THERMODYNAMIC OPTIMIZATION OF TUBULAR HEAT EXCHANGERS BASED ON MINIMUM IRREVERSIBILITY CRITERIA

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1 THERMODYNAMIC OPTIMIZATION OF TUBULAR HEAT EXCHANGER BAED ON MINIMUM IRREVERIBILITY CRITERIA As. dd. ing. Adina GHEORGHIAN, Pof. d. ing. Alexandu DOBROVICECU, As. dd. ing. Andeea MARIN,.l. d. ing. Claudia IONIȚĂ UNIVERITY POLITEHNICA BUCHARET Abstact. Foced convection heat tansfe inside cicula tubes was analyzed having as stating pointthe equation of entopy geneation pe unit volume. The entopy geneation function pofile was deived assuming Poiseuille flow conditions and consideing velocity and tempeatue pofiles. It was shown that this can be expessed as a sum of two tems: heat tansfe acoss finite tempeatue diffeences and viscos flow within tubes. A nondimensional fom of entopy geneation function was developed expessing the local entopy geneation numbe pe unit volume. Entopy geneation ate pe unit length was detemined by integation acoss tube length. Finally, an expession of optimum tube diamete was detemined based on minimum ievesibility citeia. Keywods: Entopy geneation; Themodynamic optimization; Tubula heat exchange, Minimum ievesibility citeia.. INTRODUCTION Entopy geneation quantifies the loss of available enegy o the decease of convesion capacity of an enegy fom. The poblem of entopy geneation minimization fo convective heat tansfe pocesses was appoached fo the fist time by [] fo both intenal and extenal flow. Moe ecent studies can be mentioned: [] developed a numeical solution fo entopy geneation in a cicula pipe. It was found that tempeatue and velocity gadients in the pipe wall egion detemine a highe entopy geneation ate in this egion. Anothe impotant conclusion was that themal component of the entopy geneation is dominant in compaison to the fiction component. Entopy geneation ate is an impotant paamete in the design of heat exchanges. It descibes the heat exchange pefomance in tems of the econd Law of Themodynamics. [3] eviewed heat exchanges pefomance citeia based on econd Law analysis. It was shown that it is possible to categoize the heat exchanges pefomance citeia in two classes: citeia that use entopy as evaluation paamete and the citeia that use exegy as evaluation paamete. It was demonstated that some citeia ae elated to each othe and cannot be consideed as standalone. A compehensive eview of themodynamic optimization appoaches was caied out by [4]. [5] pesented seveal examples of heat exchanges geomety optimization based on entopy geneation minimization method. Effectiveness, design and optimization of heat exchanges based on econd Law analysis have been studied extensively in the liteatue [6-]. The analysis of entopy geneation mechanism fo convective heat tansfe is pesented in detail in []. The ievesibility of convective heat tansfe is caused by the following effects: heat tansfe acoss finite tempeatue diffeence and fluid fiction. Consideing an infinitesimal fluid element in section one can expess the entopy geneation ate pe unit volume as a sum of two tems: one tem geneated by conduction diven by a tempeatue gadient and the othe is viscos dissipation. The study of foced convection inside staight cicula tubes equies to ecast the entopy geneation equation in cylindical coodinates fo convenience. The entopy geneation ate pe unit volume fo a cicula staigth tube is deived in section 3 based on equations fo the velocity and tempeatue pofiles. In ode to study the influence of viscous dissipation tem magnitude compaed to the tem that accounts fo heat tansfe the equation of entopy geneation ate pe unit volume is modified in section 4, intoducing the concept of numbe of entopy geneation units pe unit volume. The gaphical epesentation of numbe of entopy geneation units pe unit volume against elative tube adius shows the influence of the two tems on the entopy geneation ate o ievesibility. In section 5, the equation of entopy geneation ate pe tube length unit is deived fo lamina and tubulent flow. By joining all heat exchange 48 TERMOTEHNICA /3

2 THERMODYNAMIC OPTIMIZATION OF TUBULAR HEAT EXCHANGER geometical paametes into one tem design paamete, the entopy geneation ate pe tube length unit can be expessed as a function of Re, Nu and f. The numbe of entopy geneation units can be expessed as a function of Re only by using the appopiate coelations. By equating the fist deivative of this expession to, the optimum value of Re is obtained in section 6, fo which the entopy geneation eaches its minimum. The optimum tube diamete can be calculated fom optimum Re value. A woked example is pesented in section 7 of the pape. It demonstates the pocedue fo calculation of the optimum tube diamete fo a tubula cyogenic heat exchange.. ENTROPY GENERATION PER UNIT VOLUME It was consideed an incompessible fluid element without intenal heat geneation, shown in Figue. q x w x qy+dqy qy wx+dwx wy q x +dq x w x +dw x Fig.. Enegy balance of a D infintesimal fluid element. uch a two-dimensional element can be consideed as an open system with its suface cossed by mateial, enegy and entopy flows. Consideing that the themodynamic state of the fluid inside the element depends only on time vaiable and does not depend on the spatial coodinates we can expess the entopy geneation ate pe unit volume gen 3 m K as []: T T gen T x y w wy w y x w x T x y y x () gen T ( ) T T whee λ is the themal conductivity; Φ function of viscous dissipation; T tempeatue; w x, w y velocity components on x and y axes espectively. Eq. () shows that entopy geneation ate pe unit volume consists of two tems: the fist tem is themal conduction diven by a tempeatue gadient (λ) and the second is the viscous dissipation (µ). Foced convection heat tansfe inside tubes is analysed in the next section with the pupose of demonstating the intedependence between conductive and viscos effects in the entopy geneation pocess. 3. FORCED CONVECTION INIDE CIRCULAR TUBE It was consideed a Poisseule flow though a cicula tube with constant wall heat flux q. m Velocity and tempeatue distibution ae given by []: w x w x, max q 4 4 x T T x 4 whee: w x, max () (3) dp 4 dx and x w x, max Pe a (4) and x ae the adial and axial coodinates; tube adius; a themal diffusivity; T absolute tempeatue at a point of oigin located on the tube axis. Entopy geneation pe unit volume equation can be ewitten in cylindical coodinates: T T wx gen (5) T x T Tempeatue pofile equation (3) will be diffeentiated with espect to axial coodinate: 4 4q T q x x x (6) TERMOTEHNICA /3 49

3 Adina GHEORGHIAN, Alexandu DOBROVICECU, Andeea MARIN, Claudia IONIȚĂ Eq. (6) becomes taking into account Eq. (4): T 4q (7) x Pe Tempeatue pofile equation (3) will be again diffeentiated with espect to adial coodinate: T q (7 ) Using the notation R, Eq. (7 ) becomes: T q R R (8) Velocity pofile equation () will be diffeentiated with espect to adial coodinate: wx wx, max R wx, max (9) Taking into account Eqs. (7), (8) and (9), the entopy geneation ate pe unit volume (5) becomes: q 3 6 gen RR T Pe 4 wx, max R T () Eq. () descibes the entopy geneation pofile fo a cicula staight tube. It completes the foced convection heat tansfe model in cicula tubes model togethe with the tempeatue and velocity pofile equations. 4. NUMBER OF ENTROPY GENERATION UNIT PER UNIT VOLUME The expession of entopy geneation numbe pe unit volume is obtained dividing Eq. () by q / T : N q R R Pe R T 3 6 () includes the effect of ievesibility geneated by axial conduction, and, which quantifies the impotance of viscosity effect. Fo Pe values less than 4 axial conduction contibution is highe than adial conduction. A numbe of adial entopy geneation pofiles neglecting the ievesibility due to axial conduction (6/Pe << ) ae pesented in Figue. 5 TERMOTEHNICA /3 φ= φ=/4 φ=/ φ= Fig.. Entopy geneation pofile against R. It can be noticed that centeline egion of condution ( R = ) is fee of entopy geneation because the velocity and tempeatue gadients ae zeo in the tube axis. Fo = the numbe of entopy geneation units pe unit volume eaches its maximum at R. As inceases (fluid 3 viscosity inceases) the adial coodinate of the point with maximum value of shifts towads the tube wall. Fo = the coodinate of the point with maximum is equal to the tube adius. In all cases, the egion nea the inne suface acts as an ievesibility concentato. 5. ENTROPY GENERATION RATE PER UNIT TUBE LENGTH By integating Eq. () the entopy geneation ate pe unit tube length can be deived: w x, max whee 4T. () q gen gend V m K (3) It can be noticed fom Eq. () that is a Taking into account that function of adial coodinate, Peclet numbe, which dv da m πd (4)

4 THERMODYNAMIC OPTIMIZATION OF TUBULAR HEAT EXCHANGER and R, Eq. (3) becomes: gen gen RdR (5) Neglecting axial conduction (Pe >> 4) in Eq. (), the integal given by Eq. (5) becomes: q w x, max gen (6) 48 T T Taking into account the heat flux pe unit tube length expession: ql q m (7) and consideing the mean fluid velocity [] wx.5wx, max, (8) Eq. (6) takes the following fom: q l 8 m gen 48 4 T T whee m is the mass flow ate though the tube. (9) It can be noticed one moe time that ievesibility geneation gen can be expessed as a sum of two tems: heat tansfe at finite tempeatue diffeence and fluid fiction. It can be noticed as well that the heat tanfe contibution is constant if heat flux pe unit tube length is known. Fo tubulent flow thee is no analytical solution fo velocity and tempeatue pofiles. Fo lamina flow inside cicula tubes with constant wall heat flux the heat tansfe coelation is 48 given by Nu and fiction facto f 8. m By substituting these two tems in Eq. (9) the following expession of the entopy geneation ate pe unit length is obtained: 3 q l m f 5 gen () Nu T T Eq. () will be consideed valid fo any flow egime with Nu and f given by thei espective coelations fo each flow egime. 6. TUBE DIAMETER OPTIMIZATION If heat flux pe unit length q l and mass flow ate m ae known it is possible to detemine the optimum diamete of the tube fo which the entopy geneation ate eaches the minimum value. The numbe of entopy geneatio units is defined as follows: whee N gen Δ T, lamina gen is the entopy geneation ate pe unit q Δ T, lamina length (), and is the 48 T fist tem in Eq. (9). Afte substitutions the following expession of the numbe of entopy geneation units is obtained: whee Re N Nu f A () ql m A is the design paamete 5 T of the heat exchange. ( ) The tube diamete can be ewitten as: 4m D. Re Fo given q l, m and fluid natue, the optimum diamete value fo which gen eaches its minimum can be detemined () fo a constant value of A. Afte detemination of optimum Re value, the optimum diamete of the tube can be found using the following expession: 4m Dopt () Re Fo lamina flow it is possible to show that Re, opt ia N, min. (3) This esult shows that the optimum tube diamete must be lage enough to ensue that the heat tansfe acoss finite tempeatue diffeences has the main contibution to the entopy geneation ate. The following coelations fo Nu numbe and fiction facto f ae used fo tubulent flow though cicula staight tubes [3]: opt,.4.8 Nu.3P Re (4). f.46re (5) TERMOTEHNICA /3 5

5 Adina GHEORGHIAN, Alexandu DOBROVICECU, Andeea MARIN, Claudia IONIȚĂ The following expession is obtained fo a function of Re: A 5 Re N P Re The fist deivative of d dre as (6) with espect to Re is: A Re 5.779P Re d olving the equation dre of Re numbe fo which is obtained: (7) the optimum value eaches its minimum Re.3P A (8) opt The optimum tube diamete fo which the numbe of entopy geneation units eaches its minimum can be obtained by taking into account Eq. (): m Dopt.69 A P (9) The minimum value of the entopy geneation numbe can be obtained by substituting Eq. (8) in (6): 35 7, min N P A (3) The entopy geneation ate pe unit length can be obtained by intoducing Eq. (3) in (): q 35 7 gen, min 9.88 P A (3) T It can be noticed fo a given P that if the design paamete of the heat exchange A inceases, then the optimum tube diamete deceases (Re opt inceases) and the entopy geneation numbe deceases as well. 7. ORKED EXAMPLE A tubula cyogenic heat exchange with the following data is consideed: heat flux pe unit length kg q l 6, mass flow ate (ai) m.5 m s inside tubes. The mean value of ai tempeatue is T = 3.5 K, and the mean pessue p =.554 MPa. The objective is to detemine the optimum tube diamete on the condition that entopy geneation ate eaches its minimum value. To stat, the themo-physical popeties of ai at the mean tempeatue and pessue will be detemined by using the built-in functions of Engineeing Equation olve commecial softwae [4]: kg fluid density 5.59 ; m 3 themal conductivity.89 m K 6 dynamic viscosity 9.37 Pa s ; isobaic specific heat capacity: J cp 3. kg K Then, the dimensionless paametes c p 6 P.796 and A 3.63 ae calculated using Eq. ( ). The vaiation of entopy geneation numbe (6) against Reynolds numbe is pesented in Figue 3. Fig. 3. The vaiation of entopy geneation numbe against Re numbe. Re value fo which N ( Re ) eaches its minimum 6 can be calculated using Eq. (8), Reopt.688. The optimum tube diamete calculated using Eq. (9) is Dopt 4.6 mm. Finally, minimum values of the entopy geneation numbe (3) and entopy geneation ate pe unit length (3) ae detemined: N,.57 3 min and min.7 espectively. m K 5 TERMOTEHNICA /3

6 THERMODYNAMIC OPTIMIZATION OF TUBULAR HEAT EXCHANGER 8. CONCLUION The entopy geneation ate pe unit volume consists of two tems: the fist tem accounts fo heat conduction due to a tempeatue gadient and the second accounts fo viscos dissipation. The entopy geneation ate pe unit volume is positive and finite if velocity and tempeatue gadients occu. The entopy geneation pofile togethe with tempeatue and velocity pofiles completes the themodynamic desciption of the convective heat tansfe inside cicula tubes. The entopy geneation in the tube axis is zeo because velocity and tempeatue gadients ae zeo in the cente of the tube. On the othe hand, the inteio tube suface acts as a stong entopy geneato. As viscosity effects intensify, the maximum of the entopy geneation pofile shifts towads the tube wall. The optimum tube diamete fo lamina flow must be lage enough so that the entopy geneation ate is mainly dominated by heat tansfe acoss finite tempeatue difeences. In the case of tubulent flow, fo a given numbe, the optimum tube diamete deceases as the design paamete of the heat exchange inceases. The application included at the end of the pape demonstated the calculation of the optimum tube diamete on the condition that entopy geneation pe unit length is minimum. The mathematical model developed fo staight tubes heat exchanges can be extended fo othe types (helically coiled tubes, tube in tube etc.) by using appopiate coelations. REFERENCE [] F.A McClintock., The design of heat exchanges fo minimum ievesibility, AME Pape No. 5-A-8, 95. [] ahin, A.Z., econd Law Analysis of Lamina Viscous Flow though a Duct ubjected to Constant all Tempeatue, AME J. Heat Tansfe, 76-83, 998. [3] M. Yilmaz, O.N. aa,. Kasli, Pefomance evaluation citeia fo heat exchanges based on second law analysis, Exegy Int. J., (4), pp ,. [4] M.M. Awad, Y.. Muzychka, Themodynamic Optimization - chapte, Heat Exchanges - Basics Design Applications, InTech,. [5] A. Bejan, Themodynamic optimization of geomety in engineeing flow systems, Exegy Int. Jounal, (4), pp ,. [6]. Kakac, A.E. Begles, F. Mayinge, Eds., Heat Exchanges: Themal-Hydaulic Fundamentals and Design, Hemisphee, ashington, DC, 98. [7] A. Gheoghian, Al. Dobovicescu, L. Gosu, B. Popescu, C. Ioniță, A compaative analysis of cyogenic ecupeative heat exchanges based on exegy destuction, ECO - Intenational Confeence on Efficiency, Cost, Optimization, imulation and Envionmental Impact of Enegy ystems, Peugia, Italia, pp. 9-9-,. [8] Faas, A. P., Heat Exchange Design, nd Ed., iley, New Yok, 989. [9] chlunde, E. U., Ed., Heat Exchange Design Handbook, Hemisphee, New Yok, 983. [] D. tanciu, A. Dobovicescu, Optimization of intenal foced convection though a duct based on the second law, U.P.B. ci. Bull., eies D, Vol. 7, I., pp.79-94, 9. [] A. Bejan, Entopy geneation though heat and fluid flow, John iley and ons, UA, 98. [] R.B. Bid,. E. tewat, E. N. Lightfoot, Tanspot Phenomena, John iley and ons Inc., New Yok, 96. [3] A. Bejan, Advanced Engineeing Themodynamics, Thid Edition, John iley and ons, UA, 6. [4] Engineeing Equation olve commecial softwae: TERMOTEHNICA /3 53