NUCLEAR THERMAL-HYDRAULIC FUNDAMENTALS

Transcription

1 NUCLEAR THERMAL-HYDRAULIC FUNDAMENTALS Dr. J. Micael Doster Departent of Nuclear Engineering Nort Carolina State University Raleig, NC Copyrigted

2 POER CYCLES Te analysis of Terodynaic Cycles is based alost exclusively on applications of te first and second law of terodynaics to a control volue, along wit ass conservation principles. Te derivation of tese conservation equations is covered in ost eleentary terodynaic texts and as suc te equations are siply repeated ere witout proof. Generally, tese analyses provide global ass and energy balances wit no indication of coponent size. Te inerent assuption, is tat te coponent or device can be constructed to provide te required perforance. First Law of Terodynaics Te First Law is an energy conservation equation applied to a control volue. A coon for of te First Law is v g..... g g z de in c v vexit g c v + in in + + in exit exit dt g g z exit + in c c exit c c c. v. () were: c. v. Heat transfer rate into (+) or out of (-) te control volue in exit in exit in vin g + + g g z in Rate of energy convected into te control volue exit c c vexit g + + g g z c c exit Rate of energy convected out of te control volue dec. v. Tie rate of cange of energy witin te control volue dt c. v. Rate of work done on (-) or by (+) te control volue. Note, te sign convention for work and eat transfer rate is soewat arbitrary and oter sign conventions are equally valid. Most cycle analyses are perfored at steady state, suc tat dec. v. 0. dt In addition, for ost applications involving te analysis of power cycles, te kinetic and potential energy ters are negligible suc tat an effective working equation is () + + c. v. in in exit exit c. v. in exit

3 Mass Conservation Conservation of ass for a control volue is given by exit () in exit dmc. v. in dt were M c. v. is te ass of control volue. Under steady state conditions dmc. v. 0 dt and in exit () in exit or te su of te ass flow rates entering te control volue ust equal te su of te ass flow rates leaving te control volue. Second Law of Terodynaics for a Control Volue Te Second Law of Terodynaics for a control volue is given by ds dt c v / A + exit sexit insin + da (5) T c. v.... exit were s is entropy and A is te surface area of te control volue. Te equality () refers to internally reversible processes, and te greater tan sign (>) refers to internally irreversible processes. For steady state, reversible, adiabatic systes, Equation 5 reduces to in exit exit exit in wic for single-inlet/single-outlet systes can be siplified to give A s s (6) in in s exit s (7) in or s exit s (8) in A constant entropy process is said to be isentropic. Entropy is a fundaental terodynaic property of te fluid and can be related to entalpy troug te expression Tds d υ dp (9)

4 wereυ is specific volue and P is pressure. If we integrate between an arbitrary inlet and exit point along te flow strea, exit exit exit inlet inlet inlet and assue a reversible, adiabatic (isentropic) process, i.e. ten Tds d υ dp (0) exit ds 0 Tds 0 inlet exit exit d υ dp () inlet inlet or υ dp () exit inlet If we furter assue te fluid density (or specific volue) is independent of pressure (incopressible), ten υ is a constant and exit inlet υ( P P ) () exit inlet exit inlet Note, tis expression is only valid for reversible, adiabatic processes involving fluids wic can be approxiated as incopressible. Pysically, tis situation olds for te ideal puping of liquids.

5 Carnot Cycle Te Carnot Cycle is te ost efficient cycle possible. It operates between a ig teperature source and a low teperature sink and ay be caracterized by four basic processes: ) A reversible-isoteral process in wic eat is transferred to or fro a ig teperature reservoir. ) A reversible-adiabatic process in wic te teperature of te working fluid decreases fro te ig teperature to te low teperature. ) A reversible-isoteral process in wic eat is transferred to or fro te low teperature reservoir. ) A reversible-adiabatic process in wic te teperature of te working fluid increases fro te low teperature to te ig teperature. Te efficiency of te Carnot Cycle is given by net l Tl η () T were te teperatures are in absolute degrees. If we were to try and operate a nuclear power plant on tis cycle, it could be approxiated by te diagra in Figure. Hig Teperature (Reactor). Stea Generator. p Pup Turbine. t Condenser. l Low Teperature (Cooling Pond) Figure : Carnot Cycle Representation of a Pressurized ater Reactor e can represent te Carnot Cycle on a T-S diagra, were te saturated liquid and vapor lines for water ave been added for illustration purposes.

