231b

When an exothermic homogeneous reaction is carried out in tubular flow, cooling is often applied at the wall in order to prevent the attainment of an excessive temperature, and thereby a thermal runaway and/or an undesirable side reaction. Alternatively, if the reaction is endothermic, heating at the wall may be necessary to prevent self-quenching. It was first noted over 40 years that the combination of an energetic reaction and heat transfer at the wall may result in great enhancement or modest attenuation of the convective heat transfer coefficient. However, this interaction has gone unmentioned in textbooks and handbooks on either reaction engineering or heat transfer, presumably because the analyses are ancient and of questionable accuracy and validity, the experimental data are fragmentary and incoherent, and an explanation for the anomalous behavior has not been established on either physical or mathematical grounds. Detailed and coherent and numerical solutions that take radial diffusion of momentum, energy, and species into account have recently been carried out by the author and his coworkers. These solutions provide a sufficient data base to allow the construction of theoretically based correlative equations for the seemingly anomalous behavior, and, perhaps of equal importance, have lead to both physical and mathematical explanations for it. The physical explanation for the enhancement is the preferential release or absorption of the heat of reaction near the wall, thereby perturbing the radial temperature distribution relative to that for pure convection and reducing the distance over which the heat must be transferred on the mean. The mathematical explanation follows from the definition of the heat transfer coefficient, namely h = jw/(Tw –Tm). If the difference between the wall and mixed-mean temperatures approaches zero owing to the confluence of the heat of reaction and the heat flux at the wall, the coefficient may become very large. Conversely, attenuation is a physical consequence of the preferential release or absorption of the heat of reaction near the centerline, and a mathematical consequence of an increased temperature difference due to a particular confluence of the heat of reaction and the heat flux at the wall. The local release or absorption of the heat of reaction is a function of the radial distribution of the velocity, composition and temperature, and this explanation could not have been formulated on the basis of the solutions for perfect radial diffusion (plug flow) or negligible radial diffusion that pervade the current literature. Theoretically based algebraic equations are presented that predict the chemical conversion and the Nusselt number even for conditions that produce gross and chaotic enhancements and attenuations of the latter. The predictive equations for the Nusselt number and the chemical conversion each have a theoretical structure but incorporate some idealizations, and, for some conditions, one or more arbitrary coefficients. Illustrative numerical results are presented graphically and compared with the predictions.

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