Solar energy through windows

Solar radiation impinging a glazed surface can be transmitted, reflected or absorbed. Due to the energy conservation, the sum of the energy absorbed (Ia), reflected (Ir) and transmitted (It) is equal to the amount of incident energy (I):

r a

t

I I

I

I = + +

^(4.43)

Dividing by the overall energy impinging the surface, the following equation can be derived for the coefficients of sorption (α), transmission (τ) and reflection (ρ):

= 1 + + α ρ

τ

^(4.44)

The characteristics of a glazing element depend on the wavelength of the considered solar radiation. The absorption, transmission and reflection coefficients depend also on the angle of incidence of the solar radiation with respect to the normal incidence of the glazing surface (θ) [49], as reported in Figure 4.16.

Figure 4.16: Transmission, sorption and reflection coefficients as a function of the incident angle

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 30 60 90

Incident angle

αα αα ρρρρ

ττττ

86

For obtaining the overall value of a coefficient (absorption, reflection or transmission), the incident radiation and the correspondent absorbed, reflected and transmitted radiation has to be integrated on the wavelength. For this purpose the incident angle has to maintained fixed, hence usually the values are provided for an incident radiation normal to the considered surface (θ=0°). As an example, the characteristics of the solar radiation transmittance is reported in Figure 4.17. The overall value of the generic coefficient M derives by the integration within the considered range [λ1, λ2] of the response M(λ) of the glazed element with respect to the solicitation P(λ):

∫

∫

=

2

1 2

1

) (

) ( ) (

λ

λ λ

λ

λ λ

λ λ λ

d P

d P M

M (4.45)

Solar energy through windows can be considered along the whole wavelength or only in the range of visible wavelength. The first case refers to the overall energy characteristics of the glazing surface with respect to the solar radiation (between 300 nm and 2500 nm), while the second case refers to the visual transmission through a glazed element (between 380 nm and 780 nm).

Figure 4.17: Transmittance characteristics of different glazing systems with respect to wavelength

4.3.4.1 Solar transmittance

Solar transmittance (τs) depends on the angle θ and on the wavelength λ. The intergal of the solar transmission over the whole wavelength range (between 200 nm and 2500 nm) is weighted on the curve of the energy intensity of a solar radiation with AM = 2:

300 400 500 600 700 750 800 900 1000 1500 2000

Visible Infrared

Ultra-violet

Clear glass

Coloured glass

Wavelength [nm]

87

( ) ( )

∫ ( )

∫

^{⋅ ⋅}

= nm

nm nm s nm

d E

d E

2500 300 2500

300

, λ λ

λ θ λ τ λ

τ (4.46)

In figure 4.18, the different behaviours of some glazing elements are reported. As can be shown, a clear glass (float) permits a uniform transmission all over the wavelength.

Glasses used in the past with colour bronze or grey limit the solar transmission in a uniform way along the whole wavelength. Glazing elements with selective behaviour allow transmission in the range of visible radiation, while they reduce the transmission in the infrared wavelength range (above 750 nm).

The solar transmittance alone can give an idea on the possible energy transmission through the glazing, but it does not represent the energy transmitted through the glazed element. In fact the solar radiation impinging a glazing system is reflected, absorbed and transmitted. The transmitted energy enters directly in the room, while the reflected energy does not affect thermal balance of the glazed surface. The absorbed radiation heats the glazing, therefore part of the energy enters the room via convection with the air and partly through infrared radiation with the other surfaces.

Figure 4.18: Transmittance characteristics of different glazing systems with respect to wavelength

Transmittance[%]

Wavelength [nm]

visible

UV infrared

float

selective bronze

grey

0 20 40 60 80 100

200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400

88

Figure 4.19: Thermal processes involved in the energy transmitted through a glazing surface

Source: [30]

This fact explains that this aspect is more complicated to be described than the solar transmittance. There are three ways to calculate the Solar Heat Gain (SHG):

1. Solar transmission through a reference clear glass panel 2. Solar transmission compared to the overall incident radiation

3. Detailed calculation within the elements which form the glazing system 4.3.4.2 Solar gain through a reference clear glass panel

The reference value of the solar transmission is usually estimated with a normal incident radiation. If the solar radiation impinges the glass with a different angle θ, the behaviour of the glass depends, among other factors, especially on the incident angle θ, as shown in Figure 4.16. For considering the behaviour of the energy transmitted through a generic glass for different incident angles, the simplest method compares the energy transmitted through the glass (Igl) to the energy entering the room through a reference glazing surface (Iref,gl), i.e. 3 mm float glass. The parameter defining the energy transmitted through the glass is called shading coefficient (Cs):

gl ref

gl

s I

C I

,

= (4.47)

