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by: Hugh Wilkinson


Hugh Wilkinson
GPA 3.56

Michael Podowski

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Michael Podowski
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This 111 page Class Notes was uploaded by Hugh Wilkinson on Monday October 19, 2015. The Class Notes belongs to MANE 4400 at Rensselaer Polytechnic Institute taught by Michael Podowski in Fall. Since its upload, it has received 19 views. For similar materials see /class/224908/mane-4400-rensselaer-polytechnic-institute in Mechanical and Aerospace Engineering at Rensselaer Polytechnic Institute.

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Date Created: 10/19/15
MAN E44OO I UVVCI UIYOLCI I I0 En ineerin I 1quot Notes Z Podowski II Thermodynamics of Nuclear Power Plants continued Bravton Gas Cycle The present generation gascooled reactors use an indirect cycle including a gascooled primary system and a steamwater secondary system However it is also possible to manufacture a nuclear power plant in which the gas from the reactor is directly used to drive a gas turbine The resultant thermodynamic cycle is called the Brayton Cycle Gas Reactor L V Compress r Heat Rejection 3 AT s diaram for a sim Ie ideal Bra ton c cle Assuming that the working fluid can be modeled as an ideal gas we have 1 1 51 Tdsz Idhzhl h4 cpI T4 1159 4 4 2 2 51m Tdsz Idhzhz h3chT2 T3 1160 3 3 1 1 WT Itsz Idhzhl hzchTl T2 1161 2 2 4 4 w luau jdhzh4 h3chm T3 1162 3 3 Since for isentropic processes the following relation holds P1quot constant 1163 the temperatures T and T2 and T3 and T4 can be related to the pressures P1 and P2 k1 um 1164 Consequently the net work heat can be obtained from E wnet qnet cp1 k where rp is the compressor pressure ratio given by rp 191192 gt1 1166 Combining EqsII59 and 1165 yields 1167 Practically the highest and lowest temperatures which can be achieved in a power plant system are always limited Hence an important problem is concerned with determining an optimum pressure ratio for given T1 and T3 re the value or rp which maximizes the net amount of work per unit mass of the working gas which can be roduced b the lant Rearranging EqII65 into the form 1 WWI61131 rpyEEl rpk 1168 and differentiating with respect to rp yields the optimum compressor ration as k k rltzT3gtmltmgtm ltH69gt P Substituting EqII69 into EqsII65 and 1167 respectively yields VInet cpT3maX 1 T371039571armmos 1170 TImm U The Actual Bravton Cycle The net amount of work per unit mass is 1173 1174 The turbine and compressor temperature ratios respectively can be expressed as k iijk 254 1175 19311 1176 is the pressure loss ratio Now EqsII73 and 1174 respectively can be rewritten as 1179 The total heat added to the system can be calculated from 1180 Conse uent the net amount of work and the efficienc of actual Brayton cycle become 1 1181 1182 ghle dependenCIes of both and 77am on rp are shown e ow Wm CPR act MAN E44OO I UVVCI UIYOLCI I I0 En ineerin I 1quot Notes Z Podowski II Thermodynamics of Nuclear Power Plants continued Irreversible Thermodynamic Processes In actual irreversible processes the entropy always increases Consequently Eq 119 becomes an inequality In articular in a rocess with no heat exchan39e an adiabatic process we have Whereas Eqs 1111 and 1112 cannot be applied to irreversibw rHWWWS up 1113 and 1114 remain true Thermod namic Relationshi sfor Sin IePhase Fluids Expressing the specific enthalpy as a function of temperature and pressure and the s ecific internal enerw as a function of temperature and specific volume I hTP 1116 u TM 1117 EqsH16 and 1117 yield dhj dP j dTj dP c dT 1118 6P T 6T P 6P T 1 8L1 8L1 du 81 d1 j dT j dz cUdT 1119 81 8T U c 3 2 is the specific heat at a constant pressure P 00 3 is the specific heat at a constant volume I Using E lsII13 and II14 EqsII18 and 1119 respectively can be replaced by 1120 1121 Maxwell Relations 51 2 66 30 1122 8 8 Z 1124 6 ng Z ea QT 1125 The Equation of State FPUT 0 1126 For an ideal gas we have 1127 where in general 6 and CU depend on temperature but are constant for noble gases 1129 1130 The ideal gas equation of state is P1 2 RT 1131 Carnot Cycle A thermodynamic cycle can be defined as a series of processes during which a substance starts in a particular state and eventually returns to the same state The Carnot C cle is a theoretical ideal thermodynamic cycle which applies to systems consisting of four main elements heat source heat engine heat sink and pump Carnot Cycle cont gt1 lt Heat Source Heat Sink The Cycle consists of four reversible