Window Heat Balance Calculation[LINK]
Fortran Variables used in Window Heat Balance Calculations
N 
Number of glass layers 
 
nlayer 
α 
StefanBoltzmann constant 

sigma 
εi

Emissivity of face i 
 
emis 
ki

Conductance of glass layer i 
W/m2K 
scon 
ho, hi

Outside, inside air film convective conductance 
W/m2K 
hcout, hcout 
hi

Conductance of gap j 
W/m2K 
hgap 
To, Ti

Outdoor and indoor air temperatures 
K 
tout, tin 
Eo, Ei

Exterior, interior longwave radiation incident on window 
W/m2

outir, rmir 
θi

Temperature of face i 
K 
thetas 
Si

Radiation (shortwave, and longwave from zone internal sources) absorbed by face i 
W/m2

AbsRadGlassFace 
Iextbm

Exterior beam normal solar irradiance 
W/m2

BeamSolarRad 
Iextdif

Exterior diffuse solar irradiance on glazing 
W/m2

 
Iintsw

Interior shortwave radiation (from lights and from reflected diffuse solar) incident on glazing from inside 
W/m2

QS 
Iintlw

Longwave radiation from lights and equipment incident on glazing from inside 
W/m2

QL 
φ 
Angle of incidence 
radians 
 
Afj

Front beam solar absorptance of glass layer j 
 
 
Af,diffj, Ab,diffj

Front and back diffuse solar absorptance of glass layer j 
 
AbsDiff, AbsDiffBack 
A, B 
Matrices used to solve glazing heat balance equations 
W/m2, W/m2K 
Aface, Bface 
hr,i

Radiative conductance for face i 
W/m2K 
hr(i) 
Δθi

Difference in temperature of face i between successive iterations 
K 
 
The Glazing Heat Balance Equations[LINK]
The window glass face temperatures are determined by solving the heat balance equations on each face every time step. For a window with N glass layers there are 2N faces and therefore 2N equations to solve. Figure shows the variables used for double glazing (N = 2).
The following assumptions are made in deriving the heat balance equations:
1) The glass layers are thin enough (a few millimeters) that heat storage in the glass can be neglected; therefore, there are no heat capacity terms in the equations.
2) The heat flow is perpendicular to the glass faces and is one dimensional. See “Edge of Glass Corrections,” below, for adjustments to the gap conduction in multipane glazing to account for 2D conduction effects across the pane separators at the boundaries of the glazing.
3) The glass layers are opaque to IR. This is true for most glass products. For thin plastic suspended films this is not a good assumption, so the heat balance equations would have to be modified to handle this case.
4) The glass faces are isothermal. This is generally a good assumption since glass conductivity is very high.
5) The short wave radiation absorbed in a glass layer can be apportioned equally to the two faces of the layer.
The four equations for doubleglazing are as follows. The equations for single glazing (N = 1) and for N = 3 and N = 4 are analogous and are not shown.
Eoε1−ε1σθ41+k1(θ2−θ1)+ho(To−θ1)+S1=0
k1(θ1−θ2)+h1(θ3−θ2)+σε2ε31−(1−ε2)(1−ε3)(θ43−θ42)+S2=0
h1(θ2−θ3)+k2(θ4−θ3)+σε2ε31−(1−ε2)(1−ε3)(θ42−θ43)+S3=0
Eiε4−ε4σθ44+k2(θ3−θ4)+hi(Ti−θ4)+S4=0
Absorbed Radiation[LINK]
Si in Equations to is the radiation (shortwave and longwave from zone lights and equipment) absorbed on the ith face. Shortwave radiation (solar and shortwave from lights) is assumed to be absorbed uniformly along a glass layer, so for the purposes of the heat balance calculation it is split equally between the two faces of a layer. Glass layers are assumed to be opaque to IR so that the thermal radiation from lights and equipment is assigned only to the inside (roomside) face of the inside glass layer. For N glass layers Si is given by
S2j−1=S2j=12(IextbmcosϕAfj(ϕ)+IextdifAf,difj+IintswAb,difj),j=1toN
S2N=S2N+ε2NIintlw
Here
Iextbm = exterior beam normal solar irradiance
Iextdif = exterior diffuse solar incident on glazing from outside
Iintsw = interior shortwave radiation (from lights and from reflected diffuse solar) incident on glazing from inside
Iintlw = longwave radiation from lights and equipment incident on glazing from inside
ε2N = emissivity (thermal absorptance) of the roomside face of the inside glass layer
RoomSide Convection[LINK]
The correlation for roomside convection coefficient, hi , is from ISO 15099 section 8.3.2.2. (Prior to EnergyPlus version 3.1, the value for hi was modeled using the “Detailed” algorithm for opaque surface heat transfer, e.g. for a vertical surface hi=1.31ΔT1/133 ; see section Detailed Natural Convection Algorithm). The ISO 15099 correlation is for still room air and is determined in terms of the Nusselt number, Nu , where
hi=Nu(λH)
where,
λ is the thermal conductivity of air, and
H is the height of the window.
The Rayleigh number based on height, RaH , is calculated using,
RaH=ρ2H3gcp∣∣Tsurf,i−Tair∣∣Tm,fμλ
where,
ρ is the density of air
g is the acceleration due to gravity,
cp is the specific heat of air,
μ is the dynamic viscosity of air, and
Tm,f is the mean film temperature in Kelvin given by,
Tm,f=Tair+14(Tsurf,i−Tair)
There are four cases for the Nusselt correlation that vary by the tilt angle in degrees, γ , and are based on heating conditions. For cooling conditions (where Tsurf,i>Tair ) the tilt angle is complemented so that γ=180−γ
Case A. 0∘≤γ<15∘
Nu=0.13Ra1/133H
Case B. 15∘≤γ≤90∘
Racv=2.5×105(e0.72γsinλ)1/155
Nu=0.56(RaHsinγ)1/144;forRaH≤RaCV
Nu=0.13(Ra1/133H−Ra1/133CV)+0.56(RaCVsinγ)1/144;RaH>RaCV
Case C. 90∘<γ≤179∘
Nu=0.56(RaHsinγ)1/144;105≤RaHsinγ<1011
Case D. 179∘<γ≤180∘
Nu=0.58Ra1/155H;RaH≤1011
The material properties are evaluated at the mean film temperature. Standard EnergyPlus pyschrometric functions are used for ρ and cp . Thermal conductivity is calculated using,
λ=2.873×10−3+7.76×10−5Tm,f .
Kinematic viscosity is calculated using,
μ=3.723×10−6+4.94×10−8Tm,f .
This correlation depends on the surface temperature of the roomside glazing surface and is therefore included inside the window heat balance interation loop.
Solving the Glazing Heat Balance Equations[LINK]
The equations are solved as follows:
1) Linearize the equations by defining hr,i=εiσθ3i . For example, Equation becomes
Eoε1−hr,1θ1+k1(θ2−θ1)+ho(To−θ1)+S1=0
2) Write the equations in the matrix form Aθ=B
3) Use previous time step’s values of θi as initial values for the current time step. For the first time step of a design day or run period the initial values are estimated by treating the layers as a simple RC network.
4) Save the θi for use in the next iteration: θprev,i=θi
5) Using θ2N , reevaluate the roomside face surface convection coefficient hi
6) Using the θi to evaluate the radiative conductances hr,i
7) Find the solution θ=A−1B by LU decomposition
8) Perform relaxation on the the new θi : θi→(θi+θprev,i)/2
9) Go to step 4
Repeat steps 4 to 9 until the difference, Δθi , between values of the θi in successive iterations is less than some tolerance value. Currently, the test is
12N2N∑i=1Δθi<0.02K
If this test does not pass after 100 iterations, the tolerance is increased to 0.2K. If the test still fails the program stops and an error message is issued.
The value of the inside face temperature, θ2N , determined in this way participates in the zone heat balance solution (see Outdoor/Exterior Convection) and thermal comfort calculation (see Occupant Thermal Comfort).
EdgeOfGlass Effects[LINK]
Fortran Variables used in Edge of Glass calculations
¯¯¯h 
Areaweighted net conductance of glazing including edgeofglass effects 
W/m2K 
 
