IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi10.1155/2021/66242876624287Research ArticleOn the Transient Atomic/Heat Diffusion in Cylinders and Spheres with Phase Change: A Method to Derive Closed-Form Solutionshttps://orcid.org/0000-0002-3118-7125FerreiraI. L.1https://orcid.org/0000-0002-3834-3258GarciaA.2MoreiraA. L. S.1SolovjovsSergejs1Faculty of Mechanical EngineeringFederal University of Pará-UFPAAugusto Corrêa Avenue 1Belém 66075-110PABrazilufpa.br2Department of Manufacturing and Materials EngineeringUniversity of Campinas-UNICAMPCampinas 13083-860SPBrazilunicamp.br2021922021202181220202120216120219220212021Copyright © 2021 I. L. Ferreira et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de JaneiroCoordenação de Aperfeiçoamento de Pessoal de Nível SuperiorConselho Nacional de Desenvolvimento Científico e Tecnológico
1. Introduction

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 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, therefore, being important in determining the structure and physical, mechanical, and metallurgical properties of obtained products. The 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  requiring the establishment of physical and/or mathematics simplifying hypotheses from the real conditions so that such solutions may be made viable. A limited number of exact, closed-form solutions to the multiphase, diffusion-controlled, and moving-interface problem exists in the literature. The exact, closed-form solutions generally consider the outermost phases to be infinite in extent as well as assume an interfacial diffusion rate function. It is through use of such boundary conditions that closed-form solutions are readily obtained.

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  and solid-state diffusion are physically similar since both occur due to the existence of a temperature gradient and a concentration gradient, respectively, and therefore show a certain correspondence between their physical parameters, variables, mathematical equations, and boundary conditions normally assumed. Thus, many of analytical and numerical solutions proposed for the mathematically analogous solidification problem are directly applicable to diffusion problems.

The concept behind the term “moving boundary problems,” according to Crank , is associated with problems whose solutions of differential equations must satisfy certain boundary conditions of a corresponding domain, referred as boundary value problems. Despite this, in many cases of interest, the domain boundary is not known beforehand as the process develops but needs to be determined as a part of the solution. The term “moving boundary problems” is normally applied when the domain boundary is stationary, and a steady-state problem exists. On the contrary, the terminology “free boundary problems” is related to the time-dependent problems and the boundary needs to be determined as a function of time and space. The terms “free” and “moving” are carried on in different ways. Some authors prefer to encompass both kinds of problems in a singular term, i.e., “free boundary problems,” and few ones used to apply “moving boundary” as a more general case. The moving boundary problems require a solution of a parabolic-type differential equation. The practical applications are fluid flow in porous medium, diffusion, and heat conduction with chemical reactions or phase change and, additionally, to help to understand many other free and moving boundary problems, such as collapse of dams, shock waves in gas dynamics, and cracks in solid mechanics.

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  as well as on the exact solution developed by Wagner (unpublished work) reported by Furzeland  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.

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 , only an approximate numerical solution was obtained by applying the Modified Variable Time Step Method (MVTS) in a cylindrical coordinate system. In this paper, a variation of the enthalpy method proposed by Swaminathan et al.  was applied, and a more accurate numerical result was obtained allowing to demonstrate that both analytical solutions consist of closed-form solutions. Today, only approximate solutions for cylindrical and spherical phase-change problems are available. The technological appeal of energy-storage and pharmaceutical applications are the current motivations. The simplicity of both proposed solutions and their main feature as low-computational demanding resources, allow engineers and scientists to implement them directly in the microcontrollers or low-profile single-board computers. That is not the case for the numerical schemes found in the literature to deal with moving boundary problems.

