The entransy dissipation minimization principle under given heat duty and heat transfer area conditions

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1 Article Engineering Termopysics July 2011 Vol.56 No.19: doi: /s x SPECIAL TOPICS: Te entransy dissipation minimization principle under given eat duty and eat transfer area conditions GUO JiangFeng XU MingTian * & CHENG Lin Institute of Termal Science and Tecnology Sandong University Jinan Cina Received February ; accepted August ; publised online December Under given eat duty and eat transfer area conditions te equipartition of te entransy dissipation (EoED) principle te equipartition of te temperature difference (EoTD) principle and te equipartition of te eat flux (EoHF) principle are applied to te optimization design of a eat excanger wit a variable eat transfer coefficient. Te results sow tat te difference between te results obtained using te EoED and EoTD principles is very small far smaller tan tat between te results obtained using te EoED and EoHF principles. Te correct entransy dissipation minimization principle is cosen to optimize te parameters in te ot and cold fluids in a two-fluid eat excanger under given eat duty and eat transfer area conditions. Te results indicate tat te proper coice of te two alternative fluids as an important role in te successful application of te entransy dissipation minimization principle. Te fluid tat could improve te total eat transfer coefficient sould be cosen or te fluid tat makes te temperature profiles of te ot and cold fluids parallel and decreases te temperature difference between te ot and cold fluids after optimization simultaneously could be te proper one. entransy dissipation eat excanger equipartition of entransy dissipation equipartition of temperature difference equipartition of eat flux Citation: Guo J F Xu M T Ceng L. Te entransy dissipation minimization principle under given eat duty and eat transfer area conditions. Cinese Sci Bull : doi: /s x Wit te rapidly increasing price of petroleum and coal te efficient use of energy resources as become one of te most effective ways of reducing demand on tose resources. Te eat excanger as an energy utilization device is widely used in power engineering petroleum refineries and cemical and food industries. Hence reducing unnecessary energy dissipation in a eat excanger to improve its performance is an important goal. Te eat transfer occurring in te eat excange process usually involves eat conduction under a finite temperature difference and wit fluid friction and mixing. Tese are te typical irreversible non-equilibrium termodynamic processes. In recent decades te application of te second law of termodynamics in eat excangers as attracted a lot of attention [1]. Based on te principle of entropy production minimization advanced by Prigogine *Corresponding autor ( mingtian@sdu.edu.cn) [2] Bejan [34] developed te entropy generation minimization (EGM) approac to eat excanger optimization design. Tondeur and Kvaalen [5] found tat total entropy production reaces te minimum wen te local rate of entropy production is uniformly distributed along te space and/or time variables in a contacting or separation device involving a given transfer area and acieving a specified transfer duty. Tis principle is called te equipartition of entropy production (EoEP). Sauar et al. [6] sowed tat te best trade-off between te energy dissipation and transfer area is acieved wen te termodynamic driving forces are uniformly distributed over te eat transfer area wic is called te equipartition of forces (EoF). Balkan [7] revealed tat entropy production calculated wit EoEP is always smaller tan tat calculated wit EoF altoug te difference is considerably small in itself. Guo et al. [8] sowed tat a more uniform distribution of temperature difference Te Autor(s) Tis article is publised wit open access at Springerlink.com csb.scicina.com

2 2072 Guo J F et al. Cinese Sci Bull July (2011) Vol.56 No.19 along te counter flow eat excanger contributes to a better performance of te counter flow eat excanger compared wit te parallel flow type. A new approac te equipartition of temperature difference EoTD was proposed as a sort-cut for minimizing entropy production by Balkan [7]. However te entropy production minimization metod also causes some inconsistencies and paradoxes [9 11]. Guo et al. [12] pointed out tat te focus of entropy production minimization was on te eat-work conversion process wile te rate and efficiency of eat transfer related more to eat excanger designs. Inspired by te analogy between eat and electrical conduction Guo et al. [1213] defined a new pysical quantity entransy to describe eat transfer ability. It was found tat te entransy is dissipated in irreversible eat conduction processes; te more te dissipation of entransy te iger te degree of irreversibility [14]. Xu et al. [15] derived an expression for te local entransy dissipation rate for eat convection. Liu et al. [16] sowed tat te extremum principle of entransy dissipation is advantageous over te extremum principle of entropy generation wen te eat excanger is only for eating and cooling wile te latter is better tan te former wen te eat excanger works in te Brayton cycle. Guo et al. [13] indicated tat wen te volume-average termal conductivity is kept constant te temperature gradient in te domain sould be uniform to minimize te entransy dissipation. Song et al. [17] used te extremum principle of entransy dissipation to optimize one-dimensional eat excangers and proved te validity of te uniformity principle of te temperature difference field in te eat excangers. Xia et al. [18] stated furter tat te eat flux density is uniformly distributed along te eat excanger wen entransy dissipation reaces te minimum for fixed eat duty and Newton s law of te eat transfer process wit a constant eat transfer coefficient. Meanwile te temperature difference between te ot and cold fluids is uniform along te eat excanger wic is called te principle of equipartition of eat flux (EoHF) in tis paper. Guo et al. [19] pointed out tat te entransy dissipation rate reaces te minimum for a given eat duty and eat transfer area wen te local entransy dissipation rate is uniformly distributed along te eat excanger no matter weter te eat transfer coefficient is constant or variable wic is called te principle of equipartition of entransy dissipation (EoED). Tey found tat te EoED principle as better performance tan te EoTD principle wen eat transfer coefficient is variable. Considering te above under fixed eat duty and eat transfer area conditions tere are tree principles to minimize entransy dissipation: te EoTD EoHF and EoED principles. Wen te eat transfer coefficient is constant tese tree principles are equivalent [19]. However te reality is tat te eat transfer coefficient is variable in eat excangers. Tus te study of te performances of te tree principles is of important practical significance wen te eat transfer coefficient is variable. In addition in te optimization of te two-fluid eat excanger wit a fixed eat duty and eat transfer area te aim is often acieved by canging te parameters of one fluid wile fixing te parameters of te oter. However te problem of wic of te two alternative fluids sould be cosen to minimize te entransy dissipation as not been discussed in existing literature. In tis paper wen te eat transfer coefficient is variable te optimization results obtained using te tree principles are compared wit eac oter and te appropriate principle is selected to optimize te parameters in te ot and cold fluid sides of a two-fluid eat excanger. Finally te rules for coosing te proper fluid between te two alternatives are presented. 1 Comparing te EoED EoTD and EoHF principles For a simple eat excange process as sown in Figure 1 a tin metal plate separates te ot and cold fluids. Assume bot fluids are perfectly mixed in te y-direction. Te lengt of te eat excanger is l eat is conducted only in te x-direction and te bulk temperatures of bot fluids vary only in te z-direction. Guo et al. [19] stated tat te condition for minimum entransy dissipation occurs wen te local entransy dissipation is uniform along te z-direction i.e. te EoED principle. Assume l=100 m te lengts in te Figure 1 A simple eat excanger.

3 Guo J F et al. Cinese Sci Bull July (2011) Vol.56 No x-direction and y-direction are Δx=1 m and Δy=0.1 m. Water is selected as te working fluid and its termopysical properties suc as density termal conductivity and viscosity are expressed as cubic functions of temperature in order to eliminate te effect of fixed termopysical properties on te results φ = a + a T + at + a T (1) were α 1 4 are constants. Te mass flow rate of te ot fluid is m = 2kg s and te outlet temperature of te ot fluid is T o =320 K. Because te mass flow rate and te outlet temperature of te ot fluid are given wen te inlet temperature of te ot fluid is given te eat duty is fixed. According to te Dittus-Boelter correlation te eat transfer coefficient is written as K kf vd = d η were k f is te termal conductivity of fluid d te equivalent diameter v velocity η te viscosity of te fluid and Pr te Prandtl number. Because termopysical properties are functions of temperature wen te velocity of te fluid and te equivalent diameter are given te eat transfer coefficient is a function of temperature. Te termal resistance R can be regarded as te reciprocal of te eat transfer coefficient K. According to te principle of equipartition of entransy dissipation (EoED) te expressions of temperature gradient of ot fluid eat flux density local entransy dissipation rate and temperature of cold fluid in te eat excanger can be deduced as [19]: dt ΔyC 1 = dz mc R q T c p Pr 0.4 ΔyC C Δy R x y 1 1 = e = ΔΔ = T ΔyC1R Δy were C 1 is constant subscripts and c denote ot and cold fluid and c p is te specific eat. Te C 1 can be solved by te Euler metod and te Newton iteration metod wen inlet and outlet temperatures of te ot fluid are given. Accordingly te temperature of te ot fluid eat flux density local entransy dissipation rate and temperature of te cold fluid in te eat excanger can be determined. According to te principle of equipartition of temperature difference (EoTD) wen te temperature difference between te ot and cold fluids is uniform in te eat excanger te expressions for te temperature gradient of te ot fluid eat flux density local entransy dissipation rate and temperature of te cold fluid in te eat excanger can (2) (3) be deduced as [19]: 2 dt ΔyC2 RC2 = e= Tc = T RC2 dz m c Δx p were C 2 is constant and can be solved by te Euler metod and te Newton iteration metod wen te inlet and outlet temperatures of te ot fluid are given. Accordingly te temperature of te ot fluid local entransy dissipation rate and temperature of te cold fluid in te eat excanger can be determined. For te sake of comparison te entransy dissipation number is adopted in te present work and can be written as [20]: * E diss E = (5) QT T ( i ci ) were Q is te total eat transfer rate T i and T ci are te inlet temperatures of te ot and cold fluids and E diss is te total entransy dissipation rate. Te entransy dissipation number represents te ratio of actual entransy dissipation to maximum entransy dissipation in te eat excanger. Te lower te entransy dissipation number te better te eat transfer process and te two extreme cases 0 and 1 denote tat te entransy dissipation rate is zero and at te maximum. Assuming te inlet temperature of te ot fluid is T i =370 K ten te local entransy dissipation rates obtained according to te tree principles in te eat excanger are demonstrated in Figure 2. As sown in Figure 2 te local entransy dissipation rate obtained using te EoED principle remains te same in te eat excanger. In te part of z<50 m te local entransy dissipation rate obtained using te EoTD principle is larger tan tat obtained using te EoED principle wile te local entransy dissipation rate obtained using te EoTD principle is less tan tat obtained using te EoED principle in te later stage. However te local entransy dissipation rate obtained using te EoHF principle is always larger tan te rates obtained using te EoED and EoTD principles and diminises as z grows. Figure 2 Distribution of local entransy dissipation rate in te eat excanger. (4)

4 2074 Guo J F et al. Cinese Sci Bull July (2011) Vol.56 No.19 Wen T i =370 K te optimization results obtained wit te tree principles are documented in Table 1. As Table 1 sows te total entransy dissipation rate obtained using te EoED principle is te lowest followed by te total entransy dissipation rate obtained using te EoTD principle wile te total entransy dissipation rate obtained using te EoHF principle is te largest. Compared wit te total entransy dissipation rate obtained using te EoTD principle te total entransy dissipation rate obtained using te EoED principle is reduced by only 0.15% wile te total entransy dissipation rate obtained using te EoED principle is reduced by 27.4% compared wit tat obtained using te EoHF principle. Compared wit te EoHF principle te EoTD principle approximates to te EoED principle. Table 1 sows tat te mass flow rate of cold fluid obtained using te EoHF principle is te largest followed by tat obtained using te EoTD principle wile te mass flow rate of te cold fluid obtained using te EoED principle is te smallest. Tis indicates tat wen te local entransy dissipation rate is uniform in te eat excanger te eat excanger as te igest termal efficiency under te given eat duty and area conditions wic can be illustrated by te eat excanger effectivenesses obtained using te tree principles. Te effectiveness obtained using te EoED principle is te largest te effectiveness obtained using te EoTD principle comes next followed by tat obtained using te EoHF principle. As sown in Table 1 te sequence of te entransy dissipation numbers obtained using te tree principles is te opposite to te order of te effectivenesses obtained using te tree principles. Te entransy dissipation number obtained using te EoED principle is te smallest followed by te entransy dissipation number obtained using te EoTD principle wile te entransy dissipation number obtained using te EoHF principle is te largest. 2 Application of te optimization principle to two-fluid eat excangers In a practical two-fluid eat excanger te optimization is often conducted by canging te parameters of one fluid and fixing te condition of te oter fluid. Balkan [21] presented te guidelines for coosing te proper fluid between te two alternatives to be modified wic would minimize te entropy production. Tis paper discusses te rules for coosing te proper fluid between te two alternatives to Table 1 Comparison between te optimization results obtained using te tree principles (T i =370 K) Item EoED EoTD EoHF Ė diss (W K) m c(kg/s) ε E* minimize te entransy dissipation. As mentioned above te difference between te results obtained using te EoED principle and te EoTD principle is very small. For te sake of simplicity te EoTD principle is selected to qualitatively optimize te two-fluid eat excanger. Regardless of te fouling resistance te total eat transfer coefficient can be written as 1 1 U = + K Kc were K and K c are te eat transfer coefficients on te ot fluid side and cold fluid side. Wen te dimensions of te eat excanger remain te same according to te D-B correlation te eat transfer coefficient can approximately be expressed as [21]: U K m new new = Kold m old new 1 1 = + Knew K te oter Assume te eat transfer coefficients on te ot fluid side and cold fluid side are K =900 W m 2 K 1 and K c =800 W m 2 K 1 and te parameters on one side are variable wile te parameters on te oter side remain te same trougout te optimization process. Under a given eat duty and eat transfer area te two-fluid eat excangers are optimized using te EoTD principle and te optimization results are sown in Table 2. In Table 2 te subscripts i and o denote inlet and outlet and A represents te eat transfer area. From Table 2 it can be seen tat it is not always possible to obtain better performance after optimization and coosing an inappropriate fluid may lead to worse results. After observing Table 2 carefully it can be found tat te case wic increases te mass flow rate after optimization will reduce te total entransy dissipation or te case wic increases te total eat transfer coefficient after optimization will lead to a reduction of te total entransy dissipation. To furter observe te variation in te temperature in te optimization te temperature profiles of te ot and cold fluids along te eat excanger before and after optimization are sown in Figures 3 and 4. Figure 3 sows te temperature profiles of te ot and cold fluids in te eat excanger before and after optimization wen te appropriate fluid is cosen. As sown in Figure 5 te temperature profiles of te ot and cold fluids along te eat excanger are parallel after optimization wic is consistent wit te EoTD principle. Bot temperature profiles are closer to eac oter after optimization tan before optimization indicating tat te temperature difference between te ot and cold fluids is reduced after optimization. Figure 4 sows te temperature profiles of te ot and cold fluids along te eat excanger before and after optimization wen an inappropriate fluid is cosen. As sown in Figure 4 te temperature 1. (6) (7) (8)

5 Guo J F et al. Cinese Sci Bull July (2011) Vol.56 No Figure 3 Temperature profiles of bot fluids before and after optimization wen te proper coice is (a) cold fluid and (b) ot fluid. Figure 4 Temperature profiles of bot fluids before and after optimization wen te improper coice is (a) cold fluid and (b) ot fluid. Table 2 Optimization results obtained using te EoTD principle in different cases T i (K) T o (K) T ci (K) T co (K) m (kg/s) m c (kg/s) A (m 2 ) U (W m 2 K 1 ) E diss (W K) Case 1 Before Cold fluid Hot fluid Case 2 Before Cold fluid Hot fluid Case 3 Before Cold fluid Hot fluid Case 4 Before Cold fluid Hot fluid profiles of te ot and cold fluids along te eat excanger are more parallel after optimization wic satisfies te EoTD principle but te total entransy dissipation rate is not reduced after optimization as sown in Table 2. From Figure 4 one can see tat te temperature profiles of te ot and cold fluids are farter apart after optimization wic means tat te temperature difference between te ot and cold fluids increases. Contrasting Figure 3 wit Figure 4 one can see tat wen te temperature profiles of te ot and cold fluids only meet te EoTD principle after optimization te total entransy dissipation rate may not be reduced. Only wen te temperature profiles of te ot and cold fluids are parallel and te temperature difference between te ot and cold fluids lowers simultaneously after

6 2076 Guo J F et al. Cinese Sci Bull July (2011) Vol.56 No.19 optimization can te total entransy dissipation rate decrease. Terefore te proper coice between te two alternative fluids as an important role in te successful application of te entransy dissipation minimization principles. 3 Conclusions For a given eat duty and eat transfer area te comparison of te results obtained using te EoED EoTD and EoHF principles illustrates tat wen te eat transfer coefficient is variable te EoED principle as te best performance followed by te EoTD principle and ten te EoHF principle. Te difference between te results obtained using te EoED and EoTD principles is very small far smaller tan tat between te results obtained using te EoED and EoHF principles. In practice for te sake of te simplicity te EoTD principle could take te place of te EoED principle to minimize entransy dissipation. Given te eat duty and eat transfer area te results of te applications of te EoTD principle to different fluids in two-fluid eat excanger sow tat improper coice of te two alternative fluids may lead to a worse performance in terms of entransy dissipation after optimization. Terefore coosing te proper fluid as an important role in te successful application of te entransy dissipation minimization principles. Between te two alternative fluids a coice tat increases te total eat transfer coefficient or makes te temperature profiles of te ot and cold fluids parallel and lessens te temperature difference between te ot and cold fluids simultaneously is te proper one. Tis work was supported by te National Natural Science Foundation of Cina ( ). 1 Yilmaz M Sara O N Karsli S. Performance evaluation criteria for eat excangers based on second law analysis. Exergy Int J : Prigogine I. Introduction to Termodynamics of Irreversible Processes. 3rd. New York: Wiley Bejan A. Entropy Generation troug Heat and Fluid Flow. New York: Wiley Bejan A. Entropy Generation Minimization. Boca Raton: CRC Press Tondeur D Kvaalen E. Equipartition of entropy production: An optimality criterion for transfer and separation process. Ind Eng Cem Res : Sauar E Ratkje S K Lien K M. Equipartition of forces: A new principle for process design and optimization. Ind Eng Cem Res : Balkan F. Comparison of entropy minimization principles in eat excange and a sort-cut principle: EoTD. Int J Energy Res : Guo Z Y Li Z X Zou S Q et al. Principle of uniformity of temperature difference field in eat excanger. Sci Cina Ser E-Tec Sci : Bertola V Cafaro E. A critical analysis of te minimum entropy production teorem and its application to eat and fluid flow. Int J Heat Mass Transfer : Bejan A. Second law analysis in eat transfer. Energy : Hesselgreaves J E. Rationalisation of second law analysis of eat excangers. Int J Heat Mass Transfer : Guo Z Y Ceng X G Xia Z Z. Least dissipation principle of eat transport potential and its application in eat conduction optimization. Cinese Sci Bull : Guo Z Y Zu H Y Liang X G. Entransy A pysical quantity describing eat transfer ability. Int J Heat Mass Transfer : Han G Z Guo Z Y. Pysical mecanism of eat conduction ability dissipation and its analytical expression (in Cinese). Proc CSEE : Xu M T Guo J F Ceng L. Te application of entransy dissipation teory in eat convection. Front Energy Power Eng Cina : Liu X B Meng J A Guo Z Y. Entropy generation extremum and entransy dissipation extremum for eat excanger optimization. Cinese Sci Bull : Song W M Meng J A Liang X G et al. Demonstration of uniformity principle of temperature difference field for one-dimensional eat excangers. J Cem Indust Engin (Cina) : Xia S J Cen L G Sun F R. Optimization for entransy dissipation minimization in eat excanger. Cinese Sci Bull : Guo J F Xu M T Ceng L. Principle of equipartition of entransy dissipation for eat excanger design. Sci Cina Tec Sci : Guo J F Ceng L Xu M T. Entransy dissipation number and its application to eat excanger performance evaluation. Cinese Sci Bull : Balkan F. Application of EoEP principle wit variable eat transfer coefficient in minimizing entropy production in eat excangers. Energy Convers Manage : Open Access Tis article is distributed under te terms of te Creative Commons Attribution License wic permits any use distribution and reproduction in any medium provided te original autor(s) and source are credited.