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Automatic identification of flow regimes in vertical two-phase flow using differential pressure fluctuations
Nuclear Engineering and Design 95 (1986) 221-231 North-Holland, Amsterdam
221
AUTOMATIC IDENTIFICATION OF FLOW REGIMES USING DIFFERENTIAL PRESSURE FLUCTUATIONS
IN VERTICAL TWO-PHASE
FLOW
Goichi MATSUI
Institute of Engineering Mechanics, University of Tsukuba, Sakura, Ibaraki, Japan
Differential pressure fluctuations are used to identify the flow regimes of nitrogen gas-water mixtures in a vertical pipe because the fluctuations are determined to be closely connected with the flow configuration. The regimes of vertical two-phase flow are classified by the characteristic features of the statistical properties of differential pressure fluctuations measured at two kinds of rational intervals. The results have shown that it is possible to identify the flow pattern based not on visual observations but on the shape of frequency distributions, the order of variance and the average value of differential pressures, because these statistical properties depend on the flow regimes. Furthermore, to identify the flow patterns automatically, the configuration of frequency distribution is approximated by use of the Gram-Charlier series. Then it is shown that the configuration of fitted frequency curves can be discriminated by the statistical parameters associated with the coefficients of the Gram-Charlier series, such as the mean, standard deviation, coefficient of skewness, and coefficient of excess. On the basis of these data, a flow chart is constructed and an objective and automatic identification technique of flow pattern is proposed.
1. Introduction The prediction or the identification of flow regimes in two-phase flow is one of the important problems in the design and operation of two-phase flow systems. However, it has been difficult to develop an accurate method because flow regimes have been judged mostly on a basis of visual observation. Therefore, it is desirable to develop more objective methods of flow pattern identification. In this direction, Hubbard and Dukler [1] have attempted to discriminate between the flow patterns in horizontal flows using a spectral distribution of wall pressure fluctuations. They classified the spectral distributions into three categories, and corresponding to these, horizontal flow was classified into three general regimes of separated, intermittent, and dispersed flow. Jones& Zuber [2] classified the vertical flow into bubbly-, slugand annular-like flow patterns based on the characteristics of the probability density function of void fraction measured by an X-ray absorption technique. It appears that this work has suggested the possibility of objective discrimination. M a t s u i & A r i m o t o [3] and Matsui [4] suggested the possibility of flow pattern recognition for vertical flow using the probability density function of differential pressure fluctuations measured in the observation scales
based on the pipe radius. They [3] also showed that the average velocities of large bubbles can be estimated by use of the cross-correlation between two differential pressures measured at intervals. Furthermore, the lengths of gas and liquid slugs could also be estimated by use of the average velocity of gas slugs, the average void fraction, and the characteristic frequency obtained from the power spectral density function. Inoue et al. [5] applied their method to vertical boiling flows of R-113, and have also used an optical probe for flow pattern identification. Vince&Lahey [6] attempted the study of objective discrimination of flow regimes following Jones&Zuber's work. U p o n investigation of the usefulness of the first four moments associated with P D F and PSD, they found that the P D F variance of the fluctuations of chordal-average void fraction measured with powerful X-ray sources is the best indicator of flow regime transition. Differential pressure fluctuations have been used by Tutu [7,8] in addition to Matsui [3]. Tutu found the bubble-slug transition in the case of vertical air water flow. Recently, it has been suggested by Matsui [9] that the flow patterns in horizontal two-phase flow can be identified by using the statistical features of differential pressure fluctuations between the top and the bottom at the same cross-section of a horizontal tube. This is different from Weisman et al.'s method [10].
0 0 2 9 - 5 4 9 3 / 8 6 / $ 0 3 . 5 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
222
G. Matsui / Automatic identtJieation O~rio.' rcetmt'v
As Vince&Lahey have indicated, there is not a com. plete agreement among the flow pattern maps suggested by earlier researchers. Thus, Vince&Lahey's and Tutu's data little agree with one another. One of the reasons for the lack of agreement was that discrimination was based on visual and hence subjective observations. Other reasons, as Tutu has explained, are that the flow may not have been fully developed at the point of observation, and that the flow depends on the mixing method. Moreover, this discrepancy may be due to the physical properties, especially the surface tension, of the fluid used. Therefore, it may not be useful to extract a transition criterion from experimental data obtained under restricted conditions. Preferably an objective method accepted by researchers of two-phase flow should be established using information associated closely with a flow structure, independent of the fluids and the flow conditions employed. Following this approach, flow regime transitions will be determined based on many experiments involving various two-phase flows and conducted under varying conditions. An objective measurement is required to be one that can be made locally and easily, and economical efficiency and safety will be desired in use of an objective technique developed. This work represents the further development of [3-5] and is concerned with an objective and automatic method of identifying the flow patterns in vertical gas-liquid two-phase flow using differential pressure fluctuations. In this method, differential pressures are used instead of absolute pressures to eliminate the effect of static pressure fluctuations which contain information from regions outside the test section. Differential pressure fluctuations are detected with pressure taps located at one of two short intervals equivalent to the inside radius of the pipe and at one or two long intervals equivalent to ten times the diameter of the pipe. These signals are examined together. "It has been shown in [3] and [4] that the flow regimes of vertical nitrogen gas-water two-phase flows can be classified based on the characteristic features of statistical properties, especially the frequency distribution of differential pressure fluctuations; that is, the differential pressure fluctuations characterize the flow regimes in vertical two-phase flow. We can find the characteristic features in the frequency distribution such as a bimodal distribution in slug-like flow. On the basis of the specific features of the frequency distribution, namely the variance and the mean of differential pressure fluctuations, the flow patterns have been classified into four fundamental categories: bubbly, slug-like, annular, and mist flow. See [3] or [4] for more details.
Furthcrnlore. we consider an ot~teet~vc ~md ~ut~> matic identification method not based on v~ual c~+~+ servations or subjective judgement. Since the configuration of frequency distribution shows tile features of ftox~. pattern, the frequency distribution is approximated {'+y the Gram-Charlier series to obtain the fitted frequency curve. It is shown that the statistical parameters assoc++ated with the coefficients of the frequency curve, such as the mean, standard deviation, coefficient of skewness and coefficient of excess, related closely to the configuration of frequency distribution. On the basis of ex+ perimental data, an identification method without any visual observation is proposed. Fhe values of these statistical parameters can be obtained directly by processing the differential pressure fluctuations mea-+ sured without calculating the frequency distributions+ The proposed identification technique is an objective and rational one without any visual judgment, although the experimental results have been verified through comparison with those of visual observation.
2. Conception of measuring technique 2.1, Observation scales of differential pressure fluctuations Pressure fluctuations have been used to characterize various flow regimes in vertical two-phase flow. To eliminate the effect of static pressure fluctuations which contain information from regions outside the measurement section, we use differential pressures measured over relatively small vertical distances. The vertical distances should be carefully selected in order to obtain rational observation intervals which enable characterization of the flow regimes, because the frequency distribution of differential pressure fluctuations depends on the observation distance, as shown by Akagawa et al. [11]. In the present method, we use two kinds of observation scales, namely short and long scales. In order to recognize a spherical cap bubble or a cluster of small bubbles in the size of the pipe radius. the short scale, which is called in R-scale, is selected to equal the inside radius of pipe. On the other hand, the long scale, which is called an L-scale, is selected to be the distance equal to ten times the diameter of pipe, in order to identify developed gas-slugs. Thus, the test section has two kinds of observation scales mentioned above as shown in fig. 1. The observations based on these scales or four differential pressure measurements are made simultaneously to identify the flow regimes in vertical two-phase flow. An actual diagnosis of flow patterns is made mainly through the
G. Matsui / Automatic identification of flow regimes
Do
o
R - scale L - scale L - scale
"
" 2rr~
'
I
~ Apo.
R - scale
,h
Fig. 1. Arrangement of observation scales in the test section.
observations in one R-scale, and the measurements in the other R-scale and in an L-scale are used as complements of the R-scale.
2.2. Normalization of differential pressures The differential pressure Api (i = a, b, c, d) is normalized with the hydrostatic pressure drop Ap0 ~ corresponding to the measurement section.
Ap,=ApJApo,,
(i=a,b,c,d).
(1)
In the case the acceleration and friction losses are very small compared with (Apo,--Ap~), the quantity (1 - A p,) represents approximately the average void fraction in section i. Therefore it may be convenient to use ~ * = 1 - ~ P, = x,
(2)
223
scales, e.g., one pair of '~Pa and /Xpc, are used to identify accurately the flow regimes. We also note here that by using two short observation scales together, we can find the change or transition of flow pattern in the test section, as seen in fig. 2. Fig. 2 shows the frequency distributions of A P * and Ap~' for the case of transition from a developing slug flow to a fully developed one. However, two frequency distributions of A p * and A Pd* at long scales are similar and do not indicate any transition. Typical examples of frequency distribution are shown in figs. 3 to 6 for characteristic flow patterns. The variable A Pi* is essentially a continuous distribution, but is shown as a discrete distribution owing to the measurement conditions. We can obtain the probability density or frequency function of A p,* if the increment of A Pi* tends to zero. These examples are limited to cases in which the frequency distribution of A p~' exhibits similar shape to that of A p*. Experiments have been carried out using a nitrogen gas-water two-phase flow loop. The test section was made of an acrylic tube of 22 mm inside diameter. Then the R-scale and L-scale are equal to the distances of 11 mm and 200 mm, respectively. See [3] or [4] for more details. The bubbly (fig. 3), annular (fig. 5) and mist (fig. 6)
~ 0 o8
~0.08 ~O
n <~
as the statistical variable.
<~ 0.04
0o~
2.3. Statistical features for flow regimes Both the frequency distribution and the spectral distribution of differential pressure fluctuations can be used for flow regime identification. Here the former is used because the latter needs sampling of more data and more data-processing time than the former. Identification of flow regimes is conceptually feasible based on the characteristics of a frequency distribution, namely the statistical features of the fluctuations measured by using one short observation scale. However, there may be some difficulty in discrimination using only one short scale in general, except under limited situations such as incorporation of slug-like into intermittent flow, use of specified operation condition, and employment of specified fluids. Therefore, in this study one pair of short (R-) and long (L-) observation
0Apa~
°&pb*~
1
~0.20
0.20 g_
<3
K O.lO
0 ---
040
0APc*l
'
0
0&po, 1
Fig. 2. Typical example of flow pattern transition in the test section (U(; = 0.122 m/s, UL = 0.49 m/s, ~ = 0.105).
224
G. Matsui /Automatic ~dentification oj ~low regimes
,,_., 0.2C
O.30
1.0o
O.75 I !
.1(-
{3. 0.1(
13. 0.50
o
J
°o
{
A P*
[3. 0.20
0.50
0.1c
0.25
i
A pc'~
Fig. 3. Frequency distribution of differential pressure fluctuations in bubbly flow for &~. = 0.009 m/s, U e = 0.21 m / s and a = 0.0335.
1
oAp
*~a
0
'
o
,1
APe
Fig. 6. Frequency distributions of differential pressure fluctuations in mist flow for U(; = 4.06 m/s, UL = 0.003 m / s and c~= 0.994.
0.10 13_
['i ,2t
"-" 0.05
Q..
t; iii, 0Ap•l
0Apc*l
Fig. 4. Frequency distributions of differential pressure fluctuations in slug flow for UG = 0.095 m/s, UL = 0.13 m / s and a = 0.248.
flows show u n i m o d a l or single-peaked frequency distributions for A p * a n d A p * . On the other hand, the slug flow (fig. 4) shows a b i m o d a l or a twin-peaked frequency distribution for A p * and a u n i m o d a l one for A p * . These distribution configurations are similar to those for void fractions measured by Jones and Z u b e r [2]. Taking into account the variance a n d m e a n values of
v
ill
a. 0.05
2.4. Objective discrimination o f frequen£v distribution configuration
C 0.4( 0.
* o 0.10 13.
,;'lii] ,J J .... 0
1
A
differential pressure fluctuations, we found that vertical two-phase flows exhibit the following features: (a) Bubbly flow - Single-peaked frequency distribution at small A p * (average of A p,*) and small variance. (b) Slug-like flow (spherical cap bubble, slug. and churn flow) - F o r R-scale, twin-peaked frequency distribution at m e d i u m 2JP*. For L-scale, the spherical cap and slug flow cases show a single-peaked frequency distribution but the c h u r n flow case, a twin-peaked one. The variance becomes larger in the order of spherical cap bubble, slug, and c h u r n flow. The c h u r n flow shows the m a x i m u m variance. (c) Annular flow - Single-peaked frequency distribution at large A p * a n d m e d i u m variance. (d) M i s t flow - Single-peaked frequency distribution a~ A p * near unity a n d small variance. T h e above results suggest that the configuration of frequency distribution is highly useful for identifying the flow regimes.
6
Apg
Fig. 5. Frequency distributions of differential pressure fluctuations in annular flow for Uc =12.5 m/s, UL= 0 m / s and c~= 0.645.
We now consider identification of flow regimes without calculating the frequency distributions. This will make possible a u t o m a t i c j u d g e m e n t of flow patterns. In this study, we use the G r a m - C h a r l i e r series of type A to a p p r o x i m a t e the frequency distribution. One of the aims is to check whether the series shows the characteristics of frequency distributions a n d hence the statistical parameters corresponding to the coefficient of the series indicate the features of the configuration of frequency distribution. The other aim is to investigate if
G. Matsui /Automatic identification of flow regimes the series can approximate the envelope of bimodal frequency distributions because the peak heights may be used to know the structure of intermittent flow. Since the coefficients of the G r a m - C h a r l i e r series or frequency curve are associated with the distribution configuration, these statistical parameters characterize the shape of a fitted frequency curve or the original frequency distribution of differential pressure fluctuations. On the basis of the features of statistical parameters such as the mean, standard deviation, coefficient of skewness, and coefficient of excess, which can be obtained directly by processing the measurement data, an automatic identification algorithm is constructed in flow chart form. As a result, it appears that the flow patterns observed in vertical two-phase flow can be determined conceptually at least based on this flow chart.
These will apply similarly to the case of twin-peaked distribution. This is checked by fitting of the G r a m - C h a r l i e r series to frequency distributions. 3.2. A p p r o x i m a t i o n
3.1. S t a t i s t i c a l p a r a m e t e r s
Statistical parameters are as follows: variable x i = 1 - AP i mean m = Y'.xi/N standard deviation o = [~2( x i - m ) 2/ N ] 1/2 n-th moment about the mean /~, = ~ ( x i - m ) " / N (e) coeff, of skewness 71 = ]13/03 (f) coeff, of excess Y2 = ~ 4 / o 4 -- 3 where N is the number of sampled data. The relation between the parameters and the frequency function is given as follows (Cramer [12]). If the variable x i is normal,
(a) (b) (c) (d)
series
p(x)= E a'q~.(x),
(3)
n=O
where 1
__e-X
Since the configuration of the frequency distribution contains useful information for characterizing the flow regimes, we consider the automatic discrimination of distribution configuration to identify the flow regimes more objectively.
by the G r a m - C h a r l i e r
It will now be shown that the G r a m - C h a r l i e r series of type A or frequency function gives a good approximation to frequency distributions obtained by measurements. From [12], the G r a m - C h a r l i e r frequency function is given as follows:
~,0(x) = 2¢5g 3. Curve fitting of frequency distributions
225
2
/2
q,~(x) = ~ x ~ q~o(X),
a'n -
(normal frequency function),
f+5
p(x)
d x = 1,
(-1)"f _~
H, ( x ) is the Hermite polynomial of degree n. Using the standardized variable X = ( x - m ) / o corresponding to a variable with the mean m and the standard deviation o, the calculation of the coefficients of the series becomes easy. Thus eq. (3) is expressed as p ( X ) = q~0( X ) + a3q~3 ( X ) +
a4t~4
(X)
q-
+a6+6(X) + -..,
(4)
where e~o(X ) =
l~e-X2/2
1.0
UG=O. 009 m/s UL=O,21
m/s
c~ =0.0335
/~2~+I = 0 ,
/Xz~=l'3'...-(2u--l)o
2~.
