Inverse design of indoor environment using an adjoint RNG k-ε turbulence model

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1 Zho, X. nd Chen, Q Inverse design of indoor environmen using n djoin RNG k ε urbulence model, Acceped by Indoor Air Inverse design of indoor environmen using n djoin RNG k-ε urbulence model X. Zho 1 nd Q. Chen, * 1 Tinjin Key Lborory of Indoor Air Environmenl Quliy Conrol, School of Environmenl Science nd Engineering, Tinjin Universiy, Tinjin 0007, Chin School of Mechnicl Engineering, Purdue Universiy, Wes Lfyee, IN 47907, USA *Corresponding uhor s phone: (765) , emil ddress: ynchen@purdue.edu Absrc The djoin mehod cn deermine design vribles of n indoor environmen ccording o he opiml design objecive, such s miniml prediced men voe (PMV) for herml comfor. The mehod clcules he grdien of he objecive funcion over he design vribles so h he objecive funcion cn be minimized long he fses direcion using n opimizion lgorihm. Since he objecive funcion is conrolled by he Reynolds-verged Nvier-Sokes (RANS) equions wih he RNG k-ε model during he opimizion process, ll he corresponding djoin equions should be solved, rher hn he frozen urbulence ssumpion used in previous sudies. This invesigion developed djoin equions for he RNG k-ε urbulence model nd pplied i o wo-dimensionl veniled cviy nd hree-dimensionl, wo-person office. Design processes wih he djoin RNG k-ε urbulence model led o ner-zero design funcion for he wo cses, while hose wih he frozen urbulence ssumpion did no. This invesigion hs successfully used he new mehod o design wo-person office wih opiml herml comfor level round he wo occupns. KEYWORDS Inverse design, Indoor environmen, Adjoin mehod, Numericl lgorihm, Vlidion, Turbulence model Prcicl Implicions This invesigion developed n djoin RNG k-ε urbulence model for opiml design of he indoor environmen. The djoin mehod wih he djoin RNG k-ε urbulence model cn be used o design he opiml indoor environmen for buildings. Design objecive could include ir velociy disribuion, emperure disribuion, nd he combinion of vribles clculed by CFD, such s herml comfor, indoor ir quliy, ec. Nomenclure g grviy vecor J objecive funcion k urbulen kineic energy k djoin urbulen kineic energy L ugmened objecive funcion N incompressible Nvier-Sokes equions in vecor form 1

2 p ir pressure p djoin pressure R residul form of he RNG k-ε urbulence model equions T ir emperure T djoin emperure Top opering ir emperure V ir velociy djoin velociy V Subscrips djoin mehod op opere Greek symbols γ herml expnsion coefficien of ir δ smll consn se by designer ε urbulen energy dissipion re ε djoin urbulen energy dissipion re Θ design domin κ effecive herml conduciviy ν kinemic (lminr) viscosiy νeff effecive viscosiy νk effecive diffusiviy for k ν urbulen viscosiy νε effecive diffusiviy for ε ξ design vribles 1 INTRODUCTION To cree hermlly comforble nd helhy indoor environmen, convenionl designs use ril-nd-error process 1 h ssumes cerin hermo-fluid boundry condiions (B.C.), such s ir supply inle size, number, nd locions, ir supply velociy nd emperure, ec. An pproprie mehod is hen used o esime he resuling disribuions of ir emperure, velociy, relive humidiy, nd conminn concenrions. The ril-nd-error process is very ime consuming becuse he ssumed hermo-fluid B.C. my no be desirble. Recenly, inverse or opiml design processes hve emerged, such s he compuionl fluid dynmics (CFD)-bsed geneic lgorihm (GA) mehod 4, CFD-bsed proper orhogonl decomposiion (POD) mehod 5, CFD-bsed rificil neurl nework (ANN) mehod 6, nd djoin mehod 7. To design desirble indoor environmen, he CFD-bsed GA mehod mus clcule lrge number of smples, nd he number of clculions increses exponenilly wih he number of design vribles. To reduce compuing effor, Wei e l. 5 developed CFD-bsed POD mehod h cn rnsform he nonliner problem ino liner one nd build cuse-effec mpping relionship beween he objecive funcion nd design vribles. Since i is reduced-order mehod, he ccurcy of his mehod is grely reduced. The CFD-bsed ANN

