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Heat Transfer in Nanofluid Boundary Layer Near Adiabatic Wall Copyright © 2018 by American Scientific Publishers Journal of Nanofluids All rights reserved. Vol. 7, pp. 1297–1302, 2018 Printed in the United States of America (www.aspbs.com/jon) Heat Transfer in Nanofluid Boundary Layer Near Adiabatic Wall D. Hopper, D. Jaganathan, J. L. Orr, J. Shi, F. Simeski, M. Yin, and J. T. C. Liu∗ School of Engineering (ENGN 2760) and Fluids@Brown, Brown University, Providence, Rhode Island 02912, USA Heat transfer in nanofluid boundary layers has been receiving significant attention in recent years due to widespread applications and ongoing research in both academia and industry. Cooling systems in batteries, semiconductors, and other components in the hardware technology industry are just a few areas where the limits of heat transfer are being pushed. The principal region of heat exchange between the wall and fluid is within the thermal boundary layer. Intriguingly, the heat transfer of particle-laden nanofluid is greatly affected by both the appearance of the particle and the nanoscale molecular effects; hence, a thorough characterization of the boundary layer behaviour and properties is needed. Prior research has shown that nanofluids enhance the heat transfer. Based on related works (J. T. C. Liu, Proc. Royal Society A 408, 2383 (2012); J. T. C. Liu, et al., Arch. Mech. 69, 75 (2017); C. J. B. de Castilho, et al., J. Heat Transfer Eng. 1 (2018)), convective heat transfer in nanofluid boundary layers is studied by a theoretical model with first-order perturbation of variables. Small nanoparticle volume concentration is assumed. A theoretical model of nanofluid flow over an adiabatic wall is ARTICLE presented, and the corresponding temperature and velocity profiles are obtained. The cases of both a solid and a porous wall are considered. The 1D numerical results for a gold-water nanofluid with various volume fractions are presented, showing that there exists a bulk of heat accumulation near the adiabatic wall that corresponds to an enhanced heat exchange.IP: 192.168.39.210 On: Mon, 27 Sep 2021 10:14:42 Copyright: American Scientific Publishers KEYWORDS: Nanofluids, Convective Heat Transfer,Delivered Heat Transfer by Ingenta Enhancement, Adiabatic Wall Temperature, Perturbation Method. 1. INTRODUCTION previous semi-analytical work on convective heat transfer Since the introduction of nanofluids, attributed to Choi,1 2 in nanofluids,7 while Kakac and Pramuanjaroenkij review there have been widespread applications of their use as the experimental work.8 Manca et al. offer an overview of heat transfer fluids. Many industrial applications from the heat transfer enhancement in nanofluids more generally.9 hardware manufacturers to renewable energy3 consider The continuum description10 of the nanofluid bound- boundary layer heat transfer essential to their products.4 ary layer is generally in compressible form, even though The application of nanofluids in heat transfer enhancement the base fluid is incompressible. This is because of the involves flowing nanofluids in, for instance, a forced con- dependence of the nanofluid thermophysical properties vective heat transfer situation, as in the experiments of on the nanoparticle concentration, which in turn, is sub- Wen and Ding5 and of Jung, et al.6 in microchannels. They jected to nanoparticle diffusion through possible Brown- showed that the leading-edge region of the microchannels ian diffusion and thermal diffusion effects. Nanofluid heat gave significant heat transfer enhancement as a function of transfer experiments5 6 are carried out at dilute nanofluid increasing nanofluid volume fraction. In this case, it is of volume fractions, which motivated a perturbation scheme theoretical importance to understand the convective ther- to be devised for small volume concentrations.11–13 This mal boundary layer behaviour that resembles the channel scheme reduced the problem into perturbations about the leading edge over a large streamwise distance before the base fluid; the nanofluid effect becomes linear resulting in boundary layers merge. The downstream fully developed the nanofluid momentum, diffusion and thermal boundary region has much less spectacular heat transfer behaviour. layer equations to become separable. Sheikholeslami and Ganji provide a good summary of As such, both the momentum and diffusion equations are solved first, whose solutions enter the nanofluid ther- mal energy problem, in the form of enthalpy, as variable ∗Author to whom correspondence should be addressed. Email: [email protected] coefficients and as inhomogeneous terms. Thus, due to Received: 28 February 2018 linearity, the nanofluid thermal energy problem is split Accepted: 28 March 2018 into: (1) the heat transfer problem, which depends on the J. Nanofluids 2018, Vol. 7, No. 6 2169-432X/2018/7/1297/006 doi:10.1166/jon.2018.1551 1297 Heat Transfer in Nanofluid Boundary Layer Near Adiabatic Wall Hopper et al. temperature loading; and (2) the adiabatic wall with vis- boundary layer flows. The normal component of the heat cous dissipation