microstructural engineering applied to the controlled cooling of steel

Microstructural Engineering Applied to the Controlled Cooling of Steel Wire Rod Part III. Mathematical Model Formulation and Predictions

P.C. CAMPBELL, E.B. HAWBOLT, and J.K. BRIMACOMBE

In this final part of the study, a mathematical model incorporating heat flow, microstructural phenomena, and structure-composition-mechanical property relationships has been developed to compute the yield strength (YS) and ultimate tensile strength (UTS) of steel rod control cooled on a Stelmor line. The predictive capability of the model, in terms of temperature response, microstructural evolution, and strength of the rods, has been tested by comparison to measurements from an extensive set of laboratory and plant trials. Thus, the model has been shown to simulate the complex heat flow and microstructural phenomena in the steel rod very well, although improvements need to be sought in the characterization of the austenite-ferrite transformation kinetics and of pearlite interlamellar spacing. The latter variable has a significant influence on the strength of eutectoid steels. Nonetheless, the model consistently is capable of predicting the strengths of plain-carbon steel rods ranging from 1020 to 1080 to within -4-10 pct.

I. I N T R O D U C T I O N

THE ultimate goal in the microstructural engineering methodology is to develop a mathematical model which is capable of predicting heat flow, microstructural evolution, and mechanical properties for the processing op- eration considered. In the present case, the heat flow aspect of the model is required to enable accurate prediction of the thermal response of the steel rod as it is processed on the Stelmor line. Owing to the release of the latent heat of transformation for both the austenite- ferrite and austenite-pearlite reactions, the analysis of heat flow in the rod must be coupled to the transformation event utilizing an iterative technique. As outlined in Part II of this three-part paper, [291 the final microstructure evolved in the rod also is dependent on its thermal his- tory; thus, the prediction of microstructure depends ul- timately on the ability of the model to predict temperature. Finally, the quantitative link between the evolved microstructure and the mechanical properties of the steel rod must be made. Part III of this paper presents the development, sensitivity analysis, and predictions of the Stelmor mathematical model.

II. M O D E L F O R M U L A T I O N

A. Heat Flow

Transient heat conduction within the steel rod is ex- pressed in cylindrical coordinates as follows: m

__(O "] k(OT)__ qrR = --OT k OT + - + [1] Or \ Or] r Or pCp Ot

P.C. CAMPBELL, formerly Graduate Student, The University of British Columbia, is with BHP Central Research Laboratories, Wallsend, New South Wales 2287, Australia. E.B. HAWBOLT, Professor, Department of Metals and Materials Engineering and The Centre for Metallurgical Process Engineering, and J.K. BRIMACOMBE, Stelco/NSERC Professor and Director, The Centre for Metallurgical Process Engineering, are with the University of British Columbia, Vancouver, BC V6T 1Z4, Canada.

Manuscript submitted February 14, 1990.

which is valid provided that temperature gradients are negligible with respect to axial (z) and angular (0) po- sitions. The term qrR is the latent heat released by the decomposition of austenite.*

*Other symbols are defined in the Nomenclature at the end of this paper.

Boundary conditions for the solution of Eq. [1] include the assumption of symmetrical temperature distri- bution about the centerline

OT t -- 0, r = 0, - - = 0 [2]

Or

At the surface of the rod (r = r0), the conductive heat flux is matched by radiative and convective heat losses, which can be characterized by an overall heat-transfer coefficient (hov) as follows:

0T t > 0, r = r0, - k - - = hov(To - T~) [3]

Or

Determination of hov is based on empirical expressions obtained from the laboratory and plant measurements, as described in Part I. I2sl

For the initial condition, the rod is assumed to be isothermal at temperature, Tin:

t = 0 , 0 - < r - - < r 0, T = T i n [4]

In the mathematical model, the initial temperature is taken to be the laying head temperature on the Stelmor line.

Solution of Eq. [1] is not possible analytically because of the variation of k, Cp, and qTR with temperature. Hence, an implicit, finite-difference technique was applied by discretizing time and the rod volume. The system of equations obtained through application of this technique was solved utilizing the Thomas algorithm. ~2] The temperature-dependent thermal conductivity and specific heats of austenite, ferrite, and pearlite were obtained from the literature [3-6l and included in the mathematical model. Steel density was assumed constant over the temperature range of interest.

METALLURGICAL TRANSACTIONS A VOLUME 22A, NOVEMBER 1991 2791

The latent heats of the austenite-ferrite and austenite- pearlite phase transformations vary with temperature because of the dependence of the specific heat of the three phases on temperature. For austenite decomposition into pearlite, the latent heat of transformation at 1000 K has been reported to be - 7 7 kJ/kg, tT] This value is in agreement with the data of Kramer et al.[8] who investigated the effect of pearlite spacing on the enthalpy of transformation. Similarly, the enthalpy change for the austenite- ferrite transformation at 1184 K has been given as - 1 6 . 3 kJ/kg by Barin e t al . , [6] which is in agreement with Darken and Gurry . [91 Thus, from a knowledge of the latent heat at temperature, Tz, AHt, the heat of transformation at any temperature T~ can be calculated:*

*The enthalpy of mixing of carbon in austenite has been neglected.

AHi = AHt + (Cp,~ - Cp,v) dT l

[5]

where 7r denotes either ferrite (c~) or pearlite (P). Employing the polynomial equations developed for the

dependence of Cp on temperature, Eq. [5] was solved over a temperature range applicable to the formation of both ferrite and pearlite. The calculated variation with temperature of AHi for the y-a and y-P transformations in 0.23- and 0.40-pct steels is plotted in Figure 1. For comparison, enthalpy values measured at various temperatures and reported in the literature also are included in the figure. Thus, the enthalpy change for the ferrite transformation rises sharply between 740 ~ and 780 ~ resulting from the ferromagnetic transition at 770 ~ u~ The calculated heats of transformation in Figure 1 were assumed to hold over the entire carbon range and were fitted following polynomial regression techniques for in- clusion in the mathematical model. This simplification introduces an error of less than 10 pct in the AH values.

The quantity of heat released during each time step in the finite-difference calculations is dependent on the rate of transformation of austenite to ferrite and/or pearlite.

100

"~ 9 0 -

8 0 -

o "~ 70- E ~ 60-

~ 50- t- ~ 4O cl. -~ 30 4; ~ 20 m

10 550

~ ~ +

\

+, \

§247 o + Ferrite Calculated

t3 Pearlite Calculated

o Darken and Gurry [9]

Kramer et al. [8]

. . . . Regression Lines

650 750 850 950

T e m p e r a t u r e (~

Fig. 1--Latent heat of austenite-pearlite and austenite-ferrite transformations calculated as a function of temperature for 0.40- and 0.23-pct carbon steels, respectively.

Thus, for a fraction of austenite transformed, AX, over a time step, At, the heat generated is given by

AX qrR = p A H i - [6]

At

By substituting the appropriate heat of transformation and fraction transformed for each austenite decomposition reaction, the total heat released during a given time step can be calculated.

