Numerical Investigation of Heat Transfer and Fluid Flow Characteristics Inside Wavy Channels Fully Filled With Porous Media

The combined effect of waviness and porous media on the convection heat transfer and fluid flow characteristics is numerically investigated. Two models of wavy walled channel fully filled with homogenous porous material are assumed. The first was the symmetric converging-diverging channel (case A), and the second was the channel with concave-convex walls (case B). The governing equations have been solved on non-orthogonal grid, which is generated by Poisson elliptic equations, based on ADI method. Nusselt number values are used to indicate whether any cases of corrugation studied may have led to an increase in the rate of heat transferred compared with the planar surface channel which is the purpose of the study. The results show that case A gives more enhancement in heat transfer than case B. However, the thermal performance of the wavy channels (cases A & B) is better than the straight channel (simple duct).


Introduction
Heat and mass transfer through porous media is an important development and an area of very rapid growth in contemporary heat transfer researches; because of its existence in many diverse applications such as ground-water hydrology, production of oil and gas from geological structures, the gasification of coal, geothermal operations, packed-bed chemical reactors, surface catalysis of chemical reactions, filtration, adsorption, drying, compact heat exchangers and many more.Many researchers interested with enhancing heat transfer rate by using porous media.Hadim and North (2005) presented a numerical investigation of two-dimensional laminar forced convection in a sintered porous channel with inlet and outlet slots.They studied the effects of the particle diameter, particle Reynolds number, and channel dimensions on flow and heat transfer.They developed length-averaged Nusselt number and friction factor correlations for efficient design of a porous metal heat exchanger.Mohammad (2003) numerically investigated heat transfer enhancement for a flow in a pipe and a channel fully or partially filled with porous medium.The effects of porous layer thickness on the rate of heat transfer and pressure drop were investigated.He mentioned that partially filling the conduit with porous medium enhances the rate of heat transfer.Van der Sman (2002) tested the validity of the Darcy-Forchheimer-Brinkman (DFB) theory of flow through confined porous media using experimental data of pressure drop and velocity correlations which are describing the airflow through a vented box packed with horticultural produce.His Results show that the DFB model can reproduce experimental data on pressure drop quite accurately.Angirasa (2002) presented an experimental investigation to demonstrate the heat transfer enhancement with metallic fibrous heat dissipaters.He concluded that the metallic porous heat dissipaters can achieve substantial heat transfer augmentation when compared to flat plate.Kim et al. (2001) presented an experimental study to investigate the impact of the presence of aluminum foam on the flow and convective heat transfer in an asymmetrically heated channel.They presented a simple correlation of the friction factor and the average Nusselt number of aluminum foams will be sought to provide a guide in practical applications.Kaviany (1985) developed a numerical work to investigate the fluid flow and heat transfer characteristics due to laminar flow between two isothermal parallel plates.His results show that Nusselt number for fully-developed fields increases with an increase in porous media inside the channel, while the pressure drop associated with the entrance region decreases.Vafai and Tien (1981) numerically analyzed the effects of the solid boundary and the inertial forces on flow and heat transfer in porous media attached over flat plate.Their results show that these effects are more pronounced in highly permeable media, high Prandtl number, large pressure gradients, and in the region close to the leading edge of the flow layer.
The geometrical shape of the channel walls is also one of great importance in enhancing heat transfer.The waviness is a special case of corrugation that can be used to promote heat transfer.Xei et al. (2007) numerically studied the effects of wavy heights, lengths, wavy pitches and channel widths of a wavy channel on fluid flow and heat transfer characteristics.The results showed that the heat transfer may be greatly enhanced due to the wavy characteristics.Wang and Chen (2002) had numerically studied the effects of the wavy geometry, Reynolds number and Prandtl number on the skin-friction and Nusselt number for flow through a sinusoidal curved converging-diverging channel.Their results showed that the amplitudes of Nusselt number