Thermal models of buildings : determination of temperatures, heating and cooling loads : theories, models and computer programs

Transcription

1 Thermal models of buldngs : determnaton of temperatures, heatng and coolng loads : theores, models and computer programs Källblad, Kurt Publshed: Lnk to publcaton Ctaton for publshed verson (APA: Källblad, K. (1998. Thermal models of buldngs : determnaton of temperatures, heatng and coolng loads : theores, models and computer programs Department of Buldng Scence, Lund Insttute of Technology General rghts Copyrght and moral rghts for the publcatons made accessble n the publc portal are retaned by the authors and/or other copyrght owners and t s a condton of accessng publcatons that users recognse and abde by the legal requrements assocated wth these rghts. Users may download and prnt one copy of any publcaton from the publc portal for the purpose of prvate study or research. You may not further dstrbute the materal or use t for any proft-makng actvty or commercal gan You may freely dstrbute the URL dentfyng the publcaton n the publc portal Take down polcy If you beleve that ths document breaches copyrght please contact us provdng detals, and we wll remove access to the work mmedately and nvestgate your clam. L UNDUNI VERS I TY PO Box L und

2 THERMAL MODELS OF BUILDINGS Determnaton of Temperatures, Heatng and Coolng Loads. Theores, Models and Computer Programs. Kurt Källblad

3 Keywords Buldng Physcs Energy Thermal Models Buldngs Heatng Loads Ventlaton Clmate Indoor Clmate Walls Comfort Infltraton Wndows Coolng Loads Rooms Computer Programs Temperatures Copyrght Kurt Källblad and Department of Buldng Scence, Lund Unversty, Lund Insttute of Technology, Prnted by KFS AB, Lund 1998 Report TABK--98/1015 THERMAL MODELS OF BUILDINGS, Determnaton of Temperatures, Heatng and Coolng Loads. Theores and Computer Programs Lund Unversty, Lund Insttute of Technology, Department of Buldng Scence ISSN ISRN LUTADL/TABK SE Lund Insttute of Technology Department of Buldng Scence Telephone: P.O. Box 118 Fax: SE LUND E-mal: bkl@bkl.lth.se Sweden Homepage: 2

4 Abstracts The need to estmate ndoor temperatures, heatng or coolng load and energy requrements for buldngs arses n many stages of a buldngs lfe cycle, e.g. at the early layout stage, durng the desgn of a buldng and for energy retrofttng plannng. Other purposes are to meet the authortes requrements gven n buldng codes. All these stuatons requre good calculaton methods. The man purpose of ths report s to present the authors work wth problems related to thermal models and calculaton methods for determnaton of temperatures and heatng or coolng loads n buldngs. Thus the major part of the report deals wth treatment of solar radaton n glazng systems, shadng of solar and sky radaton and the computer program JULOTTA used to smulate the thermal behavor of rooms and buldngs. Other parts of thermal models of buldngs are more brefly dscussed and ncluded n order to gve an overvew of exstng problems and avalable solutons. A bref presentaton of how thermal models can be buld up s also gven and t s a hope that the report can be useful as an ntroducton to ths part of buldng physcs as well as durng development of calculaton methods and computer programs. The report may also serve as a help for the users of energy related programs. Independent of whch method or program a user select to work wth t s hs or her own responsblty to understand the lmts of the tool, else wrong conclusons may be drawn from the results. 3

5 Acknowledgment For the help wth the work presented here, I want to express my thanks to all people who have supported ths work, those not mentoned below really not forgotten. My supervsors, professor emertus Bo Adamson and professor Bertl Fredlund, head of the department, for ther great assstance and nterest n my work. Dr Bengt Eftrng who took part n the orgnal development of the JULOTTA program as well as Dr Johan Claesson and Dr Wolfgang Fest for ther valuable deas and nterestng dscussons durng the years. My research fellows at our department, those at the School of Cvl Engneerng and the School of Archtects n Lund and frends n dfferent nternatonal research projects, especally wthn the IEA have all gven many opportuntes for constructve dscussons. The Swedsh Councl of Buldng Research, wthout ther fnancal support some of the work had not been carred out. Fnally, I want to address a great thank to my wfe Juna and my chldren for ther support and understandng durng the years. Lund, Aprl 1998 Kurt Källblad 4

6 Contents Abstracts 3 Acknowledgment 4 Contents 5 Nomenclature 7 1 Preface Background Methodology Lmtatons Contrbuton and Relaton to Prevous Work Some Hstorcal Notes 11 2 Introducton Some Defntons Comfort Parameters Energy Flows n Buldngs Elements of Thermal Models Temperature Nodes Heat Flow between Nodes Thermal Models of Buldng Components Room ar Wndows LW radaton Exchange n a Room Walls and Slabs Thermal Model of a Room, an Example 27 3 Thermal Components Volumes Surfaces Walls and Slabs Wndows 34 4 Solar Radaton Solar Radaton at a Buldng 38 5

7 4.2 Co-ordnate Systems and Solar Angles The Global Co-ordnate System The Local Co-ordnate System Radaton at Opaque Surfaces wthout Shadng Incdent Beam Radaton Incdent Dffuse Radaton Absorbed and Reflected Radaton Radaton at Wndows wthout Shadng Propertes for a sngle pane Propertes for a multple pane wndow Shadng of Beam Radaton Shadng of Dffuse Radaton Opaque Surfaces Wndows Dstrbuton of Reflected Radaton 62 5 The JULOTTA Program Background Buldng Model Input Data Characterstcs of JULOTTA Applcatons of the JULOTTA Program Conclusons 68 6 Conclusons and Further Developments 69 References 70 Appendx 1 Psychometrcs 74 Appendx 2. Solar Poston Formulas 79 6

8 Nomenclature The most commonly used symbols are gven here, other symbols are defned n the chapters they are used. Roman symbols A Area (m 2 C Heat Capactance (J/K G Heat Conductance (W/K G Global rradance or solar flux densty (W/m 2 Gb Beam rradance (W/m 2 G d Dffuse sky rradance on horzontal (W/m 2 L R Radance or radant ntensty (W/m 2,sr Thermal Resstance (K/W T Temperature (K V Volume (m 3 Q c cp d e heot hstt htst Heat flow (W Specfc Heat (J/kg,K Specfc Heat for Ar at Constant Pressure (J/kg,K Thckness (m East-co-ordnate n a local system Equaton of Tme (h Standard Tme for a tme zone (h True Solar Tme for a ste (h q Heat flow per unt area (W/m 2 s South co-ordnate n a local system t Tme (s or (h 7

9 Greek symbols Alttude Tlt angle of a surface Azmuth, Orentaton Declnaton Emssvty Sphercal co-ordnate, Longtude, Tme Medan Angle Lattude, Thermal Conductvty (W/m,K Sphercal co-ordnate Hour angle ρ Densty (kg/m 3 Subscrpts aux c l m o r s sun v Auxlary Convectve Inner, Incdent Internal load Mass Outer Room Ste Sun Ventlaton 8

10 1 Preface The man purpose of ths report s to present the authors work wth problems related to thermal models and calculaton methods for determnaton of temperatures and heatng or coolng loads n buldngs. Thus the major part of the report deals wth treatment of solar radaton and heat transfer n glazng systems, shadng of solar and sky radaton and the computer programs JULOTTA and DEROB-LTH. Other parts are more brefly dscussed and ncluded n order to gve an overvew of exstng problems and avalable solutons. A bref presentaton of how thermal models can be buld up s also gven and t s a hope that the report can be useful as an ntroducton to ths part of buldng physcs as well as durng development of calculaton methods and computer programs. The report may also serve as a help for the users of energy related programs. Independent of whch method or program a user select to work wth t s hs or her own responsblty to understand the lmts of the tool, else wrong conclusons may be drawn from the results. Ths s an ncreasng problem as so many programs nowadays are avalable on the market and only few of them are well verfed and documented. Durng the last years more nterest seems to be pad to the user s nterface and less n the methods used. Ths s n a way a dangerous trend as when the programs get easer to use they are more often used by people wthout knowledge about the lmts of used methods. 1.1 Background The need to estmate ndoor temperatures, heatng or coolng load and energy requrements for buldngs arses n many stages of a buldngs lfe cycle. At the early layout stage, the bulder and hs archtect may need some basc deas of the thermal behavor of the buldng. At ths stage, very few data about the buldng may exst, thus smplfed calculatons methods may, or must, be used. Durng the desgn of a buldng, more and more data are avalable, thus enable more and more sophstcated calculatons. In the fnal desgn stage, the HVAC consultant must be sure that he can meet the bulders requrements abut ndoor clmate, energy consumpton etcetera, thus good calculaton methods are requred. In the early lfetme of a buldng the dsagreements between estmated and used energy mght be examned, especally f legal actons are taken aganst the consultants. Later, t can be of nterest to use calculatons to check the operaton of the buldng, energy retrofttng may be consdered etcetera. All these stuatons requre good calculaton methods. Other purposes are the authortes requrements. Buldng codes often requre the use of prescrbed methods to show that a buldng wll meet the code. These 9

11 methods are seldom elaborate, but some codes allow that more complex methods may be used. Wthn the buldng research establshments there are some mportant use of ths type of calculatons. One s generalzaton of results from expermental buldngs and another s to perform parametrc studes n order to get a broader understandng of dfferent desgns. Both stuatons arse as expermental buldng s rather expensve, thus t s not always possble to test deas e.g. n dfferent clmate condtons wth real buldngs. 1.2 Methodology Wthn the feld covered by ths report t s rather dffcult to defne strct hypotheses and then test them. In many cases the hypotheses wll look curous and only constructed n order to follow the scentfc tradton nstead of beng useful to structure the work. One reason for ths s that the report deals wth appled physcs rather than pure scence, a common stuaton at an Insttute of Technology. E.g. f two programs deal wth the same problem t mght be mportant to dscuss the lmts for each of them and then conclude n whch cases one of them s better or worse than the other. Ths should lead to a whole set of hypothess lke program A s better n case 1, program B n case 2 etcetera. I do not thnk ths should make the presentaton more clear. Instead a systematc presentaton s chosen where dfferent parts of thermal models, calculaton methods and computer programs are dscussed. For each part advantages and dsadvantages of used approaches are dscussed and fnally some conclusons are drawn. 1.3 Lmtatons The major part of the report deals wth treatment of solar radaton n glazng systems, shadng of solar and sky radaton and the computer program JULOTTA used to smulate the thermal behavor of rooms and buldngs. Other parts of thermal models of buldngs are more brefly dscussed and only ncluded n order to gve an overvew of exstng problems and avalable solutons. Of nterest for smulaton wth the models dscussed are perods from days to years wth boundary condtons normally known at tme steps of one hour. The report s more focused on complex dynamc models of buldngs than on smplfed calculaton methods. The latter was partly dscussed n the author's lcentate thess. The report deals wth net energy requrements for heatng or coolng of buldngs. Thus HVAC systems are not dscussed n more detals than necessary for ncludng them n total models of rooms or buldngs, e.g. performance of bolers, ventlaton systems, coolng cols etcetera are excluded. 10

