$1.($a$)$ Write a brief technical note on the lumped system analysis method for transient heat transfer applications when internal conductive resistance is negligible.

$($b$)$ It is desired to calculate the transient temperature of chicken eggs after they are placed in an incubator maintained at $38$C$.$ The natural convection heat transfer coefficient for air over the eggs in the incubator is $5\mathrm{.}2$ W m$-2$C$-1.$ Using the Lumped Systems Analysis method, calculate the temperature of the eggs after one hour in the incubator if their initial temperature is $20$C$.$ Assume the eggs to be spherical with radii of $0\mathrm{.}025$ m$.$ The density, specific heat and thermal conductivity of the eggs are $1,035$ kg m$-3,3,350$ J kg$-1$C$-1$ and $0\mathrm{.}62$ W m$-1$C$-1$ respectively.

$2.($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 $-1$