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Advances in Mechanical Engineering

Volume 2012 (2012), Article ID 139370, 13 pages

http://dx.doi.org/10.1155/2012/139370

## Numerical Investigation on Mixed Convection in Triangular Cross-Section Ducts with Nanofluids

Dipartimento di Ingegneria Aerospaziale e Meccanica, Seconda Università degli Studi di Napoli, Real Casa dell'Annunziata, Via Roma 29, 81031 Aversa (CE), Italy

Received 5 July 2011; Accepted 21 September 2011

Academic Editor: Kambiz Vafai

Copyright © 2012 Oronzio Manca et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Convective heat transfer can be enhanced passively by changing flow geometry and boundary conditions or by improving the thermal conductivity of the working fluid, for example, introducing suspended small solid nanoparticles. In this paper, a numerical investigation on laminar mixed convection in a water-Al_{2}O_{3}-based nanofluid, flowing in a triangular cross-sectioned duct, is presented. The duct walls are assumed at uniform temperature, and the single-phase model has been employed in order to analyze the nanofluid behaviour. The hydraulic diameter is equal to 0.01 m. A fluid flow with different values of Richardson number and nanoparticle volume fractions has been considered. Results show the increase of average convective heat transfer coefficient and Nusselt number for increasing values of Richardson number and particle concentration. However, also wall shear stress and required pumping power profiles grow significantly.

#### 1. Introduction

Heat transfer enhancement is a significant issue in the research and industry fields. Several thermal device applications employ ducts characterized by noncircular cross-sections, thus many investigations have concerned with this topic. Different shapes of cross-section area have been analyzed, like square, rhombic, rectangular, triangular, sinusoidal, elliptical ones, even with truncated corners [1–6]. In particular, triangular sectioned ducts are suitable for the construction of compact heat exchangers because of their compactness and cost effectiveness [7, 8]. In fact, compact heat exchangers with triangular-sectioned passages provide relative low fabrication costs because of the easy construction with thin materials and high mechanical strength [9]. Investigations have been performed in mixed and forced convection, considering mainly the flow regime laminar and also turbulent ones [10–15] and recently also microchannel have been analyzed [16]. Experimental test [3, 14] and analytical [17] and numerical methods [18, 19] are available in the literature. The use of numerical methods is nowadays widely adopted calculator powers and scheme strongly improved. Several researchers developed and validated unstructured grid meshes both in laminar and turbulent flow regime [18, 19] in order to predict the friction and heat transfer features in triangular ducts. Moreover, different fluids have been considered like gas, liquid, and non-Newtonian ones [11, 20].

Forced convection in a triangular duct is affected by several parameters, including the apex angle, the hydraulic diameter, the axial length, and the flow conditions. A comprehensive review of theoretical and experimental studies on laminar forced convection and heat transfer in ducts having noncircular sections has been performed by Shah and London [21]. In particular, they provided the flow and thermal performances of equilateral, isosceles, and right triangular ducts in developed and thermal developing flow. For the fully developed flow, a Nusselt number value equal to 2.47, 3.111, and 1.892 were reported for , and boundary conditions, respectively. A few works are reported on natural [22] or mixed convection [10] in triangular sectioned ducts. Ali and Al-Ansary [22] analyzed the natural convection in vertical triangular ducts in laminar and transition regime, giving the critical values of the modified Rayleigh number for the transition to turbulent flow. They underlined the local axial (perimeter averaged) heat transfer coefficient decrease in the laminar region and increase in the transition one. Talukdar and Shah [10] studied the effect of Rayleigh number on bulk mean temperature and Nusselt number in triangular ducts with different apex angles. They pointed out the increase of these parameters for increasing Rayleigh numbers and the higher heat transfer rate from the bottom boundary.

