Progress in Nuclear Energy

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1 Progress in Nucer Energy 51 (009) Contents ists vibe t ScienceDirect Progress in Nucer Energy journ homepge: A mode for predicting sttic instbiity in two-phse fow systems Kzem Frhdi * Engineering Science Reserch Group, Nucer Science Reserch Schoo, Nucer Science nd Technoogy Reserch Institute, AEOI, P. O. Box: , Tehrn, Irn bstrct Keywords: Sttic instbiity Pressure-drop Two-phse fow friction fctor Sfety mrgin An nytic mode for predicting the onset of Ledinegg instbiity in vertic chnne under both downfow nd upfow conditions hs been deveoped nd evuted. The mode divides the fow fied into two regions bsed upon the fuid temperture. The pressure drop is then found by soving n pproprite set of equtions for ech region. The theoretic resuts re compred to n existing set of experiment dt covering rnge of chnne dimeters nd operting conditions. A very good greement is obtined with the vibe experiment dt from the iterture for wter systems. A prmeter, the rtio between the surfce het fux nd the het fux required to chieve sturtion t the chnne exit for given fow rte, is found to be very ccurte indictor of the minimum point veocity in the demnd curve. Ó 009 Esevier Ltd. A rights reserved. 1. Introduction Fow instbiity is n importnt considertion in the design of nucer rectors becuse of the possibiity of fow excursion during postuted ccident. Genery, fow instbiity in heted chnne is cssified s either dynmic or sttic. The present study ddresses the tter type of instbiity. In the present work, fow instbiity (sttic) is defined s the occurrence of minimum in the demnd curve. This curve ttributes to the chnne pressure drop versus veocity curve. This is commony referred to s Ledinegg (1948) or excursive instbiity. The point t which this minimum occurs is defined s the onset of fow instbiity (OFI) point. Lhey nd Moody (1977) presented compete discussion of fow instbiity in heted chnnes. In system comprised mutipe fow pths, n increse in pressure drop in one fow chnne cn cuse the fow to be diverted to n ternte chnne. This fow reduction resuts in n even greter pressure drop nd more fow diversion from the ffected chnne. Eventuy, burnout cn occur in the unstbe chnne. One opertion pproch to soving this probem is to identify the veocity t which the minimum point occurs nd then to design the system so tht this condition is not reched during postuted ccidents. An nytic mode hs been deveoped to investigte this type of fow instbiity in vertic chnnes under upfow nd downfow conditions. The nytic study in the present work invoved the deveopment of pressure drop mode tht coud predict the entire * Te.: þ ; fx: þ E-mi ddress: k_frhdi37@yhoo.com fow regime ong heted chnne s it progresses from singe phse fow through the fuy deveoped nuceting boiing regime. Experiment dt were coected from the iterture on circur tubes of vrying dimeter under rnge of fow conditions nd het fuxes. These experiment dt re then used to evute the present mode.. Bckground Mrshek (1958) hs obtined some of the eriest dt for fow instbiity in downfow. Since then, number of reserchers hve studied this probem for both upfow nd downfow conditions. The pressure drop in sm dimeter tubes under subcooed boiing conditions is investigted by Dormer nd Berges (1964). These experiments were conducted in horizont chnnes under controed fow so tht dt were obtined we beyond the minimum point. A rge mount of dt ws correted using the rtio of the chnne pressure drop to the zero fux pressure drop versus the rtio of the surfce het fux to the het fux required to rise the buk fuid temperture to sturtion t the chnne exit. Empoying these prmeters, it ws shown tht the ony remining infuencing prmeter ws geometry. Fow instbiity in sm dimeter, with high chnne ength to chnne dimeter rtio ws studied by Mubetsch nd Griffith (1966). They demonstrted tht during pre fow opertion excursive fow instbiity took pce when pressure drop versus mss fow reched t minimum. The pressure drop in rectngur nd circur chnnes under subcooed boiing conditions is exmined by Whitte nd Forgn (1967). These reserchers determined tht for given chnne ength to dimeter rtio, the minimum in the demnd curve occurred t fixed vues of /$ see front mtter Ó 009 Esevier Ltd. A rights reserved. doi: /j.pnucene

