Analysis of Enthalpy Approximation for Compressed Liquid Water

Transcription

1 Analysis of Entalpy Approximation for Compressed Liqid Water Milioje M. Kostic Nortern Illinois Uniersity, DeKalb, IL It is cstom to approximate solid and liqid termodynamic properties as being a fnction of temperatre only, since tey are irtally incompressible, and d bondary work may be neglected. Frtermore, in classical literatre, for isotermal compression processes, a general improement and correction for liqid entalpy approximation is gien by adding te pressre correction, d, to te corresponding satration ale. It is sown tat sc correction gien for isotermal processes is generally alid for isentropic processes only. Analysis of water real properties, oer te satration temperatre range and a wide pressre range p to 100 Ma, sows tat te recommended corrections are only beneficial for iger pressres at smaller temperatres (below 200 C), insignificant for smaller pressres at most of te temperatres, abot te same bt opposite sign (ts nnecessary) for intermediate temperatres and pressres, and more erroneos (ts conterprodctie and misleading) for iger temperatres and pressres, tan te corresponding satration ales witot any correction. Te misconception in te literatre is a reslt of te erroneos assmption, tat de to incompressibility for liqids in general, te internal energy is less dependent on pressre tan entalpy. DOI: / Keywords: entalpy, water, termodynamic properties, termodynamic analysis, isotermal, isentropic 1 Introdction Since solids and liqids are irtally bt not exactly incompressible, ten te compression work, d, cold be neglected and ts many properties irtally will not be a fnction of pressre bt temperatre only, sc as specific internal energy,, etc. Frtermore, any process is also at te same time an isocoric, constant-olme process. Namely, te isobaric, constant-pressre process will be a simltaneosly constant-olme process for an incompressible sbstance, so tat specific eat at constant pressre, c p, and constant olme, c, are te same, or approximately te same for irtally incompressible real solids and liqids, particlarly wen compared to apors and gases, i.e.: T sat T and c p c c T 1 Een te specific entalpy for a liqid from ere on word specific will be assmed and omitted for breity, can be approximated to be independent from pressre and coneniently taken to be eqal to te corresponding satrated liqid ale at te gien temperatre:,t T sat T 2 Howeer, entalpy is niqe, since it is explicitly defined as a fnction of pressre, namely: + ts, T, = T, + sat T + 3 Terefore, it is common in most engineering references, inclding classical and widely sed termodynamics textbooks 1,2, to ealate te cange of entalpy, assming incompressibility d =0, bt taking correction for pressre increase as: Contribted by te Heat Transfer Diision of ASME for pblication in te JOUR- NAL OF HEAT TRANSFER. Manscript receied December 13, 2004; final manscript receied Noember 1, Reiew condcted by Jon H. Lienard V. aper presented at te 2004 ASME International Mecanical Engineering Congress IM- ECE2004, Noember 13 19, 2004, Anaeim, California, USA. Frtermore, for isotermal processes dt=0 and d 0, ten d d, and finally, for finite pressre difference cange from satrated pressre, sat, corresponding to te gien temperatre, T, te specific entalpy wit correction, corr T, at tat temperatre, T, and any pressre,, will be 1,2 : 4 5a 5b were, sat T and sat T are liqid satration entalpy and liqid satration specific olme at gien temperatre, T, respectiely. It is stated in many references, inclding 1,2, tat te aboe eqations 5a and 5b are recommended as te correction for isotermal, liqid entalpy dependence on pressre, and tat it is more accrate tan a simple, approximation witot correction, sat Eq. 2. It is te objectie of tis paper to point ot te erroneos general recommendations in te literatre. Te correction Eq. 5, as recommended in many references, is only sefl for iger pressres at smaller temperatres, bt is actally more erroneos ts conterprodctie and misleading for iger temperatres and pressres, and is abot te same bt opposite sign, ts not necessary for intermediate temperatres, tan te simple approximation Eq. 2 witot any correction. Corresponding analysis sing real water data 3 and pysical jstification are presented below. Jornal of Heat Transfer Copyrigt 2006 by ASME MAY 2006, Vol. 128 / 421