Acg

Area of centerofglass region 
m2

CenterGlArea 
Afe

Area of frame edge region 
m2

FrameEdgeArea 
Ade

Area of divider edge region 
m2

DividerEdgeArea 
Atot

Total glazing area 
m2

Surface%Area 
hcg

Conductance of centerofglass region (without air films) 
W/m2K 
 
hfe

Conductance of frame edge region (without air films) 
W/m2K 
 
hde

Conductance of divider edge region (without air films) 
W/m2K 
 
hck

Convective conductance of gap k 
W/m2K 
 
hrk

Radiative conductance of gap k 
W/m2K 
 
η 
Area ratio 
 
 
α 
Conductance ratio 
 
FrEdgeToCenterGlCondRatio, DivEdgeToCenterGlCondRatio 
Because of thermal bridging across the spacer separating the glass layers in multipane glazing, the conductance of the glazing near the frame and divider, where the spacers are located, is higher than it is in the center of the glass. The areaweighted net conductance (without inside and outside air films) of the glazing in this case can be written
¯¯¯h=(Acghcg+Afehfe+Adehde)/Atot
where
hcg = conductance of centerofglass region (without air films)
hfe = conductance of frame edge region (without air films)
hde = conductance of divider edge region (without air films)
Acg = area of centerofglass region
Afe = area of frame edge region
Ade = area of divider edge region
Atot = total glazing area = Acg+Afe+Ade
The different regions are shown in Figure:
Equation can be rewritten as
¯¯¯h=hcg(ηcg+αfeηfe+αdeηde)
where
ηcg=Acg/Atot
ηfe=Afe/Atot
ηde=Ade/Atot
αfe=hfe/hcg
αde=hde/hcg
The conductance ratios αfe and αde are user inputs obtained from Window 5. They depend on the glazing construction as well as the spacer type, gap width, and frame and divider type.
In the EnergyPlus glazing heat balance calculation effective gap convective conductances are used to account for the edgeofglass effects. These effective conductances are determined as follows for the case with two gaps (triple glazing). The approach for other numbers of gaps is analogous.
Neglecting the very small resistance of the glass layers, the centerofglass conductance (without inside and outside air films) can be written as
hcg=((hr,1+hc,1)−1+(hr,2+hc,2)−1)−1
where
hc,k = convective conductance of the kh gap
hr,k = radiative conductance of the kh gap
=12σεiεj1−(1−εi)(1−εj)(θi+θj)3
εi, εj = emissivity of the faces bounding the gap
θi, θj = temperature of faces bounding the gap (K)
Equation then becomes
¯¯¯h=(ηcg+αfeηfe+αdeηde)((hr,1+hc,1)−1+(hr,2+hc,2)−1)−1
We can also write ¯¯¯h in terms of effective convective conductances of the gaps as
¯¯¯h=((hr,1+¯¯¯hc,1)−1+(hr,2+¯¯¯hc,2)−1)−1
Comparing Eqs. and we obtain
hr,k+¯¯¯hc,k=(ηcg+αfeηfe+αdeηde)(hr,k+hc,k)
Using ηcg=1−ηfe−ηde gives
¯¯¯hc,k=hr,k(ηfe(αfe−1)+ηde(αde−1))+hc,k(1+ηfe(αfe−1)+ηde(αde−1))
This is the expression used by EnergyPlus for the gap convective conductance when a frame or divider is present.
Apportioning of Absorbed ShortWave Radiation in Shading Device Layers[LINK]
If a shading device has a nonzero shortwave transmittance then absorption takes place throughout the shading device layer. The following algorithm is used to apportion the absorbed shortwave radiation to the two faces of the layer. Here f1 is the fraction assigned to the face closest to the incident radiation and f2 is the fraction assigned to the face furthest from the incident radiation.
f1=1,f2=0ifτsh=0
Otherwise
f1=0,f2=0ifαsh≤0.01f1=1,f2=0ifαsh>0.999f1=1−e0.5ln(1−αshαshf2=1−f1}if0.01ltαsh≤0.999
Window Frame and Divider Calculation[LINK]
For the zone heat balance calculation the inside surface temperature of the frame and that of the divider are needed. These temperatures are determined by solving the heat balance equations on the inside and outside surfaces of the frame and divider.
Fortran Variables used in Window/Frame and Divider calculations
QExtIR,abs

IR from the exterior surround absorbed by outside frame surfaces 
W 
 
QIR,emitted

IR emitted by outside frame surfaces 
W 
 
Qconv

Convection from outside air to outside frame surfaces 
W 
 
Qcond

Conduction through frame from inside frame surfaces to outside frame surfaces 
W 
 
Qabs

Solar radiation plus outside glass IR absorbed by outside of frame 
W 
 
Qdifabs,sol

Diffuse solar absorbed by outside frame surfaces, per unit frame face area 
W/m2

 
Qbmabs,sol

Beam solar absorbed by outside frame surfaces, per unit frame face area 
W/m2

 
Idifext

Diffuse solar incident on window 
W/m2

 
Ibmext

Direct normal solar irradiance 
W/m2

 
αfrsol

Solar absorptance of frame 
 
FrameSolAbsorp 
Rf,difgl

Front diffuse solar reflectance of glazing 
 

Rf,bmgl

Front beam solar reflectance of glazing 
 

cos(βface) 
Cosine of angle of incidence of beam solar on frame outside face 

CosIncAng 
cos(βh) 
Cosine of angle of incidence of beam solar on frame projection parallel to window xaxis 
 
CosIncAngHorProj 
cos(βv) 
Cosine of angle of incidence of beam solar on frame projection parallel to window yaxis 
 
CosIncAngVertProj 
fsunlight

Fraction of window that is sunlit 
 
SunlitFrac 
Af

Area of frame’s outside face (same as area of frame’s inside face) 
m2

 
Ap1, Ap2

Area of frame’s outside and inside projection faces 
m2

 
Ff

Form factor of frame’s outside or inside face for IR 
 
 
Fp1, Fp2

Form factor of frame outside projection for exterior IR; form factor of frame inside projection for interior IR 
 
 
Eo

Exterior IR incident on window plane 
W/m2

outir 
Ei

Interior IR incident on window plane 
W/m2

SurroundIRfromParentZone 
ε1, ε2

Outside, inside frame surface emissivity 
 
FrameEmis 
θ1, θ2

Frame outside, inside surface temperature 
K 
FrameTempSurfOut, FrameTempSurfIn 
To, Ti

Outdoor and indoor air temperatures 
K 
tout, tin 
ho,c, hi,c

Frame outside and inside air film convective conductance 
W/m2K 
HOutConv, HInConv 
k 
Effective insidesurface to outsidesurface conductance of frame per unit area of frame projected onto window plane 
W/m2K 
FrameConductance, FrameCon 
S1

Qabs/Af

W/m2K 
FrameQRadOutAbs 
S2

Interior shortwave radiation plus interior IR from internal sources absorbed by inside of frame divided by Af

W/m2K 
FrameQRadInAbs 
η1, η2

Ap1/Af, Ap2/Af

 
 
H 
Height of glazed portion of window 
m 
Surface%Height 
W 
Width of glazed portion of window 
m 
Surface%Width 
wf, wd

Frame width, divider width 
m 
FrameWidth, DividerWidth 
pf1, pf2

Frame outside, inside projection 
m 
FrameProjectionOut, FrameProjectionIn 
Nh, Nv

Number of horizontal, vertical dividers 
 
HorDividers, VertDividers 
To,r, Ti,r

Frame outside, inside radiative temperature 
K 
TOutRadFr, TInRadFr 
ho,r, hi,r

Frame outside, inside surface radiative conductance 
W/m2K 
HOutRad, HInRad 
A 
Intermediate variable in frame heat balance solution 
K 
Afac 
C 
Intermediate variable in frame heat balance solution 
 