2 806 K. Frhdi / Progress in Nucer Energy 51 (009) Nomencture Symbo quntity A fuid fow cross-section (m ) C p het cpcity (KJ/kg K) C constnt used in Eq. (19) D dimeter of heted chnne (m) f friction fctor G mss veocity (kg/m s) L chnne ength (m) P pressure (P) Q voumetric fow (kg/m 3 ) Re Reynods number St Stnton number T temperture (K) v veocity (m/s) W weight fow (kg/s) f het fux (W/cm ) m dynmic viscosity (kg/m s) r density (kg/m 3 ) 3 chnne roughness (m) c tt prmeter defined in the text Subscripts c critic d dimeter e equiibrium f fuid g gs iquid p pipe st sturtion SPR singe phse, region TPR two phse, region T tot tt turbuent turbuent the rtio of the chnne temperture rise to the inet sub-cooing. The rtio defined by them is given s: ððt out T in Þ=ðT st T in ÞÞ. Whitte nd Forgn so determined tht fow direction, t est for their fow-rte regime, did not ffect the veocity t which the minimum point (OFI) occurred. Johnston (1989) conducted downfow experiments in n interny heted nnuus. Due to pprtus imittions, dt were imited to Pecet number beow 50,000. Aso, het fuxes were beow 0.75 MW/m. The onset of significnt void (OSV) Stnton number chieved ws , which ws ess thn the vue predicted by the corretion of Sh nd Zuber (1974). However, the uthor recommended tht for downfow n OSV Stnton number of be used provisiony for Pecet numbers. The purpose of the test progrm described in this pper ws to generte the required downfow dt bse. An nytic mode ws deveoped by Duffey nd Hughes (1990) to describe sttic fow instbiity in vertic chnnes. They showed tht iner retionship exists between the OFI point nd the power density. In their pper, the uthors deveoped retionship for the fow rte t the minimum point bsed upon the friction pressure drop nd the buoyncy-induced pressure drop. In their deveopment, the ssumption ws mde tht the friction fctor incuding the twophse mutipier ws independent of fow rte. Experiment resuts for fow instbiity studies in vertic tube for chnne ength to dimeter rtios from 100 to 150 ws presented by Dougherty et. (1989, 1990). These reserchers showed tht the resuts coud be correted using Q rtio prmeter defined by the foowing eqution: Q rtioðupþ fpdl=wc pf ðt st T in Þ. This rtio is the sme s tht deveoped by Dormer nd Berges (1964). In ddition, this rtio is in greement with the R coefficient defined by Whitte nd Forgn (1967). The Q rtio cn be pprecited if one considers tht the pressure drop in the chnne is composed of singe-phse component nd two-phse component. As the fow veocity in the chnne decreses, the former component, this is proportion to the chnne fow veocity, decreses. With decresing chnne fow rte, more voids re formed, nd therefore, the two-phse pressure drop component tends to increse. These competing effects cuse minimum in the pressure drop to occur. Under subcooed boiing conditions, the w temperture exceeds sturtion, nd voids re formed in the fuid. As the fuid temperture pproches sturtion, the number of subcooed void increses. The corresponding increse in Q rtio cn be used s mesure of the void popution increse. The rtio so contins the test prmeters such s the surfce het fux, the inet temperture, the fow veocity, nd the exit pressure, which determines the sturtion temperture. Vrious modes used to predict the onset of significnt voids (OSVs) under subcooed boiing conditions were exmined by Lee et. (199). Their resuts show tht in vertic downfow the OFI point is most coincident with OSV. These reserchers so concuded tht the corretion suggested by Sh nd Zuber (1974) ws the best predictor of OSV. This corretion is given by: Q rtioðdownþ 1=ð1 þ 0:5½St OSV ðl=dþš 1 Þ. In ddition, it ws shown tht the mode deveoped by Levy (1967) ws best mong nytic pproches. An nytic mode to predict the pressure drop in vertic chnne under downfow conditions ws deveoped by Bock et. (1990). This mode considers four fow regimes: singe phse, prti subcooed boiing, fuy deveoped subcooed boiing, nd sturted boiing. For ech regime, pressure drop corretion nd trnsition criteri were ppied. Comprison with imited test dt showed resonbe greement. Upon this mode, n nytic mode nd experiment test to verify the mode ws conducted by Steing et. (1996). These uthors deveoped simpe nytic mode