Estimation of densities and coefficients of thermal expansion of solid Ni-base superalloys

doi:10.1016/S0921-5093(97)00699-0

Materials Science and Engineering: A

Volume 245, Issue 1, 30 April 1998, Pages 135-141

https://doi.org/10.1016/S0921-5093(97)00699-0 Get rights and content

Abstract

To calculate the densities of solid Ni-base superalloys as functions of temperature and composition, lattice parameters of 18 alloys at 20°C and coefficients of thermal expansion (CTEs) of 17 alloys were estimated by combining available data. To estimate the lattice parameters of the alloys at 20°C, the changes in the lattice parameter of Ni caused by additions of the elements were simply summed to get the lattice parameters of alloys. This procedure results in calculated densities that are <1% in error for all 18 alloys. For the densities of the alloys at an elevated temperature, the CTEs were then estimated. The CTEs of the 17 Ni-base alloys were gathered and regressed to yield the CTE as a function of temperature and composition. The CTEs calculated from the regressions are <10% in error for all 17 alloys at 400, 700, 1000 and 1300 K. These errors of the CTEs also result in densities that are within 0.5% error or less in the overall temperature range.

Introduction

In order to model shrinkage, transport phenomena and macrosegregation during dendritic solidification of alloys, the densities of the solid and liquid must be known. In the solidification range, the temperature and the concentrations of the elements in both phases vary so we have devised methods to estimate the densities. We are particularly interested in simulating the solidification of Ni-base superalloys because of their use in castings that are used as critical components in gas-turbine engines.

In the previous paper [1], the densities of many of the liquid transition metals and aluminum, which include the alloy elements in Ni-base superalloys, and the change in density with temperature, called the temperature coefficient of liquid density (dρ/dT, where ρ is density and T is temperature), were gathered, reviewed and applied to make a simple correlation. The correlation is used to critically evaluate the temperature coefficients, which are not well established for many of the transition metals with high melting points (mp). Also, this procedure was applied to estimate the liquid densities of five Ni-base superalloys as a verification [1]. By including a regression estimate of the molar mixing volume, the estimates of the liquid densities agreed with the measured values to ±2.5%.

In this paper, densities of solid Ni-base superalloys were estimated. First, the densities of Ni-base alloys at 20°C were estimated as a function of composition. Second, the coefficient of thermal expansion (CTE) of Ni was regressed as a function of temperature and then the CTEs of the solid Ni-base superalloys were estimated by regression analyses to get functions of temperature and composition.

Section snippets

Densities of Ni-base superalloys at 20°C

In order to estimate the densities of the alloys at 20°C, the density was calculated from the lattice parameter as $ρ= 4 ∑ i=1 n_{s} M_{i} X_{i} N_{0} a^{3},$ where ρ is the density of the alloy (Ni-rich γ), g cm⁻³; M_i is the atomic weight of the element i, g mol⁻¹; X_i is the atom fraction of the element i; N₀ is Avogadro's number, mol⁻¹; a is the lattice parameter of the alloy, cm; n_s is the number of solute elements.

Abe and Tanabe [2]measured the lattice parameter of Ni–Cr–W alloys over a wide range of concentrations.

Coefficient of thermal expansion of Ni

The CTE is estimated so that the density of the solid at an elevated temperature can be calculated. First, the CTEs of Ni from several sources 12, 13, 14, 15, 16were compiled and regressed to yield CTE as a function of temperature.

The CTE is defined as $α= 1 l_{R} l−l_{R} T−T_{R},$ where l is length, T is temperature in K, T_R is the reference temperature (293 K) and l_R is the reference length at T_R. Four data sets are plotted in Fig. 1. Three sets are from compiled sources 12, 13, 14; the fourth is from an

Coefficients of thermal expansion of Ni-base superalloys

The CTEs of the Ni-base alloys in Table 4 were compiled from various sources 8, 9, 18, 19, 20, 21and subjected to regression analyses in order to obtain an equation for the CTE as a function of temperature and composition. In addition to the elements shown in Table 4, some of the alloys contain small amounts of C and Hf and impurities (Fe, Mn, Si). Only the elements shown in Table 4 were included in the regression analyses. Co is chemically similar to Ni and has almost the same atomic radius.

Acknowledgements

This work was supported by the Advanced Research Projects Agency, as part of the Micromodeling Program of the Investment Casting Cooperative Arrangement, under contract MDA972-93-2-0001. The authors very much appreciate Dr Dan Backman of General Electric Aircraft Engines, Dr Boyd Mueller of Howmet Research Center and Dr W. Boettinger of NIST for always being available to discuss our work and in administrating the contract. We also thank Dr Anthony Giamei of the United Technologies Research

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