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ABSTRACT
In the present paper, natural convection in an annulus between two confocal elliptic cylinders filled with a Cu-Al2O3/water hybrid nanofluid is investigated numerically. The inner cylinder is heated at a constant surface temperature while the outer wall is isothermally cooled. The basic equations are formulated in elliptic coordinates and developed in terms of the vorticity-stream function formulation using the dimensionless form for 2D, laminar and incompressible flow under steady-state condition. The governing equations are discretized using the finite volume method and solved by an in-house FORTRAN code. Numerical simulations are performed for various volume fractions of nanoparticles (0 ≤ ϕ ≤ 0.12) and Rayleigh numbers (103 ≤ Ra ≤ 3 × 105). The eccentricity of the inner and outer ellipses and the angle of orientation are fixed at e1 = 0.9, e2 = 0.6 and γ = 0° respectively. It is found that employing a Cu-Al2O3/water hybrid nanofluid is more efficient in heat transfer rate compared to the similar Al2O3/water nanofluid.
Nomenclature
| A1, A2 | = | major axes of the inner and outer elliptic cylinders (m) |
| B1, B2 | = | minor axes of the inner and outer elliptic cylinders (m) |
| Cp | = | specific heat at constant pressure (J · kg−1 · K−1) |
| g | = | gravitational acceleration (m · s−2) |
| h | = | metric coefficient (m) |
| H | = | dimensionless h |
| K | = | thermal conductivity (W · m−1 · K−1) |
| Nu | = | Nusselt number |
| Pr | = | Prandtl number |
| Ra | = | Rayleigh number |
| T | = | dimension temperature (K) |
| u, v | = | axial and radial velocities (m · s−1) |
| Vη, Vθ | = | velocity components in η, θ directions (m · s−1) |
| x, y | = | Cartesian coordinates (m) |
| α | = | thermal diffusivity (m2 · s−1) |
| β | = | thermal expansion coefficient (K−1) |
| γ | = | orientation angle of the annulus |
| K | = | thermal conductivity (W · m−1 · K−1) |
| μ | = | dynamic viscosity, kg/m s |
| υ | = | kinematic viscosity (m2 · s−1) |
| ρ | = | density (kg · m−3) |
| ϕ | = | volume fraction of the nanoparticles |
| η, θ | = | elliptic coordinates, (m) |
| ψ | = | stream function (m2 · s−1) |
| ω | = | vorticity (s−1) |
| e1, e2 | = | eccentricities of ellipses |
| Subscripts | = | |
| c | = | cold |
| h | = | hot |
| f | = | fluid |
| hnf | = | hybrid nanofluid |
| nf | = | nanofluid |
| p | = | solid particles |
| 1 | = | inner cylinder |
| 2 | = | outer cylinder |
| Superscript | = | |
| * | = | dimensionless parameters |
Nomenclature
| A1, A2 | = | major axes of the inner and outer elliptic cylinders (m) |
| B1, B2 | = | minor axes of the inner and outer elliptic cylinders (m) |
| Cp | = | specific heat at constant pressure (J · kg−1 · K−1) |
| g | = | gravitational acceleration (m · s−2) |
| h | = | metric coefficient (m) |
| H | = | dimensionless h |
| K | = | thermal conductivity (W · m−1 · K−1) |
| Nu | = | Nusselt number |
| Pr | = | Prandtl number |
| Ra | = | Rayleigh number |
| T | = | dimension temperature (K) |
| u, v | = | axial and radial velocities (m · s−1) |
| Vη, Vθ | = | velocity components in η, θ directions (m · s−1) |
| x, y | = | Cartesian coordinates (m) |
| α | = | thermal diffusivity (m2 · s−1) |
| β | = | thermal expansion coefficient (K−1) |
| γ | = | orientation angle of the annulus |
| K | = | thermal conductivity (W · m−1 · K−1) |
| μ | = | dynamic viscosity, kg/m s |
| υ | = | kinematic viscosity (m2 · s−1) |
| ρ | = | density (kg · m−3) |
| ϕ | = | volume fraction of the nanoparticles |
| η, θ | = | elliptic coordinates, (m) |
| ψ | = | stream function (m2 · s−1) |
| ω | = | vorticity (s−1) |
| e1, e2 | = | eccentricities of ellipses |
| Subscripts | = | |
| c | = | cold |
| h | = | hot |
| f | = | fluid |
| hnf | = | hybrid nanofluid |
| nf | = | nanofluid |
| p | = | solid particles |
| 1 | = | inner cylinder |
| 2 | = | outer cylinder |
| Superscript | = | |
| * | = | dimensionless parameters |