According to Bergles [23] there are several enhancement techniques available. Among them, passive techniques, employing special surface geometries, rough surfaces, extended surfaces or swirl flow devices, and fluid additives for the enhancement can be adopted [24]. An innovative way to improve the fluid thermal conductivity consists of the introduction of nanosize particles suspended in the working fluids. The so-called nanofluids are composed by a base fluid and solid particles with a diameter smaller than 100 nm. They are becoming more and more popular, and there is a fast growth in the number of scientific research activities in these topics in order to evaluate solutions to develop efficient heat exchangers or cooling devices [25–27]. In fact, several investigations pointed out that nanofluid heat transfer average coefficients could increase by more than 20% in comparison with the base fluid coefficients also in the case of low nanoparticle concentrations [28]. However, a critical discussion about the lack of agreement among results from different research groups must be pointed out. There are still some difficulties in formulating efficient theoretical models for the predictions [29, 30], but this topic is still incomplete, as recently underlined by Gherasim et al. [31]. It is well known that the evaluation of properties, like thermal conductivity and viscosity, differs from different research groups because of the numerical and experimental approaches and processes adopted [32, 33].

The number of theoretical and experimental investigations on convective heat transfer in confined flows with nanofluids is growing. For example, experimental tests were performed on forced convective heat transfer in laminar and turbulent flow regime of nanofluids inside tubes in [34, 35]. Correlations for the Nusselt number, using nanofluids composed of water and Cu, TiO_{2}, and Al_{2}O_{3} nanoparticles were proposed. Experimental results for the convective heat transfer of Al_{2}O_{3} (27–56 nm)/water-based nanofluids flowing through a copper tube in laminar regime was reported in [36]. An experimental investigation was carried out to study the mixed laminar convection of Al_{2}O_{3}-water nanofluid inside an inclined copper tube submitted to a uniform wall heat flux at its outer surface by Ben Mansour et al. [37]. Results showed that the experimental heat transfer coefficient decreases slightly with an increase of particle volume concentration from 0% to 4%. They developed correlations to calculate the Nusselt number in the fully developed region for horizontal and vertical tubes, for Rayleigh number from 5 × 10^{5} to 9.6 × 10^{5}, Reynolds number from 350 to 900 and particle volume concentrations up to 4%. Moreover, also the number of numerical studies on nanofluids is growing, and the analysis can be carried out by using two approaches. It is possible to consider that the continuum assumption is still valid for fluids with suspended nanosize particles at low volume fractions or to adopt two-phase models in order to describe both the fluid and the solid phase. The single-phase model with physical and thermal properties, all assumed to be constant with temperature, was employed in [38–41]. The advantages of adopting nanofluids with respect to the heat transfer mechanism were discussed in [38, 39], but it was also found that the presence of nanoparticles led to significant effects on growing wall shear stress and pumping power for heated tubes in laminar and turbulent regime. For a given Reynolds number, buoyancy force has a negative effect on the Nusselt number, while the nanoparticles concentration had a positive effect on the heat transfer enhancement and also on the skin friction reduction, as pointed out in [40], where the mixed convection in horizontal curved tubes was performed. Mirmasoumi and Behdzadmehr [41] adopted a two-phase model in order to describe the strong influence of particle concentration on the thermal parameters, above all on the tube bottom part and also at the near wall region.

In the present paper, a numerical investigation on laminar mixed convection in ducts with an equilateral triangular cross-section is presented. A constant and uniform temperature is applied on the walls. Results are given to evaluate the fluid dynamic and thermal features of the considered geometry with different Richardson number values, adopting Al_{2}O_{3}/water nanofluid as the working fluid. Different nanoparticle volume fractions are considered and the single-phase model has been assumed.

#### 2. Governing Equations

A computational analysis of a three-dimensional model concerning with the mixed convection in a triangular cross sectioned duct is considered, and its scheme is reported in Figure 1. The aim is to evaluate its thermal and fluid-dynamic behaviours and study the temperature and velocity fields in the cases with Al_{2}O_{3}/water nanofluid as working fluid. A constant uniform temperature is applied on the walls, such as the boundary condition indicated in [21], and its value is evaluated on the basis of the chosen Richardson numbers. The inlet average velocity is set in the ranges of laminar regime. Single-phase model is employed and the fluid properties are considered constant with temperature.

