200 A Simple Two-Phase Critical Flow Model for Long Pipes,,, 05 flashing, L/D flashing Abstract A simple two-phase critical flow model is developed for estimating flashing flow rates through breaches in vessels or pipeworks The model considers both subcooled and saturated conditions The model has been tested against an extensive set of data from critical flow experiments with water as the test fluid In addition, comparison of the predictions with other theoretical models is made Results show that present model adequately predicts flashing flow rates through long pipes or large L/D geometries flashing, nozzle, slit -, [] Richter [2] general drift flux (GSL) [] space-dependent model, Marviken [3] [] slit [4,5]
,, 2 (,,, ) (G c ) (T o ) [4,5] (G * ) ( T* sub ) : G G * G c ref T T sat o T * sub (2) Tsat Tref, [ 2 ( P P )] G = ( Cd ) ρref o b ref ( C d ) ref ref = + K + f L D 05 ref 05 G ref (C d (P o P b ) T ref = 20 C (discharge coefficient) G* T* sub [4,5]: 52 G * = (5) * + exp[( + 0578) / 088] T sub slit : 05 52 Gc = ( Cd [ 2ρ ( Po Pb )] ref (6) + exp ( T * + 0578) / 088 sub (,, receiving end pressure) (C d 6 : x x G = o + o 3 TP (7) 3 3 G G = 0 x o = x o () (3) (4)
G x, 6 o = 0 = 00 T* sub G : x o = 2 k + 2k P 2 2 o k k G x = = Cdg (8) o k vo k + k C dg 2, (C d C dg K f [4,6,7] (C d 4,, (C d 6 C dg (C d (C d, ( X ) ( σ ) : n X i i= X = n 2 (9) 05 n 2 ( X ) i X σ = 00 % (0) i= n Gexp Gcorr X i = 00 % () Gexp i G exp G corr, n 3 (6 )
, 3 Table 284, 0 ~ 62 MPa, 025 ~ 509 mm, 40 ~ 2,335 mm 6 Table 8 % 0 %,,, (7 ) 267, 30 ~ 00 MPa, 52 mm 762 mm, 76 mm,778 mm (Table 2 ) Table Predictions of the subcooled inlet critical flow data with the Park model Experiment Pressure (MPa) Hydraulic Diameter, D (mm) Flow Length, L (mm) L/D No of Data X (%) Amos et al [6] 4 ~ 62 025 ~ 076 635 > 83 72-44 04 Ardron et al [8] 02 ~ 04 263,05 386 3 85 2 Boivin [9] 20 ~ 0 2 ~ 50 Celata et al [0] 08 ~ 23 46 700 ~ 2,305 46 ~,380 s (%) > 37 2-76 4 > 0 60-32 60 Fincke et al [] 0 ~ 03 828 26 8 92-24 33 Jeandey et al [2] 20 ~ 20 203 363 80 88-24 68 John et al [7] 40 ~ 40 04 ~ 28 460 > 35 57 25 99 Marviken [3] 20 ~ 50 Super Mobydick [3] 30 ~ 00 200, 300, 500 52 55 > 590 > 5 > 730 76 56 > 29 > 7 > 5 46 0 386 0 57 28 28-6 56 Reocreux [4] 02~034 20 2,335 7 39-8 59 Seynhaeve [5] 03 ~ 0 25 54 433 57-20 67 Sozzi et al [6] 62 28 2285-2 32 58 Sozzi et al [6] 57 ~ 70 27 08 ~,778 > 5 49-09 05 Park [4] 05 ~ 20 0 ~ 75 40 ~ 400 > 74-24 58 Remarks Slit Down Flow Down Flow Slit Down Flow Transient Pipe Down Flow Transient, Venturi Transient,,
Fig Table 2 267 % 83 % Table 2 Predictions of the two-phase inlet critical flow data with the Park model Experiment Pressure (MPa) Hydraulic Diameter, D (mm) Flow Length, L (mm) L/D for No of Data X (%) s (%) Remarks Super Mobydick [3] 30 ~ 00 52, 55 Sozzi et al [6] 27 ~ 70 27 76 56 46 0 9 20-04 96 08 ~,778 > 50 228 06 80 Sozzi et al [6] 63 ~ 69 54,2-4 -26 39 Sozzi et al [6] 65 ~ 67 762,076-3 -93 53 Sozzi et al [6] 68 28 2285-3 42 85 Transient, Transient, Venturi Transient, Venturi Transient, Venturi
50000 Calculation (kg/m 2 -s) 40000 30000 20000 0000 0 0 0000 20000 30000 40000 50000 Measurement (kg/m 2 -s) Fig Comparison between the model predictions and the test data in Table 2,,, : () 025 ~ 762 mm, L/D 8, 40 mm, (2) 200 mm L/D 5 32 [7], Table 3,, Table 4 space-dependent model Moody model [8], Henry-Fauske model [9], Homogeneous Equilibrium model (HEM) [], space-dependent model Elias-Chambre model [20], Richter model [2], general drift flux (GSL) model [] Table 3 Selected subcooled inlet critical flow data for the analytic models Experiment Hydraulic Flow Pressure No of Diameter, Length, L/D (MPa) Data D (mm) L (mm) Remarks