12 Specal problems may occur when more extreme buldngs or part of buldngs as glazed courtyards or ndustral halls should be examned. These problems are not covered n the report. The basc physcs, mathematcs and numercal methods nvolved n varous parts of thermal models of rooms and buldngs are assumed as known from general textbooks and not ncluded n the report. 1.4 Contrbuton and Relaton to Prevous Work The author's contrbuton wthn the am of ths report may be descrbed wthout any specfc order as follows. A systematc presentaton of thermal models. Shadng of solar and sky radaton. Some hstorcal notes Treatment of solar radaton n wndows. Use of an llustraton technque for thermal models. There are of course a lot of relatons between the work presented here and prevous and present work wthn ths area of buldng physcs. Contrbutons from the author as well as from others drect related to ths report are referred to n each chapter. 1.5 Some Hstorcal Notes These notes wll manly refer to my own experence and the stuaton n Sweden but smlar experence and developments have of course occurred on other stes and n other countres. In Sweden we can look back on two specfc breakponts n the developments of buldng codes and desgn tools durng the last half century. In the 60:th the archtects, nfluenced from US, started to desgn offce buldngs wth much larger wndows than used before and the ol crss n 1973 nfluenced peoples thoughts about energy used for heatng and coolng. Other thngs that have nfluenced the development of desgn tools may be more specfc for Sweden than for other countres. The earler rather detaled buldng codes and the rules for governmental loans have forced the bulders to partcular desgns. These codes and rules have been radcally changed n later years. The development of computers has also played an mportant role for development of more advanced desgn tools and the number of researchers workng wth programs for dfferent buldng physc problems has ncreased, especally snce the ntroducton of personal computers. The Swedsh Buldng Codes The Swedsh buldng code was earler amed to force the bulders to produce safety and healthy buldngs and dd not dscuss the use of energy n any detals as the cost of energy was low. Questons related to ndoor temperatures were 11

13 focused on lowest acceptable ar and nner surface temperatures, the later solved by settng a maxmum allowed U-value for buldng components as wndows and walls. As a result of the code the consultants had to calculate the U-values by use of strct rules n the code n order to show the authortes that there constructons followed the code. The calculaton of operatve temperatures was also requred and solved by use of dagrams. Partly due to the code and partly to the costs the bulders dd not pay much attenton n the use of energy, thus the calculaton methods for heatng systems where rather smple, sometmes only roles of the thumb. In order to estmate the heatng power requred for a buldng some more serous consultants used steady state calculatons bult on the U-values of the buldng s envelope. But even more smple rules was used, e.g. n 1970 I rebult an old one famly house stuated n Malmö (n the South of Sweden and the consultant desgned the drect electrcal heatng system only on the bases of 200 W/m 2 lvng area. I changed the desgn. The ventlaton and ar condtonng consultants were normally more advanced, especal when they dealt wth coolng systems. But they often estmated the heatng or coolng loads wth the U-values and dd not used dynamc calculatons. The annual energy requrement for heatng was commonly calculated by the Degree Day method where the nfluence of solar radaton and nternal loads was assumed to ncrease the ndoor temperature by 3 Cº. After the ol cres n 1973 rules for effcent use of energy was ncluded n the code and n the code from 1980 the use of more elaborate calculatons was mentoned, e.g. for drect electrcal heated buldngs the code requred 40% less energy use than a smlar buldng wth another heatng system and ths could be shown by usng the BKL-Method, Källblad (1978. In the newer buldng codes, NR (1989 and BBR 94 (1993, the energy use has been pad more attenton and the requrements are formulated more as functons of the whole buldng than propertes of sngle components. The bulder may more freely choose dfferent methods when to show that the requrements are met whch also has nfluenced the development of new methods and computer programs. Analogue Computers Before the dgtal computers was avalable a qut dfferent technque was sometmes used, the actve and passve analogue computers. An actve analogue computer was bult wth operaton amplfers and manly used for solvng dfferental equatons, thus able to use n studes of dynamc heat transfer n e.g. walls and slabs. However, these computers were manly used for smulaton of ndustral processes etcetera. An example from Kockum s shpyard n Malmö n the begnnng of the 1960th was the smulaton of a new submarne s reactons on rudder movements. 12

14 The passve analogue computers were bult on the analogue between temperatures and heat flow wth electrcal voltage and current n a network of resstors and capactors. At the Dansh Insttute of Technology a passve analogue computer was buld and used for desgn of heatng and coolng systems for buldngs. At our department I desgned one amed to be used for educaton and another used to estmate optmal nsulaton of cold storage houses. However, the development of the dgtal computers made the analogue computer less nterestng. Program Developments n Sweden The changes n archtecture durng the 60:th gave buldngs wth larger wndows and problems was notced when some hgh rased buldngs was bult n the cty of Stockholm and a clear need of dynamc models was observed n order to desgn solar shadng and coolng equpment. At ths tme the man frame computers also became more common avalable n Sweden and the development of dynamc thermal models of buldngs begun. One of the frst dynamc computer programs for determnaton of ndoor clmate, heatng and coolng loads was developed by Brown, G. (1963. The program, named BRIS, used frst prncples of physcs ncludng non lnear heat transfer phenomena and performed the calculatons hour by hour for typcal days. The program was wrtten n a dalect of Algol developed for the expermental computer TRASK at the Royal Insttute of Technology, KTH, n Stockholm. The program s stll used but translated to a more modern programmng language and s mplemented on newer computers. Brng and Isfält at the Royal Insttute of Technology (KTH n Stockholm have also to a great extent been nvolved n the work wth BRIS. Another poneer work n Sweden whch should be mentoned was carred out at Svenska Fläkt AB where Sune Larm played a leadng role developng the program VENTAC, partly n cooperaton wth KTH and our department. Ths program has been frequently used by Svenska Fläkt for desgn of HVAC systems n many dfferent countres. Professor Bo Adamson, the former head of our department, had an early nterest n the development n ths feld. An expermental buldng wth one room equpped wth 500 thermocouples was buld at the department n order to valdate computer programs and some comparsons between the BRIS program and measurements was carred out. He also worked wth smplfed methods to predct ndoor temperatures, Adamson, (1968. In the end of 1968 I was employed at the department n order to mantan measurement equpment but became rather soon nvolved n the development of smulatons models and programs as the BKL-Method mentoned above and the more elaborate program JULOTTA whch wll be dscussed later n ths report. Our department has partcpated n several research project wthn the Internatonal Energy Agency (IEA, e.g. the very frst one, Load Energy Determnaton of Buldngs , U.S. Department of Energy (1981, n whch Professor Adamson and I were engaged. 13

15 Later the modular programs have been of nterest, e.g. IDA under development at the Insttute of Appled Mathematcs n Stockholm. The man dea of ths project s brefly to have a lbrary of modules e.g. a wall, a heater or a heat exchanger descrbed both as a numercal model and as a graphcal symbol. The user can buld up more complex modules by connectng the graphcal symbols on the screen thus formng an elaborate thermal model. After a total model has been fnshed the program wll connect the correspondng numercal models and the user wll be asked for necessary parameters where after the ncluded solver can perform the requred calculatons. Program Developments n some other Countres In the Unted States Mtalas (1967 and Stephenson (1967 were some of the poneers wth ther work wth response factors, the numercal method most program developers n US adopted for the buldng energy analyss. The response factors for heatng and coolng load as functons of outdoor temperatures, solar radaton etcetera at constant ndoor temperature was used, thus requrng lnearzaton of all heat transfer phenomena. In many cases the man nterest was to determne heatng and coolng load at constant ndoor temperatures. Among other poneers n the US Kusuda, T. (1978 and Lokmanhekm, M. (1971 should be mentoned. Both have carred out a lot of work wthn ths feld. Lokmanhekm was nvolved n several programs but especally known through hs work wth the program CAL-ERDA, later renamed to DOE, a program whch more or less became a standard n the US. Use of weghtng factors has also been added to ths program and allows t to handle floatng ndoor temperatures. Some programs n the US are bult on the frst prncples of physcs and allow non lnear treatment of heat transfer problems. Among these s the BLAST program rather well known as well as the DEROB program from the Unversty of Texas, Arum-Noa, F., (1979. The DEROB program has been used on several stes and exsts n many versons. One verson, DEROB-LTH, was adopted at our department and has been further developed. Most programs use a rather fxed thermal model of the buldng but the program TRNSYS, manly developed at the Unversty of Wsconsn wth contrbutons from many others s somewhat dfferent. The program conssts of several modules, e.g. a room, a solar collector, a storage tank etcetera. The user has to buld a thermal model for the whole buldng usng these modules and descrbe how these are connected to each other. Another modular approach s taken at Berkeley where the Energy Kernal System (EKS s under development. Some of ths work was done n connecton wth the development of IDA n Sweden. The frst Norwegan program, BYVOK, was wrtten by Larsen, B. (1970 at the Norwegan Unversty of Scence and Technology n Trondhem after the prncples used by Mtalas and Stephenson. Later, when I showed some serous errors when usng BYVOK on heavy buldngs, Börresen, B. started the development of DEBAC, a program buld on the frst prncples of physcs. Most European program developers also seem to have chosen ths approach. Among later work n Norway related to ths feld s the study of atra by Bryn, I. (

16 Attomäk, A. (1971 gave the prncples used n the Fnnsh program TASE whch later has been further developed by Kalema, T. (1991. Ths program uses response factors to solve the heat flow through walls and slabs but uses nonlnear equatons for heat balance of the surfaces and ar volumes thus enable non-lnear treatment of e.g. convectve flm coeffcents. As a curous detal t can be mentoned that the program used a dfferental method to calculate the response factors for the walls. In Denmark the man development has been concentrated on the program TEMPFO, Andersen, B. (1972, later further developed and now known as TSBI. They also made an early work on clmate data and produced a reference year for Denmark rather early, Andersen, B. et al (1974. The dea behnd ths reference year has later been adopted by the European Communty. An extensve study on thermal models of buldngs has been carred out by Fest, W. (1994 n Germany and another study was done by Clarke, J. A. (1985 who developed the ESP program n the UK. Ther reports are valuable sources for those nterested n the feld. 15

17 2 Introducton The am of ths chapter s to gve a bref ntroducton to the problems related to thermal behavor of buldngs. Some used terms wll be explaned, some comments on comfort parameters and a bref overvew of the energy flows n a buldng are gven. How a thermal model of a room or a buldng can be bult up and llustrated s also presented. 2.1 Some Defntons A Model, as the term s used n ths report, s a descrpton of a real physcal process. One can formulate a model as equatons, a mathematcal model, or llustrate t by a Fgure, e.g. as dscussed later n ths chapter. A Thermal model s n ths report used to descrbe heat transfer phenomena n and around a room or a buldng. A model may be an almost exact descrpton of the real process, e.g. the Fourer's heat transfer equaton n three dmensons used for a wall, but s often an approxmaton. The approxmatons can be of dfferent levels, e.g. the Fourer's equaton approxmated by dfference equatons wth dfferent number of layers and dfferent tme steps. A model may be very smplfed, e.g. a U-value approxmately descrbng the heat flow through a wall n a steady state case. The term Smulaton Method s used for dfferent ways to smulate a real process, e.g. calculaton methods, studes of ar movement n wnd tunnels and water tanks or studes of heat transfer problems by use of analogue electrcal networks or analogue computers. A Calculaton Method s buld up wth a mathematcal model and the formats, numercal methods, algorthm etc. used to solve the equatons n the model. Ths report wll focus on calculaton methods for heat transfer n rooms and buldngs. A Computer Program contans a calculaton method and the program code used to carry out the calculatons. As any model can be used, a computer program not necessarly ncludes a detaled model. Heat flow between two ponts may be gven as Specfc heat flow (Heat flow per unt area n W/m 2, or as total flow n W. It s common to use the further n e.g. the Fourer's equaton and the later n e.g. a heat balance for the ar n a room. However, when a thermal model for a room s buld up, ths wll e.g. lead to that the convectve heat transfer at an nner surface s represented by one value n the heat balance for the surface and by another value n the heat balance for the room ar. The rato between the values s of course only the area, but to clearly ndcate that t s the same flow, the total heat flow wll manly be used throughout ths report. Heat transfer by radaton s commonly dvded nto two groups accordng to the wavelength. Radaton manly wthn the vsble spectra s n ths report called Short Wave Radaton (SW radaton or, f the source of the radaton s mportant, Solar Radaton, Artfcal Lght etcetera. For radaton wth a wavelength above the vsble spectra, as emtted from buldng surfaces, the ground or the sky, the term Long Wave Radaton (LW radaton wll be used. 16

