Diffusionin Solids and Liquids

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Title

Diffusionin Solids and Liquids

ISBN: 0-87849-438-3

Year: 2007

Subtitle: Heat Transfer - Microstructure & Properties

Authors/Editors: Andreas Öchsner and José Grácio

Published in: Materials Science Forum, Volume 553 until 553

Proceedings: 2nd International Conference on Diffusion in Solids and Liquids, Mass Transfer - Heat Transfer - Microstructure & Properties, DSL-2006, 26-28 July 2006, University of Aveiro, Portugal

Pages: 280

Edition: softcover

Description: in preparation

TOC: Table of Contents

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Diffusionin Solids and Liquids (Table of Contents)

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Papers of Title

Diffusionin Solids and Liquids• Page 3/4 •

Table of Contents (39 papers, 10 per page listed)

Design of an Induction Glass Melting Furnace by Means of Mathematical Modelling using the Finite Element MethodIsaac Arellano, Gabriel Plascencia, Elías Carillo, Miguel A. Barrón, Adolfo Sánchez, Juliana Gutiérrez

p124

Estimation of the Heat Flux Through Furnace Side Walls Protected with Water Cooled Cooling DevicesGabriel Plascencia, Torstein A. Utigard, Juliana Gutiérrez, David Jaramillo, Vicente Mayagoitia, Marcello Hernández

p130

Estimation of the Heat Flux Through Furnace Side Walls Protected with Water Cooled Cooling DevicesMarina Dubatovskaya, Sergei Rogosin

p130

Hele-Shaw Model for Melting/Freezing with Two DendritsSergei Rogosin, Tatsyana Vaitekhovich

p143

Intermediates in Ring-Disc Electrode ProcessesCésar A.C. Sequeira

p152

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Diffusionin Solids and Liquids (Table of Contents)

Vibration Analysis of a Ni-Ti Shape Memory Alloy RodM. Ghazinejad, Ali Shokuhfar

p164

SPH as an Inverse Numerical Tool for the Prediction of Diffusive Properties in Porous MediaAntonio C.M. Sousa, Fangming Jiang

p171

Evaluation of Thermal and Mechanical Filler Gas Influence on Honeycomb Structures BehaviorMatej Vesenjak, Andreas Öchsner, Zoran Ren

p190

Thermal Post-Impact Behavior of Closed-Cell Cellular Structures with Fillers – Part IMatej Vesenjak, Andreas Öchsner, Zoran Ren

p196

Thermal Post-Impact Behavior of Closed-Cell Cellular Structures with Fillers – Part IIMatej Vesenjak, Zoran Žuni•, Andreas Öchsner, Zoran Ren

p202

© 1995-2006 by Trans Tech Publications Inc. Laubisrutistr. 24, CH-8712 Stafa-Zurich. Switzerland. Fax +41 44 922 10 33. info@ttp.net

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ESTIMATION OF THE HEAT FLUX THROUGH FURNACE SIDE WALLS PROTECTED WITH WATER COOLED COOLING DEVICES

Gabriel Plascencia 1,2, Torstein A Utigard 2, Juliana Gutiérrez 3, David

Jaramillo1, Vicente Mayagoitia1, Marcelo Hernández 4

1 CIITEC - IPN, Cerrada Cecati s/n, Col. Sta. Catarina, México, D.F. C.P. 02250, México.

2 Dept. of Materials Science & Engineering, University of Toronto,

184 College St. Toronto, ON, M5S 3E4, Canada

3 ESIME-Azcapotzalco, Av. de las Granjas 682, Col. Sta. Catarina, México, D.F., C.P. 02250, México

4 ESIQIE - IPN,

Dept. de Metalurgia y Materiales, Apdo. P. 75-874 México, D.F., C.P. 07300, México

Key words: Furnace side wall; Cooling fingers; Finite element method; Thermal conductivity; Heat flux; Copper

