Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System
Volume 4, Issue 1, March 2018, Pages: 1-13
Received: Oct. 18, 2017;
Accepted: Dec. 8, 2017;
Published: Jan. 15, 2018
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Mohammad Hassan Mohammadi, Institute of Mathematics, Department of Differential Equations, National Academy of Sciences of Armenia, Marshal Baghramyan Av., Yerevan, Armenia
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.
Mohammad Hassan Mohammadi,
Numerical Analysis of Fluid Flow and Heat Transfer Based on the Cylindrical Coordinate System, Fluid Mechanics.
Vol. 4, No. 1,
2018, pp. 1-13.
Ungan A., R. Viskanta, Three-dimensional Numerical Modeling of Circulation and Heat Transfer in a Glass Melting Tank. IEEE Transactions on Industry Applications, Vol. IA-22, No. 5, pp. 922–933, 1986.
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.
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.
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.
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.
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.
Alexiades V., A. D. Solomon, Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing Corporation, 1993.
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.
Rodrigues J. F., Variational Methods in the Stefan Problem, Lecture Notes in Mathematics, Springer-Verlag, pp. 147-212, 1994.
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.
Byron Bird R., Warren E. Stewart, Edwin N. Lightfoot, Transport Phenomena, John Wiley & Sons, Inc. 2nd Edition, 2002.
Irving H. Shames, Mechanics of Fluids, McGraw-Hill, 4th Edition, 2003.
Robert W. Fox, Alan T. McDonald, Philip J. Pritchard, Introduction to Fluid Mechanics, John Wiley & Sons, Inc. 6th Edition, 2004.
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.
Brenner S., R. Scott, the Mathematical Theory of Finite Element Methods. Springer Verlag, 1994. Corr. 2nd printing 1996.
Johnson C., Numerical Solution of Partial Differential Equations by the Finite Element Method. CUP, 1990.
Epperson J. F., an Introduction to Numerical Methods and Analysis, John Wiley & Sons, Inc. 2002.
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.
Mohammadi M. H., Three-Dimensional Mathematical Modeling of Heat Transfer by Stream Function and its Numerical Solution, Far East Journal of Mathematical Sciences (FJMS), Vol. 99, No. 7, pp. 969-981, 2016.
Babayan A. H., M. H. Mohammadi, the Mathematical Modeling of Garnissage Furnace, NPUA, Bulletin, Collection of Scientific Papers, Part 1, pp. 7-11, 2016.