In particular, 7~ = 0 and 3'2 = 0. The coefficient 71 is considered as a measure of the asymmetry or skewness of the distribution. If 3'1 is positive, the long tail will be on the positive side of the distribution, and if 3'1 is negative, the long tail on the negative side. The coefficient 3'2 is used as a measure of the degree of flattening of a frequency curve near its center. If 3'2 is positive, the frequency curve is taller and slimmer in the neighborhood of the maximum point.
x m = 0.0251 0 = 0.06 y1= 0.154 72=-0.199 as= 0.149/5! a6= 0.070/6! ~=-0. 0096 =-0.159
0
~ I . ~0
xa
Fig, 7. Bubbly flow (R-scale).
as~ 5 ( X )
226
G. Matsui / A utomatw identification ql /lo~* regimes
1.0~i
UG=0.0158 UL=0.145
a 5 :=
m/s m/s
- 10p,3)/5!.
,t~, = (/*~, - 15~4 + 3 0 ) / 6 ! .
(5)
~ =0.056
X
"K
IL 0
1.0
The curves for the derivatives 9,,(X) of the even order are symmetric a b o u t X = 0, while the derivatives of the odd order introduce an asymmetric element into the fitted frequency curves. It is desired that a g o ( ~ a p p r o x i m a t i o n be given by a small n u m b e r of lerms in the polynomial expression (3) or eq. (4) if possible. However, we see that the terms up to n = 4, which are often used for an approximation of distribution in general, do not give a sufficient approximation to bimodal distributions. Thus we consider the terms up lo n = 6 inclusive in order to approximate the bimodal distributions with asymmetry. We have tried to fit the curve of G r a m - C h a r l i e r series to the envelope of frequency distribution shown in [3] or [4]. The results are seen in figs. 7 to 12. In each figure, the flow conditions a n d the values of statistical parameters are given together with the frequency distribution. The solid curve drawn in each figure indicates
m = 0.0497 o = 0.176 yl = [.58 y2 = 3.61 as=-3.69/5! a6 = 4.61/6! B1 = 0 . 0 3 2 7 g = 0.i86
xa
Fig. 8. Spheficalcap bubble flow (R-scale).
¢.(x) = dd;-~¢o(X), a 0=1,
(~
a 1=0,
a3 = - / x 3 / 3 r
a4(#4-3)/4!
a 2=0,
('gl = - 3 r a 3 ) ,
(3'2=4!a4),
UG=0.095
m/s
UL=0.13
m/s
l°ij ~J
=0,248 X m
=
O. 3 4 9
<7 =
O. 6 3 2
y1 =
0.238
rn = 0.L92 = 0. 156 T ~=-0.077 "~2=-0. 345 as= 0.264/~! ar,=-l. 53/6 ! ~=-0.00754
v 2 = - i 117 as = 2.02/5[ a6 = 6.50/6[ ~i = 0.34 B = 0.538
U
[~ =-0. 048
"a
i ,v
1 0
X
C
Fig. 9. (a, left) Slug flow (R-scale), (b, right) Slug flow (L-scale).
1.Or
UG=2.39
m/s
UL=0.12
m/s
(~ = 0 , 6 4 5 m
=
cr =
,"'
1.0 T
X U ! jI i[ m = 0.451 o = 0. 353 yi=-1.44 y2 = i. 54 as=-2.79/5 ! a6=-4.14/6! $t=-0.165 =-0.467
0.504 0. 7 9 2
y I=-0.
929
T
[
y2 = 1 . 4 5 8 as = 0.318/5! a6=-2.2/6! BI=-0.067 B =-0. 084
0
1.0
xa
Fig. 10. (a, left) Churn flow (R-scale), (b, right) Churn flow (L-scale).
1.0
X C
G. Matsui / Automatic identification of flow regimes
1.0 T
UG=I2.5 m/s UL= 0
] .0 T
I
~Xr~ 7=,
I
m/s
Ot =0. 876 m = O. 916 (7 = 0.453 y:= 0.258 yz=-0.0266 as: 0.222/5!
a6=-0"372/6! B1= 0.037 = O. 081 0
I t
+-
m = O. 884 o = O. 029 YI=-0.0087 y2 = 0.102 as = 0.01/5! a6=-1.04/6! B i=-0. 016 B =-0.536
I! 't
]]il
I-~
- - 4
0
xa
1.0
227
1.0
X
C
fig. 11. (a, left) Annular flow (R-scale), (b, fight) Annular flow (L-scale).
1.0:
UG=4.06
m/s
].0
T
UL=0.003 m/s ¢t =0.994
"J
I m = 0.99 d = O. 0516 y1 = 0.031 y2 = 0.015 as= 0.368/5! a6=-0.24/6! B~ = 0.0004 fl : o.oo7
I
+
k .0
X
m = 0.974 (7 = 0.0149 yi=-0.993 Y2 =-i. Ol a5=-6.96/5! a6 = 5.14/6!
T
+
BI = 0.006 6=o.4
0
1 0
X
C
Fig. 12. (a, left) Mist flow (R-scale), (b, fight) Mist flow (L-scale).
the fitted or approximate frequency curve with terms up to n = 6. The frequency distribution and fitted curve are normalized by the values of the maximum point, respectively. In fig. 9(a), the G r a m - C h a r l i e r frequency curves are compared for different values of n. The fit to the distributions is very good, except that the curves have sometimes negative values or a low tertiary peak. The results shown above suggest the possibility of flow pattern discrimination based only on the statistical parameters characterizing the curve shape or the frequency distribution configuration.
4. D i s c u s s i o n of useful parameters
In figs. 7 to 12, the values of statistical parameters are also given. We shall consider the relation between the shape of the distribution and the values of the parameters. Here distributions considered to be of normal-type appear in bubbly, annular, and mist flows for the R-scale. Moreover they appear in bubbly, spherical
cap-bubble, slug, annular, and mist flow for the data of L-scale. (a) the mean, m The value of m becomes larger with the transition of flow pattern from bubbly flow (B) to mist flow (M) through slug-like flow and annular flow (A). Thus we can use m as a criterion for flow identification. The mean can be used directly for identification if the fluid and the operating conditions are fixed. Unfortunately there is a disadvantage to the use of the mean since it can sometimes shift especially for the distribution in an R-scale, depending on the operation conditions. (b) the standard deviation, o The value of a becomes larger with transition from bubbly flow to churn flow (C) through slug (S), but the value is smaller again in annular and mist flows. In the case of o < 0.1, frequency distributions resemble a normal distribution. (c) the coefficient of skewness, 3`1 Distributions with small 13'1 I will be considered nor-
228
G. Matsui /Automatic tdenttfication el flow regimes
real. On the other hand, distributions with large 17~i may be judged to be those with long tail (on the positive side for "y1 > 0, and on the negative side for Y1 < 0) or bimodal. Therefore, we may consider the distribution with the value of I "¢~ [ < 0 5 to be normal. Moreover, for a distribution of x a with large and positive 71, the flow is judged to be spherical cap bubble (SP) or slug flow. If we obtain a distribution of x a with large and negative "h, the flow is judged to be slug or churn flow. (d) the coefficient of excess, 72 Distributions with small ] Y2 ] will be considered normal. If we obtain a distribution of x, with positive and large "h, the flow is judged to be spherical cap bubble or churn flow because the fitted curve is taller and slimmer than the normal one in the neighborhood of the maximum point. On the other hand, if we get a distribution of x~ with negative and large "h, the flow is judged to be slug flow because the fitted curve is lower and flatter than the normal one.
[[ Identification using data of R- scale ( aPa or ~Pb) ]I I Calculation of m, o, x 0, ~l, Y~, ~i and
1
I
I
YES
¢
+ I
[ Identification using data of L - scale ( aPc ) ] Calculation of m, ~, % , YI, Y2, ~x and
(e) the coefficient of the fifth derit~ative, a 5 When the value of [ a 5 I is large, the distribution will be considered to have a tail. Distributions with small ['Y1 I and [a 5 [ may be considered to be of normal-type. (f) the coefficient of the sixth derivative, a 6 Distributions with small I a6 [ are similar to normaltype. (g) fll m x o The quantity/3~ may be used as a measure of the asymmetry or skewness of the distribution. A distribution with small I/3a I will be considered normal. =
- -
(h) fl = f l l / o (Pearson's measure of skewness [12]) A distribution with small I/3[ will also be considered normal.
