Calculation of the distribution of incoming solar radiation in enclosures

Applied Thermal Engineering 29 (2009) 1096–1105

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Applied Thermal Engineering

journal homepage: www.elsevier .com/locate /apthermeng

Calculation of the distribution of incoming solar radiation in enclosures

K. Chatziangelidis, D. Bouris *

Department of Engineering and Management of Energy Resources, University of Western Macedonia, Bakola and Sialvera, 50100 Kozani, Greece

a r t i c l e i n f o

Article history:Received 19 October 2007Accepted 28 May 2008Available online 5 June 2008

Keywords:Solar radiationTRNSYSView factorEnclosureThermal simulationThermal comfort

1359-4311/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.applthermaleng.2008.05.026

* Corresponding author. Tel.: +30 24610 56675; faxE-mail address: [email protected] (D. Bouris).

a b s t r a c t

Solar heat gains are an important factor in the calculation of cooling loads for buildings. This paper aimsat introducing an improved methodology to calculate the distribution of incoming solar energy on theinternal surfaces of closed spaces with multiple openings. The independent numerical methodology isbased on the view factor theory and in order to justify and prove its functionality, it has been linkedto the commercial software of TRNSYS, which normally uses a surface area ratio based algorithm forthe same process. For the simplified building structures that have been examined, there are noticeabledifferences in the spatial and temporal distribution of the absorbed solar energy. The proposed approachis indeed an improvement over the surface area ratio method, having a strong physical basis with rela-tively little extra computational effort.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The need for accurate simulation models, concerning the distri-bution of solar energy entering domestic buildings has become themain subject of many research efforts in the last few decades. Im-proved accuracy of the distribution algorithm will lead to moreaccurate prediction of the energy requirements of the simulatedbuilding and therefore valid conclusions regarding energy effi-ciency and indoor thermal comfort conditions [1].

Calculation of the distribution of incoming solar energy in en-closed spaces can be accomplished through a number of differentapproaches with increasing levels of complexity, computational ef-fort and accuracy. Wall [2] has presented an interesting study com-paring four such approaches for solar radiation distribution in aroom and concluded that a geometrical description of the enclosedspace is important and transmission through windows, reflectionand absorption must be accurately taken into account. Perhapsthe simplest approach is that of an area weighted distributionwhereby only the area of each surface (i.e. walls) is used in the dis-tribution algorithm. This is the approach currently applied by thecommercial software TRNSYS [3], with a surface absorptance factoralso being taken into account but no other geometrical relationsbetween the enclosure surfaces, e.g. view factors. The more accu-rate approach is that of the exact calculation of ‘sun patches’ thatare formed as direct solar radiation passes through windows. How-ever, this requires detailed geometrical information with regard tointernal surfaces, the borders of the enclosure’s openings and the

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time varying position of the sun. Athienitis and Stylianou [4] andCucumo et al. [5], presented analysis for estimating the solarabsorptance of a room, based on the radiosity-irradiation method(RIM) algorithm that was developed by Sparrow and Cess [6].The above mentioned algorithm (RIM) uses the view factor theoryand leads to an N * N system of equations, where N refers to thenumber of elements that the larger wall surface is divided into. La-ter on, Wen and Smith [7] developed a model which describes thedynamic thermal behavior of a building, considering its inner spaceto be surrounded by a number of elemental areas including interiorand exterior windows. The radiosity-irradiation method (RIM) wasalso used in this case in order to compute the illumination (irradi-ation) of each area. Both Wen and Smith [7] and Cucumo et al. [5]also calculated the redistribution of solar energy inside a building’srooms and the room’s effective solar absorptance, a concept whichwas initially introduced by Duffie and Beckman [8]. Trombe et al.[9] proceeded to implement a similar procedure for calculating‘sun patch’ location in a complex enclosure including an occupant.The procedure was implemented in a zone thermal simulationmodel within the TRNSYS simulation program basically focusingon the thermal comfort of the occupant. The importance of thehighest achievable accuracy in solar radiation distribution usuallybecomes evident inside highly-glazed spaces, i.e. greenhouses,sunspaces, etc. Mottard and Fissore [10], showed that the view fac-tor weighted approach is not sufficient for highly-glazed spaces,from which a large portion of the incoming solar radiation finallyescapes. In these configurations, insolation and shading becomeincreasingly important as shown by Pieters and Deltour [11],who used a semi one-dimensional climate model to investigatethe relative importance of the constructional parameters that

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Nomenclature

Latin symbolsAs area of surface s, m2

a solar altitude angle, degFi!j view factor for surfaces i and jfd;s;s fraction of diffuse or reflected solar radiation leaving

any surface s and absorbed by any other surface sGðx; y; n,gÞ view factor parameter in Eqs. (2) and (3)GSi fractions of the total incoming solar radiation absorbed

by surface iQ solar radiation, W/m2

Q th thermal load, kW hU thermal transmittance, W/m2 Kxi coordinate i on x axisyi coordinate i on y axisz distance between two parallel rectangular surfaces

Greek symbolsas solar absorptance of surface sc solar azimuth angle, deg

gi alternative Cartesian coordinate used in Fig. 1ni alternative Cartesian coordinate used in Fig. 1q solar reflectances solar transmittance

Subscriptsarea result obtained by using the absorptance-weighted area

ratio method for the distribution of total incoming di-rect solar radiation

d diffuse solar radiationdir direct solar radiations surfacesum summer day (June 2nd)tot totalv.f. result obtained by using the view factor method for the

distribution of total incoming direct solar radiationwi winter day (January 2nd)

K. Chatziangelidis, D. Bouris / Applied Thermal Engineering 29 (2009) 1096–1105 1097

influence the solar energy collecting efficiency of greenhouses un-der Western European conditions. Increased computational effortis needed in the approach of Hiller et al. [12], who developed analgorithm for shading and insolation calculations focusing mainlyon surface shapes, interactions and shading, but including the ef-fects of internal non-opaque surfaces.

