A Thermal Conductivity Model for Grey Iron

Abstract

Thermal conductivity is an important property for many iron cast components, and the lack of widely accepted thermal conductivity model for cast iron, especially grey cast iron, motivates the efforts in this research area. The present study contributes to understanding the effects alloy microstructure has on thermal conductivity. A thermal conductivity model for a pearlitic cast iron has been proposed, based on the as-cast alloy composition and microstructural parameters obtained at different solidification rates. According to the model, available parallel heat transfer paths formed by connected graphite flakes across eutectic cells are determined by the space between dendrite arms. The uncertainties both for model inputs and for validation measurements have been estimated. Sensitivity analysis has been conducted to result in better understanding of the model behaviour. The agreement between modelled and measured thermal conductivities has been achieved within 5% on the average for the investigated samples.

Introduction

The thermal conductivity of cast iron components is one of the most important thermophysical properties for the end-user of components, such as, e.g. for cylinder and engine blocks where the generated heat should be efficiently transferred to the engine coolant. Alongside the study of thermal conductivity, other related investigations also play a pivotal role. These include investigation of the austenite morphology (microstructure),¹ as it has the potential to impact thermal conductivity, and the effect of cast iron microstructure variation on thermal fatigue.² Thus, understanding these factors becomes essential for comprehending the overall thermal performance of cast iron components and optimizing their efficiency in heat transfer applications.

For the thermal conductivity, many measurements and modelling have been done in the past. Hecht et al. found a linear relationship between the thermal diffusivity and aspect ratio of graphite flakes.³ Holmgren et al. have discussed the relationship between the thermal conductivity of grey cast irons and microstructure⁴ and the thermal conductivity of cast irons have been reviewed.⁵ Kempers has illustrated the difference of the heat propagation in steel, spheroidal graphite iron (SGI) and grey cast iron by considering the heat propagation parallel to the basal planes in graphite.⁶ The influence of the alloying elements such as Si, temperature and microstructural constituents are also often discussed.^7,8,9,10

The modelling to evaluate the thermal conductivity of cast iron has also been performed by many researchers. The thermal conductivity of a multi-phase composite is of theoretical interest, and some general models to estimate the thermal conductivity of multi-phase composites have been proposed, including the Maxwell model,¹¹ the Bruggeman model,¹² etc. The modelling of the thermal (or electrical) conductivity of composite material is well developed in the field of polymer material to describe the influence of the second phase (filler) on the conductivity, including the percolation theory.¹³ Cast iron can be considered as a multi-phase composite, which consists of graphite, ferrite and/or pearlite, etc. The above-mentioned models are classical and relatively simple models, but they will give reasonable values for the SGI.⁷

Apart from the above-mentioned models, Helsing and Grimvall¹⁴ have developed a model for cast iron. In their model, the cast iron is treated as a combination of an anisotropic phase (represented by graphite), isotropic phase (represented by ferrite) and weakly anisotropic phase (represented by pearlite). To describe the deviation from the spherical shape, a shape factor is also introduced, although it is somewhat arbitrary.

The modelling of grey cast iron is a challenge due to the complex graphite shape and interconnection. Wang and Li¹⁵ recently developed a model for the grey cast iron by considering a simplified interconnected graphite shape. A relationship between thermal conductivity and tensile strength in cast irons was introduced in.¹⁶

A widely accepted thermal conductivity model for cast iron, especially grey cast iron, has not been developed yet due to the complex graphite shape and the lack of reliable thermal conductivity data for each phase. A general approach, enabling prediction of thermal conductivity for a wide range of cast iron alloys is still needed.

The present study aims at improved understanding of the effects the alloy microstructure has on thermal conductivity, which paves the way to the manufacturing of as-cast components with projected thermophysical properties via controlling the alloy composition and solidification rates. The paper introduces a thermal conductivity model for a pearlitic cast iron based on the as-cast alloy microstructure obtained at different solidification rates and alloy compositions. Input microstructural parameters can either be measured or predicted by solidification simulation.