3 mehod cn lso build he mpping relionship beween he objecive funcion nd design vribles, in his cse by selecing cerin number of smples o rin he ANN model. Wih well-rined ANN model, he design objecive cn be prediced wihou CFD clculions. However, he ccurcy of his mehod depends on he number of smples. In ddiion, he mpping relionship esblished by eiher he CFD-bsed POD mehod or he CFD-bsed ANN mehod is pplicble only o specific cse. For differen cse, he relionship would chnge. The djoin mehod cn quickly find he opiml design of n indoor environmen using n opimizion lgorihm wihou building mpping relionship for ech new problem, lhough i my become rpped in locl opim 8. The djoin mehod is he mos efficien nd suible mehod for he inverse design of hermlly comforble nd helhy indoor environmen. The djoin mehod ws developed recenly 9 for inversely designing n indoor environmen by solving se of Nvier-Sokes equions nd djoin equions, lernively. The djoin equions re derived from he coninuous Nvier-Sokes equions 10. Liu nd Chen 9 used he djoin mehod o inversely idenify he hermo-fluid B.C. required o chieve he opiml design of venilion for n enclosed environmen. Liu e l. 11 hen doped his mehod o improve he herml comfor level for n irline cbin. Zho e l. 1 lso emped o use he djoin mehod combined wih re-consrined opology nd cluser nlysis o design hermlly comforble indoor environmen. However, he resuls of hese sudies indiced h his mehod cnno mke he objecive funcions rech he idel vlues. One reson my be h he objecive funcions become rpped in locl opim. Anoher reson my be h he djoin mehod, which ws used in previous sudies 9, 11-14, derives only he djoin equions of he RANS equions wih he frozen urbulence ssumpion. Wih he ssumpion, he k nd ε will no feel chnges of he design vribles nd use he soluion (k nd ε) of he flow equions when he djoin equions re solved. The ssumpion cn reduce deriving mnully effor 14, 15 by neglecing vriions in he urbulen vribles. However, his mehod cn provide only pproxime grdiens, or even incorrec grdiens, which my no led o globl opim for he design. A vrin of he mehod is he discree djoin mehod 16, which firs discreizes he Nvier-Sokes equions nd hen derives he discree djoin equions from he discree Nvier-Sokes equions. Since he discree djoin mehod derives he complee djoin equions h include he djoin equions for clculing urbulen viscosiy, i cn provide ccure grdiens 17. However, i requires lrge moun of compuionl memory, which my no be ffordble in prcice. To obin ccure grdiens nd reduce compuionl memory, one opion is o develop coninuous djoin mehod wihou using he frozen urbulence ssumpion 18. Since he urbulen viscosiy ν in he momenum equion cn be solved by inroducing n pproprie urbulence model, some sudies hve derived he complee coninuous djoin equions wih n pproprie djoin urbulence model. Zymris e l. proposed Splr-Allmrs one-equion djoin urbulence model 19 nd sndrd k-ε djoin urbulence model 0 o minimize duc pressure losses. Ppousis-Kichgis 1 developed

