Heat transfer in turbulent tube flow

Summary

The differential equation for the temperature distribution in a laminar flow through a smooth cylindrical tube can be solved on certain assumptions by separating the variables; the result is a temperature distribution given by a linear combination of eigenfunctions. For a fully developed turbulent flow the same differential equation applies in principle with the following differences:a) the velocity distribution is only known from measurement; consequently only empirical formulae are available for the calculations;b) the thermal diffusivitya is increased by an amountA _q due to turbulent mixing, and the value of this thermal eddy diffusivity depends upon the velocity distribution. The cross-section of the tube is divided into three concentric parts, according to the magnitude of the frictional eddy diffusivityA _m : the turbulent core, where the molecular momentum diffusion is negligible in relation to the eddy diffusivity of momentum; the laminar boundary layer, where the molecular momentum diffusion prevails; and the transition layer, where eddy diffusivity and molecular diffusion of momentum are of the same order. Separately for each of these regions formulae have been drawn up for the velocity and momentum eddy diffusivityA _m . Given the hypothesis thatA _q /A _m =A is a constant,A _q follows fromA _m . In the present article, for each of these three regions a general solution is given for the differential equation valid in that area, by separating the variables. The constants in these solutions have been chosen in such a way as to make the temperatures and heat currents continuous at the boundaries between the three regions. For a constant wall temperature the first two eigenfunctions and eigenvalues are calculated for severalRe andPr values. Furthermore, the heat transfer and temperature distribution for a homogeneous entrance temperature are given. The calculated eigenfunctions can also be used for calculating the heat transfer and heat distribution for a non-homogeneous entrance temperature, provided the latter is symmetrical with respect to the axis of the tube.