**Author:** V. P. Monastyrskiy, M. Yu. Ershov.

This article presents an imitative model aimed at analyzing and predicting the formation of shrinkage cavities and macroporosity in casting processes. The model simulates the processes of shrinkage porosity formation by taking into account the capillary forces that arise in the biphasic zone of the solidifying casting, using a finite element mesh, and its implementation as part of the "PoligonSoft" computational casting process modeling system. Numerical experiments conducted confirm the model's adequacy and its ability to accurately reproduce real processes in foundry production, making it useful for optimizing technological parameters and improving the quality of castings.

Download PDF, RussianComputational modeling of casting processes is an effective tool for reducing time and resources when starting the production of cast parts. The main task in developing the technological process (achieving dense casting) is solved by predicting the porosity obtained from the modeling.

The physical schematization and the mathematical model of the formation of macroporosity and shrinkage cavities are presented in works [1, 2]. According to this model, the formation of shrinkage cavities is considered a process of liquid metal flow under the influence of pressure differences that arise in the biphasic zone of the casting during solidification. The formation of closed shrinkage cavities and macroporosity occurs as a process of creating new separation surfaces, i.e., due to the rupture of the liquid metal's continuity under tensile stresses that exceed its strength [3].

The main difficulties in modeling the solidification process of casting are associated with determining the continuously changing boundary between the metal and the environment (open shrinkage cavity) and the appearance of new separation surfaces within the metal: macroporosity and closed shrinkage cavities. Existing methods for constructing an arbitrary free surface of a moving liquid, such as front-tracking, adaptive meshes, and the volume-of-fluid (VOF) method, provide high accuracy in determining the curved surface of the liquid metal, but are complicated to implement and require substantial computational resources, hence they are more frequently used for scientific purposes [4].

To solve practical problems in foundry production, an imitative simulation model based on the step-by-step method and the casting's mass balance is widely used, where a single physical process of liquid metal flow under the influence of solidification forces is decomposed into several processes occurring independently of each other. The imitative model is significantly simpler to implement and sufficiently adequate. Many years of experience using this model as part of the "PoligonSoft" computational casting process modeling system [5] show that with the correct choice of model parameters, the prediction of porosity in casting is almost always experimentally confirmed.

Previously, a step-by-step method was proposed [6] to determine the shape of the shrinkage cavity, taking into account the capillary feeding of interdendritic spaces above the free surface of the liquid metal, applicable to the finite difference model of casting. This article is dedicated to the development of a numerical model that simulates the processes of shrinkage porosity formation by considering the capillary forces that arise in the biphasic zone of the solidifying casting, using a finite element mesh, and its implementation as part of the "PoligonSoft" computational casting process modeling system.

The formation of shrinkage porosity occurs in the presence of a forming solid structure. The sizes of the interdendritic spaces determine the capillary forces acting in the biphasic zone. Since the magnitude of these forces can exceed the metallostatic pressure and the surrounding pressure, it can be assumed that in the formation of internal shrinkage cavities and macroporosity, capillary forces play an important role. Considering the capillary forces allows for the formulation of a model in which the formation of macroporosity depends on the dispersion of the dendritic framework, making the model more adequate and opening up possibilities to predict the pore size.

Consider the crystallization process of the liquid metal poured into the mold, which cools due to heat dissipation into the environment, both through the mold walls and directly from the free surface of the liquid metal. The casting, represented by its mesh model, is divided into elemental volumes: tetrahedrons with nodes located at their vertices. This mesh model is used to solve the thermal problem: calculating the temperature distribution in the casting block using the finite element method.

Suppose the temperature in the elemental volumes associated with node iii of the mesh is known from solving the corresponding heat conduction equation. We also assume that the free surface of the liquid metal (mirror) is flat and initially located at the upper edge of the casting. The phase composition of the metal in the elemental volume is characterized by the equilibrium volumetric proportions of the liquid phase *f _{L}* and solid phase

Following the accepted representations of the model, we consider that if the liquid phase fraction satisfies the condition *f _{L}>f_{L}^{∗∗}*, the solid phase does not form an immobile dendritic structure and moves along with the liquid metal mirror [1, 5, 6]. From the moment a rigid dendritic structure forms

Fig. 1 - Schematic representation of the stages of solidification of the casting, where M is the surface of the liquid metal; L is the liquid metal; S is the solid phase; F is the mold; a) initial state; b) formation of shrinkage cavity; c) crystallization of the closed volume of liquid metal in a thermal node with the formation of an internal shrinkage cavity or dispersed porosity.

In general terms, the crystallization process of the casting goes through the following stages: formation of shrinkage cavities during the crystallization of open thermal nodes, formation of closed thermal nodes, and formation of internal shrinkage cavities or dispersed porosity (Fig. 1).

