Thermal comfort in indoor environment: Effect of the solar radiation on the radiant temperature asymmetry

Solar Energy 144 (2017) 295–309

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Thermal comfort in indoor environment: Effect of the solar radiation on the radiant temperature asymmetry Concettina Marino, Antonino Nucara ⇑, Matilde Pietrafesa Department of Civil, Energy, Environmental and Material Engineering (DICEAM), ‘‘Mediterranea” University of Reggio Calabria, Via Graziella - Feo di Vito, 89122 Reggio Calabria, Italy

a r t i c l e

i n f o

Article history: Received 23 May 2016 Received in revised form 7 July 2016 Accepted 7 January 2017

Keywords: Solar radiation Thermal comfort Local thermal discomfort Radiant temperature asymmetry

a b s t r a c t This paper analyzes the influence of solar radiation, considering both its direct and diffuse component, on the local thermal comfort of subjects in indoor environments. More precisely, an original equation for the assessment of the plane radiant temperature in presence of solar radiation is proposed. It can be used to evaluate the correspondent radiant asymmetry and therefore allows detailed analysis of local thermal discomfort conditions. Its reliability and feasibility have been tried out by means of an application to a simple case study and the obtained results proved that, in the studied environment, solar radiation affects the symmetry of the radiant field to a considerable extent, being the main cause of local discomfort due to not acceptable values of the radiant asymmetry. Therefore, the method can be used to verify the presence of zones not suited to human activities inside rooms and, in addition, it can be exploited by the procedures aimed at the classification of the comfort quality of indoor environments. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Nowadays, the increasing level of life quality towards the highest standards requires the extremely precise definition of the microclimatic conditions suited to ensuring people’s thermal comfort in spaces where human activities are performed. Moreover, comfort and energy demand in buildings are strongly interrelated and the achievement of high quality standards involves an energy cost which must be taken into account (Calvino et al., 2010; Marino et al., 2015a). In other words, modelling the mechanism that processes the human body’s thermal sensation has become crucial, owing to the increasing interest in the optimization of the energy efficiency of buildings, which implies energy conservation policies without possibly compromising comfort conditions. The human thermal comfort sensation is strictly related to the metabolic heat production, to the energy exchanges between the human body and the environment and to the consequent variations of the physiological parameters (Fanger, 1970). A too hot or too cold sensation may regard the human body as a whole, so that the subject experiences a generalized discomfort sensation: this case is called ‘‘global thermal discomfort” which is generally assessed by means of the Fanger’s PMV and PPDindexes, described in the ISO 7730 (ISO, 2005) and based on the 7-point ASHRAE thermal sensation scale (ASHRAE, 2010). ⇑ Corresponding author. E-mail address: [email protected] (A. Nucara). http://dx.doi.org/10.1016/j.solener.2017.01.014 0038-092X/Ó 2017 Elsevier Ltd. All rights reserved.

In order to asses global comfort conditions, average values of the microclimatic parameters are generally used; nevertheless, in many practical cases, this approach does not allow a correct appraisal of the environmental quality of the indoor space on the grounds that the space and time variability of the physical parameters should be taken into account (Marino et al., 2015b; Simone et al., 2013). In addition, a sensation of thermal discomfort can be also caused by an undesired local cooling or heating of a particular area of the body: this case is called ‘‘local thermal discomfort”. The main causes of local discomfort are: significant vertical air temperature differences, atypical floor temperatures, air drafts and significant radiant asymmetry (Fanger, 1970; Rizzo et al., 2004). The study reported in this paper focuses on a specific facet of the human thermal comfort issue: the asymmetry of the radiant field which is considered one of the main sources of local discomfort. Indeed, the presence of too hot or too cold surfaces or other heat sources may alter the uniformity of the environment, so that different parts of the body, not uniformly exposed, might reach different temperatures which, in turn, may be cause of discomfort sensations for the subject. ISO 7730 standard (ISO, 2005) devices a method based on the asymmetry of the plane radiant temperature, which is the difference between the plane radiant temperatures of two opposite faces of a plane element called ‘‘test area”, and the related percentage of dissatisfied to assess the environments from this point of view and it constitutes the starting point of the analysis reported in this article.

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Nomenclature a A AAT b c C dn Cs Co EQI F
f j h hsr hss Ib Ib? Ibh Iin b Iout b Id Idh Iin d Iout d Q_ AT Q_ O;AT Q_ AT$S Q_ b!AT Q_ d!AT Q_ env !AT Q_ sun!AT Q_ S T T AT T pr Dtpr

surface height [m] area of the surface [m2] area of the surface of the test area [m2] surface width [m] distance between the test area and the surface [m] day–night coefficient own-shadow coefficient obstruction coefficient environmental quality index angle factor indicator measuring the persistence in specific jth class of the values of appropriate parameters solar hour [h] sunrise solar hour [h] sunset solar hour [h] intensity of the beam solar radiation [W m
2] component of the beam solar radiation perpendicular to the surface of the test area [W m
2] direct component of the solar radiation falling on a horizontal surface [W m
2] intensity of the inner beam solar radiation [W m
2] intensity of the external beam solar radiation [W m
2] intensity of the diffuse solar radiation [W m
2] diffuse component of the solar radiation falling on a horizontal surface [W m
2] intensity of the inner diffuse solar radiation [W m
2] intensity of the outer diffuse solar radiation [W m
2] radiative net flow [W] emitted flux [W] thermal flux exchanged between the test area and the environment [W] direct contribution of solar radiation [W] diffuse contribution of solar radiation [W] long wave radiation coming from the surfaces of the environment [W] short wave solar radiation passing through the external windows [W] thermal flow coming from the environment [W] surface temperature [K] surface temperature of the test area [K] plane radiant temperature [K] radiant asymmetry [°C]

Specifically, the equation used in the ISO 7730 Standard to calculate the plane radiant temperature was modified, on a theoretical basis and following physics considerations, in order to take the solar radiation into account. As a result an original equation for the assessment of the plane radiant temperature in presence of solar radiation is proposed; it tries to model the different behavior of both the direct and diffuse component in indoor environment and can be used to evaluate the correspondent radiant asymmetry. Therefore, it allows detailed analysis of local thermal discomfort conditions as well as the application of the method featured by international Standards, which does not consider solar radiation at all, to a wider range of cases occurring in real environments.

