A new measurement method of solute diffusivities based on MHD damping of convection in liquid metals and semi-conductors

Energy Conversion and Management 43 (2002) 409±416

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A new measurement method of solute di€usivities based on MHD damping of convection in liquid metals and semi-conductors V. Botton *, P. Lehmann, R. Bolcato, R. Moreau EPM Madylam, ENSHMG, BP 95, 38402 Saint-Martin d'H eres Cedex, Grenoble, France

Abstract A newly developed method of solute di€usivity measurement in liquid metals and semi-conductors is presented. It is based on the assumption that the strong scattering observed in previous ground-based measurements is a consequence of uncontrolled convection during the experiments, which is usually not accounted for. The idea is to impose a temperature gradient along a shear-cell, in order to create a well organised buoyancy driven ¯ow and to damp it thanks to a vertical uniform magnetic ®eld. The main features of the transport in such con®guration is presented through orders of magnitude analysis, accounting for the presence of time dependent solute buoyancy driven convection. To successive phases are present: at the beginning of the experiment, a B 2 t 1=2 law is derived, B and t being respectively the intensity of the magnetic ®eld and the elapse of time since the beginning of di€usion; at latter time, a steady state is reached and the transport is governed by a B 4 law. Experimental results provide an illustration of the two phases. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Di€usivity; Impurity di€usion; Liquid metals; Semi-conductors; MHD convection; Thermophysical properties

1. Introduction Solute di€usivities are so small in liquid metals (D  10 9 m/s) that their measurement is technically very dicult. Experimental data exhibit a very strong scattering and even for one

*

Corresponding author. Tel.: +33-4-7682-5202. E-mail address: [email protected] (V. Botton).

0196-8904/02/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 1 0 5 - 4

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experiment, apparently repeated under the same conditions, experimentalists are often confronted to several tenth percent scatterings. We believe that this repeatable scattering is partly due to uncontrolled convection during di€usion experiments. Such experiments are rather simple in principle: the two liquids of interest (e.g. a metal on the one hand and this metal plus some dopant on the other hand) are put into contact in a capillary tube and let free to di€use the one in the other. The analysis of the concentration pro®le after a given duration of mixing provides the di€usion coecient, assuming that negligible convective transport has occurred. The question is: ``Was the transport really purely di€usive?''. The diculties of obtaining very ®ne isothermal conditions at the temperatures imposed by the processing of liquid metals make it very hard to avoid thermo-gravitary convection. Moreover concentration variations inherent to the process may induce soluto-gravitary convection (this is usually avoided by performing experiments in vertical small bore tubes, with the heaviest component initially in the lower part of the experiment). A quick order of magnitude analysis can convince that even very low convective levels can lead to representative transport compared to di€usive mixing. The time-scale necessary to a di€usive transport on a reasonable length of capillary, L  10 2 m, is s  L=D  105 s; in the same time, a convective transport is obtained on a comparable length scale for velocities as weak as u  L=s  10 7 m/s. It has been shown in Ref. [1] that, for experiments of reasonably long duration, one cannot detect a concentration pro®le a€ected by convective transport: the enhancement of the mixing process results in an increase of the apparent di€usion coecient, Dapp , but the shape of the concentration pro®le is still solution of a di€usion equation. This phenomenon has been ®rst observed and explained by Taylor [2] in the case of a forced water-¯ow in a capillary tube. It can be described as the combined action of quick re-homogenisation of composition by di€usion over a cross-section together with axial convective transport. Eventually, the problem is that very small convective perturbations can lead to systematic and undetectable overestimation of the di€usion coecient. Up to now the need of accurate data lead to perform di€usion experiments in a microgravity environment, where natural convection is minimised. This solution is of course very expensive and does not always allow to check for repeatability of the measurement. One can also notice that, though a strong reduction of buoyancy driven convection can be expected, ¯ows of other origin (such as vibrations or surface tension gradients) are not avoided. These ¯ows, considered of second-order in a ground based experiment, may then become signi®cant (an experimental illustration is given by Smith [3]). The well-known damping e€ect of a magnetic ®eld on natural convection ¯ows in a cylinder is of high interest in the improvement of ground-based experiments. As any velocity gives rise to a Laplace±Lorentz force tending to brake it disregarding its driving mechanism, the problem of a change in the hierarchy between the types of ¯ows involved may not appear. Here, the magnetic ®eld, B; is not really used to brake the ¯ow and try to reach nearly di€usive conditions: this would not allow a fair quanti®cation of the deviation to pure molecular di€usion. By modelling the dependence of Dapp on B, and by varying this late parameter, one can properly isolate the convective contribution to the transport. Our method is thus to deduce the ``true'' di€usivity from a set of experiments with di€erent ®eld intensities. This method is valuable for good electroconducting liquids which include, of course, semiconductors.