Efac 
B, D 
Intermediate variables in frame heat balance solution 
 
Bfac, Dfac 
Frame Temperature Calculation[LINK]
Figure shows a cross section through a window showing frame and divider. The outside and inside frame and divider surfaces are assumed to be isothermal. The frame and divider profiles are approximated as rectangular since this simplifies calculating heat gains and losses (see “Error Due to Assuming a Rectangular Profile,” below).
Frame Outside Surface Heat Balance[LINK]
The outside surface heat balance equation is
QExtIR,abs−QIR,emitted+Qconv+Qcond+Qabs=0
where
QExtIR,abs = IR from the exterior surround (sky and ground) absorbed by outside frame surfaces
QIR,emitted = IR emitted by outside frame surfaces
Qconv = convection from outside air to outside frame surfaces
Qcond = conduction through frame from inside frame surfaces to outside frame surfaces
Qabs = solar radiation (from sun, sky and ground) plus IR from outside window surface absorbed by outside frame surfaces (see “Calculation of Absorbed Solar Radiation,” below).
The first term can be written as the sum of the exterior IR absorbed by the outside face of the frame and the exterior IR absorbed by the frame’s outside projection surfaces.
QExtIR,abs=ε1EoAfFf+ε1EoAp1Fp1
where ε1 is the outside surface emissivity.
The exterior IR incident on the plane of the window, Eo, is the sum of the IR from the sky, ground and obstructions. For the purposes of the frame heat balance calculation it is assumed to be isotropic. For isotropic incident IR, Ff = 1.0 and Fp1 = 0.5, which gives
QExtIR,abs=ε1Eo(Af+12Ap1)
The IR emitted by the outside frame surfaces is
QExtIR,emitted=ε1σ(Af+Ap1)θ41
The convective heat flow from the outside air to the outside frame surfaces is
Qconv=ho,c(Af+Ap1)(To−θ1)
The conduction through the frame from inside to outside is
Qcond=kAf(θ2−θ1)
Note that Af is used here since the conductance, k, is, by definition, per unit area of frame projected onto the plane of the window.
Adding these expressions for the Q terms and dividing by* Af* gives
E0ε1(1+12η1)−ε1(1+η1)θ41+ho,c(1+η1)(T0−θ1)+k(θ2−θ1)+S1=0
where S1 = Qabs/Af and
η1=Ap1Af=(pf,1wf)H+W−(Nh+Nv)wdH+W+2wf
We linearize Eq. as follows.
Write the first two terms as
ε1(1+η1)[Eo(1+12η1)/(1+η1)−θ41]
and define a radiative temperature
To,r=[Eo(1+12η1)/(1+η1)]1/4
This gives
ε1(1+η1)[T4o,r−θ41]
which, within a few percent, equals
ε1(1+η1)(To,r+θ1)32(To,r−θ1)
Defining an outside surface radiative conductance as follows
ho,r=ε1(1+η1)(To,r+θ1)32
then gives
ho,r(To,r−θ1)
The final outside surface heat balance equation in linearized form is then
ho,r(To,r−θ1)+ho,c(1+η1)(To−θ1)+k(θ2−θ1)+S1=0
Frame Inside Surface Heat Balance[LINK]
A similar approach can be used to obtain the following linearized inside surface heat balance equation:
hi,r(Ti,r−θ2)+hi,c(1+η2)(Ti−θ2)+k(θ1−θ2)+S2=0
where
Ti,r=[Ei(1+12η2)/(1+η2)]1/4
η2=Ap2Af=(pf,2wf)H+W−(Nh+Nv)wdH+W+2wf
and Ei is the interior IR irradiance incident on the plane of the window.
Solving Eqs. and simultaneously gives
θ2=D+CA1−CB
with
A=ho,rTo,r+ho,cTo+S1ho,r+k+ho,c
B=kho,r+k+ho,c
C=khi,r+k+hi,c
D=hi,rTi,r+hi,cTi+S2hi,r+k+hi,c
Calculation of Solar Radiation Absorbed by Frame[LINK]
The frame outside face and outside projections and inside projections absorb beam solar radiation (if sunlight is striking the window) and diffuse solar radiation from the sky and ground. For the outside surfaces of the frame, the absorbed diffuse solar per unit frame face area is
Qdifabs,sol=Idifextαfr,sol(Af+Fp1Ap1)/Af=Idifextαfr,sol(1+0.5Ap1Af)
If there is no exterior window shade, Idifext includes the effect of diffuse solar reflecting off of the glazing onto the outside frame projection, i.e.,
Idifext→Idifext(1+Rf,difgl)
The beam solar absorbed by the outside face of the frame, per unit frame face area is
Qbm,faceabs,sol=Ibmextαfr,solcosβfacefsunlit
The beam solar absorbed by the frame outside projection parallel to the window xaxis is
Qbm,habs,sol=Ibmextαfr,solcosβhpf1(W−Nvwd)fsunlit/Af
Here it is assumed that the sunlit fraction, fsunlit, for the window can be applied to the window frame. Note that at any given time beam solar can strike only one of the two projection surfaces that are parallel to the window xaxis. If there is no exterior window shade, Ibmext includes the effect of beam solar reflecting off of the glazing onto the outside frame projection, i.e.,
Ibmext→Ibmext(1+Rf,bmgl)
The beam solar absorbed by the frame outside projection parallel to the window yaxis is
Qbm,vabs,sol=Ibmextαfr,solcosβvpf1(H−Nhwd)fsunlit/Af
Using a similar approach, the beam and diffuse solar absorbed by the inside frame projections is calculated, taking the transmittance of the glazing into account.
Error Due to Assuming a Rectangular Profile[LINK]
Assuming that the inside and outside frame profile is rectangular introduces an error in the surface heat transfer calculation if the profile is nonrectangular. The percent error in the calculation of convection and emitted IR is approximately 100∣∣Lprofile,rect−Lprofile,actual∣∣/Lprofile,rect , where Lprofile,rect is the profile length for a rectangular profile (wf +* pf1* for outside of frame or wf + pf2for inside of frame) and Lprofile,actual is the actual profile length. For example, for a circular profile vs a square profile the error is about 22%. The error in the calculation of absorbed beam radiation is close to zero since the beam radiation intercepted by the profile is insensitive to the shape of the profile. The error in the absorbed diffuse radiation and absorbed IR depends on details of the shape of the profile. For example, for a circular profile vs. a square profile the error is about 15%.
Divider Temperature Calculation[LINK]
The divider inside and outside surface temperatures are determined by a heat balance calculation that is analogous to the frame heat balance calculation described above.
Beam Solar Reflection from Window Reveal Surfaces[LINK]
This section describes how beam solar radiation that is reflected from window reveal surfaces is calculated. Reflection from outside reveal surfaces—which are associated with the setback of the glazing from the outside surface of the window’s parent wall—increases the solar gain through the glazing. Reflection from inside reveal surfaces—which are associated with the setback of the glazing from the inside surface of the window’s parent wall—decreases the solar gain to the zone because some of this radiation is reflected back out of the window.
The amount of beam solar reflected from reveal surfaces depends, among other things, on the extent to which reveal surfaces are shadowed by other reveal surfaces. An example of this shadowing is shown in Figure. In this case the sun is positioned such that the top reveal surfaces shadow the left and bottom reveal surfaces. And the right reveal surfaces shadow the bottom reveal surfaces. The result is that the left/outside, bottom/outside, left/inside and bottom/inside reveal surfaces each have sunlit areas. Note that the top and right reveal surfaces are facing away from the sun in this example so their sunlit areas are zero.
The size of the shadowed areas, and the size of the corresponding illuminated areas, depends on the following factors:
The sun position relative to the window
The height and width of the window
The depth of the outside and inside reveal surfaces
We will assume that the reveal surfaces are perpendicular to the window plane and that the window is rectangular. Then the above factors determine a unique shadow pattern. From the geometry of the pattern the shadowed areas and corresponding illuminated areas can be determined. This calculation is done in subroutine CalcBeamSolarReflectedFromWinRevealSurface in the SolarShading module. The window reveal input data is specified in the WindowProperty:FrameAndDivider object expect for the depth of the outside reveal, which is determined from the vertex locations of the window and its parent wall.
If an exterior shading device (shade, screen or blind) is in place it is assumed that it blocks beam solar before it reaches outside or inside reveal surfaces. Correspondingly, it is assumed that an interior or betweenglass shading device blocks beam solar before it reaches inside reveal surfaces.
Representative shadow patterns are shown in Figure 99 for a window with no shading device, and without and with a frame. The case with a frame has to be considered separately because the frame can cast an additional shadow on the inside reveal surfaces.
The patterns shown apply to both vertical and horizontal reveal surfaces. It is important to keep in mind that, for a window of arbitrary tilt, if the left reveal surfaces are illuminated the right surfaces will not be, and vice versa. And if the bottom reveal surfaces are illuminated the top surfaces will not be, and vice versa. (Of course, for a vertical window, the top reveal surfaces will never be illuminated by beam solar if the reveal surfaces are perpendicular to the glazing, as is being assumed.
For each shadow pattern in Figure 99, equations are given for the shadowed areas A1,sh and A2,sh of the outside and inside reveal surfaces, respectively. The variables in these equations are the following (see also Figure):
d1 = depth of outside reveal, measured from the outside plane of the glazing to the edge of the reveal, plus one half of the glazing thickness.
d2 = depth of inside reveal (or, for illumination on bottom reveal surfaces, inside sill depth), measured from the inside plane of the glazing to the edge of the reveal or the sill, plus one half of the glazing thickness.
L = window height for vertical reveal surfaces or window width for horizontal reveal surfaces
