Window Heat Balance Calculation[LINK]
|Mathematical variable||Description||Units||FORTRAN variable|
|N||Number of glass layers||-||nlayer|
|εi||Emissivity of face i||-||emis|
|ki||Conductance of glass layer i||W/m2-K||scon|
|ho, hi||Outside, inside air film convective conductance||W/m2-K||hcout, hcout|
|hj||Conductance of gap j||W/m2-K||hgap|
|To, Ti||Outdoor and indoor air temperatures||K||tout, tin|
|Eo, Ei||Exterior, interior long-wave radiation incident on window||W/m2||outir, rmir|
|θi||Temperature of face i||K||thetas|
|Si||Radiation (short-wave, and long-wave 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 short-wave radiation (from lights and from reflected diffuse solar) incident on glazing from inside||W/m2||QS|
|Iintlw||Long-wave 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||-||-|
|Ajf,dif, Ajb,dif||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/m2-K||Aface, Bface|
|hr,i||Radiative conductance for face i||W/m2-K||hr(i)|
|Δθi||Difference in temperature of face i between successive iterations||K||-|
: Fortran Variables used in Window Heat Balance Calculations
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 95 shows the variables used for double glazing (N=2).
The following assumptions are made in deriving the heat balance equations:
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.
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 multi-pane glazing to account for 2-D conduction effects across the pane separators at the boundaries of the glazing.
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.
The glass faces are isothermal. This is generally a good assumption since glass conductivity is very high.
The short wave radiation absorbed in a glass layer can be apportioned equally to the two faces of the layer.
The four equations for double-glazing are as follows. The equations for single glazing (N=1) and for N=3 and N=4 are analogous and are not shown.
Si in Equations to is the radiation (short-wave and long-wave from zone lights and equipment) absorbed on the ith face. Short-wave radiation (solar and short-wave 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 (room-side) face of the inside glass layer. For N glass layers Si is given by
Iextbm = exterior beam normal solar irradiance
Iextdif = exterior diffuse solar incident on glazing from outside
Iintsw = interior short-wave radiation (from lights and from reflected diffuse solar) incident on glazing from inside
Iintlw = long-wave radiation from lights and equipment incident on glazing from inside
ε2N = emissivity (thermal absorptance) of the room-side face of the inside glass layer
The correlation for room-side convection coefficient, hi, is from ISO 15099 section 126.96.36.199. (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=1.31|ΔT|13; 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
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,
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,
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∘
Case B. 15∘≤γ≤90∘
Case C. 90∘<γ≤179∘
Case D. 179∘<γ≤180∘
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,
Kinematic viscosity is calculated using,
This correlation depends on the surface temperature of the room-side 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:
- Linearize the equations by defining hr,i=εiσθ3i . For example, Equation becomes
Write the equations in the matrix form Aθ=B
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.
Save the θi for use in the next iteration: θprev,i=θi
- Using θ2N, reevaluate the room-side face surface convection coefficient hi
Using the θi to evaluate the radiative conductances hr,i
Find the solution θ=A−1B by LU decomposition
Perform relaxation on the the new θi : θi→(θi+θprev,i)/2
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
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).
