Predictive Modeling of a Paradigm Mechanical Cooling Tower Model: II. Optimal Best-Estimate Results with Reduced Predicted Uncertainties

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1 energies Article Predictive Modeling of Prdigm Mechnicl Cooling Toer Model: II. Optiml Best-Estimte Results ith Reduced Predicted Uncertinties Ruixin Fng, Dn Gbriel Ccuci * nd Mdlin Bde Center for Nucler Science nd Energy, Deprtment of Mechnicl Engineering, University of South Crolin, Columbi, SC 908, USA fngr@cec.sc.edu R.F.) Bde@cec.sc.edu M.B.) * Correspondence: ccuci@cec.sc.edu Tel.: Acdemic Editor: Erich Schneider Received: June 06 Accepted: 5 September 06 Published: 6 September 06 Abstrct: This ork uses the djoint sensitivity model of the counter-flo cooling toer derived in the ccompnying PART I to obtin the expressions nd reltive numericl rnkings of the sensitivities, to ll model prmeters, of the folloing model responses: i) outlet ir temperture ii) outlet ter temperture iii) outlet ter mss flo rte nd iv) ir outlet reltive humidity. These sensitivities re subsequently used ithin the predictive modeling for coupled multi-physics systems PM_CMPS) methodology to obtin explicit formuls for the predicted optiml nominl vlues for the model responses nd prmeters, long ith reduced predicted stndrd devitions for the predicted model prmeters nd responses. These explicit formuls embody the ssimiltion of experimentl dt nd the clibrtion of the model s prmeters. The results presented in this ork demonstrte tht the PM_CMPS methodology reduces the predicted stndrd devitions to vlues tht re smller thn either the computed or the experimentlly mesured ones, even for responses e.g., the outlet ter flo rte) for hich no mesurements re vilble. These improvements stem from the globl chrcteristics of the PM_CMPS methodology, hich combines ll of the vilble informtion simultneously in phse-spce, s opposed to combining it sequentilly, s in current dt ssimiltion procedures. Keyords: djoint sensitivity nlysis dt ssimiltion model clibrtion best-estimte predictions reduced predicted uncertinties. Introduction In the present ork, the predictive modeling of the counter-flo cooling toer presented in is further developed by pplying the predictive modeling for coupled multi-physics systems PM_CMPS) methodology recently developed in. The PM_CMPS methodology constructs prior distribution for the prmeters nd responses by using ll of the vilble computtionl nd experimentl informtion, nd by relying on the mximum entropy principle to mximize the impct of ll vilble informtion nd minimize the impct of ignornce. Subsequently, the PM_CMPS methodology constructs formlly the posterior distribution using Byes theorem, nd then evlutes symptoticlly, to first-order sensitivities, the posterior distribution using the sddle-point method to obtin explicit formuls for the predicted optiml nominl vlues for the model responses nd prmeters, long ith reduced predicted uncertinties i.e., reduced predicted stndrd devitions) for the predicted model prmeters nd responses. The PM_CMPS methodology hs been successfully pplied to the nlysis of lrge-scle experiments nd the experimentl vlidtion of rector design codes of interest to rector physics,, light ter rectors 5 nd sodium-cooled fst rectors 6. Energies 06, 9, 77 doi:0.90/en mdpi.com/journl/energies

2 Energies 06, 9, 77 of 7 The PM_CMPS methodology relies fundmentlly on the sensitivities to model prmeters of the mesured model responses, hich, in this ork, re s follos: i) the outlet ir temperture ii) the outlet ter temperture iii) the outlet ter mss flo rte nd iv) the ir outlet reltive humidity. The expressions, numericl results, nd reltive rnkings of the sensitivities of these responses re presented in Section.. These sensitivities re subsequently used in Section. for ssimilting experimentl dt in order to clibrte the model prmeters, nd for obtining best-estimte predicted results ith reduced predicted uncertinties. Section concludes this ork by discussing the significnce of the results presented herein in the context of ongoing ork imed t further pplictions nd generliztion of the djoint sensitivity nlysis nd PM_CMPS methodologies.. Results It hs been shon in the ccompnying PART I tht the totl sensitivity of model response R m, T, T, ω α) to rbitrry vritions in the model s prmeters δα δα,..., δα Nα ) nd stte functions δm, δt, δt, δω, round the nominl vlues m 0, T 0, T 0, ω 0 α 0) of the prmeters nd stte functions, is provided by the G-differentil of the model s response to these vritions. This G-differentil s denoted s DR m 0, T 0, T 0, ω 0 α 0 δm, δt, δt, δω δα ), nd s expressed in terms of the djoint sensitivity functions s follos: ) DR m 0, T 0, T 0, ω 0 α 0 N α ) R δm, δt, δt, δω δα δα i + DR indirect, ) i i here the so-clled indirect effect term, DR indirect, is given by: DR indirect µ Q + τ Q + τ Q + o Q ) nd here the vector µ, τ, τ, o + is the solution of the folloing djoint sensitivity system: A + A + A + A + B + B + B + B + C + C + C + C + D + D + D + D + Furthermore, the sources R l r ) l µ τ τ o R R R R. ),..., r I) ) l, l,,,, for the djoint sensitivity system represented by Eqution ) re the functionl derivtives of the model responses ith respect to the stte functions, i.e.: r i) R m i+) r i) R T i+) r i) R T i) r i) R i,..., I ) ω i) hile the components of the vectors Q l q ) l,..., q I) ) l, l,,,, in Eqution ) re the derivtives of the model s equtions ith respect to model prmeters, nmely: q i) l N α j ) i) N l δα j i,..., I l,,,. 5) j The explicit expressions of the vectors Q l q ) l,..., q I) ) l, l,,, re provided in Appendix A. The model responses of interest in this ork re the folloing quntities: i) the outlet ir temperture, T ) ii) the outlet ter temperture, T 50) iii) the outlet ter flo rte, m 50) nd iv) the outlet ir reltive humidity, RH ). Except for the ter outlet flo rte m 50), these responses hve been mesured experimentlly 7,8, nd the first four moments of their respective sttisticl distributions hve been quntified in.

3 Energies 06, 9, 77 of 7.. Sensitivity Anlysis Results nd Rnkings As hs been discussed in the ccompnying PART I, there re totl of 8079 mesured benchmrk dt sets for the cooling toer model ith the fn-on, ith drfted ir exit velocity t 0 m/s t the shroud. For this velocity nd corresponding ir flo rte), the Reynolds number is round 500, hich mens tht the flo ithin the cooling toer is in the trnsitionl flo nd het trnsfer regime. As hs lso been discussed in, 7668 benchmrk dt sets out of the totl of 8079 dt sets) re considered to correspond to the unsturted conditions hich re nlyzed in this ork. The nominl vlues for boundry nd tmospheric conditions used in this ork ere obtined, s described in, from the sttistics of these 7668 benchmrk dt sets corresponding to unsturted conditions. In turn, these unsturted boundry nd tmospheric conditions ere used to obtin the sensitivity results reported, belo, in this Subsection. Sub-subsections.. through.., belo, provide the numericl vlues nd rnkings, in descending order, of the reltive sensitivities computed using the djoint sensitivity nlysis methodology for the four model responses T ), T 50), nd RH ). Note tht the reltive sensitivity, RS α i ), of response R α i ) to prmeter α i is m 50) defined s RS α i ) dr α i ) /dα i α i /R α i ). Thus, the reltive sensitivities re unit-less nd re very useful in rnking the sensitivities to highlight their reltive importnce for the respective response. Thus, reltive sensitivity of.00 indictes tht chnge of % in the respective prmeter ill induce % chnge in response tht is liner in the respective sensitivity. The higher the reltive sensitivity, the more importnt the respective prmeter to the respective response.... Reltive Sensitivities of the Outlet Air Temperture, T ) The sensitivities of the ir outlet temperture ith respect to ll of the model s prmeters hve been computed using Equtions ) nd ). The numericl results nd rnking of the reltive sensitivities, in descending order of their mgnitudes, re provided in Tble belo, long ith their respective reltive stndrd devitions. Tble. Rnked reltive sensitivities of the outlet ir temperture T ). Rnk # Prmeter α i ) Nominl Vlue Reltive Sensitivity RS α i) Reltive Stndrd Devition %) Inlet ir temperture, T,in 99. K Air temperture dry bulb), T db 99. K Inlet ter temperture, T,in K De point temperture, T dp 9.05 K P vs T) prmeter, P vs T) prmeter, Inlet ir humidity rtio, ω in Fn shroud inner dimeter, D fn. m Wter enthlpy h f T) prmeter, f Wetted frction of fill surfce re, ts Nusselt number, Nu Fill section surfce re, A surf m Dynmic viscosity of ir t T 00 K, µ kg/m s) Nu prmeter,,nu Reynolds number, Re d Fill section flo re, A fill 67.9 m C p T) prmeter, 0,cp Inlet ter mss flo rte, m,in.0 kg/s h g T) prmeter, 0g D v T) prmeter,,dv Exit ir speed t the shroud, V exit 0.0 m/s Inlet ir mss flo rte, m kg/s Het trnsfer coefficient multiplier, f ht Therml conductivity of ir t T 00 K, k ir 0.06 W/m K) Mss trnsfer coefficient multiplier, f mt Sherood number, Sh D v T) prmeter,,dv h f T) prmeter, 0f,, D v T) prmeter, 0,dv Atmospheric pressure, P tm 00,586 P Kinemtic viscosity of ir t 00 K, ν m /s Prndlt number of ir t T 80 C, Pr Schmidt number, Sc h g T) prmeter, g

4 Energies 06, 9, 77 of 7 Tble. Cont. Rnk # Prmeter α i ) Nominl Vlue Reltive Sensitivity RS α i) Reltive Stndrd Devition %) 5 D v T) prmeter,,dv Nu prmeter,,nu Fill section equivlent dimeter, D h 0.08 m C p T) prmeter,,cp C p T) prmeter,,cp Sum of loss coefficients bove fill, k sum Fill section frictionl loss multiplier, f Nu prmeter, 0,Nu Nu prmeter,,nu 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 Fn shroud height, z fn.0 m Fill section height, z fill.0 m Rin section height, z rin.6 m Bsin section height, z bs.68 m Drift elimintor thickness, z de 0.5 m Wind speed, V.80 m/s As the results in Tble indicte, the first five prmeters i.e., T,in, T db, T,in, T dp, 0 ) hve reltive sensitivities beteen c. 0% nd 50%, nd re therefore the most importnt for the ir outlet temperture response, T ). The to lrgest sensitivities hve vlues of 8%, hich mens tht % chnge in T,in or T db ould induce 0.8% chnge in T ). The next to prmeters i.e., nd ω in ) hve reltive sensitivities beteen % nd 6%, nd re therefore someht importnt. Prmeters #8 through #6 i.e., D fn, f, ts, Nu, A surf, µ,,nu, Re d, A fill ) hve reltive sensitivities of the order of 0.5%. The remining 6 prmeters re reltively unimportnt for this response, hving reltive sensitivities smller thn % of the lrgest reltive sensitivity ith respect to T,in ) for this response. Positive sensitivities imply tht positive chnge in the respective prmeter ould cuse n increse in the response, hile negtive sensitivities imply tht positive chnge in the respective prmeter ould cuse decrese in the response.... Reltive Sensitivities of the Outlet Wter Temperture, T 50) The results nd rnking of the reltive sensitivities of the outlet ter temperture ith respect to the most importnt prmeters for this response re listed in Tble. Tble. Most importnt reltive sensitivities of the outlet ter temperture, T 50). Rnk # Prmeter α i ) Nominl Vlue Reltive Sensitivity RS α i) Reltive Stndrd Devition %) De point temperture, T dp 9.05 K Inlet ir temperture, T α,in 99. K Air temperture dry bulb), T db 99. K 0..9 P vs T) prmeters, P vs T) prmeters, Inlet ter temperture, T,in K Inlet ir humidity rtio, ω in Fn shroud inner dimeter, D fn. m Wter enthlpy hft) prmeter, f D v T db ) prmeter,,dv Enthlpy h g T) prmeter, 0g,005, Sherood number, Sh The lrgest sensitivity of T 50) is to the prmeter T dp, nd hs the vlue of 0.58 this mens tht % increse in T db ould induce 0.58% increse in T 50). The sensitivities to the remining 0 model prmeters hve not been listed since they re smller thn % of the lrgest sensitivity ith respect to T dp ) for this response.... Reltive Sensitivities of the Outlet Wter Mss Flo Rte, m 50) The results nd rnking of the reltive sensitivities of the outlet ter mss flo rte ith respect to the most importnt 0 prmeters for this response re listed in Tble. This response is most

5 Energies 06, 9, 77 5 of 7 sensitive to m,in % increse in this prmeter ould cuse.0% increse in the response) nd the second lrgest sensitivity is to the prmeter T,in % increse in this prmeter ould cuse 0.7% decrese in the response). The sensitivities to the remining model prmeters hve not been listed since they re smller thn % of the lrgest sensitivity ith respect to m,in ) for this response. Tble. Most importnt reltive sensitivities of the outlet ter mss flo rte, m 50). Rnk # Prmeter α i ) Nominl Vlue Reltive Sensitivity RSα i ) Reltive Stndrd Devition %) Inlet ter mss flo rte, m,in.0 kg/s Inlet ter temperture, T,in K De point temperture, T dp 9.05 K PvsT) prmeters, Air temperture dry bulb), T db 99. K Inlet ir temperture, T,in 99. K PvsT) prmeters, Inlet ir humidity rtio, ω in Fn shroud inner dimeter, D fn. m Inlet ir mss flo rte, m kg/s Reltive Sensitivities of the Outlet Air Reltive Humidity, RH ) The results nd rnking of the reltive sensitivities of the outlet ir reltive humidity ith respect to the most importnt 0 prmeters for this response re listed in Tble. The first three sensitivities of this response re quite 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 the response of 6.66% or 6.55%, respectively. On the other hnd, n increse of % in T dp ould cuse n increse of 5.75% in the response. The sensitivities to the remining model prmeters hve not been listed since they re smller thn % of the lrgest sensitivity ith respect to T,in ) for this response. Tble. Most importnt reltive sensitivities of the outlet ir reltive humidity, RH ). Rnk # Prmeter α i ) Nominl Vlue Reltive Sensitivity RS α i ) Reltive Stndrd Devition %) Inlet ir temperture, T,in 99. K Air temperture dry bulb), T db 99. K De point temperture, T dp 9.05 K Inlet ter temperture, T,in K Inlet ir humidity rtio, ω in P vs T) prmeters, Wetted frction of fill surfce re, ts Fill section surfce re, A surf, m Nusselt number, Nu Dynmic viscosity of ir t T 00 K, µ kg/m s) Nu prmeters,,nu Fill section flo re, A fill 67.9 m Reynold s number, Re D v T db ) prmeter,,dv Mss trnsfer coefficient multiplier, f mt Sherood number, Sh Atmosphere pressure, P tm 00,586 P D v T db ) prmeter,,dv D v T db ) prmeter, 0,dv P vs T) prmeters, Overll, the outlet ir reltive humidity, RH ), displys the lrgest sensitivities, so this response is the most sensitive to prmeter vritions. The other responses, nmely the outlet ir temperture, the outlet ter temperture, nd the outlet ter mss flo rte disply sensitivities of comprble mgnitudes... Experimentl Dt Assimiltion, Model Clibrtion nd Best-Estimte Predicted Results ith Reduced Predicted Uncertinties This subsection presents the results of pplying the Predictive Modeling of Coupled Multi-Physics Systems PM_CMPS) methodology to the counter-flo cooling toer model. The PM_CMPS methodology encompsses into unified conceptul nd mthemticl frmeork, the concepts of both forrd nd inverse modeling, including dt ssimiltion, model clibrtion nd prediction

6 Energies 06, 9, 77 6 of 7 of best-estimte vlues for model prmeters nd responses, ith reduced predicted uncertinties. For the simplest cse of single computtionl model, such s the counter-flo cooling toer model nlyzed in this ork, the PM_CMPS methodology considers the folloing priori informtion:. A model comprising N α imprecisely knon system model) prmeters, α n, considered s the components of column) vector, α, defined s: α {α n n,..., N α } 6) The men vlues of the model prmeters α n re denoted s α 0 n α n, nd the covrinces beteen to prmeters α i nd α j re denoted s covα i,α j ). The men vlues α 0 n re considered to be knon priori, so tht the vector α 0, defined s α 0 { α 0 } n n,..., Nα is considered to be knon priori. The covrinces covα i,α j ) re lso considered to be priori knon these covrinces re considered to be the elements of the priori knon prmeter covrince mtrix, denoted s C N α N α ) αα nd defined s: C N α N α ) αα cov ) ) α i, α j α )N α N α i α 0 i α j α 0 j i, j,..., N α 7) N α N α. Also ssocited ith the model re N r experimentlly mesured responses, r i, considered to be components of the column vector: r {r i i,..., N r } 8) The men vlues, denoted s ri m, of the mesured responses, r i, nd the covrinces, denoted s r i ri m )r j r m j ), beteen to mesured responses, r i nd r j, re lso considered to be knon priori. The men mesured vlues ri m ill be considered to constitute the components of the vector r m defined s: r m {ri m i,..., N r }, ri m r i, i,..., N r, 9) nd the covrinces r i r m i )r j r m j ) of the mesured responses re considered to be components of the priori knon mesured covrince mtrix, denoted s C N r N r ) rr, nd defined s: C N r N r ) rr ) r i ri m ) r j r m j, i, j,..., N r. 0) N r N r. In the most generl cse, correltions my lso exist mong ll prmeters nd responses. Such correltions re quntified through priori knon prmeter-response mtrices, denoted s C N α N r ) αr, nd defined s follos: C N α N r ) αr α α 0) r r m ) + C N r N α ) rα + ) To keep the nottion simple, the dimensions of the vrious vectors nd mtrices ill not be shon in subsequent formuls. For single multi-physics system, s is the cse of the cooling toer model under considertion in this ork, the quntities predicted by the PM_CMPS methodology re s follos: A. Optimlly predicted best-estimte nominl vlues, α pred, for the model prmeters: here the mtrix D rr is defined s: α pred α 0 C αα S + rα C αr ) Drr r c α 0, β 0) r m, ) D rr S rα C αα S + rα S rα C αr C + αrs + rα + C rr, )

7 Energies 06, 9, 77 7 of 7 nd the components of the mtrix S N r N α ) rα re the first-order sensitivities i.e., functionl derivtives) of ll responses ith respect to ll model prmeters, defined s follos: S N r N α rα r r Nα..... r Nr r Nr Nα. ) It is importnt to note tht the first term on the right side of Eqution ) is the covrince mtrix of the computed responses, Crr comp, hen only the first-order sensitivities re tken into ccount, i.e.: Crr comp S rα C αα S rα. + 5) B. Reduced predicted uncertinties, Cαα pred, for the predicted nominl prmeter vlues, given by the expression belo: C pred αα C αα C αα S + rα C αr ) Drr C αα S + rα C αr ) + 6) C. Optimlly predicted best-estimte nominl vlues, r pred, for the model responses, given by the expression belo: r pred r m C + αrs + rα C rr ) Drr r c α 0, β 0) r m 7) D. Reduced predicted uncertinties, Crr pred, for the predicted nominl response vlues, given by the expression belo: C pred rr C rr C + αrs + rα C rr ) Drr C + αrs + rα C rr ) + 8) E. Predicted correltions, Cαr pred, beteen the predicted model prmeters nd responses, given by the expression belo: C pred αr C αr C αα S + rα C αr ) Drr C + αrs + rα C rr ) +. 9) The expressions given in Equtions 6) through 9) cn lso be obtined from the results presented originlly in 9 for the prticulr cse of time-independent single multi-physics system. Note tht if the model is perfect hich mens tht C αα 0 nd C αr 0), Equtions 6) through 9) ould yield 0, Cαα pred 0, α pred α 0 nd r pred r c α 0,β 0 ), ithout ny ccompnying uncertinties i.e., Crr pred 0). In other ords, for perfect model, the PM_CMPS methodology predicts vlues for C pred αr the responses nd the prmeters tht ould coincide ith the model s originl corresponding prmeter nd computed responses ssumed to be perfect), nd the experimentl mesurements ould hve no effect on the predictions s ould be expected, since imperfect mesurements could not possibly improve perfect model s predictions). On the other hnd, if the mesurements ere perfect, i.e., C rr 0 nd C αr 0), but the model ere imperfect, then Equtions 6) through 9) ould yield α pred α 0 C αα S rα + S rα C αα S rα + r d α 0), Cαα pred C αα C αα S rα + S rα C αα S rα + S rα C αα, r pred r m, Crr pred 0, Cαr pred 0. In other ords, in the cse of perfect mesurements, the PM_CMPS predicted vlues for the responses ould coincide ith the mesured vlues ssumed to be perfect), hile the model s uncertin prmeters ould be clibrted by tking the respective mesurements into ccount to yield improved nominl vlues nd reduced prmeters uncertinties. The priori response-prmeter covrince mtrix, C rα, hs been lredy computed in, Eqution A5), nd is reproduced belo:

8 Energies 06, 9, 77 8 of 7 Cov T mes, out, Tmes, out, RHmes, α,..., α 5 ) Crα ) here the mesured correlted prmeters re: α T db, α T dp, α T,in, nd α P tm. The priori prmeter covrince mtrix, C αα, hs lso been lredy computed in, Eqution B) see the Appendix of PART I.), nd is lso reproduced belo: C αα Vrα ) Covα, α ) Covα, α 5 ) Covα, α ) Vrα ) Covα, α 5 ) Covα 5, α ) Vrα 5 ) The priori covrince mtrix of the computed responses, Crr comp, is obtined by using Equtions 5) nd ) together ith the sensitivity results presented in Tbles the finl result is given belo: Crr comp Cov T ) T ) T 50), T 50), RH )) S rα C αα S rα +,..., T) Nα,..., T50) Nα RH ),..., RH) Nα Vrα ) Covα, α ) Covα, α 5 ) Covα, α ) Vrα ) Covα, α 5 ) Covα 5, α ) Vrα 5 ). The priori covrince mtrix, Cov T mes nmely: the outlet ir temperture, T mes T mes,out T 50),out, Tmes,out, RHmes out,out T ) mesured, nd the outlet ir reltive humidity, RH mes out computed in, Eqution A), nd is reproduced belo: Cov T mes,out, T mes,out, RHout mes ) Crr T ) T 50),..., T) Nα,..., T50) Nα RH ),..., RH) Nα + ) ) ) Crr, of the mesured responses mesured the outlet ter temperture, RH ) mesured s lso )... Model Clibrtion: Predicted Best-Estimted Prmeter Vlues ith Reduced Predicted Stndrd Devitions The best-estimte nominl prmeter vlues hve been computed using Eqution ) in conjunction ith the priori mtrices given in Equtions 0) ) nd the sensitivities presented in Tbles. The resulting best-estimte nominl vlues re listed in Tble 5, belo. The corresponding best-estimte bsolute stndrd devitions for these prmeters re lso presented in this tble. These

9 Energies 06, 9, 77 9 of 7 vlues re the squre-roots of the digonl elements of the mtrix Cαα pred, hich is computed using Eqution 6) in conjunction ith the priori mtrices given in Equtions 0) ) nd the sensitivities presented in Tbles. For comprison, the originl nominl prmeter vlues nd originl bsolute stndrd devitions re lso listed. As the results in Tble 5 indicte, the predicted best-estimte stndrd devitions re ll smller or t most equl to i.e., left unffected) the originl stndrd devitions. The prmeters re ffected proportionlly to the mgnitudes of their corresponding sensitivities: the prmeters experiencing the lrgest reductions in their predicted stndrd devitions re those hving the lrgest sensitivities. Tble 5. Best-estimted nominl prmeter vlues nd their stndrd devitions. i Independent Sclr Prmeters α i ) Mth. Nottion Originl Nominl Vlue Originl Stndrd Devition Best-Estimted Nominl Vlue Best-Estimted Stndrd Devition 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, , Wetted frction of fill surfce re ts Sum of loss coefficients bove fill k sum Dynmic viscosity of ir t T 00 K kg/m s) µ Kinemtic viscosity of ir t T 00 K m /s) v Therml conductivity of ir t T 00 K W/m K) k ir Het trnsfer coefficient multiplier f ht Mss trnsfer coefficient multiplier f mt Fill section frictionl loss multiplier f.00 P vs T) prmeters ,cp C p T) prmeters,cp ,cp ,dv D v T) prmeters,dv ,dv ,dv h f T) prmeters 0f,,.8 5,,.7 5 f h g T) prmeters 0g,005, ,005, g ,Nu Nu prmeters,nu ,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 fn Fn shroud inner dimeter m) D fn Fill section height m) z fill 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 ) A fill Fill section surfce re m ) A surf, Prndlt number of ir t T 80 C P r Wind speed m/s) V Exit ir speed t the shroud m/s) V exit i Boundry Prmeters Mth. Nottion Originl Nominl Vlue Originl Stndrd Devition Best-Estimted Nominl Vlue Best-Estimted Stndrd Devition 5 Inlet ter mss flo rte kg/s) m,in Inlet ir temperture K) T,in Inlet ir mss flo rte kg/s) m Inlet ir humidity rtio ω in i Specil Dependent Prmeters Mth. Nottion Originl Nominl Vlue Originl Stndrd Devition Best-Estimted Nominl Vlue Best-Estimted Stndrd Devition 9 Reynold s number Re d Schmidt number Sc Sherood number Sh Nusselt number Nu

10 Energies 06, 9, 77 0 of 7... Predicted Best-Estimted Response Vlues ith Reduced Predicted Stndrd Devitions Using the priori mtrices given in Equtions 0) ) together ith the sensitivities presented in Tbles in Eqution 8) yields the folloing predicted response covrince mtrix, C pred rr : C pred rr Cov T ) be, T 50) be, RH ) ) be ) The best-estimte response-prmeter correltion mtrix, C pred αr, is obtined using Eqution 9) together ith the priori mtrices given in Equtions 0) ) nd the sensitivities presented in Tbles. The non-zero elements ith the lrgest mgnitudes re s follos: rel. cor.r, α ) 0.78 rel. cor.r, α ) rel. cor.r, α 9 ) 0.09 rel. cor.r, α ) 0.08 rel. cor.r, α ) 0.09 rel. cor.r, α ) 0. rel. cor.r, α ) 0.7 rel. cor.r, α 9 ) ) The nottion used in Eqution 5) is s follos: R T ), R T 50), R RH ), α P tm, α A sur f nd α 9 Re d. The best-estimte nominl vlues of the model responses) outlet ir temperture, T ) outlet ter temperture T 50) nd outlet ir reltive humidity, RH ), hve been computed using Eqution 7) together ith the priori mtrices given in Equtions 0) ) nd the sensitivities presented in Tbles. The resulting best-estimte predicted nominl vlues re summrized in Tble 6. To fcilitte comprison, the corresponding mesured nd computed nominl vlues re lso presented in this tble. Note tht there re no direct mesurements for the outlet ter flo rte, m 50). For this response, therefore, the predicted best-estimte nominl vlue hs been obtined by forrd re-computtion using the best-estimte nominl prmeter vlues listed in Tble 5, hile the predicted best estimte stndrd devition for this response hs been obtined by using best-estimte vlues in Eqution 5), i.e.: C comp rr be Srα be C αα be S rα + be. 6) Tble 6. Computed, mesured, nd optiml best-estimte nominl vlues nd stndrd devitions for the outlet ir temperture, outlet ter temperture, outlet ir reltive humidity, nd outlet ter flo rte responses. Nominl Vlues nd Stndrd Devitions T ) K) T 50) K) RH ) %) m 50) kg/s) Mesured Nominl vlue Stndrd devition ±.6 ±.59 ±5.89 Computed Nominl vlue Stndrd devition ±.0 ±.78 ±.90 ±. Best-estimte Nominl vlue Stndrd devition ±.59 ±.5 ±.05 ±.0 The results presented in Tble 6 indicte tht the predicted stndrd devitions re smller thn either the computed or the experimentlly mesured ones. This is indeed the consequence of using the PM_CMPS methodology in conjunction ith consistent s opposed to discrepnt) computtionl nd experimentl informtion. Often, hoever, the informtion is inconsistent, usully due to the presence of unrecognized errors. Solutions for ddressing such situtions hve been proposed in 0.

11 Energies 06, 9, 77 of 7 It is lso importnt to note tht the PM_CMPS methodology hs improved i.e., reduced, lbeit not by significnt mount) the predicted stndrd devition for the outlet ter flo rte response, for hich no mesurements ere vilble. This improvement stems from the globl chrcteristics of the PM_CMPS methodology, hich combines ll of the vilble simultneously on phse-spce, s opposed to combining it sequentilly, s is the cse ith the current stte-of-the-rt dt ssimiltion procedures,.. Discussion In the present ork, the djoint sensitivity model of the counter-flo cooling toer derived in the ccompnying PART I s used to obtin the expressions nd reltive numericl rnkings of the sensitivities, to ll model prmeters, of the folloing responses quntities of interest): i) the outlet ir temperture ii) the outlet ter temperture iii) the outlet ter mss flo rte nd iv) the ir outlet reltive humidity. These sensitivities ere subsequently used ithin the predictive modeling for coupled multi-physics systems PM_CMPS) methodology to obtin explicit formuls for the predicted optiml nominl vlues for the model responses nd prmeters, long ith reduced predicted stndrd devitions for the predicted model prmeters nd responses. These explicit formuls embody the ssimiltion of experimentl dt nd the clibrtion of the model s prmeters. The results presented in this ork indicte tht the predicted stndrd devitions re smller thn either the computed or the experimentlly mesured ones. It is lso importnt to note tht the PM_CMPS methodology hs improved i.e., reduced, lbeit not by significnt mount) the predicted stndrd devition for the outlet ter flo rte response, for hich no mesurements ere vilble. This improvement stems from the globl chrcteristics of the PM_CMPS methodology, hich combines ll of the vilble informtion simultneously in phse-spce, s opposed to combining it sequentilly, s is the cse ith the current stte-of-the-rt dt ssimiltion procedures,. This is indeed the consequence of using the PM_CMPS methodology in conjunction ith consistent s opposed to discrepnt) computtionl nd experimentl informtion. Often, hoever, the informtion is inconsistent, usully due to the presence of unrecognized errors. Solutions for ddressing such situtions hve been proposed in 0. The djoint sensitivity nlysis methodology used in PART I for computing exctly nd efficiently the st -order response sensitivities to model prmeters hs been recently extended to computing efficiently nd exctly the nd-order response sensitivities to prmeters for liner nd nonliner lrge-scle systems. As hs been shon in 5 8, the nd-order response sensitivities hve the folloing mjor impcts on the computed moments of the response distribution: ) they cuse the expected vlue of the response to differ from the computed nominl vlue of the response nd b) they contribute decisively to cusing symmetries in the response distribution. Indeed, neglecting the second-order sensitivities ould nullify the third-order response correltions, nd hence ould nullify the skeness of the response. Consequently, non-gussin fetures i.e., symmetries, long-tils) ny events occurring in response s long nd/or short tils, hich re chrcteristic of rre but decisive events e.g., mjor ccidents, ctstrophes), ould likely be missed. Ongoing ork ims t further pplictions nd generliztion of the djoint sensitivity nlysis nd the PM_CMPS methodologies, to enble the computtion of rd- nd higher-order sensitivities nd response distributions. The exct nd efficient computtion of high-order response sensitivities for lrge-scle systems is expected to dvnce significntly the res of uncertinty quntifiction, model vlidtion, reduced-order modeling, nd predictive modeling/dt ssimiltion. Acknoledgments: This ork hs been prtilly sponsored by the US Deprtment of Energy Jmes J. Peltz, Progrm mnger) ith the University of South Crolin. Author Contributions: Ruixin Fng performed ll of the numericl clcultions in this pper. Dn Ccuci conceived nd directed the reserch reported herein, nd rote the pper. Mdlin Bde contributed the progrmming of the equtions underlying the predictive modeling formlism. Conflicts of Interest: The uthors declre no conflict of interest.