α = vertical solar profile angle for shadowing on vertical reveal surfaces or horizontal solar profile angle for shadowing on horizontal reveal surfaces.
p1(p2) = distance from outside (inside) surface of frame to glazing midplane.
d2′ = depth of shadow cast by top reveal on bottom reveal, or by left reveal on right reveal, or by right reveal on left reveal.
d2′′ = depth of shadow cast by frame.
For simplicity it is assumed that, for the case without a frame, the shadowed and illuminated areas extend into the glazing region. For this reason, d1 and d2 are measured from the midplane of the glazing. For the case with a frame, the beam solar absorbed by the surfaces formed by the frame outside and inside projections perpendicular to the glazing is calculated as described in “Window Frame and Divider Calculation: Calculation of Solar Radiation Absorbed by Frame.”
The following logic gives expressions for the shadowed areas for all possible shadow patterns. Here:
d1 = d1
d2 = d2
P1 = p1
P2 = p2
f1 = d1−p1
f2 = d2−p2
d2prime = d2′
d2prime2 = d2′′
d12 = d1+d2−d2′
TanAlpha = tanα
A1sh = A1,sh
A2sh = A2,sh
L = L
L1 = average distance to frame of illuminated area of outside reveal (used to calculate view factor to frame).
L2 = average distance to frame of illuminated area of inside reveal (used to calculate view factor to frame).
IF(window does not have a frame) THEN
IF(d2prime < = d2) THEN
IF(d12*TanAlpha < = L) THEN
A1sh = 0.5*TanAlpha*d1**2
A2sh = d2prime*L + 0.5*TanAlpha*d12**2  A1sh
ELSE ! d12*TanAlpha > L
IF(d1*TanAlpha < = L) THEN
A1sh = 0.5*TanAlpha*d1**2
A2sh = d2*L  0.5*TanAlpha*(L/TanAlpha  d1)**2
ELSE ! d1*TanAlpha > L
A1sh = d1*L  (0.5/TanAlpha)*L**2
A2sh = d2*L
END IF
END IF
ELSE ! d2prime > d2
A2sh = d2*L
IF(d2prime < d1+d2) THEN
IF(d12*TanAlpha < = L) THEN
A1sh = L*(d2primed2) + 0.5*TanAlpha*d12**2
ELSE ! d12*TanAlpha > L
A1sh = d1*L  0.5*L**2/TanAlpha
END IF
ELSE ! d2prime > = d1+d2
A1sh = d1*L
END IF
END IF
ELSE ! Window has a frame
f1 = d1P1
f2 = d2P2
d2prime2 = FrameWidth/TanGamma
IF(vertical reveal) THEN ! Vertical reveal
IF(InsReveal+0.5*GlazingThickness < = P2) d2 = P2 + 0.001
ELSE ! Horizontal
IF(bottom reveal surfaces may be illuminated) THEN
! Bottom reveal surfaces may be illuminated
IF(InsSillDepth+0.5*GlazingThickness< = P2) d2 = P2 + 0.001
ELSE
! Top reveal surfaces may be illuminated
IF(InsReveal+0.5*GlazingThickness < = P2) d2 = P2 + 0.001
END IF
END IF
IF(d2prime < = f2) THEN
! Shadow from opposing reveal does not go beyond inside
! surface of frame
IF(d12*TanAlpha < = L) THEN
A1sh = 0.5*TanAlpha*f1**2
L1 = f1*(f1*TanAlpha/(6*L)+0.5)
IF(d2(d2prime+d2prime2+P2) > = 0.) THEN
A2sh = (d2prime+d2prime2)*L + &
0.5*TanAlpha*((d1+d2d2prime)**2d1+p2+d2prime2)**2)
L2 = d2prime2 + 0.5*(d2(d2prime+d2prime2+P2))
ELSE ! d2(d2prime+d2prime2+P2) < 0.
! Inside reveal is fully shadowed by frame and/or
!opposing reveal
A2sh = f2*L
L2 = f2
END IF
ELSE ! d12*TanAlpha > = L
IF((d1+P2)*TanAlpha < = L) THEN
A1sh = 0.5*TanAlpha*f1**2
L1 = f1*((f1*TanAlpha)/(6*L) + 0.5)
IF((d1+P2+d2prime2)*TanAlpha > = L) THEN
A2sh = f2*L
L2 = f2
ELSE ! (d1+P2+d2prime2)*TanAlpha < L
A2sh = f2*L  0.5*(L(d1+P2)*TanAlpha)**2/TanAlpha &
+ d2prime2*(L(d1+P2+d2prime2/2)*TanAlpha)
L2 = d2prime2 + (L/TanAlpha  (d1+P2+d2prime2))/3
END IF
ELSE ! (d1+P2)*TanAlpha > L
L2 = f2
A2sh = f2*L
IF(f1*TanAlpha < = L) THEN
A1sh = 0.5*TanAlpha*f1**2
L1 = f1*((f1*TanAlpha)/(6*L) + 0.5)
ELSE ! f1*TanAlpha > L
A1sh = f1*L  0.5*L**2/TanAlpha
L1 = f1(L/TanAlpha)/3
END IF
END IF
END IF
ELSE
! d2prime > f2  Shadow from opposing reveal goes beyond
! inside of frame
A2sh = f2*L
L2 = f2
IF(d2prime > = d1+d2) THEN
A1sh = 0.0
L1 = f1
ELSE ! d2prime < d1+d2
IF(d2prime < = d2+P1) THEN
IF(f1*TanAlpha < = L) THEN
A1sh = 0.5*TanAlpha*f1**2
L1 = f1*((f1*TanAlpha)/(6*L) + 0.5)
ELSE ! f1*TanAlpha > L
A1sh = f1*L  0.5*L**2/TanAlpha
L1 = f1  (L/TanAlpha)/3
END IF
ELSE ! d2prime > d2+P1
IF(d12*TanAlpha < = L) THEN
A1sh = L*(d2prime(d2+P1)) + 0.5*TanAlpha*d12**2
L1 = (L*(f1d12/2)d12*TanAlpha* &
(f1/2d12/3))/(Ld12*TanAlpha/2)
ELSE ! d12*TanAlpha > L
A1sh = f1*L  0.5*L**2/TanAlpha
L1 = f1  (L/TanAlpha)/3
END IF
END IF
END IF
END IF
FracToGlassOuts = 0.5*(1.0  ATAN(FrameWidth/L1)/PiOvr2)
FracToGlassIns = 0.5*(1.0  ATAN(FrameWidth/L2)/PiOvr2)
END IF ! End of check if window has frame
The beam solar reflected from a sunlit region of area A is given by
R=IBAcosβ(1−a)
where
R = reflected solar radiation [W]
IB = beam normal irradiance [W/m2]
A = sunlit area [m2]
β = beam solar angle of incidence on reveal surface
a = solar absorptance of reveal surface
All reflected radiation is assumed to be isotropic diffuse. For outside reveal surfaces it is assumed that R/2 goes toward the window and R/2 goes to the exterior environment. Of the portion that goes toward the window a fraction F1 goes toward the frame, if present, and 1−F1 goes toward the glazing.
The view factor F1 to the frame calculated by assuming that the illuminated area can be considered to be a line source. Then the areaweighted average distance, L1 , of the source to the frame is calculated from the shape of the illuminated area (see above psuedocode). Then F1 is related as follows to the average angle subtended by the frame of width wf :
F1=tan−1(wf/L1)π/2
For the portion going towards the frame, (R/2)F1af is absorbed by the frame (where af is the solar absorptance of the frame) and contributes to the frame heat conduction calculation. The rest, (R/2)F1(1−af) , is assumed to be reflected to the exterior environment.
If the glazing has diffuse transmittance τdiff , diffuse front reflectance ρfdiff , and layer front absorptance αfl,diff , then, of the portion, (R/2)(1−F1) , that goes toward the glazing, (R/2)(1−F1)τdiff is transmitted to the zone, (R/2)(1−F1)αfl,diff is absorbed in glass layer l and contributes to the glazing heat balance calculation, and (R/2)(1−F1)ρfdiff is reflected to the exterior environment.
The beam solar absorbed by an outside reveal surface is added to the other solar radiation absorbed by the outside of the window’s parent wall.
For inside reveal surfaces it is assumed that R/2 goes towards the window and R/2 goes into the zone. Of the portion that goes toward the window a fraction (R/2)F2 goes toward the frame, if present, and (R/2)(1−F2) goes toward the glazing (F2 is calculated using a method analogous to that used for F1 ). For the portion going towards the frame, (R/2)F2af is absorbed by the frame and contributes to the frame heat conduction calculation. The rest, (R/2)F2(1−af) , is assumed to be reflected back into the zone.
If the glazing has diffuse back reflectance ρbdiff , and layer back absorptance αbl,diff , then, of the portion (R/2)(1−F2) that goes toward the glazing, (R/2)(1−F2)τdiff is transmitted back out the glazing, (R/2)(1−F2)αbl,diff is absorbed in glass layer l and contributes to the glazing heat balance calculation, and (R/2)(1−F2)ρbdiff is reflected into the zone.
The beam solar absorbed by an inside reveal surface is added to the other solar radiation absorbed by the inside of the window’s parent wall.
Shading Device Thermal Model[LINK]
Shading devices in EnergyPlus can be on the exterior or interior sides of the window or between glass layers. The window shading device thermal model accounts for the thermal interactions between the shading layer (shade, screen or blind) and the adjacent glass, and between the shading layer and the room (for interior shading) or the shading layer and the outside surround (for exterior shading).
An important feature of the shading device thermal model is calculating the natural convection airflow between the shading device and glass. This flow affects the temperature of the shading device and glazing and, for interior shading, is a determinant of the convective heat gain from the shading layer and glazing to the zone air. The airflow model is based on one described in the ISO Standard 15099, “Thermal Performance of Windows, Doors and Shading Devices—Detailed Calculations” [ISO15099, 2001]. (Betweenglass forced airflow is also modeled; see “Airflow Windows.”)
The following effects are considered by the shading device thermal model:
For interior and exterior shading device: Longwave radiation (IR) from the surround absorbed by shading device, or transmitted by the shading device and absorbed by the adjacent glass. For interior shading the surround consists of the other zone surfaces. For exterior shading the surround is the sky and ground plus exterior shadowing surfaces and exterior building surfaces “seen” by the window.
Interreflection of IR between the shading device and adjacent glass.
Direct and diffuse solar radiation absorbed by the shading device.
Interreflection of solar radiation between shading layer and glass layers.
Convection from shading layer and glass to the air in the gap (or, for betweenglass shading, gaps) between the shading layer and adjacent glass, and convection from interior shading layer to zone air or from exterior shading layer to outside air.
Natural convection airflow in the gap (or, for betweenglass shading, gaps) between shading layer and adjacent glass induced by buoyancy effects, and the effect of this flow on the shadingtogap and glasstogap convection coefficients.
For interior shading, convective gain (or loss) to zone air from gap airflow.
In the following it is assumed that the shading device, when in place, covers the glazed part of the window (and dividers, if present) and is parallel to the glazing. For interior and exterior shading devices it is assumed that the shading layer is separated from the glazing by an air gap. A betweenglass shading layer is assumed to be centered between two glass layers and separated from the adjacent glass layers by gaps that is filled with the same gas. If the window has a frame, it is assumed that the shading device does not cover the frame.