|Mathematical variable||Description||Units||FORTRAN variable|
|¯h||Area-weighted net conductance of glazing including edge-of-glass effects||W/m2-K||-|
|Acg||Area of center-of-glass 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 center-of-glass region (without air films)||W/m2-K||-|
|hfe||Conductance of frame edge region (without air films)||W/m2-K||-|
|hde||Conductance of divider edge region (without air films)||W/m2-K||-|
|hck||Convective conductance of gap k||W/m2-K||-|
|hrk||Radiative conductance of gap k||W/m2-K||-|
|α||Conductance ratio||-||FrEdgeToCenterGlCondRatio, DivEdgeToCenterGlCondRatio|
: Fortran Variables used in Edge of Glass calculations
Because of thermal bridging across the spacer separating the glass layers in multi-pane 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 area-weighted net conductance (without inside and outside air films) of the glazing in this case can be written
hcg = conductance of center-of-glass 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 center-of-glass 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 96:
Equation can be rewritten as
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 edge-of-glass 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 center-of-glass conductance (without inside and outside air films) can be written as
hc,k= convective conductance of the kth gap
hr,k= radiative conductance of the kth gap
εi,εj= emissivity of the faces bounding the gap
θi,θj= temperature of faces bounding the gap (K)
Equation then becomes
We can also write ¯¯¯h in terms of effective convective conductances of the gaps as
Comparing Eqs. and we obtain
Using ηcg=1−ηfe−ηde gives
This is the expression used by EnergyPlus for the gap convective conductance when a frame or divider is present.
Apportioning of Absorbed Short-Wave Radiation in Shading Device Layers[LINK]
If a shading device has a non-zero short-wave transmittance then absorption takes place throughout the shading device layer. The following algorithm is used to apportion the absorbed short-wave 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.
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.
|Mathematical variable||Description||Units||FORTRAN variable|
|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|
|Rglf,dif||Front diffuse solar reflectance of glazing||-||-|
|Rglf,bm||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 x-axis||-||CosIncAngHorProj|
|Cos(βv)||Cosine of angle of incidence of beam solar on frame projection parallel to window y-axis||-||CosIncAngVertProj|
|fsunlit||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/m2-K||HOutConv, HInConv|
|k||Effective inside-surface to outside-surface conductance of frame per unit area of frame projected onto window plane||W/m2-K||FrameConductance, FrameCon|
|S2||Interior short-wave radiation plus interior IR from internal sources absorbed by inside of frame divided by Af||W/m2-K||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/m2-K||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|
: Fortran Variables used in Window/Frame and Divider calculations
Frame Temperature Calculation[LINK]
Figure 97 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 = 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.
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
The IR emitted by the outside frame surfaces is
The convective heat flow from the outside air to the outside frame surfaces is
The conduction through the frame from inside to outside is
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
where S1 = Qabs/Af and
We linearize Eq. as follows.
Write the first two terms as
and define a radiative temperature
which, within a few percent, equals
Defining an outside surface radiative conductance as follows
The final outside surface heat balance equation in linearized form is then
Frame Inside Surface Heat Balance[LINK]
A similar approach can be used to obtain the following linearized inside surface heat balance equation:
and Ei is the interior IR irradiance incident on the plane of the window.
Solving Eqs. and simultaneously gives
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
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.,
The beam solar absorbed by the outside face of the frame, per unit frame face area is
The beam solar absorbed by the frame outside projection parallel to the window x-axis is
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 x-axis. 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.,
The beam solar absorbed by the frame outside projection parallel to the window y-axis is
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 non-rectangular. 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 + pf2 for 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 98. 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 between-glass 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 100):
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, and 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 = d′2
d2prime2 = d′′2
d12 = d1+d2−d′2
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*(d2prime-d2) + 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 = d1-P1 f2 = d2-P2 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+d2-d2prime)**2-d1+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*(f1-d12/2)-d12*TanAlpha* & (f1/2-d12/3))/(L-d12*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 = 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 area-weighted average distance, L1 , of the source to the frame is calculated from the shape of the illuminated area (see above psuedo-code). Then F1 is related as follows to the average angle subtended by the frame of width wf :
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]. (Between-glass 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: Long-wave 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.
Inter-reflection of IR between the shading device and adjacent glass.
Direct and diffuse solar radiation absorbed by the shading device.
Inter-reflection of solar radiation between shading layer and glass layers.
Convection from shading layer and glass to the air in the gap (or, for between-glass 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 between-glass shading, gaps) between shading layer and adjacent glass induced by buoyancy effects, and the effect of this flow on the shading-to-gap and glass-to-gap 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 between-glass 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 101 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
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 - ρ4 ρsh accounts for the inter-reflection of IR radiation between glass and shading layer.
