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energies
Article
Development of a Reduced Order Model of Solar
Heat Gains Prediction
Meril Tamm 1, *, Jordi MaciàCid 2, Roser Capdevila Paramio 3, Joan Farnós Baulenas 2,
Martin Thalfeldt 1and Jarek Kurnitski 1,4
1Department of Civil Engineering and Architecture, Tallinn University of Technology, Ehitajate tee 5,
19086 Tallinn, Estonia; martin.thalfeldt@taltech.ee (M.T.); jarek.kurnitski@taltech.ee (J.K.)
2
Waste, Energy and Enivonmental Impact Unit, Eurecat, Centre Tecnol
ò
gic de Catalunya, Plaça de la Ci
è
ncia,
2, 08242 Manresa, Barcelona, Spain; jordi.macia@eurecat.org (J.M.C.); joan.farnos@gmail.com (J.F.B.)
3Department of Heat Engines, Polytechnic University of Catalonia, Carrer de Colom, 11,
08222 Terrassa, Barcelona, Spain; roser.capdevila@upc.edu
4Department of Civil Engineering, Aalto University, P.O. Box 12100, 00076 Aalto, Finland
*Correspondence: meril.tamm@taltech.ee
Received: 7 September 2020; Accepted: 18 November 2020; Published: 30 November 2020
Abstract:
The aim of this study was to elaborate and validate a reduced order model able to forecast
solar heat gains as a function of the architectural parameters that determine the solar heat gains.
The study focused on office buildings in Catalonia and Spain and their physical values were taken
from the Spanish Building Technical Code and European Union Directive 2018/844. A reduced order
model with three direct variables (solar heat gain coefficient, shade factor, window to wall ratio)
and one indirect design variable (building orientation) was obtained and validated in respect to the
International Performance Measurement and Verification Protocol. Building envelope properties
were fixed and the values were taken from the national standards of Spain. This work validates solar
heat gain coefficient as a primary variable in determining the annual solar heat gains in a building.
Further work of developed model could result in building energy need quick evaluation tool in terms
of solar heat gains for architects in building early stage as it has an advantage over detailed building
simulation programs in terms of instant calculation and the limited need for predefined input data.
Keywords: reduced order model; solar heat gains; building thermal performance
1. Introduction
Solar energy is the largest untapped energy source, which used in a smart way can result in great
active and passive energy savings in building sector. There are different ways to take advantage of
the free energy source of solar energy and methods vary between the concept of passive and active
solutions and by the investment cost. Even though solar energy is used as a passive solution that
can compensate the heating demand of a building, the thermal performance of a building should
be analyzed on an annual basis, as excessive solar heat gains in the summer period can result in
overheating and an increase of cooling demand [
1
]. In terms of new buildings, passive solar solutions
should be considered in the early design stages of the building, as with simple measures it is possible
to reduce the high amount of CO
2
emissions in building sector, as it is the largest source of emissions
in European Union [
2
]. The typical and time-consuming way of analyzing internal gains from solar
radiation is using a dynamic simulation software, which requires the knowledge of an expert. Therefore,
for making solar calculations accessible for a regular user, there is a need to develop simplified energy
calculators that require reduced order models.
In this work, a reduced order model or solar heat gains was suggested and validated. There were
4 key design variables (window to wall ratio (WWR), solar heat gain coefficient (SHGC), shade factor
Energies 2020,13, 6316; doi:10.3390/en13236316 www.mdpi.com/journal/energies
Energies 2020,13, 6316 2 of 17
and orientation of the building) chosen for creating the model, that were considered to hold the main
importance when defining the behavior of solar heat gains. While creating the reduced order model,
it was noted, that SHGC has the capacity to define the solar heat gains with acceptable precision,
while not taking convective heat transfer into consideration. SHGC, often referred to as g-value or solar
coefficient, is the sum of the solar direct transmittance and the secondary heat transfer factor towards
the inside. The second component consists of heat transfer by convection and longwave infrared
radiation of that part of the incident solar radiation, that has been absorbed by the glazing [
3
]. The value
of SHGC has to be measured when the fenestration system is under recommended standard conditions
of ISO 19467, which requires the internal temperature of +20
◦
C and external of 0
◦
C. The required
internal surface coefficient of heat transfer is 8 W/m
2·
K and external 24 W/m
2·
K. Net radiation flux of
incident radiation for the case of normal incidence is 300 W/m
2
. The tolerance for the air temperature
or environmental temperature difference between sides during the measurements shall be ±5◦C [4].
In previous studies, these design parameters, such as window-to-wall ratio, building orientation,
and glazing materials, have been studied separately from the influence of solar heat gains. It has been
well established that the orientation of the building has an impact on the energy consumption of the
building, as well as the area of the glazed surfaces, as the entering solar radiation contributes to a
heating demand through the energy absorbed in the interior furnishings and the internal partition.
Therefore, the glazing properties, mainly expressed by solar heat gain coefficient, and shading of the
windows, are holding the important place in reduction of the entering solar heat gains [
5
–
8
]. In this
study an overall conclusion was brought to a concept of a reduced order model. The current work
focused on the solar radiation that is transmitted through the windows of a building, contributing
to the energy balance by absorbing the transmitted energy in surfaces inside and reradiating the
long-wave thermal radiation, which in the end is defined as solar heat gains. The transient simulation
software Energy Plus, that was used for simulations, calculates solar heat gains with the Perez solar
model [
9
]. The purpose of the proposed reduced order model was to propose a simplified version of
the Energy Plus evaluation of solar heat gains. Some models pay special attention to the solar radiation
absorbed and reflected by the interior [
10
,
11
], but in Energy Plus, the detailed simulation software,
taken as a reference, does not simulate that phenomena when computing the solar heat gains. Hence,
the reduced model will be developed without taking into account that phenomena. Nevertheless,
leaving that phenomena out can result in an error from 3 to 8% according to [12].
