Academic Editor: Hiroshi Sekimoto Received: 18 October 2016; Accepted: 28 November 2016; Published: 8 December 2016

Transcription

1 energies Article Predictive Modeling of Buoyncy-Operted Cooling Toer under Unsturted Conditions: Adjoint Sensitivity Model nd Optiml Best-Estimte Results ith Reduced Predicted Uncertinties Federico Di Rocco nd Dn Gbriel Ccuci 2, * Deprtment of Astronutic, Electric nd Energy Engineering DIAEE, Spienz University of Rome, 0085 Rom, Itly; federicodirocco@lice.it 2 Center for Nucler Science nd Energy, Deprtment of Mechnicl Engineering, University of South Crolin, 00 Min Street, Columbi, SC 29208, USA * Correspondence: ccuci@cec.sc.edu; Tel.: Acdemic Editor: Hiroshi Sekimoto Received: 8 October 206; Accepted: 28 November 206; Published: 8 December 206 Abstrct: Nucler nd or lrge-scle energy-producing plnts must include systems tht gurntee sfe dischrge of residul het from industril process into tmosphere. This function is usully performed by one or severl cooling toers. The mount of het relesed by cooling toer into externl environment cn be quntified by using numericl simultion model of physicl processes occurring in respective toer, ugmented by experimentlly mesured dt tht ccounts for externl conditions such s outlet ir temperture, outlet ter temperture, nd outlet ir reltive humidity. The model s responses of interest depend on mny model prmeters including correltions, boundry conditions, nd mteril properties. Chnges in se model prmeters induce chnges in computed quntities of interest clled model responses, hich re quntified by sensitivities i.e., functionl derivtives of model responses ith respect to model prmeters. These sensitivities re computed in this ork by pplying generl djoint sensitivity nlysis methodology ASAM for nonliner systems. These sensitivities re subsequently used for: i Rnking prmeters in ir importnce to contributing to response uncertinties; ii Propgting uncertinties covrinces in se model prmeters to quntify uncertinties covrinces in model responses; iii Performing model vlidtion nd predictive modeling. The comprehensive predictive modeling methodology used in this ork, hich includes ssimiltion of experimentl mesurements nd clibrtion of model prmeters, is pplied to cooling toer model under unsturted conditions. The predicted response uncertinties stndrd devitions thus obtined re smller thn both computed nd mesured stndrds devitions for respective responses, even for responses here no experimentl dt ere vilble. Keyords: cooling toer; djoint sensitivity nlysis; djoint cooling toer model solution verifiction; dt ssimiltion; model clibrtion; best-estimte predictions; reduced predicted uncertinties. Introduction Nucler nd or energy-producing lrge-scle industril instlltions must include systems tht gurntee sfe dischrge of residul het from industril process into tmosphere. In cse of nucler poer plnts, residul het is dischrged into tmosphere by using cooling toers. The mount of het relesed into externl environment cn be quntified by using numericl simultion model of physicl processes tht tke plce ithin respective cooling toer, complemented by experimentlly mesured dt tht chrcterize externl conditions such Energies 206, 9, 028; doi:0.90/en92028.mdpi.com/journl/energies

2 Energies 206, 9, of 52 s outlet ir temperture, outlet ter temperture nd outlet ir reltive humidity. Bsed on flo regime occurring in fill section, cooling toers re generlly divided in to ctegories: cross-flo toers nd counter-flo toers. A model for stedy-stte simultion of both cross-flo nd counter-flo et cooling toers hs been presented in. The nturl drft cooling toer model presented in s vlidted ginst experimentl dt in 2, under prtilly nd totlly sturted conditions. This ork lso nlyses nturl drft cooling toer, but, in contrdistinction ith ork in 2,, drft cooling toer nlyzed in this ork opertes lys under unsturted conditions. Thus, just s in 2,, cooling toer nlyzed in this ork is operted ith toer s fn turned off, cusing ir mss flo rte to be n unknon stte vrible to be determined by solving underlying set of governing equtions. In contrdistinction to 2,, hoever, unsturted operting conditions nlyzed in this ork re modeled by governing equtions tht differ from ones presented in 2,. The ork in 2, used considerbly more ccurte nd efficient numericl method thn s used in, reby overcoming ll of non-convergence issues tht hd plgued some of computtions in. The ccurte nd efficient numericl method introduced in 2, is lso used in this ork for computing folloing quntities: i The ter mss flo rtes t exit of ech control volume long height of fill section of cooling toer; ii The ter tempertures t exit of ech control volume long height of fill section of cooling toer; iii The ir tempertures t exit of ech control volume long height of fill section of cooling toer; iv The humidity rtios t exit of ech control volume long height of fill section of cooling toer; v The ir reltive humidity t exit of ech control volume; nd vi ir mss flo rte t outlet of cooling toer. This ork is orgnized s follos: Section 2 presents mmticl model of mechnicl drft counter-flo cooling toer operting under unsturted conditions, long ith numericl solution for quntities stte functions mentioned bove in items i through vi. Section presents development of djoint sensitivity model for counter-flo cooling toer operting under unsturted conditions using generl djoint sensitivity nlysis methodology ASAM for nonliner systems 4 6. Using single djoint computtion enbles efficient nd exct computtion of sensitivities functionl derivtives of model responses to ll of 47 model prmeters, thus lleviting need for repeted forrd model computtions in conjunction ith finite difference methods. Sensitivities re subsequently used for rnking contributions of single model prmeters to model responses vritions, computing propgted uncertinties of model responses, nd for ppliction of predictive modeling for coupled multi-physics systems PM_CMPS methodology 7, imed t yielding best-estimte predicted nominl vlues nd uncertinties for model prmeters nd responses. Section 4 presents results of pplying PM_CMPS methodology, hich simultneously combines ll of vilble computed informtion nd experimentlly mesured dt for buoyncy-operted counter-flo cooling toer operting under unsturted conditions. The best-estimte results predicted by PM_CMPS methodology revel tht predicted vlues of stndrd devitions for ll model responses, even those for hich no experimentl dt hve been recorded, re smller thn eir computed or mesured stndrds devitions for respective responses. After discussing significnce of se predicted results, this ork concludes by indicting possible furr generliztions of djoint sensitivity nlysis nd PM_CMPS methodologies. 2. Mmticl Model of Mechnicl Drft Counter-Flo Cooling Toer Operting under Unsturted Conditions The experimentl mesurement used for this study hve been performed t Svnnh River Ntionl Lbortory SRNL for F-Are Cooling Toers in period going from April, 2004 to August, 2004; this dt collection comprises totl of 8079 benchmrk dt sets 8, ech one contining

3 Energies 206, 9, 028 of 52 Energies 206, 9, 028 of 57 folloing four mesured quntities: i Exhust ir temperture ccording to Tidbit sensor, hich Tidbit ill besensor, denoted hich in ill folloing be denoted sin T,outTidbit folloing ; ii s Exhust T ir ; ii temperture Exhust ir ccording temperture outtidbit, to Hobo ccording sensor, to hich Hobo ill besensor, denoted hich in ill folloing be denoted sin T,outHobo folloing ; iii s Outlet T ter ; iii temperture, Outlet hich ill be denoted in folloing s T,out mes outhobo, ; iv Exhust ir reltive mes ter temperture, humidity, hich ill be referred to s RH mes hich ill be denoted in folloing s T ; iv Exhust ir reltive out,. For dt set to be intended s unsturted is required tht vlue of mes humidity, hich ill be referred to s RH. For dt set to be intended s unsturted is exhust required ir tht reltive vlue humidity of exhust RH, ir computed reltive humidity by mking use of SNRL simultion code RH, computed by mking use of CTTool, is smller thn 00%, hile threshold is set on correspondence of sturtion line SNRL simultion code CTTool, is smller thn 00%, hile threshold is set on RH correspondence 00%. With of this stndrd sturtion set, line 677 RH dt 00% sets. With mong this stndrd totl 8079 set, 677 mesured dt sets hve mong mtched requirement totl 8079 bove mesured nd hve hve been mtched refore requirement identified s bove unsturted, nd hve been leding refore to include identified ms in present unsturted, study. Histogrm leding plots to include nd nlyses m in of present sttisticl study. moments Histogrm of plots nd selected nlyses 677of dt sets hve sttisticl been gred moments in Appendix of selected A. The 677 stte dt functions sets hve chosen been gred for nlysis in Appendix of A. cooling The stte toer mmticl functions chosen modelfor hereby nlysis considered of hve cooling been toer selected mmticl ccording model tohereby quntities considered comprised hve in been experimentlly selected ccording mesured to dt quntities sets. Acomprised numericl in method, experimentlly hich hsmesured been presented dt sets. ina detil in 2, numericl hs beenmethod, implemented hich hs to been improve presented bothin detil ccurcy in 2, hs ndbeen implemented efficiency into improve solution both of constitutive ccurcy equtions nd system efficiency governing solution counter-flo of constitutive cooling toer equtions model. system governing counter-flo cooling toer model. The cooling toer model nlyzed in this ork hs been originlly presented in nd hs been The cooling toer model nlyzed in this ork hs been originlly presented in nd hs thoroughly described in 2, from hich Figures hve been tken. Nturl drft ir enters been thoroughly described in 2, from hich Figures hve been tken. Nturl drft ir enters toer just toer belo just belo fill section, fill section, through through hich hich it it psses psses floing floing uprds uprdslong long height height of of toer toer finlly finlly exiting exiting t t top, top, through n n exhust comprising fn. fn. Inlet Inlet ter ter enters enters toer toer belo belo drift drift elimintor, nd nd is is spryed donrds over fill fill section, leding leding to to cretion cretion of of uniform uniform film film flo flo through fill. fill. Figure. Flo through counter-flo cooling toer. Figure. Flo through counter-flo cooling toer. The mgnitude of het nd mss trnsfer processes occurring in counter-flo cooling toer The of interest mgnitude cn be ofmmticlly het nd mss determined trnsfer by processes obtining occurring solution inof counter-flo nonliner system cooling toercomprising of interes cn folloing be mmticlly blnce equtions: determined A Liquid bycontinuity; obtiningb Liquid solution energy of blnce; nonliner C Wter system comprising vpor continuity; folloing D Air/ter blnce equtions: vpor energy A blnce; LiquidE continuity; Mechnicl B energy Liquid blnce. energy In blnce; derivtion C Wter vporprocess continuity; of se D equtions, Air/ter vpor cooling energy toer model blnce; hs E undergone Mechnicl severl energy ssumptions, blnce. such In s: derivtion process. ofir se nd equtions, ter strem tempertures cooling toer re uniform model hs t ny undergone cross section; severl ssumptions, such s: 2. cross-sectionl re of cooling toer is ssumed to be uniform;. ir. nd ter het nd strem mss trnsfer tempertures only occur re in uniform direction t nynorml cross section; to flos; 2. cross-sectionl re of cooling toer is ssumed to be uniform;. het nd mss trnsfer only occur in direction norml to flos; 4. het nd mss trnsfer through toer lls to environment is neglected;

4 Energies 206, 9, of Energies 206, het 9, 028 nd mss trnsfer through toer lls to environment is neglected; 4 of het trnsfer from cooling toer fn nd motor ssembly to ir is neglected; 6. ir nd ter vpor is considered mixture of idel gsses; 5. het trnsfer from cooling toer fn nd motor ssembly to ir is neglected; 7. flo beteen flt pltes is unsturted through fill section. 6. ir nd ter vpor is considered mixture of idel gsses; 7. It is flo orth beteen specifying flt pltes tht this unsturted ork pplies through to cooling fill toers section. of moderte size, for hich it is possible to neglect occurrence of het nd mss trnsfer phenomen in rin section. Figure It is 2 displys orth specifying verticl tht nodliztion this ork pplies chosen to for cooling fill toers section. of Figure moderte offers size, for more hich detiled it is description possible to neglect of het occurrence nd mss trnsfer of het processes nd mss occurring trnsfer t phenomen interfce beteen rin section. ter Figure film 2 nd displys ir flo verticl ithin nodliztion single control chosenvolume. for fill section. Figure offers more detiled description of As het reminded nd mss bove, trnsfer hen processes cooling occurring toer t is operted interfce nturl beteen drft/ind-ided ter film nd mode ir mss flo ithin florte single of dry control ir becomes volume. n dditionl unknon. Even ith fn off though, hot ter floing As reminded through bove, cooling hentoer cooling ill cuse toer ir is to operted continue nturl to flo drft/ind-ided through toer mode due to buoyncy, mss florte ith ofpossible dry ir becomes flo enhncements dditionl cused unknon. by Even ind ith pressure fnt off though, ir inlet hotto ter cooling floing through toer. The cooling stte functions toer ill underlying cuse ir to continue cooling to flo toer through model cf., toer Figures due to buoyncy, re s follos: ith possible flo enhncements cused by ind pressure t ir inlet to cooling toer. The stte functions underlying cooling toer model cf., Figures re s follos:. ter mss flo rtes, denoted s i m i 2,...,50, t exit of ech control volume, i,. long ter height mss flo of rtes, fill section denoted of s m cooling i i toer; 2,..., 50, t exit of ech control volume, i, 2. long ter tempertures, height of fill denoted sections of i T cooling i 2,...,50 toer;, t exit of ech control volume, i, long 2. ter tempertures, denoted s T i height of fill section of cooling i toer; 2,..., 50, t exit of ech control volume, i, long. ir height tempertures, of fill section denoted of s cooling i T i,...,49 toer;, t exit of ech control volume, i, long. ir tempertures, denoted s T i height of fill section of cooling i, toer;..., 49, t exit of ech control volume, i, long height of fill section of cooling toer; 4. humidity rtios, denoted s i ω 4. humidity rtios, denoted s ω i i,...,49, t exit of ech control volume, i, long i,..., 49, t exit of ech control volume, i, long height of fill section of cooling toer. height of fill section of cooling toer. 5. ir mss flo rtes, denoted s m 5. ir mss flo rtes, denoted s, constnt long height of fill section of cooling m, constnt long height of fill section of toer. cooling toer. Fill section nodliztion comprising 49 control volumes i i I,..., ; I both 49; both forrd forrd stte functions m stte functions i i, i, T, i i, T, ω i i, ω i, m ; i,..., I nd djoint stte functions selected m T T, m ; i,.., I nd djoint stte functions selected µ i, τ i, τ i, i, µ ; i,..., I re shon in picture. i i i i μ, τ, τ, ο, μ ; i,.., I re shon in picture. Figure Fill section nodliztion comprising 49 control volumes,.., 49

