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Heat transfer behaviours of nanofluids in a uniformly heated tube
Superlattices and Microstructures 35 (2004) 543–557 www.elsevier.com/locate/superlattices
Heat transfer behaviours of nanofluids in a uniformly heated tube Sidi El B´ecaye Ma¨ıgaa, Cong Tam Nguyena,∗, Nicolas Galanisb, Gilles Roya a Faculty of Engineering, Universit´e de Moncton, Moncton, NB, Canada E1A 3E9 b Faculty of Engineering, Universit´e de Sherbrooke, Qu´ebec, Canada J1K 2R1
Received 12 May 2003; accepted 23 September 2003 Available online 2 June 2004
Abstract In the present work, we consider the problem of the forced convection flow of water–γ Al2 O3 and ethylene glycol–γ Al2 O3 nanofluids inside a uniformly heated tube that is submitted to a constant and uniform heat flux at the wall. In general, it is observed that the inclusion of nanoparticles has increased considerably the heat transfer at the tube wall for both the laminar and turbulent regimes. Such improvement of heat transfer becomes more pronounced with the increase of the particle concentration. On the other hand, the presence of particles has produced adverse effects on the wall friction that also increases with the particle volume concentration. Results have also shown that the ethylene glycol–γ Al2 O3 mixture gives a far better heat transfer enhancement than the water–γ Al2 O3 mixture. © 2004 Elsevier Ltd. All rights reserved. Keywords: Forced convection flow; Laminar flow; Turbulent flow; Heat transfer enhancement; Heat transfer augmentation; Nanofluid; Nanoparticles; Numerical simulation
1. Introduction The thermal properties of heating or cooling fluids play a vital role in the development of new energy-efficient heat transfer equipment. However, conventional heat transfer fluids such as water, ethylene glycol and engine oils have, in general, poor heat transfer properties compared to those of most solids. In spite of considerable research and efforts deployed ∗ Corresponding author. Tel.: +1-506-858-4347; fax: +1-506-858-4082.
E-mail address: [email protected] (C.T. Nguyen). 0749-6036/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2003.09.012
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Nomenclature Cp Specific heat of the fluid D Tube diameter Nu Z Local Nusselt number, Nu Z = h D/k0 Pr Prandtl number R Radial coordinate Re Reynolds number T Fluid temperature T0 Inlet fluid temperature Average axial velocity at tube inlet V0 V R Radial velocity component Tangential velocity component Vθ Axial velocity component VZ Z Axial coordinate h Heat transfer coefficient k Thermal conductivity of the fluid p Pressure q Wall heat flux Greek letters ε Turbulent dissipation rate θ Circumferential coordinate κ Kinetic energy of turbulence µ Fluid dynamic viscosity ρ Fluid density τ Wall shear stress ϕ Particle volume concentration Indices bf Refers to base-fluid i Refers to spatial direction, i = 1, 2 and 3 nf Refers to nanofluid property p Refers to particle property r Refers to ‘nanofluid/base-fluid’ ratio 0 Refers to the reference condition
until today, major improvements in heat transfer capabilities have suffered a major lacking. As a result, an important need still exists to develop new strategies in order to improve the effective heat transfer behaviours of conventional heat transfer fluids.
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The term ‘nanofluid’ refers to a two-phase mixture with its continuous phase being generally a liquid and the dispersed phase constituted of ‘nanoparticles’ i.e. extremely fine metallic particles of size below 100 nm. It has been shown that the thermal properties of such a nanofluid appear to be well above those of the base-fluid. In fact, some available experimental data [1–3] have shown that even with a relatively low concentration of particles, say from 1 to 5% in volume, the effective thermal conductivity of the mixture has increased by almost 20% compared to that of the base-fluid. Such an increase depends mainly on several factors such as the form and size of the particles and their concentration, the thermal properties of the base-fluid as well as those of the particles. Hence, the nanofluids can constitute an interesting alternative for advanced applications in heat transfer in the future, especially those in micro scale, see for example [8]. In spite of their potentials and advantages, these very special fluids are still in their earlier development. In fact, the first experimental works were concerned only with the measuring and the determination of the effective thermal conductivity [1–7, 9]; some also provided the effective viscosity of the nanofluids [1, 4, 9] showing that the inclusion of the nanoparticles can appreciably increase the viscosity of the resulting mixture. These works considered some current fluids, water, ethylene glycol and engine oil, and metallic particles such as γ Al2 O3 , SiO2 , TiO2 and Cu particles. It is very important to note that the amount of experimental data resulting from these studies remains, surprisingly, very limited. Furthermore, there are no available data regarding the effects of the temperature on the effective thermal conductivity and viscosity of nanofluids. It appears obvious, at the present stage, that much more works will be needed on this important issue in the future. With regard to the thermal performance of nanofluids in confined flows, the only recent experimental works [9, 10] have provided the first empirical correlation for computing the Nusselt number in laminar and turbulent tube flows using nanofluids that are composed of water and Cu, TiO2 and γ Al2 O3 particles. The data as obtained from these studies have clearly shown that the suspended nanoparticles remarkably increase the heat transfer performance of the base-fluid and the suspensions have larger heat transfer coefficients than the base-fluid (pure water) under the same Reynolds number. Such an improvement becomes more important with an augmentation of the particle volume fraction. In spite of their great potentials and advantages, these new, special yet rather challenging fluids still remain in their earlier development state and much more research work will be needed in order to better understand their fluid dynamics and thermal characteristics, in particular for confined flows. In this paper, we are interested to study numerically the beneficial influence due to the inclusion of nanoparticles on the flow behaviours and the temperature field inside a uniformly heated tube. 2. Mathematical formulation and numerical method We consider in this study the problem of the forced convection flow of a nanofluid, which is composed of water or ethylene glycol and metallic γ Al2 O3 nanoparticles, flowing inside a straight tube of circular cross-section, Fig. 1. Due to the extreme size of particles, it may be reasonable to suggest that such a mixture can be easily fluidized and therefore, one may assume that the motion slip between the phases, if any, would be considered
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Fig. 1. Geometrical configuration of the problem studied.
negligible [11]. Also, by considering the local thermal equilibrium, the solid particle–liquid mixture may then be approximately considered to behave as a conventional single-phase fluid with properties that are to be evaluated as functions of those of the constituents knowing their respective concentrations. We also consider that the flow is steady, laminar and symmetrical with respect to the vertical plane passing through the tube main axis, Fig. 1. Under the above conditions, the corresponding governing equations written in a (R, θ, Z ) coordinate system are as follows [18]: ∇ · (ρV) = 0 ∇ · (ρVVi ) = −∂ p/∂ X i + ∇ · (µ∇Vi ) + Si
(1) (2)
∇ · (ρVCp T ) = ∇ · (k∇T )
(3)
where V = (V R , Vθ , V Z ) is the velocity vector; p is the pressure; X i refers to a spatial direction (X i = R, θ and Z ); ρ, µ, k and Cp are respectively the fluid density, dynamic viscosity, thermal conductivity and specific heat. The terms Si in the momentum Eq. (2), which represent the remaining viscous terms due to the choice of the coordinate system, are given as follows: – for i = 1, the radial direction: S1 = ρVθ Vθ /R − µ{V R /R 2 + (2/R 2 )∂ Vθ /∂θ }
(4)
– for i = 2, the tangential direction: S2 = µ{(2/R 2 )∂ V R /∂θ − Vθ /R 2 } − ρV R Vθ /R
(5)
– for i = 3, the axial direction: S3 = 0.
