You characterized a B$.$ Braun fermenter in your lab. It has a $10-$liter working volume and we normally aerate with $10$ std$.$ liters per minute of air. We assume it is well mixed. In order to grow to high density at a fast growth rate, we need to have a very high oxygen transfer rate so we will operate with a $2$ atm. Head pressure, gauge $($assume the hydrostatic head effect is negligible$).$ I have calibrated my $D{O}_2$ probe to $100\%$ while sparging air while maintaining the $2$atm head pressure and a temperature of $37C,$ my fermentation temperature.

a$)$ If I want to keep the $p_{O_2,\text{l}\text{i}\text{q}\text{u}\text{i}\text{d}}$ above $30\%$ of air saturation $($under standard conditions; i$.$e$.1$atm. absolute pressure$),$ what is the minimum $D{O}_2$ reading I can tolerate $(\%)?$

b$)$ I don't want to let the exit gas $O_2$ mole fraction fall below $0\mathrm{.}14.$ What is my maximum rate of oxygen delivery? $($OUR in mmoles$/$l$-$min$)$

c$)$ Under these conditions, what $k_{L}a$ do I need to transfer this much oxygen into the liquid? $($mmoles$/$l$-$atm$-$min$)$

We find that at lower agitation speeds $($up to about $500rpm),k_L$a follows the expected trend that $k_{L}a$ is proportional to $rp{m}^{1\mathrm{.}8}.$ However, at higher agitation speeds, the $k_L$a doesn't increase as fast as we expected. I suspected that we needed larger baffles, and that does indeed give us the expected $k_{L}a\text{'}$s at higher agitation speeds.

d$)$ Explain why we need to increase the baffle width to get better oxygen transfer.