COUNTER FLOW HEAT EXCHANGERS' IRREVERSIBILITY MINIMIZATION

Transcription

1 OUNTER FLOW HET EXHNGERS' RREVERSBLTY MNMZTON bbas lwi Sakhir bed Mechanical Engineering Departent llege f Engineering, l-qadisia Universit RQ dctr_abbas001@ahc BSTRT The irreversibilit iniizatin ethd f heat exchanger ptiizatin is extended t include a ter t accunt the exerg f the aterial f cnstructin f the heat exchanger The ethd perits phsicall realistic ptiizatin t be cnducted with the resulting ptiu designs prviding cnceptuall beneficial guidepsts, which d nt change with tie r lcatin Such ptia are in cntrast t the ptia btained b presentl advcated ethds, which n ne hand indicate unrealistic infinite area f the heat exchangers, and n the ther hand pint t ptia that a change draaticall with lcatin and tie ccpaning exergetic efficienc expressins using the sae tpe f aterial exerg ter shw phsicall re realistic values than the usual expressins Kewrds: heat transfer, irreversibilit, energ, exerg, heat exchanger NTRODUTON Technique fr heat exchanger analsis particularl since the wrk f Bejan (1977, 1980) where results regarding basic design variables were develped in a generalized, nn-diensinal anner The wrk in this areas has reviewed b Bejan (1987) The irreversibilit iniizatin ethd is quit useful fr evaluating design paraeters at fixed ttal heat exchanger area Hwever, ethd nl cnsiders the exerg the lsses assciated with the heat exchanger peratin, and dse nt cnsider the exerg r ecnic cst f the heat exchanger itself s a result f this,the glbal ptiu usuall results fr an engineering design view pint Therecnics, develped b Tribus & Evans (196) has been applied t heat exchangers b varius authrs, including Lndn & Shah (1983) Therecnic analsis can take int accunt all the csts assciated with building perating a heat exchanger, and glbal ptiizatin ields the realistic result f a finite area heat exchanger Therecnics als gives the designer the abilit t decpse a glbal sste int individual islated cpnent in awa such that ptiizing the in divided islated ield a gd apprxiatin t ptiu design f the glbal sste (Evans et al, 1983) The Therecnicall ptiu heat exchanger design ust change as the csts var with lcatin and tie Thus the Therecnic ptiu design fr ne lcatin and tie dse nt serve as a gd guidepst fr the ptial design in anther lcatin and tie This research presents an extensin f the iniu irreversibilit generatin analsis wherein the irreversibilit generatin equatin includes ter that exchanger aterial and the applicatin life f the heat exchanger This ethd ields ptial design that d nt change with tie r lcatin and that represent the desired ptial design in the liit f exerg cst begin dinant relative t labr and prfits The resulting ptial designs are phsicall re realistic that thse btained fr the usual irreversibilit iniizatin ethds, and althugh the designs d nt represent the ptial design fr the therecnics viewpint the represent liits that a serve as guidepsts fr design such guidepsts are quite useful cnceptuall and particularl iprtant in ties f rapidl changing and unpredictable ecnic paraeters BS METHOD The bjective functin fr the basic irreversibilit iniizatin analsis fr a heat exchanger in a nn diensinal fr (Bejan, 1977) 57

