Semi-analytic solution of the two-dimensional turbulent energy equation in round tubes using an analytic velocity profile and its experimental validation
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A semi-analytic solution of the temperature development of single-phase, turbulent viscous flows inside smooth round tubes is performed. The special feature of the theoretical analysis revolves around two single universal functions of analytic form for the accurate characterization of the turbulent diffusivity of momentum and the turbulent velocity profile in the entire cross-section of a round tube. Using this valuable information that emanates from the analytic solution of the one-dimensional momentum balance equation, the two-dimensional energy balance equation was reformulated into an adjoint system of ordinary differential equations of first–order with constant coefficients. Each equation in the system of differential equations governs the axial variation of the average temperature of a finite volume of fluid of annular shape. Exploiting the linearity of the system of differential equations, an analytic solution of it was obtained via the matrix eigenvalue method with LAPACK, a library of Fortran 77 subroutines for numerical linear algebra. Reliable series have been determined for the axial variation of the two thermal quantities of importance: (a) the time-mean bulk temperature and (b) the local Nusselt number. The semi-analytic nature of the local Nusselt number distribution is advantageous because it may be viewed as an analytic-based correlation equation. Prediction of the local Nusselt numbers for turbulent air flows compare satisfactorily with the comprehensive correlation equations and the abundant experimental data that are accessible from the literature. The air flows are regulated by a wide spectrum of turbulent Reynolds numbers.
KeywordsLocal Nusselt Number Axial Variation Momentum Balance Equation Round Tube Turbulent Reynolds Number
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