Parth Ajmera

Fick's law of diffusion

J_A = \frac{N_A}{a} = -D_{AB}\frac{dC_A}{dy}

$J_A$ — mass flux of component A (rate of diffusive mass transport per unit cross-sectional area, driven by the concentration gradient), in kilograms per square metre-second ( $\text{kg/(m}^2 \cdot \text{s)}$ )
$N_A$ — rate of mass transfer of component A, in kilograms per second ( $\text{kg/s}$ )
$a$ — cross-sectional area, perpendicular to the direction of diffusion, in square metres ( $\text{m}^2$ ) — not to be confused with the subscript $A$ , which denotes the diffusing component
$D_{AB}$ — diffusion coefficient of A through B, describing the ease of diffusion, in square metres per second ( $\text{m}^2/\text{s}$ )
$C_A$ — mass concentration of component A, in kilograms per cubic metre ( $\text{kg/m}^3$ )
$y$ — distance in the direction of diffusion, in metres ( $\text{m}$ )

Rate of mass transfer (film theory)

In notes →

N_A = k_L\, a\, (C^* - C)

$N_A$ — volumetric rate of mass transfer (how fast solute crosses the interface into the bulk liquid, driven by the gap between saturation and bulk concentration), in gram-mole per cubic metre per second ( $\text{gmol}/(\text{m}^3 \cdot \text{s})$ )
$k_L$ — (liquid) film mass-transfer coefficient (how fast a solute molecule crosses the stagnant film — sets the pace of interfacial transfer), in metres per second ( $\text{m/s}$ )
$a$ — interfacial area per unit volume, in square metres per cubic metre ( $\text{m}^2/\text{m}^3$ )
$C^*$ — saturation concentration: the maximum concentration of the solute that can dissolve in the liquid, in gram-mole per cubic metre ( $\text{gmol/m}^3$ )
$C$ — bulk concentration: the current concentration of the solute in the liquid, in gram-mole per cubic metre ( $\text{gmol/m}^3$ )

$k_L$ under film theory

In notes →

k_L = \frac{D_{AB}}{\delta}

$k_L$ — (liquid) film mass-transfer coefficient (how fast a solute molecule crosses the stagnant film), in metres per second ( $\text{m/s}$ )
$D_{AB}$ — diffusion coefficient of the solute through the liquid, in square metres per second ( $\text{m}^2/\text{s}$ )
$\delta$ — thickness of the stagnant film, in metres ( $\text{m}$ )

Gas hold-up ( $\varepsilon_g$ )

In notes →

\varepsilon_g = \frac{V_g}{V_g + V_L}

$\varepsilon_g$ — gas hold-up (fraction of the dispersion's volume occupied by gas bubbles at any instant — how much gas is retained in the liquid, not the sparging flow rate), dimensionless (fraction, $0$ – $1$ )
$V_g$ — volume of gas (bubbles) in the dispersion, in cubic metres ( $\text{m}^3$ )
$V_L$ — volume of liquid in the dispersion, in cubic metres ( $\text{m}^3$ )

Gas hold-up ( $\varepsilon_g$ )

In notes →

a = \frac{6\varepsilon_g}{d_b}

$a$ — interfacial area per unit volume (total gas-liquid contact surface available for mass transfer, per unit liquid volume), in square metres per cubic metre ( $\text{m}^2/\text{m}^3$ )
$\varepsilon_g$ — gas hold-up (fraction of dispersion volume occupied by gas), dimensionless
$d_b$ — Sauter mean bubble diameter, in metres ( $\text{m}$ )

Higbie's penetration theory

In notes →

k_L = 2\sqrt{\frac{D_{AB}}{\pi t_c}}

$k_L$ — liquid film mass-transfer coefficient (how fast a solute molecule crosses into a fresh liquid element during its brief time at the interface), in metres per second ( $\text{m/s}$ )
$D_{AB}$ — diffusion coefficient of the solute through the liquid, in square metres per second ( $\text{m}^2/\text{s}$ )
$t_c$ — contact time of a liquid element at the interface before it is replaced, in seconds ( $\text{s}$ ) — assumed identical for every element

Danckwerts' surface renewal theory

In notes →

k_L = \sqrt{D_{AB}\, s}

$k_L$ — liquid film mass-transfer coefficient (how fast a solute molecule crosses into a fresh liquid element before it is swept away), in metres per second ( $\text{m/s}$ )
$D_{AB}$ — diffusion coefficient of the solute through the liquid, in square metres per second ( $\text{m}^2/\text{s}$ )
$s$ — fractional rate of surface renewal (probability per unit time that a given liquid element gets swapped out for fresh bulk fluid), in reciprocal seconds ( $\text{s}^{-1}$ )