2. Mathematical Formulation

The mathematical formulation here introduced is found in Jost  for the solid-state diffusion and Crank  solidification of pure metals. The solution to be derived aims to analyze the changes promoted by the transient solid-state diffusion in cylindrical and spherical radial geometries and imposed by the effects of the evolving curvature, as compared to the atomic flux of Cartesian coordinates. In developing the analytical solutions, the following assumptions are taken:

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

2.1. Single-Phase Solid-State Diffusion

According to the assumptions assumed, the transient solid-state diffusion for unidirectional Cartesian  and radial cylindrical and radial spherical geometries can be represented by Fick’s second law, for single-phase solid-state diffusion with phase change, as shown in Figure 1, and the governing equation and boundary conditions can be written as follows:(1a)Ct=DPxCx+2Cx2,0<x<xξ,and t>0,for t=0,(2a)0xxξ,C=C0,for t>0,(3a)x=0,C=CS,(4a)x=xξ,C=Cξ,(5a)x>xξ,C=C0,(6a)CξC0dxξdt=DCx|xξ0,where C0 is the initial concentration, CS is the surface concentration, Cξ is the solute concentration at the moving interface xξ, D is the atomic diffusion coefficient, x is the spatial coordinate, t is the time variable, P=0 for slab, P=1 for cylinders, and P=2 for spheres.

Coordinate for single-phase solid-state diffusion.

2.2. Single-Phase Solidification

In the case of the thermal diffusion equivalent problem , it is called single-phase solidification. The reference coordinate for this problem is shown in Figure 2. For this problem at x=xξ and t>0, the temperature at the moving interface is given by T=Tξ=Tm:(1b)Tt=αSPxTx+2Tx2,0<x<xξ,and t>0,for t=0,(2b)0xxξ,T=Tm,for t>0,(3b)x=0,T=Tm,(4b)x=xξ,T=Tξ=Tm,(5b)x>xξ,T=Tm,(6b)x=xξ,ρLdxξdt=ksTx|xξ0,where ρ is the solid phase density, L is the latent heat of fusion per volume, Tm is the melting temperature, TS is the surface temperature, Tξ=Tm is the temperature at the moving interface xξ, αS is the thermal diffusivity, and, finally, ks is the thermal conductivity of the solid phase.

Coordinate for single-phase solidification.

2.3. Two-Phase Solid-State Diffusion

Considering now two-phase solid-state diffusion with phase change , the reference coordinate for this problem is shown in Figure 3:(7a)Ct=DIIPxCx+2Cx2,0<x<xξ and t>0,(8a)Ct=DIPxCx+2Cx2,xξ<x<x0 and t>0,for t=0,(9a)0xxξ,C=C0,for t>0,(10a)x=0,C=CS,(11a)x=xξ,C=CII,I,(12a)x=+xξ,C=CI,II,(13a)x=x0,C=C0,(14a)CII,ICI,IIdxξdt=DIICx|xξ0+DICx|xξ+0,where CII,I is the solute concentration at the moving boundary interface x=xξ, CI,II is the solute concentration at the interface x=+xξ, and DI and DII are diffusion coefficients of the phases I and II, respectively.

Coordinate for two-phase solid-state diffusion.

2.4. Two-Phase Solidification

In the case of two-phase solidification , the reference coordinate for this problem is shown in Figure 4. For this problem, at x=xξ and t>0, the temperature at the moving interface is given by T=Tξ=Tm. The boundary conditions will be left in the form of establishing an equivalence between the solid-state diffusion and heat conduction with phase change phenomena:(7b)Tt=αsPxTx+2Tx2,0<x<xξ and t>0,(8b)Tt=αlPxTx+2Tx2,xξ<x<x0 and t>0,for t=0,(9b)0xxξ,T=T0,for t>0,(10b)x=0,T=TS,(11b)x=xξ,T=Tm,(12b)x=+xξ,T=Tm,(13b)x=xo,T=T0,(14b)ρLdxξdt=kSTx|xξ0klTx|xξ+0,where kl is the liquid phase thermal conductivity and T0 is the initial temperature of the liquid phase.

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.

2.5. One-Dimensional Relation <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M53"><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold">V</mml:mi></mml:mrow><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="bold">A</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mfenced></mml:math></inline-formula>

Santos and Garcia proposed, for the analysis of transient solidification of metals under radial cylindrical [20, 48] and spherical  heat flux conditions, geometrical correlations to deal with the geometric curvature. These authors stated that the solidification time must be analyzed as a function of the one-dimensional correlation that considers, for each radial position, the decrease in the surface area of heat exchange at the solid/liquid interface as compared to that of the slab, which remains constant. The correlation that has been shown to be more appropriate was the relation Rξ between the volume of metal affected by transformation V in each radial position and the surface area of the metal/mold interface A0:(15)Rξ=VξA0.

