DYNAMIC MODELLING AND ELEMENTS OF VALIDATION OF SOLAR EVACUATED TUBE COLLECTORS

Transcription

1 Ninth International IBPSA Conerence Montréal, Canaa August 5-8, 005 DYNAMIC MODELLING AND ELEMENTS OF VALIDATION OF SOLAR EVACUATED TUBE COLLECTORS Jean Philie PRAENE, François GARDE, Franck LUCAS Laboratoire e Génie Inustriel, Equie Génie Civil Thermique e l Habitat IUT e Saint-Pierre, Université e La Réunion, 40 avenue e Soweto 9740, Saint-Pierre, Ile e La Réunion, France raene@univ-reunion.r ABSTRACT This aer eals with the ynamical moelling o a solar evacuate tube collector uner variable weather conitions. Ater escribing the mathematical moel, the aer resents elements o valiation. In a irst ste, the theorical moel is valiate against the exerimental results. Then a arametric sensitivity analysis is alie to the moel. Such a stuy is very interesting or moellers in orer to etermine the relative imortance an the nature o the eect o the arameters. It is imortant so as to imrove the moel by controlling this aramaters or accurately measuring it.. INTRODUCTION Solar builings technologies use the clean ower o the sun to heat cool an ower builings. The starting oint o most active solar energy systems is solar collectors. Solar cooling is one o the most attractive alications, the incience o solar energy an cooling requirements are inee aroximately in hase. Evacuate tube collectors are articularly aroriate or this alication. These collectors erorm well in both irect an iuse solar raiation an oer the avantage that they work eiciently with high absorber temeratures. The inal aim o this stuy is the moelling o a global solar cooling system rom the heat rouction to the cooling suly coule with the builing loas. The irst ste o the work resente in this aer is the moelling o the heat rouction rovie by solar collectors. The litterature contain numerous works on the moelling o solar collectors. These moels eveloe have ierents levels o comlexity. Usually, solar collectors are escribe by stationnary moels, consiering the collector working uner steay-state conitions. These aroach are generally base on the work o Klein (Klein et al., 974). The rincial avantages o this tye o moels are to be simle an have an high see calculations.however, it is well known that large overreiction may occur by stationnary moel, (Isakson an Eriksson, 99). When couling a stationnary moel with comonents whose behaviour een on time cause aitionnal errors to energy yiel reictions, (Schnieers, 997). A ynamic aroach is more interesting in several cases: control strategies, ynamic testing roceures, couling with others elements. Particulary, reict the behaviour o collectors or a time ste much than hourly ste, a ynamic moelling bring more inormations concerning the collector. The rincial iea unerlying the work reorte here is the ollowing. The collector is moelling uner a short time ste in orer take into account the variation o the meteorogical arameters. This level o escrition allows us to aly sensitivity analysis an unerstan wich aramaters have a signiicant inluence on the oulet temerature o the collector. In the ollowing, all hysical henomena are investigate searately to escribe the moel. The ynamic behaviour o the moel is veriie thanks to numerical tests an measures comarisons.. DYNAMIC DESCRIPTION OF THE COLLECTOR MODEL The moel eveloe corresons to irect low collector. It is not aroriate to vacuum tube use seciic lui in heat ies to heat the collector inlet lui by an exchanger. The tye o solar collector moele consists o six vacuum tubes. The heat transer lui lows in a coer U-tube which is wele to a narrow lat absorber. Thus, the inlet an the outlet are at the same en o the evacuate tube. In orer to moel the evacuate tube collector, a number o simliying assumtions have to be mae. Most o these have been reviously escribe by Duie & Beckmann ( Duie, 99). Perect insulation at the eges o the collector is assume. No heat is suose to be transorte in the lui moving irection, conuction is neglecte. The graients insie the glass cover an the absorber late are assume to be negligible. As the main objective lies on collector ynamical behaviour moelling, the eect o incience angle is also neglecte. As the collector stuie is a vacuum tube, ree convection insie the glass tube is not taken into account. In the ollowing sections, all heat transer