B. Phase Transformations

The technique adopted for the prediction of phase transformations in steels for nonisothermal events has been presented previously; m-14] thus, only a brief summary will be included here. The method relies on empirically determined continuous cooling transformation (CCT) start times, isothermal transformation kinetics characterized in terms of the Avrami equation, u5,16,~7] and the additivity principle to approximate the CCT event as a series of isothermal steps. The Avrami equation can be written as

I- ,q ----2- X = 1 - e x P L d , J [71

The values for the kinetic parameters n and b have been determined for three grades of steel in the present study and for a series of grades in the literature, tlz,14,18,wl The results have been presented in Part II of this paper, t29] Although no experimental work has been completed on the determination of m (austenite grain diameter expo- nent) in the present study, values of 1 for the ferrite transformation and 2 for pearlite have been proposed in the literature, t2~ These were adopted for the model.

As stated above, the additivity principle, originally due to Scheil, t21] is applied to the transformation kinetics to allow application of the isothermal kinetics to describe the nonisothermal event. This requires that the transformation rate at each temperature is a function only of the temperature and the fraction already transformed. The applicability of additivity to the austenite-ferrite and austenite-pearlite transformations in steels has been tested and validated previously, u4,22~

In the mathematical model, the prediction of phase transformation kinetics proceeds as follows. At a given node i, once the CCT start time is exceeded, the initial fraction of austenite transformed Xij over that time step j is given, according to Eq. [7], by

[-b(Ti,j) Atn] X,j = 1 - exp d~ J [8]

For the next step, j + 1, at node i, the new temperature (calculated by Eq. [1]) is Tij+l; and if the previous transformation, Xij , had occurred at the new temperature, the time taken would be

I n ( 1 ~1 ',n d"~' \1 - Xid] |

2792--VOLUME 22A, NOVEMBER 1991 METALLURGICAL TRANSACTIONS A

This is a virtual time that follows from the sole dependence of transformation rate on temperature and the fraction already transformed. Then the fraction, Xij+~, which has transformed after Oi,j+l + At, is calculated from

r Xij+l = 1 - exp L d-~-' [10]

The incremental fraction transformed over the j + 1 time step in node i is

m s i j + l -~- Xi , j+ 1 - X i , j [1 11

The release of latent heat over the time step can be estimated utilizing Eq. [6] in combination with Eq. [11]; solution of the heat-flow equations must be accom- plished iteratively, owing to the coupling between the transformation kinetics and temperature of the steel rod. Experience has shown that three to four iterations are sufficient to achieve a temperature difference between successive calculations of less than 10 -4 ~

Predictions for the fraction transformed in each time step become somewhat more complicated in hypoeutectoid steels, where the equilibrium phase fractions for both ferrite and pearlite must be considered. First, for temperatures above the A1, the incremental ferrite fraction transformed during each time step can be corrected for the actual equilibrium fraction by employing the following equation:

m x [ , j = m x i , j f a E a ( T ) [ 1 2]

where AXi'j is the real incremental fraction, AXi,j is the calculated incremental fraction based on the Avrami equation, and f , eo(T) is the equilibrium fraction of ferrite which is a function of the temperature of transformation. Values for f , Eo(T) are calculated in the model based on the phase diagram boundaries described by Kirkaldy and co-workers [23,24j which include the individual and synergistic effects of low concentrations of al- loying elements on the equilibrium concentration of carbon (in ferrite, austenite, and carbide) as a function of temperature. In the calculations, only a carbon balance is applied, and ternary sections are not utilized. A similar procedure is carried out below A1; however, the fraction of ferrite is determined from the extrapolated Acre line, as opposed to the A 3. When the total fraction of ferrite has exceeded the equilibrium fraction, the ferrite reaction is assumed to stop. The calculations for the pearlite transformation are initiated once the empirically determined CCT start time for pearlite is exceeded (provided the temperature is below A~). The fraction of pearlite formed in each time step then is corrected by the fraction of retained austenite.

The final microstructures formed in the rods are calculated based on the empirical equations presented in Part II.[29] The calculation of mechanical properties developed for the model also has been presented in Part II; and the reader is referred to Sections I I - B and C.

I I I . S E N S I T I V I T Y ANALYSIS

In order to test the influence of some of the variables considered in the mathematical model on the predicted

thermal response, microstructure, and mechanical properties of steel rod, a sensitivity analysis has been con- ducted. Three steel grades were chosen for the analysis, representative of typical low-carbon (1017), medium- carbon (1040), and eutectoid (1080) steels. The 1017 grade is at the low-carbon limit, and the 1080 steel is at the high-carbon limit of the range of steel grades examined in this study. Predictions of thermal response at the centerline of a 10-ram-diameter rod cooled by air flowing at 15 m / s for the three grades are shown in Figure 2. As can be seen, the initial transformation temperature of the 1017 steel is higher than that of either of the other grades because of its highest A3 temperature. Evident in this plot are the distinct differences in the cooling behavior among the three grades, corresponding to the austenite decomposition reactions. Although the more important variables are included in the following sections, the reader is referred to the thesis [25j from which this work is taken for a complete sensitivity analysis of the mathematical model.

A. Heat Transfer

The rod diameter and heat-transfer coefficient have a strong influence on the cooling rate of the steel rod. Di- ameters of 5, 15, and 10 mm, which correspond respectively to minimum, maximum, and average values typically processed on a Stelmor line, have been studied with the model; the predicted thermal responses for a 1080 steel are shown in Figure 3. The smallest diameter rod exhibits a lower start temperature for the pearlite transformation, which results in an acceleration of transformation kinetics and the largest relative recalescence for the reaction. These effects are reflected in the microstructure and strength of the 1080 grade, as shown in Table I; as the transformation temperature decreases, the pearlite spacing decreases, thereby increasing the strength. As can be seen in Table I, the overall effect of varying rod diameter by +5 mm results in up to + 6 pct variation in ultimate tensile strength (UTS).

Considering the two hypoeutectoid grades (Tables II

86O \ r / - - 1080 Steel \ \ /

820 \ , \ \ / .... 1040 Steel (.) \ \ |- - 1017 Steed

/

780 ", \ _

~ 740 ",, "- \

", \ 700

660 ~ . ~ . .

62O

580 ~ , ~ .

540 0 20 4'0 60

Time (s)

Fig. 2--Model-predicted thermal response at the centerline of 1017, 1040, and 1080 steel rods having 10-mm diameter cooled with air flowing at 15 m/s.

METALLURGICAL TRANSACTIONS A VOLUME 22A, NOVEMBER 1 9 9 1 - 2793

860 840 820

G 8oo 780

a~ 760

74o

720 ~.. 700

680 660 640 620 600 580 560 540

' ", . . . . Rod D i a = 1 5 m m I / ",, =

- - R o d D i a 1 0 r a m

~ " , , - - Rod Dia = 5 mm

\ l / - x \ \ ! \ % \

1080 St

\

\\ " ' " ' " . .

0 ' 20 ' 40 60 ' 80

Time (s) Fig. 3--Influence of rod diameter on the model-predicted thermal response at the centerline of a 1080 steel rod cooled with air flowing at 15 m/s.

and liD, a sizeable decrease in ferrite transformation start temperature results from the decrease in rod diameter from 15 to 5 ram, with the effect being greater in the 1040 grade. Tables II and III show that finer ferrite grain diameter and pearlite spacing, together with a smaller ferrite fraction, are predicted with the decrease in rod diameter. This is directly related to the higher cooling rate and lower transformation temperatures experienced by the smaller diameter rods. Table II shows a strength variation of - 1 0 pct for the 1040 grade, while in Table III, the 1017 steel varies in strength by -+7 pct for a similar change in rod diameter.

