ТЕПЛОФИЗИКА ВЫСОКИХ ТЕМПЕРАТУР, 2014, том 52, № 3, с. 397-401
EMPIRICAL MODEL FOR THE ESTIMATION OF THERMOPHYSICAL PROPERTIES OF LIQUID METAL ALLOYS
© 2014 D. Ceotto1, F. Miani2
1DIEGM — Universita degli Studi di Udine 2DICA — Universita degli Studi di Udine 33100 Udine (ITALY) E-mail: diego.ceotto@uniud.it Received July 19, 2013
A model for calculating the main properties of liquid metal binary alloys at standard pressure based on experimental data observation is presented. Given the characteristics of the pure metals, the model allows to calculate density, surface tension, electrical resistivity and thermal conductivity of binary alloys at various concentrations along the liquidus line. Some preliminary comparisons for Al—Si and Al—Cu systems are in satisfactory agreement with the model.
DOI: 10.7868/S0040364414030247
INTRODUCTION
Pure liquid metals have been studied since many decades and their fundamental properties may be considered on a firm basis (e.g. [1]) especially due to the studies conducted the '60 and the '70 by such authors as Grosse (1961), Ziman (1961), Edwards (1962), Allen (1963), Faber (1966), Ashcroft (1966), Ascarelli (1969), Lang (1970), and Filippov (1966) (see [1] for more details). However, the same is not true for alloys which are still under investigation. In the last years, there have been investigations of some alloys, such as Al—Cu, Pb—Bi and Pb—Sn [2], but the experimental data are sometimes contradictory. This work aims at providing a method for estimating the main physical properties at standard pressure (i.e. one atmosphere) to describe metal alloys in the liquid state in order to offer a reliable approximated approach. Density, surface tension, electrical resistivity and, consequently, thermal conductivity are the considered values. Given these literature data (i.e. [3—9]), we choose to test the model on the Al—Cu and Al—Si systems. These alloys have been investigated during the recent years with the most qualified experimental methods and the data obtained seem to be consistent.
DENSITY
Density is important because it determines the mass flow and influences the heat transfer [10]. It also determines the solidification transformation [11]. In the model we previously proposed [12], it was shown that density influences the relationship between viscosity and temperature.
On the one hand, the density of an alloy may be determined only experimentally because the models pro-
posed in literature are inadequate to satisfy the industrial needs which often involve complex chemical compositions. On the other hand, the knowledge of this property is essential for simulating and setting the design of every metallurgical process. Here, a new model is presented for estimating the density of a binary alloy and it is compared with the most recent assessed data.
We begin by considering the alloy as an ideal solution of two elements A and B. Under this hypothesis the alloy is studied as an ideal mixture of gases, where the properties of the AB solution are determined by the sum of the fraction X of the alloying elements. According to this model we consider the density of the alloy under ideal condition p AB,id to be expressed with the "linear mixture rule" as:
P ab, id = xaP a + xbP b, (1)
where pA, pB are the density of pure metals A and B, XA and XB are the components mass fraction of each pure element for which this property is valid XA + XB = 1.
Now, we compare the ideal properties with the experimental values. Given the importance for industries and the availability of experimental data in literature we choose to conduct the investigation under two well-tested systems: Al—Cu and Al—Si.
By an empirical examination one may observe that along the liquidus line this proportion holds:
pAB = Tl , (2)
pab,id T*
where pAB and TL are the density and the temperature of the alloy along the liquidus line. Using an ideal mixing ratio, the melting temperature of the alloy T* is expressed by combining the temperatures of the pure metals A and B as:
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CEOTTO, MIANI
p, kg m 3 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000
0 20 40 60 80 100
at % Al
Fig. 1. Density for Al- Cu system: 1 — Eq. (4), 2 - [3].
p, kg m 3 X 1
5000 -
4000 -
3000. -
2000" —ш— t f 1
■
1000 - 1 1 i i 1 1 1
0 24 6 8 10 12 14
wt % Si
Fig. 2. Density for Al- -Si system: 1 — [4], 2 - Eq. (4).
Managing Eqs. (1)—(3) we find that the density for an alloy along the liquidus line at the pressure of one atmosphere can be calculated as:
T
P ab = (xaP a + xbPb )).
T *
(4)
The comparison of results obtained by Eq. (4) with experimental data [3] along the liquidus line is reported in Figs. 1 and 2.
Experimental and calculated data are reported in Tables 1 and 2. The average deviation of the calculated data is equal to 9%. The average experimental error bar [3, 4] is about 3%. However, the experimental data differ from the other sources data by about 10%. Within this consideration, the accuracy of the model proposed can be considered satisfactory.
