The Mathematical Modeling of Diffusion with Reaction in the Case of Chromisation

The Mathematical Modeling of Diffusion with Reaction in the Case of Chromisation

The Mathematical Modeling of Diffusion with Reaction in the Case of Chromisation

Prof. dr. ing. Ana Veţeleanu; Assist. ing. Daniel Chiriac

“Transylvania” University of Brasov-Romania


The paper contain a study refering to the mechanism of diffusion with reaction, proper to the chromisation of the alloys Fe-C with a content of C>0,3% and modeling of the diffusion process.

The obtaining in the diffusion layer of some components of composite (mixed) carbides type ( Cr23C6; Cr7C3) as well as the existence in the layer of three different areas: the mixed carbides area, at the surface, the intermediate area, saturated with carbon and the decarbonized area at the bottom. Excludes the possibility of applying Fick’s laws in order to explain the mechanism of diffusion.

The establishment of some boundary condition (diffusion coefficient of Cr constant for each phase, the variation of concentration “Ci “ at each phase according to a certain law, the establishment of a boundary concentration at the borderline between the carbides area and the area rich in carbon) has allowed the modeling of the diffusion process.

The theoretical data, as well as the boundary conditions imposed for modeling have been cheeked experimentally through chemical analyses of qualitative optical microscopy, subjecting the ball-bearing still and the graphite nodule slit to a process of liquid chromisation.

The efficiency of the chromisation process that vas been discussed, has stood out through the mechanical characteristics of the layer, characteristics that have been obtained (fracture strength, hardness, abrasive resistance).

Introductory word

Chromizing, as process of enriching of the surface layer in chromium , applies to plain carbon steels and to alloy steels, to cast irons, to alloys having as base chromium, nickel, cobalt, niobium, molybdenum.

The chemical and mechanical characteristics obtained from the process of chromizing depend on the composition and the structure of the basic material ,on the concentration and on the characteristics of the added material, the parameters of the process of chromizing,etc.

Chromium, being an -type element, its diffusion is more rapid in steels with ferrite structure, determining the formation of the solid solution  satiated in chromium ,oxidation-resistant at high temperatures and corrosion.

In the case of iron-carbon alloys with a content of carbon >0.3% the structure of the chromized layer is formed of mixed carbides of chromium, characterized by hardness and abrasive resistance.

Function of the characteristics of the work environment there is a variety of procedures of chromizing. It is the variant of chromizing in liquid medium that has been chosen, due to its special performances, partially excluding the disadvantages of the other procedures.

The evolution of the technological process of chromizing imposes the establishing of an environment able to set free active atoms of chromium .This can be done either as a result of a process of dissociation (solid and gaseous medium) or as a result of chemical reactions.

Irrespective of the nature of the process, there have to be created the conditions that are necessary to its development, that is: the working environment should be rich in the active element, the time, the temperature and the pressure necessary to the development of the process of dissociation should be at optimal parameters, that is the chemical reactions should develop normally.

Experimental research:

In the case of liquid chromizing, the basic working environment , chosen for experimenting, consists of 60% BaCl and 10%NaCl.The active medium of the process consists of 20% CrCl2 and chromic iron, ground in proportion of 30% of the salts’ weight.

As a result of the chemical reactions taking place in the bath at a temperature of 1050˚ C we obtain the chemical chloride(CrCl3) and chromium, this being in fact the active element.

3 CrCl22 CrCl3 + Cr

CrCl3 + FeFeCl3 + Cr

The introduction in the bath of CrCl3 produced industrially is not conducive for the development of the process of chromizing, this being conditioned by the percentage of CrCl2 that is introduced and by the reaction rate, so that in the end ], the ratio:

As subject for study there have been chosen the alloys Fe-C with a content of C>0.3% due to the complexity of the process of diffusion .Due to the high content of carbon, at the surface there is formed a double layer of mixed carbides(M23C6/M7C3).Their formation in the case of nodular cast iron, for example, is favoured by the diffusion of the chromium from the working environment and the auto-diffusion of the carbon inside , contained both by the basic metallic mass as well as by the carbon nodules, determining the increase of the quantity and of the complexity of carbides.

This complexity of diffusion has imposed a more detailed analysis of this process as well as the establishing of a mathematical model that should make possible the evaluation of the parameters of the chromized layer.

If in the case of the chromizing of the alloys Fe-C with a content of C>0.3% the diffusion of chromium is realized through vacancies, in this second case , the diffusion is with reaction. The theoretical variation of the concentration in chromium along the depth of the layer is presented in figure 1, where ά represents the basic mass of the material and β the chromized layer formed of carbides.

