# Appendix 5.A11: Derivation of solar gain factors

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2 A11- Environmental design t=4 Φ et = 1/4 Σ Φ ett Φ ett = T I t t=1 A11.5 A11.4 where Φ ett is the overall gain to the environmental node from transmitted radiation at time t. However, in practice, direct and diffuse transmitted radiation must be treated separately. The gain to the environmental node due to conduction and radiation from the inner surface of the glazing is: j=n Φ eat = Σ H e j A j I t j=1 A11.7 where s the component of the radiation absorbed by the glass, subscript j denotes the number of the glazing element and n is the total number of glazing elements within the window system. Thus the total gain to the environmental node is: Φ et = Φ ett + Φ eat A11.8 Additionally, if there is an internal blind, the gain to the air node is: j=n Φ aat = Σ H aj A j I t j=1 A11.9 Thus the total gain to the air node is: Φ at = Φ aat A11.10 To simplify the calculation of these gains, solar gain factors are used. These are the ratios of the components of the gain to the incident solar radiation. The room load has both steady state and cyclic components and the space gains are to the environmental and, possibly, the air nodes. Additionally, the surface factor depends on the response time of the space. To calculate the solar gain factors, typical values are taken, as follows: for slow response space: F = 0.5; time delay = h for a fast response space: F = 0.8; time delay = 1 h The solar gain factors are as follows: S e = Φ e / I S ~ et = Φ~ et / I~ t S a = Φ a / I S ~ at = Φ~ at / I~ t A11.11 A11.1 A11.13 A11.14 Solar gain factors for generic glass and blind combinations are given in Table 5.0 repeated here as Table 5.A11.1. These have been calculated using banded solar radiation data for Kew incident on a south-west facing vertical window see chapter. The transmission t, absorption a and reflection r com ponents for thermal Table 5.A11.1 Solar gain factors and shading coefficients for generic glazing/blind combinations Description inside to outside Solar gain factor at Solar gain factor Shading coefficient, S c environmental node at air node S e S~ el S~ eh S a S~ a Shortwave Longwave Single glazing/blind combinations: clear glass absorbing glass absorbing slats/clear reflecting slats/clear generic blind/clear Double glazing/blind combinations: clear/clear clear/reflecting low emissivity/clear low emissivity/absorbing low emissivity/clear/ generic blind absorbing slats/clear/clear absorbing slats/clear/reflecting absorbing slats/low emissivity/clear absorbing slats/low emissivity/absorbing reflecting slats/clear/clear reflecting slats/clear/reflecting reflecting slats/low emissivity/clear reflecting slats/low emissivity/absorbing generic blind/low emissivity/clear Triple glazing: clear/clear/clear clear/clear/absorbing clear/clear/reflecting clear/low emissivity/clear For S ~ e, subscripts l and h denote thermally lightweight and heavyweight buildings, respectively Note: shading coefficients for windows with slatted blind or windows with inner blind are not given since these not compatible with the properties of plain glass

3 Thermal design, plant sizing and energy consumption: Additional appendices A11-3 Table 5.A11. Transmission, absorption and reflection components and emissivities for generic glass and blind combinations Description Shortwave radiation Longwave emissivity proportions of total Transmitted Reflected 1 Reflected Surface 1 Surface Glass: clear low emissivity* absorbing reflecting high performance* Slatted blind : reflecting absorbing Generic blind * Asymmetric glass properties shortwave radiation and emissivities for thermal longwave radiation for the generic glass and blind types used in calculating the solar gain factors are given in Table 5.A11.. Effectively, solar gain factors can only be calculated by means of a computer program. The following sections describe the basis of the calculation procedure. 