Rate of oxygen transfer

In notes →

N_A = k_L a\, (C^*_{AL} - C_{AL})

$N_A$ — volumetric rate of oxygen transfer (OTR — how fast oxygen crosses from gas bubbles into the broth; must keep pace with cellular consumption for aerobic growth), in gram-mole per cubic metre per second ( $\text{gmol}/(\text{m}^3 \cdot \text{s})$ )
$k_L a$ — volumetric mass transfer coefficient (lumps film coefficient and interfacial area into one measurable term — the standard parameter for bioreactor aeration design and scale-up), in reciprocal seconds ( $\text{s}^{-1}$ )
$C^*_{AL}$ — saturation concentration of oxygen in the liquid (its maximum solubility), in gram-mole per cubic metre ( $\text{gmol/m}^3$ )
$C_{AL}$ — instantaneous concentration of oxygen in the liquid, in gram-mole per cubic metre ( $\text{gmol/m}^3$ )

Henry's law and $C^*_{AL}$

In notes →

p_{O_2} = H\, C^*_{AL}

$p_{O_2}$ — partial pressure of oxygen in the gas phase, in pascals ( $\text{Pa}$ )
$H$ — Henry's law constant for oxygen in the liquid (measures how reluctant the gas is to dissolve — higher $H$ means lower solubility at the same partial pressure), in pascal-cubic-metre per gram-mole ( $\text{Pa}\cdot\text{m}^3/\text{gmol}$ ) — depends on the gas, the liquid, and temperature
$C^*_{AL}$ — saturation concentration of oxygen in the liquid, in gram-mole per cubic metre ( $\text{gmol/m}^3$ )

Rate of oxygen consumption per cell

In notes →

Q_O = q_O\, X

$q_O$ — specific oxygen uptake rate: oxygen required by one cell, in gram-mole per gram (dry cell weight) per second ( $\text{gmol}/(\text{g} \cdot \text{s})$ )
$X$ — biomass (cell) concentration, in grams (dry cell weight) per cubic metre ( $\text{g/m}^3$ )
$Q_O$ — oxygen uptake rate of the whole bioreactor (OUR), in gram-mole per cubic metre per second ( $\text{gmol}/(\text{m}^3 \cdot \text{s})$ )

1. Oxygen (gas) balance method

In notes →

N_A = \frac{1}{V_L}\Big[(F_g\, C_{AG})_{i} - (F_g\, C_{AG})_{o}\Big]

$V_L$ — volume of liquid in the fermenter, in cubic metres ( $\text{m}^3$ )
$F_g$ — volumetric gas flow rate, in cubic metres per second ( $\text{m}^3/\text{s}$ )
$C_{AG}$ — concentration of oxygen in the gas phase, in gram-mole per cubic metre ( $\text{gmol/m}^3$ )

2. Dynamic gassing-out method (most important)

In notes →

k_L a = \frac{\ln\left(\dfrac{\bar{C}_{AL} - C_{AL_1}}{\bar{C}_{AL} - C_{AL_2}}\right)}{t_2 - t_1}

$k_L a$ — volumetric mass transfer coefficient (the aeration parameter being measured, read off the slope of the reoxygenation recovery curve), in reciprocal seconds ( $\text{s}^{-1}$ )
$t_0$ — time at which the oxygen supply is cut off
$t_1$ , $t_2$ — two time points during reoxygenation, at which the instantaneous oxygen concentrations are $C_{AL_1}$ and $C_{AL_2}$
$\bar{C}_{AL}$ — oxygen level reached at steady state after reoxygenation

Fick's law of diffusion

Rate of mass transfer (film theory)

kLk_LkL​ under film theory

Gas hold-up (εg\varepsilon_gεg​)

Gas hold-up (εg\varepsilon_gεg​)

Higbie's penetration theory

Danckwerts' surface renewal theory

Rate of oxygen transfer

Henry's law and CAL∗C^*_{AL}CAL∗​

Rate of oxygen consumption per cell

1. Oxygen (gas) balance method

2. Dynamic gassing-out method (most important)

$k_L$ under film theory

Gas hold-up ( $\varepsilon_g$ )

Gas hold-up ( $\varepsilon_g$ )

Henry's law and $C^*_{AL}$