Scenario #$3$: Nutrient medium is to be heated from $10C$ to $28C$ in a single$-$pass countercurrent shell$-$and$-$tube heat exchanger before being pumped into a fed$-$batch fermenter. Medium passes through the tubes of the exchanger; the shell$-$side fluid is water that enters with flow rate $3{10}^{4}kg{h}^{-1}$ and temperature $60C.$ Preheated medium is required at a rate of $50m^3{h}^{-1}.$ The density, viscosity, and heat capacity of the medium are the same as water; the thermal conductivity of the medium is $0\mathrm{.}54W{m}^{-1}{C}^{-1}.$ It is proposed to use $30$ steel tubes with an inner diameter of $5cm$; the tubes will be arranged in line. The pipe walls are $5mm$ thick; the thermal conductivity of the metal is $50W{m}^{-1}{C}^{-1}.$

$($a$)$ What is the rate of heat transfer?

$($b$)$ Calculate the log$-$mean temperature difference.

$($c$)$ Determine the heat transfer area if overall heat transfer coefficient is $0\mathrm{.}59kJ{s}^{-1}{m}^{-2}{C}^{-1}.$

$($d$)$ What tube length is required?