Governing equations of continuity, momentum and energy, for a three-dimensional steady state laminar, incompressible flow, with temperature independent thermophysical properties and assuming the Boussinesq approximation, in rectangular coordinates are given in the following:

*continuity*:

*momentum*:

*energy*:
The assigned boundary conditions are the following:(i)inlet section: uniform velocity and temperature profiles,(ii)outlet section: outflow condition with velocity components and temperature derivatives equal to zero,(iii)duct surfaces: velocity components equal to zero and assigned temperature value.

The use of the single-phase model does not exclude the possibility to take into account for phenomena like the Brownian motion. In fact, the relations adopted to evaluate the thermal conductivity and viscosity consider the component linked to the Brownian motion as well as the particle diameter effects, and they were adopted by several authors, also in the case of nanofluid mixed convection [42, 43].

#### 3. Physical Properties of Nanofluids

The working fluid is pure water or a water/Al_{2}O_{3}-based nanofluid. The nanoparticle diameter of 30 nm and different volume fractions equal to 1%, 3%, and 5% have been considered. In Table 1, the values of density, specific heat, dynamic viscosity, thermal conductivity, and thermal expansion coefficient, given by Rohsenow et al. [44], are reported for pure water and Al_{2}O_{3} particles at the reference temperature of 293 K. The concentration of nanoparticles influences the working fluid properties. The single-phase model was adopted. Thus, fluid properties must be evaluated by employing relations, available in the literature, in order to compute the thermal and physical properties of the considered nanofluids [29, 36, 40, 45, 46]. Their values are given in Table 2. Density was evaluated by using the classical formula valid for conventional solid-liquid mixtures, while the specific heat values and thermal expansion coefficient ones were calculated by assuming the thermal equilibrium between particles and surrounding fluid [36, 40, 46]

*density*:

*specific heat:*

*thermal expansion coefficient*:
For the viscosity as well as for thermal conductivity, formula given by [29, 40, 45] were adopted, because these relations are expressed as a function of particle volume concentration and diameter

*dynamic viscosity*:

*thermal conductivity*:
where and is the Boltzmann constant, given by ; the other terms are given by:
with , , and , and the distance between nanoparticles, , obtained by
while and with is the mean free path of water (0.17 nm).

#### 4. Geometrical Configuration and Data Reduction

A numerical thermal and fluid-dynamic study on a three-dimensional duct model with an equilateral triangular section was carried out. The total length is 2.0 m, while the internal edge one is 0.017 m, as shown in Figure 1; the consequent hydraulic diameter, is equal to 0.01 m. The duct includes a developing section at the entrance region with a length of 1.5 m and a 0.4 m long test section, followed by the outlet one. The working fluid is water or a mixture of water and Al_{2}O_{3} nanoparticles with a diameter of 30 nm, at different volume fractions equal to 1%, 3% and 5%.

The dimensionless parameters of Reynolds number, Grashof number, Richardson number, Nusselt number, and friction factor are considered for the data reduction and they are expressed by: where is the average inlet velocity, is the heat flux, and represent the temperature of the surface and the fluid bulk one, respectively.

#### 5. Numerical Model

The governing equations, reported in the previous section, are solved by means of the finite volume method by means of Fluent code [47]. A steady-state solution and a segregated method are chosen to solve the governing equations, which are linearized implicitly with respect to dependent variables of the equation. A second-order upwind scheme is chosen for energy and momentum equations. The SIMPLE coupling scheme is chosen to couple pressure and velocity. The convergence criteria of 10^{−4}, 10^{−6} and 10^{−8} are assumed for the residuals of density, velocity components, and energy ones, respectively, for all the considered simulations. It is assumed that the incoming flow is laminar at ambient temperature and pressure. Reynolds number is set equal to 100 and different values of Grashof number, ranging from 0 to 50000, were considered. On the solid walls, the no-slip condition was applied and velocity inlet and outflow ones were given for inlet and outlet sections.

