First and Second-Law Efficiencies in a New Thermodynamical Diagram

J. Non-Equilib. Thermodyn. 2002 Vol. 27 pp. 239±256

First and Second-Law Ef®ciencies in a New Thermodynamical Diagram JoseÂ F. Branco1, Carlos T. Pinho2, Rui A. Figueiredo3 Dept. Mechanical Engineering, Polytechnic Institute of Viseu, Viseu, Portugal 2 CEFT DEMEGI, University of Porto, Porto, Portugal 3 Dept. Mechanical Engineering, University of Coimbra, Coimbra, Portugal

1

Registration Number 931

Abstract Graphical representations of the ®rst and second-law ef®ciencies of heat engines, refrigerators and heat-pumps are used to compare these real devices with their corresponding reversible counterparts. Other representations, such as the temperature-energy diagram, also known as the Bejan/Bucher diagram, illustrate the conservation nature of the ®rst and second-law of thermodynamics. This work intends to combine the major bene®ts of these thermodynamical representations by means of a dimensionless approach and using the concepts behind the Bejan/Bucher diagram into a new diagram. The proposed chart allows a direct reading of the ®rst and second law ef®ciencies and of the entropy generation. It may be used as well to compare different thermal machines among them, with the available thermodynamical models and with the observed performance of state of the art devices, from both ®rst and second-law viewpoints. Using simple geometrical concepts, a number of thermodynamical principles can also be easily deduced. This is illustrated, in Appendix A, through the derivation of the Curzon-Ahlborn formula for the ef®ciency of endoreversible engines; the most complex case of the Curzon-Ahlborn engine ± different cold and hot thermal resistances ± is considered. 1. Introduction The temperature-energy interaction (T-E) diagram, also known as the Bejan/Bucher diagram, illustrates both the ®rst and second laws of thermodynamics in the operation of a Carnot cycle. According to Bejan [1], it was introduced in the Russian literature by Brodianskii [2], and later independently by Bejan [3]. In the physics literature it was introduced by Bucher [4] for the reversible Carnot cycle, and Wallingford [5] in the case of irreversible systems. The use of the T-E diagram has been extended to other types of thermal processes. Yan and Chen [6] used it in the case of cycles with non-isothermic heat transfer and with non-adiabatic work delivery; they [7] also have J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3 # Copyright 2002 Walter de Gruyter Berlin New York

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treated the case of the endoreversible Carnot engine and presented a graphical derivation of the Curzon-Ahlborn [8] ef®ciency. Bucher [9] has made a generalization of the use of the diagram to the other basic reversible processes: polytropic, isochoric and isobaric, and treated the case of irreversible processes, seen as a combination of reversible ones. The free expansion and mixing of ideal gases and the contact of bodies at different temperatures were used as examples of this approach [9]. All these authors have recognized the usefulness of the T-E diagram, which also appeared in several engineering textbooks [1, 10, 11]. More recently, Chen and Andresen [12] gave detailed geometric rules to construct the Bejan/Bucher diagram for the case of endoreversible engines, to obtain a geometric derivation of the Curzon-Ahlborn results, but they did not include the case of different heat resistances between the working ¯uid and the cold and hot reservoirs. Endoreversible engines and in general endoreversible thermodynamics have been the subject of profound interest for both engineers and physicists in the last quarter of the century; a thorough review of the ®eld may be found in Hoffmann et al. [13]. Two other types of graphics representing the ®rst and second-law ef®ciencies of engines, refrigerators and heat pumps were introduced by Radcenco et al. [14] and Bejan [11], respectively. The ®rst type depicts the ®rst-law ef®ciency against the ratio T=To , where To is the ambient temperature ± for refrigerators T=To < 1 and for heat engines and heat pumps, T=To > 1. The second one represents, against the same abscissa, the second-law ef®ciency and may be seen, in the words of its author, as a way of normalizing the traditional ®gures of merit of cyclic thermal devices [11]. With this work, an attempt is made to combine the types of diagrams referred to above, allowing a more straightforward representation of the entropy generation and of the ®rst and second-law ef®ciencies. The proposed diagram condenses all the information conveyed by the T-E, the Radcenco and the Bejan second-law diagrams. The developed procedure is then used, in Appendix A, to present a new derivation of the Curzon-Ahlborn ef®ciency, from the simpler case of the Novikov [15] endoreversible engine to the case of a Curzon-Ahlborn engine with different high and low thermal resistances; the approach used only draws on geometrical principles. 2. Current Graphical Representations of the Thermodynamics Laws To ease the presentation of the new diagram, the features of the existing diagrams are brie¯y described. 2.1. The T-E Diagram Figure 1 represents the T-E diagram. The vertical axis corresponds to the absolute scale of temperatures and the length of the horizontal line segments to the absolute values of the work and of the heat ¯ows. In this case a reversible Carnot cycle [3, 4] RE, a heat engine E and a refrigerator R ± these last operating under externally irreversible Carnot cycles [3, 5] ± are presented. For comparison purposes the same thermal reservoir temperatures ± high temperature TH and low temperature TL ± and J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3

First and second-law ef®ciencies diagram

241

Fig. 1. The Bejan/Bucher diagram illustrates energy conservation and entropy generation.

the same heat ¯ow exchanged with the high-temperature reservoir QH were considered. The other energy transfers are heat transfer from/to the low-temperature reservoir QL and work done/received by the system W. The use of a heat pump P, instead of a refrigerator, would not change this analysis. The line segment representing the absolute value of QH is depicted at temperature TH , whereas at temperature TL both segments representing the absolute values of QL and W are drawn. The diagram illustrates the ®rst law of thermodynamics, which may be written as X Qi W 0; 1 i

where the index i refers to each of the thermal reservoirs, since it can be easily noted that, for the three different cycles, jQH j jQL;rev j jWrev j;

2a

jQH j jQL;E j jWE j;

2b

jQH j jQL;R j jWR j jQL;P j jWP j:

The thermal ef®ciency of the engine I;E , the coef®cient of performance COP of the refrigerator "R, or of the heat pump "P , I;E jWj=jQH j;

"R COPR jQL;R j=jWR j;

"P COPP jQH;P j=jWP j 3

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may be represented by the ratio of the corresponding horizontal segments in Figure 1. In the case of a reversible Carnot cycle I;Erev TH ÿ TL =TH ;

"Rrev TL =TH ÿ TL ;

"Prev TH =TH ÿ TL 4

and the thermal ef®ciencies can also be represented by the ratio of the vertical segments. On the other hand, the second law of thermodynamics can also be written as X Qi =Ti 0; Sgen ÿ

5

i

the generated entropy ÿSgen ± being always positive, except in the case of a reversible cycle. In the T-E diagram, the generated entropy may be represented, noting that [1, 9] SgenE tanE ÿ tanH 0;

6a

SgenR SgenP tanH ÿ tanR 0:

6b

The lost available work at temperature TL ÿ Wlost;L ÿ; as stated by the Gouy-Stodola theorem [1], is Wlost;L Sgen TL ;

7

or, in this case, Wlost;LE TL tanE ÿ tanrev ; Wlost;LR Wlost;LP TL tanrev ÿ tanR

8

and it is represented by the segments RÿRE or REÿE in Figure 1. A second law ef®ciency II can also be represented in Figure 1. Using the standard de®nitions, II;E WE =WErev ;

II;R WRrev =WR ;

II;P WPrev =WP

9

these ef®ciencies are represented by the ratios of the corresponding horizontal line segments. 2.2. The First and Second-law Diagrams of Radcenco and Bejan The Radcenco diagram represents the ®rst law ef®ciency of engines, refrigerators and heat-pumps, versus the quotient between the limiting temperatures of the heat reservoirs, the ambient temperature always being in the denominator. In the case of the second-law Bejan diagram, the ordinate axis refers to the second law ef®ciency. Figure 2 represents both these diagrams, where the available results of ten real power plants [16] and a refrigeration plant cited by Kotas [17] are also presented. J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3


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Fig. 2. First and second-law ef®ciencies of heat engines, refrigerators and heat pumps. (a) Radcenco ®rst-law ef®ciency diagram; (b) Bejan second-law ef®ciency diagram.

Up to this point the features of the existing diagrams have been summarized, following [1, 5], for the case of an irreversible thermal machine. Now, and following the same basic layout, a new diagram, allowing a straightforward representation of entropy generation and of the second law ef®ciency, will be introduced. 3. A Modi®ed Diagram In the conventional T-E diagram, entropy generation can be represented as the difference between the tangents of two angles, Eq. (6). In a diagram with unitary axes, it would be represented in a linear scale, with improved readability. To this end, the ®rst and the second laws of thermodynamics can be depicted in a dimensionless form. Choosing the maximum absolute value of the heat ¯ow and the maximum temperature of the cycle, Qmax QH and Tmax TH , as scaling variables, the following dimensionless parameters can be introduced: Q Q=Qmax ;

W W=Qmax ;

T T=Tmax ;

S S=Qmax =Tmax ; 10

where denotes a dimentionless variable. Equations (1) and (5) can then be rewritten as X Qi W 0; 11 i

ÿ Sgen

X i

Qi =Ti 0:


12

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Fig. 3. Proposed diagram for an irreversible engine. The diagram illustrates the ®rst and second law of thermodynamics and allows the direct reading of the ®rst and second law ef®ciencies as well as the entropy transfers.

Figure 3 shows the result of this approach for the case of an irreversible thermal engine, which may be represented by its cold reservoir temperature TL and the respective heat ¯ow QL at point E, signalized with a circle marker. The diagram axes extend from zero to TH 1 and QH 1, respectively, and the line segments BJ and Q AE are sketched, again using their absolute value. corresponding to QH L The work done by the engine is W EG, and its thermal ef®ciency I;E may be represented in the left axis ± DF segment. For a reversible Carnot engine with the same QH , the heat transfer to the low-temperature reservoir QL0 should fall over the OJ line ± point E0 . Hence, the second law ef®ciency, II;E EG=E0 G, can be represented by the segment CF in the left axis, since EG=E0 G CF=OF. In fact, according to the Thales theorem, EJG CJF and E0 JG OJF ( denotes a triangle whereas is the symbol for ``similar''). In the diagram, values of Q =T can also be represented by connecting with straight lines point O with the top of line segments BJ and AE and continuing to point K on



245

the right vertical axis. The segment BK represents QL =TL , the segment BJ represents QH =TH , and its difference JK according to Eq. (12), represents the entropy generation . In the case of a reversible engine, point E0 , instead of E, should be used, and the Sgen entropy transfer associated with the high and low temperature reservoirs should be the same, with the entropy generation being zero. But, using a simple geometrical construction, the entropy generation can also be represented without the need to extend the right axis beyond point J. It can be T =Q . Thereobserved that IJK OAE, so IJ=JK OA=AE and IJ Sgen L L fore, the length of the IJ segment may be seen as an alternative and indirect measure of the entropy generation, eliminating the need to expand the vertical axis out of the [0, 1] interval. (With the aid of a slightly more complex geometrical construction, the may be represented in the top axis; the presented procedure was true value of Sgen preferred because of its simplicity.) The case of a generic refrigerator or heat pump is illustrated by point G in Figure 4. The temperature scale is now on the top axis, the right axis represents again both Q and Q =T , and the bottom axis represents the COP and the second law ef®ciency, though, in the case of the former, depicting its inverse. Segment DG represents QL at at T . Irreversible refrigerators (or heat temperature TL and EL represents QH H pumps) are represented in the lower half of the diagram, whereas reversible devices are represented over the line OL.

Fig. 4. Proposed diagram for an irreversible refrigerator/heat-pump. J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3

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The COP of the heat pump, EL=GK can be found by drawing a horizontal line from point G to OL, which, again, represents the locus of reversible operation; from point F a vertical segment leads to point C on the lower horizontal axis. The segment CE represents the inverse of the heat pump COP, 1="P , since CE FH HL GK. In the case of the refrigerator, the COP is represented by the quotient DG=GK. Using similarity of triangles EFH and EJL, it can be seen that 1="R JL; hence, 1="R can be marked by drawing a line from E to F, and then to J and B. The second law ef®ciency, as de®ned in Eq. (9), is the same for the refrigerator and for the heat pump, II;R II;P G0 K=GK; considering that GHL AEL and that G0 K GH, it can be found connecting points L and G, to obtain A on the bottom axis, II;R II;P AE. To ®nish the refrigerator/heat pump analysis, the entropy generation may be represented by the segment IL, obtained directly subtracting =T . QL =TL from QH H Figure 5 shows the proposed diagram, now with all the axes needed to depict the operation of any thermal device (engine, refrigerator or heat pump) operating between two limiting temperatures. In the lower half, the working point of refrigerators

Fig. 5. Proposed diagram for an irreversible engine, refrigerator or heat pump. In this diagram any thermal device operating between two limiting temperatures may be represented; the diagram axes allow the direct reading of the ®rst and second law ef®ciencies as well as the entropy generation. Point E refers to the engine represented in Figure 3 and point G to the refrigerator of Figure 4. The dotted line represents the Curzon-Ahlborn ef®ciency and the dashed ones the models of Bejan, Eq. (13). J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3


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and heat pumps can be depicted, whereas the operation of thermal engines may be represented in the higher corner. The dotted curve describes the Curzon-Ahlborn ef®ciency, which, according to a number of sources [8], can give a rough estimate of the ef®ciency of actual engines, as can be con®rmed by comparing with the information of ten real power plants [16]. On the other hand, the dashed lines represents the models developed by Bejan [1], giving expressions for the ®rst and second law ef®ciencies of engines and refrigeration plants, respectively: I;E 1 ÿ TL TL lnTL ;

13a

II;R 1=1 Ci 1 ÿ TL :

13b

In the case of the refrigeration model, Ci represents a dimensionless internal conductance of the plant, which, from empirical results, and according to Bejan, may be taken as Ci 5. Point E on the diagram of Figure 3 represents the available values for the CANDU nuclear reactor [16] while point G on Figure 4, also marked in Figure 5, represents a refrigeration plant cited by Kotas [17]. The diagram allows the direct comparison of the ®rst and second-law ef®ciencies of the represented devices. As noted before, in the case of thermal engines, the top axis of the diagram of Figure 5 represents Sgen TL =QL intead of Sgen . Using simple geometric considerations, a relation between the entropy generation and the ®rst law ef®ciency may be obtained for each of the thermal devices. In the case of the engine, and considering Figure 3, it can be seen that JK=BJ EE0 =AE0 (OBK OAE and OBJ OAE0 ); on the other hand AE0 1 ÿ E0 G and EE0 E0 G ÿ EG. So JK=BJ E0 G ÿ EG=1 ÿ E0 G and considering that E0 G represents the Carnot ef®ciency C 1 ÿ TL and that the segment BJ has unitary length, this geometric equation may be written as Sgen;E C ÿ I;E =1 ÿ C :

14a

Similar equations for the case of refrigeration and heat pumping devices are " ÿ " =" 1 " ; Sgen;R R;C R R;C R

14b

" ÿ " =" " ÿ 1; Sgen;P P;C P P P;C

14c

where the subscript C designates the Carnot ef®ciency. These are alternative expressions for the Gouy-Stodola theorem applied to engines, refrigerators and heat pumps. 4. Conclusion A dimensionless diagram, allowing the graphical representation of the ®rst and second laws of thermodynamics was presented. It was shown that the diagram allows J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3

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the direct reading of the ®rst and second law ef®ciencies in the case of real engines, refrigerators and heat pumps; this feature may be used to rapidly compare different devices and different theoretical models. It also allows the representation of the entropy generation using a linear scale. The employed approach was also used to derive the Gouy-Stodola theorem through simple geometrical principles. In Appendix A a geometric proof of the Curzon-Ahlborn results will be also presented, from the simpler case of the Novikov engine to the case of an endoreversible engine with different cold and hot thermal resistances. Acknowledgement The ®rst author gratefully acknowledges the support of PRODEP II (Programa de Desenvolvimento Educativo para Portugal). Appendix A Geometrical Derivation of the Ef®ciency of an Endoreversible Engine at Maximum Output With the aid of the previous analysis, it is easy to derive the ef®ciency of an endoreversible engine at maximum output. This will be done ®rst for the simpler case of the Novikov engine (zero thermal resistance in the low temperature reservoir), Figure A1(a). The case of the Curzon-Ahlborn engine, with equal, Figure A1(b), and different, Figure A1(c), thermal resistances between the engine and the thermal reservoirs will be considered later, using an, although slightly more complex, entirely similar process. Figure A1, adapted from Bejan [15], represents the three models described above. A semi-graphical derivation for the simpler case can be found in Yan and Chen [7] and for the intermediate one in Chen and Andresen [12]. Here a different approach is used and the more complex case will also be treated. In this development, TH;E is the high end temperature of the engine, TL;E its low end temperature, QH;E and QL;E the corresponding heat ¯ows. For the temperature of the

Fig. A1. Models of endoreversible engines: (a) Novikov (1957), (b) Curzon-Ahlborn (1975) with RH RL , (c) Curzon-Ahlborn (1975) with RH 6 RL . J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3


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reservoirs and for the heat ¯ows in the case of the zero thermal resistance, the same designations used above, TH , TL , QH and QL , are preserved. The thermal resistances ± the inverse of the product of the overall heat transfer coef®cient and the contact area (R 1=UA) ± between the engine and the thermal reservoirs are RH and RL . The chosen scaling variables to obtain dimensionless expressions for the ®rst and second laws are TH and TH =RH . A.1. The Novikov Model In the case of an engine with a zero thermal resistance connecting to the low temperature reservoir and a ®nite one in the connection with the high temperature reservoir, the following dimensionless equations apply: QH;E TH ÿ TH;E =RH

1 ÿ T ; or QH;E H;E

A1

QH;E ÿ QL W ;

A2

Q =T : QH;E =TH;E L L

A3

Figure A2 represents the corresponding diagram. Line segment DE represents the heat ¯ow from the high temperature reservoir to the engine as TH;E varies from TH to TL . The work delivered by the engine is zero when TH;E equals TH or when TH;E

Fig. A2. Diagram used to derive the Curzon-Alhborn ef®ciency for the engine represented in Figure A1(a). The hatched area represents the product TL W and was maximized geometrically. J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3

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reaches TL . Somewhere between these limits there is a value of TH;E that maximizes the work output [8]; it can be found using the following geometrical procedure. The high-end temperature of the engine is chosen indiscriminately, TH;E TA , and the corresponding engine represented in the diagram, point A, with abscissa TA and ordinate QA . Instead of the diagonal OA, the diagonal QA TA is drawn, allowing the work delivered to be represented by the segment GTL , and QL by GJ. From now on, when referring to a point on the axes, it will only be identi®ed by the respective axis coordinate; for example TL and QA designate points (TL ; 0) and (0; QA ), respectively. Following an idea of Yan and Chen [7], there must exist another engine B, with highend temperature TB , which develops the same power. From Figure A2 it can be seen that this happens if the area SL remains constant (since SL TL W and TL does not change). It may be noted that SL SA , where SA is the rectangle GIAJ area (symbolically &GIAJ), and the choice of TB can be made noticing that SB and SA must also be equal. This implies that NB IA which, accounting for the similarity of QA OTA and QA FG, may be written as TB ÿ TL =TL 1 ÿ TA =TA :

A4

In fact, instead of NB IA, it should be written NB II 0 , as illustrated in Appendix B, but in this case IA and II 0 are equal. It is evident that the maximum work delivery, which corresponds to the maximization of the SL area, is obtained when TB TA Topt or, solving Eq. (A4), TA TB TL ;

T 1=2 Topt L

A5

where the subscript opt stands for optimum ± in the sense that maximum power is achieved. An alternative and simpler way to reach the previous result calls for the following geometrical principle: the maximum area parallelogram inscribed in a triangle may be obtained by halving its height or any of its sides (Appendix B). It was seen that the delivered work can be maximized through the maximization of &GIAJ; accordingly, this happens when the ratio between the segments GI and GI 0 equals 1/2. Considering that GI 0 1 ÿ WA ÿ TL , this may be written as ÿ T 1 ÿ W ÿ T =2; TH;E L L

A6

1 ÿ T ; W 1 ÿ TL =TH;E H;E

A7

where

from the de®nition of the Carnot ef®ciency or considering that GTL TA QA OTA . Solving these equations also leads to the result expressed by Eq. (A5), T T 1=2 , and from the de®nition of the dimensionless variables Topt TL TH 1=2 :

opt

L

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For an internally reversible engine, the ®rst law ef®ciency is given by opt CA 1 ÿ TL =TL TH 1=2 1 ÿ TL =TH 1=2 ;

A9

CA being the Curzon-Ahlborn ef®ciency. The cycle delivering maximum power, point CA, can be easily represented, noting that the length of the OM segment, obtained by intersecting line x TL with the circumference of unit diameter passing through points O and D, corresponds to the square root of TL (cf. Henderson [18]). In the case of the Novikov model, the hatched square in the diagram of Figure A2 represents the product of W and TL for the maximum output conditions. A.2. The Curzon-Ahlborn Model When the Curzon-Ahlborn engine with equal high and low end thermal resistances R is considered, (Figure A1b), the following dimensionless equations apply QH;E TH ÿ TH;E =R

1 ÿ T ; or QH;E H;E

A10

QL;E TL;E ÿ TL =R

T ÿ T ; or QL;E L;E L

A11

W ; QH;E ÿ QL;E

A12

Q =T : QH;E =TH;E L;E L;E

A13

The interaction between the engine and the high temperature reservoir is described by the line DE in the diagram of Figure A3, as in the preceding example. On the other hand, the low-end interaction can be repreented by a perpendicular line passing can vary between points T and D, and T through point TL . The temperature TH;E X L;E between points TL and TX . As in the case of the Novikov model, let point A represent the high-end of the engine. Since it is internally reversible, its low-end must be over the line OA, but also over TL X, which represents the heat transfer law between the engine and the low temperature reservoir. This de®nes the engine low-end operating T , point AL and the corresponding temperature TL;A L;E T T =2T ÿ 1: TL;A A L A

A14

HF and &HKAN have the same area. In From Figure A3 it is easy to note that &OTL;A addition, segments TL;A AL and HN also have the same length; consequently TL H 0 G AL PN. So, it is clear that the surface area of the shaded regions SL and SA are also equal. The aim is to maximize these areas, since this corresponds to the maximization of the work delivery. According to the geometrical result referred should satisfy the to previously and presented in Appendix B, the optimum TH;E equation ÿ T 1 ÿ W ÿ T =2; TH;E X X J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3

A15

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Fig. A3. Diagram used to derive the Curzon-Alhborn ef®ciency for the engine represented in Figure A1(b). The hatched area represents the product TL W and was maximized geometrically.

where TX 1 TL =2 and =T 1 ÿ T : W 1 ÿ TL;E H;E H;E

A16

Solving Eqs. (A14) to (A16) results in the following equation for TH;E 1 T 1=2 =2; Topt L

A17

and the corresponding engine low-end temperature is T 1=2 1 T 1=2 =2; TL;opt L L

A18

the resulting expression for the ®rst law ef®ciency being opt 1 ÿ TL;opt =Topt CA 1 ÿ TL =TH 1=2 ;

A19

which is again the Curzon-Ahlborn formula. In this case (RL RH ), the product of W and TL for the engine with maximum output is a rectangle whose height is half of was marked in the midpoint of its base, as can observed in Figure A3, where Topt p be the segment TL D. Following the same kind of reasoning, it is not dif®cult to consider the case of an endoreversible engine with different thermal resistances in its low and high ends. Considering that the total thermal resistance R, regarded as a ®xed constraint, is J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3


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unevenly divided between the high and low temperature reservoir connections, and following Bejan [1], one may write, 1=RH '=R

and 1=RL 1 ÿ '=R;

A20

with 0 < ' < 1. Using the mean value of the thermal conductance UA=2 1=2R as a scaling variable, Eqs. (A10) and (A11) take the form QH;E 2'TH ÿ TH;E =R

2'1 ÿ T ; or QH;E H;E

QL;E 21 ÿ 'TL;E ÿ TL =R

21 ÿ 'T ÿ T ; or QL;E L;E L

A21 A22

leading to the diagram of Figure A4. Lines DE and TL X represent Eqs. (A21) and (A22), respectively. Choosing TA as the engine high temperature, the corresponding low-end one, point AL , is 1 ÿ 'T T =T ÿ '; TL;A A L A

A23

HF and &HKAN, but also and considering again the equality of rectangles &OTL;A that &HIPN and &TL TL;A HG are equal (which can be easily proved considering that TL H 0 G AL PN) it can be found that, in this case, the maximization of W may be achieved by maximizing &IKAP or maximizing &JKAQ, which is the

Fig. A4. Diagram used to derive the Curzon-Alhborn ef®ciency for the engine represented in Figure A1(c). The hatched area represents the product TL W and was maximized geometrically. J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3

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same. This may be done by again solving Eqs. (A15) and (A16), but now with de®ned by Eq. (A23); the solution for T is TX ' 1 ÿ 'TL and TL;A opt ' 1 ÿ 'T 1=2 ; Topt L

A24

and the corresponding low-end temperature is 'T 1=2 1 ÿ 'T : TL;opt L L

A25

As a result, the ®rst law ef®ciency may once more be expressed by the CurzonAhlborn formula. In the case of an engine with different thermal resistances, RH and RL , the lines representing the heat transfer laws between the engine and the thermal reservoirs have different slopes, and the ratio between the height and width of the WTL max rectangle is 2'1 ÿ '. The corresponding point CA may be marked intersecting the high heat transfer line (DE) with the parallel to the low one that passes through the point TL 1=2 , as can be seen in Figure A4. The expression for the maximum work of the engine may be written as 2'1 ÿ '1 ÿ T 1=2 2 21 ÿ T 1=2 2 1=4 ÿ ' ÿ 1=22 ; Wopt L L A26 the last form shows immediately that, as demonstrated by Bejan [1] with a ®xed R constraint, the optimal allocation occurs when ' ÿ 1=2 0 in the previous equation RL RH . Appendix B Maximizing the Area of a Parallelogram Inscribed in a Triangle In Appendix A, and pertaining to a triangle, it was af®rmed that two inscribed rectangles have the same area when the sum of their widths and heights equal the triangle base and height, respectively. It was also said that the inscribed parallelogram of maximum area may be obtained halving any of the triangle's sides. Figure B1 provides immediate evidence of the ®rst statement, since area A1 (A1R A1L and A2 (A2R A2L are equal, and both A1 and A2 represent inscribed recangles. The case of a right triangle is a particular one (B 45 , where is the symbol for angle) and the rectangle width and height may be simply reversed to obtain a new inscribed rectangle with identical area; this is the case of constructions represented in Figure A2. The second statement may be proved considering that A1 A2 0 for y 0 and y H. This means that as y departs from zero, the rectangle area increases, attains a maximum, and then, as y goes towards H, decreases again. It is always possible to have a diferent inscribed rectangle with the same area ± ®rst statement ± so the only J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3


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Fig. B1. Geometrical construction used to show that in a triangle, the inscribed parallelogram with maximum area has half of the triangle area; this principle is used in the alternative proof of the Curzon-Ahlborn formula.

possibility of maximizing this area is when both rectangles coincide. This implies that, y H ÿ y, or y H=2, as stated before, corresponding to the rectangle acde in Figure B1. It is also clear that a number of parallelograms with maximum area can be constructed in the mentioned way; equality of triangles Abe, bBd, bde and Ced prove this statement. These results may also be shown using differential calculus [19]. References [1] Bejan, A., Entropy Generation Minimization, CRC Press, Boca Raton, 1996. [2] Brodianskii, V.M., Exergetic Method of Thermodynamic Analysis, Energia, Moscow, 1973. [3] Bejan, A., Graphic techniques for teaching engineering thermodynamics, Mech. Eng. News, 14 (2) (1977), 26. [4] Bucher, M., New diagram for heat ¯ows and work in a Carnot cycle, Am. J. Phys., 54 (1986), 850. [5] Wallingford, J., Inef®ciency and irreversibility in the Bucher diagram, Am. J. Phys., 57 (1989), 379. [6] Yan, Z., Chen, J., Modi®ed Bucher diagram for heat ¯ows and works in two class of cycles, Am. J. Phys., 58 (1990), 404. [7] Yan, Z., Chen, J., New Bucher diagram for a class of irreversible Carnot cycles, Am. J. Phys., 60 (1992), 475. [8] Curzon, F.L., Ahlborn, B., Ef®ciency of a Carnot engine at maximum power output, Am. J. Phys., 43 (1975), 22. [9] Bucher, M., Diagram of the second law of thermodynamics, Am. J. Phys., 61 (1993), 462. [10] Bejan, A., Entropy Generation Through Heat and Fluid ¯ow, pp. 25±28, Wiley, New York, 1982. [11] Bejan, A., Advanced Engineering Thermodynamics, pp. 116±123, Wiley, New York, 1988. [12] Chen, J., Andresen, B., Diagrammatic representation of the optimal performance of an endoreversible Carnot engine at maximum power output, Eur. J. Phys., 20 (1999), 21. [13] Hoffman, K.H., Burzler, J.M., Schubert, S., Endoreversible Thermodynamics, J. NonEquilib. Thermodyn., 22 (1997), 311. [14] Radcenco, V., Porneala, S., Dobrovicescu, A., Procese in Instalatii Frigori®ce, p. 21, Editura Didactica si Pedagogica, Bucharest, 1983. [15] Bejan, A., Models of power plants that generate minimum entropy while operating at maximum power, Am. J. Phys., 64 (1996), 1054. J. Non-Equilib. Thermodyn. 2002 Vol. 27 No. 3

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[16] Bejan, A., Theory of heat transfer-irreversible power plants, Int. J. Heat Mass Transfer, 31 (1988), 1211. [17] Kotas, T.J., The Exergy Method of Thermal Plant Analysis, Chap. 5, Butterworths, London, 1985. [18] Henderson, D.W., Experiencing Geometry: On Plane and Sphere, Chap. 12, Prentice Hall, New York, 1996. [19] Simmons, G.F., Calculus with Analytic Geometry, 2nd edition, Chap. 4, McGraw-Hill, New York, 1995. Paper received: 2001-11-01 Paper accepted: 2002-04-17 JoseÂ F. Branco Dept. Mechanical Engineering Polytechnic Institute of Viseu Campus PoliteÂcnico, 3504-510 Viseu Portugal E-mail: j®[email protected] Carlos T. Pinho CEFT-DEMEGI, University of Porto Rua Dr. Roberto Frias, 4200-465 Porto Portugal E-mail: [email protected] Rui A. Figueiredo Dept. Mechanical Engineering University of Coimbra Pinhal de Marrocos, 3030-000 Coimbra Portugal E-mail: rui.®[email protected]