18 Other common terms for LW radaton are nfrared radaton and low temperature radaton. 2.2 Comfort Parameters One of the man purposes of a buldng s to provde a sutable ndoor clmate. For dwellngs and offces are dfferent comfort parameters of nterest and n storage houses manly the room ar temperature. Some commonly used comfort parameters are summarzed n ths paragraph. A more complete nvestgaton of thermal comfort ndces and heat stress ndces can be found n Wang (1992, TABK--92/3004. The ar temperature s the smplest comfort parameter. However, as the human body reacts on both the ar temperature as well as radaton exchange wth the surroundng the ar temperature s not suffcent to judge the humans comfort but never the less used n many cases. E.g. heatng and coolng load are often calculated wth a gven ndoor ar temperature as the only requrement. The operatve temperature s a better comfort parameter as t ncludes the nner surface s temperature and thus takes nto account that a human exchanges heat wth the surroundng both by convecton and radaton. ISO 7730 (1994 gves followng formula for the operatve temperature. o A (1 A a where A = 0.5 for v < 0.2 m/s r (ºC ( for v = 0.2 to 0.6 m/s 0.7 for v = 0.6 to 1.0 m/s v = Ar velocty around the person (m/s o = Operatve temperature (ºC r = Mean radant temperature (ºC The ar velocty s seldom known and a common way to calculate operatve temperature s to use the arthmetc mean,.e. A = 0.5 s chosen. The mean radant temperature n relaton to a person n a gven body posture and clothng placed n a gven pont n a room, s defned as that unform temperature of black surroundngs whch gve the same radant heat loss from a person as the actual case under study, Fanger Ths defnton leads to rather elaborate calculatons and the result depends not only on the surface temperatures but also of clothng, ar velocty and the placng n the room. A commonly used smplfed formula s r j j j A A j j (

19 where Aj = Area of surface j (m 2 j = Temperature of surface j (ºC The drected operatve temperature s sometmes used, e.g. n the Swedsh buldng code SBN 80 where t s defned as the arthmetc mean of the room ar temperature and the drected mean radant temperature gven by dr where j j j j = Vew factor from an nfntesmal area to surface j dr = Drected radant temperature (ºC j = Temperature of surface j (ºC (2.2.3 The drected operatve temperature s often used to examne the asymmetry of the ndoor clmate and the Swedsh buldng code had requrements on ths temperature e.g. one meter from a wndow. The most elaborate comfort parameters are the Predcted Mean Vote (PMV and the Predcted Percentage of Dssatsfed (PPD defned by Fanger (1970 and also accepted as an nternatonal standard, ISO 7730, These parameters requre elaborate calculatons and the result depends not only on the temperatures but also of clothng, ar velocty and the placng n the room, thus normally not used n thermal models of rooms and buldngs. From the above t s obvous the at least the room ar and the surface temperatures are needed n order to estmate the ndoor clmate. When these have been determned the comfort can be studed n more detals. E.g. the PMV can be examned by the FRES program, Rømen, B.&.Frydenlund. F. (1992, who uses the smplfed Eq to determne the mean radant temperature. A more detaled study about the determnaton of the PMV was carred out by Källblad (1996, where the COMFORT program also s documented. Ths program calculates the PMV accordng to Fanger s defntons and n the report some comparsons between dfferent ways to examne the radant temperature are carred out. It s obvous from the above overvew that knowledge about the ndoor ar temperature, the temperatures of the nner surfaces and solar radaton nsde the room are needed n order to determne the ndoor comfort. Thus many smplfed thermal models are not suffcent for comfort determnaton. 18

20 2.3 Energy Flows n Buldngs Fgure 2.1 Man Energy Flows n a Buldng The man energy flows wthn a buldng are schematcally llustrated n Fgure 2.1 where the total suppled energy can be dvded nto followng areas. From Persons From Solar Radaton From Lghtng, Electrcal Equpment etcetera From Heatng Systems From Hot Tap-water Systems Energy from persons and solar radaton are often referred to as free heat. Sometmes the energy from lghtng, hot tap water system etcetera also s consdered as free heat, especally when heatng loads are dscussed. The suppled net energy may be convectve heat flow to the ndoor ar or radaton absorbed n buldng surfaces or furnture. The energy losses are of tree types: Transmsson Losses Ventlaton Losses Sewage Losses Transmsson losses depend on the heat flows through the buldng envelope and nclude heat transfer phenomena as conducton, convecton and radaton. The ventlaton losses are mass transfer phenomena and depend prmarly on how much the nlet ar must be heated or cooled. These losses are commonly 19

21 dvded nto nfltraton due to leakage n the envelope and forced ventlaton through the HVAC system. The sewage losses depend on the amount of water used and the temperatures of the nlet and outlet water. In order to calculate temperatures, heatng or coolng loads for a buldng one have to create a mathematcal model of the heat transfer phenomena n the buldng. Ths model can be very smple as e.g. n the Degree Day Method, when all losses are lumped together and the energy used for heatng s descrbed by a sngle equaton. On the other hand, a very complex model wth hundreds of dfferental equatons and non lnear equatons coupled together can be used. To make a complete model of all nvolved detals n a buldng s almost mpossble. Not only because of the elaborate calculatons ths wll lead to, but also as a lot of necessary parameters for a complete model are more or less unknown, e.g. use of furnture, equpment etcetera. In order to overcome some of these problems, exstng models often use one or more of the followng assumptons. Unform ar temperature n each room/zone Unform surface temperature on each wall or part of wall One-dmensonal heat transfer n each wall or part of wall Dffuse dstrbuton of solar radaton wthn rooms Dffuse gray surfaces accordng to LW radaton exchange All heat transfer treated as lnear phenomena The assumptons mentoned for walls also nclude slabs, roofs etcetera. Two- or three-dmensonal heat transfer are sometmes ncluded by approxmatons of dfferent levels. 2.4 Elements of Thermal Models Fgure 2.2 Illustratons of Thermal Model Elements The symbols used n ths report, of whch the symbol for one drectonal heat flow s adopted from Claesson, J. (1990, are shown n Fgure 2.2 and these wll be dscussed n followng paragraphs. 20

22 2.4.1 Temperature Nodes Each pont of nterest n a thermal model, e.g. the room ar, an sothermal part of a wall surfaces or a wndow pane s referred to as a temperature node, or shortly a node. In a node, the heat capacty may be neglected and the node s then llustrated as n Fgure 2.2a. If the heat capacty s taken nto account the symbol n Fgure 2.2b wll be used. Into each node heat flows from or to other nodes due to conducton, convecton, radaton or mass transfer may occur. Furthermore, heat sources or snks may be attached to a node. If all these flows are defned postve nto the node, the heat balance of the node s, due to the fundamental laws of thermodynamcs, Q Q c where Q = Heat flow nto the node (W Q c = Net heat flow stored n the node (W (2.4.1 In cases when the heat capacty s neglected the net heat flow becomes zero, thus Q 0 (2.4.2 For a node wth heat capacty the change n temperature by tme s descrbed by C( T dt dt Q c where C(T = Heat capactance (J/K T t = Temperature of the node (K = Tme (s E.g. a node representng a layer n a sold wall has the heat capactance (2.4.3 C c( T ( T dwl A (J/K (2.4.4 where c(t = Specfc heat (J/kg,K (T = Densty (kg/m 3 d wl = Thckness of the wall layer (m A = Area of the layer (m 2 Nodes wthn solds are, when the heat capacty s ncluded, sometmes called mass nodes. The temperature dependency of the specfc heat and densty are often neglected for mass nodes. Heat Sources and Snks Solar radaton absorbed n surfaces, nternal heat loads, auxlary heat from heatng systems etcetera represent dfferent heat sources and are llustrated as n Fgure 2.2.c. where the arrow ndcates the postve flow drecton. Wth the 21

23 arrow n the opposte drecton, or wth a negatve value of Q, the symbol wll represent a heat snk, e.g. a coolng fan-col. The man characterstc of a heat source or snk s that the heat flow s not dependent on the components connected to t. However, the heat flow may be lmted n sze or controlled by a temperature n the model. Temperature Sources An ndependent temperature, e.g. the outdoor ar temperature, s sometmes called a temperature source and represented wth the symbol n Fgure 2.2.d. The symbol s often used to dstngush nodes representng boundary varables from those representng unknown varables n the model. The man characterstc of a temperature source s that the gven temperature s not nfluenced by the components connected to t, but the temperature may be controlled by a temperature n the model Heat Flow between Nodes Between two nodes dfferent heat transfer phenomena may occur. In cases where the drecton of heat flow only depends on the temperatures of the nodes, the heat flow can be descrbed by the equaton Q1 2 G( T1, T2 ( T1 T2 (W (2.4.5 where G(T 1,T 2 = Heat conductance (W/K T = Temperatures of the nodes (K The recprocal value of the heat conductance s called the heat resstance and gven by R G 1 (K/W (2.4.6 Fgure 2.2.e llustrates a heat conductance connectng two nodes. The arrow n the fgure defnes the postve drecton of heat flow and should not be mxed up wth the real drecton of the flow. In cases when T1 < T2 Eq wll gve a negatve flow ndcatng that the flow n fact s toward T1. In cases of non lnear heat transfer, the conductance wll depend on T 1 and/or T 2 as showed below n some examples. The heat conductance can n ths case be llustrated as n Fgure 2.2.f n order to ndcate that the conductance s non lnear but also as n Fgure 2.2.e f t s obvous that a non lnear phenomena s descrbed. Example 1, Conducton E.g. between two nodes n a sold wall, the heat transfer may be assumed as lnear and the conductance s then gven by G where A d (W/K (2.4.7 = Conductvty (W/m,K 22

24 A = Area (m 2 d = Dstance between the nodes (m Example 2, LW radaton The LW radaton between two nodes may be descrbed by 4 4 Q res A1 F1,2 ( T1 T2 (W (2.4.8 where res = Stefphan Boltzmans constant = Resultng emssvty of the surfaces A1 = Area of surface 1 F1,2 = Vew factor (depends on the geometry In ths case we get 2 2 G( T1, T2 res A1 F1,2 ( T1 T2 ( T1 T2 Q /( T1 T2 (2.4.9 Example 3, Mass flows In ventlaton systems as well as between rooms n a buldng, ar s often moved n a specfc drecton between rooms and/or the outdoor ar. If ar s taken from one room to another and no ar s flowng n the opposte drecton, a conductance cannot be used n order to represent the stuaton. Here, the symbol n Fgure 2.2.g s used n order to ndcate that the heat flow only nfluences the node T 2. The heat flow can stll be descrbed by e.g. Eq but should only be ncluded n the heat balance for the node T 2. E.g. ar exchange (same amount of ar gong n both drectons between two nodes (T 1 and another T 2 may be descrbed as Q c ( T v ( p T ( T T (W ( where c p = Specfc heat at constant pressure (J/kg ρ(t 1 = Densty of ar (kg/m3 v(t 1 = Volume flow (m3/s In ths case, the conductance becomes G( T1, T2 c p ( T1 v( T1 (

25 2.5 Thermal Models of Buldng Components Usng the elements dscussed above more complex thermal models of buldng components can be buld up. In ths paragraph some example wll be gven n order to llustrate the technque. In the models sold lnes are used to show how dfferent elements are connected. If lnes are connected wth a dot they represent connectons to the same node Room ar Fgure 2.3 Thermal model of the Room Ar It s normally assumed that the buldng does not nfluence the outdoor ar, thus ths temperature s represented as a temperature source (To. The room ar s assumed to be at an unform temperature, thus represented by a sngle node (Tr, and the heat capacty of the ar s neglected. The heat from convectve nternal loads s gven by Q l and connected to the room ar node. The nlet ar from the outdoor ar s represented by Gv and the outlet ar by Gvo. As both these are drectonal only the frst wll nfluence the heat balance of the studed room. The node Tx may e.g. be another room. The nner surfaces are represented by the nodes Ts and the convectve heat transfer to these are represented by the conductance G c1 through G cn. As the heat capacty of the room ar s neglected, the heat balance equaton becomes Q l G V ( To Tr Gc ( Ts Tr 0 (2.5.1 Here t can be noted that Gvo and Tx are not ncluded as there s no ar gong from the node Tx. If the room ar s dvded nto temperature zones each zone has to be represented by a node and the heat flows between the zones by addtonal conductances. 24

26 2.5.2 Wndows Fgure 2.4 Thermal Model of a Double Pane Wndow The double pane wndow s n Fgure 2.4 modeled wth one node for each pane (T1 and T2. The heat capactes of the panes are neglected. On the outsde the wndow s connected to the outdoor ar wth a conductance Gco representng the convectve exchange wth the outdoor ar. The LW radaton exchange wth the sky (Tsky and the ground (Tgrd are represented by Gsky and Ggrd. Between the panes and on the nsde the convecton s modeled wth the conductance Gc respectve Gc. The LW radaton exchange wth the other nner surfaces s represented by Glw. The absorbed solar radaton absorbed n the panes are represented by the heat sources Q a, as the absorbed radaton s not dependent on the temperatures of the panes. The heat balances of the panes n the wndow are thus Q Q Go ( To Tw 1 ( GLW Gc ( Tw2 T 1 a, 1 w a,2 G ( G c ( T LW r T G ( T c w2 w1 G T w2 LW. ( T s, T w2 0 0 (2.5.2 ( LW radaton Exchange n a Room The LW radaton exchange between four surfaces (Ts s llustrated n Fgure 2.5 where Gj represent the heat flow between surface and j. Each of these conductances may be determned accordng to Eq The fgure also llustrates that t sometmes s necessary to allow lnes representng dfferent temperatures to cross each other n whch case they are not connected by a dot. When more complex models are used all LW radaton may be llustrated as n the rght part of Fgure

27 Fgure 2.5 Thermal Model of LW Radaton Exchange n a Room Walls and Slabs For the heat transfer by conducton n walls and slabs the heat capacty must n most cases be taken nto account and the Fourer s heat transfer equaton be used. In most cases ths equaton must be treated by dfferental approxmatons where the heat capacty e.g. s lumped nto a fnte number of nternal nodes as llustrated n ths paragraph. It s also possble to use an approxmaton that gves surface nodes wth heat capacty. Both approxmatons gve the Fourer s equaton for one dmensonal heat transfer f the number of nternal nodes goes to nfnty. Fgure 2.6 Thermal Model of a Wall An one-dmensonal model for a wall wth unform surface temperatures s shown n Fgure 2.6. In the upper part of the fgure the detals n the model are shown. In more complex models the detals may be hdden and the wall llustrated by a sngle mpedance (Z as shown n the lower part of the fgure. In ths example a homogenous wall s assumed and the wall s modeled by three layers of equal thckness. The heat capacty n each layer s placed n the center of the layer. Usng Eq and we obtan for a wall wth the thckness d w. G 6A 1 d w (W/K (2.5.5 G 2 3A d w (W/K (

28 C Ac d w 3 (J/K (2.5.7 Assumng the same surroundng for the wall as for the wndow n Fgure 2.4 the heat balance for the surfaces (Ts1 and Ts2 became Q Q G ( To Ts 1 G1 ( T1 T 1 a1 o s a2 G1 ( T3 Ts2 Gc ( Ts2 Tr Glw, j ( Tsj Ts2 0 j Accordng to Eq and the heat balances of the mass nodes are dt C dt 1 dt C dt 2 dt C dt 3 G1 ( Ts 1 T1 G2 ( T2 T m 1 G2 ( T1 T2 G2 ( T3 T2 G2 ( T2 T3 G1 ( Ts2 T3 2.6 Thermal Model of a Room, an Example 0 (2.5.8 (2.5.9 ( ( ( The models of dfferent buldng components descrbed above can now be used to buld up models of rooms or buldngs. As an example a model of a room wll be llustrated n ths paragraph and some parts of the model are shown n Fgure 2.7. The room n queston s very smple and contans four walls, a roof, a floor and a double pane wndow. The room ar (Tr as well as each surface (Tso and Ts are assumed to be at unform temperature, thus each represented by a sngle node The mpedances Z1 trough Z5 represent the roof and the walls whch on outsde are connected to the outdoor ar (To and the sky (Tsky. The LW radaton exchange between these outer surfaces and the ground s neglected. The floor Z6 s on the outsde connected to an ndependent ground temperature (Tgrd. The double pane wndow s modeled as llustrated n the last paragraph but the LW radaton between the outer pane and the ground s neglected. Between all nner surfaces heat exchanged by LW radaton s taken nto account and llustrated by the network Glw whch s a smplfed llustraton of the 21 conductances needed to gve a full llustraton. The convectve heat transfer between the room ar and the nner surfaces are represented by the conductances G c, the ventlaton losses by G v and the heat source Q l llustrates convectve nternal loads to the room ar. In order to formulate a mathematcal model of the thermal model n ths example sx dfferental equatons coupled wth sx equatons for the outer surfaces and an equaton system wth nne unknown varables (nner surfaces, 27

29 the room ar and the outer pane are needed. Furthermore the equatons and the equaton system normally are non lnear. Fgure 2.7 Part of a Thermal Model of a Room When the model then should be used, the dfferental equatons have to be approxmated e.g. wth dfference equatons and all equatons must be solved smultaneously for each tme step. The boundary condtons as outdoor ar temperature, solar radaton and nternal heat load must be gven for each tme step. Furthermore, solar angles, shadng and dstrbuton of the solar radaton must also be calculated for each tme step. When studyng the thermal behavor of a buldng t s often necessary to smulate perods of up to a whole year s usage wth a tme step of one hour or less. A computer program s obvous needed even wth a smple model as n ths example. 28

30 3 Thermal Components In ths chapter thermal model components wll be dscussed. The major am of the chapter s to pont out some problems that occur when a thermal model of a buldng s created and to gve references where the reader can fnd solutons to these problems. The paragraphs are chosen n order to get a systematc presentaton but, due to the nteractons between dfferent parts of a buldng, ths s not always possble. 3.1 Volumes Detaled treatment of ar movements n a volume leads to elaborate calculatons wth the use of Naver-Stokes' equatons. Wth programs lke PHENIX and FLOVENT and a fast PC t can take several hours to solve the equatons for one room at a sngle tme step. These calculatons are thus to complex to nclude n a model for energy calculatons as we normally want to treat longer perods as heatng seasons or years. Thus, the frst step s to approxmate the enclosures nto one or more zones, each wth unform temperature represented by a sngle node. Typcally one zone for each room s used for resdental buldngs and n large offces several zones may be defned. Then the heat balance of each zone s used to buld up the thermal model for the volume. Even f a room s treated as a sngle zone t mght be mportant to treat the temperature dfferences at dfferent heght levels n a room. The Swedsh buldng code SBN1980 gves dfferent slopes dependng on the used heatng systems whch mght be used. In Fgure 2.3 a thermal model of a sngle ar node s llustrated. If the heat capacty of the ar, Cr, s taken nto account and we assume nternal heat load, Ql and auxlary heat from a convectve heater, Qaux, we get the heat balance at the node as C r dtr dt Q l Q aux N 1 s N v G c ( T s T r 1 G v ( T v T r (3.1.1 where the frst sum represents the convectve heat flows from the room s surfaces and the second the ar nlet from other rooms or from the outdoor ar. The equaton s also vald for coolng n whch case Qaux represents the coolng load. The equaton s used n two dfferent stuatons, namely f the room ar temperature s unknown or f the heatng (or coolng load s unknown. In the frst case the auxlary heat s zero or e.g. controlled by an outdoor thermostat and thus a functon of the known outdoor ar temperature. In the other case the room ar temperature s assumed to be known, e.g. controlled by an ndoor thermostat. 29

31 Heat Capacty If the ventlaton losses are neglected the tme constant for the room ar can be defned as N s C r Gc 1 (3.1.2 The heat capacty of the room ar s rather low, e.g. n a room wth a volume of 50 m 3 about 15 Wh/K. If the area of the room surfaces s 45 m 2 and the convectve flm coeffcent 3 W/m 2,K the tme constant wll be around 7 mnutes. Takng the ventlaton nto account wll gve a much shorter tme constant. Thus the heat capacty can be neglected as the perods of nterest here are much longer. Jóhannesson, (1981, studed the possble naccuracy when ths approxmaton was used and draw the concluson The approxmaton holds down to a perod of 2 hours whch s qute suffcent f nput data are gven by hourly values. Furthermore, when heatng or coolng s studed the ndoor temperature s controlled n some way. If the control system s assumed to be perfect the temperature varaton s zero. Wth a real ndoor thermostat the temperature wll vary some degrees but the average value over the perods of nterest may stll be assumed as constant. Thus the left hand sde of Eq can be set to zero n these cases ndependent of the heat capacty. Internal Loads The nternal loads consst of heat from e.g. people and electrcal applances as lghtng etcetera. The metabolc rate from a person s between 80 to 140 W dependng on type of actvty and the convectve part of ths heat s ncluded here. Heat from lghtng, TV sets etcetera are manly convectve loads but more dffcult s to estmate heat from e.g. washng machnes where a lot of hot water s wasted. Auxlary Heat A commonly used approxmaton of the heatng system s to assume pure convectve heat sources. However, the auxlary heat s, at least n resdental buldngs, most often suppled by radators from whch the heat transfer s partly radatve. The problem to model the convectve part and the nteracton between the thermal plumes and the nner surface of a wndow has been dscussed by Kalema, T. (