The convective heat transfer from glass layer #2 to the air in the gap is
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
The convective heat transfer coefficient is given by
hc = surface-to-surface heat transfer coefficient for non-vented (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
ksh = shading layer conductance (W/m2K).
θ6 = temperature of shading layer surface facing the zone air (K).
Ssh,1 = solar radiation plus short-wave radiation from lights plus IR radiation from lights and zone equipment absorbed by the gap-side face of the shading layer (W/m2K).
The heat balance equation for the shading layer surface facing the zone air is
Ssh,2 = solar radiation plus short-wave radiation from lights plus IR radiation from lights and zone equipment absorbed by the zone-side face of the shading layer (W/m2K).
Solving for Gap Airflow and Temperature[LINK]
For interior and exterior shading devices a pressure-balance 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 102.
Pressure Balance Equation[LINK]
The pressure balance equation for airflow through the gap is
Here, ΔpT is the driving pressure difference between room air and gap air. It is given by
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
where ρ is 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 Hagen-Poiseuille law for flow between parallel plates [Munson et al. 1998]:
where μ is the viscosity of air at temperature Tgap (Pa-s).
The ΔpZ term is the sum of the pressure drops at the inlet and outlet openings:
Here, the inlet pressure drop factor, Zin, and the outlet pressure drop factor, Zout, are given by
Aeq,in = equivalent inlet opening area (m2)
Aeq,out = equivalent outlet opening area (m2)
Agap = cross-sectional area of the gap = sW (m2)
If Tgap > Tgap,in
If Tgap ≤ Tgap,in
Here, the area of the openings through which airflow occurs (see Figure 102 and Figure 103) are defined as follows:
Abot = area of the bottom opening (m2)
Atop = area of the top opening (m2)
Al = area of the left-side opening (m2)
Ar= area of the right-side 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 103 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/blind-to-glass cross-sectional 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:
Solving this gives
The choice of the root of the quadratic equation is dictated by the requirement that v = 0 if Tgap,in = Tgap.
Gap Outlet Temperature and Equivalent Mean Air Temperature
The temperature of air in the gap as a function of distance, h, from the gap inlet (Figure 104) is
is the average temperature of the glass and shading layer surfaces facing the gap (K).
H0 = characteristic height (m), given by
where Cp is the heat capacity of air.
The gap outlet temperature is given by
The thermal equivalent mean temperature of the gap air is
Solution Sequence for Gap Air Velocity and Outlet Temperature
The routine WinShadeGapFlow is called within the glazing heat balance iterative loop in SolveForWindowTemperatures to determine v and Tgap,out. The solution sequence in WinShadeGapFlow is as follows:
At start of iteration, guess Tgap as ((Tgl + Tsh)/2 + Tgap,in)/2. Thereafter use value from previous iteration.
Get still-air conductance, hc, by calling WindowGasConductance and NusseltNumber.
Get v from Equation
Get hcv from Equation
Get Tave from Equation
Get Tgap,out from Equation
Get new value of Tgap from Equation
The values of hcv and Tgap so determined are then used in the window heat balance equations to find new values of the face temperatures of the glass and shading layers. These temperatures are used in turn to get new values of hcv and Tgap until the whole iterative process converges.
Convective Heat Gain to Zone from Gap Airflow
The heat added (or removed) from the air as it passes through the gap produces a convective gain (or loss) to the zone air given by
This can also be expressed as
where the air mass flow rate in the gap is given by
Heat Balance Equations for Between-Glass Shading Device[LINK]
In EnergyPlus shading devices are allowed between the two glass panes of double glazing and between the two inner glass panes of triple glazing. Figure 105 shows the case of a between-glass shading device in double glazing.
The heat balance equations for the two glass surfaces facing the shading device are the following.
For face #2: