# A simple homogeneous numerical solution for nanofluid natural convection in an enclosure with disconnected and conducting solid blocks

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## Abstract

Following a series of discrepancies in the literature regarding the effects of nanoparticle concentration on the heat transfer rate, this paper proposes a simple homogeneous modeling for natural convection of nanofluids. The proposed modeling allows a custom choice of correlations for the nanofluid’s thermophysical properties, which proved to be very convenient in validations carried out for the case of natural convection inside a clear enclosure. The general heat transfer decreasing behavior for high concentrations of nanoparticles, previously observed in experimental and two-phase numerical works, was successfully represented. A very interesting and pioneer case study of laminar natural convection of nanofluids in a laterally heated enclosure with conductive solid blocks uniformly distributed within the enclosure was numerically investigated with the proposed modeling. To compute the nanofluids characteristics, Corcione’s correlations for thermal conductivity and dynamic viscosity for effective properties and generic correlations for density, specific heat and thermal expansion coefficient were employed. Therefore, it was possible to isolate the nanoparticle’s influence on the heat transfer rate, evaluated by the average Nusselt number. By isolating the nanoparticle’s parameters, the effects of diameter and material have been interpreted from the homogeneous solution. A parametric investigation was conducted by varying the number of blocks inside the enclosure and the effective Rayleigh number. A numerical correlation for the average Nusselt number was derived for a wide range of effective Rayleigh number and number of blocks. The model developed herein is suitable to other applications in which the total heat transfer rate needs to be computed.

## Keywords

Nanofluid Natural convection Heterogeneous media Computational fluids dynamics## List of symbols

- \(A\)
Avogadro number

- \(c_{\text{p}}\)
Specific heat at constant pressure (J/kg K)

- \({\text{CV}}\)
Control volume

- \(d\)
Mean diameter (m)

- \(g\)
Acceleration of gravity (m/s

^{2})- \(H\)
Height (m)

- \(k\)
Thermal conductivity (W/m K)

- \(K\)
Solid-to-nanofluid thermal conductivity ratio

- \(L\)
Length (m)

- \(M\)
Molar mass (kg/mol)

- \(n\)
Normal direction unit vector

- \(N\)
Number of blocks

- \(\overline{Nu}\)
Average Nusselt number

- \(p\)
Pressure (Pa)

- \(P\)
Non-dimensional pressure

- \(Pr\)
Prandtl number

- \(R\)
Coefficient of determination

- \(Ra\)
Rayleigh number

- \(Re\)
Reynolds number

- \(T\)
Temperature (K)

- \((u,v)\)
Velocity components in (x, y) directions (m/s)

- \((U,\;V)\)
Non-dimensional velocity components

- \((x,\;y)\)
Spatial coordinates (m)

- \((X,\;Y)\)
Non-dimensional spatial coordinates

## Greek symbols

- \(\alpha\)
Thermal diffusivity (m/s

^{2})- \(\beta\)
Thermal expansion coefficient (1/K)

- \(\delta\)
Distance between blocks

- \(\delta_{\text{w}}\)
Distance between blocks and the enclosure walls

*ϕ*Porosity

*φ*Nanoparticle concentration

*μ*Dynamic viscosity (Pa s)

*v*Kinematic viscosity (m/s

^{2})*θ*Non-dimensional temperature

*ρ*Density (kg/m

^{3})*ψ*Stream function (kg/s)

## Subscripts

*b*Boltzmann

*B*Brownian

- Calc
Calculated

*C*Cold

- eff
Effective

*f*Base fluid

*f*_{0}Base fluid at 0 K

*f*_{p}Base fluid at freezing point temperature

*H*Hot

- min
Minimum

- max
Maximum

- nf
Nanofluid

- np
Nanoparticles

*p*Particles

- ref
Reference

*s*Solid

*w*Wall

## Notes

### Acknowledgements

This work was supported by the Research Center for Rheology and Non-Newtonian Fluids (CERNN). The first author is grateful for the funding of National Council for Scientific and Technological Development (CNPq).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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