32 Convectve Heat Transfer The convectve heat conductance depends on the flm coeffcent hc and the area of the surface. G h A c c (W/K (3.1.2 The flm coeffcent depends on several parameters as the orentaton of the surfaces, temperature dfferences, lamnar or turbulent flow etcetera but wll not be dscussed n ths report. An extensve study on convectve heat transfer n models of buldngs has been carred out by Fest, W., (1993. That report s a valuable source for further studes. Ventlaton Ar flows n a buldng occur from nfltraton, arng and/or forced ventlaton through the ventlaton system. Ar s prmarly taken from the outdoor ar nto the buldng and s heated or cooled to acheve a requred ndoor clmate. The ar may also be heated by nternal loads. The outdoor ar contans water vapor and mosture producton can occur n the buldng, e.g. evaporaton from nhabtants, cockng or techncal processes. In some cases, the ar has to be dred or humdfed to fulfll specal requrements. A pror, desgned ar flows, mosture producton and some propertes of the outdoor ar are known. Ar flows are commonly gven as volume flows but as the volume of ar vares sgnfcantly wth the temperature t s convenent to use mass and enthalpy balances when dealng wth these problems. Commonly avalable clmate data ncludes values on dry-bulb temperature, statc pressure and humdty. The later may be gven as relatve humdty, wet-bulb temperature, dew-pont temperature, humdty by mass or volume or partal water vapor pressure. The treatment of ar flows n buldngs requres the use of psychometrc defntons and formulas and a summary of the most mportant s gven n appendx Surfaces The heat balance of an surface nvolves conducton, convectve heat transfer, absorbed SW radaton and LW radaton exchange wth other surfaces. The heat balance of a surface where no heat capacty s assumed s gven by Q abs G m ( T m T s G c ( T r T s N 1 G LW ( T s T s 0 (3.2.1 The heat source (Qabs represents the absorbed solar radaton, radaton from people, radators and lghtng. The major part s the solar radaton whch wll be dscussed n a later chapter. The conducton (Gm nto the frst mass node, (Tm, n the wall wll be dscussed n next paragraph. 31

33 The convecton (Gc between the surface and the surroundng ar was brefly dscussed above. On an outer surface the wnd speed can have a rather hgh nfluence and some expermental data s gven by Kmura (1977. The LW radaton exchange between surfaces s represented by the conductance GLW whch can be determned e.g. by Eq One way to determne the needed vew factors s dscussed n chapter 4 and there are several books avalable n ths feld, e.g. Segel and Howell. (1981. At an outer surface the LW radaton may nstead be treated as emtted and absorbed radaton n followng way. All LW radaton s assumed to be dffuse and all nvolved surfaces are assumed to be optcal dffuse-gray,.e. all emtted and reflected radaton s dffuse and the surface propertes are ndependent of the wavelength. Ths allows the use of vew factors when the LW radaton exchange s studed. The vew factor F,j gves the part of the total radaton emtted from the whole surface that wll be ncdent at the whole surface j before any reflectons has been taken nto account. If no other buldngs are n the surroundng of the studed one, the buldng s facng the sky and the ground. The LW radaton from each of the three "surfaces" wll be reflected between the surfaces and fnally absorbed. The radaton emtted from the sky, the ground and the buldng are thus E E E sky grd s s T sky grd 4 s T 4 sky T A A 4 grd s sky A grd where = Stephan Boltzman s constant ( W/m 2,K 4. (3.2.2 (3.2.3 (3.2.4 The radaton emtted from the buldng and prmarly reflected at the ground s E grd s T 4 s A F s s, g (1 grd (3.2.5 It s obvous even for a very bg buldng that A s A grd. Furthermore, the emssvtes and the temperatures of the buldng and the ground are about the same and Fs,grd ( 1 - grd 1. Thus s E'grd << E grd and can be neglected. In a smlar way t can be shown that radaton from the buldng reflected at the sky can be neglected. If the nfluence of the buldng s neglected the sky and the ground can be seen as two parallel nfnte surfaces wth Fsky,grd = Fgrd,sky = 1. The radaton emtted from the ground s reflected between these surfaces and the total radaton from the ground s thus E t grd E grd ( 1 grd Esky / 1 (1 grd (1 sky (3.2.5 The emssvty s typcally around 0.9 and thus an error of around 1% s ntroduced f the denomnator s set to 1. 32

34 The sky emssvty s seldom avalable and avalable clmate data s often measurements of the sky radaton. Thus the sky emssvty can be set to 1 and the measured values be used to estmate the sky temperature as T sky ( Emeasured / 0.25 (3.2.6 As the ground radaton reflected at the sky s ncluded n the measured radaton t s also ncluded n the sky temperature and s thus not needed to be taken nto account n any other way. The LW radaton absorbed n the outer surface can now wth Asky = Agrd and as A F,j = Aj Fj, be wrtten as Q abs s A ( s sky 3.3 Walls and Slabs F T F T (1 F s, sky sky grd s, grd grd grd s, grd sky sky T (3.2.7 Heat transfer n solds s well treated n the lterature and n ths paragraph only some few aspects on methods to handle one dmensonal heat flow wll be gven. Dscussons about two and three dmensonal heat flow n e.g. cold brdges can be found n Jóhannesson (1981 and Hagentoft (1988 has studed the heat losses from buldngs through slabs on ground. The expresson for one dmensonal heat conducton n solds s gven by the Fourer equaton a T( x, t t c a T( x, t 2 x 2 (3.3.1 (3.3.2 where a s the thermal dffusvty. Numercal treatment of ths equaton requres that the dervatves are approxmated by fnte dfferences whch can be done n dfferent ways. Two commonly used methods are the forward dfference method and the Crank-Ncholson s method. In both methods the left hand sde s approxmated by T ( x, t t T( x, t t (3.3.3 where t s the chosen tme step. The rght hand sde s approxmated by dvdng the sold nto layers of fnte length, x, and s n the forward dfference method gven by a T( x x, t T( x x, t 2T ( x, t 2 ( x (3.3.4 The equaton represents the stuaton at the begnnng of the tme step. In Crank-Ncholson s method the average heat flow durng the tme step s used and the rght hand sde becomes 33

35 2( 2( a x a x 2 2 T( x T( x x, t x, t T( x t T( x x, t 2T ( x, t x, t t 2T ( x, t t (3.3.5 Gven the ntal temperatures, the successvely use of the dfference equatons gve the new temperatures step by step. The forward dfference has the advantage of smplcty but the ncrements cannot be chosen arbtrarly. The expresson a t/( x 2 must be less than 0.5, else the results wll dverge to nfntve. Ths means that the tme step has to chosen less than the stablty tme step 2 c ( x t s 2 (3.3.6 The Crank-Ncholson s method s stable for any tme step but has the dsadvantage that all temperatures have to be solved smultaneous. Adamson (1970 showed that the best accuracy was acheved wth a tme step of 0.5 ts for the forward dfference and 2 ts for the Crank-Ncholson s method. Thus the requred amount of calculatons s about the same for both methods. 3.4 Wndows In ths paragraph the heat transfer between panes n a wndow wll be dscussed. The presented work s manly the author s mplementaton of a new wndow model n the DEROB-LTH program. A thermal model wth one temperature node n each pane s used. The heat resstance of wndow panes s neglected. Ths mght to some degree nfluence the results for sngle pane wndows but for multple pane wndows the errors ntroduced by ths approxmaton can be neglected. The LW radaton exchange between the panes and the convectve heat transfer are determned by the frst prncples of physcs. The total conductance n a gap s thus gven by G ( h h A gap r c where Aw = area of the wndow hr w = radatve heat transfer coeffcent hc = convectve heat transfer coeffcent (

36 LW Radaton The radatve conductance between two panes can be gven by Eq and as the gap between two panes s small compared wth the wndow area the vew factor between the panes can be set to 1. Thus the resultng emssvty wll be res 1/ 1 1 1/ 2 The heat transfer coeffcent s thus 1 ( hr res ( T1 T2 ( T1 T2 (3.4.3 Convecton The convectve heat transfer coeffcent s determned as h c Nu d (3.4.4 Nu f ( Ra, H, d, (3.4.5 Ra 2 3 Pr g d T / where H = Hght of the wndow (m Nu = Nusslet number Pr = Prandtl number Ra = Raylegh number 2 Tm = Average temperature n the gap (K T = Temperature dfferance over the gap (K g = Gravtatonal acceleraton (9.82 m/s 2 d = Dstance between the panes (m = Thermal expenson approxmated as 1/Tm (1/K = Tlt angle of the wndow = Conductvty (W/m,K = Vscosty (kg/m,s = Densty (kg/m 3 The tlt angle s defned as 0º Horzontal wndow wth the warmest sde downwards 90º Vertcal wndow 180º Horzontal wndow wth the warmest sde upwards (3.4.6 The temperature dependency of Pr,, and are approxmated wth lnear functons of Tm. Some examples of gas data can e.g. be found n Arasteh et al (1989. For calculatons of the Nusslet number followng chose has been done. 35

37 Formulas from ElSherbny et al (1982 for 90º (vertcal and 60º tlted wndows, shown to be vald for a wde range of Raylegh numbers. They also suggested a lnear nterpolaton between these values. Hollands et al (1976 gve a relaton for tlt angles between 0º and 60º and for Ra < 10 5, thus vald for a wde range of normal wndows. The Nusselt numbers accordng to ElSherbny and Holland dffers slghtly at a tlt angle of 60º. In order to avod ths step the numbers are slghtly changed for 0º < 60º. Compared wth measured values shown by ElSherbny (1982 ths correcton seems acceptable. For wndows tlted between 90º and 180º followng relaton from Fergusen and Wrght (1984 s used wth ElSherbny s value for Nu90. Nu 1 ( Nu 90 1sn (3.4.7 In order to study the use of the above formulas some calculatons have been carred out. The results and some conclusons are gven n ths paragraph. All exampes are for ar flled gaps wth ar propertes accordng to Arasteh (1989. Table 3.1 Nusslet numbers for dfferent ar gaps wth T = 10 C A Tm Ra ( (- ( C ( E E E E E E E E E E E E The aspect ratos A (=h/d has been chosen n order to enable comparsons wth the results reported by ElSherbny (1982 and cover a wde range of wndows. The results n Table 3.1 are close to those reported by ElSherbny. At larger gap wdth (A = 20 the Raylegh numbers are relatve hgh. Thus cauton must be taken as some of the used formulas are not valdated for these 36

38 Raylegh numbers. However, ths rato corresponds e.g. to a 50 mm gap n a wndow wth a heght of 1 m whch s not so often used. Fnally some values of the convectve heat transfer coeffcent calculated wth the formulas above are shown n Table 3.2. Table 3.2 Convectve coeffcents for dfferent ar gaps, T = 10 C A Tm Ra ( (- ( C ( E E E E E E E E E E E E

39 4 Solar Radaton Solar radaton absorbed n buldng surfaces s one of the major heat sources n a buldng, thus mportant to treat n a proper way n thermal models. In ths chapter the calculatons accordng to solar radaton at and wthn buldngs wll be dscussed. The beam radaton from the sun s sometmes called normal or drect radaton. When ths radaton s passng the atmosphere absorpton as well as scatterng, dffracton, reflecton and refracton wll occur. The beam radaton reachng the earth wll thus be decreased and have a slghtly dfferent drecton than outsde the atmosphere. It wll also result n lght travellng n all drectons, at the ground level called dffuse sky radaton. Ths radaton should not be mxed up wth LW radaton from the sky. It s n ths chapter assumed that the solar radaton at the buldng ste s known, thus detals of the mpact from the atmosphere etcetera found n e.g. Robnson (1966 wll not be dscussed. It s also assumed that the beam radaton orgns from a pont source. Ths s an acceptable approxmaton as the maxmum apparent radus of the sun s about 16 17, Robnson (1966, p31. Furthermore, the change of drecton of the beam radaton due to refracton wthn the atmosphere s only brefly dscussed. The propertes of the solar radaton as well as materal propertes related to solar radaton are almost ever dependent on the wavelength of the lght. Ths must be taken nto account e.g. when some types of wndows and daylght wthn a buldng s treated. When dealng wth heat models of buldngs t s often enough to use average propertes for the short wave radaton, as assumed n ths chapter f nothng else s mentoned. 4.1 Solar Radaton at a Buldng The total solar radaton at a buldng ste s gven by the global rradance,.e. the energy flow per unt tme per horzontal unt area caused by the beam and the dffuse sky radaton. G G G d b sn( (W/m2 (4.1.1 where G = Global rradance Gb = Beam rradance Gd = Dffuse rradance (W/m 2 α = Solar alttude The beam rradance s the energy flow per unt tme per unt area perpendcular to the beam and the dffuse rradance s the energy flow per unt tme per unt horzontal area coursed by the dffuse sky radaton. 38