SURFACE TENSION
The surface tension for a liquid metal alloy is fundamental in studying the solidification as well as the design to avoid cavitation, evaporation and boiling in the industrial processes. This property is fundamental for industrial engineering and it deserves further investigations.
In the same fashion as what has been done for the density, we obtain the relation to estimate the surface tension of the alloy yAB:
T* = TA + (TB - TA) XB, (3)
where TA and TB are the melting temperatures of each component at standard pressure.
Table 1. Data used and calculated for the Al—Cu system
at % Al Tl, °C PL, kg m-3
[3] experiment [3] calculation
0 1084 7970 7970
5 1060 7824 7670
10 1075 7321 7648
18 1032 7013 7131
25 1049 6699 7048
30 1040 6229 6838
34 1022 6013 6598
40 960 5639 6017
45 900 5420 5493
50 850 5299 5042
55 790 4938 4543
60 700 4345 3892
65 620 4238 3323
67.8 591 3881 3098
75 580 3691 2857
82.9 550 3239 2501
90 650 2952 2715
95 640 2575 2493
100 660 2374 2374
T
Y ab = (xaY a + xb Y b )). (5)
T *
The equation proposed is checked for the case of the Al—Cu systems (i.e. Fig. 3). Experimental data [5] and calculated ones are both reported in Table 3. The model is able to predict well the alloy surface tension at the melting temperature at standard pressure with an average absolute error of 7%.
ELECTRICAL RESISTIVITY AND THERMAL CONDUCTIVITY
Among the thermophysical properties of liquid metals, thermal conductivity is necessary for understanding and simulating industrial thermal processes. In foundry, for example, heat transfer drives the solidification of the metal and consequently the mass flow in the mould through the channels (e.g. [13]). In steel-making the thermal conductivity is important because it determines the grain dimension of the metal and, consequently, the mechanical characteristics of the material as the Young modulus, the yield stress and the fatigue resistance (e.g. [14]).
Actually, the experiments conducted about thermal properties have given approximated and sometimes contradictory values even if new promising methods have been recently implemented [6].
In the same manner as above, we begin by considering the metal alloy as an ideal solution of two ele-
EMPIRICAL MODEL FOR THE ESTIMATION OF THERMOPHYSICAL PROPERTIES
399
ments A and B. According to this model we propose the ideal electrical resistivity to be
pel,id = xapel,a + xbpel,b, (6)
where pelA and pel,B are the electrical resistivities of the pure metal A and B. We have tested also other possible definitions of ideal resistivity but the "linear mixture rule" has the best fit the experimental data. The investigation can be conducted also using the electrical conductivity but in this article we prefer to use the electrical resistivity because it is the parameter usually measured in the experiments and most commonly presented in literature.
Examining data for Al—Cu liquid alloys, one of the few systems for which data are available, a linear relation between two dimensionless ratios is plausible; so one may infer that along the liquidus line at standard pressure a direct proportion is valid:
Table 2. Data used and calculated for the Al—Si system
pel,ab pel,id
t*
TL'
(7)
Pl, kg m
-3
wt % Si [4] experiment [4] calculation
0 660 2374 2374
3 640 2387 2489
3.1 640 2387 2493
5.8 622 2406 2601
5.9 621 2405 2605
8.9 601 2418 2731
11.6 584 2435 2850
Table 3. Data used and calculated for the Al—Cu system
Eq. (7) is different from Eqs. (2) and (5) previously proposed for density and surface tension because, in this case, the temperature ratio in inverted: T*/TL (instead of TL/T*). Further investigations should be performed on the fact that the temperature ratio is inverted.
In Fig. 4, the direct proportion of Eq. (7) (using the experimental values of Pe^ given by Plevachuk et al. in [7]) is reported for the Al—Cu system, where the average error is about 2%. The average error of measurements reported in [7] is about 2% but, again, the data differ from the values of other recent sources by an amount of about 10%. Given these considerations, we think that the accuracy of the model proposed can be considered more than satisfactory.
In Fig. 4 it has been traced also the straight line (broken line) that indicates when the ratios pel,AB/pel,id and T*/TL are equal.
Thus, by substituting Eq. (6) into Eq. (7) we see that the electrical resistivity along the liquidus Clapey-ron line is given by:
at % Cu Tl, K Yl, N m 1
[5] experiment [5] calculation
100 1358 1.30 1.30
90 1347 1.35 1.29
83 1317 1.32 1.26
70 1304 1.19 1.24
60 1228 1.13 1.17
50 1106 1.04 1.05
40 980 1.00 0.93
30 867 0.967 0.82
17 825 0.941 0.77
10 873 0.871 0.82
0 933 0.87 0.87
pel, ab
xapel, a + xbp.
el,B
Tl
T*
Then, by taking the data of pel,A and pel,B for the pure metals from [8] and calculating the thermal co
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