AN ANALYTICAL METHOD FOR DEFINING THE PUMP'S POWER OPTIMUM OF A WATER-TO-WATER HEAT PUMP HEATING SYSTEM USING COP

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 999 Introdution AN ANALYTICAL METHOD FOR DEFINING THE PUMP'S POWER OPTIMUM OF A WATER-TO-WATER HEAT PUMP HEATING SYSTEM USING COP by a,b Jozsf NYERS a Dotoral Shool of Applid Informatis and Applid Mathmatis, Obuda Univrsity Budapst, Budapst, Hungary b Dotoral Shool of Mhanial Enginring, Sznt Istvan Univrsity, Godollo, Hungary Original sintifi papr * https://doi.org/0.2298/tsci60324n This papr analyzs th nrgy ffiiny of th hat pump and th omplt hat pump hating systm. Essntially, th maximum of th offiint of prforman of th hat pump and th hat pump hating systm ar invstigatd and dtrmind by applying a nw analytial optimization produr. Th analyzd physial systm onsists of th watr-to-watr hat pump, irulation and wll pump. In th analytial optimization produr th "first drivativ qual to zro" mathmatial mthod is applid. Th objtiv funtion is th offiint of prforman of th hat pump, and th hat pump hating systm. By using th analytial optimization produr and th objtiv funtion, as th rsult, th loal and th total nrgy optimum onditions with rspt to th mass flow rat of hot and old watr i.. th powr of irulation or wll pump ar dfind. Ky words: objtiv funtion, hat pump, hat pump hating systm, offiint of prforman, loal optimum, global optimum From th aspt of th offiint of prforman (COP) of th hating systm with spifid hating apaity of th watr-to-watr hat pump, it is important to slt th adquat apaity of th pumps. In fat, thr is an optimal apaity of th wll and irulation pump that provid th maximum COP of th hating systm. In a similar mannr, Granryd [] physially dsribd and dfind th task of nrgy optimization of th hat pumping systms, as wll as in this study, howvr, th objtiv funtion, optimization variabls and pross, h prformd on ompltly diffrnt way. Th COP of th systm has bn dfind by th idal Carnot pross, i.. by th air tmpratur of th hat sour and sink, whil in our study th COP is dfind through th hat flux in th ondnsr and th rquird powr of th omprssor and th pumps. Also, subjt of optimization was air-flow vloity in vaporator and ondnsr, whil in prsnt papr, mass flow rat of watr in th hot and old watr loop and th powr of wll and irulation pump hav bn optimizd. In both studis th mathmatial optimization produr first drivativ qual to zro is applid. In papr [] th analytial nrgy optimization, numrial simulation, and analysis ar prformd in dtail for th ooling systm, howvr, optimization of th hating systm * Author s -mail: jnyrs@gmail.om