Ardron et al [8] 02 ~ 04 263,05 386 3 Boivin - [9] 20 ~ 0 2 700 583 0 Boivin 2 [9] 20 ~ 0 30 2,305 768 6 Boivin 3 [9] 20 ~ 0 50 2240 448 5 Fincke et al [] 0 ~ 03 828 26 8 92 Jeandey et al [2] 20 ~ 20 203 363 80 5 Jeandey et al- 2 [2] 20 ~ 20 203 363 80 73 Reocreux [4] 02 ~ 034 20 2,335 7 28 Seynhaeve [5] 03 ~ 0 25 54 433 26 Seynhaeve 2 [5] 03 ~ 0 25 54 433 3 Rounded Entrance + Diffuser Rounded Entrance + Diffuser Rounded Entrance + Diffuser + Diffuser Nozzle + Nozzle + + Diffuser + Diffuser + space-dependent model Tables 3 4 Table 5 Fig 2 Elias et al [] space-dependent model Richter model GSL model Richter model [2], GSL model [] Table 4 Fig 3 4 (Space-dependent model Elias et al [] ) space-dependent model GSL model [], Table 4 Selected critical flow data for model comparison Hydraulic Flow Pressure No of Experiment Diameter, Length, L/D (MPa) Data D (mm) L (mm) Sozzi et al [6] 57 ~ 69 27 08 85 23 Remarks
Sozzi et al [6] 58 ~ 68 27 59 24 5 Sozzi et al [6] 63 ~ 69 27 235 85 2 Sozzi et al [6] 60 ~ 70 27 273 25 22 Sozzi et al [6] 57 ~ 68 27 362 285 9 Sozzi et al [6] 60 ~ 68 27 553 435 3 Sozzi et al [6] 64 ~ 69 27 679 535 96 Sozzi et al [6] 6 ~ 69 27,823 435 8 Sozzi et al [6] 60 ~ 69 27 95 89 23 Sozzi et al [6] 60 ~ 69 27 322 289 24 Sozzi et al [6] 6 ~ 69 27 53 439 24 Sozzi et al [6] 60 ~ 69 27 640 539 7 No 3 Nozzle,, No 3 Nozzle,, No 3 Nozzle,, No 3 Nozzle,, Table 5 Predictions of all the data in Table 3 using the Park and analytic models Model Moody Henry-Fauske HEM Park Mean s Mean s Mean s Mean s [8] Ardron et al 25 238 65 223 762 24 85 2 Bovin [9] -88 0-500 72-53 57-44 75 Bovin 2 [9] -40 70-752 57-44 279-26 89 Bovin 3 [9] -35 78-286 2 83 96-02 46 Fincke et al [] -29 29-8 25-29 29-24 33 Jeandey et al [2] -20 8-282 3-39 04-54 50 Jeandey et al 2 [2] -8 93-293 63 76 24-8 70 Reocreux [4] -657 08-840 264-680 20-4 63
Seynhaeve [5] -28 35-240 23-2 33-8 63 Seynhaeve 2 [5] -94 55-25 2-89 59-22 72 20 Relative X mean and σ 0 0-0 Moody X mean Moody σ Henry-Fauske X mean Henry-Fauske σ HEM X mean HEM σ Park X mean Park σ -20 0 0 20 30 40 40 50 L over D (L/D) Fig 2 Comparison of calculated relative mean differences and standard deviations between the model and the analytic models for the data in Table 4 20 Relative X mean and σ (%) 0 0-0 Elias-Chambre X mean Elias-Chambre σ GSL X mean GSL σ Park X mean Park σ -20 0 0 20 30 40 40 50 L over D (L/D) Fig 3 Comparison of calculated relative mean differences and standard deviations
between the model and the space-dependent models for subcooled inlet data in Table 4 20 Relative X mean and σ (%) 0 0-0 Elias-Chambre X mean Elias-Chambre σ Richter X mean Richter σ Park X mean Park σ -20 0 0 20 30 40 40 50 L over D (L/D) Fig 4 Comparison of calculated relative mean differences and standard deviations between the model and the space-dependent models for the two-phase inlet data in Table 4 4 flashing, L/D flashing, () 025-762 mm, L/D 8, 40 mm (2) 200 mm L/D 5 Acknowledgement This project has been carried out under the Nuclear Research and Development Program by MOST