(6)
2.1. The turbulent flow regime For the same problem, we are also interested to study the influence due to the inclusion of nanoparticles under the turbulent flow regime. Such a regime is often encountered in
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various applications in engineering. By applying the time-averaging procedure taking into account the fluctuating components of dependent variables, one can obtain the so-called ‘Reynolds-averaged’ Navier–Stokes equations that are the equations of conservation of mass and momentum. A similar Reynolds form of the energy equation can be obtained as well using the same procedure and the concept of Reynolds’ analogy to turbulent momentum transfer. With the exception of some additional terms expressing the turbulent related stresses and heat-flux quantities that have been introduced during the above timeaveraging procedure, the resulting ‘averaged equations’ have, in essence, the same general form as that of the instantaneous Navier–Stokes equations [18]. For the proper closing of these equations, one often needs to introduce empirical data or approximate models to express the turbulent stresses and heat-flux quantities to the related phenomenon. In the present study, we have adopted the well-known semi-empirical κ–ε turbulent model [19], which introduces two additional equations of conservation, namely the equation of the turbulent kinetic energy, κ, and the corresponding one for the rate of dissipation, ε. The simplifying form of these equations is given as follows for the problem under consideration here: ∇ · (ρVκ) = ∇ · [(µ + µt /σκ )∇κ)] + G κ − ρε
(7)
∇ · (ρVε) = ∇ · [(µ + µt /σε )∇ε)] + C1ε (ε/κ)G κ + C2ε ρε /κ. 2
(8)
In the above equations, G κ represents the generation of turbulence kinetic energy due to the mean velocity gradients; C1ε and C2ε are constant; σκ and σε are the turbulent Prandtl numbers for κ and ε, respectively; and µt is the turbulent (or eddy) viscosity, computed as: µt = ρCµ κ 2 /ε
(9)
where Cµ is a constant. The various constants of the model are as follows [17, 19]: C1ε = 1.44; σκ = 1.0
C2ε = 1.92; and
σε = 1.3.
Cµ = 0.09;
(10)
Note that the complete details regarding the derivation of the Reynolds form of the conservation equations as well as the κ–ε turbulent model have been very well documented elsewhere (the reader is advised to consult in particular [17–19] for further details). It is also important to note that regarding the turbulent viscosity µt , there is a clear lack of data permitting an appropriate quantification of such a property for mixtures containing nanoparticles. Since we have assumed that the nanofluid would behave as a single-phase homogenous fluid, the turbulent viscosity µt may then be evaluated as for a conventional fluid, say by Eq. (9) for the κ–ε turbulent model. As we can see later in Section 3.2 through a satisfactory comparison with experimental data in a turbulent regime, such an assumption appears justified. 2.2. Physical properties of nanofluids The following formulas have been employed to compute the thermal and physical properties of the nanofluids under consideration (the subscripts p, bf and nf refer to
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Fig. 2. Experimental data and correlation for dynamic viscosity of water–γ Al2 O3 .
the particles, the base-fluid and the nanofluid, respectively; the subscript r refers to the ratio ‘nanofluid/base-fluid’ of the quantity considered): ρnf = (1 − ϕ)ρbf + ϕρp
(11)
(C p )nf = (1 − ϕ)(C p )bf + ϕ(C p )p µnf = 123ϕ 2 + 7.3ϕ + 1 for water–γ Al2 O3 µr = µbf µnf = 306ϕ 2 − 0.19ϕ + 1 for ethylene glycol–γ Al2 O3 µr = µbf knf kr = = 4.97ϕ 2 + 2.72ϕ + 1 for water–γ Al2 O3 kbf knf = 28.905ϕ 2 + 2.8273ϕ + 1 for ethylene glycol–γ Al2 O3 . kr = kbf
(12) (13) (14) (15) (16)
Eqs. (11) and (12) are general relationships used to compute the density and specific heat for a classical two-phase mixture [9]. Eqs. (13) and (14) for computing the dynamic viscosity of nanofluids have been obtained by performing a least-square curve fitting of experimental data available for the mixtures considered [1, 3, 4]. It is interesting to mention here that for the viscosity and the water–γ Al2 O3 nanofluid in particular, the well-known formulas such as the one proposed by Einstein–Brinkman [12], as well as the one suggested by Batchelor [13] have, surprisingly, underestimated drastically the values of the mixture viscosity while compared to the experimental data, as clearly shown in Fig. 2. For the thermal conductivity of the nanofluids under consideration, Eqs. (15) and (16) have been obtained using the well-known model proposed by Hamilton and Crosser [14]. It is very interesting to note that such a model, although originally being derived for a mixture with millimetre and micrometre size particles, appears appropriate for use with nanoparticles. It is important to mention that although some data do exist in the literature for the two nanofluids considered, they were obtained at fixed reference temperatures, that is to say
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that the dependence of the fluid properties on the temperature has not yet been established clearly to date. Also, in general, for most of the nanofluids of engineering interest, such experimental data providing information on their thermal and physical properties remain rather scarce, if not to say quasi-non-existing. Much more research work will be, indeed, needed in this field. One may determine that the problem under consideration can be characterized by a set of five dimensionless parameters, namely the Reynolds number Re = V0 Dρ0 /µ0 , the Prandtl number Pr = Cp µ0 /κ0 , the particle volume concentration ϕ, and the ratios κp /κbf and (C p )p /(C p )bf (where V0 is the uniform axial velocity at the tube inlet and the subscript 0 refers to the inlet reference condition). It should be mentioned that the shape as well as the dimension of nanoparticles might have some influence on the heat transfer characteristics of the resulting nanofluids. Such influence, to our knowledge, has not been investigated to date. 2.3. Boundary conditions and numerical method The fluid has a uniform axial velocity and temperature profiles at the tube inlet section. On the tube wall, the usual non-slip condition prevails; also, the uniform heat flux boundary condition is prescribed. At the exit section, the pressure condition prevails where a known pressure is specified. The system of governing equations, Eqs. (1)–(3) for the laminar regime or their counterparts and Eqs. (7) and (8) for the turbulent regime, subject to their appropriate boundary conditions, has been successfully solved by using the numerical method that is essentially based on the ‘finite control volume approach’. Since such a method has been very well documented elsewhere (see in particular [15, 16]), only a brief review is given here. This method, as with other members of the SIMPLE-codes family [15], is based on the spatial integration of each of the conservation equations over finite control volumes using the exponential scheme for the treatment of the combined ‘convection-anddiffusion’ fluxes of mass, heat, momentum or other scalars resulting from the transport process such as turbulent related quantities κ and ε. Also, the staggered grids have been used where the velocity components are calculated at the centre of the volume interfaces while the pressure as well as other scalar quantities such as temperature and species concentration for example, are computed at the centre of the control-volume. The ‘algebraic discretization equations’ resulting from this integration process have been solved sequentially and iteratively through the physical domain considered, by combining the efficient ‘line-by-line’ procedure and the well-known TDMA technique (‘Three Diagonal Matrix Algorithm’). In order to speed up the convergence rate, we have also employed the multi-passes and alternating sweeping technique for all directions in space. On the other hand, a special ‘pressure-correction’ equation, which was obtained by combining the discretization form of the Navier–Stokes equations and the corresponding one of the continuity equation, has been employed to compute the guessed pressure field as well as to correct the velocities field, during the calculation process, in order to satisfy progressively, i.e. in an iterative manner, all the discretization equations (the reader may consult [15] for complete details regarding the above numerical algorithms and procedures).
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Fig. 3. Results of grid sensitivity study.
