$4\mathrm{.}12$ Production of $1,3-$propanediol

A process for microbial synthesis of $1,3-$propanediol is being developed for the manufacture of 'green' polyester fabric from renewable resources. Under anaerobic conditions, a selected strain of Klebsiella pneumoniae converts glycerol $(C_3{H}_8{O}_3)$ to $1,3-$propanediol $(C_3{H}_8{O}_2)$ and acetic acid $(C_2{H}_4{O}_2),$ with minimal formation of other fermentation products such as butyric acid, ethanol, and $H_2$ gas. The fermentation and cell growth equation can be written as:

$68C_3{H}_8{O}_3+3N{H}_{3}3C_4{H}_7{O}_{2}N+49C_3{H}_8{O}_2+15C_2{H}_4{O}_2+15C{O}_2+40H_{2}O$

where $C_4{H}_7{O}_{2}N$ represents the biomass.

A continuous fermenter is set up for $1,3-$propanediol production at $37C$ and atmospheric pressure. Anaerobic conditions are maintained by sparging the broth with nitrogen gas at a flow rate of $1000$ litres $mi{n}^{-1}.$ The feed rate of medium into the fermenter is $1000kg{h}^{-1}$; the medium contains ammonia and $14\%\frac{w}{w}$ glycerol. The yield of $1,3-$ propanediol is strongly affected by glycerol concentration, which must be kept above a certain level to suppress the formation of undesired by$-$products. Accordingly, the fermentation process is designed so that the product stream contains an unreacted glycerol concentration of $3\%\frac{w}{w}.$

$($a$)$ What is the volumetric flow rate and composition of the off$-$gas?

$($b$)$ What minimum concentration of $N{H}_3$ is needed in the feed stream?

$($c$)$ If the $N{H}_3$ concentration from $($b$)$ is used, what is the concentration of $1,3-$propanediol in the product stream?