The improved accuracy of the previous methodologies comes atthe cost of computational complexity and effort. It is interesting tonote that another commercial software targeting building thermalsimulation [13] includes both a simpler form of view factorweighted methodology and a more complex beam tracking oneas options.

The purpose of this paper is to present a simple and compu-tationally efficient methodology that distributes the total incom-ing solar radiation in enclosures of parallelepiped shape, takinginto account enclosure geometry, view factor theory and the po-sition of the sun throughout the day. The total incoming solarradiation from multiple openings is distributed among the enclo-sure surfaces with the use of simple distribution factors, withoutthe need to separately trace each opening’s beam radiation inci-dence on other surfaces. Analytical expressions are used for theview factors in parallelepiped geometry, this being the most

x2x

x1

y2y

y1

1

2

1

2

x

y

1

2

a

ξ

ξ

ξ

ξ

η η η A

A

Fig. 1. Schematic diagram of perpendicular (

common representative geometry in the majority of buildings:i.e. a typical building consists of surfaces that are either perpen-dicular or parallel. In order for the algorithm to be tested andverified, it was linked to the commercial simulation softwareTRNSYS and the results were compared to the absorptance-weighted area ratio distribution algorithm [3] that the softwarealready uses. As previously mentioned, more accurate distribu-tion algorithms including multiple and/or specular reflectionsmay be applied but the motivation for the present methodologyis (a) to provide improved accuracy and physical basis comparedto the absorptance-weighted area ratio method, (b) to includegeometrical characteristics of the enclosure walls and openingssuch as area, relative position and distance, (c) to account forthe incident solar radiation on each of the multiple openings,as a function of geographical location of the building and theopening’s orientation relative to the diurnally varying positionof the sun and (d) to retain simplicity in form, implementationand computational effort.

In the next section, the proposed numerical methodology is de-scribed, followed by information concerning the test case buildingconfiguration and the whole simulation process. Results andcomparison of the two approaches are presented for two different

x1

1

x2x

y

1

y2

2

y1

2

z

x

y

2

1

b

ξξ ξ

η η ηη

A

A

a) and parallel (b) rectangular surfaces.

1098 K. Chatziangelidis, D. Bouris / Applied Thermal Engineering 29 (2009) 1096–1105

simplified building models and finally, findings and conclusionsregarding implementation of the proposed algorithm are provided.

2. Numerical methodology

2.1. Overview

The purpose of this section is to describe the mathematicalexpressions that form the basis of the distribution algorithm andyield the solar energy balances through the use of time varying dis-tribution functions. The distribution functions represent the frac-tion of the total incoming direct solar radiation that reaches eachof the enclosure’s surfaces. As a result, the sum of all values ofthese functions is not allowed to exceed unity (1) within a zoneat any moment in time. The algorithm applies to any enclosure(zone) that consists of six opaque surfaces, having from one to fiveexternal openings (windows or doors), one on each surface. If morethan one opening is present on a given surface, then they are con-sidered as a single opening of area equal to the total area of theopenings. The methodology will be applied as an extension tothe commercial thermal simulation software TRNSYS [3], whichwill be responsible for calculating incident solar radiation on exter-nal surfaces, distribution of the diffuse and reflected compo-nents and performing the thermal balances on the walls of theenclosure.

Solar energy is commonly considered through its direct and dif-fuse components. One of the major simplifications assumed here isthat, after passing through an opening, direct solar radiation losesmost of its directional character and is emitted diffusely towardsall other surfaces of the enclosure. The assumption is that of a uni-form diffuse transmitter, the same as that considered in TRNSYS[3], and has been empirically found acceptable for the commoncases of shaded, diffusing or multi-layered glazings [14]. Anotherpopular thermal simulation software [13] also assumes all incidentdirect solar radiation to be transmitted as diffuse, if the glass is dif-fusing or a window shade is in place. For clear, unshaded glass,there is a portion of the direct solar radiation component that willcreate a ‘solar patch’ on the opposing internal surface but thiswould necessitate more complex geometrical calculations involv-ing the exact position of the sun and the openings’ borders (seefor example [9]). If each wall is considered as a single surface (asis the case here) then even if solar patches were calculated, the ab-sorbed heat flux would be distributed over the whole surface thusdamping the details of the sun patch position. This is actually thepractice in the advanced option in EnergyPlus [13] where sunpatches are calculated.

Data regarding the errors induced by the assumption that solarradiation is diffusely transmitted by complex glazing is difficult tofind. An indication of the effects can be derived from the work pre-sented by Wall [2] for an enclosure with an attached sunspace,where four different distribution algorithms were used. The differ-ences in the distribution algorithms involved both the diffuse and/or specular transmittance and the view factor or area weightedassumptions. For winter calculations, the differences among themodels in absorbed, transmitted and escaping radiation weresmall, i.e. a maximum of 15% but for summer calculations, theyreached 50%. For the enclosures being studied here, where theglazed areas are a small fraction of the enclosure, the errors in-duced by the assumption of diffuse behavior are expected to bemuch smaller since the percentage of solar radiation re-escapingthrough the glazing will be negligible.