In order to ensure the consistency as well as the accuracy of the numerical results, several non-uniform grids have been submitted to an extensive testing procedure. Fig. 3 shows, for example, the comparison of results as obtained by using different grids for V Z velocity component at R = 0 and other analytical and numerical data. These results have shown that for the problem of the tube flow under consideration, the 32 × 24 × 155 nonuniform grid appears to be quite satisfactory to ensure the precision and the independency of numerical results with respect to the number of nodes used. Such a grid has, respectively, 32, 24 and 155 nodes along the radial, tangential (covering all angular coordinates from θ = 0◦ to θ = 180◦) and axial directions, with highly packed grid points in the vicinity of the tube wall and also in the tube entrance region. As a convergence indicator, we have essentially based it on the residuals resulting from the integration of the governing Eqs. (1)–(3), (7) and (8) over finite control-volumes. During the calculation process, these residuals were constantly monitored and scrutinized. For all the simulations performed in this study, a converged solution was usually achieved with a very low level of these residuals, say 10−8 or less for all the governing equations considered. As starting conditions, we have employed the velocity and temperature fields corresponding to the cases of forced flow without particles. For subsequent cases, the flow data as obtained for a converged case with a given value of the particle volume concentration ϕ were used as initial conditions. 3. Results and discussion The computer model has been successfully validated by comparing the results as obtained for the development of fluid axial velocity V Z to the corresponding analytical and numerical data by others for the classical case of an isothermal developing laminar forced convection flow in a tube [18]. Fig. 3 (introduced previously) has shown such comparison where the agreement can be qualified as very good. Fig. 4 shows another comparison
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Fig. 4. Validation of the mathematical model.
performed for the Nusselt number Nu Z hence the heat transfer coefficient, in forced laminar convection flow with numerical results and data from Orfi [20] and Petukhov [21]. We can see, here again, that the agreement appears acceptable except for the region near the tube inlet where a certain discrepancy has been observed. Such a discrepancy is believed to be due to the grid size effect in this particular area. The computer code was then used to perform numerical simulations for the case under consideration here, using two different nanofluids, namely water–γ Al2 O3 and ethylene glycol–γ Al2 O3 . The heated tube has a diameter of 0.01m and a total length of 1.0 m. For the laminar cases, the uniform axial velocity V0 at the tube inlet has been adjusted in order to have a constant Reynolds number, say Re = 250; the wall heat flux was fixed to q = 10, 240 W/m2 . For the turbulent cases, q = 500,000 W/m2 and two different values of the Reynolds number, namely Re = 10,000 and Re = 50,000 have been considered (these values of Re, which represent normal conditions often encountered in practical applications, have also been adopted in order to be able to compare our results with some existing experimental data in the turbulent regime). In the following, some significant results showing the beneficial influence due to the nanoparticles are presented and discussed (unless otherwise noted, most of the results presented hereafter are for the water–γ Al2 O3 mixture). 3.1. Effect of nanoparticles in laminar flow heat transfer Results have clearly revealed that the inclusion of nanoparticles has modified considerably the thermal behaviour of the mixture. Fig. 5(a) shows the influence of the particle volume concentration ϕ on the radial temperature profile at the tube exit section. We can observe that the fluid temperature has decreased with the augmentation of ϕ, in particular in the region near the tube wall, thus indicating clearly a higher heat transfer rate with particles. Such behaviour may be better understood by scrutinizing Fig. 5(b) where one can notice at the exit section, a diminution of nearly 17 K of the wall temperature
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Fig. 5. Effect of particle loading ϕ: (a) on the radial temperature profile at tube exit and (b) on the axial development of wall temperature.
between the case ϕ = 10% and the one with ϕ = 0. It is interesting to note that such a decrease of the fluid temperature at the tube wall does exist all along the tube length and seems to be more important toward the tube end. These results have obviously indicated the beneficial effect due to the nanoparticles, an effect that may be explained by the fact that with the presence of these particles, the thermal properties of the resulting mixture have been greatly improved. For the case ϕ = 10% for example, the value of the product ρCp and the thermal conductivity k have increased approximately by 18% and 33%, respectively, with respect to the values corresponding to the case ϕ = 0. Thus, a nanofluid possesses, so far, a higher thermal capacity than a conventional one. We also note that with a higher thermal conductivity k, the heat transfer at the tube wall would be generally more important, as confirmed through the following results. Fig. 6(a) shows clearly that the presence of nanoparticles has produced a considerable improvement of the heat transfer at the tube wall. Thus, for the volume concentration of 7.5% for example, the ‘nanofluid/base-fluid’ ratio h r of the heat transfer coefficients (h r ≡ h nf / h bf ) is approximately 1.6 at the tube end, that is, the heat transfer coefficient has increased by almost 60% compared to that of the base-fluid. This ratio h r also increases appreciably with the augmentation of the parameter ϕ. Such an increase is observed all along the tube length and is clearly more important toward the tube end. We can also see that for a relatively low particle concentration, the heat transfer coefficient ratio h r increases quasi-linearly with the axial coordinate Z . Although a direct comparison with other results and experimental data was not possible because of the lack of data for the laminar regime and the nanofluids considered, it is interesting to mention that the above behaviour regarding the improvement of the heat transfer coefficient due to nanoparticles appears to be quite consistent with some recent experimental data obtained for nanofluids that contain Cu particles [10]. Fig. 6(b) shows the ‘global’ effect of the particle concentration on the averaged heat transfer coefficient (note that h r is defined as the ‘nanofluid/base-fluid’ ratio of the averaged heat transfer coefficients, h r ≡ h nf /h bf ). It is observed, at first, that the
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Fig. 6. Effect of particle loading ϕ: (a) on the axial development of the ratio h r , and (b) on the ratio h r of averaged heat transfer coefficients.
enhancement of the averaged heat transfer coefficient appears clearly more pronounced for the ethylene glycol–γ Al2 O3 mixture than for the water–γ Al2 O3 nanofluid. One can also observe that this ratio h r appears more important for a higher particle loading, say for φ > 3%; while for φ ≤ 3%, h r remains relatively low and almost identical for the two nanofluids considered. With regard to the wall friction, results have clearly shown, however, that the addition of nanoparticles into a base fluid has produced an adverse effect on the wall shear stress. Fig. 7(a) illustrates the axial development of the ‘nanofluid/base-fluid’ ratio τr (defined as τr ≡ τnf /τbf ). It has been observed that, for a given particle loading ϕ, the ratio τr reaches quite rapidly a constant value that is due to the axial development of the velocity field. In general, τr increases considerably with the particle volume fraction ϕ and this, all along the tube length. For ϕ = 7.5% in particular, τr has reached nearly 2.2 at the tube end. Complete results obtained in this study have shown that similar behaviours were also observed for the ethylene glycol–γ Al2 O3 mixture. The above adverse effects, although predictable, have also been observed on the averaged wall friction. Thus, Fig. 7(b) shows the influence of the particle loading parameter ϕ on the ‘nanofluid/base-fluid’ ratio τr (defined as the ratio of the averaged wall shear stresses, τr = τnf /τbf ). One can observe that for the ethylene glycol–γ Al2 O3 nanofluid, the increase of the averaged shear stress ratio appears clearly more pronounced than for the water–γ Al2 O3 mixture. For example for φ = 7.5% in particular, the ratio τr is approximately 2.75 for the former but only 2.2 for the second mixture. 3.2. Effect of nanoparticles in turbulent flow heat transfer The beneficial effects due to the inclusion of nanoparticles have also been clearly observed for a turbulent flow heat transfer within a tube. Fig. 8(a) shows, for example, the axial variation of the ‘nanofluid/base-fluid’ heat transfer ratio h r as obtained for the case Re = 50,000 and various values of ϕ with the water–γ Al2 O3 mixture. One can
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Fig. 7. Effect of particle loading ϕ in laminar flow regime: (a) on the axial development of the ratio τr , and (b) on the ratio τr of averaged wall shear stresses.
