- Email: [email protected]
Validity of water leak rate prediction methods
Validity of water methods
leak rate prediction
L. Friedel and F. Westphal* Technische Pruefung, Hoe&t AG, D-6230 Frankfurt/M 80, FRG *Arbeitsgruppe Sicherheitstechnik, Universitaet Dortmund, D-4600 Dortmund FRG
SO,
Models for the calculation of leakage rates through cracks in vessel walls and pipes due to overpressure are analysad. For evaluation of predictive quality, the models were applied to all the accessible (water) leakage rate data for real and model cracks. The statistical description of the differences between model predictions and experimental results reveals that none of the models can be considered universally valid, even for experiments with water, when specific data from different sources are compared. (Keywords:
pressure and leak testing:
modelling;
In the case of a failure in vessel and pipe walls, (caused by fatigue loading from bending moments or internal pressure) the assumption of a leak-before-break behaviour of the material is generally justified, particularly concerning the highly developed state of design technology and material science. Essentially, this leakbefore-break concept holds even for crack propagation in cases with superimposed corrosion effects. However, it is sometimes not possible to exclude, in the event of heavy stress corrosion, the potential occurrence of axisymmetrical circumferential breaks. The application of the leak-before-break concept to such devices exhibiting corrosive potential requires, by way of supervision, a specially-designed in-service programme’. Fracture mechanics theories reveal that, in the case of leak-before-break behaviour, a realistic approximation to the expected leak size is a slit-shaped crack with a very small hydraulic diameter. For the calculation of discharge rates through these cracks, such as is necessary for risk analysis, only methods developed from the nuclear power industry are accessible. These so-called leak rate models are generally based on interrelated semi-empirical pressure drop and critical mass flux correlations. In this paper, empirical factors have been developed with only a few experiments using subcooled water through, in most cases, model cracks within limited test parameter ranges. From experience, an extrapolation of these leak rate models to other geometries and especially to real crack configurations will result in unreliable predictions. First, an evaluation of leak rate prediction methods with a view to their general validity and predictive accuracy has been attempted. The methods are analysed Received 15 June I988
probability)
in detail with regard to the basic fluid, dynamic and thermodynamic assumptions, equation variables and original validity ranges. Next, a statistical description of the predictive accuracy of the models was made, on the basis of all accessible water/steam leak rate measurements from the literature and statistical error numbers. A relative order of rank followed, according to the individual merits of the correlations.
Leak rate models Nine calculation methods for leak rates through small cracks in vessel and pipe walls have been reported*-‘. All the models were developed for subcooled, hot water situated upstream of the crack, which will partially evaporate during outflow due to the pressure decay along the crack depth. In this case, the models consider a two-phase flow development was considered inside or at the exit of the crack and the limitation of the discharge rate by the self-adjusting critical mass flux. Calculation of critical mass flux requires prediction of the (critical) pressure in the exit plane, which can be estimated from the stagnation pressure and the predicted pressure drop. Leak rate models are, therefore, iterative calculation methods, requiring much computational before a point of intersection between pressure drop and mass flux characteristics from numerical mathematics is attained. An alternative model has been proposed3. This included an explicit calculation for mass flow quality, the pressure in the crack, and for the fluid properties on the basis of average values. An independent engineering prediction method for leakage rates should require only the stagnation state and fluid properties, the crack configurations and the
0950-4230/00/040213-00$3.00 Q 1988 Butterworth & Co. (Publishers) Ltd
J. Loss Prev.
Process
Ind.,
1988,
Vol
7, October
213
Validity
of water
leak rate prediction
methods:
L. Friedel
lation of the critical mass flux. Essentially, this mass flux model is an extended and modified Bernoulli equation. It assumes that the temperature of the subcooled fluid, is only slightly changed while flowing through the crack and that the flashing of the liquid will start just within the crack exit area. Thus, the critical pressure in the exit is set equal to the saturation pressure corresponding to the upstream stagnation temperature. The result is that this model produces physically wrong tendencies for low subcooled liquids, since a zero mass flux would be calculated for decreased subcooling. Therefore, it should be only used for stagnation states with large amounts of subcooling. The leak rate models from KfK and KWU are both based on the critical mass flux model of Pana’, which includes discharge modes as a function of the upstream fluid stagnation state. For the case of subcooled inlet flow, Pana provides two calculation schemes dependent on the extent of the initial subcooling. For high stagnation subcooling, a modified Bernoulli equation is applied. This differs from the original for subcritical single-phase flow only by the inclusion of the saturation pressure, corresponding to the stagnation temperature instead of the back pressure as downstream pressure. This follows on from the assumption that evaporation can only start, in this case, in the crack exit area, irrespective of the depth. For low stagnation subcooling, within the other scheme, it is presumed that the flashing point will be located in the crack and that it will move upstream towards the inlet with decreasing subcooling. The calculation of the actual critical mass flux results from a linear interpolation between the possible extreme cases of a single phase liquid flow through the crack, and a two-phase flow already at the crack inlet with respect to the temperature. In the latter case of a two-phase flow, the critical mass flux calculation may rely on the slip model developed by Moody lo or alterna-
back pressure as input parameters. In contrast, the three models of Amos et al. require specific entrance pressure losses and flow resistances as input variables, since no generally valid pressure drop prediction method is integrated. Another model6 is incomplete, because friction (as a function of the liquid flow Reynolds number), is also required. These models are, therefore, only suitable for post-calculation of experimental results. They are thus not included in the following analysis of the leak rate models’ qualities. The five remaining leak rate models differ with respect to the underlying (critical) mass flux models. Table 1 shows the assumptions about the location of flashing point, the beginning of two-phase flow inside or at the exit of the crack and the internally coupled correlations for the pressure drop across the leak in cases of liquid and liquid/two-phase flow. The leak rate models of Collier et al. and Abdollahian et al. (model I) rely on the critical mass flux model of Henry’, originally developed for the calculation of discharge rates of initially subcooled water through long pipes. Henry fixed the onset of flashing at a distance of 12 times the hydraulic diameter down stream of the crack inlet, in this way being independent in each flow condition from the initial subcooling. This criterion is now discredited for pipe flow by the work of Fauske & Associates Inc. on behalf of DIERS. Nevertheless, it has been retained here, since the original leak rate models were compared without any modification. In the section on two-phase flow development in the crack, Henry assumes a homogeneous flow, that is to say, there is no slip between gas and liquid phases. However, the model includes a thermodynamic non-equilibrium in the form of an empirical relationship for the decrease in local mass flow quality towards an equilibrium value. The Abdollahian er al. (model II) model relies on a simplified, and thus explicit, fluid dynamic (homogeneous) and thermodynamic equilibrium model for calcu-
Table 1
and E Westphal
Leak rate models assumptions Pressure drop
Leak rate model
Basic assumptions for twophase flow
Flashing point
R. F! Collier et el. (Leak-00) D. Abdollahian (model I)(Leak-011 D. Abdollahian ef al. (model II)
12d, after inlet 12dh after inlet Crack exit
V. Kefer er al. IKWU) H. John era/. (KfK)
Subcooling dependent Subcooling dependent
214
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1988,
Vol
Model for critical mas.s flux
Inlet pressure loss coefficient
Liquid flow friction factor
vol Karman (modified) van Karman
+
_
R. E. Henry
1.7
R. E. Henry
1.7
Explicit homogeneous equilibrium model P. Pana
1.7
van Karman
0.5
van Karman
P Pana
0.5
won Karman (modified)
1, October
Impact and form losses
+
Validity
of water