Four different unstructured mesh distributions were tested on a triangular sectioned duct at Re = 100 and 250 in order to perform the grid-independence analysis. They had 152190, 302778, 563104, and 1146232 nodes, respectively. The third grid case was adopted, because it ensured a good compromise between the computational time and the accuracy requirements. In fact, comparing the third and fourth configurations, differences of 0.58% and 0.19% at most were evaluated in terms of average Nusselt number and pressure coefficient, in the case of pure water.

The validation has been performed by comparing the numerical results for Re = 100 and Ri = 0 with literature data for boundary condition. For the local and average Nusselt number, the comparison has been accomplished with Wilbulswas results [21], presented as a function of the axial coordinate for the thermal entrance region, , defined as . For the friction factor, data from Fleming-Sparrow, Miller-Han, and Gangal [21] have been considered. Data reduction has been carried out by means of parameter, the axial coordinate for the hydrodynamic entrance region, given by , with the Peclet number, , defined as . The present numerical results in terms of average and local Nusselt number and friction factors are in good agreement with the given correlations, as shown in Figure 2. In particular, a difference of 3% is observed for local and average Nusselt number at most, while a maximum error of 1% is evaluated in terms of friction factor.

#### 6. Results and Discussion

A computational analysis of a three-dimensional model, regarding the nanofluid mixed convection in a triangular cross-sectioned duct is considered. A uniform temperature has been applied on the duct walls, according to the considered Richardson number. The inlet velocities ensure the steady laminar regime, and they correspond to Re = 100. The working fluids are pure water or a water/Al_{2}O_{3}-based nanofluid at different volume fractions. The single-phase model approach is adopted. The range of the considered Richardson numbers and volume fractions are given below:(i)Reynolds number, Re: 100,(ii)Richardson number: 0.0, 0.1, 0.5, 1, 2, 3, and 5,(iii)particle concentrations, 0%, 1%, 3% and 5%.

Results are presented in terms of average convective heat transfer coefficient, average Nusselt number, wall shear stress and required pumping power profiles in the test section.

Figure 3(a) depicts the average convective heat transfer coefficient profiles as a function of Ri for different values of nanoparticle volume concentration. The effect of buoyancy leads to an increase in terms of heat transfer coefficients, and a sharp growth is detected at low Richardson numbers. In fact, in the case of pure water is equal to about 148, 162, 178, 190, and 225 W/m^{2}K for Ri = 0, 0.1, 0.5, 1, 5, respectively. A similar behaviour is observed for the configurations with nanofluids. However, increasing values of convective heat transfer are evaluated as particle concentration increases. In fact, is equal to about 160, 170, 188, 199, 208, and 238 W/m^{2}K at % while at % is equal to about 168, 193, 230, 250, 266, and 298 W/m^{2}K for Ri = 0, 0.1, 0.5, 1, 2 and 5, respectively. Figure 3(b) describes the average convective heat transfer enhancement in comparison with the pure water case at Ri = 0. The maximum improvement is detected for the cases, characterized by %, which show heat transfer coefficient values equal to 1.14, 1.55, 1.68, and 2.01 times greater than the reference case for Ri = 0, 0.5, 1, and 5. Smaller increases are observed for decreasing values of nanoparticle volume concentration. For example, the enhancement ratio is equal to 1.06, 1.26, 1.33, and 1.60 for % at Ri = 0, 0.5, 1 and 5 while for % it is equal to 1.10, 1.32, 1.41, and 1.68.

The average Nusselt profiles as a function of Ri for different values of nanoparticle volume concentration are presented in Figure 4(a). For fully developed laminar flow at Ri = 0, the average Nusselt number is equal to 2.47 for triangular ducts with walls at a uniform temperature. In mixed convection, the fully developed condition is reached much farther upstream than for pure forced convection, depending on the importance of buoyancy effects. However, average Nusselt number in the test section tends to increase as Ri and nanoparticle fraction increase. In fact, at Ri = 5, Nu_{avg} = 3.76, 3.87, 3.90, and 4.5 for at %, 1%, 3% and 5%. The consequent enhancement in terms of Nusselt number is depicted in Figure 4(b); it is less significant if compared with Figure 3 because of the nanofluid thermal conductivity increase. In fact, for equal to 1% and 3% the average Nusselt number is 4% and 6% higher than the cases with pure water on average. A more significant enhancement is provided by the cases with %.

The heat transfer enhancement is partly due to the increase of fluid velocity, because the simulations have been carried out at a constant Reynolds number. In fact, the increase of nanofluid viscosity is not balanced by the increase of density. Another possible reason is linked to the Brownian motion which could become very important in the case of laminar flow regime. However, a further analysis about the comparison criteria among the results in terms of heat transfer coefficients for base fluid and nanofluid is necessary as observed by Prabhat et al. [48].

The working fluid flows into a triangular cross-sectioned duct with walls at a uniform temperature; thus, the heat transfer mechanism is different if the bottom wall behaviour is compared with the inclined ones when the buoyancy effects are considered. In fact, the orientation of hot surfaces and their inclination must be taken into account even in internal flows. For Ri = 0 no differences are detected among the heated walls. Figure 5 allows to describe the different behaviour of bottom surface and upper ones. In general, convective heat transfer coefficient profiles result to be higher for *wall 1* for all the considered Ri values, as shown by Figure 5(a). Profiles tend to increase as Ri increases; in fact, for pure water is equal to about 210 and 370 W/m^{2}K for Ri = 0.1 and 5, respectively. The introduction of nanoparticles leads to a significant enhancement of heat transfer coefficients even at low Ri numbers and particle concentration. For %, the highest heat transfer coefficient value is detected for Ri = 5, and it is equal to 510 W/m^{2}K. *Wall 2 *and* 3* present smaller heat transfer coefficient values, as reported in Figure 5(b). In fact, at Ri = 5 is equal to 152, 160, 170, and 194 W/m^{2}K for %, 1%, 3%, and 5%, respectively. Moreover, for Ri < 2, buoyancy determines negative effects on the heat transfer mechanism if results are compared with ones obtained in the case of forced convection. The results in terms of average Nusselt number are shown in Figure 6. The highest Nusselt numbers for *wall 1* are equal to 6.2, 6.4, 6.5, and 7.7 for %, 1%, 3%, and 5%, respectively; the profiles for *wall 2 *and *wall 3* tends to decrease as Ri increases until Ri is equal to 1 for %, 1% and 3%, and 1 for %, then they increase.

The use of nanofluids as working fluids instead of pure water provides a heat transfer enhancement, according to the particle volume fraction, but it leads to increasing values of wall shear stress, while the friction factor keeps substantially constant as reported in Figure 7, because the flow is fully developed. The average wall shear stress profiles as a function of Ri are depicted in Figure 8(a). Values grow as the buoyancy effects become more significant. In fact, the highest values are observed at Ri = 5; moreover, wall shear stress increases as particle concentration increases and the maximum values are evaluated for %. In fact, for % wall shear stress is 2.63, 2.65, 2.68, 2.75, and 2.94 times higher than the value calculated at Ri = 0 with pure water, as depicted in Figure 8(b). For lower particle concentrations, wall shear stress ratio is smaller than the ones calculated at % and, for example, it is equal to 1.18, 1.20, and 1.32 for % and Ri = 0.1, 1, and 5, respectively, while it is equal to 1.40, 1.42, and 1.58 for %. *Wall 1* is featured by higher wall shear stress values than the ones calculated for *wall 2* and *wall 3*, as observed in Figure 9. Profiles increase as Ri, and nanoparticle concentration increases if *wall 1* is considered, while they keep substantially constant in the case of *wall 2 *and *wall 3*. The friction factor keeps substantially constant as reported in Figure 7, because the flow is fully developed.