observe that for a very short distance from the inlet section, h r increases steeply due to the simultaneous development of both the velocity and thermal field. As in the laminar cases shown previously, it is observed that with the augmentation of ϕ, the heat transfer coefficient has also increased considerably with respect to that of the base-fluid. Such an increase is noticed all along the tube length and it appears to be more pronounced toward the tube end. For the cases with relatively low particle loading, say for ϕ ≤ 5%, the ratio h r is nearly constant over a large portion of the tube length. For the cases shown, h r has as values at the tube end i.e. under fully developed flow condition, 1.05, 1.15, 1.40, 1.61 and 1.89, respectively, for ϕ = 1%, 2.5%, 5%, 7.5% and 10%. Results from this study have shown that similar behaviours regarding the heat transfer improvement were also observed for the case Re = 10,000. Table 1 shows the comparison between our results as obtained for the ratio h r and the case Re = 50,000 and the corresponding values obtained by using the only recent experimental correlation proposed by Pak and Cho [9]. One can see that the agreement appears quite satisfactory with the maximum deviation estimated to be 10% with respect to the values of h r given by the correlation; this may give a clear indication about the appropriateness as well as the reliability of the mathematical model used in the present study. One may also state, although indirectly, that the assumption of a single-phase homogenous fluid as well as that of the calculation of the turbulent viscosity, Eq. (9), appear quite reasonable. Fig. 8(b) shows the influence of the particle loading parameter ϕ on the nanofluid-tobase-fluid ratio h r of the averaged heat transfer coefficients for the two Reynolds numbers studied. The beneficial effects due to the presence of particles are, once again, clearly demonstrated here. Thus, one can easily observe that with an increase of the particle volume fraction ϕ, the improvement of the average heat transfer coefficient becomes more pronounced. For the case Re = 10,000 for example, the ratio h r has reached 1.035, 1.09, 1.17, 1.33 and 1.47 respectively for ϕ = 1%, 2.5%, 5%, 7.5% and 10%. It is interesting to see that for ϕ ≤ 3%, the ratio h r remains almost identical for the cases tested; while for
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Fig. 8. Effect of particle loading ϕ in turbulent flow: (a) on the axial development of the ratio h r , (b) on the ratio h r of averaged heat transfer coefficients, and (c) on the ratio τr of averaged wall shear stresses.
a higher value of ϕ, it clearly becomes more important with the increase of the Reynolds number. For ϕ = 7.5% in particular, h r has as values, 1.33 and 1.43, respectively for Re = 10,000 and 50,000. It is very interesting to mention here that a similar trend regarding such influence of the Reynolds number for nanofluids heat transfer has been experimentally observed by other researchers (see in particular, Pak and Cho [9] and Li and Xuan [10]). Fig. 8(c) shows finally the influence of the particle concentration ϕ on the ratio τr for two different cases with Re = 10,000 and 50,000. As for the laminar cases considered before, we can observe that such a ratio clearly increases with an augmentation of the parameter ϕ. Thus, for the case Re = 10,000 in particular, τr has as values, 1.2, 1.45, 2.3, 4.2 and nearly 6.9 respectively for ϕ = 1%, 2.5%, 5%, 7.5% and 10%. It is interesting to see that the ratio τr appears to be independent with respect to the flow Reynolds number;
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Table 1 Comparison with experimental correlation for h r (Re = 50,000) Particle loading ϕ (%)
h r present study
h r by correlation from [9]
1 2.5 5 7.5 10
1.05 1.15 1.40 1.61 1.89
1.16 1.27 1.51 1.78 2.11
thus, for a particular particle loading ϕ, the corresponding values of τr are almost identical for the values considered of parameter Re. 4. Conclusion In this paper, we have numerically investigated the hydrodynamic and thermal behaviours of nanofluids flowing inside a uniformly heated tube. Results have clearly shown that the inclusion of nanoparticles has produced a considerable increase of the heat transfer with respect to that of the base liquid. Such heat transfer enhancement, which appears to be more pronounced with the augmentation of the particle volume concentration, is accompanied, however, by a major drawback on the wall shear stress. Among the nanofluids studied, it has been shown that the ethylene glycol–γ Al2 O3 offers, so far, a better heat transfer enhancement than the water–γ Al2 O3 nanofluid. It is also the one for which a more pronounced adverse effect on the wall shear stress has been observed. For the turbulent flow regime, results have also shown that the heat transfer enhancement due to the presence of nanoparticles becomes more important with the increase of the Reynolds number. Acknowledgements The authors wish to sincerely thank the Natural Sciences and Engineering Research Council of Canada, the Faculty of Graduate Studies and Research of the Universit´e de Moncton for financial support to this project, and also to the Faculty of Engineering of the Universit´e de Moncton for allocating computing facilities. References [1] H. Masuda, A.A. Ebata, K. Teramae, N. Hishinuma, Netsu Bussei 4 (4) (1993) 227. [2] S.U.-S. Choi, Developments and applications of non-Newtonian flows, ASME Publications FEDVol. 231/MD-Vol. 66, p. 99, 1995. [3] S. Lee, S.U.-S. Choi, S.S. Li, J.A. Eastman, J. Heat Transfer 121 (1999) 280. [4] X. Wang, X. Xu, S.U.-S. Choi, J. Thermophys. Heat Transfer 13 (4) (1999) 474. [5] J.A. Eastman, S.U.-S. Choi, S. Li, G. Soyez, L.J. Thompson, R.J. DiMelfi, J. Metastable Nanocryst. Mater. 2 (6) (1999) 629. [6] Y. Xuan, Q. Li, Int. J. Heat Fluid Flow 21 (2000) 58. [7] J.A. Eastmanm, S.U.-S. Choi, S. Li, W. Yu, L.J. Thompson, Appl. Phys. Lett. 78 (6) (2001) 718.