Nomenclature (C d discharge coefficient evaluated at 20 C C dg discharge coefficient of pure vapor D diameter, mm f friction factor G mass flux, kg/m 2 s G c critical mass flux, kg/m 2 s G ref mass flux evaluated at 20 C, kg/m 2 s G TP critical mass flux of two-phase inlet conditions, kg/m 2 s G * dimensionless mass flux, G /G c ref K pipe entrance loss coefficient k ratio of specific heats L (total) length of test section, mm n number of data P pressure, MPa P b back pressure, MPa P o stagnation pressure, MPa T temperature, C T o stagnation temperature, C T ref reference temperature, 20 C T sub subcooling, C T * sub dimensionless subcooling, (T sat - T o )/(T sat -T ref ) v o specific volume of steam, m 3 /kg x o quality ρ density of water, kg/m 3 Subscript b receiver system c critical o stagnation condition ref values at 20 C sat saturation condition TP two-phase condition x = 0 saturated water x o o = all vapor Superscript * dimensionless References [] E Elias and GS Lellouche, "Two-Phase Critical Flow," Int J Multi-phase Flow, Vol 20, Suppl, pp 9-68, 994 [2] HJ Richter, Separated Two-Phase Flow Model: Application to Critical Flow, EPRI NP-800, 98 [3] The Marviken Full Scale Critical-Flow Tests, NUREG/CR-267, MXC-30, 982 [4] CK Park, An Experimental Investigation of Critical Flow Rates of Subcooled Water
through Short Pipes with Small Diameters, Ph D Thesis, KAIST, Korea, 997 [5] CK Park, JW Park, MK Chung, and MH Chun, An Empirical Correlation for Critical Flow Rates of Subcooled Water Through Short Pipes with Small Diameters, J Kor Nucl Soc, Vol 29, No, pp 35-44, 997 [6] CN Amos and VE Schrock, "Two-Phase Critical Flow in Slits," NUREG/CR-3475, 983 [7] H John et al, "Critical Two-Phase Flow through Rough Slits," Int J Multiphase Flow, Vol 4, No 2, pp 55-74, 988 [8] D H Ardron and MC Ackerman, "Study of the Critical Flow of Subcooled Water in a Pipe," GEGB Report: RD/B/N4299, 978 [9] JY Boivin, "Two-Phase Critical Flow on Long Nozzles," Nuclear Technology, Vol 46, Mid-Dec, 979 [0] GP Celata et al, "Two-Phase Flow Models in Unbounded Two-Phase Critical flows," Nuclear Engineering and Design, Vol 97, pp 2-222, 986 [] JR Fincke and DR Collins, "The Correlation of Two Dimensional and Nonequilibrium Effects in Subcooled Choked Nozzle Flow," NUREG/CR-907, EGG-208, 98 [2] C Jeandey et al, "Auto Vaporization d'ecoulements Eau/Vapeur," CEN de Grenoble, Report TT No 63, 98 [3] Bethsy Team, "Selected Results from Characterization Tests of the Bethsy Break Nozzles (2" and 6 ) Conducted in the Super Moby-Dick Facility," Addendum to NOTE SETh/ LES/90-04, CEN Grenoble [4] M Reocreux, "A Contribution l'etude des Debits Critiques en Recoulement Diphasique Eau-Vapeur," Doctoral Dissertation a l'universite Scientifique et de Grenoble, 974 [5] JM Seynhaeve, "Etude Experimentale des Ecoulements Diphasiques Critiques a Faible Titre," Doctoral Dissertation, Universite Catholique de Louvain, France, 980 [6] GL Sozzi and WA Sutherland, "Critical Flow of Saturated and Subcooled Water at High Pressure," NEDO-348, 975 [7] V Ilic, S Banerjee, and S Behling, "A Qualified Data Base for the Critical Flow of Water," EPRI NP-4556, 986 [8] FJ Moody, Maximum Two-Phase Vessel Blowdown from Pipes, ASME J Heat Transfer, 966 [9] RE Henry and HF Fauske, The Two-Phase Critical Flow of One Component Mixtures in Nozzles, Orifices, and Short Pipes, ASME J Heat Transfer, Vol 93, 97 [20] E Elias and PL Chambrë, A Mechanistic Non-Equilibrium Model for Two-Phase Critical Flow, Int J Multiphase Flow, 0,, 984