Background

Ultrafiltration is a process used for the concentration and purification of macromolecular solutions, particularly in the bioprocessing industries. It is a type of membrane filtration that separates particles based on their size and shape. In the feed and bleed configuration, a portion of the product stream $($retentate$)$ is recycled back into the feed, which increases the mass transfer coefficient in the module, leading to higher permeate $($filtrate$)$ flowrates, shown in Figure $1.$ Typically, the flowrate through the module greatly exceeds the inlet flowrate, and thus the system can reasonably be assumed to be well mixed, i$.$e$.,$ the retentate concentration is taken to be the same as the mean concentration in the module itself. For this assignment, we look at the continuous concentration of a protein solution. Here c is the concentration of protein. It was found that no protein passes through the membrane at any time, i$.$e$.$ permeate$$s protein concentration $($c$\_$Per$)$ is equal to zero. The system can be assumed to be well mixed, which leads to the retentate concentration is equal to the mean module concentration.

For this assignment, we assume that the gel polarisation model applies, i$.$e$.$ the permeate flux across the membrane $($j$)$ is:

j$=$k$*$ln$($c$\_$g$/$c$\_$ret$)$

where k is the mass transfer coefficient $($assumed constant$),$ c$\_$g is the limiting or $$gel$$ concentration $($constant$),$ and c$\_$ret is the retentate concentration.

The solute balance for the system, assuming no proteins go through the membrane. where V$\_$dot$\_$in is the feed volumetric flowrate, c$\_$in is the feed concentration, c$\_$ret is the retentate concentration and V$\_$dot$\_$ret is the retentate flowrate where A is the membrane area.

Substituting V$\_$dot$\_$ret from the solute balance $($Equation $2)$

$$into solvent balance $($Equation $3)$

$$leads to the flux function:

j$=$V $\_$dot$\_$in$/$A $*(1-$c$\_$in$/$c$\_$ret $)$

Task $1$:

A protein solution with a concentration of $8$ g L$-1$ is fed into a single$-$stage continuous feed and bleed ultrafiltration system at a rate of $2$ L min$-1.$ The gel concentration, c$\_$g$,$ is $350$ g L$-1.$ The system has a mass transfer coefficient of $5$x$10-6$ m s$-1$ and an area of $3\mathrm{.}1$ m$2.$ Use the Newton method to compute the retentate concentration.

Arrange Equation $1$ and $4$ as a single equation in a form suitable for the Newton$$s method with c$\_$Ret as the unknown variable.

SOLVE THIS USING PYTHON CODE. AND THE my$\_$fsolve function below to solve for solution c$\_$ret. And graph it by importing matplotlib and numpy $($the import code is provided below$).$

Importing matplotlib and numpy into python:

# essential packages

$"$

import numpy as np # to perform vector and matrix calculations

import matplotlib.pyplot as plt # for plotting results

$\%$config InlineBackend.figure$\_$format $=$ 'retina'

$"$

And the Newton Methode python code named as my$\_$fsolve:

#my f$\_$solve

def my$\_$fsolve$($fun$,$fun$\_$prime,x$0,$args$=(),$detailed$\_$output$=$False$)$:

$"""$

find the root of a single variable non linear function

inputs

fun...function that takes at least one argument, i$.$e$.$ f$($x$,$a$,$b$)$

fun$\_$prime...first derivative of fun

x$0...$starting estimate of the root

args: tuple optional,any extra arguments to fun and fun$\_$prime

detailed$\_$output: bool, optional, if true displays Newton steps

returns

x$\_$new... the solution $($result of the last iteration$)$

exitflag..$\mathrm{.}0$ or $1(0...$no solution$)(1...$solution

msg$........$message on success

$"""$

# termination conditions

n$\_$max$=100$ # maximum iterations

xTol$=1$e$-3$ # elative error between two consecutive iterates

fTol$=1$e$-6$ # function value

exitflag$=0$

# Loop through Newton Method

for i in range$(0,$n$\_$max$)$:

# get function values for fun and fun$\_$prime

fun$\_$x$0=$fun$($x$0,*$args$)$

fun$\_$prime$\_$x$0=$fun$\_$prime$($x$0,*$args$)$

# Newton iteration

x$\_$new $=$ x$0-$ fun$\_$x$0/$ fun$\_$prime$\_$x$0$

# display iteration results

if detailed$\_$output:

# create header

if i$==0$:

print$($f$\text{'}$iteration $\backslash$t x$0\backslash$t$\backslash$t f$($x$0)\text{'})$

print$(\text{'}----------------------------------------------\text{'})$

else:

print$($f$\text{'}\{$i:$2$d$\}\backslash$t$\backslash$t $\{$x$0$:$\mathrm{.}3$f$\}\backslash$t$\backslash$t $\{$fun$\_$x$0$:$\mathrm{.}4$e$\}\text{'})$

# check if accuracy is reached

fun$\_$x$\_$new$=$fun$($x$\_$new,$*$args$)$

if np$.$abs$(($x$\_$new$-$x$0)/$x$\_$new$)$