A Mathematical Model of Interdendritic Thermometallurgical Strain for Dendritic Solidification Processes

Abstract

A mathematical model of interdendritic thermometallurgical strain (ITM) generated during dendritic solidification of steel alloys has been developed. The model consists of two main parts in which the first part represents the derivation of ITM strain based on the concept of thermal storage energy. Consequently, the second one represents the volume changes associated with dendritic solidification phenomena during the interdendritic coherent region in the mushy zone. Calculations for Fe-C binary alloys show that the straining criteria such as strain rate and accumulated strain are sensitive to the solidification behavior of different steel alloys. The results also point out that good predications of different phases, accurate alloy properties (especially thermal expansion coefficient), and precise enthalpy functions during dendritic solidification are extremely necessary. Also, the predications show that the dendrite coherency criterion is considered an essential parameter to define ITM strain of particularly long solidification interval alloys. Model predications of ITM strain are compared and discussed with other established theoretical approaches and previous modeling work.

Appendices

Appendix A. Thermal Contraction of Dendritic Solidification Element CV

Research showed that few direct data were available for the thermometallurgical contraction. The only relevant work has been produced by Jablonka et al.[27] As suggested by Jablonka et al.,[27] the thermometallurgical contraction of a dendritic solid is a function of the average density of CV. It is assumed that the average density of CV follows the mixture linear rule. Therefore, the density of a phase mixture containing liquid and various phases can be defined as follows:

$$ \rho^{e} = f_{\text{l}} \rho^{e} + \sum\limits_{k = 1}^{N} {f_{\text{s}}^{k} \rho_{\text{s}}^{k} } $$

(A1)

According to Jablonka et al.[27] the thermometallurgical contraction can be expressed as follows:

$$ \in_{ph} = \left( {{\frac{{\rho (T_{\text{coh}} )}}{\rho (T)}}} \right)^{{\frac{1}{3}}} - 1 $$

(A2)

where

$$ \rho (T) = {\frac{1}{V(T)}} $$

(A3)

and

$$ \rho (T_{\text{coh}} ) = {\frac{1}{{V(T_{\text{coh}} )}}} $$

(A4)

By substituting Eqs. [A3] and [A4] into Eq. [A2], the thermometallurgical contraction can be rewritten as follows:

$$ \in_{ph} = \left( {{\frac{V(T)}{{V(T_{\text{coh}} )}}}} \right)^{{\frac{1}{3}}} - 1 $$

(A5)

For carbon steels, the densities and specific volumes of different phases are illustrated in References 30 and 31 where they depend on temperature and carbon content. The values of these functions were tabulated in Table AI.

Table AI Thermophysical Data of Carbon Steel Alloy[27–31]

Appendix B. Liquid and Solid Phases of Dendritic Solidification Element CV

The solidification behaviors of carbon steels can be classified into four modes as shown in Figure 3, and the interdendritic and dendritic different phases can be calculated by Fredriksson’s approach.[29] Only a few explanations are provided here, and the reader is referred to the original references for the details, the assumptions of derivation, properties of steels and nomenclature made in the derivation. The liquid fraction f _Lcan be computed as follows:

$$ f_{\text{L}} = 1 - \sum\limits_{i = 1}^{N} {f_{\text{s}}^{i} } $$

(B1)

whereas the fraction of primary delta f ^δ_s for the first three modes as well as the fraction of primary gamma f ^γ_s for the fourth mode are determined by applying the following approach developed by BoRoberg[45]:

$$ f_{\text{s}}^{\gamma } = {\frac{1}{{1 - {}^{P}K_{\rm C}^{i} }}}\left\{ {1 - {\frac{{{}^{P}K_{\rm C}^{i} }}{{{\frac{{\left( {T_{\text{liq}} - T} \right)\left( {1 - {}^{P}K_{\rm C}^{i} } \right)}}{{\left( {T_{\text{liq}} - T_{\text{sol}} } \right)}}} + {}^{P}K_{\rm C}^{i} }}}} \right\} $$

(B2)

Equation [B2] was employed until the end of solidification for the first and fourth modes and until the peritectic temperature T _pr for the second and third modes (Figure 3). The partition coefficient of carbon in the P-component system ${}^{P}K_{\text{C}}^{i}$ can be approximated using the following relationship[46]:

$$ {}^{P}K_{\text{C}}^{k} = \log K_{\text{C}}^{m} + \sum\limits_{w = 1}^{P} {e_{p}^{w} \chi_{w}^{l} } $$

(B3)

The values of Eq. [B3] used in the computations are illustrated in Table AI.[46] The volume fraction of austenite f ^γ_s during δ → γ solid phase transformation in mode 1 (Figure 3(a)) can be calculated as follows:

$$ {\frac{{df_{\delta \to \gamma }^{\gamma } }}{dt}} = {\frac{{D_{\rm C}^{\gamma } }}{{Af_{\text{s}}^{\gamma } V_{m}^{\gamma } }}}\left\{ {{\frac{{V_{m}^{\delta } }}{{X_{\text{C}}^{\delta /\gamma } - X_{\text{C}}^{\delta } }}}} \right\} $$

(B4)

In the second mode and at the peritectic temperature (Figure 3(b)), the previous approach was used and was reformulated as follows:

$$ {\frac{{df_{\text{s}}^{\gamma } }}{dt}} = {\frac{{D_{\rm C}^{\gamma } }}{{Af_{\text{s}}^{\gamma } }}}\left( {{\frac{{X_{\text{C}}^{\gamma /l} - X_{\text{C}}^{\gamma /\delta } }}{{V_{m}^{\gamma } }}}} \right)\left\{ {{\frac{{V_{m}^{\delta } }}{{X_{\text{C}}^{\delta /\gamma } - X_{\text{C}}^{\delta } }}} + {\frac{{V_{m}^{l} }}{{X_{\text{C}}^{\text{L}} - X_{\text{C}}^{\gamma /l} }}}} \right\} $$

(B5)

At the peritectic temperature in third mode, f ^γ _l→γ was calculated using Eq. [B2], whereas f ^γ _δ→γ was determined by applying the previous approach (Figure 3(c)), and the following equation was employed:

$$ {\frac{{df_{\text{s}}^{\gamma } }}{dt}} = {\frac{{D_{\text{C}}^{\gamma } }}{{Af_{\text{s}}^{\gamma } }}}\left( {{\frac{{X_{\text{C}}^{\gamma /l} - X_{\text{C}}^{\gamma /\delta } }}{{V_{m}^{\gamma } }}}} \right)\left\{ {{\frac{{V_{m}^{\delta } }}{{X_{\text{C}}^{\delta /\gamma } - X_{\text{C}}^{\delta } }}}} \right\} $$

(B6)

The mole fractions of carbon for different phasesX ^k_C are taken from the equilibrium phase diagrams.[29,31] The diffusion coefficient D ^γ_C was assumed to be temperature dependent,[13] whereas molar volumes of austenite V ^γ _m , ferrite V ^δ _m , and liquid V ^l _m were assumed to be a function of both temperature and carbon content. These functions are tabulated in Table AI.[28–31]

Appendix C. Thermal Conductivity for Dendritic Solidification element CV of Iron–Carbon Alloys

The effect of the thermal conductivity of CV λ is controlled by two main factors. The first factor is the type of crystal structure, whereas the second factor is different phases and their fractions.[19] Here, it is convenient for this problem to replace the effective thermal conductivity λ of CV in Eq. [13] with equal electrical resistances. Therefore, the type of crystal structure plays a major role in defining the heat pass model. For columnar crystals, the heat pass model follows two parallel resistances of dendritic solid and interdendritic liquid such as dendritic solidification or solid phase transformation and is expressed as follows:

$$ {\frac{1}{{\lambda_{\text{col}} }}} = \sum\limits_{i = 1}^{n} {{\frac{{f_{\text{s}}^{i} }}{{\lambda_{\text{s}}^{i} }}}} + {\frac{{f_{\text{l}} }}{{\lambda_{\text{l}} }}} $$

(C1)

For equiaxed crystals, the heat pass model can be represented as two resistances in series as follows:

$$ \lambda_{\text{equ}} = \sum\nolimits_{i = 1}^{n} {f_{\text{s}}^{i} \lambda_{\text{s}}^{i} } + f_{\text{l}} \lambda_{\text{l}} $$

(C2)

However, for a mixture between columnar and equiaxed in CV, such as columnar equiaxed transition, the thermal conductivity of solidified alloys can formulated as follows:

$$ \lambda = f_{\text{col}} \lambda_{\text{col}} + f_{\text{equ}} \lambda_{\text{equ}} $$

(C3)

The values of steel thermal conductivities used in Eqs. [C1] to [C3] depend on the temperature and carbon concentration in both the interdendritic liquid and the dendritic solid. The values of the thermal conductivity of different carbon steel alloys used in these computations are summarized in Table AI.[30,47,48]

Appendix D. Dendrite Coherency for Dendritic Solidification element CV of Iron–Carbon Alloys

During dendritic solidification, the dendrite coherency is an important parameter to define the coherent region in the mushy zone and, therefore, to calculate the interdendritic strain accurately.[34] This criterion depends on the dendritic growth rate V _a and primary dendrite arm radius R _d where the V _a /R _d ratio is inversely proportional to the coherency fraction solid in the alloy. The same equation proposed by Chai et al.[34] was employed and can be computed as follows:

$$ f_{\text{coh}} = \left\{ {a - b\left( {{\frac{{V_{a} }}{{R_{\text{d}} }}}} \right)} \right\} $$

(D1)

where a and b are constants and equal to 0.8269 and –0.1091, respectively, whereas R _d is equal to half of primary dendrite arm spacing ${\frac{{\lambda_{1} }}{2}}$ as shown in Figure 2(b) and it can be calculated using the same approach as El-Bealy and Thomas in Reference 49.

Equation [D1] was used to calculate f _coh in the primary delta f ^δ_s region shown in Figures 3(a) through (c) for carbon steel alloys. But for the peritectic reaction steel alloy (Figures 3(b) and (c)), and if f _coh is greater than f _per, the following equation can be used:

$$ f_{\text{coh}} \succ f_{\text{per}} \quad f_{\text{coh}} = f_{\text{per}} $$

(D2)

This is because of the nature of the fast solidification process and the solid phase transformation rates of peritectic steels at peritectic temperature.[50]