2 422 / Vol. 128, MAY 2006 Transactions of te ASME Ma Table 1 s kj/kg K Compressed liqid water property data at 260 C 3 and different entalpy corrections C kj/kg K Cp kj/kg K c sond m/s JT K/Ma corr or Cpd 4.69sat sat - corr -

3 Jornal of Heat Transfer MAY 2006, Vol. 128 / 423 Ma Table 2 Compressed liqid water property data at different temperatres and pressres 3 T=4 C T=20 C T=50 C T=100 C T=150 C Sat Ma T=200 C T=250 C T=300 C T=350 C Sat sat = Ma sat = Ma

4 424 / Vol. 128, MAY 2006 Transactions of te ASME Table 3 Compressed liqid water entalpies,, teir approximation differences, sat Eq. 2, and corr Eq. 5, and related percentages Note: More erroneos approximations are indicated in bold. T C 4 C 20 C 50 C 100 C 150 C 200 C 250 C 300 C 350 C Ma sat corr sat corr sat corr sat corr sat corr sat corr sat corr sat corr sat corr % % % % % % % % % % % % % % % % % % Sat % 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% Vapor % % % % % % % % % % % % % % sat = Ma % 0.3% 10.0% 0.7% 3.9% 0.7% 1.7% 0.7% 0.9% 0.7% 0.4% 0.7% 0.0% 0.7% 0.1% 0.3% % 0.5% 14.3% 1.0% 5.8% 1.0% 2.6% 1.0% 1.4% 1.1% 0.7% 1.1% 0.0% 1.2% 0.5% 1.2% Vapor sat = Ma % 0.7% 18.2% 1.3% 7.6% 1.4% 3.5% 1.3% 1.9% 1.4% 0.9% 1.5% 0.1% 1.8% 0.8% 2.0% 1.5% 1.9% % 1.3% 24.9% 2.0% 10.9% 2.0% 5.1% 2.0% 2.9% 2.1% 1.5% 2.3% 0.2% 2.8% 1.2% 3.5% 3.9% 5.3% % 1.8% 30.6% 2.6% 14.0% 2.6% 6.7% 2.6% 3.8% 2.7% 2.0% 3.1% 0.4% 3.7% 1.5% 4.8% 5.2% 7.7% % 2.4% 35.4% 3.1% 16.9% 3.1% 8.3% 3.1% 4.8% 3.3% 2.6% 3.8% 0.7% 4.6% 1.6% 6.0% 6.0% 9.7% % 3.0% 39.6% 3.7% 19.6% 3.7% 9.8% 3.7% 5.7% 3.9% 3.2% 4.5% 1.0% 5.4% 1.6% 7.1% 6.6% 11.4% % 4.1% 46.4% 4.7% 24.5% 4.7% 12.6% 4.7% 7.6% 5.1% 4.4% 5.7% 1.7% 6.9% 1.5% 9.1% 7.3% 14.4% % 5.2% 51.8% 5.7% 28.8% 5.7% 15.3% 5.7% 9.4% 6.1% 5.7% 6.9% 2.5% 8.3% 1.2% 10.9% 7.5% 16.9%