heating, which depends on the Eckert flux is defined as:5 number. The Eckert number is defined as the ratio of the T ln T free stream kinetic energy to the temperature loading. The q =−k − D h − D h y p y p p T y p (4) second part strongly resembles the thermal boundary layer 14 15 problem in an incompressible fluid. The first part of where k is the nanofluid thermal conductivity, p is the the nanofluid thermal energy problem on heat transfer is nanoparticle material density, D is the Brownian diffu- 15 discussed in Refs. [11–13]. The present paper discusses the sion coefficient, DT is the thermal diffusion coefficient, nanofluid “thermometer,” i.e., the adiabatic wall tempera- is the nanoparticle phase volume fraction, and hp = cpT ture reading in a nanofluid subjected to viscous dissipation is the static enthalpy of the nanoparticle phase with heat in convective flows. For future work, it would be benefi- capacity cp. This definition includes the mechanisms of cial to expand this approach for multi-physics problems16 conduction and thermal energy transport owing to nanopar- and different geometries.4 17 ticle concentration diffusion due to Brownian motion and to thermal diffusion attributed to thermophoresis. Thermal 2. GOVERNING EQUATIONS equilibrium is assumed between the base fluid and the nanoparticle phase.10 The rate of viscous dissipation is the The fundamental equations are necessarily in the form last term in the thermal energy equation. For completeness, for a compressible fluid, even though the individual the energy transport due to thermal diffusion is included constituents, such as the base fluid and the dispersed above but is not included in the following discussions since nanoparticles, are individually incompressible. This setup detailed calculations of Buongiorno,10 which were reaf- is a direct result of the dependence of the nanofluid firmed by Liu,11 showed that it has a much weaker effect thermophysical properties on the nanoparticle volume relative to Brownian diffusion. fraction. The continuum description is synthesized by Buongiorno.10 The two-dimensional flat plate is used as an approximation for the leading edge of a semi-infinite 3. DETAILED CONSIDERATION OF microchannel or tube, in the developing region prior to the NANOFLUID THERMAL ENERGY merging influence of the side walls.IP: 192.168.39.210 The two-dimensional On: Mon,The 27 nanofluidSep 2021boundary 10:14:42 layer energy equation is recast boundary layer continuity and streamwiseCopyright: momentum American Scientificinto the followingPublishers form after substituting the heat flux from Delivered by Ingenta equations for steady flow (e.g., Refs. [12, 13]) are Eq. (4) into Eq. (3), and defining dimensionless quantities u v (although, for later convenience, the temperature is left + = dimensional): x y 0(1) ∗ ∗ ∗ T ∗ T ARTICLE u v u c u + v u + v = (2) x∗ y∗ x y y y T = 1 k∗ + c∗D∗ T where and are the nanofluid density and viscosity, ∗ ∗ ∗ ∗ Re Prf y y Re Scf y y respectively, and are functions of the nanoparticle concen- 2 ∗ 2 tration; x and y are the streamwise (measured from the U /cf u + ∗ (5) leading edge) and wall-normal coordinates; and, u and v Re y∗ are the respective velocity components. The y-momentum equation is replaced by constancy of pressure across the Here, subscript denotes conditions in the free stream. Re = UL/ boundary layer. The Reynolds and Prandtl numbers are f Pr = / f To study nanofluid effects on the adiabatic wall tem- and f f f , where the subscript pertains to the perature, the energy equation is augmented from that used base fluid. The kinematic viscosity and thermal diffusiv- = / = k / c in Refs. [12, 13] for heat transfer studies, to include the ity are given by f f f and f f f f , respec- Sc = /D rate of viscous dissipation. In terms of the nanofluid static tively. The Schmidt number is f f ref ,andthe enthalpy, it is dimensionless nanoparticle density-heat capacity prod- c∗ = c / c uct is p p f f Finally, the diffusion coef- 2 D∗ = D/D h h q u ficient is nondimensionalized as ref ,where u + v =− + (3) D = k T / r x y y y ref B avg 6 f d. The Brownian diffusion coefficient T is evaluated at the average temperature avg, such that where the nanofluid static enthalpy is h = cdT; c = D∗ = 1. The last two quantities are the Boltzmann con- dh/dT is the nanofluid heat capacity and T is the absolute stant, kB,andrd is the average nanoparticle radius. The temperature. The work done by the pressure gradients is normalized volume fraction is = /. not present due to zero streamwise pressure gradient in flat While the heat transfer problem in absence of viscous plate flow and the absence of normal pressure gradient in dissipation is thoroughly discussed in Refs. [12, 13], the 1298 J. Nanofluids, 7, 1297–1302, 2018 Hopper et al. Heat Transfer in Nanofluid Boundary Layer Near Adiabatic Wall present paper discusses the adiabatic wall problem, which study provided by Buongiorno et al.,22 which found the is an essential part of the overall study of the nanofluid heat transfer enhancement of nanofluids to be in reason- thermal boundary layer.
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