12 Energies 06, 9, 77 of 7 Appendix A. Derivtives of Cooling Toer Model Equtions ith Respect To Model Prmeters For convenience, the model prmeters re reproduced in Tble A belo from Appendix B of PART I. The independent model prmeters re used for computing vrious dependent model prmeters nd therml mteril properties, s shon in Tbles A nd A, belo. Tble A. Prmeters for SRNL f-re cooling toers. Index i of α i Independent Sclr Prmeters C + + String Mth. Nottion Nominl Vlues) Absolute Stndrd Devition Reltive Stndrd Devition %) 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 Wetted frction of fill surfce re ts ts Sum of loss coefficients bove fill ksum k sum Dynmic viscosity of ir t T 00 K kg/m s) muir µ Kinemtic viscosity of ir t T 00 K m /s) nuir v Therml conductivity of ir t T 00 K W/m K) tcir k ir Het trnsfer coefficient multiplier mlthtc f ht Mss trnsfer coefficient multiplier mltmtc f mt Fill section frictionl loss multiplier mltfil f 50 P vs T) prmeters A) 0,cp C p T) prmeters A),cp A),cp A) 0,dv D v T) prmeters A),dv A),dv A),dv h f T) prmeters 0f 0f,, f f h g T) prmeters 0g 0g,005, g g ,Nu Nu prmeters -,Nu ,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 fn Fn shroud inner dimeter m) fsid D fn Fill section height m) flht z fill 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 ) flf A fill Fill section surfce re m ) fls A surf Prndlt number of ir t T 80 C Pr P r Wind speed m/s) spd V Exit ir speed t the shroud m/s) vexit V exit Index i of α i Boundry Prmeters C + + String Absolute Mth. Nottion Nominl Vlue Stndrd Devition Reltive Stndrd Devition %) 5 Inlet ter mss flo rte kg/s) mfin m,in Inlet ir temperture K) tin T,in set to T db Inlet ir mss flo rte kg/s) min m Inlet ir humidity rtio Dependent Sclr Prmeter) hrin ω in ω rin Reynold s number Re Reh Re d Schmidt number Sc Sc Sherood number Sh Sh Nusselt number Nu Nu

13 Energies 06, 9, 77 of 7 Tble A. Dependent sclr model prmeters. Dependent Sclr Prmeters Mth. Nottion Defining Eqution or Correltion Mss diffusivity of ter vpor in ir m /s) Het trnsfer coefficient W/m K) D v T,α) hα) 0,dv T.5,dv +,dv +,dv T)T f ht N uk ir D h f mt S h D vt db,α) D h Mss trnsfer coefficient m/s) k m α) Het trnsfer term W/K) Hm,α) h α) ts A f f Mss trnsfer term m /s) Mm,α) M H Ok m α) ts A f f Density of dry ir kg/m ) ρα) P tm Air velocity in the fill section m/s) v m,α) R ir T db m ρα)a f ill Fill flling-film surfce re per verticl section m A ) A sur f ff I Rin section inlet flo re m ) 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 Z z f ill z rin Fill section control volume height m) z Fill section length, including drift elimintor m) L fill z f ill + z de Fn shroud inner rdius m) r fn 0.5D f n Fn shroud flo re m ) A out πr f n z f ill I Tble A. Therml properties dependent sclr model prmeters). Therml Properties Functions of Stte Vribles) Mth. Nottion Defining Eqution or Correltion h f T ) sturted liquid enthlpy J/kg) h f T,α) 0 f + f T H g T ) sturted vpor enthlpy J/kg) h g, T,α) 0g + g T H g T ) sturted vpor enthlpy J/kg) h g, T,α) 0g + g T C p T) specific het of dry ir J/kg K) C p T,α) 0,cp +,cp +,cp T)T P vs T ) sturtion pressure P) P vs T,α) P c e 0+ T, in hich P c.0 P P vs T ) sturtion pressure P) P vs T,α) P c e 0+ T, in hich P c.0 P Note: The prmeters α through α i.e., the dry bulb ir temperture, de point temperture, inlet ter temperture, nd tmospheric pressure) ere mesured t the SRNL site t hich the F-re cooling toers re locted. Among the 8079 mesured benchmrk dt sets 8, 7688 dt sets re considered to represent unsturted conditions, hich hve been used to derive the sttisticl properties mens, vrince nd covrince, skeness nd kurtosis) for these model prmeters, s shon in Figures B through B nd Tbles B through B7 in Appendix B of PART I. Recll tht the cooling toer model comprises conservtion blnces representing mthemticlly the folloing physicl phenomen: A. liquid continuity B. liquid energy blnce C. ter vpor continuity D. ir nd ter vpor energy blnce. For esy reference, these conservtion equtions re reproduced belo from Section of PART I : A. Liquid continuity equtions: i) Control Volume i : N ) m, T, T, ω α) m ) ii) Control Volumes i,..., I : N i) m, T, T, ω α) m i+) iii) Control Volume i I: N I) m, T, T, ω α) m I+) B. Liquid energy blnce equtions: m,in + Mm,α) R m i) + Mm,α) R m I) + Mm,α) R P vs ) T ) P i+) vs P I+) vs,α) T ) T i+),α) T i+) T I+),α) T I+) ω ) P tm T ) 0.6+ω ) ) ω i) P tm T i) 0.6+ω i) ) ω I) P tm T I) 0.6+ω I) ) 0 A) 0 A) 0 A)

14 Energies 06, 9, 77 of 7 i) Control Volume i : N ) m, T, T, ω α) m,in h f T,in, α) T ) T ) )Hm, α) m ) h ) f T ), α) m,in m ) )h ) g,t ), α) 0 A) ii) Control Volumes i,..., I : N i) m, T, T, ω α) m i) m i+) h i+) f T i+) h i) f T i), α) T i+) T i) )Hm, α), α) m i) m i+) )h i+) g, T i+), α) 0 A5) iii) Control Volume i I: N I) m, T, T, ω α) m I) m I+) h I+) f T I+) h I) f T I), α) T I+) T I) )Hm, α), α) m I) m I+) )h I+) g, T I+), α) 0 A6) C. Wter vpor continuity equtions: i) Control Volume i : m, T, T, ω α) ω ) ω ) + m.in m ) m N ) 0 A7) ii) Control Volumes i,..., I : N i) m, T, T, ω α) ω i+) ω i) + mi) m i+) m 0 A8) iii) Control Volume i I: N I) m, T, T, ω α) ω in ω I) + mi) m I+) m 0 A9) D. The ir/ter vpor energy blnce equtions: i) Control Volume i : N ) m, T, T, ω α) T ) T ) )C ) p + T) T ) )Hm,α) m + m,inm ) )h ) g,t ),α) T) +7.5, α) ω ) h ) m + ω ) h ) g, T ) g, T ), α) 0, α) A0) ii) Control Volumes i,..., I : N i) m, T, T, ω α) T i+) + Ti+) T i) )Hm,α) m + mi) m i+) T i) )h i+) g, )C i) p Ti) +7.5 T i+),α), α) ω i) h i) m + ω i+) h i+) g, g,t i), α) T i+), α) 0 A) iii) Control Volume i I: N I) m, T, T, ω α) T,in T I) + TI+) T I) )Hm,α) m + mi) m I+) )C p I) TI) +7.5 )h I+) g, T I+), α) ω I) h I),α) g,t I), α) m + ω in h g, T,in, α) 0. A)

15 Energies 06, 9, 77 5 of 7 The components of the vector α, hich ppers in Equtions A) A), comprise the model prmeters, i.e.: α α,..., α Nα ) A) here N α denotes the totl number of model prmeters. These model prmeters re described in Tble A. The folloing nottion ill be used for the derivtives of the bove equtions ith respect to the prmeters: i,j l Ni) l j) l,,, i,..., I j,..., N α. A. Derivtives of the Liquid Continuity Equtions ith Respect to the Prmeters A) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) T db re s follos: here: N i) ) N i) T i, db R D v T db, α) T db P i+) vs T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j, Mm, α) D v T db, α) Mm, α) D v T db, α) Mm,α) D v T db,α) D vt db,α) T db.5 0dvT 0.5 db D v T db, α) dv + dv T db ) dv + dv T db + dv T db A5) A6) A7) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) T dp re s follos: N i) i) N i, ) T 0 l i,..., I j. A8) dp The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) T,in re s follos: N ) ) N ), T m,in l i j, A9),in T,in here: m,in T,in T ρ T,in,in ) 5850.,ρ +,ρ T,in 7.5) +,ρ T,in 7.5) A0) nd here,ρ ,ρ.869 0,ρ N i) i) N i, ) T 0 l i,..., I j. A),in The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) P tm re s follos: N i) ) N i) + R P tm i, Mm,α) R P vs i+) T i+),α) T i+) l i,..., I j, ω i) T i) 0.6+ω i) ) ω i) P tm 0.6+ω i) )T i) Mm,α) NuRe,α) NuRe,α) m m P tm A)

16 Energies 06, 9, 77 6 of 7 here: NuRe, α) m MRe, α) NuRe, α) Mm, α) NuRe, α), 0 Re d < 00,Nu Rem, α)/m 00 Re d NuRe, α)/m Re d > 0000 A), A) m P tm V R ir T exit πd f n A5),in Note: The term on the right hnd side of Eqution A5) stems from the folloing reltion: m ρ T ) V exit πd f n P tm V R ir T exit πd f n. A6) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 5) ts re s follos: here: N i) 5) N i) ts i,5 R P i+) vs T i+),α) T i+) l i,..., I j 5, ω i) P tm 0.6+ω i) )T i) Mm,α) ts Mm, α) M HO f mt NuRe, α) νpr ) D v T db, α) A sur f ts D h I A7) A8) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 6) k sum re s follos: N i) i) N i,6 6) k 0 l i,..., I j 6. A9) sum The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 7) µ re s follos: here: N i) 7) N i) µ i,7 R Mm, α) µ P i+) vs T i+),α) T i+) l i,..., I j 7,,Nu Mm,α) Rem,α) NuRe,α) µ ω i) P tm 0.6+ω i) )T i) 0.8 Mm,α) µ Re d > 0000 Mm,α) µ A0) 0 Re d < Re d A) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 8) υ re s follos: N i) 8) N i) υ i,8 R P i+) vs T i+),α) T i+) l i,..., I j 8, ω i) P tm 0.6+ω i) )T i) Mm,α) υ A) here: Mm, α) υ Mm, α). A) υ

17 Energies 06, 9, 77 7 of 7 The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 9) k ir re s follos: N i) i) N i,9 9) k 0 l i,..., I j 9. A) ir The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 0) f ht re s follos: N i) i) N i,0 0) f 0 l i,..., I j 0. A5) ht The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) f mt re s follos: here: N i) i) N ) f mt i, R P vs i+) T i+),α) T i+) l i,..., I j, ω i) P tm 0.6+ω i) )T i) Mm,α) f mt A6) Mm, α) M HONuRe, α) νpr ) D v T db, α) ts A sur f. A7) f mt D h I The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) f re s follos: N i) i) N ) f i, 0 l i,..., I j. A8) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) 0 re s follos: N i) i) N i, ) Mm, α) 0 R here: P i+) vs T i+) P i+) vs T i+), α) 0 T i+), α) l i,..., I j, A9) 0 P vs i+) T i+), α). A0) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) re s follos: N i) i) N ) i, Mm,α) P vs i+) T i+),α) R T i+) l i,..., I j, A) here: P i+) vs T i+), α) Pi+) vs T i+), α) T i+). A) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 5) 0,cp re s follos: N i) i) N i,5 5) 0 l i,..., I j 5. A) 0,cp

18 Energies 06, 9, 77 8 of 7 The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 6),cp re s follos: N i) i) N i,6 6) 0 l i,..., I j 6. A),cp The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 7),cp re s follos: N i) i) N i,7 7) 0 l i,..., I j 7. A5),cp The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 8) 0,dv re s follos: N i) i) N 8) 0,dv i,8 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 8, here Mm,α) s defined previously in Eqution A6), nd: D v T db,α) Mm,α) D v T db,α) D vt db,α) 0,dv A6) D v T db, α) 0,dv T db.5 dv + dv T db + dv T db. A7) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 9),dv re s follos: N i) i) N 9),dv i,9 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 9, here Mm,α) s defined previously in Eqution A6), nd: D v T db,α) Mm,α) D v T db,α) D vt db,α),dv A8) D v T db, α),dv.5 0dv T db ). A9) dv + dv T db + dv T db The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 0),dv re s follos: N i) i) N 0),dv i,0 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 0, here Mm,α) s defined previously in Eqution A6), nd D v T db,α) Mm,α) D v T db,α) D vt db,α),dv A50) D v T db, α),dv.5 0dv T db ). A5) dv + dv T db + dv T db

19 Energies 06, 9, 77 9 of 7 The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ),dv re s follos: N i) i) N ),dv i, R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j, here Mm,α) s defined previously in Eqution A6), nd D v T db,α) D v T db, α),dv Mm,α) D v T db,α) D vt db,α),dv A5) 0dv T db.5 dv + dv T db + dv T db ). A5) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) 0f re s follos: N i) i) N i, ) 0 l i,..., I j. A5) 0 f The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) f re s follos: N i) i) N i, ) 0 l i,..., I j. A55) f The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) 0g re s follos: N i) i) N i, ) 0 l i,..., I j. A56) 0g The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 5) g re s follos: N i) i) N i,5 5) 0 l i,..., I j 5. A57) g The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 6) 0,Nu re s follos: N i) i) N 6) 0,Nu i,6 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 6, here Mm,α) s defined previously in Eqution A), nd NuRe,α) NuRe, α) 0,Nu Re d < Re d Re d > 0000 Mm,α) NuRe,α) NuRe,α) 0,Nu A58). A59)

20 Energies 06, 9, 77 0 of 7 The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 7),Nu re s follos: N i) i) N 7),Nu i,7 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 7, Mm,α) NuRe,α) NuRe,α),Nu A60) here Mm,α) s defined previously in Eqution A), nd: NuRe,α) NuRe, α),nu 0 Re d < 00 Rem, α) 00 Re d Re d > A6) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 8),Nu re s follos: N i) i) N 8),Nu i,8 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 8, Mm,α) NuRe,α) NuRe,α),Nu A6) here Mm,α) s defined previously in Eqution A), nd: NuRe,α) NuRe, α),nu 0 Re d < Re d Re d > 0000 A6) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 9),Nu re s follos: N i) i) N 9), Nu i,9 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 9, Mm,α) NuRe,α) NuRe,α),Nu A6) here Mm,α) s defined previously in Eqution A), nd: NuRe,α) NuRe, α),nu 0 Re d < Re d 0000 Rem, α) 0.8 Pr Re d > A65) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 0) W dkx re s follos: N i) i) N i,0 0) W 0 l i,..., I j 0. A66) dkx The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) W dky re s follos: N i) i) N i, ) W 0 l i,..., I j. A67) dky

21 Energies 06, 9, 77 of 7 The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) z dk re s follos: N i) i) N i, ) z 0 l i,..., I j. A68) dk The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) z fn re s follos: N i) i) N i, ) z 0 l i,..., I j. A69) f n The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) D fn re s follos: N i) i) N ) here Mm,α) NuRe,α) nd: D f n i, R nd NuRe,α) m P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j, Mm,α) NuRe,α) m NuRe,α) m D f n A70) ere defined previously in Equtions A) nd A), respectively, m D f n m D f n. A7) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 5) z fill re s follos: N i) i) N i,5 5) z 0 l i,..., I j 5. A7) f ill The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 6) z rin re s follos: N i) 6) N i) i,6 z 0 l i,..., I j 6. A7) rin The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 7) z bs re s follos: N i) i) N i,7 7) z 0 l i,..., I j 7. A7) bs The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 8) z de re s follos: N i) i) N i,8 8) z 0 l i,..., I j 8. A75) de The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 9) D h re s follos: N i) i) N 9) D h i,9 R P vs i+) T i+),α) T i+) l i,..., I j 9, ω i) P tm 0.6+ω i) )T i) Mm,α) D h A76)

22 Energies 06, 9, 77 of 7 here: Mm, α) D h Mm, α)/d h Re d < 00,Nu Mm,α) NuRe,α)D h 00 Re d Mm, α)/d h Re d > A77) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 0) A fill re s follos: here: N i) i) N 0) A f ill i,0 Mm, α) A f ill R P vs i+) T i+),α) T i+) l i,..., I j 0, ω i) P tm 0.6+ω i) )T i) Mm,α) A f ill 0 Re d < 00,Nu Mm,α)Rem,α) NuRe,α)A f ill 00 Re d Mm, α)/a f ill Re d > 0000 A78). A79) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) A surf re s follos: here: N i) i) N ) A sur f i, R P vs i+) T i+),α) T i+) l i,..., I j, Mm, α) A sur f Mm, α) A sur f ω i) P tm 0.6+ω i) )T i) Mm,α) A sur f A80) A8) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) Pr re s follos: N i) i) N ) Pr i, R P vs i+) T i+),α) T i+) l i,..., I j, ω i) P tm 0.6+ω i) )T i) Mm,α) Pr A8) here: Mm, α) Pr Mm, α)/ Pr) Re d Re d > A8) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) V re s follos: N i) i) N i, ) V 0 l i,..., I j. A8) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α ) V exit re s follos: N i) i) N ) V exit i, R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j, Mm,α) NuRe,α) m NuRe,α) m V exit A85)

23 Energies 06, 9, 77 of 7 here Mm,α) NuRe,α) nd nd NuRe,α) m ere defined previously in Equtions A) nd A), respectively, m V exit P tm πd f n R ir T,in A86) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 5) m,in re s follos: N ) ) N,5 5) m l i j 5, A87),in N i) i) N i,5 5) m 0 l i,..., I j 5. A88),in The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 6) T,in re s follos: N i) i) N i,6 6) T 0 l i,..., I j 6. A89),in The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 7) m re s follos: N i) i) N 7) m i,7 here Mm,α) NuRe,α) nd NuRe,α) m R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 7, Mm,α) NuRe,α) NuRe,α) m A90) ere defined previously in Equtions A) nd A), respectively. The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 8) ω in re s follos: N i) i) N i,8 8) ω 0 l i,..., I j 8. A9) in The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 9) Re d re s follos: N i) i) N 9) Re d i,9 R P vs i+) T i+),α) T i+) ω i) P tm 0.6+ω i) )T i) l i,..., I j 9, Mm,α) NuRe d,α) NuRe d,α) Re d A9) here Mm,α) s defined in Eqution A), nd: NuRe,α) NuRe d, α) Re d 0 Re d < 00,Nu 00 Re d ,Nu Re 0. d Pr / Re d > 0000 A9) The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 50) Sc re s follos: N i) i) N 50) Sc i,50 R P vs i+) T i+),α) T i+) l i,..., I j 50, ω i) P tm 0.6+ω i) )T i) Mm,α) Sc A9)

24 Energies 06, 9, 77 of 7 here: Mm, α) Sc Mm, α). A95) Sc The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 5) Sh re s follos: here: N i) i) N 5) Sh i,5 R P vs i+) T i+),α) T i+) l i,..., I j 5, Mm, α) Sh ω i) P tm 0.6+ω i) )T i) Mm,α) Sh A96) Mm, α). A97) Sh The derivtives of the liquid continuity equtions cf. Equtions A) A) ith respect to the prmeter α 5) Nu re s follos: N i) i) N 5) Nu i,5 R here Mm,α) s defined in Eqution A). NuRe,α) P vs i+) T i+),α) T i+) l i,..., I j 5, ω i) P tm 0.6+ω i) )T i) Mm,α) Nu A. Derivtives of the Liquid Energy Blnce Equtions ith Respect to the Prmeters A98) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) T db re s follos: N i) i) N i, ) T 0 l i,..., I j. A99) db The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) T dp re s follos: N i) i) N i, ) T 0 l i,..., I j. A00) dp The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) T,in re s follos: N ) ) N) T,,in m,in h ) f T,in,α) T + h ),in f l i j, T,in, α) m,in T,in h ) g,t ), α) m,in T,in A0) here m,in T,in s defined in Eqution A0), nd: N i) i) h ) f T,in, α) T,in f, A0) N i, ) T 0 l i,... I j. A0),in

25 Energies 06, 9, 77 5 of 7 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) P tm re s follos: N i) ) Ni) P tm i, T i+) l i,..., I j, T i) ) Hm,α) NuRe,α) NuRe,α) m m P tm A0) here NuRe,α) m nd m P tm ere defined in Equtions A) nd A5), respectively, nd: Hm, α) NuRe, α) Hm, α) NuRe, α). A05) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 5) ts re s follos: here: N i) N 5) i) ts i,5 T i+) Hm, α) ts T i) ) Hm, α) l i,..., I j 5, A06) ts f htnure, α)k ir A sur f D h I. A07) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 6) k sum re s follos: N i) i) N i,6 6) k 0 l i,..., I j 6. A08) sum The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 7) µ re s follos: here: N i) N 7) i) µ i,7 Hm, α) µ T i+) T i) ) Hm, α) l i,..., I j 7, A09) µ 0 Re d < 00,Nu Hm,α) Rem,α) 00 Re NuRe,α) µ d Hm,α) µ Re d > 0000 A0) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 8) υ re s follos: N i) i) N 8) υ i,8 0 l i,..., I j 8. A) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 9) k ir re s follos: N i) N 9) i) k ir i,9 T i+) T i) ) Hm, α) l i,..., I j 9, A) k ir here: Hm, α) Hm, α) f htnure, α) ts A sur f. A) k ir k ir D h I

26 Energies 06, 9, 77 6 of 7 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 0) f ht re s follos: here: N i) i) N i,0 0) f ht T i+) T i) ) Hm, α) l i,..., I j 0, A) f ht Hm, α) Hm, α) k irnure, α) ts A sur f. A5) f ht f ht D h I The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) f mt re s follos: N i) i) N i, ) f 0 l i,..., I j. A6) mt The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) f re s follos: N i) i) N ) f i, 0 l i,..., I j. A7) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) 0 re s follos: N i) i) N i, ) 0 l i,..., I j. A8) 0 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) re s follos: N i) i) N i, ) 0 l i,..., I j. A9) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 5) 0,cp re s follos: N i) i) N i,5 5) 0 l i,..., I j 5. A0) 0,cp The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 6),cp re s follos: N i) i) N i,6 6) 0 l i,..., I j 6. A),cp The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 7),cp re s follos: N i) i) N i,7 7) 0 l i,..., I j 7. A),cp

27 Energies 06, 9, 77 7 of 7 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 8) 0,dv re s follos: N i) i) N i,8 8) 0 l i,..., I j 8. A) 0,dv The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 9),dv re s follos: N i) i) N i,9 9) 0 l i,..., I j 9. A),dv The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 0),dv re s follos: N i) i) N i,0 0) 0 l i,..., I j 0. A5),dv The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ),dv re s follos: N i) i) N i, ) 0 l i,..., I j. A6),dv The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) 0f re s follos: N i) Ni) ) h i) f T i),α) 0 f h i+) f i, 0 f m i) m i+) l i,..., I j. T i+),α) 0 f m i) m i+) A7) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) f re s follos: N i) Ni) ) T i) m i) i, f m i) h i) f T i),α) f m i+) h i+) f T i+),α) f T i+) m i+) l i,..., I j. A8) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) 0g re s follos: N i) Ni) ) 0g i, m i) m i+) ) hi+) g, 0g l i,..., I j. T i+),α) m i+) m i) A9) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 5) g re s follos: N i) Ni) 5) T i+),α) g i,5 m i) m i+) ) hi+) g, g l i,..., I j 5. m i) m i+) ) T i+) A0)

28 Energies 06, 9, 77 8 of 7 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 6) 0,Nu re s follos: N i) Ni) 6) 0,Nu i,6 T i+) T i) ) Hm,α) NuRe,α) NuRe,α) 0,Nu l i,..., I j 6, A) here Hm,α) NuRe,α) s defined in Eqution A05) nd NuRe,α) s defined in Eqution A59). 0,Nu The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 7),Nu re s follos: N i) Ni) 7),Nu i,7 T i+) T i) ) Hm,α) NuRe,α) NuRe,α),Nu l i,..., I j 7, A) here Hm,α) NuRe,α) s defined in Eqution A05) nd NuRe,α) s defined in Eqution A6).,Nu The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 8),Nu re s follos: N i) Ni) 8),Nu i,8 T i+) T i) ) Hm,α) NuRe,α) NuRe,α),Nu l i,..., I j 8, A) here Hm,α) NuRe,α) s defined in Eqution A05) nd NuRe,α) s defined in Eqution A6).,Nu The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 9),Nu re s follos: here Hm,α) NuRe,α) N i) Ni) 9),Nu i,9 T i+) T i) ) Hm,α) NuRe,α) NuRe,α),Nu l i,..., I j 9, A) NuRe,α) s defined in Eqution A05) nd s defined in Eqution A65).,Nu The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 0) W dkx re s follos: N i) i) N i,0 0) W 0 l i,..., I j 0. A5) dkx The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) W dky re s follos: N i) i) N i, ) W 0 l i,..., I j. A6) dky The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) z dk re s follos: N i) i) N i, ) z 0 l i,..., I j. A7) dk The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) z fn re s follos: N i) i) N i, ) z 0 l i,..., I j. A8) f n

29 Energies 06, 9, 77 9 of 7 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) D fn re s follos: N i) Ni) ) D f n i, T i+) l i,..., I j, T i) ) Hm,α) NuRe,α) NuRe,α) m m D f n A9) here Hm,α) NuRe,α) s defined in Eqution A05), hile NuRe,α) m nd m D ere defined in Equtions f n A) nd A7), respectively. The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 5) z fill re s follos: N i) i) N i,5 5) z 0 l i,..., I j 5. A0) f ill The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 6) z rin re s follos: N i) 6) N i) i,6 z 0 l i,..., I j 6. A) rin The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 7) z bs re s follos: N i) i) N i,7 7) z 0 l i,..., I j 7. A) bs The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 8) z de re s follos: N i) i) N i,8 8) z 0 l i,..., I j 8. A) de The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 9) D h re s follos: here N i) i) N i,9 9) D h Hm, α) D h T i+) T i) ) Hm, α) l i,..., I j 9, A) D h Hm, α)/d h Re d < 00,Nu Hm,α) NuRe,α)D h 00 Re d Hm, α)/d h Re d > 0000 A5) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 0) A fill re s follos: N i) i) N i,0 0) A f ill T i+) T i) ) Hm, α) l i,..., I j 0, A6) A f ill here: Hm, α) A f ill 0 Re d < 00,Nu Hm,α)Rem,α) NuRe,α)A f ill 00 Re d Hm, α)/a f ill Re d > 0000 A7)

30 Energies 06, 9, 77 0 of 7 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) A surf re s follos: here: N i) i) N i, ) A sur f T i+) Hm, α) A sur f Hm, α) A sur f T i) ) Hm, α) l i,..., I j, A8) A sur f f htk ir NuRe, α) ts. A9) D h I The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) Pr re s follos: N i) i) N ) Pr i, T i+) T i) ) Hm, α) l i,..., I j, A50) Pr here: Hm, α) Pr 0 Re d 0000 Hm, α)/ Pr) Re d > 0000 A5) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) V re s follos: N i) i) N i, ) V 0 l i,..., I j. A5) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α ) V exit re s follos: N i) Ni) ) V exit i, T i+) l i,..., I j, T i) ) Hm,α) NuRe,α) NuRe,α) m m V exit A5) here Hm,α) NuRe,α) s defined in Eqution A05), hile NuRe,α) m nd m V nd ere defined previously exit in Equtions A) nd A86), respectively. The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 5) m,in re s follos: N ) N) 5) m,5,in h ) f T,in, α) h ) g,t ), α) T,in f g T ) + 0 f 0 g, l i j 5, A5) N i) i) N i,5 5) m 0 l i,..., I j 5. A55),in The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 6) T,in re s follos: N i) i) N i,6 6) T 0 l i,..., I j 6. A56),in

31 Energies 06, 9, 77 of 7 The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 7) m re s follos: N i) Ni) 7) m i,7 T i+) T i) ) Hm,α) NuRe,α) NuRe,α) m l i,..., I j 7, A57) ere defined previously in Equtions A) nd A05), respectively. The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 8) ω in re s follos: here Hm,α) NuRe,α) nd NuRe,α) m N i) i) N i,8 8) ω 0 l i,..., I j 8. A58) in The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 9) Re d re s follos: N i) Ni) 9) Re d i,9 T i+) T i) ) Hm,α) NuRe d,α) NuRe d,α) Re d l i,..., I j 9, A59) here Hm,α) NuRe,α) s defined in Eqution A05), nd NuRe,α) m s defined in Eqution A9). The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 50) Sc re s follos: N i) i) N 50) Sc i,50 0 l i,..., I j 50. A60) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 5) Sh re s follos: N i) i) N 5) Sh i,5 0 l i,..., I j 5. A6) The derivtives of the liquid energy blnce equtions cf. Equtions A) A6) ith respect to the prmeter α 5) Nu re s follos: N i) i) N 5) Nu i,5 T i+) T i) here Hm,α) s defined in Eqution A05). NuRe,α) ) Hm, α) l i,..., I j 5, A6) Nu A. Derivtives of the Wter Vpor Continuity Equtions ith Respect to the Prmeters The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) T db re s follos: N i) i) N i, ) T 0 l i,..., I j. A6) db The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) T dp re s follos: N i) i) N i, ) T 0 l i,..., I j, A6) dp

32 Energies 06, 9, 77 of 7 here: N I) ) N I) I, T ω in l i I j, A65) dp T dp ω in T dp T P tm e dp P tm e 0+ ). T dp T tdp A66) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) T,in re s follos: N ) ) N ), T m,in l i j, A67),in m T,in here,in T,in s defined in Eqution A0). N i) i) N i, ) T 0 l i,... I j. A68),in The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) P tm re s follos: N i) N ) N I) ) i) P tm i, mi) m i+) m N I) I, P ω in mi) tm P tm here m P tm s defined in Eqution A5) nd: m P tm l i,..., I j, A69) m i+) m m P tm l i I j, A70) 0 + T dp ω in 0.6e P tm P tm e 0+ ). T dp A7) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 5) ts re s follos: N i) i) N i,5 5) 0 l i,..., I j 5. A7) ts The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 6) k sum re s follos: N i) i) N i,6 6) k 0 l i,..., I j 6. A7) sum The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 7) µ re s follos: N i) i) N 7) µ i,7 0 l i,..., I j 7. A7)

