$2.($a$)$Write a brief technical note on the main factors influencing the rate of fouling during thermal processing in dairy processing applications. Briefly outline one technique which can be employed for on$-$line detection of fouling during processing. Illustrate your answer with neat sketches where appropriate.

$($b$)$A double$-$pipe $($shell and tube$)$heat exchanger illustrated in the schematic below is constructed from a stainless steel tube $($inner diameter $0\mathrm{.}015$m$,$outer diameter $0\mathrm{.}019$m$)$and an outer shell $($inner diameter $0\mathrm{.}032$m$).$The convection heat transfer coefficients are $800$and $1,200$W m$-2$C$-1$on the inside and outside surfaces of the tube respectively. For fouling factors of $0\mathrm{.}0004$m$2$C W$-1$and $0\mathrm{.}0001$m$2$C W$-1$on the inside and outside surfaces of the tube respectively, determine $($i$)$the thermal resistance of the heat exchanger per unit length and $($ii$)$the overall heat transfer coefficients based on the inner and outer surface areas of the tube respectively.

Data: Thermal conductivity of steel $15\mathrm{.}1$W m$-1$C $-1$