Introduction

Radiation exchange plays a significant role in the human energy budget. Steadman (1971) estimated that the apparent temperature was raised by nearly 14°C under calm conditions and by 7°C in a strong wind by the effect of solar radiation. Hodder and Parsons (2007) stated that each increase of direct beam solar radiation of around 200 Wm−2 increased predicted mean vote (PMV; Fanger 1972) by one sensation scale unit. Also, radiation exchange was revealed as the largest component of total body energy loss for people in a normal environment (Landsberg 1969) and on clear winter nights in New Zealand (Tuller 1980).

Human receipt and emission of radiation are affected by body shape, posture and clothing. Body shape and posture control the body surface area exposed to direct beam solar or other point-source radiation (projected area, A p) and the proportion of the total body surface area exposed to the surrounding radiant environment rather than to other body parts (effective radiation area, A eff). Often, factors that represent proportions of the body surface area are utilized. The projected area factor is A p /A eff (f p) or A p /A D \( \left( {f_p^{ * }} \right) \) when A D is total body surface area. The effective radiation area factor (f eff) is A eff /A D.

Many human thermal exchange models have employed f eff and/or \( {f_p}\left( {f_p^{ * }} \right) \) directly [e.g. Burt model (Burt 1979, modified by Tuller 1990), COMFA (comfort formula; Brown and Gillespie 1986, 1995), MENEX (Man-environment heat exchange; Blazejczyk 1994, 2004, 2005), OUT_SET* (Pickup and de Dear 2000), RayMan (Matzarakis et al. 2000, 2007, 2009), PT (perceived temperature; http://www.utci.org/isb/documents/perceived_temperature.pdf)] or mean radiant temperature (T mrt), which uses f eff and f p in its formula [e.g., PMV, PET (physiological equivalent temperature; Höppe 1999), UTCI (universal thermal climate index; http://www.utci.org)].

A number of studies estimating human body radiation area factors are available. However, each of these employs somewhat different methods, subjects and sample sizes. No study has \( f_p^{ * } \) or f eff values that are applicable to the entire population. Our purpose is to extend the investigation of human body radiation area factors to body shapes and postures that have received only limited attention. We begin with a review of the methods and subjects of major human body radiation area studies.

Clothing styles, color and insulation values are quite variable and will not be addressed in this study. The effect of clothing can be considered by utilizing clothing area factors dependent on various clothing types and ensembles employing information found in sources such as McCullough et al. (1985, 1989), ASHRAE (1997) and ISO9920 (2007).

The age composition of the population and human body shape in developed countries are changing. There is increasing concern about implications of the trend toward more overweight people. The mean contemporary Canadian adult population is already considered to belong to the overweight body mass index (BMI) category (CHS 1978; CHHS 1992; CCHS 2004). The population is aging. However, many reported f p and f eff values have been determined from samples of young, normal-weight adults, idealized human body shapes or cylinders.

The widely used f eff and f p results of Underwood and Ward (1966) and Fanger (1972) were determined by photographing a limited sample of standing and sitting people from a variety of angles. Steinman et al. (1988) modified Fanger’s model making it applicable to the complex enclosures found in modern architecture using mean f p values. Jones et al. (1998) applied the photographic method to a mannequin in clothed and unclothed standing postures. They studied both the whole body and individual body parts. Tanabe et al. (2000) and Kubaha et al. (2004) used three-dimensional, computerized human body models in unclothed sitting and standing postures.

Studies that have investigated the effects of gender (male or female) and body type (under-, normal-, over-weight or obese) on body area factors have usually relied on small samples. Results are not wholly consistent between studies. Bandow and Bohnenkamp (1935) used an electrical capacity technique and found that f eff slightly decreased with increasing body size for males but increased for females. However, Guibert and Taylor (1952) noted these results suffered from problems of accuracy and consistency (9% difference in f eff when the measurement was repeated). Guibert and Taylor (1952) showed 3% and 1% decreases of f eff from medium to heavy and light standing male body types, respectively. However, they studied only one male subject in each or the light, medium and heavy body type categories. Horikoshi et al. (1990) also tested only three male subjects and found a 1% difference in f eff between two under-weight subjects and no difference in f eff between under-weight and over-weight subjects. Underwood and Ward (1966) originally used 25 male and 25 female subjects 14–59 years old and found less than 1.0% difference in \( f_p^{ * } \) between genders. Hence, they focused on males only and gave no data for females. They also found less than 2.5% between body sizes (largest and smallest A D subjects). Though they compared \( f_p^{ * } \) between genders and body sizes, they did not compare \( f_p^{ * } \) between different body types. They also measured at only five different azimuth angles (0, 45, 90, 135 and 180°) and four different altitude angles (0, 30, 63 and 90°). Fanger (1972) reported no gender- and body type-related differences in f eff and no gender-, body type- and clothing-related differences in f p. His ten male and ten female college students belonged to only one body type category, the normal-weight BMI class, so his results did not confirm similarity or difference of body area factors (f eff and f p) between various body types. Therefore, the similarity or variation of body area factors among the combination of gender and body type has not yet been clearly proved.