5 Energies 206, 9, of 52 Energies 206, 9, of 57 Figure. Het nd mss trnsfer phenomen t interfce beteen donrd ter film nd uprd ir ithin single control volume of cooling toer s fill section. Here nd in folloing, bove stte functions re ssumed to be components of Here nd in folloing, bove stte functions re ssumed to be components of folloing column vectors: folloing column vectors: 2 I + m,..., m m, 2 I + T,..., T T, I T,..., T T, I ω ω,..., ω, m m m 2,..., m I+, T T 2,..., T I+, T T,..., T I, ω ω,..., ω I, m In this pper s nottion, dgger is used to denote trnsposition, nd ll vectors in this ork In this re pper s column nottion, vectors. Becuse dgger of forementioned is used to denote similrities trnsposition, ith prtilly nd ll vectors sturted in cse this ork in 2, re column constitutive vectors. eqution Becuse system of forementioned governing similrities cooling toer ith model prtilly of interest sturted is closely cse comprble in 2, constitutive ith one eqution provided system in Equtions governing 2 5 cooling in 2. toer For this model unsturted of interest nlysis, is closely governing comprble equtions ith ithin one provided totl of I in 49 Equtions control volumes 2 5 represented in 2. For in this Figure unsturted 2 re s follos nlysis, : governing equtions ithin totl of I 49 control volumes represented in Figure 2 re s A. follos Liquid : Continuity Equtions: A. i Control LiquidVolume Continuity i : Equtions: 2 2 ω i 2 Mm, P T, P vs tm N Control α α m, TVolume, T, ω, mi ; α : m m + 0;, in 2 2 R T T ω N ii Control Volumes m i, T 2,...,, T I, ω, m : ; α m 2 m,in + Mm,α P vs 2 T 2,α 0; 2 R i+ i+ i ii i i+ i Mm, P T, ω P vs tm N Control α α m, TVolumes, T, ω, mi ; α 2,..., mi : m + + 0; i i i R T T ω N i iii Control Volume m i, T I:, T, ω, m ; α m i+ m i + Mm,α P vs i+ T i+,α R 0; I+ I+ I iii I I+ I Mm, P T, ω P vs tm N Control α α m, T Volume, T, ω, m i ; α I: m m + + 0; I I I R T T ω 4 N I m, T, T, ω, m ; α m I+ m I + Mm,α P vs I+ T I+,α R 0; 4 B. Liquid Energy Blnce Equtions: B. Liquid Energy Blnce Equtions: i Control Volume i : T 2 T i+ T I+ ω P tm T ω ω i P tm T i ω i ω I P tm T I ω I N m, T, T, ω, m ; α m h T, α T T H m, α 2 2, in f, in m h T, α m m h T, α 0; f, in g, 5 ii Control Volumes i 2,..., I :

6 Energies 206, 9, of 52 i Control Volume i : N 2 m, T, T, ω, m ; α m,in h f T,in, α T 2 T Hm, α m 2 h 2 f T 2, α m,in m 2 h 2 g,t 2, α 0; 5 ii Control Volumes i 2,..., I : N i 2 m, T, T, ω, m ; α m i m i+ h i+ f T i+ h i f, α m i T i, α T i+ T i Hm, α m i+ h i+ g, T i+, α 0; 6 iii Control Volume i I: N I 2 m, T, T, ω, m ; α m I m I+ h I+ f T I+ h I f, α m I T I, α T I+ T I Hm, α m I+ h I+ g, T I+, α 0; 7 C. Wter Vpor Continuity Equtions: i Control Volume i : m, T, T, ω, m ; α ω 2 ω + m.in m 2 m N 0; 8 ii Control Volumes i 2,..., I : N i m, T, T, ω, m ; α ω i+ ω i + mi m i+ m 0 ; 9 iii Control Volume i I: N I m, T, T, ω, m ; α ω in ω I + mi m I+ m 0 ; 0 D. The Air/Wter Vpor Energy Blnce Equtions: i Control Volume i : C p T , α ω h g, T + T2 T Hm,α m + m,in m 2 h 2 g,t 2,α m + ω 2 h 2 g, T 2, α 0 ; N 4 m, T, T, ω, m ; α T 2 T, α ii Control Volumes i 2,..., I : N i 4 m, T, T, ω, m ; α T i+ + Ti+ T i Hm,α m + mi m i+ T i h i+ g, C i p Ti T i+,α 2, α ω i h i m + ω i+ h i+ g, g,t i, α T i+, α 0 ; 2 iii Control Volume i I: N I 4 m, T, T, ω, m ; α T,in T I + TI+ T I Hm,α m + mi m I+ C p I TI h I+ g, T I+,α 2, α ω I h I g,t I m + ω in h g, T,in, α 0., α

7 Energies 206, 9, of 52 E. The Mechnicl Energy Blnce Eqution: N 5 m, T, ω, T, m ; α 2ρT db,α A out α 2 A in α 2 + k sum A f ill 2 + f 2ρT tdb,α 96 L f ill α R e m,α m A 2 f ill D m h gzαρt db, α V 2 ρt db,α 2 + z rin gρt db, α + gρt, α z 4 2 α +g zα P tm R ir 2T,in + + I 2T 0; i2 T i 4 The vector α, hich ppers in Equtions 2 4, comprises s its components 47 model prmeters, hich ill be denoted in folloing s α i, i.e., α α,..., α Nα 5 here N α 47 represents totl number of model prmeters. These model prmeters vlue nd stndrd devitions hve been experimentlly derived, cusing ir sttisticl distributions not to be completely knon; first four sttisticl moments mens, vrince/covrince, skeness, nd kurtosis of ech of se prmeter distributions hve neverless been quntified, s presented in Appendix B. As discussed in 2, numericl method used in originl ork to solve Equtions 2 4 filed to chieve convergence for ll considered dt sets, nd for this reson it s substituted ith more ccurte nd efficient one, bsed on Neton s method toger ith GMRES liner itertive solver for sprse mtrices 9 comprised in NSPCG pckge 0. This method hs been thoroughly described in 2, Section 2.. The Jcobin mtrix of derivtives of Equtions 2 4 ith respect to stte functions is denoted s: J u n A B C D A 2 B 2 C 2 D 2 A B C D A 4 B 4 C 4 D 4 A 5 B 5 C 5 D 5 E E 2 E E 4 E 5, 6 The mjority of components of this block mtrix remin unltered from Jcobin mtrix in 2, Eqution 20, hose components re detiled in 2, Appendix C; components hich re cused by unsturted operting conditions to be different from study in 2 re seprtely detiled in Appendix C of this pper. The bove mtrix J u n is non-symmetric sprse mtrix of order 97 by 97; digonl storge method used in NSPCG to relevntly reduce mtrix size, toger ith pproximtion introduced by setting column vectors E,..., E 4 nd ro vectors A 5,..., D 5 to zero, re discussed in 2, Section 2.. As detiled in Appendix A, ech of se dt sets comprises mesurements of folloing quntities: i Outlet ir temperture mesured ith Tidbit sensor; ii Outlet ir temperture mesured ith Hobo sensor; iii Outlet ter temperture; iv Outlet ir reltive humidity. Accordingly to forementioned quntities contined in benchmrk dt sets, folloing responses of interest hve been chosen for this ork: vector m m 2,..., m I+ of ter mss flo rtes t exit of ech control volume i, b i,..., 49; vector T i,..., 49; c vector T i,...,49; T 2 T,..., T I+ of ter tempertures t exit of ech control volume i,,..., T I of ir tempertures t exit of ech control volume i,

8 Energies 206, 9, of 57 c vector i,...,49 ; I T,..., T T of ir tempertures t exit of ech control volume i, Energies 206, 9, of 52 I d vector RH,..., RH RH, hving s components ir reltive humidity t exit of d vector RH RH,..., RH I ech control volume i, i,...,49 ;, hving s components ir reltive humidity t exit e of ech sclr control m, representing volume i, i,..., ir 49; mss flo rte long height of cooling toer e vlue sclr of m ir, representing mss florte. ir mss flo rte long height of cooling toer vlue of ir mss florte. i i It is importnt to note tht ter mss flo rtes m, ter tempertures T, ir tempertures It is importnt T i nd to note tht ir mss ter flo rte mssm flo rtes m i, ter tempertures T i, ir tempertures T i re computed by solving Equtions 2 4, hile nd ir mss flo i ir reltive humidity vlue, rte m re computed by solving Equtions 2 4, hile ir reltive humidity vlue, RH i RH, is obtined through folloing expression:, is obtined through folloing expression: i ω P RH i P v i ω i tm i P ω, α i P tm i v RH 00 ω ω P vs T i 00 i i P T, α vs 0 i T e e 0+ T i For unperturbed bse cse, ll model prmeters α i re used in solving Equtions 2 4 ith For ir nominl unperturbed vlues, bse hich cse, re lllisted model in prmeters Appendix α B. i It re is orth used inspecifying solving Equtions tht nominl 2 4 ith vlues ir for nominl prmeters vlues, α hich re listed in Appendix B. It is orth specifying tht nominl through α 5 i.e., dry bulb ir temperture, de point temperture, vlues for prmeters α through α 5 i.e., dry bulb ir temperture, de point temperture, inlet inlet ter temperture, tmospheric pressure nd ind speed re sttisticlly computed by ter temperture, tmospheric pressure nd ind speed re sttisticlly computed by verging verging vlues of respective quntities in 677 dt sets considered for this study. vlues of respective quntities in 677 dt sets considered for this study. The br plots displyed belo in Figures 4 7 sho, t exit of ech of 49 control volumes, The br plots displyed belo in Figures 4 7 sho, t exit of ech of 49 control volumes, vlues of ter mss flo rtes i i, ter tempertures T i i, ir tempertures i vlues of ter mss flo rtes m, ter tempertures T, ir tempertures T i,, nd ir reltive humidity, RH i i nd ir reltive humidity, RH,, respectively. respectively. Figure 4. Br plot of ter mss flo rtes m i, i 2,..., 50, t exit of ech control volume long Figure 4. height Br plot of of fill ter section mss of flo cooling rtes m toer. i, i 2,...,50, t exit of ech control volume long height of fill section of cooling toer.

9 Energies 206, 9, 028 Energies 206, 028 Energies206, 206,9,9, 9, Energies 9 of of ofof i iii Figure 5. Br plot of outlet ter temperture,,tt exit of ech control Figure 5.Br Br plot of outlet outlet ter temperture T texit exit exit ofech echcontrol control 2,...,50 i2,...,50 2,...,50 Figure plot of ter temperture,,, 2, of ii..., Figure 5. 5.Br plot of outlet ter temperture T,TT i 50, t, of ech control volume long height of fill section of cooling toer. volume long height of fill section of cooling toer. volumelong long height heightof of fill fillsection sectionof of cooling coolingtoer. toer. volume i iii Figure 6. 6.Br plot of of outlet irir temperture T TT,T i,444t ech volume Figure 6. Br plot of outlet ir temperture,,,..., exit of ech control Figure 6.Br Br plot of outlet irtemperture temperture texit of exit ofcontrol echcontrol control,...,,49,...,...,,,,tt Figure plot outlet exit of ech 999, iii long long height of height fillof section of cooling. volume long height of fill section of cooling. volume long height of fill fill section of cooling. cooling. volume section of Figure Br plot of outlet ir reltive humidity t exit of ech control Figure Br Brplot plotof of outlet outletir irreltive reltivehumidity humidity RH t exit exitof ofech echcontrol control RH RH iii,,, iii,, Figure...,,...,...,444999,,,t Figure 7. Br plot of outlet ir reltive humidity RH i, i,..., 49, t exit of ech control volume long height of fill section of cooling. volume long height of fill section of cooling. volumelong long height heightof of fill fill section section of volume of cooling. cooling.

10 Energies 206, 9, of 52. Development of Cooling Toer Adjoint Sensitivity Model The development of cooling toer djoint sensitivity model follos sme pth s in 2, Section., here topic is discussed more in detil. The totl sensitivity of model response R m, T, T, ω, m ; α, ith respect to rbitrry vritions in model s prmeters δα δα,..., δα Nα nd stte functions δm, δt, δt, δω, δm, round nominl vlues m 0, T 0, T 0, ω 0, m 0 ; α 0 of prmeters nd stte functions, is obtined by mens of G-differentil of model s response to se chnges. This G-differentil is referred to s DR m 0, T 0, T 0, ω 0, m 0 ; α 0 ; δm, δt, δt, δω, δm ; δα, nd introducing djoint sensitivity functions it becomes: DR m 0, T 0, T 0, ω 0, m 0 ; α 0 N α R ; δm, δt, δt, δω, δm ; δα δα α i + DR indirect 8 i i here indirect effect term, DR indirect, is obtined s: DR indirect µ Q + τ Q 2 + τ Q + o Q 4 + µ Q 5 9 nd here vector µ, τ, τ, o, µ is required to be solution of folloing djoint sensitivity system: A A 2 A A 4 A 5 µ R B B 2 B B 4 B 5 τ R 2 C C 2 C C 4 C 5 τ D D 2 D D 4 D 5 R, 20 o R 4 E E 2 E E 4 E 5 µ R 5 The vectors R l r l,..., r I l of model responses ith respect to stte functions, i.e.,: r i R m i+ nd here R 5 is defined s follos: hile vectors Q l q l, l, 2,, 4 in Eqution 20 comprise functionl derivtives ; r i 2 R T i+ ; r i R 5 R T i model s equtions ith respect to model prmeters, i.e.,: q i l ; r i 4 R ; i,..., I. 2 ω i R m 22,..., q I l, l, 2,, 4 in Eqution 9 comprise derivtives of N α j i N l δα α j ; i,..., I; l, 2,, 4. 2 j nd here Q 5 is defined s follos: Q 5 N α N 5 δα α j. 24 j j It is orth reminding tht djoint sensitivity system in Eqution 20 is independent of prmeter vritions. This feture llos selected djoint functions µ, τ, τ, o, µ to be computed by solving djoint sensitivity system just once.