A thermal node is considered open if the liquid metal is in contact with the environment (Fig. 1b). If there is no direct contact of the liquid metal with the environment, the thermal node is considered closed (Fig. 1c).

Based on the crystallization sequence of the cast piece presented in Fig. 1, we will consider the mathematical models of shrinkage defect formation at each stage.

The crystallization of the liquid metal is accompanied by the contraction of the metal. In an open thermal node, crystallization does not cause a pressure drop if the contraction is compensated by the descent of the free surface (liquid metal mirror). The liquid metal surface is free and can move if it does not have a fixed solid phase framework, i.e., *f _{L}>f_{L}^{∗∗}* (Fig. 2).

Fig. 2 - Scheme of crystallization of an open thermal node in the presence of a liquid metal mirror, where: a) standard model of "PoligonSoft"; b) new porosity model; 1 - zone of dry dendrites; 2 - zone fed by the liquid metal due to the capillary effect; S/L - zone of continuous dendritic structure, f_{L }** ≤ **f

The descent of the liquid metal mirror must compensate for the metal contraction at each time step. The movement of the mirror is determined by the expression:

where *V _{Ω}* is the contraction volume,

The contraction volume at this time step is equal to:

where *v _{Ωj}* is the contraction volume at the node of the finite element mesh;

According to the model conditions, in the mesh nodes located above the mirror, where there is no fixed dendritic framework, i.e., *f _{L}>f_{L}^{∗∗}*, there should be no metal, and therefore, the actual fractions of the liquid phase

The volume of the casting assigned to each node of the finite element mesh is equal to one-fourth of the sum of the volumes of the elements to which that node belongs. Thus, the descent of the mirror below that node leads to the exclusion of the metal from the calculation, whose volume *V _{m}* may be greater than the contraction volume

Fig. 2 schematically shows the principle of interaction of the biphasic zone of the casting with the liquid metal mirror, implemented in the macroporosity model of the "PoligonSoft" software and the new model proposed in this work. In the "PoligonSoft" software model, the movement of the liquid metal mirror in a fixed dendritic framework leads to the drying of the interdendritic spaces and the formation of macroporosity (see Fig. 2a, zone 1), whose volumetric fraction is numerically equal to *f _{L}* [7].

In the new model, it is assumed that the dendritic framework above the liquid metal level is completely filled with liquid metal due to the capillary effect. In the capillary feeding zone *g _{L}=f_{L}* and

Fig. 3 - Scheme of macroporosity formation in the biphasic zone at the boundary of an open thermal node, where L - liquid metal; S - solid phase; F - mold; 1 - zone of dry dendrites (macroporosity); r - radius of curvature of the liquid metal surface; S/L - zone of continuous dendritic structure.

The formation of a continuous dendritic framework (zones where *f _{L }*

(1)

where *P _{atm}*

Due to the resulting depression, the liquid metal is drawn towards the center of the thermal node, drying the interdendritic spaces on its periphery, leading to the formation of macroporosity (zone 1 in Fig. 3b). The pressure drop in the thermal node and the volume of porosity formed depend on the capillary forces acting in the dendritic framework. Conditionally, the expression defining the equilibrium between the forces drawing the liquid metal from the periphery to the center of the thermal node and the capillary forces opposing this process can be written as follows:

(2)

*r* is the radius of curvature of the menisci in the interdendritic spaces. To estimate *r*, the following expression is used:

where *λ** _{II}* is the spacing between the secondary dendrite arms. Equation (2) allows determining the value of

From a certain point, due to the decrease in the liquid phase fraction, the boundaries of the thermal node become impermeable, and the metal contraction during crystallization is no longer compensated by the change in the liquid metal level in the dendritic framework, and the thermal node closes (see Fig. 4). This leads to an intense pressure drop in the thermal node, which is determined by the following expression:

(3)

The last term in the expression determines the pressure drop due to contraction. After some time, when the pressure at some point in the thermal node falls to a critical value *P _{crit}*, it becomes "energetically" favorable to form a new free surface of the liquid metal in the free liquid metal zone (

Fig. 4. Scheme of shrinkage cavity formation (b) in a closed thermal node (a), where L - liquid metal; S - solid phase; F - mold; M - free surface (mirror) of the liquid metal; V_{P} - volume of the shrinkage cavity

The appearance of a new flat separation surface fully compensates for the accumulated contraction in the thermal node from the moment of its isolation. Therefore, the location of the free surface of the liquid metal can be determined from the condition of equality between the contraction volume and the volume of the cavity formed: *V _{P}=V_{Ω}* .

With the appearance of the free surface of the liquid metal, the thermal node reopens in the sense that crystallization contraction will henceforth be compensated by the descent of the liquid metal's free surface. In the node, a shrinkage cavity will form according to the algorithm described earlier (see Fig. 2).