5.A11.3 Clear glass Transmission, absorption and reflection for direct solar radiation For clear glass the transmission, absorption and reflection tar coefficients can be derived theoretically Jones, The angle of refraction is obtained from the angle of incidence using Snell s Law: ζ r = arcsin sin ζ i / μ A11.15 The reflected beams for radiation polarised parallel to and perpendicular to the plane of incidence are determined using Fresnel s formula: tan ζ i ζ r r // = tan ζ i + ζ r sin ζ i ζ r r = sin ζ i + ζ r A11.16 A11.17 As the angle of incidence approaches 0 i.e. normal incidence: This is a useful result as it enables the calculation of the extinction coefficient k if the transmission at normal incidence T n is known. The extinction coefficient is a non-linear function of the glass thickness L and is related to the transmission coefficient by: 1 r exp k L T n = 1 r exp k L A11.0 For the beam polarised parallel to the plane of incidence, the fraction of incident energy absorbed for each beam is calculated as follows: a // = 1 exp k L / cos ζ r and similarly for the perpendicularly polarised beam. A11.1 The transmitted, absorbed and reflected coefficients are calculated separately for each beam i.e. parallel and perpendicularly polarised and the average taken to give the overall coefficients. For the beam polarised parallel to the plane of incidence: 1 r 1 a // T D// = 1 r 1 a // a // 1 r [1 + r 1 a // ] A D// = 1 r 1 a // r 1 r 1 a // R D// = + r 1 r 1 a // A11. A11.3 A11.4 tan ζ i > sin ζ i > ζ i A11.18 and similarly for the perpendicularly polarised beam. hence: Therefore: μ 1 r // > r > μ + 1 A11.19 T D = 1 / T D// + T D A11.5 and similarly for the absorption and reflection coefficients.

4 A11-4 Environmental design Note that since the transmitted, absorbed and reflected components add up to unity, only two need be calculated, the third being obtained by subtraction. Reflecting and other glasses The characteristics of such glasses differ from those for plain glass and therefore must be obtained from the manufacturers. If the characteristics are supplied as a graph of tar coefficients against angle of incidence, the appropriate values can be read-off directly or by curve-fitting techniques. Slatted blinds The analysis is the same for both horizontal and vertical slatted blinds. Radiation may be transmitted into a room by the following paths Parmelee and Vild, direct: i.e. passes through the blind without touching any surface; may be zero reflected 1: i.e. passes through the blind after one reflection from the slat surface which is directly irradiated by the sun reflected : i.e. passes through the blind after undergoing any number of reflections, the final reflection being from the slat surface opposite the one directly illuminated by the sun reflected 3: i.e. passes through the blind after undergoing any number of reflections, the final reflection being from the one directly illuminated by the sun. In order to calculate these components, up to five configuration factors are required, each of which depends on the blind geometry. The number of factors needed depends on whether all or only part of the slat is illuminated. The amount of a slat that is illuminated i.e. not shaded by the slat above it depends on the geometry of the blind and the profile angle. The profile angle β is the angle that the direct radiation beam makes with the blind in a vertical plane perpendicular to the plane of the window. For horizontal slatted blinds on a vertical window, the profile angle is the vertical shadow angle: β = σ v = arctan tan h sec γ s A11.6 For vertical slatted blinds on a vertical window, the profile angle is the wall solar azimuth: β = γ s = φ γ A11.7 In the following analysis, it is assumed that the radiation is incident on the upper surface of the slat. The width of slat that is illuminated is calculated from: D cos β M = min W, sin β + ψ The configuration factors are calculated as follows. A11.8 Radiation that is reflected by the lower slat and passes into the room when the whole width is illuminated C 1 : C 1 = 1 / {1 + D / W [1 + D / W + D sin ψ / W] 1/ } A11.9 Radiation that is reflected by the lower slat and intercepted by the upper slat when the whole width is illuminated C: C = 1 / {[1 + D / W + D sin