Effect of Heat Exchangers Connection on Effectiveness

Transcription

1 Joural of Roboics ad Mechaical Egieerig Research Effec of Hea Exchagers oecio o Effeciveess Voio W Koiaho Maru J Lampie ad M El Haj Assad * Aalo Uiversiy School of Sciece ad echology P O Box 00 FIN Filad Ausralia ollege of Kuwai P O Box Safa-05 Kuwai wwwverizoaoliepublishigcom Vol: Issue: *orrespodig auhor: Dr Mamdouh El Haj Assad Associae Professor/Head of Mechaical Egieerig Deparme Ausralia ollege of Kuwai Mechaical Egieerig Deparme PO Box Safa-05Kuwai el: Ex: 79 Fax: ; massad@aceduw Aricle ype: Research Submissio Dae: 9 February 05 Acceped Dae: 0 May 05 Published Dae: 5 May 05 Absrac iaio: Voio W Koiaho Maru J Lampie ad M El Haj Assad (05 Effec of Hea Exchagers oecio o Effeciveess J Robo Mech Eg Resr (: -7 opyrigh: 05 Dr Mamdouh El Haj Assad his is a ope-access aricle disribued uder he erms of he reaive ommos Aribuio Licese which permis uresriced use disribuio ad reproducio i ay medium provided he origial auhor ad source are credied I his wor he performace of he recuperaive ype hea exchagers coeced i series is sudied Geeral expressios for he effeciveess of couer flow ad parallel flow coecios for ay ype of hea exchagers have bee derived I is proved ha he effeciveess of he hea exchager is oly fucio of he hermal coducace ad hea capaciy flows A umerical example is give i order o fid how may cross flow hea exchagers coeced i series would give he same effeciveess as ha of sigle couerflow hea exchager Iroducio I series coecio he overall coducace of he coeced uis is equal o he sum of he coducaces of all idividual uis he mai objecive of his wor is o derive a geeral expressio for he overall effeciveess of he series coecio as a fucio of he effeciveess s of whole uis i ha coecio Moreover his sudy purpose is o obai he bes performace of he hea exchager based o heir coecio Hea exchagers are devices used o rasfer hea bewee wo fluids a differe emperaures he goal of hea exchager desig is o relae he ile ad oule emperaure he overall hea rasfer coefficie ad he geomery of he hea exchager o he rae of hea rasfer bewee he wo fluids Figure: ouer- ad parallel coecio Hea exchagers are exesively used i power plas as boilers codesers feedwaer heaers superheaers ecoomizers ad air heaers; i refrigeraio ad air-codiioig equipmes as evaporaors ad codesers; ad i may oher applicaios Effec of he imum ad miimum hea capaciace o he performace of hea exchagers from eropy geeraio poi of view has bee ivesigaed [] heoreical aalysis of couerflow hea exchagers [] ad parallel flow hea exchagers [] wih a hea source wihi he ho fluid has bee sudied Moreover hea exchagers of ay ype ca be coeced i series for cerai purposes More deails o hea exchager desig ad applicaio ca be foud i lieraure [-7] Hea exchagers ca be coeced i series i couer flow coecio as show Figure a or i parallel flow coecio as show i Figure b while he hea exchagers coeced i series i boh coecios ca be ay ype of hea exchager I order o avoid ay misudersadigs bewee coceps couer-flow ui ad couer coecio as well as bewee parallel-flow ui ad parallel coecio a couer coecio of hree parallel flow uis (parallel flow hea exchagers is preseed i Figure Figure : Parallel coecio wih parallel flow uis ouer coecio Le us desigae by ad of ho flow; ad by ad of cold flow respecively J Robo Mech Eg Resr ( Page as he emperaure ad hea capaciy as he emperaure ad hea capaciy osider he couer coecio as show i Figure ad assume firs ha > Figure is a couer coecio wih couer flow hea exchagers (couer uis However uis ca be ay ype of hea exchager sice he same mahemaical formulaio is valid for hem; his

