New approach for Finite Difference Method for Thermal Analysis of Passive Solar Systems

Transcription

1 New appoach fo Fte Dffeece Method fo hemal Aalyss of Passve Sola Systems Stako Shtakov ad Ato Stolov Depatmet of Compute s systems, South - West Uvesty Neoft Rlsk, Blagoevgad, BULGARIA, (Dated: Febuay 7, 005 Mathematcal teatmet of massve wall systems s a useful tool fo vestgato of these sola applcatos. he objectves of ths wok ae to develop (ad valdate a umecal soluto model fo pedcato the themal behavou of passve sola systems wth massve wall, to mpove kowledge of usg dect passve sola systems ad assess ts eegy effcecy accodg to clmatc codtos Bulgaa. he poblem of passve sola systems wth massve walls s modelled by themal ad mass tasfe equatos. As a bouday codtos fo the mathematcal poblem ae used equatos, whch descbe fluece of weathe data ad costuctve paametes of buldg o the themal pefomace of the passve system. he mathematcal model s solved by meas of fte-dffeeces method ad mpoved soluto pocedue. I atcle ae peseted esults of theoetcal ad expemetal study fo developg ad valdatg a umecal soluto model fo pedcato the themal behavou of passve sola systems wth massve wall. I. INRODUCION he cocept of passve sola systems s well-kow method fo use of sola eegy as a souce of heatg buldgs. hee s a vast lteatue o ths techology, but eal objects (houses wth passve sola systems ae stll ae. he majo mpedmets to cease maket peetato of passve sola systems s the lack of avalable fomato ad expeece data fo the effcecy ad costuctve paametes of passve sola elemets. he ma cocept of dect passve sola systems s ombe-mchel wall. Most expemetal ad theoetcal data, publshed o ombe wall pefomace ae fom of oveall buldg pefomace. Data of oveall pefomace s of lmted use, as t oly povdes seasoal estmates of heat gas fo specfc buldg desgs, wall pattes ad clmates. Because of the lage umbe of paametes ad the wde age of weathe codtos, whch fluece the opeato of massve walls, the assessmet of the themal behavo eques the use of themal smulato techques. Lteatue evew shows that the poblem of passve sola systems wth massve walls s odaly modelled by themal ad mass tasfe equatos [4,7]. As bouday codtos fo the mathematcal poblem must be used equatos, whch descbe fluece of weathe data ad costuctve paametes of buldg o the themal pefomace of the passve system. he mathematcal model, composed fo the massve wall pefomace, s usually vey complcated ad fo solvg the mathematcal system of equatos t s ecessay to apply a dffeet set of assumptos. he pupose of ths atcle s to peset the esults of theoetcal ad expemetal study fo developg ad valdatg a umecal soluto model fo pedcato the themal behavou of passve sola systems wth massve wall.. MAHEMAICAL MODEL he smulato scheme of typcal passve sola system wth massve wall s show Fg.. he massve wall s usually mouted o the south facade of the house. It compses thee layes: a taspaet cove (oe o two glasses o plastc plates, a massve wall (masoy, cocete ad a gap betwee taspaet cove ad massve wall. At the bottom ad the top of the massve wall, thee ae vets fo allowg a a cculato betwee a gap ad oom space. he exteal taspaet cove tasmts sola adato, but holds back heat. Wall suface s pated black at ts oute sde ad act as a absobe of sola adato. It stoes heat fom the day ad eleases t wth tme to the oom space by adatve heat tasfe. he a laye betwee taspaet cove ad massve wall s heated cotact wth the wall suface, ses ad cculates towads the oom (whe the vets ae ope.