ψ / W ] 1/ + [1 + D / W D sin ψ / W ] 1/ D / W} A11.30 Radiation reflected by the upper slat which passes into the room C 3 : C 3 = 1 / {[1 + D / W [1 + D / W D sin ψ / W ] 1/ } A11.31 Radiation reflected by the lower slat, which passes into the room when the lower slat is partially shaded C 4 : C 4 = 1 / 1 + {[W M / M ] + D / M + [ W M D sin ψ / M ]} 1/ [W / M + D / M + W D sin ψ / M ] 1/ A11.3 Radiation reflected by the lower slat, which is intercepted by the upper slat when the lower slat is partially shaded C 5 : C 5 = 1 / [W / M + D / M + D W sin ψ / M ] 1/ D / M + [1 + D / M D sin ψ / M] 1/ {[W M / M ] + D / M + [ W M D sin ψ / M ]} 1/ A11.33 If the whole of the lower slat is illuminated and some radiation may pass directly into the room, the tar coefficients for the blind are calculated from: W sin φ + ψ T D = 1 D cos φ C 1 a [C 3 + C 1 C 1 a] 1 C 1 1 a 1 C 1 a a W sin φ + ψ A D = D cos φ [1 C 1 a] R D = 1 A D T D A11.34 A11.35 A11.36

5 Thermal design, plant sizing and energy consumption: Additional appendices A11-5 Where part of the lower slat is shaded by the slat above: Roller blinds T D = C 4 1 a + {C 5 1 a C 3 + C 1 C 1 a A C 1 a C 5 1 a A D = a C 1 a R D = 1 A D T D A11.38 A11.39 The properties for roller blinds are not well defined. It is generally sufficient to assume that the tar coefficients are independent of the angle of incidence and take the values at normal incidence supplied by the manufacturers. 5.A11.4 Transmission, absorption and reflection for sky diffuse and ground reflected radiation Transmission, absorption and reflection coefficients for glasses and blinds are calculated by considering the direct properties over a range of angles appropriate to the radiation. For glass, the tar values for sky diffuse and ground reflected radiation are the same since glass has symmetrical properties. The characteristics for roller blinds can be assumed to be the same for direct and diffuse radiation. However, slatted blinds are highly asymmetrical so the two sources of diffuse radiation must be calculated separately. Glasses The standard properties are calculated on the assumption that the glass is exposed to a hemispherical source of uniform radiance therefore the transmission and absorption angles are from 0º to 90º. Mathematically, the expressions for tar could be integrated over this range, i.e.: T d = e 0 90 Td ζ i sin ζ i dζ i A11.40 In practice the direct properties are summed for angles of incidence from.5º to 87.5º at intervals of 5º, i.e.: ζ=87.5 T d = Σ {T D ζ [sin ζ i +.5 sin ζ i.5]} ζ=.5 A11.41 A d is calculated similarly and R d is obtained by subtraction from unity, see equation A Slatted blinds The direct properties are summed for profile angles from 5º to 85º at intervals of 10º for sky diffuse radiation. For ground reflected radiation, they are summed from 85º to 5º at intervals of 10º taking into account the configuration factor of the hemispherical radiating source bounded by profile angles of β + 5º and β 5º Nicol, Thus, for sky diffuse radiation: β=85 T ds = Σ {T D β [sin β +.5 sin β.5]} β=5 A11.4 For ground reflected radiation: β= 85 T dg = Σ {T D β [sin β +.5 sin β.5]} β= 5 A11.43 A ds and A dg are calculated similarly and R ds and R dg are obtained by subtraction from unity, see equation A A11.5 Properties of glass and blind combinations The properties of multiple layer windows can be calculated from the properties of the individual components. There are many glass types and many permutations; the method of calculation is demonstrated in the following for double and triple glazing using generic glass and blind types. In the same way that the properties of a single sheet of glass are calculated from the fundamental properties of the glass and an infinite number of inter-reflections at both glass/air interfaces, the properties of multiple glazing are calculated by considering the inter-reflections between the component layers Jones, 1980; Mitalas and Stephenson, 196. These calculations are performed for both direct and diffuse radiation. However, if the window incorporates a blind, the radiation reflected by or transmit ted through