The application of Osborne Reynolds' theory of heat transfer to flow through a pipe

In a recent paper Messrs. Eagle and Ferguson describe a very complete series of measurements of the conditions of heat transfer between a brass tube and water flowing through it. They base the discussion of their results on Osborne Reynolds theory of heat transfer according to which there is a complete analogy between the transfer of heat and momentum so that if a hot sheet is moved edgewise through a fluid the distributions of temperature and momentum in the water are identical. The assumption underlying the theory is that any portion of the fluid which comes sufficiently near the heated surface to be moved forward with the speed of the hot surface is also heated to the temperature of that surface, or, alternatively, a portion which is moved forward at a fraction, β, times the speed of the plate is also heated through a temperature equal to β times the difference in temperature between the plate and the fluid. In this manner Reynolds’ theoretical coefficient of heat transfer, κR , may be calculated. The observed heat transfer coefficient is represented by Messrs. Eagle and Ferguson as κ0 and their results are expressed in the form F = κR /κ0 where F is a fraction determined under a variety of different conditions of experiment. This crude form of Reynolds’ theory suffers from two possible main sources of error, (A) the heated surface may raise the velocity of any portion of the fluid near it through a greater fraction of its own velocity than it raises the temperature expressed as a fraction of its own temperature, the initial temperature of the fluid being taken as zero. This might be expected to give rise to large errors in cases where the thermal conductivity is specially low. (B) The effect of local pressure differences which are inherent in all turbulent motion and alter the momentum of the fluid at any point without altering its temperature is neglected. The essential assumption in Reynolds’ theory is that these local pressure differences have no effect on the average distribution of velocity.