In a chemical processing plant, an acidic liquid requires heating before entering the reactor. Shell and tube heat exchanger has been decided to be the most suitable type of heat exchanger to achieve the desired heating. In the heat exchanger, the acidic liquid is expected to flow in unfinned tubes and $($:$D_0=11(mm)\},$ while the hot gas flows on the shell side. The inner and outer heat transfer coefficients are $h_i=1500\frac{W}{m^2}*K$ and $h_0=200\frac{W}{m^2}*K,$ respectively.

As a process engineer who is involved in the designing of the heat exchanger, you need to decide the material of construction for the tubes. You are given with two options. The first option is Ni$-$Cr$-$Mo corrosion resistance metal alloy which has the density of $8900k\frac{g}{m^3}$ and thermal conductivity of $8\frac{W}{m}*K.$ The second choice of material is polyvinylidene fluoride $($PVDF$)$ plastic which is known to withstand a wide range of temperatures and chemicals. The plastic has a density of $1780k\frac{g}{m^3}$ and thermal conductivity of $0\mathrm{.}17\frac{W}{m}*K$ it is also known that the cost of the metal alloy is thrice the cost of the plastic.

a$.$ Determine the ratio of plastic to metal surface areas $(\frac{A_p}{A_m})$ required to transfer the same amount of heat.

b$.$ Using the information that you have calculated in $($a$)$ and considering both materials can withstand the hot gas excellently, suggest which material is best for the tubes. Using appropriate calculation, justify your selection