Heat-transfer coefficients in the model are based on the correlations for forced convect ion and radiation, as outl ined in Part I. 1281 Owing to the error associated with correlations of this type for characterization of heat-transfer coefficients, a variation of -+ 10 pct has been applied to examine the effect on the thermal response and strength of the steel rod. The predicted thermal response is shown

Table Io Predicted Rod Microstructure and Strength as a Function of Parameters Tested in the Sensitivity Analysis for 1080 Steel

Ferrite Pearlite Ferrite Diameter Spacing YS UTS

Parameter Fraction (/x m) (/z m) (MPa) (MPa)

Mean settings 0.000 - - 0.239 539.3 1046.3

Rod diameter = 15 mm 0.004 0.95 0.260 533.9 996.7 Rod diameter = 5 mm 0.000 - - 0.194 596.2 1111.7

hov x 1.1 0.000 - - 0.235 543.6 1051.3 hov • 0.9 0.000 - - 0.243 534.4 1040.7

Pct C = 0.85 0.000 - - 0.237 541.0 1048.3 Pct C = 0.75 0.005 1.48 0.240 537.4 991.7

In be + 1.0 0.000 - - 0.255 522.1 1026.6 In be - 1.0 0.000 - - 0.222 558.9 1068.9

dv = 32/zm 0.000 - - 0.226 553.7 1062.9 dv = 16/.~m 0.000 - - 0.249 528.1 1033.5

Table II. Predicted Rod Microstructure and Strength as a Function of Parameters Tested in the Sensitivity Analysis for 1040 Steel


Parameter Fraction (/zm) (/~m) (MPa) (MPa)

Mean settings 0.479 4.12 0.246 426.6 644.2

Rod diameter = 15 mm 0.500 4.74 0.275 417.5 633.9 Rod diameter = 5 mm 0.401 3.36 0.201 463.1 707.6

hov • 1.1 0.474 4.02 0.234 431.7 651.7 hov x 0.9 0.484 4.22 0.252 422.8 638.2

Pct C = 0.43 0.440 3.64 0.240 441.0 671.9 Pct C = 0.37 0.519 4.59 0.253 415.9 633.0

In bF + 0.6 0.479 4.12 0.246 426.6 644.2 In be - 0.6 0.479 4.12 0.242 427.5 645.3

In bp + 1.0 0.479 4.12 0.260 423.5 640.7 In bp - 1.0 0.479 4.12 0.224 431.8 650.2

dv = 32/xm 0.479 4.12 0.228 430.9 649.1 dv = 16/xm 0.479 4.12 0.259 423.7 641.0


Table III. Predicted Rod Microstructure and Strength as a Function of Parameters Tested in the Sensitivity Analysis for 1017 Steel

Parameter

Ferrite Pearlite Ferrite Diameter Spacing

Fraction (/zm) (/~m) YS UTS

(MPa) (MPa)

Mean settings 0.791 9.27 0.266 315.7 520.6

Rod diameter = 15 mm 0.801 9.60 0.297 311.6 514.0 Rod diameter = 5 mm 0.702 8.31 0.237 340.4 552.4

hov x 1.1 0.775 8.94 0.260 321.0 526.8 hov x 0.9 0.785 9.14 0.270 317.6 518.1

Pct C = 0.43 0.743 8.58 0.258 329.9 537.7 Pct C = 0.37 0.805 9.33 0.266 312.3 494.6

In bF + 0.6 0.780 9.03 0.273 319.3 517.1 In be - 0.6 0.780 9.03 0.275 319.3 522.3

In be + 1.0 0.780 9.03 0.281 319.2 520.3 In be - 1.0 0.780 9.03 0.253 319.5 520.9

d r = 3 2 / z m 0.780 9.03 0.253 319.5 522.1 d r = 16 /zm 0.780 9.03 0.275 319.3 519.1

in Figure 4 for the 1080 steel grade. Although there is an increase in the rod cool ing rate with increasing heat- transfer coefficient, as expected, the effect is not as strong as that predicted for the change in rod diameter of 5 to 15 m m shown in Figure 3. Referring to Tables I through 1II, this variation in heat-transfer coefficient produces only a minor change in the predicted steel microstructure and strength.

B. Steel Composition

The effect o f small changes in carbon content, which most influences microstructure and properties on a weight percent basis, was studied within the acceptable range set by SAE specifications for 1017, 1040, and 1080 plain- carbon steels. The mode l prediction of thermal response at the centerline o f a 1040 steel rod is shown in Figure 5. An increase in the carbon content to 0 .43 pct decreases

860

840 " ~ I . . . . h~ 820 ~ - - hov x 1.0 ~, 800

- - hov x 0.9 780 e,._,. ',x i 760 ',x

" 740 ', \ 720 (3,) XXx

EL 700 E 680 ", \ F-- 660 ,,, x" ~

640 " , \ 620 . , , , \ \ \ 6OO 580 ", "- 560 1080 Steel \ "-. 54O

2() 40 60 Time (s)

Fig. 4 - - I n f l u e n c e of uncertainty in the heat-transfer coefficient on the model-predicted thermal response at the centerline of a 10-mm- diameter 1080 steel rod cooled with air f lowing at 15 m / s .

0

o) El. E }-

860 840 820 800 780 760 740 720 700 680 660 640 620 600 580 560 540

~ _ _ . . . . %C 0.43 ] l - - %C = 0.40

_ I - - %C=0,37

lO,O

0 20 4O 60 Time (s)

Fig. 5 - - I n f l u e n c e of carbon content on the model-predicted thermal response at the centerline of a 10-mm-diameter 1040 steel rod cooled with air f lowing at 15 m / s .

the ferrite start temperature. Because o f the increased hardenability, a decrease in the ferrite fraction and ferrite grain diameter is evident in Table II, whi le only a slight decrease in pearlite spacing is predicted, a consequence of the lower pearlite start temperature. The net effect , as seen in Table II, is a variation in strength o f only -+2 to 4 pct. A similar influence o f carbon content is predicted for the 1017 steel (Table III). With respect to the 1080 grade, the lowes t carbon steel rod (0 .75 pct) is predicted to contain a small fraction o f proeutectoid ferrite, which decreases the UTS by 5 pct.

C. Phase Transformation Kinetics

Uncertainty of -+0.6 in the value of In b F in the Avrami equation has the predicted effect shown in Figure 6 for the 1040 steel rod. As the value o f In b v is reduced, the

METALLURGICAL TRANSACTIONS A V O L U M E 22A, N O V E M B E R 1 9 9 1 - - 2 7 9 5

860

S40 ~ j inbF+0. 6 .... 820 \ [ - - I n b F + 0.0 800

~ 740760780 ~ . i - - InbF- 0.6

720 {3- 700

680 I.-- 660 \ ~'-

640 , ' , 620 600 580 560 1040 Steel 540

0 20 40 60 Time (s)

Fig. 6 - - I n f l u e n c e of variation in In b F on the model-predicted thermal response at the centerline of a 10-mm-diameter 1040 steel rod cooled with air f lowing at 15 m / s .

ferrite reaction is seen to be slower and the associated recalescence to be diminished, as expected. Interest- ingly, the slower ferrite reaction (In bF -- 0.6) and increased transformation time to produce the same fraction of ferrite cause a reduction in the transformation temperature of pearlite, an acceleration in the pearlite reaction kinetics, and slightly greater recalescence. Similar behavior was predicted for the 1017 grade, although owing to the smaller fraction of pearlite formed, the effect on recalescence arising from the pearlite reaction is less. In terms of microstructure and strength changes with In bF, no appreciable variation is evident from Tables II and III. However, in the model, the ferrite fraction and ferrite grain diameter are calculated from the cooling rate immediately prior to the start of the transformation and, therefore, are independent of the transformation kinetics.