2 iaio: Voio W Koiaho Maru J Lampie ad M El Haj Assad (05 Effec of Hea Exchagers oecio o Effeciveess J Robo Mech Eg Resr (: -7 is because we oly cosider he emperaures of he ho ad cold flows a he ile ad exi of he ui osider ui of Figure ε δ ε R ε or i geeral form δ R ε he by usig Eq ( he effeciveess of he whole coecio for 0 R < is obaied as he hea balace ca be wrie as ad effeciveess of he ui is ε / o From hese expressios i follows ha ε o ad ( mi / o where R / / Hece o - ε o ad o - R ε o we obai uis ε ad orrespodigly for he oher ε Muliply ε ε ε we ge he oal hea balace of he coecio is ( + where + + Hece R + - ad for R ad R for R O he oher had he effeciveess of he whole coecio is ad hus ε + auxiliary variable as ε δ ε he auxiliary variable is obaied as δ Figure : ouer coecio wih couer flow uis ε ε (R R R Now le us defie a ( ε ε ε R I he same way as above a geeral expressio for he whole coecio (couer coecio effeciveess for < ie R / ca be obaied which is he same as ha of Eq ( So he hea capaciace raio R does o chage mahemaically he geeral expressio for he effeciveess For R ad hece we obai ( + + ε ad hus we obai ε he he auxiliary variable is obaied as ε + + δ + + ε sice for R O he oher had he effeciveess of ui is ε ad δ ad δ geerally δ + ε ε ε ε δ he ε + orrespodigly δ + ε ε hece we ge δ δ + δ + δ or he effeciveess of he whole coecio for R ca be wrie i geeral form as Subsiuig he expressio of / i he above equaio we ge J Robo Mech Eg Resr ( Page

3 iaio: Voio W Koiaho Maru J Lampie ad M El Haj Assad (05 Effec of Hea Exchagers oecio o Effeciveess J Robo Mech Eg Resr (: -7 ε + ε ε ε ε Whe he uis are ideical ad we assume ha he coducace of a ui does o chage if is geomery does o chage i is valid ha ε ε ε ε u ha is because i ca be prove ha he effeciveess e of a hea exchager is oly fucio of hea capaciy flows ad coducace as will be show laer ad he hea capaciy flows are he same for all he uis Hece i follows from Eqs ( ad ( ha ( From hese equaios i follows ha ε ad - ( / - Figure : Parallel coecio wih parallel flow uis u εu ε for 0 R < ( u R εu ad εu ε for R (5 + ( ε u hus - -(- - ε - ( - ε - which meas ha ε orrespodigly for he oher uis ε emperaure differece raio as ad ε Muliplyig he he followig where R is eiher / or / (he miimum hea capaciy flow is i he omiaor ad is he umber of uis I a similar way i is easy o show ha Eqs ( ad ( are valid also for R 0 Equaio (5 is obaied by alig he limi of Equaio ( as R eds o ad he usig L Hopials rule as R eds o Now we cosider he case where he hea balace of a idividual ui ca be wrie as u u G A u where G is coducace per ui hea rasfer area of he hea exchager walls A u is area of he hea exchagers walls ad is he average emperaure differece bewee cold ad ho fluids As ad he size of he coecio sill remais fiie he size of he uis becomes differeially small ad i follows ha where - Bu his is he hea balace of a differeial ui of a couer-flow exchager ha meas ha we ed up o equaios of he ordiary couer-flow exchager hus as he umber of uis is very large we ca cosider he whole coecio as oe couer-flow exchager Parallel coecio osider he parallel coecio where he idividual uis cosis of parallel flow hea exchagers as show i Figure Le us firs cosider he case where > which ie R emperaure differece raio is obaied as hea balace of he coecio is which meas ha R - - ad R + he effeciveess of he whole coecio is ε R + ( ε ( ε R + or i a geeral form R + ( ε / ( εu u sice < 0 he effeciveess of ui is ε / o / ε R + (7 J Robo Mech Eg Resr ( Page ε o ( ε he overall (6 R + I a similar way we ed up wih Eq (6 for he case where < I is easy o show ha Eq (6 is valid also whe R If all he uis of a parallel coecio are ideical wih effeciveess ε u i follows from Eq (6 ha

