mathematical modeling of inclusion dissolution processes: the grow code ernesto gutierrez-miravete...

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.