^{1}

^{1}

^{*}

Two simplified models, linear and nonlinear, were used in a cementation process on a homogeneous thin carbon steel plate. The parameters for these models, as obtained by the least squares’ method the first one in a global way while the other parameters refer to the second model—were estimated by a set of local minimums. To compare the performance of these models we used theoretical data, for the same diffusion problem obtained by a one-dimensional transient model considering the concentrations in the mean plane of the plate. The results for carbon concentrations in weight percentage in the plate (%pC) as a time-only dependent function with these simplified models to represent the analyzed diffusion process were in good agreement with those from a stricter model. The diffusion flows of these models were determined and a reasonable agreement can be seen in relation to the flow obtained by the theoretical model on the surface of the plate. This study shows that it is possible to use this methodology with the given restrictions adopted here to describe the concentration and the diffusion flow of other solutes in thin membranes.

The one-dimensional diffusion in a medium is limited by two parallel planes, for example, in x = 0 and x = L of such thin thickness, so that the entire diffusive process occurs through these sheets or membranes, while only a negligible amount occurs through the lateral faces, which is well known [

The cementation process consists of the hardening of the surface of steel to higher levels to that of its interior by the diffusion or transport of carbon atoms at high temperatures in an atmosphere rich in hydrocarbon gas such as CH_{4} methane gas [

d C d t = k A V [ C e − C ( t ) ] , (1)

where C_{e} is the concentration of the medium surrounding the membrane, and k is a membrane permeability constant, A is the surface area, V is the constant volume of the cell, while C(t) it is the concentration of the diffusing solute in the cell in t time. The k constant depends on each solution, the thickness and the structure of the membrane, so it needs to be estimated for each situation. The second model, also simplified, but with two parameters, is given by an ordinary non-linear differential equation as

d [ C ( t ) − C e ] d t = A V [ k 1 ( C e − C ( t ) ) + k 2 ( C e − C ( t ) ) 2 ] . (2)

The term in square brackets is the function for solute flowing into the cell [

According to Bassanezi and Ferreira Jr. [

The results show that there was good agreement among the models adopted in this study with the results of the carbon concentrations estimated by the one-dimensional model of transient diffusion on the plane of the plate x = 0.0005 m or membrane. Thus, the simplified models, in particular the one-parameter model, gave a good description of the problem analyzed, providing a semi-qualitative mean value for the concentration within the plate at any point and instant in the x direction.

Consider a homogeneous metal plate formed by an Iron-gamma Carbon alloy, indicated by, Fe_{γ}-C which hast to be hardened through a process of cementation [_{4}), under T = 1000 ˚ C . The homogeneous plate with the properties shown in

The thickness is so small that it will be assumed that the entire diffusion process happens through these sheets or membranes, while a negligible amount

PROCESS PARAMETERS | NOTATION | VALUES |
---|---|---|

Surface area of the plate | A | 29.4 × 10^{−4} m^{2} |

Volume of the plate | V | 29.5 × 10^{−7} m^{3} |

Board thickness | x L | L = 0.001 m |

Diffusion Rate | D o | 2.3 × 10^{−5} m^{2}/s*^{ } |

C initial concentration | C ( x , 0 ) = C o , 0 < x < L | 0.08% pC |

C concentration on faces | C ( 0 , t ) = C 1 ; C ( L , t ) = C 2 | 0.10% pC |

Diffusion temperature | T | 1000˚C*^{ } |

Activation energy | Qd | 148 kJ/mol*^{ } |

Constant of the gas | R | 8.31 J/mol∙K*^{ } |

Iron density range | ρ Fe γ | 7.86 g/cm^{3}**^{ } |

Carbon density | ρ C | 2.267 g/cm^{3}*** |

*Reference value obtained for this diffusion (Callister, 2002, p. 70,

occurs through the lateral faces [

Given a set of points ( x i , y i = f ( x i ) ) , i = 1 , ⋯ , n and a ≤ x i ≤ b , the goal is to choose g i ( x ) continuous real functions in the I = [ a , b ] interval to obtain constants or parameters λ 1 , ⋯ , λ n so that

ϕ ( x ) = ∑ i = 1 n λ i g i ( x ) ≅ f ( x ) . (3)

As the λ i coefficients appear linearly in the definition of the φ ( x ) approximation function, this model is called linear [

F ( λ ) = F ( λ 1 , ⋯ , λ n ) = ∑ i = 1 m R i 2 (4)

where m ≫ n . To get a F ( λ ) minimum point of it is necessary to solve the equation of the critical points given by

