# Thermodynamic Analysis of Entropy Generation Minimization in Thermally Dissipating Flow Over a Thin Needle Moving in a Parallel Free Stream of Two Newtonian Fluids

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## Abstract

**:**

_{∞}(the free stream velocity) is in the positive axial direction (x-axis) and the motion of the thin needle is in the opposite or similar direction as the free stream velocity. The reduced self-similar governing equations are solved numerically with the aid of the shooting technique with the fourth-order-Runge-Kutta method. Using similarity transformations, it is possible to obtain the expression for dimensionless form of the volumetric entropy generation rate and the Bejan number. The effects of Prandtl number, Eckert number and dimensionless temperature parameter are discussed graphically in details for water and air taken as Newtonian fluids.

## 1. Introduction

_{2}O

_{3}) and copper (Cu) and water was taken as base fluid. Soid et al. [23] examined the analysis of boundary layer flow in a thin needle in continuous motion in a nanofluid. All the above thin needle problems focused on the heat transfer analysis only. However, there has been no attention given to the understanding of how entropy is generated with a thin needle in a parallel motion.

## 2. Flow Analysis

_{w}and the fluid in a stress-free region is moving with velocity u

_{∞}as shown in Figure 1. Under the stated assumptions, the governing flow equations along with the boundary conditions take the following form [21,22,23,24]:

_{w}+ u

_{∞}≠ 0 and the dimensionless stream function is denoted by f(η). By taking η = a (which represents the wall of the needle) in Equation (4), we get $R(x)={\left(\frac{a\nu x}{U}\right)}^{1/2}$ which describes the size and the shape of the surface of revolution).

#### 2.1. First Law Analysis

_{p}represent Equation (4), Equation temperature distribution, thermal diffusivity, kinematic viscosity and specific heat at constant pressure. The imposed boundary conditions are:

_{w}> T

_{∞}(heated needle) and the dimensionless temperature θ(η) is given by:

_{u}= (Re

_{x})

^{−1/2}is:

#### 2.2. Second Law Analysis

_{T}(modified Brinkman number), N

_{ΔT}= θ

^{’2}(conductive irreviersibility), N

_{Fric}= 4Brf″

^{2}, ${\left({\dot{S}}_{prod}^{m}\right)}_{c}=\raisebox{1ex}{$4k{\mathsf{\Omega}}_{T}^{2}\eta {\mathrm{Re}}_{x}$}\!\left/ \!\raisebox{-1ex}{${x}^{2}$}\right.\left(characteristic\text{}entropy\right).$

_{f}is the major factor whereas when $0<\mathsf{\Phi}<1$ the heat transfer irreversibility Ns

_{h}is the dominant one. When Ф = 1, the improvement secondary to heat transfer (Ns

_{h}) and to fluid friction (Ns

_{f}) are equal.

## 3. Results and Discussions

_{t}increases with the diminishing size of the thin needle as depicted in Figure 4a,b and this is because of the increasing velocity and temperature gradients. Figure 4a,b also reveal that the total entropy generated in water is more than in air. Furthermore, for a fixed value of “a” the entropy decreases as the needle velocity decreases or the free stream velocity increases, so by increasing the free stream velocity or by decreasing the velocity of the thin needle one can minimize the entropy generation, which is the main goal of the second law analysis, the minimization of entropy generation.

## 4. Closing Remarks

- Heat transfer and fluid friction irreversibility increases with the decreasing size of the thin needle for both type of fluids air and water.
- Total entropy enhances with the reduced size needle.
- Entropy generated due to heat transfer and fluid friction in water is more than in air.
- When ε < 0, entropy can be minimized either by increasing the free stream velocity or by decreasing the needle velocity.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathit{a}$ | Constant |

$\mathit{B}\mathit{e}$ | Bejan Number $[-]$ |

$\mathit{B}\mathit{r}$ | Brinkman number $[-]$ |

${\mathit{c}}_{p}$ | Specific heat $\left[{\mathrm{JKg}}^{-1}{\text{}\mathrm{K}}^{-1}\right]$ |

$\mathit{E}\mathit{c}$ | Eckert number $[-]$ |

$\mathit{f}$ | Dimensionless stream function |

$k$ | Thermal conductivity $\left[{\mathrm{Wm}}^{-1}{\text{}\mathrm{K}}^{-1}\right]$ |

