Effects of thermoviscous flow and mechanical

stresses on nitrogen desorption during iron

transformation

Abstract. - The objective of this paper is a theoretical calculation of the mechanical stresses due to nitrogen

pressure in an iron vacancy in the temperature range of 600 to 1100 °C and its effect on the swelling

phenomenon associated with the high-temperature viscous flow. The method for quantification is theoretical

and based on the analysis of experimental data reported in the literature. Two equations related to the

variable were generated: swelling index with time and the stress due to nitrogen pressure. Both the variables

are described in increasing ways and correlated significantly at the 99% confidence level. The equations

include the value of stresses obtained from previous papers for ferrite (Feα) of 621, 2 Kgf/mm2 and for

austenite (Feγ) of 727, 6 Kgf/mm2. The overall effect when opposing these values to the average sample cold

compression strength of commercial samples, which varies with the corresponding degree of reduction, the

final value of this parameter results in the range of 35.2 and 69.6 Kgf/mm2.

Keywords: Swelling, nitrogen, reduction, stresses.

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Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

Dam G. Oscar

https://orcid.org/ 0000-0002-0594-6757

oscar.curmetals@gmail.com

Unexpo, Vicerrectorado Puerto Ordaz

Puerto Ordaz-Venezuela

Resumen: El objetivo de este trabajo es realizar un cálculo teórico de los esfuerzos mecánicos debidos a la

presión de nitrógeno en una vacante de hierro en el rango de temperatura de 600 a 1100 °C y su efecto

sobre el fenómeno de hinchamiento asociado al flujo viscoso a alta temperatura. El método de cuantificación

es teórico basado en el análisis de datos experimentales reportados en la literatura. Se generaron dos

ecuaciones que relacionan la variable índice de hinchamiento con el tiempo y con el estrés por presión de

nitrógeno, en ambas las variables se relacionan de manera creciente y se correlacionan de manera

significativa al 99% de confianza. Las ecuaciones incluyen el valor de tensiones, obtenido de trabajos

anteriores, para ferrita (Feα) de 621,2 Kgf/mm2 y para austenita (Feγ) de 727,6 Kgf/mm2. El efecto global

resultante al oponer estos valores al promedio de resistencia a la compresión en frío de las muestras

comerciales, que varía con el correspondiente grado de reducción, da como resultado el valor final de este

parámetro en el rango de 35.2 y 69.6 Kgf/mm2.

Palabras clave: Hinchamiento, nitrógeno, reducción, esfuerzos.

Efectos del flujo termoviscoso y de las tensiones mecánicas en la desorción de

nitrógeno durante la transformación del hierro

Received (3/06/2022), Accepted (27/08/2023)

https://doi.org/10.47460/athenea.v4i13.60

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Vol. 4, Issue 13, (pp. 7-16)

I. INTRODUCTION

This investigation continues the swelling of iron oxides in metallization processes in the FeO/Fe phase

transition [1]. In this theoretical work, a mechanism and a calculation method are defined to obtain the values

of the pressure exerted by the absorption of nitrogen gas molecules. Consequently, the stresses generated by

the desorption of three nitrogen gas molecules inside a point defect or vacancy of the crystal lattice need to

be estimated in the presence of iron allotropic phases of ferritic (Feα) and austenitic (Feγ) iron at temperatures

between 900 and 1100 °C, respectively, without quantifying thermoplastic effects on iron.

The objective is to calculate the stresses (pN2) and their relationship with the swelling phenomenon,

incorporating the thermoplasticity characteristics of iron in the temperature range associated with the

evolution from nascent iron to when the highest metallic iron content is obtained during the reduction

process in the temperatures range 600 to 1100 °C. The following method was used for the purpose: (a) setting

the temperature for the formation of iron; (b) calculations of the stress produced by nitrogen solubility in

metallic iron described in the previous aimed temperature range [1], (c) fixing of temperature where iron

acquires the thermoplastic property. Thus, apply the plasticity calculation method, determine the retention

time at each reduction temperature, and associate the stress values with the swelling index (HI).

