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03/05/2015 ThermalFluidsPedia | Onedimensional transient heat conduction in cylinder | ThermalFluids Central https://www.thermalfluidscentral.org/encyclopedia/index.php/Onedimensional_transient_heat_conduction_in_cylinder 1/2 Onedimensional transient heat conduction in cylinder From ThermalFluidsPedia A long cylinder with radius of ro and a uniform initial temperature of Ti is exposed to a fluid with temperature of ( ). The convective heat transfer coefficient between the fluid and cylinder is h. Assuming that there is no internal heat generation and constant thermophysical properties, obtain the transient temperature distribution in the cylinder. Since the temperature changes along the rdirection only, the energy equation is subject to the following boundary and initial conditions Defining the following dimensionless variables eqs. (1) – (4) will be nondimensionalized as Assuming that the temperature can be expressed as and substituting eq. (10) into eq. (6), one obtains which can be rewritten as the following two equations Equation is a Bessel’s equation of zero order and has the following general solution where J0 and Y0 are Bessel functions of the first and second kind, respectively. The general solution of eq. (13) is where C1,C2, and C3 are integral constants. The boundary conditions for eq. (12) can be obtained by substituting eq. (10) into eqs. (7) and (8), i.e., The derivative of Θ is Since , C2 must be zero. Substituting eqs. (14) and (18) into eq. (17) and considering C2 = 0, we have where n is an integer. The eigenvalue λn can be obtained by solving eq. (19) using an iterative procedure. The dimensionless temperature with eigenvalue λn is where Cn = C1C3. Equation (20) is a solution that satisfies eqs. (6) – (8) but not eq. (9). For a linear problem, the sum of different θn for each value of n also satisfies eqs. (6) – (8). Search ThermalFluids Central Entire Website Search Home » Encyclopedia » Index » Onedimensional_transient_heat_conduction_in_cylinder Home Encyclopedia Journals eBooks eResources Jobs Events News Who's Who Links 03/05/2015 ThermalFluidsPedia | Onedimensional transient heat conduction in cylinder | ThermalFluids Central https://www.thermalfluidscentral.org/encyclopedia/index.php/Onedimensional_transient_heat_conduction_in_cylinder 2/2 Substituting eq. (21) into eq. (9) yields Multiplying the above equation by and integrating the resulting equation in the interval of (0, 1), one obtains According to the orthogonal property of Bessel’s function, the integral on the righthand side equals zero if but it is not zero if m = n. Therefore, we have Changing notation from m to n, we get thus, the dimensionless temperature becomes References Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO. Further Reading External Links Retrieved from "https://www.thermalfluidscentral.org/encyclopedia/index.php/Onedimensional_transient_heat_conduction_in_cylinder" This page was last modified on 3 July 2010, at 07:31. About Us | Contact Us | Terms of Use | Privacy Policy | Disclaimer Copyright © 20102011 by Global Digital Central. All Rights Reserved.
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