i don't need an answer on the first 2 questions, they are jsut mentionned to give all the context

SweetFruit, Inc. has hired you to design a process to reduce the concentration of organic matter in the wastewater from one of their orange juice production plants. The process envisaged relies on the biodegradation of the organic matter by microorganisms voluntarily introduced in the water to be treated.

We write [S] (g/m3) the concentration of organic matter and [X] (g/m3) the concentration of microorganisms used for the degradation of the organic matter. As these microorganisms are voluntarily introduced into the water to be treated, their concentration is therefore controlled at the inlet of a perfectly mixed continuous reactor or at the initial operation time of a batch reactor. We write rX (g/m3/day) the production rate of microorganisms and rS (g/m3/day) the consumption rate of organic matter. We suppose that these two reaction rates follow a Monod law:

rX = max [X] [S]/(Ks + [S])

rS = -1/y rX

We assume that the mortality of the microorganisms can be neglected.

Question 1. What is the biological significance of the parameter y? What are the two phenomena compared by this parameter?

Question 2. Two preliminary experiments were carried out in a perfectly mixed batch reactor, using the wastewater (diluted or concentrated) and different initial amounts of microorganisms, in order to determine the different parameters appearing in the expressions of rS and rX.

Experiment 1: [X](t = 0) = 100 g/m3 and [S](t = 0) = 1000 g/m3 The idea is here to be able to neglect Ks in front of [S] in the balance equations. We therefore voluntarily place ourselves at an initial high concentration of organic matter. We then hope (to check later!) That [S](t = 0) is much larger than Ks. Using the data presented in the table below, determine y and max, as precisely as possible.

t(day)	[X] (g/m3 )	[S] (g/m3 )
0	100	1000
0.075	110	979
0.15	118	954
0.225	131	925
0.3	143	895
0.375	154	864
0.45	171	825
0.525	184	790
0.6	203	746
0.675	222	700
0.75	241	652

Experiment 2: [X](t = 0) = 200 g/m3 and [S](t = 0) = 5 g/m3. Unlike experiment 1, the idea is now to be able to neglect [S] in front of Ks in the balance equations (to check later!). Furthermore, having [X](t = 0) >> [S](t = 0), we can reasonably assume that [X](t) is constant over time (to be demonstrated!). Using the data presented in the table below, determine Ks, as precisely as possible.

t (day)	[S] (g/m3 )
0	5
0.005	4.5
0.01	4.0
0.015	3.6
0.02	3.3
0.025	2.9
0.03	2.6
0.035	2.4
0.04	2.1
0.045	1.9
0.05	1.7

Question 3. We still consider a perfectly mixed batch reactor, with [X](t = 0) = X0 and [S](t = 0) = S0. First, establish a linear relation between [X](t) and [S](t). The introduction of this relation in the balance equation for the organic matter makes it possible to obtain a differential equation for [S](t) no longer involving [X](t). Obtain this equation and solve it analytically. A third experiment is carried out in a perfectly mixed batch reactor, with [X](t = 0) = 50 g/m3 and [S](t = 0) = 200 g/m3. The following results are obtained:

t (day)	[X] (g/m3)	[S] (g/m3 )
0	50	200
0.15	59	179
0.30	70	152
0.45	82	128
0.6	91	92
0.75	106	60
0.9	116	31
1.05	129	7
1.2	131	2
1.35	132	1

Check that with, the values obtained previously for max, Ks and y, you would have been able to reasonably predict the results of this experiment.

Question 4. First, to treat the wastewater from the plant, you want to develop a process composed of a single perfectly mixed continuous reactor (CSTR). A wastewater flow of 500 m3/day must be treated by the installation. In the feed stream, the organic matter concentration is 200 g/m3, while the microorganism concentration is 50 g/m3. What is the volume of the reactor needed to obtain a 90% conversion of the organic matter?

Question 5. Suggest and design an improvement to the previous arrangement. Justify your answer (hint: biodegradation can be seen as an autocatalytic reaction).