2 Rooml x Massve Wall A cculato z A gap Fg.. Scheme of ombe Wall System d Glass coves Vet, a ( ( τ x z I summe, habtats close the vets ad a cculato s stopped dug the day. Dug the ght, vets uppe ed of glazg ca be opeed ad atual vetlato oom space ca be ogazed. he themal aalyss of such system s vey complcated. Mathematcal model s based o the taset pefomace of the system. It compses eegy balace equatos, wtte fo each elemet of the system. Sce the wall s take to be lage ( compaso wth wall thckess, the tempeatue vaato y - decto (wall wdth wll be eglected, ad oly two-dmesoal poblem ca be cosdeed - the hght (z decto ad thckess (x decto. he goveg eegy cosevato equato of heat tasfe massve walls s: whee efes to the tempeatue the wall, τ s tme vaable, x ad z space vaables ad a s a mateal costat. he bouday codtos, eeded fo the soluto of eq (, ae deved fom a eegy balace fo elemets of passve system. he cove glazg s assumed to absob o sola adato ad oly exchages heat by covecto ad adato wth the wall suface ad the ambet. Heat tasfe ate elemets of the massve wall s gve by: at x δ (a e wall suface d λ dx h croom ( w 0 - h at x 0: (a oute wall suface Room ( w 0 - d - λ q ( τα h cgap ( ag - w h gap g - w s e dx at e glass cove: wall ( ( h h hcgap ag g gap w0 g g g (4 at oute glass cove: h h h g a g sky g g, c (5 whee w, w0 ae tempeatues of wall sufaces (xδ ad x0 ag, g, g, a, wall,, sky - tempeatues of a the gap, glass coves, ambet a, aveages of oom s walls, a oom ad sky, espectvely h c.. - Covectve tasfe coeffcets, W/m o K, h.. - Radat tasfe coeffcets, W/m o K, h - heat tasfe coeffcet space betwee glass coves, W/m o K. Next the a gap s cosdeed. A dffeetal fomulato, whch cludes tems due to themal capacty ad covectve heat tasfe fom the wall suface ad glazg cove to the a, leads to the equato: d ag ρ G c ( h h Bdz (6 ag p cgap w0 ag cgap g ag dz whee ρ s the a desty [kg/m ]; c p - heat capacty of a [J/kg o K]; G ag - a flow ate a gap, [m /s]; B - wall wdth [m], (τα e - a absoptace-tasmttace poduct fo the total solato o a vetcal suface

3 he a flow ate fo a cculato the a cavty s detemed by the aveage a velocty. Accodg to J.A.Duffe ad W.A. Beckma [4], atual covecto cavty ca be assessed by ext expesso: V C( Ag gh / Av C m m whee C ad C ae costats that deped o hydaulc chaactestcs of the gap, A g,a v - gap aea ad vet aea [m ] ; H - wall heght, m, - aveage tempeatue of a gap ad tempeatue of a oom. System ( (7 s usteady two-dmesoal mathematcal model of massve wall system. he model has a combed system of algebac ad dffeetal equatos as a bouday codto. hs model s athe dffcult to solve due to the complcate themal ad mass tasfe pocesses system. he ma poblems whch ase dug solvg pocesses ae: - empeatue vaato decto z (accodg to the wall heght eques solvg all equatos ( (7 wth espect to ths vaato. Hece, t s ecessay to cosde vetcal tempeatue vaatos the glasses ad gap. hs ca be doe oly by cludg ew heat tasfe equatos (such as equato ( fo glass coves. hs tempeatue vaato s caused maly by a cculato the cavty (a covecto. - he cculato the a cavty depeds o the value of a velocty ad theefoe, o the tempeatue dffeece dow ad uppe pat of the a gap - eq. (7. At the same tme, the tempeatue dffeece depeds o the covectve tasfe coeffcet h cgap, whch s fucto of the a velocty. hs detemes the mathematcal model as olealy ad eques specal teato pocedues fo solvg the heat ad mass tasfe equatos. - he complcated fom of equatos fo bouday codtos pesumes may dffcultes tyg to solve the mathematcal equatos by egula umecal scheme. hs