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  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

C.Garcia de Andres and C.Capdevila-Montes,
Departamento de Metalurgia Fisica,
Avda. Gregorio del Amo 8,
Madrid 28040

H.K.D.H. Bhadeshia and S.S Babu*,
Phase Transformations Group,
Department of Materials Science and Metallurgy
University of Cambridge
Cambridge CB23QZ

*Dr Suresh Babu is now with the Oak Ridge National Laboratory, Oak Ridge, TN, USA

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Uses a finite difference method for the solution of the problem of the diffusion profile collapse during heat treatment at temperature T.

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Complete program

Product form:Source code

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The carbon composition profile build up ahead of the allotriomorphic ferrite/austenite interface in the residual austenite, is examined here using a finite difference method to stimulate any homogenisation of concentration. The main advantage of the application of finite difference method to this problem is that it can handle the changing boundary conditions during the homogenisation process.

The finite difference method is based on the technique discussed by Crank [6]. The model assumes one dimensional diffusion process ahead of a planar allotriomorphic-austenite interface. The initial profile set up at time t=0.0, is given by analytical equation calculated by Coates [3,4,5]. The composition profile is then subjected to finite difference method after normalising the variables to dimensionless quantities suitable for this method.

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  1. H.K.D.H.Bhadeshia, Metal Science, 15 477 1981
  2. H.K.D.H.Bhadeshia, Metal Science, 14 477 1980
  3. D.E.Coates, Metallurgical Transactions, 4 395 1973
  4. D.E.Coates, Metallurgical Transactions, 4 1077 1973
  5. D.E.Coates, Metallurgical Transactions, 4 2313 1973
  6. J.Crank,The mathematics of Diffusion, Clarendon Press, Oxford, 1975
  7. S.S.Babu, PhD. Disertation, University of Cambridge, 1991
  8. S.S.Babu and H.K.D.H.Bhadeshia, Mat. Sci. and Tech. 6 1005 1991
  9. S.S.Babu and H.K.D.H.Bhadeshia, Mat. Trans. JIM 36 679 1991

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Input parameters

CARBON - real
Overall concentration of carbon (in weight percent).

SI - real
Overall concentration of silicon (in weight percent).

MN- real
Overall concentration of manganese (in weight percent).

NI - real
Overall concentration of nickel (in weight percent).

MO - real
Overall concentration of molybdenum (in weight percent).

CR - real
Overall concentration of chromium (in weight percent).

V - real
Overall concentration of vanadium (in weight percent).

CO - real
Overall concentration of cobalt (in weight percent).

CU - real
Overall concentration of copper (in weight percent).

AL - real
Overall concentration of aluminium (in weight percent).

WO - real
Overall concentration of tungsten (in weight percent).

LINEAL - real
Prior austenite grain size in microns.

CTEMP - real
Isothermal heat treatment temperature (in centigrade).

TIME - real
Time at CTEMP temperature (in seconds).

HALF - real
Half-thickness of allotriomorphic ferrite (in centimeters).

TIME2 - real
Time required to form allotriomorphic ferrite of half-thickness HALF at temperature T (in seconds).

Output parameters

a) Output to the screen:

Interface Concentration - real
Concentration of carbon at the ferrite/austenite interface in mole fraction.

Time - real
Time required to reach this carbon concentration at the interface (in seconds).

b) Output to the OUT file:

TIME - real
Time to check the evolution of profile ahead the ferrite-austenite interface (in seconds).

Profile concentration ahead the interface corresponding at time selected.

Distance into austenite grain ahead the interface (in centimeters).

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Error Indicators


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A good accuracy is maintained if NSLI is higher than 200 slices.

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Further Comments

The coefficient R=D*t/x**2 of the general finite difference equation described by Cranck [3,4,5] is taken to be 0.1. This particular value was assumed after checking that variation of R coefficient from 0.1 - 0.3 made negligible difference to the results.

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1. Program text

Complete program.

