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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.

Diffusion is a process which leads to an equalization of concentrations within a single phase. The laws of diffusion connect the rate of flow of the diffusing substance with the concentration gradient responsible for this flow [

The analytical methods proposed for the study of solid-state 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. These methods are mostly limited to studying diffusion in binary systems with planar diffusion geometry, so despite the importance of the subject, 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. Although the numerical methods require a new reprocessing each time a parameter is introduced or modified and need the use of computational resources with the programming involved the more complex the greater the degree of precision desired, these methods lead in the most of the time to a greater proximity of cases observed in practice allowing, for example, more real boundary conditions to be admitted. The numerical methods are the ones that have been used most extensively for the study of solid-state diffusion in systems with cylindrical and spherical geometries. Numerical solutions demonstrate sometimes a certain complexity; however, they present as one of their fundamental advantages the fact that they represent problems of practical interest for which it would not be possible to obtain analytical solutions. It is well known that the conductive heat transfer process [

The concept behind the term “moving boundary problems,” according to Crank [

Jost [

In the preceding paragraphs, several special cases were treated by mathematical methods; however, these methods are not sufficient for all cases of practical importance. Therefore, based on the geometric correlation proposed by Moreira [

The current approach represents a method for a set of solutions for the transient diffusion (heat/mass) with phase change (reaction) for cylindrical and spherical coordinate systems, independently of the number of moving boundaries. There is no other analytical closed-form solution in the literature to deal with such problems. The proposed method remained two decades unpublished, as the numerical solution effort to solve properly the initial solid-state diffusion problem with such a high degree of accuracy imposed by the radial cylindrical coordinate system was considerably difficult to achieve due to nonlinearities arisen in the moving boundary. The numerical resolution at that time allowed us to conclude that the proposed analytical solution for the cylinder was approximate. And only recently, by chance, the authors decided to numerically solve the diffusion times and concentration profile, and the comparison with the analytical results have proved to be a close-form analytical solution for cylinder, which in this paper is extended to deal with the sphere for single- and two-phase problems. During the time of the Master of Science Thesis [

The mathematical formulation here introduced is found in Jost [

The atomic flux is unidirectional Cartesian, radial cylindrical, or radial spherical

The diffusion front is macroscopically plane, cylindrical, or spherical

The diffusion coefficient or thermal diffusivity is concentration independent

The solute concentration remains constant at the surface of the semi-infinite slab, cylinder, or sphere

The solute concentration remains constant at the moving boundary

The considered systems are isotropic

The contact resistance in the interface between the atomic flux of solute and the material surface is neglected

According to the assumptions assumed, the transient solid-state diffusion for unidirectional Cartesian [

Coordinate for single-phase solid-state diffusion.

In the case of the thermal diffusion equivalent problem [

Coordinate for single-phase solidification.

Considering now two-phase solid-state diffusion with phase change [

Coordinate for two-phase solid-state diffusion.

In the case of two-phase solidification [

Coordinate for two-phase solidification.

During the solid-state diffusion or the heat conduction with phase change in a slab, the surface at the diffusion front remains constant until the end of the diffusion species/thermal is achieved. Otherwise, in the radial cylindrical and spherical systems, due the effects imposed by the geometric curvature, the surface varies as far as the process advances. Such feature influences directly the flux of species/heat. Therefore, the development of geometrical correlations and the use of a suitable change of variables will enable the use of the partial differential diffusion equation in Cartesian coordinates to represent the radial cylindrical and spherical coordinates.

Santos and Garcia proposed, for the analysis of transient solidification of metals under radial cylindrical [

For slabs,

In 1991, Moreira [

In the radial cylindrical and spherical systems of coordinates, the quantity

Figure

Schematic representation of inward solid-state diffusion with phase change in a cylinder and a sphere.

Applying the change of variables in terms of

For

A general solution in terms of the error function, for solid-state diffusion of a semi-infinite slab, was firstly proposed by C. Wagner [

Considering equation (

Then, rearranging equation (

Consequently, the diffusion time can be expressed as

Finally, by combining equations (

Equation (

Introducing equation (

On the contrary, using equation (

Finally, by combining equations (

In the case of the single-phase solidification,

The diffusion times for radial cylinder and radial sphere coordinates can be obtained by applying into equation (

The derived equation for the diffusion times of inward diffusion in a cylinder can be expressed as

In the case of single-phase solidification,

The diffusional concentration profiles for inward diffusion in a cylinder and a sphere can be described by equation (