Abstract. A three dimensional numerical heat transfer model has been developed to estimate the heat flux trough furnace side walls protected with water cooled cooling fingers. The model was set up by means of the finite element method. Materials with different thermal conductivity were modelled and the results obtained with the mathematical model were compared with experimental data. In every case, it was found excellent agreement between the experimental data and the model computations. Introduction Current smelting and converting are characterized by high intensity and productivity. Such processing features have lead to more demanding conditions that refractory linings must withstand; as a consequence the service life of such linings is shortened. To extend the service life of those linings, several cooling systems have been designed and implemented in many copper and / or nickel making facilities [1,2]. The different cooling systems can be grouped according to their ability to remove heat from the hearth of the furnace. The amount of heat that can be removed by these devices is in the range of 1 to 100 kW/m2, depending upon of whether the cooling systems are embedded or not into the lining [1]. Proper cooling system design is necessary since not every smelter runs in the exactly same manner, as an example, if the side wall heat flux is too low, the refractory may wear back, or if the cooling is highly intense, the excessive cooling may lead to higher heat losses. Hatch and Wasmund [2] recognized that refractories are attacked by several mechanisms, such as melting, dissolution by molten metal/slag, chemical reactions between the refractory and the slag. They also acknowledged that refractory spalling may happen as a result of thermal cycling and also tapping and charging operations may promote refractory erosion due to the collision of the charging materials with the lining.

Another problem related to the linings wear is the penetration of molten material into cracks or joints. Thermal cycling not only induces stresses into the lining they also promote the freezing and re-melting of the material deposited on the cracks, enlarging them to a point where leaking of the molten material may produce run outs. Sarvines et al [3] offer similar data on the amount of heat that can be removed with different cooling systems. In this compilation Sarvines et al, include finger coolers. These cooling devices are in the mid range in terms of heat removal, they account for a heat flux capacity of nearly 100 kW/m2, which accordingly to Legget and Gray [1] is equivalent to the energy that can be removed using panel coolers. Mathematical heat transfer modelling To predict the thermal and oxidizing behavior of the cooling elements in an actual furnace, a 3-D mathematical model was set up to estimate the temperature field and the heat flux through the refractory and the cooling elements. The model was set up using the commercial FEMLAB™ software. The mathematical model was set up based on cooling elements currently in operation at Falconbridge’s facilities in Sudbury On [4]. The cooling elements configuration is sketched in Figure 1. In this arrangement, the distance between each cooling element is 50 cm. The cooling elements are long cylinders that are partially embedded into the refractory lining; the un-embedded part of the cooler is attached to cooling water through a piping system. The dimensions of each cooler are shown in Table 1

Coolingelements

Furnaceside wall(front view)

50 cm

50 cm

Refractory

Cooler

Waterchannel

Cooler embedded into the refractory lining (not to dimensions)

Lateralview

Figure 1. Furnace side wall view with cooling elements Table 1 Dimensions1 of the cooling elements modelled

Length 0.79 Diameter 0.15 Water channel length 0.51 Water channel diameter 0.04

1 Dimensions are in (m) In developing the mathematical model, it is necessary to evaluate the heat transfer coefficient between the cooling water and the cooling element as well as the heat transfer coefficient between the molten slag and the refractory lining.

Given that the cooling water is forced to run through a circular closed channel, the Dittus- Boelter correlation is the one that better correlates the heat transfer coefficient in turbulent flows [5]. The Dittus-Boelter correlation is expressed as: (1) 3.08.0 PrRe023.0 ⋅⋅=Nu Where Nu and Re are the Nusselt and Reynolds numbers respectively, Prwater is the Prandtl number for liquid water, kwater is the thermal conductivity of water (W/m/K), D is the diameter of the inner channel (m) and h is the heat transfer coefficient between the water and the cooling element (W/m2/K). Equation 1 is valid if the following conditions are satisfied:

10

10Re160Pr7.0

4

≥

≥

≤≤

DL

(2)

Table 2 Heat transfer coefficients estimated from Equation 1

Water velocity (m/s)