4 iaio: Voio W Koiaho Maru J Lampie ad M El Haj Assad (05 Effec of Hea Exchagers oecio o Effeciveess J Robo Mech Eg Resr (: -7 which is valid for all values of R (whe 0 R Le us agai cosider he case where he hea balace of oe ui ca be wrie as u u G A u As ad he size of he coecio sill remais fiie he size of he uis is differeially small ad i follows ha where - However his is he hea balace of a differeial ui of a parallel-flow exchager hece we ed up wih he equaios of he ordiary parallelflow exchager hus as he umber of uis is very large we ca cosider he whole coecio as oe parallel-flow exchager A proof ha he effeciveess of a hea exchager is a fucio of R ad NU oly his proof is based o he Bucigham s Π-heorem ad o such a reasoable assumpio ha he chage of emperaure of he smaller hea capaciy flow i he hea exchager is fucio of four variables which are emperaure differece of eerig flows o (which is he imum emperaure differece i he whole hea exchager miimum hea capaciy flow mi imum hea capaciy flow ad coducace of he exchager G Hece we ca wrie ( o mi G Mahemaically here is a fucio F ha saisfies he followig codiio F ( o G 0 he dimesios of mi he variables are: [ ] [ o ] K ad [ mi ] [ ] [G] W/K We ca choose dimesios K ad W as basic uis ha meas here are wo liearly idepede dimesios i fucio F his ca be doe eve hough W g m s - is o a basic ui i he sadard ui sysems because he liear idepedece of K ad W is all ha is demaded hus he amou of variables is 5 ad he amou of idepede dimesios is Accordig o Bucigham s Π-heorem o iformaio is los if he fucio F is derived o a fucio of dimesioless groups ha is Π-groups so ha he amou of Π-groups i he ew fucio is We ca form from he variables of fucio F for example he followig Π-groups: Π / o e Π / R ad Π G / mi NU Hece we ge a ew fucio f so ha f(π Π Π 0 Bu his ca be derived direcly o form Π Π (Π Π or e e (R NU which was o be proved ompariso bewee a couerflow exchager ad a couerflow coecio made of crossflow elemes osiderig he hree ypes of hea exchagers couer- crossad parallel flow exchagers i is well ow ha he couerflow exchager has he bes effeciveess Hece he effeciveess of he couer flow exchager ca be used as a referece he closer he effeciveess of a coecio o he referece oe he beer is he coecio A impora quesio is how may elemes mi eeded i order ha a coecio pracically performs as couer flow exchager wih he same hea rasfer area? I order o sudy his we use he followig approximae equaio for cross-flow exchagers wih boh flows umixed [8]: osider a couer flow exchager ad a couer coecio made of cross-flow uis so ha hey boh have he same G mi ad ad hece he same R ad NU Le be he umber of he cross flow uis For a cross flow ui NU u NU/ (9 Subsiuig Eq (9 io Eq (8 we ge he effeciveess ε u for a idividual cross flow ui Usig Eqs ( ad (5 we ge he effeciveess ε co for couer coecio For a couer-flow exchager [] he effeciveess is J Robo Mech Eg Resr ( Page (8 (0 he he perceage differece bewee boh effeciveesses for various values of R NU ad ca be calculaed by εcou εco 00 % εcou For example wih values R 075 NU 5 ad we ge e cro 088 (Eq(8 NU u 5 (Eq(9 e u 056 (Eq(8 e co 089 (Eq( εcou εco e co 0909 (Eq(0 ad 00 % % εcou Resuls ad Discussio For a couer coecio he effeciveess of he coecio ca be calculaed from Eq ( or Eq ( ad especially from Eq ( or Eq (5 whe he uis are ideical he correspodig equaios for a parallel-flow coecio are Eqs (6 ad (7 he daa calculaed accordig o he above procedure is show graphically i Figures 5a - 5d Effeciveesses of couerflow-ad crossflow hea exchagers ad of couer coecios made of crossflow uis wih differe values of R I each of he figures he uppermos curve represes he couer flow hea exchager ad he lowermos represes a sigle cross flow hea exchager ad bewee he upper- ad lowermos curves here are 5 curves which represe he couer coecios made of 6 8 ad 0 wih cross flow hea exchager uis he larger he umber of uis he closer he curve is o he uppermos curve