it is assumed to be diffuse whatever the nature of the source. This is because the slat surfaces are assumed to be diffusing rather than specular reflectors Parmelee and Vild, The following equations are derived from Figure 5.A11.11 where all layers are symmetrical, i.e. both surfaces of the layer have the same reflection and the specularity of the radiation is not changed by the layer. If any of the layers are asymmetrical, the equations become more complicated since they have to include the reflection of both surfaces of the layer. If any of the layers is a diffusing slatted blind, then the direct radiation equations need to include the diffuse properties of the elements for radiation that has been reflected by the blinds. Examples for some of these situations are given elsewhere Jones, Double glazing tar coefficients for double glazing, denoted by prime, are as follows: T = T i / 1 R o A11.44 = + [ / 1 R o Ri] A11.45 = / 1 R o A11.46 R = 1 T A11.47

6 A11-6 Environmental design Incident where subscript o denotes the outer glazing element and subscript i denotes the inner glazing element. Triple glazing R o tar coefficients for triple glazing, denoted by double prime, are as follows: T i T c T i T = 1 R o 1 T c R o A11.48 R o a etc. = + 1 R o Incident Outer pane Inner pane T c + A R o 1 T c R o A c A c 1 + T c A c = A R o 1 T c R o T c etc. T c = 1 R o 1 T c R o A11.51 etc. T c To T c T i R = 1 T A c A11.5 b Outer pane Centre pane Inner pane Figure 5.A11.11 Transmitted, absorbed and reflected radiation; a double glazing, b triple glazing where subscript o denotes the outer glazing element, subscript c denotes the central glazing element and subscript i denotes the inner glazing element. The heat gain to the environmental node due to conduction and radiation from the inner surface of the glazing is given by equation A11.7. If there is an internal blind, the additional heat gain to the air node is given by equation A11.9. In these equations, the transmittance factors H depend on the values taken for the thermal resistances i.e. the radiant and convective heat transfer coefficients of the layers of the window. They are calculated by considering the thermal resistance network for the window. Figure i j k o θ ai R vi θ ij R vj R vk R vo θ jk θ ao θ ei θ eo R rij R rjk θi θj θk θo Figure 5.A11.1 General thermal resistance network for a multiple-layer window

7 Thermal design, plant sizing and energy consumption: Additional appendices A A11.1 shows the general thermal resistance network for a multiple-layer window. The properties of the glazing systems are calculated using the following standard thermal resistances and heat transfer coefficients: thermal resistance between inner surface of window and environmental point i.e. inside thermal resistance: = 0.1 m K W 1 thermal resistance between outer surface of window and sol-air temperature i.e. outside thermal resistance: = 0.06 m K W 1 convective resistance between a window layer and the air: = 0.33 m K W 1 for vertical window, corresponding to h c = 3 W m K 1 radiative resistance between two layers j, k of window: R r j,k = ε j + ε k ε j ε k / h r ε j ε k A11.53 if both layers have an emissivity of 0.84 and h r = 5.7 W m K 1, R r j,k = 0.4 m K W 1 ventilation resistance across window layer between adjacent air spaces: R v = 0 m K W 1 if the layer is a blind; R v = if the layer is glass. Example A11.1: Triple glazing without blinds Figure 5.A11.13 shows the network for triple glazing and Figure 5.A11.14 shows the simplified network resulting from evalua tion of the parallel resistances. The total resistance of the network is: Σ R = + c + o + = = 0.54 m K W 1 where c is the thermal resistance between inner and central elements of the glazing m K W 1 and o is the thermal resistance between central and outer elements of the glazing m K W 1. The transmittance factors for the inner, central and outer elements of the glazing can be shown to be: H ei = c + o + / Σ R = / 0.54 = 0.78 H ec = o + / Σ R = / 0.54 = 0.44 H eo = / Σ R = 0.06 / 0.54 = 0.11 From equation A11.7, the cyclic component of the convective and longwave radiant