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Effects of nanofluid and radiative heat transfer on the double-diffusive forced convection in microreactors

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Abstract

Understanding transport phenomena in microreactors remains challenging owing to the peculiar transfer features of microstructure devices and their interactions with chemistry. This paper, therefore, theoretically investigates heat and mass transfer in microreactors consisting of porous microchannels with thick walls, typical of real microreactors. To analyse the porous section of the microchannel, the local thermal non-equilibrium model of thermal transport in porous media is employed. A first-order, catalytic chemical reaction is implemented on the internal walls of the microchannel to establish the mass transfer boundary conditions. The effects of thermal radiation and nanofluid flow within the microreactor are then included within the governing equations. Further, the species concentration fields are coupled with that of the nanofluid temperature through considering the Soret effect. A semi-analytical methodology is used to tackle the resultant mathematical model with two different thermal boundary conditions. Temperature and species concentration fields as well as Nusselt number for the hot wall are reported versus various parameters such as porosity, radiation parameter and volumetric concentration of nanoparticles. The results show that radiative heat transfer imparts noticeable effects upon the temperature fields and consequently Nusselt number of the system. Importantly, it is observed that the radiation effects can lead to the development of a bifurcation in the nanofluid and porous solid phases and significantly influence the concentration field. This highlights the importance of including thermal radiation in thermochemical simulations of microreactors.

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Abbreviations

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

Interfacial area per unit volume of porous media, m−1

\(Bi\) :

Biot number

\(c\) :

Concentration of the chemical products per unit volume, \({\text{mol}}\,{\text{m}}^{ - 3}\)

\(c_{\text{p,nf}}\) :

Specific heat of the fluid phase of the porous medium, \({\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1}\)

\(D\) :

Diffusion coefficient, \({\text{m}}^{2} \,{\text{s}}^{ - 1}\)

\(Da\) :

Darcy number

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

Thermodiffusion coefficient, \({\text{m}}^{2} \,{\text{s}}^{ - 1} \,{\text{K}}^{ - 1}\)

h 1 :

Height of the lower wall, m

h 2 :

Height of the lower boundary of the upper wall, m

h 3 :

Height of the upper boundary of the upper wall, m

h :

External heat convection coefficient, \({\text{W}}\,{\text{m}}^{ - 2} \,{\text{K}}^{ - 1}\)

h sf :

Internal heat convection coefficient, \({\text{W}}\,{\text{m}}^{ - 2} \,{\text{K}}^{ - 1}\)

k :

Solid-to-fluid effective thermal conductivity ratio

k 1 :

Reference thermal conductivity for lower solid material, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

k 2 :

Reference thermal conductivity for upper solid material, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

k e1 :

Ratio of the fluid to lower solid material thermal conductivities

k e2 :

Ratio of the fluid to upper solid material thermal conductivities

k e,nf :

Effective thermal conductivity of the nanofluid phase of the porous medium, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

k es :

Effective thermal conductivity of the solid phase of the porous medium, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

k f :

Thermal conductivity of the base fluid, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

k nf :

Thermal conductivity of the nanofluid, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

k p :

Thermal conductivity of the nanoparticles, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

k R :

Kinetic constant, \({\text{m}}\,{\text{s}}^{ - 1}\)

k s :

Thermal conductivity of the solid phase of the porous medium, \({\text{W}}\,{\text{m}}^{ - 1} \,{\text{K}}^{ - 1}\)

N c :

Dimensionless convection heat transfer (Case 2)

Nu :

Nusselt number

p :

Pressure, Pa

\(Q_{1}\) :

Dimensionless volumetric internal heat generation rate for the lower solid material

\(Q_{2}\) :

Dimensionless volumetric internal heat generation rate for the upper solid material

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

Dimensionless heat flux boundary condition (Case 2)

\(\dot{q}_{1}\) :

Volumetric internal heat generation rate for the lower solid material, \({\text{W}}\,{\text{m}}^{ - 3}\)

\(\dot{q}_{2}\) :

Volumetric internal heat generation rate for the upper solid material, \({\text{W}}\,{\text{m}}^{ - 3}\)

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

Heat flux boundary condition (Case 2), \({\text{W}}\,{\text{m}}^{ - 2}\)

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

Radiation heat flux, \({\text{W}}\,{\text{m}}^{ - 2}\)

Rd :

Dimensionless radiation parameter

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

Volumetric internal heat generation rate for the solid phase of the porous medium, \({\text{W}}\,{\text{m}}^{ - 3}\)

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

Volumetric internal heat generation rate for the fluid phase of the porous medium, \({\text{W}}\,{\text{m}}^{ - 3}\)

Sr :

Soret number

\(T\) :

Temperature, K

\(T_{1}\) :

Temperature of the lower solid material, K

\(T_{2}\) :

Temperature of the upper solid material, K

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

Outer temperature of the upper solid material, K

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

Outer temperature of the lower solid material, K

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

Temperature of the fluid phase of the porous medium, K

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

Temperature of the solid phase of the porous medium, K

U m :

Average dimensionless velocity

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

Velocity of the fluid in porous medium, m s−1

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

Dimensionless velocity

Y 1 :

Dimensionless height of the lower wall

Y 2 :

Dimensionless height of the upper wall lower boundary

\(\gamma\) :

Damköhler number

ε :

Porosity

θ :

Dimensionless temperature

θ 1 :

Dimensionless temperature of the lower solid material

θ 2 :

Dimensionless temperature of the upper solid material

θ nf :

Dimensionless temperature of the fluid phase of the porous medium

\(\theta_{\text{nf,m}}\) :

Dimensionless average temperature of the fluid phase of the porous medium

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

Dimensionless temperature of the solid phase of the porous medium

\(\theta_{\text{H}}\) :

Dimensionless temperature at outer side of the lower wall

\(\kappa\) :

Permeability, \({\text{m}}^{2}\)

\(\kappa^{*}\) :

Rosseland mean absorption coefficient

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

Dynamic viscosity of porous medium, \({\text{Kg}}\,{\text{s}}^{ - 1} \,{\text{m}}^{ - 1}\)

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

Dynamic viscosity of the se fluid, \({\text{Kg}}\,{\text{s}}^{ - 1} \,{\text{m}}^{ - 1}\)

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

Dynamic viscosity of the nanofluid, \({\text{Kg}}\,{\text{s}}^{ - 1} \,{\text{m}}^{ - 1}\)

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

Dimensionless volumetric internal heat generation rate for the solid phase of the porous medium

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

Dimensionless volumetric internal heat generation rate for the fluid phase of the porous medium

\(\rho\) :

Density of the fluid phase, \({\text{kg}}\,{\text{m}}^{ - 3}\)

\(\sigma^{*}\) :

Stefan–Boltzmann constant, \({\text{W}}\,{\text{m}}^{ - 2} \,{\text{K}}^{ - 4}\)

\(\phi\) :

Dimensionless concentration

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Acknowledgements

Lilian Govone was funded through Erasmus Programme, and Linwei Wang acknowledges the financial support of Chinese Scholarship Council.

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Authors and Affiliations

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

    Lilian Govone, Linwei Wang & Nader Karimi

  2. The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Mohsen Torabi

  3. Civil and Mechanical Engineering Department, University of Missouri-Kansas City, Kansas City, MO, 64110, USA

    Nader Karimi

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  1. Lilian Govone
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Govone, L., Torabi, M., Wang, L. et al. Effects of nanofluid and radiative heat transfer on the double-diffusive forced convection in microreactors. J Therm Anal Calorim 135, 45–59 (2019). https://doi.org/10.1007/s10973-018-7027-z

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