tively on a fluid dynamic equilibrium (homogeneous) flow model. The KWU and KfK models are based in this way on a homogeneous model, since it is easier to handle and is still considered to be adequate for the description of turbulent flow along the crack depth, and the relatively low fluid qualities. A complete or closed leak rate model necessitates a relationship for the prediction of the leak exit pressure and, hence, of the total pressure drop along the crack depth. Conceptually, this can be considered to consist of individual recoverable and irrecoverable, self-adjusting and partially superimposed pressure changes. For crack geometries, especially for the leak-before-break behaviour of the vessel or pipe wall, the total pressure drop is made up of inlet pressure losses resulting from the sharp-edged crack inlet planes; a recoverable pressure drop resulting from an acceleration of the intially stagnant fluid; friction pressure drops due to wall friction and momentum exchange between the phases; impact and form losses resulting from flow area and flow direction changes in the cracks; an acceleration pressure drop due to the inevitable decay in static pressure along the flow path; an overall acceleration or deceleration pressure change due to different geometric inlet and outlet leak area sizes and an exit pressure loss. The individual pressure drops occur in single-phase or two-phase flow conditions depending on the thermodynamic state of the inlet fluid. In the event of twophase flow, the prediction of the self-adjusting pressure changes is always based on the corresponding singlephase, liquid flow correlations, which have been corrected by the introduction of a two-phase mixture density according to a homogeneous model. The leak rate models include two constant entrance pressure loss coefficients for the calculation of inlet impact losses. Collier eC a/. and Abdollahian et al. introduce a factor of 1.7, while in the KfK and KWU models this coefficient is 0.5, as indicated for singlephase flow in sharp-edged inlet planes in the literature. The dependence on local velocity and thus the Reynolds number is not considered. On the whole, the contribution of the entrance losses to the total pressure drop is small, especially when two-phase flow occurs downstream in the crack. The (single-phase) friction pressure drop along the crack depths contributes most to the total pressure drop, particularly in the presence of wall roughnesses and small crack widths. All leak rate models include a friction pressure drop prediction method arbitrarily based on the same mathematical structure for the resistance coefficient, which originates from the Nikuradse-von Karman equation for friction during single-phase flow through sand grain roughened pipes. In some cases, this equation has been modified by re-adjusting the empirical factors to better fit the experimental data of a particular author or to overcome the effects of relative roughnesses in excess of unity. The impact losses due to sudden flow area changes in the crack cause losses resulting from turns in the flow path. Recoverable acceleration or deceleration pressure
leak rate prediction
methods:
1. Friedel
and 17 Westphel
changes due to different inlet and outlet crack area sizes would, on average, also contribute markedly to the total pressure drop. However, since only the models of Collier et al. and Abdollahian et al. (1) consider these, variables such as the number of turns or of area changes along the crack depths are required. Indeed, these details cannot be predicted in a general manner from fracture mechanics, so that any introduction of form and impact losses in the methods is, therefore, coupled with great uncertainties at the present time. In the models of Abdollahian et al. (model II), KWU and KfK form the impact losses and recoverable pressure changes resulting from locally changing flow cross-area sizes, which may have actually occurred in the experiments, are provisionally included in a so-called effective friction pressure loss. The possibility of an exit pressure loss occurring is not considered in the leak rate models. Indeed, it does not enter in the calculation as long as critical flow conditions in the exit plane are supposed. Overall, the mode1 of Abdollahian et al. (model 1) shows the most comprehensive pressure drop calculation method. An independent check of the predictive accuracy of the respective pressure drop sub-models is not yet possible. There are no reliable measurements for critical pressure in the exit plane of cracks and, therefore, even fewer for the individual pressure drop contributions in cases of two-phase flow. Hence in the leak rate models, an internally, incorrectly calculated critical pressure, in combination with an incorrect mass flux correlation, may finally result in a correct leak rate by compensating errors. The extrapolation of the correlations depends on the (complete) inclusion of the relevant, primary parameters as equation variables. In this respect, the analysis of the leak rate models reveals that none of the models are capable of accounting for a change of the liquid phase viscosity or wettability, and the surface tension. This is because the influence of all these properties on leakage rates is not yet established. In the measurements with subcooled/flashing water these properties could not be varied independently of each other. Therefore, an application of the leak rate models to other fluids may be restricted, as it is expected that these properties have a marked effect. Furthermore, the leak rate model predictions suffer from a lack of detailed knowledge on the influence of the thermodynamic non-equilibrium encountered before and during vaporization (and condensation). In view of the wall roughness and small crack width, there might only be a small boiling (and condensation) delay, since there would be several nucleation sites on the crack wall allowing for immediate evaporation. The fluid mean residence time in the crack is short, due to the high mean fluid velocities and the relatively small crack depths (wall thicknesses), so that a marked boiling (and condensation) delay could be encountered, clearly influencing the value of the leakage rate”. Studies on the discharge rate through long pipes with larger bores, show that the combined assumptions of a
J. Loss Prev. Process
Ind.,
7988,
Vol 1, October
215
Validity
of water
leak
rate prediction
methods:
L. Friedel
while John ef al. and Kefer et al. produced them by cyclic bending of a plane probe on a material testing machine. The specified crack widths are related to the crack outlet or to the outer edge of the vessel or pipe wall. The crack depths correspond, by definition, to the wall thickness of the apparatus or pipe. As a rule, the crack sizes in the inlet and exit planes are equal. Collier et al. and John et al. have also used cracks with different inlet and exit areas, which were obtained by incidental widening during the experiments due to the applied high pressure difference on the specimen. The wall roughnesses inside the model cracks depend on the way these cracks have been produced. Collier et al. claimed to have used model cracks with sand blasted wall roughnesses from 0.3 up to IOpm. In the same manner, John et al. produced wall roughnesses up to 150 pm, which are in some cases greater than the adjusted crack widths, so that the relative wall roughnesses can amount to values greater than unity. Amos el al. provide no information on the wall in their ‘smooth’ slits. The spark erosion cracks of Kefer el al. show of about randomly occurring wall roughnesses lo-30 pm. The authors’ statements about the wall roughnesses in real cracks are restricted to the material used, and to the way the crack has been generated. The documented values are in the range from 2 to 240 ,um. Generally, the measurements of the effective wall roughnesses of real cracks is subject to relatively large uncertainties. This is highly unsatisfactory, because here the wall roughness is a decisive parameter for pressure drop prediction. Another difficulty in assessing the influence of wall roughnesses on leak rates are the non-standard techniques and the uncertainty whether the peak, the mean, or any other roughnesses value is a characteristic physical parameter for self-adjusting critical mass flux and pressure drop. The respective total single-phase leak pressure drops are only reported in Refs. 4,5,7 (deduced from their
thermodynamic equilibrium between the phases and a homogeneous flow leads to acceptable predictions for the critical mass flow rate”. It is possible that both assumptions introduce errors, which tend to almost compensate each other. However, the assumption of a homogeneous (mechanical equilibrium) flow in cracks should be physically admissible, because the two-phase mixture generated is characterized by relatively low mass flow qualities and low slip between the phases. The analysis of the leak rate models with respect to the incorporated assumptions and parameters gives an indication of theoretical deficiencies, and allows an early prediction of accuracy for a particular model. Hence, a statistical analysis with all accessible water leak rate measurements has been carried out additionally to evaluate the relative models’ qualities when applied to data in a different and a wider range of real and model cracks.