The effect of varying the In bp parameter by -+ 1 on the thermal response of the 1080 steel rod is shown in Figure 7. Not surprisingly, the smallest value of In bp produces the cooling cur~e with the lowest average transformation temperature. This is reflected in a re-

duced pearlite spacing and increased strength (Table I). Similar predictions hold for the 1040 steel (Table II), although the change in strength is tempered by the presence of proeutectoid ferrite. The 1017 grade is influ- enced to the least extent compared to the other two steels because of the small fraction of pearlite in the former.

Although not a variable studied experimentally in this work, the prior austenite grain diameter has been included in the sensitivity analysis. In the present study, the prior austenite grain size in steel rod investigated in both the laboratory and plant trials has been measured as 22 p~m (ASTM 8 +- 1). The effect of prior austenite grain diameter on the transformation kinetics has been determined by utilizing the modified Avrami equation (Eq. [7]), with the values of m being taken as 1 and 2 for the ferrite and pearlite reactions, respectively, t2~ Model predictions of the influence of grain diameter varying by + 1 0 / x m and - 6 /xm about 2 2 / x m (ASTM 8 -+ 1) on the phase transformations in the 1017 steel are shown in Figure 8. As expected, the smallest prior austenite grain size favors the fastest transformation rate for both the ferrite and pearlite reactions, but for the latter, the effect is very small in the 1017 steel. In terms of rod microstructure and strength, coarser pearlite spacing results from the higher transformation temperature associated with the finer austenite grain size; this is reflected in the decrease in strength predicted for the 1080 steel (Table I). Owing to the method of calculation, no change in the ferrite grain diameter is predicted in the two hypoeutectoid grades, although it should certainly be affected by a variation in prior austenite grain size. Clearly, additional work is required to obtain a quantitative relationship between the austenite and resulting ferrite grain diameters.

D. Microstructural Effects

In order to assess the variation in rod strength as a function of individual microstructural parameters, independent of steel thermal response, the mathematical model has been run employing correction factors for the ferrite fraction, ferrite grain diameter, and pearlite spacing. The

860

8 O82o i \ [ .... ,n0p+10 80o \ / - - 'nbp+ 0'0

0 780 .0 760 740 720

cu 700 I= 680 I-- 660

640 620 6O0 580 560 1080 Steel . ~ ,

540 ' 2(3 4(3 60

Time (s)

Fig. 7 - - I n f l u e n c e of variation in In bp on the model-predicted thermal response at the centerline of a 10-ram-diameter 1080 steel rod cooled with air f lowing at 15 m / s .

880 860 840 820 800 780

2 760 740 720

E 700 r 680 F-.

660 640 620 6O0 58O 56O 54O

~ . .... G.S. = 32 ,//m - - G.S. = 22 t im

1017 Steel i ~ J J

20 40 Time (s)

60

Fig. 8 - - I n f l u e n c e of variation in prior austenite grain size on the model- predicted thermal response at the centerline of a lO-mm-diameter lO17 steel rod cooled with air f lowing at 15 m / s .


magnitude of the correction factors applied to the ferrite fraction and ferrite grain diameter has been chosen to be _4-1 standard deviation of the estimated values obtained from the regression analyses, i .e., ---0.056 and ---0.6 /xm, respectively. The error associated with the prediction of the mean interlamellar pearlite spacing is about -+20 pct based on this study (Part II[291).

The results for the predicted strengths of the 1080 steel corresponding to a ---20 pct variation in pearlite spacing are shown in Table IV, where a considerable effect can be seen. The predicted yield strength (YS) and UTS vary by - 8 to + 11 pct and - 5 to +7 pct, respectively. These changes in strength are relatively large and are greater than the effect of any one parameter examined hitherto in the sensitivity analysis. Consequently, the accurate prediction of mechanical properties in plain-carbon eutectoid steel depends importantly on obtaining valid pearlite interlamellar spacings.

The model-predicted strengths for the 1040 steel as a function of the uncertainty in the microstructural parameters are summarized in Table V. Thus, the variation in ferrite fraction accounts for a _+ l pct and -+ 1.5 pct change

in the predicted YS and UTS, respectively, which is considered to be minor compared with variations associated with other parameters. Table V also shows that the --+0.6/zm variation in ferrite grain diameter accounts for a change of about 2.5 pct in both predicted and ultimate strengths. A similar strength variation is reported as a function of change in pearlite spacing for the 1040 steel. Yield strengths and UTS values reported in Table V vary from ---2.5 to 3 pct and _+1.5 to 2.5 pct of the mean values, respectively. These levels of sensitivity are considered acceptable at this stage in the model development.

Table VI shows the effect of varying the microstructure on the strength of the 1017 steel. A change of ---0.056 in ferrite fraction has a slightly greater effect on predicted strength in the 1017 than in the 1040 steel: the YS varies by ---4 pct and the UTS by ---3 pct due to the greater fraction of ferrite present in the 1017 steel and the form of the modified Gladman equations used to predict strength. Changes in the ferrite grain diameter yielded minor variations in predicted strength owing to the large diameters already present in this steel. Both YS and UTS varied by approximately -+ 1 pct. Sensitivity to variations

Table IV. Predicted Rod Strength as a Function of Changes in Pearlite Spacing for a 1080 Steel


Parameter Fraction (/zm) (/xm) (MPa) (MPa)

Mean settings 0.000 - - 0.239 539.3 1046.3

Mean spacing x 1.2 0.000 - - 0.286 494.1 994.5 Mean spacing x 0.8 0.000 - - O. 191 600.4 1116.5

Table V. Predicted Rod Strength as a Function of Changes in Steel Microstructure for a 1040 Steel


Parameter Fraction (/z m) (/xm) (MPa) (MPa)

Mean settings

Mean fraction + 0.056 Mean fraction - 0.056

Mean ferrite diameter + 0.6/xm Mean ferrite diameter - 0.6 /zm

Mean spacing x 1.2 Mean spacing x 0.8

0.479 4.12 0.246 426.6 644.2

0.535 4.12 0.246 422.7 629.6 0.423 4.12 0. 246 430.9 660.3

0.479 4.72 0.246 416.0 628.5 0.479 3.52 0.246 439.8 664.1

0.479 4.12 0.295 417.0 633.3 0.479 4.12 0.197 439.7 659.4

Table VI. Predicted Rod Strength as a Function of Changes in Steel Microstructure for a 1017 Steel

Parameter

Ferrite Pearlite Ferrite Diameter Spacing YS

Fraction (/xm) (/zm) (MPa) UTS

(MPa)

Mean settings

Mean fraction + 0.056 Mean fraction - 0.056

Mean ferrite diameter + 0.6/xm Mean ferrite diameter - 0.6/xm

Mean spacing x 1.2 Mean spacing x 0.8

0.780 9.03 0.265 319.4 525.3

0.836 9.03 0.265 307.1 509.9 0. 724 9.03 0. 265 331.6 540.8

0.780 9.63 0.265 316.3 521.2 0.780 8.43 0.265 322.8 529.9

0.780 9.03 0.317 318.9 524.4 0.780 9.03 0.212 320.0 526.6


in pearlite spacing for this grade were almost nonexistent due to the small fractions of pearlite present.

IV. MODEL VALIDATION AND PREDICTIONS

The mathematical model has been run to predict the thermal and microstructural evolution and mechanical properties of steel rods cooled under conditions typical of the laboratory and plant tests. Results of the predictions are reported in the following sections.

A. Laboratory Tests

1. Thermal evolution Figure 9 shows the measured and model-predicted

centerline temperature for an 8-ram-diameter rod of 1070 steel (steel B) cooled with air at a velocity of 22 m/s . To test the predictive capabilities of the model fully for microstructure and mechanical properties, the heat-transfer coefficient has been adjusted to provide the best agreement with the experimental results by matching the early cooling period of each test prior to the phase transformation; the correction factor is included in the legend in the figure. The predicted thermal response shows excellent agreement with the experimental data, although the CCT start time predicted for the pearlite transformation seems to be slightly earlier than the measured value and the recalescence is smaller in magnitude.

The model-predicted and measured thermal response for a lower carbon 1038 steel (steel C) is shown in Figure 10, corresponding to an 8-mm-diameter rod cooled with an air velocity of 6 m/s . A correction factor of + 10 pet has been applied to the predicted heat-transfer coefficient to achieve good agreement early in the cooling test. It can be seen that the model satisfactorily predicts the thermal response both prior to and after the austenite decomposition reactions. However, the predicted kinetics for the austenite-ferrite reaction are ap- parently too rapid and result in a recalescence greater

860 840 ~ I o Experimental 820 " ~ I - - M o d e l P r e d i c t e d

800 ~ 1 (hov x 1.1) ~" 780 v 760

740

720 700

680 %0000 oo I-- 660 o ~

640 620 600 I T e s t C 7

580 I Rod D i a . = 8 mm I - ~ 560 I Air Vel. --6 m/s I -%~ 540 . . . .

0 20 40

Time (s)

Fig. 10- -Model -predicted and measured thermal responses at the centerline of an 8-ram-diameter 1038 steel rod (steel C) cooled with air flowing at 6 m/s .

than that measured. This behavior was observed for all laboratory tests performed on this grade of steel. How- ever, in light of the variations in predicted thermal response found in the sensitivity analysis, the agreement between model-predicted and measured thermal evolution is still considered to be reasonable.

The model-predicted and measured thermal responses for a typical cooling test with the 1020 steel (steel E) are shown in Figure 11. The overall prediction is in good agreement with the measurement, although the model consistently predicts an early austenite-ferrite transformation start time. More work is being pursued to im- prove the prediction.

2. Microstructural evolution Although only a sample of typical thermal responses

obtained during the laboratory tests was presented in the previous section, the results of the complete set of predicted and measured microstructures are reported below.

860 840 t ~ o E x p e r i m e n t a l

820 ~r~ - - M o d e l P r e d i c t e d

8O0 ~" 780 -%~ 760

740 720 700

o. E 680 I.- 660

64O 620

580 I Rod Dia. = 8 mm " ~ 560 I Air Vel. = 22 m/s 540 i i ~ , , i , i ~ i , i i , i

0 4 8 12 16 20 24 28 32

Time (s)

Fig. 9 - -Model -predic ted and measured thermal responses at the centerline of an 8-mm-diameter 1070 steel rod (steel B) cooled with air flowing at 22 m/s .

880 860 ~ o E x p e r i m e n t a l

840 " ~ , - - M o d e l P r e d i c t e d

820 ~ (hov x 1.1) 800 780 760

= 74o "~ 720 000000[3 DO00 o_ 700 % E 680

I--- 660 640 620 6 O O

580 Rod Dia. = 8 mm I ~'~n 560 Air V e l . --_6 m / s _] "<%0 540 ~

0 20 40

Time (s)

Fig. l I - -Model -pred ic ted and measured thermal responses at the centerline of an 8-mm-diameter 1020 steel rod (steel E) cooled with air flowing at 6 m/s .

2798 VOLUME 22A, NOVEMBER 1991 METALLURGICAL TRANSACTIONS A

Predicted ferrite fractions are compared to measured values in Figure 12. The shaded area in the figure indicates one standard deviation from the regression equation utilized for the prediction of ferrite fraction in the model (Part II, ~29~ Eq. [16]). Given that measured ferrite fractions for the rods included in the figure have been utilized in developing the regression equation, agreement between measured and model-predicted values is seen to be reasonable.

Model-predicted ferrite grain diameters are plotted against the measured values in Figure 13. In the model, the ferrite grain diameter is calculated as a function of the steel composition and the cooling rate prior to transformation. The shaded area in the figure represents -+ 1 standard deviation from the regression equation for ferrite diameter. Good agreement has been obtained between the model-predicted and measured data.

Fig. 12--Model-predic ted and measured ferrite fractions for steels B (1070), C (1038), and E (1020) from the laboratory experiments. The shaded area in the figure indicates ---1 standard deviation of the predicted value.

A comparison of model predictions to measurements of pearlite spacing is shown in Figure 14. Agreement is seen to be only fair, with most predicted values falling below measurements although remaining within one standard deviation. For the 1070 steel (steel B), this is not surprising, since the best-fit regression line reported in Part II ~291 (Figure 12) also underestimated the pearlite spacing. It should be noted as well that the measured values for the 1038 and 1020 steels (steels C and E) have not been determined metallographically, as in the case of steel B, but have been obtained from the measured undercooling below the A1 temperature for each sample and the regression equation reported in Part II. That the predicted pearlite spacing for the 1020 steel is relatively constant arises from the difficulty of effectively predicting CCT start times for the pearlite reaction in hypoeutectoid steels; the net result was that, despite different cooling conditions, the predicted undercooling varied little.

3. Mechanical properties Model predictions of mechanical properties, based on

steel composition and calculated microstructures, are plotted against measurements in Figure 15, where again the shaded region represents -+ 1 standard deviation, corresponding to the regression equations for strength. The predicted YSs in Figure 15(a) are in good general agreement with the measured results. However, for the 1070 steel (steel B), the predicted values are consistently higher than those measured due to the finer pearlite spacings computed for this grade. This illustrates again the strong influence of pearlite spacing on the strength of eutectoid steels. Despite this difficulty, the predicted strengths are still within -+ 10 pct of the measured values, which has been suggested as an allowable strength variation for wire rod. t261 Good agreement is observed for the two hypoeutectoid steel grades, although predicted strengths for the 1020 steel (steel E) are relatively constant.

The predicted and measured ultimate strengths for the laboratory tests are presented in Figure 15(b). Excellent agreement is seen for the lower carbon grades; however, strength levels predicted for the 1070 steel once again

Fig. 13--Model-predic ted and measured ferrite grain diameters for steels B (1070), C (1038), and E (1020) from the laboratory experiments. The shaded area in the figure indicates + 1 standard deviation of the predicted value.

Fig. 14--Model-predic ted and measured mean interlarnellar pearlite spacings for steels B (1070), C (1038), and E (1020) from the laboratory experiments. The shaded area represents approximately --+ 1 standard deviation in the predicted value.

METALLURGICAL TRANSACTIONS A VOLUME 22A, NOVEMBER 1991- 2799

840

820 ~a~ o Experimental 800 & . . . . Model : hov x 1.1

~ " 780 " , ~ a _ - - Model : hov x 0.9

76O

740 _= ,~ 720

700 ~ . . ~ \ 680 -- . . . .