to predict the onset of Ledinegg instbiity in vertic chnnes under downwrd fow conditions. Their mode divides the fow fied into three regions bsed upon the fuid temperture. The pressure drop is then found by soving n pproprite set of equtions for ech region. The theoretic resuts re compred to n extensive set of experiment dt covering rnge of chnne dimeters nd operting conditions. They observed greement is exceent, nd the prediction of the veocity t which the minimum point in the chnne curves occurs is within 1 percent over the rnge of experiment resuts. These investigtors did not incude the sturted boiing regime in their pressure drop mode, however. Stoddrd et. (00) studied the onset of fow instbiity nd critic het fux in thin horizont nnui. For OFI, the uthors used six different test sections, with inner dimeter of 6.4 mm, nnur gp widths of mm, nd heted engths of mm. These uthors deveoped theoretic mode bsed on the soution of one-dimension fuid conservtion equtions, which ssumes no voidge upstrem the OSV point, nd ccounts for the thermodynmic non-equiibrium beyond the OSV point using n empiric quity profie fit, ws shown to predict the conditions reting to OFI resonby we. In the present work, in

3 K. Frhdi / Progress in Nucer Energy 51 (009) contrry to the work done by Steing et. (1996) sturted boiing is incuded when the pressure drop mode is deveoped. Comprisons re mde between the existing experiment dt nd proposed nytic mode. 3. Ledinegg instbiity in two-phse fow systems Two-phse fows cn exhibit rnge of simir instbiities. Usuy, however, the instbiity is the resut of non-monotonic pipeine chrcteristic rther thn compex centrifug pump chrcteristic. Perhps the we-known exmpe is the Ledinegg instbiity which is shown in Fig. 1. It hd been experimenty observed tht when wter fows through heted chnne where it is initiy subcooed t the inet, the circution of wter might be due to presence of pump in the network or cused by buoyncy effect resuted from grvity force on density grdient of working fuid, the vritions of pressure-oss with mss fow rte my be of such form tht more thn one fow rte my be in equiibrium t given pressure-drop. The curve of the pressure-drop cross the heted chnne ginst mss rte of fow t uniform het fux which is known s intern chrcteristic of the chnne (demnd curve) my tke one of the two forms shown schemticy in Fig. 1. Curve DP T pump represents stbe fow sitution since it hs no mximum or minimum. The Ledinegg or fow excursion instbiity my occur with curve DP T ine (ctu) when the sope of the pressure-drop versus mss rte of fow curve of the fuid suppy system which is known s extern chrcteristic of the chnne is numericy smer thn sope of the intern chrcteristic. In this cse, the working point Q is unstbe nd ny sm disturbnce in the fow under these conditions wi resut in the mss rte of fow jumping to the stbe working point. This point corresponds to the intersection of DP T pump curve with DP T ine ( vpor) or DP T ine ( iquid). 4. Pressure drop mode In n ttempt to find n nytic mode, tenttive pproch for ccution of pressure-drop during forced circution of wter through pipeine is proposed. Mthemtic formution of the ide is s foows: Pressure-oss in singe-phse region of the heted chnne (pressure-drop vs. mss rte of fow chrcteristic curve) my be written in the form (Mrtinei nd Neson, 1948) Dp BW n (1) where B cn hve different vues for iquids nd gses. For ccuting turbuent fow pressure-drop in heted chnne, Lockhrt nd Mrtinei (1949) proposed corretion. This is written s: Dp f p D rv () On equting equtions (1) nd () it foows tht B f p DrA Wn (3) Defining new prmeter B 0 h p =DA, then B cn be written s B f r Wn B 0 (4) It is customry to use the Hgen Poiseuie formu (Streeter nd Wyie, 1975) for ccuting minr fow pressure-drop in given chnne. The formu is given s Dp 18m pq pd 4 (5) There is strong beief tht there coud be correction fctor for converting minr pressure-drop into turbuent pressure-drop (Steing et., 1996). Let it be defined by retion of the form h Dp Turbuent Dp Lminr (6) Eqution (6) is beieved to be vid in prticur operting condition of given chnne (Steing et., 1996). By combining equtions (), (5), nd (6), respectivey, h coud be written s h f 16 Re D (7) Fig. 1. Sketch iustrting the Ledinegg instbiity.