For slabs,(16)RU=VA0U=A0xA0.for cylinders:(17)RC=VA0C=R02Rξ22R0,and for spheres:(18)RS=VA0S=R03Rξ33R02,where the variable r0 is the radius of a cylinder and a sphere, rξ is the position of moving interface, Rξ is the one-dimensional relation of transformed volume in relation to the initial surface area, RU for Cartesian, RC for radial cylindrical, and RS for radial spherical coordinate systems.

2.6. Geometric Correlation <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M67"><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>

In 1991, Moreira  proposed an analytical model representing the solid-state diffusion in binary single-phase systems for radial cylindrical and spherical coordinates. The representative equations were derived by the application of geometric correlations and a corresponding change in variables able to encompass the curvatures of the radial cylindrical and spherical coordinates. These geometric correlations are introduced into the solution of the partial differential for the semi-infinite 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. These correlations obtained by Moreira for cylinders and spheres are the following. For cylinders,(19)ΘC=2RξR0R0Rξ/R0,and for spheres,(20)ΘS=32RξR0R0Rξ/R0.

2.7. Change of Variables <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M70"><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi mathvariant="bold">T</mml:mi><mml:mo>,</mml:mo><mml:mi mathvariant="bold">R</mml:mi></mml:mrow></mml:mfenced></mml:math></inline-formula>

In the radial cylindrical and spherical systems of coordinates, the quantity x in the transient solid-state diffusion for Cartesian coordinates will be exchanged by R, so to be applied to radial coordinates. On the contrary, the introduction of the geometric correlation Θ in the partial differential equation, will be made through the change of variable. From t to a modified variable T, which encompasses the geometric correlation:(21)T=tΘ.

2.8. Deriving the Analytical Solution for a Single-Phase Diffusion

Figure 5 shows a schematic representation of solid-state inward diffusion in a cylinder and a sphere.

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

Applying the change of variables in terms of R, T, and Θ in equations (15)–(21), the mathematical formulation of the problem of solid-state diffusion can be rewritten in the following form:(22)CT=D2CR2,0<R<Rξ, and T>0.

For T=0,(23)0RRξ,C=C0,and for T>0,(24)R=0,C=CS,(25)R=Rξ,C=Cξ,(26)R>Rξ,C=C0,(27)CξC0dRξdT=DCx|Rξ0,where(28)D˜=DΘ.

A general solution in terms of the error function, for solid-state diffusion of a semi-infinite slab, was firstly proposed by C. Wagner , regarding the performed change of variables, which gives(29)CR,T=B1B2erfR2DT.

Considering equation (29) and the boundary condition given by equation (24),(30)B1=CS,substituting the boundary condition of equation (25) into equation (29),(31)Cξ=CSB2erfRξ2DT.

Then, rearranging equation (31) as(32)B2erfRξ2DT=CSCξ=constant,introducing the geometric correlation Θ into the similarity variable Λ(33)Rξ2DT=ΛΘ,and rearranging equation (33) in order to obtain an expression in terms of the diffusion time, i.e., by substituting equation (21) into equation (33),(34)T=Θ214DΛ2Rξ2.

Consequently, the diffusion time can be expressed as(35)t=Θ314DΛ2Rξ2,by inserting equation (34) into equation (31):(36)B2=CSCξerfΛ/Θ.

Finally, by combining equations (30) and (36) into equation (29), an expression for the solute concentration profile can be obtained as follows:(37)CR,T=CSCSCξerfΛ/ΘerfR2DT,

Equation (37) presents the transient solute concentration profile for cartesian, cylindrical, and spherical coordinate systems. The derivative of equation (37) with respect to the spatial variable “R,” at the moving interface position, R=Rξ yields(38)CR,TR|R=Rξ=CSCξerfΛ/Θ2πexpRξ2DT2,12DT.

Introducing equation (33) in equation (38),(39)CR,TR|R=Rξ=CSCξerfΛ/Θ2πexpΛΘ212DT.

On the contrary, using equation (33) to derive Rξ with respect to T, provides(40)dRξdT=ΛΘDT.

Finally, by combining equations (39) and (40) with the equation of the moving interface, equation (27), an expression is obtained to calculate and update the roots Λ for each radial position, i.e., each position in the one-dimensional relation Vξ/A0:(41a)πΛΘexpΛΘ2erfΛΘ=CSC0CξC0.