Fig.1. The diffusion with chemical reaction

Because in the case of the diffusion with reaction , there also takes place process of auto-diffusion of the carbon inside in figure 2. there is presented the way of variation of the concentration of carbon, along the depth of the chromized layer.

Fig.2. The variation of the concentration of C along the depth of the chromized layer

At the surface, between carbon and chromium, there takes place a chemical reaction forming mixed carbides of Cr. Along the depth of the layer there can be distinguished an area of carbides (Cr 23 C 6;Cr 7 C 3) in which the concentration of Cr is of 90% , an intermediate area satiated in carbon (1) and a decarbonized area (2).

In order to solve the equation of diffusion, there are considered the following:

-the coefficient of diffusion at each stage is constant

-at each stage, the concentration “Ci” varies according to the following law:


- the interface between the area of carbides and area(1) rich in carbon is a plan of constant composition, that drifts function of according to the relation:


-the material balance at the mobile interface shows that the mass of substance that diffuses and crosses the plan of discontinuity in the interval equals the difference between the flux of diffusion from one side to the other.



Cάβ=concentration of chromium in stage β [%]

Cβά=concentration of chromium in stage ά [%]

Dβ =coefficient of diffusion in stage β [cm2/sec]

D=coefficient of diffusion in stage ά [cm2/sec]

x=depth of the layer [cm]

The relations represent the initial conditions and the imposed boundary conditions. The determination of the concentration at the interface imposes:

-the derivation of the equation(1)


-the account of the gradient of concentration in and .


Where we have noted:

We consider the case of liquid chromizing of nodular cast iron in static position ,at 1050 º C, for two hours.In this case, phase β is made of carbides and phase ά is a ferrite structure satiated in carbon.


=0; x>0; C=C0





It is determinated




In order to determine the concentration at the interface we consider the following values:

-the coefficient of diffusion of chromium in carbides

D(Cr7C3)=D0 exp (-Ea/RT)

D0=5x10-4 [cm2/sec]

Ea=40 [kcal/atom grad]

D(Cr7C3)=1,09 x10-10 [cm2/sec]

D(Cr23C6)=D0exp (-Ea/RT)

D0=6x10-5 [cm2/sec]

Ea=40 [kcal/atom grad]

D(Cr23C6) =1,38 x10-11 [cm2/sec]

-the coefficient of diffusion of chromium in ferrite

D=10-10 [cm2/sec]

-the concentration of chromium in carbides

%Cr (Cr23C6)=94


-the concentration of chromium at the surface of the layer


-the initial concentration in chromium of the material


-the depth of chromizing determined theoretically and checked experimentally


The calculated measures based on the presented data are centralized in table(1)

Table 1 The calculed measures based
Variables / C C
Calculated measures / 0,44 0,19 1,17 5,85 0,89

From the analysis of the presented data we notice that the interface is situated at about 0.19mm from the surface, that corresponds to the experimental results (x=0.2).There is an intermediate area between the carbide area of the layer and the ferrite one of the slug in which the concentration in chromium is of 0.89%.

A particular feature of liquid chromizing is represented by the fact that the activity of the satiating environment decreases in time.In the case of a decreased activity of the satiating environment , the law of layer increase becomes linear.

The increase of the activity is realized through the periodical completion of the bath with active element (CrCl 2 and chromic iron) either through inducing the bath that has as effect the entrainment of the deposed reaction products and the increase of the reaction rate.

From the analysis of the experimental data obtained in the two variants of chromizing (in static and mobile mixtures) ,keeping the working parameters constant, we notice that the activation energy decreases with 40% in the case of chromizing in mobile mixture.

The general sheet of the chromizing installation in mobile mixture is presented in figure 3 and the microstructures obtained chromized assay samples in static liquid mixture at the temperature of 1050 º C at different intervals of time are presented in figure 4

Figure 3.The general sheet of the liquid chromizing installation


Figure 4. The micrographs of the chromized assay samples at a temperature of 1050 º C at different time intervals (a-1h; b-1,5h; c-2,5h)


The liquid chromizing applied to alloys of Fe-C with a content of Cr>0.3% ensures an increase of the abrasive resistance due to the formation of mixed carbides in the layer, from this point of view ranging among the larger field of surface engineering.The main purpose of “surface engineering” is to build up a complex system formed of surface and the sublayer adherent to it, system with plainly superior performances to the two elements taken separately.

The structure of the layer and of the sublayer in the case of liquid chromizing of the alloys Fe-c with C>0.3% determines an increase with 30% of the abrasive resistance compared to other technologies of deposing the chromium .Knowing the mechanism of diffusion and the mathematical modeling allow for an easier interpretation of the structure of the layer and an easier determination of its parameters.


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