goal s complcated addtoally by uegulated vaato of ambet clmatc paametes (ambet tempeatue ad sola adato. Fom the pot of vew of egeeg applcato, fo the smulato model of passve sola system wth massve wall the followg assumptos ca be gouded: - he lteatue evew [,,5] ad calculatos, we caed out, showed that the heat tasfe by covecto wth a cculato s up to 0-5% of all heat tasfe massve wall. hs heat tasfe detemes vey small vetcal tempeatue vaato the wall ( o C, because of the tempeatue equalzato by heat coductvty ad the lage heat capacty of the wall. O the base of these esults, the model ca be smplfed to oe-dmesoal oe by assumg the dffeet layes of wall costucto to be at ufom tempeatue at ay gve tme. I ths way, equato ( ca be smplfed to oe-dmesoal poblem, efeed oly to vaable x. - hemal chaactestcs of the massve wall ad the a flow cavty ae cosdeed as costats, because of small tempeatue vaato themal ad mass tasfe pocesses. Covecto ad adato heat tasfe coeffcets eegy balace equatos ae teated as depedg o velocty ad tempeatue dffeece betwee coespodg elemets. - Lastly, a teatve calculato pocess ca be ogazed, f equato (6 has solved sepaately, by usg ufom tempeatues of wall suface ad glass cove (a umecal method fo solvg dffeetal equato ca be used wth cosdeg these tempeatues kow. he soluto wll deteme the a tempeatue vaato the a cavty. Sepaate fte dffeece method z decto has bee used fo solvg the equato (6. O the base of eceved tempeatue se the duct, equato (6 ca be ewtte as algebac scheme usg mea a tempeatue the a duct: ρ G ag whee m c p HB h cgap w ma ma h m cgap g ma ad m ad g ae the a tempeatues bottom ad top of the a gap. Afte solvg the ma task (, (, (, (4, (5 ad (8 ad ecevg values fo tempeatues of wall suface ad e glass, the equato (6 ca be solved aga ad all pocedue ca be epeated. Pocess ca be cotued utl suffcet accuacy s aved. (8 (7

4 . FINIE DIFFERENCE APPROXIMAION Outte glass - Ie glass A gap,,,-,,- he fte-dffeece fom of dffeetal equato ( s deved by tegato ove cotol volume suouded the typcal ode, soluto gd (Fg.. he dexes ad efe to the thckess (x ad the tme (τ vaable, espectvely. A mplct tme appoxmato, whch s stable fo fowad tegato tme, s developed fo taset dffeetal equatos. I ths case a set of smultaeous equatos eeds to be solved at each tme step. If the tme teval s amed τ (τ, τ τ (τ, τ, the tme devatve ca be wtte usg fowad Eule fomula fo dscetzato: X 0 - I Fg.. he mesh tme ad space - τ τ (9 Fo the space devatve s appled symmetcal Cac-Ncolso s scheme [] fo dscetzato: - - σ - (-σ - (0 x x x whee σ s a weght coeffcet. Afte substtutg (9 ad (0 ( ad eaagg, the followg geeal appoxmato s eceved: x - - ( F,,,..I ( σ τ a -σ x whee : F - - σ σ τ a hs s a system of (I- algebac equatos wth I (,...I ukow ode tempeatues (wth uppe dex. empeatues wth uppe dex ae cosdeed as kow, whch ae eceved fom calculato, made fome tme step o fom tal codtos the fst tme step. Equato ( yelds a specal dagoal matx of equatos fo the tme laye at each space pot. Bouday codtos (, (, (4, (5 ad (8 must be added to complete the system. hese equatos clude a ew ukow tempeatues ag, g, g, whch eques pelmaly solvg the bouday equatos system. Because of complcated atue of bouday codtos ad umeous dffcultes, whch appea soluto pocess, a ew pocedue fo completg the algebac system of equatos s poposed. he mesh s expaded by addg thee ew space layes, umbeed, ad, as t s show o Fg.. hese layes coespod to the elemets of ombe wall - two glass coves ad a the gap wth tempeatues ag, g, g. hs meas that, to the algebac system (, t s ecessay to add algebac equatos (, (4 ad (8. As bouday codtos ths ew system must be used oly equatos ( ad (5. I ths way, mathematcal task becomes cosdeably ease to solve, because of smplfed bouday codto system. 4. SOLUION PROCEDURE Equatos ( ca be solved by stadad algebac methods. Because of smple fom of algebac system (, the well-kow pocedue wth twofold calculato passage the space decto of mesh s used [8]. hs method s applcable fo algebac systems fom as follow: ( A - B C F