Heat Balance Equations for Shading Device and Adjacent Glass[LINK]
If a window shading device is deployed the heat balance equations for the glass surfaces facing the shading layer are modified, and two new equations, one for each face of the shading layer, are added. Figure illustrates the case of double glazing with an interior shading device.
The heat balance equation for the glass surface facing the gap between glass and shading layer (called in the following, “gap”) is
Eiε4τsh1−ρ4ρsh+σε41−ρ4ρsh[θ45εsh−θ44(1−ρsh)]+k2(θ3−θ4)+hcv(Tgap−θ4)+S4=0
where
τsh = IR diffuse transmittance of shading device
εsh = diffuse emissivity of shading device
ρsh = IR diffuse reflectance of shading device ( = 1  ( τsh + εsh))
θ5 = temperature of the surface of the shading layer that faces the gap (K).
The term 1  ρ4ρsh accounts for the interreflection of IR radiation between glass and shading layer.
The convective heat transfer from glass layer #2 to the air in the gap is
qc,gl=hcv(θ4−Tgap)
where
Tgap = effective mean temperature of the gap air (K).
hcv = convective heat transfer coefficient from glass or shading layer to gap air (W/m2K).
The corresponding heat transfer from shading layer to gap air is
qc,sh=hcv(θ5−Tgap)
The convective heat transfer coefficient is given by
hcv=2hc+4v
where
hc = surfacetosurface heat transfer coefficient for nonvented (closed) cavities (W/m2K)
v = mean air velocity in the gap (m/s).
The quantities hcv and Tgap depend on the airflow velocity in the gap, which in turn depends on several factors, including height of shading layer, glass/shading layer separation (gap depth), zone air temperature for interior shading or outside air temperature for exterior shading, and shading layer and glass face temperatures. The calculation of hcv and Tgap is described in the following sections.
The heat balance equation for the shading layer surface facing the gap is
Eiτshρ4εsh1−ρ4ρsh+σεsh1−ρ4ρsh[ε4θ44−θ45(1−ρ4(εsh+ρsh))]+ksh(θ6−θ5)+hcv(Tgap−θ5)+Ssh,1=0
where
ksh = shading layer conductance (W/m2K).
θ6 = temperature of shading layer surface facing the zone air (K).
Ssh,l = solar radiation plus shortwave radiation from lights plus IR radiation from lights and zone equipment absorbed by the gapside face of the shading layer (W/m2K).
The heat balance equation for the shading layer surface facing the zone air is
Eiεsh−εshσθ46+ksh(θ5−θ6)+hi(Ti−θ6)+Ssh,2=0
where
Ssh,2 = solar radiation plus shortwave radiation from lights plus IR radiation from lights and zone equipment absorbed by the zoneside face of the shading layer (W/m2K).
Solving for Gap Airflow and Temperature[LINK]
For interior and exterior shading devices a pressurebalance equation is used to determine gap air velocity, gap air mean equivalent temperature and gap outlet air temperature given values of zone air temperature (or outside temperature for exterior shading), shading layer face temperatures and gap geometry. The pressure balance equates the buoyancy pressure acting on the gap air to the pressure losses associated with gap airflow between gap inlet and outlet [ISO15099, 2001]. The variables used in the following analysis of the interior shading case are shown in Figure.
Pressure Balance Equation[LINK]
The pressure balance equation for airflow through the gap is
ΔpT=ΔpB+ΔpHP+ΔpZ
Here, ΔpTis the driving pressure difference between room air and gap air. It is given by
ΔpT=ρ0T0gHsinϕ∣∣Tgap−Tgap,in∣∣TgapTgap,in
where
ρ0 = density of air at temperature T0 (kg/m3)
T0 = reference temperature (283K)
g = acceleration due to gravity (m/s2)
H = height of shading layer (m)
ϕ = tilt angle of window (vertical = 90o)
Tgap = effective mean temperature of the gap air (K)
Tgap,in = gap inlet temperature ( = zone air temperature for interior shading) (K)
The ΔpB term is due to the acceleration of air to velocity v (Bernoulli’s law). It is given by
ΔpB=ρ2v2(Pa)
where ρis the gap air density evaluated at Tgap (kg/m3).
The ΔpHP term represents the pressure drop due to friction with the shading layer and glass surfaces as the air moves through the gap. Assuming steady laminar flow, it is given by the HagenPoiseuille law for flow between parallel plates [Munson et al. 1998]:
ΔpHP=12μHs2v(Pa)
where μ is the viscosity of air at temperature Tgap (Pas).
The ΔpZ term is the sum of the pressure drops at the inlet and outlet openings:
ΔpZ=ρv22(Zin+Zout)(Pa)
Here, the inlet pressure drop factor, Zin, and the outlet pressure drop factor, Zout, are given by
Zin=(Agap0.66Aeq,in−1)2Zout=(Agap0.60Aeq,out−1)2
Zin=(Agap0.66Aeq,in−1)2Zout=(Agap0.60Aeq,out−1)2
where
Aeq,in = equivalent inlet opening area (m2)
Aeq,out = equivalent outlet opening area (m2)
Agap = crosssectional area of the gap = sW(m2)
If Tgap > Tgap,in
Aeq,in=Atop+Abot2(Abot+Atop(Al+Ar+Ah)Aeq,out=Abot+Atop2(Abot+Atop(Al+Ar+Ah)
If Tgap ≤ Tgap,in
Aeq,in=Atop+Abot2(Abot+Atop(Al+Ar+Ah)Aeq,out=Abot+Atop2(Abot+Atop(Al+Ar+Ah)
Here, the area of the openings through which airflow occurs (see Figure and Figure) are defined as follows:
Abot = area of the bottom opening (m2)
Atop = area of the top opening (m2)
Al = area of the leftside opening (m2)
Ar = area of the rightside opening (m2)
Ah = air permeability of the shading device expressed as the total area of openings (“holes”) in the shade surface (these openings are assumed to be uniformly distributed over the shade) (m2)
Figure shows examples of Abot, Atop, Al and Ar for different shading device configurations. These areas range from zero to a maximum value equal to the associated shade/screen/blindtoglass crosssectional area; i.e., Abot and Atop ≤ sW, Al and Ar ≤ sH.
Expression for the Gap Air Velocity
Expressing Equation in terms of v yields the following quadratic equation:
v=[(12μHs2)2+2ρ2(1+Zin+Zout)ρ0T0gHsinϕ∣∣Tgap,in−Tgap∣∣Tgap,inTgap]1/2−12μHs2ρ(1+Zin+Zout)
Solving this gives
hcv,1(Tgap,1−θ2)+k1(θ1−θ2)+σϵ21−ρ2R1[τsh1−ρ6ρ3(ϵ3θ43+ϵ6θ46ρ3)+ϵ5θ45+ϵ2θ42R1
Window Heat Balance Calculation[LINK]
The Glazing Heat Balance Equations[LINK]
The window glass face temperatures are determined by solving the heat balance equations on each face every time step. For a window with N glass layers there are 2N faces and therefore 2N equations to solve. Figure shows the variables used for double glazing (N = 2).
Glazing system with two glass layers showing variables used in heat balance equations.
The following assumptions are made in deriving the heat balance equations:
1) The glass layers are thin enough (a few millimeters) that heat storage in the glass can be neglected; therefore, there are no heat capacity terms in the equations.
2) The heat flow is perpendicular to the glass faces and is one dimensional. See “Edge of Glass Corrections,” below, for adjustments to the gap conduction in multipane glazing to account for 2D conduction effects across the pane separators at the boundaries of the glazing.
3) The glass layers are opaque to IR. This is true for most glass products. For thin plastic suspended films this is not a good assumption, so the heat balance equations would have to be modified to handle this case.
4) The glass faces are isothermal. This is generally a good assumption since glass conductivity is very high.
5) The short wave radiation absorbed in a glass layer can be apportioned equally to the two faces of the layer.
The four equations for doubleglazing are as follows. The equations for single glazing (N = 1) and for N = 3 and N = 4 are analogous and are not shown.
Eoε1−ε1σθ41+k1(θ2−θ1)+ho(To−θ1)+S1=0
k1(θ1−θ2)+h1(θ3−θ2)+σε2ε31−(1−ε2)(1−ε3)(θ43−θ42)+S2=0
h1(θ2−θ3)+k2(θ4−θ3)+σε2ε31−(1−ε2)(1−ε3)(θ42−θ43)+S3=0
Eiε4−ε4σθ44+k2(θ3−θ4)+hi(Ti−θ4)+S4=0
Absorbed Radiation[LINK]
Si in Equations to is the radiation (shortwave and longwave from zone lights and equipment) absorbed on the ith face. Shortwave radiation (solar and shortwave from lights) is assumed to be absorbed uniformly along a glass layer, so for the purposes of the heat balance calculation it is split equally between the two faces of a layer. Glass layers are assumed to be opaque to IR so that the thermal radiation from lights and equipment is assigned only to the inside (roomside) face of the inside glass layer. For N glass layers Si is given by
S2j−1=S2j=12(IextbmcosϕAfj(ϕ)+IextdifAf,difj+IintswAb,difj),j=1toN
S2N=S2N+ε2NIintlw
Here
Iextbm = exterior beam normal solar irradiance
Iextdif = exterior diffuse solar incident on glazing from outside
Iintsw = interior shortwave radiation (from lights and from reflected diffuse solar) incident on glazing from inside
Iintlw = longwave radiation from lights and equipment incident on glazing from inside
ε2N = emissivity (thermal absorptance) of the roomside face of the inside glass layer
RoomSide Convection[LINK]
The correlation for roomside convection coefficient, hi , is from ISO 15099 section 8.3.2.2. (Prior to EnergyPlus version 3.1, the value for hi was modeled using the “Detailed” algorithm for opaque surface heat transfer, e.g. for a vertical surface hi=1.31ΔT1/133 ; see section Detailed Natural Convection Algorithm). The ISO 15099 correlation is for still room air and is determined in terms of the Nusselt number, Nu , where
hi=Nu(λH)
where,
λ is the thermal conductivity of air, and
H is the height of the window.
The Rayleigh number based on height, RaH , is calculated using,
RaH=ρ2H3gcp∣∣Tsurf,i−Tair∣∣Tm,fμλ
where,
ρ is the density of air
g is the acceleration due to gravity,
cp is the specific heat of air,
μ is the dynamic viscosity of air, and
Tm,f is the mean film temperature in Kelvin given by,
