Heat Transfer in Nanofluid Boundary Layer Near Adiabatic Wall

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 thermal 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. The dependency of the nanofluid able accordance with Maxwell’s effective medium theory. thermophysical properties on the nanofluid concentration Thus, the experimental values of heat transfer enhance- necessitates the computation of the volume concentration ment were not significantly higher than the expected values according to the diffusion equation for .10 This rela- formed from analysis of the base fluids. tion can be derived from the continuity equation for the nanoparticle species concentration in terms of its mass 3.2. Perturbation for Dilute Nanoparticle fraction and then related to the volume fraction for dilute Concentration 1 12 13 concentration. In general, nanofluid heat transfer experiments5 6 are performed for dilute nanoparticle concentration, viz. 1. 3.1. Thermophysical Properties This is used as a simplification of the nonlinear, compress- The continuum description requires input from separate ible form of the problem to one which can be analysed considerations of the thermophysical properties, which are without massive numerical computations. The thermo- here represented in terms of the volume fraction. They are physical properties expressed in Eq. (6) are already in presented to first-order dependency in the volume fraction, perturbation expansion form. The velocity components and in dimensionless form normalized by the corresponding temperature are expressed as expansions in ascending pow- quantity of the base fluid: ers of the volume fraction as ∗ = 1 + ∗ + O2 =0 q = q + q + O2 0 1 (7) ∗c∗ = + ∗c∗ + O2 1 =0 (6) where the zeroth-order, q , pertains to the base fluid devoid ∗ = + ∗ + O2 0 ARTICLE 1 =0 q of nanoparticles and 1 represent the first-order perturba- k∗ = + k∗ + O2 1 =0 tion to nanofluid quantities. The perturbation expansions Eqs. (6) and (7) are substituted into the basic equa- ∗ where =0 is the slope of the dimensionless thermo- ∗ tions of continuity (Eq. (1)) and momentum (Eq. (2)), physical quantity at the origin.IP: 192.168.39.210 The representations On: Mon, 27 Sep 2021 10:14:42 12 13 Copyright: American Scientificwhich are Publishers in compressible fluid form, and into the are in “general form,” in that they could be obtained from energy equation, Eq. (5), to obtain the resulting conser- separate considerations discussed in Refs. [12,Delivered 13] and by Ingenta vation equations in zeroth order. These equations take the will thus not be elaborated here. The preference is to incompressible form for the base fluid and are reduced use properties recently obtained from molecular dynamics to the Blasius momentum and Pohlhausen thermal energy simulations,19–21 although such simulations are performed problems.15 only for gold-water nanofluids. Denoted by subscript MD The perturbation expansion is first applied to the par- for molecular dynamics, the properties as reported in ∗ ∗ ∗ tial differential equations, incorporating the thermophysi- Refs. [19–21] are 187and c =0 MD =0 MD cal properties (which are already in perturbation form), and −237, whereas results for viscosity and thermal conduc- the results are then subjected to the similarity transforma- tivity, which are only available in Ref. [19] and subject to ∗ ∗ tion of the independent variables into Blasius-Pohlhausen interpretation,12 are 10 and k 20. =0 MD =0 MD form15 for both the zeroth-order (base fluid) and first-order The same set of properties for gold nanoparti- √ = y∗/ x∗/Re cles, if obtained from mixture theories are as follows: (nanofluid) perturbation: √ with stream- ∗ = /UL = f x∗/Re ∗ = 1830, which is close to the molecular function and transformed =0 MIX u∗ = f dynamics value. However, ∗c∗ =−042, which velocity components√ in the streamwise and =0 MIX v∗ = f − f/ x∗/Re is considerably below the molecular dynamics result. The 2 in the wall-normal direction. use of mixture theory results is accompanied by Einstein’s Here, the primes denote derivative with respect to .The ∗ = 18 advantage of the small volume fraction perturbation is that value for viscosity =0 Einstein 2 50 and Maxwell’s k∗ = it not only reduces the boundary layer equations to an estimate of thermal conductivity =0 Maxwell 3. These values are contrasted to 10 and 20, respectively, as incompressible form, but also linearizes the zeroth-order obtained from MD calculations. For a discussion of the momentum equation. The first-order nanofluid momentum use of mixture theory results and its comparison with the and diffusion problems remain unchanged for the present use of molecular dynamics results for properties on heat problem, and the reader is directed to Refs. [12, 13] for transfer and drag rise, please refer to the thorough discus- details of the solutions. sions in Refs. [12, 13]. In the present paper, only molecular dynamics simulation results for thermophysical properties 3.2.1. The Thermal Energy Problem (for gold nanoparticle fluids) are considered. The method for solving the thermal boundary layer prob- When examining the thermophysical properties of lem strongly resembles that for an incompressible fluid nanofluids, it is necessary to acknowledge the benchmark after the perturbation expansion has been obtained.15