Substituting equation (

By inserting equation (

Now, equations (

Applying the change of variables in terms of

A general solution in terms of the error function, for solid-state diffusion of a semi-infinite slab, was firstly proposed by C. Wagner [

Substituting the boundary condition equations (

Rearrange equation (

Finally, combining equations (

Equations (

Introducing equation (

Introducing equations (

On the contrary, using equation (

Finally, by combining equation (

Considering the two-phase solidification,

The diffusion times for radial cylinder and radial sphere coordinates can be obtained by applying into equation (

The derived equation for the diffusion times of inward diffusion in a cylinder can be expressed as

The diffusional concentration profiles for inward diffusion in a cylinder and a sphere can be described by equations (

Substituting equation (

By inserting equation (

For

The case of concentration profiles for spheres provides for

Now, the set of equations for the concentration profile, equations (

Aiming to verify the validity of the analytical solutions for diffusion times and concentration profiles for radial cylindrical and spherical coordinate systems, a numerical method based on the well-known enthalpy method [

The equation to deal with the concentration density field is similar to those applied for the heat conduction with phase change [

The concentration “

In this section, we will be presenting the solid-state diffusion with phase change and the equivalent solidification for single-phase and two-phase processes. The present model is general in nature for cylinders and spheres, regarding the diffusional radial flux of atoms/heat, as in the case of equations (

Figure

Comparison between numerical and analytical solutions for diffusion times as a function of one-dimensional correlation

Aiming to verify the validity of the analytical solutions of diffusion times and concentration profiles for radial cylindrical and spherical coordinates, a numerical method based on the well-known enthalpy method [

Comparison between numerical and analytical solutions for diffusion times during inward diffusion of a slab, a cylinder, and a sphere.

Comparison between numerical and analytical solutions for one-phase solidification times during inward diffusion of a slab, a cylinder, and a sphere.

Figures

Comparison of concentration profiles during inward diffusion of a slab, a cylinder, and a sphere as a function of dimensionless position

Comparison of concentration profiles during inward diffusion of a slab, a cylinder, and a sphere as a function of dimensionless position

Comparison of concentration profiles during inward one-phase solidification of a slab, a cylinder, and a sphere as a function of dimensionless position

Comparison of concentration profiles during inward single-phase solidification of a slab, a cylinder, and a sphere as a function of dimensionless position

Figure

Comparison between numerical and analytical solutions for diffusion times as a function of one-dimensional correlation

Figure

Comparison of similarity variables in relation to the concentration profiles during inward diffusion of a slab, a cylinder, and a sphere at 70% of

Figure

Comparison of similarity variables in relation to the moving interface dimensionless position

Aiming to validate the analytical method developed to obtain transient radial cylindrical and spherical diffusional flux with phase change solutions as a more general solution, once error function solution exists for a semi-infinite slab, by applying a one-dimensional correlation

Comparison between numerical and analytical solutions for diffusion times during inward diffusion of a slab, a cylinder, and a sphere.

Comparison between numerical and analytical solutions for times during inward two-phase solidification of a slab, a cylinder, and a sphere.

Figures

Comparison of concentration profiles during inward diffusion of a slab, a cylinder and a sphere as a function of dimensionless position

Comparison of concentration profiles during inward diffusion of a slab, a cylinder, and a sphere as a function of dimensionless position

Comparison of concentration profiles during inward diffusion of a slab, a cylinder and a sphere as a function of dimensionless position

Comparison of concentration profiles during inward two-phase solidification of a slab, a cylinder, and a sphere as a function of dimensionless position

In the similar way, Figure

Comparison of similarity variables in relation to the concentration profiles during inward diffusion of a slab, a cylinder, and a sphere at 70% of

Figure

Comparison of similarity variables in relation to the moving interface dimensionless position

Exact analytical solutions for cylinders and spheres were derived for transient one-phase and two-phase solid-state atomic diffusion and solidification regarding inward radial cylindrical and spherical geometries. The exact analytical solutions for both radial coordinate systems were derived from two solutions for a semi-infinite slab based on the error function, performing a suitable change in variables and the application of a geometric correlation factor

No data were used to support this study.

The authors declare that they have no conflicts of interest.

The authors acknowledge the financial support provided by FAPERJ (The Scientific Research Foundation of the State of Rio de Janeiro-Brazil), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brazil-Finance Code 001), and CNPq-Brazil (National Council for Scientific and Technological Development).