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Steady heat transfer through a radial fin with rectangular and hyperbolic profiles. (English) Zbl 1205.80035
Summary: We construct some exact solutions for the thermal diffusion in a fin with a rectangular profile and another with a hyperbolic profile. Both the thermal conductivity and the heat transfer coefficient are assumed to be temperature dependent. Moreover, the thermal conductivity and the heat transfer terms are given by the same power law in one case and distinct power laws in the other. A point transformation is introduced to linearize the problem when the power laws are equal. In the other case, classical Lie symmetry techniques are employed to analyze the problem. The exact solutions obtained satisfy the realistic boundary conditions. The effects of applicable physical parameters such as the thermo-geometric fin parameter and the fin efficiency are analyzed.

##### MSC:
 80A20 Heat and mass transfer, heat flow (MSC2010)
##### Keywords:
heat transfer; radial fin; nonlinear equations; exact solutions
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##### References:
  Kraus, A.D.; Aziz, A.; Welty, J., Extended surface heat transfer, (2001), John Wiley and Sons, Inc. New York, USA  ()  Khani, F.; Aziz, A., Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient, Commun. nonlinear sci. numer. simul., 15, 590-601, (2010)  Aziz, A.; Khani, F., Analytical solutions for a rotating radial fin of rectangular and various convex parabolic profiles, Commun. nonlinear sci. numer. simul., 15, 6, 1565-1574, (2010)  Taufiq, B.N.; Masjuki, H.H.; Mahlia, T.M.I.; Saidur, R.; Faizul, M.S.; Mohamad, E.N., Second law analysis for optimal thermal design of radial fin geometry by convection, Appl. therm. eng., 27, 1363-1370, (2007)  Heggs, P.J.; Ooi, T.H., Design charts for a radial rectangular fins in terms of performance ratio and maximum effectiveness, Appl. therm. eng., 24, 1341-1351, (2004)  Khani, F.; Ahmadzadeh Raji, M.; Hamedi-Nejad, H., Analytic solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, Commun. nonlinear sci. numer. simul., 14, 3327-3338, (2009) · Zbl 1221.74083  Khani, F.; Ahmadzadeh Raji, M.; Hamedi-Nezhad, H., A series solution of the fin problem with a temperature-dependent conductivity, Commun. nonlinear sci. numer. simul., 14, 7, 3007-3017, (2009)  Moitsheki, R.J.; Hayat, T.; Malik, M.Y., Some exact solutions for a fin problem with a power law temperature-dependent thermal conductivity, Nonlinear anal. RWA, 11, 5, 3287-3294, (2010) · Zbl 1261.35140  Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover Publication, Inc. New York · Zbl 0543.33001  Polyanin, A.D.; Zaitsev, V.F., Handbook of exact solutions for ordinary differential equations, (1995), CRC Press New York · Zbl 0855.34001  Mahomed, F.M., Symmetry group classification of ordinary differential equations: survey of some results, Math. methods appl. sci., 30, 1995-2012, (2007) · Zbl 1135.34029  Olver, P.J., Applications of Lie groups to differential equations, (1986), Springer New York · Zbl 0588.22001  Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York, USA · Zbl 0698.35001  Stephani, H., Differential equations: their solution using symmetries, (1989), Cambridge University Press Cambridge, UK · Zbl 0704.34001  Ibragimov, N.H., Elementary Lie group analysis and ordinary differential equations, (1999), Wiley New York · Zbl 1047.34001  Bluman, G.W.; Anco, S.C., Symmetry and integration methods for differential equations, (2002), Springer-Verlag New York · Zbl 1013.34004  Joneidi, A.A.; Ganji, D.D.; Babaelahi, M., Differential transformation method to determine fin efficiency of convective straight fin with temperature dependent thermal conductivity, Int. commun. heat mass transfer, 36, 757-762, (2009)  Kundu, B.; Das, P.K., Performance and optimum designs analysis of convective fin arrays attached to flat and curved primary surfaces, Int. J. refrig., 32, 430-443, (2009)
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