Re Pr Nu h (W/m2/K)

2 86792 5.83 349 4894 4 173584 5.83 607 8520 6 260375 5.83 839 11785 8 347167 5.83 1057 14835

In the case of the heat transfer coefficient between the molten phase and the lining, the value of the heat transfer coefficient was set as 1500 W/m2/K as suggested by Utigard et al. [6]. The geometry and the boundary conditions used to run the mathematical model are sketched in Figure 2. The model was run in order to predict the effect of the refractory thickness in front of the cooler on the temperature at the hot end of the cooler. Figure 3 shows some of the results from the modelling, while Table 3 summarizes the results from the model for a certain set of conditions. The model results are compared with some experimental measurements carried out in our laboratory. For more information about the experimental approach, we refer the reader to the literature [7,8]. In this table Tcold represents the temperature at the cold end of the cooler, and Thot represents the temperature at the cooler/refractory interface. To conduct our simulations, we tried different mesh sizes, and after convergence and mesh size tests, we found out that the optimum mesh consisted of 35000 elements.

Cooler Slag

hwater= 0.023 * Re0.8 * Pr0.3

Tslag = 1250 ºC

hslag = 1500 W/m2/K

(B)(A)Symmetry lines

(Adiabatic boundaries)

Brick

Figure 2. Mathematical model set up: a) Model geometry, b) Boundary conditions

Copper cooler Alloy cooler

Refractory infront of cooler = 22 cm

Refractory infront of cooler = 0 cm

(A)

(D)

(C)

(B)

Figure 3. Calculated isotherms and heat flux from the mathematical model for copper coolers: (A) when the lining in front of the cooler is new and (B) when the lining is completely corroded away. Results for the alloy cooler under the same conditions are shown in (C) and (D). [Cones are proportional to the heat flux.]

Table 3. Comparison of the numerical results with experimental data

Material Refractory in front of the cooler

(cm)

Calc. Thot (ºC)

Exp. Thot (ºC)

Calc. sidewall heat flux (kW/m2)

Exp. heat flux (kW/m2)

Copper

22.0 11.0 2.2 0.0

342 646 1099 1084

1102

117 179 526 714

574

Cu - 4wt% Al alloy

22.0 11.0 2.2 0.0

491 794 1098 1070

1092

94 144 382 733

427 Model results discussion. It can be seen that as the refractory lining corrodes away, the cooler temperature increases noticeably. When 90 % of the lining in front of the cooler is removed, the temperature at the tip of both coolers is higher than their melting point. The cooler therefore would dissolve into the molten phase, decreasing its capacity to remove heat from the hearth and possibly cause major damage to the furnace due to the possible contact between the molten phase and the cooling water. Such effect is even more pronounced when no refractory is in front of the cooler. The calculations reveal that the copper element would lose 6 % of its original length due to dissolution into the molten phase (slag), while the alloy cooler may lose 12 % of its initial length. Direct comparison of the experimental and calculated temperature at the hot end for each cooler are in good agreement, while the comparison between the calculated and experimental heat fluxes present a bigger difference. Such difference can be attributed to the material lost due dissolution, since the distance for heat transfer decreases, it results in an increased heat flux. At the same time the difference in the calculated heat fluxes when there is no refractory protection is practically zero; whereas under different lining lengths there are significant differences between the heat fluxes that can be removed using the different coolers. Since the slags used in copper making have a liquidus temperature of around 1150 ºC and the actual slag operating temperature varies between 1250 and 1300 ºC, a superheat of 150 ºC is required to start the solidification of the slag. When we back calculate the superheat from the model results, we found that having the full refractory in front of the cooler would not allow to freeze the slag (superheat = 78 ºC) onto the lining surface, indeed the refractory is likely to be eroded. Even after losing 50 % of its original length (superheat = 119 ºC), no solid crust would be formed. In order to start the solidification of a protective slag shell, the lining would need to lose around 65 % of its initial length. In such case, the temperature at the hot end of the cooler would be in the vicinity of 700 ºC. It also should be noticed that within the furnace, the actual temperature of the slag decreases towards the side walls. This will lead to decreased localized superheats, increased viscosity and decreased fluid flow. All these effects will tend to decrease the heat flux through the side walls, resulting in temperatures of the hot end of the cooler below those calculated above.

