Short-Time Oxidation Behavior of Low-Carbon, Low-Silicon Steel in Air at 850–1180 °C: II. Linear to Parabolic Transition Determined Using Existing Gas-Phase Transport and Solid-Phase Diffusion Theories

Abstract

Existing gas-phase transport and solid-phase diffusion theories are used to calculate the oxidation kinetics of low carbon and low silicon steel in flowing air at 850–1180 °C. The linear-to-parabolic transition scale thickness derived is proportional to the parabolic rate constant and inversely proportional to the linear rate constant. Calculated parabolic rate constants are consistent with experimental results, whereas calculated linear rate constants are 40% smaller and calculated linear-to-parabolic transition scale thicknesses are 20–40% greater than the experimental results. Introduction of a correction factor cannot simultaneously resolve the discrepancies. Various sources causing the discrepancies are discussed. One important issue identified is that the conditions for deriving the equation for calculating the linear rate constant are different from those in laboratory furnaces. The linear-to-parabolic transition scale thickness derived is also a function of gas composition, gas velocity and sample length. An intrinsic gas-phase mass-transfer factor, independent of gas velocity and sample size, is recommended to express the gas-phase mass-transfer coefficient for calculating the linear rate constant.

Appendix

The diffusion coefficient $D_{{{\text{O}}_{2}}}$ can be calculated using the Chapman–Enskog formula [7, 9, 20, 21],

$$ D_{{{\text{O}}_{2}}} = {\frac{{0.0018583\sqrt {T^{3} \left( {1/M_{{{\text{O}}_{2}}} + 1/M_{{{\text{N}}_{2}}} } \right)} }}{{P\sigma_{{{\text{N}}_{2} - {\text{O}}_{2} }}^{2} \Upomega_{{D,{\text{N}}_{2} - {\text{O}}_{2}}} }}}\left( {{\text{cm}}^{ 2} \;{\text{s}}^{ - 1} } \right) $$

(27)

where T, P and $M_{{{\text{O}}_{2}}}$ have been defined, $M_{{{\text{N}}_{2}}} = 28.013$ (g/mol) is the molecular weight of nitrogen (N₂), $\sigma_{{{\text{O}}_{2} - {\text{N}}_{2}}}$ is the average collision diameter (in Å) in the O₂–N₂ gas mixture which can be estimated using the arithmetic average of the individual collision diameters of oxygen and nitrogen in their respective gases, $\sigma_{{{\text{O}}_{2}}}$ and $\sigma_{{{\text{N}}_{2}}}$ [9, 20],

$$ \sigma_{{{\text{O}}_{2} - {\text{N}}_{2}}} = {\frac{{\sigma_{{{\text{O}}_{2}}} + \sigma_{{{\text{N}}_{2}}} }}{2}}$$

(28)

$\Upomega_{{D,{\text{O}}_{2} - {\text{N}}_{2}}}$ is the “collisional integral” for diffusion, which is a dimensionless quantity and a function of the dimensionless temperature $kT/\varepsilon_{{{\text{O}}_{2} - {\text{N}}_{2}}}$ where k is the boltzmann constant (k = 1.38066 × 10⁻²³ J/K) and $\varepsilon_{{{\text{O}}_{2} - {\text{N}}_{2}}}$ is the interaction energy (in J) between oxygen and nitrogen molecules, or the maximum energy of attraction between nitrogen and oxygen molecules, and can be estimated using the geometric average of the interaction energy of the two participating molecules [9],

$$ \varepsilon_{{{\text{O}}_{2} - {\text{N}}_{2}}} = \sqrt {\varepsilon_{{{\text{O}}_{2}}} \varepsilon_{{{\text{N}}_{2}}} } $$

(29)

$\sigma_{{{\text{O}}_{2}}}, \sigma_{{{\text{N}}_{2}}}, {\varepsilon_{\text{O}}{_{2}}}/k$ and $\varepsilon_{{{\text{N}}_{2}}} /k$ are also called Lennard–Jones parameters [9]. The Lennard–Jones parameters and the “collisional integral” for diffusion for oxygen and nitrogen, taken from textbooks [9, 20], are listed in Appendix Table 4.

Table 4 Lennard–Jones parameters for oxygen and nitrogen molecules

The kinematic viscosity of a gas or a gas mixture, $v_{\text{i}}$ [9, p. 58], is defined as,

$$ v_{\text{i}} = {\frac{{\eta_{\text{i}} }}{{\rho_{\text{i}} }}}\left( {i = {\text{ O}}_{ 2},{\text{ N}}_{ 2} {\text{ or O}}_{ 2}{-}{\text{N}}_{ 2} {\text{ mixture}}} \right) $$

(30)

where $\eta_{{{\text{O}}_{2} - {\text{N}}_{2}}}$ (in poise or g cm⁻¹ s⁻¹, [9, p. 35]) and $\rho_{{{\text{O}}_{2} - {\text{N}}_{2}}}$ (in g cm⁻³) are the viscosity and density of the gas mixture. Although a more complicated formula is available, a simplified formula is provided [9, p. 90] for calculating the viscosities of simple gas mixtures. For calculating $\eta_{{{\text{O}}_{2} - {\text{N}}_{2}}}$, this formula is,

$$ \eta_{{{\text{O}}_{2} - {\text{N}}_{2}}} = {\frac{{X_{{{\text{O}}_{2}}} M_{{{\text{O}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \eta_{{{\text{O}}_{2}}} + X_{{{\text{N}}_{2}}} M_{{{\text{N}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \eta_{{{\text{N}}_{2}}} }}{{X_{{{\text{O}}_{2}}} M_{{{\text{O}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + X_{{{\text{N}}_{2}}} M_{{{\text{N}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} }}}$$

(31)

where $X_{{{\text{O}}_{2}}}$ and $X_{{{\text{N}}_{2}}}$ are the mole fractions of oxygen and nitrogen in the gas mixture, and $\eta_{{{\text{O}}_{2}}}$ and $\eta_{{{\text{N}}_{2}}}$ are the viscosities of oxygen and nitrogen, respectively.

