evaluation of crack opening area and leak rate in …

NOMENCLATURE A (2c) Crack opening area for crack length 2c Ab Crack opening area due to bending load A (2c)cyl Crack opening area in cylindrical coordinates Ak Average roughness height At Crack opening area due to tensile load c Semi-crack length Cd Discharge coefficient dH Hydraulic diameter E Young's Modulus f Friction factor G Modulus of rigidity Gc Critical mass flux K Constant in FEM approach method K' Resultant stress intensity factor L Flow path length N Non equilibrium factor p Length of plastic zone ∆Pa Acceleration pressure loss Pc Critical pressure ∆Pc Entrance pressure loss ∆Pf Friction pressure loss

Po Stagnation pressure ∆Ptot Total pressure loss R Mean pipe radius t Pipe thickness V Crack opening displacement x Length along crack front xE Equilibrium quality α(λ) Bulging function δ Half crack mouth opening displacement γ Isentropic exponent ν Poisson's ratio θ ' Semi-crack angle θeff ' Effective semi-crack angle σ Applied stress σb Stress due to bending load σf Flow stress σt Stress due to tensile load σu Ultimate tensile stress σy Yeild stress

ABBREVIATIONS ASME American Society of Mechanical Engineers COA Crack Opening Area DEGB Double Ended Guillotine Break FEM Finite Element Method LBB Leak Before Break LSC Leakage size crack PDL Pump Discharge Line PHT Primary Heat Transport PHWR Pressurised Heavy Water Reactor SGI Steam Generator Inlet SGO Steam Generator Outlet

EVALUATION OF CRACK OPENING AREA AND LEAK RATE IN VARIOUS PHT PIPINGS FOR LBB ANALYSIS

OF INDIAN PHWRs

Naseem Ahmed Ansari (Heavy Water Division) Satish Patil (T&MSD)

B. Ghosh, J. Chattopadhyay S.K.Bandyopadhyay, S.K.Gupta, H.S.Kushwaha

(Reactor Safety Division) Bhabha Atomic Research Centre

Mumbai - 400085

ABSTRACT The advent of Leak Before Break (LBB) concept has replaced the traditional design basis event of Double Ended Guillotine Break (DEGB) in the design of high energy piping systems of a nuclear reactor.. The LBB analysis shows that if there is any undetected flaw on the inside surface of the pipe it will not grow throughwall during the inservice inspection / repair interval period and even if it grows throughwall it will lead to detectable leakage before any catastrophic break occurs. Determination of crack opening area which decides the Leakage Size Crack (LSC) is one of the key element in LBB analysis. In this paper, the leakage areas have been evaluated using Bartholom`e et al. Model by the integration of crack opening displacement along the crack front, considering plasticity and geometrical effects using Dugdale’s model. Crack opening area using this method has been compared with well-known Tada Paris Method and Numerical Integration Method based on Finite Element Method Analysis for various PHT piping. Also, leak flow rates using Henry’s non-equilibrium model have been evaluated for all the above leakage areas and the results have been compared with published results.

1.0 LEAK BEFORE BREAK METHODOLOGY: 1.1 INTRODUCTION

The Leak Before Break concept is nowadays widely used for the design of primary heat transport system piping systems of Nuclear Power Plants. Leak before Break aims at application of fracture mechanics technology to demonstrate that piping is very unlikely to experience Double-Ended-Guillotine-Break under all loading conditions. It shows that if there is any undetected flaw on the inside surface of pipe it will not grow throughway during the in-service inspection / repair interval period and even if it grows throughway, it will lead to detectable leakage before any catastrophic break occurs. The previous pipe rupture design requirements for Nuclear Power Plants are responsible for all the numerous and massive pipe restraints and jet-shields installed for each plant. These results in significant plant congestion, increased labour costs and radiation dosage for normal maintenance and inspection. Also the restraints increase the probability of interface between the piping and supporting structures during plant heat-up thereby potentially reducing overall plant reliability. The LBB approach to eliminate postulating ruptures in high energy piping systems is a significant improvement to former regulatory methodologies, and therefore, the LBB approach to design is gaining worldwide acceptance. The LBB essentially consists of demonstrating three level of confidence to show that the piping is very unlikely to experience any sudden, catastrophic break. Level 1 confidence is inherent in the ASME sec III design philosophy of piping system with some factor of safety, it does not, however, consider any presence of flaw in pipes. Level 2 consists of postulating a part-through crack at the inside surface of the PHT piping and then to demonstrate that it will not grow through-wall during the interval period of in-service inspection / repair and also there are enough margins against unstable extension of flaw through the pipe wall. Level 3 consists of postulating a through-wall crack that will ensure detectable leakage and then to demonstrate that the flaw will be stable under the severe most loading. The minimum leakage that can be detected depends on the sensitivity and accuracy of the leak detecting instruments. Generally it has been seen that a leak rate of 0.05 kg / sec can be easily detected in the plant. On this Nuclear Regulatory Commission recommends a factor of safety equal to 10. Hence it is assumed that minimum leak detection capability in a plant is 0.5 kg / sec. It is to be shown that the crack that ensures this much of leakage will be stable under the maximum credible loading conditions. Evaluation of leakage area which decides the Leakage Size Crack is one of the key element in Leak Before Break analysis. In this paper, the expression for crack opening displacement as given in Bartholome et al.[2,3] Model has been integrated along the crack front and final expression for crack opening area has been evaluated. Leakage area has been derived by the integration of crack opening displacement along the crack front considering plasticity and geometrical effects using Dugdale’s model [4]. The leakage areas are calculated for various PHT straight pipes and elbows of Indian PHWR's. These results are compared with the results based on Numerical Integration using FEM analysis [5,6] and Tada Paris Method [1]. Also COA is calculated for pressure tube for which R / t ratio is 10 and compared with well-known Tada-Paris method which is applicable for R / t=10. Using these values of crack opening area leak rate has been evaluated using well-known Henry's Model.