In the first part of the work, a parametric study was carried out with Energy Plus and the
correlation between solar heat gains and chosen building design parameters was obtained. In the
second part of the study a reduced order model based on the physical phenomena was developed and
validated in accordance with the International Performance Measurement and Verification Protocol [
13
].
2. Methods
The work was divided into six phases, as expressed in Figure 1. In order to create the most
adequate model possible, key design variables of a building had to be chosen. As there was no
previous work carried out by the author to define the key variables, the decision was based on previous
research carried out in building thermal analysis sector [
14
–
16
]. Before launching the simulations,
design variable ranges and simulation base cases had to be defined. Therefore 4 base buildings were
chosen, with a detailed data from building engineering guide book of Spain [
17
]. The design phase of
current work was carried out with Design Builder [
18
], where the building shape was drawn, given the
construction values, and defined the activity and mechanical systems inside. The building base cases
were simulated in 12 climate zones of Spain with transient energy simulation software EnergyPlus [
19
].
Data analyses were done in Matlab [20].
Energies 2020,13, 6316 3 of 17
Energies2020,13,xFORPEERREVIEW3of17
Figure1.Methodologyofworkcarriedout.
2.1.IdentificationoftheDesignVariables
Fordevelopingafastpredictionmodelfordeterminingtheeffectofthemaininfluencing
parametersonsolarheatgains,fourvariables(SHGC,WWR,shadefactor,andorientation)were
chosen.Physicalandsimplydeterminablevariableswerechoseninordertobebeneficialforthe
architectsfordirectuseinabuildingdesignstage.Designvariableswerechosenbasedonprevious
researchdoneinthebuildingenergyperformancesector,pointingoutthekeyaspectsthatinfluence
solarheatgainsinabuilding.
2.2.FirstDesignVariable:ShadeFactor
Shadingdevices,whetherverticalorhorizontalones,aredirectlyaffectingthesunlightentering
theroomandthereforeparticipatinginadecreaseorincreaseinthermalenergyloadsofthebuilding
[14].Detailedunderstandingofshadingfactorisofgreatimportancefordeterminingthesolarheat
gainsinabuilding.Eventhoughwindowtowallratioisthemaininfluenceroftheenergyloadsof
thebuilding[16],properevaluationofshadingcanmakeagreatdifference[21].Theimpactof
shadingisoftenunderestimatedinthebuildingdesignphase,eventhoughdetailedanalyzeof
shadingsystemscanresultinsignificantenergysavings,especiallyinsummermonthsin
compensationofcoolingloads[15].Shadingfactorisdescribedasaratiobetweentheirradiance
presenceofshadingobstaclesandirradianceinabsenceofobstacles.Shadingfactorvaluecanbe
representedasfollows:
𝐹
ₛ
𝐼ₜˎ
ₛ
𝐼
ₜ
(1)
whereFₛistheshadingfactor,
𝐼ₜˎₛisglobalirradianceonashadedsurface,W/m2
Itisglobalirradiance,thatshouldreachthesurfaceinabsenceofshading,W/m2
Toprovideaspecificunderstandingofshadingfactor,thepreviousequationcanbeexpressed
explicitlyasfollows[15]:
𝐹𝑠 𝐹, ∙𝐼 𝐹,∙𝐼 𝐼
𝐼 𝐼 𝐼(2)
where𝐹,isthegeometricshadingcoefficientfordirectradiation
𝐹,isthegeometricshadingcoefficientfordiffuseradiation
𝐼isthedirectirradianceinabsenceofshadingobstacles,W/m2
𝐼isthediffuseirradianceinabsenceofshadingobstacles,W/m2
𝐼isthereflectedirradianceinabsenceofshadingobstacles,W/m2
Bysimplifiedmeaning,theshadefactoristheratioofshadedareaofthewindow,being0,when
thereisnoshadingand1,whenthewindowisfullyshaded.Thisunderstandingallowsthearchitects
touseshadefactorasdirectandfastinputdata,beingcausedwhetherbyshadingobjectsofawindow
ornearbybuildings,trees,orotherinfluencingaspects.Forfastevaluationoftheshadefactor,for
Mediterraneanregion,valuescouldbeobtainedfromtheSpanishBaseDocumentofEnergySavings
Figure 1. Methodology of work carried out.
2.1. Identification of the Design Variables
For developing a fast prediction model for determining the effect of the main influencing
parameters on solar heat gains, four variables (SHGC, WWR, shade factor, and orientation) were
chosen. Physical and simply determinable variables were chosen in order to be beneficial for the
architects for direct use in a building design stage. Design variables were chosen based on previous
research done in the building energy performance sector, pointing out the key aspects that influence
solar heat gains in a building.
2.2. First Design Variable: Shade Factor
Shading devices, whether vertical or horizontal ones, are directly affecting the sunlight entering the
room and therefore participating in a decrease or increase in thermal energy loads of the building [
14
].
