Heat Transfer Coefficient at Cast-Mold Interface During Centrifugal Casting: Calculation of Air Gap

During the centrifugal casting of cylindrical parts, it is reasonable to assume that fields of variables and other properties are uniform in the tangential direction. In fact, also, axial variations will be often small and, therefore, could be neglected, too. This finding directly suggests using a plane stress model. Although different variations of plane stress models have been used in diverse industrial applications, such as autofrettage of gun barrels, strain-hardened pressure vessels, and multilayer seamless pipes, in the past,[48‐51] they have rarely been employed in air gap thickness calculations. Especially, concerning the centrifugal casting, to the best of our knowledge, not a single match was found in the literature survey.

Radial and tangential stresses, σ_r and σ_t, are coupled through the equilibrium equation:where ρ, Ω, and r are the density, the rotation rate, and the radial coordinate, respectively. When only elastic deformations are considered, stresses are coupled with strains via the Hooke’s law with the thermoelastic term. Such a relationship takes the following form:where ν, E, α, and T are Poisson’s ratio, Young’s modulus, the thermal expansion coefficient, and the temperature, respectively. Strains and total radial displacements are related through the following laws:

It would, however, be incorrect to only consider elastic strains. Since the casting in a semisolid state can easily yield under strong centrifugal forces, plastic deformations also must be taken into account. Then, total strains can be conveniently expressed as a sum of elastic and plastic strains as follows:where \( \varepsilon_{\text{t}}^{\text{p}} \)and \( \varepsilon_{\text{r}}^{\text{p}} \) represent plastic strains in corresponding directions. In addition, it is also physically meaningful to consider a temperature-dependent Young’s modulus. Substituting total strains in Eq. [9] with displacements from Eq. [8], and by combining the resulting equations with Eq. [6], we can arrive at the following ordinary differential equation for the total displacement u in the casting:

In the mold, Eq. [10] is considerably simplified because one assumes a constant Young’s modulus and pure elastic deformations. The differential equation for the total displacement u in the mold becomes

Note that in Eqs. [10] and [11], subscripts […] _S and […] _M denoting the mold and the casting are omitted for the sake of brevity. Equations [10] and [11] can be solved provided that plastic strains \( \varepsilon_{\text{r}}^{\text{p}} \;\;{\text{and}} \;\;\varepsilon_{\text{t}}^{\text{p}} \) are known. In order to determine them, a universal stress-strain curve, usually assumed to be equivalent to the stress-strain curve obtained from the uniaxial loading test, must be known in advance. The universal stress-strain curve relates two scalar quantities: the effective stress \( \overline{\sigma } \) and the effective plastic strain \( \overline{{\varepsilon^{\text{p}} }} \). Several models of the effective stress \( \overline{\sigma } \) exist. Here, the von Mises stress, calculated by assuming a principal stress loading, was used in the following form:

The von Mises stress \( \overline{\sigma } \) is coupled with the increment of the effective plastic strain \( \overline{{d\varepsilon^{\text{p}} }} \) by the Prandtl–Reuss (Levy–Mises) flow rule as follows:and

Knowing or assuming \( \overline{{d\varepsilon^{p} }} \), the plastic strains \( d\varepsilon_{r}^{p} \) and \( d\varepsilon_{t}^{p} \) can be easily calculated and then used in Eq. [10] to extrapolate the total displacement u. In Eqs. [13] and [14], the increment of the effective plastic strain \( \overline{{d\varepsilon^{p} }} \) is used and not the effective plastic strain \( \overline{{\varepsilon^{p} }} \), which means that the plastic strain history (or the loading path) is very important. Correspondingly, a progressive load must be also applied in the simulation. Here, such loading is automatically realized by a time-dependent temperature field and a gradual progress of solidification.

In the present study, a temperature-dependent, perfectly elastic-plastic material is used, as shown in Figure 2. At a given temperature, the material deforms elastically until a certain threshold of the effective stress, known as the yield strength, is reached, at which point the material starts yielding with no further increase of the effective stress. Instead of the elastic-perfectly plastic material, any kind of other material could be used such as a strain-hardening or a strain-softening material.

In the following text, we summarize all the facts, assumptions, and solution strategy necessary to run a successful numerical simulation and obtain a reasonable air gap.

