Modelling Study on Internal Energy Loss Due to Entropy Generation for Non-Darcy Poiseuille Flow of Silver-Water Nanofluid: An Application of Purification
<p>Poiseuille flow model of nanofluid.</p> "> Figure 2
<p>Residual error of (<b>a</b>) velocity and (<b>b</b>) temperature profiles.</p> "> Figure 3
<p>The impact of <math display="inline"><semantics> <mi>M</mi> </semantics></math> on (<b>a</b>) velocity and (<b>b</b>) temperature profiles.</p> "> Figure 4
<p>The impact of Darcy number on (<b>a</b>) velocity and (<b>b</b>) temperature profiles.</p> "> Figure 5
<p>The impact of Forchheimer number on (<b>a</b>) velocity and (<b>b</b>) temperature profiles.</p> "> Figure 6
<p>The impact of Brinkman number on (<b>a</b>) velocity and (<b>b</b>) temperature profiles.</p> "> Figure 7
<p>The impact of <math display="inline"><semantics> <mi>M</mi> </semantics></math> on (<b>a</b>) entropy generation and (<b>b</b>) Bejan number.</p> "> Figure 8
<p>The impact of Darcy number on (<b>a</b>) entropy generation and (<b>b</b>) Bejan number.</p> "> Figure 9
<p>The impact of Forchheimer number on (<b>a</b>) entropy generation and (<b>b</b>) Bejan number.</p> "> Figure 10
<p>The impact of <math display="inline"><semantics> <mrow> <mrow> <mrow> <mi>B</mi> <mi>r</mi> </mrow> <mo>/</mo> <mi mathvariant="sans-serif">Ω</mi> </mrow> </mrow> </semantics></math> on (<b>a</b>) entropy generation and (<b>b</b>) Bejan number.</p> "> Figure 11
<p>Average breakdown in entropy generation for different <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>1.0</mn> </mrow> </semantics></math>.</p> "> Figure 12
<p>Average breakdown in entropy generation due to <math display="inline"><semantics> <mi>M</mi> </semantics></math> at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>1.0</mn> </mrow> </semantics></math>.</p> "> Figure 13
<p>Average breakdown in entropy generation due to <math display="inline"><semantics> <mrow> <mi>D</mi> <mi>a</mi> </mrow> </semantics></math> at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>1.0</mn> </mrow> </semantics></math>.</p> "> Figure 13 Cont.
<p>Average breakdown in entropy generation due to <math display="inline"><semantics> <mrow> <mi>D</mi> <mi>a</mi> </mrow> </semantics></math> at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>1.0</mn> </mrow> </semantics></math>.</p> "> Figure 14
<p>Average breakdown in entropy generation due to <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>*</mo> </mrow> </semantics></math> at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>1.0</mn> </mrow> </semantics></math>.</p> "> Figure 15
<p>Average breakdown in entropy generation for <math display="inline"><semantics> <mrow> <mrow> <mrow> <mi>B</mi> <mi>r</mi> </mrow> <mo>/</mo> <mi mathvariant="sans-serif">Ω</mi> </mrow> </mrow> </semantics></math> at (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mo>−</mo> <mn>1.0</mn> </mrow> </semantics></math>.</p> ">
Abstract
:1. Introduction
2. Formulation
3. Entropy Generation Analysis
4. Analytic Solution
5. Convergence Analysis
6. Results and Discussion
7. Conclusions
- It is noticed that velocity gives the reduction flow map with increasing values of magnetic field and non-Darcy (Forchheimer) parameter, while velocity increases for large values of Darcy and Brinkman number.
- Temperature distribution increases for increasing values of and non-Darcy (Forchheimer) . On the other hand, the temperature profile decreases for various values of Darcy and Brinkman number .
- Energy loss due to entropy generation becomes stronger along the walls of the channel for the magnetic field and non-Darcy (Forchheimer) parameter , and near the center of the channel energy loss becomes zero for said parameters.
- Energy loss due to entropy generation becomes weaker at the upper wall as compared to the lower wall of the channel for Darcy number , and group parameter is also negligible near the middle of the channel.
- The Bejan number at the center of the channel attained maximum value when the magnetic field was neglected, and gained extreme value when group parameter was zero. Moreover, the Bejan number accelerated at boundaries with a large value of Darcy number and at the center of the channel increased with non-Darcy (Forchheimer) parameter.
- Non-Darcy porous media irreversibility in the average break of energy loss due to entropy generation was enhanced with enhancing nanoparticle volume fraction , non-Darcy (Forchheimer) parameter , and group parameter , but the reduction in non-Darcy porous media irreversibility was due to magnetic field parameter and Darcy number .
- A rise in entropy was evident due to an increase in the pressure gradient.
Author Contributions
Funding
Conflicts of Interest
Appendix A
References
- Darcy, H. Les Fontaines Publiques de la Ville de Dijon: Exposition et Application; Victor Dalmont: Paris, France, 1856. [Google Scholar]
- Forchheimer, P. Wasserbewegung durch boden. Z. Ver. Dtsch. Ing. 1901, 45, 1782–1788. [Google Scholar]
- Lee, S.; Yang, J. Modeling of darcy-forchheimer drag for fluid flow across a bank of circular cylinders. Int. J. Heat Mass Transf. 1997, 40, 3149–3155. [Google Scholar] [CrossRef]
- Jeong, N.; Choi, D.H.; Lin, C.-L. Prediction of darcy–forchheimer drag for micro-porous structures of complex geometry using the lattice Boltzmann method. J. Micromech. Microeng. 2006, 16, 2240. [Google Scholar] [CrossRef]
- Chol, S.U.S.; Eastman, J.A. Enhancing Thermal Conductivity of Fluids with Nanoparticles. ASME Publ. Fed. 1995, 231, 99–106. [Google Scholar]
- Darbari, B.; Rashidi, S.; Abolfazli Esfahani, J. Sensitivity analysis of entropy generation in nanofluid flow inside a channel by response surface methodology. Entropy 2016, 18, 52. [Google Scholar] [CrossRef]
- Tripathi, D.; Bég, O.A. A study on peristaltic flow of nanofluids: Application in drug delivery systems. Int. J. Heat Mass Transf. 2014, 70, 61–70. [Google Scholar] [CrossRef]
- Selimefendigil, F.; Öztop, H.F.; Abu-Hamdeh, N. Natural convection and entropy generation in nanofluid filled entrapped trapezoidal cavities under the influence of magnetic field. Entropy 2016, 18, 43. [Google Scholar] [CrossRef]
- Gireesha, B.; Mahanthesh, B.; Thammanna, G.; Sampathkumar, P. Hall effects on dusty nanofluid two-phase transient flow past a stretching sheet using KVL model. J. Mol. Liq. 2018, 256, 139–147. [Google Scholar] [CrossRef]
- Bhatti, M.M.; Abbas, T.; Rashidi, M.M.; Ali, M.E.-S. Numerical simulation of entropy generation with thermal radiation on MHD carreau nanofluid towards a shrinking sheet. Entropy 2016, 18, 200. [Google Scholar] [CrossRef]
- Mahanthesh, B.; Gireesha, B.; Shashikumar, N.; Shehzad, S. Marangoni convective MHD flow of SWCNT and MWCNT nanoliquids due to a disk with solar radiation and irregular heat source. Phys. E Low-Dimens. Syst. Nanostruct. 2017, 94, 25–30. [Google Scholar] [CrossRef]
- Feng, Q.L.; Wu, J.; Chen, G.; Cui, F.; Kim, T.; Kim, J. A mechanistic study of the antibacterial effect of silver ions on Escherichia coli and Staphylococcus aureus. J. Biomed. Mater. Res. 2000, 52, 662–668. [Google Scholar] [CrossRef]
- Oyanedel-Craver, V.A.; Smith, J.A. Sustainable colloidal-silver-impregnated ceramic filter for point-of-use water treatment. Environ. Sci. Technol. 2007, 42, 927–933. [Google Scholar] [CrossRef]
- Brown, J.; Sobsey, M.D. Microbiological effectiveness of locally produced ceramic filters for drinking water treatment in Cambodia. J. Water Health 2010, 8, 1–10. [Google Scholar] [CrossRef] [PubMed]
- Van Halem, D.; Van der Laan, H.; Heijman, S.; Van Dijk, J.; Amy, G. Assessing the sustainability of the silver-impregnated ceramic pot filter for low-cost household drinking water treatment. Phys. Chem. Earth Parts A/B/C 2009, 34, 36–42. [Google Scholar] [CrossRef]