Data and statistical examination predictive accuracy
of the
The statistical investigation of the predictive accuracy is based on nearly 1000, now accessible, leak rate data from Collier et of., Amos et al., John et al. (KfK) and Kefer et al. (KWU). The parameter range limits of the experimental data and the geometrical sizes of the leaks are listed in Table 2. The measurements are restricted to the case of subcooled hot water upstream of the crack, which will partly evaporate during outflow, thus generating a two-phase flow already inside the crack or at the outlet. The experiments were carried out on rectangular slits as model cracks and on artificially produced slits, so-called ‘real’ cracks. The model cracks’*‘*’ are made up of two plane rectangular steel blocks arranged to allow for various discrete slit widths, while Kefer et al. produced model cracks by spark erosion in steel slabs. The real cracks of Collier et al. are obtained after intergranular stress corrosion in stainless steel pipes,
Table 2
Parameter
ranges of leak rate experiments
and E Westphal
with initially subcooled water
Model cracks
Collier et al.
0.2-1.12 63.5 57 0.3-10
Crack width (mm) Crack length (mm) Crack depth (mm) Roughness (cm) Hydr. diameter Imm) Exit area (mm2) Exit/inlet area ratio Relative roughness Total pressure drop coefficient
216
Amos et al.
Kefer et a/. (KWU)
Collier et al.
John et al.
Kefer et al.
25-161 O-65 6-74
39-122 2-61 6-45
20-l 61 O-66 12-69
35-95 O-72 4-53
40-l 21 2-61 5-22
20-l 56 O-58 9-39
33-l 15 33-120 19-62
Pressure (bar) Subcooling (I<) Mass flux (1 O3 kg/m’s)
0.4-2.2 1 l-64 1 0.002-0.05
J. Loss Prev.
0.127-0.381 15-21 64
0.09-0.4 80 46 5-l 50
0.31-0.325 19-40 10-30 13.9- 30
0.16-0.75 3-8 1
0.18-0.8 20-46 l-l .9 0.01-1.7 3-38
0.61-0.65 6-13
3-72
Process
Real cracks
John et al. fKfK)
Ind.,
1988,
0.04- 0.1 l-4
Vof 1, October
0.04-0.025 0.7-28 19 2-5 0.08-0.5 0.02-6 0.04-0.2 0.007-O. _
1
0.2-0.4 80 46 240
0.097-O.
0.4-0.8 35-51
0.194-0.258 11-14
0.6-l .2 23-85
108
10.3-30 32.1-4.0.5
0.25-0.42 11-23
129
All data points 20-161 O-120 4-74 0.04-l. 12 0.7-108 1 O-64 0.3-240 0.08-2.2 0.02-64 0.04-l .9 0.002-l .7 3-85
Validity
Table
3
Characteristic
of water
leak rate prediction
x,,ln= In fm,:X,/m,jl,,ed) Mean logarithmic error ratio
deviation of logarithmic
error ratios
s,, = exp
,$ xB./fn - f- 1 I) - 1 CJ Difference between experimental and predicted result F absolute error l
Standard
l
deviation of absolute errors
s.= n = number of experimental results f = number of variables in each correlation
measurements using extremely subcooled water flowing through the crack). For the statistical evaluation in Ref. 7, the missing declarations for the wall roughness were obtained by re-calculating them from measured flow resistance coefficients, by application of the pressure drop relationship used in the respective leak rate model. The statistical evaluation of the models’ reproductive and hence predictive accuracy is founded on characteristic error numbers, Table 3. The standard deviation of the logarithmic ratio of measured and calculated leak mass fluxes and the standard deviation of the absolute difference between both values are chosen from the large number of definitions possible. Based on these two error numbers, the results from each model are assessed since, until now, there has been no criterion for an absolute and/or objective evaluation of Table 4
Accuracy
of reproduction
of the experimental
L. Friedel
leak rates Real cracks
Model cracks Collier et al.
Model
Amos et al.
R. P. Collier et a/. (Leek-001 0. Abdollahian et a/. (model I) (Leak-01) D. Abdollahian et a/. (model II) Kefer et al. (KWU) John et a/. (KfKl
46”
18”
40s
12”
41b
34*,*,=
lo*,“’
76
71
21
3Wb
68
22
3Vb
P. Pana f measured liquid flow resistance coefficients
-
Data points
438
19
27
7
56
14
25b
56
68
17
49
5
237
21
7=+”
58
12
7"b
1 64b.C
11
30
7
345
Kefer at al.
John et al. 71
96
and E Westphal
the predictive accuracy of semi-empirical correlations r3. With respect to the overall assessment and in view of the average measuring accuracy, standard deviations of logarithmic errors <40% and absolute standard deviations 8000 kg/m2s will be assumed ‘to be acceptable. Table 4 shows the best values in each column and all acceptable combinations of logarithmic and absolute standard deviations. The best results for the reproduction of the experiment by Collier et a/. in model cracks were not obtained by our method, unlike the two models of Abdollahian ei of. The standard deviations of the logarithmic and absolute errors are just acceptable with values of approximately 40% and 10000 kg/m2s. The unsatisfactory performances of the KWU and the KfK models essentially result from an inadequate reproduction of the actual pressure drop across the leak. This could originate from the rough specification of the crack wall roughnesses given by Collier et al., so that both models begin with incorrect input values for the pressure drop calculation. The models of Abdollahian et a/. (model I), KWU and KfK produce acceptable results. Abdollahian et al. (model I) shows considerably better results than when reproducing the data of Collier et al., on which it was originally based. As expected, the model crack data from John et al. are best reproduced by their KfK model. Abdollahian et al. (model I) also gives acceptable results, when applied to this data set. All other models failed to do this, and produced very high error numbers, because the underlying respective pressure drop relationships cannot account for large wail roughnesses. This is also valid for the KWU model, which was empirically adjusted using our experiments with cracks of substantially lower roughnesses, and which always predicts too large a mass flux. Again, as expected, the KWU model best reproduces
error numbers
Logarithmic ratio of experimental and predicted results F= logarithmic error ratio
Standard
methods:
9
2
359
144”
John et al.
Kefer et al.
19”
47
6
1267
28
1 O’,b
86
10
195
18
100
16”
73
8
373
18
6.0,’
137
15
231
26
306
8b
131
16
25~“~
23
5
_
_
7
23.0.c
3Lb.'
Collier et al.
60D,b
47
70
3L”” 1
61*.b
lpb
115
15
72
5
100
22
’ Numbers related to models derived from original experiments * Acceptable combinations of logarithmic and absolute standard deviations ’ Best values in each column
J. Loss Prev.
Process
lnd.,
1988,
Vat
7, October
217
Validity
”
of water
0
leak
rate prediction
ZO
i0
Experimental Figure
1
Predictive
XI, = 84%;
8,
Collier
et al.;
(KWU).
Real
er al.
= 178%; +
,
Amos
mass of
et a/.;
x
flux
the
Sa = 1 1949
A, Collier
L. Friedel
80
leak
accuracy
cracks:
methods:
er
era/.
(KfK):
o , John
, Kefer
IKfK);
Figure
model:
cracks,
&
era/.
0, et al.
the underlying experimental results from Kefer et al. with model cracks. Considering the merits of the other models, only the KfK model is also capable of an acceptable accurate reproduction of the Kefer et al. model crack data. None of the models gave adequate predictions, when applied to the Collier et 01. and Kefer et a/. real crack data. Altogether, the only acceptable result with real cracks is obtained with the KfK model, when reproducing their own experiments. But even this model fails, when applied to leak rates from experiments carried out with real cracks from other authors. Obviously, it is difficult to describe the mass flux and the
20
40
Figure 2 x,, = 13%;
21%
Predictive S,, = 44%;
leak
60 mass
flux
80 (lo3
Ind.,
1988,
Figure
accuracy
S,, = 84%:
of
80
flux
(lo3
Abdollahian
SA = 10304
kg/m’s,
1
kg/m2s) et al. (Ii) Symbols
model: as in
pressure drop characteristic of real cracks, possibly due to the cited uncertainties in the characterization of the relevant crack wall roughnesses. For illustration and explanation of the general statistical results related to the data as a whole, the predicted specific leak rates are plotted against the experimentally deduced values, Figures 1-7. In these plots the parity points should be arranged as systematically as possible along the diagonal, which would represent the ideal zero deviation between measured and computed values. Indeed, for all relationships there is a broad scatter along both sides of the diagonal. In some cases, single data sets are systematically calculated to be too low or too large by some models. The reasons for this could be
2’0
1
1, October
Figure4 S,. = 73%;
Predictive SA = 12671
60
li0
Experimental
al. (1) modei: as in Figure 1
Voi
Predictive
mass
I
kg/m2s.)
accuracy of Abdollahian et Sn = 8351 kg/m’s, Symbols
J. Loss Prev. Process
3
Xln = 29%:
60
leak
0, Kefer
(KWU)
Experimental
4ll
Experimental
al.
Model
io
“0
100
kg/m’s)
Collier
kg/m’s,
, John
era/.;
(lo3
and E Westphal
accuracy kg/m*
leak
mass of
KWU
s. Symbols
1 10
80
flux
(lo3
kg/m’s)
model: as in Figure
X,, = 1
-21%;
Validity
of water
leak
rate prediction
100
methods:
L. Friedel
and E Westphal
100
P ;; .TJ e LL
20
0 20
40
Experimental Figure
5
Sn = 7348
PredicJive
accuracy
kg/m’s,
Symbols
leak
60 mass
of KfK
Flux
model:
as in Figure
0
80 (lo3
1
PI, = 6%:
Figure
S,,, = 38%:
7
leakage
I
bols
that part of the data taken in cracks is erroneous or the relationships fail in particular experimental ranges. With respect to the mass flux data distribution over the whole range, it is evident that the smaller values, in general resulting from experiments with relatively low subcoolings and/or low stagnation pressures, predominate, Therefore, the results from the analysis are biased to the lower mass flux range. The magnitude of the obtained logarithmic and absolute deviations indicates that for the Collier et al. model and Abdollahian et al. (models I and II). the deviations are altogether too great to emphasize them as reliable predictive methods for leak rates. However, for leak
20
Figure
6
measured .Sn = 4545 Symbols
Predictive pressure kg/&s. as in Figure
60
40
Experimental accuracy drop Model 1
leak
mass
of Pana
Flux model,
coefficients: cracks:
0,
I,, A,
+.
1
80 (lo3
kg/m2s)
in combination = 10%; Real
40
20 Experimental
excess
OY 0
0
kg/m’s)
8, cracks:
with = 20%; x1
0.
Predictive rates of
20
leak
accuracy
from
experiments
K: X,. = 8%;
as in Figure
60 mass
Flux
of Abdollahian with
S,, = 48%;
80 (lo3
et a/.
stagnation Sn =
100
kg/m251
10000
(II)
model
subcoolings kg/m’s
for in Sym-
7
rates with (large) subcooling in excess of approximately 20 K, the Abdollahian et al. (model II) can be accepted as an efficient, and easily applicable approximation method, Figure 7. The large error numbers only resulted from calculating the mass flux far too low for the case of low subcoolings. The method from KfK allows for the best overall reproductive accuracy, notwithstanding that it has been empirically adjusted for almost 50% of the data used here. However, even this model shows deficiencies by reproducing single real crack data sets here i.e. those of Kefer et al. For further clarification of the partially unsatisfactory predictive performance of the KWU and KfK leak rate methods, the critical mass flux model of Pana, on which both models are based, was checked separately for its predictive capability. The original Pana models necessitates the liquid flow resistance as an input parameter in the sub-model for the calculation of the critical mass flux. In the leak rate models of KWU and KfK, this value is calculated internally from crack dimensions and wall roughnesses by an independent relationship. By introducing the measured liquid flow pressure drop, it should be now possible to independently investigate the quality of this sub-model. In this way, the predicted and measured critical mass fluxes are compared directly. From the error numbers included in Table 4 and from Figure 6, it is obvious that the Pana model reproduces the leak rates in the experiments of Amos et al., Kefer et al. and John et al. partly with good accuracy. It follows that this model is a suitable calculation method for the critical mass flux in cracks with large relative wall roughnesses, if the actual flow resistances of the cracks are known as input parameters or if they could be predicted correctly by an independent relationship. The comparison between the calculation results according to the Pana model and those by the KWU and
J. Loss Prev.
Process
Ind.,
1988,
Vol I, October
219
Validity
of water
leak
rate prediction
methods:
L. Friedel
KfK models, when applied to the experiments of John et al., gives weight to the argument that the failure of the pressure drop relationship proposed by KWU is the reason for obtaining error numbers greater than those having the pressure drop correlation proposed by KfK. In turn, the KfK pressure drop correlation causes the relatively inadequate performance of the KfK leak rate model, when applied to data from Kefer et al. In total, the results obtained using the critical mass flux model of Pana, when the original measured total liquid flow resistances of the cracks are given, indicate that the assumption of a homogeneous thermodynamic equilibrium flow included in the Pana model seems to be adequate, at least for the flow conditions in the experiments of John et al. The leak rate models from KfK and KWU, which are based on this critical mass flux model, only show deficiencies in the prediction of the actual pressure drop across the crack.
and E Westphal
sound leak rate design, it is essential to improve the pressure drop relationship in such a way that it can be used for other (or all) crack configurations and wall roughnesses. Liquid viscosity and surface and interfacial tension should be included as equation variables to the methods, to allow a more universally valid leakage rate prediction. Liquid more viscous than water could then be handled adequately.
Acknowledgement The authors acknowledge the financial support Bundesminister fiir Forschung und Technologie.
References
Conclusions At present, the statistical results allow only a preliminary statement about the general applicability of the leak rate models, since all accessible leak rate data originate from experiments with steam/water. However, even for this particular fluid system, none of the models is able to reproduce all data points within an acceptable standard deviation of 40% (which may still include single value errors in excess of 100%). An adequate reproduction of the results with an acceptable predictive accuracy is only possible for the best methods with the re-calculation of their own experimental values, since error scatters < 3OVo may occur. From the results of the statistical investigation and the correlation analysis, it can be deduced that for a
