Linear Regression Modelling

After all of the significant factors, and interactions have been identified in the factorial design, the data can be fit to a regression model

Y = β0 + β1x1 + β2x2 +...+ βkxk + ϵ
(37)

Where Y is the response variable, xk are the independent variables, or factors, βk are the unknown parameter coefficients, and ϵ is the error term. Determining the β coefficients will allow for us to describe the relationship between independent variables, and the response variable (dependent variable) through an approximate function. Since there are multiple phases that compose the 2032Nb system, a set of regressions functions will need to be determined to appropriately describe the system. After these functions have been approximated, a model for optimizing the system can be drawn, either through statistical or linear programming methods.

Using linear regression models to fit the factorial design data, assumes that the data fits linearly with the response variable. This assumption neglects any significant interactions that were uncovered in the factorial design, as they would be regarded as polynomial terms. However, a linear model may still be used by approximating these polynomial terms as new variables. For example if the significant terms in the regression model were Si,Nb, and Si × Nb, the regression function would be Y = β0 + β1Si + β2Nb + β3NbSi + ϵ. If we let x3 = NbSi this modifies the equation from containing two independent variables to incorporating three independent variables, and can now be considered a linear function.

The linear coefficient can be solved for using the Least Squares Method described in the next section. The regression function should then be tested for significance, and how well if fits with the original data. The easiest way to compare the fit of the approximated function is to calculate the coefficient of multiple determination , or R2 value. R2 is calculated as

R2 = 1- SSE-
        SST
(38)

The R2 value will be a fraction of how much of the model accounts for the variability in the original data. For example if R2 = 0.95, 95% of the variability of the response data is accounted for in the regression model. There are some short-comings of the R2 value as it keeps improving as more terms are added to the model. This is compensated for in the adjusted R2 value, but for the purposes of this study only the regular R2 value will be reported.

1.2. Least Squares Method

The least squares method is an effective method in solving a system of linear equations, and is used in both regression modelling, and fitting experimental data to Gibbs energy models. If a set of linear equations are described as

         k
y = β  + ∑  βx  + ϵ ∀i ∈ {1,2,...,n}
 i   0   j=1  j ij   i
(39)

Where xij is the ith observation of the k total independent variables xj, yi is the dependent variable of observation i, βj is the coefficient for the jth term, and ϵ is the error between the calculated ŷi and the measured yi values [26]. The best fits for the βj coefficients are determined by the minimization of the least squares function,

                      (                )
            ∑n     ∑n           ∑k       2
minw.r.t.βj =   ϵ2i =   (yi - β0 -   βjxij)
            i=1    i=1          j=1
(40)

which can be expressed as

∑n    ∂ϵi
   ϵi ⋅∂β = 0∀j ∈ {1,...,k}
i=1     j
(41)

Rearranging for yi a set of n equations can be solved for providing the best estimates for βj.

2. References

[1]    Shi, S., Lippold, J.. Microstructure evolution during service exposure of two cast, heat-resisting stainless steels – hp-nb modified and 20-32nb. Mater Charact 2008;59(8):1029–1040.

[2]    Berghof-Hasselbcher, E., Gawenda, P., Schorr, M., Schtze, M., Hoffman, J.. Atlas of Microstructures. Materials Technology Institute; 2008.

[3]    Nishimoto, K., Saida, K., Inui, M., Takahashi, M.. Changes in microstructure of hp-modified heat-resisting cast alloys with long term aging. repair weld cracking of long term exposed hp-modified heat-resisting cast alloys. (report 2). Quarterly Journal of the Japan Welding Society 2000;18(3):449–458.

[4]    Nishimoto, K., Saida, K., Inui, M., Takahashi, M.. Mechanism of hot cracking in haz of repair weldments. repair weld cracking of long term exposed hp-modified heat-resisting cast alloys. (report 3). Quarterly Journal of the Japan Welding Society 2000;18(4):590–599.

[5]    Powell, D.J., Pilkington, R., Miller, D.A.. The precipitation characteristics of 20% cr/25% ni—nb stabilised stainless steel. Acta Metall 1988;36(3):713–724.

[6]    Chen, Q.Z., Thomas, C.W., Knowles, D.M.. Characterisation of 20cr32ni1nb alloys in as-cast and ex-service conditions by sem, tem and edx. Mater Sci Eng, A 2004;374(1-2):398–408.

[7]    Danielsen, H.K., Hald, J.. On the nucleation and dissolution process of z-phase cr(v,nb)n in martensitic 12%cr steels. Mater Sci Eng, A 2009;505(1-2):169–177.

[8]    Danielsen, H., Hald, J.. Influence of z-phase on long-term creep stability of martensitic 9-12%cr steels. In: 9th Liege Conference on Materials for Advanced Power Engineering. 2010, p. 310.

[9]    Sourmail, T., Bhadeshia, H.. Microstructural evolution in two variants of nf709 at 1023 and 1073 k. Metall Mater Trans A 2005;36(1):23–34.

[10]    Erneman, J., Schwind, M., Liu, P., Nilsson, J.O., Andrén, H.O., Ågren, J.. Precipitation reactions caused by nitrogen uptake during service at high temperatures of a niobium stabilised austenitic stainless steel. Acta Mater 2004;52(14):4337–4350.

