Aggregation study of Brownian nanoparticles in convective phenomena

  • Mostafa Mahdavi
  • Mohsen Sharifpur
  • Mohammad H. Ahmadi
  • Josua P. Meyer


The explanation of abnormal enhancement of transported energy in colloidal nanoparticles in a liquid has sparked much interest in recent years. The complexity comes from the inter-particle phenomenon and cluster formation. The process of nanoparticle aggregation, which is caused by convective phenomena and particle-to-particle interaction energy in a flow, is investigated in this research. Therefore, the probability of collision and cohesion among clusters is modelled, as stated in this research. ANSYS-Fluent 17 CFD tools are employed to implement a new method of nanoparticle aggregation, new essential forces, new heat law and cluster drag coefficient. The importance of the interaction forces is compared to drag force, and essential forces are considered in coupling between nanoparticles and fluid flow. An important parameter is defined for the surface energy density regarding the attractive energy between the double layer and surrounding fluid to capture the cohesion of particles. Particles’ random migration is also presented through their angular and radial displacement. The analyses for interactions show the significance of Brownian motion in both particles’ migration and coupling effects in the fluid. However, nanoparticles are pushed away from walls due to repulsive forces, and Brownian motion is found to be effective mainly on angular displacement around the tube centreline. The attractive energy is found to be dominant when two clusters are at an equal distance. Hence, the cluster formation in convective regions should be taken into account for modelling purposes. A higher concentrated region also occurs midway between the centreline and the heated wall.


Nanoparticles Aggregation Brownian motion Cohesion Surface energy CFD 

List of symbols


Hamaker constant (J)


Particle surface area (m2)


Particle projected area (m2)


Cunningham correction factor


Drag coefficient


Rotational coefficient


Rotational drag coefficient


Specific heat (J kg−1 K−1)


Particle diameter (m)


Thermophoresis coefficient

\(f_{{{\text{B}}_{\text{i}} }}\)

Brownian force (N kg−1)


Interaction forces (N m−3)


Drag function


Gaussian weight function


Particle–particle distance (m)


Moment of inertia (kg m−2)


Thermal conductivity (W m−1 K−1)


Boltzmann constant (m2 kg K−1 s−2)


Particle mass (kg)


Particle mass flow rate (kg s−1)


Number of particles in the parcel


Possible number of collision


Poisson distribution


Particle Reynolds number


Thermal interaction between particles and fluid (W m−3)

\(\Delta t_{\text{p}}\)

Particle time step (s)

U1, U2

Uniform random number


Particle velocity (m s−1)


Particle–fluid relative velocity (m s−1)


Electric double-layer energy (J)


van der Waals energy (J)


Location (m)

\(\Delta x\)

Characteristic length of the cell


Webber number

Greek letters


Vacuum permittivity (CV−1 m−1)

\(\varepsilon {}_{\text{r}}\)

Relative permittivity


Debye–Huckel parameter (m−1)


Surface energy density (J m−2)

\(\dot{\gamma }\)

Shear rate (1/s)


Particle angular velocity (1/s)


Relative particle–liquid angular velocity (1/s)


Potential on the surface of particle (V)


Particle variable in the node

\(\bar{\theta }_{\text{parcel}}\)

Particle variables affected by nodes in the neighbourhood


Particle relaxation time (s)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\zeta }\)

