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: jianxin.pan@manchester.ac.uk

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

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

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

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

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