Problem $1(3\mathrm{.}5$ points$).$ You are a chemical engineer to develop to alloy materials composed of A and

B metals. Your team is about to develop the $1$:$1$ volumetric alloy of A and B metals to form cylindrical

heat conductor $($length: $L=5m)($i$.$e$.,$ inner radius: $r_1=0\mathrm{.}1m,$ outer radius: $r_2=2r_1).$ Inside the

cylinder is hot air at $T_1=400K,$ while the outside of the cylinder is exposed to cold air at $T_2=300K.$

The two metals have the same density, the same thermal expansion coefficient, but different heat

conductivities $($i$.$e$.,k_A=0\mathrm{.}1\frac{W}{m*K}$ and $k_B=2k_A).$ For the conductor, you can consider that

there is orientational symmetry around the axis and the cylindrical conductor length is long enough. All

the physical constants are assumed to be fixed. For this problem, you can assume that the inner surface

temperature and outer surface temperature are equal to the

respectively.

The cylinder can be considered not being thermal expanded. Now, please address following questions.

$1\mathrm{.}1(1\mathrm{.}5$ points$).$ As references samples, your team has a plan to make two samples; the first one is A

for the inner and B for the outer, and the second one is vice versa. Given the equal volumetric fractions

of A and B metals in the conductor, please calculate the overall heat transfer rates for the sample $1(Q_1)$

and sample $2(Q_1),$ respectively.

$1\mathrm{.}2(1\mathrm{.}5$ points$).$ To test the alloy composition, one of your colleagues suggests a linear composition.

To make $1$:$1$ composition of A and $B$ metals inside the cylinder, please design the composition profile

as a function of the radial direction $(r).$ Given the fact that the heat conductivity is proportional to the

composition of A metal, please calculate the overall heat fluxes for the case $1$ in which A is $100\%$ at the

inner surface, while B is $100\%$ at the outer surface and case $2$ in which the composition is vice versa of

the case $1.$