Mathematical models of air-cooled condensers for thermoelectric units

Transcription

1 Energy and Sustanablty II 399 Mathematcal models of ar-cooled condensers for thermoelectrc unts S. Bracco 1, O. Calgars 2 & A. Trucco 1 1 Department of Machnery, Energy Systems and Transportaton, Genoa Unversty, Italy 2 Department of Producton Engneerng, Thermoenergetcs and Mathematcal Models, Genoa Unversty, Italy Abstract The present paper deals wth the use of ar-cooled condensers n thermoelectrc unts, such as steam power plants or combned cycle unts. The paper descrbes the man features of ar-cooled condensers, showng the problems related to ther use, such as the presence of non-condensable gases nsde the steam flow or the condenser performances varablty due to ambent condtons. The present study refers to A-frame ar-cooled condensers characterzed by sngle-pass and multple-row arrangement. A phenomenologcal study has been done n order to calculate the man physcal parameters whch descrbe the condenser operatng condtons. In partcular, three mathematcal models have been developed: whle both the frst and the second model consder a condenser wth tubes characterzed by the same fnned surface, wth or wthout the use of throttlng valves upstream of each tube, the thrd model examnes the behavour of a condenser wth dfferent row effectveness. The three mathematcal models permt one also to nvestgate the possble vapour back flow nsde each row of tubes, by calculatng the axal coordnate value along each tube at whch the condensng steam mass flow rate becomes null; the thrd mathematcal model has been mplemented n the Matlab/Smulnk envronment n order to couple the smulator of the condenser wth the whole plant smulaton model. Keywords: ar-cooled condenser, non-condensable gases, smulaton model. 1 Introducton Nowadays ar-cooled condensers are more and more nstalled n thermoelectrc unts, takng the place of water cooled condensers, because ther use allows to WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne) do: /esu090351

2 400 Energy and Sustanablty II save a lot of coolng water, n accordance wth strct envronmental rules, and to buld power plants n stes far from rvers or the sea [1]. But, as reported by Fabbr [2 4], Kröger [5] and Larnoff et al [6], ar-cooled condensers are also characterzed by several techncal and practcal problems: the heat exchange process between the ambent ar and the condensng steam needs large heat exchange areas whch have to be cleaned perodcally; the condenser operatng pressure depends on the ste condtons such as ambent temperature, humdty and wndness; the large number of fans are characterzed by hgh electrcty consumptons and nose. Furthermore one of the major problems of a sngle-pass ar-cooled condenser s the accumulaton of non-condensable gases nsde the fnned tubes, as analysed by Fabbr [2], Kröger [5] and Berg and Berg [7]; ths undesred phenomenon determnes the reducton of the effectve heat exchange area and possble condensate freezng n several fnned tubes. The am of ths paper s that of descrbng three dfferent smplfed mathematcal models whch have been mplemented n order to predct the steady-state behavour of an A-frame ar-cooled condenser, consderng dfferent desgn data and operatng condtons. Both the frst and the second model have been mplemented usng the Fortran code and are used to study the condenser from a phenomenologcal pont vew whle the thrd smulator, tuned-up n the Matlab/Smulnk envronment, has been created and valdated takng nto account the desgn and operatng data of a condenser nstalled n a 400 MW combned cycle power plant stll operatve n the Italan electrcty market. 2 The ar-cooled condenser The ar-cooled condenser s a steam-ar heat exchanger composed by several modules such as that sketched by fg. 1. Fgure 1: The cross-secton of a sngle module of the ar-cooled condenser. WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