2 Vl 3, N 3, Nveber 01 cadeic Research nternatinal Ns (1) T ax Ns + Ns T Ns () τ (3) ax 58 Ns T (4) T illustrate the ethd need t be deterined The exerg f heat exchanger aterials can be divided int cheical and therechnical exergies Materials acquire cheical exegies when the are transfred fr the dead state int their final fr aterials als have therechaniccal exerg if the perate at teperatures r pressures different fr that f the dead state int their final fr Materials als have theechanical exerg if the stre elastic exerg as a result f defratins during the anufacture f the heat exchanger Heat exchanger aterial can be written as +, ch, ph The theral exerg is fund b calculating the axiu wrk that can be btained when the aterial under cnsideratin is brught t cheical equilibriu with the cpnents f the dead state The ethd is illustrated b Mran (198) and Ktas (1985), where a table f cheical exerg value fr cn aterials with respect t a standard dead state is included Heat exchangers perate at pressures and teperatures different fr thse f the dead state Heat exchanger aterials then have a theechanical exerg different fr zer Hwever, these exerg values are sall relative t the cheical exerge f the aterial, and can be neglected in the analsis The elastic exerg resulting fr defring a ateral int a heat exchanger pipe is estiated here b fllwing a siplified prcedure: E σ E ε r where is the radial crdinate easured fr the center f the tube The stress at the tube wall ( δ / ) fr eq (5a) is: Eδ σ w r The stress varies linearl with in the elastic regin and has a cnstant value in the plastic regin, accrding t the assued σ ε behavir f the aterials n st practical cases the wall is thick enugh s that the elastic zne is sall relative t the ttal vlue Under these assuptins, the ttal exerg in the aterial is 1 1 σ, ph σεdv σ V v E plδσ E and the elastic exerg f the aterial per unit area is δσ, ph E σ (5) (5a) (5b) (5c) (5d)

3 The wrk necessar t prduce the defratin in the aterial is greater than the value f the elastic exerg Eq (5a) This is because part f the wrk has t be spent in defring the aterial phsicall, and this part f the wrk is nt recverable Objective Functin nsider the case f a heat exchanger with negligible pressure drp irreversibilit Equatin(3) then s previusl discussed, the heat exchanger aterial has an exerg value at the ent f installatin the exerg value f the heat exchanger can be viewed as begin used up after a tie equal applicatin lift τ : The decrease in the aterial can be cnsidered as an irrevesiblit generatin rate given b Ns T ax γω (6) Where is the irrevesiblit rate per unit area f the heat exchanger and γ is the aterial exerg paraeter defined as: γ T U When ideal gas behavir is assued fr the fllwing fluid, and there are n heat lsses t the surrundings, the heat transfer irreversibilit generatin nuber is Ns T ε ω ln 1+ (1 R) + ln 1 R [ + ωε ( R 1) substitutin f (6) and (8) in t () ields the final fr f the bject functin ε Ns ω ln 1+ (1 R) + R ln 1 ) + γω [ + ωε ( R 1 This bjective functin can be used t ptiize an heat exchanger if the ε relatinship is knwn and the assuptin ade during the develpent f this equatin are satisfied The ε relatinships fr an heat exchanger cnfiguratin are given in Kas & Lndn (1984) bjective functin (9) is used next fr the design ptiizatin f a cunter flw heat exchanger Optiizatin f unter Flw Heat Exchangers When the arital irreversibilit is nt include in the analsis Ns T is equal t Ns, the value f Ns T increases fr zer with increasing, reaches a axiu, and then decreases t a iniu at infinit This iniu Ns T has a value f zer fr balanced heat exchanger, the decrease in irreversibilit n the left side f the axiu was shwn b Bejan (1980) t be due t insufficient heat transfer acrss a teperature difference f rder ( T ) T 1 Bejan (1987) this regin wuld vilate the principle f therdnaic f the cpnent being ptiized Since the value f γ is functin f verall heat transfer cefficient U, the pint f iniu irreversibilit generatin rate crrespnds t a given U n the present analsis the value f U has t be knw t calculate and This additinal degree f freed is due t the assuptin f negligible pressure drp; therefre, U shuld (7) (8) (9) 59