5. Automatic identification of flow regimes 5.1. Flow chart for identification
I "~1
'
<
!~r2 J
k5
YES
I < k10
q ~ < k12
YES
,o
Churn Flow unidentifiable i f not judged SC)
I I
Mist Flow I ( unidentifiable i f not judged BM)
Spherical Cap Bubble Flow, i f judged SP Slug Flow, i f judged S or SC Annular Flow, i f judged A Bubbly Flow, i f judged BM
I
Print or Indication of result ]
It has been shown that the frequency distribution of differential pressure fluctuations can be fitted by the frequency curve of the G r a m - C h a r l i e r series. Furthermore, the statistical parameters including the coefficients of the series give the features of the distribution configuration and hence flow patterns. On the basis of these results, a flow chart for automatic identification of flow regimes is constructed as shown in fig. 13 taking into account the features of statistical parameters. If the flow pattern identified from x a is different from that of
Fig. 13. Flow chart for flow regime identification.
Xb, the flow may be judged to be in transition in the test section. The values kl-k13 in the figure denote the threshold
G. Matsui / Automatic identification of flow regimes of judgment. These values should be determined on the basis of sufficient experimental data.
229
I .O!,T
5.2. Examples of identification Based on the data shown in figs. 7 to 12 and further experimental data, the thresholds kl-k13 are determined as follows: k l = 0.4,
k 2 = 0.4,
k 5 = 0.14,
k 6 =
k 9 = 0.5,
kl0 = 0.01,
0.25,
k 3 = 0.0l,
k 4 = 0,1,
0.4,
k 8 = 05,
k 7 =
kll = 0.1,
k12 : 0 . 1 ,
w
0
(6)
.0
x
1.o
×
a
l.OTi ! II!
kl3 = 0.95.
Typical results of automatic identification using this flow chart are shown in figs. 14 to 16. Experiments have been carried out using another nitrogen gas-water twophase flow loop. The text section was made of an acrylic tube of 20 mm inside diameter. Thus the R-scale is equal to the distance of 10 mm and the L-scale the
uT
o 1
n
Ira..L
Fig. 15. Slug flow (Uo = 0.179 m / s a n d U L = 2.71 m / s ) .
I.OT
0.51.0
(i
[ i1
xa
l.O
01.0
xa
1.0 T (.,9 X
- - o " o'.s
i'.o
xc
Fig. 14. B u b b l y flow ( U o ~ 0.146 m / s a n d U L = 0.414 m / s ) .
Fig. 16. C h u r n flow ( U o = 1.71 m / s a n d U L = 0.377 m / s ) .
01.0
x
C
230
G. Matsui / Automatic tdentification oj flow regtmex
distance 200 mm. The result of identification is indicated in the figure caption in each figure. For reference. the picture of flow and frequency distributions of differential pressure fluctuations at both short and long observation scales are shown in each figure. In the case of transient flow, the determination of flow pattern will be difficult because it may change during the measurement. Even if the results of identification for flow in the transition region ~rre different from those of visual observation, this will not always be due to misjudgment. In such cases, it may be advisable to indicate the results of identification in terms of probabilities.
tained at several points along the flow direction, i~ i~ desirable to determine the threshold of parameter~ ,,)n the basis of more sufficient data under ~arious expew mental conditions.
Acknowledgement A part of this study was supported by CASIO Science Promotion Foundation.
Nomenclature 6. Conclusions a, An automatic and objective technique of flow regime identification has been proposed for vertical two-phase flow. It is a technique using differential pressure fluctuations measured at one pair of rational observation scales, namely the short scale equal to the inside radius of a pipe and the long scale equal to ten times the inside diameter. The experimental results for nitrogen gas-water flow show that the flow regimes in vertical two-phase flow exhibit the characteristic features of the statistical properties of differential pressure fluctuations and that the flow patterns are determinable based on the classification of the statistical features. Moreover, to make the identification less subjective, the envelope of frequency distributions is fitted by the polynomial expression of the Gram-Charlier series of type A. The results of curve fitting showed that the statistical parameters associated with the coefficients of the frequency curve obtained characterize the frequency distribution or flow regime, and therefore, the automatic and objective judgment of flow regimes is feasible. Thus, an objective and automatic method of flow pattern identification for vertical two-phase flow was developed in flow chart form on the basis of the features of statistical parameters such as the mean, standard deviation, coefficient of skewness, and coefficient of excess. The values of these parameters can be obtained by processing directly the differential pressure fluctuation measurements without calculating the frequency distribution and fitting the curve. In addition, because the identification is conceptually feasible even by use of one short observation scale, the present technique using differential pressure fluctuations can be extended to identification of a developing flow pattern, if the measurements of short observation scale are oh-
a~ L m N
p(x) Ap Ap0 Ap
normalized coefficient of the n-th derivative of the Gram-Charlier series ( = a',,/a"), coefficient of the n-th derivative of the Gram-Charlier series, length of long observation scale. mean of variable, number of sampling data, frequency distribution of x or probability density or frequency function of x, differential pressure, hydrostatic pressure drop, normalized differential pressure ( = Ap / A Po ),
Ap*
=I--AP=x,
R U x x0 X a
length of short observation scale, superficial velocity, variable ( = 1 - Ap), maximum point of frequency curve, standardized variable ( = (x - m )/o ), void fraction,
~1
~g Y2 /% o q,0(x)
= m -- x0,
coefficient of skewness, coefficient of excess, n-th moment about the mean, standard deviation, "~ / normal frequency function ( = e x p ( - x - / 2 L
2¢Tg), q~,,
= d"q~o(x)/dx ~.
Suffix G i L -
gas, observation section of measurement section ( a, b, c, d), liquid, average.
G. Matsui / Automatic identification of flow regimes References [1] M.G. Hubbard and A.E. Dukler, The characterization of flow regimes for horizontal two-phase flow, Proc. Heat Transfer and Fluid Mechanics Institute (Stanford Press, 1966) pp. 100-121. [2] O.C. Jones and N. Zuber, The interrelation between void fraction fluctuations and flow patterns in two-phase flow, Int. J. Multiphase Flow 2 (1975) 273-306. [3] G. Matsui and S. Arimoto, Flow pattern estimation of gas-liquid two-phase mixture in a perpendicular pipe, ICHMT-1978 Int. Seminar on Momentum, Heat and Mass Transfer in Two-Phase Energy and Chemical System, Dubrovnik, Yugoslavia, 1978. [4] G. Matsui, Identification of flow regimes in vertical gas-liquid two-phase flow using differential pressure fluctuations, Int. J. Multiphase Flow 10 (1984) 711-720. [5} T. lnoue, M. Agui, G. Matsui and S. Arimoto, Quantitative flow-pattern discrimination of gas-liquid two-phase mixture (in Japanese), Trans. SICE 17 (1981) 231-236. [6] M.A. Vince and R.T. Lahey, Jr., On the development of an objective flow regime indicator, Int. J. Multiphase Flow 8 (1982) 93-124.
231
[7] N.K. Tutu, Pressure fluctuations and flow pattern recognition in vertical two-phase gas-liquid flow, Int. J. Multiphase Flow 8 (1982) 443-447. [8] N.K. Tutu, Pressure drop fluctuations and bubble-slug transition in a vertical air-water flow, Int. J. Multiphase Flow 10 (1984) 211-216. [9] G. Matsui, Identification of flow patterns in horizontal gas-liquid two-phase flow using differential pressure fluctuations, Preprints, Int. Symp. on Fluid Control and Measurement (Pergamon Press), Tokyo, 1985, pp. 819-824. [10] J. Weisman, D. Duncan, J. Gibson, and T. Crawford, Effects of fluid properties and pipe diameter on two-phase flow patterns in horizontal lines. Int. J. Multiphase Flow 5 (1977) 437-462. [11] K. Akagawa, H. Hamaguchi, T. Sakaguchi and T. Ikari, Studies on the fluctuation of pressure drop in two-phase slug flow, Bull. JSME 14 (1971) 447 469. [12] H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946).
