Numerical investigations on convective heat transfer enhancement in jet impingement due to the presence of porous media using Cascaded Lattice Boltzmann method

Highlights

• Cascaded collision Lattice Boltzmann Model is successfully implemented for forced convection flow through porous media.

• Cascaded collision model is found to be more stable and computationally efficient over BGK collision model.

• Jet impingement (1) over porous heat sink and (2) through a porous passage are investigated for heat transfer enhancement.

• Porous passage configuration showed appreciable enhancement in the heat transfer, while porous heat sinks deteriorates it.

• Increasing trend in the stagnation Nusselt number is seen, while lowering the Darcy number in porous passage configuration.

Abstract

The present numerical study using the Lattice Boltzmann method investigates the convective heat transfer characteristics in impinging jets due to the inclusion of porous media. A cascaded collision model is employed in the present study, and the comparison with BGK collision model shows that the cascaded collision model is more stable and computationally efficient. The local thermal non equilibrium model (LTNE) or the two energy equation model is used in the present study to account for the local thermal non-equilibrium arising between the fluid and solid phase temperatures due to the high solid to fluid thermal conductivity ratios associated with metallic porous foams saturated with air. Two jet impingement configurations with porous media are modeled: jet impingement over a porous heat sink and jet impingement through a porous passage. Jet impingement over a porous heat sink is observed to be deteriorating the convective heat transfer whereas jet impingement through a porous passage showed enhancement. A parametric study is carried out for the porous passage configuration by varying the porous passage width from twice the jet width (2W) to once the jet width (W) and varying the Darcy number from 2.75e-2 to 2.75e-4. Jet impingement through a porous passage of width equal to the slot jet width is found to enhance the stagnation Nusselt number by 60.7% and the average Nusselt number by 53.7% compared to the jet impingement configuration in the absence of porous media. Similarly, lowering the Darcy number helps in augmenting the stagnation Nusselt number, however, it did not show any influence on the Nusselt number in the wall jet region.

Introduction

The rapid surge in the power density of electronic systems due to miniaturization necessitated reliable and efficient heat dissipation techniques. Jet impingement is one of the few efficient methods for removing high heat fluxes. Apart from electronic cooling, it has a wide range of applications like cooling gas turbine blades, annealing metals, drying paper, textiles etc. [1], [2], [3], [4].

Recently, metallic porous foams drew attention from the thermal engineering community as heat sinks in jet impingement cooling due to their properties like high thermal conductivity and an unique open cellular structure. Jeng and Tzeng [1] carried out numerical investigations on slot jet impingement in a channel filled completely with a porous medium. Their results showed that the presence of the porous medium as heat sink enhanced the average Nusselt number by 2–3 times compared to the jet impingement cooling in the absence of porous media. Saeid and Mohamad [2] conducted a numerical parametric study on jet impingement cooling in a channel completely filled with porous media in the Darcy regime characterized by a Rayleigh number for the porous media, Ra^∗ (10 < Ra^∗ < 100). They concluded that for high Peclet number flows, the average Nusselt number could be increased (1) by reducing the distance between the jet and the heated surface, (2) by increasing the jet width or (3) by increasing the Rayleigh number. Wong and Saeid [3], [4] studied a similar configuration numerically under mixed convection conditions using the Brinkman Darcy model [3] and the Brinkman Forchheimer extended Darcy model [4] models, respectively. From the Brinkman Forchheimer extended Darcy model [4] they concluded that for sufficiently high Peclet numbers, the average Nusselt number decreases with increasing porosity or inertial coefficient in the non-Darcy regime.

By employing Finite Volume method and local thermal non equilibrium assumption, Wong and Saeid [5] carried out a parametric study to understand the influence of parameters like Rayleigh number, fluid to solid thermal conductivity ratio, and interfacial heat transfer coefficient over the Nusselt number in Jet impingement over fully filled porous media between two confined plates under mixed convection regime. The results show that the local thermal non-equilibrium condition between solid and fluid phases would exist when the interfacial heat transfer coefficient or/and the fluid to solid thermal conductivity ratio ($\frac{k_f}{k_s}$) is low. At higher Rayleigh number (80–200) and for Pe ≤ 300 the buoyant forces becomes dominant deteriorating the Nusselt number. Except the use of local thermal non equilibrium model, the physical domain, the geometric and porous parameters and the convection regimes investigated in the above study are quite different from the present study.