4 808 K. Frhdi / Progress in Nucer Energy 51 (009) where the vue of Re D in the trnsition region of Moody chrt (Moody, 1944) ong with corresponding vue of f coud be extrcted. Combining equtions (1), (), nd (4), respectivey nd rerrnging them wi resut in eqution (8). p DB 0rV yw (8) Aso, combining equtions (1), (3), nd (4), respectivey nd rising the power of individu terms to n nd re-rrnging them it foows tht nf m n pd4 B 0 n y n 18 p h n Wn (9) Mutipying eqution (8) by eqution (9) eds to eqution (10). m n yw B 00 rv f n y n h n W n (10) Coefficient B 00 is representtive of the geometry of concerned system. Eqution (10) cn be written for both iquid phse s we s gs phse for two-phse fow in heted-chnne, seprtey. Then, on dividing eqution of iquid by gs, one my obtin eqution (11). m m g ny W y g W g ny f f g y g n 1 hg n n V W (11) For the dt given by Mrtinei nd Neson (1948) vue of n between 0. nd 0.5 ws found. In the present work, vues in the rnge of for the exponent n re ssumed. For this cse, eqution (11) cn be written s m m g ny W y g W g n ny f f g y g h V g W g n 1 hg n V (1) The LHS of eqution (1) is equ to c tt where vue of c tt is given by Mrtinei nd Neson c n tt f gtt f tt dp d dp d g h V g Eqution (11) with use of eqution (14) nd fter re-rrnging cn be written in the form 1 n nmg f f g m n hg n S n (15) h Eqution (15) does not show the dependency of on x. The sip rtio (S) determines the degree of dependency of on x, however (E-Wki, 1971). Tking equtions (7) nd (14) into considertion, eqution (15) cn be written in simper form such s G 1 þ S n n 1 x x (16) where G is n empiric correction fctor for different vues of, which depends on the vues of S (Simir correction fctor, for fow in horizont heted chnnes, ws suggested by Stoddrd et. (00). As function of the chnne ength, S hs been found to increse rpidy t the beginning nd then more sowy s the chnne exit is pproched. At the exit itsef, turbuence seems to cuse sudden jump in the vue of S (EBWR, Ser. Rept. ANL-5607, 1957). Fig. shows correction fctor decreses s the vpor veocity increses. To pprecite this vrition of S, n eqution for the intersection hs been suggested 1 x* S x * 1 (17) This eqution hods true for system operting t no sip condition. Eqution (17) is speci use of the gener retionship given by eqution (16). The effect of sip is to decrese the vue of corresponding to certin vue of x beow tht which exists for no sip. Therefore, t constnt pressure nd quity, the fctor decreses with S. A high S is thus n dvntge from both the het-trnsfer nd moderting effect stndpoints (E-Wki, 1971). As is cer from Fig. 3 for operting points in the neighbourhood of intersection point, the vue of cn be evuted from simpe retion given by G (18) where n my be extrcted from the Bsius eqution f þ C Re n D (13) Steing et. (1996) used the friction fctor given by Zigrng nd Syvester (198) in their mode. This friction fctor is given by h h ii 3=D f 0:5 :0og 3:7 5:0 3=D Re f og 3:7 þ Re 13 f. The foowing stndrd definitions in two-phse fows re tken into considertion Coier (197). W 1 x W g x V g V S Substituting for quity (x) nd sip rtio (S) into eqution (1), it cn be shown tht y r g x y g r 1 x 1 1 S (14) Fig.. Vritions of empiric correction fctor (G) with sip rtio (S) for given twophse fow system.