Facts:

Assumptions:

In addition to Eqs. [10] and [11], we also need to solve the heat conduction for the temperature T. In the cylindrical coordinate system, it takes this form in the mold:where ρ_M, c_pM, and k_M are the density, specific heat, and thermal conductivity of the mold material, respectively. Similarly, in the casting, it can be written aswhere ρ _S , c_pS, and k _S are density, specific heat, and thermal conductivity of the casting material, respectively. The last term is a latent heat source term due to the phase change, in which L_f and g_s represent the latent heat and the solid fraction, respectively. In the present study, a simple linear relationship is considered between the solid fraction g_s and the temperature T. Other relationships, however, could also be considered (e.g., the lever rule or the Gulliver–Scheil equation). The heat conduction equations, Eqs. [15] and [16], are coupled via the heat flux at the cast-mold interface. A thin-wall model, also known as a zero-capacity model, was used to numerically simplify the situation at the interface by considering only a thermal resistance, exerted by the coating and possibly the air gap. Then, the heat flux at the interface reads as where k_ifc, k_C, and k_a denote the effective thermal conductivity, the thermal conductivity of the coating, and the air gap, respectively. Other quantities are explained in Figure 1. The air gap thermal conductivity k_a is, in fact, the sum of the thermal conductivity of air k_a,phys and a thermal conductivity, which is equivalent to the radiative heat transfer through the air gap, given bywhere \( \sigma \) is the Stefan–Boltzman constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴). The temperatures T’_S and T^* must be given in Kelvin. Note that without the air gap (d_a = 0), Eq. [18] is still valid and represents the effective thermal conductivity only in the presence of the coating. The reader should be reminded that in this study, the black body radiation model has been taken for its simplicity and convenience. However, when targeting more accurate results, considering gray bodies is better justified and Eq. [19] would then becomewhere ε_S and ε_C are emissivity coefficients of both surfaces enclosing the air gap, which belong to the casting and the coating, respectively. Using Eq. [20]. instead of Eq. [19] will naturally reduce the radiative heat transfer. In reality, the air between the casting and the shell may, as a participating gas, further reduce the radiative heat transfer. A description of the corresponding mathematical model can be found, e.g., in Reference 52.

Concerning thermal boundary conditions, both the free surface of the casting and the outer surface of the mold are considered to be adiabatic:

In simulations focused on a comparison with experimental data, thermal boundary conditions, however, should be specified more precisely, e.g., with the help of existing empirical formulas for the Nusselt number[53,54] in rotating geometries.

In addition to Eq. [21], boundary conditions have to be specified also for Eqs. [10] and [11]. Here, we have to distinguish between two cases: a perfect contact or an air gap. In the case of the contact, appropriate boundary conditions take the following form:

Otherwise (the air gap),

Differential equations for total displacements, Eqs. [10] and [11], and temperature, Eqs. [15] and [16], were all solved using the finite difference method with second-order accurate central difference schemes for derivatives (including points at the boundaries). In the radial direction, the casting and the mold were divided into N_S and N_M uniformly spaced grid points with the dimensions dr_S and dr_M of 1 mm (Figure 1). The same grids with uniform spacing were used for both quantities u and T. For Eqs. [15] and [16], the implicit backward Euler method was used for time-stepping. In addition, an iterative approach was necessary for heat conduction equations due to temperature-dependent thermophysical properties, the nonlinear heat flux \( q \) at the cast-mold interface (Eq. [17]), and especially the stiff, latent heat source term. Treatment of the latent heat source term was realized by using a semi-implicit method proposed by Voller and Swaminathan.[55] A detailed description of the discretization can be found in Reference 56. Although systems of equations are unconditionally stable, the time-step size should be small enough so that the loading rate still allows finding correct and physically meaningful increments of plastic strains. Here, the time-step of 0.1 seconds was found to be reasonable.

Solution strategy:

As the main objective of the present study is to propose and test a novel approach of calculating an air gap rather than numerically analyzing a particular process, material and mechanical properties used in the simulations only roughly correspond to those of real materials (Table II). The geometry, initial conditions, and other casting parameters are given in Table III.

First, numerical tests were performed for different coating thicknesses (0.5, 1, 2, and 4 mm) and all other parameters were fixed. Total displacements u of the mold and the casting at the interface are shown as a function of time t in Figure 5. As one would expect, initially, the mold and the casting are in contact. At a certain moment, they detach and their displacements follow different paths. This is clearly shown by bifurcating curves in Figure 5. A difference between the casting and the mold displacement corresponds to the air gap thickness d_a. Obviously, the air gap appears earlier in the case of a thin coating than that of a thick one. Consequently, the heat transfer coefficient h at the interface will drop faster in the case of a thin coating than that of a thick one, which can be seen in Figure 6. The heat transfer coefficient h was simply calculated as

For each configuration (d_C = 0.5, 1, 2, and 4 mm), the time evolution of the heat transfer coefficient h is, therefore, unique. Ultimately, it seems that after 300 seconds, the heat transfer coefficients are almost identical, close to 200 W m⁻² K⁻¹.

Similar tests with similar outputs were performed for different values of the rotation rate Ω (50, 71, and 90 rad s⁻¹). Naturally, the higher the centrifugal force, the better the contact between the casting and the mold. Concerning the elastic deformations of the mold, displacements u are larger at higher rotation rates Ω (Figure 7). Higher centrifugal forces are also responsible for stronger and longer yielding of the partly solidified casting, which delays a formation of the air gap. Consequently, at a given instance, the heat transfer coefficient h is higher in the case of a higher rotation rate Ω (Figure 8).

A similar study was carried out for different values of solidus temperature T_sol such that T_liq − T_sol = 10 K, 50 K, 100 K, and 200 K (10 °C, 50 °C, 100 °C, and 200 °C). Although it is a somewhat intuitive, one could confidently state that the smaller the difference is, the earlier the air gap occurs. Again, we provide total displacements and heat transfer coefficients at the interface in Figures 9 and 10, respectively. Since coating parameters are fixed this time, initial heat transfer coefficients are all identical, equal to 1000 W m⁻² K⁻¹. Later, they significantly deviate. While the curves referring to mold displacements at the interface have a similar trend, indicating a continuous thermoelastic expansion of the mold, those corresponding to casting displacements exhibit more complex scenarios due to the combination of plastic and elastic deformations. In Figure 9, e.g., the dash-dot line representing T_liq − T_sol = 200 K (200 °C) indicates the yielding of the casting material within the entire time span simulated. On the contrary, the dash line T_liq − T_sol = 50 K (50 °C) displays only slight yielding in the beginning, immediately followed by thermoelastic contraction.

Although the quantities calculated at the interface, such as the air gap thickness d_a and the heat transfer coefficient h, are of primary interest here, the numerical model also provides other quantities such as stresses and elastic/plastic strains. In Figure 11, a typical example of temperature and strains appears at 50 seconds. The gray zone on the right represents the mold. The rest on the left belongs to the casting. A temperature drop can be seen at the interface due to a large thermal resistance. As the time proceeds, total strains grow quite uniformly throughout the entire thickness of the mold. The same applies also to the casting but only at the early stage. Later, when the casting is partly solidified and the yield strength Y increased, total strains start dropping and the casting contracts consequently.

In addition to strains (Figure 11), stresses are shown in Figure 12. A typical distribution of stresses can be seen in the mold. While the radial stresses are exclusively compressive, the tangential stresses are compressive close to the inner surface of the mold and become tensile as they approach the outer surface of the mold. The greatest stresses the mold must withstand are naturally located at the inner surface due to a sudden temperature loading. In this particular case (Figure 12), they do, in fact, reach the yield strength of the material. After several casting cycles, a thermal loading would most likely lead to the formation of fire cracks.[61] Concerning stresses in the casting, at the early stage of solidification, they are within the temperature-dependent envelope of the yield strength. The (semi-)solid part of the casting close to the mold can already hold some stresses, whereas the liquid part remains stress free. At later stages of the casting, the stresses in the casting are well below the yield strength and only the elastic loading is present. When the casting procedure is finished and the temperature field becomes uniform, residual stresses remain in the casting. An analysis of stresses and strains is, however, beyond the scope of the present article.

In centrifugal casting simulations, exponential functions are generally used to describe the heat transfer coefficient at the cast-mold interface, varying due to the air gap formation. Such functions contain empirical constants, which must be carefully specified. Unfortunately, this is not an easy task. An experiment alone is not sufficient to determine such constants, and computationally expensive inverse methods should be employed, which is, however, rarely the case. A literature survey performed here reveals an expansive scatter of data used in current and previous research. In the present study, we offer an alternative of calculating an air gap thickness and the corresponding heat transfer coefficient at the interface. The heat transfer model is coupled with a plane stress model, taking into account thermoelastic stresses, centrifugal forces, plastic deformations, and a temperature-dependent Young’s modulus. Several numerical tests were performed for different coating thicknesses d_C, rotation rates Ω, and solidus temperatures T_sol. Results were analyzed in the sense of comparing heat transfer coefficients at the interface and air gap thicknesses as a function of time. The numerical model developed here helps demonstrate that the scenario at the interface is unique for each set of parameters. Therefore, deploying any of the exponential functions that explicitly describe the thermal resistance at the cast-mold interface will always give rise to the question about the actual value of empirical constants used in that particular function. Although the material properties taken for this study do not strictly correspond to any particular material, they are obviously not far from material properties of common steels and coatings, and the results obtained here appear to be entirely reasonable and meaningful. In the near future, we plan to verify the current numerical approach against the results obtained from the inverse task run with the experimental data. Finally, possible room for improvement of the presented model remains. For example, some kind of implicit coupling between the heat transfer model and the plane stress model would be beneficial and might even be necessary in order to maintain numerical stability (or suppress unphysical oscillations of calculated displacements), especially at higher cooling rates, e.g., when an air gap is being just formed. In addition, the optimization loop involved in the loading step, i.e., the process of calculating the plastic strains could also be further improved.