221
AUTOMATIC IDENTIFICATION OF FLOW REGIMES USING DIFFERENTIAL PRESSURE FLUCTUATIONS
IN VERTICAL TWO-PHASE
FLOW
Goichi MATSUI
Institute of Engineering Mechanics, University of Tsukuba, Sakura, Ibaraki, Japan
Differential pressure fluctuations are used to identify the flow regimes of nitrogen gas-water mixtures in a vertical pipe because the fluctuations are determined to be closely connected with the flow configuration. The regimes of vertical two-phase flow are classified by the characteristic features of the statistical properties of differential pressure fluctuations measured at two kinds of rational intervals. The results have shown that it is possible to identify the flow pattern based not on visual observations but on the shape of frequency distributions, the order of variance and the average value of differential pressures, because these statistical properties depend on the flow regimes. Furthermore, to identify the flow patterns automatically, the configuration of frequency distribution is approximated by use of the Gram-Charlier series. Then it is shown that the configuration of fitted frequency curves can be discriminated by the statistical parameters associated with the coefficients of the Gram-Charlier series, such as the mean, standard deviation, coefficient of skewness, and coefficient of excess. On the basis of these data, a flow chart is constructed and an objective and automatic identification technique of flow pattern is proposed.
1. Introduction The prediction or the identification of flow regimes in two-phase flow is one of the important problems in the design and operation of two-phase flow systems. However, it has been difficult to develop an accurate method because flow regimes have been judged mostly on a basis of visual observation. Therefore, it is desirable to develop more objective methods of flow pattern identification. In this direction, Hubbard and Dukler [1] have attempted to discriminate between the flow patterns in horizontal flows using a spectral distribution of wall pressure fluctuations. They classified the spectral distributions into three categories, and corresponding to these, horizontal flow was classified into three general regimes of separated, intermittent, and dispersed flow. Jones& Zuber [2] classified the vertical flow into bubbly-, slugand annular-like flow patterns based on the characteristics of the probability density function of void fraction measured by an X-ray absorption technique. It appears that this work has suggested the possibility of objective discrimination. M a t s u i & A r i m o t o [3] and Matsui [4] suggested the possibility of flow pattern recognition for vertical flow using the probability density function of differential pressure fluctuations measured in the observation scales
based on the pipe radius. They [3] also showed that the average velocities of large bubbles can be estimated by use of the cross-correlation between two differential pressures measured at intervals. Furthermore, the lengths of gas and liquid slugs could also be estimated by use of the average velocity of gas slugs, the average void fraction, and the characteristic frequency obtained from the power spectral density function. Inoue et al. [5] applied their method to vertical boiling flows of R-113, and have also used an optical probe for flow pattern identification. Vince&Lahey [6] attempted the study of objective discrimination of flow regimes following Jones&Zuber's work. U p o n investigation of the usefulness of the first four moments associated with P D F and PSD, they found that the P D F variance of the fluctuations of chordal-average void fraction measured with powerful X-ray sources is the best indicator of flow regime transition. Differential pressure fluctuations have been used by Tutu [7,8] in addition to Matsui [3]. Tutu found the bubble-slug transition in the case of vertical air water flow. Recently, it has been suggested by Matsui [9] that the flow patterns in horizontal two-phase flow can be identified by using the statistical features of differential pressure fluctuations between the top and the bottom at the same cross-section of a horizontal tube. This is different from Weisman et al.'s method [10].
0 0 2 9 - 5 4 9 3 / 8 6 / $ 0 3 . 5 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
222
G. Matsui / Automatic identtJieation O~rio.' rcetmt'v
As Vince&Lahey have indicated, there is not a com. plete agreement among the flow pattern maps suggested by earlier researchers. Thus, Vince&Lahey's and Tutu's data little agree with one another. One of the reasons for the lack of agreement was that discrimination was based on visual and hence subjective observations. Other reasons, as Tutu has explained, are that the flow may not have been fully developed at the point of observation, and that the flow depends on the mixing method. Moreover, this discrepancy may be due to the physical properties, especially the surface tension, of the fluid used. Therefore, it may not be useful to extract a transition criterion from experimental data obtained under restricted conditions. Preferably an objective method accepted by researchers of two-phase flow should be established using information associated closely with a flow structure, independent of the fluids and the flow conditions employed. Following this approach, flow regime transitions will be determined based on many experiments involving various two-phase flows and conducted under varying conditions. An objective measurement is required to be one that can be made locally and easily, and economical efficiency and safety will be desired in use of an objective technique developed. This work represents the further development of [3-5] and is concerned with an objective and automatic method of identifying the flow patterns in vertical gas-liquid two-phase flow using differential pressure fluctuations. In this method, differential pressures are used instead of absolute pressures to eliminate the effect of static pressure fluctuations which contain information from regions outside the test section. Differential pressure fluctuations are detected with pressure taps located at one of two short intervals equivalent to the inside radius of the pipe and at one or two long intervals equivalent to ten times the diameter of the pipe. These signals are examined together. "It has been shown in [3] and [4] that the flow regimes of vertical nitrogen gas-water two-phase flows can be classified based on the characteristic features of statistical properties, especially the frequency distribution of differential pressure fluctuations; that is, the differential pressure fluctuations characterize the flow regimes in vertical two-phase flow. We can find the characteristic features in the frequency distribution such as a bimodal distribution in slug-like flow. On the basis of the specific features of the frequency distribution, namely the variance and the mean of differential pressure fluctuations, the flow patterns have been classified into four fundamental categories: bubbly, slug-like, annular, and mist flow. See [3] or [4] for more details.
Furthcrnlore. we consider an ot~teet~vc ~md ~ut~> matic identification method not based on v~ual c~+~+ servations or subjective judgement. Since the configuration of frequency distribution shows tile features of ftox~. pattern, the frequency distribution is approximated {'+y the Gram-Charlier series to obtain the fitted frequency curve. It is shown that the statistical parameters assoc++ated with the coefficients of the frequency curve, such as the mean, standard deviation, coefficient of skewness and coefficient of excess, related closely to the configuration of frequency distribution. On the basis of ex+ perimental data, an identification method without any visual observation is proposed. Fhe values of these statistical parameters can be obtained directly by processing the differential pressure fluctuations mea-+ sured without calculating the frequency distributions+ The proposed identification technique is an objective and rational one without any visual judgment, although the experimental results have been verified through comparison with those of visual observation.
2. Conception of measuring technique 2.1, Observation scales of differential pressure fluctuations Pressure fluctuations have been used to characterize various flow regimes in vertical two-phase flow. To eliminate the effect of static pressure fluctuations which contain information from regions outside the measurement section, we use differential pressures measured over relatively small vertical distances. The vertical distances should be carefully selected in order to obtain rational observation intervals which enable characterization of the flow regimes, because the frequency distribution of differential pressure fluctuations depends on the observation distance, as shown by Akagawa et al. [11]. In the present method, we use two kinds of observation scales, namely short and long scales. In order to recognize a spherical cap bubble or a cluster of small bubbles in the size of the pipe radius. the short scale, which is called in R-scale, is selected to equal the inside radius of pipe. On the other hand, the long scale, which is called an L-scale, is selected to be the distance equal to ten times the diameter of pipe, in order to identify developed gas-slugs. Thus, the test section has two kinds of observation scales mentioned above as shown in fig. 1. The observations based on these scales or four differential pressure measurements are made simultaneously to identify the flow regimes in vertical two-phase flow. An actual diagnosis of flow patterns is made mainly through the
G. Matsui / Automatic identification of flow regimes
Do
o
R - scale L - scale L - scale
"
" 2rr~
'
I
~ Apo.
R - scale
,h
Fig. 1. Arrangement of observation scales in the test section.
observations in one R-scale, and the measurements in the other R-scale and in an L-scale are used as complements of the R-scale.
2.2. Normalization of differential pressures The differential pressure Api (i = a, b, c, d) is normalized with the hydrostatic pressure drop Ap0 ~ corresponding to the measurement section.
Ap,=ApJApo,,
(i=a,b,c,d).