4 Vl 3, N 3, Nveber 01 cadeic Research nternatinal be in a range where this assuptin is nt vilated; hwever, when the pressure drp is cnsidered, an ptiu value f U fr a given surface area can be btained fr the best balance between Ns T Ns p and The variatin f with the capacit rate rati ω fr varius values f γ is shwn in Figure 1 This figure des nt shw a value f fr sall value f ω since, as stated befre, there is n ptiu design fr sall capacit rati Figure 1 crrespnds t an inlet teperature rati R f /3 (Witt, 1983) Reducing R increases the irreversibilit due t heat transfer at a given and hence increases the, and vice versa The use f Figure 1 t btain fr cunter flw heat exchanger is illustrated b a specific applicatin in later sectin Figure 1 Optiu as a functin f ω and γ fr cunter flw heat exchanger with R/3 MODFTON OF BS METHOD urrent prcesses t prduce aterials have lw efficiencies, and the actual exerg expenditure fr given heat exerg f the aterial The bjective functin fr irreversibilit iniizatin analsis can be frulated t take int a cunt all the exerg expenses assciated with the fabricatin f the aterial cntributin t the irreversibilit generatin equatin This analsis was based n average values fr exerg expenses assciated with fabricatin f the aterial This ethd was used in case b Bd et al, (1981) t analze siplified heat exchanger del The cnsidered a aterial cntributin t the irreversibilit generatin equatin This cnsidered a aterial cntributin t the irreversibilit generatin equatin This cntributin was calculated as the exerg spent in ining, selting, refining, illing, and transprting the heat exchanger aterial The analsis was based n average values fr exerg expenses and transprting distances The aterial energ ter in the glbal energ cnsuptin equatin The aterial energ was estiated fr the cpnent diensins b used epirical equatins ppling these estiates t find ptiu heat exchanger designs fr different bjective functins based n energ, ecnics, and cbinatins f the tw nther wa f including the irreversibilites generated in the prcess f building a heat exchanger is t divided the aterial irreversibilit rate b an adequate efficienc fr verall efficiencies fr anufacture f se iprtant aterials The bjective functin expressing the glbal irreversibilites generated at the current state f technlg is Where Ns Ns + Ns (10) Ns T Ns (11) 60

5 and is the exergetic efficienc fr the verall anufacturing prcess When ideal behavir is assued fr the flwing fluid and the pressure drp irreversiblitit is sall, the bjective functin reduces t Where ε Ns ω ln 1+ (1 R) + R ln 1 ) + γ ω γ γ [ + ωε ( R 1 bth bjective functins given b equatin (9) and (1) have the sae functin fr, and are identical when the exegetic efficienc f the anufacturing prcess is unit The sae ptiizatin prcedure used fr equatin (9) is then applicable fr (1), and an ptiu fr a cunter flw heat exchanger can als be btained fr Figure 1 fr a given value f γ the ptiu value f is the ptiu when is used in the sae figure Exegetic Efficienc The exegetic efficienc f a plant cpnent is defined in a variet f was in the literature The fllwing expressin is selected fr this research (Tsatarnis & Winhld, 1985) p p f Where is the rate exerg lss b the prduct, and is the rate f exerg lss b the fuel When the cpnent is a heat exchanger designed fr heating the rate f exerg gained b the cld strea is p equal t, and the rate f exerg lss b the ht strea is equal t f When the irreversibilit cntributin fr the aterial is nt included in the analsis, the rate f exerg lss b the fuel is the rate f exerg lss designed fr heating, the rate f exerg lss b the ht strea,and is designated as f The exergetic efficiencies fr a heat exchanger with and withut the irreversibilit cntributin fr the aterial are develped next The exerg gained b the prduct is f p 1e 1 (15) (1) (13) (14) When the aterial irreversibilit rate is nt include, the exerg lss the fuel is f e When the aterial irreversibilit rate is nt include, the exerg lss the fuel is given b (16) f e + Fr an ideal gas, when pressure drp irreversibilit and heat lsses t the surrunding are negligible, the abve equatins can be written as (17) 61