6 T Saturated Liquid Saturated Vapor S Figure : T-S Diagra for te Carnot Cycle Note: Te orizontal distance fro point () to te saturated vapor line is a easure of te oisture contained witin te stea at te turbine exaust. Elevated oisture content results in increased erosion of turbine blades. Terefore, to reduce turbine wear it is desirable to aintain ig stea quality. Te cycle illustrated in Figure iplies a proibitive oisture content at te turbine exaust. To alleviate tis proble requires tat we eiter raise te condenser pressure (wic would raise T l and terefore lower te cycle efficiency), lower te stea generator pressure (wic would lower T and terefore lower te cycle efficiency) and/or supereat te stea beyond te noinal state point at (). Te addition of supereat iplies non isoteral eat addition and a deviation fro te Carnot Cycle. In addition, te Carnot Cycle illustrated above would require tat a two-pase ixture of liquid and vapor be puped fro state points () to (). Designing and building pups, wic can operate under tese conditions is usually ipractical, and as a result ost pups are designed to operate wit liquid water. Tis requires te stea to be fully condensed at (). Tese deviations fro te Carnot Cycle iply a new cycle for a practical stea power plant. 5

7 Rankine Cycle Te Rankine Cycle is te ideal cycle for a siple stea power plant. Given te power plant diagraed below, te processes involved are: - Constant pressure eat transfer in te boiler - Reversible-adiabatic expansion in te turbine - Constant pressure eat transfer in te condenser - Reversible-adiabatic puping to te boiler pressure Te line fro to allows for te option of supereating te stea in te boiler. Te efficiency of te Rankine Cycle is ost easily deterined fro te expression net η (5) were is te eat addition in te boiler and net is te net rate of work done by all coponents in te syste. Since te Rankine Cycle does not require constant teperature eat addition or rejection, in analyzing te Rankine Cycle, it is elpful to tink of efficiency as depending on te average teperature at wic eat is supplied and rejected. Any canges tat increase te average teperature at wic eat is added, or decrease te average teperature at wic eat is rejected, will increase te Rankine Cycle efficiency. As bot of tese teperatures are doinated by te saturation teperatures (and terefore saturation pressures) in te boiler and condenser, te sae efficiency arguents are true for increasing te boiler pressure and decreasing te condenser pressure. Boiler Turbine T ' ' Condenser S Figure : Ideal Rankine Cycle 6

8 Exaple: Deterine te efficiency of a Rankine Cycle utilizing stea as te working fluid operating under te following conditions Boiler Pressure 900 psia Boiler Supereat 5 F Condenser Pressure psia Since te only work done in tis cycle is by te turbine and condensate pup, te efficiency is given by η t + net p were we ave noralized te work and eat transfer rates by te total syste ass flow rate. Turbine Boiler Condenser Turbine Fro te stea tables: 0.79 Btu/lb T F s.7 Btu/R-lb In te absence of kinetic and potential energy ters, te First Law applied to te turbine is + t suc tat te turbine work per unit ass flow rate is given by t 7

9 Te entalpy at te turbine inlet is given by te boiler exit conditions. Te entalpy at te turbine exit is unknown. Since te condenser pressure alone is insufficient to specify te fluid s state at (), additional inforation is required. Tis if obtained fro te Second Law, wic for a reversible-adiabatic turbine requires s s s.7 Btu/R-lb To deterine te turbine exit conditions given te turbine exaust pressure and entropy, note: ( f fg ) s s + xs s s x s fg f P psia At psia, s f.6 Btu/R-lb and s fg.855 Btu/R-lb giving x Te entalpy at te turbine exaust is ten given by ( f fg ) + x For f 69.7 Btu/lb and fg 06. Btu/lb at psia (0.7069)(06.) 80. Btu/lb. Te turbine work per unit ass flow rate is ten Pup t Btu/lb In te absence of kinetic and potential energy ters, te First Law applied to te condensate pup gives + p suc tat te pup work per unit ass flow rate is - / - p For reversible-adiabatic processes were te density of te working fluid is approxiately incopressible + υ ( P - P ) suc tat - / ( P P ) p υ 8

10 For tis exaple: P psia P 900 psia υ 0.06 ft /lb (evaluated at te pup inlet conditions) Note: υ( Pe - Pi ) ( 0. 06)( )( ) 08. To resolve te unit inconsistency, divide by 778 ft-lbf/btu ft - lbf lb - p/ / Btu/lb For te ideal Rankine Cycle, te liquid leaving te condenser is saturated at te turbine exaust pressure, suc tat Boiler Te First Law applied to te boiler gives Cycle Efficiency f at psia 69.7 Btu/lb Btu/lb. + / / Btu/lb η net t + p % 58.8 Note, te cycle efficiency is independent of te agnitude of te ass flow rate. ile not required for te efficiency calculation, it is also of interest to exaine te eat transfer rate across te condenser Condenser cond / Btu/lb Te (-) sign indicates eat transfer out of te control volue (cooling tower, cooling pond, etc.). Of te 58.8 Btu/lb added in te boiler, only 8.69 Btu/lb, or a little over / was extracted as work in te turbine. Te reainder is duped across te condenser and discarged to te environent. It is also of interest to copare te pup work to te turbine work. For Rankine type stea cycles, te pup work is generally uc less tan te turbine work, suc tat it is a reasonable approxiation to relate te eat input in te boiler to te turbine (or generator) output by η. t 9

11 Effect of Pressure and Teperature on te Rankine Cycle ) Turbine Exaust (Condenser) Pressure and Teperature Decreasing te turbine exaust pressure decreases te saturation teperature in te condenser and terefore te average teperature at wic eat is rejected. Tis iplies an increase in te cycle efficiency. However, exaination of te T-S diagra for te Rankine Cycle indicates, tat for a given boiler pressure, lowering te condenser pressure results in an increase in te oisture content at te turbine exaust and enanced erosion of te turbine blades. ) Boiler Pressure Increasing boiler pressure results in an increased saturation teperature in te boiler and terefore an increase in te teperature at wic eat is added. Tis iplies an increase in cycle efficiency. However, as in decreasing te condenser pressure, increasing boiler pressure for a given condenser pressure results in an increase in oisture content at te turbine exaust. ) Boiler Supereat Supereating in te boiler increases te average teperature at wic eat is added and terefore would be expected to increase te cycle efficiency. Exaination of te T-S diagra for te Rankine Cycle sows tat supereating also results in a decrease in oisture content at te turbine exaust. Figures and 5 below illustrate te effect of boiler pressure on a siple Rankine Cycle in te absence of supereat. Clearly, te oisture content at te turbine exaust quickly becoes excessive. Figure 6 gives te teperature te stea ust be supereated to at te corresponding boiler pressure to aintain a oisture content of 0% Cycle Efficiency Boiler Pressure (psia) Figure : Cycle Efficiency Versus Boiler Pressure 0

12 0 8 6 Moisture Content (%) Boiler Pressure (psia) Figure 5: Moisture Content at te Turbine Exaust Versus Boiler Pressure Supereat Teperature (F) Boiler Pressure (psia) Figure 6: Stea Teperature Necessary to Maintain 0% Moisture Content

13 Reeat Cycle It as been sown, tat increasing te boiler pressure leads to increases in cycle efficiency, but at te expense of a iger oisture content at te turbine exaust. It as also been sown tat supereating is effective in reducing te oisture content at iger boiler pressures, but can lead to excessive stea teperatures. Te Reeat Cycle is a way to take advantage of te iger boiler pressures wile aintaining ig stea quality at te turbine exaust and oderate stea teperatures. Classically, tis is accoplised by expanding te stea to soe interediate pressure in te turbine and ten reeating it in te boiler before expanding troug te final turbine stages to te exaust pressure. Tis cycle is diagraed in Figure 7 below. Te efficiency of te Reeat Cycle is in general only sligtly (toug not insignificantly) iger tan te corresponding Rankine Cycle. Te cief advantage is te decreased oisture content at te turbine exaust. If te coponent aterials could witstand supereating to point 'on te T-S diagra, te Rankine Cycle would ave te sae turbine exaust quality and a iger overall cycle efficiency and te Reeat Cycle would not be necessary. Boiler HP LP Turbine 5 T ' ' Condenser 6 Pup 6 5 S Figure 7: Classic Reeat Cycle Exaple: Boiler Pressure 900 psia Boiler Supereat 5 F Reeat Pressure 00 psia Condenser Pressure psia Deterine te cycle efficiency. Hig Pressure Turbine 0.79 Btu/lb T F s.7 Btu/R-lb

14 For a reversible-adiabatic turbine s s s.7 Btu/R-lb s s s ( s f + xs fg ) x s fg f P At 00 psia, s f 0.58 Btu/R-lb and s fg.006 Btu/R-lb giving x ( f fg ) + x At 00 psia, f 55. Btu/lb and fg 8.8 Btu/lb giving (0.899)(8.8) 07.9 Btu/lb. Te turbine work per unit ass flow rate in te ig pressure stage of te turbine is ten Low Pressure Turbine For te ideal reeat cycle, it is norally assued tat T 05.5 Btu/lb T F s.660 Btu/R-lb (note, s s ) p Btu/lb T. s 5 s.660 Btu/R-lb s5 s s5 ( s f + xs fg ) x 5 5 s fg f P 5 At psia, s f 0.6 Btu/R-lb and s fg.855 Btu/R-lb giving x ( f fg ) + x For f 69.7 Btu/lb and fg 06. Btu/lb at psia (0.879)(06.) 97.5 Btu/lb. Te turbine work per unit ass flow rate in te low pressure stage of te turbine is ten lp Btu/lb 5 5

15 Pup Te pup work is deterined fro For a reversible-adiabatic pup - / - p 6 + υ ( P - P ) e i e i P i psia P e 900 psia υ 0.06 ft /lb υ( Pe - Pi ) ( 0. 06)( )( / 778).67 - p /. 67 Btu/lb Btu/lb. Boiler Te eat input fro te boiler ust now include te reeat stage suc tat / Btu/lb Cycle Efficiency η net p + lp + p % 56. An alternative to reeating in te boiler coonly eployed in nuclear power systes, is to extract soe sall aount of stea prior to entering te ig pressure turbine, and use tis stea in an external reeater prior to entering te low pressure turbine as illustrated below. Note, tat te axiu reeat teperature in tis design is te stea teperature leaving te boiler. It can be sown, tat if ipleented properly, tis cycle as te sae efficiency as tat of te classic reeat cycle. Reeater - H.P. Turbine L.P. Turbine Boiler - Figure 8: Alternate Reeat Cycle

16 Consider te Rankine Cycle witout supereat. Regenerative Cycle ' T S Figure 9: T-S Diagra for a Rankine Cycle itout Supereat Between and ', te working fluid is being eated to te saturation point. Te corresponding average teperature of te fluid is uc lower tan in te vaporization process fro 'to. Obviously, if te working fluid could enter te boiler closer to te saturation point, i.e. between and ', wile aintaining te sae condenser pressure te cycle efficiency would be iproved. Te Regenerative Cycle accoplises tis by extracting a relatively sall aount of stea fro te turbine after it as partially expanded and using it to eat te feedwater in feedwater eaters. Te cycle diagra is illustrated below.. Turbine Boiler. - ṁ ṁ F.. Heater Condenser Figure 0: Siple Regenerative Cycle 5

17 Te pysical processes involved are: a) Stea enters te turbine at () and expands to soe interediate state designated as point (). b) At (), soe of te stea is extracted and enters te feedwater eater. c) Te reaining stea is expanded to te condenser pressure at () and condensed in te condenser. d) Te condensate is puped to te feedwater eater were it ixes wit te extraction stea fro te turbine. Te aount of stea extracted at () is just enoug to eat te entering condensate to te saturation point at (7). e) Te saturated liquid leaving te feedwater eater is puped to te boiler pressure at (). Te associated T-S diagra is given below. T S Figure : T-S Diagra for a Siple Regenerative Cycle Exaple: Boiler Pressure 900 psia Boiler Supereat 5 F Tap Pressure 00 psia Condenser Pressure psia Deterine te cycle efficiency. Hig Pressure Turbine 0.79 Btu/lb T F s.7 Btu/R-lb For a reversible-adiabatic turbine s s s.7 Btu/R-lb 6

18 s s s ( s f + xs fg ) x s fg f P At 00 psia, s f 0.58 Btu/R-lb and s fg.006 Btu/R-lb giving x ( f fg ) + x At 00 psia, f 55. Btu/lb and fg 8.8 Btu/lb giving (0.899)(8.8) 07.9 Btu/lb. s s s ( s f + xs fg ) x s fg f P At psia, s f 0.6 Btu/R-lb and s fg.855 Btu/R-lb giving x ( f fg ) + x For f 69.7 Btu/lb and fg 06. Btu/lb at psia (0.7069)(06.) 80. Btu/lb. Condensate Pup P 5 psia P 6 00 psia υ 0.06 ft /lb + υ( P - P ) ( 0. 06)( 00 - )( / 778) Btu lb Feedwater Heater Apply te First Law to te feedwater eater. + ( ) 6 7 Neiter ass flow rate is known, owever as we ave seen in previous exaples, efficiency is independent of te agnitude of te ass flow rate. e can ten solve for te relative ass flow rate 7

19 7 6 6 Note: 7 f at 00 psia 55.5 Btu/lb Feed Pup Boiler P 900 psia P 7 00 psia υ 0.08 ft /lb Cycle Efficiency + υ( P - P ) ( 0. 08)( )( / 778) Btu lb / η Btu/lb net t + cp + fp Note: Since te ass flow rate is not unifor trougout te cycle, we ust account for tis is deterining te individual works and eat transfer rates. e can still noralize eac of tese ters by te total syste ass flow rate suc tat te efficiency is written η t / + cp / + net / fp / Turbine ork Pup ork ( ) + + t t ( ) ( ) ( 0. 75)( 709. ) ( 0. 75)(80. ). 79 Btu lb t ( )( ) cp 6 5 ( )( ) cp 6 5 ( 0. 75)(. 59) 0. Btu lb ( ) fp 7 8

20 ( ) fp 7. 8 Btu lb η % 87.9 Tis efficiency sould be copared to te 6.78 % efficiency of te Rankine operating between te sae boiler and condenser conditions. Te type of eater discussed in te previous exaple is called an open or deaerating feedwater eater as tere is no pysical separation of te inlet streas. Anoter coon type of feedwater eater separates te extraction stea fro te feedwater via tubes as illustrated below. Tis type of eater is referred to as a closed feedwater eater. Heat transfer is accoplised troug condensation on te tube walls. Te condensate ay be puped into te feedwater line or allowed to drain to a lower pressure eater or te condenser. Extraction Feedwater To low pressure eater Condensate Figure : Closed Feedwater Heater Open feedwater eaters ave te advantage of ceap cost and better eat transfer caracteristics tan closed eaters, owever tey require a pup for eac eater. Closed feedwater eaters can operate wit one pup per several eaters. Most power plants utilize several feedwater eaters, wit te final nuber liited by econoic considerations. Te efficiency of te Regenerative Cycle is a function of te turbine tap pressure at wic feedwater eating is to be perfored. In general, optiization tecniques are required to deterine te tap pressures and feed teperatures wic axiize te efficiency of any given cycle. However, a reasonable approxiation to te optiu configuration can be obtained wen te outlet teperature fro eac eater is equally spaced between te condenser teperature and te boiler saturation teperature. For a single eater ten, te optiu eater outlet teperature is idway between te boiler and condenser saturation teperatures. Tis is illustrated in Figure below for te Regenerative Cycle in te previous exaple. Cycle efficiency is given as a function of te turbine tap pressure wit te optiu approxiately 80 psia. Since te fluid leaves te eater as a saturated liquid, te corresponding feed teperature is ten approxiately F. For a boiler pressure of 900 psia (Tsat 5 F) and a condenser pressure of psia (Tsat 0 F), te optiu feed teperature sould be approxiately 7 F wic corresponds to a tap pressure of approxiately 85 psia. 9