Tm,f=Tair+14(Tsurf,i−Tair)
There are four cases for the Nusselt correlation that vary by the tilt angle in degrees, γ , and are based on heating conditions. For cooling conditions (where Tsurf,i>Tair ) the tilt angle is complemented so that γ=180−γ
Case A. 0∘≤γ<15∘
Nu=0.13Ra1/133H
Case B. 15∘≤γ≤90∘
Racv=2.5×105(e0.72γsinλ)1/155
Nu=0.56(RaHsinγ)1/144;forRaH≤RaCV
Nu=0.13(Ra1/133H−Ra1/133CV)+0.56(RaCVsinγ)1/144;RaH>RaCV
Case C. 90∘<γ≤179∘
Nu=0.56(RaHsinγ)1/144;105≤RaHsinγ<1011
Case D. 179∘<γ≤180∘
Nu=0.58Ra1/155H;RaH≤1011
The material properties are evaluated at the mean film temperature. Standard EnergyPlus pyschrometric functions are used for ρ and cp . Thermal conductivity is calculated using,
λ=2.873×10−3+7.76×10−5Tm,f .
Kinematic viscosity is calculated using,
μ=3.723×10−6+4.94×10−8Tm,f .
This correlation depends on the surface temperature of the roomside glazing surface and is therefore included inside the window heat balance interation loop.
Solving the Glazing Heat Balance Equations[LINK]
The equations are solved as follows:
1) Linearize the equations by defining hr,i=εiσθ3i . For example, Equation becomes
Eoε1−hr,1θ1+k1(θ2−θ1)+ho(To−θ1)+S1=0
2) Write the equations in the matrix form Aθ=B
3) Use previous time step’s values of θi as initial values for the current time step. For the first time step of a design day or run period the initial values are estimated by treating the layers as a simple RC network.
4) Save the θi for use in the next iteration: θprev,i=θi
5) Using θ2N , reevaluate the roomside face surface convection coefficient hi
6) Using the θi to evaluate the radiative conductances hr,i
7) Find the solution θ=A−1B by LU decomposition
8) Perform relaxation on the the new θi : θi→(θi+θprev,i)/2
9) Go to step 4
Repeat steps 4 to 9 until the difference, Δθi , between values of the θi in successive iterations is less than some tolerance value. Currently, the test is
12N2N∑i=1Δθi<0.02K
If this test does not pass after 100 iterations, the tolerance is increased to 0.2K. If the test still fails the program stops and an error message is issued.
The value of the inside face temperature, θ2N , determined in this way participates in the zone heat balance solution (see Outdoor/Exterior Convection) and thermal comfort calculation (see Occupant Thermal Comfort).
EdgeOfGlass Effects[LINK]
Because of thermal bridging across the spacer separating the glass layers in multipane glazing, the conductance of the glazing near the frame and divider, where the spacers are located, is higher than it is in the center of the glass. The areaweighted net conductance (without inside and outside air films) of the glazing in this case can be written
¯¯¯h=(Acghcg+Afehfe+Adehde)/Atot
where
hcg = conductance of centerofglass region (without air films)
hfe = conductance of frame edge region (without air films)
hde = conductance of divider edge region (without air films)
Acg = area of centerofglass region
Afe = area of frame edge region
Ade = area of divider edge region
Atot = total glazing area = Acg+Afe+Ade
The different regions are shown in Figure:
Different types of glass regions.
Equation can be rewritten as
¯¯¯h=hcg(ηcg+αfeηfe+αdeηde)
where
ηcg=Acg/Atot
ηfe=Afe/Atot
ηde=Ade/Atot
αfe=hfe/hcg
αde=hde/hcg
The conductance ratios αfe and αde are user inputs obtained from Window 5. They depend on the glazing construction as well as the spacer type, gap width, and frame and divider type.
In the EnergyPlus glazing heat balance calculation effective gap convective conductances are used to account for the edgeofglass effects. These effective conductances are determined as follows for the case with two gaps (triple glazing). The approach for other numbers of gaps is analogous.
Neglecting the very small resistance of the glass layers, the centerofglass conductance (without inside and outside air films) can be written as
hcg=((hr,1+hc,1)−1+(hr,2+hc,2)−1)−1
where
hc,k = convective conductance of the kh gap
hr,k = radiative conductance of the kh gap
=12σεiεj1−(1−εi)(1−εj)(θi+θj)3
εi, εj = emissivity of the faces bounding the gap
θi, θj = temperature of faces bounding the gap (K)
Equation then becomes
¯¯¯h=(ηcg+αfeηfe+αdeηde)((hr,1+hc,1)−1+(hr,2+hc,2)−1)−1
We can also write ¯¯¯h in terms of effective convective conductances of the gaps as
¯¯¯h=((hr,1+¯¯¯hc,1)−1+(hr,2+¯¯¯hc,2)−1)−1
Comparing Eqs. and we obtain
hr,k+¯¯¯hc,k=(ηcg+αfeηfe+αdeηde)(hr,k+hc,k)
Using ηcg=1−ηfe−ηde gives
¯¯¯hc,k=hr,k(ηfe(αfe−1)+ηde(αde−1))+hc,k(1+ηfe(αfe−1)+ηde(αde−1))
This is the expression used by EnergyPlus for the gap convective conductance when a frame or divider is present.
Apportioning of Absorbed ShortWave Radiation in Shading Device Layers[LINK]
If a shading device has a nonzero shortwave transmittance then absorption takes place throughout the shading device layer. The following algorithm is used to apportion the absorbed shortwave radiation to the two faces of the layer. Here f1 is the fraction assigned to the face closest to the incident radiation and f2 is the fraction assigned to the face furthest from the incident radiation.
f1=1,f2=0ifτsh=0
Otherwise
f1=0,f2=0ifαsh≤0.01f1=1,f2=0ifαsh>0.999f1=1−e0.5ln(1−αshαshf2=1−f1}if0.01ltαsh≤0.999
Window Frame and Divider Calculation[LINK]
For the zone heat balance calculation the inside surface temperature of the frame and that of the divider are needed. These temperatures are determined by solving the heat balance equations on the inside and outside surfaces of the frame and divider.
Frame Temperature Calculation[LINK]
Figure shows a cross section through a window showing frame and divider. The outside and inside frame and divider surfaces are assumed to be isothermal. The frame and divider profiles are approximated as rectangular since this simplifies calculating heat gains and losses (see “Error Due to Assuming a Rectangular Profile,” below).
Cross section through a window showing frame and divider (exaggerated horizontally).
Frame Outside Surface Heat Balance[LINK]
The outside surface heat balance equation is
QExtIR,abs−QIR,emitted+Qconv+Qcond+Qabs=0
where
QExtIR,abs = IR from the exterior surround (sky and ground) absorbed by outside frame surfaces
QIR,emitted = IR emitted by outside frame surfaces
Qconv = convection from outside air to outside frame surfaces
Qcond = conduction through frame from inside frame surfaces to outside frame surfaces
Qabs = solar radiation (from sun, sky and ground) plus IR from outside window surface absorbed by outside frame surfaces (see “Calculation of Absorbed Solar Radiation,” below).
The first term can be written as the sum of the exterior IR absorbed by the outside face of the frame and the exterior IR absorbed by the frame’s outside projection surfaces.
QExtIR,abs=ε1EoAfFf+ε1EoAp1Fp1
where ε1 is the outside surface emissivity.
The exterior IR incident on the plane of the window, Eo, is the sum of the IR from the sky, ground and obstructions. For the purposes of the frame heat balance calculation it is assumed to be isotropic. For isotropic incident IR, Ff = 1.0 and Fp1 = 0.5, which gives
QExtIR,abs=ε1Eo(Af+12Ap1)
The IR emitted by the outside frame surfaces is
QExtIR,emitted=ε1σ(Af+Ap1)θ41
The convective heat flow from the outside air to the outside frame surfaces is
Qconv=ho,c(Af+Ap1)(To−θ1)
The conduction through the frame from inside to outside is
Qcond=kAf(θ2−θ1)
Note that Af is used here since the conductance, k, is, by definition, per unit area of frame projected onto the plane of the window.
Adding these expressions for the Q terms and dividing by* Af* gives
E0ε1(1+12η1)−ε1(1+η1)θ41+ho,c(1+η1)(T0−θ1)+k(θ2−θ1)+S1=0
where S1 = Qabs/Af and
η1=Ap1Af=(pf,1wf)H+W−(Nh+Nv)wdH+W+2wf
We linearize Eq. as follows.
Write the first two terms as
ε1(1+η1)[Eo(1+12η1)/(1+η1)−θ41]
and define a radiative temperature
To,r=[Eo(1+12η1)/(1+η1)]1/4
This gives
ε1(1+η1)[T4o,r−θ41]
which, within a few percent, equals
ε1(1+η1)(To,r+θ1)32(To,r−θ1)
Defining an outside surface radiative conductance as follows
ho,r=ε1(1+η1)(To,r+θ1)32
then gives
ho,r(To,r−θ1)
The final outside surface heat balance equation in linearized form is then
ho,r(To,r−θ1)+ho,c(1+η1)(To−θ1)+k(θ2−θ1)+S1=0
Frame Inside Surface Heat Balance[LINK]
A similar approach can be used to obtain the following linearized inside surface heat balance equation:
hi,r(Ti,r−θ2)+hi,c(1+η2)(Ti−θ2)+k(θ1−θ2)+S2=0
where
Ti,r=[Ei(1+12η2)/(1+η2)]1/4
η2=Ap2Af=(pf,2wf)H+W−(Nh+Nv)wdH+W+2wf
and Ei is the interior IR irradiance incident on the plane of the window.
Solving Eqs. and simultaneously gives
θ2=D+CA1−CB
with
A=ho,rTo,r+ho,cTo+S1ho,r+k+ho,c
B=kho,r+k+ho,c
C=khi,r+k+hi,c
D=hi,rTi,r+hi,cTi+S2hi,r+k+hi,c
Calculation of Solar Radiation Absorbed by Frame[LINK]
The frame outside face and outside projections and inside projections absorb beam solar radiation (if sunlight is striking the window) and diffuse solar radiation from the sky and ground. For the outside surfaces of the frame, the absorbed diffuse solar per unit frame face area is
Qdifabs,sol=Idifextαfr,sol(Af+Fp1Ap1)/Af=Idifextαfr,sol(1+0.5Ap1Af)
If there is no exterior window shade, Idifext includes the effect of diffuse solar reflecting off of the glazing onto the outside frame projection, i.e.,
Idifext→Idifext(1+Rf,difgl)
The beam solar absorbed by the outside face of the frame, per unit frame face area is
Qbm,faceabs,sol=Ibmextαfr,solcosβfacefsunlit
The beam solar absorbed by the frame outside projection parallel to the window xaxis is
Qbm,habs,sol=Ibmextαfr,solcosβhpf1(W−Nvwd)fsunlit/Af
Here it is assumed that the sunlit fraction, fsunlit, for the window can be applied to the window frame. Note that at any given time beam solar can strike only one of the two projection surfaces that are parallel to the window xaxis. If there is no exterior window shade, Ibmext includes the effect of beam solar reflecting off of the glazing onto the outside frame projection, i.e.,
