"The time - dependent surface temperature is determined for a sphere subjected to cooling by boiling. One dimensional heat transfer is considered and the solid is assumed to be homogeneous, isotropic and opaque to thermal radiation and to have temperature independent physical properties. The initial temperature of the sphere and the coolant are assumed to be uniform and arbitrary. The boiling heat transfer coefficient at the surface of a sphere is a strong function of the surface temperature thus resulting in an extremely nonlinear transient heat conduction problem. As a practical application of the problem, the process of quenching has been cited.
In the analysis, the transient heat conduction equation with its initial and boundary conditions is transformed to a singular nonlinear Volterra integral equation of the second kind by use of the Laplace transformations. The equation is solved numerically on a digital computer by the method of modified successive approximations for the sphere. The time scale is broken into a desired number of intervals. The solution for the first time interval is obtained by successive approximations and then used in finding the solution for the next time interval. Initial guesses are obtained by the use of the ideal case of infinite thermal conductivity. The process of successive approximation is continued along the time scale till the solution for all desired time is obtained. Within the limitations of the basic assumptions, the method can be termed as exact, since any degree of accuracy can be obtained. The results are presented in graphical and tabular forms and compared with analytical and experimental results where available.
A separable kernel method is also applied for the solution of the Volterra integral equation describing the surface temperature. The kernel of the integral equation is approximated by a simple expression and substituted in the original integral equation. By suitable mathematical techniques, the integral equation is then transformed to a differential equation which is much simpler to solve than the integral equation. The better the kernel is approximated, the more the separable kernel solution tends towards the solution obtained by the successive approximation method. This method of solution is more convenient to apply than the successive approximation method, but is not very accurate for Biot numbers greater than ten. Another approximate method, termed as the modified separable kernel method, is also presented for the solution of the integral equation. The accuracy of the modified method seems to be much better than the ordinary one and is found to give results within five percent of the exact solution when the kernel is represented by a suitable number of equations.
Limiting cases of large and small Biot numbers (0.1 to 10.0) are calculated and analysed in terms of the present solutions. It is concluded that the infinite thermal conductivity approach can approximate the exact solution to within ten percent for Biot numbers approximate the exact solution to within ten percent for Biot numbers less than one"--Abstract, pages viii-ix.
Crosbie, A. L. (Alfred L.)
Grimm, L. J.
Rhea, L. G.
Mechanical and Aerospace Engineering
M.S. in Mechanical Engineering
University of Missouri--Rolla
ix, 77 pages
© 1972 Salil Kumar Bandyopadhyay (Banerjee), All rights reserved.
Thesis - Open Access
Heat -- Transmission -- Mathematical models
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Banerjee, Salil K., "Transient cooling of a sphere due to boiling" (1972). Masters Theses. 5051.