$(20$ points$)$ see Figure below $-$ To reduce the weight of a prototype vehicle, you are using

titanium for a chassis element. In order to meet a deadline and reduce costs, you are considering

moving a fabrication process to the foundry $-$ IF your analysis shows that the material will still be

hot enough to form after it spends $2$ min cooling on a foundry material handling system. Assume

$1$ D with constant properties at $300$ K tp model the transient cooling.

a$.$ Confirm that lumped capacitance is not valid for the Titanium slab.

b$.$ Determine the centerline temperature after $2$min$[C].$

c$.$ Determine the surface temperature after $2$min$[C].$

d$.$ If the centerline temperature cannot dip below $600C,$ would reducing the cooling time to

$1$ min achieve this goal? Ensure you recalculate the Fourier number $(F_0)$ in this analysis.

Titanium slab

$4$ cm thick, $2$ m long, $1$ m wide

$T(0)=1425C($initial temperature, uniform$)$

$T_{}=20C,h_{}=900\frac{W}{m^2}*K$