6 Vl 3, N 3, Nveber 01 cadeic Research nternatinal 1 1 p int1 ε 1 T0 in ln 1 + ε 1 R 1 f axt1εω 1 + T R 0 ax ln 1 1 [ εω( R) 1 f axt1 εω 1 + T0 ax ln[ 1 ωε ( 1 R) + R γt in (18) (19) (0) where T 1 is the inlet teperature f the cld strea The value f γ is released b γ t take int accunt all the irreversibilites assciated with the anufacturing prcess at the current state f technlg The exergetic efficiencies with and withut the aterial irreversibilit rate and, respectivel can nw be written as: T ( ) ε( R 1) ln[1 + ε( R 1) T0 ω T 1 1 ωε ( R 1) + ln[1 ωε ( 1 R) ) + γω T0 (1) T1 ( ) ε( R T0 ω T 1 ωε ( R T ) ln[ 1+ ε( R 1 1) + ln[1 ωε 1 1) ( 1 R) The equatins develped abve fr the exegetic efficienc are nw applied t bth cunter flw and parallel flw heat exchangers The effectiveness equatins can be taken fr Dnald(000) as fllws: ε ε pf cf [ ( 1+ ω) 1 exp 1+ ω 1 exp 1 ω exp [ ( 1 ω) [ ( 1 ω) ) () (a) (b) The variatin f exergetic efficiencies and with effectiveness fr cunter flw and parallel flw heat exchangers with the sae γ(γ001), are shwn in Figure This figure is fr abalanced capacit rate with the cld strea inlet teperature equal t the dead state teperature The exergetic efficienc is alwas less than t the infinite heat exchanger liit ( ε 0 5 fr parallel flw heat exchanger ε 1 fr the cunter flw heat exchanger) tends t zer while is a iaxiu pf The exergetic efficienc fr parallel flw heat exchanger at a given ε cf is alwas lwer than fr a cunter flw heat exchanger and this difference can be clearl seen at values f the effectiveness clse t 05 The exergetic efficiencies fr the parallel and cunter flw heat exchangers are the sae Hwever, fr a given dut (a given effectiveness in Figure ) a cunter flw heat exchanger is knwn t be re attractive than a parallel flw unit because f the reduced area, this is prperl shwn in the exegetic efficienc if the aterial irreversibilit is included in Figure The irreversibilit

7 (XY) 5 Sep 006 rate in building and peratin an infinite area heat exchanger ake the exegetic efficienc in the larger heat exchanger area liit tend t zer when the applicatin life is finite Hence; including the aterial irreversibilit result in phsicall realistic value fr the exegetic efficienc The paraeter γ is inversel prprtinal t the applicatin life f the heat exchanger τ when the applicatin life appraches zer, γ tends t infinit and the exergetic efficienc is equal t zer in the liit Hwever, if the heat exchanger has infinite applicatin life then γ is equal zer and the exergtic efficienc equal t The variatin f the exergitic efficienc with effectiveness fr a balanced cunter flw heat exchanger is shwn in Figure 3, fr different value f γ This figure is fr a heat exchanger inlet teperature rati f /3 and the cld strea inlet teperature equal t the dead state teperature 1 09 R /3 w 1 Ti/T F PF ε Figure Exergetic efficienc fr cunter flw heat exchanger with and, the effective aterial irreversibilit fr R/3, ω 1, Ti / T 0 1 and γ 001 cf Figure 3 shw that increases with decreasing γ (increasing applicatin life τ ), fr a given value f ε, and has a axiu fr the value f γ The dtted line n Figure 3 shws the lcus f the axiu exergetic efficienc, with the arrw indicating the directin f increasing applicatin life τ When the heat exchanger is reversible ( 1 ) then γ 0, and ε 1 This shws that an infinite area balanced cunter flw heat exchanger with an infinite applicatin life is reversible Hence, including the aterial irrevesibilit ter result in adding a tie cnstraint in the reversible heat exchanger liit Figure 3 Exergetic efficienc f a cunter flw heat exchanger as a functin f ε Fr different values f the aterial exerg paraeter 63

8 Vl 3, N 3, Nveber 01 cadeic Research nternatinal Therecnic nalsis n therecnics, the bjective functin beces the ttal cst f peratin This includes capital csts and irreversibilit penalt csts ssuing that the capital csts are prprtinal t the area, the bjective functin beces: T + T T + The bjective functin can be nndiensinlized as fllws: Nc T T T ax τ p p Under the assuptins f ideal gas behaviur and negligible pressure drp irreversibilit, the bjective functin beces: Where ε Nc ω ln 1 + R 1 + γ c ω γ c T Uτ T ( 1 R) + ln[ 1+ ωε ( R ) Nte that the therecnic bjective functin given b equatin (5) has the sae functinal fr as the tw previusl derived irreversibilit rate based bjective functins given b equatin (9) γ and(1) The nl difference is that the value f c has t be used instead f γ r γ This prpert siplifies the ptiizatin prcedure since Figure 1 can be used t btain the ptiu, when the heat exchanger begin ptiized is a cunter flw unit The ptiu NUT btained fr therecnic analsis is designated c The next sectin shw the relatinship f,, and c fr a specific case Specific pplicatin The evaluatin f the ptiu value f fr a cunter flw heat exchanger based n iniu irreversibilit r iniu cst f irreversibilit is illustrated b the fllwing specific case This sectin als includes discussin n the significance f the results We need t deterine,, and c fr a cunter flw heat exchanger fr values f the applicatin life τ f 5 and 10 ears, and fr U 40 and 70 W / K The capacit rate rati and the inlet teperature rati are ω 07 and R /3, respectivel The heat exchanger is shell and tubes are ade f carbn steel, and se data fr the steel tubes are given in Table 1 (3) (4) (5) (6) 64

