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MHD natural convection in an annular space between two coaxial cylinders partially filled with metal base porous layer saturated by Cu–water nanofluid and subjected to a heat flux

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Abstract

The aim of this study is to better understand the behavior of the nanofluid in a specific configuration, aiding in the creation of new models and designs for heat transfer systems, by investigating the MHD natural convection in an annular partially porous metal space between two vertical concentric cylinders, which is saturated by (Cu–water) nanofluid. The inside cylinder undergoes a regular heat flux, whereas the outer cylinder maintains a uniform temperature. The upper and lower walls are impermeable and insulated. In the upward direction, an exterior magnetic field with constant intensity is used. The nonlinear coupled conservation equations with specified boundary conditions in the vorticity-stream function form are solved using the finites differences method in conjunction with the successive over relaxation method. The numerical results obtained are presented to show the impact of a variety of control parameters depicted in Darcy number \(10^{-5} \le \textrm{Da} \le 10^{-1}\), Rayleigh number \(10^{4} \le \textrm{Ra} \le 10^{6}\), Hartmann number \(0 \le \textrm{Ha} \le 100\), heater size, the porous layer thickness \(0.25 \le \textrm{Xp} \le 0.75\), and nanoparticle concentration \(0.01 \le \phi \le 0.05\). From this study, the increase in the Ra number from \(10^{4}\) to \(10^{6}\) causes a thermal energy transmission improvement of 50%. Furthermore, a rise in the Da number from \(\textrm{Da}=10^{-5}\) to \(\textrm{Da}=10^{-1}\) enhances the thermal energy transport by approximately 30\(\%\), while it reduces by 4.8% when we increase the Hartmann number from 0 to 100. Also, the rise in nanoparticle concentration leads to an enhancement of the average Nusselt number, while the heat transfer rate is reduced by extending the heater size. The numerical results also show a significant improvement in the thermal energy transport in active walls by using an optimum thickness layer of stainless steel porous medium, according to the Da number. Furthermore, this study demonstrates that there is a critical value of porosity for a given nanoparticle concentration and porous layer thickness for better heat transfer enhancement.

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Abbreviations

AL:

Aspect ratio

\(R_{{\text{i}}}\) :

Inner radius \(\left[ {\text{m}} \right]\)

\(R_{{\text{e}}}\) :

Outer radius \(\left[ {\text{m}} \right]\)

H :

Cavity height \(\left[ {\text{m}} \right]\)

Xp:

Porous layer thickness

g:

Gravity acceleration \(\left[ {{\text{ms}}^{{ - 2}} } \right]\)

\(r,z\) :

System coordinate \(\left[ {\text{m}} \right]\)

\(\overline{r},\overline{z}\) :

Dimensionless coordinate

Ha:

Hartmann number

T :

Temperature function \(\left[ {\text{K}} \right]\)

\(\overline{T}\) :

Dimensionless temperature

U, W :

Velocity component \(\left[ {{\text{ms}}^{{ - 1}} } \right]\)

\(\Omega\) :

Vorticity function

K :

The permeability \(\left[ {{\text{m}}^{{ - 1}} } \right]\)

Da:

Da number

Pr:

Prandtl number

Ra:

Rayleigh number

\({\text{Nu}}_{{{\mathrm{aver}}}}\) :

Average Nusselt number

Q :

Heat flux \(\left[ {{\text{Wm}}^{2} } \right]\)

\(C_{{\text{p}}}\) :

Specific capacity \(\left[ {{\text{JKg}}^{{ - 1}} {\text{K}}^{{ - 1}} } \right]\)

\(\lambda\) :

Thermal conductivity \(\left[ {{\text{Wm}}^{{ - 1}} {\text{K}}^{{ - 1}} } \right]\)

\(\beta\) :

Thermal expansion \(\left[ {{\text{K}}^{{ - 1}} } \right]\)

\(\Sigma , \Lambda\) :

Nanofluid constants

\(\phi\) :

Nanoparticle concentration

\(\mu\) :

Dynamic viscosity\(\left[ {{\text{Kgm}}^{{ - 1}} {\text{s}}^{{ - 1}} } \right]\)

Ls:

Source length \(\left[ {\text{m}} \right]\)

\(\rho\) :

Density \(\left[ {{\text{Kgm}}^{{ - 3}} } \right]\)

\({\Gamma }\) :

Effective viscosity

\(\alpha\) :

Thermal diffusivity

\(B_{0}\) :

Magnetic field \(\left[ {{\text{Kgs}}^{{ - 2}} {\text{A}}^{{ - 1}} } \right]\)

\(\overline{U},\overline{W}\) :

Dimensionless velocity component

\(\overline{\Omega }\) :

Dimensionless vorticity

\(\overline{\Psi }\) :

Dimensionless stream function

p :

Porous medium

nf:

Nanofluid

eff:

Effective

c:

Cold

h:

Hot

\(b_{{\text{f}}}\) :

Base fluid

\(n_{{\text{p}}}\) :

Solid nanoparticles

\(\epsilon\) :

Porosity

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  1. Laboratory of Research in Physics and Engineering Sciences, Poly-Disciplinary Faculty, Sultan Moulay Slimane University, 23000, Beni Mellal, Morocco

    Youness Foukhari, Mohamed Sammouda & Mohamed Driouich

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  1. Youness Foukhari
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All authors contributed to the study conception and design. Data collection and analysis were performed by YF, MS and MD. The first draft of the manuscript was written by YF, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Youness Foukhari.

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Foukhari, Y., Sammouda, M. & Driouich, M. MHD natural convection in an annular space between two coaxial cylinders partially filled with metal base porous layer saturated by Cu–water nanofluid and subjected to a heat flux. J Therm Anal Calorim 149, 131–144 (2024). https://doi.org/10.1007/s10973-023-12709-w

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