Detailed understanding of shading factor is of great importance for determining the solar heat gains
in a building. Even though window to wall ratio is the main influencer of the energy loads of the
building [
16
], proper evaluation of shading can make a great difference [
21
]. The impact of shading is
often underestimated in the building design phase, even though detailed analyze of shading systems
can result in significant energy savings, especially in summer months in compensation of cooling
loads [
15
]. Shading factor is described as a ratio between the irradiance presence of shading obstacles
and irradiance in absence of obstacles. Shading factor value can be represented as follows:
Fs=I¸t,s
It(1)
where Fsis the shading factor,
I¸t,s is global irradiance on a shaded surface, W/m2
Itis global irradiance, that should reach the surface in absence of shading, W/m2
To provide a specific understanding of shading factor, the previous equation can be expressed
explicitly as follows [15]:
FS=Fs,b·Ib+Fs,d·Id+Ir
Ib+Id+Ir(2)
where Fs,bis the geometric shading coefficient for direct radiation
Fs,dis the geometric shading coefficient for diffuse radiation
Ibis the direct irradiance in absence of shading obstacles, W/m2
Idis the diffuse irradiance in absence of shading obstacles, W/m2
Iris the reflected irradiance in absence of shading obstacles, W/m2
By simplified meaning, the shade factor is the ratio of shaded area of the window, being 0,
when there is no shading and 1, when the window is fully shaded. This understanding allows the
architects to use shade factor as direct and fast input data, being caused whether by shading objects of
a window or nearby buildings, trees, or other influencing aspects. For fast evaluation of the shade
factor, for Mediterranean region, values could be obtained from the Spanish Base Document of Energy
Energies 2020,13, 6316 4 of 17
Savings [
17
], which, depending on the geometry of different shade, suggests numerical averaged
values for fast energy demand prediction.
The equations above show, that the relationship between shade factor and reduced radiation rate
I¸t,s
is proportional, which gives the leads to the conclusion that also the relationship between solar heat
gains and shade factor should be linear.
Shade factor represents the shaded area ratio of the window, varying from 0 to 1. Shade factor
does not carry information about specific shading device type, as it represents only the percentage of
shaded area of a window. Energy Plus simulations were carried out to obtain desired shade factor ratio,
while adding shade to the windows and using the theoretical approach of varying the transparency
of the shade. EnergyPlus considers shade as a parallel layer to the window, as shown in Figure 2,
independent of angle of incidence. When used, shade is assumed to cover all the glazed area, dividers
included [22].
Energies2020,13,xFORPEERREVIEW4of17
[17],which,dependingonthegeometryofdifferentshade,suggestsnumericalaveragedvaluesfor
fastenergydemandprediction.
Theequationsaboveshow,thattherelationshipbetweenshadefactorandreducedradiation
rate𝐼ₜˎₛisproportional,whichgivestheleadstotheconclusionthatalsotherelationshipbetween
solarheatgainsandshadefactorshouldbelinear.
Shadefactorrepresentstheshadedarearatioofthewindow,varyingfrom0to1.Shadefactor
doesnotcarryinformationaboutspecificshadingdevicetype,asitrepresentsonlythepercentageof
shadedareaofawindow.EnergyPlussimulationswerecarriedouttoobtaindesiredshadefactor
ratio,whileaddingshadetothewindowsandusingthetheoreticalapproachofvaryingthe
transparencyoftheshade.EnergyPlusconsidersshadeasaparallellayertothewindow,asshown
inFigure2,independentofangleofincidence.Whenused,shadeisassumedtocoveralltheglazed
area,dividersincluded[22].
Figure2.Windowshadeexamples.Firstshadefromtheviewfromthetop(a)andsecondshadeview
fromtheside(b).
2.3.SecondDesignVariable:WindowtoWallRatio(WWR)
Foraestheticsandfordaylightingpurposes,glazedbuildingfacadesareoftenusedinmodern
officebuildings.Solarradiationthatistransmittedthroughtheglassheatsupthesurfacesinsidethe
zonethataftersometimebecomeheatsources[14].Therefore,aslargescalefenestrationcanbring
greatbenefitsindaylighting,thermaleffectshouldbeconsideredwithrespect.Solarheatgainsina
buildinghasahighdependenceontransparentsurfaces.Eventhoughtheopaquesurfacescontribute
totheenergydemandofabuilding,transparentareashaveamoresignificantinfluence[16].For
describinghowbigapartofthewallisoccupiedbythewindow,theratioofwindowandwall(WWR)
isoftenusedandisexpressedasfollows:
WWR ∑𝐺𝑙𝑎𝑧𝑖𝑛𝑔 𝑎𝑟𝑒𝑎, m
∑𝐺𝑟𝑜𝑠𝑠 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑤𝑎𝑙𝑙 𝑎𝑟𝑒𝑎, m(3)
WWR=0referstoawallwithoutthewindow,andWWR=100toawall,whichhastotalglass
façade.AvisualexpressionofthiscanbeseeninFigure3below.
Figure3.Windowtowallratioexamples.WWR=10onthe(left)andWWR=100onthe(right).
Figure 2.
Window shade examples. First shade from the view from the top (
a
) and second shade view
from the side (b).