21 Cycle Efficiency Turbine Tap Pressure (psia) Figure : Regenerative Cycle Efficiency Versus Tap Pressure Plants coonly incorporate a reeat stage wit oisture separation in addition to te feedwater eaters as illustrated in Figures and 5. Te oisture separator acts to reove water droplets fro te stea resulting in a iger quality at te separator exit and terefore ore efficient reeating as tis oisture does not ave to be reevaporated. Reeater - H.P. Turbine Moisture Separator L.P. Turbine Boiler - Figure : Moister Separator and Reeater An ideal separator would result in coplete reoval of all oisture, suc tat te stea exits te separator as a saturated vapor as illustrated below. Te extracted oisture can ten be diverted to soe oter point in te cycle for feed water eating. 0

22 ṁ x Moisture Separator g xṁ f (-x)ṁ Figure 5: Ideal Moisture Separator

23 Deviations of Actual Cycles Fro Ideal Cycles Actual cycle efficiencies differ fro tose coputed for ideal cycles due to irreversible losses in piping and cycle coponents. Exaples of losses wic ay affect te overall cycle efficiency include: ) Piping Losses Piping losses result priarily fro eat loss to te environent as well as pressure loss due to friction. ) Turbine Losses Actual turbines are not reversible-adiabatic acines. Te deviation of an actual turbine's perforance fro tat of an ideal turbine is given in ters of a turbine efficiency. Tis efficiency is defined suc tat η t ta ts were: ta Actual turbine work output ts Ideal (isentropic) turbine work output ) Puping losses As in turbines, actual pups are not reversible-adiabatic acines. Pup perforance is also caracterized by a pup efficiency defined suc tat η p ps pa were: pa Actual work input of te pup ps Ideal (isentropic) work input of te pup Note, in coputing pup work, te isentropic work appears in te nuerator. ) Condenser losses Condenser losses are usually inor. One suc loss is tat due to subcooling of te working fluid prior to return to te boiler.

24 Exaple: Consider te siple Rankine Cycle illustrate below wit te following state point conditions. For tis exaple, assue a turbine efficiency of 85% and a pup efficiency of 8%. Turbine Boiler Condenser 5 6 Location Pressure (psia) Teperature (F) Turbine 7 Btu/lb s.8 Btu/R-lb (a) Ideal Turbine ork s s.8 s s s ( s f + xs fg ) xs s fg f P At P psia s f 0. s fg.855 x s ( ) + x s f fg At P psia

25 f 69.7 fg 06. s (.700)(06.) Btu/lb ts - s Btu/lb (b) Actual ork Output η t ta ( a ) ts ts a ηt ts ta ts ηt ( 0. 85)( 8. ) Btu/lb a 7 ( 0. 85)(. 8) Btu/lb Pup (a) Ideal Pup ork w υ( P P ) ps 6 5 ps w ps ( 0. 06)( 970 )( 778). 89 Btu lb (b) Actual Pup ork w w pa ps p w Btu lb pa η Note: w 6a 5 6a pa Btu lb Boiler 58.0 Btu/lb 0.79 Btu/lb Btu lb Efficiency η net ta + pa % 7.77