To quantify the values of the swelling index (HI) due to the expansion of dissolved nitrogen in a cluster of iron

vacancies. It is (convenient) necessary to highlight that obtaining solid-state iron begins by reducing its

minerals, oxides generally having fragile mechanical properties. The reduction progress starts with the

increase in temperature and the presence of reducing agents; this process initiates metallic iron formation.

The oxides and materials of known plastic properties at room and high temperatures present, and the viscous

flow predominates; consequently, it is necessary to define these properties and relate them to the nascent

iron. The thermoplastic property of iron will favor the participation of the abnormal swelling (HA) phenomenon

if the stress produced by the pressure of nitrogen is greater than the yield stress and even more significant

than the breaking stress. The value of the stresses previously obtained for Feα was 621.2 Kgf/mm2, and for

Feγ, it was 727.6 Kgf/mm2. The criteria to incorporate and determine the start temperature of the

thermoplastic effect of the metal is based on multiplying its melting temperature of the phases involved, FeO

1377ºC and 1535ºC for iron, by a specific proportionality factor, whose value varies depending on the source

consulted.

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

II. DEVELOPMENT

A. Fixing the nascent iron Nano crystal particle's temperature

In the reduction process of hematite iron oxide, with the increase in temperature, the well-known

transformation to magnetite, wustite, and metallic iron occurs; this begins with the formation of Nanocrystals.

After exceeding the critical radius, these will generate the nuclei and growth of iron Feα and become crystals,

and to reach the cubic crystalline structure centered on the body, it is necessary to group at least nine atoms,

two per cell with an edge of 0.33 nm. Thus, it turns out that the nascent iron is formed by Nanocrystals,

consistent with experimental results obtained in [2] when iron nucleation is formed on wustite reduced with

hydrogen at 800 °C and from a spheroidal nucleus with a diameter of 3.6 nm. The further growth of a

pyramidal body with a square base with sides 4-8 nm and a height of 1.2-1.8 nm, as shown in Image 1 below

determines the temperature at which the Nanocrystals appear. For this assessment, the Chaudron triple point

of the iron-oxygen equilibrium diagram was used; this is the temperature point at which the

magnetite/wustite/iron phases coexist simultaneously and, according to [3], is 590°C. For calculation purposes,

a reference temperature of 600 °C was assumed.

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Fig. 1. Semi-reduced natural ore particle reduced with 100% hydrogen, showing wustite (light gray) and several

metallic iron nuclei on the wustite surface.

Source: Author files.

B. Stress calculations in iron allotropic phases

Following the proposed calculation methodology to determine the pressure of three nitrogen molecules in

the vacancy of the iron crystal lattice, without considering thermoplastic effects, used in [1], and starting from

600 °C, the stress values were obtained, as shown in Table 1. It also got the relationship with the value of

resistance to rupture of the allotropic phases of iron at room temperature; the obtained values are 28 and

105 Kgf/mm2 for Feα and Feγ, respectively.

Table 1. Stress produced by nitrogen in a vacancy of metallic iron in the study temperature range.

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

These stresses, knowing that they exceed the breaking strength of the iron, will be indirectly related to the

expansion of the solid or agglomerate through the increase in the radius of the vacancy due to the

thermoplastic effect that occurred in the material, depending on the time of agglomerate retention at each of

the indicated temperatures.

The Nanocrystals generated at 600 °C will become crystals as the reduction process progresses and the

temperature increases since most industrial reduction processes are carried out under dynamic, not

isothermal, conditions.

C. High and low-temperature limits and the iron thermoplastic effect

The mechanical properties of materials are affected by temperature, specifically in crystalline solids such as

metals and their alloys. Thus, the resistance properties of metals decrease with the increase in temperature, in

exchange with the rise in plasticity properties and depending on the temperatures, stresses, and deformation

speed. This can be achieved by the metal adopting a behavior quasi-viscous. It behaves like a fluid, increasing

its original dimensions and limiting its use at high temperatures. In the criteria to determine the limits between

low and high temperature, the correction factor is disclosed in [4], p. 373, and [5], p. 450, which is Tlow 0.5 and

Thigh > 0.5 Tf. In an issue, it is considered to highlight that, in ferrous materials, the viscous flow begins at 420-

430 °C, p.p. 355 and 429, respectively. It would correspond to a factor of 0.39, while in [6], p. 363, an aspect of

0.33 or 0.5 is considered, and in [7], p. 5, the plastic deformation starts from T > 0.6 Tf.