2. Comfort assessment and short-wave radiation: A literature review Human thermal sensation is commonly related to how people ‘‘feel” inside an environment which, in turn, is a consequence of

Greek symbols a solar altitude [°] aAT absorption coefficient of the test area referred to infrared radiation ab absorption coefficient of the test area referred to direct solar radiation ad absorption coefficient of the test area referred to diffuse solar radiation b tilt angle between the surface and the horizontal [°] h angle between the line normal to the surface and the solar rays [°] eAT emissivity of the test area c solar azimuth [°] q albedo r Stefan–Boltzmann constant (567  10
8 W m
2 K
4) sb optical transmittances of the glass for the direct component of the solar radiation sd optical transmittances of the glass for the diffuse component of the solar radiation sn optical transmittance the actual glazed surface for normal incidence n angle between the projection of the normal to the surface on a horizontal plane and the South [°] Subscripts AT test area b beam solar radiation d diffuse solar radiation surfaces of the environment env i i-th isothermal surface ði ¼ 1 . . . NÞ j j-th transparent surface ðj ¼ 1 . . . MÞ n normal pr plane radiant S environment sun sun Superscripts in inner out outer

the response of the body to personal and environmental conditions. Therefore, models addressing this issue need the employment of thermophysiological models which are able to predict human thermal response as a function of both physiological parameters and variables defining the surrounding thermal environment. For the last few decades many human thermal response models, based on energy balance equations for the human body, have been developed (Katic et al., 2014). They have evolved from a one node model into multi-layered cylinders representing separate body parts connected by a circulatory blood flow (Holopainen, 2012). As far as the estimation of thermal sensation and comfort is concerned, several methods are currently used. The widely used international standards ISO 7730 (ISO, 2005) and ASHRAE 55 (ASHRAE, 2010) refer to Fanger’s PMV (Predicted Mean Vote) method (Fanger, 1970). The PMV index predicts the mean response of a large group of people according to the ASHRAE thermal sensation scale (ASHRAE, 2010). The thermophysiological model which the PMV is related to, is a one node model and is based on a steady state

C. Marino et al. / Solar Energy 144 (2017) 295–309

heat balance equation of the human body. In this model, the effects of the surrounding environment are explained only by the physics of heat and mass exchanges between the body and the environment, and the human being is considered as a passive recipient of thermal stimuli. Skin temperature, governing the heat exchanges with the environment, is estimated as a function of other physiological variables by means of an empirical relationship which was experimentally deduced. PMV expresses the Overall Thermal Sensation (OTS) of the subject and is referred to a fairly uniform environment also from the point of view of the radiative exchange, whereas high intensity sources are not specifically taken into account. Therefore, with a view to allowing the assessment of non-uniform environmental conditions which, acting on a restrained area of the human body, usually affect Local Thermal Sensations (LTS), ISO 7730 (ISO, 2005) standard presents methods aimed at evaluating the local discomfort. One of these methods is referred to the asymmetry of the radiant field which is the object of this paper, and is based on the calculation of the Percentage of Dissatisfied (PD) as a function of the radiant temperature asymmetry (Fanger et al., 1985, 1980). However, in order to alternatively take into account the environmental conditions variability, other models can be used (Fiala, 1998; Fiala et al., 2003; Zhang, 2003; Zhang et al., 2004). These models assess the thermal sensation, both local and overall, by using skin temperature as an input. To assess skin temperature and its variation with time thermal physiological models may be used. They represent the human body from a thermokinetic point of view by means of energy balance equations (Fengzhi and Yi, 2005; Fiala et al., 2012, 2001, 1999; Huizenga et al., 2001; Tanabe et al., 2002). Particularly, the body is modelled by means of a multi-node system obtained by splitting the human frame into various elements (e.g. head, face, neck, shoulders, arms, hands, etc) which, in turn, are multi-layered concentric cylinders or spheres (Fiala et al., 2012, 1999; Huizenga et al., 2001; Kingma, 2012; Schellen et al., 2012; Tanabe et al., 2002) in order to simulate the various body tissues (e. g. core, muscle, fat and skin layers, etc.). As regards the radiative exchange affecting the human body, most of these models consider the environment as homogeneous and the short wave radiation is seldom taken into account. In the Tanabe model, convective and radiative fluxes are assessed by means of the correspondent heat transfer coefficient, whereas the solar radiation is not specifically considered (Tanabe et al., 2002). Similarly, in the Fiala Model (Fiala et al., 1999), as well as in the mathematical thermoregulation model (ThermoSEM) (Kingma, 2012; Schellen et al., 2013), the environment is considered homogeneous from the point of view of the radiative exchange with the human body, whereas, as far as short-wave radiation is concerned, its contribute to the heat balance of superficial body element sectors is assessed considering the environment as an uniform source which sees each body sector on account of the corresponding view factor. For the purpose of creating a new Universal Thermal Climate Index (UTCI) the Fiala model was selected (Katic et al., 2014). To fulfill this purpose, the original Fiala model was adapted and expanded into new UTCI-Fiala model (Fiala et al., 2012). In this revised form, direct solar radiation was more accurately treated by assessing the corresponding heat exchange with the various body sectors by means of the projected area factors and the short-wave absorptivity of the body surface. Radiative exchange with an uniform environment is also implemented in the Human Thermal Model (HTM) which combines the human thermal modelling with the Zhang’s thermal sensation and comfort model (Zhang et al., 2004) inside of a building simulation

297

program (Holopainen, 2012). The effect of short wave radiation is not specifically explicated. The University of California at Berkeley developed a model (UC Berkeley Comfort Model) which merges a thermophysiological model (Huizenga et al., 2001) with the Zhang’s comfort model (Zhang, 2003; Zhang et al., 2004). As far as the radiative exchange is concerned, the model divides the surface of the human body into more than five thousand polygons to calculate the radiation heat transfer between the body and the environment. Therefore, the heat transfer between the body and thermally asymmetric surfaces can be calculated. The solar load on the body is also calculated based on the relative geometric position of the sun, the environment, and the polygons mapping the body surface. On balance, the multi-node models take into account the anatomical and thermalphysiological properties of the human body to simulate the human heat transfer inside the body and at its surface. Therefore, multi-segmental models are capable to predict local skin temperatures of individual body parts, which is a needed input for the assessment of local or overall thermal sensation by means of many comfort models. On the other hand, one of the flaws of these physiology models regards the level of accuracy with which the environment inhomogeneities are considered, especially with reference to the radiative exchanges occurring among the body part and the neighboring surfaces and thermal sources. The majority of the models (apart from the UC Berkeley Model), as a matter of fact, considers the environment as homogeneous from the point of view of the radiative exchanges. This fact might cause misinterpretations in environments characterized by highly asymmetric radiant fields whose effects should be specifically investigated. In addition, even when the radiant asymmetries are taken into account, the described models yield results which are strongly dependent on the subject posture, on his position inside the indoor environment and on the physiological factors used as input data or as preset model parameters which, in turn, being extremely variable among individuals, make the human thermal response to the environment not univocally predictable (Katic et al., 2014). Therefore, while there is a question regarding the development of a system that can recognize subjective difference among individuals (Haldi and Robinson, 2010; Katic et al., 2014; Marino et al., 2015a; Rijal et al., 2008), the use of the cited models is feasibly finalized to the detailed analysis of the human thermal reaction in environments where a single subject occupies fixed positions with static postures. Otherwise, in environments occupied by different subjects carrying out different activities, suitable results may be only obtained by means of a large number of simulations considering all the varying issues. In these cases, on the other hand, the statistical approach to the comfort problem, based on the percentage of dissatisfied (PD), adopted by the ISO 7730 Standard (ISO, 2005) might be a compromise solution to strike a balance among the required accuracy of results, the demand for practical methods of analysis and the need to obtain outcomes easily to be interpreted and referred to the preferences of a plurality of subjects. It could be used to perform general and preliminary analysis of the environment, so that the thermophysiological models can be more suitably employed when specific configurations need detailed examination. The models proposed in this paper, regarding the asymmetry of the radiant field and its effect on comfort, follows this statistical approach and modifies the method devised by ISO 7730 standard, based on the statistical dependence between the plane radiant temperature asymmetry and the related percentage of dissatisfied, in order to take short –wave radiation into account, distinguishing among direct and diffuse radiation.

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3. Effect of the solar radiation on the radiant asymmetry: The proposed method The radiant temperature asymmetry (Dtpr ) is the difference between the plane radiant temperature of the two opposite sides of a small plane element (ISO, 1998), whereas, in turn, the plane radiant temperature ðt pr Þ is defined as the uniform temperature of an enclosure where the radiative flux on one side of a small plane element, the so called ‘‘test area”, is the same as in the actual environment (ISO, 2001). For assessments of local discomfort, the radiant temperature asymmetry is evaluated with reference to three orthogonal directions. Therefore, the calculation of six plane radiant temperatures (Fig. 1) allows the asymmetry in the directions X, Y and Z to be assessed. As a consequence, three plane elements respectively orthogonal to the axis X, Y, Z must be considered and, hence, six test areas (one on each side of each plane element) must be taken into account. Consequently, the radiant asymmetry may be assessed, for each direction as follows:

Dtpr ¼ DT pr ¼

qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi 4 4 T 4prðþÞ
T 4prð
Þ

ð1Þ

where the signs (+) and (
) respectively refer to each face (namely a test area) of each plane correspondingly normal to the considered directions X, Y and Z. The plane radiant temperature, on the other hand, can be calculated as follows (ISO, 1998):

T pr

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u X 4 ¼ t F AT!i T 4i

ð2Þ

i¼1

where T i is the temperature of the ith isothermal surface of the environment, whereas F AT!i is the view factor between the ith isothermal surface and the test area. Experimental studies conducted on a large number of people have shown that the number of dissatisfied subjects grows rapidly as the radiant asymmetry rises (Fanger et al., 1985, 1980). This effect is even more enhanced by solar radiation, which can make the radiant field dramatically asymmetric.

Z+

Solar radiation, as a matter of fact, may shine on a single face of the test area, while the opposite one may be shielded. Furthermore, depending on the shading effect of the building envelope at the considered position, the influence of the sun rays may be remarkable for a specific direction, but negligible for the others. In addition, diffuse and direct component of solar radiation have different behavior inside indoor environment and may affect the plane temperature asymmetry with different intensity and processes. In the following paragraphs a proposal to modify Eq. (2), in order to take all these issues into account, is reported. Particularly, both the direct and diffuse component of solar radiation are considered with a view to trying to model their different behavior in indoor environments. 3.1. Plane radiant temperature in the presence of the solar radiation In order to infer an analytical relation for the plane radiant temperature in the presence of the solar radiation, the radiative heat fluxes regarding the test area must be analyzed. Inside an enclosed concave environment, the radiative net flow Q_ AT leaving the surface named ‘‘test area” is equal to the thermal

flux, Q_ AT$S , exchanged between the test area and the environment (La Gennusa et al., 2005), that is:

Q_ AT$S ¼ Q_ AT

ð3Þ

In turn, the net flow leaving the surface is equal to the difference between the emitted and the absorbed flux (Fig. 2):

Q_ AT ¼ Q_ O;AT
aAT Q_ S

ð4Þ

in which Q_ 0;AT is the emitted flux, Q_ S the thermal flow coming from the environment and aAT the absorption coefficient of the test area. Assuming that T AT is the absolute temperature of the test area, the corresponding emitted thermal flow is:

Q_ O;AT ¼ reAT AAT T 4AT

ð5Þ

where r is the Stefan–Boltzmann constant, eAT , AAT and T AT are the emissivity, the surface and the absolute temperature of the test area, respectively. Moreover, the thermal flow that reaches the test area is the sum of the long wave radiation coming from the surfaces of the environment, Q_ env !AT , and the short wave solar radiation passing through the windows, Q_ sun!AT :

up

Q_ S ¼ Q_ env !AT þ Q_ sun!AT

Y-

X-

right

back

ð6Þ

Q S

Q0 , AT

Y+ X+

le

front

AT α AT Q S

Zdown Fig. 1. Reference system feasible for the calculation of the radiant asymmetry.

Fig. 2. Radiative heat fluxes regarding the test area.

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(2) the obstruction coefficient, C o , equal to 1 when the test area is directly hit by the solar beam and equal to 0 in the other cases; (3) the own-shadow coefficient, C s , equal to 1 when the test area is in its own shadow and equal to 0 in the other cases.

Relave absorptance

1.0

0.9

Under these hypotheses, the Eq. (8) becomes:

0.8

T pr

r

i¼1

0.7

0.6

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u N M X X u 0:7  C dn 4 ¼ t F AT!i T 4i þ Id F AT!j þ C o C s Ib?

0

1000

2000

3000

4000

5000

6000

Color temperature (K)

ð9Þ

j¼1

As regards the coefficients in Eq. (9) the parameter C dn can be evaluated with reference to the sunrise time, hsr , and sunset time, hss :

(


1 ð
tan u tan dÞ
180

hsr ¼ 12
cos

15


1 ð
tan u tan dÞ
180

hss ¼ 12 þ cos

Fig. 3. Absorbance coefficient versus the temperature of the emitting source (ASHRAE, 1989).

ð10Þ

15

where u is the site latitude and d is the declination angle. The value of the coefficient C dn at the solar time h is: Therefore, in accordance with the detailed considerations reported in the Annex, the plane radiant temperature may be expressed by the following equation:

T pr

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N M X u ad X ab 4 aAT ¼t F AT!i T 4i þ Id F AT!j þ Ib?

eAT

reAT

i¼1

reAT

j¼1

ð7Þ

where F AT!j the view factor between the test area and the jth glazed surface, Id the intensity of the solar radiation diffuse from the sky, Ib? is the component of the beam solar radiation perpendicular to the surface of the test area. The three absorption coefficients aAT , ad and ab , are referred to infrared radiation, diffuse solar radiation and direct solar radiation respectively. By and large, they are supposed to assume different values depending on both the radiation’s wavelength and the temperature of the source (Fig. 3). Thus, whereas the test area may be assumed to behave as a blackbody (with aAT ¼ 1), when infrared radiation is taken into account, on the contrary, this hypothesis is not reliable when the solar radiation is considered. In this case, indeed, being about 5780 K the surface temperature of the sun, the absorption coefficients for diffuse and direct solar radiation can be assumed equal to 0.7 (Fig. 3). Moreover, the emissivity of the test area may be assumed eAT ¼ 1. All these considerations lead to the following proposed equation:

T pr

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N M X u 0:7 X 0:7 4 ¼ t F AT!i T 4i þ Id F AT!j þ Ib? i¼1

r

j¼1

r

ð8Þ

3.2. Shading effects In real environments, obstructions, like building walls or shading devices, can shade the direct component of the solar radiation on the test area; in this case the third term of Eq. (8) must be set to zero. In addition, the test area may not be facing the sun, being in its own shadow, in this case the third term must be set to zero, as well. Moreover, in nighttime period, both the second and third terms must be set to zero. With a view to considering the effect of these phenomena, three coefficients have been introduced in Eq. (8): (1) the day–night coefficient, C dn , equal to 1 during daytime and equal to 0 during the nighttime;



C dn ¼ 1

if

hsr < h < hss

C dn ¼ 0 otherwise

ð11Þ

On the other hand, the parameter C o can be evaluated with reference to the angles depicted in Fig. 4. It is:



Co ¼ 1

if

a01 < a < a02 and c01 < c < c02

C o ¼ 0 otherwise

ð12Þ

In addition, as regards the own-shadow coefficient, C s , it can be assessed as follows:



Cs ¼ 1 if cos h > 0 C s ¼ 0 otherwise

ð13Þ

where h is the angle of incidence between the normal to the surface and the solar ray direction. 3.3. Direct radiation on test area In order to calculate the plane radiant temperature for each of the six test areas considering the effect of the solar radiation, the direct radiation striking each test area, Ib? , must be reckoned. This component can be calculated provided that the angle of incidence h, which is the angle between the line normal to the surface and the solar rays (Fig. 5), is known. It depends on a set of angles characterizing the position of the sun and the inclination and orientation of the surface, namely: the solar altitude a, the solar azimuth c, the tilt angle b between the surface and the horizontal plane, the surface azimuth n, equal to the angle between the projection of the normal to the surface on a horizontal plane and the South; in fact it can be assessed as:

cos h ¼ cos a cosðc
nÞ sin b þ sin a cos b

ð14Þ

This relationship needs to be particularized for each of the six test areas which is required for assessing the radiant asymmetry. To reach this goal, for every test area, the tilt angle b, and the surface azimuth, n, must be evaluated. Assuming that n0 is the angle between the positive direction of the X axis (Fig. 1) and the South, the values of both the tilt angle b and the surface azimuth, n, and the consequent expressions of the incidence angle, h, are reported in Table 1, for each test area, which, in turn, is identified by means of the direction of its normal line (namely the axis X, Y or Z) and of the sign (+) or (
) referred to its orientation (namely front, back, etc.). Knowing the incidence angle h, the direct radiation on the test area can be assessed:

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Fig. 4. Angles needed for the evaluation of the obstruction coefficient.

4. Case study In the present section, the proposed model will be clarified through an example. Specifically, the evaluation of the radiant asymmetry in presence of solar radiation at various positions inside a building module is carried out. The structure of the procedure used in the example consists of the following steps: 1. definition of the building module; 2. determination of the solar radiation entering the module through the window; 3. assessment of the radiant temperature asymmetries.

4.1. Definition of the building module

Fig. 5. Angle of incidence.

Ib? ¼ Ib cos h

ð15Þ

being Ib the intensity of the direct normal radiation. 3.4. Proposed equation In conclusion, when the shading effects and incidence angle are considered, the equation (8) can be modified as follows:

T pr

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u N M X X u 0:7  C dn 4 4 t ¼ F AT!i T i þ Id F AT!j þ C o C s Ib cos h

r

i¼1

ð16Þ

j¼1

It represents an analytical tool for processing direct and diffuse components of the solar radiation entering a room in a given site and for assessing their effect on the plane radiant temperature. This allows the assessment of the corresponding asymmetry in very frequently occurring situations.

Table 1 Relationships expressing the incidence angle between the solar rays and the line normal to the test area. Direction

b

n

h

Front (X+) Back (X
) Left (Y+) Right (Y
) Up (Z+) Down (Z
)

90°

n0 + 0° n0 + 180° n0 + 270° n0 + 90° Any Any

cos
1 ½cos a cosðc
nÞ

0° 180°

cos
1 ðsin aÞ cos
1 ð
sin aÞ

An analysis of the spatial and temporal variations of the radiant asymmetry has been carried out at the height of 0.80 m above the floor in six points of the building module reported in Fig. 6. The module is located in Reggio Calabria (38°70 1500 North, 15°390 5300 East), a Southern Italian town characterized by a Mediterranean mild climate. The analysis has been carried out in winter, on December 21st, and has involved a period of 24 h. For each considered point, the X-axis of the reference system, reported in Fig. 1, is lined up with the North-South direction and points South (n0 =0°), the Y axis is parallel to the East-West direction and points East, the Z axis is normal to the XY plane and points upward. As regards the orientation of the test areas, the following conventions were utilized: – the South facing area and the North facing one, both normal to the X direction, were singled out with the symbols X+ and X
respectively; – the East facing area and the West facing one, both normal to the Y direction, were identified by the symbols Y+ and Y
respectively; – the upward facing area and the downward facing one, both normal to the Z direction, were identified by the symbols Z+ and Z
respectively. The temperature of all the internal surfaces of the building module has been assumed equal to 20 °C, while the optical transmittance of the glazed surface at normal incidence of the rays has been assumed equal to 0.87. The view factors between the test

301

As regards the calculation of the solar radiation entering the indoor environment through the glazed surface, a series of hypothesis were made: (a) the beam component maintains its direction crossing the glass; (b) the diffuse component is considered to cross the glass layer as a direct radiation having, an altitude angle of 60° and an azimuth angle, evaluated with respect to the normal to the surface, equal to 0; (c) the diffuse radiation entering the room through the glazed surfaces follows the Lambert’s law. In this case, the direct normal radiation and the diffuse component complying with the hypothesis (b), both shining on the external side of the glazed surface, can be assessed by means of the following equations (La Gennusa et al., 2007):

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80

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1

2-4 3-5-6

N

(

where a is the solar altitude and q is the albedo, assumed equal to 0.3. Therefore, the intensity of solar radiation entering the indoor environment can be evaluated as:

50

100

3

out Iin b ¼ sb I b

ð18Þ

out Iin d ¼ sd Id

100

2

(

where sb and sd are the optical transmittances of the glass for the direct and diffuse component respectively. They may be calculated using the SHGF procedure proposed by ASHRAE (ASHRAE, 1989), which express the optical transmittances as:

500

180

1

100

6 5

ð17Þ

q 3 Iout d ¼ 4 ðIbh þ Idh Þ þ 4 I dh

500

4

Ibh Iout b ¼ sin a

100

5 X

sj cos j h

sb ¼ Fig. 6. Building module.

j¼0

0:86

ð19Þ

sn

in which h is the angle between the solar rays and the line normal to the glass surface and sn is the optical transmittance of the glazed surface at normal incidence of the solar rays. The values assumed by the sj coefficients are reported in Table 3. At this stage of the procedure, by means of Eq. (15) combined with the equations reported in Table 1, the direct radiation hitting the test areas is able to be assessed.

area and the glazed surface, calculated with the relations reported in the ISO 7726 standard (ISO, 1998), are reported in Table 2. 4.2. Determination of the solar radiation entering the module This stage of the procedure regards the assessment of the solar path, the obstruction angles of the window, the shadow periods due to the building shape at each considered point, the related obstruction coefficients, the hourly direct and diffuse components of the solar radiation shining in the outdoor environment on a horizontal surface, the solar radiation passing through the glazed surface into the indoor environment. The hourly direct and diffuse components of the solar radiation shining on a horizontal external surface (respectively Idh and Ibh ) have been evaluated by means of the Liu and Jordan method (Liu and Jordan, 1960), knowing the monthly average daily global solar radiation on the horizontal surface, which, for the considered site, is equal to 6.9 MJ/m2day (Huld et al., 2012).

4.3. Assessment of the radiant temperature asymmetries During this last step of the procedure, solar radiation hourly data, formerly obtained, are used as input in Eq. (16) to assess the six plane radiant temperatures in correspondence of the six points inside the building module. Finally, Eq. (1) is used to evaluate, at each point, the radiant asymmetries along the three directions X, Y and Z. Obviously, as a consequence of the variation of solar radiation with time and of the variability of the shading effect, both plane radiant temperatures and the corresponding asymmetries are also time dependent.

Table 2 View factors between the test area and the glazed surface for each considered point. Point

X+

X


Y+

Y


Z+

Z


1 2 3 4 5 6

0.418 0.186 0.089 0.123 0.071 0.040

0.000 0.000 0.000 0.000 0.000 0.000

0.264 0.063 0.020 0.047 0.016 0.009

0.264 0.063 0.020 0.000 0.000 0.000

0.132 0.032 0.010 0.043 0.016 0.009

0.000 0.000 0.000 0.000 0.000 0.000

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Table 3 Coefficients for the calculation of the optical transmittance of the glass with the SHGF procedure (ASHRAE, 1989). j

sj

0 1 2 3 4 5


0.0089 2.71235
0.6206
7.0733 9.75995
3.8992

5. Results and discussion Preliminary, in the result analysis, the shading effect of the building envelope is worthy of note. It is depicted in Fig. 7, which reports the sun path during the considered day and where the gray areas point out the positions of the sun which cause the direct radiation to be shielded by the building envelope in the considered points of the room. Being present in each point for limited and variable periods of the day, direct solar radiation affects asymmetry only in correspon-

Point 2 90

90

80

80

70

70

60 50 40 30

altude angle (°)

altude angle (°)

Point 1

60 50 40 30

20

20

10

10

0 180

150

120

90

60

30

0

-30

-60

-90

-120

-150

0

-180

180

150

120

90

60

azmuth angle (°)

30

0

-30

-60

-90

-120

-150

-180

azmuth angle (°)

Point 4 90

90

80

80

70

70

60 50 40 30

altude angle (°)

altude angle (°)

Point 3

60 50 40 30

20

20

10

10

0 180

150

120

90

60

30

0

-30

-60

-90

-120

-150

0

-180

180

150

120

90

60

azmuth angle (°)

30

0

-30

-60

-90

-120

-150

-180

azmuth angle (°)

Point 5

Point 6

90

80

80

70

70

60 50 40 30

altude angle (°)

altude angle (°)

90

60 50 40 30

20

20

10

10

0 180

150

120

90

60

30

0

-30

-60

azmuth angle (°)

-90

-120

-150

-180

0 180

150

120

90

60

30

0

-30

-60

azmuth angle (°)

Fig. 7. Shadow periods due to the building shape, at each examined point.

-90

-120

-150

-180

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dence of these time intervals which, hence, must be singled out for each point. The results of this analysis are reported in Fig. 8, which depicts the time trend of the solar radiation components calculated by means of Eqs. (17) and (18) and where, as far as the indoor direct

reported in Fig. 9, where the interaction with the shadow patterns owing to the window geometry is also depicted. By and large, it is worth noting that: (a) along the X direction, solar radiation usually comes from South and hits the X+ test area, but there is no contribution from North on the X
test area; (b) along the Y direction, before 12 a.m., solar radiation comes from East, hitting the Y+ test area; afterward it comes from West and hence affects the Y
test area;

normal radiation (Iin b ) is concerned, the calculated shadow periods, characterizing each examined point, are taken into account. The results regarding the direct component of solar radiation shining on each test area, obtained by means of Eq. (15), are

Point 1 Diffuse I bin OUT

Point 2 Direct I dout IN

Diffuse I din IN

Direct I boutOUT

500

500

450

450

irradiaon (W/m2)

irradiaon (W/m2)

Direct I boutOUT

400 350 300 250 200

300 250 200 150

100

50

50 0 3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

0,00

3,00

6,00

Point 3 Direct I boutOUT

Diffuse I bin OUT

9,00

12,00

15,00

18,00

21,00

24,00

21,00

24,00

21,00

24,00

Point 4 Direct I dout IN

Diffuse I din IN

Direct I boutOUT

500

500

450

450

irradiaon (W/m2)

irradiaon (W/m2)

Diffuse I din IN

350

100

0.00

400 350 300 250 200

Diffuse I bin OUT

Direct I dout IN

Diffuse I din IN

400 350 300 250 200

150

150

100

100

50

50

0

0 0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

0,00

3,00

6,00

Point 5 Direct I boutOUT

Diffuse I bin OUT

9,00

12,00

15,00

18,00

Point 6 Direct I dout IN

Diffuse I din IN

Direct I boutOUT

500

500

450

450

irradiaon (W/m2)

irradiaon (W/m2)

Direct I dout IN

400

150

0

Diffuse I bin OUT

400 350 300 250 200

Direct I dout IN

Diffuse I din IN

400 350 300 250 200

150

150

100

100

50

50

0

Diffuse I bin OUT

0

0,00

3,00

6,00

9,00

12,00

15,00

18,00

21,00

24,00

0,00

3,00

6,00

9,00

12,00

15,00

18,00

Fig. 8. Components of solar radiation on both the outer and inner side of the glass layer and their interaction with the shadow patterns at each considered point.

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(c) along the Z direction, the radiation always comes from above and therefore only the Z+ test area is involved for the whole daylight period.

First and foremost, the diffuse component on the test areas depends on the view factor between the window and each test area; therefore:

On the other hand, as regards the contribution of the diffuse solar radiation, hypothetically coming from the window in accordance with the Lambert’s Law and irradiating the test areas located at each analyzed point (Fig. 10), the considerations reported in the following sections are worthy of note.

(a) for all the considered points, the X+ test area, which faces the window, is always irradiated by diffuse radiation, whereas the North facing X
is never affected by this phenomenon, being zero the correspondent view factors (Table 2);

Point 1 Y (+) (EST)

Y (-) (WEST)

X (+) (SOUTH)

Z (+) (UP)

400

400

350

350

300

300

Irradiaon (W/m2)

Irradiaon (W/m 2)

X (+) (SOUTH)

Point 2

250 200 150 100 50

Z (+) (UP)

250 200 150 100

0 0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

0,00

3,00

6,00

Point 3 X (+) (SOUTH)

Y (+) (EST)

9,00

12,00

15,00

18,00

21,00

24,00

21,00

24,00

21,00

24,00

Point 4 Y (-) (WEST)

X (+) (SOUTH)

Z (+) (UP)

400

400

350

350

300

300

Irradiaon (W/m2)

Irradiaon (W/m 2)

Y (-) (WEST)

50

0

250 200 150 100 50

Y (+) (EST)

Y (-) (WEST)

Z (+) (UP)

250 200 150 100 50

0

0 0.00

3.00

6.00

9.00

12.00

15.00

18.00

21.00

24.00

0,00

3,00

6,00

Point 5 X (+) (SOUTH)

Y (+) (EST)

9,00

12,00

15,00

18,00

Point 6 Y (-) (WEST)

Z (+) (UP)

X (+) (SOUTH)

400

400

350

350

300

300

Irradiaon (W/m2)

Irradiaon (W/m2)

Y (+) (EST)

250 200 150 100 50

Y (+) (EST)

Y (-) (WEST)

Z (+) (UP)

250 200 150 100 50

0

0

0,00

3,00

6,00

9,00

12,00

15,00

18,00

21,00

24,00

0,00

3,00

6,00

9,00

12,00

15,00

Fig. 9. Direct radiation on the test areas and their interaction with the shadow patterns at each considered point.

18,00

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Point 1 Z (+) (UP)

Point 2

Y (+) (EST)

Y (-) (WEST)

X (+) (SOUTH)

45

45

40

40

35

35

Irradiaon (W/m2)

Irradiaon (W/m2)

X (+) (SOUTH)

30 25 20 15

20 15

5

5 0

3,00

6,00

9,00

12,00

15,00

18,00

21,00

24,00

0,00

3,00

6,00

9,00

Point 3 X (+) (SOUTH)

Z (+) (UP)

12,00

15,00

18,00

21,00

24,00

21,00

24,00

21,00

24,00

Point 4

Y (+) (EST)

X (+) (SOUTH)

Y (-) (WEST)

45

45

40

40

35

35

Irradiaon (W/m2)

Irradiaon (W/m2)

Y (-) (WEST)

25

10

0,00

30 25 20 15

Z (+) (UP)

Y (+) (EST)

Y (-) (WEST)

30 25 20 15

10

10

5

5

0

0

0,00

3,00

6,00

9,00

12,00

15,00

18,00

21,00

24,00

0,00

3,00

6,00

9,00

Point 5 X (+) (SOUTH)

Z (+) (UP)

12,00

15,00

18,00

Point 6

Y (+) (EST)

Y (-) (WEST)

X (+) (SOUTH)

45

45

40

40

35

35

Irradiaon (W/m2)

Irradiaon (W/m2)

Y (+) (EST)

30

10

0

Z (+) (UP)

30 25 20 15

Z (+) (UP)

Y (+) (EST)

Y (-) (WEST)

30 25 20 15

10

10

5

5 0

0 0,00

3,00

6,00

9,00

12,00

15,00

18,00

21,00

24,00

0,00

3,00

6,00

9,00

12,00

15,00

18,00

Fig. 10. Diffuse radiation on the test areas at each considered point.