5 whee,...i-, wht bouday codtos: 0 a 0 b 0 ad I a I I- b I. ( hs algebac system s smla to ou poblem (, (, (, (5 ad (8 wth espect to the ukow tempeatues (supescpt dex. I the ext cosdeatos the uppe dexes of vaables fte dffeece equatos ca be omtted fo smplcty. Soluto fo above system s wated fom as follow: α, (4 whee a ad b ae ukow coeffcets. hs equato ca be wtte fo all dexes, cludg -: - α - - (5 Substtutg (4 ( ad eaagg, t ca be eceved ext fomulas fo coeffcets: C F - A - α, (6 Aα - B Aα - B If t s take to accout that fo 0: α 0 a 0 ad 0 b 0 (fom bouday codto eq.(, coeffcets α ad ca be calculated by usg ecuet equato (6. hs s the fst calculato passage the space decto of the soluto gd. wo equatos ae avalable fo the last ode I of the gd: secod pat of equato ( ad equato (4 fo I- ode: I- α I- I I-. Fom these two equatos, t s possble to deteme I : a - a I I - I I (7 I I - α b Kowg I, t ca be made secod calculato passage though the soluto gd by usg ecuet equato (4 to calculate tempeatues ( I-, I-...0 of all odes of the gd. hs s a odaly pocedue fo solvg the poblem tme step. Recevg the tempeatues fo tme step, t s possble to make ext tme step. hs smple pocedue fo solvg the algebac system of fte dffeece appoxmato ca ot be used dectly fo the mathematcal poblem of massve wall system descbed above. System of equatos (, (, (4 ad (8, whch s added to ma system (, s ot fully compatble wth the system (. Equato (4 compses fou ukow tempeatues - two tempeatues of glass coves g ad g, tempeatue of a gap ag ad suface tempeatue w. hs meas, that equato (4 ca ot be expessed fom as the equato (4. o use a techque smla to metoed above pocedue fo solvg equatos (, a modfed pocedue ca be used. Istead of eq (5, a ew tempeatue fucto (wth thee cosecutve tempeatues ca be cosdeed: α ι γ (8 hs meas that, tee ukow coeffcets α ι, ad γ ι, must be calculated fst calculatg passage. Afte dog smla steps as t has bee made above (equato 6, equatos fo ukow coeffcets α ι, ad γ ι ca be ease to eceve. Hee, these coeffcets wll ot be descbed fo the commo case, but fo the specal case of mathematcal model fo passve sola system wth massve wall. Afte eaagg equatos of bouday codtos stadad fom (equato (, wth espect to tempeatue fucto (8 ad dexes umecal gd (fg., the followg equatos fo coeffcet α, ad γ have bee eceved: - fo oute glass cove equato (5. It ca be ewtte fom lke eq.: g α. g, (9 whee the coeffcets wth dex ae: h h c α,, 0 (0 a γ h h c h h c - fo e glass cove equato (4. Substtutg g fom (9 ad eaagg, the equato (4 ca be tasfomed fom as eq.8 wth coefces (: α h h, h(-α h hcgap (-α h h γ h(- h gap gap α gap gap