11 Energies 206, 9, 028 of 52 Dimensionl nlysis llos determining units of djoint functions from Eqution 9. Nmely, units for djoint functions must stisfy folloing reltions: µ i R N ; τ i R N 2 ; τ i R N ; o i R N 4 ; µ R N 5 25 here R denotes unit of response R, hile units for respective equtions re s follos: N kg s ; N 2 J s ; N ; N 4 J kg ; N 5 J m 26 Tble belo lists units of djoint functions for five responses: R T, R T 50, R RH, R m 50 nd R m, respectively, in hich, T denotes exit ir temperture; T 50 denotes exit ter temperture; RH denotes exit ir reltive humidity; m 50 denotes exit ter mss flo rte; nd m denotes ir mss flo rte. Responses Tble. Units of djoint functions for different responses. µ i τ i τ i o i µ R T K/ kg/s K/J/s K K/ J/kg K/ J/m R T 50 K/ kg/s K/J/s K K/ J/kg K/ J/m R RH kg/s J/s J/kg J/m R m 50 J/kg kg/s kg/s / J/kg kg/s / J/m R m J/kg kg/s kg/s / J/kg kg/s / J/m Figures 8 2 belo disply br plots of djoint functions corresponding to five mesured responses of interest, nmely: i The exit ir temperture R T ; ii The outlet exit ter temperture R T 50 ; iii The exit ir humidity rtio R RH ; iv The outlet exit ter mss flo rte R m 50 ; nd v ir mss flo rte R m. Let S j denote bsolute sensitivity of response R ith respect to prmeter α j, nd is defined s: S j R I α j i µ i N i α j + τ i N i 2 α j + τ i N i + o α j i Ni 4 α j + µ N 5 α j An independent method to compute bsolute response sensitivities S j is by mking use of perturbtive finite difference methods, such s:. considering n rbitrrily smll perturbtion δα j to model prmeter α j ; 2. re-computing perturbed response R α 0 j + δα j, here α 0 j denotes unperturbed prmeter vlue;. using finite difference formul S FD j 27 R α 0 j + δα j R α 0 j + O 2 δα δα j 28 j 4. using pproximte equlity beteen Equtions 27 nd 28 to obtin independently respective vlues of djoint functions being verified. The independent verifiction methodology discussed in steps 4 bove ill be clerly illustrted in Appendix D, here djoint functions depicted in Figures 8 2 ill be verified.

12 S FD j 0 0 α + δα R j j αj R + O δα j δα j using pproximte equlity beteen Equtions 27 nd 28 to obtin independently respective vlues of djoint functions being verified. Energies 206, 9, of 52 Figure 8. Br plots of djoint functions for response R T s functions of height of cooling toer s fill fill section: 49 µ μ µ μ,,..., µ 49 μ ; b τ τ τ,...,,..., τ τ 49 ; c ; c τ τ τ,...,, τ..., τ 49 ; d; d o o 49 o o,...,,..., o o. 49. For response For response R T R T,, vlue of djoint function µ vlue of djoint function μ is is

13 Energies 206, 9, 028 of 52 Energies 206, 9, 028 of 57 Figure Br plots of djoint functions R for response R T 50 s functions of height of cooling toer s fill fill section: µ μ µ μ,..., μ, µ 49 ; b τ τ τ τ,...,, τ 49 ; ; c τ τ,..., τ,..., τ 49 ; d; d 49 o o o,..., o o,..., o. 49 For. For response response R T R 50, T 50 vlue, vlue of of djoint djoint function function μ is µ is 0.77.

14 Energies 206, 9, of 52 Energies 206, 9, of 57 Figure Figure Br Br plots plots of of djoint djoint functions functions for response s functions of height of for response R RH s functions of height of 49 cooling toer s fill fill section: µ μ µ μ,..., μ, µ 49 ; b τ τ,..., τ, τ 49 ; ; c τ τ,...,,..., τ τ 49 ; d; d 49 o o o,..., o o,..., o. 49. For response For response R RH R RH,, vlue of djoint function µ vlue of djoint function μ is is

15 Energies 206, 9, of 52 Energies 206, 9, of 57 R m, s functions of height of Figure. Br plots of of djoint functions for response R m 50, s functions of height of cooling toer s fill section: µ µ cooling toer s fill section: μ,..., µ 49 ; b τ μ,..., μ τ ; b τ,..., τ 49 ; c τ τ,..., τ ; c,..., τ τ,..., τ 49 ; ; d d o o,..., o 49. For response R m 50, vlue of djoint function µ is o o,..., o. For response R m, vlue of djoint function μ is

16 Energies 206, 9, of 52 Energies 206, 9, of 57 Figure 2. Br Br plots of of djoint functions for for response R m,, s s functions of of height of of cooling toer s fill section: µ µ cooling toer s fill section: μ,..., µ 49 ; b τ μ,..., μ τ ; b τ,..., τ 49 ; c τ τ,..., τ τ ; c,..., τ τ,..., τ 49 ; ; d d o o,..., o 49. For response R m, vlue of djoint function µ is o o,..., o. For response R m, vlue of djoint function μ is Sensitivity Anlysis Results nd Rnkings The As hs independent been discussed verifiction bove, methodology re re totl discussed of 8079 mesured in steps 4 benchmrk bove dt ill sets be for clerly illustrted cooling toer in Appendix model operted D, here in fn-off djoint regime. functions As it hs depicted lso been in Figures mentioned, ill benchmrk verified. dt sets out of totl of 8079 dt sets re considered to correspond to unsturted conditions.. hich Sensitivity re nlyzed Anlysis in this Results ork. nd The Rnkings nominl vlues for boundry nd tmospheric conditions used in this As ork hs been erediscussed obtined from bove, re sttistics re of totl se of mesured benchmrk benchmrk dt sets corresponding dt sets for to unsturted cooling toer conditions. model operted In turn, in fn-off se unsturted regime. As it boundry hs lso been nd tmospheric mentioned, 677 conditions benchmrk ere dt used sets to obtin out of sensitivity totl of results 8079 reported dt sets inre thisconsidered Subsection. to Sections correspond....5, to belo, unsturted provide conditions numericl hich vlues re nd nlyzed rnkings, in inthis descending ork. The order, nominl of vlues reltivefor sensitivities boundry computed nd tmospheric using conditions djoint sensitivity used in nlysis this ork methodology ere obtined for from five model sttistics responses of se T, 677 T 50, benchmrk m 50, RH dt, ndsets m. corresponding Note thtto unsturted reltive sensitivity conditions. RS α In i turn, of se response unsturted R α i to boundry prmeter nd αtmospheric i is defined conditions s RS α i ere dr used α i /dα to obtin i α i /R α sensitivity i. Thus, results reltive reported sensitivities in this Subsection. re unit-less Sections nd....5, re very belo, useful in provide rnking numericl prmeters vlues to highlight nd rnkings, ir reltive in descending importnce order, forof respective reltive sensitivities response.

17 Energies 206, 9, of 52 Thus, reltive sensitivity of.00 indictes tht chnge of % in respective prmeter ill induce % chnge in response tht is liner in respective sensitivity. The higher reltive sensitivity, more importnt respective prmeter to respective response.... Sensitivity Anlysis Results nd Rnkings for Outlet Air Temperture, T Tble 2 lists sensitivities, computed using Equtions 8 nd 9, of ir outlet temperture ith respect to ll of model s prmeters. The prmeters hve been rnked ccording to descending order of ir reltive sensitivities. Tble 2. Rnked reltive sensitivities of outlet ir temperture, T. Rnk # Prmeter α i Nominl Vlue Rel. Sens. RS α i Rel. Std. Dev. % Inlet ter temperture, T,in K Air temperture dry bulb, T db K Inlet ir temperture, T,in K P vs T prmeter, De point temperture, T dp K P vs T prmeter, Wind speed, V.859 m/s Fill section equivlent dimeter, D h 0.08 m Fn shroud inner dimeter, D f n 4. m Atmospheric pressure, P tm 00,588 P Wter enthlpy h f T prmeter, f Nu prmeter, 0,Nu Fill section surfce re, A sur f 422 m Wetted frction of fill surfce re, ts Fill section flo re, A f ill m Inlet ir humidity rtio, ω in Inlet ter mss florte, m,in kg/s Dynmic viscosity of ir t T 00 K, µ kg/m s Fill section frictionl loss multiplier, f Fill section height, z f ill 2.0 m D v T prmeter,,dv C p T prmeter, 0,cp Therml conductivity of ir t T 00 K, k ir W/m K Het trnsfer coefficient multiplier, f ht h g T prmeter, 0g 2,005, Mss trnsfer coefficient multiplier, f mt D v T prmeter, 2,dv D v T prmeter, 0,dv h f T prmeter, 0 f,4, Kinemtic viscosity of ir t 00 K, ν m 2 /s Prndlt number of ir t T 80 C, Pr Schmidt number, Sc h g T prmeter, g Sum of loss coefficients bove fill, k sum D v T prmeter,,dv Drift elimintor thickness, z de m C p T prmeter,,cp Cooling toer deck idth in x-dir, W dkx 8.5 m Cooling toer deck idth in y-dir, W dky 8.5 m Cooling toer deck height bove ground, z dk 0.0 m C p T prmeter, 2,cp Fn shroud height, z f n.0 m Rin section height, z rin.6 m Bsin section height, z bs.68 m Nu prmeter,,nu Nu prmeter, 2,Nu Nu prmeter,,nu As results in Tble 2 indicte, first prmeter i.e., T,in hs reltive sensitivity round 90%, nd is refore most importnt for ir outlet temperture response, T, since tht mens tht % chnge in T,in ould induce 0.9% chnge in T. The next four prmeters i.e., T db, T,in, 0, T dp hve reltive sensitivities beteen % nd 6%, nd re refore someht importnt. Prmeters #6 through #9 i.e.,, V, D h, D f n hve reltive sensitivities beteen 0.% nd 0.8%.

18 Energies 206, 9, of 52 The remining 8 prmeters re reltively unimportnt for this response, hving reltive sensitivities smller thn % of lrgest reltive sensitivity ith respect to T,in for this response. Positive sensitivities imply tht positive chnge in respective prmeter ould cuse n increse in response, hile negtive sensitivities imply tht positive chnge in respective prmeter ould cuse decrese in response...2. Sensitivity Anlysis Results nd Rnkings for Outlet Wter Temperture, T 50 The results nd rnking of reltive sensitivities of outlet ter temperture ith respect to most importnt nine prmeters for this response re listed in Tble. The lrgest sensitivity of T 50 is to prmeter T,in, nd hs vlue of ; this mens tht % increse in T,in ould induce % increse in T 50. The sensitivities to remining 8 model prmeters hve not been listed since y re smller thn % of lrgest sensitivity ith respect to T,in for this response. Tble. Most importnt reltive sensitivities of outlet ter temperture, T 50. Rnk # Prmeter α i Nominl Vlue Rel. Sens. RS α i Rel. Std. Dev. % Inlet ter temperture, T,in K Inlet ir temperture, T,in K Air temperture dry bulb, T db K De point temperture, T dp K P vs T prmeters, P vs T prmeters, Inlet ir humidity rtio, ω in Wter enthlpy hft prmeter, f Wind speed, V.859 m/s Sensitivity Anlysis Results nd Rnkings for Outlet Wter Mss Flo Rte, m 50 The results nd rnking of reltive sensitivities of outlet ter mss flo rte ith respect to most importnt 2 prmeters for this response re listed in Tble 4. This response is most sensitive to m,in % increse in this prmeter ould cuse.0% increse in response nd second lrgest sensitivity is to prmeter T,in % increse in this prmeter ould cuse 0.24% decrese in response. The sensitivities to remining 5 model prmeters hve not been listed since y re smller thn % of lrgest sensitivity ith respect to m,in for this response. Tble 4. Most importnt reltive sensitivities of outlet ter mss flo rte, m 50. Rnk # Prmeter α i Nominl Vlue Rel. Sens. RS α i Rel. Std. Dev. % Inlet ter mss flo rte, m,in kg/s Inlet ter temperture, T,in K De point temperture, T dp K Inlet ir temperture, T,in K Air temperture dry bulb, T db K P vs T prmeters, P vs T prmeters, Inlet ir humidity rtio, ω in Wind speed, V.859 m/s Fn shroud inner dimeter, D f n 4. m Fill section equivlent dimeter, D h 0.08 m Fill section flo re, A f ill m

19 Energies 206, 9, of Sensitivity Anlysis Results nd Rnkings for Outlet Air Reltive Humidity, RH The results nd rnking of reltive sensitivities of outlet ir reltive humidity ith respect to most importnt 29 prmeters for this response re listed in Tble 5. The first three sensitivities of this response re most relevnt; in prticulr, n increse of % in T db or T,in ould cuse n increse in response of 0.27% or 0.25%, respectively. On or hnd, n increse of % in T,in ould cuse decrese of 0.2% in response. The sensitivities to remining 8 model prmeters hve not been listed since y re smller thn % of lrgest sensitivity ith respect to T,in for this response. Tble 5. Most importnt reltive sensitivities of outlet ir reltive humidity, RH. Rnk # Prmeter α i Nominl Vlue Rel. Sens. RS α i Rel. Std. Dev. % Inlet ter temperture, T,in K Air temperture dry bulb, T db K Inlet ir temperture, T,in K De point temperture, T dp K D v T db prmeter,,dv Fill section equivlent dimeter, D h 0.08 m Mss trnsfer coefficient multiplier, f mt D v T db prmeter, 2,dv Wind speed, V.859 m/s D v T db prmeter, 0,dv Fill section surfce re, A sur f 422 m Wetted frction of fill surfce re, ts Nu prmeter, 0,Nu Fn shroud inner dimeter, D f n 4. m Therml conductivity of ir t T 00 K, k ir W/mK Het trnsfer coefficient multiplier, f ht C p T prmeter, 0,cp P vs T prmeters, Kinemtic viscosity of ir t 00 K, ν m 2 /s Prndlt number of ir t T 80 C, Pr Schmidt number, Sc Fill section flo re, A f ill m D v T prmeter,,dv Dynmic viscosity of ir t T 00 K, µ kg/m s Fill section frictionl loss multiplier, f P vs T prmeters, Atmosphere pressure, P tm 00,588 P Fill section height, z f ill 2.0 m Inlet ir humidity rtio, ω in Reltive Sensitivities of Air Mss Flo Rte, m The results nd rnking of reltive sensitivities of ir mss flo rte ith respect to most importnt 4 prmeters for this response re listed in Tble 6. The first three sensitivities of this response re very lrge reltive sensitivities lrger thn unity re customrily considered to be very significnt. In prticulr, n increse of % in T,in or T db ould cuse decrese in response of 8.5% or 8.49%, respectively. On or hnd, n increse of % in T,in ould cuse n increse of 6% in response. The sensitivities to remining model prmeters hve not been listed since y re smller thn % of lrgest sensitivity ith respect to T,in for this response. Overll, ir mss flo rte, m, displys lrgest sensitivities, so this response is most sensitive to prmeter vritions. The or responses, nmely outlet ir temperture, outlet ter temperture, outlet ter mss flo rte nd outlet ir reltive humidity disply sensitivities of comprble mgnitude.

