A General Thermal Equilibrium Discharge Flow Model

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1 Journal of Enrgy and Powr Enginring 1 (216) doi: / / D DAVID PUBLISHING A Gnral Thrmal Equilibrium Discharg Flow Modl Minfu Zhao, Dongxu Zhang and Yufng Lv Dpartmnt of Ractor Enginring Tchnology, China Institut of Atomic Enrgy, Bijing 12413, China Rcivd: May 9, 216 /Accptd: May 19, 216 /Publishd: July 31, 216. Abstract: Basd on isntropic flow and thrmal quilibrium assumptions, a modl was drivd to calculat discharg flow rat, which unifid th ruls of room tmpratur watr discharg, high tmpratur and high prssur watr discharg, two-phas critical flow, saturatd stam and suprhatd stam critical flow, and gav a mthod to calculat critical condition. Bcaus of th influnc of friction, th ntropy is incrasd in th actual discharg procss, and th discharg flow rat in thrmal quilibrium condition can b obtaind by th original modl multiplid by an appropriat corrction cofficint. Th modl calculatd rsults agrd wll with th xprimnt data of long nozzl critical flow. Ky words: Thrmal quilibrium, critical flow, discharg flow rat. Nomnclatur C G h P P b P cr s V cr Subscript Empirical cofficint Mass flow rat Spcific nthalpy at th nozzl inlt Spcific hat capacity ratio Initial prssur Bac prssur Critical prssur Spcific ntropy at th nozzl outlt Vlocity at th nozzl outlt Critical prssur ratio Dnsity th inlt condition b th outlt condition cr critical quilibrium 1. Introduction Whn a vssl of prssurizd fluid bras, th fluid would blowdown to th nvironmnt drastically and a discharg flow occurs. For room tmpratur watr and any othr incomprssibl fluid, th flow would not rach critical flow condition and th discharg flow rat dpnds on Corrsponding author: Minfu Zhao, associat profssor, rsarch filds: nuclar safty and thrmal-hydraulics. th prssur diffrnc of th nozzl inlt and th outlt, which can b calculatd by th Brnoulli Equation [1], as G P P (1) 2 b whr, G is mass flow rat, ρ is watr dnsity, P is initial prssur, P is bac prssur. b Du to th ffct of friction and local rsistanc, th actual flow rat is lowr than that calculatd by Eq. (1). Thrfor, an mpirical cofficint is multiplid to Eq. (1), which bcoms G C P P (2) 2 b For th comprssibl fluid, such as prfct gas and suprhatd stam, a choing or critical condition would occur, whr th flow rat no longr incrass as th downstram prssur dcrasing furthr to th critical prssur P cr. According to th thoris in arodynamics [2], th critical prssur and critical flow rat can b valuatd by th following formulas (Eq. (3)): 2 1 Pcr P cr P G P (3) 1 (4)

