Entransy analysis of open thermodynamic systems

Transcription

1 Article Engineering hermohysics August 0 Vol.57 No.: doi: 0.007/s x Entransy analysis of oen thermodynamic systems CHENG Xueao, WANG WenHua & LIANG XinGang * Key Laboratory for hermal Science and Power Engineering of Ministry of Education, Deartment of Engineering Mechanics, singhua University, Beijing 00084, China Received January, 0; acceted Aril 7, 0 he concet of entransy develoed in recent years can describe the heat transort ability. his aer extends this concet to the oen thermodynamic system and defines the concet of enthaly entransy. he entransy balance equation of steady oen thermodynamic systems, as well as the concet of entransy loss, is develoed. he entransy balance equation is alied to analyzing and discussing the air standard cycle. It is found that the entransy loss rate can describe the change in ower outut from the cycle but the entroy generation rate cannot when the heat absorbed by the working medium is from the combustion reaction of the gas fuel. When the working medium is heated by a high temerature stream, both the maximum entransy loss rate and the minimum entroy generation rate corresond to the maximum ower outut from the cycle. Hence, the concet of entransy loss is an aroriate figure of merit that describes the cycle erformance. entransy loss, enthaly entransy, entransy balance equation, oen system, air standard cycle Citation: Cheng X, Wang W H, Liang X G. Entransy analysis of oen thermodynamic systems. Chin Sci Bull, 0, 57: , doi: 0.007/s x As nearly 80% of the total energy consumtion is related to heat [], the otimization design of heat transfer and heatwork conversion has become one of imortant research toics under the increasing requirements on reducing energy consumtion. For instance, the design otimization could imrove the erformance of heat exchangers and increase heat transfer rate []. Otimizing the thermal conductance distribution of the heat exchanger grous could also increase the work outut in heat-work conversion []. Several theories have been develoed for the otimal design of heat transfer and heat-work conversion in recent years [ 7]. Bejan [4 6] develoed the constructal theory and thermodynamic otimizations for heat transfer rocesses. In heat conduction otimization of the Volume-to-Point roblem, Bejan [4] develoed the constructal theory, in which the basic, imagined structure of the high conductivity material is given, and the asect ratio of the structure is otimized through theoretical derivation and numerical simulation to decrease the highest temerature of the heated domain. he sub-structure, third degree structure, etc. are evolved to *Corresonding author ( liangxg@tsinghua.edu.cn) cover the whole domain. he constructal theory was found to be effective for heat conduction otimization and was extended to heat convection otimizations [8 0]. Furthermore, Bejan [4 6] related the otimal heat transfer to the minimum roduction of entroy generation and extended this idea into other systems. Researchers have done many investigations on heat transfer otimization based on the minimum rincile of entroy generation [,]. As the objective of this kind of otimization method is to minimize the entroy generation, it is called thermodynamic otimization. However, there are arguments on the constructal theory and thermodynamic otimization in heat transfer [6,3 7]. Ghodoossi [3,4] alied the constructal theory to analyzing the effect of the comlexity levels of the tree work of conducting aths and found that the heat flow erformance does not essentially imrove if the internal comlexity of the heat generating area increases. In heat convection, Escher et al. [6] showed that the arallel channel work achieves a more than fivefold higher erformance coefficient than the bifurcating tree-like work with a constant flow rate, while almost four times more heat can be removed for a constant ressure gradient across the works. For the minimum he Author(s) 0. his article is ublished with oen access at Sringerlink.com csb.scichina.com

2 Cheng X, et al. Chin Sci Bull August (0) Vol.57 No. 935 rincile of entroy generation, it was noted that the heat exchanger effectiveness does not always increase with decreasing entroy generation when the concet of entroy generation is used to analyze heat exchangers [6]. Instead, it may become smaller under some conditions. Shah and Skieko [7] also observed that the heat exchanger effectiveness can be maximum, intermediate or minimum at the maximum entroy generation in their discussion of 8 kinds of heat exchangers. herefore, the minimum entroy generation rincile is not always aroriate to heat transfer otimization. In the last decade, Guo et al. [7] introduced a new hysical arameter, entransy, which describes the heat transort ability. he new concet was roosed by the analogy between heat and electrical conductions [7]. Heat flow corresonds to electrical current, thermal resistance to electrical resistance, temerature to electric voltage, and heat caacity to caacitance. Entransy is actually the otential energy of the heat in a body, corresonding to the electrical energy in a caacitor. For an object whose internal energy is U and temerature is, its entransy is defined as [7] G U. () When heat is transferred from a high temerature object to a low temerature one, it was roved that entransy dissiation will always be roduced [7,8]. So, the concet of entransy could describe the irreversibility of heat transfer [7,8 0]. Furthermore, Cheng et al. [8,,] set u the thermal equilibrium criteria based on the concet of entransy, develoed a microscoic exression of entransy for a monatomic ideal gas system, related the entransy to the microstate number, indicated the microscoic hysical meaning of entransy to some extent, and extended the entransy theory to generalized flows. Xu [3] discussed entransy and entransy dissiation from the thermodynamic viewoint. Based on the concet of entransy and entransy dissiation, Guo et al. [7] derived the minimum entransy dissiation rincile for rescribed heat flux boundary conditions and the maximum entransy dissiation rincile for rescribed temerature boundary conditions. hese rinciles are named as the extremum entransy dissiation rincile. hey further defined the entransy-dissiation-based thermal resistance. hen, the extremum entransy dissiation rincile is evolved into the minimum thermal resistance rincile that is much easier for understanding. he rinciles based on the concet of entransy were also used to otimize the Volume-to-Point roblem [7]. For this roblem, a Lagrange function was formed based on the extremum entransy dissiation rincile with the limitation of finite high conductivity material. By ursing the extremum value of this Lagrange function, a rule of uniform distribution of temerature gradient was obtained mathematically. By utting the high conductivity materials at the lace where the temerature gradient is highest, the construct of high conductivity materials was formed ste by ste and the average temerature of the volume was reduced effectively. It has been shown that the average temerature of the heated domain thus obtained is lower than that by the constructal theory [7,4]. Chen et al. [5] comared the entransy theory with the minimum rincile of entroy generation, and showed that the entransy theory could lead to a lower average temerature of the volume than the minimum rincile of entroy generation. Other researches also indicate that the entransy theory is suitable for the otimizations of heat conduction [6,7], heat convection [8], thermal radiation [9,30] and heat exchangers [,3 34]. In addition, there is no aradox similar to the entroy generation when the entransy theory is used to analyze heat exchanger [,3]. For heat-work conversion, the concet of entroy has been widely used [,3,35 39]. Entroy was first defined based on the reversible cycle of the Carnot engine in 854 [40]. In any irreversible rocess, entroy generation exists. During a hysical rocess, the more entroy generation is, the more the system loses the ability of doing work [40]. With this consideration, entroy generation has been used to otimize thermodynamic rocesses so as to decrease the ability loss of doing work and to increase the work outut []. For instance, Chen et al. [] revealed that the minimum entroy generation always corresonds to the maximum work outut in the otimization of thermodynamic rocess with heat exchanger grous. When analyzing an absortion chiller, Myat et al. [39] showed that the minimization of entroy generation in absortion cycle leads to the maximization of the COP. On the other hand, the alicability of the entransy theory on the otimization of the thermodynamic rocesses with work was also discussed from different viewoints [,4]. Chen et al. [] thought that the concet of entransy dissiation could not be used to otimize heat-work conversion in closed thermodynamic systems. Wu [4] defined the conversion entransy and used the definition to otimize closed thermodynamic cycles. From above discussion, it could be found that the entransy theory is effective in heat transfer otimization, while the concet of entroy is not always. In heat-work conversion, the concet of entroy is widely used. However, there are still no definite conclusions for the alicability of the entransy theory because researchers have different oinions [,4]. he existing reorts on the alicability of the entransy theory to heat-work conversion are all about closed systems. It is worth of making an investigation on the oen thermodynamic systems. his aer exlores the alicability of entransy theory to the oen systems with heat-work conversion. Enthaly entransy For any thermodynamic rocess, the first law of thermody-

3 936 Cheng X, et al. Chin Sci Bull August (0) Vol.57 No. namics gives Q du W, () where Q is the absorbed heat, du is the change of the internal energy and δw is the outut work. Multilying eq. () by the working medium temerature,, leads to Q du W, (3) where the left term is the heat entransy flow as it is associated with heat Q, and the second term on the right hand is called work entransy flow as it is associated with work W. Considering an ideal gas system with constant secific heat caacity at constant volume, c V, the first term on the right hand could be exressed as du dcvm d cvm d U dg, (4) where m is the mass. he first term on the right hand of eq. (3) indicates the change of entransy. he entransy associated with work, heat flow and internal energy are all defined in eq. (3). However, there is no definition for entransy change related to enthaly though it is a very imortant arameter in thermodynamics. We will define the enthaly entransy below. For a thermodynamic system, the definition of enthaly is H U V, (5) where is the ressure and V is the volume of the system. Based on eq. (5), the differentiation of the enthaly is dh d U V du d V. (6) Multilying eq. (6) by temerature results in dh du d V, (7) where the left art of eq. (7) could be defined as the change of enthaly entransy as it is associated with enthaly dh. For an ideal gas system, we have dh d c m d mr V d cv R m cm H d d, where m is the gas mass, R is the gas constant, and c is the secific heat caacity at constant ressure. he integration of eq. (8) yields (8) GH H. (0) he entransy balance equation of oen ideal gas systems is set u in the following section based on eqs. (8) and (0). Entransy balance equation of oen systems A steady oen system is shown in Figure. According to the first law of thermodynamics, there is [4] Q du W W, () where Q is the absorbed heat, du is the change in the internal energy of the working medium, W is the outut work, and W f is the change in the flow work whose exression is f d V Eq. () can be exressed as [4] f W. () Q du d V W dh W. (3) Multilying eq. (3) by temerature, we have Q dh W. (4) Substituting eqs. (8) and (0) into eq. (4) yields where G dg G, (5) Q H W- G Q, (6) Q G W, (7) W - G Q is the entransy flow due to heat exchange, dg H is the change in the enthaly entransy, and G W- is the work entransy flow of the working medium. It could be found that art of the absorbed heat entransy flow goes into the working medium, making the enthaly entransy increase, while the rest is used to do work. Eq. (5) is the entransy balance for steady oen systems. H dh d H H. (9) 0 0 Based on eq. (5), the enthaly entransy of the ideal gas system could be defined as Figure Sketch of an oen thermodynamic system.

4 Cheng X, et al. Chin Sci Bull August (0) Vol.57 No Analyses of the air standard cycle he air standard cycle is widely used in engineering. As shown in Figure, the working medium is comressed through an isentroic rocess, increasing its temerature from to. hen, it absorbs heat during an isobaric rocess, and its temerature gets to 3. After that, it does work during the second isentroic rocess and its temerature decreases to 4. Finally, the used working medium is discharged into the environment, making its temerature back to the initial value after a cycle. he working medium in the cycle is the gas mixture after the combustion reaction of the gas fuel. he working medium can still be treated as the air because about 80% of the air is nitrogen which does not take art in the combustion reaction. hen, the heat generated from the combustion reaction could be treated as that absorbed from an inner heat source. Assume that the air could be treated as the ideal gas. As the working medium gets back to the initial state after a cycle, its enthaly will not change. From energy conservation, the outut ower is Q 0 4 W Q Q 0 Q Q, (8) Q 3 where Q is the heat generation rate from the combustion reaction, and Q 0 is the heat rate released to the environment by the used working medium. In the air standard cycle, there is [4] herefore 4 4. (9) 3 3 W Q. (0) On the other hand, the cycle in Figure can be analyzed from the entransy oint of view. he cycle system is comosed of the working medium and the environment. he enthaly entransy of the working medium will not change when a cycle is finished. According to eq. (5), the work entransy flow rate of the cycle is G G G, () W- Q Q0 where G Q is the absorbed heat entransy flow rate, and G Q0 is that gets out of the working medium. When the used working medium is discharged into the environment, it is cooled and its temerature decreases to the initial value. his rocess is a heat transfer rocess during which entransy dissiation is roduced [7,8]. he entransy dissiation rate is G G G, () dis Q0 Q-e where G Q e is the heat entransy flow that gets into the environment. For the whole system, it could be found that some of G Q is used to do work, some is dissiated during the cooling rocess of the used working medium, and the rest is released into the environment. herefore, the entransy loss rate of the system is defined as G G G G G G G G G.(3) loss W- dis Q Q0 Q0 Q-e Q Q-e Considering the rocesses in Figure, G Q H33 H cm 3, (4) where m is the working medium circulation rate. As there is no work during the heat transfer rocess between the used working medium and the environment, according to eq. (5), G G Q, (5) Q-e H-e 0 where G H -e is the increase rate in the environment enthaly entransy. herefore, from eq. (9), we have G loss cm 3 Q 0 Q cm cm Q Q. cm (6) Figure Sketch of the air standard cycle. Eq. (0) indicates that the ower outut increases with increasing Q and or with decreasing of. At the same time, the entransy loss rate also increases with increasing Q and or with decreasing. herefore, the entransy loss rate could describe the change of the ower for the air standard cycle. Let us analyze the entroy change rate in the air standard

5 938 Cheng X, et al. Chin Sci Bull August (0) Vol.57 No. cycle. he entroy generation rate induced by duming the used working medium into the environment should be considered as indicated by [43]. he entroy generation rate can be calculated by 3 Q0 S g cm ln cm ln, (7) 4 where the first term on the right hand is the entroy increase rate when the working medium absorbs heat, the second term is the entroy decrease rate when the working medium releases heat into the environment, and the last term is the entroy increase rate of the environment. From eqs. (8) and (9), eq. (7) reduces to 3 Q 0 Q S g cm ln. (8) 4 Eq. (8) tells that the entroy generation rate declines with increasing. However, the entroy generation rate will increase with increasing Q. When Q and are rescribed, the entroy generation rate will not change with. Considering the variation tendency of the ower, one could find that the entroy generation rate cannot describe the change of the ower in the air standard cycle. As shown in Figure 3, we consider another case in which the heat absorbed by the working medium is not from the combustion reaction but from a high temerature stream through a counter flow heat exchanger. he heat caacity flow rate, the inlet and the outlet temeratures of the high temerature stream are C H, H-in and H-out, resectively. he heat rate from the high temerature stream is [44] Q C H- min H-in H C H- min H-in H* ex NU C H* H* C ex NUC, (9) where NU is the number of the heat transfer unit of the heat exchanger, C H-min is the smaller one between C H and Figure 3 Sketch of the air standard cycle when the working medium is heated by a high temerature stream. cm, ε H is the effectiveness, and H* C CH cm CH cm min(, ) max(, ). (30) Based on eq. (9), the ower can be calculated by eq. (0). Let us analyze the cycle from the entransy viewoint. As the used high temerature stream and the used working medium are to be discharged into the environment, the entransy dissiation rate roduced in the discharging rocess should be considered. According to eq. (5), some of the heat entransy flow rate from the high temerature stream is dissiated during the heat transfer rocess between the stream and the working medium and in the discharging rocesses of the used high temerature stream and the used working medium, some is used for doing work, and the rest is released into the environment. herefore, G G G G, (3) f-h dis W- Q-e where G f-h is the heat entransy flow rate from the high temerature stream, which is exressed as G f-h CH H-in. (3) G Q-e is calculated by G Q-e Q e Q all W CH H-in W, (33) where Q e is the total heat rate released to the environment, including the heat released by the used high temerature stream; and Q all is the total heat from the high temerature stream. It could be found that some entransy flow rate is lost in heat transfer, while some other is lost when the working medium does work. herefore, G loss G dis G W - G G f-h Q-e CHH-in CHH-in W CHH-in W. (34) he entransy loss rate increases monotonously with increasing ower when C H, H and are rescribed. he maximum entransy loss rate corresonds to the maximum ower outut from the cycle. If the concet of entroy is used to analyze the cycle, there is S g C C H ln Q all W C W H-in H H-in H ln, H-in (35)

6 Cheng X, et al. Chin Sci Bull August (0) Vol.57 No. 939 where the first term on right hand exresses the entroy decrease rate of the high temerature stream, and the second term is the entroy increase rate of the environment. It could be found that the entroy generation rate reduces monotonously with increasing ower outut. herefore, the minimum entroy generation rate corresonds to the maximum ower of the cycle. A numerical examle is shown below. Assume that C H = W/K, c = 000 J/(kg K), m = 0.00 kg/s, NU =, in = 40 K and = 300 K. In this case, temerature should be otimized to increase the ower outut from the cycle as much as ossible. he variations of the ower outut, the entransy loss rate and the entroy generation rate with are calculated by eqs. (0), (34) and (35). he results are shown in Figure 4. he ower outut and the entransy loss rate both reach to the maximum value when is 355. K, while the entroy generation rate reaches the minimum value at the same time. he numerical results are in agreement with the above analyses. 4 Conclusion In this aer, the concet of enthaly entransy is defined. Based on the definition, the entransy balance equation for steady, oen thermodynamic system is develoed. It is found that art of the absorbed heat entransy flow runs into the working medium and makes its enthaly entransy increase, while the rest is used to do work. he entransy of the steady, oen thermodynamic rocesses is in balance. he air standard cycle is analyzed and discussed based on the entransy balance equation. When the heat absorbed by the working medium is from the combustion reaction of the gas fuel, the entransy loss rate could describe the change in the ower outut from the cycle, but the entroy generation rate cannot. 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