5 iaio: Voio W Koiaho Maru J Lampie ad M El Haj Assad (05 Effec of Hea Exchagers oecio o Effeciveess J Robo Mech Eg Resr (: -7 Figures 5 a - 5 d illusrae ha he larger R value he larger he differeces of e bewee idividual cases I case R 0 which aes place i a phase chage here would be o differece a all ad here would be oly a sigle curve i he figure hus we ca mae a coclusio ha he differeces are he larges for R ie whe he hea capaciy flows are he same his coclusio ca be reasoed also i he followig way: le us cosider a siuaio where mi < I his case he emperaure chage of is very small hece here is almos cosa emperaure o he oher side of he wall So i is almos isigifica o which direcio mi flows compared o hus he emperaure chage of mi is less sesiive o he relaive flow direcios ie he geomery he larger he differece bewee mi ad ie he closer he R is o zero Hece he emperaure chage of mi ad by defiiio he e is more sesiive o he geomery he closer R value o uiy hus he larges differeces bewee effeciveesses appear i he case R he larges differeces bewee a sigle couerflow exchager ad couerflow coecios wih various umber of uis of cross flow hea exchagers have bee foud from he umerical daa of Figure 5a he mai resuls are give i able he larges differeces of e bewee a sigle couerflow exchager ad couerflow coecios whe R I he able umber of uis NU ( he NU-value where he larges differece occurs % ( larges differece (% % (NU he differece of e whe NU (% he umerical daa was calculaed wih he accuracy of wo decimals ad wih some values of here appeared o be wo imum pois I mos cases he imum poi was very close o he poi NU ad for his reaso here is a exra row which shows he differece for NU he resuls idicaed by able ca be preseed shorly so ha wo uis are eeded i order ha he differece i all codiios is less ha 0 % for uis less ha 5 % for 8 uis less ha % ad for 8 uis less ha % Bu i mus be remembered ha we cosidered here he limi case R ad wih smaller values of R usually smaller amou of uis is eeded able idicaes also ha he differece of e bewee a couerflow ad a sigle crossflow exchager is ever larger ha % J Robo Mech Eg Resr ( Page 5

6 iaio: Voio W Koiaho Maru J Lampie ad M El Haj Assad (05 Effec of Hea Exchagers oecio o Effeciveess J Robo Mech Eg Resr (: NU( %( %(NU NU( %( %(NU NU( %( %(NU able : Mai Resuls I ca be prove by he dimesioal aalysis ha for ay hea exchager he effeciveess is a fucio of NU ad R oly ad does o deped o he icomig emperaures Pracically o so may crossflow uis are eeded i order ha he effeciveess of a couerflow coecio is pracically o worse ha ha of a sigle couerflow exchager I was proved ha 8 uis are eeded i order o have oly % differece i effeciveess oclusios A heoreical formulaio was preseed i order o obai he effeciveess of hea exchagers coeced i series he effeciveess was derived for couer ad parallel flow hea exchagers coeced i series If he hea exchagers are coeced i series hey should be coeced i couer flow coecio i order o achieve he bes possible effeciveess I was show ha he effeciveess of he hea exchager is oly fucio of he hea capaciace raio of he wo fluids ad he hermal coducace Usually oly a few uis are eeded i order ha he effecivees is almos he same as a sigle couerflow exchager Nomeclaure A area of hea rasfer wall m specific hea J/g c p hea capaciy rae W/K ( mc p G m coducace W/K mass flow g/s umber of uis NU umber of hea rasfer uis ( NU G / mi R raio of hea capaciace ( R / mi cold fluid emperaure K ho fluid emperaure K Gree Leers δ auxiliary variable ε effeciveess of a hea exchager emperaure differece K Subscrips cro crossflow exchager cou couerflow exchager co couerflow coecio made of crossflow uis Mi miimum imum cold fluid ho fluid u ui Refereces M El Haj Assad Effec of imum ad miimum hea capaciy rae o eropy geeraio i a hea exchager I J Eergy Research 00; (:0-08doi: 000/er67 M El Haj Assad Voio Koiaho hermal aalysis of a couerflow hea exchager wih a hea source I J Ambie Eergy 00; (:-7doi:0080/ M El Haj Assad Voio Koiaho Aalysis of a parallel flow hea exchager wih a hea source Hea rasfer Egieerig 0; (5:8-89 doi:0080/ A Beja Hea rasfer New Yor: Wiley; p J Robo Mech Eg Resr ( Page 6

7 iaio: Voio W Koiaho Maru J Lampie ad M El Haj Assad (05 Effec of Hea Exchagers oecio o Effeciveess J Robo Mech Eg Resr (: -7 5 S Kaac edior Boilers evaporaors ad codesers New Yor: Wiley; p 6 WM Kays AL Lodo ompac Hea Exchagers New Yor: McGraw- Hill; 98 7 RK Shah DP Seulic Fudameals of Hea Exchager Desig New Yor: Wiley; 00 9 p 8 FP Icropera DP DeWi Fudameals of Hea ad Mass rasfer New Yor: Joh Wiley &Sos; 990 J Robo Mech Eg Resr ( Page 7