Thermal conductivities of hypoeutectic Al-Cu alloys during solidification and cooling

HATERIALS SCIENCE & ENGINEERING

~

E LS EVI E R

A

Materials Science and Engineering A224 (I 997) 48- 52

Thermal conductivities of hypoeutectic AI-Cu alloys during solidification and cooling D . R . Poirier a, E. McBride b a The Departmem of Materials Science and Eng#wering, The University of Arizona, Tucson, AZ 85721, USA b Department of Aerospace and Mechanical Engbwering, The University of Arizona, Tucson, AZ 85721, USA

Received 3 September 1996; revised 11 October 1996

Abstract The thermal conductivity of hypoeutectic A1-Cu alloys during solidification and cooling was esthnated by combining available data on electrical resistivity and thermal conductivity. To estimate the thermal conductivity of the solid alloys, the Smith-Palmer equation was used. This equation enables one to closely estimate the thermal conductivities from known data on electrical conductivities. The electrical resistivities of AI-Cu melts were also gathered. The Smith-Palmer equation was tested successfully for pure Al-melt and assumed to apply to AI-Cu melts, so that their thermal conductivities~could be estimated. Finally simple-mixture models were applied to estimate the electrical resistivities and thermal conductivities of the alloys during solidification. © 1997 Elsevier Science S.A. Keywords: Eutectic; Thermal conductivity; Temperature coefficient

1. Introduction In order to model transport phenomena during the solidification of alloys, the mass, momentum, energy and solute conservation equations are solved numerically [1-3]. Hence, at least reasonable estimates of the relevant transport and thermodynamic properties of alloys are needed. Recently, we have embarked on research pertaining to the directional solidification of hypoeutectic A 1 - C u alloys, so attention is given to the properties required as input data in heat transfer and solidification simulators. In earlier programs on solidification, the densities [4], viscosities [5] and surface tensions [6] of hypoeutectic alloys were summarized in the forms of data-regressions and/or models. In this work, we found a paucity of data on thermal conductivity of A I - C u alloys at high temperatures, so we have resorted to utilizing electrical resistivity and estimating the thermal conductivity from electrical conductivity. 0921-5093/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PIIS0921-5093(96)10554-2

2. Electrical resistivity and thermal conductivity of solidified alloys Data on cast alloys with 4.5, 8, 12 and 14 wt.% Cu alloys and 353~
where p, the alloy Many ables in

C 2,

a3CT

(1)

electrical resistivity (~ m); C, composition of (wt.% Cu); and T, temperature (K). other combinations of the independent variEq. (1) were attempted, including terms in C 1/2, T 2, T 1/2 and (CT)~/z; the regression of Eq.

flD.R. Poirier, E. McBride~Materials Science a~zdEngineering A224 (1997) 48-52

49

Table 1 Electrical resistivity and thermal conductivity of hypoeutectic AI-Cu alloys Concentration (wt.% Cu) Temperature (K)

Electrical resistivity ( ~ m × 108)

Thermal conductivity (W (m K)-~)

Reference

14 14 14 14 14 I4 12 12 12 12 12 12 8 8 8 8 8 8 4.5 4.5 4.5 4.5 4.5 4.5 6 6 6 4.5 8 12 0 0 0 0

5.24 6.25 6.97 7.69 8.40 9.14 5.20 5.96 6.51 7.03 7.57 8. I 1 (4.06) (4.77) (5.40) 6.16 7.03 8.08 4.04 4.96 5.61 6.26 6.92 7.58 3.i0 3.69 4.28 (3.60) (4.70) (4.90) 2.50 3.62 4.78 6.00

154.8 154.8 159.0 ---15%0 163.2 167.4 ---188.3 188.3 188.3 188.3 188.3 -188.3 i88.3 188.3 188.3 192.5 ----I80.0 138.0 130.0 236.0 240.0 237.0 233.0

[7] [7] [7] [7] [7] [7] [7] [7] [7] [7] [71 [7] [7] [7] [7] [7] [7] [7] [7] [7] [7] [7] [7] [7] [8] [8] [8] [8] [8] [8] [9] [9] [9] [9]

353 423 473 523 573 623 353 423 473 523 573 623 353 423 473 523 573 623 353 423 473 523 573 623 273 323 373 273 273 273 273 373 473 573

