-
Discrepancy measure for uniformity qualification ofbubble
movement in a direct contact heat exchangerQingtai Xiao1,2, Jianxin
Pan3*, Junwei Huang1, Jianxin Xu1,3, Hua Wang1,2
ISIMet 2017
2nd InternationalSymposium
onImage basedMetrology
Maui, Hawaii
December 16-21, 2017
Abstractis presentation aims to introduce three novel methods
for assessing the distribution uniformityof bubbles and determining
their mixing time to uniformity. ese uniform design based
methodsare illustrated through image analysis in a direct-contact
heat exchanger (DCHE). e noveltiesinclude the lile constraint of
nal evaluation value and the ecient algorithm of bubble
centroid,leading to fast and accurate characterization of mixing
uniformity. To determine the eects ofresolution aspect (considering
two types of aspect ratios: 16:9 and 4:3) and local region on
themeasurement, various evolutions of bubbles movement and three
mixing measures in the DCHEare investigated experimentally. Real
experiments and simulations are conducted, showing that theproposed
methods outperform the existing methods such as Bei number. It is
also shown that thelocation eects of bubbles can be measured
successfully using these discrepancy measures, whichbrings a new
insight into the comparison of mixing state of dierent
systems.Keywordsmeasure of uniformity — bubbles distribution —
direct contact heat exchanger1State Key Laboratory of Complex
Nonferrous Metal Resources Clean Utilization, Kunming University of
Science andTechnology, Kunming, PR China2Faculty of Metallurgical
and Energy Engineering, Kunming University of Science and
Technology, Kunming, PR China3School of Mathematics, The University
of Manchester, Manchester, United Kingdom*Corresponding author:
[email protected]
INTRODUCTIONe purpose of mixing is to obtain a homogeneous
mixtureor a certain degree of uniformity of mixtures in
non-reactivesystems [1, 2, 3]. Mixing uniformity has a decisive
impact onthe overall performance of mixing processes and can
some-times serve as a surrogate for other properties, such as
thequality grade of print defects [4], the heat transfer
perfor-mance of uids [5, 6], the mechanical behavior of
materials[7], the drug content of monolithic devices [8], and so
on.ere is an increased desire for measuring and comparingmixing
uniformity which is required from a practical pointof view and for
validation of theoretical models as well invarious elds [9, 10,
11].
In the literature, the image processing technology hasbeen
widely used for feature extraction and multiphase mix-ing
quantication in single phase, gas-liquid, solid-liquidand
gas-liquid-solid systems [12, 13, 14]. A large number
ofexperimental studies have been also devoted to demonstrat-ing its
eciency in mixing time estimation for congurableoptimization [1, 2,
6, 8]. ere is however no universallyaccepted image-based method for
determination of mixingtime mainly because each one has its own
limitations, suchas conductivity method, pH probe method, dual
indicatorsystem method [15], tracer concentration method,
colorationdecoloration method [16] and box-counting method
[17],etc. Some limitations have been reported in details, e.g.,
in[5, 6, 18].
In earlier publications, Bei number was reported in [1]as one of
the most ecient methods in determining criti-
cal mixing time of mixing process [19] and acquiring morespatial
evolutions information of ow eld associated withheat transfer
performance of uids [20]. However, we foundthat mixing time and
uniformity determined by Bei numberhave an issue of spatio-temporal
limitation. Particularly, themixing uniformity of bubbles in a
rectangle area with thesame Bei number (β0 when white pixels are
the backgroundor β1 in other situations) is very dierent.
e issue of measuring the space-time uniformity of ran-dom
bubbles in a recorded image was addressed by the uni-formity
coecient (UC) method [3] and modied UC method[21] in certain
extents. But in the denition of reportedUC methods, it is
reasonable that the nal evaluation valueshould be between 0 and 1.
Otherwise, the denition may benot sensible. Fang et al. pointed out
that one should choose aset of experimental points with smallest
discrepancy amongall possible designs of a given number of factors
and exper-imental runs [22], which satises the limit condition.
Onthe other hand, the nding procedure of bubble centroidcan be made
eciently using the Image Processing Toolbox(IPT) in Matlab and its
in-house function, which performsbeer in computational complexity
than traversal operationof elements by [3] and [21].
Inspired and motivated by the previous work, we aim
toinvestigate the space-time feature by the denition of
dis-crepancy directly and present an analysis of numerical
simu-lations and experiments of the mixing process.
