NUMERICAL SIMULATION OF MIXED CONVECTIVE COOLING IN A HIGH RESIDENTIAL BUILDING DIVIDED INTO THREE STORIES

Yawovi Nougbléga, HodoAbalo Samah, Kokou N’wuitcha, Kodjo Kpode and Magolmééna Banna. Laboratoire Sur l’Energie Solaire /Groupe Phénomène de Transfert et Energétique Université de Lomé 01 BP 1515 Lomé 01–Togo. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 17 August 2018 Final Accepted: 19 September 2018 Published: October 2018

Mixed convection heat transfer in a ventilated high building is numerically studied by solving mixed convection equations with the Boussinesq approximation. The present investigation deals with velocity and temperature distribution in a high building divided into three stories; in which the left heated wall is vented by a fresh air stream introduced at various inlet openings and hot waste air exits from different outlet openings. Numerical solutions of the Navier Stokes equations and energy equation have been solved by the Thomas' algorithm. Solutions are presented for various geometrical aspect ratios and for different values of Grashof and Reynolds numbers. The results are presented in terms of streamlines, isotherms, velocity, and heat transfer versus the governing control parameters in detail.

…………………………………………………………………………………………………….... Introduction:-
Conventional ventilation systems are based on mechanical components (pumps, compressors, blowers) that consume electric power. Heating in winter is also provided by means usually related to fuel consumption. Although these systems are effective, their construction and operation is expensive. Effective ventilation is important for various buildings, including houses, shelters, mobile homes, warehouses, greenhouses. Usually, ventilation is provided by systems based on mechanical components that consume electric power is not available. Also these systems are effective, their construction, operation and maintenance are expensive. In remote and desert areas where electric power is not available, conventional methods cannot be used. For reasons of energy savings and cost reduction, it becomes necessary to explore alternative ways for effective ventilation and heating. Hence passive ventilation based on natural convection has been extensively studied in laboratories in recent years. Both experimental investigations and numerical simulations have been performed and analyzed (Druckman S. et al., 2000;Glat Y.et al, 2000;Dubovsky V et al., 2001;Mazal B et al., 2001;Ziskind G. et al., 2001;Ziskind G. et al., 2002). From these studies, one can conclude that effective ventilation is achievable in small-scale laboratory models and real size one-story buildings. During the last decade, the idea of passive ventilation based on solar heating has gained growing interest. Special attention has been paid to the shape and size of the heated element that should transfer energy to the air, causing the latter to flow through the building. (Letan et al., 2003) numerically studied a multi-story building. They showed that both ventilation and heating can be achieved by natural convection. (Oosthuizen and Paul, 1985) numerically studied mixed convection heat transfer in a cavity with uniformly heated isothermal vertical walls and horizontal adiabatic walls. Forced flow was considered either aiding or opposing the buoyancy force effect. The cold wall was provided by two openings for the admission and evacuation of the forced stream. (Papanicolaou and On the basis of the literature review, it appears that no work was reported on mixed convection in a high buildings divided into three stories. Therefore, due to its practical interest for energy saving, the subject needs to be further explored to improve knowledge in this field. Hence, the aim of the present study consists in numerically studying a mixed convection problem in a high ventilated building divided into three floors submitted to a constant heat flux along its left vertical wall. In this analysis, the forced flow enters the high building divided into three stories through the openings respectively located in the middle of the heated vertical left wall of each floor and leaves it from the openings located in the right opposite adiabatic wall. This kind of ventilation supports a double interaction between buoyancy-induced flow and forced flow. In the lower part of the high building, the forced flow injected in the high building divided into three stories promotes the natural convection motion. Thus, the combined effects of the Rayleigh number and the intensity of the imposed flow (through the Reynolds number) on the flow structure and heat transfer across the high building is examined for this specific situation.

Mathematical Formulation:-Physical Model and Governing Equations:-
The configuration under study with the system of coordinates is sketched in Fig. 1. It consists of a high ventilated building divided into three stories heated by a uniform heat flux from its vertical left wall while the remaining walls are considered perfectly insulated. The system is submitted to an imposed flow of fresh air, parallel to the horizontal walls, entering the building from three inlet openings located respectively at the middle of the left vertical wall of the stories and leaving through the outlet openings. The third dimension of the high building (direction perpendicular to the plane of the diagram) is assumed to be large enough so that fluid motion can be considered twodimensional. Flow is assumed to be laminar and incompressible with negligible viscous dissipation.

