Invariance relations for laminar forced convection in ducts with slowly varying cross-section

Laminar flow forced convective heat transfer in ducts with slowly varying cross-section is analysed. The analysis is based on a simplifying assumption concerning the velocity field, called the similarity assumption. It is assumed that the axial velocity profile is locally fully developed, irrespective of the variation of the cross-sectional area with axial position. It is shown that in the axisymmetric case (circular tube with varying diameter) for boundary conditions of the first kind thr diameter variation]las no influence at all on the heat transferred. As in the case of a tube with constant diameter, the degree of temperature equilibration depends only on the Fourier number Fo = pi aL/Q that is on the ratio of length of the tube, L, to flow rate Q. In the case of a plane channel, however, variations of the channel height 2H have a pronounced influence on the transfer coefficient. It is shown that the degree of equilibration depends only on the effective Fourier number Fo(eff) = aB/Q . integral (l)(0) dz/H(z). The heat transfer is enhanced by a narrow design of the duct. These results hold for any shape of the axial velocity distribution over the duct cross-section, provided the above-mentioned similarity assumption is justified. By analogy, all foregoing results are also applicable to the convective-diffusive mass transfer in now through ducts with slowly varying cross-section. The range of validity of the similarity assumption is discussed in a hydrodynamic section. It can be shown that for creeping flow of a Newtonian liquid in an infinite cone or wedge this assumption is a very good approximation to the exact solution of Stokes' equations for cone or wedge angles up to 30 degrees. Apart from Newtonian creeping flow, the assumption might well be applicable in the duct Row of a power-law fluid and in pure elongational flow as encountered in stretching filaments or sheets of liquid, where the axial velocity is constant over the cross-section. (C) 2001 Elsevier Science Ltd. All rights reserved.