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... 2000 THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 with a hat pump is only mntiond. In this study th main targt was just an analytial nrgy optimization of th hating systm with a hat pump. Th objtiv funtions, optimization mthods, as wll as th strutur of th hat pump systm in this papr is diffrnt rlating to othr paprs, it an b sn basd on th topis of th following paprs. Sayyaadi t al. [2] dalt with thrmodynami and thrmo onomi optimization of a vrtial ground sour hat pump systm. Esnand and Turgut, [3] fousd on th optimization of vrtial ground oupld hat pump systm. In svral paprs th onomial optimization of th hat pump systms is xamind [4-6]. Som authors optimizd th onstrutiv dtails of th systm [7-9]. In addition to th mntiond ass, optimization also b dvlopd for diffrnt struturs and for various appliations of hat pumps [0-2]. Th authors, and artils that dal with th topi of optimization, hat pumps systms and rnwabl nrgy sours ar Kalmar [3], Kalmar t al. [4], Kajtar t al. [5], and Kassai t al. [6]. In th nlosd study analytially drivd th basi rlations of th ral, nrgy optimum onditions of th hating systm with watr-to-watr hat pump. Th main goals wr: dfin xatly th objtiv funtion and basd on that, apply an analytial optimization produr to dtrmin th nrgy optimum onditions of th onsidrd systm and fill in th gap in th world sintifi litratur rlatd to th analytial nrgy optimization, (it an b sn from th rviw of litratur, th basi analytial nrgy optimization onditions wr not prformd and publishd). Th onsidrd physial systm, i.. th hating systm with watr-to-watr hat pump, onsists of th following omponnts: hat pump, irulation, and wll pump. Th hat pump is dividd into th following omponnts: vaporator, ondnsr, and omprssor. It should b mntiond, that th produr is appliabl for both air-to-air, air-to-watr hating-ooling hat pump systm. Enrgy optimization is prformd for th stablishd rgim of th hating systm, baus in th tim domain obsrvd 98% of ass ar taking pla in this mod. Nyrs t al. [7], Nyrs, J. and Nyrs, A. [8, 9], and Nyrs and Pk [20] ratd a stady-stat mathmatial modl with th aim to implmnting th analytial nrgy optimization of th hat pump and th hat pump hating systm. Th modl ontains th ovrall COP for th hat pump and hating systm with a hat pump. Th ovrall COP of th hating systm is th ratio btwn th obtaind hat flux for hating and ltri powr usd to driv th ltri motor of th omprssor, irulation and wll pumps. Part of analytial optimization is th modifiation of th basi stady-stat modl for th vaporator and th ondnsr aording to th rquirmnts of th optimization pross, [7]. In th analytial mathmatial produr of nrgy optimization, th objtiv funtion is th ovrall COP of th hating systm. Th analytial optimization produr is basd on a mthod of first drivativ qual to zro. Th ovrall COP of th hating systm with rspt to th powr of irulation and wll pumps is partially diffrntiatd. Individually, th obtaind quations rprsnt onditions of th loal nrgy optimum with rspt to th pumps powr. Togthr, ths two quations, th onditions of loal nrgy optimum, rprsnt th ondition of th global nrgy optimum. Th ommon additional ondition at loal and global optimization is th omprssor powr onstant. At loal optimization, bsids th omprssor powr, th powr of irulation pump or th powr of wll pump altrnatly is onstant. Howvr, whn loally optimizd with rspt to th powr of irulation pump, thn, th powr of th wll pump is th onstant and vi vrsa. Th global and loal onditions of nrgy optimum with rspt to th powr of th irulation and wll pump provid only trivial solutions and th optimal powr of pumps quals

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 200 to zro. Th problm is, that thr is no mathmatial rlation btwn th hat fluxs in th ondnsr and th vaporator as wll as among th powr of th irulation, th wll pump and th omprssor powr. Th triviality problm is solvd by introduing nw variabls: th mass flow rat of hot and old watr. Namly, th hat flux in th ondnsr and th powr of th irulation pump dpnds on th mass flow rat of hot watr, whil th hat flux in th vaporator and th powr of wll pump dpnds on th mass flow rat of old watr. Applying th mass flow rats of hot and old watr, w ratd nw loal and global nrgy optimum onditions. Equations of th nrgy optimum onditions ar th basis of rlations for analyti optimization of th hating systm. This part of optimization pross will b shown in th following artil. Th nw approah and th optimization produr rat a nw opportunity to dtrmin th optimal powr of irulation and wll pump. Optimal powrs provid th maximum of th ovrall COP of hat pump hating systm. Th novlty in th papr ar: th ondition quations of th loal and global nrgy optimum of th hating systm with rspt to th powr of irulation and wll pump, th ondition quations of th loal and global nrgy optimum of th hating systm with rspt to th mass flow rat of hot and old watr, and th omplt proposd analytial nrgy optimization produr. Th mthod and produr of nrgy optimization hav gnral importan, baus it is appliabl not only for watr-watr, but also for air-to-air, air-watr hat pump hating systms. This papr is an introdution to a omplt study of th analytial optimization produr of th nrgy optimum of hat pump hating systms. Th final thm will b th total nrgy optimization of th hating systm aording to thr nrgy omponnts: powr of th omprssor, irulation, and wll pump. k p Th physial modl of th watr-to-watr hat pump hating systm Strutur and funtioning From th aspt of th ovrall COP of th watr-to-watr hat pump hating systm, th systm s strutur onsists of thr loops, fig.. Cold watr loop. In hat pumps vaporator th ground watr transports th hat from th layrs of ground. Th wll pump provids nrgy for th watr irulation in th vaporator and th loop using ltrial powr. Hot watr loop. From th hat pumps ondnsr th hot watr transports th hat to hating bodis. Th irulation pump nsurs th powr for th watr irulation. Th rfrigration loop. Th rfrigrant rivs hat from th old wll watr by vaporation. Th omprssor suks th rfrigrant vapor and by applying ltrial 4 m ww t vo t fo m w t wo q k 2 m f m f q t vi t fi t f t wi P p P wp Figur. Funtional shm of th hat pump hating systm with omponnts and symbols inlt at vaporator, 2 inlt at omprssor, 3 inlt at ondnsr, 4 inlt at thrmoxpansion valv. ooling stion in ondnsr, 2 ondnsing stion in ondnsr 3 2 P