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Thus, at high temperatures, it is necessary to consider the viscous flow due to hot forming or slow creep

(Creep), as in the present work where the vacancy cavity is subjected to nitrogen pressure. When deformed at

high temperatures, the material's response presents a curve similar to plastic flow at low temperatures.

However, according to [6], these results do not allow predicting the behavior at high temperatures. However,

in the temperature range studied for the FeO/Fe transformation, at 600 °C for nascent iron, it corresponds to

the factor 0.48 (8), and for higher temperatures, values of 0.39 can be considered to 0.33, which is in the high-

temperature range for the effects of thermoplasticity, viscous behavior.

To quantify the plasticity of iron at temperatures above 420 °C, equation (1) was used [9].

l = Length of the material according to the time elapsed at the set temperature.

lo = Length at the instant of applying the effort.

β = Slope of strain rate, known as transient flow.

t = Time.

k = Slope of strain rate, called viscous flow.

According to [10], the parameters lo, β, and k are constants that must be determined experimentally since

they correspond to physical processes, being necessary to obtain them to generate the curve that represents

the thermoplastic characteristics of the material. In this regard, in his experimental results, p. 336, values of β

and k are collected for iron, lead, copper, tin, and mercury at different temperatures. From their analysis, the

following conclusions were highlighted:

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

β, in general, presents slight variation with the increase in temperature, with a tendency to decrease.

βFe, at a temperature of 444 °C (factor 0.40), although it increases with effort, tends to a constant value of

0.0210. When compared with the values of βPb, it presents a value of 0.045 at 17 °C, low temperature for

Pb, and 0.043 at 160 °C (Thigh), associating it with the main result of his work, [10], p. 332, where he

expresses "it is to show that typical metals of widely different natures obey the same general flow law," it is

assumed that βFe will be maintained at high temperatures, that is, above 444 °C, so βFe = 0 will be

used,0.0210.

The value of k increases with the increase in temperature. With the effort to determine the values that it

acquires at temperatures higher than 444 °C, it is necessary to know one of the following characteristics of

iron: (a) the curve of real deformation (ε) as a function of the residence time at the temperatures

previously set from 600 °C, (b) the constants indicated above for the same thermal range, data not

available in the literature. Based on [4] "at elevated temperature viscous creep is predominant," p. 372;

and the proof of it concerning k, p. 362, where it can be interpreted as independent of temperature, being

consistent with results of viscous creep in aluminum, where a graph taken from [11] is shown showing the

invariance of the relationship: ε = f(t), for absolute temperatures: 424, 478 and 531, p. 376, with factors

0.45, 0.46 and 0.57 respectively, the value of kFe = 0.00033 will be assumed. In the real case that it was

more significant, an increase in stretching would be obtained with an increase in volumetric expansion,

thus maintaining the accepted value of k, and a lower swelling index would be obtained.

D. Fixing the sample soaking time temperature range of 600 °C - 1100 °C

The sample soaking time will depend on the heating rate used in the reduction processes, whose values do

not vary appreciably between them, as shown in Table 2, which summarizes the results of the best-known

commercial processes.

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Table 2. Heating rate according to industrial reduction processes.

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

The 5 °C/min rate will be used, considering that by applying the relationship (1), the length obtained for the

3.8 rates and 5 °C/min, a variation of 0.76 % is generated, that is, produces little impact, and also allows to be

consistent with the search for greater productivity. For an increase of 100 °C, with a rate of 5 °C/min., the

residence time equals 20 min. To associate the heating rate with the residence time of the material at

temperatures between 600 and 1100 °C, heating bands were defined. A 100 °C increase in temperature

would allow them to be associated with the retention time, and It accumulates as the temperature rises since

the reduction process is continuous. Table 3 shows the results of this approach.