1.2 LIMITATIONS OF LEAK BEFORE BREAK CONCEPT While the application of the concept of LBB is universally accepted in designing high energy fluid piping in nuclear power plants, the concept with its present status has some limitations. One should be cautious about these limitations before applying LBB methodology in design. The United States Nuclear Regulatory Commission task group recommends that the following limitations apply to the mechanistic evaluation of pipe break in high energy fluid piping. (a) Specifying design criteria for emergency core cooling systems, containment and

other engineered safety features, loss of coolant shall be assumed in accordance with existing regulations.

(b) The LBB approach should not be considered applicable to high energy fluid system

piping, or portions thereof, that operating experience has indicated particular susceptibility of failure from the effect of corrosion (e.g. intergranular stress corrosion cracking ), water hammer.

(c) For plants for which there is an operating license or construction permit, component

( e.g. vessels, pumps, valves ) and piping support structural integrity should be maintained with no reduction in margin for the Final Safety Analysis Report ( FSAR ) or Preliminary Safety Analysis Report (PSAR) loading combination that governs their design.

(d) The LBB approach should not be considered applicable if there is a high probability

of degradation or failure of the piping from more indirect causes such as fire, missiles and damage from equipment failure (e. g. cranes) and failures of systems or components in close proximity.

(e) The LBB approach as described here is limited in application to piping systems

where the material is not susceptible to cleavage type fracture over the full range of systems operating temperatures, where pipe rupture could have significant adverse consequences.

2.0 ASSESSMENT OF CRACK OPENING AREA Crack opening area is the key element in LBB assessment. Estimates of crack opening area for postulated through-wall crack can vary widely depending on how the crack is idealized, which crack opening model is used and what material properties are assumed. A wide range of published solutions is available for idealized notch-like cracks in simple geometry subject to basic loading (pressure, membrane and bending). Their accuracy varies with geometry (e.g. R / t ratio), crack size, type of load and magnitude of load.

2.1 TADA-PARIS METHOD In this method [1] crack opening area depends on the load acting on the crack plane, material property such as Young's Modulus of the piping material, pipe dimensions, crack size and finally on the crack orientation. For circumferential crack in tension and bending, crack-opening area is calculated as follows:

)'(IERA

)'(IERA

b

2

bb

t

2

tt

θπ

σ

θπ

σ

=

=

+

−

+

−

+

+

−

+

=4323

25.1

2t

'242'5.247'7.205'755.22'

'24

'3.136.8

'1

'2)'(I

πθ

πθ

πθ

πθ

πθ

πθ

πθ

πθ

θθ

( )

4)'(I'cos43

)'(I tb

θθθ

+=

Crack opening area is the sum of areas due to tension and bending loads. A= At + Ab Where At and Ab are the crack opening areas due to tensile and bending loads respectively. Semi crack angle θ ' is modified to effective crack angle (θ'eff) to take into account the small plasticity effect.

2f

2

eff R2'K''σπ

θθ += ; where σf = (σy+σu)/

Assumptions of Tada-Paris Method: a) The formula has been derived for R / t ratio equal to 10. b) Estimation formula is expected to yield a slightly over estimate results for R / t near

10. c) For smaller R / t ratio, the degree of over estimation would increase. 2.2 (a) CRACK OPENING AREA CALCULATED FROM CRACK MOUTH OPENING DISPLACEMENT: The crack mouth opening displacement is commonly used to calculate the COA by assuming that the crack opens in to an elliptical shape. The area is calculated from the semi crack length c, and half crack mouth opening displacement δ, using the equation, A= π δ c

2.2 (b) CRACK OPENING AREA BY NUMERICAL INTEGRATION METHOD

( FEM APPROACH ) : In this method, COA is calculated from crack opening displacements (COD) along the crack length by Simpson’s 1/3 rule. Simpson’s 1/3 rule can be illustrated as follows: ∫ y.dx =h/3[y0 +4(y1+y3+y5+……yn-1)+2(y2+y4 +y6 +…+yn-2) +yn] Where, y (here, COD) =Function of x (here, circumferential crack angle position) Here y’s at different nodes has been evaluated using FEM approach [5,6] n =Even no. of sub intervals h =width of each sub interval. In the analysis, the width of each sub interval is taken as 2.5°. The total COA is obtained by multiplying the calculated area by a factor of 4, since only a quarter of cylinder had been modeled. If the crack opening were of exact elliptical shape, these two methods 2.2(a) and 2.2(b) would give identical results. However, crack opening is sometimes not of elliptical shape. In that case, these results will not match. A constant K is defined as, COA (by FEM APPROACH) =K.δ c If the crack opening is of elliptical shape, K will be equal to π. Hence more the ‘K’ deviates from π, the more the crack opening deviates from elliptical shape. It is seen that for smaller angles ‘K’ deviates more from π, indicating deviation from elliptical shaped crack opening. 2.3 COA BASED ON BARTHOLOME et. al. MODEL : In general, the method is founded on well known relationship between the displacement of the crack surface due to external load and by taking into account the plastic zone at the crack tip. The leakage area is the integral of the crack opening displacement V (x) along the crack front x, where 0 < x < (c+p):

