There is a $20\%$ increase in Riboflavin production, so the designed equipment does not have the capacity to support the new load.

a$.$ The pipe at the entrance of the centrifuge has an internal diameter $($Di$)$ of $0\mathrm{.}22$ m$.$ Calculate the speed at which the broth is flowing if the volumetric flow rate is $2\mathrm{.}5$ m$^3/$s$.$ If this speed is maintained, what should be the internal diameter of the pipe when this volumetric flow rate increases by $20\%?(1$ point$)$

b$.$ The decanter centrifuge $($P$-18)$ operates at $3500$ RPM$,$ the rotor has a length of $2$ m and a diameter of $1$ m$.$ With a $20\%$ increase in total production, it is also expected to increase the sedimentation speed of particles so that the scaled centrifuge can support the load increase. The modified Stokes' law for centrifuges allows us to calculate the sedimentation speed using the physical parameters of the particles. Based on the provided data, calculate the sedimentation speed of the current centrifuge. Q $=2\mathrm{.}5$ m$^3/$s $(1$ point$)$

Particle limit data: minimum diameter $1$m$,$ and density $1030$ g$/$L$.$

Solvent with density $1000$ g$/$L and viscosity $1\mathrm{.}5$ cP$.$

c$.$ In addition to the sedimentation speed, the residence time of the riboflavin particles in the decanter centrifuge plays an important role in the process performance; it is determined by the ratio of the rotor volume $($cylindrical shape$)$ to the inlet flow. Calculate the residence time of the current centrifuge. $(1$ point$)$

d$.$ With the $20\%$ increase in the inlet flow, it is expected to have a residence time of $0\mathrm{.}8$ s$,$ what should be the rotor radius of the scaled centrifuge? $(1$ point$)$

e$.$ Compare the sedimentation speed of the initial and the scaled centrifuge, which is greater? Argue what implications increasing the sedimentation speed has in a decanter centrifuge. $(2$ points$)$

f$.$ Argue how this new sizing would affect the costs reported in table $3.(1$ point$)$

g$.$ With the $20\%$ increase in plant production, consider the entire process to suggest a strategy to rote