2 occure in the solar collector are eine. Figure resent the general escrition o thermal transer in the solar collector. ( τα ).G T g T T a T sky temerature obtain or segment (x i- ) is the initial or inlet lui temerature or segment x i. The inal oulet temerature is obtaine by connecting the N segments o the collector. As roose by Henning (995), the artial ierential equation (3) can be written as an orinary equation using the metho o charatesristics, Hollan an Liais (983). The velocity u o the lui is assume to be constant, thus equation (3) becomes : T C = h ( T T ) (4) t Finally, the new set o equations can be illustrate with the thermal networks shown in igure. O Figure General escrition o the solar collector moel. Each comonents o the solar collector (the lui, absorber late, glass cover) are consiere searately. Each element has his own heat caicity. Collector ierential equation system The starting oint o the moel is a mathematical escrition roose by Kamminga (985). The moel consists on three noes corresoning to the lui, the absorber late an the transarent glass cover. It is consiere that the temerature o the lui is a unction o x. The lui is moving in a single channel with the velocity u, along x-axis. It results a 3-noe collector moel given by the ollowing ierential equation system : T C = εσ T T + h T T + T T t 4 4 ( ) ( ) 4 4 εσ ( ) g g g sky g g a a g g g T C = G + T T + h T T t 4 4 τα ε gσ ( g ) ( ) T T C + u = h T T t x ( ) () () (3) The system given by this three equations can be solve using Fourier transorm o the time eenent set o ierential equations () (3). We have chosen to numerically solve this system using inite ierence metho. In this case, the collector is eine as single lui channel, which is ivie into N segments. The ierential equation system is solve or each segments in the time omain using a 4 th orer Runge-Kutta metho. The inal oulet T Glass cover Absorber late Convective heat transer Raiative heat transer x T sky T a h sky T g C g Figure Thermal networks or the 3-noe moel. Thus all the comonents are escribe in the time omain. Convection heat transer rom cover ue to win The convection heat transer coeicient ue to win rom McAams (954) is generally assume: hw ( τα).g = 5, , 8. v (5) This correlation are generally use. In the case o vacuum tube collector it is also ossible to use the relation rom Hilert (933) which escribes the external lui low istribution on a cyliner. Force convection heat transer between absorber an lui low In a laminar low region, the ormalism use has been irst escribe by Colburn (933) Nu = 0.03 Re Pr (6) D In the case o turbulent low region (Re > 6000), Koo(999) recommane to use the correlation obtaine rom the relationshi o Gnielinsky : C T hg a h g h T C O x

3 Glass cover τ ( α ) τ ( α ) τρ ( α ) τρ ( α ) τρ Absorber late τα ( ) τα α ρ ( ) τα α ρ Figure 3 Absortion o solar raiation by absorber late uner cover Nu = 8 ( ) Re 000 Pr Pr (7) The riction actor o Darcy or the above tubes may be obtaine rom : ( ln Re.64) = (8) Long wave raiation transer between the glass cover an sky The emissivity o sky is assume to be equal to, thus the long wave lux may be written as: combination rather than as a rouct o two roerties. The rouct ( τα ) is the result o multile relection o iuse raiation so that the raction o the incient energy inally absorbe is given by: τα = = () ρ ( τα ) τα [( α ) ρ ] n= 0 n ( α) This escrition is illustrate in igure 3. The subscrit reresents the iuse raiation in the vacuum tube. 3. EXPERIMENTAL AND SIMULATION φ = ε σ. T (9) 4 LW sky sky For the sky temerature, Boyer (993) has roose to use a simle linear relationshi eens on air temerature: T = T a (0) sky a The constant a has been eine by otimization, it eens on the lace that the simulation is suose to reresent. Gare (997) suggeste to use a = 6, or simulation occur in Reunion. Exerimental setu uner natural conitions Data rom testing at the University Test Fiel in Reunion ( S, 55 E) have been use. The exeriments occur uner natural troical humi conitions. The collector consiere was teste on a ixe rame as illustrate by the hotograh in igure 4. Solar raiation The solar raiation heat lux absorbe by the absorber late surace is eine by: S = ( τα ) G () As suggeste by Duie (99), the transmisttanceabsortance rouct ( τα ) shoul be thought o as a symbol reresenting a roerty o the cover-absorber Figure 4 The exerimental set u

