Journal of Applied and Computational Mechanics
Free Convection Flow and Heat Transfer of Nanofluids of Different Shapes of Nano-Sized Particles over a Vertical Plate at Low and High Prandtl Numbers
Document Type : Research Paper
Author
Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria
Abstract
In this paper, free convection flow and heat transfer of nanofluids of differently-shaped nano-sized particles over a vertical plate at very low and high Prandtl numbers are analyzed. The governing systems of nonlinear partial differential equations of the flow and heat transfer processes are converted to systems of nonlinear ordinary differential equation through similarity transformations. The resulting systems of fully-coupled nonlinear ordinary differential equations are solved using a differential transformation method - Padé approximant technique. The accuracy of the developed approximate analytical methods is verified by comparing the results of the differential transformation method - Padé approximant technique with those of previous works as presented in the literature. Thereafter, the analytical solutions are used to investigate the effects of the Prandtl number, the nanoparticles volume-fraction, the shape and the type on the flow and heat transfer behaviour of various nanofluids over the flat plate. It is observed that as the Prandtl number and volume-fraction of the nanoparticles in the basefluid increase, the velocity of the nanofluid decreases while the temperature increases. Also, the maximum decrease in velocity and the maximum increase in temperature are recorded in lamina-shaped nanoparticles, followed by platelets, cylinders, bricks, and sphere-shaped nanoparticles, respectively. Using a common basefluid for all nanoparticle types, it is established that the maximum decrease in velocity and the maximum increase in temperature are recorded in TiO2 followed by CuO, Al2O3 and SWCNTs nanoparticles, respectively. It is hoped that the present study will enhance the understanding of free convection boundary-layer problems as applied in various industrial, biological and engineering processes.
Keywords
- Free convection
- Boundary layer
- Prandtl number
- Nanofluid
- Differential transformation method
- Padé-approximant technique
Main Subjects
[1] E. Schmidt and W. Beckmann. Das temperatur-und geschwindigkeitsfeld vor einer wärme abgebenden senkrecher platte bei natürelicher convention. Tech. Mech. U. Themodynamik, Bd. 1(10) (1930), 341-349; cont. Bd. 1(11) (1930), 391- 406.
[2] S. Ostrach, An analysis of laminar free-convection flow and heat transfer about a flat plate parallel to the direction of the generating body force, NACA Report, 1111, 1953.
[3]. E.M. Sparrow and J.L. Gregg, Laminar free convection from a vertical plate with uniform surface heat flux, Trans. ASME. 78 (1956) 435-440.
[4]. E.J. Lefevre, Laminar free convection from a vertical plane surface, 9th Intern. Congress on Applied Mechanics, Brussels, paper I, 168 (1956).
[5]. E.M. Sparrow and J.L. Gregg, Similar solutions for free convection from a nonisothermal vertical plate, Trans. ASME. 80 (1958) 379-386.
[6]. K. Stewartson and L.T. Jones, The heated vertical plate at high Prandtl number, J. Aeronautical Sciences 24 (1957) 379-380.
[7]. H.K. Kuiken, An asymptotic solution for large Prandtl number free convection, J. Engng. Math. 2 (1968) 355-371.
[8]. H.K. Kuiken, Free convection at low Prandtl numbers, J. Fluid Mech. 37 (1969) 785-798.
[9]. S. Eshghy, Free-convection layers at large Prandtl number, J. Applied Math. Physics 22 (1971) 275 -292.
[10]. S. Roy, High Prandtl number free convection for uniform surface heat flux, J. Heat Transfer 95 (1973) 124-126.
[11]. H.K. Kuiken and Z. Rotem, Asymptotic solution for the plume at very large and small Prandtl numbers, J. Fluid Mech. 45 (1971) 585-600.
[12] T.Y. Na, I.S. Habib, Solution of the natural convection problem by parameter differentiation, Int. J. Heat Mass Transfer 17 (1974) 457–459.
[13]. J.H. Merkin, A note on the similarity solutions for free convection on a vertical plate, J. Engng. Math. 19 (1985) 189-201.
[14] J.H. Merkin, I. Pop, Conjugate free convection on a vertical surface, Int. J. Heat Mass Transfer 39 (1996) 1527–1534.
