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Program MAP_STEEL_CONCDIF

  1. Provenance of code.
  2. Purpose of code.
  3. Specification.
  4. Description of subroutine's operation.
  5. References.
  6. Parameter descriptions.
  7. Error indicators.
  8. Accuracy estimate.
  9. Any additional information.
  10. Example of code
  11. Auxiliary subroutines required.
  12. Keywords.
  13. Download source code.
  14. Links.

Provenance of Source Code

H.K.D.H. Bhadeshia,
Phase Transformations Group,
Department of Materials Science and Metallurgy,
University of Cambridge,
Cambridge, U.K.

Added to MAP: September 1999.

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Purpose

MAP_STEEL_CONCDIF uses a numerical solution for the problem of the growth of a planar interface under the conditions of volume diffusion control and a diffusion coefficient in the matrix which varies with concentration, to obtain a value for the one-dimensional parabolic thickening rate constant for the growth of ferrite in austenite.

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Specification

Language:FORTRAN
Product form:Source code

Complete program.

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Description

The problem of a planar boundary growing under conditions of volume diffusion control, with a diffusion coefficient in the matrix phase which varies with concentration, is solved numerically to provide a value for the growth rate constant [1,2]. This numerical method is more rigorous than using a weighted average value of the diffusivity, as has been done in other programs [3] and gives more realistic results when the velocity is not constant. The model is relevant for the growth of precipitates from solid solution and is used in this program to find the parabolic rate constant for the growth of ferrite from austenite in a low alloy steel. The program solves equations 5 and 6 from reference 2:

[Equation 5 of reference 2.]

and

[Equation 6 of reference 2.]

where

[Equation 3 of reference 2.]
subject to the boundary conditions for the concentration, c:
c = Cn as eta tends to infinity (in the austenite, well away from the interface)
c = Co as eta tends to 0 (in the austenite, at the interface)

Cn is the carbon concentration in the austenite, well away from the interface (at infinity).
Co is the carbon concentration in the austenite at the interface.
C1 is the carbon concentration in the ferrite at the interface.
Do is the carbon diffusion coefficient, D, at the interface.
x is the distance from the interface in the direction of motion of the interface.
t is the time.
alpha is the parabolic growth rate constant for ferrite in austenite.

For a given steel composition, temperature and carbon concentration at the interface in the austenite, Co, the program calls MAP_STEEL_OMEGA to calculate Cn, MAP_STEEL_XALPH to obtain a value for C1, and MAP_STEEL_DIFFUS to calculate Do and an initial value for D. In the numerical analysis the concentration profile is split up into N sections. The calculations are carried out using different values of N. An initial estimate for alpha is obtained with N=10. Subsequent evaluations are made for N=60, 110 and 160. Further details of the numerical method used are given in reference [2].

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References

  1. C. Atkinson, 1967, Acta Metallurgica, 15, 1207.
  2. C. Atkinson, 1968, Acta Metallurgica, 16, 1019.
  3. MAP_STEEL_ALLSOL; MAP_STEEL_ALLL.

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Parameters

Input parameters

C - real array of dimension 8
C(1) - C(7) are the concentrations (wt%) of the alloying components carbon, silicon, manganese, nickel, molybdenum, chromium and vanadium, in that order. (C(8) is used to hold the iron concentration, assumed to be the remaining wt%.)

CTEMP - real
CTEMP is the temperature (°C) for which the parabolic rate constant is to be calculated.

XGAG - real
XGAG is the equilibrium mole fraction of carbon in the austenite at the austenite-ferrite interface.

Output parameters

XALPHA - real
XALPHA is the equilibrium mole fraction of carbon in the ferrite at the austenite-ferrite interface.

DIF0 - real
DIF0 is the diffusion coefficient at the interface (cm2s-1).

N - integer
N is the number of steps used for approximating the concentration profile in the calculations.

ALPH - real
ALPH is the one-dimensional parabolic rate constant, alpha, (cms-0.5).

RESDU1 - real
RESDU1 is equal to | In-0.5 - I*n-0.5| and gives an indication of the accuracy of the results. See reference [2] for further details.

RESDU2 - real
RESDU2 is the change in alpha/sqrt(Do) between successive iterations.

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Error Indicators

None.

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Accuracy

The iteration procedure is iterated until RESDU1 is less than 10-4 and RESDU2 is less than 10-6.
i.e.

| In-0.5 - I*n-0.5| < 10-4
and
alpha/sqrt(Do) < 10-6.

For further details see reference [2].

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Further Comments

None.

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Example

9.1 Program text

Complete program.

9.2 Program data

 Input  C,  Si,  Mn,  Ni,  Mo,  Cr,  V wt%:
      0.12 0.49 1.16 0.0  0.0  0.0  0.0

 Input temperature (deg.C) and C mole fraction in austenite at the interface:
 780 0.01

 Repeat calculations for another set of data (y/n)?
y
 Input temperature (deg.C) and C mole fraction in austenite at the interface:
 700 0.0266
 
 Repeat calculations for another set of data (y/n)?
n

9.3 Program results

**************************************************************************

 Element:       C      Si      Mn      Ni      Mo      Cr      V
 conc. wt%:  0.1200  0.4900  1.1600  0.0000  0.0000  0.0000  0.0000
 mole frac:  0.0055  0.0096  0.0117  0.0000  0.0000  0.0000  0.0000

 Carbon-carbon interaction energy in austenite =  8402.7 J/mol
 Starting mole fraction of carbon in austenite =  0.0055

 Temperature =  780.00 deg. C
 Equ. C conc. in austenite at the interface = 0.0100 mole fractions
 Equ. C conc. in ferrite at the interface   = 0.4859D-03 mole fractions
 Diffusivity of carbon in austenite (Do)    = 0.1250D-07 squ.cm/s

 No. steps  Rate constant (cm/s**0.5)  Residue 1  Residue 2
     10             0.7350D-04         0.481D-04  0.000D+00
     60             0.8405D-04         0.581D-04  0.657D-06
    110             0.8511D-04         0.431D-04  0.300D-06
    160             0.8551D-04         0.498D-04  0.545D-06

 Temperature =  700.00 deg. C
 Equ. C conc. in austenite at the interface = 0.0266 mole fractions
 Equ. C conc. in ferrite at the interface   = 0.6691D-03 mole fractions
 Diffusivity of carbon in austenite (Do)    = 0.5162D-08 squ.cm/s

 No. steps  Rate constant (cm/s**0.5)  Residue 1  Residue 2
     10             0.1163D-03         0.738D-04  0.000D+00
     60             0.1484D-03         0.176D-04  0.845D-06
    110             0.1522D-03         0.277D-04  0.848D-06
    160             0.1536D-03         0.322D-04  0.764D-06

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Auxiliary Routines

Subroutines        Functions
MAP_STEEL_OMEGA        MAP_STEEL_CG
MAP_UTIL_TRAPE        MAP_STEEL_DCG
MAP_STEEL_DIFFUS        MAP_STEEL_XALPH

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Keywords

parabolic, thickening, diffusion, planar, growth, concentration dependent, diffusion controlled, diffusion coefficient

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Download

Download source code

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