processes Isothermal heating between states 1 and 2 in the diagrams shown lsentropic expansion between 2 and quot3 Isothermal heat rejection between 3 and quot4 lsentropic compression between 4 and 1 T T11T3T4 The overall thermal efficiency of a thermodynamic cycle in general and of the Carnot cycle in particular is defined as d re Q 7 6101 6107 is the heat added to the system per unit mass of working fluid kJkg is the heat reiected from the s stem per unit mass working fluid kJkg Que Qua re is the net heat added per unit mass working fluid kJkg For reversible processes we have 1133 1134 Substituting Eqs 1133 and 1134 into Eq 1132 and taking into account that 5352 and 5411 yields 1135 The Second Law of Thermodynaw states t at it is impossible to construct a device which receives heat and converts it entirely into work That is a thermodynamic cycle must always include heat rejection which in turn means that the cycle thermal efficiency can never even theoretically reach 1Com The Carnot cycle is the most efficient but only theoretical thermodynamic cycle so that it cannot be used in actual systems The most commonly used cycle in nuclear power plants is the liquidvapor Rankine cycle Another thermodynamic cycle considered for application in gascooled reactors is the Brayton cycle MAN E44OO uclear Power Svstems Engineering III Thermal Hydraulics of Nuclear Power Plants Nomenclature and Units Pp pressure PaNm2 psi a W work J lbfftBtu Ttemperature C K F R Q heat J Btu m M mass kg lbm P Power W Btuhr V volume m3 ft3 AHT heat transfer area m2 ft2 p MV density kgm3 lbmft3 PH heated perimeter m ft 1 1p specific volume m3kg ft3lbm q internal heat source Wm3 Btuhrft3 v velocity msec ftsec q heat flux Wm2 Btuhr ftz G pv mass flux kgmZs lbmftZhr q linear heat rate Wm Btuhrft A channel cross section area m2 ft2 qq AHTq PH thermal power W Btuhr w GA mass flow rate kgs lbmhr u specific internal energy Jkg Btulbm h specific enthalpy Jkg Btulbm s specific entropy JkgK Btulbm R Conversion Factors Conservation Principles Relationships to determine the rates of creation CPs refer to the following parameters describing the thermalhydraulic state of fluids MASS ENERGY anaMcmanl wt E atons LawofThennodynan cs qu I MOM ENTUM ENTRO Second aW of Inequalitybecomesyequation Thermodynamics for reversible processes Constitutive Relationships When modeling thermalhydraulic systems mi PSs in particular the conservation principles are usually complemented by various constitutive relationships Examples Equation of State for ideal gas Pu RT Fourier Law Correlations for Friction Factor Heat Transfer Coefficient etc Frame of Reference for Conservation Equations Conservation Principles usually refer to either a given volume elements control volumes CV associated with a stationary Eulerian frame of reference or b material elements the latter with a moving Langrangian frame of reference Control Volumes Control volumes can be a finitesize lumpedparameter approach b infinitesimal differential approach leading to distributedparameter models NOTE Lumped parameter models are simpler given by algebraic eqs or ODES UUl less accurate and require using more constitutive relationships than distributedparameter models which usually take the form of PDEs Control Volumes continued Control volumes may be either fixed non deformable or may have moving boundaries Two examples of the latter are I given below Boiling 39rwo 39 Channel cry4 re 0 O Subcooled liquid VC1Z VNBIZAZIJI win Tsallt T sat Z2307 boiling boundary Conservation I rinci le 0 3 02 13 I infow 0 outflow 01 Rate of Creation Total Outflow Total Inflow Instantaneous Storage Rate 1111 Mass Conservation Equation Continuity Equation LPm Inflow and outflow terms are mass flow rates The overall rate of creation is zero 01574 ZWOW wn CquotO 1112 i dt J Example 1 NOTE wO can be either into or out of the vdume The mass conservation equation can be written as W Wi out 39 wo O 71 Example 2 The mass conversion equation becomes dm W2 W1 0 dt where Exam le 2 Continued Case A Incompressible fluid p1 2 p2 constant van W o and magma Case B Compressible fluid steady flow W2 2 W1 constant Energy Carried by a Flowing Fluid The energy carried by a unit mass of flowing fluid ie the specific energy at a certain location in a channel is given by V2 6 21 7 gZ internal killetic potential energy energy energy Energy Stored in a Control vmume The total energy stored in a control volume is 1114 Forms of Energy Transfer Energy can be added to removed from a control volume due to fluid inflow andor outflow heat transfer through walls q HHT inflow if positive internal heat generation inflow external work per unit time due to external forces resulting in control volume deformation Pom outflow if ositive ie if work is done b the fluid Energy Conservation Equation Consequently a general form of the energy conservation equation ECE becomes Ziwu gjRACL Zw gzjPiAc P0w d u V2 1115 yAhT m wiwgsqo EqIV5 can also be rewritten as d v2 v2 Anew13 82 11 821 111 61 q 39VCV 4311 ltD 1116 Zw0m j 01 g out j 2 2 wewu V ozwhV ozj 1117 p 2 2 is the total inflowoutflow of energy associated with a given flow rate w MAN E44OO I UVVCI UIYOLCI I I0 En ineerin I 1quot Notes Z Podowski II Thermodynamics of Nuclear Power Plants continued Thermodynamic Processes with Phase Change LiquidVapor Basic properties of liquidvapor in particular watersteam processes can be illustrated by using the Ts and hs diagrams given below x constant s gt notation follows The notation is as follows subcooled liquid region twophase mixture region su erheated va or reion critical point defined by the critical temperature Tc and critical pressure P vapor quality x hghf fg saturated liquid line x0 saturated vapor line x1 Simple Ideal Rankine Cvcle Consider a simplified schematic of a power plant consisting of a boiler a turbine a steam condenser and a pump Steam Turbine Boiler Steam Condenser Recirculation feedwater Pump Feedwater Assume that a liquid coolant in particular water enters the boiler in which it s Heated to the saturation temperature and then completely evaporated ie converted into saturated steam Subsequently the saturated steam is isentropically expanded in the turbine The lowpressure wet steam leaving the turbine enters the condenser where it is converted into saturated liquid water The liquid water is then compressed in the pump back to the original pw and ent t the boiler awn The resultant closed loop system yields the simplest version of an ideal Rankine c7 cle This or cle is illustrated in the T s and h s diagrams shown next The individual processes are as follows 12 isobaric heating in the boiler 23 ISObarIc evaporation In the boner 34 isentropic expansion in the turbine 45 isobaric heat rejection in the condenser 51 isentropic compression in the pump NOTE The heat addition process in the boiler is divided into two sections first 12 the temperature of feedwater is increased from the inlet value to the saturation temperature at the pressure P and then 2 3 the saturated liquid water is converted at a constant pressure and temperature into dry saturated steam ln nuclear power plants the boiler becomes either the reactor itself BWRs or is represented by steam generators in which the heat carried by the reactor coolant ie subcooled water either light PWR or heavy PHWR gas HTGR or liquid metal LMR is used to produce steam The efficienc of this Rankine C cle is Viven wqad wqre qad qre qnet and qad qad 3 2 ad ITdsITdsTIS3 S2 1 1 1138 On the other hand integrating EqII13 between states 1 and and taking into account that along the integration path P constant yields 9 1 16 h h l 1 Similarly 4 4 des jdh r24 1140 Hence the thermal efficiency of the cycle can also be expressed as 1141 Since the steam entropy is constant during the isentropic expansion in the turbine the mechanical work per unit mass of stem produced by the turbine becomes see IqH13 4 4 VT judp jdhh3 424 1142 The work of isentropic compression in the pump can be obtained in a similar way as l 1 WP judP dh hl hj 1143 D Subtracting EqII43 from EqII42 yields the net mechanical work of the cycle per unit mass of working fluid V h3 h4h1h5h3 h1 h4 h5 1181 1144 net 1145 NOTE Since the specific volume of water obtained by condensing the wet steam leaving the turbine is much smaller than the specific volume of steam the work of compression in the I um is very small compared to the mechanical output of the turbine Working fluid properties at various thermodynamic states of the cycle can be evaluated as nouows state 2 state 3 state 4 saturated liquid at the pressure PI from steamwater tables saturated steam at the pressure PI from steamwater tables P4 2P1 ZapKPH S4 2 S3 2 557031 S4 SfPII 4 ng 114 hd PH x4hw Workin39 fluid ro erties at various thermodynamic states of the Rankine cycle continued state 5 saturated liquid at the pressure PH State quot1quot P1 P S1 Z 55 l h1h5 udP T1 01UrPIah1 the last two arameters can be obtained from the tables of properties of subcooled water NOTE An approximation of the pump work can be obtained assuming incompressibility of water in the pump Le 01 U5 This assumption yields hl hs USPI P11 Actual Rankine Cvcle Actual thermodynamic processes are irreversible wc dominant effects of thermodynamic irreversibi m we Rankine cycle are related to the turbine work and pump works Tl Because of the energy loss in the turbine the actual expansion work is always ICSS man we theoretical wum given by EqII42 Introducing the turbine efficiency 77 we write 132 h3 h4 2 mil 2 77Th3 h4 1146 where h4gth4 is the actual steam enthalpy at turbine exhaust Similarly the actual pumping work can be expressed as h h up n n5 27 1147 where h1gth1 is the actual enthalpy of compressed water at pump exit Hence the net mechanical work is 43 17439 hr ks c155 1148 whereas the heat supplied to the system becomes 3 h3 kl 1149 EqsII48 and 1149 yield the following expression for thermal efficiency of the actual Rankine cycle considered here h3 h439h139 ks h hl 3 171m 2 NOTE 77m lt 77W where 77 W is the thermal efficiency of the ideal Rankine cycle Methods of Improving the Rankine Cycle 1 FeedwaterSubcoolinq For the simplest Rankine cycle saturated water enters the T feedwater pump In order to remove any steam content from the water which could cause cavitation and dramatically reduce the pump efficiency the feedwater should be slightly subcooled by a few degrees NOTE the introduction of feedwater subcooling does not change much the basic characteristics of the cycle 2 Superheat Turbine efficiency of the T1 dramatically decreases with the increasing moisture content in the exhaust steam One way of improving c situation is to superheat the steam before sending it to the turbine NOTE the errect of superheat on major cycle parameters net work thermal efficiency etc is much stronger when considering the actual rather than ideal cycle 3 Reheat Another method of reducing the steam moisture content in the turbine is to use the concept of reneat A scnematIc of a power plant system with reheat is shown on the next slide Boiler High Pressure Turbine Lp Low Pressure Turbine Steam 1 Pump 7 Condenser 1 I Feed WaterU Steam expanded in a High Pressure turbine is then returned to a second stage bonler The reheated steam is subsequently expanded in a low pressure turbine to the final exhaust pressure The concept of reheat can also be combined with steam superheat Tl 4 Regeneration Consider a pOWGF plant in WhiCII LhU llUclL IIUUUUU LU heat up the feedwater to the boiling point is taken directly from the turbine 1 Steam 30 Turbine 1 39l I 4 gt Steam Condenser Pump 5 As shown in the Ts diagram below the originally isentropic steam expansion rMoe i Mn would have to be divided into two sections the heat transfer into feedwater 3339 and the isentropic expansion thereafter 3394 Til For an ideal situativ thy following relatv quotowd W satisfied 2 3 Q12 JTdS JTdS 3393 1151 1 339 3 2 3 cjad JTdS JTCZS JTdS TIS3 S2 1152 1 1 2 4 re JTdS T11 5439 S5 T11S3 52 1153 5 EqsII52 and 1153 yield the following expessiu or the thermal efficiency of the cycle which is the Carnot cycle efficiency for the same temperatures of the heat source TI and heat sink TH unfortunately sucn nrgnefficrency cyCIe cann0t be accomplished in actual systems A more realistic approach to regeneration is via using steam bled from the turbine to heat the feedwater This concept is shown next in the plant schematic and the respective Ts diagram Steam Condenser A V Regen Heat Exchanger A reaoi Wtev t e aw vf we we we a fraction y of the total steam flow rate and the feedwater temperature at the outlet of the regenerative heat excnanger can be Wt ltten as yh7 k8 yh6 hl 1155 Then the heat added to and rejected from the system can be obtained from respectively 1157 61 1 yh4 M ks 1158 Steam Generation in Nuclear Power Plants In the case when steam is produced in the reactor core no superhe is tw cman v example in BWRs where saturated steam is separated from the steamwater mixture leaving the reactor In the plants employing an indirect cycle the steam prOduced In steam generators 56 can be either saturated or superheated depending on the SG design concept The selection of an appropriate superheat is usually related to the various thermal hydraulic limitations on the system parameters Effect of steam superheat on system pressure for a given maxrmum permitted steam eXIt temperature MAN E44OO uclear Power Svstems Engineering I Introduction Power Plants Power Plants systems in which the energy of fuel chemical in fossil plants nuclear in nuclear plants is converted into electrical energy Fuel Power Electrical Energy Conversion In the present generation of power plants using water as a working fluid the energy