(b) at points 1, 2 and 3, the diffuse radiation, coming from the window, shines with the same intensity on the two test areas normal to the Y direction; in fact, in these cases, for symmetry reasons, the view factors between the window and the East facing test area (Y+) are equal to the view factors between the window and the West facing test area (Y
) (Table 2); on the other hand, at points 4, 5, 6 the Y
test area is not irradiated by the diffuse radiation coming from the window, being zero the correspondent view factors; (c) for all the considered points, the Z+ test area is always irradiated by diffuse radiation coming from the window,

whereas Z
is never affected by this phenomenon, being zero the correspondent view factor. Finally, the time trends of the radiant asymmetries at each considered point and their connections with the shadow patterns are reported in Fig. 11. It is worth pointing out that, at every irradiated point, the presence of the solar radiation, especially of its direct component, alters considerably the symmetry of the radiant field in the environment. In fact, the radiant asymmetries along the three directions X, Y and Z rise significantly when the considered point is irradiated by the

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Point 1 Dtpr (X)

Dtpr (Y)

Point 2 Dtpr (Z)

50

Dtpr (X)

Dtpr (Y)

9,00

12,00

Dtpr (Z)

50

pr (X) 40

40

pr (Z)

30

20 10 0 0,00

3,00

6,00

9,00

12,00

15,00

18,00

21,00

24,00

asymmetry (°C)

asymmetry (°C)

30

-10

20 10 0 0,00

3,00

6,00

pr (Y)

-20

21,00

24,00

18,00

21,00

24,00

18,00

21,00

24,00

-30

Point 3 Dtpr (X)

Dtpr (Y)

Point 4 Dtpr (Z)

50

50

40

40

30

30

20 10 0 0,00

3,00

6,00

9,00

12,00

15,00

18,00

21,00

24,00

asymmetry (°C)

asymmetry (°C)

18,00

-20

-30

Dtpr (X)

Dtpr (Y)

9,00

12,00

10 0 0,00 -10

-20

-20

-30

-30

3,00

6,00

Point 5 Dtpr (Y)

30

30

20

20

10

0 6,00

9,00

12,00

15,00

18,00

21,00

24,00

asymmetry (°C)

40

3,00

15,00

Point 6 Dtpr (Z)

40

0,00

Dtpr (Z)

20

-10

Dtpr (X)

asymmetry (°C)

15,00

-10

Dtpr (X)

Dtpr (Y)

9,00

12,00

Dtpr (Z)

10

0 0,00

-10

-10

-20

-20

-30

-30

3,00

6,00

15,00

Fig. 11. Plane radiant asymmetries at each considered point.

sun, and, conversely, they decrease during the shadow periods when the direct component of solar radiation is shielded by the building envelope. In addition, the phenomenon is also evident when the conditions of irradiated and not irradiated positions are compared. For example, the radiant asymmetries along the three directions X, Y and Z at Point 1, which is the nearest to the window and always irradiated directly by the sun, are systematically greater than the correspondent ones assessed at point 3, which is never irradiated by the direct component of the solar radiation during the studied period (Fig. 9), because of the shading effects of the building envelope.

Therefore, the major changes in the plane radiant temperature values are due to the contribution of the direct component of the solar radiation, while the effect of the diffuse one can be considered as negligible. On balance, the analysis of these results shows that, generally, in indoor environment, the radiant asymmetry may present temporal and spatial variabilities which depend on a series of factors (view factors, solar radiations, shadow patterns, etc.) and may cause conditions of local thermal discomfort. However, it is the solar radiation, especially its direct component, which affects the radiant field the most, causing the higher values of the radiant

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asymmetry in the environment and restraining the zone where comfort conditions are possible.

oretically basis; however, to verify its consistency from an experimental point of view, a measurement campaign is going to be designed and developed during the future stages of the current research.

6. Conclusions Annex Solar radiation plays a pivotal role in building design, especially because its correct managing allows energy efficiency purpose to be fulfilled. On the other hand, solar radiation affects the radiant field inside the indoor environment and hence it has a remarkable influence on the comfort conditions, being possible cause of both local and global discomfort. In order to properly evaluate this phenomenon, methods allowing the assessment of the human comfort conditions in presence of solar radiation are highly needed. Generally, the methods today available involve human thermal simulation models which, albeit able to yield detailed results regarding the thermal response of a specific individual in a precise configuration, might not be easily and directly suitable for the assessment of the comfort quality of environments occupied by different subjects undertaking different tasks. In these cases, a statistical approach to the comfort problem based on the percentage of dissatisfied (PD) might be more appropriate with a view to taking into account the preferences of a plurality of subjects. This approach is adopted by ISO 7730 international Standard (ISO, 2005) and, with regard to the problem of local discomfort related to the radiant field, it is based on the calculation of the Percentage of Dissatisfied (PD) as a function of the radiant temperature asymmetry (Fanger et al., 1985, 1980). As far as this issue is concerned, this paper proposes a new analytical relationship for the evaluation of the asymmetry of the plane radiant temperature in a confined environment in presence of solar radiation, direct and diffuses as well. For the proposed relationship the radiant asymmetry is a function of: (a) the temperature of the internal surfaces of the environment; (b) the intensity of the direct and diffuse component of the solar radiation; (c) the angle factors among the opaque and glazed surfaces of the environment and the test area; (d) the incidence angle of the solar rays on the test area. It may be conveniently utilized for studies on the radiant field, in order to identify the zones of an environment which are not suitable because of not acceptable comfort conditions. In particular, with a view to evaluating local thermal discomfort, it can allow detailed computation of the radiant asymmetry, being able to take into account the spatial and temporal variability of this parameter, which always occurs in presence of solar radiation. It may be also exploited during the design phase of the indoor environment in order to properly shape the building envelope considering its effect on the occupants, without involving thermal simulation models not easily configurable at this stage of the construction process owing to the large number and variety of variable to take into account. Moreover, the proposed relationship may be used as part of the procedures aimed at ranking the indoor quality of edifices (CEN, 2007; Marino et al., 2012), in turn needed as a complement of the energy certification of buildings, as stated in Europe by the Energy Performance Building Directive (EU, 2010). The potentiality of the proposed methodology was highlighted by means of an application to a building module, sited in a southern Italian town; for the examined case, solar radiation proved to affect the radiant asymmetry to a large extent, so that the comfort quality classification was also influenced. In conclusion, the proposed method, taking into account the most important causes of the asymmetry of the radiant field, appears to be a reliable means to evaluate the presence of local discomfort in indoor confined environments. It is inferred from a the-

