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Natural convection of water-based nanofluid in a chamber with a solid body of periodic volumetric heat generation

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Abstract

A computational analysis of convective energy transport of water-based nanosuspension having variable thermal properties has been performed using finite difference method. The considered square cavity includes cold vertical walls and adiabatic horizontal boundaries. The local heater of periodic thermal production is placed on the lower border of the domain. The working fluid is water with copper oxide nanoparticles of low concentration. Control differential equations with initial and boundary conditions have been written using non-dimensional stream function, vorticity and temperature. The resulting nonlinear partial differential equations with associated boundary conditions are solved using the finite difference methodology on a uniform calculation mesh. The analyzed control parameters including volumetric heat generation frequency, initial fraction of nanoparticles, heater location and time have been studied. The physics of the problem is well-explored for the embedded material parameters through tables and graphs. The obtained data have shown that the volumetric thermal production frequency of the source and the initial concentration of nanoparticles have the greatest influence on the heat transfer performance. The energy source temperature can be reduced by up to 20% by varying the characteristics of the source and nanosuspension.

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Abbreviations

c :

Heat capacity (J kg−1 K−1)

C :

Nanoparticles volume fraction

d p :

Diameter of nanoparticles

\(\overline{D}_{{\text{B}}} = \frac{{k_{{\text{B}}} }}{{3\pi \mu_{{\text{f}}} \left( T \right)d_{{\text{p}}} }}T\) :

Dimensional Brownian diffusion coefficient

\(\overline{D}_{{\text{T}}} = C\left[ {\frac{{\mu_{{\text{f}}} \left( T \right)}}{{\rho_{{\text{f}}} }}} \right]\left( {\frac{{0.26 \cdot k_{{\text{f}}} }}{{2k_{{\text{f}}} + k_{{\text{p}}} }}} \right)\) :

Dimensional thermophoretic diffusion coefficient

D B, D T :

Additional functions

f :

Volumetric thermal production frequency (s–1)

g :

Gravity acceleration (m s–2)

k :

Thermal conductivity (W m−1 K−1)

K :

Additional function

L :

Length of the enclosure (m)

l :

Distance from the left wall of the chamber to the left wall of the heater (m)

\({\text{Le}} = {{\alpha_{{\text{f}}} } \mathord{\left/ {\vphantom {{\alpha_{{\text{f}}} } {\overline{D}_{{\text{B}}} \left( {T_{{\text{c}}} } \right)}}} \right. \kern-\nulldelimiterspace} {\overline{D}_{{\text{B}}} \left( {T_{{\text{c}}} } \right)}}\) :

Lewis number

M :

Additional function

Nu:

Nusselt number (–)

\(\overline{{{\text{Nu}}}}\) :

Mean Nusselt number (–)

\({\text{Nt}} = \overline{D}_{{\text{B}}} \left( {T_{{\text{c}}} ,C_{0} } \right) \cdot {{\Delta T} \mathord{\left/ {\vphantom {{\Delta T} {\left( {\alpha_{{\text{f}}} \cdot T_{{\text{c}}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\alpha_{{\text{f}}} \cdot T_{{\text{c}}} } \right)}}\) :

Thermophoresis parameter

p :

Static pressure (Pa)

\({\text{Pr}}_{{\text{T}}} = \frac{{\mu_{{\text{f}}} \left( T \right)}}{{\rho_{{\text{f}}} \alpha_{{\text{f}}} }}\) :

Prandtl number (–)

Q :

Volumetric heat flux (W m–3)

\({\text{Ra}} = \frac{{g \cdot \left( {\rho \beta } \right)_{{\text{f}}} \cdot \Delta T \cdot L^{3} }}{{\mu_{{\text{f}}} \left( {T_{{\text{c}}} } \right) \cdot \alpha_{{\text{f}}} }}\) :

Rayleigh number (–)

T :

Temperature (K)

T c :

Cold wall temperature (K)

t :

Time (s)

\(\overline{u}, \, \overline{v}, \, \overline{w}\) :

Velocity projections (m s−1)

u, v, w :

Non-dimensional velocity projections (–)

\(\overline{x}, \, \overline{y}, \, \overline{z}\) :

Coordinates (m)

x, y, z :

Non-dimensional coordinates (–)

α :

Thermal diffusivity (W m−2 K−1)

β :

Thermal expansion coefficient (K−1)

γ :

Dimensionless volumetric heat generation oscillation frequency

∆T :

Temperature drop (K)

δ = l/L :

Dimensionless distance from the left wall of the chamber to the left wall of the heater

θ :

Non-dimensional temperature (–)

ρ :

Density (kg m−3)

τ :

Dimensionless time (–)

φ :

Normalized nanoparticles volume fraction

\(\overline{\psi }_{{\text{x}}} , \, \overline{\psi }_{{\text{y}}} , \, \overline{\psi }_{{\text{z}}}\) :

Vector potential functions (m2 s−1)

\(\psi_{{\text{x}}} ,\psi_{{\text{y}}} ,\psi_{{\text{z}}}\) :

Non-dimensional vector potential functions (–)

\(\overline{\omega }_{{\text{x}}} , \, \overline{\omega }_{{\text{y}}} , \, \overline{\omega }_{{\text{z}}}\) :

Projections of vorticity vector (s−1)

\(\omega_{{\text{x}}} ,\omega_{{\text{y}}} ,\omega_{{\text{z}}}\) :

Dimensionless projections of vorticity vector (–)

c:

Cooled

f:

Fluid

hs:

Heat source

nf:

Nanofluid

p:

Nanoparticles

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Acknowledgements

This study was supported by the Tomsk State University Development Programme (Priority–2030). The authors would like to thank very much to the very competent Reviewers for their valuable time spent on reading the manuscript and for the valuable comments and suggestions.

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Authors and Affiliations

  1. Laboratory on Convective Heat and Mass Transfer, Tomsk State University, Tomsk, Russia, 634050

    Marina S. Astanina & Mikhail A. Sheremet

  2. Department of Mathematics, Babes-Bolyai University, 400084, Cluj-Napoca, Romania

    Ioan Pop

  3. Academy of Romanian Scientists, 3 Ilfov Street, 050044, Bucharest, Romania

    Ioan Pop

Authors
  1. Marina S. Astanina
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Correspondence to Ioan Pop.

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Astanina, M.S., Pop, I. & Sheremet, M.A. Natural convection of water-based nanofluid in a chamber with a solid body of periodic volumetric heat generation. J Therm Anal Calorim 148, 1011–1024 (2023). https://doi.org/10.1007/s10973-022-11735-4

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