(1)
In the case the acceleration and friction losses are very small compared with (Apo,--Ap~), the quantity (1 - A p,) represents approximately the average void fraction in section i. Therefore it may be convenient to use ~ * = 1 - ~ P, = x,
(2)
223
scales, e.g., one pair of '~Pa and /Xpc, are used to identify accurately the flow regimes. We also note here that by using two short observation scales together, we can find the change or transition of flow pattern in the test section, as seen in fig. 2. Fig. 2 shows the frequency distributions of A P * and Ap~' for the case of transition from a developing slug flow to a fully developed one. However, two frequency distributions of A p * and A Pd* at long scales are similar and do not indicate any transition. Typical examples of frequency distribution are shown in figs. 3 to 6 for characteristic flow patterns. The variable A Pi* is essentially a continuous distribution, but is shown as a discrete distribution owing to the measurement conditions. We can obtain the probability density or frequency function of A p,* if the increment of A Pi* tends to zero. These examples are limited to cases in which the frequency distribution of A p~' exhibits similar shape to that of A p*. Experiments have been carried out using a nitrogen gas-water two-phase flow loop. The test section was made of an acrylic tube of 22 mm inside diameter. Then the R-scale and L-scale are equal to the distances of 11 mm and 200 mm, respectively. See [3] or [4] for more details. The bubbly (fig. 3), annular (fig. 5) and mist (fig. 6)
~ 0 o8
~0.08 ~O
n <~
as the statistical variable.
<~ 0.04
0o~
2.3. Statistical features for flow regimes Both the frequency distribution and the spectral distribution of differential pressure fluctuations can be used for flow regime identification. Here the former is used because the latter needs sampling of more data and more data-processing time than the former. Identification of flow regimes is conceptually feasible based on the characteristics of a frequency distribution, namely the statistical features of the fluctuations measured by using one short observation scale. However, there may be some difficulty in discrimination using only one short scale in general, except under limited situations such as incorporation of slug-like into intermittent flow, use of specified operation condition, and employment of specified fluids. Therefore, in this study one pair of short (R-) and long (L-) observation
0Apa~
°&pb*~
1
~0.20
0.20 g_
<3
K O.lO
0 ---
040
0APc*l
'
0
0&po, 1
Fig. 2. Typical example of flow pattern transition in the test section (U(; = 0.122 m/s, UL = 0.49 m/s, ~ = 0.105).
224
G. Matsui /Automatic ~dentification oj ~low regimes
,,_., 0.2C
O.30
1.0o
O.75 I !
.1(-
{3. 0.1(
13. 0.50
o
J
°o
{
A P*
[3. 0.20
0.50
0.1c
0.25
i
A pc'~
Fig. 3. Frequency distribution of differential pressure fluctuations in bubbly flow for &~. = 0.009 m/s, U e = 0.21 m / s and a = 0.0335.
1
oAp
*~a
0
'
o
,1
APe
Fig. 6. Frequency distributions of differential pressure fluctuations in mist flow for U(; = 4.06 m/s, UL = 0.003 m / s and c~= 0.994.
0.10 13_
['i ,2t
"-" 0.05
Q..
t; iii, 0Ap•l
0Apc*l
Fig. 4. Frequency distributions of differential pressure fluctuations in slug flow for UG = 0.095 m/s, UL = 0.13 m / s and a = 0.248.
flows show u n i m o d a l or single-peaked frequency distributions for A p * a n d A p * . On the other hand, the slug flow (fig. 4) shows a b i m o d a l or a twin-peaked frequency distribution for A p * and a u n i m o d a l one for A p * . These distribution configurations are similar to those for void fractions measured by Jones and Z u b e r [2]. Taking into account the variance a n d m e a n values of
v
ill
a. 0.05
2.4. Objective discrimination o f frequen£v distribution configuration
C 0.4( 0.
* o 0.10 13.
,;'lii] ,J J .... 0
1
A
differential pressure fluctuations, we found that vertical two-phase flows exhibit the following features: (a) Bubbly flow - Single-peaked frequency distribution at small A p * (average of A p,*) and small variance. (b) Slug-like flow (spherical cap bubble, slug. and churn flow) - F o r R-scale, twin-peaked frequency distribution at m e d i u m 2JP*. For L-scale, the spherical cap and slug flow cases show a single-peaked frequency distribution but the c h u r n flow case, a twin-peaked one. The variance becomes larger in the order of spherical cap bubble, slug, and c h u r n flow. The c h u r n flow shows the m a x i m u m variance. (c) Annular flow - Single-peaked frequency distribution at large A p * a n d m e d i u m variance. (d) M i s t flow - Single-peaked frequency distribution a~ A p * near unity a n d small variance. T h e above results suggest that the configuration of frequency distribution is highly useful for identifying the flow regimes.
6
Apg
Fig. 5. Frequency distributions of differential pressure fluctuations in annular flow for Uc =12.5 m/s, UL= 0 m / s and c~= 0.645.
We now consider identification of flow regimes without calculating the frequency distributions. This will make possible a u t o m a t i c j u d g e m e n t of flow patterns. In this study, we use the G r a m - C h a r l i e r series of type A to a p p r o x i m a t e the frequency distribution. One of the aims is to check whether the series shows the characteristics of frequency distributions a n d hence the statistical parameters corresponding to the coefficient of the series indicate the features of the configuration of frequency distribution. The other aim is to investigate if
G. Matsui /Automatic identification of flow regimes the series can approximate the envelope of bimodal frequency distributions because the peak heights may be used to know the structure of intermittent flow. Since the coefficients of the G r a m - C h a r l i e r series or frequency curve are associated with the distribution configuration, these statistical parameters characterize the shape of a fitted frequency curve or the original frequency distribution of differential pressure fluctuations. On the basis of the features of statistical parameters such as the mean, standard deviation, coefficient of skewness, and coefficient of excess, which can be obtained directly by processing the measurement data, an automatic identification algorithm is constructed in flow chart form. As a result, it appears that the flow patterns observed in vertical two-phase flow can be determined conceptually at least based on this flow chart.
These will apply similarly to the case of twin-peaked distribution. This is checked by fitting of the G r a m - C h a r l i e r series to frequency distributions. 3.2. A p p r o x i m a t i o n
3.1. S t a t i s t i c a l p a r a m e t e r s
Statistical parameters are as follows: variable x i = 1 - AP i mean m = Y'.xi/N standard deviation o = [~2( x i - m ) 2/ N ] 1/2 n-th moment about the mean /~, = ~ ( x i - m ) " / N (e) coeff, of skewness 71 = ]13/03 (f) coeff, of excess Y2 = ~ 4 / o 4 -- 3 where N is the number of sampled data. The relation between the parameters and the frequency function is given as follows (Cramer [12]). If the variable x i is normal,
(a) (b) (c) (d)
series
p(x)= E a'q~.(x),
(3)
n=O
where 1
__e-X
Since the configuration of the frequency distribution contains useful information for characterizing the flow regimes, we consider the automatic discrimination of distribution configuration to identify the flow regimes more objectively.
by the G r a m - C h a r l i e r
It will now be shown that the G r a m - C h a r l i e r series of type A or frequency function gives a good approximation to frequency distributions obtained by measurements. From [12], the G r a m - C h a r l i e r frequency function is given as follows:
~,0(x) = 2¢5g 3. Curve fitting of frequency distributions
225
2
/2
q,~(x) = ~ x ~ q~o(X),
a'n -
(normal frequency function),
f+5
p(x)
d x = 1,
(-1)"f _~
H, ( x ) is the Hermite polynomial of degree n. Using the standardized variable X = ( x - m ) / o corresponding to a variable with the mean m and the standard deviation o, the calculation of the coefficients of the series becomes easy. Thus eq. (3) is expressed as p ( X ) = q~0( X ) + a3q~3 ( X ) +
a4t~4
(X)
q-
+a6+6(X) + -..,
(4)
where e~o(X ) =
l~e-X2/2
1.0
UG=O. 009 m/s UL=O,21
m/s
c~ =0.0335
/~2~+I = 0 ,
/Xz~=l'3'...-(2u--l)o
2~.