20 Energies 206, 9, of 52 Tble 6. Most importnt reltive sensitivities of ir mss flo rte, m. Rnk # Prmeter α i Nominl Vlue Rel. Sens. RS α i Rel. Std. Dev. % Inlet ir temperture, T,in K Air temperture dry bulb, T db K Inlet ter temperture, T,in K Atmosphere pressure, P tm 00,588 P Wind speed, V.859 m/s Fn shroud inner dimeter, D f n 4. m P vs T prmeters, Fill section equivlent dimeter, D h 0.08 m De point temperture, T dp K Fill section flo re, A f ill m P vs T prmeters, Dynmic viscosity of ir t T 00 K, µ kg/m s Fill section frictionl loss multiplier, f Fill section height, z f ill 2.0 m Cross-Comprison of Sensitivity Results In Tbles 7, rnked reltive sensitivities for ech response re compred side-by-side beteen three operting conditions, i.e., to subcses discussed in 2 prtilly sturted subcse I, completely sturted subcse II nd unsturted cse nlyzed in this pper. Among three operting conditions, unsturted cse is defined s orking condition in hich ir is unsturted from inlet to outlet of cooling toer; hile in sturted subcse II, on contrry, ir is sturted from inlet to outlet of cooling toer; sturted subcse I is combintion of se to cses, i.e., ir in loer portion of fill section of cooling toer is in unsturted conditions, reching sturtion t some point long height of toer nd remining sturted in upper prt of cooling toer. Cross-comprison of sensitivity results revels sensitivity vritions beteen three operting conditions. The reltive sensitivities nd corresponding prmeters listed in Tble 7 re extrcted from Tbles nd 6 in, nd Tble 2 in this pper. As shon in Tble 7, for ll three operting conditions, first most sensitive prmeters of response of ir outlet temperture, T, is sme i.e., T,in. The 2nd nd rd most sensitive prmeter re inverted in unsturted cse ith respect to to subcses of sturted cse, but ith vlues very close beteen to prmeters. The prmeters tht rnks in 4th plce for this response is sme for ll cses i.e., 0. The 5th prmeter is 0 for Subcses I nd II nd T dp for unsturted cse. For first prmeter i.e., T,in, unsturted cse displys lrgest sensitivity for this response; subcse II hs smllest sensitivity; hile subcse I hs n intermedite vlue of sensitivity beteen to. This is expected since subcse I is mixed cse beteen unsturted cse nd sturted subcse II, s explined bove. For ll remining prmeters in tble sitution is reversed, ith Subcse II shoing lrgest sensitivity vlues nd unsturted cse presenting smllest ones, ith Subcse I still in middle. Generlly, sensitivity mgnitude of subcse I is slightly closer to tht of subcse II. This cn be explined by fct tht ir remins unsturted less thn hlf of height of fill section, nd flos in sturted conditions for more thn hlf of height of fill section, s nlyzed in 2. The reltive sensitivities nd corresponding prmeters listed in Tble 8 re extrcted from Tbles 2 nd 7 in, nd Tble in this pper. As shon in Tble 8, for response of ter outlet temperture, T 50, both unsturted cse nd subcse I re most sensitive to prmeter T,in, heres subcse II is most sensitive to prmeter T,in. As comprison, response of ter outlet temperture to prmeter T,in rnks in rd plce, ith vlue comprble to or to cses. The next to most sensitive prmeters tht rnk from 2nd to rd plces of this response re lso different beteen operting conditions: for both unsturted cse nd subcse I, prmeters T,in nd T db rnk in 2nd nd rd plces, respectively; hoever, for subcse II, prmeters tht tke 2nd nd rd plces re T db nd T,in, respectively. The prmeters tht tke 4th nd 5th plces re lso different beteen operting conditions, s shon in tble. Overll, for response

21 Energies 206, 9, of 52 of ter outlet temperture, T 50, sensitivity behvior of subcse I is more similr to tht of unsturted cse. The reltive sensitivities nd corresponding prmeters listed in Tble 9 re extrcted from Tbles nd 8 in, nd Tble 4 in this pper. As shon in Tble 9, for ll three operting conditions, first to most sensitive prmeters of response of ter outlet mss flo rte, m 50, re sme i.e., m,in nd T,in, respectively. In ddition, for ech of first to prmeters, ll three operting conditions hve comprble sensitivity mgnitudes. This indictes tht sensitivities of first to prmeters re insensitive to operting condition chnge. The third most sensitive prmeter of this response is different beteen operting conditions: for both unsturted cse nd subcse I, this prmeter is T dp ; heres for subcse 2, this prmeter is T,in. Similrly, prmeters tht tke 4th nd 5th plces re lso different beteen operting conditions, s shon in tble. The reltive sensitivities nd corresponding prmeters listed in Tble 0 re extrcted from Tbles 4 nd 9 in, nd Tble 5 in this pper. As shon in Tble 0, for Subcses I nd II, first three most sensitive prmeters of response of ir outlet reltive humidity, RH, re sme i.e., T,in, T db nd T dp, respectively; order is different for unsturted cse. The next to most sensitive prmeters tht rnk 4th nd 5th plces of this response re different beteen operting conditions. For ech of first three prmeters, ll three operting conditions re sensitive to prmeter chnges. In hich, subcse II is most sensitive cse; nd unsturted cse is lest sensitive cse comprtively. For instnce, % chnge in T,in, T db or T dp ill cuse round 0.2% chnge in RH for unsturted cse, round 2% chnge in RH for subcse I; nd nerly 5% chnge in RH for subcse II, respectively. Overll, for response of ir outlet reltive humidity, RH, sensitivity behvior of subcse I is lso more similr to tht of subcse II, s lso signs of most of sensitivity vlues, inverted in unsturted cse ith respect to Subcse I nd II, sho in Tble 0. The reltive sensitivities nd corresponding prmeters listed in Tble re extrcted from Tbles 5 nd 0 in, nd Tble 6 in this pper. As shon in Tble, for ll operting conditions, first three most sensitive prmeters of response of ir mss flo rte, m, re sme i.e., T db, T,in nd T,in, respectively ith order of first to being spped for unsturted cse. P tm is 4th more sensitive prmeter in ll operting conditions, nd ith vlues comprble beteen three cses; prmeters rnking in 5th plce re different for three operting conditions. For ech of first three prmeters, ll three operting conditions re sensitive to prmeter chnges. Differently from response RH, subcse II is this time lest sensitive cse, hile unsturted cse is most sensitive cse comprtively. For instnce, % chnge in T,in, T db or T,in ill cuse round 8% chnge in m for unsturted cse, round 24% chnge in m for subcse I; nd nerly 22% chnge in m for subcse II, respectively. Overll, for response of ir mss flo rte, m, sensitivity behvior of subcse I is lso more similr to tht of subcse II. Tble 7. Cross-comprison of top five reltive sensitivities for response of ir outlet temperture, T. Rnk # 2 4 Rel. Sens. for Unsturted Conditions Bsed on 677 Unsturted Dt Sets Rel. Sens. for Sturted Conditions Subcse I Bsed on 77 Dt Sets ith Inlet Air Unsturted Subcse II Bsed on 290 Dt Sets ith Inlet Air Sturted T,in T,in T,in T db T,in T,in T,in T db T db T dp

22 Energies 206, 9, of 52 Tble 8. Cross-comprison of top five reltive sensitivities for response of ter outlet temperture, T 50. Rnk # Rel. Sens. for Unsturted Conditions Bsed on 677 Unsturted Dt Sets Rel. Sens. for Sturted Conditions Subcse I Bsed on 77 Dt Sets ith Inlet Air Unsturted Subcse II Bsed on 290 Dt Sets ith Inlet Air Sturted T,in T,in T,in T,in T,in T db T db T db T,in T dp T dp Tble 9. Cross-comprison of top five reltive sensitivities for response of ter outlet mss flo rte, m 50. Rnk # Rel. Sens. for Unsturted Conditions Bsed on 677 Unsturted Dt Sets Rel. Sens. for Sturted Conditions Subcse I Bsed on 77 Dt Sets ith Inlet Air Unsturted Subcse II Bsed on 290 Dt Sets ith Inlet Air Sturted m,in m,in m,in T,in T,in T,in T dp T dp T,in T,in 0 T db T db T,in 0 Tble 0. Cross-comprison of top five reltive sensitivities for response of ir outlet rel. humidity, RH. Rnk # Rel. Sens. for Unsturted Conditions Bsed on 677 Unsturted Dt Sets Rel. Sens. for Sturted Conditions Subcse I Bsed on 77 Dt Sets ith Inlet Air Unsturted Subcse II Bsed on 290 Dt Sets ith Inlet Air Sturted T,in T,in T,in T db T db T db T,in T dp T dp T dp T,in ω in ,dv,dv T,in

23 Energies 206, 9, of 52 Tble. Cross-comprison of top five reltive sensitivities for response of ir mss flo rte, m. Rnk # Rel. Sens. for Unsturted Conditions Bsed on 677 Unsturted Dt Sets Rel. Sens. for Sturted Conditions Subcse I Bsed on 77 Dt Sets ith Inlet Air Unsturted Subcse II Bsed on 290 Dt Sets ith Inlet Air Sturted T,in T db T db T db T,in T,in T,in T,in T,in P tm P tm P tm V D f n 0.. Experimentl Dt Assimiltion, Model Clibrtion nd Best-Estimte Predicted Results ith Reduced Predicted Uncertinties This subsection presents results of pplying Predictive Modeling of Coupled Multi-Physics Systems PM_CMPS methodology 4 to counter-flo cooling toer model. The priori covrince mtrix, Cov T,out mes, Tmes,out, RHmes out Crr, of mesured responses nmely: outlet ir temperture, T,out mes T mesured; outlet ter temperture, T,out mes T 50 mesured, nd outlet ir reltive humidity, RH mes out RH mesured, cf. Eqution A4, is reproduced belo: Cov T mes,out, T mes,out, RHout mes Crr The priori response-prmeter covrince mtrix, C rα, cf. Eqution A5, is reproduced belo: Cov T mes,out, Tmes,out, RHmes, α,..., α 47 Crα here mesured correlted prmeters re: α T db, α 2 T dp, α T,in, α 4 P tm, nd α 5 V. The priori prmeter covrince mtrix, C αα, is: C αα Vrα Covα, α 2 Covα, α 47 Covα 2, α Vrα 2 Covα 2, α 47 Covα 47, α Vrα

24 Energies 206, 9, of 52 The priori covrince mtrix of computed responses, Crr comp, is given belo: Crr comp Cov T T α T 50 α RH α, T 50, RH S rα C αα S rα,..., T α Nα,..., T50 α Nα,..., RH α Nα Vrα Covα, α 2 Covα, α 47 Covα 2, α Vrα 2 Covα 2, α 47 Covα 47, α Vrα 47 T α T 50 α RH α,..., T α Nα,..., T50 α Nα,..., RH α Nα 2... Model Clibrtion: Predicted Best-Estimted Prmeter Vlues ith Reduced Predicted Stndrd Devitions The best-estimte nominl prmeter vlues hve been computed s follos: α pred α 0 C αα S + rα C αr Drr r c α 0, β 0 r m, in conjunction ith priori mtrices given in Equtions 29 2 nd sensitivities presented in Tbles 5. The resulting best-estimte nominl vlues re listed in Tble 2, belo. The corresponding best-estimte bsolute stndrd devitions for se prmeters re lso presented in this tble. These vlues re squre-roots of digonl elements of mtrix Cαα pred. For comprison, originl nominl prmeter vlues nd originl bsolute stndrd devitions re lso listed. As results in Tble 2 indicte, predicted best-estimte stndrd devitions re ll smller or t most equl to i.e., left unffected originl stndrd devitions. The prmeters re ffected proportionlly to mgnitudes of ir corresponding sensitivities: prmeters experiencing lrgest reductions in ir predicted stndrd devitions re those hving lrgest sensitivities. Tble 2. Best-estimted nominl prmeter vlues nd ir stndrd devitions. i Independent Sclr Prmeters α i Mth. Nottion Originl Nominl Vlue Originl Absolute Std. Dev. Best-Estimted Nominl Vlue Best-Estimted Absolute Std. Dev. Air temperture dry bulb, K T db De point temperture K T dp Inlet ter temperture K T,in Atmospheric pressure P P tm 00, Wind speed m/s V Sum of loss coefficients bove fill k sum Dynmic viscosity of ir t T 00 K kg/m s Kinemtic viscosity of ir t T 00 K m 2 /s Therml conductivity of ir t T 00 K W/m K µ ν k ir Het trnsfer coefficient multiplier f ht Mss trnsfer coefficient multiplier f mt Fill section frictionl loss multiplier f P vs T prmeters ,cp C p T prmeters,cp ,cp

25 Energies 206, 9, of 52 Tble 2. Cont. i Independent Sclr Prmeters α i 8 Mth. Nottion Originl Nominl Vlue Originl Absolute Std. Dev. Best-Estimted Nominl Vlue Best-Estimted Absolute Std. Dev. 0,dv D v T prmeters,dv ,dv ,dv h f T prmeters 0 f,4, ,4, f h g T prmeters 0g 2,005, ,005, g ,Nu ,Nu Nu prmeters 28 2,Nu ,Nu Cooling toer deck idth in x-dir m W dkx Cooling toer deck idth in y-dir m W dky Cooling toer deck height bove ground m z dk Fn shroud height m z f n Fn shroud inner dimeter m D f n Fill section height m z f ill Rin section height m z rin Bsin section height m z bs Drift elimintor thickness m z de Fill section equivlent dimeter m D h Fill section flo re m 2 A f ill Fill section surfce re m 2 A sur f 4, , Prndlt number of ir t T 80 C P r Wetted frction of fill surfce re ts 0 0 i Boundry Prmeters Mth. Nottion Originl Nominl Vlue Absolute Std. Dev. Best-estimted Nominl Vlue Best-estimted Absolute Std. Dev. 44 Inlet ter mss florte kg/s m,in Inlet ir temperture K T,in ; set to T db Inlet ir humidity rtio ω in i Specil Dependent Prmeters Mth. Nottion Originl Nominl Vlue Absolute Std. Dev. Best-estimted Nominl Vlue Best-estimted Absolute Std. Dev. 47 Schmidt number Sc Predicted Best-Estimted Response Vlues ith Reduced Predicted Stndrd Devitions The predicted response covrince mtrix, Crr pred, is s follos: C pred rr Cov T be, T 50 be, RH be The non-zero elements ith lrgest mgnitudes of best-estimte response-prmeter correltion mtrix, Cαr pred, re s follos: rel. cor.r, α ; rel. cor.r, α ; rel. cor.r 2, α ; rel. cor.r 2, α ; rel. cor.r, α ; rel. cor.r, α The nottion used in Eqution 5 is s follos: R T, R 2 T 50, R RH ; α 4 P tm, α 4 A sur f.

