Technical Note Heat transfer in turbulent pipe flow revisited: similarity law for heat and momentum transport in low-Prandtl-number fluids

Abstract

The task of studying transport phenomena in turbulent flow within a completely rational framework has, so far, been beset by current lack of understanding of turbulence. Nevertheless, the need for reliable tools to design heat and mass transfer equipment has prompted considerable research effort over several decades. That has led to a number of empirical models of turbulent transport. Two approaches have been utilized to model the transport process at a wall boundary: one is based on the Prandtl mixing length concept and eddy diffusivity, the second one is based on the surface renewal concept first introduced by Danckwerts [1] . Eddy diffusivity models have proved quite useful in a large variety of cases. However, these empirical models provide little insight into the real mechanism underlying turbulent transport. As a result, the thermal eddy diffusivity, as related to the momentum eddy diffusivity, cannot be assessed a priori without further empirical assumptions. More recently, the periodic surface renewal idea has received much attention in modeling heat and mass transfer in a turbulent flow. The basic assumption considers the surface to be covered by a mosaic of laminar flowing patches of fluid, where transport occurs only by molecular diffusion. These fluid patches are supposed to be periodically replaced by new ones (surface renewal model ) [2] or to form viscous layers that periodically grow and collapse (growth-breakdown model ) [3] . These are unsteady state one dimensional models. It has also been proposed that the fluid patches are arranged in a regular repetitive pattern of steady state boundary layers developing for a given length [4] . These simple models are able to explain a number of qualitative features observed in turbulent transport. In order to improve quantitative agreement with experimental data, these early concepts have been extensively elaborated. It is not our intent to provide a critical review of the many eddy diffusivity or surface renewal models. However, it should be borne in mind that these models appeal to postulated mechanistic pictures of turbulence and that the concepts and the quantities involved have no fundamental relationship with the correlated turbulent fluctuations, the sole quantities that actually determine turbulent transport. The purpose of this work is to construct a theory able to predict turbulent heat transport from fundamental information, namely the thermal diffusivity and the normal turbulence intensity in the fluid bulk. The theory applies to heat transfer in turbulent incompressible flow for Pr 1. The theoretical predictions are compared with available experimental data and empirical correlations to heat transport in liquid metals. For the sake of simplicity we shall refer to flow through cylindrical tubes in the following analysis. However, the results apply for arbitrarily shaped ducts as well.