Materials Algorithms Project
Program Library

Subroutine MAP_STEEL_TRIVSOL

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

G.I. Rees,
Phase Transformations Group,
Department of Materials Science and Metallurgy,
University of Cambridge,
Cambridge, U.K.

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Purpose

Calculates the growth rate (as a function of the supersaturation) of a plate in the shape of a parabolic cylinder growing under carbon diffusion control.

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Specification

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Description

A fitted polynomial is used to approximate the relationship between the Peclet number p and the supersaturation Omega for a plate in the shape of a parabolic cylinder growing under carbon diffusion control [1]:

where x^{[gamma alpha]} is the carbon concentration in the austenite, rho_c is the critical plate-tip radius (for zero growth rate), and S₂ is a function of p displayed graphically in Trivedi's original paper [1].

The plate tip radius rho is also approximated from a solution given by Trivedi, and hence the growth rate G is calculated using :-

where D is the weighted average coefficient [2] of carbon for austenite between the mean alloy composition and the capillarity-affected tip radius carbon concentration [3, 4].

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References

1. R. Trivedi, Metall. Trans A, 1A, (1970), 921-927.
2. H.K.D.H. Bhadeshia, Progress in Materials Science, 29, (1985), 321-386.
3. J.W. Christian, Theory of Transformation in Metals and Alloys, Part 1, 2nd ed., Pergamon Press, Oxford, (1975).
4. H.K.D.H. Bhadeshia, Mater. Sci. and Technol., 1, (1985), 497-504.

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Parameters

Input parameters

XMAX - real

XMAX is the equilibrium carbon concentration x^{[gamma alpha} of austenite (mole fraction).

XBAR - real

XBAR is the carbon concentration x in austenite far from the alpha/gamma interface (mole fraction).

DCOEFFW - real

DCOEFFW is the weighted average diffusivity D of carbon in austenite (in cm²s^-1).

TCC - real

TCC is the temperature (in centigrade).

RADIUS - real

RADIUS is the critical plate tip radius rho_c for zero growth rate (in metres).

C - real array of dimension 6

C contains the coefficients for the polynomial approximation of the function.

C0 - real

C0 is the constant term in the polynomial approximation of the function.

PI - real

PI is pi.

R - real

R is the gas constant (in joules per mole per kelvin, Jmol^-1K^-1).

Output parameters

G - real

G is the plate growth velocity (in ms^-1).

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Error Indicators

The fit deteriorates for large OMEGA, and the program terminates if OMEGA is greater than 0.995.

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Accuracy

If p_s represents the series approximation of the Peclet number, and p_a is the true solution to Trivedi's equation, the error in the approximation of the Peclet number is given by:-

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Further Comments

Called by the routine MAP_STEEL_MICRO.

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Example

1. Program text

       DOUBLE PRECISION XMAX, XBAR, DCOEFFW, TCC, RADIUS, G,
&      C(6), C0, PI, R
       INTEGER J
       INCLUDE 'map_constants_pi.f'
       INCLUDE 'map_constants_gas.f'
       DATA (C(J), J=1,6)/-20.15, 174.0, -562.3, 912.4, -732.9, 232.4/
       C0 = -1.8756
       READ (5,*) XMAX, XBAR, DCOEFFW
       READ (5,*) TCC, RADIUS
       CALL MAP_STEEL_TRIVSOL(XMAX, XBAR, DCOEFFW, TCC, RADIUS,
&      G, C, C0, PI, R)
       WRITE (6,10) G
10     FORMAT ('Plate growth velocity = ',D13.5,' m/s')
       STOP
       END

2. Program data

0.39994D-01   0.11871D-01   0.21860D-08

3. Program results

Plate growth velocity  = 0.63712D-05 m/s

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Auxiliary Routines

MAP_STEEL_XALPH

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Keywords

Trivedi, carbon, diffusion, plate, growth rate, carbon diffusion control, supersaturation

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Download

Download source code

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