4 low-reynolds-number Lunder-Shrm k-ε djoin urbulence model o opimize duc shpe, wih he im of minimizing viscous losses. All hese pplicions hve proved h n pproprie djoin urbulence model cn provide n ccure grdien of he objecive funcion over he design vribles. However, none of he bove urbulence models is suible for inversely designing n indoor environmen. Chen, Conceição e l., Zhng e l. 4 nd Zhng e l. 5 compred differen urbulence models nd found h he RNG k-ε urbulence model shows he bes overll performnce for solving indoor irflow mong vrious RANS models. In ddiion, here do no hve he corresponding djoin equion. Therefore, his sudy imed o develop n djoin RNG k-ε urbulence model for inversely designing n indoor environmen. METHODS.1 Adjoin mehod wih djoin RNG k-ε urbulence model To design n indoor environmen using he djoin mehod, we mus firs consruc suible objecive funcion. For exmple, he objecive funcion J is desirble disribuion of ir velociy V nd emperure T in he design domin Θ: ξ ( V T ) Θ J f, d (1) where ξ is vecor h represens he design vribles, such s ir supply inle size, number, nd locions; ir supply velociy, Vinle; ir supply emperure, Tinle, ec., h could led o he desirble disribuion. In his sudy, ir velociy V nd emperure T in he design domin Θ, s shown in Eq. (1), re conrolled by he incompressible, sedy-se RANS equions, s shown in Eqs. (), (), nd (4), closed wih he RNG k-ε urbulence model 6, s expressed by Eqs. (5) nd (6). N V 0 () 1 (N,N,N ) ( V) Vp ( D( V)) g (TT ) 0 () T 4 eff op 5 N V T T 0 (4) R V k k P G 0 (5) 1 k k b where R V C1Pk CGbC k C 1 / k (6) 4

5 165 ; c eff k ;P k V xj x i xj Vi j V i ; i, j x, y, z; 1 V V i j Sk/ ; S ( SijSij ); Sij ; Gb g xj x i Pr T (7) In hese equions, p represens he ir pressure; D(V) = (V+(V) T )/ is he re of srin ensor; k is he urbulen kineic energy; ε is he urbulen energy dissipion; cμ = ; η0 = 4.8; β = 0.01; Cε1 = 1.4; Cε = 1.68; Cε = 1; Gb is he buoyncy producion res of urbulen kineic energy; Pk is he sher producion res of he urbulence kineic energy; Pr urbulence Prndl number for emperure; S is he modulus of he men re-of-srin ensor; N represens he incompressible, sedy-se RANS equions in residul form; nd R represens he residul form of he RNG k-ε urbulence model equions. To minimize he objecive funcion s shown in Eq. (1), his invesigion used he seepes decen mehod 7 s shown in Eq. (8) o upde he design vribles. Therefore, he djoin mehod ws used o clcule he grdiens used in Eq. (8). dj ξn1 ξnn dξ where ξn, ξn+1 re design vribles curren nd succeeding design cycles, respecively; n represens he design cycle nd λ is he consn sep size. The djoin mehod inroduces n ugmened objecive funcion L s shown in Eq. (9) nd rnsforms he consrined design problem ino n unconsrined opimizion problem. Ω, k 5 n L J p, V,T, N, R dω (9) where Ω represens he compuionl domin nd p, V, T, k, nd ε re he djoin pressure, djoin velociy, djoin emperure, djoin urbulence kineic energy, nd djoin re of dissipion of urbulen energy, respecively. From Eq. (9) we cn hen derive he following djoin equions of he RANS equions closed wih he RNG k-ε urbulence model. Deiled inermedie derivion process of he djoin equions cn be found in 1. Adjoin coninuiy equion: V J 0 (10) p (8)

6 Adjoin momenum equion: V V V V p D V TT kk 6 eff V V J k C V i j 1 xj k xj x i 4 4 / 0 0 x j 1 1 Adjoin energy equion: / C 0 T T γ gk V V g Pr J C1C g 0 Pr k T Adjoin urbulen kineic energy equion: Vk kk 4 DVV TT kk k Pr k k k k P k gt C Pk k k Pr k k k k 1 J CC T C k 1 g Pr k k / 0 0 C C / 1/ 1 k 1 k Adjoin equion for he dissipion re of urbulen kineic energy: Pk g Pr k V DVV T T k k Pr 1 1 J k k T k C 4 4 / / 0 C 1 / 0 0 C k 1 k k (11) (1) (1) (14)