This is the easiest way to calculate the solar heat gain. From a research investigation carried out in the 60ies [50] the following equation can be used for the k-th generic window:

(

bk bk

) (

d k g k

)(

d k d k

)

k b k gl

ref I B A I I B A

I _{, ,} = _{, ,} + _, + _, + _{, ,} + _, (4.48)

where coefficients A_b, B_b, B_d and A_d depend on the solar incident angle θ as follows:

( )

¹

6 1

cos

−

∑

=

=

j j

b bj

B θ (4.49)

89 Table 4.7 an example of shading coefficients to be used in calculation is reported.

Once calculated the entering solar energy through each window in the considered time step (I_s,k), the overall solar gain can be determined as follows:

k

Table 4.5: Values of coefficients for determining the solar heat gain of a reference glass

j bj aj

Table 4.6: Solar thermal flux Iref,gl transmitted by the reference glass on 21 July at 42° N latitude (W/m²)

90

Table 4.7: Shading coefficients for some glazing systems and shadings

Cs [-] U [W/(m² K)]

Single clear glass (3 mm) 1 6.33

Single clear glass + clear venecian blind (45°) 0.36 3.76 Single clear glass + dark venecian blind (45°) 0.50 3.76

Single clear glass + medium courtain 0.68 3.76

Double clear glazing 0.77 2.95

Double clear glazing + clear venecian blind (45°) 0.39 2.17 Double clear glazing + dark venecian blind (45°) 0.47 2.17

Double clear glazing + medium courtain 0.56 2.17

Double clear low emission glazing 0.55 1.85

Triple clear glazing 0.65 2.09

4.3.4.3 Solar gain compared to the overall incident radiation

The energy transmitted through a glass can also be defined considering, as reference, the impinging solar external radiation [51]. This parameter is called g-factor and it represents the sum of the transmitted energy through the glass and the fraction c of the energy absorbed by the glass which is transferred (via convection and infrared radiation) in the room (Figure 4.20) for a normal angle of incident radiation:

α τ c I

cI

g I^t + ^a = +

= (4.54)

Since the reference glazing has g = 0.89, it is easy to demonstrate that the solar factor and the shading coefficient are linked via the following equation:

89 .

, 0

g I

C I

ref gl

gl

s = = (4.55)

91

Figure 4.20: Determination of the g factor

In this case the solar energy entering the room is easy to determine for a generic angle of incidence by means of equation (4.48), changing equation (4.53) with the following

4.3.4.4 Detailed balance within the glazing elements

Solar heat gain is quite complex to estimate in a detailed way. The solar energy entering the room depends on the amount and characteristics of the glazing elements composing the window and on the presence of a shading device. The calculation is based on an iterative solution which tends to minimise the error of the solution [52, 53, 54]. Named i the generic glazing element, I₁(i) is the radiation leaving the glass on the external surface and I₂(i) is the radiation leaving the glass on the internal surface, which depend on the boundary conditions and on the characteristics of the considered glazed element

92

Where τ(i) is the transmission coefficient, ρ1(i) is the reflection coefficient of the outer surface, ρ2(i) is the reflection coefficient of the inner surface I_a(i) is the absorbed radiation. Applying the equations (4.57), (4.58) and (4.59) to all the glazing surfaces of the window, the solar radiation transmitted into the room is defined. Moreover the absorbed radiation I_a(i) in each glazing element is defined and is then considered in the thermal balance of the glazing elements. The generic glazing element can be transparent or a shading device with suitable values of absorption, transmission and reflection coefficients to be used in equations (4.57), (4.58) and (4.59).

Figure 4.21: Radiation processes in a general i-th glazing element

The general scheme for the thermal balance inside a window is defined byu means of a resistance network, where conduction through glazing elements, convection and infrared radiation in cavities are considered (Fig. 4.22a). For the generic i-th element, defined 1 and 2 the inner and outer surface respectively, the following equation can be written:

( ) ( )

transferred into the room via convection with air and infrared radiation with all the other surfaces is already included in the model and solved internally by means of the set of equations (4.60) for each k-th window. Therefore with this method, the solar radiation to be considered impinging the room surfaces is:

)

( _{, , , , , ,}

,k k Dtotk Dk dtotk dk gtotk gk

s S I I I

q = τ +τ +τ ^(4.61)

Usually the directions of the direct, diffuse and reflected solar radiations differ, hence in the model the mutual reflections, absorptions and transmissions have to be separated for

93

considering the different behaviour with respect to incident angles of the glazing elements. This means that equations (4.57), (4.58) and (4.59) should be calculated separately for the direct, diffuse and reflected incident radiations.

