You are assigned to develop the aerobic treatment of the wastewater of a company working in the field of petroleum processing. You have a pure bacterial strain which was shown with high capability to bioremediate the wastewater.

To overcome the growth inhibition caused by excess of substrate, you tested the chemostat technology using a fermentor with a total volume of $51$ liters. In the batch step you performed in the same reactor before starting the chemostat, you determined the following parameters:

The COD at the end of the batch was $0\mathrm{.}130\frac{g}{l}$ corresponding to a growth rate of $0\mathrm{.}294h^{-1}.$

The sludge $($bacterial biomass$)$ concentration at the end of the batch step was $15\mathrm{.}72\frac{g}{l}$

The crude wastewater COD was $45\frac{g}{l}.$

$1-$ Calculate the maximal volume of the crude wastewater which can be treated in the reactor in the process of batch step.

$2-$ Calculate the yield of production of sludge $($biomass$),$ considering that all the available volume of the reactor was used.

$3-$ Calculate the efficiency of the treatment by the strain at the fed$-$batch technology