Furthermore, it should be kept in mind that, although isotropicdiffuse solar radiation enters equally distributed through all of theenclosure’s external openings, the direct component will strikeeach opening as prescribed by the diurnally varying position of

the sun. This information is retained in the present calculation pro-cedure and it is combined with the relative geometrical position ofeach opening and the remaining internal surfaces. Therefore, thedirect solar radiation component that is transmitted through eachopening still retains a significant degree of directional character.This is the major improvement over the absorptance-weightedarea ratio method, against which comparison will be performed.

The methodology is applied only for the direct component; thediffuse component is distributed according to absorptance-weighted area ratios and this is also the method used for distribut-ing all radiation after the first reflection from a solid wall sincemost building materials can be treated as a Lambert surface: i.e.a perfect uniform and diffuse emitter, absorber and reflector ofradiant energy. This assumption is not far from reality since fornon-metallic surfaces 75% of the radiant energy is uniformly emit-ted, i.e. perfectly diffuse, within the cone angle <60� and 97% of theradiant energy is emitted within the cone angle <80� [15]. Verse-ghy and Munro [16] experimentally determined that neglectingspecular reflections in shortwave radiation leads to errors in inci-dent radiation on enclosure surfaces of less than 5–10 W/m2 (amaximum of 1% for solar radiation values up to 1000 W/m2), ex-cept in enclosures such as greenhouses or atria, which have glazingon over 20% of the enclosure area.

The solar radiation attributed to each internal surface is intro-duced into the thermal energy balance for the internal walls bythe TRNSYS software, and this is where the direct and diffuse com-ponents are finally combined.

2.2. View factor calculation

In general, the view factor between two objects can be de-scribed as the fraction of the total radiation that leaves the first ob-ject and strikes the second. The approaches for the calculation ofview factors range between complex numerical methods [9] andsimple solutions that refer to specific geometries. In the presentpaper the analytical expressions for view factors between parallelor perpendicular rectangular surfaces, given by Howell and Siegel[17], were used

F1!2 ¼1

ðx2 � x1Þðy2 � y1ÞX2

l¼1

X2

k¼1

X2

j¼1

X2

i¼1

½ð�1ÞðiþjþkþlÞGðxi; yj;gk; nlÞ�

ð1Þ

where

Gðx; y; n;gÞ ¼ 1=2pðy� gÞðx2 þ n2Þ1=2 tan�1 ðy�gÞ

ðx2þn2Þ1=2

h i

� 14 ½x2 þ n2 � ðy� gÞ2� ln½x2 þ n2 þ ðy� gÞ2�

8<:

9=;ð2Þ

for perpendicular surfaces and

Gðx; y; n;gÞ ¼ 1=2p

ðy� gÞ½ðx� nÞ2 þ z2�1=2 tan�1 y�g½ðx�nÞ2þz2 �1=2

n o

þðx� nÞ½ðy� gÞ2 þ z2�1=2 tan�1 x�n½ðy�gÞ2þz2 �1=2

n o

� z2

2 ln½ðx� nÞ2 þ ðy� gÞ2 þ z2�

0BBB@

1CCCA

ð3Þ

for parallel surfaces. Eq. (1) consists of 16 terms, which are func-tions of the x, y coordinates of the centers and corners of two rect-angular, perpendicular (2) or parallel (3) surfaces respectively. Thenotation in (1)–(3) is given in Fig. 1a and b. There are no limitationsconcerning the dimensions and the distance between the two sur-faces (A1, A2Þ, as long as the planes that contain them form a 90� an-gle in the first case and 0� angle in the second.

Weather DataFile

(Athens, Helsinki, Teheran)

Building DescriptionFile

(Geometry)

1st Calculation of:Radiation, Loads,

Temperatures(Area Ratio Distribution)

Weather DataFile

(Athens, Helsinki, Teheran)

Building DescriptionFile

(Geometry + GS)

2nd Calculation of:Radiation, Loads,

Temperatures(View Factor Distribution)

TRNSYS

TRNSYS

FORTRAN

Calculationof GS parameters

Fig. 2. Flow diagram of the whole simulation process.


2.3. Absorptance-weighted area ratios calculation

Although the approach being presented is general in nature,since it will be implemented using TRNSYS, a short description ofthe method against which it will be compared is appropriate.According to the TRNSYS manual [3], the incoming diffuse solarradiation and reflected direct solar radiation is distributed withinan enclosure with the use of absorptance-weighted area ratios. Dif-fuse or reflected direct solar radiation leaving any surface is ab-sorbed by any other surface (s) according to the fraction:

fd;s;s ¼asAs

Psurfacesð1� qd;sÞAs

ð4Þ

where as is the solar absorptance of the surface (defined in thebuilding description), As is the surface area and qd;s stands for thereflectance for diffuse solar radiation of the surface. For opaque sur-faces with no transmittance (ss= 0)

qd;s ¼ ð1� asÞ ð5Þ

For windows, the transmission losses are considered by

ss ¼ 1� as � qd;s ð6Þ

For direct solar radiation passing through an external opening, TRN-SYS calculates its distribution on the remaining internal surfaces inthe same way as for the diffuse component.

From (4)–(6), it is obvious that the only factors that can affectthe absorptance-weighted area ratios within a zone are surfacematerial properties and surface area. An opening’s relative positionwith regard to other internal surfaces and its orientation relative tothe diurnally varying position of the sun are neglected. As a result,the solar radiation distribution functions of each surface are alwaysconstant in time.