and the skin-friction coefficient curves increase as Reynolds number and the amplitude-wavelength ratio increase.Russ andBeer (1997a, 1997b) numerically and experimentally studied the heat and mass transfer for a wide range of Reynolds numbers from laminar to turbulent flow in a pipe of a wavy surface.Their results showed that frictional loss increases with the increase of amplitude for the same Reynolds number.Moreover, a maximum value of Nusselt number was determined near the reattachment point of the flow in the converging part of the wave.Tanda and Vittori (1996) presented a numerical study for fully developed flow and heat transfer in a wavy channel.Their results showed that The position of the local heat transfer coefficient is sensitive to Reynolds and Prandtl numbers and to the geometric parameters of the wall waviness.Stone and Vanka (1996) have presented an accurate numerical scheme to solve the unsteady flow and heat transfer equations in a wavy passage.They observed that the flow is steady in part of the channel and unsteady in the rest of it.Also, as Reynolds number is progressively increased, the unsteadiness is onset at a much earlier location, accompanied by increased overall heat transfer and friction coefficients.Saniei and Dini (1993) experimentally studied the heat transfer characteristics due to turbulent flow conditions in a wavy-wall channel containing from seven waves.They concluded that the local Nusselt number has the highest magnitude on the second wave.
Based on the reported importance of using porous media in a hand, and using corrugated surfaces on the other hand, in improving and increasing the heat transfer, the present work integrates these two ideas and numerically investigates their combined effect on the nature of the flow and heat transfer by forced convection.So, this study assumed two models of wavy walled channel, fully filled with homogenous porous material.The first was the  5) Where and are the permeability and porosity of the porous structure, and is a dimensionless form-drag constant, which is evaluated by using the widely used empirical correlation (Hadim & North, 2005): (7) The dimensionless forms have been rendered for the quantities with respect to the characteristic length ( ), and the characteristic velocity ( °) as the following: In this study, the pressure terms in Equations ( 4) and ( 5) had been eliminated by differentiating these equations with respect to and respectively, and then subtracting one of them from the other.By using the dimensionless groups (8), the following equations will be the final form of the governing equations in terms of voricity-stream function formula: Where ( 12) The boundary conditions must be specified to solve the partial differential equations, which govern the model of study.No slip condition is considered at the solid walls (top and bottom).The flow over the cross-section at the inlet of the channel has uniform velocity ( °), whereas at the outlet section is fully developed.Thus, the dimensionless boundary conditions will be: 1) At the inlet section: 2) At the outlet section: 3) At the bottom wall: 4) At the top wall: In order to manage the irregular boundaries in the model (wavy walls), it is important to assume new coordinates ( , ), which introduce a regular domain, instead of the original coordinates ( , ).The general transformation from the physical domain ( , ) to the computational domain ( , ) is: The governing equations are solved on a curvilinear non-orthogonal grid, which is generated by solving Poisson's elliptic equation system.This system is represented by the following two equations: Middlecoff and Thomas (1980) developed a method to evaluate the values of the control functions ( ) and ( ) by assuming the following : Where the parameters Φ and Ψ are evaluated as follows: (18) Where and respectively are the values of and along the boundaries.
Upon introducing terms P and Q from Equations (16, 17), after specifying the parameters (Φ, Ψ), the transformed form of Equations (14, 15) will be as the following: (20) By solving the Equation systems (20, 21) numerically using Line successive over relaxation (LSOR) method (Petrović & Stupar, 1996), a typical grid system would be generated for the posed model like the illustrated in Figure 3.It is important to mention that suitable clustering functions were used in mesh generation operation in order to increase the density of the grid points in the regions having high steeper gradients (Petrović & Stupar, 1996) like the furrows of the wavy surfaces.
= ( , ) , = ( , ) The Figu Figure 3 Figure 4. C Figure (6) that channel, whi ed.Hereafter, l and the posit 10 ).The app n remote areas ct er strongly aff e latter formin ss. Figure (7) s nside non-poro middle of the m along the di low inside the

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