Mathematical Modeling of Inclusion Dissolution Processes: The GROW Code

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Mathematical Modeling of Inclusion Dissolution
ProcessesThe GROW Code

• Ernesto Gutierrez-Miravete
• Rensselaer at Hartford
• Brice Cassenti
• United Technologies Research Center
• Mas Hongoh
• Pratt Whitney

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Outline

• Introduction
• Model Description
• Description of the GROW Code
• Examples Runs of the Code
• Parametric and Sensitivity Studies
• Summary

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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.

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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.

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Fig 1a
6
Fig 1b
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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.

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C
Flux of N (or O)
?
?
?
L
x
Fig 2 Concentration profile around a dissolving
inclusion.
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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

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Model Description (contd.)

• Governing Equation
• dc/dt 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)

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a
?
?
?
L
C
Fig 3 Activity-concentration relationship in Ti-N
(or Ti-O)
12
a
?
?
?
L
a
Fig 4
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Model Description (contd.)

• Solution Methodology
• Finite Difference Method
• Fixed Mesh
• Explicit Scheme
• Physico-Chemical Data
• Phase Diagrams
• Diffusivities

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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

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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.

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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

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Fig 5a
18
Fig 5b
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Fig 6a
20
Fig 6b
21
Fig 7a
22
Fig 7b
23
Fig 8a
24
Fig 8b
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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.

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Fig 9a
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Fig 9b
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Fig 10a
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Fig 10b
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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

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Fig 11a
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Fig 11b
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Fig 12a
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Fig 12b
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Fig 13a
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Fig 13b
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Fig 14a
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Fig 14b
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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).

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Fig 15
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Fig 16
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Fig 17
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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).

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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.

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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.

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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.