If a closed thermal node has a fixed solid phase framework everywhere, i.e., *f _{L }*

The rupture of the liquid metal's continuity due to the pressure drop occurs through the formation of macroporosity. The first pore forms when the pressure falls below the critical value *P _{crit}*. The most likely place for the pore to initiate is the point where the pressure is minimal, i.e., where the driving force for the formation of the separation surface Φ

Once the pore formation process has begun in the zone, it is likely that the contraction occurring in the next time step will be compensated by the growth of the existing pores. The condition for this process is *P ^{k+1}<-Pσ^{k}* in those nodes of the finite element mesh where there is porosity. Here,

As the liquid metal crystallizes, the radius of the channels in the solid phase framework decreases, the capillary pressure at the pore edge increases, and consequently, the pressure in the liquid metal decreases, which can ensure the growth of the pore (see Fig. 5). If the pressure required for the growth of the existing pore is less than *P _{crit}*, it is more likely that a new porosity zone will form. The relationship between

When *Φ _{poro}*

Fig. 5 illustrates the algorithm for modeling pore formation in the dendritic structure.

Fig. 5. Formation of macroporosity in a closed thermal node; a) scheme of pore formation in an immobile dendritic structure; b) forces acting at the pore-liquid metal interface; c) porosity distribution in the finite element nodes. L - liquid metal; S - solid phase; V_{P} - pore; r - radius of curvature of the meniscus; λ_{I} y λ_{II} - distances between the primary and secondary dendrite arms.

The pore formed in node iii initially occupies a volume *V _{1}*, assigned to that node. If this volume is less than the contraction volume that needs to be compensated, the pore "grows" at the expense of neighboring nodes in the finite element mesh with a total volume

(4)

where :

*P _{atm}* is the atmospheric pressure,

Based on equation (4), the pressure balance for node *i* at the boundary between the liquid metal and the pore can be written as:

(5)

where *g ^{k}_{L}* and

The growth of the porosity zone continues as long as the left side of equation (5) is greater than the right side. To meet this equality, in addition to the sum of the neighboring nodes of node *i *where the pore originated, the sums of the neighbors of these neighbors must be included –*ω*_{1},*ω*_{2},*...ω _{n}*:

(6)

Here ⟨f_{L}^{k+1}⟩* _{ω}* is the average value of the liquid phase fraction in the set of nodes

*V _{1}=g_{l,i}v_{i} *is the contraction volume compensated by the drying of the volume in the node where the pore originates;

*V _{3}* is calculated for the nodes that form a star for the nodes considered in

Below is one of the possible solutions to equation (6). Formalizing the sequence of pore propagation through the nodes of the finite element mesh, we write:

Equation (6) takes the form:

(7)

Based on this solution, porosity *g _{P}=g_{L}* is assigned to the nodes in the set

The model presented in this article has been implemented in the "PoligonSoft" casting simulation system for trial tests.

Figure 6 presents the results of the simulation of the solidification process of a large working blade of a gas turbine (GTU) made with the ChS70 alloy using the technology described in the work [9].

Fig. 6. Result of the porosity modeling in the working blades of the ST-20 gas turbine; a) exterior view of the casting block; b) P_{crit}= -1 MPa, *λ** _{II}* = 30 μm, E = 200 MPa;

c) P

1 - cast part; 2 - ceramic shell; 3 - thermal insulation.

The calculations were performed considering *f _{L}^{∗∗}*=0.7 and different values of the parameters

The formation of shrinkage pores occurs when the pressure in the molten metal drops to a critical value *P _{crit }*; the pressure in the melt at the time of shrinkage pore formation is negative, which can be estimated from the Laplace equation

Since the surface tension *σ* of molten nickel is approximately 1.7 N/m and the diameter of the pores observed in this casting ranges from 3.5 to 60 µm, it can be assumed that the pressure in the melt at the time of pore formation can vary from -0.1 to -1 MPa.

The meniscus radius *r* depends on the size of the interdendritic spaces, which are determined by the cooling rate of the melt in the biphasic zone. It was assumed that during the solidification of castings of this size, possible interdendritic spacings range from 20 to 300 µm.

The compressibility modulus of the melt can be estimated using the formula* E=a ^{2}ρ_{l}*

In the proposed model, the compressibility modulus characterizes the pressure drop process in the thermal node. Under ideal conditions, the rate of pressure drop in a closed thermal node is proportional to *E*. During the crystallization of a real casting, the solid metal crust surrounding the thermal node may not be airtight, which reduces the rate of pressure drop in the node. There is also the possibility of deformation of the crust under the action of the pressure difference between the environment and the inside of the thermal node, which also reduces the rate of pressure drop, as part of the crystallization contraction is compensated by deformation.