The resistivities set in parentheses are not included in the regression of Eq. (1). (1) was c h o s e n b e c a u s e it gave the least s t a n d a r d e r r o r o f fit a n d the highest s a m p l e i n d e x o f c o r r e l a t i o n . W h e n the c a l c u l a t e d resistivities were c o m p a r e d to the resistivities in T a b l e 1, however, there were six t h a t e x h i b i t e d differences o f m o r e t h a n 10%. It was decided, therefore, to exclude these six a n d base the regression on 28 resistivities. T h e resistivities n o t u s e d in the final regression are set in p a r e n t h e s e s in T a b l e 1. T h e final regression resulted in the following coefficients: a0 = - 7.433 x 1 0 - 9 ; al = 1.5594 x 1 0 - 9 ; a2 = 1.1738 x 10-1°; a n d a3 = 1.1575 x 10 -13. T h e scopes are 0 ~< C~< 14 wt.% C u a n d 273 ~ T~< 623 K. T o e s t i m a t e the t h e r m a l c o n d u c t i v i t y , a m o d i f i e d f o r m o f the L o r e n z e q u a t i o n , k n o w n as the S m i t h P a l m e r e q u a t i o n , is a p p l i e d [11]. Specifically, a l u m i n u m alloys ( w i t h o u t silicon) follow: k = 0.909Lcr~T+ 10.5

(2)

where k, t h e r m a l c o n d u c t i v i t y , W m - 1 K - 1; L, 2.45 x 10 . 8 W ~ K - 2 ( L o r e n z n u m b e r ) ; a n d o-~, electrical c o n d u c t i v i t y , f~ - 1 m - i.

F o r the alloys with b o t h k a n d ~e ( = 1/p) r e p o r t e d in T a b l e 1, the m e a s u r e d a n d c a l c u l a t e d t h e r m a l c o n d u c tivities are p l o t t e d in Fig. 1. T h e a g r e e m e n t is g o o d , so Eq. (2) with the coefficients o r i g i n a l l y d e t e r m i n e d b y K l e m e n s a n d W i l l i a m s [12] is u s e d in this w o r k for the scopes o f Eq. (1).

3. Electrical resistMty and thermal conductMty of the liquid O n l y the t h e r m a l c o n d u c t i v i t y o f liquid a l u m i n u m is available, so the e s t i m a t i o n o f the t h e r m a l c o n d u c t i v i t y o f the A I - C u liquid alloys is b a s e d largely o n the electrical resistivities t h a t were g a t h e r e d . I n the sources t h a t were c o n s u l t e d the resistivity o f a l u m i n u m a n d the alloys are consistently linear with t e m p e r a t u r e . T h e resistivity o f a l u m i n u m , as t a b u l a t e d in H a t c h [9], is well r e p r e s e n t e d b y p = ( 0 . 0 1 4 7 2 T + 10.464) × 10 - s

(3)

50

D.R. .Poirier, E. AfcBride ,/ ~lfaterials Science and Engineering A224 (1997) 48-5-7

with T in K and p in f2 m. Using the results of Poirier and Liu [10] almost the same regression results: p = (0.01534T+ 9.730) x 10 -s

(4)

and p = ao + a l C 2 + a2T

(8)

Although the coefficients in Eqs. (3) and (4) differ somewhat, the resistivities compare to within 1% in the range 800 ~< T~< 1200 K. Since the data in Hatch are more widely available, we rely on Eq. (3). Resistivities of liquid alloys are taken from Poirier and Liu (5.1 wt.% Cu) [10] and Dyos and Farrell (0.98 and 33 wt.%Cu) [13]. Since the resistivity of aluminum given by Poirier and Liu [10] agrees almost exactly to the resistivity tabulated by Hatch [9] it is assumed that their resistivities of the AI-5.1wt.%Cu alloy are also reliable. The reported resistivities of Al-0.98Cu alloy [13], however, are too high and its temperature coefficient ( d p / d T ) is almost double those of aluminum, A1-5.1wt.%Cu and A1-33wt.%Cu. Hence, the resistivities of the A1-0.98wt.%Cu are not used. There are also electrical resistivities of A I - C u melts listed as data sets 19-27 in the supplementary tables of a compilation given by Ho et al. [14]. These resistivities are extraordinarily high, however, and are not used in this article. In addition to the resistivity of aluminum, Eq. (3), the resistivities of the alloys are AI-5.1wt.%Cu:

Both Eqs. (7) and (8) yielded comparable standard errors of fit and almost perfect indices of correlation (0.9998), but the standard error in at in Eq. (7) was greater than al, itself. Hence, this term was dropped from the multilinear regression, and Eq. (8) was applied. The final regression yielded: a0 = 1.109 x 1 0 - 7 ; a i = 1 . 1 2 5 4 × 10-1°; and a2=1.4123× 10 -t°, with a standard error of the fit equal to 1.039 x 10 -9. Such excellent measures of the fit are not surprising because the temperature coefficients in Eqs. (3), (5) and (6) are similar and the value of at results from a quadratic fit to only three values of C. Lacking data in the range 5.1 < C < 33 wt.% Cu, Eq. (8) is used to calculate the electrical resistivities of the hypoeutectic A I - C u alloys in the range 800 ~ T~< 1200 K. To estimate the thermal conductivity, we examined the correlation between the electrical and thermal conductivities of aluminum. Thermal conductivities of the Al-5.1wt.%Cu and AI-33wt.%Cu alloys are not available. It is stated by Iida and Guthrie [15] and Turkdogan [16] that the Lorenz equation (also called the Wiedemann-Franz-Lorenz law) applies to liquid metals. This equation is

p = (0.0144T+ 11.14) × 10 -s

(5)

/c = L G T

(6)

which, of course, is the predecessor to the SmithPalmer equation (Eq. (2)). A linear regression between the thermal conductivities and electrical conductivities of liquid aluminum gives

and Al-33wt.%Cu: p = (0.01325T+24.22) × 10 -8

Eqs. (3), (5) and (6) were used to generate p ( C , T) at 800, 900, 1000, 1100 and 1200 K, and these resistivities were subjected to regressions of the forms (7)

p = ao + a l C + a 2 C 2 + a 3 T

300

/

~d 275 O

~: 250

J

"~ 225

:~

2oo

"~ 175 r,.) N 150

k = 0.853LGT+ 10.07

(9)

(I0)

Notice that the coefficients in Eq. (10) are approximately the same as those in Eq. (2) for the solid alloys. The thermal conductivities calculated with Eq. (10) are within 0.2% of the reported values in the range 934 ~< T~< 1273 K, whereas the differences are 4.2-5.8% when the thermal conductivities are calculated with Eq. (9). Hence, Eq. (10) was chosen. Lacking data on the thermal conductivities of the alloys, we assume that Eq. (10) also applies to them. Hence, the thermal conductivity of the liquid is calculated by applying Eqs. (8) and (10).

o

4. Electrical resistivity and thermal conductivity during solidification

N 125 100 ~ / /

100

125

150

175

200

225

250

275

300

Thermal Conductivity, W m "l K 1 Fig. 1. Calculated and reported thermal conductivities of AI-Cu alloys with 0 ~< C~< 14 wt.% Cu and 273 ~< T~< 623 K.

The solubility in the aluminum-rich solid solution at 623 K is approximately 1 wt.% Cu. Hence, Eqs. (1) and (2) are applicable to alloys with two microconstituents (the aluminum-rich primary solid plus the solid-eutectic). The relative amounts of these microconstituents, along with precipitates within the primary solid, affect

D.R. Poirier, E. McBride/3[aterials Science and Engineering A224 (I997) 48-52

the electrical and thermal conductivities. Hence, the conductivities of the primary solid during solidification must be estimated independently. The dependence of the resistivity on concentration can be written as dp p = p° +-d--C c

(11)

where p is the resistivity of the solid solution, p O is the resistivity of aluminum, C is the concentration of the solute in solid solution and d p / d C is a constant. Eq. (11) is valid for solid solutions with low solubilities [17]. According to Hatch [18] the effect of Cu in solution on the resistivity of aluminum is d p / d C = 0.344 ~tf~ cm (wt.%)- ~, which is presumably valid at 298 K. Data set 13 in Ho et al. [14] provides additional data on the effect copper in solid solution on the resistivity of aluminum at elevated temperatures. The data set is for A1-3.9wt.%Cu, which is a single-phase 767 ~ T<~ 847 K. Six data are in the single phase region and yield d p / d C = 0.404 bt~ cm (wt.%)- t at an average temperature of 803 K. The effect of Cu on the resistivity of the primary solid was estimated with Eq. (11), assuming that d p / d C varies linearly with temperature with its value known at 298 and 803 K. By tracking the resistivity of the solid plus liquid mixture during solidification and the primary solid plus eutectic constituent after solidification, the thermal conductivity of the alloy was estimated. The resistivity of the eutectic, itself, is also known as it passes through solidification [13]; the resistivities of the liquid eutectic and the solid eutectic are P~_E (823 K) = 35.2 x 10 . 8 f~ m and PSE (813 K) = 14.6x10 - s f~ m, respectively. The upper and lower bounds on the resistivity of the mixtures are (a) series model: Pm~ = g~P.~ + (1 -- g~)P,as