Ibmext→Ibmext(1+Rf,bmgl)
The beam solar absorbed by the frame outside projection parallel to the window yaxis is
Qbm,vabs,sol=Ibmextαfr,solcosβvpf1(H−Nhwd)fsunlit/Af
Using a similar approach, the beam and diffuse solar absorbed by the inside frame projections is calculated, taking the transmittance of the glazing into account.
Error Due to Assuming a Rectangular Profile[LINK]
Assuming that the inside and outside frame profile is rectangular introduces an error in the surface heat transfer calculation if the profile is nonrectangular. The percent error in the calculation of convection and emitted IR is approximately 100∣∣Lprofile,rect−Lprofile,actual∣∣/Lprofile,rect , where Lprofile,rect is the profile length for a rectangular profile (wf +* pf1* for outside of frame or wf + pf2for inside of frame) and Lprofile,actual is the actual profile length. For example, for a circular profile vs a square profile the error is about 22%. The error in the calculation of absorbed beam radiation is close to zero since the beam radiation intercepted by the profile is insensitive to the shape of the profile. The error in the absorbed diffuse radiation and absorbed IR depends on details of the shape of the profile. For example, for a circular profile vs. a square profile the error is about 15%.
Divider Temperature Calculation[LINK]
The divider inside and outside surface temperatures are determined by a heat balance calculation that is analogous to the frame heat balance calculation described above.
Beam Solar Reflection from Window Reveal Surfaces[LINK]
This section describes how beam solar radiation that is reflected from window reveal surfaces is calculated. Reflection from outside reveal surfaces—which are associated with the setback of the glazing from the outside surface of the window’s parent wall—increases the solar gain through the glazing. Reflection from inside reveal surfaces—which are associated with the setback of the glazing from the inside surface of the window’s parent wall—decreases the solar gain to the zone because some of this radiation is reflected back out of the window.
The amount of beam solar reflected from reveal surfaces depends, among other things, on the extent to which reveal surfaces are shadowed by other reveal surfaces. An example of this shadowing is shown in Figure. In this case the sun is positioned such that the top reveal surfaces shadow the left and bottom reveal surfaces. And the right reveal surfaces shadow the bottom reveal surfaces. The result is that the left/outside, bottom/outside, left/inside and bottom/inside reveal surfaces each have sunlit areas. Note that the top and right reveal surfaces are facing away from the sun in this example so their sunlit areas are zero.
Example of shadowing of reveal surfaces by other reveal surfaces.
The size of the shadowed areas, and the size of the corresponding illuminated areas, depends on the following factors:
The sun position relative to the window
The height and width of the window
The depth of the outside and inside reveal surfaces
We will assume that the reveal surfaces are perpendicular to the window plane and that the window is rectangular. Then the above factors determine a unique shadow pattern. From the geometry of the pattern the shadowed areas and corresponding illuminated areas can be determined. This calculation is done in subroutine CalcBeamSolarReflectedFromWinRevealSurface in the SolarShading module. The window reveal input data is specified in the WindowProperty:FrameAndDivider object expect for the depth of the outside reveal, which is determined from the vertex locations of the window and its parent wall.
If an exterior shading device (shade, screen or blind) is in place it is assumed that it blocks beam solar before it reaches outside or inside reveal surfaces. Correspondingly, it is assumed that an interior or betweenglass shading device blocks beam solar before it reaches inside reveal surfaces.
Representative shadow patterns are shown in Figure 99 for a window with no shading device, and without and with a frame. The case with a frame has to be considered separately because the frame can cast an additional shadow on the inside reveal surfaces.
The patterns shown apply to both vertical and horizontal reveal surfaces. It is important to keep in mind that, for a window of arbitrary tilt, if the left reveal surfaces are illuminated the right surfaces will not be, and vice versa. And if the bottom reveal surfaces are illuminated the top surfaces will not be, and vice versa. (Of course, for a vertical window, the top reveal surfaces will never be illuminated by beam solar if the reveal surfaces are perpendicular to the glazing, as is being assumed.
For each shadow pattern in Figure 99, equations are given for the shadowed areas A1,sh and A2,sh of the outside and inside reveal surfaces, respectively. The variables in these equations are the following (see also Figure):
d1 = depth of outside reveal, measured from the outside plane of the glazing to the edge of the reveal, plus one half of the glazing thickness.
d2 = depth of inside reveal (or, for illumination on bottom reveal surfaces, inside sill depth), measured from the inside plane of the glazing to the edge of the reveal or the sill, plus one half of the glazing thickness.
L = window height for vertical reveal surfaces or window width for horizontal reveal surfaces
α = vertical solar profile angle for shadowing on vertical reveal surfaces or horizontal solar profile angle for shadowing on horizontal reveal surfaces.
p1(p2) = distance from outside (inside) surface of frame to glazing midplane.
d2′ = depth of shadow cast by top reveal on bottom reveal, or by left reveal on right reveal, or by right reveal on left reveal.
d2′′ = depth of shadow cast by frame.
For simplicity it is assumed that, for the case without a frame, the shadowed and illuminated areas extend into the glazing region. For this reason, d1 and d2 are measured from the midplane of the glazing. For the case with a frame, the beam solar absorbed by the surfaces formed by the frame outside and inside projections perpendicular to the glazing is calculated as described in “Window Frame and Divider Calculation: Calculation of Solar Radiation Absorbed by Frame.”
Expression for area of shaded regions for different shadow patterns: (a) window without frame, (b) window with frame
Expression for area of shaded regions for different shadow patterns: (a) window without frame, (b) window with frame
Vertical section through a vertical window with outside and inside reveal showing calculation of the shadows cast by the top reveal onto the inside sill and by the frame onto the inside sill.
The following logic gives expressions for the shadowed areas for all possible shadow patterns. Here:
d1 = d1
d2 = d2
P1 = p1
P2 = p2
f1 = d1−p1
f2 = d2−p2
d2prime = d2′
d2prime2 = d2′′
d12 = d1+d2−d2′
TanAlpha = tanα
A1sh = A1,sh
A2sh = A2,sh
L = L
L1 = average distance to frame of illuminated area of outside reveal (used to calculate view factor to frame).
L2 = average distance to frame of illuminated area of inside reveal (used to calculate view factor to frame).
The beam solar reflected from a sunlit region of area A is given by
R=IBAcosβ(1−a)
where
R = reflected solar radiation [W]
IB = beam normal irradiance [W/m2]
A = sunlit area [m2]
β = beam solar angle of incidence on reveal surface
a = solar absorptance of reveal surface
All reflected radiation is assumed to be isotropic diffuse. For outside reveal surfaces it is assumed that R/2 goes toward the window and R/2 goes to the exterior environment. Of the portion that goes toward the window a fraction F1 goes toward the frame, if present, and 1−F1 goes toward the glazing.
The view factor F1 to the frame calculated by assuming that the illuminated area can be considered to be a line source. Then the areaweighted average distance, L1 , of the source to the frame is calculated from the shape of the illuminated area (see above psuedocode). Then F1 is related as follows to the average angle subtended by the frame of width wf :
F1=tan−1(wf/L1)π/2
For the portion going towards the frame, (R/2)F1af is absorbed by the frame (where af is the solar absorptance of the frame) and contributes to the frame heat conduction calculation. The rest, (R/2)F1(1−af) , is assumed to be reflected to the exterior environment.
If the glazing has diffuse transmittance τdiff , diffuse front reflectance ρfdiff , and layer front absorptance αfl,diff , then, of the portion, (R/2)(1−F1) , that goes toward the glazing, (R/2)(1−F1)τdiff is transmitted to the zone, (R/2)(1−F1)αfl,diff is absorbed in glass layer l and contributes to the glazing heat balance calculation, and (R/2)(1−F1)ρfdiff is reflected to the exterior environment.
The beam solar absorbed by an outside reveal surface is added to the other solar radiation absorbed by the outside of the window’s parent wall.
For inside reveal surfaces it is assumed that R/2 goes towards the window and R/2 goes into the zone. Of the portion that goes toward the window a fraction (R/2)F2 goes toward the frame, if present, and (R/2)(1−F2) goes toward the glazing (F2 is calculated using a method analogous to that used for F1 ). For the portion going towards the frame, (R/2)F2af is absorbed by the frame and contributes to the frame heat conduction calculation. The rest, (R/2)F2(1−af) , is assumed to be reflected back into the zone.