2.4. GS distribution parameters calculation

The proposed methodology modifies the above mentioned proce-dure for incoming solar radiation by altering the distribution methodfor the direct component. As commonly assumed [7,9], the reflectedcomponents are considered to be diffuse due to the diffuse behaviourof building materials and so, along with their diffuse counterparts,they are still distributed based on the absorptance-weighted area ra-tios. For clarity, the expressions will be presented for a parallelepipedwith two windows, each on a different surface. Extension to the situ-ation with windows on all surfaces is straightforward.

The total amount of incoming direct solar radiation, throughwindow 1 (Q dir;1Þ, is distributed among the remaining five internalsurfaces according to view factor theory (Fi!j is the view factorfrom surface i to surface j), as follows:

Q dir;1 ¼ F1!2Q dir;1 þ F1!3Q dir;1 þ F1!4Qdir;1 þ F1!5Qdir;1

þ F1!6Q dir;1 ð7Þ

with each term on the right representing the fraction of solar radi-ation that enters through window 1 and arrives at the correspond-ing other surface. The energy balance for the direct solar radiationcoming through the second window (Qdir;2Þ is

Q dir;2 ¼ F2!1Q dir;2 þ F2!3Q dir;2 þ F2!4Qdir;2 þ F2!5Qdir;2

þ F2!6Q dir;2 ð8Þ

If the total direct radiation entering the enclosure is defined as

Q dir;tot ¼X6

j¼1

GSjðQdir;1 þ Q dir;2Þ ð9Þ

where GSj is the distribution parameter for surface j (corresponds tothe GEOSURF parameter in TRNSYS) then

Qdir;tot ¼ Q dir;1 þ Q dir;2 )X

i¼2;...;6

F1!iQ dir;1 þX

i¼1;3;...;6

F2!iQ dir;2

¼X6

j¼1

GSjðQ dir;1 þ Qdir;2Þ ð10Þ

and each term (GSjðQdir;1 þ Qdir;2ÞÞ in the last sum of (10) is the en-ergy arriving at surface j. From the left hand side of (10), the energyarriving at surface j is (F1!jQ dir;1 þ F2!jQdir;2Þ and the calculation ofthe distribution parameter GSj of surface j, for example, is given by

GSj ¼F1!jQ dir;1 þ F2!jQ dir;2

Q dir;1 þ Q dir;2ð11Þ

As a result, (11) gives a factor that’s responsible for the distributionof the total incoming direct solar radiation, as a function of relativeposition of each opening (through the view factors). Furthermore,opening orientation is taken into account since Qdir;1 and Qdir;2

could, for example, be the incoming radiation from a southernand western window respectively and thus change depending onsolar time and latitude. The solar radiation hitting the surface ofan internal wall (j) is therefore simply

Qdir;j ¼ GSj � Qdir;tot ð12Þ

where (Qdir;totÞ is the sum of direct solar radiation entering from allopenings.

3. Simulation

Before describing the models and conditions that were used forthe simulation, it is essential to give an outline of the whole pro-cess. The flow chart presented in Fig. 2 shows the steps that weretaken in order for the view factor based distribution of direct solarradiation to be compared with the area ratio distribution method


that TRNSYS uses. During the first simulation, the weather data ofthree different cities, in combination with the geometrical and con-structional details of a single-zone and a dual-zone model wereused with the TRNSYS software. In this case, the absorptance-weighted area ratios method is applied in order to distribute theincoming solar radiation to all the internal surfaces of the zones.The data taken from each simulation step were the total solar radi-ation absorbed by each surface in (kJ/h), the incident solar radia-tion on each opening in (kJ/h), the temperature of each internalsurface in (�C) and the thermal loads arising for each zone.

N

Fig. 3. Schematic diagram of dual-zo

Zone

Zone

13.00

0.4

m

9.00

1.00 m

2.50

m

5.00

m

1.00 m

1.25

m

2.00 m

1.00

m

1.3

m

Nor

Sout

West

Nor

Sou

West

Fig. 4. Top view of dual-zone bu

A calculation algorithm, developed in FORTRAN computer lan-guage, used the external distribution of the solar radiation on theopenings (calculated from an initial run), a geometrical descriptionof each zone, including its windows and the methodology describedin Section 2.4, in order to produce a file containing the GS parame-ters for each simulation time step. The above mentioned parame-ters were inserted, via a data reader, to the TRNSYS software asdistribution parameters and a second simulation was then per-formed. After obtaining the same type of results from the secondsimulation, comparisons were made and conclusions were drawn.

ne building (simulation model 2).

1

2

m

3.00

m

10.0

0 m

0.4 m

1.00 m

2.00

m

m

1.00

m

1.50

m

1.00 m

th

h

East

th

th

East

ilding (simulation model 2).


The procedure was applied to a single-zone and a dual-zone build-ing, using three different weather data files. It should be noted thatthe second simulation is only necessary if the present algorithm isto be maintained independent. It could be written as an integralpart of the TRNSYS software in order to calculate the distributionparameters during each time step of the initial simulation.

Comparisons with other methodologies would require more de-tailed alterations to the basic thermal simulation code and escapethe scope of the present effort. Thus the standard method used inTRNSYS is considered adequate for demonstration of the differ-ences in the calculated results when using the present methodol-ogy. Furthermore, this can be considered as a comparison againstcurrent standard practice since, as already mentioned, the standardmethodology used in TRNSYS is very close to the basic option inEnergyPlus [13].