Fig.1:-Physical model
All the thermophysical properties of the coolant fluid are assumed constant except density that gives rise to the buoyancy forces (Boussinesq approximation). Taking into account the above-mentioned assumptions, the nondimensional governing equations, written in vorticitystreamlines ( -ψ) formulation, are as follows: Boundary Conditions:- Initial conditions: at τ = 0: and the aspect ratio expressed as: ; ; Evaluation Of The Model Characteristics:-From the engineering viewpoint, the most important concern is heat transfer through the heated walls. These are best represented by the Nusselt number, which is a measure of the ratio of the heat transfer by conduction to the flux convected by fluid flow. The local Nusselt numbers on the heated walls are given by: Numerical Procedure:-Method Of Solution:-The nonlinear partial differential governing equations (1-3), were discretized using a finite difference technique. The first and second derivatives of the diffusive terms were approached by central differences while a second order upwind scheme was used for the convective terms to avoid possible instabilities frequently encountered in mixed convection problems. The integration of the algebraic equations (2) and (3) was assured by the Thomas' algorithm. At each time step, the Poisson equation, Eq. (4), was treated using the Point Successive Under-Relaxation method (PSUR) with an optimum under-relaxation coefficient equal to 0.8 for the grid (121×61) adopted in the present study. Convergence of iteration for stream function solution is obtained at each time step .The following criterion is employed to check for steady-state solution. Convergence of solutions is assumed when the relative error for each variable between consecutive iterations is below the convergence criterion ε such that ∑|( ) | where stands for ψ, , , n refers to time and i and j refer to space coordinates. The time step used in the computations is Grid independency solutions are assured by comparing different grid meshes for the highest Grashof and Reynolds numbers used in this work (Gr =10 6 and Re = 200). It was found that the differences between meshes of 121 x 121 and 141x141 were not significant for all variables. The results obtained with these grids were comparable to those obtained with a non-uniform grid size of 121 x61. Thus, a non-uniform mesh of 121 x61 was selected. The vorticity computational formula of (Woods .L. C, 1954) for approximating the wall vorticity was used: ( ), where and are stream function values at the points adjacent to the boundary wall; n the normal abscise on the boundary wall. Validation:-In order to test the computer code developed for this study, the problem of a ventilated rectangular enclosure with its left and upper walls submitted to a constant heat flux, while the remaining walls are considered adiabatic was studied. Very good agreement is obtained between the test problem solution and the ventilated rectangular enclosure solutions according to the work of (A. Raji et al., 2000). The horizontal cold jet enters the enclosure from the bottom of its heated wall and leaves from the top of the other vertical wall with no slip boundary conditions applied to all the walls. The Reynolds number, Re was set at100, for Raleigh number Ra set at10 6  The effect of the Reynolds number on the flow structure and temperature distribution is shown in Fig.2 (a-c). The streamlines and the isotherms are presented for steady state flows obtained for Gr = 10 6  1466 structure in the whole building. It reveals that there is a mixed convection on the first story of the building, while the manifestation of the natural convective flow is respectively observed on the second and third floors. By increasing the Reynolds number, the closed cells and the open lines appear simultaneously on the three stories. This situation indicates the real manifestation of the mixed convection in the high building, Fig.2 (b-c).
In fact, the heated portions of the left wall, located above the inlet openings, impose a clockwise circulation on the stories. The lower closed cells will play an increasingly important role by increasing Re since the more intense is the forced flow, the greater its negative (positive) effect on the natural convection flow in the upper (lower) part of the divided high building. The corresponding isotherms are more tightened in the vicinity of the heated wall testifying to a noticeable increase in convective heat exchange. In addition, a net progression of the cold zone towards the right wall is observed on the  For different values of the Reynolds number or for the inlet jet of the fresh air in the building, the difference between the ambient temperature and the internal temperature is less equal to 5K. This behavior situation indicates that thermal comfort is attained in the building, fig.5 (a-b). In fig.6, the local Nusselt number is an increasing function along the length of the building for a fixed Reynolds number. One can show that the second floor is less heated than the first floor in the high building. Because the Nusselt number is defined as the inverse value of the dimensionless temperature along the heated walls in the building. (X)  fig.7 (a-c), the local Nusselt number decreases to the minimal value along the first part of the left heated wall on the stories and then increases to the maximal value near the inlet opening. Then, the local Nusselt number is a decreasing function along the second heated part of the left wall, for the fixed Reynolds number. This variation of the local Nusselt number indicates that the heat exchange is respectively more important before the inlet opening on the stories in the divided high building.  fig.9 (a-c). This variation of the local Nusselt number proves that the local Nussselt number is inversely proportional to the heated flux which is correlated with Grashof number. By increasing Grashof number, the dimensionless temperature decreases, consequently the local Nusselt number increases.  fig.10 (a) the coolant fluid temperature is an increasing function of the Grashof number. One can observe that the temperature decreases along the horizontal axis which relates the left heated wall to the right adiabatic wall. This situation indicates that the temperature is stratified from the left heated wall to the right adiabatic wall. Fig.10 (b) showed that temperature is stratified along the vertical axis on the stories. Air movement in the high divided building is significantly affected by the variation of the velocity components along the horizontal axis. Hence, fig.11 (a-d) showed that the vertical component of velocity increases to the maximal value before decreasing to the minimal value near the right adiabatic wall, while the horizontal component is decreasing and reaching the minimal value and then increasing to the maximal value near the right adiabatic wall. These variations of the velocity components indicate the presence of buoyancy forces in the building. Then for the fixed Reynolds number, the vertical velocity component is an increasing function of Grashof number near the left heated wall while it is a decreasing function near the right adiabatic wall fig.11 (a, b), due to the opposed variation for the horizontal velocity component as shown in fig. 11(c, d).

Conclusion:-
The numerical investigation in this study allowed the authors to know that air flow in a high divided building for the lowest value of Reynolds number is natural convection while it is forced convection for the highest value of Reynolds number. The role of air flow is to extract the excess heat along the left heated wall. Flow analysis in the high divided building has given several possibilities to reduce the energy charge for air conditioning and provide passive cooling on the stories. Within the investigated parameters ranges, the following conclusions can be drawn:  Coolant fluid temperature is an increasing function of the geometrical aspect ratio while the local Nusselt number is a decreasing function.  Openings give the possibility to decrease conventional electrical energy consumption charge for air conditioning in a divided multi-storied high building.  Openings provide passive venting and thermal comfort by natural convection on the floors in the high building.  For the lowest values of Grashof and Reynolds numbers, thermal comfort is attained in a high building divided into three stories. Air inlet velocity (m.s -1 ) x,y Coordinates defined in fig. 1