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- Open Access
Nonlinear Rosseland thermal radiation and energy dissipation effects on entropy generation in CNTs suspended nanofluids flow over a thin needle
- Muhammad Idrees Afridi^{1}Email author,
- Iskander Tlili^{2},
- Muhammad Qasim^{1} and
- Ilyas Khan^{3}
- Received: 30 May 2018
- Accepted: 10 September 2018
- Published: 25 September 2018
Abstract
In this paper, we examine thermal radiation effect of the nonlinear form on the dissipative nanofluids containing carbon nanotubes past a moving horizontal thin needle. We also perform a second law analysis with viscous dissipation. Single-wall carbon nanotube and multiple-wall carbon nanotube drop in \(H_{2}O\) base fluid. Introducing suitable dimensionless variables, we reduce the governing equations to self-similar nonlinear differential equations. Matlab in-built boundary value solver bvp4c and shooting method are applied for the solution of the reduced set of self-similar differential equations. The numerical results thus obtained are compared, which agree well with respect to desired accuracy. Various graphs are depicted and illustrate qualitatively the influence of flow controlling parameters such as Eckert number, heating parameter, radiation parameters, nanoparticles solid volume fraction, and size of thin needle on entropy generation, temperature distribution, and Bejan number.
Keywords
- Entropy generation number
- Thin needle CNTs
- H_{2}O
- Viscous dissipation
- Nonlinear Rosseland thermal radiation
- Bejan number
1 Introduction
The innovative idea of reducing the entropy generation in convective heat transfer phenomenon is introduced by Bejan [1]. Basically, entropy generation is a measure of molecular disorder or randomness generated in a thermodynamic system. In the light of the second law of thermodynamics, the quality of energy reduces with increasing molecular disorderness. Bejan [2] reported that heat transfer due to temperature difference and energy dissipation are the key sources of entropy generation. Later on, numerous researchers performed a second law analysis in the presence of different geometries and physical conditions. For example, Afridi et al. [3] reported entropy generation in a viscous fluid flow past over an inclined stretching sheet under the influence of Lorentz force. The analysis reveals that more entropy is generated in the presence of magnetic field due to nonconservative nature of the applied magnetic field. The impacts of variable viscosity and Newtonian heating on entropy generation are investigated by Makinde [4]. Gul et al. [5] reported the influence of mixed convection on entropy generation in the Poiselle flow of Jeffry nanofluid. Recently, Butt et al. [6] examined the entropy generation in a second-grade nanofluid with effects of porous medium. Besides all these mentioned studies, some of the recent investigations on entropy generation are reported in [7–11].
In heat transfer and entropy generation analysis, energy dissipation is very significant, especially in boundary layer flows where the velocity gradients are significantly high. Historically, for the first time, Gebhart [12] studied the viscous dissipation effects on natural convection flow of Newtonian fluid. Gebhart [12] reported a significant rise in temperature due to dissipation effect. The rise in temperature is because of the work done against the viscous forces, which irreversibly convert the kinetic energy of the fluid into internal energy. Mohamed et al. [13] examined the impact of viscous dissipation on heat transfer in mixed convection flow past over a circular cylinder. Recently, Afridi and Qasim [14] performed the irreversibility analysis in a three- dimensional flow with viscous dissipation. Saritha et al. [15] performed the heat transfer analysis of boundary layer flow of power law fluid with viscous dissipation. Mohamed et al. [16] numerically studied the effects of viscous dissipation on a flow of nanofluid over a moving flat plate. Some recent studies on the heat transfer analysis with and without viscous dissipation effects are reported in [17–27].
Boundary layer flow with heat transfer over different body shapes, such as a horizontal cylinder, stretching sheet, stretching disk, flat plate, a sphere, an elastic sheet with variable thickness, Riga plat, and stretching cylinder, is investigated by numerous researchers. Historically, Lee [28] introduced the boundary layer flow over a thin needle moving horizontally in a parallel free stream. Ishak et al. [29] extended the work of Lee [28] and performed the heat transfer analysis under constant wall temperature. They found that temperature rises with increasing size of the thin needle. Soid et al. [30] studied the flow of nanofluid over a thin needle with heat transfer effects by taking the Tiwari and Dass model. Liu and Chan [31] investigated a mixed convection flow over a thin axisymmetric body. The impacts of mass transfer and mixed convection on heat transfer and fluid flow over a thin needle are reported by Kafoussias [32]. Some very interesting studies on boundary layer flow over a thin needle with different effects and imposed boundary conditions are reported in [33–36].
