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Numerical study of unsteady natural convection heat transfer of Newtonian and non-Newtonian fluids in a square enclosure under oscillating heat flux

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Abstract

This paper presents a numerical study on the unsteady natural convective flow of Newtonian and non-Newtonian fluids in a square enclosure. A heat source with oscillating heat flux is located on the bottom wall of the enclosure. The top wall is thermally insulated and the other walls are at a relatively low temperature. The continuity, momentum, and energy equations for a computational domain encompassing the enclosure are solved numerically using the SIMPLE algorithm. The flow and temperature fields and the heat transfer performance are examined for different non-Newtonian fluids and heat source locations. The results are presented for different values of power-law index, Rayleigh number, and fluctuation period. It is found that the flow and temperature fields vary as the oscillating heat flux is changed. The pseudoplastic non-Newtonian fluid \((n < 1)\) is associated with a higher heat transfer, and the dilatant non-Newtonian fluid \((n > 1)\) is associated with a lower heat transfer with respect to the Newtonian fluid. The heat source oscillation period significantly affects the maximum flow temperature in the enclosure. This study provides useful information for the designers of electronic cooling systems using non-Newtonian fluids.

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Abbreviations

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

Specific heat \(\left( {{\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} } \right)\)

\(g\) :

Gravitational acceleration \(\left( {\text{m}\,\text{s}^{ - 2} } \right)\)

\(h\) :

Convection heat transfer coefficient \(\left( {{\text{W}}\,{\text{m}}^{ - 2} \,{\text{K}}^{ - 1} } \right)\)

\(k\) :

Thermal conductivity \(\left( {{\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1} } \right)\)

\(\kappa\) :

Consistency index \(\left( {{\text{Pa}}\,{\text{s}}^{n} } \right)\)

\(L\) :

Enclosure length (\(\text{m}\))

\(n\) :

Power-law index

\(Nu_{\text{s}}\) :

Local Nusselt number on the heat source, \(1/\theta_{\text{s}} (X)\)

\(Nu_{\text{m}}\) :

Average Nusselt number \(1/W_{\text{s}} \int_{{X_{\text{s}} - 0.5W_{\text{s}} }}^{{X_{\text{s}} + 0.5W_{\text{s}} }} {Nu_{\text{s}} (X){\text{d}}X}\)

\(p\) :

Fluid pressure \(\left( {\text{Pa}} \right)\)

\(\bar{p}\) :

Modified pressure \((p + \rho_{\text{c}} gy)\)

\(P\) :

Dimensionless pressure \((\bar{p}L^{2} /\rho \alpha^{2} )\)

\(Pr\) :

Prandtl number \((\nu /\alpha )\)

\(q^{{\prime \prime }}\) :

Oscillating heat flux \(\left( {\text{W}\,\text{m}^{ - 2} } \right)\)

\(q_{0}^{{\prime \prime }}\) :

Amplitude of oscillating heat flux \(\left( {\text{W}\,\text{m}^{ - 2} } \right)\)

\(Ra\) :

Rayleigh number \((g\beta L^{3} \Delta T/\nu \alpha )\)

t :

Time (s)

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

Oscillation period \(\left( \text{s} \right)\)

T :

Temperature (K)

\(u,v\) :

Velocity components in x-, y-directions \(\left( {\text{m}\,\text{s}^{{ - \text{1}}} } \right)\)

\(U,V\) :

Dimensionless velocity components \((uL/\alpha ,^{{}} vL/\alpha )\)

w S :

Heat source length \(\text{(m)}\)

W s :

Dimensionless heat source length \((w_{\text{s}} /L)\)

\(x_{\text{s}}\) :

Distance of the heat source from the left wall \((\text{m})\)

\(X_{\text{s}}\) :

Dimensionless distance of the heat source from the left wall \((x_{\text{s}} /L)\)

\(x,y\) :

Cartesian coordinates \(\text{(m)}\)

\(X,Y\) :

Dimensionless coordinates \((x/L,^{{}} y/L)\)

\(\alpha\) :

Thermal diffusivity \(\left( {\text{m}^{2} \,\text{s}^{ - 1} } \right)\)

\(\beta\) :

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

\(\Delta T\) :

Temperature difference \((q^{\prime\prime}_{0} L/k)\)

\(\mu\) :

Dynamic viscosity \(\left( {\text{N}\,\text{s}\,\text{m}^{{ - \text{2}}} } \right)\)

\(\mu_{\text{a}}\) :

Apparent viscosity \(\left( {\text{N}\,\text{s}\,\text{m}^{{ - \text{2}}} } \right)\)

\(\mu_{\text{a}}^{{\prime }}\) :

Dimensionless apparent viscosity

\(\nu\) :

Kinematic viscosity \(\left( {\text{m}^{2} \,\text{s}^{ - 1} } \right)\)

\(\theta\) :

Dimensionless temperature \(((T - T_{\text{c}} )/\Delta T)\)

\(\theta_{\hbox{max} }\) :

Maximum heat source temperature along its length

\((\theta_{\hbox{max} } )_{\hbox{max} }\) :

The highest value of \(\theta_{\hbox{max} }\) with respect to time

\(\rho\) :

Density \(\left( {\text{kg}\,\text{m}^{{ - \text{3}}} } \right)\)

\(\tau\) :

Time in dimensionless form \((\alpha t/L^{2} )\)

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

Oscillation period in dimensionless form \((\alpha t_{\text{p}} /L^{2} )\)

\(\psi_{\hbox{max} }\) :

Maximum stream function

\({\text{c}}\) :

Cold wall

\({\text{nf}}\) :

Newtonian fluid

\({\text{nnf}}\) :

Non-Newtonian fluid

\(s\) :

Heat source

i, j :

Index

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Authors and Affiliations

  1. Department of Mechanics, Payame Noor University, PO Box 19395-3697, Tehran, Iran

    Iman Pishkar

  2. Engineering Faculty, Shahrekord University, PO Box 115, Shahrekord, Iran

    Iman Pishkar, Behzad Ghasemi & Afrasiab Raisi

  3. School of Mechanical and Mining Engineering, The University of Queensland, Brisbane, QLD, 4072, Australia

    Saiied Mostafa Aminossadati

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Correspondence to Behzad Ghasemi.

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Pishkar, I., Ghasemi, B., Raisi, A. et al. Numerical study of unsteady natural convection heat transfer of Newtonian and non-Newtonian fluids in a square enclosure under oscillating heat flux. J Therm Anal Calorim 138, 1697–1710 (2019). https://doi.org/10.1007/s10973-019-08253-1

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