Conclusions Numerical calculations may lead to the conclusion that unless the refractory linings in industrial vessels are severely damaged, it would not be major safety or heat losses issues. However, the heat flux calculated in un-attacked linings imposes superheats on the slag below that required for freezing a protective shell of slag. Furthermore, in order to start the solidification of the slag onto the lining, for typical slag conditions it is required that the lining reduces its original length by at least 60 %, which may result in the oxidation of coolers made from copper. Acknowledgements The authors would like to thank CONACyT and COFAA-IPN in México as well as to NSERC in Canada for founding this research References 1. A.R. Legget, N.B. Gray: Advances in Refractories for the Metallurgical Industries II

(CIM, Canada 1996). 2. G.G. Hatch, B.O. Wasmund: U.S. patent # 3,849,587 (USA 1974). 3. J. Merry, J. Sarvinis, N. Voermann: JOM Vol. 52 (2000), p. 62. 4. R. Berryman: Private communication, May 2001 5. F.P. Incropera, D.P. DeWitt: Fundamentals of Heat and Mass Transfer (John Wiley &

sons, U.S.A 1996). 6. T.A. Utigard, A. Warczok, P. Descalaux: Met. TransB Vol. 25B (1994), p. 43. 7. G. Plascencia, T.A. Utigard, D. Jaramillo: JOM Vol. 57 (2005), p. 44. 8. G. Plascencia, T.A. Utigard: Proceedings of the 5th International Conference

Copper/Cobre 2003. Vol 4 Book 2 (CIM, Canada 2003).

Diffusion in Solids and Liquids

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Title

Diffusion in Solids and Liquids

ISBN: 3-908451-36-1

Year: 2006

Subtitle: Mass Diffusion

Authors/Editors: Andreas Öchsner and José Grácio

Published in: Defect and Diffusion Forum, Volume 258 until 260

Proceedings: 2nd International Conference on Diffusion in Solids and Liquids, Mass Transfer - Heat Transfer - Microstructure & Properties, DSL-2006, 26-28 July 2006, University of Aveiro, Portugal

Pages: 624

Edition: softcover

Description: in preparation

TOC: Table of Contents

Prices: USD: 274.00 / EUR: 238.00

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Diffusion in Solids and Liquids

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Diffusion in Solids and Liquids (Table of Contents)

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Papers of Title

Diffusion in Solids and Liquids• Page 5/9 •

Table of Contents (90 papers, 10 per page listed)

Diffusion of Critical Elements in Steel during Thermal Treatments in a Diamond Chemical Vapour Deposition AtmosphereV.F. Neto, R. Vaz, T. Shokuhfar, C.A.A. Ghumman, Orlando M.N.D. Teodoro, N. Ali, Monica S.A. Oliveira, J. Gracio

p270

Non-Hydrostatic Pressure Induced Structural Phase Transitions of Silicon Analyzed by Raman ScatteringPaulo S. Pizani, Renato G. Jasinevicius, Ricardo A. Zanatta

p276

Numerical Model to Describe the Growth of the Internal Oxidation Layer in Binary AlloysGabriel Plascencia, Torstein A. Utigard, Juliana Gutiérrez, David Jaramillo, Fernando Martínez

p282

Rate of Reduction of Chromium Oxides in a Slag by FeSi and FeMgSi FerroalloysJuliana Gutiérrez, Gabriel Plascencia, Antonio Romero, Fidel Reyes, Isaac Arellano

p288

http://www.ttp.net/3-908451-36-1/4.html (1 of 2)28/01/2007 10:18:42 a.m.