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The temperature from Eq. (5), because of the linearity after 3.2.2. The Momentum Problem the perturbation expansion, can be written as The inputs from zeroth-order momentum are the Blasius U 2 profile, recomputed for convenience from T− T = CH + P (8) c 1 2 f f + f f = 0 0 2 0 0 (13) T f = = f f → where is the free stream temperature. The tempera- 0 0 0 0 0 0 1 ture solution is generally written as a superposition of the 12 13 heat transfer solution, = + + O2 , with and the first order nanofluid momentum in terms of H H0 H1 f zero viscous dissipation and the viscous dissipation-driven the perturbation velocity 1 = + +O2 1 1 solution P P0 P1 . Linearity allows for f + f f + f f = ∗ − ∗ f f 1 0 1 1 0 =0 =0 0 0 their superposition. The zeroth-order, H0, and first-order, 2 2 ∗ H , heat transfer problems are identical to those dis- + f 1 =0 0 (14) cussed in Refs. [12, 13]. The viscous dissipation effect f = = f f → 1 0 0 1 0 1 0 enters through the particular solutions, P , for the adia- 15 ∗ batic wall problem. The nanofluid boundary layer above The term involving =0 on the right side reflects vis- an adiabatic wall is addressed in this paper. The constant cosity diffusion effects, and is included due to the C takes the form for heat transfer with viscous dissipation dependence of nanofluid viscosity on the volume fraction. T ∗ with imposed wall temperature W The term involving =0 is an inertial effect originally U 2 on the left side, which also depends on the volume frac- C = T − T − tion. The outer boundary condition is homogeneous, as the W c P 0 (9) 2 f main outer boundary condition is satisfied by the zeroth- f → The zeroth-order temperature function, which includes the order velocity, 0 1. viscous dissipation, satisfies 3.2.3. The Volume Concentration Diffusion Problem 1 + Pr f =− Pr f 2 P0 f 0 P0 2 f 0 Lastly, the input from the volume concentration is 2 (10) discussed. It satisfies the simple convective diffusion = IP: 192.168.39.210 = On: Mon, 27 Sep12 2021 13 10:14:42 P0 0 0 P0 0 Copyright:0 American Scientificequation: Publishers Delivered by Ingenta Sc for an insulated wall. It obtains an integral solution in f + f = 0 terms of the Blasius function,14 2 0 (15) = → 0 W 1 = Pr f Prf f 2−Prf d d P0 2 f 0 0 (11) which is similar to the zeroth-order convective heat trans- 0 ARTICLE + Pr / f = fer problem for H0: H0 f 2 0 H0 0 with Although the above Pohlhausen integral solves the 0 = 1and → 0, except for the boundary Pr f H0 H0 P0 f profile in terms of the Blasius function 0 , condition at the wall and the presence of the Prandtl num- the re-computation of the original differential equation ber in place of the Schmidt number. A transformation of (Eq. (10)) is numerically more convenient as an input to the form = − 1/W − 1 gives the first-order nanofluid problem. The resulting first-order Sc perturbation nanofluid thermal boundary layer problem is + f f = 0 0 obtained as 2 (16) 0 = 1 → 0 Pr + f f +f P 0 P 1 P which renders to become independent of the wall value 1 2 1 0 . It is now possible to write down the solution, follow- =−k∗ −∗c∗ −k∗ W =0 =0 P0 =0 P0 ing that for H0 as ∗ c −Pr − Pr ∗ f 2+ f f f Scf d f P0 2 f =0 0 2 0 1 0 Scf Scf = (17) f Scf d 0 0 0=0 P →0 P1 1 Again, because of convenience in the numerical implemen- (12) tation, the code solves the differential Eq. (16) directly. The perturbation satisfies the insulated wall condition and the free stream condition; the latter is already satisfied 4. THE NUMERICAL EXAMPLE by the zeroth-order solution in Eq. (11). Terms on the right FOR GOLD NANOFLUID side of Eq. (12) involving , and hence the associated Due to the nonlinearity of the equations, shooting method thermophysical properties, reflect their linear dependence is employed for obtaining a numerical solution. The numer- on the volume fraction. ical integration results for gold-water nanofluid using