If the glazing has diffuse back reflectance ρbdiff , and layer back absorptance αbl,diff , then, of the portion (R/2)(1−F2) that goes toward the glazing, (R/2)(1−F2)τdiff is transmitted back out the glazing, (R/2)(1−F2)αbl,diff is absorbed in glass layer l and contributes to the glazing heat balance calculation, and (R/2)(1−F2)ρbdiff is reflected into the zone.
The beam solar absorbed by an inside reveal surface is added to the other solar radiation absorbed by the inside of the window’s parent wall.
Shading Device Thermal Model[LINK]
Shading devices in EnergyPlus can be on the exterior or interior sides of the window or between glass layers. The window shading device thermal model accounts for the thermal interactions between the shading layer (shade, screen or blind) and the adjacent glass, and between the shading layer and the room (for interior shading) or the shading layer and the outside surround (for exterior shading).
An important feature of the shading device thermal model is calculating the natural convection airflow between the shading device and glass. This flow affects the temperature of the shading device and glazing and, for interior shading, is a determinant of the convective heat gain from the shading layer and glazing to the zone air. The airflow model is based on one described in the ISO Standard 15099, “Thermal Performance of Windows, Doors and Shading Devices—Detailed Calculations” [ISO15099, 2001]. (Betweenglass forced airflow is also modeled; see “Airflow Windows.”)
The following effects are considered by the shading device thermal model:
For interior and exterior shading device: Longwave radiation (IR) from the surround absorbed by shading device, or transmitted by the shading device and absorbed by the adjacent glass. For interior shading the surround consists of the other zone surfaces. For exterior shading the surround is the sky and ground plus exterior shadowing surfaces and exterior building surfaces “seen” by the window.
Interreflection of IR between the shading device and adjacent glass.
Direct and diffuse solar radiation absorbed by the shading device.
Interreflection of solar radiation between shading layer and glass layers.
Convection from shading layer and glass to the air in the gap (or, for betweenglass shading, gaps) between the shading layer and adjacent glass, and convection from interior shading layer to zone air or from exterior shading layer to outside air.
Natural convection airflow in the gap (or, for betweenglass shading, gaps) between shading layer and adjacent glass induced by buoyancy effects, and the effect of this flow on the shadingtogap and glasstogap convection coefficients.
For interior shading, convective gain (or loss) to zone air from gap airflow.
In the following it is assumed that the shading device, when in place, covers the glazed part of the window (and dividers, if present) and is parallel to the glazing. For interior and exterior shading devices it is assumed that the shading layer is separated from the glazing by an air gap. A betweenglass shading layer is assumed to be centered between two glass layers and separated from the adjacent glass layers by gaps that is filled with the same gas. If the window has a frame, it is assumed that the shading device does not cover the frame.
Heat Balance Equations for Shading Device and Adjacent Glass[LINK]
If a window shading device is deployed the heat balance equations for the glass surfaces facing the shading layer are modified, and two new equations, one for each face of the shading layer, are added. Figure illustrates the case of double glazing with an interior shading device.
Glazing system with two glass layers and an interior shading layer showing variables used in heat balance equations.
The heat balance equation for the glass surface facing the gap between glass and shading layer (called in the following, “gap”) is
Eiε4τsh1−ρ4ρsh+σε41−ρ4ρsh[θ45εsh−θ44(1−ρsh)]+k2(θ3−θ4)+hcv(Tgap−θ4)+S4=0
where
τsh = IR diffuse transmittance of shading device
εsh = diffuse emissivity of shading device
ρsh = IR diffuse reflectance of shading device ( = 1  ( τsh + εsh))
θ5 = temperature of the surface of the shading layer that faces the gap (K).
The term 1  ρ4ρsh accounts for the interreflection of IR radiation between glass and shading layer.
The convective heat transfer from glass layer #2 to the air in the gap is
qc,gl=hcv(θ4−Tgap)
where
Tgap = effective mean temperature of the gap air (K).
hcv = convective heat transfer coefficient from glass or shading layer to gap air (W/m2K).
The corresponding heat transfer from shading layer to gap air is
qc,sh=hcv(θ5−Tgap)
The convective heat transfer coefficient is given by
hcv=2hc+4v
where
hc = surfacetosurface heat transfer coefficient for nonvented (closed) cavities (W/m2K)
v = mean air velocity in the gap (m/s).
The quantities hcv and Tgap depend on the airflow velocity in the gap, which in turn depends on several factors, including height of shading layer, glass/shading layer separation (gap depth), zone air temperature for interior shading or outside air temperature for exterior shading, and shading layer and glass face temperatures. The calculation of hcv and Tgap is described in the following sections.
The heat balance equation for the shading layer surface facing the gap is
Eiτshρ4εsh1−ρ4ρsh+σεsh1−ρ4ρsh[ε4θ44−θ45(1−ρ4(εsh+ρsh))]+ksh(θ6−θ5)+hcv(Tgap−θ5)+Ssh,1=0
where
ksh = shading layer conductance (W/m2K).
θ6 = temperature of shading layer surface facing the zone air (K).
Ssh,l = solar radiation plus shortwave radiation from lights plus IR radiation from lights and zone equipment absorbed by the gapside face of the shading layer (W/m2K).
The heat balance equation for the shading layer surface facing the zone air is
Eiεsh−εshσθ46+ksh(θ5−θ6)+hi(Ti−θ6)+Ssh,2=0
where
Ssh,2 = solar radiation plus shortwave radiation from lights plus IR radiation from lights and zone equipment absorbed by the zoneside face of the shading layer (W/m2K).
Solving for Gap Airflow and Temperature[LINK]
For interior and exterior shading devices a pressurebalance equation is used to determine gap air velocity, gap air mean equivalent temperature and gap outlet air temperature given values of zone air temperature (or outside temperature for exterior shading), shading layer face temperatures and gap geometry. The pressure balance equates the buoyancy pressure acting on the gap air to the pressure losses associated with gap airflow between gap inlet and outlet [ISO15099, 2001]. The variables used in the following analysis of the interior shading case are shown in Figure.
Vertical section (a) and perspective view (b) of glass layer and interior shading layer showing variables used in the gap airflow analysis. The opening areas Abot, Atop, Al, Ar and Ah are shown schematically.
Pressure Balance Equation[LINK]
The pressure balance equation for airflow through the gap is
ΔpT=ΔpB+ΔpHP+ΔpZ
Here, ΔpTis the driving pressure difference between room air and gap air. It is given by
ΔpT=ρ0T0gHsinϕ∣∣Tgap−Tgap,in∣∣TgapTgap,in
where
ρ0 = density of air at temperature T0 (kg/m3)
T0 = reference temperature (283K)
g = acceleration due to gravity (m/s2)
H = height of shading layer (m)
ϕ = tilt angle of window (vertical = 90o)
Tgap = effective mean temperature of the gap air (K)
Tgap,in = gap inlet temperature ( = zone air temperature for interior shading) (K)
The ΔpB term is due to the acceleration of air to velocity v (Bernoulli’s law). It is given by
ΔpB=ρ2v2(Pa)
where ρis the gap air density evaluated at Tgap (kg/m3).
The ΔpHP term represents the pressure drop due to friction with the shading layer and glass surfaces as the air moves through the gap. Assuming steady laminar flow, it is given by the HagenPoiseuille law for flow between parallel plates [Munson et al. 1998]:
ΔpHP=12μHs2v(Pa)
where μ is the viscosity of air at temperature Tgap (Pas).
The ΔpZ term is the sum of the pressure drops at the inlet and outlet openings:
ΔpZ=ρv22(Zin+Zout)(Pa)
Here, the inlet pressure drop factor, Zin, and the outlet pressure drop factor, Zout, are given by
Zin=(Agap0.66Aeq,in−1)2Zout=(Agap0.60Aeq,out−1)2
Zin=(Agap0.66Aeq,in−1)2Zout=(Agap0.60Aeq,out−1)2
where
Aeq,in = equivalent inlet opening area (m2)
Aeq,out = equivalent outlet opening area (m2)
Agap = crosssectional area of the gap = sW(m2)
If Tgap > Tgap,in
Aeq,in=Atop+Abot2(Abot+Atop(Al+Ar+Ah)Aeq,out=Abot+Atop2(Abot+Atop(Al+Ar+Ah)
If Tgap ≤ Tgap,in
Aeq,in=Atop+Abot2(Abot+Atop(Al+Ar+Ah)Aeq,out=Abot+Atop2(Abot+Atop(Al+Ar+Ah)
Here, the area of the openings through which airflow occurs (see Figure and Figure) are defined as follows:
Abot = area of the bottom opening (m2)
Atop = area of the top opening (m2)
Al = area of the leftside opening (m2)
Ar = area of the rightside opening (m2)
Ah = air permeability of the shading device expressed as the total area of openings (“holes”) in the shade surface (these openings are assumed to be uniformly distributed over the shade) (m2)
Figure shows examples of Abot, Atop, Al and Ar for different shading device configurations. These areas range from zero to a maximum value equal to the associated shade/screen/blindtoglass crosssectional area; i.e., Abot and Atop ≤ sW, Al and Ar ≤ sH.
Examples of openings for an interior shading layer covering glass of height H and width W. Not to scale. (a) Horizontal section through shading layer with openings on the left and right sides (top view). (b) Vertical section through shading layer with openings at the top and bottom (side view).
Expression for the Gap Air Velocity
Expressing Equation in terms of v yields the following quadratic equation:
v=[(12μHs2)2+2ρ2(1+Zin+Zout)ρ0T0gHsinϕ∣∣Tgap,in−Tgap∣∣Tgap,inTgap]1/2−12μHs2ρ(1+Zin+Zout)
Solving this gives
hcv,1(Tgap,1−θ2)+k1(θ1−θ2)+σϵ21−ρ2R1[τsh1−ρ6ρ3(ϵ3θ43+ϵ6θ46ρ3)+ϵ5θ45+ϵ2θ42R1