2.3. Second Design Variable: Window to Wall Ratio (WWR)
For aesthetics and for daylighting purposes, glazed building facades are often used in modern
office buildings. Solar radiation that is transmitted through the glass heats up the surfaces inside the
zone that after some time become heat sources [
14
]. Therefore, as largescale fenestration can bring great
benefits in daylighting, thermal effect should be considered with respect. Solar heat gains in a building
has a high dependence on transparent surfaces. Even though the opaque surfaces contribute to the
energy demand of a building, transparent areas have a more significant influence [
16
]. For describing
how big a part of the wall is occupied by the window, the ratio of window and wall (WWR) is often
used and is expressed as follows:
WWR =PGlazing area, m2
PGross exterior wall area, m2(3)
WWR =0 refers to a wall without the window, and WWR =100 to a wall, which has total glass
façade. A visual expression of this can be seen in Figure 3below.
Energies2020,13,xFORPEERREVIEW4of17
[17],which,dependingonthegeometryofdifferentshade,suggestsnumericalaveragedvaluesfor
fastenergydemandprediction.
Theequationsaboveshow,thattherelationshipbetweenshadefactorandreducedradiation
rate𝐼ₜˎₛisproportional,whichgivestheleadstotheconclusionthatalsotherelationshipbetween
solarheatgainsandshadefactorshouldbelinear.
Shadefactorrepresentstheshadedarearatioofthewindow,varyingfrom0to1.Shadefactor
doesnotcarryinformationaboutspecificshadingdevicetype,asitrepresentsonlythepercentageof
shadedareaofawindow.EnergyPlussimulationswerecarriedouttoobtaindesiredshadefactor
ratio,whileaddingshadetothewindowsandusingthetheoreticalapproachofvaryingthe
transparencyoftheshade.EnergyPlusconsidersshadeasaparallellayertothewindow,asshown
inFigure2,independentofangleofincidence.Whenused,shadeisassumedtocoveralltheglazed
area,dividersincluded[22].
Figure2.Windowshadeexamples.Firstshadefromtheviewfromthetop(a)andsecondshadeview
fromtheside(b).
2.3.SecondDesignVariable:WindowtoWallRatio(WWR)
Foraestheticsandfordaylightingpurposes,glazedbuildingfacadesareoftenusedinmodern
officebuildings.Solarradiationthatistransmittedthroughtheglassheatsupthesurfacesinsidethe
zonethataftersometimebecomeheatsources[14].Therefore,aslargescalefenestrationcanbring
greatbenefitsindaylighting,thermaleffectshouldbeconsideredwithrespect.Solarheatgainsina
buildinghasahighdependenceontransparentsurfaces.Eventhoughtheopaquesurfacescontribute
totheenergydemandofabuilding,transparentareashaveamoresignificantinfluence[16].For
describinghowbigapartofthewallisoccupiedbythewindow,theratioofwindowandwall(WWR)
isoftenusedandisexpressedasfollows:
WWR ∑𝐺𝑙𝑎𝑧𝑖𝑛𝑔 𝑎𝑟𝑒𝑎, m
∑𝐺𝑟𝑜𝑠𝑠 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑤𝑎𝑙𝑙 𝑎𝑟𝑒𝑎, m(3)
WWR=0referstoawallwithoutthewindow,andWWR=100toawall,whichhastotalglass
façade.AvisualexpressionofthiscanbeseeninFigure3below.
Figure3.Windowtowallratioexamples.WWR=10onthe(left)andWWR=100onthe(right).
Figure 3. Window to wall ratio examples. WWR =10 on the (left) and WWR =100 on the (right).
Energies 2020,13, 6316 5 of 17
The formula of window to wall ratio reveals that window area is whether reduced or increased
proportionally with the WWR, therefore also the possible solar radiation that can hit the transparent
surface is influenced proportionally. This means that the expected relationship between solar radiation
and therefore the solar heat gains is linear.
In those simulations WWR was chosen as a design variable, which was given values over
WWR =10 . . . 100, which is illustrated in the following graphs (Figure 4):
Energies2020,13,xFORPEERREVIEW5of17
Theformulaofwindowtowallratiorevealsthatwindowareaiswhetherreducedorincreased
proportionallywiththeWWR,thereforealsothepossiblesolarradiationthatcanhitthetransparent
surfaceisinfluencedproportionally.Thismeansthattheexpectedrelationshipbetweensolar
radiationandthereforethesolarheatgainsislinear.
InthosesimulationsWWRwaschosenasadesignvariable,whichwasgivenvaluesoverWWR
=10…100,whichisillustratedinthefollowinggraphs(Figure4):
Figure4.Windowtowallratiosimulationcases.Examplewithfirstbuildingshape.
2.4.ThirdDesignVariable:BuildingOrientation
Buildingorientationisconsideredasanimportantbuildingdesignvariable,asdifferentfacades
aresunlitondifferenthoursandintensitiesofaday,duetothesunpaththatcausesincidentradiation
angleanddirectionvariations.Therefore,buildingorientationshouldbeanalyzedinabuilding
designstagebythearchitects.Oftenthedecisionoforientationismadebythebestaccessibilityor
landscape,eventhoughbuildingorientationaffectstheamountofsolarradiationthatentersthrough
theopaqueandtransparentsurfaces.Asbuildingorientationhasanimpactonheatingandcooling
loads,thisimportantdesignvariableshouldbeconsideredinthepreliminarybuildingdesignstage
[14].
Buildingbasecaseorientation0representstheshortersideofthebuildingfacingnorth,as
illustratedinFigure5.Inthefollowingsimulations,thebuildingwasrotatedover360degrees.
Figure5.Buildingorientationsimulationswithrelationtosolartrajectory,fourthbuilding.
Orientation90degreesonthe(left)and0onthe(right).
2.5.FourthDesignVariable:SolarHeatGainCoefficient
Figure 4. Window to wall ratio simulation cases. Example with first building shape.