Diffusion in Solids and Liquids (Table of Contents)

Reactive Diffusion in Cu-Nb-Sn Internal Tin Superconducting Wires for the ITER ProjectI. Pong, S.C. Hopkins, Xiao Wei Fu, Bartek A. Glowacki, J.A. Elliott, Alberto Baldini

p294

Effect of Deformation, Annealing and Doping with Zr on Nb Filaments Nanocrystalline Structure in High-Strength In Situ Cu-Nb CompositesE.N. Popova, V.V. Popov, E.P. Romanov, N.E. Hlebova, V.I. Pantsyrny, A.K. Shikov

p299

Diffusion Coefficients of Aqueous Solutions of Carbohydrates as Seen by Taylor Dispersion Technique at Physiological Temperature (37 ºC)Ana C.F. Ribeiro, Cecilia Isabel A.V Santos, Victor M.M. Lobo, Artur J.M. Valente, Pedro M.R.A. Prazeres, Hugh D. Burrows

p305

Thermal Conductivity in Nanoscale Lennard-Jones Systems: Size Effects in the Fluid and Solid PhasesD.F. Botelho, Paulo S. Branício, José Pedro Rino

p310

Thermal Expansion and Oxidation of Calcium Oxysulfides by DiffusionE.N. Selivanov, R.I. Gulyaeva, V.M. Chumarev, N.I. Selmenskikh

p316

Hydrogen Diffusion in Austenitic Stainless SteelsYun Chen, Diogo M.F. Santos, César A.C. Sequeira

p322

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NUMERICAL MODEL TO DESCRIBE THE GROWTH OF THE INTERNAL OXIDATION LAYER IN BINARY ALLOYS

Gabriel Plascencia1,2, Torstein A. Utigard2, Juliana Gutiérrez3, David Jaramillo1, Fernando Martínez1

1 CIITEC - IPN,

Cerrada Cecati s/n, Col. Sta. Catarina, México, D.F. C.P. 02250, México.

2 Dept. of Materials Science & Engineering, University of Toronto, 184 College St. Toronto, ON, M5S 3E4, Canada

3 ESIME-Azcapotzalco,

Av. de las Granjas 682, Col. Sta. Catarina, México, D.F., C.P. 02250, México

Key words: High temperature oxidation; Internal oxidation layer; Finite differences method; Binary alloys; Diffusion Abstract. Several analytical models have been developed through the years to describe the formation and growth of the internal oxidation layer in binary alloys. Such models are often complex and their validity strongly rely on precise measurements of molar fluxes of the different species involved in the oxidation process. The main disadvantage of such measurements is that they are difficult to made and present a high degree of uncertainties, thus some assumptions are needed to ease understanding and the applicability of them. In this paper we set up a numerical scheme (finite differences) to describe the growth of the internal oxidation layer in binary Cu-Al alloys oxidized in air at different temperatures. There is good agreement between the experimental results and the values calculated with the aid of our numerical approach. Introduction When an alloy is exposed to an oxidizing environment it develops what is known as the internal oxidation layer. In such layer some of the solute element in the alloy oxidizes preferentially over the host metal (if the solute metal is more active then the host metal). As a result of the formation of such layer, catastrophic failure is imminent; consequently, it is of relevant importance to determine under which conditions such internal oxidation layer will grow to a point where the alloy will fail. Several kinetic models have been proposed to describe and/or to estimate under which the internal oxidation layer will be formed [1-3]. In either case, each model agrees that the rate limiting step for the formation of the internal oxidation zone is the diffusion of either the solute metal or the oxygen. However, in order to quantify the precise molar flux of the different species involved in the oxidation process, accurate measurements are needed. As an example, Wagner [1] developed his model by solving Fick’s second law by means of the semi infinite solid approach. In doing so, he proposed that the thickness of the internal oxidation layers is a direct function of a dimensionless variable. Such variable is just a dummy variable that relates the