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[8] S. Lee, S.U.-S. Choi, Recent advances in solids/structures and applications of metallic materials, ASME Publications PVP-Vol. 342/MD-Vol. 72, p. 227, 1996. [9] B.C. Pak, Y.I. Cho, Exp. Heat Transfer 11 (2) (1998) 151. [10] Q. Li, Y. Xuan, Proc. 12th IHTC, August 18–22, 2002, Grenoble, France, 2002, p. 483. [11] Y. Xuan, W. Roetzel, Int. J. Heat Mass Transfer 43 (2000) 3701. [12] H.C. Brinkman, J. Chem. Phys. 20 (1952) 571. [13] G.K. Batchelor, J. Fluid Mech. 83 (1) (1977) 97. [14] R.L. Hamilton, O.K. Crosser, I & EC Fundament. 1 (1962) 182. [15] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Pub. Co, McGraw-Hill, New York, USA, 1980. [16] Fluent 6 User’s Guide, Fluent Inc., NH, USA, 2002. [17] D.A. Anderson, J.C. Tannehill, R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Hemisphere Pub. Co., New York, USA, 1984. [18] S.W. Yuan, Foundations of Fluid Mechanics, Prentice-Hall, New York, USA, 1967. [19] B.E. Launder, D.B. Spalding, Comput. Meth. Appl. Mech. Eng. 3 (1974) 269. [20] J. Orfi, Convection mixte laminaire dans un tuyau inclin´e: d´eveloppement simultan´e et ph´enom`ene de bifurcation, Ph.D. Thesis, Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, Canada, 1995. [21] B.S. Pethukhov, Heat Transfer—Sov. Res. 1 (1) (1969) 24. [22] H.L. Langhaar, J. Appl. Mech. (1942) A55–A58. [23] L. Schiller, Z. Angev. Mech. 2 (1922) 96. [24] W. Pfenninger, Further laminar flow experiments in a 40-foot long 2-inch diameter tube, Rpt. No. AM-133, Northrop Aircraft, Hawthrone, CA, 1951. [25] C.T. Nguyen, Convection mixte en r´egime laminaire dans un tuyau inclin´e soumis a` un flux de chaleur constant a` la paroi, Ph.D. Thesis, Universit´e de Sherbrooke, Qu´ebec, Canada, 1988. [26] J. Boussinesq, Comptes Rendus 113 (1891) 9. [27] W.D. Campbell, J.C. Slattery, Trans. ASME, J. Basic Eng. (1963) 41–46. [28] L. Prandtl, O.G. Tietjens, Applied Hydro-and-Aeromechanics, McGraw-Hill Book Co., New York, NY, USA, 1934, p. 27.
Heat transfer behaviours of nanofluids in a uniformly heated tube Sidi El B´ecaye Ma¨ıgaa, Cong Tam Nguyena,∗, Nicolas Galanisb, Gilles Roya a Faculty of Engineering, Universit´e de Moncton, Moncton, NB, Canada E1A 3E9 b Faculty of Engineering, Universit´e de Sherbrooke, Qu´ebec, Canada J1K 2R1
Received 12 May 2003; accepted 23 September 2003 Available online 2 June 2004
Abstract In the present work, we consider the problem of the forced convection flow of water–γ Al2 O3 and ethylene glycol–γ Al2 O3 nanofluids inside a uniformly heated tube that is submitted to a constant and uniform heat flux at the wall. In general, it is observed that the inclusion of nanoparticles has increased considerably the heat transfer at the tube wall for both the laminar and turbulent regimes. Such improvement of heat transfer becomes more pronounced with the increase of the particle concentration. On the other hand, the presence of particles has produced adverse effects on the wall friction that also increases with the particle volume concentration. Results have also shown that the ethylene glycol–γ Al2 O3 mixture gives a far better heat transfer enhancement than the water–γ Al2 O3 mixture. © 2004 Elsevier Ltd. All rights reserved. Keywords: Forced convection flow; Laminar flow; Turbulent flow; Heat transfer enhancement; Heat transfer augmentation; Nanofluid; Nanoparticles; Numerical simulation
1. Introduction The thermal properties of heating or cooling fluids play a vital role in the development of new energy-efficient heat transfer equipment. However, conventional heat transfer fluids such as water, ethylene glycol and engine oils have, in general, poor heat transfer properties compared to those of most solids. In spite of considerable research and efforts deployed ∗ Corresponding author. Tel.: +1-506-858-4347; fax: +1-506-858-4082.
E-mail address: [email protected] (C.T. Nguyen). 0749-6036/$ - see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2003.09.012
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Nomenclature Cp Specific heat of the fluid D Tube diameter Nu Z Local Nusselt number, Nu Z = h D/k0 Pr Prandtl number R Radial coordinate Re Reynolds number T Fluid temperature T0 Inlet fluid temperature Average axial velocity at tube inlet V0 V R Radial velocity component Tangential velocity component Vθ Axial velocity component VZ Z Axial coordinate h Heat transfer coefficient k Thermal conductivity of the fluid p Pressure q Wall heat flux Greek letters ε Turbulent dissipation rate θ Circumferential coordinate κ Kinetic energy of turbulence µ Fluid dynamic viscosity ρ Fluid density τ Wall shear stress ϕ Particle volume concentration Indices bf Refers to base-fluid i Refers to spatial direction, i = 1, 2 and 3 nf Refers to nanofluid property p Refers to particle property r Refers to ‘nanofluid/base-fluid’ ratio 0 Refers to the reference condition
until today, major improvements in heat transfer capabilities have suffered a major lacking. As a result, an important need still exists to develop new strategies in order to improve the effective heat transfer behaviours of conventional heat transfer fluids.
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The term ‘nanofluid’ refers to a two-phase mixture with its continuous phase being generally a liquid and the dispersed phase constituted of ‘nanoparticles’ i.e. extremely fine metallic particles of size below 100 nm. It has been shown that the thermal properties of such a nanofluid appear to be well above those of the base-fluid. In fact, some available experimental data [1–3] have shown that even with a relatively low concentration of particles, say from 1 to 5% in volume, the effective thermal conductivity of the mixture has increased by almost 20% compared to that of the base-fluid. Such an increase depends mainly on several factors such as the form and size of the particles and their concentration, the thermal properties of the base-fluid as well as those of the particles. Hence, the nanofluids can constitute an interesting alternative for advanced applications in heat transfer in the future, especially those in micro scale, see for example [8]. In spite of their potentials and advantages, these very special fluids are still in their earlier development. In fact, the first experimental works were concerned only with the measuring and the determination of the effective thermal conductivity [1–7, 9]; some also provided the effective viscosity of the nanofluids [1, 4, 9] showing that the inclusion of the nanoparticles can appreciably increase the viscosity of the resulting mixture. These works considered some current fluids, water, ethylene glycol and engine oil, and metallic particles such as γ Al2 O3 , SiO2 , TiO2 and Cu particles. It is very important to note that the amount of experimental data resulting from these studies remains, surprisingly, very limited. Furthermore, there are no available data regarding the effects of the temperature on the effective thermal conductivity and viscosity of nanofluids. It appears obvious, at the present stage, that much more works will be needed on this important issue in the future. With regard to the thermal performance of nanofluids in confined flows, the only recent experimental works [9, 10] have provided the first empirical correlation for computing the Nusselt number in laminar and turbulent tube flows using nanofluids that are composed of water and Cu, TiO2 and γ Al2 O3 particles. The data as obtained from these studies have clearly shown that the suspended nanoparticles remarkably increase the heat transfer performance of the base-fluid and the suspensions have larger heat transfer coefficients than the base-fluid (pure water) under the same Reynolds number. Such an improvement becomes more important with an augmentation of the particle volume fraction. In spite of their great potentials and advantages, these new, special yet rather challenging fluids still remain in their earlier development state and much more research work will be needed in order to better understand their fluid dynamics and thermal characteristics, in particular for confined flows. In this paper, we are interested to study numerically the beneficial influence due to the inclusion of nanoparticles on the flow behaviours and the temperature field inside a uniformly heated tube. 2. Mathematical formulation and numerical method We consider in this study the problem of the forced convection flow of a nanofluid, which is composed of water or ethylene glycol and metallic γ Al2 O3 nanoparticles, flowing inside a straight tube of circular cross-section, Fig. 1. Due to the extreme size of particles, it may be reasonable to suggest that such a mixture can be easily fluidized and therefore, one may assume that the motion slip between the phases, if any, would be considered
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Fig. 1. Geometrical configuration of the problem studied.
