted longwave radiation was also relatively small. This and the limited range between incoming longwave radiation from the sky used in the models (measured and computed) limited the magnitude of model differences in net longwave radiation. The situation can be different at other times and in other environments, e.g. very cold or hot radiant environments where environmental radiant temperature differs markedly from human surface/clothing temperature. In these types of environments, inappropriate f eff values can lead to larger errors in estimated human net longwave radiation.

All four methods of computing incoming longwave radiation from the clear sky overestimated the measured value. The Idso (1981) method used in the OUT_SET* model yielded the highest value followed by the Swinbank (1963) equation (COMFA), Brunt (1932) method (Burt model) and that found in Geiger (1965) used by MENEX. However, the differences between computation methods were small, 7.5 W m−2 for the day as a whole. The variation between the mean from all four computation methods and measured open sky longwave radiation was 14 W m−2. Measured longwave radiation data are seldom available, and estimates must be made for many applied climate studies. In most cases, the differences we found between measured and computed open sky longwave radiation and between those estimated by different formulas will not have a major impact on human net all-wave or total thermal exchange evaluations.

A key variable in model absorbed solar radiation differences is the projected area factor (f p *). The empirical MENEX and Burt models’ factors had a great deal of variation with solar altitude compared with the other models whose curves have a much lower slope (Fig. 4). Human body direct beam solar radiation estimated using these two models is more sensitive to variations in solar altitude. Moreover, the Burt model has a crossing point of its f p * line with Park and Tuller model’s f p * line near the noon solar altitude in this study so that the model had results close to those of the Park and Tuller model at noon. However, the model can produce very different noon results at other seasons and latitudes that have different solar altitudes.

The OUT_SET* model’s elliptical cylinder body model has higher f p * values than most of the other models at solar altitudes less than 80°. The major axis of the ellipse facing the sun maximizes the projected area factor compared with models that incorporate the sun’s rays intersecting all sides of the body. E.g. OUT_SET*’s f p * is between 0.07 and 0.10 greater than that of the non-elliptical cylinder used in the COMFA model (Fig. 4). This creates relatively high absorbed direct beam and total solar radiation (Figs. 5 and 6).

The RayMan model has f p * values very similar to those of the Park and Tuller model, 0.035–0.025 lower, which produced about 27 W m−2 lower absorbed total solar radiation during the daytime.

Both effective radiation area and whether or not diffuse sources of radiation measured on or from a horizontal surface are reduced by one half to account for a unit area of the vertical body surface’s exposure to only half the entire horizontal surface were important controls of model differences in estimated human body diffuse beam and reflected-from-the-ground solar radiation. The MENEX model had the highest total of human diffuse beam and reflected solar radiation simply because of its implied 1.0 f eff and no ½ function (Fig. 6d). The ½ function is quantitatively the more important than most differences in f eff. Next highest was the OUT_SET* model which had a relatively low f eff value of 0.75 but did not have the ½ function. Studies need to be very clear about the surfaces on which diffuse radiation components used as model input data are measured or estimated. The ½ function applies to input data on a horizontal surface but not to direct measurement or estimation on a vertical surface.

Human radiation exchange has widespread application and can aid in the planning of more comfortable and healthy environments for people. Several human radiation exchange models are available employing a variety of different assumptions. This study has compared a selection of these models. Several radiation streams were derived from measured data and were the same for all models. Therefore, model differences were created by the input of longwave radiation from the clear sky and human body projected area (f p) and effective radiation area (f eff) factors used in each model. Values of these human radiation area factors were determined via different methods including using samples of people or mannequins of different age, size and shape or objects such as cylinders. Model differences in human net all-wave radiation were up to about 150 W m−2 which can have an important effect on thermal sensation.

It is suggested that those estimating human radiation exchange study a variety of models and take care to select algorithms and human radiation area factors that are most applicable to the subjects of their study. We also encourage more direct comparison of available models in a variety of different environments. These types of studies will help those wishing to estimate human radiation exchange select appropriate models and methods.