4 The collector loo has been oerate with continuous low hours a ay rom sunrise to sunset an in some cases by night. The inlet an outlet temerature are measure. Environmental arameters are also measure as global solar irraiance, air temerature an win velocity. The exerimental setu is use or in situ characterization o the collector accoring to the Euroean Stanar CEN It also constitutes a atabase or the valiation o the solar collector moeling. The tests erorme allow us to have a escrition o the collector uner steay state conitions. Thus, the useul energy an the eiciency o the collector are evaluate. Collector moel reictions The solar collector moel has ive inut arameters as ambient temerature, solar irraiation, mass low, inlet temerature an the win velocity. All these arameters are rea rom ata iles. The moel calculates the outlet temerature an comares it with the measure outlet. The atabase is a minute ste ata acquisition. The global irraiance in the collector lane an ambient temerature or the examine ays are resente in igure 5. Irraiance (W /m ²) Time (min) Air tem erature ( C) Time(min) Figure 5 Weather conitions or the examine ays. The results o simulation are rawn on igure 6. The irst basic ste in valiation is a measure moel reictions comarison. The ynamic moel oes not much ier in the reiction o the outlet temerature. In the case o solar vacuum tube moel comuting, the encountere iiculty is the initialization at each time ste o the absorber late temerature. This temerature is not measure, as the collector is uner vacuum. Thus at the beginning o the simulation, there is a signiicant ierent between measures an moel. When simulating, the ynamic behavior o the solar collector, this ste have to be consier as initialization hase. Thus, the values obtaine at the beginning o each simulation will not be taken into account. Outlet temerature ( C) moel ata Time (min) Figure 6 Comarison o measure an moele outlet temeratures or the vacuum tube collector. In orer to make u to this question, two solutions aroach is actually uner testing. The irst one is a theoretical aroach, consiering artiicial neural networks to moel the ynamic behavior o the absorber late accoring to metrological ata. The other aroach is the instrumentation o a secon collector, incluing measures o absorber late temerature. In this way, it will be ossible to initialize all arameters at each time ste. SENSITIVITY ANALYSIS As eine by Saltelli (999), the objective o sensitivity analysis (SA) o moel outut is to ascertain how a given moel (numerical or otherwise) eens on its inut actors. This way it is also ossible to etermine i the moel oes not exhibit unexectely strong eenencies uon non inluential arameters This analysis is an imortant ste in the veriication an valiation o moels. Thus SA hels to unerstan the unamental mechanisms unerlying the behaviour o the moel an interactions between the ierent arameters. There are inee ierent tyes o SA, an a numerous techniques have been eveloe. In the resent work, we move rom a metho which turns on two consecutives techniques: A screening test roose by Morris (99), which allows ientiying qualitatively, the relative inluence o arameters. FAST metho (acronym o Fourier Amlitue Sensitivity Test) to etermine the inluence o actor an its nature. The Metho o Morris The basic iea o this metho is to etermine, within a reasonable uncertainty, which inut arameters coul be consiere to have a signiicant inluence on the outut. The main avantage o this metho consists in its short comuting time, regaring the