∇ F ( λ ) = 0 → . (5)

The equations obtained from Equation (5) give rise to a linear system of n × n order given by

A λ = b , (6)

where, according to Ruggiero e Lopes [

If ϕ ( x ) it is not a linear model of the parameters as in Equation (3), the equation of the critical points no longer produces a linear system as obtained in Equation (6). Consider

R : U ⊂ I R n → I R m R ( λ ) = ( R 1 ( λ ) , ⋯ , R m ( λ ) ) T . (7)

Therefore, as R ( λ ) is a vector function for residues in the R m space and φ i ( λ ) = φ ( x i , λ ) is the nonlinear model for adjustment point ( x i , y i = f ( x i ) ) , i = 1 , ⋯ , m and a ≤ x i ≤ b , where λ = ( λ 1 , ⋯ , λ n ) T is a vector of adjustable parameters of the I R n space. With this notation, the k-th residue of this approximation is defined by R k ( λ ) = f ( x k ) − ϕ ( x k , λ ) , k = 1 , ⋯ , m . The objective is to minimize F ( λ ) as in Equation (4), that is, the Equation (5) of the critical points is written thus

∑ i = 1 m R i ( λ ) ∂ ϕ ( x i , λ ) ∂ λ k = 0 , k = 1 , ⋯ , n (8)

Using a Taylor series expansion [

R i ( λ + p ) = R i ( λ ) + ∇ R i ( λ ) p , (9)

where ∇ R i ( λ ) = ( ∂ R i ( λ ) ∂ λ 1 , ⋯ , ∂ R i ( λ ) ∂ λ n ) , with i = 1 , ⋯ , m .

From Equations ((8), (9)) it is k = 1 , ⋯ , n

∑ i = 1 m [ R i ( λ ) + ∑ j = 1 n ∂ ϕ ( x i , λ ) ∂ λ j p j ] ∂ ϕ ( x i , λ ) ∂ λ k = 0 . (10)

The Jacobian matrix of the R ( λ ) transformation is an m × n order matrix given by

J ( λ ) = [ ∂ ϕ ( x 1 , λ ) ∂ λ 1 ⋯ ∂ ϕ ( x 1 , λ ) ∂ λ n ⋮ ⋱ ⋮ ∂ ϕ ( x m , λ ) ∂ λ 1 ⋯ ∂ ϕ ( x m , λ ) ∂ λ n ] . (11)

From Equations ((10), (11)) we have a linear system of n × n order given by

J T ( λ ) J ( λ ) p = − J T ( λ ) R ( λ ) . (12)

Equation (12) is the basis for an iterative process and is known as a modified Newton method [

The solution of the transient one-dimensional diffusion equation for the diffusion problem in thin membranes with constant surface concentrations and initial distribution with uniform concentration, that is,

C 1 = C ( 0 , t ) , C 2 = C ( L , t ) ; t ≥ 0 and C 0 = C ( x , 0 ) ; 0 < x < L , (13)

can be obtained by the method of separation of variables, whose solution is given according to Crank [

C = C ( x , t ) = C 1 + ( C 2 − C 1 L ) x + 2 π ∑ n = 1 ∞ ( C 2 cos ( n π ) − C 1 n ) sin ( n π x L ) e − D n 2 π 2 t L 2 + 4 C 0 π ∑ m = 0 ∞ ( 1 2 m + 1 ) sin ( ( 2 m + 1 ) π x L ) e − D ( 2 m + 1 ) 2 π 2 t L 2 (14)

Model 1 as given and reported by Equation (1) is given by

d C d t = k A V [ C e − C ( t ) ] , (15)

as proposed by Bassanezi [

∫ C 0 C d C C e − C ( t ) = k A V ∫ 0 t d t . (16)

From Equation (16) results the C ( t ) expression for given by

C = C ( t ) = ( C 0 − C e ) e − k A V t + C e , (17)

where C 0 = C ( 0 ) . Note that C ( t ) → C e when t → ∞ .

In Equation (15), an analogy with Fick’s first law for one-dimensional diffusion in a steady state [

d [ C ( t ) − C e ] d t = A V [ k 1 ( C e − C ( t ) ) + k 2 ( C e − C ( t ) ) 2 ] . (18)

Making f ( C ) = F ( C e − C ) where F ( C e − C ) it is denominated by the authors as a function for flow based on the difference of concentrations, where f ( C ) is a real function of a real variable, and where f ( C e ) = F ( C e − C e ) = F ( 0 ) = 0 .

