The determination of the form of a probability density function (PDF3) of diameters for nodular particles by using a probability density function (PDF2), which form is empirically estimated from cross-sections of these nodules in a metallographic specimen, can be regarded as a special case of Wicksell's corpuscle problem (WCP). The estimation of the PDF3 for the nodular particles provides information about the kinetics of these particles nucleation, and so about the kinetics of their growth. This information is essential for building more accurate mathematical models of the alloy crystallization. In the paper there are presented two derivations of the methods used for the estimation of the PDF3 form. The first method bases on diameters received from a planar cross-section. The second one uses also data from the planar cross-section but not the diameters only chords. Both methods provide practical rules for the analysis of the empirical diameters’ and chord’s size distribution and allow to estimate the mean value of the external surface area of the particles.
The paper presents validation tests for method which is used for the evaluation of the statistical distribution parameters for 3D particles’ diameters. The tested method, as source data, uses chord sets which are registered from a random cutting plane placed inside a sample space. In the sample space, there were individually generated three sets containing 3D virtual spheres. Each set had different Cumulative Distribution Function (CDF3) of the sphere diameters, namely: constant radius, normal distribution and bimodal distribution as a superposition of two normal distributions. It has been shown that having only a chord set it is possible, by using the tested method, to calculate the mean value of the outer sphere areas. For the sets of data, a chord method generates quite large errors for around 10% of the smallest nodules in the analysed population. With the increase of the nodule radii, the estimation errors decrease. The tested method may be applied to foundry issues e.g. for the estimation of gas pore sizes in castings or for the estimation of nodule graphite sizes in ductile cast iron.
The study presents a mathematical model of the crystallisation of nodular graphite cast iron. The proposed model is based on micro- and macromodels, in which heat flow is analysed at the macro level, while micro level is used for modelling of the diffusion of elements. The use of elementary diffusion field in the shape of an averaged Voronoi polyhedron [AVP] was proposed. To determine the geometry of the averaged Voronoi polyhedron, Kolmogorov statistical theory of crystallisation was applied. The principles of a differential mathematical formulation of this problem were discussed. Application of AVP geometry allows taking into account the reduced volume fraction of the peripheral areas of equiaxial grains by random contacts between adjacent grains. As a result of the simulation, the cooling curves were plotted, and the movement of "graphite-austenite" and "austenite-liquid” phase boundaries was examined. Data on the microsegregation of carbon in the cross-section of an austenite layer in eutectic grains were obtained. Calculations were performed for different particle densities and different wall thicknesses. The calculation results were compared with experimental data.
Theoretical problems concerning the determination of work parameters of the two-phase sand-air stream in the cores making process by blowing methods as well as experimental methods of determination of the main and auxiliary parameters of this process decisive on the cores quality assessed by the value and distribution of their apparent density are presented in the paper. In addition the results of visualisations of the core-box filling with the sand-air stream, from the blowing chamber, obtained by the process filming by means of the quick-action camera are presented in the paper and compared with the results of simulation calculations with the application of the ProCast software.
The effects of filling the core box cavity and sand compaction in processes of core production by blowing methods (blowing, shooting) depend on several main factors. The most important are: geometrical parameters of cavity and complexity of its shape, number, distribution and shape of blowing holes feeding sands as well as the venting of a technological cavity. Values of individual parameters are selected according to various criteria, but mostly they should be adjusted to properties of the applied core sand. Various methods developed by several researchers, including the authors own attempts, allow to assess core sands properties on the basis of special technological tests projecting the process into a laboratory scale. The developed criteria defining a degree or a filling ability factor provide a better possibility of assessing the core sand behavior during flowing and core box filling, which indicate the value and structure of the obtained compacting decisive – after hardening – for strength and permeability. The mentioned above aspects are analyzed – on the basis of authors’ own examinations - in the hereby paper.
The mathematical model of the globular eutectic solidification in 2D was designed. Proposed model is based on the Cellular Automaton Finite Differences (CA-FD) calculation method. Model has been used for studies of the primary austenite and of globular eutectic grains growth during the ductile iron solidification in the thin wall casting. Model takes into account, among other things, non-uniform temperature distribution in the casting wall cross-section, kinetics of the austenite and graphite grains nucleation, and non-equilibrium nature of the interphase boundary migration. Calculation of eutectic saturation influence (Sc = 0.9 - 1.1) on microstructure (austenite and graphite fraction, density of austenite and graphite grains) and temperature curves in 2 mm wall ductile iron casting has been done.