2. Program data

0.053 0.510  1.410 0.030 0.010 1.59 0.000 0.000 0.000 0.000 0.000

Chemical composition C,Si,Mn,Ni,Mo,Cr,V,Co,Cu,Al,W
Prior austenite grain size (in microns)
Temperature (in centigrade)
Time at this temperature (in seconds)
Half-thickness of allotriomorphic ferrite (in centimeters)
Time required to form allotriomorphic ferrite of previous half thickness
Anealing temperature (in centigrade)

3. Program results

a) Output to STDOUT:

           ****Chemical composition****

  Carbon= .0530               Silicon= .5100
  Manganese=1.4100            Nickel= .0300
  Molybdenum= .0100           Chromium=1.5900
  Vanadium= .0000


     Isothermal temperature= 600.0
     Holding time= 120.00
     Austenite grain size in microns= 70.00

     +++++++Initial Profile Calculation is over+++++++

     Area under the normalised curve=  .5430652E+03


          /////-Finite difference method starts-/////

              INTERFACE CONCENT.          TIME

                 .4578793E-01          .0000000E+00
                 .4422596E-01          .1962215E-01
                 .4285907E-01          .6543451E+00
                 .4164919E-01          .1306012E+01
                 .4056746E-01          .1971883E+01
                 .3959168E-01          .2651476E+01
                 .3870459E-01          .3343907E+01
                 .3789258E-01          .4044720E+01
                 .3714478E-01          .4758699E+01
                 .3645246E-01          .5480714E+01
                 .3580847E-01          .6211984E+01
                 .3520693E-01          .6950158E+01
                 .3464294E-01          .7696216E+01
                 .3411238E-01          .8451103E+01
                 .3361178E-01          .9211989E+01
                 .3313815E-01          .9979625E+01
                 .3268895E-01          .1075474E+02
                 .3226195E-01          .1153805E+02
                 .3185522E-01          .1232622E+02
                 .3146707E-01          .1311980E+02
                 .3109601E-01          .1391933E+02
                 .3074072E-01          .1472533E+02
                 .3040001E-01          .1553403E+02
                 .3007286E-01          .1634583E+02
                 .2975830E-01          .1716550E+02
                 .2945549E-01          .1798906E+02
                 .2916366E-01          .1881689E+02
                 .2888212E-01          .1964934E+02
                 .2861023E-01          .2048677E+02
                 .2834743E-01          .2132484E+02
                 .2809317E-01          .2216853E+02
                 .2784697E-01          .2301816E+02
                 .2760840E-01          .2386924E+02
                 .2737704E-01          .2472199E+02
                 .2715251E-01          .2558157E+02
                 .2693446E-01          .2644331E+02
                 .2672258E-01          .2730742E+02
                 .2651655E-01          .2817413E+02
                 .2631609E-01          .2904364E+02
                 .2612095E-01          .2991615E+02
                 .2593089E-01          .3079189E+02
                 .2574567E-01          .3167104E+02
                 .2556508E-01          .3255380E+02
                 .2538892E-01          .3344039E+02
                 .2521701E-01          .3433099E+02
                 .2504917E-01          .3522035E+02
                 .2488523E-01          .3611407E+02
                 .2472503E-01          .3701233E+02
                 .2456844E-01          .3790974E+02
                 .2441530E-01          .3881203E+02
                 .2426550E-01          .3971372E+02
                 .2411890E-01          .4062061E+02
                 .2397538E-01          .4152714E+02
                 .2383485E-01          .4243919E+02
                 .2369718E-01          .4335110E+02
                 .2356228E-01          .4426886E+02
                 .2343006E-01          .4518671E+02
                 .2330042E-01          .4610474E+02
                 .2317327E-01          .4702307E+02
                 .2304855E-01          .4794780E+02
                 .2292615E-01          .4887303E+02
                 .2280602E-01          .4979885E+02
                 .2268808E-01          .5072536E+02
                 .2257227E-01          .5165884E+02
                 .2245851E-01          .5259321E+02
                 .2234674E-01          .5352856E+02
                 .2223691E-01          .5446498E+02
                 .2212896E-01          .5540256E+02
                 .2202283E-01          .5634137E+02
                 .2191847E-01          .5728152E+02
                 .2181583E-01          .5822307E+02
                 .2171486E-01          .5916611E+02
                 .2161552E-01          .6011073E+02
                 .2151776E-01          .6105701E+02
                 .2142154E-01          .6200502E+02
                 .2132681E-01          .6295486E+02
                 .2123355E-01          .6390659E+02
                 .2114170E-01          .6486031E+02
                 .2105123E-01          .6581608E+02
                 .2096212E-01          .6677400E+02
                 .2087431E-01          .6773413E+02
                 .2078779E-01          .6869656E+02
                 .2070251E-01          .6966136E+02
                 .2061846E-01          .7062169E+02
                 .2053559E-01          .7158451E+02
                 .2045388E-01          .7254989E+02
                 .2037331E-01          .7351791E+02
                 .2029384E-01          .7448866E+02
                 .2021545E-01          .7546221E+02
                 .2013812E-01          .7643149E+02
                 .2006182E-01          .7740368E+02
                 .1998652E-01          .7837885E+02
                 .1991221E-01          .7935708E+02
                 .1983887E-01          .8033118E+02
                 .1976647E-01          .8130844E+02
                 .1969499E-01          .8228894E+02
                 .1962441E-01          .8327276E+02
                 .1955471E-01          .8425256E+02
                 .1948588E-01          .8523578E+02
                 .1941790E-01          .8622250E+02
                 .1935074E-01          .8720529E+02
                 .1928440E-01          .8819168E+02
                 .1921885E-01          .8918174E+02
                 .1915408E-01          .9016796E+02
                 .1909007E-01          .9115796E+02
                 .1902681E-01          .9215180E+02
                 .1896428E-01          .9314187E+02
                 .1890247E-01          .9413590E+02
                 .1884137E-01          .9512621E+02
                 .1878096E-01          .9612056E+02
                 .1872122E-01          .9711904E+02
                 .1866215E-01          .9811388E+02
                 .1860374E-01          .9911293E+02
                 .1854596E-01          .1001084E+03
                 .1848882E-01          .1011082E+03
                 .1843229E-01          .1021123E+03
                 .1837636E-01          .1031130E+03
                 .1832103E-01          .1041181E+03
                 .1826629E-01          .1051198E+03
                 .1821212E-01          .1061260E+03
                 .1815851E-01          .1071288E+03
                 .1810546E-01          .1081363E+03
                 .1805295E-01          .1091404E+03
                 .1800098E-01          .1101493E+03
                 .1794953E-01          .1111548E+03
                 .1789860E-01          .1121652E+03
                 .1784817E-01          .1131805E+03
                 .1779825E-01          .1141925E+03
                 .1774882E-01          .1152096E+03
                 .1769987E-01          .1162234E+03
                 .1765139E-01          .1172424E+03
                 .1760339E-01          .1182581E+03
                 .1755584E-01          .1192707E+03
                 .1750874E-01          .1202884E+03