Table 3. Sample heating ranges, average temperatures, and retention time.

III. METHODOLOGY

The quantification method used is a theoretical approach, defined in the previous section, of a mechanism

where the (a) physicochemical aspects of adsorption-desorption of nitrogen gas are associated, (b) crystalline

structure of the material, (c) experimental data reported in the literature and (d) the ideal gas equation to

facilitate algebraic calculations. For the study subjects, the phenomenon of abnormal swelling that can occur

in the reduction of iron minerals in the solid state is included in the study, and through intentional sampling,

six samples were fixed to relate the dependent variable (IH) with the reduction retention time and nitrogen

pressure at a temperature of 600 to 1100°C.

Having defined the iron retention time at each average temperature, the expanded length is estimated by

(1), considering the following values for the constants:

lo = initial length = vacancy radius of Fe crystal lattice: 0.124 nm.

β = 0.0210

k = 0.00033

IV. RESULTS

A.Swelling Index relation with iron thermoplasticity

The resulting values of the expanded length make it possible to obtain the volume change concerning the

initial volume of the vacancy left by an iron atom (8.0 10-3 nm3) and thus the swelling index (IH%), which will

be associated with temperature and time and may be modeled dynamically. Results are shown In Table 4.

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Table 4. According to residence time and temperature, the iron swelling index is associated with

nitrogen pressure.

* Calculated (1). ** Obtenidos a través de Norma ISO 4698 [15].

B. Swelling index (HI) mathematical relationship with retention time

The dispersion graph is obtained through the Excel software, Fig. 1, with the regression line, dotted, and its

equation (2) that allows estimating one variable from the other in the considered range and thus its

correlation. This determines the degree of dependence between these variables in direction and magnitude,

represented by the Pearson index (r). From the above data, it is possible to obtain a clear mathematical

expression from the curves presented in Figure 1 and shown in Equation 2 for the estimation of the volume

increase of an iron Nano crystallite.

Fig.2. Retention/soaking time effect on the swelling index by iron induces thermoplasticity by partial

pressure on N2 gas absorption desorption hysteresis.

Source: The author

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

C. Mathematical relation between the swelling index and stresses due to the nitrogen pressure

In Table 1, it is shown that the stress generated by the nitrogen pressure in an iron vacancy exceeds the

value of its resistance to breakage in a minimum ratio of 6.4 and a maximum of 22.2 times because this

pressure increases the volume of the cavity depending on whether the thermoplastic response of the iron

produces its rupture and gas escape, recovering the elastic deformation but not the plastic one, thus, utilizing

the expanded volume, Table 4, the pressure is quantified through the procedure developed in [1], the values

are recorded in Table 5.

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From the values in Table 5, a mathematical equation can be obtained through Figure 3 described in equation

Fig. 3. Effect of the stress created by the nitrogen gas absorption on metallic iron on the final

swelling index (percentage).

Source The author.

At this stage, it is worth bearing in mind that the cold compression strength of manmade iron oxide

agglomerates drops sharply with extended reduction time to 9 min; nevertheless, the compressive strength

remained at the same level as that of 1 min. The reduction degree reached 11.11% at 1 min, corresponding to

the magnetite phase, and finally increased to 32.45%rresponding to the wustite stage at 9 minutes. This effect

is associated with the porosity increase and defect generation in the sample periphery, which causes strength

loss [16]. The reduction step of hematite to magnetite causes the most extensive strength loss of 75.85% to

approximately 200 Kgf/mm2, as shown in Fig. 4.

Fig. 4. Effect of reduction time on the compressive strength of iron aggregates in the transition from

hematite to magnetite (16).

Edited by the author.

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

Table 5. The swelling index in iron is associated with the effort generated by the nitrogen pressure.

The effect shown in Fig. 4 can be represented mathematically by a polynomic equation shown in the

polyphonic relationship in equation 4.

Stress Kg/mm2 = 1.3583 t4 - 30.464 t3 + 234.82t2 - 681.12t + 777 R² = 0.9395 (4)

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This research has demonstrated and mathematically supported the triggering cause and effect of

nitrogen's atomic absorption/molecular desorption in nascent iron nanoparticles during the wustite to iron

reduction path.

Including this effect reveals this gas as a hidden, and so far, disregarded, impurity, which triggers the

abnormal swelling mechanism by inducing the internal mechanic stresses.

The magnitude of the stresses generated by the nitrogen pressure in the iron crystal lattice and the

mathematical relationship with the abnormal swelling index were determined. Considering the stresses act

as a vector working against the compression force that maintains the solid condition of the reduced

sample, it is possible to conclude that the value resulting from the stress is in the range of 69.6 and 35.2

Kg/mm2.

From the laboratory and industrial analyzed data, a direct correlation was obtained between the partial

pressure of nitrogen, the swelling index, and the reduction time.

The obtained result indicates that the swelling index increases with the increase in the reduction

temperature and that abnormal swelling, in the presence of Nitrogen gas, begins at 700 °C, reaching a

maximum effect at 900ºC.

With these two expression stress values are presented in Table 5, and considering that it is an opposite

vector working against the compression force that maintains the solid condition of the reduced sample

obtained by equation 3, the resulting stress value may be in the range of 69.6 and 35.2 Kgf/mm2.

The mathematical findings in this research pay the importance of the strength of liquid-bound granules,

which depend on at least three forces: (1) interparticle friction; (2) capillary and surface tension forces,

supposed to be thermo creep of viscous FeO/Fe material, in the liquid between the nanoparticles; (3) viscous

forces in the liquid between the particles. The former two have been addressed and quantified in this paper

[17], [18].

The high correlations obtained in the equations of Figures 2 and 3 allow the association of the values with an

increasing sense and a strong dependence between the variables. As in the present case, consider the

correlation index robust due to the low dispersion between the values obtained and the regression line. Since

the Pearson index does not give a cause-effect relationship, it is convenient to carry out a significance test. The

test was done by comparing the statistical value (tc) of Student, calculated using (4) [19], p. 118, with the

double-tailed tabulated value and degree of freedom (n-2), in the relationships: IH versus f(time) and IH versus

f(pN2), with the null hypothesis Ho of "there is no correlation between the variables. "

tc = (|r|x √(n-2))/√ (1-r^2) (5)

The significance test result allowed us to reject the null hypothesis and accept a significant correlation with a

confidence level of 99%. The relationships, IH versus f (time) and IH versus f(pN2), of Fig. 2 and 3, are

consistent with the physical phenomenon of the reduction process. The former process involves the

thermoplastic characteristics of both phases, considered FeO and iron, the time-temperature rise, and the

expansive effect of the nitrogen pressure, which allows the assumption to be valid for the stress values

presented in Table 5. These values, which are lower when compared with those of Table 2, were obtained

without quantifying the expansive effect due to the thermoplasticity of iron. The data in Table 4 was used to

establish the HI with the temperature variation relationship and obtain its linear regression and correlation.

CONCLUSIONS

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation

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ACKNOWLEDGEMENTS

The author would like to thank the Universidad Nacional Experimental Politecnica "Antonio Jose de Sucre"

UNEXPO for their support in this research, as well as to Mr. Luis Alberto Azócar for carrying out the

mathematical calculations and valuable suggestions on the definition of the defined swelling model.

CONFLICTS OF INTEREST

The author declares no conflicts of interests

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Oscar G. Dam, a Metallurgical Engineer graduated from the Central University

of Venezuela 1972. Master Science in Metallurgy and Diploma (DIC) Graduated

from the Imperial College of Science and Technology 1977, England. PhD

Engineering graduated from the University of London UK, 1983.

Dam O. Effects of thermoviscous flow and mechanical stresses on nitrogen desorption during iron transformation