9 Table 1 Data fr arbn Steel Tubes ρ kg / δ ,ch 6764 kj / kg σ Pa E Pa $ / (fr Matle, 1983 ), an estiate fr the The cst f heat exchanger area is apprxiatel 7 grss energ requireent t prduce steel tubes is J / kg (fr hapan and Rberts, 1985), and the cst fr heat transfer irreversibilit is 0 056$/ kw hr T The teperatures 98K and T 1 98K The prcedure f slutin as fllws: 1- therechanical exerg f the aterial can be calculated fr equatin (5d) as, ph δσ 649 E - the cheical exerg f the heat exchanger aterial is, ch ρδ cheical exerg (7770) The therechanical exerg f the aterial is uch saller than the cheical exerg, then the ttal aterial exeg is, ch 3- the effectiveness f the anufacturing prcess fr carbn steel tubes can be calculated as : cheical exerg 6764 grss exerg the cst f heat transfer irreversibilit is T $ / J γ, γ c 5- the values f, and γ can nw be calculated b using equatins (7),(13),and (7), respectivel, as / τ γ T U T U 98U τ Uτ ( γ γ γ 019 Uτ 8 65

10 Vl 3, N 3, Nveber 01 cadeic Research nternatinal γ c T Uτ 1556 T RESULTS ND DSUSSON (98) Uτ The results are suarized in Table The values can be btained fr Figure 1 r b slving the bjective functins as shwn in Figure 3 The results are suarized in Table 3 Table Values f γ, γ, and γ c fr heat exchanger The variatin f Ns with fr the three bjective functins cnsidered, when the verall heat transfer cefficient is 40 W / K and the applicatin life is 5 ears, is shwn in Figure 4 The agnitudes f the ptiu > > are in the rder c The value f and the crrespnding irreversibilites are cnstant, independent f tie and lcatin The irreversibilit crrespnding t this ptiu is the abslute iniu and crrespnds t the case when all the anufacturing prcesses f the heat exchanger are ideal s discussed previusl, c is a tie and lcatin dependent due t the csts and T The value f is weaker functin f tie and lcatin than c Figure 4 Values f with iniu irreversibilit generatin ( Ns ), iniu irreversibilit Ns ) and iniu therecnic cst ( Nc ) generatin at present state f technlg ( This is because is a functin f the exergetic efficiencies f the heat exchanger anufacturing prcesses which nl varies with an iprveent in the technlg and is relativel insensitive t plitical envirnent, lcatin, etc n a wrld f diinishing fuel resurces, we can expect we can 66

11 expect c t ve tward Hwever, the cst f aterial ight increase with increasing fuel cst The cllectin and analsis f heat exchanger data ver a perid f tie shuld be undertaken t establish this relatinship This wuld allw the heat exchanger design engineers t decide n the best pssible value fr the heat exchanger fr the st efficient peratin during its applicatin during its applicatin life perid Table 3 Values f fr heat exchanger ONLUSONS (1) The ethd presented in this research, which adds an irreversibilit ter due t the aterial f cnstructin f the heat exchanger in the verall irreversibilit iniizatin f the heat exchanger in the verall irreversibilit iniizatin equatin fr heat exchanger ptiizatin t be cnducted The resulting ptiu designs prvide cnceptuall beneficial guidepsts that d nt change with tie r lcatin Such ptia are in cntrast t the ptia btained b presentl advcated ethds which n ne the ther hand indicate unrealistic infinite area heat exchangers and n the ther hand pint t ptia that a change draaticall with lcatin and tie () The ptia btained b the ethd indicated here are cnceptuall siilar t the ecnic ptiizatin, and,in fact the sae nndiensinalized curves a be used, in se instance fr bth ptiizatins The ptia btained fr the basic ethd delineated here specif heat exchanger areas that are significantl larger than thse crrespnding t an ecnic ptiizatin and their use is upper liits With dificatins t represent the current state f technlg the ethd delineates ptia that are substantiall clser t thse specified b current therecnic ptiizatins (3) Exergetic efficienc expressins that siilarl include an irrevesiblit ter due t the aterial f cnstructins f the heat exchanger shw phsicall re realistic values than the usual expressins that d nt include such a ter Such exergetic efficiencies clearl shw the therdnaic advantage f cunter flw as cpared t parallel flw arrangeents, and the shwed a value f zer fr all infinite applicatin lives This is in cnstant t sae ther usual exergetic expressins that shw cunter flw and parallel flw units having equal efficiencies and that can ield values f 100 percent fr infinite area heat exchangers Nenclature : rea, : st, $, in ax : apacit rate, J / kg s E : Elasticit dulus, Pa l : length, 67