concentration of the different species as they move through a specific path, thus forming the internal oxidation layer. Therefore, the model proposed by Wagner, requires that the measurements of the concentration of either the solute metal or the oxygen within the alloy matrix must be performed almost in a perfect fashion. The problem with such measurements is that they are difficult to make and at the same time, very sophisticated equipment is needed. On the other hand, Rhines’s model [2] does not present the analytical complexities that Wagner’s model has. Instead of going through a detailed solution for the diffusion equations, Rhines proposed an approximation to the analytical solution of Fick’s second law. This model does not need the precision in the measurements of concentration as Wagner’s model needs. However, since this model was obtained in an empirical manner, some guessing of the values of concentration of the oxidizing species is needed; therefore, this model is not very useful without carrying out some experiments to confirm the guessed values inserted on it. A more recent model presented by these authors [3], approaches the diffusion problem by assuming a quasy-steady state process. By doing this basic assumption, the authors were able to equate the oxygen pick up (mass balance) by the alloy to a lineal concentration profile of oxygen within the internal oxidation layer. This model does not need complicated measurements of the different molar fluxes, it only needs to measure the depth of the internal oxidation layer, and from such measurements is possible to determine if the actual oxygen concentration in the alloy lies below or above the solubility limit of oxygen in the host metal. If the concentration of oxygen is higher than the solubility limit, then both the internal oxidation layer and an external oxide scale will form. If the concentration of oxygen lies below the solubility limit, then the alloy won’t be oxidized. Although this model gives an accurate description of the oxidation process in flat and non-flat surfaces, it cannot be used to estimate the concentration of any of the diffusing species at any particular location within the internal oxidation layer. There have been reported some modifications to Wagner’s model [4], as well as some numeric solutions to such model [5, 6]. However, in every case, the problem of determining the actual concentration of the different species still arises. In this paper, we are presenting a numerical approach to describe the oxidation process similar to that proposed by Sockel et al [7]. The description of the numerical method is given in the following section. Numerical Method In order to estimate the thickness of the internal oxidation layer we use the Finite Differences Method [8]. Since we are dealing with an unsteady state mass transfer problem, we start with Fick’s second law of diffusion in one dimension:

2

2

xCD

tC

∂∂⋅=

∂∂ (1)

Where C is the concentration of the diffusing species [mole/m3], t is the time [s], x is the spatial coordinate [m] and D is the diffusion coefficient of the diffusing species in the host metal [m2/s]. As we have proved before [3], the oxidation process takes place in one direction, in our particular case; we use cylindrical coordinates, so the actual diffusion process takes place in the radial direction. Therefore, we approach the solution of Fick’s equation in 1-D, as follows:

tCC

dtdC t

itt

i

∆−

≈∆+

(2)

And

211

2

2 2x

CCCdx

Cd ti

ti

ti

∆−+

≈ −+ (3)

Combining equation (2) & equation (3) yields:

( )ti

ti

ti

ti

tti CCC

xtCC 2112 −+⋅

∆∆

+= −+∆+ (4)

Where t= 2R

tD⋅ , x= Rr and C =

0Cc . Figure 1 shows how the numerical scheme was set up.

Un-oxidized alloy matrix

Internal oxidation layer

External oxide scale

Cu2O + traces of CuO

Some oxide precipitates

50 µm

Cu2O + CuAlO2+ traces of Al2O3

Al depletion + Al2O3 precipitates within this layer

Time (B) (A)

∆t Cit+∆t

Cit Ci+1

tCi-t

∆x

Radial position (m)

Figure 1. (A) Domain in which the oxidation process takes place, the domain has been

divided in mesh points both in the radial and time coordinates. The radial direction is subdivided in into ∆r intervals, while the time domain is divided in ∆t time steps. (B) Micrograph showing the growth of the different oxide layers in a Cu-2wt% Al alloy oxidized in air at 850 °C during 21 hrs.