First, the basic model concept is introduced, and the main factors affecting the bulk thermal conductivity are explained, including heat conduction in the connected graphite phases inside the eutectic cells and heat conduction between the eutectic cells. The influence of the earlier introduced hydraulic diameter of the inter-dendritic region¹⁷ on thermal conductivity is motivated, followed by the equivalent thermal circuit and theoretical formulation of the thermal conductivity model. The inter-dendritic region refers, in this context, to the space between dendrite arms, not to be confused with dendrite grain boundaries. Experimental routines to supply the model inputs as well as thermal conductivity of the studied cast iron alloys are described. The work is concluded by comparison between the calculated thermal conductivity and measurements as well as analysis of the model sensitivity to variation of different input microstructural parameters.

Basic Thermal Conductivity Model

This section introduces the modelling concept based on the typical lamellar graphite iron (LGI) microstructure observations, followed by the thermal resistance network to represent the identified heat transfer paths across the neighbouring eutectic cells.

Conceptual Model

It is well recognized that the heat conduction in LGI occurs mainly through connected graphite phases (flakes) arranged in the metallic matrix formed of either pearlite or ferrite or a mixture of them. The spatial distribution of the graphite forms during the solidification process in separate eutectic cells. The distribution of the graphite flakes within the eutectic cells is delimited by the primary austenite dendrite arms. The primary and eutectic austenite is generically interconnected in austenite grains but may be distinguished based on the elemental segregation of alloying elements, e.g. Si.

Assuming isotropic distribution of eutectic cells in the bulk material, consider a one-dimensional heat transfer. The heat transfer path comprises one eutectic cell with diameter d and the gap between two eutectic cells separated by distance s, Figure 1. The right-hand side of the figure represents a typical microstructure showing neighbouring eutectic cells (encircled) containing the dendrite arms of primary austenite and graphite flakes. The left-hand side of the figure reveals a schematic drawing representing the typical microstructure. The scale bars are omitted in micrographs since they are used for the conceptual model illustrative purposes.

Figure 1

Scale-free illustration of the primary dendrite arms distribution effect on conductive paths formed by connected graphite flakes in the neighbouring eutectic cells: $D_{{{\text{IR}},{\text{C}}}}^{{{\text{Hyd}}}} < D_{{{\text{IR}},{\text{A}}}}^{{{\text{Hyd}}}}$.

Research on the fracture of LGI and compacted graphite iron (CGI) has shown that eutectic cell boundaries pose additional resistance to fracture.¹⁸ This was attributed to cell boundaries featuring discontinuities in the graphite structure, also referred to as matrix bridges, which require additional load to fail. Given that the matrix has far lower thermal conductivity than graphite, these matrix bridges can be expected to also contribute to additional thermal resistance. In the present work, parameter s represents a measure of the fully pearlitic matrix bridges.

The heat conduction in graphite flakes occurs primarily in the a-direction, Figure 1, as the graphite in LGI has a pyrolytic crystal structure.^8,19,20 The heat path in the a-direction of the graphite flake has the lowest thermal resistance, and even though the flakes can be oriented differently, it is reasonable to assume that in each heat transfer direction there exist graphite flakes whose basal planes are oriented along the heat transfer direction.

The application of the simple mixture rule to evaluate the LGI thermal conductivity based merely on graphite volume fraction, and assuming pearlite as the second material, would lead to significantly overestimated values. Therefore, various modifications of graphite thermal conductivity are normally applied for evaluation of thermal conductivity of the anisotropic graphite phases. The approach introduced in the present work suggests scaling the efficiency of the heat transfer path in the LGI by the space available for continuous graphite flakes protruding between the primary austenite dendrite arms within the eutectic cells.

The space between the primary austenite dendrite arms available for graphite flakes to conduct the heat in parallel conductive paths, Figure 1a, can be characterized by a hydraulic diameter of the inter-dendritic region, again referring to the space between dendrite arms.

Hydraulic diameter of the inter-dendritic region is a microstructural parameter that can be estimated from the measurements made on a two-dimensional micrograph of the etched cast iron samples,¹⁷ and expressed as Eqn. (1):

$$D_{{{\text{IR}}}}^{{{\text{Hyd}}}} = \frac{{A_{{{\text{IR}}}} }}{{P_{{\upgamma }} }},$$

(1)

where $A_{{{\text{IR}}}}$ is the area of the inter-dendritic region and $P_{{\upgamma }}$ is the perimeter of the dendrite–eutectic interface. The micrographs in Figure 1 represent examples microstructures with larger (a) and smaller (b) hydraulic diameter.

If the space between primary dendrite arms available for graphite flakes is too narrow (smaller $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$), Figure 1b, the number of graphite flakes protruding between the primary dendrite arms decreases, as compared to the microstructure with a larger $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$, Figure 1a. In this case, primary dendrite arms present the obstacles for graphite flake growth and some of the graphite flakes terminate close to the primary dendrite surface.

The heat in the eutectic cell is conducted along multiple parallel paths. The first path is represented by a mixture of low-conductive primary dendrites and pearlite inside the eutectic cell. At room temperature, both dendrites and the non-graphite part of the eutectic are pearlite. The primary dendrites, however, have a higher Si content.²¹ The other parallel conductive paths are formed by highly conductive continuous graphite flakes within a eutectic cell.

Equivalent Thermal Circuit

An equivalent thermal resistance circuit corresponding to the above description is presented in Figure 2. The heat flow q = (T₁ – T₂)/R_tot is conducted across the total resistance R_tot which is a series combination of R_p^s, the thermal resistance of the pearlitic matrix along the path of length s between the connected graphite regions (eutectic cells), and the parallel thermal resistances R_p^d (eutectics and primary dendrite arms along the heat transfer path of length d) and R_g^d, the thermal resistance of connected graphite region along the heat transfer path of length d.

Figure 2

Equivalent thermal circuit for the heat conduction path comprising one eutectic cell and the gap between two neighbouring eutectic cells.

Thermal circuit in Figure 2 contains multiple parallel paths R_g^d to account for the effective number n($D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$) of the graphite flakes that can pass between the primary dendrite arms. This number effectively represents the scaling factor for the conduction due to continuous graphite flakes arranged in a eutectic cell between the primary dendrite arms and it depends on the hydraulic diameter of the inter-dendritic region. The value of n($D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$) ranges between 0 and 1; n = 1 gives the maximum thermal conduction of the parallel combination of R_g^d and R_p^d thereby corresponding the mixture rule applied to the graphite and pearlite based on the graphite volume fraction as explained later in the next section.

Model Formulation

Taking into account the thermal circuit in Figure 2, in terms of the area-independent thermal resistance coefficients (m² K W^-1), the total thermal resistance coefficient $R_{{{\text{tot}}}}$ is expressed as Eqn. (2):

$$R_{{{\text{tot}}}} = R_{{\text{p}}}^{{\text{s}}} + R_{||} = R_{{\text{p}}}^{{\text{s}}} + \frac{{R_{{\text{p}}}^{{\text{d}}} R_{{\text{g}}}^{{\text{d}}} }}{{n(D_{{{\text{IR}}}}^{{{\text{Hyd}}}} )R_{{\text{p}}}^{{\text{d}}} + R_{{\text{g}}}^{{\text{d}}} }},$$

(2)

where $R_{{\text{p}}}^{{\text{s}}} = \frac{s}{{k_{{\text{p}}} }}$; $R_{{\text{p}}}^{{\text{d}}} = \frac{d}{{k_{{\text{p}}} \left( {1 - f_{{\text{g}}} } \right)}}$; $R_{{\text{g}}}^{{\text{d}}} = \frac{d}{{k_{{\text{g}}} f_{{\text{g}}} }}$, and $\frac{1}{{R_{||} }} = \frac{1}{{R_{{\text{p}}}^{{\text{d}}} }} + \frac{{n(D_{{{\text{IR}}}}^{{{\text{Hyd}}}} )}}{{R_{{\text{g}}}^{{\text{d}}} }}$. Additionally, f_g is the graphite volume fraction, k_p is the thermal conductivity of pearlite, and k_g is the thermal conductivity of graphite phase.

Thermal resistance coefficients $R_{{\text{p}}}^{{\text{d}}}$ and $R_{{\text{g}}}^{{\text{d}}}$ take into account the area fraction of graphite, which is assumed to equate $f_{{\text{g}}} .$ In the volume occupied by connected graphite phases, the heat is conducted in the parallel paths, i.e. through the graphite flakes, occupying the cross-sectional area fraction $f_{{\text{g}}}$, and the pearlite occupying the cross-sectional area fraction (1−$f_{{\text{g}}}$) between the graphite flakes.

Resistance $R_{{{\text{tot}}}} { }$ can be re-written in terms of the effective thermal conductivity $k_{{{\text{eff}}}}$ of the heat transfer path, described by the equivalent circuit in Figure 2. Combining $R_{{{\text{tot}}}} = \frac{d + s}{{k_{{{\text{eff}}}} }}$ with Eqn. (2), leads to Eqn. (3):

$$\frac{d + s}{{k_{{{\text{eff}}}} }} = \frac{s}{{k_{{\text{p}}} }} + \frac{d}{{n(D_{{{\text{IR}}}}^{{{\text{Hyd}}}} )k_{{\text{g}}} f_{{\text{g}}} + k_{{\text{p}}} \left( {1 - f_{{\text{g}}} } \right)}} ,$$

(3)

resulting in the final effective thermal conductivity model described by Eqn. (4):

$$k_{{{\text{eff}}}} = \frac{1}{{\frac{{F_{{\text{s}}} }}{{k_{{\text{p}}} }} + \frac{{1 - F_{{\text{s}}} }}{{n(D_{{{\text{IR}}}}^{{{\text{Hyd}}}} )k_{{\text{g}}} f_{{\text{g}}} + k_{{\text{p}}} \left( {1 - f_{{\text{g}}} } \right)}}}},$$

(4)

where $F_{{\text{S}}} = \frac{s}{d + s}$ is the connectivity factor.

Required Modelling and Validation Parameters

The methodology to obtain the required model inputs and the validation metric (bulk thermal conductivity) are described in this section.

Model Input Parameters

The experimental castings are fully pearlitic lamellar graphite iron alloys with three different carbon equivalents, Table 1, solidified at three different rates. The casting experiment is described in more detail in.¹⁷

Table 1 Chemical Composition (wt%) and Carbon Equivalent C_Eq ¹⁷

The samples for microstructural analysis were etched in a picric acid-based solution to reveal the eutectic grains and the dendrites of primary austenite. The etching reagent contained picric acid, NaOH, KOH and distilled water at a ratio of 1:1:4:5, respectively. The etching was performed at 110 °C. Figure 3 shows typical microstructures for the different carbon content and cooling conditions. The primary dendrites were coloured black in the image manipulation software GIMP,²² and the microstructural parameters A_IR and P_γ were estimated from the binary image via the open-source software ImageJ.²³

Figure 3

Micrographs of etched samples from different solidification rates and C contents.

Connectivity factor $F_{{\text{S}}}$ is calculated from the simple ruler measurements performed on the micrograph. Since $F_{{\text{S}}}$ is the ratio of lengths, the actual scale on the micrograph is not needed. Figure 4 demonstrates a line drawing technique that is sufficient to estimate s- and d-measurements in the regions with the largest neighbouring eutectic cells. The measurement on largest eutectic cells ensures the adequate reproduction of the actual three-dimensional arrangement of the eutectic cells.

Figure 4

Demonstration of the scale-independent line drawing technique to estimate $F_{{\text{S}}}$ based on colour etching.

Measurement results regarding microstructural parameters for the studies alloys are presented in Table 2.

Table 2 Measured Thermal Conductivity and Microstructural Parameters of the Grey Iron Alloys

Thermal Conductivity

The thermal conductivity (k_m) at room temperature was calculated from Eqn. (5), based on the measured (subscript “m”) data, as follows:

$$k_{{\text{m}}} = \alpha \cdot \rho \cdot c_{{\text{p}}}$$

(5)

The thermal diffusivity (α) was measured by a Netzsch LFA 427 laser flash apparatus which is based on the principle presented in.²⁴ The specific heat (c_p) was measured in a Netzsch DSC 404C Pegasus differential scanning calorimeter and the Archimedes principle was applied for the determination of the density ($\rho )$ at the room temperature by measuring the mass of the investigated alloys in the air and in a distilled water of known temperature.

The average thermal conductivity of pearlite, k_p, was calculated considering actual Mn and Si contents in the alloy; the method proposed in¹⁴ was employed. The basal plane thermal conductivity of the pyrolytic graphite, k_g = 1950 W m⁻¹ K⁻¹ was adopted from.^{8, 25}

Thermal conductivity data are summarized in Table 2.

Model Calibration Principles

Experimental observations of bulk thermal conductivity of the studied alloys led to further modification of the basic thermal conductivity model.

Implications of Thermal Conductivity Measurements

The measured thermal conductivity as function of $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$ is presented in Figure 5. Even with three data points provided for each alloy, it is evident from the figure that thermal conductivity first increases with $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$ and then demonstrate a saturation trend at the limiting value of the hydraulic diameter, $D_{{{\text{IR}},{\text{ lim}}}}^{{{\text{Hyd}}}}$ ranging between 20 and 32 μm for the considered alloys. Some decrease in the thermal conductivity beyond the limiting value is within the measurement error discussed in the next section.

Figure 5

Measured thermal conductivity as function of hydraulic diameter: A-samples (a), B-samples (b) and C-samples (c).

The revealed behaviour of the thermal conductivity can have two possible reasons. The observed saturation trend can be related to the approximately equal f_g value in each alloy obtained at medium (sand) and slow (insulation) solidification rates. It can, however, be shown with the help of simple mixture rule, that thermal conductivity of the pearlite and graphite mixture can hardly be affected by the increase of volume fraction of graphite by 1% as in the cases corresponding to the fast (chill) and medium (sand) cooling rates. Therefore, the saturation behaviour also cannot be related to the graphite fraction. More evidence will be presented later by means of modelling. Another explanation of the thermal conductivity trends observed in Figure 5 is illustrated in Figure 6.

Figure 6

Illustration of the thermal conductivity dependence on $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$: the effective number of the graphite flakes protruding the space between primary dendrite arms does not increase at $D_{{{\text{IR}}}}^{{{\text{Hyd}}}} > D_{{{\text{IR}},{\text{ lim}}}}^{{{\text{Hyd}}}}$.

Thermal conductivity increases with $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$ up to the certain value $D_{{{\text{IR}},{\text{ lim}}}}^{{{\text{Hyd}}}}$, above which the number of graphite flakes protruding between the primary dendrite arms cannot increase any more due to the lack of sufficient amount of carbon that is necessary for the graphite flake growth. Indeed, the carbon contents decrease in the samples from A to C, Table 1, together with the limiting value $D_{{{\text{IR}},{\text{ lim}}}}^{{{\text{Hyd}}}}$, Figure 5. It means that the shortage of carbon comes earlier with the increased hydraulic diameter in samples C, and from that moment graphite flakes form no additional parallel heat transfer paths in the space between the primary dendrite arms despite increasing $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$, Figure 6.

Final Model Formulation

As the result of experimental observations regarding the limiting $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$ value, Eqn. (4) is modified, as follows:

$$k_{{{\text{eff}}}} = \frac{1}{{\frac{{F_{{\text{s}}} }}{{k_{{\text{p}}} }} + \frac{{1 - F_{{\text{s}}} }}{{F_{{\text{C}}} k_{{\text{g}}} f_{{\text{g}}} + k_{{\text{p}}} \left( {1 - f_{{\text{g}}} } \right)}}}},$$

(6)

where $F_{{\text{C}}} = n\left( {D_{{{\text{IR}}}}^{{{\text{Hyd}}}} } \right) = F_{{\text{D}}} \cdot 0.5 \cdot \left( {1 - {\text{erf}}\left( {6.0 \cdot F_{{\text{D}}} - C} \right)} \right)$ can be termed as the ”hydraulic conductivity” factor validated for the studied LGI castings (C is the fitting parameter); $F_{{\text{D}}} = \frac{{D_{{{\text{IR}}}}^{{{\text{Hyd}}}} }}{{D_{{{\text{IR}},{\text{ max}}}}^{{{\text{Hyd}}}} }}$ and the maximum hydraulic diameter value $D_{{{\text{IR}},{\text{ max}}}}^{{{\text{Hyd}}}} = 90 \;\upmu{\text{m}}$ corresponds to the longest practical solidification time.

The error function supplying the sigmoid-shaped curve, Figure 7, is designed to dampen the thermal conductivity when the hydraulic diameter increases beyond the limiting value. It is worth noting that another sigmoid-shaped function could be used to represent the saturation behaviour, with no preference given to a particular expression.

Figure 7

The function which dampens the effect of $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$ when it reaches $D_{{{\text{IR}},{\text{ lim}}}}^{{{\text{Hyd}}}}$.

The argument of the error function in Figure 7 is set to vary in the range between 0 and 1. Therefore, the relative hydraulic diameter F_D was taken as the function’s argument, i.e. the actual $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$ normalized to the maximum hydraulic diameter value corresponding to the longest practical solidification time for iron castings.²⁶

The proposed modelling approach is applicable to ferritic cast irons, which, however, would require modification of the equivalent thermal circuit shown in Figure 2 in terms of additional thermal resistance describing thermal conduction through the ferritic phase contained in the material matrix. It would of course lead to modification of Eqn. (6) to include more terms.

Finally, Eqn. (6) supplies reasonable extreme values for thermal conductivity. If s ≪ d ${\text{ then }}F_{{\text{s}}} { }$, and $k_{{{\text{eff}}}} F_{{\text{C}}} k_{{\text{g}}} f_{{\text{g}}} + k_{{\text{p}}} \left( {1 - f_{{\text{g}}} } \right)$, which corresponds to an LGI alloy with flake-type graphite “flowers” inside the eutectic cells, with minimal separation between the connected graphite regions (eutectic cells).

Conversely, if s ≫ d ${\text{ then }}F_{{\text{s}}}$ 1, and $k_{{{\text{eff}}}} k_{{\text{p}}}$ which corresponds to the isolated graphite spheroids in SGI, having the minimal effect on the thermal conductivity of the dendritic (pearlitic) matrix. The intermediate thermal conductivity should, according to Eqn. (6), be calculated for compacted graphite iron (CGI) castings. However, proving the relevance of the developed thermal conductivity model for SGI and CGI castings is out of scope of the present study.

Modelling Results and Discussion

The calculated thermal conductivity values are compared with measurements of the bulk thermal conductivity, followed by the sensitivity analysis to evaluate the model behaviour.

Model Validation

The modelling results with fitting parameter C = 3.2 were compared with the measured thermal conductivity values for 9 investigated cases. The comparison plots are provided in Figure 8. The error bars for measured thermal conductivity were estimated to be from ±1.5 to ±2.5 W m⁻¹ K⁻¹, Table 2, based on the thermal diffusivity measurement uncertainty ±3%, density measurement uncertainty ±0.1% and c_p measurement uncertainty, ±2.5% for the studied cases. The error bars for the calculated thermal conductivity values were estimated based on the uncertainty analysis of the thermal conductivity of pearlite, k_p, in Eqn. (6). More precisely, the uncertainty of the thermal conductivity of ferrite was estimated by considering the uncertainties of the Si and Mn concentrations and the coefficient for the electrical resistivity which was used for the calculation of the thermal conductivity of ferrite phase. Then, together with the thermal conductivity of cementite, the uncertainty of the thermal conductivity of the pearlite, k_p, was assessed (±1.08, ±1.24 and ±1.09 W m⁻¹ K⁻¹, for samples A, B and C, respectively). The uncertainty will propagate to the uncertainty of the effective thermal conductivity, k_eff, via Eqn. (6). The uncertainty of the k_eff is shown as error bars in Figure 8.

Figure 8

Calculated thermal conductivity values versus the experimentally obtained results as function of the measured hydraulic diameter of the inter-dendritic region: A-samples (a), B-samples (b) and C-samples (c)

As can be seen in the figure, most of the error bars are overlapped or nearly overlapped, such that the most calculated values are in reasonable agreement with the experimental values. It created the prerequisites for further investigation of the model behaviour via performing the sensitivity analysis, with the results reported in the next section.

Sensitivity Analysis

The model sensitivity to variation of different input microstructural parameters was evaluated. First, it was confirmed that variation of parameter s would result in a nonlinear change of the thermal conductivity. The modelling results are provided in Figure 9, where factor $F_{{\text{S}}} = \frac{s}{d + s}$ was set to vary in the range between 0 (s ≪ d) and 1 (s ≫ d). Increased separation between the eutectic cells expectedly results in the increased thermal resistance R_p^s, which worsens thermal connection between the eutectic cells and results in overall degradation of thermal conductivity.

Figure 9

Calculated thermal conductivity at $D_{IR}^{Hyd}$ = 12 μm, k_p = 24 W m⁻¹ K⁻¹, f_g = 9.2%. F_S varies between 0 and 1, at fixed d and increasing s.

Second, the model sensitivity to the variation of the volume fraction of graphite was evaluated. The modelling results in Figure 10 were compared to the measured thermal conductivity values for samples A, B and C. When related to the data points in Figure 8, the data provided in Figure 10 prove a larger effect of hydraulic diameter on the thermal conductivity than the effect of graphite fraction. In other words, it was confirmed that graphite morphology has a larger influence on thermal conductivity than the graphite fraction.

Figure 10

Calculated versus measured thermal conductivity showing that both the increase in thermal conductivity and the saturation behaviour cannot be related to the variation of the graphite fraction: A-samples (a), B-samples (b), and C-samples (c).

Finally, the developed thermal conductivity model includes thermal properties and microstructural parameters obtained at room temperature. For applying the model at elevated temperature, one is expected to substitute the input thermal conductivities at the specific elevated temperature. The cast iron microstructure may change due to thermal expansion of the material at elevated temperatures, which typically amounts to maximum 3%. Assuming parameters s and d would bear approximately the same percentage change, it would lead to a negligible change in $F_{{\text{S}}}$. Therefore, the thermal conductivity model is expected to be applicable at elevated temperatures, which, however, requires additional investigation.

Conclusions

The thermal conductivity model for cast iron was introduced. The modelling results were compared to the measured values for LGI alloys with different solidification rates and alloy compositions. The agreement within 5% on the average was achieved for the investigated samples.

The model utilizes microstructural parameters of as-cast alloys that can be either measured or predicted by casting simulation. The key assumption discussed in the paper was that available parallel heat transfer paths formed by connected graphite phases (flakes) across eutectic cells are determined by hydraulic diameter of inter-dendritic region. This microstructural parameter is believed to be relevant for understanding graphite morphology in terms of its ability to conduct heat. The proposed approach represents a feasible complement to the effective-value approaches for evaluation of thermal conductivity of the anisotropic graphite phases found in cast irons.

It was found that larger hydraulic diameter may have a significantly higher effect on the thermal conductivity than the graphite fraction in the alloy. At the same time the saturation behaviour of thermal conductivity as function of hydraulic diameter was experimentally observed and studied by modelling.

The thermal conductivity behaviour in relation to $D_{{{\text{IR}}}}^{{{\text{Hyd}}}}$ should be further investigated. The choice of the mathematically fitted function for “hydraulic conductivity” factor F_C should be better motivated from the material physics perspective. Influence of Si segregation on the pearlite thermal conductivity has still to be addressed. The developed model is expected to be general. The next steps should include experimental validation towards nodular CGI castings.