3 Energy and Sustanablty II 401 The exhaust steam, leavng the low pressure turbne, comes up to the horzontal man duct and then enters the parallel-flow modules flowng down nsde banks of fnned tubes where ts condensaton occurs owng to the heat exchange wth the ambent ar moved by axal fans; the condensate accumulates nto two outlet headers, n order to be wthdrawn, whle the remanng condensng steam goes to the dephlegmator characterzed by the steam and condensate n counterflow arrangement. The dephlegmator modules are also equpped wth steam jet ar ejectors for the removal of non-condensable gases. Each fan s usually moved by a two-speed motor and the fnned tubes are nstalled n staggered rows; tubes can be round, ellptcal or flat and several types of fns are on the market wth the am of optmsng the techncal performance of condensers subjected to dfferent operatng condtons. 3 The mathematcal models The developed mathematcal and smulaton models are based on the NTU effectveness method whch permts to determne the condenser operatng parameters begnnng from the calculaton of the rows effectveness and the condenser effectveness, as suggested by Fabbr [2]: T T Erow = +1 Tar _ out Tar _ n _, Econdenser = (1) Ts T Ts Tar _ n where T s the ar temperature upstream the th row whle T s s the condensaton temperature. Furthermore the system thermal balance equaton s: M ar c p ( Tar out Tar n ) = M s r xs (2) where ar M ar and M s denote the ar and steam flow rates, c p ar s the ar average specfc heat at constant pressure, r the condensaton latent heat and x s the steam qualty at the condenser nlet. In accordance wth the model proposed by Fabbr [2], the steam condensaton rate per unt of tube length s assumed constant along each tube: M ar c p ( T T ) ar +1 m s = (3) r xs L where L denotes the tubes length. The frst two models analyse the steady-state behavour of a sngle module of the condenser, coupled wth the correspondng fan, and characterzed by four rows of fnned tubes whle the thrd one refers to a condenser composed of 21 modules (18 parallel-flow and 3 dephlegmator) wth three tubes rows. The frst step of the analyss has been the desgn calculaton of the condenser man operatng parameters: the steam mass flow rate enterng each row, the pressure drop nsde the tubes, the ar ntermedate temperatures T, the rows effectveness and the axal coordnate value a along each tube at whch the condensng steam mass flow rate becomes null and the pressure reaches the mnmum value. All models assume the pressure drop per unt of tube length proportonal to the square of the current steam flow rate and consder the same pressure drop WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

4 402 Energy and Sustanablty II between the nlet and the outlet of each fnned tube, the man duct and the lower headers beng n common. 3.1 The frst model The man nputs of the frst smulator are: the ar nlet temperature, the exhaust steam flow rate at the turbne outlet, the pressure n the man duct and the geometrcal data of the condenser. The model consders four tubes rows havng the same effectveness. For ths case, n absence of non-condensable gases, the frst rows, whch are those n contact wth the coldest ar, have pressure mnmum values nsde the tubes nstead of at ther outlet (a < L); as a consequence, there s vapour back flow from the other rows to the frst ones. If the presence of non-condensable gases s consdered, these gases accumulate along the frst rows from the a axal coordnate to the tubes outlet and determne the reducton of the heat exchange effectveness. The followng consderatons refers to the smulaton of a parallel-flow module belongng to a condenser charactersed by the next desgn data: - Total thermal flux = MW th ; - Total steam mass flow rate = kg/s; - Nomnal condensaton temperature = C; - Steam qualty at the condenser nlet = ; - Ambent temperature = 15 C, ambent pressure = kpa; - 20 modules each composed of 6 bundles; - 4 rows per bundle; - Condensng steam mass flow rate per bundle = 0.81 kg/s. Fgure 2: Condensng steam flow rates (E 1 =E 2 =E 3 =E 4 ). Assumng that each row has an effectveness equal to 35%, the followng ar ntermedate temperatures have been calculated by means of eqn. (1): T 2 = 24.2 C, T 3 = C, T 4 = C, T 5 = C. The condensng steam mass flow rate and pressure along each tube row are plotted by fg. 2 and fg. 3, consderng the presence of non-condensable gases nsde the flow. It s possble to notce WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

5 Energy and Sustanablty II 403 that the steam condenses completely nsde both row 1 and row 2, whch operate n better condtons n terms of ar lower temperatures and hgher steam mass flow rates; the condensng steam flow rate becomes null at the end of the 3 rd row whle the steam whch enters the 4 th row does not condensate completely nsde the tubes and goes to the dephlegmator unt. If non-condensable gases had not been consdered, the steam extng the 4 th row would have gone back nto the tubes of the frst two rows. So t s clear that the vapour back-flow does not occur nsde the tubes where non-condensable gases accumulate. Fgure 3: The pressure along each tube (E 1 =E 2 =E 3 =E 4 ). Beng the operatng condtons depcted by fg. 2 and fg. 3 not optmal, n terms of heat exchange effectveness and steam flow rate dstrbuton between the tubes rows, t has been necessary to fnd better desgn data for the condenser n order to optmse ts techncal performance; t would be better to have the same flow rate at each tube nlet, the same ar temperature ncrease through each tubes row and the steam complete condensaton at the end of tubes (a L), n order to avod any vapour back flow and non-condensable gases accumulaton. 3.2 The second model Dfferent solutons have been adopted n order to optmse the condenser operatng condtons. The frst s relatve to a condenser wth tubes characterzed by the same fnned surface and conssts n nstallng a gauged throttle upstream of each tube wth the am of completng the steam condensaton not before the tube end. The condensng steam mass flow rate and pressure along each tube row are plotted by fg. 4 and fg. 5, consderng gauged pressure drops upstream of each tube and row effectveness equal to 35%. The results show that the steam flow rate at the tubes nlet s not the same for all rows and t s hgher for row 1 n contact wth cold ar; as a consequence, ths soluton does not seem to be the most adequate. It appears clear that the best WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

6 404 Energy and Sustanablty II Fgure 4: Condensng steam flow rates (E 1 =E 2 =E 3 =E 4, gauged pressure drops). Fgure 5: The pressure along each tube (E 1 =E 2 =E 3 =E 4, gauged pressure drops). soluton s that of desgnng condensers wth tubes rows characterzed by dfferent effectveness values, as nowadays proposed by manufacturers. 3.3 The rows effectveness optmzaton From the consderatons drawn n 3.1 and 3.2 t derves that a well-desgned mult-row ar-cooled condenser needs dfferently fnned tube rows; n partcular, t s necessary to ncrease the fn surface of the more external rows whch are those exposed to a hotter ar flow. The optmzaton of the rows effectveness for the condenser descrbed n 3.1 has determned the followng best values: E 1 = 20.7%, E 2 = 26.2 %, E 3 = 35.5 %, E 4 = 55.0%. Consderng these nomnal values, the condensng steam mass flow rate and pressure along each tube become the same for all rows, as shown by fg. 6. Furthermore, there s not vapour back flow and the ar temperature ncrease s the same through each row. WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

7 Energy and Sustanablty II 405 Fgure 6: The pressure and steam flow rate along each tube (E 1 E 2 E 3 E 4 ). 3.4 The thrd model The thrd smulaton model, mplemented n the Matlab/Smulnk envronment, s called operatve because t s gong to be lnked to the dynamc smulator of the whole combned cycle power plant; t s used to calculate off-lne the condenser performance parameters n off-desgn operatng condtons. In ths case the steam flow rate s an mportant datum whle the pressure downstream the turbne s unknown. In the followng the man equatons used to smulate a sngle module of the condenser are reported. The man nputs of the smulator are: - the ambent temperature (T 1 ) and pressure (p 1 ); - the exhaust steam mass flow rate and qualty at the condenser nlet; - the condenser desgn operatng parameters: condensaton temperature (T s-nom = C), ambent temperature and pressure (T 1-nom = 15 C, p 1-nom = kpa), exhaust steam mass flow rate and qualty ( m s nom = 5.52 kg/s, x s-nom = ), ar volume flow rate and densty ( Q ar nom = 581 m 3 /s, ρ ar-nom = kg/m 3 ), ar ntermedate temperatures (T 2-nom = C, T 3-nom = C, T 4-nom = C), rows effectveness (E 1-nom = 25.07%, E 2-nom = 33.45%, E 3-nom = 50.27%), condenser global effectveness (E condenser = 75.2%); - the condenser geometrcal data. The heat balance equaton for the th row can be wrtten as: M ar c p ( T + 1 T1 ) = U S T ar lm _ + 1 (4) where U s the global transmttance, S the heat exchange area and T lm the logmean temperature dfference. Snce the NTU (Number of Transfer Unts) corresponds to the rato [8]: U S NTU =, (5) M c ar p ar WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

8 406 Energy and Sustanablty II after a lttle algebra one gets: NTU E row _ =1 e. (6) Consderng constant the average values of U, S and c p ar, t derves that the product ( M ar NTU ) must be kept constant and so the effectveness values can be calculated from the current ar mass flow rate, whch depends on the fan rotatonal speed, as follows: M ar nom NTU nom M ar E row _ =1 e. (7) The next step s the calculaton of the condensaton temperature T s : t s determned by eqn. (2), after havng expressed the condensaton latent heat as a polynomal functon of T s and the ar outlet temperature T 4 as a functon of E 1, E 2, E 3, T 1 and T s. The T s calculated value permts to determne the condensaton pressure, by means of the water-steam propertes, and the ar ntermedate temperatures T 2, T 3 and T 4, by means of eqn. (1). As a consequence the steam condensaton rates per unt of tube length can be determned by eqn. (3). As proposed by Fabbr [2], t s necessary to dstngush the tubes characterzed by vapour back flow (bf) from those where the steam does not condense completely (nbf). In the frst case, the steam mass flow rate at the axal coordnate x along the th tube s calculated by: ( ) = m ( a x) M (8) s bf where a s the length of the tube segment n whch the steam flows from the man duct to the lower header. The steam flow rate crossng the outlet end of each tube s equal to: Π s = m s ( L a ). (9) On the other hand the steam flow rate at the axal coordnate x along the tubes wthout back flow s calculated by: M = m L x + Π. (10) s ( s ) ( ) s s nbf The calculaton of the pressure drop between the nlet and the outlet of each tube has been done referrng to the model reported by Fabbr [2]. In partcular, for the tubes wth back flow the followng equaton has been adopted: 2 k m s p = 2a 3a L + 3a L L (11) ( ) ( ) bf 3 k denotng a frcton coeffcent whch depends on the tubes materal. On the other hand, the tubes wthout back flow are characterzed by a pressure drop equal to: 2 k m s 3 2 ( p ) = ( L 3a L + 3a L) nbf. (12) 3 Snce all tubes rows have the nlet and outlet plena n common, the pressure drops have to be the same: p = p. (13) ( ) bf ( ) nbf WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

9 Energy and Sustanablty II 407 In order to calculate pressure drops wthn the Matlab/Smulnk model effcently, eqn. (12) and (13) have been mplemented n non-dmensonal form, as done by Fabbr [2] by proper choce of reference varables, as a functon of the steam flow rates defned by eqn. (8) and (10). Assumng the steam total flow rate at the ext of tubes wthout back flow equal to the one flowng back to the other tubes, that s equvalent to not consderng a dephlegmator module downstream, the model calculates the pressure drop between the nlet and outlet plena. Once ths value has been determned, t s possble to solve eqn. (11) and (12) n order to calculate the axal coordnate a along each tube at whch the condensng steam mass flow rate becomes null. As a consequence, the steam flow rate at each row nlet can be evaluated by the followng equaton: M s = m s a. (14) In order to consder the presence of a dephlegmator unt downstream of the parallel-flow module, t s necessary to modfy the tubes length as follows: * 1 L = α L, α < 1 (15) where the coeffcent α s connected wth the steam flow rate to the dephlegmator: M s ( ) dph M _ 1 α s. (16) Consequently, the steam total flow rate at the parallel-flow module nlet s equal to: M s = m s a = M s dph + M _ s _ c (17) where: * s M _ c = α L m s (18) s the steam flow rate whch condenses nsde the parallel-flow module. The rows length ncrease s equvalent to mpose the steam complete condensaton nsde the system composed of one parallel-flow module and one dephlegmator unt connected n seres. 3.5 The condenser performance curves The am of the present analyss has been that of determnng, by the thrd smulator, the condenser performance curves whch show, on the pressure steam flow rate plane, how the condensaton pressure p s s nfluenced by both the ar nlet temperature T 1 and the fans rotatonal speed. Fgure 7 reports the performance curves consderng all fans runnng at full speed. It s clear that, for the same value of steam flow rate, the hgher s T 1 the hgher s the condenser pressure; on the other hand, consderng the same temperature T 1, the lower s the steam flow rate the lower s the condenser pressure. The curves n fg. 7 are not parallel lnes but ther slope ncreases wth the ar nlet temperature. Snce ths type of ar-cooled condensers are equpped wth fans runnng at two dfferent speed values, respectvely 100% and 50% of the nomnal rotatonal WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

10 408 Energy and Sustanablty II speed, t has been nterestng to calculate the performance curves also for the case of all fans at half speed, that s an operatve condton to whch the condenser s subjected both durng startup and cold days. The results are plotted n fg. 8: the curves have the same shape as those depcted by fg. 7 but are characterzed by a hgher slope. In fact n ths case, consderng the same values of ar nlet temperature T 1 and steam flow rate, the ar flow rate s lower and so the ar temperature T 4 at the condenser outlet s hgher; as a consequence, the condensaton temperature ncreases as well as the pressure. Fgure 7: The condenser performance curves wth all fans at full speed. Fgure 8: The condenser performance curves wth all fans at half speed. WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

11 Energy and Sustanablty II 409 The curves obtaned by the thrd smulaton model have been compared wth those suppled by Ansaldo Energa SpA, an Italan company whch nstalls arcooled condensers n ts power plants stes. The model valdaton has been done wth reference to dfferent steady-state operatng condtons and t s mportant to remark that the calculated performance curves are n good agreement wth manufacturers ones. 4 Conclusons The mathematcal and smulatons model descrbed n ths paper have been developed to study a mult-row A-frame ar-cooled condenser for combned cycle power plants. The analyss has been focused on the smulaton of the condenser behavour under dfferent steady-state operatng condtons wth the am of evaluatng the nfluence of the ar nlet temperature on the system performance. The problem of the non-condensable gases accumulaton nsde the fnned tubes rows has been analysed from a phenomenologcal pont of vew and some techncal solutons have been proposed n order to reduce the rsk of ther formaton, as well as the vapour back flow nsde the tubes. The condenser performance curves have been determned consderng two dfferent fans operatng condtons, full or half speed, and varyng both the ar nlet temperature and the saturated exhaust steam flow rate. The calculated curves valdaton phase has confrmed the model valdty over a wde range of operatng condtons. The next step of the study s gong to be the development of a dynamc smulaton model to study the condenser behavour under dfferent transent condtons; ths model wll be ntegrated wth the combned cycle power plant dynamc smulator and valdated wth expermental data comng from the plant. References [1] Nagel, P. & Wurtz, W., Dry coolng for power plants: an nnovatve modularzaton concept. PowerGen Europe SPX Conference, Cologne, [2] Fabbr, G., Analyss of vapor back flow n sngle-pass ar-cooled condensers. Int. J. of Heat and Mass Transfer, Vol. 40, No. 16, pp , [3] Fabbr, G., Effect of dsunformtes n vapour saturaton pressure and coolant velocty on vapour back flow phenomena n sngle-pass ar-cooled condensers. Int. J. of Heat and Mass Transfer, Vol. 43, pp , [4] Fabbr, G., Analyss of the noncondensable contamnant accumulaton n sngle-pass ar-cooled condensers. Heat Transfer Engneerng, Vol. 18, No. 2, pp , [5] Kröger, D.G., Ar-cooled Heat Exchangers and Coolng Towers, Penn Well Corporaton, [6] Larnoff, M.W., Moles, W.E. & Rechhelm, R., Desgn and specfcaton of ar-cooled steam condensers. Texas Chemcal Engneerng, WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)

12 410 Energy and Sustanablty II [7] Berg, W.F. & Berg, J.L., Flow patterns for sothermal condensaton n one pass ar cooled heat exchanger. Heat Transfer Engneerng, Vol. 1, No. 4, pp , [8] Rcard, J., Equpement Thermque des Usnes Génératrces d Energe Electrque, Dunod, Pars, WIT Transactons on Ecology and the Envronment, Vol 121, 2009 WIT Press ISSN (on-lne)