negligible [11]. Also, by considering the local thermal equilibrium, the solid particle–liquid mixture may then be approximately considered to behave as a conventional single-phase fluid with properties that are to be evaluated as functions of those of the constituents knowing their respective concentrations. We also consider that the flow is steady, laminar and symmetrical with respect to the vertical plane passing through the tube main axis, Fig. 1. Under the above conditions, the corresponding governing equations written in a (R, θ, Z ) coordinate system are as follows [18]: ∇ · (ρV) = 0 ∇ · (ρVVi ) = −∂ p/∂ X i + ∇ · (µ∇Vi ) + Si
(1) (2)
∇ · (ρVCp T ) = ∇ · (k∇T )
(3)
where V = (V R , Vθ , V Z ) is the velocity vector; p is the pressure; X i refers to a spatial direction (X i = R, θ and Z ); ρ, µ, k and Cp are respectively the fluid density, dynamic viscosity, thermal conductivity and specific heat. The terms Si in the momentum Eq. (2), which represent the remaining viscous terms due to the choice of the coordinate system, are given as follows: – for i = 1, the radial direction: S1 = ρVθ Vθ /R − µ{V R /R 2 + (2/R 2 )∂ Vθ /∂θ }
(4)
– for i = 2, the tangential direction: S2 = µ{(2/R 2 )∂ V R /∂θ − Vθ /R 2 } − ρV R Vθ /R
(5)
– for i = 3, the axial direction: S3 = 0.
(6)
2.1. The turbulent flow regime For the same problem, we are also interested to study the influence due to the inclusion of nanoparticles under the turbulent flow regime. Such a regime is often encountered in
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various applications in engineering. By applying the time-averaging procedure taking into account the fluctuating components of dependent variables, one can obtain the so-called ‘Reynolds-averaged’ Navier–Stokes equations that are the equations of conservation of mass and momentum. A similar Reynolds form of the energy equation can be obtained as well using the same procedure and the concept of Reynolds’ analogy to turbulent momentum transfer. With the exception of some additional terms expressing the turbulent related stresses and heat-flux quantities that have been introduced during the above timeaveraging procedure, the resulting ‘averaged equations’ have, in essence, the same general form as that of the instantaneous Navier–Stokes equations [18]. For the proper closing of these equations, one often needs to introduce empirical data or approximate models to express the turbulent stresses and heat-flux quantities to the related phenomenon. In the present study, we have adopted the well-known semi-empirical κ–ε turbulent model [19], which introduces two additional equations of conservation, namely the equation of the turbulent kinetic energy, κ, and the corresponding one for the rate of dissipation, ε. The simplifying form of these equations is given as follows for the problem under consideration here: ∇ · (ρVκ) = ∇ · [(µ + µt /σκ )∇κ)] + G κ − ρε
(7)
∇ · (ρVε) = ∇ · [(µ + µt /σε )∇ε)] + C1ε (ε/κ)G κ + C2ε ρε /κ. 2
(8)
In the above equations, G κ represents the generation of turbulence kinetic energy due to the mean velocity gradients; C1ε and C2ε are constant; σκ and σε are the turbulent Prandtl numbers for κ and ε, respectively; and µt is the turbulent (or eddy) viscosity, computed as: µt = ρCµ κ 2 /ε
(9)
where Cµ is a constant. The various constants of the model are as follows [17, 19]: C1ε = 1.44; σκ = 1.0
C2ε = 1.92; and
σε = 1.3.
Cµ = 0.09;
(10)
Note that the complete details regarding the derivation of the Reynolds form of the conservation equations as well as the κ–ε turbulent model have been very well documented elsewhere (the reader is advised to consult in particular [17–19] for further details). It is also important to note that regarding the turbulent viscosity µt , there is a clear lack of data permitting an appropriate quantification of such a property for mixtures containing nanoparticles. Since we have assumed that the nanofluid would behave as a single-phase homogenous fluid, the turbulent viscosity µt may then be evaluated as for a conventional fluid, say by Eq. (9) for the κ–ε turbulent model. As we can see later in Section 3.2 through a satisfactory comparison with experimental data in a turbulent regime, such an assumption appears justified. 2.2. Physical properties of nanofluids The following formulas have been employed to compute the thermal and physical properties of the nanofluids under consideration (the subscripts p, bf and nf refer to
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Fig. 2. Experimental data and correlation for dynamic viscosity of water–γ Al2 O3 .
the particles, the base-fluid and the nanofluid, respectively; the subscript r refers to the ratio ‘nanofluid/base-fluid’ of the quantity considered): ρnf = (1 − ϕ)ρbf + ϕρp
(11)
(C p )nf = (1 − ϕ)(C p )bf + ϕ(C p )p µnf = 123ϕ 2 + 7.3ϕ + 1 for water–γ Al2 O3 µr = µbf µnf = 306ϕ 2 − 0.19ϕ + 1 for ethylene glycol–γ Al2 O3 µr = µbf knf kr = = 4.97ϕ 2 + 2.72ϕ + 1 for water–γ Al2 O3 kbf knf = 28.905ϕ 2 + 2.8273ϕ + 1 for ethylene glycol–γ Al2 O3 . kr = kbf
(12) (13) (14) (15) (16)
Eqs. (11) and (12) are general relationships used to compute the density and specific heat for a classical two-phase mixture [9]. Eqs. (13) and (14) for computing the dynamic viscosity of nanofluids have been obtained by performing a least-square curve fitting of experimental data available for the mixtures considered [1, 3, 4]. It is interesting to mention here that for the viscosity and the water–γ Al2 O3 nanofluid in particular, the well-known formulas such as the one proposed by Einstein–Brinkman [12], as well as the one suggested by Batchelor [13] have, surprisingly, underestimated drastically the values of the mixture viscosity while compared to the experimental data, as clearly shown in Fig. 2. For the thermal conductivity of the nanofluids under consideration, Eqs. (15) and (16) have been obtained using the well-known model proposed by Hamilton and Crosser [14]. It is very interesting to note that such a model, although originally being derived for a mixture with millimetre and micrometre size particles, appears appropriate for use with nanoparticles. It is important to mention that although some data do exist in the literature for the two nanofluids considered, they were obtained at fixed reference temperatures, that is to say
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that the dependence of the fluid properties on the temperature has not yet been established clearly to date. Also, in general, for most of the nanofluids of engineering interest, such experimental data providing information on their thermal and physical properties remain rather scarce, if not to say quasi-non-existing. Much more research work will be, indeed, needed in this field. One may determine that the problem under consideration can be characterized by a set of five dimensionless parameters, namely the Reynolds number Re = V0 Dρ0 /µ0 , the Prandtl number Pr = Cp µ0 /κ0 , the particle volume concentration ϕ, and the ratios κp /κbf and (C p )p /(C p )bf (where V0 is the uniform axial velocity at the tube inlet and the subscript 0 refers to the inlet reference condition). It should be mentioned that the shape as well as the dimension of nanoparticles might have some influence on the heat transfer characteristics of the resulting nanofluids. Such influence, to our knowledge, has not been investigated to date. 2.3. Boundary conditions and numerical method The fluid has a uniform axial velocity and temperature profiles at the tube inlet section. On the tube wall, the usual non-slip condition prevails; also, the uniform heat flux boundary condition is prescribed. At the exit section, the pressure condition prevails where a known pressure is specified. The system of governing equations, Eqs. (1)–(3) for the laminar regime or their counterparts and Eqs. (7) and (8) for the turbulent regime, subject to their appropriate boundary conditions, has been successfully solved by using the numerical method that is essentially based on the ‘finite control volume approach’. Since such a method has been very well documented elsewhere (see in particular [15, 16]), only a brief review is given here. This method, as with other members of the SIMPLE-codes family [15], is based on the spatial integration of each of the conservation equations over finite control volumes using the exponential scheme for the treatment of the combined ‘convection-anddiffusion’ fluxes of mass, heat, momentum or other scalars resulting from the transport process such as turbulent related quantities κ and ε. Also, the staggered grids have been used where the velocity components are calculated at the centre of the volume interfaces while the pressure as well as other scalar quantities such as temperature and species concentration for example, are computed at the centre of the control-volume. The ‘algebraic discretization equations’ resulting from this integration process have been solved sequentially and iteratively through the physical domain considered, by combining the efficient ‘line-by-line’ procedure and the well-known TDMA technique (‘Three Diagonal Matrix Algorithm’). In order to speed up the convergence rate, we have also employed the multi-passes and alternating sweeping technique for all directions in space. On the other hand, a special ‘pressure-correction’ equation, which was obtained by combining the discretization form of the Navier–Stokes equations and the corresponding one of the continuity equation, has been employed to compute the guessed pressure field as well as to correct the velocities field, during the calculation process, in order to satisfy progressively, i.e. in an iterative manner, all the discretization equations (the reader may consult [15] for complete details regarding the above numerical algorithms and procedures).