∫=c

0dx)x(V4)c2(A

Applying a plastic zone correction on the crack tip according to Dugdale model [4 ] the crack opening displacements have been given by Bartholome et al [2,3] in a general form for the two relevant conditions: Under loading (maximum displacement and area)

[ ]

+

==

−++

=++

=

pcxcosarc;lengthcrackresultingtheonpositionx

x)pc(G4

)k1(sin)pc(G4

)k1()x(V 2122

φ

σφσ

After loading (minimum displacement and area)

c1cos

1p;2

;)1(2

EG

conditionstrainplanefor43k;conditionstressplanefor13kwhere

sinsinsinsinlncos

)sin()sin(lncos)pc(

G4)k1()x(V

f

f

−==

+=

−=+−

=

−+

+

+−

++

=

θσσπ

θν

ννν

φθφθ

θφθφθ

φσπ

The leakage areas in cylindrical components (pipes and vessels) are estimated using the bulging functions α(λ), which takes into account the curvature of the cylinder and crack orientations:

tRc)1(12where

cracksaxialfor)16.01.01()(

cracksntialcircumferefor)117.01()(

)c2(A)()c2(A

222

2

2

cyl

νλ

λλλ

λλα

λα

−=

++=

+=

=

α

EVALUATION OF THE INTEGRAL:

∫= + )(0 )(4)2( pc dxxVcA

dx)sin(sin)sin(sinlncos

)(sin)(sinlncos)pc(

G8)k1(4

2

2

2

2

f

)pc(

0

−+

+

+−

++

= ∫+

φθφθ

θφθφθ

φσπ

{ }

[ ]G8

)pc()k1(4mwheresay,IIm

d)sin(sin)sin(sinlnsincosd

)(sin)(sinlnsincos

G8)pc()k1(4

dsin)pc()sin(sin)sin(sinlncos

)(sin)(sinlncos

G8)pc()k1(4

)c2(A

0pcpccos)pc(xatand

2pc0cos0xatAlso

dsin)pc(dxso,cos)pc(xasNow

2f

21

0

2

0

22

2

2

22f

0

22

2

2

2f

11

πσ

φφθφθ

φθφφθφθ

φφπ

σ

φφφθφθ

θφθφθ

φπ

σ

φπ

φ

φφφ

π π

π

++=+−

−+

+

+−++−

=

+−

−+

+

+−++

=

=

++

=+==

+

==

+−=+=

∫ ∫

∫

−−

φφ

φθφθ

φφθφθφ

φθφθ

φθφθ

φφθφθ

φφφθφθ

φφ

ππ

ππ

d2

2cos)sin()sin(

dd

)sin()sin(

22cos

)sin()sin(ln

21I

functionfirstas)sin()sin(lntakingpartsbygIntegratin

d)sin()sin(ln2sind

)(sin)(sinlnsincosI

separatelyIandIcalculateweNow

0

2

0

2

2

1

0

2

0

22

2

1

21

−

+−

−+

−

−

+−

=

+−

+−

=

+−

=

∫

∫∫

{ }

−

+−=

−

+−−

−=

−+−

−=−

−=

−+−

−++−−=

−

++−−−+−

−+

−=

∫ ∫∫ ∫

∫∫

∫

∫

0

2

0

2

0

2

0

2

0

2

0

2

0

2

2

0

21

)2cos2(cosd2cos

d2sin)2cos2(cos

d2cosd

)2cos2(cos)2cos2(cos2sin

d)2cos2(cos

)2cos2cos2(cos2sin)2cos2(cos

d2cos2sin

d2cos)sin()sin(2

)sin(

d2

2cos)(sin

)cos()sin()cos()sin()sin()sin(I

so,zerobetooutturnsbracketsfirsttheLimitstheputtingBy

π ππ π

ππ

π

π

θφφθ

φθθφ

φθφ

θφθφ

θ

φθφ

θθφθ

θφφφ

θ

φφφθφθ

φθφθ

φφ

φθφθφθφθφθ

φθφθ

−

+−−=

−

+−−=

−

+−−=

+−−

+−−=

−−−

+−−=

+−−

+−−=

∫

∫

∫

∫

∫

∫

0

222

22

0

222

22

0

22222

0

222

0

22222

0

22222

tantandsec

2sec2cos

22sin

)tan(tandsecsec

22cos

22sin

d)cos2sinsin2(cos

12cos2

2sin

)2cos1(sin)2cos1(cosd2cos

22sin

d)sin2coscos2cossin(cos

12cos2

2sin

)sin(cos2cossincosd2cos

22sin

π

π

π

π

π

π

φθφφθθπ

θ

φθφθφθπ

θ

φθφθφ

θπ

θ

θφθφφ

θπ

θ

φφθφθφφ

θπ

θ

φφθφφφ

θπ

θ

)1.....(..............................2sin22

2sinI

,sozerotovanishesttanttanlnitslimputtingBy

xaxaln

a21

xadxthatknowweAs

ttanttanln

tan21

21

2sec2cos

22sin

)t(tantan222

1

2

22

022

θππ

θ

θθ

θθ

θθθπ

θ

θθ

=

−−=

−+

−+

=−

−+

+−−=

−

∫

∞

∞

)t(tanln1sec2cos2sin

t)(tandt

2sec2cos

22sinI

,sodtdsecthatso,ttanputtingbyNow

02

0

22

2

1

2

θθθπθ

θθθπ

θ

φφφ

+

+−−=

−

+−−=

==

∫∞

,dsin)sin(sin)sin(sinlncos2d

)sin(sin)sin(sinlnsincosI

IcalculatetohaveweNow

0

2

0

22

2

2

,2

φφφθφθ

θφφθφθ

φθππ ∫∫

−+

=

−+

=

,getwefunctionfirstas)sin(sin)sin(sinlntakingpartsbygIntegratin

−+

φθφθ

−−

−+−−+−

−

−

−+

=

∫0

22

0

22

d)cos()sin(sin

)cos)(sin(sincos)sin(sinsinsinsinsin

)cos()sin(sin)sin(sinln

cos2I

π

π

φφφθ

φφθφφθφθφθ

φφθφθ

θ

,so2)2cos2(cos

2cosalso)2cos2(cos

d)2cos2(cos

d2cos2sin2

)2cos2(cosd)12(cos

2sin2)2cos2(cos

d)11cos2(2sin2

)2cos2(cosdcos22sin2

)2cos1()2cos1(dcos22sin2

)sin(sindcossincos4

)sin)(sinsin(sindcoscossin2cos2

d)cos()sin(sin

cossincossincossincossin)sin(sin

1cos2I

so,vanishestermfirsttheLimitsputtingBy

0

2

0

2

0

2

0

2

0

2

2

0

2

20

2

2

0

222

20

2

0

22

πθφ

φθφ

φθφ

φφθ

θφφφ

θθφ

φφθ

θφφφ

θφθ

φφθ

φθφφ

θθφθφθ

φφφθθ

φφφθ

φφφθφφφθφθ

θ

ππ π

ππ

ππ

ππ

π

−=−

−

−−

=

−−

=−

−+=

−=

−−−=

−=

−+

=

−

−++−

+−=

∫∫ ∫

∫∫

∫∫

∫∫

∫

+−−

+−=

+−−

+−=

−

−−=

∫

∫

∫

2

0 22

2

0 2222

0

22

)2cos1(sin)2cos1(cosd

22sin2

)sin(cos2cos)sin(cosd

22sin2

)2cos2(cosd

22sin2I

π

π

π

θφθφφπ

θ

φφθφφφπ

θ

θφφπ

θ

−

+−=

−

+−=

∫

∫

2

0 22

22

2

0 2222

tantandsecsec

21

22sin2

cossinsincosd

21

22sin2

π

π

φθφθφπ

θ

θφθφφπ

θ

)2........(........................................2sin2

2sin2I

,sovanishesIntegralthethatearlierseenhaveweAs

2 θππ

θ −=

−=

( )

G42sin)pc()k1(

areaopeningcrackfinally

G42sin)pc()k1(

2sin21

G8)pc()k1(4

)2sin(2sin2G8

)pc()k1(4II

G8)pc()k1(4

)c2(A,So

2f

2f

2f

2f

21

2f

θσ

θσθπ

πσ

θπθπ

πσ

πσ

++=

++=

−

++−=

−+

++−=+

++−=

2.4 INPUT PARAMETERS FOR VARIOUS PHT STRAIGHT PIPES AND ELBOWS: The input parameters for various PHT straight pipes and Elbows that has been used to calculate crack opening area and consequently leak rate based on these values of crack opening areas are given in the tables (2.1) and (2.2). These specifications are for 500 MWe Indian PHWR.

Table (2.1) Input parameters for various straight PHT pipes.

Parameters Steam generator outlet

Pump discharge line

Pressure tube

Mean radius ( r m ) in mm

280.00 211.0 41.25

Pipe thickness (t) in mm

50.00 35.0 4.00

Internal pressure ( P ) in MPa

9.51 11.4 10.06

Bending moment ( M ) in KN-m

564.00 242.0 0.00

Young's Modulus ( E ) in GPa

179.00 179.0 89.60

Flow stress ( σf ) in MPa

349.00 349.0 377.00

Table (2.2) Input parameters for various PHT elbows

Parameters Steam generator outlet

Pump discharge line

Steam generator inlet

Mean radius ( r m ) in mm

280.00 211.00 234.00

Pipe thickness (t) in mm

50.00 35.00 40.00

Internal pressure ( P ) in MPa

9.51 11.40 9.81

Bending moment ( M ) in KN-m

417.00 102.00 251.00

Young's Modulus ( E ) in GPa

179.00 179.00 179.00

Flow stress (σf) in MPa

349.00 349.00 349.00

2.5 TABLES SHOWING COA USING DIFFERENT METHODS: Tables (2.3) to (2.8) show the leakage areas as evaluated by different methods for various straight pipes and elbows for the input parameters given in tables (2.1) and (2.2).The geometry of circumferentially cracked straight pipe and elbow is as shown in figure (1.1) and (1.2) respectively. In the following tables crack opening areas have been evaluated using Tada-Paris method [1] and Bartholome et al model [2,3]. The data of crack opening area based on FEM approach has been taken from references [5] and [6] for comparison with the above two methods. In all these calculations circumferential crack has been assumed. For pressure tube, more probable type of crack is axial, but circumferential crack has been assumed for COA evaluation only to compare the results obtained by Bartholom'e model and Tada-Paris method at R/t = 10 as pressure tube has a R/t nearly equal to 10.