To determine the extent of diffusion of aluminium to the surface, an unsteady state model was developed using the following expression for the diffusion coefficient (effective) of aluminium in copper [9]

)TR

,exp(.)s/m(D×

−×⋅= − 0001651054 62 (5)

Where R is 8.314 J/mole/K and T is the temperature (K). It was assumed that independent of the bulk concentration, the surface concentration of metallic aluminium was fixed to be zero due to reactions with oxygen in the air or traces in the argon used. For a 2 wt% Al alloy at a heating rate of 10 oC/min, the flux of aluminium to the surface and the resulting alumina thickness are given in Figure 2 as a function of temperature. It is seen that when reaching 600 oC, the resulting alumina layer thickness is less than one nano-meter, while at 1000 oC its thickness is about 0.6 µm. Since the argon (99.999%) used to heat the samples will have

some traces of oxygen, this diffusion of aluminium to the surface may form a noticeable alumina layer when the temperature reaches 900 and 1000 oC.

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

500 600 700 800 900 1000

Temperature (oC)

Al 2O

3 lay

er ( µ

m)

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

Al Flux (m

ol/m2/m

in)

10 oC/min

Figure 2. Calculated diffusion flux of aluminium and the resulting alumina thickness as

a function of temperature. As seen in Figure 2, the numerical approach allows the estimation of the depth of the internal oxidation layer. Since aluminium oxidizes preferentially over copper due to its higher affinity for oxygen, the low solubility product and high thermodynamic stability of the formed alumina allows for an easy determination of such internal oxidation layer. However, as has been reported [5,7], the present calculation scheme has to be modified for systems where the solubility product of the oxides formed is relatively high so the alloying element and the oxygen may remain dissolved in the host metal. The model developed in this paper was verified experimentally. Experimental details as well as a discussion of the results obtained are presented below. Experimental High purity copper and aluminium were melted in a graphite crucible under an inert atmosphere to prevent oxygen pick up. Once melted, the alloys were cast into small cylinders. To ensure clean surfaces, before each test every cylinder was pickled in an acidic solution, then rinsed in water and ground with grit paper grade 800. After this procedure, the samples were measured and weighed individually, then placed into a thermal gravimetric analysis unit (Setaram TGA-92), to be oxidized at temperatures up to 1000 ºC in air at a rate of 10 °C/min. The samples used in the TGA tests had a diameter of 0.17 cm and a length of 0.40 cm, within the TGA unit the sample weight was continuously recorded. Once cooled, the samples were weighed again, and then mounted for metallographic examination. In every case the mounted samples were ground with grinding paper (grades 180 to 1200) and polished with chromia to avoid alumina contamination. Discussion To determine how much of the actual experimental weight gain is due to the formation of alumina relative to copper oxide, the alloys after being heated in air to 1000 oC were

subsequently cooled in Ar and then reduced in 80%H2-20%Ar gas at 600 oC. Since at 600 oC hydrogen will not reduce Al2O3, any mass loss is assigned to the reduction of copper oxide. It was found that for the 2 wt% Al alloy, the most oxidized alloy; the mass loss during reduction was nearly identical to the mass gain during oxidation. This means that nearly all the mass gain during the oxidation process was due to Cu2O with basically no Al2O3 formation. This is reasonable since the experimental mass gain of 4.3 wt% corresponds to a dense 70 µm thick Cu2O layer while the maximum calculated by the diffusion of Al to form Al2O3 is only 0.6 µm. This confirms that the penetration of oxygen and the diffusion of copper outwards, are faster than the diffusion of aluminium towards the surface. For alloys with 2.25 wt% Al or more, the mass loss during hydrogen reduction corresponded to between 40 and 50 % of the gain during the previous oxidation. This suggests that during oxidation upon heating to 1000 oC at 10 oC/min, slightly more than 50% of the mass gain is due to oxidation of aluminium. This means that with these alloys during heating in air, the aluminium provides some oxidation resistance but a single dense layer is not formed and mixed oxidation takes place. To confirm and extend the oxidation rates obtained in the TGA apparatus, other tests were conducted with larger samples exposed to air for extended periods of time in a muffle furnace. Figures 3 and 4 show results of these tests at 800 and 1000 oC in air. The results at 800 oC presented in Fig. 3, show that the 1 wt% Al alloy oxidizes at a rate comparable to that of pure copper, while the 2 wt% Al alloy has a rate of oxidation 5 times slower than pure copper. This lower rate appears to contradict the results from the TGA showing that the 2 wt% Al alloy oxidizes at about the same rate as copper. However, a more detailed look at Figure 3 shows that in the early oxidation stage, the 2 wt% Al alloy oxidizes similarly to copper, just as in the TGA. This initial rate increment may be due to the development of the internal oxidation layer. Once such a layer has formed, it seems that the alumina that has precipitated may serve as a diffusion barrier, nearly stopping the further oxidation. Figure 4 confirms that at 1000 ºC the 1, 2 and 2.5 wt% Al alloys no longer are able to prevent the oxidation of copper. Indeed, the rates of oxidation of the 1 and 2 wt% Al alloys are faster than that of pure copper. The 2.5 wt% Al alloy has a rate comparable to that of pure copper, but as more aluminium is added to the copper, the oxidation rate decreases quite drastically. Both the 3 and 4 wt% Al alloys oxidize at a rate three orders of magnitude slower than pure copper. In terms of the oxidation rates, it is the solubility limit of oxygen in copper the limiting parameter to develop or not the internal oxidation layer [3,10]. In terms of the diffusion mechanism, it is the bulk diffusion of oxygen and aluminium into the alloy the one that controls the oxidation rates at the tests temperature. Grain boundary diffusion is observed at temperatures as high as 500 °C [10]. Conclusions A simple one dimensional finite differences approach is able to accurately estimate the depth of the internal oxidation layer in binary alloys. If the alloying element forms a highly stable oxide, the numerical method is enough to predict the extent of the internal oxidation layer. If the oxidation product is not as stable or has a high solubility product, the numerical approach to predict the depth of the internal oxidation layer has to be modified. Experimental results are in excellent agreement with the results predicted with the numerical method.

0.001

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Figure 3. (A) Mass gain versus time during oxidation in the TGA for copper and Cu- Al alloys at 800 ºC in air. (B) Mass gain during oxidation tests in a muffle furnace for copper and Cu- Al alloys at 800 ºC.

Figure 4. (A) Mass gain versus time during oxidation in the TGA for copper and Cu- Al

alloys at 1000 ºC in air. (B) Mass gain during oxidation tests in a muffle furnace for copper and Cu- Al alloys at 1000 ºC.

Acknowledgements. The authors would like to thank CONACyT, COFAA-IPN and NSERC (Canada) for founding this project. References 1. C. Wagner: Z. Elektrochem Vol. 63 (1959), p. 772. 2. F.N. Rhines, W.A. Johnson, W.A. Anderson: Trans. Amer. Inst. of Mining and

Metallurgical Engs. Vol. 147 (1942), p. 205. 3. G. Plascencia, T. Utigard: Corr. Sci. Vol. 47 (2005), p. 1149. 4. F. Gesmundo, P. Castello, F. Viani, C. Roos: Oxid. Met. Vol. 49 (1998), p. 237. 5. D. Huin, P. Flauder, J.B. Leblond: Oxid. Met. Vol. 64 (2005), p. 131. 6. A. Katsman, H.J. Grabke, L. Levin: Oxid. Met. Vol. 46 (1996), p. 313. 7. H.G. Sockel, H.J. Christ: Mat Sci Eng Vol. 87 (1987), p. 119. 8. B. Carnahan: Applied Numerical Methods (Wiley, USA 1961). 9. D.R. Askeland: The Science and Engineering of Materials (PWS Publishing Co, USA

1993). 10. G. Plascencia: Ph. D. dissertation (University of Toronto, Canada 2004).

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