Currently available body models consider only sitting and standing postures. In outdoor areas and many indoor situations, walking is another common posture. Standing and sitting postures have the arms and legs in consistent positions. Walking is a series with different positions of each arm and leg in relation to other parts of the body. PET and UTCI are based on an adult’s walking posture. However, no detailed study of actual walking posture has been published. Only one study, Ward and Underwood (1967), modeled one position of a walking stride among three male subjects using a photographic method. However, measured angle variation was too limited, and the results seem unrealistically high. Also, they did not define the effective radiation area factor (f eff). Roller and Goldman (1968) measured a projected area factor \( \left( {f_p^{ * }} \right) \) of 0.24 at only one solar altitude, 60°, which was 0.02 greater than standing posture. Steadman (1979) assumed 0.80 as a walking subject’s f eff and presented a formula for estimating \( f_p^{ * } \) as a function of altitude angle adding 0.02 to the mean f p values of Fanger’s (1972) standing posture. de Freitas et al. (1985) modelled moving people (runners). They estimated runners’ f eff as 0.82, from Fanger’s (1972) f p values, and \( f_p^{ * } \) values from Taylor’s (1956) formula. Their estimate of f eff has not been tested, and Taylor’s formula is not for a moving body posture but for a cylindrical standing posture (Pugh and Chrenko 1966).

The purpose of this study is to fill some of the gaps in currently available human body radiation area factors, f eff and f p, and expand their application to a wider range of body shapes and postures. Many of the studies noted above were done several years ago on a very limited number of subjects. Our study sampled a relatively larger number of present-day adults. This is the first detailed investigation of walking posture to include the complete stride cycle. The study includes both genders and investigates whether there are any radiation area factor differences between normal- and over-weight people. Comparisons with body area factors given in previous studies are included. Examples of possible effects of differences in radiation area factors on modelled human absorbed solar and longwave radiation are given.

Methods

Analytical theory

According to the reciprocity theorem (Fanger 1972),

$$ {A_1}{F_{{A1 - A2}}} = {A_2}{F_{{A2 - A1}}} $$
(1)

A 1 is the effective radiation area of the human body surface (A eff), F A1-A2 is the angle factor between the person and the sphere (A 2), A 2 = 4πr 2 is the spherical surface area and F A2-A1 is the angle factor between the sphere and the person.

For a small part of the spherical surface area, dA 2, Eq. 1 will be (ASHRAE 1997),

$$ d{F_{{A1 - dA2}}} = d{F_{{A1 - \cos \beta 2dA2}}} $$
(2)
$$ {A_1}d{F_{{A1 - \cos \beta 2dA2}}} = \cos {\beta_2}d{A_2}{F_{{\cos \beta 2dA2 - A1}}} $$
(3)

Therefore, the angle factor between A 1 and A 2 would be (Oguro et al. 2001),

$$ {F_{{A1 - A2}}} = \int\limits_{{A2}} {d{F_{{A1 - dA2}}} = \int\limits_{{A2}} {d{F_{{A1 - \cos \beta 2dA2}}} = \int\limits_{{A2}} {\left( {\frac{{{F_{{\cos \beta 2d{A_2} - {A_1}}}}}}{{{A_1}}}} \right)} \cos {\beta_2}d{A_2}} } $$
(4)

From the definition of angle factor (ASHRAE 1997),

$$ {F_{{\cos \beta 2d{A_2} - {A_1}}}} = \int\limits_{{A1\,}} {\frac{{\cos {\beta_1}\cos {\beta_2}d{A_1}}}{{\pi {r^2}}}} $$
(5)

β 1 and β 2 are incident angles between central points of dA 1 and dA 2. If the size of the body part dA 1 and the portion dA 2 is very small compared to the distance r between dA 1 and dA 2, it could be considered that cosβ 2 ≈ 1.0. Then Eq. 5 can be written as,

$$ {F_{{\cos \beta 2dA2 - A1}}} = \int\limits_{{A1}} {\frac{{\cos \beta_1dA_1}}{{\pi {r^2}}}} = \frac{{{A_p}}}{{\pi {r^2}}} $$
(6)
$$ {A_p} = \int\limits_{{A1}} {\cos \beta_1dA_1} $$
(7)

By combining Eqs. 4 and 6,

$$ {F_{{{A_1} - {A_2}}}} = \frac{1}{\pi }\int\limits_{{{A_2}}} {\frac{{{f_p}}}{{{r^2}}}\cos {\beta_2}d{A_2} = \frac{1}{\pi }} \sum\limits_{{i = 1}}^n {\frac{{{f_{{pi}}}}}{{{r^2}}}\cos {\beta_2}d{A_2}} $$
(8)

(Kubaha et al. 2003)

$$ {f_p} = \frac{{{A_p}}}{{{A_1}}} = \frac{{{A_p}}}{{{A_{{eff}}}}} $$
(9)

n is the number of equal areas dA 2 comprising the entire spherical surface area.

