Mass Transfer

Key points

1. 01

   Fick's law ($J_A = -D_{AB}\, dC_A/dy$) always runs high concentration to low; film theory assumes a linear concentration profile across each stagnant film — steady-state diffusion, no accumulation, same reasoning as linear conduction through a wall.

2. 02

   Film theory gives $k_L \propto D_{AB}^{1.0}$; penetration (Higbie) and surface renewal (Danckwerts) theories both give $k_L \propto D_{AB}^{0.5}$ — the two square-root theories differ only in their contact-time assumption: fixed and equal (Higbie) vs. randomly distributed (Danckwerts).

3. 03

   At the phase boundary itself there is no resistance — instantaneous equilibrium — and it's the partition coefficient, not diffusion, that decides the concentration jump between the two phases there.

4. 04

   $k_L$ and $a$ move together and can't be measured separately, so they're lumped into $k_La$, the volumetric mass transfer coefficient — in aerobic fermentation this is the reactor's oxygen transfer capacity, the standard scale-up parameter.

5. 05

   Of the eight resistances between a gas bubble and a cell's reaction site, diffusion through the liquid film around the bubble is the largest — which is why $k_La$ (built around that film) governs overall O$_2$ transfer.

6. 06

   At steady state, OTR = OUR: $k_La(C^*_{AL} - C_{AL}) = q_O X$. Setting $C_{AL} = C_{crit}$ gives the minimum $k_La$ a reactor needs to design for; setting $C_{AL} = 0$ gives the maximum biomass a given aeration setup can sustain.

7. 07

   Temperature is a trade-off for $k_La$: it speeds up the kinetics (viscosity↓, diffusivity↑ → $k_L$↑) but shrinks the driving force (solubility $C^*$↓, since dissolution is exothermic).

8. 08

   The dynamic gassing-out method is the standard way to measure $k_La$: cut the air supply, let dissolved O$_2$ fall, restart it, and read $k_La$ off the slope of the recovery curve between two timepoints.