b) Output to file out:

        Profile corresponding at time of   21.32 seconds


                 .2834743E-01          .1000000E-02
                 .2204003E-01          .1083333E-02
                 .1524192E-01          .1166667E-02
                 .9650050E-02          .1250000E-02
                 .5939514E-02          .1333333E-02
                 .3898470E-02          .1416667E-02
                 .2951983E-02          .1500000E-02
                 .2577544E-02          .1583333E-02
                 .2450031E-02          .1666667E-02
                 .2412389E-02          .1750000E-02
                 .2402702E-02          .1833333E-02
                 .2400520E-02          .1916667E-02
                 .2400089E-02          .2000000E-02
                 .2400013E-02          .2083333E-02
                 .2400002E-02          .2166667E-02
                 .2400000E-02          .2250000E-02
                 .2400000E-02          .2333333E-02
                 .2400000E-02          .2416667E-02
                 .2400000E-02          .2500000E-02
                 .2400000E-02          .2583333E-02
                 .2400000E-02          .2666667E-02
                 .2400000E-02          .2750000E-02
                 .2400000E-02          .2833333E-02
                 .2400000E-02          .2916667E-02
                 .2400000E-02          .3000000E-02
                 .2400000E-02          .3083333E-02
                 .2400000E-02          .3166667E-02
                 .2400000E-02          .3250000E-02
                 .2400000E-02          .3333333E-02
                 .2400000E-02          .3416667E-02

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Auxiliary Routines



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finite difference, allotriomorphic ferrite, profile collapse

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Download source code

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