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The Stability of the Moving Boundary in Spherical

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  • "Coupled heat and mass diffusion equations are set up and solved for various Stefan numbers.A stability criterion is developed for the moving interface in the general MBP, of importancein many fields, particularly in directional solidification. The a..

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  • "Coupled heat and mass diffusion equations are set up and solved for various Stefan numbers.A stability criterion is developed for the moving interface in the general MBP, of importancein many fields, particularly in directional solidification. The analysis is applied to thehomogenous nucleation and growth of a spherical particle. Traditional analyses have reliedon energy balances. Other formulations are reviewed in Vafai [ ]. Carslaw and Jaeger [ ]claim only certain solutions for the MBP are known, it being difficult to find solutions for thegeneral case. The difficulty is due to the extreme non-linear nature of the problem comprisingdiscontinuous material properties across the liquid and solid regions, with unknown positionsof the liquid-solid phase boundary. An exact solution is analyzed for appropriate boundaryconditions here. The present derivation presents unpublished analyses using perturbation andconsideration of the unknown moving boundary of the nucleating particle. The problem of solidification and the reverse, of melting, is of interest in such diverseareas as geology, metallurgy, food processing and cryosurgery, to name a few. The problemwas first tackled by Stefan [4] in the analysis if the melting of polar ice in the late 1800’s.The problem is a highly nonlinear one since the properties of the medium change abruptly atthe phase boundary, and further, the position of the boundary is not known, but must besolved for.Few exact solutions exist for the moving boundary problem (MBP), even for the one- dimensional case. It has been more than 100 years since the problem was first analysed, yetno method is known to solve the problem with all types of boundary conditions. Carslaw andJaeger [ ] have listed some solutions and claimed these to be the only known ones. Most ofthe known solutions for the M.B.P are given in Carslaw and Jaeger [ ], very few additionalsolutions have been published in closed form. In particular, for the spherical freezing case,series solutions were developed early on, by means of perturbation expansions by PedrosoDomoto [ ]. The moving heat source melting is treated with various transformations together witha decoupling for the heat and mass transfer terms. Some of the earlier works on the interfaceboundary are by Mullins Sekerka 2, and Pedroso Domoto 3. The classic work of MullinsSekerka 2 dealt with a perturbation analysis of the moving phase interface Here, the coupled diffusive heat and mass transfer equations in porous media aresolved for convective boundary condition, both by perturbation methods and exact methods.A stability criterion is derived for the moving interface in the convective case, withappropriate linearisations. Specifically, the stability of the interface is examined andcompared with other studies such as Mullins Sekerka(2). The nucleating phase is treated ashaving a moving boundary, the phenomenon treated as an MBP and stability analyzed fromthis point of view.The ice-vapor and naphtha – vapor sublimation cases which are of theoretical and practicalinterest are proposed to be simulated. Sublimation involves the direct transformation fromsolid to vapor. No liquid phase is produced; the latent heat of sublimation can be estimated bysumming the heats for melting, thermal heat to go up to vaporization and the latent heat ofvaporization. The cases of self-sublimation without any heat source are also examined. Practical instances of sublimation exist in space vehicles operating in cold regions of spaceunder near vacuum, production of whiskers and preservation of food and artifacts bytreatment in vacuum.Naphtha is of interest as it is used in experimental flow studies as a tracer, as itsublimes at room temperature. A further interesting feature of Naphtha is the observation thatit condenses in the form of needles. This fact prompts the subsequent modeling and simulation of sublimation from a general surface (plane or spherical) and condensation assolid in the form of cylinders. One further difficulty is the determination of the physicalconstants of heat transfer for Naphtha. The heat and mass transfer analogy is used as one wayout of this difficulty, Cho ( ). In particular, papers appearing on self-freezing and sublimation are rare. One of theseminal papers dealing with the general problem is by Paterson (5), where the generalsolution is given for the linear, cylindrical and spherical cases. In addition, Paterson lists outvarious solutions and eliminates several on the basis of the flux balance at the movinginterface. By transforming the problem to rectilinear coordinates it is seen that this signchange in L merely shifts the sublimating surface from one end to the other end of the slab.Here, it depends on the sign of the terms in eqn[5]. Focussing on the case of self-sublimation( ice-vapour), heat source strength q is set to zero, and using the appropriate values for iceand vapour, the results are given in the following figures. Further simulations using theboundary temperature have been done for the three geometries ( sphere, cylinder, and plane)using zero applied heat source ( self-driven by latent heat alone). The results have beensummarized in the figures. In the case of water, results can be obtained for both sublimationand accretion ( L has different signs).For water, a difficulty arises for cylindricaltransformation where zero velocities are obtained due to the peculiar properties of the Ei(exponential integral) function. Hence, the comparison for water of the three geometries isnot performed. A plot of the data given for positive L?/2 is added below to give an idea of the typeof variation of the phase interface velocity with latent heat, and a similar figure is adduced forthe reverse (accretion from the vapour) although the transformation is an order of magnitudeless. FIGURE : sublimation velocity water ice(solid-vapour)FIGURE : Accretion from vapour to solid ( vapour ice) Applications for the methods described above have been suggested for the formationof nano composites with micro spheres embedded in a matrix ( Wu (17)), while the problemof the exact solution of spherical phase change has been occupying researchers since the pastfour decades at least (Soward, McCue, Stewartson (18,19,20 ). It is seen that unless theboundary conditions are formulated correctly with relation to the physics, the solution isdifficult to obtain even in approximate form. Soward (18) has given a perturbation expressionwhich is applicable to either the cylindrical or spherical cases by changing the exponent ofthe terms in the series. While the other works have tried to analyse the series for the problemwith an insulated boundary, it needs to be asserted that such a problem is physically andmathematically inconsistent with the boundary conditions, which is perhaps the reason for thecomplexity of the analysis where attempts are made to match the various terms in the solutionregimes. "

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