ON DETERMINATION OF THE DIFFUSION LAYER AND PENETRATION OF SUBSTITUTIVE ELEMENT IN WELDED JOINT OF TWO DIFFERENT STEELS

Transkript

1 Acta Metallurgica Slovaca,, 006, 4 ( ) 399 O DETERMIATIO OF THE DIFFUSIO LAYER AD PEETRATIO OF SUBSTITUTIVE ELEMET I WELDED JOIT OF TWO DIFFERET STEELS Řeháčková L., Kalousek J., Dobrovská J., Stránský K. FMME VŠB TU Ostrava, Czech Republic, lenka.rehackova@vsb.cz FME VUT Brno, Czech Republic KE STAOVEÍ DIFÚZÍ VRSTVY A PEETRACE SUBSTITUČÍCH PRVKŮ VE SVAROVÉM SPOJI DVOU RŮZÝCH OCELÍ Řeháčková L., Kalousek J., Dobrovská J., Stránský K. FMMI VŠB TU Ostrava, Česká republika, lenka.rehackova@vsb.cz FSI VUT Brno, Česká republika Abstrakt Článek představuje výpočet difúzní vrstvy a penetrace substitučních prvků ve svarovém spoji dvou různých ocelí, výpočet je založen na výsledcích měření koncentračních profilů. Oceli jsou strukturně stejného typu feritické a z hlediska chemického složení představují polykomponentní systém. Základní výpočetní metoda je založena na řešení nestacionární difúze podél přímky, jak v kladném, tak záporném směru. Výchozím bodem je druhý Fickův zákon. Vstupní data byla získána experimentálním měřením pomocí disperzní spektrální mikroanalýzy. Hodnocení koncentračních profilů na obou stranách svarového spoje bylo provedeno numerickými metodami. Byl použit optimalizační model založený na nelineární regresi. Levenberg-Marquardtova metoda byla upravena a odvozena pro tento účel. Parametry modelu, tj. koeficient difúze, efektivní koncentrace substitučního prvku ve velké vzdálenosti od svarového spoje, stejně jako nepřesnost v umístění kontaktu obou kovů byly odhadnuty. Výstupy zahrnují také statistické výsledky vyjadřující chyby zmíněných veličin. Postup je demonstrován numericky i graficky na difúzi chrómu. Získané výsledky mohou být potenciálně využity pro praktické informace o heterogenitě chrómu v daném nebo podobném typu svarového spoje. Je nutné zdůraznit, že pojetí tloušťky difúzní vrstvy a penetrace závisí na vhodné definici, což je důležité pro srovnání chování různých substitučních prvků. Tento článek je úzce vztažen k předchozí publikaci zabývající se stejným problémem pro prvek křemík. Abstract The article presents the calculation of diffusion layer and penetration of substitutive elements in welded joint of two different steels, based on results of measurement of concentration profile. Steels have identical structure of ferritic type; from the viewpoint of chemical composition they represent a polycomponent system. Fundamental calculation method is developed on the solution of unsteady diffusion along the straight line, both in positive and negative direction. The starting point is the second Fick s law. The input data have been obtained by experimental measurements using the method of energy dispersive (ED) X-ray microanalysis. The evaluation of concentration profiles on both sides of welded joint has been performed by means of numerical method. on-linear regression algorithm optimising model parameters had to be used. The Levenberg Marquardt s method has been adapted and developed for this purpose. The parameters of the model, i.e. coefficient of diffusion, the effective concentrations of substitutive element in great distance from the welded join as well as

