Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System

Mohammad Hassan Mohammadi

doi:doi:10.11648/j.fm.20180401.11

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Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System

Mohammad Hassan Mohammadi

Published in Fluid Mechanics (Volume 4, Issue 1)

Received: 18 October 2017 Accepted: 8 December 2017 Published: 15 January 2018

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Abstract

In this work we will apply the three-dimensional mathematical modelling of fluid flow and heat transfer inside the furnaces based on the cylindrical coordinate system to describe the behavior of the transport phenomena. This modelling is constructed by using the mass, momentum, and energy conservation laws to achieve the continuity equation, the Navier-Stokes equations, and the energy conservation equation. Due to the moving boundary between the solid and melted materials inside of the furnaces we will impose the Stefan condition to describe the behavior of the free boundary between two phases. We will derive the variational formulation of the system of transport phenomena, then we will discrete the domain to complete the finite element method stages and we will obtain the system of nonlinear equations in 256 equations in 256 unknowns. To get the numerical solution of the large-scale system we will prepare a convenient mathematical work and gain some diagrams where they would be applicable in the design process of the furnaces shapes.

Published in	Fluid Mechanics (Volume 4, Issue 1)
DOI	10.11648/j.fm.20180401.11
Page(s)	1-13
Creative Commons	This is an Open Access article, 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 or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2018. Published by Science Publishing Group

Keywords

Fluid Flow, Heat Transfer, Mathematical Modeling, Stefan Condition, Cylindrical Coordinate

References

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[2]	Ungan A, R. Viskanta, Three-dimensional Numerical Simulation of Circulation and Heat Transfer in an Electrically Boosted Glass Melting Tank. Part. 2 Sample Simulations, Glastechnische Berichte, Vol. 60, No. 4, pp. 115–124, 1987.
[3]	Sadov S. YU., P. N. Shivakumar, D. Firsov, S. H. Lui, R. Thulasiram, Mathematical Model of Ice Melting on Transmission Lines, Journal of Mathematical Modeling and Algorithms, Vol. 6, No. 2, pp. 273-286, 2007.
[4]	Pilon L., G. Zhao, and R. Viskanta, Three-Dimensional Flow and Thermal Structures in Glass Melting Furnaces. Part I. eﬀects of the heat ﬂux distribution, Glass Science and Technology, Vol. 75, No. 2, pp. 55–68, 2002.
[5]	Pilon L., G. Zhao, and R. Viskanta, Three-Dimensional Flow and Thermal Structures in Glass Melting Furnaces. Part II. Eﬀect of Batch and Bubbles. Glass Science and Technology, Vol. 75, No. 3 pp. 115-124, 2006.
[6]	Choudhary Manoj K., Raj Venuturumilli, Matthew R. Hyre, Mathematical Modeling of Flow and Heat Transfer Phenomena in Glass Melting, Delivery, and Forming Processes. International Journal of Applied Glass Science, Vol. 1, No. 2, pp. 188–214, 2010.
[7]	Alexiades V., A. D. Solomon, Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing Corporation, 1993.
[8]	Henry Hu, Stavros A. Argyropoulos, Mathematical Modelling of Solidification and Melting: a Review, Modelling and Simulation in Materials Science and Engineering, Vol. 4, pp. 371-396, 1996.
[9]	Rodrigues J. F., Variational Methods in the Stefan Problem, Lecture Notes in Mathematics, Springer-Verlag, pp. 147-212, 1994.
[10]	Vuik C., A. Segal, F. J. Vermolen, a Conserving Discretization for a Stefan Problem with an Interface Reaction at the Free Boundary, Computing and Visualization in Science, Springer-Verlag, Vol. 3, pp. 109-114, 2000.
[11]	Byron Bird R., Warren E. Stewart, Edwin N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc. 2nd Edition, 2002.
[12]	Irving H. Shames, Mechanics of Fluids, McGraw-Hill, 4^th Edition, 2003.
[13]	Robert W. Fox, Alan T. McDonald, Philip J. Pritchard, Introduction to Fluid Mechanics, John Wiley & Sons, Inc. 6th Edition, 2004.
[14]	Xu Quan-Sheng, Zhu You-Lan, Solution of the Two-Dimensional Stefan Problem by the Singularity-Separating Method, Journal of Computational Mathematics, Vol. 3, No. 1, pp. 8-18, 1985.
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[17]	Epperson J. F., an Introduction to Numerical Methods and Analysis, John Wiley & Sons, Inc. 2002.
[18]	Mohammadi M. H., Mathematical Modeling of Heat Transfer and Transport Phenomena in Three-Dimension with Stefan Free Boundary, Advances and Applications in Fluid Mechanics, Vol. 19, No. 1, pp. 23-34, 2016.
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Mohammad Hassan Mohammadi. (2018). Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System. Fluid Mechanics, 4(1), 1-13. https://doi.org/10.11648/j.fm.20180401.11

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ACS Style

Mohammad Hassan Mohammadi. Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System. Fluid Mech. 2018, 4(1), 1-13. doi: 10.11648/j.fm.20180401.11

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AMA Style

Mohammad Hassan Mohammadi. Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System. Fluid Mech. 2018;4(1):1-13. doi: 10.11648/j.fm.20180401.11