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2. A model directed by the intensity of solutal convection The goal here is to describe the evolution of the concentration pro®le in the capillary tube by a 1D transport equation for C…x; s†, the mean concentration in the cross-section located at abscissa x, at time s. The initial condition for C…x; s† is a step function of height DC0 ; the mixing process makes this pro®le smoother and smoother until, asymptotically, the concentration ®eld is homogeneous in the whole capillary (see Fig. 2). We use a horizontal capillary, of length 2L and diameter H  L, and a vertical magnetic ®eld, which provides the most ecient electromagnetic braking of natural convection ¯ow (see Fig. 1). In this case the longitudinal density gradients generate a well-organised ®rst-order ¯ow; this density gradients are due to the variation of composition along the capillary or possibly to an imposed constant temperature gradient …DT =L†. The characteristic of the ¯uid are denoted r, q, v, bC , bT for respectively its conductivity, density (in the Boussinesq approximation), viscosity, and solutal and thermal expansion coecient. The variables are reduced using the scales H, L, m=H , DC0 , L2 =D for respectively lateral and longitudinal lengths, velocities, concentration and time. We describe the problem using the thermal and solutal Grashof numbers, the Hartmann number and the Schmidt number de®ned as: gbT DT =LH 4 m2 r r Ha ˆ BH  1 qm GrT ˆ

and GrS ˆ and Sc ˆ

gbC DC0 =LH 4 ; m2

m  1: D

Fig. 1. Con®guration of the experiment.

Fig. 2. Shape of the concentration pro®le initially (Ð) and at time s (- - -).

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In such con®guration, in the high Hartmann limit, and neglecting inertia, the longitudinal velocity exhibits a linear pro®le along the magnetic ®eld lines. This result is easily retrievable in a plane 2D con®guration (see Ref. [9]): the electric potential is then uniform, consequently assuming a quasi-permanent parallel ¯ow u…z† (dimensionally), the Laplace±Lorentz force reduces to rB2 u. In the high Hartmann limit, the viscous term is negligible except in very thin Hartmann layers, the two mechanisms balancing each others in the core are thus buoyancy and electromagnetic braking. Neglecting temperature and composition ¯uctuations in the cross-section then leads to a uniform vertical velocity gradient. (The fact that horizontal gradients of density create vertical gradients of velocity via the pressure shows that the actual balance to be considered is the one of the curl of the two force ®elds.) The result in a cylindrical geometry di€ers only by a factor two from the one in a plane geometry:   GrT GrS oC ‡ …X ; t† Z: …1† U …Z; t† ˆ 2 Ha2 Ha2 oX The thermal term is stationary, whereas the solutal term decreases monotonically with time (for X ˆ 0) as the spreading of the concentration gradient weakens it (see Fig. 2). Given this velocity pro®le, some simple algebra drives to derive the 1D transport equation for the mean concentration C…X ; t†: (" )  2 # p GrT Sc p GrS Sc oC oC o oC ‡ a…Ha† …X ; t† ˆ 1‡ a…Ha† …X ; t† …X ; t† …2† ot oX Ha2 Ha2 oX oX as presented in detail in Refs. [4,6]. The coecient a…Ha† can account for transport in the Hartmann layer or by secondary ¯ows within the cross-section. An approximate solution of this non linear equation is presented in Ref. [5]. The evolution of the concentration pro®le with time can be analysed as a two phase process, keeping in mind that the intensity of the concentration gradient is very strong at the beginning of the experiment and decreases monotonically. This can be easily recovered by order of magnitude analysis. Let us ®rst re-write Eq. (2) in a simpler form, introducing two non dimensional numbers: T ˆ

p GrT Sc a…Ha† Ha2

and S ˆ

p GrS Sc a…Ha† ; Ha2

so our transport equation becomes: (" )  2 # oC o oC oC …X ; t† ˆ 1‡ T ‡S …X ; t† …X ; t† : ot oX oX oX

…3†

…4†

Using the analogy with a di€usion process, we can compute orders of magnitude considering a length scale of di€usive evolution. r Dapp oX  t; oC  1 and ot  t: …5† D

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Then two asymptotic phases can be considered in the evolution of the concentration pro®le, depending of the relative importance of …S oC=oX †2 and 1 ‡ T 2 . By substitution of Eq. (5) in these terms, the transition between these phases is characterised by: Dapp S2 t D 1 ‡ T2

…6†

in which the right-hand side is a constant in each experiment, for it depends only on the materials involved and in the magnetic ®eld. The ®rst phase corresponds to the case in which the mixing is mainly due to solutal convection (this phase always exists at least at the early stages of the experiment); Eq. (4) then reduces to: (  3 ) oC o 2 oC : …7† …X ; t† ˆ S ot oX oX The evolution of the apparent di€usivity can be found making use of Eq. (5): Dapp  St D

1=2

;

…8†

which is dimensionally speaking a B 2 s 1=2 dependence. Now that this tendency is determined, we can provide an order of magnitude of the date of transition, t0 , between the two phases: from Eq. (6), t0 

S2 2

…1 ‡ T 2 †

:

…9†

The second phase is reached for durations far greater than t0 . In this phase, the solutal term in Eq. (4) can be neglected (S  0) so that obviously: Dapp ˆ 1 ‡ T 2: D