3.1. Description of building models

For the whole simulation process to be completed, two geomet-rical models were used (Figs. 3 and 4). The first simulation modelrefers to a single-zone building of parallelepiped shape, with its

Single zone building (A

26.0%

27.0%

28.0%

29.0%

9.0%

10.0%

11.0%

12.0%

13.0%

14.0%

15.0%

16.0%

17.0%

18.0%

33 34 35 36 37 38 39 40 41

3654

3655Simulation Ti

Per

cen

tag

e o

f in

com

ing

so

lar

rad

iati

on

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

33 34 35 36 37 38 39 40 4136

5436

5536

Simulation

Floor

North

South

East

West

WINTER

Fig. 5. Absorbed solar energy percentages o

front wall facing the south and having two external windows cen-trally placed on its south and east walls. The building is 3 m high,5 m wide and 10 m long. As far as its openings are concerned, thedimensions of the south window are 4.00 m � 1.50 m (height) andof the east window 2.00 m � 1.50 m (height). For economy ofspace, only the second, dual-zone building model is shown in Figs.3 and 4; the first, single zone building is essentially the southernzone (zone 2) without the western window.

The second model is a dual-zone building, with openings oneach of its external walls, excluding the ceiling. Fig. 3 shows a threedimensional drawing of the model and Fig. 4 presents a top viewdiagram with all the dimensions needed. Both the first and secondmodel are orientated on a north–south axis and their openings arecentrally placed on each zone’s walls. In the first simulation, thesingle-zone model was used and the dual-zone model was usedin the second and third simulation. The difference between the lasttwo simulations is that the internal wall of the building initiallyhas a U-value of 3.14 W/m2 K, consisting of two layers of plaster(thickness: 0.025 m, thermal capacity: 1 kJ/kg K and density:2000 kg/m3) and one of bricks (thickness: 0.10 m, thermalcapacity: 1 kJ/kg K and density: 1800 kg/m3) and during the last

thens, Greece)

Qarea

Qv.f.

3656

3657

3658

3659

3660

3661

3662

3663

3664

3665

3666

3667

3668me (hrs)

5636

5736

5836

5936

6036

6136

6236

6336

6436

6536

6636

6736

68

Time (hrs)

Qarea

Qv.f.

SUMMER

f all internal walls (simulation model 1).


simulation the same wall is considered as a mass wall, consisting ofone 0.60 m layer of heavy reinforced concrete, having a U-value of2.26 W/m2 K, thermal capacity of 0.84 kJ/kg K and density equal to2400 kg/m3 This leads to a thermal capacity ratio of �5 betweenthe two walls.

3.2. Conditions and process

In both buildings, the windows consist of double glazing, with a10 mm air space between them, having a U-value of 2.83 W/m2 K.In addition, natural ventilation is considered equal to 2.2 ach/h. Ini-tial conditions inside both structures are considered to be 20 �Cand 50% relative humidity. The first building is almost insensitiveto outer conditions by applying highly insulated external wallswith a U-value of 0.045 W/m2 K. In order to study only the incom-ing solar radiation that’s distributed among the inside surfaces,there are no HVAC systems, which leads to a varying internal zonetemperature, and the internal gains from people, lighting, equip-ment etc. are considered to be zero. The dual-zone building is nor-mally insulated; with external walls having a U-value of 0.595 W/m2 K. The heating thermostat is set to 22 �C and the cooling ther-mostat to 26 �C and no internal gains, other than solar, are takeninto account.

In order to test the methodology in different climates, the sim-ulations where repeated for weather data concerning the cities ofAthens (Greece), Helsinki (Finland) and Teheran (Iran). In each casethe thermal loads of the building and the percentage of the totalincoming solar radiation, absorbed by each wall, were calculated.The results were initially obtained using the area ratio distributionmethod and then the view factor distribution method was applied.Because of the great number of simulation time steps (dt = 1 h,8760 h in one year), two representative winter and summer timeperiods for each year are presented for comparison. The first one

Dual zone building

24.0%

25.0%

26.0%

27.0%

28.0%

29.0%

30.0%WINTER

Floor

14.0%

15.0%

16.0%

17.0%

18.0%

Boundary (South

North

2.5%

3.5%

4.5%

5.5%

33 35 37 39 4136

5436

Simulatio

East

WestPer

cen

tag

e o

f in

com

ing

so

lar

rad

iati

on

Fig. 6. Absorbed solar energy percentages of all

from the 33rd until the 41st simulation time step and referringto the 2nd day of January, while the other lasting from the3654th until the 3668th simulation time step and referring tothe 2nd day of June. To be more specific, on January 2nd the sunrises at 7:41 a.m. and sets at 5:17 p.m. (local time), but due to TRN-SYS round off errors, the first and last time steps to show any en-ergy values are 8:00–9:00 a.m. and 4:00–5:00 p.m. These are thetime values shown for winter in Figs. 5–7. The solar azimuth (cÞand the maximum solar altitude (a) angles were analytically calcu-lated [8] for the city of Athens during both days of the simulations.For the 2nd of January the solar azimuth angle for sunrise/sunset(cwiÞ is 60.55� while the maximum solar altitude (awiÞ is 29.49�.For the 2nd of June csum ¼118.61� and asum ¼74.72�.