Discrepancyconcept and thresholding algorithm are applied to
quantifythe space-time mixing uniformity of random bubbles in a
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Measures of uniformity in DCHE — 2/5
direct contact heat exchanger (DCHE) in this paper.
1. METHODSSuppose there are n bubbles of interest (i.e., black
or whiteelements) over a binary imageC2 = [1, P]×[1,Q] (i.e., a
P×Qmatrix mathematically). e goal here is to assess whetheror not
the bubbles are uniformly scaered on the processedimage. Let Fu(x)
= x1x2 be the uniform distribution functionwith a given point x =
(x1, x2) where 1 6 x1 6 P and1 6 x2 6 Q, and FX(x) be the empirical
distribution functionof bubbles set X =
(xT1 , x
T2 , · · · , xTn
)T, i.e.,FX(x) =
1n
n∑i=1
I[xi,∞](x) =](X⋂[1, x])
n(1)
where I[xi,∞](x) is the indicator function and ](X⋂[1, x])
denotes the number of bubbles of X falling into the localregion
[1, x] = [1, x1] × [1, x2]. e L∞-star discrepancy of Xon C2 (star
discrepancy for short) is dened as
D∗∞(X) =‖ FX(x) − Fu(x) ‖∞= supx∈C2
��FX(x) − Fu(x)�� (2)where x decides the area and controls the
number of the givenpoints within the area (i.e., the number of
local regions), andsup denotes the superior/maximum of the local
discrepancyfunctions (LDFs).
Statistically, the centered discrepancy (CD) and the wrap-around
discrepancy (WD) exhibit some advantages such aspermutation
invariance, rotation invariance (reection in-variance) and the
ability to measure projection uniformity,but star discrepancy does
not. Assume that xi1, xk1 are thex-axis values and xi2, xk2 are the
y-axis values of ith, kthbubbles,respectively, the analytical
expressions for CD andWD are given respectively as follows:
CD(X) ={ (
1312
)2− 2
n
n∑i=1
2∏j=1
(1 +
12|xi j −
12|
− 12|xi j −
12|2)+
1n2
n∑i=1
n∑k=1
2∏j=1
(1
+12|xi j −
12| + 1
2|xk j −
12|
− 12|xi j − xk j |
)} 12(3)
and
WD(X) ={−
(43
)2+1n
(32
)2+
2n2
n−1∑i=1
n∑k=i+1
2∏j=1
( 32− |xi j − xk j | + |xi j − xk j |2
)} 12 (4)Obviously, the greater the value of the discrepancy,
the
more non-uniform the bubbles in a given region. In order to
measure the eects of robustness and assess the performanceof
three measures, experiments are carried out in a DCHE. Asan eective
way to use energy, DCHE has been applied andresearched extensively
in energy recovery from industrialwaste. It can transfer the heat
of continuous phase (i.e., heatconduction oil) to dispersed phase
(i.e., organic uid), whichcan be inuenced by four factors with
three levels. Let E1-E9denote nine dierent experimental levels
based on orthogonaldesign table L9(34).
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
0 . 1 0
0 . 1 2
0 . 1 4
0 . 1 6
4 : 3
Star d
iscrep
ancy
[-]F r a m e [ - ]
1 6 : 9
I n i t i a l s t a g e o fe x p e r i m e n t E 4
Figure 1. Eect of dierent resolution on star discrepancy
e images of bubbles are captured using a high-speedvideo camera
with brand PRAKTICA of Germany and resolu-tion 4 million pixels
with no LED light. Generally speaking,the objective is bubble
(i.e., gas phase) and background rep-resents the liquid phase. For
the convenience of engineeringcalculation and programming, we set
P=1280 and Q=720while adopting equal interval sampling of 300 which
cor-responds to 5 min. A processed bubbles image is
selectedrandomly for representing the binary mixture, as shown
inthe inserted illustration of Fig. 1
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
0 . 1 0
0 . 1 2
0 . 1 4
0 . 1 6
Star d
iscrep
ancy
[-]
F r a m e [ - ]
s m a l l n u m b e r s g r e a t n u m b e r s
E x p e r i m e n t E 4
Figure 2. Eect of dierent local regions on D∗∞(X)
e novelty of the presented contribution are two-fold:using the
mathematical measure of discrepancy and applying
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Measures of uniformity in DCHE — 3/5
the IPT function regionprops directly, leading to
lowcomputational cost and high degree of accuracy. us, theeects of
resolutions aspects and numbers of local regionson the denitions of
three discrepancies in this paper areall investigated. To save
space, Fig. 1 only gives the D∗∞(X)trends over time when the pixels
size reduces from 16:9 to4:3. e reason why the results are inuenced
by the aspectratio of the image is that the LDFs ratio relies on
the arearatio of images with dierent sizes. In terms of the
mixingtime determination, it shows that the inuence of
homog-enization curve by our method does not change a lot withthe
resolution size. Likewise, CD(X) and WD(X) are bothdependent on the
coordinate axes established for the bub-bles image. As magnication
factor (px/mm) is still similar,the aspect ratio does not have an
inuence on the resultsif adequate equipment is used. As shown in
Fig. 2, there issubtle inuence of number of local regions on the
trend ofuniformity curve.
