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Natural Convective Heat Transfer Flow of Nanofluids Inside a Quarter-Circular Enclosure Using Nonhomogeneous Dynamic Model

  • Research Article - Mechanical Engineering
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Abstract

In this paper, we have studied the problem of two-dimensional transient convective flow and heat transfer in nanofluids in a quarter-circular-shaped enclosure using newly proposed nonhomogeneous dynamic mathematical model. The round wall of the enclosure is maintained at constant low temperature and the variable thermal condition on the bottom heated wall is considered, whereas the vertical wall is regarded as adiabatic. The enclosure is permeated by an inclined uniform magnetic field. The Galerkin weighted residual finite element method has been employed to solve the governing nondimensional partial differential equations. The result shows that 1- to 10-nm-sized nanoparticles are uniform and stable in the solution. The external magnetic field and its direction control the flow pattern of nanofluid significantly. The average Nusselt number increases significantly, as nanoparticle volume fraction, magnetic field inclination angle and Rayleigh number increase. Average Nusselt number of cobalt–kerosene nanofluid is much higher than other 17 types of nanofluids which are studied in the present analysis. The flow behaviors and the heat transfer rates of 18 types of nanofluids have been presented for the scientific and engineering community to become familiar with such nanofluids and apply them in practical applications.

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Abbreviations

B :

Applied magnetic field (Nm\(^{-1}\) A\(^{-1}\))

C :

Concentration of nanofluid \((\hbox {mol}\,\hbox {m}^{-3})\)

\(C_\mathrm{C}\) :

Reference concentration \(\hbox {(mol m}^{3})\)

\(c_\mathrm{p}\) :

Specific heat \(\hbox {(J}\,\hbox {kg}^{-1} \hbox {K}^{-1})\)

\(D_\mathrm{B}\) :

Brownian diffusion coefficient \( \hbox {(m}^{2}\,\hbox {s}^{-1})\)

\(d_\mathrm{p}\) :

Diameter of nanoparticle \((\hbox {nm)}\)

\(D_\mathrm{T}\) :

Thermal diffusion coefficient \(\hbox {(m}^{2}\,\hbox {s}^{-1})\)

g :

Acceleration due to gravity \((\hbox {m}\,\hbox {s}^{-2})\)

h :

Heat transfer coefficient of nanofluid \(\hbox {(W}\,\hbox {m}^{-2}\,\hbox {K}^{-1})\)

\(k_\mathrm{B}\) :

Boltzmann constant \(\hbox {(J}\,\hbox {K}^{-1})\)

L :

Length of the bottom wall \(\hbox {(m)}\)

Le :

Lewis number

m :

Mass \(\hbox {(kg)}\)

n :

Empirical nanoparticle shape factor

\(N_\mathrm{TBTC} \) :

Dynamic diffusion parameter

\(N_\mathrm{TBT}\) :

Dynamic thermo-diffusion parameter

Nu :

Nusselt number

p :

Dimensional modified pressure \(\hbox {(Pa)}\)

P :

Dimensionless modified pressure

Pr :

Prandtl number

\(Ra_\mathrm{T} \) :

Thermal Rayleigh number

\(Ra_\mathrm{C}\) :

Solutal Rayleigh number

Sc :

Schmidt number

Ha :

Hartmann number

Sr :

Shear rate \(\hbox {(s}^{-1})\)

t :

Dimensional time \(\hbox {(s)}\)

T :

Nanofluid temperature \(\hbox {(K)}\)

\(T_\mathrm{C} \) :

Reference temperature \(\hbox {(K)}\)

U, V :

Dimensionless nanofluid velocity

(u, v):

Dimensional nanofluid velocity \(\hbox {(m}\,\hbox {s}^{-1})\)

\(V_\mathrm{T}\) :

Thermophoretic velocity \(\hbox {(m}\,\hbox {s}^{-1})\)

X, Y :

Dimensionless coordinates

\(\alpha \) :

Thermal diffusivity \(\hbox {(m}^2\,\hbox {s}^{-1})\)

\(\beta \) :

Thermal expansion coefficient \(\hbox {(K}^{-1}) \quad \)

\(\Omega \) :

Angle between hot and cold walls (radian)

\(\beta ^{*}\) :

Mass expansion coefficient \(\hbox {(mol)}^{-1}\)

\(\gamma \) :

Magnetic field inclination angle

\(\phi \) :

Nanoparticles volume fraction

\(\Phi \) :

Dimensionless concentration

\(\Psi \) :

Sphericity of the nanoparticle

\(\psi \) :

Stream function

\(\mu \) :

Viscosity \(\hbox {(kg}\,\hbox {m}^{-1}\,\hbox {s}^{-1})\)

\(\nu \) :

Kinematic viscosity \(\hbox {(m}^2\,\hbox {s}^{-1})\)

\(\kappa \) :

Thermal conductivity \(\hbox {(W}\,\hbox {m}^{-1}\,\hbox {K}^{-1})\)

\(\sigma \) :

Electric conductivity \((\hbox {S}\,\hbox {m}^{-1})\)

\(\Delta C\) :

Film concentration drop \(\hbox {(mol}\,\hbox {m}^{-3})\)

\(\Delta T\) :

Film temperature drop \(\hbox {(K)}\)

\(\theta \) :

Dimensionless temperature

\(\theta _\mathrm{C} \) :

Dimensionless temperature at cold wall

\(\theta _\mathrm{h} \) :

Dimensionless temperature at hot wall

\(\rho \) :

Density \(\hbox {(kg}\,\hbox {m}^{-3})\)

:

Correction factor

\(\xi \) :

Dimensionless time

bf:

Base fluid

p:

Nanoparticles

nf:

Nanofluid

ave:

Average

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Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, P.O. Box 36, P.C. 123, Al-Khod, Muscat, Sultanate of Oman

    M. J. Uddin, M. S. Alam & M. M. Rahman

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  1. M. J. Uddin
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  2. M. S. Alam
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  3. M. M. Rahman
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Correspondence to M. M. Rahman.

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M. S. Alam: On leave from the Department of Mathematics, Jagannath University, Dhaka, Bangladesh.

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Uddin, M.J., Alam, M.S. & Rahman, M.M. Natural Convective Heat Transfer Flow of Nanofluids Inside a Quarter-Circular Enclosure Using Nonhomogeneous Dynamic Model. Arab J Sci Eng 42, 1883–1901 (2017). https://doi.org/10.1007/s13369-016-2330-0

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