33 Energies 06, 9, 77 of 7 The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 8) υ re s follos: N i) i) N 8) υ i,8 0 l i,..., I j 8. A75) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 9) k ir re s follos: N i) i) N i,9 9) k 0 l i,..., I j 9. A76) ir The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 0) f ht re s follos: N i) i) N i,0 0) f 0 l i,..., I j 0. A77) ht The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) f mt re s follos: N i) i) N i, ) f 0 l i,..., I j. A78) mt The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) f re s follos: N i) i) N ) f i, 0 l i,..., I j. A79) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) 0 re s follos: N i) i) N i, ) 0 l i,..., I j, A80) 0 here: N I) I) N I, ) ω in l i I j, A8) 0 0 ω in 0 + T dp 0.6P tme 0 P tm e 0+ ) T dp A8) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) re s follos: N i) i) N i, ) 0 l i,..., I j, A8) N I) I) N I, ) ω in l i I j, A8)

34 Energies 06, 9, 77 of 7 here: ω in 0.6P tm e 0+ T dp T dp P tm e 0+ ). T dp A85) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 5) 0,cp re s follos: N i) i) N i,5 5) 0 l i,..., I j 5. A86) 0,cp The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 6),cp re s follos: N i) i) N i,6 6) 0 l i,..., I j 6. A87),cp The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 7),cp re s follos: N i) i) N i,7 7) 0 l i,..., I j 7. A88),cp The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 8) 0,dv re s follos: N i) i) N i,8 8) 0 l i,..., I j 8. A89) 0,dv The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 9),dv re s follos: N i) i) N i,9 9) 0 l i,..., I j 9. A90),dv The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 0),dv re s follos: N i) i) N i,0 0) 0 l i,..., I j 0. A9),dv The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ),dv re s follos: N i) i) N i, ) 0 l i,..., I j. A9),dv The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) 0f re s follos: N i) i) N i, ) 0 l i,..., I j. A9) 0 f

35 Energies 06, 9, 77 5 of 7 The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) f re s follos: N i) i) N i, ) 0 l i,..., I j. A9) f The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) 0g re s follos: N i) i) N i, ) 0 l i,..., I j. A95) 0g The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 5) g re s follos: N i) i) N i,5 5) 0 l i,..., I j 5. A96) g The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 6) 0,Nu re s follos: N i) i) N i,6 6) 0 l i,..., I j 6. A97) 0,Nu The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 7),Nu re s follos: N i) i) N i,7 7) 0 l i,..., I j 7. A98),Nu The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 8),Nu re s follos: N i) i) N i,8 8) 0 l i,..., I j 8. A99),Nu The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 9),Nu re s follos: N i) i) N i,9 9) 0 l i,..., I j 9. A00),Nu The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 0) W dkx re s follos: N i) i) N i,0 0) W 0 l i,..., I j 0. A0) dkx The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) W dky re s follos: N i) i) N i, ) W 0 l i,..., I j. A0) dky

36 Energies 06, 9, 77 6 of 7 The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) z dk re s follos: N i) i) N i, ) z 0 l i,..., I j. A0) dk The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) z fn re s follos: N i) i) N i, ) z 0 l i,..., I j. A0) f n The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) D fn re s follos: N i) i) N i, ) D f n mi) m i+) m m D f n l i,..., I j, A05) here m D f n s defined in Eqution A7). The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 5) z fill re s follos: N i) i) N i,5 5) z 0 l i,..., I j 5. A06) f ill The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 6) z rin re s follos: N i) 6) N i) i,6 z 0 l i,..., I j 6. A07) rin The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 7) z bs re s follos: N i) i) N i,7 7) z 0 l i,..., I j 7. A08) bs The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 8) z de re s follos: N i) i) N i,8 8) z 0 l i,..., I j 8. A09) de The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 9) D h re s follos: N i) i) N i,9 9) D 0 l i,..., I j 9. A0) h

37 Energies 06, 9, 77 7 of 7 The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 0) A fill re s follos: N i) i) N i,0 0) A 0 l i,..., I j 0. A) f ill The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) A surf re s follos: N i) i) N i, ) A 0 l i,..., I j. A) sur f The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) Pr re s follos: N i) i) N ) Pr i, 0 l i,..., I j. A) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) V re s follos: N i) i) N i, ) V 0 l i,..., I j. A) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α ) V exit re s follos: N i) i) N i, ) V exit mi) m i+) m m V exit l i,..., I j, A5) here m V exit s defined in Eqution A86). The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 5) m,in re s follos: N ) ) N,5 5) m l i j 5, A6),in m N i) i) N i,5 5) m 0 l i,..., I j 5. A7),in The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 6) T,in re s follos: N i) i) N i,6 6) T 0 l i,..., I j 6. A8),in The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 7) m re s follos: N i) i) N i,7 7) m mi) m i+) l i,..., I j 7. A9) m

38 Energies 06, 9, 77 8 of 7 The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 8) ω in re s follos: N i) i) N i,8 8) ω 0 l i,..., I j 8, A0) in N I) I) N I,8 8) ω l i I j 8. A) in The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 9) Re d re s follos: N i) i) N i,9 9) Re 0 l i,..., I j 9. A) d The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 50) Sc re s follos: N i) i) N 50) Sc i,50 0 l i,..., I j 50. A) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 5) Sh re s follos: N i) i) N 5) Sh i,5 0 l i,..., I j 5. A) The derivtives of the ter vpor continuity equtions cf. Equtions A7) A9) ith respect to the prmeter α 5) Nu re s follos: N i) i) N 5) Nu i,5 0 l i,..., I j 5. A5) A. Derivtives of the Air nd Wter Vpor Energy Blnce Equtions ith Respect to the Prmeters The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) T db re s follos: N i) i) N i, ) T 0 l i,..., I j, A6) db N I) ) ) T I, db C I) T I) +7.5 h I+) g, p, α + ω in T ),in C I) T I) +7.5 p, α + ω in g l i I j. N I) T,in,α) A7) Note: The vlue of the inlet ir temperture is set equl to dry-bulb temperture, lthough these quntities re treted s to different prmeters in the model. The dry-bulb temperture is used in mss diffusivity clcultions. The reltion beteen the to prmeters, i.e., T,in T db, needs to be ccounted for hen computing the respective derivtives: the derivtive of Eqution A) ith respect to the dry-bulb temperture must be the sme s the derivtive of Eqution A) ith respect to the inlet ir temperture.

39 Energies 06, 9, 77 9 of 7 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) T dp re s follos: N i) i) N i, ) T 0 l i,..., I j, A8) dp N I) ) N I) I, T ω in ) g T dp T,in + 0g l i I j, A9) dp here ω in T dp s defined in Eqution A66). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) T,in re s follos: N ) ) N ) T,,in h) g,t ) m l i j,,α) m,in T gt ) + 0g,in m m,in T,in A0) here ω,in T,in s defined in Eqution A0), nd N i) i) N i, ) T 0 l i,..., I j. A),in The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) P tm re s follos: N i) ) N I) ) N i) P tm i, T i+) T i) ) m )h i+) g, mi) m i+) m T i+),α) N I) P tm I, T i+) T i) ) m )h i+) g, mi) m i+) m T i+),α) Hm,α) NuRe,α) NuRe,α) m m P tm Hm,α) m m P tm l i,..., I j, Hm,α) NuRe,α) NuRe,α) m m P tm Hm,α) m m P tm m P tm A) m P tm + ω in P tm g T,in + 0g ) l i I j, A) here Hm,α) NuRe,α) nd NuRe,α) m ere defined in Equtions A) nd A05), respectively, hile m P tm nd ω in P tm ere defined in Equtions A5) nd A7), respectively. The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 5) ts re s follos: N i) N 5) here Hm,α) ts i) ts i,5 Ti+) T i) ) Hm, α) m ts l i,..., I j 5, A) s defined in Eqution A07). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 6) k sum re s follos: N i) i) N i,6 6) k 0 l i,..., I j 6. A5) sum The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 7) µ re s follos: N i) N 7) i) µ i,7 Ti+) T i) ) Hm, α) m µ l i,..., I j 7, A6)

40 Energies 06, 9, 77 0 of 7 s defined in Eqution A0). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 8) υ re s follos: here Hm,α) N i) i) N 8) υ i,8 0 l i,..., I j 8. A7) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 9) k ir re s follos: N i) N 9) here Hm,α) k ir i) k ir i,9 Ti+) T i) ) Hm, α) m k ir l i,..., I j 9, A8) s defined in Eqution A). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 0) f ht re s follos: N i) i) N i,0 0) f ht here Hm,α) f ht Ti+) T i) ) Hm, α) m f ht l i,..., I j 0, A9) s defined in Eqution A5). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) f mt re s follos: N i) i) N i, ) f 0 l i,..., I j. A0) mt The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) f re s follos: N i) i) N ) f i, 0 l i,..., I j. A) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) 0 re s follos: N I) N i) i) N i, ) 0 l i,..., I j, A) 0 I) N I, ) ω in ) g T 0,in + 0g l i I j, A) 0 here ω in s defined in Eqution A8). 0 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) re s follos: here ω in N I) N i) i) N i, ) 0 l i,..., I j, A) I) N I, ) ω in ) g T,in + 0g l i I j, A5) s defined in Eqution A85).

