Compact models for transient conduction or viscous transport in non-circular geometries with a uniform source

Transient heat conduction in solid prismatic bars of constant cross-sectional area having uniform heat generation and unsteady momentum transport in infinitely long ducts of arbitrary but constant cross-sectional area are examined. In both cases the solutions are mathematically modeled using a transient Poisson equation. By means of scaling analysis a general asymptotic model is developed for an arbitrary non-circular cross-section. Further, by means of a novel characteristic length scale, the solutions for a number of fundamental shapes are shown to be weak functions of geometry. The proposed models can be used to predict the dimensionless mean flux at the wall and the area averaged temperature or velocity for the tube, annulus, channel and rectangle for which exact series solutions exist. Due to the asymptotic nature of the proposed models, it is shown that they are also applicable to other shapes at short and long times for which no solutions or data exist. The root mean square (RMS) error based on comparisons with exact results is between 2.2–7.6 percent for all data considered.