220
J. Loss Prev.
Process
lnd.,
1988,
Vol 1, October
of the
4 5 6 7 8 9 10 11
12 13
Barrachin, B., et al. French regulatory practice and reflections on the leak-before-break concept, Rapport DAS No. 222 e, 1986 Collier, R. P.. ef al. Two-phase flow through intergranular stress corrosion cracks, EPRI-NP-3540-LD, 1984 Abdollahian, D., et al. AIChE Symposium Series 1984, 80 (236), 19-23 Kefer, V., ef al. Leckraten bei unterkritischen Rohrleitungsrissen, Ausstrijmraten und ihre Gergusche, BWK 1987, 39 (a), 310-315 John, H., et al. Critical two-phase tlow through rough slits. European Two-Phase Flow Group Meeting, Miinchen. 1986 Schrock, V. E., et al. A computational model for critical flow through intergranular stress corrosion cracks. Univ. of California, Report LBL-21%7. 1986 Amos, C. N., et al. Critical discharge of initially sub-cooled water through slits, NUREG/CR-3475, 1983 Henry, R. E.. Nucl. Sci. Engng. 1970. 41, 336-342 Pana. P., Berechnung der stationsren Massenstromdichte van Wasserdampfgemischen und der auftretenden RilckstoRkrBfte, Wiss, Bericht IRS-W-24, 1976 Moody, F. J., J. Heat Tratufer, Series C 1965, 87 (2). 134-142 Reimann, J., Vergleich van kritischen Massenstrom-Modellen im Hinblick auf die Strdmung durch Leeks, KfK-Primiirbericht PNS-Nr 742183, 1983 Fauske, H. K.. PIanr/Operarions Progress 1984, 3 (3). 145 Friedel, L., Chem.-Ing.-Tech. 1981 53 (l), 59
leak rate prediction
L. Friedel and F. Westphal* Technische Pruefung, Hoe&t AG, D-6230 Frankfurt/M 80, FRG *Arbeitsgruppe Sicherheitstechnik, Universitaet Dortmund, D-4600 Dortmund FRG
SO,
Models for the calculation of leakage rates through cracks in vessel walls and pipes due to overpressure are analysad. For evaluation of predictive quality, the models were applied to all the accessible (water) leakage rate data for real and model cracks. The statistical description of the differences between model predictions and experimental results reveals that none of the models can be considered universally valid, even for experiments with water, when specific data from different sources are compared. (Keywords:
pressure and leak testing:
modelling;
In the case of a failure in vessel and pipe walls, (caused by fatigue loading from bending moments or internal pressure) the assumption of a leak-before-break behaviour of the material is generally justified, particularly concerning the highly developed state of design technology and material science. Essentially, this leakbefore-break concept holds even for crack propagation in cases with superimposed corrosion effects. However, it is sometimes not possible to exclude, in the event of heavy stress corrosion, the potential occurrence of axisymmetrical circumferential breaks. The application of the leak-before-break concept to such devices exhibiting corrosive potential requires, by way of supervision, a specially-designed in-service programme’. Fracture mechanics theories reveal that, in the case of leak-before-break behaviour, a realistic approximation to the expected leak size is a slit-shaped crack with a very small hydraulic diameter. For the calculation of discharge rates through these cracks, such as is necessary for risk analysis, only methods developed from the nuclear power industry are accessible. These so-called leak rate models are generally based on interrelated semi-empirical pressure drop and critical mass flux correlations. In this paper, empirical factors have been developed with only a few experiments using subcooled water through, in most cases, model cracks within limited test parameter ranges. From experience, an extrapolation of these leak rate models to other geometries and especially to real crack configurations will result in unreliable predictions. First, an evaluation of leak rate prediction methods with a view to their general validity and predictive accuracy has been attempted. The methods are analysed Received 15 June I988
probability)
in detail with regard to the basic fluid, dynamic and thermodynamic assumptions, equation variables and original validity ranges. Next, a statistical description of the predictive accuracy of the models was made, on the basis of all accessible water/steam leak rate measurements from the literature and statistical error numbers. A relative order of rank followed, according to the individual merits of the correlations.
Leak rate models Nine calculation methods for leak rates through small cracks in vessel and pipe walls have been reported*-‘. All the models were developed for subcooled, hot water situated upstream of the crack, which will partially evaporate during outflow due to the pressure decay along the crack depth. In this case, the models consider a two-phase flow development was considered inside or at the exit of the crack and the limitation of the discharge rate by the self-adjusting critical mass flux. Calculation of critical mass flux requires prediction of the (critical) pressure in the exit plane, which can be estimated from the stagnation pressure and the predicted pressure drop. Leak rate models are, therefore, iterative calculation methods, requiring much computational before a point of intersection between pressure drop and mass flux characteristics from numerical mathematics is attained. An alternative model has been proposed3. This included an explicit calculation for mass flow quality, the pressure in the crack, and for the fluid properties on the basis of average values. An independent engineering prediction method for leakage rates should require only the stagnation state and fluid properties, the crack configurations and the
0950-4230/00/040213-00$3.00 Q 1988 Butterworth & Co. (Publishers) Ltd
J. Loss Prev.
Process
Ind.,
1988,
Vol
7, October
213
Validity
of water
leak rate prediction
methods:
L. Friedel
lation of the critical mass flux. Essentially, this mass flux model is an extended and modified Bernoulli equation. It assumes that the temperature of the subcooled fluid, is only slightly changed while flowing through the crack and that the flashing of the liquid will start just within the crack exit area. Thus, the critical pressure in the exit is set equal to the saturation pressure corresponding to the upstream stagnation temperature. The result is that this model produces physically wrong tendencies for low subcooled liquids, since a zero mass flux would be calculated for decreased subcooling. Therefore, it should be only used for stagnation states with large amounts of subcooling. The leak rate models from KfK and KWU are both based on the critical mass flux model of Pana’, which includes discharge modes as a function of the upstream fluid stagnation state. For the case of subcooled inlet flow, Pana provides two calculation schemes dependent on the extent of the initial subcooling. For high stagnation subcooling, a modified Bernoulli equation is applied. This differs from the original for subcritical single-phase flow only by the inclusion of the saturation pressure, corresponding to the stagnation temperature instead of the back pressure as downstream pressure. This follows on from the assumption that evaporation can only start, in this case, in the crack exit area, irrespective of the depth. For low stagnation subcooling, within the other scheme, it is presumed that the flashing point will be located in the crack and that it will move upstream towards the inlet with decreasing subcooling. The calculation of the actual critical mass flux results from a linear interpolation between the possible extreme cases of a single phase liquid flow through the crack, and a two-phase flow already at the crack inlet with respect to the temperature. In the latter case of a two-phase flow, the critical mass flux calculation may rely on the slip model developed by Moody lo or alterna-
back pressure as input parameters. In contrast, the three models of Amos et al. require specific entrance pressure losses and flow resistances as input variables, since no generally valid pressure drop prediction method is integrated. Another model6 is incomplete, because friction (as a function of the liquid flow Reynolds number), is also required. These models are, therefore, only suitable for post-calculation of experimental results. They are thus not included in the following analysis of the leak rate models’ qualities. The five remaining leak rate models differ with respect to the underlying (critical) mass flux models. Table 1 shows the assumptions about the location of flashing point, the beginning of two-phase flow inside or at the exit of the crack and the internally coupled correlations for the pressure drop across the leak in cases of liquid and liquid/two-phase flow. The leak rate models of Collier et al. and Abdollahian et al. (model I) rely on the critical mass flux model of Henry’, originally developed for the calculation of discharge rates of initially subcooled water through long pipes. Henry fixed the onset of flashing at a distance of 12 times the hydraulic diameter down stream of the crack inlet, in this way being independent in each flow condition from the initial subcooling. This criterion is now discredited for pipe flow by the work of Fauske & Associates Inc. on behalf of DIERS. Nevertheless, it has been retained here, since the original leak rate models were compared without any modification. In the section on two-phase flow development in the crack, Henry assumes a homogeneous flow, that is to say, there is no slip between gas and liquid phases. However, the model includes a thermodynamic non-equilibrium in the form of an empirical relationship for the decrease in local mass flow quality towards an equilibrium value. The Abdollahian er al. (model II) model relies on a simplified, and thus explicit, fluid dynamic (homogeneous) and thermodynamic equilibrium model for calcu-
Table 1
and E Westphal
Leak rate models assumptions Pressure drop
Leak rate model
Basic assumptions for twophase flow
Flashing point