Through their experimental studies, Shih et al. [6] explained the effect of a non-dimensional porous heat sink height on the heat transfer characteristics. They concluded that reducing the porous heat sink height from a non-dimensional height of 0.92 to 0.15, at first increases the average Nusselt number due to the reduction in the resistance offered by the porous media. However, after a certain height reducing the porous heat sink height further deteriorates the average Nusselt number because of the decrease in heat transfer area. Fu and Huang [7] performed a numerical investigation on the convective heat transfer enhancement using three differently shaped porous heat sinks, rectangular, concave and convex. They showed that for a low porosity block, all three shapes showed heat transfer enhancement. However, for a high porosity block only the concave shaped porous block showed enhancement. The above studies show that one of the most important parameters, which has a major influence on the convective heat transfer enhancement, is the amount of the fluid that is able to reach the target plate.

Marafie et al. [8] conducted a numerical parametric study on jet impingement with a partially filled porous block. They showed that the maximum Nusselt number occurred at a non-dimensional porous block height of 0.05. Further increase in the porous block height resulted in a reduction of the local Nusselt number value asymptotically. De Lemos and Fischer [9] conducted a numerical parametric study on jet impingement cooling of a heated plate with and without the presence of porous media. They found that for a certain parametric range the presence of porous media as a heat sink actually deteriorated the heat transfer rate. They also showed that the local thermal equilibrium assumption gave accurate results only for porous foams having solid to fluid thermal conductivity ratios below 10.

Recently, Kumar and Pattamatta [10] showed through their numerical studies that in the forced convection regime, the presence of porous media as heat sink has deteriorated the convective heat transfer. They attributed the enhancement in the Nusselt number shown by Jeng and Tzeng [1] and Marafie et al. [8] to the inconsistencies observed in the geometric parameters of the compared configurations, but not to the influence of the porous heat sink. Hsieh et al. [11] experimentally investigated the heat transfer enhancement in the presence of a porous heat sink with a restricted outlet. They explained that the porous heat sink adds resistance to the incoming jet due to which the jet could penetrate only partially and the remaining flow escapes through the lateral sides. They used flow-restricting masks on the lateral sides of the porous heat sink and observed enhancement in the Nusselt number.

The literature study carried out above showed contrasting results with heat transfer enhancement due to the inclusion of porous medium as a heat sink in jet impingement cooling. Most of the numerical studies above modeled porous heat sinks with low solid to fluid thermal conductivity ratio and used the local thermal equilibrium approximation for the porous heat sink. However, in general, porous metallic foams possess high stagnant solid to fluid thermal conductivity ratios. It is shown in literature [12], [13] that using the local thermal equilibrium approximation for high thermal conductivity porous foams will lead to unacceptable errors. Therefore, the local thermal non-equilibrium (LTNE) approximation is adopted for the porous medium in this present study.

Over the last two decades, the Lattice Boltzmann Method has emerged as a powerful approach for simulating fluid and energy transport problems [14], [15], [16], [17]. This method has the distinct advantages of easy implementation of complex interface and boundary conditions, programming ease, and parallelization of the algorithm. The present study employs the Lattice Boltzmann method (LBM) as the numerical solver. The Bhatnagar-Gross-Krook (BGK) model is the simplest collision model and is used extensively for many fluid flow problems [18], [19]. However, at higher Reynolds numbers this model requires enormous computational effort due to the high resolution required by this model. Recently Geier et al. [20] [21] proposed the cascaded LBM, in which the collision operator relaxes the moments in a co-moving reference frame and requires less computational effort compared to BGK collision operator.

The main objectives of the present study are: (1) To validate the cascaded Lattice Boltzmann model along with the local thermal non-equilibrium approximation for porous media. (2) To conduct a numerical investigation on convective heat transfer enhancement in jet impingement using porous heat sinks (3) To investigate whether porous media as a flow passage in jet impingement would preferably enhance the convective heat transfer.