In the case of the single-phase solidification,(41b)πΛΘexpΛΘ2erfΛΘ=CpsTmTSL,where Cps is the specific heat of the solid phase.

2.9. Diffusion Times

The diffusion times for radial cylinder and radial sphere coordinates can be obtained by applying into equation (35) the one-dimensional relation Vξ/A0 of volume affected by the transformation and initial surface area, equations (17) and (18), and the geometric correlation Θ for cylinder and sphere, equations (19) and (20), respectively:(42)t=Θ314DΛ2Rξ2=Θ314DΛ2VξA02.

The derived equation for the diffusion times of inward diffusion in a cylinder can be expressed as(43a)tCyl=2RξR0R0Rξ/R0314DΛ2R02Rξ22R02.

In the case of single-phase solidification,(43b)tCyl=2RξR0R0Rξ/R0314αSΛ2R02Rξ22R02,and for the atomic diffusion in a sphere,(44a)tSph=32RξR0R0Rξ/R0314DΛ2R03Rξ33R022,and, finally, for single-phase solidification,(44b)tSph=32RξR0R0Rξ/R0314αSΛ2R03Rξ33R022.

2.10. Concentration Profiles

The diffusional concentration profiles for inward diffusion in a cylinder and a sphere can be described by equation (37), and multiplying the erf function argument by the term Rξ/Rξ provides(45)CR,T=CSCSCξerfΛ/ΘerfΛΘRRξ.

Substituting equation (15) for R and Rξ, the following expression can be determined as a function of one-dimensional relations V/A0 and Vξ/A0, respectively:(46)CR,T=CSCSCξerfΛ/ΘerfΛΘV/A0Vξ/A0.

By inserting equation (17) for the cylinder and equation (18) for the sphere in equation (46), the concentration profiles for cylinders and spheres can be derived as(47a)CCylR,T=CSCSCξerfΛ/ΘCerfΛΘCr02r2r02rξ2,for single-phase solidification inward a cylinder,(47b)TCylR,T=TSTSTmerfΛ/ΘCerfΛΘCr02r2r02rξ2,for diffusion inward a sphere, the solute profile is given by(48a)CSphR,T=CSCSCξerfΛ/ΘEerfΛΘSr03r3r03rξ3,and, for the single-phase solidification,(48b)TSphR,T=TSTSTmerfΛ/ΘEerfΛΘSr03r3r03rξ3.

Now, equations (43) and (44) for diffusion times and equations (47) and (48) for the solute concentration profiles for cylinders and spheres, respectively, can be carried out by solving equation (41) for each radial position. In other words, for n positions in the solution domain, which represents n one-dimensional correlation positions V/A0, it implies now there will be n roots to be calculated from equation (41).

2.11. Deriving the Analytical Solution for Two-Phase Diffusion

Applying the change of variables in terms of R, T, and Θ in equations (7a)–(14a), the mathematical formulation of the problem of solid-state diffusion can be rewritten in the following form:(49)CT=DII2CR2,0<R<Rξ and T>0,and(50)CT=DI2CR2,Rξ<R<R0 and T>0.For T=0,(51)0RR0,C=C0,and for T>0,(52)R=0,C=CS,(53)R=Rξ,C=CII,I,(54)R=+Rξ,C=CI,II,(55)R>Rξ,C=C0,(56)CII,ICIdRξdT=DIICx|Rξ0+DICx|Rξ+0.

A general solution in terms of the error function, for solid-state diffusion of a semi-infinite slab, was firstly proposed by C. Wagner , regarding the performed change of variables, which gives(57)CR,T=CSBIIerfR2DIIT,and(58)CR,T=C0+BI1erfR2DIT.

Substituting the boundary condition equations (53) and (54) into equations (57) and (58) as(59)CII,I=CSBIIerfRξ2DIIT,and(60)CI,II=C0+BI1erfRξ2DIT,introducing the geometric correlation Θ into the similarity variable Λ,(61)Rξ2DIIT=ΛΘ,and rearranging equation (61) in order to obtain an expression in terms of the diffusion time,(62)T=Θ214DIIΛ2Rξ2,and, by substituting equations (21) into (62) provides(63)t=Θ314DIIΛ2Rξ2.