Table (2.3) Steam generator outlet straight pipe crack opening area

CRACK OPENING AREA ( mm2 )

FEM APPROACH

BARTHOLOME et al

MODEL

TADA PARIS

METHOD

SEMI-

CRACK ANGLE

(θ degree)

INSIDE

OUTSIDE

INSIDE

OUTSIDE MEAN COA

5

0.96

1.52

1.29

1.51

2.38

15

11.12

17.23

12.30

15.30

24.39

25

39.09

61.52

38.20

47.50

78.26

35

88.13

145.00

84.00

106.00

177.10

Table (2.4) Pump discharge line straight pipe crack opening area

CRACK OPENING AREA ( mm2 )

FEM APPROACH

BARTHOLOME MODEL

TADA-PARIS METHOD

SEMI-CRACK ANGLE (θ deg)

INSIDE

OUTSIDE

INSIDE

OUTSIDE

MEAN COA

5 0.50 0.75 0.86 1.05 1.12

15 5.81 9.35 8.17 10.24 11.46

25 18.95 31.40 25.50 32.01 36.88

35 46.18 71.93 56.90 71.17 83.86

45 100.68 154.02 107.10 136.30 160.79

Table. (2.5) Steam generator outlet elbow crack opening area

CRACK OPENING AREA ( mm2 ) FEM APPROACH

BARTHOLOME MODEL

TADA-PARIS METHOD

CRACK ANGLE (2 θ deg)

INSIDE

OUTSIDE

INSIDE

OUTSIDE

MEAN COA

18 0.31 10.03 3.52 4.5 6.33

36 10.51 34.36 14.90 18.8 28.68

54 34.44 75.99 36.80 47.8 73.60

72 72.78 146.6 73.00 95.8 149.20

Table (2.6) Pump discharge line elbow crack opening area


BARTHOLOME MODEL

TADA-PARIS METHOD

CRACK ANGLE (2 θ deg )

INSIDE

OUTSIDE

INSIDE

OUTSIDE

MEAN COA

18 0.00 7.6 1.72 2.50 2.96

36 5.78 23.87 7.44 10.30 13.49

54 20.77 49.37 18.51 25.90 34.87

72 48.11 91.03 36.73 52.00 71.32

Table (2.7) Steam generator inlet elbow crack opening area.


BARTHOLOME MODEL

TADA-PARIS METHOD

CRACK ANGLE (2 θ deg )

INSIDE

OUTSIDE

INSIDE

OUTSIDE

MEAN COA

18 0.00 8.11 2.60 3.50 4.76

36 6.98 23.13 11.40 14.50 21.58

54 24.32 55.59 28.30 36.20 55.39

72 56.44 105.20 56.20 73.00 112.29

Table (2.8). Pressure tube crack opening area

CRACK OPENING AREA BASED ON MEAN RADIUS (cm2 )

SEMI CRACK ANGLE (θ) ( deg )

TADA PARIS METHOD

BARTHOLOME MODEL

5

0.00048

0.0005

10

0.00208

0.0021

15

0.00506

0.0050

20

0.00977

0.0096

25

0.01666

0.0162

30

0.02625

0.0254

35

0.03919

0.0377

2.6 GRAPHS SHOWING VARIATION OF COA WITH CIRCUMFERENTIAL SEMI-

CRACK ANGLE FOR VARIOUS PHT STRAIGHT PIPES: Using the values of crack opening areas as given in tables (2.3) and (2.4) various graphs have been plotted to show the variation of crack opening area with circumferential semi crack angle for various primary heat transport system straight pipes. These graphs are plotted as shown below in fig (2.1) to (2.4).

5 10 15 20 25 30 35-20

0

20

40

60

80

100

120

140

160

180

200

M ean radius=280 m m , P ipe thickness=50 m m , Internal pressure=9.51 M P aB ending M om ent=564 K N m , Y oung's M odulus=179 G P a,

F ig.2.1 V ariation of inside C rack opening area w ith sem i crack angle for S G O straight pipe

F E M A pproach B artholom e et al. M odel T ada-P aris M odel

Crack

ope

ning

area ( m

m

2 )

S em i crack angle ( degree )

5 10 15 20 25 30 35-20

0

20

40

60

80

100

120

140

160

180

200

M ean radius= 280 m m , P ipe thickness= 50 m m , Internal pr.= 9.51 M P aB ending M om ent= 564.0 K N -m , Y oung's M odulus= 179.0 G P a

F ig 2.2 V ariation of outside crack opening area w ith sem i crack angle for S G O straight pipe

F E M A pproach B artholom e m odel T ada P aris M ethod

Crack

ope

ning

area (m

m

2 )

S em i crack angle (degree)

0 10 20 30 40 50-20

0

20

40

60

80

100

120

140

160

180

M ean radius=211.0 m m , P ipe thickness= 35.0 m m , Internal pr.=11.4 M P a,B ending M om ent= 242.0 K N -m , Y oung's M odulus=179.0 G P a.

F ig.2. 3 V ariation of inside crack opening area w ith sem i crack angle for P D L straight pipe

F E M A pproach B artholom e M odel T ada P aris M ethod

Crac

k op

ening area

(mm

2 )

S em i crack angle (degree)

0 10 20 30 40 50

0

50

100

150M ean radius= 211 m m , P ipe thickness= 35 m m , Internal pr.= 11.4 M P a,B ending M om ent= 242.0 K N -m , Y oung's m odulus= 179.0 G P a.