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Fig. 3. Results of grid sensitivity study.
In order to ensure the consistency as well as the accuracy of the numerical results, several non-uniform grids have been submitted to an extensive testing procedure. Fig. 3 shows, for example, the comparison of results as obtained by using different grids for V Z velocity component at R = 0 and other analytical and numerical data. These results have shown that for the problem of the tube flow under consideration, the 32 × 24 × 155 nonuniform grid appears to be quite satisfactory to ensure the precision and the independency of numerical results with respect to the number of nodes used. Such a grid has, respectively, 32, 24 and 155 nodes along the radial, tangential (covering all angular coordinates from θ = 0◦ to θ = 180◦) and axial directions, with highly packed grid points in the vicinity of the tube wall and also in the tube entrance region. As a convergence indicator, we have essentially based it on the residuals resulting from the integration of the governing Eqs. (1)–(3), (7) and (8) over finite control-volumes. During the calculation process, these residuals were constantly monitored and scrutinized. For all the simulations performed in this study, a converged solution was usually achieved with a very low level of these residuals, say 10−8 or less for all the governing equations considered. As starting conditions, we have employed the velocity and temperature fields corresponding to the cases of forced flow without particles. For subsequent cases, the flow data as obtained for a converged case with a given value of the particle volume concentration ϕ were used as initial conditions. 3. Results and discussion The computer model has been successfully validated by comparing the results as obtained for the development of fluid axial velocity V Z to the corresponding analytical and numerical data by others for the classical case of an isothermal developing laminar forced convection flow in a tube [18]. Fig. 3 (introduced previously) has shown such comparison where the agreement can be qualified as very good. Fig. 4 shows another comparison
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Fig. 4. Validation of the mathematical model.
performed for the Nusselt number Nu Z hence the heat transfer coefficient, in forced laminar convection flow with numerical results and data from Orfi [20] and Petukhov [21]. We can see, here again, that the agreement appears acceptable except for the region near the tube inlet where a certain discrepancy has been observed. Such a discrepancy is believed to be due to the grid size effect in this particular area. The computer code was then used to perform numerical simulations for the case under consideration here, using two different nanofluids, namely water–γ Al2 O3 and ethylene glycol–γ Al2 O3 . The heated tube has a diameter of 0.01m and a total length of 1.0 m. For the laminar cases, the uniform axial velocity V0 at the tube inlet has been adjusted in order to have a constant Reynolds number, say Re = 250; the wall heat flux was fixed to q = 10, 240 W/m2 . For the turbulent cases, q = 500,000 W/m2 and two different values of the Reynolds number, namely Re = 10,000 and Re = 50,000 have been considered (these values of Re, which represent normal conditions often encountered in practical applications, have also been adopted in order to be able to compare our results with some existing experimental data in the turbulent regime). In the following, some significant results showing the beneficial influence due to the nanoparticles are presented and discussed (unless otherwise noted, most of the results presented hereafter are for the water–γ Al2 O3 mixture). 3.1. Effect of nanoparticles in laminar flow heat transfer Results have clearly revealed that the inclusion of nanoparticles has modified considerably the thermal behaviour of the mixture. Fig. 5(a) shows the influence of the particle volume concentration ϕ on the radial temperature profile at the tube exit section. We can observe that the fluid temperature has decreased with the augmentation of ϕ, in particular in the region near the tube wall, thus indicating clearly a higher heat transfer rate with particles. Such behaviour may be better understood by scrutinizing Fig. 5(b) where one can notice at the exit section, a diminution of nearly 17 K of the wall temperature
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Fig. 5. Effect of particle loading ϕ: (a) on the radial temperature profile at tube exit and (b) on the axial development of wall temperature.
between the case ϕ = 10% and the one with ϕ = 0. It is interesting to note that such a decrease of the fluid temperature at the tube wall does exist all along the tube length and seems to be more important toward the tube end. These results have obviously indicated the beneficial effect due to the nanoparticles, an effect that may be explained by the fact that with the presence of these particles, the thermal properties of the resulting mixture have been greatly improved. For the case ϕ = 10% for example, the value of the product ρCp and the thermal conductivity k have increased approximately by 18% and 33%, respectively, with respect to the values corresponding to the case ϕ = 0. Thus, a nanofluid possesses, so far, a higher thermal capacity than a conventional one. We also note that with a higher thermal conductivity k, the heat transfer at the tube wall would be generally more important, as confirmed through the following results. Fig. 6(a) shows clearly that the presence of nanoparticles has produced a considerable improvement of the heat transfer at the tube wall. Thus, for the volume concentration of 7.5% for example, the ‘nanofluid/base-fluid’ ratio h r of the heat transfer coefficients (h r ≡ h nf / h bf ) is approximately 1.6 at the tube end, that is, the heat transfer coefficient has increased by almost 60% compared to that of the base-fluid. This ratio h r also increases appreciably with the augmentation of the parameter ϕ. Such an increase is observed all along the tube length and is clearly more important toward the tube end. We can also see that for a relatively low particle concentration, the heat transfer coefficient ratio h r increases quasi-linearly with the axial coordinate Z . Although a direct comparison with other results and experimental data was not possible because of the lack of data for the laminar regime and the nanofluids considered, it is interesting to mention that the above behaviour regarding the improvement of the heat transfer coefficient due to nanoparticles appears to be quite consistent with some recent experimental data obtained for nanofluids that contain Cu particles [10]. Fig. 6(b) shows the ‘global’ effect of the particle concentration on the averaged heat transfer coefficient (note that h r is defined as the ‘nanofluid/base-fluid’ ratio of the averaged heat transfer coefficients, h r ≡ h nf /h bf ). It is observed, at first, that the
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Fig. 6. Effect of particle loading ϕ: (a) on the axial development of the ratio h r , and (b) on the ratio h r of averaged heat transfer coefficients.
