School of Aeronautics and Astronautics, Purdue University West Lafayette, Indiana, 47907, USA.
An empirical model is evaluated that is in a very simple form and is often used in automobile industry to relate the pressure drop and mass flow rate in internal flows. Despite the simplicity of the model, it is remarkably accurate when it is used in a wide range of internal flows. Such accuracy and the theoretical basis of the model is not well understood, and this work aims to provide such an understanding. The theoretical basis of the empirical model is sought by performing an integral analysis based on the Navier-Stokes equation in a laminar developing channel flow. The analysis successfully yields a model that is in the same form as the empirical model. The accuracy and sensitivity of the model is then thoroughly examined through the computational studies of several internal flows. Two regimes of the model behavior in internal flows are identified, a convection dominated flow regime and a diffusion dominated flow regime. In each regime, the sensitivity of the model accuracy to the model parameters is found to be substantially different. Finally, the empirical model is applied to several more complicated internal flows to demonstrate the applicability of the model in general flows.