A crescent-shaped papermaking machine has been widely applied in recent years, due to its high efficiency and energy conservation. However, the pulp waste liquid can be sprayed towards the guard plate of the water absorption box at high speed after paper production. Huge mechanical impact and damage to the guard plate can also result in the waste of the kinetic energy carried by the pulp waste liquid. Therefore, this article aims to recover and utilize the residual pressure energy carried by the pulp waste liquid through a crossflow turbine. Numerical studies were also performed on two schemes of crossflow turbines at multiple rotational speeds（Scheme I featured a single deflector guiding flow to impact the left side of the runner (inducing counter-clockwise rotation), with an additional deflector near the forming roller to prevent water splash affecting study quality. Scheme II was utilized as a dual-deflector system to direct flow onto the right side of the runner (clockwise rotation), steering effluent away from the forming roller. A systematic investigation was made to explore the influence of the deflector form and the rotational speed of the runner on the internal flow and energy conversion of the double-impulse turbine. The results indicate that in Scheme I: The deflector exhibited the ‌weak constraint effect on the pulp waste liquid, while both the area between the deflector and forming roller and the first work phase zone of the runner generated the high turbulent kinetic energy‌, leading to the energy dissipation and significant hydraulic losses‌. There was a dispersion ‌ of the water flow along the circumferential direction, particularly with the comparable circulation consumption capacity in the two work phases. Multiple circulation abrupt peaks‌ occurred circumferentially, due to the uneven energy transfer‌. The scheme II with the double deflectors well guided the water flow to the runner, significantly reducing the high turbulent kinetic energy area, and the Hydraulic loss of the runner. The energy of the crossflow turbine was improved to relatively concentrate the working area of the runner, where about 85% of the circulation was transformed in the first impact working area from 330° to 30°. The second working area of the crossflow turbine gradually approached the first working area with the increase of the runner rotational speed. Especially in the case of high rotational speed, the water flow of the two working times was mixed and interfered with each other, leading to large hydraulic losses. The maximum efficiency of the crossflow turbines was reached under the different schemes with the increase of the rotational speed. In Scheme I, there was the highest efficiency of 39.2% at 250 r/min. In scheme II, the highest efficiency of 56.4% was found at 300r/min. As such, Scheme II's dual-deflector guided the flow to impact the runner's right side for the clockwise rotation. An optimal speed of 300 r/min was represented as a highly effective solution to recover the residual energy from the paper machine. Therefore, Scheme II demonstrated the superior hydraulic efficiency. The finding can also provide a strong reference to optimize the energy recovery turbine for papermaking machines.

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