Analytical Regression and Geometric Validation of the Blade Arc Segment BC⌢ in a Michell–Banki Turbine

This study introduces a systematic methodology for modelling the radius of curvature of the arc-shaped section BC⌢ in a Michell–Banki cross-flow turbine blade. The method combines geometric modeling in polar coordinates with nonlinear regression, using both two- and three-parameter formulations estimated through the Ordinary Least Squares (OLS) method. Model performance is assessed through two complementary criteria: the coefficient of determination (R2) and the computed arc length, ensuring that statistical accuracy aligns with geometric fidelity. The methodology was validated on digital measurements obtained from CATIA, using datasets with N=187 and a reduced subset of N=48 points. Results demonstrate that even with fewer data points, the regression model maintains high predictive accuracy and geometric consistency. The best-performing three-parameter model achieved R2=0.958, with a five-point Gauss–Legendre quadrature yielding an arc length of approximately 145mm, representing 98.8% agreement with the reference value of 146.78mm. By representing the arc as a single smooth exponential function rather than a piecewise mapping, the approach simplifies analysis and enhances reproducibility. Coupling regression precision with arc-length verification provides a robust and reproducible basis for curvature modeling. This methodology supports turbine blade design, manufacturing, and quality control by ensuring that the blade geometry is validated with high statistical confidence and physical accuracy. Future research will focus on deriving analytical arc-length integrals and integrating the procedure into automated design and inspection workflows.