To justify this fact we assume the plausibility where if there is no difference of concentration there should be no flow between the means. Assuming f ( C ) is at least C 3 ( I , R ) class function; where I = [ 0 , t 0 ] [

f ( C e − h ) = f ( C e ) + f ′ ( C e ) ( C e − C ( t ) ) + f ″ ( C e ) 2 ( C e − C ( t ) ) 2 , (19)

or,

f ( C ) = F ′ ( 0 ) ( C e − C ( t ) ) + F ″ ( 0 ) 2 ( C e − C ( t ) ) 2 . (20)

From Equation (15) we have seen that J = k ⋅ [ C e − C ( t ) ] represents a flow of molecules into the cell, then replacing that term with the given f ( C ) flow function as in Equation (18), we obtain the two-parameter formulation for cell diffusion only reported by Bassanezi and Ferreira Jr. [

d d t ( C − C e ) = A V [ k 1 ( C e − C ( t ) ) + k 2 ( C e − C ( t ) ) 2 ] , (21)

where, k 1 = k F ′ ( 0 ) and k 2 = k F ″ ( 0 ) 2 . For the physical sense of the problem, the unity of k 1 is the same as of k, that is, m/s, while the unity k_{2} of can be given as, m^{4}/kg∙s. Making y = C − C e the Equation (21) rewrite in the form

d y d t + k 1 A V y = k 2 A V y 2 . (22)

Equation (22) is a Bernoulli equation [

d u d t − k 1 A V u = − k 2 A V . (23)

Equation (23) can be solved using the μ ( t ) = e − k 1 A V t integral factor to obtain

u = K k 1 e k 1 A V t + k 2 k 1 . (24)

Using the fact that u = y − 1 and y ( 0 ) = C 0 − C e follows from Equation (23) the solution to Equation (21) given by

C = C e + k 1 ( C 0 − C e ) k 2 ( C 0 − C e ) + e k 1 A V t [ k 1 − k 2 ( C 0 − C e ) ] . (25)

Note what C ( 0 ) = C 0 and C ( t ) → C e when t → ∞ . In particular, if F ′ ( 0 ) = 1 and F ″ ( 0 ) = 0 and only then, the differential equation describing model 1, as given by Equation (13) is retrieved. However, if F ′ ( 0 ) ≠ 1 and F ″ ( 0 ) = 0 , the constant k 1 = k F ′ ( 0 ) ≠ k , and in this sense, Model 2 is not exactly an extension of Model 1, but the analytical form of Model 2 is an extension of Model 1. To estimate the parameters of Equations (17) and (24) we will use the theoretical data obtained from the cementation process, whose parameters are shown in

The permeability parameter (Model 1) can be obtained through the global minimum with the linearization of Equation (15) according to the procedure

Time (hours) | % pC (MD) |
---|---|

0.0 | 0.0008000 |

0.5 | 0.0008231 |

1.5 | 0.0009089 |

2.0 | 0.0009353 |

3.0 | 0.0009674 |

5.0 | 0.0009917 |

7.0 | 0.0009979 |

9.0 | 0.0009990 |

described in Section 2.1. doing y = C − C e , and therefore Equation (15) is rewritten thus

− y = ( C e − C 0 ) e A k V t . (26)

In applying the Neperian Logarithm to Equation (26) we obtain

z = ln ( − y ) = ln ( C e − C 0 ) − k A V t . (27)

Thus, using the discrete points in

z i = B + λ t i , (28)

where z i = ln ( C e − C ( t i ) ) , B = ln ( C e − C 0 ) and λ = − k A V . As B is constant, Equation (28) can be reduced to a linear form given by

w i = λ t i , (29)

which w i = z i − B . Therefore Equation (28) can be solved by rewriting Equation (1) as, φ ( t , λ ) = λ t being g 1 ( t ) = t . Thus, the matrix of the system given by Equation (4) is of an order of 1 × 1 , that is, a 11 = 〈 g ¯ 1 , g ¯ 1 〉 = ∑ i = 1 7 t i 2 , b 1 = 〈 y ¯ , g ¯ 1 〉 = ∑ i = 1 7 t i w i . Hence, Equation (6) gives the first-order linear equation given by

a 11 λ = b 1 ⇒ λ = − 104.4298 170.50 = − 0.612491 . (30)

