On the Nusselt Number for Thermally Developed Flow Between Isothermal Parallel Plates With Dissipation

Abstract

A recent paper announced that most textbooks, graduate and undergraduate, present an incorrect value for the Nusselt number in thermally developed laminar flow between isothermal parallel plates. The stated cause is flow work and/or dissipation that acts as a persistent source of heat generation in the channel. The purpose of this paper is to rehabilitate the textbook literature. I show that the commonly reported value of the thermally developed Nusselt number, 7.541, is quite acceptable for commonly encountered situations. In particular, for ideal gases, the wall heat flux is predicted exactly without accounting for these effects because they cancel one another. For liquids, I derive the channel length within which dissipation makes a negligible contribution to heat flux. This length will often span the entire range within which the bulk temperature changes in response to a wall temperature change. The residual bulk temperature rise from dissipation can amount to only millikelvin. The Nusselt number following a change in wall temperature should be calculated after separating the temperature rise and heat flux caused by dissipation. Failure to do so gives a Nusselt number that can be zero, negative, or singular. The effects of dissipation on flux and temperature can be added to the ordinary Graetz solution if they are not negligible. The present results show that the Seban–Shimazaki criterion for thermally developed flow is misleading when dissipation is considered. Instead, the flow may be called thermally developed when the Graetz series is well approximated by its first term.