[15] F. M. Ali, R. Nazar, and N. M. Arifin, Numerical investigation of free convective boundary layer in a viscous fluid, The American Journal of Scientific Research 5 (2009) 13-19.
[16] S. S. Motsa, S. Shateyi, and Z. Makukula, Homotopy analysis of free convection boundary layer flow with heat and mass transfer, Chemical Engineering Communications 198(6) (2011) 783-795.
[17] S. S. Motsa, Z. G. Makukula and S. Shateyi. Spectral Local Linearisation Approach for Natural Convection Boundary Layer Flow. Mathematical Problems in Engineering (2013) Article ID 765013, 7 pages.
[18] A.R. Ghotbi, H. Bararnia, G. Domairry, A. Barari, Investigation of a powerful analytical method into natural convection boundary layer flow, Communications in Nonlinear Science and Numerical Simulation 14(5) (2009) 2222-2228.
[19] S. Mosayebidorcheh, T. Mosayebidorcheh. Series solution of convective radiative conduction equation of the nonlinear fin with temperature dependent thermal conductivity. International Journal of Heat and Mass Transfer 55 (2012) 6589-6594.
[20] S. Mosayebidorcheh, M. Farzinpoor, D.D. Ganji. Transient thermal analysis of longitudinal fins with internal heat generation considering temperature-dependent properties and different fin profiles, Energy Conversion and Management 86 (2014) 365–370.
[21] M. Hatami a, S. Mosayebidorcheh, D. Jing, Thermal performance evaluation of alumina-water nanofluid in an inclined direct absorption solar collector (IDASC) using numerical method, Journal of Molecular Liquids 231 (2017) 632–639.
[22] S. Mosayebidorcheh, Solution of the Boundary Layer Equation of the Power-Law Pseudoplastic Fluid Using Differential Transform Method, Mathematical Problems in Engineering (2013) Article ID 685454, 8 pages.
[23] S. Mosayebidorcheh, M. Hatami, D.D. Ganji, T. Mosayebidorcheh, S.M. Mirmohammadsadeghi. Investigation of Transient MHD Couette flow and Heat Transfer of Dusty Fluid with Temperature-Dependent properties, Journal of Applied Fluid Mechanics, 8(4) (2015) 921-929.
[24] S. Mosayebidorcheh. Analytical Investigation of the Micropolar Flow through a Porous Channel with Changing Walls, Journal of Molecular Liquids 196 (2014) 113-119.
[25] M. Sheikholeslami, S.A. Shehzad. CVFEM for influence of external magnetic source on Fe3O4-H2O nanofluid behavior in a permeable cavity considering shape effect, International Journal of Heat and Mass Transfer 115(A) (2017) 180-191
[26] M. Sheikholeslami, S.A. Shehzad. Magnetohydrodynamic nanofluid convective flow in a porous enclosure by means of LBM. International Journal of Heat and Mass Transfer, 113 (2017) 796-805
[27] M. Sheikholeslami, M. Seyednezhad. Nanofluid heat transfer in a permeable enclosure in presence of variable magnetic field by means of CVFEM. International Journal of Heat and Mass Transfer 114 (2017) 1169-1180
[28] M. Sheikholeslami, H.B. Rokni. Simulation of nanofluid heat transfer in presence of magnetic field: A review. International Journal of Heat and Mass Transfer 115(B) (2017) 1203-1233
[29] M. Sheikholeslami, T. Hayat, A. Alsaedi. On simulation of nanofluid radiation and natural convection in an enclosure with elliptical cylinders. International Journal of Heat and Mass Transfer 115(A) (2017) 981-991
[30] M. Sheikholeslami, Houman B. Rokni. Melting heat transfer influence on nanofluid flow inside a cavity in existence of magnetic field. International Journal of Heat and Mass Transfer 114 (2017) 517-526
[31] M. Sheikholeslami, M.K. Sadoughi. Simulation of CuO-water nanofluid heat transfer enhancement in presence of melting surface. International Journal of Heat and Mass Transfer 116 (2018) 909-919
[32] M. Sheikholeslami, Mohammadkazem Sadoughi. Mesoscopic method for MHD nanofluid flow inside a porous cavity considering various shapes of nanoparticles. International Journal of Heat and Mass Transfer 113 (2017) 106-114.
[33] M. Sheikholeslami, M.M. Bhatti. Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles. International Journal of Heat and Mass Transfer 111 (2017) 1039-1049.
[34] M. Sheikholeslami, M.M. Bhatti. Active method for nanofluid heat transfer enhancement by means of EHD. International Journal of Heat and Mass Transfer 109 (2017) 115-122.
[35] M. Sheikholeslami, Houman B. Rokni. Nanofluid two phase model analysis in existence of induced magnetic field. International Journal of Heat and Mass Transfer 107 (2017) 288-299.
[36] M. Sheikholeslami, S.A. Shehzad. Magnetohydrodynamic nanofluid convection in a porous enclosure considering heat flux boundary condition. International Journal of Heat and Mass Transfer 106 (2017) 1261-1269.
[37] M. Sheikholeslami, T. Hayat, A. Alsaedi. Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using lattice Boltzmann method. International Journal of Heat and Mass Transfer 108(B) (2017) 1870-1883.
[38] M. Sheikholeslami, T. Hayat, A. Alsaedi. Numerical study for external magnetic source influence on water based nanofluid convective heat transfer. International Journal of Heat and Mass Transfer 106 (2017) 745-755.
[39] M. Sheikholeslami, T. Hayat, A. Alsaedi, S. Abelman. Numerical analysis of EHD nanofluid force convective heat transfer considering electric field dependent viscosity. International Journal of Heat and Mass Transfer 108(B) (2017) 2558-2565.
[40] J.K. Zhou. Differential Transformation and Its Applications for Electrical Circuits. Huazhong University Press, Wuhan, China (1986) (in Chinese).
[41] L.T. Yu, C.K. Chen, The solution of the Blasius equation by the differential transformation method, Math. Comput. Modell. 28 (1998) 101–111.
[42] B.L. Kuo, Thermal boundary-layer problems in a semi-infinite flat plate by the differential transformation method, Appl. Math. Comput. 153 (2004) 143–160.
[43] B. L. Kuo. Application of the differential transformation method to the solutions of the free convection problem. Applied Mathematics and Computation 165 (2005) 63–79.
[44] M. M. Rashidi, N. Laraqi, S. M. Sadri, A Novel Analytical Solution of Mixed Convection about an Inclined Flat Plate Embedded in a Porous Medium Using the DTM-Pade. International Journal of Thermal Sciences 49 (2010), 2405-2412.
[45] N. S. Akbar and A. W. Butt. Ferro-magnetic effects for peristaltic flow of Cu-water nanofluid for different shapes of nano-size particles. Appl. Nanosci. 6 (2016) 379-385.
[46] M. Sheikholesmi, M. M. Bhatti. Free convection of nanofluid in the presence of constant magnetic field considering shape effects of nanoparticles. International Journal of Heat and Mass Transfer 111 (2017) 1039-1049.
[47] R. Ul Haq, S. Nadeem, Z.H. Khan, N.F.M. Noor. Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes, Physica B 457 (2015) 40–47.
[48] L.D. Talley, G.L. Pickard, W.J. Emery, J.H. Swift. Descriptive Physical Oceanography, Physical Properties of Sea Water, sixth ed., Elsevier Ltd, 2011, 29–65.
[49] M. Pastoriza-Gallego, L. Lugo, J. Legido, M. Piñeiro. Thermal conductivity and viscosity measurements of ethylene glycol-based Al 2O3 nanofluids. Nanoscale Res. Lett. 6 (221) 1–11.
[50] S. Aberoumand, A. Jafarimoghaddam. Experimental study on synthesis, stability, thermal conductivity and viscosity of Cu–engine oil nanofluid, Journal of the Taiwan Institute of Chemical Engineers 71 (2017) 315–322.
[51] G.A. Baker, P. Graves-Morris. Pade Approximants, Cambridge U.P., 1996.
[52] H.K. Kuiken. On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small. IMA Journal of Applied Mathematics 27(4) (1981) 387–405.
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