A modified L2-star discrepancy method for measuring mixing uniformity in a direct contact heat exchanger

There is a linear relationship between the flow patterns of a bubble swarm and heat transfer from the viewpoint of experimental analysis. To investigate the space-time features of the mixing process in a direct contact heat exchanger, uniformity coefficient (UC) method based on L2-star discrepancy (UC-LD) for assessing the mixing time of bubbles behind the viewing windows is effective. Although the UC makes a distinction between the local and global uniform with the same Betti number, the properties of UC have not been explored and there was a great influence of calculating initial position on the original UC, namely UC-LD. In this paper, we extend the UC-LD to the modified uniformity coefficient based on modified L2-star discrepancies such as centered discrepancy and wrap-around discrepancy. Uniformity coefficient based on centered discrepancy (UC-CD) and uniformity coefficient based on wrap-around discrepancy (UC-WD) theoretically satisfy a Koksma-Hlawka type inequality. Moreover, they exhibit some advantages such as permutation invariance, rotation invariance (reflection invariance) and the ability to measure projection uniformity. Real experimental cases are analyzed and simulations are conducted to evaluate the performance of the proposed novel method. The study also find that UC-CD presents more sensitive performance than UC-WD finally, so we recommend using the former industrially.