QUESTION $2$

$1000kg$ of fruit juice containing $5\%$ dry matter is concentrated to $25\%$ dry matter in single stage evaporator. The entering temperature of fruit juice is $28C,$ while the pressure in the vapor space of evaporator is $0\mathrm{.}25$ bar and the steam supplied is saturated at $2\mathrm{.}70$ bar. Before packaging process the concentered fruit juice temperature is raised to $90C$ in a jacketed $1\mathrm{.}5m$ diameter agitated vessel without baffle in order to ensure food safety. The agitator is $0\mathrm{.}5m$ in diameter and is a flat$-$blade turbine rotating at $60rpm.$ Heating in the vessel is applied only by side surface. If the inner surface temperature of vessel is greater than $95C,$ a scrape should be used to avoid deposition on the surface.

$($The density, specific heat, thermal conductivity, viscosity of fruit juice is $1045k\frac{g}{m^3},4\mathrm{.}05k\frac{J}{k}g*K,0\mathrm{.}562\frac{W}{m}*K,0\mathrm{.}894$mPa$*s,$ respectively. Assume that these physical properties of fruit juice are not changed with temperature.$)$

a$)$ Calculate amount of steam used in evaporator. $(17$ points$)$

b$)$ Prove mathematically that a scrape should be used or not. $(18$ points$)$

You must SHOW ALL CALCULATIONS IN DETAIL and use correct mathematical symbol representation.