Golden section heat engines and heat pumps

Transcription

1 Golden setion eat engines and eat pumps Mamoud uleil Aademi Institute for training arab teaers AIA. (Reeived 5 July 01, aepted 13 September 01) Abstrat In tis study te effiieny of eat engines and te oeffiient of performane of eat pumps are reonsidered following te idea of te golden setion ratio. wo models are analyzed: te golden setion arnot yle and te golden setion endo-reversible yle. e observed expressions for effiieny and oeffiient of performane are asted in terms of te golden setion ratio. Single (steam turbines [40ilfried, U., OMMI, Vol., issue 3, pp. 1-9, 003]) and double eat engine yles (ombined yles [Isikawa, M., eraui, M., Komori,., Yasuraoka, J., Mitsubisi eavy Industries, td., enial Review Vol. 45 No. 1, pp , 008]) are onsidered. Expressions for te working fluid temperatures at te ot side and te old side are derived and it was found tat te relative differenes ompared to te endoreversible results falls in te range -4% only wit relative power redution of 15% from te maximum power point. Finally, it is sown tat te system effiieny differs approximately by 3.6% from te effiieny derived based on te golden setion. Keywords: eat engines, eat pumps. Resumen En este estudio, la efiienia de los motores de alor y el oefiiente de rendimiento de las bombas de alor son onsideradas siguiendo la idea de la proporión de la seión de oro. Dos modelos an sido analizados: El ilo de arnot la seión de oro y el ilo de la seión de oro endo-reversible. as expresiones observadas para la efiienia y el oefiiente de rendimiento están difundidas en términos de la proporión de la seión de oro. Únio (turbinas de vapor [40ilfried, U., OMMI, Vol., issue 3, pp. 1-9, 003]) y dobles ilos de motor térmio (ilos ombinados [Isikawa, M., eraui, M., Komori,., Yasuraoka, J., Mitsubisi eavy Industries, td., Revisión énia Vol. 45 No. 1, pp , 008]) se onsideran. Expresiones para la temperaturas del fluido en el lado aliente y el lado frío se derivan y se enontró que las diferenias relativas omparadas on los resultados endo-reversibles ae en el rango -4% solo on reduión de potenia relativa de 15% desde la máxima power point. Finalmente, se muestra que la efiienia del sistema difiere aproximadamente un 3.6% de la efiienia derivada basada en la seión de oro. Palabras lave: Máquinas de alor, bombas de alor. PAS: d, a ISSN I. INRODUION e arnot engine onstitutes te basi eat engine model aving reversible proesses and yields upper bound of termal effiieny for eat engines working between two eat reservoirs at onstant temperatures. Pratially a arnot engine yields zero power. en te power output is limited by te eat transfer rates between te eat reservoirs and te working substane, te eat engine is defined as endo-reversible. Using te endo-reversible yli model, Novikov [1] and urzon & Alborn [] ave obtained te effiieny A 1 of a arnot engine at maximum power output in wi and are temperatures of eat soure and sink, respetively. Following tese studies, te metods of finite time termodynamis (F) ave been applied to a wide range of termodynami systems [3-7]. In a different approa, Popov and Sipitsyn onsidered te Golden setion in te arnot yle [8]. In teir study, some aspets of lassial termodynamis are analyzed for te presene of duality and of te golden setion. Markowsky in is book review [35], gave some properties of te golden setion. e number (1 + 5)/ = is widely known as te golden ratio, and pi. Pi appears in many different equations and formulas and as many interesting properties. e laims tat many people ave eard marvelous tales about pi and ow it permeates art and nature. retanu and rasmareanu, [9] laims tat te golden ratio (wi is also known as te golden setion) is a fasinating topi tat ontinually generates new ideas. e main purpose of teir paper was to point out and find some appliations of te golden ratio and of Fibonai numbers in differential geometry. ey studied a struture defined on a lass of Riemannian manifolds, alled by tem a at. Am. J. Pys. Edu. Vol. 6, No. 3, Sept ttp://

2 golden struture. A Riemannian manifold endowed wit a golden struture was alled a golden Riemannian manifold. atson [30] states tat te golden setion is intimately related to growt and nature. More ever e laims tat Eggs, an apple blossom, a uman fae, a seasell all embody golden setion proportions. e believes tat te golden setion is a simple tool tat may be used to enane te meaning and beauty of an aritetural work. Based on tat, e says tat designing buildings wit tis knowledge automatially reates armonizing, uplifting effets on tose wo experiene tem. More tan tat e laims tat wen te strutural lines of a building are designed aording to te priniples of armoni proportions, a natural aesteti beauty results beauty tat an benefit tose wo work, live, and play witin tose environments. On te oter and, Kak S. [31], reported in is artile tat te golden mean,, as been applied in diverse situations in art, ariteture and musi, and altoug some ave laimed tat it represents a basi aesteti proportion, oters ave argued tat it is only one of a large number of su ratios. efebvre and Efremov onsidered te problem of double geometri progression and golden setion in te deay profiles of bursts [3]. ey found tat for bursts lasting less tan 5se and if two signifiant peaks are observed in te power spetrum, te ratio of teir entroid frequenies is approximately onstant and equal to ey found tat tis ratio is very lose to te Golden Setion ratio Smirnov [33] in is paper, reviews tat Kepler analyzed planet movement parameters wit musi proportions. ere exists no ontradition in su a metod. It is known tat te same different equations of osillatory movement, for example, desribe diverse pysial penomena. e matematial basis of su a musial desription of astronomial penomena was te so-alled God s proportion or golden setion. e golden setion is found in bioengineering problems too. Bejan [34] laims tat sapes wit lengt/eigt ratios (/) lose to 3/ are everywere and give te impression tat tey are being designed to mat te golden ratio ( = 1.618). In is artile e sowed tat tese sapes emerge as part of an evolutionary penomenon tat failitates te flow of information from te plane to te brain, in aordane wit te onstrutal law of generation and evolution of design in nature. e golden setion is of interest to matematiians and among tem tey are interested in finding new onstrution metods. Kaplan et al. [36] onstruted te golden ratio by using a bisetor area of a trapezoid. ofstetter [37] gave a simple 5-step division of a segment into golden setion, using ruler and rusty ompass. In a different study [38] ofstetter onstruted, in 4 steps using ruler and ompass, tree points two of te distanes between wi bear te golden ratio. A different way of onstrution is given by ong and Kung [39]. ey onstrut te golden ratio by using an area bisetor of a trapezoid. Golden setion eat engines and eat pumps In tis study te effiieny of arnot and endoreversible eat engines and te oeffiient of performane of arnot eat pumps are ast wit respet to te golden setion ratio. e new expressions for effiieny are ompared to te urzon & Alborn effiieny. Altoug te observed effiienies resulted from pure matematial basis, it is interesting to note tat te state of te art effiieny of steam turbines [40] and of te ombined gas yles [41] relatively differs by few perentages only. e order of te artile is as follows: In setion te golden setion arnot eat engine and eat is analyzed, in setion 3 te golden setion endo-reversible eat engine wit Newtonian eat transfer is onsidered and a numerial example is given in setion 4. Finally onlusion and disussion are given in setion 4. II. E GODEN SEION ARNO EA ENGINE AND EA PUMP onsider a arnot eat engine tat work between two eat reservoirs, a ot reservoir wit temperature and old reservoir wit temperature. e semati of te arnot eat engine is given by Fig. 1. e eat input to te engine is and te eat output from te engine is. e work output is alulated based on te energy equation and is given by:. (1) FIGURE 1. Sematis of te arnot eat engine tat working between two eat reservoirs, te ot reservoir at and te old reservoir at. e eat input to te engine is always divided into two parts. Due to losses of different types (finite eat transfer rates, frition, et ) te amount of work output ould be smaller or larger tan te eat rejetion. itout performing any massive alulations, one ould follow te golden setion division of a line and dedue expressions for te effiieny of te eat engine. e semati of te golden setion division of te eat input is depited in Fig.. e golden setion ratio for te ase depited in Fig. a is: at. Am. J. Pys. Edu. Vol. 6, No. 3, Sept ttp://

3 Mamoud uleil. () If we reverse te arrows, we get a arnot eat pump instead of a eat engine. e performane of a eat pump is usually given by te oeffiient of performane. ere are two oeffiients of performane: one for eating and te oter one for ooling. e oeffiient of performane for eating OP is given by: OP. (6) 1 FIGURE. e golden setion division if te eat input is done in two different ways: a. te eat rejetion is larger tan te work output and b. te work output is larger tan te eat rejetion. Equation ould be arranged in te following way after some manipulation: (3) e pysial solution of Eq. gives te effiieny of golden setion arnot eat engine for te ase tat te eat rejetion is larger tan te work output. is effiieny is omparable wit single yle eat engines [40]. Performing te alulation leads to: ere sin gle (4) sin gle is te single yle of golden setion arnot eat engines asted in terms of te golden setion ratio, and is te golden setion ratio ( ). For te arnot eat engine tere are six pysial quantities: te amount of eat input to te engine, te amount of eat rejeted from te engine, te work output, te reservoir's temperatures and effiieny. ese quantities are related to ea oters wit tree equations: e energy onservation, te entropy onservation and te effiieny definition. For a given value of effiieny, eat input, te old reservoir temperature and te two onservation equations, is it possible to solve for te oter unknowns: te eat output, te work output and te ot reservoir temperature. After some matematial manipulation, te ot reservoir temperature is given by:. (5) And te oeffiient of performane for ooling given: OP 1 1 OP is. (7) For te oter ase, wen te larger part of te eat input is turned to work output (as depited in Fig. b), te golden setion ratio is given by:. (8) Rearranging Eq. 8 leads to te following quadrati equation: 1 0. (9) e pysial solution of Eq. 9 gives te effiieny of golden setion arnot eat engine for te ase tat te eat rejetion is smaller tan te work output. is effiieny is omparable wit double yle eat engines 41]. Performing te alulation leads to: double (10) ere is te double yle of golden setion arnot double eat engines asted in terms of te golden setion ratio. For tis ase te temperature if te ot reservoir is given by:. (11) 1 III. E GODEN SEION ENDO- REVERSIBE EA ENGINE AND EA PUMP In tis setion we onsider an endo-reversible eat engine model. e model is desribed sematially by Fig. 3. e endo-rersible eat engine runs between two eat reservoirs, ot reservoir at temperature and old reservoir at temperature. e eat engine runs at finite at. Am. J. Pys. Edu. Vol. 6, No. 3, Sept ttp://

4 rates, and te eat input and te eat rejetion follow a Newtonian law. e eat input is given by:. (1) ere is te eat ondutane at te ot side. Similarly, te eat rejetion is given by: ere. (13) is te eat ondutane at te old side. Golden setion eat engines and eat pumps. (17) Equations (1, 13, 14, 15, 16, 17) ould be manipulated and te different pysial amounts ould be alulated expliitly from te following expressions. e eat input is given by: (1 ). (18) 1 1 e work output is given by: 1 (1 ) 1. (19) From Eqs. (18-19) it is obvious tat for a given eat ondutane at bot sides ot and old, te temperatures of te eat reservoirs and te effiieny of te eat engine; it is possible to alulate te eat input and te work output. urzon & Alborn derived te effiieny at maximum power operation and it is given by (te urzon & Alborn results are summarized for omparison purposes): FIGURE 3. Semati drawings for te endo-reversible eat engine. e ot reservoir temperature, is te working temperature at te ot side of te engine, is te working temperature at te old side of te eat engine, is te temperature of te old reservoir, is te eat ondutane at te ot side, is te eat ondutane at te old side, is te eat input, is te eat rejetion and is te work output from te eat engine. e work output is given by Eq. 1 wi is alulated from te energy equation: E 0. (14) e entropy for te endo-reversible eat engine is onserved and is given by: S 0. (15) A 1. (0) And te maximum power for te ase is given by: 1. (1) e working temperature of te urzon & Alborn eat engine at te ot side is given by:. () And te working temperature at te old side is given by:. (3) Equation 15 ould be reast expliitly as follows: (16) astly, te effiieny of te endo-reversible eat engine is defined as te work output divided by te eat input and is given by: In te following paragraps te golden setion endoreversible eat engine is onsidered. Instead of seeking te effiieny at maximum power output, te effiieny is assumed to be given by Eq. 4. en te work output of te golden setion endo-reversible eat engine (for te ase of equal eat ondutane values at bot sides of te engine) is given by: at. Am. J. Pys. Edu. Vol. 6, No. 3, Sept ttp://

5 Mamoud uleil 1 1, (4) e resulting working temperature at te ot side given by: is. (5) And te resulting working temperature at te old side is given by:. (6) Based on te equations given above, and for omparison purposes between te golden setion endo-reversible eat engine and te endo-reversible eat engine, te following ratios are developed. e work output ratio r is defined as te ratio between te work output of te golden setion endoreversible eat engine and te maximum work output of te endo-reversible engine is given by: r 1 1 1, (7) e working temperature ratio at te ot side r wi is te ratio between te working temperature at te ot side of te golden setion endo-reversible eat engine and te working temperature of te endo-reversible eat engine at te ot side is given by: r 1 1. (9) 1 Finally, te effiieny of te system is given by: sys as defined in [1] A sys. (30) IV. NUMERIA EXAMPE In tis setion we onsider a numerial example for illustration and omparison purposes. onsider a arnot eat engine wit ot reservoir at 600K and old reservoir at 300K. For tis ase, te arnot effiieny is 50%. e golden setion effiieny (as given by Eq. 4 equals 38.% and te urzon & Alborn effiieny (as given by Eq. 0) equals 9.9% approximately. e system effiieny (alulated from Eq. 40) is 39.65% approximately. us te relative differene between te system effiieny and te golden setion effiieny is approximately 3.6%. For te golden setion arnot eat engine, te ot side reservoir temperature as estimated by Eqs. (5) and (11), equals 484.4K for te single yle and for te double yle, respetively. e endo-reversible eat engine predits working temperature at te ot side as 710K and te working temperature at te old side is predited to be 410K. On te oter and, te results predited from te golden setion endo-reversible eat engines are 693K (Eq. 5) and 48K (Eq. 6) respetively. us te relative differenes fall in te range of -4% approximately. Finally, te work ratio (Eq. 39) predits 85%. e golden setion endo-reversible eat engine introdues only 15% redution of te maximum work output. r 1 1. (8) e working temperature ratio at te old side r wi is te ratio between te working temperature at te old side of te golden setion endo-reversible eat engine and te working temperature of te endo-reversible eat engine at te old side is given by: V. ONUSION AND DISUSSION In tis study a golden setion arnot eat engine and eat pump models were onsidered in setion. Expression for te effiieny and te ot reservoir temperature were observed and vast as a funtion of te golden setion ratio. For te arnot eat engines two ases were onsidered: a single yle and double yle. e effiieny of te single yle is observed to be 38.%. e ot side temperature of te arnot eat engine is predited to be 484.4K based on 300K old reservoir. For te double yle ase, te golden setion effiieny of te arnot yle is 61.8% wit 785.4K ot reservoir temperature. e golden setion endo-reversible eat engine model was onsidered in setion 3. e observed effiieny of te at. Am. J. Pys. Edu. Vol. 6, No. 3, Sept ttp://

6 eat engine working between 600K and 300K eat reservoirs is 38.% wi differs by about 3.6% relative to te system effiieny 39.65% (wi is given by Eq. 40). e working fluid temperatures for te golden setion endo-reversible eat engine and e endo-reversible engine were alulated and te relative differene between tem falls in te range -4% at te prie of 15% of te work output. Similar analysis was performed for te arnot eat pump and expressions for te oeffiients of performane for eating and ooling were given by Eqs. (6) and (7). Altoug te golden setion ratio is a pure matematial issue, it is interesting to note tat te observed effiienies of te single yle and double yle differs by small amount from te state of te art effiienies of steam turbines and gas ombined yles. is an be seen from te studies of ilfried [40] and Isikawa et al., [41]. ilfried [40] reports tat te state of te art of steam turbines (single yle) provided by steam parameters wit superritial pressures and temperatures up to 6100 rea iger yle effiienies. e improvement in effiieny is due to progress in material tenology (for asing, rotors, REFERENES [1] Novikov, I. I., e effiieny of atomi power stations, J. Nul. Energy 7, (1958). [] urzon, F.. and Alborn, B., Effiieny of a arnot engine at maximum power output, Am. J. Pys. 43, -4 (1975). [3] Andresen, B., Salamon, P. and Berry, R. S., ermodynamis in finite time, Pys. oday 37, 6-70 (1984). [4] Andresen, B., Salamon, P. and Berry, R. S., ermodynamis of finite time: Extremals for imperfet eat engines, J. em. Pys. 66, (1977). [5] Salamon, P., Nitzan, A., Andresen, B. and Berry, R. S., Minimum Entropy...Optimization of eat Engines, Pys. Rev. A 1, (1980). [6] Salamon, P. and Berry, R. S., Pys. Rev. ett. 51, (1983). [7] Rubin, M.., Optimal onfiguration of a lass of irreversible eat engines, Pys. Rev. A 19, (1979). [8] Mozurkewi, M. and Berry, R. S., Finite-time termodynamis: Engine performane improved by optimized piston motion, Pro. Natl. Aad. Si. U.S.A. 78, (1981). [9] Mozurkewi, M. and Berry, R. S., Optimal pats for termodynami system: te ideal Otto yle, J. Appl. Pys. 53, 34-4 (198). [10] offmann, K.., atowit, S. J. and Berry, R. S., J. Appl. Pys. 58, (1985). [11] Gordon, J. M. and uleiil, M., On optimizing maximum-power eat engines, J. Appl. Pys. 69, 1-7 (1991). Golden setion eat engines and eat pumps and turbine blades) and omputers (FD simulations). Optimizing te turbine blades led to redue inner losses wit 96% turbine effiieny. us te system effiieny (net effiieny) ould rea 43%. Isikawa et al., [41] Gas turbine ombined yle power plants (double yle) mainly using MI D, F and G-type gas turbines are urrently in operation in large numbers, and tere are many under onstrution or planned bot in Japan and abroad. ey report tat te effiieny of te ombined yle plant wit a 1,700 lass gas turbine wit its inlet temperature improved to 1,700 ompared wit te F-type and G-type wit inlet temperatures of 1,500 as reaed te level of 6% - 65% (V base), and is expeted to stand at a iger level tan te effiieny of te onventional ombined yle plant. e golden setion as been reported in te literature as a measure of beauty in different fields of siene and nature. In tis study te effiieny of eat engines evolved over te years to rea te level of golden setion ratio. e big question ere - Did eat engines rea te beauty level or te observed expressions for te effiieny are just fortuitous?! [1] Gordon, J. M. and uleiil, M., General performane arateristis of real eat engines, J. Appl. Pys. 7, (199). [13] Nulton, J. D., Salamon, P. and Patria, R. K., amotlike proesses in. finite time. I. eoretial limits, Am. J. Pys. 61, (1993). [14] Nulton, J. D., Salamon, P. and Patria, R. K., amotlike proesses in. finite time. II. Appliations to model yles, Am. J. Pys. 61, (1993). [15] en, J. and Yan, Z., Optimal performane of an endoreversible-ombined refrigeration yle, J. Appl. Pys. 63, (1988). [16] Bejan, A., eory of eat transfer-irreversible refrigeration plants, Int. J. eat Mass ransfer 3, (1989). [17] Gordon, J. M. and Ng, K.., ermodynami modelling of reiproating illers, J. Appl. Pys. 75, (1994). [18] offmann, K.., Andresen, B. and Salamon, P., Measures of dissipation, Pys. Rev. A 39, (1989). [19] Gordon, J. M. and Ng, K.., ool ermodynamis, (ambridge International Siene, UK, 000). [0] u,., en,. and en, J., Reent Advanes in Finite-ime ermodynamis, edited by, (Nova Siene, New York, 1999). [1] Bejan, A., Entropy Generation Minimization R, (Boa Raton, F, 1996). [] Amelkin, S. A., Andresen, B., Burzler, J. M., offmann, K.. and sirlin, A. M., ermo-meanial systems wit several eat reservoirs: maximum power proesses, J. Non-Equilib. ermodyn. 30, (005). [3] Andresen, B. and Salamon, P., ermodynamis of Energy onversion and ransport, edited by S. Sieniutyz and A. De Vos, (Springer, New York, 000), pp at. Am. J. Pys. Edu. Vol. 6, No. 3, Sept ttp://

7 Mamoud uleil [4] Sön, J.. and Andresen, B., J. Pys. em. 100, 8843 (1996); Bak,. A., Salamon, P. and Andresen, B., J. Pys. em. A 106, (00). [5] uleiil, M., Andresen, B., onvetive eat transfer law for an endoreversible engine, J. Appl. Pys. 100, , 1-4 (006). [6] uleiil, M., Andresen, B., Optimal piston trajetories for adiabati proesses in te presene of frition, J. Appl. Pys. 100, , 1-6 (006). [7] uleiil, M., Maximum windmill effiieny in finite time, J. Appl. Pys. 105, (009). [8] Popkov, V. V., Sipitsyn, E. V., Golden setion in te arnot yle, PYS-USP 43, (000). [9] retanu,. E., rasmareanu, M.., Appliations of te Golden Ratio on Riemannian Manifolds, urk J. Mat., 33, (009). [30] atson, S., Nature s proportion system, Impliations (4), 1-5 (006). [31] Kak, S., e Golden Mean and te Pysis of Aestetis, Arive of Pysis: Pysis/ , 004, Foarm magazine 5, (006). [3] efebvre, V. A. and Efremov, Y. N., Possible analogues of ognitive proesses in te patterns of te burster x-ray variability of te rapid, Astronomial and Astropysial ransations 18, (1999). [33] Smirnov, V. A., Musi teory and te armony metod in J. Kepler s work te armony of te universe, Astronomial and Astropysial ransations 18, (1999). [34] BEJAN, A., e golden ratio predited: Vision, ognition and loomotion as a single design in nature, Int. J. of Design & Nature and Eodynamis 4, (009). [35] Markowsky, G., Book review: e golden ratio, Notis of te AMS 5(3), (005). [36] Kaplan, A., ortumlu, N. and izari, S., A Simple onstrution of te Golden Ratio, orld Applied Sienes Journal 7(7), (009). [37] ofstetter, K., Divison of a Segment in te Golden Setion wit Ruler and Rusty ompass, Forum Geometriorum 5, (005). [38] ofstetter, K., Divison of a Segment in te Golden Setion wit Ruler and Rusty ompass, Forum Geometriorum 6, (006). [39] ong, J. and Kung, S., A Simple onstrution of te Golden Ratio, Forum Geometriorum 7, 31 3 (007). [40] ilfried, U., e situation in steam turbine onstrution and urrent development trends, OMMI, 1-9 (003). [41] Isikawa, M., eraui, M., Komori,., Yasuraoka, J., Development of ig effiieny gas turbine ombined yle power plant, Mitsubisi eavy Industries, td., enial Review 45, (008). at. Am. J. Pys. Edu. Vol. 6, No. 3, Sept ttp://