$\mathit{P}\mathit{r}$ | Prandtl number $[-]$ |

$\mathit{R}\left(\mathit{x}\right)$ | Shape and size of the needle [m] |

$\mathit{R}{\mathit{e}}_{\mathit{x}}$ | Local Reynold number $[-]$ |

$\mathit{T}$ | Temperature field [K] |

$\mathit{U}$ | Composite velocity [${\mathrm{m}\text{}\mathrm{s}}^{-1}$] |

${\mathit{u}}_{\mathit{w}},{\mathit{u}}_{\infty}$ | Velocity of needle and free stream, respectively [${\mathrm{m}\text{}\mathrm{s}}^{-1}$] |

$\mathit{u},\mathit{v}$ | Velocity components in axial and radial directions, respectively [m] |

$\mathit{x},\mathit{r}$ | Spatial coordinates measured in axial and radial directions, respectively [m] |

$\mathsf{\theta}$ | Non-dimensional temperature $[-]$ |

$\mathsf{\rho}$ | Density of fluid $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-3}\right]$ |

$\mathsf{\mu}$ | Dynamic viscosity $\left[{\mathrm{kg}\text{}\mathrm{m}}^{-1}{\text{}\mathrm{s}}^{-1}\right]$ |

$\mathit{\Psi}$ | Dimensional stream function $\left[{\mathrm{m}}^{2}\text{}{\mathrm{s}}^{-1}\right]$ |

$\mathit{\epsilon}$ | Velocity ratio parameter |

## References

- Bejan, A. A study of entropy generation in fundamental convective heat transfer. J. Heat Transf.
**1979**, 101, 718–725. [Google Scholar] [CrossRef] - Bejan, A. Entropy Generation through Heat and Fluid Flow; John Wiley and Sons: New York, NY, USA, 1982. [Google Scholar]
- Mahmud, S.; Fraser, R.A. Thermodynamics analysis of flow and heat transfer inside channel with two parallel plates. Exergy
**2002**, 2, 140–146. [Google Scholar] [CrossRef] - Makinde, O.D. Irreversibility analysis for a gravity driven non-Newtonian liquid film along an inclined isothermal plate. Phys. Scr.
**2006**, 74, 642–645. [Google Scholar] [CrossRef] - Makinde, O.D.; Maserumule, R.L. Thermal criticality and entropy analysis for a variable viscosity Couette flow. Phys. Scr.
**2008**, 78, 1–6. [Google Scholar] [CrossRef] - Makinde, O.D. Second law analysis for variable viscosity hydromagnetic boundary layer flow with thermal radiation and Newtonian heating. Entropy
**2011**, 13, 1446–1464. [Google Scholar] [CrossRef] - Aziz, A.; Khan, W.A. Entropy Generation in an Asymmetrically Cooled Slab with Temperature-Dependent Internal Heat Generation. Heat Transf.-Asian Res.
**2012**, 41, 260–271. [Google Scholar] [CrossRef] - Mkwizu, M.; Makinde, O.D. Entropy generation in a variable viscosity channel flow of nanofluids with convective cooling. Compt. Rendus Mecanique
**2015**, 343, 38–56. [Google Scholar] [CrossRef] - Makinde, O.D.; Khan, W.A.; Aziz, A. On inherent irreversibility in Sakiadis flow of nanofluids. Int. J. Exergy
**2013**, 13, 159–174. [Google Scholar] [CrossRef] - Rashidi, M.M.; Ali, M.; Freidoonimehr, N.; Nazari, F. Parametric analysis and optimization of entropy generation in unsteady MHD flow over a stretching rotating disk using artificial neural network and particle swarm optimization algorithm. Energy
**2013**, 55, 497–510. [Google Scholar] [CrossRef] - Butt, A.S.; Ali, A. Investigation of entropy generation effects in magnetohydrodynamic three-dimensional flow and heat transfer of viscous fluid over a stretching surface. J. Braz. Soc. Mech. Sci. Eng.
**2015**, 37, 211–219. [Google Scholar] [CrossRef] - Das, S.; Sarkar, B.C.; Jana, R.N. Entropy Generation in MHD free convection boundary layer flow past an inclined flat plat embedded in a porous medium with hall currents. Int. J. Comput. Appl.
**2013**, 8, 36–46. [Google Scholar] - Govindaraju, M.; Saranya, S.; Hakeem, A.K.A.; Jayaprakash, R.; Ganga, B. Analysis of slip MHD nanofluid flow on entropy generation in a stretching sheet. Procedia Eng.
**2015**, 127, 501–507. [Google Scholar] [CrossRef] - Rashidi, M.M.; Mohammadi, F.; Abbasbandy, S.; Alhuthali, M.S. Entropy generation analysis for stagnation point flow in a porous medium over a permeable stretching surface. J. Appl. Fluid Mech.
**2015**, 8, 753–765. [Google Scholar] [CrossRef] - Sheikholeslami, M. New computational approach for exergy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media. Comput. Methods Appl. Mech. Eng.
**2019**, 344, 319–333. [Google Scholar] [CrossRef] - Hakeem, A.K.A.; Govindaraju, M.; Ganga, B.; Kayalvizhi, M. Second law analysis for radiative MHD slip flow of a nanofluid over a stretching sheet with non-uniform heat source effect. Sci. Iran.
**2016**, 23, 1524–1538. [Google Scholar] [Green Version] - Butt, A.S.; Ali, A.; Mehmood, A. Numerical investigation of magnetic field effects on entropy generation in viscous flow over a stretching cylinder embedded in a porous medium. Energy
**2016**, 99, 237–249. [Google Scholar] [CrossRef] - Afridi, M.I.; Qasim, M.; Khan, I.; Shafie, S.; Alshomrani, A.S. Entropy Generation in Magnetohydrodynamic Mixed Convection Flow over an Inclined Stretching Sheet. Entropy
**2016**, 19, 10. [Google Scholar] [CrossRef] - Lee, L.L. Boundary Layer over a thin needle. Phys. Fluids
**1967**, 10, 820–822. [Google Scholar] [CrossRef] - Chen, J.L.S.; Smith, T.N. Forced confection heat transfer from non-isothermal thin needles. J. Heat Transf. Trans. ASME
**1978**, 100, 1–5. [Google Scholar] [CrossRef] - Ishaq, A.; Nazar, R.; Pop, I. Boundary layer flow over a continuously moving thin needle in a parallel free stream. Chin. Phys. Lett.
**2007**, 24, 2895. [Google Scholar] [CrossRef] - Grosan, T.; Pop, I. Forced convection boundary layer flow past non isothermal thin needles in nanofluids. J. Heat Transf.
**2011**, 113, 054503-1. [Google Scholar] [CrossRef] - Soid, S.K.; Ishak, A.; Pop, I. Boundary layer flow past a continuously moving thin needle. Appl. Therm. Eng.
**2017**, 114, 58–64. [Google Scholar] [CrossRef] - Afridi, M.I.; Qasim, M. Entropy generation and heat transfer in boundary layer flow over a thin needle moving in a parallel stream in the presence of nonlinear Rosseland radiation. Int. J. Ther. Sci.
**2018**, 123, 117–128. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Khan, I.; Khan, W.A.; Qasim, M.; Afridi, I.; Alharbi, S.O.
Thermodynamic Analysis of Entropy Generation Minimization in Thermally Dissipating Flow Over a Thin Needle Moving in a Parallel Free Stream of Two Newtonian Fluids. *Entropy* **2019**, *21*, 74.
https://doi.org/10.3390/e21010074

**AMA Style**

Khan I, Khan WA, Qasim M, Afridi I, Alharbi SO.
Thermodynamic Analysis of Entropy Generation Minimization in Thermally Dissipating Flow Over a Thin Needle Moving in a Parallel Free Stream of Two Newtonian Fluids. *Entropy*. 2019; 21(1):74.
https://doi.org/10.3390/e21010074

**Chicago/Turabian Style**

Khan, Ilyas, Waqar A. Khan, Muhammad Qasim, Idrees Afridi, and Sayer O. Alharbi.
2019. "Thermodynamic Analysis of Entropy Generation Minimization in Thermally Dissipating Flow Over a Thin Needle Moving in a Parallel Free Stream of Two Newtonian Fluids" *Entropy* 21, no. 1: 74.
https://doi.org/10.3390/e21010074