2.4. Third Design Variable: Building Orientation
Building orientation is considered as an important building design variable, as different facades
are sunlit on different hours and intensities of a day, due to the sun path that causes incident radiation
angle and direction variations. Therefore, building orientation should be analyzed in a building
design stage by the architects. Often the decision of orientation is made by the best accessibility or
landscape, even though building orientation affects the amount of solar radiation that enters through
the opaque and transparent surfaces. As building orientation has an impact on heating and cooling
loads, this important design variable should be considered in the preliminary building design stage [
14
].
Building base case orientation 0 represents the shorter side of the building facing north, as illustrated
in Figure 5. In the following simulations, the building was rotated over 360 degrees.
Energies2020,13,xFORPEERREVIEW5of17
Theformulaofwindowtowallratiorevealsthatwindowareaiswhetherreducedorincreased
proportionallywiththeWWR,thereforealsothepossiblesolarradiationthatcanhitthetransparent
surfaceisinfluencedproportionally.Thismeansthattheexpectedrelationshipbetweensolar
radiationandthereforethesolarheatgainsislinear.
InthosesimulationsWWRwaschosenasadesignvariable,whichwasgivenvaluesoverWWR
=10…100,whichisillustratedinthefollowinggraphs(Figure4):
Figure4.Windowtowallratiosimulationcases.Examplewithfirstbuildingshape.
2.4.ThirdDesignVariable:BuildingOrientation
Buildingorientationisconsideredasanimportantbuildingdesignvariable,asdifferentfacades
aresunlitondifferenthoursandintensitiesofaday,duetothesunpaththatcausesincidentradiation
angleanddirectionvariations.Therefore,buildingorientationshouldbeanalyzedinabuilding
designstagebythearchitects.Oftenthedecisionoforientationismadebythebestaccessibilityor
landscape,eventhoughbuildingorientationaffectstheamountofsolarradiationthatentersthrough
theopaqueandtransparentsurfaces.Asbuildingorientationhasanimpactonheatingandcooling
loads,thisimportantdesignvariableshouldbeconsideredinthepreliminarybuildingdesignstage
[14].
Buildingbasecaseorientation0representstheshortersideofthebuildingfacingnorth,as
illustratedinFigure5.Inthefollowingsimulations,thebuildingwasrotatedover360degrees.
Figure5.Buildingorientationsimulationswithrelationtosolartrajectory,fourthbuilding.
Orientation90degreesonthe(left)and0onthe(right).
2.5.FourthDesignVariable:SolarHeatGainCoefficient
Figure 5.
Building orientation simulations with relation to solar trajectory, fourth building. Orientation
90 degrees on the (left) and 0 on the (right).
Energies 2020,13, 6316 6 of 17
2.5. Fourth Design Variable: Solar Heat Gain Coefficient
Heat gains from the transparent surfaces of a building have an important part in building’s cooling
load, especially in buildings with greater transparent areas [
23
]. One part of the solar radiation is
being transmitted to inner space, a smaller portion of solar radiation gets absorbed in the fenestration,
and part of this is transferred to the room as infrared radiation and the other part by convection [
24
].
SHGC consists of two parts, which are expressed below [25]:
SHGC =Ts+As·Ni(4)
where SHGC is the solar heat gain coefficient
Tsis the solar transmittance
Asis the solar absorptance
Niis the fraction of absorbed solar radiation that enters the room.
Both of the components of SHGC are proportional to solar radiation [
26
] and therefore also to
solar heat gains. This allows to predict the relationship between SHGC and solar heat gains to be linear.
Fourth building design variable was chosen to be SHGC and in order to obtain it, window solar
transmissivity was varied from 0 to 0.8.
2.6. Reduced Order Model (ROM)
The main purpose of this study was to elaborate and validate a reduced order model of solar heat
gains, to be applicable in monthly method energy calculation. The idea of creating reduced order
model relies on simplifying and optimizing the calculations in time and complexity. When determining
the key aspects in a complex dynamic system, the model could be created with only the predominant
variables. The idea of this approach, called the model order reduction, has the goal to reduce the
original degrees of freedom to very small number, while keeping the input-output data accuracy the
same [27].
2.7. Simulation
The purpose of the simulations was to observe the numerical data and correlate solar heat gains
behavior to the chosen building design variables, to compare the outcome of the simulations with the
set hypothesis. Current work was carried out with energy simulation program EnergyPlus, which was
fed by the input data created with Design Builder. Energy Plus is a free software developed by the
Department of Energy (DOE, USA) which is validated for energy simulation. The software contains
tools in order to run a large number of simulations, as well as parametric simulations.
In order to obtain general overview of building thermal performance in Spain, simulations were
carried out in all twelve climate zones of Spain. Spanish climate zones are categorized by the summer
and winter severity rates. Climate zone A3 represents the climate with the mildest summer and winter,
while zone E1 with the roughest ones [17].
Simulations were carried out with four reference buildings with different floor shapes and number
of floors, to ensure the approach was independent from building shape. Each building shape was
simulated with respect to the chosen variables.
2.8. Description of Simulation Cases
The base window to wall ratio was chosen to be typical for the office buildings, being 40%.
Floor height was chosen 3.5 m in all the cases. Every floor consisted of 1 zone. All the buildings had
the shortest side facing the north. The four simulation base cases are presented in Figure 6.
Energies 2020,13, 6316 7 of 17
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Figure6.Simulationbasecases.