Rearrange equation (59),(64)BII=CSCII,IerfΛ/Θ,and equation (60),(65)BI=CI,IIC01erfφΛ/Θ,where(66)φ=DIIDI.

Finally, combining equations (64) and (57) provides an expression for the solute concentration profile for 0 <R<Rξ and T>0 which can be obtained as follows:(67)CR,T=CSCSCII,IerfΛ/ΘerfR2DIIT,and, similarly, considering equations (65) and (66) and equation (60),(68)CR,T=C0+CI,IIC01erfφΛ/Θ1erfRξ2DIT,

Equations (67) and (68) represent the transient solute concentration profile for Cartesian, cylindrical, and spherical coordinate systems. The derivative of equations (67) and (68) with respect to the spatial variable “R,” at the moving interface position, R=Rξ, yields(69)CR,TR|R=Rξ=CSCII,IerfΛ/Θ2πexpRξ2DIIT212DIIT.

Introducing equation (61) into equation (69),(70)CR,TR|R=Rξ=CSCII,IerfΛ/Θ2πexpΛΘ212DIIT,(71)CR,TR|R=+Rξ=CI,IIC01erfΛ/Θ2πexpφRξ2DIT212DIT.

Introducing equations (61) and (66) into equation (71),(72)CR,TR|R=+Rξ=CI,IIC01erfΛ/Θ2πφexpφΛΘ212DIIT.

On the contrary, using equation (61) to derive Rξ with respect to T provides(73)dRξdT=ΛΘDIIT.

Finally, by combining equation (70), equations (72) and (73) with the equation of the moving interface, and equation (56), an expression is obtained to calculate and update the roots Λ for each radial position Θ, i.e., each position in the one-dimensional relation Vξ/A0:(74a)CII,ICI,II=CSCII,IexpΛ/Θ2πΛ/ΘerfΛ/ΘCI,IIC0expφΛ/Θ2πφΛ/Θ1erfφΛ/Θ.

Considering the two-phase solidification,(74b)LCpS=TSTmexpΛ/Θ2πΛ/Θ erfΛ/Θ+klkSTmT0expφΛ/Θ2πφΛ/Θ1erfφΛ/Θ.

2.12. Diffusion Times for Two-Phase Diffusion

The diffusion times for radial cylinder and radial sphere coordinates can be obtained by applying into equation (63) the one-dimensional relation Vξ/A0 of volume affected by the transformation and initial surface area, equations (17) and (18), and the geometric correlation Θ for cylinder and sphere, equations (19) and (20), respectively:(75)t=Θ314DIIΛ2Rξ2=Θ314DIIΛ2VξA02.

The derived equation for the diffusion times of inward diffusion in a cylinder can be expressed as(76a)tCyl=2RξR0R0Rξ/R0314DIIΛ2R02Rξ22R02,and, for two-phase solidification,(76b)tCyl=2RξR0R0Rξ/R0314αsΛ2R02Rξ22R02,and for inward diffusion in a sphere,(77a)tSph=32RξR0R0Rξ/R0314DIIΛ2R03Rξ33R022,and, in similar manner for two-phase solidification,(77b)tSph=32RξR0R0Rξ/R0314αsΛ2R03Rξ33R022.

2.13. Concentration Profiles for Two-Phase Diffusion

The diffusional concentration profiles for inward diffusion in a cylinder and a sphere can be described by equations (67) and (68), and multiplying the erf function argument by the term Rξ/Rξ provides(78)CR,T=CSCSCII,IerfΛ/ΘerfΛΘRRξ,(79)CR,T=C0+CI,IIC01erfφΛ/Θ1erfφΛΘRRξ.

Substituting equation (15) for R and Rξ, the following expression can be determined as a function of one-dimensional relations V/A0 and Vξ/A0, respectively:(80)CR,T=CSCSCII,IerfΛ/ΘerfΛΘV/A0Vξ/A0,and, similarly for equation (79),(81)CR,T=C0+CI,IIC01erfφΛ/Θ1erfφΛΘV/A0Vξ/A0.

By inserting equation (18) for the cylinder and equation (19) for the sphere in equations (80) and (81), the concentration profiles for cylinders can be derived as follows.

