Q$2$: A brewery company in Shah Alam, Malaysia has received a batch of barley for new beer product formulation and the barley stock is to be discharged out from a hopper to the subsequent unit operation. As the process engineer of this company, you are being tasked to design the hopper for particle transportation. Answer the following questions:

$($a$)$ According to the laboratory results, the particle size distribution of the barley stock is shown in Table $2.$ Estimate the particle mean diameter of the barley stock using MATLAB with appropriate mean$/$average model. Justify the selection of the model used.

Table $2$: Particle size distribution of newly received barley.

$\backslash$table$[[$Diameter $($mm$),3\mathrm{.}2,4\mathrm{.}8,6\mathrm{.}6,9\mathrm{.}3,13\mathrm{.}7,27\mathrm{.}3,42\mathrm{.}9,63\mathrm{.}1,86\mathrm{.}3,214\mathrm{.}2],[$Mass$\%$ undersize,$0\mathrm{.}3,0\mathrm{.}8,4\mathrm{.}2,11\mathrm{.}6,24\mathrm{.}6,68\mathrm{.}2,75\mathrm{.}7,88\mathrm{.}1,99\mathrm{.}8,100]]$

$($b$)$ Shear cell tests have been performed on the received barley and it is proposed to design a stainless$-$steel conical hopper for process delivery. The semi$-$included angle of hopper conical section, $,$ is $30$ and the bulk density, ${}_b,$ is $900kg{m}^{-3}.$ The test results indicate that the angle of wall friction, ${}_w,$ is $10$ and the corresponding pairs of ${}_y$ and ${}_c$ are shown in Table $3.$

$\backslash$table$[[$Table $3$: Shear test experimental data.,$],[{}_c(kN{m}^{-2}),2\mathrm{.}4,2\mathrm{.}0,1\mathrm{.}6,1\mathrm{.}3],[{}_y(kN{m}^{-2}),0\mathrm{.}97,0\mathrm{.}91,0\mathrm{.}85,0\mathrm{.}78]]$

$($i$)$ Determine the critical yield stress, ${}_{\text{c}\text{r}\text{i}\text{t}},$ via graphical approach with MATLAB. The flow factor, $ff,$ can be determined from Figure $2.($Hint: ${}_{\text{c}\text{r}\text{i}\text{t}}$ is the intersection of the plot of shear test results in Table $3$ and ${}_y$ versus $($:$\frac{{}_c}{f}f\}.$

$($ii$)$ Determine the minimum diameter of the circular hopper outlet required to enable the barley to flow when the outlet slide valve is opened at full opening. The minimum outlet diameter is given as

Minimum outlet diameter, $B=\frac{H(){}_{\text{c}\text{r}\text{i}\text{t}}}{{}_{B}g}$

where $H()$ is the factor based on the hopper wall slope, $g$ is the gravitational acceleration $(9\mathrm{.}81(m)s^{-2}).$ The value of $H()$ can be estimated using the following expression:

$H()=2\mathrm{.}0+\frac{}{60}.$