L$-$Tryptophan is an essential amino acid that is produced as a product in a fed$-$batch fermentation using the bacteria $E.$ coll. Using glucose as the carbon substrate, ammonium sulphate as the nitrogen source, and by bubbling air through the reactor to provide oxygen, the reactor is run for $24$ hours as a batch, and $30$ hours as fed$-$batch. The reactor is initially filled with $3000L$ of media containing $7\mathrm{.}5\frac{g}{L}$ glucose and a certain concentration of ammonium sulphate. After $24$ hours of batch operation, there is $0\mathrm{.}5\%$ by volume glucose remaining and a feed is started at a rate of $170\frac{L}{hr}$ containing $800\frac{g}{L}$ glucose but it also contains ammonium sulphate. At the end of the fed$-$batch, the reactor is harvested with $45\frac{g}{L}$ of L$-$Tryptophan and $670,680$grams of cells, with $10\mathrm{.}29\%$ by volume glucose remaining. The fermentation temperature is $37C$ and the stirrer power was $8kW.$ Assume that no carbon dioxide was present in either the initial batch media or the feed solution. Also assume no tryptophan was present initially and starting cell concentration and trace nutrients were negligible.

POSSIBLY USEFUL INFO

General Stoichiometric Equation Including Product:

$C_w{H}_x{O}_4{N}_2+a{O}_2+b{H}_2{O}_{4}NcC{H}_a{O}_3{N}_a+dC{O}_2+e{H}_{2}O+f{C}_3{H}_k{O}_2{N}_m$

Escherichia Coli $C{H}_{1\mathrm{.}77}{O}_{0\mathrm{.}49}{N}_{0\mathrm{.}24}$ with $7\%$ ash

Tryptophan $C_{11}{H}_{12}{N}_2{O}_{2}MW=204$

Glucose $C_6{H}_{12}{O}_{6}MW=120$

Ammonium Sulphate $(N{H}_4{)}_{2}S{O}_{4}MW=132$

Oxygen $O_{2}MW=32$

Carbon Dioxide $C{O}_{2}MW=44$

Useful densities:

${}_{Glucose}=156\frac{g}{ml}$

${}_{\text{A}\text{i}\text{r}}=1\mathrm{.}135\frac{g}{ml}$

${}_{\text{O}\text{x}\text{y}\text{g}\text{e}\text{n}}=1\mathrm{.}331\frac{g}{ml}$

${}_{\text{W}\text{a}\text{t}\text{e}\text{r}}=1\mathrm{.}00\frac{g}{ml}$

${}_{\text{A}\text{m}\text{m}\text{o}\text{n}\text{i}\text{u}\text{m}\text{S}\text{u}\text{l}\text{p}\text{h}\text{a}\text{t}\text{e}}=1\mathrm{.}77\frac{g}{ml}$

Overall energy balance for bioreactor: $,-H_{rxn}-M_v{h}_v-Q+W_s=0$

$h_v$ of water at $37C=2413\mathrm{.}1\frac{kJ}{kg}$

$H_{rxn}$ for aerobic reactions $=460\frac{kJ}{\text{m}\text{o}\text{l}}{O}_2$