Working fluids such as water, engine oil, kerosene oil, and so on have low thermal conductivity. To enhance the thermal conductivity of working fluids, different types of nanoparticles such as copper, silver, Graphene, gold, and CNTs are added in the working fluids. Historically, the term nanofluid was introduced by Choi [37], who reported that the addition of nanoparticles in base fluids enhances the thermal conductivity to great extent. For the very first time, Khan and Pop [38] reported a boundary layer flow of nanofluid. In 1991, Lijima [39] introduced carbon nanotubes. Xue [40] proposed a model for the effective thermal conductivity of CNTs suspended nanofluids. CNTs have tube-shape nanostructure and are composed of carbon allotropes. Two types of CNTs are reported in the literature [41, 42]; one is a single-walled carbon nanotube (SWCNT), and the second one is a multiple-walled carbon nanotube (MWCNT). The impacts of CNTs on boundary layer flow and heat transfer are studied in [43–45].
After a careful review of the literature, we have come to conclusion that heat transfer and entropy generation of CNTs suspended nanofluids over a thin needle in the presence of viscous dissipation and nonlinear thermal radiation is important but has never been reported. Therefore, to fill this gap, in this study, we perform heat transfer and irreversibility (second law) analysis by taking the effects of nonlinear Rosseland thermal radiation and viscous dissipation. A thin needle is supposed to be moving in a parallel free stream. We utilize Matlab bvp4c solver to obtain the numerical solutions of a reduced set of self-similar equations. Variations of temperature distribution \(\theta ( \xi ) \), entropy generation number Ns, and irreversibility ratio (Bejan number Be) with physical flow parameters are depicted graphically and discussed physically in detail.
2 Mathematical formulation
3 Second law analysis
4 Results and discussions
Thermophysical properties of nanoparticles and base fluid
Properties | Base fluid | ||
---|---|---|---|
water | SWCNT | MWCNT | |
\(c_{p}\) [J/Kg⋅K] | 4179 | 425 | 796 |
k [W/Km] | 0.613 | 6600 | 300 |
ρ [Kg m^{−3}] | 997.1 | 2600 | 1600 |
Pr | 6.8 |
Comparison of numerical values of \(g''(0)\) and \(- \theta ' ( 0 ) \) for validation of numerical codes corresponding to the different values of a when \(\theta_{r} = 2.0\), \(N _{r} = 10\), \(\varepsilon = 0.3\), \(\phi = 0.04\), and \(Ec = 0.3\)
a | Shooting Method | bvp4c | ||
---|---|---|---|---|
\(g''(0)\) | \(- \theta '(0)\) | \(g''(0)\) | \(- \theta '(0)\) | |
SWCNT | ||||
0.2 | 0.297233 | 1.310988 | 0.297243 | 1.310988 |
0.1 | 0.502498 | 2.026587 | 0.502496 | 2.026588 |
0.01 | 3.259774 | 9.999569 | 3.259778 | 9.999570 |
0.001 | 23.895819 | 56.449154 | 23.895820 | 56.449157 |
MWCNT | ||||
0.2 | 0.297359 | 1.309687 | 0.297357 | 1.309688 |
0.1 | 0.502686 | 2.019083 | 0.502689 | 2.019081 |
0.01 | 3.260616 | 9.876561 | 3.260614 | 9.876565 |
0.001 | 23.900446 | 55.257994 | 23.900444 | 55.257991 |
Numerical values of \(g''(a)\) when \(\varepsilon = 0.0\)
Numerical values of skin friction coefficient \(C_{fx} ( \operatorname{Re}_{x} ) ^{1/2}\) for different values of a, ϕ, and ε
a | ε | ϕ | SWCNT | MWCNT |
---|---|---|---|---|
\(C_{fx} ( \operatorname{Re}_{x} ) ^{1/2}\) | \(C_{fx} ( \operatorname{Re}_{x} ) ^{1/2}\) | |||
0.2 | 0.3 | 0.01 | 1.118923 | 1.119043 |
0.1 | 1.333363 | 1.333487 | ||
0.01 | 2.716088 | 2.716263 | ||
0.01 | 0.2 | 0.1 | 4.969901 | 4.972249 |
0.4 | 1.629730 | 1.630798 | ||
0.6 | −1.598538 | −1.599574 | ||
0.01 | 0.1 | 0.05 | 5.984571 | 5.986487 |
0.1 | 6.673508 | 6.677828 | ||
0.2 | 8.487619 | 8.498633 |
Numerical values of local Nusselt number \(Nu_{x} ( \operatorname{Re}_{x} ) ^{ - 0.5}\) for different values of physical flow parameters when \(\Pr = 6.8\)
a | ε | ϕ | Ec | \(\theta _{r}\) | \(N_{r}\) | SWCNT | MWCNT |
---|---|---|---|---|---|---|---|
\(Nu_{x} ( \operatorname{Re}_{x} ) ^{ - 0.5}\) | \(Nu_{x} ( \operatorname{Re}_{x} ) ^{ - 0.5}\) | ||||||
0.2 | 0.3 | 0.01 | 0.2 | 2.0 | 8.0 | 2.943883 | 2.784078 |
0.1 | 3.119112 | 2.946959 | |||||
0.01 | 4.352717 | 4.099008 | |||||
0.01 | 0.1 | 0.1 | 0.2 | 2.0 | 8.0 | 12.583547 | 11.54771 |
0.2 | 13.578897 | 12.52149 | |||||
0.4 | 14.624973 | 13.55412 | |||||
0.01 | 0.3 | 0.05 | 0.2 | 2.0 | 8.0 | 8.345254 | 8.222671 |
0.1 | 14.257931 | 14.03552 | |||||
0.2 | 28.520924 | 28.22232 | |||||
0.01 | 0.3 | 0.1 | 0.3 | 2.0 | 8.0 | 13.961114 | 13.72888 |
0.5 | 13.367821 | 13.11598 | |||||
0.7 | 12.774984 | 12.50358 | |||||
0.01 | 0.3 | 0.1 | 0.2 | 3.0 | 8.0 | 15.409649 | 14.00310 |
5.0 | 15.722904 | 14.42210 | |||||
7.0 | 19.602963 | 18.21462 | |||||
0.01 | 0.3 | 0.1 | 0.2 | 2.0 | 5.0 | 14.924475 | 13.70168 |
10 | 13.873075 | 12.87780 | |||||
15 | 13.152421 | 12.27827 |
5 Concluding remarks
- *
The temperature distribution enhances with increasing values of the Eckert number, heating parameter, and nanoparticle solid volume fraction for both SWCNT suspended nanofluid and MWCNT suspended nanofluid.
- *
High temperature profile is observed for SWCNT suspended nanofluid.
- *
Increasing values of the thermal radiation parameter and decreasing the size of a thin needle result in reducing the temperature for both SWCNT and MWCNT suspended nanofluids.
- *
Increasing values of the heating parameter, the Eckert number and nanoparticle solid volume fraction enhances entropy generation number, but it is higher for SWCNT suspended nanofluid as compared to MWCNT suspended nanofluids.
- *
The quality of energy increases by reducing the size of a thin needle and increasing the thermal radiation parameter.
- *
The contribution of fluid friction in the entropy generation enhances with increasing values of the thermal radiation parameter and the Eckert number and decreasing the size of a needle.
- *
The Bejan number increases with heating parameter and nanoparticle solid volume fraction for both SWCNT and MWCNT suspended nanofluids.
Declarations
Acknowledgements
Not applicable.
Availability of data and materials
All data are fully available without restriction.
Authors’ information
Muhammad Idrees Afridi is a PhD scholar at COMSATS University, Islamabad, Pakistan, in the department of mathematics. Dr. Iskander Tlili is working as a researcher at University of Monastir, Monastir, Tunisia in the Energy department. Dr. Muhammad Qasim is assistant professor at COMSATS University, Islamabad, Pakistan, in the department of mathematics. Dr. Ilyas Khan is working as an assistant professor at Majmaah University, Saudi Arabia.
Funding
No funding was received.
Authors’ contributions
MIA and MQ formulated the problem. MIA, MQ, and IK solved the problem. MIA and IT computed the results. All the authors equally contributed in writing and proof reading of the paper. All the authors reviewed the manuscript. All authors approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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