12 Vl 3, N 3, Nveber 01 cadeic Research nternatinal Nc : Ecnic bjective functin t be iniized Ns : rreversibilit generatin nuber : Nuber f transfer units p : Width f the plate, T : Teperature, K U : Overall heat transfer cefficient, W / K R : nlet teperature rati : Exerg, J : rreversibilit rate, W τ : pplicatin life, s γ : Material exerg paraeter, T U ε : Heat exchanger effectiveness σ : Yield stress, Pa : Exergetic efficienc ω : apacit rate rati, δ : Thickness, Subscripts : Dead state 1 : ld strea : Ht strea c : Therecnic value cf : cunterflw ch : heical e : Exit f : Fuel in /ax : Value at current state f technlg : Material p : Prduct ph : Therphsical 68

13 T : Heat transfer : Yield pint Superscripts *: Optiu value ': nventinal values nt including aterial exerg REFERENES Bejan, (1977) The ncept f rreversibilit in Heat Exchanger Design: unter Flw Heat Exchangers fr Gas-Gas pplicatins, se Jurnal f Heat Transfer, Vl 99, pp Bejan, (1980) Secnd Law nalsis in Heat Transfer, Energ, Vl 5, pp Bejan, (1987) Therdnaics f Transfer Devices, Secnd Law nalsis f Theral Sstes, Prc Vth ntel Sp On nd Law nalsis f Theral Sstes, Mran, MJ and Sciubba, E eds, SME, New Yrk, pp 1-15 Bed, JM, Blueel, V, Keil, TH, Kuncinkas, GR and Mlinari, S (1981) The Secnd Law f Therdnaics as a riterin fr Heat Exchanger Design, Energ Vl 6, pp hapan, PF and Rberts, F (1985) Metal Resurces and Energ, Lndn: Butterwrth, Dnald, RP (000) Ther and Prbles f Heat Transfer, New Yrk: McGraw-Hill Evans, RB, Kadaba, PV and Hendrix, W (1983) Exergetic Functinal nalsis fr Prcess Design and Snthesis, Efficienc and csting, S Spsiu Series, pp Kas, WM and Lndn, L (1984) pact Heat Exchangers, New Yrk: McGraw-Hill Ktas, TJ (1985) The Energ Methd f Theral Plant nalsis, Lndn: Butterwrths Lndn, L and Shah, SRK (1983) sts f rreversibilities in Heat Exchanger Design, Heat Transfer Engineering, Vl 4, N, pp Matl, J (1984) Mdern st Engineering: Methds and Data, Vl, New Yrk: McGraw-Hill Mran, MJ (198) vailabilit nalsis: Guide t Efficient Energ Use, New Yrk: Prentice-Hall Tribus, M and Evans, R (196) Therecnics, UL Reprt N 6-63 Tsatsarnis, G, and Winhld, M 1985, Exergecnics nalsis and Evaluatin f Energ nversin Plant New General Methdlg, Energ Vl 10, pp Witte, L and Shasundar, N (1983) Therdnaic Efficienc ncept fr Heat Exchange Devices, se Jurnal Of Enginering Fr Pwer, Vl 105, pp