41 Energies 06, 9, 77 of 7 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 5) 0,cp re s follos: N i) i) N 5) 0,cp i,5 T i+) l i,..., I j 5. ) T i) C i) T i) p +7.5,α ) 0,cp T i+) T i) A6) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 6),cp re s follos: N i) 6) N i) 0.5 T i+),cp i,6 T i+) T i) ) T i) C i) T i) p +7.5,α ),cp ) T i) + 7.5) l i,..., I j 6. A7) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 7),cp re s follos: N i) 7) N i) 0.5 T i+) ) C i) T i) p +7.5,α ),cp i,7 T i+) T i),cp A8) T i) ) T i) l i,..., I j 7. The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 8) 0,dv re s follos: N i) i) N i,8 8) 0 l i,..., I j 8. A9) 0,dv The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 9),dv re s follos: N i) i) N i,9 9) 0 l i,..., I j 9. A50),dv The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 0),dv re s follos: N i) i) N i,0 0) 0 l i,..., I j 0. A5),dv The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ),dv re s follos: N i) i) N i, ) 0 l i,..., I j. A5),dv The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) 0f re s follos: N i) i) N i, ) 0 l i,..., I j. A5) 0 f

42 Energies 06, 9, 77 of 7 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) f re s follos: N i) i) N i, ) 0 l i,..., I j. A5) f The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) 0g re s follos: N i) i) N i, ) 0g ω i+) ω i) + mi) m i+) m l i,..., I j. A55) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 5) g re s follos: N i) i) N 5) g i,5 ω i+) T i+) l i,..., I j 5. ω i) T i) m i) m i+) ) T i+) + m A56) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 6) 0,Nu re s follos: N i) i) N 6) 0,Nu i,6 Ti+) m T i) ) l i,..., I j 6, Hm,α) NuRe,α) NuRe,α) 0,Nu A57) here Hm,α) NuRe,α) nd NuRe,α) 0,Nu ere defined previously in Equtions A59) nd A05), respectively. The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 7),Nu re s follos: N i) i) N 7),Nu i,7 Ti+) m T i) ) l i,..., I j 7, Hm,α) NuRe,α) NuRe,α),Nu A58) ere defined previously in Equtions A6) nd A05), respectively. The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 8),Nu re s follos: here Hm,α) NuRe,α) nd NuRe,α),Nu N i) i) N 8),Nu i,8 Ti+) m T i) ) l i,..., I j 8, Hm,α) NuRe,α) NuRe,α),Nu A59) ere defined previously in Equtions A6) nd A05), respectively. The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 9),Nu re s follos: here Hm,α) NuRe,α) nd NuRe,α),Nu N i) i) N 9),Nu i,9 Ti+) m T i) ) l i,..., I j 9, Hm,α) NuRe,α) NuRe,α),Nu A60) here Hm,α) NuRe,α) nd NuRe,α) ere defined previously in Equtions A65) nd A05), respectively.,nu

43 Energies 06, 9, 77 of 7 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 0) W dkx re s follos: N i) i) N i,0 0) W 0 l i,..., I j 0. A6) dkx The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) W dky re s follos: N i) i) N i, ) W 0 l i,..., I j. A6) dky The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) z dk re s follos: N i) i) N i, ) z 0 l i,..., I j. A6) dk The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) z fn re s follos: N i) i) N i, ) z 0 l i,..., I j. A6) f n The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) D fn re s follos: N i) i) N ) D i, f n T i+) )h i+) g, mi) m i+) m T i) ) m T i+),) Hm,α) NuRe,α) NuRe,α) m m D Hm,α) f n m m D f n l i,..., I j, m D f n A65) here Hm,α) NuRe,α) nd NuRe,α) m NuRe,α) m ere defined previously in Equtions A05) nd A), respectively, hile m D f n s defined in Eqution A7). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 5) z fill re s follos: N i) i) N i,5 5) z 0 l i,..., I j 5. A66) f ill The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 6) z rin re s follos: N i) 6) N i) i,6 z 0 l i,..., I j 6. A67) rin The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 7) z bs re s follos: N i) i) N i,7 7) z 0 l i,..., I j 7. A68) bs

44 Energies 06, 9, 77 of 7 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 8) z de re s follos: N i) i) N i,8 8) z 0 l i,..., I j 8. A69) de The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 9) D h re s follos: N i) i) N i,9 9) D h here Hm,α) D h Ti+) T i) ) Hm, α) m D h l i,..., I j 9, A70) s defined in Eqution A5). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 0) A fill re s follos: N i) i) N i,0 0) A f ill Ti+) T i) ) Hm, α) m A f ill l i,..., I j 0, A7) here Hm,α) A s defined in Eqution A7). f ill The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) A surf re s follos: N i) i) N i, ) A sur f Ti+) T i) ) Hm, α) m A sur f l i,..., I j, A7) here Hm,α) A s defined in Eqution A9). sur f The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) Pr re s follos: N i) i) N ) Pr i, Ti+) T i) ) Hm, α) m Pr l i,..., I j, A7) s defined in Eqution A5). The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) V re s follos: here Hm,α) Pr N i) i) N i, ) V 0 l i,..., I j. A7) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α ) V exit re s follos: N i) i) N ) V i, exit T i+) )h i+) g, mi) m i+) m T i) ) m T i+),α) Hm,α) NuRe,α) NuRe,α) m m V Hm,α) exit m m V exit l i,..., I j, m V exit A75) here Hm,α) NuRe,α) nd NuRe,α) m ere defined previously in Equtions A05) nd A), respectively, hile m V exit s defined in Eqution A86).

45 Energies 06, 9, 77 5 of 7 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 5) m,in re s follos: N ) ) N,5 5) m,in h) g,t ) m, α) ) gt + 0g l i j 5, A76) m N i) i) N i,5 5) m 0 l i,..., I j 5. A77),in The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 6) T,in re s follos: N i) i) N i,6 6) T 0 l i,..., I j 6, A78),in N I) I) N 6) C I) p ) T I,6,in C I) T I) +7.5 h I+) g, p, α + ω in T ),in T I) +7.5, α + ω in g l i I j 6. T,in,α) A79) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 7) m re s follos: + T i+) N i) i) N 7) T i) ) m m i,7 Hm,α) NuRe,α) NuRe,α) m m i) ) h i+) g, T i+),α) m m i+) l i,..., I j 7, Hm,α) m A80) here Hm,α) NuRe,α) nd NuRe,α) m ere defined previously in Equtions A) nd A05), respectively. The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 8) ω in re s follos: N i) i) N i,8 8) ω 0 l i,..., I j 8, A8) in N I) I) N I,8 8) ω h I+) g, T,in, α) l i I j 8. A8) in The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 9) Re d re s follos: N i) i) N 9) Re d i,9 Ti+) m T i) ) l i,..., I j 9, Hm,α) NuRe d,α) NuRe d,α) Re d A8) here Hm,α) NuRe,α) s defined in Eqution A05) nd NuRe,α) Re s defined in Eqution A9). d The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 50) Sc re s follos: N i) i) N 50) Sc i,50 0 l i,..., I j 50. A8)

46 Energies 06, 9, 77 6 of 7 The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 5) Sh re s follos: N i) i) N 5) Sh i,5 0 l i,..., I j 5. A85) The derivtives of the ir/ter vpor energy blnce equtions cf. Equtions A0) A) ith respect to the prmeter α 5) Nu re s follos: N i) i) N 5) Nu i,5 Ti+) T i) ) Hm, α) m Nu here Hm,α) s defined in Eqution A05). NuRe,α) l i,..., I j 5, A86) References. Ccuci, D.G. Fng, R. Predictive modelling of prdigm mechnicl cooling toer. I: Adjoint sensitivity model. Energies 06, 9, 78. CrossRef. Ccuci, D.G. Predictive modeling of coupled multi-physics systems: I. Theory. Ann. Nucl. Energy 0, 70, CrossRef. Ccuci, D.G. Bde, M.C. Predictive modeling of coupled multi-physics systems: II. Illustrtive pplictions to rector physics. Ann. Nucl. Energy 0, 70, CrossRef. Ltten, C. Ccuci, D.G. Predictive modeling of coupled systems: Uncertinty reduction using multiple rector physics benchmrks. Nucl. Sci. Eng. 0, 78, CrossRef 5. Ccuci, D.G. Arsln, E. Reducing uncertinties vi predictive modeling: FLICA clibrtion using BFBT benchmrks. Nucl. Sci. Eng. 0, 76, 9 9. CrossRef 6. Arsln, E. Ccuci, D.G. Predictive modeling of liquid-sodium therml hydrulics experiments nd computtions. Ann. Nucl. Energy 0, 6, CrossRef 7. Alemn, S.E. Grrett, A.J. Opertionl Cooling Toer Model CTTool V.0) Svnnh River Ntionl Lbortory: Svnnh River, SC, USA, Jnury Grrett, A.J. Prker, M.J. Vill-Alemn, E. 00 Svnnh River Site Cooling Toer Collection SRNL-DOD Atmospheric Technologies Group Svnnh River Ntionl Lbortory: Aiken, SC, USA, Ccuci, D.G. Ionescu-Bujor, M. Model clibrtion nd best-estimte prediction through experimentl dt ssimiltion: I. mthemticl frmeork. Nucl. Sci. Eng. 00, 65, 8. CrossRef 0. Ccuci, D.G. Ionescu-Bujor, M. On the evlution of discrepnt scientific dt ith unrecognized errors. Nucl. Sci. Eng. 00, 65, 7. CrossRef. Frgó, I. Hvsi, A. Zltev, Z. Advnced Numericl Methods for Complex Environmentl Models: Needs nd Avilbility Benthm Science Publishers: Ok Prk, IL, USA, 0.. Ccuci, D.G. Nvon, M.I. Ionescu-Bujor, M. Computtionl Methods for Dt Evlution nd Assimiltion Chpmn & Hll/CRC: Boc Rton, FL, USA, 0.. Ccuci, D.G. Second-order djoint sensitivity nlysis methodology nd-asam) for computing exctly nd efficiently first- nd second-order sensitivities in lrge-scle liner systems: I. Computtionl methodology. J. Comput. Phys. 05, 8, CrossRef. Ccuci, D.G. Second-order djoint sensitivity nlysis methodology nd-asam) for lrge-scle nonliner systems: I. Theory. Nucl. Sci. Eng. 06, in press. 5. Ccuci, D.G. Second-order djoint sensitivity nlysis methodology nd-asam) for computing exctly nd efficiently first- nd second-order sensitivities in lrge-scle liner systems: II. Illustrtive ppliction to prdigm prticle diffusion problem. J. Comput. Phys. 05, 8, CrossRef 6. Ccuci, D.G. Second-Order Adjoint sensitivity nd uncertinty nlysis of benchmrk het trnsport problem: I. Anlyticl results. Nucl. Sci. Eng. 06, 8,. CrossRef

47 Energies 06, 9, 77 7 of 7 7. Ccuci, D.G. Ilic, M. Bde, M.C. Fng, R. Second-order djoint sensitivity nd uncertinty nlysis of benchmrk het trnsport problem II: Computtionl results using GM rector therml-hydrulics prmeters. Nucl. Sci. Eng. 06, 8, 8. CrossRef 8. Ccuci, D.G. Second-order djoint sensitivity nlysis methodology nd-asam) for lrge-scle nonliner systems: II. Illustrtive ppliction to prdigm nonliner het conduction benchmrk. Nucl. Sci. Eng. 06, in press. CrossRef 06 by the uthors licensee MDPI, Bsel, Sitzerlnd. This rticle is n open ccess rticle distributed under the terms nd conditions of the Cretive Commons Attribution CC-BY) license