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Optimization and sensitivity analysis of heat transport of hybrid nanoliquid in an annulus with quadratic Boussinesq approximation and quadratic thermal radiation

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Abstract

The quadratic convective flow of hybrid nanoliquid in an annulus subjected to quadratic thermal radiation is studied for the first time. The impact of suction/injection and the uniform movement of the rings are considered. Nonlinear equations are handled numerically by adopting the shooting technique. An optimization procedure is performed by using response surface methodology. The maximum heat transport is observed for chosen values of effective parameters (thermal radiation parameter \( (5 \le Rt \le 15) \), temperature ratio parameter \( (1.1 \le \theta_{w} \le 5.1) \) and nanoparticle volume fraction of copper \( (1\% \le \phi_{\text{Cu}} \le 3\% )) \) at three different levels (low(− 1), middle(0) and high(+ 1)). In addition, a slope of the data point is evaluated for the friction coefficient and the Nusselt number. The results showed that the impact of quadratic thermal radiation on velocity and temperature distributions is more significant than linear thermal radiation. Further, an increase in quadratic convection and quadratic thermal radiation leads to an improvement in the friction coefficient of the skin on the inner surface of the outer annulus. Furthermore, the sensitivity of the friction coefficient is positive for the appearance of quadratic thermal radiation.

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Abbreviations

\( A \) :

Coefficient of velocity for the inner annulus

\( B \) :

Coefficient of velocity for outer annulus

\( a \) :

The radius of the inner annulus \( \left( {\text{m}} \right) \)

\( b \) :

The radius of the outer annulus \( \left( {\text{m}} \right) \)

\( g \) :

Acceleration due to gravity \( \left( {{\text{m}}\,{\text{s}}^{ - 2} } \right) \)

\( p \) :

Pressure \( \left( {\text{Pa}} \right) \)

\( P \) :

Dimensionless pressure

\( \Pr \) :

Prandtl number

\( S \) :

Suction/injection parameter

\( \text{Re} \) :

Reynolds number

\( C_{p} \) :

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

\( k \) :

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

\( n \) :

Shape factor

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

Nusselt number

\( D_{\text{h}} \) :

Hydraulic diameter \( \left( {\text{m}} \right) \)

\( r \) :

Axis in cylindrical coordinates

\( T \) :

Temperature \( \left( {\text{K}} \right) \)

\( T_{0} \) :

Outer wall temperature \( \left( {\text{K}} \right) \)

\( T_{1} \) :

Inner wall temperature \( \left( {\text{K}} \right) \)

\( Rt \) :

Thermal radiation parameter

\( u_{z} \) :

Velocity along \( z \) direction \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

\( u_{r} \) :

Velocity in \( r \) direction \( \left( {{\text{m}}\,{\text{s}}^{ - 1} } \right) \)

\( \theta_{w} \) :

Temperature ratio parameter

\( \alpha \) :

Quadratic convection parameter

\( \beta \) :

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

\( \eta \) :

Mixed convection parameter

\( \theta \) :

Dimensionless temperature

\( \lambda \) :

Radius ratio

\( \mu \) :

Dynamic viscosity \( \left( {{\text{kg}}\,{\text{m s}}^{ - 1} } \right) \)

\( \rho \) :

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

\( \nu \) :

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

\( \tau \) :

Skin friction coefficient

\( \phi \) :

The total volume concentration of \( {\text{Cu}} \) and \( {\text{Al}}_{2} {\text{O}}_{3} \)

\( 1,\lambda \) :

Value on the inner and outer wall

\( {\text{bl}} \) :

Bulk temperature

\( {\text{l}} \) :

Base liquid

\( {\text{hnl}} \) :

Hybrid nanoliquid

\( {\text{Cu}},{\text{Al}}_{2} {\text{O}}_{3} \) :

Nanoparticles

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Acknowledgements

The authors are grateful to the Management of CHRIST (Deemed to be University), India, for their support in completing this work. The authors also thank the Editors and Reviewers for their constructive suggestions on the manuscript.

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  1. Department of Mathematics, CHRIST (Deemed to be University), Bangalore, Karnataka, 560029, India

    K. Thriveni & B. Mahanthesh

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Thriveni, K., Mahanthesh, B. Optimization and sensitivity analysis of heat transport of hybrid nanoliquid in an annulus with quadratic Boussinesq approximation and quadratic thermal radiation. Eur. Phys. J. Plus 135, 459 (2020). https://doi.org/10.1140/epjp/s13360-020-00484-8

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