5 2 Analysis Compressed liqid water properties 3, for different pressre at 260 C, are presented in Table 1 and selected properties for different temperatres in Table 2. In addition, te corresponding corrections,, and, see Eqs. 4 8 are tablated along wit differences of te approximated entalpies, witot sat and wit correction corr from te real entalpy ales,, in te last two colmns in Table 1, respectiely, for satrated liqid water and compressed liqid water p to 100 Ma. Cmlatie water entalpy data wit te corresponding differences and related percentages are presented in Table 3 for a wide range of temperatres between triple and critical points, and, a wide range of pressres from satration p to 100 Ma. Note tat te tablated entalpy differences wit correction is based on Eq. 5, i.e., sing correction corr only, as generally recommended in te literatre, wile corrections, and are neglected 1,2. Te corrections in Tables 1 and 3 are calclated sing te following eqations: = =,T sat T N 1 = d i=1sat N 1 = d i=1sat 1 2 i + i+1 i+1 i 1 2 i + i+1 i+1 i corr Note tat in Eq. 5 te flid is assmed to be incompressible, wile in Eqs. 6 8 ariability in, altog small, is taken into accont, ts te differences in corr and ales in Table 1. Frtermore, for te isotermal compression, te correction is mc smaller tan te corrections and, and ts it may be neglected. Howeer, corrections negatie and corrections positie are comparable in magnitde bt opposite in sign, so it is better not to take te corrections, as in Eq. 2, tan to take only te correction corr,asineq. 5. Itwill be sown below tat te recommended entalpy correction for te isotermal compression 1,2, is actally alid for isentropic compression. It appears from real data ales in Tables 1 and 2 tat te magnitde of te negatie correction i.e., increases wit bot pressre and temperatre, wile positie correction i.e., d depends mostly on pressre since te specific olme,, does not cange significantly. As is eident from data in Table 3, te correction, Eq. 5 1,2, as a general improement for compressed liqid entalpy calclation, is not jstified in general, bt for smaller temperatres only less tan 200 C. Howeer, it is more erroneos and ts conterprodctie and misleading for iger temperatres aboe 200 C and particlarly at iger pressres, were a simple approximation Eq. 2, witot any correction, is more accrate. In Fig. 1, te compression of satrated liqid water, state f sat at 260 C, is presented for isotermal compression to state T50 at 50 Ma, and for isentropic compression to state s50 to te same pressre of 50 Ma see Table 1 for property data. Since d= q w were w= d, and real liqids in tis case water are not exactly incompressible, ten dring isentropic compression q=tds=0 tere will be some negatie work d and an increase of internal energy and temperatre in tis case for 11.7 C, from 260 C to C 3. Howeer, dring te isotermal compression, to cool and maintain constant water temperatre, tere mst be some eat transfer ot, q, and in te process te internal energy,, will be decreased wit pressre increase at constant temperatre see te corresponding data in Tables 1 and 2; note = 0 if 0. At ig temperatres, 300 C and aboe, een specific entalpy is decreasing wit Fig. 1 Isotermal and isentropic compression of satrated liqid water pressre increase de to a strong decrease of internal energy, making te recommended positie correction, corr, to be erroneos and ts conterprodctie and misleading, see Tables 2 and 3. Te correction is not inclded in Eq. 5 een tog its magnitde may be, and sometimes is, larger tan te inclded corr correction. Terefore, te recommended entalpy correction for isotermal compression in te literatre is appropriate for te isentropic processes see additional jstification below, bt not appropriate for isotermal processes, altog it may sometimes be beneficial, de to an erroneos assmption tat internal energy is not, and entalpy is, dependent on pressre. It is qite te opposite in Table 1, see ow te corresponding ales and cange wit pressre at constant temperatre of 260 C. Te aboe pysical jstifications cold be confirmed sing te corresponding differential property correlations obtained sing te Maxwell s relations 1 : 10 From Eq. 9 it is eident tat cange in internal energy for te isotermal process is zero d=0 only for ideal incompressible flids d=0, bt for real liqids d 0 te bracketed term wit d in Eq. 9 is not zero since real liqids are not exactly incompressible. Eqation 10 confirms tat pressre correction d is always appropriate for isentropic processes d=tds+d=d for s =const, bt not for isotermal processes as gien in many refer- 9 Jornal of Heat Transfer MAY 2006, Vol. 128 / 425

6 ences 1,2. For example, for an isentropic compression of liqid water from satration at 260 C to 50 Ma, see Fig. 1 and data in Table 1 sat =4.69 Ma, sat = kj/kg, s sat = kj/kg K, te temperatre and entalpy will increase to C and s50 = kj/kg, respectiely 3. Te isentropic entalpy increase, s50 sat = kj/kg=56.7 kj/ kg is irtally identical to te corresponding entalpy correction = d=56.05 KJ/kg corr =57.82 kj/kg, were te small differences in Table 1 are de to an integral approximation wit discrete smmation Eq. 8 or assming = sat =const in Eq. 5 dring te process. For te isotermal process te entalpy pressre-correction cold be ealated sing te tree corrections, Eqs. 6 8, or integrating Eq. 9 or 10, or combined Eqs. 9 and 10, i.e.: 11 Since te Jole-Tomson coefficient is defined as JT = T/, ten Eq. 11 cold be expressed as: T =,T sat T T = + + C JT d 1 2 C JT,i + C JT,i+1 i+1 i = sat N 1 i=1sat 12 Note tat real data ales for bot, te JT and C are aailable 3, and for te isotermal entalpy correction,, Eq. 12 was sed instead of Eqs. 6 8, see Table 1. Compressed liqid water entalpies, different corrections, teir approximation differences sat, and corr and te corresponding error percentages, wit regard to te satration ale, are presented in Table 3, witot any correction Eq. 2 and te correction recommended in te literatre corr, Eq. 5, for a wide range of temperatres and pressres. Te corrections, as recommended in 1,2 Eq. 5, are only beneficial for iger pressres at smaller temperatres, insignificant for smaller pressres at most of te temperatres, abot te same bt opposite sign ts nnecessary for intermediate temperatres and pressres, and more erroneos ts conterprodctie and misleading for iger temperatres and pressres, tan te simple approximation witot any correction Eq. 2, as seen from Tables 1 and 3. 3 Conclsion An analysis wit pysical jstification, spported by water real entalpy data, regarding liqid entalpy approximation, is presented ere. Te analysis sows, and a conclsion is drawn, tat te recommendation in te classical reference textbooks for improement of entalpy calclation of compressed liqids, by acconting for pressre dependence, is not generally alid for isotermal processes. Te misconception in te literatre is a reslt of te erroneos assmption tat, de to incompressibility for liqids in general, te internal energy is less dependent on pressre tan entalpy. Te literatre recommendations may be erroneos and ts conterprodctie and misleading, as is te case for liqid water at iger temperatres and pressres. For intermediate pressres and temperatres, te entalpy corrections recommended in te literatre are nnecessary, since te errors are abot te same in magnitde bt opposite in sign as if te corresponding satrated entalpy ales witot any corrections were sed. Te isotermal corrections are only beneficial for ery ig pressres at smaller temperatres below 200 C for liqid water. In smmary, te recommended pressre corrections in te classical reference textbooks for isotermal, liqid entalpy approximation are not appropriate and often insignificant, nnecessary or more erroneos tan te simple approximation sing te corresponding satrated ales. Frtermore, it is sown ere tat te recommended pressre correction for isotermal processes are actally alid in general for entalpy correction for isentropic processes. Nomenclatre c,c p,c specific eat, at constant pressre, or constant olme c sond speed of sond entalpy correction inclding all releant corrections, Eq. 12 corr entalpy correction de to te cange of sat = sat, Eq. 5a entalpy correction de to te cange of, Eq. 6 entalpy correction de to te cange of, Eq. 7 entalpy correction de to te cange of, Eq. 8 liqid entalpy at any and T corr liqid entalpy approximation wit sat sat correction as recommended in te literatre sat satration liqid entalpy pressre q eat transfer per nit of mass s entropy T temperatre internal termal energy specific olme w bondary work per nit of mass JT Te Jole-Tomson coefficient Sbscripts s50 isentropic compression from satrated to 50 Ma sat for satrated liqid T50 isotermal compression from satrated to 50 Ma References 1 Cengel, Y. A., and Boles, M. A., 2002, Termodynamics, An Engineering Approac, 4t ed., McGraw-Hill, New York, Sec , p Moran, M. J., and Sapiro, H. N., 2000, Fndamentals of Engineering Termodynamics, 4t ed., Wiley, New York, Sec , p Lemmon, E. W., McLinden, M. O., and Friend, D. G., 2005, Termopysical roperties of Flid Systems, NIST Cemistry WebBook, NIST Standard Reference Database No. 69,. J. Linstrom, and W. G. Mallard, eds., National Institte of Standards and Tecnology, Gaitersbrg, MD ttp:// webbook.nist.go/cemistry/flid. 426 / Vol. 128, MAY 2006 Transactions of te ASME