For 0<R<Rξ,(82a)CCylR,T=CSCSCII,IerfΛ/ΘC erfΛΘCR02R2R02Rξ2,and, for one phase-solidification for the region 0<R<Rξ,(82b)TCylR,T=TSTSTmerfΛ/ΘCerfΛΘCR02R2R02Rξ2,and, for Rξ<R<R0,(83a)CCylR,T=C0+CI,IIC01erfφΛ/ΘC1erfφΛΘCR02R2R02Rξ2,and, for one phase-solidification for the region 0<R<Rξ,(83b)TCylR,T=T0+TmT01erfφΛ/ΘC1erfφΛΘCR02R2R02Rξ2.

The case of concentration profiles for spheres provides for 0<R<Rξ,(84a)CSphR,T=CSCSCII,IerfΛ/ΘS erfΛΘSR03R3R03Rξ3,and, for two-phase solidification,(84b)TSphR,T=TSTSTmerfΛ/ΘSerfΛΘSR03R3R03Rξ3,and, Rξ<R<R0,(85a)CSphR,T=C0+CI,IIC01erfφΛ/ΘS1erfφΛΘSR03R3R03Rξ3,and, finally for two-phase solidification,(85b)TSphR,T=T0+TmT01erfφΛ/ΘS1erfφΛΘSR03R3R03Rξ3.

Now, the set of equations for the concentration profile, equations (82) and (83), for cylinder and, equations (84) and (85), for sphere, and diffusion times and equations (76) and (77) for the solute concentration profiles for cylinders and spheres, respectively, can be carried out by solving equation (74) for each radial position, that is, for n positions in the solution domain, which represents n one-dimensional correlation positions V/A0. By consequence, regarding the V/A0 correlation, now there will be n roots to be calculated from equation (74).

3. The Numerical Method

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  were applied to a slab, a cylinder, and a sphere. The complexity of solving the moving interface and its high nonlinearity, due to the rise of a discontinuity in the concentration field between the regions Rξ and Rξ+, was circumvented by rewriting the solid-state diffusion partial differential equation in terms of a concentration density field ρC , e.g.,(86)ρCt=DPxCx+2Cx2,0<x<xξ, and t>0,for t=0,(87)0xxξ,C=C0,for t>0,(88)x=0,C=CS,(89)x=xξ,C=Cξ,(90)x>xξ,C=C0,(91)ρCξρC0dxξdt=D¯Cx|xξ0,where P=0, for the slab, P=1, for the cylinder, and P=2, for the sphere. The diffusion coefficient, in this case, is now taken as D¯=D/ρ in kgm1s1.

The equation to deal with the concentration density field is similar to those applied for the heat conduction with phase change [6, 18] as follows:(92)ρC=ρCC<Cξ,ρC0C>Cξ.

The concentration “C” can be extracted from the concentration density field ρC via(93)C=ρCρρC>ρCξ,C0ρC<ρCξ.

4. Results and Discussion

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 (1)–(14) which were considered for atomic and thermal diffusion with phase change. The solution regarding two moving boundaries can also be derived. Two-phase diffusion was chosen as an additional application example. If an analytic solution for diffusion problem exists for semi-infinite slab in terms of error function, for instance, one that considers convective boundary conditions , this method can be applied with no-restriction.

4.1. Single Phase

Figure 6 presents the geometric correlations derived by Moreira  for solid-state diffusion of slabs, cylinders, and spheres as a function of the dimensionless interface position undergone to diffusion. Ferreira  applied the geometric correlation “ΘC” to derive an exact analytical solution for inward diffusion in a cylinder. As can be noticed, at the end of the diffusional process, these correlations assume ΘU=1 for slabs, ΘC=2 for cylinders, and finally ΘC=3 for spheres.

Comparison between numerical and analytical solutions for diffusion times as a function of one-dimensional correlation V/A0 for a slab, a cylinder, and a sphere.

