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Modeling of Soret and Dufour’s Convective Heat Transfer in Nanofluid Flow Through a Moving Needle with Artificial Neural Network

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Abstract

In this study, forced convective heat and mass transfer of a nanofluid using the Buongiorno model and moving radially through a thin needle has been analyzed using the Runge–Kutta fourth-order technique with shooting approach. In order to analyze the thermo-diffusion and diffusion-thermoeffects on the flow, Dufour and Soret effects have been investigated and the mass transport phenomenon has also been investigated by activation energy. Partial differential systems of the flow model have been obtained with the boundary-layer approach and modified by using the appropriate transformations to be connected to nonlinear ordinary differential systems. The Runge–Kutta technique is the most popular methodology for obtaining the numerical results to solve the differential equations. It can evaluate higher-order numerical solutions and provide answers that are as close to correct solution. Therefore, using the Runge–Kutta fourth-order strategy with a shooting strategy, a data set has been created for different flow scenarios of the interesting and comprehensive model for nanofluid (Boungiorno’s model), which incorporates Brownian motion and thermophoresis. Using this data set, an artificial neural network model has been developed to predict skin friction coefficient, Sherwood number and Nusselt number values. Seventy percentage of the data used in ANN models developed with different numbers of datasets have been used for training, \(15\%\) for validation and \( 15\%\) for testing. The results show that ANN models can predict skin friction coefficient, Sherwood number and Nusselt number values with error rates of \(-\,0.33\%\), \(0.08\%\) and \(0.03\%\), respectively.

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Abbreviations

\(\tilde{T}\) :

Nanofluid’s temperature (K)

\(\tilde{T}_{\tilde{\mathrm{m}}}\) :

Fluid mean temperature

\(\tilde{K}\) :

Fluid’s thermal conductivity (kg m s\(^{-3}\) K\(^{-1}\))

\(\tilde{H}\) :

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

\(\tilde{\sigma }^{*}\) :

Stefan–Boltzmann constant

\({\tilde{v}}_{1},\tilde{u}_{1},\) :

Velocity components (m s\(^{-1}\))

\(\tilde{D}_{\tilde{\mathrm{m}}}\) :

Mass diffusivity coefficient (m \(^{2}\) s\(^{-1}\))

\(\left( \tilde{x}_{1},\tilde{r}_{1}\right) \) :

Coordinate system

\(\tilde{u}_{\infty }\) :

Free stream velocity

\(\tilde{q}_\mathrm{r}\) :

Radiative flux

\(\tilde{H}_{\infty }\) :

Ambient concentration (mol  m\(^{-3}\))

\(N_\mathrm{b}\) :

Brownian motion parameter

\(k_{1}^{*}\) :

Mean absorption coefficient

\(P_{1}\) :

Prandtl number

\(M_{1}\) :

Magnetic parameter

\(N_\mathrm{t}\) :

Thermophoresis parameter

\(R_\mathrm{a}\) :

Radiation parameter

\(\alpha _{1}\) :

Relaxation time

\(\alpha _{2}\) :

Temperature difference ratio

\(\left( w,b\right) \) :

Weights and bias, respectively

\(B_{1}\) :

Biot number

\(\tilde{T}_{n}\) :

Wall’s temperature (K)

\(\breve{D}_{\breve{\mathrm{B}} } \) :

Brownian diffusivity coefficient (m\(^{2}\) s\(^{-1}\))

\(\tilde{D}_{\check{\mathrm{T}}}\) :

Thermophoretic diffusivity coefficient (m\(^{2}\) s\(^{-1}\))

\(q_\mathrm{r}\) :

Radiative flux (W m\(^{-2}\))

\(S_\mathrm{c}\) :

Schmidt number

\(\breve{T}_{\infty }\) :

Ambient fluid’s temperature (K)

\( \tilde{H}_\mathrm{n}\) :

Wall’s nanoparticle volume fraction (mol m\(^{-3}\))

\(\tilde{\tau }^{*}\) :

Fraction of effective heat capacity

\(\tilde{u}_\mathrm{n}\) :

Needle’s velocity

\(\bar{U}\) :

Composite velocity between needle and free stream

\(S_\mathrm{r}\) :

Soret number

\(\tilde{K}_{\tilde{\mathrm{T}}}\) :

Thermal diffusion ratio

\(\lambda _{1}\) :

Velocity ratio number

\(\tilde{c}_\mathrm{s}\) :

Nanoparticle concentration susceptibility (kg\(^{-1}\) J K\(^{-1}\))

\(\tilde{c}_\mathrm{p}\) :

Heat at uniform pressure (kg\(^{-1}\) J K\(^{-1}\))

\(E_{2}\) :

Activation energy

\(D_\mathrm{u}\) :

Dufour number

f :

Transfer/activation function

GMDH:

Group method of data handling

\(\Psi \) :

Stream function

\(\phi \) :

Dimensionless concentration

\(\sigma _{0}^{*}\) :

Stefan–Boltzmann constant

\(\tilde{\alpha }\) :

Fluid thermal diffusivity (m s\(^{-1}\))

\(\tilde{\sigma }^{*}\) :

Coefficient of mean adsorption

\(\sigma \) :

Electrical conductivity (\(\Omega ^{-1}\) m\(^{-1}\))

\(\xi \) :

Similarity variable

\(\theta \) :

Dimensionless temperature

\(\rho _{0}\) :

Density of fluid (kg m\(^{-3}\))

\(\nu \) :

Dynamic viscosity (m\(^{2}\) s\(^{-1}\))

\(\mu \) :

Dynamic viscosity of base fluid (kg m\(^{-1}\) s\(^{-1}\))

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Anum Shafiq

  2. Mechanical Engineering Department, Niğde Ömer Halisdemir University, Niğde, Turkey

    Andaç Batur Çolak

  3. Department of Statistics, Quaid I Azam University, Islamabad, 45320, Pakistan

    Tabassum Naz Sindhu

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  1. Anum Shafiq
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  2. Andaç Batur Çolak
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  3. Tabassum Naz Sindhu
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Shafiq, A., Çolak, A.B. & Sindhu, T.N. Modeling of Soret and Dufour’s Convective Heat Transfer in Nanofluid Flow Through a Moving Needle with Artificial Neural Network. Arab J Sci Eng 48, 2807–2820 (2023). https://doi.org/10.1007/s13369-022-06945-9

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