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ABSTRACT
A two-phase model based on the double-diffusive approach is used to perform a numerical study of natural convection in differentially heated vertical cavities filled with water-based nanofluids, assuming that Brownian diffusion and thermophoresis are the only slip mechanisms by which the solid phase can develop a significant relative velocity with respect to the liquid phase. The system of the governing equations of continuity, momentum, and energy for the nanofluid, and continuity for the nanoparticles, is solved through a computational code, which incorporates three empirical correlations for the evaluation of the effective thermal conductivity, the effective dynamic viscosity, and the thermophoretic diffusion coefficient, all based on the literature experimental data. The pressure–velocity coupling is handled using the SIMPLE-C algorithm. Numerical simulations are executed for three different nanofluids, using the diameter and the average volume fraction of the suspended nanoparticles, as well as the cavity width, the average temperature of the nanofluid, and the temperature difference imposed across the cavity, as independent variables. It is found that the heat transfer performance of the nanofluid relative to that of the base fluid increases notably with increasing the average temperature, showing a peak at an optimal particle loading. Conversely, the other controlling parameters have moderate effects.
Nomenclature
| c | = | specific heat at constant pressure, J/(kg K) |
| DB | = | Brownian diffusion coefficient, m2/s |
| DT | = | thermophoretic diffusion coefficient, m2/s |
| dp | = | nanoparticle diameter, m |
| g | = | gravity vector, m/s2 |
| I | = | unit tensor |
| Jp | = | nanoparticle diffusion mass flux, kg/(m2 s) |
| k | = | thermal conductivity, W/(m K) |
| kB | = | Boltzmann constant = 1.38066 · 10−23 J K−1 |
| m | = | nanoparticle mass fraction |
| Nu | = | Nusselt number |
| p | = | pressure, Pa |
| Pr | = | Prandtl number |
| Q | = | heat transfer rate, W |
| q | = | heat flux, W/m2 |
| Ra | = | Rayleigh number |
| ST | = | thermophoresis parameter |
| T | = | temperature, K |
| t | = | time, s |
| U | = | horizontal velocity component, m/s |
| V | = | velocity vector, m/s |
| V | = | vertical velocity component, m/s |
| VT | = | thermophoretic velocity vector, m/s |
| W | = | width of the enclosure, m |
| x | = | horizontal Cartesian coordinate, m |
| y | = | vertical Cartesian coordinate, m |
| Greek symbols | = | |
| φ | = | nanoparticle volume fraction |
| μ | = | dynamic viscosity, kg/( m s) |
| ρ | = | mass density, kg/m3 |
| τ | = | stress tensor, kg/( m s2) |
| Subscripts | = | |
| av | = | average |
| c | = | cooled wall, at the temperature of the cooled wall |
| f | = | base fluid |
| h | = | heated wall, at the temperature of the heated wall |
| max | = | maximum value |
| min | = | minimum value |
| n | = | nanofluid |
| opt | = | optimal value |
| s | = | solid phase |
Nomenclature
| c | = | specific heat at constant pressure, J/(kg K) |
| DB | = | Brownian diffusion coefficient, m2/s |
| DT | = | thermophoretic diffusion coefficient, m2/s |
| dp | = | nanoparticle diameter, m |
| g | = | gravity vector, m/s2 |
| I | = | unit tensor |
| Jp | = | nanoparticle diffusion mass flux, kg/(m2 s) |
| k | = | thermal conductivity, W/(m K) |
| kB | = | Boltzmann constant = 1.38066 · 10−23 J K−1 |
| m | = | nanoparticle mass fraction |
| Nu | = | Nusselt number |
| p | = | pressure, Pa |
| Pr | = | Prandtl number |
| Q | = | heat transfer rate, W |
| q | = | heat flux, W/m2 |
| Ra | = | Rayleigh number |
| ST | = | thermophoresis parameter |
| T | = | temperature, K |
| t | = | time, s |
| U | = | horizontal velocity component, m/s |
| V | = | velocity vector, m/s |
| V | = | vertical velocity component, m/s |
| VT | = | thermophoretic velocity vector, m/s |
| W | = | width of the enclosure, m |
| x | = | horizontal Cartesian coordinate, m |
| y | = | vertical Cartesian coordinate, m |
| Greek symbols | = | |
| φ | = | nanoparticle volume fraction |
| μ | = | dynamic viscosity, kg/( m s) |
| ρ | = | mass density, kg/m3 |
| τ | = | stress tensor, kg/( m s2) |
| Subscripts | = | |
| av | = | average |
| c | = | cooled wall, at the temperature of the cooled wall |
| f | = | base fluid |
| h | = | heated wall, at the temperature of the heated wall |
| max | = | maximum value |
| min | = | minimum value |
| n | = | nanofluid |
| opt | = | optimal value |
| s | = | solid phase |