**Acknowledgement**

of the mixture can be characterized by the Betti numbers for binary images of the pat-

terns, leading to a useful parameter to characterize the mixture homogeneity. The first

novel method may be applied for studying a variety of multiphase mixing problems in

**2.**  In a DCHE, Betti number can estimate the number of bubbles assembling in flow patterns and to get the pseudo-homogeneous time. Experimental analysis constructs a simple linear model representing a bubble swarm and the heat transfer performance of a DCHE. In addition, the Betti number average and the pseudo-homogeneous time *t* define a new index. A better fitting curve about and the volumetric heat transfer coefficient average is received and its correlation coefficient is 0.95. A paradigm is established on the basis of this novel method for the study of flow patterns and heat transfer performance. And the paradigm offers an optional route to study the relationship of flow patterns and heat

**3.**  A novel method relying on image analysis and statistics was developed to estimate the mixing time accurately in a DCHE. The three sigma method researches the critical point determination of the pseudo-homogeneous process, which satisfies approximately normal distribution and surpasses the range of occurring twice. The mean value method, slope method and standard deviation method make quantitative comparisons of the mixing time. In addition, time intervals between in-homogeneous time and mixing time quantify the quasi-steady state. Neglecting critical point could make substantial errors in

ancy (UC-LD), presented for assessing the uniformity and mixing time of bubbles behind the viewing windows in a DCHE is effective. An imaging technique processed in the MATLAB software tracks the evolution of bubbles movement. The local discrepancy of a set of bubbles seems to be helpful to judge the difference between theory and empirical distribution. The UC links to a discrepancy, leading to a useful parameter which expresses the mixture homogeneity and mixing time. A comparison was made between the mixing time and uniformity obtained by UC method and the data obtained by Betti numbers method. Discussing the simulation and experiments conducted between local and global uniform (with the same Betti numbers) and examples are given for illustration. UC method calculates the space-time features of the mixing process successfully. The UC curves can study and compare mixing efficiency of different systems with the novel method, which can generate accurate mixing information and has a well reliability.

**5.**  The properties of UC have been explored and there was a great influence of calculating the initial position on the original UC, namely UC-LD. The UC-LD method

A Koksma-Hlawka-type inequality is applicable to uniformity coefficient based on wrap-around discrepancy (UC-WD) as well as uniformity coefficient based on centred discrepancy(UC-CD) theoretically. In addition, they show some advantages includ-

applies to the modified uniformity coefficient based on modified *L*<sup>2</sup>

**4.**  A straightforward method, uniformity coefficient (UC) method based on *L*<sup>2</sup>

which multiphase components or tracers are visually distinguishable.

are introduced to characterize the non-homogeneity of the mixture. This

are used to estimate the numbers of pieces in the pat-



terns. The zeroth Betti numbers *β*<sup>0</sup>

170 Heat Exchangers– Design, Experiment and Simulation

transfer in other heat transfer processes.

mixing time estimation, which is proved.

Betti numbers *β*<sup>1</sup>

This work is supported by the National Natural Science Foundation of China (51666006, 51406071) and Scientific and Technological Leading Talent Projects in Yunnan Province (2015HA019).
