Analytical solution and numerical simulation of the thermal entrance region problem for laminar flow through a circular pipe

In this paper, the assumptions implicited in Leveque’s approximation are re-examined, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This report gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form, and presents a similarity solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth order (RK4) method. Finally, other important thermal results obtained from this analysis, such as; approximate Nusselt number in the thermal entrance region was discussed in detail. A comparison with the previous study available in literature has been done and found an excellent agreement with the published data.

Similarity solution and Runge Kutta method to a thermal boundary layer model at the entrance region of a circular tube

Purpose The purpose of this paper is to re-examine the assumptions implicit in Leveque’s approximation, and the variation of the temperature and the thickness of the boundary layer were illustrated using the developed solution. The analytical solutions are then checked against numerical solution programming by FORTRAN code obtained via using Runge–Kutta fourth-order (RK4) method. Finally, other important thermal results obtained from this analysis, such as approximate Nusselt number in the thermal entrance region, was discussed in detail. After that, the analytical results of the present paper are validated with certain previous investigations which were found in the specialized literature. Design/methodology/approach By defining a similarity variable, the governing equations are reduced to a dimensionless equation with an analytic solution in the entrance region. This paper gives justification for the similarity variable via scaling analysis, details the process of converting to a similarity form and presents a similarity solution. The calculation methodology for numerical resolution is based on the RK4 technique. Findings The profiles of the solutions are provided from which the authors infer that the numerical and exact solutions agreed very well. Another result that the authors obtained from this paper is the number of Nusselt in the thermal entrance region for which a parametric study was carried out and discussed well for the impact of scientific contribution. Originality/value The novelty of this paper is the application of the RK4 with a step size control, as a sequential numerical method of a ODEs system compared with the exact similarity solution of the thermal boundary layer problem.

Analytical Solution of the Thermal Entrance Heat Transfer Problem for Viscous Flow in Annulus

The Graetz–Nusselt problem with Brinkman extension is considered for steady-state laminar Newtonian flow in annuli. To solve the problem, a separation of variables method is used. In the limiting cases, the eigenvalues are in full agreement with the eigenvalues corresponding to flat channel and circular pipe. Useful formulas are represented to calculate the length of the thermal entrance region and Nusselt numbers in annuli.

Analysis of the Forced Convection in a Porous Channel Saturated by a Nanofluid: Effects of Brownian Diffusion and Thermophoresis

The Darcy-Graetz problem for a channel filled by a nanofluid saturated porous medium is studied. The flow is assumed to be fully developed, and a boundary temperature linearly varying with the longitudinal coordinate in the thermal entrance region is prescribed. A study of the thermal behaviour of the nanofluid is performed by solving numerically the fully–elliptic coupled equations. For the model of the nanofluid, both thermophoresis and Brownian diffusion are taken into account. The governing equations have been solved separately for the fully developed region and for the thermal entrance region. With reference to the fully developed region the solution has been obtained analytically, while for the thermal entrance region it has been obtained numerically, by a Galerkin finite element method implemented through the software package Comsol Multiphysics (© Comsol, Inc.). The analysis shows that for physically interesting values of the Péclet number the concentration field depends very weakly on the temperature distribution. Indeed, the homogeneous model could be employed effectively, since the effects of thermophoresis and Brownian diffusion are negligible.