The viscosities of oxygen and nitrogen required in Eq. 31 can be calculated using the Chapman–Enskog formula [9, p. 85],

$$ \eta_{\text{i}} = 2.6693 \times 10^{ - 5} {\frac{{\sqrt {M_{\text{i}} T} }}{{\sigma_{\text{i}}^{2} \Upomega_{\mu } }}}\left( {{\text{i}} = {\text{O}}_{2} \;\;{\text{or}}\;\;{\text{ N}}_{2} } \right) $$

(32)

where Ω _μ is a collision integral for viscosity and thermal conductivity, which is different from the collision integral for diffusion $\Upomega_{{D,{\text{O}}_{2} - {\text{N}}_{2}}}$ shown in Eq. 27. Ω _μ is also a function of the dimensionless temperature, $kT/\varepsilon_{\text{i}}$ (i = O₂ or N₂). Its values are tabulated in textbooks [9, 20].

The kinematic viscosity of air, where the molar fractions of oxygen and nitrogen are 0.233 and 0.767, respectively, can be calculated,

$$ \begin{gathered} \eta_{\text{air}} = {\frac{{X_{{{\text{O}}_{2}}} M_{{{\text{O}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \eta_{{{\text{O}}_{2}}} + X_{{{\text{N}}_{2}}} M_{{{\text{N}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \eta_{{{\text{N}}_{2}}} }}{{X_{{{\text{O}}_{2}}} M_{{{\text{O}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + X_{{{\text{N}}_{2}}} M_{{{\text{N}}_{2} }}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} }}} = {\frac{{0.233 \times \sqrt {31.999} \eta_{{{\text{O}}_{2}}} + 0.767 \times \sqrt {28.013} \eta_{{{\text{N}}_{2}}} }}{{0.233 \times \sqrt {31.999} + 0.767 \times \sqrt {28.013} }}} \hfill \\ = {\frac{{1.318\eta_{{{\text{O}}_{2}}} + 4.060\eta_{{{\text{N}}_{2}}} }}{5.378}} \hfill \\ \end{gathered} $$

(33)

From Eq. 32, it can be seen that the viscosity of each individual gas is a function of temperature. Therefore, the viscosity of air is also a function of temperature and their values at the experimental temperatures are listed in Appendix Table 5. From Eq. 33 and Appendix Table 5, we can see that the viscosity of an O₂–N₂ gas mixture with an oxygen concentration less than 21% should fall within the values for air and nitrogen.

Table 5 Relevant parameters and calculated diffusion coefficient of oxygen in binary O₂–N₂ gases and kinematic gas viscosity of air at various temperatures

For an ideal gas, the density of an O₂–N₂ gas mixture at the reaction temperature can be calculated using the following formula,

$$ \rho_{{{\text{O}}_{2} - {\text{N}}_{2},T}} = V_{{{\text{O}}_{2}}} \rho_{{{\text{O}}_{2},T}} + V_{{{\text{N}}_{2}}} \rho_{{{\text{N}}_{2},T}}$$

(34)

where $V_{{{\text{O}}_{2}}}$ and $V_{{{\text{N}}_{2}}}$ are the volume fractions of oxygen and nitrogen in the gas mixture and $\rho_{{{\text{O}}_{2},T}}$ and $\rho_{{{\text{N}}_{2}, T}}$ are the densities of oxygen and nitrogen at temperature T (in K), respectively, determined by,

$$ \rho_{{{\text{O}}_{2},T}} = \rho_{{{\text{O}}_{2},300{\text{K}}}} \times {\frac{300}{T({\text {K}})}} = 1.301 \times 10^{ - 3} \times {\frac{300}{T({\text {K}})}} = {\frac{0.3903}{T({\text {K}})}}\left( {{\text{g}}/{\text{cm}}^{ 3} } \right) $$

(35)

$$ \rho_{{{\text{N}}_{2},T}} = \rho_{{N_{2},300{\text{K}}}} \times {\frac{300}{{T({\text{K}})}}} = 1.139 \times 10^{ - 3} \times {\frac{300}{{T({\text{K}})}}} = {\frac{0.3417}{{T({\text{K}})}}}\left( {{\text{g}}/{\text{cm}}^{ 3} } \right) $$

(36)

The density of air is therefore,

$$ \rho_{{{\text{air}},T}} = 0.21 \times {\frac{0.3903}{{T({\text{K}})}}} + 0.79 \times {\frac{0.3417}{{T({\text{K}})}}} = {\frac{0.3519}{{T({\text{K}})}}}\left( {{\text{g}}/{\text{cm}}^{ 3} } \right) $$

(37)

Using Eq. 30 and those values calculated from Eqs. 31–37, the kinematic viscosities of oxygen, nitrogen and air can be calculated (Appendix Table 5).