Momentum, heat and mass transfer in turbulent pipe flow: The extended random surface renewal model

Abstract

A model is presented for a quantitative prediction of the transfer coefficients of momentum, heat and mass, and the radial profiles of the axial velocity, the temperature, and the concentration in the near-wall fluid of a turbulent pipe flow. In this model, the tube wall is assumed to be covered by a mosaic of fluid elements of random age and laminar flow with unsteady profiles of velocity, temperature, or concentration. This model, which is an extension of the RSR model developed by Fortuin and Klijn, is the extended random surface renewal model, and is referred to as the ERSR model. The distribution and the mean value of the ages of the fluid elements at the tube wall govern the local time-averaged transfer coefficients and radial profiles in the wall region. The mean age of the fluid elements is derived from the friction factor and the Reynolds number using the ERSR model. Both the distribution and the mean ages of the fluid elements at the tube wall agree quantitatively with the experimental results obtained from velocity signals measured with a laser-Doppler anemometer in turbulent pipe flow at thirteen Reynolds numbers between 5 × 10 3 and 43 × 10 3. The equations derived from the ERSR model for the local time-averaged heat and mass transfer coefficients in turbulent pipe flow, and the radial profiles of the axial velocity, the temperature and the concentration in the wall region, agree with correlations or experimental data presented in literature. The analogy between momentum, heat and mass transfer in turbulent pipe flow is elucidated by introducing a dimensionless number for momentum transfer. This so-called Fanning number for momentum transfer ( Fa = k m d/v), is comparable with the Nusselt number for heat transfer and the Sherwood number for mass transfer. Furthermore, the ERSR model provides a basis for both explaining the Chilton-Colburn analogy and the relationship between transfer rates and measured mean ages of fluid elements at the wall in turbulent pipe flow.