2.9.SolarRadiationSimulations
Theanalysisofthefourdesignvariableswerecarriedoutandpredictedhypothesisofthe
behaviorwerecomparedtothesimulatedresultinordertodetermineiftheresultscouldbe
implementedtothenumericalmodel.
Fourdesignvariables,namelyshadefactor,windowtowallratio,orientation,andSHGC,were
studiedtobeapplicableforthereducedordermodelforquickestimationofsolarheatgainsina
building.Asallthedesignvariablesprovedtheirimportancethoughtheirlinearrelation(Figure7)
tosolarheatgains,allfourvariableswereimplementedtothedevelopednumericalmodel.Shade
factor,SHGC,andwindowtowallratiowereaddedasdirectmultiplicatorsofsolarradiationand
orientationindirectlybeingattachedtoeverysinglesurfaceplacementthroughsolarangles,as
representedinthedirectbeamradiationcalculation.
Figure 6. Simulation base cases.
2.9. Solar Radiation Simulations
The analysis of the four design variables were carried out and predicted hypothesis of the behavior
were compared to the simulated result in order to determine if the results could be implemented to the
numerical model.
Four design variables, namely shade factor, window to wall ratio, orientation, and SHGC,
were studied to be applicable for the reduced order model for quick estimation of solar heat gains in a
building. As all the design variables proved their importance though their linear relation (Figure 7) to
solar heat gains, all four variables were implemented to the developed numerical model. Shade factor,
SHGC, and window to wall ratio were added as direct multiplicators of solar radiation and orientation
indirectly being attached to every single surface placement through solar angles, as represented in the
direct beam radiation calculation.
Energies 2020,13, 6316 8 of 17
Energies2020,13,xFORPEERREVIEW8of17
Figure7.Shadefactor/annualsolarheatgains,firstbuilding(up,left);SHGC/annualsolarheatgains
correlation,thirdbuilding(up,right);Orientation/solarheatgains,firstbuilding(down,left);
WWR/solarheatgains,firstbuilding(down,right).
2.10.SolarHeatGainModel
Asthecorrelationsobtainedinthesimulationpartshowedtheindependenceofbuildingshape
andclimatezones,generalconclusionsasprovidingamodel,couldbedone.Chosendesignvariables,
thatshowedtohavelinearcorrelationwithsolarheatgains,wereusedasaninputdataforthe
designablemodel,asdescribedinfollowingsections.Afterthecreationofthenumericalmodel,
validationwascarriedoutinordertoprovetheaccuracyofthemodel.
Theexpectationsweretoobservelinearcorrelationbetweensimulatedsolarheatgainsand
architecturaldesignparameterswhichintheendwerecomparedtothemathematicalcalculationof
theoreticalsolarheatgains.Thedeterminablemodelwasexpectedtoshowthecorrelationbetween
theoreticalsolarheatgainsandsimulatedsolarheatgains,whichcouldallowtodetermineempirical
model,withthedesignvariablesasaninputdata.
2.11.ModelDevelopment
Numericalmodelforestimatingsolarheatgainswereobtainedthroughthecomparisonof
simulatedresultsofsolarheatgainstothemathematicalcalculationofsolarheatgains.Itwas
expectedthatsimilarbehaviorinbothcaseswouldbeobservedand,inthecaseofanoffset,a
correctionfactorwouldbeprovided.Solarradiationcouldbecalculatedwithknownmathematical
formulasandtheintentionwasbymultiplyingthemathematicalresultwithchosendesignvariables
toreachthesamevaluesthatweresimulated.Thisapproachwouldallowtoestimatesolarheatgains
fasterthanusinganenergysimulationprogram,whichrequiresspecialistknowledgeforadequate
analysis.
Figure 7.
Shade factor/annual solar heat gains, first building (
up
,
left
); SHGC/annual solar heat
gains correlation, third building (
up
,
right
); Orientation/solar heat gains, first building (
down
,
left
);
WWR/solar heat gains, first building (down,right).
2.10. Solar Heat Gain Model
As the correlations obtained in the simulation part showed the independence of building shape
and climate zones, general conclusions as providing a model, could be done. Chosen design variables,
that showed to have linear correlation with solar heat gains, were used as an input data for the
designable model, as described in following sections. After the creation of the numerical model,
validation was carried out in order to prove the accuracy of the model.
The expectations were to observe linear correlation between simulated solar heat gains and
architectural design parameters which in the end were compared to the mathematical calculation of
theoretical solar heat gains. The determinable model was expected to show the correlation between
theoretical solar heat gains and simulated solar heat gains, which could allow to determine empirical
model, with the design variables as an input data.
2.11. Model Development
Numerical model for estimating solar heat gains were obtained through the comparison of
simulated results of solar heat gains to the mathematical calculation of solar heat gains. It was expected
that similar behavior in both cases would be observed and, in the case of an offset, a correction factor
would be provided. Solar radiation could be calculated with known mathematical formulas and the
intention was by multiplying the mathematical result with chosen design variables to reach the same
values that were simulated. This approach would allow to estimate solar heat gains faster than using
an energy simulation program, which requires specialist knowledge for adequate analysis.
Energies 2020,13, 6316 9 of 17
The developed model was created while using mathematical expression of solar heat gains,
as explained in the next chapter, while integrating chosen simulated building design variables as
multipliers of total incident solar radiation.