4. Results and findings

The results for the single-zone building are shown in Fig. 5. Atthis point it is essential to stress that Qarea refers to the distributionof incoming solar radiation using the area ratio method and Q v:f : re-fers to the current view factor based distribution method. The totalamount of incoming solar radiation is equal in both cases, becausethe same solar radiation calculation algorithm (the one providedby TRNSYS) is used. The only difference is the way that this amountis being distributed among the zone surfaces. During the whole daythe diffuse part of the total solar radiation that enters the zonethrough the openings is distributed among all internal surfaces.The direct component of solar radiation will be non-zero only ifthe sun’s position is such to permit incidence on the external sur-face of the opening. Then the GS factors will lead to a direct solarradiation component being calculated in addition to the diffusecomponent. This explains the time variation of the Qv:f :. distribu-tion results as opposed to the area distribution method which isbased solely on geometrical characteristics, invariable in time.

(Zone 1, Athens)

SUMMER

Qarea

Qv.f.)

5636

5836

6036

6236

6436

6636

68

n Time (hrs)

internal walls (simulation model 2, Zone 1).


The structure is the fully insulated parallelepiped and theweather data refer to the city of Athens, Greece. Apart from thefloor, the north wall has a constantly greater percentage of ab-sorbed solar radiation mainly because of its position across the

Dual zone building

21.0%

22.0%

23.0%

24.0%

25.0%

26.0%

27.0%

28.0%

29.0%

30.0%

33 34 35 36 37 38 39 40 4136

5436

5536

5

Simulatio

14.0%

15.0%

16.0%

17.0%

18.0%

19.0%

33 34 35 36 37 38 39 40 4136

5436

5536

5

Simulatio

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

33 34 35 36 37 38 39 40 4136

5436

5536

5

Simulatio

Per

cen

tag

e o

f in

com

ing

so

lar

rad

iati

on

South

Floor

Boundary (North)

East

West

WINTER

Fig. 7. Absorbed solar energy percentages of all

Table 1Annual thermal loads in kWh, for simulation model 2 with simple internal wall (+ heating

Athens Helsinki

Zone 1 Zone 2 Zone 1

Q th:area Q th:v:f : Q th:area Q th:v:f Q th:area Q th

January 2083 2080 1849 1842 4907 490February 1751 1748 1552 1546 4420 441March 1541 1538 1384 1380 4021 401April 776.9 774.2 650.7 647.1 2799 279May 161.2 159.7 83.5 80.9 1603 159June �247.6 �249.3 �382.5 �386 693.5 689July �642.9 �644.9 �845.6 �849.1 458.2 454August �578.1 �581.8 �847.9 �855.5 833.5 829September �126.1 �128.3 �337.6 �346 1809 180October 445.1 440.8 298.1 292.3 2787 278November 1190 1187 985,2 979,6 3642 364December 1800 1797 1615 1609 4533 453

south window and its greater surface area, compared to that ofthe south wall which includes a window. The Q area distributionyields a steady percentage of 9.8% for the south and 16.3% for thenorth walls during both summer and winter simulation periods.

(Zone 2, Athens)

636

5736

5836

5936

6036

6136

6236

6336

6436

6536

6636

6736

68

n Time (hrs)

636

5736

5836

5936

6036

6136

6236

6336

6436

6536

6636

6736

68

n Time (hrs)

636

5736

5836

5936

6036

6136

6236

6336

6436

6536

6636

6736

68

n Time (hrs)

Qarea

Qv.f.

SUMMER

internal walls (simulation model 2, Zone 2).

, � cooling)

Teheran


:v:f : Q th:area Q th:v:f: Q th:area Q th:v:f: Q th:area Q th:v:f:

6 5034 5032 3109 3105 2859 28498 4413 4408 2416 2412 2177 21697 3874 3865 1725 1721 1502 14946 2589 2583 616.2 613.1 498.7 4948 1393 1385 �40.5 �42.1 �116.1 �118.8.5 541.6 536.4 �725.3 �728.1 �871.5 �876.8.3 313.5 308.7 �1253 �1256 �1447 �1453

661.1 654.9 �1022 �1026 �1289 �12975 1689 1682 �319.9 �323.5 �605 �615.44 2712 2704 415.3 410.3 172.5 163.50 3735 3731 1611 1605 1281 12711 4675 4672 2737 2733 2470 2460


On the other hand, results that come from the Q v:f : distributionmethod are diurnally and annually variable. In the winter, thenorth wall and the floor and roof (not shown due to symmetry withthe floor) absorb more of the incoming solar radiation than thatcalculated by the area ratio method and this difference becomesgreater during the midday hours. The physical explanation is thatthe sun is low in the horizon during the winter (awi ¼29.49�) andwill reach maximum penetration into the zone during the middayhours when it is due south (the orientation of the major zone open-ing). Since the solar azimuth angle at sunrise and sunset iscwi ¼60.55�, there is minimal direct solar radiation incident onthe openings and so it is the diffuse component that determinesthe radiation distribution at these times of the day. This compo-nent is calculated in the same way (area weighted ratios) for bothmethodologies and therefore, early and late in the day, the samevalues are calculated for both methods. During midday, the east,west and south walls will absorb less solar radiation than the northwall, the floor and the roof, clearly due to the direct solar radiationcomponent that dominates incident radiation on the southwindow.

In the summer, the variations for both the north and the southwalls are smaller, 16.3–16.8% and 9.3–10% respectively. The highervalues at the beginning of the day are due to the direct radiationcoming from the east window while the absence of a west windowleads to the lower afternoon values. This is also evident from thehigher value of absorbed radiation on the southern wall duringthe first part of the day, when the sun is still low and in the east(csum ¼ �118.61�). According to the solar path during summer(asum ¼74.72�), the direct component on the southern window isexpected to be minimal at midday and this is reflected in the rela-tively small variation in the north wall’s absorbed radiation. Theradiation absorbed on the ceiling (not shown) is the same as thatof the floor due to the symmetric placement of the openings onthe walls and it is interesting to note that these two surfaces re-ceive the major part of the incoming radiation (28.8%), at middayin winter and during the morning in summer. Because of its largesurface area, the floor absorbs the greatest part of the diffuse solarradiation entering the zone (weighted area ratio method) and be-cause of the high view factor values (0.336 between south windowand floor and 0.314 between east window and floor), over 30% ofthe direct solar radiation that comes from the windows reachesthe floor (view factor distribution parameters).

Fig. 6 shows results for Zone 1 and Fig. 7 for Zone 2 of the dualzone building, again for Athens, Greece. In these figures, the southwall of Zone 1 and the north wall of Zone 2, refer to the two sides ofthe intermediate wall that separates the building into two zones,as shown in Figs. 3 and 4. The direct solar radiation entering both

Table 2Annual thermal loads in kWh, for simulation model 2 with internal mass wall (+ heating, �

Athens Helsinki


Q th:area Q th:v:f : Q th:area Q th:v:f Q th:area Q th

January 2101 2098 1848 2490 4902 490February 1758 1756 1542 1535 4424 442March 1553 1551 1377 1073 4029 402April 783.7 781.3 635.8 631.6 2808 280May 171 169.9 85.4 83.2 1620 161June �237.2 �238.5 �383.3 �386.6 702.7 699July �631.3 �633.4 �851.2 �855.8 466.8 463August �566.5 �569.1 �859.1 �866 841.6 837September �124.2 �125.7 �353.6 �361.5 1810 180October 437.5 433.7 271.27 265.33 2789 278November 1199 1197 971.8 965.5 3636 363December 1808 1806 1606 1599 4526 452

zones from their west windows, which did not exist in the single-zone model, is expected to affect the afternoon absorbed solar radi-ation, especially in the summer (csum ¼118.61�). In Fig. 6, the dis-tribution of solar radiation on the north and intermediate (south)walls of Zone 1, almost coincide during the winter, no matter whatdistribution method is used. This is due to the symmetry of the twowalls, the absence of a southern opening and the sun rising andsetting at an azimuth angle of cwi ¼ �60.55�, so the effect of thediffuse solar radiation component is dominant. The only exceptionis near the end of the day when there is some direct solar incidenceon the larger surface area of the west window. This cannot bereproduced in the weighted area ratio method since solar geome-try is not taken into account. During the summer, the effects aremore prominent because of the wider sun path (csum ¼ �118.61�)causing more direct incidence on the east and west windows.Again, it is the floor and ceiling that gather most of the incomingsolar radiation with the view factor distribution method predictinga 5% increase in the summer early morning and late evening, ascompared to the weighted area ratio method. Because of the smallview factor values between the east and west walls and theiropposing openings, the fraction of total solar radiation being ab-sorbed by them is reduced by as much as 20% when compared tothe absorptance-weighted area ratio distribution method. Duringwinter, the Q v:f : curves are almost of the same shape for these wallsbut during the summer, when there is more direct solar radiationreaching the east and west openings, there is a larger reductionfor the east wall in the early morning and for the west wall inthe late afternoon.

Fig. 7 refers to Zone 2 which is geometrically similar to simulationmodel 1 except that Zone 2 has a west facing opening. In Fig. 7, thegeneral distribution is the same as in Fig. 5 except that in the after-noon, the west facing opening directly affects the results, especiallyduring the summer, when the azimuth angle at sunset is further tothe west and the direct solar radiation component is incident onthe western and eastern walls for a longer time. The southern win-dow has the biggest surface area, among the other openings of thezone, thus resulting in the intermediate (north-boundary) wallabsorbing the second greatest percentage of solar radiation (17–18.5%), as shown in Fig. 7. Between the Q area and Q v:f : curves andreferring to the intermediate (north) wall, there is a noticeable var-iation due to the solar path, most prominent in the winter.

For further comparison, surface temperatures and thermal loadswere also calculated during simulation, when using both the arearatio and the view factor distribution methods. The maximum tem-perature variation between any two surface temperatures, for thewhole simulation process was 0.1 �C. As far as the thermal loadsare concerned, Tables 1 and 2 show the annual values in kWh for

cooling)

Teheran


:v:f : Q th:area Q th:v:f: Q th:area Q th:v:f: Q th:area Q th:v:f:

2 5038 5036 3126 3123 2854 28443 4415 4410 2429 2426 2170 21616 3868 3859 1736 1734 1491 14835 2579 2569 622.8 620.3 484.7 480.18 1388 1379 �27.5 �29.4 �113.2 �116.4.1 526.3 520.6 �716.9 �719.5 �877.3 �882.8.4 294.3 288.9 �1244 �1246 �1457 �1463.9 641.2 634.6 �1014 �1017 �1307 �13157 1673 1665 �312.2 �315 �622.5 �633.46 2706 2798 412.1 407.9 148.8 138.94 3735 3731 1620 1615 1255 12435 4678 4674 2749 2745 2459 2448


both zones of simulation model 2, under different climate condi-tions. A positive value means heating load and negative standsfor cooling loads. It is important to underline that two calculationshave been performed: for the first (Table 1), the internal wall has aU-value of 3.14 W/m2 K and consists of two layers of plaster andone of bricks while for the second case the same wall is consideredas a mass wall, made of one layer of heavy reinforced concrete,which gives a U-value of 2.26 W/m2 K