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... 2002 THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 powr omprsss it. During omprssion th prssur and tmpratur of suprhatd vapor ris. Th ondition of th hating systm opration: all thr irulation loops must b in opration. Th mathmatial modl Th hating systm, i.. th systm for th transport of hat is nrgy optimizd, so it is important to dfin and analyz th nrgy ffiiny of th omponnts sparatly and th nrgy ffiiny of th whol systm. It is suitabl to do it through th COP sparatly, for th hat pump and for th ntir hat pump hating systm. Th COP of th hat pump Th COP of th individual hat pump in hating rgim is th ratio btwn hat pump hating apaity and th omprssor powr: qk ηhp () P Th hat pump's hating apaity is qual to th hat, whih th suprhatd vapor transfrs in th ondnsr to th hot watr. Hat transfrrd in th ondnsr is th sum of th hat pump ooling apaity and th omprssor powr: Aftr substitution: qk q P (2) q P q hp (3) P P It an b sn that th hat pump COP dpnds only on th hat pump ooling apaity and omprssor powr. Th powr onsumption of th irulation and wll pump or fans has not bn takn into aount. Th COP of th hat pump hating systm For th opration of th whol hat pump hating systm, it is nssary to nabl th simultanous irulation of th old and hot watr. Whn th systm COP is dfind, bsids th omprssor powr, th irulation and wll pump powr should b takn into aount as wll. Th ovrall COP of th hating systm is th total hating apaity of th systm dividd by th sum of all onsumrs powr dmand: qt (4) 3 P Th hat transfrrd in th ondnsr is th sum of th ooling apaity and th omprssor powr: q q P (5) k Th total hating apaity of th systm is hat transfrrd in th ondnsr plus th powr of th irulation pump: t k p p i q q P q P P (6)

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 2003 Th onsumd powr ontains th powr of th omprssor, irulation pump, and wll pump: 3 P i P P P wp p (7) Aftr substitution, th xprssion of th ovrall COP of th hat pump hating systm is obtaind in two forms. from th point of ondnsr: q k Pp (8) P P P from th point of vaporator: Th analytial optimization produr wp p q P Pp P P P wp p Th goal of th nrgy optimization is that th ovrall COP of th hat pump hating systm ahivs th maximum valu. For a rtain hating systm with dfind hating powr of th watr-to-watr hat pump, th maximum valu of th ovrall COP is ahivd by optimizing th powr of irulation and wll pump. Applying th mathmatial modl and th analytial or numrial optimization produr, th optimum valu of th mass flow rat of hot and old watr an b dtrmind. Thn, on th basis of th optimum valu of th mass flow rats, th optimum powr of irulation and wll pump ar dfind. Th obtaind optimum valus ar th loal optimums. From th aspt of mathmatis, thr is only on global nrgy optimum, i.. global maximum ovrall COP of th hating systm. Th two loal optimum onditions togthr prsnt th ondition of th global nrgy optimum. On of th loal optimum is, whn optimization produr dtrmins th optimum mass flow rat of hot watr. Furthrmor, on th basis of th mass flow rat, th optimum powr of irulation pump is dtrmind. Th othr loal optimum is whn th optimization produr dtrmins th optimum mass flow rat of wll watr. Again, on th basis of th mass flow rat, th optimum powr of wll pumps is dtrmind. Th applid analytial mthod of optimization is th first diffrntial qual to zro. Dirt optimization produr In dirt produr, th optimization is don dirtly through th powr of irulation and wll pump. It is th first stp in th analytial optimization. Th goal of optimization is th objtiv funtion, i.. th ovrall COP of th hating systm rahs th maximum valu. Objtiv funtion In th nrgy optimization produr th objtiv funtion is th ovrall COP of hat pump hating systm: q P Pp max P P P (0) wp p q P k p max P P wp P () p (9)

