Carnot Factor of a Vapour Power Cycle with Regenerative Extraction

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1 Journal of Modern Pysics, 2017, 8, ttp:// ISSN Online: X ISSN Print: arnot Factor of a Vapour Power ycle wit Regenerative Extraction Duparquet Alain Engineering Energy Equipment, Bourgoin-Jallieu, France ow to cite tis paper: Alain, D. (2017) arnot Factor of a Vapour Power ycle wit Regenerative Extraction. Journal of Modern Pysics, 8, ttps://doi.org/ /jmp Received: September 14, 2017 Accepted: October 21, 2017 Publised: October 24, 2017 opyrigt 2017 by autor and Scientific Researc Publising Inc. Tis work is licensed under te reative ommons Attribution International License ( BY 4.0). ttp://creativecommons.org/licenses/by/4.0/ Open Access Abstract Te present paper describes te energy analysis of a regenerative vapour power system. Te regenerative steam turbines based on te Rankine cycle and comprised of vapour extractions ave been used industrially since te beginning of te 20 t century, particularly regarding te processes of electrical production. After aving performed worked in te first stages of te turbine, part of te vapour is directed toward a regenerative excanger and eats feedwater coming from te condenser. Tis process is known as regeneration, and te eat excanger were te eat is transferred from steam is called a regenerator (or a feedwater eater). Te profit in te output brougt by regenerative rakings is primarily enabled by te lack of excange of te tapped vapour reeating water wit te low-temperature reservoir. Te economic optimum is often fixed at seven extractions. One knows te arnot relation, wic is te best possible teoretical yield of a dual-temperature cycle; in a arnot cycle, one makes te assumption tat bot compressions and expansions are isentropic. Tis article studies an ideal teoretical macine comprised of vapour extractions in wic eac cycle partial of tapped vapour obeys tese same compressions and isentropic expansions. Keywords Termodynamic, arnot Factor, Rankine ycle, Power Plant, Energy, Efficiency, Entropy, Second Law Analysis, Irreversibility, Regenerative ycle, Termal ycle 1. Introduction Te work output is maximised wen te process between te two specified states is executed in a reversible manner. owever, according to te second law of termodynamics, suc a reversible process is not possible in practice. A system DOI: /jmp Oct. 24, Journal of Modern Pysics

2 delivers te maximum possible work as it undergoes a reversible process from te specified initial state to te state of its environment, tat is, te dead state. According wit M.Pandey, T.K Gogoi [1], te study of termodynamic cycles applied to power stations is of great importance due to te increasing energy consumption, te opening of electricity markets and te increasingly strict environmental restrictions (specifically regarding te carbon dioxide emissions issue). Power plants tat use steam as teir working fluid operate on te basis of te Rankine cycle (1). Te first stage in designing tese power plants is performing te termodynamic analysis of te Rankine cycle. According wit Da una, A., Fraidenraic, N. and Silva, [2] and Baumann, K., [3], te regenerative cycle is a modified Rankine cycle wic intends to improve te efficiency of te system. Baumann (1930) analyzed te development of steam cycle power during years, using ig steam pressure and temperature, wit te objective to improve te efficiency of te power plants. Tere are some advantages of regenerative power cycle compared of Rankine cycle. Te termal stresses in te collectors are minimized because of te water being otter. Te steam condenser size can be reduced, te pumps energy to cool te steam condenser is reduced in te same proportion. But also tere are some disadvantages as te plant become more complicated, more expensive and increase te number of maintenance. Many industries use steam based termodynamic cycles. Water steam is generated in a boiler, wic brings te steam to a ig pressure and a ig temperature. Te steam expands in a turbine to produce work. Te steam ten passes troug a condenser. In tis cycle, referred to as te Rankine or irn cycle, te eat referred to as waste is te eat excanged in te condenser. A regenerative cycle is a cycle in wic part of te waste eat is used for eating te eat-transfer fluid. Since te early 20 t century, steam turbines ave been most frequently used in regenerative cycles, e.g., in termal and nuclear power plants. Tese steam engines are equipped wit five to seven steam extractions. Part of te steam tat performed work in te turbine is drawn off for use in eating te water. Regeneration or te eating up of te feedwater by te steam extracted from te turbine as a marked effect on te cycle efficiency. Figure 1 presents diagram of a regenerative steam extraction. Te mass of steam generated for te given flow rate of flue gases is determined by te energy balance. In fact, te drawn-off steam is proportional to te flow of water to be eated to conserve bot mass and energy. 2. Simple Rankine ycle 2.1. Real ycle Te termodynamics first principle stipulates tat over a cycle, W + Q = 0. W + Q = 0 (1) W + Q Q = 0 (2) Q Q = W (3) DOI: /jmp Journal of Modern Pysics

3 Figure 1. Diagram of a regenerative steam extraction cycle. were, Q = mass entalpy of te ig-temperature reservoir. Q = mass entalpy of te low-temperature reservoir. Te transferred energy is equal to te differential specific entalpy multiplied by te fluid mass E = m (4) Te energy efficiency of te engine power is defined as W η = (5) Q Te efficiency is rewritten using (1), (3) and (5) as follows: η Q Q Te following is deduced from (4) and (6): = (6) Q Q = m (7) S W Q = m (8) S W were: m = fluid mass. S = specific entalpy of te ot vapour to te admission of te turbine. W = specific entalpy in liquid form. S = specific entalpy of te cold vapour to te exaust of te turbine. Figure 2 presents te real Rankine cycle in T-s diagram. Te cycle performance is deduced from (5)-(8) as 2.2. Ideal arnot ycle ( S W ) ( S W ) m( ) m m η = S Let us consider now tat te cycle is performed wit ideally isentropic com- W (9) DOI: /jmp Journal of Modern Pysics

4 Figure 2. Te real Rankine cycle (sown in red). pressions and isentropic expansions. According wit Lucien Borel [4], te eat excanged wit te ig-temperature reservoir is equal to were T Q = ds = mt s s (10) S W 0 T = temperature of te ig-temperature reservoir. s = cold vapour specific entropy to te exaust of te turbine. S s = cold condensate water specific entropy. W Figure 3 presents te real Rankine cycle in T-s diagram in red en te coloured surface corresponds to te total energy of a ig-temperature reservoir tat would be necessary if te cycle obeyed te assumptions of isentropic compressions and isentropic expansions (10). Te eat excanged wit te low-temperature reservoir is equal to T = d = S W 0 Q s mt s s (11) were T = temperature of te low-temperature reservoir. Figure 4 presents te Real cycle in red. Te coloured surface corresponds to te total energy of low-temperature reservoir tat would be necessary if te cycle DOI: /jmp Journal of Modern Pysics

5 Figure 3. Te real cycle is represented in red. Te coloured surface corresponds to te total energy of a ig-temperature reservoir tat would be necessary if te cycle obeyed te assumptions of isentropic compressions and isentropic expansions (10). Figure 4. Te Real cycle is represented in red. Te coloured surface corresponds to te total energy of low-temperature reservoir tat would be necessary if te cycle obeyed te assumptions of isentropic compressions and expansions (11). DOI: /jmp Journal of Modern Pysics

6 obeyed te assumptions of isentropic compressions and expansions (11). Te cycle efficiency is deduced from (6), (10) and (11) as ( ) ( ) mt ( s s ) mt s s mt s s η = S W S W S W (12) wic is equal to te arnot factor. T T T η = = 1 (13) T T 3. Termodynamics Analysis of Steam Power ycle wit One Feed Watereater 3.1. Real ycle wit One Feedwater eater We study ere te evaluation of te output of a cycle comprised of only 1 steam extraction. To determine te output of te cycle comprising an extraction, we separate tis cycle into 2 partial cycles. Te eat of te ot reservoir in te cycle is equal to te sum of te eat of te ot reservoir of te principal cycle and te eat of ot reservoir of te cycle partial of te extraction vapour: Te eat of ot reservoir of te principal cycle is equal to (7). Te eat of ot reservoir of te cycle partial of te extraction vapour is equal to Q = m (14) EX ex S EXW were Q EX = entalpy of te ot reservoir of te cycle partial of extraction. m ex = extraction vapour mass. EXW = extracted steam specific entalpy in liquid form. One form of te deduced total eat from te ot reservoir: Q = m + m (15) S W ex S EXW were m = vapour mass excanging wit te low-temperature reservoir in te condenser Te eat of te low-temperature reservoir in te cycle is equal to te sum of te eat of te low-temperature reservoir of te principal cycle and te eat of te low-temperature reservoir of te partial cycle of te tapped vapour: Te eat of low-temperature reservoir of te principal cycle is equal to (8) Q = m S W Te eat of low-temperature reservoir of te cycle partial of te tapped vapour is equal to: Q = m (16) were EX ex EXS EXW DOI: /jmp Journal of Modern Pysics

7 Q EX = entalpy of te low-temperature reservoir of te partial cycle of extraction EXS = te mass vapour entalpy arnessed by te extraction side. One form of te deduced total eat from te low-temperature reservoir: Q = m S W + mex EXS EXW (17) Te cycle efficiency is deduced from (6), (15) and (17) ( S W ) + ex ( S EXW ) ( S W ) + ex ( EXS EXW ) m( ) + m ( ) m m m m η = S W ex S EXW Te output of a regenerative cycle comprising an extraction is clearly iger tan tat of a non-regenerative simple cycle. It is enoug to compare te expressions of te output in (18) and in (9). Te mass of te steam generated for te given flow rate of flue gases, m ex, is obtained from te energy balance. eat gained by te steam = eat lost by te flue gases. ( ) = ( ) m m ex EXW EXS EXW W EXW W mex = m EXW EXS (18) (19) Figure 5 presents te cycle represented in red is te principal cycle traversed Figure 5. Te cycle represented in red is te principal cycle traversed by te vapour mass m. Te cycle represented in blue is te cycle traversed by te mass m ex of te tapped vapour. DOI: /jmp Journal of Modern Pysics

8 by te vapour mass m. Te cycle represented in blue is te cycle traversed by te mass m ex of te tapped vapour Ideal ycle omprised of One Feedwater eater Let us now consider tat te two cycles are ideally performed via compressions and expansions tat are isentropic. Let us separate te cycles into two partial cycles. Te eat of te ot reservoir in te cycle is equal to te sum of te eat of te ot reservoir of te principal cycle and te eat of te ot reservoir of te partial cycle of te tapped vapour: Te eat of ot reservoir of te principal cycle is equal to (10). T Q = ds = mt s s S W 0 Figure 6 presents te real principal cycle traversed by te vapour m mass is represented in red. Te coloured surface corresponds to te total energy of te ot reservoir tat would be necessary if te cycle obeyed te assumptions of isentropic compressions and isentropic expansions (10). Figure 6. Te real principal cycle traversed by te vapour m mass is represented in red. Te coloured surface corresponds to te total energy of te ot reservoir tat would be necessary if te cycle obeyed te assumptions of isentropic compressions and isentropic expansions (10). DOI: /jmp Journal of Modern Pysics

9 T QEX = ds = mext ss sw (20) Figure 7 presents te cycle represented in blue te real cycle traversed by te mass m ex of te tapped vapour. Te coloured surface corresponds to te total energy of te ot reservoir tat would be necessary if te 2 cycles obeyed te assumptions of isentropic expansions and isentropic compressions (20). One deduces te total eat from te ot reservoir as follows: Q = mt s s + m T s s (21) S W EX S W Te eat of te low-temperature reservoir in te cycle is equal to te sum of te eat of te low-temperature reservoir of te principal cycle and te eat of te low-temperature reservoir of te partial cycle of te tapped vapour: Te eat of te low-temperature reservoir of te principal cycle is equal to (11). T = d = S W 0 Q s mt s s Figure 8 presents te cycles in T-s diagram. Te cycle represented in red is Figure 7. Te cycle represented in blue is te real cycle traversed by te mass m ex of te tapped vapour. Te coloured surface corresponds to te total energy of te ot reservoir tat would be necessary if te 2 cycles obeyed te assumptions of isentropic expansions and isentropic compressions (20). DOI: /jmp Journal of Modern Pysics

10 Figure 8. Te cycle represented in red is te real cycle traversed by te mass m of te vapour. Te coloured surface corresponds to te total energy of te low-temperature reservoir tat would be necessary if te 2 cycles obeyed te assumptions of isentropic compressions and isentropic expansions. te real cycle traversed by te mass m of te vapour. Te coloured surface corresponds to te total energy of te low-temperature reservoir tat would be necessary if te 2 cycles obeyed te assumptions of isentropic compressions and isentropic expansions. Te eat of te low-temperature reservoir of te extracted steam partial cycle is equal to: T EX Q = ds = m T s s (22) EX EX EX S W 0 Figure 9 presents te cycles in T-s diagram. Te eat of te low-temperature reservoir of te partial cycle of extraction is represented by te coloured surface. Note tat te difference in entropy mass s is equal to te difference between te mass entropy of te cold vapour and te mass entropy of condensate of te principal cycle. Q = mt s s + m T s s (23) S W EX EX S W Te energy efficiency is deduced from (6), (22), and (23): DOI: /jmp Journal of Modern Pysics

11 Figure 9. Te eat of te low-temperature reservoir of te partial cycle of extraction is represented by te coloured surface. Note tat te difference in entropy mass s is equal to te difference between te mass entropy of te cold vapour and te mass entropy of condensate of te principal cycle. ( S W )( + EX ) ( S W ) + EX EX ( S W ) T ( s s )( m+ m ) T s s m m mt s s m T s s η = wic can be expressed as: S W EX T m + m mt + m T mt + m T η = = 1 T m+ m T m+ m EX EX EX EX EX EX EX Applying Equation (19), one determines te m ex mass value according to te m mass: T m = m (25) T EX T T mt + m T + η = 1 = 1 T T T m+ m T m+ m η = 1 T T 1+ EX (24) (26) DOI: /jmp Journal of Modern Pysics

12 4. Generalised Termodynamics Analysis of a Steam Power ycle wit N Number of Feedwater eaters 4.1. Real ycle wit Two Feedwater eaters Te termodynamic cycles of steam turbines are comprised of between five and eigt extractions. We treat ere te case of a macine comprised of two extractions; tis metodology can be extended to account for a iger number of extractions. Te output is deduced from (18) ( S W ) + ex1( EX 1S EX 1W ) + ex2( EX 2S EX 2W ) m( ) + m ( ) + m ( ) m m m η = 1 S W ex S EXW ex2 S EX 2W were: m ex1 = vapour mass of te first extraction. (Recall tat te classification numbering of conventional feedwater eaters starts from te condenser.) EX 1W = specific entalpy of te steam tat eats feedwater eater number 1. m EX 2 = mass of condensate from extraction number 2. EX 1S = specific entalpy of extraction number 1 vapour. EX 2W = specific entalpy of te condensed water into feedwater eater number 1. EX 2S = specific entalpy of extraction number 2 vapour. alculation of m EX 2 One deduces m EX 2 from (19) EX 2W EX 1S mex 2 = ( m+ mex 1) (28) EX 2S EX 2W 4.2. Ideal ycle n Number of Feed Water eaters (27) One deduces te output from te ideal cycle from (30) and (26): η = 1 [ mt + mex 1 1+ mex 2 2] T ( m+ m + m ) ex1 ex2 (29) were: T EX 1 = temperature of te condensate wit extraction vapour number 1 T EX 2 = temperature of te condensate wit extraction vapour number 2 One generalises te following: exi exi i= 1 η = 1 n n mt + m T T m+ m i= 1 One deduces m EX 2 from (26) and (28) T T T T mex 2 = exi EX 1 EX 2 EX 1 One deduces from (29) and (31) te following: (30) (31) DOI: /jmp Journal of Modern Pysics

13 η = 1 T 1 T ( 2 1) 1 T T T T T T EX 1 EX 1 EX 2 EX 1 Te above result (32) can be generalised as follows: η = 1 T n1 i 1 T ( x + 1x ) i= 1 1 x= 1 n1 i 1T 1T x + 1T EXx i= 1 1 x= 1 x + 1 (32) (33) 5. onclusions Te arnot factor, as is universally known, is a typical case limited to te cycles tat do not involve feedwater eaters. In te case of regenerative cycles, te arnot factor of a macine is given by te following relationsip: η = 1 T n1 i 1 T ( x+ 1x ) i= 1 1 x= 1 n1 i 1T 1T x+ 1T EXx i= 1 1 x= 1 x+ 1 Tis relationsip obeys te second principle of termodynamics: te variation of te entropy of an unspecified termodynamic system, due to te internal operations, can be only positive or wortless. Watever te fluid, if it is possible to use a part of its mass and not to excange wit te low-temperature reservoir, it is advantageous. An example to consider te following issue is about combined cycles. A part of designers of electrical generating units using te combined cycles of a gas turbine and a steam turbine do not use te regenerative Rankine cycle. Tis is because tey consider tat tey ave acieved a yield close to arnot s performance. Tis article demonstrates tat it is irrelevant. Te advantage in tis case is above all to reduce te steam condenser size. Te pumps energy to cool te steam condenser is reduced in te same proportion. If an atmosperic refrigerant is used, te economy would be significant. References [1] Pandey, M. and Gogoi, T.K. (2013) International Journal of Emerging Tecnology and Advanced Engineering, 3, [2] Da una, A., Fraidenraic, N. and Silva, L. (2017) Journal of Power and Energy Engineering, 5, ttps://doi.org/ /jpee [3] Baumann, K. (1930) Proceedings of te Institution of Mecanical Engineers, 2, ttps://doi.org/ /pime_pro_1930_119_024_02 [4] Borel, L. Termodynamique et énergétique. [Termodynamics and Energetics.] presses polytecniques et universitaires Romandes 3 ème édition revue et corrigée ISBN Lausanne. DOI: /jmp Journal of Modern Pysics

14 Nomenclature T Temperature (K) s Specific Entropy (J/kg K) P Pressure (Pa) Specific Entalpy (J/kg) W Work (J) Q eat (J) Entalpy (J) m Main Steam ycle Mass (kg) m ex Extraction Steam Feedwater Mass (kg) m Extraction Steam Feedwater Number 1 Mass (kg) ex1 m ex2 Extraction Steam Feedwater Number 2 Mass (kg) η Energy Efficiency (%) Σ Π Sum Product Integral DOI: /jmp Journal of Modern Pysics