26 Energies 206, 9, of 52 The resulting best-estimte predicted nominl vlues re summrized in Tble. To fcilitte comprison, corresponding mesured nd computed nominl vlues re lso presented in this tble. Note tht re re no direct mesurements for outlet ter flo rte, m 50. For this response, refore, predicted best-estimte nominl vlue hs been obtined by forrd re-computtion using best-estimte nominl prmeter vlues listed in Tble 2, hile predicted best estimte stndrd devition for this response hs been obtined s follos: C comp rr be Srα be C αα be S rα be 6 The results presented in Tble indicte tht predicted stndrd devitions re smller thn eir computed or experimentlly mesured ones. This is indeed consequence of using PM_CMPS methodology in conjunction ith consistent s opposed to discrepnt computtionl nd experimentl informtion. Often, hoever, informtion is inconsistent, usully due to presence of unrecognized errors. Solutions for ddressing such situtions hve been proposed in 0. It is lso importnt to note tht PM_CMPS methodology hs improved i.e., reduced, lbeit not by significnt mount predicted stndrd devition for outlet ter flo rte response, for hich no mesurements ere vilble. This improvement stems from globl chrcteristics of PM_CMPS methodology, hich combines ll of vilble simultneously on phse-spce, s opposed to combining it sequentilly, s is cse ith current stte-of--rt dt ssimiltion procedures,2. Tble. Computed, mesured, nd optiml best-estimte nominl vlues nd stndrd devitions for outlet ir temperture, outlet ter temperture, outlet ir reltive humidity, outlet ter mss flo rte nd ir mss flo rte responses. Norminl Vlues nd Stndrd Devitions T K T 50 K RH % m 50 kg/s m kg/s Mesured nominl vlue stndrd devition ±2.84 ±.9 ±.62 Computed nominl vlue stndrd devition ±.67 ±.96 ±.7 ±2.20 ±2.20 Best-estimte nominl vlue stndrd devition ±.57 ±.8 ±.09 ±2.9 ± Discussion The originl numericl method presented in for model solution hs been replced in this ork ith considerbly more ccurte nd efficient one hich gurntees convergence of computtions for ll of vilble dt sets. The djoint model of cooling toer hs been implemented to compute exctly nd efficiently sensitivities of model responses to ll 47 model prmeters. The djoint sensitivity model yields djoint stte functions hich re used to compute sensitivities of ech model response to ll of 47 model prmeters by mens of just one djoint model computtion. These djoint stte functions hve been computed nd ir numericl ccurcy hs been independently verified. The response sensitivities to ll model prmeters hve been computed for folloing responses: i outlet ir temperture; ii outlet ter temperture; iii outlet ter mss flo rte; iv ir outlet reltive humidity; nd v ir mss flo rte. Thes sensitivities hve been subsequently used ithin predictive modeling for coupled multi-physics systems PM_CMPS methodology 4 to obtin: optiml best-estimte prediction for model prmeter vlues; b optiml best-estimte nominl vlues of model responses;

27 Energies 206, 9, of 52 c reduced predicted stndrd devitions for best-estimte clibrted model prmeter vlues; nd d reduced predicted stndrd devitions for best-estimte predicted response vlues. The results presented in this ork sho tht PM_CMPS methodology reduces predicted stndrd devition to vlues tht re smller thn both stndrd devitions of mesured nd computed response, respectively, even for responses, such s for ir mss flo rte, for hich no Energies experimentlly 206, 9, 028 mesured vlues re vilble. This reduction stems from fct tht PM_CMPS 0 of 57 methodology simultneously combines ll vilble dt in phse-spce; customry dt cooling ssimiltion toer methodologies under unsturted,2 conditions. only lloongoing sequentil ork combintion ims using ofsecond-order vilblesensitivities, informtion. to Allbe incomputed ll, ppliction by pplying of PM_CMPS 2nd-ASAM methodology presented in,4. hs produced The vilbility nd improved, of second-order clibrted response nd vlidted sensitivities model ill for simulting enble computtion functioning of non-gussin of buoyncy-operted fetures, such cooling s skeness toer under nd kurtosis, unsturted of conditions. response distributions Ongoing ork of ims interest. t using second-order sensitivities, to be computed by pplying 2nd-ASAM presented in,4. The vilbility of second-order response sensitivities Acknoledgments: ill enble computtion This ork of hs non-gussin been sponsored fetures, by US such Deprtment s skeness of Energy nd kurtosis, under contrct of response ith University of South Crolin. distributions of interest. Author Contributions: Federico Di Rocco performed ll of numericl clcultions in this pper. Dn Gbriel Ccuci Acknoledgments: conceived nd This directed ork hsreserch been sponsored reported by herein, US nd Deprtment rote pper. of Energy under contrct ith University of South Crolin. Conflicts of Interest: The uthors declre no conflict of interest. The founding sponsors hd no role in Author Contributions: Federico Di Rocco performed ll of numericl clcultions in this pper. Dn Gbriel design Ccuciof conceived study; nd in directed collection, reserch nlyses, reported interprettion herein, ndof rote dt; in pper. riting of mnuscript, nd in decision to publish results. Conflicts of Interest: The uthors declre no conflict of interest. The founding sponsors hd no role in design of study; in collection, nlyses, or interprettion of dt; in riting of mnuscript, nd in Appendix decision to publish A. Sttisticl results. Anlysis of Experimentlly Mesured Responses for SRNL F-Are Cooling Toers Appendix A. Sttisticl Anlysis of Experimentlly Mesured Responses for SRNL F-Are Cooling Histogrm Toersplots of 677 mesurement sets considered in this ork ech set contining mes mes mesurements of T, T, T, nd RH, toger ith sttisticl nlyses reof α, out Tidbit α, out Hobo out, Histogrm plots of 677 mesurement sets considered in this ork ech set contining re mesurements presented in of T reminder α,outtidbit, T of this Appendix. α,outhobo, T,out mes, nd RHmes, toger ith sttisticl nlyses reof mes re presented The mesured in outlet reminder exit ofir this reltive Appendix. humidity, RH, s obtined using Hobo humidity sensors. The The mesured ccurcy outlet of se exit ir sensors reltive is depicted humidity, in RHFigure mes, s A, obtined hich indictes using Hobo folloing humidity tolernces sensors. The stndrd ccurcydevitions: of se sensors ±2.5% is depicted for reltive in Figure humidity A, hich from 0% indictes to 90%; folloing beteen ±2.5% tolernces nd ±.5% stndrd for reltive devitions: humidity ±2.5% from for reltive 90% to 95%; humidity nd ±.5% ±4.0% from 0% to 90%; from beteen 95% to 00%. ±2.5% Hoever, nd ±.5% hen for exposed reltive humidity to reltive from humidity 90% tobove 95%; nd 95%, ±.5% ±4.0% mximum from sensor 95% error to 00%. my Hoever, temporlly hen increse exposed by n to dditionl reltive humidity %, so bove tht 95%, error mximum cn rech sensor vlues error beteen my ±4.5% temporlly nd ±5.0% increse for byreltive n dditionl humidity %, from so tht 95% to error 00%. cn rech vlues beteen ±4.5% nd ±5.0% for reltive humidity from 95% to 00%. Figure A. Humidity sensor ccurcy plot dopted from specifiction of HOBO Pro v2. As shon in this Figure A2, lthough computed reltive humidity for ech of 677 dt sets is less thn 00%, mesured reltive humidity RH mes ctully spns rnge from.0% to 04.%; in this rnge, 4925 dt sets hve ir respective RH mes less thn 00% hile or mes 792 dt sets hve ir respective RH over 00%. This sitution is neverless consistent ith rnge of sensors hen ir tolernces stndrd devitions re tken into ccount, hich mes ould mke it possible for mesurement ith RH 05% to be neverless unsturted. Consequently, ll 677 benchmrk dt sets plotted in Figure A, ere considered s

28 Energies 206, 9, of dt sets hve ir respective RH mes over 00%. This sitution is neverless consistent ith rnge of sensors hen ir tolernces stndrd devitions re tken into ccount, hich ould mke it possible for mesurement ith RH mes 05% to be neverless unsturted. Consequently, ll 677 benchmrk dt sets plotted in Figure A, ere considered s unsturted, Energies 206, 9, 028 Energies since ir 206, respective 9, 028 RH mes of 57 s less thn 05%. of 57 Figure A2. Histogrm plot of mesured ir outlet reltive humidity, ithin 677 dt sets A2. plot of ir collected Figure A2. by Histogrm SRNL from plot F-Are of cooling mesured toers. ir outlet reltive humidity, ithin 677 dt sets collected by SRNL from F-Are cooling toers. The sttisticl properties of mesured ir outlet reltive humidity distribution shon in The sttisticl properties of mesured ir outlet reltive humidity distribution shon in Figure The A2 sttisticl hve been properties computed ofusing mesured stndrd pckges, ir outlet nd reltive re humidity presented in distribution Tble A. shon These Figure A2 hve been computed using stndrd pckges, nd re presented in Tble A. These sttisticl in Figure properties A2 hve been ill be computed needed for using stndrd uncertinty pckges, quntifiction nd rend presented predictive in modeling Tble A. sttisticl properties ill be needed for uncertinty quntifiction nd predictive modeling computtions These sttisticl presented properties in ill min be needed body for of this ork. uncertinty quntifiction nd predictive modeling computtions presented in min body of this ork. computtions presented in min body of this ork. Tble A. Sttistics of ir outlet reltive humidity distribution %. Tble A. Sttistics of ir outlet reltive humidity distribution %. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Minimum 8.2 Mximum 04. Rnge Men Std..6 Dev. Vrince Skeness.0 Kurtosis The histogrm plots nd ir corresponding sttisticl chrcteristics of 677 dt sets for The histogrm plots nd ir corresponding sttisticl chrcteristics of 677 sets for or or The mesurements, mesurements, histogrm plots nmely nmely ndfor: ir for: corresponding outlet ir temperture outlet ir temperture sttisticl chrcteristics T mesured mesured of 677 using using dt sets Tidbit outtidbit, Tidbit for outtidbit, sensors; or mesurements, outlet ir temperture nmely for: T outlet ir mesured temperture using T sensors; outlet ir temperture mesured using,outtidbit Hobo sensors; mesured nd using outlet Tidbit ter outhobo, Hobo sensors; nd outlet ter outhobo, sensors; outlet mes ir temperture T,outHobo mesured using Hobo sensors; nd outlet temperture T mes re ter temperture temperture re reported belo in Figures A A6, nd Tbles A2 A5, respectively. out, T,out mes reported belo in Figures A A6, nd Tbles A2 A5, respectively. out, re reported belo in Figures A A6, nd Tbles A2 A5, respectively. Figure A. Histogrm plot of ir outlet temperture mesured using Tidbit sensors, ithin Figure A. Histogrm plot of ir outlet temperture mesured using Tidbit sensors, ithin 677 dt sets collected by SRNL from F-Are cooling toers. 677 dt sets collected by SRNL from F-Are cooling toers.

29 Energies 206, 9, of 52 Energies 206, 9, 028 Energies 206, 9, of 57 2 of 57 Figure A4. Histogrm plot of ir outlet temperture mesured using Hobo sensors, ithin ithin 677 dt sets collected by SRNL from F-Are cooling toers. Figure A5. Histogrm plot of ter outlet temperture mesurements, ithin 7688 dt sets collected by SRNL from F-Are cooling toers. Figure Figure A6. A6. Histogrm Histogrm plot of ir outlet tempertures verged from FiguresA And nda4. A4. Tble A2. Sttistics of ir outlet temperture distribution K, mesured using Tidbit sensors.