2 Acta Metallurgica Slovaca,, 006, 4 ( ) 400 the inaccuracy in location of the contact of both metals have been estimated. The output comprises also statistical results expressing the errors of the above mentioned quantities. The procedure is demonstrated numerically and graphically on diffusion of chromium. Obtained results can be potentially used for practical information about chromium heterogeneity in the given or similar type of welded joint. It is necessary to emphasize that the conceptions of the thickness of diffusion layer and the penetration depend on the appropriate definition, what is important for comparison of different substitutive elements. This paper is close related to the former publication dealing with the same problem of silicon. []. Keywords: diffusion, diffusion layer, modelling, welded joint of steels.. Introduction Welded joints of steels with increased contents of alloying elements are almost always accompanied by higher chemical and structural heterogeneity than initial materials, from which the given joint was made of. High chemical and structural heterogeneity is characteristic particularly for transition zone of the welded joint. High chemical heterogeneity of constitutive and admixture elements, which is bound to a transition zone of the welded joint and is acompanied usually with high structural heterogeneity, has strongly statistical impact on structural stability of the welded joint. Unfavourable impacts of chemical and structural heterogeneity manifest themselves in technical practice mostly in welded joints of refractory steels, which are exploited for a long time under high temperatures. There have been recorded cases of catastrophic failures of steam piping of power plants, which had been caused namely by high structural instability of transition zones of welded joints.. Experiment The welded joint formed by refractory low-alloyed steel CriMoV (marked as P) and low-alloyed silicon steel (steel of the type 80C.34Si, (marked as V) was used as experimental material. Chemical composition of the steels is given in Table. The welded joint was given the heat treatment (diffusion annealing) of different time in the temperature interval C. Marking of samples and regime of their heat treatment is given in the first column of Table. Table Chemical composition of steels of welded joint [weight %] Steel C Mn Si P S Cr i Cu Mo V Al Fe P rest V rest 3. Theoretical basis 3. Model of diffusion Model in the system of welded joint made of steels P/V of structural type ferrite/ferrite - is based on solution of the nd Fick s law of semi-infinite one-dimensional diffusion, (0 < x < ), ( < x < 0). The result valid for both sides of the welded joint has the form:

3 = ( 0, t) Acta Metallurgica Slovaca,, 006, 4 ( ) 40 x,. () Dt ( x t) = + 0,5( ) erfc Basic solution of problem can be found in Ref. [], and also with respect to analogy between diffusion and heat conduction in Ref. [3]. Final form of eq() can be found in Refs [4,5]. The values,, substitute concentration in weight percentages,, correspond theoretically to initial concentrations at time t = 0, t [s] time, x [µm] coordinate, D [cm /s] diffusion coefficient. Boundary condition expresses the equality of diffusion flux density at x=0. It is presumed that diffusion coefficient in both zones is independent on concentration. If D =D, the concentration profile for t > 0 at x = 0 is continuous. This assumption is used in the following calculation. 3. Efficient thickness of diffusion layer and estimation of depth of penetration. The values of the depth of penetration and the thickness of diffusion layer, used in the area of transport phenomena of heat conduction and diffusion are not always defined in uniform manner. Their values depend on manner of definition. Solution of semi infinite unsteady diffusion for constant boundary condition at x = 0: = (0,t) and initial condition: =( x,0) has the following form: ( x, t) = erf x Dt. () This is valid for both sides of the welded joint ((0 < x < ) or ( < x < 0)), if they are considered separately. Depth of penetration then is then: = δ 4 Dt. (3) This equation follows from analogy with heat conduction according to Ref. [6], assuming the value of complementary error function approximately The consequence of used boundary condition is that the concentration for any time t>0 at x=0 is arithmetic mean of initial concentration,. Therefore is possible to regard the diffusion into the diffusion region with lower concentration as solution of the nd Fick s law with constant, above mentioned boundary condition, with result: ( x, t) = erf x Dt. (4) If the left side of the equation (4) should also be equal to according to analogy with the equation (), it is then possible to derive from the course of the error function the following relation for the depth of penetration: = δ 3.70 Dt (5)

4 Acta Metallurgica Slovaca,, 006, 4 ( ) 40 The efficient thickness of diffusion layer δ for a semi infinite solid is often determined as an intersection of a tangent at a point x=0 with an asymptote of concentration profile for x. Concentration profile in the close proximity of the welded joint is very steep (Fig. ). As the function () for D = D is continuous at x = 0, is then possible to consider its linearization and to derive the efficient thickness of diffusion layer from differentiation of the function () in a point x = 0, substituting the differentials d and dx by their finite differences - and δ: δ =. (6) d [ ( x, t) ] x= 0 dx It follows from the equation (6): d dx ( x, t) It follows from the equations (6), (7): x =. (7) Dt ( π ) / ( ) ( ) exp / πdt δ = Dt. (8) Significance of the relation (8) can be implied also from the Fig.. In the zone between the coordinates (x, x ), the distance of which is δ, occurs the extensive change of concentration of diffusing element. 3,5 3,5 Cr [wt.% ],5 0,5 0 x ,5 δ - -,5 x [0-6 m ] Fig. Estimation of thickness of diffusion layer (sample 0 (900 C/8 h)) x

5 Acta Metallurgica Slovaca,, 006, 4 ( ) 403 It can be proven from the equations (), (8) and from boundary condition, that coordinates x, x are for t = const. symmetrical in respect to the axis of concentrations. Due to the fact that (7) is related only to semi infinite region ((0 < x < ), ( < x < 0)), it is possible to make a comparison with only half value of the thickness of diffusion layer expressed by eq. (8): δ π 0.5 = δ 3.70 (9) Then eq.(9) represents the relation of results of efficient thickness of diffusion layer δ and penetration depth δ according to envisaged definitions. Calculation of δ is given in the last column of the table. Table Chromium concentration in the steel P, V, its diffusion coefficients and thickness of diffusion layer Sample o.(t[k], t[s]) 0 [wt.%] 0 [wt.%] 0 5 D Cr [cm /s] δ [µm] sample mean 4.3 ± ±.5.5 ± (500 C, 860h) mean 4.6 ± ± ± sample mean 6. ± ± ± (550 C, 59h) mean 6. ± ± ± sample 3 mean.8 ± ± ± (600 C, 383h) mean.7 ± ± ±. 3.0 sample 4 mean 6.4 ± ± ± (650 C, 80h) mean 5.9 ± ±.0.4 ± sample 5 mean 56.0 ± ± ± (700 C, 49h) mean 5.9 ± ± ± sample 6 mean 55.7 ± ± ± (700 C, 40h) mean 56.4 ± ± ± sample 7 mean 5.9 ± ± ± (750 C, 7h) mean 5.9 ± ± ± sample 8 mean 8. ± ±.7 4. ± (800 C, 30h) mean 8. ± ± ± sample 9 mean 4.9 ± ± ± (850 C, 6h) mean 4. ± ± ± sample 0 mean.5 ± ± ± (900 C, 8h) mean.4 ±. 8.8 ± ± sample mean 0.4 ± ± ± (950 C, 4h) mean 9.7 ±. 5. ± ± sample mean 6.8 ± ± ± (000 C, h) mean 6.8 ± ± ± ote: Mean is the value determined as the average from calculated values for areas 3. Mean is the value calculated from the averaged measured concentrations.

6 Acta Metallurgica Slovaca,, 006, 4 ( ) Conclusion The paper presents original results of methodology for evaluation of the experimental data for calculation of diffusion layer and penetration of diffusing elements in the welded joint. This methodology is illustrated by examples for chromium in steels (polycomponent system of elements). Behaviour of diffusion layer and penetration depth of elements in steels forming the envisaged welded joint has not only theoretical, but also practical significance. Both quantities change generally with time and temperature, by virtue of diffusion coefficient expressed by Arrhenius equation. This is of importance for steam piping tubes of classical and nuclear power plants, which are in the long run exposed to thermal loads, both static under normal operation, and dynamic during individual operational breaks. Acknowledgement This work was created under the support of the project of the Ministry of Education, Youth and Sports of the Czech Republic, o. MSM The authors would like to thank Ing. Antonín Rek, CSc. for the accomplished concentration measurements. Literature [] Řeháčková L., Kalousek J., Dobrovská J., Stránský K.: Zeszyty naukowe, Mechanika o 38/006, pp [] Crank J.: The Mathematics of Diffusion, Oxford University Press, nd Ed., London, 975, pp [3] Carlslaw H. S., Jaeger J. C.: Conduction of Heat in Solids, nd Ed., Oxford Clarendon Press, London, 959. [4] Řeháčková L.: et al. On methodology of concentration data processing at mathematical modelling of substitution element diffusion in the zone of welded joint of steels. Acta Metalurgica Slovaca (in print) [5] Pilous V., Stránský K.: Structure stability of weld deposits and welded joints in energetic engineering (in Czech), Academia Praha, 989. [6] Bird R. B., Steward W. S., Lightfoot E. L.: Transport phenomena, John Wiley and Son Inc., 6 th Ed., ew York, 965, p. 354.