26 Exaple: Saturated stea at 000 psia enters a ig pressure turbine wit an efficiency of 85%. Te stea expands to 00 psia were it enters a oisture separator and all entrained oisture is reoved. Saturated stea at 00 psia is ten expanded troug a low pressure turbine of 85% efficiency to te condenser at psia. Saturated liquid fro te oisture separator is sent to a closed feedwater eater were it is cooled to 0 F before being sent to te condenser. For a pup efficiency of 85% calculate te overall plant efficiency. SOLUTION oisture separator 5 turbine turbine 6 boiler condenser 7 reeater 8 9 Turbine 000 psia 9.9 Btu/lb s s 000 psia.9 Te entalpy at te turbine exaust and actual work done by te ig pressure turbine is given by η t s tpa tps suc tat η ( ) t s To deterine te quality and entalpy at te ig pressure turbine exaust x s s s s fg P 00 psia s f 0.7 s fg.8 5

27 f 98.5 Btu/lb fg Btu/lb x s P 00psia s f s fg s s ( 0. 8)( ) 00. η ( ) t s ( ) 06. w w w tpa tpa tpa Te true quality at te turbine exaust is ten x f fg Moisture Separator Assuing te oisture separator to be ideal ( x ) / x Low Pressure Turbine 5 P 00 psia s 5 s P 00 psia Btu/lb s Te entalpy at te turbine exaust and actual work done by te low pressure turbine is given by η t s tlpa tlps suc tat η ( ) 6 5 t 5 6s To deterine te entalpy at te low pressure turbine exaust 6

28 x 6s s5 s s fg P 6 psia s f 0.6 s fg.855 f 69.7 Btu/lb fg 06. Btu/lb x6 s P psia 6s f 6s fg 6 6s 6s ( )( 06. ) 895. η ( ) 6 5 t 5 6s ( ) w w w tlpa tlpa tlpa Condensate Pup w υ( P P ) cps w ( 0. 06)( 000 )( / 778) cps w. 98 cps 7 w w cpa cpa wcp η p s Btu / lb w cp a Reeater Application of te first law to te reeater gives Since te ass flow rate ratios are known, we solve for te boiler inlet entalpy 7

29 P 00 psia 98.5 Btu/lb Btu/lb ( ) + / ( ) ( )( ) 089. Boiler q q q Cycle Efficiency + + η tp ( + + ) / / tp tlp cp tp tlp cp w tpa / w tp tpa tp / 6. 6Btu / lb ( ) w tlp tlpa / ( / ) w tlp tlpa / ( )( 8. ) tlp tlp / 08.9 Btu / lb cp w cpa / w cp cpa cp / 5Btu. / lb q / q / Btu / lb η. 5%

30 Exaple: A stea power plant based on te regenerative cycle is illustrated below. Assuing te ig pressure turbine is 90 % efficient and te low pressure turbine is 85 % efficient: a) Deterine te cycle efficiency. You ay assue te condensate and low pressure feed pups to be 90 % efficient. b) Deterine te teperature cange across eac of te feed water eaters. c) Deterine te power output of te turbines. d) Deterine te efficiency of te ig pressure boiler feed pup. 6 Turbine Turbine 8 Boiler 5 7 Open Heater Condenser Point Pressure (psia) Teperature (F) Mass Flow Rate (lb/r) , , , , ,

31 SOLUTION Hig Pressure Turbine 8.75 Btu/lb s.585 Hig Pressure Tap At P 0 psia and an entropy of.585, te stea is supereated. Fro te supereat tables s 8.9 Btu/lb ηt s η ( ) t s (0.90)( ) 96.6 Btu/lb Interediate Pressure Tap At P 0 psia and an entropy of.585, te stea is supereated. Fro te supereat tables s 9.9 Btu/lb ηt s η ( ) t s (0.90)( ) 8.9 Btu/lb Low Pressure Tap x 5s s s s fg P psia 5 + P psia s f s fg s f 0.08 s fg.50 f 8. fg x s 5s 8. + (0.99)(95.) 7. Btu/lb ηt 5 5s η ( ) 5 t 5s 0

