(140k) An Analogy between Chemical Reaction and Heat Transfer

The most accurate predictions of forced convection in fully developed turbulent flow are produced by modifications and generalizations of the Reichardt analogy between momentum and energy transfer. That analogy was originally derived by combining the differential energy and momentum balances, making several ingenious mathematically based approximations, introducing an algebraic model for turbulent transport, and finally integrating over the radius in closed form. The subsequent improvements have resulted primarily from the introduction of a better model for transport by the turbulent fluctuations. A comparable analogy between convective heat transfer and an energetic chemical conversion has been devised by a completely different process of modeling. First, an exact algebraic solution was derived for convection in fully developed laminar flow with a volumetrically uniform rate of reaction and no radial diffusion of species. That solution, which relates the Nusselt number to the ratio of the heat flux at the wall to the heat of reaction, was then generalized for developing reaction and convection, including diffusion, at the expense of the introduction of one functional idealization and one arbitrary coefficient. The error due to the functional idealization, which consists of expressing the rate of reaction in terms of the mixed-mean temperature, appears to be completely negligible. The arbitrary coefficient, which was evaluated by means of essentially exact numerical computations, proved to vary regularly and moderately from condition to condition and thereby to be susceptible to correlation and generalization. The new analogy incorporates an explicit dependence on Re, Pr, Sc, as well as on a number of other dimensionless parameters involving the frequency factor, the energy of activation, and the thermicity of the reaction, the heat flux density or the temperature at the wall. It is applicable for both laminar and fully turbulent flow. In combination with a complementary expression for the chemical conversion, the analogy becomes predictive in a numerical sense. The resulting predictions have been tested by comparison with essentially exact numerical solutions, and have proven satisfactory even for conditions that produce gross and seemingly chaotic enhancement or attenuation of the Nusselt number. The success of this analogy as well as the improved one of Reichardt suggests that similar theoretically structured expressions may be possible for other processes involving transport and/or reaction.