The laminar Forced Heat and Momentum Transfer inDeveloping flow in the Entrance region for Isothermaland Iso-Heat Flux walls of circular and flat rectangularducts

Abstract

The velocity and the temperature profiles development from initial uniform distribution at the entrance of a channel towards the fully developed profiles downstream causes a desirable heat transfer enhancement associated with an increase of the pressure drop due to the entrance effect which requires more power to overcome the increased power consumption. The use of short flow passages became very important in the design of the compact heat exchangers to have higher heat transfer rates over the whole flow passage length thus decreasing the dimensions of the heat exchanger. Laminar heat transfer and flow development in the entrance regions of circular tubes and parallel-plate ducts is of considerable practical significance, and not surprisingly, there exists a large number of references in the literature on this topic, especially for incompressible laminar flow .Most of the investigations are based on the boundary layer model which approximates the actual flow case only at high Reynolds numbers on a portion of the developing length, away from the entrance, whose length depends on the value of the Reynolds and Prandtl numbers. In recent years some numerical solutions of the complete Navier-Stokes and the energy equations have been carried out numerically with finite difference, finite element and finite volume methods. In this work numerical solutions of the full Navier Stokes equations with no approximations in the form of the vorticity and stream function, the Poisson type pressure equation and the full energy equations for parallel plate ducts and circular tubes are carried on, presenting the hydrodynamic and thermal developing lengths, the pressure drop, the local and the average friction factors , the velocity , the pressure and the temperature as well as the local and average Nusslet number distributions showing the effect of the axial diffusion of momentum and heat. The laminar incompressible fluid flow field in the entrance region of parallel plate ducts and circular tubes is solved using the transient approach procedure. The unsteady state two-dimensional Navier Stokes equations are transformed into the transient vorticity transport equation. The stream function and the vorticity relationship, which is a Poisson type equation, is used to calculate the stream function. The velocity components U and V are obtained from the stream function field using their relation with the stream function. The boundary conditions for the two velocity components U and V are set to satisfy the nonslip condition at the walls, the irrotational flow condition at the entrance for which both of the velocity components U and V at the entrance are uniform, the symmetry with respect to the axes of the conduits and the fully developed flow condition beyond the developing lengths. The vorticity transport and the stream function-vorticity equations are cast in finite difference forms which satisfy both of the governing differential equations and the boundary conditions. The chosen finite-difference scheme is validated through the comparison of the results obtained by it for the natural convection in a two-dimensional square cavity with a previously published solution for this case obtained using a series expansion of orthogonal functions for different Rayleigh numbers. The pressure drop is obtained from the axial momentum equation as usually done by various procedures in boundary layer solutions from the solution of the core region but by applying Newton’s second law of motion on the whole flow cross section. It is also obtained by solving the Poisson type pressure equation whose finite difference form represented the differential governing equation in the interior of the flow domain. At the walls and the boundaries the pressure Poisson type equation is cast in a finite difference form that satisfies the differential governing equation and the corresponding boundary conditions.