gain from the glazing to the environmental node is calculated as follows: H ~ e A = H ei A~ i + H ec A~ c + H eo A~ o i c o c o θ eo Figure 5.A11.14 Simplified thermal resistance network for triple glazing i c o θ i R ric R rco θeo Figure 5.A11.13 Thermal resistance network for triple glazing Table 5.A11.3 Example A11.1: components of radiation Time / h Solar Radiation absorbed/transmitted Gains to space / W m irradiance by glazing system / W m / W m Radiation absorbed by Directly Cyclic component Cyclic component of transmitted inner, central and outer transmitted of absorbed radiation for lightweight l and glazing elements radiation, T radiation, H e A heavyweight h buildings A c T ~ L T~ H Mean:

8 A11-8 Environmental design Table 5.A11.3 summarises the steps in the calculation of the solar gain to the space by means of an example. The calculation was carried out as follows. For 1:00 h: H ~ e A = = 36.6 The gain at other times is calculated similarly. The mean gain is calculated using the mean, rather than the cyclic, absorption values. The cyclic component of the directly transmitted shortwave radiation is attenuated by the surface factor F, which is appropriate to the thermal weight of the building and corresponding time delay, see Table 5.A11.4. For a lightweight building, F = 0.8 and the time delay is 1 hour, i.e.: T ~ L = 0.8 T t+1 T and for a heavyweight building, F = 0.5 and the time delay is hours, i.e.: T ~ H = 0.5 T t+ T where subscript L denotes thermally lightweight building and subscript H denotes thermally heavyweight building. The mean solar gain factor is given by: mean transmitted radiation plus mean absorbed radiation S e = daily mean incident radiation Hence: S e = / 179 = 0.51 The cyclic solar gain factors are calculated using the gains appropriate to a time one or two hours after the time of peak radiation, depending on the thermal weight of the structure, i.e.: total swing in gain to space S ~ e = swing in external gain Peak solar irradiance occurs at 14:00 h; hence, for a thermally lightweight structure i.e. 1-hour delay: S ~ el = / = 0.48 and for a thermally heavyweight structure i.e. -hour delay: S ~ eh = / = 0.4 Example A11.: Single glazing with internal absorbing blind Figure 5.A11.15 shows the network for single glazing with an internal blind and Figure 5.A11.16 shows the simplified net work resulting from evaluation of the parallel resistances. In this case, there are transmittance factors to both the air and environmental nodes, which are calculated as follows: x = R rio + [ / + ] = [ / ] = 0.8 x / + x H ei = + [ x / + x ] / = = [ / ] x / + x H ai = + [ x / + x ] / + + R ix + [ / + ] + = 0.4 R ox = R rio + [ / + ] = 0.3 / + H eo = R ox + / + + = 0.10 R ox / + R ox H ao = + [ R ox / + R ox ] / + + R ox + [ / + ] + = 0.17 Table 5.A11.4 Thermal response Thermal Typical features of Response Response to short-wave radiation Time lead for response construction factor, f r Average surface Time delay, admittance, ω /h factor, F φ / h Slow Masonry external walls and > Internal partitions, bare solid floors and ceilings Fast Lightweight external cladding, de-mountable partitions, suspended ceilings, solid floors with carpet or wood block finish or suspended floors

9 Thermal design, plant sizing and energy consumption: Additional appendices A11-9 i o i c θ io θ ai R vi θ ai θei R rio θeo θ ei R rio θ eo θi Figure 5.A11.15 Thermal resistance network for single glazing with internal blind θo θi Figure 5.A11.16 Simplified thermal resistance network for single glazing with internal blind θo Table 5.A11.5 Example A11.: components of radiation Time h Solar Radiation absorbed/transmitted Gains to space / W m irradiance by glazing system / W m / W m Radiation absorbed Directly Cyclic components of Cyclic components of transmitted by inner, and outer transmitted absorbed radiation radiation for lightweight L and glazing elements radiation, T heavyweight H buildings H ea H aa ~ T L T~ H Mean: Table 5.A11.5 summarises the steps in the calculation of the solar gain to the space. The cyclic component of the convective and longwave radiant gain from the glazing