Random function


Random number between 0 and 1


  1. 1.
    Estellé P, Mahian O, Maré T, Öztop HF. Natural convection of CNT water-based nanofluids in a differentially heated square cavity. J Therm Anal Calorim. 2017;128:1765–70.CrossRefGoogle Scholar
  2. 2.
    Rashidi S, Mahian O, Languri EM. Applications of nanofluids in condensing and evaporating systems. J Therm Anal Calorim. 2018;131:2027–39.CrossRefGoogle Scholar
  3. 3.
    Meibodi SS, Kianifar A, Mahian O, Wongwises S. Second law analysis of a nanofluid-based solar collector using experimental data. J Therm Anal Calorim. 2016;126:617–25.CrossRefGoogle Scholar
  4. 4.
    Aberoumand S, Jafarimoghaddam A. Mixed convection heat transfer of nanofluids inside curved tubes: an experimental study. Appl Therm Eng. 2016;108:967–79.CrossRefGoogle Scholar
  5. 5.
    Xiao B, Yang Y, Chen L. Developing a novel form of thermal conductivity of nanofluids with Brownian motion effect by means of fractal geometry. Powder Technol. 2013;239:409–14.CrossRefGoogle Scholar
  6. 6.
    He Y, Men Y, Zhao Y, Lu H, Ding Y. Numerical investigation into the convective heat transfer of TiO2 nanofluids flowing through a straight tube under the laminar flow conditions. Appl Therm Eng. 2009;29:1965–72.CrossRefGoogle Scholar
  7. 7.
    Esfe MH, Saedodin S, Bahiraei M, Toghraie D, Mahian O, Wongwises S. Thermal conductivity modeling of MgO/EG nanofluids using experimental data and artificial neural network. J Therm Anal Calorim. 2014;118:287–94.CrossRefGoogle Scholar
  8. 8.
    Esfe MH, Saedodin S, Mahian O, Wongwises S. Thermal conductivity of Al2O3/water nanofluids: measurement, correlation, sensitivity analysis, and comparisons with literature reports. J Therm Anal Calorim. 2014;117:675–81.CrossRefGoogle Scholar
  9. 9.
    Yu W, France DM, Timofeeva EV, Singh D, Routbort JL. Thermophysical property-related comparison criteria for nanofluid heat transfer enhancement in turbulent flow. Appl Phys Lett. 2010;96:13–6.Google Scholar
  10. 10.
    Aybar HŞ, Sharifpur M, Azizian MR, Mehrabi M, Meyer JP. A review of thermal conductivity models for nanofluids. Heat Transf Eng. 2014;36:1085–110.CrossRefGoogle Scholar
  11. 11.
    Hwang KS, Jang SP, Choi SUS. Flow and convective heat transfer characteristics of water-based Al2O3 nanofluids in fully developed laminar flow regime. Int J Heat Mass Transf. 2009;52:193–9.CrossRefGoogle Scholar
  12. 12.
    Bianco V, Chiacchio F, Manca O, Nardini S. Numerical investigation of nanofluids forced convection in circular tubes. Appl Therm Eng. 2009;29:3632–42.CrossRefGoogle Scholar
  13. 13.
    Kumar N, Puranik BP. Numerical study of convective heat transfer with nanofluids in turbulent flow using a Lagrangian-Eulerian approach. Appl Therm Eng. 2017;111:1674–81.CrossRefGoogle Scholar
  14. 14.
    Rashidi S, Bovand M, Esfahani JA, Ahmadi G. Discrete particle model for convective Al2O3–water nanofluid around a triangular obstacle. Appl Therm Eng. 2016;100:39–54.CrossRefGoogle Scholar
  15. 15.
    Tahir S, Mital M. Numerical investigation of laminar nanofluid developing flow and heat transfer in a circular channel. Appl Therm Eng. 2012;39:8–14.CrossRefGoogle Scholar
  16. 16.
    Krishnamurthy S, Bhattacharya P, Phelan PE, Prasher RS. Enhanced mass transport in nanofluids. Nano Lett. 2006;6:419–23.CrossRefGoogle Scholar
  17. 17.
    Ganguly S, Sarkar S, Kumar Hota T, Mishra M. Thermally developing combined electroosmotic and pressure-driven flow of nanofluids in a microchannel under the effect of magnetic field. Chem Eng Sci. 2015;126:10–21.CrossRefGoogle Scholar
  18. 18.
    Gupta A, Kumar R. Role of Brownian motion on the thermal conductivity enhancement of nanofluids. Appl Phys Lett. 2007;91:223102.CrossRefGoogle Scholar
  19. 19.
    Gharagozloo PE, Eaton JK, Goodson KE. Diffusion, aggregation, and the thermal conductivity of nanofluids. Appl Phys Lett. 2008;93:2006–9.CrossRefGoogle Scholar
  20. 20.
    Veilleux J, Coulombe S. A dispersion model of enhanced mass diffusion in nanofluids. Chem Eng Sci. 2011;66:2377–84.CrossRefGoogle Scholar
  21. 21.
    Putnam SA, Cahill DG. Transport of nanoscale latex spheres in a temperature gradient. Langmuir. 2005;21:5317–23.CrossRefGoogle Scholar
  22. 22.
    Eslamian M, Saghir MZ. On thermophoresis modeling in inert nanofluids. Int J Therm Sci. 2014;80:58–64.CrossRefGoogle Scholar
  23. 23.
    McNab GS, Meisen A. Thermophoresis in liquids. J Colloid Interface Sci. 1973;44:339–46.CrossRefGoogle Scholar
  24. 24.
    Talbot L, Cheng RK, Schefer RW, Willis DR. Thermophoresis of particles in a heated boundary layer. J Fluid Mech. 1980;101:737–58.CrossRefGoogle Scholar