6 (, whee h gap h cgap h gap - fo a gap equato (8. Usg aalogous tasfomato, t ca also be eceved equato fom as eq.8 wth coeffcets (: ( BHh cgap ( γ ρ Gc p B BHh cgap α,, γ 0 D x Dx whee D x BH h cgap (g G c p B - fo wall suface equato (. Hee s ecessay to appoxmate the x decto devatve of tempeatue by fte dffeece. he appopate coeffcets fo algebac equato ae: (4: λσ/ F x -σ hgapα -σhgap -σ hcgap α 4, 4, γ 4 0 E x E x whee: λσ λ E x σ h gap α α σ h gap γ σ h cgap α σ h cgap - - a 4 τ F x λ( - σ λ - q s ( τα e - ( - σ h cgap - 4 h gap a 4 τ - fo oday wall laye - Hee s vald the stadad tasfomato (6 fo equato ( (: α, ( - - F, γ 0 α (4 x - α - σ τ a - empeatue of the last wall's laye (e wall suface ca be defed by followg equato: σλ x I - FF I (5 σλ ρ c p σλ σ hroom - α I τ ( - 4 whee empeatue [C] σ h (- σ λ (- σ - h - - ρ c τ FF ROOM I ROOM I I p I Ambet tempeatue emp. of e suface emp. of mddle laye emp of outsde suf. emp. of e glass Hous ad h ROOM h croom h ROOM Kowg tempeatue I, othe ode tempeatues ca be calculated wth the ecuet fomula (8. It s possble, because coeffcet γ fo oday wall layes s zeo. hs s the secod calculato passage o the gd. o solve the mathematcal poblem by poposed algothmc scheme, the heat tasfe coeffcets ad the a velocty gap, must be kow advace. hese ca be calculated wth egadg tempeatues ad othe eeded vaables fom pevous tme-step calculato o fom tal codtos statg the calculatos. Fo mpovg the pecso of umec calculatos, a teatve calculato have bee ogased utl eeded accuacy has aved. Fg.. empeatue dstbuto elemets of massve wall

7 5. NUMERICAL EXAMPLES o vefy the applcablty of the above-poposed techque, a lage umbe of umecal examples have bee caed. I Fg. ad 4 s show example of calculatos fo passve sola system wth massve wall. Clmatc data ae fo Sofa, Bulgaa. Passve system s wth south facg cocete wall wth dmesos: heght - m; wdth -.5 m ad thckess - 0.m. Clmatc data (sola adato ad ambet tempeatue ae fo Febuay. he set of moth s daly dstbuto of sola adato ad ambet tempeatue, estmated hou-by-hou peod have bee used. Fve days peod of smulato calculatos was eeded to exclude fluece of tal codtos. Followg paametes ae show Fg. ad 4: ambet tempeatue a, sola adato q s, e Eegy [w] Sola adato Heat load Heat by adato Heat by covecto Hous Fg.4. Eegy balace massve vall glass tempeatue g, oute suface tempeatue w, wall tempeatue mddle laye wm, e suface tempeatue w0, heat losses to the ambet a q t, heat tasfeed by covecto q f, heat tasfeed by adato q. 6. CONCLUSIONS I ths pape, has bee peseted a mathematcal model fo passve sola system wth massve wall. A fte dffeece soluto scheme, based o mplct method was developed fo solvg the combed system of algebac ad dffeetal equatos. A ew soluto pocedue, sutable fo the peseted mathematcal model was suggested. Compute pogam fo smulato calculatos was ceated, ad wth extesve umecal expemets, the applcablty of peseted model was vefed. he esults fom peseted mathematcal model would help eseaches feld of passve sola systems to cease the kowledge ad expeece fo themal ad mass tasfe pocesses. he desges of passve sola systems ca use ths model to select optmal costuctve paametes of the massve wall. REFERENCES [] Akbazadeh A., W.S.Chates ad D.A. Lessle, hemocculato chaactestcs of a ombe wall passve test cell. Sola Eegy, 8(6, (98. [] Boges.R, H.Akba, Fee covectve tubulet flow wth the ombe wall chael. Sola Eegy, 5(, -4 (985. [] Cac J., P.Ncolso. A pactcal method fo umecal evaluato of soluto of patal dffeetal equatos of heat-coducto type. Poc.Cambdge Phlos.Sos.,974, N 4 [4] Duffe J.A. ad W.A. Beckma, Sola Eegy hemal Pocesses. Wley, New Yok (974 [5] Duff R.J., G.Kowles, A smple desg method fo the ombe wall. Sola Eegy, 4(, 69-7 (985. [6] Duff R.J., A passve wall desg to mmze buldg tempeatue swgs. Sola Eegy, (/4, 7-4 (984. [7] Kowels.R., Popotog compostes fo effcet themal stoage walls. Sola Eegy, (, 9-6 (98 [8] Rchtmaye R.D.,K.W. Moto, Dffeece method fo tal-value poblems. Itescece Publshes, 967.