2 A Gnral Thrmal Equilibrium Discharg Flow Modl 393 whr, P cr is critical prssur, cr is critical prssur ratio, is spcific hat capacity ratio (spcific hat capacity at constant prssur/spcific hat capacity at constant volum), is th fluid dnsity at th inlt of th nozzl. For th prfct gas, = 1.4; for th suprhatd stam, = 1.3, for th saturatd watr vapor, = Considring th ffct of friction and local rsistanc, an mpirical cofficint should b applid to th original quation to obtain th actual flow rat, as following G C P 1 (5) For high tmpratur and high prssur watr or stam-watr mixtur, th phas chang would occur during th discharg procss, which vntually turns to stam-watr mixtur at lowr prssur and tmpratur. Th phas chang mas th two-phas discharg flow pattrn vry complicatd, which diffrs from room tmpratur watr and watr vapor, and th flow may rach critical condition or not [3]. Svral tns of critical flow modls hav bn proposd, but still thr is no crtain mathmatical mthod [4, 5]. So far th currnt discharg flow thoris bas on diffrnt assumptions and th scop of application is rlativly narrow. Ths xplain why th two-phas critical flow modls ar quit diffrnt and can hardly b gnrally accptd. If a gnral modl could b drivd with fw assumptions, but can b applid to th rang from room tmpratur watr to suprhatd stam, and furthr applying this modl to two phas condition, thn th problm to valuat two-phas discharg flow would b solvd radically. Basd on th abov considration, a gnral modl has bn proposd with isntropic flow and thrmal quilibrium assumptions, which is applicabl for both subcoold watr and suprhatd vapor. Furthrmor, considring th friction ffct, a corrction factor is introducd to th original modl, thus th isntropic flow assumption is liminatd and th thrmal quilibrium gnral modl is obtaind. Th modl stimatd rsults agr wll with th xprimnt data for long nozzls. Th modl also provids furthr insight into critical prssur. 2. Modl Drivation Thr ar two assumptions: (1) Th friction ffct is ignord and th flow is considrd isntropic; (2) Th vapor phas and liquid phas ar in thrmal quilibrium, and th vlocity is th sam. Th mass flow rat at th nozzl outlt is dfind as: G V (6) whr, is th fluid dnsity at th nozzl outlt, V is th vlocity at th nozzl outlt. Bcaus th friction ffct is ignord, th outlt vlocity can b obtaind by nrgy consrvation quation: V 2( h h ) (7) Substituting Eq. (7) to Eq. (6), th outlt mass flow rat can b obtaind: G 2( h h ) ( P, s ) 2( h h( P, s )) (8) whr, h is th spcific nthalpy at th nozzl inlt, s is th spcific ntropy at th nozzl outlt, h is th spcific nthalpy at th nozzl outlt, P is th nozzl outlt prssur. For any crtain inlt condition, th mass flow rat undr diffrnt outlt prssurs can b calculatd by Eq. (8), among which th largst is th critical mass flow rat, and th corrsponding outlt prssur is th critical prssur. Eqs. (8) and (4) ar totally consistnt in principl, xcpt that: during th drivation of Eq. (4), is calculatd from th assumption that th rlationship 1 1 P P is satisfid during th isntropic procss, thn th prfct gas stat quation is applid to calculat th outlt tmpratur, according to th inlt and outlt tmpratur diffrnc and

3 394 A Gnral Thrmal Equilibrium Discharg Flow Modl spcific hat capacity, th nthalpy diffrnc is obtaind, thrfor, Eq. (4) is only applicabl to fluid of which stat is clos to prfct gas; whil Eq. (8) compltly adopts watr and watr vapor proprty quation for drivation, which is mor clos to ral conditions and can b applid to any stat. In othr words, if w substitut th physical proprtis such as dnsity, spcific nthalpy to th prfct gas proprty in Eq. (8) and lt P Pcr, Eq. (8) bcoms to Eq. (4). For room tmpratur watr, th dnsity is almost constant, combining th dfinition of ntropy, during th isntropic procss, w can obtain: dh 1 ds dp T T P P dh dp h h 1 Thus, Eq. (8) may b simplifid to Eq. (1): P P G 2( h h ) 2 2P P 2 It can b concludd that Eq. (8) is a gnral modl which is applicabl to both subcoold watr and suprhatd stam. Taing th friction and local rsistanc ffct into account, th actual discharg flow is not isntropic, rfrring to th approach usd in Eqs. (2) and (5), by multiplying an mpirical cofficint C, Eq. (8) bcoms: G C( P, s ) 2( h h( P, s )) (9) During th drivation of Eq. (9), only th thrmal quilibrium assumption is applid, thrfor, it is a gnral thrmal quilibrium modl. 3. Modl Prdiction and Analysis 3.1 Discharg Flow Charactristics Analysis In this sction, Eq. (8) is usd to invstigat th discharg flow phnomna and ruls undr diffrnt fluid stat. St th initial prssur P = 1 MPa, for xampl, th variation of discharg mass flow rats with nozzl outlt prssur for diffrnt fluid stat from th rang of room tmpratur watr (2 o C) to suprhatd stam (5 o C) ar obtaind, which ar shown in Figs. 1-3, rspctivly. Fig. 1 illustrats th subcoold watr discharg condition, th tmpratur of which is 2 o C, 2 o C, 25 o C, 3 o C and 31 o C, rspctivly; Fig. 2 shows th suprhatd stam discharg condition, th tmpratur of which is 315 o C, 35 o C, 4 o C and 5 o C, rspctivly; Fig. 3 shows th vapor and watr mixtur discharg condition, th thrmal quilibrium quality is.2,.4,.6,.8,.1,.2,.3,.4 and.5, rspctivly. Fig. 1 Variation of mass flow rat with outlt prssur undr subcoold conditions.

4 A Gnral Thrmal Equilibrium Discharg Flow Modl 395 Fig. 2 Variation of mass flow rat on outlt prssur undr suprhatd stam conditions. Fig. 3 Variation of mass flow rat on outlt prssur undr saturatd conditions. It can b obsrvd that, whn th outlt prssur dcrass gradually from 1 MPa, th mass flow rats of all conditions xcpt room tmpratur watr condition incras at first and thn rach a maximum valu, corrsponding to a crtain prssur, aftr which th mass flow rat dcrass as th outlt prssur gos down. Actually, whn th bac prssur is low nough, th outlt prssur would stay at a crtain valu which corrsponds to th maximum mass flow rat, rathr than dcrasing to th bac prssur, this phnomnon is calld critical flow. Thrfor, th maximum valu of ach curv is th critical mass flow rat for a crtain prssur and tmpratur (or quality) and th corrsponding outlt prssur is th critical prssur, th partial curv whr outlt prssur is lowr than th critical prssur is artificial and actually nonxistnt. It can also b concludd from th abov figurs that: for subcoold watr, whn th subcooling is larg, as th tmpratur incrass, th critical prssur incrass and th critical mass flow rat dcrass, it is found that th critical prssur is qual to th saturatd prssur

5 396 A Gnral Thrmal Equilibrium Discharg Flow Modl Tabl 1 Calculatd critical prssur for initial prssur 1 MPa. Tmpratur ( o C) Quality Critical prssur (MPa) Critical prssur ratio corrsponding to th initial tmpratur. Whn th tmpratur is clos to th saturatd tmpratur corrsponding to th initial prssur, th critical prssur drops a littl, dviats from th saturatd prssur corrsponding to th initial tmpratur. For suprhatd stam, th critical prssur for diffrnt tmpratur is almost constant, th critical flow rat dcrass as th tmpratur incrass bcaus of th dnsity chang. For two phas mixtur, as th thrmal quilibrium quality incrass, critical prssur dcrass and critical mass flow rat dcrass. Tabl 1 lists th critical prssur for diffrnt conditions. Whn th initial prssur P quals 13 MPa and 16 MPa, rspctivly, th calculatd discharg flow charactristic is similar with th condition that initial prssur P = 1 MPa. 3.2 Modl Validation To assss th accuracy of th gnral modl, th rsults calculatd by Eq. (8) and th xprimnt data [6] conductd for lngth to diamtr 2, round-dg inlt nozzl ar compard, which is shown in Figs. 4 and 5. It can b obsrvd that th calculatd mass flux (mard with cal.-1 in th figurs) xhibits a similar trnd with th xprimnt rsults in gnral, but th calculatd mass flux is highr in quantitativ. Th rason is that th modl is basd on th isntropic flow assumption, but in rality, th ffct of friction and form rsistanc is larg and cannot b nglctd. If w calculat th mass flow rat using Eq. (9), and choos th mpirical cofficint C =.82 for subcoold condition and C =.9 for th saturatd condition, th nwly calculatd rsults (mard with cal.-2 in th figurs) agr wll with th xprimnt rsults, which is shown in Figs. 4 and 5. It could b concludd that th thrmal quilibrium gnral modl givs a good prdiction of th critical mass flow rat in long nozzl. Fig. 4 Comparison of modl prdiction with xprimnt data undr subcoold condition.

6 A Gnral Thrmal Equilibrium Discharg Flow Modl 397 Fig. 5 Comparison of modl prdiction with xprimnt data undr saturatd condition. Fig. 6 Comparison of modl prdiction with xprimnt data undr subcoold condition. Whn th initial prssur P qual to 13 MPa and 16 MPa, rspctivly, th thrmal quilibrium gnral modl prdictd rsults and long nozzl xprimnt rsults ar in good agrmnt, which is shown in Figs. 6 and 7; And th valu of mpirical cofficint C is th sam with that of initial prssur P = 1 MPa. 3.3 Dtrmination of Empirical Cofficint C Th mpirical cofficint C rflcts th ffct of friction and local rsistanc, thrfor, it is rlatd to th dtaild structur of th nozzl. Furthrmor, as discussd in Sction 3.1, though th valu of mpirical cofficint C is rlatd to th fluid stat, but whthr undr subcoold condition or saturatd condition, C is almost constant. Thrfor, th CFD mthod could b applid to simulat room tmpratur watr and suprhatd stam discharg [7, 8], rspctivly, thus th mpirical cofficint C is obtaind.

7 398 A Gnral Thrmal Equilibrium Discharg Flow Modl Fig. 7 Comparison of modl prdiction with xprimnt data undr saturatd condition. Rfrring to th nozzl actual structur, th room tmpratur watr discharg flow is simulatd, and th computd mpirical cofficint C =.84; Th discharg flow for suprhatd stam with tmpratur 5 o C is simulatd, and th computd mpirical cofficint C =.9. It is found that simulatd rsults ar clos to th xprimnt data, which indicat th CFD mthod is fasibl to dtrmin th mpirical cofficint C. 4. Conclusions (1) With th thrmal quilibrium assumption, a gnral thrmal quilibrium modl was proposd. This modl unifid th flow ruls of room tmpratur watr discharg, high tmpratur and high prssur watr discharg, two-phas critical flow, saturatd stam and suprhatd stam critical flow, and could b applid to calculat critical prssur and critical mass flow rat. (2) Th modl prdiction rsults agrd wll with th xprimnt data of long nozzl critical flow, illustrating that th modl is accurat; th mpirical cofficint C in th modl can b obtaind through th CFD simulation, and th availability of th modl is thus nhancd. (3) Th modl is only applicabl to long nozzls, and thrmal quilibrium condition, if a quantitativ rlational xprssion btwn thrmal quilibrium dgr and lngth to diamtr ratio could b complmntd, thn all assumptions would b rmovd, and a mor gnral discharg flow modl would b obtaind. Acnowldgmnts Th prsnt rsarch was supportd by th Larg Advancd Prssurizd Watr Ractor National Scinc and Tchnology Ky Projct (211ZX ). Rfrncs [1] Mo, N. 2. Enginring Fluid Mchanics. Wuhan: Huazhong Univrsity of Scinc and Tchnology Prss. [2] Shn, D. W., Jiang, Z. M., and Tong, J. G. 21. Enginring Thrmodynamics. 3rd Edition. Bijing: Highr Education Prss. [3] Chn, Y. Z., Zhao, M. F., Yang, C. S., Bi, K. M., Du, K. W., and Zhang, S. M Rsarch on Critical Flow of Watr undr Suprcritical Prssurs in Nozzls. Enrgy and Powr Enginring 6 (2): [4] Lvy, S., Abdollahian, D., Halzr, J., t al Critical Flow Data Rviw and Analysis. Rport for ris and safty managmnt program.

8 A Gnral Thrmal Equilibrium Discharg Flow Modl 399 [5] Chn, Y. Z., Yang, C. S., Zhang, S. M., Zhao, M. F., Du, K. W., and Bi, K. M. 29. Evaluation of Existing Physical Modls on Critical Flow Basd on Exprimnt with a Nozzl. Atomic Enrgy Scinc and Tchnology 43 (6): [6] Chn, Y. Z., Zhao, M. F. t al Exprimntal Study of Critical Flow at Stady Stat. Annual rport of China institut of atomic nrgy. [7] Aly, N. H., Karamldin, A., and Shamloul, M. M Modling and Simulation of Stam Jt Ejctor. Dsalination 123 (1): 1-8. [8] Pianthong K., Shanam, W., Bhnia, M., Srivraul, T., and Aphornratana, S. 27. Invstigation and Improvmnt of Ejctor Rfrigration Systm Using Computational Fluid Dynamics Tchniqu. Enrgy Convrsion and Managmnt 48 (9):

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