to the environmental node is calculated as for triple glazing, see example A11.1, i.e.: H ~ e A = H ei A~ i + H ec A~ c + H eo A~ o The instantaneous component of the convective and longwave radiant gain from the glazing to the air node is calculated as follows: H a A = H ai + H ao Hence, for 1:00 h: H a A = = 5.66 The gain at other times is calculated similarly, see Table 5.A11.5. The mean gain is calculated using the mean, rather than the cyclic, absorption values. The cyclic component of the directly transmitted shortwave radiation and attenuated by the surface factor F appropriate to the thermal weight of the building and delayed by a time corresponding to the thermal weight, see Table 5.A11.4. For a lightweight building, F = 0.8 and the time delay is one hour, i.e.: T ~ L = 0.8 T t+1 T For a heavyweight building, F = 0.5 and the time delay is two hours, i.e.: T ~ H = 0.5 T t+ T where subscript L denotes thermally lightweight building and subscript H denotes thermally heavyweight building. As for example A11.1, the mean solar gain factor at the environmental node is given by: Hence: mean transmitted radiation plus mean absorbed radiation S e = daily mean incident radiation S e = / 179 = 0.3 Again, as for example A11.1, the cyclic solar gain factors at the environmental node are calculated using the gains appropriate to a time depending on the thermal weight of the structure, i.e.: total swing in gain to space S ~ e = swing in external gain Peak solar irradiance occurs at 14:00; hence, for a thermally lightweight structure i.e. 1-hour delay: S ~ el = / = 0.8

10 A11-10 Environmental design and for a thermally heavyweight structure i.e. -hour delay: S ~ eh = / = 0.7 The mean solar gain factor at the air node is given by: Hence: mean gain to air node S a = mean incident radiation S a = / 179 = 0.1 The cyclic solar gain factor at the air node is given by: total swing in gain to air node S ~ a = swing in external gain There is no time delay associated with the air node, hence: S ~ a 5.A11.6 = 74 / = 0.19 Shading coefficients In addition to solar gain factors, Table 5.A11.1 also gives the shortwave and longwave shading coefficients S c. These correspond to the direct and indirect transmission to the space for direct radiation at normal incidence see 5.6. divided by 0.87 total incident radiation, where 0.87 is transmission coefficient of nominal 4 mm plain glass at normal incidence. However, it should be noted that the room gains, and hence the solar gain factors, depend on the direct and diffuse components of the incident radiation and the angle of incidence of the direct radiation. The G-value does not distinguish between the two components of the transmission nor make reference to clear glass. Numerically, therefore, it is equal to the sum of the shortwave and longwave shading coefficients multiplied by References for Appendix 5.A11 BSI 006a BS EN ISO : 006: Thermal performance of windows, doors and shutters. Calculation of thermal transmittance. General London: British Standards Institution BSI 006b BS EN 13947: 006: Thermal performance of curtain walling. Calculation of thermal transmittance London: British Standards Institution BSI 011 BS EN 673: 011: Glass in building. Determination of thermal transmittance U-value. Calculation method London: British Standards Institution BSI 01 BS EN ISO : 01: Thermal performance of windows, doors and shutters. Calculation of thermal transmittance. Numerical method for frames London: British Standards Institution Jones RHL 1980 Solar radiation through windows theory and equations Building Serv. Eng. Res. Technol Parmelee GV and Vild DJ 1953 Design data for slat-type sun shades for use in load estimating ASHVE Trans Nicol JF 1966 Radiation transmission characteristics of louver systems Building Science Mitalas GP and Stephenson DG 196 Absorption and transmission of thermal radiation by single and double glazed windows Research Paper No. 173 National Research Council of Canada, Division of Building Research Parmelee GV and Aubele WW 195 ASHVE Trans The shading of sunlit glass

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