Can you answer part $($c$)$ and $($d$)$

Ouestion $4$

Ni$-$based superalloys are used for a range of applications for temperatures in excess of $1000$ K $,$ with large concentrations of chromium added to improve oxidation resistance.

$($a$)$ Using the Hume$-$Rothery rules, comment on the solubility of Cr in Ni alloys if the atomic radius is $0\mathrm{.}149$ nm for fcc Ni and $0\mathrm{.}166$ nm for bcc Cr $.$

Inconel $600$ is a candidate material for metallic interconnects in solid oxide fuel cells $($SOFC$)$ due to its excellent high temperature mechanical performance and oxidation resistance. The nominal chemical composition $($in wt$.\backslash \%)$ of Inconel $600$ is presented in Table Q$4$ and the equilibrium phase diagram is presented in Figure Q$4.$

$($b$)$ Briefly comment on the potential advantages and disadvantages of the addition of alloying elements, including the potential role of trace and tramp elements in high temperature and harsh environments and the role of the Materials Engineer in the design of such materials.

$($c$)$ The Inconel $600$ material undergoes service in the SOFC at $970$ K $.$ Determine the $\backslash (\backslash$mathrm$\{$M$\}\_\{7\}\backslash$mathrm$\{$C$\}\_\{3\}\backslash )$ precipitate radius after $2,800$ hours of service, for an initial radius of $20$ nm prior to service. Assume that the carbide coarsening rate is $\backslash (2\backslash$times $10^\{-16\}\backslash$mathrm$\{$~mm$\}^\{3\}\backslash$cdot $\backslash$mathrm$\{$hr$\}^\{-1\}\backslash ).$

$($d$)$ Determine the percentage decrease in precipitate strengthening during service, using the Orowan looping model. The shear modulus of Inconel $600$ can be taken as $49$ GPa and magnitude of Burgers vector as $0\mathrm{.}249$ nm $.$ Assume an even distribution of carbides throughout the microstructure, with a mean particle spacing defined as:

$\backslash [$

$\backslash$lambda$=$r$\backslash$left$(\backslash$frac$\{4\backslash$pi$\}\{3$ f$\}\backslash$right$)^\{1/3\}$

$\backslash ]$

where $\backslash ($ f $\backslash )$ is precipitate volume fraction and $\backslash ($ r $\backslash )$ is mean precipitate radius.

$($e$)$ Comment on the assumptions made regarding the mean spacing of precipitates in this model.

Table Q$4$: Chemical composition $($in wt$.\backslash \%)$ of Inconel $600.$

Figure Q$4$: Phase equilibrium diagram for Inconel $600.$