7 We lso give finl grdien of he ugmened objecive funcion over he design vribles: dl S p +S inle,inle V inle inle V,inle +S inle V inle V,inle S eff inle d dv n inle,i +S inle T inle T,inle S inle k inle k,inle S inle inle,inle dl dt V,inle d gv Vol S ( V T T gk n,inle inle inle inle,inle,inle,inle inle dn dn Pr CC g ) 1,inle Pr k e i (15) where Vinle, i is he inle ir velociy in he i h direcion, i = x, y, z; Sinle is he ouwrd-poining fce re vecor of inle cell; p, inle, V, inle, T, inle, k, inle nd ε, inle re he djoin pressure, djoin velociy, djoin emperure, he djoin urbulen kineic energy nd he djoin urbulen energy dissipion re he cell djcen o he corresponding boundry fce, respecively; kinle nd εinle re he urbulen kineic energy nd he urbulen energy dissipion re inle boundry fce, respecively; dn is he direcion vecor beween n inle cell cener nd he boundry fce cener; ei is he uni vecor in he i h direcion, i = x, y, z; Volinle is he inle cell volume. When he RANS equions closed wih he RNG k-ε urbulence model nd he djoin equions re numericlly solved in succession, ll se fields nd djoin fields needed for clculing he grdien of he objecive funcion over he design vribles re vilble.. Adjoin boundry condiions The djoin inle boundry condiions re h zero p, V, T, k, nd ε long he inle. V, T, k, nd ε re se o zero nd zero grdien boundry condiion of p he wll. While he djoin oule boundry condiions cn be deermined by: n V V V Vn n V p ntntk nk n eff V V i j k C1 n k x j x i / Cn / j (16) J (17) V j 0 J n n n n+ (18) T V T k g C1C g 0 Pr Pr k T 1 1 J knk knv TnT V 4 DVn+ 0 (19) Pr k k k 7

8 J DVV 0 n Vn+ n + (0) where n is he norml componen nd JΓ is he objecive funcion defined on boundry.. Numericl mehod The djoin mehod wih he frozen urbulence ssumpion nd he djoin RNG k-ε urbulence model were previously implemened in OpenFOAM (Open Field Operion And Mnipulion) 9. The convecion nd diffusion erms of he RANS equions closed wih he urbulence model nd djoin equions were discreized by he firs-order upwind scheme nd he cenrl difference scheme, respecively. Previous sudies 1,7-9, 11-1 ll used he firs-order upwind scheme o discreize he convecion erms of boh se of equions nd none of he sudies hve no repored ny ccurcy issues. Thus, we used he sme scheme nd did no explore high-order numericl scheme. The Boussinesq pproximion 0 ws used o simule he herml plume genered by he emperure difference. The convergence crierion ws se s Jn-Jn-1 < δ (where n nd δ = 10 - ), where Jn-1, Jn re he objecive funcion previous nd curren design cycles, respecively. RESULTS In order o verify he performnce of he djoin RNG k-ε urbulence model for inverse design of n indoor environmen, his sudy firs esed he proposed mehod by pplying i o wo-dimensionl veniled cviy 1 nd wo-person office hrough sep-by-sep process. The wo-dimensionl veniled cviy is simple cse wih mixed venilion, while he hree-dimensionl veniled office represens complee indoor environmen wih complex geomeries nd displcemen venilion. To prove he ccurcy of he proposed mehod nd he necessiy of developing new mehod, he djoin mehod wih he frozen urbulence ssumpion ws conduced s comprison. Finlly, his sudy used he vlided mehod o inversely design n opiml indoor environmen for he wo-person office..1 Two-dimensionl veniled cviy The firs cse is simple wo-dimensionl veniled cviy, s shown in Figure 1, wih experimenl d (i.e., velociy nd emperure) long he green cener lines nd deiled informion bou ll B.C. vilble 1. The experimen supplied ir hrough he inle he op of he lef wll nd exhused ir hrough he oule he boom of he righ wll. Wih he excepion of he floor, which ws heed o 5.5 ºC, he emperures of he ceiling nd wlls were ll conrolled 15.0 ºC. 8