I-- 660 , �9 \

640 " \

620 ", \ \ 600 Test C57 " , , \ \

580 Rod Dia. = 15 mm \ \

560 Edge - FFFF - , -,,

540 0 20 40 60 80 1 O0

Time (s)

Fig. 16- -Model -predicted and measured thermal responses at the centerline of a 15-mm-diameter 1038 steel rod (steel C) cooled at the edge of the bed with air "on."

Fig. 15- -Model -predicted and measured strengths from the laboratory tests: (a) YS and (b) UTS.

slightly exceed measured values for the reason given above.

B. Plant Trials

1. Thermal evolution As outlined in Part I, t28] the method of predicting the

heat-transfer coefficient for the plant trials differed from that utilized for the laboratory tests in order to estimate the radiative component of heat transfer. In light of the scatter exhibited by the plant heat-transfer data and recognizing the difficulty associated with conducting rod- cooling experiments on the Stelmor bed, the fol lowing model predictions include a --- 10 pct variation in the predicted heat-transfer coefficient. This variation is also reflected in the microstructure and mechanical properties predictions. Owing to space limitations, only a representative number of temperature predictions will be included.

A typical thermal response of the largest diameter rod (15 mm) encountered in the plant trials is shown in Figure 16. The rod of 1038 steel was cooled at the edge

of the Stelmor bed with "air on" fan settings. As was the case for the laboratory model predictions, the CCT start times are earlier and the reaction kinetics for both transformations seem to be greater than the measured re- suits, although this difference is considered minor in light of the deviations observed in the sensitivity analysis. Similar results are reported in Figure 17 for the thermal response o f a 15-mm-diameter 1020 rod (steel E) located at the center of the bed under "air on" conditions. Once again, a slight overprediction of the austenite-ferrite transformation kinetics can be observed, but good agreement is exhibited for the pearlite reaction. In general, predictions of the thermal response of the two 15-mm- diameter rods shown in Figures 16 and 17 are in good agreement with measured values.

Model-predicted and measured thermal responses for a 9.1-mm-diameter 1080 steel rod (steel F) cooled at the center of the Stelmor cooling line under "air on" conditions are shown in Figure 18. Good agreement can be

860

840 ~ o Experimental 820 , ~ . . . . Model : hov x 1.1 ~- 800 , ~ . , 780 - - Model : hov x 0.9

760 '" ~ ~

740

720 ,~ �9

E 7o0 680 " ' , _~"

I-" 660 ", \

640 "" ""

620 \

600 ", "" ,,,

580 I Rod Dia. = 15 mm " ' , \ \ 560 [ Center - FFFF " ' , , \ ,,,

540 0 20 40 60 80 100 120

Time (s)

Fig. 17--Model-predicted and measured thermal responses at the centerline of a 15-mm-diameter 1020 steel rod (steel E) cooled at the center of the bed with air "on."


860

O

~t. E

I.-

840 820 8O0 780 760 740 72O 7O0 680 660 640 62O 60O 58O 56O 54O

0

' , oN

o_ Experimental

-- - Model : hov x 1.1 - - Model : hov x 0.9

~ O \ "~ON . - -

Test F 5 6 " - - - " ' , , ,

Rod Dia. = 9.1 mm " ' -_ ~ Center,- FFFF "',, ~ ~ ,

~0 ~b 60

Time (s)

Fig. 18- -Model -predicted and measured thermal responses at the centerline of a 9 . 1 - m m - d i a m e t e r 1080 steel rod (steel F) cooled at the center of the bed with air " o n . "

O

E I--

86O

840 t 820 800 ,~ 780 ~ 760 , , ~ , ~ 740 720 700 680 660 640 620 600 580 560 540

0

[] Experimental

. . . . Model : hov x 1.1 - - Model : hov x 0.9

,..,~.%ooO "., "- TOOo , \ \ DaD

[ Test F71 ", \ \ ~ o

Rod Dia. = 7.5 mm " ' , , \ \ \ ~176 o [Center- FFFF "" \ [3[3

\ Ooi

20 ' 40 Time (s)

Fig. 20- -Model -predic ted and measured thermal responses at the centerline of a 7.5-mm-diameter 1080 steel rod (steel F) cooled at the center of the bed with air " o n . "

seen throughout the cooling period including recalescence due to the pearlite reaction. The second 9. I-ram- diameter rod included for comparison is of 1017 steel (steel I). The model-predicted and measured thermal responses of this steel cooled at the edge of the Stelmor bed under "air on" conditions are plotted in Figure 19. Good agreement is observed, although the kinetics of the ferrite transformation are slightly faster than indicated from the measurements.

The fastest cooling rates measured during the plant trials were obtained in 7.5-mm-diameter rods cooled under "air on" conditions on the Stelmor line. Figure 20 shows the measured and model-predicted thermal responses of a 1080 steel rod (steel F) cooled at the center of the bed. As can be seen, the model-predicted cooling rates are greater than those measured, although the temperature and recalescence for the pearlite transformation show good agreement. The results for a 1035 steel rod (steel H) cooled with "air on" at the center of the Stelmor line are shown

in Figure 21. The model-predicted cooling rate is seen to be faster than that measured, and the computed ferrite transformation kinetics again appear to be faster than actual. Recognizing this to be a similar situation to that in the laboratory tests, it would seem that the coefficients employed for calculation of the austenite-to-ferrite transformation kinetics need to be adjusted or the form of the equation changed.

2. Microstructural evolution The mathematical model has been employed to predict

the microstructure evolved during rod cooling under plant conditions for the complete range of steel grades and rod diameters included in the study. Predicted and measured ferrite fractions are plotted in Figure 22, where it should be noted that each measured value has two corresponding predictions, based on -+ 10 pct of the mean predicted heat-transfer coefficient. The results show excellent agreement for the 1038, 1017, and 1022 steels, with slight underpredictions for the 1037 and 1035 steels. The 1080

860 84o Ex e , enta, 820 - - Model : hov x 1.1

~ - 800 Model : hov x 0.9 780 760

73 \ -.~ 740 ~ ! ~

720 \ ('3 , \ \

700

I-- 680 660 640 , \ 620 \ ,oo F 'est'S3 "', % \ \ 580 |Rod Dia. = 9.1 mm ",, ~ \ \

560 " "" 540 ~ ~ ' " ~ \"

20 40 60

Time (s)

Fig. 19- -Model -predicted and measured thermal responses at the centerline of a 9 . 1 - m m - d i a m e t e r 1017 steel rod (steel I) cooled at the edge of the bed with air " o n . "

860

840 tt~ a Experimental 820 . . . . Model : hov x 1.1

~ " 800 'P','o - - Mode : hov x 0.9 780 ,'~

o 760 ",~u = ,,~,o[]

740 , x o �9 720 ', x []

oE 700 , , , ",t~t~ ~ . . . . . L%~ ~176176 ~176 [] I-- 680 ,~ o

660 640 "'" \ ~176 620 "" \ \ ~176 600 F Test H55 ",, \ o[] 580 /Rod Dia. = 7.5 mm ", \ \ ~ ~

", \ o o 560 LCenter - F F F F ", \ 0% 540

20 40 Time (s)

Fig. 21- -Model -predic ted and measured thermal responses at the centerline of a 7.5-mm-diameter 1035 steel rod (steel H) cooled at the center of the bed with air " o n . "

METALLURGICAL TRANSACTIONS A VOLUME 22A, NOVEMBER 1991 - - 2801

Fig. 22--Model-predic ted and measured ferrite fractions for steels C (1038), E (1020), F (1070), G (1037), H (1035), I (1017), and J (1022) from the plant trials. The shaded area in the figure indicates -+ 1 standard deviation of the predicted value.

