Computational Assessment of Thermal Conductivity of Compacted Graphite Cast Iron

Two-dimensional FE models of CGI with different pearlite contents for thermal conductivity analysis were established according to the real metallographic images obtained by Pro/E and ANSYS. Meanwhile, thermal conductivity of CGI with different pearlite contents was tested through the laser flash method. It is indicated that the thermal conductivity of CGI declines with the increase of pearlite. When pearlite is increased from 10% to 80%, the experimental values decline from 46.63W/m·K to 36.86W/m·K, reducing by 21%. However, this declining tendency becomes gentle and slight when pearlite is more than 40%. In addition, the calculation results with the consideration of interfacial contact thermal conductance (ICTC) and pearlite are much close to experimental values; especially when pearlite is 80%, the difference between them is only about 2%. It can be concluded that the FE models are convenient and reasonable to analyze thermal conductivity of CGI.


Introduction
As a preferred material for engine components at high temperature, such as cylinder head, compacted graphite cast iron (CGI) exhibits a super combination of excellent mechanical properties and outstanding thermal conductivity [1,2].With the increase in power density, the cylinder head material endures very high temperature; therefore, the material needs superior thermal conductivity in order to transfer the heat quickly and avoid thermomechanical deformation [3][4][5].
It is generally accepted that graphite has the best thermal conductivity in cast iron.According to data [6,7], thermal conductivity of graphite along the prism planes at room temperature is 5.7 W/m•K, while the value along basal planes is 1950 W/m•K.When graphite grows along the basal planes preferentially, it becomes lamellar and constitutes eutectic cells, which build 3D structure and make heat to spread through graphite eutectic cells quickly.When graphite grows along the c-axis direction preferentially, it is spherical and distributes in the matrix separately, which leads to poor connectivity between graphite particles and compel most of the heat to pass through the matrix [8,9].Although the growth direction of vermicular graphite is a mess, graphite is mainly along basal planes and particles gather as eutectic cells, which makes better heat conduction than that of nodular graphite [10].
Recently, many researches have analyzed the thermal conductivity of cast iron.Holmgren et al. [11] estimated the relationship between graphite growth direction and thermal conductivity of cast iron.
ey pointed out that thermal conductivity of flake graphite is better than that of vermicular graphite followed by that of nodular graphite because heat diffuses faster in the a-axis direction of graphite than that in the c-axis direction.Although chunky graphite has large interconnected eutectic cells, there is no beneficial effect on heat transferring due to its c-axis direction of hexagonal crystal.Hecht et al. [12] also found out the thermal diffusivity of cast iron increases with size of A-type flake graphite.However, these studies mainly focused on experiments to analyze thermal conductivity of cast iron, which consumes much time and it is difficult to represent heat conduction process vividly.
ere are many models to calculate thermal conductivity, such as Helsing model [13,14], Maxwell model, Bruggeman model, and Hasin-Shtrikman model [15], which are convenient to calculate and apply.But these models only suit cast iron with ferric matrix.Modern engine materials need excellent mechanical property, which needs pearlite, a reinforced phase, to enhance the cast iron strength.Because of pearlite, thermal conductivity of matrix decreases.erefore, above models are no longer inadequate.Recently, finite element (FE) simulation has been used for analyzing the thermal conductivity of cast iron [16,17].It is a feasible technical method.Ma [17] pointed out that thermal conductivity of CGI increases with vermicularity through FE simulation.e calculation values are close to reality when considering interfacial contact thermal conductance (ICTC).
To our best knowledge, thermal conductivity of CGI analysis through the FE method and data with different pearlite contents are rare and scarce.erefore, it is great indispensable to discern the variation of thermal conductivity of CGI with different pearlite contents.In the present study, there are two purposes: (1) attempting to build a twodimensional heat conduction model of CGI by using FE methods and (2) analyzing the influence of pearlite and ICTC on the thermal conductivity of CGI.

Building of FE Models
Microstructure of cast iron was observed by using a Nicon300 metallographic microscope and was traced by using Pro/E.en an IGES file was generated and imported into ANSYS to establish the FE models [16,17].Cast iron material can be seen as a kind of composite material [14,18,19].Graphite, ferrite, and pearlite can been seen as three phases in the microstructure.ermal Solid Quad 8 node PLANE77 element is used to mesh the physical model.
e meshes along the interface between graphite and casting matrix are more refined in order to increase the accuracy of calculations.

FE Models with Different Graphite
Morphologies.FE models with three graphite morphologies (lamellar, vermicular, and nodular) were built, respectively, according to the microstructure, as shown in Figure 1. e corresponding FE model of grey iron is shown in Figure 2.

FE Models of CGI with Different Pearlite Contents.
e metallographic specimen after etching by using 4% nital solution was observed with an optical microscope, and the area percentages of pearlite were measured by using DT 2000 metallographic software.Figures 3(a), 3(c), 3(e), 3(g), and 3(i) present the microstructure of CGI with pearlite contents from 10% to 80%, respectively.Figures 3(b), 3(d), 3(f ), 3(h), and 3(j) present the corresponding FE models.For simplicity, it is assumed that graphite is an isotropic material and its thermal conductivity is set as 130 W/m•K [20].In addition, ignoring solid solution of alloying elements in matrix, thermal conductivity values of ferrite and pearlite are regarded as 78.5 W/m•K [21] and 52 W/m•K, respectively [22].

Boundary Conditions.
Under the steady condition, three types of boundary conditions can be used to simulate the heat transfer problems: e first one is that temperature on the boundary of T x is fixed and the simple case is that T x is constant: e second one is that heat flux on the boundary is fixed and the simple case is that heat flux is constant: where n is the normal direction of the surface and k is the thermal conductivity.e third one is that the heat transfer coefficient and the surrounding temperature between boundary and surrounding are fixed.In this paper, the first and third types of boundary conditions are used in the calculation.e upper and lower sides of the FE model are set up adiabatic.Convective heat transfer and fixed temperature boundary condition are defined on the left and right sides of the model, respectively.
According to the Fourier law of heat conduction, heat conduction is directly proportional to the temperature gradient, heat transfer area, and time.It can be expressed by equation (3); therefore, the thermal conductivity k can be calculated by equation ( 4): where Φ is the heat flux, A is the heat conduction area, and dT/dx is the temperature gradient.e heat flux Φ is a constant in the two-dimensional FE model of this paper.
erefore, the heat flux density q can be expressed by (5).e heat flux q can be calculated by equations ( 6) and ( 7): where L is the length of the FE model in the direction of heat conduction and ΔT is the temperature difference from the left side to right side of the FE model.Heat flux q, length of FE model L, and temperature difference ΔT can be obtained through FE calculation results.Hence, the thermal conductivity k can be calculated according to the following equation: 2 Advances in Materials Science and Engineering

Interfacial Contact ermal Conductance (ICTC).
According to Yang et al. [16], ICTC can be seen as a thermal barrier resistance between diamond and aluminium matrix.Ma [17,19] pointed out that interfacial thermal conductance cannot be ignored in the analysis of heat conduction.As ascribed in the former section, CGI can be seen as a simple composite material.erefore, ICTC was established in the FE models of this paper, as shown in Figure 4. e value of ICTC was set as 0.1 × 10 6 W/m 2 •K.

Thermal Conductivity Test
ermal conductivity can be tested by using the laser flash method.Firstly, thermal diffusion coefficient of CGI was measured by using the NETZSCH LFA type 457 unsteady laser thermal conductivity meter as described in the following equation: where α is the thermal diffusion coefficient, l is the thickness of the specimen, C � 0.1388, and t 0.5 is the time that the sample temperature on the back rise 50%.According to equation (10), thermal conductivity can be calculated: where λ is the thermal conductivity, ρ is the density, and C p is the specific heat.e specific heat of material can be calculated by the following equation: where C p and C PR are the specific heat of the test sample and reference sample, respectively; m p and m R are the masses of the test sample and reference sample, respectively; and ΔT P and ΔT R are the temperature changes of the test sample and reference sample, respectively.e density of samples is measured by using the hydrostatic method, and mass and volume of the sample are measured by using the METTLER TOLEDO balance.

Effect of Graphite Morphologies on ermal Conductivity.
Figure 5 shows the distribution of temperature and heat flux of the grey cast iron.It can be seen that the temperature decreases from left to right in the model, and the temperature field is relatively homogeneous.Due to excellent thermal conductivity, heat flux of graphite is higher than that of matrix, which indicates that the heat transmission would pass through the graphite preferentially.Table 1 shows the calculated thermal conductivity in different graphite morphologies.It can be seen that the values in three graphite morphologies are relatively close and the difference between them is only about 0.13% to 0.17%.Meanwhile, the calculated values are obviously greater than the actual thermal conductivity of cast irons (about 37-53 W/m•K).
ere are some reasons that can explain the large error.Firstly, the matrix of models is ferrite.It has the second best thermal conductivity in the microstructure.
ere is little obstruction during heat spread through matrix.FE models are different from reality because pearlite is out of consideration.Secondly, anisotropy of graphite is ignored.For simplicity, an assumption was made in simulation.Graphite can be seen as an isotropic material, and thermal conductivity value was set as 130 W/m•K.However, nodular graphite mainly grows along the c-axis direction, and the thermal conductivity value is only about 5.7 W/m•K.Hence, the thermal conductivity of nodular graphite iron is amplified virtually during simulation.
irdly, ICTC was out of consideration in the FE model, which has a negative effect on heat conduction, as described by Ma and Yang.From the above analysis, calculation results are much higher than reality.

Effect of ICTC and Pearlite on ermal Conductivity of CGI.
In order to make the FE models more close to the actual situation, pearlite and ICTC were applied.
Temperature distribution and heat flux with 10% pearlite of CGI without consideration of ICTC and with consideration of ICTC are described in Figures 6 and 7, respectively.It can be seen that temperature field and heat flux become uneven and irregular when ICTC is applied.Heat conduction in the matrix increases, and the efficiency of heat passing through the matrix is greater than that of without consideration of ICTC.
ermal conductivity of CGI with different pearlite contents is described in Figure 8.It can be seen that the thermal conductivity of CGI decreases with increase in pearlite.When pearlite is increased from 10% to 80%, experimental values decline from 46.63 W/m•K to 36.86 W/ m•K, reducing by 21%.In contrast, FE simulation results without consideration of ICTC decline from 81.32 W/m•K to 56.70 W/m•K, reducing by 30% and thermal conductivity values with consideration of ICTC declines from 54.78 W/ m•K to 38.87 W/m•K, reducing by 29%.
ermal conductivity decline is obvious and evident when pearlite is less than 40%.However, this declining tendency becomes slight when pearlite is more than 40%.Advances in Materials Science and Engineering is phenomenon implies that the negative influence of pearlite in heat conduction recedes with increase of pearlite.
In addition, calculation results without consideration of ICTC are much higher than test results and difference between them is about 33.8% to 39.3%.When pearlite and ICTC are applied in the models at the same time, the calculation results are relatively close to the test values; especially when pearlite is 80%, the difference between them is only about 2%.It proves that application of ICTC and pearlite in models is reasonable.6

Conclusions
Advances in Materials Science and Engineering pearlite is less than 40%; however, it becomes slight as pearlite is more than 40%.e negative influence on thermal conductivity reduces with the increase of pearlite.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.