9 FIGURE 1 Schemic of wo-dimensionl veniled cviy CFD simulion: Before sring he inverse design process, we needed o prove h we hd he biliy o conduc he CFD simulion correcly. This invesigion conduced CFD simulion using experimenl B.C. Experimenl d long he green cener lines ws used o verify he resuls of he simulion. Figure compres he ir velociy nd emperure profiles prediced by CFD wih he RNG k-ε urbulence model nd he experimenl d x = 0.5 l. The comprison indices h we were ble o predic he ir disribuion ccurely in he veniled cviy. However, he simulion resuls did no coincide compleely wih he experimenl resuls. The men relive error of he ir emperure beween he CFD simulion resuls nd he experimenl d is.%. However, he men relive error of he ir velociy is lile higher, h is becuse he velociy he cener of he cviy is very smll. Thus, here were errors beween he experimenl d nd he resuls prediced by CFD. The discrepncies my hve risen from mesuremen errors nd unconrollble fcors in he experimen nd numericl errors in he CFD simulion, especilly he RNG k-ε model. Such errors my ffec he inverse design resuls. If he experimenl d long he green lines were used s rge funcion for inverse modeling, he soluion would no be idel due o he inheried errors from he CFD in he djoin equions. Thus, i is beer o use CFD resuls long he green lines s rge funcion becuse he numericl errors in he CFD simulion re he sme s hose in he inversed simulion. 97 9

10 FIGURE Comprison of ir velociy nd emperure profiles prediced by CFD wih he RNG k-ε urbulence model nd experimenl d from Bly e l. 1 long he x = 0.5 l secion. Inverse design process: Since our purpose ws o verify he performnce of he proposed mehod using he numericl mehod, we se he CFD simulion resuls long he wo mid-secions (green lines) s rge vlues o elimine he influence of experimenl errors nd numericl errors, nd he ir supply prmeers s he design vribles o consruc he objecive funcion. If he ir supply prmeers idenified were found o be consisen wih he experimenl ir supply prmeers, he proposed mehod would be verified. For his cse, we used he prediced ir velociy V0, ii (vecor) nd ir emperure T0, ii in design domin (he vlues long he wo mid-secions), Θ, s he rge vlues o consruc he objecive funcion, which cn be expressed s: where m m 1 norm ii 0,ii norm ii 0, ii J ξ W V V V W T T T (1) ii1 1 1 V ;T ii1 norm V norm inle, x Tmx Tm in where W1 nd W re he weighing fcors, ssumed o be 0.5 in his sudy; Vnorm nd Tnorm re he normlizion fcors; nd Tmx nd Tmin re equl o 5.5ºC nd 15.0ºC, respecively; m is he ol number d in he design domin; Vii nd Tii re he inversely designed resuls in he design domin, in his cse, respecively. Wih he bove objecive funcion, he djoin mehod wih he djoin RNG k-ε urbulence model sred is inverse design process from he iniil inle B.C. These B.C. were Vinle = (0.8, 0) m/s nd Tinle =.0ºC. For ech design cycle, boh he RANS equions closed wih he urbulence model nd he djoin equions were clculed wih he use of,000 ierions o ensure convergence. To sudy he ccurcy of he grdien clculed by he proposed mehod nd rule ou he influence of sep size, we doped he seepes descen mehod wih proper consn sep size in Eq. (8) o upde he design vribles. Since n improper consn sep size could cuse he objecive funcion o become rpped in locl opim or he clculion o diverge, his invesigion es differen sep sizes for proper sep sizes (0.016 for upding Vinle is nd 00 for upding Tinle). Wih he proper sep sizes nd he sme iniilized ir supply prmeers, we lso used he djoin mehod wih he frozen urbulence ssumpion o inversely idenify he opiml design vribles for he veniled cviy, in order o illusre he need for he new mehod. Figure compred he k wih k (djoin k) of hese wo mehods when solving he djoin () 10

