الفهرس 
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Abstract
In the present work , natural –convective heat transfer in vertical annual filled with saturated porous media were investigated numerically and experimentally. The effect of the governing physical and geometrical parameters of the problem on heat transfer and fluid flow were investigated throughout the present work.
For an annulus of which the inner wall is heated at constant temperature and the outer wall is isothermally cooled, the top and bottom walls being insulated, heat transfer and fluid flow results were obtained numerically for a wide range of the governing parameters. Those, and their respective ranges of values, considered here, are; ISASIO, 2sk≤25, and Darcy-Rayleigh number Ra” up to 10000, where, A is the aspect (high-to-gap width) ratio and k is the radius ratio.
The finite difference approximation was performed on the basic steady-state governing equations (mass, momentum, and energy equations) and the Successive Overrelaxation point iterative technique (SOR) was used as the solution method. Acomputer program was designed for this purpose, where a total of 560 computational run was performed for various values of A, K, and Ra”.
To improve the understanding of connective heat transfer in vertical annual and to verify the numerical results, an experimental investigation has been conducted for an isothermally heated vertical annulus of radius ratio k=4.16 for the aspect ratios A=1, 3 and 5. Water and sand grains of 0.0081 m average diameters were used as the porous media. The Darcy-Rayleigh number ranged from 22 to 7387.
from the least-squares fit for both the experimental and numerical results, correlations for heat transfer rates in terms of the average Nusselt number (based on the gap width, D for the boundary layer regime (Ra*≥300).
Comparisons of the present numerical and experimental results with earlier published results and with each other were performed. Generally, good agreement was observed and this assures the reliability of the present experimental apparatus, the measuring instruments, and the present mathematical model. Moreover, the following results were obtained from the numerical and experimental study.
Generally, the temperature and velocity fields, and consequently the heat transfer rates, are not only functions of the Darcy-Rayleigh number Ra” and aspect ratio A, but depend strongly on the radius ratio k. The average Nusselt number always increases as the radius ratio, k, increases, through the rate of increase diminishes with the increase in radius ratio, particularly at high Darcy-Rayleigh numbers, Ra*.
Further. an increase in aspect ratio results in a decrease of the average Nusselt number and the rate of decrease becomes less at low Darcy-Rayleigh numbers. Also, an increase in the Darcy-Rayleigh number results in a stably stratified temperature distribution, and hence, producing a thermal boundary layer on the vertical walls and indicating strong natural convection flow which enhances the heat transfer coefficient in terms of the average Nusselt number.
In addition, the experimental values of Nusselt number are in good agreement with the numerical values especially for sand-A (0.0027 m dial) and sand –B (0.0056 m dial) except for Ra*<300(asymptotic flow regime) the experimentally obtained Nusselt numbers are lower than the numerical values. The maximum deviation in the higher range of Ra* is 30.7 percent from the numerical predictions, while, in the lower range of Ra*, the maximum difference between the two values is 22 percent , but mostly the experimental values are within 20 percent of the numerical values.
Finally, the good agreement between the experimental values of the average Nusselt number and the numerical predictions, also, the agreement between the measured temperatures at various locations and the numerically established temperature fields, clearly indicate that the formulation based on Darcy`s law holds good for natural convection in porous media, even for gigh Darcy-Rayleigh numbers, provided that the heat transfer rates are not dominated by the wall channeling of saturating fluid. Also, on the basis of the good agreement between the experimental and numerical results, it seems possible to determine heat transfer parameters for natural convection in vertical porous annulus using either method. However, the numerical analysis gives more information, including the velocity field, which is difficult to obtain experimentally.