Following model was suggested:
SGBuilding =m
P
i=1
SG sur f ace
=m
P
1RIT(t,θ)·WWR·Swall·SHGCwindow·(1−SFwindow)·dt
(5)
which, assuming the timestep being 1 h (t=1), being integrated, takes a form of:
SGBuilding =m
P
i=1
SG sur f ace
=m
P
1
WWR·Swall·SHGCwindow·(1−SFwindow)·Rn
1I(t,θ)·dt
=m
P
1
WWR·Swall·SHGCwindow·(1−SFwindow)·t·
n
P
1
I(θ1...n)
(6)
where SGsur f ace is the annual solar heat gains over a surface (kWh),
ITis total incident radiation on a tilted surface (kWh/m2), found with HDKR model,
nis hours of the year,
mis surfaces of the building,
Swall is the area of a wall (window included, if present) (m2), and
Swindow is the window area on a surface (m2).
The design variables:
SHGCwindow is the solar heat gain coefficient of the window,
SFwindow is the shade factor of the window, and
WWR is the window to wall ratio.
As is visible from previous equations, 3 out of 4 design variables were directly taken into
account—the window to wall ratio, SHGC and shade factor (SF). Now, the indirect information of
orientation needs to be taken into account, and this could be done through the solar angles, which
means that solar radiation incident and zenith angle need to be calculated.
2.12. The Indirect Calculation of the Third Design Variable: Orientation
The calculation of orientation was obtained through the calculations of incident solar radiation
and rotation of the building.
The angle of incidence,
θ
, is the angle between the solar beam and the normal to the surface.
Zenith angle,
θz
, is the angle between solar beam and the vertical [
28
]. Both angles are represented in
Figure 8.
In general, orientation information will be taken into account through the beam radiation, as it
will be expressed through the cosine ratio of incidence and zenith angle as follows:
Rb=cos(θ)
cos (θz)(7)
Energies 2020,13, 6316 10 of 17
Energies2020,13,xFORPEERREVIEW10of17
Figure8.Incident(θ),zenith(θ_z),tilt(β)andazimuth(γ)angles.
Ingeneral,orientationinformationwillbetakenintoaccountthroughthebeamradiation,asit
willbeexpressedthroughthecosineratioofincidenceandzenithangleasfollows:
𝑅𝑐𝑜𝑠𝜃
𝑐𝑜𝑠 𝜃(7)
Therefore,thecosinesofbothanglesneedtobefound.Asincidentandzenithanglesarea
complexsystemonmanyothersolarangles,anglesofdeclination,latitude,tilt(orslopeofthe
surface),azimuth,andsolarhourneedtobecalculated.Latitude,tilt,andazimuthwillbeconstants,
ifwedon’tchangethelocation,slope,ororientationofthewindow,whichleavesonlydeclination
andsolarhourasunknowns.Declination(𝛿)istheangularlocationofthesunnorthofsouthofthe
celestialequator,whichcanvary23.45° 𝛿 23.45°,dependingonlyonthedayoftheyear(n),
asexpressedbelow[28]:
𝛿 23.45 𝑠𝑖𝑛 284 𝑛360
365 (8)
Forcomputationalmethods,thereisyetanothermethodavailable,whichrequirescalculationof
twointermediatevariablesBandEinordertocalculatesolartimefromstandardtime.
𝐵 𝑛1360
365 (9)
wherenisthenthdayoftheyearandBisexpressedinradians.Eistheequationoftimeinminutes,
expressedasfollows:
𝐸229.20.0000750.001868𝑐𝑜𝑠𝐵0.032077𝑠𝑖𝑛𝐵
0.014615𝑐𝑜𝑠2𝐵0.04089𝑠𝑖𝑛2𝐵(10)
whichresultsinthefinalrelationshipbetween𝑆𝑜𝑙𝑎𝑟 𝑡𝑖𝑚𝑒and𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑡𝑖𝑚𝑒asfollows:
𝑆𝑜𝑙𝑎𝑟 𝑡𝑖𝑚𝑒 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑡𝑖𝑚𝑒 4 𝜆 𝜆 𝐸(11)
where𝜆isthestandardmedianofthetimezoneand𝜆isthelongitude
Thoseanalysisresultsinamoreprecisedeclinationcalculation,whichwasalsousedforthe
currentwork,andisexpressedasfollows:
𝛿 0.006918 0.399912 𝑐𝑜𝑠𝐵 0.070257 𝑠𝑖𝑛𝐵 0.006758 𝑐𝑜𝑠 2𝐵
0.000907 𝑠𝑖𝑛 2𝐵 0.002697 𝑐𝑜𝑠 3𝐵 0.00148 𝑠𝑖𝑛3𝐵 (12)
Figure 8. Incident (θ), zenith (θ_z), tilt (β) and azimuth (γ) angles.
Therefore, the cosines of both angles need to be found. As incident and zenith angles are a
complex system on many other solar angles, angles of declination, latitude, tilt (or slope of the surface),
azimuth, and solar hour need to be calculated. Latitude, tilt, and azimuth will be constants, if we don’t
change the location, slope, or orientation of the window, which leaves only declination and solar hour
as unknowns. Declination (
δ
) is the angular location of the sun north of south of the celestial equator,
which can vary
−
23.45
◦≤δ≤
23.45
◦
, depending only on the day of the year (n), as expressed
below [28]:
δ=23.45·sin (284 +n)·360
365 (8)
For computational methods, there is yet another method available, which requires calculation of
two intermediate variables Band Ein order to calculate solar time from standard time.