As the angle factor F A1-A2 should be 1.0, the effective radiation area of the body A 1 (=A eff) can be estimated from Eq. 10 by combining Eqs. 8 and 9,

$$ {A_{{eff}}} = \frac{1}{\pi }\int\limits_{{{A_2}}} {\frac{{{A_p}}}{{{r^2}}}\cos {\beta_2}d{A_2} = \frac{1}{\pi }} \sum\limits_{{i = 1}}^n {\frac{{{A_{{pi}}}}}{{{r^2}}}\cos {\beta_2}d{A_2}} $$
(10)
$$ {f_{{eff}}} = \frac{{{A_{{eff}}}}}{{{A_D}}} $$
(11)

The following sections describe the method used to obtain the information needed to compute f p and f eff from a sample of adults.

Subjects

Body data were collected for a sample of both normal- (male, n = 31; female, n = 40) and over-weight (male, n = 48; female, n = 20) adults (age 18–65 years) at Saanich Commonwealth Place recreation center in Victoria, BC, Canada (Table 1). This study was approved by the University of Victoria ethics committee. Informed consent was obtained from all subjects. The mean BMI, 24.8, of this study was at the border between the normal-weight category, 18.5–24.9, and over-weight category, 25.0–29.9.

Table 1 Subjects’ mean basic body data categorized by body mass index (BMI) class. NW_M Normal-weight male, NW_F normal-weight female, OW_M over-weight male, OW_F over-weight female. SD standard deviation
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All subjects wore swim suits (male: triangle or box style, female: one piece or bikini style). They were instructed to stand and walk naturally. Their age and height data were collected using a survey. Heights (m) were confirmed using photometric comparison with several reference heights. Weights (kg) for all subjects were measured with a digital electronic scale manufactured by Taylor (http://www.taylorusa.com) and calibrated with several reference weights. Standing posture was obtained by taking pictures with Sony Cybershot 3.2 and Nikon Coolpix 8700 cameras, and walking posture by recording videos with Sony DCR-TRV22 and Canon ZR45 camcorders. Pictures and videos of each person were taken one each from the front and side (Fig. 1a). The pictures were taken at the median height of the torso (chest and abdomen), 1.2 m, instead of using the weighting height of human body, 1.1 m, because the torso has the largest body surface area among all body parts. All photos were taken at a distance of 10 m to reduce image distortion.

Fig. 1
figure 1

The process for creating three-dimensional (3D) computer body models of standing and walking postures. a Taking pictures and videos, created in Vectorworks 2008 (Nemetschek Vectorworks®, http://www.nemetschek.net). b Creating body frames. c Front and side views of 3D computer standing models. d Positions of a stride for 3D computer walking models

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Total body surface area

Two methods were used for determining total body surface area (A D, m2): the DuBois and DuBois (1916) formula (A Du) and 3DS Max 9.0 software (A 3DS; Autodesk®, http://www.autodesk.com). The DuBois and DuBois (1916) formula overestimated A D in both male and female models compared with the 3DS Max method. The overestimation was less than 1% in male models but reached 7.4% in female models. Also, the overestimation in overweight models was almost twice that in normal weight models (Table 1).

DuBois and DuBois (1916) noted their formula could produce maximum ±5% errors. Many other researchers have also found some errors in the formula and modified it (e.g., Mosteller 1987; Mattar 1989; Livingston and Lee 2001). Recently, 3D scanning technology was adopted in this field (Tikuisis et al. 2001; Yu et al. 2003). The formula estimates of Tikuisis et al. (2001) were within 0.3% of our A 3DS values for both of the normal- and overweight male models. The formula results of Yu et al. (2003) were within 0.9% for the normal-weight female model and 4.1% for the overweight female model. The measured values of A 3DS were used as A D in this study.

Data processing

Before analyzing collected data, image distortion was tested with reference images from pictures and videos in AutoCad 2002 (Autodesk®, http://www.autodesk.com). It was found that there was no centroid distortion (i.e., between two same size objects, an object located on the center is bigger than an object located on the edge in photographs). Only horizontal/vertical rotation correction was required.

The digital body shape images were imported into AutoCad 2002 after rotation correction using ACDSee Pro. 1.5 (ACDSee®, http://www.acdsee.com). Edge-of-body lines were digitized. Widths and lengths of important body parts (m) and the angles between them (°) were measured (e.g., width: neck, shoulder, chest, abdomen, hip; length: between neck and shoulder, between shoulder and chest, upper and lower arm, between hip and knee, between knee and ankle; angle: between shoulder and elbow, between elbow and wrist, between hip and knee, between knee and ankle). The mean values of body parts were used to make front and side body frames (Fig. 1b). Four 3D computerized standing body models (normal- and over-weight male and female models) were created in Poser 6 & 7 (SmithMicro®, http://www.smithmicro.com) using the frames (Fig. 1c). Using existing body models in the Poser program, each body part’s width and angles were adjusted with measured side and front body frames. More details on these adjustments can be found in the Poser program tutorials (http://poser.smithmicro.com/tutorials.html). The two male models consisted of 62,298 small surface elements and female models 194,206 as the female models were created in the advanced version, Poser 7. This would yield greater micro-details for females. However, our study is concerned with more general body images and thus additional micro-details are not important (see Fig. 1c, d).

The mean durations of walking strides of normal-weight male and female models were 0.62 [standard deviation (SD) = 0.07)]and 0.57 (SD = 0.05) seconds, respectively. Four images within a complete walking stride, the 1/4, 2/4, 3/4 and 4/4 positions, of normal-weight male and female models were extracted from the videos using Adobe Premiere Pro 1.5 (Adobe®, http://www.adobe.com). Angles among upper body, arms and legs were measured in AutoCad 2002. The mean angles were used to construct walking body models in Poser 7 (Fig. 1d). The 1/4 starting position of a stride had the right arm and left leg located in front of the torso in this study. The opposite walking positions (e.g., 1/4 position: the left arm and right leg in front) were assumed to have the same A p and f p values in the opposite azimuth angles. Angles of 2/4 and 4/4_Front used mean values of 1/4 and 3/4_Front because their positions were transitional between the latter two.

The 3D models were imported into AutoCad and rotated regularly as shown in Fig. 2. The more frequent the measurements, the more detailed the representation of the human body and the more accurate the determination of A p and A eff. The initial investigation started with standing posture, normal-weight male and female models. Four different angle increments; 5°, 10°, 15° and 30°; were compared to assess the effects of number of observations on estimated A eff and f eff. The number of observations are \( {36} \times {18} = {648}\ \left( {{\hbox{per}}\;{5}^\circ } \right) \), \( {18} \times {9} = {162}\ \left( {{\hbox{per}}\;{1}0^\circ } \right) \), \( {12} \times {6} = {72}\ \left( {{\hbox{per}}\;{15}^\circ } \right) \) and \( {6} \times {3} = {18}\ \left( {{\hbox{per}}\;{3}0^\circ } \right) \), respectively. Also, the 1/4 spherical surface area was taken at the mid-point of the angle measurement, e.g., for every 5° at azimuth angle (α): 2.5, 7.5······172.5, 177.5°; altitude angle (β): 2.5, 7.5······82.5, 87.5°. The number of per 5° measurements, 648, was almost 4 times the number of per 10° measurements, 162, but the f eff differences were small, 0.005. Per 15° and 30° measurements created greater differences, 0.016 and 0.067, respectively. Because of the small differences, further analyses employed measurements taken every 10°. Further testing for walking posture and over-weight subjects is recommended for further studies.

Fig. 2
figure 2

a,b Variation of azimuth (α) and altitude (β) angles for fp values. a A quarter sphere (0° ≤ α ≤ 180°, 0° ≤ β ≤ 90°) for fp values of standing posture since the body shape is symmetrical in the posture, b half a sphere (0° ≤ α ≤ 180°, −90° ≤ β ≤ 90°) for walking posture because of asymmetrical body shape (created in Vectorworks 2008; http://www.nemetschek.net)

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Rotated images were exported to Photoshop 7.0 (Adobe®, http://www.adobe.com). To keep the same scale for exporting the images, the same scale value in the zoom function was used during the entire process in each category in the AutoCad program. The pixel values (1 pixel ≈ 0.056 cm2) in Photoshop were converted to the real A p values.

Ap data were measured from only 1/4 of the entire spherical surface area (α: 0–180°, β: 0–90°) for standing posture (Fig. 2a) and from half a spherical surface area (α: 0–180°, β: -90–90°) for walking posture (Fig. 2b) in this study. Therefore, Aeff can be calculated with:

$$ {A_{{eff}}} = 4 \times \frac{1}{\pi }\sum\limits_{{i = 1}}^{{{n_{{1/4}}}}} {\frac{{{A_{{pi}}}}}{{{r^2}}}\cos {\beta_2}d{A_2}\,\,\,{\hbox{for}}\,{\hbox{standing}}\,{\hbox{posture}}} $$
(12)
$$ {A_{{eff}}} = 2 \times \frac{1}{\pi }\sum\limits_{{i = 1}}^{{{n_{{1/2}}}}} {\frac{{{A_{{pi}}}}}{{{r^2}}}\cos {\beta_2}d{A_2}\,\,\,{\hbox{for}}\,{\hbox{walking}}\,{\hbox{posture}}} $$
(13)

when Ap data are collected at a variety of α and β angle increments in 1/4 (n1/4) and 1/2 (n1/2) of the spherical surface area. β2 and r are an incident angle and distance between a human body surface and a small portion dA2 of the entire surrounding spherical surface area, respectively.

Results

Standing posture

Projected area factor

Projected area factor can be presented in two ways. The first expresses projected area as the proportion of effective radiation area, \( f_p = A_p/A_{eff} \). This is then used in a formula to find the mean radiant temperature (T mrt) in PMV (Fanger 1972) and PET (Höppe 1999). The second is the proportion of total body surface area, \( f_p^{ * } = {A_p}/{A_D} = {A_p}/{A_{{eff}}} \times {A_{{eff}}}/{A_D} = {f_p} \times {f_{{eff}}} \). This is used for calculating quantities of absorbed direct beam solar radiation or other point source radiation (e.g., infrared heater) per unit area of the entire body surface. The purpose of the use of f p and \( f_p^{ * } \) is the same, i.e., to input the effect of direct beam solar radiation or other point source radiation into human thermal exchange models.

The f p difference between normal-weight male and female models reversed around 60–90° of azimuth angle (α) (Fig. 3). Female f p values were up to 0.017 greater before α = 60–90°, and male f p values were up to 0.014 greater after. Females’ breasts and males’ more open stance between the two legs seem to create this phenomenon (see Fig. 1c). In the f p comparison of all four body type models, the maximum difference was 0.017 between normal-weight male and female models. The maximum difference between other combinations was 0.015. Fanger (1972) reported only a very small male–female difference in f p. The results of this study produce the same conclusion.

Fig. 3
figure 3

Comparison of projected area factors (f p) of normal-weight male and female models (standing posture)

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Depending on the application, we can consider two extremes of human body orientation. The first is the directionless orientation used when people are facing a variety of different directions. Body orientation is essentially random. This mode is also used in general modeling studies when we do not know the actual body orientation. The second is the directional orientation used when people face a known, consistent direction.

Directionless f p values of the four body types were very close (Table 2). When altitude angle (β) was up to 65°, the maximum difference was only 0.002. When β was 75° and 85°, more differences occurred between normal- and over-weight female models, 0.005 and 0.01 respectively. Therefore, the mean f p values can be used to represent the contemporary Caucasian adult population in Canada (from the formula for mean of standing posture in Fig. 4).

Table 2 Directionless f p values of normal- and over-weight male and female models
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Fig. 4
figure 4

Directional projected area factors (f p) dependent on solar altitude angles (β) of the mean male and female body type models and best fit equation for azimuth angles (α) between 5° and 175°

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If the exact direction of standing people is known, more precise estimation of f p can be obtained from Fig. 4. The maximum directional difference in mean f p from all body type models occurs between the front and side views of the body taken from a low β angle perpendicular to the body surface. In this study with α and β angles measured in 10° increments beginning at 5°, the maximum difference was 0.128 between α = 5° and α = 95° when β = 5°. The differences decreased with increasing β angles.

The directional projected area factors of this study were compared with those of Underwood and Ward (1966), Jones et al. (1998), Fanger (1972), Tanabe et al. (2000) and Kubaha et al. (2004). The \( f_p^{*} \) values in Jones et al. (1998) and Underwood and Ward (1966) are from the ratio A p/A D instead of A p/A eff so their \( f_p^{*} \) values are expected to be lower than their unknown f p values because A D is larger than A eff. For this comparison, adjusted \( f_p^{*} \) values of the normal-weight male model were used because the results reported by Jones et al. and Underwood and Ward came from males only. The greatest difference between this study and that of Jones et al. occurred at α = 180° and β = 0°, 0.173, and with that of Underwood and Ward at α = 0° & 180° and β = 0°, 0.149 (Fig. 5).

Fig. 5
figure 5

Comparison of directional projected area factors \( \left( {f_p^{ * }} \right) \) of the normal-weight male model for standing posture between Underwood and Ward (1966), Jones et al. (1998) and this study

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The mean directional f p value of all four body type models combined was compared with the results of Fanger (1972), Tanabe et al. (2000) and Kubaha et al. (2004). The differences with Fanger’s values were up to 0.02 at the medium altitude angle (β = 45°) and from the oblique back azimuth angle, α = 135° (Fig. 6). Values reported by Kubaha et al. were up to 0.022 different at the lowest altitude angle (β = 15°) and the lowest azimuth angle (α = 15°). The maximum difference with Tanabe et al.’s values was only 0.008. An interesting result was that the greater differences with Fanger and Tanabe et al. occurred more at the higher azimuth angles (from side to back: 90° ≤ α ≤ 180°) when β is increasing and with Kubaha et al. more at the lower azimuth angles (α < 90°) when β is increasing. Probably, the location of arms and hands and the stance width affect these phenomena. The 3D computer body models of Kubaha et al. and this study had more relaxed and more front-located arms and hands as well as a wider stance compared with thigh-side located hands and closer legs of the Fanger and Tanabe et al. models.

Fig. 6
figure 6

Comparison of directional projected area factors (f p) of standing posture between Fanger (1972), Tanabe et al. (2000), Kubaha et al. (2004) and this study

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When the directionless f p values of the four studies are compared, the greatest differences between this study and the others were: 0.012 at β = 45° with Fanger, 0.011 at β = 90° with Tanabe et al., and 0.01 at β = 75° with Kubaha et al. All differences are small. However, differences in directionless \( f_p^{*} \) values are greater than those of f p.

The differences between this study and those of Fanger and Tanabe et al. increase if f p is multiplied by each study’s different f eff value (see Table 4). The maximum differences increased to 0.033 at β = 40° with Fanger and 0.025 at β = 0° and 45° with Tanabe et al. Differences with Kubaha et al. remained the same. Differences with the frequently used Fanger’s values are nearly constant at β angles less than 60°, around 0.03.

Effective radiation area factor

AD values used to compute effective radiation area factor (feff) were obtained from 3D computer body models in 3DS Max 9 (male = 2.01 m2, female = 1.68 m2). The mean feff value was 0.826. The maximum difference between all four body type models was only 0.009 (Table 3). Therefore, feff was not related to gender or body type for our sample of people.

Table 3 Comparison of standing posture effective radiation area factor (f eff) of BMI categories
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The mean height, 1.74 m, (male = 1.81 m, female = 1.67 m) of this study was similar to subjects’ heights in previous studies (Table 4). The A D value, 1.845 m2, was close to Kubaha et al.’s (2004) and Guibert and Taylor’s (1952) studies but much greater than those of Fanger (1972) and Tanabe et al. (2000). The f eff value of this study, 0.826, lies closest to that of Miyazaki et al. (1995). The absolute range between Kubaha et al.’s (2004) value, 0.84, and Fanger’s (1972) widely used value, 0.725, is over 0.11.

Table 4 Comparison of standing posture f eff with previous studies
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Walking posture

Projected area factor

People’s arms spread farther from the trunk and legs are located farther from each other during most of a walking stride compared with a standing posture. This will produce somewhat greater body area factors than in a standing posture.

The normal-weight male and female models had very similar directional f p values (Fig. 7). The maximum difference between them was only 0.025 at the 2/4 position when α = 195° and β = 25°. The difference was close to the maximum difference between the genders for standing posture, 0.017. When α was around 180° (back of the body), the normal-weight male model had slightly higher f p values than the normal-weight female model, up to 0.0246. The opposite results happened when α was close to 0° (front of the body), up to 0.0189. This seems to be created by females’ breasts and males’ wider arm angles from a body. This phenomenon is very similar in standing posture.

Fig. 7
figure 7

Projected area factors of walking posture. a Normal-weight male. b Normal-weight female (Pose 1, 2, 3 and 4: 1/4, 2/4, 3/4 & 4/4 positions in a walking stride)

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Directional mean f p values from all stride positions dependent on β angles can be found in Fig. 8. The maximum f p difference among α angles was 0.072 between front (α = 5° & 185°) and side (α = 95° & 275°) of the body when the altitude angle was the lowest (β = 5°), and the differences decreased with increasing β angles.

Fig. 8
figure 8

Directional projected area factors (f p) of walking posture dependent on both azimuth and altitude angles and best-fit equation

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Directionless f p values can be used for simple modeling when walking direction cannot be defined or is not important. Both male and female models had very small differences between stride positions, up to 0.007, until β = 65° (Table 5). The greatest difference, 0.026, between the positions occurred at β = 85° for the normal-weight female model. At β = 85°, the overhead view where f p is small, variation in the horizontal projection of the legs and arms has the greatest relative effect. However, the differences of mean directionless f p values between the two gender models were only up to 0.002. Therefore, only minor differences in walking posture directionless f p values occurred between genders and stride positions at most β angles.

Table 5 The directionless f p values of normal-weight male and female models
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The previous studies for walking posture, Ward and Underwood (1967) and Steadman (1979), were compared with this study after converting f p to \( f_p^{ * } \) because they used A D to find \( f_p^{ * } \) for direct beam solar radiation analysis and Ward and Underwood did not define f eff. Steadman’s \( f_p^{ * } \) formula (\( f_p^{ * } \) = 0.386–0.0032β, 25° ≤ β ≤ 85°) for walking posture and Ward and Underwood’s values were up to 0.068 and 0.207, respectively, higher than in this study (Fig. 9). The effect of the differences can be substantial for direct beam solar radiation analysis.

Fig. 9
figure 9

Comparison of the mean directionless projected area factors \( \left( {f_p^{ * }} \right) \) of normal-weight male and female models with those of previous studies, walking posture

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Effective radiation area factor

In the normal-weight male model, there was a 0.027 difference in effective radiation area factor (f eff) between the four positions of a walking stride (Table 6). The greatest f eff value was 0.848 at the 1/4 position where the arms were spread farthest from the torso and legs were farthest apart (Fig. 1d). The lowest was 0.821 at the 3/4 position when the legs were close together and arms were near the torso. The normal-weight female model had more variation, 0.033. The greatest f eff was 0.868 at the 1/4 position, and the lowest was 0.835 at the 3/4 position. The representative f eff of walking posture was 0.846 which was about 0.02 higher than f eff of standing posture, 0.826. Our walking posture values are somewhat greater than the 0.80 and 0.82 assumed by Steadman (1979) for walking subjects and de Freitas et al. (1985) for runners, respectively.

Table 6 Effective radiation area factors (f eff) of walking postures of the normal-weight male and female models
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The greatest f eff, which occurred at the 1/4 position of the stride, was only 0.018 higher than the f eff of standing posture in the normal-weight male model, but for the normal-weight female model it was 0.044 higher. Moreover, the normal-weight female model had higher f eff values in all stride positions than for standing posture, but f eff of the 3/4 position in the normal-weight male model was 0.009 lower than for standing posture. In standing posture, the normal-weight male model had a much wider stance between the legs than the normal-weight female model (Fig. 1c). Even though the 3/4 position in the normal-weight male model has more body movement, the legs are located closer than the wide open stance in standing posture (Fig. 1d).

Discussion

Radiation area factors have a direct effect on computed human-environment radiation exchange. Our analysis indicated that differences in radiation area factors between studies found in the available literature are greater than those between genders or body types. Brief examples of the effects of study differences on computed solar and longwave radiation are given in this section.

A comparison between two groups of studies divided on the basis of f eff values [this study and Kubaha et al. (2004) vs Fanger (1972) and Tanabe et al. (2000)] using this study’s and Fanger’s widely used f eff and \( f_p^{ * } \) values illustrates some of the human radiation exchange effects of differences in body area factors. Absorptivities of the human body surface were assumed to be 0.7 for incoming solar radiation (R) and 0.97 for incoming longwave radiation (L).

Effects of the different body area factors are indicated by an analysis of the sensitivity of human absorbed radiation to differences in incoming radiation (Fig. 10a). Slopes of the linear regression lines are created by a combination of f eff and \( f_p^{ * } \) differences and the assumed absorptivity of the body for the radiation type. Longwave radiation (L) and diffuse beam and reflected solar radiation (K d + K r) are affected only by f eff, which has a greater difference (0.1) than \( f_p^{ * }\left( {0.0{2}-0.0{3}} \right) \), which affects direct beam solar radiation (K b). Hence, the slopes were greater for L and K d + K r than for K b (Fig. 10a). The assumed body absorptivity for L (0.97) is greater than that for R (0.7). This is the sole cause of the difference in sensitivity between L and K d + K r.

Fig. 10
figure 10

Effects on modeled absorbed radiation on a body surface created by differences in body area factors between this study and Fanger (1972). a Differences in absorbed K b by solar altitude, K d + K r and L dependent on amounts of incoming radiation. b Slope of absorbed K b difference (radiation differential/incoming radiation) by solar altitude

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Absorbed longwave radiation was the radiation stream most sensitive to the f eff and \( f_{\text p}^{ * } \) differences between this study and Fanger (Fig. 10a); the least was absorbed K b. Absorbed K b depends on solar altitude (β). It is most sensitive to variations in incoming K b at around β = 40°, and after β = 50° the slopes of the linear regression lines decline rapidly (Fig. 10b).

Gagge et al. (1969) found mean skin temperature was between 27°C and 36.5°C under steady state conditions. Within this range, differences in emitted longwave radiation from the body surface created by the 0.1 f eff difference between this study and Fanger were in a very narrow range, 6 Wm−2 (45–51 Wm−2).

