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Analytical Solution for Fully Developed Flows of Nanofluids in Mixed-Convection Zone Within Vertical Channels

  • Research Article - Mechanical Engineering
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Abstract

In this paper, a closed-form analytical solution is presented for a fully developed mixed-convection laminar flow of nanofluids between two vertical parallel plates. The Buongiorno model, which considers the Brownian motion and thermophoresis force, is employed to investigate the hydrodynamic and heat transfer behavior of the nanofluid flow. The equations for the conservation of mass, momentum, energy, and the nanoparticle concentration field have been analytically solved, and expressions for the velocity, temperature, and nanoparticle concentration profiles as well as for the Nusselt number are given. The results show that in addition to the mixed-convection buoyancy parameter (Gr/Re), the immersed-particle buoyancy parameter additionally enriches the momentum and enhances the heat transfer inside the channel. Moreover, in the mixed-convection regime, in contrast to the case of forced convection, the heat transfer rate decreases sharply and then gradually as the solid/fluid thermal conductivity ratio increases. The present results contradict the prevailing perception that higher thermal conductivities of nanoparticles are always desirable and boost heat transfer. The study findings will be helpful in selecting an appropriate nanoparticle material that would provide a high heat transfer rate based on the application’s thermal conditions.

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Abbreviations

b :

Channel spacing

\(C_{\mathrm{p}}\) :

Specific heat at constant pressure

\(d_{\mathrm{p}}\) :

Nanoparticle diameter

\(D_{\mathrm{B}}\) :

Brownian diffusivity, \(= {K}_{\mathrm{BO}} {T}/3\pi \mu _{\mathrm{bf}} {d}_{\mathrm{p}}\)

\(D_{\mathrm{T}}\) :

Thermophoresis diffusivity, \(= 0.26{k}_{\mathrm{bf}} /({2k}_{\mathrm{bf}} + {k}_{\mathrm{p}} )*\mu _{\mathrm{bf}} /\rho _{\mathrm{bf}} *\phi _0\)

Gr :

Grashof number, \(\frac{{g}\beta _{\mathrm{bf}} {q}_1 {b}^{4}}{\upsilon ^{2}{K}_{\mathrm{bf}}}\)

K :

Thermal conductivity

\(K_{\mathrm{BO}}\) :

Boltzmann constant

Kr:

Solid/fluid thermal conductivity ratio, \({K}_{\mathrm{p}} /{K}_{\mathrm{bf}}\)

\({N}_{\mathrm{BT}}\) :

Ratio of Brownian and thermophoretic diffusivities, \(= {D}_{\mathrm{B}}/{D}_{\mathrm{T}}\)

Nu :

Nusselt number

p :

Fluid pressure at any cross section

\({p}^{\prime }\) :

Pressure defect at any cross section, \({p-p}_{\mathrm{s}}\)

\({p}_0\) :

Fluid pressure at channel entrance

\({p}_{\mathrm{s}}\) :

Hydrostatic pressure, \(-\rho _{0} \, {gz}\)

P :

Dimensionless pressure at any cross section, \(\frac{{p}^{\prime }-{p}_0}{\rho _0 {u}_0^2}\)

Pr :

Prandtl number

\(r_{{q}}\) :

Heat flux ratio, \(\frac{{q}_2}{{q}_1}\)

Re :

Reynolds number, \(=\frac{{u}_{\mathrm{o}} {b}}{\upsilon }\)

T :

Temperature at any point

\(T_0\) :

Inlet temperature

\(T_{\mathrm{m}}\) :

Mean temperature in each cross section \({T}_{\mathrm{m}}(z)\)

\(u_{\mathrm{o}}\) :

Entrance axial velocity

u :

Longitudinal velocity component at any point

U :

Dimensionless longitudinal velocity, \(= {u}/{u}_{\mathrm{o}}\)

y :

Horizontal coordinate

Y :

Dimensionless horizontal coordinate, y / b

z :

Vertical coordinate

Z :

Dimensionless vertical coordinate, z / (bRe)

\(\upsilon \) :

Kinematic fluid viscosity

\(\rho \) :

Fluid density

\(\mu \) :

Dynamic fluid viscosity

\(\theta \) :

Dimensionless temperature at any point, \([{=\;\frac{{k}_{\mathrm{bf}} ({T-T}_{\mathrm{m}})}{{q}_1 {b}}}]\)

