WSEAS Transactions on Heat and Mass Transfer

Print ISSN: 1790-5044
E-ISSN: 2224-3461

Volume 12, 2017

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.

Several Intensive Steel Quenching and Wave Power Models

AUTHORS: Andris Buikis, Margarita Buike, Raimonds Vilums

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ABSTRACT: In this paper we develop mathematical models for 3-D, 2-D and one-dimensional hyperbolic heat equations (wave equation or telegraph equation) and construct their analytical solutions for the determination of the initial heat flux for rectangular samples. In some cases we give expression of wave energy. Some solutions of time inverse problems are obtained in the form of first kind Fredholm integral equation, but others has been obtained in closed analytical form. Finally, writes in one dimension intensive steel quenching model numerical results. We viewed both direct and inverse problems at the time. Are given some of the wave energy results.

KEYWORDS: Hyperbolic Equation, Ocean Energy, Steel Quenching, Green Function, Exact Solution, Inverse Problem, Fredholm integral equation,Series.


[1] Kobasko N.I. Steel Quenching in Liquid Media under Pressure. – Kyiv, Naukova Dumka, 1980.

[2] Kobasko, N. I. Intensive Steel Quenching Methods, Handbook “Theory and Technology of Quenching”, Springer-Verlag, 1992.

[3] Totten G.E., Bates C.E., and Clinton N.A. Handbook of Quenchants and Quenching Technology. ASM International, 1993.

[4] Aronov M.A., Kobasko N., Powell J.A. Intensive Quenching of Carburized Steel Parts, IASME Transactions, Issue 9, Vol. 2, November 2005, p. 1841-1845.

[5] Kobasko N.I. Self-regulated thermal processes during quenching of steels in liquid media. – International Journal of Microstructure and Materials Properties, Vol. 1, No 1, 2005, p. 110- 125.

[6] Buikis A., Guseinov Sh. Solution of Reverse Hyperbolic Heat equation for intensive carburized steel quenching. Proceedings of ICCES’05 (Advances in Computational and Experimental Engineering and Sciences, December 1-6, 2005, Madras, India. 6p.

[7] Buike M., Buikis A. Approximate Solutions of Heat Conduction Problems in Multi- Dimensional Cylinder Type Domain by Conservative Averaging Method, Part 1. Proceedings of the 5th IASME/WSEAS Int. Conf. on Heat Transfer, Thermal Engineering and Environment, Vouliagmeni, Athens, August 25 -27, 2007, p. 15 – 20.

[8] Buike M., Buikis A. Hyperbolic heat equation as mathematical model for steel quenching of Lshape samples, Part 1 (Direct Problem). Applied and Computational Mathematics. Proceedings of the 13th WSEAS International Conference on Applied Mathematics (MATH’08), Puerto De La Cruz, Tenerife, Canary Islands, Spain, December 15-17, 2008. WSEAS Press, 2008. p. 198-203.

[9] Buike M., Buikis A. Several Intensive Steel Quenching Models for Rectangular Samples. Proceedings of NAUN/WSEAS International Conference on Fluid Mechanics and Heat&Mass Transfer, Corfu Island, Greece, July 22-24, 2010. p.88-93.

[10] Bobinska T., Buike M., Buikis A. Hyperbolic Heat Equation as Mathematical Model for Steel Quenching of L-Shape Samples, Part 2 (Inverse Problem). Proceedings of 5th IASME/WSEAS International Conference on Continuum Mechanics (CM’10), University of Cambridge, UK, February 23-25, 2010. p. 21-26.

[11] Bobinska T., Buike M., Buikis A. Hyperbolic Heat Equation as Mathematical Model for Steel Quenching of L-and T-Shape Samples, Direct and Inverse Problems. Transactions of Heat and Mass Transfer. Vol.5, Issue 3, July 2010. p. 63-72.

[12] Blomkalna S., Buikis A. Heat conduction problem for double-layered ball. Progress in Industrial Mathematics at ECMI 2012. Springer, 2014. p. 417-426.

[13] Joseph D.D., Preziosi L. Heat waves. Reviews of Modern Physics, vol. 61, No. 1, 1989, p. 41- 73.

[14] Joseph D.D., Preziosi L. Addendum to the paper “Heat waves”. Reviews of Modern Physics, vol. 62, No. 2, 1990, p. 375-391.

[15] Wang L., Zhou X., Wei X. Heat Conduction. Mathematical Models and Analytical Solutions. Springer, 2008.

[16] Ekergard B., Castellucci V., Savin A., Leijon M. Axial Force Damper in a Linear Wave Energy Convertor. Develepment and Applications of Oceanic Engineering. Vol. 2, Issue 2, May 2013.

[17] Wikipedia:

[18] Buikis A. Problems of Mathematical Physics with Discontinuous Coefficients and their Applications. Riga, 1991, 385 p. (In Russian, unpublished book).

[19] Buikis, A. Conservative averaging as an approximate method for solution of some direct and inverse heat transfer problems. Advanced Computational Methods in Heat Transfer, IX. WIT Press, 2006. p. 311-320.

[20] Vilums, R., Buikis, A. Conservative averaging method for partial differential equations with discontinuous coefficients. WSEAS Transactions on Heat and Mass Transfer. Vol. 1, Issue 4, 2006, p. 383-390.

[21] Bobinska T., Buike M., Buikis A. Comparing solutions of hyperbolic and parabolic heat conduction equations for L-shape samples. Recent Advances in Fluid Mechanics and Heat@Mass Transfer. Proceedings of the 9th IASME/WSEAS Internacional Conference on THE’11. Florence, Italy, August 23-25, 2011. p. 384-389.

[22] A. Piliksere, A. Buikis. Analytical solution for intensive quenching of cylindrical sample. Proceedings of 6th International Scientific Colloquium “Modelling for Material Processing”, Riga, September 16-17, 2010, p. 181-186.

[23] Blomkalna S., Buike M., Buikis A. Several intensive steel quenching models for rectangular and spherical samples. Recent Advances in Fluid Mechanics and Heat & Mass Transfer. Proceedings of the 9th IASME/WSEAS International Conference on THE’11. Florence, Italy, August 23-25, 2011. p. 390-395.

[24] M. Lencmane and A. Buikis, Analytical solution of a two-dimensional double-fin assembly, in “Recent Advances in Fluid Mechanics and Heat & Mass Transfer”, Proceedings of the 9th IASME/WSEAS International Conference on Heat Transfer, Thermal Engineering and Environment (HTE’11), Florence, Italy, August 23 – 25, 2011, pp. 396 – 401.

[25] M. Lencmane and A. Buikis, Analytical solution for steady stable and transient heat process in a double-fin assembly. International Journal of Mathematical Models and Methods in Applied Sciences 6 (2012), no. 1, 81 – 89.

[26] M. Lencmane, A. Buikis. Some new mathematical models for the Transient Hot Strip method with thin interlayer, Proc. of the 10th WSEAS Int. Conf. on Heat Transfer, thermal engineering and environment (HTE '12)”, WSEAS Pres, 2012. p. 283-288

[27] Roach G.F. Green’s Functions. Cambridge University Press, 1999.

[28] Carslaw, H.S., Jaeger, C.J. Conduction of Heat in Solids. Oxford, Clarendon Press, 1959.

[29] Stakgold I. Green’s Functions and Boundary Value Problems. John Wiley&Sons, Inc., 1998.

[30] Polyanin A.D. Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman&Hall/CRC, 2002. (Russian edition, 2001).

[31] Debnath L. Nonlinear Partial Differential Equations for Scientists and Engineers. 2nd ed. Birkhäuser, 2005.

[32] Salter S.H. Apparatus for use in the extraction of energy from waves on water. US Patient 4,134,023. January 14, 1977.

[33] Carroll C.B. Piezoelectric rotary electrical energy generator. US Patient 6194815 B1. February 27, 2001.

[34] Carroll C.B., Bell M. Wave energy convertors utilizing pressure differences. US Patient 20040217597 A1. November 4, 2004.

WSEAS Transactions on Heat and Mass Transfer, ISSN / E-ISSN: 1790-5044 / 2224-3461, Volume 12, 2017, Art. #14, pp. 107-121

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