5 K. Frhdi / Progress in Nucer Energy 51 (009) represents the tent het of the system nd ssuming tht the tot pressure-drop to be negigibe ginst bsoute pressure of the system, then Q GhA x e LGA Assuming x ð1=þx e, then one my obtin eqution (1). Q GhA x (1) GLA P represents the tot ength of the chnne nd stnds for twophse fow region, then it is possibe to write eqution (). p Q GhA Q () Considering eqution (18), nd ppying n verge vue over tot ength of the pipe it foows tht G= (3) Fig. 3. Vritions of void frction () versus quity of stem (x) ong heted chnne. First the vue of G is found from Fig., which corresponds to given S, then on substituting for G the vue of is ccuted. For wter (S 1), Gy1:75 (E-Wki, 1971). In heted tube of this nture, the fuid hs four possibe regions of fow: singe phse, prtiy deveoped subcooed boiing, fuy deveoped subcooed boiing, nd sturted boiing. The vrious regions cn be identified by vpor content nd temperture of the fuid. The proposed nytic mode divides the tube into the pproprite regions by fuid temperture. The pressure drop for ech region is then found by soving n pproprite set of equtions for ech region. The region pressure drops re then summed to determine tot pressure drop cross the tube. Eqution of momentum for two-phse fow system Coier (197) is given by dp dz ð1 Þ C DRe n r V 1 þðþ C g D DRe n g V Dgr g þ G d dz x y g þ ð1 xþ y 1 (19) Eqution (19) represents pressure grdient ong given chnne ength, singe nd two-phse fow, wi be integrted in the twophse fow region. This wi resut in n eqution of the form DP TPR ð1 Þ C G DRe n 1 þðþ C g D r x e y g þ ð1 x eþ y 1 DRe n Dg y 1 G g but G g xg nd G ð1 xþg, therefore DP TPR C DRe n+ D G ð1 xþ r þ C g DRe n+ Dg x e y g þ ð1 x eþ y y 1 þ G r g G x r g þ G (0) where h represents the inet wter sub-cooing nd Q the verge power ppied ong the tot ength of heted chnne. Aso, L Aso, 1 m LðA=DÞ Re n+ D LAG þ GhA Q 1 Re n+ Dg mg LðA=DÞ Q GhA n+ n+ (4) (4b) Equtions (4) nd (4b) re for singe phse prt of the heted chnne. Tking into ccount eqution (1) through (4), eqution (0) cn be written s DP TPR 1 G C p m n+ Q GhA Dr D Q GLA þ GhA Q n+ G Cg p mg n+ þ LA Dr g D Q GhA Q GhA n+ þ Q GhA y g Q LA G LA þ G " LGA # þ GhA Q G y LA (5) However, the pressure-drop for singe-phse region is given by DP SPR C p m n+ GhA G Dr n+ (6) D Q The tot system pressure-drop is equ to DP T DP SPR þ DP TPR (7) It is impied tht eqution (7) represents third-order function between tot pressure-drop nd tot mss rte of fow of the system. It is worthy to note tht eqution (7) ws derived on the bsis of seprted-fow mode. 5. Resuts nd discussion The mode comprised writing down eqution for tot pressuredrop cross the heted chnne nd then differentiting it in order to ccute the sope of pressure-drop vs. mss rte of fow. Then the octions of mximum nd minimum points on the pressuredrop vs. mss rte of fow curve (demnd curve) cn be found. The

6 810 K. Frhdi / Progress in Nucer Energy 51 (009) vdp SPR vg þ vdp TPR vg 0 (8) After mthemtic mnipution, one my get " G 6C p ham n+ r QD n+þ1 1 G 6C p m n+ ha r QD n+þ1 1 G L þ h # " G 6Cg p h L 3 Am n+ g ðl þ hþ L þ 3 4r g QD n+þ1 L þ G h C p m n+ L D n+þ1 r 3C p h m n+ 1 G g D n+þ1 r g QL " L " þ h C p m ## n+ þ L D 1þn+ r 3C g p m n+ g hq G D 1þn+ r g L A 4hy gq 1 G 4h y g GL A þ 1 G Qh GL A 4y Q ð GÞLA 4y G 4L A Q L þ L AL L # L þ h 0: (9) L Fig. 4. Vritions of ð vdp vg Þ versus mss veocity (G) for given (h/l) rtio. Eqution (9) cn be written s D 1 G þ D G þ D 3 0 (30) criterion for uncondition stbiity, in other words, the bsence of negtive sope on the pressure-drop vs. mss rte of fow curve then becomes evident. vdp T vg 0; or For instbiity, re roots of eqution (30) must be known. Therefore, D 4D 1D 3 0 D 4D 1D 3 (31) Fig. 5. Vritions of ð vdp vg Þ versus mss veocity (G) for different vues of (h/l).

