Determining heat transfer correlations for transition and turbulent flow in ducts

The objective of the paper is to develop correlations for the Nusselt number Nu in terms of the friction factor ξ (Re) and also Reynolds number Re and Prandtl number Re, which is valid for transitional and fully developed turbulent flow. After solving the equations of conservation of momentum and the energy for turbulent flow in a circular tube subject to a uniform heat flux, the Nusselt number values were calculated for different values of Reynolds and Prandtl numbers. Then, the form of the correlation Nu = f (Re, Pr) was selected which approximates the results obtained in the following ranges of Reynolds and Prandtl numbers: 2300 ≤ Re ≤ 1000000, 0.1 ≤ Pr ≤ 1000. The form of the correlation was selected in such a way that for the Reynolds number equals to Re=2300, i.e. at the point of transition from laminar to transitional flow the Nusselt number should change continuously. Unknown coefficients  x₁, …, x_n  appearing in the heat transfer correlation expressing the Nusselt number as a function of the Reynolds number and Prandtl number were determined by the method of least squares. To determine the values of the coefficients at which the sum of the difference squares is a minimum, the Levenberg-Marquardt method is used.

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