The required pumping power ratio profiles, referred to the water cases at Ri = 0, are carried out in Figure 10. Pumping power is defined as , and its profiles tend to increase as grows, while a very little dependence on Ri is observed. The pumping power ratio is equal to 1.26, 1.61, and 4.0 for %, 3%, and 5%, respectively.

Figures 11 and 12 describe the fully developed flow regime in terms of velocity and dimensionless temperature profiles for the vertical symmetry axis of the channel at = 125 and 150. Figure 11 shows that profiles substantially overlap each other for different values of particle concentration and Richardson number. The highest values are detected at equal to about 0.30, while the maximum values of dimensionless temperature, defined as , is evaluated at equal to about 0.20. The temperature profiles keeps similar if they are compared at the same values of Ri and volume particle concentration, as shown in Figure 12.

#### 7. Conclusions

In this paper, a numerical investigation about the laminar mixed convection in water/Al_{2}O_{3}-based nanofluids flowing into a triangular sectioned duct is carried out. The laminar flow regime was considered at Re = 100, and Ri numbers ranging from 0 to 5 were assumed. A constant and uniform temperature is applied on the duct walls, depending on the Ri number. The single-phase model was adopted in order to analyze the behaviour in the case of nanofluids as working fluid. Thus, the considered nanoparticle volume concentrations were equal to 0%, 1%, 3%, and 5%.

The introduction of nanoparticles significantly raises the convective heat transfer coefficients as particle concentration grows as well as Ri number. This effect is very significant at low Ri numbers. In fact, the maximum improvement is detected for the cases, characterized by %, which show heat transfer coefficient values equal to 1.14, 1.55, 1.68, and 2.01 times greater than the pure water case at Ri = 0. However, the wall shear stress and the required pumping power increase, and their values become very high at high concentrations. This effect is amplified at low Ri numbers, and an increase of about 30% in terms of wall shear stress is evaluated at Ri = 0 and 0.1 in comparison with the pure water results. Moreover, the pumping power ratio, referred to the pure water cases at Ri = 0, is equal to 1.26, 1.61, and 4.0 for %, 3%, and 5%, respectively.

It should be remarked that further investigations need to be accomplished in order to understand the main physical reasons of the heat enhancement using the nanofluids in laminar regime, particularly, in mixed convection, as underlined in [48].

#### Nomenclature

: | Cross-section area (m^{2}) |

: | Specific heat (J/kgK) |

: | Diameter (m) |

: | Friction factor (17) |

: | Gravitational acceleration (m/s^{2}) |

: | Grashof number (14) |

: | Duct height (m) |

: | Heat transfer coefficient (W/m^{2}K) |

: | Duct internal edge length (m) |

: | Duct length (m) |

: | Nusselt number(16) |

: | Pressure (Pa) |

: | Peclet number |

: | Required pumping power (W) |

: | Target surface heat flux (W/m^{2}) |

: | Reynolds number(13) |

: | Richardson number(15) |

: | Temperature (K) |

: | Velocity components (m/s) |

: | Average velocity (m/s) |

: | Volume flow rate (m^{3}/s) |

: | Spatial coordinates (m) |

: | Thermal diffusivity (m^{2}/s) |

: | Volumetric expansion coefficient (1/K) |

: | Distance between nanoparticles (m) |

: | Thermal conductivity (W/mK) |

: | Dynamic viscosity (Pas) |

: | Kinematic viscosity (m^{2}/s) |

: | Density (kg/m^{3}) |

: | Nanoparticle volumetric concentration |

avg: | Average |

: | Base fluid |

: | Fluid |

: | Hydraulic |

: | Mass |

: | Nanofluid |

: | Particle |

: | Wall. |

#### Acknowledgment

This work was supported by SUN with a 2010 grant and by MIUR with Articolo D.M. 593/2000 Grandi Laboratori “EliosLab.”

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