7 K. Frhdi / Progress in Nucer Energy 51 (009) Eqution (31) represents the criterion for instbiity in two-phse fow systems. A cose ttention to D s, reves the fct tht of them contin dimensioness groups. These re h L ; Lþh L nd Lþh L, respectivey. The effect of this dimensioness group ð h LÞ on the pressure-drop versus mss rte of fow wi be studied crefuy. When given system stisfy eqution (31), s redy expined, the system wi ed towrds instbiity. In other words, the pressure-drop versus mss rte of fow chrcteristic curve wi be of we-known S-shped. It is worthy to note tht the terms D 1 ; D ; nd D 3 re compicted enough mthemticy. However, the other terntive woud be to investigte function form such s vdp vg /G. For common two-phse fow system, ðg c 546:5kg=m sþ nd ð vdp vg j Gc P kg 1 sþ. A typic behvior of this function form is shown in Fig. 4. Critic mss rte of fow ðg c Þ is defined s G c Q. When the inet ha mss rte of fow of wter into the heted chnne is ess thn this vue, it is most certin tht the phenomenon of boiing wi tke pce inside the heted chnne. However, when the sope of the pressure-drop versus mss rte of fow chrcteristic curve becomes positive t this prticur point, monotonic behvior of the chrcteristic curve is to be expected nd the system eds towrds stbiity. Therefore, it is expected tht vdp vg j Gc 0. A stbiity mp for typic two-phse fow system (see Fig. 4) where its geometry nd working fuid re known is obtined. Fig. 4 indictes n interesting rnge of stbiity SM G c /G 0 where SM stnds for stbiity mrgin. An increse in stbiity mrgin mens tht the system is we wy from OSV. In other words, when disturbnce tkes pce in the working fuid mss rte of fow in the rnge G 0 /G c. Then, due to wide rnge of this region ðg 0 /G c Þ, the disturbed working fuid mss rte of fow wi be dmped nd remins in the stbiity mrgin region. Aso, the sope of concerned chrcteristic curve is positive nd the disturbnce wi dmp out rpidy (Dehye et., 1981). Therefore to study instbiity of given system, it is vit to know vritions of vdp vg versus G c. Consequenty, the mgnitude of resistnce shown by the system ginst disturbnce cn be determined from the SM. To pprecite the present mode, it is convenient to provide stbiity mp in tht the effect of physic quntities h/l nd Q/L on pressure drop versus mss rte of fow chrcteristic curve re shown. In ddition to this, geometry nd operting conditions then the stbiity or instbiity of the concerned system wi be known. For given Q/L rtio, the effect of dimensioness prmeter h/l on the stbiity mrgin for different vues of ð vdp vg Þ is shown in Fig. 5 The figure shows incresing inet sub-cooing of wter into the heted chnne woud ffect sope of the pressure-drop versus mss rte of fow chrcteristic curve t the critic point ðg c Þ or vdp vg j Gc in such mnner tht the chrcteristic curve tends towrds more negtive sope. This sitution, obviousy by Ledinegg definition, demonstrtes n excursive instbiity of the two-phse fow system. Fig. 6 shows (for given h/l rtio) decrese in the sope ð vdp vg Þ of such sitution for different vues of ðq LÞ s prmeter. Fig. 7 shows comprison between Q rtio (upfow) experiment dt of Sh Zuber, Q rtio (downfow) experiment dt of Steing et. (1996) nd the present mode. In the present nytic mode this prmeter does not pper expicity; however, it cn be Fig. 6. Vritions of vdp vg j Gc versus (h/l) for different vues of (Q/L).

8 81 K. Frhdi / Progress in Nucer Energy 51 (009) Qrtio ccuted for ech point in the demnd curve. The figure so shows the effect of ength-to-dimeter rtio on the Q rtio. As the fuid temperture pproches sturtion, the number of subcooed voids increses. The corresponding increse in Q rtio cn be used s mesure of the void popution increse. This rtio so contins the test prmeters such s the surfce het fux, the inet temperture, the veocity, nd the exit pressure, which determines the sturtion temperture. 