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... 2004 THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 In th applid analytial optimization produr, th ondition of xtrm, th first diffrntial with rspt to th indpndnt variabl quals to zro. Extrm is maximum if th sond diffrntial is lss than zro. In our as, th indpndnt variabls ar th powr of th irulation and wll pump. Th mathmatial ondition of th loal optimum, i.. xtrm, th first diffrntial of th ovrall hating systm COP with rspt to th powr of irulation pump and wll pump quals to zro: 0 or 0 (2) Pp Pwp Th mathmatial ondition of th maximum is that th sond drivativ is lss than zro. Th sond drivativ of th COP, with rspt to th powr of irulation pump, wll pumps or ombind is: 2 2 2 η η η 0 or 0 or 0 (3) 2 2 Pp Pwp PpPwp Rmark: In th givn as, in th optimization produr, it is not nssary to dtrmin th sond drivativ, baus, on th basis of th simulations and masurmnts, w know that th xtrm is always maximum. Th first drivativ with rspt to th powr of th irulation pump, aftr substituting th xprssion of COP, is: qk P p 0 (4) Pp P p P P wp P p Th first drivativ with rspt to th powr of wll pump is: q P P p 0 (5) Pwp P wp P P wp P p Rmark: From th aspt of th irulation pump, th partial drivativ of th first form is mor onvnint. From th aspt of th wll pump, th sond form of th COP drivativ is mor suitabl. Loal nrgy optimum onditions with rspt to th pump powr Th first drivativ with rspt to th irulation pump powr is th first loal nrgy optimum ondition, i.. th first loal maximum. Aftr partial drivation: q P k p P P wp P p P Pwp Ppqk P p 0 Pp P p Pp Pp P p (6) Th first drivativ, with rspt to th wll pump powr, is th sond loal nrgy optimum ondition, i.. th sond loal maximum. Aftr partial drivation: q P P P P P 0 P Pwp Pp q P Pp p wp p Pwp Pwp P wp Pwp Pwp P wp (7)

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 2005 Substitutions Th irulation pump and th wll pump ar th autonomous, indpndnt omponnts of th hating systm and thr is no dirt onntion btwn thm. In traditional ontrol systms, in on-off ontrol stratgy, th pumps ar startd up at full powr. In modrn pumps thr is frqunt powr, spd ontrol. If th ontrol algorithm is adaptiv, thn thr is th possibility of harmonizing th pump powr, so as to ahiv optimum prforman. Sin, btwn th irulation and wll pump thr is no dirt intration, thus, mathmatially, th mutual partial drivativs ar qual to zro. Th own drivativ is qual to. Pwp Pp 0 and 0 (8) P P p wp P p P p Pwp and P wp (9) Aftr substituting th partial drivativs (8) and (9), th loal nrgy optimum onditions obtain a simplr form. Th loal nrgy optimum ondition with rspt to th irulation pump powr is: q P 0 P Pwp Pp qk Pp k P p P p P onstant,p onstant, and P optimum wp p (20) Loal nrgy optimum ondition with rspt to th wll pump powr is: q P P 0 P+ P wp+ PpP q + P+ Pp Pwp P wp P wp P onstant,p onstant, and P optimum p wp (2) Global nrgy optimum ondition with rspt to th powr of pumps In prinipl: th global nrgy optimum ondition of th hating systm is obtaind if th qs. of th loal nrgy optimum ondition (20) and (2) ar solvd aording to th powr of irulation and wll pump. But, it is not possibl to ahiv th dirt analytial or numrial solution of qs. (20) and (2), baus thr is no mathmatial rlation btwn hat flux and pumps, and btwn th powrs thmslvs. Consquntly, it is not possibl to dtrmin th partial drivativs. Howvr, introduing th mass flow rats as nw indpndnt variabls, w an obtain th solution of quations of loal nrgy optimum ondition. Indirt optimization produr Introdution Enrgy optimum onditions obtaind in dirt produr hav to b furthr dvlopd. Instad of th powr of irulation and wll pump, it is nssary to introdu nw indpndnt variabls, i.. th mass flow rat of th hot and old watr. Th rplamnt is justifiabl baus:

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... 2006 THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 Th hat flux in th ondnsr and th powr of irulating pump dpnd on th mass flow rat of hot watr. q = q m and P = P m (22) k k ww p p ww Th hat flow in th vaporator and th powr of wll pump dpnd on th mass flow rat of old watr. q = q m and P = P m (23) w wp wp w Th powr of th omprssor dpnds on th mass flow rat of rfrigrant and th rfrigrant mass flow rat dpnds on th mass flow rat of old and hot watr. f ww w Partial diffrntials with rspt to watr mass flow rats P= P[ m m, m ] (24) Th transition to th nw indpndnt variabls rquirs som mathmatial oprations. Instad of th powr of pumps, a nw variabl of th mass flow rats of watr appars. Mathmatially, th transformations ar: th hat flux in th ondnsr, instad of th powr of irulation pump, is diffrntiatd with rspt to th mass flow rat of hot watr: qk qk m ww qk (25) P P p mww Pp m ww p m th hat flux in th vaporator, instad of th powr of wll pump, is diffrntiatd with rspt to th mass flow rat of old watr: q q m w q (26) P P wp mw Pwp m w wp m th powr of omprssor, instad of th powr of irulation pump, is diffrntiatd with rspt to th mass flow rat of hot watr: P P m f m ww P m f (27) P P p m f mww Pp m f m ww p m th powr of omprssor, instad of th powr of wll pump, is diffrntiatd with rspt to th mass flow rat of old watr: P P m f m w P m f (28) P P wp m f mw Pwp m f m w wp m Loal nrgy optimum onditions of th hating systm with rspt to watr mass flow rats Aftr substituting th drivativs of th nw indpndnt variabls th loal nrgy optimum ondition with rspt to th mass flow rat of hot watr is: ww w ww w

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 2007 k m q P f P+P wp+ppqk Pp 0 m P ww p m P f m ww p m m ww ww P onstant,p onstant, and m optimum wp ww Th loal nrgy optimum ondition with rspt to th mass flow rat of old watr is: (29) m q P P f P +P wp+ppq +P +Pp 0 m P w wp P P wp m f m w wp m m w w P onstant,p onstant, and m optimum p w (30) Global nrgy optimum ondition of th hating systm with rspt to watr mass flow rats Th global nrgy optimum ondition of th hating systm obtaind by dirt or indirt produr is ssntially th sam. Th diffrn is that, instad of powr of pumps, th mass flow rat of th hot and old watr appar as indpndnt variabls. Mathmatially, th global nrgy optimum is obtaind, if th two quations of loal nrgy optimum ondition (29) and (30) ar solvd aording to mass flow rats. Th solution givs th optimum of mass flow rat of hot and old watr at th sam tim. Th two loal nrgy optimums rprsnt th global optimum and provid th maximum valu of COP of th hating systm. Dtrmining th optimal valu of th mass flow rat of hot and old watr an dtrmin th optimal valu of th powr of irulation and wll pump, whil th powr of th omprssor is assumd onstant. P = onstant, m = optimum, P = optimum, m = optimum, and P = optimum ww p w wp Th nrgy optimum of th hat pump Introdution It is possibl to find th nrgy optimum of hat pump mathmatially, but pratially, it has no any partiular importan. This is xplaind by th fat that th mass flow rat of hot and old watr is assumd onstant, th typ and th surfa ara of th vaporator is a known valu. Dirt optimization produr Th nrgy optimum of th hat pump in dirt optimization produr is obtaind by diffrntiating th hat pump COP with rspt to th powr of th omprssor. Th first diffrntial is qual to zro. dη dq dq dp P q 0 (3) dp dp dp dp Th obtaind diffrntial quation sparats th variabls, and it an b solvd: dq dp (32) q P

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... 2008 THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 dq q dp Th solution of th diffrntial quation is: lnq lnp q P (33) P Th solution of diffrntial quation is trivial and is not usful in prati. Howvr, th pross ontinus and a nw variabl is introdud, i.. th mass flow rat of th rfrigrant, instad of th omprssor powr. Indirt optimization produr Th hat flux in th vaporator and th omprssor powr dpnds on th mass flow rat of th rfrigrant: (33) q ( ) q m f (34) P = P ( m ) (35) f Th transition of th omprssor powr to th mass flow rat of rfrigrant is: dq dq dm f dq dp d dm f dp dm P f dm f (36) dp d P (37) Th quation of nrgy optimum ondition of th hat pump is: dq P q 0 dm dp f dm f Th optimum mass flow rat of th rfrigrant an b obtaind by solving th quation of th nrgy optimum ondition (38), and on that basis to obtain th optimum omprssor powr. ww w m f optimum P optimum m onst., m onst., F onst., and vaporator typ assumd onstant Th optimum mass flow rat of th rfrigrant and th optimum omprssor powr is valid undr th ondition that th mass flow rat of hot and old watr as wll as th siz and typ of th vaporator surfa ar known, i.. assumd onstants. Summary In this study, th aim was to st-up a physial and mathmatial basis for analytial nrgy optimization of hat pump and hat pump hating systm. In othr words, dfin th (38)