Another example employs field data collected on a typical clear summer day (10 August 2002) around noon at the University of Guelph, Ontario (latitude and longitude: 43°32′N, 80°14′W, ψ sky : 0.88, β: 60.9°, T a: 29.9°C, K b: 759 Wm−2, K d + K r: 340 Wm−2, L: 479 Wm−2), the differences of f eff (0.1) and \( f_p^{*}\left( {0.0{22}} \right) \) between this study and Fanger made a gap of 24.6 Wm−2 in human net radiation.

The mean f p values of the 1/4 through 4/4 stride positions for walking posture were compared with those for standing posture (Fig. 11). The directional f p values of walking posture had less variation between front/back (α = 0° and 180°) and side (α = 90°) α angles with increasing β angles than those of standing posture. Also, the actual directional f p differences between α angles of walking posture were much lower, maximum difference 0.072, than those of standing posture, 0.128. More open walking body posture reduced the f p differences dependent on α angles.

Fig. 11
figure 11

Comparison of directional projected area factors (f p) of standing and walking postures for a selection of altitude angles (β) between 5° and 85°

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These directional f p differences would create up to 16 Wm−2 difference in absorbed K b in walking posture and up to 29 Wm−2 difference in standing posture in both low and moderate solar altitude angle simulations (β = 25°, K b = 350 Wm−2 and β = 55°, K b = 700 Wm−2).

If walking direction is not important or unknown and percentages of standing and walking people are unknown (e.g., square, plaza or open field), directionless f p values for K b analysis can be estimated from a formula for combined standing and walking postures (Fig. 12). If walking direction is known (e.g., corridors in subway stations and buildings, along sidewalks or park trails), directional f p values for walking posture can be found in Fig. 8.

Fig. 12
figure 12

Directionless projected area factors (f p) of standing and walking postures and best-fit polynomial equations

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For f eff values in modeling, the f eff difference from mean standing and walking postures (0.836) to standing posture (0.826) and walking posture (0.846) was only 0.01, which is not important in radiation analysis. Therefore, the mean f eff value, 0.836, can be used in most applications.

When directionless f p values for walking posture from half a sphere (0 ≤ α ≤ 360, 0 ≤ β ≤ 90) and a quarter sphere (0 ≤ α ≤ 180, 0 ≤ β ≤ 90) are compared; theoretically, standing posture is symmetrical, so f p values from a quarter sphere would be analyzed to find total f p values from an entire sphere. Walking posture is asymmetrical, so f p values from half a sphere should be analyzed. However, the differences in walking posture directionless f p values between the two ways were negligible, only up to 0.001. Therefore, analysis for directionless f p values within a quarter sphere (0 ≤ α ≤ 180, 0 ≤ β ≤ 90) can be adequate to find the total f p values from an entire sphere even though the walking body postures are not symmetrical.

A limitation of this study was that other body types and postures were not included, i.e., under-weight and obese BMI body type categories and sitting and running postures. These body types and postures will be investigated in further studies.

Conclusions

Computation of human radiation exchange requires effective radiation area and projected area factors of the modeled subjects. Our results improve the ability to select appropriate values by: including a larger sample of people than in previous studies, focusing on both normal-weight and over-weight, present-day adults, and presenting the first detailed analysis of walking posture.

Key findings include:

  1. 1.

    When determining human body effective radiation area and projected area factors via computer modeling, measurements every 10° are recommended as a good compromise between accuracy and data processing time.

  2. 2.

    Differences in f p and f eff of standing posture between the four body types (normal- and over-weight men and women) were relatively small. These results indicate that for most general modeling studies, a single value can be used regardless of gender or body type.

  3. 3.

    Differences between values presented in the wide variety of available studies are much greater than those found for body type. Hence, differences come more from the methods employed than from the human figures from which the body area factors were determined. Our results, in general, tend to be closer to those of studies using methods similar to ours.

  4. 4.

    There were only minor differences in directionless f p values between standing posture genders and body types, and between walking posture genders and stride positions. However, directional f p values varied with azimuth angle (α). Large differences occurred at low altitude angles (β) that are closest to perpendicular to the vertical body surface, and differences decrease with increasing altitude angles.

  5. 5.

    feff values between standing (0.826) and walking (0.846) postures were not too different, thus the mean feff (0.836) could be applied in most general modeling studies where applications to a variety of people both standing and walking are of interest. Standing posture has been used in many applications where walking posture might be appropriate. The small difference in feff suggests this has not caused any major errors or invalid conclusions. However, fp values should be selected carefully because directional and directionless fp differences were quite large at some azimuth angles.

Digital photography and computer processing have eased the assessment of human area factors. This will allow future research on a wide variety of body figures and postures to test whether our findings of only small differences between normal- and over-weight people and standing and walking postures extend to other body types and postures. The results can be used in a variety of human thermal exchange studies and expand their applicability to a wider range of activities. These new results will be compared with diverse body area factors used in radiation components of existing human thermal exchange models in further studies.