5 number o simulations. However this statistical metho oes not allow arranging the arameters in orer o inluence. In this work, the screening metho was use to hel us on choosing an ocusing on the most imortant arameters in the secon art by alying FAST metho. The Morris metho is base on exerimental lans that are comose o iniviually ranomize oneactor-at-a-time (OAT). Thus each actor will take only ossible values. Consiering a -imensional actor vector X o the moel, the outut isy( X,..., X,..., X ). i For a given value o X, the eect o the ith inut actor is eine as: () i Y( X,..., X +,... X ) Y( X ) = () () () () () i i () i (3) The irst ste o the metho is the stanarization o the actors. I X [ u, l i i i], thus the stanarize exression o X i is x = ( X b )/ a, with i i i i b = ( u + l )/ an ai ( u l )/ i i i = i i. Alying this stanarization, all arameters are in the same interval or examle [0,]. The secon ste is the choice o k i values between [-,], so that x i.k i >, here the non linearity o the resonse is taken into account. Then a ranomly selecting values o x () allows the calculation o the corresoning X () an inally the resonse Y () associate. The arameter X is ranomly moiie between the k ossible values an its eect is evaluate by equation (3). (r) The roceure is reeate r times. Finally a i matrix is obtaine: () () () () () () () () ( r ) ( r ) ( r ) ( r) ( r) ( r) ( r) The ientiication o the relative imortance o actors are shown by a grahical analysis o the stanar eviation σ i versus the mean o the eects µ i ue to X i : µ µ µ σ σ σ Figure 7 shows the results obtaine on the solar collector moel. The total comutational cost o the exeriment is n=r(+) runs. The ata analysis o igure 7 shows that there are our rincial arameters which have a signiicant inluence: transmittance-absorbance, absorber surace, caacity o lui an absorber. The eect o these actors are not correlate an non linear. As we can see on igure 7, the other arameters have a mean eect value which is 0. Generally, the screening hase is very helul when the moel have a lot o inut actors; it allows eliminating arameters that o not have any inluence. As we have a short number o arameter or the FAST metho all o them will be use. Thus we coul veriy that the two methos on the same signiicant arameters. The FAST Metho This metho is an exerimental lan in the sectral omain, eveloe by Cukier & al. (973). It allows the comutation o the raction o the variance o a given moel outut which is ue to each inut variable. The guiing iea unerlying the metho is to aly the ergoic theorem as emonstrate by Weyl (938). Consiering a one outut (y) moel with inut arameters y = (x,x,,x ), the arameter are samling in their own range o variation. Each arameter x h inclues a erioical unction G h characterize by a requency w h. The requency is the signature o the arameter. Thus, the samling o the arameter x h can be exresse by the ollowing ormula: x h,k = G h (sin(w h s k )) (4) The transormation unction G h is generally chosen to assure a goo reresentation o the wie range o arameters. That means that the variable x h has to be samle ollowing a recise given ensity robability (corresoning to the uncertainty on its value). Mara (00) roose to samle the arameters with the ollowing manner: x h,k = x h,0 + δ h sin(w h s k ), with s k = πk/ns (5) Where k reresents the simulation number (k = à Ns), xh,0 is the basis value o the arameter h an δh is chosen such as x h,k Є [x h,0 - δ h, x h,0 + δ h ], Ns is the number o simulations. The Fourier transorm o the outut o the moel y is calculate an the sectrum is rawn. We ientiy the requencies which aear at each ste o the grahical analysis. The requency assigne to each arameter is inexe in table. The next ste is the ientiication accoring to their requency o the most imortant arameters

6 Absorber late surace τ g α Stanar Deviations C C Means Figure 7 Distribution o elementary eects o the inut actors Factors Range o x h,0 Frequency variation C [7000,9000] C [5000,7000] C g [4500,5500] h - [00,40] 0 63 h g-a [5,5] 0 8 ε g [0.8, ] ε sky [0.8, ] α g [0,0.] S [,].3 8 τ g α [0.6,] Table Frequency associate to the inut arameters moel. Figure 8 shows the results o FAST metho. Choosing o requency allows taking into account trigonometric roerties o sinus unction. The eaks generate at even requencies are ue to o orer interaction, quaratic eect or even rincial eect. The o eaks requencies are obtaine o rincial eect, or even orer interaction between arameters. The imortance o a arameter is correlate with the intensity o the eak. The visual analysis o the sectrum shows clearly that the most imortant arameters are the transmission-absortion (4) an the surace o the absorber late (8). The eak at the requency 60 is ue to a secon orer interaction between (τ g α ) an S. Concerning the heat caacity o absorber an the lui, they seem to have a little inluence uon the observe outut. What we observe with SA conirms the reality o the hysical henomenon. The act that the eect o the (τ g α ) rouct has the biggest art on the global variance o outut, has to be associate to the quantity o solar energy receive by the absorber late. Each arameter eect coul be associate to be the hysical henomenon. In this way, it is ossible to veriy i all imortant henomena are really taking into account. Thus, mixing the two analysis o the screening metho, it is ossible to classiy the ierent actors in orer o imortance. CONCLUSION In this aer a mathematical moel was eveloe an use or simulation o the ynamic behavior o evacuate tube collector. For the simulation, an ODE system was establishe an comute in MATLAB. To ientiy the system variables, a