[11]    Pickering, F.B., Keown, S.. Niobium in stainless steels. In: Stuart, H., editor. Niobium, Proceedings of the International Symposium. Warrendale, PA: Met. Soc. AIME; 1981, p. 1113–1142.

[12]    Sourmail, T.. Literature review precipitation in creep resistant austenitic stainless steels. Mater Sci Technol 2001;17(January):1–14. URL http://www.thomas-sourmail.org/papers_html/precipitation_review/precipitation_review.pdf.

[13]    Xiao, B., Xing, J.D., Feng, J., Zhou, C.T., Li, Y.F., Su, W., et al. A comparative study of cr 7 c 3 , fe 3 c and fe 2 b in cast iron both from ab initio calculations and experiments. Journal of Physics D: Applied Physics 2009;42(11):115415.

[14]    Jo, T.S., Lim, J.H., Kim, Y.D.. Dissociation of cr-rich m23c6 carbide in alloy 617 by severe plastic deformation. J Nucl Mater 2010;406(3):360–364.

[15]    Holman, K.L., Morosan, E., Casey, P.A., Li, L., Ong, N.P., Klimczuk, T., et al. Crystal structure and physical properties of mg6cu16si7-type m6ni16si7, for m = mg, sc, ti, nb, and ta. Mater Res Bull 2008;43(1):9–15.

[16]    Hans Lukas, S.G.F., Sundman, B.. Computational Thermodynamics. Cambridge University Press; 2007.

[17]    Hillert, M.. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis. Cambridge University Press; 2007.

[18]    Liu, Z.K.. Computational thermodynamics using thermo-calc; 2010. ThemoCalc & Dictra Training Course.

[19]    Song, Y.Y.. Thermodynamic study on b and fe substituted cr23c6 using first-principles calculations. Ph.D. thesis; Pohang University of Science and Technology; 2010.

[20]    Hoffman, J., Magnan, J.. Cast 20cr32ni1nb alloy aged mechanical property improvements via chemistry modifications. In: Corrosion 2003. no. 3469; NACE International; 2003,.

[21]    Vitek, J.. G-phase formation in aged type 308 stainless steel. Metallurgical and Materials Transactions A 1987;18(1):154–156.

[22]    Danielsen, H.K., Hald, J.. A thermodynamic model of the z-phase cr(v, nb)n. Calphad 2007;31(4):505–514.

[23]    Shi, S., Lippold, J., Ramirez, J.. Hot ductility behavior and repair weldability of service-aged, heat-resistant stainless steel castings. Weld J 2010;89(10):210–217.

[24]    Shibasaki, T., Mohri, T., Takemura, K.. Experience with cast material for steam reformer furnaces. Ammonia Plant Saf Relat Facil 1994;34:166–176.

[25]    Hoffman, J.. High temperature aging characteristics of 20cr32ni1nb castings. In: Corrosion 2000. no. 512; NACE International; 2000,.

[26]    Montgomery, D., Runger, G.. Applied Statistics and Probability for Engineers, 4th Edition. John Wiley & Sons; 2006.

[27]    Spliid, H.. Design and analysis af experiments with k factors having p levels; 2002.

[28]    Minitab, . Technical support document rank deficiency; 2010. http://www.minitab.com/support/documentation/Answers /RankDeficiency.pdf.

 

3.Glossary *

constituent
Element, or species, that occupies a specific sublattice of a specific phase. A phase can also be considered as a constituent of the total system.. 5, 14

 

end-member
The final chemical formula of a stable or metastable phase whose sublattice(s) are occupied by single constituents. For example M23C6 is an end member of (Cr,Ni,Fe,Nb)23C6.. 8

 

factor
The independent variable of a factorial design. 23

 

main effect
How much the change in an individual factor effects the change in the response variable of a factorial design.. 24

 

replicate
Independent repetition of a treatment in a factorial experiment. 26
response variable
The dependent variable of a factorial experiment, or a regression model.. 24

 

ThermoCalc
A computational thermodynamics program that can calculate equilibrium phase diagrams for multicomponent systems, as well as Scheil simulations, and various thermodynamic properties (Cp, ΔHm, ΔGm etc...). 3
treatment
A specific level of a factor in a factorial design.. 24

4. Acronyms *

ANOVA
Analysis of variance. 24, 25

 

CEF
Compound-energy formalism. 8

 

LRO
Long Range Ordering. 8

 

5. Nomenclature *

βj
the effect of the ith level of factor ‘B’
ϵijk
random error component
for all instances of ...
in a set ...
μ
Overall mean effect
μi
Chemical potential of component or end-member i
ψk
the effect of the ith level of factor ‘C’
τi
the effect of the ith level of factor ‘A’
G
total Gibbs energy; G = αmα ·Gmα
Giα
partial Gibbs energy of component i in phase α; Giα = (  α)
 ∂∂GNi-T,P,Nj
Gmα
integral molar Gibbs energy of a phase
Ii
constituent array of order i
LI
interaction parameter of compound I
mα
fraction of a phase
Ni
moles of component i
R
gas constant, 8.314Jmol-1K-1
R2
coefficient of multiple determination
Smα
molar entropy of a phase
T
Temperature (K)
xi
total mol fraction of component i; xi = αmα · xiα
xiα
mole fraction of component i in phase α

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