R. F! Collier et el. (Leak-00) D. Abdollahian (model I)(Leak-011 D. Abdollahian ef al. (model II)
12d, after inlet 12dh after inlet Crack exit
V. Kefer er al. IKWU) H. John era/. (KfK)
Subcooling dependent Subcooling dependent
214
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1988,
Vol
Model for critical mas.s flux
Inlet pressure loss coefficient
Liquid flow friction factor
vol Karman (modified) van Karman
+
_
R. E. Henry
1.7
R. E. Henry
1.7
Explicit homogeneous equilibrium model P. Pana
1.7
van Karman
0.5
van Karman
P Pana
0.5
won Karman (modified)
1, October
Impact and form losses
+
Validity
of water
tively on a fluid dynamic equilibrium (homogeneous) flow model. The KWU and KfK models are based in this way on a homogeneous model, since it is easier to handle and is still considered to be adequate for the description of turbulent flow along the crack depth, and the relatively low fluid qualities. A complete or closed leak rate model necessitates a relationship for the prediction of the leak exit pressure and, hence, of the total pressure drop along the crack depth. Conceptually, this can be considered to consist of individual recoverable and irrecoverable, self-adjusting and partially superimposed pressure changes. For crack geometries, especially for the leak-before-break behaviour of the vessel or pipe wall, the total pressure drop is made up of inlet pressure losses resulting from the sharp-edged crack inlet planes; a recoverable pressure drop resulting from an acceleration of the intially stagnant fluid; friction pressure drops due to wall friction and momentum exchange between the phases; impact and form losses resulting from flow area and flow direction changes in the cracks; an acceleration pressure drop due to the inevitable decay in static pressure along the flow path; an overall acceleration or deceleration pressure change due to different geometric inlet and outlet leak area sizes and an exit pressure loss. The individual pressure drops occur in single-phase or two-phase flow conditions depending on the thermodynamic state of the inlet fluid. In the event of twophase flow, the prediction of the self-adjusting pressure changes is always based on the corresponding singlephase, liquid flow correlations, which have been corrected by the introduction of a two-phase mixture density according to a homogeneous model. The leak rate models include two constant entrance pressure loss coefficients for the calculation of inlet impact losses. Collier eC a/. and Abdollahian et al. introduce a factor of 1.7, while in the KfK and KWU models this coefficient is 0.5, as indicated for singlephase flow in sharp-edged inlet planes in the literature. The dependence on local velocity and thus the Reynolds number is not considered. On the whole, the contribution of the entrance losses to the total pressure drop is small, especially when two-phase flow occurs downstream in the crack. The (single-phase) friction pressure drop along the crack depths contributes most to the total pressure drop, particularly in the presence of wall roughnesses and small crack widths. All leak rate models include a friction pressure drop prediction method arbitrarily based on the same mathematical structure for the resistance coefficient, which originates from the Nikuradse-von Karman equation for friction during single-phase flow through sand grain roughened pipes. In some cases, this equation has been modified by re-adjusting the empirical factors to better fit the experimental data of a particular author or to overcome the effects of relative roughnesses in excess of unity. The impact losses due to sudden flow area changes in the crack cause losses resulting from turns in the flow path. Recoverable acceleration or deceleration pressure
leak rate prediction
methods:
1. Friedel
and 17 Westphel
changes due to different inlet and outlet crack area sizes would, on average, also contribute markedly to the total pressure drop. However, since only the models of Collier et al. and Abdollahian et al. (1) consider these, variables such as the number of turns or of area changes along the crack depths are required. Indeed, these details cannot be predicted in a general manner from fracture mechanics, so that any introduction of form and impact losses in the methods is, therefore, coupled with great uncertainties at the present time. In the models of Abdollahian et al. (model II), KWU and KfK form the impact losses and recoverable pressure changes resulting from locally changing flow cross-area sizes, which may have actually occurred in the experiments, are provisionally included in a so-called effective friction pressure loss. The possibility of an exit pressure loss occurring is not considered in the leak rate models. Indeed, it does not enter in the calculation as long as critical flow conditions in the exit plane are supposed. Overall, the mode1 of Abdollahian et al. (model 1) shows the most comprehensive pressure drop calculation method. An independent check of the predictive accuracy of the respective pressure drop sub-models is not yet possible. There are no reliable measurements for critical pressure in the exit plane of cracks and, therefore, even fewer for the individual pressure drop contributions in cases of two-phase flow. Hence in the leak rate models, an internally, incorrectly calculated critical pressure, in combination with an incorrect mass flux correlation, may finally result in a correct leak rate by compensating errors. The extrapolation of the correlations depends on the (complete) inclusion of the relevant, primary parameters as equation variables. In this respect, the analysis of the leak rate models reveals that none of the models are capable of accounting for a change of the liquid phase viscosity or wettability, and the surface tension. This is because the influence of all these properties on leakage rates is not yet established. In the measurements with subcooled/flashing water these properties could not be varied independently of each other. Therefore, an application of the leak rate models to other fluids may be restricted, as it is expected that these properties have a marked effect. Furthermore, the leak rate model predictions suffer from a lack of detailed knowledge on the influence of the thermodynamic non-equilibrium encountered before and during vaporization (and condensation). In view of the wall roughness and small crack width, there might only be a small boiling (and condensation) delay, since there would be several nucleation sites on the crack wall allowing for immediate evaporation. The fluid mean residence time in the crack is short, due to the high mean fluid velocities and the relatively small crack depths (wall thicknesses), so that a marked boiling (and condensation) delay could be encountered, clearly influencing the value of the leakage rate”. Studies on the discharge rate through long pipes with larger bores, show that the combined assumptions of a
J. Loss Prev. Process
Ind.,
7988,
Vol 1, October
215
Validity
of water
leak
rate prediction
methods:
L. Friedel
while John ef al. and Kefer et al. produced them by cyclic bending of a plane probe on a material testing machine. The specified crack widths are related to the crack outlet or to the outer edge of the vessel or pipe wall. The crack depths correspond, by definition, to the wall thickness of the apparatus or pipe. As a rule, the crack sizes in the inlet and exit planes are equal. Collier et al. and John et al. have also used cracks with different inlet and exit areas, which were obtained by incidental widening during the experiments due to the applied high pressure difference on the specimen. The wall roughnesses inside the model cracks depend on the way these cracks have been produced. Collier et al. claimed to have used model cracks with sand blasted wall roughnesses from 0.3 up to IOpm. In the same manner, John et al. produced wall roughnesses up to 150 pm, which are in some cases greater than the adjusted crack widths, so that the relative wall roughnesses can amount to values greater than unity. Amos el al. provide no information on the wall in their ‘smooth’ slits. The spark erosion cracks of Kefer el al. show of about randomly occurring wall roughnesses lo-30 pm. The authors’ statements about the wall roughnesses in real cracks are restricted to the material used, and to the way the crack has been generated. The documented values are in the range from 2 to 240 ,um. Generally, the measurements of the effective wall roughnesses of real cracks is subject to relatively large uncertainties. This is highly unsatisfactory, because here the wall roughness is a decisive parameter for pressure drop prediction. Another difficulty in assessing the influence of wall roughnesses on leak rates are the non-standard techniques and the uncertainty whether the peak, the mean, or any other roughnesses value is a characteristic physical parameter for self-adjusting critical mass flux and pressure drop. The respective total single-phase leak pressure drops are only reported in Refs. 4,5,7 (deduced from their
thermodynamic equilibrium between the phases and a homogeneous flow leads to acceptable predictions for the critical mass flow rate”. It is possible that both assumptions introduce errors, which tend to almost compensate each other. However, the assumption of a homogeneous (mechanical equilibrium) flow in cracks should be physically admissible, because the two-phase mixture generated is characterized by relatively low mass flow qualities and low slip between the phases. The analysis of the leak rate models with respect to the incorporated assumptions and parameters gives an indication of theoretical deficiencies, and allows an early prediction of accuracy for a particular model. Hence, a statistical analysis with all accessible water leak rate measurements has been carried out additionally to evaluate the relative models’ qualities when applied to data in a different and a wider range of real and model cracks.
Data and statistical examination predictive accuracy
of the
The statistical investigation of the predictive accuracy is based on nearly 1000, now accessible, leak rate data from Collier et of., Amos et al., John et al. (KfK) and Kefer et al. (KWU). The parameter range limits of the experimental data and the geometrical sizes of the leaks are listed in Table 2. The measurements are restricted to the case of subcooled hot water upstream of the crack, which will partly evaporate during outflow, thus generating a two-phase flow already inside the crack or at the outlet. The experiments were carried out on rectangular slits as model cracks and on artificially produced slits, so-called ‘real’ cracks. The model cracks’*‘*’ are made up of two plane rectangular steel blocks arranged to allow for various discrete slit widths, while Kefer et al. produced model cracks by spark erosion in steel slabs. The real cracks of Collier et al. are obtained after intergranular stress corrosion in stainless steel pipes,
Table 2
Parameter
ranges of leak rate experiments
and E Westphal
with initially subcooled water
Model cracks
Collier et al.
0.2-1.12 63.5 57 0.3-10
Crack width (mm) Crack length (mm) Crack depth (mm) Roughness (cm) Hydr. diameter Imm) Exit area (mm2) Exit/inlet area ratio Relative roughness Total pressure drop coefficient
216
Amos et al.
Kefer et a/. (KWU)
Collier et al.
John et al.
Kefer et al.
25-161 O-65 6-74
39-122 2-61 6-45
20-l 61 O-66 12-69
35-95 O-72 4-53
40-l 21 2-61 5-22
20-l 56 O-58 9-39
33-l 15 33-120 19-62
Pressure (bar) Subcooling (I<) Mass flux (1 O3 kg/m’s)
0.4-2.2 1 l-64 1 0.002-0.05
J. Loss Prev.
0.127-0.381 15-21 64
0.09-0.4 80 46 5-l 50
0.31-0.325 19-40 10-30 13.9- 30
0.16-0.75 3-8 1
0.18-0.8 20-46 l-l .9 0.01-1.7 3-38
0.61-0.65 6-13
3-72
Process
Real cracks
John et al. fKfK)
Ind.,
1988,
0.04- 0.1 l-4
Vof 1, October
0.04-0.025 0.7-28 19 2-5 0.08-0.5 0.02-6 0.04-0.2 0.007-O. _
1
0.2-0.4 80 46 240
0.097-O.
0.4-0.8 35-51
0.194-0.258 11-14
0.6-l .2 23-85
108
10.3-30 32.1-4.0.5
0.25-0.42 11-23
129
All data points 20-161 O-120 4-74 0.04-l. 12 0.7-108 1 O-64 0.3-240 0.08-2.2 0.02-64 0.04-l .9 0.002-l .7 3-85
Validity
Table
3
Characteristic
of water
leak rate prediction
x,,ln= In fm,:X,/m,jl,,ed) Mean logarithmic error ratio
deviation of logarithmic
error ratios
s,, = exp
,$ xB./fn - f- 1 I) - 1 CJ Difference between experimental and predicted result F absolute error l
Standard
l
deviation of absolute errors
s.= n = number of experimental results f = number of variables in each correlation
measurements using extremely subcooled water flowing through the crack). For the statistical evaluation in Ref. 7, the missing declarations for the wall roughness were obtained by re-calculating them from measured flow resistance coefficients, by application of the pressure drop relationship used in the respective leak rate model. The statistical evaluation of the models’ reproductive and hence predictive accuracy is founded on characteristic error numbers, Table 3. The standard deviation of the logarithmic ratio of measured and calculated leak mass fluxes and the standard deviation of the absolute difference between both values are chosen from the large number of definitions possible. Based on these two error numbers, the results from each model are assessed since, until now, there has been no criterion for an absolute and/or objective evaluation of Table 4
Accuracy
of reproduction
of the experimental
L. Friedel
leak rates Real cracks
Model cracks Collier et al.
Model
Amos et al.
R. P. Collier et a/. (Leek-001 0. Abdollahian et a/. (model I) (Leak-01) D. Abdollahian et a/. (model II) Kefer et al. (KWU) John et a/. (KfKl
46”
18”
40s
12”
41b
34*,*,=
lo*,“’
76
71
21
3Wb
68
22
3Vb
P. Pana f measured liquid flow resistance coefficients
-
Data points
438
19
27
7
56
14
25b
56
68
17
49
5
237
21
7=+”
58
12
7"b
1 64b.C
11
30
7
345
Kefer at al.
John et al. 71
96
and E Westphal
the predictive accuracy of semi-empirical correlations r3. With respect to the overall assessment and in view of the average measuring accuracy, standard deviations of logarithmic errors <40% and absolute standard deviations 8000 kg/m2s will be assumed ‘to be acceptable. Table 4 shows the best values in each column and all acceptable combinations of logarithmic and absolute standard deviations. The best results for the reproduction of the experiment by Collier et a/. in model cracks were not obtained by our method, unlike the two models of Abdollahian ei of. The standard deviations of the logarithmic and absolute errors are just acceptable with values of approximately 40% and 10000 kg/m2s. The unsatisfactory performances of the KWU and the KfK models essentially result from an inadequate reproduction of the actual pressure drop across the leak. This could originate from the rough specification of the crack wall roughnesses given by Collier et al., so that both models begin with incorrect input values for the pressure drop calculation. The models of Abdollahian et a/. (model I), KWU and KfK produce acceptable results. Abdollahian et al. (model I) shows considerably better results than when reproducing the data of Collier et al., on which it was originally based. As expected, the model crack data from John et al. are best reproduced by their KfK model. Abdollahian et al. (model I) also gives acceptable results, when applied to this data set. All other models failed to do this, and produced very high error numbers, because the underlying respective pressure drop relationships cannot account for large wail roughnesses. This is also valid for the KWU model, which was empirically adjusted using our experiments with cracks of substantially lower roughnesses, and which always predicts too large a mass flux. Again, as expected, the KWU model best reproduces
error numbers
Logarithmic ratio of experimental and predicted results F= logarithmic error ratio
Standard
methods:
9
2
359
144”
John et al.
Kefer et al.
19”
47
6
1267
28
1 O’,b
86
10
195
18
100
16”
73
8
373
18
6.0,’
137
15
231
26
306
8b
131
16
25~“~
23
5
_
_
7
23.0.c
3Lb.'
Collier et al.
60D,b
47
70
3L”” 1
61*.b
lpb
115
15
72
5
100
22
’ Numbers related to models derived from original experiments * Acceptable combinations of logarithmic and absolute standard deviations ’ Best values in each column
J. Loss Prev.
Process
lnd.,
1988,
Vat
7, October
217
Validity
”
of water
0
leak
rate prediction
ZO
i0
Experimental Figure
1
Predictive
XI, = 84%;
8,
Collier
et al.;
(KWU).
Real
er al.
= 178%; +
,
Amos
mass of
et a/.;
x
flux
the
Sa = 1 1949
A, Collier
L. Friedel
80
leak
accuracy
cracks:
methods:
er
era/.
(KfK):
o , John
, Kefer
IKfK);
Figure
model:
cracks,
&
era/.
0, et al.
the underlying experimental results from Kefer et al. with model cracks. Considering the merits of the other models, only the KfK model is also capable of an acceptable accurate reproduction of the Kefer et al. model crack data. None of the models gave adequate predictions, when applied to the Collier et 01. and Kefer et a/. real crack data. Altogether, the only acceptable result with real cracks is obtained with the KfK model, when reproducing their own experiments. But even this model fails, when applied to leak rates from experiments carried out with real cracks from other authors. Obviously, it is difficult to describe the mass flux and the
20
40
Figure 2 x,, = 13%;
21%
Predictive S,, = 44%;
leak
60 mass
flux
80 (lo3
Ind.,
1988,
Figure
accuracy
S,, = 84%:
of
80
flux
(lo3
Abdollahian
SA = 10304
kg/m’s,
1
kg/m2s) et al. (Ii) Symbols
model: as in
pressure drop characteristic of real cracks, possibly due to the cited uncertainties in the characterization of the relevant crack wall roughnesses. For illustration and explanation of the general statistical results related to the data as a whole, the predicted specific leak rates are plotted against the experimentally deduced values, Figures 1-7. In these plots the parity points should be arranged as systematically as possible along the diagonal, which would represent the ideal zero deviation between measured and computed values. Indeed, for all relationships there is a broad scatter along both sides of the diagonal. In some cases, single data sets are systematically calculated to be too low or too large by some models. The reasons for this could be
2’0
1
1, October
Figure4 S,. = 73%;
Predictive SA = 12671
60
li0
Experimental
al. (1) modei: as in Figure 1
Voi
Predictive
mass
I
kg/m2s.)
accuracy of Abdollahian et Sn = 8351 kg/m’s, Symbols
J. Loss Prev. Process
3
Xln = 29%:
60
leak
0, Kefer
(KWU)
Experimental
4ll
Experimental
al.