\(\beta \) :

Thermal expansion coefficient

\(\phi \) :

Particle volume fraction

\(\varPhi \) :

Rescaled nanoparticle volume fraction, \([{=\frac{\phi }{\phi _0}}]\)

\(\gamma \) :

Immersed-particle buoyancy parameter, \(= 0.78\pi \frac{\mu _{\mathrm{bf}}^2}{\rho _{\mathrm{bf}}}\frac{{\mathrm{d}}_{\mathrm{p}}}{{K}_{\mathrm{BO}} \beta _{\mathrm{bf}} {T}_0^2}(\frac{\rho _{\mathrm{p}}}{\rho _{\mathrm{bf}}}-1\))

bf:

Base fluid

nf:

Nanofluid

m:

Mean

w:

Wall

p:

Particle

1:

Duct wall at \(Y = 0\)

2:

Duct wall at \(Y = 1\)

0:

Condition at the entrance

References

  1. Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles. ASME Fluids Eng. Div. 231, 99105 (1995)

    Google Scholar 

  2. Farbod, M.; Ahangarpour, A.; Etemad, S.G.: Stability and thermal conductivity of water-based carbon nanotube nanofluids. Particuology 22, 59–65 (2015)

    Article  Google Scholar 

  3. Hwang, K.S.; Jang, S.P.; Choi, S.U.S.: Flow and convective heat transfer characteristics of water-based \(\text{ Al }_{2} \text{ O }_{3}\) nanofluids in fully developed laminar flow regime. Int. J. Heat Mass Transf. 52(1–2), 193–199 (2009)

    Article  MATH  Google Scholar 

  4. Hu, Z.S.; Dong, J.X.: Study on antiwear and reducing friction additive of nanometer titanium oxide. Wear 216(1), 92–96 (1998)

    Article  Google Scholar 

  5. Li, W.; Nakayama, A.: Temperature dependency of thermophysical properties in convective heat transfer enhancement in nanofluids. J. Thermophys. Heat Transf. 29(3), 504–512 (2015)

    Article  Google Scholar 

  6. Xing, M.; Yu, J.; Wang, R.: Experimental study on the thermal conductivity enhancement of water based nanofluids using different types of carbon nanotubes. Int. J. Heat Mass Transf. 88, 609–616 (2015)

    Article  Google Scholar 

  7. Nayak, R.K.; Bhattacharyya, S.; Pop, I.: Numerical study on mixed convection and entropy generation of a nanofluid in a lid-driven square enclosure. J. Heat Transf. 138(1), 012503 (2015)

    Article  Google Scholar 

  8. Hwang, K.S.; Jang, S.P.; Choi, S.U.S.: Flow and convective heat transfer characteristics of water-based \(\text{ Al }_{2}\text{ O }_{3}\) nanofluids in fully developed laminar flow regime. Int. J. Heat Mass Transf. 52(1–2), 193–199 (2009)

    Article  MATH  Google Scholar 

  9. Maïga, S.E.B.; Nguyen, C.T.; Galanis, N.; Roy, G.: Heat transfer behaviors of nanofluids in a uniformly heated tube. Superlattices Microstruct. 35(3–6), 543–557 (2004)

    Article  Google Scholar 

  10. Mokmeli, A.; Saffar-Avval, M.: Prediction of nanofluid convective heat transfer using the dispersion model. Int. J. Therm. Sci. 49(3), 471–478 (2010)

    Article  Google Scholar 

  11. Serna, J.: Heat and mass transfer mechanisms in nanofluids boundary layers. Int. J. Heat Mass Transf. 92, 173–183 (2016)

    Article  Google Scholar 

  12. Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. 128(3), 240–250 (2005). https://doi.org/10.1115/1.215083

    Article  Google Scholar 

  13. Chen, C.-K.; Chen, B.-S.; Liu, C.-C.: Heat transfer and entropy generation in fully-developed mixed convection nanofluid flow in vertical channel. Int. J. Heat Mass Transf. 79, 750–758 (2014)

    Article  Google Scholar 

  14. Heris, S.Z.; Esfahany, M.N.; Etemad, G.: Numerical investigation of nanofluid laminar convective heat transfer through a circular tube. Numer. Heat Transf. Part A Appl. Int. J. Comput. Methodol. 52(11), 1043–1058 (2007)