Fig. 23- -Model -predic ted and measured ferrite grain diameters for steels C (1038), E (1020), F (1070), G (1037), H (1035), I (1017), and J (1022) from the plant trials, The shaded area in the figure indicates - 1 standard deviation of the predicted value.

steel, although included in the plot, has such a small ferrite fraction that comparison is not realistic on the scale employed in the figure. Also of note in Figure 22 is the insensitivity of the ferrite fraction to changes in the predicted cooling rate; the predicted fractions corresponding to -+ 10 pet of the mean heat-transfer coefficient are barely distinguishable.

Ferrite grain diameters predicted by the mathematical model are plotted in Figure 23 against measurements where, again, the two predictions at each measured value correspond to +-10 pet of the mean heat-transfer coefficient. Comparing these results with those for ferrite fraction, an identical variation in steel cooling rate seems to exert a greater influence on the grain diameter. Figure 23 shows reasonable agreement between model- predicted and measured ferrite diameters, although for the 1020 steel (steel E), the predictions are consistently low.

Owing to the method employed for the prediction of ferrite grain diameter and ferrite fraction in the model, reasonable estimates of these parameters will be made only if predictions of the average steel cooling rate prior to the start of the ferrite transformation are satisfactory. A major flaw in this method is encountered when these variables are to be predicted under conditions where the cooling rate changes significantly during the ferrite transformation. Whereas an increased cooling rate would be expected to result in a decrease in the ferrite fraction and corresponding grain diameter, the model as pres- ently formulated would predict the same relative fraction and grain size. Fortunately, most blower settings employed on Stelmor lines result in uniform cooling conditions throughout the ferrite transformation. The strong link between steel cooling rate immediately prior to transformation and the microstructure formed has been recognized by other researchers, t271

Predictions of the interlamellar spacing of pearlite are compared to measurements in Figure 24. As was the case with the laboratory results in Figure 14, the agreement

Fig. 2 4 - - Model-predicted and measured pearlite spacing for steels C (1038), E (1020), F (1070), G (1037), H (1035), I (1017), and J (1022) from the plant trials. The shaded area in the figure indicates - 1 standard deviation of the predicted value.

is relatively poor for the reasons given earlier. In par- ticular, the method employed for the determination of the mean measured spacing in a majority of the grades involved insertion of the average undercooling below A ~, obtained from the recorded temperature in each experi- ment, in Eq. [18] (Part Iit291). The variation in average undercooling predicted during the course of the pearlite transformation, owing to a change in the heat-transfer coefficient at the rod surface, can be estimated from the figure. For the lower carbon grades, the --- 10 pet change in the heat-transfer coefficient produces minimal variation in the predicted pearlite spacing. However, as the carbon level is increased, the effect of the change in cooling conditions becomes more pronounced. Thus, es- pecially for higher carbon grades, an accurate prediction of steel microstructure will be directly affected by a valid prediction of the rod thermal response.


3. Mechanical properties The final test for the mathematical model is its ability

to predict the mechanical properties of steel cooled on the Stelmor line. Figure 25(a) compares model-predicted and measured YSs of the steels from the plant trials. For the most part, the predicted strengths are within the range of --+ 1 standard deviation, with the exception of the 1080 steel (steel F), for which the maximum difference between computed and measured values is - 1 2 pct. The poor agreement is related to the prediction of pearlite spacing, as discussed earlier for high-carbon steels. Note that in Figure 24, a majority of the model-predicted pearlite spacings for this grade were larger than the measured values, and as a result, the predicted strengths are lower. The pairs of predictions shown in the figure, corresponding to - 10 pct of the mean predicted heat-transfer coefficient, reveal a relatively minor effect, within 2 pct for the low- and medium-carbon steels and _+2.5 pct for the higher carbon grades. Thus, the YS for this range of grades is relatively insensitive to minor changes in cooling conditions on the Stelmor line.

A comparison of model predictions to measurements of UTSs is shown in Figure 25(b). Agreement is excellent for the low- and medium-carbon steel grades. How- ever, the 1080 steel (steel F) once again displays a wider range of predicted strengths than the others. The large variation in UTS of several of the 1080 steel rods is due to the presence of ferrite, predicted with the heat-transfer coefficient reduced by 10 pct whereas for the higher cooling conditions, the predicted microstructure con- sisted only of pearlite. The presence of ferrite in these grades results in a significantly lower predicted strength than a totally pearlitic steel. For the worst case shown in Figure 25(b), the predicted UTS is 11 pct lower than the measured value.

C. Rods from Normal Plant Production

In order to provide an independent validation of the predictive capabilities of the mathematical model, the UTS of several industrial steel rods was measured during the campaign of plant trials. The method employed for re- trieving and testing the tensile samples was identical to that utilized for routine inspection of Stelmor-cooled rod. Loops of the desired grades were cut from the coils as they passed over the final Stelmor cooling zone. Tensile testing was performed by plant personnel on 450-mm- long samples without prestraightening. Yield tests are

Fig. 25--Model-predicted and measured slrengths from the plant trials: (a) YS and (b) UTS.

usually not performed on such samples. Table VII presents a summary of the grades and conditions encountered during the tests, and as can be seen, they cover a range similar to that of the plant and laboratory investigations.

Inputting the steel compositions and line settings for

Table VII. Summary of Stelmor Line Cooled, Industrial Rod Grades and Diameters Employed for Comparison with Model UTS Predictions

Grade Rod Diameter (mm) Pct C Pct Mn Pct Si Air Settings*

1065 7.1 0.63 0.78 0.23 FFFF 1022 7.5 0.21 0.98 0.017 OFFF 1022 7.5 0.22 0.91 0.29 OFFF 1070 7.5 0.70 0.77 0.23 FFFF

Torque rod 9.1 0.58 0.95 0.25 FFFF 1060 9.1 0.61 0.75 0.24 FFFF 1015 9. I 0.16 0.54 0.02 OFFF 1038 12.7 0.38 0.77 0.27 OOOO

Spring 12.7 0.66 0.96 0.25 FFFF

*O = off; F = full on.

METALLURGICAL TRANSACTIONS A VOLUME 22A, NOVEMBER 1991--2803

Fig. 26--Model-predicted and measured UTS of industrial Stelmor- cooled rod. The shaded area in the figure indicates -+ 1 standard deviation of the predicted value.

these grades, as well as • 10 pct of the mean heat-transfer coefficient, the mathematical model has been run to predict UTSs. The computed and measured UTS values of the steel rods are compared in Figure 26. Agreement is seen to be good, with the exception of a few medium- carbon grades ( - 9 2 0 MPa measured UTS) which are in the range of composition not specifically studied in this work. Grades falling between 1045 and 1065 rely on ex- trapolations either from 1020 and 1038 or from 1070 in order to calculate a number of parameters necessary for microstructure and strength predictions.