…10†

Note that, in this case, the equality is exact and the apparent di€usivity is a constant; Eq. (2) is thus a classical di€usion equation whereas in the other cases, where S 6ˆ 0 , it was a di€usion equation with variable di€usivity (Dapp was function of both time and space). When such negligible solutal convection phase is reached, the measurement method is particularly easy to understand: the plot of Dapp vs. a…Ha†=B4 is a straight line of origin D, the coecient we expect to measure. Experimental results corresponding to this case are plotted on Fig. 3. At last, let us present the simpler case of isothermal experiments (T 2  1), in which the order of magnitude analysis of Eq. (4) with Eq. (5) leads to the relation: Dapp S2t 1 1‡ ; D Dapp =D

…11†

without any consideration on the strength of solutal convection. This order of magnitude relation is used in the present paper to plot experimental results on di€usion of Bi in SnBi (0.5%), but more detailed and accurate calculations can be found in Refs. [4,5].

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Fig. 3. Experimental results for SnIn(1 at.%) at 275°C with a 173°C/m temperature gradient.

3. Experimental illustration of the two regimes Several experimental apparatus can be used to the measurement of di€usivities; we use the shear cell method, exposed for example in M uller-Vogt and K ossler [10], which will not be developed in the present paper. The range of magnetic ®eld used here (0.3±2 T) corresponds to Hartmann numbers ranging from 20 to nearly 150. Two metallic alloys has been chosen to illustrate the two regimes described above. In the two cases the Schmidt number is of the order of 100. The inter-di€usion in the couple Sn±SnIn(1 at.%) allowed us to observe the low solute buoyancy driven convection case. Sn and In have indeed very similar density and this alloy exhibit an extremely low solutal expansion coecient; as a consequence the solutal Grashof number is fairly small: GrS ˆ 6:5. Two temperature gradients were applied along the capillaries: DT =L ˆ 68 K/m and DT =L ˆ 173 K/m, to provide respectively GrT ˆ 21 and GrT ˆ 53. Two averaged temperatures were investigated: T0 ˆ 250°C and 275°C. Fig. 3 shows our experimental results in the case 173 K/m with T0 ˆ 275°C these results correspond to 36 h duration of di€usion. The linearity of Dapp as a function of a=B4 is very well veri®ed, however one observes two lines depending on the direction of the temperature gradient relatively to the residual solutal buoyancy forces. As indium is lighter than tin, when the rich liquid is on the hotter side of the capillary, the convection is slightly enhanced compared to when it is on the cold side. The slope of these lines corresponds to the theoretical prediction and their origin, 2:6  10 9 m2 /s is in very good agreement with recent microgravity data [7,8]. The di€usion in the couple Sn/SnBi(0.5 at.%) illustrates the case of strong solute buoyancy driven convection: in this case GrS ˆ 90. As the solutal term in Eq. (2) cannot be neglected, the experiment were performed in an isothermal furnace (GrT ˆ 0) for the sake of simplicity. The presence of a well-organised predominant ¯ow is insured by solutal convection. The experimental measurements plotted in Fig. 4 have been obtained with di€erent durations (from 13 to 26 h). Eq. (8) was ®rst used to analyse these results, the B 2 tendency was very well observed and the time

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Fig. 4. Experimental results for SnBi(0.5 at.%) at 275°C in an isothermal shear cell.

exponent checked with three data points only was 0:4 instead of 0:5, however it appeared that axial molecular di€usion was not negligible so that the full analytical solution or more simply Eq. (11) became useful to the description of these data. The coordinate system in Fig. 4 is chosen from Eq. (11), as this equation provides orders of magnitude only, this plot cannot be looked upon as providing an accurate measurement but the origin of the ®tted straight line is an indicator of the value of the di€usion coecient. The value obtained here (2.18 10 9 m/s) is in very good agreement with the recent data issued from the Foton 12 space experiment [7] with similar temperature (300°C).

4. Conclusion This ground-based method to measure solute di€usivities in liquid metals and semi-conductors is based on a simple strategy: · create a ®rst-order well-organised ¯ow (by natural convection); · damp it, together with all existing ¯ows, by the imposition of a constant homogeneous magnetic ®eld, which keeps unchanged their hierarchy; · measure apparent di€usion coecients, Dapp …B), and extrapolate accordingly to our analytical model to in®nite magnetic ®eld to obtain the molecular di€usivity value. The feasibility of the method has been demonstrated with low fusion point, easy to handle, alloys (tin based). These experiments are in very good agreement with the predictions of our model. The comparison with existing microgravity data points is also favourable to this ground based method.

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This method may be limited to dilute alloys and to not-too-low di€usivities (>5  10 10 m/s), though further work must be done to derive a real conclusion. Two problems arise when investigating non dilute alloys: the solute buoyancy driven motion becomes very strong and it is not clear at the moment if our model is able to provide an accurate measurement of D in this case, but the main matter is that the properties of the ¯uid becomes function of its composition. A better description of the ¯ow and of the transport in the case where solute and thermal buoyancy forces oppose each others would also be interesting.

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