Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation

Asim Aziz; Wasim Jamshed; Yasir Ali; Moniba Shams

doi:10.3934/dcdss.2020142

Article Contents

2020, Volume 13, Issue 10: 2667-2690. Doi: 10.3934/dcdss.2020142

This issue Previous Article Conservation laws and line soliton solutions of a family of modified KP equations Next Article Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation

1.
College of Electrical and Mechanical Engineering, National University of Sciences and Technology, Rawalpindi, 46070, Pakistan
2.
Department of Mathematics, Capital University of Science and Technology, Islamabad 44000, Pakistan
3.
Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, 44000, Pakistan

^* Corresponding author: Asim Aziz
^* Corresponding author: Asim Aziz

Received: January 31, 2019

Revised: June 30, 2019

Early access: November 2019

Published: July 2020

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Abstract

In this numerical study, researchers explore the flow, heat transfer and entropy of electrically conducting hybrid nanofluid over the horizontal penetrable stretching surface with velocity slip conditions at the interface. The non-Newtonian fluid models lead to better understanding of flow and heat transfer characteristics of nanofluids. Therefore, non-Newtonian Maxwell mathematical model is considered for the hybrid nanofluid and the uniform magnetic field is applied at an angle to the direction of the flow. The Joule heating and thermal radiation impact are also considered in the simplified model. The governing nonlinear partial differential equations for hybrid Maxwell nanofluid flow, heat transfer and entropy generation are simplified by taking boundary layer approximations and then reduced to ordinary differential equations using suitable similarity transformations. The Keller box scheme is then adopted to solve the system of ordinary differential equations. The Ethylene glycol based Copper Ethylene glycol ($ Cu $-$ EG $) nanofluid and Ferro-Copper Ethylene glycol ($ Fe_3O_4-Cu $-$ EG $) hybrid nanofluids are considered to produce the numerical results for velocity, temperature and entropy profiles as well as the skin friction factor and the local Nusselt number. The main findings indicate that hybrid Maxwell nanofluid is better thermal conductor when compared with the conventional nanofluid, the greater angle of inclination of magnetic field offers greater resistance to fluid motion within boundary layer and the heat transfer rate act as descending function of nanoparticles shape factor.

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Mathematics Subject Classification: Primary: 76A05, 76W05; Secondary: 82D80.

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Figure 1. Flow geometry

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Figure 2. Flow Sheet of Keller Box method

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Figure 3. Velocity distribution against $ \beta $

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Figure 4. Temperature distribution against $ \beta $

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Figure 5. Entropy generation against $ \beta $

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Figure 6. Velocity distribution against $ M $

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Figure 7. Temperature distribution against $ M $

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Figure 8. Entropy generation against $ M $

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Figure 9. Velocity distribution against $ \Gamma $

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Figure 10. Temperature distribution against $ \Gamma $

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Figure 11. Entropy generation against $ \Gamma $

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Figure 12. Velocity distribution against $ \phi, \phi_{hnf} $

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Figure 13. Temp distribution against $ \phi, \phi_{hnf} $

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Figure 14. Entropy generation against $ \phi, \phi_{hnf} $

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Figure 15. Temperature distribution against $ Nr $

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Figure 16. Entropy generation against $ Nr $

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Figure 17. Temperature distribution against $ Ec $

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Figure 18. Entropy generation against $ Ec $

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Figure 19. Temperature distribution against $ Bi $

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Figure 20. Entropy generation against $ Bi $

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Figure 21. Temperature distribution against $ m $

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Figure 22. Entropy generation distribution against $ m $

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Figure 23. Entropy generation distribution against the parameter $ Re $

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Figure 24. Entropy generation distribution against the parameter $ Br $

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Table 1. Comparison results of -$ \theta^{'}(0) $ for different Prandtl number when $ \beta = 0 $, $ A = 0 $, $ M = 0 $, $ \Gamma = 0 $, $ \phi_{1} = 0 $, $ \phi_{2} = 0 $, $ \Lambda = 0 $, $ Nr = 0 $, $ Ec = 0 $, $ S = 0 $, $ m = 3 $ and $ B_i = 0 $

$ Pr $	$ Ishak $	$ Nazar $	$ Abolbashari $	$ Das $	$ Present $
	Results[27]	Results[28]	Results[2]	Results[16]	Results
0.72	0.8086	0.8086	0.80863135	0.80876122	0.80876181
1.0	1.0000	1.0000	1.00000000	1.00000000	1.00000000
3.0	1.9237	1.9236	1.92368259	1.92357431	1.92357420
7.0	3.0723	3.0722	3.07225021	3.07314679	3.07314651
10	3.7207	3.7006	3.72067390	3.72055436	3.72055429

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Table 2. Thermo-physical properties of Base Fluid and Nanoparticles

Thermo-physical	$ \rho({kg}/{m^{3}}) $	$ c_p({J}/{kgK}) $	$ k({W}/{mK}) $	$ \sigma({S}/{m}) $
Ethylene glycol $ (EG) $	1114	2415	0.252	$ 5.5 \times 10^{-6} $
Pure water $ (H_2O) $	997.1	4179	0.613	$ 0.05 $
Copper $ (Cu) $	8933	385.0	401.00	$ 5.96 \times 10^{7} $
Ferro $ (Fe_3O_4) $	5180	670	9.7	$ 0.74 \times 10^{6} $
Copper oxide $ (Cu_O) $	6510	540	18	$ 5.96 \times 10^{7} $
Alumina $ (Al_{2}O_{3}) $	3970	765.0	40.000	$ 3.5 \times 10^{7} $
Titanium oxide $ (T_iO_{2}) $	4250	686.2	8.9538	$ 2.38 \times 10^{6} $

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Table 3. Values of Skin Friction $ = C_fRe_x^\frac{1}{2} $ and Nusselt Number $ = N_uRe_x^\frac{-1}{2} $ for $ P_{r} = 6.2 $ and $ m = 3 $

$ \beta $	$ A $	$ M $	$ \Gamma $	$ \phi $	$ \phi_{2} $	$ \Lambda $	$ Nr $	$ Ec $	$ Bi $	$ C_fRe_x^\frac{1}{2} $	$ C_fRe_x^\frac{1}{2} $	$ N_uRe_x^\frac{-1}{2} $	$ N_uRe_x^\frac{-1}{2} $
										$ Cu $-$ EG $	$ Fe_3O_4-Cu/EG $	$ Cu $-$ EG $	$ Fe_3O_4-Cu/EG $
0.01	0.6	0.6	$ \pi/4 $	0.18	0.09	0.1	0.2	0.2	0.1	1.1918	1.8311	0.0818	0.0824
0.1										2.0011	2.0673	0.0813	0.0819
0.3										2.1039	2.0894	0.0804	0.0808
	0.6									1.1918	1.8311	0.0818	0.0824
	1.6									2.2308	2.1736	0.0839	0.0879
	2.6									2.4621	2.3211	0.0896	0.0899
		0.6								1.1918	1.8311	0.0818	0.0824
		1.6								2.2661	2.1068	0.0809	0.0820
		2.6								2.4149	2.2922	0.0800	0.0815
			$ \pi/4 $							1.1918	1.8311	0.0818	0.0824
			$ \pi/3 $							1.2100	2.0138	0.0810	0.0821
			$ \pi/2 $							1.8610	2.0771	0.0802	0.0820
				0.09						2.1314	-	0.0801	-
				0.15						2.0229	-	0.0810	-
				0.18						1.1912	-	0.0818	-
					0.0					-	1.9597	-	0.0811
					0.06					-	1.9021	-	0.0818
					0.09					-	1.8311	-	0.0824
						0.0				2.0121	2.0065	0.0852	0.0858
						0.1				1.1918	1.8311	0.0818	0.0824
						0.2				1.1728	1.6216	0.0810	0.0820
							0.0			1.1918	1.8311	0.0818	0.0824
							0.2			1.1918	1.8311	0.0921	0.1269
							0.4			1.1918	1.8311	0.0996	0.2106
								0.2		1.1918	1.8311	0.0706	0.0580
								0.4		1.1918	1.8311	0.0818	0.0824
								0.6		1.1918	1.8311	0.0881	0.0829
									0.1	1.1918	1.8311	0.0828	0.0824
									0.2	1.1918	1.8311	0.013	0.0821
									0.6	1.1918	1.8311	0.0806	0.0815

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