  25. 25.
    Koo J, Kleinstreuer C. Laminar nanofluid flow in microheat-sinks. Int J Heat Mass Transf. 2005;48:2652–61.CrossRefGoogle Scholar
  26. 26.
    Vladkov M, Barrat JL. Modeling transient absorption and thermal conductivity in a simple nanofluid. Nano Lett. 2006;6:1224–9.CrossRefGoogle Scholar
  27. 27.
    Gao JW, Zheng RT, Ohtani H, Zhu DS, Chen G. Experimental investigation of heat conduction mechanisms in nanofluids. Clue on clustering. Nano Lett. 2009;9:4128–32.CrossRefGoogle Scholar
  28. 28.
    Babaei H, Keblinski P, Khodadadi JM. A proof for insignificant effect of Brownian motion-induced micro-convection on thermal conductivity of nanofluids by utilizing molecular dynamics simulations. J Appl Phys. 2013;113:084302.CrossRefGoogle Scholar
  29. 29.
    Laín S, Sommerfeld M. Numerical calculation of pneumatic conveying in horizontal channels and pipes: detailed analysis of conveying behaviour. Int J Multiph Flow. 2012;39:105–20.CrossRefGoogle Scholar
  30. 30.
    Ozturk S, Hassan YA, Ugaz VM. Interfacial complexation explains anomalous diffusion in nanofluids. Nano Lett. 2010;10:665–71.CrossRefGoogle Scholar
  31. 31.
    Kumar R, Milanova D. Effect of surface tension on nanotube nanofluids. Appl Phys Lett. 2009;94:073107.CrossRefGoogle Scholar
  32. 32.
    Mokhtari Moghari R, Akbarinia A, Shariat M, Talebi F, Laur R. Two phase mixed convection Al2O3–water nanofluid flow in an annulus. Int J Multiph Flow. 2011;37:585–95.CrossRefGoogle Scholar
  33. 33.
    Yang C, Peng K, Nakayama A, Qiu T. Forced convective transport of alumina–water nano fluid in micro-channels subject to constant heat flux. Chem Eng Sci. 2016;152:311–22.CrossRefGoogle Scholar
  34. 34.
    Zhang H, Shao S, Xu H, Tian C. Heat transfer and flow features of Al2O3–water nanofluids flowing through a circular microchannel—experimental results and correlations. Appl Therm Eng. 2013;61:86–92.CrossRefGoogle Scholar
  35. 35.
    Oesterle B, Dinh TB. Experiments on the lift of a spinning sphere in a range of intermediate Reynolds numbers. Exp Fluids. 1998;25:16–22.CrossRefGoogle Scholar
  36. 36.
    Li A, Ahmadi G. Dispersion and deposition of spherical particles from point sources in a turbulent channel flow. Aerosol Sci Technol. 1992;16:209–26.CrossRefGoogle Scholar
  37. 37.
    Zhao B, Chen C, Lai ACK. Lagrangian stochastic particle tracking: further discussion. Aerosol Sci Technol. 2011;45:901–2.CrossRefGoogle Scholar
  38. 38.
    Box GEP, Muller ME, et al. A note on the generation of random normal deviates. Ann Math Stat. 1958;29:610–1.CrossRefGoogle Scholar
  39. 39.
    Marshall JS, Li S. Adhesive particle flow. Cambridge: Cambridge University Press; 2014.CrossRefGoogle Scholar
  40. 40.
    Israelachvili JN. Intermolecular and surface forces: revised. 3rd ed. Cambridge: Academic Press; 2011.Google Scholar
  41. 41.
    Zheng X, Silber-Li Z. The influence of Saffman lift force on nanoparticle concentration distribution near a wall. Appl Phys Lett. 2009;95:24–7.Google Scholar
  42. 42.
    Haider A, Levenspiel O. Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol. 1989;58:63–70.CrossRefGoogle Scholar
  43. 43.
    Michaelides EE, Feng Z. Heat transfer from a rigid sphere in a nonuniform flow and temperature field. Int J Heat Mass Transf. 1994;37:2069–76.CrossRefGoogle Scholar
  44. 44.
    O’Rourke PJ. Collective drop effects on vaporizing liquid sprays. Los Alamos Natl Lab, NM. Technical report; 1981.Google Scholar
  45. 45.
    Bhuiyan MHU, Saidur R, Amalina MA, Mostafizur RM, Islam A. Effect of nanoparticles concentration and their sizes on surface tension of nanofluids. Proc Eng. 2015;105:431–7.CrossRefGoogle Scholar
  46. 46.
    Chinnam J, Das DK, Vajjha RS, Satti JR. Measurements of the surface tension of nanofluids and development of a new correlation. Int J Therm Sci. 2015;98:68–80.CrossRefGoogle Scholar
  47. 47.
    Van Oss CJ, Chaudhury MK, Good RJ. Interfacial Lifshitz-van der Waals and polar interactions in macroscopic systems. Chem Rev. 1988;88:927–41.CrossRefGoogle Scholar
  48. 48.
    Apte SV, Mahesh K, Lundgren T. Accounting for finite-size effects in simulations of disperse particle-laden flows. Int J Multiph Flow. 2008;34:260–71.CrossRefGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Department of Mechanical and Aeronautical EngineeringUniversity of PretoriaPretoriaSouth Africa
  2. 2.Faculty of Mechanical EngineeringShahrood University of TechnologyShahroodIran

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