V. S U M M A R Y A N D C O N C L U S I O N S

A mathematical model has been developed, based on the concept of microstructural engineering, for the prediction of mechanical properties of steel rod subjected to Stelmor cooling. The model relies on one-dimensional heat conduction and incorporates phase transformation kinetics as well as relationships for ferrite fraction, ferrite grain diameter, pearlite interlamellar spacing, and mechanical properties. Based on a comparison of predicted temperature response, microstructural evolution, and mechanical properties, with measurements made in the laboratory and on an operating Stelmor line, the following conclusions can be drawn:

1. The mathematical model has demonstrated the ability to predict accurately the thermal response of steel rod cooled in the laboratory and under plant conditions on the Stelmor line.

2. Predictions of phase transformation kinetics for the austenite-pearlite reaction during continuous cooling showed excellent agreement with measured values in eutectoid steels. Agreement between measured and predicted austenite-ferrite kinetics was reasonable, but consistently, predictions of an early ferrite reaction start time and rapid reaction kinetics were observed.

3. Model predictions of ferrite fraction and ferrite grain diameter agreed well with values from regression equations developed in the study. The predicted val-

ues of pearlite spacing did not agree well with measurements and did not show a strong relationship to the thermal response.

4. Strengths calculated for rods from both the laboratory and plant tests showed excellent agreement with measured values. The results revealed the importance of the pearlite spacing for the strength of eutectoid steel and the relative insensitivity of strength to steel cooling rate over defined limits for all grades.

5. Agreement between the predicted and measured UTS for the independent group of rods taken from the Stelmor line clearly demonstrates the capability of the model to predict microstructure and mechanical properties of continuously cooled plain-carbon steel rod for industrial conditions.

b

bF

bp

c . d dr f~ f~Eo h hov A H i

k m

nF

np

qrR

r

r0 ri t T Ta TA~

T~,

Tac~

Tin To X

P 0

N O M E N C L A T U R E

kinetic parameter in Avrami equation, Eq. [7] kinetic parameter for ferrite in Avrami equation, Eq. [7] kinetic parameter for pearlite in Avrami equation, Eq. [7] specific heat, J kg-1 o C- grain diameter, m prior austenite grain diameter, m volume fraction of ferrite equilibrium ferrite fraction heat-transfer coefficient, W m 2 ~ overall heat-transfer coefficient, W m -2 ~ enthalpy of the austenite-ferrite or austenite- pearlite phase transformation, J kg- thermal conductivity, W m 1 oC-1 kinetic parameter in modified Avrami equation, Eq. [7] kinetic parameter in Avrami equation, Eq. [7] kinetic parameter for ferrite in Avrami equation, Eq. [7] kinetic parameter for pearlite in Avrami equation, Eq. [7] heat released during phase transformation, W m -3

radial position, m radius of rod, m radial position of node i, m time, s temperature, ~ ambient temperature, ~ austenite-ferrite equilibrium transformation temperature, ~ austenite-pearlite equilibrium transformation temperature, ~ austenite-carbide equilibrium transformation temperature, ~ initial temperature, ~ temperature at rod surface, ~ fraction transformed symbol denoting ferrite symbol denoting austenite density, kg /m 3 virtual time, s


A C K N O W L E D G M E N T S

The authors are grateful to the Natural Sciences and Engineering Research Council o f Canada and to the University of British Columbia for financial support of the modeling study.

R E F E R E N C E S

1. F. Kreith and W.Z. Black: Basic Heat Transfer, Harper and Row, New York, NY, 1980, p. 49.

2. B. Carnahan, H.A. Luther, and W.O. Wilkes: Applied Numerical Methods, John Wiley & Sons, New York, NY, 1979, p. 466.

3. British Iron and Steel Research Association: Physical Constants of Some Commercial Steels at Elevated Temperatures, Butterworth's Scientific Publications, Guilford, Surrey, United Kingdom, 1953, pp. 3-14.

4. L.S. Darken and R.W. Gurry: Physical Chemistry of Metals, McGraw-Hill, New York, NY, 1953, p. 397.

5. JANAF Thermochemical Tables, The Thermal Research Laboratory, Dow Chemical Company, Midland, MI, 1960.

6. I. Barin, O. Knacke, and O. Kubaschewski: Thermomechanical Properties of Inorganic Substances; Supplement, Springer-Verlag, New York, NY, 1977, pp. 245-46.

7. S. Taniguchi, T. Murakami, A. Watanabe, and A. Kikuchi: Tetsu- to-Hagan~, 1988, vol. 74, pp. 318-25.

8. J.J. Kramer, G.M. Pound, and R.F. Mehl: Acta Metall., 1958, vol. 6, pp. 763-71.

9. L.S. Darken and R.W. Gurry: Physical Chemistry of Metals, McGraw-Hill, New York, NY, 1953, p. 415.

10. R.E. Smallman: Modern Physical Metallurgy, Butterworth's, London, 1983, pp. 104-06.

11. P.K. Agarwal and J.K. Brimacombe: Metall. Trans. B, 1981, vol. 12B, pp. 121-33.

12. E.B. Hawbolt, B. Chau, and J.K. Brimacombe: Metall. Trans. A, 1983, vol. 14A, pp. 1803-15.

13. J. Iyer, J.K. Brimacombe, and E.B. Hawbolt: Mechanical Working and Steel Processing Conf. XXll, ISS, Pittsburgh, PA, 1984, pp. 47-58.

14. E.B. Hawbolt, B. Chau, and J.K. Brimacombe: Metall. Trans. A, 1985, vol. 16A, pp. 565-78.

15. M. Avrami: J. Chem. Phys., 1939, vol. 7, pp. 1103-12. 16. M. Avrami: J. Chem. Phys., 1940, vol. 8, pp. 212-24. 17. M. Avrami: J. Chem. Phys., 1941, vol. 9, pp. 177-83. 18. E.B. Hawbolt, B. Chau, and J.K. Brimacombe: The University

of British Columbia, Vancouver, unpublished research, 1983. 19. R. Kamat, B. Chau, E.B. Hawbolt, and J.K. Brimacombe: Phase

Transformations '87, The Institute of Metals, London, 1987, pp. 522-25.

20. I. Tamura: Trans. Iron Steel Inst. Jpn., 1987, vol. 27, pp. 763-79. 21. E. Scheil: Arch. Eisenhuttenwes., 1938, vol. 8, pp. 565-67. 22. M.B. Kuban, R. Jayaraman, E.B. Hawbolt, and J.K. Brimacombe:

Metall. Trans. A, 1986, vol. 17A, pp. 1493-1503. 23. J.S. Kirkaldy and E.A. Baganis: Metall. Trans. A, 1978, vol. 9A,

pp. 495-501. 24. J.S. Kirkaldy, B.A. Thomson, and E.A. Baganis: in Hardenability

Concepts with Application to Steels, D.V. Doane and J.S. Kirkaldy, eds., AIME, 1978, pp. 82-125.

25. P.C. Campbell: Ph.D. Thesis, The University of British Columbia, Vancouver, 1989.

26. M. Wells: Stelco Inc., Hamilton, ON, Canada, private communication, 1988.

27. P. Hodgson: BHP Melbourne Research Laboratories, Melbourne, Australia, private communication, 1989.

28. P.C. Campbell, E.B. Hawbolt, and J.K. Brimacombe: Metall. Trans. A, 1991, vol. 22A, pp. 2769-78.

29. P.C. Campbell, E.B. Hawbolt, and J.K. Brimacombe: Metall. Trans. A, 1991, vol. 22A, pp. 2779-90.


microstructural engineering applied to the controlled cooling of steel

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