Nyrs, J.,: An Analytial Mthod for Dfining th Pump's Powr... THERMAL SCIENCE, Yar 207, Vol. 2, No. 5, pp. 999-200 2009 maximum COP aording to th optimum mass flow rat of th hot and old watr. As a final goal, on th basis of th optimum mass flow rats, to dfin th optimum powr of irulation and wll pump. Enrgy ffiiny of th onsidrd hat pump and hat pump hating systm is analyzd through th COP. In th pross of nrgy optimization of th hating systm, th objtiv funtion is takn as COP of th omplt systm. In th mathmatial optimization produr for obtaining th maximum COP th prinipl of first drivativ qual to zro is applid. Basd on th mntiond prinipl, in th first stp, th loal optimum is dtrmind with rspt to th powr of irulation and wll pump. In othr words, th partial drivation of th objtiv funtion was arrid out with rspt to th wll and irulation pump powr. Th obtaind first drivativs quald to zro. Th quations of th COP drivativ ar impliit and ontain partial drivativs of th irulation and wll pump powr. Th obtaind quations ar th analytial loal nrgy optimum onditions. Togthr, th two loal optimum onditions provid th global nrgy optimum ondition. By dirt produr, it is not possibl to obtain th partial drivativs for dfining th onditions of th loal and th global nrgy optimum. Th produr should b ontinud with th indirt dtrmination of th partial drivativs that mans, instad of th pump powr, th mass flow rat of old and hot watr should b introdud as nw variabls. This is possibl, baus th ooling apaity, omprssor powr and powr of irulation and wll pump dpnd on th mass flow rats. Th mntiond rlations hav a mathmatial dsription, that is, thr ar mathmatial rlations. In th papr th basi quations of th loal and th global nrgy optimum onditions ar dfind by dirt and indirt produr. Th following studis will dfind h mathmatial rlations of ooling apaity, omprssor powr and th powr of irulation and wll pump, dpnding on th mass flow rats. Nomnlatur F vaporator surfa, [m 2 ] kp hydrauli rsistan, [Nm 2 s 2 kg 2 ] ṁ mass flow rat, [kgs ] P powr, [W] q hat flux, [W] t tmpratur, [ o C, K] Grk symbols offiint of prforman - COP Subsripts omprssor p irulation pump w old watr vaporator f rfrigrant hp hat pump k ondnsr wp wll pump ww hot watr Rfrns [] Granryd, E., Analytial Exprssions for Optimum Flow Rats in Evaporators Condnsrs of Hat Pumping Systms, Intrnational Journal of Rfrigration, 33 (200), 7, pp. 2-220 [2] Sayyaadi, H., t al., Multi-Objtiv Optimization of a Vrtial Ground Sour Hat Pump using Evolutionary Algorithm, Enrgy Convrsion and Managmnt, 50 (2009), pp. 2035-2046 [3] Esn, H., Turgut, E., Optimization of Oprating Paramtrs of a Ground Coupld Hat Pump Systm by Taguhi Mthod, Enrgy and Buildings, 07 (205), Nov., pp. 329-334 [4] Mu, W., t al., Disussion of an Optimization Shm for th Ground Sour Hat Pump Systm of HVAC, Rnwabl Enrgy Rsours and a Grnr, Futur VIII-3-3 (2006), Nov., ESL-IC-06--35 [5] Shi, Z., Li, Z., Thrmo-Eonomi Optimization of a Sawatr Sour Hat Pump Systm for Rsidntial Buildings, Advand Matrials Rsarh Onlin, 354-355 (20), Ot., pp. 794-797 [6] Ahmadi, M. H., t al., Thrmo-Eonomi Optimization of Stirling Hat Pump by Using Non-Dominatd Sorting Gnti Algorithm, Enrgy Convrsion and Managmnt, 9 (205), Fb., pp. 35-322

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