2. RESULTS AND DISCUSSION2.1 antification of mixing stateAs
shown in Fig. 3, quantitative comparisons of the uni-formity curves
determined by the proposed techniques areconducted with our
previous experimental data. Specially,here there is a clear and
distinctive dierent at the early stageof mixing process of
experimental cases. Obviously, any dif-ference in numerical
performance has a deep signicant rolein quantifying the mixing
state.
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00 . 0 0
0 . 0 5
0 . 1 0
0 . 1 5
0 . 2 0
0 . 2 5
0 . 3 0
0 . 3 5
Discre
panc
y [-]
F r a m e [ - ]
S t a r d i s c r e p a n c y C e n t e r e d d i s c r e p a n
c y W r a p - a r o u n d d i s c r e p a n c y
E 2
Figure 3. Comparison of three discrepancies curve for E2.
e results show that good agreements of the criticalpoint
obtained by existing and proposed methods are made,see Table 1. It
is observed that the eectiveness of thosetechniques are valid.
However, the Bei number methodfails to take into account the
location eects of bubbles withrespect to mixing transient.
2.2 Account for the location eectsIt may not be correct to
conclude that the best transientsare that present the highest Bei
number for special cases,
Table 1. Mixing time obtained by dierent methods.
methods β1 D∗∞(X) CD(X) WD(X)E1 156 151 151 151E2 93 97 95 96E3
168 172 170 170E4 225 224 225 224E5 122 120 120 120E6 84 84 82 83E7
262 259 195 198E8 117 118 118 118E9 128 127 127 127
as the purely numerical approach does not take positiondetails
into account. In practical applications, one can moreaccurately
calculate the value of discrepancy for choosingthe best
experimental condition for space-time feature ofbubbles, which is
the main advantage of our methods.
In order to verify the feasibility of our methods, two
pro-cessed images which both present 194 bubbles are taken froman
integrated experiment, as shown in Fig. 4 (a) and (b). eright-hand
plot in Fig. 4 (c) presents the dierence of mixinguniformity in the
special transient with the same Bei num-ber β1=194. Comparisons
show that dierent experimentalcases with the same Bei numbers can
be detected by thediscrepancy measure. e combined results in Fig. 4
indicatethat the proposed evaluations may be enough to obtain agood
determination and quantication of mixing state ofgas-liquid
two-phase ow.
Figure 4. Mean absolute discrepancy evolutions of
dierentexperimental images with the same β1=194.
In addition, the movement rules of bubbles in DCHEare also
discussed by the method of numerical simulation.According to Fig.
5, the green regions in the three insertedimages refer to the
bubbles in DCHE. In particular, threesets were randomly generated
and xed with β1 = 234 bymeans of the Matlab soware. Just like
usual, each imagehas the dimension of 1280×720. Fig. 5 also depicts
that thestar discrepancy would be inuenced by the initial
positionfor calculating local discrepancy function, including
top-le(TL), boom-le (BL), boom-right (BR) and top-right (TR),but CD
and WD do not. It is also interesting to point outthat when one
wants to assess the mixing uniformity moreaccurately, the CD may
outperform the WD and is more
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Measures of uniformity in DCHE — 4/5
sensitive to practical engineering application in some
sense.
Figure 5. Dierent discrepancy measures for quantifyingthe
particle mixing state in three dierent synthetic imageswith the
same β1=234.
