Question $1(20$ marks$)$

a$)$ The cut size for the centrifuge is given by Equation $1.$ Discuss what $K$ and $K_2$ represent and their

significance in relation to centrifuges.

$d_{s0}^2=\frac{1}{2K{K}_2}In(\frac{2r_3^2}{r_3^2+r_i^2})$

Equation $1$

b$)$ A low concentration suspension of basalt $($density $2640k\frac{g}{m^3})$ in water with viscosity of $0\mathrm{.}001N\frac{s}{m^2}$

is to be separated by centrifugal sedimentation. Laboratory experiments on a $0\mathrm{.}2m$ long tubular bowl

centrifuge were performed to produce a satisfactory clarified overflow at a throughput of $9\mathrm{.}3{10}^{-5}$

$\frac{m^3}{s}.$ The tubular bowl centrifuge has an internal radius of $0\mathrm{.}0250m$ and the radius of the liquid

surface is $0\mathrm{.}0133m$ operating at $25000rpm$

i$)$ What would be the design flow rate for an industrial scale tubular bowl centrifuge $0\mathrm{.}675m$ long with

an internal radius of $0\mathrm{.}0555m$ and air core radius of $0\mathrm{.}0265m,$ operating at $12000rpm$ with the same

overflow clarity?

ii$)$ What would be the maximum flow rate if a stack disc centrifuge containing $35$ channels with half

inclined angle of $60$ degrees, internal radius of $0\mathrm{.}0555m$ and air core radius of $0\mathrm{.}0265m$ operated at

$12000rpm.$ Sigma, that represents the equivalent area of the stack disc centrifuge is calculated using

Equation $2.$

$=\frac{2}{3}\frac{{}^2}{g}mn(r_2^3-r_1^3)cot$

$=\frac{1}{2}$ angle of the disc

Equation $2$

Where $$ is the angular speed, $n$ the number of channels and $r_1$ and $r_2$ are the inner and outer

inched

separation boundaries for the channels, and $$ is half the included angle of the discs.

iii$)$ Which of the two centrifuges do you think will give a higher separation efficiency and why?