    Article  Google Scholar 

  15. You, X.-C.; Xu, H.; Pop, I.: Analysis of fully developed opposing mixed convection flow in an inclined channel filled by a nanofluid. J. Heat Transf. 136(12), 124502 (2014)

    Article  Google Scholar 

  16. Xu, H.; Pop, I.: Fully developed mixed convection flow in a vertical channel filled with nanofluids. Int. Commun. Heat Mass Transf. 39(8), 1086–1092 (2012)

    Article  Google Scholar 

  17. Grosan, T.; Pop, I.: Fully developed mixed convection in a vertical channel filled by a nanofluid. J. Heat Transf. 134(8), 082501 (2012)

    Article  Google Scholar 

  18. di Schio, E.R.; Celli, M.; Barletta, A.: Effects of Brownian diffusion and thermophoresis on the laminar forced convection of a nanofluid in a channel. J. Heat Transf. 136(2), 022401 (2014)

    Article  Google Scholar 

  19. Elcock, D.: Potential impacts of nanotechnology on energy transmission applications and needs. Environmental Science Division, Argonne National Laboratory, United States, ANL/EVS/TM/08-3 (2007)

  20. Yu, W.; France, D.M.; Choi, S.U.S.; Routbort, J.L.: Review and assessment of nanofluid technology for transportation and other applications. Argonne National Laboratory, United States, ANL/ESD/07-9, 10.2172/919327 (2007)

  21. Al-Amri, F.; Mallick, T.K.: Alleviating operating temperature of concentration solar cell by air active cooling and surface radiation. Appl. Therm. Eng. 59, 348–354 (2013)

    Article  Google Scholar 

  22. Escher, W.; Brunschwler, T.; Shalkevich, N.; Shalkevish, A.; Burgi, T.; Michel, B.; Poulikakos, D.: On the cooling of electronics with nanofluids. J. Heat Transf. 133, 051401 (2011)

    Article  Google Scholar 

  23. Turkyilmazoglu, M.: Anomalous heat transfer enhancement by slip due to nanofluids in circular concentric pipes. Int. J. Heat Mass Transf. 85, 609–614 (2015)

    Article  Google Scholar 

  24. Mahdavi, M.; Sharifpur, M.; Meyer, J.P.: CFD modelling of heat transfer and pressure drops for nanofluids through vertical tubes in laminar flow by Lagrangian and Eulerian approaches. Int. J. Heat Mass Transf. 88, 803–813 (2015)

    Article  Google Scholar 

  25. Mital, M.: Semi-analytical investigation of electronics cooling using developing nanofluid flow in rectangular microchannels. Appl. Therm. Eng. 52, 321e327 (2013)

  26. Cimpean, D.S.; Pop, I.: Fully developed mixed convection flow of a nanofluid through an inclined channel filled with a porous medium. Int. J. Heat Mass Transf. 55, 907–914 (2012)

    Article  MATH  Google Scholar 

  27. Barletta, A.; Zanchini, E.: On the choice of the reference temperature for fully-developed mixed convection in a vertical channel. Int. J. Heat Mass Transf. 42(16), 3169–3181 (1999)

    Article  MATH  Google Scholar 

  28. Yang, C.; Li, W.; Sano, Y.; Mochizuki, M.; Nakayama, A.: On the anomalous convective heat transfer enhancement in nanofluids: a theoretical answer to the nanofluids controversy. J. Heat Transf. 135(5), 054504 (2013). https://doi.org/10.1115/1.4023539

    Article  Google Scholar 

  29. Xuan, Y.; Roetzel, W.: Conceptions for heat transfer correlation of nanofluids. Int. J. Heat Mass Transf. 43, 3701–3707 (2000)

    Article  MATH  Google Scholar 

  30. Avci, M.; Aydin, O.: Mixed convection in a vertical parallel plate microchannel with asymmetric wall heat fluxes. J. Heat Transf. 129(8), 1091–1095 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical and Energy Engineering, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 31441, Saudi Arabia

    Fahad G. Al-Amri

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Correspondence to Fahad G. Al-Amri.

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Al-Amri, F.G. Analytical Solution for Fully Developed Flows of Nanofluids in Mixed-Convection Zone Within Vertical Channels. Arab J Sci Eng 44, 739–752 (2019). https://doi.org/10.1007/s13369-018-3260-9

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  • DOI: https://doi.org/10.1007/s13369-018-3260-9

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