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Numerical study of nonlinear mixed convection inside stagnation-point flow over surface-reactive cylinder embedded in porous media

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Abstract

Nonlinear mixed convection of heat and mass in a stagnation-point flow of an impinging jet over a solid cylinder embedded in a porous medium is investigated by applying a similarity technique. The problem involves a heterogenous chemical reaction on the surface of the cylinder and nonlinear heat generation in the porous solid. The conducted analysis considers combined heat and mass transfer through inclusions of Soret and Dufour effects and predicts the velocity, temperature and concentration fields as well as the average Nusselt and Sherwood number. It is found that intensification of the nonlinear convection results in development of higher axial velocities over the cylinder and reduces the thickness of thermal and concentration boundary layers. Hence, consideration of nonlinear convection can lead to prediction of higher Nusselt and Sherwood numbers. Further, the investigation reveals that the porous system deviates from local thermal equilibrium at higher Reynolds numbers and mixed convection parameter.

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Abbreviations

a :

Cylinder radius (m)

\(a_{\text{sf}}\) :

Interfacial surface area per unit volume of the porous medium (m−1)

\({\text{Bi}}\) :

Biot number \({\text{Bi}} = \frac{{h_{\text{sf}} a_{\text{sf}} \cdot a}}{{4k_{\text{f}} }}\)

\(C\) :

Fluid concentration (kg m−3)

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

Specific heat at constant pressure (J K−1 kg−1)

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

Concentration (kg m−3)

\(D\) :

Molecular diffusion coefficient (m2 s−1)

\({\text{Df}}\) :

Dufour number \({\text{Df}} = \frac{{D \cdot k_{\text{f}} }}{{C_{\text{s}} \cdot C_{\text{p}} }}\frac{{C_{\infty } }}{{\left( {T_{\text{w}} - T_{\infty } } \right)\upsilon }}\)

\(f(\eta )\) :

Function related to u-component of velocity

\(f^{\prime } (\eta )\) :

Normalised velocity related to w component

\(h\) :

Heat transfer coefficient (W K−1 m−2)

\(h_{\text{sf}}\) :

Interstitial heat transfer coefficient (W K−1 m−2)

\(k\) :

Thermal conductivity (W K−1 m−2)

\(\bar{k}\) :

Freestream strain rate (s−1)

\(k_{1}\) :

Permeability of the porous medium (m2)

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

Mass transfer coefficient (m s−1)

\(k_{\text{R}}\) :

Kinetic constant (kg m−2 s−1)

\(k_{\text{T}}\) :

Thermal diffusion ratio

\(N^{*}\) :

Ratio of concentration to thermal buoyancy forces \(N^{*} = \frac{{\beta_{3} \cdot C_{\infty } }}{{\beta_{1} \left( {T_{\text{w}} - T_{\infty } } \right)}}\)

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

Nusselt number

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

Nusselt number averaged over the surface of the cylinder

\(p\) :

Pressure (Pa)

\(P\) :

Dimensionless fluid pressure

\(P_{0}\) :

The initial pressure (Pa)

\({ \Pr }\) :

Prandtl number

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

Heat source parameter \(Q_{\text{h}} = \frac{{Q_{1} a^{2} }}{{4k_{\text{f}} }}\)

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

Mass flow at the wall (kg m−2 s−1)

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

Heat flow at the wall (W m−2)

\(r\) :

Radial coordinate

\({\text{Re}}\) :

Freestream Reynolds number \({\text{Re}} = \frac{{\bar{k} \cdot a^{2} }}{2\upsilon }\)

\({\text{Sc}}\) :

Schmidt number \({\text{Sc}} = \frac{\upsilon }{D}\)

\({\text{Sr}}\) :

Soret number \({\text{Sr}} = \frac{{D \cdot k_{\text{f}} }}{{T_{\infty } }}\frac{{\left( {T_{\text{w}} - T_{\infty } } \right)}}{{C_{\infty } \cdot \alpha }}\)

\({\text{Sh}}\) :

Sherwood number

\({\text{Sh}}_{\text{m}}\) :

Average Sherwood number

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

Mean fluid temperature (K)

\(u, w\) :

Velocity components along (\(r - \varphi - z\))-axis (m s−1)

\(z\) :

Axial coordinate

\(\alpha\) :

Thermal diffusivity (m2 s−1)

\(\beta_{\text{C}}\) :

Nonlinear mixed convection parameter for concentration \(\beta_{\text{C}} = \frac{{\beta_{4} \cdot C_{\infty } }}{{\beta_{3} }}\)

\(\beta_{\text{t}}\) :

Nonlinear mixed convection parameter for temperature \(\beta_{\text{t}} = \frac{{\beta_{2} \left( {T_{\text{w}} - T_{\infty } } \right)}}{{\beta_{1} }}\)

\(\gamma\) :

Modified conductivity ratio \(\gamma = \frac{{k_{\text{f}} }}{{k_{\text{s}} }}\)

\(\gamma^{*}\) :

Damköhler number \(\gamma^{*} = \frac{{k_{\text{R}} \cdot a}}{2D}\frac{1}{{C_{\infty } }}\)

\(\eta\) :

Similarity variable, \(\eta = \left( {\frac{r}{a}} \right)^{2}\)

\(\theta (\eta )\) :

Non-dimensional temperature

\(\lambda\) :

Permeability parameter, \(\lambda = \frac{{a^{2} }}{{4k_{1} }}\)

\(\varepsilon\) :

Porosity

\(\varLambda\) :

Dimensionless temperature difference \(\varLambda = \frac{{\left( {T_{\text{w}} - T_{\infty } } \right)}}{{T_{\infty } }}\)

\(\mu\) :

Dynamic viscosity of fluid (N s m−2)

\(\upsilon\) :

Kinematic viscosity of the fluid (m2 s−1)

\(\rho\) :

Density of fluid (kg m−3)

\(\phi\) :

Dimensionless concentration

\(\varphi\) :

Angular (circumferential) coordinate

\(w\) :

Condition on the surface of the cylinder

\(\infty\) :

Far field

\(f\) :

Fluid

\(s\) :

Solid

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Acknowledgements

This work was supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (17KJA530001) and Foundation of Huaian Municipal Science and Technology Bureau (HAA201734). Dr. Hong thanks the support of Six Talent Peaks Project of Jiangsu Province (2018-XNY-004).

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Authors and Affiliations

  1. National and Local Joint Engineering Research Center for Mineral Salt Deep Utilization, Key Laboratory for Palygorskite Science and Applied Technology of Jiangsu Province, Huaiyin Institute of Technology, Huai’an, 223003, China

    Kun Hong & Qingang Xiong

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

    Rasool Alizadeh

  3. Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

    Mostafa Valizadeh Ardalan

  4. Department of Mechanical Engineering, Bu-Ali Sina University, Hamedan, Iran

    Amireh Nourbakhsh

  5. School of Engineering, University of Glasgow, Glasgow, G12 8QQ, UK

    Nader Karimi

  6. Stanley Black & Decker, Inc., Townson, MD, 21286, USA

    Yang Yang

  7. IT Innovation Center, General Motors, Warren, MI, 48092, USA

    Qingang Xiong

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  1. Kun Hong
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Hong, K., Alizadeh, R., Ardalan, M.V. et al. Numerical study of nonlinear mixed convection inside stagnation-point flow over surface-reactive cylinder embedded in porous media. J Therm Anal Calorim 141, 1889–1903 (2020). https://doi.org/10.1007/s10973-019-09245-x

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