PART A: Assume that you are the chief engineer in charge of polystyrene cylindrical rod extrusion unit.

Your company is producing these PS cylindiral rods with a length of $3m$ and diameter of $5mm.$ Normally,

your processing temperature for this operation is at $160C$ where the viscosity is $1\mathrm{.}5{10}^3$ poises for

polystyrene whose weight average degree of polymerization $(D{P}_w)$ is $4000.$ However, the company that

provides you the raw material is out of stock this week. Yet you have to continue your production in

order to fulfill your supply to your customers. An engineer under your supervision suggests to use the

PS in the company's inventory. However you realized that there is not enough one type monodisperse

polystyrene to satisfy the demand and you decided to mix them in appropriate amounts that will provide

a MW good enough for the production. In order to determine the right conditions, a polydisperse sample

of polystyrene is prepared by mixing three monodisperse samples to produce cylindrical plastic rods via

a lab$-$scale mini extruder in the following proportions.

Extrusion is planned to be performed with a shear rate ${}^{}=35s^{-1}$ with the melt viscosity $1\mathrm{.}5{10}^3$ Poise

which shows almost Newtonian behavior. The dependence of the melt viscosity of polystyrene to weight

average molecular weight is given by

${}_0=KL(2DPw{)}^{1\mathrm{.}0}$ where $D{P}_w<mn\; id="EditorElement\_3764943">3</mn><mn\; id="EditorElement\_3764944">0</mn><mn\; id="EditorElement\_3764945">0</mn>$ and ${}_0=K{H}_H(2DPw{)}^{3\mathrm{.}4}$ where $D{P}_w>300$ where $KL$ and $KH$ are

constants for low and high degrees of polymerization.

The glass transition temperature of PS is given as $T_g=100C.$ Williams$-$Landel$-$ Ferry $($WLF$)$ relationship

constants $C_1$ and $C_2$ are respectively $13\mathrm{.}7$ and $50\mathrm{.}0$ for polystyrene. What will be the appropriate

processing temperature for this mixture of PS in order to have the same melt viscosity as before?

What must the temperature be to reduce the same viscosity ${}_0$ of this PS by one half? Eps$=24\mathrm{.}887$

kca$\frac{l}{m}ol.$