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  were applied to a slab, a cylinder, and a sphere, as shown in Figure 7 for single-phase atomic diffusion with phase change and Figure 8 for single-phase solidification. The space grid used were divided in 100 representative elementary volumes (REVs) and dt=1,5x103s. A good agreement is observed for all cases between analytical results and numerical scatters. As the values for diffusion time equation (94), solute concentration, and temperature profiles, and equation (95) are the same in dimensionless form, there is no need to provide a further set of comparisons with the numerical method for the case one-phase solidification. That is the reason why the behavior of Figures 7 and 8 are the same, only the scale of the problems differs from one another:(94)Τ=Dtr02=αstr02,(95)Ξ=CCSCξCS=TTSTmTS,where Τ and Ξ are the dimensionless time and concentration or temperature, respectively.

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 9 and 10 present analytical simulations for a slab, a cylinder, and a sphere at dimensionless interface positions of 30% of R0 for atomic diffusion and single-phase solidification. In similar manner, Figure 11 present for atomic flux and Figure 12 present for solidification at a position of 70% of R0. As expected, considering Figures 8 and 11, the lowest diffusion rate is observed for the slab and the highest for the sphere, as the amount of material for an atom to be diffused decrease as the diffusion thickness of the cylinder and sphere increase. This fact is directly related to the geometric correlation factor Θ, which according to equation (20) changes the diffusion kinetics to conform the geometric curvature to the radial cylindrical and spherical coordinate systems.

Comparison of concentration profiles during inward diffusion of a slab, a cylinder, and a sphere as a function of dimensionless position Rξ/R0 at 30% of R0.

Comparison of concentration profiles during inward diffusion of a slab, a cylinder, and a sphere as a function of dimensionless position Rξ/R0 at 70% of R0.

Comparison of concentration profiles during inward one-phase solidification of a slab, a cylinder, and a sphere as a function of dimensionless position Rξ/R0 at 30% of R0.

Comparison of concentration profiles during inward single-phase solidification of a slab, a cylinder, and a sphere as a function of dimensionless position Rξ/R0 at 70% of R0.

Figure 13 shows the diffusion times as a function of the one-dimensional relation Vξ/A0 representing the diffusion interface position for the three geometries. Both cylinder and sphere provide the higher times as compared to that of the slab. This observation can be easily explained in terms of the thickness of material subjected to the diffusion process. For the same value of the one-dimensional relation Vξ/A0, the slab has the greatest amount of material remaining, followed by the cylinder and sphere, respectively.

Comparison between numerical and analytical solutions for diffusion times as a function of one-dimensional correlation V/A0 inward a slab, a cylinder, and a sphere.

Figure 14 provides the similarity variable equations (24)–(33) for slab, cylinder, and sphere. In the case of the slab, the root is unique for all the domains, as it does not present any geometric curvature to exert influence on the diffusion kinetics. Nevertheless, for cylinders and spheres, this is not the case. For inward diffusional process, considering both radial geometries, the curvature imposes an acceleration of the process as far as the amount of remaining material not undergone to the atomic diffusion decreases as the radius decreases. Then, for each one-dimensional relation Vξ/A0 for the radial coordinate systems, the root must be updated at each radial position.

Comparison of similarity variables in relation to the concentration profiles during inward diffusion of a slab, a cylinder, and a sphere at 70% of R0.

Figure 15 shows similarity variables for the slab, cylinder, and sphere as a function of the dimensionless interface position Rξ/R0. As can be observed, Figure 7 is very similar to Figure 6. The reason is that, in equation (41), where the roots are updated at each radial position, the only difference among the three geometries is the geometric correlation Θ. This observation permits to state that equation (33) changes the kinetics of the diffusion process to encompass the geometric curvatures of the radial geometries. This is an important remark of the solution, as far as, once provided a one-dimensional relation Vξ/A0 and a geometric correlation Θ and an analytical solution for a constant cross-section geometry, an exact analytical solution for a generalized coordinate geometry problem, based on these assumptions can be derived.

Comparison of similarity variables in relation to the moving interface dimensionless position Rξ/R0 during inward diffusion of a slab, a cylinder, and a sphere at 70% of R0.

4.2. Two Phases

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 Vξ/A0 to encompass the kinetic of radial geometries by the geometric update of the roots provided by equation (74) for each radial position of the moving interface. By considering the proposed solution, a solid-state diffusion in the two-phase system and a two-phase solidification problem was chosen, as shown by the governing equations (7) to (14) and the solution derived from equations (49) to (85). The diffusion times for the atomic diffusion and for the solidification problems are presented in Figures 16 and 17, where it can be noticed that atomic diffusion times and solidification times have the same features, but the solid-state diffusion is a thousand times faster than solidification. While performing upward solidification of pure Al in a 100 mm height mold, Ferreira et al. , even considering there was a high transient global heat transfer coefficient at the bottom, not assumed here, but instead a Dirichlet (prescribed temperature) boundary condition at the heat cooling surface (chill), times around 70 seconds were observed for pure aluminum, what agrees for the times provided by the Neumann solution (semi-infinite slab) represented by the black line in Figure 17.

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 18 and 19 present analytical simulations for a slab, a cylinder, and a sphere at dimensionless interface positions of 30% of R0 for two-phase atomic diffusion and two-phase solidification. On the contrary, Figures 20 and 21 represent the phenomena at a position of 70% of the moving interface. Considering that, in the solid-state diffusion case, atoms are being transported into the sample from the surface, while in the solidification problem, heat is extracted from the bulk to the surface. In the first case, the slab concentration profile is the highest, while in the second case it is the lowest.

Comparison of concentration profiles during inward diffusion of a slab, a cylinder and a sphere as a function of dimensionless position Rξ/R0 at 30% of R0.

Comparison of concentration profiles during inward diffusion of a slab, a cylinder, and a sphere as a function of dimensionless position Rξ/R0 at 70% of R0.

Comparison of concentration profiles during inward diffusion of a slab, a cylinder and a sphere as a function of dimensionless position Rξ/R0 at 30% of R0.

Comparison of concentration profiles during inward two-phase solidification of a slab, a cylinder, and a sphere as a function of dimensionless position Rξ/R0 at 70% of R0.

In the similar way, Figure 22 provides the similarity variable equation (61) for the slab, cylinder, and sphere for the case of two-phase diffusion and solidification phenomena. The same behavior is noticed for the slab, as shown in Figure 14. The root is unique for all the domain, as it does not present any geometric curvature to influence the diffusion kinetics. Nevertheless, for cylinders and spheres, this is not the case. The curvatures at positions close to 0.01 concentration, for the cylindrical and spherical solutions are due to the existence of a profile in the second phase. For inward diffusional process considering both radial geometries, both the curvatures impose an acceleration of the process, as far as the amount of remaining material not reached by the moving boundary interface decreases with the radius. This implies that, for each one-dimensional relation Vξ/A0 for the radial coordinate systems, the root must be updated at each radial position of the moving interface, for equations similar to equation (74), acting as a transcendental equation.

Comparison of similarity variables in relation to the concentration profiles during inward diffusion of a slab, a cylinder, and a sphere at 70% of R0.

Figure 23 shows similarity variables for the slab, cylinder, and sphere as a function of the dimensionless interface position Rξ/R0 for the two-phase diffusion. The behavior is similar to that of Figure 15 for one-phase diffusion. The kinetic implies for the radial geometries that the similarity variable is two folds that of the slab for a cylinder and three folds for a sphere at the end of the process. That is why it is normally observed that the Neumann solution is enough to be applied providing accurate predictions at positions very close the surface of cylinders and spheres, as far as the influence of geometry on the kinetic can be neglected.

Comparison of similarity variables in relation to the moving interface dimensionless position Rξ/R0 during inward diffusion of a slab, a cylinder, and a sphere at 70% of R0.

5. Conclusions

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 Θ and the corresponding one-dimensional volume-surface area relations Vξ/A0 in order to achieve a closed-form solution, in this case, for single-phase and two-phase phase-change problems. The geometric correlation changes the basic partial differential equation for space and time variables, in order modify the diffusion coefficient for each one-dimensional relation R=Vξ/A0 position to encompass the effect of the curvature of the radial geometries on the kinetics of the semi-infinite slab solution. The model proved to be general, as far as, a semi-infinite solution for diffusion problems with phase-change exists for a slab, the kinetic of the radial geometries enables exact analytical solutions of inward diffusion in radial geometries to be derived, by updating the root for each radial position of the moving boundary interface by equations similar to equations (41) and (74), acting exactly as a transcendental equation, providing an update for the roots, as observed in the behavior of similarity variable as a function of the solute concentration for single-phase and two-phase diffusion.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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).