11 equions he firs design cycle h ws becuse hese wo mehods hve he sme ir supply prmeers only in his design cycle during he inverse design process. The k is he urbulen energy, while he k is djoin urbulen energy. The urbulen energy represens he degree of chos of he flow, while he djoin urbulen energy is he opposie. They re very differen becuse wo ses of equions re differen. As resul, he grdiens clculed by hese wo mehods re differen. When he objecive funcion minimized by he djoin mehod wih he frozen urbulence ssumpion me he convergence crieri, s shown in Figure 4, he objecive funcion minimized by he djoin RNG k-ε urbulence model reched he sme vlue lmos he sme ime. However, he new mehod cn cuse he objecive funcion o coninue o decrese. The design vribles idenified by he djoin RNG k-ε urbulence model for Cse A- were lmos he sme s he known experimenl d, s shown in Tble 1. This indices h he djoin RNG k-ε urbulence model hs higher clculion ccurcy. k k () (b) FIGURE Comprison of () k soluion of he flow equion nd (b) k clculed by he djoin RNG k-ε urbulence model he firs design cycle FIGURE 4 Vriion in he objecive funcion wih he design cycle for he wo-dimensionl veniled cviy 11

12 TABLE 1 Design vribles idenified by wo mehods compred wih he experimenl B.C. Vinle, x Vinle, y Tinle (m/s) (m/s) (K) J Wih he frozen urbulence ssumpion E Wih he djoin RNG k-ε urbulence model E E-4 Experimenl d Afer he opiml ir supply prmeers, s shown in Tble 1, hd been idenified by he wo mehods, we conduced CFD simulion wih he RNG k-ε urbulence model o deermine wheher he opimized ir disribuion ws consisen wih he rge vlues (red line/dos). Figure 5 illusres he velociy vecor nd emperure profiles in he design domin prediced wih he ir supply prmeers idenified by he djoin mehod wih he frozen urbulence ssumpion (green line/dos) nd wih he djoin RNG k-ε urbulence model (blue lines/dos). The ir disribuion in he design domin opimized wih he djoin RNG k-ε urbulence model ws closer o he rge vlues hn he disribuion opimized wih he frozen urbulence ssumpion, especilly in he cener of he design domin. The ir emperures x = 0.5 l prediced wih he frozen urbulence ssumpion were lso much lower hn he cul vlues. Thus, he performnce of he djoin RNG k-ε urbulence model ws beer. Inle Wll Design domin 75 y x Floor Oule FIGURE 5 Comprison of ir velociy vecors in he design domin nd emperure profiles x = 0.5 l prediced by he experimenl B.C. from Bly e l. 1 nd he ir supply prmeers idenified by he djoin mehod wih he frozen urbulence ssumpion nd wih he djoin RNG k-ε urbulence model 1

13 Three-dimensionl veniled office The second cse is hree-dimensionl veniled office, s shown in Figure 6, wih experimenl d. In he experimen, he office ws veniled by displcemen venilion sysem wih n ir supply velociy of Vinle = (0.09, 0, 0) m/s nd ir supply emperure of Tinle = 17 ºC. The ir ws supplied hrough wll diffuser floor level nd exhused hrough n oule ceiling level. This office ws more compliced hn he wo-dimensionl veniled cviy nd closer o reliy, lhough he occupns, compuers, cbines, nd lighs were ll simplified s recngulr boxes. Therefore, i ws more prcicl o use his office for inversely idenifying he opiml ir supply prmeers. Similr o he previous cse, we used CFD simulion resuls long lines 1,, 5, 7, nd 9 o consruc he objecive funcion s expressed by Eqs. (1) nd (). We used srucured grid of hexhedrl elemens in his geomeric model, wih 7,51 cells ccording o our grid-independence ess FIGURE 6 Schemic of wo-person office, nd he design domin locion where he numbers 1 o 9 indice locions where ir velociy nd emperure profiles were mesured in he experimen. To sr he inverse design process, he inle B.C. were iniilized s Vinle = (0.5, 0, 0) m/s nd Tinle = 0ºC. Noe h he sep sizes for upding Vinle nd Tinle were 0.0 nd 100, respecively. Figure 7 shows he vriion in he objecive funcion wih he design cycle during he inverse design process, wih he frozen urbulence ssumpion nd wih he djoin RNG k-ε urbulence model. The convergence speed wih he frozen urbulence ssumpion ws en imes fser hn h he djoin RNG k-ε urbulence model. This occurred becuse he grdiens clculed wih he frozen urbulence ssumpion were lrger hn hose clculed wih he djoin RNG k-ε urbulence model in he firs wo design cycles. However, wih he djoin RNG k-ε urbulence model, one could obin n objecive funcion close o zero. 1

14 FIGURE 7 Vriion in he objecive funcion wih he design cycle for he hree-dimensionl veniled office cse TABLE Design vribles idenified by wo mehods compred wih experimenl B.C. Vinle, x Vinle, y Vinle, z Tinle (ºC) (m/s) (m/s) (m/s) Wih he frozen urbulence ssumpion E-4.5E Wih he djoin RNG k-ε urbulence model E-6.1E Experimenl d As shown in Tble, he ir supply prmeers finlly idenified by he djoin RNG k-ε urbulence model were close o he experimenl d. We hen used hese prmeers o conduc CFD simulions o deermine wheher he idenified ir supply prmeers formed he sme ir disribuion s he resuls wih experimenl B.C. Figure 8 quniively compres he prediced ir velociy nd ir emperure profiles wih he experimenl profiles he cener of he room (line ) s n exmple. The ir supply prmeers idenified by he djoin RNG k-ε urbulence model were much beer hn hose wih he frozen urbulence ssumpion. 14

15 FIGURE 8 Comprison of ir velociy nd emperure profiles prediced by he experimenl B.C. from Yun e l. nd he ir supply prmeers idenified by he djoin mehod wih he frozen urbulence ssumpion nd wih he djoin RNG k-ε urbulence model wih he experimenl d in he cener of he room. Inverse design of comforble indoor environmen for he hree-dimensionl veniled office Using he vlided djoin RNG k-ε urbulence model, his invesigion nex conduced n opiml design of he indoor environmen for he wo-person office. In he opiml design process, he design domin ws he surfces disnce of 0.1 m wy from he occupns, s shown in Figure 9. We used he prediced men voe (PMV) 4 o evlue he herml comfor level in he domin. The rnge of PMV vlues is from - o, nd PMV = 0 signifies high comfor level. Our design gol ws o idenify he opiml ξ (ir supply velociy Vinle nd ir supply emperure Tinle) h would mke he verge PMV in he design domin close or equl o zero. Thus, he objecive funcion cn be expressed by PMV d Jξ () d For comforble indoor environmen, he ir quliy mus sisfy he design crieri. According o he ASHRAE Hndbook 5, he generl design crierion used for office buildings is les 4 ir chnges per hour, nd he corresponding fce velociy for his office ws m/s. Therefore, we dded his consrin (Vinle, x m/s) o insure indoor ir quliy 15

16 during he inverse design process FIGURE 9 Schemic of wo-person office nd inverse design domin Figure 10 depics he vriion in he objecive funcion wih he design cycle during he inverse design process. The inverse design process chieved he convergence crierion fer 18 design cycles. The finl objecive funcion minimized by he djoin mehod wih he djoin RNG k-ε urbulence model ws 0.1 nd did no chieve zero. The reson my be h he objecive funcion ws rpped in locl opim. The finl B.C. idenified were Vinle = (0.184, -5.0E-04, 1.E-04) m/s nd Tinle = ºC FIGURE 10 Vriion in he objecive funcion wih he design cycle during he opiml design processes wih he djoin RNG k-ε urbulence model. Figure 11 depics he PMV disribuions round he occupns inversely designed by he djoin mehod wih he djoin RNG k-ε urbulence model. The men PMV vlue round he occupn is 0.1 nd PMV disribuions re ll ner zero. Hence, he djoin mehod wih he djoin RNG k-ε urbulence model hs good performnce in he opiml design of he indoor environmen. 16