B=(n−1)·360
365 (9)
where nis the nth day of the year and Bis expressed in radians. Eis the equation of time in minutes,
expressed as follows:
E=229.2·(0.000075 +0.001868·cos(B)−0.032077·sin(B))−0.014615·cos(2B)−0.04089·sin(2B))(10)
which results in the final relationship between Solar time and Standard time as follows:
Solar time =Standard time +4·(λst −λloc )+E(11)
where λst is the standard median of the time zone and λloc is the longitude.
Those analysis results in a more precise declination calculation, which was also used for the
current work, and is expressed as follows:
δ=0.006918 −0.399912·cos(B)+0.070257·sin(B)−0.006758·cos (2B)
+0.000907·sin (2B)−0.002697·cos (3B)+0.00148·sin(3B)(12)
For next, the solar hour angle could be calculated, considering hour angle as angular displacement
of sun east or west of the local meridian due to the rotation of the Earth on its axis at 15 degrees per
hour, on mornings negative, on the afternoons positive, and was found as follows: [28]
ω=(min −770)·0.25 (13a)
Energies 2020,13, 6316 11 of 17
where min is the time in minutes in a given day.
When all the solar angles defined, finally the incident and zenith angle could be calculated and
equations are provided below respectively [28]:
cos (θ)=sin (δ)·sin (φ)·cos (β)−sin (δ)·cos (φ)·sin (β)·cos (γ)
+cos (δ)·cos (φ)·cos (β)·cos (ω)+cos (δ)·sin (β)
·sin (γ)·sin (ω)+cos (δ)·sin (φ)·sin (β)·cos (γ)·cos (ω)
(13b)
cos (θz)=cos (φ)·cos (δ)·cos (ω)+sin (φ)·sin (δ)(14)
Found cosine values are the input data for beam radiation calculation, which will be taken
into account in the HDKR incident radiation calculation and therefore will be reflected also in the
developed model.
2.13. Calculation of the Total Incident Solar Radiation
In order to calculate solar heat gains, calculation for tilted surfaces had to be found. The incident
solar radiation is a combination of beam radiation, diffuse radiation, which has three components,
and radiation that has been reflected from the surfaces, that reflect the radiation to the observed surface.
Total incident radiation on a tilted surface can be expressed as follows [28]:
IT=IT,b+IT,d,iso +IT,d,cs +IT,d,hz +IT,re f (15)
where IT,bis the total beam radiation, W/m2
IT,d,iso is the total diffuse isotropic radiation, W/m2
IT,d,cs is the total diffuse circumsolar radiation, W/m2
IT,d,hz is the total diffuse horizon radiation, W/m2
IT,re f is the total reflected radiation stream, W/m2
Isotropic radiation is this part of diffuse radiation, that is received uniformly from the entire sky
dome. Circumsolar radiation, which is resulting from forward scattering of solar radiation and is being
concentrated in the part of the sky around the sun. Horizon brightening radiation is concentrated near
the horizon, being the most pronounced in clear skies. Those different radiation parts are expressed in
Figure 9[28].
Energies2020,13,xFORPEERREVIEW11of17
Fornext,thesolarhouranglecouldbecalculated,consideringhourangleasangular
displacementofsuneastorwestofthelocalmeridianduetotherotationoftheEarthonitsaxisat15
degreesperhour,onmorningsnegative,ontheafternoonspositive,andwasfoundasfollows:[28]
ω min 770 0.25(13a)
whereministhetimeinminutesinagivenday
Whenallthesolaranglesdefined,finallytheincidentandzenithanglecouldbecalculatedand
equationsareprovidedbelowrespectively[28]:
𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝛿 𝑠𝑖𝑛 𝜙 𝑐𝑜𝑠 𝛽 𝑠𝑖𝑛 𝛿 𝑐𝑜𝑠 𝜙 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 𝛾
𝑐𝑜𝑠 𝛿 𝑐𝑜𝑠 𝜙 𝑐𝑜𝑠 𝛽 𝑐𝑜𝑠 𝜔 𝑐𝑜𝑠 𝛿 𝑠𝑖𝑛 𝛽
𝑠𝑖𝑛 𝛾 𝑠𝑖𝑛 𝜔 𝑐𝑜𝑠 𝛿 𝑠𝑖𝑛 𝜙 𝑠𝑖𝑛 𝛽 𝑐𝑜𝑠 𝛾 𝑐𝑜𝑠 𝜔
(13b)
𝑐𝑜𝑠 𝜃𝑧 𝑐𝑜𝑠 𝜙 𝑐𝑜𝑠 𝛿 𝑐𝑜𝑠 𝜔 𝑠𝑖𝑛 𝜙 𝑠𝑖𝑛 𝛿(14)
Foundcosinevaluesaretheinputdataforbeamradiationcalculation,whichwillbetakeninto
accountintheHDKRincidentradiationcalculationandthereforewillbereflectedalsointhe
developedmodel.
2.13.CalculationoftheTotalIncidentSolarRadiation
Inordertocalculatesolarheatgains,calculationfortiltedsurfaceshadtobefound.Theincident
solarradiationisacombinationofbeamradiation,diffuseradiation,whichhasthreecomponents,
andradiationthathasbeenreflectedfromthesurfaces,thatreflecttheradiationtotheobserved
surface.Totalincidentradiationonatiltedsurfacecanbeexpressedasfollows[28]:
𝐼 𝐼, 𝐼,, 𝐼,, 𝐼,, 𝐼,(15)
where 𝐼,isthetotalbeamradiation,W/m2
𝐼,,isthe