6. Concusion L/D Experiment dt (upfow) Sh - Zuber (1974) Present mode Experiment dt (downfow) Steing et. (1996) Fig. 7. Comprison of Q rtio versus L/D for experiment dt nd nytic resuts. A new representtion of the criteri for predicting Ledinegg fow instbiity hs been derived. The importnt vribes tht my infuence the demnd curve of given two-phse fow system re found to be h/l nd Q/L, respectivey. The mode coud possiby predict fow conditions from singe-phse fow through the sturted boiing. The mode so provides prediction of the mss veocity t which the critic point in the sfety mrgin rnge occurs. This is n importnt fctor for rector sfety. The experiment dt show tht the occurrence of Ledinegg fow instbiity depends on the ength-to-dimeter rtio of the heted chnne, the inet temperture, exit pressure, nd surfce het fux. These prmeters re combined in the Q rtio defined in the text. The Q rtio ccuted from the present nytic mode exhibits fir greement with the experiment dt for both upfow s we s downfow. The Q rtio(up) corretion nd Q rtio(down) corretion contin the significnt prmeters ffecting the onset of Ledinegg fow instbiity. References Bock, J.A., Crowey, C.J., Don, F.X., Sm, R.G., Stoedefke, B.H., Nucete boiing pressure drop in n nnuus. CREARE TN 499, Oct. Coier, J.G., 197. Convective Boiing nd Condenstion. McGrw-Hi, New York. Dehye, J.M., Giot, M., Riethmuer, M.L., Thermodynmics of Two-Phse Systems for Industri Design nd Nucer Engineering. McGrw-Hi Book Compny (Chpter 17), p Dormer, J., Berges, A.E., Pressure drop with surfce boiing in sm dimeter tubes. Report No , Deprtment of Mechnic Engineering, MIT, Cmbridge, MA. Dougherty, T., Fighetti, C., McAssey, E., Reddy, G., Yng, B., Chen, K., Qureshi, Z.C., Fow instbiity in vertic down-fow t high het fuxes. ASME HTD-vo. 199, pp Dougherty, T., Fighetti, C., Reddy, G., Yng, B., Jfri, T., McAssey, E., Qureshi, Z., Fow boiing in vertic downfow. In: Proc. Ninth Interntion Het Trnsfer Conference, vo., Jerusem, Isre. Duffey, R.B., Hughes, E.D., Sttic fow instbiity in tubes, chnnes, nnui, nd rod bundes. ASME HTD-vo. 150, pp EBWR, The Experiment Boiing Wter Rector. U.S. Atomic Energy Comm. Nucer Techno. Ser. Rept. ANL-5607, prepred by Argonne Ntion Lbortory, My E-Wki, M.M., Nucer Het Trnsport. Interntion Text Book Compny, New York, p. 331, fig. 1-5, (Chpter 1). Johnston, B.S., Subcooed boiing of downwrd fow in vertic nnuus. ASME HTD-vo. 109, pp Lhey, R.T., Moody, E.J., The Therm Hydruics of Boiing Wter Nucer Rector. Americn Nucer Society. Lee, S.C., Dorr, H., Bndoff, S.G., 199. A critic review of predictive modes for the onset of significnt void in forced-convection subcooed boiing, fundments of subcooed fow boiing. ASME HTD-vo. 17, pp Ledinegg, M., Die Wrme 61, 891. Levy, S., Forced convection subcooed boiing prediction of vpor voumetric frction. J. Het Mss Trnsfer 10, Lockhrt, R.W., Mrtinei, R.C., Proposed corretion of dt for isotherm two-phse two-component fow in pipes. Chem. Eng. Prog. 45, 39. Mrshek, S., 1958, Trnsient fow of boiing wter in heted tubes. Svnnh River Lbortory Report DP-310 TL. Mrtinei, R.C., Neson, D.B., Trns. Am. Soc. Mech. Eng. 70, 695. Mubetsch, J.S., Griffith, P., A study of system induced instbiities in forcedconvection fows with sub-cooed boiing. Report No , Deprtment of Mechnic Engineering, MIT, Cmbridge, MA. Moody, L.F., Friction fctors for pipe fow. ASME Trns. 66, Sh, P., Zuber, N., Point of net vpor genertion nd vpor void frction subcooed boiing. In: Proc. Fifth Interntion Het Trnsfer Conference, B4.7, Tokyo, Jpn. Steing, R., Mcssey, E.V., Dougherty, T., Yng, B.W., The onset of fow instbiity for downwrd fow in vertic chnnes. J. Het Trnsfer 118 (3), Stoddrd, R.M., Bsick, A.M., Ghisin, S.M., Abde-khik, S.I., Jeter, S.M., Dowing, M.F., Apri 00. Onset of fow instbiity nd critic het fux in thin horizont nnui. Exp. Therm Fuid Sci. 6 (1), Streeter, V.L., Wyie, E.B., Fuid Mechnics, sixth ed. McGrw-Hi, New York, pp Whitte, R.H., Forgn, R., A corretion for the minim in pressure drop versus fow-rte curves for sub-cooed wter fowing in nrrow heted chnnes. Nuc. Eng. Design 6, Zigrng, D.J., Syvester, N.D., 198. Expicit pproximtion to the soution of Coebrook s friction eqution. AIChE J. 8 (3),