7 measurement uner natural conitions was carrie out I Amlitue o Fourier I *49-4* Frequency Figure 8 Ientiication accoring their requency o the inluential arameters The moeling results achieve showe a airly goo coincience with the measurements. This measure moel reiction comarison was the irst ste o elements o valiation. The secon aroach or valiation was the use o sensitivity analysis which aims to quantiy the relative imortance o inut arameters or actors in etermining their intensity. This analysis has irst etermine qualitatively which arameters have no imact on the outut resonse. Then a sectral analysis establishe that the most imortant arameters is the transmission absortion rouct. SA allows us to be sure that hysical henomena that are not inluential in theory are not taking into account, ue to the act that all arameters can be associate to a hysical henomenon. This analysis constitutes a crucial ste in roviing elements o valiation or the moel. ACKNOWLEGDMENTS This work was suorte by the ADEME (the French agency or environment an energy management) an the Regional Council o la Reunion. This exerimental setu was artly inance by GIORDANO INDUSTRIES. NOMENCLATURE C lui heat caacity (J/m².K) C g Heat caacity o glass cover (J/m².K) C Heat caacity o absorber (J/m².K) Friction actor o Darcy G Global solar irraiance in the lane o the collector (W/m²) h - heat transer coeicient lui absorber (W/m².K) h g-a heat transer coeicient glass ambient (W/m².K) h sky heat transer coeicient glass sky (W/m².k) h w win convection heat transer coeicient (W/m².k) T a Ambient temerature ( C) T Flui temerature ( C) T g Temerature o glass cover ( C) T Absorber temerature ( C) T sky Sky temerature ( C) u lui velocity (m/s) v win velocity (m/s)

8 Greek symbols α absortivity coeicient variation on inut arameters ε emissivity σ Stean-Boltzmann constant (5.67x0-8 W.m -.K -4 ) σ i Stanar eviation Φ LW Long wave lux (W/m²) ρ Diuse relectance o the cover system µ i means o arameters values Subscrits lui g glass absorber late REFERENCES Boyer H. Concetion thermo-aéraulique e bâtiments multizones. Proosition un outil à choix multiles e moèles. Ph thesis. : INSA e Lyon, (993). Colburn A.P., Trans. AlChE, Vol. 9,.74, (933). Cukier R.I., & al. Stuy o the sensitivity o coule reaction systems to uncertainties in rate coeicients, Part I Theory. Journal o chemical Physics. Vol. 59, (973). Duie J.A. an W.A Beckman, Solar Engineering o thermal rocesses, n eition (99). Gare F. Valiation et éveloement un moèle thermo-aéraulique e bâtiments en climatisation assive et active. Intégration multimoèle e système., PhD thesis. : Université e La Réunion, (997). Henning H.M. A collector harware simulator theoretical analysis an exerimental results. Solar Energy, Vol. 55, N, , (995). Hollan & Liais - "Comuter methos or solving ynamic searation roblems" - Mc Graw Hill, (983). Isakson P., L.O. Eriksson, A ynamic solar collector moel or TRNSYS, revise version o aer resente at Noric Solar Energy R&D Meeting, Borlange. (99). Kamminga W. Exeriences o a solar collector test metho using Fourier transer unctions. International Journal Heat Mass Transer, Vol. 8, N 7, , (985). Klein S.A, J.A. Duie an W.A Beckman, Transient consierations o a late-late solar collectors, ASME, J.Eng. Power 96A, 09-3 (974). Koo J.A, Develoment o a lat-late solar collector esign rogram, Master o science University o Wisconsin-Maison, (999). Mara T. Contribution à la valiation un logiciel e simulation e thermo-aéraulique u bâtiment: Proosition e nouveaux outils aie à la valiation. PhD Thesis, Université e la Réunion. (000). McAams W.H., Heat Transmission, 3 r en, Mc Graw Hill, New York (954). Morris M.D., Factorial samling lans or reliminary comutational exeriments, Technometrics, vol. 33, n, (99). Saltelli A., Tarantola S., Chan K.P-S, A quantitative Moel-Ineenent Metho or global sensitivity analysis o moel outut, Technometrics,, vol. 4, n, February (999). Schnieers J., Comarison o the energy yiel reictions o stationary an ynamic solar collector moels an the moel s accuracy in the reiction o a vacuum tube collector, Solar Energy, Vol. 6, n 3, 79-90, (997). Weyl H., Mean motion, Am. J. Math 60, , (938)