(12)

and (b) parallel model: 1

1

1 -

-

Prnix -- g~ 77 + (1 -- g~) Pee

(13)

where Pm~ is the resistivity of the mixture; P,e, resistivity of the liquid eutectic (i = L) or solid eutectic (i = S); p~, resistivity of the primary solid; and g~, volume fraction of the primary solid. Hence, these equations can be used to estimate Pm,~ provided the relative amount of the primary solid in the microstructure is known. To estimate the weight fraction ( f : ) of the primary solid and its average concentration of Cu, the Scheil equation was applied. For an alloy containing 5.1 wt.% Cu, f.~ = 0.894. Since the densities of the microconstituents in the A1-Cu alloys are known [4], the respective weight fractions of the microconstituents were converted to volume fractions for use in Eqs. (12) and (13).

5i

A major assumption in the Scheii equation is that there is no diffusion in the solid during solidification. Admitting that there is diffusion increases f , and the concentration of Cu in the primary solid. Since there is limited diffusion in the solid during solidification, then the average concentration is in the range 1.8 < C < 3.3 wt.% Cu, although the maximum solubility of Cu in the primary solid is 5.65 wt.% Cu. Hence, the calculated thermal conductivity of hypoeutectic A1-Cu alloys is only weakly dependent on the value of dp/dC. For Al-15wt.%Cu alloy, calculated thermal conductivities during solidification differ by less than 1.5% depending on whether we use the high-temperature or the lowtemperature value of dp/dC. This is probably well below the error in our ability to measure thermal conductivity in solidifying alloys. For more dilute alloys, the differences are smaller yet because the concentration of Cu in the primary solid is less. Hence, although the value of d p / d C is uncertain, the estimation of the thermal conductivity of the alloy during solidification is almost unaffected by its value.

5. Recommended thermal conductivities The thermal conductivity of hypoeutectic A I - C u melts can be calculated by Eqs. (8) and (10) with G = P - 1. The thermal conductivity of the solidified alloys in the range 273 ~< T~< 623 K can be calculated by Eqs. (1) and (2). The thermal conductivity during solidification is predicted by using Eq. (11) with d___p_p= (1.19 × 1 0 - 4 T + 0 . 3 0 8 ) × 10 - s dC

(14)

and T in K, p in f~ m and C in wt.% Cu. Eq. (14) is for the primary solid, and Eqs. (8) and (10) are for the liquid. The thermal conductivity of the solid plus liquid mixture is bounded by kp = g,k~ + (1 - g~)/~L

(15)

and k s t = g,k:] 1 + (1 - g~.)kL z

(16)

where kp, k s are the calculated thermal conductivities based on a parallel model (kp) and a series model (ks), respectively, and the subscripts c~ and k refer to the primary solid and the liquid, respectively. The recommended thermal conductivity of the solidifying alloy is the average of kp and k s. The aforementioned methodology was used to construct Fig. 2. Eutectic solidification occurs at 821 K where there is a discontinuity in the thermal conductivity. When the alloys are completely solidified at 821 K, they contain two microconstituents, a primary solid

52

D.R. Poirier, E. McBride/Materials Science and Engineering A224 (1997) 48-52

3°°I

I

References

-------l+a~0w~

Cu/

i~ i50 100 50

300

500

700

900

1100

Temperature, K

Fig. 2. The thermal conductivities of A1-Cu alloys during solidification and cooling.

a n d the solid eutectic. Eqs. (15) a n d (16) are applicable p r o v i d e d kL is replaced with losE, the t h e r m a l conductivity o f the solid eutectic. T h e discontinuities o f the t h e r m a l c o n d u c t i v i t y at 821 K occur because the electrical resistivity o f the eutectic constituent decreases f r o m 35.2 × 10 - 8 to 14.6 × 10 - 8 fl m u p o n solidification. As the alloys cool f r o m 821 to 623 K, we r e c o m m e n d the i n t e r p o l a t i o n s i n d i c a t e d by the b r o k e n lines.