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16 obtained through further applications of molecular dynam- φ W=0 ics methods to other combinations of nanoparticles and 14 φ W=1 φ base ﬂuids, or, via experimentally obtained properties. W=2 12

10 NOMENCLATURE

,0 xy Stream-wise and normal-to-wall coordinates P

θ θ 8 P,1 u v Stream-wise and normal-to-wall velocity

and 6 components ,0

P h Static enthalpy, [J] θ − 4 q Heat flux, [W m 2] θ T Absolute temperature, [K] P,0 2 D Brownian diffusion coefficient, [m2s−1] 2 −1 0 DT Thermal diffusion coefficient, [m s ] c Heat capacity, [J K−1] –2 Re UL/ 012345678 Reynolds number, η Pr Prandtl number, / Sc Schmidt number, /D Fig. 1. Dimensionless temperature profiles (solid line) P0 U −1 and (dashed lines) versus . Adiabatic wall temperatures Free stream velocity of fluid, [m s ] P1 −23 −1 = = = kB Boltzmann constant, 138 × 10 JK P1 0 : P1 0 0 9 46, P1 0 1 11 87, P1 0 2 14 86. L Physical length scale, [m] f Dimensionless stream function. Table I. Non-dimensional parameters for Figure 1. ARTICLE Pr Sc k∗ ∗ ∗c∗ f f =0 MD =0 MD =0 MD Subscripts Au-water 7 2 × 104 20 10 −2.37 p Nanoparticle f Base fluid IP: 192.168.39.210 On: Mon, 27 SepFree 2021 stream 10:14:42 molecular dynamics properties are shownCopyright: in Figure American 1. The Scientific0 Zeroth-orderPublishers perturbation to base fluid quantities Delivered by Ingenta diffusion layer for is much thinner than the thermal 1 First-order perturbation to nanofluid boundary layer due to the large Schmidt number, Sc = H Heat transfer solution f 2 × 104 (for an average nanoparticle radius of 10 nm) P Particular solution, i.e., viscous dissipation-driven and Pr = 7. These and other relevant non-dimensional solution f parameters are given in Table I. Referring to Figure 1, w Wall the = 1 case is equivalent to a solid wall for which _ MD Molecular dynamics theory W there is no nanoparticle flux at the wall, i.e., 0 = 0. _ MIX Mixture theory ‘’ Represents any quantity. The W = 0 case corresponds to a porous wall, and hence, nanoparticles are removed by a magnitude equal to the Superscripts free stream concentration , resulting in a lowered adi- = ∗ Non-dimensionalised quantity, abatic wall temperature. Finally, the W 2 case is one d in which nanoparticles are injected at the wall by a magni- d tude equal to the free stream concentration , resulting in d an increased adiabatic wall temperature. This effect is due . =0 d to the dependence of transport properties on the volume =0 concentration near the wall. The bulge of the temperature Greek Symbols profiles P is due to the fact that most of the viscous 1 −3 dissipation occurs in the interior of the boundary layer. Density, [kg m ] Dynamic viscosity, [kg m−1 s−1] Thermal conductivity, [W m−1 K−1] 5. CONCLUDING REMARKS Nanoparticle phase volume fraction The present particular solution P can always be com- Normalized volume fraction, / bined with the homogeneous heat transfer solution H Similarity independent variable for scaling boundary obtained in Refs. [12, 13] to form the general solution layer thickness, y∗/ x∗/Re for heat transfer with viscous dissipation. The possibili- Stream function ties of applying the continuum framework to estimate heat Dimensionless temperature, T − T/T − Tw transfer capabilities, relative to skin friction rise, depend Transformed normalized volume fraction, ( − / on the availability of nanofluid properties. These could be − w.

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