A special product in a food plant is a non$-$Newtonian fluid which behaviors changing with temperature due to partial micro scale crystallization, as described bellow:

a$)$ At room temperature it shows Newtonian behavior with constant viscosity of $80$ centi$-$Poise $($cP$).$

b$)$ At lower temperature it acts like a non$-$Newtonian fluid $($dilatant$)$ where the flow index of $"n=1\mathrm{.}4"$ at a share rate of $150s^{-1},$ while apparent viscosity is $80$ Poise.

c$)$ At higher temperature it acts like a pseudoplastic non$-$Newtonian fluid with flow index of $"n={0\mathrm{.}6}^s$ at the same shear rate with apparent viscosity of $60cP.$

The product is to be delivered from the reactor to a storage tank which is $200m$ away $($Horizontally$).$ The pipe is Nominal $=2$ inches with Sch$.$ # $40.$ There are four $90$ Deg. elbows and two gate valves. Calculate the power requirement by a centrifugal pump for the delivery of the product at the three conditions. Assume all pipes are at ground level, hydrostatic pressure is negligible. Compare and contrast the results and discuss the differences. The pump efficiency may be assumed to be $75\%.$ There is no cavitation as the boiling point if the product is much higher the delivery temperature.

Make your assumption if you need $($please justify$).$

The pressure drop and power can be calculated using the following equation:

$P_f=(\frac{fL}{D,}+K,)\frac{V^2}{2}$