Model
io
“0
100
kg/m’s)
Collier
kg/m’s,
, John
era/.;
(lo3
and E Westphal
accuracy kg/m*
leak
mass of
KWU
s. Symbols
1 10
80
flux
(lo3
kg/m’s)
model: as in Figure
X,, = 1
-21%;
Validity
of water
leak
rate prediction
100
methods:
L. Friedel
and E Westphal
100
P ;; .TJ e LL
20
0 20
40
Experimental Figure
5
Sn = 7348
PredicJive
accuracy
kg/m’s,
Symbols
leak
60 mass
of KfK
Flux
model:
as in Figure
0
80 (lo3
1
PI, = 6%:
Figure
S,,, = 38%:
7
leakage
I
bols
that part of the data taken in cracks is erroneous or the relationships fail in particular experimental ranges. With respect to the mass flux data distribution over the whole range, it is evident that the smaller values, in general resulting from experiments with relatively low subcoolings and/or low stagnation pressures, predominate, Therefore, the results from the analysis are biased to the lower mass flux range. The magnitude of the obtained logarithmic and absolute deviations indicates that for the Collier et al. model and Abdollahian et al. (models I and II). the deviations are altogether too great to emphasize them as reliable predictive methods for leak rates. However, for leak
20
Figure
6
measured .Sn = 4545 Symbols
Predictive pressure kg/&s. as in Figure
60
40
Experimental accuracy drop Model 1
leak
mass
of Pana
Flux model,
coefficients: cracks:
0,
I,, A,
+.
1
80 (lo3
kg/m2s)
in combination = 10%; Real
40
20 Experimental
excess
OY 0
0
kg/m’s)
8, cracks:
with = 20%; x1
0.
Predictive rates of
20
leak
accuracy
from
experiments
K: X,. = 8%;
as in Figure
60 mass
Flux
of Abdollahian with
S,, = 48%;
80 (lo3
et a/.
stagnation Sn =
100
kg/m251
10000
(II)
model
subcoolings kg/m’s
for in Sym-
7
rates with (large) subcooling in excess of approximately 20 K, the Abdollahian et al. (model II) can be accepted as an efficient, and easily applicable approximation method, Figure 7. The large error numbers only resulted from calculating the mass flux far too low for the case of low subcoolings. The method from KfK allows for the best overall reproductive accuracy, notwithstanding that it has been empirically adjusted for almost 50% of the data used here. However, even this model shows deficiencies by reproducing single real crack data sets here i.e. those of Kefer et al. For further clarification of the partially unsatisfactory predictive performance of the KWU and KfK leak rate methods, the critical mass flux model of Pana, on which both models are based, was checked separately for its predictive capability. The original Pana models necessitates the liquid flow resistance as an input parameter in the sub-model for the calculation of the critical mass flux. In the leak rate models of KWU and KfK, this value is calculated internally from crack dimensions and wall roughnesses by an independent relationship. By introducing the measured liquid flow pressure drop, it should be now possible to independently investigate the quality of this sub-model. In this way, the predicted and measured critical mass fluxes are compared directly. From the error numbers included in Table 4 and from Figure 6, it is obvious that the Pana model reproduces the leak rates in the experiments of Amos et al., Kefer et al. and John et al. partly with good accuracy. It follows that this model is a suitable calculation method for the critical mass flux in cracks with large relative wall roughnesses, if the actual flow resistances of the cracks are known as input parameters or if they could be predicted correctly by an independent relationship. The comparison between the calculation results according to the Pana model and those by the KWU and
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Validity
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KfK models, when applied to the experiments of John et al., gives weight to the argument that the failure of the pressure drop relationship proposed by KWU is the reason for obtaining error numbers greater than those having the pressure drop correlation proposed by KfK. In turn, the KfK pressure drop correlation causes the relatively inadequate performance of the KfK leak rate model, when applied to data from Kefer et al. In total, the results obtained using the critical mass flux model of Pana, when the original measured total liquid flow resistances of the cracks are given, indicate that the assumption of a homogeneous thermodynamic equilibrium flow included in the Pana model seems to be adequate, at least for the flow conditions in the experiments of John et al. The leak rate models from KfK and KWU, which are based on this critical mass flux model, only show deficiencies in the prediction of the actual pressure drop across the crack.
and E Westphal
sound leak rate design, it is essential to improve the pressure drop relationship in such a way that it can be used for other (or all) crack configurations and wall roughnesses. Liquid viscosity and surface and interfacial tension should be included as equation variables to the methods, to allow a more universally valid leakage rate prediction. Liquid more viscous than water could then be handled adequately.
Acknowledgement The authors acknowledge the financial support Bundesminister fiir Forschung und Technologie.
References
Conclusions At present, the statistical results allow only a preliminary statement about the general applicability of the leak rate models, since all accessible leak rate data originate from experiments with steam/water. However, even for this particular fluid system, none of the models is able to reproduce all data points within an acceptable standard deviation of 40% (which may still include single value errors in excess of 100%). An adequate reproduction of the results with an acceptable predictive accuracy is only possible for the best methods with the re-calculation of their own experimental values, since error scatters < 3OVo may occur. From the results of the statistical investigation and the correlation analysis, it can be deduced that for a
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Barrachin, B., et al. French regulatory practice and reflections on the leak-before-break concept, Rapport DAS No. 222 e, 1986 Collier, R. P.. ef al. Two-phase flow through intergranular stress corrosion cracks, EPRI-NP-3540-LD, 1984 Abdollahian, D., et al. AIChE Symposium Series 1984, 80 (236), 19-23 Kefer, V., ef al. Leckraten bei unterkritischen Rohrleitungsrissen, Ausstrijmraten und ihre Gergusche, BWK 1987, 39 (a), 310-315 John, H., et al. Critical two-phase tlow through rough slits. European Two-Phase Flow Group Meeting, Miinchen. 1986 Schrock, V. E., et al. A computational model for critical flow through intergranular stress corrosion cracks. Univ. of California, Report LBL-21%7. 1986 Amos, C. N., et al. Critical discharge of initially sub-cooled water through slits, NUREG/CR-3475, 1983 Henry, R. E.. Nucl. Sci. Engng. 1970. 41, 336-342 Pana. P., Berechnung der stationsren Massenstromdichte van Wasserdampfgemischen und der auftretenden RilckstoRkrBfte, Wiss, Bericht IRS-W-24, 1976 Moody, F. J., J. Heat Tratufer, Series C 1965, 87 (2). 134-142 Reimann, J., Vergleich van kritischen Massenstrom-Modellen im Hinblick auf die Strdmung durch Leeks, KfK-Primiirbericht PNS-Nr 742183, 1983 Fauske, H. K.. PIanr/Operarions Progress 1984, 3 (3). 145 Friedel, L., Chem.-Ing.-Tech. 1981 53 (l), 59