Abstract
Analytical solutions for the transient single-phase and two-phase inward solid-state diffusion and solidification applied to the radial cylindrical and spherical geometries are proposed. Both solutions are developed from the differential equation that treats these phenomena in Cartesian coordinates, which are modified by geometric correlations and suitable changes of variables to achieve closed-form solutions. The modified differential equations are solved by using two well-known closed-form solutions based on the error function, and then equations are obtained to analyze the diffusion interface position as a function of time and position in cylinders and spheres. These exact correlations are inserted into the closed-form solutions for the slab and are used to update the roots for each radial position of the moving boundary interface. The predictions provided by the proposed analytical solutions are validated against the results of a numerical approach. Finally, a comparative study of diffusion in slabs, cylinders, and spheres is also presented for single-phase and two-phase solid-state diffusion and solidification, which shows the importance of the effects imposed by the radial cylindrical and spherical curvatures with respect to the Cartesian coordinate system in the process kinetics. The analytical model is proved to be general, as far as, a semi-infinite solution for diffusion problems with phase change exists in the form of the error function, which enables exact closed-form analytical solutions to be derived, by updating the root at each radial position the moving boundary interface.
Highlights
E analytical methods proposed for the study of solidstate diffusion are generally based on infinite series which may offer some difficulty in the face of practical applications since their mathematical resolutions normally occur through nontrivial processes. ese methods are mostly limited to studying diffusion in binary systems with planar diffusion geometry, so despite the importance of the subject, International Journal of Mathematics and Mathematical Sciences exact analytical solutions for cylindrical and spherical geometries have not yet been obtained since they present greater mathematical difficulties arising from the complexity of the mathematical equations as well as from the assumed boundary conditions
It is well known that the conductive heat transfer process [2,3,4,5] and solid-state diffusion are physically similar since both occur due to the existence of a temperature gradient and a concentration gradient, respectively, and show a certain correspondence between their physical parameters, variables, mathematical equations, and boundary conditions normally assumed. us, many of analytical and numerical solutions proposed for the mathematically analogous solidification problem are directly applicable to diffusion problems
Erefore, based on the geometric correlation proposed by Moreira [16] as well as on the exact solution developed by Wagner reported by Furzeland [47] for a semi-infinite slab, in this study is presented an exact solution which is able to estimate diffusion times and position of the moving boundary, as well as the solute concentration profiles during inward transient solid-state diffusion in radial cylindrical and spherical geometries
Summary
Diffusion is a process which leads to an equalization of concentrations within a single phase. e laws of diffusion connect the rate of flow of the diffusing substance with the concentration gradient responsible for this flow [1]. e need to study this physical phenomenon may be justified by the great influence it has on the solute enrichment rate and concentration profiles of this element in the solid, being important in determining the structure and physical, mechanical, and metallurgical properties of obtained products. e diffusion rate, generally in a transient regime, becomes the mathematical analysis of this phenomenon difficult since it leads to differential equations to present nonlinear boundary conditions making it difficult to obtain exact analytical solutions [2] requiring the establishment of physical and/or mathematics simplifying hypotheses from the real conditions so that such solutions may be made viable. Erefore, based on the geometric correlation proposed by Moreira [16] as well as on the exact solution developed by Wagner (unpublished work) reported by Furzeland [47] for a semi-infinite slab, in this study is presented an exact solution which is able to estimate diffusion times and position of the moving boundary, as well as the solute concentration profiles during inward transient solid-state diffusion in radial cylindrical and spherical geometries. Ese geometric correlations are introduced into the solution of the partial differential for the semiinfinite slab, based on the well-known error function, in order to permit the diffusion interface position and the evolution of the solute concentration profiles during inward solid-state diffusion in cylinders and spheres to be determined. By consequence, regarding the (V/A0) correlation, there will be n roots to be calculated from equation (74)
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