30 Energies 206, 9, of 52 Tble A2. Sttistics of ir outlet temperture distribution K, mesured using Tidbit sensors. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Tble A. Air outlet temperture distribution sttistics K, mesured using Hobo sensors. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Tble A4. Wter outlet temperture distribution sttistics K. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Tble A5. Sttistics of verged ir outlet temperture distribution K. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Ordering bove-mentioned four mesured responses s follos: i outlet ir temperture T,outTidbit ; ii outlet ir temperture T,outHobo ; iii outlet ter temperture T,out mes ; nd iv outlet ir reltive humidity RHout mes, yields folloing mesured response covrince mtrix, denoted s Cov T,outTidbit, T,outHobo, T,out mes, RHmes out : Cov T,outTidbit, T,outHobo, T,out, mes RHout mes For purposes of uncertinty quntifiction, dt ssimiltion, model clibrtion nd predictive modeling, tempertures mesurements provided by Tidbit nd Hobo sensors cn be combined into n verged dt set of mesured ir outlet tempertures, hich ill be denoted s T,out mes. The histogrm plot nd corresponding sttisticl chrcteristics of this verged ir outlet temperture re presented in Figure A6 nd Tble A5, respectively. Computing covrince mtrix, denoted s Cov T,out mes, Tmes,out, RHmes out, for ll of dt relevnt experimentl dt for verged outlet ir temperture T,out mes, outlet ter temperture T mes,out, nd outlet ir reltive humidity RH mes out, yields folloing result: Cov T mes,out, T mes,out, RHout mes dt A comprison beteen results in Equtions A nd A2 mkes cler tht elimintion of second column nd ro in Eqution A yields -by- mtrix hich hs entries bsiclly equivlent to covrince mtrix shon in Eqution A2. Therefore, this mens tht temperture distributions mesured by Tidbit nd Hobo sensors do not need to be delt ith s seprte dt sets for purposes of uncertinty quntifiction nd predictive modeling. The stndrd devition of humidity sensor utilized for mesurements σ sensor 5.0% for response RH hve been lredy considered by including in ctegory of unsturted.. A A2

31 Energies 206, 9, 028 of 52 dt sets those tht hve ir respective mesured reltive humidity, RH mes, up to 05.0%. In ddition to tht, respective uncertinties of temperture sensors stndrd devitions, σ sensor 0.2K for both responses T nd T 50 must lso be tken into considertion for 677 dt sets. The mesuring methods nd devices re not dependent ith respect to ech or, refore dt stndrd devition σ sttistic, stemming from sttisticl nlysis of 677 benchmrk dt sets, nd sensor stndrd devition, σ sensor, stemming from instrument s uncertinty, must stck ccording to ell-knon formul of ddition of vrinces of uncorrelted vrites, i.e.,: Cov T mes,out, T mes σ,out, RHout mes σ sttistic 2 + σ sensor 2. Coupling bove reltion ith result presented in Eqution A2 ill led to incremented vlues of vrinces on digonl of respective mesured covrince mtrix ; this ne form of covrince mtrix hich ill be denoted s Cov T,out mes, Tmes,out, RHmes out. The obtined result is:. A In predictive modeling formlism hich includes uncertinty quntifiction, dt ssimiltion, nd model clibrtion covrince mtrix beteen mesured prmeters nd responses is required s n input. In cse of interest, ll prmeters nd responses cn be considered s uncorrelted, except for mesured responses considered in this Appendix nd mesured prmeters listed in Appendix B. The prmeter-response covrince mtrix in Eqution A5, indicted s Cov T,out mes, Tmes,out, RHmes, α,..., α 47, refers to bove mentioned prmeters nmely: dry-bulb ir temperture, T db ; de-point ir temperture, T dp, inlet ter temperture, T,in, tmospheric pressure, P tm, nd ind speed V nd responses i.e., verge outlet ir temperture, outlet ter temperture, nd outlet ir reltive humidity: Cov T mes,out, Tmes,out, RHmes, α,..., α Appendix B. Model Prmeters for SRNL F-Are Cooling Toers The men vlues nd stndrd devitions for independent model prmeters α i, i,..., N α 47,presented in Tble B, belo, hve been derived in collbortion ith Dr. Sebstin Alemn of SRNL privte communictions, 206. Tble B. Prmeters for SRNL F-re cooling toers A A5 Index i of α i Independent Sclr Prmeters C++ String Mth. Nottion Nominl Vlues Absolute Std. Dev. Rel. Std. Dev. % Air temperture dry bulb K tdb T db De point temperture K tdp T dp Inlet ter temperture K tin T,in Atmospheric pressure P ptm P tm 00, Wind speed m/s spd V Sum of loss coefficients bove fill ksum k sum Dynmic viscosity of ir t T 00 K kg/m s Kinemtic viscosity of ir t T 00 K m 2 /s Therml conductivity of ir t T 00 K W/m K muir µ nuir ν tcir k ir

32 Energies 206, 9, of 52 Tble B. Cont. Index i of α i Independent Sclr Prmeters C++ String Mth. Nottion Nominl Vlues Absolute Std. Dev. 0 Het trnsfer coefficient multiplier mlthtc f ht Mss trnsfer coefficient multiplier mltmtc f mt Fill section frictionl loss multiplier mltfil f Rel. Std. Dev. % P vs T prmeters A 0,cp C p T prmeters A2,cp A 2,cp A 0,dv D v T prmeters A2,dv A 2,dv A4,dv h f T prmeters 0f 0 f,4, f f h g T prmeters 0g 0g 2,005, g g ,Nu ,Nu Nu prmeters 28-2,Nu ,Nu Cooling toer deck idth in x-dir. m dkx W dkx Cooling toer deck idth in y-dir. m dky W dky Cooling toer deck height bove ground m dkht z dk 0 0. Fn shroud height m fsht z f n Fn shroud inner dimeter m fsid D f n Fill section height m flht z f ill Rin section height m rsht z rin Bsin section height m bsht z bs Drift elimintor thickness m detk z de Fill section equivlent dimeter m deqv D h Fill section flo re m 2 flf A f ill Fill section surfce re m 2 fls A sur f 4, Prndlt number of ir t T 80 C Pr P r Wetted frction of fill surfce re ts ts 0 0 Index i of α i Boundry Prmeters C++ String Mth. Nottion Nominl Vlue Absolute Std. Dev. 44 Inlet ter mss florte kg/s mfin m,in Rel. Std. Dev. % 45 Inlet ir temperture K tin T,in set to T db Inlet ir humidity rtio Dependent Sclr Prmeter hrin ω in ; ω rin Index i of α i Specil Dependent Prmeters C++ String Mth. Nottion Nominl Vlue Absolute Std. Dev. Rel. Std. Dev. % 47 Schmidt number Sc Sc The bove independent model prmeters re used for computing vrious dependent model prmeters nd rml mteril properties, s shon in Tbles B2 nd B, belo.

33 Energies 206, 9, 028 of 52 Tble B2. Dependent sclr model prmeters. Dependent Sclr Prmeters Mth. Nottion Defining Eqution or Correltion Mss diffusivity of ter vpor in ir m 2 /s D v T, α 0,dv T.5,dv + 2,dv +,dv TT Het trnsfer coefficient W/m 2 K hα f ht N u k ir D h Mss trnsfer coefficient m/s k m α f mt S h D v T db,α D h Het trnsfer term W/K Hm, α h α ts A f f Mss trnsfer term m /s Mm, α M H2 Ok m α ts A f f Density of dry ir kg/m ρα P tm R ir T db Air velocity in fill section m/s v m, α m ραa f ill Fill flling-film surfce re per verticl section m 2 A A sur f f f I Rin section inlet flo re m 2 A in W dkx W dky Height for nturl convection m Z z dk + z f n z bs Height bove fill section m z 4 2 Z z f ill z rin z Fill section control volume height m z f ill I Fill section length, including drift elimintor m L f ill z f ill + z de Fn shroud inner rdius m r f n 0.5D f n Fn shroud flo re m 2 Energies 206, 9, 028 A out π r 2 f n 7 of 57 Tble B. Therml Properties Dependent Sclr Model Prmeters. HgT sturted vpor enthlpy J/kg h T, α g, + T 0g g + T Therml Properties Functions of Stte Vribles Mth. Nottion Defining Eqution or Correltion HgT sturted vpor enthlpy J/kg h T, α g, 0g g h f T sturted liquid enthlpy J/kg h f T, α 0 f + f HCpT g specific sturtedhet vpor of enthlpy dry ir J/kg K Ch g, T, T α, α + 0g + g T T p 0, cp, cp 2, cp H g T sturted vpor enthlpy J/kg h g, T, α 0g + g T 0 + C p PvsT T specific sturtion het of dry pressure ir J/kgP P T, α T K vs C p T, α P e 0,cp, in+ hich,cp P+.0 2,cp P TT c P vs T sturtion pressure P P vs T, α P c e 0+ c T, in hich P c.0 P 0 + PvsT sturtion pressure P P T, α T vs P vs T sturtion pressure P P vs T, α P c e 0+ P e, in T hich, in hich P.0 c c P c P.0 P Note Note : : The The mesurements mesurements of of prmeters prmeters α α α α 5 5 i.e., i.e., dry dry bulb bulb ir ir temperture, temperture, de de point point temperture, inlet ter temperture, tmospheric pressure nd nd ind ind speed speed ere ere tken tken t t SRNL SRNL site, here site, here F-re cooling F-re cooling toers re toers locted. re locted. Out of Out 8079 of totl 8079 benchmrk totl benchmrk dt sets 8, dt 677 sets dt 8, 677 sets hve dt been sets hve considered been considered in this study, in this since study, unsturted ; since unsturted ; through se through dtse sets dt sttisticl sets sttisticl properties properties mens, vrince mens, nd covrince, nd skeness covrince, ndskeness kurtosis for nd se kurtosis modelfor prmeters se model hve prmeters been derived, hve s shon been derived, in Figures s shon B B5 in ndfigures TblesB B5 B4 B8. nd Tbles B4 B8. Figure B. Histogrmplot plot of dry-bulb of dry-bulb ir temperture ir temperture dt collected dt collected by SRNL by from SRNLF-Are from cooling F-Are toers. cooling toers.

34 Figure B. Histogrm plot of dry-bulb ir temperture dt collected by SRNL from F-Are cooling Energies 206, 9, of 52 toers. Figure B2. Histogrmplot plot of de-point of de-point ir temperture ir temperture dt collected dt collected by SRNL by from SRNLF-Are from cooling F-Are Energies toers. cooling 206, 9, toers of 57 Figure B. Histogrm plot plot of inlet of inlet ter ter temperture temperture dt collected dt collected by SRNL by from SRNLF-Are from cooling F-Are toers. cooling toers. Figure B4. Histogrm plot plot of of tmospheric tmospheric pressure pressure dt dt collected collected by SRNL by SRNL from F-Are from F-Are cooling cooling toers. toers.

35 Figure B4. Histogrm plot of tmospheric pressure dt collected by SRNL from F-Are cooling Energies 206, 9, of 52 toers. Figure Figure B5. Histogrm B5. Histogrm plot plot of ind of ind speed speed dt dt collected collected by SRNL by SRNL from from F-Are F-Are cooling cooling toers. toers. Tble Tble B4. B4. Sttistics Sttisticsof of dry-bulb temperture set set to to ir ir inlet inlet temperture distribution K. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Tble B5. Sttistics of de-point temperture distribution K. Minimum Tble B5. Sttistics of de-point temperture distribution K. Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Tble B6. Sttistics of inlet ter temperture distribution K. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis Tble B7. Sttistics of tmospheric pressure distribution P. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis , , Tble B8. Sttistics of ind speed distribution m/s. Minimum Mximum Rnge Men Std. Dev. Vrince Skeness Kurtosis The 5-by-5 covrince mtrix for bove experimentl dt hs lso been computed nd is provided belo, ith four model prmeters ordered s follos: dry-bulb ir temperture T db, de-point ir temperture T dp, inlet ter temperture T,in, tmospheric ir pressure P tm, nd ind speed V. Cov T db ; T dp ; T,in ; P tm ; V B

36 Energies 206, 9, of 52 The covrince mtrix bove neglects uncertinty ssocited ith sensor redings throughout dt collection period. When combining uncertinties by dding vrinces, contribution from sensors is 0.04 K for ech of first three prmeters, hich ccounts for mximum of c. % of totl vrince for inlet ter temperture, specificlly. The uncertinty in tmospheric pressure sensor is t this time unknon. For se resons, ir contribution to overll uncertinty is considered insignificnt t this time. Appendix C. Derivtive Mtrix Jcobin of Model Equtions ith Respect to Stte Functions As mentioned in Section 2, Jcobin mtrix presents similrities ith Jcobin mtrix detiled in 2, Eqution C. More precisely, sub-mtrices A i, B i, C i, D i, E i ; i 2,, 4, 5 hose elements represents derivtives of Equtions 6 4 ith respect to vector vlued stte function u m, T, T, ω, m remin sme s in 2; for resons of brevity, y hve not been reported in this pper. On or side, sub-mtrices A i, B i, C i, D i, E i ; i hose elements represents derivtives of Equtions 2 4 ith respect to vector vlued stte function u m, T, T, ω, m, re different from ir respective formultions in 2, nd refore y ill be hereby detiled ith folloing nottion: Ni i,j b i,j m j+ T j+ ; i,..., I; j,..., I; C Ni ; i,..., I; j,..., I; C2 Ni ; i,..., I; j,..., I; C c i,j d i,j T j Ni ; i,..., I; j,..., I; C4 ω j e i Ni ; i,..., I. C5 m The derivtives of liquid continuity equtions cf., Equtions 2 4 ith respect to m j re s follos: N i m j+ i,j N i m i 0; i,..., I; j,..., I; j i, i; C6 i,i ; i 2,..., I; j i ; C7 N i m i+ i,i ; i,..., I; j i. C8 For subsequent use, bove quntities re considered to be components of I I mtrix A defined s follos: A i,j C9 I I

37 Energies 206, 9, of 52 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to T j s follos: N i T i+ N i T j+ b i,i Mm, α R b i,j 0; i,..., I; j,..., I; j i; C0 P vs i+ T i+, α T i+ 2 { T i+ + } re ; i,..., I; j i. C For subsequent use, bove quntities re considered to be components of I I mtrix B defined s follos: B b i,j I I b 2, C b I,I b I,I b, The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to T j s follos: N i T i N i T j c i,j c i,i Mm, α R 0; i,..., I; j,..., I; j i; C T i re ω i P tm ; i,..., I; j i. C4 + ω i For subsequent use, bove quntities re considered to be components of I I mtrix C defined s follos: C c i,j I I c 2, C c I,I c I,I c, The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to ω j re s follos: N i di,i ω i N i Mm, α R di,j ω j P tm ω i T i 0; i,..., I; j,..., I; j i; C6 { } ω i ω i ; i,..., I; j i. C7 For subsequent use, bove quntities re considered to be components of I I mtrix D defined s follos: D d i,j I I d 2, C d I,I d I,I d,

38 Energies 206, 9, of 52 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to m re s follo follos: For Re d < 200 N i e i 0; i,..., I; C9 m 2 For 200 Re d 0, 000 N i e i m P vs T i+, α ω R T i+ i P tm R T i ω i For Re d > 0, 000 N i e i m P vs T i+, α ω R T i+ i P tm R T i ω i M 2m, α m ; i,..., I; C20 M m, α m ; i,..., I; C2 For subsequent use, bove quntities re considered to be components of I column vector E defined s follos: E e i I e e 2. e I e I. C22 Appendix D. Derivtives of Cooling Toer Model Equtions With Respect to Model Prmeters The differences beteen governing equtions for this study nd for subcse I in 2 concern only liquid continuity equtions. Or governing Equtions i.e., liquid energy blnce equtions; ter vpor continuity equtions; nd ir/ter vpor energy blnce equtions re sme for both cses. For this reson, derivtives of Equtions 6 4 ith respect to model prmeters remin sme s in, Eqution A; for resons of brevity, y hve not been reported in this pper. On or side, derivtives of Equtions 2 4 ith respect to model prmeters re different from ir respective formultions in, nd refore y ill be hereby detiled ith folloing nottion: Ni α j ; i,..., I; j,..., N α. i,j For ske of brevity, only nonzero derivtives hve been reported in this ppendix. The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α : T db re s follos: N i α N i T i, db R P i+ vs T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j, Mm,α D v T db,α D vt db,α T. db D D2 here: D v T db, α T db Mm, α D v T db, α 2 Mm, α D v T db, α.5 0dvT 0.5 db D v T db, α 2dv + 2 dv T db dv + 2dv T db + dv T 2 db D D4

39 Energies 206, 9, of 52 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 4 : P tm re s follos: N i α 4 N i P tm i,4 Mm,α R T i l ; i,..., I; j 4, ω i ; ω i D5 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 7 : µ re s follos: here: N i α 7 N i µ i,7 R Mm, α µ P i+ vs T i+,α T i+ l ; i,..., I; j 7. ω i P tm ω i T i Mm,α µ ; 0 Re d < 200,Nu Mm,α Rem,α 200 Re NuRe,α µ d 0, Mm,α µ Re d > 0, 000 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 8 : υ re s follos: here: N i α 8 N i υ i,8 R P i+ vs T i+,α T i+ l ; i,..., I; j 8. Mm, α υ ω i P tm ω i T i Mm,α υ ; D6 D7 D8 Mm, α. D9 υ The derivtives of liquid continuity equtions cf. Equtions A A4 ith respect to prmeter α : f mt re s follos: here: N i i N α f mt i, R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j. Mm,α f mt ; D0 Mm, α M H2ONuRe, α νpr D v T db, α 2 ts A sur f. D f mt D h I The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α : 0 re s follos: here: N i i N i, α Mm, α 0 R P i+ vs T i+ P i+ vs T i+, α 0 T i+, α ; l ; i,..., I; j. D2 0 P vs i+ T i+, α. D The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 4 : re s follos: N i i N i,4 α 4 Mm, α R T i+ P i+ vs T i+, α ; l ; i,..., I; j 4. D4

40 Energies 206, 9, of 52 here: P i+ vs T i+, α Pi+ vs T i+, α T i+. D5 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 8 : 0,dv re s follos: N i i N α 8 0,dv i,8 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 8. here Mm,α s defined previously in Eqution D, nd D v T db,α Mm,α D v T db,α D vt db,α ; 0,dv D6 D v T db, α 0,dv T db.5 dv + 2dv T db + dv T db 2. D7 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 9 :,dv re s follos: N i i N α 9,dv i,9 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 9. here Mm,α s defined previously in Eqution D, nd D v T db,α Mm,α D v T db,α D vt db,α ;,dv D8 D v T db, α,dv 0dv T.5 db dv + 2dv T db + dv T 2 db 2. D9 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 20 : 2,dv re s follos: N i i N α 20 2,dv i,20 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 20. here Mm,α s defined previously in Eqution D, nd D v T db,α Mm,α D v T db,α D vt db,α ; 2,dv D20 D v T db, α 2,dv 0dv T 2.5 db dv + 2dv T db + dv T 2 db 2. D2 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 2 :,dv re s follos: N i i N α 2,dv i,2 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 2. here Mm,α s defined previously in Eqution D, nd D v T db,α Mm,α D v T db,α D vt db,α ;,dv D22 D v T db, α,dv 0dv T.5 db dv + 2dv T db + dv T 2 db 2. D2

41 Energies 206, 9, of 52 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 26 : 0,Nu re s follos: N i i N α 26 0,Nu i,26 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 26. Mm,α NuRe,α NuRe,α ; 0,Nu D24 here NuRe, α 0,Nu MRe, α NuRe, α Mm, α NuRe, α Re d < Re d 0, Re d > 0, 000 D25 D26 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 27 :,Nu re s follos: N i i N α 27,Nu i,27 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 27. Mm,α NuRe,α NuRe,α ;,Nu D27 here Mm,α s defined previously in Eqution D25, nd NuRe,α NuRe, α,nu 0 Re d < 200 Rem, α 200 Re d 0, Re d > 0, 000 D28 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 28 : 2,Nu re s follos: N i i N α 28 2,Nu i,28 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 28. Mm,α NuRe,α NuRe,α ; 2,Nu D29 here Mm,α s defined previously in Eqution D25, nd NuRe,α NuRe, α 2,Nu 0 Re d < Re d 0, Re d > 0, 000 D0 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 29 :,Nu re s follos: N i i N α 29,Nu i,29 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 29. Mm,α NuRe,α NuRe,α ;,Nu D here Mm,α s defined previously in Eqution D25, nd NuRe,α NuRe, α,nu 0 Re d < Re d 0, 000 Rem, α 0.8 Pr Re d > 0, 000 D2

42 Energies 206, 9, of 52 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 9 : D h re s follos: here N i i N α 9 D h i,9 Mm, α D h R P vs i+ T i+,α T i+ l ; i,..., I; j 9. ω i P tm ω i T i Mm,α D h ; Mm, α/d h Re d < 200 2,Nu Mm,α NuRe,αD h 200 Re d 0, Mm, α/d h Re d > 0, 000 D D4 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 40 : A f ill re s follos: here N i i N α 40 A f ill i,40 Mm, α A f ill R P vs i+ T i+,α T i+ l ; i,..., I; j 40. ω i P tm ω i T i Mm,α A f ill ; 0 Re d < 200,Nu Mm,αRem,α NuRe,αA f ill 200 Re d 0, Mm, α/a f ill Re d > 0, 000 D5 D6 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 4 : A sur f re s follos: here N i i N α 4 A sur f i,4 R P vs i+ T i+,α T i+ l ; i,..., I; j 4. Mm, α A sur f Mm, α A sur f ω i P tm ω i T i Mm,α A sur f ; D7 D8 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 42 : Pr re s follos: N i i N α 42 Pr i,42 R P vs i+ T i+,α T i+ l ; i,..., I; j 42. ω i P tm ω i T i Mm,α Pr ; D9 here Mm, α Pr Mm, α/ Pr Re d 0, Re d > 0, 000 D40 The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 4 : ts re s follos: N i i N α 4 ts i,4 R P vs i+ T i+,α T i+ ω i P tm ω i T i l ; i,..., I; j 4. Mm,α ts ; D4

43 Energies 206, 9, of 52 here Mm, α M H2O f mt NuRe, α νpr D v T db, α 2 A sur f. D42 ts D h I The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 44 : m,in re s follos: N N,44 α 44 m ; l ; i ; j 44, D4,in N i i N i,44 α 44 m 0; l ; i 2,..., I; j 44. D44,in The derivtives of liquid continuity equtions cf. Equtions 2 4 ith respect to prmeter α 47 : Sc re s follos: N i i N α 47 Sc i,47 R P vs i+ T i+,α T i+ l ; i,..., I; j 47. ω i P tm ω i T i Mm,α Sc ; D45 here Mm, α Sc Mm, α. D46 Sc Appendix E. Verifiction of Model Adjoint Functions This ppendix provides complete disply of procedure folloed to verify numericl ccurcy of djoint functions computed. Five specific djoint functions µ ; o 49 ; τ 49 ; τ ; µ hve been selected for ech of five responses of model T ; T 50 ; RH ; m 50 ; m in such y tht, once those hve been verified, ll or djoint functions ould be consequently verified s ell. For clrity resons, djoint functions hve been grouped bsed on response y refer to. Appendix E.. Verifiction of Adjoint Functions for Outlet Air Temperture Response T When R T, quntities r i l defined in Equtions 2 22 ll vnish except for single component, nmely: r R/ T. Thus, djoint functions corresponding to outlet ir temperture response T re computed by solving djoint sensitivity system given in Eqution 20 using r R/ T s only non-zero source term; for this cse, solution of Eqution 20 hs been depicted in Figure 8. Verifiction of djoint function µ Note tht vlue of djoint function µ obtined by solving djoint sensitivity system given in Eqution 20 is µ K/ J/m, s indicted in Figure 8. No select vrition δv in ind speed V, nd note tht Eqution 27 yields folloing expression for sensitivity of response R T to V : 49 S 5 V R i µ i N i V + τ i N i 2 V + τ i N i V + o 0 µ N 5 V µ V ρt tdb, α. i Ni 4 V N + µ 5 V E

44 Energies 206, 9, of 52 Re-riting Eqution E in form µ S 5 N 5 V E2 indictes tht vlue of djoint function µ could be computed independently if sensitivity S 5 ere vilble, since quntity N 5 / V J/ m 4 /s is knon. To first-order in prmeter perturbtion, finite-difference formul given in Eqution 28 cn be used to compute pproximte sensitivity S5 FD ; subsequently, this vlue cn be used in conjunction ith Eqution E2 to compute finite-difference sensitivity vlue, denoted s µ SFD, for respective djoint, hich ould be ccurte up to second-order in respective prmeter perturbtion: µ SFD SFD 5 N 5 / V,nom T,pert T δv N5 E V Numericlly, ind speed V hs nominl bse-cse vlue of V m/s. The corresponding nominl vlue T,nom of response T is T,nom K. Consider next perturbtion δt,in 0.0 V, 0 for hich perturbed vlue of inlet ir temperture becomes V pert V 0 δv.8404 m/s. Re-computing perturbed response by solving Equtions 2 4 ith vlue of V pert yields perturbed response vlue T,pert K. Using no nominl nd perturbed response vlues toger ith prmeter perturbtion in finite-difference expression given in Eqution 28 yields corresponding finite-difference-computed sensitivity S5 FD T,pert T,nom δv K m/s. Using this vlue toger ith nominl vlues of or quntities ppering in expression on right side of Eqution E yields µ SFD K/ J/m. This result compres ell ith vlue µ K/ J/m obtined by solving djoint sensitivity system given in Eqution 20, cf., belo Figure 8. The sme prmeter perturbtion s utilized to perform sme verifiction procedure for djoint function µ ith respect to or four responses; Tble E displys obtined results, hich compre ell ith vlues in br plots in Figures 9 2. Tble E. Verifiction Tble for djoint function µ ith respect to responses T, T 50, RH, m 50 nd m. Response of Interest T Response of Interest T 50 Response of Interest V T S5 FD µ SFD µ m/s K K/m/s K/ J/m Bse cse Perturbed cse V 50 T S5 FD µ SFD µ m/s K K/m/s K/ J/m Bse cse Perturbed cse V RH S5 FD µ SFD µ m/s % m/s J/m RH Bse cse Perturbed cse

45 Energies 206, 9, of 52 Response of Interest m 50 Response of Interest Tble E. Cont. V m 50 m/s kg/s Bse cse Perturbed cse m Bse cse Perturbed cse S5 FD µ SFD µ kg/s m/s kg/s / J/m V m S5 FD µ SFD µ kg/s m/s kg/s m/s kg/s / J/m b Verifiction of djoint function o 49 Note tht vlue of djoint function o 49 obtined by solving djoint sensitivity system given in Eqution 20 is o K/ J/kg, s indicted in Figure 8. No select vrition δt,in in inlet ir temperture T,in, nd note tht Eqution 27 yields folloing expression for sensitivity of response R T to T,in : 49 S 45 T R,in µ i N i T + τ i N i 2 i,in T + τ i N i,in T + o,in 0 o49 N49 4 N T + µ 5,in T,in o 49 Cp T49 +tk 2 + ω in α g R µ ir 2 P tm m m + 96 f Re A 2 out A 2 in + k sum A 2 f ill Re-riting Eqution E4 in form i Ni 4 T,in L f ill A 2 f ill D h N + µ 5 T,in + g P tm Z + V2 R ir T,in 2 2g z rin z 2. E4 o 49 S 45 + µ N 5 T,in N 49 4 T,in E5 indictes tht vlue of djoint function o 49 could be computed independently if sensitivity S 45 ere vilble, since quntities N 49 4 / T,in J/ kg K nd N 5 / T,in J/ m K re knon. To first-order in prmeter perturbtion, finite-difference formul given in Eqution 28 cn be used to compute pproximte sensitivity S45 FD ; subsequently, this vlue cn be used in conjunction ith Eqution E5 to compute finite-difference sensitivity vlue, denoted s o 49 SFD, for respective djoint, hich ould be ccurte up to second-order in respective prmeter perturbtion: o 49 SFD S FD 45 + µ N 5 T,in N 49 4 T,in T,pert T,nom δt,in + µ N 5 T,in 49 N 4 E6 T,in Numericlly, inlet ir temperture T,in T db hs nominl bse-cse vlue of T,in K. The corresponding nominl vlue T,nom of response T is T,nom K. Consider next perturbtion δt,in T,in 0, for hich perturbed vlue of inlet ir temperture becomes T pert,in T0,in δt,in K. Re-computing perturbed response by solving Equstions 2 4 ith vlue of T pert,in yields perturbed

