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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.

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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.

 

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.

 

Specification

 Language:    FORTRAN
 Product form:Source code

Complete program.

 

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 = C_n as eta tends to infinity (in the austenite, well away from the
     interface)
     c = C_o as eta tends to 0 (in the austenite, at the interface)

C_n is the carbon concentration in the austenite, well away from the
interface (at infinity).
C_o is the carbon concentration in the austenite at the interface.
C_1 is the carbon concentration in the ferrite at the interface.
D_o 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, C_o, the program calls MAP_STEEL_OMEGA to
calculate C_n, MAP_STEEL_XALPH to obtain a value for C_1, and MAP_STEEL_DIFFUS
to calculate D_o 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].

 

References

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

 

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 (deg. 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 (cm**2/s).

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 | I_(n-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(D_o) between successive iterations.

 

Error Indicators

None.

 

Accuracy

The iteration procedure is iterated until RESDU1 is less than 10(-4) and
RESDU2 is less than 10(-6).
i.e.
                         | I_(n-0.5) - I*_(n-0.5)| < 10(-4)
and
                           alpha/sqrt(D_o) < 10(-6).
For further details see reference [2].

 

Further Comments

None.

 

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


 

Auxiliary Routines

   Subroutines            Functions
MAP_STEEL_OMEGA        MAP_STEEL_CG
MAP_UTIL_TRAPE         MAP_STEEL_DCG
MAP_STEEL_DIFFUS       MAP_STEEL_XALPH

 

Keywords

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

 

Download

Download source code

 

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