F ig. 2.4 V ariation of outside crack openeng area w ith sem i crack angle for P D L straight pipe

F E M A pproach B artholom e et al M odel T ada P aris M ethod

Crack

ope

ning

area (m

m

2 )

S em i crack angle(degree)

2.7 VARIATION OF COA WITH CIRCUMFERENTIAL CRACK ANGLE FOR

VARIOUS PHT ELBOWS : Figures (2.5) to (2.10) show variation of crack opening area with crack angle using the data of crack opening area as given in tables (2.5), (2.6) and (2.7) for SGO, PDL and SGI elbows respectively. All these figures are for circumferentially cracked elbows.

10 20 30 40 50 60 70 80

0

20

40

60

80

100

120

140

160M ean radius= 280 m m , P ipe thickness= 50 m m , Internal pr.= 9.51 M P a,B ending M om ent= 417.0 K N -m , Y oung's M odulus= 179.0 G P a.

F ig 2.5 V ariation of inside crack opening area w ith crack angle for S G O E lbow

F E M A pproach B artholom e et al M odel T ada-P aris M ethod

Crack

ope

ning

area (m

m2 )

C rack angle (degree)

10 20 30 40 50 60 70 80

0

20

40

60

80

100

120

140

160 Mean radius=280 mm, Pipe thickness=50 mm, Internal Pr.=9.51 MPa,Bending moment=417.0 KN-m, Young's modulus=179.0 GPa.

Fig 2.6 Variation of outside crack opening area with crack angle for SGO Elbow

FEM Approach Bartholome et al. Model Tada-Paris Method

Cra

ck o

peni

ng a

rea

(mm

2 )

Crack angle (degree)

10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

M ean radius = 211 m m , P ipe thickness= 35 m m , Internal P r. = 11.40 M P a,B ending M om ent= 102.0 K N -m , Y oung's m odulus= 179.0 G P a.

F ig 2.7 V ariation of inside crack opening area w ith crack angle for P D L E lbow

F E M A pproachB artholom e et al. M odel T ada-P aris M ethod

Crack

ope

ning

area (m

m2 )


10 20 30 40 50 60 70 80

0

20

40

60

80

100

M ean radius= 211 m m , P ipe thickness= 35 m m , Internal pr.= 11.40 M P a,B ending M om ent= 102.0 K N -m , Y oung's M odulus= 179.0 G P a.

F ig 2.8 V ariation of outside crack opening area w ith crack angle for P D L E lbow

F E M A pproach B artholom e et al M odel T ada-P aris M ethod

Crack

ope

ning

area (m

m2 )


10 20 30 40 50 60 70 80

0

10

20

30

40

50

60

70

80

90

100

110

120 Mean radius =234 mm, Pipe thickness=40 mm, Internal pr.=9.81 MPa,Bending Moment=251.0KN-m, Young's modulus =179.0 GPa.

Fig 2.9 Varition of inside crack opening area with crack angle for SGI Elbow

FEM Approach Bartholome et al.Model Tada-Paris Method


2 )

ng a

rea

(mm

o

peni

k

C

rac

10 20 30 40 50 60 70 80

0

20

40

60

80

100

120 Mean radius=234 mm, Pipe thickness=40 mm, Internal pr.=9.81 MPa,Bending Moment=251.0 KN-m, Young's Modulus= 179.0 GPa.

Fig 2.10 Variation of outside crack opening area with crack angle for SGI Elbow

FEM Approach Bartholome et al. Model Tada-Paris Method

Cra

ck o

peni

ng a

rea

(mm

2 )


2.8 VARIATION OF COA WITH CIRCUMFERENTIAL SEMI CRACK ANGLE FOR PRESSURE TUBE: Figure (2.11) shows variation of crack opening area with circumferential semi-crack angle for a pressure tube. For pressure tube ratio of mean radius to thickness is nearly 10. As Tada Paris method has been derived mainly for R/t ratio of 10 and results as calculated using this method approaches that of other methods as bending moment goes on decreasing.In this case bending moment has been assumed to be zero. So as seen from the graph that the two curves are very close for this pipe.

5 10 15 20 25 30 35

0.00

0.01

0.02

0.03

0.04 Mean radius=41.25 mm, Pipe thickness=4 mm, Internal pr.=10.06 MPa,Bending Moment=0.00, Young's modulus=89.6 GPa.

Fig 2.11 Variation of crack opening area with semi crack angle based on mean radius for Pressure Tube

Tada-Paris Method Bartholome et al. Model

Cra

ck o

peni

ng a

rea

(mm

2 )

Semi crack angle angle (degree)

3.0 LEAKAGE RATE BASED ON HENRY'S MODEL For the successful application of LBB methodology, an accurate estimation of leak rate through cracked pipes is required. This leak rate is used for determining the limited crack size. Fracture mechanics analyses are then performed to prove that the crack remains stable without pipe rupture under the given loads and material conditions. 3.1 HENRY’S MODEL: In Henry's model (7), thermodynamic non-equilibrium effects are introduced in an expression of non-equilibrium vapour generation rate that contains an empirical parameter N, which is a function of the equilibrium quality and the flow path length / hydraulic diameter ratio. The total pressure drop along the flow path through the crack is the sum of the pressure drop component as )1.......(..........PPPPPP kaaafetot ∆+∆+∆+∆+∆=∆

Where 2d

lo2C

e C2vG

P =∆ …………(2)

is the entrance pressure loss term. The discharge coefficient between 0.61 to 0.95 is chosen based on the judgement of the user as to how round the entrance edges are in comparison to the crack opening displacement. The friction pressure drop is obtained by separately integrating the Darch-Weisbach expression of the friction factor between the liquid region and the two-phase region, which results in

[ ] )3.........(..........)vv(xvG2

12d/LfvGf

212P lgl

2C

Hlo

2Cf −+

−+=∆

With the assumption of the flashing location at L / dH=12 and the friction factor f calculated from modified Karman correlation.