46 Energies 206, 9, of 52 response vlue T,pert K. Using no nominl nd perturbed response vlues toger ith prmeter perturbtion in finite-difference expression given in Eqution 28 T,pert T,nom yields corresponding finite-difference-computed sensitivity S45 FD δt,in Using this vlue toger ith nominl vlues of or quntities ppering in expression on right side of Eqution E6 yields o 49 SFD K/ J/kg. This result compres ell ith vlue o K/ J/kg obtined by solving djoint sensitivity system given in Eqution 20, cf., Figure 8. When solving this djoint sensitivity system, computtion of o 49 depends on previously computed djoint functions o i, i,..., I ; hence, forgoing verifiction of computtionl ccurcy of o 49 lso provides n indirect verifiction tht functions o i, i,..., I, ere lso computed ccurtely. The sme prmeter perturbtion s utilized to perform sme verifiction procedure for djoint function o 49 ith respect to or four responses; Tble E2 displys obtined results, hich compre ell ith vlues in br plots in Figures 9 2. Tble E2. Verifiction Tble for djoint function o 49 ith respect to responses T, T 50, RH, m 50 nd m. Response of Interest T Response of Interest T 50 Response of Interest T,in T Bse cse Perturbed cse S FD 45 o 49 SFD K K K/ J/kg T,in T 50 o S FD 45 o 49 SFD K K K/ J/kg Bse cse Perturbed cse T,in RH S45 o FD SFD o 49 K % K J/kg o 49 RH Bse cse Perturbed cse Response of Interest m 50 Response of Interest T,in m 50 K kg/s Bse cse Perturbed cse S45 FD kg/s K T,in m S45 FD K kg/s kg/s K m Bse cse Perturbed cse o 49 SFD kg/s / J/kg o o 49 SFD kg/s / J/kg o c Verifiction of Adjoint Function τ 49 Note tht vlue of djoint function τ 49 obtined by solving djoint sensitivity system given in Eqution 20 is τ K, s indicted in Figure 8. No select vrition δω in in

47 Energies 206, 9, of 52 inlet ir humidity rtio ω in, nd note tht Eqution 27 yields folloing expression for sensitivity of response R T to ω in : S ω in i R 0 τ 49 µ i N 49 ω in N i ω in + o Re-riting Eqution E7 in form + τ i 49 N49 4 N i 2 ω in ω in + τ i N i ω + o i Ni 4 N in ω + µ 5 in ω in τ 49 + o 49 h 50 g, T,in, α. E7 τ 49 S 46 o 49 h 50 g, T,in, α E8 indictes tht vlue of djoint function τ 49 could be computed independently if sensitivity S 46 ere vilble, since o 49 hs been verified in previous Appendix E. b nd quntity h 50 g, T,in, α is knon. To first-order in prmeter perturbtion, finite-difference formul given in Eqution 28 cn be used to compute pproximte sensitivity S46 FD ; subsequently, this vlue cn be used in conjunction ith Eqution E8 to compute finite-difference sensitivity vlue, denoted SFD, s for respective djoint, hich ould be ccurte up to second-order in respective τ 49 prmeter perturbtion: τ 49 SFD S FD 46 o 49 h 50 g, T,in, α E9 Numericlly, inlet ir humidity rtio ω in hs nominl bse-cse vlue of ωin The corresponding nominl vlue T,nom of response T is T,nom K. Consider next perturbtion δω in ωin 0, for hich perturbed vlue of inlet ir humidity rtio becomes ω pert in ωin 0 δω in Re-computing perturbed response by solving Eqsutions 2 4 ith vlue of ω pert in yields perturbed response vlue T,pert K. Using no nominl nd perturbed response vlues toger ith prmeter perturbtion in finite-difference expression given in Eqution 28 yields T,pert T,nom corresponding finite-difference-computed sensitivity S46 FD δω in.878 K. Using this vlue toger ith nominl vlues of or quntities ppering in expression on right side of Eqution E9 yields τ 49 SFD K. This result compres ell ith vlue τ K obtined by solving djoint sensitivity system given in Eqution 20, cf. Figure 8. When solving this djoint sensitivity system, computtion of τ 49 depends on previously computed djoint functions τ i ccurcy of τ 49, i,..., I ; hence, forgoing verifiction of computtionl lso provides n indirect verifiction tht functions τ i, i,..., I ere lso computed ccurtely. The sme prmeter perturbtion s utilized to perform sme verifiction procedure for djoint function τ 49 hich compre ell ith vlues in br plots in Figures 9 2. ith respect to or four responses; Tble E displys obtined results,

48 Energies 206, 9, of 52 Tble E. Verifiction Tble for djoint function τ 49 ith respect to responses T, T 50, RH, m 50 nd m. Response of Interest T Response of Interest T 50 Response of Interest ω in T S FD 46 τ 49 % K K K Bse cse Perturbed cse ω in T 50 SFD τ S FD 46 τ 49 % K K K Bse cse Perturbed cse ω in RH S FD 46 RH Bse cse Perturbed cse Response of Interest m 50 Response of Interest SFD τ τ 49 % % ω in m 50 SFD τ S FD 46 τ 49 SFD τ 49 % kg/s kg/s kg/s Bse cse Perturbed cse ω in m S FD 46 m Bse cse Perturbed cse τ 49 SFD τ 49 % kg/s kg/s kg/s d Verifiction of djoint functions τ Note tht vlues of djoint function τ given in Eqution 20 is s follos: τ obtined by solving djoint sensitivity system K/J/s, s indicted in Figure 8. No select vrition δt,in in inlet ter temperture T,in, nd note tht Eqution 27 yields folloing expression for sensitivity of response R T to T,in : 49 S T R,in 0 τ i N 2 µ i N i T,in 0 τ Re-riting Eqution E0 in form N i 2 N i T + τ i,in T + τ i,in T + o,in m,in f. i Ni 4 T,in N + µ 5 T,in E0 τ S E m,in f indictes tht vlue of djoint function τ could be computed independently if sensitivity S ere vilble, since quntity m,in f is knon. To first-order in prmeter perturbtion, finite-difference formul given in Eqution 28 cn be used to compute pproximte sensitivity S FD ; subsequently, this vlue cn be used in conjunction ith Eqution E to compute finite-difference sensitivity vlue, denoted s τ ccurte up to second-order in respective prmeter perturbtion: SFD, for respective djoint, hich ould be

49 Energies 206, 9, of 52 τ SFD S FD m,in f E2 Numericlly, inlet ter temperture, T,in, hs nominl bse-cse vlue of T,in K. As before, corresponding nominl vlue T,nom of response T is T,nom K. Consider no perturbtion δt,in T,in 0, for hich perturbed vlue of inlet ir temperture becomes T pert,in T0,in δt,in K. Re-computing perturbed response by solving Equtions 2 4 ith vlue of T pert,in yields perturbed response vlue T,pert K. Using no nominl nd perturbed response vlues toger ith prmeter perturbtion in finite-difference expression given in Eqution 28 yields corresponding finite-difference-computed sensitivity S FD δt,in Using this vlue toger ith nominl vlues of or quntities ppering in expression on right SFD side of Eqution E2 yields K/J/s. This result compres ell ith vlue τ τ T,pert T,nom K/J/s obtined by solving djoint sensitivity system given in Eqution 20, cf. Figure 8. The sme prmeter perturbtion s utilized to perform sme verifiction procedure for djoint function τ hich compre ell ith vlues in br plots in Figures 9 2. ith respect to or four responses; Tble E4 displys obtined results, Tble E4. Verifiction Tble for djoint function τ ith respect to responses T, T 50, RH, m 50 nd m. Response of Interest T Response of Interest T 50 Response of Interest T,in T Bse cse Perturbed cse S FD τ SFD τ K K K/J/s T,in T S FD τ SFD τ K K K/J/s Bse cse Perturbed cse T,in RH S FD τ SFD τ K % K J/s RH Bse cse Perturbed cse Response of Interest m 50 Response of Interest T,in m 50 K kg/s S FD τ SFD τ kg/s K J/kg Bse cse Perturbed cse T,in m S FD τ SFD τ K kg/s kg/s K J/kg m Bse cse Perturbed cse

50 Energies 206, 9, of 52 e Verifiction of djoint function µ Note tht vlues of djoint function µ given in Eqution 20 is s follos: µ obtined by solving djoint sensitivity system K/ kg/s, s indicted in Figure 8. No select vrition δm,in in inlet ter mss flo rte m,in, nd note tht Eqution 27 yields folloing expression for sensitivity of response R T to m,in : 49 S 44 m R,in µ i N i m + τ i N i 2 i,in m + τ i N i,in m + o,in 0 µ N m + τ N 2,in m + τ N,in m + o N 4,in µ + τ i Ni 4 m,in m,in N + µ 5 m,in T,in f g T f 0 g + τ m + o g T 2 + 0g m. E Since djoint functions τ 49 nd o 49 hve been lredy verified s described in Appendix E. b nd c, it follos tht computed vlues of djoint functions τ K o K/ J/kg cn lso be considered s being ccurte, since y constitute strting point for solving djoint sensitivity system in Eqution 20; τ Appendix E. d. Re-riting Eqution E in form µ S 44 + τ T,in f g T f 0 g + τ m s proved being ccurte in + o gt 2 m + 0g E4 indictes tht vlue of djoint function µ could be computed independently if sensitivity S 44 ere vilble, since ll or quntities re knon. To first-order in prmeter perturbtion, finite-difference formul given in Eqution 28 cn be used to compute pproximte sensitivity S44 FD ; subsequently, this vlue cn be used in conjunction ith Eqution E4 to compute finite-difference sensitivity vlue, denoted s µ ccurte up to second-order in respective prmeter perturbtion: µ SFD µ SFD, for respective djoint, hich ould be S44 FD + τ T,in f g T f 0 g + τ m + o g T 2 + 0g m Numericlly, inlet ter mss flo rte, m,in, hs nominl bse-cse vlue of m 0,in kg/s. As before, corresponding nominl vlue T,nom of response T is T,nom K. Consider no perturbtion δm,in m 0,in, for hich perturbed vlue of inlet ir temperture becomes m pert,in m0,in δm,in kg/s. Re-computing perturbed response by solving Equtions 2 4 ith vlue of m pert,in yields perturbed response vlue T,pert K. Using no nominl nd perturbed response vlues toger E5 ith prmeter perturbtion in finite-difference expression given in Eqution 28 yields corresponding finite-difference-computed sensitivity S44 FD T,pert T,nom δm,in K kg/s. Using this vlue toger ith nominl vlues of or quntities ppering in expression on right side of Eqution E5 yields ith vlue µ µ SFD K/kg/s. This result compres ell K/ kg/s obtined by solving djoint sensitivity system given in Eqution 20, cf. Figure 8. The sme prmeter perturbtion s utilized to perform sme verifiction procedure for djoint function µ hich compre ell ith vlues in br plots in Figures 9 2. ith respect to or four responses; Tble E5 displys obtined results,

51 Energies 206, 9, of 52 Tble E5. Verifiction Tble for djoint function µ ith respect to responses T, T 50, RH, m 50 nd m. Response of Interest T Response of Interest T 50 Response of Interest m,in T kg/s K Bse cse Perturbed cse m,in T 50 kg/s K S FD K kg/s µ SFD µ K kg/s S FD K kg/s µ SFD µ K kg/s Bse cse Perturbed cse m,in RH S FD µ SFD µ kg/s % kg/s kg/s RH Bse cse Perturbed cse Response of Interest m 50 Response of Interest m,in m S FD µ SFD µ kg/s kg/s Bse cse Perturbed cse m,in m S FD m Bse cse Perturbed cse µ SFD µ kg/s kg/s References. Alemn, S.E.; Grrett, A.J. Opertionl Cooling Toer Model CTTool v.0; SRNL-STI , Revision 0; Svnnh River Ntionl Lbortory: Svnnh River Site, SC, USA, Ccuci, D.G.; Di Rocco, F. Predictive modeling of buoyncy-operted cooling toer operting under sturted Conditions. I: Adjoint sensitivity model. Nucl. Sci. Eng. 206, in press.. Di Rocco, F.; Ccuci, D.G.; Bde, M.C. Predictive modeling of buoyncy-operted cooling toer operting under sturted Conditions. II: Optiml best-estimte results ith reduced predicted uncertinties. Nucl. Sci. Eng. 206, in press. 4. Ccuci, D.G. Sensitivity ory for nonliner systems: I. Nonliner functionl nlysis pproch. J. Mth. Phys. 98, 22, CrossRef 5. Ccuci, D.G. Sensitivity nd Uncertinty Anlysis: Theory; Chpmn & Hll/CRC: Boc Rton, FL, USA, Ccuci, D.G.; Ionescu-Bujor, M.; Nvon, I.M. Sensitivity nd Uncertinty Anlysis: Applictions to Lrge Scle Systems; Chpmn & Hll/CRC: Boc Rton, FL, USA, Ccuci, D.G. Predictive modeling of coupled multi-physics systems: I. ory. Ann. Nucl. Energy 204, 70, CrossRef 8. Grrett, A.J.; Prker, M.J.; Vill-Alemn, E Svnnh River site Cooling Toer Collection; SRNL-DOD ; Atmospheric Technologies Group, Svnnh River Ntionl Lbortory: Aiken, SC, USA, Sd, Y.; Schultz, M.H. GMRES: A generlized miniml residul lgorithm for solving nonsymmetric liner systems. SIAM J. Sci. Stt. Comput. 986, 7, CrossRef

52 Energies 206, 9, of Oppe, T.C.; Joubert, W.D.; Kincid, D.R. A Pckge for Solving Lrge Sprse Liner Systems by Vrious Itertive Methods; NSPCG User s Guide, Version.0; Center for Numericl Anlysis, University of Texs t Austin: Austin, TX, USA, Frgó, I.; Hvsi, A.; Zltev, Z. Advnced Numericl Methods for Complex Environmentl Models: Needs nd Avilbility; Benthm Science Publishers: Ok Prk, IL, USA, Ccuci, D.G.; Nvon, I.M.; Ionescu-Bujor, M. Computtionl Methods for Dt Evlution nd Assimiltion; Chpmn & Hll/CRC: Boc Rton, FL, USA, Ccuci, D.G. Second-order djoint sensitivity nlysis methodology 2nd-ASAM for lrge-scle nonliner Systems: I. Theory. Nucl. Sci. Eng. 206, 84, Ccuci, D.G. Second-Order djoint sensitivity nlysis methodology 2nd-ASAM for lrge-scle nonliner Systems: II. Illustrtive ppliction to prdigm nonliner het conduction benchmrk. Nucl. Sci. Eng. 206, 84, 52. CrossRef 206 by uthors; licensee MDPI, Bsel, Sitzerlnd. This rticle is n open ccess rticle distributed under terms nd conditions of Cretive Commons Attribution CC-BY license