In particular, 7~ = 0 and 3'2 = 0. The coefficient 71 is considered as a measure of the asymmetry or skewness of the distribution. If 3'1 is positive, the long tail will be on the positive side of the distribution, and if 3'1 is negative, the long tail on the negative side. The coefficient 3'2 is used as a measure of the degree of flattening of a frequency curve near its center. If 3'2 is positive, the frequency curve is taller and slimmer in the neighborhood of the maximum point.
x m = 0.0251 0 = 0.06 y1= 0.154 72=-0.199 as= 0.149/5! a6= 0.070/6! ~=-0. 0096 =-0.159
0
~ I . ~0
xa
Fig, 7. Bubbly flow (R-scale).
as~ 5 ( X )
226
G. Matsui / A utomatw identification ql /lo~* regimes
1.0~i
UG=0.0158 UL=0.145
a 5 :=
m/s m/s
- 10p,3)/5!.
,t~, = (/*~, - 15~4 + 3 0 ) / 6 ! .
(5)
~ =0.056
X
"K
IL 0
1.0
The curves for the derivatives 9,,(X) of the even order are symmetric a b o u t X = 0, while the derivatives of the odd order introduce an asymmetric element into the fitted frequency curves. It is desired that a g o ( ~ a p p r o x i m a t i o n be given by a small n u m b e r of lerms in the polynomial expression (3) or eq. (4) if possible. However, we see that the terms up to n = 4, which are often used for an approximation of distribution in general, do not give a sufficient approximation to bimodal distributions. Thus we consider the terms up lo n = 6 inclusive in order to approximate the bimodal distributions with asymmetry. We have tried to fit the curve of G r a m - C h a r l i e r series to the envelope of frequency distribution shown in [3] or [4]. The results are seen in figs. 7 to 12. In each figure, the flow conditions a n d the values of statistical parameters are given together with the frequency distribution. The solid curve drawn in each figure indicates
m = 0.0497 o = 0.176 yl = [.58 y2 = 3.61 as=-3.69/5! a6 = 4.61/6! B1 = 0 . 0 3 2 7 g = 0.i86
xa
Fig. 8. Spheficalcap bubble flow (R-scale).
¢.(x) = dd;-~¢o(X), a 0=1,
(~
a 1=0,
a3 = - / x 3 / 3 r
a4(#4-3)/4!
a 2=0,
('gl = - 3 r a 3 ) ,
(3'2=4!a4),
UG=0.095
m/s
UL=0.13
m/s
l°ij ~J
=0,248 X m
=
O. 3 4 9
<7 =
O. 6 3 2
y1 =
0.238
rn = 0.L92 = 0. 156 T ~=-0.077 "~2=-0. 345 as= 0.264/~! ar,=-l. 53/6 ! ~=-0.00754
v 2 = - i 117 as = 2.02/5[ a6 = 6.50/6[ ~i = 0.34 B = 0.538
U
[~ =-0. 048
"a
i ,v
1 0
X
C
Fig. 9. (a, left) Slug flow (R-scale), (b, right) Slug flow (L-scale).
1.Or
UG=2.39
m/s
UL=0.12
m/s
(~ = 0 , 6 4 5 m
=
cr =
,"'
1.0 T
X U ! jI i[ m = 0.451 o = 0. 353 yi=-1.44 y2 = i. 54 as=-2.79/5 ! a6=-4.14/6! $t=-0.165 =-0.467
0.504 0. 7 9 2
y I=-0.
929
T
[
y2 = 1 . 4 5 8 as = 0.318/5! a6=-2.2/6! BI=-0.067 B =-0. 084
0
1.0
xa
Fig. 10. (a, left) Churn flow (R-scale), (b, right) Churn flow (L-scale).
1.0
X C
G. Matsui / Automatic identification of flow regimes
1.0 T
UG=I2.5 m/s UL= 0
] .0 T
I
~Xr~ 7=,
I
m/s
Ot =0. 876 m = O. 916 (7 = 0.453 y:= 0.258 yz=-0.0266 as: 0.222/5!
a6=-0"372/6! B1= 0.037 = O. 081 0
I t
+-
m = O. 884 o = O. 029 YI=-0.0087 y2 = 0.102 as = 0.01/5! a6=-1.04/6! B i=-0. 016 B =-0.536
I! 't
]]il
I-~
- - 4
0
xa
1.0
227
1.0
X
C
fig. 11. (a, left) Annular flow (R-scale), (b, fight) Annular flow (L-scale).
1.0:
UG=4.06
m/s
].0
T
UL=0.003 m/s ¢t =0.994
"J
I m = 0.99 d = O. 0516 y1 = 0.031 y2 = 0.015 as= 0.368/5! a6=-0.24/6! B~ = 0.0004 fl : o.oo7
I
+
k .0
X
m = 0.974 (7 = 0.0149 yi=-0.993 Y2 =-i. Ol a5=-6.96/5! a6 = 5.14/6!
T
+
BI = 0.006 6=o.4
0
1 0
X
C
Fig. 12. (a, left) Mist flow (R-scale), (b, fight) Mist flow (L-scale).
the fitted or approximate frequency curve with terms up to n = 6. The frequency distribution and fitted curve are normalized by the values of the maximum point, respectively. In fig. 9(a), the G r a m - C h a r l i e r frequency curves are compared for different values of n. The fit to the distributions is very good, except that the curves have sometimes negative values or a low tertiary peak. The results shown above suggest the possibility of flow pattern discrimination based only on the statistical parameters characterizing the curve shape or the frequency distribution configuration.
4. D i s c u s s i o n of useful parameters
In figs. 7 to 12, the values of statistical parameters are also given. We shall consider the relation between the shape of the distribution and the values of the parameters. Here distributions considered to be of normal-type appear in bubbly, annular, and mist flows for the R-scale. Moreover they appear in bubbly, spherical
cap-bubble, slug, annular, and mist flow for the data of L-scale. (a) the mean, m The value of m becomes larger with the transition of flow pattern from bubbly flow (B) to mist flow (M) through slug-like flow and annular flow (A). Thus we can use m as a criterion for flow identification. The mean can be used directly for identification if the fluid and the operating conditions are fixed. Unfortunately there is a disadvantage to the use of the mean since it can sometimes shift especially for the distribution in an R-scale, depending on the operation conditions. (b) the standard deviation, o The value of a becomes larger with transition from bubbly flow to churn flow (C) through slug (S), but the value is smaller again in annular and mist flows. In the case of o < 0.1, frequency distributions resemble a normal distribution. (c) the coefficient of skewness, 3`1 Distributions with small 13'1 I will be considered nor-
228
G. Matsui /Automatic tdenttfication el flow regimes
real. On the other hand, distributions with large 17~i may be judged to be those with long tail (on the positive side for "y1 > 0, and on the negative side for Y1 < 0) or bimodal. Therefore, we may consider the distribution with the value of I "¢~ [ < 0 5 to be normal. Moreover, for a distribution of x a with large and positive 71, the flow is judged to be spherical cap bubble (SP) or slug flow. If we obtain a distribution of x a with large and negative "h, the flow is judged to be slug or churn flow. (d) the coefficient of excess, 72 Distributions with small ] Y2 ] will be considered normal. If we obtain a distribution of x, with positive and large "h, the flow is judged to be spherical cap bubble or churn flow because the fitted curve is taller and slimmer than the normal one in the neighborhood of the maximum point. On the other hand, if we get a distribution of x~ with negative and large "h, the flow is judged to be slug flow because the fitted curve is lower and flatter than the normal one.
[[ Identification using data of R- scale ( aPa or ~Pb) ]I I Calculation of m, o, x 0, ~l, Y~, ~i and
1
I
I
YES
¢
+ I
[ Identification using data of L - scale ( aPc ) ] Calculation of m, ~, % , YI, Y2, ~x and
(e) the coefficient of the fifth derit~ative, a 5 When the value of [ a 5 I is large, the distribution will be considered to have a tail. Distributions with small ['Y1 I and [a 5 [ may be considered to be of normal-type. (f) the coefficient of the sixth derivative, a 6 Distributions with small I a6 [ are similar to normaltype. (g) fll m x o The quantity/3~ may be used as a measure of the asymmetry or skewness of the distribution. A distribution with small I/3a I will be considered normal. =
- -
(h) fl = f l l / o (Pearson's measure of skewness [12]) A distribution with small I/3[ will also be considered normal.