ƒ = 2

H

174.1

K2d

log2−

+

)](x[GP logcc2

a ν−ν=∆ ……………….(4) ∆Paa and ∆PK are the pressure drops due to the area change and due to protrusions in the crack length. Considering all these losses, the critical pressure is obtained as PC = P0 -∆Ptot …………………(5) In accordance with Henry's homogeneous non-equilibrium model, critical mass flux is given by

GC2 = 1

t

Elog

C

g

dPdx

N)vv(Pv

x −−−

γ

……………(6)

where subscript t denotes throat quantity

Where equilibrium quality is calculated as,

Elg

l0E ss

ss

−−

=x

And N = 20 x if xE E < 0.05 and N = 1 if xE ≥ 0.05 For the given stagnation conditions and crack geometry, the critical mass flux can be calculated by the iterative solution of eq (5) and eq (6). The values of GC and PC are assumed to be correct when the relative errors between the successive iterations are less than 10-4.

3.2 TABLES SHOWING LEAK RATES: Based on crack opening areas as given in tables (2.3) to (2.8) leak rates have been evaluated by computer code formulated using Henry's homogeneous non-equilibrium model and shown in the tables from (3.1) to (3.6).

TABLE (3.1) SGO STRAIGHT PIPE LEAK RATE

LEAK RATE USING HENRY'S MODEL ( Kg/sec ) SEMI CRACK ANGLE

( Degrees ) LEAK RATE

USING COA BY FEM APPROACH

LEAK RATE USING COA BY

BARTHOLOME et al MODEL

LEAK RATE USING COA BY TADA- PARIS

METHOD 5 0.02264 0.03745 0 .0874

15 0.3516 0.41109 0.9799

25 1.5786 1.5442 3.4761

35 3.8614 3.6488 8.1745

TABLE (3.2) PDL STRAIGHT PIPE LEAK RATE

LEAK RATE USING HENRY'S MODEL (Kg/ sec) SEMI CRACK ANGLE ( Degrees )

LEAK RATE USING COA BY FEM APPROACH

LEAK RATE USING COA BY BARTHOLOME et al MODEL

LEAK RATE USING COA BY TADA-PARIS METHOD

5 0.01132 0.02767 0.03895

15 0.17447 0.2859 0.4444

25 0.7519 1.0798 1.5789

35 2.0311 2.5773 3.8878

45 4.6739 5.0076 7.6493

TABLE (3.3) LEAK RATE PRESSURE TUBE

LEAK RATE USING HENRY'S MODEL IN (kg / sec) SEMI-CRACK ANGLE ( DEGREES )

USING COA BY TADA-PARIS METHOD


5 0.00068 0.00071

10 0.00524 0.00530

15 0.01780 0.01743

20 0.04394 0.04268

25 0.09091 0.08733

30 0.17190 0.16280

35 0.30900 0.29600

TABLE (3.4) SGI ELBOW LEAK RATE

LEAK RATE USING HENRY'S MODEL IN ( kg / sec ) CRACK ANGLE ( DEGREES) LEAK RATE



LEAK RATE USING COA BY TADA PARIS METHOD

18 0.0000 0.0950 0.2011

36 0.1757 0.3926 0.8778

54 0.9469 1.1415 2.4553

72 2.4510 2.4391 5.1635

TABLE (3.5) SGO ELBOW LEAK RATE

LEAK RATE USING HENRY'S MODEL (kg/sec) CRACK ANGLE ( Degrees ) LEAK RATE



LEAK RATE USING COA BY TADA PARIS METHOD

18 0.0000 0.1254 0.2528

36 0.2779 0.4930 1.1464

54 1.3336 1.4450 3.2193

72 3.5380 3.5380 6.7913

TABLE (3.6) PDL ELBOW LEAK RATE

LEAK RATE USING HENRY'S MODEL CRACK ANGLE

( Degrees ) LEAK RATE



BARTHOLOME et al MODEL


TADA PARIS METHOD (kg/sec)

18 0.0000 0.05859 0.11670

36 0.15047 0.23200 0.52030

54 0.8262 0.71340 1.51720

72 2.1197 1.55490 3.26200

3.3 LEAK RATES Vs CIRCUMFERENTIAL CRACK ANGLE FOR VARIOUS PHT STRAIGHT PIPES AND ELBOWS USING HYENRY'S NON-EQUILIBRIUM MODEL: Figure (3.1) to (3.6) show variation of leak rates with circumferential crack angle for various PHT straight pipes and elbows using data as given in tables (3.1) to (3.6).