40 Radant ntensty s defned as the energy flow per unt tme n a sngle drecton per unt area normal to the flow drecton nto a unt sold angle centered around the drecton,.e. n W/sr,m 2. The sky radaton s often assumed to be sotropc,.e. the ntensty s ndependent of the drecton. Ths assumpton s here used f nothng else s mentoned. Fgure 4.1 Isotropc sky radaton The relaton between dffuse rradance and the ntensty for sotropc sky radaton s llustrated n Fgure 4.1. As the ntensty s ndependent of the drecton we have 1 Gd dap ( d da (W/m 2 (4.1.2 where da = Infntesmal horzontal area (m 2 dap( = The projecton of da normal to = Intensty of ncdent radaton (W/sr,m 2 = The hemsphere above da = Sold angle (sr The projecton of da and the sold nfntesmal angle are gven by dap ( da cos (m 2 (4.1.3 d sn( d d (sr (4.1.4 Thus the dffuse sky radaton on horzontal becomes G d 2 0 / 2 cos( sn( d d (W/m 2 (

41 Fgure 4.2 Solar radaton at a buldng The solar radaton at an outer surface of a buldng s schematcally llustrated n fgure 4.2. Beam and dffuse radaton wll fall onto the buldng as well as on the surroundngs. From the surroundngs some radaton wll be reflected onto the buldng. The reflecton of the beam radaton may be specular or dffused but n ths chapter, as well as n most thermal models and programs, all reflected radaton s assumed to be sotropc. In some cases, e.g. a calm sea surface or ce covered ground, specular reflecton mght be necessary to consder. At an opaque outer surface some radaton wll be absorbed and some reflected and at a wndow some wll be transmtted nto the buldng. The transmtted radaton wll partly be absorbed n the nner surfaces and partly undergo multple reflectons before t s absorbed or transmtted out through the wndows. As for the ground reflected radaton, all reflected beam radaton s assumed to be sotropc f nothng else s mentoned. 4.2 Co-ordnate Systems and Solar Angles In order to carry out calculatons concernng beam radaton we need the solar poston related to a ste and to the buldng surfaces. The used co-ordnate systems and equatons are descrbed n ths paragraph where the earth s assumed to be sphercal The Global Co-ordnate System The global co-ordnate system s a postve orentated, sphercal system defned by the equator plane and the axs from the center through the Greenwch medan and the North pole as llustrated n fgure

42 A ste s stuaton s tradtonally gven by ts lattude and longtude. The lattude ( s s counted from the equator plane, postve to the North and negatve to the South. Thus we get the frst sphercal co-ordnate as s s 2 (4.2.1 Fgure 4.3 Global co-ordnate system The longtude or medan ( s s counted postve to the East of the Greenwch medan thus equal to the second sphercal co-ordnate. However n e.g. n the US t s common to defne the longtude for a ste postve to the West. Poston of the sun The sun's poston vares due to the earth s ellpsodal orbt, ts tlt from the eclptc and ts rotaton. In the global system s the poston at a gven day and hour defned by a unt vector pontng toward the sun, here called the sun vector. r sun ( 1, sun, sun (4.2.2 It s common to use the declnaton ( to descrbe the angle between the equator plane and the sun vector. The declnaton s defned postve to the North and we have sun sun 2 (4.2.3 The eccentrcty of the earth s orbt s small ( so the declnaton vares almost snusodal wth the ampltude of about and wth ts maxmum at mdsummer. Exact values can be found n e.g. nautcal calendars and a numercal formula s gven n appendx 2. The maxmum varaton over 24 h s about 0.40, thus daly values s accurate enough for our calculatons. 41

43 The angle sun depends on the hour of the day and the day of the year. The stuaton s llustrated n fgure 4.4, a projecton on the equator plane seen from the North. The angles ndcated n clockwse drecton are defned ths way accordng to the defntons of tme. Fgure 4.4 Hour angles n the global co-ordnate system The standard tme (h stt n a tme zone s referred to the yearly mean solar poston at the zone's tme medan ( tm. Some tme zones and correspondng tme medans are lsted n appendx 2. The true solar tme (h tst s referred to each ste's medan ( s and the true solar poston. The Equaton of Tme (h eot depends on the ellpsodal orbt of the earth around the sun and gves the dfference between true solar and standard tme on the tme medan. It s common to gve the equaton of tme n mnutes and the varaton durng the year s between and Exact values can be found n e.g. nautcal calendars and a numercal formula s gven n appendx 2. The maxmum varaton over 24 h s about 29 (0.12, thus daly values s accurate enough for our calculatons. From fgure 4.4 we get followng relatons. h tst h stt h eot s tm (4.2.4 sun s h tst (4.2.5 Sometmes the hour angle ( s used and defned as h tst (

44 4.2.2 The Local Co-ordnate System The local co-ordnate system at a ste s defned as a postve orentated, Cartesan co-ordnate system wth ts axs toward South, East and Zenth. In e.g. the US t s nstead common to use West, South and Zenth. The unt vectors for the axs n the local system are gven by sˆ (1, s / 2, s (4.2.7 eˆ (1, / 2, s / 2 (4.2.8 zˆ (1, s, s (4.2.9 Fg 4.5 Solar poston n the local system The local system s strctly not defned on the poles. However, no formulas n ths chapter wll fal at the poles but we mght want to name the axs as somethng else than South and East. The solar poston n the local system s llustrated n fgure 4.5 and the components of the sun vector n the local system are acheved by the scalar products wth the unt vectors of the axs: s sun r sun sˆ cos( cos( sn( s sn( cos( s ( esun rsun ê sn( cos( ( z sun r sun zˆ cos( cos( cos( s sn( sn( s ( Tradtonally the sun's poston s gven by the alttude ( sun, the angle between the sun vector and the horzontal plane, and the azmuth ( sun, the angle counted postve clockwse from North to the projecton of the sun vector on the horzontal plane. In some lterature s the azmuth counted from the South. If the alttude and azmuth are known, the components of the sun vector can be acheved by s sun cos( sun cos( sun ( e sun cos( sun sn( sun (

45 z sun sn( sun ( When dealng wth the nverse relatons the computer functon atan2 s useful. Ths functon gves the angle (- accordng to the values and sgns of ts arguments and we get sun arctg( e sun / s sun atan 2( e sun, s sun ( Specal care must be taken f the sun s close to zenth where the azmuth s not defned. E.g. the functon atan2 s normally undefned f both argument are zero. 4.3 Radaton at Opaque Surfaces wthout Shadng Incdent Beam Radaton A non shaded part of an outer surface s llustrated n fgure 4.6. The beam rradance s gven for the unt area da perpendcular to the beam drecton and wll ht the area das of the wall. Fgure 4.6 Beam radaton ncdent at a surface. The ncdent angle ( s defned as the angle between the sun vector and the vector normal to the surface. Thus the energy flow per tme unt and unt area of the surface s equal to E G cos( f 0, else 0 (W/m 2 (4.3.1 b, s b If we know the unt vector normal to the surface n ( s, e, z (4.3.3 s s s s the cosne for the ncdent angle can be acheved by the scalar product between ths vector and the sun vector cos( n s r sun s s s sun e e s sun z s z sun (4.3.4 The poston of a surface may be specfed n many ways, e.g. by the wall azmuth as the angle to the South, counted postve to West, and to defne the tlt angle as zero for an upward horzontal surface. However, t seems more naturally to use the compass orentaton and let a normal vertcal wall have a zero tlt angle,.e. the same conventon for the surface s normal vector as for the sun vector. Thus the poston of a surface s here defned by ts azmuth or orentaton ( s counted postve clockwse from the North and ts tlt angle ( s defned as the angle between the horzontal plane and the surface s normal vector,.e. zero for a 44

46 vertcal surface and postve for an upward tlted surface, e.g. a roof. The components of the normal unt vector of the surface are then s s cos( s cos( s (4.3.5 e s sn( s cos( s (4.3.6 z s sn( s ( Incdent Dffuse Radaton Sky radaton Fgure 4.7 Radaton at a tlted surface For a tlted surface only a part of the sky can be seen from the surface, thus the ntensty of the sky radaton at the surface can be wrtten G d (, f < /2 and < /2, else 0. ( where s the ncdent angle for radaton from the drecton (,. The energy flow ncdent at a tlted surface (A s thus E 1, sky, s (, cos( d (W/m 2 ( da d Solvng the last equaton gves E d, s Fs, skygd (W/m 2 ( wth the vew factor from the surface to the sky gven by F s,sky (1 sn 2 ( The sotropc approach can sometmes be avoded even f only G d s known. Threnkeld (1962 gves for clear sky the followng relaton for the ncdent sky radaton on a vertcal, non shaded surface. E d, v G f cos( < -0.2, else d E, cos( cos( cos( G d v d (

47 If radaton accordng to Brown and Isfält (1974 for Malmö, Sweden (55.6 N, E, March 21 at noon s used we have G = 589, G d = 93 W/m 2, G b = 882 W/m 2 and cos( = If an sotropc sky radaton s assumed and the ground reflectance s set to 0.2 and all reflected radaton s assumed to be dffuse the total rradance on the vertcal south facng surface s 833 W/m 2. If we nstead use Threnkeld's formula we get 890 W/m 2. The total ncdent radaton s n ths case ncreased by about 7 per cent ndcatng that the sotropc approach can underestmate the rradance. However, n many stuatons we have to deal wth cloudy sky and shadng and thus Threnkeld s formula s of lmted use. Ground reflected radaton If the beam radaton reflected at the ground s assumed as sotropc, the ntensty of all ground reflected radaton wll be grdg / grd (W/m 2,sr ( In a smlar way as for the sky radaton the vew factor from the surface to the ground can be determned. The result s F s,grd (1 sn 2 ( and the energy flow ncdent at a tlted surface coursed by ground reflected radaton becomes E, grd, s Fs grd grdg (W/m 2 ( d, Absorbed and Reflected Radaton The ncdent radaton at an opaque surface wll partly be absorbed and partly reflected accordng to the propertes of the surface. The absorptance and reflectance are often dependent on the ncdent angle but mght be assumed as constants for common opaque buldng materals, further detals can be found n e.g. Segel (1981. It s also common to assume that all radaton reflected at opaque buldng surfaces s sotropc. Wth the propertes ndependent of the ncdent angle the same propertes are vald for both beam and dffuse radaton and we have E E E E (W/m 2 (4.3.8 a, s ( b, s d, sky, s d, grd, s E E E E (W/m 2 (4.3.9 r, s ( b, s d, sky, s d, grd, s where Ea,s = Absorbed radaton (W/m 2 Er,s = Reflected radaton (W/m 2 α = The absorptance = The reflectance 46

48 4.4 Radaton at Wndows wthout Shadng Propertes for a sngle pane In ths paragraph a sngle pane wthout any nteractons wth other panes s dscussed. The word pane s used for all knd of layers n a wndow, e.g. normal panes, curtans etcetera. Fgure 4.8 Solar radaton at a sngle pane Symbols I Drect radaton Absorptance Dffuse radaton Reflectance D Dffused radaton Transmttance Indces a D b Dd f d w Absorbed Drect ncdent, drect reflected or transmtted Backward Drect ncdent, dffused reflected or transmtted Forward Dffuse ncdent, dffuse reflected or transmtted Wndow property 47

