© Strojni{ki vestnik 49(2003)2,100-110 © Journal of Mechanical Engineering 49(2003)2,100-110 ISSN 0039-2480 ISSN 0039-2480 UDK 536.75:536.24 UDC 536.75:536.24 Pregledni znanstveni ~lanek (1.02) Review scientific paper (1.02) Entropijska analiza soto~nih prenosnikov toplote Entropy Analysis of Parallel-Flow Heat Exchangers Antun Galovi} - Marija @ivi} - Mladen Andrassy V prispevku obravnavamo povečanje entropije v ločilnih sotočnih prenosnikih toplote. Med delovanjem menjalnika se pojavljata dva vira entropije, vsled padca tlaka zaradi trenja in zaradi prenosa toplote. Obravnavajmo samo prenos toplote. Analiza je opravljena z matematičnim modelom, ki uporablja ista brezrazsezna števila, kakor jih je pred leti uvedel Bošnjakovič pri energijski analizi prenosnikov toplote. Ovrednoten je vpliv posameznih brezrazseznih spremenljivk (delovni pogoji sotočnega prenosnika toplote) na povečanje entropije. Izsledki so prikazani brezrazsezno v diagramih, kar jim daje bolj splošen pomen. Posebej so opisane delovne razmere sotočnega rekuperatorja, ki so povsem enake tudi pri protitočnih in križnih menjalnikih. © 2003 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: prenosniki toplote, povečevanje entropije, učinkovitost, rešitve analitične) The entropy generation in a parall-flow recuperative heat exchanger is analysed in this paper. During the operation of a heat exchanger two sources of entropy generation normally exist: the pressure drop friction) source and the heat-exchange source. Here, only the heat-exchange source is considered. The analysis is performed using an analytical mathematical model and the same non-dimensional numbers that were introduced into the energy analysis of heat exchangers by Bošnjakovič, many years ago. The influence of the individual non-dimensional variables (the operating points of the parallel-flow heat exchanger) on the entropy generation is quantified. The results are presented non-dimensionally in diagrams, which gives them a more universal meaning. Special conditions, i.e. the boundary operating conditions, of the parallel-flow recuperator, which are identical to those for counter-flow and cross-flow exchangers, are also discussed. © 2003 Journal of Mechanical Engineering. All rights reserved. (Keywords: heat exchangers, entropy generation, effectiveness, analytical solutions) 0 UVOD V splošnem sta dva vzroka za povečevanje entropije v vsaki vrsti prenosnikov toplote: razlika temperatur med obema pretokoma in padec tlaka zaradi trenja v tokovih skozi prenosnik. Podrobno in sistematično analizo vpliva znižanja tlaka in temperaturne razlike dobimo v [1] in [2]. V tem prispevku ne bomo obravnavali vpliva padca tlaka na povečevanje entropije. Podobne analize najdemo tudi v novejših virih [3] in [4], ki med drugim ocenjujejo povezanost povečanja entropije in izkoristka z namenom pridobitve najugodnejših delovnih pogojev prenosnika toplote. Namen tega prispevka je prikazati povezavo med povečanjem entropije in izkoristkom sotočnih prenosnikov toplote kot odvisnost znanih brezrazseznih števil n1, n2 in n3, ki jih je v energijsko 0 INTRODUCTION There are generally two causes of entropy generation in any type of heat exchanger: the temperature difference between the flows and the pressure drop induced by the friction in the flows streaming through the exchanger. A detailed and systematic analysis of the influence of pressure drop and temperature difference of the flows may be found in [1] and [2]. In this paper the pressure drop influence on the entropy generation is neglected. Similar analyses may also be found in recent works [3] and [4] that analyse, amongst other factors, the relationship between entropy generation and the effectiveness in order to determine the optimum operating parameters of the heat exchanger. The objective of this paper is to show the relation-ship between entropy generation and the effectiveness of parallel-flow heat exchangers as a function of the known non-dimensional numbers p1, p2 and p3, introduced into VH^tTPsDDIK stran 100 Galovi} M., @ivi} M., Andrassy M.: Entropijska analiza - Entropy Analysis A= O A A = An Sl. 1. Potek temperature vzdolž površin sotočnega prenosnika toplote Fig. 1. Temperature change along the surface area of a parallel-flow heat exchanger analizo uvedel Bošnjakovič [4], kakor tudi pred kratkim uvedeno brezrazsežno število pT za potrebe predstavljene entropijske analize. Dobljeni model tako lahko neposredno uporabimo v optimizacijski postopek sotočnih prenosnikov toplote. 1 MATEMATIČNI ZAPIS PROBLEMA Enačbo povečanja entropije v odvisnosti od toplotne zmogljivosti ter vstopnih in izstopnih temperatur pretokov zlahka dobimo z integracijo vzdolž tokovnic obeh tekočin: the energy analysis by Bošnjakovič [4], and the newly introduced non-dimensional number pT, for the purposes of the present entropy analysis. The model obtained may thus be directly included into the parallel-flow heat-exchanger optimisation process. 1 THE MATHEMATICAL FORMULATION OF THE PROBLEM The equation for entropy generation as a function of thermal capacities and the inlet and outlet temperatures of the flows is easily obtained by an integration along the streamlines of both flows: D S& T1'' T2' ' C1ln ' + C2ln i TT (1), kjer indeksa 1 in 2 pomenita šibkejši in močnejši pretok. Pretok z večjo toplotno zmogljivostjo C = q . c imejmo za močnejšega. Z delitvijo s toplotno zmogljivostjo močnejšega pretoka C zgornja enačba postane brezrazsežna: where the indices 1 and 2 designate the weaker and the stronger flows respectively. The flow with the larger heat capacity C = qm. cp is considered to be stronger. Dividing by the heat capacity of the stronger flow C2 the above equation becomes non-dimensional: D S& C ln ln (2). Člen p v enačbi (2) je po Bošnjakovičevi zasnovi razmerje toplotnih zmogljivosti šibkejšega in močnejšega pretoka: The member p in Equation (2), according to Bosnjakovic’s concept, is the ratio of the heat capacities of the weaker and stronger flows: qc m1 p1 qc 2 m2p2 (3). Z uporabo obrazca za brezrazsežno število p1 prenosnikov toplote (kar je tudi enako izkoristku menjalnika e [4]) Using the formula for the non-dimensional number p1 of heat exchangers (which is also identical to the effectiveness of the exchanger e [4]): kakor zaradi enačbe ohranitve energije: T1' -T1'' T1' -T2' as well as the energy-conservation equation: enačbo (2) preoblikujemo v: C1 (T1' -T1'' ) =C2 (T2' ' -T2' ) Equation (2) is transformed into: D S& C p3 ln ln (4), (5), (6), | IgfinHŽslbJlIMlIgiCšD I stran 101 glTMDDC Galovi} M., @ivi} M., Andrassy M.: Entropijska analiza - Entropy Analysis Vrednost p1 lahko izrazimo kot odvisnost med brezrazsežnima številoma p2 in p3, kakor navajajo Bošnjakovičevi učbeniki (izpeljavo funkcijske odvisnosti najdemo v [6]): The value p1 may be expressed as a function of the non-dimensional numbers p2 and p3 as it is stated in Bosnjakovic’s textbooks (the derivation of the functional relation may be found in [6]): kjer je 1-exp(-(1 + p3)p2) 1 + p3 where: C1 1 (7), (8). Z vstavitvijo (7) v (6) in označitvijo razmerja med vstopnimi temperaturami: Inserting (7) into (6), and denoting the ratio of the inlet temperatures with: pt =2' T2' T1' dobimo enačbo (6) v njeni končni obliki Equation (6) takes its final form: DS" C& p ln 1-exp(-(1 + p3)p2) 1 + p 1-p + ln 1+p 1-exp(-(1+p3)p2)r 1 1 1 + p (9), (10). Enačba (10) pove, da povečanje entropije v sotočnem prenosniku toplote lahko izrazimo z uporabo istih brezrazsežnih števil p2 in p3, kakor jih je uporabil Bošnjakovič na ravni energetske analize. Uporabiti moramo dodatno brezrazsežno število pT. Preden podamo grafično ponazoritev enačbe (10), zapišimo nekaj pripomb o posebnih primerih. a) Ko je pT = 1, enačba (10) daje D S genC2 =0, kar je pravilno, saj pomeni primer enakih vstopnih temperatur, ko ni ne spremembe toplote in ne povečanja entropije. b) V primeru izmišljenega prenosnika toplote z neskončno veliko menjalno površino, ko p2 -> oo, enačba (10) postane: Equation (10) indicates that the entropy generation of a parallel-flow exchanger may be expressed by means of the same non-dimensional numbers p2 and p3, as introduced by Bošnjakovič in the energy-level analysis. However, the additional non-dimensional number p must be used. Before the graphical presentation of Equation (10) is given, here are some comments on its special cases: a) If pT = 1, eq. (10) yields D S g /C2 =0. This is physically correct, because it represents the case of equal inlet temperatures, where there is no exchange of heat and thus no entropy generation. b) For the case of a hypothetical exchanger with an infinite heat-exchange surface area, where p2 -> oo, Eq (10) becomes: D S ge p ln 1 + p ln pT ( 1 + p3 (11), kar pomeni pri določenih vrednostih p in p povečanje entropije do končne vrednosti (vodoravna asimptota). b1) Dodatno, v primeru pretokov enakih toplotnih zmogljivosti, ko je p3 =1, postane enačba (11): indicating that the entropy generation for defined values of p and pT tends to a finite value (horizontal asymptote). b1) Additionally, in the case of flows of equal heat capacities, where p3 = 1, eq. (11) becomes: gen C ln Ap2 (1 + pt)2 4p (12). c) Kadar eden od pretokov kondenzira ali se uparja (fazna premena, npr. C = oo; p konst), postane enačba (10): 0; t2 = t2 c) When one of the flows condenses or evaporates (phase change: e.g. C2 = oo; p const), where eq. (10) becomes: 0; t2 = t2 D S dobimo nedoločeno vrednost: the undetermined value occurs: D Sgen=0-C2=0-ao (13), (14). V tem primeru je nedoločen tudi drugi člen v enačbi (1) in ga moramo spremeniti v skladu z drugim In this case the second member in Eq. (1) is also undetermined and has to be modified according VH^tTPsDDIK stran 102 Galovi} M., @ivi} M., Andrassy M.: Entropijska analiza - Entropy Analysis zakonom termodinamike, glede na nespremenljivo to the Second law of thermodynamics, regarding the temperaturo močnejšega pretoka: constant temperature of the stronger flow: D Sgen=C1ln T t1' ' C1 ( t1'-t1'' ) (15), ki se, po izločitvi t1'' in uporabi enačb (4), (7) in (9), spremeni v želeno obliko: D S which, after eliminating T1'' and using Eq. (4), (7) and (9), is transformed into the required form: C1 ln(exp ( -p2 ) + pT(1-exp ( -p2 ) ))+-----1 (1-exp ( -p2 ) ) Če enačbi (10) in (16) delimo z izkoristkom e = p1, dobimo razmerja D S genC2 e in D S g/C1 e, ki podajajo obseg pozitivnega povečanja entropije glede na hitrost prenosa toplote: (16). If the Eq. (10) and (16) are divided by the effectiveness e = p1, the ratia D S genC2e and DS /C1e are obtained, as relevant indicators of the range where the entropy generation is acting positively on the exchanged-heat flow rate: D S n 1 p1 p3 ln f 1 V C2e D S en 1-exp(-(1 + p3)p2) 1 + p 1-p ln 1 + p 1-exp(-(1 + p3)p2)f 1 + p C1e ln(exp ( -p2 ) +pT(1-exp ( -p2 ) ))+-----1 (1-exp ( -p2 ) ) (17), (18) , kjer p1 v (17) izračunamo po (7) oziroma v (18) z: where p1 in (17) is calculated according to (7), and in (18) according to: p1 =1-exp(-p2 (19). 2 GRAFIČNA PONAZORITEV IZRAČUNA POVEČANJA ENTROPIJE Podani matematični postopek smo prelili v računalniški zapis z uporabo Fortrana, rezultate pa prikažimo grafično. Seveda mora biti graf enačbe (10) prostorski, saj je odvisnost brezrazsežne vrednosti D S g /C2 funkcija p2, p 3 in p. Ustrezno diagram na sliki 2 kaže povečanje entropije v odvisnosti od p2 in p3 za ravni ploskev pt = 2, 4, 6, 8 in 10, medtem ko diagram na sliki 3 predstavlja isto odvisnost za pt = 0,2 , 0,4 , 0,6 , 0,8 in 1,0. Iz diagramov lahko razberemo, da vse parametrične krivulje pT kažejo ničelno povečanje entropije pri p3 = 0, kar smo že poprej opisali pod c) med pripombami o posebnih primerih. Povečanje entropije v tem primeru je prikazan v diagramih na slikah 4 in 5. Diagrama na sliki 4a in 4b sta narisana po enačbi (16) za vse vrednosti parametra p= 0,2 do 1,0 in 2,0 do 10,0. Vrednosti odvisnosti iz (18) so prav tako podane v teh diagramih. Diagram na sliki 5, narisan za p = 1,0 in p3 = 0, jasno kaže vpliv pT na brezrazsežno povečanje entropije. Diagrami na slikah 6 in 7 podajajo vrednosti, dobljene z uporabo enačb (10) in (17) za p = 0,5 in pT 0,2 do 1,0 oziroma za p3 = 0,5 in pT = 2,0 do 10,0. Diagrami na slikah 8 in 9 ovrednotijo iste spremenljivke za p = 1,0. 2 GRAPHICAL PRESENTATION OF THE ENTROPY-GENERATION CALCULATION The presented mathematical procedure was put into a calculation algorithm using Fortran, and the results are presented graphically. It is obvious that the graph of Eq. (10) must be three dimensional, because the non-dimensional value D S gen /C2 is the function of p2, p3 and pT Accordingly, the diagram in Fig. 2 displays the entropy generation as a function of p2 and p3 through the niveau surfaces p = 2, 4, 6, 8 and 10, while the diagram in Fig. 3 represents the same relationship for p = 0.2, 0.4, 0.6, 0.8 and 1.0. It can be seen from the diagrams that all the parametric curves pT give a zero-entropy generation for p3 = 0, which is explained above in part c) of the comments on special cases. The entropy generation for this case is presented in the diagrams in Figures 4 and 5. The diagrams in Figure 4a and 4b are plotted according to Eq. (16) for all the parametric curves pT = 0.2 to 1.0 and 2.0 to 10.0 respectively. The values of the function defined by (18) are shown in the same diagrams. The diagram in Fig. 5, drawn for p2 = 1.0 and p3 = 0, clearly indicates the influence of pT on the non-dimensional entropy generation. The diagrams in Figure 6 and 7 represent the values obtained by using equations (10) and (17) for p3 = 0.5 and pT = 0.2 to 1.0, and for p3 = 0. 5 and pT = 2.0 to 10.0, respectively. The diagrams in Figure 8 and 9 quantify the same variables for p3 = 1.0. | IgfinHŽslbJlIMlIgiCšD I stran 103 glTMDDC Galovi} M., @ivi} M., Andrassy M.: Entropijska analiza - Entropy Analysis D S gen C2 pT =10 pT =8 pT =6 pT =2 p 3 1 4 Sl. 2. Brezrazsežno povečanje entropije v sotočnem prenosniku toplote v odvisnosti od p2 in p3 pri pT = 2, 4, 6, 8 in 10 kot parametri Fig. 2. Non-dimensional entropy generation of a parallel-flow heat exchanger as a function of p2 and p3 with pT = 2, 4, 6, 8 and 10 as parameters pT =0,2 pT =0,4 pT =0,6 pT =0,8 p 3 Sl. 3. Brezrazsežno povečanje entropije v sotočnem prenosniku toplote v odvisnosti od p2 in p3 pri pT = 0,2 , 0,4 , 0,6 , 0,8 in 10,0 kot parametri Fig. 3. Non-dim f p2 andp3 4 3.5 3 2 1.5 1 0.5 DS C --------?--------;tpT = --------A--------ttpT = --------v— pT = ----------¦--------- pSGT 1= -----------*----------- pSGT 1= ----------¦---------- pSGt 1= ------------»------------ pSGT 1= ----------T---------- pSGT 1= ________ e 1 E0P,2 1 E0P,4 1 E0P,6 1 E0P,8 1 0,2 0,4 0,6 0,8 1 01234 p2 Sl. 4a. Brezrazsežno povečanje entropije in razmerje brezrazsežnega povečanja entropije s toplotnim izkoristkom v odvisnosti od p2 in parametrične krivulje pT = 0,2 do 1,0 za p3 = 0 Fig. 4a. Non-dimensional entropy generation and the ratio of non-dimensional entropy generation to heat transfer effectiveness as a function of p2 and the parametric curves pT = 0.2 to 1.0 for p3 = 0 VBgfFMK stran 104 lysis D S gen (C1e DS gen/C 1 4 3.5 3 2.5 2 1.5 1 0.5 ------m-^-a— DS gen( C e) SpGT 1= E 2P ------A------ SpGT 1= E 4P ------0------ SpGT 1= E 6P —>— SpGT 1= E 8P —v----- SpGT 1= E 1P0 DS C, SpGT1 = 2 » SpGT1 = 4 SpGT1 = SpGT1 = SpGT1 = 6 ------T------ 10 0 &=-~—,—,—i—,—,—,—,—i———,—i,,—. 01234 Sl. 4b. Brezrazsežno povečanje entropije in razmerje brezrazsežnega povečanja entropije s toplotnim izkoristkom v odvisnosti od p2 in parametrične krivulje pT = 2,0 do 10,0 za p3 = 0 Fig. 4b. Non-dimensional entropy generation and the ratio of non-dimensional entropy generation to heat transfer effectiveness as a function of p2 and the parametric curves pT = 2.0 to 10.0 for p3 = 0 5 -•— D S gen(C1e) DS/C 1 4 " 3 - 2 - I I 1 - i 0 -¦— p2=1 — SDGS 1/C 1 1 2 3 4 5 6 7 8 910 pT Sl. 5. Brezrazsežno povečanje entropije in razmerje brezrazsežnega povečanja entropije s toplotnim izkoristkom v odvisnosti od pT za p2 = 1,0 in p3 = 0 Fig. 5. Non-dimensional entropy generation and the ratio of non-dimensional entropy generation to heat transfer effectiveness as a function of pT for p2 = 1.0 and p3 = 0 1.6 1.4 1.2 0.8 0.6 0.4 0.2 DS /C 2 D S gen(C2e) 3= 0,2 pTS = 0,4 ~* pt = 0,6 *-------pt S =G 0,8 "* pTS = 1 e 0 Qr:-r-T=^-, 01234 p2 Sl. 6. Brezrazsežno povečanje entropije in razmerje brezrazsežnega povečanja entropije s toplotnim izkoristkom v odvisnosti od p2 in parametrične krivulje pT = 0,2 do 1,0 za p3 = 0,5 Fig. 6. Non-dimensional entropy generation and the ratio of non-dimensional entropy generation to heat transfer effectiveness as a function of p2 and the parametric curves pT = 0.2 to 1.0 for p3 = 0.5 ^vmskmsmm 03-2 stran 105 | ^BSSITIMIGC Galovi} M., @ivi} M. D S gen (C2e) DS /C 2 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 <5==——¦—¦ D S gen ( C2e) ------D---- -----A---- DS /C 2 GpET P= 2 GpET P= 4 GpET P= 6 GpET P= 8 pSTG = 2 pSTG = 4 pSTG = 6 pSTG = 8 0 1234 Sl. 7. Brezrazsežno povečanje entropije in razmerje brezrazsežnega povečanja entropije s toplotnim izkoristkom v odvisnosti od p2 in parametrične krivulje pT = 2,0 do 10,0 za p3 = 0,5 Fig. 7. Non-dimensional entropy generation and the ratio of non-dimensional entropy generation to heat transfer effectiveness as a function of p2 and the parametric curves pT = 2.0 to 10.0 for p3 = 0.5 3 2.5 2 1.5 1 0.5 D S gen(C2e) DSJC 2 ------D------ e) pT = SGEP 0,2 ------A------ SGpET P= 0,4 ___o___ SGpET P= 0,6 —t— SGpET P= 0,8 —v— SGpET P= 1 D S genC 2 STG p = 0,2 STG p = 0,4 STG p = 0,6 STG p = 0,8 STG p e Ep 0 — SGpET P= 8 ---------v--------- p = 10 SGET P DSJC 2 pSTG = 2 pSTG = 6 pSTG = 8 pSTG = 10 Ep e 0 oo. Izračunane so iz enačbe (18). Če p2 = 0, tj. C2 -> oo vstavimo v (18), dobimo nedoločeno vrednost 0/0. Z uporabo L’Hospitalovega pravila zlahka pokažemo, da je: 2.1 Interpretation of the diagrams The three-dimensional diagrams in Figure 2 and 3 indicate that the entropy generations for given values of pT effectively reach their asymptotic values when p2 becomes larger than one. These asymptotic values are quantified by Equations (11) and (12). It can be seen that the non-dimensional entropy generation becomes smaller with the rise of pt from 0 to 1.0. For pT = 1.0 the entropy generation is zero. (This is physically justified because for pT = 1.0 the inlet temperatures of the flows are equal, thus yielding no entropy generation). However, further increasing pT above the value of 1 increases the entropy generation again. The entropy generation is zero for each pT when p3 = 0. Thus, the defined non-dimensional entropy generation DS* C2 is not suitable for the solution of this special case because it gives the undefined value 0 oo for DSgen. The solution of this problem is presented in the diagrams in Figure 4a, 4b and 5, obtained by using Equations (16), (18) and (19). The dotted lines in the diagrams give the values of D S,gen/(eC1), and the full lines the ones of D S gC1. It is interesting to note the change of the value D S gen /(eC1). It is clear that each curve for a given value of pT drops monotonously from the value for p2 = 0 to the asymptotic value (horizontal asymptote), when p2 -> oo. They are calculated from Equation (18). If p2 = 0, i.e. C2 -> oo is inserted into (18), the undefined form 0/0 is obtained. Using the L’Hospital rule it is easily shown that: D S& eC (p =0) Kaj pravzaprav predstavlja desna stran enačbe (20)? Odgovor zlahka dobimo, če zapišemo enačbo brezrazsežne spremembe entropije zaradi menjave toplote pri različnih temperaturah, na različnih prenosnih površinah: d S& Z vstavitvijo T1=T1= konst. in T2=T2 = konst. v (21), ob uporabi (9), postane prvi člen na desni strani (21) povsem enak desni strani (20). To pomeni, da enačba (20) dejansko predstavlja največje povečanje entropije, ki je neposredno povezano z vstopnima temperaturama obeh pretokov. Praktično uporabo tega najdemo v prenosnikih toplote, kjer oba pretoka prestaneta fazni premeni. Tam imata tokova neskončno toplotno zmogljivost, saj eden kondenzira in se drugi uparja. Asimptotično vrednost D S gen f(eC1) na slikah 4a in 4b dobimo z vstavitvijo p J oo v enačbo (18): D S& (p -1) (20). What is actually represented on the right-hand side of Equation (20)? The answer is easily obtained by establishing the non-dimensional equation of entropy generation due to heat exchange at different temperatures on a differential exchange surface area: (T1 -T2 TT k dA (21). By inserting T1=T1 = const. and T2=T2 = const. into (21), and by using (9), the first term on the right-hand side of (21) becomes identical to the right-hand side of (20). This means that equation (20) in fact represents the maximum entropy generation, which is directly related to the inlet temperatures of both flows. The practical application of this may be found in heat exchangers, where both flows undergo a phase change. There the flows have an infinite heat capacity because one condenses and the other evaporates. The asymptotic value of DS* /(eC) in Figure 4a and 4b are obtained by inserting p2 1 oo into Equation (18): eC1 ln (p 1-p (22). | IgfinHŽslbJlIMlIgiCšD I stran 107 glTMDDC Galovi} M., @ivi} M., Andrassy M.: Entropijska analiza - Entropy Analysis Diagram na sliki 5 predstavlja brezrazsežno spremembo entropije D S gen/C in DS* (eC1) v odvisnosti od nT za n2 = 1,0 in n = 0. Jasno se vidi iz diagrama, ob rasti nT v območju 0 < n < 1 se obe izračunani vrednosti strmo znižata in se izničita pri n = 1. Za nT > 1 je povečanje izračunane vrednosti počasnejše, saj se sprememba entropije DS* en/C1 nagiba v neskončnost, ko se tt nagiba k nič ali v neskončnost. To zlahka dokažemo z analizo enačbe (16). Kadar prenosnik toplote deluje v območju 0 1 the rise of the ordinate values is slower because the entropy generation D S g IC1 tends to infinity when ji tends to zero or to infinity. This is easily proven by analysing Equation (16). If the heat exchanger is operated in the range 0 <7t< 1, a slight decrease of7tT significantly increases the entropy generation. Although the diagrams in Figure 4a, 4b and 5 were obtained according to the equations for parallel-flow heat exchangers, the same solutions are valid for counter-flow and cross-flow heat exchangers because of tl = 0. The diagrams in Figure 6 and 7 represent the non-dimensional values DS* /C1 and DS* /(eC1) as functions ofmform= 0.5. In Figure 6 the curves formT = 0.2 to 1 are plotted 3 nd in Figure 7 form= 2.0 to 10.0. In both diagrams the curve for the effectiveness: p1 = 1-exp -1,5p2 23( ()) (23) prav tako narisane. V obeh primerih kažejo diagrami, da se pri takšnih prenosnikih toplote pojavi največja vrednost izkoristka, praktično pri n = 3,0 (s = 0,659). Nadaljnje večanje n2 pomeni povečanje entropije, ob zanemarljivem izboljšanju izkoristka. Nazadnje, diagrami na slikah 8 in 9 kažejo brezrazsežne vrednosti D S ge /C1 in DS* /(eC1 )v odvisnosti od n2 za n = 1,0 oziroma vrednosti parametrov n= 0,2 do 1,0 in n= 2,0 do 10,0. Krivulja izkoristka is also drawn. For both cases the diagrams indicate that for such heat exchangers the maximum effectiveness value is practically achieved forTr = 3.0 (s = 0.659). A further increase of n2 only results in entropy generation, with an insignificant improvement of the effectiveness. Finally, the diagrams in Figure 8 and 9 represent the non-dimensional values D S g C1 and D S gen ( eC1 ) as functions of n2, for n = 1.0 and parametric values of TT = 0.2 to 1.0 and*T = 2.0 to 10.0 respectively. The effectiveness curve: e =p1= 1-exp -2p2 12( ()) (24) je prav tako vnesena v diagramih. Slike jasno kažejo, da je povečanje entropije zelo blizu svoje asimptotične vrednosti pri n2 = 2,5. Nadaljnje večanje n2 daje enakomerno rast entropije ob zanemarljivem vplivu na izboljšanje izkoristka. Glede na sliko 8 smemo poudariti, da povečanje nT v območju 0,2 do 0,8 enakomerne rasti entropije pomeni bistven vpliv na povečanje s v širokem območju spremenljivke n2. Po sliki 9 velja podoben izrek za zmanjšanjemT od 10,0 do 2,0. 3 SKLEP Izvedena analiza entropije sotočnih prenosnikov toplote je dokazala, da povečanje entropije lahko podamo z istimi brezrazsežnimi veličinami jt1, n2 in n3 , kakor pri energijski analizi, z dodatnim brezrazsežnim številom nT. Entropijska analiza kaže neposredno povezanost med izkoristkom e in povečanjem entropije. Prikazali smo, is also plotted in the diagrams. The figures clearly indicate that the entropy generations are very close to their asymptotic values for p2 = 2.5. A further increase of p2 yields constant entropy generation with negligible influence on the improvement of effectiveness. Referring to Figure 8, it should be emphasized that increasing pT in the range 0.2 to 0.8 the constant entropy generation has a significant influence on the increase of e in a wide range of the variable p2. According to Figure 9, the same can be said for decreasing pT from 10.0 to 2.0. 3 CONCLUSION The performed entropy analysis of parallel-flow heat exchangers has proven that entropy generation can be represented using the same non-dimensional characteristics p1, p2 and p3, used in the energy analysis, and the additional non-dimensional number pT. The entropy analysis reveals a direct connection between the effectiveness e and the VH^tTPsDDIK stran 108 Galovi} M., @ivi} M., Andrassy M.: Entropijska analiza - Entropy Analysis da je povečanje entropije veliko, kadar je razlika med toplotnima zmogljivostima obeh pretokov velika. Največja je, kadar en pretok spremeni fazo (p3 = 0), ter najmanjša kadar sta toplotni zmogljivost enaki (p = 1). Razmere so prav nasprotne glede izkoristka e prenosa toplote. Prav tako je prikazano, da za 0 < p < 1 povečanje brezrazsežne značilnice menjalne površine p2 = kA0/C1 učinkuje samo do določene vrednosti, nato pa se entropija le povečuje ob praktično nespremenljivem izkoristku. Brezrazsežne značilnice p p in 12 p3, uporabljene v tem prispevku, je uvedel že Bošnjakovič za svoje energijske izračune prenosnikov toplote. Sedanji način izračuna entropije z uporabo dodatne značilnice p dodaja več splošnosti pri problemih sotočnih prenosnikov toplote. entropy generation. It was shown that the entropy generation is large when the difference between the heat capacities of the heat exchanging flows is big. It is largest when one of the flows changes its phase (p3 = 0), and smallest when the flow heat capacities are equal (p3 = 1). The situation is just the opposite for the heat-transfer effectiveness e. Further, it was shown that for 0 < p3 < 1, increasing the non-dimensional exchanger area characteristic p2 = kA/C1 is useful only to a limited value, after which only entropy is produced with a practically constant effectiveness. The non-dimensional characteristics p1, p2 and p3 used in this paper were introduced by Bošnjakovič for his energy calculation of heat exchangers. This approach to the entropy calculation, using the additional characteristic pT, adds more generality to the problem of parallel heat-exchanger analysis. površina toplotne menjave specifična toplota pri p=konst. šibkejšega oz. močnejšega pretoka toplotna zmogljivost šibkejšega oz. močnejšega pretoka izkoristek prenosnika toplote celotni količnik prenosa toplote brezrazsežno razmerje šibkejšega oz. močnejšega pretoka vstopne temperature brezrazsežna temperaturna značilnica prenosnika toplote brezrazsežna površinska značilnica prenosnika toplote brezrazsežna značilnica razmerja zmogljivosti šibkejšega oz. močnejšega pretoka pretočna količina šibkejšega oz. močnejšega pretoka celotno povečanje entropije v prenosniku toplote absolutna vstopna in izstopna temperatura šibkejšega pretoka absolutna vstopna in izstopna temperatura močnejšega pretoka 4 OZNAKE 4 NOMENCLATURE A c, c P1 P2 m2 J/(kg K) C1,C2 W/K e k pT -W/(m2K) - p1 - p2 - p3 - q, q m1 m2 kg/s D S&gen W/K T1' ,T1'' K T2',T2' ' K 5 LITERATURA 5 REFERENCES heat-exchange surface area specific heat capacity at p=const. of the weaker and stronger flow respectively heat capacity of the weaker and stronger flow respectively heat-exchanger effectiveness overall heat-transfer coefficient non-dimensional relation of the weaker and stronger flow input temperatures non-dimensional temperature characteristic of the heat exchanger non-dimensional heat-exchanger area characteristic non-dimensional characteristic of the weaker and stronger flow heat-capacity ratio mass flow rate of the weaker and stronger flow respectively overall entropy generation of the heat exchanger absolute input and output temperatures of the weaker flow absolute input and output temperatures of the stronger flow [1] [2] [3] [4] [5] [6] Bejan, A. (1996) Entropy generation minimization, CRC Press, New York.. Bejan, A. (1988) Advanced engineering thermodynamics, John Willey & Sons, New York. Guo, Z.Y., S. Zhou, Z. Li, L. Chen (2001) Theoretical analysis and experimental confirmation of the uniformity principle of temperature difference field in heat exchanger, International Journal of Heat and Mass Transfer 45 (10), 2119-2127 Can, A., E. Buyruk, D. Eriner (2002) Exergoeconomic analysis of condenser type heat exchanger, Exergy, an International Journal 2, 113-118 Bošnjakovič, F. (1950) Nauka o toplini, dio prvi, Tehnička knjiga, Zagreb. Galovič, A. (1997) Nauka o toplini II, Sveučilište u Zagrebu, Fakultet strojarstva i brodogradnje, Zagreb. Galovi} M., @ivi} M., Andrassy M.: Entropijska analiza - Entropy Analysis Naslovi avtorjev: prof.dr. Antun Galovič prof.dr. Mladen Andrassy Fakulteta za strojništvo in ladjedelništvo Univerza v Zagrebu Ivana Lučiča 5 10000 Zagreb, Hrvaška antun.galovic@fsb.hr mladen.andrassy@fsb.hr dr. Marija Živič Fakulteta za strojništvo v Sl. Brodu Univerza v Osijeku Authors’ Address: Prof.Dr. Antun Galovič Prof.Dr. Mladen Andrassy Faculty of Mechanical Eng. and Naval Architecture University of Zagreb Ivana Lučiča 5 10000 Zagreb, Hrvatska antun.galovic@fsb.hr mladen.andrassy@fsb.hr Dr. Marija Živič Faculty of Mech. Eng. Sl. Brod University of Osijek Prejeto: Received: 7.10.2002 Sprejeto: Accepted: 29.5.2003 Odprt za diskusijo: 1 leto Open for discussion: 1 year VH^tTPsDDIK stran 110