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| Posted By: | Mohamed Ouzzane |
| Date: | Thu, 7 Feb 2002, 10:12 p.m. |
Ph.D ABSTRACT
Title:
² Simultaneous development of laminar mixed convection in a tube with non-uniform heat flux applied to its external surface: (tubes with and without external fins)².
Mohamed Ouzzane
Abstract:
Developing mixed convection inside tubes with non-uniform heat flux on its outer surface for cases with and without fins has been studied numerically. The flow is assumed to be steady and laminar. The fluid is Newtonian, incompressible with constant properties except for the density in the expression of the gravity force where Boussinesq's hypothesis is adopted. Viscous dissipation is neglected. Similarly, axial diffusion of heat and momentum in both fluid and wall are neglected based on the evidence provided in the literature which indicates that, for the conditions of interest in our study, flow reversal is highly unlikely. This last assumption has been used successfully to model developing mixed convection in many previous publications. It leads to a considerable simplification of the fundamental equations, which become parabolic in the axial direction while remaining elliptical in the radial and tangential directions. Finally, in accordance with the argument presented by Patankar and Spalding, the pressure is expressed as the sum of a cross-sectional average value p' which depends on the axial position only and an in plane perturbation p'' which depends on both radial and tangential directions (r and *). The coupled non-linear partial differential equations were integrated along the control volume and discretised using the staggered grid approach proposed by Patankar. The SIMPLEC procedure is used for the linkage of velocities and pressure while the iterative method proposed by Raithby and Schneider is used to calculate the axial pressure gradient. The set of linearized difference equations is solved with the tridiagonal matrix algorithm (T.D.M.A.). It should be noted that the same equations are used in the region occupied by the fluid and within the solid wall. In order to ensure that the velocity in the wall is zero, the corresponding diffusion coefficients for momentum were set equal to 1020. The heat flux collected or evacuated by longitudinal fins, given by the analytical expression is applied as boundary condition at the external surface of the tube. Comparisons with several experimental, numerical and analytical results have successfully validated the numerical code written in Fortran.
Four different chapters present the results. In the first and second one, heat losses and fins are not considered. The simultaneous effects of wall conduction and non-uniform heating on developing mixed convection have been studied. For this, four different cases of the thermal boundary condition are considered: a uniform heat flux is applied over the entire circumference or its top half at either the outer tube surface or at the fluid-solid interface, while the bottom half is thermally insulated. The conduction through the tube affects considerably the thermal and hydrodynamic fields especially when the uniform heat flux is applied at the top half of the tube section. It should be taken into consideration, particularly when the ratio of the solid to fluid conductivity kp is not very small. When the heat flux is imposed at the top half of the solid-fluid interface, temperature stratification is observed and the secondary motion, which consists of a vortex, is weak and limited to the upper half of the cross-section. A considerable part of fluid in the lower half of the domain remains at the entry temperature. On the other hand, when the heat flux is imposed at the top half of the outer surface, a considerable fraction of the energy supplied is conducted circumferentially through the wall and reaches the fluid from the bottom half of the solid-fluid interface. The fluid in the bottom half of the cross section is then considerably warmer; the secondary motion is more intense, especially at a large distance from the tube entrance, and extends over the entire cross-section. Thus, far from the entrance, the fraction of the externally supplied heat reaching the fluid from the bottom half of the interface is very important for material with high thermal conductivity (72% for copper, 60% for steel) and small for glass tube (30%).
In the third and fourth chapters, longitudinal fins are considered (cases of solar collector and heat exchangers). In the previous chapters, convection heat losses are not taken into account and the bulk temperature is the same at given axial position for any tube material. While, in these last two chapters, the heat losses are considered and lead to the variation of the useful energy with the problem parameters. The fins influence considerably the flow and the secondary motions. Because of the stratification phenomenon of the temperature in the fluid, the circumferential distribution of the transmitted heat flux by the fins is not uniform. For both heating and cooling, it was shown that the heat flux evacuated by the bottom fin is about half of that transmitted by the corresponding one in the top. It is important to optimise the number and the width of fins as well as the length of the tube by considering the total heat, the mass of the tube and the pressure drop.
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