32 (0.9)( ) 8. Btu/lb Hig Pressure Turbine Exaust x 6s s6 s s fg P6 0 psia + P psia 6s f 6s fg 6 0 s f 0.58 s fg.96 f 96.7 Btu/lb fg 960. Btu/lb x s 6s (0.898)(960.) 055. Btu/lb ηt 6 6s η ( ) 6 t 6s (0.9)( ) 09.7 Btu/lb Low Pressure Turbine Btu/lb s 6 s f + x 6a s fg x 6a 6 f fg s (0.97)(.96).69 Low Pressure Tap x 7s s6 s s fg P7 psia + P psia 7s f 7s fg 7 s f s fg.97 f 65.5 Btu/lb fg Btu/lb

33 x s 7s (0.909)(979.6) 05.7 Btu/lb ηt s η ( ) 7 6 t 6 7s (0.85)( ) Btu/lb Low Pressure Turbine Exaust x 8s s6 s s fg P psia 8 + P psia s f s fg s f 0.6 s fg.895 f Btu/lb fg 0. Btu/lb x s 8s (0.8057)(0.) Btu/lb ηt s η ( ) 8 6 t 6 8s (0.85)( ) 97.6 Btu/lb Condensate Pup wcps υ( P0 P9 ) ( 0. 06)( )( / 778) 0. Btu/lb wcps 0. wcp Btu/lb η p 0 9 w cp Btu/lb Low Pressure Feed Heater

34 Since all te ass flow rates are known, we can solve for te fluid entalpy leaving te eater as 7 + ( 7 ) ,000 6,000 6,000,000 55,000 lb/r P 7 psia 65.5 Btu/lb (5/55)( ) 5.85 Btu/lb Open Feed Heater + + ( + ) Again te ass flow rates are known, suc tat te entalpy leaving te open eater is + + ( + ) Btu/lb 5.85 Btu/lb 5 P 0 psia 9.0 Btu/lb (/700)(8.)+(55/700)(5.85)+(/700)(9) 0. Btu/lb Interediate Pressure Pup wips υ( P P ) Te specific volue is taken as tat of a saturated liquid corresponding to a liquid entalpy of 0. Btu/lb. υ ft /lb w ips ( )( )( / 778). 0 Btu/lb wips. wip. Btu/lb 0. 9 η p w ip Btu/lb Interediate Pressure Feed Heater ( + ) 7 6 5

35 Solving for 6 gives ( + ) Btu/lb.6 Btu/lb 5 P 0 psia 9.0 Btu/lb 7 P 0 psia 0.8 Btu/lb (6/700)( 8.9)+(6/700)(0.8)-(/700)(9) 97. Btu/lb Hig Pressure Heater Solving for 8 gives + ( 7 ) Btu/lb Btu/lb Btu/lb (6/700)( ) 75. Btu/lb Hig Pressure Boiler Feed Pup wps υ( P P8 ) Te specific volue is taken as tat of a saturated liquid corresponding to a liquid entalpy of Btu/lb. υ ft /lb w ps ( )( )( / 778). 99 Btu/lb η p w w ps p wp Btu/lb Btu/lb w p Btu/lb

36 η p Boiler % 7. q Btu/lb Cycle Efficiency η net tp tlp cp ip p tp ( ) ( 700, 000)( 8. 75) ( 6, 000)( 96.6) - (6,000)(8.9) - (,000)(8.) - (55,000)(09.7).8 0 Btu/r tlp ( ) (55, 000)( 09.7) - (5,000)(057.9) - (500,000)(97.6) 8. 0 Btu/r cp 0 wcp ip (55,000)( 0.58) 875 Btu/r w ip ( 700, 000)(. ) 85, 000 Btu/r p wp (700,000)( 7.) q 7 Btu/r ( 700, 000)( 06. 5) Btu/r. 0 η 9.9 % ,5 85, Te teperatures corresponding to te eater inlet and outlets are approxiately T 0 9 F T 78 F 5

37 T F T 6 7 F T 8 00 F Te cange in teperature across te individual eaters is ten ) Low pressure eater, T F ) Interediate pressure eater, T 7-8 F ) Hig pressure eater, T F 6