A simplest version of the model can be carried out by means of the overall resistance of the window Rwindow (without overall heat exchange coefficients with the indoor and outdoor environments), together with its overall absorption, transmission and reflection coefficients (Figure 4.22b). In this way, the overall absorption solar radiation can be split in two, hence it is included in the external and internal surface equations, as shown in paragraph 4.3.6.

a b

Figure 4.22: Possible schemes for determining window thermal balance 4.3.4.5 Detailed thermal balance within double skin facades

Double skin facades are increasingly used in architecture. The thermal behaviour to solar radiation is quite complex, since air is flowing inside the cavity where a screen is installed for stopping part of the solar radiation entering the room.

The simplest way to determine the behaviour of these elements with respect to the solar radiation is to use fixed values of transmittance U and solar factor g, as declared by manufacturers [55].

Another possibility is to solve directly the thermal balance inside the double skin [53].

For considering the temperature gradient inside a ventilated glazing cavity, the cavity itself should be discretised into n vertical elements, each of them with the same length of the room, as shown in Figure 4.23.

For each vertical layer there are two possible schemes for the thermal balance inside the ventilated element. One possibility is to calculate the same temperature of the air before and after the shading element (Fig. 4.24a); the other possibility is to assume different temperatures before and after the shading element (Fig. 4.24b). These two approaches depend if the screen is permeable to the air or not.

In the first approach (one unique temperature before and after the shading device in the air cavity) the equation describing the thermal balance for each vertical layer is:

( ) ( ) ( )

equations can be used:

94

where A and B mean the temperatures of the air after and before the solar shading. The temperature of the air entering the air layer from the previous air node layer is named ta(j-1). The temperature of the previous node ta(j-1) is determined once set the mass flow rate balance and the continuity equations.

Figure 4.23: Vertical discretisation of a double skin facade

a b

Figure 4.24: Possible schemes for the thermal balance within a double skin facade

In this case, the solar radiation to be used in the thermal balance of the room is again described by equation (4.61). This means that the solar radiation behaviour of the

95

window is described in terms of solar radiation transmission. The part of the absorbed radiation which is transferred into the room via convection with air and infrared radiation with all the other surfaces is already included in the model and solved internally by means of the set of equations (4.60) for each k-th window.

4.3.4.6 Visual transmittance

Visual transmittance (τv) is estimated not only by means of the optical properties of the glazed element, but also by means of the effect that the visible component of the solar radiation (380-780 nm) has on human eyes. Human vision sensibility depends on the spectrum of solar radiation; as a matter of fact the peak of sensitivity of human eyes is related to 480 nm (colour blue), as shown in Figure 4.25. The function describing the sensitivity of the human eye to the visible radiation is called V(λ). The visual transmittance can be thus estimated as:

( ) ( ) ( )

In practice, visual transmittance is estimated by means of the response of two filters, i.e.

the glazed element and the human eye, and it represents already the effective perceived visible solar radiation passing through the considered glazing surface for a normal incident radiation.

Figure 4.25: Response of human eye to the visible solar radiation spectrum 4.3.5 Solar radiation modelling in the rooms

There are different models for dealing with solar radiation inside a room. Four possible models are here presented. Two of the models contain the standard criteria which are used in building simulation techniques while the other two are detailed. In the latest two

0.00

380 480 580 680 780

Wavelength [nm]

Normalisedsensitivity[-]

96

models the surfaces of the room have to be discretised in sub-regions, i.e. in surface elements.

4.3.5.1 Standard simplified models

In the simplest model, solar radiation is distributed in a uniform way, without taking into account the absorption coefficients of the surface elements or the colour of the surfaces.

This is the most common model used in room simulations, where the solar radiation is not assigned to the glazed elements. The room is a black body that absorbs all of the solar radiation with no radiation exiting through the windows. The general equation to be used in the room balance is the equation (4.29).

An alternative model has been proposed in CEN TC 89WG 6 [56], in order to calculate the design performance of the radiant systems under cooling conditions. The overall radiation entering the room is distributed depending on the type of surface (floor, ceiling, wall or window) and the colour of the floor (tables 4.8 and 4.9). Usually, with this model the incoming solar radiation is supposed to remain in the room, i.e. it does not exit through windows. Internal radiant gains are usually also shared between surfaces with the same criterion.

Table 4.8: The solar energy distribution coefficients for the dark-coloured floor

Floor Ceiling Wall Window

tot f

f

S S

S +

⋅ 2

tot f

c

S S

S

+ _{f tot}

op

S S

S +

0

97

Table 4.9: The solar energy distribution coefficients for the bright-coloured floor

Floor Ceiling Wall Window

In each time step of the first detailed model, depending on the reciprocal position of the surface element and the Sun, the surface elements that are struck by direct solar radiation through the window are determined (Figure 4.26). Such surface elements have a non-null reflection coefficient; thus, the direct solar radiation is redistributed to the surrounding elements. The multiple reflections between the surface elements can be expressed by (Figure 4.27):

∑

= − ⋅ − ⋅

which leads to the following linear system:



The direct solar radiation is calculated at each time step under the hypothesis of a clear sky, as proposed by Wen and Smith [57]. Windows are assumed to have an influence on energy transmission but not on solar direction. In this way, it is not possible to couple windows with internal shading devices because the shading device itself acts as a diffuse radiation source.

