An industry that produces apple juices has installed a clarification stage using tubular UF membranes from the KOCH SUPERCOR series. The

design includes tubular membrane modules of $7\mathrm{.}6cm$ internal diameter and a length of $3m.$ The juice feed stream circulates through the interior of

the tubular membrane at a speed of $1\mathrm{.}5\frac{m}{s}.$ Of the components in the juice, the person responsible for turbidity is pectin $(30-50$kDa, which is

present in a concentration of $3\frac{g}{L}$ and that accumulates on the surface of the membrane forming a gel layer. According to ultrafiltration studies of

the literature, values of $37\frac{g}{L}$ of pectin can be reached in the gel layer. The rejection factor for membrane pectin is $96\%.$

a$)$ Propose an equation to calculate the permeate flow and determine the main constants of the model $(3\mathrm{.}5$ points$).$

b$)$ If working in recirculation mode with a concentration factor of $10,$ estimate the maximum and average flow that can be obtained in the

clarification process $(3\mathrm{.}5$ points$)$

c$)$ If you want to obtain a production of $1m\frac{3}{h}$ of clarified juice what will be the number of modules needed knowing that the module chosen based

on PVDF can accommodate $9$ tubes per module $(2$ points$).$

d$)$ What molecular weight cutoff value would you use when selecting the tubular membrane? $(0\mathrm{.}5$ points$)$

e$)$ Indicates an alternative treatment to use based on membranes to achieve the objective of clarifying the juice and in turn concentrate its sucrose

content $(10\%$ mass$).$ Indicate what type of module you would choose $(0\mathrm{.}5$ points$)$

Data:

Juice density $1,06K\frac{g}{m^3}$

Juice viscosity: $8,6\frac{g}{m}.s$

Pectine Diffusion Coefficient in water: $9{10}^{-11}\frac{m^2}{s}$