5. Automatic identification of flow regimes 5.1. Flow chart for identification
I "~1
'
<
!~r2 J
k5
YES
I < k10
q ~ < k12
YES
,o
Churn Flow unidentifiable i f not judged SC)
I I
Mist Flow I ( unidentifiable i f not judged BM)
Spherical Cap Bubble Flow, i f judged SP Slug Flow, i f judged S or SC Annular Flow, i f judged A Bubbly Flow, i f judged BM
I
Print or Indication of result ]
It has been shown that the frequency distribution of differential pressure fluctuations can be fitted by the frequency curve of the G r a m - C h a r l i e r series. Furthermore, the statistical parameters including the coefficients of the series give the features of the distribution configuration and hence flow patterns. On the basis of these results, a flow chart for automatic identification of flow regimes is constructed as shown in fig. 13 taking into account the features of statistical parameters. If the flow pattern identified from x a is different from that of
Fig. 13. Flow chart for flow regime identification.
Xb, the flow may be judged to be in transition in the test section. The values kl-k13 in the figure denote the threshold
G. Matsui / Automatic identification of flow regimes of judgment. These values should be determined on the basis of sufficient experimental data.
229
I .O!,T
5.2. Examples of identification Based on the data shown in figs. 7 to 12 and further experimental data, the thresholds kl-k13 are determined as follows: k l = 0.4,
k 2 = 0.4,
k 5 = 0.14,
k 6 =
k 9 = 0.5,
kl0 = 0.01,
0.25,
k 3 = 0.0l,
k 4 = 0,1,
0.4,
k 8 = 05,
k 7 =
kll = 0.1,
k12 : 0 . 1 ,
w
0
(6)
.0
x
1.o
×
a
l.OTi ! II!
kl3 = 0.95.
Typical results of automatic identification using this flow chart are shown in figs. 14 to 16. Experiments have been carried out using another nitrogen gas-water twophase flow loop. The text section was made of an acrylic tube of 20 mm inside diameter. Thus the R-scale is equal to the distance of 10 mm and the L-scale the
uT
o 1
n
Ira..L
Fig. 15. Slug flow (Uo = 0.179 m / s a n d U L = 2.71 m / s ) .
I.OT
0.51.0
(i
[ i1
xa
l.O
01.0
xa
1.0 T (.,9 X
- - o " o'.s
i'.o
xc
Fig. 14. B u b b l y flow ( U o ~ 0.146 m / s a n d U L = 0.414 m / s ) .
Fig. 16. C h u r n flow ( U o = 1.71 m / s a n d U L = 0.377 m / s ) .
01.0
x
C
230
G. Matsui / Automatic tdentification oj flow regtmex
distance 200 mm. The result of identification is indicated in the figure caption in each figure. For reference. the picture of flow and frequency distributions of differential pressure fluctuations at both short and long observation scales are shown in each figure. In the case of transient flow, the determination of flow pattern will be difficult because it may change during the measurement. Even if the results of identification for flow in the transition region ~rre different from those of visual observation, this will not always be due to misjudgment. In such cases, it may be advisable to indicate the results of identification in terms of probabilities.
tained at several points along the flow direction, i~ i~ desirable to determine the threshold of parameter~ ,,)n the basis of more sufficient data under ~arious expew mental conditions.
Acknowledgement A part of this study was supported by CASIO Science Promotion Foundation.
Nomenclature 6. Conclusions a, An automatic and objective technique of flow regime identification has been proposed for vertical two-phase flow. It is a technique using differential pressure fluctuations measured at one pair of rational observation scales, namely the short scale equal to the inside radius of a pipe and the long scale equal to ten times the inside diameter. The experimental results for nitrogen gas-water flow show that the flow regimes in vertical two-phase flow exhibit the characteristic features of the statistical properties of differential pressure fluctuations and that the flow patterns are determinable based on the classification of the statistical features. Moreover, to make the identification less subjective, the envelope of frequency distributions is fitted by the polynomial expression of the Gram-Charlier series of type A. The results of curve fitting showed that the statistical parameters associated with the coefficients of the frequency curve obtained characterize the frequency distribution or flow regime, and therefore, the automatic and objective judgment of flow regimes is feasible. Thus, an objective and automatic method of flow pattern identification for vertical two-phase flow was developed in flow chart form on the basis of the features of statistical parameters such as the mean, standard deviation, coefficient of skewness, and coefficient of excess. The values of these parameters can be obtained by processing directly the differential pressure fluctuation measurements without calculating the frequency distribution and fitting the curve. In addition, because the identification is conceptually feasible even by use of one short observation scale, the present technique using differential pressure fluctuations can be extended to identification of a developing flow pattern, if the measurements of short observation scale are oh-
a~ L m N
p(x) Ap Ap0 Ap
normalized coefficient of the n-th derivative of the Gram-Charlier series ( = a',,/a"), coefficient of the n-th derivative of the Gram-Charlier series, length of long observation scale. mean of variable, number of sampling data, frequency distribution of x or probability density or frequency function of x, differential pressure, hydrostatic pressure drop, normalized differential pressure ( = Ap / A Po ),
Ap*
=I--AP=x,
R U x x0 X a
length of short observation scale, superficial velocity, variable ( = 1 - Ap), maximum point of frequency curve, standardized variable ( = (x - m )/o ), void fraction,
~1
~g Y2 /% o q,0(x)
= m -- x0,
coefficient of skewness, coefficient of excess, n-th moment about the mean, standard deviation, "~ / normal frequency function ( = e x p ( - x - / 2 L
2¢Tg), q~,,
= d"q~o(x)/dx ~.
Suffix G i L -
gas, observation section of measurement section ( a, b, c, d), liquid, average.
G. Matsui / Automatic identification of flow regimes References [1] M.G. Hubbard and A.E. Dukler, The characterization of flow regimes for horizontal two-phase flow, Proc. Heat Transfer and Fluid Mechanics Institute (Stanford Press, 1966) pp. 100-121. [2] O.C. Jones and N. Zuber, The interrelation between void fraction fluctuations and flow patterns in two-phase flow, Int. J. Multiphase Flow 2 (1975) 273-306. [3] G. Matsui and S. Arimoto, Flow pattern estimation of gas-liquid two-phase mixture in a perpendicular pipe, ICHMT-1978 Int. Seminar on Momentum, Heat and Mass Transfer in Two-Phase Energy and Chemical System, Dubrovnik, Yugoslavia, 1978. [4] G. Matsui, Identification of flow regimes in vertical gas-liquid two-phase flow using differential pressure fluctuations, Int. J. Multiphase Flow 10 (1984) 711-720. [5} T. lnoue, M. Agui, G. Matsui and S. Arimoto, Quantitative flow-pattern discrimination of gas-liquid two-phase mixture (in Japanese), Trans. SICE 17 (1981) 231-236. [6] M.A. Vince and R.T. Lahey, Jr., On the development of an objective flow regime indicator, Int. J. Multiphase Flow 8 (1982) 93-124.
231
[7] N.K. Tutu, Pressure fluctuations and flow pattern recognition in vertical two-phase gas-liquid flow, Int. J. Multiphase Flow 8 (1982) 443-447. [8] N.K. Tutu, Pressure drop fluctuations and bubble-slug transition in a vertical air-water flow, Int. J. Multiphase Flow 10 (1984) 211-216. [9] G. Matsui, Identification of flow patterns in horizontal gas-liquid two-phase flow using differential pressure fluctuations, Preprints, Int. Symp. on Fluid Control and Measurement (Pergamon Press), Tokyo, 1985, pp. 819-824. [10] J. Weisman, D. Duncan, J. Gibson, and T. Crawford, Effects of fluid properties and pipe diameter on two-phase flow patterns in horizontal lines. Int. J. Multiphase Flow 5 (1977) 437-462. [11] K. Akagawa, H. Hamaguchi, T. Sakaguchi and T. Ikari, Studies on the fluctuation of pressure drop in two-phase slug flow, Bull. JSME 14 (1971) 447 469. [12] H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946).