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Conjugate natural convection of nanofluids inside an enclosure filled by three layers of solid, porous medium and free nanofluid using Buongiorno’s and local thermal non-equilibrium models

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Abstract

The natural convective heat transfer of nanofluids was addressed inside a square enclosure filled by three different layers: solid, porous medium and free fluid. The behavior of the porous layer has been simulated using local thermal non-equilibrium model. The Buongiorno’s model was utilized to evaluate the distribution of nanoparticles inside the enclosure that arose from the thermophoresis and Brownian motion. The governing equations were solved by the Galerkin finite element method in a non-uniform grid. The governing parameters are Rayleigh number Ra = 103–106, porosity ε = 0.3–0.9, Darcy number Da = 10−5–10−2, interface parameter Kr = 0.1–10, H = 0.1–1000; ratio of wall thermal conductivity to that of the nanofluid, Rk = 0.1–10, dimensionless length of the heater B = 0.2–0.8; dimensionless centre position height of the heater Z = 0.3–0.7 and Lewis number Le = 10–100. A considerable concentration gradient of nanoparticles was found inside the enclosure. In some studied cases, the non-dimensional volume fraction of nanoparticles is about 10% higher than the average volume fraction of nanoparticles at the region near the cold wall. The variability of Darcy and the Rayleigh numbers indicated significant effects on heat transfer rate and the concentration patterns of the nanoparticles and inward the cavity. The increase in Le and Nr amplifies and decreases the heat transfer rates through fluid and solid phases, respectively. In addition, it can be seen that the increment in heat transfer rates with Le increases as Nr increases.

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Abbreviations

b :

Length of the heater (m)

B :

Dimensionless length of the heater

C :

Nanoparticle volume fraction

C 0 :

Ambient nanoparticle volume fraction

d :

Wall thickness (m)

D :

Dimensionless wall thickness

Da :

Darcy number

D B :

Brownian diffusion coefficient

D T :

Thermophoresis diffusion coefficient

g :

Gravitational acceleration vector (m s−2)

h nfs :

Volumetric heat transfer coefficient between the nanofluid and solid porous matrix (W m−3 K−1)

H :

Interface heat transfer coefficient parameter

k :

Thermal conductivity (W m−1 K−1)

K :

Permeability of the porous medium (m2)

K r :

Nanofluid to solid porous matrix thermal conductivity ratio parameter

L :

Square cavity size (m)

Le :

Lewis number

n :

Normal vector (m)

N :

Dimensionless normal vector

Nb :

Brownian motion parameter

Nr :

Buoyancy ratio parameter

Nt :

Thermophoresis parameter

Nu :

Local Nusselt number

\( \overline{Nu} \) :

Average Nusselt number

p :

Pressure (Pa)

P :

Dimensionless pressure

Pr :

Prandtl number

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

Total interfacial heat flux (W m−2)

Q w :

Dimensionless local heat transfer through the wall

\( \overline{{Q_{\text{w}} }} \) :

Dimensionless average heat transfer through the wall

Ra :

Rayleigh number

R k :

Wall to nanofluid thermal conductivity ratio parameter

s :

Porous layer thickness (m)

S :

Dimensionless porous layer thickness

Sh :

Local Sherwood number

T :

Temperature (K)

u, v :

Velocity components along x, y directions, respectively (m s−1)

U, V :

Dimensionless velocity components along x, y directions, respectively

x, y :

Cartesian coordinates (m)

X, Y :

Dimensionless Cartesian coordinates

z :

Center position height of the heater (m)

Z :

Dimensionless center position height of the heater

α :

Effective thermal diffusivity (m2 s−1)

β :

Thermal expansion coefficient of the fluid (K−1)

Δ:

Difference value

ε :

Porosity of the porous medium

θ :

Dimensionless temperature

μ :

Dynamic viscosity (kg m−1 s−1)

ν :

Kinematic viscosity (m2 s−1)

ρ :

Density (kg m−3)

(ρc):

Effective heat capacity (J K−1 m−3)

τ :

Parameter defined by τ = (ρc)p/(ρc)nf

ϕ :

Relative nanoparticle volume fraction

Ψ :

Dimensionless stream function

0:

Ambient property

c:

Cold

eff:

Effective

h:

Hot

max:

Maximum

nf:

Nanofluid

p:

Nanoparticle

s:

Solid porous matrix

w:

Wall

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Acknowledgements

Mohammad Ghalambaz is thankful to Dezful Branch Islamic Azad University of the financial support of the present study. The authors are tankful to Iran Nanotechnology Initiative Council (INIC) for its crucial support.

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Authors and Affiliations

  1. Young Researchers and Elite Club, Yasooj Branch, Islamic Azad University, Yasooj, Iran

    S. A. M. Mehryan

  2. Department of Mechanical Engineering, Dezful Branch, Islamic Azad University, Dezful, Iran

    Mohammad Ghalambaz

  3. Mechanical Engineering Department, Faculty of Engineering, Lorestan University, Khorramabad, Iran

    Mohsen Izadi

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  1. S. A. M. Mehryan
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Correspondence to Mohammad Ghalambaz.

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Mehryan, S.A.M., Ghalambaz, M. & Izadi, M. Conjugate natural convection of nanofluids inside an enclosure filled by three layers of solid, porous medium and free nanofluid using Buongiorno’s and local thermal non-equilibrium models. J Therm Anal Calorim 135, 1047–1067 (2019). https://doi.org/10.1007/s10973-018-7380-y

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