enhancement of the averaged heat transfer coefficient appears clearly more pronounced for the ethylene glycol–γ Al2 O3 mixture than for the water–γ Al2 O3 nanofluid. One can also observe that this ratio h r appears more important for a higher particle loading, say for φ > 3%; while for φ ≤ 3%, h r remains relatively low and almost identical for the two nanofluids considered. With regard to the wall friction, results have clearly shown, however, that the addition of nanoparticles into a base fluid has produced an adverse effect on the wall shear stress. Fig. 7(a) illustrates the axial development of the ‘nanofluid/base-fluid’ ratio τr (defined as τr ≡ τnf /τbf ). It has been observed that, for a given particle loading ϕ, the ratio τr reaches quite rapidly a constant value that is due to the axial development of the velocity field. In general, τr increases considerably with the particle volume fraction ϕ and this, all along the tube length. For ϕ = 7.5% in particular, τr has reached nearly 2.2 at the tube end. Complete results obtained in this study have shown that similar behaviours were also observed for the ethylene glycol–γ Al2 O3 mixture. The above adverse effects, although predictable, have also been observed on the averaged wall friction. Thus, Fig. 7(b) shows the influence of the particle loading parameter ϕ on the ‘nanofluid/base-fluid’ ratio τr (defined as the ratio of the averaged wall shear stresses, τr = τnf /τbf ). One can observe that for the ethylene glycol–γ Al2 O3 nanofluid, the increase of the averaged shear stress ratio appears clearly more pronounced than for the water–γ Al2 O3 mixture. For example for φ = 7.5% in particular, the ratio τr is approximately 2.75 for the former but only 2.2 for the second mixture. 3.2. Effect of nanoparticles in turbulent flow heat transfer The beneficial effects due to the inclusion of nanoparticles have also been clearly observed for a turbulent flow heat transfer within a tube. Fig. 8(a) shows, for example, the axial variation of the ‘nanofluid/base-fluid’ heat transfer ratio h r as obtained for the case Re = 50,000 and various values of ϕ with the water–γ Al2 O3 mixture. One can
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Fig. 7. Effect of particle loading ϕ in laminar flow regime: (a) on the axial development of the ratio τr , and (b) on the ratio τr of averaged wall shear stresses.
observe that for a very short distance from the inlet section, h r increases steeply due to the simultaneous development of both the velocity and thermal field. As in the laminar cases shown previously, it is observed that with the augmentation of ϕ, the heat transfer coefficient has also increased considerably with respect to that of the base-fluid. Such an increase is noticed all along the tube length and it appears to be more pronounced toward the tube end. For the cases with relatively low particle loading, say for ϕ ≤ 5%, the ratio h r is nearly constant over a large portion of the tube length. For the cases shown, h r has as values at the tube end i.e. under fully developed flow condition, 1.05, 1.15, 1.40, 1.61 and 1.89, respectively, for ϕ = 1%, 2.5%, 5%, 7.5% and 10%. Results from this study have shown that similar behaviours regarding the heat transfer improvement were also observed for the case Re = 10,000. Table 1 shows the comparison between our results as obtained for the ratio h r and the case Re = 50,000 and the corresponding values obtained by using the only recent experimental correlation proposed by Pak and Cho [9]. One can see that the agreement appears quite satisfactory with the maximum deviation estimated to be 10% with respect to the values of h r given by the correlation; this may give a clear indication about the appropriateness as well as the reliability of the mathematical model used in the present study. One may also state, although indirectly, that the assumption of a single-phase homogenous fluid as well as that of the calculation of the turbulent viscosity, Eq. (9), appear quite reasonable. Fig. 8(b) shows the influence of the particle loading parameter ϕ on the nanofluid-tobase-fluid ratio h r of the averaged heat transfer coefficients for the two Reynolds numbers studied. The beneficial effects due to the presence of particles are, once again, clearly demonstrated here. Thus, one can easily observe that with an increase of the particle volume fraction ϕ, the improvement of the average heat transfer coefficient becomes more pronounced. For the case Re = 10,000 for example, the ratio h r has reached 1.035, 1.09, 1.17, 1.33 and 1.47 respectively for ϕ = 1%, 2.5%, 5%, 7.5% and 10%. It is interesting to see that for ϕ ≤ 3%, the ratio h r remains almost identical for the cases tested; while for
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Fig. 8. Effect of particle loading ϕ in turbulent flow: (a) on the axial development of the ratio h r , (b) on the ratio h r of averaged heat transfer coefficients, and (c) on the ratio τr of averaged wall shear stresses.
a higher value of ϕ, it clearly becomes more important with the increase of the Reynolds number. For ϕ = 7.5% in particular, h r has as values, 1.33 and 1.43, respectively for Re = 10,000 and 50,000. It is very interesting to mention here that a similar trend regarding such influence of the Reynolds number for nanofluids heat transfer has been experimentally observed by other researchers (see in particular, Pak and Cho [9] and Li and Xuan [10]). Fig. 8(c) shows finally the influence of the particle concentration ϕ on the ratio τr for two different cases with Re = 10,000 and 50,000. As for the laminar cases considered before, we can observe that such a ratio clearly increases with an augmentation of the parameter ϕ. Thus, for the case Re = 10,000 in particular, τr has as values, 1.2, 1.45, 2.3, 4.2 and nearly 6.9 respectively for ϕ = 1%, 2.5%, 5%, 7.5% and 10%. It is interesting to see that the ratio τr appears to be independent with respect to the flow Reynolds number;
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Table 1 Comparison with experimental correlation for h r (Re = 50,000) Particle loading ϕ (%)
h r present study
h r by correlation from [9]
1 2.5 5 7.5 10
1.05 1.15 1.40 1.61 1.89
1.16 1.27 1.51 1.78 2.11
thus, for a particular particle loading ϕ, the corresponding values of τr are almost identical for the values considered of parameter Re. 4. Conclusion In this paper, we have numerically investigated the hydrodynamic and thermal behaviours of nanofluids flowing inside a uniformly heated tube. Results have clearly shown that the inclusion of nanoparticles has produced a considerable increase of the heat transfer with respect to that of the base liquid. Such heat transfer enhancement, which appears to be more pronounced with the augmentation of the particle volume concentration, is accompanied, however, by a major drawback on the wall shear stress. Among the nanofluids studied, it has been shown that the ethylene glycol–γ Al2 O3 offers, so far, a better heat transfer enhancement than the water–γ Al2 O3 nanofluid. It is also the one for which a more pronounced adverse effect on the wall shear stress has been observed. For the turbulent flow regime, results have also shown that the heat transfer enhancement due to the presence of nanoparticles becomes more important with the increase of the Reynolds number. Acknowledgements The authors wish to sincerely thank the Natural Sciences and Engineering Research Council of Canada, the Faculty of Graduate Studies and Research of the Universit´e de Moncton for financial support to this project, and also to the Faculty of Engineering of the Universit´e de Moncton for allocating computing facilities. References [1] H. Masuda, A.A. Ebata, K. Teramae, N. Hishinuma, Netsu Bussei 4 (4) (1993) 227. [2] S.U.-S. Choi, Developments and applications of non-Newtonian flows, ASME Publications FEDVol. 231/MD-Vol. 66, p. 99, 1995. [3] S. Lee, S.U.-S. Choi, S.S. Li, J.A. Eastman, J. Heat Transfer 121 (1999) 280. [4] X. Wang, X. Xu, S.U.-S. Choi, J. Thermophys. Heat Transfer 13 (4) (1999) 474. [5] J.A. Eastman, S.U.-S. Choi, S. Li, G. Soyez, L.J. Thompson, R.J. DiMelfi, J. Metastable Nanocryst. Mater. 2 (6) (1999) 629. [6] Y. Xuan, Q. Li, Int. J. Heat Fluid Flow 21 (2000) 58. [7] J.A. Eastmanm, S.U.-S. Choi, S. Li, W. Yu, L.J. Thompson, Appl. Phys. Lett. 78 (6) (2001) 718.
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