This indicates that the effective compressibility modulus of the melt must be significantly lower than the theoretical estimate. Within the framework of this model, the listed phenomena are not considered, and therefore, the compressibility modulus, like the critical pressure, is a fitting parameter that must be determined based on experimental data.

As expected, the volumetric fraction of porosity depends on the choice of model parameters *P _{crit}* and

In the variants shown in Figures 6a and 6b, the porosity in the turbine blade does not exceed 0.2%. In Figure 6c, the porosity in the blade reaches 1.5%. The location of the porosity zone practically does not depend on the model settings and is determined by the casting geometry and the pouring technology.

In general, the simulation results are quite consistent with the results of the metallographic study of the castings and with the simulation in the standard model of the "PoligonSoft" system [9].

To assess the sensitivity of the model to the parameters *P _{crit} *,

Fig. 7. Result of porosity simulation in a casting made of St.3. a) exterior view of the casting block; 1 - casting; 2 - St.3 metal mold; 3 - sand; b) Standard PoligonSoft model;

c) P_{crit}= -1 MPa, *λ** _{II}* = 300 μm, E = 2000 MPa;

d) P

e) P

f) P

g) P

The thermophysical properties of the casting were calculated using the Computherm thermodynamic database [11] based on the alloy's chemical composition. The initial temperature of the metal was 1550°C, and the mold and sand were at 500°C. Cooling was conducted in ambient air at 20°C through convection with a heat transfer coefficient of 10 W/m²/K and radiation, with an emissivity factor for both the mold and the metal of 0.8.

For comparison, calculations were performed in the "PoligonSoft" system with standard configurations used for simulation in the casting production of the FGUP NPCG "Salut" company.

The simulation results are presented in Figure 7. The solidification of the casting in all presented variants corresponds to the scheme shown in Figure 1. In the first stage, the crystallization of the open thermal node occurs, compensating for contraction by the displacement of the casting mirror. In the second stage, a closed thermal node is formed where, depending on the set model parameters, a closed cavity (Figures 7c and 7d) or dispersed macroporosity (Figures 7e-7g) is formed.

The realization of one of these variants depends on the rate of two processes: the pressure drop in the thermal node due to crystallization contraction and the growth of the solid phase fraction leading to the formation of a continuous dendritic structure.

A rapid pressure drop in the thermal node to a critical value *P _{crit} * before the formation of a continuous dendritic structure leads to the formation of a free surface and an internal shrinkage cavity. This mode of porosity formation is favored by a large compressibility modulus

With a low compressibility modulus, the pressure in the thermal node drops slowly and reaches a critical value when the continuous dendritic structure has already formed. In this case, the accumulated crystallization contraction manifests as dispersed macroporosity (Figures 7e, 7g). The volumetric fraction and size of the formed pores depend on the amount of liquid phase and the dendritic structure parameters, i.e., the spacing between the secondary dendrite arms.

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[5] Casting simulation system "PoligonSoft", trademark of "CSoft Development", http://csdev.ru.

[6] Monastyrsky V.P. Modeling of macroporosity and shrinkage cavity formation in castings // Russian Foundryman. No. 10, 2011, pp. 16-21.

[7] Tikhomirov M.D. Fundamentals of casting process modeling. Shrinkage problem. - Moscow: Appendix to the journal "Foundry Production". - 2001. No. 12, pp. 8-14.

[8] Kent D. Carlson, Zhiping Lin, C. Beckermann, G. Mazurkevich, and Marc C. Schneider. Modeling porosity formation in aluminum alloys. Proceedings MCWASP-XI, ed. Ch.-A. Gandin, M. Bellet (Warrendale, PA: The Minerals, Metals & Materials Society, 2006), pp. 627-634.

[9] Monastyrsky V.P., Monastyrsky A.V., Levitan E.M. Development of casting technology for large gas turbine blades for power plants using "Poligon" and ProCAST systems // Foundry Production. No. 9, 2007, pp. 29-34.

[10] Physical quantities: Reference manual / Babichev A.P., Babushkina N.A., Bratkovsky A.M. et al.; Ed. Grigoriev I.S., Meilikhov E.Z. - Moscow: Energoatomizdat, 1991. - 1232 pp.

[11] Thermodynamic database for iron-based superalloys: PanIron 5.0, CompuTherm, LLC, USA.

Translated by A.J. Camejo Severinov

Original text in Russian

Development of Quenching Technology for Steel Parts Using Computer Modeling

Numerical Simulation of the Formation Conditions of Castings with Exothermic Sleeves

Application of a new porosity model for predicting shrinkage defects in castings

Imitative model of the formation of shrinkage cavities and macroporosity

Quantitative Estimation of Formation of Shrinkage Porosity by the Niyama Criterion

Development of Promising Technology for Manufacturing Parts of Gas Turbine Engines