Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties
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Ryoichi Chiba
Abstract
The transient natural convection of a viscous fluid in a heated vertical tube is studied using the two-dimensional differential transform method (DTM). A time-dependent Dirichlet boundary condition is imposed for tube wall temperature. The partial differential equations for the velocity and temperature fields within the tube are solved by the DTM while considering temperature-dependent viscosity and thermal conductivity of the fluid. As a result, tractable solutions in double-series form are derived for the temperature and flow velocity. The transformed functions included in the solutions are obtained through a simple recursive procedure. Numerical results illustrate the effects of temperature-dependent properties on transient temperature and flow behaviour, including the Nusselt number and volumetric flow rate. The DTM gives accurate series solutions without any special functions for nonlinear transient heat transfer problems which are advantageous in finding the derivative or integral.
References
[1] R. J. Goldstein and D. G. Briggs, J. Heat Transfer 86, 490 (1964).10.1115/1.3688728Suche in Google Scholar
[2] M. A. Al-Nimr, Int. J. Heat Mass Transfer 36, 2385 (1993).10.1016/S0017-9310(05)80122-XSuche in Google Scholar
[3] M. A. Al-Nimr and M. A. I. El-Shaarawi, Heat Mass Transfer 30, 241 (1995).10.1007/BF01602770Suche in Google Scholar
[4] B. K. Jha, A. K. Samaila, and A. O. Ajibade, Math. Comput. Model. 54, 2880 (2011).10.1016/j.mcm.2011.07.008Suche in Google Scholar
[5] K. T. Yang, J. Appl. Mech. 27, 230 (1960).Suche in Google Scholar
[6] T. H. Schwab and K. J. De Witt, AIChE J. 16, 1005 (1970).10.1002/aic.690160624Suche in Google Scholar
[7] M. A. I. El-Shaarawi and M. Al-Attas, JSME Int. J. 36b, 156 (1993).10.1299/jsmeb.36.156Suche in Google Scholar
[8] B. L. Kuo, Appl. Math. Comput. 165, 63 (2005).10.1016/j.amc.2004.04.090Suche in Google Scholar
[9] C. K. Chen, H. Y. Lai, and C. C. Liu, Int. Commun. Heat Mass Transfer 38, 285 (2011).10.1016/j.icheatmasstransfer.2010.12.016Suche in Google Scholar
[10] H. A. Peker and G. Oturanc, arXiv 1212.1706 (2012) 11 pages.Suche in Google Scholar
[11] R. Chiba, Int. J. Thermophys. 33, 363 (2012).10.1007/s10765-011-1153-1Suche in Google Scholar
[12] M. Hatami and D. D. Ganji, Case Studies Thermal Eng. 2, 14 (2014).10.1016/j.csite.2013.11.001Suche in Google Scholar
[13] M. Hatami, J. Hatami, M. Jafaryar, and G. Domairry, J. Brazil. Soc. Mech. Sci. Eng. (2015) in press.Suche in Google Scholar
[14] J. C. Umavathi and M. Shekar, Meccanica (2015) in press.Suche in Google Scholar
[15] P. L. Ndlovu and R. J. Moitsheki, Commun. Nonlinear Sci. Numer. Simul. 18, 2689 (2013).10.1016/j.cnsns.2013.02.019Suche in Google Scholar
[16] R. Chiba, Abst. Appl. Anal. 2014 (2014), Article ID 684293.Suche in Google Scholar
[17] R. Chiba, Appl. Mech. Mater. 627, 145 (2014).10.4028/www.scientific.net/AMM.627.145Suche in Google Scholar
[18] R. Chiba, Nonlinear Eng. 3, 215 (2014).10.1515/nleng-2014-0020Suche in Google Scholar
[19] M. J. Jang, C. L. Chen, and Y. C. Liu, Appl. Math. Comput. 121, 261 (2001).10.1016/S0096-3003(99)00293-3Suche in Google Scholar
[20] M. J. Jang, Y. L. Yeh, C. L. Chen, and W. C. Yeh, Appl. Math. Comput. 216, 2339 (2010).10.1016/j.amc.2010.03.079Suche in Google Scholar
[21] E. R. Menold and K. T. Yang, J. Appl. Mech. 29, 124 (1962).10.1115/1.3636443Suche in Google Scholar
[22] I. Pop and T. Y. Na, Warme-und Stoffubertragung 18, 37 (1984).10.1007/BF01461488Suche in Google Scholar
[23] Y. Joshi and B. Gebhart, Int. J. Heat Mass Transfer 27, 1573 (1984).10.1016/0017-9310(84)90269-2Suche in Google Scholar
[24] A. K. Singh, Defence Sci. J. 38, 35 (1988).10.14429/dsj.38.4823Suche in Google Scholar
[25] T. Paul, B. K. Jha, and A. K. Singh, Heat Mass Transfer 32, 61 (1996).10.1007/s002310050092Suche in Google Scholar
[26] B. K. Jha and A. O. Ajibade, J. Process Mech. Eng. 224, 247 (2010).10.1243/09544089JPME319Suche in Google Scholar
[27] K. Mizukami, Int. J. Heat Mass Transfer 20, 981 (1977).10.1016/0017-9310(77)90069-2Suche in Google Scholar
[28] Rizwan-uddin, Numer. Heat Transfer 33b, 269 (1998).10.1080/10407799808915033Suche in Google Scholar
[29] D. Holdsworth and R. Simpson, Thermal Processing of Packaged Foods, Springer, New York 2007.10.1007/978-0-387-72250-4Suche in Google Scholar
[30] M. Mishra, P. K. Das, and S. Sarangi, Int. J. Heat Mass Transfer 51, 2583 (2008).10.1016/j.ijheatmasstransfer.2007.07.054Suche in Google Scholar
[31] Q. Rubbab, D. Vieru, C. Fetecau, and C. Fetecau, PLoS ONE 8, e78352 (2013).10.1371/journal.pone.0078352Suche in Google Scholar PubMed PubMed Central
[32] R. K. Deka, N. Kalita, and A. Paul, Proc. Int. Conf. Frontiers Math., Guwahati (2015).Suche in Google Scholar
[33] W. T. Lawrence and J. C. Chato, J. Heat Transfer 88, 214 (1966).10.1115/1.3691518Suche in Google Scholar
[34] K. E. Torrance and D. L. Turcotte, J. Fluid Mech. 47, 113 (1971).10.1017/S002211207100096XSuche in Google Scholar
[35] S. H. Chang and I. L. Chang, Appl. Math. Comput. 215, 2486 (2009).10.1016/j.amc.2009.08.046Suche in Google Scholar
[36] M. N. Ozisik, Boundary Value Problems of Heat Conduction, Dover, New York 1989.Suche in Google Scholar
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Artikel in diesem Heft
- Frontmatter
- Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation
- The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations
- Application of the Reverberation-Ray Matrix to the Non-Fourier Heat Conduction in Functionally Graded Materials
- Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations
- Structural, Electronic, Magnetic and Optical Properties of Ni,Ti/Al-based Heusler Alloys: A First-Principles Approach
- Ab Initio Calculations on the Structural, Mechanical, Electronic, Dynamic, and Optical Properties of Semiconductor Half-Heusler Compound ZrPdSn
- Qualitative Behaviour of Generalised Beddington Model
- Studying Nuclear Level Densities of 238U in the Nuclear Reactions within the Macroscopic Nuclear Models
- Total π-Electron Energy of Conjugated Molecules with Non-bonding Molecular Orbitals
- Investigation of Thermal Expansion and Physical Properties of Carbon Nanotube Reinforced Nanocrystalline Aluminum Nanocomposite
- Bistable Bright Optical Spatial Solitons due to Charge Drift and Diffusion of Various Orders in Photovoltaic Photorefractive Media Under Closed-Circuit Conditions
- Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties
Artikel in diesem Heft
- Frontmatter
- Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation
- The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations
- Application of the Reverberation-Ray Matrix to the Non-Fourier Heat Conduction in Functionally Graded Materials
- Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations
- Structural, Electronic, Magnetic and Optical Properties of Ni,Ti/Al-based Heusler Alloys: A First-Principles Approach
- Ab Initio Calculations on the Structural, Mechanical, Electronic, Dynamic, and Optical Properties of Semiconductor Half-Heusler Compound ZrPdSn
- Qualitative Behaviour of Generalised Beddington Model
- Studying Nuclear Level Densities of 238U in the Nuclear Reactions within the Macroscopic Nuclear Models
- Total π-Electron Energy of Conjugated Molecules with Non-bonding Molecular Orbitals
- Investigation of Thermal Expansion and Physical Properties of Carbon Nanotube Reinforced Nanocrystalline Aluminum Nanocomposite
- Bistable Bright Optical Spatial Solitons due to Charge Drift and Diffusion of Various Orders in Photovoltaic Photorefractive Media Under Closed-Circuit Conditions
- Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties
