Improvement of unsteady friction model of transient flow in pressurized pipeline

Water hammer can often occur in a pressurized pipeline flow system, leading to the pipeline explosions and even casualties. Therefore, it is of great significance to accurately predict the water hammer in the pressure pipelines for the safe operation. Among them, the weighting function-based (WFB) unsteady friction can be represented by the ZIELKE model in the simulation of water hammer, in order to effectively describe the frictional effects in the transient flow. However, the flow rates are required from all previous time steps at each time. In computational efficiency, the ZIELKE weighting function has been utilized to approximately sum a finite number of exponential terms, and then combine with the recursive formulas for the unsteady friction terms. But, the existing approximation models have also been limited to the low range of applicability or high computational costs. In this study, a two-stage least squares fitting (TSLSF) has been introduced to simulate the unsteady friction of transient flow in the pressurized pipeline. The initial parameters were obtained to fit the segmentation-based least squares segmentation of the ZIELKE weighting function in dimensionless time. After that, the new approximate weighting function was followed by a global least square fitting over the entire dimensionless time. Finally, 19 exponential terms were included in the TSLSF approximate weighting function. The TSLSF approximate weighting function was better fitted the ZIELKE weighting function than before, indicating the relative percentage error close to 0. Moreover, a broader range of applicability was also achieved to effectively simulate the unsteady flows. The experiment of water hammer was carried out in a reservoir-pipeline-valve system, in order to compare the computational efficiency and accuracy of the TSLSF friction model with the existing WFB. The computational accuracy of TSLSF model was closely aligned with the ZIELKE in the pressure wave curves at the midpoint and valve. Furthermore, the maximum relative percentage errors of the pressure head were 0.15% and 0.17%, respectively, similar to the URBANOWICZ model with the value of 0.15%. Therefore, the TSLSF and URBANOWICZ model exhibited the high computational accuracy. The computational efficiency was then compared in the calculation time of different friction models at different time steps. Specifically, the calculation time of the TSLSF model was smaller than that of the ZIELKE and the URBANOWICZ model. The computational efficiency of TSLSF model was improved by about 90% and 30%, respectively, compared with the ZIELKE and URBANOWICZ. In summary, the TSLSF friction model demonstrated the better performance in the applicability, computational accuracy, and computational efficiency. The broader range of applicability was achieved to improve the computational efficiency and accuracy. The accurate and efficient simulation of water hammer was realized in the pressurized pipelines. This finding can also provide the theoretical support and reference for the accurate and efficient simulation of transient flow in the pressurized pipelines