10 20 30 40 50 60 70-1

0

1

2

3

4

5

6

7

8

9

Leak detection capability

Fig.3.1 Variation of Leak rate with circum ferential crack angle for SGO straight pipe

Surface Roughness=18.53E-6 cm, Equilibrium Qualit y= -0.09 B ased on COA by FEM approach Based on COA b y Bartholome et al. Model Based o n COA by Tad a-Paris Method

Leak

rate

( Kg

/ sec

)

Circumferential crack angle ( degree )

0 20 40 60 80 100-1

0

1

2

3

4

5

6

7

8


Surface roughness=18.53E-6 cm, Equilibrium quality= -0.09

Fig.3.2 Variation of leak rate with circumferential crack angle for PDL straight pipe

Base on COA by FEM approachBased on COA by Bartholome et al. ModelBased on COA by Tada-Paris Method


)

ec

g/s

(k

e

at

r

Leak

0 0 0 0 0 0 0 0

5 10 15 20 25 30 35

.00

.05

.10

.15

.20

.25

.30

.35

Surface roughness = 8.53E-6 cm, Equilibrium quality = -0.09

Fig.3.3 Variation of leak rate with semi crack angle for Pressure Tube

Based on COA by Bartholome et al. Model Based on COA by Tada-Paris Method

Leak

rate

(kg/

sec)

Semi crack angle (degree)

0 10 20 30 40 50 60 70 80-1

0

1

2

3

4

5

6

7


Fig.3.4 Variation of leak rate with circumferential crack angle for SGO elbow

Surface roughness= 18.53E-6 cm, Equilibrium quality= -0.09Based on COA by FEM approachBased on COA by Bartholome et al. ModelBased on COA by Tada-Paris Method

Leak

rate

( Kg

/ sec

)


10 20 30 40 50 60 70 80-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5


F ig 3.5 V ariation of leak rate w ith circum ferential crack angle for P D L elbow

S urface roughness= 18.53E -6 cm , E quilibrium quality= -0.09B ased on C O A by F E M approachB ase on C O A by B artholom e et al. M odelB ased on C O A by T ada-P aris M ethod

Leak

rate ( Kg / s

ec )

C ircum ferential crack angle ( K g / sec )

10 20 30 40 50 60 70 80-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Leak dection capability

S urface roughness = 18.53E -6 cm ., E quillibrium quality = -0.09

F ig .3.6 V ariation of leak rate w ith circum ferential crack angle for S G I elbow

B ased on C O A by F E M A pproach B ased on C O A by B artholom e et al M odel B ased on C O A by T ada-P aris M ethod

Leak rate (Kg/sec)

C ircum ferential crack angle (degree)

4.0 DISCUSSION AND CONCLUSIONS: Figures (2.1) to (2.10) show the variation of crack opening area with circumferential crack angle or semi crack angle for various PHT straight pipes and elbows. Observing these graphs it is interpreted that crack opening areas as calculated by FEM approach method and Bartholome et.al. model match well within limits or in other words the difference is not wide. On the other hand COA as calculated by Tada Paris equation overestimates rather largely. The overestimation grows with increase in crack opening length and bending moment. Tada- Paris equation has been derived taking R/t = 10 and it also overestimates the result with pipes having R/t ratio less than 10. Figure (2.11) shows the variation of COA with circumferential semi-crack angle for pressure tube, where R/t= 10 and also bending moment is taken as zero and it is observed that the results of Tada Paris and Bartholome et.al. model match very closely. Similarly figure (3.1) to (3.6) show the variation of leak rate with circumferential crack angle using Henry’s non-equilibrium model. In these graphs leak rates have been evaluated using leakage areas by three different methods i.e. FEM approach, Bartholome et. al. model and Tada-Paris model. It is observed from these graphs that leakage rate increases with circumferential crack angle in parabolic manner. Also it is observed that leakage rates as calculated using COA by FEM approach and Bartholome et.al. Model are in agreement. It is also observed that leak rate as calculated using COA by Tada-Paris method overestimates the results due to the reasons mentioned above. The leakage size crack is determined for different PHT straight pipes and elbows as shown in these figures.

REFERENCES

(1) P.C.Paris and H.Tada (1983): The application of Fracture Proof Design Meth- od using tearing instability theory to nuclear piping postulating circumferential throughwall crack", NUREG-CR-3464.

(2) G. Bartholome, W. Kastner, E.Keim and G. Senski (1993): "LBB analysis, Verification of Leakage Area and Leakage Rate evaluation by tests", 12th SmiRT, paper G06 / 1, Stuttgart, Germany.

(3) G. Bartholome, W. Kastner, E.Keim (1993): "Design and calibration of Leak

Detection Systems by thermal hydraulics and Fracture Mechanics analyses", Nuclear Engg and Design, 142, 1-13.

(4) D.S. Dugdale (1960): "Yielding of steel sheets containing slits", J. Mech. Physics

Solids, vol. 8, pp100-104.

(5) J. Chattopadhyay, B.K.Dutta and H.S.Kushwaha (1997): "Leak Before Break qualification of 500 MWe PHWR PHT straight pipes by J-Integral- Tearing Modulus and Limit Load Method". BARC external report, BARC / 1997/ E/ 017.

(6) J. Chattopadhyay, B.K.Dutta and H.S.Kushwaha, "Leak Before Break qualification of primary heat transport elbows of 500 MWe Tarapur Atomic Power Plant". BARC external report, BARC/ 1998/ E/ 0

(7) Henry. R.E (1970),"The two phase critical discharge of initially saturated or subcooled liquid", Nuclear Science and Engineering, vol. 41, pp 336-342.

(8) E Keim," Determination of Leakage Areas in Nuclear piping", LBB-95, NE-

OECD-IAEA-NRC, Specialist meeting on Leak Before Break in Reactor Piping and vessels, Lyon -France, Oct 9-11.

(9) C. Maricchiolo and P.P. Milella (1989): "Prediction of Leak Areas and

experimental verification on carbon and stainless steel pipes", Nuclear Engg. and Design, vol-111, pp 47-54.

evaluation of crack opening area and leak rate in …

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