17 FIGURE 11 PMV disribuions round he occupns inversely designed by he djoin mehod wih he djoin RNG k-ε urbulence model 4 DISCUSSION The djoin RNG k-ε urbulence model improved he opiml design ccurcy, bu i could no overcome he inheren disdvnges of he djoin mehod. The objecive funcion could lso become rpped in locl opim. If he iniil design vribles were fr wy from he opiml vlues or he sep size ws no pproprie, he clculion migh no led o he opiml design. This sudy used only PMV o evlue he herml comfor level nd o dd consrin o ensure h he indoor ir quliy me he sndrd. Wih he djoin mehod, one could dd furher design objecives wihou incresing he compuing coss. The design vribles deermined in Secion. did no mke he objecive funcion equl o zero. This resul my imply h he consrins were oo resricive. For exmple, he design process did no llow chnges in he posiion, size, or shpe of he ir supply inles. For he sme cse wih differen objecive funcions in secions. nd., he convergence speed is no comprble. This ws becuse he relionship beween he objecive funcion nd he design vribles ws no explici. This invesigion only used he firs-order upwind scheme nd he cenrl difference scheme o discreize he convecion nd diffusion erms of he RANS equions closed wih he urbulence model nd djoin equions, respecively. While, he influence of differen discreizion schemes on he inverse design resuls is no cler he momen nd our fuure reserch will conduc n in-deph sudy nd nlysis of his issue. 5 CONCLUSIONS This invesigion developed n djoin RNG k-ε urbulence model for he djoin mehod for opiml design of he indoor environmen. The following conclusions cn be drwn from his sudy: The design process wih he djoin RNG k-ε urbulence model idenified design 17

18 vribles h were more ccure hn hose idenified wih he frozen urbulence ssumpion. However, he design vribles idenified wih he frozen urbulence ssumpion were more sble during he inverse design process. The djoin mehod wih he djoin RNG k-ε urbulence model cn be used o design he opiml indoor environmen for n office. Design objecive could include ir velociy disribuion, emperure disribuion, nd he combinion of vribles clculed by CFD, such s herml comfor, indoor ir quliy, ec. The design vribles mus no be overly resricive. Oherwise, he objecive funcions my no rech zero, even fer numerous design cycles. ACKNOWLEDGEMENTS This reserch ws prilly suppored by he Nionl Key R&D Progrm of he Minisry of Science nd Technology, Chin, on Green Buildings nd Building Indusrilizion hrough Grn No. 016YFC nd by he Nionl Nurl Science Foundion of Chin hrough Grn No REFERENCES 1. Liu W, Zhng T, Xue Y, e l. Se-of-he-r mehods for inverse design of n enclosed environmen. Build Environ. 015; 91: Chen Q, Zhi Z, You X, Zhng T. Inverse Design Mehods for Buil Environmen. Rouledge, Oxford, Englnd; Chen Q. Venilion performnce predicion for buildings: A mehod overview nd recen pplicions. Build Environ. 009; 44: Xue Y, Zhi ZJ, Chen Q. Inverse predicion nd opimizion of flow conrol condiions for confined spces using CFD-bsed geneic lgorihm. Build Environ. 01; 64: Wei Y, Zhng TT, Wng S. Promp design of he ir-supply opening size for commercil irplne bsed on he proper orhogonl decomposiion of flows. Build Environ. 016; 96: Zhng T, You X. A simulion-bsed inverse design of prese ircrf cbin environmen. Build Environ. 014; 8: W. Liu, M. Jin, C. Chen, Q. Chen. Opimizion of ir supply locion, size, nd prmeers in enclosed environmens using compuionl fluid dynmics-bsed djoin mehod. J Build Perform Simu. 016; 9(): Liu W, You R, Zhng J, Chen Q. Developmen of fs fluid dynmics-bsed djoin mehod for he inverse design of indoor environmens. J Build Perform Simu. 016; 10(): Liu W, Chen Q. Opiml ir disribuion design in enclosed spces using n djoin mehod. Inverse Probl Sci En. 015; (5): Ndrjh S, Jmeson A. A comprison of he coninuous nd discree djoin pproch o uomic erodynmic opimizion. In 8h Aerospce Sciences Meeing nd Exhibi. 000; Liu W, Dun R, Chen C, e l. Inverse design of he herml environmen in n irliner cbin by use of he CFD-bsed djoin mehod. Energ Buildings. 015; 104(): Zho X, Liu W, Li D, Chen Q. Opiml design of n indoor environmen by he 18

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