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Mathematical Modeling of Inclusion Dissolution Processes:
The GROW Code
Ernesto Gutierrez-MiraveteRensselaer at Hartford
Brice CassentiUnited Technologies Research Center
Mas Hongoh Pratt & Whitney
Outline
• Introduction
• Model Description
• Description of the GROW Code
• Examples Runs of the Code
• Parametric and Sensitivity Studies
• Summary
Introduction
• Undetected N- and/or O-containing particles in Ti alloys (hard-alpha) can result in catastrophic failure of aircraft engine components.
• The process metallurgy of Ti alloys provides many potential sources of N and/or O.
• Better understanding of the dissolution behavior of N- and/or O containing Ti inclusions in Ti alloys during thermal processing is required.
Model Description• When N and/or O come in contact with Ti
several different phases can form depending on composition and temperature.– The Ti-N phase diagram (Fig 1a).– The Ti-O phase diagram (Fig 1b).
• If an isolated N-rich or O-rich seed particle is embedded in a Ti matrix, the various phases appear as concentric layers on the original particle.
Fig 1a
Fig 1b
Model Description (contd.)
• The concentration of impurity decreases with distance from the center of the seed particle.
• Dissolution of the resulting layers involves mass transport of N and/or O away from the seed particle.
• See Figure 2.
C
x
L
Fig 2 Concentration profile around a dissolving inclusion.
Flux of N (or O)
Model Description (contd.)
• Assumptions and Limitations– Binary Systems (Ti-N or Ti-O)
– Chemical Equilibrium at all Interfaces
– All Phases form Ideal Solutions
– Temperatures restricted to within beta transus of pure Ti and first peritectic
• 882 - 2020 C for Ti-N and 882- 1720 C for Ti-O
– All Necessary Diffusivity Data readily Available
– Porosity is Neglected
Model Description (contd.)
• Governing Equation
c/t = div ( D grad a)where
c = concentration of N (or O)
D = diffusivity of N (or O)
a = activity of N (or O) (Figs 3 and 4)
a
C
L
Fig 3 Activity-concentration relationship in Ti-N (or Ti-O)
a*
a
L
Fig 4
Model Description (contd.)
• Solution Methodology: – Finite Difference Method– Fixed Mesh– Explicit Scheme
• Physico-Chemical Data:– Phase Diagrams– Diffusivities
The GROW code• Derived from earlier code MICRO developed at
UTRC.• FORTRAN program embedded in a UNIX
wrapper.• Inputs:
– Inclusion size and geometry– Inclusion and matrix concentration– Thermal history– Mesh
The GROW Code (contd.)
• Outputs– Concentration profiles around inclusion at
selected times during specified temperature history
– Extent of the various layers as functions of time.
– Extent of the diffusion zone surrounding the inclusion as function of time.
Example Runs (Ti-N)
• Isothermal Hold at 1200 C (Figs. 5a and 5b)
• Isothermal Hold at 1600 C (Figs. 6a and 6b)
• Isothermal Hold at 2020 C (Figs. 7a and 7b)
• Sample Thermal History (Figs. 8a and 8b) t (s) 0 1 5 10 12 13 15
T(C) 2000 1670 1000 1000 1300 1500 1000
Fig 5a
Fig 5b
Fig 6a
Fig 6b
Fig 7a
Fig 7b
Fig 8a
Fig 8b
Example Runs (Ti-N) (contd.)
• Two-dimensional system (250 by 1000 micron inclusion). Figs. 9a and 9b.
• Three-dimensional system (250 by 500 by 1000 micron inclusion). Figs. 10a and 10b.
Fig 9a
Fig 9b
Fig 10a
Fig 10b
Example Runs (Ti-O)
• Isothermal Hold at 1200 C (Figs. 11a and 11b)• Isothermal Hold at 1600 C (Figs. 12a and 12b)• Isothermal Hold at 1720 C (Figs. 13a and 13b)• Sample Thermal History (Figs. 14a and 14b) t (s) 0 1 5 10 12 13 15
T(C) 2000 1670 1000 1000 1300 1500 1000
Fig 11a
Fig 11b
Fig 12a
Fig 12b
Fig 13a
Fig 13b
Fig 14a
Fig 14b
Example Runs (Ti-N) (contd.)
• Two alternative calculation methods of phase thickness under thermal history (Figs. 15 and 16)
• Two alternative calculation methods of phase thickness under isothermal hold at 2020 C (Fig. 17).
Fig 15
Fig 16
Fig 17
Parametric and Sensitivity Studies
• Effect of Initial Seed Particle Size on Extent of Diffusion Zone under Specified Thermal History (Triple Melt VAR).
• Effect of Initial Seed Particle Concentration on Extent of Diffusion Zone under Specified Thermal History (Triple Melt VAR).
Summary (contd.)
• A mathematical model and associated computer code are now available to investigate the spread of diffusion zones around N- or O-rich inclusion particles in Ti as a function of thermal history, inclusion geometry and composition.
Summary (contd.)
• Once fully validated, the GROW code can help process engineers, designers, NDT and quality assurance personnel to achieve their goal of producing hard-alpha free aircraft engine components.
Summary (contd.)
• Although the results of calculation are in reasonably good agreement with at least some of the existing empirical data on dissolution rates, full validation of the model requires comparison against results of carefully conducted experiments on selected systems.