49 Table 4.1 Sngle pane propertes accordng to Fgure 4.8 Forward ncdent radaton Drect Dffuse Boundary If,0 > 0 f,0 > 0 condtons f,0 = Ib,n = b,n = 0 If,0 = Ib,n = b,n = 0 Reflectance Df = Ib,n-1 / If,n-1 Ddf = Db,n-1 / If,n-1 df = b,n-1 / f,n-1 Transmttance Df = If,n / If,n-1 Ddf = Df,n / If,n-1 df = f,n / f,n-1 Absorptance Df = a,n / If,n-1 df = a,n / f,n-1 Backward ncdent radaton Drect Dffuse Boundary Ib,n > 0 b,n > 0 condtons If,0 = f,0 = b,n = 0 If,0 = f,0 = Ib,n = 0 Reflectance Db = I'f,n / Ib,n Ddb = 'Df,n / Ib,n db = 'f,n / b,n Transmttance Db = I'b,n-1 / Ib,n Ddb = 'Db,n-1 / Ib,n db = 'b,n-1 / b,n Absorptance Db = 'a,n / Ib,n db = 'a,n / b,n 48

50 Homogeneous (.e. uncoated glass When thckness, refracton ndex and absorpton coeffcent for a homogeneous glass s gven Fresnel's equatons and Snell's law can be used n order to get the propertes for a sngle pane. These can be formulated as b t r r 1 2 e n 2 2 dn/ b cos( cos( t( 1 r t r r ( 1 t cos ( 2 b b 2 n cos( 2 n cos( 2 b b 1 2 where n = Real part of the refractve ndex = Incdent angle t = Transmsson for a sngle path through the glass = Absorpton coeffcent d = Thckness of the glass r = Reflecton at the surfaces = Transmttance for the panes = Reflectance for the pane = 1, 2 for perpendcular resp. parallel polarzaton (4.4.1 When the thckness, the reflectance and transmttance at normal ncdent are gven the refracton ndex and absorpton coeffcent are acheved wth followng formulas obtaned from the equatons above. b r n where b b r 1 r r ln( / s r = Transmttance for the glass at normal ncdent. = Reflectance for the glass at normal ncdent (

51 Coated and tnted glass Modern glazng systems often uses coated or tnted glass and n these cases a more elaborate approach must be taken e.g. n order to handle optcal thn layers where nterference occur. Pfrommer et al (1995 dscuss a method to handle the calculatons n these cases and gve a good ntroducton to the problems. Roos (1997 suggests nstead a smple polynomal wth only two terms to predct the angular varatons because of the low precson of expermental results and the complex nature of exact Fresnel calculatons Propertes for a multple pane wndow In order to get the solar radaton absorbed n each pane of a glazng system the absorptvtes for drect and dffuse ncdent radaton from both sdes of the wndow are needed. Ths paragraph descrbes how these propertes can be determned when the sngle pane propertes are known. The method was prmarly used n the JULOTTA program and s later also mplemented n the DEROB-LTH program.. A glazng system wth N panes and drect and dffuse ncdent radaton on both sdes s llustrated n Fgure 4.9 where all radaton are totals after all reflectons etcetera has been taken nto account. These are used n Table 4.2 where the wndow propertes are defned. In order to determne the wndow propertes all radaton n all gaps must be known and dfferent methods can be used to calculate these. Below one method s descrbed for forward ncdent radaton as the applcaton on backward ncdent radaton then s trval. Fgure 4.9 Total radaton n a glazng system 50

52 Table 4.2 Wndow propertes accordng to Fgure 4.9 Forward ncdent radaton Drect Dffuse Boundary If,0 > 0 f,0 > 0 condtons f,0 = Ib,N = b,n = 0 If,0 = Ib,N = b,n = 0 Reflectance w Df = Ib,0 / If,0 w Ddf = b,0 / If,0 w df = b,0 / f,0 Transmttance w Df = If,N / If,0 w Ddf = f,n / If,0 w df = f,n / f,0 Absorptance w Df,n = a,n / If,0 w df,n = a,n / f,0 Backward ncdent radaton Drect Dffuse Boundary Ib,N > 0 b,n > 0 condtons If,0 = f,0 = b,n = 0 If,0 = f,0 = Ib,N = 0 Reflectance w Db = If,N / Ib,N w Ddb = f,n / Ib,N w db = f,n / b,n Transmttance w Db = Ib,0 / Ib,N w Ddb = b,0 / Ib,N w db = b,0 / b,n Absorptance w Db,n = a,n / Ib,N w db,n a,n / b,n Fgure 4.10 A pane together wth a fctve pane 51

53 Fgure 4.10 llustrates a part of a glazng system where all panes to the left of pane n are totally neglected. All panes to the rght of pane n form a fctve pane n + 1. All reflectons etcetera n those panes are ncluded n the propertes of the fctve pane. Wth forward ncdent radaton there wll be no backward radaton to the rght of the fctve pane, thus the llustrated radaton only depends on the propertes of pane n and the fctve pane. Propertes for dffuse ncdent radaton The prmarly through pane n transmtted radaton s reflected between the panes and decreased by db,n respectve 'df,n+1 (the reflectance of the fctve pane at each reflecton. The total dffuse radaton n the gap s thus f, n df, n f, n 1 /( 1 db, n cf, n 1 (4.4.3 b n cf, n f, n, (4.4.4 Totally reflected from the combnaton and totally absorbed n pane n are b n 1 df, n f, n 1 db, n b, n, (4.4.5 a n df, n f, n 1 db, n b, n, (4.4.6 The propertes for dffuse radaton ncdent on the new fctve pane n are then defned as df, n f, n / f, n 1 df, n b, n 1 / f, n 1 df, n a, n / f, n 1 (4.4.7 (4.4.8 (4.4.9 Wth n = N (the last pane n the wndow these propertes are obvous equal to the pane's propertes. For n = N - 1 the propertes can be determned wth the above equatons. Ths s then repeated untl the frst pane where the reflectance s equal to the wndow's reflectance. w df df,1 ( Then the forward ncdent radaton can be set to an arbtrarly postve value (e.g. 1 and traced through the wndow. At each pane the wndow's absoptvty and the total forward radaton n next gap can be determned. w df, n df, n f, n 1 / f,0 f, n df, n f, n 1 t Fnally the wndow's transmttance s acheved. w df, n f, N / f,0 ( ( (

54 Propertes for drect ncdent radaton The same method as for dffuse radaton s here used but as the drect radaton may be dffused at any pane the calculatons are more elaborate. The prmarly transmtted drect radaton through pane n s reflected between the two panes and decreased by Db,n respectve 'Df,n+1 at each reflecton. The total drect radaton n the gap s thus I I /( 1 Db, n Df, 1 f, n Df, n f, n 1 n I b n Df, n I f, n (4.4.13, ( If some of the radaton s dffused there may be three sources of dffuse radaton n the gap. Df I, n Ddf, n f, n 1 Df n Ddb, n Ib, n (4.4.15, ( Db n Ddf, n 1 I f, n, ( The total dffused radaton n the gap s thus Df, n ( Df, n Df, n db, n Db, n/( 1 db, n df, n 1 ( Df, n Df, n Db, n /(1 db, n df, 1 Db, n df, n 1 n ( Drect and dffused reflected radaton from the combnaton become I b n 1 Df, n I f, n 1 Db, n Ib, n (4.4.19, ( Db n 1 Ddf, n I f, n 1 Ddb, n Ib, n db, n Db, n, ( and the totally absorbed radaton n pane n I a, n I a, n Df, n I a, n I f, n 1 I a, n Db, n I b, n db, n Db, n The propertes for drect radaton at the fctve pane n are then defned as I I Df, n f, n / f, n 1 I I Df, n b, n / f, n 1 I I Df, n a, n / f, n 1 Ddf, n Df, n / f, n 1 Ddf, n Dbf, n 1 / f, n 1 I I ( ( ( ( ( ( Wth n = N these propertes are obvous equal to the pane's propertes and for n = N - 1 the propertes can be determned wth the above equatons. 53

55 Ths s then repeated untl the frst pane where the reflectance s equal to the wndow's reflectvtes. w Df Df,1 ( w Ddf Ddf,1 ( Then the forward ncdent drect radaton can be set to an arbtrarly postve value (e.g. 1 to avod some dvsons and the forward ncdent dffused radaton to zero. These are then traced through the wndow. At each pane the absobtance and the total forward radaton n next gap can be determned. I w Df, n ( Df, n I f, n 1 df, n Df, n 1 / I f,0 I f, n Ddf, n f, n 1 df, n f, n 1 I f, n Df, n f, n 1 Fnally the wndows transmttances are acheved. I I w Df, n f, N / f,0 I w Ddf, n f, N / f,0 Numercal consderatons ( ( ( ( ( Numercal problems can arse n some equatons above f (1 - db,n 'df,n+1 or (1 - Db,n 'Df,n+1 are close to zero. Ths can only be the case f db,n 1 or Db,n 1 and the problems can be avoded wth followng consderatons. If db,n 1 must db,n be close to zero. Ths mples that all transmsson of dffuse radaton through the pane can be neglected. Nether can any drect radaton be transmtted as dffuse radaton conssts of drect radaton wth all ncdent angles. Thus all radaton n the gap can be set to zero and the dvsons wth (1 - db,n 'df,n+1 and (1 - Db,n 'Df,n+1 are not needed. Even wth db,n < 1 may Db,n be close to 1. But ths requre that Db,n 0 whch mples that all drect radaton n the gap can be neglected and set to zero and the dvson wth (1 - Db,n 'Df,n+1 s not needed. Dependences on wavelength, polarzaton and ncdent angle In order to take these dependences completely nto account when the propertes for a wndow are determned followng calculatons are needed. For each ncdent angle (or nterval, each wavelength (or nterval and two polarzaton drectons all the above calculatons for drect radaton have to be carred out. For each ncdent angle (or nterval the averages of the propertes for the two polarzaton drectons must then, weghted wth the wavelength dependent ntensty of the solar radaton, be ntegrated (or summed over the whole solar 54

56 spectra to gve the wndow propertes for drect ncdent radaton at that ncdent angle. The wndow propertes for dffuse radaton are fnally acheved by ntegraton of the propertes for drect ncdent radaton over the hemsphere. The mportance of treatng the polarzaton n a correct way s shown n Fgure 4.11 where some propertes for a trple pane wndow wth 3 mm clear glass are llustrated. Fgure 4.11 Propertes calculated wth dfferent polarzaton consderatons In the frst case (Detaled polarzaton calculaton s carred out as above. In the other case the propertes for each pane have frst been acheved as the average of the two polarzaton drectons and then these values have been used to determne the wndow propertes. Often t's not possble to get the data needed for all the above calculatons. Furthermore, f some drect radaton s dffused n a glazng system, the propertes for dffuse radaton are needed before the drect propertes are determned. Ths may be solved wth a much more elaborate calculaton than shown above. 4.5 Shadng of Beam Radaton Detaled calculaton of shadng s elaborate. In order to reduce the calculaton efforts t s convenent to separate objects far away from the buldng from those close to t as far objects can be treated n a smplfed way. The term horzon shadng s used for shadng by far objects and for close objects ray tracng and geometrcal methods wll be dscussed. Horzon shadng If the whole buldng can be assumed as shaded when the solar alttude s below a certan value we talk about horzon shadng. The horzon lne may n ths case be descrbed as a horzon angle ( h as a functon of solar azmuth so that G b 0 f sun h ( sun (4.5.1 If the beam rradance s measured on ste, ths type or shadng should not be necessary as no beam radaton wll be detected when the sun s below the horzon. However, many tmes we need to work wth clmate data from meteorologcal statons some dstance away from the buldng ste or we work 55

57 wth theoretcal values for the solar radaton, thus ths type of shadng occurs. There s also the stuaton when the radaton s measured on the roof of the buldng and lower parts of the buldng are shaded. In some cases a surface n a buldng may be assumed as totally shaded or not shaded at all, e.g. by a hll or a large buldng far away compared wth the dmensons of the surface n queston. In these cases a specal horzon lne for that surface and can be used n order to avod more elaborate calculatons Ray tracng A common technque to deal wth shadng screens s to dvde the recevng area nto grds and then trace the ray from the sun to the center of each grd. If a ray s shaded by any screen the whole grd s assumed to be shaded and vce versa. If each surface that the ray passes s transparent wth the transmttance k (,k 0 where the ncdent angle ( can be dfferent at dfferent surfaces, the rradance on the grd can be expressed as E, s Gb cos(, s k (, k (W/m 2 (4.5.2 k The dsadvantages of ths method are the rather elaborate calculatons nvolved f the accuracy s mportant and the extreme error that can occur f we try to reduce the calculaton work by usng too few grds. E.g. f we want to have an accuracy of 1%, we need 100 grds and for each of these we have to examne f the beam s passng nsde or outsde the surfaces that mght shade the recevng area. A typcal example of error that can arse wth ths approach s found n some versons of the DEROB program. Each surface s dvded nto nne grds and a very small surface whch shades the center of one grd gves the result that about 11 % of the surface s shaded. In the same way could a very small wndow result n beam radaton on 11 % of an nner surface, thus the solar heat gan s consderably overestmated. In DEROB-LTH the stuaton s to some extent mproved, 25 grds are used and the solar radaton absorbed at the nner surfaces s adjusted accordng to the radaton transmtted through the wndows. Geometrcal methods Better accuracy and faster calculatons than wth ray tracng technque can sometmes be acheved by geometrcal methods. An early example of a geometrcal method s the shadng routne n TRNSYS, the subroutne type 34, where two sde wngs and an overhang are treated. To get a more general procedure to deal wth shadng screens, the followng method was suggested by the author. The method has been mplemented n the DEROB-LTH program and seems to work satsfactory. A smlar method s also descrbed by Johnsen, K., (

58 Fgure 4.12 A shaded surface Fgure 4.12 llustrate a plane wth a recevng area R shaded by screens gvng the shadows Sj. The shadows are projectons of the screens for a gven drecton of radaton and an ncdent angle on the recevng area less then /2. The shaded part of the recevng area s the ntersecton between the area and the unon of the shadows. If N shadows are consdered, the sunlt area can be expressed by SLA( R, S A( R A( R N (m 2 (4.5.3 N S j j 1 where A(X s the area of the regon X. The unon of the shadows may be complex to represent. As seen n fgure 4.12 the unon can consst of several regons of whch some can contan holes. In order to avod unons, the last term can be developed as follows. A( R N S j j 1 A ( R S ( R S N N j 1 1 j A( R S A ( R S ( R S A( R S N N j j 1 N 1 N 1 SLA ( R S N, S N 1 A( R S j (m 2 (4.5.4 j 1 Repeated development of the last term gves the fnal result SLA( R, S N A( R SLA( R S j, S j 1 where, as S0 s empty SLA( X S, S A( X S 1 0 J N 1 1 N j 1 1 j (4.5.5 (4.5.6 We can also observe the followng two relatons, whch are useful when the recursve formula should be used. SLA( X S j, S j 1 0 f X S j 0 (4.5.7 SLA ( R, S N 0 f R S j R for any j. (

59 In order to get a smple representaton of all ntersectons between two regons t s useful to allow only convex polygons,.e. polygons wth all nner angles less than 180. Ths restrcton s not crtcal, most buldng parts are of convex shape. In some cases however, a surface may be dvded nto two or more regons to fulfll ths requrement. In other cases we have a recevng area wth holes, e.g. a facade wall wth wndows. Here, the wndows can be treated as shadows on the wall. It s obvous that the shadow from a convex screen s convex, thus a shadow lke Sj n fgure 4.12 cannot exst wth ths restrcton. It s also obvous that a non empty ntersecton between two convex regons s convex. Followng strategy can thus be used n order to determne the ntersecton between two regons. If the ntersecton s not determned by one test, the next test must be carred out. 1. If the maxmum x-value (y-value for the vertces n one regon s less than or equal to the mnmum x-value (y-value for the vertces n the other, the ntersecton s empty. 2. If all vertces of one regon are placed on or outsde the same sde, or ts extenson, of the other regon, the ntersecton s empty. 3. If all vertces of one regon are nteror or boundary ponts of the other regon, the ntersecton s equal to the frst regon. 4. If any vertex n one regon s an nternal or boundary pont of the other regon, ths vertex s chosen as the frst vertex of the ntersecton. Else the sdes of one polygon are checked one by one untl an ntersecton between that sde and the boundary of the other polygon s found, thus gvng the frst vertex. Then the leftmost way of the boundares are followed n the postve drecton untl the ntersecton s closed. There are no major problems wth the recursve formula, the maxmum recursve levels needed s equal to the number of shadows. For each level t s only necessary to stack one area and one polygon and f all shadows fallng totally outsde the recevng area are excluded, the numbers of shadows s normally small. 4.6 Shadng of Dffuse Radaton Opaque Surfaces A common method to deal wth shadng of dffuse radaton s to use vew factors to estmate the ncdent radaton. The vew factor s defned as the fracton of the dffuse radaton leavng a surface that arrves at another surface when no reflectons are assumed. If a sphercal coordnate system s related to surface 1 the vew factor between ths surface and another can be wrtten 58

60 F A 1 A1 cos( d da where A1 = Area of surface 1 (m 2 F1-2 = Vew factor between surface 1 and 2 1 = The angle over da1 where surface 2 can be seen (sr The sky radaton ncdent at a surface s then gven by (4.6.1 E d, s Fs, skygd (W/m 2 (4.6.2 where Fs,sky s the vew factor between the surface and the sky. The absorbed and reflected radaton can be determned wth Eq and f the absorpton and reflecton s assumed to be ndependent of the ncdent angle Wndows When the radaton at a wndow s treated t s common to calculate the ncdent radaton as n the last paragraph and then use the propertes determned for a not shaded, vertcal or horzontal wndow. A problem wth ths method s llustrated for two horzontal wndows shaded by a cone as llustrated n Fgure In the frst case the wndow s placed below the cone and n the other nsde the cone. In both cases the wndow area s nfntesmal, thus no shadng s assumed n the frst case when > /4 or n the other when < /4. Fgure 4.13 Incdent sotropc sky radaton at two shaded wndows 59

61 Wth use of Eq the vew factors become F sky 1 2 1, cos( sn( d d 0 / 2 / 4 1/ 2 (4.6.3 F sky 1 2 2, cos( sn( d d 0 / 4 0 1/ 2 (4.6.4 For a sngle pane wndow wth clear 3 mm glass havng a refracton ndex of 1.52 and an absorpton coeffcent of 19.6 m -1 the dffuse transmttance determned for a not shaded horzontal wndow s Thus the estmated flow transmtted through each of the wndows wll be qt IdH IdH (W/m 2 (4.6.5 However, all ncdent angles n the frst case are greater than /4 and n the second less than /4. Ths ndcates that less radaton s transmtted n the frst case than n the other and the result seems to be erroneous. In order to examne the error, a more exact calculaton s performed. The energy flow transmtted through a wndow n Fg can be wrtten qt t ( cos( d D ( cos( d (W/m 2 (4.6.6 where t( = The ntensty of transmtted radaton (W/sr,m 2 D( = The transmttance for drect radaton A numercal soluton of Eq for the two cases wth the same type of wndow as above gves q q t t, , I I dh dh (W/m 2 (4.6.7 (W/m 2 (4.6.8 The errors when usng Eq are n these examples around 9%. The result shows that t s not enough to establsh the fracton of dffuse radaton reachng a shaded wndow, the drecton s also mportant even when the sky radaton s assumed to be sotropc. The same type of error wll, to a less degree, occur for a not vertcal or horzontal wndow. The shadng calculatons for dffuse radaton are only needed to be carred out once for each wndow and the results can then be used at each tme step durng a smulaton. Thus elaborate calculatons are not mportant to avod and n order to mprove the calculatons followng method can be used. 60

62 Fgure 4.14 Dffuse sky radaton at a tlted wndow An eventually shaded and tlted wndow s llustrated n Fgure The ntensty of the dffuse sky radaton now depends on the drecton as t may be shaded or not. G d (, f < /2, < /2 and non shaded, else 0. (4.6.9 where s the ncdent angle. The energy flow ncdent at the wndow can thus be wrtten Q (, cos( (, d da A where A The area of the wndow (m 2 (W ( The ncdent angle for radaton from the drecton (, (rad The part of the sky from where radaton can reach the wndow The radaton transmtted through the wndow becomes Q (, ( (, cos( (, d da (W ( t A where ( (, s the transmttance of the wndow for drect radaton. If As(, s the sunlt area as defned n paragraph 4.5 Eq can be wrtten Q t G d 2 0 / 2 As, ( (, cos( (, sn( 0 1 ( d d ( The reflected and absorbed radaton can be determned n smlar ways and the equatons can be used for any surface, shaded or not. E.g. an non shaded wndow has As(, = A f < /2 and < /2. The method has been mplemented n the DEROB-LTH program and seems to work satsfactorly. 61

63 4.7 Dstrbuton of Reflected Radaton For each buldng surface, all other surfaces may be assumed as shadng screens, eventually partly transparent. Thus the prmarly ncdent beam radaton on all surfaces can be determned by the methods descrbed above. In order to determne the dstrbuton wthn a buldng, we also have to deal wth the radaton reflected from the surfaces. It s common to assume all reflected radaton as dffuse and use the treatment dscussed n the followng paragraphs. In many cases, ths approxmaton s acceptable, but cauton has to be taken n some cases, e.g. a room glassed on more than one sde. If the solar alttude s low, most of the radaton wll obvous be transmtted out from the buldng but assumng dffuse reflecton leads to the concluson that most wll reman nsde the room. The treatment of specular reflectons n a room wll not be dscussed n ths report but s partly dscussed n e.g. Segel and Howell (1981. If all surfaces are assumed to opaque and optcally black or dffuse-gray,.e. all emtted or reflected radaton s assumed to be dffuse followng method from Gebhert (1971 can be used to determne the dstrbuton of the reflected radaton. Fgure 4.15 Reflectons between surfaces Fgure 4.15 llustrates the dstrbuton of the total radaton Q from surface to other surfaces. Ths radaton may be sky radaton prmarly reflected at the surface or emtted LW radaton and wll after all reflectons be absorbed n dfferent surfaces. The dstrbuton factor Dj s defned as the fracton of Q that after all reflectons s absorbed n surface j. Ths factor can be determned n the followng way. Q wll before any reflectons has been taken nto account reach other surfaces but not necessarly all. The vew factors can be used n order to determne how much wll reach each surface, thus FjQ respectve FkQ wll reach surface j and k. 62