Modification and Improvement of the Churchill Equation for Friction Factor Calculation in Pipes

Abstract

Accurate prediction of the friction factor is fundamental for designing and calibrating fluid transport systems. While the Colebrook–White equation is the benchmark for precision due to its physical basis, its implicit nature hinders practical applications. Explicit correlations like Churchill’s equation are commonly used but often sacrifice accuracy. This study introduces two novel modifications to Churchill’s equation to enhance predictive capabilities. Developed through a rigorous analysis of 240 test cases and validated against a dataset of 21,000 experiments, the proposed Churchill B(Re) and Churchill B(V,ε) models demonstrate significantly improved accuracy compared to the original Churchill equation. The development of these functions was achieved through generalized reduced gradient (GRG) nonlinear optimization. This optimized equation offers a practical and precise alternative to the Colebrook–White equation. The mean relative errors (MRE) for the modified models, Churchill B(Re) and Churchill B(V,ε), are 0.025% and 0.807%, respectively, indicating a significant improvement over the original equation introduced by Churchill in 1973, which exhibits an MRE of 0.580%. Similarly, the mean absolute errors (MAE) are 0.0008% and 0.0154%, respectively, compared to 0.0291% for the original equation. Beyond practical applications, this research contributes to a deeper understanding of friction factor phenomena and establishes a framework for refining other empirical correlations in the field.