conversion process occurs in steps nuclear nuclear pp boiler turbine electrical generator Power Plant Schematic A veneral sim lified schematic of a ower plant is shown below Steam 2 Electrical Generator AJ Feedwater39 Nuclear Power Cycles Nuclear power plants a erate in er direct cycle or indirect cycle R t I bi grfr Reactor Prlmary Steam COO39ant Generator boiler E a l Feedwater Direct Cycle Indirect Cycle Steam Nuclear Power Cycles continued Indirect Cycle Steam Interim HE Reactor Condenser Q4 Primary Intermediate Secondary System System System Classification of Nuclear Power Reactors 1 Coolant 2 Fuel 3 Moderator thermal reactors only Coolant H2O D20 GAS COZ He Liquid Metals Na Fuel fissionablefissile materials U235 U233 from Th232 Pu239 from U238 Fission Energy 200 MeVfission 1ev 1 6x103919 J U235 U233 Pu239 Enrichment Natural U Slightly Enriched Highly Enriched Umet U02 UC uc2 PuO2 07 U235 U02 2 4 in UC gt90 in HTGR Moderator Coolant H20 Graphite D20 Tvloes of Nuclear Power Plants LWR PWR BWR HWR PHWR Candu GCR British French old HTGR FBR Produces more fuel than uses PWR BWR PHWR HTGR LMFBR uo2 2 4 enrichment uo2 2 3 enrichment uo2 u nat UC enrichment gt90 U02 enrichment gt20 Moderator VVGLCI VVGLCI D20 Graphite None Coolant Light Water Light Water He Analysis of nuclear power plant systems THERMODYNAMICS Thermodynamic cycle boiler FLUID FLOW NUCLEAR POWER GENERATION HEAT TRANSFER turbinecondenserpumpboiler lncompressible fluids liquids com ressible fluids wases two phase watersteam mixture Fission Energy Neutron flux distribution Reactivity feedback effects Conduction in reactorfuel convection and phase change in working fluids radiation MAN E44OO I UVVCI UIYOLCI I I0 En ineerin I 1quot Notes Z Podowski II Thermodynamics of Nuclear Power Plants 111 Principles of Thermodynamics The First Law of Thermodynamics In general terms the first law of thermod namics states that energy can neither be created nor destroyed The most commonly used mathematical formulation of this law is the energy conservation equation For open systems ie systems in which mass and energy can be transformed across the system boundaries this equation can be written as E th Q12 W12 AE ow 111 I IS IIUL dLIUn uscu Ill qH1 iS as fOIIOWS mechanical potential and kinetic energy internal thermal energy also denoted by U flow energy net heat added into the system net mechanical work done by the system Closed systems are usually defined as systems in which there is no mass transfer across the system boundaries For such systems there is no change In the mechanical energy and Iq 111 becomes U1 Q U2 W 112 or AUU2UIQW 113 The Enthalpy Consider a certain quantity of a substance in a volume V and under a pressure P For a given thermal energy of the substance U the substance enthalpy is defined as H U PV 114 lfthe mass of the substance is m its specific enthalpy hHm can be expressed in terms of the specific internal energy may and the specific volume uVm h u F0 115 EqII5 can be differentiated to obtain db du dPU du PdU UdP 116 where the symbol dy refers to an incremental value change of a given parameter from the original state y to a new state ydy For a thermodynamic process transforming a fixed mass of a substance from state 1 to state the resultant change in the specific enthalpy of the substance can be obtained from Ah 2 h2 h1 Tali1 2JPU 2I01 12 712J PU2J UP 1 1 1 1 1 117 Two articular cases of E II7 are given below a an isobaric process b an isochoric process constant pressure constant volume hzhl LIZMl PUZ Ul h2hl u2ullPzPl Reversible Thermodynamic Processes and the Concept of Entropv A thermodynamic process is called reversible ideal if it can reverse itself by following exactly the same path so that the amount of heat and energy exchanged in each case is the same Although all actual physical processes are irreversible the concept of reversibility provides a convenient vehicle for the analysis of various thermodynamic systems Heat exchanged during an incremental reversible process can be expressed as de TdS 118 where S is the entropy of a given substance Equation 118 can also be rewritten in terms of the specific heat and the specific entropy qum T ds 119 where T is the absolute temperature Integrating EqII9 between states 1 and yields 2 612 ITdS 1110 1 Examples a an isothermal process b an isentropic process Ti Tgtllt For reversible rocesses the followin39 relation holds d m du PdU 1111 Combining Eq 111 with Eq 16 we obtain dh d rev UdP 1112 Also dh Tds udP 1113 1 24 2 J39a s Fa u 1114


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