Numerical Simulation of Natural Convection of a Nanofluid in an Inclined Heated Enclosure Using Two-Phase Lattice Boltzmann Method: Accurate Effects of Thermophoresis and Brownian Forces

Ahmed and Eslamian Nanoscale Research Letters (2015) 10:296 DOI 10.1186/s11671-015-1006-0 NANO EXPRESS Open Access Numerical Simulation of Natural Convection of a Nanofluid in an Inclined Heated Enclosure Using Two-Phase Lattice Boltzmann Method: Accurate Effects of Thermophoresis and Brownian Forces Mahmoud Ahmed1 and Morteza Eslamian2* Abstract Laminar natural convection in differentially heated (β = 0°, where β is the inclination angle), inclined (β = 30° and 60°), and bottom-heated (β = 90°) square enclosures filled with a nanofluid is investigated, using a two-phase lattice Boltzmann simulation approach. The effects of the inclination angle on Nu number and convection heat transfer coefficient are studied. The effects of thermophoresis and Brownian forces which create a relative drift or slip velocity between the particles and the base fluid are included in the simulation. The effect of thermophoresis is considered using an accurate and quantitative formula proposed by the authors. Some of the existing results on natural convection are erroneous due to using wrong thermophoresis models or simply ignoring the effect. Here we show that thermophoresis has a considerable effect on heat transfer augmentation in laminar natural convection. Our non-homogenous modeling approach shows that heat transfer in nanofluids is a function of the inclination angle and Ra number. It also reveals some details of flow behavior which cannot be captured by single-phase models. The minimum heat transfer rate is associated with β = 90° (bottom-heated) and the maximum heat transfer rate occurs in an inclination angle which varies with the Ra number. Keywords: Natural convection; Nanofluids; Thermophoresis; Inclined enclosure; Differentially heated enclosure; Bottom-heated enclosure; Two-phase lattice Boltzmann method Background A nanofluid is a mixture of a small quantity of conducting nanoparticles suspended in a base fluid, such as water. A nanofluid is particularly known for its high, non-linear, and anomalous thermal conductivity, compared to the base fluid. Most recent studies also show an increase in heat transfer rate when a nanofluid is used in natural or forced convection in cavities, channels, etc. Despite the overwhelming number of publications, the heat transfer augmentation in several cases, such as natural convection in an enclosure utilizing a nanofluid, is not fully understood. Some of the results are contradictory to one another, due * Correspondence: 2 University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai 200240, China Full list of author information is available at the end of the article to several reasons, such as the lack of reliable experimental data and fundamental theoretical studies and accurate numerical simulations. While a nanofluid flow in general is a non-homogenous two-phase flow with a significant relative drift or slip velocity between particles and the base fluid, many works have assumed a homogenous mixture to simplify the simulation. This simplification may still provide a general picture and understanding of the problem, but in some cases, such as laminar natural convection, it may be a source of significant errors in estimating the Nu number and the convection heat transfer coefficient. In this paper, the state-of-the-art modeling approach, i.e., the lattice Boltzmann method (LBM) is employed. In recent years, the powerful LBM has been used to simulate heat transfer in nanofluids in various geometries, such as cavities and channels, e.g., [1–6]. When the LBM is used for nanofluids, © 2015 Ahmed and Eslamian. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Ahmed and Eslamian Nanoscale Research Letters (2015) 10:296 the external forces acting on nanoparticles need to be considered, separately. The important external forces that are responsible for creating a slip velocity on the surface of the suspended nanoparticles are discussed by Buongiorno [7]. In addition, the present authors [5, 6], among others, have shown that at certain conditions, slip velocity develops in natural convection in bottom-heated and differentially heated enclosures, as a result of external forces, such as thermophoresis and Brownian forces. These forces as well as the gravitational force play a significant role in the flow and heat transfer characteristics. This paper is an attempt to contribute to the physical understanding of the velocity slip mechanisms, and in particular the role of thermophoresis using a two-phase non-homogenous model. The thermophoresis role in nanofluids has been either neglected or in some works has been estimated inaccurately or erroneously. Some workers have used thermophoresis models that are only applicable to gases to model thermophoresis in liquids, causing errors as large as several orders of magnitude. The objective of this work is twofold: first, to use a two-phase lattice Boltzmann method that can model a slip velocity on particle surfaces which causes mixing and heat transfer augmentation and, second, to investigate the accurate contribution of thermophoresis as an external force and a mechanism that causes velocity slip and heat transfer augmentation. Laminar and turbulent natural convection of nanofluids have been extensively studied in the bottom-heated and differentially heated enclosures. But very few works have been performed concerning inclined enclosures. Earlier studies on natural convection of pure fluids, such as air or water in enclosures with two parallel walls insulated and the other two walls kept at different temperatures, indicate that heat transfer rate in pure fluids changes with the inclination angle, where the lowest heat transfer rate occurs when the heated surface is on the top [8], if the enclosure rotation span is 360°. Similar observations are expected, when a nanofluid is used. Some workers have used single-phase or homogenous models to study heat transfer and fluid flow in nanofluids in inclined and other geometries, e.g., [9–14]. In most cases, the external forces are neglected. In a differentially heated inclined enclosure, the heat transfer rate increases as the inclination angle increases up to an optimum inclination angle (45° for Ra = 104, 30° for Ra = 105 and 106), beyond which the heat transfer rate decreases, e.g., [10, 12, 13]. Aminossadati and Ghasemi [11] studied heat transfer characteristics in an inclined square cavity with and without a central solid block. The inclination had no effect on heat transfer rate at low Ra numbers. At high Rayleigh numbers, with a central block, inclination enhanced heat transfer rate. This was not the case in the absence of the block. Page 2 of 12 The magneto-hydrodynamic (MHD) nanofluid in an inclined enclosure (...truncated)