Acknowledgements The a u t h o r s are grateful to the M i c r o g r a v i t y Science a n d A p p l i c a t i o n D i v i s i o n o f N A S A for the s u p p o r t p r o v i d e d b y G r a n t NCC8-96, which is a d m i n i s t e r e d by N A S A - M S F C . S.N. L i u o f the D a l i a n University o f T e c h n o l o g y , D a l i a n , China, c o n t r i b u t e d by m e a s u r i n g the electrical resistivity o f a few alloys while he was on leave at T h e University o f A r i z o n a . M. C r o m w e l l t y p e d the m a n u s c r i p t ; we are t h a n k f u l for her usual excellent preparation.

[1] S.D. Felicelli, J.C. Heinrich, and D.R. Poirier, Simulation of freckles during vertical solidification of binary alloys, Metall. Trans. B, 22B (1991) 847-859. [2] D.R. Poirier and J.C. Heinrich, Continuum model for predicting macrosegregation in dendritic alloys, Mater. Charact., 32 (1994) 287-298. [3] H.-W. Huang, J.C. Heinrich, and D.R. Poirier, Simulation of directional solidification with steep thermal gradients, Model. Simul. Mater. Sci. Eng., 4 (1996) 245-259. [4] S. Ganesan and D.R. Poirier, Densities of aluminum-rich aluminum-copper alloys during solidification, Metall Trans. A, i8,4 (1987) 721-723. [5] D.R. Poirier, S. Ganesan, and R. Speiser, Viscosities of aluminum-rich AI-Cu liquid alloys, Metall. Trans. B, 18B (1987) 421 - 424. [6] D.R. Poirier and R. Speiser, Surface tension of aluminum-rich A1-Cu liquid alloys, Metall. Trans. A, 18.4 (1987) 1156-1160. [7] Y.S. Touloukian, R.W. Powell, C.Y. Ho, and P.G. Klemens, Thermal Conductivity--Metallic Elements and Alloys, IFIPlenum Press, New York, 1970, pp. 471-472. [8] E.A. Brandes and G.B. Brook (eds.), Smithells Metals Reference Boo#, 7th edn, Butterworth-Heinemann, Oxford, UK, I992, pp. 14-14, 19-3. [9] J.E. Hatch (ed.), Alumimon, Properties and Ptzysieal Metallurgy, American Society of Metals, Metals Park, OH, 1984, pp. 7-10. [10] D.R. Poirier and S.N. Liu, Solidification behavior of A1 and AI-Cu alloys probed by electrical resistivity, Department of Materials Science and Engineering, The University of Arizona, Tech. Rep. for ALCOA Technical Center, Dec. 1994. [i1] D.R. Poirier and G.H. Geiger, Transport Phenomena in Materials Processing, Mineral, Metals and Materials Society, Warrendale, PA, I994, pp. 196-198. [12] P.G. Klemens and R.K. Williams, Thermal conductivity of metals and alloys, Int. Met. Rev., 31 (1986) 197. [13] G.T. Dyos and T. Farrell (eds.), Electrical Resistivity Handbook, Peter Peregrinus, London, UK, I992, p. 42. [14] C.Y. Ho, M.W. Ackerman, K.Y. Wu, T.N. Havill, R.H. Bogaard, R.A. Matala, S.G. Oh, and H.M. James, Electrical resistivity of ten selected binary alloy systems, J. Phys. Chem. Ref Data, I2 (1983) I83-322. [I5] T. Iida and R,I.L. Guthrie, The Physical Properties of Liquid Metals, Oxford University Press, Oxford, UK, 1988, p. 241. [16] E.T. Turkdogan, Physical Chemistry of High Temperature Technology, Academic Press, New York, 1980, pp. 124-125. [t7] D.R. Askeland, The Science and Engineering of Materials, 3rd edn, PWS Publishing, Boston, MA, 1994, p. 605. [18] J.E. Hatch (ed.), Alumimon, Properties and Physical Metallurgy, American Society of Metals, Metals Park, OH, 1984, p. 205.