Contents PrefaceIX Part 1General Aspects1 Chapter 1Thermodynamic Optimization3 M.M. Awad and Y.S. Muzychka Chapter 2Analytical Solutionof Dynamic Response of Heat Exchanger53 D. Gvozdenac Chapter 3Self-Heat Recuperation: Theory and Applications79 Yasuki Kansha, Akira Kishimoto, Muhammad Azizand Atsushi Tsutsumi Chapter 4Development of High EfficiencyTwo-Phase Thermosyphons for Heat Recovery97 Ignacio Carvajal-Mariscal, Florencio Sanchez-Silvaand Georgiy Polupan Chapter 5Impact of a Medium Flow Maldistributionon a Cross-Flow Heat Exchanger Performance117 Tomasz Bury Chapter 6Control of LNG Pyrolysis and Applicationto Regenerative Cooling Rocket Engine143 R. Minato, K. Higashino, M. Sugioka and Y. Sasayama Chapter 7Numerical Analysis of the StructuralStability of Heat Exchangers The FEM Approach165 Agnieszka A. Chudzik Part 2Micro-Channels and Compact Heat Exchangers187 Chapter 8Microchannel Simulation189 Mohammad Hassan Saidi, Omid Asgari and Hadis Hemati VIContents Chapter 9Compact Heat Exchange ReformerUsed for High Temperature Fuel Cell Systems221 Huisheng Zhang, Shilie Weng and Ming Su Chapter 10Single-Phase Heat Transfer and FluidFlow Phenomena of Microchannel Heat Exchangers249 Thanhtrung Dang, Jyh-tong Teng, Jiann-cherng Chu, Tingting Xu,Suyi Huang, Shiping Jin and Jieqing Zheng Chapter 11Heat Exchangers for Thermoelectric Devices289 David Astrain and lvaro Martnez Part 3Helical Coils and Finned Surfaces309 Chapter 12Helically Coiled Heat Exchangers311 J. S. Jayakumar Chapter 13Fin-Tube Heat Exchanger Optimization343 Piotr Wais Chapter 14Thermal Design of Cooling and Dehumidifying Coils367 M. Khamis Mansour and M. Hassab Part 4Plate Heat Exchangers395 Chapter 15The Characteristics of Brazed Plate HeatExchangers with Different Chevron Angles397 M. Subbiah Part 5Energy Storage Heat Pumps Geothermal Energy425 Chapter 16PCM-Air Heat Exchangers: Slab Geometry427 Pablo Dolado, Ana Lzaro, Jos Mara Marnand Beln Zalba Chapter 17Ground-Source Heat Pumps and Energy Saving459 Mohamad Kharseh Chapter 18The Soultz-sous-Forts Enhanced GeothermalSystem: A Granitic Basement Usedas a Heat Exchanger to Produce Electricity477 Batrice A. Ledsert and Ronan L. Hbert Part 6Fouling of Heat Exchangers505 Chapter 19Fouling and Fouling Mitigationon Heat Exchanger Surfaces507 S. N. Kazi ContentsVII Chapter 20Fouling in Plate Heat Exchangers:Some Practical Experience533 Ali Bani Kananeh and Julian Peschel Chapter 21Self-Cleaning Fluidised Bed Heat Exchangers for Severely Fouling Liquids and Their Impact on Process Design551 Dick G. Klaren and Eric F. Boer de Preface Asmotiveforceofprocesses,heatmustbetransferredfromonefluidtoother,task thatisperformedbymeansofheatexchangers.Fromthispointofview,heat exchangersrepresentanimportantelementofthermalfacilitiesthathassubstantially contributed to technical development of the society. Today it is impossible to imagine any branch of process engineering and energy technology without involvement of heat exchangers. Advanced models of these apparatus were proposed in the middle of the 18th century, while theoretical backgrounds have been completed a century later.Correspondingtopracticalimportanceofheatexchangers,innumerablestudiesand treatisesaredevotedtoprocessestakingplaceinthesedevicesandtheirconstructive shaping. The actual development trend in this field is guided by the ideas of reduction of thermaltransportresistancesandtheraiseofenergyconversionefficiency.Theseideas havealsoguidedtheconceptionofthepresentbook.Itisacollectionofcontributions prepared by the specialists. It consists of 21 Chapter that are arranged in 6 Sections:Section 1: General Aspects,Section 2: Micro-Channels and Compact Heat Exchangers,Section 3: Plate Heat Exchangers,Section 4: Helical Coils and Finned Surfaces,Section 5: Energy Storage, Heat Pumps and Geothermal Energy, Section 6: Fouling of Heat Exchangers. Section 1 - General Aspects Thispartcomprises7Chaptersdealingmainlywiththequestionsoffluidflowand heattransferinheatexchangers.Chapter1byAwadandMuzychkaaddressesthe entropygenerationarisingfromheattransferandfluidflowandprovidesabasisfor thermodynamicoptimisationofheatexchangers.GvozdenacgivesinChapter2a detailedanalysisofconvectiveheattransferinheatexchangersatdifferentflow arrangements under transient conditions. Kansha et al. present in Chapter 3 a self-heat recuperation technology for transport of latent and sensible heat of the process streams without heat addition and introduce a theoretical analysis of this technology. Carvajal-Mariscal et al. recommend in Chapter 4 combinations of process parameters that give highefficiencyoftwo-phasethermosyphons.Thesedevicesareusedfortransportof highheatflowratesfromheatsourcetoheatsinkbyconnectingtheevaporatorand XPreface the condenser. The heat flow rate is reduced, if the elements of the heat exchangers are not evenly supplied with the fluid. Bury examined the issue in Chapter 5 and reports anaveragedeteriorationfactorof15%foracross-flowheatexchanger.InChapter6, Minatoetal.dealwiththeLNGpyrolysisinconnectionwithregenerativecoolingof rocket engines. The processes occurring are exceedingly complex, not only because of high process temperatures, which usually cause large temperature gradients. Thermal stressesthusinducedmayimpairthestructuralstabilityofconstructions,asis exemplified by A. Chudzik in Chapter 7 for a shell-and-tube heat exchanger.Section 2 - Micro-Channels and Compact Heat Exchangers Section2clustersthecontributionsdealingwiththeprocessesoccurringinmicro-channelsandapparatuscomposedofsuchelements.Byusingmicro-channels,one pursues the idea of shortening the heat transfer paths, thereby trying to copy solutions evolved in the nature. In Chapter 8, Asgari analyses numerically the heat transfer in a heatexchanger,consistingofanumberofrectangularmicro-channels,connectedin parallel,atlowRenumberswithfullydevelopedflow.Theappliedheatfluxisor-thogonal and uniform on the bottom plane of the apparatus. Zhang et al., in Chapter 9, simulate dynamically a compact heat exchange reformer for high temperature fuel cell systemsundercatalyticconditions.Themodellingtechniqueissuitedforquickand realtimecalculations.Withsinglephaseflow,Dangetal.showinChapter10the hydraulicdiameterofthemicrochanneltobethechiefparametergoverningthe thermo-fluidcharacteristicoftheapparatus.Bothexperimentalandnumericaltreat-mentsconfirmadvantagesofcounter-currentfluidflowarrangement.Thecontribu-tioninChapter11dealswiththethermoelectricdevices,whichareusedeitherto generateelectricpotentialinatemperaturegradient(Seebeckeffect)ortogeneratea temperaturedifferencebymeansofelectriccurrent(Peltiereffect).Astrainand Martnez analysed there the efficiency of the device, mainly focussing on heat transfer. Thethermalperformancesofheattransfermodulesaredemonstratedtodecisively affect the efficiency of the whole system. Section 3 - Plate Heat Exchangers Section3comprisesthecontributionsdealingwiththeheatexchangersconsistingof helical coils and finned surfaces. Helical coils provide the simplest construction of heat exchangerswhilefiningofsurfacesshouldcompensateforthelowheattransfer.In Chapter 12 Jayakumar presents a detailed analysis of hydrodynamic and heat transfer of single-phase and two-phase water flow in helical pipes, for various coil parameters and boundary conditions. Basing on the results, correlations for the average and local Nusseltnumbersweredeveloped.WaispursuesinChapter13thepossibilitiesof finding the fin shape that should maximize the heat transfer and reduce the fin mass. Theresultsofnumericalexperimentsareusedfordevelopingofheattransfer correlations. Chapter 14 by Khamis Mansour deals with the thermal design of cooling anddehumidifyingcoils.Theusedrow-by-rowcalculationmethodprovidesabetter reliability than the common averaging method. This is of particular importance when local data are required as in case of air dehumidifiers.PrefaceXI Section 4 - Helical Coils and Finned Surfaces Plateheatexchangersconsistofanumberofplatesassembledinparallelnexttoone another thus forming flow channels such that each plate separates hot from cold fluid stream.Platesareprovidedwithmacrostructuresandtheneighbouringplatestouch each other on the crests of the structures. The channels are gasketed or brazed on the circumference.InChapter15Muthuramanpresentsresultsoftheexperimental condensationstudiesofR410Ainbrazedplateheatexchangersforvariousplate structures and provides correlations for heat transfer and pressure drop.Section 5 - Energy Storage, Heat Pumps and Geothermal Energy Part5bundlesthecontributionsdealingwiththestorageandconversionofthermal energy, focussing on alternative energy sources and clean energy. Thermal radiation of the Sun counts to the cleanest energies; being availably mainly seasonally, its storage is becomingincreasinglyimportant.Ifstoredasinternalenergyofasubstance,the substanceshouldundergoanendothermicphasetransitionduringstoringwhile exothermicwhenreleasingtheheat.Chapter16byDoladoetal.dealswiththeheat exchangerscomprisingphasechangematerials,illustratingtheprocessesmainlyin formoftemperaturehistorydiagrams.Thermalenergystoredinthegroundmaybe utilizedbymeansofheatpumps.Chapter17byKharsehillustratesanexamplewith surface geothermal energy, while Ledsert and Hbert give in Chapter 18 an overview on the exploitation of deep geothermal energy. The high temperature of energy source in later case allows transformation of geothermal energy in other energy forms. Section 6 - Fouling of Heat Exchangers Dissolvedsolidsubstancesandimpuritiescontainedinprocessstreamsinteractwith theheattransfersurfaces,attractivelyorrepulsively.Incaseofattraction,the concentrationofthedissolvedmayreachthesolubilityboundaryandinitiate crystallisation. Starting from this initial state, a solid layer grows on the surface during operation;itdiminishesthethermalcapacityofheatexchanger,ifitsthermal conductivity is low. This is referred to as fouling.The contribution devoted to fouling ofheatexchangersaregroupedinthisPart.Chapter19byKaziaddressesgeneral questionsoffouling,therebyanalysingitsimpactsonheattransfer.Somepractical insightsintofoulinginplateheatexchangersareprovidedinChapter20byBani KananehandPeschel,whileinChapter21KlarenanddeBoerreportontheself-cleaning fluidised bed heat exchangers. Prof. Dr. Ing. Jovan Mitrovic Thermodynamics and Thermal Process Engineering Germany Part 1 General Aspects 1 Thermodynamic Optimization* M.M. Awad1 and Y.S. Muzychka2 1Mechanical Power Engineering Department,Faculty of Engineering, Mansoura University,2Faculty of Engineering and Applied Science, St. John's, NL,Memorial University of Newfoundland,1Egypt 2Canada 1. IntroductionSecondlawanalysisinthedesignofthermalandchemicalprocesseshasreceived considerable attention since 1970s. For example, Gaggioli and Petit (1977) reviewed the first and second laws of thermodynamics as an introduction to an explanation of the thesis that energy analyses of plants, components, and processes should be made by application of the second law that deals with the availability of energy or the potential energy. They illustrated theirmethodologysuggestedbyapplyingittoananalysisoftheKoppers-Totzek gasification system. Optimization of heat exchangers based on second-law rather than first-law considerations ensures that the most efficient use of available energy is being made.Second-law analysis has affected the design methodology of different heat and mass transfer systems to minimize the entropy generation rate, and so to maximize system available work. Manyresearchersconsideredtheseprocessesintermsofoneoftwoentities:exergy (availableenergy)andirreversibility(entropyproduction).Forinstance,McClintock(1951) described irreversibility analysis of heat exchangers, designed to transfer a specified amount of heat between the fluid streams. He gave explicit equations for the local optimum design offluidpassagesforeithersideofaheatexchanger.Totheknowledgeofauthors, McClintock(1951)wasthefirstresearcherwhoemployedtheirreversibilityconceptfor estimating and minimizing the usable energy wasted in heat exchangers design. Bejan (1977) introduced the concept of designing heat exchangers for specified irreversibility rather than specifiedamountofheattransferred.Manyauthorsusedthistechniqueinthefieldof cryogenicengineering(BejanandSmith(1974,1976),Bejan(1975),andHilalandBoom (1976)). Oneofthefirstexaminationsofentropygenerationinconvectiveheattransferwas conductedbyBejan(1979)foranumberoffundamentalapplications.Muchoftheearly *ThepartofthischapterwaspresentedbyY.S.Muzychkainfall2005asPartIIIduringtheshort course: Adrian Bejan, Sylvie Lorente, and Yuri Muzychka,Constructal Design ofPorous andComplex FlowStructures,MemorialUniversityofNewfoundland,FacultyofEngineeringandAppliedScience, St. John's, NL, Canada, September 21-23, 2005. Heat Exchangers Basics Design Applications 4 workiswelldocumentedinhisbooks(Bejan,1982aand1996a).Sincethepublicationof (Bejan,1996a),entropygenerationininternalstructurehasbeenexaminedbynumerous researchers.Inthissection,wewillexaminethesestudiesthatincludetheoptimizationof heat exchangers, and enhancement of internal flows. Also, we will proceed to develop some of the basic principles and examine selected results from the published literature. 1.1 Optimization of heat exchangers Inthepastthirtyfiveyears,muchworkrelatingtoheatexchangerdesignbasedonthe second law of thermodynamics was presented by researchers (Bejan, 1988). Heat exchangers haveoftenbeensubjectedtothermodynamicoptimization(orentropygeneration minimization)inisolation,i.e.,removedfromthelargerinstallation,whichusesthem. Examplesincludetheparallelflow,counterflow,crossflow,andphase-changeheat exchanger optimizations. We will talk in details about this in this section. Bejan(1977)presentedaheatexchangerdesignmethodforfixedorforminimum irreversibility (number of entropy generation units, Ns). The researcher obtained the number ofentropygenerationunits(Ns)bydividingentropygenerationratebythesmallestheat capacityrateofthefluids.ThevalueofNscanrangebetween0-.Theheatexchanger would have a better performance if the entropy generation was at its minimum (Ns0). This dimensionless number can clearly express how a heat exchanger performance is close to an ideal heat exchanger in terms of thermal losses. He showed that entropy generation in a heat exchanger is due to heat transfer through temperature gradient and fluid friction. In contrast with traditional design procedures, the amount of heat transferred between streams and the pumpingpowerforeverysidebecameoutputsoftheNsdesignapproach.Also,he proposedamethodologyfordesigningheatexchangersbasedonentropygeneration minimization.Toillustratetheuseofhismethod,thepaperdevelopedthedesignof regenerativeheatexchangerswithminimumheattransfersurfaceandwithfixed irreversibility Ns. Thethermaldesignofcounterflowheatexchangersforgas-to-gasapplicationsisbasedon the thermodynamic irreversibility rate or useful power no longer available as a result of heat exchangerfrictionalpressuredropsandstream-to-streamtemperaturedifferences.The irreversibility(entropyproduction)conceptestablishesadirectrelationshipbetweenthe heatexchangerdesignparametersandtheusefulpowerwastedduetoheatexchanger nonideality.Bejan(1978)demonstratedtheuseofirreversibilityasacriterionforevaluationofthe efficiencyofaheatexchanger.Theresearcherminimizedthewastedenergyusingthe optimumdesignoffluidpassagesinaheatexchanger.Hestudiedtheinterrelationship between the losses caused by heat transfer across the stream-to-stream due to differences in temperatures and losses caused by fluid friction. He obtained the following relation for the entropy generation rate per unit length as follows: . . .2 201 1gen dS dq dq m dP T m dP TT T dx T dx dx T dx dxT TT T A A | | | |= + ~ + > ||A A | | | |\ . \ .+ + ||\ . \ .(1) Thermodynamic Optimization 5 The first term in expression (1) is the entropy production contribution due to fluid friction in thefluidduct.Thesecondterminexpression(1)representsthecontributionduetoheat transfer across the wall-fluid temperature difference. These two contributions were strongly interrelatedthroughthegeometric characteristicsoftheheatexchanger.Itshouldbenoted that the use of density () instead of the inverse of specific volume (v) in the first term on the righthandside.Also,thedenominatorofthesecondtermontherighthandsidewas simplified by assuming that the local temperature difference (AT) was negligible compared with the local absolute temperature (T). Heat transfer losses could be reduced by increasing theheattransferarea,butinthiscasepressuredropsinthechannelsincreased.Bothheat transfer losses and frictional pressure drops in channels determined the irreversibility level of heat exchanger.AremarkablefeatureofEq.(1)andofmanylikeitforothersimpledevicesisthata proposeddesignchange(forinstance,makingthepassagenarrower)induceschangesof opposite signs in the two terms of the expression. Then, an optimal trade-off exists between the fluid friction irreversibility and the heat transfer irreversibility contributions, an optimal designforwhichtheoverallmeasureofexergydestructionisminimum,whilethesystem continuestoserveitsspecifiedfunction.Inordertoillustratethistrade-off,usethe definitionoffrictionfactor(f),Stantonnumber(St),massflux(G),Reynoldsnumber(Re), and hydraulic diameter (dh): 22hd dPfdx G | |= |\ .(2) 1pdqStdx p TcG=A(3) .mGA= (4) RehGd= (5) 4hAdp= (6) In Eq. (3), the quantity (dq/dx)/(pAT) is better known as the average heat transfer coefficient. The entropy generation rate, Eq. (1) becomes 3. .2. 2 2224genhhpdS dq m f ddx dx TdAT mcSt| |= + |\ .(7) Whereheattransferrateperunitlengthandmassflowratearefixed.Thegeometric configurationoftheexchangerpassagehastwodegreesoffreedom,theperimeter(p)and the cross-sectional area (A), or any other pair of independent parameters, like (Re; dh) or (G; Heat Exchangers Basics Design Applications 6 dh). If the passage is a straight pipe with circular cross-section, p and A are related through thepipeinnerdiameterdthatistheonlydegreeoffreedomleftinthedesignprocess. Writing 2, /4,hd d A d and p d t t = = = (8) Equation (7) becomes 3. .22 2 2 532 genhdS dq m f ddx dx TkNu Td t t | |= + |\ .(9) Where Re = 4 m/td. The Nusselt number (Nu) definition, and the relation betweenNu, St, Re, and the Prandtl number (Pr = v/o) .Re.Pr ..av hh dNu St St Pek= = = (10) Introducing two classical correlations for fully developed turbulent pipe flow (Bejan, 1993), 0.8 0.4 40.023Re Pr ( 0.7 Pr 160 : Re 10 ) Nu = ( ( ) (11) -0.2 4 60.046 Re (10 Re 10 ) f = ( ( (12) andcombiningthemwithEq.(9),yieldsanexpressionforexergydestruction,which dependsonlyonRe.DifferentiatingtheexergydestructionwithrespecttotheReynolds number(Re)andequalingtheresultwithzero,wefindthattheentropygenerationrateis minimized when the Reynolds number (or pipe diameter) reaches the optimal value (Bejan, 1982a) -0.071 0.358optRe2.023Pr B = (13) Equation(13)showshowtoselecttheoptimalpipesizeforminimalirreversibility. ParameterBisaheatandfluidflowdutyparameterthataccountsfortheconstraintsof heat transfer rate per unit length, and mass flow rate: .5/2 1/2( )dq pB mdx kT | |= |\ .(14) Additional results may be obtained for non-circular ducts using the appropriate expressions for the geometry A and p, and appropriate models for heat transfer and friction coefficients. TheReynoldsnumber(Re)effectontheexergydestructioncanbeexpressedinrelative terms as . 0.8 4.8.min/ Re Re0.856 0.144Re Re( / )genopt optgendS dxdS dx| | | | || = + ||\ . \ .(15) Thermodynamic Optimization 7 wheretheratioontheleft-handsideisknownastheentropygenerationnumber(Ns), (Bejan,1982a).InthedenominatorofthelefthandsideofEq.(15),theminimumexergy destructioniscalculatedattheoptimumReynoldsnumber(Reopt).Also,Re/Reopt=dopt/d becausethemassflowrateisfixed.UsingEq.(15),itisclearthattherateofentropy generationincreasessharplyoneithersideoftheoptimum.Thelefthandsideofthe optimum represents the region in which the overall entropy generation rate is dominated by heat transfer effects. The right hand side of the optimum represents the region in which the overallentropygenerationrateisdominatedbyfluidfrictioneffects.Thelefthandsideof Eq. (15) is used to monitor the approach of any design relative to the best design that can be conceivedsubjecttothesameconstraints.Bejan(1982a,1988)usedthisperformance criterionextensivelyintheengineeringliterature.Also,Mironovaetal.(1994)recognized this performance criterion in the physics literature. Bejan(1978)alsomadeaproposaltousethenumberofentropyproductionunits(Ns)asa basicyardstickindescribingtheheatexchangerperformance.Thisdimensionlessnumber was defined as the entropy production rate or irreversibility rate present in a heat exchanger channel. When Ns 0, this implied an almost ideal heat exchanger channel. According to his study,itwasenoughtoincreasetheeffectivenessbyusingdesigncriterionslikethe minimizationofdifferencewalltemperatureormaximizationoftheratioofheattransfer coefficient to fluid pumping power. Bejan (1979) illustrated the second law aspects of heat transfer by forced convection in terms offourfundamentalflowconfigurations:pipeflow,boundarylayeroverflatplate,single cylinderincross-flow,andflowintheentranceregionofaflatrectangularduct.The researcher analyzed in detail the interplay between irreversibility due to heat transfer along finite temperature gradients and, on the other hand, irreversibility due to viscous effects. He presented the spatial distribution of irreversibility, entropy generation profiles or maps, and thoseflowfeaturesactingasstrongsourcesofirreversibility.Heshowedhowtheflow geometricparametersmightbeselectedtominimizetheirreversibilityassociatedwitha specific convective heat transfer process. Bejan(1980)usedthesecondlawofthermodynamicsasabasisforevaluatingthe irreversibility(entropygeneration)associatedwithsimpleheattransferprocesses.Inthe first part of his paper, he analyzed the irreversibility production from the local level, at one point in a convective heat transfer arrangement. In the second part of his paper, he devoted toalimitedreviewofsecondlawanalysisappliedtoclassicengineeringcomponentsfor heatexchange.Inthiscategory,thepaperincludedtopicslikeheattransferaugmentation techniques,heatexchangerdesign,andthermalinsulationsystems.Theresearcher presentedanalyticalmethodsforevaluatingandminimizingtheirreversibilityassociated with textbook-type components of heat transfer equipment. Also, he obtained an expression for the entropy generation rate in a balanced counterflow heat exchanger with zero pressure drop irreversibility as follows: 1 22 121 1ln(1 )sT TNTU NTUT TNNTU| | | |+ + ||\ . \ .=+(16) Heat Exchangers Basics Design Applications 8 UsingEq.(16),Ns =0atbothc=0(oratNTU=0)andc=1(oratNTU=),andhadits maximum value at c = 0.5 (or at NTU = 1). The maximum Ns increases as soon as T1/T2 goes above or below 1: 1 2,max2 11 1ln2 4sT TNT T ( | |= + + (| ( \ . (17) NsincreaseswiththeabsolutetemperatureratioT2/T1.WhenNs>1,theirreversibility decreasessharplyasc1.OntheleftsideofthemaximumNs 0 represented poorer quality. Thermodynamic Optimization 9 Also,hedescribedtherelativeimportanceofthetwoirreversibilitymechanismsusingthe irreversibility distribution ratio (|) that was defined as: .,.,fluid - flow irreversibility heat transfer irreversibilitygen Pgen TSS|AA= = (19) For example, the irreversibility distribution ratio (|) varies along with the V-shaped curve of entropygenerationnumber(Ns),orrelativeentropygenerationrateinasmoothpipewith heat transfer (Bejan, 1980), increasing in the direction of large Reynolds numbers (small pipe diameters because the mass flow rate is fixed) in which the overall entropy generation rate is dominatedbyfluidfrictioneffects.Attheoptimum(correspondingtoNs=1),the irreversibilitydistributionratio(|)assumesthevalue|opt=0.168.Thismeansthatthe optimaltrade-offbetweentheirreversibilityduetoheattransfereffectsandthe irreversibilityduetofluidfrictioneffectsdoesnotcoincidewiththedesignwherethe irreversibility mechanisms are in perfect balance, even though setting| = 1 is a fairly good way of locating the optimum.Substituting Eq. (19) into Eq. (18) yields . ., (1 ) gen gen T S S | A = + (20) In addition, augmentation entropy generation number (Ns,a) was given by .,,.,gen asagen oSNS= (21) Thisdefinitionrepresentstheratiooftheaugmentedtobasechannelentropygeneration rates.Underparticularflowconditionsand/orconstraints,Ns,a

= == = = =(2) Heat Exchangers Basics Design Applications 56Theseconditionsassumethatonlyfluid1inletconditionisperturbed.Thestepchangeof inlettemperatureoffluid1iscertainlythemostimportantfromphysicalpointofview. Other inlet temperature changes can be analyzed using described mathematical model and procedures for their analytical solution.In equations 1 and 2, the convention of index 1 referring to weaker fluid flow and index 2 to stronger fluid flow is introduced. Fluid undergoing higher temperature changes because of smallervalueofthethermalcapacity pW mc = iscalledweaker?Theotherflowisthen strongeranditislesschangedintheheatexchanger.Theproductofmassflowrateand isobaricspecificheatoffluidistheindicatoroffluidsflowstrengthandrepresentsits essentialcharacteristic.Therefore,itisnecessarytomakestrictdistinctionbetweenweaker andstrongerflow.Onlytheweakerfluidflowcanchangethestateformaximum temperaturedifference.Therefore, ( )' 'max 1 2minpQ mc T T = .Thisisvalidinsteadystate conditionsalthoughflowdesignationconventionisalsoapplicabletounsteadystate analysis. Generally, the heat exchangers effectiveness is defined in the relation of actually exchanged heatandmaximumpossibleoneanditisthemeasureofthermodynamicqualityofthe device.Inthisway,theeffectivenessofallheatexchangerscanbeanumbertakenfroma closed interval[0, 1] c =Anotherconventionisusefulforfurtheranalysis.Ifweakerandstrongerfluidflowsare designatedwithindices1and2,respectively,thenstandardizedrelationbetweenheat capacities of fluids is: 12(0 1)WWe e = s s (3) The value0 e always designates that the stronger fluid flow tends to isothermal change intheheatexchangersince( )2pmc .WithfinalQ,implying '' '2 20 T T ,thismeans thattheflow2changesthephase(condensationorevaporation).Onthecontrary,1 e =refers to well balanced flows, i.e. the temperatures from inlet to outlet change equally. Inordertodefinedimensionlesstemperatures,itisappropriatetochoosereference temperature Tr and a characteristic temperature difference T* - Tr so that: *( , )( , ) ( 1, 2, )i rirTXt TXt i wT Tu= =(4) Itissuitablethatreferencetemperaturesareminimumandmaximumones,i.e.T*andTr, respectively. If the weaker flow is designated with index 1 and if * '1T T =and '2 rT T =then, the weaker flow enters the heat exchanger with '11 u = and the stronger flow with '20 u = .Forthepurposeofsimplifyingthemathematicalmodelthedimensionlessdistanceand dimensionless time are introduced: *,X tx NTU zL t= = (5) Analytical Solution of Dynamic Response of Heat Exchangers 57 The number of heat transfer units is: 1 21 2 1( ) ( ) 1( ) ( )hA hANTUhA hA W= +(6) and time parameter *1 2( ) ( )w wc MthA hA=+(7) Further, the relation for the product of heat transfer coefficient and heat transfer area of each fluid and the sum of these products is as follows: 11 2 11 2( ), 1( ) ( )hAK K KhA hA= = +(8) Finally, complex dimensionless parameter is: 1( 1, 2)iiw w i iWC L ic M K U= = (9) Itisinverselyproportionaltothefluidspeedinheatexchangerflowchannels.Thehigh fluid velocity with other unchanged values in the equation (9) means that0iC and that fluid dwell time in the heat exchanger is short. As the fluid velocity decreases, the value of parameters Ci increases and the time of fluid dwell time in the core of the heat exchanger is prolonged. Fluid velocity in heat exchangers is:( , 1, 2)iii imU fluid velocity iF = =(10) Now, the system of equations (1) can be written in the following form : 1 1 2 2wwK Kzuu u uc+ = + c 1 11 2 1 wC Kz xu uu uc c + = c c 2 1 22 2 wKCz xu uu uec c + = c c(11) The initial and inlet conditions (Eqs. 2) become: 121 20 0(0, )1 0(0, ) 0( , 0) ( , 0) ( , 0) 0wfor zzfor zzx x xuuu u u< = >== = =(12) Heat Exchangers Basics Design Applications 58Theequation(11)and(12)definetransientresponseofparallelflowheatexchangerwith finitewallcapacitance.Mathematicalmodelisvalidforthecasewhen 1 2W W s and temperature of fluid 1 is perturbed (unit step change). Outlet temperatures of both fluids in steady state ( z ) are: "1"2( , ) 1( , )NTUNTUu cu e c = = (13) wherec iseffectivenessofheatexchanger.Effectivenessofparallelheatexchangerisas follows: | | 1 exp (1 )0 11NTUforec ee += < s+(14) For the case0 e =the effectiveness is( ) 1 exp NTU c = (15) and is valid for all types of heat exchangers.Forthecasewhenstrongerfluid(fluid2)isperturbed,theinletconditionofthe mathematical problem is changed and is as follows: 121 2(0, ) 00 0(0, )1 0( , 0) ( , 0) ( , 0) 0wzfor zzfor zx x xuuu u u=< = >= = =(16) In this case, outlet temperatures in the conditions of steady state are equal: "1"2( , )( , ) 1NTUNTUu e cu c = = (17) Inthisway,resolvingofthismathematicalproblemfortwoinletconditionsincludesall possible cases of fluid flow strength, i.e. 1 2 1 2W W and W W s > . Only the case 1 2W W sis analyzedinthispaperbecauseoflimitedspace.However,thepresentedprocedurefor resolvingmathematicalmodelforalltypesofheatexchangersgivesopportunitiestoget easily to the solution in case when1 2W W > . 2.2 Counter flowInthesamewayasinthecaseofparallelflowheatexchanger,itispossibletosetup mathematicalmodelofcounterflowheatexchanger(Fig.2).Theessentialdifference between these two heat exchangers is in inlet conditions. Analytical Solution of Dynamic Response of Heat Exchangers 59 h1, A1h2, A2 21X L 0F2F1x NTU 0' '1 1m , T' '2 2m , T dX Fig. 2. Schematic Description of Counter Flow Heat Exchanger Proceduresimilartotheaboveforparallelflowdeliversthefollowingmathematical formulation for the transient behavior of counter flow heat exchanger: 1 1 2 2wwK Kzuu u uc+ = + c 1 11 2 1 wC Kz xu uu uc c + = c c 2 1 22 2 wKCz xu uu uec c = c c(18) The initial and inlet conditions are: 121 20 0(0, )1 0( , ) 0( , 0) ( , 0) ( , 0) 0wfor zzfor zNTUzx x xuuu u u< = >== = =(19) If the system of equations (11) and (18) is compared, it can be observed that the difference is onlyinthesignbeforethesecondmemberontherightsideofthethirdequation.Ifwe compareequations(12)and(19)(inletandinitialconditions),thedifferenceisonlyinthe secondequation.However,theseseeminglysmalldifferencesmakesubstantialdifferences in the solution of the problem which will be shown later on. Outlet temperatures of both fluids in steady state ( z ) are as in the case of parallel flow heatexchangerbuttheeffectivenessisincaseofcounterflowheatexchangerdesignedas follows: | || |1 exp (1 )0 11 exp (1 )NTUforNTUec ee e = s < (20) and 11 NTUforNTUc e = =+(21) Heat Exchangers Basics Design Applications 60When stronger fluid (fluid 2) is perturbed, the inlet condition of the mathematical problem is changed and is as follows: 121 2(0, ) 00 0( , )1 0( , 0) ( , 0) ( , 0) 0wzfor zNTUzfor zx x xuuu u u=< = >= = =(22) The problem formulated in this way is valid for W1 W2. For the case W1 W2, the problem is very similar and because of that it will not be elaborated in details. 2.3 Cross flow (both fluids unmixed) The drawing of cross flow heat exchanger which is used for mathematical analysis is shown in Fig. 3. It contains the necessary system of designation and coordinates which will be used in this paper. The fluid 1 flows in the X direction and the fluid 2 in the Y direction. The fluid flows are not mixed perpendicularly to their flow.Basedontheseassumptionsandbyapplyingenergyequationstobothfluids,three simultaneouspartialdifferentialequationscanbeobtainedinthecoordinatesystemas shown in Fig. 3. ( ) ( ) ( ) ( )1 21 2www w wTMc h A T T h A T Ttc= c ( ) ( )1 11 1 1111p o wT Tm c X h A T TX U t| | c c+ = |c c\ . ( ) ( )2 22 2 2221p o wT Tm c Y h A T TY U t| | c c+ = |c c\ .(23) Independentvariablesinspaceandtime(X,Yandt)varyfrom0tothelengthofheat exchangersXoandYo,i.efrom0to .Bycomparingthesystemofequations(1),itcanbe noticedthatthereisthepresenceofthespacecoordinate(Y)andtheexistenceoftwo dimensions of heat exchangers (Xo and Yo).=e y NTU= x NTU Fig. 3. Schematic Description of Cross Flow Heat Exchanger. Analytical Solution of Dynamic Response of Heat Exchangers 61 Initial and inlet conditions of analyzed problem are as follows: 1*0(0, , )0T for tT YtT for t < = > 2( , 0, ) T X t T const = = 1 2( , , 0) ( , , 0) ( , , 0)wT X Y T X Y T X Y T const = = = = (24) By introducing new dimensionless variable: *, ,o oX Y tx NTU x NTU zX Y t= = = (25) the set of equations (23) is as follows: 1 1 2 2wwK Kzuu u uc+ = + c 1 11 2 1 wC Kz xu uu uc c + = c c 2 22 1 2 wC Kz yu uu uc c + = c c(26) and initial and inlet conditions (Eq. 24) as: 10 0(0, , )1 0for zyzfor zu< = > 2( , 0, ) 0 x z u = 1 2( , , 0) ( , , 0) ( , , 0) 0wxy xy xy u u u = = = (27) Outlettemperaturesofbothfluidsinsteadystate( z )aredefinedbyEq.(13)butthe effectiveness in the case of cross flow heat exchanger is defined as follows (Bali, 1978): | |( ) ( ) ( )/20 121 exp (1 )12 2 2nnnNTUI NTU I NTU I NTUc eee e e e ee== + ( + ( (28) and | | ( ) ( )0 11 exp 2 2 2 1 NTU I NTU I NTU for c e = + ( = (29) In Eqs. (28 and 29), the( )nI is modified Bessel function. Heat Exchangers Basics Design Applications 62Forthecasewhenstrongerfluid(fluid2)isperturbed,theinletconditionofthe mathematical problem is changed and it is as follows: 1( , 0, ) 0 x z u = 20 0(0, , )1 0for zyzfor zu< = > 1 2( , , 0) ( , , 0) ( , , 0) 0wxy xy xy u u u = = = (30) Asopposedtoparallelandcounterflowheatexchangerswhereoutletfluidtemperatures areconstantoverthewholelengthofoutletedges,itisnotthecaseforcrossflowheat exchangers.Then,outlettemperaturefromtheheatexchangerisobtainedasmean temperature at the outlet edge of the heat exchanger.Specialcasesofcrossflowheatexchangerswhenoneorbothfluidflowsaremixed throughout will not be elaborated in this paper.IntheSectionthatfollows,definedmathematicalproblemsfordeterminingtemperature fieldsandoutlettemperatureswillberesolvedforthreebasictypes:parallel,counterand cross flow heat exchangers. 3. General solution The set of three partial differential equations for all types of heat exchanger are linear (Eqs. 11,18and26).ThesesystemscanbesolvedbyusingmultifoldLaplacetransform.Inthe caseofparallelandcounterflowheatexchangers,itisdouble-foldandinthecaseofcross flow it is three-fold Laplace transform.3.1 Parallel flow By applying this transform over the equations (11) and initial and inlet condition (Eq.16), the following algebraic equations are obtained: 1 1 2 21wK Kpu uu + =+ 112 21 1 1wC psK K pu u| | ++ = + |\ . 221 1( 1)wC psK Ke eu u| | ++ = |\ . (31) From this set of equations, the outlet and wall temperatures are as follows: + +u = + + ++ +e1 22 1w1 212 11K KK s C p 1 1K Kpp 1KK s C p 1s C p 1(32) Analytical Solution of Dynamic Response of Heat Exchangers 63 212 1 2 111 1wKK s C p K s C p puu = + + + + +(33) 2111wKs C puue=+ +(34) Afterperformingsomemathematicaltransformationsandbyusingsomewellknown relations: ( )101 111nnnx x+== + ;( )0nnm n mmna b a bm=| |+ = |\ .(35) the temperatures can be expressed in the following form which is convenient for developing the inverse Laplace transform: 1111 12 012 2111 212 1 1 01 11 22 2 1 11 1( 1)11 1 1( 1)1nwn nnm n mnn mn m n mKKK p pCs pK KnK KKm K Kp pC Cs p s pK K K Kuee e++ +=+ = =+ + | |= + | +\ . | |+ + |\ .| | | | | | |||\ . \ . \ . +| | | | + + + + ||\ . \ . (36) 1111 212 011 12 2111 22 1 1 01 21 22 2 1 11 1 1 11( 1)11 1 1( 1)1nn nnm n mnn mn m n mKCp K p pCs ps pK KK KnK Km K Kp pC Cs p s pK K K Kuee e++ +=+ = =+ + | |= + + | +\ . | |+ ++ + |\ .| | | | | | |||\ . \ . \ . +| | | | + + + + ||\ . \ . (37) 1 11 222 1 0 01 1 11 22 2 1 11 1 1( 1)1m n mnn mn m n mnK Km K Kp pC Cs p s pK K K Keue e+ += =+ + +| | | | | | = |||\ . \ . \ . +| | | | + + + + ||\ . \ . (38) Heat Exchangers Basics Design Applications 64From the techniques of Laplace transformation (convolution and translation theorems)and usingtheLaplacetransformsofspecialfunctionsFn(x,c)andIn,m(x,c,d),definedinthe Appendix, one can obtain the inverse Laplace transformation of Eqs. 36-38, and the transient temperature distributions for the parallel flow heat exchanger: 11 12 1 1, 12 2 2 0111 222 1 1 01 21 1, 12 1 2 101( , ) , , 0, 11, , ( ) , 0,nw n nnm n mnn mxm n m nK Cxz K F x I z xK K KnK KKm K KC CF x u F u I z x u uK K K Kuee e++ +=+ = =+ +| | | | | |= + |||\ . \ . \ .| | | | | | + |||\ . \ . \ .| | | | | | + |||\ . \ . \ . }1 du ( ( ( (39) 11 12 211 12 1, 12 2 2 0111 22 1 1 02 1, 12 101( , ) ,1, , 0, 11, ,nn nnm n mnn mxm n m nCxz z x F xK KK CF x I z xK K KnK Km K KCF x u F u I zK Ku kee++ +=+ = =+ +| | | |= + ||\ . \ .| | | | | |+ + |||\ . \ . \ .| | | | | | + |||\ . \ . \ .| | | | ||\ . \ . }1 22 1( ) , 0, 1Cx u u duK Ke ( | | + (| ( \ . (40) 1 11 222 1 0 01 21 1 1, 12 1 2 10( , )1, , ( ) , 0, 1m n mnn mxm n m nnK Kxzm K KC CF x u F u I z x u uK K K Keue e+ += =+ + +| | | | | | = |||\ . \ . \ . ( | | | | | | + (||| ( \ . \ . \ . }(41) Outlet temperatures of both fluids are obtained for x = NTU.In the practical use of solutions, the computation of integrals in this paper is done through collocationatnineChebishevspoints:0.0000000000;0.1679061842;0.5287617831; 0.6010186554; 0.9115893077, for the given integration interval.Special case = 0 In this case, 2( , ) 0 xz u =resulting in reduced Eq. (31): 11 12 2 211( 1)wKp C Kp s pK K Ku = | |+ + + |\ .(42) After some mathematical manipulations, using already mentioned techniques, this equation can be transformed into: Analytical Solution of Dynamic Response of Heat Exchangers 65 11212 0112 211( 1)nwnnnKKKCp p s pK Ku++=+| |= |\ . | | + + + |\ .(43) The inverse two-fold Laplace transform of Eq. 43 gives: 11 12 1 1,12 2 2 01( , ) , , 1, 1nw n nnK C xxz K F x I zK K Ku++ +=| | | | | | = |||\ . \ . \ .(44) and Eq. 32 gives: 11 1 11 1 2 1,12 2 2 2 2 01 1( , ) , , , 1, 1nn nnC x K C xxz z F x F x I zK K K K Ku k++ +=| | | | | | | | | | = + |||||\ . \ . \ . \ . \ . (45) This solution is valid for all types of heat exchangers with = 0. 3.2 Counter flow Averysimilarprocedurecanbeappliedforresolvingthemathematicalmodelofcounter flowheatexchanger.Thesetofalgebraicequationsobtainedaftertwo-foldLaplace transform of Eqs. (18) and initial and inlet conditions (Eq (19)) is as follows: 1 1 2 2( 1)wp K K u u u + = + (46) 112 2 21 1 1wCs pK K K pu u| |+ + = + |\ . (47) ( )22 21 1 110,wCs p pK K Ke eu u u| | = |\ . (48) The procedure will be explained in more details here since this case is much more complex than the previous one. By introducing designations: 12 1 22 21( , ) 1Csp K s C p K s pK Ko| |= + + = + + |\ .,(49) 1 1 221 1( , ) 1K K Csp s C p s pK Ke e|e e| | = + + = |\ .,(50) 1 2( , ) 1( , ) ( , )K KA sp psp sp o |= + ,(51) the both fluids and wall temperatures of the counter flow heat exchanger are as follows: Heat Exchangers Basics Design Applications 66 ( )1 2 1 220,wK K K Kpp A Au uo e | = (52) ( )2 1 2 1 21 220,K K K K Kpp A p Au uo e o | o = + (53) ( )1 2 1 22 21 0,K K K Kpp A Au uo | e | || | = + | \ . (54) It is very simple to prove that: 1 210 01 1 1( 1)nm n mn m n mn mnK Km A p o |+ = =| |= |+ \ . ,(55) andthatinverseLaplacetransformationsofthefunctions1/m+1(s,p)and1/m+1(s,p) (m=1,2,3,) with respect to the complex parameter s are: 1 111 12 2 21 1 1, exps x mm mCL F x x pK K K o ++ +| | | | = `|| )\ . \ .,(56) 11 1 2111 1 11( 1) , expmms x mmCL F x x pK K Ke e e|+ + ++ | | | | | | = `||| \ . \ . \ . ).(57) The essential problem in resolving dynamic behavior of the counter flow heat exchanger is in the use of other inlet conditions (Eq. 19). IftheEq.54iscollocatedintox=NTUthen,INLETtemperatureofthefluid2isobtained which is according to given inlet conditions 2( , ) 0 NTUz u = , therefore: ( )1 1 1 1 2 1 2220,s NTU s NTUK K K K KL p Lp A Aue | o | e | | | + = | ` ` \ . ) )(58) ThisisFredholmsintegralequationofthesecondorder.Theproblemisreducedtoits solving.The collocation method is used for solving this equation. Perhaps, it is the simplest one. The trial function is: ( ) ( )2 210, 0, 1 exp( ) exp( )!= (= ( ( k NCPkkzz z a zku u (59) Inequations(58)andfurtheron, 2(0, ) u isthesteady-statefluid2outlettemperaturefor thecounterflowheatexchanger.ItcanbecalculatedbyusingthesecondofEq.13and effectiveness of counter flow heat exchanger (Eqs. 20 and 21). It follows that: Analytical Solution of Dynamic Response of Heat Exchangers 67 | || |211(0, )1 exp (1 )0 11 exp (1 )NTUforNTUNTUforNTUeuee ee e=+ = s < (60) Laplace transform of trial function (Eq. 59) is: ( ) ( )92 2111 10, 0,( 1) ( 1)kkkp ap p pu u+= (= (+ + ( (61) Thetrialfunctionchoseninthiswaysatisfiescompletelytheequation(58)inpointsz=0 and z . Within the interval 0 < z < , it is necessary to determine collocation points and coefficients ak (k = 1, 2, 3, ... , NCP). Here, the NCP is the number of collocation points. The accuracyinwhichtheoutlettemperaturesoffluid2versustimearedetermineddepends directlyonNCP.Inthismodelofheatexchanger,therearemanyinfluentialfactorsand determinationofthenumberofcollocationpointsforthegivenaccuracyofoutlet temperatureissimplestthroughpracticaltestingofthesolution.Fortheheatexchangers parameters appearing in practice, it can be said that NCP varying from 5 to 7 is sufficient for the accuracy of four significant figures and for z 15.Substituting the equation (61) in the equation (58) and collocating resulting equation in the NCPpoint,asetoflinearalgebraicequationsisobtainedandtheirsolvinggenerates unknownconstantsak.Thesetofalgebraicsolutionscanalsobewritteninthefollowing form : 1NCPk k Rka= A = A(62) Substituting the equation (61) in (58) and using Eqs. (55), (56) and (57), it is obtained: ( ) ( ) ( )( )122 1 2 21 011 222 1 2 1 101 222 1(0, ) ( ), 1 ( 1) , 0 ( ), 11( 1) , , 0( ) ,nnk k n n knm n mNTUnn mm n mn mn kKF z r F NTU F z rKnK KF u F NTU um K K KC CF z r uK Keuee++ + + += + += =+ + | | A = | \ . | | | | | | | | ||||\ . \ . \ . \ .| | + |\ . }11 exp ( , 1, 2,... ) u du kr NCPKe ( | | = ( `| ( \ . )(63) ( ) ( )( )122 1,1 2 2,11 011 222 1 2 1 10(0, ) ( ( ), 1, 1) ( 1) , 0 ( ), 1, 11( 1) , , 0 ...nnR n nnm n mNTUnn mm n mn mKI z r F NTU I z rKnK KF u F NTU um K K Keue++ += + += = | | A = | \ . | | | | | | | | ||||\ . \ . \ . \ . } Heat Exchangers Basics Design Applications 68( )1 22,12 1 11 11 1 21 12 1 2 0 001 21,12 1... ( ) , 1, 1 exp1( 1) , , 0( ) , 1, 1nm n mNTUnn mm n mn mnC CI z r u u duK K KnK KF u F NTU um K K KC CI z r uK Ke eee++ + ++ += =+ ( | | | | + (|| ( \ . \ . | | | | | | | | ||||\ . \ . \ . \ . | | + |\ . }1exp u duKe( | | ( `| ( \ . ) (64) Theequations(63)and(64)definemembersinthesetofalgebraicequations(62).For determining constants ak, it is possible to use any of the well known methods.Thetemperaturedistributionofbothfluidsandtheseparatingwallcanbecalculatedby usingEqs.(52-54)andbysubstitutingtheLaplacetransformoffluid2outlettemperature givenbyEq.(59).Constantsakarenowknownandarevalidforallvaluesofzwithinthe close interval where the collocation is performed.Temperatures of fluid and wall are as follows: 11 12 211 12 1,12 2 2 0111 22 1 1 02 1,2 101( , ) ,1, , 1, 1( 1)1, ,nn nnn mmnn mn mxm n m nCxz z x F xK KK CF x I z xK K KnK Km K KF u F x u IK Ku kee++ +=+ = =+ +| | | |= + ||\ . \ .| | | | | | + |||\ . \ . \ .| | | | | | |||\ . \ . \ .| | | | ||\ . \ . }1 212 11 11 2 1 22 1 0 01 21 1 2,12 1 2 10( ) , 1, 1(0, )( 1)1, , ( ) , 1, 1m n mnn mn mxm n m nC Cz u x u duK KnK Km K KC CF u F x u I z u x uK K K Keu eee e+ + += =+ + +| |( | ( | \ .| | | | | | |||\ . \ . \ . ( | | | | | | (||| ( \ . \ . \ . }91 222 1 1( ) , 1k n kkC Ca F z u x u duK Ke+ += ( | | (| ( \ . (65) 11 12 1 1,12 2 2 0111 22 1 1 01 21 1,12 1 2 101( , ) , , 1, 1( 1)1, , ( )nw n nnm n mnn mn mxm n m nK Cxz K F x I z xK K KnK Km K KC CF u F x u I z u x uK K K Kuee e++ +=+ = =+ +| | | | | |= +|||\ . \ . \ . | | | | | | |||\ . \ . \ .| | | | ||\ . \ . }, 1, 1 ... du | | ( | `( | \ . ) Analytical Solution of Dynamic Response of Heat Exchangers 69 11 1 2 21 2,11 1 1 01 22 1 221 2 1 1 1 12... ( 1) , , 1, 11, 1 ( 1)1,nnn nnm n mK nn mk n kk n mm n mK K CF x I z xK K KnC K Ka F z xm K K KF u FKe e eee ee+++ +=+ ++ += = = + | | | | | | + ||| \ . \ . \ . ( | | | | | | | | + + (||||( \ . \ . \ . \ . | | |\ . 1 21 2,11 2 101 222 1 1, ( ) , 1, 1( ) , 1xnKk n kkC Cx u I z u x uK K KC Ca F z u x u duK Ke ee++ +=( | | | | ( || ( \ . \ . ( | | (`| (\ . )}(66) 1 11 1 222 1 0 01 21 1 1,12 1 2 1022 1 1,11 1( , ) ( 1)1, , ( ) , 1, 1(0, ) , , 1,m n mnn mn mxm n m nnK Kxzm K KC CF u F x u I z u x u duK K K KCF x I z xK Keue ee eu+ + += =+ + +| | | | | | = |||\ . \ . \ . | | | |( + `||( \ . \ . )| | + |\ . }9221 1122 21 1 092 22,1 21 1 1122 1 11 , 1(0, ) ( 1) ,, 1, 1 , 1(0, ) ( 1)k n kknnnnn k n kknn mn mCa F z xKKF xK KC CI z x a F z xK KKKee eue eu+ +=++=+ + +== = ( | | | | + (|| ( \ . \ . | | | | ||\ . \ . ( | | | | + + (|| ( \ . \ . | 1211 22 2,12 1 2 1091 222 1 11, , ( ) , 1, 1( ) , 1m n mxm n m nk n kknKm KC CF u F x u I z u x uK K K KC Ca F z u x u duK Kee ee + + ++ +=| | | | | |||\ . \ . \ .( | | | | | | ( ||| ( \ . \ . \ . ( | | (`| ( \ . )}(67) 3.3 Cross flow Theequations(25)arelinearper1(x,y,z),w(x,y,z),and2(x,y,z).Ifthree-foldLaplace transform of above equations is taken in relation to x, y and z with complex parameters s, q, andp,respectively,andifinletandinitialconditionsareused(equation15),asetof algebraic equations is generated : 1 1 2 2( 1)wp K K u u u + = + (68) ( )22 1 11wKK s C ppqu u + + = + (69) Heat Exchangers Basics Design Applications 70( )1 2 21wK q C p u u + + = (70) Solving the set of algebraic set (equations (16)-(18)) is as follows: ( )1 22 11 22 1 1 2111 1wK Kpq K s C pK KpK s C p K q C pu + +=+ + + + +(71) ( ) ( )212 1 2 11 1wKK s C p pq K s C puu = ++ + + +(72) ( )21 21wK q C puu = + +(73) Afterperformingcertainmathematicaltransformationsasdoneinpreviouscases,the algebraic equation (71) can be expressed in the following form: ( ) ( ) ( )1 11 21 10 02 1 1 21 1 1m n m nwn m n mn mnK Kmp p K s C p q K q C pu+ + + + = =| | = | + + + + + \ . (74) whichisverysuitableforinverseLaplacetransformsbymeansoffunctions( , )nF x c and ,( , , )n mI x cddefined in the Annex. However, for the case n = m in the equation (74) and later on, the twofold sum will be separated into two (single and double) sums so that: ( ) ( )( ) ( ) ( )11 21 102 11 1 11 21 11 02 1 1 21 11 1 1nwn nnm n m nn m n mn mK Kp p q K s C pnK Kmp p K s C p q K q C pu+ + +=+ + + + = == + + + +| | | + + + + + \ . (75) Theinsertionoftheequation(74)inequations(72)and(73)generatesthefollowing algebraic equations: ( )( ) ( )( ) ( ) ( )12 1 211 22 1 02 11 1 11 21 21 02 1 1 211 11 1 1nn nnm n m nn m n mn mK K Kp q K s C pp p q K s C pnK Kmp p K s C p q K q C pu+ + +=+ + + + = == + + + + + + +| | | + + + + + \ . (76) ( ) ( ) ( )1 11 221 1 10 02 1 1 21 1 1m n m nn m n mn mnK Kmp p K s C p q K q C pu+ + + + += =| | = | + + + + + \ . (77) Now it is possible to get the inverse Laplace transform equation (75)-(77), so that: Analytical Solution of Dynamic Response of Heat Exchangers 71 111 1 1,1 12 2 0111 2 12 1 0/1,1 1 21 2 1 10( , , ) ,1 ,1,1,1,1 ,1,1nw n nnnm n mmn my Kn m nx xx yz K F I z CK KnxK K Fm Kv x v dvF I z C CK K K Ku++ += + += = +| | | |= + ||\ . \ .| | | | ||\ . \ .| | | | ||\ . \ .}(78) 11 1 12 211 2 1,1 12 2 0111 2 22 1 0/1,1 1 21 2 1 10( , , ) ,1,1 ,1,1,1,1 ,1,1nn nnnm n mmn my Kn m nx xx yz z C FK Kx xK F I z CK KnxK K Fm Kv x v dvF I z C CK K K Ku k++ += + += = +| | | |= + ||\ . \ .| | | | + ||\ . \ .| | | | ||\ . \ .| | | | ||\ . \ .}(79) 112 1 2 12 0 0/1 1,1 1 21 2 1 10( , , ) ,1,1 ,1,1nm n mmn my Kn m nnxx yz K K Fm Kv x v dvF I z C CK K K Ku+ += =+ +| | | |= ||\ . \ .| | | | ||\ . \ .}(80) Theequations(78)-(80)areanalyticalexpressionsfortemperaturefieldsoffluids1and2 and separating wall of cross heat exchanger dependant on time. At the beginning, the inlet temperatureoffluid1isinstantlyraisedfrom0to1,andflowvelocitiesofbothfluidsare constant. Outlettemperaturesofbothfluidsareobtainedbyintegratingtemperaturesalongoutlet edges of the heat exchanger. This is how outlet temperatures become equal; "1 101( ) ( , , )bz ayz dybu u = } (81) "2 201( ) ( , , )az xbz dxau u = } ,(82) where a = NTU and b = NTU. Substitutingequations(79)and(80)inequations(81)and(82)generatesaccurateexplicit expressions for mean outlet temperatures: Heat Exchangers Basics Design Applications 72 1"1 1 12 211 2 1,1 12 2 0121 2 22 1 0/1,1 1 21 1 2 10( ) , 1, 1 , 1, 11, 1, 1 , 1, 1nn nnnm n mmn mb Kn m na az z C FK Ka aK F I z CK KnaK K Fm b Kb v v a vF I z C CK K K Ku k++ += + += = +| | | |= + ||\ . \ .| | | | + ||\ . \ .| | | | ||\ . \ .| | | | ||\ . \ . }1dvK(83) 1 2" 12 1 20 0/ /1 1 1,1 1 22 1 2 1 2 10 01( ), 1 , 1 , 1, 1nm n mn my K a Km n m nnz K Km au v u v du dvF F I z C CK K K K K Ku+ = =+ + +| |= |\ .| | | | | | |||\ . \ . \ . } }(84) Above solutions are also valid for the case of indefinite fluid velocities (C1 = C2 = 0).4. Calculation results Themainpurposeofthispaperistoprovideexactanalyticalsolutionsbywhich performancesofparallel,counterandcrossflowheatexchangerscanbecalculatedand compared.Manyparametersareinvolvedintemperaturedistributionsofbothfluidsand thewalland,therefore,itisnotpossibletopresentquantitativeinfluencesofallthese parameters in this paper. However, there is enough space to give particular results showing main characteristics of solutions. Programmingofequationsexpressingtemperaturefieldsandoutlettemperaturesfor consideredtypesofheatexchangerscanbeverytiresome.Therefore,thewebsite www.peec.uns.ac.rspresentsprogramsinMSEXCELforcalculations.Programscanbe modified and improved as required.The example of a heat exchanger where NTU = 1, = 0.5, K1 = 0.25 (K2 = 1 K1 = 0.75), C1 = 4.0andC2=0.5willbediscussedbelow.Thetemperaturedistributionsofbothfluidsand thewallofPARALLELflowheatexchangerareplottedversusdimensionlessheat exchanger length (distance x) for z = 2 and 4 in Figure 4.The occurrence of heating up of separating wall and fluid 1 by fluid 2 is typical for parallel flowheatexchanger.Thiscanhappenatthebeginningofanon-steadystateprocesswhen thevelocityofthefluid2flowishigherthanthevelocityoffluid1.Thiswillbeexplained somewhat later when comparing outlet temperatures for all three types of heat exchangers. TheFigure5showstemperaturedistributionfortheCOUNTERflowheatexchanger.The parametersofthisheatexchangerarethesameasfortheparallelone.Differencesof temperature distribution between parallel and counter flow heat exchangers are evident. Analytical Solution of Dynamic Response of Heat Exchangers 73 Fig. 4. Temperature Distribution of Both Fluids and the Wall of Parallel Flow Heat Exchanger for z = 2 and 4. Heat Exchangers Basics Design Applications 74Fluid 1WallFluid 2z = 2Dimensionless Distance, xDimensionless Temperature,1, 2and w0.00.10.20.30.40.50.60.70.80.91.00.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Fluid 1WallFluid 2z = 4Dimensionless Distance, xDimensionless Temperature,1, 2and w0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.00.10.20.30.40.50.60.70.80.91.0 Fig. 5. Temperature Distribution of Both fluids and the Wall of Counter Flow Heat Exchanger for z = 2 and 4. As an example of the use of presented solutions for cross flow heat exchanger,temperature fields for both fluids and separating wall are given for the same case (NTU = 1, = 0.5, K1 = 0.25,C1=4,andC2=0.5).Temperaturefieldsofbothfluidsandthewallareshownfor dimensionless lengths of heat exchangers at dimensionless time z = 6 (Figure 6). At the time z=6,thefrontofbothfluidshasleftboundariesoftheheatexchanger.Alongtheoutlet Analytical Solution of Dynamic Response of Heat Exchangers 75 fluidedge,walltemperaturehasbeensignificantlyraisedbutwalltemperaturealongthe outletedgeofthefluid1isverymodest.Theperturbationofthefluid1hasjustleftthe outlet edge of the heat exchanger. For the fluid 2, the perturbation has moved far away from the outlet edge. Since the the perturbation front of the fluid 1 has just left the outlet edge of the heat exchanger, wall temperature at this edge are low. The same conclusion is also valid for fluid 2 temperature. However, it should be noted that the strength of the fluid 2 flow is two times higher that the strength of the fluid 1.iu Fig. 6. Temperature Fields of Both Fluids and The wall of CROSS Flow Heat Exchanger for z = 2 and 4. Fig.7showsoutlettemperaturesofbothfluidflowsforallthreetypesofheatexchangers. ThesizeofthesethreeheatexchangersisNTU=1.0and=0.5.Thecharacteristicsof transient heat are also equal for all three types of heat exchangers and they are defined by K1 =0.25,i.e.,K2=1-K1=0.75.Thevelocityoffluidflow1(C1=4.0,i.e.,1 11 / U C )isless thanthevelocityoftheflow2(C2=0.5,i.e.,2 21 / U C ).Thismeansthatthefluid1flows longerthroughtheflowchannelsthanfluid2.Intheanalyzedcase,theratiooffluid velocities is U1/U2 = 0.04167. For the fluid 2, the time from z=0 to 1 is necessary to pass the wholelengthoftheheatexchangeratitssideoftheseparatingwall.Thetimez=5.33is required for the fluid 1.The change curve of outlet temperature of fluid 2 is continuous for all three cases (Fig. 7). It is logical that the highest outlet temperature is achieved in the counter flow heat exchanger for which the effectiveness (steady-state) is also the highest for the same values of NTU and . It is followed by the cross flow and then by the parallel flow heat exchanger as the worst amongthethree.Inallcases,thefinaloutlettemperature( ) z isequalto 1 ( , , ) NTU flow arrangement c e = .Also,intransientregime,differencesregardingthe quality of exchangers are retained.Itisoppositeforthefluid1.Thelowesttemperatureisobtainedforthecounterflowheat exchangerandthehighestfortheparallelone.Finaloutlettemperaturesareequal Heat Exchangers Basics Design Applications 76( , , ) NTU flow arrangement e c e = .Itislogicalthattheoutlettemperatureofthefluid1isa discontinued function. After the step unit increase of the temperature of the fluid 1 at z = 0, the temperature of the fluid 1 falls due to heating of the wall of the heat exchanger and then heatingofthefluid2.However,inthecaseoftheparallelflowheatexchanger,inthe beginningafterperturbation,theoutlettemperatureofthefluid1growsevenbeforethe perturbationreachestheoutletoftheexchanger.Thismeansthatatonetimeofthenon-steadystatepartoftheprocess,thefluid2heatsuptheflowofthefluid1,aswellasthe wall instead of vice versa. Namely, ahead of the front, there is the fluid flow 2 heated up by the fluid flow 1. Since the velocity of the fluid flow 2 is higher than the velocity of the fluid flow1therefore,itheatsuplaternon-perturbedpartoftheflow1whichisaheadofthe movingfrontoftheperturbation.Byallmeans,thisindicatesthatbeforetheoccurrenceof theperturbationallnon-dimensionlesstemperaturesareequaltozero(initialcondition). Afterthetimez=5.33,theperturbationofthefluid1hasreachedtheoutletedgeofthe exchangerwhichisregisteredbythestepchangeoftheoutlettemperature.Incaseofthe cross and counter flow heat exchangers, there is not heating up of the fluid flow 1 ahead of theperturbationfront(Fig.7). Thefluidflow1coolsdown inthebeginningbyheatingup the wall of the heat exchanger and the part of the fluid flow 2 in case of the cross flow heat exchanger and the whole fluid flow 1 in the case of counter flow but, it cannot happen that the fluid flow 2 gets ahead of the perturbation front and causes a reversal process of the heat transfer which is possible in case of the parallel flow heat exchanger.

Fig. 7. Outlet Temperature of Both Fluids for Parallel, Counter and Cross Flow Heat Exchangers. Analytical Solution of Dynamic Response of Heat Exchangers 77 5. Conclusion A method providing exact analytical solutions for transient response of parallel, counter and crossflowheatexchangerswithfinitewallcapacitanceispresented.Solutionsarevalidin thecasewherevelocitiesaredifferentorequal.Thesesolutionsprocedureprovides necessary basis for the study of parameters estimated, model discriminations and control of all analyzed heat exchangers. Generallyspeaking,theanalyticalmethodissuperiortonumericaltechniquesbecausethe finalsolutionalsopreservesphysicalessenceoftheproblem.Testingofsolutionsgivenin thispaperindicatesthattheycanbeusedinpracticeefficientlywhendesigningand managing processes with heat exchangers. 6. Appendix Functions( , )nF x c and ,( , , )n mI x cd andtheirLaplacetransformsaregivenasdescribed below (x 0,, cd < < , and n, m = 1, 2, 3,....). For x < 0, both functions are equal to zero. ( )( )( )11( , ) exp1 !nnnxF x c c xns c= +(A.1) ( ) ( ),111( , , ) ( , , )jn m n m jn mjm jI x cd d F x cdjs c s c d+ +=+ | |= |+ + \ .(A.2) Some additional details about these functions can be found in an earlier paper (Gvozdenac, 1986). 7. Nomenclature A1, A2total heat transfer area on sides 1 and 2 of a heat exchanger, respectively, [m2] F1, F2cross-section area of flow passages 1 and 2, respectively, [m2] cpisobaric specific heat of fluid, [J/(kg K)] cwspecific heat of core material, [J/(kg K)] hheat transfer coefficient between fluid and the heat exchanger wall, [W/(K m2)] Mwmass of heat exchanger core, [kg] m mass flow rate, [kg/s] NTUnumber of heat transfer units, [-] (Eq. ) Ttemperature, [K] ttime, [s] Wthermal capacity rate of fluid, pmc = , [W/K] Wminlesser of W1 and W2, [W/K] X, Ydistance from fluid entrances, [m] Ufluid velocity, [m/s] density, [kg/m3] unit step function dimensionless temperature x, y, zdimensionless independent variables, (Eqs. ) Heat Exchangers Basics Design Applications 78Subscripts: 1fluid 1 2fluid 2 wwall 8. AcknowledgmentThisworkwasperformedasapartoftheresearchsupportedbyProvincialSecretariatfor Science and Technological Development of Autonomous Province of Vojvodina. 9. References Profos,P.(1943).DieBehandlungvonRegelproblemenvermittelsdesFrequenzgangesdes Regelkreises, Dissertation, Zurich, 1943 Tahkahashi, Y. (1951). Automatic control of heat exchanger, Bull. JSME, 54, pp 426-431 Kays,W.M.&London,A.L.(1984).Compactheatexchangers(3rded),NewYork,McGraw-Hill Liapis, A. I. & McAvoy, T. J. (1981). Transient solutions for a class of hyperbolic counter-current distributed heat and mass transfer systems, Trans. IChemE, 59, pp 89-94 Li,Ch.H.(1986).Exacttransientsolutionsofparallel-currenttransferprocesses,ASMEJ.Heat Transfer, 108, pp 365-369 Romie,F.E.(1985).Transientresponseofcounterflowheatexchanger,ASMEJournalofHeat Transfer, 106, pp 620-626 Romie,F.E.(1986).Transientresponseoftheparallel-flowheatexchanger,ASMEJ.Heat Transfer, 107, pp 727 -730 Gvozdenac,D.D.(1987).Analyticalsolutionoftransientresponseofgas-to-gasparalleland counterflow heat exchangers, ASME J. Heat Transfer, 109, pp 848-855 Romie,E.E.(1983).Transientresponseofgas-to-gascrossflowheatexchangerswithneithergas mixed, ASME J. Heat Transfer, 105, pp 563-570 Gvozdenac, D. D. (1986). Analytical solution of the transient response of gas-to-gas crossflow heat exchanger with both fluids unmixed, ASME J. Heat Transfer, 108, pp 722-727 Spiga,G.&Spiga,M.(1987).Two-dimensionaltransientsolutionsforcrossflowheatexchangers with neither gas mixed, ASME J. Heat Transfer, 109, pp 281-286 Spiga, M. & Spiga, G. (1988). Transient temperature fields in crossflow heat exchangers with finite wall capacitance, ASME J. Heat Transfer, 110, pp 49-53 Gvozdenac,D.D.(1990).Transientresponseoftheparallelflowheatexchangerwithfinitewall capacitance, Ing. Arch., 60, pp 481 -490 Gvozdenac,D.D.(1991).Dynamicresponseofthecrossflowheatexchangerwithfinitewall capacitance, Wrme- und Stoffbertragung, 26, pp 207-212 Roetzel,W.&Xuan,Y.(1999).DynamicBehaviorofHeatExchangers(DevelopmentsinHeat Transfer, Volume 3, WITpress/Computational Mechanics Publications Bali,B.S.(1978),ASimplifiedFormulaforCross-FlowHeatExchangerEffectiveness,ASME Journal of Heat Transfer, 100, pp 746-747 3 Self-Heat Recuperation:Theory and Applications Yasuki Kansha1, Akira Kishimoto1,Muhammad Aziz2 and Atsushi Tsutsumi1 1Collaborative Research Center for Energy Engineering, Institute of Industrial Science,The University of Tokyo 2Advanced Energy Systems for Sustainability, Solution Research Laboratory Tokyo Institute of Technology Japan 1. Introduction Sincethe1970s,energysavinghascontributedtovariouselementsofsocietiesaroundthe worldforeconomicreasons.Recently,energysavingtechnologyhasattractedincreased interest in many countries as a means to suppress global warming and to reduce the use of fossilfuels.Thecombustionoffossilfuelsforheatingproducesalargeamountofcarbon dioxide(CO2),whichisthemaincontributortoglobalgreenhousegaseffects(Eastop& Croft1990,Kemp2007).Thus,thereductionofenergyconsumptionforheatingisavery importantissue.Todate,toreduceenergyconsumption,heatrecoverytechnologysuchas pinch technology, which exchanges heat between the hot and cold streams in a process, has beenappliedtothermalprocesses(Linnhoffetal.1979,Cerdaetal.1983,Linnhoffetal. 1983, Linnhoff 1993, Linnhoff & Eastwood 1997, Ebrahim & Kawari 2000). A simple example ofthistechnologyistheapplication ofa feed-effluentheatexchangerinthermalprocesses, wherein heat is exchanged between feed (cold) and effluent (hot) streams to recirculate the self-heatofthestream(Seideretal.2004).Toexchangetheheat,anadditionalheatsource mayberequired,dependingontheavailabletemperaturedifferencebetweentwostreams forheatexchange.Theadditionalheatmaybeprovidedbythecombustionoffossilfuels, leading to exergy destruction during heat production (Som & Datta 2008). In addition, many energysavingtechnologiesrecentlydevelopedareonlyconsideredonthebasisofthefirst lawofthermodynamics,i.e. energyconservation.Hence,process designmethodsbasedon these technologies are distinguished by cascading heat utilization. Simultaneously, many researchers have paid attention to the analysis of process exergy and irreversibility through consideration of the second law of thermodynamics. However, many oftheseinvestigationsshowonlythecalculationresultsofexergyanalysisandthe possibilityoftheenergysavingsofsomeprocesses,andfewclearlydescribemethodsfor reducing the energy consumption of processes (Lampinen & Heillinen 1995, Chengqin et al 2002,Grubbstrm2007).Toreduceexergyreduction,aheatpumphasbeenappliedto thermal processes, in which the ambient heat or the process waste heat is generally pumped toheattheprocessstreambyusingworkingfluidcompression.Althoughitiswell-known thataheatpumpcanreduceenergyconsumptionandexergydestructioninaprocess,the Heat Exchangers Basics Design Applications 80heatloadandcapacityoftheprocessstreamareoftendifferentfromthoseofthepumped heat.Thus,anormalheatpumpstillpossiblycauseslargeexergydestructionduring heating.Inheatrecoverytechnologies,vaporrecompressionhasbeenappliedto evaporation,distillation,anddrying,inwhichthevaporevaporatedfromtheprocessis compressedtoahigherpressureandthencondensed,providingaheatingeffect.The condensationheatofthestreamisrecirculatedasthevaporizationheatintheprocessby using vapor recompression. However, many investigators have only focused on latent heat andfewhavepaidattentiontosensibleheat.Asaresult,thetotalprocessheatcannotbe recovered,indicatingthepotentialforfurtherenergysavingsinmanycases.Recently,an energy recuperative integrated gasification power generation system has been proposed and a design method for the system developed (Kuchonthara & Tsutsumi 2003, Kuchonthara et al. 2005, Kuchonthara & Tsutsumi 2006). Kansha et al. have developed self-heat recuperation technologybasedonexergyrecuperation(2009).Themostimportantcharacteristicsofthis technologyarethattheentireprocessstreamheatcanberecirculatedintoaprocess designedbythistechnologybasedonexergyrecuperation,leadingtomarkedenergy savings for the process. In this chapter, an innovative self-heat recuperation technology, in which not only the latent heat but also the sensible heat of the process stream can be circulated without heat addition, andthetheoreticalanalysisofthistechnologyareintroduced.Then,severalindustrial applicationcasestudiesofthistechnologyarepresentedandcomparedwiththeir conventional counterparts. 2. Self-heat recuperation technology Self-heatrecuperationtechnology(Kanshaetal.2009)facilitatesrecirculationofnotonly latent heat but also sensible heat in a process, and helps to reduce the energy consumption of the process by using compressors and self-heat exchangers based on exergy recuperation. Inthistechnology,i)aprocessunitisdividedonthebasisoffunctionstobalancethe heatingandcoolingloadsbyperformingenthalpyandexergyanalysisandii)thecooling loadisrecuperatedbycompressorsandexchangedwiththeheatingload.Asaresult,the heat of the process stream is perfectly circulated without heat addition, and thus the energy consumptionfortheprocesscanbegreatlyreduced.Inthissection,first,thetheoryofthe self-heatrecuperationtechnologyandthedesignmethodologyforself-heatrecuperative processesareintroducedforabasicthermalprocess,andthenself-heatrecuperative processes applied to separation processes are introduced. 2.1 Self-heat recuperative thermal process Exergy loss in conventional thermal processes such as a fired heater normally occurs during heattransferbetweenthereactionheatproducedbyfuelcombustionandtheheatofthe processstream,leadingtolargeenergyconsumptionintheprocess.Toreducetheenergy consumptionintheprocessthroughheatrecovery,heatingandcoolingfunctionsare generallyintegratedforheatexchangebetweenfeedandeffluenttointroduceheat circulation.Asysteminwhichsuchintegrationisadoptediscalledaself-heatexchange system. To maximize the self-heat exchange load, a heat circulation module for the heating and cooling functions of the process unit has been proposed, as shown in Figure 1 (Kansha et al. 2009). Self-Heat Recuperation: Theory and Applications 81 Figure1(a)showsathermalprocessforgasstreamswithheatcirculationusingself-heat recuperationtechnology.Inthisprocess,thefeedstreamisheatedwithaheatexchanger (12)fromastandardtemperature,T1,toasettemperature,T2.Theeffluentstreamfrom the following process is pressurized with a compressor to recuperate the heat of the effluent stream(34)andthetemperatureofthestreamexitingthecompressorisraisedtoT2 throughadiabaticcompression.Stream4iscooledwithaheatexchangerforself-heat exchange (45). The effluent stream is then decompressed with an expander to recover part oftheworkofthecompressor.Thisleadstoperfectinternalheatcirculationthroughself-heat recuperation. The effluent stream is finally cooled to T1 with a cooler (67). Note that thetotalheatingdutyisequaltotheinternalself-heatexchangeloadwithoutanyexternal heating load, as shown in Fig. 1 (b). Thus, the net energy required of this process is equal to the cooling duty in the cooler (67). To be exact, the heat capacity of the feed stream is not equaltothatoftheeffluentstream.However,theeffectofpressuretotheheatcapacityis small. Thus, two composite curves in Fig. 1 (b) seem to be in parallel. In addition, the exergy destructionoccursonlyduringtheheattransferintheheatexchanger.Theamountofthis exergy destruction is illustrated by the gray area in Fig. 1 (b). Inthecaseofidealadiabaticcompressionandexpansion,theinputworkprovidedtothe compressorperformsaheatpumpingroleinwhichtheeffluenttemperaturecanachieve perfectinternalheatcirculationwithoutexergydestruction.Therefore,self-heat recuperationcandramaticallyreduceenergyconsumption.Figure1(c)showsathermal processforvapor/liquidstreamswithheatcirculationusingtheself-heatrecuperation technology.Inthisprocess,thefeedstreamisheatedwithaheatexchanger(12)froma standard temperature, T1, to a set temperature, T2. The effluent stream from the subsequent processispressurizedbyacompressor(34).Thelatentheatcanthenbeexchanged between feed and effluent streams because the boiling temperature of the effluent stream is raised to Tb by compression. Thus, the effluent stream is cooled through the heat exchanger forself-heatexchange(45)whilerecuperatingitsheat.Theeffluentstreamisthen depressurizedbyavalve(56)andfinallycooledtoT1withacooler(67).Thisleadsto perfectinternalheatcirculationbyself-heatrecuperation,similartotheabovegasstream case. Note that the total heating duty is equal to the internal self-heat exchange load without an external heating load, as shown in Fig. 1 (d). It is clear that the vapor and liquid sensible heatofthefeedstreamcanbeexchangedwiththesensibleheatofthecorresponding effluentstreamandthevaporizationheatofthefeedstreamisexchangedwiththe condensation heat of the effluent stream. Similar to the thermal process for gas streams with heatcirculationusingself-heatrecuperationtechnologymentionedabove,thenetenergy requiredofthisprocessisequaltothecoolingdutyinthecooler(67)andtheexergy destructionoccursonlyduringheattransferintheheatexchangerandtheamountofthis exergy destruction is indicated by the gray area in Fig. 1 (d). As well as the gas stream, the effect of pressure to the heat capacity is small. Thus, two composite curves in Fig. 1 (b) are closedtobeinparallel.Asaresult,theenergyrequiredbytheheatcirculationmoduleis reduced to 1/221/2 of the original by the self-heat exchange system in gas streams and/or vapor/liquid streams.To use the proposed self-heat recuperative thermal process in the reaction section of hydro-desulfurizationinthepetrochemicalindustryasanindustrialapplication,Matsudaetal. (2010)reportedthattheadvancedprocessrequires1/5oftheenergyrequiredofthe Heat Exchangers Basics Design Applications 82conventionalprocessonthebasisofenthalpyandexaminedtheconsiderablereductionof theexergydestructionsinthisprocess.Theotherrelatedindustrialapplicationsofthe proposedself-heatrecuperativethermalprocessarethepreheatingsectionsofthefeed streams for reaction to satisfy the required physical conditions. Fig. 1. Self-heat recuperative thermal process a) process flow of gas streams, b) temperature-entropy diagram of gas streams, c) process flow of vapor/liquid streams, d) temperature-entropy diagram of vapor/liquid streams. 2.2 Self-heat recuperative separation processes Expanding the self-heat recuperative thermal process to separation processes (Kansha et al. 2010a),asystemincludingnotonlytheseparationprocessitselfbutalsothe preheating/cooling section, can be divided on the basis of functions, namely the separation moduleandtheheatcirculationmodule,inwhichtheheatingandcoolingloadsare balanced, as shown in Fig. 2. To simplify the process for explanation, Fig. 2 shows a case that has one feed and two effluents. In this figure, the enthalpy of inlet stream (feed) is equal to thesumoftheoutletstreams(effluents)enthalpiesineachmodule,givinganenthalpy Self-Heat Recuperation: Theory and Applications 83 difference between inlet and outlet streams of zero. The cooling load in each module is then recuperatedbycompressorsandexchangedwiththeheatingloadusingself-heat recuperationtechnology.Asaresult,theheatoftheprocessstream(self-heat)isperfectly circulatedwithoutheatadditionineachmodule,resultinginperfectinternalheat circulation over the entire separation process. Fig. 2. Conceptual figure for self-heat recuperative separation processes. 2.2.1 Self-heat recuperative distillation process Althoughdistillationcolumnshavebeenwidelyusedinseparationprocessesbasedon vapor/liquid equilibria in petroleum refineries and chemical plants, the distillation process consumesamassiveamountofenergyrequiredforthelatentheatofthephasechange, resulting in the emission of a large amount of CO2. To prevent the emission of CO2 through useofself-heatrecuperationtechnology(Kanshaetal.2010b),adistillationprocesscanbe dividedintotwosections,namelythepreheatinganddistillationsections,onthebasisof functionsthatbalancetheheatingandcoolingloadbyperformingenthalpyandexergy analysis,andtheself-heatrecuperationtechnologyisappliedtothesetwosections.Inthe preheating section, one of the streams from the distillation section is a vapor stream and the stream to the distillation section has a vaporliquid phase that balances the enthalpy of the feed streams and that of the effluent streams in the section. In balancing the enthalpy of the feed and effluent streams in the heat circulation module, the enthalpy of the streams in the distillationmoduleisautomaticallybalanced.Thus,thereboilerdutyisequaltothe condenserdutyofthedistillationcolumn.Therefore,thevaporandliquidsensibleheatof thefeedstreamscanbeexchangedwiththesensibleheatofthecorrespondingeffluent streams,andthevaporizationheatcanbeexchangedwiththecondensationheatineach module. Figure 3 (a) shows the structure of a self-heat recuperative distillation process consisting of two standardized modules, namely, the heat circulation module and the distillation module. Notethatineachmodule,thesumoftheenthalpyofthefeedstreamsandthatofthe effluent streams are equal. The feed stream in this integrated process module is represented bystream1.Thisstreamisheatedtoitsboilingpointbythetwostreamsindependently recuperatingheatfromthedistillate(12)andbottoms(13)bytheheatexchanger(12).A distillationcolumnseparatesthedistillate(3)andbottoms(9)fromstream2.Thedistillate (3) is divided into two streams (4, 12). Stream 4 is compressed adiabatically by a compressor andcooleddownbytheheatexchanger(2).Thepressureandtemperatureofstream6are adjustedbyavalveandacooler(678),andstream8isthenfedintothedistillation Heat Exchangers Basics Design Applications 84column as a reflux stream. Simultaneously, the bottoms (9) is divided into two streams (10, 13).Stream10isheatedbytheheatexchangerandfedtothedistillationcolumn(1011). Streams12and13aretheeffluentstreamsfromthedistillationmoduleandreturntothe heat circulation module. In addition, the cooling duty of the cooler in the distillation module is equal to the compression work of the compressor in the distillation module because of the enthalpy balance in the distillation module. Theeffluentstream(12)fromthedistillationmoduleiscompressedadiabaticallybya compressor(1214).Streams13and14aresuccessivelycooledbyaheatexchanger.The pressure of stream 17 is adjusted to standard pressure by a valve (1718), and the effluents arefinallycooledtostandardtemperaturebycoolers(1516,1819).Thesumofthe cooling duties of the coolers is equal to the compression work of the compressor in the heat circulation module. Streams 16 and 19 are the products. Figure3(b)showsthetemperatureandheatdiagramfortheself-heatrecuperative distillation process. In this figure, each number corresponds to the stream numbers in Fig. 3 (a),andTstdandTbarethestandardtemperatureandtheboilingtemperatureofthefeed stream,respectively.Boththesensibleheatandthelatentheatofthefeedstreamare subsequentlyexchangedwiththesensibleandlatentheatofeffluentsinheatexchanger1. Thevaporizationheatofthebottomsfromthedistillationcolumnisexchangedwiththe condensationheatofthedistillatefromthedistillationcolumninthedistillationmodule. The heat of streams 4 and 12 is recuperated by the compressors and exchanged with the heat in the module. It can be seen that all the self-heat is exchanged. As a result, the exergy loss of the heat exchangers can be minimized and the energy required by the distillation process is reducedto1/61/8ofthatrequiredbytheconventional,heat-exchangeddistillation process.Toexaminetheenergyrequired,thetemperaturedifferenceofheatexchangers betweencoldandhotstreamsisanimportantparameter.Infact,toincreasethis,theheat transfersurfaceareacanbedecreased.Toachieveindustrialself-heatrecuperative distillationprocesses,furtherinvestigationoftheminimumtemperaturedifferenceinthe heat exchangers is required, especially the difference of the heat types of the streams in the heat exchanger (e.g. sensible heat and latent heat). Asindustrialapplicationsofthisself-heatrecuperativedistillationprocesses,Kanshaetal. (2010c) examined the energy saving efficiency of an integrated bioethanoldistillation process usinganazeotropicdistillationmethodascomparedwiththeconventionalazeotropic distillation processes. In this paper, the energy required for the proposed integrated processes using self-heat recuperative distillation was only 1/8 of the conventional process, leading to a dramatic reduction in the production cost of bioethanol. They also applied it to the cryogenic airseparationprocessandexaminedtheenergyrequiredcomparedwiththeconventional cryogenic air separation for an industrial feasibility study (Kansha et al. 2011a). In that paper, the conventional cryogenic air separation was well integrated on the basis of the heat required to decrease the temperature to near -200 C, especially, and they pointed out that a cryogenic air separation is a kind of multi-effect distillation column. However, there was potential for a 40%energyreductionbyusingself-heatrecuperativedistillation.Furthermore,theauthors appliedittoawell-knownandrecentlydevelopedenergysavingdistillationprocess,an internally heat integrated distillation column (HIDiC). In HIDiC, the distillation column can be dividedintotwosections(therectificationsectionandthestrippingsection)andthe condensationheatisexchangedwiththevaporizationheatbetweenthesetwosectionsusing Self-Heat Recuperation: Theory and Applications 85 thepressuredifference.Designingthisbasedonself-heatrecuperationtechnologyshows further energy saving (Kansha et al. 2011b). From these three industrial case studies, self-heat recuperationtechnologycanbeappliedtorecentlydevelopedheatrecoverydistillation processessuchasheatintegrateddistillationprocesses,multi-effectdistillationprocessesand HIDiCprocesses.Finally,toexaminethefeasibilityofself-heatrecuperationforindustrial processesinthepetrochemicalindustry,Matsudaetal.(2011)applieditusingpractical industrial dataandmodified the stream lines to enablepractical processesand examinedthe energyrequired,exergydestructionandeconomicalefficiency.Fromthesestudies,itcanbe concludedthattheself-heatrecuperativedistillationprocessisverypromisingforsaving energy. Fig. 3. Self-heat recuperative distillation process a) process flow diagram, b) temperature-heat diagram. Heat Exchangers Basics Design Applications 862.2.2 Self-heat recuperative drying process Dryingisusuallyconductedtoreducetransportationcostsbydecreasingproductweight andsize,givinglong-termstoragestabilityandincreasingthethermalefficiencyin thermochemicalconversionprocesses.Unfortunately,dryingisoneofthemostenergy intensiveprocessesowingtothehighlatentheatofwaterevaporation.Theoretically, assuminganambienttemperatureof15C,theenergyrequiredforwaterevaporation ranges from 2.5 to 2.6 MJ per kg evaporated water, depending on the wet bulb temperature (Brammer&Bridgwater1999).Therearetwoimportantpointsregardingreductionof energyconsumptionduringdrying:(i)intensificationofheatandmasstransferinsidethe dryerand(ii)efficientheatrecoveryandenergyutilization(Strumilloetal.2006). Concerningthelatter,severalmethodshavebeendevelopedtoimproveenergysaving during drying, including heat recovery with and without flue gas recirculation, heat pumps, and pinch technology. However, these systems cannot effectively recover all the heat of the drying medium, the evaporated water, and the dried products.Toimprovetheenergyefficiencyindrying,Azizetal.(2011a,2011b)haverecently developed a drying process based on self-heat recuperation technology. In this technology, thehotstreamisheatedbycompressiontoprovideaminimumtemperaturedifference requiredforheatpairingandexchangewiththecoldstreamandalloftheself-heatofthe processstreamisrecirculatedbasedonexergyrecuperation.Asaresult,alloftheheat involvedindryingcanberecuperatedandreusedasaheatsourceforthesubsequent dryingprocess.Thisincludesrecuperationofsensibleheatfromthegasservingasthe drying medium, both sensible and latent heat of the evaporated water and the sensible heat ofthedriedproducts.Aprocessdiagramforbrowncoaldryingbasedonself-heat recuperation technology is shown in Fig. 4 (a). A fluidized bed dryer with an immersed heat exchangerisselectedastheevaporatorowingtoitshighheattransfercoefficient,excellent solidmixing,anduniformtemperaturedistribution(WanDaud,2008,Law&Mujumdar 2009).Wetbrowncoalisfedandheatedthroughapre-heater(dryer1a)toagiven temperature.Subsequently,themaindryingstage(waterevaporation)isperformedinside the fluidized bed dryer (dryer 2), where evaporation occurs. The immersed heat exchangers, whicharefilledbyacompressedmixtureofairandsteam,areimmersedinsidethe fluidized bed, providing the heat required for water removal. The exhausted mixture of air and steam is then compressed to achieve a higher exergy rate before it is circulated back and utilized as the heat source for evaporation (dryer 2) and pre-heating (dryer 1a, dryer 1b), in thatorder.Inaddition,thesensibleheatofthehot,driedbrowncoalisrecoveredbythe drying medium, to further reduce drying energy consumption (dryer 1c).The heat exchange inside the fluidized bed dryer is considered to be co-current because the bediswellmixedandtheminimumtemperatureapproachdependsontheoutlet temperatureofthehotstreams(compressedair-steammixture)andthetemperatureofthe bed.Figure4(b)showsatemperature-enthalpydiagramfortheself-heatrecuperativebrown coal drying. Almost all of the heat is recovered, leading to a significant reduction in the total energyconsumption.Thelargestamountofheatrecuperationoccursindryer2,which involvestheheatexchangebetweenthecondensationheatofthecompressedair-steam mixture and the evaporation heat of the water in the brown coal. The heat curves of the hot Self-Heat Recuperation: Theory and Applications 87 and cold streams, especially in dryer 2, are almost parallel owing to the efficient heat pairing within the dryer. Fig. 4. Self-heat recuperative brown coal drying (a) process flow diagram, (b) temperature-heat diagram.This drying process can reduce the total energy consumption to about 75% of that required forhotairdryingusingconventionalheatrecovery.Furthermore,astheheatrequiredfor water evaporation is provided by the condensation of the compressed air-steam mixture, the inlet air temperature is considerably lower, leading to safer operation due to reduced risk of fire or explosion. Heat Exchangers Basics Design Applications 88In addition, the thermodynamic model of heat exchange inside the fluidized bed is shown in Fig. 5. The compressed air-steam mixture flows inside a heat transfer tube immersed in the fluidized bed dryer. Thus, in-tube condensation occurs and heat is transferred to the bed via the tube wall and is finally transferred from the bed to the brown coal particles. Fig. 5. Model of heat transfer inside the fluidized bed dryer. The heat transfer rate from the compressed vapor inside the heat transfer tube to the drying sample in FBD, qs, can be approximated as: s v s( ) q UA T T = (1) Also,becausetheheatexchangeinsidethefluidizedbeddryerinvolvesconvectionand conduction, the product of the overall heat transfer coefficient, U, and surface area, A, may be approximated by equation (2). ( )c c t t tln1 1 12RrUA A L A o t o= + +(2) The first term of the right side of equation (2) represents the heat transfer resistance of vapor condensationinsidethetube.Acandocaretheinnersurfaceareaofthetubeandtheheat transfercoefficient,respectively.Thesecondtermcorrespondstotheconductiveheat transferthroughthetubewallhavingthethermalconductivity,innerradiusandouter radius of t, r and R, respectively. Convective heat transfer from the outer tube surface to the browncoalparticlesinsidethebedisexpressedbythethirdterm,inwhichtheconvective heat transfer coefficient and the outer surface area of the tube are ot and At, respectively. Theheattransfercoefficientonahorizontaltubeimmersedinsidethefluidizedbedhas been reported by Borodulya (1989, 1991): ( )( )0.14 0.24 2320.1 s s3tg g10.74 1 0.46CNu Ar Re PrCc c c| | | | || = + ||\ . \ .(3) Self-Heat Recuperation: Theory and Applications 89 t stgdNuo= (4) Theheattransfercoefficientofthecondensingvaporiscalculatedusingageneral correlation proposed by Shah (1979): ( )( )( )0.040.76 0.8 0.40.8l l lc0.38crit3.8 1 0.02312x x Re Prxrp po ( ( = + ( (5) 2.2.3 Self-heat recuperative CO2 absorption process Carbon capture and storage (CCS) has attracted significant attention in the past two decades toreducegreenhousegasemissionsandmitigateglobalwarming.CCSconsistsofthe separationofCO2fromindustrialandenergy-relatedsources,transportationofCO2toa storage location and long-term isolation of CO2 from the atmosphere (Rubin et al. 2005).It is reported that the most significant stationary point sources of CO2 are power generation processes. In fact, the amount of CO2 emission from power generation processes comprises 40%ofglobalCO2emissions(Rubinetal.2005,Toftegaard,2010).Forpowergeneration, therearethreedifferent typesforCO2 captureprocesses: post-combustion,pre-combustion and oxy-fuel combustion (Rubin et al. 2005). In this section, the CO2 absorption process for post-combustion is used as a case study (Fig. 6). Post-combustioncaptureinpowerplantsisgenerallyusedforpulverized-coal-firedpower plants. The CO2 concentration in post-combustion is low compared with the other two CO2 capture processes: around 10% (wet base). The CO2 capture is generally performed through chemical absorption with monoethanolamine (MEA).Electricity & Heat generationCO2captureDehydration, Compression, Transportation and StorageCoal, GasAirFlue GasCO2N2, O2 Fig. 6. Post-combustion capture. Figure 7 shows a diagram of the conventional CO2 absorption process, which consists of an absorber,aheatexchanger(HX)forheatrecoveryandastripper(regenerator)witha reboiler. The flue gas and a lean CO2 concentration' amine solution (lean amine) are fed into theabsorber,andCO2gasisabsorbedintotheleanamine.Thisaminesolutioncontaining absorbedCO2iscalledthe'richCO2concentration'aminesolution(richamine).Exhaust gasesaredischargedfromthetopoftheabsorber.Therichamineisfedintothestripper through the HX and then lean amine is regenerated and the CO2 gas is stripped by heating inthereboilerofthestripper.IntheconventionalabsorptionprocessusingMEA,theheat (4.1GJ/t-CO2)issuppliedbythereboilerinthestripper.Theratioofthisheatfor regeneration and vaporization is 1:1. From Fig. 7, it can be understood that a part of sensible heat is recovered from lean amine using the HX. However, the heat of vaporization cannot berecoveredfromheatofsteamcondensationforstrippinginthereboilerbecauseofthe Heat Exchangers Basics Design Applications 90temperaturedifferencebetweenthecondenserandthereboiler.Thus,CO2captureisthe mostcostlyandhighenergyconsumptionprocessofpowergeneration,leadingtohigher CO2emissions.Infact, itisreportedthatthisprocessdropsthenetefficiencyofthepower plant by about 10% (Damen 2006, Davison 2007). Fig. 7. Conventional CO2 absorption process. Ifallprocessheat(sensibleheat,latentandreactionheat)canberecirculatedintothe process,theenergyrequiredforCO2capturecanbegreatlyreduced.Toachieveperfect internalheatcirculation,aself-heatrecuperationtechnologywasappliedtotheCO2 absorptionprocessandaself-heatrecuperativeCO2absorptionprocesswasproposed,as showninFig.8(a)(Kishimotoetal.2011).Inthisprocess,theaforementionedself-heat recuperative distillation module in 2.2.1 can be applied to the stripping section (A) in Fig. 8 (a).AmixtureofCO2andsteamisdischargedfromthetopofstripperandcompressed adiabaticallybyacompressortorecuperatethesteamcondensationheat.Thisrecuperated heatisexchangedwiththeheatofvaporizationforstrippinginthereboiler,leadingtoa reduction in the energy consumption for stripping.InthesectionBinFig.8(a),theaforementionedheatcirculationmodulein2.2.1canbe applied, and furthermore the heat of the exothermic reaction generated at low temperature intheabsorberistransportedandreusedasreactionheatforsolutionregenerationathigh temperatureusingareactionheattransformer(RHT).ThisRHTisatypeofclosed-cycle compression system with a volatile fluid as the working fluid and consists of an evaporator toreceiveheatfromtheheatofexothermicreactionintheabsorber,acompressorwith drivingenergy,acondensertosupplyheattothestripperasheatoftheendothermic reaction,andanexpansionvalve.Theheatoftheexothermicabsorptionreactionatthe evaporatorintheabsorberistransportedtotheendothermicdesorptionreactioninthe condenser of the stripper by the RHT. Therefore, both the heat of the exothermic absorption reactionintheabsorberandtheheatofsteamcondensationfromthecondenserinthe Self-Heat Recuperation: Theory and Applications 91 stripperarerecuperatedandreusedasthereactionheatforsolutionregenerationandthe vaporization heat for CO2 stripping in the reboiler of the stripper.Asaresult,theproposedself-heatrecuperativeCO2absorptionprocesscanrecirculatethe entire process heat into the process and reduce the total energy consumption to about 1/3 of the conventional process. Fig. 8. Self-heat recuperative CO2 absorption process, (a) process flow diagram, (b) temperature-heat diagram. Heat Exchangers Basics Design Applications 923. Conclusion In this chapter, a newly developed self-heat recuperation technology, in which not only the latentheatbutalsothesensibleheatoftheprocessstreamcanbecirculatedwithoutheat addition, and the theoretical analysis