As a result, λ = − k A V and A, V are given as shown in

To estimate the Model 2 parameters, we used the non-linear least squares’ method described in Section 2.2. From Equation (24) two models were proposed: the model named MD21, where only the k 2 parameter was allowed to vary while k 1 = k and Model MD22, where both parameters varied. The initial parameter vector for MD21 was p 21 0 = 0.01 with a tolerance t o l = 10 − 4 and convergence occurring in only two cycles for the value of k 2 2 = 1.9224 , where ‖ R ( p 21 2 ) ‖ 2 = ∑ i = 1 m R i ( p 21 2 ) 2 = 1.0567 × 10 − 9 . For MD22, the initial parameter vector was taken as p 22 0 = ( 0.0007 , 1.0 ) T with a tolerance t o l = 10 − 5 and convergence also occurring in two cycles for the p 22 2 = ( k 1 2 = 0.0004 , k 2 2 = 3.4907 ) T , the vector being ‖ R ( p 22 2 ) ‖ 2 = ∑ i = 1 m R i ( p 22 2 ) 2 = 1.0544 × 10 − 9 . Please note that in terms of quadratic residues, there is no significant difference between these iterative models.

The relative percentage error for MD1, ε MD1 in relation to the estimates of concentration in %pC estimated by the theoretical model MD, is below 3.7%.

Similarly, for MD21, ε MD21 < 6.2 % while for the MD22 model, ε MD22 < 6.5 % . The highest divergence between concentrations took place in the t ≤ 2.0 h time interval. Based on the criteria analyzed, and because the MD1 model has its parameter estimated in a global way, and due to its greater simplicity, it is at first the model to be adopted to estimate the mean carbon concentration in the thin plate.

One of the advantages of these simplified models is that their expressions for C ( t ) are analytical, as opposed to the solution obtained by Equation (14) which is given in terms of an infinite series.

If we consider the x = 0.00025 m plane of the plate, the maximum percentage

difference between the concentrations estimated by the MD1 model for the x = 0.0005 m plate plane are lower than 2% when compared with the theoretical data (not shown here), for the estimates of the concentrations in x = 0.00025 m obtained by the MD model. This allows us to assume that the simplified model to a parameter can estimate, in a semi qualitative way, the %pC concentrations inside the plate or thin membrane, and without the need to use iterative methods where the global minimum cannot be obtained. Unlike the MD model, the diffusion of the simplified models lies in the analytical functions. That is, from Equation (15) the diffusion flow function is given by J 1 = k ⋅ [ C e − C ( t ) ] , while in Equation (20), the diffusion flow is given by J 2 = k 1 ( C e − C ( t ) ) + k 2 ( C e − C ( t ) ) 2 . From Equation (12) the diffusion flow across the face is given using [

J = − D ∂ C ( 0 , t ) ∂ t = − D { 2 L ∑ m = 1 ∞ [ C 2 cos ( m π ) − C 1 ] e − D m 2 π 2 L 2 t + 4 C 0 L ∑ n = 0 ∞ e − D ( 2 n + 1 ) 2 π 2 L 2 t } (31)

The flow functions from simple models are dependent on the difference in concentration between the media, so it is to be expected that the J 2 flow will be less pronounced than the J 1 flow.

to the same terms of the MD1 model, indicating that the diffusion for models MD21 and MD22 is “less apparent” in relation to the J 1 model.

After two hours of the cementation process, the carbon transfer rates through the flat section on the surface of the plate were in good agreement when compared with the data provided by the theoretical model. A possible explanation for the discrepancy of flows estimated by the simplified models for times under two hours can be credited to the fact that the parameters estimated for the models were based on the data of the theoretical concentrations obtained for the mean plane, and not the surface plane where the concentration was kept fixed during the cementation process.

The analyzes showed that the simplified diffusion models in cell membranes analyzed in this study may be an alternative to the transient one-dimensional models used for the description of membrane diffusion processes. The simplest models depend on parameters that can be obtained globally using known function adjustment methods, such as that of the least squares. In particular, they were used to obtain percentages by weight of carbon in a cementation process with restricted thickness conditions. The results obtained were in good agreement when compared with the estimates of the theoretical model used for this purpose. The simplified model with one parameter was shown to be the best option to represent the average estimate of the concentration of carbon solute in the diffusion process due to the concentration difference on the plate or membrane, and this is due to the fact that this model uses only one parameter and that it can be obtained non-iteratively; that is, in a global way.

The authors declare no conflicts of interest regarding the publication of this paper.

de Araújo, J.C. and Márquez, R.G. (2019) Simple Models for Diffusion in Thin Plates or Membranes. Journal of Applied Mathematics and Physics, 7, 1547-1559. https://doi.org/10.4236/jamp.2019.77105