- Shahverdi, A.R.; Fakhimi, A.; Shahverdi, H.R.; Minaian, S. Synthesis and effect of silver nanoparticles on the antibacterial activity of different antibiotics against Staphylococcus aureus and Escherichia coli. Nanomed. Nanotechnol. Boil. Med. 2007, 3, 168–171. [Google Scholar] [CrossRef] [PubMed]
- Godson, L.; Raja, B.; Mohan Lal, D.; Wongwises, S. Experimental investigation on the thermal conductivity and viscosity of silver-Deionized water nanofluid. Exp. Heat Transf. 2010, 23, 317–332. [Google Scholar] [CrossRef]
- Rashidi, M.M.; Freidoonimehr, N. Analysis of entropy generation in MHD stagnation-point flow in porous media with heat transfer. Int. J. Comput. Methods Eng. Sci. Mech. 2014, 15, 345–355. [Google Scholar] [CrossRef]
- Bhatti, M.M.; Abbas, T.; Rashidi, M.M.; Ali, M.E.-S.; Yang, Z. Entropy generation on MHD Eyring–Powell nanofluid through a permeable stretching surface. Entropy 2016, 18, 224. [Google Scholar] [CrossRef]
- Marin, M. An approach of a heat-flux dependent theory for micropolar porous media. Meccanica 2016, 51, 1127–1133. [Google Scholar] [CrossRef]
- Othman, M.I.; Marin, M. Effect of thermal loading due to laser pulse on thermoelastic porous medium under GN theory. Results Phys. 2017, 7, 3863–3872. [Google Scholar] [CrossRef]
- Abbas, T.; Ayub, M.; Bhatti, M.M.; Rashidi, M.M.; Ali, M.E.-S. Entropy generation on nanofluid flow through a horizontal riga plate. Entropy 2016, 18, 223. [Google Scholar] [CrossRef]
- Qasim, M.; Hayat Khan, Z.; Khan, I.; Al-Mdallal, Q.M. Analysis of entropy generation in flow of methanol-based nanofluid in a sinusoidal wavy channel. Entropy 2017, 19, 490. [Google Scholar] [CrossRef]
- Ellahi, R.; Zeeshan, A.; Shehzad, N.; Alamri, S.Z. Structural impact of kerosene-Al2O3 nanoliquid on MHD poiseuille flow with variable thermal conductivity: Application of cooling process. J. Mol. Liq. 2018, 264, 607–615. [Google Scholar] [CrossRef]
- Ellahi, R.; Alamri, S.Z.; Basit, A.; Majeed, A. Effects of MHD and slip on heat transfer boundary layer flow over a moving plate based on specific entropy generation. J. Taibah Univ. Sci. 2018, 12, 476–482. [Google Scholar] [CrossRef]
- Zeeshan, A.; Ijaz, N.; Abbas, T.; Ellahi, R. The sustainable characteristic of Bio-bi-phase flow of peristaltic transport of MHD Jeffery fluid in human body. Sustainability 2018, 10, 2671. [Google Scholar] [CrossRef]
- Rashidi, S.; Esfahani, J.A.; Ellahi, R. Convective heat transfer and particle motion in an obstructed duct with two side by side obstacles by means of DPM model. Appl. Sci. 2017, 7, 431. [Google Scholar] [CrossRef]
- Moshizi, S.A. Forced convection heat and mass transfer of MHD nanofluid flow inside a porous microchannel with chemical reaction on the walls. Eng. Comput. 2015, 32, 2419–2442. [Google Scholar] [CrossRef]
- Matin, M.H.; Pop, I. Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall. Int. Commun. Heat Mass Transf. 2013, 46, 134–141. [Google Scholar] [CrossRef]
- Nield, D.A.; Bejan, A. Mechanics of fluid flow through a porous medium. In Convection in Porous Media; Springer: New York, NY, USA, 2013; pp. 1–29. [Google Scholar]
- Tiwari, R.K.; Das, M.K. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 2007, 50, 2002–2018. [Google Scholar] [CrossRef]
- Lauriat, G.; Prasad, V. Non-darcian effects on natural convection in a vertical porous enclosure. Int. J. Heat Mass Transf. 1989, 32, 2135–2148. [Google Scholar] [CrossRef]
- Liao, S. Beyond Perturbation: Introduction to the Homotopy Analysis Method; CRC Press: Boca Raton, FL, USA, 2003. [Google Scholar]
- Einstein, A. Eine neue bestimmung der molekuldimensionen. Ann. Phys. 1911, 34, 591–592. [Google Scholar] [CrossRef]
- Maxwell, J.C. A Treatise on Electricity and Magnetism; Clarendon Press: Oxford, UK, 1881; Volume 1. [Google Scholar]
Property | Water (H2O) | Silver (Ag) |
---|---|---|
9.877 × 102 | 10,500 | |
4.066 × 103 | 235 | |
5.0 × 10−2 | 6.30 × 107 | |
6.44 × 10−1 | 429 |
0.3% | 1.0065 | 1.0286 | 0.0090 | 0.9998 | 0.9988 | 0.9537 |
0.6% | 1.0080 | 1.0572 | 0.0181 | 0.9997 | 0.9976 | 0.9566 |
0.9% | 1.0095 | 1.0858 | 0.0272 | 0.9995 | 0.9963 | 0.9595 |
Order of Approximation | Time | ||
---|---|---|---|
05 | 8.2651 | 1.3340 × 10−3 | 2.3980 × 10−3 |
10 | 35.1732 | 7.4001 × 10−5 | 3.2385 × 10−6 |
15 | 67.9793 | 1.5624 × 10−8 | 3.5705 × 10−9 |
20 | 187.6291 | 1.6199 × 10−12 | 4.7723 × 10−14 |
25 | 296.1218 | 1.7193 × 10−16 | 1.7037 × 10−17 |
0.5 | 0.0 | 0.3111 | 0.6329 | 1.3898 | −1.069 |
0.5 | 0.3836 | 0.5634 | 1.2572 | −0.9680 | |
1.0 | 0.3928 | 0. 5483 | 1.0021 | −0.7155 | |
1.5 | 0.5067 | 0. 4428 | 0.7424 | −0.4595 | |
1.0 | 0.0 | 0.6497 | 0.3028 | 1.5772 | −1.2715 |
0.5 | 0.6854 | 0.2655 | 1.5600 | −1.2552 | |
1.0 | 0.8509 | 0.0987 | 1.6243 | −1.3209 | |
1.5 | 1.1791 | −0.2298 | 1.9714 | −1.6666 | |
2.0 | 0.0 | 0.7814 | 0.1738 | 1.7266 | −1.4126 |
0.5 | 0.9048 | 0.0496 | 1.8225 | −1.5089 | |
1.0 | 1.1602 | −0.2057 | 2.1024 | −1.7891 | |
1.5 | 1.5454 | −0.5894 | 2.7338 | −2.4202 | |
10.0 | 0.0 | 0.9064 | 0.0517 | 1.8619 | −1.5410 |
0.5 | 1.0892 | −0.1309 | 2.0655 | −1.7442 | |
1.0 | 1.3881 | −0.4285 | 2.5071 | −2.1850 | |
1.5 | 1.7725 | −0.8098 | 3.2985 | −2.9743 |
0.2 | 0 | 0.4768 | 0.4768 | 1.9264 | −1.8003 |
1 | 1.1824 | −0.2266 | 1.8695 | −1.7435 | |
2 | 1.8639 | −0.9062 | 1.8104 | −1.6846 | |
3 | 2.5192 | −1.5601 | 1.7491 | −1.6234 | |
0.5 | 0 | 0.4768 | 0.4768 | 2.2683 | −1.9541 |
1 | 1.1602 | −0.2057 | 2.1024 | −1.7891 | |
2 | 1.7575 | −0.8043 | 1.9234 | −1.6110 | |
3 | 2.2464 | −1.3067 | 1.7315 | −1.4199 | |
0.7 | 0 | 0.4768 | 0.4768 | 2.4907 | −2.0519 |
1 | 1.1301 | −0.1795 | 2.2375 | −1.8004 | |
2 | 1.6391 | −0.6957 | 1.9601 | −1.5246 | |
3 | 1.9802 | −1.0481 | 1.6583 | −1.2244 | |
1.0 | 0 | 0.4768 | 0.4768 | 2.8159 | −2.1914 |
1 | 1.0622 | −0.1220 | 2.4125 | −1.7912 | |
2 | 1.3923 | −0.4735 | 1.9632 | −1.3449 | |
3 | 1.4183 | −0.5289 | 1.4679 | −0.8527 |
Einstein [34] | Godson et al. [17] | Absolute Error | |||||
---|---|---|---|---|---|---|---|
At Lower Wall | At Upper Wall | ||||||
0.0% | 0.5 | 2.1334 | −1.8122 | 2.1247 | −1.8050 | 0.0087 | 0.0072 |
1.0 | 0.8328 | −0.5086 | 0.8404 | −0.5157 | 0.0076 | 0.0071 | |
1.5 | −1.4790 | 1.8133 | −1.4429 | 1.7756 | 0.0361 | 0.0377 | |
2.0 | −4.7799 | 5.1249 | −4.6860 | 5.0293 | 0.0939 | 0.0956 | |
0.3% | 0.5 | 2.1395 | −1.8206 | 2.1419 | −1.8227 | 0.0024 | 0.0021 |
1.0 | 0.8417 | −0.5175 | 0.8461 | −0.5217 | 0.0044 | 0.0042 | |
1.5 | −1.4898 | 1.8224 | −1.4752 | 1.8081 | 0.0146 | 0.0143 | |
2.0 | −4.8487 | 5.1925 | −4.8034 | 5.1474 | 0.0453 | 0.0451 | |
0.6% | 0.5 | 2.1453 | −1.8287 | 2.1601 | −1.8414 | 0.0148 | 0.0127 |
1.0 | 0.8485 | −0.5265 | 0.8523 | −0.5281 | 0.0038 | 0.0016 | |
1.5 | −1.5010 | 1.8318 | −1.5088 | 1.8418 | 0.0078 | 0.0100 | |
2.0 | −4.9202 | 5.2628 | −4.9270 | 5.2718 | 0.0068 | 0.0090 | |
0.9% | 0.5 | 2.1508 | −1.8364 | 2.1791 | −1.8608 | 0.0283 | 0.0244 |
1.0 | 0.8554 | −0.5354 | 0.8590 | −0.5350 | 0.0036 | 0.0004 | |
1.5 | −1.5125 | 1.8416 | −1.5438 | 1.8770 | 0.0313 | 0.0354 | |
2.0 | −4.9945 | 5.3359 | −5.0571 | 5.4026 | 0.0626 | 0.0667 |
Maxwell Model [35] | Godson et al. [17] | Absolute Error | |||||
---|---|---|---|---|---|---|---|
At Lower Wall | At Upper Wall | ||||||
0.0% | 0.5 | 1.2131 | −0.1831 | 1.2367 | −0.2059 | 0.0236 | 0.0228 |
1.0 | 1.2806 | −0.2283 | 1.2991 | −0.2457 | 0.0185 | 0.0174 | |
1.5 | 0.2233 | 0.8759 | 0.2110 | 0.8901 | 0.0123 | 0.0142 | |
2.0 | −5.4104 | 6.5961 | −5.5297 | 6.7189 | 0.1193 | 0.1228 | |
0.3% | 0.5 | 1.2166 | −0.1873 | 1.2528 | −0.2221 | 0.0362 | 0.0348 |
1.0 | 1.2924 | −0.2405 | 1.3144 | −0.2606 | 0.0220 | 0.0201 | |
1.5 | 0.2174 | 0.8820 | 0.1745 | 0.9279 | 0.0429 | 0.0459 | |
2.0 | −5.5766 | 6.7639 | −5.8279 | 7.0202 | 0.2513 | 0.2563 | |
0.6% | 0.5 | 1.2199 | −0.1913 | 1.2693 | −0.2388 | 0.0494 | 0.0475 |
1.0 | 1.3042 | −0.2528 | 1.3297 | −0.2757 | 0.0255 | 0.0229 | |
1.5 | 0.2109 | 0.8886 | 0.1353 | 0.9683 | 0.0756 | 0.0797 | |
2.0 | −5.7487 | 6.9375 | −6.1410 | 7.3365 | 0.3923 | 0.3990 | |
0.9% | 0.5 | 1.2230 | −0.1951 | 1.2863 | −0.2558 | 0.0633 | 0.0607 |
1.0 | 1.3161 | −0.2652 | 1.3452 | −0.2909 | 0.0291 | 0.0257 | |
1.5 | 0.2040 | 0.8957 | 0.09331 | 1.0115 | 0.1107 | 0.1158 | |
2.0 | −5.9269 | 7.1172 | −6.4696 | 7.6682 | 0.5427 | 0.5510 |
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Shehzad, N.; Zeeshan, A.; Ellahi, R.; Rashidi, S. Modelling Study on Internal Energy Loss Due to Entropy Generation for Non-Darcy Poiseuille Flow of Silver-Water Nanofluid: An Application of Purification. Entropy 2018, 20, 851. https://doi.org/10.3390/e20110851
Shehzad N, Zeeshan A, Ellahi R, Rashidi S. Modelling Study on Internal Energy Loss Due to Entropy Generation for Non-Darcy Poiseuille Flow of Silver-Water Nanofluid: An Application of Purification. Entropy. 2018; 20(11):851. https://doi.org/10.3390/e20110851
Chicago/Turabian StyleShehzad, Nasir, Ahmed Zeeshan, Rahmat Ellahi, and Saman Rashidi. 2018. "Modelling Study on Internal Energy Loss Due to Entropy Generation for Non-Darcy Poiseuille Flow of Silver-Water Nanofluid: An Application of Purification" Entropy 20, no. 11: 851. https://doi.org/10.3390/e20110851