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@article{10.11648/j.fm.20180401.11,
  author = {Mohammad Hassan Mohammadi},
  title = {Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System},
  journal = {Fluid Mechanics},
  volume = {4},
  number = {1},
  pages = {1-13},
  doi = {10.11648/j.fm.20180401.11},
  url = {https://doi.org/10.11648/j.fm.20180401.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.fm.20180401.11},
  abstract = {In this work we will apply the three-dimensional mathematical modelling of fluid flow and heat transfer inside the furnaces based on the cylindrical coordinate system to describe the behavior of the transport phenomena. This modelling is constructed by using the mass, momentum, and energy conservation laws to achieve the continuity equation, the Navier-Stokes equations, and the energy conservation equation. Due to the moving boundary between the solid and melted materials inside of the furnaces we will impose the Stefan condition to describe the behavior of the free boundary between two phases. We will derive the variational formulation of the system of transport phenomena, then we will discrete the domain to complete the finite element method stages and we will obtain the system of nonlinear equations in 256 equations in 256 unknowns. To get the numerical solution of the large-scale system we will prepare a convenient mathematical work and gain some diagrams where they would be applicable in the design process of the furnaces shapes.},
 year = {2018}
}

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TY  - JOUR
T1  - Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System
AU  - Mohammad Hassan Mohammadi
Y1  - 2018/01/15
PY  - 2018
N1  - https://doi.org/10.11648/j.fm.20180401.11
DO  - 10.11648/j.fm.20180401.11
T2  - Fluid Mechanics
JF  - Fluid Mechanics
JO  - Fluid Mechanics
SP  - 1
EP  - 13
PB  - Science Publishing Group
SN  - 2575-1816
UR  - https://doi.org/10.11648/j.fm.20180401.11
AB  - In this work we will apply the three-dimensional mathematical modelling of fluid flow and heat transfer inside the furnaces based on the cylindrical coordinate system to describe the behavior of the transport phenomena. This modelling is constructed by using the mass, momentum, and energy conservation laws to achieve the continuity equation, the Navier-Stokes equations, and the energy conservation equation. Due to the moving boundary between the solid and melted materials inside of the furnaces we will impose the Stefan condition to describe the behavior of the free boundary between two phases. We will derive the variational formulation of the system of transport phenomena, then we will discrete the domain to complete the finite element method stages and we will obtain the system of nonlinear equations in 256 equations in 256 unknowns. To get the numerical solution of the large-scale system we will prepare a convenient mathematical work and gain some diagrams where they would be applicable in the design process of the furnaces shapes.
VL  - 4
IS  - 1
ER  -

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Author Information

Mohammad Hassan Mohammadi

Institute of Mathematics, Department of Differential Equations, National Academy of Sciences of Armenia, Marshal Baghramyan Av., Yerevan, Armenia

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Mohammad Hassan Mohammadi. (2018). Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System. Fluid Mechanics, 4(1), 1-13. https://doi.org/10.11648/j.fm.20180401.11

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ACS Style

Mohammad Hassan Mohammadi. Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System. Fluid Mech. 2018, 4(1), 1-13. doi: 10.11648/j.fm.20180401.11

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AMA Style

Mohammad Hassan Mohammadi. Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System. Fluid Mech. 2018;4(1):1-13. doi: 10.11648/j.fm.20180401.11

Copy | Download

@article{10.11648/j.fm.20180401.11,
  author = {Mohammad Hassan Mohammadi},
  title = {Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System},
  journal = {Fluid Mechanics},
  volume = {4},
  number = {1},
  pages = {1-13},
  doi = {10.11648/j.fm.20180401.11},
  url = {https://doi.org/10.11648/j.fm.20180401.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.fm.20180401.11},
  abstract = {In this work we will apply the three-dimensional mathematical modelling of fluid flow and heat transfer inside the furnaces based on the cylindrical coordinate system to describe the behavior of the transport phenomena. This modelling is constructed by using the mass, momentum, and energy conservation laws to achieve the continuity equation, the Navier-Stokes equations, and the energy conservation equation. Due to the moving boundary between the solid and melted materials inside of the furnaces we will impose the Stefan condition to describe the behavior of the free boundary between two phases. We will derive the variational formulation of the system of transport phenomena, then we will discrete the domain to complete the finite element method stages and we will obtain the system of nonlinear equations in 256 equations in 256 unknowns. To get the numerical solution of the large-scale system we will prepare a convenient mathematical work and gain some diagrams where they would be applicable in the design process of the furnaces shapes.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System
AU  - Mohammad Hassan Mohammadi
Y1  - 2018/01/15
PY  - 2018
N1  - https://doi.org/10.11648/j.fm.20180401.11
DO  - 10.11648/j.fm.20180401.11
T2  - Fluid Mechanics
JF  - Fluid Mechanics
JO  - Fluid Mechanics
SP  - 1
EP  - 13
PB  - Science Publishing Group
SN  - 2575-1816
UR  - https://doi.org/10.11648/j.fm.20180401.11
AB  - In this work we will apply the three-dimensional mathematical modelling of fluid flow and heat transfer inside the furnaces based on the cylindrical coordinate system to describe the behavior of the transport phenomena. This modelling is constructed by using the mass, momentum, and energy conservation laws to achieve the continuity equation, the Navier-Stokes equations, and the energy conservation equation. Due to the moving boundary between the solid and melted materials inside of the furnaces we will impose the Stefan condition to describe the behavior of the free boundary between two phases. We will derive the variational formulation of the system of transport phenomena, then we will discrete the domain to complete the finite element method stages and we will obtain the system of nonlinear equations in 256 equations in 256 unknowns. To get the numerical solution of the large-scale system we will prepare a convenient mathematical work and gain some diagrams where they would be applicable in the design process of the furnaces shapes.
VL  - 4
IS  - 1
ER  -

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