In this annex the analytical steps that allow the determination of the proposed equation are reported in detail. The first equation describing the thermal flow on the test area is the following:

Q_ AT ¼ Q_ O;AT
aAT Q_ S

ð20Þ

in which Q_ 0;AT is the emitted flux, Q_ S the thermal flow coming from the environment and aAT the absorption coefficient of the test area. The emitted thermal flow may be expressed by:

Q_ O;AT ¼ reAT AAT T 4AT

ð21Þ

where r is the Stefan–Boltzmann constant, eAT , AAT and T AT are the emissivity, the surface and the absolute temperature of the test area, respectively. Otherwise, the thermal flow coming from the environment is the sum of the long wave radiation coming from the surfaces of the environment, Q_ env !AT , and the short wave solar radiation passing through the windows, Q_ sun!AT :

Q_ S ¼ Q_ env !AT þ Q_ sun!AT

ð22Þ

The relationships able to evaluate the terms of the Eq. (22) will be derived in the follow. With reference to Fig. 12, assuming that the N walls of the environment can be considered as black bodies, the total radiant flow coming from the environment and reaching the test area may be expressed by the following equation:

Q_ env !AT ¼

N N X X Q_ i!AT ¼ r F i!AT Ai T 4i i¼1

ð23Þ

i¼1

where Ai and T i are the area and the temperature of the ith isothermal surface of the environment respectively, whereas F i!AT is the view factor between the ith isothermal surface and the test area. Considering the reciprocity rule of view factors, which is:

F i!AT Ai ¼ F AT!i AAT

ð24Þ

Eq. (23) can be written as:

Q_ env !AT ¼ rAAT

N X

F AT!i T 4i

ð25Þ

i¼1

2

1 Q2

Qi

Q1

3

i AT AT

Q3

AT

AT

AT

Fig. 12. Radiative exchanges between a confined environment and the test area.

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C. Marino et al. / Solar Energy 144 (2017) 295–309

Whereas view factors among environment and human body depend on the shape of the subject and need appropriate approaches to be evaluated (Calvino et al., 2005; La Gennusa et al., 2008; Rizzo et al., 1992; Tanabe et al., 2000), on the contrary, view factors between the test area and the surfaces of the environment may be calculated by means of the typical geometrical approach which leads to a fairly simple analytical formula. As a matter of fact, with reference to Fig. 13, view factors between the test area and the rectangular surfaces of the environment can be evaluated using the following relationships (ISO, 1998): Case (a) Parallel surfaces

On the other side, the contribution due to the direct radiation can be calculated by means of:

Q_ b!AT ¼ AAT Ib?

where Ib? is the component of the beam solar radiation perpendicular to the surface of the test area. Using the relationships previously derived in Eq. (20), the net heat flow from the test area, Q_ AT , can be obtained:

Q_ AT ¼ reAT AAT T 4AT
aAT

2

F AT!i ¼

ð31Þ

rAAT

F AT!i T 4i þ Id AAT

i¼1

1 6 a=c b=c 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p 2 1 þ ða=cÞ 1 þ ða=cÞ2 3

! ð32Þ

j¼1

Q_ AT$S ¼ Q_ AT

ð26Þ

ð33Þ

Therefore the flow exchanged between the test area and the environment may be expressed by:

Case (b) Orthogonal surfaces

2

3

Q_ AT$S ¼ reAT AAT T 4AT

1 6
1 1 c=b 1 7 F AT!i ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 4tan 2p ðc=bÞ 2 2 2 2 ða=bÞ þ ðc=bÞ ða=bÞ þ ðc=bÞ


aAT

N M X X rAAT F AT!i T 4i þ Id AAT F AT!j þ AAT Ib? i¼1

ð27Þ

ation, Q_ b!AT , respectively:

Q_ sun!AT ¼ Q_ d!AT þ Q_ b!AT

Q_ AT$S ¼ reAT AAT ðT 4AT
T 4pr Þ

In the hypothesis that the diffuse radiation entering the environment through glazed surfaces follows the Lambert’s law, its contribution to the radiative flux on the test area is given by:

ð29Þ

ð35Þ

Q AT

j¼1

being F j!AT the view factor between the jth glazed surface and the test area, Aj the area of the jth glazed surface, and Id the intensity of the diffuse solar radiation on the test area. Hence, exploiting the reciprocity rule, the Eq. (29) can be expressed as: M X F AT!j

ð34Þ

j¼1

follows:

ð28Þ

M X F j!AT Aj Id

!

Indeed, on the basis of the definition of plane radiant temperature, in the hypothesis that the test area may be considered as a small surface with respect to the environment dimensions (Fig. 14), the thermal heat flow Q_ AT$S can also be calculated as

Then, the contribution of solar radiation, Q_ sun!AT , consists of two components related to diffuse radiation, Q_ d!AT , and to direct radi-

Q_ d!AT ¼ Id AAT

M X F AT!j þ AAT Ib?

Furthermore, the radiative net flow leaving the test area is equal to the thermal flux exchanged between the same test area and the environment (La Gennusa et al., 2005):

b=c a=c 7 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 2 2 1 þ ðb=cÞ 1 þ ðb=cÞ

Q_ d!AT ¼

N X

S

AT

ð30Þ

j¼1

with the view factor F AT!j calculated with the Eqs. (26) and (27).

Fig. 14. Radiative exchanges between a black enclosure and the test area.

Case a) Parallel surfaces

b

case b) Orthogonal surfaces

b

a

a

i

i 90°

90° c

n

n

AT

AT

c

Fig. 13. Geometric configurations suitable for the calculation of the angle factor.

C. Marino et al. / Solar Energy 144 (2017) 295–309

Therefore, considering Eqs. (33)–(35), the following equation can be written:

reAT AAT T 4AT
aAT rAAT

N M X X F AT!i T 4i þ Id AAT F AT!j þ AAT Ib? i¼1

!

j¼1

¼ reAT AAT ðT 4AT
T 4pr Þ

ð36Þ

Which, considering that the absorption coefficient depends on the radiation’s wavelength and temperature of the source, yields:

T pr

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N M X u ad X ab 4 aAT ¼t F AT!i T 4i þ Id F AT!j þ Ib?

eAT

i¼1

reAT

j¼1

reAT

ð37Þ

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