2.3 Verification of three propertiesAssume centroid coordinates
of bubbles are wrien as,
X =©«
x11 x12...
...xn1 xn2
ª®®¬ (5)where n is the number of bubbles in an image processed
byclassical thresholding algorithm (for example, Otsu
method)mentioned earlier. As mentioned early, the rst column
ele-ments x11, x21, · · · , xn1 of X correspond to the x-axis
valuesand the second column elements x12, x22, · · · , xn2 of X
cor-respond to the y-axis values. Let CD(X) and WD(X) denotethe two
dierent measures of uniformity individually.
eoretically, CD(X) and WD(X) have their own advan-tages. To
begin with, they are both invariant to disruptedorder of the
experimental points. Second, they are invariant ifxi1 and xi2 are
replaced individually by 1-xi1 and 1-xi2, 1≤i≤n.ird, the projection
uniformity over all sub dimensions istaken into account, and CD(X)
andWD(X) are both invarianton the subspaces. e resulting data
verication are depictedin Table 2, 3, and 4, respectively. Hence,
we conclude that thedierent experimental paerns with the same Bei
numberscan be identied by the proposed discrepancy measures.
In-terestingly, those provides a tool to investigate the dierenceof
mixing state quantication between 2-dimensional areaand
3-dimensional spaces.
3. CONCLUSIONSIn this paper, the mixing state of gas-liquid
two-phase owin a DCHE for waste-heat utilization is quantied
throughthree novel statistical measures with reference to
discrep-ancy. e proposed methods do not rely on the
perceiveduniformity (resulting from relaxing the range constraint
of
nal evaluation values, and the nding algorithm of bubblecentroid
based on [3] and [21]. Summarily, three assertionsfor this
narrating work are list as follows.
(1) With respect to the star discrepancy, the local discrep-ancy
function of a set of bubbles seems to be a useful conceptto measure
the irregularity of the distribution of bubbleswithin a rectangle
region. e scheme is based on partition-ing of image data obtained
from a high-speed video camera,thus leading to a new consideration
in the interpretation ofow visualization.
(2) Using the three image analysis techniques (star
dis-crepancy, CD and WD) processed in the ImageJ or Matlabsoware,
the evolution of bubbles movement is tracked ex-perimentally. e
inuences of expressions, iteration stepsand pixels on the mixing
eciency are also discussed. Com-pared to [3] and [21], the proposed
approaches are easy toimplement and are computationally low
cost.
(3) e CD and WD are more reasonable than the stardiscrepancy in
terms of mixing state quantication in thiscurrent study. at is to
say, the former two could satisfythe three properties including
invariance to permutation,invariance under reection, and projection
uniformity, butthe later one does not.
Based on the aforementioned observations, it is believedthat the
proposed approaches can be applied to measuremixing uniformity with
a good accuracy in pharmaceutical,chemical, metallurgical, printing
and medical industries, etc.
Table 2. Verication of invariance to permutation
Discrepancy originalorderdisordered
bubble coordinates bothCD(X) 0.0249 0.0249 0.0249 0.0249WD(X)
0.0277 0.0277 0.0277 0.0277
Table 3. Verication of invariance under reection
Discrepancy noreectedreected
x = 12 y =12 both
CD(X) 0.0249 0.0249 0.0249 0.0249WD(X) 0.0277 0.0277 0.0277
0.0277
Table 4. Verication of projection uniformity
Discrepancy noprojected y=0 x=0projectedto origin
CD(X) 0.0249 0.6025 0.6009 0.9446WD(X) 0.0277 0.4720 0.4717
0.6812
ACKNOWLEDGMENTSe authors wish to extend special thanks to
anonymousreviewers for numerous detailed questions and
constructive
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Measures of uniformity in DCHE — 5/5
comments that greatly improved the presentation. is workwas
nancially supported by National Natural Science Foun-dation of
China (Nos. 51666006, 51406071 and 51706195),Joint Funds of the
National Natural Science Foundation ofChina (No. U1602272),
Scientic and Technological Lead-ing Talent Projects in Yunnan
Province (No. 2015HA019)and Academician Workstation of ZHANGWenhai
in YunnanProvince (No. 2015IC005).
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IntroductionMethodsResults and DiscussionQuantification of
mixing stateAccount for the location effectsVerification of three
properties
ConclusionsAcknowledgmentsReferences