The internal distribution of the solar radiation is affected by the discretisation of the room. In this case, if a surface element is only partially subject to direct solar radiation, it is assumed that no solar radiation impinges on the surface element and that the related part of the direct solar radiation is added to the diffuse radiation.

After the distribution of the direct solar radiation in the room has been determined, the diffuse radiation and the albedo solar radiation are distributed as follows:

( )

The sum of I and ID is the overall incident solar radiation on the single-surface element.

To determine the dynamic thermal behaviour of the room, the calculation of the radiation that is absorbed by each surface element is important because it is a boundary condition for the calculation of the thermal balance of the room. Therefore, for the generic i-th element, the following equation can be written:

( )

Ia i =αi ⋅Ii + Id (4.69)

If the internal radiant gains are present, the load is distributed and added to the absorbed solar radiation that is expressed by equation (4.69).

98

Figure 4.26: Identification of building elements that are subject to direct solar radiation

Figure 4.27: Schematic of the distribution of solar radiation inside of the room in the detailed model

There is an alternative way to build up a detailed model. In this case surface elements that are subject to direct solar radiation are determined as in the previous model but, in this case, the direct radiation that is absorbed by the surface also depends on the incidental angle of the solar radiation on the surface, i.e. on the angle between the impinging radiation and the normal to the surface. Such a model allows the assignment of direct solar radiation considering the mutual orientation of the surfaces based on the position of the centre of the elements. Therefore, the discretisation may affect the calculations in this model and errors may occur in two cases:

• the area of the whole illuminated surface is null (no centre of the elements is lighted) but the power entering from the window is not zero;

Fik⋅(1-αk)⋅Ik

Ii

Fij⋅(1-αj)⋅Ij

IBi

αi⋅Ii

(1-αi)⋅Ii

k-th surface element

j-th surface element

i-th surface element Sun

Window

99

• the surface of the window that is projected onto the plane perpendicular to the incident angle of solar radiation differs from the sum of the projections of the single surface elements on the same plane.

The correction procedure changes depending on the case. In the first case, the whole direct solar radiation entering the room is added to the diffuse radiation and is distributed as described hereafter. In the second case, the remaining radiation is added to the impinged surface elements via the equation:

∑

=

The diffuse radiation that passes through the window is assigned to the surface elements of the window, which acts as a radiant source for the other internal surface elements, i.e.

the window is a perfect diffuse radiation surface source [58]. This model can be summarised using the following equation:

[ ]

where Idk is not null for surfaces that are not windows.

4.3.5.3 Remarks on the models of solar radiation in a room

In a recent work [37] it has been shown that if one aims to accurately predict localised thermal behaviour of a room, the detailed method, which takes into account the re-irradiation, must be utilised. If the evaluation is based on average values (temperatures, heat flows and comfort), then the rough models are also valid. Furthermore, when solar radiation is considered to be uniformly distributed comfort conditions at a single, central point of the room can be calculated. Whereas, in the case of the accurate models, comfort conditions at specific points in the room can be predicted, thus aiding space distribution in the design phase.

Once a model with only one surface and no element discretisation is considered, it is unnecessary to split the radiation into different methods in order to take into account the colour of the surfaces or their position. More detailed models do not achieve results which are significantly different from those of simplified models which assume solar radiation entering a room to be diffuse.

4.3.6 Example of a balance in steady state conditions

In this paragraph a simplified method for a first sizing of a cooling system is presented for a better understanding of the thermal balance of a room. In this case (Figure 4.28) one external surface (the glazed roof) and one internal surface (the floor) are considered as surface elements. The solar radiation is partly absorbed by the glazed roof (Ia) and the remaining transmitted part is absorbed by the floor (It). Internal gains are partly

100

convective (qc,int) and partly radiant (ql). Handled air enters the room at temperature Ta,in.

As for the absorbed solar radiation, a simple approach is to fix the thermal characteristics of the glazed surface (thermal resistance Rceil) and to split the absorbed solar energy into

As for the absorbed solar radiation, a simple approach is to fix the thermal characteristics of the glazed surface (thermal resistance Rceil) and to split the absorbed solar energy into

In document Simulation and numerical methods Energy modeling for buildings and components (Pldal 85-0)