Beginning with load variations between both solar radiationdistribution methods, they are small and range from 1 to10 kW h throughout the year. As expected and because of its geo-graphic position, Helsinki exhibits the maximum annual heatingload compared to the other cities and has almost no cooling load.Comparison of Tables 1 and 2 shows that the construction of amass wall, as a boundary for the two zones, results in an increaseof the annual heating loads and a corresponding decrease of thecooling load for Zone 1, with the opposite happening for Zone 2.The above are a direct consequence of the mass wall’s ability tostore heat because of its large specific heat capacity 7.92 (kJ/h m K). The solar gains are greater in Zone 2, thus allowing themass wall’s south side to store more heat than the north sidethroughout the day. Similar observations regarding the importanceof thermal mass and direct solar radiation distribution were notedby Yohanis and Norton [1]. As the sun sets, the air temperature in-side the zone drops, so the stored heat is released. During winterthis process leads to the reduction of the heating load needed bythe zone but during the summer it results in an increase of thecooling load. When the wall’s mass is less (Table 1), Zone 1 alsobenefits from the solar gains of Zone 2 and the heating loads areslightly smaller.

5. Conclusions

The purpose of this study is the development of a method thatdistributes the total direct solar radiation entering through multi-ple openings of a building, among its internal surfaces, based onview factor theory. For its effectiveness to be tested, the methodwas compared with the standard absorptance-weighted area ratiodistribution method, used by the TRNSYS software. Comparisonwas performed based on inner surface solar energy absorptionand thermal loads for a single and dual-zone building as well asfor various climate conditions. Results showed that of all parame-ters, solar radiation absorption by internal surfaces is the one beingaffected the most by the choice of the solar radiation distributionmethod. The view factor based method introduces a distributionfunction dependent upon the building geometry and each open-ing’s orientation relative to the temporally varying sun position.

This leads to a time dependent distribution of the direct solar radi-ation component incident on each opening and the method can beapplied for multiple (1–5) openings. For the conventional buildingmodels that were used, the variation in the absorbed radiation byan internal wall ranged between 0% and 2% of the total solar radi-ation entering the zone. In some cases, this corresponds to a vari-ation of 20% of the total radiation being absorbed by the specificsurface. Although surface temperatures and thermal loads do notseem to be significantly affected by the use of the view factor baseddistribution method, its sound physical basis and the relativelysmall extra computational effort justify its use.

References

[1] Y.G. Yohanis, B. Norton, Useful solar heat gains in multi-zone non-domesticbuildings as a function of orientation and thermal time constant, RenewableEnergy 27 (2002) 87–95.

[2] M. Wall, Distribution of solar radiation in glazed spaces and adjacentbuildings. A comparison of simulation programs, Energy and Buildings 26(1997) 129–135.

[3] TRNSYS 16, A TRaNsient SYstem Simulation program. Volume 6: MultizoneBuilding modeling with Type 56 and TRNBuild, Solar Energy Laboratory,University of Wisconsin-Madison, TRANSSOLAR Energietechnik GmbH, CSTB– Centre Scientifique et Technique du Bâtiment, TESS – Thermal EnergySystems Specialists, 2005.

[4] A.K. Athienitis, M. Stylianou, Method and global relationship for estimation oftransmitted solar energy distribution in passive solar rooms, Energy Sources13 (1991) 319–336.

[5] M. Cucumo, D. Kaliakatsos, V. Marinelli, Estimating effective solarabsorptances in rooms, Energy and Buildings 23 (1995) 117–120.

[6] E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, Hemisphere, Washington,1978.

[7] J. Wen, T.F. Smith, Development and validation of a dynamic simulation modelfor thermal response of a building. Presented at 8th International Conferenceon Building Envelopes, 2001.

[8] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, second ed.,Wiley Publications, UK, 1991.

[9] A. Trombe, L. Serres, M. Moisson, Solar radiation modelling in a complexenclosure, Solar Energy 67 (1999) 297–307.

[10] J.M. Mottard, A. Fissore, Thermal simulation of an attached sunspace and itsexperimental validation, Solar Energy 81 (2007) 305–315.

[11] J.G. Pieters, J.M. Deltour, Modelling solar energy input in greenhouses, SolarEnergy 67 (1999) 119–130.

[12] M.D. Hiller, W.A. Beckman, J.W. Mitchell, TRNSHED – a program for shadingand insolation calculations, Building and Environment 35 (2000) 633–644.

[13] EnergyPlus, Engineering Reference, The Reference to EnergyPlus Calculations(2007), University of Illinois, Ernest Orlando Lawrence Berkeley National Lab.and U.S.D.O.E. <www.eere.energy.gov/buildings/energyplus>.

[14] ASHRAE, Fundamentals Handbook, ASHRAE, 2001.[15] A. Mbiock, R. Weber, Radiation in Enclosures, Springer-Verlag, 1999.[16] D.L. Verseghy, D.S. Munro, Sensitivity studies on the calculation of the

radiation balance of urban surfaces: I. Shortwave radiation, Boundary-LayerMeteorology 46 (1989) 343–365.

[17] J.R. Howell, R. Siegel, Thermal Radiation Heat Transfer, fourth ed., Taylor andFrancis, New York, 2001.

http://www.eere.energy.gov/buildings/energyplus

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Calculation of the distribution of incoming solar radiation in enclosures